The chiral phase transition and the axial anomaly

To date numerical simulations of lattice QCD have not found a chiral phase transition of first order which is expected to occur for sufficiently light pions. We show how the restoration of an exact global chiral symmetry can strongly decrease the breaking of the approximate, anomalous $U_A(1)$ symmetry. This is testable on the lattice through simulations for one through four flavors. In QCD a small breaking of the $U_A(1)$ symmetry in the chirally symmetric phase generates novel experimental signals.

One of the most beautiful phenomena in quantum field theory is the axial anomaly of Adler, Bell, and Jackiw [1][2][3][4].In four spacetime dimensions massless fermions are chiral, whose spin is either opposite or along the direction of motion, and so respectively left or right handed.For chirally symmetric interactions, as with a gauge field, the current for the total number of fermions, left plus right, is always conserved.In contrast, the axial current, equal to the difference of the left and right handed currents, is conserved classically but not quantum mechanically.Instead, the divergence of the axial current is proportional to the density of the topological charge for the gauge field.
The relationship between the divergence of the axial vector current and topological charge density, computed at one loop order, is exact [1,3,4].Even so, this does not tell one how large the topologically nontrivial fluctuations are [19,20].At zero temperature they must be large in order to make the η ′ heavy.In contrast, at high temperature instantons are the dominant topologically nontrivial fluctuations [21,22].In this limit the density of instantons can be computed semiclassically, which implies that the magnitude of the topological charge susceptibility vanishes as a high power of the temperature T , as T → ∞.Even though it vanishes at high T , it is natural to expect that the density of topologically nontrivial fluctuations is nonzero for any finite T .
This leaves the relationship between the restoration of the exact chiral symmetry, and the approximate, anomalous U A (1) symmetry, obscure.Based upon extensive results from numerical simulations in lattice QCD, in this Letter we outline how the restoration of an exact chiral symmetry strongly affects the approximate restoration of the anomalous U A (1) symmetry.This can be tested in lattice QCD with different numbers of flavors, especially for a single flavor.The approximate restoration of the anomalous U A (1) symmetry has dramatic implications for the collisions of heavy ions, and surely implications for condensed matter systems as well.

I. EFFECTIVE LAGRANGIANS
We consider QCD-like theories, with a SU (N c ) gauge field coupled to N f flavors of massless quarks in the fundamental representation.As massless fields, the Lagrangian is invariant under the global chiral rotations q L,R → e i(θ V ∓θ A )/2 U L,R q L,R , where q L and q R are left and right handed quarks, and U L and U R elements of the global symmetry groups SU L (N f ) and SU R (N f ), respectively.There are two U (1) groups, one for quark number, θ V , and one for axial quark number, θ A .
We assume that in vacuum, the exact global chiral symmetry is is characterized by an expectation value for a color singlet, spin-zero field Φ, Φ = q L q R , where Φ transforms under the fundamental representation of the global symmetry group of (1), this symmetry can be ignored.
Through the axial anomaly, the U A (1) symmetry is violated quantum mechnically by topologically non-trivial fluctuations such as instantons.The exact chiral symmetry which remains is just In vacuum, it is expected that chiral symmetry breaks to the maximal diagonal subgroup of SU V (N f ), ⟨Φ ab ⟩ = ϕ 0 δ ab , where a, b = 1 . . .N f are the indices for the SU L (N f ) and SU R (N f ) groups.Phenomenologically, this pattern certainly occurs in QCD, where N c = 3 and N f = 2 or 3. Coleman and Witten proved that it arises in the limit of large N c and small N f [23].
The appropriate effective Lagrangian for chiral symmetry breaking is well known [5-7, 11, 19, 24-41].There are two types of terms which enter.The first type are terms invariant under G cl .Up to terms of sixth order in Φ, these are Our trace is normalized, so tr 1 = 1.For a gauge theory in 3 + 1 dimensions, a phase transition at a nonzero temperature T is characterized by an effective theory in three dimensions.Couplings to sixth order then represent the relevant operators: the mass squared, m 2 ; two quartic coupling constants, λ 1 and λ 2 , with dimensions of mass; and the six point couplings: κ 1 , κ 2 , and κ 3 , with dimensionless coupling constants.Terms of eighth and higher order are irrelevant operators, whose coupling constants have negative mass dimension.
The second class of terms are invariant under G qu but not U A (1), and so are generated by topologically nontrivial fluctuations [5-7, 11, 12]: For three flavors, these are terms to third, fifth, and sixth order in Φ.
The Atiyah-Singer index theorem relates the change in the axial fermion number to the topological charge as while for an anti-instanton with Q < 0, the quark zero modes are right-handed [42].Thus the first two terms, ∼ det Φ, arise from instantons with charge one, [5][6][7]24], while the last term, ∼ (det Φ) 2 , is due to instantons with charge two [11,12].
The anomalous couplings in Eq. ( 2) are the first terms in an infinite series, (3) Terms with couplings ∼ ξ (j,k) i are generated by fluctuations with topological charge |Q| = i.For ease of notation, we have denoted ξ (0,0) i ≡ ξ i .

