Field dependence of the Yukawa coupling in the three flavor quark--meson model

We investigate the renormalization group flow of the field dependent Yukawa coupling in the framework of the three flavor quark--meson model. In a conventional perturbative calculation, given that the field rescaling is trivial, the Yukawa coupling does not get renormalized at the one loop level if it is coupled to an equal number of scalar and pseudoscalar fields. Its field dependent version, however, does flow with respect to the scale. Using the functional renormalization group technique, we show that it is highly nontrivial how to extract the actual flow of the Yukawa coupling as there are several new chirally invariant operators that get generated by quantum fluctuations in the effective action, which need to be distinguished from that of the Yukawa interaction.


I. INTRODUCTION
One of the merits of the functional renormalization group (FRG) technique is that it allows for calculating the flows of coupling constants non-perturbatively. Via the FRG one has the freedom to evaluate the scale dependence of them in nonzero background field configurations, which is thought to be essential once spontaneous symmetry breaking occurs [1][2][3]. For the sake of an example, in scalar (φ) theories quantum corrections to the wave function renormalization (Z) typically vanish at the one-loop level, but using the FRG one gets a nonzero contribution once one generalizes the corresponding kinetic term in the effective action as ∼ Z(φ)∂ µ φ∂ µ φ, and evaluates Z at a symmetry breaking stationary point of the effective action. This procedure is essential e.g. in two dimensional systems that undergo topological phase transitions, as the wave function renormalization is known to be diverging in the low temperature phase, which cannot be described in terms of perturbation theory [4]. Similarly, in four dimensions, it has recently been shown that in the three flavor linear sigma model the coefficient of 't Hooft's determinant term also receives substantial contributions when evaluated at nonzero field [5].
A similar treatment should be in order for Yukawa interactions, whose renormalization group flow vanishes at the one loop level, if complex scalar fields are coupled to the fermions and the field strength renormalization is trivial [6]. In phenomenological investigations of the two and three flavor quark-meson models several papers have dealt with the non-perturbative renormalization of the Yukawa coupling. The essence of the corresponding calculations is that one determines the RG flow of the fermion-fermion-meson proper vertex, defined as δ 3 Γ/δ Lψ δ R ψδφ in a given background (Γ being the effective action), and then associates it with the flow of the Yukawa coupling itself. This obviously works for one flavor models [7,8], and even for the two flavor case [9,10], in particular for models restricted to the σ − π subsector, i.e. with O(4) symmetry in the meson part of the theory. * gfejos@caesar.elte.hu † patkos@galaxy.elte.hu The reason why the approach has legitimacy in the latter case is that it turns out that the only way fluctuations generate new Yukawa-like operators in the quantum effective action is that the original Yukawa term is multiplied by powers of the quadratic O(4) invariant of the meson fields. That is to say, one is allowed to choose e.g. a vacuum expectation value for which all pseudoscalar (π) fields vanish, but the scalar (s) one is nonzero, and associate ∼ s 2 with the aforementioned quadratic invariant leading to the appropriate determination of the field dependence of the Yukawa coupling.
The situation is not that simple in case of three flavors, i.e. chiral symmetry of U L (3) × U R (3). Since the effective action automatically respects linearly realized symmetries of the Lagrangian, field dependence of any coupling involves possible functional dependence on all chirally invariant operators. Since there are more invariants compared to the two flavor case, as a result, if one chooses a specific background for the evaluation of the RG flow of the fermion-fermion-meson vertex, one can easily lose sight of chiral symmetry and can end up with results that do not respect it. That is to say that other operators than that of the Yukawa term (multiplied by powers of the quadratic invariant) can get generated, which are presumably not even considered neither in the Lagrangian nor in the ansatz of the effective action in question. These, however, cannot be distinguished from the former if the background is too simple. For the sake of an example, the seminal paper of Jungnickel and Wetterich [11] derives the RG equation of the Yukawa coupling for general number of flavors in a diagonal symmetry breaking background, but neither this nor later papers touch upon the possible generation of new fermion-fermion-meson couplings that are independent of the original Yukawa term. As it will be shown, a naive calculation squeezes inappropriate terms into the Yukawa flow, which should have been separated via the choice of a suitable background and projected out.
