∎

e5Editors e4Corresponding author e-mail<EMAIL_ADDRESS>giessen.de<EMAIL_ADDRESS>
11institutetext: Justus Liebig Universität Gießen, 35390 Gießen, Germany
22institutetext: University of Connecticut, Storrs, CT 06269, USA
33institutetext: Kyungpook National University, Daegu 41566, Korea
44institutetext: Thomas Jefferson National Accelerator Facility, Newport News,
VA 23606, USA 55institutetext: Institut für Theoretische Physik, Universität
Regensburg, 93040 Regensburg, Germany 66institutetext: Institute of Physics,
Academia Sinica, Taipei 11529, Taiwan 77institutetext: Physics Department,
Temple University, Philadelphia, PA 19122-1801, USA 88institutetext: Nuclear
Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA 99institutetext: Department of Physics, Pukyong National University
(PKNU), Busan 48513, Korea 1010institutetext: University of Wuppertal, D-42097
Wuppertal, Germany 1111institutetext: Department of Mathematics, Physics, and
Computer Science, Faculty of Science, Japan Women’s University, Tokyo
112-8681, Japan and Theory Center, Institute of Particle and Nuclear Studies,
KEK, Tsukuba, Ibaraki, 305-0801, Japan 1212institutetext: Department of
Physics, Pusan National University, Busan 46241, Korea and Asia Pacific Center
for Theoretical Physics, POSTECH, Pohang 37673, Korea 1313institutetext:
Physics Department, University of Virginia, Charlottesville, VA 22904-4714,
USA 1414institutetext: University College Dublin, School of Physics, Science
Centre Belfield Dublin 4, Ireland 1515institutetext: Department of Physics,
Inha University, Incheon 22212, Korea 1616institutetext: National Centre for
Nuclear Research (NCBJ), 02-093 Warsaw, Poland 1717institutetext: University
of Regina, Regina SK S4S 0A2 Canada 1818institutetext: Universidad de Alcalá,
28801 Alcalá de Henares, Spain 1919institutetext: Institut für Kernphysik and
$\text{PRISMA}^{+}$ Cluster of Excellence, Johannes Gutenberg Universität,
55099 Mainz, Germany 2020institutetext: DPhN/Irfu, CEA Saclay, 91191 Gif sur
Yvette, France

# Exploring Baryon Resonances with Transition Generalized Parton
Distributions: Status and Perspectives

S. Diehladdr1,addr1a,e5,e4  K. Jooaddr1a,e5  K. Semenov-Tian-Shanskyaddr10,e5
C. Weissaddr5,e5  V. Braunaddr2  W.C. Changaddr4  P. Chatagnonaddr5  M.
Constantinouaddr6  Y. Guoaddr7  P. T. P. Hutaurukaddr8  H.-S. Joaddr10  A.
Kimaddr1a  J.-Y. Kimaddr5  P. Krolladdr14  S. Kumanoaddr15  C.-H. Leeaddr16
S. Liutiaddr17  R. McNultyaddr18  H.-D. Sonaddr22  P. Sznajderaddr23  A.
Usmanaddr24  C. Van Hulseaddr25  M. Vanderhaeghenaddr26  M. Winnaddr28

###### Abstract

QCD gives rise to a rich spectrum of excited baryon states. Understanding
their internal structure is important for many areas of nuclear physics, such
as nuclear forces, dense matter, and neutrino-nucleus interactions.
Generalized parton distributions (GPDs) are an established tool for
characterizing the QCD structure of the ground-state nucleon. They are used to
create 3D tomographic images of the quark/gluon structure and quantify the
mechanical properties such as the distribution of mass, angular momentum and
forces in the system. Transition GPDs extend these concepts to $N\rightarrow
N^{\ast}$ transitions and can be used to characterize the 3D structure and
mechanical properties of baryon resonances. They can be probed in high-
momentum-transfer exclusive electroproduction processes with resonance
transitions $e+N\rightarrow e^{\prime}+M+N^{\ast}$, such as deeply-virtual
Compton scattering ($M=\gamma$) or meson production ($M=\pi,K$, etc.), and in
related photon/hadron-induced processes.

This White Paper describes a research program aiming to explore baryon
resonance structure with transition GPDs. This includes the properties and
interpretation of the transition GPDs, theoretical methods for structures and
processes, first experimental results from JLab 12 GeV, future measurements
with existing and planned facilities (JLab detector and energy upgrades,
COMPASS/AMBER, EIC, EicC, J-PARC, LHC ultraperihperal collisions), and the
theoretical and experimental developments needed to realize this program.

††journal: Eur. Phys. J. A

###### Contents

  1. 1 Introduction
  2. 2 Overview
  3. 3 Transition GPDs
    1. 3.1 GPDs of the ground state nucleon
    2. 3.2 $N\rightarrow\Delta$ transition GPDs
    3. 3.3 Second resonance region
  4. 4 Baryon resonance structure
    1. 4.1 3D tomography of baryon resonances
    2. 4.2 Spin and angular momentum in baryon resonances
    3. 4.3 Mechanical properties of baryon resonances
    4. 4.4 Tensor charge and anomalous magnetic moment
  5. 5 Theoretical methods for transition GPDs
    1. 5.1 Chiral dynamics in $N\rightarrow\pi N$ transitions
    2. 5.2 $1/N_{c}$ expansion and $N\rightarrow\Delta$ transitions
    3. 5.3 Light-cone sum rules
    4. 5.4 Lattice QCD calculations
  6. 6 Transition processes: electron scattering
    1. 6.1 $N\to\Delta,\,N^{*}$ DVCS and transition GPDs
    2. 6.2 $N\to N^{*}$ DVMP and transition GPDs
    3. 6.3 Near-threshold pion production
  7. 7 First experimental results from JLab 12 GeV
    1. 7.1 $\pi^{-}\Delta^{++}$ electroproduction with CLAS12
    2. 7.2 Further $N\to N^{*}$ DVMP channels with CLAS12
    3. 7.3 $\pi^{+}\Delta^{0}$ electroproduction in JLab Hall C
    4. 7.4 $N\to\Delta^{+}$ DVCS with CLAS12
  8. 8 Transition processes: hadron scattering
    1. 8.1 Meson-induced exclusive Drell-Yan process
    2. 8.2 Proton-induced 2 $\rightarrow$ 3 processes
    3. 8.3 Exotic hadron production using GPDs
  9. 9 Transition processes: diffractive scattering
    1. 9.1 Inelastic diffraction and quantum fluctuations
    2. 9.2 GPDs at HERA
    3. 9.3 Transition GPDs in ultraperipheral collisions
  10. 10 Future experimental facilities
    1. 10.1 CLAS12 luminosity upgrade and JLab 22 GeV
    2. 10.2 Muon and meson beams at COMPASS/AMBER
    3. 10.3 Electron-Ion Collider EIC
    4. 10.4 Hadron beams at J-PARC and FAIR
    5. 10.5 LHC fixed-target and ultraperipheral collisions
  11. 11 Challenges of GPD analysis
  12. 12 Future transition GPD program
  13. 13 Acknowledgements

##  1 Introduction

Hadrons are emergent phenomena of Quantum Chromodynamics (QCD), see e.g.
Gross:2022hyw . The elementary dynamics is expressed in both the spectrum and
the structure of the hadronic states. Exploring both expressions together is
essential for a complete understanding of strong interactions. There are many
examples of connections between the internal motion and the excitation
spectrum in nonrelativistic quantum systems (atoms, nuclei). Similar
connections are expected in the much more complex relativistic quantum systems
presented by the hadrons in QCD.

A rich spectrum of baryon resonances is known to emerge from QCD. They are
observed in scattering experiments with electron, photon, and hadron beams at
energies $\sim$ few GeV. These are excited states of the nucleon (proton or
neutron), the basic building block of nuclei and of visible matter in the
universe. The spectrum of these excited states was essential for establishing
the quark model as a precursor to QCD. The baryon resonances play an important
role in the theory of nuclear forces (e.g. the $\Delta$ isobar), in the
behavior of matter at high densities and temperatures (early universe, stellar
structure), and in the existence of hypothetical strange matter (neutron
stars) Marquez:2022gmu . They are also needed for describing the interactions
of neutrinos with nuclei at energies $\sim$ few GeV (neutrino detection in
oscillation experiments) Amaro:2019zos . Understanding the internal structure
of the baryon resonances is thus of fundamental interest and practical
importance.

Some information on the structure of baryon resonances is available from the
electromagnetic transition form factors measured in electroproduction
experiments. They describe the spatial distribution of charge and current in
the dynamical system. Further information from other probes is needed to
characterize the excited baryon states.

The study of ground-state nucleon structure has made significant advances in
the last two decades. The concept of generalized parton distributions (GPDs)
Muller:1994ses ; Radyushkin:1997ki ; Ji:1996nm  has provided a rigorous
formulation of the spatial distributions of quarks and gluons in the nucleon
and offered new opportunities for characterizing nucleon structure, see e.g.
Diehl:2023nmm  for an overview. The GPDs unify the concepts of the elastic
nucleon form factors and the quark/gluon particle densities. They enable the
construction of 3D tomographic images of the distribution of quarks and
gluons, which have great potential for the visualization of nucleon structure
and the discussion of the internal motion. They also quantify the distribution
of energy, momentum, angular momentum, and forces in the nucleon, which allows
one to discuss the mechanical properties of the quantum system in analogy with
classical systems. The GPDs are extracted using a combination of theoretical
methods and experimental data from various scattering processes, particularly
exclusive processes at energy and momentum transfers $\gg 1$ GeV. Programs of
GPD measurements are under way at Jefferson Lab (JLab) with 12 GeV electron
beam Burkert:2020akg ; PAC42hallC , CERN COMPASS COMPASS:2018pup , Large
Hadron Collider (LHC) Klein:2020fmr , and Japan Proton Accelerator Research
Complex (J-PARC) Aoki:2021cqa , and planned with the future Electron-Ion
Collider (EIC) AbdulKhalek:2021gbh ; Burkert:2022hjz  and Electron-ion
collider in China (EicC) Anderle:2021wcy .

Recent advances have made it possible to extend the framework of GPDs to
excited baryon states and use it to characterize baryon resonance structure.
Theoretical studies have formulated the concept of
$N\rightarrow\Delta,\,N^{*}$ transition GPDs and extended the method of
quark/gluon tomography to baryon resonances. Experimental efforts have
produced the first measurements of exclusive processes with
$N\rightarrow\Delta$ transitions and demonstrated their feasibility. The
transition GPDs allow one to ask and answer new questions about baryon
resonance structure:

  * •

What is the spatial distribution of quarks in excited baryon states, and how
does it differ from the ground state? Can we construct tomographic images of
the baryon resonances?

  * •

What are the distributions of energy, momentum, and angular momentum carried
by quarks and gluons in baryon resonances? Can we quantify the mechanical
properties of the baryon resonances?

  * •

What is the distribution of quark tensor charge in baryon resonances? What is
the gluonic structure of resonances? Can we excite baryon resonances using
transition operators with quantum numbers other than the vector current
operator used in traditional photo/electroexcitation?

These questions greatly expand the scope of baryon resonance structure studies
and raise it to the same level as current ground-state nucleon structure
studies.

