M Piai

M Round

Mass-degenerate heavy vector mesons at hadron colliders

We study the LHC phenomenology of two mass-degenerate heavy gauge bosons with the same quantum numbers as the Z and γ. We give a leading-order estimate of the number of events expected in Drell-Yan processes in terms of the parameters of the model (mass and coupling) and of the LHC machine specifications (integrated luminosity and energy). We consider the feasibility of measuring a forwardbackward asymmetry for various choices of the parameters and estimate the potential reach. We comment on how the results may affect future collider design and apply our results to a specific model of electro-weak symmetry breaking by way of example. ‡ In some strongly coupled models, with approximate conformal symmetry, a light dilaton might be present with interactions very similar to the Higgs of the SM [4]. We do not consider this possibility.

Introduction

The main purpose of the LHC is to discover the origin of electro-weak (EW) symmetry breaking and any associated new physics at the TeV scale.

One clean signal for the LHC would be the discovery of new, higher-mass copies of the standard-model (SM) gauge fields. For such a scenario, the LHC reach in observable masses and couplings is of interest. There are theoretical motivations for such searches. Many weakly-coupled extensions of the SM gauge group yield new heavy vector fields [1]. Such new fields also appear in little Higgs models [2]. Even strongly-coupled extensions, such as technicolor (TC), predict the existence of towers of heavy spin-1 fields [3].

In TC models EW symmetry is broken by a condensate, along the lines of chiral symmetry breaking in QCD. Thus these models do not contain a light elementary Higgs particle ‡. As a result the most promising test of these models at the LHC is a search for higher-mass spin-1 resonances. In particular, a copy of the full SU( 2)×U (1) realized as heavy gauge bosons would be distinctive. If the new bosons are nearly degenerate in mass then the model will satisfy the precision EW constraints [5,6], without suppressing the coupling to SM fermions. This situation would imply that the new boson resonances overlap at the LHC.

Once the LHC reaches its design specification, proton-proton collisions will scan over the energy range current data allow higher resonances to occur at. In generic models this is in the 1 TeV region, but in many concrete realizations the bound is above the 2 -3 TeV range [7,[START_REF] Csaki | For bounds in the context of the Randall-Sundrum models[END_REF].

In this paper, we consider the scenario in which a complete set of four spin-1 copies of the SM gauge bosons with heavy degenerate masses are produced in pp collisions. We concentrate on detection of the neutral heavy-Z and heavy-γ through their decay into muons. The phenomenology of the degenerate scenario has been largely ignored to date in the literature. This is in contrast to the single Z situation, see for example [START_REF]ATLAS detector and physics performance[END_REF] and references therein. For simplicity, we model the decay width of these particles by assuming it to be dominated by the decay into SM fermions. This avoids complications (and model-dependencies) arising when including the decay into light scalars and gauge bosons. If the underlying dynamics is based on an SU (N c ) gauge theory with large values of N c then the boson to boson decays are suppressed by the N c counting.

At LHC energies the SM background is mostly hadronic, so by looking at the neutral sector of higher resonances we exploit the very low background for muon and electron pair production. Within these assumptions we can study the event rate to leading order. Our aim is to identify what portion of parameter space will be testable at the LHC as a function of integrated luminosity and energy.

In the SM, a forward-backward asymmetry arises from interference terms between Z and γ exchange diagrams. It is a sizable effect, even in the TeV energy range. Measuring such an effect at a hadron collider is onerous but a method and its use at the LHC was surveyed for instance by Dittmar [START_REF] Dittmar | [END_REF]. It was shown that a deviation of the measured asymmetry from the SM prediction is a possible signature of the presence of a new, very broad resonance with the quantum numbers of a Z .

In contrast we will study a relativity narrow resonance. In our case, near the pole of some new resonance well into the TeV region the number of events is well in excess of the SM. So much so that the study of the di-lepton invariant-mass distribution will contain a clear peak. This will be a suitable discovery signature. Furthermore for all practical purposes the SM contribution to this peak can be ignored. In this case, one expects no significant asymmetry to be measurable at the peak from the SM.

If such a narrow peak is discovered in the data, but it results from the existence of a γ , in addition to the Z , then the interference between these two new states yields a large asymmetry. Measuring this asymmetry is hence a clean way to identify the presence of two distinct particles in the peak region, with masses so close together that they cannot be resolved in the invariant-mass distributions [11].

The organization of our paper is the following -We begin by introducing a model that encompasses the phenomenology of many specific models of EWSB. We then summarize some known results about decay widths and cross-sections. Then we proceed to study the Drell-Yan production of spin-1 resonances at the LHC. To illustrate the dependence of the event rate on model parameters we make a series of approximations to give a simple formula for the total number of events near a peak. We provide estimates of the discovery reach for two collider energies, 10 and 24 TeV.

We then derive a simple formula for computing the reach of forward-backward asymmetry measurement at a collider. At a 10 TeV machine the ability to search for a forward-backward asymmetry, of the type we are interested in, is limited to already excluded parameter space. Therefore we present data at 14 TeV (which is close to the border of feasibility) and 24 TeV. To illustrate the results we finish by concentrating on a particular model. Individual models give relations between the free parameters in our model-independent analysis and restrict the parameter space. This allows us to evaluate a typical reach scenario.

A few words of explanation about these choices are needed. The 10 TeV centre of mass energy analysis is representative of the LHC experimental program. As we will see, large integrated luminosities are needed for discovery and hence comparing to the 7 TeV -with low luminosity -initial phase of LHC running is not very useful for the cases we study. By 10 TeV energy, however, the reach is quite substantial. This is the reason for our conservative choice of energy. The 24 TeV energy analysis is somewhat speculative, but provides a way to gauge how an energy upgrade compares to a luminosity upgrade of the apparatus, in reference to the specific process we study.

Set-up and Rationale

This section reviews cross-section and asymmetry calculations as well as defining the notations used throughout the paper. Most of the material in this section is well-known, but it is useful to remind the reader about it, and to fix the notation used in later sections. We model the new neutral resonances by adding an extra set of SU(2) × U (1) gauge fields into the SM electro-weak Lagrangian. We assume that some mechanism spontaneously breaks the SU(2) 2 × U (1) 2 gauge symmetry to the SM SU(2) L × U (1) Y above the TeV scale, and that in the process a copy of the W , Z and γ acquire a common mass m. In general, the latter is not rigorously correct: the new heavy bosons would acquire different masses in any realization. However, the idea that the mass splitting is parametrically small is suggested by the precision electro-weak parameters. We will provide a simple illustration of this in the next sub-section. To parametrize the couplings of the new heavy gauge bosons to the SM currents we introduce a parameter R. This is defined so that if the coupling of a given SM boson to a given SM current is g SM then the corresponding new heavy boson has a coupling of

√

Rg SM to the same current. For example if the SM Z coupling to an electron current is g Z then the corresponding new boson, the Z , would have a coupling of

√

Rg Z to the electron current. The common mass m and the universal ratio R of the couplings are the only new parameters in the model.

It must be noted that in the paper we focus our attention on the neutral vectors only, setting aside the charged W . This choice is due to the fact that for the processes we consider, the kinematics of the leptonic final state can be reconstructed precisely when the intermediate state is neutral, while this is difficult for charged states, in which case one has either to deal with missing energy, or to consider final states with hadronic jets.

Mass Degeneracy

As previously noted, in this work we assume that the new heavy gauge fields are degenerate in mass. As an example, illustrating the rationale behind this assumption, we examine the precision parameter Ŝ [5,6]. Recall that on general grounds one can write the precision parameter Ŝ as

Ŝ = 1 4 g 2 f 2 v m 2 v - f 2 a m 2 a , (1)
with g the SU(2) L coupling in the SM, f and m being the decay constant and mass of the vector (v) and axial (a) mesons. In models where vector dominance holds the lightest among the states dominate the sum. Denoting them states as the techni-ρ meson and the techni-a 1 meson we truncate the sum to these two states.

Under this truncation, we write the parameters of these two states in terms of a constant a 1 and a mass splitting ∆m 2 ,

f 2 a 1 = f 2 ρ + a 1 f 2 ρ m 2 ρ ∆m 2 , (2)
m 2 a 1 = m 2 ρ + ∆m 2 . ( 3
)
Working with these definitions to leading order in ∆m 2 /m 2 ρ we see that,

Ŝ = 1 4 g 2 f 2 ρ m 2 ρ ∆m 2 m 2 ρ (1 -a 1 ) . (4)
Experimentally, Ŝ is found to be O(10 -3 ) [6]. This requires to implement some mechanism to obtain an acceptable value from eq.( 4). Choosing a 1 = 1 would require some sort of fine-tuning, hence we assume that a 1 be some generic O(1) number. Another possibility, that f 2 ρ /m 2 ρ 1, is realized in QCD. However, this ratio scales as N c in the large-N c limit (for N c the number of colors of the confining gauge theory). This makes it an unsuitable mechanism in a large-N c theory. Also, allowing for f 2 ρ /m 2 ρ 1 to be present would mean that the (techni-)rho mesons coupling to the fermions is heavily suppressed (the parameter R is very small), and its main decay is into longitudinally polarized standard-model vector mesons. This is in analogy to what happens in QCD where the rho meson decays into pions. Finally, in this case the rho mesons are actually strongly coupled, in the sense that their width is large. For all of these reasons we prefer not to use this suppression mechanism. Another possibility to satisfy the electro-weak precision constraints is to assume that ∆m 2 /m 2 ρ 1, small enough that Ŝ 10 -3 . Interestingly, this observation extends beyond the vector-dominance approximation, at least at large-N c , in the sense that Ŝ is suppressed if the mass splitting between new axial and vector meson states is small. This is the regime which inspires this present work.

