# Constituent quark axial current couplings to light vector mesons in the
vacuum and with a weak magnetic field

Fabio L. Braghin
Instituto de Física, Federal University of Goias
Av. Esperança, s/n, 74690-900, Goiânia, GO, Brazil

###### Abstract

Unusual constituent quark axial current couplings to light vector mesons,
$\rho$ and $\omega$, 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, suppressed
nearly by $1/{M^{*}}^{2}$ with respect to the vector mesons minimal coupling
to the quark vector current. Some three leg meson vertices are also presented:
$\pi-\rho-A_{1}$ and $V_{1}V_{2}A$ (where $V_{1},V_{2}$ are vector mesons and
$A$ an axial meson). A vector and axial-vector mesons mixing is identified at
non zero magnetic field which however can contribute only in the presence of a
third particle or in a medium. Numerical results are presented for different
effective gluon propagators.

##  1  Introduction

Pions, neutrinos and eventually the light axial mesons can be considered to
probe the axial content of the nucleon or their corresponding constituent
quarks and of other hadrons. Nucleon’s axial form factor provides a measure of
its spin content and the interplay of strong and weak charges and
interactions. It has been extensively investigated theoretically and
experimentally, for few works [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. The
decomposition of the nucleon, or correspondingly constituent quark, axial
vertex involves eight tensor structures [10, 11]. The physical processes and
particles that can probe them can be attached to each of these structures.
Mesons form factors, however, are far more difficult to be measured. Some
calculations for spacelike or timelike rho-meson form factors can be found,
for example, in [13, 14, 15, 16].At the same time, the understanding of hadron
dynamics depend on different types of interactions. For instance, it becomes
important to understand how these mesons not only couple to the nucleon but
also how interactions among mesons can manifest in experimental situations.
Besides that, the way these couplings might be related to each other might be
of relevance for establishing connections to QCD, eventually as manifestation
of fundamental symmetries. It is expected that eventual breakdowns of
fundamental symmetries should be observed with higher precision experiments.

Low energies (Strong) couplings of the constituent quark (or nucleon) axial
current involve either the pion or the light axial mesons. Among these axial
mesons the $A_{1}$ and the $f_{1}$ are usually considered to be chiral
partners to the $\rho$ and $\omega$ [17, 18]. There are however difficulties
to determine axial mesons properties, in particular because they are highly
unstable. Therefore there are uncertanties about their structures [19, 20]. In
Ref. [20] several couplings and decays of the $A_{1}$ were analyzed with
possible mixing angles. To understand further these mesons it becomes
important to take into account how they interact with other hadrons before
decaying. Light axial mesons couplings and decays, including its decay to the
pion and a vector meson, have also been found to be relevant to describe the
$\tau$-decay by the axial current [21, 22, 23, 24]. It has also been
considered for the $\mu$ (g-2) problem [25] wherein the VVA correlators are
important. Three-leg mesons vertices, many time representing decays, can also
be searched experimentally and this is planned in different facilities as for
example at FAIR/GSI [26]. Eventually, associated information from finite
energy density medium should be compatible with their dynamics in the vacuum.
As an example, the $\rho-\omega-A_{1}$ and $\rho-\pi-A_{1}$ couplings give
rise to the in medium mixing $\rho-A_{1}$ [27, 28, 23, 29]. Eventually this
mixing contributes for the dilepton spectrum in hot and/or dense matter [30,
27]. The role of vector and axial mesons at finite energy density up to the
Chiral phase transition has been investigated in different approaches [31,
32]. Besides that, further meson interactions might lead to modifications in
the nuclear potential [33]. From the strict theoretical point of view, three
leg couplings, such as the $\rho-\pi-A_{1}$ or $\rho-\omega-\pi$, are
suppressed in the $1/N_{c}$ counting. Nevertheless, they have been found to
arise in different approaches such as in the massive Yang Mills gauge theory,
or even as Wess-Zumino-Witten (WZW) terms [34, 35], in [36, 37, 38, 39, 40,
23]. Facilities currently working and being built are expected to probe
several aspects of vector and axial mesons and their interactions. High
density investigations on the restoration of chiral symmetry and meson
spectroscopy are planned in FAIR/GSI, NICA and J-PARC. Vector meson
photoproduction and production mostly in ultraperipheral collisions have been
investigated in HERMES, JLAB, FAIR/GSI and SLAC and currently in LHC besides
plans for LHC and EIC. Polarized vector mesons are currently produced and
investigated in JLAB for example in [41, 42]. Besides that axial meson
production in central collisions has also been envisaged [43].

The constituent quark model (CQM) remains one of the leading pictures for
global properties of hadron structure around which other contributions can be
evaluated, for example in [44, 45, 46]. Several nucleon properties are
described directly, and mostly, in terms of constituent quarks that carry
masses and charges of hadrons and they are directly responsible for nucleon’s
couplings to mesons and other particles. This association might somewhat
manifest in baryons form factors. A more precise relation between this two
levels of description can be helpful to the detailed understanding of the
hadrons structure and dynamics. The lack of understanding of the confinement
mechanism does not prevent a good theoretical description of hadron properties
and interactions. It becomes then interesting to understand further
predictions of these CQM. Global Color (GCM) [47, 48] and Nambu-Jona-Lasinio
(NJL) [49, 50] type models go along the CQM description in terms of
constituent quarks. They have been shown to be suitable for the description of
many aspects of light meson spectrum and properties. In these models,
Dynamical Chiral Symmetry Breakdown (DChSB) leads to the formation of chiral
condensates, directly associated as sea quarks states [51]. DChSB endows
quarks with a large mass providing an important link between fundamental
symmetries at the QCD and the hadron levels. In the CQM, the pion, as a
Goldstone boson, has the axial coupling to (constituent) quarks whose coupling
constant $g_{A}$ is usually given by $g_{A}=3/4$, $g_{A}=5/3$ or $g_{A}=1$
[50, 52, 53]. Light mesons couplings to constituent quarks and form factors
have been derived dynamically and analytically [16, 54] along with the same
spirit of the CQM. By starting with a leading term of QCD quark-effective
action, based in one non-perturbative gluon exchange (GCM), standard
techniques have been applied and they will be considered in the present work.
Background quarks, dressed by a sort of gluon cloud, give rise to constituent
quarks. Mesons interactions with constituent quarks arise and provide, for
example, a pion cloud. This is seen explicitly in the Weinberg’s large Nc
effective field theory (EFT), that copes constituent quark picture with the
large Nc expansion [53]. This EFT has also been derived with symmetry breaking
corrections with the same analytical techniques considered in the present work
in the vacuum and under weak magnetic fields [55, 56]. Furthermore, this
approach might lead to a straightforward extension to finite baryon densities
or finite temperatures. Moreover, the approach considered directly provides
relative strength (ratios) of different coupling constants or form factors.
These ratios lead to reasonable estimates of their relative importance. This
is usually important also for planning experiments or for the interpretation
of results.

Besides the interest in calculating hadron couplings in the vacuum, strong
magnetic fields have been estimated to appear in non central relativistic
heavy ions collisions (r.h.i.c.) and magnetars and they can produce many
different effects [57, 58]. Although these magnetic fields can be as large as
$10^{15}$ T, they are not so large if compared to an hadron mass scale such as
an effective (constituent) quark mass - or even the pion mass. One has
typically $(eB_{0})\sim 10^{-2}{M^{*}}^{2}-10{M^{*}}^{2}$, where $M^{*}$ is a
quark effective mass typical from the CQM. It becomes interesting to identify
changes in the quark and hadron dynamics due to the external magnetic fields.
Under (relatively) weak magnetic fields, pions, vector and axial mesons
develop additional couplings to (constituent) quark currents [59, 60]. These
couplings may have reduced strengths as compared to the couplings to quark
vector and axial currents. At energies in which magnetic fields can be
expected to show up in non-central heavy ions collisions, vector and axial
mesons can also be more copiously produced. Therefore the investigation of the
effects of magnetic fields in the mesons dynamics and couplings becomes of
further interest.

In the present work, light vector mesons (rho and omega) anomalous couplings
to the constituent quark axial current will be derived in the vacuum and under
a constant weak magnetic field by considering the same framework used in [16,
54, 59, 60, 61]. These anomalous couplings can be considered as corrections to
the rho/omega ($A_{1}/f_{1}$) form factors that correspond to a very small
axial (vector) component. As such, this small axial component should be at the
origin of the rho-$A_{1}$ mixing expected to occur in the medium. The same can
be expected for a $\omega-f_{1}$ mixing. They can be seen as Wess-Zumino-
Witten type terms. These coupling functions are considerably smaller than the
known rho vector meson coupling to the nucleon (constituent quark) vector
current. Besides that, anomalous three-leg mesons couplings that might
correspond or contribute for vector and/or axial mesons mixings will also be
presented. The work is organized as follows. In the next section, the main
steps of the approach are briefly reminded and the quark determinant will be
exhibited. A large quark effective mass expansion is performed and the (next
leading order) anomalous terms will be presented. The next leading terms, and
form factors, are presented in sections (2) and (3). These couplings are
shortly compared to leading couplings of the pion, light vector and axial
mesons previously derived. The action term for the vector meson coupling to
the axial current, as a WZW type term, is shown to be topologically conserved.
Numerical results are presented in the section (4) for the low momentum
regime, $Q^{2}<1$ GeV2, by considering three different effective gluon
propagators. On-shell values for some of the coupling constants are provided
and estimations for the axial averaged quadratic radius (a.q.r.) of the vector
mesons are calculated. In the last section there is a summary and final
remarks.

