#  $1/N_{c}$ approximation and universality of vector mesons

M. K. Volkov1<EMAIL_ADDRESS>A. A. Osipov1<EMAIL_ADDRESS>A. A.
Pivovarov1<EMAIL_ADDRESS>K. Nurlan1,2,3<EMAIL_ADDRESS>1Bogoliubov
Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,
Dubna, 141980, Russia  2 Institute of Nuclear Physics, 1 Ibragimova, 050032,
Almaty, Kazakhstan  3L. N. Gumilyov Eurasian National University, 2 Satpaev,
010000, Nur-Sultan, Kazakhstan

###### Abstract

In this work, we argue that the observed differences in the value of the
vector coupling constant extracted from the decays $\rho\to\pi\pi$
($g_{\rho}=6.0$), $\rho\to l^{+}l^{-}$ ($g_{\rho}=5.0$) and $\omega\to
l^{+}l^{-}$ ($g_{\rho}=5.7$), where $l=e,\mu$, are an indication of the
important role played by the $1/N_{c}$ corrections in the description of these
processes. We show that an emission of a photon by charged meson loops in the
$\rho^{0},\omega,\phi\to\gamma$ transitions is a key process that allows to
describe above vector meson decays into two leptons with a single value
$g_{\rho}=6.0$. Our result supports the idea of universality of neutral vector
mesons and clarifies the role of accounting of $1/N_{c}$ corrections to its
fulfilment.

##  I Introduction

It is well-known that the hypothesis of vector-meson dominance (VMD)
Sakurai:69 ; Feinman:72  possesses one to compute the lepton-pair decay rate
of the neutral vector meson $V=\rho^{0},\omega,\phi$ in terms of the vector-
meson coupling constant $f_{V}$ appearing in a current-field identity
$J_{\mu}^{em}=\sum_{V}\frac{m^{2}_{V}}{f_{V}}V_{\mu}$. This yields

$\Gamma_{V\to
l^{+}l^{-}}=\frac{4\pi\alpha^{2}}{3f_{V}^{2}}\left(1+\frac{2m_{l}^{2}}{m_{V}^{2}}\right)\sqrt{m_{V}^{2}-4m_{l}^{2}},$
(1)

where $\alpha=e^{2}/4\pi=1/137$. Current experimental data on $V\to
l^{+}l^{-}$ decay widths PDG:20  allow to determine from (1) the
phenomenological values of these constants. For instance, considering the
$V\to e^{+}e^{-}$ mode one finds $f_{\rho}\equiv
g_{\rho}=4.96,\,f_{\omega}=17.06,\,f_{\phi}=-13.44$. On the other hand, the
complete $\rho$ dominance of the electromagnetic form factor of $\pi^{\pm}$
implies $g_{\rho}=g_{\rho\pi\pi}$ Sakurai:68 , where the coupling constant of
the $\rho\to\pi\pi$ decay is known to be $g_{\rho\pi\pi}=5.96$, as it follows
from theoretical decay width of the $\rho$-meson

$\Gamma_{\rho\to\pi\pi}=\frac{g^{2}_{\rho\pi\pi}}{48\pi}\,m_{\rho}\left(1-\frac{4m^{2}_{\pi}}{m^{2}_{\rho}}\right)^{\frac{3}{2}},$
(2)

and the phenomenological value $\Gamma_{\rho\to\pi\pi}^{exp}=149.1\pm 0.8$
MeV. We see that describing the strong decay $\rho\to\pi\pi$, one obtains the
value $g_{\rho}\simeq 6.0$. At the same time, when describing electromagnetic
decays $\rho\to e^{+}e^{-}$, one obtains $g_{\rho}\simeq 5.0$, and when
describing a related process $\omega\to e^{+}e^{-}$, one finds
$g_{\rho}=f_{\omega}/3\simeq 5.7$. What is the reason of such differences in
extracted values of $g_{\rho}$?

Our expectation is based on the Sakurai idea of vector meson universality,
which suggests to consider the vector mesons as a gauge bosons of a local
isospin symmetry Sakurai:60  with universal coupling $g_{\rho}$ of these gauge
boson to conserved currents. In accord with this idea, kinetic terms and
couplings of the rho-mesons with matter fields have the Yang-Mills form,
except a non-zero mass of vector mesons. This idea was extensively studied at
the tree-level order in the framework of the ”massive Yang-Mills”
Gasiorowicz:69 ; Schechter:84a ; Schechter:84b ; Schechter:85  and ”hidden-
gauge” theories Bando:88 . There are also reviews on the subject Meissner:88 ;
Birse:96 .

We believe that the discrepancies between the predictions of VMD model and
experiment should rather be attributed to its approximate nature than to the
idea of universality that lies behind it. In particular, we argue that the
above problem can be solved by going beyond the tree-level approximation,
namely, by considering the one-loop meson diagrams accounting for $1/N_{c}$
corrections to a tree-level result. Thus, we assume (and it will be argued in
the following) that the $1/N_{c}$ expansion Hooft:74a ; Hooft:74b ; Witten:79
, where $N_{c}$ is a number of colors, is a relevant approximation to the
issue.

Indeed, it is known that a tree-level $V\to\gamma\to\l^{+}l^{-}$ decay
amplitude, hadronic part of which is described by the matrix element $\langle
0|J|V\rangle$ of the vector quark current $J_{\mu}=\bar{q}\gamma_{\mu}q$, is
of order $\sqrt{N}_{c}$ Witten:79 . The one-loop meson corrections are
suppressed by a factor $1/N_{c}$. So, the $V\to\pi^{+}\pi^{-}\to\gamma\to
l^{+}l^{-}$ amplitude is of order $1/\sqrt{N}_{c}$. This is precisely the
order of the tree-level diagram describing the $\rho\to\pi\pi$ decay. Thus,
the tree-level $\rho\to\pi\pi$ amplitude has the same weight as the $\rho\to
l^{+}l^{-}$ amplitude obtained with allowance for meson loops. From that we
conclude that the coupling constant $f_{\rho}$ extracted from formula (1)
should not be directly compared with $g_{\rho\pi\pi}$. Before such comparison
one should take into account that $f_{\rho}$ contains both the tree-level and
one-meson-loop contributions. Only after separating them one can judge if the
hypothesis of universality is fulfilled.

Further, from the set of all $1/\sqrt{N}_{c}$ order one-loop amplitudes
contributing to $V\to l^{+}l^{-}$ decay one may exclude diagrams of the vector
meson self-energy type. These diagrams were considered in Weise:96  with the
conclusion that the coupling constant $f_{V}$ remains unaltered. The argument
is as follows. Expanding the self-energy contribution
$\Pi_{\mu\nu}(p^{2})=g_{\mu\nu}\Pi(p^{2})+\mbox{longitudinal part}$ around the
physical mass of vector state $m_{V}$

$\Pi(p^{2})=\Pi(m_{V}^{2})+(p^{2}-m_{V}^{2})\frac{\partial\Pi}{\partial
p^{2}}|_{p^{2}=m_{V}^{2}}+\ldots\,,$ (3)

where $p$ is the four-momentum carried by the vector meson, one can require
that $\frac{\partial}{\partial p^{2}}\mbox{Re}\Pi|_{p^{2}=m_{V}^{2}}=0$. Then,
the first term $\Pi(m_{V}^{2})$ will determine the physical mass of the vector
state through the equation
$m_{V}^{2}=\mbox{\r{m}}_{V}^{2}+\mbox{Re}\Pi(m_{V}^{2})$, where
$\mbox{\r{m}}_{V}$ is a ”bare” mass of the vector meson, which it would have
if one-meson-loop contributions to its self-energy were turned off. Thus,
assuming that the meson masses correspond to their empirical values, we
implicitly take into account all self-energy contributions. In the following
we assume this. Since we deal with the on-mass-shell vector mesons, terms of
higher degrees in $(p^{2}-m_{V}^{2})$ in (3) vanish, as does the entire
longitudinal part of the self-energy diagram. As a consequence, only the one-
loop diagrams of two types remain, namely $V\to\pi^{+}\pi^{-}\to\gamma$ and
$V\to K^{+}K^{-}\to\gamma$.

