# Light-front approach to axial-vector quarkonium $\gamma^{*}\gamma^{*}$ form
factors

Izabela Babiarz<EMAIL_ADDRESS>Institute of Nuclear Physics,
Polish Academy of Sciences, ul. Radzikowskiego 152, PL-31-342 Kraków, Poland
Roman Pasechnik<EMAIL_ADDRESS>Department of Astronomy and
Theoretical Physics, Lund University, SE-223 62 Lund, Sweden  Wolfgang Schäfer
<EMAIL_ADDRESS>Institute of Nuclear Physics, Polish Academy of
Sciences, ul. Radzikowskiego 152, PL-31-342 Kraków, Poland  Antoni Szczurek
<EMAIL_ADDRESS>Institute of Nuclear Physics, Polish Academy of
Sciences, ul. Radzikowskiego 152, PL-31-342 Kraków, Poland  Faculty of
Mathematics and Natural Sciences, University of Rzeszów, ul. Pigonia 1,
PL-35-310 Rzeszów, Poland

###### Abstract

In this work, we perform a detailed study of transition form factors for
axial-vector meson production via the two-photon fusion process
$\gamma^{*}\gamma^{*}\to 1^{++}$, with space-like virtual photons in the
initial state and a $P$-wave axial-vector quarkonium in the final state. In
this analysis, we employ the formalism of light-front quarkonium wave
functions obtained from a solution of the Schrödinger equation for a selection
of interquark potentials for $Q\bar{Q}$ interaction. We found the helicity
structure and covariant decomposition of the matrix elements that can be
generically applied for any $q\bar{q}$ axial-vector meson
$\gamma^{*}\gamma^{*}\to 1^{++}$ transition, while our numerical results are
given for the phenomenologically relevant charmonium $\chi_{c1}$ state. We
present the helicity form factors as functions of both photon virtualities. We
also obtain, that $QF_{\rm LT}(Q^{2},0)/F_{\rm TT}(Q^{2},0)={\rm const.}$

###### pacs:

12.38.Bx, 13.85.Ni, 14.40.Pq

##  I Introduction

Exclusive production in $e^{+}e^{-}$ collisions in $\gamma\gamma$ fusion has
been considered as a probe for internal structure of QCD bounds states
quantified by meson-photon transition form factors Chernyak and Eidelman
(2015); Köpp  _et al._ (1974); Poppe (1986); Schuler  _et al._ (1998); Li  _et
al._ (2022). Besides the QED production mechanism, such form factor can be
used also in color-singlet QCD production mechanism via gluon-gluon fusion.
The latter has significant implications in physics of vector and axial-vector
meson production in hadronic and nuclear collisions, which recently attracts a
lot of attention Brambilla  _et al._ (2011).

In Refs. Babiarz  _et al._ (2019, 2020a, 2020b), we have analysed production
of scalar and pseudo-scalar mesons in the light-cone formulation which are
$P$\- and $S$-wave $Q\bar{Q}$ states, respectively. We present the results
including the intrinsic (anti)quark motion in the meson wave function, and
also taking the non-relativistic QCD (NRQCD) limit which in our case of the
$P$-wave meson is proportional to the derivative of the wave function at the
origin.

In this work, we continue our studies of the light-cone approach to the
transition form factors focusing on $\gamma\gamma$ fusion into axial-vector
$J^{PC}=1^{++}$ mesons. We utilise several well-known models for the
interquark potentials in determination of the axial-vector meson wave function
on the light cone. Our consideration is general for quarkonium states while
our numerical analysis would be concentrated on $\chi_{c,J=1}$ state. The
corresponding form factors are derived for various meson polarisation states.
Provided that, due to the Landau-Yang theorem Landau (1948); Yang (1950), the
axial-vector meson does not decay into a pair of real photons, the
corresponding transition form factors and hence the production helicity
amplitudes become non-trivial only in the case of virtual initial-state
photons.

Our formulas could easily be extended to light $f_{1}$ or $a_{1}$ mesons,
adopting an appropriate light-front quark model. Here we restrict ourselves to
heavy quarkonia in the numerical predictions as for light mesons the
production is not fully under control Szczurek (2020).

The paper is organised as follows. In Sect. II we recall the basics of the LF
approach to the axial-vector quarkonium and derive the
$\gamma^{*}\gamma^{*}\to A$ amplitude in terms of the LFWF of the bound state.
In Sect. III we present a general covariant description of the
$\gamma^{*}\gamma^{*}\to 1^{++}$ amplitude in terms of invariant helicity form
factors which we relate to the transition form factors of our light cone
amplitude. In Sect. IV we show our numerical results for the relevant form
factors for the case of the $\chi_{c1}$ meson. We will also dicuss the reduced
width for the $\gamma^{*}\gamma$ decay and the $Q^{2}$ dependence of the
$\gamma^{*}\gamma$ cross section relevant to single tagged $e^{+}e^{-}$
collisions. Finally, concluding remarks and a summary are given in Sect. V.

##  II Light-front formulation of the $\gamma^{*}\gamma^{*}\to 1^{++}$ process

###  II.1 Axial-vector meson wave function

In order to construct the light-front wave function (LFWF) of the $Q\bar{Q}$
bound state of good angular momentum quantum numbers, we follow the procedure
based on the Melosh transformation Melosh (1974); Jaus (1990) and Terent’ev
substitution Terentev (1976) starting from the quark-model rest-frame wave
function (WF). In this work, we are particularly focused on the axial-vector
$J^{PC}=1^{++}$ quarkonium state (known as ${}^{2S+1}L_{J}={}^{3}P_{1}$),
whose WF depends on the meson spin projections $\lambda_{A}=\pm 1,0$, as well
as on polarizations $\tau,\bar{\tau}$ of the heavy quark and antiquark (with
mass $m_{Q}$):

$\displaystyle\Psi^{(\lambda_{A})}_{\tau\bar{\tau}}(\vec{k})$ $\displaystyle=$
$\displaystyle\sum_{L_{z}+S_{z}=\lambda_{A}}Y_{1L_{z}}(\hat{k})\Big{\langle}{\frac{1}{2}\frac{1}{2}\tau\bar{\tau}}|{1S_{z}}\Big{\rangle}\langle{11L_{z}S_{z}}|{1\lambda_{A}}\rangle{u(k)\over
k}$ (1) $\displaystyle=$ $\displaystyle{1\over 2}\sqrt{3\over
4\pi}\,\xi^{\tau\dagger}_{Q}\,\,\Big{(}\vec{\sigma}\cdot{\vec{k}\times\vec{E}(\lambda_{A})\over
k}\Big{)}i\sigma_{2}\,\xi^{\bar{\tau}*}_{\bar{Q}}\,\,{u(k)\over k}\,.$

Here, $\vec{k}$ is the three-momentum of the heavy quark in the rest frame of
the $Q\bar{Q}$ pair, $k=|\vec{k}|$, $\vec{E}(\lambda_{A})$ is the polarization
vector of the axial-vector meson defined in Appendix A, and $u(k)$ is the
radial WF which we obtain from solutions of the Schrödinger equation for
several different models of the quark-antiquark potential.

The Melosh transform of the spin-orbital part of the WF relies on the
transition to LF spinors $\chi_{Q}^{\lambda},\chi_{\bar{Q}}^{\bar{\lambda}*}$,
by means of a rotation:

$\displaystyle\xi_{Q}$ $\displaystyle=$ $\displaystyle
R(z,\mbox{\boldmath$k$})\chi_{Q}\,,\qquad\xi^{*}_{\bar{Q}}=R^{*}(1-z,-\mbox{\boldmath$k$})\chi^{*}_{\bar{Q}}\,,$
(2)

with the unitary matrix $R(z,\mbox{\boldmath$k$})$ defined as

$\displaystyle
R(z,\mbox{\boldmath$k$})={m_{Q}+zM_{Q\bar{Q}}-i\vec{\sigma}\cdot(\vec{n}\times\vec{k})\over\sqrt{(m_{Q}+zM_{Q\bar{Q}})^{2}+\mbox{\boldmath$k$}^{2}}}={m_{Q}+zM_{Q\bar{Q}}-i\vec{\sigma}\cdot(\vec{n}\times\vec{k})\over\sqrt{zM_{Q\bar{Q}}(M_{Q\bar{Q}}+2m_{Q})}}\,.$
(3)

Here, $z$ and $1-z$ are the fractions of the meson’s LF plus-momentum carried
by the quark and antiquark, respectively, while their transverse momenta are
denoted by $\pm\mbox{\boldmath$k$}$. We also introduced the unit vector
$\vec{n}=(0,0,1)$ such that the vector product reads,
$\vec{n}\times\vec{k}=(-k_{y},k_{x},0)$. Furthermore, $M_{Q\bar{Q}}$ is the
invariant mass of the $Q\bar{Q}$ system obtained from

$\displaystyle M_{Q\bar{Q}}^{2}={\mbox{\boldmath$k$}^{2}+m_{Q}^{2}\over
z(1-z)}\,.$ (4)

The LFWF then is obtained as follows

$\displaystyle\Psi^{(\lambda_{A})}_{\lambda\bar{\lambda}}(z,\mbox{\boldmath$k$})=\chi_{Q}^{\lambda\dagger}\,{\cal
O}^{\prime}_{\lambda_{A}}\,i\sigma_{2}\,\chi_{\bar{Q}}^{\bar{\lambda}*}\,\,\psi(z,\mbox{\boldmath$k$})\,\sqrt{2(M_{Q\bar{Q}}^{2}-4m_{Q}^{2})}\,,$
(5)

where we pull out a square-root factor to simplify formulas further on. The
spin-orbital part is encoded in the $2\times 2$-matrix,

$\displaystyle{\cal O}^{\prime}_{\lambda_{A}}=\sqrt{3\over
2}\,R^{\dagger}(z,\mbox{\boldmath$k$})\Big{(}\vec{\sigma}\cdot{\vec{k}\times\vec{E}(\lambda_{A})\over\sqrt{2}k}\Big{)}R(1-z,-\mbox{\boldmath$k$})\,.$
(6)

The momentum three-vector in the rest frame of the pair is obtained in terms
of LF variables as

$\displaystyle\vec{k}=(\mbox{\boldmath$k$},k_{z})=\Big{(}\mbox{\boldmath$k$},{1\over
2}(2z-1)M_{Q\bar{Q}}\Big{)}\,.$ (7)

Finally, the “radial” LFWF is related to its rest-frame counterpart as

$\displaystyle\psi(z,\mbox{\boldmath$k$})={\pi\sqrt{M_{Q\bar{Q}}}\over
2\sqrt{2}}\,\,{u(k)\over k^{2}}\,.$ (8)

Explicit expressions for the LFWF of Eq. (5) are found in Appendix A.

###  II.2 Master formula for the light-front helicity amplitude

We now turn to a calculation of the $\gamma^{*}\gamma^{*}$-fusion amplitude.
Analogously to our previous works Babiarz  _et al._ (2020a, 2019, b), we write
down the amplitude in the frame where the photon momenta are
$q_{1\mu}=q_{1+}n^{+}_{\mu}+q^{\perp}_{1\mu}$ and
$q_{2\mu}=q_{2-}n^{-}_{\mu}+q^{\perp}_{2\mu}$, with the light-like vectors
$n^{+}_{\mu},n^{-}_{\mu}$, which fulfill $n^{+}\cdot n^{-}=1$. It proves
convenient to choose the off-shell polarization-vectors $n^{+}_{\mu}$ and
$n^{-}_{\mu}$ for the first and second photon, respectively. Our amplitude can
be obtained from the convolution of the $\gamma^{*}\gamma^{*}\to Q\bar{Q}$
amplitude with the LFWF of the axial-vector meson with polarization
$\lambda_{A}$:

$\displaystyle n^{+\mu}n^{-\nu}{\cal M}_{\mu\nu}(\gamma^{*}\gamma^{*}\to
A(\lambda_{A}))={\cal
N}\int\frac{dzd^{2}\mbox{\boldmath$k$}}{z(1-z)16\pi^{3}}\sum_{\lambda\bar{\lambda}}\Psi^{(\lambda_{A})*}_{\lambda\bar{\lambda}}(z,\mbox{\boldmath$k$})n^{+\mu}n^{-\nu}{\cal
A}_{\mu\nu}^{\lambda\bar{\lambda}}\,,$ (9)

where we pulled out the fine structure constant $\alpha_{\rm em}$ and a
normalization factor from the sum over colors, such that

$\displaystyle{\cal N}=4\pi\alpha_{\rm
em}e_{f}^{2}\,{\mathrm{Tr}\,\openone_{\mathrm{color}}\over\sqrt{N_{c}}}=4\pi\alpha_{\rm
em}e_{f}^{2}\sqrt{N_{c}}\,.$ (10)

Here, $N_{c}=3$, and the electric charge $e_{f}=2/3$ for the charm quark of
interest in this work.

