# Elucidating the effect of intermediate resonances in the quark interaction
kernel on the time-like electromagnetic pion form factor

Ángel S. Miramontes López<EMAIL_ADDRESS>Instituto de Física y
Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Morelia,
Michoacán 58040, Mexico  Hèlios Sanchis Alepuz  helios.sanchis-alepuz@silicon-
austria.com
hsanchisalepuz@gmail Silicon Austria Labs GmbH, Inffeldgasse 25F, 8010 Graz,
Austria  Reinhard Alkofer<EMAIL_ADDRESS>Institute of Physics,
University of Graz, NAWI Graz, Universitätsplatz 5, 8010 Graz, Austria

###### Abstract

An exploratory study of the time-like pion electromagnetic form factor in a
Poincaré-covariant bound state formalism in the isospin symmetric limit is
presented. Starting from a quark interaction kernel representing gluon-
intermediated interactions for valence-type quarks, non-valence effects are
included by introducing pions as explicit degrees of freedom. The two most
important qualitative aspects are, in view of the presented study, the opening
of the dominant $\rho$-meson decay channel and the presence of a multi-
particle branch cut setting in when the two-pion threshold is crossed. Based
on a recent respective computation of the quark-photon vertex, the pion
electromagnetic form factor for space-like and time-like kinematics is
calculated. The obtained results for its absolute value and its phase compare
favorably to the available experimental data, and they are analyzed in detail
by confronting them to the expectations based on an isospin-symmetric version
of a vector-meson dominance model.

###### pacs:

11.10.St, 13.30.Eg, 13.40.Gp, 14.40.n

##  I Introduction

Hadronic time-like form factors will be measured in upcoming experiments to an
unprecedented precision. An understanding of these quantities which are
displaying pronounced structures originating from hadron resonances will
contribute significantly to our knowledge on the relation between the hadrons’
substructure and the hadron spectrum. On the one hand, based on respective
experimental and theoretical progress in the last decades, it is by now
evident that hadron resonances can be, at least in principle, described in
terms of quarks and gluons. The latter, being the QCD degrees of freedom, are
considered to be complete in the sense that they allow for a computation of
every hadronic observable. Changing the perspective in an attempt to
understand the Strong Interaction starting from the low-energy regime, a
possible way of phrasing the phenomenon of confinement in QCD is the statement
that all possible hadronic degrees of freedom will form also a complete set of
physical states. Therefore, the equivalence of descriptions of observables in
either quark and glue or hadronic degrees of freedom is a direct consequence
of confinement and unitarity. This is the gist of a chain of arguments which
can be sophisticated and applied to many different phenomena involving
hadrons. The related picture is known under the name “quark-hadron duality”,
and its consequences have been verified on the qualitative as well as the
semi-quantitative level, for a review see, e.g., Ref. Melnitchouk:2005zr . A
verification of this duality is, beyond the trivial fact of the absence of
coloured states, the clearest experimental signature for confinement. To
appreciate the scope of such a scenario it is important to note that a perfect
orthogonality of the quark-glue degrees of freedom on the one hand and
hadronic states on the other hand, and thus the perfect absence of “double-
counting” in any of the two “languages”, is nothing else but another way to
express confinement.

Gaining insight into the interplay between formation of hadronic bound states,
consisting of quarks and gluons, and the open decay channels of the respective
resonance is an essential element of every study of time-like form factors in
kinematic regions close to a resonance. Here, attention should be paid to the
fact that the hadron whose form factor is investigated and the hadronic
resonance which is apparent in the form factor are both to be described as
composite objects of quarks and gluons. The same is true for the hadronic
decay products in a hadronic or semi-leptonic decay of the resonance. This
makes evident that an approach to calculate a time-like form factor from QCD,
or from a microscopic model based on QCD degrees of freedom, faces the
challenging task to treat all elements appearing in the calculation on the
same footing and to a sufficient degree of sophistication if the result is
intended to allow for conclusions on the dynamics underlying such a form
factor.

Herein, we will report on an exploratory study of the time-like pion
electromagnetic form factor using functional methods. More precisely, we will
employ a combination of Bethe-Salpeter and Dyson-Schwinger equations (for
recent reviews on this and related approaches see, e.g., Cloet:2013jya ;
Eichmann:2016yit ; Huber:2018ned ; Sanchis-Alepuz:2017jjd ). Although such an
approach is capable of allowing a first-principle calculation (see, e.g., the
computation of the glueball spectrum reported in ref. Huber:2020ngt ), for the
task at hand this is yet out of reach. To grasp all essential features of the
pions’ time-like form factor111There are several investigations of the space-
like pion electromagnetic form factor in the Dyson-Schwinger–Bethe-Salpeter
approach, early examples include Langfeld:1989en ; Roberts:1994hh ;
Maris:2000sk . A calculation of this form factor spanning the entire domain of
space-like momentum transfers is described in ref. Chang:2013nia .  one needs
to describe at least (i) the pion as bound state of quark and antiquark
thereby at the same time taking into account its special role as would-be
Goldstone boson of the dynamically broken chiral symmetry of QCD, (ii) the
mixing, respectively, the interference of the $\rho$-meson, being described
also as a quark-antiquark bound state, with a virtual photon when this photon
is in turn coupled to a quark-antiquark pair via the fully renormalized quark-
photon vertex, and (iii) the dominant decay channel of the $\rho$-meson,
namely, $\rho\to\pi\pi$. The study presented here is now in two aspects
exploratory. First of all, the interaction between quarks and antiquarks is
modelled in such a way that the essential features, as implied by QCD and
phenomenology, are taken into account but that it is on the other hand still
manageable in such an involved calculation. Second, in several places we will
make technical simplifications, especially when the such introduced error can
for good reasons assumed to be small and the reduction in the computing time
needed is substantial. We thus aim here more for an understanding of how the
different features of the form factor arise from the QCD degrees of freedom
than for a quantitative agreement with the experimental data.

In the chosen model for the quark-antiquark interaction, besides a gluon-
mediated interaction also pions will be included explicitly. The reason for
this is as follows: if one were able to take into account the fully
renormalized quark-gluon vertex exactly within this approach, hadronic degrees
of freedom will effectively emerge and thus be included in the interaction
between quarks and antiquarks, respectively, they will back-feed on the
quarks’ dynamics. Due to the pions’ Goldstone boson nature, and especially due
to the implied small pion mass, the pions are the most important low-energy
degrees of freedom within the Strong Interaction, as, e.g., also elucidated by
chiral perturbation theory. And as in order to describe the physics of decays
non-valence effects need to be taken into account, it is some minimal
requirement for the investigation reported here to include pions as the most
important non-valence-type interaction mediator in the sub-GeV region.

The interaction model herein is also chosen in view of a possible
generalisation to the study of baryon form factors, and hereby especially the
nucleons’ time-like form factor. In this respect one can build on existing
calculations of space-like form factors from bound state amplitudes, see,
e.g., refs. Eichmann:2016yit ; Nicmorus:2010sd ; Eichmann:2011vu ;
Eichmann:2011pv ; Sanchis-Alepuz:2013iia ; Sanchis-Alepuz:2017mir  for some
recent respective work. A thorough understanding of the proton time-like form
factor at very low $Q^{2}$ is a very timely subject as the upcoming PANDA
experiment possesses the unique possibility to measure the proton’s
electromagnetic form factors in the so-called unphysical region through the
process $\bar{p}p\to l^{+}l^{-}\pi^{0}$, $l=e,\mu$ Fischer:2021kcr . At large
$Q^{2}$ the question of the onset of the convergence scale between the space-
like and the time-like form factors arises.

However, also the pions’ time-like electromagnetic form factor will be studied
further by upcoming experiments, among other reasons because recently the
consistency of the available data sets has been questioned
Ananthanarayan:2020vum . Based on the long-known fact that the $\tau$
radiative decay allows to extract the pion form factor Kim:1979hx  and that a
very large number of $\tau$-leptons are produced at B-meson factories further
high-precision data in sub-GeV region will become available.222In this work we
will compare to the dataset of ref. Akhmetshin:2006bx  available at
https://www.hepdata.net/record/ins728302.