Neglecting the imaginary part of the self-energy vector meson diagrams we
neglect the relatively large width of the $\rho$ meson. The finite width
effects are relevant for the electromagnetic form factors of mesons, and are
not important for the rho-meson decays. This point has been addressed in
Sakurai:68 , where the authors, assuming the universality condition
$g_{\rho}=g_{\rho\pi\pi}$, derived the electromagnetic form factor of the pion
with the finite width effect of the $\rho$-meson. Its application to an
optimal description of the unified BABAR-BESIII data Lees:12 ; Ablikin:16  at
the elastic region leads to the value of $\Gamma_{\rho}=(126.51\pm
0.13)\,\mbox{MeV}$ Bartos:17  that differs by as much as 15% from experimental
data on the $\rho$-width. This indicates that the finite width is not only
effect which should be taken into account. Indeed, as it has been recently
reported Achasov:11 , the $\pi^{+}\pi^{-}$ and $K\bar{K}$ loops yield a
consistent ground to a description of the latest experimental data on the
production of the $\pi^{+}\pi^{-}$ pair in $e^{+}e^{-}$ annihilation at
$\sqrt{s}<1\,\mbox{GeV}$. Notice that when considering the pion form factor,
meson loops correct the extracted value of the $\rho$ width to its
phenomenological value. As we will show below, in the case of $V\to
l^{+}l^{-}$ decays, meson loops $V\to\Phi\Phi\to\gamma$ lead to the correct
description of two-lepton decay modes with the original value of the coupling
constant $g_{\rho}=6.0$.

The importance of taking one-loop meson diagrams into account when describing
the electromagnetic form factor of the pion was also noted in Weise:96 .
Importantly, the dominant role of the kaon loop in the electromagnetic decays
of scalar mesons $f_{0}(980)\to\omega(\rho)\gamma$ has been proved by
calculations made in Kuraev:09 . It should be stressed that in spite of the
great work done on the study of the pion form factor, the role of meson loops
in the $V\to l^{+}l^{-}$ decays has not yet been addressed.

Our consideration is based on the effective Lagrangian of the extended
$SU(3)\times SU(3)$ chiral symmetric Nambu-Jona-Lasinio (NJL) model Nambu:61a
; Nambu:61b ; Volkov:86 ; Ebert:82 ; Ebert:83 ; Volkov:84 ; Volkov:86 ;
Ebert:86 ; Osipov:92 ; Volkov:06 , where the $\rho\to\pi\pi$ decay coupling
constant $g_{\rho\pi\pi}=g_{\rho}=6.0$ is one of the main input parameters of
the theory. This symmetry is spontaneously and explicitly broken to its
isospin $SU(2)$ subgroup. The electromagnetic interactions are chosen to have
a gauge invariant form

${\cal L}_{\gamma
V}=\frac{e}{2}F^{\mu\nu}\sum_{V}\frac{V_{\mu\nu}}{f_{V}}+e\mathcal{A}^{\mu}J^{(mes)}_{\mu},$
(4)

where $V_{\mu\nu}=\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}$,
$F_{\mu\nu}=\partial_{\mu}\mathcal{A}_{\nu}-\partial_{\nu}\mathcal{A}_{\mu}$,
$\mathcal{A}_{\mu}$ is an electromagnetic field, and $J^{(mes)}_{\mu}$ is
electromagnetic current of mesons (see eq.(44) for details). Otherwise, the
theory will not have the direct $\pi\pi\gamma$ or $KK\gamma$ couplings. This
form proves to be significantly better than the standard VMD form without
derivatives in the description of the electromagnetic form factor of the pion
Weise:96 .

In the case of an exact $SU(3)$ flavor symmetry,
$m_{\rho}=m_{\omega}=m_{\phi}$, there is only one independent coupling
constant $f_{V}$, e.g., $g_{\rho}=6.0$. Two other constants can be expressed
through $g_{\rho}$ as it follows
$f_{\omega}=3g_{\rho}=18.0,f_{\phi}=-\frac{3}{\sqrt{2}}g_{\rho}=-12.7$. Using
a phenomenological value of $m_{\rho}=775.26\pm 0.25\,\mbox{MeV}$, one finds
from (1)

$\displaystyle\Gamma_{\rho\to e^{+}e^{-}}$ $\displaystyle=$ $\displaystyle
4.8\,\mbox{keV},\quad(7.04\pm 0.06\,\mbox{keV}),$ (5)
$\displaystyle\Gamma_{\omega\to e^{+}e^{-}}$ $\displaystyle=$ $\displaystyle
0.53\,\mbox{keV},\quad(0.60\pm 0.02\,\mbox{keV}),$ (6)
$\displaystyle\Gamma_{\phi\to e^{+}e^{-}}$ $\displaystyle=$ $\displaystyle
1.07\,\mbox{keV},\quad(1.26\pm 0.01\,\mbox{keV}),$ (7)

where in the parentheses the corresponding experimental data are shown PDG:20
. These simple estimates show qualitatively how the $SU(3)$ symmetry, VMD and
universality work in the absence of the one-loop contributions.

The material of the paper is distributed as follows. In Section II, we discuss
the relationship between dynamic symmetries and universality. In Section III,
we consider how universality arises in chiral theories. The important role of
the $1/N_{c}$ expansion for realistic theories is emphasized. These two
sections are introductory and help to understand the essence of further
calculations. In Section IV, the contribution of one-loop quark diagrams is
calculated, and, in Section V, calculations of one-loop meson diagrams are
presented. Here the main results of our calculations are contained. The
Section VI summarizes the results.

##  II Dynamic symmetries and universality

The idea of universality is associated with dynamic symmetries. These are
specified either by local gauge groups or by nonlinear and inhomogeneous
realizations of algebraic symmetries. As is known, dynamic symmetries not only
determine the form of the Lagrangian, but also yield the low-energy theorems
for massless (or very light) bosons. These theorems impose restrictions on the
response of physical systems to slowly varying gauge fields and, as a result,
determine the universality of the interaction in the sense that the first
orders of the expansion in the coupling constant begin to coincide with the
first orders of the expansion in powers of energy. This makes it possible,
regardless of the value of the coupling constant, to use the corresponding
effective field theory to obtain reasonable results in the low-energy region.
It is important that any theory having the same low-energy spectrum of
particles will have the same answer for the leading corrections independent of
what the high-energy completion might turn out to be.

To illustrate above statements let us give the classic example from the
quantum electrodynamics (QED) Euler:36 . In QED, at very low energies of
photons $\omega\ll m_{e}$, where $m_{e}$ is the electron mass, one can
integrate out the non essential fermion degrees of freedom obtaining the
effective Lagrangian for constant electromagnetic field

$\mathcal{L}_{\gamma}=-\frac{1}{4}F^{2}+\frac{\alpha^{2}}{360m_{e}^{2}}\left[4F^{4}+7\left(F\tilde{F}\right)^{2}\right]+\ldots\,,$
(8)

where the following notations are used $F^{2}=F^{\mu\nu}F_{\mu\nu}$,
$F\tilde{F}=F^{\mu\nu}\tilde{F}_{\mu\nu}$,
$\tilde{F}^{\mu\nu}=1/2\,e^{\mu\nu\rho\sigma}F_{\rho\sigma}$. Since the photon
energy is small, the tensor of electromagnetic field is slowly varying. From
(8), one reads off that the effective Lagrangian is an expansion in powers of
derivatives and coupling constant $\alpha$. The quantum correction to the
leading term is suppressed both by powers of $(\omega/m_{e})^{4}$ and
$\alpha^{2}\sim e^{4}$, i.e., the expansions in energy and in coupling
constant coincide. It is easy to understand that the structure of the
expansion is a consequence of the covariance of the theory and its gauge
symmetry. Due to covariance, the result depends on the tensor of the
electromagnetic field, and the gauge nature of the photon-electron interaction
makes the degree of charge to be always equal to the degree of the 4-potential
$\mathcal{A}_{\mu}$. This result is universal: if the problem was considered
in which the electromagnetic field would interact with vacuum fluctuations of
proton-antiproton pairs, then one would get the same formula, but with a
replacement $m_{e}\to m_{p}$. Moreover, it is not at all necessary that the
charged pairs be fermions. The result will remain the same for bosons. This
becomes especially clear when using the proper-time method of Fock and
Schwinger Schwinger:51 , where, even in the fermionic case, a transition is
made from the Dirac operator to its quadratic form - the second-order Klein-
Gordon differential operator to obtain (8).