A workshop “Exploring resonance structure with transition GPDs” was held at
the European Center for Theoretical Studies in Nuclear Physics and Related
Areas (ECT*) in Trento, Italy, on August 21-25, 2023, supported jointly by
ECT* and the Asia Pacific Center for Theoretical Physics (APCTP)
[https://indico.ectstar.eu/event/176/]. Twenty-seven presentations were given
over five days, followed by extensive discussions. This white-paper summarizes
the status of theoretical and experimental studies of transition GPDs and
baryon resonance structure, the prospects for future experimental programs
using electromagnetic and hadronic probes, and the developments needed for
their realization.

The article is organized as follows: Section 2 presents a general introduction
to the concepts and an overview of the scientific program. Section 3 provides
a theoretical introduction to transition GPDs, while section 4 shows their
connection to higher level observables of the baryon resonance structure. In
section 5 and overview on the methods for the theoretical treatment of
transition GPDs is provided. Section 6 introduces the description of lepto-
production processes sensitive to transition GPDs and section 7 presents first
experimental results from 12 GeV JLab on deeply virtual $N\to N^{*}$ meson
production and $N\to N^{*}$ Compton scattering. Sections 8 and 9 introduce the
description and measurement of transition GPDs in hadron scattering and
diffractive scattering processes. Section 10 provides an overview of future
perspectives at JLab, but also at hadron facilities like J-PARC and other
places. Finally, the strategy for the extraction of transition GPDs from
experimental observables is discussed in section 11.

##  2 Overview

Hadron structure in QCD is expressed in the matrix elements of QCD operators
between hadronic states. These gauge-invariant operators are composed of the
fundamental quark and gluon fields and measure certain features of their
distribution in the hadronic state. Two basic types of QCD operators are
accessible in scattering experiments. One type are the vector and axial-vector
currents of the quark fields, which are probed in elastic lepton-nucleon or
neutrino-nucleon scattering mediated by electromagnetic or weak interactions.
These operators are local and measure the distribution of electric and axial
charge and current in the hadronic state (form factors). Another type are QCD
operators arising from the factorization of deep-inelastic scattering
processes at energy and momentum transfers much larger than the hadronic scale
($\gg$ 1 GeV). These operators are non-local in space-time (light-like
separation of fields) and measure the momentum distributions of quarks and
gluons in the hadronic state (parton distributions). Inclusive deep-inelastic
lepton-nucleon scattering and related processes have provided extensive
information on the quark and gluon parton densities (PDFs).

Exclusive processes in lepton-nucleon scattering at energy and momentum
transfers $W^{2},Q^{2}\gg$ 1 GeV2 are another class of processes to which the
factorization method can be applied. Examples include the electroproduction of
real photons (deeply-virtual Compton scattering, or DVCS) $e+N\rightarrow
e^{\prime}+\gamma+N^{\prime}$, electroproduction of light mesons
$e+N\rightarrow e^{\prime}+M+N^{\prime}$, or photo/electroproduction of heavy
quarkonia, $e+N\rightarrow e^{\prime}+\mathcal{Q}+N^{\prime}$. In the
asymptotic regime, the production process involves individual quarks and
gluons, whose coupling to the nucleon is described by QCD operators with
light-like separation (see Fig. 1a). The matrix elements of these operators
between the nucleon states are parameterized by the GPDs; see Refs.
Goeke:2001tz ; Diehl:2003ny ; Belitsky:2005qn  for a review. These are non-
forward matrix elements, involving a longitudinal and transverse momentum
transfer of the order of the hadronic scale between the initial and final
nucleon, resulting in a rich structure. GPDs thus unify the concepts of
elastic form factors and parton densities, and contain both as limiting cases.

Figure 1: (a) QCD factorization of hard exclusive processes $e+N\rightarrow
e^{\prime}+\\{\gamma N,MN,\mathcal{Q}+N^{\prime}$. The production process
couples to the nucleon through a QCD operator. The $N\rightarrow N$ matrix
element of the operator is described by the GPD. (b) Processes with transition
$N\rightarrow\pi N,\Delta,N^{\ast}$. The $N\rightarrow\pi N,\Delta,N^{\ast}$
matrix element of the operator is described by the transition GPD.

GPDs provide new information on nucleon structure beyond what is available
from conventional probes (form factors, PDFs) (see Fig. 2). (i) The transverse
Fourier transform of the GPDs (transverse momentum transfer
$\Delta_{T}\rightarrow$ transverse coordinate $b$) describes the transverse
spatial distribution of quarks and gluons with given longitudinal momentum
fraction $x$ Burkardt:2000za ; Burkardt:2002hr . It provides a 2 + 1
dimensional “tomographic image” of the nucleon. This spatial representation is
appropriate for the nucleon as a relativistic quantum system and allows one to
visualize it as an extended object in space-time. The spatial structure can be
connected with the internal motion of the quarks and gluons and their
polarization and provides a new framework for the discussion of structure and
dynamics. “3D imaging” of the nucleon based on GPDs, which requires
measurements with high precision, is the object of experimental programs at
JLab and EIC. (ii) The moments of the GPDs (weighted integrals over the
momentum fraction $x$) represent matrix elements of local QCD operators of
spin $n>1$ (so-called generalized form factors). This makes it possible to
probe nucleon structure with local operators beyond the spin-1 operators
accessible with the electroweak currents. The spin-2 operators obtained from
GPDs contain the QCD energy-momentum tensor, the matrix elements of which
describe the distribution of momentum, angular momentum, forces, and mass and
are of fundamental interest for nucleon structure. Characterizing the
“mechanical properties” of hadrons based on the QCD energy-momentum tensor has
emerged as a field of study in its own right, with many theoretical and
experimental results; see Refs. Polyakov:2018zvc ; Lorce:2018egm  for a
review.

Figure 2: Applications of GPDs to nucleon structure: (a) Transverse spatial
distributions of partons (tomographic imaging). (b) Form factors of QCD
energy-momentum tensor describing distributions of angular momentum, mass, and
forces (mechanical properties). The concepts and structures can be extended to
$N\rightarrow\pi N,N^{\ast}$ transitions.

QCD factorization can be applied not only to exclusive processes with
$N\rightarrow N$ transitions on the target side, but also to processes in
which the nucleon undergoes a transition to a low-mass hadronic state,
$N\rightarrow\pi N,\pi\pi N,N^{\ast},...$ (see Fig. 1b) Goeke:2001tz .
Examples are DVCS with $N\rightarrow\pi N$ transitions, $e+N\rightarrow
e^{\prime}+\gamma+\pi N(\textrm{low-mass})$; or exclusive pion production with
$N\rightarrow\pi N$ transitions, $e+N\rightarrow e^{\prime}+\pi+\pi
N(\textrm{low-mass})$. The QCD operators representing the high-energy
processes are the same as in the case of $N\rightarrow N$ transitions. The
matrix elements are now taken between the initial $N$ and the final $\pi N$
state, $\langle\pi N|\mathcal{O}_{\rm QCD}|N\rangle$, and are parameterized by
so-called transition GPDs. This extends the concepts of GPDs to transitions
between low-mass hadronic states. The final $\pi N$ state here can be non-
resonant or resonant. Matrix elements for transitions to a baryon resonance
$N\rightarrow N^{\ast}$ can be defined rigorously through analytic
continuation in invariant mass of the $\pi N$ system (resonance pole). This
allows one to establish the concept of GPDs for baryon resonance transitions.

The nucleon is known to possess a rich spectrum of excited states
ParticleDataGroup:2022pth . They are observed in production and decay
processes induced by electromagnetic and hadronic probes. The spectrum has
been explained using models based on effective degrees of freedom (quark
model, chiral soliton) and is increasingly being confirmed by lattice QCD
(LQCD) calculations. Exploring the structure of the excited states in terms of
QCD degrees of freedom is the next step in their study. In the hadronic
picture the excited states are as fundamental as the ground state, and the QCD
structure of all the states is needed for a complete understanding of strong
interaction dynamics. Some information on $N^{\ast}$ structure is available
from the transition form factors of the vector current measured in
electroexcitation processes. Much more information on $N^{\ast}$ can be
obtained from the transition GPDs measured in $N\rightarrow N^{\ast}$
exclusive processes and the concepts derived from them (see Fig. 2). The
transition GPDs allow one to construct tomographic images of the $N^{\ast}$ at
the same level as the $N$ and discuss the QCD structure of resonances in these
terms. They provide access to the transition matrix elements of the QCD EMT
and allow one to discuss the mechanical properties of the resonances. The
factorization of exclusive processes provides QCD operators with quantum
numbers that are not easily accessible otherwise, such as chiral-odd quark
operators (pion production) and gluonic operators (heavy-quarkonium
production), and these operators can be used for resonance excitation and
structure studies. While extracting the structures from experiments poses
considerable challenges, the concepts derived from transition GPDs can also be
explored using dynamical models and LQCD results and enrich $N^{\ast}$
structure studies in this way. Altogether, transition GPDs have the potential
to greatly expand the range of baryon structure studies in QCD.

After first studies of associated electroproduction of real photons,
$e+p\rightarrow e^{\prime}\gamma\pi N$, in the $\Delta(1232)$-resonance region
with HERMES Duren:2014ola , recent experiments at JLab have shown the
feasibility of measuring exclusive processes with $N\rightarrow N^{\ast}$
transitions with significantly increased precision and background separation
capabilities. Measurements of exclusive electroproduction $e+p\rightarrow
e^{\prime}+\pi^{-}+\Delta^{++}$ have been performed at JLab CLAS12 and are
being interpreted in terms of transition GPDs CLAS:2023akb . Similar
measurements of $e+p\rightarrow e^{\prime}+\pi^{+}+\Delta^{0}$ have been
performed at JLab Hall C and are being analyzed. Measurements of DVCS with
$N\rightarrow\Delta$ and other $N\rightarrow N^{\ast}$ transitions will be
possible with the forthcoming CLAS12 data.

A program for exploring resonance structure using transition GPDs is emerging.
To realize it, it is necessary to develop theoretical and experimental
methods, define the scientific objectives, simulate the proposed measurements,
and optimize the analysis and extraction procedure.

Theoretical methods needed for the transition GPD program include the
structural decomposition of the $N\rightarrow N^{\ast}$ matrix elements of the
QCD operators and their parametrization in terms of transition GPDs. The
structure of the $N\rightarrow\Delta$ matrix elements of the chiral-even QCD
operators was discussed in Refs. Goeke:2001tz ; Belitsky:2005qn . Recent work
revisited the definition of the chiral-even $N\rightarrow\Delta$ transition
GPDs and extended the analysis to the chiral-odd sector Kroll:2022roq . The
$N\rightarrow\Delta$ transition matrix elements of the QCD energy-momentum
tensor were studied in Ref. Ozdem:2019pkg ; Polyakov:2020rzq ; Kim:2022bwn .

Also needed is the physical interpretation of the transition matrix elements,
especially the $N\rightarrow N^{\ast}$ transition matrix elements of the QCD
energy-momentum tensor and the mechanical properties derived from it. Recent
work has extended the concept of QCD angular momentum to $N\rightarrow\Delta$
transitions, using a formulation in terms of light-front densities appropriate
for transitions between baryon states with different masses and quantum
numbers Kim:2023xvw .

