# Dipole polarizabilities of light pseudoscalar mesons within the Domain Model
of QCD vacuum

Sergei Nedelko<EMAIL_ADDRESS>Vladimir Voronin<EMAIL_ADDRESS>Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia

###### Abstract

Dipole polarizabilities of light pseudoscalar mesons are calculated in the
framework of the mean field approach to QCD vacuum and bosonization based on
the statistical ensemble of almost everywhere homogeneous Abelian (anti-)self-
dual gluon fields, the domain model of QCD vacuum. In this approach, a
nonlocal effective action of meson fields is derived which describes all
possible strong, weak and electromagnetic interactions of meson fields
including their excited states. The considered mean field implements
confinement and chiral symmetry, which manifests itself both in the properties
of quark and gluon fields, as well as, upon bosonization, in the mass
spectrum, decay constants and form factors of nonlocal colorless hadrons, and
leads to the qualitatively distinctive features of the effective meson action.
Particularly relevant to the subject of the present paper are the nonlocality
of meson-quark-antiquark vertices and the absence of poles at real momenta in
the propagators of scalar meson fields composed of light quark-antiquark
pairs. In view of this, studying the role of manifest nonlocality of mesons
and contribution of intermediate scalar meson fields in formation of the
polarizabilities is of special interest. It turns out that for charged pions
and kaons, this contribution is substantial, but not the largest one. Nonlocal
nature of mesons provides additional contribution, so that calculated
polarizabilities are in reasonable agreement with COMPASS experimental data
and Chiral Perturbation Theory.

##  I Introduction

Polarizabilities of hadrons characterize their response to applied
electromagnetic field which cannot be attributed to point-like particles, and
are of fundamental interest for low-energy QCD. Experimental measurement of
polarizabilities is challenging, so most data are available for lightest
mesons $\pi^{\pm},\pi^{0}$, but there is long-standing discrepancy in the
values (see papers [1, 2, 3] for review of theoretical and experimental status
of the meson polarizability problem). Among the reported experimental results,
only the most recent data on charged pion polarizability by COMPASS
collaboration at CERN [4] is consistent with Chiral Perturbation Theory
(ChPT). The leading-order result of ChPT [5] is equivalent to the value found
in Ref. [6] based on the hypothesis of partially conserved axial-vector
current.

The polarizabilities were investigated theoretically within Chiral
Perturbation Theory up to two loops [7, 8, 9, 10, 11], with the methods of
Lattice QCD [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] , within various
phenomenological models [23, 24, 25, 26, 27, 28, 29, 30] and with the help of
dispersion relations [31, 32, 33]. Several studies [27, 32, 28, 30] found that
dominating part of pion polarizabilities is due to $\sigma$ meson. In the
present study, polarizabilities are extracted from the nonlocal effective
meson action deduced within the Domain Model of QCD vacuum and hadronization
(see Refs. [34, 35, 36, 37, 38]).

In this action meson fields appear as collective colorless excitations of
confined dynamical quark-antiquark, heavy and light ones, and gluon fields. A
highly nonlocal effective meson action contains information about the strong,
weak and electromagnetic interactions of mesons as well as their two-point
correlation functions. In particular, the model systematically describes
various phenomena related to confinement and chiral symmetry realization, the
heavy quark limit. Meson masses, including Regge spectrum of excited states of
mesons, their decay constants and form factors are in good agreement with
experimental values [39, 40, 34, 35, 36, 37, 38].

The specific feature of this approach is that mesons appear as extended
composite fields due to nonlocal meson-quark vertices. Meson masses are
identified as the poles of the nonlocal meson propagators. As it turns out,
there is no poles at real momenta in the nonlocal propagators of the scalar
meson-like composite fields, and therefore light scalar mesons as quark-
antiquark states are absent in the physical spectrum of the stable collective
excitations. This property relates to the peculiarities of realization of
chiral symmetry in the presence of Abelian (anti)-self-dual gluon mean field
and agrees with expectation that the lightest scalar state $\sigma$ is not
intuitively made of a quark and an antiquark [41]. At the same time, scalar
meson-like fields contribute to the amplitudes of various processes. In has to
be noted that physical scalar states occur in the hyperfine splitting of the
orbital excitations of the vector mesons with the masses above 1 GeV.

We describe the formalism and its features in Section II. In Section III we
calculate polarizabilities and find that charged pion polarizability is
consistent with ChPT [7] and COMPASS data [4]. The contribution of
intermediate scalar meson fields to polarizability of pion and kaon is
substantial, especially for the neutral ones, while for charged pion and kaon
polarizability intermediate scalar meson turns out to be less important.

##  II Effective meson action

The mean-field approach based on the random Abelian (anti-)self-dual vacuum
gluon fields allows to deduce a generating functional via bosonization of one-
gluon exchange interaction of quark currents which has the form [39, 34, 35,
36, 37, 38]:

$\displaystyle Z$ $\displaystyle={\cal N}\int D\phi_{\cal
Q}\exp\left\\{-\frac{\Lambda^{2}}{2}\frac{h^{2}_{\cal
Q}}{g^{2}C^{2}_{\mathcal{Q}}}\int d^{4}x\phi^{2}_{\cal
Q}(x)-\sum\limits_{k=2}^{\infty}\frac{1}{k}W_{k}[\phi]\right\\},$ (1)
$\displaystyle W_{k}[\phi]$ $\displaystyle=\sum\limits_{{\cal Q}_{1}\dots\cal
Q_{k}}h_{{\cal Q}_{1}}\dots h_{{\cal Q}_{k}}\int d^{4}x_{1}\dots\int
d^{4}x_{k}\Phi_{{\cal Q}_{1}}(x_{1})\dots\Phi_{{\cal
Q}_{k}}(x_{k})\Gamma^{(k)}_{{\cal Q}_{1}\dots\cal Q_{k}}(x_{1},\dots,x_{k}),$
(2) $\displaystyle\Phi_{{\cal Q}}(x)$
$\displaystyle=\int\frac{d^{4}p}{(2\pi)^{4}}e^{ipx}{\mathcal{O}}_{{\mathcal{Q}}{\mathcal{Q}}^{\prime}}(p)\tilde{\phi}_{{\mathcal{Q}}^{\prime}}(p),\quad
C_{\mathcal{Q}}=C_{J},\quad C^{2}_{S/P}=2C^{2}_{V/A}=\frac{1}{9}.$ (3)

The condensed index $\mathcal{Q}\equiv\\{aJLn\\}$ includes all quantum numbers
of a meson, $\Lambda$ is a scale related to the strength of the vacuum gluon
field, and finally to the value of gluon condensate $\langle
g^{2}F^{2}\rangle$. The physical color neutral meson fields
$\phi_{{\mathcal{Q}}}$ are obtained by means of orthogonal transformation
${\mathcal{O}}_{{\mathcal{Q}}{\mathcal{Q}}^{\prime}}$ of fields
$\Phi_{{\mathcal{Q}}}$. The quadratic part of the action for
$\phi_{{\mathcal{Q}}}$ is diagonal with respect to all quantum numbers. The
masses of mesons correspond to the poles of nonlocal propagators

$D_{\mathcal{Q}}(p^{2})=h_{\mathcal{Q}}^{-2}\left(\frac{\Lambda^{2}}{g^{2}C^{2}_{\mathcal{Q}}}+\tilde{\Gamma}^{(2)}_{\mathcal{Q}}(p^{2})\right)^{-1}$
(4)

and can be found as zeroes of inverse propagator from equation

$\displaystyle 0$
$\displaystyle=\frac{\Lambda^{2}}{g^{2}C^{2}_{\mathcal{Q}}}+\tilde{\Gamma}^{(2)}_{\cal
Q}(-M^{2}_{\cal Q}),$ (5)

where $\tilde{\Gamma}^{(2)}_{\cal Q}$ is two-point correlation function
diagonalized with respect to all quantum numbers. Constants $h_{\mathcal{Q}}$
are defined by equation

$\displaystyle 1$ $\displaystyle=h^{2}_{\cal
Q}\left.\frac{d}{dp^{2}}\tilde{\Gamma}^{(2)}_{\cal
Q}(p^{2})\right|_{p^{2}=-M^{2}_{\cal Q}}$

which ensure that residue at the pole of the propagator is equal to unity. The
results of calculation of the masses of various mesons as well as analytical
expressions of $\tilde{\Gamma}^{(2)}_{\cal Q}(p^{2})$ can be found in Ref.
[35]. In the one-loop approximation, the meson propagators given by Eq. (4)
are real, and therefore mesons are stable with respect to decay into quarks by
virtue of the optical theorem. It is also quite plausible that correlation
functions of greater number of external mesons are suppressed as number of
colors $N_{c}$ approaches infinity because the functional (1) is deduced from
QCD. Thorough investigation of this limit is an interesting topic to be
studied in the future.