Our master formula Babiarz (2022) for the light-front helicity amplitude of
the $\gamma^{*}\gamma^{*}\to 1^{++}$ process reads

$\displaystyle{\cal
T}(\lambda_{A})\equiv\int\frac{dzd^{2}\mbox{\boldmath$k$}}{z(1-z)16\pi^{3}}\sum_{\lambda\bar{\lambda}}\Psi^{(\lambda_{A})*}_{\lambda\bar{\lambda}}n^{+\mu}n^{-\nu}{\cal
A}_{\mu\nu}^{\lambda\bar{\lambda}}=(-2)\int{dz\,d^{2}\mbox{\boldmath$k$}\over\sqrt{z(1-z)}16\pi^{3}}$
$\displaystyle\times\Big{\\{}-m_{Q}\Big{[}{1\over\mbox{\boldmath$l$}_{A}\,^{2}+\varepsilon^{2}}-{1\over\mbox{\boldmath$l$}_{B}\,^{2}+\varepsilon^{2}}\Big{]}\Big{(}\sqrt{2}(\mbox{\boldmath$e$}(-)\mbox{\boldmath$q$}_{1})\Psi^{(\lambda_{A})*}_{++}(z,\mbox{\boldmath$k$})+\sqrt{2}(\mbox{\boldmath$e$}(+)\mbox{\boldmath$q$}_{1})\Psi^{(\lambda_{A})*}_{--}(z,\mbox{\boldmath$k$})\Big{)}$
$\displaystyle+\Big{(}2z(1-z)Q_{1}^{2}+(1-2z)(\mbox{\boldmath$k$}\cdot\mbox{\boldmath$q$}_{1})\Big{)}\Big{[}{1\over\mbox{\boldmath$l$}_{A}\,^{2}+\varepsilon^{2}}-{1\over\mbox{\boldmath$l$}_{B}\,^{2}+\varepsilon^{2}}\Big{]}\Big{(}\Psi^{(\lambda_{A})*}_{+-}(z,\mbox{\boldmath$k$})+\Psi^{(\lambda_{A})*}_{-+}(z,\mbox{\boldmath$k$})\Big{)}$
$\displaystyle-(1-2z)(\mbox{\boldmath$q$}_{1}\cdot\mbox{\boldmath$q$}_{2})\Big{[}{1-z\over\mbox{\boldmath$l$}_{A}\,^{2}+\varepsilon^{2}}+{z\over\mbox{\boldmath$l$}_{B}\,^{2}+\varepsilon^{2}}\Big{]}\Big{(}\Psi^{(\lambda_{A})*}_{+-}(z,\mbox{\boldmath$k$})+\Psi^{(\lambda_{A})*}_{-+}(z,\mbox{\boldmath$k$})\Big{)}$
$\displaystyle+i[\mbox{\boldmath$k$},\mbox{\boldmath$q$}_{1}]\Big{[}{1\over\mbox{\boldmath$l$}_{A}\,^{2}+\varepsilon^{2}}-{1\over\mbox{\boldmath$l$}_{B}\,^{2}+\varepsilon^{2}}\Big{]}\Big{(}\Psi^{(\lambda_{A})*}_{+-}(z,\mbox{\boldmath$k$})-\Psi^{(\lambda_{A})*}_{-+}(z,\mbox{\boldmath$k$})\Big{)}$
$\displaystyle+i[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}]\Big{[}{1-z\over\mbox{\boldmath$l$}_{A}\,^{2}+\varepsilon^{2}}+{z\over\mbox{\boldmath$l$}_{B}\,^{2}+\varepsilon^{2}}\Big{]}\Big{(}\Psi^{(\lambda_{A})*}_{+-}(z,\mbox{\boldmath$k$})-\Psi^{(\lambda_{A})*}_{-+}(z,\mbox{\boldmath$k$})\Big{)}\Big{\\}}\,.$
(11)

Here, we have introduced the shorthand notation,

$\displaystyle\mbox{\boldmath$l$}_{A}=-\mbox{\boldmath$k$}+(1-z)\mbox{\boldmath$q$}_{2}\,,\qquad\mbox{\boldmath$l$}_{B}=\mbox{\boldmath$k$}+z\mbox{\boldmath$q$}_{2}\,,\qquad
Q_{1}^{2}\equiv\mbox{\boldmath$q$}_{1}^{2}\,,$ (12)

and

$\displaystyle\varepsilon^{2}=m_{Q}^{2}+z(1-z)Q_{1}^{2}\,.$ (13)

We now wish to derive a representation of the helicity amplitudes in terms of
the radial LFWFs, which can be readily used in a numerical analysis. To this
end, we introduce an abbreviated notation for integrals over the radial LFWF:

$\displaystyle\langle{f(z,\mbox{\boldmath$k$})}\rangle\equiv(-2){\sqrt{3\over
2}}\int{dzd^{2}\mbox{\boldmath$k$}\over
z(1-z)16\pi^{3}}\psi(z,\mbox{\boldmath$k$})\,f(z,\mbox{\boldmath$k$})\,,$ (14)

as well as

$\displaystyle{\cal
A}={1\over\mbox{\boldmath$l$}_{A}^{2}+\varepsilon^{2}}-{1\over\mbox{\boldmath$l$}_{B}^{2}+\varepsilon^{2}}\,,\qquad{\cal
B}={1-z\over\mbox{\boldmath$l$}_{A}^{2}+\varepsilon^{2}}+{z\over\mbox{\boldmath$l$}_{B}^{2}+\varepsilon^{2}}\,.$
(15)

###  II.3 Longitudinally polarised meson: $\lambda_{A}=0$

Starting from the master formula (11), one straightforwardly obtains the
helicity amplitude for $\lambda_{A}=0$ state production:

$\displaystyle{\cal T}(0)$ $\displaystyle=$ $\displaystyle(-2i)\sqrt{{3\over
2}}\int{dzd^{2}\mbox{\boldmath$k$}\over
z(1-z)16\pi^{3}}\psi(z,\mbox{\boldmath$k$})\Big{\\{}{4m_{Q}^{2}\over
M_{Q\bar{Q}}}[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$k$}]{\cal
A}+{4\mbox{\boldmath$k$}^{2}\over
M_{Q\bar{Q}}}[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$k$}]{\cal A}$ (16)
$\displaystyle+$
$\displaystyle[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}]{-4\mbox{\boldmath$k$}^{2}\over
M_{Q\bar{Q}}}{\cal
B}\Big{\\}}=i[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}]I_{0}+i[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$I$}_{1}]\,,$

where

$\displaystyle I_{0}=-4\,\left\langle{{\mbox{\boldmath$k$}^{2}\over
M_{Q\bar{Q}}}\,{\cal
B}}\right\rangle\,,\qquad\mbox{\boldmath$I$}_{1}=4\left\langle{\mbox{\boldmath$k$}\,{m_{Q}^{2}+\mbox{\boldmath$k$}^{2}\over
M_{Q\bar{Q}}}\,{\cal A}}\right\rangle\,.$ (17)

Now, by rotational invariance, we can write

$\displaystyle\mbox{\boldmath$I$}_{1}=\mbox{\boldmath$q$}_{2}\,I_{1}\,,\qquad
I_{1}={\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$I$}_{1}\over
Q_{2}^{2}}={4\over
Q_{2}^{2}}\left\langle{(\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$k$})\,{m_{Q}^{2}+\mbox{\boldmath$k$}^{2}\over
M_{Q\bar{Q}}}\,{\cal A}}\right\rangle\,,\qquad
Q_{2}^{2}\equiv\mbox{\boldmath$q$}_{2}^{2}\,.$ (18)

We remind the reader that $M_{Q\bar{Q}}$ depends both on $z$ and
$\mbox{\boldmath$k$}^{2}$. Finally, our amplitude reads

$\displaystyle{\cal
T}(0)=i[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}]\Big{(}I_{0}(Q_{1}^{2},Q_{2}^{2})+I_{1}(Q_{1}^{2},Q_{2}^{2})\Big{)}\equiv
i[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}]\,\Phi_{0}(Q_{1}^{2},Q_{2}^{2})\,.$
(19)

We can write out the explicit expressions for integrals $I_{0},I_{1}$:

$\displaystyle I_{0}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle=$ $\displaystyle
8\sqrt{3\over 2}\int{dzd^{2}\mbox{\boldmath$k$}\over
z(1-z)16\pi^{3}}\,\psi(z,\mbox{\boldmath$k$})\,{\mbox{\boldmath$k$}^{2}\over
M_{Q\bar{Q}}}\Big{(}{1-z\over\mbox{\boldmath$l$}_{A}^{2}+\varepsilon^{2}}+{z\over\mbox{\boldmath$l$}_{B}^{2}+\varepsilon^{2}}\Big{)}\,,$
$\displaystyle I_{1}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle=$
$\displaystyle-8\sqrt{3\over 2}{1\over
Q_{2}^{2}}\int{dzd^{2}\mbox{\boldmath$k$}\over
z(1-z)16\pi^{3}}\psi(z,\mbox{\boldmath$k$}){\mbox{\boldmath$k$}^{2}+m_{Q}^{2}\over
M_{Q\bar{Q}}}\,(\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$k$})\Big{(}{1\over\mbox{\boldmath$l$}_{A}^{2}+\varepsilon^{2}}-{1\over\mbox{\boldmath$l$}_{B}^{2}+\varepsilon^{2}}\Big{)}\,.$

###  II.4 Transversely polarised meson: $\lambda_{A}=\pm 1$

In the case of transversely polarised axial-vector meson, the master formula
(11) leads to

$\displaystyle{\cal T}(\pm)$ $\displaystyle=$ $\displaystyle(-4i)\sqrt{3\over
2}\int{dzd^{2}\mbox{\boldmath$k$}\over
z(1-z)16\pi^{3}}\psi(z,\mbox{\boldmath$k$})\Big{\\{}[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$E$}^{*}]m_{Q}^{2}(1-2z){\cal
A}+2Q_{1}^{2}z(1-z)[\mbox{\boldmath$k$},\mbox{\boldmath$E$}^{*}]{\cal A}$ (21)
$\displaystyle-$
$\displaystyle(1-2z)(\mbox{\boldmath$q$}_{1}\cdot\mbox{\boldmath$q$}_{2})[\mbox{\boldmath$k$},\mbox{\boldmath$E$}^{*}]{\cal
B}-(1-2z)[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}](\mbox{\boldmath$k$}\cdot\mbox{\boldmath$E$}^{*}){\cal
B}+(1-2z)(\mbox{\boldmath$k$}\cdot\mbox{\boldmath$q$}_{1})[\mbox{\boldmath$k$},\mbox{\boldmath$E$}^{*}]{\cal
A}$ $\displaystyle-$
$\displaystyle(1-2z)[\mbox{\boldmath$k$},\mbox{\boldmath$q$}_{1}](\mbox{\boldmath$k$}\cdot\mbox{\boldmath$E$}^{*}){\cal
A}\Big{\\}}\,.$

Here, we have integrals that involve the vector $k$ as well as the tensor
$k_{i}k_{j}$.

Let us introduce the following integrals,

$\displaystyle J_{0}$ $\displaystyle=$ $\displaystyle
4m_{Q}^{2}\langle{(1-2z)\,{\cal
A}}\rangle\,,\qquad{\mbox{\boldmath$J$}_{1}}=8\langle{z(1-z)\,\mbox{\boldmath$k$}\,{\cal
A}}\rangle\,,$ $\displaystyle\mbox{\boldmath$J$}_{2}$ $\displaystyle=$
$\displaystyle 4\langle{(1-2z)\,\mbox{\boldmath$k$}\,{\cal B}}\rangle\,,\qquad
J_{ij}=4\langle{(1-2z)k_{i}k_{j}\,{\cal A}}\rangle\,.$ (22)

Then, our amplitude takes the form,

$\displaystyle{\cal T}(\pm)$ $\displaystyle=$
$\displaystyle[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$E$}^{*}]\,J_{0}+\mbox{\boldmath$q$}_{1}^{2}[\mbox{\boldmath$J$}_{1},\mbox{\boldmath$E$}^{*}]-(\mbox{\boldmath$q$}_{1}\cdot\mbox{\boldmath$q$}_{2})[\mbox{\boldmath$J$}_{2},\mbox{\boldmath$E$}^{*}]-[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}](\mbox{\boldmath$J$}_{2}\cdot\mbox{\boldmath$E$}^{*})$
(23) $\displaystyle+$ $\displaystyle
J_{ij}\varepsilon_{jl}\Big{(}q_{1i}E^{*}_{l}-E^{*}_{i}q_{1l}\Big{)}\,.$

Again, we can simplify this structure by introducing scalar integrals:

$\displaystyle J_{1}$ $\displaystyle=$
$\displaystyle{\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$J$}_{1}\over
Q_{2}^{2}}={8\over
Q_{2}^{2}}\,\langle{z(1-z)\,(\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$k$})\,{\cal
A}}\rangle\,,$ $\displaystyle J_{2}$ $\displaystyle=$
$\displaystyle{\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$J$}_{2}\over
Q_{2}^{2}}={4\over
Q_{2}^{2}}\,\langle{(1-2z)\,(\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$k$})\,{\cal
B}}\rangle\,.$ (24)

The tensor integral can be decomposed into two scalar functions,

$\displaystyle J_{ij}=\Big{(}\delta_{ij}-{q_{2i}q_{2j}\over
Q_{2}^{2}}\Big{)}\,J_{3}+{q_{2i}q_{2j}\over Q_{2}^{2}}\,J_{4}\,,$ (25)

where

$\displaystyle J_{3}={4\over
Q_{2}^{2}}\,\left\langle{(1-2z)\Big{(}Q_{2}^{2}\mbox{\boldmath$k$}^{2}-(\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$k$})^{2}\Big{)}\,{\cal
A}}\right\rangle\,,\quad J_{4}={4\over
Q_{2}^{2}}\,\left\langle{(1-2z)(\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$k$})^{2}\,{\cal
A}}\right\rangle\,.$ (26)

