Bled Workshops in Physics Vol. 14, No. 1 p. 11 Hot and dense QCD medium and restoration of UA(1) symmetry* Dubravko Klabucara, Sanjin Benica, Davor Horvatica and Dalibor Kekezb a Physics Department, Faculty of Science, Zagreb University, Bijenička č. 32, Zagreb 10000, Croatia b Rudjer BoSkovic Institute, Bijenicka c. 54,10000 Zagreb, Croatia Abstract. Recent RHIC results on n' multiplicity in heavy-ion collisions are of great importance because they clearly signal a partial restoration of UA (1) symmetry at high temperatures, and thus provide an unambiguous signature of the formation of a new state of matter. To explain these experimental results of STAR and PHENIX collaborations, a minimal generalization of the Witten-Veneziano relation to finite temperatures was proposed. The present paper provides a detailed, pedagogical discussion and explanation thereof. After explaining why these results show that the zero-temperature Witten-Veneziano relation cannot be straightforwardly extended to temperatures T too close to the chiral restoration temperature TCh and beyond, we find the quantity which should replace, at T > 0, the Yang-Mills topological susceptibility appearing in the T = 0 Witten-Veneziano relation, in order to avoid the conflict with experiment at T > 0. This is illustrated through concrete T -dependences of pseudoscalar meson masses in a chirally well-behaved, Dyson-Schwinger approach, but our results and conclusions are of a more general nature and, essentially, model-independent. 1 Introduction and survey QCD excitations of our "ordinary", low-energy world are hadrons, namely baryons and mesons, wherein the fundamental QCD degrees of freedom - quarks and gluons - are confined. At low energies, meaning of the order of the typical hadronic scale - 1 GeV and below, QCD is strongly nonperturbative [1]. The confinement of quarks and gluons is not the only important nonperturbative phenomenon of QCD. Another one is the spontaneous, dynamical symmetry breaking of the axial subgroup SUA(Nf) of the chiral symmetry, where Nf = 3 is the number of the light quark flavors f = u, d, s. The most conspicuous manifestation of this breaking is the small mass of the octet of the light pseudoscalar mesons: pions (n0, n±), kaons (K0, K0, K±) and the n-meson. The smallness of their masses in comparison of the typical hadronic mass scales (such as the vector meson masses, and the nucleon mass mN - 1 GeV) illustrates that the chiral symmetry is a reasonable, although rough, approximate symmetry of the physics of light hadrons. In the chiral limit, i.e., in the limit of strictly vanishing Lagrangian quark masses * Talk delivered by D. Klabucar (which in reality slightly violate explicitly the chiral symmetry for light quark flavors q = u, d, s), these light pseudoscalar mesons would be strictly massless Goldstone bosons of the dynamical chiral symmetry breaking (DChSB). Nevertheless, the above leaves unexplained why the n '-meson is not the ninth light pseudoscalar meson, i.e., the ninth almost-Goldstone boson of DChSB of the axial UA(1) symmetry of the QCD Lagrangian. Instead, the n '-meson is somewhat more massive than nucleons due to the breaking of the classical UA (1) symmetry of QCD by the quantum effects known as the (gluon, or non-Abelian) axial anomaly of QCD. This completes our understanding of the pseudoscalar meson nonet composed of light quarks, which is one of characteristic emergent manifestations [1] of low-energy QCD. However, this low-energy QCD phase changes if hadrons get sufficiently hot or dense. Because of the asymptotic freedom of QCD at high energies, it is expected that sufficiently high temperatures (T), as well as densities, bring about not only deconfinement of quarks and gluons but also the [flavor SUA(3)] chiral symmetry and the UA(1) symmetry. Such circumstances were in the early universe, and maybe also in the compact stars. Also, heavy ion collisions in terrestrial laboratories, at RHIC and LHC, seem to find "melting" of hadrons under such extreme conditions into a new phase of matter - the quark-gluon