II. A CONJECTURE ABOUT ANOMALOUS COUPLINGS
At the outset we recognize that especially in vacuum, the topologically nontrivial configurations are surely truly quantum objects, and far from any semiclassical approximation [20].For ease of discussion, we refer to the dominant configurations in vacuum as quantum instantons, and those which dominate when T → ∞ as semiclassical instantons.The contribution of a single semiclassical instanton to the partition function is ∼ exp(−8π 2 /g 2 (T )), so by asymptotic freedom this falls off as a high power of the temperature [46].Numerical simulations of lattice QCD indicate that the topological susceptibility falls off close to this power down to temperatures of T qu ∼ 300 MeV [47].While astonishingly low, this is still about twice the temperature for the chiral crossover in QCD, at T χ ∼ 156 MeV [48][49][50][51][52]. Thus we can take T qu as an estimate of the change from quantum to semiclassical instantons.[53].
The essential question is what is the relative magnitude of the anomalous coupling constants in vacuum, and as the temperature increases?The standard assumption with effective Lagrangians is that the couplings with the highest mass dimension dominate.For the U A (1) symmetric Lagrangian of Eq. ( 1), that is the mass squared, followed by the quartic couplings, etc.In the standard Wilsonian paradigm, this is inescapable, because the only way of differentiating these different operators is through their mass dimension.
Of course some operators have a larger symmetry than others: m 2 tr(Φ † Φ) and λ 1 (tr(Φ † Φ)) 2 are invariant under O(2N 2 f ), while the coupling λ 2 tr(Φ † Φ) 2 is only invariant under G cl .But this is standard, and doesn't affect the renormalization group flow [54].The only time that couplings of sixth order need to be included is at isolated points where both λ 1 and λ 2 vanish; then there is a tricritcal point, controlled by the evolution of the six-point coupling constants, κ 1 , κ 2 , and κ 3 .
For the anomalous coupling constants, the operator with the lowest mass dimension is ξ 1 det Φ.Thus naively one expects that this operator dominates the infrared behavior near the chiral phase transition [19].
However, there is something special about the anomalous couplings, which is not true in standard effective theories.Terms ∼ det Φ are due, uniquely, to the zero modes of an instanton with charge one; those ∼ (det Φ) 2 , to the zero modes of an instanton with charge two, etc. [5-7, 11, 12].
In vacuum, when chiral symmetry breaking occurs the effective coupling for the first anomalous coupling, ∼ det Φ, is a sum of an infinite number of terms: As indicated, all of the anomalous coupling constants, the ξ , and the expectation value of the scalar field, ϕ 0 , are functions of temperature.At very high temperature, the anomalous coupling constants ξ (j,k) i can be computed semiclassically, and are all nonzero [5-7, 11, 12].
Why should our conjecture be valid?Consider forming an effective Lagrangian for chiral symmetry breaking from the underlying gauge theory.We integrate out quarks and gluons to form an effective theory for Φ, over some volume V χ .The essential question is then, what is the distribution of quantum instantons which contribute in V χ ?
If in V χ quantum instantons with net charge one dominate, then so will the operator ∼ ξ 1 det Φ.If instead V χ predominately contains quantum instantons with net charge two, then the operator ∼ ξ 2 (det Φ) 2 will be more important.We suggest, then, that in vacuum quantum instantons with charge two and greater dominate V χ .Of course in all, the topological charge of the vacuum vanishes.But it need not within a finite volume V χ [55].
We now discuss the implications of our conjecture, beginning with the case of three flavors, which motivated it.