Then our amplitude becomes

$\displaystyle{\cal T}(\pm)$ $\displaystyle=$
$\displaystyle[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$E$}^{*}](J_{0}+2J_{3})+[\mbox{\boldmath$q$}_{2},\mbox{\boldmath$E$}^{*}]Q_{1}^{2}J_{1}$
(27) $\displaystyle+$
$\displaystyle\Big{(}(\mbox{\boldmath$q$}_{1}\cdot\mbox{\boldmath$q$}_{2})[\mbox{\boldmath$q$}_{2},\mbox{\boldmath$E$}^{*}]+[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}](\mbox{\boldmath$q$}_{2},\mbox{\boldmath$E$}^{*})\Big{)}{1\over
Q_{2}^{2}}\Big{(}(J_{4}-J_{3})-Q_{2}^{2}J_{2}\Big{)}\,.$

It can be further simplified by noticing that

$\displaystyle
q_{1i}q_{2j}q_{2k}E^{*}_{l}\Big{(}\delta_{ij}\varepsilon_{kl}+\delta_{jl}\varepsilon_{ik}\Big{)}=q_{1i}q_{2j}q_{2k}E^{*}_{l}\Big{(}\delta_{ij}\varepsilon_{kl}-\delta_{lj}\varepsilon_{ki}\Big{)}\,.$
(28)

The tensor in brackets is antisymmetric in $il$ and only the symmetric part in
$kj$ enters. But in two dimensions, any antisymmetric in $il$ tensor must be
proportional to $\varepsilon_{il}$. One can easily check that in the above
expression one can replace

$\displaystyle\delta_{ij}\varepsilon_{kl}-\delta_{lj}\varepsilon_{ki}\rightarrow\,\varepsilon_{il}\delta_{jk}\,,$
(29)

thus, yielding a useful identity

$\displaystyle(\mbox{\boldmath$q$}_{1}\cdot\mbox{\boldmath$q$}_{2})[\mbox{\boldmath$q$}_{2},\mbox{\boldmath$E$}^{*}]+[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}](\mbox{\boldmath$q$}_{2},\mbox{\boldmath$E$}^{*})=[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$E$}^{*}]Q_{2}^{2}\,.$
(30)

The latter enables us to represent our amplitude in the following compact
form,

$\displaystyle{\cal
T}(\pm)=[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$E$}^{*}]\,\Phi_{1}(Q_{1}^{2},Q_{2}^{2})+[\mbox{\boldmath$q$}_{2},\mbox{\boldmath$E$}^{*}]\,\Phi_{2}(Q_{1}^{2},Q_{2}^{2})\,,$
(31)

with two form factors

$\displaystyle\Phi_{1}=J_{0}+J_{3}+J_{4}-Q_{2}^{2}J_{2}\,,\qquad\Phi_{2}=Q_{1}^{2}\,J_{1}\,,$
(32)

which, using the expressions in Appendix B can be represented in the following
integral form

$\displaystyle\Phi_{1}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle=$
$\displaystyle-4\sqrt{3\over 2}\int{dzd^{2}\mbox{\boldmath$k$}\over
z(1-z)16\pi^{3}}\psi(z,\mbox{\boldmath$k$})(1-2z)\Big{\\{}(\mbox{\boldmath$k$}^{2}+m_{Q}^{2})\Big{(}{1\over\mbox{\boldmath$l$}_{A}^{2}+\varepsilon^{2}}-{1\over\mbox{\boldmath$l$}_{B}^{2}+\varepsilon^{2}}\Big{)}$
(33)
$\displaystyle-(\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$k$})\Big{(}{1-z\over\mbox{\boldmath$l$}_{A}^{2}+\varepsilon^{2}}+{z\over\mbox{\boldmath$l$}_{B}^{2}+\varepsilon^{2}}\Big{)}\Big{\\}}\,,$
$\displaystyle\Phi_{2}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle=$
$\displaystyle-8\sqrt{3\over 2}{Q_{1}^{2}\over
Q_{2}^{2}}\int{dzd^{2}\mbox{\boldmath$k$}\over
z(1-z)16\pi^{3}}\psi(z,\mbox{\boldmath$k$})z(1-z)(\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$k$})\Big{(}{1\over\mbox{\boldmath$l$}_{A}^{2}+\varepsilon^{2}}-{1\over\mbox{\boldmath$l$}_{B}^{2}+\varepsilon^{2}}\Big{)}\,.$

We thus have completed our first task and expressed our amplitude for the
$\gamma^{*}\gamma^{*}\to A(\lambda_{A})$ process in terms of form factors that
are calculated as integrals involving the LFWF.

###  II.5 NRQCD limit

In the nonrelativistic QCD (NRQCD) limit, we expand the integrand of our
amplitude to linear order in $\xi=z-{1\over 2}$ and $k$. Then,

$\displaystyle z={1\over 2}+\xi\,,\quad 1-z={1\over 2}-\xi\,,\quad
z(1-z)={1\over 4}-\xi^{2}\sim{1\over 4}\,.$ (34)

We introduce $\mu^{2}\equiv(Q_{1}^{2}+Q_{2}^{2}+4m_{Q}^{2})/4$, and expand the
relevant combinations of denominators:

$\displaystyle{\cal A}$ $\displaystyle=$
$\displaystyle{1\over\mbox{\boldmath$l$}^{2}_{A}+\varepsilon^{2}}-{1\over\mbox{\boldmath$l$}^{2}_{B}+\varepsilon^{2}}={2\mbox{\boldmath$q$}_{2}\cdot(\mbox{\boldmath$k$}+\xi\mbox{\boldmath$q$}_{2})\over\mu^{4}}\,,\qquad{\cal
B}={1-z\over\mbox{\boldmath$l$}^{2}_{A}+\varepsilon^{2}}+{z\over\mbox{\boldmath$l$}_{B}^{2}+\varepsilon^{2}}={1\over\mu^{2}}\,.$

Inserting these expressions into our master formula, we obtain the amplitude
in the following useful form,

$\displaystyle{\cal T}$ $\displaystyle=$
$\displaystyle{-2\over\mu^{4}}\int{dzd^{2}\mbox{\boldmath$k$}\over\sqrt{z(1-z)}16\pi^{3}}\Big{\\{}i[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}]\mu^{2}\,\Big{(}\Psi^{(\lambda_{A})*}_{+-}(z,\mbox{\boldmath$k$})-\Psi^{(\lambda_{A})*}_{-+}(z,\mbox{\boldmath$k$})\Big{)}$
$\displaystyle+$
$\displaystyle\Big{(}Q_{1}^{2}(\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$k$})+\xi
Q_{1}^{2}Q_{2}^{2}-2\xi\mu^{2}(\mbox{\boldmath$q$}_{1}\cdot\mbox{\boldmath$q$}_{2})\Big{)}\Big{(}\Psi^{(\lambda_{A})*}_{+-}(z,\mbox{\boldmath$k$})+\Psi^{(\lambda_{A})*}_{-+}(z,\mbox{\boldmath$k$})\Big{)}$
$\displaystyle-$ $\displaystyle
2m_{Q}(\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$k$}+\xi\mbox{\boldmath$k$}^{2})\Big{(}\sqrt{2}(\mbox{\boldmath$e$}^{*}(-)\cdot\mbox{\boldmath$q$}_{1})\Psi^{(\lambda_{A})*}_{++}(z,\mbox{\boldmath$k$})+\sqrt{2}(\mbox{\boldmath$e$}^{*}(+)\cdot\mbox{\boldmath$q$}_{1})\Psi^{(\lambda_{A})*}_{--}(z,\mbox{\boldmath$k$})\Big{)}\Big{\\}}\,.$

After somewhat lengthy calculations, we can reduce our result to a simple
form:

$\displaystyle{\cal T}(0)$ $\displaystyle=$ $\displaystyle
i[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}]\,t_{0}\,{Q_{1}^{2}+Q_{2}^{2}\over[Q_{1}^{2}+Q_{2}^{2}+4m_{Q}^{2}]^{2}},$
$\displaystyle{\cal T}(\pm)$ $\displaystyle=$ $\displaystyle
i\Big{(}Q_{1}^{2}[\mbox{\boldmath$e$}^{*},\mbox{\boldmath$q$}_{2}]-Q_{2}^{2}[\mbox{\boldmath$e$}^{*},\mbox{\boldmath$q$}_{1}]\Big{)}\,t_{0}\,{2m_{Q}\over[Q_{1}^{2}+Q_{2}^{2}+4m_{Q}^{2}]^{2}}\,,$
(37)

with

$\displaystyle t_{0}\equiv 16\sqrt{3\over 2}\,{R^{\prime}(0)\over\sqrt{\pi
M^{3}}}\,=4\,R^{\prime}(0)\sqrt{3\over\pi m_{Q}^{3}}\,,$ (38)

where the mass of the resonance $M=2m_{Q}$ in the NRQCD approximation. We can
now easily read off the explicit expressions for the form factors
$\Phi_{0,1,2}(Q_{1}^{2},Q_{2}^{2})$ in the NRQCD limit:

$\displaystyle\Phi_{0}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle=$ $\displaystyle
t_{0}{Q_{1}^{2}+Q_{2}^{2}\over[Q_{1}^{2}+Q_{2}^{2}+4m_{Q}^{2}]^{2}}\,,$
$\displaystyle\Phi_{1}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle=$ $\displaystyle
t_{0}\,{Q_{2}^{2}\,2m_{Q}\over[Q_{1}^{2}+Q_{2}^{2}+4m_{Q}^{2}]^{2}}\,,$
$\displaystyle\Phi_{2}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle=$ $\displaystyle
t_{0}\,{-Q_{1}^{2}\,2m_{Q}\over[Q_{1}^{2}+Q_{2}^{2}+4m_{Q}^{2}]^{2}}\,.$ (39)

##  III Tensor decomposition of the $\gamma^{*}\gamma^{*}\to 1^{++}$ amplitude

###  III.1 Helicity form factors

Let us start by parametrizing the covariant $\gamma^{*}\gamma^{*}\to 1^{++}$
amplitude by three separately gauge invariant tensor structures with the
corresponding invariant form factors $F_{\rm TT}$, $F_{\rm LT}$, and $F_{\rm
TL}$:

$\displaystyle{1\over 4\pi\alpha_{\rm em}}{\cal M}_{\mu\nu\rho}$
$\displaystyle=$ $\displaystyle
i\Big{(}q_{1}-q_{2}+{Q_{1}^{2}-Q_{2}^{2}\over(q_{1}+q_{2})^{2}}(q_{1}+q_{2})\Big{)}_{\rho}\,\tilde{G}_{\mu\nu}\,{M\over
2X}F_{\rm TT}(Q_{1}^{2},Q_{2}^{2})$ (40) $\displaystyle+$ $\displaystyle
ie_{\mu}^{L}(q_{1})\tilde{G}_{\nu\rho}\,{1\over\sqrt{X}}F_{\rm
LT}(Q_{1}^{2},Q_{2}^{2})+ie_{\nu}^{L}(q_{2})\tilde{G}_{\mu\rho}\,{1\over\sqrt{X}}F_{\rm
TL}(Q_{1}^{2},Q_{2}^{2})\,.$

Above we introduced

$\displaystyle\tilde{G}_{\mu\nu}=\varepsilon_{\mu\nu\alpha\beta}q_{1}^{\alpha}q_{2}^{\beta}\,,$
(41)

and the polarization vectors of longitudinal photons

$\displaystyle e_{\mu}^{L}(q_{1})$ $\displaystyle=$
$\displaystyle\sqrt{{-q_{1}^{2}\over X}}\Big{(}q_{2\mu}-{q_{1}\cdot q_{2}\over
q_{1}^{2}}q_{1\mu}\Big{)}\,,\qquad e_{\nu}^{L}(q_{2})=\sqrt{{-q_{2}^{2}\over
X}}\Big{(}q_{1\nu}-{q_{1}\cdot q_{2}\over q_{2}^{2}}q_{2\nu}\Big{)}\,.$ (42)

We are interested in the spacelike virtualities of photons only, i.e.
$Q_{i}^{2}=-q_{i}^{2}=\mbox{\boldmath$q$}_{i}^{2}>0,\,i=1,2$, and we denote
the standard kinematic variable

$\displaystyle X=(q_{1}\cdot q_{2})^{2}-q_{1}^{2}q_{2}^{2}={1\over
4}\Big{(}M^{4}+2M^{2}(Q_{1}^{2}+Q_{2}^{2})+(Q_{1}^{2}-Q_{2}^{2})^{2}\Big{)}>0\,.$
(43)

In Eq. (40) we have introduced kinematic factors in such a way, that the form
factors $F_{\rm TT},F_{\rm LT},F_{\rm TL}$ all have the same mass-dimension as
the invariant amplitude and are related to helicity amplitudes in the
$\gamma^{*}\gamma^{*}$-c.m. frame in a straightforward manner without
additional kinematic factors, for a more detailed discussion, see Appendix C.
Different conventions and tensor structures can be useful depending on the
application, see e.g. Refs. Poppe (1986); Pascalutsa  _et al._ (2012); Köpp
_et al._ (1974); Milstein and Rudenko (2020); Hoferichter and Stoffer (2020).
Then, the amplitude for the final-state meson with polarization $\lambda_{A}$
is obtained as

$\displaystyle{\cal M}_{\mu\nu}(\lambda_{A})={\cal
M}_{\mu\nu\rho}E^{\rho*}(\lambda_{A})\,,$ (44)

in terms of the meson polarization four-vector $E^{\rho}(\lambda_{A})$. All
three tensor structures in (40) are separately orthogonal to the four-momentum
$P_{\rho}=(q_{1}+q_{2})_{\rho}$ of the meson. Here we are interested only in
the case of the on-shell meson, $P^{2}=M^{2}$. For the case of $P^{2}\neq
M^{2}$, the form factors must be regarded as a function of all virtualities,
$Q_{1}^{2},Q_{2}^{2},P^{2}$ and in addition an independent tensor structure
$\propto P_{\rho}\tilde{G}_{\mu\nu}$ appears. The associated form factor is
the “anomaly form factor” of the vector-vector-axial correlator. As we have
$P\cdot E^{*}(\lambda_{A})=0$ for the physical state, it does not contribute
to our problem.

