η ′ and η mesons at high T when the U_A(1) and chiral symmetry breaking are tied
ηη (cid:48) and η mesons at high T when the U A (1) and chiral symmetry breaking are tied Davor Horvati´c, Dalibor Kekez, and Dubravko Klabuˇcar Physics Department, Faculty of Science, University of Zagreb, Bijeniˇcka cesta 32, 10000 Zagreb, Croatia Rugjer Boˇskovi´c Institute, Bijeniˇcka cesta 34, 10000 Zagreb, Croatia (Dated: December 19, 2018)The approach to the η (cid:48) - η complex employing chirally well-behaved quark-antiquark bound statesand incorporating the non-Abelian axial anomaly of QCD through the generalization of the Witten-Veneziano relation, is extended to finite temperatures. Employing the chiral condensate has led toa sharp chiral and U A (1) symmetry restoration; but with the condensates of quarks with realisticexplicit chiral symmetry breaking, which exhibit a smooth, crossover chiral symmetry restorationin qualitative agreement with lattice QCD results, we get a crossover U A (1) transition, with smoothand gradual melting of anomalous mass contributions. This way we obtain a substantial drop ofthe η (cid:48) mass around the chiral transition temperature, but no η mass drop. This is consistent withpresent empirical evidence. I. INTRODUCTION
The experiments on heavy-ion collider facilities, suchas RHIC, LHC, FAIR, and NICA, aim to produce a newform of hot and/or dense QCD matter [1, 2]. Clear signa-tures of its production are thus very much needed. Themost compelling such signal would be a change of a per-tinent symmetry, i.e. , restoration - in hot and/or densematter - of the symmetries of the QCD Lagrangian whichare broken in the vacuum, notably the [ SU A ( N f ) flavor]chiral symmetry for N f = 3 = 2 + 1 light quark flavors q ,and the U A (1) symmetry. This provides a lot of motiva-tion to establish that experiment indeed shows it, as wellas to give theoretical explanations of such phenomena.The first signs of a (partial) restoration of the U A (1)symmetry were claimed to be seen in 200 GeV Au+Aucollisions [3, 4] at RHIC by Cs¨org˝o et al. [5]. They an-alyzed the η (cid:48) -meson data of PHENIX [3] and STAR [4]collaborations through several models for hadron multi-plicities, and found that the η (cid:48) mass ( M η (cid:48) = 957 . η (cid:48) is, comparatively, so very mas-sive since it is predominantly the SU V ( N f )-flavor singletstate η . Its mass M η receives a sizable anomalous con-tribution ∆ M η due to the U A (1) symmetry violation bythe non-Abelian axial Adler-Bell-Jackiw anomaly (‘gluonanomaly’ or ‘ U A (1) anomaly’ for short), which makes thedivergence of the singlet axial quark current ¯ qγ µ γ λ q nonvanishing even in the chiral limit of vanishing cur-rent masses of quarks, m q →
0. The said mass drop isthen the sign of a partial U A (1) symmetry restoration inthe sense of diminishing contribution of U A (1) anomalyto the η (cid:48) mass, which would drop to a value readily un-derstood in the same way [6] as the masses of the octetof the light pseudoscalar mesons P = π , ± , K , ± , ¯ K , η ,which are exceptionally light almost-Goldstone bosons ofDynamical Chiral Symmetry Breaking (DChSB).Now, there is a new experimental paper [7] on 200 GeVAu+Au collisions. Although a new analysis of the limitson η (cid:48) and η masses was beyond the scope of Ref. [7], thedata contained therein make it possible, and preliminary considerations [8] confirm the findings of Refs. [5].The first explanation [9] of these original findings [5]was offered by conjecturing that the Yang-Mills (YM)topological susceptibility, which leads to the anomalouslyhigh η (cid:48) mass, should be viewed through the Leutwyler-Smilga (LS) [10] relation (12). This ultimately impliesthat the anomalous part of the η (cid:48) mass falls together withthe quark-antiquark ( q ¯ q ) chiral-limit condensate (cid:104) ¯ qq (cid:105) ( T )as the temperature T grows towards the chiral restorationtemperature T Ch and beyond. This tying the U A (1) sym-metry restoration with the chiral symmetry one, was justa conjecture until our more recent paper [11] strength-ened the support for this scenario. Nevertheless, therewas also a weakness: our approach predicted the drop ofnot only the η (cid:48) mass, but also (even more drastically) ofthe η mass M η , and signs for that have not been seen inany data, including the new [7] and the newest [12]. Inthe present paper, we show that the predicted [9] dropof M η was the consequence of employing the chiral-limitcondensate (cid:104) ¯ qq (cid:105) ( T ), since it falls too fast with T afterapproaching T ∼ T Ch . We then perform T > et al. [11], where LS relation (12) is replaced by the full QCDtopological charge parameter (18) [13–15]. There one canemploy q ¯ q condensates for realistically massive u, d and s -quarks, with much smoother T -dependence. As a re-sult, the description of the η - η (cid:48) complex of Ref. [9] issignificantly improved, since our new T -dependences ofthe pseudoscalar meson masses do not exhibit a drop ofthe η mass, while a considerable drop of the η (cid:48) mass stillexists, consistently with the empirical findings [5]. II. A SURVEY OF THE η - η (cid:48) COMPLEX
The light pseudoscalar mesons are both q ¯ q (cid:48) bound states( q, q (cid:48) = u, d, s ), and, simultaneously, (almost-)Goldstonebosons of DChSB of nonperturbative QCD. The ap-proach which simultaneously implements both , is the onethrough the Dyson-Schwinger (DS) equations for Greenfunctions of QCD. (See, e.g., Refs. [16–19] for reviews.)Presently pertinent is the gap equation for dressed quark a r X i v : . [ h e p - ph ] D ec propagators S q ( p ) with DChSB-generated self-energiesΣ q ( p ): S − q ( p ) = S free q ( p ) − − Σ q ( p ) , ( q = u, d, s ) , (1)(while S free q are free ones), and the Bethe-Salpeter equa-tion (BSE) for the q ¯ q (cid:48) meson bound-state vertices Γ q ¯ q (cid:48) :Γ q ¯ q (cid:48) ( k, p ) ef = (2) (cid:90) [ S q ( (cid:96) + p q ¯ q (cid:48) ( (cid:96), p ) S q (cid:48) ( (cid:96) − p gh K ( k − (cid:96) ) hgef d (cid:96) (2 π ) , where K is the interaction kernel, and e, f, g, h represent(schematically) collective spinor, color and flavor indices.This nonperturbative and covariant bound-state DSapproach can be applied at various degrees of trunca-tions, assumptions and approximations, ranging from abinitio QCD calculations and sophisticated truncations( e.g., see [16–22] and references therein) to very simpli-fied modeling of hadron phenomenology, such as utiliz-ing Nambu–Jona-Lasinio point interaction. For applica-tions in involved contexts such as nonzero temperatureor density, strong simplifications are especially neededfor tractability. This is why the separable approximation[23] is adopted presently [see more between Eqs. (4)-(5)].However, for describing pseudoscalar mesons (including η and η (cid:48) ), reproducing the correct chiral behavior of QCDis much more important than dynamics-dependent de-tails of their internal bound-state structure.As a rarity among bound-state approaches, the DS onecan achieve the correct QCD chiral behavior – also re-gardless of details of modeling dynamics, but under thecondition of a consistent truncation of DS equations, re-specting pertinent