A Lacour

J A Oller

U.-G Meißner

The chiral quark condensate and pion decay constant in nuclear matter at next-to-leading order

Making use of the recently developed chiral power counting for the physics of nuclear matter [1, 2], we evaluate the in-medium chiral quark condensate up to next-to-leading order for both symmetric nuclear matter and neutron matter. Our calculation includes the full in-medium iteration of the leading order local and one-pion exchange nucleon-nucleon interactions. Interestingly, we find a cancellation between the contributions stemming from the quark mass dependence of the nucleon mass appearing in the in-medium nucleon-nucleon interactions. Only the contributions originating from the explicit quark mass dependence of the pion mass survive. This cancellation is the reason of previous observations concerning the dominant role of the long-range pion contributions and the suppression of short-range nucleon-nucleon interactions. We find that the linear density contribution to the in-medium chiral quark condensate is only slightly modified for pure neutron matter by the nucleon-nucleon interactions. For symmetric nuclear matter the in-medium corrections are larger, although smaller compared to other approaches due to the full iteration of the lowest order nucleon-nucleon tree-level amplitudes. Our calculation satisfies the Hellmann-Feynman theorem to the order worked out. Also we address the problem of calculating the leading in-medium corrections to the pion decay constant. We find that there are no extra in-medium corrections that violate the Gell-Mann-Oakes-Renner relation up to next-to-leading order.

Introduction

The QCD ground state, or vacuum, is characterized by the presence of a strong condensate 0|qq|0 of scalar quark-antiquark pairs -the chiral quark condensate -which represents an order parameter for spontaneous chiral symmetry breaking in QCD. It is accepted on phenomenological grounds that the lightest pseudoscalar mesons, the pions, are identified with the Goldstone bosons of the spontaneously broken chiral symmetry [3]. Spontaneous chiral symmetry breaking results because the axial charge, Q i A = d 3 xA i 0 (x), does not annihilate the ground state. The coupling of the Goldstone bosons to the axial-vector charge is given in terms of the pion decay constant f π , which determines the chiral scale 4πf π ∼ 1 GeV that governs the size of the quantum corrections in Chiral Perturbation Theory (χPT) [START_REF] Georgi | Weak interactions and modern particle theory[END_REF]. Chiral symmetry is also explicitly broken due to the small masses of the lightest quarks, u and d. An interesting issue for the discussion of the QCD phase diagram is the dependence of the chiral quark condensate on temperature and density. It is expected that with growing temperature, the quark condensate melts [START_REF] Gerber | [END_REF][6][7]. There is an indication that this is also the case at zero temperature and increasing density as follows from the direct application of the Hellmann-Feynman theorem [8] to the energy density of a Fermi sea. A restoration of chiral symmetry is believed to be linked to a phase transition in QCD. A non-vanishing chiral quark condensate represents a sufficient condition for spontaneous breakdown of chiral symmetry. Note, that it is not a necessary one, such one would be the Goldstone boson (pion) decay constant, therefore chiral symmetry might as well be broken with a vanishing quark condensate. As long as the matrix element Ω|Q i A |π a (p) = iδ ia (2π) 3 f t p 0 δ(p) is not zero, the ground state is not left invariant by the action of the axial charge Q i A and spontaneous chiral symmetry breaking happens. In the previous equation |π a (p) denotes a pion state with Cartesian coordinate a, three-momentum p, energy p 0 and f t is the temporal weak pion decay coupling. Many recent calculations in nuclear matter share the assumption that spontaneous chiral symmetry breaking still holds for finite density nuclear systems. The form of the chiral Lagrangians changes depending whether the chiral quark condensate Ω|ūu + dd|Ω is large or small. In the former case we have standard χPT [9][10][11] and in the latter generalized χPT [12,13] should be employed. It has been shown that the first case holds [14] in vacuum for SU (2) χPT. For modern applications of chiral symmetry to nuclear systems we refer to [15]. The in-medium chiral quark condensate for symmetric nuclear matter in the linear approximation [16][17][START_REF] Wirzba | Hirschegg '95: Dynamical Properties of Hadrons in Nuclear Matter[END_REF][START_REF] Kirchbach | [END_REF][20][21] decreases with density as 1 -(0.35 ± 0.09)ρ/ρ 0 , with the error governed by that of the knowledge in the pion-nucleon sigma term, σ = m∂m/∂ m = 45 ± 8 MeV [22,23], with m = (m u + m d )/2 the mean of the u and d quark masses, m the nucleon mass and ρ 0 0.16 fm -3 the nuclear matter saturation density. Thus, if the low energy linear decrease in the quark condensate is extrapolated to higher densities the quark condensate would vanish for ρ = (2.9 ± 0.7)ρ 0 and standard χPT would not be appropriate. Of course, higher order corrections could spoil this tendency. Hence, the calculation of the in-medium quark condensate is of great interest beyond the linear approximation. The quark condensate in the nuclear medium and the quest for restoration of chiral symmetry at finite baryon density (and temperature) has a long and outstanding history, see e.g. [16][17][START_REF] Wirzba | Hirschegg '95: Dynamical Properties of Hadrons in Nuclear Matter[END_REF][START_REF] Kirchbach | [END_REF][20][21][24][25][26][27][28][29][30][31][32].

In this work we concentrate first on the application of in-medium baryon χPT to the calculation of the quark condensate and go beyond the linear density approximation. However, our approach [1] compared with previous works offers two novel features: i) It follows a strict chiral power counting that takes into account both long-and short-range (multi-)nucleon interactions. It is applicable both in vacuum and in the medium so that a clear connection between the two cases is established and used. ii) We do not take as starting point the Hellmann-Feynman theorem but directly apply the power counting mentioned to the problem of the calculation of the quark condensate. The fulfilment of the Hellmann-Feynman theorem is a consequence of the consistency of the approach order by order. In this way, the dependence on the quark mass of the input parameters in the theory, like f π , g A , nucleon and pion masses, etc, is built in. This power counting has been previously applied to the problems of the pion self-energy in the nuclear medium and the equation of state of nuclear matter. For the former issue [1] we found a cancellation of the next-to-leading order (NLO) contributions, including those stemming from the inmedium nucleon-nucleon interactions. This suppression is interesting since it allows to understand from first principles the phenomenological success of fitting data on pionic atoms with only meson-baryon interactions [33][34][35]. Regarding the equation of state of nuclear matter it is interesting to point out that saturation for symmetric nuclear matter and repulsion for neutron matter is obtained in ref. [2] at NLO. The resulting curves of the energy per particle as a function of density for neutron matter agrees with sophisticated many-body calculations [36] in terms of a subtraction constant g 0 that acquires its natural expected value, around -0.55 m 2 π . For the case of symmetric nuclear matter the experimental properties of the minimum for the bounding energy are reproduced fitting g 0 -m 2 π , while the analogous parameter g 0 keeps its just mentioned natural value. As a result of the present and the previous applications [1,2,21] the calculation of the in-medium corrections to the coupling of the pion to the axial-vector current up to NLO is straightforward. As we show below, the linear reduction with density of f t as obtained in ref. [21] holds up to NLO.

After this introduction, we present the power counting formula used and the Feynman diagrams contributing to the in-medium chiral quark condensate to NLO in section 2. The contributions stemming from pion-nucleon chiral dynamics are the object of section 3. The terms due to the nucleon-nucleon interactions are calculated in section 4. Results and discussions thereof are given in section 5. The inmedium pion decay constant and the Gell-Mann-Oakes-Renner relation are discussed in section 6. The last section is dedicated to offer our conclusions.

In-medium chiral power counting and diagrams

For determining the set of diagrams to be calculated in the evaluation of the quark condensate in the nuclear medium up-to-and-including NLO contributions we make use of the chiral power counting developed in [1]

ν = 4 -E + Vπ i=1 ( i + n i -4) + V i=1 (d i + v i + ω i -2) + V ρ . (2.1)
In this way, E is the total number of external pion lines and V π , V and V ρ are the meson-meson, mesonbaryon and in-medium generalized vertices [37], in this order. In simple terms, an in-medium generalized vertex corresponds to a closed nucleon loop that could contain an arbitrary number of meson-baryon vertices inserted. In the first summation on the right-hand-side of the previous equation the symbols i and n i are the chiral order and number of pionic lines of the i th meson-meson vertex, respectively. In the sum over the meson-baryon vertices, d i is the chiral dimension of the i th meson-baryon vertex, v i is the total number of mesons lines attached to it, including both pion and heavy meson lines, while ω i is the number of the latter ones only. The heavy meson lines correspond to auxiliary fields responsible for the local multi-nucleon interactions when taking their masses to infinity [1,38]. Eq. (2.1) counts every nucleon propagator as O(p -2 ) instead of O(p -1 ), as in the standard counting used in Baryon χPT [39,40]. In this way, the infrared enhancements associated to the large nucleon mass are taken into account from the onset. Despite baryon propagators are counted as O(p -2 ) it is important to stress that eq. (2.1) is bounded from below [2]. According to eq. (2.1) the number of lines in a diagram can be augmented without increasing the chiral power [1] by adding i) pionic lines attached to lowest order mesonic vertices, i = n i = 2, ii) pionic lines attached to lowest order meson-baryon

V ρ = 1 V ρ = 1 V ρ = 2 O(p 5 ) O(p 6 ) O(p 6 )
Figure 1: Contributions to the in-medium chiral quark condensate up to NLO or O(p 6 ). The scalar source with zero momentum is indicated by the wavy line and pions by the dashed ones. A wiggly line corresponds to the nucleon-nucleon interaction kernel (given in fig. 3) whose iteration is denoted by the ellipsis. vertices, d i = v i = 1 and iii) heavy mesonic lines attached to lowest order bilinear vertices, d i = 0, ω i = 1. In this way, resummations of infinite strings of diagrams are required, and these resummations signal the appearance of non-perturbative physics. As a major difference between this counting and the Weinberg one [41,42], the former applies directly to the physical amplitudes while the latter does only to the potential. It is also important to stress that ν in eq. (2.1) is a lower bound for the actual chiral power of a diagram, µ, with µ ≥ ν. The chiral order might be higher than ν because nucleon propagators, always counted as O(p -2 ) in eq. (2.1), could actually count as O(p -1 ) in some cases. If this is so, every O(p -1 ) baryon propagator increases the actual chiral order by one unit as compared with ν eq. (2.1). The important point is that all the diagrams with a chiral order µ are a subset of the diagrams with ν ≤ µ. We refer to [1] for more details on eq. (2.1), including its derivation.

