Therefore, the detection capabilities of neutral pions, originating from a
resonance decay, are very limited with CLAS12 due to the missing
electromagnetic calorimeter in the region of $\theta>35^{\circ}$, leading to a
limitation of the studies to the $\gamma~{}n~{}\pi^{+}$ final state for a
proton target and to the $\gamma~{}p~{}\pi^{-}$ final state for a neutron
target.

Based on data taken with a 10.2 GeV and a 10.6 GeV longitudinally polarized
electron beam and a hydrogen target, a first study has been performed with
CLAS12. A series of exclusivity cuts on the missing mass, missing energy,
missing transverse momentum, and missing cone angle are applied to select
exclusive events. In addition, kinematic cuts on W $>$ 2 GeV, $Q^{2}$ $>$ 1.5
GeV2, y $<$ 0.8 and $E_{\gamma-DVCS}$ $>$ 2 GeV have been applied for the
preliminary study. Furthermore, a cut on $M(\pi^{+}\gamma)$ $>$ 1 GeV is
applied to reduce the dominant background from exclusive $\rho^{+}$ production
($\gamma^{*}~{}p~{}~{}\rightarrow~{}~{}\rho^{+}~{}n$), with the $\rho^{+}$
decaying into $\pi^{0}\pi^{+}$, with one undetected photon, and the much less
frequent decay into $\gamma\pi^{+}$.

Figure 18 shows the resonance spectrum of the $n\pi^{+}$ invariant mass after
the listed kinematic cuts for two bins of $-t$, measured with CLAS12.

Figure 18: Preliminary measured resonance spectrum in the $n\pi^{+}$ invariant
mass of the $e~{}p~{}~{}\rightarrow~{}~{}e^{\prime}~{}\gamma~{}n~{}\pi^{+}$
process from CLAS12 (blue). The kinematic and selection cuts described in the
text are applied, and the $-t$ range is stated on top of the sub-figures. For
comparison, MC simulations for the $N\to N^{*}$ DVCS process with the
production of a $\Delta(1232)$ resonance (red) and for the production of the
resonances in the second resonance region (green), following the model
prediction in Fig. 9, as well as MC simulations for the non-resonant
contributions (brown), are shown. The magenta histogram provides the sum of
all MC contributions.

A clear peak from the delta resonance as well as from the second resonance
region can be observed in both ranges of $-t$, while for the higher $-t$ bins,
also the third resonance region exceeds the production threshold and becomes
visible.

Figure 19: Preliminary results of the $A^{\sin\phi}_{LU}$ moment of the beam-
spin asymmetry from the
$e~{}p~{}~{}\rightarrow~{}~{}e^{\prime}~{}\gamma~{}n~{}\pi^{+}$ process, with
the cuts described in the text, as a function of the resonance mass
$M_{n\pi^{+}}$ in two bins of $-t$, as stated above the figures (upper row),
and as a function of $-t$ in the region around the $\Delta(1232)$ resonance
(1.13 GeV $<M_{n\pi^{+}}<$ 1.33 GeV, lower row left), the Roper resonance
(1.33 GeV $<M_{n\pi^{+}}<$ 1.45 GeV, lower row center) and the second
resonance region (1.45 GeV $<M_{n\pi^{+}}<$ 1.60 GeV, lower row right).
Backgrounds from the
$e~{}p~{}~{}\rightarrow~{}~{}e^{\prime}~{}\pi^{0}~{}N^{*}~{}~{}\rightarrow
e^{\prime}~{}\pi^{0}~{}n~{}\pi^{+}$ process, with one photon from the
$\pi^{0}$ decay misidentified as the DVCS photon are not subtracted yet. The
red line shows the prediction for the $N\to N^{*}$ DVCS process from the
transition GPD based model described in section 6.1, integrated over the
CLAS12 kinematic distributions.

Figure 19 shows a preliminary extraction of the $A^{\sin\phi}_{LU}$ moment of
the BSA as a function of the resonance mass in different bins of $-t$ and as a
function of $-t$ in different regions of the resonance mass $M_{n\pi^{+}}$, in
comparison to the transition GPD-based theoretical predictions described in
section 6.1.

The study is currently ongoing, and further backgrounds like the $N\to
N^{*}~{}\pi^{0}$ DVMP process, with one photon from the $\pi^{0}$ decay
misidentified as the DVCS photon, need to be considered and subtracted. The
contamination from $N\to N^{*}~{}\pi^{0}$ DVMP was found to be in the overall
order of $\approx$ 30 % for the shown data sample and shows an asymmetry with
the same sign but a smaller magnitude than the $N\to N^{*}$ DVCS process. A
subtraction is therefore expected to lead to a slight increase of the measured
$A^{\sin\phi}_{LU}$ moments. Considering these uncertainties, the preliminary
results for distributions and beam spin asymmetries show a promising agreement
with the transition GPD based theory predictions described in section 6.1 and
shown in Fig. 19.

Similar to the $N\to N^{*}$ DVMP processes described in sections 7.1 and 7.2,
also for the $N\to N^{*}$ DVCS processes, studies with a deuterium target,
with further target polarizations as well as the extraction of the unpolarized
cross-section terms are possible based on CLAS12 data.

##  8 Transition processes: hadron scattering

###  8.1 Meson-induced exclusive Drell-Yan process

Besides the discussed exclusive lepton scattering reactions, the exclusive
hard reactions with the hadron beam are expected to be sensitive to the GPDs.
The most promising reaction is the meson-induced exclusive Drell-Yan process
Berger:2001zn ; Sawada:2016mao , which can be studied at J-PARC or CERN
COMPASS/AMBER with the pion and kaon beams. In addition, the primary proton
beam at J-PARC could be used for the GPDs Kumano:2009he . These processes are
described in detail below.

Recently, JLab CLAS reported the first measurement of timelike Compton
Scattering (TCS) process CLAS:2021lky . The comparison of the measured
polarization asymmetry of TCS with model predictions suggests the universality
of GPDs in both spacelike and timelike processes. In the same analogy between
TCS and DVCS processes, the meson-induced exclusive process is the
complementary timelike process to access the nucleon GPDs, with respect to the
DVMP process.

The exclusive meson-induced Drell-Yan process
$mN\rightarrow\ell^{+}\ell^{-}B$, as shown in Fig. 20, describes the
interaction of a meson beam ($m$) with a nucleon target ($N$) during which a
lepton pair ($\ell^{\pm}$) and a ground state nucleon or a baryon resonance
($B$) are produced.

Figure 20: Schematic drawing of the exclusive Drell-Yan process. For a ground-
state nucleon in the final state, the process is sensitive to the ordinary
GPDs, while a baryon resonance in the final state provides sensitivity to the
transition GPDs.

The process is expected to factorize if the momentum transfer squared $t$
between the initial and final state baryon is small and the four momentum
squared $Q^{\prime 2}$ of the final state virtual photon is large. However,
potential large contributions of non-factorized processes still need to be
evaluated in more detail and considered for the analysis and interpretation of
the measurements, as shown in Refs. Tanaka:2017ccr .

The exclusive Drell-Yan process
$\pi(p_{m})N(p)\to\gamma^{*}(q^{\prime})N(p^{\prime})\ (\to\ell^{+}\ell^{-}N)$
is described by the following variables: the virtual-photon momentum squared
$Q^{\prime 2}=q^{\prime 2}$, the invariant momentum transfer
$t=(p^{\prime}-p)^{2}$, the scaling variable $\tau$ given by

$\tau=\frac{Q^{\prime 2}}{2p\cdot p_{m}}\simeq\frac{Q^{\prime
2}}{s-M_{N}^{2}}\simeq\frac{Q^{\prime 2}}{s},$ (81)

where $s=(p+p_{m})^{2}$ and $M_{N}$ is the nucleon mass. The variable $\tau$
is similar to the Bjorken variable in the DVCS and DVMP, and it is related to
the pion momentum as $q\simeq s/(2M_{N})\simeq Q^{\prime 2}/(2M_{N}\tau)$. The
skewness variable $\xi$ is approximately given by $\tau$ as

$\xi\simeq\frac{Q^{\prime 2}}{2s-Q^{\prime 2}}=\frac{\tau}{2-\tau}.$ (82)

Using these kinematical variables, we write the $\pi^{-}N\to\ell^{+}\ell^{-}N$
cross section up to twist-4 terms as Goloskokov:2009ia ; Goloskokov:2015zsa

$\displaystyle\frac{d\sigma}{dtdQ^{\prime 2}d\cos\theta d\phi}$
$\displaystyle=\frac{3}{8\pi}\biggl{(}\sin^{2}\theta\frac{d\sigma_{L}}{dtdQ^{\prime
2}}+\frac{1+\cos^{2}\theta}{2}\frac{d\sigma_{T}}{dtdQ^{\prime 2}}$
$\displaystyle\ +\frac{\sin
2\theta\cos\phi}{\sqrt{2}}\frac{d\sigma_{LT}}{dtdQ^{\prime
2}}+\sin^{2}\theta\cos 2\phi\frac{d\sigma_{TT}}{dtdQ^{\prime 2}}\biggr{)},$
(83)

where the $\theta$ and $\phi$ are polar and azimuthal angles of a lepton
direction in the final state. The cross section $d\sigma_{L}$ ($d\sigma_{T}$)
is for the longitudinally (transversely)-polarized virtual photon. The
$d\sigma_{LT}$ and $d\sigma_{TT}$ are the longitudinal-transverse interference
and transverse (helicity $h=1$)-transverse ($h=-1$) interference terms. The
cross section $d\sigma_{L}$ is leading twist, $d\sigma_{LT}$ is twist three,
and $d\sigma_{T}$ and $d\sigma_{TT}$ are twist-four terms.