Besides earlier lattice QCD calculations of the pions’ space-like
electromagnetic form factor Brommel:2006ww ; Frezzotti:2008dr ; Boyle:2008yd ;
Aoki:2009qn ; Nguyen:2011ek ; Brandt:2013dua ; Koponen:2013boa  recently
calculations of the time-like pion form factor have become available
Meyer:2011um ; Feng:2014gba ; Bulava:2015qjz ; Erben:2019nmx . These results
typically show a good agreement with the experimental data. In those
calculations, the extraction of the time-like form factor employs a
parameterisation based on the vector meson dominance (VMD) picture to extract
from the lattice data at discrete energies the time-like pion form factor.

The exploratory calculation presented herein is done in the isospin symmetric
limit. One of the effects of isospin breaking clearly visible in the time-like
pion form factor is $\rho$-$\omega$ mixing, see, e.g., the review
OConnell:1995nse . To this end we will employ the VMD-based fit given in
OConnell:1995nse  and modify it such that an expected form of the pion form
factor without this mixing effect is extracted.333In ref. Jegerlehner:2011ti
another method has been used to remove the $\rho$-$\omega$ mixing effects from
the data. Furthermore, we will present a simplified but numerically quite
accurate VMD parameterisation of the time-like form factor which then serves
as a basis for a detailed analysis of our results. Here, the focus is more on
a comparison of our results with the form expected on the basis of the VMD
parameterisation than on numerical agreement.444A precise representation of
the time-like form factor requires parameterisations including excited
$\rho$-mesons, respective examples can be found in refs. Schael:2005am ;
Davier:2005xq . Two-photon effects, on the other hand, can very likely be
safely neglected, for a corresponding study of the form factor at large
momentum transfer see Chen:2018tch .

This paper is organized as follows: In Sect. II we review some facts about the
electromagnetic pion form factor, employ the VMD-based fit given in
OConnell:1995nse  to remove the $\rho$-$\omega$ mixing effects, and provide an
expected form of the pion form factor. In Sect. III we present our approach
based on Bethe-Salpeter and Dyson-Schwinger equations. Our results are
presented and analyzed in Sect. IV. In Sect. V we present conclusions and an
outlook. Some technical details are deferred to two appendices.

##  II  The time-like electromagnetic pion form factor

$\gamma^{\star}$$e^{-}$$e^{+}$$F_{\pi}$$\pi^{+}$$\pi^{-}$ Figure 1: Electron-
positron pair annihilating to a virtual photon with time-like momentum which
then decays to a pion pair.

The pion, being a composite object, does not have a point-like interaction
with the electromagnetic field, and the related substructure, the pion being a
pseudoscalar, is related to one form factor. Considering, for example, the
scattering of an electron off a pion $\pi^{+}$ one can describe the leptonic
part of the interaction quite precisely in lowest-order perturbation theory,
i.e., one considers the process in which the electron emits a virtual photon,
and the latter couples to the pion. Defining the form factor $F_{\pi}(s)$ via
the relation

$\langle\pi^{+}(p_{1})|j^{\mu}_{e.m.}|\pi^{+}(p_{2})\rangle=e(p_{1}+p_{2})^{\mu}F_{\pi}(q^{2})\,,$
(1)

where $q^{\mu}=p_{1}^{\mu}-p_{2}^{\mu}$ is the virtual photon momentum, and
$e$ is the elementary electric charge. The $S$-matrix element for electron-
pion scattering is then proportional to the form factor,

$\displaystyle i{\cal M}_{e\pi\to e\pi}$ $\displaystyle=$ $\displaystyle
e(p_{1}+p_{2})^{\mu}F_{\pi}(q^{2})D_{\rm photon}^{\mu\nu}(q)$ (2)
$\displaystyle\qquad\left(ie\bar{u}(k_{1},s_{1})\gamma_{\nu}u(k_{2},s_{2})\right)\,,$

where $D_{\rm photon}^{\mu\nu}(q)$ is the photon propagator, and
$u(k_{i},s_{i})$ is the electron spinor, see, e.g., Sect. 8.4 of
Aitchison:2003tq  for more details. The kinematics of this scattering process
is such that the photon momentum is space-like, and without loss of generality
one can assume the form factor $F_{\pi}(s)$ to be real. Charge conservation
requires that for a real photon one has $F_{\pi}(0)=1$.

Turning to the process of electron-positron annihilation into a pion pair the
corresponding $S$-matrix element is again proportional to the form factor, see
, e.g., Sect. 8.5 of Aitchison:2003tq ,

$\displaystyle i{\cal M}_{e^{+}e^{-}\to\pi^{+}\pi^{-}}$ $\displaystyle=$
$\displaystyle e(p_{1}+p_{2})^{\mu}F_{\pi}(q^{2})D_{\rm photon}^{\mu\nu}(q)$
(3)
$\displaystyle\qquad\left(-ie\bar{v}(k_{1},s_{1})\gamma_{\nu}u(k_{2},s_{2})\right)\,,$

with appropriately redefined momenta. Especially, the virtual photon momentum
is now time-like, cf. Fig. 1, and one measures in such an annihilation process
to a pion pair the form factor for time-like momenta. Above the two-pion
production threshold, i.e., in the physical region, the corresponding cut in
the amplitude (3) necessitates to treat the time-like pion form factor as a
complex quantity, it fulfils the dispersion relation

$F_{\pi}(q^{2})=1+\frac{q^{2}}{\pi}\int_{4m_{\pi}^{2}}ds\frac{{\cal
I}m\,F_{\pi}(s)}{s(s-q^{2}-i\epsilon)}\,.$ (4)

Especially, it is expected that the phase of the pion form factor varies
strongly in a two-pion resonance region. Below the inelastic threshold, i.e.,
for $s<4m_{\pi}^{2}$, the time-like pion form factor is, via Watson’s final
state theorem, related to the isovector P-wave scattering phase shift
$\delta_{1,1}(s)$:

$\displaystyle{\cal I}m\,F_{\pi}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2i}\left(1-e^{-2i\delta_{1,1}(s)}\right)F_{\pi}(s+i\epsilon)$
(5) $\displaystyle=$
$\displaystyle\sin(\delta_{1,1}(s))e^{-i\delta_{1,1}(s)}F_{\pi}(s+i\epsilon)\,.$

As depicted in Fig. 1, the pion form factor contains, for time-like as well as
for space-like photon virtualities, all kind of interaction processes turning
a photon into a pion pair. As strong-interaction processes dominate the
corresponding amplitude, and as gluons do not couple directly to photons, one
decisive element of the pion form factor is the amplitude describing how the
photon couples to a quark, taking hereby all possible contributing QCD
processes into account. This amplitude is exactly the full quark-photon
vertex. As we will see in the following, this correlation function carries
information about the virtual photon’s hadronic substructure. And as will be
described in detail in the next section, the other quantities needed then for
calculating the pion form factor are the fully renormalized quark propagator
and the pion bound state amplitude, the latter describing how an antiquark and
a quark form a pionic bound state.

While our calculation is based on QCD degrees of freedom it is capable of
providing an understanding why in the resonance region a vector-meson
dominance (VMD) picture provides very good results for the time-like pion form
factor. In addition, it will elucidate to which extent a VMD picture might be
applicable in other kinematic regions.

In the VMD picture, the hadronic contribution to the photon propagator is
given by mixing with electrically neutral vector mesons. Restricting to light-
quark mesons, the corresponding vector meson is the $\rho^{0}$, i.e., the
uncharged member of the isotriplet of vector mesons. In a would-be isospin
symmetric world this would be the only vector meson below one GeV with which a
virtual photon mixes because the isosinglet $\omega$ will not mix with the
photon due to $G$-parity.

In the real world, isospin symmetry is broken, and one of the many effects of
isospin breaking is $\rho$-$\omega$ mixing, see, e.g., the review
OConnell:1995nse . This mixing and the resulting interference of states lead
to quite some pronounced structure in the time-like pion form factor around
$m_{\omega}^{2}$. Indeed, it is by now well understood that the sharp dip in
the experimental data stems from a combination of $\rho$-$\omega$ mixing and
interference effects between the decays $\rho\rightarrow\pi\pi$ and
$\omega\rightarrow\pi\pi$, the latter being isospin breaking, see, e.g., Ref.
OConnell:1995nse . Moreover, it has been estimated that this combination of
mixing and interference effects decreases the height of the bump of the order
of up to $10\%$ (see Jegerlehner:2011ti  and references therein).

