P Aurenche
email: aaurenche@lapth.cnrs.fr

J Ph

Guillet
email: guillet@lapth.cnrs.fr

THE UNIFIED MODEL OF ELECTROWEAK INTERACTIONS AND APPLICATION TO NEUTRINO PHENOMENOLOGY

The unified model of electroweak interactions (Glashow-Weinberg-Salam model) is constructed step by step. As an application, the phenomenology of neutrino mixing, in the framework of the three-family model, is discussed in detail. Numerous formulae for appearance or disappearance of neutrinos in vacuum or in matter are derived. These notes should provide a self-contained introduction to the GWS model and to neutrino mixing and oscillations. Recent experimental results on neutrino oscillations are briefly reviewed.

Introduction

From the experimental point of view the world of "elementary particles" consists in:

-leptons: they have spin 1 2 and come in three doublets, (e -, ν e ) the electron and its associated neutrino, (µ -, ν µ ) the muon and its neutrino, (τ -, ν τ ) the tau and its neutrino.

-vector bosons: they have spin 1 and there is a massless boson, the photon, and three massive ones, the W + , W -and the Z.

-hadrons: one distinguishes mesons of integer spin (S = 0, 1, . . . ) from baryons of half-integer spin (S = 1 2 , 3 2 , . . . ). The hadrons have been known for a long time to have a finite size (typically of the order of 1 fm) and there exist so many hadrons (about 150 mesons and 120 baryons) that they cannot be considered as elementary. At high energy they appear as composite objects made up of quasi-free point-like fields : the quarks and the gluons. Like the leptons and the vector bosons, the quarks and the gluons are structureless down to a scale of about 10 -3 to 10 -4 fermi, i.e. 10 -18 to 10 -19 m according to the most recent experimental results obtained at the CERN Large Hadron Collider (LHC). They are treated as elementary fields appearing in the lagrangian which describes the dynamics of their interactions. Three types of forces have been identified acting on these fields: the strong force which affects only the quarks and the gluons, and the electromagnetic and weak forces. The basic principle which guides the construction of models of particle physics is that of local gauge invariance according to which the physical properties do not depend on the phases of the fields. The Standard Model is a (highly successful) example of a minimal model based on the local gauge group SU (3) ⊗ SU (2) L ⊗ U (1) Y i.e. the direct product of three simple groups. The main features of these groups are :

-The SU (3) gauge group or colour group is the symmetry group of strong interactions. This group acts on the quarks and the interaction force is mediated by the gluons which are the gauge bosons of the group. The quarks and the gluons are coloured fields. The "coupling" (fine structure constant) between quarks and gluons is denoted by α s which can be large. Under some conditions, however, α s becomes very small and perturbation theory applies. The SU (3) colour symmetry is exact and consequently the gluons are massless. The theory of strong interactions based on colour SU (3) is called Quantum Chromodynamics ; -The SU (2) L ⊗ U (1) Y is the gauge group of the unified weak and electromagnetic interactions, where SU (2) L is the weak isospin group, acting on left-handed fermions, and U (1) Y is the hypercharge group. At "low" energy, below 250 GeV, the SU (2) L ⊗ U (1) Y symmetry is "spontaneously" broken and the residual group is U (1) em whose generator is a linear combination of the U (1) Y generator and a generator of SU (2) L : the corresponding gauge boson is of course the photon and the associated "coupling" is α ≃ 1 137 . Symmetry breaking implies that the other gauge bosons acquire a mass: they are the heavy W ± , Z bosons discovered at CERN in the mid '80's. The symmetry breaking mechanism is associated to the names of Brout, Englert, Higgs, Guralnik, Hagen, Kibble and Sudarshan and it is known now as the BEH mechanism after the names of the authors (Brout, Englert, Higgs) who published their results first. Higgs emphasized the existence of a massive scalar field as a consequence of the spontaneously broken symmetry and this field is traditionally refered to as the Higgs boson, but it is sometimes called also the "BEH boson" or simply the "massive scalar boson". Unlike the strong and electromagnetic interactions, the weak interactions violate parity. The electroweak theory, based on spontaneously broken SU (2) L ⊗ U (1) Y gauge invariance, is known as the Glashow-Salam-Weinberg (GSW) model.

A specific feature of the electroweak model is the generation mixing occuring at the Born level independently for the quark and the lepton sectors. The corresponding formalism is associated to the Cabibbo-Kobayashi-Maskawa (CKM) matrix for the quarks and the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix for the leptons. In the latter case the consequences are neutrino oscillations the phenomenology of which will be the object of the second part of these notes.

Before entering the description of the unified theory of electroweak interactions, based on broken gauge invariance, it is useful to briefly review the Fermi theory of weak interactions and its phenomenological extensions: this will serve to motivate the choice of the gauge group SU (2) L ⊗ U (1) Y as well as illustrate the features related to the presence of massive gauge bosons.

The Fermi theory and its extensions

At the beginning was the Fermi theory of muon decay :

µ -→ e -ν e ν µ .
and neutron decay :

n → p e -ν e .
In the latter case we work in the quark/parton model where we assume the nucleon is made of three quarks : neutron = (udd) and proton = (uud). For neutron decay, charge conservation allows only the transition d → u e -ν e , the other two quarks being spectators.

Contact interactions

These transitions are described by a local current-current (4 fermion) interaction parameterised by the Lagrangian:

L = G F √ 2 J ν (x)J † ν (x).
(2.1)

The current has a leptonic part and a hadronic part, J ν (x) = l ν (x) + h ν (x),

l ν (x) = ψ e γ ν (1 -γ 5 )ψ νe + ψ µ γ ν (1 -γ 5 )ψ νµ + ψ τ γ ν (1 -γ 5 )ψ ντ h ν (x) = ψ d γ ν (1 -γ 5 )ψ u + ψ s γ ν (1 -γ 5 )ψ c + ψ b γ ν (1 -γ 5 )ψ t , (2.2)
where, for simplicity, the argument of the fermion fields are not shown, ψ e instead ψ e (x), • • • . The γ 5 matrix anticommutes with all γ ν 's (see the appendix for the properties of the γ 5 matrix). The particular V -A (vector (γ ν ) -axial (γ ν γ 5 )) form of the current is dictated by experiment, in particular the angular distribution of particles in the final state 1 . The Fermi constant G F is universal, 1 The γνγ5 interaction breaks parity maximally, see sec.B.2 in the appendix.

i.e. it is the same for the hadronic sector and the leptonic sector and its value has been measured to be :

G F = 1.6639(2)10 -5 GeV -2 .

(2.3) Thus the transition matrix element for µ decay is an element in (G/ √ 2)l ν (x)l † ν (x) constructed from the first two terms of l ν (x):

M = G F √ 2 (ψ e γ ν (1 -γ 5 )ψ νe )(ψ µ γ ν (1 -γ 5 )ψ νµ ) † = G F √ 2 (ψ e γ ν (1 -γ 5 )ψ νe )(ψ νµ γ ν (1 -γ 5 )ψ µ ) (2.4)
and that of neutron decay (d quark decay) is an element of (G/ √ 2)l ν (x)h † ν (x):

M = G F √ 2 (ψ e γ ν (1 -γ 5 )ψ νe )(ψ d γ ν (1 -γ 5 )ψ u ) † = G F √ 2 (ψ e γ ν (1 -γ 5 )ψ νe )(ψ u γ ν (1 -γ 5 )ψ d ).
(2.5)

Introducing the expansion of a spinor ψ i in terms of plane waves with annihilation operators b where the u iα (p) and v iα (p) are, respectively, the wave functions of the annihilated fermion (positive energy) and the created antifermion (negative energy). Injecting eq. (2.6) into the matrix element above, we see that eq. (2.5) describes several processes related by crossing symmetry such as: d → u e -νe (term in ūu

• • • u d ūe • • • v νe ) or d ū → e -νe (term in vu • • • u d ūe • • • v νe ) or ν e d → e -u (term in ūu • • • u d ūe • • • u νe ) or • • • .
Considering the last process which is the dominant mechanism for the deep inelastic scattering of a neutrino on a proton one can easily calculate the cross section at the partonic level. Defining the momenta by ν e (p 1 ) d(p 2 ) → e -(p 3 ) u(p 4 ), the invariants are (p 1 + p 2 ) 2 = s, (p 1 -p 3 ) 2 = t = q 2 = (s/2)(1 -cos θ), (p 1 -p 4 ) 2 = u = (s/2)(1 + cos θ). (2.7) Supposing all fermions massless, the matrix element is (ignoring the polarisation indices):

M = G F √ 2
[ū e (p 3 )γ µ (1 -γ 5 )u νe (p 1 )] [ū u (p 4 )γ µ (1 -γ 5 )u d (p 2 )],

(2.8) and the matrix element squared summed/averaged over polarisation is (2.9)

Σ|M| 2 = 1 4 G 2
The traces and their product can be easily evaluated using eqs. (A.9) and (A.10) in appendix A. There is no mixing between the trace with a γ 5 matrix and that without since the former is antisymmetric in µν and the latter is symmetric. After reduction the result is simple:

Σ|M| 2 = 4 G 2 F [(s 2 + u 2 ) + (s 2 -u 2 )] = 8 G 2 F s 2 , (2.10)
where the first term in the square brackets corresponds to the first term (V -A interaction) in eq. (2.9).

The differential cross section is 2 :

dσ dΩ νed→e -u = 1 2s
d 3 p 3 (2π) 3 2E 3 d 3 p 4 (2π) 3 2E 4 (2π) 4 δ (4) (p 1 + p 2 -p 3 -p 4 ) Σ|M| 2 = 1 (2π) 2 1 16s (8 G 2 F s 2 ) = G 2 F 8π 2 s, (2.11)
independent of the polar angle. This result is in agreement with the data at not too high s. It is also interesting to consider in the Fermi model the diffusion of antineutrinos on the proton. In the quark/parton model, because of charge conservation, the νe interacts only with the u quarks via the transition νe (p 1 ) u(p 2 ) → e + (p 3 ) d(p 4 ). The matrix element can easily be constructed and it is:

M = G F √ 2 [v νe (p 1 )γ µ (1 -γ 5 )v e (p 3 )] [ū d (p 4 )γ µ (1 -γ 5 )u u (p 2 )].
(2.12)

Taking the square of the matrix element one obtains eq. (2.9) with p 1 and p 3 interchanged. Because

Tr( p 3 γ µ p 1 γ ν ) is symmetric and Tr( p 3 γ µ p 1 γ ν γ 5 ) antisymmetric under this interchange, one sees im-

mediately that Σ|M| 2 = 4 G 2 F [(s 2 + u 2 ) -(s 2 -u 2 )] = 8 G 2
F u 2 and consequently the differential cross section is found to be:

dσ dΩ νeu→e + d = G 2 F 8π 2 u 2 s = G 2 F 8π 2 s 4 (1 + cos θ) 2
(2.13) also in agreement with experimental observations where the positron is produced mainly in the direction of the initial νe quark. 2 The term in {• • • } is the phase space factor for massless particles.

If instead of the V -A form of the currents we had used only the vector part the results would be in disagreement with data since for both cross sections above, eqs (2.11) and (2.13), the result would have been :

dσ dΩ = G 2 F 64π 2 s 2 + u 2 s , (2.14)
a prediction not supported by experiments because of the wrong angular distribution for both reactions.

If one had tried "scalar currents" of the form:

l(x) = ψ e ψ ν e + ψ µ ψ νµ + • • • h(x) = ψ d ψ u + ψ s ψ c + • • • , , (2.15)
both ν and ν cross sections would have been proportional to s : this prediction is correct for ν e d → e -u but incorrect for νe u → e + d. In summary, all low energy data support the V -A form to describe weak interactions.

However the Fermi theory is not satisfactory at high energy. Indeed, from eq. (2.11) one obtains for the total cross section σ νed→e -u = G 2 F s/2π. However such a rapid rise of the cross section with energy cannot be asymptotically true as it violates the famous Froissart unitarity bound which requires σ ≤ ln 2 s as s → ∞. Note that the linear rise in s of a 2 → 2 cross section integrated over all final state variables could have easily been guessed on dimensional grounds. Indeed, in Fermi theory, such a cross section is proportional to G 2 F of dimension GeV -4 but a cross section3 is measured in units of GeV -2 . Since, after integrating over the final state phase space, the only scale available in the problem is s, of dimension GeV 2 , one necessarily has σ ∝ G 2 F s.

Vector boson mediated interactions

The rapid rise of cross sections is related to the locality of the current-current interaction. One can make the 4-fermion interaction nonlocal by postulating a massive charged particle coupling to the J µ (x) current similarly to the coupling of a photon to the fermionic current ψ(x)γ µ ψ(x) to mediate the interaction between the currents. It must be a vector particle because of the γ µ coupling in eq.

(2.2) as shown below

W exchange Fermi e-u d u d W e-

Denoting M W the mass of this particle and g W its dimensionless coupling to the currents, the matrix element eq. (2.5) becomes :

M = g 2 W [ψ e γ µ (1 -γ 5 )ψ νe ] g µν -q µ q ν /M 2 W q 2 -M 2 W [ψ u γ ν (1 -γ 5 )ψ d ] (2.16)
where q is the momentum transfer from the d quark to the u quark. Coming back to the reaction ν e (p 1 ) d(p 2 ) → e(p 3 ) u(p 4 ) studied above, the matrix element eq. (2.8), in momentum space is (we do not write explicitely the polarisation index of the fermions):

M = g 2 W [ū e (p 3 )γ µ (1 -γ 5 )u νe (p 1 )] g µν -q µ q ν /M 2 W q 2 -M 2 W [ū u (p 4 )γ ν (1 -γ 5 )u d (p 2 )] = g 2 W q 2 -M 2 W
[ū e (p 3 )γ µ (1 -γ 5 )u νe (p 1 )] [ū u (p 4 )γ µ (1 -γ 5 )u d (p 2 )],

(2.17) with q = p 1 -p 3 = p 4 -p 2 and where we have used Dirac equation for massless fields p i u(p i ) = 0.

This equation is identical to eq (2.8) provided we make the substitution:

G F √ 2 = g 2 W q 2 -M 2 W → g 2 W M 2 W when q 2 → 0, (2.18)
which allows to obtain the matrix element squared summed/averaged over polarisation from eq. (2.8):

Σ|M| 2 = 16 g 4 W s 2 (q 2 -M 2 W ) 2 ,
(2.19) and the differential cross section :

dσ dΩ νed→e -u = g 4 W 4π 2 s (q 2 -M 2 W ) 2 ,
(2.20)

with q 2 = -s(1 -cos θ)/2. The integrated cross section is easily calculated to be:

σ νed→e -u = 1 π g 4 W M 2 W s s + M 2 W .
(2.21)

At low energy we recover the Fermi model prediction provided, g 2 W /M 2 W = G F √ 2, while at high energy the Froissart bound is satisfied.

Still more problems!

However this is not the end of the story ! The W particle can be produced, and has been produced at LEP2, in the reaction e -e + → W -W + but the corresponding cross section, in our model, violates

Froissart bound. To see this, let us consider instead the unrealistic, but simpler, case 4 of the scattering ν e (p 1 ) νe (p 2 ) → W + (p 3 ) W -(p 4 ) the amplitude of which is given by only one Feynman diagram with the exchange of an electron:

ν e νe W + W - p 1 p 2 p 3 p 4 e -
To illustrate the problem we first define the kinematics and make some comments on the polarisation states of a massive vector particle. We work at very high energy in the center of mass frame of the e -e + system :

(p 1 + p 2 ) µ = ( √ s, 0), p µ 1 = ( √ s 2 , 0, 0, √ s 2 ), p µ 2 = ( √ s 2 , 0, 0, - √ s 2 ).
(2.22)

We take p 3 and p 4 in the xOz plane:

p µ 3 = (E 3 , p 3 sin θ, 0, p 3 cos θ),

E 3 = √ s/2, p 3 = s/4 -M 2 W (2.23)
Unlike the photon which has two transverse polarisation states the W particle being massive has three degrees of polarisation.

• Polarisation of a massive spin 1 particle

In the rest frame of a massive particle, p = (M, 0) the polarisation is described by space-like vectors.

A basis of such vectors is given by ε (1)µ = (0, 1, 0, 0), ε (2)µ = (0, 0, 1, 0), ε (3)µ = (0, 0, 0, 1), (2.24) satisfying ε (i) .ε (j) = -δ ij as well as p.ε (i) = 0 for i, j = 1, 2, or 3.

4 For e -e + there are two diagrams

Important remark

For a boson W with momentum p, the polarisation vectors become functions of p, ε (i)µ (p), satisfying the same conditions as above, namely ε (i) (p).ε (j) (p) = -δ ij as well as p.ε (i) (p) = 0. One often needs, in the propagator for example,

P µν = i ε (i)µ (p) ε (i)ν (p) = -g µν - p µ p ν M 2 W .
(2.25)

The last equality is easily derived knowing that the rank 2 tensor P µν depends only on the vector p µ so that it is of the form ag µν + bp µ p ν : the conditions p 2 = M 2 W , p µ P µν = p ν P µν = 0 and P µ µ = -3 then determine a and b as given in eq. (2.25).

If the boson W has its momentum along the z-axis, p = (E, 0, 0, p) the polarisation vectors are boosted to:

ε (1)µ = (0, 1, 0, 0), ε (2)µ = (0, 0, 1, 0) transverse polarisations (2.26)

ε (3)µ = 1 M W (p, 0, 0, E) longitudinal polarisation.
(2.27)

For a boson W with a momentum making an angle θ in the zOx plane one simply has to make a rotation around the Oy axis, p = (E, p sin θ, 0, p cos θ), and the polarisation vectors become:

ε (1)µ (p) = (0, cos θ, 0, -sin θ), ε (2)µ (p) = (0, 0, 1, 0) transverse polarisations (2.28)

ε (3)µ (p) = 1
M W (p, p sin θ, 0, E cos θ) longitudinal polarisation.

(2.29)

In the high energy limit, in the frame of eqs. (2.22), E ≃ p ≃ √ s/2 ≫ M W , the longitudinal polarisation vector simplifies to:

ε (3)µ (p) ≈ 1 M W ( √ s/2, √ s/2 sin θ, 0, √ s/2 cos θ) ≈ p µ M W .
(2.30)

We use this approximation in the calculaion below. For convenience we introduce the notation ε (1)µ or ε (2)µ = ε µ T , and ε (3)µ = ε µ L .

In contrast, we recall that a massless spin 1 particle has only two states of transverse polarisation.

• Production of massive vector bosons

After all these kinematic preliminaries we turn to the evaluation of the matrix element. Remembering the γ µ (1 -γ 5 ) coupling of the W boson to the fermions, the matrix element for the diagram above with the electron exchange is:

M ij = -2ig 2 W v(p 2 ) ε (j) (p 4 )( p 1 -p 3 ) ε (i) (p 3 )(1 -γ 5 )u(p 1 ) (p 1 -p 3 ) 2 (2.31)
where we have pushed the (1 -γ 5 ) factors to the right, hence the factor 2. Without doing the calculation explicitely one can guess that the matrix element squared will contain terms of the form :

|M ij | 2 ∝ g 4 W ((p 1 -p 3 ) 2 ) 2 {(p 1 .ε (i) (p 3 ) p 2 .ε (j) (p 4 )) 2 , • • • , (p k .p l ) (p 1 .ε (j) (p 3 )) 2 ε (j) 2 (p 4 ), • • • , (p k .p l ) (p m .p n ) (ε (i) (p 3 ).ε (j) (p 4 )) 2 , • • • }, (2.32)
with p k , p l , • • • any of the external momenta. In the limit √ s ≫ M W , it is easy to see that, if both polarisation vectors are transverse, all expressions such as:

p 1 .ε T (p 3 ) p 2 .ε T (p 4 ) ∝ (p 1 .ε T (p 3 )) 2 ∝ s, (2.33)
since all components of the transverse polarisation vectors are of order 1 and the momenta are generically of order √ s. If, on the contrary, both W ′ s are longitudinally polarised, the components of the polarisation vectors being of O( √ s/M W ) one finds:

p 1 .ε L (p 3 ) p 2 .ε L (p 4 ) ∝ ((p 1 .ε L (p 3 )) 2 ∝ s 2 M 2 W .
(2.34)

In consequence (p 1 .p 3 ∝ s),

|M T T | 2 ∝ g 4 W , and
|M LL | 2 ∝ g 4 W s 2 M 4 W .
(2.35)

Asymptotically the matrix element squared for the production of longitudinal bosons grows very fast while it is bounded in the case of transverse bosons. Since integrating over phase space to obtain the total cross section brings a factor 1/s (see eq. (2.11)) we expect the production of two longitudinal W 's to violate unitarity. To calculate effectively this cross section, one has to be a bit more refined and to go back to eq. (2.31) using the form eq. (2.30) for the polarisation vectors:

M LL = -i g 2 W M 2 W v(p 2 ) p 4 p 4 -p 2 (p 2 -p 4 ) 2 + p 1 -p 3 (p 1 -p 3 ) 2 p 3 (1 -γ 5 )u(p 1 ),
(2.36)

where we have used the trivial equality p 1 -p 3 = p 4 -p 2 . Anticommuting the matrices so as to bring p 1 close to u(p 1 ) and use the Dirac equation p 1 u(p 1 ) = 0 and similarly for p 2 and v(p 2 ) we end up with:

M LL = -i g 2 W M 2 W v(p 2 ) M 2 W -2p 2 .p 4 (p 2 -p 4 ) 2 p 3 + p 4 2p 1 .p 3 -M 2 W (p 1 -p 3 ) 2 (1 -γ 5 )u(p 1 ) = -i g 2 W M 2 W v(p 2 ) [ p 3 -p 4 ] (1 -γ 5 )u(p 1 ) (2.37)
Averaging on the initial polarisations one finds:

Σ|M LL | 2 = 1 4 g 4 W M 4 W 2 Tr( p 2 ( p 3 -p 4 ) p 1 ( p 3 -p 4 )(1 -γ 5 )) = 2 g 4 W M 4 W Tr( p 2 p 3 p 1 p 3 (1 -γ 5 )) = 16 g 4 W M 4 W p 1 .p 3 p 2 .p 3 = g 4 W M 4 W s 2 (1 -cos 2 θ),
(2.38) in the limit s ≫ M 2 W . It is then easy to obtain the differential cross section using the phase space factor of eq. (2.11) and then the integrated cross section for ν e νe → W L W L :

σ(ν e νe → W + L W - L ) = g 4 W 24π s M 4 W .
(2.39)

In contrast one can estimate the cross section of ν e νe → W T W T (but it is more tedious and is left as an exercise):

σ(ν e νe → W + T W - T ) ∝ g 4 W M 2 W when s → ∞.
(2.40)

We thus find that the production of longitudinally polarised vector bosons violates the unitarity limit while that of transverse bosons is well behaved at high energies. Several ways have been tried to cure this problem: among them one can mention the hypothesis of a new heavy lepton (fig. 1b) and choose its couplings to enforce a proper behaviour of the cross section at high energies. It turns out that another possibility, namely that of a heavy neutral vector boson, denoted Z (fig. 1c), is realised in Nature. Assuming the Z coupling to the fermions of type g Z γ µ (a -bγ 5 ) and to the charged W bosons of type 5 g ′ Z (p ρ W + -p ρ W -)g µν + • • • , they can be chosen to make cross sections such as ν e νe → W + W -, e + e -→ W + W -, • • • asymptotically well behaved. However, this patch up job is not yet sufficient to have a satisfactory model. Indeed, keeping fermion masses and considering for exemple e + e -→ Z Z scattering one finds an interference piece in the cross section ∼ g 4 m e √ s/M 4 Z which again violates unitarity! Similar problems arise in the W W scattering process, e.g. W + W -→ W + W -which are studied at LHC or will be in the future e + e -high energy linear colliders: the cross sections for these processes diverge linearly in s. These problems can be solved by supposing the existence of a scalar particle which interacts with the bosons as well as the fermions with appropriately chosen couplings.

ν e νe W + W - p 1 p 2 p 3 p 4 e - ν e νe W + W - E - W + W - ν e νe Z (a) (b) (c)
One can thus construct a viable electroweak theory in the pedestrian way described above, carefully choosing masses and couplings of the newly introduced particles so as to ensure the correct behaviour of all cross sections. It is more instructive however to assume that these relations among masses and couplings arise from some symmetry property. This is what is done next. Before doing that, one should discuss the implications of the γ µ (1 -γ 5 ) coupling in the weak interactions compared to the γ µ coupling of electrodynamics Then we describe in some details the symmetry group assuming global then local gauge invariance. At this level, the chosen group requires all fields to be massless. The theory is renormalisable (well behaved at asymptotic energies) being a non-abelian field theory. Then, by the mechanism of "spontaneous symmetry breaking" whereby the symmetry of the lagrangian is preserved but the choice of a ground state breaks the symmetry, fermions and gauge bosons acquire a mass. After symmetry breaking, the theory remains renormalisable as a consequence of the underlying gauge invariance which imposes the required relations between couplings. One is left however with a large number of parameters (at least 18 for the Standard Model with massless neutrinos and 25 with massive neutrinos) which gives a motivation for a (still unsuccessful!) search of a deeper symmetry.

5 Dimensional arguments and gauge invariance lead to such a choice.

3 Fermions, chirality, helicity

Fermions : chirality

We saw that the Fermi model involves charged transitions such as ψd γ µ (1 -γ 5 )ψ u ou ψe γ µ (1 -γ 5 )ψ νe , i.e. charged currents of a particular type : the fermion interacts only through the combination (1 -γ 5 )ψ. We can always write :

ψ = ψ -+ ψ + , with ψ -= 1 -γ 5 2 ψ, ψ + = 1 + γ 5 2 ψ (3.1)
The spinors ψ -and ψ + have a definite chirality defined by their transformation when applying γ 5 :

γ 5 ψ -= -ψ -, γ 5 ψ + = ψ + . (3.2)
ψ -, ψ + have negative, positive chirality respectively. The combinations

P -= 1 -γ 5 2 , P + = 1 + γ 5 2 (3.3)
are projection operators satisfying:

P + + P -= 1, P + P -= 0, (P + ) 2 = P + , (P -) 2 = P -. (3.4)
Only negative chirality fermions are sensitive to the weak interactions. It is useful to note that:

ψ -= ψ 1 + γ 5 2 , ψ + = ψ 1 -γ 5 2 .
(3.5)

Fermions : positive and negative energy solutions

When using the plane wave decomposition of the spinor, eq. (2.6), the free Dirac equation (i ∂ -m)ψ = 0 implies:

( p -m) u α (p) = 0, ( p + m) v α (p) = 0 (3.6)
on the positive (u α exp(-ipx), see eq. (2.6)) and negative (v α exp(ipx)) energy component respectively.

At rest, p = 0, and using the Dirac representation of γ µ matrices given in appendix, they reduce to:

m(γ 0 -1)u α ⇒ 0 0 0 -21 1 2 χ α 0 = 0 m(γ 0 + 1)v α ⇒ 21 1 2 0 0 0 0 χ α = 0 (3.7)
where we have introduced the 2-component spinors :

χ 1 = 1 0 , χ 2 = 0 1 and u α = χ α 0 , v α = 0 χ α , α = 1, 2. (3.8)
Since one has τ 3 χ 1 = χ 1 , τ 3 χ 2 = -χ 2 one says that χ 1 has spin up and χ 2 spin down and6

1 + τ 3 2 and 1 -τ 3 2 (3.9) are respectively the spin up and spin down projection operators for the 2-component spinors. When p = 0, to obtain the spinors u α (p) and v α (p) one can apply a Lorentz boost to the solution in the rest frame or, more simply, observe that:

u α (p) = 1 √ ω + m ( p + m)u α , v α (p) = 1 √ ω + m (-p + m)v α , (3.10)
satisfy eqs. (3.6) respectively. The factor 1/ √ ω + m is the chosen normalisation factor such that:

ūα (p) u β (p) = 2m δ αβ , u † α (p) u β (p) = 2ω δ αβ , vα (p) v β (p) = -2m δ αβ , v † α (p) v β (p) = 2ω δ αβ . (3.11)
Explicitely, one has in terms of two component spinors:

u α (p) = 1 √ ω + m (ω + m) χ α p.τ χ α , v α (p) = 1 √ ω + m p.τ χ α (ω + m) χ α .
(3.12)

The solution u α (p) is the positive energy spinor while v α (p) is called the negative energy one with momentum (-ω, -p). In particular, for a boost of magnitude η in the z direction, the positive energy spinors have momentum p = (ω, 0, 0, p z ), with ω = m cosh η, p z = m sinh η, and they become:

u α (p) = 1 √ ω + m (ω + m)χ α p z τ 3 χ α ⇒ u 1 (p) = 1 √ ω + m     ω + m 0 p z 0     , u 2 (p) = 1 √ ω + m     0 ω + m 0 -p z     , (3.13)
while the negative energy solutions with momentum -p are:

v α (p) = 1 √ ω + m p z τ 3 χ α (ω + m)χ α ⇒ v 1 (p) = 1 √ ω + m     p z 0 ω + m 0     , v 2 (p) = 1 √ ω + m     0 -p z 0 ω + m     . (3.14)
In general, it is useful to introduce operators which project out positive and negative energy states.

They are defined by:

Λ ± = ± p + m 2m , (3.15)
and they satisfy the required relations:

Λ -(p) + Λ + (p) = 1 , Λ -(p)Λ + (p) = Λ + (p)Λ -(p) = 0 , (Λ -(p)) 2 = Λ -(p) , (Λ + (p)) 2 = Λ + (p) .
(3.16) Thus Λ ± ψ(p, x) respectively project the positive and negative energy solutions of ψ(x) in eq. (2.6).

We discuss in appendix B.1 the interpretation of the negative energy solution as a positive energy antiparticle.

Fermions : helicity

When applying a boost along the z-axis one does not change the orientation of the fermion spin, as shown in the figure below, so that the projection of the fermion spin along the momentum is positive

v 1 (p) v 2 (p)

helicity-helicity+ -p -p

For a spinor of momentum p one defines the helicity projection operator

S ± (p) = 1 ± T . p 2 , T = τ 0 0 τ , p = p |p| (3.17)
with p the unit vector in the direction of the momentum. Applying these operators to the positive energy spinors one finds ((τ p) 2 = 1):

S ± (p)u α (p) = 1 2 √ ω + m (ω + m)(1 ± τ p)χ α p(±1 + τ p)χ α . (3.18)
If p is in the direction of Oz, the helicity projection operators applied on the spinors reduce to

S ± (p)u α (p) = 1 2 √ ω + m (ω + m)(1 ± τ 3 )χ α p(±1 + τ 3 )χ α , (3.19)
showing that u 1 (p) is right-handed, and u 2 (p) is left-handed as found before. For negative energy spinors, since they have momentum -p, S + (-p) projects out positive helicity and S -(-p) projects our negative helicity. The operators S ± are helicity projection operators and satisfy:

(S ± (p)) 2 = S ± (p), S + (p) S -(p) = 0, S + (p) + S -(p) = 1 1 2 (3.20)
• Massless spinors : helicity and chirality

In the Standard Model, at high energy, quarks of light flavours and neutrinos are often treated as massless. Considering massless spinors with a generic momentum p one has:

u α (p) = √ ω χ α p.τ χ α , v α (p) = √ ω p.τ χ α χ α , α = 1 or 2. (3.21)
When acting on positive energy spinors u α (p), the helicity projection operator and P ± , the chirality projection operators of eq. (3.3), give the same result:

S ± (p) u α (p) = P ± u α (p) = √ ω 2 (1 ± τ p)χ α (±1 + τ p)χ α , α = 1, 2 ,
This shows that positive chirality and right-handed helicity are the same and likewise for negative chirality and left-handed helicity. For spinors v α (p) one finds instead: To summarise, in the massless case, from the definition of ψ(x) in eq. (2.6), the combination

S ± (-p) v α (p) = P ∓ v α (p) = √ ω 2 (∓1 + τ p)χ α (1 ∓ τ p)χ α , α = 1, 2 ,
ψ L (x) = 1 -γ 5 2 ψ(x) (3.22)
-destroys a left-handed fermion, with wave function u L (p) and creates a right-handed antifermion with wave function v R (p), eqs. (3.28), (3.30),

ψ L (x) = d 3 p (2π) 3 2ω b L (p) u L (p) e -ip.x + d † R (p) v R (p) e ip.x (3.23)
and mutatis mutandis:

ψ R (x) = 1 + γ 5 2 ψ(x) (3.24)
-destroys a right-handed fermion, with wave function u R (p) and creates a left-handed antifermion with wave function v L (p).

ψ R (x) = d 3 p (2π) 3 2ω b R (p) u R (p) e -ip.x + d † L (p) v L (p) e ip.x (3.25)
Thus, the Fermi interaction, discussed in the previous section, concerns only left-handed fermions and right-handed antifermions.

• Massless chiral spinors

It is easy and amusing (as well as useful for neutrino physics) to find the explicit form of massless chiral spinors of arbitrary momentum. For instance, for positive energy spinors one has, using expressions (3.12) :

γ 5 u R (p) = u R (p) ⇒ p.τ χ R = χ R γ 5 u L (p) = -u L (p) ⇒ p.τ χ L = -χ L , (3.26)
for right-handed and left-handed spinors respectively. Solving for p.τ χ = ±χ, we get the 2-component spinors after proper normalisation:

χ R = cos θ 2 e -i φ 2 sin θ 2 e i φ 2 χ L = -sin θ 2 e -i φ 2 cos θ 2 e i φ 2 , (3.27) and thus, u R (p) = √ ω χ R χ R u L (p) = √ ω χ L -χ L , (3.28)
One follows the same procedure for negative energy spinors, but since their momentum is -p they satisfy

γ 5 v R (p) = -v R (p), γ 5 v L (p) = v L (p) (3.29)
and, compared to the u(p) spinors, the role of χ R and χ L is interchanged so that:

v R (p) = √ ω -χ L χ L v L (p) = - √ ω χ R χ R , (3.30)
The relations

χ † R χ R = χ † L χ L = 1, χ † R χ L = χ † L χ R = 0 ensure that eqs.
(3.11) are satisfied.

• Massive spinors : helicity and chirality

In general, if in the rest-frame of the fermion the polarisation direction is given by the vector s = (0, s)

with s 2 = -1, s.p = 0, the spin projection operators along or opposite s are given, in a covariant form, by

Σ ± (s) = 1 ± γ 5 s 2 . (3.31)
Specifying to the helicity, the spin projection along or opposite the fermion momentum, one defines

s = ( p m , ω m p), with p = |p| and p = p p , (3.32)
(which satisfies the conditions s 2 = -1, s.p = 0) and Σ ± (s) takes the form:

Σ ± (s) = 1 2m m ± ωp.τ ∓p ±p m ∓ ωp.τ . (3.33)
The form of the projectors Σ ± (s) is different from the helicity projection operators defined in eq.

(3.17) but when acting on positive energy spinors u(p), one shows that:

Σ ± (s) u α (p) = S ± (p) u α (p), α = 1, 2 (3.34)
Thus, for positive energy spinors, Σ + projects out right-handed states and Σ -the left-handed ones.

Similarly, when acting on negative energy spinors v(p), one finds that,

Σ ± (s) v α (p) = S ± (-p) v α (p), α = 1, 2 (3.35)
related to the fact that negative energy spinors carry momentum -p. Thus, again, Σ + projects out the right-handed helicity state and Σ -the left-handed ones.

For massive spinors at very high energy if one uses (1 ± γ 5 )/2 as helicity projection operators rather than Σ ± (s), with s as defined in eq. (3.32), the error made is of O(m/ω)7 .

In summary, it is easy to see that the fermion wave-functions:

ψ f R (p, x) = Σ + (s) p + m 2m ψ(p, x)
destroys a right-handed fermion

ψ f L (p, x) = Σ -(s) p + m 2m ψ(p, x) destroys a left-handed fermion ψ f R (p, x) = Σ + (s) -p + m 2m ψ(p, x)
creates a right-handed antifermion

ψ f L (p, x) = Σ -(s) -p + m 2m ψ(p, x)
creates a left-handed antifermion.

(3.36) This summary will prove useful when discussing C and CP violation later.