For the reaction $\pi^{-}N\rightarrow\ell^{+}\ell^{-}N$, Refs. Berger:2001zn ;
Sawada:2016mao  show that the longitudinal part of the cross section
$\sigma_{L}$ can be related to the GPDs $\widetilde{H}$ and $\widetilde{E}$.
The transverse part of the cross sections $\sigma_{T}$ and the $LT$ and $TT$
interference terms $\sigma_{LT}$ and $\sigma_{TT}$ are also sensitive to
transversity GPDs as shown in Ref. Goloskokov:2015zsa . For reactions with a
baryon resonance in the final state, this can be related to the corresponding
transition GPDs. However, a formalism is not available at the present stage.

In this process, the relevant amplitude is expressed as the convolution of the
short-distance partonic annihilation processes with the two parts of long-
distance nature, associated with the nucleon GPD and the pion DA. Here, the
GPDs for the nucleon are the functions of the light-cone momentum fractions
$x$ of the average momentum, the skewness
$\xi\equiv(p-p\,^{\prime})^{+}/(p+p\,^{\prime})^{+}$, and the momentum
transfer squared $t=\Delta^{2}$ with $\Delta\equiv p\,^{\prime}-p$ where the
$p$ and $p\,^{\prime}$ are initial and final nucleon momenta, respectively.

The quark GPDs relevant to the processes without the quark-helicity flip are
given by (see, e.g., Diehl:2003ny ; Belitsky:2005qn )

$\displaystyle\int$
$\displaystyle\frac{dz^{-}}{4\pi}e^{ixP^{+}z^{-}}\left<p^{\prime}\left|\bar{q}(-z/2)\gamma^{+}q(z/2)\right|p\right>\Big{|}_{z^{+}=\vec{z}_{\perp}=0}$
$\displaystyle=$
$\displaystyle\frac{1}{2P^{+}}\,\bar{u}(p^{\prime})\left[H^{q}(x,\xi,t)\gamma^{+}+E^{q}(x,\xi,t)\frac{i\sigma^{+\alpha}\Delta_{\alpha}}{2\,M_{N}}\right]u(p),$
(84)

and

$\displaystyle\int$
$\displaystyle\frac{dz^{-}}{4\pi}e^{ixP^{+}z^{-}}\left<p^{\prime}\left|\bar{q}(-z/2)\gamma^{+}\gamma_{5}q(z/2)\right|p\right>\Big{|}_{z^{+}=\vec{z}_{\perp}=0}$
$\displaystyle=$
$\displaystyle\frac{1}{2P^{+}}\,\bar{u}(p^{\prime})\left[\tilde{H}^{q}(x,\xi,t)\gamma^{+}\gamma_{5}+\tilde{E}^{q}(x,\xi,t)\frac{\gamma_{5}\Delta^{+}}{2\,M_{N}}\right]u(p),$
(85)

for each quark flavor $q$, where $\left|p\right>$ denotes the proton state
with momentum $p$ and mass $M_{N}$, $u(p)$ denotes the Dirac spinor for the
proton, $\sigma^{\alpha\beta}$ is given by
$\sigma^{\alpha\beta}=(i/2)[\gamma^{\alpha},\gamma^{\beta}]$, and the average
momentum is denoted as $P$ $(\,\equiv(p+p\,^{\prime})/2\,)$. The
$H^{q}(x,\xi,t)$ and $E^{q}(x,\xi,t)$ are the unpolarized quark GPDs, and
$\tilde{H}^{q}(x,\xi,t)$ and $\tilde{E}^{q}(x,\xi,t)$ are the polarized ones.

The factorization has been proven for the DVMP processes at the leading twist,
including the exclusive electroproduction of pion, $\gamma^{*}N\to\pi N$
Collins:1996fb . The amplitude can be written in terms of the hard-scattering
processes at the parton level, combined with the pion distribution amplitude
(DA), $\phi_{\pi}$, and also the nucleon GPDs, $\tilde{H}$ and $\tilde{E}$. By
interchanging the initial $\gamma^{*}$ and final $\pi$ in the exclusive
electroproduction of pion, and replacing the spacelike momentum of
$\gamma^{*}$ by the timelike momentum, the factorization at twist-two is
argued to be applicable to the exclusive Drell-Yan process,
$\pi(q)N(p)\to\gamma^{*}(q^{\prime})N(p^{\prime})$, with the same universal
non-perturbative input Berger:2001zn .

The appropriate kinematical region is of large timelike virtuality $Q^{\prime
2}=q^{\prime 2}$ at fixed $t=(p^{\prime}-p)^{2}$ and fixed scaling variable
$\tau\equiv Q^{\prime 2}/(2p\cdot q)$ where the $q$, $p$ and $p^{\prime}$ are
the momenta of the pion, initial, and final nucleons, respectively. At the
large $Q^{\prime}$ scaling limit, the corresponding leading-twist cross
section of $\pi^{-}p\to\gamma^{*}n$ as a function of $t$ and $Q^{\prime 2}$ is
expressed in terms of convolution integrals $\tilde{\cal H}^{du}$ and
$\tilde{\cal E}^{du}$, as follows Berger:2001zn

$\displaystyle\left.\frac{d\sigma_{L}}{dtdQ^{\prime 2}}\right|_{\tau}$
$\displaystyle=\frac{4\pi\alpha_{\rm em}^{2}}{27}\frac{\tau^{2}}{Q^{\prime
8}}f_{\pi}^{2}\,\Bigl{[}(1-\xi^{2})|\tilde{\cal H}^{du}(\tilde{x},\xi,t)|^{2}$
$\displaystyle-2\xi^{2}\mbox{Re}\ \bigl{(}\tilde{\cal
H}^{du}(\tilde{x},\xi,t)^{*}\tilde{\cal E}^{du}(\tilde{x},\xi,t)\bigr{)}$
$\displaystyle-\xi^{2}\frac{t}{4M_{N}^{2}}|\tilde{\cal
E}^{du}(\tilde{x},\xi,t)|^{2}\Bigr{]},$ (86)

where $\alpha_{\rm em}$ is the fine structure constant, $f_{\pi}$ is the pion
decay constant, and the variable $\tilde{x}$ is

$\tilde{x}=-\frac{(p_{m}+q^{\prime})^{2}}{2(p+p^{\prime})\cdot(p_{m}+q^{\prime})}\simeq-\frac{Q^{\prime
2}}{2s-Q^{\prime 2}}=-\xi.$ (87)

The convolution integral $\tilde{\cal H}^{du}$ involves two soft objects: the
nucleon GPD $\tilde{H}^{du}$ for the $p\to n$ transition and the twist-two
pion DAs $\phi_{\pi}$. Using
$\tilde{H}^{du}(x,\xi,t)=\tilde{H}^{u}(x,\xi,t)-\tilde{H}^{d}(x,\xi,t)$ to
relate the transition GPD with the usual proton GPDs $\tilde{H}^{q}$ for quark
flavor $q=u,d$, the expression of $\tilde{\cal H}^{du}$ is given, at the
leading order in $\alpha_{s}$, by Berger:2001zn

$\displaystyle\tilde{\cal H}^{du}(\tilde{x},\xi,t)$
$\displaystyle=\frac{8}{3}\alpha_{s}\int_{-1}^{1}dz\,\frac{\phi_{\pi}(z)}{1-z^{2}}$
$\displaystyle\times\int_{-1}^{1}dx\Bigl{(}\frac{e_{d}}{\tilde{x}-x-i\epsilon}-\frac{e_{u}}{\tilde{x}+x-i\epsilon}\Bigr{)}$
$\displaystyle\bigl{(}\tilde{H}^{d}(x,\xi,t)-\tilde{H}^{u}(x,\xi,t)\bigr{)},$
(88)

where $e_{u,d}$ are the electric charges of $u,d$ quarks in units of the
elementary charge, and $\alpha_{s}$ is the running coupling constant of QCD.
The corresponding expression of $\tilde{\cal E}^{du}$ is given by (88) with
$\tilde{H}^{q}$ replaced by the proton GPDs $\tilde{E}^{q}$. Because of the
pseudo-scalar nature of the pion, the cross sections (86) receive the
contributions of $\tilde{H}$ and $\tilde{E}$ only, among the GPDs in Eqs. (84)
and (85). The pion distribution function $\phi_{\pi}(z)$ is defined in the
region $-1<z<1$ as $\int_{-1}^{1}dz\phi_{\pi}(z)=1$. Use of the same pion beam
also provides an opportunity to access pion-to-nucleon transition distribution
amplitudes Pire:2021hbl  through investigation of backward charmonium
production in pion-nucleon collisions $\pi^{-}p\to J/\psi\,n$ Pire:2016gut .