Parton-level processes

In this section we begin the computation of the relevant cross-section, following standard textbook procedures. Initially this is done for a lepton machine then the results are extended to a hadron machine. This choice is made to highlight the physics of the model considered.

We find it useful to introduce the function P V ,

P V = (ŝ -m 2 V + im V Γ V ) -1 , (5)
where √ ŝ is the total energy, in the centre of mass frame, for the collision whose crosssection we compute, m V is the mass of the particle V and Γ V is the decay width of the particle V for V = γ, Z, γ , Z . This function enters the propagator of the boson V . Note that this requires computing the decay widths for the two new resonances. The new heavy gauge bosons can also decay into SM gauge boson pairs (e.g Z → W + W -). However the rate of such processes is very model-dependent, and at the same time assuming it to dominate the decay width might imply that the new gauge bosons be strongly coupled, as anticipated. In this study we assume such channels are negligible. This is equivalent to setting the 3-point and 4-point meson vertices to zero. This assumption is justified by the 1/N c counting if the underlying dynamics is similar to that of a large-N c gauge theory and the heavy spin-1 states are mesons of the strongly coupled sector, in analogy with the ρ and a 1 of a (large-N c ) QCD-like theory.

Within these assumptions, the γ and Z have decay widths proportional to those of the γ and Z in the SM. The constant of proportionality is the ratio between the SM and new gauge boson couplings. This is the parameter R introduced earlier on. Denoting the SU( 2) coupling of the standard model by g, the Weinberg angle by θ w , the fine structure constant by α (and the new heavy boson mass by m) the decay widths into SM fermions of the new states are,

Γ γ = 8 3 Rαm, (6)
Γ Z = Rg 2 m 4π cos 2 θ w 1 -2 sin 2 θ w + 8 3 sin 4 θ w , (7)
where we summed over all the SM fermions, including the top, neglecting the fermion masses. Typical values of the new heavy masses are O(1) TeV, thus the error introduced by neglecting fermion masses in the phase-space factors is negligible.

As an illustration of the model, we begin by removing the complications of pp collisions and take the case of e + e -→ µ + µ -. There are four possible diagrams. In addition to the SM channels of γ and Z exchange there is also γ and Z exchange. Following standard notation, introduce c V and c A for the vector and axial couplings defined as

c V ≡ T 3 -2Q sin 2 θ w , (8)
c A ≡ T 3 , (9)
where Q denotes the electric charge and T 3 the third component of weak isospin for the relevant fermion. We denote

Π γ (ŝ) = 4παQ i Q f (P γ + RP γ ), (10)
Π Z (ŝ) = g 2 4 cos 2 θ w (P Z + RP Z ) . (11)
for Q i and Q f the charges of the initial and final fermions in the Feynman diagram. Using this notation the angular 2 → 2 partonic cross-section can be written out as,

d 2 σ d Ω = ŝ 64π 2 A(1 + cos 2 θ) + B cos θ , (12)
for θ the angle of inclination with respect to the beam and Ω the solid angle of the scattering process. The A and B are defined as,

A = |Π γ | 2 + 2c 2 V (Π γ Π * Z ) + (c 2 V + c 2 A ) 2 |Π Z | 2 (13)
B = 4c 2 A (Π γ Π * Z ) + 8c 2 V c 2 A |Π Z | 2 (14)
where denotes the real part.

Similarly we can write the forward-backward asymmetry A F B as

A F B = 1 0 - 0 -1 d cos θ dσ/d cos θ 1 0 + 0 -1 d cos θ dσ/d cos θ . ( 15
)
We obtain,

A F B = 3B 8A . (16)
It is useful to combine the above expressions and rewrite the cross-section in terms of an angular function g( θ):

d 2 σ d Ω = 8ŝA 192π 2 g( θ) , (17)
with

g( θ) = 3 8 (1 + cos 2 θ) + A F B cos θ . ( 18
)
This form of the partonic cross-section, in terms of g( θ), is the most useful form for our discussion of asymmetry. Above an energy of 500 GeV the SM contribution becomes negligible for our purposes, as anticipated in the introduction. Events near the resonance of the new heavy states, ŝ m 2 , are dominated by the new states as long as mΓ V ŝ, since in this case

|P V | ŝ-1 , (19)
|P V | (mΓ V ) -1 . ( 20
)
Therefore unless R is very small the new resonances will dominate. The definitions of Π V given can now be approximated by dropping the P V terms.

Π V ∝ P V . (21)
We use this approximation to re-write the asymmetry near the new resonance (ŝ m 2 ) as

A F B = 3 2 c 2 A Γ γ Γ Z + 2c 2 V c 2 A Γ 2 γ Γ 2 Z + 2c 2 V Γ γ Γ Z + (c 2 V + c 2 A ) 2 Γ 2 γ . ( 22
)
This approximation is used in computing the numerical values of A F B that we work with later. This method avoids the complications introduced by accounting for the weak dependence of A F B on ŝ . Now we quote the correct formulae in the case of a hadron collider. The formulae for asymmetry and cross-section at partonic level can be amended by adding a factor for the number of colors N c , and changing to quark couplings. Denoting the quark/lepton couplings with a q/l superscript, A is modified to

A q→l = |Π γ | 2 + 2c q V c l V (Π γ Π * Z ) ( 2 3 ) + (c q V 2 + c q A 2 )(c l V 2 + c l A 2 )|Π Z | 2 .
Including the appropriate color factor, the partonic cross-section for

q q → l + l -is σ = 1 N c ŝ 12π A q→l , (24)
where the angular dependence has been integrated over. Adjusting B to account for quarks gives

B q→l = 4c q A c l A (Π γ Π * Z ) + 8c q V c l V c q A c l A |Π Z | 2 . ( 25
)
The asymmetry at the partonic level is hence

A F B = 3B q→l 8A q→l . (26)
In summary, our study of event rates focuses around the partonic cross-section eq.( 24). Our asymmetry calculations makes use of the angular partonic cross-section

d 2 σ d Ω = 8ŝA q→l 192N c π 2 g( θ) ( 2 7 )
The forward-backward asymmetry required in g( θ) is computed from eq.( 26) under the approximation of no SM background. Explicitly this means that

A F B = 3 2 c q A c l A Γ γ Γ Z + 2c a V c l V c q A c l A Γ 2 γ Γ 2 Z + 2c a V c l V Γ γ Γ Z + (c q V 2 + c q A 2 )(c l V 2 + c l A 2 )Γ 2 γ . ( 28
)

Drell-Yan processes

We will follow the standard recipe to model processes at a hadron collider. We consider the incident protons to be composed of partons holding fraction ξ of the total proton momentum p. Numerous sets of parton distribution functions (PDF) are available through Fortran code [14]. For this paper we choose the Fermi02 PDF set [15]. For practical purposes, one can treat the proton and its constituent partons as massless.

The two beams of protons each have energy √ s/2. The energy of a parton-parton collision ŝ is given, in terms of the two fractions ξ 1 and ξ 2 , by ŝ = ξ 1 ξ 2 s.

(29)

If, for a parton x, the PDF is denoted by f x (ξ) we can follow convention and write the pp → l + l -cross-section as,

d 2 σ dξ 1 dξ 2 = q [f q (ξ 1 )f q (ξ 2 ) + f q (ξ 2 )f q(ξ 1 )]σ. ( 30
)
To fix the total energy √ s of the pp collision, we impose the condition of Eq. ( 29). This means we may define the parton luminosity L q to be

L q ≡ 1 0 δ(ŝ -ξ 1 ξ 2 s)[f q (ξ 1 )f q(ξ 2 ) + ξ 1 ↔ ξ 2 ]dξ 1 dξ 2 , = 2 s 1 ŝ/s 1 ξ f q (ξ)f q ŝ sξ dξ (31)
and write the full cross-section summed over all quarks as, dσ dŝ = q σL q .

(32)

Finally we can write the total number of events N in some range of ŝ at a machine with integrated luminosity L,

N = L dσ dŝ dŝ. (33)
We allow √ s to vary between 10 and 28 TeV. The coupling ratio R will be allowed to vary between 0 and 1, where the gauge boson coupling is certainly perturbative. Then the mass range will be taken as lying between 1 and 4 TeV. Analysis will also require the assumption that decay widths are small compared to the resonance mass. Inspection of the decay widths, under the limits on R and m, shows this condition is satisfied.

Asymmetry measurements at a hadron collider are made more difficult (in comparison with a leptonic machine) by the unknown directions of the initial quark and anti-quark. However, at high rapidity there is a much greater chance that the quark carries the bulk of the momentum. Thus if data near a resonance in the invariant-mass distribution are cut so that only events with a rapidity Y greater than some value Y c (|Y | > Y c ) are used, then there is a corresponding degree of certainty regarding the quark directions. This method is used and explained more fully by Dittmar [START_REF] Dittmar | [END_REF]. In summary, the magnitude of the rapidity controls the probability of misidentification of the quark/anti-quark directions, while the sign of the rapidity represents whether the quark was from the left or right in the collider. This methodology of inserting cuts, Lorentz boosting with the rapidity modulus to the muon centre of mass frame and finally using the sign of Y to define the angle consistently allows one to infer an asymmetry. Such an asymmetry is the remnant of the partonic asymmetry.