Ward-Takahashi identities [16–19]. Aconsistent DS truncation, where DChSB is very well un-derstood, is the rainbow-ladder approximation (RLA).Since it also enables tractable calculations, it is still themost usual approximation in phenomenological applica-tions, and we also adopt it here. In RLA, the BSE(2) employs the dressed quark propagator solution S ( p )from the gap equation (1)&(4), which in turn employsthe same effective interaction kernel as the BSE. It hasthe simple gluon-exchange form, where both quark-gluonvertices are bare:[ K ( k )] hgef = i g D abµν ( k ) eff [ λ a γ µ ] eg [ λ b γ ν ] hf , (3)so that the quark self-energy in the gap equation isΣ q ( p ) = − (cid:90) d (cid:96) (2 π ) g D abµν ( p − (cid:96) ) eff λ a γ µ S q ( (cid:96) ) λ b γ ν , (4)where D abµν ( k ) eff is an effective gluon propagator.These simplifications should be compensated by mod-eling the effective gluon propagator D abµν ( k ) eff in order toreproduce well the relevant phenomenology; here, pseu-doscalar ( P ) meson masses M P , decay constants f P , andcondensates (cid:104) ¯ qq (cid:105) , including T -dependence of all these. Inthe present paper, we use the same model as in Ref. [9], whose approach to the T -dependence of U A (1) anomalywe now seek to improve. All details on the functionalform and parameters of this model interaction can befound in the subsection II.A of Ref. [24]. Such mod-els, so-called rank-2 separable, are phenomenologicallysuccessful (see, e.g. , Refs. [23–27]), except they havethe well-known drawback of predicting a somewhat toolow transition temperature: the model we use presentlyand in Refs. [9, 24, 26, 27], has T Ch = 128 MeV, i.e. ,some 17% below the now widely accepted central valueof 154 ± T Ch and the temperaturescales characterizing signs of effective disappearance ofthe U A (1) anomaly, for which the present model is ade-quate. In addition, Ref. [31] shows that coupling to thePolyakov loop can increase T Ch , while qualitative featuresof the T -dependence of the model are preserved. Thus,separable model results at T >
T /T Ch ,as in Refs. [9, 24].Anyway, regardless of details of model dynamics, i.e., of a choice of D abµν ( k ) eff , but just thanks to the consistenttruncation of DS equations, the BSE (2) yields the masses M q ¯ q (cid:48) of pseudoscalar P ∼ q ¯ q (cid:48) mesons which satisfy theGell-Mann-Oakes-Renner–type relation with the currentmasses m q , m q (cid:48) of the corresponding quarks: M q ¯ q (cid:48) = const ( m q + m q (cid:48) ) , ( q, q (cid:48) = u, d, s ) . (5)While this guarantees all M q ¯ q (cid:48) → U A (1)-anomalous contribution responsible for ∆ M η . Thatis, RLA gives us only the non-anomalous part ˆ M NA of the squared-mass matrix ˆ M = ˆ M NA + ˆ M A of thehidden-flavor ( q = q (cid:48) ) light ( q = u, d, s ) pseudoscalarmesons. In the basis { u ¯ u, d ¯ d, s ¯ s } , ˆ M NA is simply ˆ M NA =diag[ M u ¯ u , M d ¯ d , M s ¯ s ]. The anomalous part ˆ M A arisesbecause the pseudoscalar hidden-flavor states q ¯ q arenot protected from the flavor-mixing QCD transitions(through anomaly-dominated pseudoscalar gluonic inter-mediate states), depicted in Fig. 1. They are obviouslybeyond the reach of RLA and horrendously hard to cal-culate. Nevertheless, they cannot be neglected, as canbe seen in the Witten-Veneziano relation (WVR) [32, 33]which remarkably relates the full-QCD quantities ( η (cid:48) , η and K -meson masses M η (cid:48) ,η,K and the pion decay con-stant f π ), to the topological susceptibility χ YM of the(pure-gauge) YM theory: M η (cid:48) + M η − M K = 2 N f χ YM f π = M U A (1) . (6)Namely, its chiral-limit-nonvanishing right-hand-side(RHS) is large, roughly 0.8 to 0.9 GeV , while Eq.(5) leads basically to the cancellation of all chiral-limit-vanishing contributions on the left-hand-side (LHS) [9].RHS is the WVR result for the total mass contributionof the U A (1) anomaly to the η - η (cid:48) complex, M U A (1) .The ˆ M A matrix elements generated by the U A (1)-anomaly-dominated transitions q ¯ q → q (cid:48) ¯ q (cid:48) (see Fig. 1)can be written [34] in the flavor basis { u ¯ u, d ¯ d, s ¯ s } as (cid:104) q ¯ q | ˆ M A | q (cid:48) ¯ q (cid:48) (cid:105) = b q b q (cid:48) , ( q, q (cid:48) = u, d, s ) . (7)Here b q = √ β for both q = u, d since we assume m u = m d ≡ m l , i.e., isospin SU (2) symmetry, which isan excellent approximation for most purposes in hadronicphysics. For example, M u ¯ u = M d ¯ d ≡ M l ¯ l = M u ¯ d ≡ M π obtained from BSE (2) is our RLA model pion mass for π + ( π − ) = u ¯ d ( d ¯ u ) and π = ( u ¯ u − d ¯ d ) / √
2, so thatˆ M NA = diag[ M π , M π , M s ¯ s ]. It still contains M s ¯ s , themass of the unphysical, but theoretically very useful s ¯ s pseudoscalar obtained in RLA. However, thanks to Eq.(5), it can also be expressed through the masses of phys-ical mesons, M s ¯ s = 2 M u ¯ s − M u ¯ d = 2 M K − M π , in a verygood approximation [24, 27, 34–37]. Its decay constant f s ¯ s is calculated in the same way as f π and f K .Since the s -quark is much heavier than u and d ones,in Eq. (7) we have b q = X √ β for q = s , with X < s -quarksare suppressed, and the quantity X expresses this influ-ence of the SU (3) flavor symmetry breaking. The mostusual choice for the flavor-breaking parameter had been[9, 24, 27, 34–37] the educated estimate X = f π /f s ¯ s ,but we found [11] it necessarily follows in the variant ofour approach relying on Shore’s [13, 14] generalization ofWVR (6) – see Sec. III.The anomalous mass matrix ˆ M A , which is of the pair-ing form (7) in the hidden-flavor basis { u ¯ u, d ¯ d, s ¯ s } , inthe octet-singlet basis { π , η , η } of hidden-flavor pseu-doscalars becomesˆ M A = β (1 − X ) √ (2 − X − X )0 √ (2 − X − X ) (2 + X ) , (8)showing that the SU (3) flavor breaking, X (cid:54) = 1, is nec-essary for the anomalous contribution to the η mass P q q ′ ¯ q ¯ q ′ P ′ FIG. 1. Axial-anomaly-induced, flavor-mixing transitionsfrom hidden-flavor pseudoscalar states P = q ¯ q to P (cid:48) = q (cid:48) ¯ q (cid:48) include both possibilities q = q (cid:48) and q (cid:54) = q (cid:48) . All lines andvertices are dressed. The gray blob symbolizes all possibleintermediate gluon states enabling this. Three bold dots sym-bolize an even [36], but otherwise unlimited number of addi-tional gluons. As pointed out in Ref. [36], the diamond graphis just the simplest example of such a contribution. squared, ∆ M η = β (2 / − X ) . In the flavor SU (3)symmetric case, X = 1, only the η mass receives a U A (1)anomaly contribution: M U A (1) = ∆ M η = 3 β in thislimit. Otherwise, M U A (1) ≡ Tr ˆ M A = (2 + X ) β .The SU (3) breaking, X (cid:54) = 1, causes ˆ M A (8) to be off-diagonal, but in this basis also the { η , η } -submatrix ofˆ M NA gets strong, negative off-diagonal elements, M = √ M π − M s ¯ s ) / e.g. , [34]). Eq. (8) thus shows thatthe interplay of the flavor symmetry breaking ( X < M = ˆ M NA + ˆ M A , i.e., for getting the physical isoscalarsin a rough approximation as η ≈ η and η (cid:48) ≈ η . Howthis changes with diminishing U A (1)-anomaly contribu-tions is exhibited in Secs. IV and V.Since the isospin-limit π decouples from the anomalyand mixing, only the isoscalar-subspace 2 × M needs to be considered. Even though ˆ M is stronglyoff-diagonal in the isoscalar NS-S basis { η NS , η S } , η NS η S ≡ √ ( u ¯ u + d ¯ d ) s ¯ s ≡ √ (cid:113) − (cid:113)
