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Thermal evolution of the chiral condensate in SU(2) and SU(3) Chiral Perturbation Theory

The temperature evolution of the chiral condensates in a gas made of pions, kaons and etas is studied within the framework of SU(2) and SU(3) Chiral Perturbation Theory. We describe the temperature dependence of the quark condensates by using the meson meson scattering phase shifts in a second order virial expansion. We find that the SU(3) formalism yields an extrapolated melting temperature for the non-strange condensates which is lower by about 20-30 MeV than within SU(2). In addition our results show that the strange condensate melting is slower than that of the non-strange, due to the different strange and non-strange quark masses.


Introduction
We review here our recent study of the thermal evolution of the chiral strange and non-strange condensates [1]. Our motivation is the study of the the phase diagram of QCD at zero baryonic densities, which is a state that is expected to occur in the central rapidity regions of Relativistic Heavy Ion Collisions. Our purpose is to obtain a model independent description of the condensate and its thermal evolution. To that aim we will turn to the virial expansion and to Chiral Perturbation Theory.
The virial expansion is a simple and successful approach to describe many thermodynamic features of dilute gases made of interacting pions [2] and other hadrons [3]. Let us note that for most thermodynamic properties it is enough to know the low energy scattering phase shifts of the particles, which, in principle, could be taken from experiment, avoiding any model dependence. However, we are interested in the chiral symmetry restoration, whose order parameters are the quark condensates, which are defined as derivatives of the pressure with respect to the quark masses. These derivatives cannot be measured experimentally and therefore a model independent theoretical description of the scattering amplitudes mass dependence is required. That is why we need the second ingredient of our approach: Chiral Per-turbation Theory (ChPT) [4][5][6], which provides a model-independent and systematic description of low energy hadronic interactions. ChPT follows from the identification of the pions, kaons and the eta as the Goldstone Bosons of the QCD spontaneous chiral symmetry breaking (pseudo Goldstone bosons indeed, due to the small light-quark masses). Then, the ChPT Lagrangian is built as the most general derivative and mass expansion, over 4πF ≃ 1.2 GeV, (the symmetry breaking scale) respecting the symmetry constraints. At one loop any calculation can be renormalized in terms of just an small set of parameters, L k (µ), H k (µ), (µ being the renormalization scale) which can be determined from a few experiments and used for further predictions.
In particular, within SU(2) ChPT the leading coefficients of the low temperature expansion for the pressure and the non-strange quark condensate have been calculated for a hadron gas whose only interacting hadrons were the pions [7]. Adding other more massive particles (like kaons, etas, rhos, protons, etc...) as free states, it was possible to obtain an estimate, T ≃ 190 − 200 MeV, of the phase transition critical temperature. It was also shown that the perturbative calculation of the pressure was analogous to the second order virial expansion when the interacting part of the second virial coefficient is obtained from the one loop ππ ChPT scattering lengths. Other works have studied the applicability of the virial expansion with a pionic chemical potential [8], or the critical temperature in generalized scenarios [9].
The interest of extending the previously commented work to SU(3) is that the SU(2) approach is limited by the absence of other interacting particles, like kaons or etas, whose densities at T ≃ 150 MeV are significant [7]. In addition, let us recall that the chiral phase transition can be different for the SU(2) and SU(3) cases, and that several QCD inequalities and lattice results suggest an stronger chiral condensate temperature suppression with an increasing number of light flavors [10]. Intuitively this is due to the fact that the existence of other states (strange quarks, or strange mesons) makes it easier to create entropy, that is, disorder, and therefore it is easier to melt the ordered state, that is, the condensates. This effect has been observed in lattice calculations [11], but only in the chiral limit, and it seems to lower the chiral critical temperature down by roughly 20 MeV. Note, however, that none of these results have been obtained from the hadronic phase and realistic masses, which is what we have achieved with our approach.