The explicit chiral symmetry breaking due to the quark masses is incorporated by s(x) = M + δs(x) with M = diag(m u , m d ) the quark mass matrix. From the generating functional in the presence of external sources, Z(v, a, s, p), the quark condensate is obtained by partial functional differentiation

Ω|q i q j |Ω = - δ δs ij (x) Z(v, a, s, p)| v,a,s,p=0 , (2.2)
with q 1 = u and q 2 = d quarks. Notice that the quark condensate has a global minus sign compared to diagrams calculated by using ordinary Feynman rules. See ref. [37] for a derivation of the generating functional Z(v, a, s, p) in the nuclear medium making use of functional methods, although keeping only pion-nucleon interactions. We also refer to [21] for more details on the use of external sources in relation with in-medium χPT calculations.

The quantity we will calculate in the following is the in-medium correction Ξ to the chiral quark condensate given by m q Ω|q i q j |Ω = m q 0|q i q j |0 + m q Ξ , (

where m q is the mass of a certain quark flavor.

πN eq. (2.6). These are depicted in fig. 2. However, these diagrams are zero because of the antisymmetric isospin structure of the Weinberg-Tomozawa vertex, proportional to ε lmc τ c . The same type of pion is involved in the tadpole loop. This is clear for the diagram a) but also for diagram b) due to the diagonal structure of the vertex coupling the scalar source with two pions, eq. (2.10). As a result, both diagrams are zero when the pion indices are contracted with the antisymmetric tensor.

The in-medium nucleon propagator reads [START_REF] Fetter | Quantum Theory of Many-Particle Systems[END_REF],

G 0 (k) i 3 = θ(|k| -ξ i 3 ) k 0 -E(k) + i + θ(ξ i 3 -|k|) k 0 -E(k) -i = 1 k 0 -E(k) + i + 2πi δ(k 0 -E(k))θ(ξ i 3 -|k|) , (2.4)
with E(k) = k 2 /2m the non-relativistic nucleon kinetic energy. In this equation the subscript i 3 is the third component of the isospin of the nucleon, so that i 3 = +1/2 corresponds to the proton and i 3 = -1/2 to the neutron. The symbol ξ i 3 is the Fermi momentum of the Fermi sea for the pertinent nucleon. We consider that in vacuum isospin symmetry is conserved, so that all the (vacuum) nucleon and pion masses are equal. The proton and neutron propagators can be combined in a common expression

G 0 (k) = i 3 1 2 + i 3 τ 3 G 0 (k) i 3 . (2.5)
We make use of the Heavy Baryon χPT (HBχPT) pion-nucleon Lagrangian of [START_REF] Fettes | [END_REF] up to O(p 2 )

L (1) πN = N iD 0 - g A 2 σ • u N , L (2)
πN = N 1 2m D • D + c 1 χ + + c 2 - g 2 A 8m u 2 0 + c 3 u µ u µ + c 5 χ + - χ + 2 + . . . N (2.6)
where . . . denotes the trace in isospin space and the ellipses represent terms that are not needed here. The covariant derivative is given by

D µ = ∂ µ +Γ µ , where the chiral connection is Γ µ = 1 2 [u † , ∂ µ u]-i 2 u † (v µ + a µ )u -i 2 u(v µ -a µ )u † . The chiral vielbein is given by u µ = i{u † , ∂ µ u} + u † (v µ + a µ )u -u(v µ -a µ )u † .
The coupling of a scalar source to pions is also required. At lowest order in the chiral counting this coupling arises from

L (2) ππ = f 2 π 4 χ + + . . . (2.7)
The matrix field of scalar sources s(x) ≡ ||s ij (x)||, is included in the operator χ + [10],

χ + = u † χu † + uχ † u χ = 2Bs(x) . (2.8)
The parameter B is related to the strength of the vacuum chiral quark condensate and the weak pion decay constant f π , both in the chiral limit, via

Bδ ij = -0|q i q j |0 /f 2 π . (2.9)
The vertex coupling the scalar source s ij to two pions of Cartesian coordinates a and b can be evaluated straightforwardly from -L

ππ , eq. (2.7), with the result Bδ lm δ ab .

(2.10)

This vertex is diagonal both in the scalar source as well as in the pion fields. The nucleon-nucleon scattering amplitude at lowest order, O(p 0 ), is needed to match with our aim of calculating the NLO contributions to the chiral quark condensate in the nuclear medium. Note that the nucleon-nucleon scattering amplitudes first appear in diagrams with V ρ = 2 so that it starts to give contributions to the quark condensate already at NLO. These amplitudes originate from the Lagrangian with four nucleons, without quark masses or derivatives [42]

L (0) NN = - 1 2 C S ( N N)( NN) - 1 2 C T ( N σN )( N σN ) , (2.11)
and from the one-pion exchange with the lowest order pion-nucleon coupling. Its sum is represented diagrammatically in the following by the exchange of a wiggly line as in fig. 3. For explicit expressions of these amplitudes see [2], where we also compare with vacuum nucleon-nucleon scattering data. need the vertex with the scalar source s ij coupling to a pair of nucleons l, m. It can be readily worked out from -L

V ρ = 1 contributions

πN eq. (2.6), with the result

-2B 2c 1 δ ij δ lm + c 5 τ ji • τ lm , (3.1)
where we have taken into account that 2δ il δ jm -

δ ij δ lm = τ ji • τ lm . The diagram 1 then yields Ξ 1 = -2B [2c 1 δ ij (ρ p + ρ n ) + c 5 (τ 3 ) ij (ρ p -ρ n )] . = Ξ is 1 + Ξ iv 1 , (3.2)
where the ρ p and ρ n are the proton and neutron densities given by ρ p(n) = ξ 3 p(n) /3π 2 . Notice that the isospin breaking contribution proportional to c 5 only involves the Pauli matrix τ 3 , so that for i = j the contribution vanishes, as required. The contribution in eq. (3.2) proportional to c 1 is denoted in the following by Ξ is 1 and that proportional to c 5 by Ξ iv 1 . The superscripts is and iv refer to the isoscalar and isovector character of these contributions, respectively.

Let us proceed to the evaluation of the NLO diagrams 2 and 3 of fig. 1. Diagram 2 originates by dressing the in-medium nucleon propagator with the one-pion loop nucleon self-energy, Σ π . It is given by

Ξ 2 = -2iB d 4 k (2π) 4 e ik 0 η Tr 2c 1 δ ij + c 5 τ ji • τ G 0 (k) Σ π G 0 (k) . = Ξ is 2 + Ξ iv 2 , (3.3)
with the convergence factor e ik 0 η , η → 0 + , associated with any closed loop made up by a single nucleon line [START_REF] Fetter | Quantum Theory of Many-Particle Systems[END_REF]. The trace, indicated by Tr, acts both in spin and isospin spaces. Similarly as in eq. (3.2) the term proportional to c 1 is denoted by Ξ is 2 and that proportional to c 5 by Ξ iv 2 . The contribution from diagram 2 with only free parts in all the nucleon propagators involved, including that in Σ π , vanishes. This is discussed in detail in ref. [2] for similar diagrams that appear in the calculation of the in-medium pion self-energy and nuclear matter energy. Briefly, it follows just by closing the integration contour on the complex k 0 half-plane opposite to that where the poles lie. Another important point to keep in mind is that the contributions with only Fermi sea insertions in all nucleon propagators is part of the V ρ = 2 contribution of the crossed exchange part of diagram 4 in fig. 1. This is shown diagrammatically in fig. 3 of [2], where the two external pion sources should be replaced by the scalar source for the case at hand (similarly, see fig. 4 for diagram 4.) The different V ρ = 2 contributions are evaluated in section 4. Consequently, only those terms that involve simultaneously free-space as well as density-dependent parts of the nucleon propagators are considered here. For the isovector part the calculation of the contribution Π 5 in ref. [2] applies straightforwardly, just by replacing the vertex function. The same procedure applied to the isoscalar part drives to its cancellation, Ξ is 2 = 0. For the same reason as in [2] Ξ iv 2 turns to be of O(p 6 ) or N 2 LO. This is due to the appearance of the derivative of the free part of the nucleon one-pion loop, Σ π f , with respect to energy. This derivative is finally suppressed by one chiral order [2]. For diagram 3 in fig. 1 the scalar source is attached to the pion propagator involved in the one-pion loop nucleon self-energy. In the same way as discussed above for diagram 2, the contribution with only the free part for all nucleon propagators vanishes, while that with all nucleon propagators as in-medium insertions are accounted for by the exchange part of diagram 6 of fig. 1 in section 4. Then, we consider here the part of diagram 3 having one free nucleon propagator and the density dependent part of the other. Let us recall that the coupling of a scalar source to two pions was already given in eq. (2.10). There is a close relationship between diagram 3 of fig. 1 and the contribution to the nuclear matter energy from the one-pion loop nucleon self-energy, diagram 2 in fig. 17 of [2], denoted by E 2 . Of course, this is a requirement from the Hellmann-Feynman theorem. It is straightforward to check that

Ξ 3 = iBδ ij 1 2 d 4 k (2π) 4 e ik 0 η Tr G 0 (k) ∂Σ π (k) ∂m 2 π = Bδ ij ∂E 2 ∂m 2 π , (3.4)
with the derivative affecting only the explicit dependence of Σ π f on the pion propagator, and not including the implicit one from the nucleon mass dependence on it. One has to subtract the value of the one-pion loop nucleon self-energy at k 0 = 0 since we are using the physical nucleon mass. After performing the k 0 integration, one has the expression

Ξ 3 = -2Bδ ij d 3 k (2π) 3 θ(ξ p -|k|) + θ(ξ n -|k|) ∂Σ π f (ω) ∂m 2 π , (3.5)
where ω = E(k) and the partial derivative is given by

∂Σ π f (ω) ∂m 2 π = 3g 2 A 64π 2 f 2 π m 2 π 2ω 3 -4ωm 2 π -3πm 3 π + 3m 2 π √ b π + i ln ω + i √ b -ω + i √ b , (3.6
)

with b = m 2 π -ω 2 -i and → 0 + . Notice that Ξ 3 is an O(p 7 ) or N 2 LO contribution because ∂Σ π f /∂m 2 π = O(p 2
). The O(p) contribution from the term -3πm π is cancelled by that coming from 3π √ b. We then conclude that the only contribution with V ρ = 1 up-to-and-including NLO is given by Ξ 1 , eq. (3.2).