In what follows, we choose a basis of polarization states for the axial-vector
which is convenient for our light-front calculation of the relevant
amplitudes. Such polarization states read explicitly:

$\displaystyle E_{\rho}(0)$ $\displaystyle=$ $\displaystyle{1\over
M}P_{\rho}-{M\over P_{+}}n^{-}_{\rho}\,,\qquad
E_{\rho}(\pm)=E^{\perp}_{\rho}(\pm)-{E^{\perp}(\pm)\cdot P\over
P_{+}}n^{-}_{\rho}\,.$ (45)

Here, the vector $E^{\perp}_{\mu}$ has only transverse components111We use the
light-front parametrization of four-momenta
$a=(a_{+},a_{-},\mbox{\boldmath$a$}),a^{2}=2a_{+}a_{-}-\mbox{\boldmath$a$}^{2}$.
:

$\displaystyle E^{\perp}(\pm)=(0,0,\mbox{\boldmath$E$}(\pm))\quad{\rm
with}\quad\mbox{\boldmath$E$}(\lambda)=-{1\over\sqrt{2}}(\lambda\mbox{\boldmath$e$}_{x}+i\mbox{\boldmath$e$}_{y})\,.$
(46)

Symmetry under exchanging $q_{1}\leftrightarrow
q_{2}\,,\,\mu\leftrightarrow\nu$ (crossing symmetry) entails that our form
factors must satisfy

$\displaystyle F_{\rm TT}(Q_{1}^{2},Q_{2}^{2})=-F_{\rm
TT}(Q_{2}^{2},Q_{1}^{2})\,,\qquad F_{\rm LT}(Q_{1}^{2},Q_{2}^{2})=-F_{\rm
TL}(Q_{2}^{2},Q_{1}^{2})\,.$ (47)

Due to its antisymmetry, the form factor $F_{\rm TT}$ vanishes on the diagonal
$F_{\rm TT}(Q^{2},Q^{2})=0$, and in particular for two on-shell photons
$F_{\rm TT}(0,0)=0$, thereby fulfilling the Landau–Yang theorem. The latter
has no bearing on the behaviour of $F_{\rm LT},F_{\rm TL}$. However, when the
virtuality of the longitudinal photon goes to zero, we have

$\displaystyle F_{\rm LT}(Q_{1}^{2},Q_{2}^{2})\propto Q_{1}\,,\qquad{\rm
for}\,\,Q_{1}^{2}\to 0\,,$ (48)

and accordingly for $F_{\rm TL}$, in order to cancel a spurious singular
behaviour of the tensor decomposition of ${\cal M}_{\mu\nu\rho}$. Note, that
the limit

$\displaystyle f_{\rm LT}=\lim_{Q^{2}\to 0}{F_{\rm LT}(Q^{2},0)\over Q}\,,$
(49)

is therefore finite and generally nonzero. In absence of a finite decay width
to $\gamma\gamma$ it serves as a measure of the two-photon coupling strength
of the axial-vector meson.

###  III.2 Projection on the light-front directions

We now want to calculate the projections of the amplitude on the light-front
directions $n^{+\mu}n^{-\nu}$. We are interested in the frame, where photon
momenta have the form

$\displaystyle q_{1\mu}=q_{1}^{+}n^{+}_{\mu}+q^{\perp}_{1\mu}\,,\qquad
q_{2\mu}=q_{2}^{-}n^{-}_{\mu}+q^{\perp}_{2\mu}\,,$ (50)

such that $Q_{i}^{2}=-q_{i}^{\perp 2}=\mbox{\boldmath$q$}_{i}^{2}$. Let us
collect a few useful contractions:

$\displaystyle n^{+\mu}n^{-\nu}\tilde{G}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\varepsilon^{\perp}_{\alpha\beta}q_{1}^{\alpha}q_{2}^{\beta}=[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}]\,,\quad
n^{+}\cdot e^{L}(q_{1})={Q_{1}\over\sqrt{X}}q_{2}^{-}\,,\quad n^{-}\cdot
e^{L}(q_{2})={Q_{2}\over\sqrt{X}}q_{1}^{+}\,.$ (51)

Our amplitude now takes the form

$\displaystyle{1\over 4\pi\alpha_{\rm em}}n^{+\mu}n^{-\nu}{\cal
M}_{\mu\nu\rho}$ $\displaystyle=$ $\displaystyle
i[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}]\Big{(}q_{1}-q_{2}+{Q_{1}^{2}-Q_{2}^{2}\over
M^{2}}(q_{1}+q_{2})\Big{)}_{\rho}\,{M\over 2X}F_{\rm TT}(Q_{1}^{2},Q_{2}^{2})$
(52) $\displaystyle+$ $\displaystyle i{q_{1}^{+}q_{2}^{-}\over
X}\Big{(}\varepsilon^{\perp}_{\rho\alpha}q^{\perp\alpha}_{2}Q_{1}F_{\rm
LT}(Q_{1}^{2},Q_{2}^{2})+\varepsilon^{\perp}_{\rho\alpha}q^{\perp\alpha}_{1}Q_{2}F_{\rm
TL}(Q_{1}^{2},Q_{2}^{2})\Big{)}$ $\displaystyle+$ $\displaystyle
i{[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}]\over
X}\Big{(}Q_{2}q_{1}^{+}n^{+}_{\rho}F_{\rm
TL}(Q_{1}^{2},Q_{2}^{2})-Q_{1}q_{2}^{-}n_{\rho}^{-}F_{\rm
LT}(Q_{1}^{2},Q_{2}^{2})\Big{)}\,.$

Furthermore, we obtain for the helicity amplitudes

$\displaystyle{1\over 4\pi\alpha_{\rm em}}{\cal M}(0)$ $\displaystyle=$
$\displaystyle n^{+\mu}n^{-\nu}{\cal M}_{\mu\nu\rho}E^{\rho*}(0)$ (53)
$\displaystyle=$
$\displaystyle-i[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}]\Big{\\{}\Big{(}M^{2}+Q_{1}^{2}-Q_{2}^{2}\Big{)}{F_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})\over 2X}+{Q_{2}M\over X}F_{\rm
TL}(Q_{1}^{2},Q_{2}^{2})\Big{\\}}\,.$ $\displaystyle{1\over 4\pi\alpha_{\rm
em}}{\cal M}(\pm)$ $\displaystyle=$ $\displaystyle
i[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}](\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$E$}^{*}(\pm)){MF_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})\over
X}+i[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}](\mbox{\boldmath$P$}\cdot\mbox{\boldmath$E$}^{*}(\pm)){Q_{2}\over
X}F_{\rm TL}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle+$
$\displaystyle{M^{2}+\mbox{\boldmath$P$}^{2}\over
2X}\Big{(}i[\mbox{\boldmath$E$}^{*}(\pm),\mbox{\boldmath$q$}_{2}]Q_{1}F_{\rm
LT}(Q_{1}^{2},Q_{2}^{2})+i[\mbox{\boldmath$E$}^{*}(\pm),\mbox{\boldmath$q$}_{1}]Q_{2}F_{\rm
TL}(Q_{1}^{2},Q_{2}^{2})\Big{)}\,.$

For the extraction of form factors from our amplitude, it is convenient to
rewrite the transverse amplitude using the identities of Eq. (30). We can then
represent our helicity amplitudes as follows:

$\displaystyle{X\over 4\pi\alpha_{\rm em}}{\cal M}(0)$ $\displaystyle=$
$\displaystyle-i[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}]\Big{\\{}{1\over
2}\Big{(}M^{2}+Q_{1}^{2}-Q_{2}^{2}\Big{)}F_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})+MQ_{2}F_{\rm TL}(Q_{1}^{2},Q_{2}^{2})\Big{\\}}\,,$
(54) $\displaystyle{X\over 4\pi\alpha_{\rm em}}{\cal M}(\pm)$ $\displaystyle=$
$\displaystyle
i[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$E$}^{*}]\Big{(}Q_{1}^{2}MF_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})-{1\over 2}(M^{2}+Q_{1}^{2}-Q_{2}^{2})Q_{2}F_{\rm
TL}(Q_{1}^{2},Q_{2}^{2})\Big{)}$
$\displaystyle-i[\mbox{\boldmath$q$}_{2},\mbox{\boldmath$E$}^{*}]\Big{(}Q_{1}^{2}Q_{2}F_{\rm
TL}(Q_{1}^{2},Q_{2}^{2})+{1\over 2}(M^{2}+Q_{1}^{2}+Q_{2}^{2})Q_{1}F_{\rm
LT}(Q_{1}^{2},Q_{2}^{2})\Big{)}$
$\displaystyle-[\mbox{\boldmath$q$}_{2},\mbox{\boldmath$E$}^{*}](\mbox{\boldmath$q$}_{1}\cdot\mbox{\boldmath$q$}_{2})\Big{(}MF_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})+Q_{2}F_{\rm TL}(Q_{1}^{2},Q_{2}^{2})+Q_{1}F_{\rm
LT}(Q_{1}^{2},Q_{2}^{2})\Big{)}\,.$

It is useful to write down symmetric ($F_{S}$) and antisymmetric ($F_{A}$)
combinations of form factors,

$\displaystyle F_{S}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle\equiv$
$\displaystyle Q_{2}F_{\rm TL}(Q_{1}^{2},Q_{2}^{2})-Q_{1}F_{\rm
LT}(Q_{1}^{2},Q_{2}^{2})\,,$ $\displaystyle F_{A}(Q_{1}^{2},Q_{2}^{2})$
$\displaystyle\equiv$ $\displaystyle Q_{2}F_{\rm
TL}(Q_{1}^{2},Q_{2}^{2})+Q_{1}F_{\rm LT}(Q_{1}^{2},Q_{2}^{2})\,.$ (55)

Now we wish to relate the form factors $\Phi_{i}$ of our LF amplitude to the
invariant form factors of the covariant amplitude in Eq. (40). To this end, it
is instructive to turn to the helicity amplitudes corresponding to _linear_
polarizations of the meson. These can then be expanded in a Fourier series in
$\cos n\phi,\sin n\phi$, where $\phi$ is the angle between the photon
transverse momenta $\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}$. Here, we
use the following frame: the longitudinal polarization defines the
$z$-direction, and the $x$-direction is taken along the meson transverse
momentum
$\mbox{\boldmath$P$}=\mbox{\boldmath$q$}_{1}+\mbox{\boldmath$q$}_{2}$. For a
covariant definition of this particular helicity frame, see Appendix D. We can
now project the amplitude onto the linear polarizations and obtain the
following results:

$\displaystyle(-i){\cal T}_{z}$ $\displaystyle=$ $\displaystyle
Q_{1}Q_{2}\sin\phi\,\Phi_{0}\ ,$
$\displaystyle(-i)|\mbox{\boldmath$P$}|\,{\cal T}_{x}$ $\displaystyle=$
$\displaystyle Q_{1}Q_{2}\sin\phi\,\Big{(}\Phi_{1}-\Phi_{2}\Big{)}\,,$
$\displaystyle(-i)|\mbox{\boldmath$P$}|\,{\cal T}_{y}$ $\displaystyle=$
$\displaystyle
Q_{1}^{2}\Phi_{1}+Q_{2}^{2}\Phi_{2}+Q_{1}Q_{2}\cos\phi\Big{(}\Phi_{1}+\Phi_{2}\Big{)}\,.$
(56)

We have collected rather lengthy expressions for our covariant amplitude in
Appendix D. Let us now make some observations on these results. Firstly, we
see that in the amplitudes of Eq. (56), there are no terms proportional to
$\cos 2\phi$ or $\sin 2\phi$. These do appear, however, in the decomposition
of the general invariant amplitude ${\cal M}$ – they obviously are related to
the last term in Eq. (III.2). These terms are all multiplied by the
combination,

$\displaystyle f_{A}(Q_{1}^{2},Q_{2}^{2})\equiv MF_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})+F_{A}(Q_{1}^{2},Q_{2}^{2})\,,$ (57)

which, for consistency with Eq. (56), has to vanish identically, i.e.

$\displaystyle f_{A}(Q_{1}^{2},Q_{2}^{2})\equiv 0\,.$ (58)

This appears to be a peculiar identity related to the $\gamma^{*}\gamma^{*}\to
Q\bar{Q}$ helicity amplitudes. Consequently, the number of independent form
factors is reduced to two, say, $F_{A},F_{S}$. In effect, the form factor for
two transverse photons is entirely determined through a mixture of
longitudinal/transverse ($F_{\rm LT},F_{\rm TL}$) form factors.