23 1 √ η η , (9)in this basis it has the simplest form:ˆ M ≡ M M M
2S NS M = M π + 2 β √ βX √ βX M s ¯ s + βX , (10)which also shows that when the U A (1)-anomaly contri-butions vanish ( i.e., β → η → η NS and η (cid:48) → η S , but that their respectivemasses become M π and M s ¯ s .Our experience with various dynamical models (at T = 0) shows [27, 34–37] that after pions and kaons arecorrectly described, a good determination of the anoma-lous mass shift parameter is sufficient for Eq. (10) to givegood η (cid:48) and η masses, since M s ¯ s = 2 M K − M π holds well.Nevertheless, calculating the anomalous contributions( ∝ β ) in DS approaches is a very difficult task. Ref.[38] explored it by taking the calculation beyond RLA,but had to adopt extremely schematic model interactions(proportional to δ -functions in momenta) for both theladder-truncation part (3) and the anomaly-producingpart. Another approach [39] obtained a qualitative agree-ment with lattice on χ YM (and consequently, acceptablemasses of η (cid:48) and η ) by assuming that contributions toFig. 1 are dominated by the simplest one, the diamondgraph, if it is appropriately dressed – in particular, by anappropriately singular quark-gluon vertex.We, however, take a different route, since our goal is not to figure out on a microscopic level how breakingof U A (1) comes about, but to phenomenologically modeland study the high- T behavior of masses of the realistic η (cid:48) and η , along with other light pseudoscalar mesons. InDS context, the most suitable approach is then the onedeveloped in Refs. [27, 34–37] and extended to T > U A (1) anomaly is suppressed in thelimit of large number of QCD colors N c [32, 33]. So, inthe sense of 1 /N c expansion, it is a controlled approx-imation to view the anomaly contribution as a pertur-bation with respect to the (non-suppressed) results ob-tained through RLA (3)-(4). While considering mesonmasses, it is thus not necessary to look for anomaly-induced corrections to RLA Bethe-Salpeter wavefunc-tions, which are consistent with DChSB and with thechiral QCD-behavior (5) essential for description of pionsand kaons. The breaking of nonet symmetry by U A (1)anomaly can be introduced just on the level of the massesin the η (cid:48) - η complex, by adding to the RLA-calculatedˆ M NA the anomalous contribution ˆ M A . Its anomaly massparameter β can be obtained by fitting [36] the empiri-cal masses of η and η (cid:48) , or better – because then no newfitting parameters are introduced – from lattice resultson YM topological susceptibility χ YM . Employing WVR(6) yields [9, 34] β = β WV , while Shore’s generalizationgives (see Sec. III) β = β Sho [11]: β WV = 6 χ YM (2 + X ) f π , β Sho = 2 Af π ≈ χ YM f π , (11)where A is the QCD topological charge parameter, givenbelow by Eq. (18) in terms of q ¯ q condensates of massivequarks, which turns out to be crucial for a realistic T -dependence of the masses in the η (cid:48) - η complex. III. EXTENSION TO T ≥ Extending our treatment [27, 34–37] of the η (cid:48) - η com-plex to T >
T > T -dependent versions. InWVR, these are the full-QCD quantities M η (cid:48) ( T ), M η ( T ), M K ( T ) and f π ( T ), but also χ YM ( T ), which is a pure-gauge, YM quantity and thus much more resistant to hightemperatures than QCD quantities containing also quarkdegrees of freedom. Indeed, lattice calculations indicatethat the fall of χ YM ( T ), from which one would expect thefall of the anomalous η (cid:48) mass, starts only at T some 100MeV (or even more) above the (pseudo)critical temper-ature T Ch for the chiral symmetry restoration of the full It is instructive to recall [35, 40] that nonet symmetry or brokenversion thereof is in fact assumed, explicitly or implicitly, byall approaches using the simple hidden-flavor basis q ¯ q , e.g., toconstruct the SU (3) states pseudoscalar meson states η and η without distinguishing between the q ¯ q states belonging to thesinglet from those belonging to the octet. An independent aposteriori support for our approach is also that η and η (cid:48) → γγ ( ∗ ) processes are described well [34–37]. QCD, around where decay constants already fall appre-ciably. It was then shown [24] that the straightforwardextension of the T -dependence of the YM susceptibilitywould predict even an increase of the η (cid:48) mass around andbeyond T Ch , contrary to experiment [5].It could be expected that at high T , original WVR(6) will not work since it relates the full-QCD quantitieswith a much more temperature-resistant YM quantity, χ YM ( T ). However, this problem can be eliminated [9] byusing, at T = 0, the (inverted) Leutwyler-Smilga (LS)relation [10]: χ YM = χ χ ( m u + m d + m s ) (cid:104) ¯ qq (cid:105) ≡ (cid:101) χ (12)to express χ YM in WVR (6) through the full-QCDtopological susceptibility χ and the chiral-limit conden-sate (cid:104) ¯ qq (cid:105) . The zero-temperature WVR is so retained,while the full-QCD quantities in (cid:101) χ do not have the T -dependence mismatch with the rest of Eq. (6). Thus, in-stead of χ YM ( T ), Ref. [9] used at T > (cid:101) χ ( T ) (12), where the QCD topological susceptibility χ inthe light-quark sector can be expressed as [10, 15, 41]: χ = − m u + m d + m s ) (cid:104) ¯ qq (cid:105) + C m . (13)This implies that the (partial) restoration of U A (1) sym-metry is strongly tied to the chiral symmetry restora-tion, since not χ YM ( T ), but (cid:104) ¯ qq (cid:105) ( T ), through (cid:101) χ ( T ) (12),determines the T -dependence of the anomalous parts ofthe masses in the η - η (cid:48) complex [9]. The dotted curvein Fig. 2 illustrates how (cid:104) ¯ qq (cid:105) ( T ) falls steeply to zeroas T → T Ch , indicative of the 2 nd order phase tran-sition. This behavior is followed closely by (cid:101) χ ( T ), andtherefore also by the anomaly parameter β WV ( T ) (11).This makes the mass matrix (10) diagonal immediatelyafter T = T Ch , which marks the abrupt onset of the NS-Sscenario M η (cid:48) ( T ) → M s ¯ s ( T ), M η ( T ) → M π ( T ) [9].In Eq. (13), C m denotes corrections of higher ordersin small m q , but should not be neglected, as C m (cid:54) = 0 isneeded to have a finite χ YM with Eqs. (12)-(13). They inturn give us the value C m at T = 0 in terms of q ¯ q conden-sate and the YM topological susceptibility χ YM . How-ever, to the best of our knowledge, the functional formof C m is not known. Ref. [9] thus tried various param-eterizations covering reasonably possible T -dependencesof C m ( T ), but this did not affect much the results for the T -dependence of the masses in the η (cid:48) - η complex.An alternative to WVR (6) is its generalization byShore [13, 14]. There, relations containing the massesof the pseudoscalar nonet mesons take into account that η and η (cid:48) should have two decay constants each [42]. Ifone chooses to use the η - η basis, they are f η , f η (cid:48) , f η , f η (cid:48) ,and can be equivalently expressed through purely octetand singlet decay constants ( f , f ) and two mixing an-gles ( θ , θ ). This may seem better suited for usageswith effective meson Lagrangians than with q ¯ q (cid:48) substruc-ture calculations starting from the (flavor-broken) nonet . . . . . . . . . T/T Ch A ll v a l u e s i n [ M e V ] A / χ / −h ¯ uu i / −h ¯ ss i / f π f s ¯ s −h ¯ qq i / FIG. 2. The relative-temperature