V ρ = 2 contributions

We now consider those next-to-leading order contributions to the in-medium chiral quark condensate that involve the nucleon-nucleon interactions. They are depicted in diagrams 4-6 of the last two rows of fig. 1.

Contributions Ξ 4 and Ξ 5

Diagrams 4 and 5 are analogous to diagrams a) and c) of fig. 4 in [1]. There it has been shown that those diagrams cancel each other. This argument was developed in [2] within a specific method for resumming the iteration of the wiggly lines for the non-perturbative nucleon-nucleon interactions. We expect that this cancellation also takes place for the case of the in-medium chiral quark condensate. However, the vertex coupling two nucleons with the scalar source, eq. (3.1), has both an isoscalar and an isovector term while for the aforementioned cancellation in the case of the pion self-energy only an isovector vertex was involved. Here we want to show on general grounds that the cancellation also takes place for the in-medium chiral quark condensate, following similar arguments as those in [1].

a) b) k 1 k 1 k 2 k 2 k 1 -q k 1 -q k 2 + q k 2 + q
Ξ 4 = -2iB d 4 k 1 (2π) 4 e ik 0 1 η Tr 2c 1 δ ij + c 5 τ ji • τ G 0 (k 1 )Σ NN G 0 (k 1 ) . (4.1)
The expression for Σ α 1 ,N N , corresponding to the self-energy of a nucleon with isospin

α 1 , is [1] Σ α 1 ,N N = -i α 2 ,σ 2 d 4 k 2 (2π) 4 G 0 (k 2 ) α 2 T σ 1 σ 2 α 1 α 2 (k 1 , k 2 )e ik 0 2 η . (4.2)
Here α 2 , σ 2 correspond to the running third components of isospin and spin, respectively, while σ 1 is the third component of the spin of the external nucleon. In addition, T σ 1 σ 2 α 1 α 2 (k 1 , k 2 ) refers to the elastic in-medium nucleon-nucleon scattering amplitude for

N α 1 ,σ 1 (k 1 )N α 2 ,σ 2 (k 2 ) → N α 1 ,σ 1 (k 1 )N α 2 ,σ 2 (k 2 ).
We also make use of the identity ∂G 0 (k)/∂k 0 = -G 0 (k) 2 . As a result, eq. (4.1) can be expressed as

Ξ 4 = -2B α 1 ,α 2 σ 1 ,σ 2 d 4 k 1 (2π) 4 d 4 k 2 (2π) 4 e ik 0 1 η e ik 0 2 η G 0 (k 1 ) α 1 2c 1 δ ij + c 5 (τ 3 ) ij (τ 3 ) α 1 α 1 G 0 (k 2 ) α 2 ∂T σ 1 σ 2 α 1 α 2 (k 1 , k 2 ) ∂k 0 1 , (4.3)
where in the last equation an integration by parts in k 0 1 has been performed. In order to see the cancellation between diagrams 4 and 5 of fig. 1 let us proceed similarly as in ref. [1], for the case of the in-medium pion self-energy, and consider first the case with only one reducible two-nucleon diagram, fig. 5. The contribution of diagram a), Ξ L 4 , can be readily worked from eq. ( 4.3) by reducing

T σ 1 σ 2 α 1 α 2 (k 1 , k 2 ) to its one-loop calculation. It results Ξ L 4 = -2B α 1 ,α 2 σ 1 ,σ 2 d 4 k 1 (2π) 4 d 4 k 2 (2π) 4 e ik 0 1 η e ik 0 2 η G 0 (k 1 ) α 1 2c 1 δ ij + c 5 (τ 3 ) ij (τ 3 ) α 1 α 1 G 0 (k 2 ) α 2 × ∂ ∂k 0 1 -i 2 α 1 ,α 2 d 4 q (2π) 4 V α 1 α 2 ;α 1 α 2 (q)G 0 (k 1 -q) α 1 G 0 (k 2 + q) α 2 V α 1 α 2 ;α 1 α 2 (-q) , (4.4)
where V αβ;γδ corresponds to the wiggly line with the first pair of labels belonging to the outgoing nucleons and the second pair to the in-going ones. Note that a symmetry factor 1/2 is included because V contains both the direct and exchange terms. In addition, V also will depend generally on spin. Both isospin and spin indices are globally indicated in the labels of V with Greek letters. The quantity in squared brackets in the previous equation is

T σ 1 σ 2 α 1 α 2 (k 1 , k 2 )
at the one-loop level. It is not necessary to consider T σ 1 σ 2 α 1 α 2 at the tree-level, consisting of the exchange of one wiggly line, fig. 3, because it is then independent on k 0 1 so that the derivative with respect to k 0 1 is zero. One can similarly write down the contribution from the diagram b) of fig. 5, denoted by Ξ L 5 . It reads

Ξ L 5 = 2B α 1 ,α 2 σ 1 ,σ 2 d 4 k 1 (2π) 4 d 4 k 2 (2π) 4 e ik 0 1 η e ik 0 2 η G 0 (k 1 ) α 1 G 0 (k 2 ) α 2 ∂ ∂k 0 1 -i 2 α 1 ,α 2 d 4 q (2π) 4 V α 1 α 2 ;α 1 α 2 (q) × 2c 1 δ ij + c 5 (τ 3 ) ij (τ 3 ) α 1 α 1 G 0 (k 1 -q) α 1 G 0 (k 2 + q) α 2 V α 1 α 2 ;α 1 α 2 (-q) , (4.5)
with the global symmetry factor 1/2 from closing the lines. The appearance of the derivatives with respect to k 0 1 is again due to the fact that the propagator attached to the scalar source appears squared. The loop integrals between the squared brackets in eqs. (4.4) and (4.5) are typically divergent. Nevertheless, the parametric derivative with respect to k 0 1 can be extracted out of the integral of eq. (4.5) as soon as it is regularized. Summing over all isospin states makes clear that the position of the vertex associated with the coupling of the scalar source to two nucleons, either inside or outside the squared brackets, does not yield a difference. For that one has to keep in mind that

T σ 1 σ 2 α 1 α 2 (k 1 , k 2 ) = T σ 2 σ 1 α 2 α 1 (k 2 , k 1 )
, due to the Fermi-Dirac statistics for a pair of two-nucleon states. Furthermore, notice that all the indices and four-momenta associated with the nucleons 1 and 2 are summed and integrated, respectively. In this way eqs. (4.4) and (4.5) mutually cancel. This process of mutual cancellation can be generalized to any number of two-nucleon reducible loops, using the same argument as given in [1] for the case of the in-medium pion self-energy. An n + 1 iterated #1 Alternatively, note that the term proportional to c1 in eq. (4.5) is the same independently of its location because is proportional to the identity matrix. That part proportional to c5 is purely isovector, as those terms stemming from the Weinberg-Tomozawa vertex in refs. [1,2]. As shown there only the states with I3 = ±1 give contribution so that c5(τ3) α 1 α 1 in eq. (4.5) gives rise to c5I3, which then trivially factorizes for the separate contributions of the proton-proton and neutronneutron states.

wiggly line exchange implies n two-nucleon reducible loops. The scalar source can be attached to any of them for Ξ 5 , while for Ξ 4 the derivative with respect to k 0 1 can also act on any of the loops. This is exemplified in fig. 6 for the case with two two-nucleon reducible loops. Hence, Ξ 4 + Ξ 5 = 0 .

(4.6)

The basic simple reason for such cancellation is that while for Ξ 5 there is a derivative acting onto the scattering amplitude, Ξ 4 involves an integration by parts in order to do so, which then introduces an extra minus sign. The previous cancellation is also explicitly obtained making use of Unitary χPT applied to nuclear matter as developed in ref. [2]. The intermediate result of [2] can be used straightforwardly by considering that instead of the Weinberg-Tomozawa vertex and pion-nucleon Born terms, used in the problem of the in-medium pion self-energy, one has the nucleon vertex of eq. (3.1) with the scalar source. The partial wave decomposition of Ξ 4 reads

Ξ 4 = -2B J, ,S,I α 1 ,α 2 (2J + 1)χ(S I) 2 d 4 k 1 (2π) 4 d 4 k 2 (2π) 4 e ik 0 1 η e ik 0 2 η 2c 1 δ ij + I 3 c 5 (τ 3 ) ij × G 0 (k 1 ) α 1 G 0 (k 2 ) α 2 m ∂T I 3 JI ( , , S) ∂A , (4.7)
with α 1 + α 2 = I 3 , the third component of the total isospin I of the two-nucleon state. Other symbols used are J the total angular momentum, the orbital angular momentum and S the total spin of the nucleon-nucleon pair. We also employ the kinematic variable A = 2ma 0 -a 2 , where a = (k 1 +k 2 )/2 and a is the three-vector made by the spatial components of a. For on-shell scattering A = p 2 , with p the threemomentum in the two-nucleon rest frame. For the term proportional to c 5 due to its isovector character only the difference between the proton-proton and neutron-neutron contributions survives (I 3 = 0 for proton-neutron pairs). For the expressions at O(p 6 ) and O(p 7 ) we plug into eq. (4.7) the derivative ∂T JI /∂A at leading and next-to-leading order, according to eqs. (5.24) and (5.25) of [2], respectively. For the contribution Ξ 5 the general partial wave decomposition yields

Ξ 5 = - 1 2 J, ,S,I α 1 ,α 2 (2J + 1)χ(S I) 2 d 4 k 1 (2π) 4 d 4 k 2 (2π) 4 e ik 0 1 η e ik 0 2 η G 0 (k 1 ) α 1 G 0 (k 2 ) α 2 D I 3 JI -1 • ξ I 3 JI • D I 3 JI -1 . (4.8)
The expressions for D I 3 JI at LO and NLO are given in eq. (5.25) of [2] and ξ I 3 JI is calculated order by order. For that one has to work out the partial wave decomposition of the two-nucleon reducible diagram with the scalar source attached to one of the nucleon propagators inside the loop. Following those calculations one obtains for the leading order (LO) and next-to-leading order (NLO)

ξ JI LO = -N (0) JI 2 • DL 10 , ξ JI NLO = DL (1) JI -L (1) JI + N (0) JI 2 L 10 , N (0) JI DL 10 , DL 10 = -4B 2c 1 δ ij + I 3 c 5 (τ 3 ) ij m∂L 10 ∂A , DL (1)
JI = -4B 2c 1 δ ij + I 3 c 5 (τ 3 ) ij m∂L (1) JI ∂A . (4.9)
The contribution Ξ 5 at O(p 6 ) and O(p 7 ) results when the just given expressions for ξ JI , in order, are inserted in eq. (4.8). It is straightforward to check that they exactly cancel with Ξ 4 accordingly.

