$1/N_c$ Nambu -- Jona-Lasinio model: $\pi^0$, $\eta$ and $\eta'$ mesons

We continue to study the properties of the light pseudoscalar nonet within the combined framework of Nambu -- Jona-Lasinio model and $1/N_c$ expansion, assuming that current quark masses count of order $\mathcal O(1/N_c)$. The masses, mixing angles and decay constants of the $\pi^0$, $\eta$ and $\eta'$ are calculated. The role of the $U(1)_A$ anomaly is emphasized. It is shown that the gluon anomaly suppresses the leading order effects that might otherwise be substantial for the $\eta\to 3\pi$ amplitude. A detailed comparison with the known results of $1/N_c$ chiral perturbation theory is made.


I. INTRODUCTION
In the world of massless up, down and strange quarks, the Lagrangian of quantum chromodynamics (QCD) is symmetric under U (3) L × U (3) R chiral transformations.This symmetry, however, is violated spontaneously by the non-zero quark condensate, and by the axial U (1) A anomaly.The response of the quark-gluon vacuum to the spontaneous symmetry breaking is the excitation of eight Goldstone modes π, K and η, while the ninth Goldstone mode η receives a large mass due to the U (1) A anomaly.In the real world, the π, K and η mesons acquire their masses because chiral SU (3) L × SU (3) R symmetry is broken explicitly by the non-zero quark masses m u = m d = m s .As a consequence, the physics of the pseudo Goldstone bosons is based on three pillars: the value of the quark condensate, the strong U (1) A anomaly, and the pattern of the light quark masses.
These three essential elements of pseudoscalars dynamics are deeply correlated.Indeed, if U (1) A were a good symmetry in nature, one would have a light isoscalar particle L with the mass m 2 L ≤ 3m 2 π [1].Moreover, if the ratio (m d −m u )/(m d +m u ) 0.3 were appreciable, i.e., if it were not hidden by the peculiar features of chiral dynamics indicated above, the isotopic spin symmetry would be substantially violated so that the mass eigenstates of neutral pseudoscalar mesons would be pure, each containing only one quark flavor pair: ūu, dd, and ss [2].Another manifestation of the correlation is the surprisingly large mass of pseudoscalars compared with the light quark masses.As we learned from current algebra, the masses of pseudoscalars are proportional to the current quark masses.In the case of the pion, the formula reads m 2 π = B(m u + m d ), where the constant B is non-zero in the chiral limit B 0 = − qq 0 /F 2 .The quark condensate and the pion decay constant imply a very large factor B 0 2.5 GeV (we give here the estimate obtained in the framework of the Nambu and Jona-Lasinio (NJL) model, chiral perturbation theory gives a relatively smaller, but still pretty large value B 0 1.4 GeV) which significantly enhances the effect of light quark masses.
The consequences of explicit and spontaneous violation of chiral symmetry are most interestingly reflected in the physical properties of neutral pseudoscalars π 0 , η and η .It is this aspect of chiral dynamics that this article is devoted to.The π ± and K mesons were considered in our previous work [3].The study is based on the effective meson Lagrangian originated from the effective U (3) L × U (3) R symmetric four-quark interactions of the NJL type [4,5], where we, following Leutwyler's idea [6,7], count the light quark masses to be of order O(1/N c ).
To succeed in the quantitative description of effects related to the explicit chiral symmetry breaking, we use the asymptotic expansion of the quark determinant [8][9][10], which is based on the Fock-Schwinger proper time method and the Volterra series.This powerfull tool allows not only to isolate divergent parts of quark loop diagrams, but also accurately reproduce their flavor structure.The latter circumstance is fundamental in studying the explicit violation of chiral symmetry in the NJL model.
A huge number of papers have been devoted to the study of the properties of the neutral pseudo Goldstone particles.Therefore, we consider it necessary at the beginning of our presentation to answer the question of what is the novelty of the results presented here in comparison with the already well-known achievements in this actively developed area [11].
In answering this question one should stress that the NJL model has not previously been used for the theoretical study of neutral pseudoscalar states under the assumption that m i = O(1/N c ) (except for a short letter [12]).We think that the implementation of this idea may allow us to look at the results of 1/N c chiral perturbation theory (1/N c χPT) [6,7,[13][14][15][16] from a new angle, since the prospect opens up to directly relate the low-energy constants of 1/N c χPT with the parameters of the NJL model, i.e., with the main characteristics of the hadronic vacuum.
