One-loop constituent quark contributions to the vector and axial-vector meson curvature mass

The renormalized contribution of fermions to the curvature masses of vector and axial-vector mesons is derived with two different methods at leading order in the loop expansion applied to the (2+1)-flavor constituent quark-meson model. The corresponding contribution to the curvature masses of the scalar and pseudoscalar mesons, already known in the literature, is rederived in a transparent way. The temperature dependence of the curvature mass of various (axial-)vector modes obtained by decomposing the curvature mass tensor is investigated along with the (axial-)vector--(pseudo)scalar mixing. All fermionic corrections are expressed as simple integrals that involve at finite temperature only the Fermi-Dirac distribution function modified by the Polyakov-loop degrees of freedom. The renormalization of the (axial-)vector curvature mass allows us to lift a redundancy in the original Lagrangian of the globally symmetric extended linear sigma model, in which terms already generated by the covariant derivative were reincluded with different coupling constants.


I. INTRODUCTION
The extension of the linear sigma model with vector and axial-vector degrees of freedom has a long history (see e.g. [1][2][3]). In recent years, much effort was invested in the study of the phenomenological applicability of various formulations of the model. It turned out, for example, that the gauged version of the model cannot reproduce the correct decay width of the ρ and a 1 mesons [4], and therefore the interest shifted toward versions of the model which are based on the global chiral symmetry: originally constructed for two flavors in [5] the extended linear sigma model (ELσM) was formulated for three flavors in [6].
The parametrization of the three-flavor ELσM in relation with hadron vacuum spectroscopy was thoroughly investigated in [6]. Constituent quarks were incorporated in the ELσM in [7] and their effect on the parametrization, through the correction induced in the curvature masses of the scalar and pseudoscalar mesons, was investigated along with the chiral phase transition at finite temperature and density. It is interesting to know how the model parameters and the results obtained in [7] are influenced by coupling the constituent quarks to the (axial-)vector mesons. The effect of the (axial-)vector mesons on the chiral transition was studied in [8] in the gauged version of the purely mesonic linear sigma model with chiral U (2) L × U (2) R symmetry, by using a rather crude approximation for the Lorentz tensor structure of the (axial-)vector curvature mass matrix, which was as- * kovacs.gyozo@wigner.hu † kovacs.peter@wigner.hu ‡ szepzs@achilles.elte.hu sumed to have the vacuum form even at finite temperature. Further investigations in the above-mentioned directions require the calculation of the mesonic and/or fermionic contribution to the (axial-)vector curvature mass matrix and its proper mode decomposition, as was done in many models dealing with the description of hot and/or dense nuclear matter [9][10][11]. Such a calculation within the linear sigma model would allow for a comparison with in-medium properties of the (axial-)vector mesons obtained with functional renormalization group (FRG) techniques in [12][13][14][15]. The curvature masses of the scalar and pseudoscalar mesons were derived in the U (3) L × U (3) R symmetric constituent quark model in [16]. The method used there involved taking the second derivative with respect to the fluctuating bosonic field of the ideal gas formula for the partition function in which the quark masses depend on these bosonic fields. The result was subsequently used in a plethora of publications, even when it does not apply, as was the case of Ref. [17], which seemingly uses incorrectly the result of [16] to study the effect of the temperature and chemical potential on the vector and axial-vector masses. The result derived in [16] for (pseudo)scalar mesons cannot be directly applied for (axial-)vector mesons, simply because it is not enough to consider only the boson fluctuation-dependent fermion masses: due to their Lorentz index the momentum and (axial-)vector fields couple to form a Lorentz scalar in the fermion determinant resulting from the fermionic functional integral. Due to such terms, derivatives of the fermionic functional determinant with respect to the (axial-)vector fields give additional contributions, compared to the bosonic case.
Although the calculation of the leading order fermionic contribution to the (axial-)vector curvature mass matrix can be done by taking the second field derivative of the arXiv:2105.12689v2 [hep-ph] 17 Sep 2021 functional determinant, it is much easier to take an equivalent approach and compute the self-energy at vanishing momentum with standard Feynman rules. The technical issues that need to be addressed are the mode decomposition and renormalization of the self-energy and the mixing between the (axial-)vector and (pseudo)scalar mesons.
We also mention that while our focus here is on the curvature mass, the pole mass and screening mass can also be obtained from the analytic expression of the selfenergy calculated at nonzero momentum using the usual definitions given in Eq. (6) of [18], where the relation between the pole and curvature masses of the mesons was investigated with FRG techniques within the twoflavor quark-meson model. This difference depends on the approximation used to solve the O(N ) and quarkmeson models and it is typically larger for the sigma than the pion [18][19][20][21].
The organization of the paper is as follows. In Sec. II an approximation scheme is presented for a consistent computation of the effective potential (pressure) in the ELσM which is based on curvature masses that include the fermionic correction at one-loop level. In Sec. III we compute in the one-flavor case, N f = 1, the curvature mass matrix of the mesons, with both methods mentioned above. This allows for the introduction of the relevant integrals used also in Sec. IV, where the self-energy of all the mesons is calculated at vanishing momentum for N f = 2 + 1 flavors. In this case a direct calculation of the curvature masses from the functional determinant, although completely straightforward, is made cumbersome by the large number of fields and the dimension of the matrix involved. This calculation is relegated to Appendix D. Based on the mode decomposition of the (axial-)vector self-energy, presented in detail in Appendix E, the curvature masses of the physical modes are given in terms of simple integrals. We also show in Sec. IV how to connect the expressions of the (pseudo)scalar curvature masses derived here with existing ones obtained with the alternative method of Ref. [16]. In Sec. V we discuss the renormalization of the (axial-)vector curvature masses in the isospin symmetric case. Dimensional regularization was used in order to comply with the property of the vacuum vector self-energy observed for some flavor indices, which is related to current conservation, as discussed in Appendix B. The renormalization process revealed that the Lagrangian of the ELσM can be written more judiciously compared to the form used in the literature, such that each term allowed by the chiral symmetry is included only once, in accordance with the generally accepted procedure. By looking from a new perspective at the wave-function renormalization factor related to the (axial-)vector-(pseudo)scalar mixing, we discuss in Sec. VI how the self-energy corrections modify its tree-level expression. Section VII contains numerical results concerning the temperature evolution of the meson masses obtained in a new vacuum parametriza-tion of the model which takes into account the oneloop fermionic correction in the curvature mass of all the mesons. Section VIII is devoted to conclusions and an outlook. The appendixes not mentioned here contain some further technical aspects used in the calculations.