Interestingly, after exploiting the cancellation of Eq. (58), all our
amplitudes have simple symmetry properties, namely:

$\displaystyle{\cal M}_{z}$ $\displaystyle=$ $\displaystyle{-iQ_{1}Q_{2}\over
2X}\cdot\sin\phi\cdot f_{S}(Q_{1}^{2},Q_{2}^{2})\,,$
$\displaystyle|\mbox{\boldmath$P$}|{\cal M}_{x}$ $\displaystyle=$
$\displaystyle{-iQ_{1}Q_{2}\over 2X}\cdot\sin\phi\cdot
Mf_{S}(Q_{1}^{2},Q_{2}^{2})\,,$ $\displaystyle|\mbox{\boldmath$P$}|{\cal
M}_{y}$ $\displaystyle=$ $\displaystyle{i\over
2X}\Big{\\{}\Big{(}2Q_{1}^{2}Q_{2}^{2}+\nu(Q_{1}^{2}+Q_{2}^{2})\Big{)}F_{A}(Q_{1}^{2},Q_{2}^{2})+\nu(Q_{1}^{2}-Q_{2}^{2})F_{S}(Q_{1}^{2},Q_{2}^{2})$
(59) $\displaystyle+$ $\displaystyle
Q_{1}Q_{2}\cos\phi\Big{[}(2\nu+Q_{1}^{2}+Q_{2}^{2})F_{A}(Q_{1}^{2},Q_{2}^{2})+(Q_{1}^{2}-Q_{2}^{2})F_{S}(Q_{1}^{2},Q_{2}^{2})\Big{]}\Big{\\}}\,.$

Here, we defined

$\displaystyle\nu\equiv q_{1}\cdot q_{2}={1\over
2}(M^{2}+Q_{1}^{2}+Q_{2}^{2})\,,$ (60)

as well as

$\displaystyle f_{S}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle\equiv$
$\displaystyle(Q_{1}^{2}-Q_{2}^{2})F_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})+MF_{S}(Q_{1}^{2},Q_{2}^{2})$ (61) $\displaystyle=$
$\displaystyle-{Q_{1}^{2}-Q_{2}^{2}\over
M}F_{A}(Q_{1}^{2},Q_{2}^{2})+MF_{S}(Q_{1}^{2},Q_{2}^{2})\,.$

Let us now proceed and try to extract form factors $F_{A},F_{S}$. To this end,
we chose the $y$-component of the amplitude, i.e. the linear polarization
orthogonal to the “production plane”. As a short-hand notation, we define the
form factors

$\displaystyle G_{1}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle\equiv$
$\displaystyle
Q_{1}^{2}\Phi_{1}(Q_{1}^{2},Q_{2}^{2})+Q_{2}^{2}\Phi_{2}(Q_{1}^{2},Q_{2}^{2})$
$\displaystyle=$ $\displaystyle{1\over
2}(Q_{1}^{2}+Q_{2}^{2})\Phi_{A}(Q_{1}^{2},Q_{2}^{2})+{1\over
2}(Q_{1}^{2}-Q_{2}^{2})\Phi_{S}(Q_{1}^{2},Q_{2}^{2})\,,$ $\displaystyle
G_{2}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle\equiv$
$\displaystyle\Phi_{1}(Q_{1}^{2},Q_{2}^{2})+\Phi_{2}(Q_{1}^{2},Q_{2}^{2})=\Phi_{A}(Q_{1}^{2},Q_{2}^{2})\,,$
(62)

where

$\displaystyle\Phi_{S}\equiv\Phi_{1}-\Phi_{2}\,,\qquad\Phi_{A}\equiv\Phi_{1}+\Phi_{2}\,,$
(63)

are symmetric and antisymmetric in $Q_{1}^{2}\leftrightarrow Q_{2}^{2}$
combinations. We then derive a system of linear equations:

$\displaystyle e_{f}^{2}\sqrt{N_{c}}\,\begin{pmatrix}G_{1}\\\
G_{2}\end{pmatrix}={1\over
2X}\begin{pmatrix}2Q_{1}^{2}Q_{2}^{2}+\nu(Q_{1}^{2}+Q_{2}^{2})&\nu(Q_{1}^{2}-Q_{2}^{2})\\\
2\nu+Q_{1}^{2}+Q_{2}^{2}&Q_{1}^{2}-Q_{2}^{2}\end{pmatrix}\begin{pmatrix}F_{A}\\\
F_{S}\end{pmatrix}\,.$ (64)

Here, the determinant of the $2\times 2$ matrix is
$\det(\dots)=-2X(Q_{1}^{2}-Q_{2}^{2})$ so that we can invert the matrix to
obtain

$\displaystyle\begin{pmatrix}F_{A}\\\
F_{S}\end{pmatrix}={-e_{f}^{2}\sqrt{N_{c}}\over
Q_{1}^{2}-Q_{2}^{2}}\begin{pmatrix}Q_{1}^{2}-Q_{2}^{2}&-\nu(Q_{1}^{2}-Q_{2}^{2})\\\
-(2\nu+Q_{1}^{2}+Q_{2}^{2})&2Q_{1}^{2}Q_{2}^{2}+\nu(Q_{1}^{2}+Q_{2}^{2})\end{pmatrix}\begin{pmatrix}G_{1}\\\
G_{2}\end{pmatrix}\,.$ (65)

This finally gives us

$\displaystyle F_{A}$ $\displaystyle=$ $\displaystyle
e_{f}^{2}\sqrt{N_{c}}\Big{\\{}(\nu-Q_{1}^{2})\Phi_{1}+(\nu-
Q_{2}^{2})\Phi_{2}\Big{\\}}$ (66) $\displaystyle=$
$\displaystyle{e_{f}^{2}\sqrt{N_{c}}\over
2}\Big{\\{}M^{2}\Phi_{A}-(Q_{1}^{2}-Q_{2}^{2})\Phi_{S}\Big{\\}}\,,$

and

$\displaystyle F_{S}$ $\displaystyle=$ $\displaystyle
e_{f}^{2}\sqrt{N_{c}}\Big{\\{}(\nu+Q_{1}^{2})\Phi_{1}-(\nu+Q_{2}^{2})\Phi_{2}\Big{\\}}$
(67) $\displaystyle=$ $\displaystyle{e_{f}^{2}\sqrt{N_{c}}\over
2}\Big{\\{}(M^{2}+2Q_{1}^{2}+2Q_{2}^{2})\Phi_{S}+(Q_{1}^{2}-Q_{2}^{2})\Phi_{A}\Big{\\}}\,,$

or, finally, the original form factors $F_{\rm TL},F_{\rm LT}$:

$\displaystyle Q_{1}F_{\rm LT}$ $\displaystyle=$
$\displaystyle{e_{f}^{2}\sqrt{N_{c}}\over 2}\Big{\\{}(\nu-
Q_{1}^{2})\Phi_{A}-(\nu+Q_{1}^{2})\Phi_{S}\Big{\\}}\,,$ $\displaystyle
Q_{2}F_{\rm TL}$ $\displaystyle=$ $\displaystyle{e_{f}^{2}\sqrt{N_{c}}\over
2}\Big{\\{}(\nu-Q_{2}^{2})\Phi_{A}+(\nu+Q_{2}^{2})\Phi_{S}\Big{\\}}\,,$ (68)

and

$\displaystyle F_{\rm TT}=-{1\over M}\Big{\\{}Q_{1}F_{\rm LT}+Q_{2}F_{\rm
TL}\Big{\\}}\,.$ (69)

A final comment on the determination of $F_{\rm TT},F_{\rm TL},F_{\rm LT}$ is
in order. Evidently, we have at our disposal three helicity amplitudes, and we
start out with three form factors. However, as it turns out, a cancellation
that relates $F_{\rm TT}$ with $F_{\rm TL}$ and $F_{\rm LT}$ makes our system
over determined. In fact, we could have obtained our form factors
$F_{A},F_{S}$ just as well from the comparison of ${\cal M}_{x}$ and ${\cal
M}_{y}$ or from ${\cal M}_{z}$ and one of the transverse polarizations. If we
stay within the transverse polarizations, there is no ambiguity, and the
expressions in term of $\Phi_{1,2}$ will be the same. However, involving the
longitudinal polarization involves the form factor $\Phi_{0}$, which is not
related in any obvious way to $\Phi_{1},\Phi_{2}$. This issue is evidently
related to the problems with rotational symmetry in LF quantization. We
believe that it is an artefact of the treatment of the bound state as a pure
$Q\bar{Q}$ state, however, a detailed discussion of the latter goes beyond the
scope of this work.

Before we come to our full numerical results including the integrals over the
quarkonium wave function, we present the NRQCD limits of the invariant form
factors. They are all related to the derivative of the rest frame WF at the
origin, $R^{\prime}(0)$, as follows:

$\displaystyle F_{\rm TT}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle=$
$\displaystyle 2e_{f}^{2}\sqrt{6N_{c}\over\pi
M^{3}}R^{\prime}(0)\,{Q_{1}^{2}-Q_{2}^{2}\over\nu}\,,$ $\displaystyle F_{\rm
LT}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle=$
$\displaystyle-2e_{f}^{2}\sqrt{6N_{c}\over\pi
M}R^{\prime}(0){(\nu+Q_{2}^{2})Q_{1}\over\nu^{2}}\,,$ $\displaystyle F_{\rm
TL}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle=$ $\displaystyle
2e_{f}^{2}\sqrt{6N_{c}\over\pi
M}R^{\prime}(0){(\nu+Q_{1}^{2})Q_{2}\over\nu^{2}}\,,$ (70)

where, within the accuracy of the NRQCD approximation, one should use
$M=2m_{Q}$.

##  IV Form factor visualization and observables

We now come to the presentation of our numerical results. First we will
discuss our results for the form factors $\Phi_{A,S}$ of the light front
amplitude from which all invariant form factors are calculated.

Then, we will recall the main observables which can be extracted in
$e^{+}e^{-}$ reactions and how they are related to the invariant form factors.

All our results are obtained for the $\chi_{c1}$ meson. Below we use the mass
$M=3.410\,\rm{GeV}$. The light front wave functions were obtained from the
radial wave functions in the $1P$ partial wave. More details on the WF and the
$c\bar{c}$ potentials used can be found in Ref. Babiarz  _et al._ (2020b).

###  IV.1 Form factors of the LF amplitude

Let us first discuss the form factors $\Phi_{0,1,2}$ introduced in Sec. II.
These are the primary results of our numerical calculations. First let us have
a look at the combinations $\Phi_{S}$ and $\Phi_{A}$ of Eq. (63). As we have
shown on general grounds, these form factors have good symmetry properties
under the interchange $Q_{1}^{2}\leftrightarrow Q_{2}^{2}$ – they are
respectively antisymmetric ($\Phi_{A}$) and symmetric ($\Phi_{S}$). These
symmetry properties are not obvious from the expressions Eq. (II.4), and hence
are a good check on the numerics. To this end, we introduce the average of
virtualities, $\overline{Q^{2}}$, and asymmetry, $\omega$,

$\displaystyle\overline{Q^{2}}={Q^{2}_{1}+Q^{2}_{2}\over
2}\,,\qquad\omega={Q^{2}_{2}-Q^{2}_{1}\over Q^{2}_{1}+Q^{2}_{2}}\,.$ (71)

Figure 1: Antisymmetric $\Phi_{A}$ and symmetric $\Phi_{S}$ form factors for
$\gamma^{*}\gamma^{*}\to\chi_{c}(1^{++})$ as a function of the asymmetry of
photon virtualities (left and middle panels) and the ratio of the
$\Phi_{A}/\Phi_{S}$ form factors again as a function of asymmetry of photon
virtualities (right panel).

Figure 2: Antisymmetric $\Phi_{A}$ and symmetric $\Phi_{S}$ form factors in
the nonrelativistic limit with the specific value of the quark mass $m_{c}$
and the first derivative $R^{\prime}(0)$ depending on each potential model,
see Tab. 1.

Figure 3: Invariant form factor $\Phi_{0}$ as a function of asymmetry
parameter with fixed $\bar{Q^{2}}=10\,{\rm GeV}^{2}$, on the l.h.s. within
LFWF approach, and on the r.h.s. in the nonrelativistic limit.

In Fig. 1 we present the $\Phi_{A}$ and $\Phi_{S}$ form factors as a function
of asymmetry parameter $\omega$ for fixed value of $\overline{Q^{2}}$ = 10
GeV2. We observe that the expected symmetry behaviour is well borne out by our
numerical results. In Fig. 2 we show the corresponding NRQCD results given by
Eq. (39). In the NRQCD limit the relation $\Phi_{A}=\omega\Phi_{S}$ holds
true, as can be seen from the ratio plotted in the third panel of Fig 2. There
is no reason for this relation to remain satisfied also in the full LFWF
approach, but the rightmost panel of Fig. 1 shows that it is a surprisingly
good approximation, independently of the potential used.

For completeness, in Fig. 3 we show the form factor $\Phi_{0}$ of the
amplitude in Eqs. (19,LABEL:eq:Integrals_Phi_0) as a function of $\omega$. In
the NRQCD limit $\Phi_{0}$ does not depend on $\omega$, while the full result
shows a very weak dependence on $\omega$.

###  IV.2 Form factors for two virtual photons

Figure 4: Antisymmetric $F_{A}$ and symmetric $F_{S}$ form factors as
functions of asymmetry parameter $\omega$ for several values of
$\overline{Q^{2}}$ in the range $(0.1-100)\,{\rm GeV}^{2}$ for the power-like
potential.

In Fig. 4 we show the antisymmetric $F_{A}$ and symmetric $F_{S}$ form factors
as functions of photon virtuality asymmetry for different values of
$\overline{Q^{2}}$. These plots were obtained for the power-law potential.
Both form factors decrease in magnitude when $\overline{Q^{2}}\to 0$. In Fig.
5 we show maps of helicity form factors $F_{\rm TT}$ and $F_{\rm LT}$ in the
$(Q_{1}^{2},Q_{2}^{2})$-plane. The form factor $F_{\rm TT}$ changes its sign
when crossing the $Q_{1}^{2}=Q_{2}^{2}$ line, while $F_{\rm LT}$ has no
symmetry properties. Also these plots were obtained for the power-law
potential. Qualitatively, the plots for different potentials look similar.

Figure 5:  Dependence of form factors $F_{\rm TT}(Q^{2}_{1},Q^{2}_{2})$ and
$F_{\rm LT}(Q^{2}_{1},Q^{2}_{2})$ on the two photon virtualities. Here we used
the LFWF obtained from the power-like potential model.