The universality of electromagnetic interactions can also be formulated as a
statement about the non-renormalizability of the electric charge by the
effects of strong interactions. It is a simple consequence of conservation of
the currents and some commutation properties of charges, defined by these
conserved currents DeAlfaro:73 .

##  III Large $N_{c}$ limit and universality

In thinking about the low-energy regime of QCD, at the first stage one can
neglect the small bare masses of the up, down and strange quarks. In this
case, the left- and right-handed quarks do not interact with each other and
the whole theory admits an $U(3)_{L}\times U(3)_{R}$ chiral symmetry. The
$U(1)_{A}$ axial anomaly reduces this to the $SU(3)_{L}\times SU(3)_{R}\times
U(1)_{L+R}$ group. However, the chiral symmetry is not realized in the Wigner-
Weyl mode, the ground state is asymmetric under $SU(3)_{L}\times SU(3)_{R}$
Nambu:61a ; Nambu:61b . The result is that chiral symmetry is spontaneously
broken down to the vectorial subgroup of flavor and hypercharge, generated by
the vector currents of $SU(3)_{L+R}\times U(1)_{L+R}$ group. In accord with
Goldstone theorem Goldstone:61  the spectrum of massless QCD must therefore
contain $N_{f}^{2}-1=8$ massless pseudoscalar bosons.

Goldstone particles arise in systems with a degenerate vacuum, the symmetry of
which is lower than the symmetry of the original Lagrangian. Qualitatively,
this can be interpreted as a reaction of the system aimed at restoring the
lost algebraic symmetry. In this case, the transformation properties of
Goldstone particles are determined by nonlinear transformations, which,
together with the requirement of invariance of the effective Lagrangian with
respect to these chiral transformations, correspond to a symmetry of dynamic
type Weinberg:68 .

Of course, chiral symmetry is approximate, because in the real world the quark
masses are not exactly zero. This gives rise to an explicit chiral symmetry
breaking effects. As a result, the low-energy theorems following from the
chiral dynamics for Goldstone particles are not exactly correspond to the
nature: a careful study of the explicit symmetry breaking effects is required.
This is a subject of the chiral perturbation theory Gasser:84 ; Gasser:85a ;
Gasser:85b .

Another distinction of chiral dynamics is the non-fundamental character of
pseudo-Goldstone states. Their quark - antiquark structure significantly
distinguishes these states from fundamental gravitational or electromagnetic
fields.

That mesons and nucleons are not fundamental states was particularly
emphasized by Sakurai Sakurai:69 ; Sakurai:60  when formulating his idea of
the universality of vector mesons. How does this circumstance affect the
predictions based on the idea of universality of strong interactions? We
suppose that this question can be answered within the framework of $1/N_{c}$
expansion. In various ways this expansion is reminiscent of known
phenomenology of hadron physics, indicating that an expansion in powers of
$1/N_{c}$ may be a good approximation even at $1/N_{c}=1/3$. This conclusion
can also be reached on the basis of the QCD sum rules, provided the channels
considered correspond to modest values of the critical mass $M_{crit}^{2}$
Shifman:81 . For light mesons and quarks $M_{crit}^{2}\sim 2\,\mbox{GeV}^{2}$.
It will be argued here that it is from the point of view of the $1/N_{c}$
expansion the application of universality to the chiral dynamics makes the
most sense.

To spice up our discussion, it is useful to turn to the theory with local
four-quark interactions of the NJL type, which provides a general framework
for constructing both linear and nonlinear effective meson Lagrangians
describing the dynamics of composite quark-antiquark states. Let us consider
the model with the Lagrange density Osipov:00

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\bar{q}(i\gamma^{\mu}\partial_{\mu}-\mathcal{M})q+\frac{G_{S}}{2}\left[(\bar{q}q)^{2}+(\bar{q}i\gamma_{5}\vec{\tau}q)^{2}\right]$
(9) $\displaystyle-$
$\displaystyle\frac{G_{V}}{2}\left[(\bar{q}\gamma^{\mu}\vec{\tau}q)^{2}+(\bar{q}\gamma^{\mu}\gamma_{5}\vec{\tau}q)^{2}\right].$

Here $q$ is an isodoublet spinor of light quarks $\bar{q}=(\bar{u},\bar{d})$;
summation over color indices is implicit, $\vec{\tau}$ are the isospin Pauli
matrices; coupling constants $G_{S}$ and $G_{V}$ have the same dimension
$[G_{S}]=[G_{V}]=M^{-2}$; a current quark mass matrix
$\mathcal{M}=\mbox{diag}(\hat{m}_{u},\hat{m}_{d})$ is chosen in a form
($\hat{m}_{u}=\hat{m}_{d}\equiv\hat{m}$) preserving the isospin symmetry.

Without a quark mass term ${\cal L}$ would possess a continuous
$SU(2)_{V}\times SU(2)_{A}$ symmetry. The global transformations of this group
can be parameterized by six real parameters: $\alpha_{i}$ and $\beta_{i}$,
where $i=1,2,3$. The infinitesimal transformation of quark fields $\delta
q=q^{\prime}-q$ is

$\displaystyle\delta
q=i(\alpha+\gamma_{5}\beta)q,\quad\delta\bar{q}=i\bar{q}(-\alpha+\gamma_{5}\beta),$
$\displaystyle\alpha=\alpha_{i}\frac{\tau_{i}}{2},\quad\beta=\beta_{i}\frac{\tau_{i}}{2}.$
(10)

Notice that $SU(2)_{V}\times SU(2)_{A}$ symmetry is realized here as a
homogeneous linear transformation that includes the standard isospin
transformation $\alpha$ and the chiral transformation $\gamma_{5}\beta$.

The description of collective modes can be facilitated if one introduces the
new Lagrangian with the same dynamical content

$\displaystyle Z=\\!\\!\int\\!\\!{\cal D}q{\cal D}\bar{q}{\cal D}s{\cal
D}\vec{p}{\cal D}\vec{v}_{\mu}{\cal D}\vec{a}_{\mu}$ $\displaystyle\exp
i\\!\\!\int\\!\\!d^{4}x\left[\bar{q}Dq-\frac{(s+\hat{m})^{2}+\vec{p}^{\,2}}{2G_{S}}+\frac{\vec{v}_{\mu}^{\,2}+\vec{a}_{\mu}^{\,2}}{2G_{V}}\right].$
(11)

Here $D$ is the Dirac operator in the external boson fields

$D=i\gamma^{\mu}\partial_{\mu}+s+i\gamma_{5}p+\gamma^{\mu}v_{\mu}+\gamma^{\mu}\gamma_{5}a_{\mu},$
(12)

where $s=s\cdot
1,\,p=\vec{p}\,\vec{\tau},\,v_{\mu}=\vec{v}_{\mu}\vec{\tau},\,a_{\mu}=\vec{a}_{\mu}\vec{\tau}$
are scalar, pseudoscalar, vector and axial-vector auxiliary fields.