II. LOCALIZED GAUSSIAN APPROXIMATION IN THE YUKAWA MODEL
In order to motivate our interest in the curvature mass, we present an improved calculational scheme for the effective potential of the ELσM compared to that used in [7]. This scheme, which we call the localized Gaussian approximation, uses the curvature mass of the various mesons. To keep the notation simple, we consider the simplest chirally symmetric Yukawa model, defined by the Lagrangian where ψ and ϕ are fermionic and bosonic fields and U cl (ϕ) = m 2 0 ϕ 2 /2 + λϕ 4 /24 is the classical potential. We use Minkowski metric g µν = diag(1, −1, −1, −1) and the conventions of Ref. [22].
Integrating over the fermions in the partition function 1 leads to the action ( x ≡ d 4 x) where Tr ≡ tr D d 4 x denotes the functional trace, with the subscript "D" referring to the Dirac space, and is the inverse fermion propagator. Shifting the field with an x-independent background φ, ϕ(x) → φ + ϕ(x), the effective potential can be constructed along the lines of Ref. [23]. Several approximations of the effective potential are considered in the literature.
a. Mean-field approximation The bosonic fluctuating field is neglected altogether, leading to where iS −1 f (K) = / K −m f is the tree-level fermion inverse propagator with mass m f = g S φ. Here we introduced the notation K ≡ d 4 K (2π) 4 for the momentum integral with 4-momentum K µ = (k 0 , k). The field equation used in [7] was derived in this approximation.
b. Ideal gas approximation The bosonic fluctuating field is neglected in the fermion determinant (Tr log) appearing in Eq. (3) and kept only to quadratic order in the terms coming from the expansion of U cl (φ + ϕ). The Gaussian functional integral over ϕ leads to where iD −1 (K; φ) = K 2 −m 2 (φ) is the tree-level boson propagator withm 2 (φ) = d 2 U cl (φ)/dφ 2 being the classical curvature mass. This approximation was used in a nonsystematic way in [7] to include mesonic corrections in the pressure. c. Ring resummation or Gaussian approximation The fermion determinant is expanded in powers of ϕ and keeping in Eq. (3) the term quadratic in the fluctuating mesonic field, the Gaussian functional integral over ϕ results in where the boson self-energy represents the one-loop contribution of the fermions. Expanding in Eq. (7) the logarithm one recognizes the integrals of the ring resummation. The ring resummation is widely used in the Nambu-Jona-Lasinio model, where it goes by the name of random-phase approximation [24]. In that context the integral in Eq. (7) requires no renormalization and was evaluated using cutoff regularization in [25,26]. To spare the trouble of renormalizing this integral in a linear sigma model, one can approximate the self-energy with its zero momentum limit. In this localized approximation the dressed bosonic inverse propagator appearing in Eq. (7) is of tree-level type, just that the treelevel mass is replaced by the one-loop curvature masŝ M 2 (φ) ≡m 2 (φ) + Π(K = 0; φ). Since with a homogeneous scalar background the curvature mass does not depend on the momentum, the renormalization of the integral becomes an easy task, as discussed in [27] (see also Eq. (58) in Sec. V).
Note that one can define a curvature mass in each of the above approximations, by taking the second derivative of the potential in Eq. (5), (6), or (7) with respect to the field φ. The curvature mass we investigate in this paper contains the fermionic contribution from the second field derivative of the Tr log in Eq. (3). This represents the purely fermionic one-loop contribution to the curvature mass which can be derived in principle in the localized Gaussian approximation using the background field method.
In order to compute the pressure, we need to evaluate the effective potential at the minimum. In the localized approximation the extremum, is determined as the solution of the field equation We mention that the second term in the square brackets is nothing but the fermionic correction to the three-point vertex function evaluated at vanishing momentum. This vertex function is obtained by expanding the fermionic determinant in powers of the bosonic field The expansion gives a series of one-loop diagrams in which the nth term has n external fields (see e.g. [28] or Ch. 9.5 of [22]). Using the background field method, the expansion of such a fermionic functional determinant was considered recently in [29,30] in order to derive effective couplings between constituent quarks, (axial-)vector mesons, and the photon. The second field derivative of the functional determinant, taken at vanishing mesonic field, is nothing but the one-loop self-energy associated with the bosonic field with respect to which the derivative is taken, as the contribution of diagrams not having exactly two external fields of this type vanishes. Based on this observation one can obtain the lowest order fermionic correction to the bosonic curvature mass by computing the one-loop self-energy with standard Feynman rules.

III. CURVATURE MASS IN THE N f = 1 CASE
We generalize now Eq. (1) and consider the chirally symmetric Lagrangian 2 in which a fermionic field ψ interacts through a Yukawa term to scalar (S), pseudoscalar (P ), vector (V µ ), and axial-vector (A µ ) fields (11) The mesonic part of the Lagrangian is of the form 2 The one-loop curvature mass formulas derived here can be easily modified when, in the absence of chiral symmetry, P and A can have different Yukawa couplings than S and V , respectively.
We shall return to the unspecified interacting part in the N f = 2 + 1 case in relation to the renormalization of the one-loop curvature masses.
