Perturbative RG analysis of the condensate dependence of the axial anomaly in the three flavor linear sigma model

Coupling of `t Hooft's determinant term is investigated in the framework of the three flavor linear sigma model as a function of the chiral condensate. Via perturbation theory, we calculate the renormalization group flow of the coupling of the first correction to the conventional $U_A(1)$ breaking determinant term and find that at low temperatures it drives the anomaly to increase when the chiral condensate decreases. As an application, we analyze the effect at the zero temperature nuclear liquid--gas transition.


Introduction
The U A (1) subgroup of approximate U L (3) × U R (3) chiral symmetry is anomalously broken in quantum chromodynamics (QCD). In effective theories, such as the Nambu-Jona-Lasinio or linear sigma models this is taken into account by 't Hooft's determinant term. Coefficients of all operators in the Lagrangians of the models, including the determinant term, are considered to be (coupling) constants, without any field or environment dependence. In the quantum version of the action, however, fluctuations introduce temperature (T ), baryochemical potential (µ B ) and also field dependence as they become coefficient functions. When talking about field dependence of a given coupling one has in mind the resummation of higher dimensional operators that can reappear when Taylor expanding the coefficient functions in terms of the field variable(s) around a conveniently chosen expansion point.
In QCD, it is well established that the anomalous breaking of U A (1) symmetry should gradually disappear beyond the critical temperature, as at high T the instanton density causing the anomaly exponentially vanishes [1,2]. At lower temperatures, however, the situation is far from being understood in a satisfactory fashion. One has also great interest in gaining results at finite µ B as lattice simulations of QCD suffer from the sign problem in those directions of the phase diagram.
The finite temperature and/or density behavior of the U A (1) anomaly represents an active direction of research. More conservative results usually argue that the evaporation of the anomaly should follow that of the chiral condensate and thus the U A (1) symmetry restores around the critical temperature (T C ) of the chiral transition [3,4,5,6,7]. There are also several arguments and results that indicate that it might be reasonable to assume that it is visible even beyond T C [8,9,10,11,12]. Effective restoration of the anomaly has, e.g., a consequence regarding the order of the chiral transition [13], the axion mass [14], and the fate of η ′ meson, whose mass if substantially drops [15,16,17,18,19,20], in a nuclear medium could lead to the formation of an η ′ -nucleon bound state.
The goal of this paper is to calculate the first order correction to the determinant term in the effective action perturbatively, i.e., we determine the first Taylor coefficient of the anomaly function, which allows for obtaining the behavior of the anomaly strength as the vacuum expectation value of the field varies. Fluctuations will be included using the functional variant of the renormalization group (FRG) [21], in the so-called local potential approximation. Even though renormalizable, we think of the model as an effective field theory, therefore, an ultra violet (UV) cutoff is inherently part of the system, which we set to Λ = 1 GeV . Basically our task is to integrate out all fluctuations below this scale.
The paper is organized as follows. In Section 2, we introduce the model and the corresponding method of the FRG. Section 3 is devoted for calculating the effective action and discussing the problem of the expansion point of the Taylor series. After appropriate parametrization of the model, in Section 4, as an application, we show how the anomaly strengthens at the zero temperature nuclear liquid-gas phase transition. Section 5 contains the summary.