Investigations of the three flavor case including the effects of the U A (1) anomaly were started by Mitter and Schaefer. In a first paper no RG evolution of the Yukawa coupling was taken into account [12]. This approximation was carried over in several directions, including investigations regarding magnetic [13] and topological [14] 2 susceptibilities, mass dependence of the chiral transition [15], or quark stars [16].
A strategy for extracting the RG flow of the field dependent Yukawa coupling was presented by Pawlowski and Rennecke, explicitly realized for the two flavor case [10]. In this paper interaction vertices arising from the expansion of the field dependent Yukawa coupling were also analyzed and it was found that higher order mesonic interactions to two quarks are gradually less important. These results were further used in recent ambitious computations of the meson spectra from effective models directly matched to QCD [17,18]. The method of [10] was then applied for extracting the field dependence of the Yukawa coupling in the 2+1 flavor (realistic) quarkmeson model [19], without further discussions on the possibility of the appearance of new operators allowed by chiral symmetry in the infrared. In our understanding the authors did not study the unavoidable emergence of non-Yukawa like fermion-fermion-meson couplings, and they applied the same prescription as of Ref. [10] to the full quantum correction of the light and strange quark 2-point functions (see Eq. (29) of [19]).
The aim of this paper is to show how to separate the field dependent Yukawa interaction from those new fermion-fermion-multimeson couplings that unavoidably arise as three flavor chiral symmetry allows for a much larger set of operators in the infrared than considering only a field dependent Yukawa coupling. For this the right hand side (rhs) of the evolution equation of the effective action will be evaluated in such a background that allows for expressing it in terms of explicitly invariant operators, as it has been demonstrated for purely mesonic theories some time ago [20][21][22][23].
The paper is organized as follows. In Sec. II, we present the model emphasizing its symmetry properties.
We explicitly show what kind of new operators can be generated, and set an ansatz for the scale dependent effective action accordingly. Sec. III contains the explicit calculation of the flows generated by the field dependent Yukawa coupling, and Sec. IV is devoted to an extended scenario where at the lowest order the newly established operators are also taken into account. The reader finds the summary in Sec. V.

II. MODEL AND SYMMETRY PROPERTIES
We are working with the U A (1) anomaly free three flavor quark-meson model, which is defined through the following Euclidean Lagrangian: where M stands for the meson fields, M = (s a + iπ a )T a (T a = λ a /2 are generators of U (3) with λ a being the Gell-Mann matrices, a = 0...8), q T = (u d s) are the quarks, and M 5 = (s a + iπ a γ 5 )T a . As usual, m 2 is the mass parameter andλ 1 ,λ 2 refer to independent quartic couplings. The fermion part contains / ∂ ≡ ∂ i γ i , γ 5 = iγ 0 γ 1 γ 2 γ 3 with {γ i } being the Dirac matrices and the Yukawa coupling is denoted by g.
Concerning the meson fields, for vector and axialvector transformations, respectively, . This also implies that where A 5 = exp(iθ a A T a γ 5 ). As for the quarks, the transformation properties are Note that, since {γ i , γ 5 } = 0, These transformation properties guarantee that all terms in (1) are invariant under vector and axialvector transformations.