By integrating over the fermions in the partition function, done after the usual shifts S(x) → φ + S(x) (φ is independent of x) introduced in order to deal with the spontaneous symmetry breaking (SSB), one obtains a correction to the classical mesonic action in the form of a functional determinant. The expansion of the functional determinant in powers of mesonic field derivatives, the so-called derivative expansion [31,32], leads to an effective bosonic action of the form with the leading order term of the expansion being the one-loop fermionic effective potential which depends on all the fluctuating mesonic fields collectively denoted by ξ(x). We introduced for the inverse fermion propagator, in which m f = g S φ is the tree-level (classical) fermion mass. Hereafter the x dependence of the mesonic fields will not be indicated. The second derivative of U f (φ, ξ) with respect to the mesonic fields gives an additive correction to the classical mesonic curvature mass obtained from L m . Since later we will investigate the N f = 2 + 1 case, where the fields have flavor indices a = 0, . . . 8, we give the more general formulas of these corrections In this section the flavor indices should be disregarded. The sign difference between the above definitions is due to the different signs of the corresponding classical mass terms in Eq. (12). Accordingly, for the (pseudo)scalar field one has an additive correction to the classical mass squaredm , while for the (axial-)vector field the second derivative is a Lorentz tensor, and therefore the correction tô m This is needed in a parametrization of the model that is based on the one-loop curvature masses. For the curvature mass at T = 0 one needs the mode decomposition of the tensor ∆m Writing the determinant in this form facilitates the derivation of the curvature masses, as the contribution to the scalar and the pseudoscalar comes only from the first term, while only the second and the third terms contribute in the case of the vector and the axial-vector, respectively. Also, note that D( The second derivative of the determinant with respect to a particular field denoted by ϕ is calculated using For ϕ ∈ {S, P, V µ } this is applied writing D =D 2 + R, whereD 2 is either the first or the second term on the right-hand side (rhs) of Eq. (18), while the remnant R does not contribute in Eq. (19). Introducing the notation G f (K) = 1/(K 2 − m 2 f ) one obtains for the scalar and the pseudoscalar fields, and for the vector field. For the axial-vector one applies Eq. (19) with D = D + R, whereD is the third term on the rhs of Eq. (18), to obtain For scalar and vector fields there are contributions from both the first and the second derivative ofD, while in the case of the pseudoscalar and axial-vector fields only the second derivative ofD contributes. Using Eqs. (14), (16), (18), and (20) the fermion corrections to the curvature masses of the scalar and pseudoscalar fields are obtained as where the (vacuum) tadpole integral is In the case of the vector and the axial-vector fields, one evaluates the trace in Eq. (17), using Eqs. (16) and (21) together with g µ µ = 4 and

Integrals at finite temperature
The expressions in Eqs. (22) and (24a), which were formally derived at vanishing temperature (T = 0), can be easily generalized to T = 0, where the tadpole integral consists of vacuum and matter parts: The superscripts indicate the absence or the presence of statistical factors in the respective integrands. In a covariant calculation the vacuum part T (0) (m f ) is the integral in Eq. (23), while in a noncovariant calculation it is as obtained with the usual conventions of the imaginary time formalism [33], namely (µ is the chemical potential) after doing the summation over the fermionic Matsubara frequencies ν n = (2n + 1)πT . The matter part is where f ± f (k) = 1/(exp((E f (k) ∓ µ)/T ) + 1) are the Fermi-Dirac distribution functions for particles and antiparticles and E 2 f (k) = k 2 + m 2 f with k = |k|.
For the mass derivative of the matter part of the tadpole integral (Euclidean bubble integral at vanishing momentum) one uses

and an integration by parts to obtain
.
(29) The fact that even at finite temperature the trace of the second derivative appearing in Eq. (21a) is the only relevant quantity determining the curvature mass of the vector boson is due to current conservation. For the axial-vector this is not the case and one needs the mode decomposition of the tensor in Eq. (21b). This is discussed in Appendix E.
B. Curvature mass from the self-energy As mentioned at the end of Sec. II, the one-loop curvature mass can also be obtained by computing the corresponding zero momentum self-energy. For example, for the self-energy of the vector field, the Feynman rules applied with the conventions of [22] give In order to obtain the curvature mass of the physical modes at finite temperature, we need the standard decomposition of the momentum-dependent self-energy tensor reviewed in Appendix E. The self-energy is decomposed into vacuum and matter parts. The former is evaluated using a covariant calculation performed at T = 0 in a regularization scheme compatible with the consequence of current conservation, namely that the self-energy is 4- µν,vac (Q = 0) = 0. Therefore, only the matter part contributes to the selfenergy components determining the curvature masses of the 3-longitudinal and 3-transverse vector modeŝ The components are obtained as (see Ch. 1.8 of [34]) For the axial-vector, which does not couple to a conserved current, the tensor structure of the self-energy is more complicated and it is discussed in Appendix E.
The interested reader can find in Appendix A a discussion on the matter part of the self-energy components. For the vacuum part see the discussion in Sec. V and the calculation presented in Appendix C.
The fermionic part of the chiral-invariant Lagrangian of the extended linear sigma model, whose mesonic part can be found in [7], has the form given in Eq. (11), only that the fermionic field is the triplet of constituent quarks, ψ ≡ (u, d, s) T , while the mesonic fields are nonets. For the scalars S = S a T a = S a λ a /2, a = 0, . . . , 8 and similarly for the other mesons (λ a =0 are the Gell-Mann matrices and λ 0 = 2 3 1). The integration over the fermionic field in the partition function results in a functional determinant involving a N × N matrix, where N = 3 × 4 × N c with N c being the number of colors. This matrix structure makes tedious a brute force calculation of the curvature mass similar to the one shown in Sec. III A, even in the case of a trivial color dependence (see Eq. (D1)). Therefore, we proceed as in Sec. III B by calculating the self-energy at vanishing momentum and relegate to Appendix D the sketch of a direct calculation.
A. Curvature mass from the self-energy For simplicity, we consider the case when only the scalar fields, namely S 0 , S 3 and S 8 , have expectation values, denoted by φ 0 , φ 3 , and φ 8 . It proves convenient to work in the N − S (nonstrange-strange) basis, which for a generic quantity Q is related to the (0, 8) basis by Applying the above relation to the matrices λ 0 and λ 8 , one obtains λ N = diag(1, 1, 0) and λ S = √ 2 diag(0, 0, 1), which give the antisymmetric structure constants 3, S, one obtains, using also λ 3 = diag(1, −1, 0), the tree-level inverse fermion propagator matrix in flavor space as iS −1  (37).
The one-loop self-energy of a generic field X a , with a being a flavor index, can be written as where N c is the number of colors, L = K − Q, s X = 1 for X = V, A and s X = −1 for X = S, P. The propagator matrixS 0 = diag(S u , S d , S s ) has as elements the tree-level propagators of the constituent quarks. Furthermore, Γ X contains Dirac matrices that carries a Lorentz index when X ∈ {V, A}, in which case the prime on Γ X indicates that its Lorentz index is different from that of Γ X . The matrices are explicitly given in Table I, along with the constant c X proportional to the Yukawa coupling. The trace is to be taken in Dirac and flavor spaces. We assumed a trivial color dependence and we postpone to Sec. IV B the discussion of a more complicated one.
The flavor space trace in Eq. (37) can be easily performed. Since in the N − S basis the λ a matrices have, with the exceptions of a = S, two nonzero matrix elements, one generally obtains two terms which can cancel each other for some flavor combinations. The nonzero contributions are listed in Table II. In the case of the first three entries, the factor of 2 is the consequence of the identity (38) which is applied inside the integral in Eq. (37) with Y = X, followed by the shift K → −K. This identity can be proven using the cyclicity of the trace and that, given the charge conjugation operator C = iγ 2 γ 0 , the matrices Γ X of Table I We see from Table II that after the trace in flavor space is performed, depending on the indices ab, the self-energy (37) can be expressed either in terms of integrals involving two different propagators or using integrals of the types already encountered in the one-flavor case (see Eq. (30)), obtained from Eq. (39) as  (37) from the flavor space trace, tr λaS0λ bS0 , for φ3 = 0 and their reduction in the isospin symmetric case, where l denotes the light quarks with equal masses m l ≡ mu = m d .