###  IV.3 Form factors for one real and one virtual photon

For a quantitative representation of the differences between the underlying
$c\bar{c}$-potentials it is more convenient to turn to one-dimensional form
factors with only one off–shell photon, which we wish to discuss now.

The photon helicity form factors for one virtual and one real photon are shown
in Fig. 6. Here the top row shows the result for the full LCWF, while the
bottom row displays the NRQCD limit results. In the rightmost panel, we show
$F_{\rm LT}(Q^{2},0)/Q$. As we discussed in Sec. III it has a finite limit at
$Q^{2}\to 0$. This value at $Q=0$ is directly related to the so-called reduced
width which we will introduce below. In general, we observe a strong
difference in the $Q^{2}$–dependence of form factors between the full
calculation and the NRQCD limit. Also different potentials give rise to
different shapes of form factors. This makes a future measurement in single-
tagged $e^{+}e^{-}$ collisions very interesting. We now turn to the
observables of such a measurement.

Figure 6: Form factors $F_{\rm TT}(Q^{2},0)$, $F_{\rm LT}(Q^{2},0)$ for one
virtual photon (left and middle panels) and $F_{\rm LT}(Q^{2},0)/Q$ (right
panel). The top panels represent our results in the LFWF approach and the
bottom panels correspond to the nonrelativistic limit.

###  IV.4 Virtual photon cross sections

The transition form factors obtained in this work can be used to make
predictions for the production of axial-vector mesons in the $e^{+}e^{-}\to
e^{+}e^{-}A$ reaction which proceeds via the $\gamma\gamma$ mode. Here one
distinguishes between double-tagged processes, where both final state leptons
are registered and single-tagged production, where only one of the leptons is
measured. The former reactions allow access to the full dependence on
$Q_{1}^{2},Q_{2}^{2}$ of form factors, while for the latter ones one of
photons will be quasi-real, and only the one related to the tagged lepton will
have finite virtuality $Q^{2}$.

It is a standard procedure to write the cross section for the $e^{+}e^{-}$
reaction in terms of a density matrix of photon fluxes and virtual photon
cross sections as well as response functions related to interferences, see
e.g. Ref. Budnev  _et al._ (1975); Poppe (1986).

The non-vanishing virtual photon cross sections for the production of an
axial-vector resonance read

$\displaystyle\sigma_{\rm TT}$ $\displaystyle=$ $\displaystyle{1\over
4\sqrt{X}}\,\Big{(}W(++,++)+W(+-,+-)\Big{)}={1\over
4\sqrt{X}}\,W(++,++)\,,\,\,$ $\displaystyle\ \sigma_{\rm LT}$ $\displaystyle=$
$\displaystyle{1\over 2\sqrt{X}}W(0+,0+)\,,\qquad\sigma_{\rm TL}={1\over
2\sqrt{X}}W(+0,+0)\,,$ (72)

where for the resonance of mass $M$ and width $\Gamma$ we can write

$\displaystyle
W(\lambda_{1}\lambda_{2},\lambda_{3}\lambda_{4})={M\Gamma\over(\hat{s}-M^{2})^{2}+M^{2}\Gamma^{2}}\sum_{\lambda_{A}}{\cal
M}^{*}_{\lambda_{1}\lambda_{2}}{\cal M}_{\lambda_{3}\lambda_{4}}\,.$ (73)

Here $\sqrt{\hat{s}}$ is the $\gamma^{*}\gamma^{*}$ cm energy, and ${\cal
M}_{\lambda_{1}\lambda_{2}}$ are the helicity amplitudes in the
$\gamma^{*}\gamma^{*}$ cm frame. We have collected the latter in Appendix C.
For the narrow resonance as the $\chi_{c1}$ of this work, the Breit-Wigner
prefactor above can be replaced by $\pi\delta(\hat{s}-M^{2})$.

Current experiments are not able to perform the $\rm TT$ vs $\rm LT,TL$
separation, and when tagging, say, the first lepton, one is sensitive to the
total $\gamma^{*}\gamma$ cross section

$\displaystyle\sigma_{\rm tot}(Q^{2},0)\equiv\sigma_{\rm
TT}(Q^{2},0)+\epsilon\sigma_{\rm LT}(Q^{2},0)=\epsilon\sigma_{\rm
LT}(Q^{2},0)\Big{(}1+{1\over\epsilon}R_{\rm TL}(Q^{2})\Big{)}\,.$ (74)

with the photon polarization parameter $\epsilon$, which in practice is close
to one, $\epsilon\approx 1$. Now notice, that we have shown that for the
$Q\bar{Q}$ bound state $F_{\rm TT}$ is calculable from $F_{\rm LT}$ and
$F_{\rm TL}$. In particular, for the case of one real photon, we have

$\displaystyle F_{\rm TT}(Q^{2},0)=-{Q\over M}F_{\rm LT}(Q^{2},0)\,,$ (75)

so that we obtain the universal result

$\displaystyle R_{\rm TL}={F^{2}_{\rm TT}(Q^{2},0)\over 2F^{2}_{\rm
LT}(Q^{2},0)}={Q^{2}\over 2M^{2}}\,,$ (76)

which does not depend on the wave function parameters/potential model used.

While the $1^{++}$ meson does not decay into two real photons, it is common
practice to introduce (convention-dependent) effective virtual photon widths,
see e.g. Refs. Aihara  _et al._ (1988); Olsson (1988). These are defined
through their relation to virtual photon cross sections _defined_ through Eq.
( 72). We follow the convention222We note that different experiments
historically have used definitions of $\Gamma^{\rm LT}$ differing by a factor
of two Olsson (1988). of Ref. Aihara  _et al._ (1988), where for a resonance
of spin $J$, mass $M$ and width $\Gamma$:

$\displaystyle\sigma_{ij}={32\pi(2J+1)\over N_{i}N_{j}}{\hat{s}\over
2M\sqrt{X}}\,{M\Gamma\over(\hat{s}-M^{2})^{2}+M^{2}\Gamma^{2}}\,\Gamma^{ij}_{\gamma^{*}\gamma^{*}}(Q_{1}^{2},Q_{2}^{2},\hat{s})\,,$
(77)

where $\\{i,j\\}\in\\{\rm{T,L}\\}$, and $N_{\rm T}=2,N_{\rm L}=1$ are the
numbers of polarization states of photons. In terms of our helicity form
factor, we obtain for the $\rm LT$ configuration, putting at the resonance
pole $\hat{s}\to M^{2}$, and $J=1$ for the axial-vector meson:

$\displaystyle\Gamma^{\rm
LT}_{\gamma^{*}\gamma^{*}}(Q_{1}^{2},Q_{2}^{2},M^{2})={\pi\alpha_{\rm
em}^{2}\over 3M}\,\,F_{\rm LT}^{2}(Q_{1}^{2},Q_{2}^{2})\,.$ (78)

Of particular interest is the so-called reduced width defined as333Note, that
if one wants to define the reduced width by calculating a $A\to\gamma^{*}_{\rm
L}\gamma$ “decay”, one ought to introduce an additional factor of $1/2$ in
order to agree with the convention of Ref. Aihara  _et al._ (1988), see Ref.
Pascalutsa  _et al._ (2012); Hoferichter and Stoffer (2020).

$\displaystyle\tilde{\Gamma}(A)=\lim_{Q^{2}\to 0}{M^{2}\over Q^{2}}\Gamma^{\rm
LT}_{\gamma^{*}\gamma^{*}}(Q^{2},0,M^{2})={\pi\alpha^{2}_{\rm em}M\over
3}\,f^{2}_{\rm LT}\,,\,\,\,{\rm with}\,\,f_{\rm LT}=\lim_{Q^{2}\to 0}{F_{\rm
LT}(Q^{2},0)\over Q}\,,$ (79)

which provides a useful measure of size of the relevant $e^{+}e^{-}$ cross
section in the $\gamma\gamma$ mode.

The coupling strength $f_{\rm LT}$ can be calculated from the LFWF as

$\displaystyle f_{\rm LT}=-e_{f}^{2}4M^{2}\sqrt{3N_{c}\over
2}\int{dzd^{2}\mbox{\boldmath$k$}\over
16\pi^{3}}\psi(z,\mbox{\boldmath$k$}){\mbox{\boldmath$k$}^{2}\over[\mbox{\boldmath$k$}^{2}+m_{Q}^{2}]^{2}}\,.$
(80)

Using the relation of the radial LFWF to the radial WF $u(k)$, one can reduce
this further to a one-dimensional integral

$\displaystyle f_{\rm LT}=-e_{f}^{2}M^{2}{\sqrt{3N_{c}}\over
8\pi}\int_{0}^{\infty}{dk\,k^{2}u(k)\over(k^{2}+m_{Q}^{2})^{2}}{1\over\sqrt{M_{Q\bar{Q}}}}\,\Big{\\{}{2\over\beta^{2}}-{1-\beta^{2}\over\beta^{3}}\,\log\Big{(}{1+\beta\over
1-\beta}\Big{)}\Big{\\}}\,,$ (81)

with

$\displaystyle\beta={k\over\sqrt{k^{2}+m_{Q}^{2}}}\,,\qquad
M_{Q\bar{Q}}=2\sqrt{k^{2}+m_{Q}^{2}}\,.$ (82)

In the NRQCD limit of $k\ll m_{Q}$, and hence $\beta\ll 1$, we obtain

$\displaystyle{f_{\rm LT}|}_{\rm NRQCD}=-e_{f}^{2}{\sqrt{3N_{c}M^{3}}\over
8\pi}\,{4\over 3}{1\over
m_{Q}^{4}}\int_{0}^{\infty}dk\,k^{2}u(k)=-e_{f}^{2}\sqrt{3N_{c}\over\pi
m_{Q}^{5}}R^{\prime}(0)\,,$ (83)

so that the reduced width expressed in terms of the derivative of the WF at
the origin reads:

$\displaystyle\tilde{\Gamma}(A)={2\alpha^{2}_{\rm em}e_{f}^{4}N_{c}\over
m_{Q}^{4}}\,|R^{\prime}(0)|^{2}\,,$ (84)

which is in agreement with the result of Ref. Danilkin and Vanderhaeghen
(2017). Notice, that in the NRQCD limit we have substituted $M=2m_{Q}$
everywhere.

The reduced widths for different potentials used are listed in Table 1. We
show result in our relativistic treatment as well as in the commonly used
NRQCD approximation. These two values differ considerably and strongly depend
on the potential used. In general, the full result is smaller than the NRQCD
limit. For all potentials $\tilde{\Gamma}(\chi_{c1})$ is substantially smaller
than 1 keV. Considerably larger values of $\tilde{\Gamma}(\chi_{c1})$ are
quoted in the literature. For example, in Ref. Danilkin and Vanderhaeghen
(2017) a value of $\tilde{\Gamma}(\chi_{c1})\approx 1.6\,\rm{keV}$ is reported
from a sum rule analysis. In Ref. Li  _et al._ (2022) a value of
$\tilde{\Gamma}(\chi_{c1})\approx 3\,\rm{keV}$ is obtained from a LFWF
approach444Comparing with the helicity amplitudes found in Ref. Hoferichter
and Stoffer (2020) it appears however that Ref. Li  _et al._ (2022) uses a
definition of the reduced width which is a factor of two larger than ours.. A
measurement of the reduced width would therefore be very valuable.

A comment on the total cross section (see Eq. (74)) is in order. Using the
result of Eq. (76), and putting $\epsilon=1$, one obtains

$\displaystyle\sigma_{\rm tot}(Q^{2},0)$ $\displaystyle=$ $\displaystyle
16\pi^{3}\alpha_{\rm em}^{2}\delta(\hat{s}-M^{2})\,{Q^{2}\over
Q^{2}+M^{2}}\Big{(}1+{Q^{2}\over 2M^{2}}\Big{)}\,\Big{(}{F_{\rm
LT}(Q^{2},0)\over Q}\Big{)}^{2}\,$ (85) $\displaystyle\equiv$ $\displaystyle
16\pi^{3}\alpha_{\rm em}^{2}\delta(\hat{s}-M^{2})\,{\texttt{F}}^{2}_{\rm
eff}(Q^{2})\,.$

Here, the dependence on $Q^{2}$ is controlled by the effective form
factor555See Ref. Schuler  _et al._ (1998) for a discussion in the NR quark
model. ${\texttt{F}}_{\rm eff}(Q^{2})$:

$\displaystyle{\texttt{F}}_{\rm eff}(Q^{2})=\sqrt{Q^{2}\over
Q^{2}+M^{2}}\,\sqrt{1+{Q^{2}\over 2M^{2}}}\,{F_{\rm LT}(Q^{2},0)\over Q}\,.$
(86)

We show plots of the effective form factor squared in Fig. 7. In the left
panel we show the results of the full LFWF approach for different potentials,
while in the right panel we show the NRQCD results. The two approaches lead to
different shapes of the effective formfactor. Also, in the NRQCD limit the
formfactor is substantially larger.

Table 1: Reduced width, see Eq. (79) potential model |  $m_{c}$ (GeV) |  $|R^{\prime}(0)|$ (GeV5/2) |  $\tilde{\Gamma}(\chi_{c1})_{\rm NRQCD}$ (keV) |  $\tilde{\Gamma}(\chi_{c1})$ (keV)
---|---|---|---|---
power-law | 1.33 | 0.22 | 0.97 | 0.50
Buchmüller-Tye | 1.48 | 0.25 | 0.82 | 0.30
Cornell | 1.84 | 0.32 | 0.56 | 0.09
harmonic oscillator | 1.4 | 0.27 | 1.20 | 0.53
logarithmic | 1.5 | 0.24 | 0.72 | 0.27

Figure 7: The square of the effective form factor Eq. (86) as a function of
photon virtuality within LFWF approach (on the l.h.s.) and in the
nonrelativistic limit (on the r.h.s.).