To ensure proper factorization and sensitivity to the partonic structure,
$Q^{\prime}$ has to be larger than 1 GeV and $|t|$ is large enough. Both of
these require a large center-of-mass energy ($\sqrt{s}$) for the reactions.
Nevertheless, the estimated hard exclusive Drell-Yan cross sections drop
dramatically with an increase of $\sqrt{s}$. Furthermore, the experimental
determination of exclusiveness for the measured reactions through a missing-
mass technique favors the measurement of lepton tracks from the Drell-Yan
process in an open apparatus, and thus the charged multiplicity has to be low
enough for good tracking. Considering various experimental factors and
constraints, it is found that carrying out the measurement of the exclusive
Drell-Yan process in the coming high-momentum beamline at J-PARC with a 10-20
GeV $\pi^{-}$ beam ($\sqrt{s}=$ 4-6 GeV) is unique and optimized.

The E50 experiment at J-PARC plans to investigate the charmed-baryon
spectroscopy via the measurement of $\pi^{-}+p\to Y_{c}^{*}+D^{*-}$ reaction
in the high-momentum beam line Noumi:2013yxa ; Shirotori:2015eqa  with a
spectrometer system of a large acceptance and good momentum resolutions.
Thanks to the relatively low track density in the energy regime of J-PARC and
the high-granularity tracking chambers of the E50 experiment, the measurement
of the Drell-Yan process could be operated without the installation of a
hadron absorber in front of the spectrometer. Without the multiple-scattering
effect in the hadron absorber, a good momentum determination of muon tracks
can be achieved so that the exclusive Drell-Yan process can be characterized
via the missing-mass technique.

To identify the muon track, a muon identification ($\mu$ID) system after the
spectrometer is necessary. The current design of the $\mu$ID system consists
of hadron absorber layers made of 20-cm concrete and 230-cm iron to absorb
incoming hadrons, a large-size tracker resistive plate chambers (RPC) upstream
of the absorber (2.4x1.8 m2 active area with a few mm spatial resolutions),
and another RPC downstream of the absorber (3.5x2.5 m2 active area with 5cm
spatial resolution). The thickness of concrete and iron shall be optimized
considering the stopping power for low-momentum tracks and penetrating
efficiency for high-momentum ones.

Using GK2013 GPDs Kroll:2012sm  for the exclusive Drell-Yan process, a
feasibility study for 50-day beam time was done in Ref. Sawada:2016mao . The
simulated invariant mass $M_{\mu^{+}\mu^{-}}$ and missing-mass $M_{X}$ spectra
of the $\mu^{+}\mu^{-}$ events with $M_{\mu^{+}\mu^{-}}>1.5$ GeV and
$|t-t_{0}|<0.5$ GeV2 for $P_{\pi}$=10, 15, and 20 GeV are shown in Fig. 21,
where $t_{0}$ ($=-4M_{N}^{2}\xi^{2}/(1-\xi^{2})$) is the limiting value of $t$
corresponding to the scattering angle in the center-of-mass system
$\theta^{CM}=0$. Lines with different colors denote the contributions from
various sources: exclusive Drell-Yan (red, dashed), inclusive Drell-Yan (blue,
dotted), $J/\psi$ (cyan, dash-dotted), and random background (purple, solid),
respectively. Signals of $J/\psi$ are only visible in the invariant mass
distributions for $P_{\pi}$=15 and 20 GeV.

Figure 21: The Monte-Carlo simulated invariant mass $M_{\mu^{+}\mu^{-}}$ and
missing-mass $M_{X}$ spectra of the $\mu^{+}\mu^{-}$ events with
$M_{\mu^{+}\mu^{-}}>1.5$ GeV and $|t-t_{0}|<0.5$ GeV2 for $P_{\pi}$=10, 15,
and 20 GeV. Lines with different colors denote the contributions from various
sources. The GK2013 GPD parametrization was used as an input for the
evaluation of the exclusive Drell-Yan process.

The study shows that the exclusive Drell-Yan events could be identified by the
signature peak at the neutron mass ($M_{n}\sim 0.940$ GeV) in the missing-mass
spectrum for all three pion beam momenta, and the differential cross sections
as a function of $t$ are good enough to differentiate two different GPD
parametrizations. In terms of statistics and missing mass resolution, it was
found that the measurement with a 15-GeV pion beam would be optimal.

###  8.2 Proton-induced 2 $\rightarrow$ 3 processes

GPDs can also be investigated by using the primary proton beam through the
$2\to 3$ process $NN\to NMB$ Kumano:2009he , where $M$ is a meson and $B$ is a
baryon. It is especially interesting that this process probes the kinematical
region of the ERBL (Efremov-Radyushkin-Brodsky-Lepage) in the GPDs.

There are three kinematical regions for GPDs, depending on the values of the
scaling variable $x$ and the skewness variable $\xi$ as shown in Fig. 22. The
region $\xi<x<1$ ($-1<x<-\xi$) indicates a quark (antiquark) emission of
momentum fraction $x+\xi$ ($\xi-x$) with a quark (antiquark) absorption of
momentum fraction $x-\xi$ ($-\xi-x$), so that the process indicates a quark
(antiquark) distribution amplitude. These are called DGLAP (Dokshitzer-Gribov-
Lipatov-Altarelli-Parisi) regions. The region $-\xi<x<\xi$ is the ERBL region,
and the process is a quark emission of momentum fraction $x+\xi$ with an
antiquark emission of momentum fraction $\xi-x$, so that it is a quark-
antiquark (meson) distribution amplitude.

Figure 22: ERBL and DGLAP regions of the GPDs.

The cross sections for $NN\to N\pi B$ are estimated by using the pion- and
rho-pole contributions to the GPDs in the ERBL region Kumano:2009he . Here,
$B$ is a nucleon, or $\Delta$. We consider the process where the pion and the
final nucleon have large and nearly opposite transverse momenta and a large
invariant mass. Namely, the Mandelstam variables satisfy the hard condition
$s^{\prime},|t^{\prime}|,|u^{\prime}|\gg m_{N}^{2}$ with
$t^{\prime}/s^{\prime}={\rm const}$. Under such a kinematical condition, an
intermediate exchange could be considered as a $q\bar{q}$ state. The
$q\bar{q}$ attached to the nucleon is described by the GPDs in the ERBL
region.

The cross section $NN\to N\pi B$ is described by the process in Fig. 23. The
process is factorized into two terms, the GPD part and the scattering of the
intermediate hadronic $q\bar{q}$ state $h$ with the nucleon, as ${\cal
M}_{NN\to N\pi B}={\cal M}_{N\to hB}\,{\cal M}_{hN\to\pi N}$. The transverse
sizes of the projectile nucleon ($N$) and the outgoing nucleon ($N$) and pion
($\pi$) near the interaction point are of the order of $\sim
1/\sqrt{|t^{\prime}|}$. Therefore, the factorization and the color
transparency phenomena are closely related Collins-1998 . The factorization
was shown for the DVMP process Collins-1998 , and such studies were also done
recently, including $2\to 3$ processes Qiu:2022bpq ; Qiu:2022pla . Detailed
factorization studies would be needed for the $NN\to N\pi B$ processes.

Figure 23: Factorized process for the $NN\to N\pi B$ reaction under the
kinematical condition $s^{\prime},|t^{\prime}|,|u^{\prime}|\gg M_{N}^{2}$ with
$t^{\prime}/s^{\prime}={\rm const}$.

The $NN\rightarrow N\pi B$ cross section is expressed in the factorized form,
as Kumano:2009he

$\displaystyle\frac{d\sigma_{NN\rightarrow N\pi
B}}{dt\,dt^{\prime}}=\int_{y_{\min}}^{y_{\max}}dy\
\frac{s}{16\,(2\pi)^{2}\,M_{N}\,p_{N}}$ $\displaystyle\ \
\times\sqrt{\frac{(ys-t-
M_{N}^{2})^{2}-4M_{N}^{2}t}{(s-2M_{N}^{2})^{2}-4M_{N}^{4}}}\,\frac{d\sigma_{MN\to\pi
N}(s^{\prime}=ys,t^{\prime})}{dt^{\prime}}$ $\displaystyle\ \
\times\sum_{\lambda_{N},\,\lambda_{B}}\frac{1}{[\phi_{M}(z)]^{2}}|{\cal
M}_{N\to B}|^{2}.$ (89)

where the variable $y$ is defined by $y=s^{\prime}/s$, and $d\sigma_{MN\to\pi
N}/(dt^{\prime})$ is the meson-nucleon elastic scattering cross section. This
part can be estimated by using the Brookhaven National Laboratory (BNL) data
by the E755 and E838 collaborations. The GPDs are contained in the matrix
element ${\cal M}_{N\to B}$. If the final baryon $B$ is a nucleon, the matrix-
element part is given by the vector and axial terms as $|{\cal M}_{N\to
N^{\prime}}|^{2}=|{\cal M}_{N}^{V}|^{2}+|{\cal M}_{N}^{A}|^{2}$. They are
given by