T /T Ch -dependences of thepertinent order parameters calculated in our usual [9, 24] sep-arable interaction model. The odd man out is (3 rd root ofthe absolute value of) the chiral condensate (cid:104) ¯ qq (cid:105) ( T ) fallingsteeply at T = T Ch and dictating similar behavior [9] to (cid:101) χ ( T )(12). All other displayed quantities exhibit smooth, crossoverbehaviors, the smoother the heavier the involved flavor is: thehighest curve (dash-dotted) and the second one from above(dashed) are (3 rd roots of the absolute values of) the con-densates (cid:104) ¯ ss (cid:105) ( T ) and (cid:104) ¯ uu (cid:105) ( T ), respectively, and the resultingtopological susceptibility χ ( T ) / (the thin solid curve, start-ing as the lowest) and topological charge parameter A ( T ) / (the upper, thick solid curve). The decay constants f π ( T ) and f s ¯ s ( T ) are, respectively, the lower dashed and dash-dottedcurves. (Colors online.) symmetry, such as ours. Nevertheless, Shore’s approachwas adapted also to the latter bound-state context, andsuccessfully applied there – in particular, to our DS ap-proach in RLA [27]. This was thanks to applying thesimplifying scheme of Feldmann, Kroll and Stech (FKS)[43, 44]. They showed that this “2 mixing angles for 4decay constants” formulation in the NS-S basis, althoughin principle equivalent to the η - η basis formulation,can in practice be more simplified down to one-mixing-angle scheme using plausible approximations based onthe Okubo-Zweig-Iizuka (OZI) rule. Namely, the decay-constant mixing angles in this basis are mutually close, φ S ≈ φ NS , and both approximately equal to the state mixing angle φ rotating the NS-S basis states into thephysical η and η (cid:48) mesons, η = cos φ η NS − sin φ η S , η (cid:48) = sin φ η NS +cos φ η S , (14)which diagonalizes the mass (squared) matrix (10).So, Ref. [27] solved numerically Shore’s equations(combined with the FKS approximation scheme) for me-son masses for several dynamical DS bound-state models[24, 34, 36]. Then, Ref. [11] presented analytic solutionsthereof, for the masses of η and η (cid:48) and the state NS-S mixing angle φ . These are longish, but closed-form ex-pressions in terms of non-anomalous meson masses M π , M K and their decay constants f π , f K , but also f NS and f S , the decay constants of the unphysical η NS and η S ,and, most notably, of the full QCD topological chargeparameter A . This is the quantity, taken over [13, 14]from Di Vecchia and Veneziano [15], which in the massrelations of Shore’s generalization has the role of χ YM inWVR. A will be considered in detail for the T > /N c [13, 14],Shore himself took advantage of A = χ YM + O ( 1 N c ) (at T = 0) , (15)and approximated A , as shall we at T = 0, by the latticeresult χ YM = (0 .
191 GeV) [45].Further, one should note that since the FKS schemeneglects OZI-violating contributions, that is, gluoniumadmixtures in η NS and η S , it is consistent to treat themas pure q ¯ q states, accessible by our BSE (2) in RLA.Then f NS = f π , and f S = f s ¯ s , the decay constant of theaforementioned “auxiliary” RLA s ¯ s pseudoscalar. Wecalculate its mass M s ¯ s through BSE, but at T = 0 it canalso be related to the measurable pion and kaon masses, M s ¯ s ≈ M K − M π , due to Eq. (5). Similarly, f s ¯ s can alsobe approximately expressed by these measurable quanti-ties as f s ¯ s ≈ f K − f π . Thus, up to taking A ≈ χ YM from lattice, Ref. [11] could calculate the η - η (cid:48) complexusing in its analytic solutions both the model-calculated,and also the empirical M π , M K , f π and f K . So, it [11]checked (independently of any model) the soundness ofour approach at T = 0.The analytic solutions of Ref. [11] also lead to thesimple elements of the mass matrix (10): M = M π + 4 Af π , M = 2 √ Af π f s ¯ s (16) M = M s ¯ s + 2 Af s ¯ s , (17)implying X = f π /f s ¯ s , M U A (1) = 4 A/f π + 2 A/f s ¯ s and β Sho in Eq. (11). The approximation A = χ YM (15)with χ YM = (0 .
191 GeV) from lattice [45] then yields M η (cid:48) = 997 MeV and M η = 554 MeV at T = 0.Since the adopted DS model enables the calculationsof non-anomalous q ¯ q masses and decay constants also for T >
0, the only thing still missing is the T -dependence ofthe full QCD topological charge parameter A , as χ YM ( T )is inadequate. But, A is used to express the QCD suscep-tibility χ through the “massive” condensates (cid:104) ¯ uu (cid:105) , (cid:104) ¯ dd (cid:105) and (cid:104) ¯ ss (cid:105) , i.e. , away from the chiral limit, in contrast to re-lations (12) and (13), e.g. , see Eq. (2.12) in Ref. [13]. Itsinverse, expressing A , thus also contains the q ¯ q conden-sates out of the chiral limit for all light flavors q = u, d, s , A = χ χ ( m u (cid:104) ¯ uu (cid:105) + m d (cid:104) ¯ dd (cid:105) + m s (cid:104) ¯ ss (cid:105) ) , (18)and so should χ in (18). That is, the light-quark expres-sion for the QCD topological susceptibility in the contextof Shore’s approach should be expressed by the currentmasses m q multiplied by respective condensates (cid:104) ¯ qq (cid:105) re-alistically away from the chiral limit: χ = − m u (cid:104) ¯ uu (cid:105) + m d (cid:104) ¯ dd (cid:105) + m s (cid:104) ¯ ss (cid:105) + C m . (19)As before [9], the small-magnitude and necessarily nega-tive correction term C m is found by assuming A = χ YM at T = 0. This large- N c approximation also recovers theLS relation (12) easily: by approximating the realisti-cally massive condensates with (cid:104) ¯ qq (cid:105) everywhere in Eq.(18), the QCD topological charge parameter A reducesto (cid:101) χ , justifying the conjecture of Ref. [9] tying the U A (1)symmetry restoration with the chiral symmetry one.This connection between the two symmetries is stillpresent. However, with the massive condensates wealso get a more realistic, crossover T -dependence of themasses, depicted in Figs. 3 and 4, and presented in Sec.IV.The two Figs. 3 and 4 correspond to two variationsof the unknown T -dependence C m ( T ) of the correctionterm in Eq. (19). As in Ref. [9], the simplest Ansatz isconstant, C m ( T ) = C m (0), which is most reasonable for T < T Ch , where the condensates, and thus also the lead-ing term in χ ( T ), change little. But above some higher T , the negative C m (0), although initially much smaller inmagnitude than the leading term, will make χ ( T ) (19),and therefore also A ( T ), change sign. Concretely, thislimiting T above which there is no meaningful descrip-tion is found a little above 1 . T Ch .For another, non-constant C m ( T ) that would not havesuch a limiting temperature, we now have a lead from lat-tice where the high- T asymptotic behavior of the QCDtopological susceptibility has been found to be a powerlaw, χ ( T ) ∝ T − b [46, 47]. The high- T dependence ofour model-calculated condensates is also, without fitting,such that the leading term of our χ ( T ) in Eq. (19) hasthe similar power-law behavior, with b = 5 .