Application

The helicity arguments above and conservation of angular momentum are useful to understand/predict the angular dependence of a process governed by the γ µ (1 -γ 5 ) interaction which carries total angular momentum 1 (L = 0, S = 1). For example, coming back to the processes ν e d → e -u and ν e u → e + d, eqs. (2.8) and (2.12), the leptonic transition is given by ψ e γ µ (1 between the incoming and outgoing leptons in the ν d center of mass frame, the spin projection of the system along the axis of motion of the particles is 0 because each particle has a negative helicity and they move in opposite directions (see the figure). Therefore we expect no angular dependence for the cross section, in agreement with eq. (2.11). On the contrary, for the scattering ν e u → e + d the antileptons being right-handed and the quarks left-handed the spin projection of the antileptonquark system along the direction of motion of the antilepton is always 1 : for a forward produced e + the angular momentum projection along the z axis is 1 for both initial and final states and thus is conserved while for a backward produced e + (θ = π) the spin projection of the final system along the z axis is -1, and angular momentum is not conserved, consequently the matrix element vanishes.

-γ 5 )ψ νe = 2 ψ e L γ µ ψ νe L or its hermitian conjugate ψ νe γ µ (1 -γ 5 )ψ e = 2 ψ νe L γ µ ψ e L .
From Clebsh-Gordan tables8 the associated angular distribution is proportional to d 1 11 (θ) ≃ 1 + cos θ, in agreement with eq. (2.13). We note the useful relations :

ψ L γ µ ψ L = 1 2 ψγ µ (1 -γ 5 )ψ ψγ µ ψ = ψ L γ µ ψ L + ψ R γ µ ψ R , ψ L γ µ ψ R = ψ R γ µ ψ L = 0 ψψ = ψ R ψ L + ψ L ψ R , ψ R ψ R = ψ L ψ L = 0. ( 3

Global gauge invariance and Noether theorem

One starts from the lagrangian density L(ψ(x), ∂ µ ψ(x)), which is a fonction of the field and its first derivatives, and from the action defined by

S = d 4 xL(ψ(x), ∂ µ ψ(x)).
The action has no dimension. The Maupertuis principle (least action principle) states that, "in Nature", the action is stationary under a variation of the field and this leads to the Euler-Lagrange

equations δL δψ(x) -∂ µ δL δ∂ µ ψ(x) = 0 (4.1)
Now, assume that the lagrangian density is invariant under the rigid transformation

ψ(x) → e iα ψ(x) ψ(x) → e -iα ψ(x), (4.2)
where α is a real arbitrary constant, independent of the space-time coordinate x. Considering rather

an infinitesimal transformation δψ(x) = iαψ(x) δψ(x) = -iαψ(x), (4.3)
the variation of the lagrangian is9 (note the relative position of the derivative term such as δL/δψ and the δψ)

δL(ψ, ∂ µ ψ) = δL δψ δψ + δL δ∂ µ ψ δ∂ µ ψ + δψ δL δψ + (δ∂ µ ψ) δL δ∂ µ ψ . (4.4) But, δ∂ µ ψ = ∂ µ δψ = iα∂ µ ψ, δ∂ µ ψ = ∂ µ δψ = -iα∂ µ ψ
, and using the Euler-Lagrange equations to eliminate δL/δψ and δL/δψ one finds

δL(ψ, ∂ µ ψ) = iα∂ µ δL δ∂ µ ψ ψ -ψ δL δ∂ µ ψ . (4.5)
Since δL(ψ, ∂ µ ψ) = 0 under the variation of the fields, eqs. (4.3), the current defined by

J µ (x) = δL δ∂ µ ψ ψ -ψ δL δ∂ µ ψ (4.6)
is conserved, i.e. ∂ µ J µ (x) = 0 . For a fermion field with the langrangien density

L(ψ, ∂ µ ψ) = ψ(i ∂ -m)ψ
the conserved current is simply

J µ (x) = ψγ µ ψ. (4.7)
One defines the charge by the space integration of the 0th component of the current, and specifying

x = (t, x), d 4 x = dtd 3 x, one has Q(t) = d 3 xJ 0 (t, x) = d 3 xψ † (t, x)ψ(t, x). (4.8)
Using current conservation, ∂ µ J µ (x) ≡ ∂ t J 0 (t, x) + ∇ • J(t, x) = 0, it is easy to prove that the charge is time independent since

dQ(t) dt = - Ω d 3 x∇ • J(t, x) = - ∂Ω ds • J(t, x) = 0, (4.9)
where the last equality is realised when we assume the fields are suppressed at infinity.

Thus the Noether theorem states that to an invariance under a set of continuous transformations corresponds a conserved current. The results eqs. (4.6), (4.7) are easily extended to the case of non abelian symmetries such as SU (2), • • • , SU (N ) or to a lagrangian density involving several fields ψ i .

Then if

L(ψ, ∂ µ ψ) = i ψ i (i ∂ -m i )ψ i (4.10)
is invariant under the set of transformations

δψ i = i α y i ψ i , ∀i δψ i = -i α y i ψ i , ∀i, (4.11)
with α a common real parameter and y i the charge of field ψ i , the conserved current is

J µ (x) = i y i ψ i γ µ ψ i , (4.12)
a result to be used later. The charge of fermion ψ i is then defined as y i .

The lagrangian density

As discussed above, the weak interactions induce a transition between left-handed fermions of different charges. It is then natural to group them into doublets

ν e L e - L , ν µ L µ - L , ν τ L τ - L leptons ; u L d L , c L s L , t L b L quarks (4.13)
We introduce the left handed doublets:

Ψ e L = ν e L e L , • • • Ψ q L = u L d L , • • • , (4.14)
and the right-handed singlets

ψ e R = e R , • • • , ψ q R = q R , • • • .
We assume all fermions are massless. The free massless fermion lagrangian is then written,

-iL F = ē ∂ e + νe L ∂ ν e L + ū ∂ u + d ∂ d (4.15)
where we have kept the first family of fermions ν e , e, u, d only and where we have ignored the righthanded neutrino ν e R not observed experimentally. Using the second of eq. (3.37) and regrouping the members of a doublet one finds

-iL F = Ψ e L ∂ Ψ e L + Ψ q L ∂ Ψ q L + e R ∂ e R + u R ∂ u R + d R ∂ d R (4.16)
For massless fermions the charged current introduced by Fermi, eq. (2.2), is conserved so it is tempting to introduce a global symmetry associated to this current.

The global SU(2) L gauge invariance

It is obvious that the lagrangian above is invariant under a global SU(2) phase change of the left handed fermion fields, i.e. under the transformation,

Ψ L → e i α•τ /2 Ψ L , Ψ L → Ψ L e -i α•τ /2 (4.17)
where the 2 × 2 Pauli matrices τ = (τ 1 , τ 2 , τ 3 ) satisfy the algebra

τ i 2 , τ j 2 = i ǫ ijk τ k 2 , (4.18)
and have the following properties

τ = τ † , Tr(τ i τ j ) = 2δ ij (4.19)
The parameter α = (α 1 , α 2 , α 3 ) is a set of 3 arbitrary constants. As discussed above to a global symmetry is associated a conserved current. The SU (2) group has three generators and there are three conserved currents. Following the reasoning leading to eq. (4.6) they are identified to

J µ i (x) = Ψ e L γ µ τ i 2 Ψ e L + Ψ q L γ µ τ i 2 Ψ q L . (4.20)
They are called the "weak isospin currents". The first two, J µ 1 (x), J µ 2 (x), are related to the currents introduced by E. Fermi to describe the weak interaction: for example, using the first of the eq. (3.37) identities, J µ 1 (x) is written

J µ 1 (x) = 1 2 (e L γ µ ν e L + d L γ µ u L + h.c.), = 1 4 (eγ µ (1 -γ 5 )ν e + dγ µ (1 -γ 5 )u + h.c.),
which together with J µ 2 (x) allows to reconstruct eq. (2.2). The third one is new, it is a neutral current,

J µ 3 (x) = 1 2 [ν e L γ µ ν e L -e L γ µ e L + u L γ µ u L -d L γ µ d L ]. (4.21)
The corresponding weak isopin charge is given by,

I 3 = d 3 xJ 0 3 (x) = 1 2 d 3 x(ν † e L ν e L -e † L e L + u † L u L -d † L d L ), (4.22)
which allows to assign a charge I 3 = +1/2 to the neutrino and the u quark and I 3 = -1/2 to the electron and the d quark. Obviously J µ 3 (x) cannot be the current coupling to the photon field otherwise the neutrino would interact with the photon ! J µ 3 (x) is a neutral current since it does not change the charge of the fermion.

The global U(1) Y gauge invariance

The lagrangian L F of eq. (4.15) is invariant under a U (1) global transformation acting on all fields, left and right. It is called called the U (1) Y group, where Y refers to the hypercharge. A transformation is defined by :

Ψ e L → e i βy e L /2 Ψ e L , Ψ q L → e i βy q L /2 Ψ q L e R → e i βy e R /2 e R , u R → e i βy u R /2 u R , d R → e i βy d R /2 d R , (4.23)
where the y e L , y q L , y e R , y u R , y d R are the hypercharges of the corresponding fields. The associated conserved current writes (see eq. (4.12))

J µ Y (x) = y e L Ψ e L γ µ Ψ e L + y q L Ψ q L γ µ Ψ q L + y e R e R γ µ e R + y u R u R γ µ u R + y d R d R γ µ d R . (4.24)
Since the sum of conserved currents is also a conserved current we can construct the electromagnetic current, J µ emg (x) = e e eγ µ e + e u uγ µ u + e d dγ µ d, (4.25)

as the sum of the weak isospin and hypercharge currents 10 :

J µ emg (x) = J µ 3 (x) + J µ Y (x) 2 , (4.26)
with the hypercharges of the fields chosen so as to construct their correct electric charges (which are normalised here to the charge of the proton). For the lepton sector, for exemple, one finds -1

for the left-handed doublet and -2 for the right-handed electron partner to get a charge of -1 for both left-handed and right-handed component of the electron and 0 for the neutrino. The results are summarised in the following table :

I I 3 Y Q ν e 1/2 1/2 -1 0 e L 1/2 -1/2 -1 -1 e R 0 0 -2 -1 u L 1/2 1/2 1/3 2/3 d L 1/2 -1/2 1/3 -1/3 u R 0 0 4/3 2/3 d R 0 0 -2/3 -1/3 (4.27)
which shows that the relation between the charge, hypercharge and weak isospin satisfies, by construction, the famous Gell-Mann/Nishijima relation :

Q = I 3 + Y 2 (4.28)
10 the facteur 1/2 associated to the hypercharge current is historically conventional.

• Application

In general, for a SU (2) doublet Φ T = (φ 1 , φ 2 ) of fields of hypergharge y Φ and electric charges (e 1 , e 2 ), the Gell-Mann/Nishijima relation yields

e 1 -e 2 =
Ψ ′ L → U (x)Ψ L = e i g α(x)•τ /2 Ψ L , Ψ ′ L → Ψ L U † (x) = Ψ L e -i g α(x)•τ /2 , (5.1)
with U U † = 1, or, for an infinitesimal transformation,

δΨ L = i g α(x) • τ 2 Ψ L , δΨ L = -i g ψ L α(x) • τ 2 , (5.2)
where the 3 components of the real parameter α(x) are functions of the space-time coordinates. We have introduced a coupling g associated to this transformation. Under the local transformation the lagrangian density (4.15) is no longer invariant because of the derivative term in ∂ µ α(x)

δL F = Ψ e L {-g(∂ µ α(x)) • τ 2 }γ µ Ψ e L + Ψ q L {-g(∂ µ α(x)) • τ 2 }γ µ Ψ q L (5.3)
To recover the invariance of L F under this transformation one introduces a multiplet (a triplet) of gauge vector fields

W µ (x) = (W µ 1 (x), W µ 2 (x), W µ 3 (x)
) and defines the covariant derivative operating only on the left-handed fields :

D µ L = ∂ µ -igW µ (x), with W µ (x) = τ 2 • W µ (x).
(5.4)

The transformation properties of W µ (x) are chosen such that the lagrangian density

L F = Ψ e L D L Ψ e L + Ψ q L D L Ψ q L + ψ e R ∂ ψ e R + ψ u R ∂ ψ u R + ψ d R ∂ ψ d R (5.5)
is invariant under an SU (2) transformation. Since the right-handed fields are not affected by the transformation it is enough to impose that D µ L Ψ(x) transforms as Ψ(x) to achieve the invariance of the lagrangian:

(D µ L Ψ(x)) ′ = U (x)(D µ L Ψ(x)).
(5.6) Therefore,

(D µ L Ψ(x)) ′ = (D µ L ) ′ U (x)Ψ(x)) = U (x)D µ L Ψ(x), (5.7) implies (D µ L ) ′ = U (x)D µ L U -1 (x), (5.8)
since it should hold for all Ψ(x). Consequently, using

∂ µ U -1 (x) = (∂ µ U -1 (x)) + U -1 (x)∂ µ , one finds (D µ L ) ′ = ∂ µ + U (x)(∂ µ U -1 (x)) -igU (x)W µ (x)U -1 (x), (5.9)
which can be written as

(D µ L ) ′ = ∂ µ -igW ′ µ (x) with W ′ µ (x) = i g U (x)(∂ µ U -1 (x)) + U (x)W µ (x)U -1 (x) (5.10)
Restricting to the infinitesimal transformations eq. (5.2), one obtains

W ′ µ (x) -W µ (x) = δW µ (x) = ∂ µ α(x) • τ 2 + ig [α(x) • τ 2 , W µ (x)],
(5.11) which, in terms of SU (2) components, is equivalent to

δ W µ i (x) = ∂ µ α i (x) -g ǫ ijk α j (x) W µ k (x).
(5.12)

spinor we have D ′ µ L U = U D µ L , hence eq. (5.8).

To construct the kinetic term of the gauge bosons W µ i (x) we first consider, as in QED, the tensor

F µν (x) = [D µ L (x), D ν L (x)] (5.13) Using Leibnitz rule ∂ µ W ν i (x) = (∂ µ W ν i (x)) + W ν i (x)∂ µ it
is easy to show that the tensor is given by

F µν (x) = ∂ µ W ν (x) -∂ ν W µ (x) -ig[W µ (x), W ν (x)]
(5.14) or in components

F µν i (x) = ∂ µ W ν i (x) -∂ ν W µ i (x) + g ǫ ijk W µ j (x)W ν k (x).
(5.15)

The transformation property of F µν (x) is obviously the same as that of D µ L , eq. (5.8), and we have then

F ′ µν (x) = U F µν (x)U -1 so that TrF µν (x)F µν (x) = 1 2 F µν i (x)F iµν (x) (5.16)
is a Lorentz scalar invariant under a gauge transformation by the property of cyclicity of the trace.

Furthermore it has the right dimension to be the kinetic term of the W µ i bosons. The lagrangian of left-handed fields becomes then :

L F L = - 1 4 F µν i (x)F iµν (x) + Ψ e L iD µ L γ µ Ψ e L + Ψ q L iD µ L γ µ Ψ q L .
(5.17)

where each of the three terms is invariant under a local SU (2) transformation. We note at this point the perfect analogy between the construction of the "weak" lagrangian with that of QCD: the differences are in the choice of group which requires here only three vector bosons while for SU (3) symmetry eight bosons had to be introduced. Also, the SU (2) group acts only on the left handed components of the fields and consequently the W µ i (x) gauge bosons do not couple to the right handed fermion components.

We now make the U (1) Y gauge transformation local. It is defined by

δΨ e L = ig ′ y e L 2 β(x) Ψ e L , δΨ q L = ig ′ y q L 2 β(x) Ψ q L δe R = ig ′ y e R 2 β(x) e R , δu R = ig ′ y u R 2 β(x) u R , δd R = ig ′ y d R 2 β(x) d R , (
D µ ψ R = ∂ µ -i g ′ y ψ R 2 B µ , (5.19)
while for the left handed fields the covariant derivative eq. (5.4) acquires a new piece and becomes :

D µ ψ L = ∂ µ -i g τ 2 • W µ -i g ′ y ψ L 2 B µ .
(5.20)

The stress-energy tensor of the new vector field is simply :

K µν (x) = ∂ µ B ν (x) -∂ ν B µ (x) (abelian field).
(5.21)

In summary, the initial free lagrangian eq. (4.15) becomes, after imposing a SU (2) local symmetry on the left-handed fields and an appropriate U (1) invariance on both the left-handed fields and a right-handed ones,

L = L G + L F = - 1 4 F iµν (x) F µν i (x) - 1 4 K µν (x) K µν (x) + Ψ e L i D e L Ψ e L + Ψ q L i D q L Ψ q L + + e R i D e R e R + u R i D u R u R + d R i D d R d R (5.22)
where only the (e, ν e ) and (u, d) quark family has been specified. It is important to point out that the SU (2) L ⊗ U (1) Y invariance imposes that all fermions are massless. Indeed a fermion mass term 11 The left and right covariant derivatives generically defined as

D µ L , D µ R are now denoted D µ ψ L , D µ ψ R
since they depend on the quantum numbers of the fermion fields ψ L , ψ R .

in the lagrangian would have the form

L mass = m ψψ = m(Ψ L ψ R + ψ R Ψ L ).
(5.23)

But since Ψ L is a doublet and ψ R a singlet under SU (2), the mass term cannot be invariant under a gauge transformation! It is useful to separate the lagrangian density eq. (5.22) into a free part

L 0F + L 0G = Ψ e L i ∂ Ψ e L + Ψ q L i ∂ Ψ q L + e R i ∂ e R + u R i ∂ u R + d R i ∂ d R (5.24) - 1 4 [(∂ µ W ν (x)-∂ ν W µ (x)) • (∂ µ W ν (x)-∂ ν W µ (x)) + (∂ µ B ν (x)-∂ ν B µ (x)) (∂ µ B ν (x)-∂ ν B µ (x))],
and an interacting part containing all terms depending on the couplings g and g ′ . It contains two classes of terms : one describing the fermion-gauge bosons interactions (which can be expressed very easily in terms of the currents introduced above) and the other the W boson self interactions

L IF +L IG = g J µ (x) • W µ (x) + g ′ J µ Y (x) 2 B µ (x) (5.25) - g 2 ǫ ijk (∂ µ W iν (x) -∂ ν W iµ (x))W µ j (x)W ν k (x) - g 2 4 ǫ ijk W jµ (x)W kν (x) ǫ ilm W µ l (x)W ν m (x)
with J µ the weak isospin current of eq. (4.20) and J µ Y the hypercharge current of eq. (4.24). One recognizes in the sum of these two terms the expression which lead to the construction of the electromagnetic current in eq. (4.25).

Fermion-boson interactions, construction of the photon and the Z boson

We turn first to the fermion-W µ interaction. It is read off from L IF and is simply

g J µ (x) • W µ (x) = g 2 Ψ e L γ µ τ i Ψ e L + Ψ q L γ µ τ i Ψ q L W iµ .
(5.26)

Defining the charged vector fields

(W ± ) µ (x) = (W µ 1 (x) ∓ iW µ 2 (x)) √ 2 , with (W + * ) µ (x) = (W -) µ (x) (5.27)
their interaction with the fermions can be easily obtained from the charge changing part of the currents (J µ 1 (x), J µ 2 (x)) in eq. (5.26) and we find

L IF (charged current) = g √ 2 (ν e L γ µ e L W + µ + u L γ µ d L W + µ + h.c.) (5.28) = g 2 √ 2 (ν e γ µ (1 -γ 5 ) e W + µ + u γ µ (1 -γ 5 ) d W + µ + h.c.), (5.29)
which is now expressed in terms of the usual fermion fields ν e , e, u, d. One can thus read off the W ± coupling to fermions : using standard techniques it is found to be -i(g/2 √ 2)γ µ (1 -γ 5 ), coupling with the same strength to all fermion species. (Note the relation g/2 √ 2 = g W of eq. (2.16)).

Turning now to the neutral vector bosons sector one has two pieces : one originates from the SU (2) L invariance, namely gJ µ 3 W 3µ contained in eq. (5.26), and the other one from the U (1) Y invariance, g ′ J µ Y B µ . From eq. (5.25) we can read off the neutral current interaction lagrangian which is

L IF (neutral currents) = gJ µ 3 W 3µ + g ′ 1 2 J µ Y B µ (5.30)
Note that the photon cannot be identified to the W 3µ field because of the γ 5 term in the coupling nor to the B µ boson because of the different charge assignment for the left and right component of a fermion field. The photon will be constructed as a linear combination of both. Thus, introducing the fields A µ and Z µ such that

B µ = cos θ A µ -sin θ Z µ W µ 3 = sin θ A µ + cos θ Z µ , (5.31)
with θ an adjustable parameter, one finds

L IF (neutral currents) = (g sin θJ µ 3 + g ′ cos θ 1 2 J µ Y )A µ + (g cos θJ µ 3 -g ′ sin θ 1 2 J µ Y )Z µ .
(5.32)

To construct the field A µ as the photon field we should adjust the parameters to be such that

g sin θJ µ 3 + g ′ cos θ 1 2 J µ Y = eJ µ emg (5.33)
where, by convention, e is taken as the charge of the proton. This can be achieved if we choose

g sin θ = g ′ cos θ = e (5.34)
since, then, we recover eq. (4.26) which lead to eq. (4.25) for J µ emg . With this choice, we have J µ Y /2 = J µ emg -J µ 3 which is used to eliminate in the coefficient of Z µ the hypercharge current so that the interaction lagrangien reads

L IF (neutral currents) = eJ µ emg A µ + e sin θ cos θ (J µ 3 -sin 2 θJ µ emg )Z µ , (5.35)
defining the couplings of the photon A µ (x) and the neutral Z µ (x) boson to the fermions. Concerning the Z µ couplings we can be more explicit and derive them for a pair of fermions ψ 1 , ψ 2 of charge e 1 , e 2 (normalised to the proton charge e) respectively, such that (ψ 1L , ψ 2L ) forms a SU (2) doublet (I = 1/2) and ψ 1R , ψ 2R are singlets (I = 0). Writing explicitely the currents J µ 3 and J µ emg , we have from eq. (5.35):

e sin θ cos θ (ψ 1 L ψ 2 L ) 1/2 -e 1 sin 2 θ 0 0 -1/2 -e 2 sin 2 θ Z ψ 1 L ψ 2 L +(ψ 1 R ψ 2 R ) -e 1 sin 2 θ 0 0 -e 2 sin 2 θ Z ψ 1 R ψ 2 R = e sin θ cos θ (ψ 1 ψ 2 ) 1/2 -e 1 sin 2 θ 0 0 -1/2 -e 2 sin 2 θ Z (1 -γ 5 ) 2
ψ 1 ψ 2 +(ψ 1 ψ 2 ) -e 1 sin 2 θ 0 0 -e 2 sin 2 θ Z (1 + γ 5 ) 2
ψ 1 ψ 2 = e sin θ cos θ (ψ 1 ψ 2 ) 1/4 -e 1 sin 2 θ 0 0 -1/4 -e 2 sin 2 θ Z ψ 1 ψ 2 -(ψ 1 ψ 2 ) 1/4 0 0 -1/4 Z γ 5 ψ 1 ψ 2 (5.36)
The full neutral current interaction lagrangian density eq. (5.35) can then be written for one generation of quarks and leptons L IF (neutral currents) = -e e A e + e sin θ cos θ l=ν,e l Z (a l -b l γ 5 ) l + e q=u,d e q q A q + e sin θ cos θ q=u,d q Z (a q -b q γ 5 )q

(5.37)

with

a i = I 3 2 -e i sin 2 θ, b i = I 3 2 .
(5.38)

Contrary to the photon which has a purely vector coupling to the fermions, the neutral gauge boson Z µ has both vector and axial-vector couplings. We recall that with the choice of g = e/ cos θ the charged W µ couplings are

L IF (charged current) = e 2 √ 2 sin θ (ν e γ µ (1 -γ 5 ) e W + µ + u γ µ (1 -γ 5 ) d W + µ + h.c.), (5.39)
These couplings are in agreement with those of the physical Z boson once the "weak mixing" or Weinberg angle θ (in fact introduced by Glashow!) is taken from experiment to be :

sin 2 θ ∼ .2313 .

(5.40)

We herafter denote the weak mixing angle by θ W .

• The covariant derivative in terms of the A µ , Z µ , W ± µ fields It is useful, for later use, to have an explicit representation of the covariant derivatives eqs. (5.19) and (5.20) in terms of the W ± µ , A µ and Z µ gauge bosons. Although they can be read off the previous discussion based on defining the electromagnetic current we construct them directly. For instance, the covariant derivative eq. (5.20) acting on a SU (2) doublet of fields with hypercharge y φ , the components of which having electric charge (ee 1 , ee 2 ), contains the piece

-ig τ 3 2 W 3µ -ig ′ y φ 2 B µ = -i (g sin θ W τ 3 2 + g ′ cos θ W y φ 2 )A µ + (g cos θ W τ 3 2 -g ′ sin θ W y φ 2 )Z µ (5.41)
For A µ to be the photon one imposes the conditions

1 2 (g sin θ W + g ′ y φ cos θ W ) = e e 1 g ′ y φ cos θ W = e(e 1 + e 2 ) g ′ cos θ W = e ⇒ ⇒ 1 2 (-g sin θ W + g ′ y φ cos θ W ) = e e 2 g sin θ W = e(e 1 -e 2 ) = e g sin θ W = e, (5.42)
where the rightmost equalities are a consequence, eq. (4.29), of the Gell-Mann/Nishijima relation.

Eliminating g, g ′ , y φ in favour of e, θ W and the charges one finds

-ig τ 3 2 W 3µ -ig ′ y φ 2 B µ = -ie e 1 A µ 0 0 e 2 A µ - i e sin θ W cos θ W 1 2 -e 1 sin 2 θ W Z µ 0 0 -1 2 -e 2 sin 2 θ W Z µ
Going back to the full expression, eq. (5.20), including the W ± µ contribution, the covariant derivative on a doublet field is

D µ = ∂ µ -i e √ 2 sin θ W 0 W + µ W - µ 0 -ie e 1 A µ 0 0 e 2 A µ -i e sin θ W cos θ W ( 1 2 -e 1 sin 2 θ W )Z µ 0 0 (-1 2 -e 2 sin 2 θ W )Z µ (5.43)
Since, by definition, W - * µ = W + µ , from now on we use the notation W - µ = W µ and W + µ = W * µ to respectively represent the wave functions of the W -and W + gauge bosons.

The covariant derivative acting on a singlet φ is simply

D µ = ∂ µ -ig ′ y φ 2 B µ = ∂ µ -ie e φ A µ + i e e φ sin 2 θ W sin θ W cos θ W Z µ (5.44)

Gauge bosons and their self-interactions

We already identified in eq. (5.24) the free gauge boson pieces L 0G and in eq. (5.25) the interacting terms L IG . We now reformulate these expressions in terms of the "physical" fields W * µ , W µ , Z µ and A µ . For this purpose we rewrite L 0G by doing an integration by part and neglecting, as usual, the terms which are total derivatives, we find

L 0G = 1 2 W iµ (x)D µν W iν (x) + 1 2 B µ (x)D µν B ν (x), (5.45)
with D µν = g µν -∂ µ ∂ ν . This is rewritten in a matrix form

L 0G = 1 2 (W 1µ W 2µ ) D µν 0 0 D µν W 1µ W 2µ + 1 2 (W 3µ B µ ) D µν 0 0 D µν W 3µ B µ .
(5.46)

We go from the W 3µ , B µ coordinates to the A µ , Z µ coordinates by a rotation matrix R, eq. (5.31), and since R T R = 1, we can immediately replace (W 3µ B µ ) by (A µ Z µ ) in the equation above. Now we go from the W 1µ , W 2µ components to the charged W 's ones via the matrix O defined by

W 1µ W 2µ = 1 √ 2 1 √ 2 i √ 2 -i √ 2 W * µ W µ , (5.47)
which satisfies O T O = 0 1 1 0 so that we can immediately write

L 0G = 1 2 [W * µ (x)D µν W ν (x) + W µ (x)D µν W * ν (x)] + 1 2 Z µ (x)D µν Z ν (x) + 1 2 A µ (x)D µν A ν (x) = - 1 4 K * µν K µν - 1 4 K µν K * µν - 1 4 K Z µν K µν Z - 1 4 K A µν K µν A , (5.48)
where in the last line we have dropped a total derivative and where the K * µν , K µν , K µν Z , K µν A are respectively the abelian-like stress-energy tensors, eq. (5.21), of the W ± , Z, A gauge bosons.

We turn now to the interaction lagrangian density L IG eq. (5.25). Permuting µ ↔ ν, j ↔ k in the

term ǫ ijk ∂ ν W iµ (x)W µ j (x)W ν k (x) one obtains L IG = -g ǫ ijk ∂ µ W iν (x)W µ j (x)W ν k (x) - g 2 4 ǫ ijk W jµ (x)W kν (x) ǫ ilm W µ l (x)W ν m (x).
(5.49)

The term linear in g can be written

-g det ∂ µ W 1ν W µ 1 W ν 1 ∂ µ W 2ν W µ 2 W ν 2 ∂ µ W 3ν W µ 3 W ν 3 .
(5.50)

Adding i×the second line to te first one to reconstruct W * µ and taking into account the fact that a determinant is invariant when adding or subtracting lines (eventually multiplied by a constant factor) one obtains for the expression (5.50)

-g det √ 2∂ µ W * ν √ 2W * µ √ 2W * ν ∂ µ W 2ν W µ 2 W ν 2 ∂ µ W 3ν W µ 3 W ν 3 = - g 2 i det √ 2∂ µ W * ν √ 2W * µ √ 2W * ν 2 i∂ µ W 2ν 2 iW µ 2 2 iW ν 2 ∂ µ W 3ν W µ 3 W ν 3 = i g det ∂ µ W * ν W * µ W * ν ∂ µ W ν W µ W ν ∂ µ W 3ν W µ 3 W ν 3 .
(5.51)

The last equality is obtained by subtracting the first line from the second. Then using W 3ν = sin θ W A µ + cos θ W Z µ and the relation e = g sin θ W (eq. (5.34)), the above expression becomes

i e det ∂ µ W * ν W * µ W * ν ∂ µ W ν W µ W ν ∂ µ A ν A µ A ν + i e cos θ W sin θ W det ∂ µ W * ν W * µ W * ν ∂ µ W ν W µ W ν ∂ µ Z ν Z µ Z ν .
(5.52)

Expanding the determinant we find for the γW + W -vertex

-i e [∂ µ W * ν (W µ A ν -A µ W ν ) -∂ µ W ν (W * µ A ν -A µ W * ν ) + ∂ µ A ν (W * µ W ν -W -µ W * ν )], (5.53)
By assigning a definite index to each field, e.g. A λ , W ρ , W * σ , the expression takes the usual form

i e [A λ g ρσ (W ρ ∂ λ W * σ -W * σ ∂ λ W ρ ) + W ρ g σλ (W * σ ∂ ρ A λ -A λ ∂ ρ W * σ ) + W * σ g λρ (A λ ∂ σ W ρ -W ρ ∂ σ A λ )], (5.54)
and similarly for the ZW + W -vertex with the coupling e cos θ W / sin θ W instead of e. This defines all tri-linear couplings among gauge bosons.

The term in g 2 in the interaction lagrangian density eq. (5.49) is rather boring to expand. Using the

relation ǫ ijk ǫ ilm = δ jl δ km -δ jm δ kl , it becomes - e 2 4 sin 2 θ W [W µ (x) • W µ (x)W ν (x) • W ν (x) -W µ (x) • W ν (x)W ν (x) • W µ (x)] = - e 2 4 sin 2 θ W W µ i W iρ W ν j W jσ [g ρ µ g σ ν -g σ µ g ρ ν ]
(5.55)

with the notation W µ • W ν = Σ i W µ i W iν .
One obtains the vertex for the physical fields using

W µ (x) • W ρ (x) = W µ W * ρ + W * µ W ρ + (sin θ W A µ + cos θ W Z µ )(sin θ W A ρ + cos θ W Z ρ )],
(5.56) so that eq. (5.55) becomes

- e 2 2 sin 2 θ W W µ W * ρ W σ W * ν + W µ W * ρ (sin θ W A σ +cos θ W Z σ )(sin θ W A ν +cos θ W Z ν ) [g µρ g νσ -g µσ g νρ ]
The antisymmetry of the [g µρ g νσ -g µσ g νρ ] tensor combination kills the terms with only photons and/or Z bosons. The self-couplings of gauge bosons are thus given by

L IG = -i e [A λ g ρσ (W ρ ∂ λ W * σ -W * σ ∂ λ W ρ ) + W ρ g σλ (W * σ ∂ ρ A λ -A λ ∂ ρ W * σ ) + W * σ g λρ (A λ ∂ σ W ρ -W ρ ∂ σ A λ )] + {A λ → Z λ , e → e cos θ W / sin θ W } (5.57) - e 2 2 sin 2 θ W W µ W * ρ W σ W * ν + W µ W * ρ (sin θ W A σ +cos θ W Z σ )(sin θ W A ν +cos θ W Z ν ) [g µρ g νσ -g µσ g νρ ]
In conclusion, from eq. (5.49) one has two three-boson vertices W -W + γ, W -W + Z with derivative couplings and four four-boson vertices

W -W + W -W + , W -W + γγ, W -W + ZZ, W -W + γZ.
The absence of vertices involving only γ's and/or Z's has its origin in the fact that they would arise from the term

g 2 ǫ i33 W 3µ (x)W 3ν (x) ǫ i33 W ρ 3 (x)W σ 3 (x)
, in eq. (5.49), which is of course 0. Using "standard methods" one can, from the expressions above, extract the Feynman rules for the couplings between fermions and gauge bosons. It will not be done here as they can be found in books.

To summarize this rather technical section we count at this point 15 couplings in the model (for one generation of fermions). One has:

-9 fermion-fermion-boson vertices: ēeγ, ēeZ, νe ν e Z, νe eW + , ūuγ, ūuZ, ddγ, ddZ, ūdW + -2 trilinear gauge bosons vertices : W + W -γ, W + W -Z -4 quadrilinear gauge bosons vertices :

W -W + W -W + , W -W + γγ, W -W + ZZ, W -W + γZ.
They depend only on two parameters e and θ W (and, of course, the fermion charges). It is obvious that the symmetry properties of the lagrangien is quite constraining. The important fact is that the relations between couplings derived above will be preserved by the mechanism of "spontaneous symmetry breaking" we are going to discuss. This is an important difference with a mechanism of explicit symmetry breaking where these relations would have been lost.

Progress status and problems

Considering what has been achieved until now, one finds that the model based on the SU (2) L ⊗ U (1) Y symmetry contains four gauge bosons: two charged ones with (V -A) couplings to fermions and two neutral ones with couplings such that these bosons can be interpreted as the photon and the Z boson.

The "only" difference with the real world is that in the present state of development of the model the gauge bosons are massless, because of the assumed exact gauge invariance and the fermions are also massless because of the left-right asymmetry of the gauge group. Counting the bosonic degrees of freedom of the model one realizes that three degrees of freedom are "missing", associated to the longitudinal polarisation states of the heavy vector bosons as summarised in the table.

Model

Real World degrees of freedom degrees of freedom transverse longitudinal

W - 2 0 W + 2 0 Z 2 0 γ 2 0 transverse longitudinal W - 2 1 W + 2 1 Z 2 1 γ 2 0
In order to complete the model one should therefore introduce at least three new fields in the lagrangian. This will be done through a multiplet of scalar fields and it will be seen that, by the mechanism of spontaneous symmetry breaking of local gauge invariance, some of the scalar fields become the longitudinal polarisation states and correlatively the vector bosons acquire a mass.

Spontaneous symmetry breaking under a global phase change

We proceed in steps and discuss, first, the case of a global symmetry and state the Golstone theorem.

In the next sections we deal with the case of a spontaneously broken local U (1) symmetry, leading to a massive gauge boson, and then we turn to the Glashow-Weinberg-Salam model based on a broken

SU (2) L ⊗ U (1) Y symmetry.