plasma (QGP). It is however still nonperturbative and strongly interacting, and thus designated sQGP. The ongoing experiments at RHIC and LHC, and the future experiments at FAIR and NICA, will explore in detail the properties of the QGP phase in wide intervals of temperatures and densities, especially where the critical point of the QCD phase diagram is expected. One should stress that the studies of the QGP phase are intricate and difficult, and that clear, compelling signals for the creation of this new form of matter are very much needed. Such an unambiguous, "smoking gun" signal would be the change of a symmetry obeyed by the strong interaction, such as the restoration of the [SUa(3) flavor] chiral symmetry, or the UA(1) symmetry. The nonperturbativity of the low-energy QCD limits theoretical calculations to those on lattice and in effective models. Among the latter, those within the Schwinger-Dyson (SD) approach to the physics of quarks and gluons are especially prominent, since they have a strong basis in QCD. An example of a strong connection with the fundamental theory is that SD approach has the correct chiral behavior of QCD and also reproduces the pseudoscalar meson quark-antiquark (qq) bound states as the (almost-)Goldstone bosons of DChSB, as in QCD. The SD approach is able to achieve this even with the usage of simplified model interactions, such as the one in the Nambu-Jona Lasinio (NJL) model and its nonlocal generalizations, suitable for model studies of the QCD medium at high temperatures and densities. Using DS approach, we achieved successful description of pseudoscalar mesons at T = 0 using various models for nonperturba-tive QCD interactions (e.g., see [2,3] and refs. therein), notably including also the isoscalar complex of n-n' mesons [4-8]. The extension of the description of the whole meson nonet (i.e., including n and n' ) to T > 0 was also successfully achieved [9,10] using the class of models employing the separable approximation to gluon-exchange interactions. However, we noticed that the behavior of the complex of n-n' mesons at high T depends crucially on the interrelation between the dynamically broken chiral symmetry and the UA (1) symmetry and their restoration. The plan of this paper is as follows: the pertinent experimental situation is reviewed in the next section, Sec. 2. Then, Sec. 3 introduces two peculiar relations between quantities of two different theories, namely the full QCD and the pure-gauge, Yang-Mills (YM) theory. Thanks to these relations, Sec. 4 presents how recent RHIC results on increased n' multiplicity can be theoretically explained [11, 12] if the restoration of UA (1) symmetry is directly linked with the chiral symmetry restoration. The concrete mechanism proposed to accomplish this, amounts to expressing the Yang-Mills topological susceptibility through quark condensate. We conclude in Sec. 5. 2 Experimental status of UA(1) symmetry The ultrarelativistic heavy-ion collider facilities like RHIC at BNL and LHC at CERN strive to produce a new form of hot QCD matter. The experiments show [13,14] that it has very intricate properties and presents a big challenge especially for theoretical understanding. While above the (pseudo)critical temperature Tc - 170 MeV this matter is often called the quark-gluon plasma (QGP), it cannot be a perturbatively interacting quark-gluon gas (as widely expected before RHIC results [13,14]) until significantly higher temperatures T ^ Tc. Instead, the interactions and correlations in the hot QCD matter are still strong (e.g., see Refs. [15,16]) so that its more recent and more precise name is strongly coupled QGP (sQGP) [16]. One of its peculiarities seems to be that strong correlations in the form of qq bound states and resonances still exist [15,17] in the sQGP well above Tc. In the old QGP paradigm, even deeply bound charmonium (cc) states such as J/¥ and nc were expected to unbind at T « Tc, but lattice QCD simulations of mesonic correlators now indicate they persist till around 2Tc [18,19] or even above [20]. Similar indications for