The hypothesis m i = O(1/N c ) should not be taken literally, i.e., as a direct use of the Taylor expansion in powers of quark masses.It is well known that the chiral series is not of the Taylor type.It contains non-analytic terms, so-called chiral logarithms.The assumption to count the light quark masses of order 1/N c shifts the contributions of chiral logarithms to the next-to-next-to leading order (NNLO).Correspondingly, at the next, NNNLO, step, it is necessary to take into account the contribution of two-loop meson diagrams, and so on.Thus, a full account of current quark masses by the naive summation of the Taylor series is a misleading procedure, because it does not account for essential contributions of chiral log-arithms arising at higher powers of light quark masses.Hence only the leading order (LO) result and the first 1/N c correction (NLO) to it has a polynomial form in the current quark masses.It is this approximation that is used here to study the π 0 -η-η system.
Our paper also reports on some progress in describing explicit chiral symmetry breaking in comparison with previous schemes developed on the basis of the NJL model [17][18][19][20][21][22][23].In particular, we show a deep connection between the results obtained here and similar results known from the 1/N c χPT (previous NJL approaches have been less successful in this).
But there are also differences.In the NJL model, the kinetic term of the free meson Lagrangian is the result of calculating the self-energy meson diagram with a quark loop.We show that this leads to a redefinition of the original meson fields collected in the matrix U = e iφ ∈ U (3).As a result, the neutral states in the octet-singlet basis φ a (a = 0, 3, 8) are not pure, namely φ a = b F −1 ab φ R b is a superposition of rescaled eigenfunctions φ R a for which the kinetic part of the Lagrangian is diagonal.The appearance of such impurities in φ a is the result of the explicit violation of chiral symmetry.This has physical consequences: the U (1) A anomaly contributes at the next-to-leading order (NLO) to the masses of π 0 , η, and η mesons suppressing effects of flavor and isospin symmetry breaking [24].
One example of such suppression is found in the calculation of the η-π 0 mixing angle .It is known that the interference with η , in the LO of the 1/N c expansion, strongly affects the amplitude of the η → 3π decay which is proportional to .This effect was discussed by Leutwyler [7], who pointed out on its similarity with the other effect occurring in the mass spectrum of η-η .He has found that chiral symmetry implies that the same combination of effective coupling constants which determines the small deviation from the Gell-Mann-Okubo formula also specifies the symmetry breaking effects in the decay amplitude and thus ensures that these are small.Indeed, below we show that the NLO correction significantly suppresses the isospin symmetry breaking effect observed at the LO.As a result, one can not only obtain the phenomenological values of the η and η masses, but also reduce the isospin breaking angle to the value established early in [2].We find a second example of the suppression effect of the gluon anomaly calculating the η-η mixing angle.It is known that in 1/N c χPT this angle is dramatically reduced to about −10 • from its LO value of −18.6 • [15].We show that in the 1/N c NJL model the magnitude of the NLO corrections is rather small: its LO value −15 • is corrected to −15.8 • after the NLO contributions are taken into account.
We also consider the scheme with two mixing angles, which is widely discussed in the literature [25][26][27][28][29], and demonstrate that in the NJL model it arises, as an approximation, after the NLO corrections are included.Unfortunately, within the framework of 1/N c NJL model, we fail to find a rigorous theoretical justification for this mixing scheme.The latter is possible only in the presence of off-diagonal terms in the kinetic part of the free meson Lagrangian, which explicitly violates Zweig's rule [25].Since quark loop diagrams contributing to the selfenergy meson graph satisfy this rule, off-diagonal vertices do not arise in the NJL model.Nonetheless, we show that the scheme with one mixing angle guarantees the fulfillment of the well-known relations between weak decay constants [30].
The article is organized as follows.In Sec.II, we present the form of the free Lagrangian for the η-η -π 0 fields, which arises as a result of the asymptotic expansion of the quark determinant.Additionally, the contributions of the gluon anomaly and the interaction that violates the Okubo-Zweig-Iizuka (OZI) rule are considered.In Sec.III, we calculate the coupling constants f 0 , f 3 and f 8 of neutral pseudoscalars and discuss their connection with already known results.In Sec.IV, the masses and mixing angles are calculated.In Sec.V, we consider the octet-singlet basis and calculate the weak decay constants.In particular, it is detailed here how the NLO corrections effectively lead to the well-known scheme with two mixing angles.The physical content of the initial fields φ a is discussed in Sec.VI.In Sec.VII, we shortly discuss the strange-nonstrange mixing scheme.The comments about regularization dependence of our results are given in Sec.VIII.The latter may be useful in order to have some idea of the theoretical uncertainties behind the results presented in the paper.Our conclusions are collected in Sec.IX.In order not to clutter up the text with technical details, we put them in three Appendixes.