The infinitesimal action of the chiral group on quarks (III) helps to obtain a
transformation law for quark-antiquark auxiliary fields

$\displaystyle\delta s$ $\displaystyle=$ $\displaystyle\\{\beta,p\\},$
$\displaystyle\delta p$ $\displaystyle=$ $\displaystyle
i[\alpha,p]-\\{\beta,s\\},$ $\displaystyle\delta v_{\mu}$ $\displaystyle=$
$\displaystyle i[\alpha,v_{\mu}]+i[\beta,a_{\mu}],$ $\displaystyle\delta
a_{\mu}$ $\displaystyle=$ $\displaystyle i[\alpha,a_{\mu}]+i[\beta,v_{\mu}].$
(13)

Spontaneous symmetry breaking ($s=\sigma-m,\langle\sigma\rangle=0$) leads to
the mixing between pseudoscalar and axial-vector fields already in the one-
quark-loop approximation, i.e. in the same order at which the effective
potential develops the non-symmetric ground state. To avoid the mixing term
one should define a new axial-vector field $A_{\mu}$

$a_{\mu}=A_{\mu}+\kappa m\partial_{\mu}p.$ (14)

Constant $\kappa$ must be fixed to avoid $pA_{\mu}$ mixing. The replacement
(14) adds to the Lagrange density (III) a Yukawa-type vertex $\kappa
m\,\bar{q}\gamma^{\mu}\gamma_{5}\vec{\tau}q\,\partial_{\mu}\vec{p}$ known from
the theory of pion-nucleon forces.

Integrating over the quark fields in (III), and keeping only the leading terms
in the inverse constituent quark mass expansion of the corresponding heat
kernel (it is assumed that the Fock-Schwinger proper time technique
Schwinger:51  is applied to generate $1/m^{2}$ expansion), one arrives to the
effective Lagrange density describing the strong dynamics of collective boson
fields.

$\displaystyle\mathcal{L}^{\prime}=-\frac{\hat{m}\,\mbox{tr}\,(\sigma^{2}+p^{2})}{4mG_{S}}+\frac{\mbox{tr}\,\left[v_{\mu}^{\,2}+(A_{\mu}+\kappa
m\partial_{\mu}p\,)^{2}\right]}{4G_{V}}$
$\displaystyle+I_{2}(m^{2})\,\mbox{tr}\left\\{(\bigtriangledown_{\mu}\sigma)^{2}+(\bigtriangledown_{\mu}p)^{2}-(\sigma^{2}-2m\sigma+p^{2})^{2}\right.$
$\displaystyle\left.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\frac{1}{3}(v_{\mu\nu}^{2}+A_{\mu\nu}^{2})\right\\}.$ (15)

Here the trace is taken over isospin matrices. The main elements of (III) are
$SU(2)\times SU(2)$ covariant objects

$\displaystyle\bigtriangledown_{\mu}\sigma$ $\displaystyle=$
$\displaystyle\partial_{\mu}\sigma-\\{a_{\mu},p\\},$
$\displaystyle\bigtriangledown_{\mu}p$ $\displaystyle=$
$\displaystyle\partial_{\mu}p-i[v_{\mu},p]+\\{a_{\mu},\sigma-m\\},$
$\displaystyle v_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}v_{\nu}-\partial_{\nu}v_{\mu}-i[v_{\mu},v_{\nu}]-i[a_{\mu},a_{\nu}],$
$\displaystyle A_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-i[v_{\mu},a_{\nu}]-i[a_{\mu},v_{\nu}].$
(16)

The regularized mass-dependent factor $I_{2}(m^{2})$ is

$I_{2}(m^{2})=\frac{N_{c}}{16\pi^{2}}\left[\ln\left(1+\frac{\Lambda^{2}}{m^{2}}\right)-\frac{\Lambda^{2}}{\Lambda^{2}+m^{2}}\right],$
(17)

where $\Lambda$ is a covariant cutoff of the corresponding one-quark-loop
integral.

The gap equation arises in the form of the condition which cancels a
$\sigma$-tadpole contribution in (III). This equation accounts for
contributions of quarks of different flavours $N_{f}=2$ and colors $N_{c}=3$,
as well as the current quark mass $\hat{m}$

$\frac{4\pi^{2}}{N_{f}N_{c}\Lambda^{2}G_{S}}\left(1-\frac{\hat{m}}{m}\right)=1-\frac{m^{2}}{\Lambda^{2}}\ln\left(1+\frac{\Lambda^{2}}{m^{2}}\right).$
(18)

The Lagrangian density $\mathcal{L}^{\prime}$ does not contain
$pA_{\mu}$-mixing. This is because of the cancellation which occurs between
three different contributions to the nondiagonal $pA_{\mu}$-mixing term in
$\mathcal{L}^{\prime}$. It gives the numerical value of $\kappa$

$\frac{1}{2\kappa}=m^{2}+\frac{1}{16G_{V}I_{2}(m^{2})}.$ (19)

The free part of the Lagrangian $\mathcal{L}^{\prime}$ will have a canonical
form after the rescaling of meson fields

$\sigma=g_{\sigma}\bar{\sigma},\quad\vec{p}=g_{\pi}\vec{\pi},\quad\vec{v}_{\mu}=\frac{g_{\rho}}{2}\vec{\rho}_{\mu},\quad\vec{A}_{\mu}=\frac{g_{\rho}}{2}\vec{a}_{1\mu}.$
(20)

As a result, we have

$\displaystyle g_{\sigma}^{2}=\frac{1}{4I_{2}(m^{2})},\quad
g_{\pi}^{2}=Zg_{\sigma}^{2},\quad g_{\rho}^{2}=6g_{\sigma}^{2},$
$\displaystyle m_{\pi}^{2}=\frac{\hat{m}g_{\pi}^{2}}{mG_{S}},\quad
m_{\bar{\sigma}}^{2}=4m^{2}+Z^{-1}m_{\pi}^{2},$ $\displaystyle
m_{\rho}^{2}=\frac{3}{8G_{V}I_{2}(m^{2})},\quad
m_{a_{1}}^{2}=m_{\rho}^{2}+6m^{2},$ (21)

where $Z^{-1}=1-2\kappa m^{2}=g_{A}$.

What one can learn from this simple example? Let us first indicate that as it
follows from formulas (20) and (III) there is a precise relation between
coupling constants $g_{\sigma},g_{\pi},g_{\rho}$.

Next, let us consider the vector mesons. As can be verified from (III), the
following equalities hold

$g_{\rho}=g_{\rho\pi\pi}=g_{\rho\rho\rho}=g_{\rho qq}.$ (22)

These are the well-known universality relations of Sakurai Sakurai:69  (up to
the last term, where the rho-quark-quark coupling replaces the rho-nucleon-
nucleon coupling).

As one of the arguments in favor of universality relations (22), a hypothesis
was put forward about the invariance of the interaction Lagrangian under the
local action of the group of the isospin symmetry. The role of gauge fields
was assigned to an isotriplet of vector rho-mesons Sakurai:69 . Indeed, four-
quark interactions in (9) possess a higher symmetry. They are invariant under
local $SU(2)_{V}\times SU(2)_{A}$ transformations. It is easy to see that the
term $\bar{q}Dq$ in (III) is also symmetric under the action of this group.
Thus, dynamic symmetry (if one neglects the $\rho$-meson mass term) is hidden
behind the relations (22), which makes them more significant.

Despite the fact that mesons in the NJL model are composite quark-antiquark
states, in the Lagrangian (III) them correspond the local fields. Of course,
if one would take into account the higher order terms (in derivative expansion
of the effective action) neglected in (III), one would reveal the composite
structure of meson states. In momentum space, this would lead to the momentum
dependence of the coupling constants in (22), i.e., $g_{\rho}\to
g_{\rho}(q^{2})$. The corresponding technique has been developed in Osipov:94
; Osipov:96 . For the slowly changing form factor, $g_{\rho}(q^{2})$, the
principle of universality may still have sense even for the on-shell coupling
constants. On the other hand, the large $N_{c}$ limit teaches us that meson
physics at $N_{c}=\infty$ is described by the tree diagrams of an effective
local Lagrangian with the local meson fields. If we adhere to this point of
view (supposing that $N_{c}=3$ is a good approximation to the $N_{c}=\infty$
case), then the calculation of the momentum dependence of the coupling
constants based on one-quark-loop diagrams does not make sense. The point is
that the higher order terms of the expansion of the effective action
contribute at the same order in $1/N_{c}$, and therefore the model comes into
the conflict with QCD at large $N_{c}$. In this case, one should account for
meson loops instead, which are suppressed as $1/N_{c}$. Thus, limiting
ourselves to the Lagrangian (III), we assume that $g_{\rho}$ is a constant on
the interval $0<q^{2}<\Lambda^{2}$. Actually, this assumption is not new. What
is new is the observation that this hypothesis is true at large $N_{c}$.