light-quark mesonic bound states are also accumulating from lattice QCD [21] and from other methods [15,22,23]. This agrees well with the findings on the lattice (e.g., see Ref. [24] for a review) that for realistic explicit chiral symmetry breaking (ChSB), i.e., for the physical values of the current quark masses, the transition between the hadron phase and the phase dominated by quarks and gluons, is not an abrupt, singular phase transition but a smooth, analytic crossover around the pseudocritical temperature Tc. It is thus not too surprising that a clear experimental signal of, e.g., deconfinement, is still hard to find and identify unambiguously. The most compelling signal for production of a new form of QCD matter, i.e., sQGP, would be a restoration - in hot and/or dense matter - of the symmetries of the QCD Lagrangian which are broken in the vacuum. One of them is the [SUA(Nf) flavor] chiral symmetry, whose dynamical breaking results in light, (almost-)Goldstone pseudoscalar (P) mesons - namely the octet P = n0,n±, K0, K0, K±,n, as we consider all three light-quark flavors, Nf = 3. The second one is the UA(1) symmetry. Its breaking by the non-Abelian axial Adler-Bell-Jackiw anomaly ('gluon anomaly' for short) makes the remaining pseudoscalar meson of the light-quark sector, the n', much heavier, preventing its appearance as the ninth (almost-)Goldstone boson of dynamical chiral symmetry breaking (DChSB) in QCD. The first experimental signature of a partial restoration of the UA(1) symmetry seems to have been found in the ^JsNN = 200 GeV central Au+Au reactions at RHIC. Namely, Csorgo et al. [25,26] analyzed combined data of PHENIX [27] and STAR [28] collaborations very robustly, through six popular models for hadron multiplicities, and found that at 99.9% confidence level, the n' mass, which in the vacuum is Mn / = 957.8 MeV, is reduced by at least 200 MeV inside the fireball. It is the sign of the disappearing contribution of the gluon axial anomaly to the n' mass, which would drop to a value readily understood together with the (flavor-symmetry-broken) octet of qq' (q, q' = u, d, s) pseudoscalar mesons. This is the issue of the "return of the prodigal Goldstone boson" predicted [29] as a signal of the UA(1) symmetry restoration. Another related but less obvious issue to which we want to draw attention, concerns the status, at T > 0, of the famous Witten-Veneziano relation (WVR) [30,31] Mn, + Mn - 2Mk = fM (i) 1 n between the n', n and K-meson masses Mn ',n,K, pion decay constant fn, Yang-Mills (YM) topological susceptibility xYM, and the number of the light quark flavors Nf = 3. WVR was obtained in the limit of large number of colors Nc [30,31]. It is well satisfied at T = 0 for xYM obtained by lattice calculations (e.g., [32-35]). Nevertheless, the T-dependence of xYM is such [9] that the straightforward extension of Eq. (1) to T > 0 [9], i.e., replacement of all quantities1 therein by their respective T-dependent versions Mn/ (T), Mn (T), MK(T), fn(T) andxYM(T), leads to a conflict with experiment [25,26]. Since this extension of Eq. (1) to T > 0 was studied in Ref. [9] before the pertinent experimental analysis [25,26], one of the purposes of this paper is to revisit the implications of the results of Ref. [9] for WVR at T > 0, and demonstrate explicitly that they are practically model-independent. The other, more important purpose is to propose a mechanism which can enable WVR to agree with experiment at T > 0. 