The quantum effective action, Γ, built upon the theory defined in (1) also has to respect chiral symmetry. That is to say, only chirally invariant combinations of the fields can emerge in Γ. For three flavors, there exist three independent invariants made out of the M fields, where the last one is absent in (1) due to perturbative UV renormalizability but nothing prevents its generation in the infrared. In principle, the chiral effective potential, V , defined via homogeneous field configurations, Γ| hom = x V has to be of the form V = V (ρ, τ, ρ 3 ). Obviously, the generalized Yukawa term, can (and does) also appear in Γ, but in this paper we restrict ourselves to This is motivated from a symmetry breaking point of view, that is, without explicit symmetry breaking terms in (1), chiral symmetry breaks as [28], and for any background field respecting vector symmetries, τ ≡ 0 ≡ ρ 3 . We can think of g(ρ) as which shows that it actually resums operators of the form Note that, however, as announced in the introduction, there are several other invariant combinations at a given order that are allowed by chiral symmetry and contribute to quark-meson interactions. Take the lowest nontrivial order, that is dimension 6. We have two invariant combinations: Obviously the first one is to be incorporated into the field dependent Yukawa coupling, but not the second one! Note that one may think of the fifth Dirac matrix as the difference between left and right handed projectors, γ 5 = P R − P L , and therefore, e.g. the first expression is equivalent toq L M q R +q R M † q L , and the second tō Thus, one can trade the M 5 dependence to M dependence in the invariants via working with left and right handed quarks separately. Going to next-to-leading order we see the emergence of the following new terms [note that for general configura- and we might even have ∼ τ ·qM 5 q, though we intend to drop such contributions, see approximation (8). If we are to extract the renormalization group flow of g(ρ), then we need a strategy to distinguish each operator at a given order, in order to be able to resum only the Yukawa interactions, and not the aforementioned new quark-quarkmultimeson operators. Obviously this procedure cannot be done through a one component background field, e.g. the vector-like condensate M ∼ s 0 · 1, as it mixes up the corresponding operators, and one loses the chance to recombine the field dependence into actual invariant tensors.
The flow of the effective action is described by the Wetterich equation [24,25], where Γ (2) k is the second functional derivative matrix of Γ k , and R k is the regulator function. One typically chooses it to be diagonal in momentum space, and set R k as R k (p, q) = (2π) 4 R k (q)δ(q + p). Evaluating (12) in Fourier space leads to (13) In this paper the equation will be evaluated only in spacetime independent background fields, therefore one assumes that Γ In such backgrounds it is sufficient to work with the effective potential, V k , via Γ k | hom = x V k , and (13) leads to where we have assumed that the regulator matrix R k is meant to replace everywhere p with p R , where p R is the regulated momentum through the R k function, i.e.
The ansatz we choose for Γ k is called the local potential approximation, which consists of the usual kinetic terms and a local potential (in this case, specifically, a mesonic potential plus the Yukawa term): Using the earlier notation, V k = V ch ,k +g k (ρ)qM 5 q. Note that the chiral potential will not be considered in its full generality, but rather as V ch ,k = U k (ρ) + C k (ρ)τ . Flows of U k and C k can be found in [23]. Note that this construction could be easily extended with 't Hooft's determinant term describing the U A (1) anomaly, ∼ (det M + det M † ), and the corresponding coefficient function. Investigations in this direction are beyond the scope of this paper.

III. FLOW OF THE FIELD DEPENDENT YUKAWA COUPLING
In this section we evaluate (13) in homogeneous background fields to obtain the flow of g k (ρ), defined in (15). Calculating Γ (2) k using the ansatz (15) one gets the following hypermatrix in the {s a , π a ,q T , q} multicomponent space: where the mass matrices are defined through the following relations: For practical purposes it is worth to separate Γ (2) k into three parts [26,27], Γ k, mix , where the respective terms are defined as Using these notations, and by introducing the differential operator∂ k , which, by definition acts only on the regulator function, R k , (14) can be conveniently reformulated. We may use the matrix identity and arrive at where G k,B = (Γ k,F ) −1 and the negative sign of the pure fermionic term is understood. We are interested in the flow of the Yukawa coupling, thus we need to identify in the rhs of (20) the operator ∼qM 5 q. Diagrammatically, the commonly known term that contributes is seen in Fig. 1, which is generated from the leading term of the last contribution of the rhs of (20): More specifically, it reads where S k (p) = (i / p + g k M 5 ) −1 is the fermion propagator without doubling, while G k,s , G k,π , and G k,sπ are the respective 9 × 9 submatrices of G k,B in a purely bosonic background (see Appendix B). In principle, the evaluation of (22) goes as follows. First, one evaluates Γ k,B , then inverts it to obtain G k,s , G k,π and G k,sπ . Second, one inverts the fermion propagator, and finally performs all matrix multiplications to get the operator that is sandwiched byq and q. Note that the procedure turns out to be too complicated to be carried out in a general background, but as we explain in the next subsections, fortunately we will not need to evaluate (22) in its full generality.
