The presence of baryonic matter leads to the mixing between transverse $ρ$ and $a_1$ mesons through a set of $ωρa_1$-type interactions, which results in the modification to the dispersion relation. We show that a clear enhancement of the vector spectral function appears below $\sqrt{s}=m_ρ$ for small three-momenta of the $ρ$ meson, and thus the vector spectrum exhibits broadening. We also discuss its relevance to dilepton measurements.
In this write-up we summarize main result of our recent analysis on the mixing between transverse $ρ$ and $a_1$ mesons through a set of $ωρa_1$-type interactions in dense baryonic matter. In the analysis, we showed that a clear enhancement of the vector spectral function appears below $\sqrt{s}=m_ρ$ for small three-momenta of the $ρ$ meson, and thus the vector spectrum exhibits broadening.
We discuss the importance of the axialvector meson in chiral phase transition at finite temperature. We show that there exists a significant contribution in the vector spectral function from the axialvector meson through the vector-axialvector mixing in hot matter.
We study the manifestations of chiral symmetry restoration which have a significance for the parity mixing. Restricting to pions and nucleons we establish a formalism for the expression of the vector correlator, which displays the mixing of the axial correlator into the vector one and unifies the cases of the heat bath and the dense medium. We give examples of mixing cross-sections. We also establish a link between the energy integrated mixing cross-sections and the pion scalar density which governs the quenching factors of coupling constants, such as the pion decay one, as well as the quark c…
The effect of mixing of the vector and axial vector correlation functions in the nuclear medium arising from the interaction of nuclear pions with corresponding interpolating currents is considered. It is shown that the mass difference between $ρ$ and $a_1$ meson gets smaller with the increase of the nuclear density reflecting the phenomena of partial restoration of chiral symmetry whereas the absolute values of meson masses may both decrease and increase in nuclear medium depending on the model used for the phenomenological spectral density.
We study the vector-axialvector mixing in a hot medium and its evolution toward the chiral phase transition using different symmetry restoration scenarios based on the generalized hidden local symmetry framework. We show that the presence of the $a_1$ meson reduces the vector spectral function around $ρ$ meson mass and enhances it around $a_1$ meson mass. The coupling strength of $a_1$ to $ρ$ and $π$ vanishes at the critical temperature due to the degenerate $ρ$-$a_1$ masses. This feature holds rigorously in the chiral limit and still stays intact to good approximation for the physical pion ma…
We study the decays of the lightest axial-vector and pseudovector mesons, interpreted as quark-antiquark states, into a vector and a pseudoscalar meson. We show that the quarkonium assignment delivers a good description of the decays and allows also to make further testable predictions. In the kaonic sector, the physical resonances $K_{1}(1270)$ and $K_{1}(1400)$ emerge as mixed objects of an axial vector state $K_{1,A}$ and a pseudovector state $K_{1,B}$. We determine the mixing angle as $\left\vert θ_{K}\right\vert=\left(33.6\pm4.3\right) ^{\circ}$, a value compatible with previous studies b…
We present a systematic study of the vector--axial-vector mixing (V-A mixing) in the current correlation functions and its evolution with temperature within an effective field theory. The $a_1$-$ρ$-$π$ coupling vanishes at the critical temperature $T_c$ and thus the V-A mixing also vanishes. A remarkable observation is that even for finite $m_π$ the $ρ$ and $a_1$ meson masses are almost degenerate at $T_c$. The vanishing V-A mixing at $T_c$ stays approximately intact.
Weak magnetic field induced corrections to effective coupling constants describing light vector mesons mixings and vector meson dominance (VMD) are derived. The magnetic field must be weak with respect to an effective quark mass $M^*$ such that: $eB_0/{M^*}^2< 1$ or $eB_0/{M^*}^2<<1$.For that, a flavor SU(2) quark-quark interaction due to non perturbative one gluon exchange is considered. By means of methods usually applied to the Nambu Jona Lasinio (NJL) and Global Color Models (GCM), leading light vector/axial mesons couplings to a background electromagnetic field are derived. The correspond…
We investigate what can be learnt about the $η$ -$η'$ mixing by means of dispersive representation of axial anomaly. We show that our method leads to the strong bounds for the $η-η'$ mixing angle: $θ= -15.3^o \pm 1^o$. Moreover, our result manifests also a dramatic dependence of the width $Γ_{η\to 2γ}$ on the mixing angle $θ$. This property explains how the relatively small mixing strongly effects the decay width.