$\displaystyle\sum_{\lambda_{N},\lambda_{N^{\prime}}}|{\cal
M}_{N}^{V}|^{2}=I_{N}^{\ 2}\bigg{[}8(1-\xi^{2})\\{H(x,\xi,t)\\}^{2}$
$\displaystyle+16\xi^{2}H(x,\xi,t)E(x,\xi,t)-\frac{t}{M_{N}^{2}}(1+\xi)^{2}\\{E(x,\xi,t)\\}^{2}\bigg{]},$
(90) $\displaystyle\sum_{\lambda_{N},\lambda_{N^{\prime}}}|{\cal
M}_{N}^{A}|^{2}=I_{N}^{\
2}\bigg{[}8(1-\xi^{2})\\{\widetilde{H}(x,\xi,t)\\}^{2}$ $\displaystyle\ \ \
+18\xi^{2}\widetilde{H}(x,\xi,t)\widetilde{E}(x,\xi,t)-\frac{2\,t\,\xi^{2}}{M_{N}^{2}}\\{\widetilde{E}(x,\xi,t)\\}^{2}\bigg{]}.$
(91)

Here, $I_{N}$ is the isospin factor for the nucleon defined by $I_{N}=\langle
1/2||\widetilde{T}||1/2\rangle\langle\frac{1}{2}m_{N},\,1m\big{|}\frac{1}{2}m_{N}^{\prime}\rangle/\sqrt{2}$,
and its actual values are $I_{N}=1,\ \sqrt{2}\ \ \text{for $p\rightarrow p$,
$n$}$, respectively. The GPDs $H$, $E$, $\tilde{H}$, and $\tilde{E}$
contribute to the cross section. In particular, they are estimated by
considering the pion and rho as for the intermediate state $h$. The $NN\to
N\pi\Delta$ cross section is calculated in the same way with the transition
GPDs of $N\to\Delta$ Kumano:2009he .

Figure 24: The cross sections $d\sigma/(dtdt^{\prime})$ are shown for
$p+p\rightarrow p+\pi^{+}+n$, $p+p\rightarrow p+\pi^{-}+\Delta^{++}$, and
$p+p\rightarrow p+\pi^{+}+\Delta^{0}$ as a function of $t^{\prime}$.

The GPD measurements by the $2\to 3$ processes are possible at hadron
accelerator facilities, for example, at J-PARC. It is a unique opportunity to
investigate the ERBL kinematical region of the GPDs, so such experimental
measurements are complementary to the DVCS and DVMP experiments at lepton
accelerator facilities. The $pp\to p\pi^{+}B$ ($B=\Delta^{0},\,n$) cross
sections were calculated by considering the J-PARC kinematics. In Fig.24, the
differential cross sections $d\sigma/(dtdt^{\prime})$ are shown for the three
processes, $p+p\rightarrow p+\pi^{+}+n$, $p+p\rightarrow
p+\pi^{-}+\Delta^{++}$, and $p+p\rightarrow p+\pi^{+}+\Delta^{0}$, as a
function of $t^{\prime}$ by taking the incident proton-beam energy 30 GeV and
the momentum transfer $t=-0.3$ GeV2 Kumano:2009he . There are advantages of
studying these reactions. The cross sections are large, and they are of the
order of $\mu\,$barn/GeV2, these processes are simple to measure
experimentally, the vacuum quantum number $0^{+}$ does not contribute in the
intermediate state, and the $\Delta$ cross sections are as large as the
nucleonic ones for measuring the $N\to\Delta$ transition GPDs in addition to
the nucleon GPDs.

###  8.3 Exotic hadron production using GPDs

In recent years, there have been reports on exotic hadrons. However, it is not
straightforward to find a firm evidence on the exotic nature. The GPDs of
exotic hadrons, including transition GPDs, could provide clear information on
the exotic structure by considering the following points. The simplest form of
the quark GPD $H$ could be written in the separable form, as Guidal:2004nd

$\displaystyle H_{q}^{h}(x,\xi=0,t)=f_{n}(x)\,F_{n}^{h}(t,x),$ (92)

where $n$ indicates the number of valence quarks, $h$ is a hadron, $f_{n}(x)$
is a quark distribution function, and $F_{n}^{h}(t,x)$ is a transverse form
factor at $x$.

The exotic nature should be reflected in both the PDF and the transverse form
factor Kawamura:2013wfa . For the PDFs, a simple function of $x$

$\displaystyle f_{n}(x)=C_{n}\,x^{\alpha_{n}}\,(1-x)^{\beta_{n}},$ (93)

is often used. For valence-quark distributions, the parameters $C_{n}$,
$\alpha_{n}$, and $\beta_{n}$ are determined by the valence-quark number
$\int_{0}^{1}dx\,f_{n}(x)=n$, and the quark momentum
$\int_{0}^{1}dx\,x\,f_{n}(x)=\langle x\rangle_{q}$. The parameter $\beta_{n}$
is determined the constituent counting rule in QCD as $\beta_{n}=2n-3+2\Delta
S_{z}$ with the spin factor $\Delta S_{z}=|S_{z}^{q}-S_{z}^{h}|$. Using these
constraints together with $\langle x\rangle_{q}=0.47$, we obtained the
valence-quark distribution functions of the pion, proton, tetraquark hadron,
and pentaquark hadron in Fig. 25 Kawamura:2013wfa . The pion and proton
distributions roughly agree with the current PDF parametrizations at $Q^{2}=2$
GeV2. These PDFs have a clear indication of their exotic nature because the
peak of the valence-quark distributions moves toward the smaller-$x$ region as
$n$ increases.

The transverse form factor $F_{n}^{h}(t,x)$ also reflects the exotic nature. A
simple exponential function could be used for the form factor, with the cutoff
parameter $\Lambda$ for the transverse momentum, as Guidal:2004nd

$\displaystyle F_{n}^{h}(t,x)=e^{(1-x)t/(x\Lambda^{2})}.$ (94)

The transverse spacial density is given by its Fourier transform, and then the
root-mean-square (rms) radius is given by the parameter $\Lambda$ as
$\left<r_{\perp}^{2}\right>=4(1-x)/(x\Lambda^{2})$. The transverse form factor
is shown in Fig. 25 at $x=0.4$ by taking $\Lambda$=0.5 and 1.0 GeV as
examples. These choices mean that the rms radii are $\sqrt{\langle
r_{\perp}^{2}\rangle}=0.48$ and 0.97 fm for $\Lambda$=1.0 and 0.5 GeV,
respectively. The “ordinary” hadrons with the $q\bar{q}$ and $qqq$
configurations would have compact spatial distributions, which are shown as
the hard form factor ($\Lambda$=1.0 GeV, $\sqrt{\langle
r_{\perp}^{2}\rangle}=0.48$ fm), whereas the exotic tetra- and pentaquark
hadrons, or the molecular-type hadrons, should have wide spatial distributions
and they have the soft form factor ($\Lambda$=0.5 GeV, $\sqrt{\langle
r_{\perp}^{2}\rangle}=0.97$ fm) in Fig. 25.

Figure 25: Valence-quark distributions and transverse form factors of ordinary
and exotic hadrons Kawamura:2013wfa .

It is interesting to measure the GPDs for exotic hadron candidates, but their
spacelike GPDs would not be measured easily because the exotic hadrons are
unstable particles. However, the timelike GPDs should be measured even for
unstable exotic hadrons as investigated at the KEK B factory for the pion
Belle:2015oin ; Kumano:2017lhr .

On the other hand, the exotic-hadron GPDs could be measured in the spacelike
region as transition GPDs. For example, the $\Lambda$(1405) production process
$K^{-}p\to\ell^{+}\ell^{-}\Lambda(1405)$ in Fig. 26 could be investigated at
J-PARC by using the future kaon beam, as an extension project of the pion
Drell-Yan process $\pi^{-}N\to\ell^{+}\ell^{-}N$ for measuring the nucleon
GPDs Sawada:2016mao ; J-PARC-GPD-LoI . The $\Lambda$(1405) cannot be described
by constituent-quark models, so that it is considered as a $\bar{K}N$ molecule
or a pentaquark ($uud\bar{u}s$) hadron. Such transition GPDs have not been
investigated both theoretically and experimentally, so future efforts are
needed for these kind of studies.

Figure 26: Transition GPDs for the exotic candidate $\Lambda$(1405).

There are also possibilities for studying the exotic nature in high-energy
reactions by using the constituent-counting rule in perturbative QCD
Kawamura:2013iia ; Chang:2015ioc  because the constituent number should be
different between ordinary and exotic hadrons. This rule indicates that the
cross section scales by the total number of constituents ($n$) as
$d\sigma/dt\sim 1/s^{n-2}$ where $s$ is the center-of-mass-energy squared. For
example, the high energy $\pi^{-}p\to K^{0}\Lambda(1405)$ reaction could be
measured at J-PARC Kawamura:2013iia . There is an interesting indication that
$\Lambda(1405)$ could be a penta-quark state at medium energies and a three-
quark one at high energies by the analysis of photo-production data on $\gamma
p\to K^{+}\Lambda(1405)$ Chang:2015ioc .