17. Also,the values of our leading term are, qualitatively, for all T roughly in the same ballpark as the lattice results [46, 47].We thus fit the quickly decreasing power-law C m ( T ) forhigh T requiring: (i) that this more or less rough consis-tency with lattice χ ( T )-values is preserved, (ii) that thewhole χ ( T ) has the high- T power-law dependence as theleading term (with b = 5 . (iii) that C m ( T ) joinssmoothly with the low- T value C m (0) determined from χ YM at T = 0.Our non-constant choice of C m ( T ) yields the masses inFig. 3 (and χ ( T ) and A ( T ) in Fig. 2), but these resultsturn out very similar to the ones with C m ( T ) = C m (0) (ofcourse, only up to the limiting T a little above 1 . T Ch ),in Fig. 4. Thus, we present Fig. 4 on a different scalefrom Fig. 3, i.e., only the mass interval between 0.55GeV and 1.05 GeV to zoom on the η - η (cid:48) complex anddiscern better its various overlapping curves, including M U A (1) ( T ). . . . . . . . T/T Ch . . . . . . . A ll v a l u e s i n [ G e V ] M π M s ¯ s M η M η M η M η M NS M S A / M U A (1) / FIG. 3. T -dependence, relative to T Ch , of various η (cid:48) - η com-plex masses described in the text, π mass (thick, lower dash-dotted curve) for reference, the halved (to avoid crowding ofcurves) total U A (1)-anomaly-induced mass M U A (1) (lowershort-dashed curve), and topol. ch. parameter A / as thelowest solid curve. The straight line is 2 × lowest fermionMatsubara frequency 2 πT . (Colors online.) The second choice of C m ( T ) enables in principle thecalculation of χ ( T ) and A ( T ) without any limiting T .Nevertheless, Fig. 3 does not reach higher than T =1 . T Ch , because the model chosen for the RLA part ofour calculations seems to become unreliable at higher T ’s. Namely, mass eigenvalues seem increasingly toohigh, since they tend to cross the sum of lowest q +¯ q Mat-subara frequencies. Fortunately, by
T /T Ch = 1 .
8, theasymptotic scenario for the anomaly has been reached,as explained in the next section giving the detailed de-scription of all pertinent results at T ≥ IV. RESULTS AT T ≥ IN DETAIL
Fig. 2 shows how various magnitudes of current quarkmasses m q influence the T -dependence and size of q ¯ q condensates (cid:104) ¯ qq (cid:105) and pseudoscalar decay constants f q ¯ q calculated in our adopted model. Defined, e.g. , in thesubsection II.A of Ref. [24], it employs the parametervalues m u = m d ≡ m l = 5 .
49 MeV and m s = 115 MeV. . . . . . . T/T Ch . . . . . A ll v a l u e s i n [ G e V ] M π M s ¯ s M η M η M η M η M NS M S M U A (1) FIG. 4.
T /T Ch -dependence of pseudoscalar meson masseszoomed to the area important for the η (cid:48) - η complex, for thesimplest Ansatz C m ( T ) = constant = C m (0), which limitstemperatures to T (cid:46) . T Ch . Both for condensates and decay constants, larger cur-rent quark masses lead to larger “initial” ( i.e. , T = 0)magnitudes, and, what is even more important for thepresent work, to smoother and slower falloffs with T . Themagnitude of (the third root of) the strange quark con-densate is the highest, dash-dotted curve in Fig. 2. Its T = 0 value |(cid:104) ¯ ss (cid:105)| / = 238 .
81 MeV remains almost un-changed till T = T Ch , and falls below 200 MeV, i.e., bysome 20%, only for T ≈ . T Ch . On the other hand,the T = 0 value of the isosymmetric condensates of thelightest flavors, (cid:104) ¯ uu (cid:105) = (cid:104) ¯ dd (cid:105) ≡ (cid:104) ¯ l l (cid:105) = ( − .
69 MeV) is quite close to the chiral one, (cid:104) ¯ qq (cid:105) = ( − .
25 MeV) ,showing how well the chiral limit works for u and d fla-vors in this respect. Still, the small current masses of u and d quarks are sufficient to lead to a very different T -dependence of the lightest condensates, depicted bythe dashed curve. It exhibits a typical smooth crossoverbehavior around T = T Ch , and while the falloff is muchmore pronounced than in the case of (cid:104) ¯ ss (cid:105) , it differs quali-tatively from the sharp drop to zero exhibited by the chi-ral condensate (and thus also by anomaly-related quan-tity (cid:101) χ ( T ) defined by LS relation (12)).The isosymmetric pion decay constant f π ( T ) ≡ f l ¯ l ( T )is the lower dashed curve in Fig. 2, starting at T = 0 fromour model-calculated value f π = 92 MeV. It is quite fast- falling, in contrast to f s ¯ s ( T ) (starting at f s ¯ s ( T = 0) =119 MeV), the decay constant of the unphysical, RLA ¯ ss pseudoscalar. It exhibits much “slower” T -dependence,in accordance with the s -quark condensate (cid:104) ¯ ss (cid:105) ( T ).The behavior of m l (cid:104) ¯ l l (cid:105) ( T ) largely determines that ofthe full QCD topological charge parameter A ( T ), de-picted in Fig. 2 by the thick solid curve, and in Fig.3 by the lowest solid curve. Namely, A is dominated bythe lightest flavor, just like χ and (cid:101) χ , as shown by theirrelated defining expressions (18)-(19) and (12)-(13).The smooth, monotonic fall of A ( T ) after T ∼ . T Ch reflects the degree of gradual, crossover restoration ofthe U A (1) symmetry with T . How this is reflected on themasses in the η - η (cid:48) complex, depends also on the ratios of A ( T ) with f π ( T ), f π f s ¯ s ( T ) and f s ¯ s ( T ) in Eqs. (16)-(17). M ∝ A ( T ) / [ f π ( T ) f s ¯ s ( T )] decreases comparably to A ( T ) / , and 2 A ( T ) /f s ¯ s ( T ) even faster. Thus M S ( T )(17) goes monotonically into the anomaly-free M s ¯ s ( T )basically in the same way as in Ref. [9], except now thisprocess is not completed at T = T Ch , but, due to the A ( T ) crossover, it is drawn-out till T ≈ . T Ch .In contrast, β Sho ( T ) = 2 A ( T ) /f π ( T ) even grows for T < . T Ch and 1 . T Ch (cid:46) T (cid:46) . T Ch . By making M NS ( T ) > M S ( T ) it causes the increase of the mixingangle φ (look at Figs. 5, 3 and 4 together). Note thatthis makes the η - η state mixing angle θ ( ≈ φ − ◦ ) lessnegative, i.e. , closer to zero, and brings η and η in aneven better agreement with, respectively, η (cid:48) and η , thanat T = 0.These two limited increases of A ( T ) /f π ( T ) may bemodel dependent and are not important, but what issystematic and thus important is that the “light” decayconstant f π ( T ) is making (cid:112) A ( T ) /f π ( T ) more resilientto T