Contribution Ξ 6

Let us now consider the calculation of diagram 6 of fig. 1, where the scalar source is attached to an exchanged wiggly line. Since the scalar source coupling to a local term is of higher order, it only couples to the pion exchange lines here. The appearance of the pion propagator squared with a zero momentum scalar source leads to its derivative with respect to the pion mass. The local term is independent of the pion mass, so we may act the derivative onto the explicit dependence on the pion-mass of the full nucleon-nucleon scattering amplitude. The implicit dependence of the latter on m 2 π due to that of the nucleon mass is the content of the diagrams 4 and 5 of fig. 1, which have been shown above to cancel mutually. We can write

Ξ 6 = 1 2 Bδ ij α 1 ,α 2 σ 1 ,σ 2 d 4 k 1 (2π) 4 d 4 k 2 (2π) 4 e ik 0 1 η e ik 0 2 η G 0 (k 1 ) α 1 G 0 (k 2 ) α 2 ∂T σ 1 σ 2 α 1 α 2 (k 1 , k 2 ) ∂m 2 π = Bδ ij ∂E 3 ∂m 2 π . (4.10)
Where E 3 is the contribution to the nuclear matter energy due to the in-medium nucleon-nucleon interaction at leading order, evaluated in ref. [2]. In order to proceed we have to evaluate the derivative of the nucleon-nucleon scattering amplitude with respect to m 2 π . We decompose the in-medium nucleonnucleon scattering amplitude in a sum over nucleon-nucleon partial waves T JI ( , , S), as previously done for Ξ 4 , eq. (4.7). We distinguish now between and , corresponding to the the initial and final orbital angular momentum, in this order. In terms of the nucleon-nucleon interaction kernel N JI ( , , S), the nucleon-nucleon partial wave T JI in Unitary χPT is given by [2] T JI ( , , S)

= 1 + N JI ( , , S) • L 10 -1 • N JI ( , , S) ≡ D -1 JI • N JI , (4.11)
with L 10 the nucleon-nucleon unitarity scalar function [2]. This function does not depend explicitly on the pion mass. We have also introduced D JI = 1 + N JI • L 10 . Eq. (4.11) can also be rewritten as

T JI = N JI -N JI • L 10 • T JI . (4.12)
Taking the derivative with respect to the explicit dependence on m 2 π on both sides of the previous expression leads to

∂T JI ∂m 2 π = [D JI ] -1 • ∂N JI ∂m 2 π • [D JI ] -1 . (4.13)
In terms of this equation the partial wave decomposition of Ξ 6 at NLO reads

Ξ 6 = 1 2 Bδ ij J, ,S,I α 1 ,α 2 (2J + 1)χ(S I) 2 d 4 k 1 (2π) 4 d 4 k 2 (2π) 4 e ik 0 1 η e ik 0 2 η G 0 (k 1 ) α 1 G 0 (k 2 ) α 2 ∂T I 3 JI ∂m 2 π LO . (4.14)
On the problem of evaluating and regularizing this amplitude we refer to the related detailed discussion in [2] for the calculation of E 3 .

Discussion and results

It follows from our calculation for the in-medium corrections shown in fig. 1 that the chiral quark condensate up-to-and-including NLO is given by m q Ω|q i q j |Ω = m q 0|q i q j |0 + m q (Ξ 1 + Ξ 6 ) .

(5.1)

Here we have taken into account that Ξ 2 and Ξ 3 are indeed one order higher (NNLO), while Ξ 4 + Ξ 5 = 0. Both Ξ 1 and Ξ 6 are clearly connected with the corresponding contribution to the nuclear matter energy, as required by the Hellmann-Feynman theorem [8,27,45],

m q Ω|q i q j |Ω -m q 0|q i q j |0 = m q 2 δ ij d d m + (τ 3 ) ij d d m (ρ m + E) , (5.2)
with m = (m u -m d )/2 and E the energy density of the nuclear matter system. This constraint is fulfilled in our case, where Ξ 1 , eq. (3.2), is the leading derivative with respect to m 2 π of the nucleon mass. #2 In turn, Ξ 6 corresponds to the explicit derivative of the nuclear matter energy density due to the nucleonnucleon interactions with respect to m 2 π , eq. (4.10). Notice, that the implicit dependence on m 2 π of the in-medium nucleon-nucleon interactions does not give any contribution to Ξ because of the discussed above mutual cancellation between Ξ 4 and Ξ 5 at NLO.

The term Ξ is 1 can be written directly in terms of the pion-nucleon σ term with the result

Ξ is 1 = -0|q i q j |0 σ(ρ p + ρ n ) f 2 π m 2 π , (5.3)
using the Gell-Mann-Oakes-Renner relation (GMOR) [46], m 2 π f 2 π = -2 m 0|q i q j |0 , valid at lowest order in the chiral expansion, see e.g. [47]. Regarding the numerical value of σ one has the earlier extraction σ = 45 ± 8 MeV of [22,23] or the one from the more recent partial wave analysis of pion-nucleon scattering [48] σ = 64 ± 7 MeV. In both cases the dispersion analysis of the nucleon scalar form factor of [22] to estimate the departure between the sigma-term and the πN scattering amplitude evaluated at the (unphysical) Cheng-Dashen point is used. In asymmetric nuclear matter there are additional isospin symmetry breaking terms proportional to c 5 stemming from diagrams 1 and 2, giving rise to the NLO contribution Ξ iv 1 , eq. (3.2). This contribution, which distinguishes between the ūu and dd quark condensates, are suppressed because they are proportional to the numerically small low-energy constant c 5 = -0.09 ± 0.01 GeV -1 [49]. For symmetric nuclear matter this contribution is zero since it is proportional to the difference of proton and neutron densities.

The energy density and energy per particle, E/ρ, have been calculated in [2]. In this reference, saturation of symmetric nuclear matter is obtained from the consistent chiral power counting, eq. (2.1). In addition, a remarkably good agreement with sophisticated many-body calculations [50] was obtained for the equation of state for neutron matter in terms of just one free parameter, a subtraction constant called g 0 . The appearance of this subtraction constant is due to the purely free part of the scalar twonucleon reducible loop, L 10 already introduced in eq. (4.11). Its free part, denoted by g(A), is a divergent function which requires a subtraction [2]. Taking D as the subtraction point one has

g(A) = g(D) - m(A -D) 4π 2 ∞ 0 dk 2 k (k 2 -A -i )(k 2 -D -i ) = g(D) - im 4π √ A -i |D| = g 0 -i m √ A 4π
.

(5.4) E 3 , eq. (7.36) in [2], depends on g 0 both implicitly, due to the dependence of the partial wave amplitudes T JI ( , , S) on g 0 , cf.eq. (4.11), and explicitly, in a linear manner. For the latter dependence the symbol g 0 was introduced in ref. [2] (to which we refer for a more detailed discussion on g 0 and g 0 ). For pure neutron matter both g 0 and g 0 are obtained with the value g 0 = g 0 -0.6 m 2 π , very close to their expected #2 At lowest order in the chiral expansion m 2 π = 2B m so that in eq. ( 5.2) one can make use of the operator m 2 π d/dm 2 π instead of md/d m. Figure 7: (Color online.) E/ρ for neutron matter (left panel) and for symmetric nuclear matter (right panel). The two (magenta) dotted lines correspond to ref. [36] and the solid ones are our calculation.

natural size g 0 -0.55 m -2 π . The case of symmetric nuclear matter requires some fine-tuning of g 0 , with a final value of g 0 -m 2 π , while g 0 is kept on its expected value of -0.55 m 2 π . With that, results in perfect agreement with experiments were obtained for the saturation density, energy per particle and compression modulus [2], see fig. 7 where the equations of state for neutron (left panel) and symmetric nuclear matter (right panel) are shown. The magenta lines are from the many-body calculations of refs. [36,50] employing to so-called realistic nucleon-nucleon potentials. The agreement between these references and our results is remarkable. We use our previous evaluation of E 3 for calculating Ξ 6 , eq. (4.10), by taking the derivative with respect to the explicit dependence of E 3 on m 2 π . It is important to stress that the in-medium quark condensate does not depend on g 0 because this parameter multiplies a quantity which is independent of the pion mass, denoted by Σ ∞ in ref. [2]. The latter originates from the limit p 2 → ∞ of the one-pionexchange tree level partial waves at lowest order and this limit is independent of m 2 π . Making use again of the GMOR relation, Ξ 6 eq. (4.10) can be written as

Ξ 6 = -0|q i q j |0 1 f 2 π ∂E 3 ∂m 2 π (1 + O(m 2 π )) . (5.5)
Putting together the previous equation with Ξ is 1 , eq. ( 5.3), and Ξ iv 1 , eq. (3.2), the ratio between the in-medium and vacuum quark condensate to NLO reads ρ (fm -3 ) ρ (fm -3 ) Figure 8: (Color online.) The ratio between the in-medium and vacuum chiral quark condensate, Ω|qq|Ω / 0|qq|0 , eq. (5.6), for neutron matter, left panel, and symmetric nuclear matter, right panel. Left panel: The (red) solid and (cyan) dot-dashed lines are our full results for Ω|ūu|Ω and Ω| dd|Ω , respectively. The (black) dashed and (blue) dotted lines correspond in the same order to the linear density approximation (Ξ 1 ) only. Right Panel: The (red) solid and (black) dashed lines are the full results (with g 0 = -0.97 m 2 π ) and the linear approximation, respectively. The (blue) dot-dashed line is the calculation for g 0 = -0.5 m 2 π . All the curves shown employ σ = 45 MeV [22].