Model Independent Study

Event Rate

In this section we introduce a set of approximations that are useful in order to study the dependence of observable quantities (such as number of expected events and forwardbackward asymmetry) on the parameters of the model and on the machine specifications. The numerical part of the analysis will, however, use the complete equations and PDF distributions without approximations. From the approximations made we gain insight into how event rates behave as a function of model parameters.

Using standard electro-weak relations the coupling g can be written in terms of α and sin 2 θ w 0.22. Hence, near the new resonance:

Π γ Π Z - 3 2 Q q 1 -2 sin 2 θ w + 8 3 sin 4 θ w -Q q . ( 34
)
Using Eqs. ( 23)-( 24) we can comment on the relative strength of the γ and Z contributions to the total cross-section. Because c l v 1, the cross term in Eq. ( 24) is suppressed. Also, in the last term of this equation

c f 2 V c f 2 A 1/4,
and hence the contribution of the γ is a factor of ∼ 16Q 2 q larger than the contribution coming from the Z . All of this, together with the fact that the PDF for the up-quark and downquarks are the most important, and that the coupling of the up quarks to γ is four times stronger than the coupling of down quarks, means that the most significant contribution to the total event rate is given by the subprocess uū → γ → µ + µ -. The next largest contribution to the event rate is from the cross term between the new photon and Zlike states (which is approximately half the value of the leading contribution) and is responsible for the asymmetry we discuss later.

Focusing on the u-quark PDF, we may attempt to approximate it with some form of expansion. The only relevant dimensionless combinations are ŝ/s and m 2 /s. As we are looking at the peak we can replace the square mass with ŝ so that only one relevant dimensionless combination remains. From the definition of L the overall dimension is 1/s. If the mass of the resonance were to increase one would expect the event rate to decrease. It is also reasonable to think increasing the machine energy s will increase the event rate. We use an approximate form of,

L u ∼ 1 s ŝ s n , (35)
and consider various n so as to obey the above criteria and then numerically fit the constant of proportionality. The minimal n would be -2 to fulfill our expectations for the behavior of L . Viewing our expression as the leading term in some expansion a polynomial can be added to increase accuracy. We find that an acceptable approximation is,

L u ∼ s ŝ2 • 1 1 + a 1 (ŝ/s) 2 + a 2 (ŝ/s) 4 + . . . , (36)
provided √ ŝ ∈ [1,4] TeV. We then go through a numerical fit of the constants a 1 and a 2 . In the range √ s ∈ [START_REF] Dittmar | [END_REF]28] TeV just those two terms are needed. We are interested in the event rate N in the peak region

N ∝ q dσ dŝ dŝ, L u σu dŝ. ( 37
)
Approximating L u with its value at the pole (which is allowed since the resonances are very narrow), and using

ŝ (ŝ -m 2 ) 2 + m 2 Γ 2 γ dŝ 2π m Γ γ . ( 38
)
yields

L u dσ u dŝ dŝ L u (m 2 )R 2 2π m Γ γ (39) ∝ RL u (m 2 ) , (40)
where the SM background from γ and Z has been neglected. Finally, the contribution of up-quarks to the total rate is,

N u ∝ sLR m 4 • 1 1 + a 1 m 4 /s 2 + a 2 m 8 /s 4 , (41)
up to a proportionality constant that contains the electro-weak couplings and the normalization of the PDF distribution.

To correct for the neglected contributions of d d events and Z exchange we can consider the ratio of the found rate N u to a more complete expression involving uū and d d events with γ and Z exchange. For notational simplicity let us absorb the coupling coefficients into the definition of P V briefly and write the ratio η as,

1 η ≡ L u ŝ|P γ | 2 dŝ L u ŝ|P γ + P Z | 2 dŝ + L d ŝ|P γ + P Z | 2 dŝ . ( 42
)
An expression similar to Eq. ( 36) describes, with different choices of the coefficients, L d . Then we find that a very good approximation is given by

1 η = 1 + b 1 (ŝ/s) 2 + b 2 (ŝ/s) 4 1 + a 1 (ŝ/s) 2 + a 2 (ŝ/s) 4 (1 + c 1 m + c 2 R) ( 4 3 )
for some b i and c i that are combinations of the up and down coefficients. The total event rate can be written, up to an overall constant, as

N N u η. ( 44
)
We fix the parameters, a 1 , a 2 , b 1 , b 2 , c 1 , c 2 plus an overall normalization with the direct numerical computation using the PDFs. We estimate the error hence introduced between the estimated event rate and the direct computation to be about 5%. Discovery of the new resonance requires detecting a minimum number of events N at a machine integrated luminosity L. The approximate event rate can be solved for R, given m at various machine energies and integrated luminosities, hence producing Figs. 1 and2. These two plots show how the experimental reach changes when improving the energy of a hadron collider, in comparison with increasing only its luminosity. We conclude this subsection with a couple of examples, to help the reader in interpreting the plots Suppose one considers N = 10 events to be necessary for discovery of a new resonance. Setting N = 10, if one has data corresponding to a 40 fb -1 integrated luminosity, then the line N/L = 1/4 fb indicates the minimum coupling R required to see such an effect. In the case of a 10 TeV machine, for a new resonance with mass m = 3 TeV, then the minimum R is around 0.4. Therefore any new boson with coupling ratio R > 0.4 and m < 3 TeV will produce at least 10 events in the data set obtained with 40 fb -1 integrated luminosity.

Changing the machine energy from 10 TeV to 24 TeV means moving from Fig. 1 to Fig. 2. At 10 TeV, by improving the integrated luminosity by a factor of 20, one may move from a reach in masses below 2.2 TeV to roughly 3.3 TeV, provided R = 1 (see the leftmost and rightmost lines in Fig. 1). Whilst improving the collision energy takes the reach to well above 4.0 TeV, even with the no improve in integrated luminosity, for the same R = 1 (see the leftmost curve in Fig. 2.

Forward-Backward Asymmetry

In the present study we limit ourselves to theoretical considerations only, so that we may ascertain how feasible asymmetry measurements are from a theoretical viewpoint. In fact one may argue that, due to large PDF uncertainties, a full scale simulation of the experiments is overkill before LHC running. It would be interesting to see a detailed simulation once LHC data becomes available, and new data about the PDFs is available.

We restrict our attention to two main sources of error in the asymmetry extraction algorithm: the potential for misidentification of the quark direction and finite number of events passing the rapidity cut Y > Y c . We set aside experimental errors, coming for instance from the measurement of the energy of the final leptons.

Consider the angular distribution in Eq. ( 18) as a probability distribution function. This distribution has a mean defined as

θ = θg( θ) sin θd θ = π 4 (2 -A F B ) , (45)
and a variance defined by

σ 2 p ≡ θ2 -θ 2 g( θ) sin θd θ . ( 46
)
Interestingly, A F B turns out to be just proportional to θ . It is possible to estimate the deviation of the mean θ from its true value as σ p / √ n -1 where n is the number of events used to study the angular distribution. Hence we estimate the difference between the true and measured asymmetry due to statistics to be,

∆A stat F B = 4σ p π √ n -1 . ( 47
)
Here n should be thought of as representing the number of events remaining after a cut in rapidity is applied to the total number of events N .

Secondly there is a probability of quark misidentification introduced by the methodology. The assumption that the initial quark direction be always the one of the out-coming γ or Z , implies a possible misidentification θ ↔ π -θ. Let the probability of misidentification be p(Y c ), then in a sample of n events np(Y c ) are misidentified. Let n F (n B ) events travel in the forward (backward) direction in the collider. Then n = n F + n B and the forward-backward asymmetry is defined as (n F -n B )/n in this notation. The worst case scenario would be that all np(Y c ) events were placed in n F but should have been placed in n B or vice versa. In this case the difference in true and measured asymmetry would be,

∆A sys F B = ± 2p(Y c ) ( 4 8 )
Adding the misidentification and the statistical together we can define a 'total misidentification' of,

∆A F B = 4σ p π √ n -1 + 2p(Y c ), ( 49
)
We conservatively add the two linearly. Now we define P (Y c ) to be the fraction of events with |Y | > Y c , i.e. the probability an event survives the chosen rapidity cut. Then we can relate the number of events in the peak, N , to the number used for the asymmetry extraction, n, through P (Y c ) by,

n = P (Y c )N. ( 50
)
Suppose we are interested in measuring an asymmetry at some level of significance. To do this introduce E and require that ∆A F B < E . We must impose E > 2p(Y c ) so that the chosen cut value is capable of producing data of sufficiently low misidentification. This implies that we require,

n > 4σ p π(E -2p(Y c ) 2 + 1 (51)
or equivalently,

N > 1 P (Y c ) 4σ p π(E -2p(Y c ) 2 + 1 (52)
total events. For the model considered, at the partonic level the asymmetry is A F B -0.5 in the case of uū → µ + µ -, while for d d events the asymmetry would be A F B -0.15. (For comparison, SM exchange plus only one new resonance the asymmetry vanishes up to 3 decimal places.) The measured asymmetry will then be some average of the two. Using what we showed before, namely that the uū events give a larger contribution to the total rate, we estimate the combined asymmetry to be A F B ∼ -0.42. Therefore seeing an asymmetry at a collider would indicate that there are two particles in the new neutral resonance.