III. THREE FLAVORS
In QCD there is no true phase transition, only a crossover (albeit with a large increase in the pressure).If ξ 1 (T χ ) ̸ = 0, however, for three flavors the operator ∼ det Φ is a cubic operator.The presence of a cubic operator implies that the standard effective Lagrangian for a second-order phase transition, with only terms quartic and quadratic in the fields, cannot be reached, and so the transition is of first order.Hence a chiral phase transition of first order must emerge for sufficiently light pions, m π < m crit π [19].For simplicity we discuss the case of three degenerate quark flavors.
How large m crit π is depends upon the magnitude of ξ 1 (T χ ).We suggest that in vacuum the η ′ is heavy not because ξ 1 is large, but because the higher order terms, such as ξ 2 , ξ 3 , etc., contribute and overwhelm ξ 1 .At the chiral phase transition, however, ϕ 0 = 0, and one is left In mean field theory, it is customary to assume that ξ 1 (T ) is independent of temperature.Since the η ′ is so heavy at zero temperature, in vacuum ξ 1 (0) must be large, and m crit π should also be large.In a quark meson model, one finds m crit π ≈ 150 MeV if the vacuum fluctuations of quarks are ignored [56], and m crit π ≈ 86 MeV if they are included [57].Similarly, using mean field theory in a chiral matrix model yields m crit π ≈ 110 MeV [58].
Going beyond mean field theory, mesonic fluctuations can be included by using the functional renormalization group.This gives rise to a critical mass which is dramatically smaller but still nonzero, m crit π ≈ 17 MeV [57].Presumably this occurs because the functional renormalization group is including, at least in part, higher-order anomalous contributions as in Eq. ( 4) [59].
In contrast, no simulation of lattice QCD has ever found evidence of a first order transition.Instead, they only place upper bounds on m crit π , which are much smaller than the values in mean field theory.This includes: m crit π < 50 MeV in Ref. [60]; m crit π < 100 MeV in Ref. [61]; m crit π < 90 MeV in Ref. [62].By considering the position of the tricritical point as a function of N f , it has been asserted in Ref. [63] that even for three flavors, the chiral transition is of second order in the chiral limit.
We note that a small value of ξ 1 (0) is perfectly consistent with hadronic phenomenology at zero temperature, for both hadronic masses and decay widths.In fact, these quantities can be reproduced successfully in low-energy models even with ξ 2 as the only anomalous coupling [30,31,[43][44][45].This will be analyzed in greater detail in future work [64].
While we assume that ξ 1 (T χ ) ̸ = 0, we stress that we can not exclude the possibility that ξ 1 (T χ ) = 0. From the viewpoint of effective Lagrangians, this is most unnatural, as then two parameters -m 2 (T ) and ξ 1 (T ) -vanish as one thermodynamic parameter, the temperature, is varied.
If the result of Ref. [63] holds and the chiral transition is of second order, then we speculate that not just ξ 1 (T χ ), but all of the anomalous couplings vanish at the critical temperature: This implies that the anomalous U A (1) symmetry is restored at T χ .This can only happen precisely at T χ , since as T → ∞ semiclassical instantons are present and give ξ (j,k) i (T ) ̸ = 0. Perhaps Eq. ( 5) is a consequence of the 't Hooft anomaly condition [65,66].
If ξ 1 (T χ ) = 0, the universality class of a second order chiral phase transition is that of G cl symmetry.There has been extensive work on this possibility, including using the ϵ-expansion [19,67,68], perturbation theory in three dimensions [69], Monte Carlo simulations in three dimensions [70,71], the functional renormalization group [41], and the conformal bootstrap [72][73][74].As opposed to earlier results [19,[67][68][69][70][71], recent studies with the functional renormalization group [41] and conformal bootstrap [72][73][74] find infrared stable fixed points for G cl in three dimensions.Thus if ξ 1 (T χ ) = 0, we assume that three massless flavors could have a chiral transition of second order.This is in accord with recent results using Dyson-Schwinger equations [75], where a second-order chiral transition is found in the chiral limit.In this case scaling analysis shows that the universal physics is described by mean-field behavior without further external input.A second order transition then arises if ξ 1 (T χ ) = 0, e.g., Refs.[57,64], providing strong indications that this is also true in Ref. [75].