The new BES measurement on the two-body decays of J/psi and psi' into an axial-vector meson and a pseudoscalar meson is analyzed with the axial-K mixing including the one-photon annihilation contribution. A somewhat puzzling pattern of the K_1^+ K^- decay channels can be understood with no tight constraint on the mixing angle. The branching fractions of the K_1^0 K-bar^0 channels will be the cleanest source of information to determine the mixing angle from the 1^+ 0^- decays of J/psi and psi'.
We derive the expressions of the vector and axial current from a chiral Lagrangian restricted to nucleons and pions. They display mixing terms between the axial and vector currents. We study the modifications in the nuclear medium of the coupling constants of the axial current, namely the pion decay constant and the nucleonic axial one due to the requirements of chiral symmetry. We express the renormalizations in terms of the local scalar pion density. The latter also governs the quark condensate evolu- tion and we discuss the link between this evolution and the renormaliza- tions. In the case…
It is shown that the QCD anomaly may lead to an abnormal mixing behavior of the axial vector mesons similar to the pseudoscalar mesons. These mixing effects, involving a gluonic axial vector state, generate a non-vanishing strange quark component in the nucleon. They reduce the matrix element of the singlet axial vector in comparison to the value obtained in a naive quark model. The results are in agreement with the data obtained in the polarized lepton-nucleon scattering experiments.
In a nonrelativistic constituent quark model we find a constraint on the mixing angle of the strange axial-vector mesons, $35^o\stackrel{<}{\sim } θ_K\stackrel{<}{\sim }55^o,$ determined solely by two parameters: the mass difference of the $a_1$ and $b_1$ mesons and the ratio of the constituent quark masses.
The key assumption used recently by Feldmann, Kroll and Stich [Phys.Rev. D58, 114006 (1998)] that the decay constants f_η, and f_η' of the respective eta and eta ' mesons in the quark flavor basis follow the pattern of strange and non--strange quarkonia mixing in their wave functions, is reproduced in identifying the non-isotriplet part of the strong neutral axial current with the genuine axial hypercharge current J_{μ, 5}^Y =\bar q γ_μγ_5 Y/2 q, where Y=C+S+B is defined by the Gell-Mann-Nakano-Nishijima relation as the sum of charm (C), strangeness (S), and baryon (B) quark quantum numbers. T…
In this work the analysis of mixing parameters of the system involving eta, eta' mesons and some third massive state G is carried out. We use the generalized mixing scheme with three angles. The framework of the dispersive approach to Abelian axial anomaly of isoscalar non-singlet current and the analysis of experimental data of charmonium radiative decays ratio allow us to get a number of quite strict constraints for the mixing parameters. The analysis shows that the equal values of axial current coupling constants f_8 and f_0 are preferable which may be considered as a manifestation of SU(3)…
In this write-up we summarize main results of our recent analyses on the mixing between transverse rho and a1 mesons in hot and/or dense matter. We show that the axial-vector meson contributes significantly to the vector spectral function in hot matter through the mixing. In dense baryonic matter, we include a mixing through a set of omega rho a1 -type interactions. We show that a clear enhancement of the vector spectral function appears below \sqrt{s}=m_ρfor small three-momenta of the rho meson, and thus the vector spectrum exhibits broadening.
The axial anomaly is responsible for the masses and mixing of the mesons $η$ and $η'$. An open question is if (and to what extent) it affects also other hadrons. We show that anomalous terms can be important to understand the spectroscopy of the pseudotensor mesons $η_2 (1645)$ and $η_2 (1870)$. In fact, pseudotensor mesons belong to a so-called heterochiral multiplet, for which a quadratic mixing term between nonstrange and strange isoscalar members arises. On the contrary, for so-called homochiral multiplets, such as the ground-state (axial-)vector and tensor mesons, this mixing is not possi…
We study mixing between conventional and hybrid mesons in vector and axial vector charmonium using QCD Laplace sum-rules. We compute meson-hybrid cross correlators within the operator product expansion, taking into account condensate contributions up to and including those of dimension-six as well as composite operator renormalization-induced diagrams. Using measured masses of charmonium-like states as input, we probe known resonances for nonzero coupling to both conventional and hybrid meson currents, a signal for meson-hybrid mixing.