To measure such an effect at a collider one would require a 'total misidentification' E < 0.14 in order to conclude, at the 3σ level, an asymmetry is present. Consider the case that all events happen to fall precisely at the rapidity cut Y c . Then the probability p(Y c ) is calculable, yielding a conservative estimate: with real data having |Y | > Y c one can in principle use a larger value of p(Y c ). The fraction P (Y c ) can also be computed.

Optimal choice of Y c for a minimal N can be achieved by writing a simple minimization algorithm. A high Y c gives high quality data (i.e. contaminated by small misidentification systematics) with low p(Y c ) but few events will pass such the cut, reducing also P (Y c ). The converse will be true for a low cut Y c . We used an optimization algorithm finding a Y c such that the required number of events has been minimized.

We compute the minimum number of events required for a 3σ uncertainty at given mass. Using the approximate event rate given by eqs. ( 43) and ( 44) we convert the number of events into a statement about R. The results of this process are shown in Figs. 3 and4. Figure 3 shows that large increases in integrated luminosity lead to modest gains in reach, compared to the gain obtained by going to larger energies, as shown in Fig. 4. One can also see from fig. 3 that in order to test the interesting range m > 2 TeV, one requires a very high integrated luminosity of 100 fb -1 . At a 10 TeV machine it is not possible to measure an asymmetry of a neutral heavy gauge boson in this range of most interest, and hence we omit showing the related plot.

PDF Uncertainties

So far the uncertainty on the PDFs themselves has been completely ignored. The Fortran module supplied by [14] has the capability to compute PDF uncertainties. As the computational cost to produce uncertainties with the package is very large, we use an approximate expression which captures the essential features of the PDF uncertainties. Namely, the quark PDFs are well known up to fractions of around 0.5, then they become poorly understood while the anti-quark PDFs are poorly understood across all of the fraction range. Two piecewise functions, one for the u and one for the ū, approximate this behavior

∆u(ξ) u(ξ) = 0.05 if ξ < 0.5 0.8ξ -0.35 otherwise (53) ∆ū(ξ) ū(ξ) = 2.25ξ if ξ < 0.8 25ξ -18.2 otherwise ( 54
)
These uncertainties are approximations to the results given by using the package [14] for our chosen PDF set [15] and the results broadly agree with those published by the CTEQ collaboration for their PDF sets [16]. Even if an event is not identified as being uū or d d in our analysis, the uū dominates events and hence the uncertainty indicated will offer a good guide as to the situation.

As the asymmetry extraction methodology involves cutting the data for large Y , it probes the region with a smaller anti-quark fraction and a larger quark fraction. If the quark fraction gets above 0.5 the uncertainty grows rapidly. At a 10 TeV collider this The minimum ratio R between couplings of the new and SM boson to SM currents, such that at the 3σ level (see text for definition) a forward-backward asymmetry of -0.42 is detectable, plotted for various integrated luminosities at a collision energy of √ s =24 TeV and as a function of the heavy boson mass. The leftmost line is for an integrated luminosity of 1 fb -1 , the middle line for 10 fb -1 and the rightmost 100 fb -1 .

effect will come in faster as high fractions are required to make large invariant masses while the effect will be slower to appear in a 24 TeV collider.

In general, most events in a peak will have low rapidity. As a result we expect far lower uncertainties for event rate measurements compared to asymmetry measurements.

An Example Model

In order to illustrate the results of the model-independent study it is useful to apply them to a specific model. For example in the AdS/TC model in [17] a 5D model is formulated in AdS space and bounds on the (R, m) parameter space are discussed.

In this work, several qualitatively different regions of the (R, m) parameter space are identified. Firstly there is a region where the dominant decay mode of the techniρ particles is into SM gauge bosons, and not SM fermions. This region is marked as invisible in figs. 5 & 6 because the branching ratio into fermions would strongly suppress the Drell-Yan process of interest here. There is then a second region already excluded by precision measurements. Finally the constraint R 0.65 comes from specific details of the model.

Putting this information into a plot and overlaying the search capabilities of a collider produces Figs. 5 and6. At the very high integrated luminosity of Fig. 5 most of the parameter space will be tested for the existence of new resonances. However, there is no reach in terms of asymmetry. Thus it will be hard to conclude whether there is more than one particle responsible for the peak. In contrast, with the higher collision energy of Fig. 6, a very large fraction of the parameter space presents a testable asymmetry.

Both Figs. 5 & 6 include a 1σ PDF uncertainty marked by thin red lines and a shading between. The lower of these two lines falls below the axis and is barely visible. It is clear that the event rate uncertainty is much smaller in the 24 TeV case with the cone barely visible on the plot. As for the asymmetry, the uncertainty at 14 TeV grows much faster for those masses where the 14 and 24 TeV lines are both able to reach. In the case of the 14 TeV curve, the uncertainty in coupling is 10 times the coupling estimate by 2 TeV. In comparison the coupling uncertainty will not reach a factor of 10 until 3.5 TeV in the 24 TeV machine.

For the representative point m = 2.6 TeV and R = 0.4 we simulate the asymmetry and see the rapidity cut in action. For simplicity, we assume that only uū events contribute at the partonic level. As explained, this is the dominant subprocess, although as a check simulations including d d were run which confirmed expectations or lowering the asymmetry mildly. In this limit, the asymmetry is A F B -0.5.

First let us look at the events with √ s = 24 TeV. We apply a cut rapidity Y > Y c = 1.0. In this way the misidentification due to the methodology is relatively small ( 0.1). Also, the number of events that pass the cut is large enough as to make the statistical element very small (∼ 0.03). As a result, the simulated set plotted in Fig 7 clearly shows a large asymmetry, compatible with the true one, and ∼ 4σ away TeV and an integrated luminosity of 100 fb -1 for the specific model in the text. Orange (shaded) regions of the plot represent areas of the specific model that are either already excluded, where Drell-Yan production is suppressed or not accessible by the specific model. The (thick) blue line represents the minimum coupling ratio R required for at least 10 events to be observed. (Thin) blue lines follow the contours of an upper and lower 1σ PDF uncertainty . The region between the upper and lower 1σ uncertainties is shaded blue. The (thick) red line represents the minimum coupling ratio R for an asymmetry to be visible at the 3σ level (see text for definition). (Thin) red lines follow the contours of an upper and lower 1σ PDF uncertainty (the lower line largely falls below the axis). The region between the upper and lower 1σ uncertainties is shaded red. from zero.

By comparison, in Fig. 8 we show a set of simulated events with the same choices of parameters, but √ s = 14 TeV. In this case we apply a lower cut of Y > Y c = 0.75. As a result, the asymmetry of the simulated data is A F B = -0.45 ± 0.07(stat) ± 0.14(sys), lower than the expected true value, and affected by large uncertainties.

A final comment is in order. In this numerical analysis we did not take into account the effect of experimental errors, nor any kinematical cuts that might be applied to extract analyzable data. For example, when | cos θ| 1 the experimental extraction of the kinematics of the two final leptons might be problematic. In order to understand how important this effect is, we extracted the asymmetry from the simulated data by imposing the further constraint | cos θ| < 0.8. This procedure effectively reduces the forward-backward asymmetry to A F B = -0.37 (A F B = -0.28) in the 24 (14) TeV case.

These two examples show how the method works in practice. However caution must be noted that in performing such simulations there is a degree of sensitivity to TeV and an integrated luminosity of 100 fb -1 for the specific model in the text. Orange (shaded) regions of the plot represent areas of the specific model that are either already excluded, where Drell-Yan production is suppressed or not accessible by the specific model. The (thick) blue line represents the minimum coupling ratio R required for at least 10 events to be observed. (Thin) blue lines follow the contours of an upper and lower 1σ PDF uncertainty . The region between the upper and lower 1σ uncertainties is shaded blue. The (thick) red line represents the minimum coupling ratio R for an asymmetry to be visible at the 3σ level (see text for definition). (Thin) red lines follow the contours of an upper and lower 1σ PDF uncertainty (the lower line largely falls below the axis). The region between the upper and lower 1σ uncertainties is shaded red.

the choice of PDF sets. We illustrate here that the method functions but very different results may be obtained depending on how the large-ξ distributions are modelled, and ultimately this problem will be resolved only once actual data are collected.

Non-degenerate Couplings

The region at low rapidity exhibits an interesting feature. Due to the differences in the up and down PDF distributions, the rapidity distributions of these two types of partonic events have very distinct shapes: up-type events peak at finite Y , while down-type peak at Y = 0. In principle, this might allow to discriminate between different values of the couplings R γ and R Z of the new resonances to the up and down quarks.

To investigate the feasibility of such a measurement we examine the two extreme cases, that R γ = 0 versus R Z = 0. We normalize the distributions to the same area, hence fixing the event number. The result is shown in Fig. 9. We see the largest effect is around Y = 0 where there is a 1 part in 25 difference, or 5%. The chosen situation is the An example simulation of the angular distribution for the predicted number of events at a √ s =14 TeV machine with L = 100 fb -1 , for boson mass m = 2.6 TeV and coupling ratio R = 0.4. The simulated data has been restricted to rapidity Y > Y c = 0.75 and the angular distribution in the parton-parton centre of mass frame prediction based on the detector angular distribution is plotted. The observed asymmetry is A F B = -0.45 ± 0.07(stat) ± 0.14(sys). This is the remnant of initial asymmetry of -0.42 that the data was simulated with. most extreme, a light mass in terms of what the precision constraints allow and a high energy collider. We can conclude that the effect of a non-degenerate R will be so small as not to be measurable, even assuming perfect detector efficiency and no kinematic cuts.