Correlation functions include “connected” and “disconnected” contributions of
quark loops in the background field. For example, the two-point nonlocal
vertex function $\tilde{\Gamma}^{(2)}_{\cal QQ^{\prime}}(p)$ is given by

$\Gamma^{(2)}_{{\cal Q}_{1}{\cal Q}_{2}}=\overline{G^{(2)}_{{\cal Q}_{1}{\cal
Q}_{2}}(x_{1},x_{2})}-\Xi_{2}(x_{1}-x_{2})\overline{G^{(1)}_{{\cal
Q}_{1}}G^{(1)}_{{\cal Q}_{2}}},$ (6)

where $\Xi$ is correlation function of the background field which belongs to
the statistical ensemble of the almost everywhere homogeneous Abelian
(anti-)self-dual fields. Quark loops $G^{(k)}_{{\cal Q}_{1}\dots\cal Q_{k}}$
are averaged over the ensemble of background field configurations with measure
$d\sigma_{B}$:

$\displaystyle\overline{G^{(k)}_{{\cal Q}_{1}\dots\cal
Q_{k}}(x_{1},\dots,x_{k})}=\int d\sigma_{B}{\rm Tr}V_{{\cal
Q}_{1}}\left(x_{1}\right)S\left(x_{1},x_{2}\right)\dots V_{{\cal
Q}_{k}}\left(x_{k}\right)S\left(x_{k},x_{1}\right),$ (7)
$\displaystyle\overline{G^{(l)}_{{\cal Q}_{1}\dots\cal
Q_{l}}(x_{1},\dots,x_{l})G^{(k)}_{{\cal Q}_{l+1}\dots\cal
Q_{k}}(x_{l+1},\dots,x_{k})}=$ $\displaystyle\int d\sigma_{B}{\rm
Tr}\left\\{V_{{\cal Q}_{1}}\left(x_{1}\right)S\left(x_{1},x_{2}\right)\dots
V_{{\cal Q}_{k}}\left(x_{l}\right)S\left(x_{l},x_{1}\right)\right\\}\times$
$\displaystyle{\rm Tr}\left\\{V_{{\cal
Q}_{l+1}}\left(x_{l+1}\right)S\left(x_{l+1},x_{l+2}\right)\dots V_{{\cal
Q}_{k}}\left(x_{k}\right)S\left(x_{k},x_{l+1}\right)\right\\}.$

Here $S(x,y)$ is the quark propagator and $V_{{\cal Q}}$ are nonlocal meson-
quark-antiquark vertices.

The quark propagator and meson-quark vertices in the presence of the almost
everywhere homogeneous fields are approximated by those in the homogeneous
(anti-)self-dual Abelian background field. The averaging over mean field
ensemble is achieved by averaging the quark loops over configurations of the
homogeneous background fields, supplemented by taking into account $n$-point
correlators of the mean fields $\Xi_{n}$. The averaging is performed over
self-dual and anti-self-dual Abelian (anti-)self-dual configurations and their
directions in Euclidean and color spaces. Averaging over spatial directions in
$R^{4}$ is performed with the help of generating formula

$\int
d\sigma_{B}\exp(if_{\mu\nu}J_{\mu\nu})=\langle\exp(if_{\mu\nu}J_{\mu\nu})\rangle=\frac{\sin\sqrt{2\left(J_{\mu\nu}J_{\mu\nu}\pm
J_{\mu\nu}\widetilde{J}_{\mu\nu}\right)}}{\sqrt{2\left(J_{\mu\nu}J_{\mu\nu}\pm
J_{\mu\nu}\widetilde{J}_{\mu\nu}\right)}},$ (8)

where $J_{\mu\nu}$ is an arbitrary antisymmetric tensor. Tensor $f_{\mu\nu}$
is an appropriately normalized Abelian (anti-)self-dual background field with
strength $\Lambda$:

$\displaystyle\hat{B}_{\mu}=-\frac{1}{2}\hat{n}B_{\mu\nu}x_{\nu},\
\hat{n}=t^{3}\cos\xi+t^{8}\sin\xi,$
$\displaystyle\tilde{B}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}B_{\alpha\beta}=\pm
B_{\mu\nu},\
\hat{B}_{\rho\mu}\hat{B}_{\rho\nu}=4\upsilon^{2}\Lambda^{4}\delta_{\mu\nu},$
(9) $\displaystyle
f_{\alpha\beta}=\frac{\hat{n}}{2\upsilon\Lambda^{2}}B_{\alpha\beta},\
\upsilon=\mathrm{diag}\left(\frac{1}{6},\frac{1}{6},\frac{1}{3}\right),\
f_{\mu\alpha}f_{\nu\alpha}=\delta_{\mu\nu},$

where the upper sign in “$\pm$” should be taken for self-dual field, and the
lower for anti-self-dual field. Nonlocal vertices
$V^{aJln}_{\mu_{1}\dots\mu_{l}}$ are given by formulas

$\displaystyle V^{aJln}_{\mu_{1}\dots\mu_{l}}={\cal
C}_{ln}\mathcal{M}^{a}\Gamma^{J}F_{nl}\left(\frac{\stackrel{{\scriptstyle\leftrightarrow}}{{\cal
D}}^{2}\\!\\!\\!(x)}{\Lambda^{2}}\right)T^{(l)}_{\mu_{1}\dots\mu_{l}}\left(\frac{1}{i}\frac{\stackrel{{\scriptstyle\leftrightarrow}}{{\cal
D}}\\!(x)}{\Lambda}\right),$ (10) $\displaystyle{\cal
C}^{2}_{ln}=\frac{l+1}{2^{l}n!(n+l)!},\quad
F_{nl}(s)=s^{n}\int_{0}^{1}dtt^{n+l}\exp(st),$
$\displaystyle{\stackrel{{\scriptstyle\leftrightarrow}}{{\mathcal{D}}}}\vphantom{D}^{ff^{\prime}}_{\mu}=\xi_{f}\stackrel{{\scriptstyle\leftarrow}}{{\mathcal{D}}}_{\mu}-\
\xi_{f^{\prime}}\stackrel{{\scriptstyle\rightarrow}}{{\mathcal{D}}}_{\mu},\ \
\stackrel{{\scriptstyle\leftarrow}}{{\mathcal{D}}}_{\mu}(x)=\stackrel{{\scriptstyle\leftarrow}}{{\partial}}_{\mu}+\
i\hat{B}_{\mu}(x),\ \
\stackrel{{\scriptstyle\rightarrow}}{{\mathcal{D}}}_{\mu}(x)=\stackrel{{\scriptstyle\rightarrow}}{{\partial}}_{\mu}-\
i\hat{B}_{\mu}(x),$
$\displaystyle\xi_{f}=\frac{m_{f^{\prime}}}{m_{f}+m_{f^{\prime}}},\
\xi_{f^{\prime}}=\frac{m_{f}}{m_{f}+m_{f^{\prime}}}.$