Global symmetry breaking

Consider the very simple case of a complex scalar field

ϕ = 1 √ 2 (ϕ 1 + iϕ 2 ) (6.1)
which has two degrees of freedom ϕ 1 (x), ϕ 2 (x). The lagrangian

L = ∂ µ ϕ * ∂ µ ϕ -V (ϕ) with the potential V (ϕ) = -µ 2 |ϕ| 2 + h|ϕ| 4 . (6.2)
is invariant under a rigid U(1) phase transformation ϕ(x) → e iα ϕ(x) where α is constant. The potential has the well-known "Mexican hat" or "cul-de-bouteille" shape (depending on your cultural background!). The hamiltonian is

H = π ∂ 0 ϕ -L, with π = δL δ∂ 0 ϕ = ∂ 0 ϕ * = | ∇ϕ| 2 H kinetic + V (ϕ). (6.3)
The (positive) kinetic part vanishes for static configurations and the full hamiltonian is minimal for constant values of the field given by

|ϕ 0 | = µ √ 2h = v √ 2 (6.4)
which defines the so-called vacuum expectation value v of the field ϕ in terms of the parameters of the lagrangian. Indeed, the quantum theory should be constructed from the lowest energy classical state which, in this case, is characterised by having its norm constrained by the above equation. One immediately notices that the vacuum is degenerate since the application of a gauge transformation (phase change) does not affect the norm of the state. There is an infinite number of classical vacuum states, namely all states of type |ϕ 0 |e iα . However to construct the quantum theory one needs to choose a particular vacuum, by imposing, for example, the classical vacuum field to be real i.e.

ϕ 0 = v √ 2 (6.5)
This obviously amounts to breaking the symmetry of the vacuum since ϕ 0 is no more invariant under a gauge transformation, but the dynamical laws are still unbroken because they are given by the gauge invariant lagrangian eq. (6.2). This is the basis of "spontaneous symmetry breaking" in contradistinction to "explicit symmetry breaking" where the lagrangian itself would loose gauge invariance. To study the theory, we translate the original field by its vacuum expectation value

ϕ(x) = 1 √ 2 (v + ϕ 1 (x) + iϕ 2 (x)) (6.6)
and, neglecting constant terms, the lagrangian becomes

L = 1 2 (∂ µ ϕ 1 ) 2 -hv 2 ϕ 2 1 + 1 2 (∂ µ ϕ 2 ) 2 -hvϕ 1 (ϕ 2 1 + ϕ 2 2 ) - h 4 (ϕ 2 1 + ϕ 2 2 ) 2 (6.7)
After spontaneous symmetry breaking, we are left with a model of two interacting real fields ϕ 1 and ϕ 2 . The free theory is given by the first line of the equation above which shows that ϕ 1 has a mass

m ϕ 1 = √ 2hv 2 while ϕ 2 is massless: ϕ 2 is called the Goldstone boson.
The interaction part is all contained in the second line of eq. (6.7) and there are cubic and quartic interactions between ϕ 1 and ϕ 2 . Since, the initial lagrangian contained only two parameters, there are necessarily relations between the three parameters m ϕ 1 , the coefficient of the cubic coupling term g 3 and the coefficient of the quartic coupling g 4 e.g.

g 2 3 = 2 m 2 ϕ 1 g 4 . (6.8)
Such a relation reflects the symmetry property of the lagrangian density. These features are a simple illustration of very general properties of spontaneous breaking of larger (non-abelian) group symmetry.

They are a particular case of the Goldstone theorem.

The Goldstone theorem

This theorem reads :

When a global symmetry is spontaneously broken there appear as many massless scalar modes (called the Goldstone bosons) as there are broken degrees of symmetry.

A proof of this theorem is now sketched. Consider ϕ, a collection of n scalar fields

ϕ i , i = 1, • • • , n
written as a column vector so that

ϕ T = (ϕ 1 , • • • , ϕ n ) (6.9)
The lagrangian density is formally written as

L = L(ϕ, ∂ µ ϕ) kin -V (ϕ). (6.10)
The vacuum of the model is defined by the conditions

δV δϕ i = 0, ⇒ vacuum: ϕ 0 T = (ϕ 0 1 , • • • , ϕ 0 n ) (6.11)
One perturbs around the vacuum state

ϕ = ϕ 0 + ϕ ′ , i.e. ϕ i = ϕ 0 i + ϕ ′ i (6.12)
so that the lagrangian (neglecting constant terms) is re-written

L = L(ϕ ′ , ∂ µ ϕ ′ ) kin - 1 2 ij δV δϕ i δϕ j ϕ 0 ϕ ′ i ϕ ′ j ⊕ (ϕ ′3 ) ⊕ (ϕ ′4 ) (6.13)
where it is not necessary for our present purposes to specify the cubic nor the quartic couplings. By construction, there are no terms linear in the fields because we are expanding around the minimum of the potential. The quantity of interest is the quadratic term which defines the mass matrix

m 2 ij = δV δϕ i δϕ j ϕ 0 . (6.14)
Consider now the action of an infinitesimal global gauge transformation. Its action on the fields is

δϕ = i α J T J ϕ, J = 1, • • • , N, (6.15)
where the T J are the N generators (n × n matrices) of the group and the α J are the N associated arbitrary parameters. If for some field configuration ϕ we have for a particular generator T J ,

T J ϕ = 0, ⇒ δϕ = iα J T J ϕ = 0, (6.16)
then we say that this configuration ϕ is invariant under the sub-group generated by T J : the corresponding symmetry is unbroken. If, on the contrary, T J ϕ = 0 the corresponding degree of symmetry is said to be spontaneously broken. Let us suppose now that the vacuum satisfies

T J ϕ 0 = 0 for J = 1, • • • , N ′ T J ϕ 0 = 0 for J = N ′ + 1, • • • , N, (6.17)
i.e. that the vacuum state breaks N ′ degrees of symmetry. The invariance of the potential V (ϕ) under the gauge transformation

δϕ i = i α J T J ik ϕ k yields δV (ϕ) = δV δϕ i δϕ i = i α J δV δϕ i T J ik ϕ k = 0. (6.18)
Since this true for any α J one has

δV δϕ i T J ik ϕ k = 0. (6.19)
Taking the derivative of this relation at ϕ = ϕ 0 , it comes out

δ 2 V δϕ j δϕ i ϕ 0 T J ik ϕ 0 k + δV δϕ i ϕ 0 T J ik δ kj = 0 ⇒ m 2 ji T J ik ϕ 0 k = 0, (6.20)
where the last equality is true because ϕ 0 defines the minimum of the potential. Since this relation is automatically satisfied for J = N ′ + 1, • • • , N one concludes that the mass matrix must have N ′ vanishing eigenvalues. Thus, N ′ fields ϕ ′ i will be massless which are the Golstone bosons associated to the N ′ degrees of broken symmetry (qed).

Spontaneous local U(1) symmetry breaking

We impose now that the lagrangian density eq. (6.2) is invariant under the local phase change ϕ(x) → e igα(x) ϕ(x). For this purpose we introduce a vector field B µ (x) and a covariant derivative

D µ = ∂ µ -igB µ (x) such that D µ ϕ(x) → igα(x)D µ ϕ(x) under an infinitesimal phase change. This is realised if B µ (x) transforms as B µ (x) → B µ (x) + g∂ µ α(x). Since D µ ϕ * (x) → -igα(x)D µ ϕ * (x) the
locally invariant version of the scalar field lagrangian density is

L S + L G = D µ ϕ * D µ ϕ + µ 2 ϕ * ϕ -h(ϕ * ϕ) 2 - 1 4 K µν K µν (7.1)
where we have also included the kinetic term, see eq. (5.21), of the gauge boson B µ (x). As in the study of the breaking of the global symmetry we choose as the lowest energy state ϕ 0 = v/ √ 2, (eq. (6.5), and we expand the field around this vacuum expectation value as in eq. (6.6).

Unitary gauge

We take advantage of the freedom of choice of the gauge to find a function α(x) such that e igα(x) applied to eq. (6.6) gives

ϕ(x) = 1 √ 2 (v + H(x)), (7.2)
i.e. we absorb the imaginary part in a change of phase and we are left with one real field H(x). This choice defines the unitary gauge. Applying the covariant derivative on ϕ(x) one obtains

D µ ϕ(x) = 1 √ 2 ∂ µ H(x) -igB µ (x) 1 √ 2 (v + H(x)) (7.3)
Injecting this in the lagrangian density, taking the potential part from eq. (6.7) with ϕ 1 = H, ϕ 2 = 0, and reshuffling the terms we find

L S + L G = 1 2 (∂ µ H(x)) 2 -hv 2 H 2 (x) + - 1 4 K µν (x)K µν (x) + g 2 v 2 2 B µ (x)B µ (x) + g 2 vH(x)B µ (x)B µ (x) + g 2 2 H 2 (x)B µ (x)B µ (x) -hvH 3 (x) - h 4 H 4 (x). (7.4)
The terms in the first line are those from which we build the propagators of the H and B µ fields respectively, while the second line contains the couplings between the fields. Applying the Euler-Lagrange equation (4.1) we obtain for the H field (

∂ µ ∂ µ = ) (--2hv 2 )H(x) = 3hvH 2 (x) + hH 3 (x) -g 2 vB µ (x)B µ (x) -g 2 H(x)B µ (x)B µ (x), (7.5)
and for the gauge boson

( g µν -∂ µ ∂ ν + (gv) 2 )B ν (x) = -g 2 H 2 (x)B µ (x) -2g 2 vHB µ (x). (7.6)
To get the free propagators one solves the Green's functions

(--2hv 2 ) G(x -y) = iδ (4) (x -y) ( g µρ -∂ µ ∂ ρ + (gv) 2 g µρ ) G ρν (x -y) = ig ν µ δ (4) (x -y), (7.7)
in Fourier space. For the scalar field one parameterises

G(x -y) = (d 4 k/(2π) 4 ) exp(-ik(x -y))G(k)
and one easily get the H field propagator

G(k) = i k 2 -M 2 H + iǫ with M H = v √ 2h, (7.8)
with the iǫ prescription required by causality. Similarly, for the gauge field we write

G µν (x -y) = (d 4 k/(2π) 4 ) exp(-ik(x -y))G µν (k) to get (-k 2 g µρ + k µ k ρ + (gv) 2 g µρ )G ρν (k) = ig ν µ . (7.9)
We look for the solution under the form G ρν (k) = ag ρν + bk ρ k ν which is the most general rank 2 tensor which can be constructed from a vector k µ . One obtains finally

G µν (k) = -i k 2 -M 2 B + iǫ (g µν - k µ k ν M 2 B ) with M B = gv. (7.10)
The mass of the scalar H field is M H = √ 2hv and the mass of the gauge field M B = gv : both are proportional to the vacuum expectation value of the scalar field but the latter is proportional to the gauge coupling while the former depends on the quartic coupling in the potential. The term giving rise to the gauge boson mass originates from the covariant derivative acting on ϕ(x) after symmetry breaking while the mass of the H field comes from the potential V (ϕ).

• Remark on the polarisation of a massive vector boson

The propagator of B µ (x) is that of a massive scalar field which has three states of polarisation. Indeed one can easily verify, from eq. (2.25), that the numerator of eq. (7.10) is

-g µν - k µ k ν M 2 B = i ε (i) µ (k) ε (i) ν (k), (7.11)
the trace of which is -3.

Counting the degrees of freedom in the model we have after symmetry breaking one real scalar field H(x) and the three polarisation states of the gauge boson while before symmetry breaking one had two scalar fields ϕ 1 (x), ϕ 2 (x) and the two polarisation states of the massless gauge boson : it appears that the massless Golstone boson ϕ 2 (x) has become the longitudinal polarisation of B µ (x). The gauge used in this derivation is called the unitary gauge. With this choice the vector boson propagator may lead, as we have seen, to divergences when calculationg Feynman diagrams because of the k µ k ν /m 2 B term and therefore may ruin the renormalisability of the model.

Renormalisable gauges : 't Hooft R ξ gauges

To study this is more detail we go back to the lagrangiann density eq. ( 7.1) with the general form, eq. (6.6), of the scalar field after symmetry breaking. The covariant derivative is then

D µ φ(x) = (∂ µ -igB µ (x)) 1 √ 2 (v + φ 1 (x) + iφ 2 (x)) = 1 √ 2 [∂ µ φ 1 (x) + gB µ (x)φ 2 (x)] + i √ 2 [∂ µ φ 2 (x) -gB µ (x)(v + φ 1 (x))] (7.12)
The lagrangian density takes then the form, keeping explicitely only the terms quadratic in the fields,

L S + L G = 1 2 (∂ µ φ 1 (x)) 2 -hv 2 φ 2 1 (x) + - 1 4 K µν (x)K µν (x) + g 2 v 2 2 B µ (x)B µ (x) + 1 2 (∂ µ φ 2 (x)) 2 -gvB µ (x)∂ µ φ 2 (x) + L int . (7.13)
The first line is identical to that of eq. (7.4) with a massive scalar field φ 1 (x) (φ 1 (x) = H(x) is the Higgs field) and a massive gauge boson. In the second line one has the massless φ 2 (x) scalar (the Goldstone boson) coupling to the gauge field. The function L int ,

L int = g(ϕ 2 ← → ∂ µ ϕ 1 )B µ + g 2 vϕ 1 B µ B µ + g 2 2 (ϕ 2 1 + ϕ 2 2 )B µ B µ -hvϕ 1 (ϕ 2 1 + ϕ 2 2 ) - h 4 (ϕ 2 1 + ϕ 2 2 ) 2 , (7.14)
contains the couplings between φ 1 , φ 2 and B µ .

Clearly φ 2 (x) is not independent on B µ (x) since it oscillates into the gauge boson with a derivative coupling as can be seen from eq. (7.13). In fact if we consider the polarisation tensor of the B µ field, treating both the mass term and the B µ ∂ µ φ 2 as vertices we find (gv

= M B ) = + iM 2 B g µν -M B k µ M B k ν iM 2 B g µν + (-M B k µ ) i k 2 M B k ν = iM 2 B (g µν - k µ k ν k 2 ) (7.15)
which is transverse as it should be. From eq. (2.25) one sees that the tensor structure is equivalent to summing over transverse and longitudinal polarisations of B µ : this shows that φ 2 (x) builds up the longitudinal polarisation of the originally transverse B µ (x) field. One may suspect that iterating the self-energy bubble on the B µ field propagator will reconstruct the propagator of a massive field. This is disccussed more precisely below.

We follow here a procedure familiar from QED. To quantise QED, it is necessary to break the gauge invariance and this is done by adding to the lagrangian a "gauge fixing" term. Here the gauge fixing term is chosen to be

L GF = - 1 2ξ (∂ µ B µ (x) + ξ gv φ 2 (x)) 2 . (7.16)
This choice (instead of the traditional term

-(∂ µ B µ (x)) 2 /2ξ of QED) is made to eliminate the mixed term gvB µ (x)∂ µ φ 2 (x)
in the lagrangian. This class of gauge conditions is known under the name of 't Hooft's gauges or R ξ gauges where ξ is an arbitrary real number. One considers the new lagrangian density L S + L G + L GF which then becomes

L S + L G + L GF = 1 2 (∂ µ φ 1 (x)) 2 -hv 2 φ 2 1 (x) + 1 2 (∂ µ φ 2 (x)) 2 -ξ(gv) 2 φ 2 2 (x) + - 1 4 K µν K µν + (gv) 2 2 B µ (x)B µ (x) + 1 2ξ (∂ µ B µ (x)) 2 + L int (7.17)
By the specific choice of the gauge condition the mixed term in B µ ∂ µ φ 2 in L S + L G combines with the term φ 2 ∂ µ B µ in L GF to give a total derivative which can be safely ignored in perturbation theory.

However the Goldstone boson acquires a mass from the gauge fixing lagrangian density. Following the procedure used when working in the unitary gauge one derives the Green's equation for the fields φ i and B µ , the solution of which gives the free propagators. Thus one obtains

(--2hv 2 )G φ 1 (x -y) = iδ (4) (x -y) (--ξ(gv) 2 )G φ 2 (x -y) = iδ (4) (x -y) ( g µν -(1 - 1 ξ )∂ µ ∂ ν + (gv) 2 g µν )G νρ (x -y) = ig ρ µ δ (4) (x -y). (7.18)
For the scalar fields we obtain easily for the field

φ 1 = H G H (k) = i k 2 -M 2 H + iǫ with M H = v √ 2h (7.19) for the Goldstone field φ 2 G φ 2 (k) = -i k 2 -ξM 2 B + iǫ with M B = gv . (7.20)
For the gauge fields, introducing

G νρ (x -y) = (d 4 k/(2π) 4 ) exp(-ik(x -y))G νρ (k) one has to solve (k 2 g µν -(1 - 1 ξ )k µ k ν -M 2 B g µν )G νρ (k) = -ig ρ µ (7.21)
One looks for the solution in the form of a νρ + bk ν k ρ and one finds

G νρ (k) = - i k 2 -M 2 B + iǫ g νρ -(1 -ξ) k ν k ρ k 2 -ξM 2 B .
(7.22)

One observes that for any value of ξ finite all propagators have the right asymptotic behavior i.e. they behave like 1/k 2 , k 2 → ∞ which is a necessary condition for the model to be renormalisable. However both the Goldstone and the gauge boson propagators have a spurious pole at k 2 -ξm 2 B which should cancel when calculating a physical process. It is interesting to compare the gauge boson propagator in the general 't Hooft gauge with its form in the unitary gauge. One proves easily

- i k 2 -M 2 B + iǫ g νρ -(1 -ξ) k ν k ρ k 2 -ξM 2 B = - i k 2 -M 2 B + iǫ g νρ - k ν k ρ M 2 B - i M 2 B k ν k ρ k 2 -ξM 2 B (7.23)
One recognises on the right-hand side the propagator in the unitary gauge, eq. (7.10), plus a term which has the the same pole structure as the Goldstone boson. An exemple will be given later, on how such a cancellation occurs between this extra piece and the Goldstone contribution.

Fermion masses

We now include a fermion in our toy model. We assume one massless fermion ψ(x) and impose a local U (1) gauge invariance only on the left-handed component of ψ(x) : δψ L (x) = igα(x)ψ L (x), δψ R (x) = 0. The fermion part of the lagrangian density takes the form

L F = ψL i Dψ L + ψR i ∂ψ R , (7.24)
with the covariant derivative acting on ψ L (x) defined by

Dψ L = ( ∂ -ig B)ψ L . (7.25)
We parameterise the U (1) invariant interaction between the scalar field and the fermion by the Yukawa type lagrangian density

L Y = -λ f ( ψL φψ R + ψR φ * ψ L ). (7.26)
After symmetry breaking, using the parameterisation eq. (6.6) of the scalar field, only L Y is affected

L Y = - λ f √ 2 (v + H)( ψL ψ R + ψR ψ L ) -i λ f √ 2 φ 2 ( ψL ψ R -ψR ψ L ) = - λ f v √ 2 ψψ - λ f √ 2 H ψψ -i λ f √ 2 φ 2 ψγ 5 ψ, (7.27)
where we have recombined the left-handed and right-handed fields. Regrouping all fermion terms we have

L F + L Y = ψ(i ∂ - λ f v √ 2 )ψ + g 2 ψ B(1 -γ 5 )ψ - λ f √ 2 H ψψ -i λ f √ 2 φ 2 ψγ 5 ψ. (7.28)
We read off the fermion mass

m f = λ f v √ 2 (7.29)
and the couplings of the fermion -to the gauge field : -i(g/2)γ µ (1 -γ 5 ) ;

-to the Higgs field :

iλ f / √ 2) ; -to the Goldstone boson : (λ f / √ 2)γ 5 .
The coupling of the Higgs to the fermion can be written in terms of "physical parameters", masses and the gauge coupling, and one finds coupling Higgs-fermion-fermion :

iλ f / √ 2) = i g m f M B , (7.30)
which illustrates an important feature of spontaneous symmetry breaking, namely that the coupling is proportional to the fermion mass.

In the unitary gauge, the Higgs and gauge boson couplings to the fermion are as above, while the Goldstone boson φ 2 is absorbed by the gauge choice and does not couple to the fermion.

Gauge invariance at the Born level: an exemple

Putting everything together, the lagrangien density of our model in a general R ξ gauge is

L S + L G + L GF + L F + L Y (7.31)
with L S + L G + L GF from eq. (7.17) and L F + L Y from eq. (7.28). We are now in a position to calculate the scattering amplitude for the collision

ψ 1 + ψ 2 → ψ 3 + ψ 4 .
- g 2 4 ψ3 γ µ (1 -γ 5 )ψ 1 (-i) k µ k ν M 2 B (k 2 -ξM 2 B ) ψ4 γ ν (1 -γ 5 )ψ 2 . (7.32)
Using Dirac equation this term can be considerably simplified. For instance with

k µ = p µ 1 -p µ 3 ψ3 γ µ (1 -γ 5 )ψ 1 k µ = ψ3 ( p 1 -p 3 )ψ 1 + ψ3 γ 5 p 1 ψ 1 + ψ3 p 3 γ 5 ψ 1 = 2 m f ψ3 γ 5 ψ 1 , (7.33)
where to obtain the last line we have used Dirac equation p 1 ψ 1 = m f ψ 1 and ψ3 p 3 = m f ψ 3 . The same trick can be used at the other vertex to obtain

ψ4 γ µ (1 -γ 5 )ψ 2 k µ = ψ4 ( p 4 -p 2 )ψ 2 -ψ4 p 4 γ 5 ψ 2 -ψ4 γ 5 p 2 ψ 2 = -2 m f ψ4 γ 5 ψ 2 . (7.34)
This shows that after symmetry breaking the axial current ψγ µ γ 5 ψ is not conserved since, when contracted with the gauge field momentum, it gives a term proportional to the mass of the fermion.

Thus eq. (7.32) reduces to

- g 2 m 2 f M 2 B i k 2 -ξM 2 B ψ3 γ 5 ψ 1 ψ4 γ 5 ψ 2 (7.35)
The contribution of the Goldstone boson exchange is simply

λ 2 F 2 i k 2 -ξM 2 B ψ3 γ 5 ψ 1 ψ4 γ 5 ψ 2 (7.36)
Using the relation g 2 m 2 f /M 2 B = λ 2 F /2, eq. (7.30), one easily verifies the compensation of the ξ dependant part of the gauge propagator by the Goldstone boson. Needless to say that, for this to occur, the mass term of the gauge boson and that of the fermion should have the same origin and be both related to the vacuum expectation value v. The gauge invariance can be checked on other processes notably those involving the triple gauge couplings, however the discussion is more tricky since it implies the coupling of the Goldstone field to the vector boson as given in (7.14).

The broken SU(2) L ⊗ U(1) Y symmetry

In this case, the generators of the symmetry group will be T J = (τ 1 , τ 2 , τ 3 , Y ), i.e. the generators of the weak isospin group and of the U (1) Y hypercharge gauge group. We introduce a complex scalar field Φ(x), which is a doublet of SU (2

) (I Φ = 1 2 ), Φ = 1 √ 2 ϕ 1 -i ϕ 2 ϕ 3 -i ϕ 4 (8.1)
and the standard scalar lagrangian

L S = ∂ µ Φ † ∂ µ Φ -V (Φ), V (Φ) = -µ 2 Φ † Φ + h (Φ † Φ) 2 . (8.2)
which is invariant under the rigid transformation

Φ → Φ ′ = e iτ •α/2 e iy Φ β/2 Φ. (8.3)
The minimum of the potential is obtained for (see eq. (6.4))

Φ † Φ = |Φ| 2 = µ 2 2h = v 2 2 . (8.4)
There is an infinite number of vacua states : all states with the norm v/ √ 2 obtained by a gauge transformation. We choose the physical vacuum to be

Φ 0 = 0 v √ 2 with v = µ √ h . (8.5)
Since we require the electric charge to be conserved after symmetry breaking, following the reasoning in sec. 6.2, we have to enforce that the charge generator acting on the vacuum state should vanish.

Using the Gell-Mann/Nishijima relation eq. (4.28) the charge operator acting on Φ 0 is

Q Φ 0 = (I 3 + Y 2 ) Φ 0 = 1 2 (τ 3 + Y ) Φ 0 = 1 2 + y Φ 2 0 0 -1 2 + y Φ 2 0 v √ 2 = 0 (8.6)
implying that the hypercharge of the scalar field must be y Φ = 1 to ensure charge conservation in the broken theory: the charge of the classical vacuum is 0. As in the abelian case, we can study the system around the classical minimum and expand the scalar field around its vacuum expectation value

Φ = 1 √ 2 (ω 1 (x) -iω 2 (x)) 1 √ 2 (v + ω 0 (x) -iω 3 (x)) = ω * (x) 1 √ 2 (v + ω 0 (x) -iω 3 (x))
.

(8.7)

The complex field ω * has a positive electric charge while ω 0 and ω 3 are neutral. In terms of the new variables the scalar potential V (Φ) becomes

V (Φ) = hv 2 ω 2 0 + hv ω 0 (ω 2 0 + ω 2 ) + h 4 (ω 2 0 + ω 2 ) 2 (8.8)
showing that the triplet ω of ω i fields is massless while the neutral ω 0 field acquires a mass

M ω 0 = √ 2hv 2 . (8.9)
All these fields are coupled together with a strength which can be read off the equation above.

Thus, in our model, in agreement with Noether theorem, three degrees of freedom are broken leading to three massless Goldstone bosons, and the vacuum is still left invariant under the combination

Q = I 3 + Y /2.
There is still an abelian symmetry left, namely the U (1) emg group.

Local symmetry breaking and the Brout-Englert-Higgs mechanism

Armed with this lengthy preliminaries we now turn to spontaneous breaking of the local gauge symmetry SU (2) L ⊗ U (1) Y down to U (1) emg in the framework of the Standard Model. Let us state the results before diving into an ocean of technicalities. The case of a global symmetry has just been analysed and led to the appearance of three massless (Goldstone) bosons and a massive one. When the symmetry is made local these massless bosons turn out to be unphysical (two charged ones, ω and ω * , and a neutral one ω 3 ), in the sense that they can be gotten rid off by a gauge transformation, but instead, three gauge bosons (a neutral one and the two charged ones) become massive and therefore acquire longitudinal polarisation states which are the Goldstone modes in disguise.

To implement the breaking of the local SU (2) L ⊗ U (1) Y symmetry we first have to extend the electroweak lagrangian eq. (5.22) to include the scalar field contribution L S eq. (8.2) in its locally gauge invariant form (see eq. (8.12) below) as well as the interaction of the scalar field with the fermions L Y (where Y stands for Yukawa; see eq. (8.26) below) so that the complete electroweak lagrangian density is

L = L F + L G + L S + L Y . (8.10)
In the following we work in the unitary gauge.

The Higgs and gauge bosons sector : masses and couplings

We concentrate for the moment on L S which drives the spontaneous breaking of the local electroweak symmetry. Only neutral scalar fields can acquire a vacuum expectation value : other fields, such as fermions or gauge bosons, cannot do so otherwise the physical vacuum would have some angular momentum or other non-vanishing quantum numbers. We impose now the invariance of L S under a change of the local phases

Φ(x) → Φ ′ (x) = e igα(x)•τ /2 e ig ′ y Φ β(x)/2 Φ(x). (8.11)
To keep gauge invariance requires substituting the covariant derivative to the partial derivative in L S which then takes the form

L S = D µ Φ † D µ Φ -µ 2 Φ † Φ + h(Φ † Φ) 2 (8.12)
with the definition, eq. (5.20),

D µ = ∂ µ -i g τ 2 • W µ -i 1 2 g ′ B µ (8.13)
and the choice, eq. (8.6), y Φ = 1 for the hypercharge. This can be easily checked using the same line of reasoning as used in sec. 5.

To study the system around the classical vacuum we parameterise the scalar field as in eq. (8.7).

However we note that by an appropriate gauge transformation we can find α(x), β(x) such that :

e ig ′ y Φ β(x)/2 e igτ •α(x)/2 Φ(x) = 0 v+H(x) √ 2 , (8.14)
showing that the fields ω i (x) can be removed from the lagrangian altogether and therefore are not physical. Of course, explicit gauge invariance of the vacuum state will be lost since a particular gauge has been chosen. To analyse the effects of symmetry breaking we work with the "physical" A µ and Z µ fields of eq. (5.31) rather than with W 3µ and B µ . For this purpose we use the expression eq. (5.43)

for the covariant derivative which, applied to the form eq. (8.14) of Φ (with e 1 = 1, e 2 = 0), yields

D µ 0 v+H(x) √ 2 = ∂ µ -i e √ 2 sin θ W 0 W * µ W µ 0 -ie A µ 0 0 0 -i e sin θ W cos θ W 1 2 -sin 2 θ W Z µ 0 0 -1 2 Z µ 0 v+H(x) √ 2 = -i e 2 sin θ W W * µ (v + H(x)) ∂ µ H(x) √ 2 + i e 2 √ 2 sin θ W cos θ W Z µ (v + H(x)) (8.15)
It is then trivial to get D µ Φ † D µ Φ and write the scalar lagrangian density

L S L S = 1 2 (∂ µ H(x)) 2 + e 2 4 sin 2 θ W (v + H(x)) 2 W * µ W µ + e 2 8 sin 2 θ W cos 2 θ W (v + H(x)) 2 Z µ Z µ -hv 2 H 2 -hv H 3 - h 4 H 4 (8.16)
In the last line we have used eq. (8.8) for the scalar potential dropping of course the spurious ω(x) fields which have been gauged away. The above equation contains a lot of information since it gives masses to the gauge and the Higgs fields as well as defines the couplings between them.

• Masses

Combining the terms proportional to v 2 in the equation above with the stress-energy terms of L 0 G , eq. (5.48), we have the pieces in the lagrangien density which lead to the free propagators of the H and gauge bosons,

L OS + L OG = 1 2 (∂ µ H(x)) 2 -hv 2 H 2 - 1 4 K A µν K µν A - 1 2 K * µν K µν + e 2 v 2 4 sin 2 θ W W * µ W µ - 1 4 K Z µν K µν Z + e 2 v 2 8 sin 2 θ W cos 2 θ W Z µ Z µ (8.17)
Using the same method as in sec. 7.1 we can derive the propagators of the H scalar and the gauge bosons

G(k) = i k 2 -M 2 H + iǫ G µν A (k) = -i k 2 + iǫ g µν G µν W (k) = -i k 2 -M 2 W + iǫ (g µν -k µ k ν /M 2 W ) G µν Z (k) = -i k 2 -M 2 Z + iǫ (g µν -k µ k ν /M 2 Z ) (8.18)
We recover a massive H field with M H = √ 2h v as in eq. (7.8), while the W and Z bosons acquire the masses

M W = e v 2 sin θ W , M Z = e v 2 sin θ W cos θ W , (8.19)
and the photon remains massless as no quadratic term in A µ appears in the lagrangian. The vanishing of the photon mass is a consequence of the surviving exact gauge symmetry U (1) emg . Note the important relation

M W = M Z cos θ W (8.20)
We have the relation v = sin θ W M W / √ πα between the vacuum expectation value of the scalar field and the physical parameters and, plugging in numerical values, we find v ∼ 250 GeV, which is the basis for the claim, made in the introduction, that the non-abelian symmetry is broken at the scale of 250 GeV.

• Couplings

We consider now all the terms of L S , eq. (8.16), not contained in L 0S to define the interaction lagrangian of the Higgs boson

L IS = e 2 v 2 sin 2 θ W HW * µ W µ + e 2 4 sin 2 θ W H 2 W * µ W µ + e 2 v 4 sin 2 θ W cos 2 θ W HZ µ Z µ + e 2 8 sin 2 θ W cos 2 θ W H 2 Z µ Z µ -hv H 3 - h 4 H 4 (8.21)
One notes that the trilinear couplings of the H boson to a pair of gauge bosons have the dimension of a mass, proportional to the vacuum expectation value v, while the quadrilinear couplings are dimensionless proportional to e 2 . One can show that, in terms of Feynman diagrams,

-the vertex HW + W -is : -i e 2 v 2 sin 2 θ W = -i e sin θ W M W ; -the vertex HZZ is : -i e 2 v 2 sin 2 θ W cos 2 θ W = -i e sin θ W cos θ W M Z ; (8.22) -the vertex H 2 W + W -is : -i e 2 2 sin 2 θ W -the vertex H 2 ZZ is : -i e 2 2 sin 2 θ W cos 2 θ W .
There are furthermore the H boson self-couplings proportional respectively to hv and h. These variables are easily eliminated in favour of the observables M W , M H and one finds,

-the vertex H 3 : i 6hv = i 3 2 e sin θ W M 2 H M W -the vertex H 4 : i 6h = i 3 4 e 2 sin 2 θ W M 2 H M 2 W . (8.23)
It is interesting to remark that the triple and the quartic H boson vertices vary as the square of the Higgs boson mass (for fixed W mass). As an indication of the strength of the Higgs boson couplings one finds 0.2 for the vertex H 2 W + W -and 0.12 for the quartic H 4 term.

To complete this section we recall the gauge boson self-couplings defined in L IG , eq. (5.57) : they are not affected by the spontaneous breaking of the symmetry, eventhough three gauge bosons have acquired a mass.

The Yukawa lagrangian L Y and fermion masses and couplings

The scalar field Φ couples to fermions. The requirement for such couplings to exist is that the corresponding terms in the lagrangian density be Lorentz invariant as well as invariant under a SU (2) L ⊗ U (1) Y transformation, before the spontaneous breaking of this symmetry is implemented.

Let us recall that Ψ e L and Ψ q L of eq. (4.14) and Φ of eq. (8.1) are 2 under SU (2) i.e. they transform as

δΦ = i τ 2 α Φ, • • • , δΨ = i τ 2 α Ψ, • • • , δΨ = -i Ψ τ 2 α, • • • (8.24)
so that Ψ e L Φ, Ψ q L Φ are invariant under a SU (2) L transformation and so are the hermitian conjugates

γ 0 Φ † Ψ e L , γ 0 Φ † Ψ q L .
δ Φ ≡ δ(iτ 2 Φ * ) = iτ 2 δΦ * = iτ 2 (-i τ * 2 α) Φ * = (i τ 2 α) Φ (8.25)
where one has used for the last equality the property iτ 2 τ * = -τ (iτ 2 ). The combinations Ψ e L iτ 2 Φ * and Ψ q L iτ 2 Φ * are invariant under a SU (2) L transformation and have hypercharge 0 and -4/3 (y Φ * = -y Φ = -1), respectively. Thus Ψ q L iτ 2 Φ * u R = Ψ q L Φ u R is invariant under a group transformation. Had we included a right-handed neutrino the contribution Ψ e L iτ 2 Φ * ν R = Ψ e L Φ ν R would satisfy the conditions but we will ignore it here (see sec. 12). Thus the Yukawa lagrangian then takes the form

L Y = -c d Ψ q L Φ d R -c u Ψ q L Φ u R -c e Ψ e L Φ e R + h.c. + other families, (8.26)
where we have explicitely written out the terms involving the first family of fermions (ν, e; u, d). Six other parameters should be similarly introduced for the couplings of the second and third families so that nine new parameters appear in the model.

Implementing spontaneous symmetry breaking, in the unitary gauge, i.e. substituting in L Y the expression of Φ as given in the right-hand side of eq. (8.14), we derive

L Y = -c d v + H √ 2 dd -c u v + H √ 2 uu -c e v + H √ 2 ee + other families. (8.27)
From this expression we relate the mass of a fermion f to the vacuum expectation value v via

m f = c f v √ 2 . (8.28)
This is not a prediction of the theory since the parameters c f are unknown and will be adjusted so as to obtain the "physical" mass of the corresponding fermion. Furthermore, no relation is expected between the masses of partners of a given family since one parameter is introduced for each of the fermion type in a family. One may remark that the only "prediction" is that the neutrino remains massless as a consequence of the absence a right-handed neutrino. On the other hand, the Higgs couplings to the fermions are predicted, if the fermion masses are known, e -e + → W -W + we leave it to the reader to check that, keeping fermion mass terms and including all diagrams in the unitary gauge, as shown in Fig. 2, the corresponding cross section is asymptotically finite. At higher orders, loop diagrams involve massive gauge boson propagators: in the unitary gauge

g f = c f √ 2 = m f v = e 2 sin θ W m f M W , ( 8
e - e + W - W + p 1 p 2 p 3 p 4 ν e γ, Z W - W + H W - W +
Figure 2: The e -e + → W -W + diagrams at lowest order in the unitary gauge.

they do not converge to 0 when k 2 → ∞ and this leads to an apparently non-renormalisable theory.