##  2  Quark determinant and next leading anomalous couplings

A QCD-based model for quark-antiquark interaction mediated by a gluon is the
GCM, that takes into account non perturbative/non-Abelian gluon effects by
means of an effective gluon propagator. It is given by:

$Z[\eta,\bar{\eta}]=N\int{\cal
D}[\bar{\psi},\psi]\exp\;i\int_{x}\left[\bar{\psi}\left(i\not{D}-m\right)\psi-\frac{g^{2}}{2}\int_{y}j_{\mu}^{\beta}(x){\tilde{R}}^{\mu\nu}_{\beta\alpha}(x-y)j_{\nu}^{\alpha}(y)+\bar{\psi}\eta+\bar{\eta}\psi\right],$
(1)

Where the color quark current is
$j^{\mu}_{\alpha}=\bar{\psi}\lambda_{\alpha}\gamma^{\mu}\psi$, $\int_{x}$
stands for $\int d^{4}x$, $i,j,k=0,...(N_{f}^{2}-1)$ will be used for,
SU($N_{f}=2$), isospin indices and $\alpha,\beta...=1,...(N_{c}^{2}-1)$ stands
for color in the adjoint representation. The sums in color, flavor and Dirac
indices are implicit and $\eta,\bar{\eta}$ are the quark sources.
$D_{\mu}=\partial_{\mu}-ieQA_{\mu}$ is the covariant quark derivative with the
minimal coupling to a background electromagnetic field, with the diagonal
matrix $\hat{Q}=diag(2/3,-1/3)$ for up and down electromagnetic charges. To
account for the non-Abelian structure of the gluon sector the gluon propagator
$\tilde{R}^{\mu\nu}_{\alpha\beta}(x-y)$ must be non perturbative. As an
external input for the model, it will be required to have enough strength to
yield Dynamical Chiral Symmetry Breaking (DChSB) with a given strength of the
(running) quark-gluon coupling constant. DChSB has been found in several works
with different approaches, for few examples [62, 63, 64]. It is one of the
mechanisms that endows hadrons with large masses with respect to the
(measured) quark masses [65]. Other terms from QCD effective action, such as
genuine three and four quark interactions, due the non Abelian gluon
structure, are not considered. In several gauges the gluon propagator
$\tilde{R}^{\mu\nu}_{\alpha\beta}(k)$ can be written in momentum space as:
$\tilde{R}^{\mu\nu}_{\alpha\beta}(k)=\delta_{\alpha\beta}\left[\left(g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\right)R_{T}(k)+\frac{k^{\mu}k^{\nu}}{k^{2}}R_{L}(k)\right]$,
where $R_{T}(k),R_{L}(k)$ are transversal and longitudinal components.
Contributions from extra terms due to confinement and gauge boson dynamics
proportional to $\delta(p^{2})$ [66] can be shown to be smaller or even
vanishing [67] for the observables calculated below.

By means of a Fierz transformation, Dirac and isospin structures can be
suitable introduced such as to provide the correct quantum numbers for leading
low energies quark-antiquark states corresponding to mesons. The background
field method and the auxiliary field method are then used to calculate the
quark determinant in the presence of DChSB by considering the auxiliary fields
associated with quark-antiquark mesons states. Besides that, background quark
currents give rise to constituent quark currents. In particular, the auxiliary
field method (AFM) was employed explicitly in Refs. [16, 54, 55, 61] for
bilocal auxiliary fields. These auxiliary fields can be expanded in a complete
basis of structureless mesons fields. Field renormalization constants can be
introduced explicitly with which the unit integral of the AFM can be written
as:

$\displaystyle 1$ $\displaystyle=$ $\displaystyle N\int
D[V_{\mu}^{i},\bar{A}_{\mu}^{i},V_{\mu},\bar{A}_{\mu}]e^{-\frac{i\alpha}{4}\int_{x,y}(\bar{R}^{\mu\nu})^{-1}((Z_{V}^{\frac{1}{2}}V^{i}_{\mu}-gZ_{g}Z_{\psi}{\bar{R}_{\mu\nu}}{j^{V,\nu}}^{i})(Z_{V}^{\frac{1}{2}}V_{i}^{\mu}-gZ_{g}Z_{\psi}{\bar{R}_{\mu\nu}}{j_{V,\nu}}^{i})}$
(2) $\displaystyle\times$ $\displaystyle
e^{-\frac{i\alpha}{4}\int_{x,y}(\bar{R}^{\mu\nu})^{-1}(Z_{A}^{\frac{1}{2}}\bar{A}^{i}_{\mu}-gZ_{g}Z_{\psi}{\bar{R}_{\mu\nu}}{j^{A,\nu}}^{i})(Z_{A}^{\frac{1}{2}}\bar{A}_{i}^{\mu}-gZ_{g}Z_{\psi}{\bar{R}_{\mu\nu}}{j_{A,\nu}}^{i})}$
$\displaystyle\times$ $\displaystyle
e^{-\frac{i\alpha}{4}\int_{x,y}(\bar{R}^{\mu\nu})^{-1}((Z_{V}^{\frac{1}{2}}V_{\mu}-gZ_{g}Z_{\psi}{\bar{R}_{\mu\nu}}{j^{V,\nu}})(Z_{V}^{\frac{1}{2}}V^{\mu}-gZ_{g}Z_{\psi}{\bar{R}_{\mu\nu}}{j_{V,\nu}})}$
$\displaystyle\times$ $\displaystyle
e^{-\frac{i\alpha}{4}\int_{x,y}(\bar{R}^{\mu\nu})^{-1}(Z_{A}^{\frac{1}{2}}\bar{A}_{\mu}-gZ_{g}Z_{\psi}{\bar{R}_{\mu\nu}}{j^{A,\nu}})(Z_{A}^{\frac{1}{2}}\bar{A}^{\mu}-gZ_{g}Z_{\psi}{\bar{R}_{\mu\nu}}{j_{A,\nu}})}$
$\displaystyle\times$ $\displaystyle\int
D[S_{i}]D[P_{i}]e^{-\frac{i}{2}\int_{x,y}R(x-y)\alpha\left[(Z_{S}^{\frac{1}{2}}S-gZ_{g}Z_{\psi}j^{S}_{(2)})^{2}+(Z_{P}^{\frac{1}{2}}P_{i}-gZ_{g}Z_{\psi}j^{P}_{i,(2)})^{2}\right]},$

where

$\displaystyle\bar{R}^{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\bar{R}^{\mu\nu}(x-y)=g^{\mu\nu}(R_{T}(x-y)+R_{L}(x-y))+2\frac{\partial^{\mu}\partial^{\nu}}{\partial^{2}}(R_{T}(x-y)-R_{L}(x-y)),$
(3) $\displaystyle R$ $\displaystyle\equiv$ $\displaystyle
R(x-y)=3R_{T}(z-y)+R_{L}(x-y).$ (4)

The scalar and pseudoscalar fields give rise to a non-linear representation in
terms of the (Goldstone boson) pion field by a usual chiral rotation. The
renormalization constants will not be carried on in the calculations below,
being therefore omitted. However, they allow for a systematic elimination of
the ultraviolet divergences in the resulting couplings of the effective
action.

The Gaussian integration of the quark field can now be performed in the
presence of background quark currents and mesons fields that are arranged in a
chiral invariant way. By making use of the identity $\det A=\exp\;Tr\;\ln(A)$,
it yields:

$\displaystyle S_{eff}$ $\displaystyle=$
$\displaystyle-i\;Tr\;\ln\;\left\\{i\left[{S_{0}}^{-1}(x-y)+\Xi_{v}(x-y)+\Xi_{s}(x-y)+\sum_{q}a_{q}\Gamma_{q}j_{q}(x,y)\right]\right\\},$
(5)

where $Tr$ stands for traces of all discrete internal indices and integration
of space-time coordinates. The quantity $\Xi_{v}(x-y)$ encodes the vector and
axial mesons contributions and $\Xi_{s}(x-y)$ encodes the pion field
contribution that arise from the scalar and pseudoscalar auxiliary fields. In
the last term of Eq. (5) there is a sum of background quark currents with
Dirac, flavor and color structures among which only the (isosinglet and
isotriplet) axial currents will be kept:

$\displaystyle\sum_{q}a_{q}\Gamma_{q}j_{q}(x,y)$ $\displaystyle\to$
$\displaystyle-\alpha
g^{2}\bar{R}^{\mu\nu}(x-y)\gamma_{\mu}\left[\sigma_{i}\gamma_{5}\bar{\psi}(y)\gamma_{5}\gamma_{\nu}\sigma_{i}\psi(x)+\gamma_{5}\bar{\psi}(y)\gamma_{5}\gamma_{\nu}\psi(x)\right],$
(6)

where $\sigma_{i}$ are the Pauli matrices,
$\bar{R}^{\mu\nu}=2[3R_{T}(x-y)+R_{L}(x-y)]$ and $\alpha=4/9$. The remaining
terms can be written as:

$\displaystyle\Xi_{v}(x-y)$ $\displaystyle=$
$\displaystyle-\frac{\gamma^{\mu}}{2}\left[F_{v}\sigma_{i}\left(V_{\mu}^{i}(x)+\gamma_{5}\bar{A}_{\mu}^{i}(x)+V_{\mu}(x)+\gamma_{5}\bar{A}_{\mu}(x)\right)\right]\delta(x-y),$
(7) $\displaystyle\Xi_{s}(x-y)$ $\displaystyle=$ $\displaystyle
F(P_{R}U+P_{L}U^{\dagger})\;\delta(x-y),$ (8)

where $U=e^{i\vec{\pi}\cdot\tau/2}$, $P_{R/L}=(1\pm\gamma_{5})/2$ are the
chirality right/left hand projectors, $F_{v}$ and $F$ provide the canonical
field definitions meson fields, for both iso-triplet (rho and A1) and iso-
singlet ($\omega$ and $f_{1}$) [18, 68], and for the pion. $F_{v}$ will be
incorporated into the vector and axial mesons fields to provide their
canonical dimensions. The other quark-antiquark mesons, their auxiliary
fields, will be neglected. The gap equations for the auxiliary fields are
derived as saddle point equations and solved as usually done for the model (1)
and NJL-type models. The scalar field equation will be the only one with non
trivial non zero solution and the resulting quark-antiquark scalar condensate
($\bar{S}$) becomes responsible for the increased quark mass
$M^{*}=m+\bar{S}$. The quark propagator can then be written as:
$S_{0}^{-1}(x-y)=\left(i\not{D}-M^{*}\right)\delta(x-y)$.

A large quark mass expansion [69] with a zero order derivative expansion [70]
of the determinant (5) will be implemented now. It is suitable for the long
wavelength limit of the model and they have been performed in previous works
of the author mentioned above. The leading terms of the expansion with
vector/axial currents and the light mesons have been collected and
investigated previously by the author in the vacuum and under weak magnetic
fields. Next leading couplings of (iso-triplet $\rho$ and iso-singlet
$\omega$) light vector mesons to constituent quark axial current also appear
as non-local interactions. With explicit momentum dependencies they can be
written as:

$\displaystyle{\cal L}_{vja}$ $\displaystyle=$ $\displaystyle
i\delta_{ij}\epsilon^{\sigma\rho\mu\nu}F^{vja}(K,Q)K_{\sigma}{\cal
F}^{i}_{\rho\mu}(Q)j^{A,j}_{\nu}(K,K+Q)$ (9) $\displaystyle+$ $\displaystyle
i\epsilon^{\sigma\rho\mu\nu}F^{vja}(K,Q)K_{\sigma}{\cal
F}_{\rho\mu}(Q)j^{A}_{\nu}(K,K+Q),$

where $Q$ is the vector meson 4-momentum, $K$ is the quark momentum. The iso-
triplet and iso-singlet axial currents were defined as
$j^{A,i}_{\mu}(K,K+Q)=\bar{\psi}(K+Q)\gamma_{\mu}\gamma_{5}\sigma^{i}\psi(K)$
and $j^{A}_{\mu}(K,K+Q)=\bar{\psi}(K+Q)\gamma_{\mu}\gamma_{5}\psi(K)$. The
(Abelian limit of the) stress tensors were defined for the isotriplet and
isosinglet states:

$\displaystyle{\cal F}_{\rho\mu}^{i}(Q)$ $\displaystyle=$ $\displaystyle
Q_{\rho}V_{\mu}^{i}(Q)-Q_{\mu}V_{\rho}^{i}(Q),\;\;\;\;\;\;\;\;{\cal
F}_{\rho\mu}(Q)=Q_{\rho}V_{\mu}(Q)-Q_{\mu}V_{\rho}(Q).$ (10)

The non-Abelian contribution was neglected although it can be incorporated and
it leads to three and four-vector or axial mesons-quark vertices.