Conclusions

We performed a model-independent study of the Drell-Yan processes at the LHC for two new neutral gauge bosons with the quantum numbers of the photon and Z boson. We assumed the behavior of the new states are analogous to their standard-model cousins, but with couplings to SM fermions rescaled by a common factor of R < 1. We also chose to look at the case where the peaks of the two new states were so close as to be indistinguishable in the invariant mass plots. We assumed perfect detector performance, omitting experimental errors, such as energy resolution, and without including the effect of kinematical cuts. We did not include any QCD corrections to the tree-level partonic calculation of pp → µ + µ -and pp → e + e -. In relation to the forward-backward asymmetry, we included in the analysis the main sources of theoretical uncertainty: a statistical element related to the number of expected events, a systematic due to the misidentification of the direction of the initial (anti-)quark, and the uncertainty due to the PDF distributions.

The results can be interpreted as a best case scenario at the LHC, limited only by theoretical considerations. We compared different choices of the integrated luminosity and center-of-mass energy of the pp collisions. For illustration purposes, we also referred to one specific model, in which such mass-degeneracy of the new weakly-coupled gauge bosons is required by imposing the bounds on precision electro-weak parameters.

For a 10 TeV collider, and luminosities in the 2-40 fb -1 , bounds on new resonances no higher than 3 TeV can be obtained. With a 24 TeV collider the region well above 4 TeV can be explored, depending on the specific coupling R. The uncertainty imparted upon this prediction from the PDFs is quite modest for a high energy collider, while at lower energies, like 10 TeV, the PDF uncertainty at large momentum fraction is an important limiting factor. This is in substantial agreement with the literature.

We studied in detail the forward-backward asymmetry as a method of telling a two particle resonance from a one particle peak. The collider ability to measure an asymmetry at the 3σ level is heavily restricted to light masses. Even at very high luminosities measuring the asymmetry for masses of 3 TeV is not possible at a 14 TeV machine. Factoring in the PDF uncertainty, it is hard to conceive of any possible asymmetry measurements in a 14 TeV (or lower) collider, in relation to the scenarios considered here. By contrast, at a 24 TeV machine the PDF uncertainty is markedly lower and the reach far larger. At such a machine it may be possible to measure asymmetries and greatly restrict currently open parameter space.

In many phenomenological models one would wish to measure precisely the physics of heavy gauge bosons. Measuring the forward-backward asymmetry in events with neutral intermediate states provides a theoretically clean signal allowing one to discriminate the presence of more than one boson in a heavy resonance. Such a measurement is very difficult in practice from an experimental viewpoint. On top of the practicalities there are the large PDF uncertainties at large momentum fraction. Together with the scaling of the event rate with s, the situation suggests upgrading the LHC to a higher energy (besides increasing the luminosity) to learn about such new bosons, if the new bosons are heavy as is suggested by the precision tests.

Figure 1 .

 1 Figure 1.The ratio R between the couplings of the new and SM boson to SM currents, such that the number of events N divided by the integrated luminosity L is fixed, and plotted as a function of the heavy boson mass at a collision energy of √ s = 10 TeV. The leftmost line is for N/L = 5 fb then reading from left to right the lines are, N/L = 3, 1, 1/2 fb, finishing on the rightmost line of N/L = 1/4 fb. The events N are for the production of muon pairs only.

Figure 2 .

 2 Figure 2.The ratio R between the couplings of the new and SM boson to SM currents, such that the number of events N divided by the integrated luminosity L is fixed, and plotted as a function of the heavy boson mass at a collision energy of √ s = 24 TeV. The uppermost line is for N/L = 5 fb then reading from top to bottom the lines are, N/L = 3, 1, 1/2 fb, finishing on the lowest line of N/L = 1/4 fb. The events N are for the production of muon pairs only.

Figure 3 .

 3 Figure3. The minimum ratio R between couplings of the new and SM boson to SM currents, such that at the 3σ level (see text for definition) a forward-backward asymmetry of -0.42 is detectable, plotted for various integrated luminosities at a collision energy of √ s =14 TeV and as a function of the heavy boson mass. The leftmost line is for an integrated luminosity of 1 fb -1 , the middle line for 10 fb -1 and the rightmost 100 fb -1 .

  Figure 4. The minimum ratio R between couplings of the new and SM boson to SM currents, such that at the 3σ level (see text for definition) a forward-backward asymmetry of -0.42 is detectable, plotted for various integrated luminosities at a collision energy of √ s =24 TeV and as a function of the heavy boson mass. The leftmost line is for an integrated luminosity of 1 fb -1 , the middle line for 10 fb -1 and the rightmost 100 fb -1 .

Figure 5 .

 5 Figure 5.The discovery reach at a machine energy of √ s =14 TeV and an integrated luminosity of 100 fb -1 for the specific model in the text. Orange (shaded) regions of the plot represent areas of the specific model that are either already excluded, where Drell-Yan production is suppressed or not accessible by the specific model. The (thick) blue line represents the minimum coupling ratio R required for at least 10 events to be observed. (Thin) blue lines follow the contours of an upper and lower 1σ PDF uncertainty . The region between the upper and lower 1σ uncertainties is shaded blue. The (thick) red line represents the minimum coupling ratio R for an asymmetry to be visible at the 3σ level (see text for definition). (Thin) red lines follow the contours of an upper and lower 1σ PDF uncertainty (the lower line largely falls below the axis). The region between the upper and lower 1σ uncertainties is shaded red.

Figure 6 .

 6 Figure 6.The discovery reach at a machine energy of √ s =24 TeV and an integrated luminosity of 100 fb -1 for the specific model in the text. Orange (shaded) regions of the plot represent areas of the specific model that are either already excluded, where Drell-Yan production is suppressed or not accessible by the specific model. The (thick) blue line represents the minimum coupling ratio R required for at least 10 events to be observed. (Thin) blue lines follow the contours of an upper and lower 1σ PDF uncertainty . The region between the upper and lower 1σ uncertainties is shaded blue. The (thick) red line represents the minimum coupling ratio R for an asymmetry to be visible at the 3σ level (see text for definition). (Thin) red lines follow the contours of an upper and lower 1σ PDF uncertainty (the lower line largely falls below the axis). The region between the upper and lower 1σ uncertainties is shaded red.

Figure 7 .

 7 Figure 7. An example simulation of the angular distribution for the predicted number of events at a √ s =24 TeV machine with L = 100 fb -1 , for boson mass m = 2.6 TeV and coupling ratio R = 0.4. The simulated data has been restricted to rapidity Y > Y c = 1.0 and the angular distribution in the parton-parton centre of mass frame prediction based on the detector angular distribution is plotted. The observed asymmetry is A F B = -0.51 ± 0.03(stat) ± 0.1(sys). This is the remnant of initial asymmetry of -0.42 that the data was simulated with.

Figure 8 .

 8 Figure 8.An example simulation of the angular distribution for the predicted number of events at a √ s =14 TeV machine with L = 100 fb -1 , for boson mass m = 2.6 TeV and coupling ratio R = 0.4. The simulated data has been restricted to rapidity Y > Y c = 0.75 and the angular distribution in the parton-parton centre of mass frame prediction based on the detector angular distribution is plotted. The observed asymmetry is A F B = -0.45 ± 0.07(stat) ± 0.14(sys). This is the remnant of initial asymmetry of -0.42 that the data was simulated with.

Figure 9 .

 9 Figure9. The expected rapidity distribution of a sample of events in the case that a single new Z (blue) and a single γ (red) is the only exchange particle present in a new resonance. The new boson has a mass set at 2.6 TeV and a machine energy of √ s = 24 TeV is used. The same number of events is assumed for both.

Acknowledgments

We would like to thank S. de Curtis and M. Fabbrichesi for useful discussions. The work of MR is supported by the STFC Doctoral Training Grant ST/F00706X/1, and MP is supported in part by the Wales Institute of Mathematical and Computational Sciences and also by the STFC Grant ST/G000506/1.


T Boeckel

M Hempel

I Sagert

G Pagliara

B Sa'd †

J Schaffner-Bielich

Strangeness in Astrophysics and Cosmology

HAL is

Introduction

There is no attempt in this contribution to discuss all the recent advances in the very active research area of studying the impacts of strange quark matter in astrophysics and cosmology. We refer to the other contributions of this conference for topics not covered in this contribution to the proceedings. Signals for the QCD phase transition to strange quark matter in core-collapse supernovae are addressed in the contribution by Irina Sagert et al. (see also [1]). For the appearance of a new strange phase in the evolution of proto-neutron stars we refer to the contribution by Giuseppe Pagliara et al. (see also [2]). The issues of how to nucleate strange quark matter in astrophysical systems has been studied in detail in the contribution of Bruno Mintz et al. (see also [3]). Strange quark matter, colour superconductivity and the relation to quarkyonic matter are discussed in the contribution by David Blaschke et al. (see also [4]).

In the following we give attention to a few other recent developments in the field. First we reconsider the maximum mass constraint for neutron stars using causality arguments for the nuclear equation of state. Constraints on the nuclear equation of state using heavy-ion data, here subthreshold production of kaons, can be adopted to give a new maximum mass limit of about 2.7 solar masses. Then we are studying the gravitational wave emission from rotating neutron stars and the relation to the viscosity of strange quark matter. The recent simulation of the merger of two strange stars, the release of strange quark matter nuggets to the interstellar medium and the possible impact for the flux of strangelets in cosmic rays are another issue being highlighted. Finally, we are concerned with the cosmological QCD phase transition which happens around 10 -5 seconds after the big bang. If one allows for an inflationary period at a strong first order phase transition, the early universe can actually pass through the QCD phase diagram at large baryochemical potential in contrast to the standard model of cosmology.