Here $\mathcal{M}^{a}$ and $\Gamma^{J}$ are flavor and Dirac matrices
corresponding to a given meson field, $\xi_{f},\xi_{f^{\prime}}$ provide that
$x$ is the center of mass of a meson, $n,l$ are radial and orbital quantum
numbers, respectively. Radial part $F_{nl}$ is defined by the propagator of
the gluon fluctuations charged with respect to the Abelian background,
$T^{(l)}$ are irreducible tensors of four-dimensional rotation group.
Propagator of the quark with mass $m_{f}$ in the presence of the homogeneous
Abelian (anti-)self-dual field has the form

$\displaystyle S_{f}(x,y)$
$\displaystyle=\exp\left(-\frac{i}{2}\hat{n}x_{\mu}B_{\mu\nu}y_{\nu}\right)H_{f}(x-y),$
(11) $\displaystyle\tilde{H}_{f}(p)$
$\displaystyle=\frac{1}{2\upsilon\Lambda^{2}}\int_{0}^{1}dse^{(-p^{2}/2\upsilon\Lambda^{2})s}\left(\frac{1-s}{1+s}\right)^{m_{f}^{2}/4\upsilon\Lambda^{2}}$
$\displaystyle\quad\times\left[\vphantom{\frac{s}{1-s^{2}}}p_{\alpha}\gamma_{\alpha}\pm
is\gamma_{5}\gamma_{\alpha}f_{\alpha\beta}p_{\beta}+m_{f}\left(P_{\pm}+P_{\mp}\frac{1+s^{2}}{1-s^{2}}-\frac{i}{2}\gamma_{\alpha}f_{\alpha\beta}\gamma_{\beta}\frac{s}{1-s^{2}}\right)\right],$

where anti-Hermitean representation of Dirac matrices is used, and “$\pm$”
signs are arranged in accordance with formula (9). The translation-invariant
part $H_{f}$ of the propagator is an analytical function in the finite complex
momentum plane and matches the behavior of free Dirac propagator at large
Euclidean momentum. The analyticity of quark propagator is interpreted as
confinement of dynamical quarks.

Overall, the mass spectrum of the ground state and excited mesons composed of
light and heavy quarks is described rather accurately, in complete agreement
with expectations based on confinement (Regge mass spectrum of radially and
orbitally excited states) and chiral symmetry breaking (light pseudoscalar and
heavy vector nonets, etc.) as well as asymptotic heavy-quark relations.

A peculiar property of the quark propagator in the mean gluon field under
consideration is that, unlike the case of the pseudoscalar and vector ground-
state mesons, there are no real solutions to equation (5) for their parity
partners, the ground-state scalar and axial mesons. Axial and scalar mesons
appear with a mass above $1$ GeV in the hyperfine splitting of the orbital
excitations. The inverse propagators of pseudoscalar and scalar meson fields
are shown in Fig. 1 which manifestly illustrates the absence of a pole for the
scalar field propagator. This feature is particularly relevant to the present
study. Contributions of intermediate scalar meson-like fields to various
processes are available and can be computed but a controversial issue of
existence of the light scalar mesons does not occur.

Figure 1: Left-hand side of Eq. 5 (which is proportional to inverse
propagator) for pion and ground-state scalar quark-antiquark field with
respect to Euclidean momenta $p^{2}$. The zero of inverse pion propagator
(correspondingly, the pole of propagator $D_{\pi}(p^{2})$) is located at
$-M_{\pi}^{2}$, while inverse propagator $D_{S}^{-1}(p^{2})$ has no zeroes at
real momenta.

Though the meson-quark coupling constant $h_{\mathcal{Q}}$ is obviously
undefined for meson-like composite fields if corresponding Eq. (5) has no
solutions, it is convenient to keep universal notation. Such fields can only
be virtual, and $h_{\mathcal{Q}}$ cancels out in final expressions
($h^{2}_{\mathcal{Q}}$ for two vertices cancel $h_{\mathcal{Q}}^{-2}$ in
propagator $D_{\mathcal{Q}}$).

Electromagnetic interactions are included in gauge-invariant way using the
prescription of Ref. [43] which yields expansions (see [35, 42])

$\displaystyle S_{f}(x,y|A)$
$\displaystyle=S_{f}(x,y)+\sum_{n=1}^{\infty}\left(Q_{f}e\right)^{n}\int
dz_{1}\cdots\int
dz_{n}S_{f}(x_{1},z_{1})\gamma_{\mu_{1}}A_{\mu_{1}}(z_{1})\cdots
S_{f}(z_{i-1},z_{i})\gamma_{\mu_{i}}A_{\mu_{i}}(z_{i})\cdots S_{f}(z_{n},y),$
$\displaystyle V_{\mathcal{Q}}(x|A)$
$\displaystyle=V_{\mathcal{Q}}(x)+\sum_{n=1}^{\infty}e^{n}\int
dz_{1}\cdots\int
dz_{n}V_{\mathcal{Q}\mu_{1}\dots\mu_{n}}(x,z_{1},\dots,z_{n})A_{\mu_{1}}(z_{1})\cdots
A_{\mu_{n}}(z_{n}),$

where $Q$ is a diagonal matrix of quark charges in units of electron charge
$e$, and meson-photon vertices appear due to nonlocality of meson-quark
interactions. One-photon and two-photon meson vertices are given by

$\displaystyle
V_{\mathcal{Q}\mu}(x;q)=\int_{0}^{1}d\tau\frac{1}{\tau}\frac{\partial}{\partial
q_{\mu}}\left\\{-Q_{f}V_{\mathcal{Q}}\left(\stackrel{{\scriptstyle\leftrightarrow}}{{\mathcal{D}}}(x)-iq\tau\xi\right)+Q_{f^{\prime}}V_{\mathcal{Q}}\left(\stackrel{{\scriptstyle\leftrightarrow}}{{\mathcal{D}}}(x)+iq\tau\xi^{\prime}\right)\right\\},$
(12)
$\displaystyle\begin{split}V_{\mathcal{Q}\mu\nu}(x;q_{1},q_{2})=\frac{1}{2}\int_{0}^{1}d\tau_{1}\int_{0}^{1}d\tau_{2}\frac{1}{\tau_{1}\tau_{2}}\frac{\partial}{\partial
q_{1\mu}}\frac{\partial}{\partial
q_{2\nu}}&\left[Q_{f}Q_{f^{\prime}}V_{\mathcal{Q}}\left(\stackrel{{\scriptstyle\leftrightarrow}}{{\mathcal{D}}}(x)-iq_{1}\tau_{1}\xi-
iq_{2}\tau_{2}\xi\right)\right.\\\
&-Q_{f}Q_{f^{\prime}}V_{\mathcal{Q}}\left(\stackrel{{\scriptstyle\leftrightarrow}}{{\mathcal{D}}}(x)-iq_{1}\tau_{1}\xi+iq_{2}\tau_{2}\xi\right)\\\
&-Q_{f^{\prime}}Q_{f}V_{\mathcal{Q}}\left(\stackrel{{\scriptstyle\leftrightarrow}}{{\mathcal{D}}}(x)+iq_{1}\tau_{1}\xi-
iq_{2}\tau_{2}\xi\right)\\\
&\left.+Q_{f^{\prime}}Q_{f^{\prime}}V_{\mathcal{Q}}\left(\stackrel{{\scriptstyle\leftrightarrow}}{{\mathcal{D}}}(x)+iq_{1}\tau_{1}\xi+iq_{2}\tau_{2}\xi\right)\right],\end{split}$
(13)

where $Q_{f}$ is electric charge of a quark with flavor $f$.