Being interested in the curvature mass, we evaluate the zero momentum self-energy, Π These are calculated in Appendix C, where, using partial fractioning and simple algebraic manipulations, they are reduced to a combination of simple integrals.
In the case of (pseudo)scalars the result is summarized in Table III, where we where the vacuum part needs renormalization and the matter part is finite and determined by T (1) f (and its mass derivative, for some flavor indices). In some flavor cases Eq. (42) does not represent the physical curvature mass of the (pseudo)scalars, due to their mixing with (axial-)vectors. This issue is addressed in Sec. VI, where we will see that the mixing affects all the pseudoscalars, but only the scalars with flavor indices 4 − 7.
In the case of the (axial-)vectors, the evaluation of the self-energy requires some care. The self-energy is split into vacuum and matter parts, as indicated in Table III. For some flavor indices, namely a = 3, N, S for φ 3 = 0 and additionally a = 1, 2 for φ 3 = 0, the vacuum part of the vector self-energy requires as in the N f = 1 case, a covariant calculation in a regularization scheme that complies with the requirement Π (V ),µν vac (Q = 0) = 0, which is familiar from QED. This requirement is investigated in Appendix B, where we relate it to a symmetry of the classical Lagrangian, which is manifest for a specific field background.
For simplicity, we use dimensional regularization to calculate the (axial-)vector self-energy, irrespective of the flavor index. The vacuum integral determining the self-energy can be reduced to tadpole integrals (see Eq. (C12)). Its finite and divergent pieces are given in Eqs. (C9) and (C10). For the matter part we only need to consider purely temporal (00) and spatial (ij) components, as mixed (0i) components vanish due to symmetric integration. The matter part of the relevant integrals, given in Eqs. (C16) and (C17), contains also an integral whose mass derivative is proportional to the tadpole, ). In the equal mass limit this relation considerably simplifies the result. A further complication with the (axial-)vectors is related to the fact that one needs to consider the mode decomposition of the dressed propagator. This is done in Appendix E, using the usual set of tensors that includes three-and four-dimensional projectors. As shown there, each mode has its own one-loop curvature mass, determined by the Lorentz components of the self-energy tensor in the Q → 0 limit. Using the form of the self-energy given in Table III in the expression (E11) that gives the contribution to a given mode p ∈ {t, l, L}, one sees that the curvature mass has the structurê where i refers either to flavor indices (e.g. ab = 44) or to the particle (e.g. K 1 ). ∆m 2 i is the contribution of the vacuum part ∝ I The 't' and 'l' modes are, respectively, 3-transverse and 3-longitudinal, while the 'L' mode is 4-longitudinal. We will see in Sec. VI that the 'L' mode (43) influences the physical curvature mass of the (pseudo)scalars.
For the flavor indices appearing in the last three rows of Table III (and also for ab = 11, 22 for φ 3 = 0, when m u = m d ) one has only a matter fermionic contribution to the vector curvature mass and only in the case of the 'l' mode. This is because the single mass integral is such that mat (m f ) = 0, as shown in Appendix C. In the isospin symmetric case the matter contributions to the curvature mass of the modes are (note that due to the absence of mixing ω N ≡ ω(782) and ω S ≡ φ(1020)) for the vectors and   III. Fermionic contribution to the zero momentum one-loop self-energy of the scalar (S), pseudoscalar (P ), vector (V ) and axial-vector (A) fields in the φ3 = 0 case. We indicate by f and f the quark type whose mass has to be taken into account in the formula of the self-energy having flavor indices ab. The matter part of the tadpole integral T (m) is given in Eq. (28) and the finite piece of the vacuum part in Eq. (56). The vacuum integral I V /A vac is given in Appendix C, together with the 00 and 11 components of the matter integral I V /A,µν mat . The constants are C S/V = 2Ncg 2 S/V , tS = 1, tP = 0, and s u/d = ±1 .
for the axial-vectors, where C V = 2N c g 2 V and for the f 1S meson the contributions are as for f 1N with m l replaced by m s . The integrals are explicitly given in Appendix C.
The vacuum contributions need renormalization and their finite part is given for φ 3 = 0 in Eqs. (67) and (68).

B. Connection to previous calculations
The fermionic correction to the (pseudo)scalar curvature masses was calculated first in Ref. [16] in the isospin symmetric case (φ 3 = 0). The Polyakov-loop degrees of freedom were incorporated in Ref. [35]. Bringing the expression in Eq. (B12) of [16] and in Eq. (25) of [35] in a form containing the tadpole and the bubble integral at vanishing momentum is not mandatory, but it reveals the structure behind the obtained result for the curvature mass. Also, integration by parts shows that the result can be given in terms of a single function: the Fermi-Dirac distribution or the modified Fermi-Dirac distribution (48), when Polyakov-loop degrees of freedom Φ andΦ are included. This simple observation makes superfluous the introduction of B ± f and C ± f , used also in [7] following [35], and allows for a slight simplification of the formulas used so far in the literature.
Using the method of Ref. [16] we show below how to obtain the expressions of the (pseudo)scalar curvature masses given in Table III. The method assumes that in the ideal gas contribution of the three quarks to the grand potential we can use quark masses that depend on the fluctuating mesonic fields, as in Eq. (14) of the N f = 1 case. The method works because for g V = 0 and K = 0 the eigenvalues of the mass matrix in Eq. (D1) correspond to the u, d, and s quark sectors. In case of the (axial-)vectors, it is not enough to concentrate on the mass matrix, as explained in Sec. I. Taking (axial-)vector field derivatives of the eigenvalues of the mass matrix, as in Ref. [17], results in a curvature mass tensor which breaks Lorentz covariance, as it is not proportional to g µν at T = 0.
Concentrating on the matter part of the grand potential, we start from its expression given in the ideal gas approximation in Eq. (27) Here M f are the eigenvalues of the matrix in Eq. (D1), which depend not only on the scalar background, but also on the fluctuating (pseudo)scalar fields, generically denoted by ϕ a , with a being the flavor index. After taking the second derivative with respect to ϕ a , the fluctuating field is set to zero, in which case The modified Fermi-Dirac distribution functions Then one uses that the dependence on ϕ a is through M 2 f , which only appears in the combination The above relation and integration by parts results in where the integral T (1) f ≡ T (1) (m f ) and its mass derivative, defined in Eqs. (28) and (29), now contain the modified Fermi-Dirac distribution functions (48).