Ω|q i q j |Ω 0|q i q j |0 = 1 - σ(ρ p + ρ n ) f 2 π m 2 π + 2c 5 (τ 3 ) ij (ρ p -ρ n ) f 2 π - 1 f 2 π ∂E 3 ∂m 2 π . ( 5
g 0 = -0.62 m -2 π , respectively. In fig. 8 we show the ratio Ω|qq|Ω / 0|qq|0 eq. (5.6) for pure neutron (left panel) and for symmetric nuclear matter (right panel). From the figure we observe that the leading order correction Ξ 1 , eq. (3.2), is the dominant contribution. This gives rise to the linear density approximation represented by the dashed and dotted lines in the left panel and by the dashed line in the right one. The NLO corrections rising from Ξ 6 , eq. (5.5), for the case of neutron matter correspond to the difference between the solid and dashed lines and between the dot-dashed and dotted lines in the left panel of fig. 8. The former couple of lines refer to Ω|ūu|Ω and the latter to Ω| dd|Ω . In this case the NLO corrections are small and amount to 4.5% of the LO ones at ρ = 0.3 fm -3 . The corrections increase with density, as expected, since higher three-momenta are available for larger Fermi momentum. For the case of symmetric nuclear matter the NLO contributions are larger, though still mild. They correspond to the difference between the solid and dashed lines in the right panel of fig. 8 and they are 10% and 20% of the leading correction for ρ = 0.3 and 0.5 fm -3 , respectively. The NLO corrections, Ξ 6 , tend to speed up the tendency towards a vanishing quark condensate in the nuclear medium (a signal of a possible restoration of chiral symmetry in nuclear matter). It is worth pointing out that the dependence on the subtraction constant g 0 of the in-medium quark condensate to NLO is just at the level of a few per cent. This is shown in fig. 8 by the dot-dashed line where g 0 = -0.5 m 2 π is used. For the reasons discussed, the quark condensate is significantly less dependent on g 0 than E/A. Were σ = 64 MeV used [48], the dominant linear density contribution will lead to a faster decrease of the in-medium quark condensate. E.g. at this level of approximation, the dashed line in the right panel of fig. 8 would cross the zero at already ρ ≈ 2ρ 0 . To this result one should add the difference between the solid and dashed lines corresponding to the contributions from the in-medium nucleon-nucleon contributions, Ξ 6 , which are independent of the value of σ taken.

The one-pion exchange plus the contact nucleon-nucleon interaction terms from L (0) NN , eq. (2.11), are fully iterated, as represented by the ellipsis in the diagrams 4-6 of fig. 1. In ref. [30] one-pion exchange is iterated only once, and no nucleon-nucleon contact interactions at the same chiral order are included. However, the contribution of ref. [30] from once-iterated one-pion exchange is notoriously larger than ours. Keeping only this extra contribution, together with the linear density one, ref. [30] finds that the quark condensate vanishes already at ρ 0.24 fm -3 , while in our case this happens for the larger ρ = 0.41 fm -3 (this values coincides very closely to that for which E/A becomes positive in the right panel of fig. 7). This indicates that iterating the interaction kernel and the inclusion of the nucleon-nucleon local terms from L (0) NN are significant. Being more specific, if for symmetric nuclear matter we keep only the first iteration of the interaction kernel N JI ( , , S), eq. ( 4.11), one observes little changes in the results for the in-medium quark condensate with g 0 = -0.97 m 2 π , but much larger ones for g 0 = -0.5 m 2 π . Thus, the importance of further iterating the interaction kernel depends on g 0 , while the full result is rather independent on g 0 as shown in fig. 8. #3 Note that in our approach, by construction, both one-pion exchange and local terms come together at the same order as required by the power counting. The iteration process also requires both contributions, as explained in ref. [2], in order to have a g 0 invariant nucleon-nucleon partial wave amplitudes for the exact solution. Other mechanisms for the calculation of the in-medium quark condensate were included in ref. [30]. They indicate a tendency towards a stabilization of the quark condensate in nuclear matter for densities above ρ 0 . However, let us mention that no chiral power counting is followed by the authors in ref. [30] but an expansion in the number of loops. The latter does not coincide with a chiral power counting expansion because of the infrared enhancement associated with the nucleon propagators, as discussed above and also referred to in [51]. In addition, the local nucleon-nucleon interactions are not treated consistently within a chiral power counting either. Then, we consider as an interesting future task to include the higher orders needed within our power counting and confirm, if possible, the far reaching results of ref. [30]. Other papers also find that the linear decrease with density of the in-medium quark condensate is softened for ρ ρ 0 [27,32,52]. However, important differences persist at the quantitative level. E.g. ref. [32] finds for densities above ρ 0 a much milder positive correction to the linear density decrease of the quark condensate than refs. [27,30,52]. Note that we obtain a realistic equation of state for symmetric nuclear matter and the quark condensate keeps its trend to vanish for higher densities, as also observed in ref. [32]. We agree on the observation performed in refs. [30,32] that the short-range nucleon-nucleon interactions effects are suppressed for the evaluation of the in-medium quark condensate which is dominated by long-ranged pion physics. This is a consequence of eq. (4.13). At the order we are working, if only the local nucleon-nucleon interactions from L (0) NN were kept the derivative of T JI with respect to m 2 π would be zero and Ξ 6 → 0. In the same way, the limit p 2 → ∞ is independent of the pion mass which suppresses the influence of the short-range one-pion exchange part. It follows from the left panel in fig. 8 that for the case of pure neutron matter the linear tendency of the quark condensate is only weakly modified by the higher order corrections. Similar results are obtained in ref. [31] as well. Let us mention in passing that the approach of the Munich group [31,53,54] has difficulties in providing a good reproduction of the equation of state for neutron matter while in our approach [2] it emerges in a quite straightforward way at NLO. taken into account by the diagrams 5 and 6. The remaining contributions, with one density dependent part for one nucleon propagator and a free one for the other, is suppressed by one order. The reason is because it implies the derivative of Σ π f with respect to energy. This already occurred for the diagram 2 of fig. 1 in section 3, see also refs. [1,2] for the case of the pion self-energy. For the diagrams 5 and 6 we find a mutual cancellation. In the same way as we already found that for the case of the isovector contributions to the pion self-energy of diagrams (b) and (d) in fig. 4 of [1]. Because of this there are no new contributions from the pion wave function renormalization (diagram indicated by π-WFR in fig. 9) beyond those already considered in ref. [21]. The driving mechanism for such cancellation has been explained here in section 4.1 for the case of the chiral quark condensate involving the mutual cancellation of Ξ 4 and Ξ 5 . Due to suppression and cancellation of new contributions, we find that there are no additional contributions to those already given in [21] eq. (4.24) for the in-medium pion decay.

Together with the relation for the chiral quark condensate, eq. ( 5.6), we can write down the Gell-Mann-Oakes-Renner (GMOR) relation for symmetric nuclear matter [21]

m2 π f 2 t = -m Ω|ūu + dd|Ω + δ 0 , (6.1)
where mπ is the in-medium pion mass and δ 0 corresponds to corrections that start at O(p 4 ) in vacuum χPT, which implies that the GMOR relation is only exact at lowest order. The stability of the GMOR relation under the in-medium corrections as well as the fact that it is the temporal coupling f t , and not the spatial one f s , the one involved in the GMOR relation has previously been reported in [17][START_REF] Wirzba | Hirschegg '95: Dynamical Properties of Hadrons in Nuclear Matter[END_REF][START_REF] Kirchbach | [END_REF][20][21] within the mean field approximation, and in [59] in the framework of QCD sum rules. Expanding the in-medium contributions to the pion mass, the pion decay constant and the chiral quark condensate explicitly we find

m 2 π f 2 π 1 + δ (2) m 2 π + δ (3) m 2 π + . . . 1 + 2δ (2)
fπ + 2δ

fπ + . . . =m 0|ūu + dd|0 1 + δ

Ξ + δ

Ξ + . . . + δ 0 , (

where δ

(i) m 2 π , δ (i)
fπ and δ

Ξ denote the in-medium corrections of the pion mass squared, pion decay constant and chiral quark condensate, respectively, and the superscript indicates the corresponding chiral order. The relative O(p 2 ) corrections on the left-hand-side of eq. (6.2), δ fπ , only appear for the charged pions and correspond to isospin breaking due to different proton and neutron densities in asymmetric nuclear matter [21]. These in-medium corrections to the pion mass and the pion decay constant vanish for symmetric nuclear matter. They also cancel if we take the average of the pion masses or accordingly the pion decay constant. For these reasons, we can neglect the relative corrections at O(p 2 ) for the consideration of the GMOR relation. As previously discussed, up to next-to-leading order the leading nucleon-nucleon contributions to the pion decay, and therefore to the decay constant, cancel in δ

(3) fπ . The same observation was already made for the pion self-energy in [1] and, therefore, contributions due to nucleon-nucleon interactions are absent in δ

Ξ has been calculated in section 3 and corresponds to Ξ 1 , eq. (3.2). In contrast to m2 π and f t there is a non-vanishing NLO in-medium correction to the quark condensate due to the in-medium nucleonnucleon interactions and given by Ξ 6 , eq. (4.14). Thus, δ

Ξ = 0. However, this is O(p 6 ), one order above the contributions discussed for the left-hand-side of eq. (6.2). Whence, it would be needed a full N 2 LO calculation for f t and m2 π in order to ascertain the stability of the in-medium corrections to the GMOR relation up to O(p 6 ), which is beyond the scope of the present work. On the other hand, as shown in [21], the pion-nucleon dynamics that gives rise to δ

(3) m 2 π , δ (3)
fπ and δ

Ξ for symmetric nuclear matter does not violate the in-medium GMOR relation, eq. (6.2). Then, we conclude that the in-medium corrections, including nucleon-nucleon interactions, do not spoil the validity of the Gell-Mann-Oakes-Renner (GMOR) relation up-to-and-including O(p 5 ) (or NLO) in our in-medium power counting scheme.