not only than A ( T ) / itself, but also than otheranomalous mass contributions in Eqs. (16)-(17).Indeed, β Sho ( T ) = 2 A ( T ) /f π ( T ) falls only after T ≈ . T Ch (contributing over a half of the η (cid:48) mass drop)and then again rises somewhat after T ≈ . T Ch , tostart definitively falling only after T ≈ . T Ch , but eventhen slower than other anomalous contributions. Thismakes M NS ( T ) larger enough than M S ( T ) to rise φ ( T )to around 80 ◦ , and keep it there as far as T ∼ . T Ch ,see Fig. 5.This explains how the masses of the physical mesons η (cid:48) and η (thick and thin solid curves in Figs. 3, 4), M η (cid:48) ( η ) = M + M − ) (cid:115)(cid:18) M − M (cid:19) + M , (20)exhibit the mass drop of the heavier partner η (cid:48) whichis almost as strong as in the case [9] of the abrupt dis-appearance of the anomaly contribution, while on thecontrary the lighter partner η now does not show anysign of the mass reduction around T = T Ch , let alonean abrupt degeneracy with the pion. The latter happensin the case with the sharp phase transition because the .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 . T /T Ch ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ φ FIG. 5. Relative T -dependence of the NS-S mixing angle φ ( T ). fast disappearance of the whole M U A (1) around T Ch canbe accommodated only by the sharp change of the statemixing ( φ →
0) to fulfill the asymptotic NS-S scenarioimmediately after T Ch . (See esp. Fig. 2 in Ref. [9]. Notethat in our approach M η (cid:48) ( T ) cannot drop much morethan a third of M U A (1) , since RLA M s ¯ s ( T ) is the lowerlimit of M η (cid:48) ( T ) both in Ref. [9] and here.)In the present crossover case, however, T = T Ch doesnot mark the drastic change of the mixing of the isoscalarstates, but η (cid:48) stays mostly η and η stays mostly η .Then, ∆ M η = 4 A (1 /f π − /f s ¯ s ) / η , with ( − ) in Eq. (20), anomalous contributions cancelto a large extent anyway. Thus, the mass of η behavesmostly like the masses of other q ¯ q (cid:48) (almost-)Goldstonebosons after losing their chiral protection at T Ch : it justsuffers the thermal rise towards 2 πT .Nevertheless, in M η (cid:48) (20), the anomalous contributionsfrom Eqs. (16)-(17) all add. The partial restoration of U A (1) symmetry around T Ch , where around a third ofthe total U A (1)-anomalous mass M U A (1) goes away, isconsumed almost entirely by the drop of the η (cid:48) mass overthe crossover.After T ≈ . T Ch , M η (cid:48) ( T ) starts rising again, butthis is expected since after T ≈ T Ch light pseudoscalarmesons start their thermal rise towards 2 πT , twice thelowest Matsubara frequency of the free quark and anti-quark. This rather steep joint rise brings all the masscurves M P ( T ) quite close after T ∼ . T Ch . The kaonmass M K ( T ) is not shown in Figs. 3 and 4 to avoidcrowding of curves, but at this temperature of the char-acteristic η - η (cid:48) anticrossing, M K ( T ) is roughly in between M π ( T ) and the η mass, only to be soon crossed by M η ( T )tending to become degenerate with M π ( T ) as detailed inthe following passage.The rest of M U A (1) ( T ), melting as 2 (cid:112) A ( T ) /f π ( T ), isunder 1 . T Ch sufficiently large to keep M NS ( T ) > M S ( T )and φ ≈ ◦ . So large φ makes θ positive, but not very far from zero, so that still η (cid:48) ≈ η and η ≈ η there. Thisis a fairly good approximation also for T > . T Ch ,but there, an even better approximation is η (cid:48) ≈ η NS , M η (cid:48) ( T ) ≈ M NS ( T ) and η ≈ η S , M η ( T ) ≈ M S ( T ). Fi-nally, when at T ≈ . T Ch the anomalous mass contribu-tion becomes so small that M NS ( T ) = M S ( T ), Eq. (20)enforces anticrossing: M NS ( T ) and M S ( T ) switch, andafter this, the η - η (cid:48) complex enters the NS-S asymptoticregime of the vanishing anomaly influence: M η (cid:48) ( T ) → M S ( T ) → M s ¯ s ( T ), and M η ( T ) → M NS ( T ) → M π ( T ),and φ ( T ) → V. SUMMARY, DISCUSSION ANDCONCLUSIONS
We have studied the temperature dependence of themasses in the η (cid:48) - η complex in the regime of the crossoverrestoration of chiral and U A (1) symmetry. We relied onthe approach of Ref. [11], which demonstrated the sound-ness of the approximate way in which the U A (1)-anomalyeffects on pseudoscalar masses were introduced and com-bined [24, 27, 34–37] with chirally well-behaved DS RLAcalculations in order to study η (cid:48) and η . For T = 0, thiswas demonstrated [11] model-independently, with onlyinputs being the experimental values of pion and kaonmasses and decay constants, and the lattice value of YMtopological susceptibility. However, at T >
0, dynamicalmodels are still needed to generate the temperature de-pendence of non-anomalous quantities through DS RLAcalculations, and in this paper we use the same chirallycorrect and phenomenologically well-tried model as in nu-merous earlier T ≥ e.g. , see [9, 24, 31] andreferences therein).Presently, we adopt from Ref. [11] that the anoma-lous contribution to the masses is related to the fullQCD topological charge parameter (18), which containsthe massive quark condensates. They give us the chiralcrossover behavior for high T . This is crucial, since lat-tice QCD calculations established that for the physicalquark masses, the restoration of the chiral symmetry oc-curs as a crossover ( e.g., see [29, 48, 49] and refs. therein)characterized by the pseudocritical transition tempera-ture T Ch .Nevertheless, what happens with the U A (1) restora-tion is still not clear [48, 50–52]. Whereas, e.g. , Ref.