Chiral and $U(1)_A$ restoration for the scalar/pseudoscalar meson nonets

We analyze the restoration pattern of the members of the scalar and pseudoscalar meson nonets under chiral $O(4)$ and $U(1)_A$ symmetries. For that purpose, we exploit QCD Ward Identities (WI), which allow one to relate susceptibilities with quark condensates, as well as susceptibility differences with meson vertices. In addition, we consider the low-energy realization of QCD provided by $U(3)$ Chiral Perturbation Theory (ChPT) at finite temperature to perform a full analysis of the different correlators involved. Our analysis suggests $U(1)_A$ partner restoration if chiral symmetry partners are also degenerated. This is also confirmed by the ChPT analysis when the light chiral limit is reached. Partner degeneration for the $I=1/2$ sector, the behavior of $I=0$ mixing and the temperature scaling of meson masses predicted by WI are also studied. Special attention is paid to the connection of our results with recent lattice analyses.

susceptibilities in this formalism are derived here for the first time and collected in Appendix A. Our ChPT results, albeit limited at temperatures well below the transition, will essentially capture and confirm the main results obtained formally from the analysis of WI. They will be particularly useful in the chiral limit and will provide new insights on this problem for future lattice and theoretical analysis. Finally, in Section VI we will present our main conclusions.

II. GENERAL WARD IDENTITIES
In order to clarify partner degeneration in terms of different symmetry restoration patterns, we consider the pseudoscalar P a = iψγ 5 λ a ψ and scalar S a =ψλ a ψ quark bilinears, with ψ a three-flavor fermion field with components ψ u,d,s , λ a=1,...8 the SU (3) Gell-Mann matrices and λ 0 = 2/3 1. We follow the same notation as in [47,48].
The relevant transformations to study the restoration of the chiral and U (1) A symmetries are those of the paritychanging U A (3) group, i.e., the infinitesimal transformations δψ = iα a A λa 2 γ 5 ψ and δψ = iα a Aψ λa 2 γ 5 . Note that a SU V (3) transformation would always allow one to rotate between members of the same octet, i.e., without change of parity. Under such axial transformations, the expectation value of an arbitrary pseudoscalar local operator O P (x 1 , · · · , x n ) in terms of the transformed fields leads to the following generic WI [47,48] δO P (x 1 , · · · , x n ) δα a A (x) where is the anomalous divergence of the U (1) A current [49][50][51], with J µ 5 =ψγ 5 γ µ ψ. Generally speaking, applying (1) to an n-point operator O P gives an identity relating correlators of n and n + 1 points.
In the same way, WI obtained from a isovector transformation δψ = iα a V λa 2 ψ, δψ = −iα a Vψ λa 2 on a scalar operator O S read: The analysis to follow in the next sections exploits the above two classes of WI for particular choices of the operators O P and O S . In particular, the choices O P = P a in (1) and O S = S a in (4) will lead to identities between different combinations of quark condensates and susceptibilities, whereas choosing O P = P a S b in (1) will allow one to relate differences of correlators of degenerated partners with three-point functions related to physical interaction processes.
Those identities will involve P and S correlators and their corresponding susceptibilities, defined as: where T dx ≡ β 0 dτ d 3 x at finite temperature T = 1/β and T · · · denotes the time-ordered vacuum expectation value in Minkowski space-time, which corresponds to a thermal average in Euclidean space-time. Note that in the scalar case, the subtraction of the quark-bilinear expectation values S a S b , which are non-zero for a, b = 0, 8, allows one to express the susceptibilitiesχ ab S (T ) as mass derivatives of the corresponding quark condensate, as used customarily within the ChPT framework [52][53][54] and also to analyze the critical behavior [8]. However, the study of partner degeneration in the lattice is formally investigated through the analysis of unsubtracted scalar susceptibilities [4]. In the following we will denote byχ S the subtracted susceptibilities and by χ S the unsubtracted ones.
The above transformations imply then a formal degeneration of the bilinears π/σ and η l /δ if the chiral symmetry SU (2) V × SU (2) A ∼ O(4) was completely restored. In other words, these bilinears would become chiral partners. In addition, η s and σ s fields are invariant under SU A (2), as one can see from their definition (8) and the transformations of the octet and singlet fields (12). In this way, P ls and S ls transform into δη s and πσ s respectively, which should vanish by parity conservation. More details about particular choices of chiral rotations that implement these transformations are given in [48].
We will use the symbol O(4) ∼ to denote the above chiral partner equivalence. As commented in the introduction, this would actually be an exact equivalence only for two massless flavors at the phase transition. For N f = 2 + 1 flavors and physical masses, it would become approximate near the crossover transition, although the equivalence is expected to be more accurate as the light chiral limitm → 0 + is approached. Summarizing, at exact chiral restoration one has: and so on for their corresponding susceptibilities. Therefore, the full O(4) nonet partner-degeneration picture given by the four conditions in (14), leave four independent not degenerated correlators (or susceptibilities) in this pattern, namely P ππ , P ll , P ss and S ss . On the other hand, under octet and singlet axial rotations, i.e., α 0,8 A = 0, I = 0 states transform as: δS 0 (y)/δα 0 A (x) = 2/3δ(x − y)P 0 (x), δS 0 (y)/δα 8 A (x) = 2/3δ(x − y)P 8 (x), (15) which allow one to mix π − δ and σ − η states: Therefore, π − δ and σ − η would become degenerate partners if the U (1) A symmetry was restored. Similarly, in a fully restored U (1) A scenario, the U (1) A rotations in (15) and (16) allow one to degenerate all pseudoscalar correlators into their scalar partners [48]. As explained in the introduction, such restoration is only asymptotic and in general is not fully achieved in a physical N f = 2 + 1 scenario. Nevertheless, here we are concerned with U (1) A restoration understood as approximate partner degeneration and in that sense, we will use the symbol Thus, under U (1) A restoration the following relations hold: which leaves again four independent correlators, for instance P ππ , P ll , P ss , P ls or their corresponding scalar partners. Therefore, if U (1) A restoration is effective at the chiral transition, i.e., if O(4) × U (1) A is the restoration pattern, the four states π − δ − σ l − η l would degenerate at the transition. Thus, the O(4) and U (1) A partner equivalences in (14) and (17) combine to P ππ ∼ S δδ ∼ S ll ∼ P ll , which are the correlators usually analyzed in lattice works. Moreover, the relation P ss ∼ S ss becomes an additional signal to be analyzed. Hence, since the crossed ls correlators vanish (14), there are only two independent correlators in the O(4) × U (1) A pattern.
The parameter customarily used in lattice works to parameterize the O(4) × U (1) A degeneracy is which vanishes at O(4)×U (1) A restoration and is directly related to the topological susceptibility [4], i.e., the correlator of the anomaly operator (2) encoding the U (1) A breaking Actually, as we are about to see, the connection between χ 5,disc and χ top is a consequence of the WI analyzed here 1 . Furthermore, in a fully SU (3) restored scenario not only α 0 A but also α 8 A transformations would allow one to degenerate π and δ bilinears, hence leading to the degeneration of all other members of the scalar and pseudoscalar octet.