Theoretical efforts also focus on making quantitative predictions of the
transition GPDs using methods of nonperturbative QCD. Particularly useful are
methods that can connect the $N\rightarrow N$ and $N\rightarrow\pi N,\Delta$
etc. matrix elements in a systematic fashion. The $1/N_{c}$ expansion of QCD
is a general method for analyzing matrix elements of QCD operators in meson
and baryon states. It is based on the dynamical spin-flavor symmetry of QCD in
the large-$N_{c}$ limit, where $N$ and $\Delta$ states are in the same
multiplet, and transitions between them are connected by symmetry. The method
can classify and predict the spin-flavor components of $N\rightarrow\Delta$
transition GPDs and express them in terms of the $N\rightarrow N$ GPDs. The
$1/N_{c}$ expansion was applied to chiral-even transition GPDs Goeke:2001tz ,
and was recently extended to study chiral-odd GPDs Schweitzer:2016jmd ;
Kroll:2022roq . Methods based on chiral dynamics (soft-pion theorems, chiral
perturbation theory) can connect $N\rightarrow N$ and $N\rightarrow\pi N$
transition matrix elements near threshold and predict transition GPDs in this
regime Pobylitsa:2001cz ; Chen:2003jm ; Guichon:2003ah ; Birse:2005hh . They
can also be extended to processes with $N\rightarrow\Delta$ transitions
Alharazin:2023zzc ; Alharazin:2023uhr . Models based on the holographic
representation of QCD (gauge-string duality) posit a close connection between
hadron structure and the spectrum of excited states and are useful for the
study of GPDs and transition GPDs Mamo:2022jhp .

LQCD methods for computing nucleon matrix elements of local QCD operators are
well developed. They have recently been extended to enable the computation of
$x$-dependent partonic structure, using matrix elements of nonlocal operators
with spacelike separation in fast-moving hadronic states (quasi/pseudo
distributions). Applications to GPDs are in progress Alexandrou:2020zbe ;
Bhattacharya:2022aob ; Bhattacharya:2023jsc . At the same time, LQCD methods
have been adapted to extract excited hadronic states, using arrays of
Euclidean correlation functions of operators projecting on ground and excited
states (generalized eigenvalue problem, distillation) Edwards:2011jj ;
xQCD:2019jke . While the combination of both techniques is challenging and
requires major development, there is the long-term prospect of LQCD
calculations of transition GPDs.

Theory developments currently focus on reaction theory and observables for
DVCS with $N\rightarrow\Delta$ and $N\rightarrow N^{\ast}$ transitions
Semenov-Tian-Shansky:2023bsy  as well as $\pi$ electroproduction with
$N\rightarrow\Delta$ transitions Kroll:2022roq .

Experimental developments at JLab focus on improving the measurement of pion
production with $N\rightarrow\Delta$ transitions (CLAS12, Hall C), and
preparing measurements of DVCS with $N\rightarrow\Delta$ and other
$N\rightarrow N^{\ast}$ transitions (CLAS12). It is planned to extend the
studies to the strangeness sector. The luminosity upgrade of CLAS12, which is
currently in preparation, will help to increase the statistics for these
studies. Further detector modifications to increase the detection efficiency
of the processes with CLAS12 will be investigated. A potential energy upgrade
of JLab Accardi:2023chb  would help to increase the accessible $Q^{2}$ range
(factorization of the processes) and the phase space of the invariant masses
above the background rejection cuts (detection efficiency).

The future EIC will enable an extended program of GPD measurements on the
proton at high energies ($10^{-3}\lesssim x_{B}\lesssim 0.1$). The final-state
proton or neutron moves forward in the ion beam direction and will be detected
with the far-forward detection system (magnetic spectrometer with several
detector subsystems for proton, zero-degree calorimeter for neutron)
AbdulKhalek:2021gbh . The possibility of $\pi N$ and other transition GPD
measurements with this setup is being explored and presents interesting
challenges for the far-forward detection and event reconstruction.

Ultraperipheral proton-nucleus collisions and central exclusive production in
proton-proton collisions at LHC enable measurements of photon-proton
scattering at energies $W\sim 10^{3}$ GeV, the highest energies available for
electromagnetic scattering (an order of magnitude larger than at the electron-
proton HERA collider) Baltz:2007kq . The setup is used to measure gluon GPDs
in the photoproduction of heavy quarkonia, $\gamma+p\rightarrow\bar{Q}Q+p$,
reaching momentum fractions $x\sim 10^{-6}$ ALICE:2014eof ; ALICE:2018oyo .
This exclusive scattering process is diffractive (vacuum exchange) and can be
discussed using concepts of diffractive scattering. The scattering either
leaves the proton intact, $p\rightarrow p$, or causes it to dissociate into a
low-mass hadronic state, $p\rightarrow X(\textrm{low-mass})$. The latter
process measures the gluon GPD for $p\rightarrow X$ transitions, which can be
connected with the quantum fluctuations of the gluon density in the proton
Frankfurt:2008vi .

Hadron-induced scattering processes can also be used to access GPDs under
certain conditions, providing a complementary method to electromagnetic
processes. Exclusive dilepton production (Drell-Yan pair production)
Berger:2001zn ; Sawada:2016mao  as well as 2 $\to$ 3 processes Kumano:2009he ,
have been proposed as a way of measuring GPDs and transition GPDs. These
hadron-induced processes have a natural connection to transition GPDs.

##  3 Transition GPDs

###  3.1 GPDs of the ground state nucleon

GPDs encode the 3-dimensional partonic structure of the nucleon by correlating
the internal transverse position of the partons with their longitudinal
momentum fraction Muller:1994ses ; Ji:1996ek ; Radyushkin:1996nd . For a
nucleon with arbitrary spin, there are in total 8 independent GPDs of which 4
do not flip the parton helicity, while the other 4 flip the parton helicity
and are also known as transversity GPDs (indicated by $T$ in the subscript)
Collins:1996fb ; Diehl:2003ny ; Diehl:2001pm . These GPDs can be related to
the polarization of the nucleon ($N$) and the parton ($q$), which can both be
unpolarized (U), longitudinally polarized (L) or transversely polarized (T)
and are summarized in Tab. 1.

N$\textbackslash$q | U | L | T
---|---|---|---
U | $H$ |  | $\bar{E}_{T}$
L |  | $\widetilde{H}$ | $\widetilde{E}_{T}$
T | $E$ | $\widetilde{E}$ |  $H_{T}$, $\widetilde{H}_{T}$
Table 1: GPDs for the ground state nucleon. With
$\bar{E}_{T}=2\widetilde{H}_{T}+E_{T}$.

GPDs typically depend on the kinematic variables $x$, $\xi$, and $t$. The
introduced variable $x$ can be physically interpreted in a way, that during
the scattering process, a quark is removed from the nucleon with a momentum
fraction $x+\xi$ and reinserted with a reduced momentum fraction $x-\xi$ at a
certain transverse distance, parameterized by the conjugate of the Mandelstam
variable $t$ Dupre:2016mai .

###  3.2 $N\rightarrow\Delta$ transition GPDs

Transition GPDs describe transitions from a ground-state nucleon to a
resonating meson-nucleon system and occur in a description of several classes
of hard-exclusive reactions within the collinear factorization framework.
First studies are focused on the nucleon transitions to $\Delta$ and nucleon
resonances with the lowest masses. A possible more general description in
terms of $N\to\pi N$ transition GPDs Polyakov:1998sz ; Polyakov:2006dd  was
sketched in Goeke:2001tz . In the threshold region, the $N\to\pi N$ GPDs are
related to the nucleon GPDs through the soft-pion theorems, allowing the
parameter-free predictions for the $N\to\pi N$ DVCS at the pion production
threshold Guichon:2003ah .

For the special case of the $N\to$ $\Delta$ transition, a total of $16$
(twist-$2$) transition GPDs Belitsky:2005qn ; Kroll:2022roq  can be defined.
Several equivalent definitions are available in the literature. Here, for the
cases of vector and axial-vector operators, we present the definitions of the
set of GPDs of Ref. Belitsky:2005qn  for the proton-to-$\Delta^{++}$
transition employed in the recent analysis Kroll:2022roq . There are four
even-parity transition GPDs,

$\displaystyle\int\frac{d\tau}{2\pi}e^{i\tau(n\cdot P)x}$
$\displaystyle\langle\Delta^{++}(p_{R},\lambda_{R})|\bar{u}(-\tau
n/2)\gamma^{+}d(\tau n/2)|N^{p}(p,\lambda_{N})\rangle$
$\displaystyle=\frac{1}{(n\cdot P)}\,\bar{\cal
U}_{\delta}(p_{R},\lambda_{R})\left\\{\frac{\Delta^{\delta}n^{\mu}-\Delta^{\mu}n^{\delta}}{M_{N}}\Big{(}\gamma_{\mu}G_{1}(x,\xi,t)\right.$
$\displaystyle\left.+\frac{P_{\mu}}{M_{N}}G_{2}(x,\xi,t)+\frac{\Delta_{\mu}}{M_{N}}G_{3}(x,\xi,t)\Big{)}\right.$
$\displaystyle\left.+\frac{\Delta^{+}\Delta^{\delta}}{M_{N}^{2}}G_{4}(x,\xi,t)\right\\}\gamma_{5}u(p,\lambda_{N});$
(1)

and four odd-parity ones,

$\displaystyle\int\frac{d\tau}{2\pi}e^{i\tau(n\cdot P)x}$
$\displaystyle\langle\Delta^{++}(p_{R},\lambda_{R})|\bar{u}(-\tau
n/2)\gamma^{+}\gamma^{5}d(\tau n/2)|N^{p}(p,\lambda_{N})\rangle$
$\displaystyle=\frac{1}{(n\cdot P)}\,\bar{\cal
U}_{\delta}(p_{R},\lambda_{R})\left\\{\frac{\Delta^{\delta}n^{\mu}-\Delta^{\mu}n^{\delta}}{M_{N}}\Big{(}\gamma_{\mu}\tilde{G}_{1}(x,\xi,t)\right.$
$\displaystyle+\frac{P_{\mu}}{M_{N}}\tilde{G}_{2}(x,\xi,t)\Big{)}+n^{\delta}\tilde{G}_{3}(x,\xi,t)$
$\displaystyle\left.+\frac{\Delta^{+}\Delta^{\delta}}{M_{N}^{2}}\tilde{G}_{4}(x,\xi,t)\right\\}u(p,\lambda_{N}).$
(2)

Here, $n^{2}=0$ is the light-cone vector; and for simplicity, the light-cone
gauge $A^{+}\equiv A\cdot n=0$ is adopted; $P$ is the average hadron momentum
$P\equiv\frac{p+p_{R}}{2}$ and $\Delta\equiv p_{R}-p$ is the momentum
transfer; $p(\lambda_{N},M_{N})$ and $p_{R}(\lambda_{R},M_{\Delta})$ denote
the proton and the $\Delta(1232)$ momenta (helicities and masses),
respectively. The quantity ${\cal U}_{\delta}(p_{R},\lambda_{R})$ stands for
the Rarita-Schwinger spinor describing the $\Delta$ particle, and
$u(p,\lambda_{N})$ is the nucleon’s Dirac spinor. The skewness parameter $\xi$
is defined by the ratio of light-cone plus components of $\Delta$ and $P$:

$\xi=-\frac{\Delta^{+}}{2P^{+}}\,.$ (3)

The Mandelstam variable $t$ is given by

$t=\Delta^{2}=t_{0}-\frac{\mathbf{\Delta}_{\perp}^{2}}{1-\xi^{2}},$ (4)

where $t_{0}$ is the minimal value of $-t$ implied by the positivity of the
transverse squared momentum transfer $\mathbf{\Delta}_{\perp}^{2}$

$t_{0}=-\frac{2\xi}{1-\xi^{2}}\Big{[}(1+\xi)(M^{2}_{\Delta}-M^{2}_{N})+2\xi
M_{N}^{2}\Big{]}\,.$ (5)

So far the $\Delta(1232)$ is considered to be a stable particle. Therefore,
one can apply the usual time-reversal invariance arguments to show that the
$p-\Delta$ GPDs turn to be real-valued functions of the momentum fraction $x$,
the skewness and $t$ Belitsky:2005qn . The GPDs also depend on the
factorization scale. For convenience, this dependence is suppressed here.