As the here presented exploratory calculation is performed in the isospin
limit555Isospin breaking and thus the effect of $\rho$-$\omega$ mixing on the
pion form factor is currently under investigation, the corresponding results
will be published elsewhere. and thus $\rho$-$\omega$ mixing is neglected we
estimate its effect by comparing the VMD-based fit to the pion form factor
given in ref. OConnell:1995nse  with a plot of the same expression but the
mixing matrix element put to zero, $\Pi_{\rho\omega}=0$, see Fig. 2. As
expected the resulting curve is much smoother than the one including the
$\rho$-$\omega$ interference effect. Around the $\omega$ mass the deviation of
the two curves can be as large as almost ten percent, however, this effect is
limited to a small interval, and the two curves are practically
indistinguishable outside this small interval. Therefore we expect our
calculation to reproduce all qualitative features of the curve representing
the case with $\rho$-$\omega$ mixing switched off, and to be in a reasonable
quantitative agreement with it.

Figure 2: Absolute value of the pion form factor in the time-like ($Q^{2}<0$)
domain from the VMD-based fit given in ref. OConnell:1995nse  (full line) in
comparison to the experimental data Ananthanarayan:2020vum . The dashed line
is based on the same expression but the mixing matrix element put to zero,
$\Pi_{\rho\omega}=0$.

It is instructive to analyze the momentum behaviour of a simplified version of
the fit given in ref. OConnell:1995nse . In order to be in agreement with the
notation in the following presentation we introduce the photon virtuality with
the convention that $Q^{2}<0$ corresponds to the time-like region. In the
above used fit a momentum dependent expression for the width of the
$\rho$-meson is used. It displays the two-pion cut,

$\Gamma_{\rho}(Q^{2})\propto\left(-Q^{2}-4m_{\pi}^{2}\right)^{3/2}\Theta(-Q^{2}-4m_{\pi}^{2})\,,$
(6)

which is important for a qualitatively correct analytic structure of the pion
form factor. However, especially close to the maximum of the pion form factor,
the pion mass is quantitatively negligible, and for vanishing pion mass the
momentum dependent width assumes for time-like $Q^{2}<0$ the relatively simple
form

$\Gamma_{\rho}(Q^{2})=\bar{\Gamma}_{\rho}|Q^{2}|/m_{\rho}^{2}\,,$ (7)

where $\bar{\Gamma}_{\rho}=\Gamma_{\rho}(-m_{\rho}^{2})$. This then leads to
the simplified but still quite accurate form for the fit666In contrast to a
constant width approximation the pole is in this parameterization and thus in
the employed fit not located at
$Q^{2}=-m_{\rho}^{2}+im_{\rho}\bar{\Gamma}_{\rho}$ but at
$Q^{2}=(-m_{\rho}^{2}+im_{\rho}\bar{\Gamma}_{\rho})/(1+\bar{\Gamma}_{\rho}^{2}/m_{\rho}^{2})$,
i.e., real and imaginary part of the pole position are decreased by 3.7% if
the same values for the $\rho$’s mass and width are used.

$\displaystyle F_{\pi}(Q^{2}+i\epsilon)$ $\displaystyle=$ $\displaystyle
1-\frac{g_{\rho\pi\pi}}{g_{\rho}}\frac{Q^{2}(Q^{2}+m_{\rho}^{2})}{(Q^{2}+m_{\rho}^{2})^{2}+Q^{4}\bar{\Gamma}^{2}/m_{\rho}^{2}}$
(8) $\displaystyle+$ $\displaystyle
i\frac{g_{\rho\pi\pi}}{g_{\rho}}\frac{Q^{4}\bar{\Gamma}/m_{\rho}}{(Q^{2}+m_{\rho}^{2})^{2}+Q^{4}\bar{\Gamma}^{2}/m_{\rho}^{2}}\,,$

where $\epsilon\to 0^{+}$ has been introduced to fix the sign of the imaginary
part. Hereby, the coupling constant $g_{\rho\pi\pi}$ is determined from
$\Gamma_{\rho\pi\pi}=\bar{\Gamma}=149$ MeV to be $g_{\rho\pi\pi}\approx 6$,
and $g_{\rho}$ is a parameter reflecting the strength of the
$\gamma$-$\rho^{0}$-mixing, which is then described by the effective
Lagrangian

${\cal
L}_{\rho\gamma}=-\frac{e\,m_{\rho}^{2}}{g_{\rho}}\rho_{\mu}^{0}A^{\mu}\,.$

From the partial width $\Gamma_{\rho e^{+}e^{-}}=7$ keV one infers
$g_{\rho}\approx 5$.

The form given in Eq. (8) motivates to compare the obtained results for the
real and the imaginary part of the time-like form factor from our calculation
to a rational, resp., Padé fit. Phrased otherwise, Eq. (8) formalises the
expectation for the time-like form factor based on the VMD picture, and our
results based on the quark-photon vertex function and the pion bound state
amplitude will be analysed by discussing how much they extend beyond to this
form.

##  III Dyson-Schwinger and Bethe-Salpeter formalism

We determine the necessary input for the calculation of the pion
electromagnetic form factor using a combination of Bethe-Salpeter (BSE) and
Dyson-Schwinger equations (DSE). To make this presentation self-contained we
summarise in this section the most relevant aspects of the approach, for more
details see, e.g., the recent reviews Cloet:2013jya ; Eichmann:2016yit ;
Huber:2018ned ; Sanchis-Alepuz:2017jjd  as well as references therein. All
expressions in the following are understood to be formulated in Euclidean
momentum space, i.e., after a Wick rotation.

In the DSE/BSE formalism, the fully-dressed quark-photon vertex
$\Gamma^{\mu}$, which describes the interaction between quarks and photons in
a quantum field theory, can be obtained as the solution of an inhomogeneous
BSE

$\displaystyle\left(\Gamma^{\mu}\right)_{a\alpha,b\beta}\left(p,Q\right)$
$\displaystyle=$ $\displaystyle
Z_{2}\left(\gamma^{\mu}\right)_{ab}t_{\alpha\beta}$ $\displaystyle+$
$\displaystyle\int_{q}K^{r\rho,s\sigma}_{a\alpha,b\beta}\left(Q,p,q\right)\,S_{r\rho,e\epsilon}\left(k_{1}\right)$
$\displaystyle\times$
$\displaystyle\left(\Gamma^{i,\mu}\right)_{e\epsilon,n\nu}\left(Q,q\right)S_{n\nu,s\sigma}\left(k_{2}\right)\,.$

Here, $Q$ is the photon momentum, $p$ is the relative momentum between quark
and antiquark, $q$ is an internal relative momentum which is integrated over,
the internal quark and antiquark momenta are defined as $k_{1}=q+Q/2$ and
$k_{2}=q-Q/2$, respectively, such that $Q=k_{1}-k_{2}$ and
$q=(k_{1}+k_{2})/2$. Latin letters represent Dirac indices, and Greek letters
represent flavour indices. The isospin structure of the vertex is given by
$t_{\alpha\beta}=\textrm{diag}\left(\nicefrac{{2}}{{3}},\nicefrac{{-1}}{{3}}\right)$.
The Dirac structure of the vertex can be expanded in a basis consisting of
twelve elements Miramontes:2019mco , and all of them are considered in our
calculation.

Similarly, mesons as bound states of two quarks are described in this
framework by Bethe-Salpeter amplitudes $\Gamma$ which are obtained as
solutions of a homogeneous BSE,

$\left(\Gamma\right)_{a\alpha,b\beta}\left(p,P\right)=\int_{q}K^{r\rho,s\sigma}_{a\alpha,b\beta}\left(P,p,q\right)\times
S_{r\rho,e\epsilon}\left(k_{1}\right)\left(\Gamma\right)_{e\epsilon,n\nu}\left(q,P\right)S_{n\nu,s\sigma}\left(k_{2}\right)~{},$
(10)

where for clarity we have here used $P$ for the total meson momentum (instead
of $Q$ as above). For pions, the Dirac part of the Bethe-Salpeter amplitude
$\Gamma$ can be expanded in a tensorial basis with four elements.