As remarked in ref. [1], our results cannot be applied when extrapolated to densities such that f t → 0, since than the corrections would be as large as the leading order term. The discussed result on f t , which confirms the one of Ref. [21], suggests that the vanishing of f t in nuclear matter happens for densities rather high, around 5ρ 0 0.8 fm -3 , considerably higher than those shown in figs. 7 and 8. Indeed, this density corresponds to a Fermi momentum ξ i ∼ 600 MeV, around the value of the chiral symmetry breaking scale.

Conclusions and outlook

Employing the in-medium power counting of [1] and the methods of Unitary χPT for taking into account the resummation of non-perturbative effects [2] we have calculated the chiral quark condensate up-to-and-including next-to-leading order, O(p 6 ) in nuclear matter. We have found an interesting partial cancellation between the diagrams involving the nucleon-nucleon interactions. In this way, those contributions that arise due to the leading quark mass dependence of the nucleon mass mutually cancel. This corresponds to the diagrams 4 and 5 in fig. 1. As a result, only the diagrams 6 in the figure, that stem from the quark mass dependence of the pion mass, survive. This is the reason why previous calculations have found that short range nucleon-nucleon interactions are suppressed for the calculation of the inmedium quark condensates, which is then dominated by the long-range pion contributions. Note that we did not exploit the Hellmann-Feynman theorem, but obtained our results explicitly from the generating functional, eq. (2.2). We conclude that the corrections are small for the quark condensate in the case of pure neutron matter. These corrections are more significant for the the case of symmetric nuclear matter. It is also worth pointing out that the full iteration of the nucleon-nucleon interactions reduces the force of such extra damping of the quark condensate as compared with other references. The dependence of our results on the subtraction constant g 0 is at the level of a few percent, much smaller than for the case of the binding energy [2]. On top of these nuclear effects a reduction of the present uncertainty in the pion-nucleon sigma-term would be most welcome.

We have also addressed the calculations of the pion decay constant and tested the stability of the GMOR relation against in-medium NLO in our chiral counting, that is, up to O(p 5 ). The nucleonnucleon contributions for the in-medium corrections vanish at this order, not only for the calculation of the pion mass but also for the in-medium pion decay constant. This implies that the expressions in ref. [21] for the coupling of the pion to the axial-vector current are right up to NLO, despite having considered the inclusion of nucleon-nucleon corrections. Because of this result we also find that inmedium contributions do not spoil the GMOR relation up to O(p 5 ), as in ref. [21]. The worked out NLO contributions to the quark condensate give rise to a new contribution to the GMOR relation at O(p 6 ) due to the nucleon-nucleon interactions that would require a N 2 LO calculation for the pion mass and weak pion decay constant in symmetric nuclear matter in order to check the stability of the GMOR relation. This would be an interesting future task.

Higher order calculations are of most interest as they would provide the important two-pion exchange and multi-nucleon forces. They would furthermore merge meson-baryon mechanisms with novel multinucleon contributions that can be worked out systematically within our EFT. The large impact of the ∆-isobar excitation to symmetric nuclear matter in [30], which is then partially compensated by the three-body interactions terms proportional to the low-energy constant c 1 , freezes the dropping of the inmedium chiral quark condensate at ∼ 2ρ 0 . This interesting fact requires confirmation within our power counting so as to keep all the terms contributing at the same chiral order, while keeping the full iteration of lowest order local and one-pion exchange diagrams. As commented above, the latter has a significant impact in the contribution at NLO. In addition, one should keep in mind that the proper way to address the issue of chiral symmetry restoration in the nuclear medium is the calculation of the temporal pion decay constant in the nuclear medium [21], which should be pursued for higher orders.

Figure 2 :

 2 Figure 2: These diagrams are zero because of the antisymmetric structure of the Weinberg-Tomozawa vertex coupling in the pion indices. For more details see the text.

First+Figure 3 :

 3 Figure 3: Leading order interaction kernel: the exchange of a wiggly line between two nucleons indicates in the following the sum of the O(p 0 ) local and the one-pion exchange contributions.

Figure 4 :

 4 Figure 4: The equivalence between diagram 3 and the crossed part of the one-pion exchange reduction of diagram 6 of fig. 1 is shown. The diagram in the middle is an intermediate step in the continuous transformation of the diagram on the left-hand-side to the one on the right-hand-side.

Figure 5 :

 5 Figure 5: Contribution to the chiral quark condensate with a two-nucleon reducible loop. The scalar source couples outside the loop for diagram a) and inside it for diagram b).

# 1

 1

Figure 6 :

 6 Figure6: After performing the integration by parts in eq. (4.3) the derivative with respect to k 0 1 acts onto the scattering amplitude. This gives a sum of derivatives acting on two-nucleon reducible loops, indicated by the crosses. When the derivative acts on a baryon propagator the latter becomes squared. In this way, the first diagram in the second row of the figure equals the one of the first row but with opposite sign and they cancel each other. The same applies to the second diagrams in both rows.

  ρ (fm -3 ) ρ (fm -3 )

. 6 )

 6 The input parameters are g A = 1.26, f π = 92.4 MeV, m π = 138 MeV, m N = 939 MeV and σ = 45 MeV. In addition the values of the subtraction constants g 0 that reproduce better the equation of state of symmetric nuclear matter and neutron matter are used. These values are[2] g 0 = -0.97 and 0

m 2 π

 2 as well. Regarding the right-hand-side of eq. (6.2), δ

  Furthermore, the nuclear matter ground state is denoted by |Ω , to distinguish it from the vacuum ground state |0 . The resulting contributions are denoted by Ξ i and are shown in fig.1. To determine the set of diagrams needed for the calculation of the in-medium chiral quark condensate up to NLO we proceed by increasing V ρ step by step in eq. (2.1). For each V ρ we then determine the possible configurations of vertices and lines according to eq. (2.1). The resulting diagrams are shown in fig.1. They can easily be identified by considering the corresponding vacuum diagrams. For V ρ = 1 the first diagram is the lowest-order contribution to the nucleon sigma-term. The rest of diagrams in fig.1are NLO. The diagrams 2 and 3 stem from the one-pion loop self-energy with the scalar source attached to a nucleon or pion propagator, respectively. For V ρ = 2 one can conveniently think of the nucleon-nucleon scattering in the presence of a scalar source in vacuum. Then proceed by closing the diagrams, which corresponds to sum over the states in the Fermi seas of the nucleons. The upper row of diagrams 4, 5 and 6 in fig.1involves the direct nucleon-nucleon interactions while the lower row originates from the exchange part. In the following we indicate by Ξ i the contribution to the in-medium quark condensate due to the diagram i in fig.1. One could think of similar Feynman

#3 However, for these two values of g0, the changes for E/A in symmetric nuclear matter when restricting to just one iteration of the interaction kernel are large as compared with the corresponding full result.

Acknowledgements

AL wants to thank Andreas Wirzba for useful discussions. This work is partially funded by grants MEC FPA2007-6277, Fundación Séneca grant 11871/PI/09 by BMBF grants 06BN411 and 06BN9006, EU-Research Infrastructure Integrating Activity "Study of Strongly Interacting Matter" (HadronPhysics2, grant n. 227431) under the Seventh Framework Program of EU and HGF grant VH-VI-231 (Virtual Institute "Spin and strong QCD").

... ... ...

Next-to-Leading Order

Next-to-Leading Order 2a 2b 3

...

5

... ... 6 4 Leading Order 1

Pion decay constant and the Gell-Mann-Oakes-Renner relation

Let us now consider the axial-vector current A i µ = q(x)γ µ γ 5 (τ i /2)q(x), with q(x) a two-dimensional vector corresponding to the light quarks fields and τ i the Pauli matrices. For notation we refer to ref. [21].

Due to the presence of the nuclear medium one should distinguish between the couplings of the pion to the spatial and temporal components of the axial-vector current, denoted by f s and f t , respectively. The fact that f t = f s in nuclear matter and some consequences thereof have already been discussed in [17][START_REF] Wirzba | Hirschegg '95: Dynamical Properties of Hadrons in Nuclear Matter[END_REF][START_REF] Kirchbach | [END_REF][20][21][55][56][57][58][59]. The calculations in [21] indicate a linear decrease of f t with density, f t = f π (1 -(0.26 ± 0.04)ρ/ρ 0 ), where f π = 92.4 MeV is the weak pion decay constant in vacuum and ρ 0 is the nuclear matter saturation density. This result indicates that it makes sense to use chiral Lagrangians in the nuclear medium up to central nuclear densities. However, the result so far has been obtained by considering pion-nucleon dynamics only, without including nucleon-nucleon interactions.

We depict the contributions to the process of the in-medium pion decay Ω|A j µ |π i in fig. 9. Diagrams 1-3 were already considered in ref. [21]. Regarding diagram 4 one has to take into account the same comments as previously given in section 3 concerning diagrams 2 and 3 of fig. 1. The contribution with only the free part of the nucleon propagators vanishes and that with only the density dependent part is


R Lazauskas
email: rimantas.lazauskas@ires.in2p3.fr

C Volpe
email: volpe@ipno.in2p3.fr

Low energy neutrino scattering measurements at future Spallation Source facilities

Keywords: numbers: 25.30.Pt, 97.60.Bw, 26.30.Jk, 26.30.Hj

In the future several Spallation Source facilities will be available worldwide. Spallation Sources produce large amount of neutrinos from decay-at-rest muons and thus can be well adapted to accommodate state-of-the-art neutrino experiments.