3 Relations connecting two theories, QCD and YM Both issues pointed out before Eq. (1) and around it, are best understood in a model-independent way if one starts from the chiral limit of vanishing current quark masses (mq = 0) for all three light flavors, q = u, d, s. Then not only pions and kaons are massless, but is also n, which is then (since the situation is also SU(3)-flavor-symmetric) a purely SU(3)-octet state, n = n8. In contrast, n' is then purely singlet, n' = no; since the divergence of the singlet axial quark current qY^Ys-A°q is nonvanishing even for mq = 0 due to the gluon anomaly, 1 Throughout this paper, all quantities are for definiteness assumed at T = 0 unless their T-dependence is specifically indicated in formulas or in the text. the n' mass squared receives the anomalous contribution AM^ / (= A4/f / in the notation of Ref. [29]) which is nonvanishing even in the chiral limit: f4 = AM^0 = AMn, = + O( N-). (2) However, A4 and fn' are known accurately2 only in the large Nc limit. There, in the leading order in 1/Nc, A4 is given by the YM (i.e., "pure glue") topological susceptibility xYM times 2Nf = 6 [30,31], and the "n' decay constant" fn' is the same as fn [37]. Thus, keeping only the leading order in 1/Nc, the last equality is WVR in the chiral limit. The consequences of Eq. (2) remain qualitatively the same realistically away from the chiral limit. This will soon become clear on the basis of, e.g., Eq. (3) below. Namely, due to DChSB in QCD, for relatively light current quark masses mq (q = u, d, s), the qq' bound-state pseudoscalar meson masses (including the nonanomalous parts of the n' and n masses) behave as M^q' = const (mq + mq'), (q,q' = u,d,s). (3) The pseudoscalar mesons (including n') thus obtain relatively light nonanomalous contributions Mqq' to their masses MP, allowing them to reach the empirical values. That is, instead of the eight strictly massless Goldstone bosons, n°,n±, K0, K0, K± and n are relatively light almost-Goldstones. Among them, in the limit of isospin symmetry (mu = md), only n now receives also the gluon-anomaly contribution since the explicit SU(3) flavor breaking between the nonstrange (NS) u, d-quarks and s-quarks causes the mixing between the isoscalars n and n'. For mq = 0, Eq. (2) is replaced by the usual WVR (1) containing also the nonanomalous contributions to meson masses. Nevertheless, these contributions largely cancel due to the approximate SU(3) flavor symmetry and to DChSB [i.e., Eq. (3)]. This can be seen assuming the usual SU(3) qq content of the pseudoscalar meson nonet with well-defined isospin3 quantum numbers, in particular the iso-scalar (I = 0) octet and singlet etas, n8 = —p (uu + dd — 2ss), n° = —p (uu + dd + ss), (4) v 6 v 3 whose mixing yields the physical particles n and n'. Since the nonanomalous parts of the n° and n8 masses squared, M^ and M^, are respectively M°° « 3M^ + 3 Mn and M88 « f M2 — 1M^ (see, e.g., Ref. [7]), and since M^ + M^ = Mn + Mn', the nonanomalous parts of the n and n' masses are canceled by 2M^ in WVR (1). Another way of seeing this is expressing the nonanomalous parts of Mn + Mn' = Mn8 + Mn0 by Eq. (3). Thus again Mn + Mn' — 2M2 « AMn0, 2 Also note that a unique "n' decay constant" fn' is, strictly speaking, not a well-defined quantity, as two n' decay constants are actually needed: the singlet one, fn', and the octet one, fn'; e.g., see an extensive review [36] or the short Appendix of Ref. [5]. 3 The effects of the small difference between mu and md are not important for the present considerations. We thus stick to the isospin limit throughout the present paper. showing again that already WVR's chiral-limit-nonvanishing part (2) reveals the essence of the influence of the gluon anomaly on the masses in the n '-n complex. This is important also for the presently pertinent finite-T context because thanks to this, below it will be shown model-independently that WVR (1) containing the YM topological susceptibility x^m implies T-dependence of n' mass in conflict with the recent experimental results [25,26]. Namely, the anomaly contribution (2) is established at T = 0 but it is not expected to persist at high temperatures. Ultimately, n' should also become a massless Goldstone boson at sufficiently high T, where xym(T) —> 0. However, according to WVR, AMn/ (T) falls only for T where fn(T)2 does not fall faster than 6xym(T), as stressed in Ref. [9]. The WVR's chiral-limit version (2) manifestly points out the ratio XYM(T)/fn(T)2 as crucial for the anomalous n ' mass, but the above discussion shows that this remains essentially the same away from the chiral limit. In the present context, it is important for practical calculations