##  V Conclusions

We have studied the $\gamma^{*}\gamma^{*}\to 1^{++}$ quarkonium transition
form factors in a light front approach. The quarkonium LFWFs have been
obtained from solutions of the Schrödinger equation for $c\bar{c}$ interaction
transformed to the LF via the Terent’ev prescription for the radial part and a
Melosh transform for the spin-orbit part. Different interaction potentials
from the literature have been used to generate the charmonium wave function.

In our derivation we started from the $\gamma^{*}\gamma^{*}$-fusion amplitude
for the light-like photon polarizations $n^{+}_{\mu},n^{-}_{\mu}$.
Subsequently we have demonstrated how to extract invariant form factors
$F_{\rm TT}$, $F_{\rm LT}$, $F_{\rm TL}$, which refer to photon polarizations
in the $\gamma^{*}\gamma^{*}$ c.m.-frame. We have obtained these form factors
as a function of both photon virtualities. We have found that for the
$Q\bar{Q}$–bound state the form factor $F_{\rm TT}$ is determined through
$F_{\rm LT}$ and $F_{\rm TL}$.

We have also analyzed the situation where only one photon is virtual and the
second photon is assumed to be real, which is relevant for single-tagged
$e^{+}e^{-}$ collisions. Here, the ratio $F_{\rm TT}/F_{\rm LT}$ turned out to
be universal and independent of the potential used.

Our predictions for $\gamma^{*}\gamma^{*}\to\chi_{c1}$ are ready for
experimental verification. This could be done for example by the Belle-2
collaboration in $e^{+}e^{-}\to e^{+}e^{-}\chi_{c1}$ reaction, by measuring
e.g. $\chi_{c1}\to J/\psi\gamma$ with one and two electron tagging (${\rm
Br}(\chi_{c1}\to J/\psi\gamma)$ = $(34.3\,\pm\,1.0)$ % Workman (2022)). A
similar study has already been initiated in Ref. Teramoto  _et al._ (2021) for
the axial-vector meson $\chi_{c1}(3872)$ (or X(3872)), using the $X(3872)\to
J/\psi\pi^{+}\pi^{-}$ decay. This meson is supposed to have a large
non-$c\bar{c}$ component, and is still rather enigmatic (see e.g. Ref. Cisek
_et al._ (2022) for a consideration of its production in the gluon-gluon
fusion process including a $c\bar{c}$ admixture).

Acknowledgements

This work was partially supported by the Polish National Science Center grant
UMO-2018/31/B/ST2/03537 and by the Center for Innovation and Transfer of
Natural Sciences and Engineering Knowledge in Rzeszów. R.P. is partially
supported by the Swedish Research Council grant No. 2016-05996, by the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No 668679), as well as by
the NKFI grant K133046 (Hungary).

##  Appendix A Axial-vector meson polarisation states, wave functions

Here, we would like to summarise the basic formulas for axial-vector meson
wave functions for different meson polarisations.

The polarization vectors $\vec{E}(\lambda_{A})$ introduced in Eq. (1) read

$\displaystyle\vec{E}(\pm)=(\mbox{\boldmath$E$}(\pm),0)\,,\qquad\vec{E}(0)=\vec{n}\,,\qquad\mbox{\boldmath$E$}(\lambda_{A})=-{1\over\sqrt{2}}\Big{(}\lambda_{A}\mbox{\boldmath$e$}_{x}+i\mbox{\boldmath$e$}_{y}\Big{)}\,.$
(87)

The axial-vector meson WF in the $Q\bar{Q}$ rest frame as given in Eq. (1) and
its radial part $u(k)$ are normalized as

$\displaystyle\int
d^{3}\vec{k}\,\sum_{\tau\bar{\tau}}\Psi^{(\lambda^{\prime}_{A})*}_{\tau\bar{\tau}}(\vec{k})\Psi^{(\lambda_{A})}_{\tau\bar{\tau}}(\vec{k})=\delta_{\lambda^{\prime}_{A}\lambda_{A}}\,,\qquad\int_{0}^{\infty}dk\,u^{2}(k)=1\,.$
(88)

The normalization condition for the LFWF reads accordingly

$\displaystyle\int{dzd^{2}\mbox{\boldmath$k$}\over
z(1-z)16\pi^{3}}\,\sum_{\lambda\bar{\lambda}}\Psi^{(\lambda^{\prime}_{A})*}_{\lambda\bar{\lambda}}(z,\mbox{\boldmath$k$})\Psi^{(\lambda_{A})}_{\lambda\bar{\lambda}}(z,\mbox{\boldmath$k$})=\delta_{\lambda^{\prime}_{A}\lambda_{A}}\,,$
(89)

while the “radial WF” is normalized as

$\displaystyle\int{dzd^{2}\mbox{\boldmath$k$}\over
z(1-z)16\pi^{3}}2(M_{Q\bar{Q}}^{2}-4m_{Q}^{2})\,\psi^{2}(z,\mbox{\boldmath$k$})=1\,.$
(90)

Finally, we recall that the derivative of the radial wave function at the
spatial origin is related to the momentum-space WF $u(k)$ via

$\displaystyle\int_{0}^{\infty}dk\,k^{2}u(k)=3\sqrt{\pi\over
2}R^{\prime}(0)\,.$ (91)

The Melosh transform in Eq. (6) is performed straightforwardly e.g. by using
Eq. (A.15) from Ref. Babiarz  _et al._ (2020b).

Below, we collect the explicit representation of the LFWF for different meson
polarizations in the following matrix form:

$\displaystyle\Psi^{(\lambda_{A})}_{\lambda\bar{\lambda}}(z,\mbox{\boldmath$k$})=\begin{pmatrix}\Psi^{(\lambda_{A})}_{++}(z,\mbox{\boldmath$k$})&\Psi^{(\lambda_{A})}_{+-}(z,\mbox{\boldmath$k$})\\\
\Psi^{(\lambda_{A})}_{-+}(z,\mbox{\boldmath$k$})&\Psi^{(\lambda_{A})}_{--}(z,\mbox{\boldmath$k$})\end{pmatrix}\,.$
(92)

###  A.1 Transversely polarized states, $\lambda_{A}=\pm 1$

$\displaystyle\Psi^{(\lambda_{A})}_{\lambda\bar{\lambda}}(z,\mbox{\boldmath$k$})$
$\displaystyle=$
$\displaystyle{\psi(z,\mbox{\boldmath$k$})\over\sqrt{z(1-z)}}\sqrt{{3\over
2}}{\mbox{$\displaystyle\begin{pmatrix}m_{Q}(1-2z)\sqrt{2}i[\mbox{\boldmath$e$}(-),\mbox{\boldmath$E$}(\lambda_{A})]&-(1-2z)(\mbox{\boldmath$E$}(\lambda_{A})\mbox{\boldmath$k$})+i[\mbox{\boldmath$E$}(\lambda_{A}),\mbox{\boldmath$k$}]\\\
(1-2z)(\mbox{\boldmath$E$}(\lambda_{A})\mbox{\boldmath$k$})+i[\mbox{\boldmath$E$}(\lambda_{A}),\mbox{\boldmath$k$}]&m_{Q}(1-2z)\sqrt{2}i[\mbox{\boldmath$e$}(+),\mbox{\boldmath$E$}(\lambda_{A})]\end{pmatrix}$}},$

Let us add some useful combinations:

$\displaystyle\Psi^{*(\lambda_{A})}_{+-}(z,\mbox{\boldmath$k$})-\Psi^{*(\lambda_{A})}_{-+}(z,\mbox{\boldmath$k$})$
$\displaystyle=$ $\displaystyle\sqrt{{3\over
2}}{\psi(z,\mbox{\boldmath$k$})\over\sqrt{z(1-z)}}\,2(2z-1)(\mbox{\boldmath$E$}^{*}(\lambda_{A})\mbox{\boldmath$k$})\,,$
$\displaystyle\Psi^{*(\lambda_{A})}_{+-}(z,\mbox{\boldmath$k$})+\Psi^{*(\lambda_{A})}_{-+}(z,\mbox{\boldmath$k$})$
$\displaystyle=$ $\displaystyle\sqrt{{3\over
2}}{\psi(z,\mbox{\boldmath$k$})\over\sqrt{z(1-z)}}\,(-2i)[\mbox{\boldmath$E$}^{*}(\lambda_{A}),\mbox{\boldmath$k$}]\,,$
$\displaystyle\sqrt{2}(\mbox{\boldmath$e$}(-)\mbox{\boldmath$q$}_{1})\Psi^{*(\lambda_{A})}_{++}(z,\mbox{\boldmath$k$})+\sqrt{2}(\mbox{\boldmath$e$}(+)\mbox{\boldmath$q$}_{1})\Psi^{*(\lambda_{A})}_{--}(z,\mbox{\boldmath$k$})$
$\displaystyle=$ $\displaystyle\sqrt{{3\over
2}}{\psi(z,\mbox{\boldmath$k$})\over\sqrt{z(1-z)}}\,2m_{Q}(2z-1)i[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$E$}^{*}(\lambda_{A})]\,.$

###  A.2 Longitudinally polarized states, $\lambda_{A}=0$

$\displaystyle\Psi^{(0)}_{\lambda\bar{\lambda}}(z,\mbox{\boldmath$k$})$
$\displaystyle=$
$\displaystyle{\psi(z,\mbox{\boldmath$k$})\over\sqrt{z(1-z)}}\sqrt{{3\over
2}}{1\over
M_{Q\bar{Q}}}\begin{pmatrix}i2m_{Q}\sqrt{2}[\mbox{\boldmath$e$}(-),\mbox{\boldmath$k$}]&-2\mbox{\boldmath$k$}^{2}\\\
2\mbox{\boldmath$k$}^{2}&i2m_{Q}\sqrt{2}[\mbox{\boldmath$e$}(+),\mbox{\boldmath$k$}]\end{pmatrix}\;.$
(95)

We list here the following combinations relevant for calculation of the
production amplitude:

$\displaystyle\Psi^{*(\lambda_{A})}_{+-}(z,\mbox{\boldmath$k$})-\Psi^{*(\lambda_{A})}_{-+}(z,\mbox{\boldmath$k$})$
$\displaystyle=$ $\displaystyle\sqrt{{3\over
2}}{\psi(z,\mbox{\boldmath$k$})\over\sqrt{z(1-z)}}{1\over
M_{Q\bar{Q}}}(-4\mbox{\boldmath$k$}^{2})\,,$
$\displaystyle\Psi^{*(\lambda_{A})}_{+-}(z,\mbox{\boldmath$k$})+\Psi^{*(\lambda_{A})}_{-+}(z,\mbox{\boldmath$k$})$
$\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle\sqrt{2}(\mbox{\boldmath$e$}(-)\mbox{\boldmath$q$}_{1})\Psi^{*(\lambda_{A})}_{++}(z,\mbox{\boldmath$k$})+\sqrt{2}(\mbox{\boldmath$e$}(+)\mbox{\boldmath$q$}_{1})\Psi^{*(\lambda_{A})}_{--}(z,\mbox{\boldmath$k$})$
$\displaystyle=$ $\displaystyle\sqrt{{3\over
2}}{\psi(z,\mbox{\boldmath$k$})\over\sqrt{z(1-z)}}{2m_{Q}\over
M_{Q\bar{Q}}}\,(-2i)[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$k$}]\,.$

##  Appendix B Integrals for numerical evaluation

For transverse polarizations of the meson, the integrals appearing in Eqs.
(23), (24) and (26) are given in the following explicit form:

$\displaystyle J_{0}(\mbox{\boldmath$q$}_{1}^{2},\mbox{\boldmath$q$}_{2}^{2})$
$\displaystyle=$ $\displaystyle-4\sqrt{3\over
2}\int{dzd^{2}\mbox{\boldmath$k$}\over
z(1-z)16\pi^{3}}\psi(z,\mbox{\boldmath$k$})m_{Q}^{2}(1-2z)\Big{(}{1\over
D_{A}}-{1\over D_{B}}\Big{)}\,,$ $\displaystyle
J_{1}(\mbox{\boldmath$q$}_{1}^{2},\mbox{\boldmath$q$}_{2}^{2})$
$\displaystyle=$ $\displaystyle-8\sqrt{3\over
2}\int{dzd^{2}\mbox{\boldmath$k$}\over
z(1-z)16\pi^{3}}\psi(z,\mbox{\boldmath$k$})z(1-z){\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$k$}\over\mbox{\boldmath$q$}_{2}^{2}}\Big{(}{1\over
D_{A}}-{1\over D_{B}}\Big{)}\,,$ $\displaystyle
J_{2}(\mbox{\boldmath$q$}_{1}^{2},\mbox{\boldmath$q$}_{2}^{2})$
$\displaystyle=$ $\displaystyle-4\sqrt{3\over
2}\int{dzd^{2}\mbox{\boldmath$k$}\over
z(1-z)16\pi^{3}}\psi(z,\mbox{\boldmath$k$})(1-2z){\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$k$}\over\mbox{\boldmath$q$}_{2}^{2}}\Big{(}{1-z\over
D_{A}}+{z\over D_{B}}\Big{)}\,,$ $\displaystyle
J_{3}(\mbox{\boldmath$q$}_{1}^{2},\mbox{\boldmath$q$}_{2}^{2})$
$\displaystyle=$ $\displaystyle-4\sqrt{3\over
2}\int{dzd^{2}\mbox{\boldmath$k$}\over
z(1-z)16\pi^{3}}\psi(z,\mbox{\boldmath$k$})(1-2z){\mbox{\boldmath$q$}_{2}^{2}\mbox{\boldmath$k$}^{2}-(\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$k$})^{2}\over\mbox{\boldmath$q$}_{2}^{2}}\Big{(}{1\over
D_{A}}-{1\over D_{B}}\Big{)}\,,$ $\displaystyle
J_{4}(\mbox{\boldmath$q$}_{1}^{2},\mbox{\boldmath$q$}_{2}^{2})$
$\displaystyle=$ $\displaystyle-4\sqrt{3\over
2}\int{dzd^{2}\mbox{\boldmath$k$}\over
z(1-z)16\pi^{3}}\psi(z,\mbox{\boldmath$k$})(1-2z){(\mbox{\boldmath$q$}_{2}\cdot\mbox{\boldmath$k$})^{2}\over\mbox{\boldmath$q$}_{2}^{2}}\Big{(}{1\over
D_{A}}-{1\over D_{B}}\Big{)}\,,$ (97)

and should be evaluated numerically.