Using Table III of [16] for the derivatives of the masses (Table II of [7] to also get the wave-function renormalization factors due to the shift of the (axial-)vector fields) one recovers the result obtained in [7] in the isospin symmetric case, where one has m u,d = m l = g S φ N /2. For example, in the ab = 11 scalar sector, which is not affected by the mixing, one has (M s does not contribute) so that the matter contribution of the fermions to the curvature mass obtained from Eq. (51) has the form in accordance with Table III in view of (29). The above simple calculation shows that in the presence of Polyakov degrees of freedom the fermionic contribution to the curvature mass can be given in terms of the modified Fermi-Dirac distribution functions (48). Based on this, one can safely replace in our previous matter integrals f ± f (p) with F ± f (p).

V. RENORMALIZATION OF THE CURVATURE MASS
For simplicity, we discuss the renormalization of the fermionic correction to the curvature masses only in the isospin symmetric case (φ 3 = 0) where m l ≡ m u = m d . Then, according to Table II, the contribution in the last  row of Table III vanishes, while for 1 − 3, N flavor indices one has to use the equal mass formula of the a = S case with the replacement m s → m l .
Since the renormalization of the (pseudo)scalar curvature masses poses no problem and was already done in the literature, using dimensional regularization [36][37][38] or cutoff regularization [7], we will only sketch an alternative method, which can be used in a localized approximation, that is when the self-energy is evaluated at vanishing momentum.
The divergence(s) of a vacuum integral can be separated by expanding a localized propagator around the auxiliary function G 0 (K) = 1/(K 2 − M 2 0 ), where M 0 plays the role of a renormalization scale. For the tadpole integral, iterating once the identity where the first and second terms are the overall divergence and the subdivergence given in terms of and the last term in Eq. (54) is finite, With the above renormalization procedure the finite part of the tadpole is independent of whether covariant or noncovariant calculation, cutoff or dimensional regularization is used (provided the cutoff is sent to infinity in Eq. (56)). In a noncovariant calculation, Eq. (56) is obtained from Eq. (26) by , and subtracting from the vacuum piece of the tadpole the first two terms obtained by expanding 1/E f (k) in powers of ∆m 2 f . Subtracting also the O (∆m 2 f ) 2 term when renormalizing the integral which determines the one-loop fermionic contribution to the effective potential in Eq. (5) (and, with the replacement m 2 f →M 2 , also the contribution of the ring integrals with localized self-energy in Eq. (7)), results in the following finite vacuum part F (m f ), that is, the relation also holding between the unrenormalized integrals (57) and (26). Now we turn our attention to the renormalization of the (axial-)vector curvature masses (43). The relevant terms of the ELσM Lagrangian introduced in Eq. (2) of Ref. [6] are those proportional to the coupling h i , i = 1, 2, 3 and the term containing the covariant derivative. In dimensional regularization, used here with d = 4 − 2 , no overall divergence is encountered, and thus the mass term of the (axial-)vectors ∝ m 2 1 is not needed. The treelevel mass squared of the vector and axial-vector fields depend on the strange and nonstrange scalar condensates φ N and φ S , as a result of the coupling of these fields to the scalars, which acquire an expectation value. We have to ensure that the subdivergence of the vacuum contribution to the curvature mass in Eq. (43) is removed by the environment-dependent terms (that is, proportional with φ N and φ S ) present in the tree-level mass formulas.
Using Eqs. (C9) and (C10) in the expressions of Table III with m u,d = m l , we see that the vacuum piece of the vector curvature mass is divergent only for flavor indices 4 − 7, corresponding to the K meson, while for the axial-vectors divergence is present in all the flavor sectors where the a 1 (K 1 ) meson correspond to the 1 − 3 (4 − 7) flavor indices. The above subdivergence structure means that in the tree-level mass formulas given in [6] in Eqs. (27)-(34), we have to look for terms which are only present for K and the axial-vectors. There is indeed such a term, the one proportional with g 2 1 , and by using the mass formulas (36) of the quarks and also (m l ± m s ) 2 = 2 , we see that the environmentdependent part in the tree-level mass formulas matches the form of the subdivergence, which therefore can be removed. The only problem is that, since g 1 is squared, absorbing the subdivergence in the counterterm of g 1 would result in an awkward renormalization, as the term quadratic in the counterterm should be dropped.
A close inspection of the structure of the terms in Eqs. (27)-(34) of [6] shows that one can achieve renormalization by assigning counterterms to the couplings h i , i = 1, 2, 3 instead of g 1 . Namely, splitting the bare coupling into renormalized one and counterterms, h i = h iR + δh i , one sees that the subdivergences can be eliminated with the counterterms: The fact that renormalization can be achieved without referring to the counterterm of g 1 raises the question of why g 2 1 is present at all in the tree-level mass formulas. A closer look into the origin of the terms proportional to h 2 , h 3 , and g 2 1 in the mass formulas reveals that some terms included in the Lagrangian through the terms proportional to h 2 and h 3 are also generated by the covariant derivative term which contains g 2 1 . Namely, using the covariant derivative of [6], where M = S + iP as in [7], the coefficient of the O(g 2 1 ) term in Tr (D µ M ) † D µ M )] is The above two traces were added to the Lagrangian with coefficients h 2 and h 3 , respectively. Therefore, using L † µ = L µ and the shorthand |L µ M | 2 ≡ L µ M † L µ M , the Lagrangian used in [6] is in fact where the relations between the parameters are from which h 2 + h 3 =h 2 +h 3 . Applying these relations, g 2 1 can be eliminated from the tree-level mass formulas of the (axial-)vectors in which h 2,3 is replaced byh 2,3 .
To avoid duplication of terms in the Lagrangian, it is a better practice to use a covariant derivative containing only the electromagnetic field, and write the Lagrangian that contains the mass terms of the (axial-)vectors and their interaction with the (pseudo)scalars in the form although this form is less compact than the one in [6]. After all these considerations, we give for completeness the vacuum curvature masses containing the renormalized one-loop level contribution of the fermions. The vector curvature masses arê while the axial-vectors ones arê where the classical curvature masseŝ are written, omitting the symmetry breaking terms considered in [6], using the constants These types of mixings come from the last line of Eq. (66) after performing the trace and shifting the scalar fields with their background values. Doing also a symmetrization using integration by parts in the classical action, one obtains, in Fourier space and at quadratic order in the fluctuating fields, the last four mixing (crossed) terms in Eq. (9) of [6] (up to an unnecessary factor of i in the V − S mixing terms and the wrong sign of the last two terms): At the classical level, the usual way to eliminate the mixing term is by shifting (in direct space) the (axial-)vector field by the derivative of the (pseudo)scalar field with an appropriately chosen wave-function renormalization constant as prefactor [1,2,6].