More precise conclusions can be drawn from the WI in (1). The simplest choice is O P = P a . Taking O P = π b , η l , η s one arrives to the following WI relating pseudoscalar susceptibilities and quark condensates analyzed in [47,48]: where qq l = ψ l ψ l and we denote χ ab P = δ ab χ π P . The identities (20) and (22) have been recently checked in the lattice [4]. In the case of (22) the term proportional to χ 8A P can be ignored since it is suppressed by am/m s correction. Nevertheless, it would be interesting to have a lattice check of the other WI found in [47,48] as well as those from the present work that we will discuss below. In particular, WI involving crossed ls correlators, I = 1/2 states and three-point functions. For the WI (21), although the crossed χ 8A P correlator is not measured in the lattice either, we will show below that this identity can be indirectly examined in the lattice with currently measured observables.
In particular, the above identities imply that χ 5,disc in (18) can be written as: where we have used the relation between χ 8A P and χ AA P derived in [47], i.e., from the WI in (1) taking O P = A, the anomaly operator in (2). The relation (23) between χ 5,disc and χ top , also mentioned in [4], allows one to express χ 5,disc as a pure anomalous contribution, confirmed by the cancellation of the quark condensate contributions in the χ π P − χ ll P difference in (20)- (21). Actually, since both χ 5,disc and χ top are measured in the lattice, although with great difficulty in the case of the topological susceptibility [4,55,56], checking the relation between them in (23) is an indirect way to check the WI (21), or, more precisely, the combination of that identity with (20) (also checked in the lattice) and the identity connecting χ 8A P and χ AA P in [47], namely Such verification of (23) is actually performed in [4] and it holds reasonably well taking into account the difficulty to measure χ top . Now, let us turn to a very interesting relation regarding the chiral pattern, already discussed in [48], which can be obtained by analyzing the mixed ls correlators in the pseudo-scalar sector. Using (11) and the relations obtained in [47] for the susceptibilities χ 88 P , χ 00 P and χ 08 P , we get which combined with (23), implies: The importance of the above relation is that it connects a quantity vanishing at O(4) degeneration, χ ls according to (14), with χ 5,disc and χ top , signaling O(4) × U (1) A degeneration. Therefore, (14) and (26) imply that if O(4) partners are degenerated, so there must be the O(4) × U (1) A ones. In other words, the chiral pattern should be (14) is a consequence of the δ − η l O(4) degeneration [48]. Thus, more precisely, Several additional comments are in order here: first, the previous conclusion (27) is valid in the ideal chiral restoration regime, since it relies on the O(4) partner degeneration on the l.h.s. Nevertheless, it can be understood also in a weaker sense, as a consequence only of δ − η l degeneration, which might take place approximately in a crossover scenario.
Second, although the light chiral limitm → 0 + would certainly favor exact O(4) degeneration at T c and hence the realization of (27), one must not be misled by the apparent vanishing of the χ 5,disc term in (26) whenm → 0 + for any T . This is an incorrect statement, consequence of the singular behavior of χ 5,disc withm. Namely, at T = 0 the results in [47] show that χ 8A P has a finite limit form → 0 + , which together with (23) imply χ 5,disc ∼ 1/m and χ top ∼m away from T c . The latter behavior for χ top is actually supported in the recent work [29], where it is argued that χ top ∼m qq l in the chiral limit. More discussion about the chiral limit of the different susceptibilities will be carried out within ChPT in Section V.
Therefore, the vanishing of χ 5,disc and χ top in (27) are true consequences of chiral restoration. Similar conclusion can be drawn considering other bilinear rotations. Namely, since A is invariant under a SU A (2) transformation, the rotation χ 8A P → χ δA suggests χ 8A P to vanish at exact chiral restoration by parity. Consequently, through (23), χ 5,disc and χ top should also vanish in this limit.
The same conclusion about the vanishing of χ top for any temperature above chiral restoration has been reached in [29]. The main argument in [29] relies on the identity which is nothing but the combination of (21), (24) and (19). In turn, note that (22) gives for the pure strange contribution which corresponds the one-flavor version of the same identity [29].
Let us now comment in detail how the previous ideas are realized in present lattice simulations. As explained in the introduction, there is still some controversy regarding the chiral pattern and its nature. In Fig. 1a we show the behavior of the four susceptibilities corresponding to the π − σ − δ − η l correlators discussed above for the lattice data in [4]. In that work, the O(4) partner degeneration corresponding to the first two equations in (14) is approximately realized at T c 160 MeV (corresponding to chiral restoration signaled by the peak in χ ll S ) while the degeneration of the four correlators which would favor the O(4)×U (1) A pattern according to (17) takes place asymptotically at higher temperatures. At this point it is worth mentioning that in a previous work [46], π − σ chiral partner degeneration in the light sector was also identified exploiting the WI (20) by analyzing available lattice data for the (subtracted) quark condensate and for the scalar susceptibility.
In addition, regarding U (1) A partner degeneration, the ss correlators given by the third equation in (17) are also compared with the lattice data of [4] in Fig. 1b. We see that the degeneration of those U (1) A partners is reached also asymptotically, consistently with (17) and Fig. 1a. As for the ls correlators, there are no direct available data at the moment, as far as we are concerned.
Nevertheless, as already mentioned in the Introduction, there is currently no full agreement in the lattice regarding partner degeneration and the corresponding chiral pattern. In [42], the difference between π and δ screening masses are found to be compatible with zero at the chiral transition, hence pointing out to a O(4) × U (1) A pattern even for massive light quarks. Since the screening masses are extracted from the two-point correlators, their degeneracy is a consequence of partner degeneration. In the chiral limit, the O(4) × U (1) A pattern is also supported in the analysis of [40], which suggests π − δ − σ − η l degeneration close to the chiral transition through the analysis of the correlators for those states in the overlap fermion lattice formulation. A recent analysis by the same group [41] confirms this result, showing U (1) A restoration in the chiral limit just above the transition.
At this point one may wonder about the compatibility of our result (27) with these lattice results. Naively, one would conclude that we are consistent with the results in [40][41][42] but not with [4]. However, some considerations should be taken into account. The analysis in [4] includes N f = 2 + 1 flavors and nearly physical light quark masses, which may enhance U (1) A breaking effects and distort the ideal partner degeneration given in (27). Moreover, our result (27) relies explicitly on δ − η l degeneration at chiral restoration. However, examining in detail the numerical results in [4], one actually observes that the difference χ ll P − χ δ S is much less reduced near T c than χ π P − χ ll S , as it can be seen in Fig. 1a. In particular, from the data in Table IV in [4] with T 0 = 139 MeV, the lowest temperature available in [4]. The error bars for the latter difference are also quite large, making this quantity compatible with zero for the whole temperature range considered. Nevertheless, the central values of χ ll P − χ δ S remain sizable up to the region where the U (1) A is approximately restored, i.e., where χ π P and χ δ S almost degenerate. In this sense, the numerical results in [4] are at odds with the expected chiral partner degeneration picture.