Maximum mass of neutron stars: constraints from heavy-ion data

Neutron stars are produced in core-collapse supernova explosions and are compact, massive objects with radii of 10-15 km and masses of 1-2 M . Pulsars are rotationpowered neutron stars and have been observed in binary systems, i.e. with a companion star, with a white dwarf or even with another neutron star. To create such a binary pulsar one starts with two ordinary stars, one with at least 8M . The heavier star explodes first in a supernova and leaves a neutron star as a remnant. The neutron star is spun up by accreting matter from the companion star. Then also the companion star might become a white dwarf or another neutron star depending on its initial mass (see [5] for a review).

For binary pulsars masses can be pretty nicely determined. At present more than 1800 pulsars are known with 140 binary pulsars. From those binary pulsars, the pulsar PSR J1903+0327 is particular interesting [6,7]. By measuring the post-Keplerian parameters from pulsar timing, here the Shapiro delay parameters r and s alone, constrain the mass to M = (1.67 ± 0.11)M . Combined with the periastron advance ω the mass is constraint to be M = (1.67 ± 0.01)M . This mass measurement would put some severe constraints on soft nuclear equations of state which give a smaller maximum mass [8].

Kaons are produced by associated production: NN→ NΛK and NN→NNKK in elementary collisions. In the medium as generated in central heavy-ion collisions new processes emerge through rescattering as πN → ΛK, πΛ → NK so that kaons can be even produced below the (elementary) threshold bombarding energy. In heavy-ion collisions of 1-2 AGeV nuclear matter can compressed up to 3n 0 . As kaons have a long mean-free path, they can escape the high density region serving as a messenger of the properties of high-density nuclear matter (for a review see e.g. [START_REF] Aichelin | The quest for the nuclear equation of state[END_REF]). One defines the double ratio as the multiplicity of produced kaons per mass number for Au+Au collisions relative to the one for C+C collisions. This observable turns out to be rather insensitive to input parameters but strongly dependent on the nuclear equation of state (to be more precise the in-medium potential of nucleons which can be related to the nuclear equation of state). Using a simple Skyrme parametrization, only simulations with a compression modulus of K N ≈ 200 MeV [START_REF] Fuchs | [END_REF]11] can describe the energy dependence of the kaon production data as measured by the KaoS collaboration [12,13]. Hence, the nuclear equation of state would be soft around 2-3 n 0 .

Causality arguments can then be adopted to constrain the maximum possible mass of neutron stars. Above a fiducial density as determined from the data analysis of the KaoS data the stiffest possible equation of state is taken which gives the maximum pressure for a given energy density allowed. Causality demands then that at most p =c above the fiducial density f (see [14,15]). There is a general scaling relation for the maximum possible mass of compact stars which is given by

M max = 4.2M ( 0 / f ) 1/2
where 0 is the energy density of normal nuclear matter (the numerical prefactor depends slightly on the low-density nuclear equation of state, see [16]). For a fiducial density of f = 2 0 as provided by the heavy-ion data the upper mass limit is about 2.7M [17,18].

Gravitational wave emission from rotating neutron stars: the viscosity of strange quark matter

There exists a special class of binary neutron stars, so called x-ray burster. In these astrophysical systems, the neutron star is accreting matter from an ordinary star or a white dwarf. A low-mass x-ray binary (LMXB) is a neutron star with a low mass companion star or a white dwarf. The accreting matter falls onto the neutron star surface and thereby releases x-rays in a burst. The neutron star surface is heated up to typical temperatures of T = 10 -100 keV.

For rotating neutron stars, there exists an instability, the r-mode instability. Oscillations brings the neutron star matter out of β-equilibrium which is restored by weak processes affecting the bulk and shear viscosity [19]. The viscosity of nuclear matter is quite low in certain regions of frequency and temperature so that the oscillations will not be damped, gravitational waves are emitted and the star's rotation slows down. The contributions from quarks increase the stability window in temperature and frequencies substantially. In color-superconducting phases the viscosity of quark matter can be drastically different [20,21,22,23,24]. In the 2SC phase, with the pairing of two quark flavours and two colours, there are still unpaired quarks left and one obtains similar results compared to the free case [25,26]. Both cases are compatible with the data on low-mass x-ray binaries. On the contrary, quarks of all color and flavor are paired in the CFL phase. The reaction rates are suppressed, the instability window is considerably broader which could leave an observable impact on the evolution of young pulsars and accreting neutron stars. Recently, it was also found that there is a new process which takes away the energy from the r-mode oscillations by the emission of neutrinos. It acts effectively as a viscosity which would help in damping the r-mode instability by a substantial factor [27].

Strange star mergers: the flux of strangelets in cosmic rays

According to the Bodmer-Witten hypothesis absolutely stable strange quark matter would be more bound than ordinary nuclear matter. If one allows for absolutely stable strange quark matter, then strange stars will exist which are compact stars consisting of absolutely stable strange quark matter being bound by strong interactions (selfbound) and not by gravity as ordinary neutron stars. Strange stars can coexist with neutron stars, but in principle a neutron star would collapse to strange star in a spectacular astrophysical event. Recently, the collisions of two strange stars have been simulated in 3D relativistic smoothed particle hydrodynamics with approximate treatment of effects from General Relativity [28]. Typical timescales involved are milliseconds, so that matter is in β-equilibrium. The simulations have been performed for two bag constants of B = 60 and 80 MeV/fm -3 , in both cases one gets absolutely stable strange quark matter, and for different initial masses of the two strange stars. In principle, two different scenarios have been found: either there is an immediate collapse to a black hole or a hypermassive object stabilized by differential rotation is formed. The implications of these results are quite striking: the ejected mass of strange quark matter (strangelets) released to the interstellar medium are estimated to be (1.4 -2.8) • 10 -4 M for the lower bag constant and zero for the larger one. As strange star mergers are considered to be the prime source for strangelets in cosmic rays, the latter result opens the possibility that no strangelets are seen in cosmic rays although strange stars exist. The Alpha Magnetic Spectrometer AMS will measure the strangelet flux in cosmic rays (planned to be installed at the International Space Station ISS in 2010). Interestingly, the gravitational wave pattern of compact star mergers can be used to distinguish between neutron star and strange star mergers [29].

Early universe: phase transition from strange quark matter

Let us briefly outline the history of the early universe. In a radiation dominated universe the temperature scales inversely with the scale parameter a. Big-bang nucleosynthesis happens at times of t = 1s to 3 minutes which corresponds to temperatures of T = 1 to 0.1 MeV. The QCD phase transition occurs at a temperature of about T ≈ 170 MeV, i.e. at a time of t ≈ 10 -5 s after the big bang. The early universe passes through the electroweak phase transition at t ≈ 10 -10 s (T ≈ 100 GeV). In the standard cosmology, one knows from the measurement of the microwave background radiation with WMAP and from the observed light element abundance from the big bang nucleosynthesis that the ratio of the number of baryons to the number of photons (which is related to the total entropy of the universe) is tiny:

n B /s ∼ n B /n γ ∼ µ/T ∼ 10 -9
Note that the baryon number per entropy is a conserved quantity throughout the evolution of the universe unless there is a non-equilibrium process as a first order phase transition. From the above considerations one concludes that the early universe evolves along µ ∼ 0 which implies that the early universes crosses the QCD phase transition in a region of the QCD phase diagram where one expects a crossover transition as seen in recent lattice gauge calculations. A crossover transition implies that there is nothing spectacular happening, as one expects no cosmological signals from such a kind of transition.

The Friedmann equation for radiation dominated universe reads

H 2 = 8πG 3 ρ ∼ g(T ) T 4 M 2 p
where g(T ) is effective number of relativistic degrees of freedom at a given temperature T and M p is the Planck mass. The Hubble time is defined by the Hubble parameter and is related to the true time by t = 3t H for a radiation dominated universe. By using the Friedmann equation one finds that t H = 1/H ∼ g -1/2 M P /T 2 so that there is a simple relation between the time after big bang and the temperature of the radiation dominated early universe

t 1 sec ∼ 1 MeV T 2
We see that if the number of degrees of freedom changes smoothly with the temperature, as it would be the case for a crossover phase transition, the temperature changes continuously with time.

The interesting question is what would happen if the early universe passes through a first order phase transition? We argue in the following that this is a viable scenario for the QCD phase transition and not in contradiction to our present knowledge of QCD. Furthermore, we demonstrate that it would leave an observable imprint on the gravitational wave background [30].