An alternative parametrization Pascalutsa:2006up  relying on a different set
of spin-tensor structures involves $4$ unpolarized GPDs
$h_{M,\,E,\,C,\,4}(x,\xi,t)$ and $4$ polarized GPDs
$C_{1,\,2,\,3,\,4}(x,\xi,t)$; and is employed in the analysis of Ref. Semenov-
Tian-Shansky:2023bsy . The relation of this transition GPD set to that of the
heritage parametrization of Ref. Goeke:2001tz  is specified in Ref.
Pascalutsa:2006up .

The set of $8$ transversity GPDs $G_{T1,\ldots,T8}(x,\xi,t)$ has been defined
in Kroll:2022roq :

$\displaystyle\int\frac{d\tau}{2\pi}\,e^{i\tau(n\cdot P)x}$
$\displaystyle\langle\Delta^{++}(p_{R},\lambda_{R})|\bar{u}(-\tau
n/2)i\sigma^{+j}d(\tau n/2)|N^{p}(p,\lambda_{N})\rangle$
$\displaystyle=\frac{1}{(n\cdot P)}\bar{\cal
U}_{\delta}(p_{R},\lambda_{R})\left\\{G_{T1}\frac{p^{\delta}}{M_{N}}i\sigma^{+j}\right.$
$\displaystyle+G_{T2}\,p^{\delta}\,\frac{P^{+}\Delta^{j}-\Delta^{+}P^{j}}{M_{N}^{3}}+G_{T3}\,p^{\delta}\,\frac{\gamma^{+}\Delta^{j}-\Delta^{+}\gamma^{j}}{2M_{N}^{2}}$
$\displaystyle+G_{T4}\,p^{\delta}\,\frac{\gamma^{+}P^{j}-P^{+}\gamma^{j}}{M_{N}^{2}}+G_{T5}\,(n^{\delta}\gamma^{j}-\gamma^{\delta}n^{j})$
$\displaystyle\left.+G_{T6}\,\frac{n^{\delta}\Delta^{j}-\Delta^{\delta}n^{j}}{M_{N}}\right\\}\,\gamma_{5}\,u(p,\lambda_{N})$
$\displaystyle+\frac{1}{(n\cdot P)}\,\left\\{G_{T7}\,(\bar{\cal
U}^{+}(p_{R},\lambda_{R})\gamma^{j}-\bar{\cal
U}^{j}(p_{R},\lambda_{R})\gamma^{+})\right.$
$\displaystyle+\left.G_{T8}\,\frac{\bar{\cal
U}^{+}(p_{R},\lambda_{R})\Delta^{j}-\bar{\cal
U}^{j}(p_{R},\lambda_{R})\Delta^{+}}{M_{N}}\right\\}\,\gamma_{5}\,u(p,\lambda_{N})\,.$
(6)

Isospin symmetry relates the $p\to\Delta^{++}$ GPDs to those of other
$\Delta(1232)$ states Belitsky:2005qn :

$G^{ud}_{p\Delta^{++}}=-\frac{\sqrt{3}}{2}G^{uu-
dd}_{p\Delta^{+}}=-\sqrt{3}G^{du}_{p\Delta^{0}},$ (7)

where the flavor content of the GPDs is indicated.

The definitions (1), (2) and (6) also hold for any other octet-decuplet
transition GPDs. Moreover, the flavor symmetry relates the various octet-
decuplet GPDs to each other Belitsky:2005qn .

The 8 GPDs defined in Eqs. (1), (2) are helicity-non-flip transition GPDs.
Four of them ($G_{1}$ \- $G_{4}$) are unpolarized (parity even) and can be
related to the Jones-Scadron electromagnetic form factors for the $N\to\Delta$
transition Jones:1972ky ; Goeke:2001tz :

$\begin{split}\int^{1}_{-1}dxG_{1}(x;\xi;t)&\propto
G^{*}_{M}(t),~{}~{}~{}\int^{1}_{-1}dxG_{2}(x;\xi;t)\propto G^{*}_{E}(t),\\\
\int^{1}_{-1}dxG_{3}(x;\xi;t)&\propto
G^{*}_{C}(t),~{}~{}~{}\int^{1}_{-1}dxG_{4}(x;\xi;t)=0,\end{split}$ (8)

with the magnetic dipole, electric quadrupole, and the Coulomb quadrupole form
factors $G^{*}_{M,E,C}(t)$. The other four ($\widetilde{G}_{1}$ \-
$\widetilde{G}_{4}$) are polarized (parity odd) and can be related to the
Adler form factors of the $N\to\Delta$ transition ($C^{A}(t)$) Goeke:2001tz ;
Adler:1968tw ; Adler:1975mt :

$\begin{split}\int^{1}_{-1}dx\widetilde{G}_{1}(x;\xi;t)&\propto
C^{A}_{5}(t),~{}~{}~{}\int^{1}_{-1}dx\widetilde{G}_{2}(x;\xi;t)\propto
C^{A}_{6}(t),\\\ \int^{1}_{-1}dx\widetilde{G}_{3}(x;\xi;t)&\propto
C^{A}_{3}(t),~{}~{}~{}\int^{1}_{-1}dx\widetilde{G}_{4}(x;\xi;t)\propto
C^{A}_{4}(t).\end{split}$ (9)

###  3.3 Second resonance region

The extension of the transition GPD formalism to the second $\pi N$ resonance
region requires introducing parametrizations for the twist-$2$ $N\to N^{*}$
transition GPDs for the case of the vector bilinear and axial-vector quark
operators on the light cone for the isospin-$\frac{1}{2}$ resonances
$P_{11}(1440)$, $D_{13}(1520)$ and $S_{11}(1535)$ with and the spin-parity
quantum numbers, respectively,
$J^{P}=\frac{1}{2}^{+},\frac{3}{2}^{-},\frac{1}{2}^{-}$.

The parametrizations of $N\to P_{11}(1440)$, $D_{13}(1520)$, $S_{11}(1535)$
transition GPDs introduced in Ref. Semenov-Tian-Shansky:2023bsy  for the case
of the vector bilinear quark operator were designed with the sets of spin-
tensor structures consistent with the parametrizations of corresponding
transition electromagnetic form factors detailed in Ref. Tiator:2011pw .
Similarly, for the axial-vector bilinear quark operator, the parametrization
of Ref. Semenov-Tian-Shansky:2023bsy  tends to ensure the simple connection
with the FFs occurring in the parametrizations of the transition matrix
elements of the isoscalar $A_{0}^{\nu}(0)$ and isovector $A_{3}^{\nu}(0)$axial
currents:

$\displaystyle
A_{\\{0,\,3\\}}^{\nu}(0)\equiv\frac{1}{2}\left[\bar{u}(0)\gamma^{\nu}\gamma_{5}u(0)\pm\bar{d}(0)\gamma^{\nu}\gamma_{5}d(0)\right].$
(10)

This provides a simple form of the normalization constraint for the first
Mellin moments from determining the dominant $N\to N^{*}$ axial FFs with help
of the PCAC relation

$\displaystyle\partial_{\nu}\,A^{\nu}_{3}=-f_{\pi}m_{\pi}^{2}\,\Pi_{3},$ (11)

where the pion decay constant $f_{\pi}\simeq 92.4$ MeV; and $\Pi_{3}$ stands
for the $\pi^{0}$ field operator.

As an example, we present the case of GPDs for the $N\to P_{11}$ transition.
Due to the identical spin-parity, the parametrization has a form similar to
the nucleon GPD case, with two unpolarized and two polarized twist-$2$ GPDs.
The matrix element of the vector bilinear quark operator along the light-cone
can be parameterized as:

$\displaystyle\int\frac{d\tau}{2\pi}e^{i\tau(n\cdot P)x}$
$\displaystyle\sum_{q}e_{q}^{2}\langle R(p_{R},\lambda_{R})|\bar{q}\left(-\tau
n/2\right)\gamma^{+}q\left(\tau n/2\right)|N(p,\lambda_{N})\rangle$
$\displaystyle=\frac{1}{(n\cdot
P)}\bar{R}\left(p_{R},\lambda_{R}\right)\left\\{H_{1}^{pP_{11}}(x,\xi,\Delta^{2})\right.$
$\displaystyle\times\left(n^{\nu}-\frac{n\cdot\Delta}{\Delta^{2}}\Delta^{\nu}\right)\gamma_{\nu}$
$\displaystyle\left.+H_{2}^{pP_{11}}(x,\xi,\Delta^{2})\frac{i\sigma_{\nu\kappa}n^{\nu}\Delta^{\kappa}}{(M_{R}+M_{N})}\right\\}u\left(p,\lambda_{N}\right),$
(12)

where the convectional DVCS combination is weighted by the quadratic quark
charges $e_{q}$; and $\bar{R}\left(p_{R},\lambda_{R}\right)$ denotes the Dirac
spinor of the final state $P_{11}$. This definition results in the following
normalization of the first Mellin moment of GPDs to the proton ($N=p$) and
neutron ($N=n$) to $P_{11}$ transition electromagnetic FFs:

$\displaystyle\int_{-1}^{1}dx\,H_{1,2}^{pP_{11}}(x,\xi,\Delta^{2})$
$\displaystyle=$ $\displaystyle
F_{1,2}^{pP_{11}}(\Delta^{2})+\frac{2}{3}F_{1,2}^{nP_{11}}(\Delta^{2}).$

The transition electromagnetic FFs $F_{1}^{NP_{11}}$ and $F_{2}^{NP_{11}}$ are
defined from the matrix element

$\displaystyle\langle
R(p_{R},\lambda_{R})|J_{\text{em}}^{\nu}(0)|N(p,\lambda_{N})\rangle$
$\displaystyle=\bar{R}\left(p_{R},\lambda_{R}\right)\Gamma_{\gamma
NP_{11}}^{\nu}(p_{R},p)u\left(p,\lambda_{N}\right),$ (14)

where the vertex $\Gamma_{\gamma NP_{11}}^{\nu}$ is parameterized as
Tiator:2008kd :

$\displaystyle\Gamma_{\gamma NP_{11}}^{\nu}(p_{R},p)$ $\displaystyle=$
$\displaystyle
F_{1}^{NP_{11}}(\Delta^{2})\,\left[\gamma^{\nu}-\frac{(\gamma\cdot\Delta)\Delta^{\nu}}{\Delta^{2}}\right]$
(15) $\displaystyle+$ $\displaystyle
F_{2}^{NP_{11}}(\Delta^{2})\,\frac{i\,\sigma^{\nu\kappa}\Delta_{\kappa}}{(M_{R}+M_{N})}.$

For the phenomenological applications, we rely on the empirical MAID2008
analysis for the proton and the MAID2007 analysis for the neutron, as detailed
in Ref. Tiator:2011pw .