On the one hand, the reasons above could explain numerically the apparent discrepancy between (27) and the results in [4]. On the other hand, the absence of the strange quark corrections in the N f = 2 lattice analysis [40][41][42] may explain why the O(4) × U (1) A pattern is more clearly seen in those works, even for a finite pion mass as in [42]. A quantitative measure of the departure of the results in [4] from the prediction (27) can be achieved by comparing the temperature scaling of χ 5,disc with a typical chiral-restoring order parameter. Actually, as we have commented above, the analysis in [29] supports χ 5,dis to scale with T as the (subtracted) quark condensate. Thus, in Fig. 2 we plot χ 5,disc normalized to its lowest value, versus the subtracted condensate ∆ ls (T ; T 0 ) = qq l (T ) − 2m ms ss (T ) which is free of lattice finite-size divergences q i q i ∼ m i /a 2 , with a the lattice spacing, and it is one of the typical order parameters used in lattice simulations. We can see in the plot a clear correlation between the scaling of both quantities, especially near the critical region. For comparison, we have also represented the scaling √ ∆ ls , which is motivated as follows: the WI (20) is compatible with the formal scaling π ∼ − qq l G −1 π (p = 0)/m [48], which together with (54), to be derived in Sect. IV, and (26), would lead to such square root scaling if the pion self-energy dependence with temperature is considered smooth compared to that of the quark condensate.

B. Chiral partners and mixing angles
We will explore here two interesting limits related to the mixing of the P 0 /P 8 and S 0 /S 8 states, namely the vanishingmixing and ideal-mixing angles. As we will see, these two limits are also intimately connected to the discussion of chiral partners. The mixing angle is formally defined at leading order as:  (30), with respect to the reference temperature T0 = 139 MeV. Data are taken from [4] for 32 3 × 8 lattice size andm/ms = 0.088. We include also the comparison with ∆ l,s for the reasons explained in the main text and so on in the scalar sector with the replacements θ P → θ S , η → f 0 (500), η → f 0 (980). The mixing angle is defined to cancel the crossed ηη terms in the Lagrangian, so that the correlator P ηη = 1 2 (P 88 − P 00 ) sin 2θ P + P 08 cos 2θ P = 0, where both, the correlators and the mixing angle, are temperature dependent. Let us remark that higher-order corrections introduce further mixing terms, which require additional mixing angles to be canceled. For instance at NLO in U (3) ChPT two mixing angles are required [30][31][32]. Nevertheless, the simplified picture above is enough for our present purposes. Consider first a vanishing-mixing scenario, i.e., θ P,S = 0. In the pseudoscalar sector, this occurs in the pure SU (3) limit, i.e., when m K = m π , but keeping fixed M 0 , the anomalous contribution to the η mass [30][31][32]. In that limit, m η → m π and m η → m π + M 0 . From (32), θ P → 0 asymptotically would imply then P 08 → 0, and so on for the scalar sector. It is important to remark that the reverse is not necessarily true. If P 08 → 0 in a certain regime, we can only conclude that it implies θ P → 0 if P 00 and P 88 remain not degenerate. According to (11), that means P ls = 0. Translating these conditions to the lattice basis we conclude that in a regime of vanishing mixing angle the following conditions must hold: In the pseudoscalar sector we can translate this result to the susceptibilities. Using (21), (22), (23) and (26) we have where in the second line the WI (20) has been used. This equation vanishes in the SU (3) degenerate limit, i.e., when m s →m and ss → qq l /2. This is consistent with our previous comment since in that limit θ P → 0 and P 08 → 0. In addition, taking only the leading order in them m s expansion, the r.h.s. of (34) becomes: , where we also plot −χ ls P according to (26). (b): Partner degeneration in the scenario of small X ls and X08 with X = P, S discussed in the text, according to (37).
Since qq l is small around the chiral transition, − ss is positive and smoothly decreasing with T and χ ll P is positive and increasing, it is plausible to expect that (34) might be small or even vanish near the chiral transition, although it is unclear that it should remain asymptotically small for higher temperatures. In addition, as we have commented above, the vanishing-mixing scenario requires P ls = 0 as well, which from (26) can be directly linked to O(4) × U (1) A restoration. In a scenario where the chiral O(4) pattern is well separated from U (1) A restoration, for instance in [4], it would be then possible to find an intermediate region, roughly between chiral restoration and the U (1) A one, where the pseudoscalar mixing-angle vanishes.
In Fig. 3a we plot the susceptibility combination in the l.h.s of (34), signaling a vanishing of θ P , from the lattice analysis [4], where we have used the WI in (26) for χ ls P . In addition, we plot in the same figure −χ ls =m ms χ 5,disc , which according to (33) should remain nonzero to guarantee that this is a region where θ P ∼ 0. Unfortunately, there is no way to check an analogous behavior for the scalar sector as long as χ ls S data are not provided by lattice collaborations. Consistently with our previous arguments, we see a clear signal of the vanishing of the mixing angle, which happens to be very close to chiral restoration for those lattice data. Qualitatively, from the simplifiedm m s expression (35), the positive −2 ss /m s term dominates for low temperatures. As T increases, χ ll P grows, as shown in Fig. 1a, until it compensates the strange condensate contribution. The decreasing/increasing rate of ss and χ η l P changes for higher temperatures, so that this susceptibility combination starts to grow again from around T ∼ 165 MeV, where it develops a minimum. Presumably, after that point the mixing angle changes from zero to the ideal one, which should be reached asymptotically at O(4)×U (1) A restoration, consistently with the vanishing of 2m ms χ 5,disc (T ), as explained below.
Consider now the ideal mixing limit θ = θ id = − arcsin 2/3 , which implies that η ∼ η l , η ∼ √ 2η s and so on for the scalar f 0 (500)/f 0 (980) sector. In a recent model analysis [28] it has been suggested that this limit can be reached from the transition temperature onwards, with a more dramatic effect for the η − η sector than for the scalar one. In that work, the scalar mixing remains close to ideal one for almost the entire temperature range. In the pseudoscalar sector, ideal mixing is reached when M 0 , the anomalous contribution to the η mass, vanishes [30][31][32]. In that limit, m η → m π and m 2 η → 2m 2 K − m 2 π . Thus, this limit is linked to O(4) × U (1) A restoration, where the π degenerates with the η l ∼ η, e.g. through the vanishing of χ 5,disc . The strong reduction of the η mass observed experimentally at finite temperature [20] supports that this limit is reached.
From (32) and (11), we can see that θ P → θ id P implies P ls → 0 and S ls → 0. However, as before, P ls ∼ 0 and S ls ∼ 0 are necessary but not sufficient conditions to have ideal mixing. Inserting P ls = 0 in (32) leads to sin 2θ P − 2 √ 2 cos 2θ P P 08 = 0, so one recovers θ P = θ id for sin θ P < 0 and cos θ P > 0 only if P 08 = 0. Therefore, in a ideal mixing regime, the following conditions must hold: In the pseudoscalar case P ls ∼ 0 is expected at O(4)×U (1) A restoration from (26), provided that chiral partners are ideally degenerated (formally in the chiral limit). Unlike the vanishing mixing scenario discussed above, which can be reached locally around some given temperature, ideal mixing would be reached at O(4) × U (1) A restoration and will remain like that asymptotically. Thus, ideal mixing is another signal of the O(4) × U (1) A pattern. In addition, (17) implies that both the scalar and pseudoscalar mixings become ideal asymptotically. Note that as long as U (1) A is not fully restored, θ P and θ S can take different values.
Considering now the lattice results in [4], according to our previous argument the vanishing of χ 5,disc in Fig. 2 signals the ideal mixing regime. In this regard, although one would expect to find a θ P ∼ 0 region around O(4) restoration, the mixing angle should turn into the ideal one as T increases towards the O(4) × U (1) A regime.