A first order phase transition in general allows for a false metastable vacuum state where the universe could be trapped for some time. For a constant vacuum energy, one arrives at the de Sitter solution of the Friedmann equations which results in an (additional small) inflationary period:

H = ȧ/a ∼ M -1 p ρ 1/2 v = H v = const. → a ∼ exp(H v • t)
Remarkably, just a few e-folds are enough (standard inflation needs N ∼ 50) for our purposes. The baryon to photon ratios before and after the inflationary period should scale with the ratio of the scale parameters cubed as entropy is produced during reheating while baryon number is conserved so that

µ T f ≈ a i a f 3 µ T i
The final ratio should be 10 -9 as observed so that we need just a boost of N = ln (a f /a i ) ∼ ln(10 3 ) ∼ 7, i.e. seven e-folds, to get an initial ratio of (µ/T ) i ∼ O(1). Note that for standard inflation one needs about N = 50 e-folds, so that the little inflation at the QCD phase transition can not replace it but could be present as an additional small inflationary period in the early universe. So our scenario is as follows: the early universe is initially at large baryochemical potentials µ/T ∼ 1. Such a large value is possible for e.g. Affleck-Dine baryogenesis which involves scalar supersymmetric fields carrying baryon number. The early universe reaches the first order phase transition line of QCD at high baryochemical potentials and is trapped in the false vacuum. Now the inflationary period starts with supercooling and dilution with µ/T = const.. The decay to the true vacuum state will release latent heat so that the universe is reheated to T ∼ T c . Afterwards, the final baryon to photon ratio is given by µ/T ∼ 10 -9 due to the entropy produced during the transition. Finally, the universe evolves along the standard cosmological path to big-bang nucleosynthesis and so on. The path through the QCD phase diagram is depicted in Fig. 1.

However, there are observable differences to the standard model of cosmology. The first order phase transition produces perturbations, scalar and tensor ones, by colliding bubbles. The tensor modes correspond to gravitational waves with a frequency scale given by the (redshifted) horizon scale at the transition point

ν peak ∼ H • T γ,0 /T QCD ∼ T QCD /M p • T γ,0 ∼ 10 -7 Hz
The maximum amplitude of the gravitational waves is then given by the ratio of the scale factors today and at the QCD phase transition and is h ∼ a QCD /a 0 ∼ 10 -12 . For superhorizon modes ν < H the amplitude scales as h(ν) ∝ ν -1/2 which corresponds to white noise. The modes within the horizon ν > H are determined by multi bubble collisions and could scale as h(ν) ∝ ν n where n is between -1 and -2 [31,32]. The gravitational waves from a first-order QCD phase transition are observable with pulsar timing, as with the Parkes Pulsar Timing Array and in the future with the Square Kilometre Array SKA (see figure 2).

Other cosmological implications of a first-order QCD phase transition are that the cold dark matter density is diluted by the same factor as the baryon number density, The Parkes Pulsar Timing Array PPTA already gives limits on the maximum amplitude of those gravitational waves. In the future the Square Kilometre Array SKA will push these limits down by a few orders of magnitude. For a flat spectrum at high frequencies, LISA would be also sensitive.

i.e. by up to a factor 10 -9 . To explain the present cold dark matter relic density within the standard WIMP scenario, one needs a different WIMP annihilation cross section σ ann as the cold dark matter density scales as Ω CDM ∼ σ weak /σ ann or by a substantially larger WIMP mass. Both cases can be probed by the LHC! The large-scale structure power spectrum responsible for galaxy and galaxy cluster formation would be modified up to a mass scale of M ∼ 10 9 M which is the typical mass scale of dwarf galaxies. Without QCD inflation, effects can be only up to the horizon mass ∼ 10 -9 M at the phase transition point. Also, seeds of (extra)galactic magnetic fields can be generated by charged bubble collisions during the phase transition. The observed magnetic field today in our galaxy is B ∼ 10 -5 G, extragalactic ones are in the range of B ∼ 10 -7 G. One needs primordial seed fields of B = 10 -30 . . . 10 -10 G. As the electroweak phase transition is a crossover for large Higgs masses, a first order QCD phase transition in the early universe would provide then a possible explanation of those cosmological magnetic fields within the standard model again (see [33] for a detailed calculation).

Figure 1 .

 1 Figure 1. The evolution through the QCD phase diagram for the little inflation scenario. The universe is trapped in a metastable vacuum at the first order phase transition line, supercools and is reheated back close to the critical temperature at tiny values of the baryochemical potential.

Figure 2 .

 2 Figure 2.The gravitational wave spectrum from a first order QCD phase transition. The Parkes Pulsar Timing Array PPTA already gives limits on the maximum amplitude of those gravitational waves. In the future the Square Kilometre Array SKA will push these limits down by a few orders of magnitude. For a flat spectrum at high frequencies, LISA would be also sensitive.

Acknowledgments

This work is supported by BMBF and DFG also within the framework of the excellence initiative through the Heidelberg Graduate School of Fundamental Physics, the Gesellschaft für Schwerionenforschung GSI Darmstadt, the Helmholtz Research School for Quark Matter Studies, the Helmholtz Graduate School for Heavy-Ion Research HGS-HIRe, the Graduate Program for Hadron and Ion Research (GP-HIR), the Helmholtz Alliance Program of the Helmholtz Association, contract HA-216 "Extremes of Density and Temperature: Cosmic Matter in the Laboratory", and CompStar, a research networking program of the European Science Foundation.


Gunnar Gräf

Elvira Santini

Hannah Petersen

Jan Steinheimer

Michael Mitrovski

Marcus Bleicher
email: bleicher@th.physik.uni-frankfurt.de

Strangeness production in hadronic models and recombination models

We present recent results on the production, spectra and elliptic flow of strange particles in dynamic simulations employing hadronic degrees of freedom and from recombination models. The main focus will be on the Ultra-relativistic Molecular Dynamics (UrQMD) Boltzmann approach to relativistic heavy ion collisions and a hybrid approach with intermediate hydrodynamic evolution based on UrQMD (available for download as UrQMD v3.3). Compared to the standard binary collision approach, an enhancement of the strange particle particle yields is found in the hybrid approach due to the assumption of local equilibration. The production origins of the φ-meson in the hybrid approach are studied in further detail. We also present results on the transverse momentum spectra of baryon to meson ratios of strange particles. Due to the approximate energy independent scaling of this ratio as a function of p T we argue, that a maximum in these spectra may not be a unique sign for quark coalescence but can be understood in terms of flow and fragmentation.

Introduction

Heavy ion collisions offer the unique possibility to explore the properties of nuclear matter under extreme conditions in the laboratory. To achieve high temperatures and particle densities, accelerators like the Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) or the planned FAIR facility are used to collide heavy nuclei and to detect the remnants of their interaction. During the collision of these nuclei it is expected that a hot and dense zone of free quarks and gluons, known as the quark gluon plasma (QGP) is created.

Many years ago, strangeness has been proposed as a good tool to investigate this new state of matter [1], because the strange (s) and anti-strange (s) quarks are not present in the colliding nuclei and have to be produced during the collision. An additional advantage of strange quarks is that they decay by weak interactions, which means that they have a lifetime on the order of 10 -10 seconds which is much longer than the timescale of a heavy ion collision and allows for a direct reconstruction of strange hadrons in the detectors.

On the theoretical side, the production of multi-strange (anti-)baryons and φmesons is still a difficult task. Hadronic transport approaches based on binary particle scatterings (i.e. without a QGP transition or additional effects like strong color fields [2] or multi-particle interactions [3,4]) are usually not able to reproduce the multiplicities of multi-strange hadrons. This observation is related to the long equilibration times and high mass threshold in the hadronic environment compared to a QGP state.

Hydrodynamics has been suggested as an alternative tool to describe the hot and dense stage of heavy ion collisions [5,6,7,8,9,10,11]. Here one assumes local equilibration of the QCD-matter, however, one has to deal with complications concerning the decoupling stage and potential off-equilibrium effects, like viscosities or heat conductivity. In these proceedings we present a novel integrated hybrid approach that dynamically links a non-equilibrium Boltzmann dynamics approach for the initial and final stage of the reaction to an intermediate ideal hydrodynamic stage. This integrated hybrid approach is called UrQMD v3. 3 [12] and is based on the known Ultrarelativistic Quantum Molecular Dynamics. For a detailed description of the model and comparison to data the reader is referred to [13,14,15,16].

Hybrid approaches

Hybrid approaches to unite hydrodynamics and transport equations where proposed 10 years ago [17,18] and have since then been employed for a wide range of investigations [19,20,21,22]. The hybrid approach presented here is based on the integration of a hydrodynamic evolution into the UrQMD approach [23]. During the first phase of the evolution the particles are described by UrQMD as a purely hadronic cascade. Once the two colliding nuclei have passed through each other the hydrodynamic evolution starts at the time t start = 2R/ √ γ 2 -1. At this time the spectators continue to propagate in the cascade, while all other particles are mapped to the hydrodynamic grid. By doing so one explicitly assumes a local thermal equilibrium for each cell. The hydrodynamic evolution is performed using the SHASTA [6] algorithm with a hadron gas equation of state. At the end of this phase the hydrodynamic fields are mapped to particle degrees of freedom using the Cooper-Frye equation. Two possible freeze-out criteria procedures where tested. The first is the isochronous freeze-out (IF). In this approach, all hydrodynamic cells are mapped onto particles at the same time, once the energy drops below five times the ground state energy density in all cells. The second approach is called gradual freeze-out (GF). In this approach the freeze-out happens simultaneously in a transverse slice of thickness 0.2 fm once all cells in this slice have an energy density lower than five times the ground state energy. This mimics an iso-eigentime freeze-out that accounts for time dilation effects seen in the freeze-out temperature in cells with a high rapidity. After this the final state interactions and decays of the particles happen within the UrQMD framework. In Fig. 1 the excitation functions of the Λ, Σ, Ω, π, K and p yields are shown.