Furthermore, in the case of the axial-vector operator, the parametrization for
the $N\to P_{11}$ transition reads

$\displaystyle\int\frac{d\tau}{2\pi}e^{i\tau(n\cdot P)x}$
$\displaystyle\sum_{q}e_{q}^{2}\langle R(p_{R},\lambda_{R})|\bar{q}\left(-\tau
n/2\right)\gamma^{+}\gamma_{5}q\left(\tau n/2\right)|N(p,\lambda_{N})\rangle$
$\displaystyle=\frac{1}{(n\cdot
P)}\bar{R}\left(p_{R},\lambda_{R}\right)\left\\{\tilde{H}_{1}^{pP_{11}}(x,\xi,\Delta^{2})\;\gamma\cdot
n\gamma_{5}\right.$
$\displaystyle\left.+\tilde{H}_{2}^{pP_{11}}(x,\xi,\Delta^{2})\;\frac{\Delta\cdot
n}{(M_{R}+M_{N})}\gamma_{5}\right\\}u\left(p,\lambda_{N}\right).$ (16)

To work out the normalization, we need to consider the transition matrix
elements of the isovector and isoscalar axial currents (10). For the isovector
axial current $N\to P_{11}$ transition it reads

$\displaystyle\langle
R(p_{R},\lambda_{R})|A_{3}^{\nu}(0)|N(p,\lambda_{N})\rangle$
$\displaystyle=\bar{R}\left(p_{R},\lambda_{R}\right)\bigg{\\{}G_{A}^{NP_{11}}(\Delta^{2})\,\gamma^{\nu}\gamma_{5}$
$\displaystyle+G_{P}^{NP_{11}}(\Delta^{2})\,\frac{\Delta^{\nu}\gamma_{5}}{(M_{R}+M_{N})}\bigg{\\}}\,\frac{\tau_{3}}{2}u\left(p,\lambda_{N}\right),\quad\quad$
(17)

with $\tau_{3}$ the third isospin Pauli matrix., Here, $G_{A}^{NP_{11}}$ and
$G_{P}^{NP_{11}}$ are the corresponding isovector axial transition FFs. And,
similarly, for the isoscalar axial current $N\to P_{11}$ transition:

$\displaystyle\langle
R(p_{R},\lambda_{R})|A_{0}^{\nu}(0)|N(p,\lambda_{N})\rangle$
$\displaystyle=\bar{R}\left(p_{R},\lambda_{R}\right)\bigg{\\{}G_{A,0}^{NP_{11}}(\Delta^{2})\,\gamma^{\nu}\gamma_{5}$
$\displaystyle+G_{P,0}^{NP_{11}}(\Delta^{2})\,\frac{\Delta^{\nu}\gamma_{5}}{(M_{R}+M_{N})}\bigg{\\}}\,\frac{1}{2}u\left(p,\lambda_{N}\right),\quad\quad$
(18)

with $G_{A,0}^{NP_{11}}$ and $G_{P,0}^{NP_{11}}$ the corresponding isoscalar
axial transition FFs.

The relations between the first moments of the polarized GPDs (16) for a given
quark flavor entering the definition in Eq. (16) and the axial transitions FFs
defined in Eqs. (17) and (18) are then obtained as:

$\displaystyle\int_{-1}^{1}dx\tilde{H}_{1}^{u,pP_{11}}(x,\xi,\Delta^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(G_{A}^{NP_{11}}+G_{A,0}^{NP_{11}}\right)(\Delta^{2}),$
$\displaystyle\int_{-1}^{1}dx\tilde{H}_{1}^{d,pP_{11}}(x,\xi,\Delta^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(-G_{A}^{NP_{11}}+G_{A,0}^{NP_{11}}\right)(\Delta^{2}),$
(19)

and analogous relations for $\tilde{H}_{2}^{u,pP_{11}}$ and
$\tilde{H}_{2}^{d,pP_{11}}$ in terms of $G_{P}^{NP_{11}}$ and
$G_{P,0}^{NP_{11}}$.

The physical normalization for the $N\to P_{11}$ transition isovector axial FF
can be obtained by use of the PCAC relation (11) together with an effective
$\pi NP_{11}$ vertex derived from the effective Lagrangian

$\displaystyle{\cal L}_{\pi NP_{11}}=\left(\frac{f_{\pi
NP_{11}}}{m_{\pi}}\right)\bar{R}\gamma^{\mu}\gamma_{5}\tau_{i}N\left(\partial_{\mu}\Pi_{i}\right)+\mathrm{h.c.},$
(20)

with $\pi NP_{11}$ coupling constant $f_{\pi NP_{11}}$. This results in the
generalized Goldberger-Treiman relation for the $N\to P_{11}$ isovector axial
FF:

$\displaystyle G_{A}^{NP_{11}}(0)=\left(\frac{f_{\pi
NP_{11}}}{m_{\pi}}\right)2f_{\pi}.$ (21)

Furthermore, using the pion-pole dominance for the axial FF $G_{P}^{NP_{11}}$
at small values of $\Delta^{2}$ yields:

$\displaystyle
G_{P}^{NP_{11}}(\Delta^{2})\approx\frac{(M_{R}+M_{N})^{2}}{-\Delta^{2}+m_{\pi}^{2}}G_{A}^{NP_{11}}(\Delta^{2}).$
(22)

The isoscalar $N\to P_{11}$ axial FFs $G_{A,0}^{NP_{11}}$ and
$G_{P,0}^{NP_{11}}$ are not known. As a possible best guess one may use for
$G_{A,0}^{NP_{11}}$ the same quark model relation as for the nucleon isoscalar
axial FF:

$\displaystyle
G_{A,0}^{NP_{11}}(\Delta^{2})\approx\frac{3}{5}G_{A}^{NP_{11}}(\Delta^{2}),$
(23)

and parameterize $G_{A}^{NP_{11}}$ by a dipole form as in the nucleon case:

$\displaystyle G_{A}^{NP_{11}}(\Delta^{2})=1/(1-\Delta^{2}/M_{A}^{2})^{2},$
(24)

with dipole mass $M_{A}\simeq 1.0$ GeV. Furthermore, the isoscalar FF is set
to zero

$\displaystyle G_{P,0}^{NP_{11}}\approx 0,$ (25)

in line with pion-pole dominance. A possible cross check for such estimates
can be provided by calculating these FFs within dynamical quark-diquark
approaches, which have proven to provide a good understanding of the data for
the vector FFs $F_{1,2}^{NP_{11}}$ for the $N\to P_{11}$ transition
Burkert:2017djo .

The cases of $N\to D_{13}(1520)$ and $N\to S_{11}(1535)$ in Semenov-Tian-
Shansky:2023bsy  are treated in a similar manner. The parametrization of
leading twist vector and axial-vector GPDs for $N\to D_{13}(1520)$ involves
$4+4$ invariant GPDs, while $N\to S_{11}(1535)$ involves $2+2$ invariant GPDs.
The phenomenological normalization for the first Mellin moment vector
transition GPDs is provided by the relation to the transition electromagnetic
FFs available e.g. from the empirical MAID2008 analysis for the proton and the
MAID2007 analysis for the neutron detailed in Ref. Tiator:2011pw . The
normalization of the axial vector can be established relying on the PCAC,
following a line similar to the case of $N\to P_{11}(1440)$ described above.

##  4 Baryon resonance structure

###  4.1 3D tomography of baryon resonances

GPDs provide a new set of tools for characterizing hadron structure in QCD.
Extending these concepts and methods from $N\rightarrow N$ to $N\rightarrow
N^{\ast}$ transitions and using them to resonance structure is an objective of
the transition GPD program.

One major application of $N\rightarrow N$ GPDs is the 3D imaging of the
quark/gluon structure of the nucleon. In a frame where the momentum transfer
between the nucleon states is in the transverse direction, $\xi=0$ and
$\bm{\Delta}_{T}\neq 0$, the $N\rightarrow N$ GPD
$H_{q}(x,\xi=0,t=-\bm{\Delta}_{T}^{2})$ ($q$ denotes the quark flavor) can be
represented as the Fourier transform of a transverse coordinate distributions
as (here $x>0$)

$\displaystyle H_{q}(x,\xi=0,t=-\bm{\Delta}_{T}^{2})=\int
d^{2}b\,e^{i\bm{\Delta}_{T}b}\,f_{q}(x,\bm{b}).$ $\displaystyle-
H_{q}(-x,\xi=0,t=-\bm{\Delta}_{T}^{2})=\int
d^{2}b\,e^{i\bm{\Delta}_{T}b}\,f_{\bar{q}}(x,\bm{b}).$ (26)

The functions $f_{q}(x,\bm{b})$ and $f_{\bar{q}}(x,\bm{b})$ describe the
distributions of quarks and antiquarks with momentum fraction $x$ as a
function of transverse position $\bm{b}$, in a nucleon state localized in
transverse space at the origin Burkardt:2000za ; Burkardt:2002hr . The
representation provides a “tomographic image” of the nucleon’s quark/antiquark
structure and reveals new information about the dynamical system, such as the
average transverse radius of quarks/antiquarks and the change of the
transverse distribution with $x$. Similar distributions describe the
transverse distribution gluons Kumericki:2009uq ; Strikman:2003gz , of
polarized quarks and gluons (helicity, transversity), and the distortions of
the spatial distributions induced by transverse nucleon polarization (spin-
orbit phenomena) Burkardt:2000za ; Burkardt:2002hr ; Burkardt:2005hp .

Extending the transverse coordinate representation to $N\rightarrow N^{\ast}$
transition GPDs is a subject of theoretical research. The definition of
localized transverse coordinate states in light-front quantization does not
refer to the particle mass, so localized states can be defined $N\rightarrow
N^{\ast}$ transitions in the same way as for $N\rightarrow N$. The coordinate
space densities associated with $N\rightarrow\Delta$ GPDs can be studied using
methods based on the large-$N_{c}$ limit of QCD (see Sec. 5.2) Kim:2023yhp .

A special case of the transverse coordinate representation are the transverse
densities of electric charge and magnetization in the nucleon Burkardt:2000za
; Burkardt:2002hr ; Miller:2007uy ; Miller:2010nz . The transverse charge
density in the nucleon is given by the Fourier transform of the $N\rightarrow
N$ Dirac electromagnetic form factor,

$\displaystyle F_{1}(t=-\bm{\Delta}_{T}^{2})=\int
d^{2}b\,e^{i\bm{\Delta}_{T}b}\,\rho_{1}(\bm{b}).$ (27)

The Dirac form factor is the integral of the GPD $H_{f}$ over $x$ (first
moment), weighted with the quark charges and summed over quark flavors

$\displaystyle F_{1}(t)=\sum_{q=u,d}e_{q}\int_{-1}^{1}dxH_{q}(x,\xi=0,t),$
(28)

and the transverse charge density is the integral of the difference of quark
and antiquark distributions

$\displaystyle\rho_{1}(\bm{b})=\sum_{q=u,d}e_{q}\left[f_{q}(x,\bm{b})-f_{\bar{q}}(x,\bm{b})\right].$
(29)

In this way the transverse charge density provides summary information on the
transverse coordinate distributions of quarks and antiquarks in the nucleon.
In a similar way, the transverse magnetization density representing the Pauli
form factor provides information on the transverse coordinate representation
of the nucleon GPD $E$. The transverse charge and magnetization densities can
be extracted directly from the experimentally measured electromagentic form
factors and provide interesting insight into nucleon structure; see Refs.
Venkat:2010by ; Miller:2011du ; Granados:2013moa ; Alarcon:2022adi  for
empirical and theoretical studies. The concepts can be generalized from
$N\rightarrow N$ transitions to $N\rightarrow N^{\ast}$ transitions. The
$N\rightarrow N^{\ast}$ transition charge/magnetization densities can be
extracted directly from the transition form factors Wang:2024byt  and provide
insight into the transverse spatial structure of the $N\rightarrow N^{\ast}$
transition.