The effective Lagrangian of pseudo-Goldstone particles, which we arrive at by
excluding from (III) the scalar field $\sigma$ (see, for instance, Osipov:00
), collects all soft-pion low-energy theorems (at tree-level order). It gives
us the well-known result of the nonlinear sigma model Donoghue:92 , which, due
to its universality, is used as a leading approximation in the chiral
perturbation theory.

##  IV $SU(3)$ violation: quark-loop level

Let us now turn to our main task. For this we will use a more advanced version
of the NJL model, which contains, in addition to the up and down quarks, a
heavy strange quark Volkov:93 . In the NJL model, the constituent masses of
the up, down and strange quarks $m_{u},m_{d},m_{s}$ are the result of the
spontaneous and explicit chiral symmetry breaking. A gap equation (18)
illustrates how this happens. Here we consider the model with $SU(2)$ isospin
symmetry $m_{u}=m_{d}\neq m_{s}$. The origin of the $SU(3)$ flavor symmetry
violation is the nonzero values of the bare current quark masses
$\hat{m}_{u}=\hat{m}_{d}\neq\hat{m}_{s}$. The model estimation of these values
are $\hat{m}_{u}=\hat{m}_{d}=3\,\mbox{MeV}$ and $\hat{m}_{s}=90\,\mbox{MeV}$
Volkov:93 . The current quarks get heavier in the phase with broken chiral
symmetry, where they are constituent quarks (with the following masses
$m_{u}=m_{d}=270\,\mbox{MeV}$, and $m_{s}=420\,\mbox{MeV}$) inside the
colorless mesons.

The effective local vertices of the induced meson Lagrangian are obtained from
the corresponding quark one-loop diagrams with external meson fields by the
series expansion in the inverse powers of heavy quark masses. The details of
the model can be found in Volkov:93 . Here we only remind the main steps which
are essential for fixing parameters of the model.

The decay process of a neutral vector meson, for instance the
$\rho^{0}$-meson, is described by the diagram shown in Fig. 1.

Figure 1: The tree-level diagram of $\rho\to e^{+}e^{-}$ decay. The
$\rho$-meson emits the photon through the quark loop.

In the NJL model, the calculation of the $V\to\gamma$ transition through the
quark loop, besides the standard photon-quark-antiquark vertex, requires the
Lagrangian density describing the strong interactions of vector mesons

$\displaystyle\mathcal{L}_{\bar{q}qV}$ $\displaystyle\Rightarrow$
$\displaystyle\frac{g_{\rho}}{2}\rho^{0}_{\mu}(\bar{u}\gamma^{\mu}u-\bar{d}\gamma^{\mu}d)$
(23) $\displaystyle+$
$\displaystyle\frac{g_{\omega}}{2}\omega_{\mu}(\bar{u}\gamma^{\mu}u+\bar{d}\gamma^{\mu}d)+\frac{g_{\phi}}{\sqrt{2}}\phi_{\mu}\bar{s}\gamma^{\mu}s,$

where couplings $g_{\rho},g_{\omega},g_{\phi}$ are chosen to redefine the
kinetic part of vector meson free Lagrangian to its standard form. If isospin
symmetry is fulfilled, $g_{\rho}=g_{\omega}$ the calculation of $V\to\gamma$
transition of Fig. 1 leads to the Lagrangian density (4), where
$f_{\rho}=g_{\rho}$, $f_{\omega}=3g_{\rho}$, and
$f_{\phi}=-\frac{3}{\sqrt{2}}g_{\phi}$. The coupling constants are

$g_{\rho}=\sqrt{\frac{3}{2I_{2}(m_{u}^{2})}},\quad
g_{\phi}=\sqrt{\frac{3}{2I_{2}(m_{s}^{2})}}.$ (24)

The logarithmically divergent integrals $I_{2}(m^{2}_{1},m^{2}_{2})$ are
regularized through the covariant cutoff $\Lambda$

$I_{2}(m_{1}^{2},m^{2}_{2})=-i\frac{N_{c}}{(2\pi)^{4}}\\!\int\\!\mathrm{d}^{4}k\,\frac{\theta(\Lambda^{2}+k^{2})}{(m_{1}^{2}-k^{2})(m_{2}^{2}-k^{2})}.$
(25)

The shorthand $I_{2}(m^{2}_{u})$ in (24) is introduced for the case of equal
masses $I_{2}(m^{2}_{u})\equiv I_{2}(m^{2}_{u},m^{2}_{u})$. In this specific
case, the expression coincides with (17).

In the case of charged pions and kaons, the quark-meson vertices are described
by the Lagrangian density

$\mathcal{L}_{\bar{q}q\Phi}\Rightarrow\sqrt{2}\bar{u}i\gamma_{5}\left(g_{\pi}\pi^{+}d+g_{K}K^{+}s\right)+h.c.$
(26)

The coupling constants $g_{\pi}$ and $g_{K}$ are given by

$g_{\pi}=\sqrt{\frac{Z_{\pi}}{4I_{2}(m_{u}^{2})}},\quad
g_{K}=\sqrt{\frac{Z_{K}}{4I_{2}(m^{2}_{u},m^{2}_{s})}}.$ (27)

There are two types of contributions here. Firstly, the integrals $I_{2}$
determine the redefinition of pseudoscalar fields to make that kinetic part of
their free Lagrangian to have standard form. Secondly, due to spontaneous
symmetry breaking, the partial Higgs mechanism takes place. The axial-vector –
pseudoscalar mixing term, $A_{\mu}\partial^{\mu}\Phi$, appears in the free
meson Lagrangian. This non-diagonal term must be canceled by an appropriate
redefinition of the longitudinal component of the massive axial-vector field
$A_{\mu}=A^{\prime}_{\mu}+\kappa m\partial_{\mu}\Phi$. The result is that the
kinetic term of a free pseudoscalar field get an additional factor, which must
also be compensated by the specific choice of constants $Z_{\pi}$ and $Z_{K}$.
They are

$Z_{\pi}^{-1}=1-6\frac{m^{2}_{u}}{m^{2}_{a_{1}}},\quad
Z_{K}^{-1}=1-\frac{3}{2}\frac{(m_{u}+m_{s})^{2}}{m^{2}_{K_{1A}}}.$ (28)

The coupling $Z_{\pi}$ originated by the mixing between the pion and the
$a_{1}(1260)$ mesons ($m_{a_{1}}=1230\pm 40$ MeV). The factor $Z_{K}$ is due
to the mixing between the kaon and the $K_{1}$ fields. There are two
candidates, known as $K_{1}(1270)$ and $K_{1}(1400)$. These physical states
are the mixture of $K_{1A}$ and $K_{1B}$ fields. The latters have different
types of couplings with quark-antiquark pair, namely
$K_{1A}\propto\gamma_{5}\gamma_{\mu}$,
$K_{1B}\propto\gamma_{5}\partial_{\mu}$. The constant $m^{2}_{K_{1A}}$ is
determined through the mixing angle $\alpha=57^{\circ}$ of these states
Osipov:85 ; Suzuki:93

$m^{2}_{K_{1A}}=\left(\frac{\sin^{2}{\alpha}}{m^{2}_{K_{1}(1270)}}-\frac{\cos^{2}{\alpha}}{m^{2}_{K_{1}(1400)}}\right)^{-1}.$
(29)

A general mathematical framework to deal with axial-vector–pseudoscalar mixing
in the effective Lagrangians has been developed in Osipov:17 .