Model and method
The model we are working with in this paper is the three flavor linear sigma model, which is defined via the following Euclidean Lagrangian: where M contains the meson fields, M = (s a + iπ a )T a [T a = λ a /2 are generators of the U (3) group with λ a being the Gell-Mann matrices, a = 0, ..., 8], m 2 is the mass parameter and g 1 , g 2 refer to independent quartic couplings. As discussed in the previous section, the determinant term and the corresponding a parameter is responsible for the U A (1) anomaly. We also have explicit symmetry breaking terms containing h 0 and h 8 , which represent finite quark masses. Our main goal is to calculate the quantum effective action, Γ, built upon the theory defined via (1). As announced in the introduction, we think of (1) as an inherently effective model, which is only valid up to the scale Λ = 1 GeV , therefore, one needs to take into account fluctuations with a cutoff Λ. The scale dependent quantum effective action, Γ k , which includes fluctuations whose momenta is larger than k, obeys the following flow equation: where R k is an appropriately chosen regulator function freezing fluctuations with momenta smaller than k, and Γ (2) k is the second functional derivative of Γ k in a constant background field. There are various choices for R k , in this paper we will stick to Our aim is to calculate scale dependent effective action, Γ k , in an approximation that takes into account the evolution of the anomaly at the next-to-leading order, i.e., we wish to determine in Γ k the coefficient of the operator Tr (M † M ) · (det M + det M † ). Therefore, our ansatz for Γ k is as follows: Note that (3) is compatible with (1), but all couplings come with k-dependence (except for h 0 and h 8 ), and notice the new term proportional to a 1,k . In terms of operator dimensionality, the latter one is the leading order correction for the U A (1) anomaly. Our task is to calculate Γ (2) k from (3), plug it into (2), and then identify the individual differential equations for m 2 k , g 1,k , g 2,k , a k , and a 1,k . Then, these equations need to be integrated from k = 1 GeV to k = 0 to obtain Γ ≡ Γ k=0 . In the ansatz (3), obviously the actual strength of the anomaly is not described by the parameter a, but rather a + a 1 · Tr (M † M )| min , where we need to evaluate the chiral condensates in the minimum point of the effective action. Therefore, what we are basically after is the sign of a 1 at k = 0 to decide whether the anomaly strengthens or weakens as the chiral condensate evaporates.

Calculation of the effective action
The first step is to calculate Γ (2) k . In principle it is a 18 × 18 matrix in the s a − π a space, and there is not much hope that one can invert such a complicated expression analytically. Luckily, it is not necessary at all, as in (3) we kept the field dependence up to the order of O(M 5 ). By working with a restricted background, Γ (2) k is easily invertable and by expanding the rhs of (2) in terms of the field variables it still allows for identifying each operator that are being kept in (3).
A convenient choice is to work with M = s 0 T 0 + s 8 T 8 . In such a background, the operators that need to be identified are as follows: The Γ (2) k matrix elements in the scalar sector read while the pseudoscalar components are Using that ∂ k R k (q) = 2kΘ(k 2 − q 2 ) and Γ (2) k (q) + R k (q) = Γ (2) k (k) for q < k, from (2) we get Plugging in the matrix elements calculated in the M = s 0 T 0 + s 8 T 8 background, we can expand the rhs of (15) in terms of s 0 and s 8 . After this step, using (4) we identify the ρ, τ and ∆ operators as where Note that we treated the anomaly as perturbation and dropped every term beyond O(a k , a 1,k ). Our task now is to solve these equations starting from k = Λ ≡ 1 GeV down to k = 0. One quickly realizes that via (17) this is not possible. The reason is that if we are to work with phenomenologically reasonable parameters, m 2 has to be negative. That is to say, there exists an intermediate scale k, for which all denominators blow up and the flow equations lose their meaning. The way out is to realize is that one actually has the choice to determine the flow equations in the minimum point of the effective action, Γ k | v0,min,v8,min , i.e., one should think of the m 2 , g 1 , g 2 , a and a 1 quantities as field dependent parameters and extract their renormalization group flow at v 0,min , v 8,min . The field dependence of the parameters has to reflect chiral symmetry, i.e., they can only appear in chirally invariant combinations. We will assume that this dependence is realized through the chiral invariant ρ = Tr (M † M ) [10]. One, therefore, repeats the calculations starting from (15), but this time expands only in terms of s 8 so that the ρ dependence of the parameters can be traced via s 0 . A long but straightforward calculation leads once again to the possibility of identifying the invariants appear in (16), whose coefficients now read as where we have denoted the expansion point by ρ 0 , which is to be set to the value of ρ corresponding to the minimum point of the effective action. Note that by choosing ρ 0 = 0, (18) leads to the earlier results, (17). Our task is to integrate the system of equations (18) from k = Λ ≡ 1 GeV down to k = 0 with the boundary conditions m 2 Λ = m 2 , g 1,Λ = g 1 , g 2,Λ = g 2 , a Λ = a, a 1,Λ = 0, where m 2 , g 1 , g 2 , a are such constants that reproduce (partially) the mesonic spectrum in the infrared. Here we have also chosen to make use of the fact that at the UV scale the coefficient of the operator ρ∆ is set to zero due to perturbative renormalizability. This might be questionable once the UV scale is not high enough, but we do not discuss here the possibility of having a nonzero initial value for a 1 .
Before solving the coupled system of equations (18), we need to fix the explicit symmetry breaking terms, i.e., the values for h 0 , h 8 . Instead of h 0 and h 8 , we will work in the nonstrange-strange basis, i.e.,