On top of the above, via a field dependent Yukawa coupling there is also contribution from the first term of (20), as meson masses get modified by a fermionic background (see Appendix B) leading to the possiblity of generating a ∼qM 5 q term in the effective action, see Fig. 2. Generically this contributes to the rhs of (20) as A. Flows in the symmetric phase As a first step, we are interested in the RG flows in the symmetric phase, i.e., where they are obtained at zero field. Note that, however, this still necessitates the evaluation of the rhs of the flow equation (22) in a nonzero background, but if we are interested in the flowing couplings of dimension 6 operators, it is sufficient to perform all computations at the cubic order in the meson fields. That is to say, when all propagators are expanded around M = 0 (see Appendix B for the general formulas), one is allowed to work with them at the following accuracy: where the coefficient functions (i.e. U k , U k , C k ) are evaluated at zero field. Note that in this subsection all computations are performed without specifying any background field and keeping its most general form. Terms that contain higher derivatives of C k are left out since their coefficients would lead to subleading (higher than cubic power) contributions in the background fields in (22) and (23).
Note that the scalar and pseudoscalar meson propagators do not mix in the symmetric phase and are degenerate, therefore, there is a relative (−1) factor between their contributions in (22), which exactly cancels the term linear in the mesonic fields. In the piece of the integrand that is quadratic in them the quark momentum is odd, therefore, upon integration it also vanishes. The leading contribution of (22) is determined by the mass terms: Exploiting various identities of the U (3) algebra (see Appendix A), one can perform the algebraic evaluation of the integrands to arrive at: Eq. (26) clearly shows the appearance of a new dimension 6 operator (∼qM 5 M † 5 M 5 q), which was absent in the ansatz of the effective action. Before discussing this result, let us turn to the contribution of (23). Using, again, expressions (24), a straightforward calculation leads to where the coefficient functions (i.e. U k , g k , g k , g k , C k ) are, once again, evaluated at zero field. The flow of the effective potential is the sum of (26) and (27). At this point it is clear that the flow equation does not close in the sense that the ansatz (15) does not contain the operator ∼qM 5 M † 5 M 5 q, therefore, for consistency reasons, it has to be dropped also in the rhs of (20). Instead of doing so, one may choose to project out the ∼q(M 5 M † 5 − ρ/3)M 5 q term, because in the symmetry breaking pattern U L (3) × U R (3) → U V (3) that is the actual combination, which vanishes. This choice should lead to the appropriate definition of the field dependent flowing Yukawa coupling if no explicit symmetry breaking terms are present. Note, however, that, once finite quark masses and the corresponding explicit breaking terms are introduced, in the minimum point of the effective potential M 5 M † 5 = ρ/3, and the aforementioned projection can be regarded as somewhat arbitrary. A more appropriate treatment is to include ∼qM 5 M † 5 M 5 q in the ansatz of the effective action in the first place (see Sec. IV).
We close this subsection by listing the coupled flow equations for the Yukawa coupling and its derivative evaluated at zero field, i.e. in the symmetric phase, when ∼q(M 5 M † 5 − ρ/3)M 5 q is projected out. Dropping g k in order to close the system of equations, and by using the definitions of (9), (26) and (27) we are led to

B. Flows in the broken phase
Now we explore how the flowing couplings depend on the fields, more specifically, as described in (8), their ρ dependence will be determined.
At this point, we specify a background in which we will evaluate the flow equation, as in case of general {M , q, q} configurations, there is no hope that the broken symmetry phase calculations can be done explicitly. The nice thing, however, is that it is sufficient to work in a restricted background. We, again, definitely need to assume nonzeroq, q, and then we specify the M = s 0 T 0 + s 8 T 8 "physical" condensate. This is one of the minimal choices, which allows for a unique restoration of the ∼ qM 5 M † 5 M 5 q and ∼qM 5 q operators in the rhs of the flow equation (a one component condensate would definitely not allow to do so). We have checked explicitly with other backgrounds the uniqueness of the results, and as expected, found agreement.