It is shown that the diagonalization of the axial-vector -- pseudoscalar transitions in the effective meson Lagrangian in presence of electromagnetic interactions leads to a deviation from the vector meson dominance picture which usually arises in the Nambu - Jona-Lasinio model. The essential features of such modification of the theory are studied. Some important examples are considered in detail.
We suggest a rigorous generalization of the pseudoscalar mesons mixing description in SU(3) basis. It is shown that the appearance of extra massive state nicknamed in the paper as glueball is unavoidable in any scheme with more than one angle. In this framework we develop the dispersive approach to Abelian axial anomaly of isoscalar non-singlet current. Combining it with the analysis of experimental data of charmonium radiative decays ratio we get the number of quite strict constraints for mixing parameters. Our analysis favors the equal values of axial currents coupling constants which may be…
We discuss pure $q\bar{q}$ axial-vector mesons in the Tamm-Dancoff approximation of the Coulomb-gauge QCD model from NCSU. While recent studies have put emphasis in configuration mixing with open meson-meson channels, we here concentrate on the simpler closed-channel problem and follow the $1^+$ mixing through a wide range of quark masses. We also examine their radial excitations and discuss with them the concept of insensitivity to chiral symmetry breaking.
The effects of SU(3) symmetry breaking and configuration mixing have been investigated for the weak vector and axial-vector form factors in the chiral constituent quark model (χCQM) for the strangeness changing as well as strangeness conserving semi-leptonic octet baryon decays in the nonperturbative regime. The results are in good agreement with existing experimental data and also show improvement over other phenomenological models.
The relation for transition form factors of eta and eta' mesons is obtained by combining the exact nonperturbative QCD sum rule, following from the dispersive representation of axial anomaly, and quark-hadron duality. It is valid at all virtual photon momenta and allows one to express the transition form factors entirely in terms of meson decay constants. This relation is in a good agreement with experimental data.
We construct the spectral functions for light vector mesons at finite density and temperature in the presence of a novel mixing between parity partners, induced by baryon density via the Wess-Zumino-Witten action. As the main origin of in-medium broadening, a set of baryon resonances that strongly couple to the vector mesons and the modifications of kaons and anti-kaons due to the Kaplan-Nelson term are included. It is shown that the vector spectra, even with the broadening effects, exhibit sizable signatures of chiral symmetry restoration thanks to the chiral mixing depending on three-momenta…
Interplay between axial anomaly and quark-hadron duality in the presence of strong mixing is considered. The anomaly sum rule for meson transition form factors based on the dispersive representation of axial anomaly and quark-hadron duality in octet channel is analyzed. The comparison of this sum rule to the experimental data on $η$ and $η'$ mesons transition form factors shows that the interval of duality in this channel is rather small, contradicting the usual understanding of quark-hadron duality. The same values of interval of duality are supported by considering the two-point correlator i…
The implications of Chiral Quark Model with SU(3) and axial U(1) symmetry breakings as well as configuration mixing generated by one gluon exchange forces ($χ$QM$_{gcm}$) are discussed in context of proton flavor and spin structure as well as hyperon $β$-decay data. Apart from reproducing the success of $χ$QM with symmetry breaking, it is able to improve upon the agreement with data in several cases such as, $G_A/G_V, Δ_8, <A_1^p>$ dependent on spin polarization functions and ($\frac{2 \bar s}{u+d}$), ($\frac{2 \bar s}{\bar u+\bar d}$) and f_s involving the quark distribution functions.
We propose a new eta-eta' mixing scheme where we start from the quark flavor basis and assume that the decay constants in that basis follow the pattern of particle state mixing. On exploiting the divergences of the axial vector currents - which embody the axial vector anomaly - all basic parameters are fixed to first order of flavor symmetry breaking. That approach naturally leads to a mass matrix, quadratic in the masses, with specified elements. We also test our mixing scheme against experiment and determine corrections to the first order values of the basic parameters from phenomenology. Fi…
The masses of the $K_1(^3P_1)$ and $K_1(^1P_1)$ are considered in a nonrelativistic constituent quark model, and the absolute value of the $K_1(^3P_1)-K_1(^1P_1)$ mixing angle is determined to be about $59.29^\circ$. Comparison of the theoretical predictions on the strong decay widths of the $K_1(1270)$ and $K_1(1400)$ in the $^3P_0$ decay model as well as the production ratio of these two states in the $τ$ decay between the available experimental data strongly favors that the $K_1(^3P_1)-K_1(^1P_1)$ mixing angle is about $+59.29^\circ$.