As explained in detail for the abelian case, the way out is to work in a "renormalisable" gauge (the 't Hooft gauges) where the gauge boson propagators have the form eq. (7.22) and the Golstone modes ω are explicitely kept in the calculation.

The Higgs boson discovery

As an application we consider Higgs production in proton-proton colliders at the LHC at a center of quarks and antiquarks of the initial hadrons, q + q → H, but such a coupling, eq. (8.29), is suppressed by a factor m f /v ≃ m f /250 with m f , the mass of the quark, measured in GeV. The direct process tt → H is possible but it is, of course, suppressed because of the negligibly small density of top quarks in the proton. For a Higgs mass below about 500 GeV it turns out that the dominant process is gluon-gluon fusion where the effective Higgs coupling to the gluon-gluon system is via a top quark loop as indicated in Fig. 3. The discovery channels of the Higgs boson have been H → Z Z * → 4

Conclusions

At this point one has, in a first approximation, a complete model for the electroweak interactions. It contains a massive scalar particle, a massless and three massive gauge bosons, with propagators as in eqs. (8.18). All couplings between bosons and bosons to fermions are given assuming no mixing between the three generations of matter fields. The generation mixing is dealt with in sec. 11.

14 Higgs Physics at the HL-LHC and HE-LHC, arXiv:1902.00134, [hep-ph].

9 Exercise : study of the reaction proton + proton → H + X

The mass of the Higgs boson is large enough to justify the use of the parton model and perturbative QCD to study the production of a Higgs boson in proton-proton collisions.

The gluon-gluon fusion mechanism

In this framework, considering only the dominant process via gluon-gluon fusion, the hadronic cross section of the inclusive reaction p(k 1 ) + p(k 2 ) → H(p 3 ) + X can be written as:

σ H = 1 0 dx 1 1 0 dx 2 F P g (x 1 , M 2 ) F P g (x 2 , M 2 ) σg g→H , (9.1)
where F P g (x, M 2 ) stands for the gluon density in the proton, the gluon carrying a fraction x of the proton four-momentum, evolved at the factorisation scale M . The quantity σg g→H is the cross section of the partonic reaction g(p 1 ) + g(p 2 ) → H(p 3 ). The 4-momentum of the initial gluons are such that p 1 = x 1 k 1 and p 2 = x 2 k 2 . The partonic cross section itself is given by:

σg g→H = 1 4 p 1 .p 2 d 3 p 3 (2 π) 3 2 E 3 (2 π) 4 δ 4 (p 1 + p 2 -p 3 ) | T | 2 , (9.2)
with | T | 2 the matrix element squared averaged over initial polarisations and colours. Transform-

ing d 3 p 3 /(2 E 3 ) in d 4 p 3 δ + (p 2 3 -M 2 H
), the integration on p 3 can be performed easily with the Dirac distribution and we get:

σg g→H = 1 2 x 1 x 2 S (2 π) δ + (x 1 x 2 S -M 2 H ) | T | 2 , (9.3)
with the total energy squared S = (k 1 + k 2 ) 2 = 2 k 1 .k 2 (the proton mass is neglected). Injecting eq. (9.3) in eq. (9.1), we get for the hadronic cross section:

σ H = 1 0 dx 1 x 1 1 0 dx 2 x 2 F P g (x 1 , M 2 ) F P g (x 2 , M 2 ) π S δ + (x 1 x 2 S -M 2 H ) | T | 2 . (9.4)
The integration over x 2 can be performed with the help of the remaining Dirac distribution to find:

σ H = π M 2 H S 1 M 2 H /S dx 1 x 1 F P g (x 1 , M 2 ) F P g M 2 H x 1 S , M 2 | T | 2 (9.5)
The bounds on integration are obtained from the constraints 0 < x 1 , x 2 ≤ 1 and

M 2 H /S ≤ x 1 x 2 ≤ 1 which lead to M 2 H /(x 1 S) ≤ x 2 ≤ 1 and M 2 H /S ≤ x 1 ≤ 1. We compute now | T | 2 .
At the lowest order, three diagrams contribute to the partonic process g(p 1 ) + g(p 2 ) → H(p 3 ):

g(p 1 ) g(p 1 ) g(p 1 ) g(p 2 ) g(p 2 ) g(p 2 ) k k -p 1 k + p 2 H(p 3 ) H(p 3 ) H(p 3 ) T 1 T 2 T 3
The amplitude T 3 vanishes because it is proportional to T r[T a ], T a traceless, a generator of the SU (3) colour algebra. Applying the Feynman rules in n = 4 dimensions to tame potential ultraviolet divergencies (µ is the arbirary mass introduced when going to n dimensions) and taking into account the factor -1 for a fermion loop, the amplitude T 1 is given by:

T 1 = -g 2 s m q v T r T a T b µ 4-n d n k (2 π) n T r γ ν ( k + m q ) k 2 -m 2 q + i ǫ γ µ ( k -p 1 + m q ) (k -p 1 ) 2 -m 2 q + i ǫ ( k + p 2 + m q ) (k + p 2 ) 2 -m 2 q + i ǫ × ǫ a µ (p 1 ) ǫ b ν (p 2 ) = - g 2 s e m q 4 sin θ W M W µ 4-n δ ab ǫ a µ (p 1 ) ǫ b ν (p 2 ) d n k (2 π) n × T r [γ ν ( k + m q ) γ µ ( k -p 1 + m q ) ( k + p 2 + m q )] (k 2 -m 2 q + i ǫ) ((k -p 1 ) 2 -m 2 q + i ǫ) ((k + p 2 ) 2 -m 2 q + i ǫ) (9.6)
where g s T a is the strong interaction coupling of a gluon of colour a to a quark, m q /v (see eq. (8.29)), the coupling of the quarks to the Higgs boson. In the second equation the relation 1/v = e/2 sin θ W M W , eq. (8.19) is used and T r T a T b = δ ab /2 takes care of the sum on the quark colours in the loop.

Setting

N µ ν (k) = T r [γ ν ( k + m q ) γ µ ( k -p 1 + m q ) ( k + p 2 + m q )]
and computing the trace on the Dirac matrices as usual, we get:

N µ ν (k) = 4 m q {g µ ν (k -p 1 ).(k + p 2 ) + (k + p 2 ) ν (k -p 1 ) µ -(k -p 1 ) ν (k + p 2 ) µ + k ν (k + p 2 ) µ +(k + p 2 ) ν k µ -g µ ν k.(k + p 2 ) + k ν (k -p 1 ) µ + k µ (k -p 1 ) ν -g µ ν k.(k -p 1 ) + m 2 q g µ ν = 4 m q g µ ν (m 2 q -k 2 -p 1 .p 2 ) + 4 k µ k ν -2 k ν p µ 1 + 2 k µ p ν 2 + p ν 1 p µ 2 -p ν 2 p µ 1 . (9.7)
In eq. (9.7), all the terms proportional to p µ 1 and p ν 2 can be dropped because they will vanish after contraction with the gluon polarisation vectors. The quantity N µ ν (k) becomes:

N µ ν (k) = 4 m q g µ ν (m 2 q -k 2 -p 1 .p 2 ) + 4 k µ k ν + p ν 1 p µ 2 . (9.8)
Then, two Feynman parameters x and y are introduced to linearize the denominator:

T 1 = 2 K µ ν 1 0 dy y 1 0 dx d n k (2 π) n N µ ν (k) × (1 -y) (k 2 -m 2 q + i ǫ) + x y ((k -p 1 ) 2 -m 2 q + i ǫ) + (1 -x) y ((k + p 2 ) 2 -m 2 q + i ǫ) -3 = 2 K µ ν 1 0 dy y 1 0 dx d n k (2 π) n N µ ν (k) × (k + (p 2 (1 -x) -p 1 x) y) 2 + 2 y 2 x (1 -x)p 1 .p 2 -m 2 q + i ǫ -3
(9.9)

with

K µ ν = - g 2 s e m q 4 sin θ W M W µ 4-n δ ab ǫ a µ (p 1 ) ǫ b ν (p 2 )
We shift the loop four-momentum k = l -(p 2 (1 -x) -p 1 x) y. The factor N µ ν (k) contains terms of the type k 2 and k µ k ν which transform under the shift as:

k 2 ≃ l 2 -2 y 2 x (1 -x) p 1 .p 2 k µ k ν ≃ g µ ν /n l 2 -y 2 x (1 -x) p µ 2 p ν 1
All odd powers of l will vanish after the integration over l, so they have been removed. Eq. (9.8) becomes:

N µ ν (k) = 4 m q g µ ν 4 n -1 l 2 + m 2 q + 2 p 1 .p 2 y 2 x (1 -x) - 1 2 + p ν 1 p µ 2 1 -4 y 2 x (1 -x) (9.10)
The amplitude T 1 is then:

T 1 = 2 K µ ν 1 0 dy y 1 0 dx d n l (2 π) n 4 m q (A 1 l 2 + A 2 ) g µ ν + B p ν 1 p µ 2 (l 2 -R 2 + i ǫ) 3 (9.11) with: R 2 = m 2 q -2 y 2 x (1 -x) p 1 .p 2 A 1 = 4/n -1 A 2 = 2 m 2 q -p 1 .p 2 -R 2 B = 1 -4 y 2 x (1 -x) = 1 -2(m 2 q + R 2 )/p 1 .p 2
The integration over the four-momentum l yields the following result 15 :

T 1 = i (4 π) n/2 4 m q K µ ν 1 0 dy y 1 0 dx n 2 4 -n n Γ 2 - n 2 R 2 -i ǫ -2+n/2 g µ ν -Γ 3 - n 2 R 2 -i ǫ -3+n/2 (A 2 g µ ν + B p ν 1 p µ 2 ) (9.12)
15 The general formula is:

d n k (2π) n k 2 r [k 2 -R 2 + iǫ] m = i (R 2 -iǫ) r-m+ n 2 (-1) r-m (4π) n 2 Γ(r + n 2 ) Γ( n 2 ) Γ(m -r -n 2 ) Γ(m) = i (-1) r-m (4π) 2 4π R 2 -iǫ ε (R 2 ) 2+r-m Γ(2 + r -ε) Γ(2 -ε) Γ(m -r -2 + ε) Γ(m) .
The coefficient in front of the ultraviolet divergence Γ 2 -n 2 vanishes for n = 4, more precisely:

n 2 4 -n n Γ 2 - n 2 = 2 - n 2 Γ 2 - n 2 = Γ 3 - n 2
So, actually, there is no divergence in this amplitude and we can now take safely n = 4 so that Γ 3 -n 2 reduces to 1. In addition, using p 1 .p 2 = M 2 H /2, we get:

T 1 = i (4 π) 2 4 m q K µ ν g µ ν - 2 p ν 1 p µ 2 M 2 H 1 0 dy y 1 0 dx × 2 + M 2 H 2 1 - 4 m 2 q M 2 H 1 m 2 q -y 2 x (1 -x) M 2 H -i ǫ) (9.13)
To perform the integration on the Feynman parameters, let us introduce the function:

J(z) = 1 0 dx 1 0 dy y 1 1 -y 2 x (1 -x)/z -i ǫ (9.14) with z = m 2 q /M 2 H positive.
The integration over y can be easily performed to get:

J(z) = - z 2 1 0 dx 1 x (1 -x) ln 1 - x (1 -x) z -i ǫ = - z 2 1 0 dx 1 x + 1 (1 -x) ln 1 - x (1 -x) z -i ǫ = -z 1 0 dx x ln 1 - x (1 -x) z -i ǫ . (9.15)
The roots of the argument of the logarithm are given by:

0 < z < 1 4 x 1, 2 = 1 2 ± 1 2 √ 1 -4 z ± i ǫ z > 1 4 x 1, 2 = 1 2 ± i 2 √ 4 z -1 , so ln 1 - x (1 -x) z -i ǫ = ln 1 z + ln(x -x 1 ) + ln(x -x 2 ) ,
but ln(1/z) = -ln(x 1 x 2 ) = -ln(-x 1 ) -ln(-x 2 ) because x 1 and x 2 are complex conjugate. The two terms ln(x -x 1 ) and ln(-x 1 ) can be grouped because the imaginary parts of the two arguments are the same and similarly for the terms in x 2 . So we get for J(z):

J(z) = -z 1 0 dx x ln 1 - x x 1 + 1 0 dx x ln 1 - x x 2 = z Li 2 1 x 1 + Li 2 1 x 2 . (9.16)
It can be shown, c.f. sec. 9.3, that J(z) can be written using only the logarihtm function whatever the value of z is:

J(z) = - z 2    ln 1- √ 1-4 z 1+ √ 1-4 z -i π 2 z ≤ 1/4 ln 2 i √ 4 z-1-1 i √ 4 z-1+1
z > 1/4 . (9.17)

So the amplitude T 1 is given by:

T 1 = i (4 π) 2 4 m q K µ ν g µ ν - 2 p ν 1 p µ 2 M 2 H M 2 H 2 m 2 q × 2 m 2 q M 2 H + 1 -4 m 2 q M 2 H J m 2 q M 2 H . (9.18)
For the following, we set:

F(z) = 2 z + (1 -4 z) J (z) (9.19)
The function F(z) can be complex or real following the ratio z = m 2 q /M 2 H . In fig. 7, we draw the real and imaginary parts of F(z) with respect to z (see sec. 9.3). It can be shown that this function has the limit 1/3 when z → ∞. To show that let us come back to eq. (9.14) which gives the integral representation of the function J(z), when z → ∞, the denominator cannot vanish so we can take safely the limit ǫ → 0 and write:

1 1 -y 2 x (1 -x)/z ≃ 1 + y 2 x (1 -x)/z .
So in this limit, the function J(z) behaves as:

J(z) ≃ 1 0 dx 1 0 y dy + 1 z 1 0 dx x (1 -x) 1 0 dy y 3 ≃ 1 2 + 1 24 z , (9.20)
and therefor the function F(z) → 1/3 when z → ∞. Since the amplitude T 2 can be obtained from the amplitude T 1 by changing ǫ(p 1 ), p 1 ↔ ǫ(p 2 ), p 2 , it is clear from eq. (9.18) that T 2 = T 1 . So the total amplitude T = T 1 + T 2 is:

T = - i 4 π α s e sin θ W M 2 H M W δ ab ǫ a µ (p 1 ) ǫ b ν (p 2 ) g µ ν - 2 p ν 1 p µ 2 M 2 H F m 2 q M 2 H , (9.21)
where the notation α s = g 2 s /(4 π) has been introduced for the strong interaction coupling. Note that in eq. (9.21), if we replace ǫ(p 1 ) (respectively ǫ(p 2 )) by p 1 (resp. p 2 ), the amplitude T vanishes because:

p 1 µ g µ ν - 2 p ν 1 p µ 2 M 2 H = 0
Let us now compute the modulus squared of the amplitude averaging over the initial spins and colours:

| T | 2 = 1 4 (N 2 -1) 2 polarisations colours |T | 2
For the average over the initial spins, we have to compute something like:

S = polarisations ǫ a µ (p 1 ) ǫ b ν (p 2 ) ǫ c ⋆ ρ (p 1 ) ǫ d ⋆ σ (p 2 ) g µ ν - 2 p ν 1 p µ 2 M 2 H g ρ σ - 2 p σ 1 p ρ 2 M 2 H = δ a c δ b d (-g µ ρ ) (-g ν σ ) g µ ν - 2 p ν 1 p µ 2 M 2 H g ρ σ - 2 p σ 1 p ρ 2 M 2 H = 2 δ a c δ b d
Note that we have taken pol. ǫ a µ (p 1 ) ǫ c ⋆ ρ (p 1 ) = -δ a c g µ ρ this is justified in this case because the replacement of ǫ µ (p 1 ) by p µ 1 (ǫ ν (p 2 ) by p 2 ) gave zero. Now, for the average over the initial colours, we have to compute:

C = a,b,c,d δ a b δ c d δ a c δ b d = a δ a a = N 2 -1
where N is the number of colours. Finally we obtain for the matrix element squared:

| T | 2 = α 2 s α M 4 H 8 π (N 2 -1) (sin θ W M W ) 2 F m 2 q M 2 H 2 , (9.22)
where the fine structure constant α = e 2 /4π is introduced. So far, we considered only one quark flavour in the loop, in principle we need to sum over all the possible flavours of quarks so that eq. (9.5) becomes:

σ H = 1 8 (N 2 -1) α 2 s α M 2 H (sin θ W M W ) 2 S   q=d,u,s,c,b,t F m 2 q M 2 H 2   × 1 M 2 H /S dx 1 x 1 F P g (x 1 , M 2 ) F P g M 2 H x 1 S , M 2 (9.23)
In practice, we can content ourselves to keep only the top quark since the function F is vanishingly small for other quark species. The scale M which appears in the partonic densities of eq. (9.23) must be taken of the order of the Higgs boson mass (M H ) because this is the only "hard" energy scale (much greater than Λ QCD ) which remains.

Function Li 2

The Li 2 function is defined as:

Li 2 (z) = - y 0 dt ln(1 -t) t = - 1 0 dt ln(1 -z t) t (9.24)
with z complex. From its definition, the function Li 2 has a cut in the complex plan on the real axis [1, ∞[. Furthermore, we have the following property:

Li 2 (1) = π 2 6 = ∞ k=1 1 k 2 .
In the case where z has an infinitesimal imaginary part z = x ± i ǫ and a real part x > 1, from the definition of the function Li 2 , we can show that:

Li 2 (x ± i ǫ) ǫ→0 = -Li 2 1 x - 1 2 ln 2 1 x + π 2 3 ∓ i π ln 1 x . (9.25)
This equation gives us the prescription for x > 1. More generally, if z is a complex number with a non vanishing imaginary part (it is always the case if we carefuly keep track of the small imaginary part ǫ), we have the following relations:

Li 2 1 z = -Li 2 (z) - π 2 6 - 1 2 ln 2 (-z) (9.26) Li 2 (1 -z) = -Li 2 (z) + π 2 6 -ln(1 -z) ln(z) .
(9.27)

Different rewritting of the function J(z)

We can apply the relations in the section above to simplify J(z) (eq. (9.16)). Let us start with the case where z ≤ 1/4. In this case the real parts of x 1 and x 2 are between 0 and 1. We can use eq. (9.25)

and we obtain that:

Li 2 1 x 1 + Li 2 1 x 2 = -Li 2 (y) -Li 2 (1 -y) - 1 2 ln 2 (y) - 1 2 ln 2 (1 -y) + 2 π 2 3 -i π (ln(1 -y) -ln(y)) , (9.28)
with y the real part of x 1 , y = 1/2 (1 + √ 1 -4 z). Then, we can apply the relation (9.27) to the equation (9.28), this gives:

Li 2 1 x 1 + Li 2 1 x 2 = - 1 2 ln 2 (y) - 1 2 ln 2 (1 -y) + ln(y) ln(1 -y) + π 2 2 -i π (ln(1 -y) -ln(y)) = - 1 2 (ln(1 -y) -ln(y)) 2 + π 2 2 -i π (ln(1 -y) -ln(y)) . (9.29)
As 0 ≤ y ≤ 1, we can group the logarithms and we get:

Li 2 1 x 1 + Li 2 1 x 2 = - 1 2 ln 1 y -1 + i π 2 = - 1 2 ln 2 1 - 1 x 1 . (9.30)
In the case where z > 1/4, x 1 and x 2 are complex conjugate but with an imaginary part which is not infinitesimal. We will use the relation (9.26) to write that:

Li 2 1 x 1 + Li 2 1 x 2 = -Li 2 (x 1 ) -Li 2 (1 -x 1 ) - 1 2 ln 2 (-x 1 ) - 1 2 ln 2 (x 1 -1) - π 2 3 . (9.31)
Then, applying eq. (9.27), the sum of the dilogarithms becomes:

Li 2 1 x 1 + Li 2 1 x 2 = - 1 2 ln 2 (-x 1 ) - 1 2 ln 2 (x 1 -1) - π 2 2 + ln(x 1 ) ln(1 -x 1 ) = - 1 2 (ln(x 1 -1) -ln(-x 1 )) 2 - π 2 2 -ln(x 1 -1) ln(-x 1 ) + ln(x 1 ) ln(1 -x 1 ) . (9.32)
Let us remark that x 1 has a real part and an imaginary part which are both positive, it lies then in the first quadrand. With the convention that the cut of the logarithm is along the negative real axis, then the phase of a complex number in the main Rieman sheet is between -π and π, if x 1 is parametrised like ρ e i θ then -x 1 = ρ e i (θ-π) and so the relation between the logarithms of x 1 and -x 1 is: ln(-x 1 ) = ln(x 1 ) -i π .

In the same way, 1 -x 1 has a positive real part and a negative imaginary part, so if 1 -x 1 = ρ e i θ then x 1 -1 = ρ e i (θ+π) and we have that:

ln(x 1 -1) = ln(1 -x 1 ) + i π .
Using that, we write: ln(x 1 ) ln(1 -x 1 ) = ln(-x 1 ) ln(x 1 -1) + i π (ln(x 1 -1) -ln(-x 1 )) + π 2 , (9.33) so the sum of the two dilogarithms can be written:

Li 2 1 x 1 + Li 2 1 x 2 = - 1 2 (ln(x 1 -1) -ln(-x 1 )) 2 + π 2 2 + i π (ln(x 1 -1) -ln(-x 1 )) = - 1 2 [ln(x 1 -1) -ln(-x 1 ) -i π] 2 . (9.34)
The term i π can be reabsorbed by writing ln(1-x 1 ) instead of ln(x 1 -1) and remarking that 1-x 1 = x 2 and -x 1 have a same sign imaginary part, then we finally get:

Li 2 1 x 1 + Li 2 1 x 2 = - 1 2 ln 2 1 - 1 x 1 . (9.35)
Thus, J(z) can be simplified such that only the logarithmic function is used for buth cases:

J(z) = - z 2    ln 2 √ 1-4 z-1+i ǫ √ 1-4 z+1+i ǫ z ≤ 1/4 ln 2 i √ 4 z-1-1 i √ 4 z-1+1 z > 1/4 . (9.36)
To conclude these technical remarks, we show how to rewrite J(z) to make easy the comparison with the results which can be found in the litterature. In the case z ≤ 1/4, it is easy to show that:

ln √ 1 -4 z -1 + i ǫ √ 1 -4 z + 1 + i ǫ = ln √ 1 -4 z -1 √ 1 -4 z + 1 + i ǫ = -ln 1 + √ 1 -4 z 1 - √ 1 -4 z + i π . (9.37)
For the case z > 1/4, we write:

ln i √ 4 z -1 -1 i √ 4 z -1 + 1 = ln   1 -1 4 z + i 1 4 z 1 -1 4 z -i 1 4 z   . (9.38)
Remarking that the complex number 1 -1/4 z + i 1/4 z has a modulus which is equal to 1 and it lies in the first quadrand, we show:

ln i √ 4 z -1 -1 i √ 4 z -1 + 1 = ln   1 - 1 4 z + i 1 4 z 2   = 2 ln 1 - 1 4 z + i 1 4 z = 2 i arcsin 1 4 z . (9.39)
Then, the function J(z) becomes:

J(z) = 4 z 2    -1 4 ln 1+ √ 1-4 z 1- √ 1-4 z -i π 2 z ≤ 1/4 arcsin 2 1 4 z z > 1/4 . (9.40)
10 Exercises: Higgs boson decays

We consider in the following various two body decays of a Higgs boson.

Kinematics

Let the Higgs boson of mass M H and momentum q decay into particles A and B of masses m 1 and m 2 and momenta p 1 and p 2 respectively: H(q) → A(p 1 ) + B(p 2 ). The decay rate summed over final polarisations and colours is:

d Γ = 1 2 M d 3 p 1 (2 π) 3 2 E 1 d 3 p 2 (2 π) 3 2 E 2 (2 π) 4 δ 4 (q -p 1 -p 2 ) T 2 , (10.1)
with T 2 the invariant matrix element squared, summed over final colours and polarisations. Mo-

mentum conservation imposes p 1 .p 2 = (M 2 H -m 2 1 -m 2 2 )/2 with p 2 1 = m 2 1 et p 2 2 = m 2 2 .
Thus T 2 depends only on the external masses T (m 2 1 , m 2 2 , M 2 H ) 2 and the integral in eq. ( 10.1) can be done independently of the decay channel. Using

d 3 p 2 /2 E 2 = d 4 p 2 δ + (p 2 2 -m 2
2 ) and carrying out the d 4 p 2 integration it comes out

d Γ = 1 2 M H T 2 (2 π) 2 d 3 p 1 2 E 1 δ + (q -p 1 ) 2 -m 2 2 . (10.2)
Going to the rest frame of the Higgs boson, q = (M, 0, 0, 0), one finds that the argument of the δ + function reduces to (M 2 -2M E 1 + m 2 1 -m 2 2 ) independent of the angles. Since all cases we consider have m 1 = m 2 the expressions will simplify. Using p 1 dp 1 = E 1 dE 1 all integrations are easily done to get:

Γ = 1 16 π M H T 2 1 - 4 m 2 M 2 , (10.3)
with m the common mass of the decay products.

Higgs decay into a fermion anti-fermion pair

This channel has only one diagram with the Higgs fermion-antifermion coupling, m f /v given in eq. (8.29):

¡ H(q) f (p 2 ) f (p 1 )
The coresponding amplitude T is:

T = -i m f v ū(p 1 ) v ( p 2 ) (10.4)
leading to:

| T | 2 = m 2 f v 2 (N ) T r [ p 1 p 2 ] -m 2 f T r[1] = 2 m 2 f M 2 H v 2 1 - 4 m 2 f M 2 H (N ) .
(10.5)

In the above result the colour factor N has been put in parentheses to indicate that, if the final fermions are quarks then we keep this factor, while if they are leptons it should be ignored. Getting rid of the vacuum expectation value v in favor of physical quantities via eq. (8.19), 1/v = e/(2 sin θ W M W ) and e 2 = 4πα the decay rate is:

Γ H→f f = (N ) α 8 sin 2 (θ W ) M H m 2 f M 2 W 1 - 4 m 2 f M 2 H 3/2 (10.6)
where eq. ( 10.3) has been used.

Higgs decay into a W + W -pair

Here again only one diagram contributes:

¢ H(q) W -(p 2 ) W + (p 1 )
The amplitude for this transition is:

T = i e M W sin θ W g α β ε λ 1 * α (p 1 ) ε λ 2 * β (p 2 ) , (10.7)
with ε λ α (p) the polarisation vector of a gauge boson and the coupling given in eq. (8.22). The sum over polarisations is done using:

λ ε λ α (p) ε λ * β (p) = -g α β + p α p β M 2 W , (10.8)
so that:

| T | 2 = e M W sin θ W 2 -g α µ + p 1 α p 1 µ M 2 W -g α µ + p α 2 p µ 2 M 2 W = e 2 4 sin 2 θ W M 2 W 12 M 4 W + M 4 H -4 M 2 W M 2 H . (10.9)
Finally the decay rate is:

Γ H→W + W -= α 16 sin 2 (θ W ) M 3 H M 2 W 1 - 4 M 2 W M 2 H 1 -4 M 2 W M 2 H + 12 M 4 W M 4 H (10.10)

Higgs decay in a γ γ pair

As seen in sec. 8.4 this transition goes via two types of loop diagrams, one involving fermions and the other charged gauge bosons.

W boson loop

In the unitary gauge three types of diagrams contribute:

£ k + p 1 k k + q H(q) γ(p 2 ) γ(p 1 ) T 1 ¤ k + p 2 k k + q H(q) γ(p 1 ) γ(p 2 ) T 2 ¥ k + q k H(q) γ(p 2 ) γ(p 1 ) T 3
All these diagrams have a common structure, namely the HW W vertex and the two adjacent W propagators of momentum k and k+q respectively. Each amplitude T i is written as

T i = T µ 1 µ 2 i ε * µ 1 (p 1 ) ε * µ 2 (p 2 )
where we drop for simplicity the photon polarisation indices. Furthermore we introduce the tensor T µ 1 µ 2 i :

T µ 1 µ 2 i = d n k (2 π) n T µ 1 µ 2 i , with T µ 1 µ 2 i = R α 1 α 2 M µ 1 µ 2 α 1 α 2 i
, R α 1 α 2 containing the part common to all three digrams. Applying the Feynman rules, using the unitary gauge for the W propagators, it comes out:

M µ 1 µ 2 α 1 α 2 1 = (i e) g α 2 µ 2 (k + q + p 2 ) β 1 + g µ 2 β 1 (-p 2 + k + p 1 ) α 2 + g β 1 α 2 (-k -p 1 -k -q) µ 2 × (-i) g β 1 β 2 - (k + p 1 ) β 1 (k + p 1 ) β 2 M 2 W 1 (k + p 1 ) 2 -M 2 W + i ǫ × (i e) g β 2 µ 1 (k + p 1 + p 1 ) α 1 + g µ 1 α 1 (-p 1 + k) β 2 + g α 1 β 2 (-k -k -p 1 ) µ 1 (10.11) M µ 1 µ 2 α 1 α 2 2 = M µ 1 µ 2 α 1 α 2 1 (µ 1 ↔ µ 2 , p 1 ↔ p 2 )
(10.12)

M µ 1 µ 2 α 1 α 2 3 = i e 2 [g α 1 µ 1 g α 2 µ 2 + g α 1 µ 2 g α 2 µ 1 -2 g µ 1 µ 2 g α 1 α 2 ] , (10.13)
and for the common structure of the amplitudes:

R α 1 α 2 = -i e M W sin(θ W ) g α 1 α 2 - (k + q) α 1 (k + q) α 2 M 2 W - k α 1 k α 2 M 2 W + k α 1 (k + q) α 2 k.(k + q) M 4 W × 1 (D 0 + i ǫ) (D 3 + i ǫ) . (10.14)
The quantities D 0 and D 3 are the denominators of propagators,

D 0 = k 2 -M 2 W , D 3 = (k + q) 2 -M 2 W , (10.15)
and we will need later,

D 1 = (k + p 1 ) 2 -M 2 W , D 2 = (k + p 2 ) 2 -M 2 W .
(10.16)

The diagrams T 1 , T 2 et T 3 are highly divergent in the ultraviolet region:

T 1 et T 2 ≃ d 4 k k 8 k 6 ≃ dk k 5 T 3 ≃ d 4 k k 6 k 6 ≃ dk k 3 ,
but working in n space-time dimensions regularizes the divergencies. Rather than evaluating these integrals by brute force we try to arrange the terms to make possible cancellations obvious in the integrands. One thus defines:

T µ 1 µ 2 = T µ 1 µ 2 1 + T µ 1 µ 2 2 + T µ 1 µ 2 3 = d n k (2 π) n T µ 1 µ 2 1 + T µ 1 µ 2 2 + T µ 1 µ 2 3 .
(10.17)

After integration on the loop momentum k the tensor T µ 1 µ 2 depends only on the external momenta p 1 , p 2 and it can be parameterised as:

T µ 1 µ 2 = A p 1 .p 2 p µ 2 1 p µ 1 2 + B p 1 .p 2 p µ 1 1 p µ 2 2 + C g µ 1 µ 2 . (10.18)
The aim is to calculate the expressions A, B and C. For this purpose we construct the following scalars:

g µ 1 µ 2 T µ 1 µ 2 = A + B + n C p 1 µ 2 p 2 µ 1 T µ 1 µ 2 = p 1 .p 2 (B + C) p 1 µ 1 p 2 µ 2 T µ 1 µ 2 = p 1 .p 2 (A + C) ,
where the property g µ 1 µ 2 g µ 1 µ 2 = n has been used since we work in n dimensions. The system of equations is easily solved to find:

C = 1 2 (n -2) g µ1 µ 2 T µ 1 µ 2 - p 1 µ 2 p 2 µ 1 p 1 .p 2 T µ 1 µ 2 - p 1 µ 1 p 2 µ 2 p 1 .p 2 T µ 1 µ 2 (10.19) B = p 1 µ 2 p 2 µ 1 p 1 .p 2 T µ 1 µ 2 -C (10.20) A = p 1 µ 1 p 2 µ 2 p 1 .p 2 T µ 1 µ 2 -C . (10.21)
The various contractions of the tensor T µ 1 µ 2 are calculated with the help of a form program 16,17 . By reconstructing systematically the quantities D 0 , • • • , D 3 in the numerators and cancelling them with the denominators, we get rid of the k dependence in the numerators so that only scalar integrals have to be evaluated. There are two 3-point integrals:

I 013 = d n k (2 π) n 1 (D 0 + i ǫ) (D 1 + i ǫ) (D 3 + i ǫ) I 023 = d n k (2 π) n 1 (D 0 + i ǫ) (D 2 + i ǫ) (D 3 + i ǫ)
three 2-points integrals:

I 03 = d n k (2 π) n 1 (D 0 + i ǫ) (D 3 + i ǫ) I 13 = d n k (2 π) n 1 (D 1 + i ǫ) (D 3 + i ǫ) I 23 = d n k (2 π) n 1 (D 2 + i ǫ) (D 3 + i ǫ)
and four 1-point integrals:

I 0 = d n k (2 π) n 1 (D 0 + i ǫ) I 1 = d n k (2 π) n 1 (D 1 + i ǫ) I 2 = d n k (2 π) n 1 (D 2 + i ǫ) I 3 = d n k (2 π) n 1 (D 3 + i ǫ)
.

16 For an on line documentation on form see http://www.nikhef.nl/ ~form/maindir/documentation/reference/online/ 17 The code for the evaluation of A, B and C is found at https://lectures.lapth.cnrs.fr/standard_model/cours/hgaga.frm 79 Note that the last two sets of integrals would be ultraviolet divergent in 4 dimensions, but, working in n dimensions, they are regular and we can do translations on the loop momentum to evaluate them.

For example:

I 1 = d n k (2 π) n 1 (k + p 1 ) 2 -M 2 W + i ǫ = d n k ′ (2 π) n 1 (k ′ ) 2 -M 2 W + i ǫ with k ′ = k + p 1 , (10.22)
then I 1 = I 0 . In the same way one shows that:

I 3 = I 2 = I 1 = I 0
All 2-point integrals can be written in the following form:

J 2 = d n k (2 π) n 1 (k 2 -M 2 W + i ǫ) ((k + p) 2 -M 2 W + i ǫ) , (10.23)
is reduced to:

J 2 = 1 0 dx d n k (2 π) n 1 ((k + p x) 2 -R 2 + i ǫ) 2 , (10.24)
after introduction of the Feynman variable x, with R 2 = M 2 W -p 2 x (1 -x). Doing the change of variable k to l = k + p x and using the usual formulae (see sec. 9.1) one obtains:

J 2 = i (4 π) n/2 1 0 dx R 2 -i ǫ -2+n/2 Γ(2 -n/2) Γ(2) . (10.25)
Introducing ε through n = 4 -2 ε, and expanding around ε = 0, it comes out:

J 2 = i (4 π) 2-ε Γ(1 + ε) ε 1 -ε Ĩ(p 2 ) , (10.26) with: Ĩ(p 2 ) = 1 0 dx ln M 2 W -p 2 x (1 -x) -i ǫ . (10.27)
The pole in ε is the consequence of the ultraviolet divergence of the 2-point functions. It turns out that, in our calculation, the 2-point integrals are all multiplied by ε which allows us to take the ε → 0 limit to find finally:

ε J 2 = i (4 π) 2 .
(10.28)

For the 3-point integrals, both I 013 et I 023 can be written as:

J 3 = d n k (2 π) n 1 (k -r 1 ) 2 -M 2 W + i ǫ 1 k 2 -M 2 W + i ǫ 1 (k + r 2 ) 2 -M 2 W + i ǫ (10.29)
with r 1 = p 1 and r 2 = p 2 for I 013 , and r 1 = p 2 et r 2 = p 1 for I 023 . Introducing the Feynman parameters and using l = k + (r 2 (1 -x) -r 1 x) y rather than k as integration variable one finds:

J 3 = 2 1 0 y dy 1 0 dx d n l (2 π) n 1 (l 2 -R 2 + i ǫ) 3 , (10.30) with R 2 = M 2 W -y 2
x (1 -x) q 2 and q = p 1 + p 2 = r 1 + r 2 . Integrating on l yields:

J 3 = - i (4 π) n/2 Γ 3 - n 2 1 0 y dy 1 0 dx (R 2 -i ǫ) -3+n/2 , (10.31)
which is regular. Taking n = 4 and doing the y integration J 3 can be written as:

J 3 = - i (4 π) 2 1 M 2 W J M 2 W q 2 , (10.32)
with the function J defined in eq. (9.16) of the previous section. The result depends only on r 1 + r 2 which implies I 013 = I 023 .