The above form factor is given by:

$\displaystyle F_{vja}(K,Q)$ $\displaystyle=$ $\displaystyle
4d_{2}N_{c}\;(\alpha
g^{2})\;\int_{k}\;((\tilde{S}_{0}(k+K)\tilde{S}_{0}(k+K+Q)\bar{\bar{R}}(-k))),$
(11)

where $\int_{k}=\int\frac{d^{4}k}{(2\pi)^{4}}$ in the Euclidean momentum
space, $\bar{\bar{R}}(-k)=2R(-k)$, $d_{n}=\frac{(-1)^{n+1}}{2n}$. The double
parenthesis was used for denoting the order of the original kernels and it
implicitly contains the momentum structure resulting from the trace in Dirac
indices. This integral, for infrared (IR) regular gluon propagators, is
finite. The following function was defined:
$\tilde{S}_{0}(k)=\frac{1}{k^{2}+{M^{*}}^{2}}$. These interactions (9) emerge
due to the following anomalous trace of Dirac matrices:

$\displaystyle
Tr_{D}(\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}\gamma_{5})=4i\epsilon_{\mu\nu\rho\sigma},$
(12)

where $\epsilon^{\rho\mu\sigma\nu}$ is the Levi-Civita tensor. Along this
work, the momentum dependence of the form factors is written as $F(K,Q)$ that
is a shorthanded notation for $F(K^{2},Q^{2},K\cdot Q)$. The momentum
dependent form factors in eq. (9) have dimensions of mass-2. Both vector
mesons, rho and omega, couple to the corresponding axial current with the same
strength $F_{vja}(K,Q)$.

The above interactions (9) have chiral counterparts of the light axial mesons
$A_{1}$ and $f_{1}$, denoted by $\bar{A}_{\mu}^{i}$ and $\bar{A}_{\mu}$
respectively, interacting with the vector quark currents. These couplings are
given by:

$\displaystyle{\cal L}_{vja-A}$ $\displaystyle=$ $\displaystyle
i\epsilon^{\sigma\rho\mu\nu}F^{vja}(K,Q)K_{\sigma}{\cal
G}^{i}_{\rho\mu}(Q)j^{V,i}_{\nu}(K,K+Q)$ (13) $\displaystyle+$ $\displaystyle
i\epsilon^{\sigma\rho\mu\nu}F^{vja}(K,Q)K_{\sigma}{\cal
G}_{\rho\mu}(Q)j^{V}_{\nu}(K,K+Q),$

where $j^{V,i}_{\mu}(K,K+Q)=\bar{\psi}(K+Q)\gamma_{\mu}\sigma^{i}\psi(K)$ and
$j^{V}_{\mu}(K,K+Q)=\bar{\psi}(K+Q)\gamma_{\mu}\psi(K)$. The (isotriplet and
isosinglet) axial mesons (Abelian) stress tensors in coordinate space are the
following:

$\displaystyle{\cal G}_{\mu\nu}^{i}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\bar{A}_{\nu}^{i}-\partial_{\nu}\bar{A}_{\mu}^{i},\;\;\;\;\;\;{\cal
G}_{\mu\nu}=\partial_{\mu}\bar{A}_{\nu}-\partial_{\nu}\bar{A}_{\mu}.$ (14)

Non-Abelian terms can be incorporated [37, 71].

The above anomalous interactions (9) and (13) are strongly anisotropic with
respect to the quark and vector mesons momenta, and the vector meson
polarization. They arise in the same way the Wess-Zumino-Witten type-terms
that were also obtained for example in the bosonized version of the NJL-model,
sigma models and other approaches [36, 39, 47, 40]. The anti-symmetric tensor
$\epsilon_{\sigma\rho\mu\nu}$ obviously prevents the contributions of many
components of quark and vector meson momenta. Moreover, it makes important to
select a particular vector (or axial) meson polarization. As a whole, the
vertices $Q_{\mu}K_{\nu}F_{vja}(K,Q)$ probe an anisotropy of the spatial
variation of the axial (vector) current and charge distribution in eq. (9)
(eq. (13)). These couplings correspond to a small anomalous axial (vector)
component of the vector (axial) meson structure. As such, it may be
responsible for vector-axial mesons mixings. This type of mixing could only
occur in the presence of a third particle or in a medium to conserve momenta.

###  2.1  Comparison with other couplings of the axial current

The leading light mesons (pion, omega, rho and axial $A_{1}(1260)$)
interactions with the axial isotriplet quark current had been obtained by the
same method used above [55, 16, 54]. By adding to the anomalous coupling
above, they can be written as:

$\displaystyle{\cal
L}_{j_{A}}=\left[G_{A}(Q,K)Q_{\mu}\pi^{i}(Q)+G_{\bar{A}}(Q,K)\bar{A}_{\mu}^{i}(Q)+i\;F_{vja}(Q,K)\epsilon_{\mu\nu\rho\sigma}K^{\nu}Q^{\rho}V^{\sigma}_{i}(Q)\right]j_{A,i}^{\mu}(K,Q),$
(15)

where a dimensionless pion field was written in the first term. The analogous
terms for the isosinglet axial current were omitted. Other couplings with
different - higher order - momentum dependence might arise being outside of
the scope of this work. The axial meson coupling to the axial current has been
considered recently in different approaches [72]. The following form factors
were used in eq. (15)

$\displaystyle G_{A}(K,Q)=G_{V}(K,Q)$ $\displaystyle=$ $\displaystyle
4F\;C_{A,V}\;\int_{k}(({M^{*}}\tilde{S}_{0}(k+K)\tilde{S}_{0}(k+Q+K)\bar{\bar{R}}(-k))),$
(16) $\displaystyle G_{\bar{A}}(K,Q)$ $\displaystyle=$
$\displaystyle\;C_{A,V}\;\int_{k}\frac{(k+K)\cdot(k+K+Q)+{M^{*}}^{2}}{((k+K)^{2}+{M^{*}}^{2})((k+K+Q)^{2}+{M^{*}}^{2})}\bar{\bar{R}}(-k),$
(17)

where $C_{A,V}=4N_{c}d_{2}(\alpha g^{2})$ and $F$ is the pion decay constant.
The form factor $F_{vja}(K,Q)$, eq. (11), is suppressed by $\sim 1/M^{*}$ or
$\sim 1/{M^{*}}^{2}$ with respect to $G_{A}$ and $G_{\bar{A}}$ and also with
respect to the rho or omega-coupling to the quark vector current. Besides
that, it has a tensor structure dependent on $KQ$ and it presents an intrinsic
anisotropy in momentum space that contributes to make him smaller. Note that
$G_{\bar{A}}(K,Q)$ (as well as the vector meson coupling) is dimensionless for
the canonical normalization of axial-vector fields.

It is also interesting to note that, for constant quark effective masses
$M^{*}$, the pion axial coupling, $G_{A}(K,Q)$, is directly proportional to
the anomalous coupling constant or form factor $F_{vja}(K,Q)$. They can be
related by:

$\displaystyle\frac{F_{vja}(K,Q)}{G_{A}(K,Q)}=\frac{1}{4M^{*}F}.$ (18)

Therefore the anomalous coupling constant $F_{vja}$ can be considered to be
suppressed with respect to $G_{A}\sim 1$.

###  2.2  Some three-leg mesons couplings

The meson sector of the quark determinant (5) has been investigated
extensively by many groups, for example in [37, 39]. Next, the leading
coupling terms that involve a vector meson and an axial meson are exhibited.
Although not all of them are non-zero in the vacuum they are all displayed and
their possible contributions discussed below. The leading vector-axial mesons
couplings with a pion can be written as:

$\displaystyle{\cal L}_{mix}$ $\displaystyle=$ $\displaystyle
G_{v-a-\pi}(K,Q)\left(i\epsilon_{ijk}\pi_{i}(Q+K)V_{\mu}^{j}(Q)A^{\mu}_{k}(K)+\pi_{i}(Q+K)V_{\mu}^{i}(Q)A^{\mu}(K)\right)$
(19) $\displaystyle+$ $\displaystyle
G_{v-a-\pi}(K,Q)\left(\pi_{i}(Q+K)V_{\mu}(Q)A^{\mu}_{i}(K)\right)$
$\displaystyle+$ $\displaystyle
G_{kv-a-\pi}(K,Q)i\epsilon_{ijk}\left(\partial^{\nu}\pi_{i}(Q+K)Q_{\mu}V_{\nu}^{j}KA^{\mu}_{k}(K)+\partial_{\nu}\pi_{i}(Q+K)V_{\mu}^{i}(Q)K^{\mu}A^{\nu}(K)\right)$
$\displaystyle+$ $\displaystyle
G_{kv-a-\pi}(K,Q)i\epsilon_{ijk}\pi_{i}(Q+K)Q_{\mu}V_{\nu}^{j}(Q)K^{\mu}A^{\nu}_{k}(K)+{\cal
O}_{k}$

where ${\cal O}_{k}$ contains further couplings with higher order momentum
dependence, $Q,K$ are the momenta carried by the vector and axial mesons,
respectively. In this equation the pion field is canonically normalized. The
coupling functions are given by:

$\displaystyle G_{v-a-\pi}(K,Q)$ $\displaystyle=$ $\displaystyle
d_{2}N_{c}6\int_{k}((M^{*}T(K,Q)\tilde{S}_{0}(k)\tilde{S}_{0}(k+K)\tilde{S}_{0}(k+K+Q))),$
(20) $\displaystyle G_{kv-a-\pi}(K,Q)$ $\displaystyle=$ $\displaystyle
d_{2}N_{c}6\int_{k}((M^{*}\tilde{S}_{0}(k)\tilde{S}_{0}(k+K)\tilde{S}_{0}(k+K+Q))),$
(21)

where $T(K,Q)\equiv(2k\cdot(k+K)+K\cdot(K+Q)-{M^{*}}^{2})/2$. Similar
couplings to the $\rho-\pi-A_{1}$ have been proposed in other works within
different or similar approaches, for example in [73, 74, 37, 21, 22, 23, 40,
75] and references therein. The integral of the coupling function
$G_{v-a-\pi}$ is ultraviolet (UV) logarithmic divergent and $G_{kv-a-\pi}$ is
UV finite. They have respectively dimensions of mass and mass-1.