$N\rightarrow\Delta$ transition densities have been extracted from the
$N\rightarrow\Delta$ transition form factors Carlson:2007xd . An interesting
aspect is the appearance of a quadrupole deformation in the transverse charge
density due to the presence of a quadrupole form factor ($\Delta J=2$) in the
$N\rightarrow\Delta$ transition. The $N\rightarrow\Delta$ densities have been
studied theoretically in holographic and light-front quark models
Chakrabarti:2016lod . The transverse densities in hyperon states have been
studied using dispersion theory and EFT methods Alarcon:2017asr .

The transverse coordinate representation Eq. (26) is defined at $\xi=0$, where
the transverse coordinate distributions represent proper particle densities in
the context of light-front quantization. The kinematics $\xi=0$ is not
accessible in experiment, as actual scattering processes involve a
longitudinal momentum transfer between the initial and final baryon; this
applies to $N\rightarrow N$ as well as $N\rightarrow N^{\ast}$ transitions. It
is generally not possible to extract $x$-dependent transverse distributions
directly from experiment. Experimental information can be used to probe GPDs
in other regions of the partonic variables. The program of nucleon tomography
with GPDs therefore relies essentially on theoretical models
predicting/constraining the behavior of GPDs. Lattice QCD calculations using
the quasi/pseudo-distribution formulation can potentially provide direct
information about the $x$-dependent GPDs at $\xi=0$ for use in hadron
tompagraphy (see Sec. 5.4).

###  4.2 Spin and angular momentum in baryon resonances

Another major application of $N\rightarrow N$ GPDs is the characterization of
the angular momentum carried by the quark and gluon fields in the nucleon,
including spin and orbital angular momentum. This is possible because the GPDs
contain information on the nucleon matrix elements of the QCD energy momentum
tensor (EMT), which enters into the definition of field-theoretical angular
momentum in QCD. Expanding the nonlocal QCD operator in powers of the light-
like separation of the quark fields, one obtains

$\displaystyle n_{\alpha}\,\bar{\psi}_{q}(-\tau
n/2)\gamma^{\alpha}[\textrm{gauge link}]\psi_{q}(\tau n/2)$
$\displaystyle=n_{\alpha}\,\bar{\psi}_{q}(0)\gamma^{\alpha}\psi_{q}(0)$
$\displaystyle+\frac{1}{2}n_{\alpha}n_{\beta}\,\bar{\psi}(0)\gamma^{\alpha}\overleftrightarrow{\nabla}^{\beta}\bar{\psi}(0)+\ldots$
(30)

where
$\overleftrightarrow{\nabla}^{\beta}\equiv\frac{1}{2}(\overrightarrow{\partial}^{\beta}-\overleftarrow{\partial}^{\mu})-igA^{\beta}$
is the QCD covariant derivative. The local spin-2 operator in the second term
is the quark part of the EMT ($q$ denotes the quark flavor),

$\displaystyle T_{q}^{\alpha\beta}(0)\equiv
i\bar{\psi}_{q}(0)\gamma^{\alpha}\overleftrightarrow{\nabla}\bar{\psi}_{q}(0).$
(31)

The matrix elements of this operator between nucleon states can be obtained
from the $x$-weighted integrals (second moments) of the $N\rightarrow N$ GPDs.
The angular momentum of quarks (and gluons) can be computed from its field-
theoretical definition in terms of the EMT. As a result, the angular momentum
of quarks and gluons in the nucleon can be expressed as certain $x$-weighted
integrals of the $N\rightarrow N$ GPDs (spin sum rules) Ji:1996ek . Several
versions of these sum rules have been proposed, using different but equivalent
definitions versions of the field-theoretical angular momentum; the relation
between them is now well understood Leader:2013jra . A field of study has
emerged, exploring the angular momentum content of the nucleon based on GPDs
using theoretical methods Goeke:2001tz ; Granados:2019zjw , LQCD simulations
Gockeler:2003jfa ; LHPC:2007blg ; LHPC:2010jcs ; Bali:2018zgl ;
Alexandrou:2019ali , and experimental data.

The extension of the notion of QCD angular momentum to $N\rightarrow N^{\ast}$
transitions is a subject of ongoing theoretical research. Some principal
questions need to be addressed when generalizing the concept of angular
momentum to transitions between states with different masses and quantum
numbers (spin, isospin). The invariant formulation of the spin sum rule by Ji
Ji:1996ek  uses specific properties of $1/2\rightarrow 1/2$ transitions, and
the Breit frame formulation by Polyakov Polyakov:2002yz  assumes transitions
between baryon states of equal mass.

Figure 3: Transition angular momentum derived from transition transverse
density of energy-momentum tensor, Eq.(LABEL:J_momentum), for
$N\rightarrow\Delta$ transitions Kim:2023xvw .

A framework that is well suited for the generalization to $N\rightarrow
N^{\ast}$ transitions is the formulation of QCD angular momentum as a
transverse density at fixed light-front time Adhikari:2016dir ; Lorce:2017wkb
; Granados:2019zjw . A definition of $N\rightarrow N^{\ast}$ transition
angular momentum based on this formulation has been proposed in Ref.
Kim:2023xvw . One considers the transition matrix elements of the EMT between
general baryon states with masses $m$ and $m^{\prime}$ and 4-momenta $p$ and
$p^{\prime}$,

$\displaystyle\langle
B^{\prime}(p^{\prime},\lambda^{\prime})|\hat{T}^{\alpha\beta}_{q}(0)|B(p,\lambda)\rangle.$
(32)

The baryons can have different spin $S^{\prime}\neq S$. The spin states are
described as light-front helicity states; the light-front helicities $\lambda$
and $\lambda^{\prime}$ are directly related to the spin projection along the
the 3-axis in the baryon rest frame; $\lambda\leftrightarrow
S_{3},\lambda^{\prime}\leftrightarrow S_{3}^{\prime}$. One analyzes the
transition matrix element Eq. (32) in a class of frames where the momentum
transfer has $\Delta^{+}=0$ and the average momentum has $\bm{P}_{T}=0$
(generalized Drell-Yan-West frame). In these frames the light-cone components
of the baryon momenta are given by (using the notation
$p=[p^{+},p^{-},\bm{p}_{T}]$ with $p^{\pm}\equiv p^{0}\pm p^{3}$)

$\displaystyle
p=\left[p^{+},\frac{m^{2}+|\bm{\Delta}_{T}|^{2}/4}{p^{+}},-\frac{\bm{\Delta}_{T}}{2}\right],$
$\displaystyle p^{\prime}=\left[p^{+},\frac{m^{\prime
2}+|\bm{\Delta}_{T}|^{2}/4}{p^{+}},\frac{\bm{\Delta}_{T}}{2}\right],$
$\displaystyle\Delta=\left[0,\frac{m^{\prime
2}-m^{2}}{p^{+}},\bm{\Delta}_{T}\right].$ (33)

The matrix element in Eq. (32) thus becomes a function of $\Delta_{T}$, with
$t=-\bm{\Delta}_{T}^{2}<0$. Taking the light-front component $+i$ of the EMT
($i$ = 1, 2) and performing the two-dimensional Fourier transform, one obtains
the transverse coordinate density

$\displaystyle T^{+i}_{q}(\bm{b}|S_{3}^{\prime},S_{3})$ (34)
$\displaystyle=\int\frac{d^{2}\Delta_{T}}{(2\pi)^{2}}e^{-i\bm{\Delta}_{T}\cdot\bm{b}}\langle
B^{\prime}(p^{\prime},\lambda^{\prime})|\hat{T}^{+i}_{q}(0)|B(p,\lambda)\rangle.$
(35)

The transition AM in the $z$-direction is then defined as

$\displaystyle 2S^{z}(S_{3}^{\prime},S_{3})\,J_{B\rightarrow B^{\prime}}^{q}$
$\displaystyle\equiv\frac{1}{2p^{+}}\int
d^{2}b\;\left[\bm{b}\times\bm{T}^{+T}_{q}(\bm{b}|S_{3}^{\prime},S_{3})\right]^{z},$
(36)

where $S^{z}(S_{3}^{\prime},S_{3})$ accounts for the kinematic spin dependence
and $J_{B\rightarrow B^{\prime}}^{q}$ contains the dynamical information
(reduced matrix element). This definition of the AM in the transverse
coordinate representation has the form of a vector product of position and
momentum and permits a simple mechanical interpretation (see Fig. 3); its
properties and the equivalence with other definitions were established for
$N\rightarrow N$ transitions in Refs. Adhikari:2016dir ; Lorce:2017wkb ;
Granados:2019zjw ). In the momentum representation of the matrix element the
transition AM is expressed as

$\displaystyle 2S^{z}(S^{\prime}_{3},S_{3})\,J_{B\rightarrow B^{\prime}}^{q}=$
(37)
$\displaystyle\frac{1}{2p^{+}}\left[-i\frac{\partial}{\partial\bm{\Delta}_{T}}\times\langle
B^{\prime}(p^{\prime},\lambda^{\prime})|\hat{\bm{T}}^{+T}_{q}(0)|B(p,\lambda)\rangle\right]^{z}_{\bm{\Delta}_{T}=0}.$
(38)

The kinematic spin vector is defined as (for transitions between baryon states
with $|S^{\prime}-S|=0,1$)

$\displaystyle
S^{z}(S^{\prime}_{3},S_{3})=\sqrt{S(S+1)}\,\sqrt{\frac{2S+1}{2S^{\prime}+1}}\,\langle
SS_{3},10|S^{\prime}S^{\prime}_{3}\rangle,$ (40)

where $\langle SS_{3},10|S^{\prime}S^{\prime}_{3}\rangle$ are the SU(2)
Clebsch-Gordan coefficients. Equations (36) and (LABEL:J_momentum) provide a
general definition of the QCD AM associated with baryon transitions. Its
implications should be explored in further research.

The transition AM definition Eq. (36) has been used to study the AM in
$N\rightarrow\Delta$ transitions Kim:2023xvw . Because of the isospin
difference between the states, these transition probe the isovector component
of the quark AM ($u-d$). The $N\rightarrow\Delta$ transition matrix elements
of the EMT can be analyzed using the $1/N_{c}$ of QCD, where $N$ and $\Delta$
states appear in the same representation of the emergent SU(2$N_{f}$) spin-
flavor symmetry and transitions between them are connected by the symmetry
(see Sec. 5.2). The main results of this analysis are:

(i) The isovector AM in the nucleon is leading in $1/N_{c}$; the isoscalar is
subleading:

$\displaystyle J^{u+d}_{N\to N}=\mathcal{O}(N_{c}^{0}),\hskip
10.00002ptJ^{u-d}_{p\to p}=\mathcal{O}(N_{c}).$ (41)

This explains the observed large flavor asymmetry of the quark AM in the
nucleon. Note that this scaling is consistent with that of the quark spin
contribution to the nucleon spin as given by the axial coupling,
$g_{A}^{u+d}=\mathcal{O}(N_{c}^{0})$ and $g_{A}^{u-d}=\mathcal{O}(N_{c}^{1})$.