##  9 Transition processes: diffractive scattering

###  9.1 Inelastic diffraction and quantum fluctuations

Figure 27: Exclusive vector meson production in diffractive channels at small
$x$ ($V=\rho^{0},\omega,\phi,J/\psi,\Upsilon$). The amplitude is proportional
to the gluon GPD. Upper graph: $N\rightarrow N$ transitions (elastic
diffractive scattering). Lower graph: $N\rightarrow X(\textrm{low-mass})$
transitions (inelastic diffractive scattering).

In $eN/\gamma N$ scattering at high energies $W\sim 10^{1}-10^{2}$ GeV, the
dominant exclusive meson production channels are vector mesons with the same
quantum numbers as the (virtual) photon,
$V=\rho^{0},\omega,\phi,J/\psi,\Upsilon$. In these channels, the production
process does not exchange quantum numbers between the nucleon and the produced
meson and has the characteristics of diffractive scattering. If the momentum
transfer is large or the produced meson is heavy ($Q^{2},M_{Q\bar{Q}}^{2}\gg$
1 GeV2), the amplitude of such high-energy exclusive processes can be
factorized (see Sec. 2) and samples the gluon GPD of the nucleon at $x,\xi\ll
1$ (see Fig. 27) Frankfurt:2005mc . At such values of $x$ and $\xi$ the gluon
GPD is close to diagonal and can be obtained from the gluon density ($\xi=0$)
in a controlled approximation Frankfurt:1997ha ; Shuvaev:1999ce .

Now, in such high-energy exclusive processes, the nucleon can remain intact in
the final state, $N\rightarrow N$ (elastic diffractive scattering) or
dissociate into a low-mass state with nucleon quantum numbers, $N\rightarrow
X$ (inelastic diffractive scattering) (see Fig. 27). According to the general
theory of diffractive scattering, the elastic cross section is proportional to
the quantum average squared of the strength of interaction of the high-energy
probe with the target configurations, while the inelastic cross section is
proportional to the quantum fluctuations, i.e., the variation of the strength
of interaction between the various configurations Good:1960ba ;
Miettinen:1978jb . Applying these concepts to the factorized processes at high
$Q^{2}$, one can show that elastic scattering probes the quantum average of
the gluon density in the nucleon, while inelastic scattering probes its
quantum fluctuations Frankfurt:2008vi . Specifically,

$\displaystyle\omega_{g}\equiv\frac{\langle G^{2}\rangle-\langle
G\rangle^{2}}{\langle
G\rangle^{2}}=\left[\frac{d\sigma_{\text{inel}}}{dt}\\!\right/\\!\left.\frac{d\sigma_{\text{el}}}{dt}\right]_{t=0}^{\gamma^{\ast}_{L}N\rightarrow
VX}.$ (95)

where $G$ denotes the gluon density operator and $\langle...\rangle$ the
quantum average over configurations in the nucleon (the relation is valid at
small $x$ and $\xi$, where the gluon GPD can be obtained from the gluon
density; see Ref. Frankfurt:2008vi  for details). The relation Eq. (95) allows
one to extract the quantum fluctuation of the gluon density from the ratio of
inelastic and elastic diffractive scattering. Note that the relation is valid
only at $t=0$; the inelastic cross section can be measured also at $t<0$ but
is connected with the quantum fluctuations only at $t=0$.

The concept of fluctuations of the gluon density and their connection with
inelastic diffractive scattering are derived here in the context of collinear
factorization of exclusive processes Frankfurt:2008vi . In this way, they are
directly connected with the transition GPDs describing the individual
amplitudes for $N\rightarrow X$ inelastic diffractive processes. A similiar
relation between inelastic diffraction and the fluctuations of the gluon
density has been derived in the context of the dipole model of $eN/\gamma N$
scattering Schlichting:2014ipa ; Mantysaari:2016ykx ; Mantysaari:2016jaz

###  9.2 GPDs at HERA

Measurements sensitive to GPDs in lepton-proton collisions have been performed
at the H1 and ZEUS HERA collider experiments. The covered photon-proton
centre-of-mass energies of these experiments ranges from 30 GeV to 300 GeV,
providing access hereby to $10^{-}4\leq xB\leq 10^{-}2$. Triggers from far-
forward detectors and calorimeters allowed for the separation of elastic and
dissociative diffractive events and, depending on the period of data
collection, the experiments were equipped with far-forward proton
spectrometers able to detect, with a coverage in acceptance of a few percent,
directly the proton scattered in the elastic process. Hard diffractive
production of the light mesons $\rho^{0}$ H1:2009cml ; ZEUS:2007iet  and
$\phi$ H1:2009cml ; ZEUS:2005bhf , of pion pairs ZEUS:2011tzw  and of the
quarkonia $J/\psi$ H1:1996kyo ; H1:2002voc ; H1:2005dtp ; H1:2010udv ;
H1:2013okq ; ZEUS:1997wrc ; ZEUS:2002src ; ZEUS:2004yeh ; ZEUS:2009qug ;
ZEUS:2012qog , $\psi(2S)$ H1:2002yab ; ZEUS:2002src ; ZEUS:2012qog , and
$\Upsilon$ ZEUS:2009asc ; ZEUS:2011spj  have been studied in both the elastic
and the dissociative channels, while the hard exclusive production of photons
was limited to the elastic channel H1:2005gdw ; H1:2007vrx ; H1:2009wnw ;
ZEUS:2003pwh ; ZEUS:2008hcd . A weak dependence on $W_{\gamma p}$ of the
elastic diffractive cross section is observed in absence of a hard scale. In
contrast, the cross section is seen to rise with increasing $W_{\gamma p}$ in
the presence of a hard scale, either through large $Q^{2}$ or large heavy-
quark mass, where the rise is steeper for increasing values of the hard scale.
This behaviour reflects the increase in gluon density with smaller $x_{B}$ (or
larger $W_{\gamma p}$). The measurements of the elastic channel also show a
$t$ slope, $b$, evolving from large values in the absence of a hard scale to a
small, constant value once a sufficiently hard scale is reached, and this
irrespective of the produced final-state particle. In the dipole picture this
can be understood as $b$ being a reflection of the transverse dipole size and
the transverse proton size, where at large scales the dipole size becomes
negligible. For the inelastic channel, a very different $t$ dependence is
observed. Spin-density matrix elements (SDMEs), which are related to the GPDs,
have been extracted for DVMP process from the H1 and ZEUS data through the
measurement of the angular distribution of the meson decay products. It is
observed that for quarkonia $s$-channel helicity is conserved and that, apart
from some slight deviations for some SDMEs, the same is true for the light
vector mesons. This is in contrast with the low-energy, fixed-target
experiments, where for some SDMEs a strong violation of s-channel helicity
conservation is observed, see e.g. HERMES:2009oim . The extracted SDMEs at the
collider experiments also indicate that the interaction between the virtual
photon and beam proton proceeds through the exchange of particles of natural
parity, which can be at first order equated to the exchange of two gluons.
Also here, differences are seen with the fixed-target experiments, where at
low energy unnatural-parity exchange also seems to contribute HERMES:2017qwt .
Finally, an increase in dominance of the longitudinal cross section over the
transverse cross section with increasing values of $Q^{2}$ has been observed
both for the light and the heavy mesons. In relation to the measurement of
processes involving transitions of the beam proton to another state, the H1
experiment performed a measurement of $e+p\rightarrow\rho^{0}+n+\pi^{+}$ with
$Q^{2}<2$ GeV, where the $\rho^{0}$ meson is reconstructed in the central
detector, the neutron is tagged in a forward detector system and the $\pi^{+}$
escapes along the beam line without being detected H1:2015bxa . The
interpretation of this interaction can be attributed to scattering off the
proton pion cloud or off the neutron core, but also to the creation of a
$N^{*}$ resonance, which subsequently decays into $\pi^{+}n$. The measurement
of this process exemplifies one possibility to study transition GPDs at
electron-proton colliders.

###  9.3 Transition GPDs in ultraperipheral collisions

Ultra-peripheral collisions of hadrons at very high energies can be used to
access GPDs. Hard exclusive production, in particular, the photoproduction of
heavy quarkonium ($J/\psi$, $\psi(2S)$, $\Upsilon({\rm n}S)$) states, have
been studied in $\gamma-p$ and $\gamma-$lead nucleus collisions at RHIC and at
the LHC Klein:2020fmr , providing unique access to the low $x_{B}$ region,
down to $x_{B}=10^{-6}$. We focus here on $\gamma-$p reactions. They can be
studied in proton-proton collisions and in proton-nucleus collisions at these
two running hadron colliders.