The generating functional and the effective meson action take the form

$\displaystyle Z$ $\displaystyle={\cal N}\int D\phi_{\cal Q}\int
DA_{\mu}\exp\left\\{-\frac{1}{4}\int d^{4}x\
F_{\mu\nu}F_{\mu\nu}-\frac{\Lambda^{2}}{2}\frac{h^{2}_{\cal
Q}}{g^{2}C^{2}_{\mathcal{Q}}}\int d^{4}x\phi^{2}_{\cal
Q}(x)-\Gamma(A)-\sum\limits_{k=2}^{\infty}\frac{1}{k}W_{k}[\phi|A]\right\\},$
(14) $\displaystyle W_{k}[\phi|A]$ $\displaystyle=\sum\limits_{{\cal
Q}_{1}\dots\cal Q_{k}}h_{{\cal Q}_{1}}\dots h_{{\cal Q}_{k}}\int
d^{4}x_{1}\dots\int d^{4}x_{k}\Phi_{{\cal Q}_{1}}(x_{1})\dots\Phi_{{\cal
Q}_{k}}(x_{k})\Gamma^{(k)}_{{\cal Q}_{1}\dots\cal
Q_{k}}(x_{1},\dots,x_{k}|A),$

where

$\Gamma(A)=\int
d\sigma_{B}\mathrm{Tr}\log\left[1+Qe\gamma_{\mu}A_{\mu}(x)S(x,y)\right],$

and $\Gamma^{(k)}_{{\cal Q}_{1}\dots\cal Q_{k}}(x_{1},\dots,x_{k}|A)$ are
obtained from $\Gamma^{(k)}_{{\cal Q}_{1}\dots\cal Q_{k}}(x_{1},\dots,x_{k})$
by substitutions

$S_{f}(x,y)\to S_{f}(x,y|A),\quad V_{\mathcal{Q}}(x)\to V_{\mathcal{Q}}(x|A).$

The free parameters of the model are scale $\Lambda$ (scalar gluon
condensate), infrared limits of dynamical quark masses and strong coupling
$\alpha_{s}$ which has been determined by fitting to the masses of mesons
$\pi,\rho,K,K^{*},J/\psi,\Upsilon,\eta^{\prime}$ (for details see [35]).

As it has been mentioned, diagonalization of the quadratic part of the
effective action (2) with respect to the radial quantum number is a part of
the calculation procedure. In practice some finite number of excited states
can be taken into account. As it has been analyzed in [35], though typically
about five lowest radial states have to be taken into account for robust
stability of the computation, a consistent overall description of the mass
spectrum of mesons is achieved irrespective to a number of accounted radial
excitations. Just the values of the free parameters have to be adjusted when
the number of accounted radial excitation changes.

##  III Evaluation of polarizabilities

The polarizabilities are defined by the Compton scattering amplitude

$P(p)+\gamma(q,\varepsilon)\to
P(p^{\prime})+\gamma(q^{\prime},\varepsilon^{\prime})$

of a pseudoscalar meson $P$

${}_{\text{out}}\langle
P(p^{\prime})\gamma(q^{\prime},\varepsilon^{\prime})|P(p)\gamma(q,\varepsilon)\rangle_{\text{in}}=i(2\pi)^{4}\delta^{(4)}(p^{\prime}+q^{\prime}-p-q)\varepsilon^{\mu}(q)\varepsilon^{*\nu}(q^{\prime})M_{\mu\nu}.$

We concentrate on the electric $\alpha_{\text{E}}$ and magnetic
$\beta_{\text{M}}$ dipole polarizabilities which appear in expansion of the
amplitude in small photon momenta as

$\varepsilon^{\mu}(q)\varepsilon^{*\nu}(q^{\prime})M_{\mu\nu}=-2e^{2}\vec{\varepsilon}\cdot\vec{\varepsilon}{\,}^{\prime*}+8\pi
M(\alpha_{\text{E}}\omega\omega^{\prime}\,\vec{\varepsilon}\cdot\vec{\varepsilon}{\,}^{\prime*}+\beta_{\text{M}}\left(\vec{\varepsilon}\times\vec{q}\,\right)\cdot\left(\vec{\varepsilon}{\,}^{\prime*}\times\vec{q}{\,}^{\prime}\right))+\dots,$

where $M$ is the mass of a pseudoscalar meson. The tensor $M_{\mu\nu}$ can be
separated into two parts

$M_{\mu\nu}=M_{\mu\nu}^{\text{Born}}+M_{\mu\nu}^{\text{NB}},$ (15)

where the part $M_{\mu\nu}^{\text{NB}}$ describes the response of a meson as a
composite system to the applied electromagnetic field. The term
$M_{\mu\nu}^{\text{Born}}$ given by

$M_{\mu\nu}^{\text{Born}}=e^{2}\left[2g_{\mu\nu}-\frac{(2p_{\mu}+q_{\mu})(2p^{\prime}_{\nu}+q^{\prime}_{\nu})}{(p+q)^{2}-M^{2}}-\frac{(2p_{\nu}-q^{\prime}_{\nu})(2p^{\prime}_{\mu}-q_{\mu})}{(p-q^{\prime})^{2}-M^{2}}\right]$
(16)

describes real Compton scattering of a structureless pseudoscalar particle.

In the case of real Compton scattering ($q^{2}={q^{\prime}}^{2}=0$,
$\varepsilon^{\mu}(q)q_{\mu}=\varepsilon^{*\mu}(q^{\prime})q^{\prime}_{\mu}=0$),
the tensor $\mathcal{M}_{\mu\nu}$ contains only two independent tensor
structures [44, 45, 46]

$\displaystyle\varepsilon^{\alpha}(q)\varepsilon^{*\beta}(q^{\prime})M_{\alpha\beta}$
$\displaystyle=\varepsilon^{\alpha}(q)\varepsilon^{*\beta}(q^{\prime})\left(AT_{1\alpha\beta}+BT_{2\alpha\beta}\right),$
(17) $\displaystyle T_{1\alpha\beta}$
$\displaystyle=-\frac{t}{2}g_{\alpha\beta}-q_{\beta}q^{\prime}_{\alpha},$
$\displaystyle T_{2\alpha\beta}$
$\displaystyle=-\frac{t}{2}P_{\alpha}P_{\beta}+\nu^{2}g_{\alpha\beta}-\nu(P_{\alpha}q_{\beta}+P_{\beta}q^{\prime}_{\alpha}),$

where

$P=\frac{p+p^{\prime}}{2},\quad s=(p+q)^{2},\quad t=(q-q^{\prime})^{2},\quad
u=(p-q^{\prime})^{2},\quad\nu=\frac{1}{4}(s-u).$

In accordance with Eqs. (15) and (16), amplitudes $A$ and $B$ can be split in
two parts

$\displaystyle A(\nu^{2},t)$
$\displaystyle=A^{\text{Born}}(\nu^{2},t)+A^{\text{NB}}(\nu^{2},t),$
$\displaystyle B(\nu^{2},t)$
$\displaystyle=B^{\text{Born}}(\nu^{2},t)+B^{\text{NB}}(\nu^{2},t),$

where $A^{\text{Born}},B^{\text{Born}}$ are given by

$A^{\text{Born}}(\nu^{2},t)=-\frac{e^{2}t}{(s-M_{\pi}^{2})(u-M_{\pi}^{2})},\quad
B^{\text{Born}}(\nu^{2},t)=-\frac{8e^{2}}{(s-M_{\pi}^{2})(u-M_{\pi}^{2})}.$

The electric $\alpha_{\text{E}}$ and magnetic $\beta_{\text{M}}$
polarizabilities are related to $A^{\text{NB}},B^{\text{NB}}$ by means of
equations (see [47, 48])

$\displaystyle\alpha_{\text{E}}+\beta_{\text{M}}$
$\displaystyle=-\frac{M}{8\pi}B^{\text{NB}}(0,0),$ (18)
$\displaystyle\alpha_{\text{E}}-\beta_{\text{M}}$
$\displaystyle=-\frac{1}{8\pi
M}\left(2A^{\text{NB}}(0,0)+M^{2}B^{\text{NB}}(0,0)\right).$ (19)

The diagrams that contribute to Compton tensor are shown in Fig.2.

|  |  |
---|---|---|---
(a) | (b) | (c) | (d)
Figure 2:  Diagrams contributing to Compton scattering tensor $M_{\mu\nu}$.
The grey circles denote all possible one-particle irreducible contributions.