Anomaly strengthening at the nuclear liquid-gas transition
In this section we apply our results to the zero temperature nuclear liquid-gas transition. We assume that the nucleon field couples to the mesons via Yukawa interaction, L int = gψM 5 ψ, ψ T = (p, n), M 5 = a= ns ,1,2,3 (s a + iπ a γ 5 )T a , where the nonstrange generator is T ns = 2/3T 0 + 1/ √ 3T 8 , while γ 5 is the fifth Dirac matrix. In principle one would also need to include the dynamics of an ω vector particle into the system [22,23] that models the repulsive interaction between nucleons, but as we will see in a moment, for our purposes it plays no role.
First, we exploit some of the zero temperature properties of nuclear matter. Note that, in the current model, the nucleon mass entirely originates from the spontaneous breaking of chiral symmetry, and since m N (f π ) ≈ 939 MeV in the vacuum, we arrive at g Y ≈ 20.19. Normal nuclear density, n N ≈ 0.17 fm −3 ≈ (109.131 MeV ) 3 leads to the Fermi momentum, p F , of the nucleons, since in the mean field approximation, at T = 0 we have therefore, p F ≈ 267.9 MeV ≈ 1.36 fm −1 . This yields the nonstrange condensate in the liquid phase, s ns, liq , because the Landau mass, which is defined as is known to be M L ≈ 0.8m N (f π ) ≈ 751.2 MeV , and thus s ns,liq ≈ 69.52 MeV [22,23]. This shows that as we increase the chemical potential, the nonstrange chiral condensate, s ns , jumps: f π → s ns, liq . This will definitely be accompanied by a jump in the strange condensate, but it has been shown to be significantly smaller [24]. Neglecting the change in s s , the ρ chiral invariant jumps as (f 2 π + s 2 s,min )/2 → (s 2 ns, liq + s 2 s ,min )/2. As discussed in the previous section, the anomaly strength is A = a k=0 + a 1,k=0 · ρ, which also jumps accordingly, and the change in A becomes where ∆ρ = (s 2 ns , liq − f 2 π )/2. Solving (18) one gets a k=0 ≈ −0.472 GeV and a 1,k=0 ≈ 4.9 · 10 −3 GeV −1 , therefore, the relative change in the anomaly at the liquid-gas transition is which is rather small. At this point we once again wish to emphasize that we have neglected the change in the strange condensate, and also, the present analysis is based on perturbation theory. In principle higher order operators that break the U A (1) subgroup should also be resummed, e.g. terms such as ∼ Tr (M † M ) n (det M + det M † ) could be of huge importance. As non-perturbative RG analyses show, the actual (relative) change can be more than one order of magnitude higher [24]. The lesson we wish to point out here is that the present, rather simple perturbative calculation can also capture the phenomenon of strengthening anomaly as the chiral condensate (partially) evaporates.

Conclusions
In this paper we have investigated how the U A (1) anomaly behaves as a function of the chiral condensate.
We have chosen to work with the three flavor linear sigma model, and calculated the leading correction to the conventional anomaly term caused by quantum fluctuations. We have found that the coefficient of the aforementioned operator, ∼ Tr (M † M ) · (det M + det M † ), causes the actual strength of the anomaly to become larger once the chiral condensate evaporates. For the sake of an example, we have demonstrated that at the zero temperature nuclear liquid-gas transition, where on top of a jump in the nuclear density, the chiral condensate also partially restores, the actual strength of the anomaly increases. This could also happen toward the full restoration of chiral symmetry, where quark dynamics also play a significant role. Note that our findings are based solely on calculating mesonic fluctuations, and no instanton effects have been taken into account. Our study calls for an extension via a non-perturbative treatment, where fluctuations are taken into account beyond the O(a) order, and the coefficient function of the determinant term is obtained in a functional fashion, rather than at the lowest order of its Taylor series. Competition between instanton effects and mesonic fluctuations should also be investigated. The former could be implemented via an environment dependent bare anomaly coupling at the UV scale, which, in principle is determined by the underlying theory of QCD. These directions are under progress and will be reported elsewhere.