to go realistically away from the chiral limit, in which the chiral restoration is a sharp phase transition at its critical temperature TCh where the chiral-limit pion decay constant vanishes very steeply, i.e., as steeply as the chiral quark condensate. In contrast, for realistic explicit ChSB, i.e., mu and md of several MeV, this transition is a smooth crossover (e.g., see Ref. [24]). For the pion decay constant, this implies that fn(T) still falls relatively steeply around pseudocritical temperature TCh, but less so than in the chiral case, and even remains finite, enabling the usage of WVR (1) for the temperatures across the chiral and UA(1) symmetry restorations. WVR is very remarkable because it connects two different theories: QCD with quarks and its pure-gauge, YM counterpart. The latter, however, has much higher characteristic temperatures than QCD with quarks: the "melting temperature" Tym where XYM(T) starts to decrease appreciably was found on lattice to be, for example, Tym - 260 MeV [38,39] or even higher, Tym - 300 MeV [40]. In contrast, the pseudocritical temperatures for the chiral and deconfinement transitions in the full QCD are lower than TYM by some 100 MeV or more (e.g., see Ref. [24]) due to the presence of the quark degrees of freedom. This difference in characteristic temperatures, in conjunction with XYM(T) in WVRs (1) and (2) would imply that the (partial) restoration of the UA(1) symmetry (understood as the disappearance of the anomalous no/n' mass) should happen well after the restoration of the chiral symmetry. But, this contradicts the RHIC experimental observations of the reduced n' mass [25,26] if WVRs (1), (2) hold unchanged also close to the QCD chiral restoration temperature TCh, around which fn(T) decreases still relatively steeply4 [9] for realistic explicit ChSB, thus leading to the increase of 6xYM (T)/fn(T)2 and consequently also of Mn /. There is still more to the relatively high resistance of XYM(T) to temperature: not only does it start falling at rather high TYM, but XYM(T) found on the lattice 4 Relative to decay constants of mesons containing a strange quark; e.g., compare fss (T) of the unphysical ss pseudoscalar with fn (T) in Fig. 1. is falling with T relatively slowly. In some of the applications in the past (e.g., see Refs. [41,42]), it was customary to simply rescale a temperature characterizing the pure-gauge, YM sector to a value characterizing QCD with quarks. (For example, Refs. [41,42] rescaled Tym = 260 MeV found by Ref. [38] to 150 MeV). However, even if we rescale the critical temperature for melting of the topological susceptibility xYM(T) from TYM down to TCh, the value of 6xYM(T)/fn(T)2 still increases a lot [9] for the pertinent temperature interval starting already below TCh. This happens because xYM(T) falls with T more slowly than fn(T)2. (It was found [9] that the rescaling of TYM would have to be totally unrealistic, to less than 70% of TCh, in order to achieve sufficiently fast drop of the anomalous contribution that would allow the observed enhancement in the n' multiplicity.) These WVR-induced enhancements of the n' mass for T - TCh were first noticed in Ref. [9]. This reference used a concrete dynamical model (with an effective, rank-2 separable interaction, convenient for computations at T > 0) [43] of low energy, nonperturbative QCD to obtain mesons as q q' bound states in SD approach [44-46], which is a bound-state approach with the correct chiral behavior (3) of QCD. Nevertheless, this concrete dynamical SD model was used in Ref. [9] to get concrete values for only the nonanomalous parts of the meson masses, but was essentially not used to get model predictions for the mass contributions from the gluon anomaly, in particular xYM(T). On the contrary, the anomalous mass contribution was included, in the spirit of 1/Nc expansion, through WVR (1). Thus, the T-evolution of the n '-n complex in Ref. [9] was not dominated by dynamical model details, but by WVR, i.e., the ratio 6xym(T)/fn (T)2. Admittedly, fn(T) was also calculated within this model, causing