In the equations above, the interaction kernel $K$ describes the interaction
between quark and antiquark, and $S$ is the fully-dressed quark propagator. We
will discuss in detail the interaction kernels below. The quark propagator
$S(p)$ is obtained as the solution of the quark DSE,

$S^{-1}=S_{0}^{-1}-Z_{1f}\int_{q}\gamma_{\mu}S(q)\Gamma^{qgl}_{\nu}(q,k)D_{\mu\nu}(k)~{},$
(11)

with $S_{0}^{-1}$ the renormalised bare propagator,

$S_{0}^{-1}(p)=Z_{2}\left(i\ooalign{\hfil/\hfil\crcr$p$}+Z_{m}m\right)~{},$
(12)

and $Z_{1f}$, $Z_{2}$ and $Z_{m}$ are renormalisation constants, $m$ is the
(renormalisation-point dependent) current quark mass, $\Gamma^{qgl}$ is the
full quark-gluon vertex and $D_{\mu\nu}$ is the full gluon propagator which,
in the Landau gauge, is parametrised as

$D_{\mu\nu}(k)=\left(\delta_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}\right)\frac{Z(k^{2})}{k^{2}}~{},$
(13)

with $Z(p^{2})$ being the gluon dressing function. For simplicity, we have
suppressed the color indices.

Figure 3: Diagrams relevant for the calculation of the pion form factor in the
truncation employed herein, as determined by the gauging method. The impulse
approximation implies considering the first diagram and the corresponding
permutation only.

###  III.1 Interaction Kernels

The interaction kernel $K$ in eqs. (III) and (10) encodes all possible
interactions processes between a quark and an antiquark. In a diagrammatic
representation, it contains a sum of infinitely many terms. In practical
calculations, the expansion of the interaction kernel must be truncated to a
sum of a finite number of terms, chosen such that the relevant dynamics and
global symmetries are correctly implemented. Chiral symmetry and its dynamical
breaking ensures that pions are massless bound states in the chiral limit as a
consequence of Goldstone’s theorem. On the other hand, U(1) vector symmetry
ensures charge conservation in electromagnetic processes. Chiral symmetry will
be correctly implemented in the DSE/BSE formalism only if the kernel fulfills
the axial-vector Ward-Takahashi identity (Ax-WTI)

$i\Sigma_{ar}\left(p_{+}\right)\gamma^{5}_{rb}t^{i}_{\alpha\beta}+i\gamma^{5}_{ar}\Sigma_{rb}\left(p_{-}\right)t^{i}_{\alpha\beta}=\int_{q}K^{r\rho,s\sigma}_{a\alpha,b\beta}\left(Q,p,q\right)\left[i~{}t^{i}_{\rho\nu}\gamma^{5}_{rn}S_{n\nu,s\sigma}\left(q_{-}\right)+i~{}S_{r\rho,e\epsilon}\left(q_{+}\right)\gamma^{5}_{es}~{}t^{i}_{\epsilon\sigma}\right]~{},$
(14)

with $\Sigma$ the quark self-energy and $v_{\pm}=v\pm Q/2$. Similarly, vector
symmetry will be correctly implemented if the kernel satisfies the vector
Ward-Takahashi identity (V-WTI)

$i\Sigma_{ab}\left(p_{+}\right)t^{i}_{\alpha\beta}-i\Sigma_{ab}\left(p_{-}\right)t^{i}_{\alpha\beta}=\int_{q}K^{r\rho,s\sigma}_{a\alpha,b\beta}\left(Q,p,q\right)\left[i~{}t^{i}_{\rho\nu}S_{r\nu,s\sigma}\left(q_{-}\right)-i~{}S_{r\rho,s\epsilon}\left(q_{+}\right)~{}t^{i}_{\epsilon\sigma}\right]~{}.$
(15)

In DSE/BSE studies the most widely used truncation is the so-called rainbow-
ladder (RL) truncation, whereby the BSE kernel consists of a vector-vector
gluon exchange, namely (omitting again color indices)

$K^{r\rho,s\sigma}_{a\alpha,b\beta}\left(Q,p,q\right)=\alpha\left(k^{2}\right)\gamma^{\mu}_{ar}\gamma^{\nu}_{sb}D^{\mu\nu}\left(k\right)\delta^{\alpha\rho}\delta^{\sigma\beta}~{},$
(16)

with $k=p-q$ the gluon momentum. In order to preseve the Ax-WTI and V-WTI the
kernel (16) is used in combination with a truncated quark DSE, defined by the
replacement

$Z_{1f}\gamma_{\mu}Z(k^{2})\Gamma_{\nu}^{\text{qgl}}(q,p)\rightarrow
Z_{2}^{2}\gamma_{\mu}4\pi\alpha(k^{2})\gamma_{\nu}$ (17)

such that $\alpha(k^{2})$ provides an effective coupling that describes the
strength of the quark-antiquark interaction. To parametrise this effective
interaction we use the Maris-Tandy model Maris:1997tm ; Maris:1999nt

$\alpha(q^{2})=\pi\eta^{7}\left(\frac{q^{2}}{\Lambda^{2}}\right)^{2}e^{-\eta^{2}\frac{q^{2}}{\Lambda^{2}}}+\frac{2\pi\gamma_{m}\big{(}1-e^{-q^{2}/\Lambda_{t}^{2}}\big{)}}{\textnormal{ln}[e^{2}-1+(1+q^{2}/\Lambda_{QCD}^{2})^{2}]}~{},$
(18)

where the second term on the right-hand side reproduces the one-loop QCD
behavior of the quark propagator in the ultraviolet, and the Gaussian term
provides enough interaction strength for dynamical chiral symmetry breaking to
take place. The model parameters $\Lambda$ and $\eta$ are determined as
explained in the next section. The scale $\Lambda_{t}=1$ GeV is introduced for
technical reasons and has no impact on the results. For the anomalous
dimension we use $\gamma_{m}=12/(11N_{C}-2N_{f})=12/25$ with $N_{f}=4$
flavours and $N_{c}=3$ colours. For the QCD scale we use $\Lambda_{QCD}=0.234$
GeV.

In the RL truncation, bound states cannot develop a decay width,
correspondingly their masses are real numbers. Note that bound states,
determined as solutions of BSEs, appear as poles in Green’s functions with the
corresponding quantum numbers. In the RL approximation these poles occur for
real (and in the convention employed here, negative) momentum-squared values
in certain kinematic configurations. In particular, the photon being described
by a vector field, electrically neutral vector mesons appear as poles of the
quark-photon vertex. In the RL approximation these poles are located at
negative and real values of $Q^{2}$ (for which $M^{2}=-Q^{2}$ with $M$ being
the mass of the vector meson). Such poles in the quark-photon vertex also
manifest as poles in the calculation of time-like form factors, in
contradiction with phenomenology. Generally speaking, any physical phenomenon
that is triggered by the presence of virtual intermediate particles, as, e.g.,
decays, will be absent from any calculation using the RL truncation only.

It is possible to improve the RL truncation in this respect by re-
introducing777Note that effects of intermediate virtual states like the decay
of hadrons are in principle present in the full quark-gluon vertex which,
however, is drastically simplified in the RL truncation. the presence of
intermediate particles explicitly. The simplest implementation of such an idea
was introduced in Fischer:2007ze ; Fischer:2008sp 888See, however, ref.
Alkofer:1993gu  for considering pion loop contributions to the electromagnetic
pion radius in the DSE/BSE approach. An alternative way is to introduce two-
pion states via an explicit two-pion component in the bound state amplitude,
see ref. Santowsky:2020pwd  and references therein., where, based on the role
of the pion as lightest hadron, explicit pion-quark interactions were
introduced in the truncated quark DSE and in the BSE kernel $K$, with the
pion-quark interaction vertex given by the pion Bethe-Salpeter amplitude
$\Gamma$, calculated consistently via a truncated BSE. The corresponding
additional BSE kernels are given in Appendix A and shown in Fig. 8, and the
technical difficulties arising for time-like momenta when those kernels are
used in BSEs have been described in detail in Miramontes:2019mco . This type
of kernels enable the possibility of intermediate virtual decays in the BSE
interaction kernel to occur. As a consequence, certain BSE solutions signal a
finite decay width and thus represent (a) hadron resonance(s). E.g., the
description of the $\rho$-meson as a finite-width resonance is then mostly due
to the intermediate process $\rho\rightarrow\pi\pi$ (see Williams:2018adr  for
a treatment in the here discussed approach as well as Jarecke:2002xd ;
Mader:2011zf  and references therein for respective calculations based on DSEs
and BSEs), the partial decay width to the latter process representing more
than 99% of the total $\rho$ decay width. Additionally, and thereby completing
the physical effects of intermediate virtual $\pi\pi$-states, in this
truncation the quark-photon vertex develops a multi-particle branch cut along
the negative real $Q^{2}$ axis, starting as expected at the two-pion
production threshold Miramontes:2019mco . Clearly, including these two effects
of the intermediate virtual $\pi\pi$-states is especially important when it
comes to the calculation of the pion form factor for time-like momenta in the
sub-GeV kinematic region.