II. BASIC ELEMENTS
Let us first make a remark regarding the original form of four-quark interactions considered in this and our previous paper [3].They include the U (3) L × U (3) R chiral invariant terms describing the scalar, pseudoscalar, vector and axial-vector nonets.This reflects the the symmetry of QCD at N c → ∞.Actually this set of four-quark interaction channels is not complete in the spirit of Fierz transformations.The symmetry of massless QCD allows the Fierz-invariant terms, describing the singlet vector and axial-vector four-quark couplings [31].We neglect them here as it has also been done in [31] when describing the meson spectrum.The reason is that singlet-octet degeneracy is quite well realized in the empirical mass data on spin-1 mesons.Theoretically, it can be understood as an indication that corresponding couplings are 1/N c suppressed in comparison with the channels considered in our paper.
To study the main characteristics of the neutral pseudoscalars (masses, mixing angles and decay constants) we need only a part of the effective Lagrangian describing non-interacting π 0 , η and η fields.Recall that in the NJL model the effective meson Lagrangian results from the evaluation of the one-loop quark diagrams.On the one hand, this requires a redefinition of the initial field functions, and on the other hand, it allows one to calculate the meson coupling constants appearing as a result of such a redefinition.The details of the one-loop calculations have been presented in our previous work [3], so let us write out only the final result arising for the diagonal pure flavor states of the pseudoscalar nonet, φ i (i = u, d, s), Here the coupling constants G S and G V characterize the strength of the U (3) L × U (3) R chiral symmetric fourquark interactions with spin zero and one correspondingly.Their dimension is (mass) −2 , and at large N c they are of order O(1/N c ). M i is the mass of the constituent i-quark.The heavy constituent masses arise through the dynamic breaking of chiral symmetry and are related to the masses of light quarks m i by the gap equation.The diagonal elements of the matrix κ A is obtained in the result of eliminating the mixing between pseudoscalar and axial-vector states.They can be expressed through the main parameters of the NJL model where (3) Here, Λ is the cutoff characterizing the scale of spontaneous symmetry breaking.The values of the parameters were fixed in [3].We collect them in Table I.
As one can see from (1), the quark one-loop diagrams generating the kinetic part of the free Lagrangian lead to a diagonal quadratic form in the flavor basis, which, after redefining the fields takes the conventional form 1 4 i=u,d,s The new field φ R i has the dimension of mass because the coupling constant f i has this dimension and the initial field φ i is a dimensionless quantity.The transition from the flavor components φ R i to the octet-singlet ones φ R a is described by the matrix O given by Eq. (A4).Due to this transformation, the transition to the octet-singlet basis φ R a does not destroy the diagonal form of Eq. ( 5).As a consequence, the unscaled field φ a has admixture of scaled components φ R b , with b = a and vice versa φ R a = b F ab φ b (see Appendix A for details).The nondiagonal elements of the symmetric matrix F ab given by Eq. (A10) violate flavor symmetry.Both the SU (3) breaking term F 08 and isospin breaking terms F 03 , F 38 are 1/N c -suppressed.A bit later we will dwell on the connection of the elements F ab with the decay constants of pseudoscalars.
For the mass term in Eq. ( 1) we find where M 2 1 is a symmetric matrix with the elements Here and below, for the convenience of writing formulas, we use the notation Now it is necessary to take into account two important points -the U (1) A anomaly and the violation of the OZI rule -both explained within the 1/N c expansion [32][33][34][35][36][37].The U (1) A anomaly contributes to pseudoscalar masses given by Eqs.(7) already at leading order (notice that we count m i to be of order O(1/N c )).The OZI-violating interactions are responsible for the 1/N c correction to the leading order result.The Lagrangians corresponding to these processes have the form of the product of two traces.At the quark-gluon level, such a contribution comes from diagrams with quark loops coupled through the pure gluon exchange.
The Lagrangian breaking the U (1) A symmetry was obtained in [37][38][39][40].Using this result, we set where U = e iφ , and φ = r φ r λ r , r = 0, 1, . . ., 8, the matrix λ 0 = 2/3 and λ 1 , . . ., λ 8 are the eight Gell-Mann matrices of SU (3).The dimensional constant λ U = O(N 0 c ) is the topological susceptibility of the purely gluonic theory, [λ U ] = M 4 .TABLE I.The six parameters of the model Λ, GS, GV , mu, m d , and ms are fixed by using the meson masses m π 0 , m π + , m K 0 , m K + , the pion decay constant fπ and the cutoff Λ as an input.The electromagnetic corrections to the masses of charged mesons are estimated taking into account the violation of Dashen's theorem at next to leading order in 1/Nc.To do this, we additionally used the value of fK and the phenomenological data on the η → 3π decay rate.All units, except dimensionless quantities δM , a, [Λ] = GeV, and [GS,V ] = GeV −2 , are given in MeV.This Lagrangian implies the following contributions to the matrix elements of the η -η-π 0 mass matrix where the couplings f 0 and f 3 are given in Eqs.(A14).