(ii) The isoscalar component of the AM in the nucleon and $\Delta$ are related
by

$\displaystyle J^{{u+d}}_{N\to N}=J^{{u+d}}_{\Delta\to\Delta}.$ (42)

This provides insight into the spin structure of $\Delta$ resonance. Note that
this relation is consistent with the spin sum rule for the $\Delta$ state.

(iii) The isovector AM in the nucleon, the AM in the $N\rightarrow\Delta$
transitions, and the isovector AM in the $\Delta$ are related by

$\displaystyle J^{u-d}_{p\to
p}=\frac{1}{\sqrt{2}}J^{{u-d}}_{p\to\Delta^{+}}=5J^{u-d}_{\Delta^{+}\to\Delta^{+}}.$
(43)

This suggests that the $N\rightarrow\Delta$ transition AM is large and
provides a way to probe the isovector nucleon AM with $N\rightarrow\Delta$
transition measurements.

Lattice QCD | $J^{u+d}_{p\to p}$ | $J^{u+d}_{\Delta^{+}\to\Delta^{+}}$ | $J^{u-d}_{p\to p}$ | $J^{u-d}_{p\to\Delta^{+}}$ | $J^{u-d}_{\Delta^{+}\to\Delta^{+}}$
---|---|---|---|---|---
Gockeler:2003jfa  $\mu^{2}=4\,\mathrm{GeV}^{2}$ | $0.33^{\ast}$ | $0.33$ | $0.41^{\ast}$ | $0.58$ | $0.08$
LHPC:2007blg  $\mu^{2}=4\,\mathrm{GeV}^{2}$ | $0.21^{\ast}$ | $0.21$ | $0.22^{\ast}$ | $0.30$ | $0.04$
LHPC:2010jcs  $\mu^{2}=4\,\mathrm{GeV}^{2}$ | $0.24^{\ast}$ | $0.24$ | $0.23^{\ast}$ | $0.33$ | $0.05$
Bali:2018zgl  $\mu^{2}=1\,\mathrm{GeV}^{2}$ | $-$ | $-$ | $0.23^{\ast}$ | $0.33$ | $0.05$
Alexandrou:2019ali  $\mu^{2}=4\,\mathrm{GeV}^{2}$ | $-$ | $-$ | $0.17^{\ast}$ | $0.24$ | $0.03$
Table 2: Estimates of the isoscalar and the isovector AM for $p\to p$,
$p\to\Delta^{+}$ and $\Delta^{+}\to\Delta^{+}$ obtained from lattice QCD data
on $J^{u+d}_{p\to p}$ and $J^{u-d}_{p\to p}$ and the relations provided by the
leading-order $1/N_{c}$ expansion. Input values are marked by an asterisk ∗.

Furthermore, within the $1/N_{c}$ expansion one can obtain numerical estimates
of the $N\rightarrow\Delta$ transition AM, using Eq. (43) and lattice QCD
results for the isovector EMT $N\rightarrow N$ matrix elements Kim:2023xvw .
Results are summarized in Table 2. The dominance of the isovector component of
the quark angular momentum in $N\rightarrow N$ transitions was also confirmed
by a calculation using the light-cone wave function of the chiral soliton in
the large-$N_{c}$ limit Kim:2023yhp .

The $N\rightarrow\Delta$ transition matrix element of the EMT can be connected
with the $N\rightarrow\Delta$ transition GPDs of the parity-even quark
operator, Eq. (1) Goeke:2001tz ; Kim:2023xvw . In the context of the $1/N_{c}$
expansion, the dominant $N\rightarrow\Delta$ transition GPD is the GPD
$G_{1}$111In the $h_{M,E,C,4}$ convention employed in Ref. Semenov-Tian-
Shansky:2023bsy  the dominant in $1/N_{c}$ unpolarized $N\to\Delta$ GPD is
$h_{M}$, which in leading order in $1/N_{c}$ corresponds to
$\frac{4}{3}G_{1}$., which scales as

$\displaystyle G_{1}(x,\xi,t)$ $\displaystyle\sim
N_{c}^{3}\times\mathrm{function}(N_{c}x,N_{c}\xi,t).$ (44)

In leading order of $1/N_{c}$, one obtains

$\displaystyle\int^{1}_{-1}dx\,x\,G_{1}(x,\xi,0)$
$\displaystyle=\frac{8}{3}J^{u-d}_{p\to\Delta^{+}}.$ (45)

The $N_{c}$ scaling of the second moment implied by Eq. (44) agrees with Eq.
(41).

###  4.3 Mechanical properties of baryon resonances

A further major application of $N\rightarrow$ GPDs is the study of the so-
called “mechanical properties” of the nucleon. The components of the EMT with
3-dimensional tensor and scalar character describe the distributions of
mass/energy and forces/pressure in a mechanical system Polyakov:2002yz ;
Polyakov:2018zvc ; Lorce:2018egm ; Burkert:2023wzr . The nucleon matrix
elements of these EMT components can again be connected with the $x$-weighted
integrals of the GPDs (second moments). The second moment of the $N\rightarrow
N$ GPD $H$ contains two EMT form factors222Because the EMT describes the
coupling of matter to gravity, these form factors are also known as the
“gravitational form factors” of the nucleon.

$\int dx~{}x~{}H(x,\xi,t)=M_{2}(t)+\frac{4}{5}\xi^{2}d_{1}(t).$ (46)

$M_{2}(t)$ describes the distribution of mass/energy in the nucleon;
$d_{1}(t)$ describes the distribution of shear forces and pressure. An
interpretation of these structures has been developed in terms of
3-dimensional spatial densities in the Breit frame ($\Delta^{0}=0$).
Determining these form factors and spatial distributions using theoretical and
experimental methods data has become a major goal of nucleon structure
physics.

The extension of these concepts to $N\rightarrow N^{\ast}$ transitions would
provide new tools for resonance structure physics. The distributions of
mass/energy and pressure/forces would allow for new assessments of the size
and spatial structure of resonances, correlated with their mass and spin. One
present limitation is that the interpretation of the EMT form factors in terms
of spatial densities uses 3D densities in the Breit frame and cannot readily
be generalized to transitions between states with different masses. This could
be overcome by using the framework of transverse densities at fixed light-
front time, as in the definition of the transition AM, Eq. (36) Kim:2023xvw .

In the large-$N_{c}$ limit the baryon masses are $\mathcal{O}(N_{c})$ and the
baryon motion becomes effectively non-relativistic, so that the baryon mass
does not play a role in transition matrix elements. In this situation the
$N\rightarrow N^{\ast}$ transition EMT form factors (and even the
$N^{\ast}\rightarrow N^{\ast}$ ones) permit an interpretation in terms of
Breit frame densities in the same way as the $N^{\ast}\rightarrow N$ form
factors.

The $d_{1}(t)$ form factor (the first coefficient of the Gegenbauer expansion
of the so-called D-term Polyakov:1999gs ) is of special interest because it
makes a determining contribution into the subtraction constant of the
fixed-$t$ dispersion relation for the DVCS amplitude within the leading-twist
approximation Goeke:2001tz . As such it can be extracted from the experimental
data on the beam spin asymmetry (imaginary part of DVCS amplitude) and cross
section (real part) without need for a GPD parametrization. Several
extractions have been performed using existing experimental data, in
particular from JLab CLAS 6 GeV Burkert:2018bqq ; Kumericki:2019 ;
Moutarde:2019tqa . It would be interesting to explore whether the DVCS
dispersion relations could be generalized to $N\rightarrow\Delta$ transitions,
and whether there would be a connection with an $N\rightarrow\Delta$ D-term.
It should be noted that the amplitude of $N\rightarrow\Delta$ DVCS is
suppressed in the high-energy limit (non-diffractive process, quantum number
exchange), so that the need for subtraction in a dispersion relation is not
apparent. It is also known that the isovector D-term is suppressed in the
$1/N_{c}$ expansion Goeke:2001tz . In $N\rightarrow N^{\ast}$ transitions to
states with the same quantum numbers as the nucleon, the high-energy behavior
of DVCS is the same as in $N\rightarrow N$, and a transition D-term might
appear in the dispersion relation.

###  4.4 Tensor charge and anomalous magnetic moment

For the GPDs of the ground state nucleon, the first moments of the
transversity GPDs $\bar{E}_{T}$ and $H_{T}$ can be related Burkardt:2005hp ;
Burkardt:2006ev ; Ahmad:2008hp  to the anomalous tensor magnetic moment
$k_{T}$ of the nucleon for the single quark flavors ($u$, $d$):

$\displaystyle k_{T}^{u,d}=\int dx\bar{E}^{u,d}_{T}(x,\xi=0,t=0)$ (47)

and to the tensor charge $\delta_{T}$:

$\displaystyle\delta_{T}^{u,d}=\int dxH^{u,d}_{T}(x,\xi=0,t=0).$ (48)

Since relations between the transversity transition GPDs and the ground state
transition GPDs can be found in the large $N_{c}$ limit Kroll:2022roq , it is
assumed that similar relations exist for the transversity transition GPDs and
allow an extraction of the equivalent properties for baryon resonances.

##  5 Theoretical methods for transition GPDs

###  5.1 Chiral dynamics in $N\rightarrow\pi N$ transitions

The simplest case of transition GPDs are transitions between $N$ and $\pi N$
states in the near-threshold region. If the center-of-mass momentum of the
$\pi N$ system in the final state is “small” of the order $k\sim m_{\pi}$, the
emission of the pion is governed by chiral dynamics and can be computed from
first principles (so-called soft-pion theorems) (see Fig. 4). Chiral dynamics
is the effective long-range dynamics emerging from the spontaneous breaking of
chiral symmetry in QCD, where the pion appears as a Goldstone boson, the form
of its interaction with other hadrons is dictated by the chiral symmetry. In
particular, chiral symmetry implies that the QCD isovector axial current
$J_{5V}^{\mu}$ produces low-momentum pions out of the vacuum and can be used
as an interpolating QCD operator for pion states. In nucleon transition matrix
elements, this allows one to derive relations between the $N\rightarrow\pi N$
transition matrix elements of a QCD operator $\mathcal{O}$ and the
$N\rightarrow N$ matrix elements of the commutator
$\mathcal{O}^{\prime}=[\mathcal{O},J_{5V}^{\mu}]$; schematically

$\displaystyle\langle\pi N|\mathcal{O}|N\rangle\;\leftrightarrow\;\langle
N|\mathcal{O}^{\prime}|N\rangle,\hskip
10.00002pt\mathcal{O}^{\prime}=[\mathcal{O},J_{5V}^{\mu}]$ (49)

These relations become predictive if the commutator can be reduced to an
operator whose matrix elements are known using the field equations.
Application of this method to near-threshold $N\rightarrow\pi N$ transitions
is demonstrated in Ref. Pobylitsa:2001cz . The $N\rightarrow\pi N$ transition
GPDs measured in DVCS and meson production are considered in Refs. Chen:2003jm
; Guichon:2003ah ; Birse:2005hh . Higher-order corrections in the pion
momentum can be computed using chiral EFT Kivel:2004bb . Applications to hard
exclusive pion production with $N\rightarrow\pi N$ transitions are discussed
in Ref. Polyakov:2006dd . The soft-pion theorems are practically applicable in
S-wave of the $\pi N$ system. In the P-wave, the strong $\Delta$ resonance
limits the useful range of the chiral expansion. In this channel other methods
can be applied, such as extensions of chiral EFT including $\Delta$ degrees of
freedom Alharazin:2023zzc ; Alharazin:2023uhr  or the $1/N_{c}$ expansion (see
Sec. 5.2).