[29] finds its breaking as high as T ∼ . T Ch , Ref. [53]finds that above the critical temperature U A (1) is re-stored in the chiral limit , and JLQCD collaboration [52]discusses possible disappearance of the U A (1) anomalyand point out the tight connection with the chiral sym-metry restoration. Hence the need to clarify “ if, how(much), and when ” [48] U A (1) symmetry is restored. Insuch a situation, we believe instructive insights can befound in our study on how an anomaly-generated massinfluences the η - η (cid:48) complex, although this study is noton the microscopic level.Since JLQCD collaboration [52] has recently stressedthat the chiral symmetry breaking and U A (1) anomalyare tied for quark bilinear operators, we again recall howRef. [11] provided support for the earlier proposal ofRef. [9] relating DChSB to the U A (1)-anomalous masscontributions in the η (cid:48) - η complex. This adds to the mo-tivation to determine the full QCD topological chargeparameter (18) on lattice from simulations in full QCDwith massive, dynamical quarks [besides the original mo-tivation [13, 14] to remove the systematic O (1 /N c ) un-certainty of Eq. (15)]. More importantly, this ties the U A (1) symmetry breaking and restoration to the chiralsymmetry ones. It ties them in basically the same way inthe both references [9] and [11] (and here), except thatthe full QCD topological charge parameter (18) enablesthe crossover U A (1) restoration by allowing the usage ofthe massive quark condensates. But, if the chiral con-densate ( i.e. , of massless quarks) is used in extendingthe approach of Ref. [11] to finite temperatures, the T > U A (1) symmetry restoration at T Ch (consistentlywith Ref. [53]), which causes not only the empiricallysupported [5] drop of the η (cid:48) mass, but also an even larger η mass drop; if M U A (1) ( T ) ∝ β ( T ) → T → T Ch , Eq. (10) mandates M η ( T → T Ch ) → M π ( T Ch )equally abruptly (as in Ref. [9]). However, no experi-mental indication for this has ever been seen, althoughthis is a more drastic fall than for the η (cid:48) -meson.The present paper predicts a more realistic behavior of M η ( T ) thanks to the smooth chiral restoration, which inturn yields the smooth, partial U A (1) symmetry restora-tion (as far as the masses are concerned) making variousactors in the η - η (cid:48) complex behave quite differently fromthe abrupt phase transition (such as that in Ref. [9]). Inparticular, the η mass is now not predicted to drop, butto only rise after T ≈ T Ch , just like the masses of other(almost-)Goldstone pseudoscalars, which are free of the U A (1) anomaly influence. Similarly to T = 0, η agreesrather well with the SU (3) flavor state η until the anti-crossing temperature, which marks the beginning of theasymptotic NS-S regime, where the anomalous mass con-tributions become increasingly negligible and η → η NS .In contrast to η , the η (cid:48) mass M η (cid:48) ( T ) does fall almostas in the case of the sharp phase transition, where itslower limit, namely M s ¯ s ( T ), is reached at T Ch [9]. Now, M η (cid:48) ( T ) at its minimum (which is only around 1 . T Ch because of the rather extended crossover) is some 20 to 30MeV above M s ¯ s ( T ), after which they both start to growappreciably, and M η (cid:48) ( T ) is reasonably approximated by M η ( T ) up to the anticrossing. Only beyond the anti-crossing at T ≈ . T Ch , the effective restoration of U A (1) regarding the η - η (cid:48) masses occurs, in the sense of reach-ing the asymptotic regime M η (cid:48) ( T ) → M s ¯ s ( T ). Another,less illustrative qualitatively, but more quantitative cri-terion for the degree of U A (1) restoration is that there,at T ≈ . T Ch , M U A (1) is still slightly above 40%, and at T ≈ . T Ch still around 14% of its T = 0 value. Thus,the drop to the minimum of M η (cid:48) ( T ) around 1 . T Ch inany case signals only a partial U A (1) restoration.This M η (cid:48) ( T ) drop is around 250 MeV, which is consis-tent with the present empirical evidence claiming that itis at least 200 MeV [5]. For comparison with some otherapproaches exploring the interplay of the chiral phasetransition and axial anomaly, note that the η (cid:48) mass droparound 150 MeV is found in the functional renormaliza-tion group approach [54]. A very recent analysis withinthe framework of the U (3) chiral perturbation theoryfound that the (small) increase of the masses of π , K and η after around T ∼
120 MeV, is accompanied by thedrop of the η (cid:48) mass, but only by some 15 MeV [55].Admittedly, the crossover transition leaves more spacefor model dependence, since some model changes whichwould make the crossover even smoother would reduceour η (cid:48) mass drop. Nevertheless, there are also changeswhich would make it steeper, and those may, for exam-ple, help M η (cid:48) ( T ) saturate the M s ¯ s ( T ) limit. Exploringsuch model dependences, as well as attempts to furtherreduce them at T > model current u - and d -quarkmass of 5.49 MeV. Since it is essentially a phenomenolog-ical model parameter, it cannot be quite unambiguouslyand precisely related to the somewhat lower PDG val-ues m u = 2 . +0 . − . MeV and m d = 4 . +0 . − . MeV [56].Still, their ratio m u /m d = 0 . +0 . − . is quite instructivein the present context, since the QCD topological sus-ceptibility χ (19) and charge parameter A (18) containthe current quark masses in the form of harmonic aver-ages of m q (cid:104) ¯ qq (cid:105) ( q = u, d, s ). Since a harmonic averageis dominated by its smallest argument, our χ (19) and A (18) are dominated by the lightest flavor, providing themotivation to venture beyond the precision of the isospinlimit and in the future work explore the maximal isospinviolation scenario [57] within the present treatment ofthe η - η (cid:48) complex. Acknowledgment:
This work was supported in partby the Croatian Science Foundation under the projectnumber 8799, and by STSM grants from COST Ac-tions CA15213 THOR and CA16214 PHAROS. D. Kl.thanks for many helpful discussions with T. Cs¨org˝o andD. Blaschke. [1] Y. Akiba et al. , arXiv:1502.02730 [nucl-ex].[2] A. Dainese et al. , Frascati Phys. Ser. (2016)[arXiv:1602.04120 [nucl-ex]]. [3] S. S. Adler et al. [PHENIX Collabora-tion], Phys. Rev. Lett. , 152302 (2004)doi:10.1103/PhysRevLett.93.152302 [nucl-ex/0401003]. [4] J. Adams et al. [STAR Collaboration], Phys. Rev. C ,044906 (2005) doi:10.1103/PhysRevC.71.044906 [nucl-ex/0411036].[5] T. Cs¨org˝o, R. Vertesi and J. Sziklai, Phys. Rev. Lett. , 182301 (2010) doi:10.1103/PhysRevLett.105.182301[arXiv:0912.5526 [nucl-ex]]; R. Vertesi, T. Cs¨org˝oand J. Sziklai, Phys. Rev. C , 054903 (2011)doi:10.1103/PhysRevC.83.054903 [arXiv:0912.0258[nucl-ex]]; M. Vargyas, T. Cs¨org˝o and R. Vertesi, CentralEur. J. Phys. , 553 (2013) doi:10.2478/s11534-013-0249-6 [arXiv:1211.1166 [nucl-th]].[6] J. I. Kapusta, D. Kharzeev and L. D. McLerran, Phys.Rev. D , 5028 (1996) doi:10.1103/PhysRevD.53.5028[hep-ph/9507343].