Finally, let us fix the parameters of the model. The value of $g_{\rho}$ is
determined from the $\rho\to\pi\pi$ decay, that gives $g_{\rho}=6.0$. Then,
using eq. (24), one finds $I_{2}(m_{u}^{2})=3/(2g_{\rho}^{2})=0.042$. The
charged pion weak decay constant $\pi^{+}\to\mu^{+}\nu_{\mu}$,
$f_{\pi}=92.2\,\mbox{MeV}$, and the quark-level Goldberger-Treiman relation
$g_{\pi}=m_{u}/f_{\pi}$ allow us to establish an equation for the non-strange
quark mass

$m_{u}=\frac{1}{2}f_{\pi}\sqrt{\frac{Z_{\pi}}{I_{2}(m_{u}^{2})}}=f_{\pi}g_{\rho}\sqrt{\frac{Z_{\pi}}{6}}.$
(30)

This quadratic equation can be easily solved in terms of known
phenomenological parameters. As a result we have

$m_{u}^{2}=\frac{m^{2}_{a_{1}}}{12}\left[1\pm\sqrt{1-\left(\frac{2g_{\rho}f_{\pi}}{m_{a_{1}}}\right)^{2}}\right].$
(31)

It allows us to obtain the value of constituent quark masses
$m_{u}=m_{d}=270\,$MeV (for the negative sign in the solution (31)). Now from
$I_{2}(m_{u}^{2})$ we find the quark loop cutoff parameter
$\Lambda=1265\,$MeV.

To establish the value of strange quark mass we use the Goldberger-Treiman
relation for the kaon $g_{K}$

$g_{K}=\frac{m_{u}+m_{s}}{2f_{K}},$ (32)

where $f_{K}\simeq 1.2f_{\pi}$ Beringer:12 . Using eq. (27), one comes to the
formula, where $m_{s}$ is the only unknown parameter

$f_{K}=(m_{u}+m_{s})\sqrt{\frac{I_{2}(m_{u}^{2},m_{s}^{2})}{Z_{K}}}.$ (33)

Solving numerically this equation we obtain the value of the constituent
strange quark mass $m_{s}=420$ MeV. For above values of the quark masses and
the cutoff $\Lambda=1265\,\mbox{MeV}$, the condensates are
$\langle\bar{u}u\rangle^{1/3}=\langle\bar{d}d\rangle^{1/3}=-304\,\mbox{MeV}$,
$\langle\bar{s}s\rangle^{1/3}=-335\,\mbox{MeV}$. These values can be compared
with the lattice QCD results
$\langle\bar{l}l\rangle^{\overline{MS}}(2\mbox{GeV})=(-283(2)\,\mbox{MeV})^{3}$
and
$\langle\bar{s}s\rangle^{\overline{MS}}(2\mbox{GeV})=(-290(15)\,\mbox{MeV})^{3}$,
where $l$ is a light quark with mass equal to the average of the $u$ and $d$
quarks Bazavov:13 .

It is known that the gap-quation of the NJL model leads to underestimated
values of the current quark masses, which is reflected in the magnitude of the
quark condensates. The latter is due to the fact that the value of the chiral
condensate is related with masses of pseudoscalars through the well-known
Gell-Mann, Oakes, Renner relation

$m_{\pi}^{2}f_{\pi}^{2}=-\frac{m^{0}_{u}+m_{d}^{0}}{2}\left(\langle\bar{u}u\rangle+\langle\bar{d}d\rangle\right).$
(34)

So, all is arranged in such a way that model describes correctly the masses of
pions and kaons (it is interesting to note that this parametrization of the
model leads to a perfect agreement with the experimental value for the kaon
mass, giving $m_{K}=494$ MeV).

Now, we can estimate $g_{\phi}=7.5$, and, as a consequence, this predicts
$\Gamma_{\phi\to e^{+}e^{-}}=0.9\,\mbox{keV}$. One can see that $SU(3)$
symmetry breaking at the level of quark loops leads to an even greater
disagreement with the experimental data on the $\phi\to e^{+}e^{-}$ decay
mode.

##  V $SU(3)$ violation: meson-loop level

Let us consider now the contributions to the amplitudes due to the meson
loops. The corresponding diagrams for the $\rho\to e^{+}e^{-}$ decay are shown
in Fig. 2.

Figure 2: Diagrams describing contributions from meson loops to $\rho\to
e^{+}e^{-}$ decay. The dotted line indicates pions or kaons.

The vertices, describing $V\to\Phi\Phi$ decays in the case of exact $SU(3)$
symmetry are given by the Lagrangian density

$\displaystyle{\cal
L}_{V\Phi\Phi}=-i\frac{g_{\rho}}{4}\,\mbox{tr}\left(V_{\mu}[\Phi,\partial^{\mu}\Phi]\right)$
$\displaystyle=-i\frac{g_{\rho}}{2}\left[\rho^{0}_{\mu}\left(2\pi^{+}\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\partial^{\mu}}}\\!\pi^{-}+K^{+}\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\partial^{\mu}}}\\!K^{-}+\bar{K}^{0}\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\partial^{\mu}}}\\!K^{0}\right)\right.$
$\displaystyle+\left.(\omega_{\mu}-\sqrt{2}\phi_{\mu})(K^{+}\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\partial^{\mu}}}\\!K^{-}-\bar{K}^{0}\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\partial^{\mu}}}\\!K^{0})\right].$
(35)

The left-right derivative is
$\phi_{1}\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\partial^{\mu}}}\\!\phi_{2}=\phi_{1}\partial^{\mu}\phi_{2}-(\partial^{\mu}\phi_{1})\phi_{2}$,
and nonets of the pseudoscalar $\Phi$ and vector $V$ fields are given by the
following matrices

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\pi^{0}&\sqrt{2}\pi^{+}&\sqrt{2}K^{+}\\\
\sqrt{2}\pi^{-}&-\pi^{0}&\sqrt{2}K^{0}\\\
\sqrt{2}K^{-}&\sqrt{2}\bar{K}^{0}&0\end{array}\right)+\sum_{a=0,8}\eta_{a}\lambda_{a},$
(39) $\displaystyle V$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\omega+\rho^{0}&\sqrt{2}\rho^{+}&\sqrt{2}K^{*+}\\\
\sqrt{2}\rho^{-}&\omega-\rho^{0}&\sqrt{2}K^{*0}\\\
\sqrt{2}K^{*-}&\sqrt{2}\bar{K}^{*0}&\sqrt{2}\phi\end{array}\right),$ (43)

where the last term in (39) contains neutral singlet and octet components
$\eta_{0}$ and $\eta_{8}$, the superposition of which describes physical
$\eta,\eta^{\prime}$ states; $\lambda_{0}=\sqrt{2/3}\,I$, with $I$ the
$3\times 3$ identity, and $\lambda_{8}=1/\sqrt{3}\,\mbox{diag}(1,1,-2)$. We
assume in (43) that vector $\rho^{0},\omega,\phi$ mesons have the following
quark structure: $\rho^{0}=(u\bar{u}-d\bar{d})/\sqrt{2}$,
$\omega=(u\bar{u}+d\bar{d})/\sqrt{2}$, and $\phi=s\bar{s}$ consists only of
strange quarks. Therefore, in the following we neglect the small $\phi-\omega$
mixing.

To take into account the effects of $SU(3)$ symmetry breaking, we, as in the
previous section, should consider the quark triangle diagrams leading to the
effective Lagrangian (V) by taking into account the strange quark effects. The
only consequence of such calculations is that one should replace the coupling
$g_{\rho}\to g_{\phi}$ in the $\phi\bar{K}K$ vertex of (V).

The $U(1)$ gauge invariant electromagnetic interactions of the pseudoscalar
$\Phi$ mesons in the NJL model are given by the effective Lagrangian density

$\displaystyle\mathcal{L}_{\gamma\Phi\Phi}$ $\displaystyle=$
$\displaystyle\frac{ie}{2}\mathcal{A}^{\mu}\mbox{tr}\left(Q[\partial_{\mu}\Phi,\Phi]\right)$
(44) $\displaystyle=$ $\displaystyle-
ie\mathcal{A}^{\mu}\left(\pi^{+}\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\partial_{\mu}}}\\!\pi^{-}+K^{+}\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\partial_{\mu}}}\\!K^{-}\right),$

where $Q=\mbox{diag}(2/3,-1/3,-1/3)$ is the quark charge matrix in relative
units of the proton electric charge $e>0$. This is the usual form which can be
obtained independently by the standard $U(1)$ gauging of the free meson
Lagrangian.