In addition, within SU(2) it is not possible to study the ss condensate. This could be interesting because, in the chiral phase diagram, quark masses play the same role as magnetic fields in ferromagnets: Intuitively, we need a higher temperature to disorder a ferromagnet when there is a magnetic field aligned along the direction of the magnetization. Analogously, it has been found that the SU(2) chiral condensate melts at a lower temperature in the massless limit [7]. In particular, we have obtained the temperature dependence of ss in the hadronic phase. Since the strange quark mass m s is much larger than m u , m d , we have indeed found a sizable "ferromagnetic" effect which translates into a slower thermal ss evolution. This effect had already been studied in the large N c limit with the use of QCD-motivated effective Lagrangian [12].
Thus, the second order relativistic virial expansion of the pressure for an inhomogeneous gas made of three species: i = π, K, η is [13,2]: where β = 1/T and ξ i = exp(−βm i ). Expanding up to the second order in ξ i means that we consider only binary interactions. For a free boson gas, B ij = 0 for i = j, whereas: The degeneracy is g i = 3, 4, 1 for π, K, η, respectively. The interactions appear through [13,7]: where K 1 is the first modified Bessel function and: δ ij I,J,S being the ij → ij phase shifts (chosen so that δ = 0 at threshold) of the elastic scattering of a state ij with quantum numbers I, J, S (J being the total angular momentum and S the strangeness). The virial expansion breaks around T ≃ 200 − 250 MeV [8]. Fortunately, the physics we are interested in occurs already at T < 250 MeV, and ξ π > ξ K ≃ ξ η , and it is enough to consider ij = ππ, πK and πη in B ij .
At this point we recall that the quark mass appears in the QCD Lagrangian, and therefore in the partition function, as m qαqα q α , where q α = u, d, s. Thus, in order to obtain the condensate we simply have to differentiate the partition function with respect to m qα . At finite temperature, the partition function is substituted by the free energy density z, so that [7] Note that we have separated the T = 0 part from the temperature dependent part, which is nothing but the pressure P = ǫ 0 − z, ǫ 0 being the vacuum energy density. Of course, at T = 0 we recover q α q α = 0|q α q α |0 = ∂ǫ 0 /∂m qα . Once more, we emphasize that in contrast with most thermodynamic quantities, for the chiral condensate it is not enough to know the δ(E), but we also need their dependence with the quark masses as well as a value for the T = 0 vacuum expectation value. We will obtain this information from ChPT. Nevertheless, ChPT does not deal with quarks, and thus we first have to rewrite the condensate, eq.(5), in terms of meson masses: where, as before, i = π, K, η, and we have defined: In the isospin limit, when m u = m d , the u and d condensates are equal, and we define 0|qq|0 ≡ 0|ūu +dd|0 . It is tedious but straightforward to obtain the c parameters above, whose explicit expressions can be found in [1]. The only relevant comment is that the c coefficients depend on the chiral parameters L k , for k = 4...8, and H 2 .
In Table I, we show two L k determinations from meson data, but it makes little difference to use other parameter sets. For H 2 we will use H r 2 (M ρ ) = (−3.4 ± 1.1)10 −3 , obtained as explained in [14] but using a more recent estimation of 0|ss|0 / 0|qq|0 = 0.75 ± 0.12 [15]. For instance, the c parameters obtained when using the L k in the first column of Table I are: The above cq q π is in good agreement with the SU(2) estimates: 0.85 and 0.90 ± 0.05 [7].
Finally, we have to introduce the meson-meson interactions in the second order virial coefficients through the phase shifts in eq.(4). These are nothing but the complex phases of the mesonmeson amplitudes, once they are projected in partial waves of definite isospin and angular momentum. The one loop ChPT amplitudes have been µ = M ρ Refs. [5,21] IAM  Table 1 Different sets of chiral parameters ×10 3 . In the first column L r 1 , L r 2 , L 3 are taken from [21] and the rest from [5] (L r 4 and L r 6 are estimated from the Zweig rule). The last column is the IAM fit [17] to meson-meson scattering up to 1.2 GeV.
given in [16,17]. As already commented within SU(2) ChPT it has been shown [7] that using just the amplitudes at threshold (scattering lengths) in the virial expansion is equivalent to expanding the partition function to third order in T /F or T /m π . This approach yields a fairly good representation of the pion gas thermodynamics at low temperatures, T ≪ 150 MeV [7]. Let us recall that ChPT provides a good low energy description (E < 500 MeV) of the meson-meson amplitudes [16].