We wish to stress here that, for computational feasibility, for the pion
vertices in (23)–(26) we used the leading $\gamma_{5}$ component of the pion
Bethe-Salpeter amplitude in the chiral limit, given by $B/f_{\pi}$ with $B$
one of the quark’s dressing functions, see Eq. (27). On the other hand, the
pion amplitudes used in the form factor calculation of Eq. (19) are considered
in full, including their leading and sub-leading contributions. In that
regard, our calculations contain two types of treatments of pions.

Even though the “pionic” kernels (23)–(26) are phenomenologically justified
and, as we will see in the next section, constitute a first step in the
correct direction, it must be noted that they have not been (yet) rigorously
derived from QCD. Lacking a solid quantum-field theoretical basis the use of
these kernels comes with some shortcomings, especially it implies that the Ax-
WTI and V-WTI are not fulfilled simultaneously. Indeed, one can choose to
preserve either the Ax-WTI (and hence chiral symmetry) or the V-WTI (and hence
charge conservation), but not both Fischer:2007ze ; Miramontes:2019mco . It
turns out, however, that the respective violation of either of them induce
typically only small errors in physical observables, as we will demonstrate
for some quantities in the next section.

###  III.2 Form factor calculation

Meson form factors are extracted from a current $J^{\mu}$ encoding the
coupling of a meson to an external electromagnetic current. In the BSE
framework, the current is calculated by means of the coupling of an external
photon to each of the constituents of the bound state, as specified by a
procedure known as gauging and developed in Haberzettl:1997jg ;
Kvinikhidze:1998xn ; Kvinikhidze:1999xp ; Oettel:1999gc ; Oettel:2000jj . The
conserved current $J^{\mu}$ that describes the coupling of a single photon
with a quark-antiquark, a three-quark or other multi-quark system is given by,

$J^{\mu}=\bar{\Psi}_{f}G_{0}(\mathbf{\Gamma}^{\mu}-K^{\mu})G_{0}\Psi_{i}~{},$
(19)

with $\Psi_{i,f}$ the incoming and outgoing Bethe-Salpeter amplitudes of the
meson, the baryon or some other multi-quark state, and $G_{0}$ represents the
appropriate product of dressed quark propagators. This equation is shown
diagrammatically for the quark-antiquark–meson case in Fig. 3. The term
$\Gamma^{\mu}$ represents the impulse approximation diagrams where the photon
couples to the valence quarks only

$\mathbf{\Gamma}^{\mu}=\left(S^{-1}\otimes
S^{-1}\right)^{\mu}=\Gamma^{\mu}\otimes S^{-1}+S^{-1}\otimes\Gamma^{\mu}~{},$
(20)

with $\Gamma^{\mu}$ the quark-photon vertex. The term $K^{\mu}$ describes the
interaction of the photon with the Bethe-Salpeter kernel, which in our
truncation includes the coupling of the photon to the quark-pion vertex and to
the propagating pions (second and third diagram in Fig. 3). Including both
terms in (19) is necessary in order to implement current conservation
precisely. Note that the s- and u-channel pion decay terms (most right diagram
in Fig. 8) do not contribute to the term $K^{\mu}$, also not via seagulls, the
reason being that trying to include them would leave a $\pi$ $\pi$ $\pi$
amplitude on one side of the respective full diagram. However, the new
vertices appearing in the term $K^{\mu}$, the coupling of the photon to the
quark-pion vertex and to the propagating pions, represent an enormous
computational challenge. Given the exploratory purpose of the present
calculation, we thus decided to omit the coupling of the photon to the quark-
pion vertex and to the propagating pions, and to consider the impulse diagram
only (first diagram in Fig. 3). As we will show in the next section, the
thereby implied violation of charge conservation is at the level of
approximately one percent.

##  IV  Results

Following the formalism sketched above, we have calculated the pion
electromagnetic form factor in the time-like $Q^{2}<0$ domain. For comparison
with previous calculations we will also show results in the space-like
$Q^{2}>0$ domain.

In our discussion in the preceding section, it remained to be explained how
the parameters $\eta$ and $\Lambda$ of our interaction model (18) as well as
the value of the current quark mass are fixed. It is customary in studies
using the RL truncation to adjust those parameters such that the pion decay
constant agrees with the experimental value999Note that, for this observable,
the result is quite independent of the value of $\eta$ around $\eta=1.8$ and
thus, effectively, only the parameter $\Lambda$ has to be adjusted.. In ref.
Miramontes:2019mco  the quark-photon vertex has been calculated for the first
time with the above discussed interaction kernels taken into account and also
used herein. The parameters, including the isospin symmetric light quark
current mass $m_{q}$, were adjusted such that the pion mass and decay constant
as well as the $\rho$-meson mass have been correctly reproduced in the
employed approximation. Here, only the gluon- and pion-exchange kernels need
to be used for fixing the parameters because the pion decay kernels (25) and
(26) do not contribute to the pion BSE.

For the present exploratory calculation, we choose to adjust the parameters in
a slightly different and simpler manner, especially as we aim at a qualitative
understanding of the physical mechanisms involved in determining the shape of
the pion form factor, and not so much at achieving an accurate quantitative
agreement with experiment. First, we set initially $\eta=1.5$. Second,
although we assume (confirmed by our calculation) that the pion-decay kernels
will not only move the $\rho$-meson pole into the complex plane but also shift
down its real value, we nevertheless adjust the parameter $\Lambda$ such that
we obtain a $\rho$-meson mass close to the phenomenological value already in
the calculations with gluon- and pion-exchange kernels only. Third, we require
to reproduce a quite accurate value for the pion decay constant. Finally, we
adjust $m_{q}$ to obtain the correct value for the pion mass as well. In this
way we chose the model parameters to be $\eta=1.5$, $\Lambda=0.78$ GeV and
$m_{q}=6.8$ MeV at a renormalisation scale $\mu=19$ GeV. As a rudimentary test
of model dependence we additionally perform the calculations for $\eta=1.6$ as
well (keeping $\Lambda$ and $m_{q}$ unchanged). The results for the pion mass
and decay constant as well as for the $\rho$-meson and $\omega$-meson masses,
without the pion decay kernels being taken into account, are shown in Tab. 1.

Table 1:  The pion mass $m_{\pi}$, the pion decay constant $f_{\pi}$, the $\rho$-meson and $\omega$-meson masses $m_{\rho}$ and $m_{\omega}$ for the two different parameterizations of the model used herein and for the case with rainbow-ladder and pion-exchange kernels but without decay kernels are shown. The light quark mass has been set to $m_{q}=0.0068~{}$GeV. The rightmost column displays the extracted $\rho$-meson pole position defined as $M_{pole}^{2}=M_{\rho}^{2}-iM_{\rho}\Gamma_{\rho}$, as discussed in the text. All values for dimensionful quantities are given in GeV. $\Lambda=0.78$ | $m_{\pi}$ | $f_{\pi}$ | $m_{\rho}$ | $m_{\omega}$ | $M_{\rho}$ | $\Gamma_{\rho}$
---|---|---|---|---|---|---
$\eta=1.5$ | 0.139 | 0.138 | 0.768 | 0.778 | 0.750 | 0.100
$\eta=1.6$ | 0.126 | 0.138 | 0.774 | 0.784 | 0.759 | 0.105