Using these Lagrangians in the calculations of diagrams in Fig. 2 one obtains
the following amplitudes

$\displaystyle T_{\rho\to e^{+}e^{-}}\\!$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{g_{\rho}}\\!\left[1+\frac{g^{2}_{\rho}}{6m^{2}_{\rho}}\left(2I_{\rho\pi}+I_{\rho
K}\right)\right]\\!\epsilon_{\mu}(p)l^{\mu},$ (45) $\displaystyle T_{\omega\to
e^{+}e^{-}}\\!$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{3g_{\rho}}\\!\left(1+\frac{g^{2}_{\rho}}{2m^{2}_{\omega}}I_{\omega
K}\right)\\!\epsilon_{\mu}(p)l^{\mu},$ (46) $\displaystyle T_{\phi\to
e^{+}e^{-}}\\!$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{3}\frac{e^{2}}{g_{\phi}}\left(1+\frac{g^{2}_{\phi}}{2m^{2}_{\phi}}I_{\phi
K}\right)\\!\epsilon_{\mu}(p)l^{\mu}.$ (47)

Here the first term in the parentheses represents the tree-level contribution
of order $\sqrt{N}_{c}$, the other terms accounts for $1/\sqrt{N}_{c}$
corrections. We also assume that $m_{\rho}=m_{\omega}$ ($SU(2)$ symmetry),
$\epsilon_{\mu}(p)$ is a polarization 4-vector of the corresponding vector
meson with 4-momentum $p$, $l^{\mu}$ is a vector lepton current
$l^{\mu}=\bar{e}\gamma^{\mu}e$. The integral over the pion-loop for the decay
$\rho\to e^{+}e^{-}$ after regularization (the covariant cutoff
$\Lambda_{\pi}$ is introduced) has the form

$\displaystyle
I_{\rho\pi}(m_{\rho}^{2})=\frac{1}{16\pi^{2}}\left[2\Lambda_{\pi}^{2}+(m_{\rho}^{2}-6m_{\pi}^{2})\ln\left(1+\frac{\Lambda_{\pi}^{2}}{m_{\pi}^{2}}\right)\right.$
(48) $\displaystyle+$ $\displaystyle
2(2\Lambda_{\pi}^{2}+m_{\rho}^{2}-4m_{\pi}^{2})D(\Lambda^{2}_{\pi})\arctan\frac{1}{D(\Lambda^{2}_{\pi})}$
$\displaystyle+$
$\displaystyle\left.2m_{\rho}^{2}D(0)^{3}\arctan\frac{1}{D(0)}\right]-I_{tp}(m^{2}_{\pi},\Lambda^{2}_{\pi}),$

where

$D(\Lambda^{2}_{\pi})=\sqrt{\frac{4(\Lambda^{2}_{\pi}+m^{2}_{\pi})}{m^{2}_{\rho}}-1},$
(49)

and the last integral in (48) represents the contribution of the tadpole
diagram, shown in Fig. 2

$I_{tp}(m^{2}_{\pi},\Lambda^{2}_{\pi})=\frac{3}{8\pi^{2}}\left[\Lambda_{\pi}^{2}-m_{\pi}^{2}\ln\left(1+\frac{\Lambda_{\pi}^{2}}{m_{\pi}^{2}}\right)\right].$
(50)

If expression (48) is expanded in a series in terms of the square of the mass
of the $\rho$-meson, then such an expansion begins with a term that is
completely canceled out by the contribution of the tadpole
$I_{tp}(m^{2}_{\pi},\Lambda^{2}_{\pi})$. Thus, the integral $I_{\rho\pi}$
contains only a logarithmically divergent (at $\Lambda_{\pi}\to\infty$) part.
The other consequence is that $I_{\rho\pi}(0)=0$ at fixed $\Lambda_{\pi}$.

The integrals $I_{\rho K},I_{\omega K}$, and $I_{\phi K}$ are obtained from
$I_{\rho\pi}$ by replacing the masses $m_{\pi}\to m_{K}$, $m_{\rho}\to
m_{\omega},m_{\phi}$ and the cutoff $\Lambda_{\pi}\to\Lambda_{K}$
correspondingly. The cutoff $\Lambda_{K}=750\,\mbox{MeV}$ is fixed from the
decay width $\phi\to e^{+}e^{-}$. Then, the cutoff parameter
$\Lambda_{\pi}=850\,\mbox{MeV}$ is fixed from the experimental width of the
processes $\rho\to e^{+}e^{-}$. As a result, the process $\omega\to
e^{+}e^{-}$ and three $\mu^{+}\mu^{-}$ decay modes have no arbitrary
parameters, and the theoretical widths for all of them agree with the
experimental data. The results are shown in the Table I.

Table 1: Decay widths of neutral vector mesons into lepton pairs (in keV). The tree-level NJL results are given in the second column. The full results which take into account the one-meson-loop corrections are shown in the third column. Modes |  NJL |  NJL+mes. loops |  Experiment PDG:20
---|---|---|---
$\rho\to e^{+}e^{-}$ |  4.8 | 6.99 | 6.98 $\pm$ 0.12
$\omega\to e^{+}e^{-}$ |  0.53 | 0.62 | 0.62 $\pm$ 0.02
$\phi\to e^{+}e^{-}$ |  0.90 | 1.26 | 1.26 $\pm$ 0.02
$\rho\to\mu^{+}\mu^{-}$ |  4.28 | 6.22 | 6.72 $\pm$ 0.46
$\omega\to\mu^{+}\mu^{-}$ |  0.48 | 0.56 | 0.63 $\pm$ 0.16
$\phi\to\mu^{+}\mu^{-}$ |  0.84 | 1.18 | 1.22 $\pm$ 0.08

The first column shows the modes of two-lepton decays of neutral vector
mesons, and the second contains estimates made on the bases of the tree
approximation ($\sqrt{N}_{c}$-order). One can see that the tree level result
underestimates the decay widths considered. It corresponds the following
values of coupling constants: $f_{\rho}=6,f_{\omega}=18,f_{\phi}=-15.9$.
Notice, that in the case of decays of $\rho$ and $\omega$ mesons, we deal with
the case of unbroken $SU(3)$ symmetry, and for the $\phi$ meson our result
takes into account the $SU(3)$ breaking effect arising from the heavier quark
mass of the strange quark, $m_{s}>m_{u}$. The numbers presented in the third
column include the one-meson-loop effects. They correct the results to their
phenomenological values. The decay rates of similar decays of neutral vector
mesons into a muon pair are also in satisfactory agreement with experiment.

It seems necessary to discuss in more detail the use of the cutoff
$\Lambda_{\Phi}$ in the integrals $I_{V\Phi}$. Usually, in such cases,
dispersion methods are used Weise:96 ; Achasov:11 . Our remark is that since
the vector mesons are on the mass-shell, the two methods are equivalent.
Indeed, the dispersion approach for the meson loop gives the following result
for the integral with two subtractions

$I_{V\Phi}(p^{2})=c_{0}+c_{\Phi}p^{2}+\frac{p^{4}}{\pi}\\!\\!\int\limits_{4m_{\Phi}^{2}}^{\infty}\\!\\!ds\,\frac{\mbox{Im}I_{V\Phi}(s)}{s(s-p^{2}-i\epsilon)},$
(51)

where the subtraction constants $c_{0}$ and $c_{\Phi}$ need to be determined.
Due to gauge symmetry $c_{0}$ should vanish. In our case $c_{0}$ is also
vanish because of a tadpole contribution. It is clear now that on the mass
shell $p^{2}=m_{V}^{2}$ the value of this integral depends on the only free
parameter $c_{\Phi}$. Therefore, one can always establish the one-to-one
correspondence between $c_{\Phi}$ and cutoff $\Lambda_{\Phi}$ by equating the
integrals.