Here we adopt a different strategy and show that the wave-function renormalization constant is recovered when one identifies the contribution of the physical modes to the partition function evaluated in the ideal gas approximation, discussed in Sec. II. In this approximation the bosonic fluctuations are neglected in the fermionic determinant obtained by integrating out the fermions in the partition function and, keeping only quadratic terms in the mesonic Lagrangian, the Gaussian functional integral is done over the (axial-)vector and (pseudo)scalar fields. Then, we apply the same method at finite T in the Gaussian approximation, that is when the quadratic part of the mesonic Lagrangians is corrected by the field expansion of the fermionic determinant. Considering self-energies at vanishing momentum, we find that the form of the wave-function renormalization constant, resulting from the mixing of the (pseudo)scalar with the nonpropagating 4-longitudinal (axial-)vector mode, is unchanged at T = 0, only that it involves one-loop curvature masses instead of the treelevel ones that appear in the ideal gas approximation.
A. Classical level mixing

S − V mixing
We start with the mixing in the 4 − 5 flavor sectors. Using Eq. (70) and Eq. (9) of Ref. [6] one obtains where the 5 × 5 matrix is with The calculation is simplified by the identity which gives in the present case Using Eq. (73) we have the projector decomposition of the remaining 4 × 4 matrix, hence computing its determinant is an easy task. Given that P µ Tµ = 3P µ Lµ = 3, one obtains The physical mass squared of the scalar mode arises as a result of its mixing with the nonpropagating 4-longitudinal vector mode. Z K ± 0 is the wave-function renormalization constant.
The momentum-independent prefactor C 2 45 in Eq. (77) reflects the existence of the nonpropagating 4-longitudinal vector mode. When dimensional regularization is used to perform the momentum integral in Eq. (74), the logarithm of the partition function receives contributions only from the propagating modes, represented by the two brackets in Eq. (77), i.e. there is no contribution from ln C 2 45 , which depends on the scalar backgrounds.
A similar calculation in the 6 − 7 flavor sector gives We mention that in the isospin symmetric case, φ 3 = 0, one has C 2 67 = C 2 45 andm 2 as given in [6] in the last line of Eq. (14).

P − A mixing
We start with the P a − A a , a = 1, 2 mixing, given by where the 5 × 5 matrices are with , respectively.
The functional integral over A a , P a , a = 1, 2 and steps paralleling those leading to Eq. (77) give (C 2 11 =m 2 with the physical mass of the pseudoscalar mode and the associated wave-function renormalization constant: A similar calculation involving fields with flavor indices a = 4, 5 and a = 6, 7 gives a determinant as in Eq. (82), with some obvious replacements: where c 44/66 = g 1 (±φ 3 +φ N + √ 2φ S )/2. Again, for φ 3 = 0, one has a single wave-function renormalization constant, Z K , given in Eq. (13) of [6]. Now we turn our attention to the mixing in the 3−N−S sectors, given by ([P, where c NN = c 11 = g 1 φ N and c SS = g 1 √ 2φ S . For φ 3 = 0 the complete quadratic Lagrangian involves a 15 × 15 matrix. In this case, the appearance of the wave-function renormalization constant is nontrivial and will be presented elsewhere [39]. Here we consider the isospin symmetric case (φ 3 = 0), in which the (P 3 , A µ 3 ) fields decouple. Their treatment is completely analogous to that of the (P 1 , A µ 1 ) sector, giving when φ 3 = 0, and thus Z π 0 = Z π ± .
The remaining P − A mixing in the N − S sectors is described by the 10 × 10 matrix where N NN µν and N SS µν are 5 × 5 matrices of the form given in Eq. (81), but with appropriate masses in the diagonal elements and constants c NN/SS in the off-diagonal ones.
The functional integral over the strange and nonstrange fields present in (85) results in and C 2 aa =m 2 f1a − c 2 aa , a = N, S. The classical pseudoscalar curvature masses used in Eq. (89) contain the wave-function renormalization constants: In the second line of Eq. (89), one recognizes the elements of the mass squared matrix of the mixing P N − P S sector. In terms of the physical eigenvalues of this matrix, namelŷ one obtains the final result, (92) This contains the contribution of the propagating 4transverse vector and physical pseudoscalar modes.

B. Mixing in the Gaussian approximation
We consider only the isospin symmetric case, φ 3 = 0, in the localized approximation, in which the self-energies have vanishing momentum. In this case there is no correction to the off-diagonal elements of the 5 × 5 matrices (also having Lorentz indices) considered in the previous subsection, while the diagonal elements are replaced by . For the flavor indices involved in the mixing, vector and axial-vector self-energies have the same decomposition, given in Eq. (E10). Basically what happens at T = 0 is that in the inverse propagator the 4-transverse part encountered previously splits into 3-transverse and 3-longitudinal parts, with projectors P t µν and P l µν , so that whereM 2 l/t/L =m 2 + Π l/t/L (0). The components Π l/t/L (0) are given in terms of the Lorentz components of the self-energy Π µν (0) in Appendix E, where details on the tensor decomposition can also be found.
Comparing Eq. (93) to Eq. (73) and using that det(LP L + lP l + tP t ) = −Llt 2 one immediately sees how to modify our previous results, obtained in the ideal gas approximation: instead of three 4-transverse (axial-)vector modes one has the contribution of two 3transverse modes and one 3-longitudinal mode with oneloop curvature massesM 2 l,t , while the mixing between the 4-longitudinal (axial-)vector mode and the scalar mode involves the respective one-loop curvature massesM 2 L andM 2 , all with appropriate flavor indices.
Taking as an example the V 4/5 −S 5/4 mixing, one starts from Eq. (77), writes the classical curvature masses with flavor indices, instead of physical meson indices, and corrects them with the appropriate self-energy. In terms of physical modes, one has where d t = 2d l = 2,

VII. NUMERICAL RESULTS
In this section we put to work the formulas derived so far and present in the isospin symmetric case (φ 3 = 0) the temperature dependence of the one-loop curvature masses obtained for nonvanishing (axial-)vector Yukawa coupling. In order to achieve this, we minimally extend the parametrization used in Ref. [7] and solve the model using the field equations derived there in the mean-field approximation (see Eq. (40) there). In that article the model parameters were determined based on one-loop curvature masses for (pseudo)scalar mesons and treelevel ones for (axial-)vector mesons. A parametrization and solution of the model in the proposed localized Gaussian approximation will be presented elsewhere.