This UV divergence in eq. (20) is the same of the momentum dependent free
vector and axial mesons terms. The free and self interacting light vector and
axial mesons -sector have been investigated in very similar approaches, see
for example in [37, 76, 71]. This divergence is directly eliminated by the
vector and axial fields renormalization constants in the same way some
couplings were made finite in [61]. The Abelian contributions were found to be
given by:

$\displaystyle{\cal L}_{free}=-\frac{g_{f}^{(0)}}{4}\left({\cal
F}^{\mu\nu}_{i}{\cal F}_{\mu\nu}^{i}+{\cal G}^{\mu\nu}_{i}{\cal
G}_{\mu\nu}^{i}+{\cal F}^{\mu\nu}{\cal F}_{\mu\nu}+{\cal G}^{\mu\nu}{\cal
G}_{\mu\nu}\right),$

where the following effective parameter have been defined in the long
wavelength and zero momentum limit considered before:
$g_{f}^{(0)}=d_{1}4N_{c}\;Tr^{\prime}\;((\tilde{S}_{0}^{2}(k)))$. This
parameter can be set finite as a renormalization condition, the one for
$Z_{V}$ and $Z_{A}$, and this makes all the coupling constants and coupling
functions in eq. (19) finite. The mass terms for the vector and axial mesons
can also be found from this approach with complementary contributions in
similar developments [77, 78]. However the masses does not provide important
information for the coupling constants and form factors addressed in the
present work. To obtain terms consistent with the massive Yang Mills approach,
one must impose as renormalization point: $g_{f}^{(0)}=1$. It follows that:

$\displaystyle
G_{v-a-\pi}(0,0)\sim\frac{g_{f}^{(0)}M^{*}}{2}+\frac{1}{e}g_{F\rho\omega},$
(22)

where $g_{F\rho\omega}$ is the (finite) coupling constant of the of the
neutral rho meson coupling to a photon and to the omega meson: ${\cal
L}_{F\rho\omega}=-g_{F\rho\omega}F_{\mu\nu}{\cal F}^{\nu\rho}_{3}{\cal
F}_{\rho}^{\mu}$ for $F_{\mu\nu}$ the background photon stress tensor.

Finally, there are other three-leg couplings that can also be associated to
mixing of $\rho$ and other vector and axial mesons, in particular the
$\rho-\omega-A_{1}$ vertex [27, 79]. Also couplings that yield fusion of
vector mesons into an axial mesons $g_{VVA}$. It can be written as:

$\displaystyle{\cal L}_{3-v}$ $\displaystyle=$ $\displaystyle g_{\omega\rho
A_{1}}(K,Q)2\epsilon^{\mu\nu\alpha\beta}V_{\mu}(K+Q)\;\left({\cal
F}_{\nu\alpha}^{i}(K)A_{\beta}^{i}(Q)+{\cal
G}_{\nu\alpha}^{i}(Q)V_{\beta}^{i}(K)\right)+{\cal O}_{\cal F}$ (23)
$\displaystyle+$ $\displaystyle g_{\omega\omega
f_{1}}(K,Q)2\epsilon^{\mu\nu\alpha\beta}V_{\mu}(K+Q)\;\left({\cal
F}_{\nu\alpha}(K)A_{\beta}(Q)+{\cal G}_{\nu\alpha}(Q)V_{\beta}(K)\right)+{\cal
O}_{\cal F}$ $\displaystyle+$ $\displaystyle g_{\rho\rho
A_{1}}(K,Q)2\epsilon^{\mu\nu\alpha\beta}i\epsilon_{ijk}V_{\mu}^{i}(K+Q)\;\left({\cal
F}_{\nu\alpha}^{j}(K)A_{\beta}^{k}(Q)+{\cal
G}_{\nu\alpha}^{j}(Q)V_{\beta}^{k}(K)\right)+{\cal O}_{\cal F}$

where ${\cal O}_{\cal F}$ contains a different momentum structures. These
other structures are obtained by exchanging the roles of the rho and the
omega, or the $A_{1}$ meson, i.e. ${\cal F}\leftrightarrow{\cal
F}_{i}\leftrightarrow{\cal G}_{i}$ and correspondingly
$V_{\mu}^{i}\leftrightarrow V_{\mu}\leftrightarrow A_{\mu}^{i}$. The coupling
function $g_{\omega\rho A_{1}}$, that is symmetrized due to the possible
different order of external lines, was defined as:

$\displaystyle g_{\omega\rho A_{1}}(K,Q)$ $\displaystyle=$ $\displaystyle
C_{3}\int_{k}((S_{0}(k)S_{0}(k+K)\tilde{S}_{0}(k+K+Q)+\tilde{S}_{0}(k){S}_{0}(k+K){S}_{0}(k+K+Q))),$
(24) $\displaystyle g_{\omega\rho A_{1}}(K,Q)$ $\displaystyle=$ $\displaystyle
g_{\omega\omega f_{1}}(K,Q)=g_{\rho\rho A_{1}}(K,Q),$ (25)

where $C_{3}=6N_{c}d_{2}$, the momentum integral has the same UV divergence as
eq. (20) and they can be renormalized exactly in the same way as discussed
above. The right hand side ratio (25) is also obtained from the naive quark
model relation with a VMD hypothesis [26]. For constant quark effective mass,
it can be written:

$\displaystyle g_{\omega\rho A_{1}}(Q,K)=\frac{1}{M^{*}}G_{v-a-\pi}(Q,K).$
(26)

These coupling constants are also directly proportional to coupling constants
found in the framework of the Skyrme model or massive Yang Mills model [80,
37, 38].

###  2.3  Possible quantization of meson couplings to axial and vector
couplings

The coupling $F_{vja}(K,Q)$ in the action can be written in the coordinate
space and, with the same construction of Witten [35, 81], it can be expressed
as a five dimensional closed surface of a total divergence by means of the
Stoke’s theorem. In this case a quantization condition emerges. By writing it
back in the momentum space, eq. (9) can be written as:

$\displaystyle n\Gamma=-\frac{i}{240\pi^{2}}\int
d^{4}K\;d^{4}Q\;\epsilon_{\sigma\rho\mu\nu}F^{vja}(K,Q)K_{\sigma}{\cal
F}^{i}_{\rho\mu}(Q)j^{A,i}_{\nu}(K,K+Q),$ (27)

where $n$ is an integer. This integral, as a term in the action, corresponds
therefore to a topologically conserved quantity assuming integer multiple
values of $\Gamma$. The axial current however has specific properties. For
instance, the axial charge is only partially conserved due to both quark
masses and the ABJ anomaly. So the following question arise: how can a
topologically conserved coupling involving the axial current be related to the
partially conservation of the axial current? Seemingly the vector meson
anomalous coupling selects a component of $j_{\nu}^{A,i}$ that would be
quantized and conserved. By decomposing the above integral into the different
Lorentz components, i.e.
$n\Gamma=\epsilon^{\sigma\rho\mu\nu}\Gamma_{(\sigma\rho\mu\nu)}$, one can
write explicitly one of these terms as:

$\displaystyle\Gamma_{(xyz0)}$ $\displaystyle=$
$\displaystyle-\frac{i}{240\pi^{2}}\int
d^{4}K\;d^{4}Q\;F^{vja}(K,Q)K_{x}Q_{y}$ $\displaystyle\times$
$\displaystyle\left[\rho^{-}_{z}(Q)\bar{u}(K+Q)\gamma_{0}\gamma_{5}d(K)+\rho^{+}_{z}(Q)\bar{d}(K+Q)\gamma_{0}\gamma_{5}u(K)\right],$
(28)

where $\rho^{\pm}_{z}(Q)$ are the z-polarization component profile of the
charged rho field. The quantization condition however involves the sum of all
the couplings of all the components $\Gamma_{(xyz0)}$.

Besides that, for constant effective mass, these equations can be rewritten in
terms of the pion axial coupling to constituent quarks, eq. (16). From eq.
(18) it can be written that:

$\displaystyle n\Gamma=-\frac{i}{240\pi^{2}\times 4M^{*}F}\int
d^{4}K\;d^{4}Q\;\epsilon^{\sigma\rho\mu\nu}G_{A}(K,Q)K_{\sigma}{\cal
F}^{i}_{\rho\mu}(Q)j^{A,j}_{\nu}(K,K+Q).$ (29)

Note however, that the momenta $K$ and $Q$ are orthogonal to each other,
differently from the pion axial coupling to the constituent quark current.
This eq. (29) has a higher order dependence on the constituent quark momentum
$K_{\sigma}$ but it is of the same order of the meson momentum (pion or vector
meson) $Q_{\rho}$.

By applying the same reasoning to the axial meson coupling to the constituent
quark vector current (13) the following quantization conditions are obtained:

$\displaystyle m_{t}\bar{\Gamma}$ $\displaystyle=$ $\displaystyle\int
d^{4}x_{1}d^{4}x_{2}\;i\epsilon^{\sigma\rho\mu\nu}F^{vja}(x_{1},x_{2}){\cal
G}^{i}_{\rho\mu}(x_{1})\partial_{\sigma}j^{V,i}_{\nu}(x_{2},x_{1}),$ (30)
$\displaystyle m_{s}\tilde{\Gamma}$ $\displaystyle=$ $\displaystyle\int
d^{4}x_{1}d^{4}x_{2}\;i\epsilon^{\sigma\rho\mu\nu}F^{vja}(x_{1},x_{2}){\cal
G}_{\rho\mu}(x_{1})\partial_{\sigma}j^{V}_{\nu}(x_{2},x_{1}),$ (31)

where $m_{s},m_{t}$ are integers for the isosinglet and isotriplet axial
mesons interactions. Notwithstanding the vector current is conserved only for
degenerate quark masses these conditions should not disappear in the case of
non degeneracy of quark masses. A similar question to the one raised above
arises: how can a topologically conserved quantity, involving the constituent
quark vector current, be related to the non conservation of the (global)
vector current that is due to the non degenerate quark masses? Similarly to
the axial current coupling to the vector meson, the axial meson profile and
coupling might "select" a topologically preserved component of the vector
current. The effect of quark mass non-degeneracy will be investigated in
another work.