##  VI Conclusions

We have shown that one-loop meson diagrams with the emission of the photon,
i.e., the $V\to\pi^{+}\pi^{-}\to\gamma$ and $V\to K^{+}K^{-}\to\gamma$
transitions, can be an efficient mechanism for linking the hypothesis of
vector meson universality with the empirical data on the electromagnetic
vector meson decays into the lepton pair. The problem considered is well-
known. We argue that it can be solved in the framework of $1/N_{c}$ expansion.
An important step of our study is the gauge-invariant form of the Lagrangian
describing the transition of the vector meson to a photon. Since in the
processes under consideration the photon is far from its mass shell, this
interaction does not vanish. One should recall that the standard VMD picture
does not have the $V\to\gamma$ transitions through the meson loop.

In our estimates, we have not calculated the self-energy diagrams of vector
mesons. It is implied that the renormalization of vector-meson masses are
already performed to their physical values, as it has been also assumed in
Weise:96 .

Our result is based on the effective meson Lagrangian of the NJL model and the
$SU(3)\to SU(2)$ symmetry breaking mechanism which is implemented in the
model, as like as in QCD, i.e., through the corresponding mass term of the
free quark Lagrangian. The gap equation, which relates the masses of the
current and constituent quark fields, makes it possible to trace the
consequences of such a violation up to the vertices of the effective meson
Lagrangian. We stress the importance of this step for the problem considered
here.

To summarize: we have demonstrated that in the framework of the $1/N_{c}$
approximation, the $\rho\to\pi\pi$ decay constant $g_{\rho}=6$. It is this
value that should be also used in the theoretical description of two-lepton
decays of neutral vector mesons. This finding supports the idea of vector
meson universality, showing the importance of taking into account the
contributions of the $1/N_{c}$ suppressed meson one-loop diagrams. These
contributions do not exceed 30% in comparison with the tree result for the
$V\to l^{+}l^{-}$ decays and thereby bring the results of theoretical
calculations into full agreement with experimental data and universality.

## Acknowledgments

The authors are grateful to A. B. Arbuzov for his interest in this work and
useful discussions. The work of A. A. Osipov is supported by Grant from
Fundação para a Ciência e Tecnologia (FCT) through the Grant No. CERN/FIS-
COM/0035/2019, and the European Cooperation in Science and Technology
organisation through the COST Action CA16201 program. The work of K. Nurlan is
supported by the JINR grant for young scientists and specialists No.
21-302-04, and the fund from the Science Committee of the Ministry of
Education and Science of the Republic of Kazakhstan, Grant No. AP09057862.

## References

  * (1) J. J. Sakurai, _Currents and Mesons_ , University of Chicago Press, Chicago, IL, 1969.
  * (2) R. P. Feynman, _Photon-Hadron Interactions_. W. A. Benjamin, Inc. Reading, Massachusetts 1972.
  * (3) P. A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020)
  * (4) G J. Gounaris, J. J. Sakurai, Phys. Rev. Lett. 21, 244 (1968)
  * (5) J. J. Sakurai, Ann. Phys. 11, 1 (1960)
  * (6) S. Gasiorowicz, D. A. Geffen, Rev. Mod. Phys. 41, 531 (1969)
  * (7) Ö. Kaymakcalan, S. Rajeev and J. Schechter, Phys. Rev. D 30, 594 (I984)
  * (8) H. Gomm, Ö. Kaymakcalan, J. Schechter, Phys. Rev. D 30, 2345 (1984)
  * (9) Ö. Kaymakcalan, J. Schechter, Phys. Rev. D 31, 1109 (I985)
  * (10) M. Bando, T. Kugo, K. Yamawaki, Phys. Rep. 164, 217 (1988)
  * (11) U.-G. Meissner, Phys, Rep. 161, 213 (1988)
  * (12) M. C. Birse, Z. Phys. A 355, 231 (1996)
  * (13) G. ’t Hooft, Nucl. Phys. B 72, 461 (1974)
  * (14) G. ’t Hooft, Nucl. Phys. B 75, 461 (1974)
  * (15) E. Witten, Nucl. Phys. B 160, 57 (1979)
  * (16) F. Klingl, N. Kaiser, W. Weise, Z. Phys. A 356, 193 (1996)
  * (17) J. P. Lees et al., Phys. Rev. D 86, 032013 (2012)
  * (18) M. Ablikin et al., Phys. Lett. B 753, 629 (2016)
  * (19) E. Bartos, S. Dubnička, A. Z. Dubničková, R. Kamiński, A. Liptaj, Phys. Rev. D 96, 113004 (2017)
  * (20) N. N. Achasov, A. A. Kozhevnikov, Phys. Rev. D 83, 113005 (2011)
  * (21) M. K. Volkov, E. A. Kuraev, Y. M. Bystritskiy, Phys. Atomic Nucl. 72, 1513 (2009)
  * (22) Y. Nambu, G. Jona-Lasinio, Phys. Rev. 122, 345 (1961)
  * (23) Y. Nambu, G. Jona-Lasinio, Phys. Rev. 124, 246 (1961)
  * (24) D. Ebert and M. K. Volkov, Yad. Fiz. 36, 1265 (1982)
  * (25) D. Ebert and M. K. Volkov, Z. Phys. C 16, 205 (1983)
  * (26) M. K. Volkov, Ann. Phys. 157, 282 (1984)
  * (27) M. K. Volkov, PEPAN 17, 433 (1986)
  * (28) D. Ebert and H. Reinhardt, Nucl. Phys. B 271, 188 (1986)
  * (29) V. Bernard, A. A. Osipov, Ulf-G. Meißner, Phys. Lett. B 285, 119 (1992)
  * (30) M. K. Volkov, A. E. Radzhabov, Phys. Usp. 49, 551 (2006)
  * (31) W. Heisenberg, H. Euler, Z. Phys. 98, 714 (1936)
  * (32) J. Schwinger, Phys. Rev. 82, 664 (1951)
  * (33) V. De Alfaro, S. Fubini, G. Furlan and C. Rossetti, _Currents in hadron physics_ , North Holland, Amsterdam-London, (1973)
  * (34) J. Goldstone, Nuovo Cim. 19, 154 (1961)
  * (35) S. Weinberg, Phys. Rev. 166, 1568 (1968)
  * (36) J. Gasser, H. Leutwyler, Ann. Phys. 158, 142 (1984)
  * (37) J. Gasser, H. Leutwyler, Nucl. Phys. B 250, 465 (1985)
  * (38) J. Gasser, H. Leutwyler, Nucl. Phys. B 250, 517 (1985)
  * (39) V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 191, 301 (1981)
  * (40) A. A. Osipov, B. Hiller, Phys. Rev. D 62, 114013 (2000)
  * (41) V. Bernard, Ulf-G. Meißner, A. A. Osipov, Phys. Lett. B 324, 201 (1994)
  * (42) V. Bernard, A. H. Blin, B. Hiller, Yu. P. Ivanov, A. A. Osipov, and Ulf-G. Meißner, Ann. Phys. 249, 499 (1996)
  * (43) J. F. Donoghue, E. Golowich and B. R. Holstein, _Dynamics of the standard model_ , Cambridge University Press, Cambridge, 1992.
  * (44) M. K. Volkov, Phys. Part. Nucl. 24 35 (1993)
  * (45) M. K. Volkov, A. A. Osipov, Sov. J. Nucl. Phys. 41, 500 (1985) [Yad. Fiz. 41, 785 (1985)]
  * (46) M. Suzuki, Phys. Rev. D 47, 1252 (1993)
  * (47) J. Morais, B. Hiller, A. A. Osipov, Phys. Lett. B 773, 277 (2017)
  * (48) J. Beringer et al., Phys. Rev. D 86, 010001(2012)
  * (49) C. McNeile, A. Bazavov, C. T. H. Davies, R. J. Dowdall, K. Hornbostel, G. P. Lepage, and H. D. Trottier, Phys. Rev. D 87, 034503 (2013).