As outlined above, the first step is to evaluate the 2point functions Γ (2) k,s , Γ (2) k,π (note that in the current background there is no s − π mixing), see details in Appendix B, in particular (B10) and (B11). Then one inverts these matrices to obtain G k,s and G k,π . They are diagonal except for the 0-8 sectors. Similarly, for the fermion propagator, one has S −1 k (p) = (i / p + g k (ρ)M 5 ), which leads to the inverse Since in the background in question [S k , M 5 ] = 0, Eq.
Similarly, exploiting the choice of the simplified background, (23) becomes It is important to mention that one does not need to work with general s 0 , s 8 background values, but can safely assume that s 8 s 0 , and thus expand both (30)  g k (ρ) 3 p 2 R + U k (ρ) As in the previous subsection, we may drop the last term in (32) due to consistency and thus the first two terms provide the result for the fully non-perturbative flow of the Yukawa coupling. One can check explicitly that expanding (32) around ρ = 0 leads back to the earlier results of (27). The expression obtained for the flow of the Yukawa coupling g k (ρ) is thought to be a resummation of operators as described below (9). As such, one is able to determine how strong the interaction is between quarks and mesons as e.g. the chiral condensate evaporates at high temperature. In principle g k (ρ) also depends explicitly on the temperature but one expects that the decrease in ρ is more important [5]. For phenomenology, e.g. parametrization, or mass spectrum calculation, one needs to evaluate g k (ρ) and its derivatives at the minimum point of the effective potential, ρ = ρ min .
Motivated by the nonuniqueness of the definition of the field dependent Yukawa coupling in the earlier setting, now we investigate the case when the term x g 53,kq M 5 M † 5 M 5 q is added to the ansatz (15) of the effective action (without substracting ρqM 5 q/3 from it). Here g 53,k is a new running coupling constant, and we will not be considering its field dependence. Also, we restrict ourselves to calculations in the symmetric phase, therefore, a similar treatment as of in Sec. IIIA is in order. All calculations presented there are still valid, but there is one more term contributing to rhs of the flow equation of the effective action, coming from the tadpole diagram of Fig. 2 [see the mass matrices in (B9)]: Substituting the propagators from Eqs. (24), without specifying an actual background field, straightforward calculations lead to Note that all couplings (i.e. U k , U k , C k ) are evaluated at zero field as we have worked in the symmetric phase. One observes that even though the original (field independent) Yukawa coupling does not flow in the symmetric phase, once one includes the g 53 coupling, it does run with respect to the scale. Using (28a), (28b) and (34), we arrive at the following system of equations for g 0,k , g 1,k and g 53,k : We wish to note that based on Sec. IIIB, a broken phase calculation can also be done. This, however, leads to such complicated formulas that we do not go into the details.

V. SUMMARY
In this paper we raised the question of determining the field dependence of the Yukawa coupling in the three flavor quark-meson model. Our main motivation was to clarify the problem of consistency between chiral symmetry and the Yukawa term. As opposed to two flavors, in case of three flavors, the existence of non Yukawa-like interactions and their generation in the infrared scales of the quantum effective action prevents a straightforward generalization of the two flavor approach [10]. That is to say, one cannot consistently work with e.g. a scalar chiral condensate and associate the complete field dependence of the quark-quark-meson vertex with that of the Yukawa coupling itself. One needs great care to project out those operators that are not included in the ansatz of the effective action, which, by construction, need to respect chiral symmetry. We believe that such a consistent approach was missing in the literature so far.
We have explicitly calculated the renormalization group flow of the field dependent Yukawa coupling separately in the symmetric and broken phases. We have showed that at the order of dimension 6 operators new type of terms arise, and argued how to project them out required by consistency. Motivated by the very definition of the flow of the field dependent Yukawa coupling, for the sake of a complete treatment of dimension 6 operators, we have determined the symmetric phase flows when these new (non-renormalizable) interactions are also embedded into the ansatz of the quantum effective action (Γ k ).
It would be very important to see what effects the current approach has from a phenomenological point of view. One is typically interested in mapping the details of the chiral phase transition at finite temperature and/or density, such as the transition point, mass spectrum, interaction strengths, etc. To do so one also needs to include t' Hooft's determinant term into the system describing the U A (1) breaking, and it would be of particular interest to check the interplay between the anomaly coupling and that of the Yukawa interaction. These directions represent future studies to be reported elsewhere.