Notice that, because of the Eq.(A13), an additional mixing is induced between the rescaled neutral components, which is associated with violations of the isospin and SU (3) f symmetries beyond the leading order.Here only the terms which are responsible for leading and nextto-leading order contributions (in 1/N c counting) are retained.
The Lagrangian violating the OZI rule has the form [6] where As we will see, the counting rule λ Z ∼ N 0 c leads to a coherent picture for the masses and decay constants of the pseudoscalar nonet.
The quadratic part of the Lagrangian ( 11) is and contributes only at next to leading order 1/N 2 c in the matrix elements which interfere with the next-to-leading order contribution of the gluon anomaly.

III. DECAY CONSTANTS fi
Our next task is to isolate the leading order contribution together with the first 1/N c correction to it in the formulas above.Here we realize this plan for the decay couplings of neutral states For that we need the nontrivial solution of the gap equation M i (m i ).The latter, in the considered approximation, can be written as a sum [3] where M 0 is the mass of the constituent quark in the chiral limit m i → 0, and Then, from Eq. ( 15), Eqs.(A14) and (A10) we find where m = (m u + m d )/2 and Here F and κ A0 are the values of the pion decay constant f π and κ Aii at m i = 0. Further, according to Leutwyler [30], 1/N c χPT provides the relation among the decay constants: which is valid to first nonleading order.It can easily be verified that, on using Eq. ( 18) and expressions for the kaons decay couplings obtained in [3] relation ( 20) is also satisfied in our approach.
The other relation which is a direct consequence of the approach developed here is This result is known from [27], where the authors used a different method to obtain it.
For the constants f 0 and f 8 , in addition to the above quadratic relations, one can establish the following linear relations with the constants f π and f K In contrast to the formulas ( 20) and ( 22), there are no higher-order terms in these relations which must be systematically discarded.
The non-diagonal elements of the matrix F ab in the considered approximation are They are negative.The first two are associated with the isospin symmetry breaking, and the last one with the violation of SU (3) symmetry:

IV. MASSES AND MIXING ANGLES
Let us now consider meson mass relations.For that we expand the elements of the resulting mass matrix M 2 = i=1,2,3 M 2 i (see Eqs. ( 7), ( 10) and ( 14)) in powers of 1/N c retaining only the first two terms.
The leading 1/N c -order result is where λ 2 η ≡ λ U /F 2 , and It coincides with the formulas obtained by Leutwyler [6], but in the case under consideration, all parameters (except for λ U ) are related to the main parameters of the four-quark dynamics.
It is clear from (26), that mixing of φ R 3 with φ R 0 and φ R 8 is due to the breaking of isospin symmetry.In the first order in the mass difference m d − m u , this mixing is removed by rotating to small angles and , respectively.The φ R 0 -φ R 8 mixing is due to the breaking of SU (3) symmetry and can be removed by rotating to the angle θ.With an accuracy to the first order in the breaking of isospin symmetry, the transformation of the neutral components to the physical π 0 , η, and η states has the form This orthogonal transformation diagonalizes the mass matrix M 2 , giving the eigenvalues (the mass squares) and eigenvectors of physical states (for more details see Appendix B).
The result of the diagonalization of the mass-matrix ( 26) is well known: The predicted mass of the η meson m η = 494 MeV is much smaller than its phenomenological value m η = 548 MeV and the angle θ is θ −20 • .The numerical values of the parameters used are given in Table II (see set (a)).
Recall that the difference between the masses of the charged and neutral pions is due primarily to the electromagnetic interaction.The contribution of the strong interaction is proportional to (m d − m u ) 2 and is thereby negligibly small.The model estimate is Here, the overline indicates that the masses were obtained without taking into account electromagnetic corrections.Now, we do the next step and calculate the first correction ∆µ 2  ab to the leading term.This correction includes the contributions from Eqs. ( 7), (10) and (14).Note that when calculating these corrections, we systematically neglect the terms (m d − m u ) 2 , replacing, for example, the sum only by its first term.This is also in agreement with the accuracy with which the rotation matrix (28) is defined.As a result, we find where Here it is appropriate to make a few remarks about formula (30).Let us first comment on the origin of different contributions.Corrections caused by Eq. ( 7) are the terms that remain after we put ∆ N = 0.In fact, they coincide up to a common factor with the known result of 1/N c χPT [15].The correspondence between factors is Next, the corrections proportional to λ U in (30) are related with the U (1) A anomaly: The corresponding contribution to ∆µ 2 00 arises due to the NLO correction to the coupling f 0 in (10).The other two contributions to ∆µ 2 08 and ∆µ 2 03 are the result of an admixture of rescaled neutral octet components in the singlet field φ 0 described by the Eq.(A13).Both account for the symmetry breaking corrections due to the U (1) A anomaly.Such corrections interfere with the OZI violating contributions of Lagrangian (11) and, as a result, the effective coupling constant ∆ N arises.If we compare the formulas for ∆µ 2  08 , ∆µ 2  03 with the analogous expressions obtained in [15], one can establish a correspondence where on the right-hand side we have retained the notation of work [15], so one should not confuse the notation M 0 (singlet mass) adopted there with the constituent quark mass M 0 used here.The only difference between the NJL approach considered here and the 1/N c χPT is the absence of the NLO term −M 2 0 Λ 1 in our expression for ∆µ 2 00 .Probably it is this circumstance that leads to different estimates for the mixing angle θ in the compared approaches.