##  Appendix C Helicity amplitudes in the $\gamma^{*}\gamma^{*}$-c.m. frame
and virtual photon cross sections

In the $\gamma^{*}\gamma^{*}$ cm frame, we take the momenta of colliding
photons along the $z$-axis:

$\displaystyle q_{1\mu}=\omega_{1}\hat{t}_{\mu}+q_{z}\hat{z}_{\mu}\,,\qquad
q_{2\mu}=\omega_{2}\hat{t}_{\mu}-q_{z}\hat{z}_{\mu}\,,$ (98)

with

$\displaystyle\omega_{1}={M^{2}+Q_{2}^{2}-Q_{1}^{2}\over
2M}\,,\qquad\omega_{2}={M^{2}+Q_{1}^{2}-Q_{2}^{2}\over 2M}\,,\qquad
q_{z}={\sqrt{X}\over M}\,,$ (99)

and for the covariantly defined time and $z$-directions:

$\displaystyle\hat{t}_{\mu}={q_{1\mu}+q_{2\mu}\over
M}\,,\qquad\hat{z}_{\mu}={M\over
2\sqrt{X}}\Big{(}q_{1\mu}-q_{2\mu}+{Q_{1}^{2}-Q_{2}^{2}\over
M^{2}}(q_{1\mu}+q_{2\mu})\Big{)}\,.$ (100)

Let $\hat{x}_{\mu},\hat{y}_{\mu}$ be two spacelike unit vectors
$\hat{x}^{2}=\hat{y}^{2}=-1$ orthogonal to $\hat{z}_{\mu}$. Then we can write

$\displaystyle\tilde{G}_{\mu\nu}=-\sqrt{X}\epsilon_{\mu\nu\rho\sigma}\hat{t}_{\rho}\hat{z}_{\sigma}=-\sqrt{X}\Big{(}\hat{x}_{\mu}\hat{y}_{\nu}-\hat{x}_{\nu}\hat{y}_{\mu}\Big{)}\,.$
(101)

Therefore, our amplitude of Eq. (40) takes the intuitively obvious form

$\displaystyle{1\over 4\pi\alpha_{\rm em}}{\cal M}_{\mu\nu\rho}$
$\displaystyle=$
$\displaystyle\,\,-i\Big{(}\hat{x}_{\mu}\hat{y}_{\nu}-\hat{x}_{\nu}\hat{y}_{\mu}\Big{)}\hat{z}_{\rho}F_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle-$ $\displaystyle
ie_{\mu}^{L}(q_{1})\,\Big{(}\hat{x}_{\nu}\hat{y}_{\rho}-\hat{x}_{\rho}\hat{y}_{\nu}\Big{)}F_{\rm
LT}(Q_{1}^{2},Q_{2}^{2})-ie_{\nu}^{L}(q_{2})\Big{(}\hat{x}_{\mu}\hat{y}_{\rho}-\hat{x}_{\rho}\hat{y}_{\mu}\Big{)}F_{\rm
TL}(Q_{1}^{2},Q_{2}^{2})\,.$

For completeness, we introduce the polarization vectors of the axial-vector
meson $E_{\mu}(\lambda_{A})$ and photons $e_{\mu}^{1,2}(\lambda_{1,2})$ in the
$\gamma\gamma$-frame as

$\displaystyle E_{\mu}(0)=\hat{z}_{\mu}\,,\qquad
E_{\mu}(\pm)=e^{1}_{\mu}(\pm)=-{1\over\sqrt{2}}(\pm\hat{x}_{\mu}+i\hat{y}_{\mu})\,,\qquad
e^{2}_{\mu}(\pm)=e^{1}_{\mu}(\mp)\,,$ (103)

and for longitudinal photons

$\displaystyle e^{1}_{\mu}(0)=e^{L}_{\mu}(q_{1})={1\over
Q_{1}}(q_{z}\hat{t}_{\mu}+\omega_{1}\hat{z}_{\mu})\,,\qquad
e^{2}_{\mu}(0)=e^{L}_{\mu}(q_{2})={1\over
Q_{2}}(-q_{z}\hat{t}_{\mu}+\omega_{2}\hat{z}_{\mu})\,.$ (104)

Then, the helicity amplitudes in the $\gamma\gamma$-frame are obtained from

$\displaystyle{\cal
M}_{\lambda_{1}\lambda_{2}}=e^{1\mu}(\lambda_{1})e^{2\nu}(\lambda_{2})\,{\cal
M}_{\mu\nu\rho}E^{\rho*}(\lambda_{A})\,,\qquad\lambda_{A}=\lambda_{1}-\lambda_{2}\,.$
(105)

We can now easily read off the helicity amplitudes as

$\displaystyle{\cal M}_{++}=4\pi\alpha_{\rm em}F_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})\,,\qquad{\cal M}_{0+}=-4\pi\alpha_{\rm em}F_{\rm
LT}(Q_{1}^{2},Q_{2}^{2})\,.$ (106)

This is the easiest way to relate our form factors to other conventions in the
literature.

Finally, let us for convenience quote the results in the NRQCD limit:

$\displaystyle{\cal M}_{++}$ $\displaystyle=$ $\displaystyle 8\,\alpha_{\rm
em}e_{f}^{2}\sqrt{6N_{c}\pi\over
M^{3}}\,R^{\prime}(0){Q_{1}^{2}-Q_{2}^{2}\over\nu}\,,\qquad{\cal
M}_{0+}=8\alpha_{\rm em}e_{f}^{2}\sqrt{6N_{c}\pi\over
M}\,R^{\prime}(0)\,{Q_{1}(\nu+Q_{2}^{2})\over\nu^{2}}\,.$

These results are in agreement with Schuler  _et al._ (1998); Danilkin and
Vanderhaeghen (2017).

##  Appendix D Projection on linear polarizations

We define the covariant tetrad associated with the LF polarizations as

$\displaystyle\hat{t}_{\mu}={P_{\mu}\over
M}\,,\quad\hat{z}_{\mu}=E_{\mu}(0)\,,\quad\hat{x}_{\mu}={P^{\perp}_{\mu}\over|\mbox{\boldmath$P$}|}+{|\mbox{\boldmath$P$}|\over
P_{+}}n^{-}_{\mu}\,,\quad\hat{y}_{\mu}=-\varepsilon_{\mu\nu\rho\sigma}\hat{x}^{\nu}\hat{t}^{\rho}\hat{z}^{\sigma}\,.$
(108)

The projection of the amplitude on $z$-direction then reads:

$\displaystyle{\cal M}(0)\equiv{\cal M}_{z}$ $\displaystyle=$
$\displaystyle{-iM\over
X}[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}]\Big{\\{}{M^{2}+Q_{1}^{2}-Q_{2}^{2}\over
2M}F_{\rm TT}(Q_{1}^{2},Q_{2}^{2})+Q_{2}F_{\rm
TL}(Q_{1}^{2},Q_{2}^{2})\Big{\\}}$ (109) $\displaystyle=$
$\displaystyle{-i\over
2X}[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}]\Big{\\{}(M^{2}+Q_{1}^{2}-Q_{2}^{2})F_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})+F_{A}(Q_{1}^{2},Q_{2}^{2})+F_{S}(Q_{1}^{2},Q_{2}^{2})\Big{\\}}$
$\displaystyle=$ $\displaystyle{-i\over
2X}Q_{1}Q_{2}\sin\phi\Big{\\{}(Q_{1}^{2}-Q_{2}^{2})F_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})+MF_{S}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle+$
$\displaystyle M^{2}F_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})+MF_{A}(Q_{1}^{2},Q_{2}^{2})\Big{\\}}\,.$

We notice that it has both a symmetric and antisymmetric piece under
$Q_{1}^{2}\leftrightarrow Q_{2}^{2}$ interchange.

Now we turn to the projection on the $x$-axis:

$\displaystyle{\cal M}_{x}$ $\displaystyle=$ $\displaystyle{i\over
X|\mbox{\boldmath$P$}|}[\mbox{\boldmath$q$}_{1},\mbox{\boldmath$q$}_{2}]\Big{\\{}Q_{2}^{2}MF_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})+(Q_{1}^{2}+Q_{2}^{2}){1\over
2}F_{A}(Q_{1}^{2},Q_{2}^{2})-{1\over 2}M^{2}F_{S}(Q_{1}^{2},Q_{2}^{2})$ (110)
$\displaystyle+$
$\displaystyle(\mbox{\boldmath$q$}_{1}\cdot\mbox{\boldmath$q$}_{2})\Big{(}MF_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})+F_{A}(Q_{1}^{2},Q_{2}^{2})\Big{)}\Big{\\}}$
$\displaystyle=$ $\displaystyle i{Q_{1}Q_{2}\over
2X|\mbox{\boldmath$P$}|}\Big{\\{}\sin\phi\Big{(}2Q_{2}^{2}MF_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})+(Q_{1}^{2}+Q_{2}^{2})F_{A}(Q_{1}^{2},Q_{2}^{2})-M^{2}F_{S}(Q_{1}^{2},Q_{2}^{2})\Big{)}$
$\displaystyle+$ $\displaystyle Q_{1}Q_{2}\sin(2\phi)\Big{(}MF_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})+F_{A}(Q_{1}^{2},Q_{2}^{2})\Big{)}\Big{\\}}$
$\displaystyle=$ $\displaystyle i{Q_{1}Q_{2}\over
2X|\mbox{\boldmath$P$}|}\Big{\\{}\sin\phi\Big{[}-M(Q_{1}^{2}-Q_{2}^{2})F_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})-M^{2}F_{S}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle+$
$\displaystyle(Q_{1}^{2}+Q_{2}^{2})\Big{(}MF_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})+F_{A}(Q_{1}^{2},Q_{2}^{2})\Big{)}\Big{]}$
$\displaystyle+$ $\displaystyle Q_{1}Q_{2}\sin(2\phi)\Big{(}MF_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})+F_{A}(Q_{1}^{2},Q_{2}^{2})\Big{)}\Big{\\}}\,.$

Finally, the projection onto the $y$-axis (normal to the production plane) is
given by

$\displaystyle{\cal M}_{y}$ $\displaystyle=$ $\displaystyle{i\over
2X|\mbox{\boldmath$P$}|}\Big{\\{}Q_{1}^{2}Q_{2}^{2}\Big{(}F_{A}(Q_{1}^{2},Q_{2}^{2})-MF_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})\Big{)}$ (111) $\displaystyle+$ $\displaystyle{1\over
2}(M^{2}+Q_{1}^{2}+Q_{2}^{2})\Big{[}(Q_{1}^{2}-Q_{2}^{2})F_{S}(Q_{1}^{2},Q_{2}^{2})+(Q_{1}^{2}+Q_{2}^{2})F_{A}(Q_{1}^{2},Q_{2}^{2})\Big{]}$
$\displaystyle+$ $\displaystyle
Q_{1}Q_{2}\cos\phi\Big{[}(M^{2}+2Q_{1}^{2}+2Q_{2}^{2})F_{A}(Q_{1}^{2},Q_{2}^{2})+(Q_{1}^{2}-Q_{2}^{2})F_{S}(Q_{1}^{2},Q_{2}^{2})\Big{]}$
$\displaystyle+$ $\displaystyle
Q^{2}_{1}Q^{2}_{2}\cos(2\phi)\Big{[}F_{A}(Q_{1}^{2},Q_{2}^{2})+MF_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})\Big{]}\Big{\\}}\,.$

The $x,z$ components contain both symmetric and antisymmetric parts. Indeed,
these components contain only two combinations of form factors:

$\displaystyle f_{S}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle=$
$\displaystyle(Q_{1}^{2}-Q_{2}^{2})F_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})+MF_{S}(Q_{1}^{2},Q_{2}^{2})\,,$ $\displaystyle
f_{A}(Q_{1}^{2},Q_{2}^{2})$ $\displaystyle=$ $\displaystyle MF_{\rm
TT}(Q_{1}^{2},Q_{2}^{2})+F_{A}(Q_{1}^{2},Q_{2}^{2})\,,$ (112)

which are symmetric and antisymmetric w.r.t. the interchange
$Q_{1}^{2}\leftrightarrow Q_{2}^{2}$, respectively. The $x$ and $z$ components
therefore are not enough to fix the three functions $F_{\rm TT},F_{A},F_{S}$.

This is different, however, for the $y$-component. We note that the
$y$-component is fully antisymmetric w.r.t. interchange
$Q_{1}^{2}\leftrightarrow Q_{2}^{2}$. Here, we have three orders in the
Fourier expansion in $\cos n\phi$, which would give us three equations for the
three form factors.

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