For the access to GPDs through meson production, the quasi-real nature of the
exchanged photon, is limiting the useful studies to the production of
quarkonia, while there are also proposals to study exclusive photon-meson pair
production at sufficiently high invariant mass Duplancic:2022ffo ;
Duplancic:2023kwe  as to introduce the needed hard scale. Exclusive production
of the quarkonia $J/\psi$, $\psi(2S)$ and $\Upsilon$ has been successfully
studied in proton - proton LHCb:2013nqs ; LHCb:2014acg ; LHCb:2015wlx ;
LHCb:2018rcm , proton lead ALICE:2014eof ; CMS:2018bbk ; ALICE:2018oyo  and
lead-lead collisions ALICE:2012yye ; ALICE:2013wjo ; CMS:2016itn ;
ALICE:2019tqa ; ALICE:2021gpt ; ALICE:2021tyx ; LHCb:2021hoq ; LHCb:2022ahs .
The photon-proton cross sections for exclusive $J/\psi$ production extracted
from the measurements in lepton-proton, proton-proton, and lead-proton
collisions are compatible with each other in the common $W_{\gamma p}$ range,
which hints at the universality of the underlying probed physics. At present,
the exclusive quarkonium production cross section has been used in the study
of standard PDFs Flett:2020duk , but not yet to constrain GPDs.

A measurement of the DVCS process is not possible due to the small virtuality
of the initial state photon that originates from the electrostatic field of
the colliding ultra-relativistic ions. Time-like Compton scattering
measurements are in principle possible Pire:2008ea . A $t$-integrated cross
section result on exclusive dilepton production in $\gamma-p$ collisions
already exists in a kinematic domain of interest (dilepton mass between 1 and
2.5 GeV$/c^{2}$) ALICE:2023mfc .

In addition to hard exclusive photoproduction, ultra-peripheral collisions
allow the study of hard dissociative photoproduction processes at high
energies, where the target proton breaks up. This observable class is within
the Good-Walker-formalism sensitive to the fluctuations of the proton
structure PhysRevD.18.1696 , whereas the exclusive production is sensitive to
the average. A first measurement of dissociative $J/\psi$ photoproduction in
$\gamma-$p collisions has been published recently ALICE:2023mfc . Future
measurements at the LHC can be expected based on the interest of the community
and the luminosity increase in the 2020s compared to previous data taking
campaigns, thanks to the HL-LHC upgrades Citron:2018lsq .

Ultra-peripheral collisions can also be studied in the fixed-target mode of
the LHC ($\sqrt{s}\approx$ 100 GeV). A few observables and corresponding
publications are summarized in Ref. Hadjidakis:2018ifr . LHCb performed with
the SMOG gas target originally designed for luminosity measurements via beam
imaging Barschel:2014iua , a first set of low luminosity cross section
measurements, e.g., on charm and on antiproton production LHCb:2018jry ;
LHCb:2018ygc .

Transition GPDs can, in principle, be accessed within dissociative production
provided that the final state nucleon resonance can be at least partially
reconstructed. At the existing hadron colliders, this is a difficult
undertaking given the existing forward instrumentation. Dedicated simulation
studies are required to assess the feasibility with the current experimental
set-ups or with dedicated additional instrumentation.

##  10 Future experimental facilities

The data from already existing measurements allows the first extraction of
observable, sensitive to transition GPDs. However, for more detailed studies,
higher statistics and also higher beam energies are favorable. In addition,
hadron beams at J-PARC will allow the study of transition GPDs in hadronic
reactions. The following section will summarize upcoming opportunities with
existing and planned experiments.

###  10.1 CLAS12 luminosity upgrade and JLab 22 GeV

The $N\rightarrow N^{*}$ DVCS, as well as the $N\rightarrow N^{*}$ DVMP
processes will both strongly profit from a luminosity upgrade of CLAS12 as
well as from an energy upgrade of JLab.

From the statistics point of view, the low efficiency for the detection of the
multi-particle final states, in combination with the background suppression
cuts, strongly limit the available statistics of the final sample. At the
moment, the statistics that can be collected with CLAS12 are mainly limited by
the rate capability of the detector, which is determined primarily by the
forward tracking in the drift chambers, while the CEBAF accelerator could
deliver around two orders of magnitude more beam current. Therefore, a
luminosity upgrade of CLAS12, which is currently in progress, will help
collect data more efficiently and increase the amount of data that can be
collected in a certain time. Based on this upgrade, it is expected that
sufficient data can be collected in a reasonable time frame to perform a fully
differential study of the $N\rightarrow N^{*}$ DVCS and DVMP processes, which
will allow a partial wave decomposition of the resonance spectrum.

From the beam energy point of view, the currently available beam energy of
10.6 GeV allows a study of the lower lying nucleon and $\Delta$ resonances in
a limited $Q^{2}$ range. However, especially for higher-mass resonances, the
factorization requirement $Q^{2}\gg m^{2}_{N^{*}}$ strongly limits the options
based on a 10.6 GeV electron beam. Here, a 22 GeV upgrade of JLab would enable
the investigation of higher-mass resonances and extend the accessible $Q^{2}$
range for the lower mass resonances. Based on this extended range, a detailed
study of the scaling behavior of the different observables will become
possible. Fig. 28 shows the available phase space, accessible with the present
CLAS12 setup, in $Q^{2}-x_{B}$ for the $\pi^{-}\Delta^{++}$ process under
forward kinematics and the $\pi^{+}\pi^{-}$ invariant mass of the same process
for a 10.6 GeV, 18 GeV, and 22 GeV electron beam. The corresponding
distributions of the $N\rightarrow\Delta$ DVCS process show similar
characteristics.

Figure 28: Comparison of the available phase space, accessible with the
present CLAS12 setup, in $Q^{2}-x_{B}$ or the $\pi^{-}\Delta^{++}$ process
under forward kinematics ($-t<$ 1.5 GeV2) [up] and for the $\pi^{+}\pi^{-}$
invariant mass of the same process, which is used to suppress the dominant
$\rho$ production background by the cut on $M(\pi^{+}\pi^{-})>$ 1.1 GeV,
indicated by the yellow line [down] for a 10.6 GeV, 18 GeV and 22 GeV electron
beam.

It can be seen that a 22 GeV upgrade of JLab would provide a significantly
increased $Q^{2}$ range for a fixed value of $x_{B}$. This leads to a big
advantage for the study of these processes, since the factorization of the
$N\rightarrow N^{*}$ DVCS and DVMP processes requires a high virtuality
$Q^{2}$ to be above the resonance mass squared. While this condition can
already be fulfilled with a 10.6 GeV electron beam for lower mass resonances,
like the $\Delta(1232)$, an energy upgrade to 22 GeV will be a big advantage
to ensure the factorization of the process for higher mass resonances and to
study the scaling behavior of the observables. As shown in the lower part of
Fig. 28 the increase in phase space for the different invariant mass
combinations will allow a more efficient suppression of non-resonant
background from exclusive meson production and also from other (differently
charged) nucleon resonance production channels, which is mostly expected at
lower masses. Higher beam energies will therefore also provide a more
efficient event selection and a better suppression of the non-resonant
background. While the presently available 10.6 GeV electron beam already
allows a study of the 3D structure of lower-mass nucleon resonances, a 22 GeV
upgrade of JLab in combination with a luminosity upgrade of CLAS12 will
provide ideal conditions for the study of the 3D structure of nucleon
resonances via transition GPDs. Also, future studies with the high resolution
two arm spectrometer in JLab hall C will strongly benefit from an energy
upgrade.

###  10.2 Muon and meson beams at COMPASS/AMBER

So far, no physics program for the measurement of transition GPDs at COMPASS
COMPASS:2007rjf  or AMBER Adams:2018pwt  exists. However, the existing COMPASS
data based on a 160 GeV muon beam will provide a potential access to lower
$x_{B}$ (0.01 $<x_{B}<$ 0.2) and higher $Q^{2}$ values. It can therefore help
to constrain the impact of sea quarks on the resonance properties and also
provide valuable inputs for an extrapolation to the gluonic regime, accessible
with the EIC.

Since the main spectrometer of COMPASS is focused at very forward angles and
provides relatively high momentum thresholds, it can only be used to detect
the scattered muons and the DVCS photon or DVMP pion. The decay products of
the resonance have to be detected by the CAMERA detector, which consists of
two regions of plastic scintillator bars, covering the region around the
target. Since only charged tracks can be detected by this device, it mainly
limits the channels that can be studied with COMPASS to two $N\to N^{*}$ DVMP
channels:

  * •

$\mu^{\pm}~{}p~{}\to~{}\mu^{\pm}~{}\pi^{-}~{}\Delta^{++}~{}~{}~{}~{}~{}\to~{}\mu^{\pm}~{}\pi^{-}~{}[p\pi^{+}]$

  * •

$\mu^{\pm}~{}p~{}\to~{}\mu^{\pm}~{}\pi^{+}~{}\Delta^{0}(N^{*})~{}\to~{}\mu^{\pm}~{}\pi^{+}~{}[p\pi^{-}]$

Currently, feasibility studies for these two channels are ongoing based on
existing data taken with COMPASS in 2016. With AMBER also the high-intensity
pion and kaon beams with up to 190 GeV/c are expected to provide an excellent
basis for the study of meson induced transitions within the exclusive Drell-
Yan process. Feasibility studies for potential future muon and meson based
measurements with AMBER, including potential detector upgrades, are planned.