However, it is worth mentioning that in that work the combination (34) (blue squares in Fig. 3a) still remains numerically small for the temperature range explored, compared with the typical values reached by χ ll P and χ ss P in that combination (see Fig. 1a and 1b respectively). Thus, as (m/m s )χ 5,disc becomes negligible, the relation P ll ∼ 2P ss still holds approximately. Moreover, this condition can be combined with P ll ∼ S δδ , holding at O(4) restoration. Note however that, as we discussed in Section III A, the latter equivalence is not so accurately satisfied in [4]. In conclusion, the following two additional partner degeneration conditions would be satisfied approximately in the intermediate region between O(4) and U (1) A restoration: Near U (1) A restoration, the four correlators 2P ss ∼ S δδ ∼ 2S ss ∼ P ππ would become degenerate. In Fig. 3b we check the degeneration (37), which holds reasonably well given the approximations considered and the lattice uncertainties. In fact, if the susceptibility combination in Fig. 3a would keep on growing for higher T , the degeneration in Fig. 1d would not be maintained.
The scenario depicted in Fig. 3 is clearly a consequence of the O(4) and U (1) A neat separation in that particular lattice analysis. However, for a O(4) × U (1) A chiral pattern, as that observed in [40][41][42], there would be no room for a vanishing mixing region since U (1) A restoration is already activated around the O(4) transition, where the ideal mixing would be operating.

C. Including isospin breaking: connected and disconnected scalar susceptibilities
In this section, we derive additional results in the form of WI, which become useful for the discussion of the role of the connected and disconnected parts of the scalar susceptibilities regarding chiral partners and patterns. For that purpose, let us consider the general isovector isovector WI in (4) with a scalar operator O b = S b satisfying If we also take into account isospin breaking effects m u = m d in the quark mass matrix, i.e., the WI in (4) becomes after integration in the Euclidean space-time where the charged χ δ,ch [53]. In fact, it allows one to relate the present analysis with the standard decomposition of the subtracted scalar susceptibility into its quark-diagram connected and disconnected contributions, which are relevant for lattice studies [4,6]. Assuming m u = m d one has [53] χ dis S =χ ud S , In this way, comparing with (38), one gets consistently with recent lattice studies [4]. The relation (40) is also consistent with the SU (3) ChPT isospin-breaking analysis in [53]. Our current WI derivation is completely general and then it is also valid for the U (3) scenario, which will be analyzed in Section V. Actually, combining (40) with (39) allows one to obtain the connected and disconnected parts fromχ ll S and χ δ S , quantities which can be directly derived from the ChPT Lagrangian formulation. In principle, the connected part of the scalar susceptibility is expected to have a softer T -dependence than the disconnected one in the relevant temperature range studied here. This is observed for instance in the lattice analysis in [4] and is confirmed in SU (3) ChPT, where one findsχ dis S ∼ T /m π andχ con S ∼ T 2 /m 2 η [53]. That is, the infrared (IR) m π → 0 + part ofχ ll S is carried only by its disconnected part, which is the perturbative counterpart of the chiral transition peak observed in the lattice for this quantity. Conversely, the growth of the connected piece is controlled by the heavier scale m 2 η coming from π 0 η mixing andKK loops. However, it is important to remark that the above picture may change if the U (1) A symmetry is restored close to the O(4) transition. First, since χ δ S grows with T and χ π P decreases like qq l from (20), their degeneration would give rise to a maximum for χ δ S at U (1) A restoration. Such possible maximum is not really seen from Fig. 1a, since higher T data points in [4] would be needed to appreciate correctly that region. However, going back to earlier papers of the same collaboration, the observed maximum ofχ con S = χ δ S /2 at around 190 MeV [3] can be understood in this way. Another signal of this behavior would be a minimum of the screening mass in the δ channel (see our discussion about screening masses in Section III D). Such minimum is clearly observed for instance in [42] and it takes place at chiral restoration. Note that the O(4) and U (1) A transition almost coexist in [42]. A minimum for the screening mass in the δ-channel is also seen in an earlier work [37]. In this work, which we will refer to in Section III D, the full SU (3) degeneration is also visible at higher temperatures, where all the screening masses for different octet channels become degenerate.
From the ChPT point of view, the connected susceptibility peak, linked to U (1) A restoration, can be naively understood by taking the m η → m π limit. This case is reached only when the anomalous part of the η mass vanishes, corresponding parametrically to U (1) A restoration [30][31][32]. This m η → m π limit generates an IR behavior for m π → 0 + , which will discussed in more detail in Section V within the U (3) ChPT framework.
Finally, as pointed out in [4], from (18), (39) and (40) one finds Since the second and third terms in the r.h.s of (41) should vanish at exact O(4) restoration, then, if U (1) A is also restored χ 5,disc = 0 ⇒χ dis S = 0, which is an apparent contradiction with the peak forχ dis S observed in the lattice. However, there are two possible complementary ways to address this argument: first, from the theoretical point of view, in an ideal restoration regime only the total subtracted scalar susceptibilityχ ll S should be divergent at the O(4) transition [8]. Thus, it may happen that the peak of the connected contribution at O(4) × U (1) A transition discussed above could compensate an absent peak in the disconnected part. Second, in an approximate scenario where O(4) and U (1) A restoration are close but still separated by a finite gap, the third term in (41) may remain small while both χ 5,disc andχ dis S keep a peaking behavior scaling as (T − T c ) −γ /m π in the light chiral limit, with γ some critical exponent [6]. However, at U (1) A restoration the divergent parts ofχ dis S and −χ ll S /4 (second term in the r.h.s of (41)) may cancel, which is compatible with a vanishing χ 5,disc . We will actually obtain a explicit realization of this second scenario in Section V in the IR limit m π → 0 + , where the gap between O(4) and U (1) A is also vanishing with m π . D. I = 1/2: WI, partner degeneration and lattice screening masses Consider now transformations of the I = 1/2 components of the octets, i.e., P a ≡ K a and S a ≡ κ a with a = 4, . . . , 7, which correspond to the kaon (pseudoscalar) and the κ (scalar), respectively. Following similar steps as before, under SU A (2) and U (1) A transformations we have: with a, c = 4, . . . , 7 and b = 1, 2, 3. Since there are non-vanishing d abc coefficients for those a, b values and c = 4, . . . , 7, both SU (2) and U (1) A transformations would make the I = 1/2 S/P octet partners degenerate.