The representation of the mass matrix M 2 as the sum of the leading contribution and the 1/N c correction to it implies a similar representation for all parameters of the transformation that is used to diagonalize the mass matrix: θ = θ 0 +∆θ, = 0 +∆ , = 0 +∆ .Accordingly, the eigenvalues obtained should have a similar form (see Appendix B for details).
To obtain numerical values, we fix the main parameters of the model as it was in the case of the charged particles (see the Table I).Additionally, the phenomenological values of the masses of η and η mesons are used to fix the topological susceptibility λ U and the OZI-violating coupling constant λ Z .As a result, we obtain the values of the mixing angles θ, and (see set (b) in Table II).
It should be also noted that the angle θ obtained here differs noticeably from the estimate θ −10 • worked out in the framework of 1/N c χPT [15].We have already pointed out the reason for this discrepancy above.Here we note that the 1/N c NJL model does not lead to a huge effect from taking into account NLO contributions observed in [15].As one can see from the Table II, the LO result θ 0 receives only a 5% NLO correction.
Numerical estimates show that the mixing angles and are substantially modified at NLO.The corrections account for around 35% of the LO result.In particular, the mixing angle is found to be = 0.65 • while the LO result is 0 = 1.0 • .This result can be compared with the estimate = 0.56 • that arises in χPT when only octet degrees of freedom are included [44].The similar behaviour is found for the angle = 0.12 • which is equal Since the NJL model in the LO reproduces analytically the mixing angles and known from [7] where the angle ¯ 0 has been obtained by Gross, Treiman, and Wilczek [2] disregarding the η − η mixing we observe the known effect: The mixing with the η increases significantly the value of the angle 0 compared to ¯ 0 .The LO estimate 0 = 0.018 we found is consistent with the estimates 2¯ 0 , for θ 0 −22 • made in [7], and = 0.017 ± 0.002 in [28], both obtained under the same assumptions: The use of Daschen's theorem and η − η mixing.This is frustrating because enters the amplitude η → 3π, making the width unacceptably large.This effect was considered in [7], where it was indicated that the problem lies in the accuracy of the LO result.Deviations of order 20 − 30% are to be expected and this does not indicate that the 1/N c expansion fails.It was claimed that the effect can be resolved by taking into account the higher order corrections.Our calculations show that this is exactly what happens.The 1/N c correction ∆ leads to complete agreement of our result = 0.011, both with the result of the current algebra and with the result of χPT.From this we conclude that the LO effect of η-η mixing on the angle is completely offset by the NLO corrections.
Equation ( 31) must be discussed in a little more detail due to its relation to the low energy constants of 1/N c χPT given by Eq. (33).Recall that ∆ N is treated as a small parameter, because it represents a term of order 1/N c .Indeed, it is reasonably small, our estimate is ∆ N = −0.46+ 0.82 = 0.36.It can be seen that the contribution of the gluon anomaly (the second term) differs in sign from the OZI-rule violating contribution (the first term) and dominates.This way the gluon anomaly suppresses the effects of SU (3) and isospin symmetry breaking in ∆µ 2 08 and ∆µ 2 03 .Of course, the opposite can also be said: the OZI rule violating interaction (11) reduces the effect of the gluon anomaly in these channels.
Further, since the following relations hold we obtain the following estimates for the couplings on the right-hand side of these relations, namely, Λ 1 /2 − Λ 2 = −0.46 and 4L 5 M 2 0 /F 2 0 = 0.82.These values are noticeably lower than the estimates obtained in [15], where, for instance, set (NLO No. 1) gives the values −0.65 and 1.12 correspondingly.In this case, however, it would be naive to expect complete agreement between the approaches, since the Λ 1 is also responsible for NLO correction to ∆µ 2 00 in 1/N c χPT, which, as we have already noted above, is not the case in the 1/N c NJL model.