Consider contribution of diagrams (a),(b),(c) in Fig.2 which form gauge-
invariant combination.

|  |  |  |
---|---|---|---|---
(a) | (b) | (c) | (d) | (e)
|  |  |  |
(f) | (g) | (h) | (i) | (j)
Figure 3: The one-loop diagrams contributing to $M_{\alpha\beta}^{(a)}$. Wavy
filling represents the vacuum gluon field (9). The diagrams related by
crossings are not shown.

The diagrams (b) and (c) contain kinematic singularities that are canceled by
corresponding Born terms in Eq. (16). One can notice that among these
diagrams, only diagram (a) contains tensors proportional to $g_{\alpha\beta}$
and can be parametrized as

$\displaystyle M^{(\text{a})}_{\alpha\beta}$
$\displaystyle=\left(-\frac{t}{2}A(\nu^{2},t)+\nu^{2}B(\nu^{2},t)\right)g_{\alpha\beta}+\text{other
tensor structures},$ $\displaystyle M^{(\text{a})\text{NB}}_{\alpha\beta}$
$\displaystyle=\left(-\frac{t}{2}A(\nu^{2},t)+\nu^{2}B(\nu^{2},t)-2\right)g_{\alpha\beta}+\text{other
tensor structures},$ (20)

according to formulas (16) and (17). The amplitudes $A,B$ that appear in
definition of polarizabilities (18),(19) can be extracted from coefficient of
$g_{\alpha\beta}$ in $M_{\alpha\beta}^{\text{NB}}$ with the help of formulas

$\displaystyle A^{\text{NB}}(0,0)$ $\displaystyle=-2\frac{\partial}{\partial
t}\left.\left(-\frac{t}{2}A(\nu^{2},t)+\nu^{2}B(\nu^{2},t)-2\right)\right|_{t=0,\nu=0},$
$\displaystyle B^{\text{NB}}(0,0)$
$\displaystyle=\frac{1}{2}\frac{\partial^{2}}{\partial^{2}\nu}\left.\left(-\frac{t}{2}A(\nu^{2},t)+\nu^{2}B(\nu^{2},t)-2\right)\right|_{t=0,\nu=0}.$

It is therefore sufficient to calculate only diagrams (a) of gauge-invariant
combination of (a),(b) and (c) in order to extract electric and magnetic
dipole polarizabilities. This is more straightforward because diagram (a) does
not contain kinematic poles. One-loop contributions of this type are shown in
Fig. 3 (the Feynman rules in Euclidean space are given by formulas (2),(6) and
(7) for loops, Eqs. (10),(12),(13) describe non-local vertices which should be
multiplied by corresponding $h_{\mathcal{Q}}$, the local vertices are the same
as in QED, the quark propagators are defined by Eq.(11)). For example, diagram
(e) in Fig.3 corresponds to

$\begin{split}(2\pi)^{4}\delta^{(4)}(p^{\prime}+q^{\prime}-p-q)M_{\mu\nu}^{\text{(a,e)}}=e^{2}h_{P}^{2}&\int\\!d\sigma_{B}\int\\!d^{4}x\int\\!d^{4}y\int\\!d^{4}z\exp(ipx+iqx-
ip^{\prime}y-iq^{\prime}z)\\\
&\times(-1)\text{Tr}V_{\mu}^{aP00}(x;-q)S(x,y)V^{bP00}(y)S(y,z)Q\gamma_{\nu}S(z,y)\\\
&+\text{crossed term},\end{split}$

for two external pseudoscalar ground-state mesons (see Appendix A for
details).

The contribution to the amplitude corresponding to diagram (d) in Fig. 2 with
intermediate scalar fields is separately gauge-invariant and can be
represented as

$\varepsilon^{\alpha}(q)\varepsilon^{*\beta}(q^{\prime})M^{(d)}_{\alpha\beta}=\varepsilon^{\alpha}(q)\varepsilon^{*\beta}(q^{\prime})\sum_{S}\Gamma_{\alpha\beta}^{S\gamma\gamma}D_{S}h_{P}^{2}\Gamma_{SPP},$
(21)

where $\Gamma_{S\gamma\gamma}$ parametrizes $S\to\gamma\gamma$ subprocess,
$D_{S}$ is a scalar meson field propagator, $h_{P}^{2}\Gamma_{SPP}(t)$
describes transition of scalar field to a couple of pseudoscalar mesons
(corresponding one-loop diagrams are shown in Fig. 4, the formulas are given
in Appendix A). Diagram $M^{(d)}$ does not contain tensor $T_{2\alpha\beta}$
(see formula (17)) and hence contributes only to amplitude $A$.

|  |  |
---|---|---|---
(a) | (b) | (c) | (d)
Figure 4:  One-loop diagrams contributing to $M_{\alpha\beta}^{S}$ in formula
(21) (crossed diagrams are not shown). Diagrams (a),(b),(c) are related to
$\Gamma_{S\gamma\gamma}$, diagram (d) results in $h_{P}^{2}\Gamma_{SPP}$.

The computational complexity of the amplitudes which we need to evaluate in
order to extract polarizabilities quickly grows with increasing the radial
number $n$ taken into account for diagonalization of the quadratic part of the
action. In the present paper, for numerical calculation only $n=0$ states are
taken into account, such that the matrix $\mathcal{O}$ in Eq. (3) is reduced
to the unity matrix, and the higher radial states are neglected. The values of
parameters in this lowest approximation with respect to the “radial excitation
mixing” are given in Table 1 (for detailed discussion see [35]).

$m_{u/d}$(MeV) |  $m_{s}$(MeV) |  $\Lambda$(MeV) | $\alpha_{s}$
---|---|---|---
$174.8$ | $393.3$ | $439.7$ | $6.23$
Table 1: Values of parameters used for calculations of polarizabilities which
were extracted from the experimental values of masses of $\pi,\rho,K,K^{*}$
via Eq. (5). The values of quark masses are largely affected by the presence
of background gluon field, and they are not in exact one-to-one correspondence
with the values of masses in models without background gluon field, see e.g.
Ref. [35].