[7] A. Adare et al. [PHENIX Collaboration],Phys. Rev. C , no. 6, 064911 (2018)doi:10.1103/PhysRevC.97.064911 [arXiv:1709.05649[nucl-ex]].[8] Tamas Cs¨org˝o, HBT overview - with an emphasis onmulti-particle correlation , XLVII Internat. Symposiumon Multiparticle Dynamics (ISMD2017), Sep. 11-15,2017, Tlaxcala, Mexico.[9] S. Beni´c, D. Horvati´c, D. Kekez andD. Klabuˇcar, Phys. Rev. D , 016006 (2011)doi:10.1103/PhysRevD.84.016006 [arXiv:1105.0356[hep-ph]].[10] H. Leutwyler and A. V. Smilga, Phys. Rev. D , 5607(1992). doi:10.1103/PhysRevD.46.5607[11] S. Beni´c, D. Horvati´c, D. Kekez andD. Klabuˇcar, Phys. Lett. B , 113 (2014)doi:10.1016/j.physletb.2014.09.029 [arXiv:1405.3299[hep-ph]].[12] C. Aidala et al. [PHENIX Collaboration],[arXiv:1805.04389 [hep-ex]].[13] G. M. Shore, Nucl. Phys. B , 34 (2006)doi:10.1016/j.nuclphysb.2006.03.011 [hep-ph/0601051].[14] G. M. Shore, Lect. Notes Phys. , 235 (2008) [hep-ph/0701171]; G. M. Shore, Nucl. Phys. B , 107 (2000)doi:10.1016/S0550-3213(99)00623-9 [hep-ph/9908217].[15] P. Di Vecchia and G. Veneziano, Nucl. Phys. B , 253(1980). doi:10.1016/0550-3213(80)90370-3[16] R. Alkofer and L. von Smekal, Phys. Rept. ,281 (2001) doi:10.1016/S0370-1573(01)00010-2 [hep-ph/0007355].[17] C. D. Roberts and S. M. Schmidt, Prog. Part. Nucl. Phys. , S1 (2000) doi:10.1016/S0146-6410(00)90011-5 [nucl-th/0005064].[18] A. H¨oll, C. D. Roberts and S. V. Wright, nucl-th/0601071.[19] C. S. Fischer, J. Phys. G , R253 (2006)doi:10.1088/0954-3899/32/8/R02 [hep-ph/0605173].[20] G. Eichmann, R. Williams, R. Alkofer and M. Vu-jinovi´c, Phys. Rev. D , no. 10, 105014 (2014)doi:10.1103/PhysRevD.89.105014 [arXiv:1402.1365 [hep-ph]].[21] D. Binosi, L. Chang, J. Papavassiliou, S. X. Qin andC. D. Roberts, Phys. Rev. D , no. 9, 096010 (2016)doi:10.1103/PhysRevD.93.096010 [arXiv:1601.05441[nucl-th]].[22] S. x. Qin, Few Body Syst. , no. 11, 1059 (2016).doi:10.1007/s00601-016-1149-2[23] D. Blaschke, G. Burau, Y. L. Kalinovsky, P. Maris andP. C. Tandy, Int. J. Mod. Phys. A , 2267 (2001)doi:10.1142/S0217751X01003457 [nucl-th/0002024]. [24] D. Horvati´c, D. Klabuˇcar and A.E. Radzhabov,Phys. Rev. D , 096009 (2007)doi:10.1103/PhysRevD.76.096009 [arXiv:0708.1260[hep-ph]].[25] D. Blaschke, Y. L. Kalinovsky, A. E. Radzhabov andM. K. Volkov, Phys. Part. Nucl. Lett. , 327 (2006).doi:10.1134/S1547477106050086[26] D. Horvati´c, D. Blaschke, D. Klabuˇcar andA. E. Radzhabov, Phys. Part. Nucl. , 1033 (2008)doi:10.1134/S1063779608070095 [hep-ph/0703115 [HEP-PH]].[27] D. Horvati´c, D. Blaschke, Y. Kalinovsky, D. Kekezand D. Klabuˇcar, Eur. Phys. J. A , 257 (2008)doi:10.1140/epja/i2008-10670-x [arXiv:0710.5650 [hep-ph]].[28] A. Bazavov et al. , Phys. Rev. D , 054503 (2012)doi:10.1103/PhysRevD.85.054503 [arXiv:1111.1710 [hep-lat]].[29] V. Dick, F. Karsch, E. Laermann, S. Mukherjee andS. Sharma, Phys. Rev. D , no. 9, 094504 (2015)doi:10.1103/PhysRevD.91.094504 [arXiv:1502.06190[hep-lat]].[30] A. Bazavov et al. , Phys. Rev. D , no. 5, 054504 (2017)doi:10.1103/PhysRevD.95.054504 [arXiv:1701.04325[hep-lat]].[31] D. Horvati´c, D. Blaschke, D. Klabuˇcar andO. Kaczmarek, Phys. Rev. D , 016005 (2011)doi:10.1103/PhysRevD.84.016005 [arXiv:1012.2113[hep-ph]].[32] E. Witten, Nucl. Phys. B , 269 (1979).doi:10.1016/0550-3213(79)90031-2[33] G. Veneziano, Nucl. Phys. B , 213 (1979).doi:10.1016/0550-3213(79)90332-8[34] D. Kekez and D. Klabuˇcar, Phys. Rev. D ,036002 (2006) doi:10.1103/PhysRevD.73.036002 [hep-ph/0512064].[35] D. Klabuˇcar and D. Kekez, Phys. Rev. D ,096003 (1998) doi:10.1103/PhysRevD.58.096003 [hep-ph/9710206].[36] D. Kekez, D. Klabuˇcar and M. D. Scadron, J. Phys. G , 1335 (2000) doi:10.1088/0954-3899/26/9/305 [hep-ph/0003234].[37] D. Kekez and D. Klabuˇcar, Phys. Rev. D ,057901 (2002) doi:10.1103/PhysRevD.65.057901 [hep-ph/0110019].[38] M. S. Bhagwat, L. Chang, Y. X. Liu, C. D. Robertsand P. C. Tandy, Phys. Rev. C , 045203 (2007)doi:10.1103/PhysRevC.76.045203 [arXiv:0708.1118[nucl-th]].[39] R. Alkofer, C. S. Fischer and R. Williams, Eur.Phys. J. A , 53 (2008) doi:10.1140/epja/i2008-10646-x[arXiv:0804.3478 [hep-ph]].[40] F. J. Gilman and R. Kauffman, Phys. Rev. D ,2761 (1987) Erratum: [Phys. Rev. D , 3348(1988)]. doi:10.1103/PhysRevD.37.3348, 10.1103/Phys-RevD.36.2761[41] S. D¨urr, Nucl. Phys. B , 281 (2001)doi:10.1016/S0550-3213(01)00325-X [hep-lat/0103011].[42] See, e.g. , the extensive review [44], or the Appendix inRef. [36].[43] T. Feldmann, P. Kroll and B. Stech, Phys. Rev. D ,114006 (1998) doi:10.1103/PhysRevD.58.114006 [hep-ph/9802409]; T. Feldmann, P. Kroll and B. Stech,Phys. Lett. B , 339 (1999) doi:10.1016/S0370- , 159 (2000)doi:10.1142/S0217751X00000082 [hep-ph/9907491].[45] L. Del Debbio, L. Giusti and C. Pica, Phys. Rev. Lett. ,032003 (2005) doi:10.1103/PhysRevLett.94.032003 [hep-th/0407052].[46] P. Petreczky, H. P. Schadler and S. Sharma, Phys. Lett.B , 498 (2016) doi:10.1016/j.physletb.2016.09.063[arXiv:1606.03145 [hep-lat]].[47] S. Borsanyi et al. , Nature , no. 7627, 69 (2016)doi:10.1038/nature20115 [arXiv:1606.07494 [hep-lat]].[48] S. Aoki, H. Fukaya and Y. Taniguchi, Phys. Rev.D , 114512 (2012) doi:10.1103/PhysRevD.86.114512[arXiv:1209.2061 [hep-lat]].[49] M. I. Buchoff et al. , Phys. Rev. D , no. 5, 054514 (2014)doi:10.1103/PhysRevD.89.054514 [arXiv:1309.4149 [hep-lat]].[50] S. Sharma [HotQCD Collaboration], arXiv:1801.08500[hep-lat].[51] F. Burger, E. M. Ilgenfritz, M. P. Lombardo andA. Trunin, arXiv:1805.06001 [hep-lat]. [52] H. Fukaya [JLQCD Collaboration], EPJ Web Conf. , 01012 (2018) doi:10.1051/epjconf/201817501012[arXiv:1712.05536 [hep-lat]].[53] A. Tomiya, G. Cossu, S. Aoki, H. Fukaya, S. Hashimoto,T. Kaneko and J. Noaki, Phys. Rev. D , no. 3, 034509(2017) Addendum: [Phys. Rev. D , no. 7, 079902(2017)] doi:10.1103/PhysRevD.96.034509, 10.1103/Phys-RevD.96.079902 [arXiv:1612.01908 [hep-lat]].[54] M. Mitter and B. J. Schaefer, Phys. Rev. D ,no. 5, 054027 (2014) doi:10.1103/PhysRevD.89.054027[arXiv:1308.3176 [hep-ph]].[55] X. W. Gu, C. G. Duan and Z. H. Guo, Phys. Rev. D ,no. 3, 034007 (2018) doi:10.1103/PhysRevD.98.034007[arXiv:1803.07284 [hep-ph]].[56] M. Tanabashi et al. [Particle Data Group],Phys. Rev. D , no. 3, 030001 (2018).doi:10.1103/PhysRevD.98.030001[57] D. Kharzeev, R. D. Pisarski and M. H. G. Tyt-gat, Phys. Rev. Lett.81