Including the Yukawa coupling g V among the fitting parameters, we determined the ELσM parameters using the χ 2 minimization described in Ref. [7]. We used the same physical quantities as in that article, but replaced the tree-level (axial-)vector curvature mass formulas with the vacuum one-loop level ones. The renormalization scale was fixed to the value used in Ref. [7], while for the Polyakov-loop potential we used the parameters given in Table IV and Fig. 1 of that article. The parameters corresponding to the lowest χ 2 value were found from a fit started in 10 5 random initial points of the 15dimensional parameter space, representing the parameters of the ELσM Lagrangian. Their values are given in Table IV and can be compared to those appearing in Ta-TABLE IV   ble IV of Ref. [7]. Both parameter sets are compatible with the constraints among m 2 0 , λ 1 , and λ 2 required by the spontaneous symmetry breaking, which were derived in Ch. 44.13 of Ref. [40] from the classical potential.
In Fig. 1 we compare the T dependence of the (pseudo)scalar masses obtained with a parametrization that takes into account the one-loop contribution of the quarks in the vacuum masses of all the mesons (g V = 0) to the previous result of Ref. [7] (g V = 0). In the inset we plot the wave-function renormalization constants that correspond to the two cases and are computed with the formulas of Secs. VI B and VI A, respectively. Given that the field equations are the same in both cases and that the parameter values are not much different, we see similar behaviors as the temperature increases. The temperature evolution of the scalar condensates and of the Polyakov-loop expectation values is almost identical to that shown in Fig. 1 of Ref. [7], as can be explicitly seen here in Fig. 2.
The mass of the pseudoscalars is more affected by the change in the parametrization than the mass of the scalars, especially around the pseudocritical temperature T c and above it. This is expected because all the pseudoscalar mesons mixes with an axial-vector meson with matching quantum numbers, while from the vector mesons only the mass of K is directly affected by the mixing with the scalar meson K 0 . Interestingly, the decrease of the η and η masses around T c is more prominent for the parametrization with g V = 0. For both parametrizations the a 0 meson becomes degenerate with [GeV] the η meson at large T . Such a pattern was observed also within the FRG formalism, but only when one goes beyond the local potential approximation [41]. We also mention that if the model is solved at nonzero temperatures with unchanged parameter values but with all the Z factors set to 1, then a 0 degenerates with η . The drop of the η mass around T c seen in Fig. 1, which is observed experimentally in [42], is accompanied in our case by a drop of the η mass. This behavior is related only to the decrease of the φ N,S condensates, as in [41], and not to the restoration of the U (1) A symmetry, which in our case would require a temperature-dependent 't Hooft coupling c 1 . The effect of such a coefficient that decreases exponentially with T 2 was studied in [43] within the (2 + 1)-flavor Polyakov-loop quark-meson model. In [44] mesonic fluctuations were incorporated into the axial anomaly in the N f = 2 + 1 flavor linear sigma model using the FRG method in the local potential approximation. The chiral-condensate-dependent anomaly coefficient is subject to its own flow equation, and it was shown that under certain circumstances the thermal evolution of the condensate could induce a reduction of the axial anomaly. However, a careful parametrization of the model done later in [45] showed that the anomaly actually increases around T c . While in that paper m η increases monotonically with the temperature, m η has a nonmonotonic thermal evolution, showing a slight decrease above T c , before becoming equal with m a0 at high T . A direct link between the restoration of U (1) A symmetry and the drop in the m η (T ), without a drop in m η (T ), was reported in [46]. A recent model-independent analysis done in [47] suggests that the axial symmetry is restored when the chiral partners become degenerate. In Fig. 3 we show the temperature dependence of the one-loop curvature mass of various (axial-)vector modes. In the case of ρ and ω vector mesons only the mass of the 3-longitudinal mode acquires fermionic correction, and the mass of the other modes remains the tree-level one. In the case of all (axial-)vector mesons this is the mode whose mass increases with increasing temperature deep in the symmetric phase, similarly to the mass of the (pseudo)scalar mesons. Compared to the N f = 2 version of the model studied with FRG techniques in [12,14,15], where all the chiral partners degenerate basically at the same temperature, the light vector and axial-vector chiral partners ρ and a 1 degenerate at slightly higher temperatures than the (pseudo)scalar ones, f L 0 and π. The chiral partners K and K 1 having both light and strange quark content degenerates at a higher temperature than those containing only light quarks, as the strange condensate is still large around the temperature where the nonstrange condensate φ N melts (see Fig. 2). The purely strange chiral partners ω S and f 1S degenerate at even higher temperatures, where φ S also melts. The degeneracy of the chiral partners is displayed also by the masses of 4-longitudinal and 3-transverse modes. The mass gap between the 3-longitudinal and 3-transverse modes increases with T as a result of the violation of the Lorentz symmetry.

VIII. CONCLUSIONS AND OUTLOOK
We investigated the one-loop fermionic contribution to the curvature masses of (pseudo)scalar and (axial-)vector mesons in the framework of a U (3) L × U (3) R linear sigma model with a Yukawa type interaction between mesons and constituent quarks. These corrections were calculated by evaluating the self-energy of the mesons at zero external momentum. It was showed explicitly that this is equivalent to the direct calculation of the second field derivative of the fermionic functional determinant. The one-loop curvature masses of the (pseudo)scalars agree with those derived in Ref. [16] with an alternative method that uses fluctuation-dependent quark masses. We pointed out that this alternative method cannot be used for the (axial-)vector mesons due to the presence of the momentum-dependent Lorentz scalars V µ Q µ and A µ Q µ in the fermion determinant.
The renormalization of the curvature masses was discussed in detail. The divergencelessness of the vector current, which occurs on a specific scalar background for certain flavor indices (e.g. for a = 4 − 7 in the isospin symmetric case), has the consequence that the corresponding vector self-energy is 4-transverse and vanishes at zero momentum. To comply with this property a suitable regularization scheme is needed. To keep the discussion uniform, dimensional regularization was used in the renormalization of both the vector and the (axial-)vector self-energies for all flavor indices. Additionally, the renormalization revealed that a chiral-invariant term appeared twice in the ELσM Lagrangian [5,6]. This can be cured with the appropriate redefinition of some couplings.