The values of polarizabilities found in the present work are presented in
Table 2. Since no small-momentum expansion is employed, we can also calculate
polarizabilities of kaons. In contrast with results obtained in several
distinct quark-meson models [27, 28, 30], the main contribution in the model
under consideration comes from one-loop diagrams, while contribution of
intermediate scalar field is less important. However, this can be considered
as a rearrangement of contributions because only their sum is observable.

|  | Diagrams in Fig. 3 | Diagrams in Fig. 4 | Total | Experiment | ChPT
---|---|---|---|---|---|---
$\pi^{\pm}$ | $\alpha_{\text{E}}+\beta_{\text{M}}$ | $0.13$ | 0 | $0.13$ |  $0.5\pm 0.5_{\text{stat}}$ [4] |  $0.16$ [7]
$\alpha_{\text{E}}-\beta_{\text{M}}$ | $4.82$ | $1.48$ | $6.3$ |  $4.0\pm 1.2_{\text{stat}}\pm 1.4_{\text{syst}}$ [4] |  $5.7$ [7]
$\pi^{0}$ | $\alpha_{\text{E}}+\beta_{\text{M}}$ | $0.71$ | 0 | $0.71$ |  $0.98\pm 0.03$ [32] |  $1.15$ [11]
$\alpha_{\text{E}}-\beta_{\text{M}}$ | $-0.26$ | $1.48$ | $1.22$ |  $-1.6\pm 2.2$ [32] |  $-1.9$ [11]
$K^{\pm}$ | $\alpha_{\text{E}}+\beta_{\text{M}}$ | $0.41$ | 0 | $0.41$ |  |
$\alpha_{\text{E}}-\beta_{\text{M}}$ | $1.22$ | $0.47$ | $1.69$ |  |
$K^{0},\bar{K}^{0}$ | $\alpha_{\text{E}}+\beta_{\text{M}}$ | $0.62$ | 0 | $0.62$ |  |
$\alpha_{\text{E}}-\beta_{\text{M}}$ | $-0.29$ | $0.17$ | $-0.12$ |  |
Table 2: Numerical results for polarizabilities of pseudoscalar mesons in
Gaussian units of $10^{-4}\ \text{fm}^{3}$. Column “Diagrams in Fig. 3”
corresponds to leading-order contribution of diagrams (a),(b),(c) in Fig. 2
given by diagrams in Fig. 3 which is extracted with formula (20). “Diagrams in
Fig. 4” labels leading-order contribution of diagram (d) in Fig. 2 with
intermediate scalar quark-antiquark fields given by formula (21).

##  IV Discussion

We investigated dipole polarizabilities of the light pseudoscalar mesons in
the framework of the nonlocal effective meson action obtained within the mean-
field approach to QCD vacuum. The model described by the functional (14)
allows consistent treatment of various phenomena of low-energy hadronic
physics: spectra of mesons, their decay constants and form-factors. Comparison
of the present formalism with other approaches like Functional Renormalization
Group, Dyson–Schwinger Equations, Lattice QCD and AdS/QCD is outlined in paper
[35].

The values of charged pion polarizabilities calculated in the present study
are in agreement with COMPASS data and most recent two-loop ChPT calculation
[7]. The pion mass and leptonic decay constant evaluated in the same framework
earlier [35] agree with experimental data, and these values serve as
phenomenological input for the basic Lagrangian of ChPT. The agreement with
ChPT is then follows from identification of pion as pseudo Goldstone boson of
broken chiral symmetry. Moreover, an effective low-energy Lagrangian for pions
can be obtained from generating functional (14) if one integrates out heavier
fields and performs an expansion in small momenta of pions. It is clear that
such an analysis would be technically complicated, it deserves a separate
investigation which would be interesting to perform, and we hope to do it in
due course.

Prediction of Lattice QCD for polarizabilities depends on parameters such as
the lattice volume, lattice spacings, quark masses. The value of dipole
magnetic polarizability of charged pion found in paper [16] with the finest
lattice is $\beta_{\pi^{\pm}}=-2.06\pm 0.76\times 10^{-4}\ \text{fm}^{3}$
supports data of COMPASS collaboration [4], ChPT [7] and findings of this
paper.

The distinctive feature of the present approach is that mesons are extended
collective excitations of quark-antiquark and gluon fields in the confining
gluon background field. The structure of meson is encoded in the nonlocal
meson-quark vertices (10) which are straightforwardly calculated. The
nonlocality of meson-quark vertices leads to meson-quark-photon interactions
given by Eqs. (12),(13). Another feature of the present approach is that
intermediate scalar quark-antiquark field cannot be identified with physical
light scalar meson because corresponding propagator has no pole at real
momenta. Even though there is no light scalar quark-antiquark particles, the
corresponding field contributes to dipole polarizabilities. As a result of
these features, the contributions to polarizabilities are arranged differently
from other quark-meson models [27, 28, 30], and the main contribution to
polarizabilities in the model under consideration comes from one-loop diagrams
in Fig.3.

Besides ground-state scalar fields, the effective meson action (14) contains
other scalar fields. For instance, it includes the scalar component of
orbitally excited vector meson field emerging from hyperfine splitting, with
meson-quark vertices given by

$V^{a01n}_{\mu\mu}=\frac{1}{4}{\cal
C}_{1n}\mathcal{M}^{a}\gamma^{\mu}F_{n1}\left(\frac{\stackrel{{\scriptstyle\leftrightarrow}}{{\cal
D}}^{2}\\!\\!\\!(x)}{\Lambda^{2}}\right)\frac{1}{i}\frac{\stackrel{{\scriptstyle\leftrightarrow}}{{\cal
D}}_{\mu}\\!(x)}{\Lambda}.$

The inverse propagator of corresponding isosinglet field in ground radial
state $n=0$ at real momenta is shown in Fig. 5 One expects that the
contribution to dipole polarizabilities of these fields via diagram (d) in
Fig. 2 is smaller than contribution of ground-state scalar quark-antiquark
field (the inverse propagator is shown in Fig. 1) if for no other reason than
their propagator is also smaller at $p^{2}=0$. Thorough investigation of this
contribution, however, is even more complex than contribution of ground-state
scalar quark-antiquark field. The zero of inverse propagator shown in Fig. 5
is located at $\sqrt{p^{2}}=1252-i\ 203\ \text{MeV}$. In contrast, it was
found that the inverse propagator of ground-state scalar quark-antiquark field
has no zeroes in complex plane in physically relevant region
$\left|p^{2}\right|<\left(2\ \text{GeV}\right)^{2}$.

Figure 5: The inverse propagator for scalar field emerging from hyperfine
splitting of orbital excitation of vector meson field with respect to
Euclidean momenta $p^{2}$.

The computation has been performed in the lowest approximation with respect to
the mixing of radially excited states in the functional (14), and it would be
interesting and important to check the stability of obtained results in this
respect by accounting the higher radial excitations, which will also allow one
to estimate the polarizabilities of radially excited pion and kaon states.
However, the latter has mostly purely theoretical importance as experimental
measurement seems to be hardly achievable. More detailed discussion about
relation of the present approach to ChPT is an interesting issue which we
expect to address in future work.

##  V Acknowledgements

We are grateful for resources provided by “Govorun” supercomputer at the Joint
Institute for Nuclear Research.

##  Appendix A Evaluation of diagrams

###  A.1 Formulas for diagrams in Figure 3

The notations are given in Sections II and III. Indices $a,b$ correspond to
combination of flavor matrices for a given pseudoscalar ground-state meson.
The one-loop contribution to $M_{\mu\nu}^{(a)}$ in Fig. 2 is given by

$M_{\mu\nu}^{\text{(a)}}=\sum_{\text{k}=\text{a,b,c,d,e,f,g,h,i,j}}M_{\mu\nu}^{\text{(a,k)}}+\text{crossed
diagrams}.$

Here $M_{\mu\nu}^{\text{(a,k)}}$ are given by diagrams in Fig. 3:

$\displaystyle\begin{split}(2\pi)^{4}\delta^{(4)}(p^{\prime}+q^{\prime}-p-q)M_{\mu\nu}^{\text{(a,a)}}=e^{2}h_{P}^{2}&\int\\!d\sigma_{B}\int\\!d^{4}x\int\\!d^{4}y\exp(ipx+iqx-
iq^{\prime}x-ip^{\prime}y)\\\
&\times(-1)\text{Tr}V_{\mu\nu}^{aP00}(x;-q,q^{\prime})S(x,y)V^{bP00}(y)S(y,x),\end{split}$
(a)
$\displaystyle\begin{split}(2\pi)^{4}\delta^{(4)}(p^{\prime}+q^{\prime}-p-q)M_{\mu\nu}^{\text{(a,b)}}=e^{2}h_{P}^{2}&\int\\!d\sigma_{B}\int\\!d^{4}x\int\\!d^{4}y\exp(ipx+iqy-
iq^{\prime}y-ip^{\prime}y)\\\
&\times(-1)\text{Tr}V^{aP00}(x)S(x,y)V_{\mu\nu}^{bP00}(y;-q,q^{\prime})S(y,x),\end{split}$
(b)
$\displaystyle\begin{split}(2\pi)^{4}\delta^{(4)}(p^{\prime}+q^{\prime}-p-q)M_{\mu\nu}^{\text{(a,c)}}=e^{2}h_{P}^{2}&\int\\!d\sigma_{B}\int\\!d^{4}x\int\\!d^{4}y\int\\!d^{4}z_{1}\int\\!d^{4}z_{2}\exp(ipx-
ip^{\prime}y+iqz_{1}-iq^{\prime}z_{2})\\\
&\times(-1)\text{Tr}V^{aP00}(x)S(x,y)V^{bP00}(y)S(y,z_{1})Q\gamma_{\mu}S(z_{1},z_{2})Q\gamma_{\nu}S(z_{2},x),\end{split}$
(c)
$\displaystyle\begin{split}(2\pi)^{4}\delta^{(4)}(p^{\prime}+q^{\prime}-p-q)M_{\mu\nu}^{\text{(a,d)}}=e^{2}h_{P}^{2}&\int\\!d\sigma_{B}\int\\!d^{4}x\int\\!d^{4}y\int\\!d^{4}z_{1}\int\\!d^{4}z_{2}\exp(ipx-
ip^{\prime}y+iqz_{1}-iq^{\prime}z_{2})\\\
&\times(-1)\text{Tr}V^{aP00}(x)S(x,z_{1})Q\gamma_{\mu}S(z_{1},z_{2})Q\gamma_{\nu}S(z_{2},y)V^{bP00}(y)S(y,x),\end{split}$
(d)
$\displaystyle\begin{split}(2\pi)^{4}\delta^{(4)}(p^{\prime}+q^{\prime}-p-q)M_{\mu\nu}^{\text{(a,e)}}=e^{2}h_{P}^{2}&\int\\!d\sigma_{B}\int\\!d^{4}x\int\\!d^{4}y\int\\!d^{4}z\exp(ipx+iqx-
ip^{\prime}y-iq^{\prime}z)\\\
&\times(-1)\text{Tr}V_{\mu}^{aP00}(x;-q)S(x,y)V^{bP00}(y)S(y,z)Q\gamma_{\nu}S(z,y),\end{split}$
(e)
$\displaystyle\begin{split}(2\pi)^{4}\delta^{(4)}(p^{\prime}+q^{\prime}-p-q)M_{\mu\nu}^{\text{(a,f)}}=e^{2}h_{P}^{2}&\int\\!d\sigma_{B}\int\\!d^{4}x\int\\!d^{4}y\int\\!d^{4}z\exp(ipx+iqx-
ip^{\prime}y-iq^{\prime}y)\\\
&\times(-1)\text{Tr}V_{\mu}^{aP00}(x;-q)S(x,y)V_{\nu}^{bP00}(y;q^{\prime})S(y,x),\end{split}$
(f)
$\displaystyle\begin{split}(2\pi)^{4}\delta^{(4)}(p^{\prime}+q^{\prime}-p-q)M_{\mu\nu}^{\text{(a,g)}}=e^{2}h_{P}^{2}&\int\\!d\sigma_{B}\int\\!d^{4}x\int\\!d^{4}y\int\\!d^{4}z\exp(ipx+iqx-
iq^{\prime}z-ip^{\prime}y)\\\
&\times(-1)\text{Tr}V_{\mu}^{aP00}(x;-q)S(x,z)Q\gamma_{\nu}S(z,y)V^{bP00}(y)S(y,x),\end{split}$
(g)
$\displaystyle\begin{split}(2\pi)^{4}\delta^{(4)}(p^{\prime}+q^{\prime}-p-q)M_{\mu\nu}^{\text{(a,h)}}=e^{2}h_{P}^{2}&\int\\!d\sigma_{B}\int\\!d^{4}x\int\\!d^{4}y\int\\!d^{4}z\exp(ipx+iqy-
ip^{\prime}y-iq^{\prime}z)\\\
&\times(-1)\text{Tr}V^{aP00}(x)S(x,y)V_{\mu}^{bP00}(y;-q)S(y,z)Q\gamma_{\nu}S(z,x),\end{split}$
(h)
$\displaystyle\begin{split}(2\pi)^{4}\delta^{(4)}(p^{\prime}+q^{\prime}-p-q)M_{\mu\nu}^{\text{(a,i)}}=e^{2}h_{P}^{2}&\int\\!d\sigma_{B}\int\\!d^{4}x\int\\!d^{4}y\int\\!d^{4}z_{1}\int\\!d^{4}z_{2}\exp(ipx-
ip^{\prime}y+iqz_{1}-iq^{\prime}z_{2})\\\
&\times(-1)\text{Tr}V^{aP00}(x)S(x,z_{1})Q\gamma_{\mu}S(z_{1},y)V^{bP00}(y)S(y,z_{2})Q\gamma_{\nu}S(z_{2},x),\end{split}$
(i)
$\displaystyle\begin{split}(2\pi)^{4}\delta^{(4)}(p^{\prime}+q^{\prime}-p-q)M_{\mu\nu}^{\text{(a,j)}}=e^{2}h_{P}^{2}&\int\\!d\sigma_{B}\int\\!d^{4}x\int\\!d^{4}y\int\\!d^{4}z\exp(ipx+iqz-
ip^{\prime}y-iq^{\prime}y)\\\
&\times(-1)\text{Tr}V^{aP00}(x)S(x,z)Q\gamma_{\mu}S(z,y)V_{\mu}^{bP00}(y;q^{\prime})S(y,x).\end{split}$
(j)

The trace is taken with respect to flavor, color and spinor indices. Crossed
diagrams can be obtained by $q\leftrightarrow
q^{\prime},\mu\leftrightarrow\nu$. The vertex operator

$V^{aP00}(x)=\mathcal{M}^{a}i\gamma_{5}\int_{0}^{1}dt\
\exp\left(\frac{\stackrel{{\scriptstyle\leftrightarrow}}{{\cal
D}}^{2}\\!\\!\\!(x)}{\Lambda^{2}}t\right)$

is a function of $\stackrel{{\scriptstyle\leftrightarrow}}{{\cal D}}_{\mu}$
which acts as

$\displaystyle S_{f}(y-x)\stackrel{{\scriptstyle\leftrightarrow}}{{\cal
D}}_{\mu}\\!(x)\ S_{f^{\prime}}(x-z)$ $\displaystyle=$ $\displaystyle
e^{-\frac{i}{2}y_{\mu}\hat{B}_{\mu\nu}x_{\nu}}\int\frac{d^{4}p}{(2\pi)^{4}}e^{-ip(y-x)}\tilde{H}_{f}(p)\stackrel{{\scriptstyle\leftrightarrow}}{{\cal
D}}_{\mu}\\!(x)\
e^{-\frac{i}{2}x_{\mu}\hat{B}_{\mu\nu}z_{\nu}}\int\frac{d^{4}q}{(2\pi)^{4}}e^{-iq(x-z)}\tilde{H}_{f^{\prime}}(q)$
$\displaystyle=$
$\displaystyle\int\frac{d^{4}p}{(2\pi)^{4}}\int\frac{d^{4}q}{(2\pi)^{4}}e^{-\frac{i}{2}y_{\mu}\hat{B}_{\mu\nu}x_{\nu}}e^{-ip(y-x)}\tilde{H}_{f}(p)\left\\{\xi_{f}\left[ip_{\mu}-\frac{i}{2}\hat{B}_{\mu\nu}\left(x_{\nu}-y_{\nu}\right)\right]+\xi_{f^{\prime}}\left[iq_{\mu}-\frac{i}{2}\hat{B}_{\mu\nu}\left(x_{\nu}-z_{\nu}\right)\right]\right\\}$
$\displaystyle\times
e^{-\frac{i}{2}x_{\mu}\hat{B}_{\mu\nu}z_{\nu}}e^{-iq(x-z)}\tilde{H}_{f^{\prime}}(q).$

The loop integrals are finite due to non-local meson vertices, so no
regularization is needed. With meson vertices and quark propagators given by
formulas (10) and (11), the space and momentum integrals are Gaussian and can
be computed analytically. The averaging over background field is performed
with the help of formula (8) where tensor $J_{\mu\nu}$ is a combination of
external momenta of mesons and photons.