IV. TWO AND FOUR FLAVORS
For two flavors, the term ∼ ξ eff 1 is a mass term which splits the η meson from the pions.The couplings ∼ ξ (1,1) 1 and ∼ ξ 2 are of quartic order.Thus in the chiral limit, ξ 1 ̸ = 0 implies that the η meson is massive at T χ , and the universality class is that of G qu = SU L (2) × SU R (2) ≡ O(4).Numerical simulations using Wilson fermions by Brandt et al. [76] find that the mass of the η meson is much smaller near T χ than at T = 0, in accord with our conjecture.If the speculation of Eq. ( 5) holds, then the η meson is massless at T χ , and the universality class is then O(4) × O(2).
For four flavors, the coupling ∼ ξ 1 det Φ is of quartic order, and a relevant quartic coupling, of the same mass dimension as the couplings ∼ λ 1 and ∼ λ 2 .The critical behavior of G qu for N f = 4 is unknown.
If our conjecture is correct, then while there may be no true chiral phase transition, there could well be a sharp crossover from a low temperature phase, dominated by quantum instantons with large ξ 1 (T ) and ϕ 0 (T ), to a phase dominated by semiclassical instantons, with small ξ 1 (T ) and ϕ 0 (T ).As T → ∞, ξ 1 (T ) and ϕ 0 (T ) → 0.
If the speculation of Eq. ( 5) is true, only λ(T χ ) ̸ = 0, with m 2 (T χ ) and all ξ (j,k) i (T χ ) = 0.There is then a chiral phase transition of second order for an emergent U A (1) symmetry at T χ .This would be most dramatic.

VI. IMPLICATIONS FOR QCD
We have worked exclusively in the chiral limit.What are the implications for QCD, where numerical simulations on the lattice find no true phase transition, but crossover [77][78][79]?
If QCD is close to the chiral limit for three massless flavors, then the restoration of the axial U A (1) symmetry at T χ surely implies that the approximate restoration of the axial U A (1) symmetry is closely tied to the crossover temperature.
Our analysis also applies to nonzero quark chemical potential, µ.For a theory at T ̸ = 0, the effective theory is three dimensional.If T ≪ µ, though, the relevant effective theory is then in four dimensions.Assuming that confinement gaps the quarks and gluons, the effective theory is again that of Eqs. ( 1) and (2).While the mass dimensions of the coupling constants change, the conclusion remains that if ξ 1 (T χ ) ̸ = 0, the chiral phase transition is of first order in the chiral limit.
Our analysis predicts that the breaking of the anomalous U A (1) symmetry is uniformly small in a chirally symmetric regime.The η ′ meson, which is heavy is vacuum, must become light.
There is an interesting possibility which arises.Like the U A (1) invariant coupling constants, the anomalous coupling constants are all functions of both temperature and chemical potential, ξ (j,k) i (T, µ).Analogous to the critical endpoint, where for two light flavors the O(4) invariant quartic coupling constant vanishes, λ(T cr , µ cr ) = 0 [97][98][99][100], since we have two thermodynamic parameters to vary, it is possible that there is a single point in the phase diagram where ξ 1 (T A , µ A ) = 0.About this point, instead of SU V (3) flavor eigenstates, the π 0 , η, and η ′ are eigenstates of flavor, and there is a large violation of isospin [19].It is very intriguing that such a violation has been reported by the NA61/SHINE collaboration recently [101,102].
If ξ 1 (T, µ) vanishes at a point in the plane of T and µ, then perhaps there is a region where ξ 1 (T, µ) if of opposite sign to that in the vacuum.If chiral symmetry is broken, then instead of the σ meson condensing, the η ′ does.This implies that CP symmetry is spontaneously broken by an η ′ condensate.
Other signals which have been suggested include: Hanbury-Brown-Twiss correlations [103][104][105], possibly confirmed by the PHENIX experiment [106], and an excess of soft dileptons [107].The HADES experiment finds that the η meson is about twice as abundant as expected from a statistical distribution [108,109].Certainly when the η ′ meson becomes light, so does the η meson [110].
Besides the other implications of our results, it is also natural to wonder how the suppression of topologically nontrivial fluctuations in a chirally symmetric phase affects baryogenesis in the early universe [111].