In this paper low energy neutrino scattering experiments that can be performed at such facilities are reviewed. Estimation of expected event rates are given for several nuclei, electrons and protons at a detector located close to the source.

A neutrino program at Spallation Sources comprises neutrino-nucleus cross section measurements relevant for neutrino and core-collapse supernova physics, electroweak tests and lepton-flavor violation searches.

Introduction

Spallation sources can be well adapted to accommodate innovative neutrino experiments.

Large amounts of neutrinos are produced at such facilities, from the decay of pions π + → µ + + ν µ and the subsequent decay of muons µ + → e + + ν e + νµ . On the other hand, very few electron anti-neutrinos are emitted via π -decay (typically at the level of 10 -5 ), due to the strong π -absorption in the spallation target. Spallation sources have already been used to perform a very interesting neutrino physics program in the past.

In particular, crucial results on neutrino oscillations have been obtained by the LSND [1,2,3] and KARMEN [4] collaborations at LANSCE in Los Alamos and ISIS at Rutherford Appleton Laboratory respectively.

These experiments have used a 167 (LSND) and 65 (KARMEN) tons scintillator detector located at about 17 meters from the source.

The same experimental setups have also been exploited to perform measurements of neutrino scattering on 12 C [5,6,7,8,9,10,11,12], 13 C and 56 Fe [13] as well as electroweak tests (lepton flavor universality [11], V-A structure in muon decay [14], Weinberg angle [15]) and lepton-flavor violation searches [16].

New spallation sources are being constructed, planned or under study. These include the Spallation Neutron Source (SNS) facility at Oak Ridge [17], the European Spallation Source (ESS) facility in Lund, the Japanese Spallation Neutron Source (JSNS) at JPARC [18] and the SPL at CERN [19]. Table 1 presents a comparison of some key characteristics of these sources.

Although their physics program is focussed on neutron and material science, these facilities will be ideal locations to host a low energy neutrino physics program of interest for particle physics ‡, neutrino astrophysics and nuclear physics.

The only technical alternative to produce controlled neutrino fluxes in the 100 MeV energy range is a low energy beta-beam facility [21]. This is based on the novel method of the beta-beams [22] that exploits the beta-decay of boosted radioactive ions. Detailed investigations of the low energy beta-beam physics potential have pointed out § a variety of interesting applications in nuclear physics, in the study of fundamental interactions and corecollapse supernova physics (for a review of beta-beams see [23]). Beta-beams have two main advantages compared to spallation sources: i) they produce collimated beams of both ν e and νe ; ii) the average energy of the neutrino beams can be varied. On the other hand, spallation sources will reach significantly larger neutrino production rates.

The determination of neutrino cross sections in a broad range of energies (several hundred MeV to multi-GeV range) is currently the focus of many experiments such as Minerνa [24] and Sciboone [25].

The MiniBOONE experiment is also providing interesting cross section measurements, like the quasi-elastic differential neutrino-carbon one [26]. Note that the MiniBOONE results have revealed an excess of the electron-like events for the several hundred MeV neutrino energy range [27].

Neutrino scattering experiments at Spallation Sources are complementary to such experiments since the neutrino fluxes are in the 50 MeV energy range (Figure 1).

Neutrino-nucleus cross section measurements with low energy neutrinos have timely applications, that go from the precise calibration of neutrino detectors to constraining neutrino-less double beta decay calculations.

Neutrino detectors based on nuclei are used for the observation of neutrinos from a future core-collapse supernova explosion, or from past explosions (the diffuse supernova neutrino background). Indeed neutrino-nucleus scattering is necessary for the identification of the electron neutrinos; while electron anti-neutrinos can be detected through proton scattering in water Cherenkov and scintillator detectors.

For example, a lead-based supernova observatory -the HALO project -is now planned at SNOLAB. The measurement of neutrino-scattering on lead in coincidence with 1 or 2 neutrons would be necessary since this observable has an interesting sensitivity upon the third neutrino mixing angle [28].

Large-size detectors are currently under study (GLACIER, MEMPHYS/UNO/Hyper-K and LENA) [29]. In these observatories the ν e from a core-collapse supernova or of the diffuse supernova neutrino background can be measured through scattering on argon [30], on oxygen and on carbon [31]. Concerning the observation of supernova neutrinos from past explosions in our galaxy, an original strategy is based on the geochemical measurement of 97 Tc produced by the interaction of electron neutrinos in 98 Mo ore [32].

To extract unambiguously such a contribution the ν-98 Mo and ν-97 Tc cross sections need to be precisely known, as pointed out in [33]. Neutrino-nucleus cross sections on stable and radioactive nuclei are also necessary to understand stellar nucleosynthesis and, in particular, to unravel the site of the r-process (see e.g. [34,35]), one of the major open questions in nuclear astrophysics (see e.g. [36]).

Finally a precise knowledge of the nuclear response to neutrinos is also crucial in order to calibrate some ingredients of the microscopic approaches currently used to predict neutrinoless double-beta decay (e.g. forbidden multipole contribution to the half-lives and their possible quenching) [37,38].

The observation of this lepton-flavour violating process represents a major discovery. Key information on the CP Majorana phases, on the electron neutrino effective mass and mass hierarchy can be extracted if this rare process is observed [39,40]. However this requires reducing the current discrepancies among the half-life predictions. In this respect a step forward can be made through neutrino-nucleus measurements, combined with other measurements on the candidate emitters (beta-decay, muon capture, charge-exchange reactions and the two-neutrino double beta decay) [41].

The theoretical description of neutrino-nucleus cross sections in the low energy range benefits from a variety of sophisticated models, including Effective Field Theories [42,43], the Continuum Random-Phase-Approximation (CRPA) [44,45], the Quasi-particle RPA (QRPA) [46] and projected QRPA [47], relativistic RPA [49], the Shell Model (SM) [46,48] and the Shell Model in the complex energy plane [50]. Regardless the high degree of sophistication achieved, important discrepancies remain between the predicted neutrino-nucleus cross sections. For example, in [47] it is shown that the calculations of the energy dependent cross section on 56 Fe differ as much as by factor 3-4 at a given energy, whereas the convoluted neutrino cross sections turn to be in agreement with the experimental data [13]. In [49] the comparison of the flux-averaged neutrino-lead cross sections, associated to supernova neutrinos, show very similar discrepancies.

The realization of precise measurements of the neutrino scattering cross sections for an ensemble of nuclei would help to pin down differences among the models, enabling accurate and reliable description of the isospin and spin-isospin nuclear response.

Coherent neutrino-nucleus scattering has never been measured as well. Such an experiment is a useful test of the Standard Model and to establish the feasibility of supernova neutrino detectors and dark matter searches based on this process. The corresponding cross section in the 50 MeV energy range can be as large as 10 -39 cm 2 . Its measurement is now planned at SNS (the CLEAR experiment) [51,52]. Finally let us mention the neutrino capture on radioactive nuclei that might open a completely new window on the Universe. This process could indeed be used to observe the cosmological neutrino background. This exciting possibility as pointed out in [53] and further investigated in [54].

Note that one can calibrate the corresponding cross sections using the inverse process, namely the beta-decay of the nuclei of interest, since the impinging neutrino energy is extremely low (the neutrino background has a temperature of 1.96 K).

The physics scope of spallation source facilities covers neutrino physics, neutrino astrophysics and tests of the Standard Model. Detailed investigations can be found in the literature (see e.g. [55]).

Ref.

[17] is a proposal made for the SNS facility presenting also an in-depth study of possible backgrounds and of detectors' design. In this paper, we present new predictions for neutrino-nucleus, neutrino-proton and electron scattering rates, motivated by the recently approved European Spallation Source facility. We emphasize general aspects, relevant for the cross section measurements and for low energy tests of the Standard Model. In particular we focus on a Weinberg angle measurement and Lepton-Flavour Violation (LFV) searches.

Note that the experimental uncertainties are in this case at the level of 50%. [ Table 2. Results on the number of events at a neutrino experiment based at a spallation source facility.

The events are calculated assuming 10 15 ν e /s, in a year (3 10 7 s), with a fully efficient 1 ton cubic detector. The columns correspond to the considered targets (first column), the rates at different distances d (meters) from the source, the material density (fourth column) and the flux-averaged cross sections in units 10 -40 cm 2 (last column).

Results on expected rates

We assume here a neutrino flux at the source of n ν =10 15 ν e /s and a fully efficient 1 ton cubic detector.

At this early stage, we present only very general estimates for the expected neutrino event rates, neglecting statistical and systematic errors coming from possible backgrounds. Therefore, our numbers of events can be easily scaled, for any neutrino production rate and experiment running time.

Nuclear excitations: Neutrino-nucleus charged-current event rates are given in Table 2, assuming different target nuclei. The detector location is varied, its front being set at distance d =10, 20 and 50 meters from the source. It is obvious that the detector should be placed as close as possible to the source in order to increase the rates, since the neutrino flux decreases as the inverse distance squared. At such short distances the detector finite size also impacts the number of events. To demonstrate this effect we present in Figure 2 the ratio of the neutrino flux averaged over the detector volume V , i.e. < φ > det = V dV φ(E ν )dE ν /V over the neutrino flux at the detectors front side (n ν /4πd 2 ), as a function of the distance from the source. We see that the bigger is the detector the larger is its finite size effect which disappears only at distances as far as about 60 meters from the source. We present rates for nuclei that can be found in detectors based on various technologies, namely carbon (detectors made of liquid scintillators, e.g. C 16 H 18 ), oxygen (for water Cherenkov), argon, iron, molybdenum and lead.

The corresponding neutrino-nucleus cross sections, averaged over the decay-at-rest (DAR) muon flux (Michel spectrum, Figure 1), are chosen as follows. We take the experimental flux-averaged cross sections for 12 C [13,10]. Our QRPA calculations with the Skyrme interaction are used for 16 O, 56 Fe, 100 Mo and 208 Pb from Ref. [56].

The 208 Pb cross sections with neutron(s) emission are from Ref. [56]. CRPA calculations of the 40 Ar cross sections are taken from [44].