some quantitative model dependence of the anomalous mass in WVR, but this cannot change the qualitative observations of Ref. [9] on the n' mass enhancement. Namely, our model fn(T), depicted as the dash-dotted curve in Fig. 1, obviously has the right crossover features [24]. It also agrees qualitatively with fn(T)'s calculated in other realistic dynamical models [22,45]. Various modifications were tried in Ref. [9] but could not reduce much the n ' mass enhancement caused by this ratio, let alone bring about the significant n' mass reduction found in the RHIC experiments [25,26]. One must therefore conclude that either WVR breaks down as soon as T approaches TCh, or that the T-dependence of its anomalous contribution is different from the pure-gauge xYM(T). We will show that the latter alternative is possible, since WVR can be reconciled with experiment thanks to the existence of another relation which, similarly to WVR, connects the YM theory with full QCD. Namely, using large-Nc arguments, Leutwyler and Smilga derived [37], at T = 0, xYM = 1 , x x Nf (= x) ' (5) 1 + x m (q q) o the relation (in our notation) between the YM topological susceptibility xYM, and the full-QCD topological susceptibility x, the chiral-limit quark condensate (qq}0, and m, the harmonic average of Nf current quark masses mq. That is, m is Nf times the reduced mass. In the present case of Nf = 3, q = u, d, s, so that * = L -• (6) m ^— m _ mm q=u,d,s q T/Toh Fig. 1. The relative-temperature dependences, on T/TCh, of X1/4, (q q)<°/3, fn and fss, i.e., the T/Tch-dependences of the quantities entering in the anomalous contributions to various masses in the r| '-r| complex - see Eq. (10) and formulas below it. The solid curve depicts X1/4 for 5 = 0 in Eq. (10), and the short-dashed curve is X1/4 for 5 = 1. At T = 0, the both X's are equal to x\m = (0.1757 GeV)4, the weighted average [9] of various lattice results for Xym. The dotted (red) curve depicts — (qq)0/3, the dash-dotted (blue) curve is fn, and the long-dashed (blue) curve is fss. (Colors in the electronic version.) Leutwyler-Smilga relation, Eq. (5), is a remarkable relation between the two pertinent theories. For example, in the limit of all very heavy quarks (mq —» oo, q = u, d, s), it correctly leads to the result that x^m is equal to the value of the topological susceptibility in quenched QCD, xYM = x(mq = oo). This holds because x is by definition the vacuum expectation value of a gluonic operator, so that the absence of quark loops would leave only the pure-gauge, YM contribution. However, the Leutwyler-Smilga relation (5) also holds in the opposite (and presently pertinent) limit of light quarks. This limit still presents a problem for getting the full-QCD topological susceptibility x on the lattice [47], but we can use the light-quark-sector result [37,48] x = - mf + C- , (7) where C— stands for corrections of higher orders in small mq, and thus of small magnitude. The leading term is positive (as (qq)o < 0), but C— is negative, since Eq. (5) shows that x < min(-m (qq)0/Nf,xYM). Although small, C— should not be neglected, since C— = 0 would imply, through Eq. (5), that xYM = oo. Instead, its value (at T = 0) is fixed by Eq. (5): C- = C-(0) = ^ 0 - XYM ^-V1 . (8) Nf V m (q qW 4 n—n' complex at high temperatures All this starting from Eq. (5) has so far been at T = 0. If the left- and right-hand side of Eq. (5) are extended to T > 0, it is obvious that the equality cannot hold at arbitrary temperature T > 0. The relation (5) must break down somewhere close to the (pseudo)critical temperatures of full QCD (~ TCh) since the pure-gauge quantity x^m is much more temperature-resistant than the right-hand side, abbreviated as X. The quantity X, which may be called the effective susceptibility, consists of the full-QCD quantities x and (qq)o, the quantities of full QCD with quarks, characterized by TCh, just as fn(T). As T —» TCh, the chiral quark condensate (qq)0(T) drops faster than the other DChSB parameter in the present problem, namely fn(T) for realistically small explicit ChSB. (See