###  10.3 Electron-Ion Collider EIC

The planned electron-ion collider (EIC) Accardi:2012qut  will allow an
extension of the kinematic regime to the gluonic sector at low $x_{B}$ and
towards higher $Q^{2}$ values (see Fig. 29), which are especially important
for the factorization of the process in relation to higher mass resonances.

Figure 29: Comparison of the available phase space in $Q^{2}-x_{B}$,
accessible with a 12 GeV electron beam at JLab (orange), a 160 GeV muon beam
at COMPASS (blue) and different CM energies of the EIC (green, red).

The upcoming data from the EIC will provide insights into the contributions of
the gluons to the excitation process and to the characteristics of baryon
resonances. The planed configuration of the far-forward detectors will provide
excellent opportunities to study the low $-t$ region of hard exclusive
processes. So far, no theoretical predictions exist for the low $x_{B}$ regime
of the $N\to N^{*}$ DVCS and DVMP processes, but following the predictions for
the ground-state DVCS and DVMP processes in this regime, a measurement is
expected to be feasible from the cross section point of view.

###  10.4 Hadron beams at J-PARC and FAIR

As discussed in section 8, theoretical models also allow the study of
transition GPDs with hadronic beams in $2\rightarrow 3$ processes and based on
the exclusive Drell-Yan process. The high-momentum hadron beam line of J-PARC
Aoki:2021cqa  provides excellent opportunities for such studies. Further
opportunities with proton and antiproton beams in a similar energy region will
become available at the Facility for Antiproton and Ion Research (FAIR)
FAIRrep22 .

The feasibility of measuring the $2\rightarrow 3$ process in the ongoing E16
experiment in the high-momentum beamline of J-PARC is currently investigated.
Equipping additional detectors in the forward direction will become necessary
to increase the acceptance of these exclusive hard events. Also studies of
this process at FAIR may be considered and further investigated.

For the measurement of the exclusive Drell-Yan process in the E50 experiment,
a letter of intent J-PARC-GPD-LoI  was submitted to J-PARC 2019 PAC. A full
proposal with the design of the required mu-ID system and dimuon trigger
setting is currently being prepared. The “$\pi 20$” project of delivering
10-20 GeV meson beam in the high-momentum beamline of J-PARCs hadron hall has
been included in the hadron hall extension project Aoki:2021cqa . This six-
year project has been selected as the top priority in the KEK mid-term plan
(KEK-PIP2022) and is scheduled to start in 2024. After the upgrade, high
resolution pion beams with a momentum of up to 20 GeV will be available.

###  10.5 LHC fixed-target and ultraperipheral collisions

In 2024, results from the first unpolarised fixed-target set-up for high-
luminosity data takings at the LHC are expected from LHCb SMOG-2
Bursche:2649878 ; 2707819 . This set-up, installed in 2022 will allow to run
LHCb in fixed-target and collider mode in parallel throughout the full data
taking period and with up to two orders of magnitude larger instantaneous
luminosities than SMOG, depending on the gas. An effort towards a polarized
gas target in LHCb is ongoing Aidala:2019pit .

With the HL-LHC upgrades Citron:2018lsq  also future measurements of
dissociative $J/\psi$ photoproduction in $\gamma-p$ collisions at the LHC can
be expected with significantly increased statistics. However, the relevant
luminosity increase strongly depends on the experimental set-up, collision
system, and operational decisions.

##  11 Challenges of GPD analysis

It is instructive to compare the phenomenology of transition GPDs with that of
diagonal counterparts, the latter of which has been extensively studied over
the last three decades. Both types of objects naturally shares many similar
features, however studies of transition GPDs come with a number of unique
difficulties.

Both cases require the simultaneous extraction of several multidimensional
objects from observables for exclusive processes, presenting a non-trivial
task. Specifically, distinguishing among many contributing objects can only be
achieved by measuring a variety of exclusive processes (DVCS, TCS, DDVCS,
DVMP, etc.) and observables (cross-sections, beam / target / charge
asymmetries, etc.). This motivates a global experimental program conducted in
laboratories covering complementary kinematic domains, and diverse setups
equipped with sophisticated detection systems capable of determining the
exclusivity condition. In the case of diagonal GPDs, the experimental programs
carried out in laboratories such as JLab, DESY, and CERN have already resulted
in numerous measurements, fueling phenomenological analyses like
Kumericki:2019ddg ; Moutarde:2019tqa ; Guo:2023ahv . As indicated in Sec. 7,
so far only the first data sensitive to transition GPDs have been collected,
allowing for exploratory studies that help to prove the correctness and
usefulness of the formalism. We stress again that processes sensitive to both
diagonal and transition GPDs can, in principle, be measured together. However,
the latter typically come with smaller cross-sections and are more prone to
detector effects threatening the reconstruction of all particle states.

Similarly to diagonal counterparts, transition GPDs also enter observables via
convolutions with coefficients describing the hard part of a given process,
and, in the case of, for example, DVMP, also with additional non-perturbative
objects describing produced hadrons. This requires the development of
sophisticated deconvolution methods, which in practice for processes like DVCS
and lowest-order coefficient functions turns out to be an arduous task (for a
full discussion, see both Refs. Bertone:2021yyz  and Moffat:2023svr ). Despite
this difficulty, the phenomenology of diagonal GPDs demonstrates that analyses
of amplitudes (not requiring deconvolution) still allow us to learn a lot
about the structure of hadrons. In particular, one can directly study the
nuclear tomography at low $\xi$ thanks to certain similarities with the
optical theorem ZEUS:2008hcd ; H1:2009wnw ; COMPASS:2018pup , mechanical
properties via the subtraction constant appearing in the dispersion relation
Burkert:2018bqq ; Kumericki:2019ddg ; Dutrieux:2021nlz , and response of
hadrons on probes carrying different angular momenta thanks to the recent
development of techniques based on the Froissart-Gribov projections Semenov-
Tian-Shansky:2023ysr . The same can be done in the non-diagonal case, assuming
experimental data allow for it.

Another common challenge is defining models that fulfill a maximum of theory-
driven requirements. This proves to be a challenge, particularly due to the
non-trivial task of combining polynomiality with positivity constraints. Many
modelling techniques have been proposed in the diagonal case, including those
involving Radyushkin’s double distribution Ansatz Goloskokov:2005sd ;
Vanderhaeghen:1998uc ; Mezrag:2013mya , conformal moment expansion
Kumericki:2015lhb ; Guo:2023ahv , dual parametrization Polyakov:2002wz ;
Polyakov:2009xir ; Muller:2014wxa  and light front wave functions
Mezrag:2023nkp . This selection is supplemented by non-parametric models based
on machine learning techniques Dutrieux:2021wll  used to study model
dependency. All these modelling techniques can be re-used in the non-diagonal
case. The issue of incorporating diagonal GPDs into models of non-diagonal
objects relying on the large $N_{c}$ limit, as discussed in Ref. Kroll:2022roq
, remains an open and interesting phenomenological issue. Additional
difficulty arises from the limited knowledge of forward limits and elastic
form factors, which are used as basic ingredients in the modeling procedures;
see, for instance, Eqs. (8) and (9).

It is also very important to generalize for non-diagonal hard exclusive
reaction the consistent methods for accounting the higher-order kinematical
effects relying on the approaches developed in Refs. Braun:2012hq ;
Guo:2021gru ; Braun:2022qly .

Assuming we have experimental data and a model we would like to constrain, and
we understand well the connection between these data and the model, the last
non-trivial task is the creation of an extraction framework. In this
framework, all elements needed for the extraction must be implemented and
well-tested. Since we expect data to span a broad kinematic domain, to fully
facilitate their potential, it is required to use evolution equations, but
also a suitable description of exclusive processes, particularly in terms of
$\alpha_{S}$ and twist expansions. Evolution also acts as a useful tool,
allowing us to distinguish between various contributions. For instance, in the
analysis of the subtraction constant, evolution allows us to decompose this
object to access the part related to the energy-momentum tensor form factor
$C$ Dutrieux:2021nlz . In the GPD field, two open-source platforms dedicated
to diverse phenomenological studies exist: Gepard Kumericki:2007sa  and
PARTONS Berthou:2015oaw . Both of them can be adapted to the non-diagonal
case.

The constrained models can be employed in Monte Carlo generators to support
ongoing and future experimental campaigns, such as those planned at JLab
Accardi:2023chb  and EIC AbdulKhalek:2021gbh . The generators should
incorporate radiative corrections, the understanding of which is crucial for
interpreting data, and ideally, they should be capable of generating multiple
exclusive processes. A potential candidate for such a generator is EpIC
Aschenauer:2022aeb , which, due to its modular architecture, can easily
accommodate elements required to generate events related to non-diagonal GPDs.

##  12 Future transition GPD program

Transition GPDs provide new tools for quantifying and interpreting the
structure of baryon resonances in QCD. The transition GPDs describe matrix
elements of nonlocal partonic QCD operators between ground and excited baryon
states and contain much more information than vector/axial transition current
traditionally used for resonance excitation. They provide a tomographic image
of spatial structure of resonances and contain information about mechanical
properties such as angular momentum and forces. Systematic theoretical methods
are available for characterizing the $N\rightarrow N^{\ast}$ transition GPDs
and connecting them with the $N\rightarrow N$ ground state GPDs.