Extending previous work we study the constraints of QCD sum rules on mass and width of light vector and axial-vector mesons in vacuum and in a medium with finite nuclear density. For the latter case especially the effect of nuclear pions leading to vector-axial-vector mixing is included in the analysis.
Three-quark nucleon interpolating fields in QCD have well-defined SU_L(2) x SU_R(2) and U_A(1) chiral transformation properties. Mixing of the [(1,1/2) + (1/2,1)] chiral multiplet with one of [(1/2,0) + (0,1/2)] or [(0,1/2) + (1/2,0)] representation can be used to fit the isovector axial coupling g_A(1) and thus predict the isoscalar axial coupling g_A(0) of the nucleon, in reasonable agreement with experiment. We also use a chiral meson-baryon interaction to calculate the masses and one-pion-interaction terms of J=1/2 baryons belonging to the [(0,1/2) + (1/2,0)] and [(1,1/2) + (1/2,1)] chiral…
We study a chiral quark model of $π$, $σ$, $ρ$, and $a_1$ mesons in the large-$N_c$ limit. We show that the quadratic $a_1-π$ mixing can be set to zero at zero momentum, thus protecting the $g_{σππ}$ and $g_{ρππ}$ couplings from being contaminated by the corresponding vertices with one or both pions replaced by the axial-vector meson. We further require the chiral-quark Lagrangian to feature an approximate classical scale invariance, the latter being only broken by dimension-two mass terms, and that the longitudinal vector meson scattering amplitudes grow at most like $s$, where $\sqrt{s}$ is …
It is shown, that the correlators of vector and axial-vector currents at finite temperature T in order T^2 reduce to mixing of vacuum VV and AA correlators with universal mixing coefficient given by the parameter ε= T^2/6 F^2_π, unlike the claim in the recent paper by Mallik and Sarkar [1].
We propose to measure the $τ^-\to K_1^-ν_τ\to (K^-ω) ν_τ\to (K^- π^+π^-π^0)ν_τ$ decay in order to determine the $K_1$ axial vector mixing angle $θ_{K_1}$. We derive, for the first time, the differential decay rate formula for this decay mode. Using the obtained result, we perform a sensitivity study for the Belle (II) experiment. We will show that the $K^-π^+π^-π^0$ spectrum of the $τ^-\to K_1^-ν_τ\to (K^-ω) ν_τ\to (K^- π^+π^-π^0)ν_τ$ decay can discriminate the two solutions $θ_{K_1}=\sim 30^{\circ}$ or $\sim 60^{\circ}$ observed in the other measurements.
If dark matter couples predominantly to the axial-vector currents with heavy quarks, the leading contribution to dark-matter scattering on nuclei is either due to one-loop weak corrections or due to the heavy-quark axial charges of the nucleons. We calculate the effects of Higgs and weak gauge-boson exchanges for dark matter coupling to heavy-quark axial-vector currents in an effective theory below the weak scale. By explicit computation, we show that the leading-logarithmic QCD corrections are important, and thus resum them to all orders using the renormalization group.