##  3  Anomalous vector mesons couplings to the quark axial current under weak
magnetic field

The effect of a magnetic field, weak with respect to the constituent quark
mass $M^{*}$, will be presented due to two different mechanisms described in
detail in Ref. [60]. The validity of the (semi)classical approximation for the
magnetic field to describe observables in heavy ion collisions has been object
of attention in the last years [82, 83]. The dependence of the quark
propagator on the weak magnetic field is considered and, secondly, the overall
photon couplings to the legs of the meson-current coupling. For the degenerate
quark effective mass $M^{*}$, the quark propagator, with the leading
contribution from the weak magnetic field aligned in the $z$-direction, can be
written for degenerate quark masses as [84, 85]:

$\displaystyle G(k)$ $\displaystyle=$ $\displaystyle
S_{0}(k)+S_{1}(k)(eB_{0})=\frac{\not{k}+{M^{*}}}{k^{2}-{M^{*}}^{2}+i\epsilon}+i\gamma_{1}\gamma_{2}\frac{(\gamma_{0}k^{0}-\gamma_{3}k^{3}+{M^{*}})}{(k^{2}-{M^{*}}^{2}+i\epsilon)^{2}}\hat{Q}(eB_{0}).$
(32)

The anomalous light vector mesons couplings to the axial current in the
presence of this constant weak magnetic field provide similar expressions in
both mechanisms mentioned above that can be added. Below some of resulting
corrections for the couplings, with the form factors, of the previous section
are shown for the isotriplet and isosinglet vector and axial mesons. By
writing them with a constant small multiplicative factor
$(eB_{0})/{M^{*}}^{2}$, they can be written as:

$\displaystyle{\cal L}_{vjaB}$ $\displaystyle=$
$\displaystyle\frac{(eB_{0})}{{M^{*}}^{2}}\epsilon_{ij3}\frac{F_{vja}^{B}(K,Q)}{{M^{*}}^{2}}\left[\epsilon^{12\rho\mu}K_{\rho}Q_{\nu}\cdot
V_{\nu}^{i}(Q)+2\epsilon_{12\rho\nu}K^{\rho}{\cal
F}^{\mu\nu}_{i}(Q)\right]\;j^{A,j}_{\mu}(K,K+Q)$ (33) $\displaystyle+$
$\displaystyle\frac{(eB_{0})}{{M^{*}}^{2}}\frac{F_{vja}^{B}(K,Q)}{3{M^{*}}^{2}}\left[\epsilon^{12\rho\mu}K_{\rho}Q_{\nu}\cdot
V_{\nu}(Q)+2\epsilon_{12\rho\nu}K^{\rho}{\cal
F}^{\mu\nu}(Q)\right]\;j^{A,3}_{\mu}(K,K+Q),$ $\displaystyle{\cal L}_{mix,B}$
$\displaystyle=$
$\displaystyle\frac{(eB_{0})}{{M^{*}}^{2}}G_{v-a}^{B,1}(K,Q)i\epsilon_{12\mu\nu}{M^{*}}^{2}i\epsilon_{ij3}V^{\mu}_{i}(Q)\bar{A}^{\nu}_{j}(K)\delta(Q+K)$
$\displaystyle+$
$\displaystyle\frac{(eB_{0})}{{M^{*}}^{2}}G_{v-a-\pi}^{B,1}(Q,K)i\epsilon_{12\mu\nu}{M^{*}}\left(\pi_{3}(Q)V^{\mu}(K)\bar{A}^{\nu}(Q+K)+T_{ijk}\pi_{i}(Q)V^{\mu}_{j}(K)\bar{A}^{\nu}_{k}(K+Q)\right),$

where $T_{ijk}=tr_{F}(Q\sigma_{i}\sigma_{j}\sigma_{k})$. The coupling
functions can be written in the Euclidean momentum space as:

$\displaystyle F_{vja}^{B}(K,Q)$ $\displaystyle=$ $\displaystyle
4d_{2}N_{c}(\alpha
g^{2}){M^{*}}^{4}\;\int_{k}\;((\tilde{S}_{0}(k+K)\tilde{S}_{0}(k+K)\tilde{S}_{0}(k+K+Q)\bar{R}(-k))),$
(35) $\displaystyle G_{v-a}^{B,1}(K,Q)\delta(K+Q)$ $\displaystyle=$
$\displaystyle
2d_{2}N_{c}\;\int_{k}(({M^{*}}^{2}\tilde{S}_{0}(k+Q)\tilde{S}_{0}(k+Q)\tilde{S}_{0}(k))),$
(36) $\displaystyle G_{v-a-\pi}^{B,1}(Q,K)$ $\displaystyle=$ $\displaystyle
3d_{2}N_{c}{M^{*}}^{2}$
$\displaystyle\times\;\int_{k}(((k\cdot(k+Q)-{M^{*}}^{2})\tilde{S}_{0}(k+Q)\tilde{S}_{0}(k+Q)\tilde{S}_{0}(k+K+Q)\tilde{S}_{0}(k))),$

where the vector mesons are the canonically normalized ones. These coupling
functions (coupling constants) are all finite and they are dimensionless. The
couplings $F_{vja}^{B}(K,Q)$ might be seen as magnetic field corrections to
the anomalous vector meson form factor.

The mixing $G_{v-a}^{B,1}(K,Q)$ disappears in the absence of magnetic field.
Note that the similar term for isosinglet vector and axial fields does not
show up. This term, however, is trivially zero in the absence of other
particles due to conservation of momentum, and this is made explicit in the
delta function $\delta(K+Q)$. In the presence of another particle, or in a
finite density medium, the conservation of momentum can be satisfied and this
anomalous mixing can contribute because vector and axial mesons propagate in
orthogonal directions. This coupling should contribute for in medium vector-
axial mixing besides the mixing induced by the pion of eq. (19) [28, 29]. All
these interaction terms emerge due to the anomalous trace of five Dirac
matrices, eq. (12). There is a similar momentum anisotropy to the case of zero
magnetic field, being also dependent on the vector or axial mesons
polarization. The magnetic field contributions are suppressed by factors
$1/{M^{*}}$ or $1/{M^{*}}^{2}$ with respect to the zero magnetic field
couplings, besides the factors $(eB_{0})/{M^{*}}^{2}$ were included to.
Although the explicit magnetic field contribution is factorized and suppressed
by a factor $(eB_{0})/{M^{*}}^{2}$, the effective quark masses also depend on
the magnetic field in the gap equations. These anomalous form factors can be
simply added to the anomalous form factors (9,13) and (19) in the vacuum as it
follows:

$\displaystyle F(K,Q;B_{0})=F(K,Q)+f\frac{(eB_{0})}{{M^{*}}^{2}}F^{B}(K,Q),$
(38)

where $f$ is a factor that depends on the momentum components in eq. (33) or
(3). $F_{vja}(K,Q)$ and $F_{vja}^{B}(K,Q)$ also can receive magnetic field
corrections from the gluon propagator and running coupling constants
dependencies on $B_{0}$ being however neglected in the present work. Axial
vector mesons couplings to the quark vector currents, eq. (13), receive
analogous corrections due to the weak magnetic field to those shown above.

##  4  Numerical results

For the numerical estimations two different effective gluon propagators will
be considered. The first one will be a transversal obtained from Schwinger-
Dyson equations (SDE) hat reproduce many hadron observables [9, 62]. It can be
written as:

$\displaystyle D_{I}(k)=h_{I}g^{2}R_{T}(k)$ $\displaystyle=$
$\displaystyle\frac{8\pi^{2}}{\omega^{4}}De^{-k^{2}/\omega^{2}}+\frac{8\pi^{2}\gamma_{m}E(k^{2})}{\ln\left[\tau+(1+k^{2}/\Lambda^{2}_{QCD})^{2}\right]},$
(39)

where $h_{I}$ is the factor that normalizes the coupling constant of the
vector meson to the vector current, $\gamma_{m}=12/(33-2N_{f})$, $N_{f}=4$,
$\Lambda_{QCD}=0.234$GeV, $\tau=e^{2}-1$,
$E(k^{2})=[1-exp(-k^{2}/[4m_{t}^{2}])/k^{2}$, $m_{t}=0.5GeV$,
$D=0.55^{3}/\omega$ (GeV2) and $\omega=0.5$GeV.

The second type is based in a longitudinal effective confining
parameterization by Cornwall [64] that can be written as:

$\displaystyle R_{L}(k)=D_{II,z}(k)$ $\displaystyle=$ $\displaystyle
h_{II,z}\frac{K_{F}}{(k^{2}+M_{z}^{2})^{2}},$ (40)

where $K_{F}=(2\pi M_{z}/3k_{e})^{2}$ was considered in previous works [16,
54]. In this eq., $k_{e}\simeq 0.15$, the normalization factor $h_{II,k}$ is
considered to reproduce a reasonable value for the vector meson (rho or omega)
coupling constant to the nucleon (or constituent quarks). This type of gluon
effective propagator exhibits features of lattice QCD calculations [86]. Two
choices were made for the effective gluon mass $M_{z}^{2}$: as a function of
momentum ($z=6$ and $z=7$) for $M_{6}=0.5/(1+k^{2})$ GeV and
$M_{7}=0.5/(1+k^{2}/5)$ GeV.

In the figures below the following normalized space-like form factors with
respect to the usual vector meson coupling to the vector current will be
drawn:

$\displaystyle
G_{vja}(K,Q)=\frac{F_{vja}(K,Q)}{G_{V}(0,0)},\;\;\;\;\;\;\;G_{vja}^{B}(K,Q)=\frac{F_{vja}^{B}(K,Q)}{G_{V}(0,0)},$
(41)

where the following normalizing parameters where chosen: $h_{I}=1.4$
($M^{*}=0.33$GeV), $h_{I}=1.6$ ($M^{*}=0.45$GeV), and $h_{II,6}=h_{II,7}=16$
($M^{*}=0.33$GeV). These values lead to $G_{V}(0,0)=12$ [16] This way we can
extract relative strengths of the couplings in an uniform way to make
comparisons. Besides that, only one component of $K$ and $Q$ contribute, and,
due to this, we considered (in a spacelike spherical coordinate basis)
$Q=|Q|/4$ and $K\simeq|K|/4$.