The virial coefficients and ∂P/∂m i have been calculated numerically. For simplicity, in the figures, we have represented the chiral condensate over its vacuum expectation value, q α q α / 0|q α q α |0 , so that all of them are normalized to 1 at T = 0. As a matter of fact, since there is always an small explicit chiral symmetry breaking due to the quark masses, the condensate should only vanish completely in the T → ∞ limit (following with our analogy between the quark masses and the magnetic field, our ferromagnet above the Curie point becomes paramagnetic, but as if it was in the presence of a magnetic field, which still produces some magnetization.). Of course, with a virial expansion truncated at second order we cannot generate such an analytic behavior, and our curves become negative above some T , where the approach becomes clearly unreliable. Nevertheless, since the u and d masses are so small compared with the size of the nonstrange condensate at T=0 (less than 3%), it is a fairly good approximation to speak about a melting temperature in that case. For the ss condensate, we will only consider an extrapolated melting temperature to ease the comparison with the non-strange case, but that number does not have an actual physical meaning. In Figure 1 we show the results of using the one-loop ChPT phase shifts obtained using the central values of the parameters in the first column of Table I. For illustration, we have separated the different contributions. In particular, the thin continuous line represents the interacting pion gas in SU(2), (see [6] for the conversion of SU(3) into SU(2) parameters), whereas the thindashed line is the corresponding one-loop SU(3) result also for a pion gas. We have checked that the tiny difference between them comes only from the pure O(p 4 ) phase shift contribution, ( the amplitude, once the parameters are translated, is the same for SU(2) and SU(3) only up to s/M 2 K or s/M 2 η terms coming from the expansion of kaon or eta loops, which, of course, are not present in SU(2) [6]). This amounts to a 4 MeV decrease of the pion gas extrapolated melting temperature: T Our SU(3) result for a π, K, η gas, where also the kaons and etas interact with each other and the pions, is the thick continuous line, where we see that the additional decrease, basically due to the πK, πη interactions, amounts to 10-11 MeV. On a first glance, it may seem surprising that the πK, πη interaction effect comes out comparable or larger than that of free kaons and etas, since it is thermally suppressed by another exp(−m π /T ) factor. Note, however, that the exp(−m π /T ) suppression only amounts to a factor of 6, 4, 2.5 at T= 80,100,150 MeV, respectively. In contrast, the πK and πη interactions depend strongly on m π , which is much more sensitive tom than m K or m η , which carry the onlym dependence of the free K and η terms. In particular, it is easy to see that ∂m 2 π /∂m ≃ 2 ∂m 2 K /∂m and ∂m π /∂m ≃ 6 ∂m K /∂m, and this "temperature independent enhancement" competes with the thermal suppression, making the πK and πη interaction effect comparable to the free one, already at T = 80 − 100 MeV and even larger if T > 140 MeV.
Putting all the pieces together, the T qq m decrease in SU (3) is roughly 20 MeV, in agreement with the chiral limit lattice results quoted in [11]: T c = 173 MeV for SU (2) and T c = 154 MeV, for SU (3). The effect of using the real masses would be to increase both temperatures.
Furthermore, it can be noticed that ss melts much slower than qq , since m s ≫m. Indeed, there is still 70% left of the ss condensate at the qq melting point. Let us recall that both effects are already sizable at low temperatures T ≃ 100 MeV, where we can still trust pure ChPT. As commented above, only for illustration and to ease the comparison between curves and with previous works, we have extrapolated the condensates down to zero.