We will now obtain more quantitative statements studying the WI of this sector. On the one hand, starting with a one-point pseudoscalar operator, i.e., O b = P b with b = 4, . . . , 7, both sides of (1) vanish but for a = 4, . . . , 7, for which one gets [47] − (m + m s )χ K P (T ) = qq l (T ) + 2 ss (T ), (42) already obtained in [47]. On the other hand, considering the isovector WI in (4) with O b = S b (b = 4, . . . , 7) and taking into account that δO b (y)/δα a V (x) = δ(x − y)f abc S c , we obtain: where we have considered the isospin limit, i.e., m u = m d =m and S 3 = ūu − d d = 0, and χ ab S = χ κ S δ ab . This new identity (43) has interesting consequences and provides a first hint towards the fate of I = 1/2 partners at chiral restoration, which has not been explored yet in lattice analysis. Actually, the combination of (42) and (43) gives rise to [48]: which states that in the strict light chiral limit, i.e., for a second-order chiral phase transition withm = 0 and qq l = 0 but m s = 0 and ss = 0, K and κ become degenerate partners. Moreover, in the real crossover scenario where the light quark mass and condensate are not zero, (44) provides a measure of the I = 1/2 partner degeneracy since with ∆ ls defined in (30) and, as explained above, very well determined in the lattice. Roughly speaking, lattice predicts ∆ ls (T ; 0) ∼ 0.5 at the chiral transition [3]. Hence, according to (45), in the physical case K and κ would only be degenerate around 50% of their T = 0 value at the O(4) transition. This result provides then a way to extract information on K − κ degeneration from lattice data without measuring directly the corresponding correlators. It is important to remark that K − κ correlators also degenerate at U (1) A restoration [48] and then, according to the results above, they do so at O(4) × U (1) A restoration. A confirmation of the previous results will be obtained also in our ChPT analysis in Section V. The other important consequence of the identity (43) is that it allows one to explain the behavior of lattice screening masses in the κ channel, in a similar way as it was done in [47] for the π, K andss ones. Actually, the only available lattice data of correlators in this sector are the results for K and κ screening masses in [37]. This result shows that both screening masses degenerate beyond the chiral transition, consistently with our previous result based on (44). The observed asymptotic degeneration would be a consequence of the U (1) A asymptotic restoration.
Following the analysis in [47], the lattice result for the κ screening mass in [37] can also be used to check the WI in (43). If we assume a smooth temperature dependence for the residue of the κ correlator as well as for the ratio between pole and screening masses, we can use the WI in (43) to obtain a prediction for the T scaling of the (spatial screening) mass ratio: since the susceptibilities correspond to zero momentum correlators and hence to inverse square masses [47].
To test the scaling law in (46), together with those for the π, K andss channels analyzed in [47], we take lattice data for screening masses and quark condensates from the same lattice group. As mentioned above and to the best of our knowledge, the more recent available results for screening masses in the I = 1/2 sector are those in [37]. The corresponding condensate data of the same group with the same lattice conditions (p4 action, N τ = 6, m s = 10m) are given in [57]. Nevertheless, as pointed out in [47] and in Section III A, lattice results for quark condensates are affected by finite size divergences of the type q i q i ∼ m i /a 2 . Thus, in order to check (46), we have to consider subtracted condensates free of lattice divergences. Following [47] and [3], we replace qq l (T ) → qq l (T ) − qq l (0) + qq  [3]. We proceed as in [47] and consider qq ref l and ss ref as fit parameters, used to minimize the squared difference between the relative screening masses and subtracted condensates. We remark that we cannot just take the reference value provided in [3] since we are taking older lattice results with very different lattice conditions. Thus, with only two free parameters, we can test the validity of our scaling laws based on WIs using lattice data in the four channels. In addition, we use for the condensates the dimensionless quantity r 3 1 qq , where r 1 0.31 fm is defined in lattice analysis to set the physical scale [3,57]. An important difference when including the κ channel is that in [57] the data are not given relative to their T = 0 value. Therefore, we have taken the lowest temperature point T 0 as the reference value for the screening masses in that channel, so that, according to (46), we define: and then we should compare M sc κ (T )/M sc κ (T 0 ) with ∆ κ (T ; T 0 ) −1/2 . In Fig. 4 we show our results for the four channels. The definitions of ∆ l , ∆ K and ∆ s are given in [47] and correspond to the subtracted condensate combinations predicted by the WI with respect to the T = 0 values. It is important to point out that we have not included in the fit the points above T c in the κ channel. We do not expect that the smoothness assumptions we are using to justify the scaling law can be maintained above T c . In particular, the deviations between pole and screening masses can be sizable, as commented in [47] and confirmed by recent model analysis [28]. Nevertheless, we include those points in the plot to emphasize the minimum around T c exhibited by the κ screening mass. The results below T c show an excellent agreement with the predicted WI scaling, the maximum deviation being of 5.2% (second point in the κ channel). Moreover, the reference values qq ref l , ss ref are very similar to those obtained in in [47] for a three-channel fit. In addition, we remark that the scaling law in (46) explains qualitatively the observed minimum of M sc κ near the transition, which arises from the relative behavior of (subtracted) light and strange condensates. Near the chiral transition the inflection point of qq l signals an abrupt decreasing with respect to ss , which remains smoothly decreasing.

IV. IDENTITIES RELATING CORRELATOR DIFFERENCES WITH THREE-POINT VERTICES
A. I = 0, 1 Further relations can be obtained from the axial WI in (1) when two-point field operators are chosen. In particular, the evaluation of (1) with O b (y) = σ l (y)π b (0) and O b = δ b (y)η l (0) gives rise to the identities: These are particular combinations of the operators O(y) = S 8,0 (y)π(0) and O(y) = P 8,0 (y)δ(0), which using (12) yield: Note that, due to the η − η mixing, the above WIs contain the nonzero 08 correlator, albeit it disappears in the light sector WI in (49). Moreover, eliminating in (50)-(53) the δδ and ππ correlators, we get two new WIs: which as we have seen in Sections III A and III B, play a crucial role for the discussion of the chiral pattern, partner degeneration and mixing angles. These identities can be translated to WIs for susceptibilities, once the integration in the y variable is performed (p = 0 in Fourier space): which can be also checked in the lattice or using different model analysis in terms of the p = 0 three-point functions.
The WIs in (48)- (49) and (54)- (55) parametrize the degeneration of chiral partners in terms of three-point functions. If SU A (2) is exactly restored, i.e., in the light chiral limit and for a vanishing light-quark condensate, the r.h.s. of these equations should vanish and hence the analysis of those three-point correlators provide alternative ways to study chiral symmetry restoration. More precisely, according to (26) and (54): The importance of the WIs (48)- (49) and (54)- (55) is that they provide precise and direct information about the relevant interaction vertices and physical processes responsible for the breaking of the degeneracy in (14) in the finite mass case and T < T c . In this way, the analysis of the mass and temperature dependence of the three-point functions in the r.h.s would be very relevant to analyze the evolution towards degeneration. In particular, (48) and (49) imply that π/σ l and η l /δ partner degeneration are driven by the σππ and a 0 πη l vertices, respectively, whereas a 0 πη s and σ s ππ vertices enter in the crossed correlators (54)- (55).
We could also construct WI relating three point functions in the r.h.s. of (48)- (49) and (54)- (55) with four-point pseudoscalar operators. This would be a much manageable scenario within an effective theory description (like ChPT), and it would not require to introduce explicitly the f 0 (500)/(σ) degree of freedom in the Lagrangian. Looking in more detail at the isoscalar case, the σ l and σ s bilinears in (48) couple to the scalar source s(x) in the QCD Lagrangian [58], which on the meson Lagrangian translates into a contribution from the ππ,KK and ηη channels at leading order. Therefore, the r.h.s. of the identity (48) is directly related to ππ → ππ scattering in the I = J = 0 (σ) channel, as well as toKK → ππ and ηη → ππ, where the σ/f 0 (500) resonance can also be generated. Thus, this identity states that the σ/f 0 (500) resonance produced in ππ scattering plays a fundamental role for the O(4) degeneration of partners. This is fully consistent with the recent analysis in [46], where it is shown that the critical crossover behavior ofχ ll S can be achieved including the thermal pole of the σ/f 0 (500), as generated in unitarized ππ scattering [59]. Similarly, the δ bilinear translates into a contribution from the πη andKK channels. In this way, the r.h.s. of (49) connects with the a 0 (980) resonance, which is produced in πη → πη andKK → πη scattering and motivates a future finite temperature analysis of this resonance.