In Figure (1) the form factor $G_{vja}(K,Q)$, calculated with the effective
gluon propagator $D_{I}(k)$, is presented as a function of the vector meson
momentum $Q^{2}$ for different values of the modulus of the constituent quark
momentum $K=|K|/4$. Two different values of the quark effective mass, typical
from the NJL-model, are considered $M^{*}=0.33$GeV and $M^{*}=0.45$ GeV. The
momentum dependencies in $K$ and $Q$ are not very different, as it can be seen
in eq. (11). There is a slight increase up to $Q^{2}\sim(0.2GeV)^{2}$ or
$K^{2}\sim(0.2GeV)^{2}$ and then a decrease with both $Q^{2}$ and $K^{2}$. For
the zero or very low quark momenta $K\sim 0$ limit the behavior with $Q^{2}$
is almost monotonically decreasing. A larger quark effective mass leads to a
suppression of the interaction and to a less strong dependence on momenta.
This figure, and the following ones, show that there might have a weak
coupling of the light vector mesons, $\rho$ and/or $\omega$, with the axial
constituent quark current. The relative strength of the coupling
$F_{vja}(K,Q)$ and the usual vector coupling close to $Q\sim K\sim 200-500$MeV
can be estimated to be:

$\displaystyle\left.\frac{F_{vja}(K,Q)\times|K||Q|}{G_{V}(K,Q)}\right|_{Q\sim
K\sim 200-500MeV}\sim 0.1.$ (42)

This coupling function also provides an (anomalous) axial contribution for the
vector meson form factor. Therefore only precision measurements of the vector
meson interactions and form factors would identify such contributions. It is
important to emphasize that $F_{vja}(K,Q)$ is strongly anisotropic, being that
it selects a particular meson polarization. Furthermore, $F_{vja}(K,Q)$ for
larger momenta is suppressed stronger than the vector coupling $G_{V}(K,Q)$.
Its (anisotropic) effects could also be searched in the strength of the vector
meson dominance (VMD) [87]. Besides that, these couplings correspond to
anomalous axial corrections to the rho or omega ($A_{1}$ or $f_{1}$) form
factors for which there are some estimations [88, 89, 16].

Figure 1:  Anomalous form factor $G_{vja}(K,Q)$ for the effective gluon
propagator $D_{I}(k)$ as a function $Q^{2}$ for different values of $K$. Two
different quark effective masses are used $M^{*}=0.33$GeV and $M^{*}=0.45$
GeV.

The same form factor $G_{vja}(K,Q)$ as a function of $Q^{2}$ is exhibited, for
different values of $K=0,200,500$MeV, for the effective gluon propagators
$D_{II,6}(k)$ and $D_{II,7}(k)$ in figure (2) for the quark effective mass
$M^{*}=0.33$GeV. The behavior is very similar to the one found in figure (1)
for the gluon propagator $D_{I}(k)$. The very small difference between the
results from the gluon propagators only shows up for low-intermediary momenta,
and it tends to zero for higher momenta $K>400$MeV and $Q^{2}>1$GeV2.

Figure 2:  Anomalous axial form factors $G_{vja}(K,Q)$ for the two effective
gluon propagators $D_{6}(k)$ and $D_{7}(k)$, and with $M^{*}=0.33$ GeV.

###  4.1  Magnetic field induced correction

In figure (3) the form factor $G_{vja}^{B}(K,Q)$ is drawn for the gluon
propagator $D_{I}(k)$ and two different quark effective masses,
$M^{*}=0.33$GeV and $0.45$GeV, as a function of the vector meson momenta
$Q^{2}$ for the same quark momentum as the previous figures. The effect of
modifying the quark effective mass is much larger at low momenta $Q$ and $K$.
For higher momenta, the difference between the form factors with the different
effective masses tends to become smaller with $Q^{2}$ and with $K^{2}$. Also,
analogously to the zero magnetic field case exhibited above, there is a small
increase with low momenta up to $K\sim 200$MeV or $Q\sim 200$MeV. In figure
(4) the same form factor $G_{vja}^{B}(K,Q)$ is presented for the same values
of $K=0,200,500$MeV, with $M^{*}=0.33$ GeV and for the effective gluon
propagators $D_{II,6}(k)$ and $D_{II,7}(k)$. The behavior with external
momenta is basically the same as the form factor $G_{vja}$ and the difference
between results of the two different gluon propagators shows up in
intermediary momenta. This anomalous constituent quark and vector meson
coupling function could manifest in the low/intermediary vector meson energy
regime in ultra-peripheral collisions.

Figure 3:  Magnetic field induced correction to the anomalous axial form
factors probed by the rho vector meson for the effective gluon propagator
$D_{I}(k)$ and for two different quark effective masses $M^{*}=0.33$GeV and
$M^{*}=0.450$ GeV. Modulus of quark momentum was chosen to be the same of the
previous figures: $K=0,200,500$MeV.  Figure 4:  The same as figure (3) for the
two effective gluon propagators $D_{6}(k)$ and $D_{7}(k)$, with $M^{*}=0.33$
GeV and $K=0,200,500$ MeV.

###  4.2  Averaged quadratic radius and on-shell estimates for coupling
constants

Vector (and axial) mesons couple on shell although constituent quarks cannot
be really said to do so. Below some estimates are provided for coupling
constants with on shell vector mesons. Because of the involved analytic
structure, only gluon effective propagator (40) will be considered for which
we adopted the value $M_{6}=500$ MeV. The momentum integral of the coupling
constant $g_{\omega\rho A_{1}}$ is ultraviolet divergent. Because of that, a
regularized truncated version of the quark propagator was considered [16, 60]:
$S_{0}(k)\to M^{*}/(k^{2}-{M^{*}}^{2})$. It may yield an effective momentum
dependence close to the ones obtained from SDE. With that, the integral
becomes finite and we denote the result by $G_{\omega\rho A_{1}}^{tr}(K,Q)$.
All the singularities were taken into account by considering the average of
the integration over the up and down complex semi-planes. The following ranges
of values were obtained for a range of quark effective masses
$M^{*}=0.33-0.35$ GeV:

$\displaystyle G_{vja}(K_{0}=M^{*},Q_{0}=M_{\rho})$ $\displaystyle=$
$\displaystyle 0.6-0.4\;GeV^{-2},$ (43) $\displaystyle G_{\omega\rho
A_{1}}^{tr}(K_{0}=M_{\rho},Q_{0}=M_{\omega})$ $\displaystyle=$ $\displaystyle
0.2-0.3,\;\;\;\;$ (44) $\displaystyle
G_{kv-a-\pi}(K_{0}=M_{\rho},Q_{0}=M_{A_{1}})$ $\displaystyle=$ $\displaystyle
0.9-1.2\;GeV^{-1},\;\;\;\;$ (45) $\displaystyle
G_{v-a}^{B_{1}}(K_{0}=M_{\rho})$ $\displaystyle=$ $\displaystyle 0.1-0.3,$
(46)

Note that: $G_{v-a-\pi}^{tr}={M^{*}}^{2}G_{kv-a-\pi}$. Also, from eq. (25),
one has for the vector mesons fusion channel: $g_{\omega\rho
A_{1}}(K_{0},Q_{0})\simeq g_{\omega\omega f_{1}}(K_{0},Q_{0})\simeq
g_{\rho\rho A_{1}}(K_{0},Q_{0})$, since the mass difference
$M_{\rho}-M_{\omega}\sim 12$MeV is very small

For the sake of comparison, few values for the coupling $G_{\omega\rho A_{1}}$
or $G_{VA\pi}$ considered in the literature are quoted next. The coupling
constant $G_{\omega\rho A_{1}}$ has been considered for the investigations (of
vector/axial mesons mixing) at finite density [27, 28]. The following values
were used: $g_{\omega\rho A_{1}}<\omega_{0}>=C\simeq 0.1\to 0.3$ GeV, at the
saturation density $\rho_{0}$ where the quantity $<\omega_{0}>$ is the omega
mean field in the medium. From Ref. [20] the following values were considered
for the dimensionless three-leg coupling given in (26): $G_{kv-a-\pi}\sim
M^{*}G_{AVP}\sim 2M^{*}\sim 0.7$.

A simply estimation for the contribution of the form factor $F_{vja}(K,Q)$ to
the rho meson averaged quadratic radius can be also provided It corresponds to
a small axial component. For that, we can define a normalized dimensionless
coupling function (form factor) as: $\bar{G}_{vja}=\bar{K}\bar{Q}G_{vja}(K,Q)$
where $\bar{K},\bar{Q}\sim 200$ MeV are averaged constituent quark and meson
momenta. The usual definition of averaged quadratic radius (a.q.r.) will be
adopted:

$\displaystyle\Delta_{A}<r^{2}_{\rho}>=-6\;\left.\frac{d\bar{G}_{vja}}{dQ^{2}}\right|_{Q=0}.$
(47)

Results are shown in fig. (5) for two gluon propagators $D_{I}$ and $D_{II,6}$
with $M_{6}=500$ MeV. These values can be compared to estimations of the rho
a.q.r.: $<r^{2}_{\rho}>\simeq 0.28-0.56$fm2 [16, 88, 89, 90, 13]. It very
small, i.e. $\sqrt{\Delta_{A}<r^{2}_{\rho}>}$ can be one order of magnitude
smaller than the rho vector meson radius. The contribution of the weak
magnetic field is very small, it is one order of magnitude smaller than the
above contribution, i.e.
$\Delta_{A}^{B}<r_{\rho}^{2}>\sim\frac{\Delta_{A}<r_{\rho}^{2}>}{10}$.

Figure 5:  The axial a.q.r., $r_{A}\equiv\sqrt{\Delta_{A}<r^{2}_{\rho}>}$,
obtained with $D_{I}$ and $D_{II,6}$ (with $M_{6}=500$MeV) as a function of
the quark effective mass.

##  5  Final remarks

Unusual couplings of light vector mesons, rho and omega, to the constituent
quark axial current were derived in this work. Their momentum dependencies
were presented for different effective gluon propagators in the low momentum
regime. The resulting coupling constants and form factors are suppressed at
low momenta by $(K_{\mu}Q_{\nu})/{M^{*}}^{2}$ in comparison with the minimal
vector mesons couplings to the quark vector current [16]. $K_{\mu}$ and
$Q_{\nu}$ are respectively the quark and vector meson momenta, being that the
coupling functions are strongly anisotropic. The momentum dependencies of
$F_{vja}$ on $Q$ and $K$ are basically the same. The couplings are also
dependent on the polarization of the vector meson. Although they go to zero
for very large, infinite, meson momenta, the resulting equations are
suppressed fast with respect to the leading form factors at higher energies.
Besides that, at higher energies, quark effective masses should decrease and
this might, to some extent, invalidate the large quark mass expansion. The aim
of the present work is to provide a dynamical derivation for the low energies
mesons couplings. As such, the resulting equations can be written in terms of
structureless (local) meson fields. When analyzing topological-based equations
from section (2.3) there is a need to integrated over all the meson field
profile, or their momentum dependencies. However, since the structure of
$F_{vja}(K,Q)$ decreases quite strongly with increasing momenta, the overall
high energy contributions from the mesons profiles should be suppressed. This
was not investigated further in the present work.