One observes an enhancement of the multiplicities for all particles with strange content compared to the non-hybrid results. This enhancement leads to an improved description of the experimental data. This is especially true for the gradual freeze-out, while the isochronous freeze-out suffers from relativistic effects at higher beam energies. For further discussion on spectra, the reader is referred to [13].

φ-production

Apart from multi-strange baryons, φ-mesons pose a second problem in many conventional hadronic approaches. This difficulty is not restricted to ultra-relativistic collision energies, but is also present at lower reaction energies, see e.g. [24]. Recently, the present hybrid approach has been used to investigate φ → μ + μ -production in In+In collisions at E lab =158AGeV [25]. It was found that the hybrid model naturally provides a realistic description of the φ yields and their transverse mass spectra. In Fig. 2 (left), the hybrid model calculations are compared to pure cascade calculations, in which no assumption of an intermediated locally equilibrated phase is made, and to NA60 data [26]. The two extreme cases of central and peripheral collisions are considered. We observe that differences in the transverse dynamics resulting from the two approaches become stronger especially for central collisions, with NA60 data favouring the hybrid approach to the pure transport calculation. Small deviations between the hybrid approach calculations and the experimental data are found for peripheral collisions where this kind of hybrid models seem to reach their limit of applicability [14]. The transverse dynamics resulting from pure transport calculations is comparable to the experimental data for the peripheral reaction, though the yield remains underestimated. In this respect, however, it should be mentioned that the UrQMD model underestimates φ meson production at high energies already in p+p collisions. This might partially explain that the main deviations between the experimental data and the transport calculations for peripheral reactions are essentially related to the yield. Let us now discuss more in detail the hybrid model calculations and, in particular, the emission during the cascade stage. In Fig. 2 (right) a decomposition of the various contributions from the cascade emission is shown. This stage contains pre-and postequilibrium emission. The pre-equilibrium di-muon emission is negligible, as expected from the shortness of this stage. The post-equilibrium emission can be divided in two categories: (i) the emission from φ particles produced via Cooper-Frye at the transition point and (ii) the emission from φ particles produced during the cascade.

In the first case, the particles have a momentum distribution that reflects the thermal properties of the transition point, although their di-muon decay occurs later in a non-thermal environment. In this sense, this copious emission, though not specifically thermal (i.e. not described by thermal rate equations) still carries information about the preceding thermal phase.

The second contribution, on the contrary, can be labeled as a "purely cascade" one. This is the contribution of φ particles produced in the non-equilibrium environment line) is compared to the pure UrQMD transport calculation (dashed line) and to experimental data [26]. Bin-widths which coincide with the ones of the experimental data have been used. Right: Decomposition of the post-equilibrium φ → μ + μ - production in: (i) emission from φ mesons which are merged into the cascade at the transition point via the Cooper-Frye equation and emit in the cascade stage of the evolution (dashed line); (ii) emission from φ mesons which are produced and emit in the cascade stage (dotted-dashed line). The pre-equilibrium emission is denoted by the dotted line. The full line represents the total cascade emission. Due to the smallness of the pre-equilibrium emission the latter practically coincides with the total post-equilibrium emission.

on the way to final decoupling. This second contribution is characterised by steeper transverse mass spectra. the shape of the total spectra is found to be composed by the interplay of both emissions. Emission during the hydrodynamical phase of the evolution has been investigated too and was found to be smaller than the post-cascade emission [25].

The analysis supports the picture of a φ meson being emitted from an equilibrium source. The mechanism behind the thermalization of the φ meson cannot be inferred by this analysis and remains an open question. The possible mechanism for the (eventual) thermalization of the φ in medium are still under debate. In transport calculations the dynamics of the φ meson is typically characterised by small cross sections and early transport calculations [27,28] supported the idea of a large φ mean free path in the hot hadronic fireball (see also [29]). In the UrQMD model, more specifically, the φ meson couples mainly to K K and ρπ channels and its elastic and inelastic scattering with mesons and baryons are modelled within the Additive Quark Model. Values for the cross sections for such reactions can be found in Table IX of Ref. [30]. Over the years, however, some mechanisms have been suggested that could be responsible for a stronger φ coupling to the high density high temperature phase than expected. Alvarez-Ruso and Koch, for example, studied the interactions of the φ meson with other pseudo-scalar and vector mesons in a hadronic gas at finite temperature [31]. Within the Hidden Local Symmetry Lagrangian they calculate the φ collision rate and mean free path including an extended family of mechanisms that are allowed by symmetries but are not present in calculations that rely on observed decays. In such a scenario a significantly smaller mean free path (between 2.4 and 1 fm at temperatures above T =170 MeV) was found. Recently, the hypothesis of a catalytic φ meson production in heavy-ion collisions has been suggested [32]. When implemented within a schematic fireball evolution scenario, such an hypothesis can account for the increase of the φ rapidity distribution width with the collision energy. In a recent calculation of spectral density of the φ meson in a hot bath of nucleons and pions [33] a significant increase of the width of the φ meson in medium, which again supports the scenario of a φ not at all weakly coupled to the medium. Another possibility within the hadronic scenario might be multi-particle interaction that could as well contribute to increase the degree of thermalization and lead to thermalized φ production [34].

Baryon to meson ratios

At the Relativistic Heavy Ion Collider (RHIC) at Brookhaven an unexpected enhancement of the baryon to meson ratios in central Au+Au reactions at √ s = 200GeV in the transverse momentum region about 2 GeV/c had been observed. Fries et al. looked in to the baryon to meson ratios [35] using a combined recombination and fragmentation picture. Comparing their results with statistical models and data they suggest that the maximum in the p ⊥ dependence of p/π 0 and Λ/K 0 ratios is a strong sign for a transition of a recombination dominated regime to a fragmentation dominated regime at higher p ⊥ .

Let us investigate this effect in Fig. 3(left). Here, the results of pure UrQMD calculations without a hydrodynamic phase are shown. One clearly observes a maximum of the baryon to meson ratio at about 2 GeV/c. However, it seems that the magnitude of the enhancement is less pronounced than in the data. To test, if the enhancement is unique to the high p ⊥ tail or if it results from an overall normalisation, we normalise the p ⊥ spectra to their integral.

The normalised (Λ + Σ 0 )/K 0 s ratios are shown in Fig. 3 (right). The surprising result is a collapse of all (Λ + Σ 0 )/K 0 s ratios at various center of mass energies from 3.8 GeV (lower AGS) to 5.5 TeV (LHC) to a single line. I.e. the apparent maximum of the baryon to meson ratio at p ⊥ ≈ 2 GeV is present at all investigated energies. We thus conclude, that the maximum in the baryon to meson ratios, may not be a unique sign for quark coalescence, but rather a natural consequence of a flow plus fragmentation picture.

In summary, we have discussed recent developments in the area of hadron transport simulations with emphasis on strangeness production. The most important improvements over the last decade include the wide range employment of hybrid approaches to simulate the hot and dense reaction stage by ideal (and also dissipative) hydrodynamics coupled to fluctuating non-equilibrium initial conditions and followed by a decoupling stage modelled via hadronic cascades. These combined approaches have proven their power over a wide range of applications ranging from strange particle production, over (elliptic) flow analysis to leptonic and photonic probes. At the end of these proceedings we have also shortly presented a critical view on the baryon to meson ratios and their potential as a signal for parton recombination.

Figure 1 .

 1 Figure 1. Excitation functions of the Λ, Σ, Ω, π, K and p multiplicities (4π) in central (b < 3.f fm) Au+Au/Pb+Pb collisions. The results for the hybrid model with isochronous freeze-out (full lines), the hybrid model with gradual freeze-out (squares) and pure UrQMD-2.3 (dotted lines) are compared to experimental data from various experiments.

φ

  is produced at the transition φ is produced in the cascade sum central In-In φ-meson reconstructed from φ -> μμ decomposition of post-hydro cascade emission pre-hydro emission

Figure 2 .

 2 Figure 2. Left: Transverse mass distributions of the φ meson in central (top) and peripheral (bottom) indium-indium collisions. The hybrid model calculation (fullline) is compared to the pure UrQMD transport calculation (dashed line) and to experimental data[26]. Bin-widths which coincide with the ones of the experimental data have been used. Right: Decomposition of the post-equilibrium φ → μ + μ - production in: (i) emission from φ mesons which are merged into the cascade at the transition point via the Cooper-Frye equation and emit in the cascade stage of the evolution (dashed line); (ii) emission from φ mesons which are produced and emit in the cascade stage (dotted-dashed line). The pre-equilibrium emission is denoted by the dotted line. The full line represents the total cascade emission. Due to the smallness of the pre-equilibrium emission the latter practically coincides with the total post-equilibrium emission.

Figure 3 .

 3 Figure 3. Left: Transverse momentum distribution at midrapidity |y| ≤ 0.5 of the (Λ + Σ 0 )/K 0 s ratio from √ s = 3.8 GeV (top) to √ s = 5.5 TeV (bottom). The values were calculated with pure UrQMD-2.3 for 0-5% most central Au+Au collisions. Right:Transverse momentum distribution at midrapidity of the (Λ + Σ 0 )/K 0 s ratio scaled with the yield at midrapidity |y| ≤ 0.5 for center of mass energies from 3.8 GeV to 5.5 TeV.

Acknowledgements

This work was supported by the Helmholtz International Center for FAIR within the framework of the LOEWE program launched by the State of Hesse, GSI, BMBF and DESY. The hydrodynamic model is provided by Dirk Rischke. The calculations have been computed at the Center for Scientific Computing at Goethe University.