Furthermore, at first glance the identities (48)- (49) and (54)- (55) suggest the degeneration conditions in (14) once the light chiral limitm → 0 is taken, albeit this could be only possible at temperatures close to T c . In fact, at T = 0, qq l is O(1) in the light chiral limit and the scalar and pseudoscalar susceptibilities satisfy [8,54], hence in contradiction with partner degenerations. Similarly, for the δ − η l identity (57), χ δ S = O(1) at T = 0 [53] while χ η l P diverges at least as O(m −1 ) (21). Thus, the three-point functions in the r.h.s. of (48)-(49) and (54)-(55) should scale as 1/m at T = 0 in the light chiral limit. As T increases, χ π P drops proportionally to qq l as given by (20) whileχ ll S increases. Hence, they will eventually match consistently with partner degeneration around T c . According to (48) such degeneration, expressed in term of two-point correlators, is driven by the σππ vertex, which becomes the physically relevant interaction. The same happens in the δ channel, where χ ll P drops, hence tending to match with χ δ S , driven by a 0 πη interaction through (49). Further identities can be derived considering diagonal rotations α 0 A . On the one hand, considering O bc (y) = π b (y)δ c (0) and O(y) = σ(y)η l (0) in (1), one gets for a = 0: whereη On the other hand, from the transformation in (12), taking the combinations O = P 8,0 S 8,0 , one obtains: The identities (64) can also be combined to give for the ls and ss correlators: Like in the previous discussion, the above identities show the different vertices responsible for the symmetry breaking of the expected U (1) A degenerated patterns, i.e., π − δ and η l,s − σ l,s degeneration, which are now related with additional three-point vertices. Compared to the previous identities (48)- (55), there are two new terms. Namely, one proportional to m s and an anomalous term proportional to A(x) in (63). The latter corresponds to the U (1) A breaking contributions in (3).
Further relations for the K and κ correlators can be obtained taking the two-point operator O bc = P b (y)S c (0). Considering a SU A (2) transformations in (1), i.e., taking a = 1, 2, 3, one obtains for the KK and κκ correlators: where we denote P a P b = δ ab P KK and S a S b = δ ab P κκ for a, b = 4, . . . , 7.
The above identity provides information of the physical processes responsible for such degeneration. The possible values for d abc = ±1/2 account for the different combinations of allowed κ → Kπ processes, which, within a pure light or NGB theory, are Kπ → Kπ and Kη → Kπ. Hence, (67) highlights the relevant role of the controversial κ resonance at finite T for the chiral symmetry restoration in the I = 1/2 channel.
Finally, we will also consider the effect of U (1) A transformations in this sector. Taking O bc as before but now with a = 0, (1) gives which corresponds to κ → Kη and κ → Kη decays including the anomalous contribution, or Kη(η ) → Kπ and Kη(η ) → Kη(η ) meson scattering processes in the κ channel. Note that the l.h.s. of (67) and (68) are the same except for the d abc = ±1/2 factor, which allows one to connect the different scattering processes involved. Thus, the vanishing of the r.h.s. of equations (67) and (68) would be consistent with the K − κ degeneration at chiral and U (1) A transitions described in Section III D.

V. EFFECTIVE THEORY ANALYSIS WITHIN U (3) CHIRAL PERTURBATION THEORY
The WI studied in this work have been derived within the QCD generating functional. Thus, in principle, they are subject to renormalization ambiguities related to the fields and vertices involved [60,61]. It is therefore important to provide a specific low-energy realization of WI and the observables entering them, such as the scalar and pseudoscalar susceptibilities that we have been analyzing in previous sections. We will carry out such analysis in this section, where we provide a thorough ChPT U (3) analysis, hence extending the work in [47] to include the relevant chiral and U (1) A partners. As we are about to see, this study will confirm our previous findings based on WI and symmetry arguments.
The U (3) ChPT formalism provides a consistent framework for calculating low-energy physical processes related to the pseudoscalar nonet. With respect to standard SU (3) ChPT, where pions, kaons and the octet η 8 state are the pseudo-Goldstone bosons, it incorporates also the singlet η 0 as a ninth pseudo-Goldstone boson. However, due to the U A (1) anomaly, the mass of the η 0 is too heavy to be included in the standard chiral power counting in terms of meson masses, energies and temperatures. Nevertheless, the axial anomaly vanishes in the N c → ∞ limit, in which the singlet field η 0 would become the ninth Goldstone boson in the chiral limit. For that sake, the large N c limit framework must be considered [16][17][18], so that the chiral counting is extended to include the 1/N c counting. In this way, the expansion is performed in terms of a parameter δ such that M 2 , E 2 , T 2 ,m, m s = O(δ) and 1/N c = O(δ), where M, E are typical meson masses and energies. In this counting, the tree-level pion decay constant F 2 = O(N c ) = O(1/δ), which hence suppresses loop diagrams. The counting of the different Low-Energy Constants (LECs), according to their O(N c ) trace structure, is given in detail in [17,[30][31][32].
In [47], one-point WI involving pseudoscalar susceptibilities and quark condensates were verified within U (3) ChPT and the explicit expressions for those susceptibilities and condensates were given up to NNLO in the δ counting. Here, we will extend that work to the scalar sector, which will allow us to check our previous results based on WI for the nonet partners under O(4) and U (1) A restoration. For that purpose, we consider the Lagrangian up to NNLO, namely L = L δ 0 + L δ + L δ 2 in the notation of [30][31][32]. Besides, the η − η mixing angle has to be properly incorporated. The explicit expressions for lagrangians, self-energies and the mixing angle up to the relevant order we are considering here can be found in [31].
Within this U (3) framework and including scalar sources in the effective Lagrangian as dictated by chiral symmetry [17,18,[30][31][32], we have calculated all the scalar susceptibilities involved in our present analysis, namelyχ ll S (T ), χ ss S (T ), χ ls S (T ), χ δ S (T ) and χ κ S (T ) up to the NNLO O(δ 0 ). Their explicit expressions are collected in Appendix A. With those expressions, we have checked that the WI (43) holds to the order considered. Therefore, together with the analysis in [47] of the identities (20)- (22) and (42), we complete the check of all the one-point WI. Recall that the LO O(δ −2 ) vanishes for the scalar susceptibilities (it contributes to the pseudoscalar ones). Note also that, since we work within the Dimensional Regularization scheme, the differencesχ ll S − χ ll S andχ ss S − χ ss S formally vanish as δ (D) (0) in the ChPT calculation.
As in the SU (3) calculation of scalar susceptibilities [52,54], our present calculation involves tree level terms, as well as one-loop corrections. Temperature effects show up on three type of topologies: 1. Tadpole contributions coming from the Euclidean tree-level propagator G i (x = 0), whose finite part reads where i = π, K, η, η , m 0i are the tree level masses and µ is the renormalization scale.