The occurrence of the S − V and P − A mixing already showed the importance of the mode decomposition of the (axial-)vector self-energy, which was investigated in detail at both T = 0 and T = 0, as the 4-longitudinal mode of the (axial-)vectors mixes with the (pseudo)scalars. As a result, in the case of the Gaussian approximation, the one-loop curvature mass of the (pseudo)scalar mesons is modified by a wave-function renormalization constant determined in terms of the one-loop curvature mass of the 4-longitudinal (axial-)vector mode. In a simpler approximation we recovered the already known versions of these constants appearing in [6].
The vacuum parametrization of the model was redone based on curvature masses that include one-loop fermionic contributions for all the mesons. The temperature dependence of all these masses was investigated. The (axial-)vector tensor splits up into 3-transverse modes (which turns out to have the same contribution as the 4-longitudinal one) and a 3-longitudinal mode. In the isospin symmetric case the mass of 3-transverse modes of the vector mesons ρ, ω (or ω N ) and φ (or ω S ) coincides with the corresponding tree-level mass, while for the other particles the mass of the 3-transverse modes is slightly different from the tree-level mass. For all (axial-)vector particles the mass of the 3-longitudinal mode significantly deviates from the tree-level one. It increases with increasing temperature, similarly to the (pseudo)scalar curvature mass, while the mass of the 3transverse components decreases with increasing temperature. The particle masses of the two modes become de-generate separately as the chiral symmetry restores with increasing temperature and the mass gap increases between them as a reflection of Lorentz symmetry violation.
As a side benefit of the new parametrization of the model, the value of the vector meson Yukawa coupling g V was determined. This value influences the equation of state used to describe properties of the compact star, where it has a prominent role in determining the maximal value of the compact star mass of the M -R curves (see e.g. Ref. [48]).
The curvature masses of the various (axial-)vector modes determined here can be used not only in the localized Gaussian approximation proposed in Sec. II, but also in the localized version of the two-particle irreducible formalism in which in [8] the gauged version of the purely mesonic model was solved at two-loop level for N f = 2. In the latter context the mode decomposition of selfenergy presented here would allow for an improved approximation, as there the complexity of the numerical problem was reduced by using even at finite temperature a curvature mass tensor of a vacuum form, that is, proportional to g µν . as one can check using the values of the antisymmetric structure constant.
We show below that at quantum level the divergencelessness of the current has the consequence that the vacuum part of the one-loop self-energy defined in Eq. (37) Table II we know that the self-energy is nonzero only when b = a (for the implication above see p. 233 of Ref. [50]).
Considering the meson fields as classical external fields, we start by relating the expectation value of the current and its divergence with the fermion propagator matrix (see p. 66 in [51]) The trace in Eq. (B6a) is to be taken in color, flavor, and Dirac spaces. In the SSB case, when the fields are shifted with their expectation values, the full propagator obeys with M (x) given in Eq. (B2) andM in Eq. (B3). Next, we expand the full propagator about the treelevel propagator introduced in Sec. IV, which obeys This gives at one-loop level Taking the derivative of Eq. (B6a) and using Eq. (B9) we obtain where the contribution ofS 0 (x, y) from Eq. (B9) vanishes due to translational invariance.
It would be tempting to say that the left-hand side of Eq. (B10) vanishes as result of Eq. (B4), but the usual proof using the invariance of the functional integral with respect to the vector U (3) V transformation does not go through because we neglected the mesonic fields in Eq. (B1) and the current vanishes only on a specific scalar background. What is easy to prove however, is that Eq. (B6b) vanishes at linear order in M (x), i.e. the order at which Eq. (B10) was derived. Indeed, using the first term in Eq. (B9), one has in the cases listed in Eq. (B4) (B12) Going to momentum space and using the definition (37) of the self-energy, one easily obtains Eq. (B5), which holds in the cases listed in Eq. (B4).
where ξ = {S a , P a , V µ a , A µ a |a = 1, . . . , 7, N, S} denotes the set of fields contained in the nonets, ⊗ is the Kronecker product, m f , f = u, d, s, is the constituent quark mass given in Eq. (36), while k 0 f = iν n + µ f , with ν n the Matsubara frequency and µ f the chemical potential.
We calculate the determinant of S −1 E,f (K; ξ) with the symbolic program MAPLE keeping only those (pseudo)scalar or (axial-)vector fields which are used for differentiation in Eq. (16) and setting to zero the remaining set of fields, denoted as ξ = ξ \ {X a }. This simplified determinant is evaluated in Dirac and flavor spaces and denoted as D (Xa) := det S −1 E,f (K; ξ) | ξ =0 . We found that it can have two forms: for the mixing sector involving the fields X 3 , X N , and X S the three quark sectors completely factorize, while for fields with other flavor indices there is a mixing between two quark sectors.
The contribution of the scalar (X = S) and pseudoscalar (X = P ) mixing sectors can be written with X ± = (X N ± X 3 )/ √ 2 in the following factorized form In this appendix we consider at zero and finite temperature the decomposition into physical modes of the oneloop fermionic contribution to the momentum-dependent self-energy tensor of massive vector and axial-vector bosons, generically denoted by Π µν (Q). Special interest is devoted to the curvature mass of the modes, obtained from the self-energy in the limit Q → 0, which at T = 0 represents the limit q 0 → 0, followed by q → 0.
a. T = 0 case. The vacuum self-energy Π µν vac (Q) can be decomposed as (E4) It is evident that the curvature masses of the propagating (T) and nonpropagating (L) modes are: vac,L/T =m 2 + Π vac,L/T (0).
In the N f = 1 case, due to the fermion number (current) conservation, the vector boson self-energy not only is transverse, that is, Q µ Π µν (Q) = 0, but also satisfies Π µν (Q = 0) ≡ 0, and therefore Π vac,L/T (0) ≡ 0, just like in the case of the photon polarization tensor in the QED. In the N f = 2+1 case the above relations hold due to Eq. (B12) for the vector boson self-energy with flavor indices listed in Eq. (B4). These indices correspond to the last three entries of Table II (also for the first entry in the φ 3 = 0 case), when the integrals involve fermion propagators with identical masses. For the first three entries of the table (except for the first one in the φ 3 = 0 case) the vector polarization tensor is alike the axialvector one, that is, Q µ Π µν (Q) = 0 and Π µν (Q = 0) = 0, so that, using Π µ µ,vac (Q) = 3Π vac,T (Q) + Π vac,L (Q) and Π µν vac (Q = 0) ∝ g µν , one obtains Π vac,T (0) = Π vac,L (0) = Π 00 vac (0) = −Π 11 vac (0).