After contraction of the tensor T µ 1 µ 2 with the photons polarisation vectors we obtain:

T W = T µ 1 µ 2 ε * µ 1 (p 1 ) ε * µ 2 (p 2 ) = C g µ 1 µ 2 + A p 1 .p 2 p µ 1 1 p µ 2 2 ε * µ 1 (p 1 ) ε * µ 2 (p 2 ) (10.33)
one observes that the B term has disappeared as it is multiplied by 0!. The form code gives A = -C and:

C = i (4 π) 2 e 2 e M W sin(θ W ) 6 + 1 z W + J(z W ) -12 + 6 z W (10.34) with z W = M 2 W /M 2 H .
Putting everything together it comes out:

T W = i (4 π) 2 e 2 e M W sin(θ W ) 6 + 1 z W + J(z W ) -12 + 6 z W g µ 1 µ 2 - p µ 1 1 p µ 2 2 p 1 .p 2 × ε * µ 1 (p 1 ) ε * µ 2 (p 2 ) = i α 4 π e sin(θ W ) M 2 H M W G M 2 W M 2 H g µ 1 µ 2 - 2 p µ 1 1 p µ 2 2 M 2 H ε * µ 1 (p 1 ) ε * µ 2 (p 2 ) (10.35)
where:

G(z) = [6 z + 1 + 6 J (z) (1 -2 z)]
Some remarks are in order.

1. All ultraviolet divergences have disappeared: it was necessary to go to n dimensions in the intermediate steps of the calculation to give a mathematical meaning to individual integrals and allow for the momentum translations in the loops, but after combining all terms one takes the limit to 4 dimensions since the final result is regular;

2. T µ 1 µ 2 is transverse, which means p 1 µ 1 T µ 1 µ 2 = p 2 µ 2 T µ 1 µ 2 =0 .

Fermion loops

This part is very similar to the calculation of Higgs boson production via gluon-gluon fusion in sec. 9

and the result eq. (9.21) can be used with appropriate changes. First the strong coupling is replaced by e Q f and α s then becomes α Q 2 f with Q f = -1, 2/3 or -1/3. Since the photons are colour neutral the colour factor T r T a T b = δ ab /2 becomes 1 (see eq. (9.6)). The result is:

T f = -i α Q 2 f e 2 π sin θ W M 2 H M W F(z f ) g µ 1 µ 2 - 2 p µ 1 1 p µ 2 2 M 2 H ε * µ 1 (p 1 ) ε * µ 2 (p 2 ) , (10.36)
with

z f = m 2 f /M 2 H .
Eventhough heavy fermions only will contribute (Higgs coupling proportional to the fermion mass) it is necessary to recall that we have to sum over all fermions. The function F is defined in eq. (9.19) and we recall it here:

F(z) = [2 z + (1 -4 z) J (z)] .
The amplitude for the decay of a Higgs boson into two photons is then:

T f = -i α e 2 π sin θ W M 2 H M W g µ 1 µ 2 - 2 p µ 1 1 p µ 2 2 M 2 H ε * µ 1 (p 1 ) ε * µ 2 (p 2 ) × l Q 2 l F m 2 l M 2 H + N q Q 2 q F m 2 q M 2 H (10.37)
where the sum over l stands for leptons and q for quarks. In the latter case an extra factor N obviously appears from the colour sum in the loop.

Final result

The final amplitude will be the sum of the amplitudes T f and T W . To calculate its square one has to sum on the photon polarisation and evaluate the expression:

S = λ 1 λ 2 ε λ 1 * µ 1 (p 1 ) ε λ 2 * µ 2 (p 2 ) ε λ 1 ν 1 (p 1 ) ε λ 2 ν 2 (p 2 ) g µ 1 µ 2 - 2 p µ 2 1 p µ 1 2 M 2 H g ν 1 ν 2 - 2 p ν 2 1 p ν 1 2 M 2 H = g µ 1 µ 2 - 2 p µ 1 1 p µ 2 2 M 2 H g µ 1 µ 2 - 2 p 1 µ 1 p 2 µ 2 M 2 H = 2 (10.38)
thus:

|T W + T f | 2 = α 2 e 2 8 π 2 sin θ W M 4 H M 2 W |Y | 2
(10.39) with:

Y = G M 2 W M 2 H -2 l F m 2 l M 2 H -2 N q Q 2 q F m 2 q M 2

H

Using eq. ( 10.3), the decay rate of a Higgs boson in two photons is:

Γ H→γ γ = α 3 32 π 2 sin 2 (θ W ) M 3 H M 2 W |Y | 2 (10.40)
11 Family mixing and the Kobayashi-Maskawa matrix

The above discussion has been considerably simplified since it completely ignored mixing between the three fermion families. For example, if the coupling u → d + W + is allowed the couplings u → s + W + for the three families constructed in the previous sections are eigenstates of the mass matrix but the charged electroweak transition is not diagonal in these states but rather it is diagonal in a (u ′ i ) , (d ′ i ) basis called the flavour basis. The two bases are related as follows:

u i L = S u L ij u ′ j L , d i L = S d L ij d ′ j L , u i R = S u R ij u ′ j R , d i R = S d R ij d ′ j R , i, j = 1, 2, 3. (11.1)
Eq. (11.1) can be written in a matrix form u L = S u L u ′ L and similarly for the right-handed up sector as well as the left-handed and right-handed down sectors. As will be shown later the matrices S u L , S u R , • • • are unitary. The quark sector of the SU (2) L ⊗ U (1) Y lagrangian is written in general18 (see eq. (5.22))

L F = i (u ′ i L d ′ i L ) i D q L u ′ i L d ′ i L + i u ′ i R i D u R u ′ i R + i d ′ i R i D d R d ′ i R , (11.2)
so that the electroweak interactions are diagonal in the "primed" flavour basis. After symmetry breaking, the most general Yukawa lagrangian takes the form in the "primed" basis, (see eqs. (8.26), (8.27)), ignoring for the moment the H boson couplings to the fermions

L Y = - v √ 2 ij (u ′ i L c u ij u ′ j R + d ′ i L c d ij d ′ j R + h.c.) = - v √ 2 (u ′ L C u u ′ R + d ′ L C d d ′ R + h.c.) (11.3)
where the complex, 3 × 3, C u , C d matrices are the generalized Yukawa couplings. Including the common normalisation factor v/ √ 2 with the C u and C d matrices, the most general such matrices can be written as a product :

v √ 2 C u = M u T u , v √ 2 C d = M d T d (11.4)
with M u a hermitian matrix (M u = M † u ) and T u a unitary matrix (T -1 u = T † u ). The hermitian matrix can be diagonalized by a unitary transformation, M u = S -1 u m u S u = S † u m u S u where m u is diagonal with real eigenvalues, and similarly for the d sector. The Yukawa lagrangian reduces to the very simple diagonal form :

L Y = -u L m u u R -u R m u u L -d L m d d R -d R m d d L = -u m u u -d m d d (11.5)
when written in terms of the mass eigenstate basis related to the original one by :

u L = S u u ′ L , u R = S u T u u ′ R d L = S d d ′ L , d R = S d T d d ′ R , (11.6)
which defines the matrices S u L , S u R , • • • introduced above. We remark that, as advertized before, the transformation from the primed basis to the unprimed one is unitary since such are the S, T matrices.

The components of u in which the mass matrix is diagonal are, by definition, the "physical" quark fields (u, c, t) of definite mass eigenstate (idem for the d sector). The same is obviously true for the Higgs couplings which are diagonal in the u and d bases. Having achieved a simple form for the Yukawa lagrangian we re-write now the gauge part L F in terms of these physical fields. Singling out the neutral current interactions we have

L F (neutral current) = i (u ′ i L i D u L u ′ i L + u ′ i R i D u R u ′ i R ) + d ′ sector = (u ′ L i D u L u ′ L + u ′ R i D u R u ′ R ) + d ′ sector, ( 11
L F (neutral current) = u L i D u L u L + u R i D u R u R + d L i D u L d L + u R i D u R u R . (11.8)
This equation is the basis for the slogan that, in the Standard Model and in agreement with experiments, there is "no flavour-changing neutral current", in other words the neutral current is diagonal in flavour space as well as in the mass eigenstates : it does not induce transition between the c quark and the u quark or between the b quark and the d quark for example. The case of the charged current pieces is more involved because it couples the up sector and the down sector which do not transform with the same unitary matrices and, as a consequence, there is no reason for the charged current interactions to be diagonal in the basis which diagonalises the mass matrix. Indeed we have, from eqs. (11.2), (5.43):

L F (quark charged current) = e √ 2 sin θ W (u ′ L W * d ′ L + d ′ L W u ′ L ) = e √ 2 sin θ W (u L W * S u S † d d L + d L S d S † u W u L ) (11.9)
where the last equation is obtained from eq. (11.6). The matrix CKM = S u S † d is the famous Cabibbo-Kobayashi-Maskawa matrix which parameterises the flavour changing content of charged current transitions, i.e transitions between left-handed up spinors (u L , c L , t L ) of definite mass to down spinors (d L , s L , b L ) of definite mass. Its matrix elements are often written as

CKM =   V ud V us V ub V cd V cs V cb V td V ts V tb   .
(11.10)

The matrix CKM is unitary, since both S u , S † d are such, and its matrix elements must satisfy relations of type matrix is a 3 × 3 unitary matrix and from its definition20 it depends on nine parameters which can be chosen as three real parameters and six phases. Changing the phase of u iL and d jL respectively by e iφ ui and e iφ dj the CKM matrix elements (S u S † d ) ij becomes e i(φ dj -φ ui ) (S u S † d ) ij . The five arbitrary phase differences can be used to absorb as many phases of the CKM matrix leaving one CP violating phase. One should of course shift the phases of the right-handed fields and the left-handed ones by the same amount to leave the mass terms eq. (11.5) invariant. The independent parameters of the CKM matrix are chosen as three angles (c ij = cos θ ij , s ij = sin θ ij with 0 < θ ij < π/2) and a phase δ (0 < δ < 2π) and one writes: transition (eq. (5.28)) is given by the term (g/ √ 2) (e L W ν ′ L + ν ′ L W * e L ), diagonal in flavour where

V * ub V ud + V * cb V cd + V * tb V td = 0 or V * ub V ub + V * cb V cb + V * tb V tb =
CKM =   1 0 0 0 c 23 s 23 0 -s 23 c 23   .
CKM =   1 -λ 2 /2 λ Aλ 3 (ρ -iη) -λ 1 -λ 2 /2 Aλ 2 Aλ 3 (1 -ρ -iη) -Aλ 2 1   , ( 11
e L =   e L µ L τ L   (12.2)
is the triplet of left-handed charged leptons. We emphasize that e L , µ L , τ L are the mass eigenstates of the charged leptons. In analogy with the case of quarks, after symmetry breaking, the Dirac mass term for neutrinos is of the form

L Y D = - v √ 2 ν ′ L C ν ν ′ R + h.c. (12.3)
Following the steps leading from eq. ( 11.3) to eq. (11.6), we introduce the notation (M ν hermitian,

T ν unitary), v √ 2 C ν = M ν T ν , (12.4)
and diagonalize the hermitian matrix by the transformation

M ν = S -1 ν m ν S ν = S † ν m ν S ν (S ν unitary). Defining ν L = S ν ν ′ L and ν R = S ν T ν ν ′ R , (12.5)
the Yukawa lagrangian becomes diagonal,

⇒ L Y D = -ν L m ν ν R -ν R m ν ν L = -ν m ν ν (12.6)
with m 1 , m 2 , m 3 the three real eigenvalues of m ν and ν 1 , ν 2 , ν 3 the three neutrino mass eigenstates

ν = ν L + ν R =   ν 1 ν 2 ν 3   . (12.7)
Then, using eq. (12.5), the charged current transition is written

L(leptonic charged current) = g √ 2 (e L W ν ′ L + ν ′ L W * e L ) = e √ 2 sin θ W (e L W S † ν ν L + ν L S ν W * e L ).
(12.8)

Similarly to the CKM mixing matrix one introduces the Pontecorvo-Maki-Nakagawa-Sakata matrix21 PMNS = S † ν the matrix elements of which are often written as22 :

PMNS =   U e1 U e2 U e3 U µ1 U µ2 U µ3 U τ 1 U τ 2 U τ 3   (12.9)
where e, µ, τ refer to flavour states and 1, 2, 3 to mass eigenstates.

It is easy to check that all terms in the lagrangian density are invariant under the global phase change of all fields

e L → e i λ e L , ν L → e i λ ν L , ν R → e i λ ν R . ( 12

Neutrino survival and oscillation

The space-time evolution of a state of given mass is ( = c = 1)

|ν i (x) >= e -i(E t-k x) |ν i > .
(12.12)

As will be seen below it is justified to assume the neutrinos to be ultrarelativistic particles so that

E ≈ k + m 2 i /2 k

91

We consider a neutrino of type α produced in a charged current interaction with momentum k. It is a coherent superposition of neutrino states of definite mass27 ,

|ν α >= i U * αi |ν i >, α = e, µ, τ : i = 1, 2, 3. (12.14)
The space-time evolution of this neutrino is given at a later time t and at a distance x = t by

|ν α (x) >= i U * αi e -ix (m 2 i /2 k) |ν i > . (12.15)
The probability for this neutrino, initially of flavour α, to be observed as a neutrino of flavour β at the distance x from the emission point is

P (ν α → ν β ) = | < ν β |ν α (x) > | 2 = i,j U * αi U αj U βi U * βj exp i x δm 2 ji 2k , (12.16)
where the symbol δm 2 ji = m 2 j -m 2 i . Separating the real and imaginary part of the phase factor and using the unitarity of the U matrix (U * αi U βi = δ αβ ) this expression can be written as28 :

P (ν α → ν β ) = δ αβ -4 i>j Re(U * αi U αj U βi U * βj ) sin 2 x δm 2 ij 4k + 2 i>j Im(U * αi U αj U βi U * βj ) sin x δm 2 ij 2k ,
(12.17)

For the time reversed transition P (ν β → ν α ) permuting α and β in eq. (12.16) is equivalent to permuting i and j so that it comes out

P (ν β → ν α ) = δ αβ -4 i>j Re(U * αi U αj U βi U * βj ) sin 2 x δm 2 ij 4k -2 i>j Im(U * αi U αj U βi U * βj ) sin x δm 2 ij 2k , (12.18)
which exhibits the violation of T invariance due to the phase factor in the PMNS matrix. One finds the same result, eq. (12.18), for P (ν α → ν β ), exhibiting this time the CP violation of the model.

Then CPT is conserved because P (ν β → ν α ) = P (ν α → ν β ).
P (ν α → ν β ) = ν β =νe,νµ,ντ P (ν α → ν β ), for any ν α , ν α . (12.19)
It is important to remark that in case of a disappearance probability, P (ν α → ν α ), the last term in the eqs. (12.17) or (12.18) disappears since terms such as U * αi U αj U αi U * αj are real and therefore a disappearance probability cannot depend on the imaginary part of the PMNS matrix elements.

Summary of results

It turns out, as will be discussed below, that the last factor in eqs. (12.17) and (12.18) is small. Then, oscillations, as a function of x, in the probability for the neutrino to change flavour (or to remain in the same flavour) are essentially induced by the factors sin 2 (x δm 2 ij /4k). For the oscillation to be seen this factor should be of O(1). To be quantitative, we have to inject the and c factors to make the argument of the sin 2 factor dimensionless. One finds 29

δm 2 ij x 4 k ⇒ 1.27 10 -18 δm 2 ij [GeV 2 ] x [km] k [GeV] = 1.27 δm 2 ij [eV 2 ] x [km] k [GeV] = 1.27 δm 2 ij [eV 2 ] x [m] k [MeV]
, (12.20)

where we have given this expression in terms of the units commonly used. One defines the oscillation length associated to a given mass squared difference by the condition may remain small and its contribution to the oscillation pattern becomes negligible. In other cases on the contrary, it stays very large and the oscillating sin 2 term averages out to 1/2. These facts simplify the analysis of the oscillations as will be seen below in the discussion of several experiments. We give here the values of the parameters, with the PMNS matrix written as in eq. (11.12), obtained from of 29 In eq. (12.12) the dimensionless phase should be -i(Et -k x)/ , with the energy E measured in GeV and k in GeV/c as appropriate for neutrino experiments. It can be written -i(Ect -k x)/( c), with both E and k as well as the mass measured in GeV and [ct] = [x] in km if c is expressed in km/sec. We have (see the PDG tables) c = 197.3267 10 -21 GeV•km; using the approximate form eq. (12.13), the oscillation factor in eq. (12.17) becomes

δm 2 ij x 4 k = π ⇒ x [m] = 2.47 k [MeV] δm 2 ij [eV 2 ] or x [km] = 2.47 k [GeV] δm 2 ij [eV 2 ] . ( 12
δm 2 ij [GeV 2 ] x [km]/(4 k [GeV] c[GeV•km]) = 1.27 10 -18 δm 2 ij [GeV 2 ] x [km]/k [GeV].
a recent global analysis of data 30 δm 2 21 = (6.92 -7.91) 10 -5 eV 2 , δm 2 31 = (2.392 -2.594) 10 -3 eV 2 sin 2 θ 12 = 0.265 -0.346, sin 2 θ 23 = 0.430 -0.602, sin 2 θ 13 = 0.0190 -0.0239.

(12.22)

By convention the mass m 2 is chosen larger than m 1 but there are two possibilities for m 3 : either m 1 < m 2 < m 3 , labeled normal hierarchy, or m 3 < m 1 < m 2 , labeled inverted hierarchy. The above results are obtained assuming a normal hierarchy. In the other case the values of the parameters are very similar except of course for the sign of δm 2 3i . For example, the best fit value for δm 2 32 is 2.418 10 -3 eV 2 (normal hierarchy) and δm 2 32 = -2.478 10 -3 eV 2 (inverted hierarchy). There are two independent δm 2 ij and given the relative smallness of δm 2 21 , one is justified to take |δm 2 32 | ≈ |δm 2 31 |. Concerning the mass hierarchy and the CP violating phase δ, they are difficult to extract because, as will be seen, they enter the observables with small coefficients. One notes that the angle θ 13 is much smaller than the other mixing angles and small θ 13 approximations will often be used. At present fits to data seem to indicate a value δ ≈ 3π/2, with large error bars, for both mass hierarchies and a preference for normal hierarchy.

One does not know the absolute scale of neutrino masses. If we assume m 1 ≪ m 2 one gets m 2 ≈ 8.6 10 -3 eV and m 3 ≈ 5.1 10 -2 eV, while in the inverted hierarchy case, assuming m 3 ≪ m 1 the result is m 1 ≈ m 2 ≈ 5.1 10 -2 eV. One way to experimentally access the mass scale of neutrinos is through nuclear β decay which allows to give a direct limit on the ν e mass. For these purposes, several past and ongoing experiments study tritium decay 31 , 3 H → 3 He + e -+ ν e . The electron energy spectrum is sensitive to the neutrino mass near the upper end of the spectrum. Denoting E 0 the total energy release in the decay, the maximum value of the electron energy is E < E 0 ≈ 18 keV, if the neutrino is massless. A non vanishing neutrino mass will slightly reduce the bound to E < E 0 -m νe and will modify the shape of the spectrum near this end point of the distribution. Near the end point the electron spectrum behaves as m νe < 2.05 eV at 95% c.l., quite a bit higher than the tentative scales suggested above. The KATRIN experiment which started operation in 2018, in Karlsruhe, quotes now an upper limit of 1.1 eV at 90% c.l. 33 . By 2024 the collaboration expects to reach 0.2 eV (90% c.l.) or 0.35 eV (5 σ). Finally astrophysical and cosmological limits are available on the sum of neutrino masses and a recent result reported by the Planck collaboration is 34 j m j < .12 eV, (12.24) but this result is model dependent. In the following we use for the PMNS matrix, eq. (12.9), the representation, eq. (11.12).

dN/dE ∝ E νe k νe = (E 0 -E)((E 0 -E) 2 -m 2 νe ) 1/

Survival probabilities in vacuum

Since, in experiments, both survival and oscillation probabilities can be measured we quote below the general form of these expressions for 3 flavoured neutrinos 35 In the analysis of results it will turn out that different approximations can be made, depending on the experimental set-up, which simplify considerably the general expressions. The reduced forms will be easily obtained from the results given in this section and the next. The simplest case is the electron survival probability, the exact expression of which is: insensitive to δ (thus P (ν e → ν e ) = P (ν e → ν e )) and to the hierarchy of mass. The muon survival is given by: More precisely, in eq. (12.26), we have neglected very small terms of type sin 4 (θ 13 ) sin 2 (xδm 2 21 /4k), sin 3 (θ 13 ) cos(δ) and sin 2 (θ 13 ) cos 2 (δ). The τ survival probability is obtained from this equation, by exchanging sin 2 (θ 23 ) and cos 2 (θ 23 ) and reversing the sign of the cos(δ) term.

P (ν e → ν e ) =
P (ν µ → ν µ ) = 1 -[sin 2 (

Oscillation in vacuum, CP asymmetries, mass hierarchy and δ

It is important to obtain the dependence on the phase δ of the oscillation probabilities as it is related to the CP asymmetries and to the mass hierarchy. In fact, all oscillation probabilities have, up to a sign, the same dependence on sin(δ) which is relatively easy to obtain. Injecting the parameterisation eq. (11.12) in the U matrices in eq. (12.16) one finds without approximations From the expressions given in eqs. (12.25) to (12.32) and with the help of the relations given in sec.

P (ν e → ν µ ) = sin 2 (
12.1 we can obtain all survival or oscillation probabilities of neutrinos and antineutrinos. For instance, one obtains P (ν µ → ν e ) from P (ν e → ν µ ) by reversing the sign of δ in eq. ( 12.29) and one derives

P (ν µ → ν τ ) = 1 -P (ν µ → ν e ) -P (ν µ → ν µ )
from the sum rule. For completeness we quote it at the same level of approximation as the previous rates with the further simplification of dropping all terms proportional to sin 2 (θ 13 ) in the first line:

P (ν µ → ν τ ) = - 1 4 sin 2 (
A(ν α → ν β ) = P (ν α → ν β ) -P (ν α → ν β ), (12.35)
then the following relations hold true: Atmospheric (anti)neutrinos observed after crossing the earth or neutrinos produced in the sun propagate through matter and interact with it before reaching the detector. The scattering on protons, neutrons and electrons in matter will modify the oscillation patterns. This is the Mikheyev-Smirnov-Wolfenstein (MSW) effect 39 . The important parameters in this effect are the electron density in matter and the neutrino energy. For some values of the parameters large resonance effects enhance the neutrino conversion rate compared to what is expected in vacuum.

A(ν e → ν µ ) = -A(ν µ → ν e ) = -A(ν e → ν τ ) = A(ν µ → ν τ ) = 4 J sin(δ) SIN = 4 J sin(δ)
To illustrate this point it is sufficient to consider a two-flavour model with mass eigenvectors |ν 1 > and |ν 2 > with the 1 state being the lightest one. From eq. (12.13) the evolution of the doublet of |ν i (t)> states, in vacuum, is given by (t = x):

i d dt |ν 1 (t)> |ν 2 (t)> = H 0 |ν 1 (t)> |ν 2 (t)> = m 2 1 /2k 0 0 m 2 2 /2k |ν 1 (t)> |ν 2 (t)> , (13.1)
with H 0 the free hamiltonian. A global phase change on the |ν i (t)> states does not affect the physics but shifts the hamiltonian by a matrix proportional to the unit matrix. For instance, a phase change im 2 1 t/2k on both states leads to the evolution equation:

i d dt |ν 1 (t)> |ν 2 (t)> = 0 0 0 δm 2 /2k |ν 1 (t)> |ν 2 (t)> , (13.2)
with δm 2 = m 2 2 -m 2 1 taken to be positive. The evolution of the flavour states |ν e (t)> and |ν x (t)> (|ν x (t)> can be a combination of |ν µ (t)> and |ν τ (t)>) 40 , is easily obtained from the relation:

|ν e (t)> |ν x (t)> = R(θ) |ν 1 (t)> |ν 2 (t)> with the matrix R(θ) = cos(θ) sin(θ) -sin(θ) cos(θ) . (13.3)
We then have:

i d dt |ν e (t)> |ν x (t)> = R(θ) 0 0 0 δm 2 /4k R T (θ) |ν e (t)> |ν x (t)> (13.4) = (δm 2 /2k) sin 2 (θ) (δm 2 /4k) sin(2θ) (δm 2 /4k) sin(2θ) (δm 2 /2k) cos 2 (θ) |ν e (t)> |ν x (t)> = H fl 0 |ν e (t)> |ν x (t)> , (13.5)
with H fl 0 is the free hamiltonian in the flavour basis. The interaction of neutrinos with matter can preserve or destroy the coherence of the system. In the latter case, the state of the particles (momentum and spin) is modified and it can be shown that incoherent interactions are negligible. 39 L. Wolfenstein, Phys. Rev. D17 (1978) 2369; S.P. Mikheyev, A.Yu. Smirnov, Prog. Part. Nucl. Phys. 23 (1989) 41. 40 We have in mind solar neutrinos but the discussion applies to any two flavour system. 100

Incoherent scattering

For example, for neutrinos up to a GeV, scattering on nucleons ν x + n → x + p is the dominant process and the cross section41 can be parameterised as:

σ ≈ 10 -43 E ν MeV 2 cm 2 , (13.6)
with E ν the energy in the frame where the nucleon is at rest. The scattering length of the neutrino in matter is l matter = 1/N N σ where N N is the number of nucleons per cm 3 . In the core of the sun, the density is 150 gr/cm 3 , so approximately 10 26 nucleons per cm 3 . The scattering length is then:

l sun ≈ 10 17 E ν MeV -2 cm ≈ 10 12 E ν MeV -2
km.

(13.7)

The typical energy of solar neutrinos being .1 MeV < E ν < 10 MeV, the corresponding scattering length is 10 14 km > l sun > 10 10 km, to be compared to the sun radius of 7 10 5 km. Incoherent neutrino scattering in the sun is negligible.

The range of energy of neutrinos crossing the earth is much larger, from .1 MeV for solar neutrinos to TeV's for atmospheric or cosmic ones. At high energy the charged current ν-nucleon cross section behaves as σ ≈ 6.7 10 -39 E ν GeV cm 2 .

The matter density in the earth ranges from 4 gr/cm 3 in the mantle to, on the average, 11 gr/cm 3 in the core. This leads respectively to N N = 2.4 10 24 to 6.6 10 24 nucleons per cm 3 . Then, the scattering length of 100 GeV neutrinos l earth varies from 6. 10 6 km in the mantle to 2 10 6 km in the inner core. This is to be compared to the mantle thickness of 2.9 10 3 km and the core radius of 3.4 10 3 km. Thus the effect of the earth matter is negligible for neutrinos of energy up to hundreds of GeV. On the contrary, for neutrinos around 100 TeV and above the earth becomes opaque since the cross section grows linearly with energy.

Coherent scattering

Coherence of the neutrino system is preserved by forward elastic scattering of the neutrino on matter.

This can go via neutral current interactions, on protons, neutrons or electrons, ν e,x + N → ν e,x + N and ν e,x + e -→ ν e,x + e -, which are universal for all neutrinos species or via charged current exchange which is specific to ν e scattering on electrons (see fig. 15 in sec. 14.4). These interactions add a piece to the hamiltonian which becomes

H = H fl 0 + H fl int (13.8)
where H fl int is diagonal in flavour. Implementing a phase change on the states amounts to shifting the hamiltonian by a matrix proportional to unity and one can thus subtract the universal neutral current contribution leaving the charged current one which affects only the element <ν e |H fl int |ν e >= <ν e |H fl cc |ν e >. This interaction is given by eqs. (2.1), (2.2) in sec. 2.1,

G F √ 2 ψ e (x)γ µ (1 -γ 5 )ψ νe (x) ψ νe (x)γ µ (1 -γ 5 )ψ e (x) = 2 √ 2 G F ψ e L (x)γ µ ψ νe L (x) ψ νe L (x)γ µ ψ e L (x) = 2 √ 2 G F ψ νe L (x)γ µ ψ νe L (x) ψ e L (x)γ µ ψ e L (x), (13.9)
where a Fierz transformation has been made to obtain the second line. The effective interaction hamiltonian of the neutrinos in matter is obtained by summing over all electrons in matter42 :

H fl cc = 2 √ 2 G F dp 3 e f (p e ) <e L (p e )|ψ νe L (x)γ µ ψ νe L (x) ψ e L (x)γ µ ψ e L (x)|e L (p e )> = 2 √ 2 G F ψ νe L (x)γ µ ψ νe L (x) dp 3 e f (p e ) <e L (p e )|ψ e L (x)γ µ ψ e L (x)|e L (p e )>, (13.10)
where the electron energy distribution f (p e ) in matter is homogeneous, isotropic and is normalised to dp 3 e f (p e ) = 1. Assuming the electron approximately at rest in the medium, the space components γ i can be neglected and the combinations ψγ µ ψ reduce to ψγ 0 ψ = ψ † ψ, so that

H fl cc = 2 √ 2 G F ψ † νe L (x)ψ νe L (x) dp 3 e f (p e ) <e L (p e )|ψ † e L (x)ψ e L (x)|e L (p e )>, = √ 2 G F ψ † νe L ψ νe L N e , (13.11)
with N e the density of electrons in the medium (N e L = N e /2). The evolution equation will then be of the form

i d dt |ν e (t)> |ν x (t)> = (δm 2 /2k) sin 2 (θ) + √ 2 G F N e (δm 2 /4k) sin(2θ) (δm 2 /4k) sin(2θ) (δm 2 /2k) cos 2 (θ) |ν e (t)> |ν x (t)> = H |ν e (t)> |ν x (t)> .
(13.12)

Due to the charged current interaction the mass eigenstates |ν i (t)> of eq. (13.1) no longer diagonalize the hamiltonian. Let us denote ω 1 and ω 2 the eigenvalues of the above matrix and |ν m 1 (t) > and |ν m 2 (t)> the corresponding mass eigenstates related to the flavour states |ν e (t)> and |ν x (t)> at time

t by |ν e (t)> = cos θ m |ν m 1 (t)> + sin θ m |ν m 2 (t)> |ν x (t)> = -sin θ m |ν m 1 (t)> + cos θ m |ν m 2 (t)> .
(13.13)

Matter of constant density

If N e is independent of t, so are θ m and the eigenvalues given by:

ω 1,2 = G F N e √ 2 + δm 2 4k ∓ 1 2 ( √ 2G F N e -cos(2θ)δm 2 /2k) 2 + (sin(2θ)δm 2 /2k) 2 = δm 2 4k  + 1 ∓ sin 2 (2θ) + (cos(2θ) -Â) 2 , (13.14)
with ω i the eigenvalue of the state |ν m i (t)>. The important parameter  is defined by:

 = 2 √ 2G F kN e δm 2 , (13.15)
which is the ratio of the interaction energy in matter to the vacuum energy. The matrix H in eq.

(13.12) is diagonalised by R T (θ m ) H R(θ m ) = diag(ω 1 , ω 2 ) (see eq. (13.3)) and one finds:

tan(θ m ) = Â -cos(2θ) + sin 2 (2θ) + (cos(2θ) -Â) 2 sin(2θ) (13.16)
from which we derive (for δm 2 positive):

cos(2θ m ) = cos(2θ) -Â sin 2 (2θ) + (cos(2θ) -Â) 2 sin(2θ m ) = sin(2θ) sin 2 (2θ) + (cos(2θ) -Â) 2 , ( 13.17)
To obtain the oscillation probabilities we use eqs. (12.43):

P (ν e → ν e ) = 1 -sin 2 (2θ m ) sin 2 δM 2 t 4k , P (ν e → ν x ) = sin 2 (2θ m ) sin 2 δM 2 t 4k , (13.18)
where43

δM 2 = δm 2 sin 2 (2θ) + (cos(2θ) -Â) 2 . (13.19)
The corresponding oscillation length in matter is given by (see eq. (12.21):

l mat = 4πk δM 2 = 2π ω 2 -ω 1 (13.20)
Several cases can be distinguished assuming N e constant in the medium (with δm 2 positive).

• If  ≪ 1, then sin(2θ m ) ≈ sin(2θ)(1 +  cos(2θ)), δM 2 ≈ δm 2 (1 - cos(2θ)): the interaction with matter is small and the neutrino system evolves almost as in empty space, l mat ≈ l vac with a small correction;

• If  ≫ | cos(2θ)|, interaction with matter is dominant: then sin(2θ m ) ≈ sin(2θ)/  ≈ 0 and cos(2θ m ) ≈ -1, hence θ m ≈ π/2: from eq. (13.13) the electron neutrino tends to a pure mass eigenstate |ν m 2 >, the heaviest state (ω 2 ≈ √ 2G F N e ); it propagates without oscillations independent of the value of the mixing angle in vacuum;

• If  ≈ cos(2θ), this is the resonant regime: it occurs only if cos(2θ) is positive (0 < θ < π/4), then cos(2θ m ) ≈ 0, sin(2θ m ) ≈ 1, θ m ≈ π/4, l mat ≈ l vac / sin(2θ); the electron neutrino is an equal combination of |ν m 1 > and ν m 2 >, independent of the initial mixing angle, the amplitude of oscillations is maximal, since sin(2θ m ) ≈ 1, as well as the oscillation length. For π/4 < θ < π/2 there is no resonance effect possible and θ m is always larger than π/4.

Remarks

• When applying eq. (13.11) to antineutrinos states one will obtain an extra -sign44 , thus giving a contribution -√ 2 G F N e to H. Then, the sign of  for antineutrinos is opposite to that for neutrinos. If the resonance condition  ≈ cos(2θ) can be reached for neutrinos, it cannot occur for antineutrinos and vice-versa. For antineutrinos the resonance condition requires π/4 < θ < π/2.

• The evolution of neutrinos in matter violates the CP symmetry, which is obvious since matter is not CP symmetric.

Application to solar neutrinos

Electron neutrinos are produced in the core of the sun where N e can be as large as 6. 10 25 cm -3 . It is useful to define the quantity N Res by

N Res = δm 2 cos(2θ) 2 √ 2G F k , ( 13
E ν MeV -1 MeV 3 ≈ 10 26 E ν MeV -1 cm -3 , (13.23) so N 21
Res 10 25 cm -3 for E ν 10 MeV. In that case, the condition N e ≫ N 21 Res (equivalently  ≫ cos(2θ 12 )) is realised and the neutrino is produced in a mass eigenstate. On the contrary, neutrinos of energy E ν ≈ .1 MeV evolve as in vacuum since they satisfy N e ≪ N 21

Res . The range of values of θ 12 given in eqs. (12.22 ), 30 o < θ 12 < .38 o implies cos(2θ 12 ) > 0 so that the resonance regime  ≈ cos(2θ 12 ) can be satisfied for neutrinos of intermediate energies. In the sun, however, N e is a decreasing function of x, the distance from the center, and taking this effect into account requires a special treatment to which we turn in the next section. We can also consider oscillations to the third generation and estimate N 31

Res . Using the values of θ 13 and δm 2 31 from eq. (12.22) one finds N e /N 31 Res ≈ 6.10 -3 (E ν /MeV), so that 6.10 -4 < N e /N 31 Res < 6.10 -2 in the E ν range [.1, 10.] MeV, making matter effects negligible in this case. When studying oscillations in the sun, working in the 2 family oscillation model will be a good enough approximation.