Various phenomenological studies indicate that the mixing angle $θ_{K_1}$ of $K_{1A}$ and $K_{1B}$, the strange partners of the axial-vector mesons $a_1(1260)$ and $b_1(1235)$, respectively, lies in the vicinity of $35^\circ$ or $55^\circ$, but whether this angle is larger or smaller than $45^\circ$ still remains controversial. When the $f_1(1285)$-$f_1(1420)$ mixing angle $θ_{^3P_1}$ and the $h_1(1170)$-$h_1(1380)$ mixing angle $θ_{^1P_1}$ are determined from the mass relations, they depend on the masses of $K_{1A}$ and $K_{1B}$, which in turn depend on the mixing angle $θ_{K_1}$. We show tha…
Analyses of various experimental measurements all indicate that the mixing angle $θ_{K_1}$ of $K_1(1270)$ and $K_1(1400)$ is in the vicinity of $33^\circ$ or $57^\circ$. However, whether $θ_{K_1}$ is greater or less than $45^\circ$ is still quite controversial. For example, there were two very recent studies of the strong decays of $K_1$ mesons. One group claimed that $θ_{K_1}\approx 60^\circ$, while the other group obtained $θ_{K_1}=(33.6\pm4.3)^\circ$. Since the determination of the mixing angles $α_{^3\!P_1}$ and $α_{^1\!P_1}$ with the former (latter) being the mixing angle of $f_1(1285)$ (…
Based on schematic two-level models extended to $a_1$-meson degrees of freedom, we investigate possible mechanisms of chiral restoration in the vector/axialvector channels in cold nuclear matter. In the first part of this article we employ the massive Yang-Mills framework to construct an effective chiral Lagrangian based on low-energy mesonic modes at finite density. The latter are identified through nuclear collective excitations of `meson'-sobar type such as $π\leftrightarrow [Δ(1232)N^{-1}]\equiv\hatπ$, $ρ\leftrightarrow [N^* (1520)N^{-1}]\equiv\hatρ$, etc.. In a mean-field type treatment t…
The weak vector and axial-vector form factors have been investigated for the semi-leptonic octet baryon decays in the chiral constituent quark model with configuration mixing (\chiCQM_{config}). The effects of SU(3) symmetry breaking and configuration mixing have also been investigated and the results are not only in good agreement with existing experimental data but also show improvement over other phenomenological models.
Eta-eta' mixing is discussed in the quark-flavor basis with the hypothesis that the decay constants follow the pattern of particle state mixing. On exploiting the divergences of the axial vector currents - which embody the axial vector anomaly - all mixing parameters are fixed to first order of flavor symmetry breaking. An alternative set of parameters is obtained from a phenomenological analysis. We also discuss mixing in the octet-singlet basis and show how the relevant mixing parameters are related to those in the quark-flavor basis. The dependence of the mixing parameters on the strength o…
The rates and spectra of the anomalous $η\toπ^+π^-γ$ and $η' \toπ^+π^-γ$ decays are calculated. The approach is based on the effective meson Lagrangian obtained in the Nambu-Jona-Lasinio model with vector and axial-vector mesons by integrating out quark fields. The resulting action is affected by mixing between members of pseudoscalar $J^{PC}=0^{-+}$ and axial-vector $1^{++}$ nonets that violates some low-energy theorems. In this note we point out that a gauge covariant procedure to diagonalize this mixing allows for consistent description of the $η\toπ^+π^-γ$ and $η' \toπ^+π^-γ$ decays.
We report on recent work relating the medium effects observed in dilepton spectra in heavy-ion collisions to potential signals of chiral symmetry restoration. The key connection remains the approach to spectral function degeneracy between the vector-isovector channel with its chiral partner, the axialvector-isovector channel. Several approaches are discussed to elaborate this connection, namely QCD and Weinberg sum rules with input for chiral order parameters from lattice QCD, and chiral hadronic theory to directly evaluate the medium effects of the axialvector channel and the pertinent pion d…
In this article, we separate the vector and axialvector components of the tensor diquark operators explicitly, construct the axialvector-axialvector type and vector-vector type scalar tetraquark currents and scalar-tensor type tensor tetraquark current to study the scalar, vector and axialvector tetraquark states with the QCD sum rules in a consistent way. The present calculations do not favor assigning the $Z_c(4100)$ to be a scalar or vector tetraquark state. If the $Z_c(4100)$ is a scalar tetraquark state without mixing effects, it should have a mass about $3.9\,\rm{GeV}$ or $4.0\,\rm{GeV}$…
It is well known that because of the axial anomaly in QCD, mesons with $J^P = 0^-$ are close to $SU(3)_{\mathrm{V}}$ eigenstates: the $eta^\prime(958)$ meson is largely a singlet, and the $η$ meson an octet. In contrast, states with $J^P = 1^-$ are flavor diagonal: \textit{e.g.}, the $ϕ(1020)$ is almost pure $\bar{s} s$. Using effective Lagrangians, we show how this generalizes to states with higher spin, assuming that they can be classified according to the unbroken chiral symmetry of $G_{\mathrm{fl}} = SU(3)_{\mathrm{L}} \times SU(3)_{\mathrm{R}}$. We construct effective Lagrangians from ter…