After these straightforward transformations one arrives at integrals over
proper times $s_{i},t_{i}$ that originate from vertices and propagators. These
integrals are computed numerically. Unfortunately, the analytical expressions
are too cumbersome to be presented here.

###  A.2 Formulas for diagrams in Figure 4

The one-loop contribution to $\Gamma_{\mu\nu}^{S\gamma\gamma}$ is given by

$\Gamma_{\mu\nu}^{S\gamma\gamma}=\Gamma_{\mu\nu}^{\text{(a)}S\gamma\gamma}+\Gamma_{\mu\nu}^{\text{(b)}S\gamma\gamma}+\Gamma_{\mu\nu}^{\text{(c)}S\gamma\gamma}+\text{crossed
diagrams},$

where $\Gamma_{\mu\nu}^{\text{(k)}S\gamma\gamma}$ are given by diagrams
(a),(b),(c) in Fig. 4. The one-loop contribution to $\Gamma^{SPP}$ is given by
diagram (d) in Fig. 4:

$\displaystyle\begin{split}(2\pi)^{4}\delta^{(4)}(p^{\prime}+q^{\prime}-p-q)\Gamma_{\mu\nu}^{\text{(a)}S\gamma\gamma}=e^{2}&\int\\!d\sigma_{B}\int\\!d^{4}x\exp(ipx-
ip^{\prime}x+iqx-iq^{\prime}x)\\\
&\times(-1)\text{Tr}V_{\mu\nu}^{aS00}(x;-q,q^{\prime})S(x,x),\end{split}$ (a)
$\displaystyle\begin{split}(2\pi)^{4}\delta^{(4)}(p^{\prime}+q^{\prime}-p-q)\Gamma_{\mu\nu}^{\text{(b)}S\gamma\gamma}=e^{2}&\int\\!d\sigma_{B}\int\\!d^{4}x\int\\!d^{4}z\exp(ipx-
ip^{\prime}x+iqx-iq^{\prime}z)\\\
&\times(-1)\text{Tr}V_{\mu}^{aS00}(x;-q)S(x,z)Q\gamma_{\nu}S(z,x),\end{split}$
(b)
$\displaystyle\begin{split}(2\pi)^{4}\delta^{(4)}(p^{\prime}+q^{\prime}-p-q)\Gamma_{\mu\nu}^{\text{(c)}S\gamma\gamma}=e^{2}&\int\\!d\sigma_{B}\int\\!d^{4}x\int\\!d^{4}z_{1}\int\\!d^{4}z_{2}\exp(ipx-
ip^{\prime}x+iqz_{1}-iq^{\prime}z_{2})\\\
&\times(-1)\text{Tr}V^{aS00}(x)S(x,z_{1})Q\gamma_{\mu}S(z_{1},z_{2})Q\gamma_{\nu}S(z_{2},y),\end{split}$
(c)
$\displaystyle\begin{split}(2\pi)^{4}\delta^{(4)}(p^{\prime}+q^{\prime}-p-q)h_{P}^{2}\Gamma^{SPP}=h_{P}^{2}\int\\!d\sigma_{B}&\int\\!d^{4}x\int\\!d^{4}y\int\\!d^{4}z\exp(iqx-
iq^{\prime}x-ip^{\prime}y+ipz)\\\
&\times(-1)\text{Tr}V^{aS00}(x)S(x,y)V^{bP00}(y)S(y,z)V^{cP00}(z)S(z,y).\end{split}$
(d)

The scalar two-point correlation function is an example where the final
formula used for numerical computation has concise form:

$\displaystyle\Gamma_{abS}^{(2)}(x-y)=$
$\displaystyle\int\\!d\sigma_{B}\text{Tr}V^{aS00}(x)S(x,y)V^{bS00}(y)S(y,x),$
$\displaystyle\tilde{\Gamma}_{abS}^{(2)}(p)=$
$\displaystyle\mathrm{Tr}\left(\mathcal{M}^{a}\mathcal{M}^{b}\right)\frac{\Lambda^{2}}{4\pi^{2}}\
\textrm{Tr}_{\upsilon}\int\limits_{0}^{1}\\!dt_{1}\\!\int\limits_{0}^{1}\\!dt_{2}\\!\int\limits_{0}^{1}\\!ds_{1}\\!\int\limits_{0}^{1}\\!ds_{2}\\!\left(\frac{1-s_{1}}{1+s_{1}}\right)^{m_{f}^{2}/4\upsilon\Lambda^{2}}\left(\frac{1-s_{2}}{1+s_{2}}\right)^{m_{f^{\prime}}^{2}/4\upsilon\Lambda^{2}}\times$
$\displaystyle\times\frac{1}{\Phi_{2}^{2}}\left[\frac{p^{2}}{\Lambda^{2}}\frac{F_{1}}{\Phi_{2}^{2}}+\frac{m_{f}m_{f^{\prime}}}{\Lambda^{2}}\frac{F_{2}}{(1-s_{1}^{2})(1-s_{2}^{2})}+\frac{F_{3}}{\Phi_{2}}\right]\exp\left\\{-\frac{p^{2}}{2\upsilon\Lambda^{2}}\frac{\Phi_{1}}{\Phi_{2}}\right\\},$

where

$\displaystyle\Phi_{1}=s_{1}s_{2}+2\left(\xi_{f^{\prime}}^{2}s_{1}+\xi_{f}^{2}s_{2}\right)(t_{1}+t_{2})v,$
$\displaystyle\Phi_{2}=s_{1}+s_{2}+2(1+s_{1}s_{2})(t_{1}+t_{2})\upsilon+16(\xi_{f^{\prime}}^{2}s_{1}+\xi_{f}^{2}s_{2})t_{1}t_{2}\upsilon^{2},$
$\displaystyle
F_{1}=(1+s_{1}s_{2})\left[2(\xi_{f^{\prime}}s_{1}+\xi_{f}s_{2})(t_{1}+t_{2})\upsilon+4\xi_{f}\xi_{f^{\prime}}(1+s_{1}s_{2})(t_{1}+t_{2})^{2}\upsilon^{2}+s_{1}s_{2}(1-16\xi_{f}\xi_{f^{\prime}}t_{1}t_{2}\upsilon^{2})\right],$
$\displaystyle F_{2}=(1+s_{1}s_{2})^{2},$ $\displaystyle
F_{3}=4\upsilon(1+s_{1}s_{2})(-1+16\xi_{f}\xi_{f^{\prime}}t_{1}t_{2}\upsilon^{2}).$

Analogous formulas for $\Gamma_{V,P}$ obtained in Ref. [35] describe the
spectrum of radially excited mesons: light, heavy-light mesons and heavy
quarkonia.

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