Neutrinos through the earth

The electron density in the earth is much less than in the sun and it remains approximately constant in the core 45 (N e ≈ 3.3 10 24 cm -3 ) and in the mantle (N e ≈ 1.2 10 24 cm -3 ). It is then expected that solar neutrinos with E ν < 10 MeV will be little affected by coherent interactions when traversing the earth. However this will not the case for higher energy neutrinos in the GeV and multi-Gev range. Furthermore, in the 3-ν model, 13 oscillations will become important since

N e /N 31 Res = 2 √ 2G F k/(δm 2 31 cos(2θ 13
)) can be of order 1 in the GeV range. This will be discussed later.

Matter of varying density: ν e in the sun

When the density of electrons decreases from the core to the surface, as it is the case in the sun, the angle θ m (t) becomes a function of x = t. The variation of θ m (x) should bring a dθ m (x)/dx = θ ′ m (x) dependence in the evolution equations of the neutrino system. From eq. (13.13) written as

|ν e (x)> |ν x (x)> = R(θ m (x)) |ν m 1 (x)> |ν m 2 (x)> , (13.24)
we derive

i d dx |ν e (x)> |ν x (x)> = i d dt R(θ m (x)) |ν m 1 (x)> |ν m 2 (x)> + R(θ m (x)) i d dt |ν m 1 (x)> |ν m 2 (x)> = R(θ m (x)) R T (θ m (x)) i d dt (R(θ m (x)) + ω 1 (x) 0 0 ω 2 (x) |ν m 1 (x)> |ν m 2 (x)> = R(θ m (x)) ω 1 (x) iθ ′ m (x) -iθ ′ m (x) ω 2 (x) R T (θ m (x)) |ν e (x)> |ν x (x)> , (13.25)
similar to eq. (13.4) except for the off-diagonal term

iθ ′ m (x). If |2 θ ′ m (x)/(ω 2 (x) -ω 1 (x)| ≪ 1, then ω 1 (x)
and ω 2 (x) will remain approximate eigenvalues of the system and the |ν m i (x)> will be approximately the mass eigenstates. Intuitively, one expects this to happen if the rate of change of the electron density (1/N e )dN e /dx is very slow compared to the oscillation length in matter. This rate of change is measured by (1/N e )dN e /dx = 1/r 0 , where a large value of r 0 corresponds to a small variation of N e and if

r 0 /l mat ≫ 1, (13.26)
with l mat given by eq. (13.20), then the variation of N e will have a small effect on the neutrino mass eigenstates. More precisely, this condition is :

ω 2 (x) -ω 1 (x) 2 |θ ′ m (x)| ≫ 1. (13.27)
From eqs. (13.17) one derives

2 θ ′ m (x) = d  dx sin(2θ) sin 2 (2θ) + (cos 2 (2θ) -Â) 2 , ( 13.28)
and from eq. (13.15) one has,

d  dx =  r 0 . (13.29)
Using then the relations

sin 2 (2θ) + (cos 2 (2θ) -Â) 2 sin 2 (2θ) = 1 + tan -2 (2θ m ) , (13.30)
the condition (13.27) can be written:

1 Â r 0 δm 2 2k sin 2 (2θ)(1 + tan -2 (2θ m )) 3/2 = 2πr 0 l mat N Res N e tan(2θ)(1 + tan -2 (2θ m )) ≫ 1 . (13.31)
If this condition is satisfied the evolution of the neutrino system in matter is said to be adiabatic.

The flavoured neutrinos related, at the initial time, to the mass eigenstates |ν m i (x 0 )> by the angle θ m = θ m (x 0 ) as in eq. (13.24), will be, at each point of the evolution, related to the mass eigenstates |ν m i (x)> by the angle θ m (x), until they exit from matter in vacuum, at a distance R where the mixing angle is θ and the mass eigenstates |ν i >. The assumed adiabatic evolution does not mix the |ν m 1 (x)> and |ν m 2 (x)> states which evolve respectively to the |ν 1 > and |ν 2 > states of the vacuum when the neutrino exit from the medium. Thus, for

|ν e (x 0 )>= cos(θ m (x 0 ))|ν m 1 (x 0 )> + sin(θ m (x 0 ))|ν m 2 (x 0 )>, (13.32)
at some initial time, one has at time x,

|ν e (x)>= cos(θ m (x))|ν m 1 (x)> + sin(θ m (x))|ν m 2 (x)>, (13.33)
and when the neutrino reaches the surface of the sun,

|ν e (R)>= cos(θ)|ν 1 > + sin(θ)|ν 2 >, (13.34)
The probability to find a ν e at the surface will be | <ν e (R)|ν e (x 0 )> | 2 , i.e.:

P (ν e → ν e ; x 0 , R) = [cos(θ) cos(θ m (x 0 )) <ν 1 |ν m 1 (x 0 )> + sin(θ) sin(θ m (x 0 )) <ν 2 |ν m 2 (x 0 )>] 2 = 1 2 [1 + cos(2θ) cos(2θ m (x 0 ))] + oscillating term ≈ sin 2 (θ) + cos(2θ) cos 2 (θ m (x 0 )), (13.35)
where we have supposed that the oscillating term averages out to 0. As a special case, if at x 0 the neutrino is produced in a pure mass eigenstate |ν m 2 (x 0 ) > (θ m (x 0 ) = π/2), then the neutrino will remain in this pure mass eigenstate |ν m 2 (x)> during its propagation until it reaches the surface where

|ν m 2 (R)>= |ν 2 > in vacuum.
The probability to find a ν e at the surface will then be

P (ν e → ν e ; x 0 , R) = sin 2 (θ). (13.36)
On the contrary, one may consider the extreme non-adiabaticity case of the evolution in matter: in that case a ν e produced in the |ν m 2 (x 0 )> state ends up as the |ν 1 (R)> when exiting from the medium, and if this occurs P (ν e → ν e ; x 0 , R) = cos 2 (θ). (13.37) in contrast with eq. (13.36). The general treatment of a non adiabatic evolution is given by Petcov 46 .

It is easy to check that, in the sun, the adiabaticity condition is satisfied.

46 S.T. Petcov, Phys. Lett. 200 (1988) 373.

Neutrinos through the earth

As mentioned above, for energetic neutrinos traversing the earth N e /N 21 Res is very large and N e /N 31 Res may be of order 1 for E ν 1 GeV: indeed, in that case, N e ≈ 1.2 to 3.3 10 24 cm -3 compared

N 21
Res ≈ 10 23 (E ν /GeV) -1 cm -3 and N 31 Res ≈ 10 25 (E ν /GeV) -1 cm -3 . It is then necessary to work with the full 3-ν model. The free hamiltonian when acting on the mass eigenstates is

H 0 =   m 2 1 /2k 0 0 0 m 2 2 /2k 0 0 0 m 2 3 /2k   . (13.38)
After a change of phase on the states it can be put in the form

H 0 =   0 0 0 0 δm 2 21 /2k 0 0 0 δm 2 31 /2k   (13.39) with δm 2 ij = m 2 i -m 2 j .
Going to the flavour basis,

  ν e ν µ ν τ   = U   ν 1 ν 2 ν 3   , (13.40)
the hamiltonian is written H fl 0 = U H 0 U † where U is parameterised 20 as in eq. (11.12), U = U 23 U 13 (δ)U 12 . Since the interaction in matter affects only the electron the interacting hamiltonian is written

H fl = U   0 0 0 0 δm 2 21 /2k 0 0 0 δm 2 31 /2k   U † +   √ 2G F N e 0 0 0 0 0 0 0 0   (13.41) = U 23 U (δ)   U 13 U 12   0 0 0 0 δm 2 21 /2k 0 0 0 δm 2 31 /2k   U † 12 U † 13 +   √ 2G F N e 0 0 0 0 0 0 0 0     U † (δ) U † 23 .
Several comments are in order. The matrix U 23 does not affect the interaction matrix which can then be multiplied by U 23 on the left and U † 23 on the right. Furthermore, writing U 13 (δ) = U (δ) U 13 U † (δ) with

U (δ) =   1 0 0 0 1 0 0 0 e iδ   , (13.42)
the δ dependence can be factored out as indicated above. We know that δm 2 21 ≪ δm 2 31 and we have seen that, in the earth, for neutrinos in the GeV range and above, the ratio δm 2 21 /2 √ 2G F N e k is very small which justifies the approximation δm 2 21 = 0 which is now done. This will considerably simplify the discussion47 . The hamiltonian in the flavour basis can then be written:

H fl = U 23 U (δ)   U 13   0 0 0 0 0 0 0 0 δm 2 31 /2k   U † 13 +   √ 2G F N e 0 0 0 0 0 0 0 0     U † (δ) U † 23 . (13.43)
The matrix U 12 plays no role because of our choice δm 2 21 = 0, so we take θ 12 = 0, U 12 = 1. Then this equation becomes:

H fl = U 23 U (δ)   (δm 2 31 /2k) sin 2 (θ 13 ) + √ 2G F N e 0 (δm 2 31 /4k) sin 2 (2θ 13 ) 0 0 0 (δm 2 31 /4k) sin 2 (2θ 13 ) 0 (δm 2 31 /2k) cos 2 (θ 13 )   U † (δ) U † 23 . (13.44)
The diagonalisation of the interacting hamiltonian follows the procedure of sec. 13.3. Here one eigenvalue ω 2 is 0 while the other two, ω 1,3 , are given by

ω 1,3 = δm 2 31 4k  + 1 ∓ sin 2 (2θ 13 ) + (cos(2θ 13 ) -Â) 2 , (13.45)
identical to the eigenvalues given in eq. (13.14) with the substitution θ → θ 13 and δm 2 → δm 2 31 . As in the work of M. Freund 47 Â is now

 = 2 √ 2G F N e k/δm 2 31 .
(13.46)

The 3 × 3 matrix in eq. (13.44) is diagonalised via the matrix U m 13 and H fl is then written:

H fl = U 23 U (δ) U m 13   ω 1 0 0 0 0 0 0 0 ω 3   U m † 13 U † (δ) U † 23 , (13.47)
with the matrix U m 13 of the same form as U 13 but function of the angle θ m 13 . This angle is given by eqs. (13.16) or (13.17) with the appropriate change of notation. Finally the matrix U m which relates the flavour eigenstates and the mass eigenstates (with eigenvalues ω 1 , 0, ω 3 ) of the interacting theory is of the usual form To reconstruct the various ν e transition probabilities, one needs to define the oscillating factors given by x(ω i -ω j )/2. They are, in the small θ 13 approximation (see eqs. (12.22)), and using  < 1:

U m = U m 23 U m 13 (δ) U m 12 = U 23 U m 13 (δ) U 12 , (13.48)
x (ω 2 -ω 1 ) 2 = -x ω 1 2 ≈ - Â δm 2 31 4k x x (ω 3 -ω 2 ) 2 = x ω 3 2 ≈ δm 2 31 4k x x (ω 3 -ω 1 ) 2 = x Ĉ δm 2 31 4k ≈ |1 -Â| δm 2 31 4k
x.

(13.51)

The oscillation probabilities, eqs. (12.25), (12.29) and (12.32) considerably simplify because of the vanishing of θ 12 : the only oscillating factor to be kept is sin 2 (x (ω 3 -ω 1 )/2) = sin 2 (x Ĉ δm 2 31 /4k) all others are multipled by sin(θ 12 ) and disappear. One finds:

P (ν e → ν µ ) ≈ sin 2 (θ 23 ) sin 2 (2θ 13 ) Ĉ 2 sin 2 x Ĉ δm 2 31 4k , (13.52)
Changing sin(θ 23 ) to cos(θ 23 ), one obtains P (ν e → ν τ ). In the small θ 13 approximation Ĉ ≈ |1 -Â| and

P (ν e → ν µ ) ≈ sin 2 (θ 23 ) sin 2 (2θ 13 ) (1 -Â) 2 sin 2 x(1 -Â) δm 2 31 4k , (13.53)
As a result of neutrino interaction with matter, both the amplitude and the frequency of oscillations are modified.

Going beyond the δm 2 21 = 0 approximation leads to much more complicated expressions for the different parameters which are given in the work of Martin Freund 47 . All parameters in eqs. (13.49) receive a correction proportional to α = δm 2 21 /δm 2 31 . However, in a realistic and often used limit, drastic simplifications are possible. This is the case if one keeps only leading terms in α and sin(θ 13 ). In practice if one keeps, in the probability functions, only terms up to O(α 2 ), O(sin 2 (θ 13 )), O(α sin(θ 13 )), the only correction to the parameters in eqs. (13.49) to take into account is a modification of θ 12 to θ m 12 . To derive it, we turn back to eq. (13.41) and consider, assuming now θ 13 ≈ 0, U 13 ≈ 1, the diagonalisation by the matrix U m 12 of

U m 12   U 12   0 0 0 0 δm 2 21 /2k 0 0 0 δm 2 31 /2k   U † 12 +   √ 2G F N e 0 0 0 0 0 0 0 0     U m † 12 .
(13.54) This is done in sec. 13.3, the only difference being here that we define ω 1 as the largest eigenvalue and ω 2 the smallest. This amounts to exchanging ω 1 and ω 2 , hence reversing the sign of the square root factor in eq (13.16). This leads to a negative θ m 12 , and in the large Â21 = Â/α ≫ 1 limit, to sin(2θ To obtain the terms in sin(δ) and cos(δ) we use respectively eqs. (12.37) and (12.39) with J as defined in eq. (12.27). We recall this expression is valid in the small δm 2 21 /δm 2 31 and sin(θ 13 ) approximation. The effect of matter is contained in  = 2 √ 2G F N e k/δm 2 31 which changes the relative weights of the terms compared to vacuum and the magnitude of the change is energy dependent since  ∝ k. Taking α = 0 one recovers a previously derived result but it is not allowed in this expression to make  = 0, the vacuum limit, since the derivation was done assuming  = N e cos(2θ 13 /N 31 Res > α. With the present value of δm 2 21 this condition is, for neutrinos traversing the earth, E ν > .3 GeV. The results above thus do not apply to solar neutrinos but it does apply to atmospheric and accelerator neutrinos.

The time reversed probability P (ν µ → ν e ) is obtained from the above equation by reversing the sign of δ while for P (ν e → ν µ ) one reverses both the sign of  and δ. From eq. (12.33) and the above results one can obtain the oscillation probability P (ν µ → ν τ ) in matter which are stronger than ν µ → ν e , the dominant term being proportional to sin 2 (2θ 23 ) rather than sin 2 (2θ 13 ).

• Discussion and order of magnitude of the parameters

We summarize here for later use the value of the parameters and the order of magnitude of the

|α| Â xδm 2 21 /4k xδm 2 31 /4k
x Âδm 2 31 /4k

3. 10 -2 .125 (k/GeV) 10 -4 (x/km)(k/GeV) -1 3.2 10 -3 (x/km)(k/GeV) -1 4 10 -4 (x/km) positive or negative ? Although the derivation above was done assuming this quantity positive it also holds with δm 2 31 < 0 keeping δm 2 21 > 0. In that case,  is also negative but the combinations α/  and  δm 2 31 remain positive. Similarly to the oscillations in vacuum the difference between the two hypothesis is the sign of the cos δ term but this term is very small if δ ≈ 3π/2 (see eq. (13.56)).

In matter however, since the magnitude of the oscillation depends on  one can use the energy as a parameter to probe the hierarchy hypothesis. For example, all terms with a normalisation factor in 1/(1 -Â) will be sensitive to the sign of δm 2 31 provided of course that the associated oscillating factor x(1-Â)δm 2 31 /4k be large enough so as not to compensate the normalisation otherwise one can expand sin(x(1 -Â)δm 2 31 /4k) ≈ (1 -Â) sin(xδm 2 31 /4k) and then get back the vacuum oscillation result.

Neutrino experiments

In • At accelerators, π ± 's produced in hadronic collisions decay predominently in ν µ and ν µ while K ± 's decay also ν e and ν e . The average energy < E ν > ≈ 1 GeV and the flux is measured at a distance of 295 km (T2K), 735 km (MINOS), 810 km (NOνA). For OPERA the incident neutrino energy is much higher <E νµ > ≈ 17 GeV and the detector is 730 km away from the source. All these are long baseline experiments.

• Atmospheric neutrinos are produced in cosmic ray showers from π + → ν µ µ + followed by µ + → e + ν e ν µ (and similarly with π -) so that they are a mixture (ν µ + ν µ ) and (ν e + ν e ) in proportion 2 : 1 at low energy < 1 Gev. Before being detected the neutrinos travel 1 to 30 km (above the Earth, "downward flux") or 1.3 10 4 km (through the Earth, "upward flux") (SNO, Super-Kamiokande).

• For the solar neutrinos, the flux from 8 B ( 8 B → 7 Be * + e + + ν e ) is particularly useful. It has a relatively large energy, 1.5 MeV < E ν < 15 MeV, and the 8 B is the only source of ν e 's in this energy range. The neutrinos travel 1.5 10 8 km before being detected in mines on Earth (SNO).

Previous experiments (GALLEX, GNO, SAGE) measured the flux of lower energy ν e 's: .1 MeV < E ν < .4 MeV.

• Ultra-high energy or cosmic or cosmogenic neutrinos have energies in the range of 100 TeV to several PeV: they are produced by collisions of ultra-high energy cosmic rays on protons or photons, for example on photons from the Cosmic Microwave Background (CMB), and by sources such as Active Galactic Nuclei (AGN). Their flux is very small and they require huge detectors (telescopes) to be observed (IceCube, ANTARES, KM3net, Baikal-GVD).

14.1 Nuclear reactors : KamLAND, Double-Chooz, Daya Bay, RENO

Nuclear reactors produce dominantly electron antineutrinos and, assuming three flavours, we recall that their survival probability at a distance x is from eq. (12.25), KamLAND, a long baseline experiment (< x >=180 km) with the detector in Kamioka mine in Gifu, Japan, receives ν e 's from 56 nuclear power reactors48 . The average neutrino energy is < k >= 3. MeV so that the factors x δm 2 31 /4 < k > ≈ x δm 2 32 /4 < k > ≈ 190, and integrating over the energy of the neutrino, averages the value of the factors sin 2 (x δm 2 31 /4k) ≈ sin 2 (x δm 2 32 /4k) ≈ 0.5. The equation above reduces to P (ν e → ν e ) ≈ 1 -cos 4 (θ 13 ) sin 2 (2θ 12 ) sin 2 (δm 2 21 x/4 k) -0.5 sin 2 (2θ 13 ) ≈ cos 4 (θ 13 ) P (2) (ν e → ν e ) + sin 4 (θ 13 ) (14.2)

P (ν e → ν e ) =
where one has introduced the oscillation probability in a two flavour neutrino world, eq. (12.43), P (2) (ν e → ν e ) = 1 -sin 2 (2θ 12 ) sin 2 (δm 2 21 x/4 k).

(14.3)

Taking advantage of the smallness of sin 2 (θ 13 ), it is reasonable to make the further approximation (appropriate for long baseline experiments), neglecting sin 4 (θ 13 ) terms, With the high statistics available these short baseline experiments are well suited to constrain the small θ 13 mixing angle. For instance, the Daya Bay collaboration reports a precise determination of the angle θ 13 , sin 2 2θ 13 = 0.0856 ± 0.0029. They also quote the value for the mass-squared difference for normal ordering δm 2 32 = (2.471 +.0068 -.0070 ) 10 -3 eV 2 . Recently the result from Double Chooz 53 is sin 2 2θ 13 = 0.105 ± 0.0014. In a simplified form (δm 2 31 = δm 2 32 ), the ν µ survival probability is written (eq. (12.26)): For the T2K configuration, the sin x δm 2 21 /4k term is very small (≈ 0.048) compared to sin x δm 2 32 /4k which justifies the neglect of terms in sin 2 x δm 2 21 /4k in the cofficient of J (see eqs. (12.38) and (12.40)). Since sin 2 (2θ 13 ) ≈ .084 is small, we drop such terms in the coefficient of sin for the peak energy of 0.6 GeV. Furthermore the variation of P (ν µ → ν e ) as a function of δ is opposite to that of P (ν µ → ν e ). The amplitude of variation is about 0.026 when going from δ = π/2 to δ = 3π/2 as illustrated in fig. 9. From the oscillation data the collaboration quotes the following results, at a 1 σ confidence level, taking into account the reactor constraints on θ 13 : δ = 4.56 + 0.81 -0.85 (1.45 π + 0.26 π -0.27 π ) for normal mass order, δ = 4.83 + 0.68 -0.73 (1.54 π + 0.22 π -0.23 π ) for inverted mass order.

P (ν e → ν e ) ≈ (1 -2 sin 2 (θ 13 )) P (2) (ν e → ν e ) ( 14
P ( ν (-) µ → ν (-) µ ) = 1 -sin 2 (
The correlation δ -sin(θ 13 ) is illustrated in fig. 10. The best fit value of the CP violating angle is δ ≈ 3π/2, which means cos(δ) ≈ 0 and, consequently, it will be difficult to solve the hierarchy problem from any oscillation experiment in vacuum. In principle, since the neutrinos propagate through the Earth crust on a distance of about 300 km, matter effects should be taken into account when extracting the values of parameters. However, for a peak energy E ν = 0.6 GeV and a density of electrons in the Earth crust around N e = 8 10 23 cm -3 , the relevant parameter  = 2 √ 2G F N e E ν /δm 2 31 is very small,  ≈ 0.05, leading to negligible matter effects.

NOνA is another long baseline accelerator experiment, optimised to study ν µ ↔ ν e oscillations, which started publishing results recently 56 . It is a ν µ disappearance ν e appearance experiment for both neutrinos and antineutrinos, with a beam of peak energy E ν ≈ 2 GeV from Fermilab with a far detector 810 km away in Minnesota. With this choice of parameters, the value of sin 2 (xδm 2 32 /E ν ) is near its maximum which maximizes the disappearance of ν µ and the appearance of ν e . NOνA has collected an equivalent of 8.85 10 20 protons on target for neutrinos and 6.9 10 20 for antineutrinos. It should be more sensitive to matter effects than T2K with a value of  ≈ .18. A preliminary analysis, for normal hierarchy (with δm 2 31 ≈ δm 2 32 ), yields δm 2 32 = 2.51 +0.12 -0.08 10 -3 eV 2 with a mixing angle, sin 2 (θ 23 ) = .58 ± .03.

OPERA is a τ appearance experiment : it is the only detector designed to identify τ leptons in a ν µ beam on an event-by-event basis. The ν µ source is the CNGS (CERN Neutrinos to Gran Sasso) beam directed at the Grand Sasso underground facility 730 km away. Compared to other accelerator experiments the ν µ energy is very high, <E νµ >= 17 GeV to overcome the τ production threshold, E th = 3.55 GeV. The observed number of τ leptons is written 57

N τ = A E th Φ νµ (E) P (ν µ → ν τ ) σ CC τ (E) ε(E) dE, (14.12)
where A is a normalisation constant taking account of the detector mass, Φ νµ (E) the neutrino flux, σ CC τ (E) the charged-current ν τ cross section and ε(E) the ν τ detection efficiency. As for P (ν µ → ν τ ) the oscillation rate given in eq. (12.33), it simplifies considerably since the sin 2 (δm 2 12 x/4k) term with δm 2 12 x/4k ≈ 4.1 10 -3 gives a negligeable contribution, P (ν µ → ν τ ) ≈ sin 2 (2θ 23 ) sin 2 (δm 2 32 x/4k) (14.13) ignoring furthermore sin 2 (θ 13 ) pieces. For the OPERA configuration the number of observed τ leptons is given 58

N τ ≈ A ′ sin 2 (2θ 23 ) (δm 2 32 [eV 2 ] L[km]) 2 E th Φ νµ (E) σ CC τ (E) ε(E) dE E 2 . (14.14)
In 2010 the first observation of a τ lepton in a ν µ beam 59 was made. According to the final results 60 10 ν τ candidate events have been reported, for an expected no oscillation background of 2 events, which allows to claim for the discovery of ν µ → ν τ oscillations with a significance level of 6.1 σ. A value of δm 2 32 = 2.7 +0.7 -0.6 10 -3 eV 2 is obtained, consistent with the world average. In 1998, the collaboration provided the first experimental evidence of neutrino oscillations 61 . Super-Kamiokande is an underground detector of 50 kilotons of ultra-pure water located in Gigu prefecture in Japan. It records the µ ± and e ± produced in ν and ν induced reactions. In a first analysis it is difficult to tell ν µ (ν e ) from ν µ (ν e ) so that the results are given for ν µ + ν µ and ν e + ν e fluxes. One distinguishes the downward going flux (zenithal angle θ z ≈ 0) with the neutrinos interacting (primary vertex) in the detector after a path length of 1 to 30 km in the atmosphere, from the upward going flux (zenithal angle θ z ≈ π) where the neutrinos, after travelling up to 1.3 10 4 km through the Earth, are interacting in the rocks outside Super-K producing a muon energetic enough to enter the detector 62 .

In a first approximation (e.g. x/k < 10 3 ) one ignores the oscillation terms in sin(δm 2 21 x/4k) and take δm 2 32 ≈ δm 2 31 . The relevant rates of oscillations (in vacuum) are obtained from secs. 12.3 and 12.4: One checks easily that P (ν µ → ν µ ) = 1 -P (ν µ → ν e ) -P (ν µ → ν τ ). We quote here very simplified formulae which are sufficient to understand the global features of the data but, in their analysis, the Super-K collaboration uses the full model including the CP violating phase δ as well as matter effects. From eqs. (14.15) it is expected that ν µ will fluctuate dominantly in ν τ (sin 2 (2θ 23 ) ≈ .99 vs sin 2 (2θ 13 ) ≈ .1) and the ν µ disappearance will be less important for downward neutrinos since they do not have time to oscillate unlike those crossing the Earth. Because of the small value of sin 2 (2θ 13 ) ν e oscillation is less effective.

P (ν e ↔ ν µ ) ≈ sin 2 (2θ 13 ) sin 2 (θ 23 ) sin 2 (δm 2 31 x/4 k) P (ν e → ν τ ) ≈ sin 2 (2θ 13 ) cos 2 (θ 23 ) sin 2 (δm 2 31 x/4 k) P (ν µ → ν τ ) ≈ sin 2 (
The Super-K collaboration has given the most precise measurements of the atmospheric neutrino fluxes in a large energy range 63 : 0.15 < E ν [GeV ] < 65 for ν e + ν e and 0.25 < E ν [GeV ] < 2500 for ν µ + ν µ (see fig. 11 which also displays model predictions with and without oscillations). At high energies, the spectrum is dominated by ν µ + ν µ and, for kinematical reasons, the ν τ flux is negligible.

As expected the ν e + ν e flux is globally not sensitive to oscillations while the ν µ + ν µ flux below 100 GeV is reduced. Above this energy the factor (x δm 2 31 /4 k) is small and oscillations become irrelevant.

= 2 √ 2G F N e E ν δm 2 31 ≈ .1 E ν [GeV ]
,

for N e ≈ 1. 10 24 cm -3 . Panel c) illustrates, as a function of energy, the survival pattern of an upward muon neutrino exiting from the Earth: because of the 1/k dependence of the oscillating factor, oscillations are much more rapid at low energy. In data, an average over a large energy range is performed so that the oscillating factor sin 2 (xδm 2 31 /4k) reduces to .5. The distribution of events as a function of the zenith angle is given in fig. 13: for events labelled "Multi-GeV µ-like" (middle panel) the increase in the number of events when cos θ z decreases from 1 to 0 is due to the increase of the effective thickness of the atmosphere, then at cos θ z = -1 the oscillations reduce the ν µ + ν µ flux by a factor 2 compared to the no oscillation expectation. Concerning ν e 's, the disappearance (left panels) is much less pronounced. One notices however that energetic upgoing Figure 13: Superkamiokande zenithal oscillations : "Sub-GeV" refers to events with E vis < 1.33 Gev while "Multi-GeV refers to neutrinos with E vis > 1.33 GeV. The 4 left most panels have a reconstruted vertex in the SK detector while the 2 right most panels show the sample of upward-going muons produced by neutrinos in the rock surrounding the detector. The blue lines show the nonoscillated prediction and the red lines the oscillated ones. From R. Wendell for the Super-Kamiokande collaboration, arXiv:1412.5234 [hep-ex].

(anti)neutrinos are less suppressed than downgoing (cos θ z ≈ 1) ones: at high energy the atmospheric ν µ + ν µ flux is much larger than the ν e + ν e flux and, furthermore, between 2 and 10 GeV the ν µ + ν µ to ν e + ν e resonant enhancement in the Earth is possible as discussed in sec. 13.3. The resonant enhancement is sensitive to the sign of δm 2 31 and affects ν e 's for normal hierarchy and ν e 's for inverted hierarchy. Separating neutrinos from antineutrinos would allow to determine the sign of δm 2 31 . For this purpose the collaboration is constructing ν e and ν e enriched samples.

In recent analyses of their data64 , keeping the δ dependence and matter effects as in eq. (13.56), for example, the Super-Kamiokande collaboration constrains several mixing parameters. The analyses are constrained, i.e. fixing sin 2 (θ 13 ), or unconstrained. In the latter case the best fit for normal hierarchy gives sin 2 (θ 13 ) = .008 + 0.025 -0.005 and:

δm 2 31 = (2.63 + 0.10 -0.21 ) 10 -3 eV 2 , sin 2 (θ 23 ) = 0.588 + 0.030 -0.062 , δ = 3.84 + 2.00 -2.14 ( 1.22π + 0.63π -0.68π ), (14.16) where ν x stands for ν e , ν µ or ν τ . The Cherenkov light emitted by the electron in the final state is used to detect the first and third reactions and the second one is seen via the emission of a photon of 6.25 MeV emitted in the capture of the neutron on deuterium. The first reaction (CC), mediated by a W boson exchange, is only sensitive to electron neutrino while the second one (NC), mediated by Z boson exchange, receives an equal contribution from all three flavours

σ N C (ν e ) = σ N C (ν µ ) = σ N C (ν τ ) (14.21)
For the third one, ν e has a higher cross section since it can go both by charged or neutral current as shown in fig. 15, and one has with a good approximation The collaboration measures the flux of neutrinos in the various channels and finds (in units of 10 6 cm -2 s -1 ) φ CC = φ(ν e ) = 1.76 + 0.06 -0.05 (stat.) + 0.09 -0.09 (syst.) φ ES = φ(ν e ) + 0.156 (φ(ν µ ) + φ(ν τ )) = 2.39 + 0.24 -0.23 (stat.) + 0.12 -0.12 (syst.) φ N C = φ(ν e ) + φ(ν µ ) + φ(ν τ ) = 5.09 + 0.44 -0.43 (stat.) + 0.46 -0.43 (syst.) (14.23)

σ ES (ν µ ) = σ ES (ν τ ) ≈ 0.156 σ ES (ν e ) ( 14
The result of φ N C is in very good agreement with the Standard Solar Neutrino Model 73 . From these results the collaboration derives (in the same units) φ(ν µ ) + φ(ν τ ) = 3.41 + 0.45 -0.45 (stat.) + 0.48 -0.45 (syst.), (14.24) which is clear evidence for the disappearance of solar ν e 's. In later stages, the SNO collaboration improved the detection efficiency of neutrons by adding an array of 3 He proportional counters in the D 2 O volume and they obtain the most precise estimate of active neutrino flux (in units of 10 6 cm -2 s -1 ) φ N C = 5.25 + 0.16 -0.16 (stat.) + 0.11 -0.13 (syst.) φ(ν e ) φ N C = .317 ± 0.016 (stat.) ± 0.009 (syst.), at E ν = 10 MeV, independent on E ν . (14.25) Because SNO observes ν e and ν µ + ν τ only, neither the mixing angle θ 23 nor the CP violating phase play a role (see eqs. (12.25), (12.29), (12.32)). Furthermore, given the distance involved, 1.5 10 9 km, the argument of the oscillating factors are so large that the corresponding sin 2 terms reduce to 1/2. In vacuum, the ν e survival rate is then neutrino is produced as the heaviest mass eigenstate and will emerge from the sun in a |ν 2 > state with a probability P (ν e → ν e ; x 0 , R) = sin 2 (θ 12 ) as in eq. (13.36). Being a pure eigenstate of the vacuum it will propagate without oscillation to Earth and will give φ(ν e ) φ N C = sin 2 (θ 12 ) ≈ .325, (14.30) in good agreement, within errors, with the experimental result of eq. (14.25). Based on their flux measurements the SNO collaboration performs a two flavour and a three flavour neutrino oscillation analysis. However SNO data alone are not sufficient to give tight constraints on the parameters δm 2 12 , θ 12 so an analysis is performed using also other solar data as well as KamLAND reactor data. Fig. 16 shows the constraints provided by SNO alone as well as various combinations of data. The best fit to the joint data, in the three flavour analysis, yields: δm 2 21 = (7.46 + 0.20 -0.19 ) 10 -5 eV 2 , tan 2 θ 12 = 0.443 + 0.030 -0.025 , sin 2 θ 13 = (2.49 + 0.20 -0.32 ) 10 -2 , (14.31) in very good agreement with eq. (12.22). Coming back to the case of low energy neutrinos from reaction eq. (14.18), the adiabatic condition is still verified but, in this case, the resonance condition cannot be satisfied since  < cos(2θ 12 ), or equivalently N e (x 0 ) < N Res , and interaction with matter becomes weaker. One expects from eq. (13.35) to have a larger ratio for φ(ν e )/φ N C as is found by the collaborations GALLEX, GNO, SAGE and Borexino. In fact, for E ν ≈ .2 MeV, one finds N e /N Res ≈ .1 from eq. (13.23) and, with a good approximation, the ν e 's should propagate as in vacuum with the result P (ν e → ν e ) = .56 as in eq. (14.27) (see fig. 17).

P (ν e → ν e ) = 1 - 1 2 sin 2 (2θ 12 ) cos 4 (θ 13 ) - 1 2 sin 2 (2θ 13 ) = sin 4 (θ 13 ) + (1 - 1 2 sin 2 (2θ 12 )) cos 4 (θ 13 ) ≈ 1 - 1 2 sin 2 (2θ 12 ), ( 14
The parameters δm 2 21 and θ 12 are sometimes referred to as solar oscillation parameters and indexed with the symbol ⊙.

Ultra-high energy or cosmic neutrinos

It is expected that, in the multi-TeV energy range and above, neutrinos from astrophysical or cosmic origin, will dominate over the atmospheric neutrinos. They can be produced in violent phenomena such as those occuring in Active Galactic Nuclei (AGN) or in collisions of ultra-high energy (UHE) cosmic rays on nucleons or photons, in particular photons from the cosmic microwave background (CMB).

Neutrinos produced in a supernova event or in the merging of stars or black holes are expected to have energies in the MeV/GeV range. Unlike other cosmic messengers such as cosmic rays or photons the universe is transparent to neutrinos 75 . Cosmic rays (protons, nuclei) are deflected by extra-galactic and galactic fields so that it is not possible to identify the source which produced them. They also loose energy when scattering on CMB photons, gaz and dust. Concerning photons, if their energy is high enough, they are absorbd on their way to Earth by e + e -pair production on CMB to UV Figure 18:

The photon horizon. Photons emitted in the grey domain do not reach the Earth because of annihilation into e + e -pairs. A redshift z = 1 corresponds to a distance of 14 Gly from the Earth. From J.G. Learned, K. Mannheim, Annual Rev. Nuc. Part. Sci. 50 (2000) 679.

background photons via γ HE + γ bkgrd → e + + e -. The threshold for such a process is obtained by solving the constraint (p γ HE + p γ bkgrd ) 2 > 4 m 2 e . Because of their high density (∼ 400 cm -3 ) the CMB photons (E γ CM B ≈ .23 meV) are particularly efficient in this respect cutting the high energy photon flux above 10 15 eV: even those emitted nearby in the galactic center do not reach the Earth, as seen in Fig. 18. This figure illustrates the depth of the photon horizon as a function of the photon energy: for example a 10 12 eV photon emitted by an object with a redshift z = .1 (i.e. roughly 1 Gly away) is absorbed before reaching the Earth. On the contrary, neutrinos are expected to travel undisturbed once they are emitted.