Numerically we have found $G_{vja}=F_{vja}/G_{V}\sim 0.1-0.3$ GeV-2, i.e. this
coupling might produce small (anisotropic) effects in observables associated
to the vector meson coupling to constituent quarks or to the nucleon. The
corresponding light axial mesons couplings to the constituent quark vector
current were also derived and they have the same strength and overall momentum
dependence. When comparing the behavior of the form factors $F_{vja}(K,Q)$,
calculated with different effective gluon propagators, very small (negligible)
differences were found mostly for momenta 0.15 GeV${}^{2}\leq Q^{2}\leq$ 0.7
GeV2. Modifications in the value of the quark effective masses lead to shifts
in the low momenta region of $F_{vja}(K,Q)$ and to its weak magnetic field
induced correction $F_{vja}^{B}(K,Q)$. These form factors correspond to axial
(vector) components of the vector (axial) mesons. Correspondingly a simple
estimation of the axial averaged quadratic radius for the rho meson was
provided. It is, at most, one order of magnitude smaller than the estimations
for the rho a.q.r. There might appear other consequences. For example, a small
(anisotropic) contribution for the so-called vector meson dominance (VMD) and
related form factors [87].

Other three-leg meson vertices often considered, such as $\rho-A_{1}-\pi$,
$\rho\rho-A_{1}$ and $\omega-\rho-A_{1}$, were also derived being the
estimated values in quite good agreement with other investigations. These
three-meson coupling constants, or coupling functions, might be finite or
logarithmic UV divergent. In this last case, they are renormalized by the same
renormalization constants as the free vector mesons kinetic terms [61].
Finally, mixings between light vector and axial mesons were also found.
However the corresponding anomalous mixing only can provide dynamical
contributions in the presence of a third particle or in a finite density
medium. This is due to the conservation of linear momentum. Incoming (or
outgoing) vector meson and outgoing (or incoming) axial meson must propagate
in orthogonal directions. These couplings and mixings are strongly anisotropic
and dependent on both mesons momenta and polarizations.

Besides that, weak magnetic field contributions for the anomalous couplings
were also presented with their momentum dependencies in the form factors,
$F_{vja}^{B}(K,Q)$. These corrections to the anomalous vector mesons couplings
to the axial currents are reasonably similar to the zero magnetic field case,
$F_{vja}$. These corrections are, however, numerically suppressed by the
factor $eB/{M^{*}}^{2}$ that was assumed to be small. Two different mechanisms
that generate weak magnetic field contributions for hadron couplings were
considered. Firstly he leading correction to the quark propagator - eq. (32) -
and secondly the background photon overall coupling to the quark-vector meson
vertex. Both mechanisms provide similar contributions that add to each other
in eqs. (33) and (3). Vector-axial mesons mixing induced by the magnetic field
was also found.

By following the same method considered by Witten [35] for Wess-Zumino terms,
conditions for the quantization of the anomalous couplings $F_{vja}$ were
found. The total axial current, however, is only partially conserved because
of Lagrangian quark masses and of the ABJ anomaly. The vector current is not
conserved due to the quark mass non degeneracy. So, it is not clear how to
cope these small symmetry breakings, dictated by well known low energy
theorems, with the topologically conserved terms in the action given by eqs.
(27) or (30). They correspond to quantization conditions that contain the
axial or vector currents, or a components of them. One way that, maybe, solve
this apparent contradiction would be to consider that the light vector or
axial mesons select components of, respectively, the axial or vector
constituent quark currents that remain (topologically) preserved. This is not
investigated further in the present work. For degenerate quark masses the
quantization condition for the flavor singlet axial meson coupling to the
vector current, with $m_{s}$, is the same as the flavor triplet one, with
$m_{t}$. The non degeneracy of quark masses does not prevent the quantization
conditions to emerge, such as (27), (30) or (31), it slightly changes its
shape. This case of non degenerate quark mass will be addressed in another
work.

## Acknowledgments

F.L.B. thanks short discussions with G.I. Krein, C.D. Roberts and F.S.
Navarra. The author is member of INCT-FNA, Proc. 464898/2014-5, and he
acknowledges partial support from CNPq-312072/2018-0 and CNPq-421480/2018-1.