Contributions arising from Wick contractions of two pairs of meson fields at different space-time points, proportional to
whose finite part can be written in terms of 3. Loop contributions coming from mixed contributions of the type: which reduces to (71) for m 2 i → m 2 j . An important consistency check of our calculation is that all the results are finite and scale independent. Together with the χ P susceptibilities already calculated in [47], these results will allow us to examine how our previous results on partner degeneration are realized within ChPT. Although the ChPT framework is limited to a low temperature description, we are going to see that the thermal extrapolation of the ChPT curves provides useful model-independent results confirming our previous analysis for partner degeneration. In addition, this framework will allow us to examine the chiral limit consistently.
Let us start by analyzing in U (3) ChPT the susceptibilities in Section III A regarding the O(4) vs O(4) × U (1) A pattern and the corresponding partner degeneration in the I = 0, 1 sector. The results for the four susceptibilities involved are plotted in Fig. 5 for the physical value of the pion mass. The numerical values of the LECs involved are taken from [31] and the bands in the figure cover the uncertainties of those LEC quoted also in [31]. We consider Let us define T c as the (pseudo-critical) O(4) restoration temperature for which degeneration of the chiral partner states σ/π takes place, i.e., χ π P (T c ) = χ ll S (T c ). Note that this temperature is more advisable than the standard definition in terms of the vanishing quark condensate, since the latter is meant to remain nonzero at the chiral transition for physical masses. Recall that throughout this section, what we really mean by degeneration of partners is the matching of their corresponding susceptibilities, since ChPT is not able to reproduce neither a true degeneration, nor a crossover or a phase transition behavior. Numerically, for the physical pion mass and for the LECs in [31], we obtain T c ∼ 264 MeV and T 0 1.09 MeV (for the central values in Fig. 5) where T 0 is defined as qq l (T 0 ) = 0. We stress that the particular numerical value for T c is not important, i.e. the ChPT expansion is limited at low temperatures so it is not supposed to provide a quantitative description of the transition. Nevertheless, as we are about to see, the main qualitative features in terms of partner degeneration and the relation between different pseudocritical temperatures obtained from the extrapolation of the ChPT results are consistent with lattice and with our previous WI analysis.
In addition, the results in Fig. 5 show that χ π P matches χ δ S above T c . This crossing point can be considered as an estimate of U (1) A degeneration with a critical temperature T c2 defined as χ π P (T c2 ) = χ δ S (T c2 ). Using physical pion masses one finds T c2 1.07 T c (for the central values) i.e., quite close to T c . Nevertheless, the numerical difference lies within the ChPT uncertainty range, as seen in the figure. The behavior of χ ll P (T ) shown in Fig. 5 is not so reliable as the other susceptibilities. In this case the O(δ 0 ) ChPT corrections at T = 0 turn out to be of the same order as the leading O(δ −1 ) ones. This effect is worsened as T increases. Nevertheless, taking this caveat in mind, we can still see that the difference between χ π P (T ) and χ ll P (T ) does vanish close to (and above) T c2 . Once more, this value can be taken as the pseudo-critical temperature characteristic of O(4) × U (1) A restoration, which according to (18) we define as χ 5,disc (T c3 ) = 0. In the physical case depicted in Fig. 5 we get T c3 1.13T c . As a summary, from the results plotted in Fig. 5, we conclude that the U (3) ChPT analysis yields O(4) × U (1) A partner degeneration close and above O(4). Recall that we may have different pseudo-critical temperatures in terms of partner degeneration, both for O(4) and for U (1) A partners, in the physical mass case.
In Fig. 5 we also show the K and κ susceptibilities for I = 1/2. They match at χ K P (T c4 ) = χ κ S (T c4 ) with T c4 T c2 . This behavior is compatible with the pattern predicted in Section III D, i.e., K − κ degeneration takes place at U (1) A restoration. Furthermore, as we will see below, this temperature approaches O(4) restoration in the chiral limit, consistently with (44).
More revealing results are obtained from our ChPT expressions when we approach the chiral limit. In that regime, we would expect that the two pseudo-critical temperatures corresponding to the chiral transition, T 0 and T c , should tend to coincide. In addition, from the analysis in Section III A, we would also expect the U (1) A and O(4) × U (1) A pseudo-critical temperatures to approach the chiral O(4) ones. This is indeed what we obtain, as it is shown in Fig. 6, where the hierarchy T c3 > T 0 > T c2 > T c is maintained as the chiral limit is approached.
As explained above, T 0 > T c is expected from chiral restoration arguments, while we expect T c2 > T c and T c3 > T c since U (1) A partners are meant to degenerate after O(4) ones. It is also natural that T c3 > T c2 since the restoration of χ 5,disc requires the vanishing of both χ π P − χ δ S and χ δ S − χ ll P . In any case, from our present ChPT approach, given the decreasing behavior obtained for χ ll P in Fig. 5, the condition T c3 > T c2 clearly holds. Finally, there is no a priori reason on how T c3 or T c2 should be related to T 0 .
As for the I = 1/2 K − κ matching, we see from Fig. 6 that T c4 remains almost identical to T c3 for all values of m π , approaching the other restoration temperatures in the chiral limit. This is consistent with what we expect from the WI (44).
Moreover, the leading order in the chiral limit for the susceptibilities is actually quite useful for our present purposes. We obtain from the expressions in Appendix A: where a, b, c 0 , c 1 , d 0 and d 1 are positive dimensionless constants independent of T and m π . One has a = 3 while exponentially suppressed contributions of order exp(−m K /T ) have been neglected.
From the previous expressions and the definitions of pseudo-critical temperatures explained before, we get which is consistent with the numerical results showed in Fig. 6 and with the T c3 > T 0 > T c2 > T c hierarchy. In addition, the gap between the U (1) A pseudocritical temperatures T c3 and T c2 is O(m 2 π ), which is also the gap between them and T 0 . On the contrary, the gap between T 0 , T c2 or T c2 and the O(4) T c is O(m π ), i.e., larger in the chiral limit expansion.
The chiral expansion of the U (3) ChPT results is also particularly useful to disentangle the behavior of the connected and disconnected parts of the scalar susceptibility, which we have discussed in a general context in Section III C. The ChPT expansion, by construction, is not able to generate a peak for the scalar susceptibility as T → T c . However, we can learn about the critical behavior of the different susceptibilities involved by examining their infrared (IR) chiral limit m π → 0 + behavior, for which ChPT does capture the expected behavior for condensates and susceptibilities [6,8].
Thus, consider the behavior of the different susceptibilities involved in the relation (41) in the chiral limit at O(4) and O(4) × U (1) A restoration, i.e., at T c and T c3 . On the one hand, we have at T = T c which stem from (75) and (77) withχ dis S = 1 4 χ ll S − χ δ S according to the discussion in Section III C. Therefore, at T c the IR divergent behavior of χ 5,disc in the l.h.s. of (41) is carried entirely byχ dis S in the r.h.s.. Note that the second term in the r.h.s. of (41) vanishes by definition at T c and the third term in the r.h.s. is regular in the IR limit.