At this point we can also consider the uncertainties in the chiral parameters in Table I. By performing a Montecarlo gaussian sampling of the parameters within their errors, we can obtain a rather conservative estimate of our uncertainties (since the sampling is done assuming uncorrelated errors). We thus find: Note that the ss extrapolation is beyond the reliability region of the second order virial expansion, is just illustrative, and really only vanishes in the T → ∞ limit. Let us also remark that the extrapolated melting temperatures are strongly correlated, so that their difference has less uncertainty than simply adding in quadrature their respective errors. In particular, we find We remark once more that the ss melting temperature is, of course, only a crude extrapolation for illustrative purposes and to ease the comparison with the non-strange condensate evolution. Furthermore, we have estimated the effect of other, more massive, hadrons. In the SU(2) case [7,8] these states also included the kaons and etas, although assuming a rather large uncertainty in ∂M i /∂m qα . All in all this effect reduced T qq m by approximately 10-20 MeV. In the SU(3) case we have indeed calculated explicitly the kaon and etâ m dependence in the cq α qα parameters, thus reducing considerably the above uncertainty. Since the kaons and eta are treated explicitly, the other massive hadrons are heavier than m η , have a very low density, and their main contribution to the pressure comes from the first virial coefficient, i.e. the free gas. The only uncertainty is on ∂M h /∂m qα , conservatively estimated to lie within the number of valence quarks N qα and 2N qα . Thus, we have found that their contribution decreases T qq m by 7-12 MeV, respectively. Finally, and in order to estimate the effects of higher energies we can extend ChPT up to E ≃ 1.2 GeV by means of unitarization models [18,19,17]. These techniques resum the ChPT series respecting unitarity but also the low energy expansion, including the mass dependency. In particular, it has been shown that the coupled channel Inverse amplitude Method (IAM) provides a remarkable and accurate description of the complete meson-meson interactions below 1.2 GeV, generating dynamically six resonances from the one-loop ChPT expansion and a set of fitted L k compatible with other previous determinations. This approach can be extended sys-tematically to higher orders (for SU (2) case, see [20]).
Hence, in Figure 2 we show the results of using the one-loop coupled channel IAM fitted phase shifts. Its corresponding L i parameters are given in the last column of Table I, with two errors, the first, very small, is purely statistical, and the second covers the uncertainty in the parameters depending on what systematic error is assumed for the experimental data. Let us remark that, although the L k are highly correlated, the second, larger error, ignores completely these correlations, and should be considered as a very conservative range. The continuous line corresponds to the central values, and the dark shaded areas cover the one standard deviation uncertainty due to the small errors in the parameters. These areas have been obtained from a Montecarlo Gaussian sampling. The conservative ranges are covered by the light gray areas. We now find: The IAM results show that higher energy effects do not affect very much our conclusions and that the ChPT extrapolation from low temperatures seems robust.
To summarize, we have studied the SU(2) and SU(3) temperature evolution of the chiral condensates in the hadronic phase. The thermodynamics of the meson gas has been obtained from the virial expansion and Chiral Perturbation Theory. Our results clearly show a significant decrease, about 20-30 MeV, of the non-strange condensate melting temperature, from the SU(2) to the SU(3) case, similar to lattice results. Of these, about 6 MeV had already been explained with free kaons and etas, but the rest are mainly due to πK and πη interactions. We have also estimated the effect of heavier hadrons and of the third order virial coefficient. All in all we find T qq m = 201 +23 −11 MeV, In addition, our results show an slower temperature evolution of the strange condensate, shifted by about 80 MeV with respect to the non-strange, due to the different quark masses. More quantitatively, qq does not show a sizable melting up to T ≃ 150 MeV and it still remains about 80% when the non-strange is basically zero.
Both effects are clearly visible already at low temperatures. However, we have checked that their size is completely similar when using unitarized models.
These techniques should be easily extended to Heavy Baryon Chiral Perturbation Theory, in order to study the condensates with non-zero baryon density. In general the whole approach could be used with any effective Lagrangian formalism, in particular to study other QCD phase transitions like those of the color superconducting phases.
The author thanks A. Dobado