Fig. 1 for the results of the dynamical model adopted here from Ref. [9], and, e.g., Refs. [22,45] for analogous results of different SD models). Thus, the troublesome mismatch in T-dependences of fn(T) and the pure-gauge quantity xym(T), which causes the conflict of the temperature-extended WVR with experiment at T > Ch, is expected to disappear if xym(T) is replaced by X(T), the temperature-extended effective susceptibility. The successful zero-temperature WVR (1) is, however, retained, since Xym = X at T = 0. Extending Eq. (7) to T > 0 is something of a guesswork as there is no guidance from the lattice for x(T) [unlike xym(T)]. Admittedly, the leading term is straightforward as it is plausible that its T-dependence will simply be that of (qq)0(T). Nevertheless, for the correction term Cm such a plausible assumption about the form of T-dependence cannot be made and Eq. (8), which relates YM and QCD quantities, only gives its value at T = 0. We will therefore explore the T-dependence of the anomalous masses using the following Ansatz for the T > 0 generalization of Eq. (7): X(T )=- + Cm(0) N f (qq)o(T) (q q)o(T = 0) (9) where the correction-term T-dependence is parametrized through the power 6 of the presently fastest-vanishing (as T —» TCh) chiral order parameter (qq)0(T). The T > 0 extension (9) of the light-quark x, Eq. (7), leads to the T > 0 extension of X: ) = m (qq)o(T) (1 - m (qq)o(T) X( ) Nf 1 NfCm(0) (q q)o(T = 0) (q q)o(T) We now use X(T) in WVR instead of xYM(T) used by Ref. [9]. This gives us the temperature dependences of the masses in the n-n' complex, such as those in Fig. 2 and in Fig. 3, illustrating the respective cases 6 = 0 and 6 = 1. It is clear that X(T) (10) blows up as T —» TCh if the correction term there vanishes faster than (qq)o(T) squared. Thus, varying 6 between 0 and 2 covers the cases from the T-independent correction term, to (already experimentally excluded) enhanced anomalous masses for 6 noticeably above 1, to even sharper mass blow-ups for 6 —» 2 when T —» TCh. On the other hand, it does not seem b b ^ym =(0.1757 GeV)4, 5=0 1.0 H 2nT " 0.8 Hs 1 0.6 ss H8 _ _ CL ^ / 0.4 / 0.2 n ____ 0.6 0.8 1.0 1.2 T/Toh Fig. 2. The relative-temperature dependence, on T/Tch, of the pseudoscalar meson masses for x(T), namely Eq. (10), with 5 = 0. The meaning of the symbols is as follows: the masses of n' and n are, respectively, the upper and lower solid curve, those of the pion and nonanomalous ss pseudoscalar are, respectively, the lower and upper dash-dotted curve, Mn 0 and Mn 8 are, respectively, the short-dashed (red) and long-dashed (red) curve, Mn NS is the medium-dashed (blue), and Mns is the dotted (blue) curve. (Colors in the electronic version.) The straight line 2nT is twice the lowest Matsubara frequency. natural that the correction term vanishes faster than the fastest-vanishing order parameter (qq)o(T). Indeed, already for the same rate of vanishing of the both terms (5 = 1), one can notice in Fig. 3 the start of the precursors of the blow-up of various masses in the q'-q complex as T —» TCh although these small mass bumps are still experimentally acceptable. Thus, in Fig. 3 we depict the 5 = 1 case, and the case with 5 = 0 (T-independent correction term) is depicted in Fig. 2 as the other acceptable extreme. Since they turn out to be not only qualitatively, but also quantitatively so similar that the present era experiments cannot discriminate between them, there is no need to present any 'in-between results', for 0 < 5 < 1. To clarify completely how the results in Figs. 2 and 3 were obtained, we now give some additional explanations. Using x(T) in WVR instead of xYM(T) used by Ref. [9], does not change anything at T = 0, where X(T) = xYM(0), which remains an excellent approximation even well beyond T = 0. Nevertheless, this changes drastically as T approaches TCh. For T - TCh, the behavior of X(T) is dominated by the T-dependence of the chiral condensate, tying the restoration of the UA(1) symmetry to the chiral symmetry restoration. > = ^[M2NS - M2S]2 + 8p2X2 , p = 1 6x 2 + X2 f2 M2 = Mn + 2p , M2 = M2S + px2 , X ^ f 1 2 M^ = M0O + 3(2 + X)2 p, < = M2S + 3(1 - X)2 p, 2 m25 +1 mn , m2o =1 m25 + 3 In all expressions after Eq. (11), the T-dependence is understood. In both cases considered for the topological susceptibility (9) [6 = 0, i.e., the constant correction term, and 6 = 1, i.e., the strong T-dependence 0. We have confirmed the results of Ref. [9] on WVR where the ratio xw(T)/fn(T)2 dominates the T-dependence, and clarified that these results are practically model-independent. It is important to note the difference between our approach and those that attempt to give model predictions for topological susceptibility, such as Refs. [49,50]. By contrast, in Refs. [9] and here, as well as earlier works [4-7] at T = 0, a SD dynamical model is used (as far as masses are concerned) to obtain only the nonanomalous part of the light pseudoscalar meson masses (where the model dependence is however dominated by their almost-Goldstone character), while the anomalous part of the masses in the n '-n complex is, through WVR, dictated by 6xYM/fn In this ratio, fn(T) is admittedly model-dependent in quantitative sense, but other realistic models yield qualitatively similar crossover behaviors [51] of fn(T) for mq = 0, as exemplified by our Fig. 1, and Fig. 2 in Ref. [9], and by Fig. 6 in Ref. [22]. Such fn(T) behaviors are also in agreement with the T-dependence expected of the DChSB order parameter on general grounds: a pronounced fall-off around TCh - but exhibiting, in agreement with lattice [24], a smooth crossover pattern for nonvanishing explicit ChSB, a crossover which gets slower with growing mq [e.g., compare fn(T) with fs5 (T) in Fig. 1]. In contrast to the QCD topological susceptibility x, the YM topological susceptibility and even its T-dependence xym(T), including its "melting" temperature TYM, can be extracted [9] reasonably reliably from the lattice [34,38]. Thus, it was not modeled in Ref. [9]. Hence our assertion that the results of Ref. [9] unavoidably imply that the straightforward extension of WVR to T > 0 is falsified by experiment [25,26], especially if one recalls that even the sizeable T-rescaling [41,42] TYM —» TCh was among the attempts to control the n' mass enhancement [9]. Nevertheless, we have also shown that there is a plausible way to avoid these problems of the straightforward, naive extension of WVR to T > 0, and this is the main result of the present paper. Thanks to the existence of another relation, Eq. (5), connecting the YM quantity xYM with QCD quantities x and (qq}°, it is possible to define a quantity, x, which can meaningfully replace x^m (T) in finite-T WVR. It remains practically equal to xYM up to some 70% of TCh, but beyond this, it changes following the T-dependence of (qq}° (T). In this way, the successful zero-temperature WVR is retained, but the (partial) restoration of UA(1) symmetry [understood as the disappearing contribution of the gluon anomaly to the n' (n°) mass] is naturally tied to the restoration of the SUA(3) flavor chiral symmetry and to its characteristic temperature TCh, instead of TYM. It is very pleasing that this fits in nicely with the recent ab initio theoretical analysis using functional methods [52], which finds that the anomalous breaking of UA(1) symmetry is related to DChSB (and confinement) in a self-consistent manner, so that one cannot have one of these phenomena without the other. Of course, the most important thing is that this version of the finite-T WVR, obtained by xYM(T) —> x(T), is consistent with experiment [25,26] for all reasonable strengths of T-dependence [0 < 6 < 1 in Eq. (9)]. Namely, the both Figs. 2 and 3 show, first, that n' mass close to TCh suffers the drop of more than 200 MeV with respect to its vacuum value. This satisfies the minimal experimental requirement abundantly. Second, Figs. 2 and 3 show an even larger drop of the n° mass, to some 400 MeV, close to the "best" value of the in-medium n' mass (340 MeV, albeit with large errors) obtained by Csorgo et al. [25,26]. 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