Transition GPDs are sampled in hard exclusive electroproduction processes with
$N\rightarrow N^{\ast}$ transitions, such as DVCS $e+N\rightarrow
e^{\prime}+\gamma+N^{\ast}$ and meson production $e+N\rightarrow
e^{\prime}+M+N^{\ast}$. QCD factorization is applied to the amplitudes along
the same lines as for $N\rightarrow N$ processes. First experimental results
from JLab 12 GeV in $\pi^{-}\Delta^{++}$ and $\pi^{0}\Delta^{+}$ meson
production, as well as $p\rightarrow\Delta^{+}$ DVCS, demonstrate the
feasibility of the measurements and provide a first test of the applicability
of framework. Much more extensive data are expected from the ongoing JLab 12
GeV program. Further measurements of transition processes in electron/muon
scattering could be performed with COMPASS, EIC and the possible JLab 22 GeV
upgrade. Complementary information comes from hadron-induced processes such as
exclusive dilepton production $\pi+N\rightarrow(\ell^{+}\ell^{-})+N^{\ast}$,
which will be measured at J-PARC. The concept of transition GPDs can also be
applied to gluon-mediated diffractive scattering at small $x$, with
measurements performed in ultraperipheral collisions at LHC and
photo/electroproduction at the future EIC.

This opens the prospect of a new scientific program exploring baryon resonance
structure, using transition GPDs as a unifying concept and quantitative
analysis tool. To realize this program, further efforts are needed to develop
the theoretical framework, extend the experimental studies at existing
facilities, and simulate measurements with future facilities. On the
theoretical side, this includes:

  * •

Revisit and standardize the structural decomposition of the $N\to\Delta$
transition matrix elements; establish the connections between the various
definitions of transition GPDs used in the literature; clarify the
manifestation of their basic properties in the different parametrizations.
Extend the structural decomposition to other $N\to N^{*}$ transition GPDs.

  * •

Construct a general formalism for non-diagonal hard exclusive reactions based
on $N\to\pi N$ transition GPDs in the full $\pi N$ resonance domain, using
methods of amplitude analysis to separate resonant and non-resonant
contributions. To avoid complications associated with the spins of the target
and produced hadrons, a first step might be to treat the toy example of
$\pi\to\pi\pi$ transition GPDs.

  * •

Develop the interpretation of the EMT transition form factors and the
mechanical properties of baryon resonances, using representations of the
transition matrix elements in terms of 2-dimensional light-front densities or
3-dimensional Breit frame densities.

  * •

Apply the $1/N_{c}$ expansion to achieve a complete classification of the
$N\rightarrow\Delta$ transition GPDs, including subleading structures and
$1/N_{c}$ corrections to leading structures, using algebraic methods based on
the spin-flavor symmetry group. Extend the $1/N_{c}$ analysis to $N\rightarrow
N^{\ast}$ transitions with mass difference $\mathcal{O}(N_{c}^{0})$ and
orbital excitations.

  * •

Explore quantitative features of transition GPDs in dynamical models of
baryons such as light-front quark models, the chiral quark-soliton model based
on the large-$N_{c}$ limit, relativistic bound state models based on Dyson-
Schwinger equations, or GPD models based on holographic QCD.

  * •

Classify and compute kinematic power corrections and dynamic higher-twist
corrections in non-diagonal DVCS process

  * •

Develop and implement the transition GPD formalism for hard exclusive
reactions with strangeness production $\gamma^{*}N\to\\{\Lambda,\,\Sigma\\}K$,
using methods of SU(3) flavor symmetry. Study the feasibility of strange
baryon production in the kinematic conditions of JLab, EIC and EicC.

  * •

Develop dynamical models of diffractive $N\rightarrow X$ gluon transition GPDs
at small $x$ based on quantum fluctuations of the gluon fields in the nucleon;
explore sensitivity and impact of measurements of inelastic diffraction in
ultraperipheral $pA$ collisions at LHC and in photo/electroproduction at EIC
and EicC.

  * •

Explore applications of non-diagonal hard exclusive reactions with $N\to\pi N$
transitions to resonance spectroscopy, particularly the production of exotic
states enriched by large gluonic or multiquark components. This requires
employing theoretical tools developed for conventional partial wave analysis,
such as the $K$-matrix approach, $N/D$-method, dispersive methods etc.

  * •

Calculate and interpret polarization observables depending on the decay angles
of the meson-nucleon system, as a necessary step towards hadron spectroscopy
applications.

On the experimental side, this includes:

  * •

Perform high statistics, fully differential measurements of the $N\to N^{*}$
DVMP and DVCS processes with CLAS12, enabling a full partial wave
decomposition of the resonance spectrum. For this purpose, an upgrade of the
rate capability of CLAS12 will be essential to fully exploit the luminosity
which can be provided by CEBAF at JLab. Adding a calorimeter to the central
detector of CLAS12 would allow for the detection of further final sates for
both processes.

  * •

Perform cross section measurements of the $p\to\Delta$ transition and a
potential L-T separation of the unpolarized cross section for certain $N\to
N^{*}$ DVMP processes in JLab Hall C.

  * •

Explore the options for transition GPD measurements with a possible 22 GeV
upgrade of JLab, extending the phase space and kinematic coverage for hard
exclusive processes, especially in $Q^{2}$, and allowing for more efficient
background separation.

  * •

Investigate the possibilities for measuring DVMP reactions with hyperon
transitions at JLab 12 GeV, to probe the nucleon-to-hyperon transition GPDs.

  * •

Explore options for studies of transition GPDs with muon and meson beams at
CERN COMPASS/AMBER.

  * •

Simulate transition GPD measurements with EIC using the far-forward detectors,
especially $N\rightarrow\Delta$ transition processes in EIC kinematics. Far-
forward $\Delta$ detection can use complementary decay channels, e.g.
$\Delta^{+}\rightarrow\pi^{+}n$ or $\pi^{0}p$ (= one charged and one neutral
particle), $\Delta^{0}\rightarrow\pi^{-}p$ or $\pi^{0}n$ (= all charged or all
neutral particles); each of these channels present particular challenges or
opportunities. The detection of far-forward pions is also needed for the
$u$-channel exclusive processes.

  * •

Study opportunities for transition GPD measurements at J-PARC, which will
provide access to the time-like regime and are complementary to lepton
scattering experiments.

  * •

Simulate measurements of diffractive transition GPDs $p\rightarrow X$ in
ultraperipheral collisions at the LHC.

Altogether, the available theoretical concepts and methods and the existing
and planned experimental facilities provide excellent opportunities for
measurements of transition GPDs and next-generation studies of baryon
resonances.

##  13 Acknowledgements

The workshop on “Exploring resonance structure with transition GPDs” has been
supported by STRONG-2020 “The strong interaction at the frontier of knowledge:
fundamental research and applications” which received funding from the
European Union’s Horizon 2020 research and innovation program under grant
agreement No 824093, by the European Centre for Theoretical Studies in Nuclear
Physics and Related Areas (ECT*) and by the Asia Pacific Center for
Theoretical Physics (APCTP).

S. Diehl was partially supported by Deutsche Forschungsgemeinschaft Project
No. 508107918 and the German Bundeministerium für Bildung und Forschung
(BMBF).

S. Kumano was partially supported by Japan Society for the Promotion of
Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) Grant Numbers
19K03830 and 24K07026.

K. Semenov-Tian-Shansky was supported by Basic Science Research Program
through the National Research Foundation of Korea (NRF) funded by the Ministry
of Education RS-2023-00238703; and under Grants No. NRF-2018R1A6A1A06024970
(Basic Science Research Program); and by the Foundation for the Advancement of
Theoretical Physics and Mathematics “BASIS”.

M. Vanderhaeghen was supported by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation), in part through the Cluster of Excellence
[Precision Physics, Fundamental Interactions, and Structure of Matter]
(PRISMA+ EXC 2118/1) within the German Excellence Strategy (Project ID
39083149).

The work presented by A. Usman was supported by Natural Science and
Engineering Research Council of Canada (NSERC) Grant No. SAPIN-2021-00026 and
the National Science Foundation of USA (NSF) Grant No. PHY2012430 and
PHY2309976

Parada T. P. Hutauruk was supported by the National Research Foundation of
Korea Grant Nos. 2018R1A5A1025563, 2022R1A2C1003964, 2022K2A9A1A0609176, and
2023R1A2C1003177.

C.-H. Lee was supported by the National Research Foundation of Korea(NRF)
grant funded by the Korea government(MSIT) (No. 2023R1A2C1005398).

H.-D. Son was supported by the National Research Foundation of Korea(NRF)
grant funded by the Korea government(MSIT) (RS-2023-00210298).

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177.

The research reported here takes place in the context of the Topical
Collaboration “3D quark-gluon structure of hadrons: mass, spin, tomography”
(Quark-Gluon Tomography Collaboration) supported by the U.S. Department of
Energy, Office of Science, Office of Nuclear Physics.

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