As mentioned in the previous section, the full QCD quark-photon vertex
possesses poles reflecting the masses and widths of the electrically neutral
vector meson resonances. Correspondingly, and as discussed in detail in refs.
Miramontes:2019mco ; Williams:2018adr , a solution for the quark-photon vertex
in the DSE/BSE framework allows to extract the $\rho$-meson mass and width via
the position of the poles of the vertex dressing functions. For the RL
truncation as well as for the RL plus pion exchange approximation this pole is
located on the real negative $Q^{2}$ axis indicating that the $\rho$ mesons
were stable for those truncations. Including the $s$\- and $u$-channel decay
kernels (25) and (26), the pole of the dressing functions moves into the
complex plane and can be extracted from the data on the real axis via a Padé
fit. Parametrising the pole position as
$M_{pole}^{2}=M^{2}_{\rho}-iM_{\rho}\Gamma_{\rho}$, we extract the
corresponding results for the $\rho$-meson mass and width, $M_{\rho}$ and
$\Gamma_{\rho}$ for this truncation (see Tab. 1) in reasonable agreement with
the experimental values. Hereby, it has to be noted that underestimating the
$\rho$-meson width does not come unexpected because taking into account only
the leading $\gamma_{5}$ component of the pion Bethe-Salpeter amplitude in the
kernels (23)–(26) misses some strengths therein. Whether considering in
addition also the sub-leading pion amplitudes will provide a much better
result for the $\rho$-meson width can only be answered by performing the
corresponding calculation. This computation is then, however, an order of
magnitude more expensive than the present exploratory calculation.

It is interesting to note that the pion exchange kernels lift the degeneracy
in between the isovector $\rho$\- and the isosinglet $\omega$-meson present at
the level of RL calculations the interaction kernels of which are flavour
blind and thus flavour U(2) (resp., flavour U$(N_{f})$) symmetric. The pion
exchange kernels lead to a splitting such that $m_{\omega}-m_{\rho}=10$ MeV
which compares favorably with the experimental splitting of 7 - 8 MeV.

We turn now to the calculation of the pion form factor.

As already indicated in the previous section, there are, besides restricting
to the leading pion amplitude in the kernels(23)–(26), two major
approximations that we must perform in order to keep the calculation
technically manageable. First, using the decay kernels as described in this
work entails that one must choose whether the axial-vector or the vector WTI
are preserved while the other one is violated. Following Miramontes:2019mco
we choose to preserve the vector identity. A violation of the axial-vector WTI
is manifested, among others, in the pion not being massless in the chiral
limit, and therefore the value of the current mass for which the pion becomes
massless allows for a quantification of the violation of the axial-vector WTI.
In Fig. 4 we therefore show the evolution of the pion mass with varying
$m_{q}$ in the employed truncation. First, the relation is linear as expected
from the Gell-Mann–Oakes–Renner relation which is a direct consequence of the
dynamical breaking of chiral symmetry. Second, the pion does not become
massless in the chiral limit but for a value of the current mass
$m_{q}^{(0)}$($\mu=19$ GeV) = 3 MeV. On the one hand, this explains the
relatively large value of $m_{q}$($\mu=19$ GeV) = 6.8 MeV we needed to obtain
the correct pion mass: The related explicitly chiral-symmetry-breaking term is
$m_{q}-m_{q}^{(0)}$ = 3.8 MeV, and thus much closer to what is expected from
the known parameters of QCD. Second, as the masses of the vector mesons depend
linearly on the current mass the induced error on $m_{\rho}$ and $m_{\omega}$
is of the order of $m_{q}^{(0)}$ = 3 MeV, and thus it is as small or even
smaller than other uncertainties in our calculation of the vector meson
masses.

Figure 4: The pion mass squared $m_{\pi}^{2}$ vs. the current mass
$m_{q}$($\mu=19$ GeV) for the employed truncation.

Second, in the calculation of the form factor we use the impulse approximation
which, in this context, implies neglecting the second and third diagrams in
Fig. 3. The consequence of discarding diagrams is the violation of charge
conservation or, equivalently, a deviation from $F_{\pi}(0)=1$. As can be seen
in the inset in Fig. 5, this effect is of the order of $\sim 1\%$ only.

Figure 5: Pion form factor in the spacelike $Q^{2}>0$ domain for the model
parameters $\eta=1.5$ and $\eta=1.6$ as described in the text and compared to
experimental data.

Figure 6: Absolute value of the pion form factor in the time-like $Q^{2}<0$
domain for the model parameters $\eta=1.5$ and $\eta=1.6$ as described in the
text and compared to experimental data.

As can be also seen from Fig. 5, the results for the pion form factor in the
space-like $Q^{2}>0$ regime are practically independent from the value of the
$\eta$ parameter of the model. Even more remarkable is the fact that our
calculation shows a very good agreement with experimental data in the space-
like domain even though, as evident from the above discussion, we aimed at
including all physical effects which are important in the time-like regime. An
interpretation of this result in view of the dispersion relation (4) provides
an indication that the imaginary part in the time-like region is precisely
enough reproduced to provide very good results for the space-like form factor.

Figure 7: Phase of the pion form factor in the time-like $Q^{2}<0$ domain for
the model parameters $\eta=1.5$ and $\eta=1.6$ as described in the text and
compared to experimental data on pion-pion phase shift.

We show our results for the pion form factor for time-like ($Q^{2}<0$)
virtualities in Figs. 6 and 7. As discussed in the previous section, as a
consequence of the decay kernels in our truncation, the pion form factor
develops a branch cut along the real negative axis starting from the two-pion
threshold $Q^{2}=-4m_{\pi}^{2}$, induced by the corresponding cut in the
quark-photon vertex Miramontes:2019mco . Hence, in that region the (complex)
form factor is defined from its analytic continuation as $F(Q^{2}+i\epsilon)$.
In the numerical calculations we have typically chosen $\epsilon=0.0001$ GeV2
after verifying that this value is small enough to not disturb the presented
results. In Fig. 6 we present the absolute value for the model parameter
$\eta=1.5$ and $\eta=1.6$, with the remaining model parameters kept constant,
as discussed above. As a manifestation of the fact that the $\rho$-meson pole
in the quark-photon vertex moves into the complex plane when the decay kernels
are included in the calculation, the pion form factor develops a bump on the
real and negative $Q^{2}$ axis with an approximately correct height and width.
Therefore our calculation overcomes a major deficiency of the RL truncation,
without or even with the pion exchange term, for which the form factor
diverges instead at the $Q^{2}$-value corresponding to the $\rho$-meson mass
in those truncations, see e.g. Maris:1999bh ; Krassnigg:2004if ;
Eichmann:2019tjk . This constitutes already one main result of the here
presented investigation.

We note, however, that, contrary to the results for space-like regime, the
position and height of the bump of the form factor depends strongly on the
value of the $\eta$ parameter shape of the form factor. Of course, this
reflects the different positions of the $\rho$-meson pole, cf. Tab. 1.
Nevertheless, there are features that appear to be independent of $\eta$, most
prominently that the height of the bump is underestimated. As expected from
the discussion in Sec. II the form factor behaves smoothly after the bump, in
contrast to the sharp dip in the experimental data. Even though an unambiguous
analysis of the origin of such discrepancies can only result from the
inclusion of all relevant physical mechanisms in our calculations, it is
evident from the analysis performed in Sec. II that one of the main missing
elements is the $\rho$-$\omega$ mixing due to isospin breaking, which is
completely absent in the present study.

Particularly sensitive to the deficiencies of our truncation is the phase of
the form factor, shown in Fig. 7. Even though our data shows the expected
behaviour near the resonance value, it severely underestimates the
experimental data, particularly in the elastic region. This is a manifestation
of the absence of some hadronic effects in our approximation scheme, which
only includes those stemming from the resonance complex pole in the quark-
photon vertex and the $\rho\rightarrow\pi\pi$ induced branch cut. In addition
to the isospin breaking effects discussed above, and which would be more
relevant in the region above the resonance, the impulse approximation used in
our calculation of the form factor entails that effects coming from the
coupling of the photon to the intermediate hadrons, via its coupling to the
exchanged pion or to the quark-pion vertex (see Fig. 3), are missing. It has
been shown Cotanch:2002vj  that considering only impulse-like diagrams leads
to a very small pion-pion scattering amplitude in the elastic region in the
isospin $I=1$ channel. This, due to unitarity, implies a very small value of
the imaginary part of the pion form factor in the elastic region (which is, in
fact, what we observe) which translates into a very small phase, as seen in
Fig. 7 (and as could be inferred from Watson’s theorem).