V. WEAK-DECAY COUPLING CONSTANTS IN THE OCTET-SINGLET BASIS
To find the decay constants of pseudoscalars we should relate the fields φ a (a = 0, 8, 3) to the physical eigenstates P = η , η, π 0 .As we have already learned, a transition to the physical fields P is carried out in two steps At the first step, the dimensionless field φ = φ a λ a arising in the effective meson Lagrangian through the exponential parametrization U = ξ 2 = exp(iφ) is replaced by the dimensional variable φ R given in the same basis (see Eq. (A7)).The symmetric matrix F f (A9) is worked out in such a way that the kinetic part of the free Lagrangian takes the standard form.Then, at the second step, diagonalizing the mass part of the free Lagrangian by the rotation U θ (for definition of matrix U θ see Eq. ( B2)), we come to the physical fields P = η , η, π 0 .
The first step of the described procedure generalizes the standard construction of the effective Lagrangian of pseudo Goldstone fields to the case of explicitly broken flavor symmetry.Here [7], the pseudo Goldstone field φ is also represented by the exponent U = exp(iφ), and the pion decay constant F appears in the kinetic part of the effective Lagrangian to make the field φ dimensional by redefining F φ = φ R .
In the NJL model the factor at the kinetic part of the Lagrangian arises from a direct calculation of the quark one-loop diagrams.In the case of broken flavor symmetry m u = m d = m s , the place of the factor F is taken by the matrix F f .As a result, to redefine the field φ, it is necessary to use the matrix F f , i.e., F f φ = φ R , and not a simple factor F .Obviously, in the chiral limit F f is a diagonal matrix F f = F diag(1, 1, 1).
To lowest order in 1/N c , we have F 08 = F 03 = F 38 = 0 and then in (38) we arrive to the standard pattern with one mixing angle θ.
In the formulas above, it is necessary to take into account only the terms that do not exceed the accuracy of our calculations here.Hence, expanding in powers of 1/N c and retaining only the first two terms, we find where To obtain the numerical values of weak decay constants we use parameter set (b) given in Table II.As a result, we find 1.16 −0.54 −0.0054 0.12 1.20 −0.016 −0.001 0.011 1.02 The numerical estimations show that the η -meson contains a noticeable (∼ 50%) admixture of the octet component φ 8 .On the contrary, the η meson is nearly a pure octet: The admixture of φ 0 is an order of magnitude lower than φ 8 .Note, that the analysis done in the work [30] led to the same conclusion.The neutral pion is a pure φ 3 -state, the admixture of which in the η meson is three times greater than in the η state.We also get the following estimates for ratios which perfectly agrees with the values f 8 = 1.27(2)f π and f 0 = 1.14(5)f π obtained exclusively on the transition form factors of η and η , reanalyzed in view of the BESIII observation of the Dalitz decay η → γe + e − in both space-and time-like regions [45].The 1/N c -corrections to the leading order result allows one to distinguish two mixing angles ϑ 8 and ϑ 0 , which are often used in the phenomenological analysis of η-η data [27].Indeed, from Eq. ( 40) one infers That gives ϑ 8 = θ 0 + F ∆θ − /f 8 = −24.2• , and ϑ 0 = θ 0 + F ∆θ + /f 0 = −6.0• .Further, if we restrict ourselves only to the first correction, we get This result agrees with a low energy theorem [30], which states that the difference between the two angles ϑ 0 − ϑ 8 is determined by f K − f π .The numerical values of the angles again can be compared with the result of [45]: ϑ 8 = −21.2(1.9)• and ϑ 0 = −6.9(2.4)• .

VI. PHYSICAL CONTENT OF φa
Let us establish a connection between the octet-singlet components φ 0 , φ 8 , and φ 3 and the physical eigenstates P = η , η, π 0 .For that one needs to know the matrix F P a in the inverse to the (38) relation Its dimension is [F P a ] = M −1 .The entries of the matrix can be found with the use of formulas (A13) and (28).
It is not difficult to establish that the matrix elements F P a have the form (39), where one should substitute F a P → F P a , and F f → F 1/f .Then, expanding the result in the 1/N c series, we find to first nonleading order where The elements of matrix F P a numerically are 0.76 0.42 0.011 −0.0074 0.73 0.01 0.0012 −0.0071 0.98 Thus, we see that the singlet component, in addition to the leading contribution of η , has a noticeable admixture of η.The latter dominates in the octet component φ 8 , while the neutral pion dominates in φ 3 .
Since matrices F a P and F P a are mutually inverse, one would expect that the numerical result (50) should be reasonably close (up to the approximations made) to the result obtained by the direct inversion of matrix (42).This is indeed the case for all elements of matrix (50) except F η 8 .The latter turns out to be an order of magnitude smaller than such a qualitative estimate gives: Here there is a significant compensation between two terms: F η 8 = (0.175 − 0.182)/F .The first term is the SU (3) breaking contribution with cosine, the second one is (F/f 8 ) sin θ 0 .In the limit of exact SU (3) symmetry this term gives −0.258.The SU (3)breaking correction in f 8 makes it to be equal −0.182.This combined effect of mixing angle θ 0 and SU (3) breaking pushes η out of the octet.