Unusual constituent quark axial current couplings to light vector mesons, $ρ$ and $ω$, are derived in the vacuum and under weak magnetic field by considering a quark-antiquark interaction mediated by a non perturbative gluon exchange. Similarly, light axial mesons are found to couple anomalously with the constituent quark vector current. These interactions are of the type of the Wess-Zumino-Witten terms, being strongly anisotropic and dependent on the vector (or axial) meson polarization. They also provide axial (vector) form factors for the vector (axial) mesons and are quite small, suppresse…
The implications of the $f_1(1285)-f_1(1420)$ mixing for the $K_1(^3P_1)-K_1(^1P_1)$ mixing angle is investigated. Based on the $f_1(1285)-f_1(1420)$ mixing angle $\sim 50^\circ$ suggested from the analysis for a substantial body of data concerning the $f_1(1420)$ and $f_1(1285)$, the masses of the $K_1(^3P_1)$ and $K_1(^1P_1)$ are determined to be $\sim 1307.35\pm 0.63$ MeV and $1370.03\pm 9.69$ MeV, respectively, which therefore suggests that the $K_1(^3P_1)-K_1(^1P_1)$ mixing angle is about $\pm (59.55\pm 2.81)^\circ$. Also, it is found that the mass of the $h^\prime_1(^1P_1)$ (mostly of $s…
It is found that in presence of electroweak interactions the gauge covariant diagonalization of the axial-vector -- pseudoscalar mixing in the effective meson Lagrangian leads to a deviation from the vector meson and the axial-vector meson dominance of the entire hadronic electroweak current. The essential features of such a modification of the theory are investigated in the framework of the extended Nambu-Jona-Lasinio model with explicit breaking of chiral $U(2) \times U(2)$ symmetry. The Schwinger-DeWitt method is used as a major tool in our study of the real part of the relevant effective a…
Low energy effective couplings of baryons' constituent quarks to light vector and axial mesons are derived by considering quark polarization for a dressed one gluon exchange quark interaction. The quark field is splitted into two components, one for background constituent quarks and another one for quark-antiquark states, light mesons and the scalar chiral condensate. By considering a large quark effective mass derivative expansion, several effective coupling constants are resolved as functions of the original model parameters and of components of the quark and gluon propagators. Besides the l…
Using a recently proposed gauge covariant diagonalization of $πa_1$-mixing we show that the low energy theorem $F^π=e f_π^2 F^{3π}$ of current algebra, relating the anomalous form factor $F_{γ\to π^+π^0π^-} = F^{3π}$ and the anomalous neutral pion form factor $F_{π^0 \to γγ}=F^π$, is fulfilled in the framework of the Nambu-Jona-Lasinio (NJL) model, solving a long standing problem encountered in the extension including vector and axial-vector mesons. At the heart of the solution is the presence of a $γπ{\bar q} q $ vertex which is absent in the conventional treatment of diagonalization and lead…
A quarkonium-gluonium mixing scheme previously developed to describe the characteristic of the pseudoscalar mesons is applied to axial and tensor mesons. The parameters of the model are determined by fitting the eigenvalues of a mass matrix. The corresponding eigenvectors give the proportion of light quarks, strange quarks and glueball in each meson. However the predictions of the model for branching ratios and electromagnetic decays are incompatible with the experimental results. These results suggest the absence of gluonic components in the states of axial and tensor isosinglet mesons analyz…
The singlet pseudoscalar sector of light mesons is investigated in the soft-wall model, a bottom-up approach to the AdS/QCD correspondence. The $η^\prime$ mass results from the mixing among the fields dual to the axial current, pseudoscalar current and the $G\tilde G$ operator. The topological susceptibility is computed for any quark mass, and the Witten-Veneziano relation is obtained in the large $N_c$ limit.
Utilizing in-medium vector spectral functions which describe dilepton data in ultra-relativistic heavy-ion collisions, we conduct a comprehensive evaluation of QCD and Weinberg sum rules at finite temperature. The starting point is our recent study in vacuum, where the sum rules have been quantitatively satisfied using phenomenological axial-/vector spectral functions which describe hadronic tau-decay data. In the medium, the temperature dependence of condensates and chiral order parameters is taken from thermal lattice QCD where available, and otherwise estimated from a hadron resonance gas. …
We are calculated the expectation value of the axial-vector current induced by the vacuum polarization effect of the Dirac field in constant external electromagnetic field. In calculations we use Schwinger's proper time method. The effective Lagrangian has very simple Lorenz invariant form. Along with the anomaly term, it also contains two Lorenz invariant terms. The result is compared with our previous calculation of the photon - Z boson mixing in the magnetic field.