Last but not least, we are comparing Padé fits, resp. rational fits of the
order (3,3), for the real part and for the imaginary part of the form factor
to the expression (8). Trying first,

$\displaystyle Re\,F_{\pi}(Q^{2})-F_{\pi}(0)$ $\displaystyle\approx$
$\displaystyle-\frac{a_{0}+a_{1}Q^{2}+a_{2}(Q^{2})^{2}+a_{3}(Q^{2})^{3}}{b_{0}+b_{1}Q^{2}+b_{2}(Q^{2})^{2}+b_{3}(Q^{2})^{3}}$
$\displaystyle Im\,F_{\pi}(Q^{2})$ $\displaystyle\approx$
$\displaystyle\frac{c_{0}+c_{1}Q^{2}+c_{2}(Q^{2})^{2}+c_{3}(Q^{2})^{3}}{d_{0}+d_{1}Q^{2}+d_{2}(Q^{2})^{2}+d_{3}(Q^{2})^{3}}\,,$

we obtain tiny values for the coefficients $a_{0}$, $a_{3}$, $b_{3}$, $c_{0}$,
$c_{3}$ and $d_{3}$. Note that this confirms the structure expected from the
VMD form (8). We repeated the fits for

$\displaystyle Re\,F_{\pi}(Q^{2})-F_{\pi}(0)$ $\displaystyle\approx$
$\displaystyle-\frac{a_{1}Q^{2}+a_{2}(Q^{2})^{2}}{b_{0}+b_{1}Q^{2}+b_{2}(Q^{2})^{2}}$
$\displaystyle Im\,F_{\pi}(Q^{2})$ $\displaystyle\approx$
$\displaystyle\frac{c_{1}Q^{2}+c_{2}(Q^{2})^{2}}{d_{0}+d_{1}Q^{2}+d_{2}(Q^{2})^{2}}\,.$
(22)

The coefficients resulting from these fits as well as the ones resulting from
expression (8) are given in table 2. From this we conclude that the expression
based on the VMD is an astonishingly good representation of our results.
Therefore, our investigation makes it plausible that the VMD picture can be
derived from QCD. At least, the results of the here presented microscopic
approach give a strong hint into this direction.

Table 2:  The coefficients of the rational fits to the pion form factor as discussed in the text. All values for dimensionful quantities are given in GeV. |  $\eta$ =1.5 |  $\eta$ =1.6 | Eq. (8)
---|---|---|---
$a_{1}$ | 0.5587 | 0.4149 | 0.72
$a_{2}$ | 0.8828 | 0.6827 | 1.2
$b_{0}$ | 0.3600 | 0.3600 | 0.36
$b_{1}$ | 1.2307 | 1.2517 | 1.2
$b_{2}$ | 1.0722 | 1.1000 | 1.037
$c_{1}$ | 0.0591 | 0.0997 | 0
$c_{2}$ | 0.1295 | 0.2383 | 0.2308
$d_{0}$ | 0.3600 | 0.3600 | 0.36
$d_{1}$ | 1.1924 | 1.2464 | 1.2
$d_{2}$ | 0.9973 | 1.0916 | 1.0037

##  V  Conclusions and outlook

In this work we have presented an exploratory study of the pion form factor in
the DSE/BSE approach. Our focus has been to explore how the interplay between
hadron structure (as described by form factors) and the hadron spectrum (as
described by resonance masses and widths) can be realised in the microscopic
approach presented herein. In particular, we focused on the effect of
intermediate pions in the BSE interaction kernel, the inclusion of which is
sufficient to describe the $\rho$ meson as a resonance Williams:2018adr . As
elucidated by the detailed analysis in the last section our calculation
represents a verification of the vector meson dominance picture and provides
an explanation how at the quark level vector meson dominance becomes
effective. A more complete calculation than the one presented here might then
actually provide a derivation of vector meson dominance from QCD.

Despite the fairly drastic approximations used in this preliminary study, the
agreement with experiment is remarkable. On the space-like side our
calculations agree with experimental data at the quantitative level. For time-
like momentum transfers the agreement is mostly qualitative and consistent
with the fact that in our approximation scheme time-like physics is dominated
by the lowest-lying $\rho$-meson resonance only. The absolute value of the
calculated form factor features a bump at approximately the correct $Q^{2}$
region as caused by a resonance pole. Our result lacks, however, other
features such as those caused by the isospin-breaking $\rho$-$\omega$ mixing
and interference. The phase of the form factor also shows deficiencies caused
by the employed impulse approximation. However, the overall qualitatively
correct behaviour is, nevertheless, very encouraging for future studies on
time-like phenomenology with BSE methods as it shows that the necessary
computational techniques are getting more and more under control, and that
within a functional-method-based bound-state approach to QCD direct
calculations in the time-like regime are becoming feasible.

Among the different physical mechanisms absent in our calculation, the most
relevant one appears to be isospin breaking by the light quarks’ masses and
electric charges, and the different phenomena associated with it. The
presented exploratory calculation paved the way to include in a bound-state
approach formulated in QCD degrees of freedom the effects of isospin
violation, and hereby most prominently $\rho$-$\omega$ mixing, on the time-
like pion electromagnetic form factor. Thus including isospin violation in a
BSE approach is the topic of ongoing work. A further related topic is the
study of the form factor for the coupling of a photon to three pions. On the
one hand, this process is of special theoretical interest because the related
form factor is at the soft point completely determined by the Abelian chiral
anomaly. On the other hand, data of the COMPASS experiment are currently
analysed Dominik , and therefore experimental data for this form factor in the
time-like region will become available. Combing previous studies in the
DSE/BSE approach for the space-like $\gamma\pi\pi\pi$ form factor
Alkofer:1995jx ; Bistrovic:1999dy ; Cotanch:2003xv ; Benic:2011rk ;
Eichmann:2011ec  with the techniques of the here presented calculation will
thus enable a respective investigation of this form factor.

Another aspect we want to investigate is how to realise the idea of decay
kernels like (25) and (26) in a baryon bound state equation. This is a
necessary step in order to tackle time-like nucleon form factors in the
DSE/BSE approach, which is one of our major goals due to the increased effort
and interest from the experimental side in highly precise measurements over a
wide kinematical domain of the nucleon form factors.

## Acknowledgements

We thank Gernot Eichmann and Christian Fischer for a critical reading of the
manuscript and helpful discussions.

This work was partially supported by the the Austrian Science Fund (FWF) under
project number P29216-N36.
A.S. Miramontes acknowledges CONACyT for financial support.

The numerical computations have been performed at the high-performance compute
cluster of the University of Graz.

##  Appendix A Interaction kernels

Figure 8: Truncations used herein for the BSE interaction kernel $K$ (upper
diagram) and the quark DSE one (lower diagram). In the upper diagram, the
terms on the right-hand side correspond to the rainbow-ladder, pion exchange,
and s- and u-channel pion decay contributions to the truncation, respectively.
The s- and u-channel pion decay terms do not contribute to the quark DSE.

The BSE kernel representing the exchange of an explicit pionic degrees of
freedom as defined in Fischer:2007ze ; Fischer:2008sp  reads

$\displaystyle K^{(t)~{}ut}_{rs}(q,p;P)=$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{C}{4}[\Gamma_{\pi}^{j}]_{ru}\left(\frac{p+q-P}{2};p-q\right)[Z_{2}\gamma^{5}]_{ts}D_{\pi}(p-q)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{C}{4}[\Gamma_{\pi}^{j}]_{ru}\left(\frac{p+q-P}{2};q-p\right)[Z_{2}\gamma^{5}]_{ts}D_{\pi}(p-q)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{C}{4}[Z_{2}\gamma^{5}]_{ru}[\Gamma_{\pi}^{j}]_{ts}\left(\frac{p+q+P}{2};p-q\right)D_{\pi}(p-q)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{C}{4}[Z_{2}\gamma^{5}]_{ru}[\Gamma_{\pi}^{j}]_{ts}\left(\frac{p+q+P}{2};q-p\right)D_{\pi}(p-q)~{},$
(23)

in combination with the following truncation of the quark DSE

$\displaystyle S^{-1}(p)$ $\displaystyle=$ $\displaystyle
S^{-1}(p)^{RL}-\frac{3}{2}\int_{q}\Bigg{[}Z_{2}\gamma_{5}S(q)\Gamma_{\pi}\left(\frac{p+q}{2},q-p\right)$
(24)
$\displaystyle+Z_{2}\gamma_{5}S(q)\Gamma_{\pi}\left(\frac{p+q}{2},p-q\right)\Bigg{]}\frac{D_{\pi}(k)}{2}~{},$

with $S^{-1}(p)^{RL}$ being the right-hand-side of the quark DSE in the RL
truncation with the gluon-mediated interaction as described in Sect. III. In
Eqs. (23) and (24) the pion propagator is taken as
$D_{\pi}(k)=(k^{2}+m_{\pi}^{2})^{-1}$. The factor $3/2$ in Eq. (24) originates
from the flavour traces, and in the same way the factor $C$ in (23) should be
obtained. When done in the quark-photon vertex, the flavour factor leads to
$C=+3/2$. However, such value of $C$, violates the Ax-WTI while preserving the
V-WTI. Since the V-WTI is related to charge conservation in electromagnetic
form factors, we use $C=+3/2$ at the expense of violating the Ax-WTI. Herein,
the quark-pion vertex $\Gamma_{\pi}$ is taken to be the pion Bethe-Salpeter
amplitude.