However the flux of UHE neutrinos is very low and to observe them requires huge detectors such as the km 3 IceCube detector 76 at the South Pole, the projected KM3NET 77 with a volume of 5 km 3 in the Mediterranean Sea which builds up on the ANTARES telescope 78 or the Giga Volume Detector 79 (GVD) which is an upgrade of the Lake Baikal experiment. As neutrino cross sections increase with 75 The "Glashow resonance", i.e. the reaction νe + e -→ W -→ X should affect the νe flux above Eν e > 6. energy the Earth will become opaque to neutrinos for E ν > 100 TeV. The UHE neutrinos will then be searched for in the downward neutrino fluxes, but, in that case, the cosmic ray shower background will be enormous and must be vetoed.

IceCube recently extended the measurement of the ν µ + ν µ flux above the domain shown in Fig. 11, up to more than 2 PeV 80 . The results are displayed in Fig. 19 where a hardening of the spectrum is observed above 100 TeV. Using a parameterisation of the cosmic ray flux and models of interactions of cosmic rays with the atmosphere they estimate the flux of atmospheric neutrinos : model and observation are in very good agreement up to around 100 TeV, energy above which the atmospheric neutrino flux falls below the data. The excess is interpreted as the flux of "astrophysical neutrinos" i.e. neutrinos directly emitted by sources such as AGN or produced in collisions of cosmic rays with dust, gaz or CMB photons.

On 22 September 2017 a high-energy neutrino-induced muon track event was detected by IceCube: the muon energy loss was estimated at 23.7 ± 2.8 TeV corresponding to a probable parent neutrino energy of 290 TeV (event labelled IceCube-170922A) 81 . Furthermore the reconstructed neutrino direction appeared to be pointing at the known blazar TXS 0506+56 (redshift z = .3365). An automatic alert was activitated and led to the subsequent observation of very high energy gamma rays by the Earth it is then possible to estimate, after taking account of the relevant boost factor, the energy of the proton in the frame of the emission zone. An important feature is that hadronic models predict also the emission, from neutral pion decays, of ultra energetic photons in the same energy range as that of the neutrinos, namely hundreds of TeV. If these photons escape from the emission zone they are not seen on Earth because of e + e -pair production which would cut-off their flux (see the "photon horizon" cut-off on Fig. 18). Most of the ultra-high energy photons however are expected to be absorbed by e + e -pair creation in the emission zone and the e ± ′ s radiate, create electromagnetic cascades ending in the UV, X-ray or soft gamma regimes. In conclusion, in this model, the rate of emission of neutrinos is strongly constrained by the spectral energy density in the UV and X-ray range, but no very high energy photons are expected to be seen in association with ν ′ s observations. A model of SED spectra of TXS 0506+056 is shown in Fig. 23 : it is seen that the hadronic component Grenoble, a high flux reactor using a 93% enriched 235 U with no time evolution on the ν e flux, has a segmented detector taking data at distances betwwen 9 and 11 m from the core. In PROSPECT 96 , at the High Flux Isotope Reactor at Oak Ridge National Laboratory, the detector is 7.4 m from the core.

All these experiments reduce the domain of sterile neutrino parameters obtained in previous reactor data 90 or global fits 92 and already exclude some best fits, as illustrated in Fig. 25 from the STEREO collaboration: the best RAA fit is already excluded at 99% C.L. More data are being accumulated and could reduce further the allowed domain of θ 14 , δm 2 41 .

Neutrinos: conclusions

The work for more precision on the determination of neutrino oscillation parameters is continuing.

The present experiments will increase the precision even more and this is crucial for the determination

Majorana mass term for neutrinos

Coming back to the Standard Model with three generations, we recall that we have defined, eq. (12.1),

ν ′ L =   ν e L ν µ L ν τ L   ,
a triplet of left-handed neutrino flavour eigenstates. A Yukawa mass term is defined by100 ,

L Y M = - 1 2 (ν ′c L M ν ′ L + ν ′ L M † ν ′c L ) (15.8)
where M is a 3×3 complex matrix. One shows first that this matrix is symmetric, M = M T . Indeed, from eq. (B.11) in the appendix, and using iγ T 2 = iγ 2 ,

ν ′c L M ν ′ L = ν ′T L iγ 2 γ 0 M ν ′ L = -ν ′T L M T γ 0 iγ 2 ν ′ L = ν ′T L i γ 2 γ 0 M T ν ′ L (15.9)
where the second line is obtained from the first by transposition with a change of sign due to the anticommutation of fermions, which proves the symmetry property of the mass matrix. The complex matrix M is diagonalised by the matrix S M = S T m S, (15.10) with m is diagonal with real eigenvalues. Plugging this expression in L Y M , the mass term is written

L Y M = - 1 2 (ν ′c L M ν ′ L + ν ′ L M † ν ′c L ) = - 1 2 (ν ′T L iγ 2 γ 0 S T m S ν ′ L + ν ′ † L γ 0 S † m S * iγ 2 ν ′ * L ) = - 1 2 (ν c L m ν L + ν L m ν c L ), (15.11)
where we have defined

ν L = S ν ′ L ⇔ ν * L = S * ν ′ * L ⇔ ν † L = ν ′ † L S † .
(15.12)

The Yukawa mass term can then be simplified to

L Y M = - 1 2 ν m ν = - 1 2 i m i ν i ν i (

Neutrino masses and the see-saw mechanism

We restrict the discussion to a one generation model and postulate a very massive right-handed neutrino singlet, N R , under the gauge group (sterile neutrino since non-interacting with gauge bosons).

The Yukawa lagrangian is assumed to have both a Dirac mass term (arising from the usual symmetry breaking mechanism with a scalar field doublet) and a Majorana mass term, coupling the right-handed neutrino to its charge conjugate, of the following form,

L Y = -m D N R ν L - 1 2 M R N R N c R + h.c. = - 1 2 (ν c L N R ) 0 m D m D M R ν L N c R + h.c., (15.15)
where we have used ν c L N c R = N R ν L to recover the first line from the matrix expression of the second one. We assume m D ≪ M R , m D being of the order of the electroweak symmetry breaking scale and M R much larger (of the order of a grand unification scale?). The symmetric mass matrix can be diagonalised and, taking into account the hierarchy of the two mass scales, one finds eigenvalues approximately equal to -m 2 D /M R and M R . To make both eigenvalues positive we rather write

0 m D m D M R ≈ i ρ -iρ 1 ρ 2 M R 0 0 M R i -iρ ρ 1 (15.16)
with ρ = m D /M R ≪ 1, so that the eigenstates of the mass matrix are (15.17) and the Yukawa term can be written

ν 1L = i(ν L -ρN c R ) N 1L = ρν L + N c R ≃ N c R ,
L Y = - 1 2 (ν c 1L N 1L ) ρ 2 M R 0 0 M R ν 1L N c 1L + h.c. = - 1 2 ρ 2 M R ν 1 ν c 1 - 1 2 M R N 1 N c 1 , (15.18)
after introducing the Majorana neutrinos

ν 1 = ν 1L + ν c 1L , N 1 = N 1L + N c 1L . (15.19)
To summarize, from a light left-handed "Dirac" neutrino and a heavy right-handed "Majorana" neutrino, the symmetric mass matrix of type eq. (15.15), can be diagonalised leading to two Majorana neutrinos, a light one, ν 1 with mass ρ 2 M R , and a heavy one with mass M R . The light left-handed neutrino ν 1L has a small mixing component with the heavy neutrino which could in principle be produced at the LHC, if its mass is not too high.

The procedure above can be generalised to the three generations of the Standard Model. One of the simplest way (type I see-saw) is to introduce three right-handed heavy neutrinos, N R i , singlets under the gauge group, similar to what is done for charged leptons, and add in the Yukawa Lagrangian, besides the term coupling to the scalar doublet field Φ, a Majorana mass term for the N R i 's: and 3 heavy ones. The associated eigenstates are Majorana neutrinos. Introducing these mass eigenstates in the charged current interactions term will yield a PMNS mixing matrix exactly as before, in the case of Dirac neutrinos. There is a difference however since it is not allowed to rotate away the phases of the neutrino fields 102 : the phase of a Majorana neutrino is fixed by the condition ν c i = Cγ 0 ν * i = ν i . In the expression of the charged current lagrangian, eq. (12.8), only the phases of the charged lepton fields can be changed, e α L → e -iφe α e α L , so that the PMNS matrix elements (S † ν ) αj become e -iφe α (S † ν ) αj (see the discussion after eq. (11.11)). These three arbitrary phases are used to absorb three phases of the PMNS matrix, which can be written in the form, see eq.

L Y = - 1 2 i=1,2,3 M Ri N c Ri N Ri -

Conclusions

Putting all together, the lagrangian which contains the dynamics of particle physics, as described by the Standard Model, is decomposed into (not including gauge fixing terms)

L = L QCD + L G + L F + L S + L Y
where each piece has been previously defined. If one attempts to count the number of parameters we arrive at:

-SU (3) (QCD) gauge invariance: 1 coupling g s or α s which is rather precisely determined by an enormous amount of data in deep-inelastic scattering, proton-proton, proton-antiproton or more generally hadron-hadron collisions, e + + e -→ jets as well as, at low energy, hadronic τ decays i e 2 cos θ W sin θ W g αδ g βγ + g αγ g βδ -2 g αβ g γδ (C.11)

(u L ) c (p) = Cγ 0 u * L (p) = iγ 2 u * L (p) = √ ω iγ 2 χ * L -χ * L = - √ ω χ R χ R = v L (p) (u R ) c (p) = C γ 0 u * R (p) = iγ 2 u * R (p) = √ ω iγ 2 χ * R χ * R = √ ω -χ L χ L = v R (p
W + -W --W + -W -vertex ¬ W -(β) W + (α) W -(δ) W + (
Z-W + -W -vertex W + (β) k 2 Z(α) k 1 W -(γ) k 3 i e cos θ W sin θ W g αβ (k 1 -k 2 ) γ + g βγ (k 2 -k 3 ) α + g γα (k 3 -k 1 ) β (C.19) γ-W + -W -vertex W + (β) k 2 γ(α) k 1 W -(γ) k 3 i e g αβ (k 1 -k 2 ) γ + g βγ (k 2 -k 3 ) α + g γα (k 3 -k 1 ) β (C.20)

i

  (p) for a positive energy and negative energy particle respectively (α is the polarisation index):ψ i (x) = d 3 p (2π) 3 2ω ψ i (p, x) = d 3 p (2π) 3 2ω α b (α) i (p) u iα (p) e -ip.x + d (α) † i (p) v iα (p) e ip.x , p.x = ωtp.x, (2.6)

F 2 4

 4 Tr( p 3 γ µ p 1 γ ν (1 -γ 5 ))Tr( p 4 γ µ p 2 γ ν (1 -γ 5 )) p 3 γ µ p 1 γ ν )Tr( p 4 γ µ p 2 γ ν ) + Tr( p 3 γ µ p 1 γ ν γ 5 )Tr( p 4 γ µ p 2 γ ν γ 5 )].

,

  and this complicates the discussion.

Figure 1 :

 1 Figure 1: Possible Feynman diagrams for ν e ν e → W + W -scattering. (a): e exchange; (b) hypothetical heavy electron E exchange; (c) neutral vector boson Z exchange.

  thus a right-handed negative energy spinor has negative chirality and a left-handed one positive chirality. Thus if one constructs a massless spinor u(p) as a linear combination of u α , α = 1, 2, then u L (p) = (1-γ 5 ) 2 u(p) and u R (p) = (1+γ 5 ) 2 u(p) are respectively left-handed and right-handed spinors, while v L (p) = (1+γ 5 ) 2 v(p) is left-handed and v R (p) = (1-γ 5 ) 2 v(p) right-handed, so helicity = chirality for positive energy spinors but helicity = -chirality for negative energy ones .

  From eq. (3.23), we see that these transitions involve only left-handed leptons or right-handed antileptons. Likewise, from the ψ d γ µ (1 -γ 5 )ψ u or ψ u γ µ (1 -γ 5 )ψ d interactions, only left-handed quarks or right-handed antiquarks are allowed. In the scattering ν e d → e -u only left-handed leptons and quarks are involved. If θ denotes the angle

  Similar arguments can be applied to ν/ν scattering on quarks or antiquarks and, then, one can easily derive eqs. (2.11), (2.13).

  Special choices of ξ can be made: -ξ = 0 (Landau gauge) : the Golstone boson is massless and the gauge boson propagator is transverse i.e. k ν G νρ = 0; -ξ = 1 (Feynman gauge) : the Golstone boson has the same mass as the gauge boson but one looses the transversity property of the gauge boson propagator; -ξ → ∞ : the Goldstone boson does not propagate and one keeps only the physical degrees of freedom in the model: one recovers the unitary gauge already considered.

  .29) where eqs. (8.28) and (8.19) have been used: the Higgs particle couples to a fermion flavour in proportion to the fermion mass, implying that the top quark could play a major role in the production and/or decay of the Higgs particle (m t ∼ 175 GeV) while the electron and light fermion contributions can be safely neglected. It is a puzzle why one observes such a large spectrum of masses from m e = .511 10 -3 GeV to m t = 173.21 GeV! No model naturally "explains" this fact.• RemarkIn sec. 2.3 we mentioned a problem related to massive gauge bosons namely the bad asymptotic behavior of the cross section of W pair production in e -e + colliders. This was illustrated on the simpler case ν ν → W -W + showing that the longitudinal polarisation states yield a cross section violating the Froissart bound if one keeps only the neutrino exchange diagram. Coming back to

Figure 3 :Figure 4 :

 34 Figure 3: Higgs production mechanism at hadron-hadron colliders : the dominant contribution arises from the subprocess where two gluons couple to the Higgs via a top quark loop. Another diagram with the fermion arrow reversed should be added.

Figure 5 :Figure 6 :

 56 Figure 5: ATLAS (Phys. Rev. D90 (2014) 112015) results on Higgs observation through its decay into 2 photons.

Figure 7 :

 7 Figure 7: Real and imaginary part of the function F(z) with respect to z = m 2 q /M 2 H

  and u → b + W + are not possible in the model. However it turns out that the transitions the hierarchy as indicated. They can be summarised by defining a transition u → d ′ with d ′ a linear superposition of the d, s, b quarks. The quark states (u i ) and (d i ) with i = 1, 2, 3

  .7) in which we keep only the diagonal (in SU (2)) part of the operator D L . Because of the unitarity of the transformations within the left-handed bases and the right-handed bases the above lagrangian immediately reduces itself to

  1. The determination of the V ij is a very active area of particle physics phenomenology at present and it is one of the aims of the LHCb collaboration at CERN and Belle II at KEK in Japan. If V ud is mainly constrained from nuclear β decays the others are essentially determind from K decays and heavy flavour decays. The 2018 edition of the particle data group 19 quotes the following values |V ud | = 0.97446 ± 0.00010, |V us | = 0.22452 ± 0.00044, |V ub | = (3.65 ± 0.12)10 -03 |V cd | = 0.22438 ± 0.00044, |V cs | = 0.97359 +0.00010 -0.00011 , |V cb | = (4.214 ± 0.076)10 -02 (11.11) |V td | = (8.96 +0.24 -0.23 )10 -03 , |V ts | = (4.133 ± 0.074)10 -02 , |V tb | = 0.999105 ± 0.000032 The CKM matrix generalizes to three families the Cabibbo angle, sin θ C = λ ∼ .22 ≃ V us introduced long ago to deal with the mixing of two families. The Cabibbo-Kobayashi-Maskawa quark mixing 19 M. Tanabashi et. al. (Particle Data Group), Phys. Rev. D98 (2018) 030001 (http://pdg.lbl.gov).

c

  12 c 13 s 12 c 13 s 13 e -iδ -s 12 c 23 -c 12 s 23 s 13 e iδ c 12 c 23 -s 12 s 23 s 13 e iδ s 23 c 13 s 12 s 23 -c 12 c 23 s 13 e iδ -c 12 s 23 -s 12 c 23 s 13 e iδ c 23 c 13   . (11.12) This last form is not very illuminating and in view of the relative smallness of |s 13 | ≃ |V ub | ≃ 3.57 10 -03 , |s 23 | ≃ |V cb | = 4.11 10 -02 the approximation c 13 ≈ c 23 ≈ 1 is justified. With |s 12 | ≈ |V us | = 0.225, Wolfenstein introduced the convenient and often used parameterisation (see the PDG review 19 ):

  .13) with s 12 ≈ λ = 0.22453 ± 0.00044, A = 0.836 ± 0.0015 ρ = 0.122 +0.018 -0.017 , η = 0.355 +0.012 -0.011 (11.14) This parameterisation shows that the charged current transition, for example, of a u quark to d, s, b quarks takes place with amplitudes which are proportional to (1 -λ 2 /2), λ, Aλ 3 (ρ -iη) respectively. The phase factor η, or equivalently δ in eq. (11.12), is responsible for CP violation in the Standard Model (see appendix B, in particular B.3). The measurement of this CP violating parameter, in kaon and B meson systems, for example, is of great theoretical interest in order to understand the origin of CP violation and of great practical importance since it may be related to the origin of the baryon asymmetry in the universe. A very important point is to check experimentally the unitarity of the CKM matrix : indeed any violation of one of the unitarity relations may indicate the existence of a new quark or a new family of quarks. Present data are consistent with the unitarity of the CKM matrix within a 3% accuracy.

  .10) To this invariance corresponds the conservation of the total lepton number defined as L = α L α , α = e, µ, τ . It follows that such transitions as µ -→ e -+ γ or µ -→ e -+ e + + e -are allowed in the model. A recent fit to available data shows that the mixing pattern is quite different from that of the quark 23 |U e1 | = 0.800 → 0.844, |U e2 | = 0.515 → 0.581, |U e3 | = 0.139 → 0.155 |U µ1 | = 0.229 → 0.516, |U µ2 | = 0.438 → 0.699, |U µ3 | = 0.614 → 0.790 |U τ 1 | = 0.249 → 0.528, |U τ 2 | = 0.462 → 0.715, |U τ 3 | = 0.595 → 0.776. (12.11) As for quarks, no satisfactory model can account for this mixing pattern. The phenomenology of neutrino mixing is discussed below in the framework of Dirac neutrinos. There are several recent reviews on this topic, in particular by Nakamura and Petcov 24 and by Giganti, Lavignac and Zito 25 . The case of Majorana neutrinos is discussed in sec. 15 and by Bilenky and Petcov 26 .

  Finally one has the sum rule, valid in the three family model 1 = ν β =νe,νµ,ντ

  23 ) = sin(θ 23 ), δ m = δ cos(2θ m 13 ) = cos(2θ 13 ) -Â sin 2 (2θ 13 ) + (cos(2θ 13 ) -Â) 13 )sin 2 (2θ 13 ) + (cos(2θ 13 ) -Â) 2 (2θ 13 ) + (cos(2θ 13 ) -Â) 2 . (13.50)

  the following we discuss how the values of the PMNS matrix elements are extracted from data. The first experiments were "disappearance" experiments where one measured the neutrino flux of a given flavour near the emission point and compared it to the flux of neutrinos of the same flavour measured at a distance. More recently several collaborations are able to carry out "appearance" experiments where one measures, near or far from the emission point, the flux of neutrino of a flavour different from the emitted one. The source of (anti)neutrinos are varied: • Nuclear reactors produce ν e of typical energy < E νe > ≈ 3. MeV which are measured close to the reactors ∼ 100 m or ∼ 1 km (Double Chooz, Daya Bay, RENO) or far away 180 km (KAMLAND).

. 4 )Figure 8 :

 48 Figure 8: KamLAND oscillation pattern and fits in the 2-ν and 3-ν models.

14. 2

 2 Neutrinos from accelerators: T2K, NOνA and OPERA ; δm 2 32 , θ 23 , δ T2K is a long baseline experiment with a muon neutrino beam with a peak energy of 0.6 GeV produced at the J-PARC (Japan Proton Accelerator Research Complex in Tokai) facility and observed in a near detector at 280 m and in the Super-Kamiokande detector at a distance x = 295 km from the production source. This is both a ν µ disappearance and a ν e appearance experiment. In 2011 the collaboration gave the first indication of ν e appearance in a ν µ beam 54 . Based on the small number of ν e observed, a non vanishing value of θ 13 is obtained for the first time: sin θ 13 = .11 with a large error however.Results analysing both ν and ν oscillations based on a ν µ beam generated by 7.48 10 20 POT ("protons on target") and a ν µ beam from 7.47 10 20 POT have been published in 2017 55 . Comparing ν µ → ν e and ν µ → ν e transitions is very useful to extract a precise measurement of the CP violating parameter.

Figure 9 :Figure 10 :

 910 Figure 9: Comparison of the oscillation rate ν e in a ν µ beam (right) with that of ν e in a ν µ beam (left) for differente hypothesis on the CP violation parameter δ. Note that δ = -π/2 in the figure corresponds to δ = 3π/2 in the text. From Y. Oyama, for T2K Collaboration, PoS PLANCK2015 (2015) 094, arXiv:1510.07200 [hep-ex].

Figure 11 :

 11 Figure 11: Energy spectra of ν e + ν e and ν µ + ν µ atmospheric neutrinos by the Super-K collaboration in comparison with other measurements. The solid (dashed) lines are model predictions with (without) oscillations. From Super-Kamiokande collaboration, E. Richard et al., Phys. Rev. D94 (2016) 052001, arXiv:1510.08127 [hep-ex].

  Fig. 12 displays details of ν oscillations in the Earth. Panel b) illustrates the survival pattern of an upward (cos θ z = -1) 4 GeV ν µ as a function of the distance travelled in the Earth. After crossing the Earth (x ≈ 1.28 10 4 km) the neutrinos have undergone 3 cycles of oscillations i.e. (xδm 2 31 /4k) ≈ 3 π. Notice that, in the model illustrated in fig. 12-b, the oscillation strength is enhanced as the muon neutrino crosses the Earth indicating a modification of the mixing angles (see eqs. (13.49)). A naive estimate of the effect of matter is obtained by calculating the factor Â, eq. (13.46): Â

Figure 12 :

 12 Figure 12: Neutrino oscillation patterns in the Earth. a) definition of the zenith angle θ, denoted θ z in the text, and multilayer structure of the Earth ; the average density of electrons in the core (grey areas) is N core e ≈ 3.3 10 24 cm -3 , and in the mantle (red area) N mantle e ≈ 1.2 10 24 cm -3 ; b) survival probability of an upward (cos θ = -1) E = 4 GeV muon neutrino crossing the Earth (red) and correlated appearance probability of an electron neutrino (green); c) survival probability of an upward going muon neutrino having crossed the Earth as a function of energy. From C. Rott, A. Taketa, D. Bose, Nature Scientific Reports: 15225, www.nature.com/articles/srep15225.

Figure 15 :

 15 Figure 15: Feynman diagrams for the elastic diffusion of a neutrino on an electron: on the left for ν e , on the right for ν µ or ν τ .

Figure 16 :

 16 Figure 16: Three flavour neutrino oscillation analysis : the blue lines are obtained using all solar neutrino experiments, the black ones are from KamLAND data and the colored potatoes are from a joint analysis. (From SNO collaboration, B. Aharmim et al., Phys. Rev. C88 (2013) 025501, arXiv:1109.0763 [nucl-ex].)

Figure 17 :

 17 Figure 17: Summary of solar ν e survival probabilities as a function of the average neutrino energy. ν e 's produced in the p p reactions eq. (14.18): red point; in the 7 Be + e -→ 7 Li + ν e channel: blue point; in the p + e -+ p → D + ν e channel: light blue point; in the 8 B channel eq. (14.19): black and grey points. The band is the theoretical prediction from the standard solar model with the MSW effect. The figure is from Borexino Collaboration, M. Agostini et al., arXiv:1709.00756 [hep-ex].

  3 10 15 eV. 76 IceCube collaboration, Science 342 (2013) no. 6161. 77 KM3NET Collaboration, Maarten De Jong, PoS NEUTEL2015 (2015) 055 78 ANTARES Collaboration, Maurizio Spurio, PoS NEUTEL2015 (2015) 054. 79 BAIKAL-GVD Collaboration, A.D. Avrorin et al. (2015), DOI: 10.1142/9789814663618 0019.

Figure 19 :

 19 Figure 19: The high energy ν µ + ν µ flux in IceCube, [arXiv:1705.07780].

Figure 23 :

 23 Photons Muon Neutrinos

  gives a major contribution to the spectral energy density in the X-ray range.The above discussion illustrates how a multi-messenger analysis can constrain models and thereby help understand the physics of astrophysical objects.85 Coming back to neutrinos IceCube can, to some extent, determine the neutrino flavor. Using this85 No neutrinos have been observed in correlation with the detection of gravitational waves emitted in the merging of black holes or neutron stars: ANTARES, IceCube and the Pierre Auger Observatory, Astrophys.J. 850 (2017) L35, [arXiv:1710.05839]; IceCube collaboration[arXiv:1908.07706].

Figure 24 :

 24 Figure 24: The figure illustrates the range of various oscillation parameters as a function the reactordetector distance: a sterile neutrino with parameters as given in the text does not affect long baseline experiments, from the 2013 presentation of STEREO experiment by S. Kox et al., on the site lpsc.in2p3.fr/trac/neutrino/wiki/.

(

  Figure 26: The measured or expected flux of neutrinos originating from different sources, from C. Spiering, Eur. Phys. J. H37 (2012) 515, [arXiv:1207.4952]. The range in energy covers 24 orders of magnitude, from µeV to EeV

  15.13) with ν = ν L + ν c L , a triplet of Majorana neutrinos diagonalising the mass term. This shows that one could, in principle, give a mass to neutrinos solely from left-handed neutrinos.To generate in the Standard Model a mass term by spontaneous symmetry breaking, coupling a lefthanded neutrino to its conjugate ν c L and a SU (2) scalar doublet field Φ, we turn to sec. 8.3. Hovever a possible candidate like ψ e L Φ ν c L is not acceptable since is not a singlet under SU (2) L ⊗ U (1) Y : ψ e L Φ is indeed a singlet but ν L is a SU (2) L doubletwith a non-vanishing y quantum number. One could introduce a more complicated stucture 101 , L Y M = c ψ e L Φ ( Φψ e L ) c , (15.14)which after symmetry breaking gives a mass to neutrinos of order m ν ≃ c v 2 ≃ c G -1 F . Since this term has dimension 5, the coupling c is necessarily of the form 1/Λ, with Λ a large scale introduced to keep the neutrinos very light. But this interaction is not renormalizable and it would require the introduction of new particles to render the theory finite in analogy with what is done to go from the Fermi model to the Standard Model. Therefore, it seems difficult to generate massive neutrinos without introducing new degrees of freedom.

c

  αi ψ α L ΦN Ri + h.c. . (15.20)The second term c αi (ν L Φ 0 -e L Φ -)N Ri , after symmetry breaking, generates Dirac mass parametersm Dαi = c αi v/ √ 2.The Yukawa term can be written in a matrix form identical to eq. (15.15) where(ν T L N cT R ) = (ν T Le ν T Lµ ν T Lτ N cT R1 N cT R2 N cT R3 ) (15.21)is the transpose of a six-component spinor and the mass matrixM = 0 m D m T D M R(15.22) is a 6 ⊗ 6 matrix contructed from the 3 ⊗ 3 matrix m D with elements m Dαi and M R a 3 ⊗ 3 diagonal matrix with elements M Ri . This symmetric matrix M can be diagonalised yielding 3 light eigenvalues of order m ν ≃ -m D M -1

  13 s 12 c 13 s 13 e -iδ -s 12 c 23 -c 12 s 23 e iδ c 12 c 23 -s 12 e iδ s 23 s 23 c 13 s 12 s 23 -c 12 c 23 e iδ -c 12 s 23 -s 12 c 23 e iδ c 23 c 13 and three phases: the CP phase δ and the 2 Majorana phases 0 < α 1 < 2π and 0 < α 2 < 2π. The pattern of oscillations of Majorana neutrinos is the same as that of Dirac neutrinos since the combination which controls the change of flavour (see eq. (12.9)),U αi U * αj U * βi U βj ,in eq. (12.17) is independent of the form of the PMNS matrix, eqs. (11.12) or (15.24), and it is not possible from the study of oscillations to distinguish Majorana from Dirac neutrinos.If one considers a global phase change on all left-handed fieldsν ′ i L (x) = e i Λ ν i L (x) ⇔ χ ′ i L (x) = e i Λ χ i L (x); l ′ (x) = e i Λ l(x),(15.25)the gauge interaction part remains invariant but this is not the case for the Yukawa term in the lagrangian since the right-handed fields are not independent and one hasν ′c L (x) = e i Λ ν c L (x). (15.26) Invariance under the global phase change eq. (15.25) is associated to lepton number conservation L = L e + L ν + L τ . In the presence of a Yukawa mass term, the invariance is lost and the lepton number is not conserved: it is possible to have nuclear transitions with emission of two electrons without neutrino (A, Z) → (A, Z + 2) + e -+ e -, (15.27) i.e. a neutrinoless double-beta decay {0νββ}. This is illustrated in fig. 27. In a first beta decay in a nucleus, a ν e is produced which turns into a ν e via the Majorana Yukawa mass term eq. (15.8) followed by the reaction ν e + n → e -+ p. The rate of transition is minute : proportional the G 4 F m 2 ν the fourth power of Fermi constant and the square of the neutrino mass ! Several experiments (CUORE 103 ,

  19 : α s (M 2 Z ) = 0.1181 ± 0.0011 in the M S renormalisation scheme with 5 active flavours. -SU (2) L ⊗ U (1) Y gauge invariance: two couplings g, g ′ , and the weak mixing angle θ W : in fact one coupling e, the charge of the proton/electron and the angle because of the relation g sin θ W = g ′ cos θ W = e. One has α = 1/(137.035999139 ± 0.000000031) and sin θ W = M W /M Z with M W = 80.385 ± 0.015 GeV, M Z = 91.1876 ± 0.0021 GeV. -spontaneous symmetry breaking from L S : two parameters µ and h or rather the vacuum expectation value v and h determined, for example, from v = M W sin θ W / √ πα and h = 0.5M 2 H /v 2 with M H = 125.09 ± 0.24 GeV 109 . -Yukawa couplings in L Y : nine couplings, i.e. one coupling per lepton and quark species and four CKM parameters for the mixing between quark generations. The Yukawa couplings are determined from the masses m e = 0.510998946 MeV, m µ = 105.6583745 MeV, m τ = 1.7768 GeV, m u = 2.2 MeV, m d = 4.7 MeV, m s = 96 MeV, m c = 1.27 GeV, m b = 4.18 GeV, m t = 173.2 GeV. Massive neutrinos of Dirac type require seven new parameters. There are thus 25 parameters assuming Dirac neutrinos (27 with Majorana neutrinos), most of them related to the fermions, which is not a satisfactory situation for a minimal model! However the model is strongly constrained since there are no less than 68 vertices of various types expressed in terms of the above parameters. Any persistent deviation from the predicted values in the Standard Model will indicate new physics. Checking experimentally the value of these couplings is one of the tasks of particle physicists. satisfy the condition (B.6) which is equivalent to: γ 2 γ µ γ 2 = γ * µ . (B.9) Using the relations eqs. (A.4) and (A.5), it is easy to prove C = -C -1 = -C † = -C T and Cγ 5 C -1 = γ 5 ⇔ γ 2 γ 5 γ 2 = γ 5 . (B.10) Under charge conjugation, the wave function ψ which satisfies eq. (B.1) becomesψ c = Cγ 0 ψ * = Cψ T = iγ 2 ψ * ⇔ ψ c = iψ T γ 2 γ 0 ,(B.11) solution of eq. (B.2). Let us first discuss free massless chiral spinors, eqs. (3.28) and (3.30), important in the construction of the Standard Model. The application of C parity yields:

  ), (B.12) and thus, the C operator transforms the wave-function of a positive energy spinor (electron) into the wave-function of a negative energy one (positron) of the same helicity (similar relations exist for (v L ) c and (v R ) c ). Recalling the definition of ψ L (x), eq. (3.23),ψ L (x) = d 3 p (2π) 3 2ω b L (p) u L (p) e -ip.x + d † R (p) v R (p) e ip.xits C transformed is:(ψ L ) c (x) = d 3 p (2π) 3 2ω d R (p) u R (p) e -ip.x + b † L (p) v L (p) e ip.x , (B.13)which destroys a right-handed antifermion with wave-function u R (p) and creates a left-handed fermion with v L (p). Equivalently, in a compact form, if one writes ψ L = (1 -γ 5 )ψ/2, its charge conjugate is:(ψ L ) c = iγ 2 ψ * L = 1 + γ 5 2 iγ 2 ψ * = 1 + γ 5 2 ψ c = (ψ c ) R , (B.14) a right-handed wave-function. Likewise the C conjugate of a right-handed wave-function is left-handed (ψ R ) c = 1 -γ 5 2 ψ c = (ψ c ) L . (B.15) C Feynman rules of the Glashow-Weinberg-Salam model C.1 Propagators Massless spin 1 boson (Feynman gauge)¡ k -i g µ ν k 2 + i ǫ (C.1)Massive spin 1 boson¢ k -i g µν -k µ k ν /M 2 k 2 -M 2 + i ǫ (C.2) Fermion £ p i p + m p 2 -m 2 + i ǫ (θ W cos θ W γ µ I f 3 -2 Q f sin 2 θ W -I f cos 2 θ W sin 2 θ Wg αδ g βγ + g αγ g βδ -2 g αβ g γδ (C.9) γ-γ-W + -W -vertex ª g αδ g βγ + g αγ g βδ -2 g αβ g γδ (

  Before entering the details of the model it is useful to recall the relation between rigid (global) gauge transformations and conserved currents since, as we shall see, the construction of the Weinberg-Salam model is made more transparent when using this notion. The choice of SU (2) L is motivated by the structure of currents building up the Fermi interaction. For massless particles these currents are conserved, hence from Noether theorem, they are the consequences of a global SU (2) invariance. The assumed U (1) Y global invariance is the minimal group necessary to construct the electromagnetic current : indeed by an appropriate choice of the hypercharges Y, one constructs the electromagnetic

.37) 4 The global SU(2) L ⊗ U(1) Y gauge invariance : conserved currents current as the sum of the neutral SU (2) one and the U (1) current. Following the spectacular success of QED the SU (2) L ⊗ U (1) Y invariance is made local to generate the interactions. It works !

  The point is to check that the first two diagrams lead to a gauge independent contribution since the

	3	4	3	4	3	4
	1	2	1	2	1	2
	gauge boson			Goldstone boson	Higgs boson
					The diagrams to be considered
	are

Higgs exchange diagram is independent of the gauge choice ξ. Using the decomposition eq. (7.23) of the gauge boson propagator in the general 't Hooft gauge it is enough to prove that the rightmost term in eq. (7.23) is cancelled by the Goldstone exchange diagram. From the gauge boson exchange we have

  and the equation becomes (x denotes now the length travelled by the neutrino)

	|ν i (x) >= e -ix (m 2 i /2 k) |ν i > .	(12.13)
	23 I. Esteban, M.C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, T. Schwetz, JHEP 1701 (2017) 087,
	arXiv:1611.01514 [hep-ph]. The variation of the coefficients is given for a 3 σ range.
	24 K. Nakamura, S.T. Petcov, in Particle Data Group (PDG), M. Tanabashi et. al., Phys. Rev. D98 (2018) 030001
	(http://pdg.lbl.gov).

25 C. Giganti, S. Lavignac, M. Zito, Prog. Part. Nucl. Phys. 98 (2018) 1, arXiv:1710.00715 [hep-ex]. 26 S.M. Bilenky, S.T. Petcov, Rev. Mod. Phys. 59 (1987) 671.

Table 1 :

 1 Sensitivity in terms of δm 2 ij of the different types of neutrino experiments characterised by the energy k of the neutrino and the distance x between the ν source and the detector.