VII. THE STRANGE-NONSTRANGE MIXING SCHEME
Instead of a flavor octet-singlet basis, one could choose a scheme with a mixture of strange and non-strange com-ponents.It is also often used in the literature (the socalled Feldmann-Kroll-Stech scheme [27]).Therefore, we will briefly focus on it here.To do this, just as we did in Appendix A, let us represent the field φ R by its components in the orthogonal basis λ α = (λ S , λ q , λ 3 ) where tr(λ α λ β ) = 2δ αβ , and It follows then where the ideal mixing angle θ id = arctan √ 2 54.7 • .Now, it is easy to establish from (28) that the physical states P = (η , η, π 0 ) are the linear combinations of states φR = (φ R S , φ R q , φ R 3 ), namely where the angle ϕ = θ + θ id , or numerically ϕ = 39.0 • , and the matrix U is defined in (B2).This result agrees with phenomenological estimates ϕ = 39.3 • ± 1.0 • in [27], and lattice data obtained by ETM Collaboration [46] , where the second errors refer to uncertainties induced by chiral extrapolations to the physical point.The anomaly sum rule approach to the transition form factors with a systematic account of the η-η mixing and quark-hadron duality gives a bit smaller result: ϕ = (38.1 ± 0.5) • [47].

VIII. DEPENDENCE ON THE REGULARIZATION SCHEME
The results presented above are based on a well-defined regularization scheme, proper-time regularization.In this case, two integrals with quadratic J 0 (M i ) and logarithmic J 1 (M i ) divergences are regulated by subtracting off suitable counterterms at the scale Λ, ρ t,Λ = 1 − (1 + tΛ 2 )e −tΛ 2 .The choice of the regularizing kernel is made in such a way that the expressions for regularized integrals coincide with the one obtained in the covariant four-dimensional Euclidean regularization scheme.This implies, in particular, the following form of the gap equation Let us recall that the choice of regularization essentially determines the NJL model.A reasonable regularization must satisfy two main criteria: the minimum effective potential condition should lead to a gap equation, and the vacuum corresponding to the nontrivial solution should possess the Goldstone modes.It is also desirable that the regularization does not break the symmetry of the model.All these requirements are met by the Fock-Schwinger method used here.It is one of the standard regularization prescriptions often used in the literature [48,49].
The gap equation for m i = 0 and a fixed value of G S , has a solution M 0 (Λ) that changes drastically with Λ.
i.e., a 1% change in the value of the cutoff Λ leads to about 7% changes in the value of M 0 .This can be a source of theoretical uncertainties associated with the regularization scheme used.Since all other vacuum characteristics are expressed in terms of these two parameters (and G V = 7.4 GeV −2 ), it is easy to establish that a = 3.50 −0.28 +0.33 , δ M = 0.67 +0.10 −0.12 , F = 90.54+2.82 −3.10 MeV, | qq It should be noted that current knowledge about the constant f π = 92.277(95)MeV [50] imposes such a strict boundaries on the range of values admissible for the cutoff Λ (or alternatively on the coupling of four-quark interactions G S ), that one could reach such accuracy in the NJL model only by the precise tuning of Λ = 1100.00+0.55 −0.16 MeV.It is also interesting to check how successful the NJL model predictions are for the low-energy constants L 5 and L 8 of the 1/N c χPT which are scale independent at the order considered.Here the 1/N c NJL model gives The values of these constants are consistent with their phenomenological estimates made in χPT: L 5 = (2.2 ± 0.5)10 −3 , L 8 = (1.1 ± 0.3)10 −3 [44], however these values have to be redetermined using the logic of the mixed expansion scheme adopted in the 1/N c χPT.Such calculations were made, for example, in [15].Neglecting the contributions of chiral logarithms, the authors obtained the following estimates: 2L 5 + L 8 = (5.26± 0.01)10 −3 , 2L 8 −L 5 = (0.8±0.9)10 −5 .Our result (62) for these combinations leads to the values 2L 5 + L 8 = (5.47−0.73 +0.89 )10 −3 , 2L 8 − L 5 = (0.50 +0.06 −0.09 )10 −5 .The NLO analysis of the η-η mixing in [51] gives L 5 = (1.86 ± 0.06)10 −3 , and L 8 = (0.78 ± 0.05)10 −3 .As pointed out in [52], L 8 cannot be determined on purely phenomenological grounds.As a consequence, the sign of the difference 2L 8 − L 5 is fixed only in the framework of a specific model.We see that in the 1/N c NJL model the sign is positive in contrast to the estimates in [51].It is clear that inequality (63) is ensured by a positive value of δ M = 0.67 (with a 15% theoretical uncertainty).This analysis can be extended to all the estimates we obtained above.This will be done a little later, when we have accumulated enough theoretical material on various physical properties and processes involving pseudoscalar mesons.