## References

  * [1] K.L. Miller et al., Study of the reaction $\nu_{\mu}d\to\mu^{-}pp_{s}$, Phys. Rev. D 26, 537 (1982); T. Kitagaki et al., High-energy quasielastic $\nu_{\mu}n\to\mu^{-}p$ scattering in deuterium, Phys. Rev. D 28, 436 (1983).
  * [2] J-M. Gaillard, G. Sauvage, Hyperon Beta Decays, Ann. Rev. Nucl. Part. Sci. 34, 351 (1984).
  * [3] S. Choi, et al, Axial and Pseudoscalar Nucleon Form Factors from Low Energy Pion Electroproduction, Phys. Rev. Lett. 71, 3927 (1993).
  * [4] G. Bardin et al  Measurement of the ortho para transition rate in the p mu p molecule and deduction of the pseudoscalar coupling constant $g_{p}^{\mu}$, Phys. Lett. 104 B, 320 (1981).
  * [5] V.A. Andreev, et al., (MuCap Collaboration), Measurement of the Muon Capture Rate in Hydrogen Gas and Determination of the Proton’s Pseudoscalar Coupling $g_{P}$, Phys. Rev. Lett. 99, 032002 (2007)
  * [6] E.J. Beise,The Axial Form Factor of the Nucleon, Eur.Phys.J. A24S2, 43 (2005).
  * [7] D. Drechsel, Th. Walcher, Hadron structure at low Q2, Rev.Mod.Phys. 80, 731 (2008).
  * [8] V. Bernard, L. Elouadrihiri, Ulf-G. Meissner, Axial structure of the nucleon, J. Phys. G: Nucl. Part. Phys. 28, R1 (2002).
  * [9] P. Maris, C. D. Roberts, Dyson-Schwinger equations: A Tool for hadron physics, Int. J. Mod. Phys. E 12, 297 (2003). P. Tandy, Hadron physics from the global color model of QCD, Prog.Part.Nucl.Phys. 39, 117 (1997).
  * [10] G. Eichmann, C. S. Fischer, Nucleon axial and pseudoscalar form factors from the covariant Faddeev equation, Eur. Phys. J. A 48, 9 (2012).
  * [11] J.S. Ball, T.W. Chiu, Analytic properties of the vertex function in gauge theories. I Phys. Rev. D22, 2542 (1980).
  * [12] Chen Chen, C. S.Fischer, C. D.Roberts, J. Segovia Form factors of the nucleon axial current, Phys. Lett. B815, 136150 (2021).
  * [13] H.L.L. Roberts, et al, $\pi$ and $\rho$ mesons, and their diquark partners, from a contact interaction, Phys. Rev. C83, 065206 (2011).
  * [14] M.E. Carrillo-Serrano, W. Bentz, I. Cloet, A.W. Thomas, $\rho$ meson form factors in a confining Nambu-Jona-Lasinio model, Phys. Rev. C92, 015212 (2015). E.P. Biernat, W. Schweiger, Electromagnetic $\rho$-meson form factors in point-form relativistic quantum mechanics, Phys. Rev. C89, 055205 (2014).
  * [15] V.Yu. Haurysh, V.V. Andreev, $\rho$-Meson Form-factors in Point form of Poincaré-Invariant Quantum Mechanics, Few-Body Syst. 62, 29 (2021). J.P.B.C. de Melo, T. Frederico, Covariant and light-front approaches to the $\rho$-meson electromagnetic form factors, Phys. Rev. C55, 2043 (1997)
  * [16] F.L. Braghin, Light vector and axial mesons effective couplings to constituent quarks, Phys. Rev. D 97, 054025 (2018); Phys. Rev. D 101, 039901(E) (2020).
  * [17] D. Parganlija, P. Kovacs, G. Wolf, F. Giacosa, D.H. Rischke, Meson vacuum phenomenology in a three-flavor linear sigma model with (axial-)vector mesons. Phys. Rev. D 87(1), 014011 (2013).
  * [18] P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 083C, 01 (2020). K. Nakamura et al (Particle Data Group),
  * [19] K. Chen, C.-Q. Pang, X. Liu, T. Matsuki, Light axial vector mesons, Phys. Rev. D91, 074025 (2015).
  * [20] M.-C. Du, Q. Zhao, Comprehensive study of light axial vector mesons with the presence of triangle singularity, Phys. Rev. D104, 036008 (2021).
  * [21] D. Gomez Dumm, P. Roig, A. Pich, J. Portoles, Hadron structure in $\tau\to KK\pi\nu_{\tau}$ decays, Phys. Rev. D81, 034031 (2010).
  * [22] D. Gomez Dumm, P. Roig, A. Pich, J. Portoles, $\tau\to\pi\pi\pi\nu_{\tau}$ decays and the a1(1260) off-shell width revisited Physics Letters B 685, 158 (2010) .
  * [23] M. Wagner, S. Leupold, Information on the structure of the $a_{1}$ from $\tau$-decay, Phys. Rev. D78, 053001 (2018).
  * [24] M. Mikhasenko et al, Pole position of the A1(1260) from tau decay, Phys. Rev. D98, 096021 (2018).
  * [25] L. Cappiello, et al, Axial-vector and pseudoscalar mesons in the hadronic light-by-light contribution to the muon $(g-2)$, Phys. Rev. D102, 016009 (2020). and references therein.
  * [26] P. Lebiedowicz, Otto Nachtmann, P. Salabura, A. Szczurek, Exclusive $f_{1}(1285)$ meson production for energy ranges available at the GSI-FAIR with HADES and PANDA, Phys. Rev. D104, 034031 (2021).
  * [27] C. Sasaki, Signatures of chiral symmetry restoration in dilepton production, Phys. Lett. B 801, 135172 (2020).
  * [28] E. Marco, R. Hofmann, W. Weise, Note on finite temperature sum rules for vector and axial-vector spectral functions, Phys. Lett. B 530, 88 (2002). M. Urban, M. Buballa, J. Wambach, Temperature Dependence of $\rho$ and $a_{1}$-Meson Masses and Mixing of Vector and Axial-Vector Correlators, Phys. Rev. Lett. 88, 042002 (2002).
  * [29] M. Harada, C. Sasaki, W. Weise, Vector–axial-vector mixing from a chiral effective field theory at finite temperature, Phys. Rev. D78, 114003 (2008). M. Harada, C. Sasaki, Novel spectral broadening from vector–axial-vector mixing in dense matter Phys. Rev. C80, 054912 (2009).
  * [30] M. Harada, C. Sasaki, Effect of vector–axial-vector mixing to dilepton spectrum in hot and/or dense matter, arXiv:1003.0331v1 [hep-ph].
  * [31] R.A. Tripolt, et al, Vector and axial-vector mesons in nuclear matter, Phys. Rev. D104, 054005 (2021).
  * [32] Peter Kovacs, Zsolt Szep, Gyorgy Wolf, Existence of the critical endpoint in the vector meson extended linear sigma model Phys. Rev. D 93, 114014 (2016).
  * [33] For example in: G. Chanfray, J. Magueron, Contribution of the $\rho$ meson and quark substructure to the nuclear spin-orbit potential Phys. Rev. C102, 024331 (2020).
  * [34] S.L. Adler, W.A. Bardeen, Absence of Higher-order Corrections in the Anomalous Axial-Vector Divergence equation, Phys. Rev. 182, 1517 (1969). W.A. Bardeen, Anomalous Ward Identities in Spinor Field Theories, Phys. Rev. 184, 1848 (1969).
  * [35] E. Witten, Global aspects of Current Algebra, Nucl. Phys. B223, 422 (1983).
  * [36] H. Gomm, O. Kaymakcalan, J. Schechter, Anomalous spin-1-meson decays from the gauged Wess-Zumino term, Phys. Rev. D30, 2345 (1984).
  * [37] U.-G. Meissner, Low-energy hadron physics from effective chiral Lagrangians, Phys. Rept. 161, 213 (1988).
  * [38] P.M. Hohler, R. Rapp, Realistic implementation of massive Yang-Mills theory for $\rho$ and $a_{1}$ mesons, Phys. Rev. D89, 125013 (2014).
  * [39] D. Ebert, H. Reinhardt, Effective chiral hadron lagrangian with anomalies and skyrme terms from quark flavour dynamics, Nucl. Phys. B 271, 188 (1986).
  * [40] P. Ko, S. Rudaz, Phenomenology of scalar and vector mesons in the linear $\sigma$ model, Phys. Rev. D50, (1994).
  * [41] V. Mathieu, et al, Vector meson photoproduction with a linearly polarized beam, Phys. Rev D97, 094003 (2018).
  * [42] P. Roy, PhD dissertation, Measurement of polarization observables in vector meson photonproduction using a transversely polarized frozen-spin target at CLAS, JLAB, Florida State University, (2016).
  * [43] P. Lebiedowicz, et al, Central exclusive diffractive production of axial-vector $f_{1}(1285)$ and $f_{1}(1420)$ mesons in proton-proton collisions, Phys. Rev. D102, 114003 (2020).
  * [44] T. J. Hobbs, M. Alberg, G. A. Miller, A Euclidean bridge to the relativistic constituent quark model, Phys. Rev. C 95, 035205 (2017)
  * [45] Chen Chen et al, Nucleon axial-vector and pseudoscalar form factors, and PCAC relations, arXiv:2103.02054
  * [46] J.-B. Wang, The axial charges of proton within an extended chiral constituent quark model, arXiv:2106.00866v1 [hep-ph].
  * [47] C.D. Roberts, R.T. Cahill, J. Praschifka, The effective action for the Goldstone modes in a global colour symmetry model of QCD, Ann. of Phys. 188, 20 (1988).
  * [48] P.C. Tandy, Hadron Physics from the Global Color Model of QCD, Prog.Part.Nucl.Phys. 39, 117 (1997).
  * [49] S. P. Klevansky The Nambu-Jona-Lasinio model of quantum chromodynamics, Rev. Mod. Phys. 64, 649 (1992).
  * [50] U. Vogl, W. Weise, The Nambu and Jona-Lasinio model: Its implications for Hadrons and Nuclei, Progr. in Part. and Nucl. Phys. 27, 195 (1991).
  * [51] S.J. Brodsky, C.D. Roberts, R. Schrock, P.C. Tandy, New perspectives on the quark condensate, Phys. Rev. C82, 022201 (2010).
  * [52] S. Weinberg, Why do quarks behave like bare Dirac particles?, Phys. Rev. Lett. 65, 1181 (1990). S. Weinberg, Axial vector coupling of the quark, Phys. Rev. Lett. 67, 3473 (1991).
  * [53] S. Weinberg, Pions in Large N Quantum Chromodynamics, Phys. Rev. Lett. 105, 261601 (2010).
  * [54] F.L. Braghin, Pion Constituent Quark Couplings strong form factors: A dynamical approach, Phys. Rev. D 99, 014001 (2019).
  * [55] F.L. Braghin, Quark and pion effective couplings from polarization effects, Eur. Phys. Journ. A 52, 134 (2016).
  * [56] F.L. Braghin, Low energy constituent quark and pion effective couplings in a weak external magnetic field, Eur. Phys. J. A54, 45 (2018).
  * [57] V.A. Miransky, I.A. Shovkovy, Quantum field theory in a magnetic field: from quantum chromodynamics to graphene and Dirac semimetals, Phys. Rep. 576, 1 (2015)
  * [58] J.O. Andersen, W.R. Naylor. A. Tranberg, Phase diagram of QCD in a magnetic field: a review, Rev. Mod. Phys. 88, 025001 (2016).
  * [59] F.L. Braghin, Constituent quark-light vector mesons effective couplings in a weak background magnetic field, Phys. Rev. D 97, 014022 (2018); Phys. Rev. D 101, 039902(E) (2020)
  * [60] W.F. de Sousa, F.L.Braghin, Form factors for pions couplings to constituent quarks under weak magnetic field, Journ. of Phys. G 47, 045110 (2020).
  * [61] F.L. Braghin, Weak magnetic field corrections to light vector or axial mesons mixings and vector meson dominance, J. Phys. G: Nucl. Part. Phys. 47, 115102 (2020).
  * [62] D. Binosi, L. Chang, J. Papavassiliou, C.D. Roberts, Bridging a gap between continuum-QCD and ab initio predictions of hadron observables, Phys. Lett. B 742, 183 (2015) and references therein.
  * [63] K.-I. Kondo, Abelian-projected effective gauge theory of QCD with asymptotic freedom and quark confinement Phys. Rev. D 57, 7467 (1998) .
  * [64] J. M. Cornwall, Entropy, confinement, and chiral symmetry breaking, Phys. Rev. D 83, 076001 (2011).
  * [65] X.-D. Ji, QCD Analysis of the Mass Structure of the Nucleon, Phys. Rev. Lett. 74, 1071 (1995), arXiv:hep-ph/9410274. W. Kou, R. Wang, X. Chen, Trace Anomaly of Proton Mass with Vector Meson Near-Thresholds Photoproduction Data,. arXiv:2103.10017v2 [hep-ph]. The ideas of EHM: for example in C.D. Roberts, S.M. Schmidt, Reflections upon the emergence of hadronic mass, Eur. Phys. J. Special Topics 229, 3319 (2020).
  * [66] P. Lowdon, Nonperturbative structure of the photon and gluon propagators, Phys. Rev. D96, 065013 (2017).
  * [67] F.L. Braghin, Flavor-dependent U(3) Nambu–Jona-Lasinio coupling constant, Phys. Rev. D103, 094028 (2021).
  * [68] D. Parganlija, P. Kovacs, G. Wolf, F. Giacosa, and D. H. Rischke, Meson vacuum phenomenology in a three-flavor linear sigma model with (axial-)vector mesons, Phys. Rev. D 87, 014011 (2013).
  * [69] A. Paulo Jr., F.L. Braghin, Vacuum polarization corrections to low energy quark effective couplings, Phys. Rev. D90, 014049 (2014).
  * [70] M. Gaillard, The Effective One Loop Lagrangian With Derivative Couplings, Nucl. Phys. B 268, 669 (1986). L.-H. Chan, Derivative Expansion for the One Loop Effective Actions With Internal Symmetry, Phys. Rev. Lett. 57, 1199 (1986). O. Cheyette, Effective Action for the Standard Model With Large Higgs Mass, Nucl. Phys. B 297, 183 (1988).
  * [71] C. A. Ballon Bayona et al, Form factors of vector and axial-vector mesons in holographic D4-D8 model, Journal of High Energy Physics 2010, 52 (2010).
  * [72] Y. Unal, A. Kuçukarslan, S. Scherer, Contribution of the a1 meson to the axial nucleon-to-delta transition form factors, Phys. Rev. 90, 014012 (2019).
  * [73] G. Ecker, J. Gasser, J. Leutwyler, A. Pich, E. de Rafael, Chiral Lagrangians for massive spin 1 fields, Phys. Lett. B223, 425 (1989).
  * [74] L. Roca, J.E. Palomar, E. Oset, Decay of axial-vector mesons into VP and P$\gamma$, Phys. Rev. D 70, 094006 (2004).
  * [75] M.C. Birse, Effective chiral Lagrangians for spin-1 mesons, Z. Phys. A 355, 231–246 (1996).
  * [76] P. Maris, P.C. Tandy, Bethe-Salpeter study of vector meson masses and decay constants, Phys. Rev. C 60, 055214 (1999).
  * [77] G. Kovács, P. Kovács, Z. Szép One-loop constituent quark contributions to the vector and axial-vector meson curvature mass Phys. Rev. D104, 056013 (2021).
  * [78] J. Kim, S.H. Lee, Vector meson mass in the chiral symmetry restored vacuum, Phys. Rev. D103, 103, L051501 (2021)
  * [79] N. Kaiser, U.-G. Meissner, Generalized hidden symmetry for low-energy hadron physics, Nucl. Phys. A519, 671 (1990).
  * [80] I. Zahed, G.E. Brown, The Skyrme model, Phys. Rept. 142, 1 (1986). Balachandran et al, A.P. Balachandran, V.P. Nair, S.C. Rajeev and A. Stern, Phys. Rev. Lett. 49 (1982) 1124; Phys. Rev. D27 (1983) 1153
  * [81] S. Weinberg, The Quantum Theory of Fields Vol. II Cambridge University Press, Cambridge, (1996).
  * [82] I. Danhoni, F. S. Navarra, Magnetic field in relativistic heavy ion collisions: testing the classical approximation, Phys. Rev. C 103, 024902 (2021).
  * [83] I. Danhoni, F. S. Navarra, Magnetic excitation in relativistic heavy ion collisions, Phys. Lett. B 805, 135463 (2020).
  * [84] T.-K. Chyi et al, Weak-field expansion for processes in a homogeneous background magnetic field, Phys. Rev. D 62 105014 (2000).
  * [85] E.V. Gorbar, V.A. Miransky, I.A. Shovkovy, X. Wang, Radiative corrections to chiral separation effect in QED, Phys. Rev. D 88 025025 (2013).
  * [86] P. Costa, O. Oliveira, P. J. Silva, What does low energy physics tell us about the zero momentum gluon propagator, Phys. Lett. B 695, 454 (2011).
  * [87] See for example: S. Pacetti, R. B. Ferroli, E. Tomasi-Gustafsson, Proton electromagnetic form factors: Basic notions, present achievements and future perspectives, Phys. Rep. 550-551, 1 (2015), and references therein.
  * [88] A. Ballon-Bayona, G. Krein, and C. Miller, Phys. Rev. D 96, 014017 (2017).
  * [89] A. F. Krutov, R. G. Polezhaev, and V. E. Troitsky, Phys. Rev. D 93, 036007 (2016) and references therein.
  * [90] M.S. Bhagwat, P. Maris, Vector meson form factors and their quark-mass dependence, Phys. Rev. C77, 025203 (2008).