The pions in the kernel can also appear in the s- and u- channels
Fischer:2007ze . We used here a version of the kernels slightly different to
the one in Fischer:2007ze  in order to be consistent with the construction for
the t-channel, where one of the pion vertices is kept bare and the kernel is
then symmetrised. They read

$\displaystyle K^{(s)~{}he}_{da}(q,p,r;P)=~{}$
$\displaystyle\frac{C}{2}~{}D_{\pi}\left(\frac{P+r}{2}\right)D_{\pi}\left(\frac{P-r}{2}\right)~{}\left[[Z_{2}\gamma_{5}]_{dc}S_{cb}\left(p-\frac{r}{2}\right)[Z_{2}\gamma_{5}]_{ba}\right.$
$\displaystyle~{}~{}~{}~{}\times[\Gamma_{\pi}^{j}]_{hg}\left(q-\frac{P}{4}-\frac{r}{4};\frac{r-P}{2}\right)S_{gf}\left(q-\frac{r}{2}\right)[\Gamma_{\pi}^{j}]_{fe}\left(q+\frac{P}{4}-\frac{r}{4};-\frac{P+r}{2}\right)$
$\displaystyle+~{}[\Gamma_{\pi}^{j}]_{dc}\left(p+\frac{P}{4}-\frac{r}{4};\frac{P+r}{2}\right)S_{cb}\left(p-\frac{r}{2}\right)[\Gamma_{\pi}^{j}]_{ba}\left(p-\frac{P}{4}-\frac{r}{4};\frac{P-r}{2}\right)$
$\displaystyle~{}~{}~{}~{}\left.\times[Z_{2}\gamma_{5}]_{hg}S_{gf}\left(q-\frac{r}{2}\right)[Z_{2}\gamma_{5}]_{fe}\right]~{},$
(25)

and

$\displaystyle K^{(u)~{}he}_{da}(q,p,r;P)=~{}$
$\displaystyle\frac{C}{2}~{}D_{\pi}\left(\frac{P+r}{2}\right)D_{\pi}\left(\frac{P-r}{2}\right)~{}\left[[Z_{2}\gamma_{5}]_{dc}S_{cb}\left(p+\frac{r}{2}\right)[Z_{2}\gamma_{5}]_{ba}\right.$
$\displaystyle~{}~{}~{}~{}\times[\Gamma_{\pi}^{j}]_{hg}\left(q-\frac{P}{4}-\frac{r}{4};\frac{r-P}{2}\right)S_{gf}\left(q-\frac{r}{2}\right)[\Gamma_{\pi}^{j}]_{fe}\left(q+\frac{P}{4}-\frac{r}{4};-\frac{P+r}{2}\right)$
$\displaystyle+~{}[\Gamma_{\pi}^{j}]_{dc}\left(p+\frac{P}{4}+\frac{r}{4};\frac{P-r}{2}\right)S_{cb}\left(p+\frac{r}{2}\right)[\Gamma_{\pi}^{j}]_{ba}\left(p-\frac{P}{4}+\frac{r}{4};\frac{P+r}{2}\right)$
$\displaystyle~{}~{}~{}~{}\left.\times[Z_{2}\gamma_{5}]_{hg}S_{gf}\left(q-\frac{r}{2}\right)[Z_{2}\gamma_{5}]_{fe}\right]~{},$
(26)

where now $r$ is an additional integration momentum in the BSE (cf. Eqs.(III)
or (10)). The resulting truncation of the BSE kernel and the quark DSE is
shown in Fig. 8.

The inclusion of the two kernels given in equations (25) and (26) generates a
highly non-trivial analytic structure of the integrand of the BSE, induced by
the intermediate pions going potentially on-shell as well as by singularities
in the quark propagators, see ref. Miramontes:2019mco  for details where also
the techniques for finding viable contour deformations for performing the
numerical integrations in a mathematically correct way are described.101010
Further details on applying contour deformations for performing the numerical
integrations in the context of DSEs and BSEs can be found in refs.
Santowsky:2020pwd ; Windisch:2013dxa ; Windisch:2014lce ; Eichmann:2019dts
and references therein.

The pion Bethe-Salpeter amplitude possesses four tensor components. Hereby,
only one is generically small such that for a precise calculation of pion
properties it would be necessary to take into account three of them, namely
the leading pseudoscalar term and two subleading ones related to the
axialvector structure. However, using in the above described kernels three
amplitudes for every pion Bethe-Salpeter amplitude in these expressions is
numerically by an order of magnitude more expensive and far beyond the scope
of the present study. Having restricted to the leading component of the pion
amplitude in the kernels (23) and (26) one can further exploit that this
leading amplitude may be well approximated using the chiral limit value of the
quark dressing function $B(p^{2})$ and normalize the amplitude by dividing
through the pion decay constant $f_{\pi}$,

$\Gamma_{\pi}^{i}(p;P)=\tau^{i}\gamma_{5}\frac{B(p^{2})}{f_{\pi}}\,.$ (27)

For light quarks the difference between calculated leading order amplitude and
this approximation is at the level of a few percent, see, e.g., Fischer:2008wy
and references therein. Therefore we use this simplified form.

##  Appendix B Poles of the quark propagator

In the truncations of the quark DSE used in this paper, the quark propagator
features pairs of complex conjugate poles in the complex plane (see, e.g.
Windisch:2016iud ). In order to facilitate the use of the quark propagators
and easily identify the analytic structures generated by those poles in the
form factor and vertex calculations, it is useful to parametrise the quark
propagator simply as a sum of poles (see e.g. El-Bennich:2016qmb )

$\displaystyle S(p)$ $\displaystyle=$
$\displaystyle-i\ooalign{\hfil/\hfil\crcr$p$}\sigma_{v}(p^{2})+\sigma_{s}(p^{2})\,,$
(30) $\displaystyle\sigma_{v}(p^{2})$ $\displaystyle=$
$\displaystyle\sum_{i}^{n}\left[\frac{\alpha_{i}}{p^{2}+m_{i}}+\frac{\alpha_{i}^{\ast}}{p^{2}+m_{i}^{\ast}}\right]\,,$
$\displaystyle\sigma_{s}(p^{2})$ $\displaystyle=$
$\displaystyle\sum_{i}^{n}\left[\frac{\beta_{i}}{p^{2}+m_{i}}+\frac{\beta_{i}^{\ast}}{p^{2}+m_{i}^{\ast}}\right]~{},$
(31)

where the parameters $m_{i}$, $\alpha_{i}$, $\beta_{i}$ can be obtained by
fitting the corresponding quark DSE solution along the $p^{2}$ real axis or,
alternatively, on a parabola in the complex plane that does not enclose the
poles.

In the numerical solution of the quark DSE in the complex plane we tested fits
with one real and one pair of complex conjugated poles, with two pairs of
complex conjugated poles, and with three pairs of complex conjugated poles.
Based on this tests we concluded that $n=2$ pairs of complex conjugated poles
provided a precise fit, and as the use of three pairs of poles did not provide
any further improvement, the reported calculations have been performed based
on fits with two pairs of poles.

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