IX. CONCLUSIONS
We continued studying the properties of the nonet of pseudoscalar mesons in the NJL model, where we changed the counting rule for the masses of current quarks.In the recent work [3] the charged states were considered, and in this work the same tool has been applied to calculate the main characteristics of the neutral members of the nonet.
The realistic description of neutral modes is impossible without the use of Lagrangians responsible for breaking the axial U (1) A symmetry and OZI rule.They are well known, so we did not set ourselves the goal of obtaining them on the basis of corresponding multi-quark interactions.
Four-quark interactions generate the kinetic part of the free Lagrangian and make a major contribution to the particle masses.We calculated these one-loop diagrams and demonstrated that the singlet component φ 0 (as well as other two φ 3 and φ 8 ) is a superposition of three eigenvectors φ R a , a = 0, 3, 8 which diagonalize the kinetic part of free Lagrangian in the octet-singlet basis.We have shown that this is a direct result of explicit symmetry breaking in the superconducting model.
Since the singlet field φ 0 acquires a mass term due to the gluon anomaly, the following transition to the eigenstates φ R a induces contributions to the off-diagonal elements ∆µ 2 08 and ∆µ 2 03 of the mass matrix already in the next order of the 1/N c expansion.We have shown that due to this mechanism, the gluon anomaly substantially weakens the effects of explicit chiral symmetry breaking observed in the leading order.
It is interesting to note that this mechanism copes with the problem that Leutwyler pointed out at the time: The value of the π 0 -η mixing angle turns out to be too large if we restrict ourselves to only the LO contribution.This will have a catastrophic effect on the width of the η → 3π decay.We have shown that taking into account the NLO correction eliminates this difficulty.
The mass matrix of neutral states is diagonalized by a rotation parameterized by three mixing angles θ, and .In particular, rotation through the angle θ eliminates the η-η mixing.However, when considering the decay constants of η and η mesons, one angle θ is not enough.Therefore, a picture with two mixing angles is often used.We have shown that NLO corrections effectively lead to a two-angle pattern, but unfortunately could not give its full theoretical justification.
It should be emphasized that the above results would be more complete if calculations of electromagnetic decays of pseudoscalar mesons were carried out.We are currently working on this issue.
and other well-known relation Both of them are valid to first nonleading order.
If in these formulas we put Taking here into account that we arrive to the modified Leutwyler formula [30] sin from which it is possible to determine the value of the difference ϑ 0 − ϑ 8 = 25 • .
The problem with using the formulas (C5) and (C6) is that when they are substituted into the linear relations (20) we are forced to conclude that ϑ 0 = ϑ 8 = θ.
The reason is clear.Both linear (C1) and quadratic (C2) relations, when used, imply the rejection of higherorder terms.This means that the formulas (C5)-(C6) also need to be limited to terms of the required precision.In the main text of the paper, we showed how this can be realized.

Fig. 1
shows three curves corresponding to the values of Λ deviating from the value chosen in our work Λ = 1.1 GeV by 10%.Such behavior is associated with the original quadratic divergence of the integral J 0 (M 0 ), and is typical for any of the regularization schemes commonly used in the NJL model.In particular, one can obtain that M 0 (1.0 GeV) = 19.7 MeV, M 0 (1.1 GeV) = 274 MeV, M 0 (1.2 GeV) = 468 MeV for G S = 6.6 GeV −2 .It shows that M 0 changes a lot with Λ, and it does make sense to consider only 1% (or less) deviations in the value of Λ.This case is shown in Fig.2.It can be seen that the corresponding solutions of the gap equation differ only marginally Λ = 1.10 ± 0.01 GeV, M 0 = 274 ± 20 MeV,

− 7
MeV. (60) Note that a 1% change in the value of Λ leads to a 3% change in the value of the constant F , i.e., the physical constant f π coincides with F within the error.Indeed, our estimates give f π = 92.22+2.54−2.74 .

TABLE II .
In the first row, set (a), we show the leading order result for mixing angles θ, and .Quark masses are given in MeV, θ in degrees, small angles and in radians.The numerical value of λ 2 η (in GeV 2 ) is extracted from the experimental value of m η .Set (b) describes a fit which takes into account the first 1/Nc-correction.In this case the light quark masses are given up to NLO corrections included.The two input parameters λ 2 η and ∆N , are fixed by the phenomenological masses of mη, m η .