On the other hand, at T = T c3 one finds Note that T c3 is defined as the temperature for which χ 5,disc (T c3 ) = χ π P (T c3 ) − χ ll P (T c3 ) = 0. This vanishing is compatible with the fact thatχ dis S (T c3 ) in the r.h.s of (41) is IR divergent, as given by (79). Namely, such divergence is exactly cancelled by that of −χ ll S (T c3 )/4. The remaining terms in (41) are IR regular and their sum vanishes exactly. As a summary, it is perfectly compatible from a ChPT point of view to have a divergentχ dis S and a vanishing χ 5,disc at T c3 while both diverge at T c , with T c3 − T c = O(m π ). These features can be appreciated in Fig. 7, where we plot those susceptibilities very close to the chiral limit. At T = T c , χ π P −χ ll S vanishes while χ 5,disc andχ dis S are both large and of the same order, which arises from their 1/m π behavior (compare with the typical numerical values of susceptibilities in the physical case in Fig. 5). At T = T c3 , χ 5,disc vanishes and the large positive value ofχ dis is compensated by the large negative contribution of -χ ll S /4, as discussed above. In the above discussion, the connected susceptibility, i.e.,χ con S = χ δ S /2, remains regular in the chiral limit. Nevertheless, as already discussed in Section III C, general arguments indicate thatχ con S could actually peak near U (1) A restoration. A hint of that behavior can be seen also in U (3) ChPT by taking simultaneously the limits m π → 0 + and M 0 → 0 + . Note that M 0 is the anomalous part of the η mass, which should vanish in a U (1) A restoring scenario. The contributions to χ δ include mixed loop terms of the form (74) with i = π, j = η. In the M 0 → 0 + limit, we have m η → m + π , leading to which, according to (76), generates an additional IR divergent term not present in the m π → 0 + for a fixed m η . In more detail, in the m π → 0 + and M 0 → 0 + limit we obtaiñ with α = M 0 /m π . We see that the connected scalar susceptibility above contains an IR divergent part in this combined limit, whose strength is parameterized by α. On the one hand, taking α → ∞ we recover in (81) the results given in (75), corresponding to m π → 0 + and M 0 = 0. On the other hand, the α → 0 + limit would correspond to the maximum U (A) 1 restoration in this parameterization. In Fig. 8 we plot the ratioχ con S /χ ll S at leading order in T /m π as a function of α. We see that for α → 0 + a maximum finite value of 1/2 is reached for that ratio. For reference, the value of α corresponding to the physical values of m π and M 0 is α 5.99, which corresponds in Fig. 8 toχ con S /χ ll S 0.21. Following the discussion in Section III A, let us now compare the temperature scaling of χ 5,disc (T ) and the light quark condensate qq l (T ). In Fig. 9 we plot χ 5,disc (T )/χ 5,disc (0) and qq l (T )/ qq l (0) as the pion mass is reduced. It is clear that their temperature scaling is almost identical as the chiral limit is approached, consistently with [29] and with our analysis in Section III A. The reason can be understood again from the chiral limit expressions (75). In the chiral limit the η l contribution χ ll P is parametrically negligible with respect to χ π P , so that their difference given by χ 5,disc is dominated by χ π P , which vanishes exactly like qq l due to the WI (20). Finally, we will analyze the behavior of the scalar and pseudoscalar mixing angles. With the mixing angle defined through (32), we solve for every T the equations using the U (3) ChPT expressions for the susceptibilities. The result is showed in Fig. 10. First, as commented in Section III B, the degeneration of the scalar and pseudoscalar mixing angles takes place at about T 1.05T c , i.e., around O(4) × U (1) A degeneration. In addition, they coincide in a value close to the ideal mixing θ id , also consistently with the discussion in Section III B. In the case of θ S , the variation with respect to its T = 0 value is small and close to ideal mixing. These findings are in fair agreement with the results in [27] obtained within the framework of the Polyakov-loop extended NJL model. Note that we do not see in this U (3) ChPT analysis a region of vanishing mixing, since that would require a larger gap between O(4) and O(4) × U (1) A restoration.

VI. CONCLUSIONS
In this work we have performed a detailed analysis of the correlators and susceptibilities corresponding to the scalar and pseudoscalar meson nonets, both from general arguments in terms of Ward Identities and from the modelindependent description provided by U (3) Chiral Perturbation Theory. Our main physical motivation has been the study of partners and patterns of chiral and U (1) A restoration.
In particular, we have showed that in the limit of exact O(4) restoration, understood in terms of δ − η partner degeneration, the WI analyzed yield also O(4) × U (1) A restoration in terms of π − η degeneration, i.e., from the vanishing of χ 5,disc . Our analysis also provides a connection between χ 5,disc and the topological susceptibility χ top , which is defined from the correlator of the anomaly operator. The results we obtain using ChPT are consistent with this analysis. Namely, one finds that the pseudo-critical temperatures for restoration of O(4) and O(4) × U A (1) tend to coincide in the chiral limit. In the real physical world with massive quarks, our conclusions agree with N f = 2 lattice results for partner degeneration. The large gap between O(4) and O(4) × U (1) A partner degeneration observed in N f = 2 + 1 simulations can be explained by the distortion in δ − η degeneration, presumably induced by strange quark mass effects. The large NLO corrections for the η l susceptibility that we obtain within U (3) ChPT support this conclusion.
In addition, including isospin breaking m u = m d effects, we have recovered the formal connection of the δ and σ susceptibilities with the connected and disconnected scalar ones, customarily measured in lattice analysis. The behavior of the connected and disconnected contributions to the scalar susceptibility have been studied within ChPT near O(4) and O(4) and O(4) × U (1) A restoration. In that context, we have shown that a vanishing χ 5,disc at O(4) × U (1) A restoration is compatible with a divergent χ disc S . Moreover, the ChPT behavior for a vanishing M 0 (the anomalous part of the η mass) is a hint towards a possible peaking behavior of the connected χ con S . Regarding scalar and pseudoscalar mixing angles, our analysis shows that the WI are consistent with θ P ∼ θ S ∼ θ id degeneration around O(4) × U (1) A restoration, where θ id is the ideal mixing angle.That conclusion is supported also by the U (3) ChPT analysis, where θ S remains close to ideal mixing for all temperatures, consistently with recent analyses. In the N f = 2+1 lattice data, an intermediate range between O(4) and O(4)×U (1) A restoration, compatible with vanishing pseudoscalar mixing is present.
Our analysis shows also that in the I = 1/2 sector, the K and κ states degenerate both at exact O(4) and U (1) A restoration. Moreover, the degree of degeneracy of these two patterns is directly determined by the subtracted condensate ∆ l,s measured in the lattice. These results are confirmed also within the U (3) ChPT analysis. In addition, we have also showed in this sector that the temperature behavior of the screening mass in the κ channel measured in the lattice can be explained with the corresponding WI relating χ κ S with the difference of light and strange quark condensates, which we have checked in ChPT. Such analysis extends a previous work for the π, K, η channels. We have also showed that the four channels can be simultaneously described with a two-parameter fit.
Our U (3) ChPT analysis allows one to obtain all the nonet scalar susceptibilities up to NNLO in the chiral power counting for finite temperature, thus completing previous calculations of the pseudoscalar ones. The explicit expressions for those scalar susceptibilities are also provided here.
In addition, we have discussed additional WI relating two and three-point functions, which may become useful to relate O(4) and U (1) A partner degeneration quantities with meson vertices and scattering amplitudes. A detailed analysis of those WI is left for future investigation.
As a summary, our study provides new theoretical insight for the understanding of the nature of the chiral and U (1) A transitions in terms of the degeneration of the meson nonet states, which is meant to be useful for lattice, phenomenological and experimental analyses. The picture emerging both from a general Ward Identity framework and from ChPT is robust and provides model-independent conclusions that could guide future work on this subject.