As can be seen from Table 2 convolved DAR cross sections as well as the event rates are lowest for carbon and oxygen nuclei. This is mainly due to the high reaction threshold but, for oxygen, also due to the closed shell structure.

The most interesting event rates are obviously from detectors based on heavy nuclei. In order to understand the nuclear structure response, involved in the DAR fluxaveraged 3. Fraction (in %) of the flux-averaged cross section associated to states of a given multipolarity, with respect to the total flux-averaged cross section, i.e. σ J π / σ tot . Results are given for all positive and negative states, having total angular momentum J between 0 and 3. The first column gives the considered nucleus [56].

A X 0 + 1 + 2 + 3 + 0 - 1 - 2 - 3 - 16 O 0.
. cross sections, it is necessary to identify the different multipole contributions. Note that a precise knowledge of the nuclear transition matrix elements of states of high multipolarity is crucial to better constrain neutrino-less double beta-decay predictions [38].

The forbidden contributions are usually located at higher energies and can be best studied when one can vary the average energy of the impinging neutrinos.

In this respect low energy beta-beams constitute a powerful tool since the average neutrino energy can be increased by accelerating the ions at different boost factors.

On the other hand DAR neutrino fluxes from Spallation Sources have much higher intensities than the ones that will be achieved by low energy betabeams.

In [58] a comparative analysis of the multipole contributions to the neutrino-lead cross section at spallation sources and low energy beta-beams is made, showing that measurements at these facilities are complementary. Ref. [59] has first investigated neutrino-nucleus cross section measurements at a low energy beta-beam facility while the multipole decomposition of the corresponding flux-averaged cross sections are given in [56].

In Figure 3 and Table 3 we show the decomposition of the total cross section over the states of different multipolarity, that contribute to the cross sections.

For a closed shell nucleus as oxygen, the forbidden 1 -, 2 -excitations dominate the cross section.

The case of oxygen is exceptional: the allowed (Gamow-Teller and Fermi) transitions accounts for less than 2% of the total cross section. In most cases allowed excitations are the dominant contribution; while the forbidden one can be significant. For example it is at the level of 30% for 100 Mo and 208 Pb.

A Weinberg angle measurement: Neutrino detectors can also be used to study neutrinoelectron scattering. The expected number for ν e,µ -e and ν µ -e scattering are given in Table 4. Although the ν-e cross section is much weaker compared to those on nuclei, such events can be separated in the detector by considering forward scattering.

If the systematic errors are kept sufficiently low, one can use this measurement to extract non-standard contributions to the weak interaction (e.g. flavor-changing neutral currents effects) and perform a precise measurement of the Weinberg angle at low momentum transfer.

The current best value for ν e -e scattering is σ = 10.1 ± 1.1(stat) ± 1.0(sys)E νe × 10 -45 cm 2 [15]. Note that a possible measurement with low energy beta-beams can determine the Weinberg angle with a precision of 10 % if the systematic errors are kept below 10% [60].

The time-structure of the beam pulse can be crucial for the accurate determination

ν Figure 4.
Energy distributions of the ν-e events, in relative units, for a Weinberg angle sin 2 θ W = 0.21 (dashed), 0.23 (full lines). The upper (two) curves represent the total number of electron events, the next lines correspond to: the ν e -e events, followed by the ν µ -e and the ν µ -e ones.

of the Weinberg angle at spallation sources. Indeed, positive pion (π + ) decay produce neutrinos of ν µ , ν e and ν µ flavors, whose scattering on electrons have a very different reaction cross section dependence on the Weinberg angle, as can be seen in Figure 4. The ν e -e and ν µ -e cross sections increase for an increasing value of the Weinberg angle, while ν µ -e cross section decreases. Let us define the quantity : S=-dN(sin 2 θ W )

dsin 2 θ W | sin 2 θ W N(sin 2 θ W ) , (1)
that we will call the sensitivity, where N stands for the experimental observable to be used in determining the Weinberg angle sin 2 θ W .

It turns out that the measurement of the total number of neutrino-electron events has a sensitivity of S = 0.5 only. If the pulse length is made much shorter than the muon life-time, the ν µ -e events can be separated from ν µ -e and ν e -e ones. This enables one to study the quantity -as proposed for the SNS facility in [55,17] -

O=N(ν µ -e) N (νµ-e)+N (νe-e), (2)
whose sensitivity to the Weinberg angle is as much as S = 1.68. If however one disposes of a detector with a decent energy resolution the determination of the Weinberg angle can be improved even in the case of long pulses.

To this aim one has to determine the size of the jump in the electron energy window at E e = 29.7 MeV, the end point of the ν µ -e events.

In practice, this method can be realized by counting the number of events and fitting the electron energy distributions in two separate windows ¶).: one containing electron energies below 29.7 MeV, and the other comprising the events whose energies exceed 29.7 MeV.

Although such a procedure increases the statistical error in each window by roughly a factor of 2, compared to the total cross section measurement, the jump height turns out to have a sensitivity to the Weinberg angle as large as S =2.

Therefore the impact of the increased statistical error in determining the Weinberg angle is reduced by a factor of 4 compared to an experiment measuring the total cross section, due to the significant improvement of S.

A similar conclusion can be drawn on the effect of the systematic errors, related e.g. to the finite energy resolution of the detector. ¶ In general, there are almost 4-times more events in the energy window below E e = 29.7 MeV than in the above window. However in practice an important part of the low energy events are cut out due to backgrounds and to the detector sensitivity. The measurement made by the LSND collaboration have identified about 200 events as ν-e scattering using 167 tons of liquid scintillator at 17 meters from the target. If a 5 ton scintillator detector can be placed as close as 10 meters from the ESS source for example, one can expect in a running time of 1 year almost 5 times more events (Table 4).

Lepton-Flavour-Violation: Finally let us discuss the case of ν e scattering. If the amount of anti-neutrinos is as low as expected at the SNS facility, namely 10 -5 , one could use this very pure source of neutrinos to search for rare processes such as the Lepton-Flavour-Violating decay µ + → e + ν e ν µ . This search has been performed, e.g. by the KARMEN Collaboration using 65 tons of scintillator at 17 meters from the source which has given the current best limit for the branching ratio of 0.9 × 10 -3 [16]. This experiment requires a precise identification of the associated events, measuring the neutron through neutron capture on Gadolinium or on protons as done in [16]. Table 5 gives the expected ν e -p scattering rates assuming that as many ν e as 10 15 ν e /s are produced, these numbers must be scaled according to the suppression factor, that depends on the specific spallation source considered. The electron anti-neutrino-proton cross section has been calculated using the exact formula, valid to all orders in the anti-neutrino energy (E ν /m p ) expansion, as derived by Ya. I. Azimov and V. M. Shekhter [57].

Conclusions

New Spallation Source facilities are under construction, planned or under study. They will produce an intense flux of neutrinos. If these facilities are designed to accommodate a neutrino detector(s) close to the source, they will offer an unique opportunity to perform low energy neutrino scattering experiments. Neutrino detectors with different active material can be used to perform neutrino nucleus cross section measurements that have a variety of innovative and timely applications going from nuclear physics to neutrino astrophysics.

Neutrino-nucleus scattering data would create conditions for the breakthrough of the theoretical models, permitting to pin down some of their phenomenological ingredients. Besides, physics beyond the standard model can be tested through the searches for nonstandard contributions to neutrino electron scattering; while the identification of electron anti-neutrino scattering on protons can be used to improve established limits on rare lepton flavor violating decays, in particular µ + → e + + νe + ν µ . We have presented new predictions for these processes and emphasized some general aspects, hoping one day such measurements to be realized.

Figure 1 .

 1 Figure 1. Neutrino fluxes from muon and pion decay-at-rest at a spallation source.

Figure 2 .

 2 Figure 2. Effect of the finite size of a fully efficient cubic water Cherenkov detector: the curves show the ratio of the neutrino flux averaged over the detector volume < φ > det , normalized by the flux n ν /(4πd 2 ) at distance d, as a function of the distance from the spallation source (see text). The quantity n ν is the number of electron neutrinos per unit time at the source. The curves correspond to a 1 ton, a 10 ton and a 50 ton detector.

Figure 3 .

 3 Figure 3. Contributions of the positive and negative parity states, in percentage, to the number of events for oxygen (left) and lead (right figure) nuclei, as examples.

  /cm 3 ) < σ > DAR ν e -e 230 (223) 60 (58) 10 1.0 (0.992) 2.99•10 -3 ν µ -e 35 (34) 9.15 (9.0) 1.52 (1.5) 1.0 (0.992) 4.58•10 -4 ν µ -e 38 (37) 9.9 (9.6) 1.6 (1.5) 1.0 (0.992) 4.93•10 -4

10 20 50 ρ

 50 (g/cm 3 ) < σ > DAR(10

Table 1 .

 1 Comparison of characteristics of the past, present, and future Spallation Source Facilities in different regions of the world.

	Facility	Power	Proton energy Time structure Repetition rate
	LANSCE (USA) 56 kW	0.8 GeV	Continuous	N/A
	ISIS (UK)	160 kW	0.8 GeV	200 ns	50 Hz
	SNS (USA)	> 1 MW	1 GeV	380 ns	60 Hz
	JSNS (Japan)	1 MW	3 GeV	1 µs	25 Hz
	SPL (CERN)	4 MW	3.5 GeV	0.76 ms	50 Hz
	ESS (Sweden)	5 MW	1.3 GeV	2 ms (1.4 µs)	17 Hz (50 Hz)

  23 1.38 0.70 0.74 0.79 42.3 53.7 0.1 56 Fe 13.0 66.7 1.09 1.12 0.24 9.45 8.27 0.06 100 Mo 16.7 53.3 2.12 2.08 0.30 12.2 12.6 0.29 208 Pb 12.9 43.3 4.95 4.76 .317 14.1 17.1 1.14

	Table

Table 4 .

 4 Same as Table 2 (anti)neutrino scattering on electrons in water. The values given in parenthesis correspond to the results for electrons in C 16 H 18 as another example.

Table 5 .

 5 Same as Table2but for electron anti-neutrino scattering on protons.

Acknowledgements

The authors thank G. Drexlin and M. Mezzetto for useful discussions.