	Source	type of ν	k [GeV] x [km] <δm 2 > [eV 2 ]
	Reactors (short baseline)	ν e	10 -3	1	10 -3
	Reactors (long baseline)	ν e	10 -3	100	10 -5
	Accelerators (short baseline)	ν µ , ν µ	1	1	1
	Accelerators (long baseline)	ν µ , ν µ	1	10 3	10 -3
	Atmospheric	ν e , ν e , ν µ , ν µ	1	10 4	10 -4
	Sun	ν e	10 -2	1.5 10 8	10 -10

.21) Conversely, we can use this formula to estimate the sensitivity of typical neutrino experiments to mass squared differences as shown in the table below. In some experimental conditions, a factor (x δm 2 ij /4k)

  2 , (12.23) which has a non-zero derivative, in fact -∞, if the neutrino is massive. The effect is very hard to measure since the rate of energetic electrons is very low. A limit established some years ago 32 was 30 F. Capozzi, E. Lisi, A. Marrone, A. Palazzo, Prog. Part. Nucl. Phys. 102 (2018) 48, arXiv:1804.09678 [hepph]; see also P.F. de Salas, S. Gariazzo, O. Mena, C.A. Ternes, M. Tórtola, Front. Astron. Space Sci. 5 (2018) 36; arXiv:1806.11051 [hep-ph]; P.F. de Salas, D.V. Forero, C.A. Ternes, M. Tórtola, J.W.F. Valle, Phys. Lett. B782 (2018) 633, arXiv:1708.01186 [hep-ph]; NuFIT webpage, http://www.nu-fit.org/.

31 G. Drexlin, V. Hannen, S. Mertens, and C. Weinheimer, Adv.High Energy Phys. (2013) 293986, arXiv:1307.0101. 32 Troitzk Collaboration, V.N. Aseev et al. Phys. Rev D84 (2011) 112003.

  2θ 12 ) cos 4 (θ 23 ) + sin 2 (2θ 23 )[cos 4 (θ 12 ) + sin 4 (θ 12 )] sin 2 (θ 13 )] sin 2 x δm 2 (2θ 23 ) cos 2 (θ 13 ) cos 2 (θ 12 ) + sin 2 (2θ 13 ) sin 4 (θ 23 ) sin 2 (θ 12 )] sin 2 x δm 2

	with the Jarlskog factor 36 J:
	J =	1 8	sin(2θ 12 ) sin(2θ 23 ) sin(2θ 13 ) cos(θ 13 ),		(12.27)
	and the expression COS νµ :
	COS νµ = cos 2 (θ 23 ) cos(2θ 12 ) sin 2 x	δm 2 21 4k	-sin 2 (θ 23 ) sin 2 x	δm 2 32 4k	-sin 2 x	δm 2 31 4k	(12.28)
								21
								4k
	-[sin 2 32 4k
	-[sin 2 (2θ 23 ) cos 2 (θ 13 ) sin 2 (θ 12 ) + sin 2 (2θ 13 ) sin 4 (θ 23 ) cos 2 (θ 12 )] sin 2 x	δm 2 31 4k
	-8 J cos(δ) COS νµ + O(sin 3 (θ 13 )),			(12.26)
	Phys. E12 (2003) 569, hep-ph:0308123.

33

G. Drexin for the

KATRIN Collaboration, 16th

TAUP International Conference, Toyama, Japan, Sept. 2019. 34 Planck 2018 results. VI. Cosmological parameters, N. Aghanim et al., in Astronomy and & Astrophysics, 2018, arXiv:1807.06209 [astro-ph.CO]. 35 Exact expressions in a somewhat different form are found in V. Barger, D. Marfatia, K. Whisnant, Int. J. Mod.

  2θ 12 ) cos 2 (θ 13 )[cos 2 (θ 23 ) -sin 2 (θ 23 ) sin 2 (θ 13 )] sin 2 x P (ν e → ν τ ) = sin 2 (2θ 12 ) cos 2 (θ 13 )[sin 2 (θ 23 ) -cos 2 (θ 23 ) sin 2 (θ 13 )] sin 2 x

	Using the sum rule eq. (12.19) or by direct calcultation it comes out:
									δm 2 21
									4k
	+ sin 2 (2θ 13 ) cos 2 (θ 23 ) sin 2 (θ 12 ) sin 2 x		δm 2 32 4k
	+ sin 2 (2θ 13 ) cos 2 (θ 23 ) cos 2 (θ 12 ) sin 2 x	δm 2 31 4k
	-4 J cos(δ) COS -2 J sin(δ) SIN,			(12.32)
									δm 2 21
									4k
	+ sin 2 (2θ 13 ) sin 2 (θ 23 ) sin 2 (θ 12 ) sin 2 x	δm 2 32 4k
	+ sin 2 (2θ 13 ) sin 2 (θ 23 ) cos 2 (θ 12 ) sin 2 x		δm 2 31 4k
	+ 4 J cos(δ) COS + 2 J sin(δ) SIN,			(12.29)
	with
	COS = cos(2θ 12 ) sin 2 x	δm 2 21 4k	-sin 2 x	δm 2 32 4k	+ sin 2 x	δm 2 31 4k	(12.30)
	SIN = sin x	δm 2 21 2k	+ sin x	δm 2 32 2k	+ sin x	δm 2 13 2k	.	(12.31)
	36 C. Jarlskog, Z. Phys. C29 (1985) 491. Due to the unitarity of the PMNS matrix one shows that Im(U * αi Uαj U βi U * βj ) with α = β, α, β = e, µ, τ , i = j, i, j = 1, 2, 3, is up to a sign an invariant, thus Im(U * e3 Ue2Uµ3U * µ2 ) = J sin(δ).
					96

  If one defines a measure of the CP asymmetry in the oscillation ν α → ν β by

						2 x	δm 2 31 4k
	+ 4 J cos(δ) COS τ + 2 J sin(δ) SIN,				(12.33)
	with
	COS τ = cos(2θ 23 ) cos(2θ 12 ) sin 2 x	δm 2 21 4k	+ sin 2 x	δm 2 32 4k	-sin 2 x	δm 2 31 4k	(12.34)

2θ 23 ) sin 2 (2θ 12 ) sin 2 x δm 2 21 4k + sin 2 (2θ 23 )[cos 2 (θ 12 ) -sin 2 (θ 12 ) sin 2 (θ 13 )] cos 2 (θ 13 ) sin 2 x δm 2 32 4k + sin 2 (2θ 23 )[sin 2 (θ 12 ) -cos 2 (θ 12 ) sin 2 (θ 13 )] cos 2 (θ 13 ) sin

  The difference between normal and inverted hierarchy occurs only in the sign of the cos(δ) coefficient, all other terms being insensitive to the sign of δm 2 32 . If the present experimental value of δ around 3π/2 (with large error bars) is confirmed, it will be very difficult to solve the mass hierarchy problem from oscillation experiments in vacuum. More on this later.Sometimes, it is sufficient to consider only a two neutrino system, ν e and ν x say, in which case the

	Since the δm 2 ij factors are not independent, δm 2 31 = δm 2 32 + δm 2 21 , one can eliminate m 2 31 , for example, oscillation formulae simplify considerably:
	and obtain:	SIN = 4 sin x P (ν e → ν e ) = 1 -sin 2 (2θ 12 ) sin 2 x δm 2 21 4k sin x δm 2 31 4k sin x δm 2 32 4k δm 2 21 4k = 4 sin x δm 2 21 4k sin 2 x δm 2 32 4k + O sin 2 x P (ν e → ν x ) = sin 2 (2θ 12 ) sin 2 x δm 2 21 4k	δm 2 21 4k	.	(12.37) (12.38) (12.43)
	where the last relation is valid when x δm 2 21 /4k is small compared to x δm 2 32 /4k. Coming back to the
	oscillation probabilities, the coefficient of the cos(δ) piece in the equations can likewise be simplified
	and one finds 37
	COS = 2 sin x	δm 2 21 4k	sin x	δm 2 31 4k	cos x	δm 2 32 4k	-2 sin 2 (θ 12 ) sin 2 x	δm 2 21 4k	(12.39)
		= 2 sin x	δm 2 21 4k	sin x	δm 2 32 4k	cos x	δm 2 32 4k	+ O sin 2 x	δm 2 21 4k	.	(12.40)
				2 x	δm 2 21 4k	+ sin 2 (2θ 13 ) sin 2 (θ 23 ) sin 2 x	δm 2 32 4k
		+ 8 J sin x	δm 2 21 4k	sin x	δm 2 32 4k	cos(δ) cos x	δm 2 32 4k	+ sin(δ) sin x	δm 2 32 4k	(12.41)
				2 x	δm 2 21 4k	+ sin 2 (2θ 13 ) cos 2 (θ 23 ) sin 2 x	δm 2 32 4k
		-8 J sin x	δm 2 21 4k	sin x	δm 2 32 4k	cos(δ) cos x	δm 2 32 4k	+ sin(δ) sin x	δm 2 32 4k	(12.42)
	sin x and similarly for other probabilities. 37 One has also COSτ = -2 cos(2θ23) [sin xδm 2 δm 2 21 2k 21 /4k sin xδm 2 + sin x δm 2 32 2k 32 /4k cos xδm 2 + sin x 31 /4k + sin 2 (θ12) sin 2 xδm 2 13 2k . δm 2 21 /4k ]. (12.36)
							98 99

Under these simplifications 38 , and neglecting small sin 2 (θ 13 ) corrections in the coefficients of terms in sin 2 x δm 2 21 /4k , the oscillation probabilities for ν e → ν µ and ν e → ν τ take the form:

P (ν e → ν µ ) ≈ sin 2 (

2θ 12 ) cos 2 (θ 23 ) sin P (ν e → ν τ ) ≈ sin 2 (2θ 12 ) sin 2 (θ 23 ) sin 38 They are particularly useful in oscillation experiments with accelerator neutrinos. Note that one can use indifferently δm 2 32 or δm 2 31 in eqs. (12.38) and (12.40).

  Taking for θ and δm 2 the values θ 12 and δm 2 21 from eq. (12.22) below, one obtains

	N 21 Res ≈ .8 10 -6
				.21)
	related to the parameter  previously introduced by
	N e N Res	=	Â cos(2θ)	(13.22)

  Using this result together with eqs. (13.49) and (13.51) one reconstructs the various probability functions. All oscillatory factors now enter the formulae and, from eq. (12.29), one finds for the oscillation ν e → ν µ :

	m 12 ) ≈ -	sin(2θ 12 ) Â21	= -α	sin(2θ 12 ) Â	.	(13.55)
	from eq. (13.17). P (ν e → ν µ ) ≈ sin 2 (θ 23 ) + α 8J cos(δ) sin 2 (2θ 13 ) (1 -Â) 2 cos x sin 2 x(1 -Â) δm 2 31 4k sin x δm 2 31 4k  δm 2 31 4k Â(1 -Â) + α 8J sin(δ) Â(1 -Â) sin x δm 2 31 4k sin x  δm 2 31 4k	+ α 2 cos 2 (θ 23 ) sin x(1 -Â) sin 2 (2θ 12 ) Â2 δm 2 31 4k sin x(1 -Â) δm 2 31 4k .	sin 2 x	31  δm 2 4k (13.56)

Table 2 :

 2 Value of the parameters controling the neutrino oscillations in the earth mantle: |α| = δm 2 21 /|δm 2 31 |, Â = 2 √ 2G F kN e /|δm 2 31 | with N e = 1.25 10 24 cm -3 , δm 2 21 is positive and δm 2 32 ≈ δm 2 31 is assumed. The value of the masses are taken from eq. (12.22). oscillating factors. One of the experimentally unsolved question is the mass ordering, i.e. is δm 2 31

  1 -sin 2 (2θ 12 ) cos 4 (θ 13 ) sin 2 (δm 2 21 x/4 k) -sin 2 (2θ 13 ) sin 2 (θ 12 ) sin 2 (δm 2 32 x/4 k) -sin 2 (2θ 13 ) cos 2 (θ 12 ) sin 2 (δm 2 31 x/4 k).

	(14.1)
	14.1.1 Long baseline: KamLAND, δm 2 21 , θ 12

  2θ 12 ) cos 4 (θ 23 ) sin 2 x

							δm 2 21 4k			(14.7)
	-[sin 2 (2θ 23 ) cos 2 (θ 13 ) + sin 2 (2θ 13 ) sin 4 (θ 23 )] sin 2 x	δm 2 32 4k
	-16 J sin 2 (θ 23 ) cos(δ) sin x	δm 2 21 4k	sin x	δm 2 32 4k	cos x	δm 2 32 4k	,
	and the oscillation probability is (see eq. (12.41):
	P ( ν (-) µ → ν (-) e ) = sin 2 (2θ 13 ) sin 2 (θ 23 ) sin 2 x	δm 2 32 4 k	+ sin 2 (2θ 12 ) cos 2 (θ 23 ) sin 2 x	δm 2 21 4 k	(14.8)
	+ 8 J sin x	δm 2 21 4k	sin x	δm 2 32 4k	cos(δ) cos x	δm 2 32 4 k	± sin(δ) sin x	δm 2 32 4 k	.
	where the -sign is for neutrino and the + sign for antineutrinos.

53 H. de Kerret et al., arXiv:1901.09445 [hep-ex]. 54 T2K collaboration, K.Abe et al., Phys. Rev. Lett. 107 (2011) 041801, arXiv:1106.2822 [hep-ex]. 55 T2K collaboration, K.Abe et al., Phys. Rev. D 96 (2017) 092006, arXiv:1707.01048 [hep-ex].

  2 x δm 2 21 /4k but keep them in the coefficient of sin 2 x δm 2 32 /4k . The survival probabilities are dominated by the sin 2 (2θ 23 ) sin 2 x δm 2 32 /4k piece and lead to a good determination of θ 23 and δm 2 32 . Based on data collected until 2016 the T2K collaboration quotes the values, at a 1 σ confidence level: GeV and the base line of 295 km one finds cos x δm 2 32 /4k ≈ 0 which means that the cos(δ) term has almost no contribution to the survival or oscillation probabilities. Since it is the only term which changes sign when going from normal to inverted hierarchy, T2K is not sensitive to the sign of δm 2 32 ≈ δm 2 31 . The δ dependence of P (ν µ → ν e ) is therefore almost entirely given by the sin(δ) piece which is

	for normal mass ordering, and
	δm 2 32 = (2.51 ± 0.08) 10 -3 eV 2 , sin 2 (θ 23 ) = 0.55 + 0.05 -0.08		(14.10)
	for inverted mass ordering. For this value of δm 2 32 and for a peak energy of .6 -8 J sin x δm 2 21 4k sin 2 x δm 2 32 4k sin(δ)
	≈ -sin(2θ 12 ) sin(2θ 23 ) sin(2θ 13 ) cos(θ 13 ) sin x	δm 2 21 4k	sin 2 x	δm 2 32 4k	sin(δ)
	≈ -0.013 sin(δ)				(14.11)
	δm 2 32 = (2.54 ± 0.08) 10 -3 eV 2 , sin 2 (θ 23 ) = 0.55 + 0.05 -0.09		(14.9)

  OPERA Collaboration, N. Agafonova et al. Phys. Lett. B 691 (2010) 138, arXiv:1006.1623. 60 N. Agafonava et al. Phys. Rev. Lett. 120 (2018) 211801, arXiv:1804.04912 [hep-ex]. 14.3 Atmospheric neutrinos: Super-Kamiokande ; δm 2 32 , θ 23 , δ

	57 OPERA Collaboration, S. Dusini, AIP Conference Proc. 1666 (2015) 110003; doi: 10.1063/1.4915575.
	58 In this expression the approximation sin L δm 2 32 /4E ≈ 1.27 δm 2 32 [eV 2 ] L[km]/E[GeV] is justified.

59

  2θ 23 ) cos 4 (θ 13 ) sin 2 (δm 2

	31 x/4 k)	(14.15)

P (ν e → ν e ) ≈ 1 -sin 2 (2θ 13 ) sin 2 (δm 2 31 x/4 k)

P (ν µ → ν µ ) ≈ 1 -[sin 2 (

2θ 23 ) cos 2 (θ 13 ) + sin 2 (2θ 13 ) sin 4 (θ 23 )] sin 2 x δm 2 31 4k .

  It is essentially the only ν e source in the energy range 1.5 MeV < E νe < 15. MeV but the solar ν e 's can convert to ν µ 's and ν τ 's on their way to the detector. The SNO collaboration 72 in Canada conducted an elaborate study of 8 B solar neutrinos. SNO is a detector using 1000 tons of ultra-pure heavy water (D 2 O) surrounded by an ultra-pure water (H 2 O) shield. Three types of reactions are studied ν

Figure 14: The solar neutrino spectrum from the sun (from J.N. Bahcall, A.M. Serenelli, S. Basu, Astrophys.J. 621 (2005) L85.

8 B decay into an excited beryllium state: 8 B → 8 Be * + e + + ν e . (14.19) e + D → p + p + e -, via charged current (CC) ν x + D → p + n + ν x , via neutral current (NC) ν x + e -→ ν x + e -, elastic scattering (ES), (14.20)

  .26) where the last approximate equality is a consequence of the smallness of θ 13 . It is then justified to use a two neutrino model. Assuming the validiy of the oscillation model in vacuum to explain the SNO ≈ 6. 10 25 and r 0 ≈ .1 R ⊙ ≈ .7 10 5 km (valid for x 0 .05 R ⊙ ). If one uses for δm 2 and θ

	data, one would obtain	φ(ν e ) φ N C ≈ 1 -	1 2	sin 2 (2θ 12 ) ≈ 0.56,	(14.27)
	in contradiction with the SNO result of 0.317. The obvious conclusion is that neutrinos interact with
	matter in the sun.
	• Neutrinos in the sun The electron density in the sun is parameterised as 74
		N e (x) = N e (x 0 ) exp	x -x 0 r 0	(14.28)
	with N e (0) the values δm 2 21 and θ 12 given by eq.(12.22 ) the adiabaticity condition eq. (13.31) will be satisfied if
		1 Â r 0 δm 2 12 2E ν	sin 2 (2θ 12 ) ≈ 2.7 10 4	E ν MeV	-2	≫ 1,	(14.29)

where  is taken from eq. (13.15). The inequality is satisfied for the SNO range of 5. MeV < E ν < 15. Mev. Besides  remaining large (see secs. 13.3 and 13.4) one is justified to assume that the 74 J.N. Bahcall, Neutrino Astrophysics, Cambridge University Press, 1989.

80

  IceCube collaboration, M.G. Aarsten et. al., EPJ C77 (2017) 692, [arXiv:1705.07780]. 81 IceCube collaboration, M.G. Aartsen et. al. Science 361 (2018) 347, [arXiv:1807.08794].

  γ) i e 2 sin 2 θ W g αδ g βγ + g αγ g βδ -2 g αβ g γδ (C.12)

	H-H-W + -W -vertex
			H	W -(β)
							i	e 2 2 sin 2 θ W	g αβ	(C.16)
			H	W + (α)
	H-Z-Z vertex
		H
				Z	i	e sin θ W cos θ W	M Z g αβ	(C.17)
		Z
	H-H-H vertex
	H-H-Z-Z vertex		H
		H	H	H Z(β)	i	-i 2 sin 2 θ W cos 2 θ W 3 2 e sin θ W M 2 H M W e 2 g αβ	(C.13) (C.18)
	H-H-H-H vertex	H	H H AE H H Z(α)		-i	3 4	e 2 sin 2 θ W	M 2 H M 2 W	(C.14)
	H-W + -W -vertex		± H W	W ±		i	e sin θ W	M W g αβ	(C.15)

We work in the system where = c = 1.

The Pauli matrices τ i are given in appendix A.

A negative chirality massive fermion at very high energy will be mainly left-handed with a small admixture, of O(m/ω), of the right-handed component, and vice-versa.

See, Clebsh-Gordan coefficients, spherical harmonics and d-functions in Particle Data group, C. Patrignani et. al., Chin. Phys. C40 (2016) 100001 (http://pdg.lbl.gov).

To lighten the notation and when no ambiguity arises one simply writes in the following ψ for ψ(x) and δψ for δψ(x).

ATLAS Collaboration, M. Aaboud et al., Phys. Lett. B786 (2018) 59, arXiv:1808.08238 [hep-ex].

CMS Collaboration,Phys. Rev. Lett. 121 (2018) 121801, arXiv:1808.08242 [hep-ex].

We could in fact identify flavour and mass eigenstates of the up sector and take for simplicity S u L = S u R = 1.

A unitary matrix U can be written U = exp(i a α a T a ), with T a the generators of the SU (3) group for a = 1, • • • , 8 and T 9 = 1 1. The α a are real parameters. A matrix U parameterised as in eq. (11.12) is often written U = U23U13(δ)U12.

In 1952, B. Pontevorvo was the first to mention the possibility of νe -νe oscillations. In 1962, the year when νµ was discovered, Ziro Maki, Masami Nakagawa and Shoichi Sakata, assuming two kinds of neutrinos proposed a "particle mixture theory of neutrino", Prog. Theor. Physics 28 (1962), 870.

The PMNS matrix appears simpler than the CKM one since the gauge interaction eq. (12.8) is written directly in terms of the charged lepton mass eigenstates.

From eqs. (12.5), (12.9) a flavour field να L is related to the fields νi L of given masses by να L = i (S † )αiνi L = i Uαiνi L , but the state |να > is created by the field να L , hence eq. (12.14).

We use cos xδm 2 ij /2k = 1 -2 sin 2 x δm 2 ij /4k , the factor 1 then leading to the δ αβ term.

J.A. Formaggio, G.P. Zeller, Rev. Mod. Phys. 84 (2012) 1307.

M.C. Gonzalez-Garcia, M. Maltoni, Phys. Rep. 460 (2008) 1.

The physics depends only on the difference ω2 -ω1 and θ m , which are functions of the difference of the diagonal elements of H, in agreement with the fact that one can modify H by adding to it a matrix proportional to unity.

νe → νe scattering is obtained from νe → νe by crossing symmetry which implies a relative -sign when crossing fermions.

One assumes an equal number of neutrons and protons hence Np = Ne = NN /2, with NN given above.

The full treatment, which is applied here in a simplified form, is given in M. Freund, Phys. Rev. D64 (2001) 053003, [arXiv:hep-ph/0103300].

KamLAND collaboration, A. Gando et al., Phys.Rev D83 (2011) 052002, arXiv:1009.4771, [hep-ex].

Atsuto Suzuki, Eur.Phys.J. C74 (2014) 3094, arXiv:1409.4515 [hep-ex].

NOνA collaboration, P. Adamson, Phys. Rev. Lett. 118 (2017) 231801, arXiv:1703.03328 [hep-ex]; Jianming Bian, for the NOνA collaboration, arXiv:1812.09585 [hep-ex].

Super-Kamiokande collaboration, K. Abe et al., Phys.Rev. D97 (2018) 072001, arXiv:1710.09126 [hep-ex].

SNO collaboration, B. Aharmim et al., Phys. Rev. C88 (2013) 025501, arXiv:1109.0763 [nucl-ex].

A.S. Brun, S. Turck-Chièze, J.P. Zahn, Astrophys. J. 525 (2001) 1032; J.N. Bahcall, M.H. Pinsonneault, S. Basu, Astrophys. J. 555 (2001) 990.

An AGN consists typically in a supermassive rotating black hole in the center (10 6 M⊙ to 10 10 M⊙), an accretion disk, clouds of ionized gaz, a dust ring, two jets extending on 10 ′ s of parsecs and lobes extending on 100 ′ s of parsecs.

For the DUNE collaboration, N. Grant, PoS(NuFact2017) 052 (2017). DUNE is a long base line oscillation experiment (1300 km) with a highly pure νµ beam from FERMILAB and 4 10kt Liquid Argon Time Projection Chambers deep underground in South-Dakota, expected to start operation in 2026.

Note the factor 1/2, compared to a Dirac mass term, because ν ′ and ν ′ contain the same degrees of freedom.

S. Weinberg, Phys. Rev. Lett. 43 (1979) 1566.

J. Bernabeu, P. Pascual, Nucl. Phys. B228 (1983) 21; S.T. Petcov, Adv. High Energy Phys. 2013 (2013) 852987, arXiv:1303.5819 [hep-ph] .

CUORE Collaboration, K. Alfonso et al., Phys. Rev. Lett. 115 (2015) 102502.

Combined Measurement of the Higgs Boson Mass with the ATLAS and CMS Experiments, G. Aad, et al., arXiv:1503.07589.

Neutrinos and the Pontecorvo-Maki-Nakagawa-Sakata matrix

The absence of a CKM mixing matrix for the leptonic sector requires a comment. We assumed that the right-handed neutrinos decouple from the observed world. As a consequence, as mentioned above, the neutrinos ν e , ν µ , ν τ remain massless even after spontaneous symmetry breaking since there are, in the lagrangian density, no terms coupling left-handed and right-handed fields like in eq. (8.26).

Therefore no mass matrix can be constructed from which the "physical" neutrino states are defined.

When studying the weak-current transition from charged leptons to neutrinos we are thus free to define the neutrino physical states as those for which the charged weak current is diagonal in lepton flavour.

However, recent experiments have shown that neutrinos oscillate i.e. they change flavour when propagating from their emission point to their detection point. This can be explained if neutrinos are massive. If one follows the same procedure as for the quarks one introduces right-handed fields and this leads to Dirac type massive neutrinos. There is another possibility which relies on the fact that, being neutral, neutrinos can be their own antiparticles and this leads to Majorana type neutrinos. In the first case the total lepton number L = L e + L µ + L τ is conserved, while, in the latter case, it is not. In this section, we deal with Dirac neutrinos, the Majorana case being treated in sec. 15.

We assume that, like quarks, the neutrinos are of Dirac type with both left-handed and right-handed components. In the flavour basis, besides the triplet of left-handed fields ν ′ L one introduces a triplet of right-handed fields ν ′ R , singlets under SU (2) L ,

In this notation ν e L is the neutrino produced by an electron in a charged current interaction and similarly for ν µ L and ν τ L . Assuming the SU (2) L ⊗ U (1) Y symmetry holds true, the right handed neutrinos cannot be produced or interact in reactions mediated by gauge bosons: they do not couple to the SU (2) L gauge bosons nor to the U (1) Y boson since being neutral y ν R = 2e ν R = 0, thus they do not couple to W ± , Z or γ gauge bosons. They can be produced in Higgs decays but, given that the Higgs couplings are proportional to the masses, it is fair to assert that the production rate via this channel is not measurable. Their only effect is to give masses to neutrinos. The charged current while, fixing sin 2 (θ 13 ) = 0.0210 ± 0.0011 (from the Daya Bay, RENO and Double-Chooz) the results are:

δm 2 31 = (2.53 + 0.22 -0.12 ) 10 -3 eV 2 , sin 2 (θ 23 ) = 0.425 + 0.046 -0.037 , δ = 3.14 + 2.67 -1.35 ( π + 0.85π -0.43π ). (14.17)

Very similar numbers are obtained for the inverted hierarchy hypothesis but the data indicate a weak preference for the normal mass hierarchy. One observes that, when constraining θ 13 , the θ 23 angle is in the first octant θ 23 < π/4 but, for the other case, θ 23 is in the second octant θ 23 > π/4 65 . This illustrates the strong correlations between parameters as shown by eqs. (14.15), as well as the difficulty to obtain a precise determination of the mixing angles.

The upper end of the ν + ν spectra in fig. 11 do not play any role in the physics of oscillations but, as will be seen in sec. 14.5, they carry information on cosmic sources.

The parameters δm 2 32 and sin 2 (θ 23 ) are often called atmospheric oscillation parameters.

14.4 Solar neutrinos: SNO ; δm 2 12 , θ 12

Since the mid sixties solar neutrinos presented a nagging problem : the measured flux 66 was two to three time smaller than the predicted one by the standard solar neutrino model 67 . Several explanations were proposed to account for this discrepancy 68 but now it has been shown that the correct explanation lies in the incoherent interactions of neutrinos with matter in the sun.

According to the standard solar neutrino model, the production modes of neutrinos are given in fig. 14.

The most abondant one is Fermi-LAT satellite and the Magic telescope by this blazar in a flaring state. Radio, optical, and X-ray observations were carried out and pointed to an increase of radio emission and variability in months before the alert and of X-ray emission a week after 82 . This event is interesting as it is, at present, the only example of an identified neutrino emission from a blazar. radio/microwave IR/UV X-ray γ-ray In more details one has the photoproduction of π 0 via γ + p → π 0 + p followed by π 0 → γ + γ and also production of π ± , e.g.

Hadronic models then imply, from charged pion decays, the production of ν ′ e s and ν ′ µ s carrying on the average 5% of the energy of the initiating proton. Knowing the energy of the neutrino detected on possibility and taking into account oscillations, the observations will then give precious information on the flavor composition in the production zone which in turn helps distinguish between production models 86 .

Problems?

The three neutrino oscillation model can account, at present, for almost all data. However two collaborations, LSND 87 and MiniBooNE 88 , claim results in strong disagreement with the above experiments.

To add to the confusion LSND results are not confirmed by KARMEN 89 where very similar technics are used. MiniBooNE considers ν µ → ν e and ν µ → ν e in short baseline experiments with .2 < E ν [GeV] < 1.25 and a ratio x/E ν in the range .25 < x/E ν [m/MeV] < 2.5. In a 2-neutrino oscillation model involving a sterile neutrino, the oscillations are best fitted with the parameters [δm 2 , sin 2 (2θ)] = [3.14 eV 2 , 0.002] for ν's, and [0.043 eV 2 , 0.88] for ν's. MiniBooNE results are summarised saying that "the data are consistent with neutrino oscillations in the 0.01 < δm 2 [eV 2 ] < 1.0 range" and they "have some overlap with the evidence for antineutrino oscillations from LSND".

In the last few years, the ν e flux from nuclear reactors has raised a puzzle. In short baseline experiments (10 < x [m] < 100) there is a 6% deficit in the observed ν e compared to model expectations: this is the Reactor Antineutrino Anomaly (RAA) 90 . Several explanations have been proposed. In a recent study the Daya Bay collaboration 91 observes correlations between the time evolution of the fuel in the core (the composition in U and Pu isotopes varies with time) and changes in the ν e flux and energy spectrum. A detailed study of these correlations shows a 7.8% discrepancy between the observed and predicted 235 U yields which suggests that this isotope is the main contributor to the RAA. An alternative explanation has been to assume a fourth (sterile) neutrino to account for the ν e deficit in short baseline nuclear reactor experiments. This is illustrated in Fig. 24 which shows that short and very short (less than 10 m) baseline reactor measurements are not sensitive to the three family neutrino parameters as given in eqs. (12.22), but would be affected by a fourth neutrino according the disappearance probability (see eq. (14.3)): The large number of parameters has prompted an intensive continuing search for higher hidden symmetries such as supersymmetry, for example, and Minimal Supersymmetric Standard Models and Next-to-Minimal Supersymmetric Standard Models and • • • have been constructed. The very unfortunate situation is that in such models the number of fields is more than doubled compared to the Standard Model as no supersymmetric multiplets can be filled with only known particles. Furthermore, while in principle supersymmetry breaking can in turn trigger electroweak symmetry breaking, one does not know yet how supersymmetry is dynamically broken. Thus one is led to describe it in a effective way which requires many more parameters than in the Standard Model. It seems that, at present, the remedy is worse than the disease but there are still hopes ......

Appendices A Properties of γ µ matrices

In Dirac representation γ µ matrices are defined by:

where the τ i are the Pauli matrices:

The Pauli matrices are hermitian and they satisfy:

The matrices γ µ have the following properties:

They satify anticommutation relations:

The matrix γ 5 is defined by:

It anticommutes with γ µ matrices:

One proves easily:

For the evaluation of traces of products γ α γ β ... one has the following relations :

Tr(γ α γ β ...) = 0 for an odd number of matrices (A.9)

Tr(γ 5 γ α γ β ...) = 0 for an odd number of γ α matrices Tr(γ 5 γ α γ β ) = 0

where ǫ αβδλ is the totally antisymmetric tensor under permutation of its indices with ǫ 0123 = +1. One has ǫ αβδλ = -ǫ αβδλ and in particular ǫ 0123 = -1. A useful relation is:

There exists other representations due to Weyl and to Majorana which satisfy the relations eq. (A.5) to eq. (A.9). In general, when doing calculations, the explicit form of γ µ matrices is not necessary.

2)

The solution ψ c can be constructed in the following way. From the first equation above one has

We look for ψ c under the form

where C is a 4 × 4 matrix. Then eq. (B.3) yields after multiplication on the left by Cγ 0 :

and, if one finds a matrix C such that:

then we recover eq. (B.2). In our representation of γ µ matrices, we have

so that the choice of the real matrix

Going back to the general case, it is easy to show that under C parity the helicity projection operators satisfy:

and the energy projection operators satisfy:

where one has used the relations eq. (B.7). Thus, a solution of the Dirac equation of positive (resp.

negative) energy and given helicity becomes a solution of negative (resp. positive) energy of the same helicity:

It is useful to list the transformation of fermion bilinears under C. They easily derived from eqs.(B.9) to (B.11), remembering the -sign (due to Fermi statistics) when transposing the expressions to obtain the right hand-side, and one finds:

B.2 Space reflection P

The space reflection, or parity transformation is defined by :

The transformation is parameterised in the following way

Knowing ψ(x 0 , x) satisfying the free Dirac equation, we look for the form of the solution obtained under a space reflection. We write

From the free Dirac equation

we obtain, using

which leads to

If we find a matrix P such that a ν µ Pγ µ = γ ν P, then ψ ′ (x 0 , x ′ ) will be a solution. Such a matrix should commute with γ 0 and anticommute with γ. Obviously

has such a property. Thus

Note that the parity operator reverses the fermion helicity. Thus a massless left-handed fermion becomes right-handed:

(where for simplicity we ignore an irrelevant phase). In terms of Dirac spinors one has:

as can be immediatly verified from eqs (3.12). It is easy to check the behavior of the fermion bilinears under a parity transformation:

a pseudovector or axial vector

B.3 Variance and invariance of the lagrangien under C and CP

From the above discussion, it is easy to obtain the transformation properties of the lagrangien. The easiest case is that of QED:

Under P all vectors such as

reduces to the lagrangien above. The transformation is also very simple under C. The U (1) gauge transformation, ψ ′ (x) → exp(-ieα(x))ψ(x), implies ψ c′ (x) → exp(ieα(x))ψ c (x) and the U (1) gauge invariance of L QED leads to A c µ (x) = -A µ (x) (use eqs. (B.17) to prove the invariance). For the derivative term it is a bit more tricky since

One goes from the first to the second line by transposing the expression keeping in mind the -sign for the anticommutation of the fermions and from the second line to the last one by a partial integration neglecting, as usual, a total derivative. This proves the invariance of the QED lagrangian under C, P and therefore CP transformations.

On the contrary a theory with an interaction term of the form ψ(x)γ µ (1 -γ 5 )ψ(x) is not invariant under C or P since this term becomes, up to an overall sign, ψ(x)γ µ (1 + γ 5 )ψ(x) (use eqs. (B.17) and (B.29)), and one can say that there is maximum violation of these symmetries. However it is invariant under CP. The case of the Standard Model with three generations is a bit more subtle. Consider the charged current piece eq. (11.9) written in the mass eigenstate basis. Denoting V the CKM matrix,

where the index i, j run over the number of fermion generations. If ψ is in the fundamental representation of the unitary group G, with generators τ a the generators operating on the ψ c fields are τ a * and the conjugate of the gauge boson is W c µ = -W a µ τ a * = -W * µ , so that W µ ↔ -W * µ under C parity (with the definition of W µ given after eq. (5.43)). Under charge conjugation, L F becomes:

where we have used eqs. (B.17). If we do furthermore a P transformation on this expression we obtain:

since, following eqs. (B.29), the term in W µ γ µ is invariant while W µ γ µ γ 5 changes sign. This is identical to eq. (B.33) except for the v ij ↔ v * ij factors interchanged between the two terms of the expression: if the CKM matrix were real then the lagrangian would be invariant under CP, in other words the phase of the CKM matrix is at the origin of CP violation in the Standard Model since all other terms in the lagrangien are invariant under CP.

B.4 Time reflection T

The time-reflection transformation takes the coordinate x = (x 0 , x) to x ′ = (-x 0 , x). This transformation can be written Recalling that γ µ * = γ µ for µ = 0, 1, 3 and γ 2 * = -γ 2 , the above conditions reduce to T γ i = γ i T for i = 0, 2 and T γ j = -γ j T , j = 1, 3. The matrix