Figure 4: $N\rightarrow\pi N$ transition GPD in the chiral regime

###  5.2 $1/N_{c}$ expansion and $N\rightarrow\Delta$ transitions

The $1/N_{c}$ expansion is a general method for analyzing QCD in the
nonperturbative domain and connecting it with meson and baryon properties. The
large-$N_{c}$ limit corresponds to a semi-classical limit of the quantum field
theory, in which dynamics simplifies in characteristic ways (“string-like”)
yet remains strongly coupled and generates meson/baryon spectra close to those
observed in experiment tHooft:1973alw ; Witten:1979kh . In the baryon sector,
a dynamical spin-flavor symmetry emerges, whose ground-state representation
consists of states with equal spin and isospin, $S=I$ Dashen:1993jt ;
Dashen:1994qi ; Jenkins:1998wy . The $N$ and $\Delta$ appear as the states
with $S=I=1/2$ and $3/2$ in this representation. Matrix elements of QCD
operators between $N$ and $\Delta$ states are therefore constrained by the
spin-flavor symmetry (see Fig. 5). The transformation properties of QCD
operators under the emergent spin-flavor symmetry can be inferred from model-
independent considerations and are determined by their spin-flavor quantum
numbers. The transition matrix between ground-state baryons are then connected
by group-theoretical coefficients, similar to the matrix elements of spherical
tensor operators between angular momentum states. Schematically,

$\displaystyle\langle
N(S_{3}^{\prime},I_{3}^{\prime})|\mathcal{O}|N(S_{3},I_{3})\rangle=C(\textstyle{\frac{1}{2}}\textstyle{\frac{1}{2}}|S_{3}^{\prime}I_{3}^{\prime},S_{3}I_{3})\langle\mathcal{O}\rangle,$
(50)
$\displaystyle\langle\Delta(S_{3}^{\prime},I_{3}^{\prime})|\mathcal{O}|N(S_{3},I_{3})\rangle=C(\textstyle{\frac{3}{2}}\textstyle{\frac{1}{2}}|S_{3}^{\prime}I_{3}^{\prime},S_{3}I_{3})\langle\mathcal{O}\rangle,$
(51)
$\displaystyle\langle\Delta(S_{3}^{\prime},I_{3}^{\prime})|\mathcal{O}|\Delta(S_{3},I_{3})\rangle=C(\textstyle{\frac{3}{2}}\textstyle{\frac{3}{2}}|S_{3}^{\prime}I_{3}^{\prime},S_{3}I_{3})\langle\mathcal{O}\rangle,$
(52)

where $S_{3}/I_{3}$ denote the spin/isospin projections, $C$ are group-
theoretical coefficients given by products of SU(2) vector coupling
coefficients, and $\langle\mathcal{O}\rangle$ is the reduced matrix element
containing the dynamical information.

Figure 5: $N\rightarrow N$ and $N\rightarrow\Delta$ transition GPDs in the
large-$N_{c}$ limit

The relations Eqs. (50)–(52) can be used to derive the $N\rightarrow\Delta$
and $\Delta\rightarrow\Delta$ transition matrix elements of QCD operators from
the $N\rightarrow N$ matrix elements if the latter are known (measured,
calculated in Lattice QCD). The method has extensively been applied to matrix
elements of local QCD operators such as vector and axial vector currents, with
very successful results Buchmann:2002mm ; Jenkins:2002rj ; Pascalutsa:2007wz .
It can also be applied to matrix elements of nonlocal light-ray operators, to
predict the $N\rightarrow\Delta$ transition GPDs in terms of the $N\rightarrow
N$ GPDs. Large-$N_{c}$ relations for the chiral-even transition GPDs are
discussed in Ref. Frankfurt:1999xe ; Goeke:2001tz  and applied to
$N\rightarrow\Delta$ DVCS processes. The extension to chiral-odd GPDs is
discussed in Refs. Schweitzer:2016jmd ; Kroll:2022roq  and applied to hard
exclusive pion production with $N\rightarrow\Delta$ transitions.

The $1/N_{c}$ expansion has other interesting applications to GPDs besides
connecting $N$ and $\Delta$ transition matrix elements. It predicts the
$N_{c}$-scaling of the reduced matrix element $\langle\mathcal{O}\rangle$ in
dependence on the spin-flavor quantum numbers of the operator and allows one
to classify the matrix elements in leading and subleading ones. It also
permits the calculation of $1/N_{c}$ corrections to the leading-order
relations of Eqs. (50)–(52). The $1/N_{c}$ expansion can also be combined with
chiral dynamics to enable quantitative predictions. One formulation is given
by effective field theories (EFTs) based on a combined chiral and $1/N_{c}$
expansion CalleCordon:2012xz ; Fernando:2019upo . Another formulation is the
chiral quark-soliton model of baryons based on the dynamics of chiral
constituent quarks Diakonov:1987ty ; this framework has been applied
extensively to study GPDs and transition GPDs Petrov:1998kf ; Goeke:2001tz .

###  5.3 Light-cone sum rules

Helicity-conserving elastic and transition form factors at very large momentum
transfers can be calculated in QCD in terms of the light-cone distribution
amplitudes (LCDAs) that describe momentum fraction distributions of valence
quarks at small transverse distances. The problem is that the onset of the
pQCD regime is postponed to very large $Q^{2}$ because the factorizable
contribution involves a small factor $\sim(\alpha_{s}/(2\pi))^{2}$ and has to
win over nonperturbative “soft” or “end-point” contributions that are
suppressed by an extra power of $Q^{2}$ but do not involve small coefficients.
The light-cone sum rule (LCSR) technique Balitsky:1989ry  makes use of quark-
hadron duality and dispersion relations to calculate such non-factorizable
“soft” contributions in terms of the same LCDAs that enter the pQCD
calculation and avoid double counting. Thus, the LCSRs provide one with the
most direct relation between the hadron form factors and LCDAs that is
available at present, with no other nonperturbative parameters. This approach
was initially developed to describe the weak decay $\Sigma\to p\gamma$
Balitsky:1989ry . It is very flexible, and has been applied to light- and
heavy-meson decays (see, e.g., Khodjamirian:2023wol  for a recent review), to
elastic and transition baryon form factors Braun:2005be ; Braun:2006hz ;
Anikin:2013aka ; Anikin:2015ita ; Anikin:2016teg  and pion electroproduction
near threshold Braun:2007pz . The current state-of-the art LCSRs have NLO
accuracy. The LCSR technique proves to be very powerful, especially if used in
combination with lattice QCD inputs on the LCDAs, which are gradually becoming
more and more accurate, see e.g. RQCD:2019hps .

###  5.4 Lattice QCD calculations

Aspects of GPDs have been investigated in lattice QCD for decades. Most of
these calculations are based on the Mellin moments of GPDs, such as the
electromagnetic and axial form factors of the proton, mainly because these are
the most reliably extracted. Lattice calculations of these quantities have
advanced significantly and now take into account sources of systematic
uncertainties, such as physical pion mass calculations, excited states,
infinite volume, and extrapolation to the continuum limit. Another crucial
improvement is the inclusion of disconnected-diagram contributions that enable
the extraction of the individual flavor quantities. Despite notable
advancements, theoretical and computational constraints hinder the calculation
of higher Mellin moments of GPDs, making the reconstruction of GPDs inherently
challenging. Rather than relying solely on Mellin moments, alternative methods
for determining the $x$-dependence of various distribution functions have
emerged over time Liu:1993cv ; Ji:2013dva ; Ji:2014gla ; Detmold:2005gg ;
Braun:2007wv ; Radyushkin:2017cyf ; Orginos:2017kos ; Chambers:2017dov ;
Ma:2017pxb . In the past decade, significant progress has been made in
extending these methods to calculate $x$-dependent GPDs Alexandrou:2020zbe ;
Lin:2020rxa ; Alexandrou:2021bbo ; CSSMQCDSFUKQCD:2021lkf ; Lin:2021brq ;
Bhattacharya:2022aob ; Bhattacharya:2023ays ; Bhattacharya:2023nmv ;
Bhattacharya:2023jsc . Comprehensive reviews of recent progress in the field
can be found in Refs Cichy:2018mum ; Ji:2020ect ; Constantinou:2020pek ;
Cichy:2021lih . This progress can also be extended to the calculation of
transition GPDs, which offer another avenue for probing hadronic structure.
There are only a few lattice QCD calculations limited to the transition form
factors (see, e.g., Refs. Leinweber:1992pv ; Alexandrou:2010uk ;
Alexandrou:2018jbt ; ExtendedTwistedMass:2023hin ).

##  6 Transition processes: electron scattering

Similar to the GPDs of the ground-state nucleon, transition GPDs are
accessible in different lepton scattering reactions as well as hadronic
reactions. In leptoproduction experiments, transition GPDs can be accessed by
the $N\to N^{*}$ deeply virtual Compton scattering (DVCS) and the $N\to N^{*}$
deeply virtual meson production (DVMP) processes.

###  6.1 $N\to\Delta,\,N^{*}$ DVCS and transition GPDs

The non-diagonal DVCS reaction with a nucleon-to-resonance-transition can be
accessed through the study of the hard exclusive process

$\displaystyle e^{-}(k)+N(p)$ $\displaystyle\rightarrow$ $\displaystyle
e^{-}\left(k^{\prime}\right)+\gamma\left(q^{\prime}\right)+R\left(p_{R}\right)$
$\displaystyle\rightarrow$ $\displaystyle
e^{-}\left(k^{\prime}\right)+\gamma\left(q^{\prime}\right)+\pi\left(p_{\pi}\right)+N\left(p^{\prime}\right),$

in which the produced nucleon resonance $R$ further decays into a $\pi N$
system. The first studies of $N\to\Delta(1232)$ non-diagonal DVCS were
reported in Refs. Guichon:2003ah ; Guidal:2003ji  and a generalization for the
second nucleon resonance region was recently considered in Ref. Semenov-Tian-
Shansky:2023bsy .

The process (LABEL:ND_hard_electroproducton_of_gamma), see Fig. 6 for the
specification of kinematic quantities, is described in terms of $8$
kinematical invariants, which can be chosen as

$\displaystyle s=(k+p)^{2},\quad Q^{2}=-q^{2}\equiv(k^{\prime}-k)^{2},\quad
x_{B}=\frac{Q^{2}}{2p\cdot q},$ (54) $\displaystyle
t=\Delta^{2}\equiv(p_{R}-p)^{2},\quad M_{\pi
N}^{2}=p_{R}^{2}\equiv(p^{\prime}+p_{\pi})^{2},$
$\displaystyle\Phi,\quad\theta_{\pi}^{*},\quad\phi_{\pi}^{*}.$

Here $\Phi$ is the angle between the leptonic plane and the production plane
spanned by the vectors $\vec{q}$ and $\vec{q}^{\prime}$ defined in the
$\gamma^{*}(q)N(p)$ Center-of-Mass System; and the angles $\theta_{\pi}^{*}$,
$\phi_{\pi}^{*}$ denote, respectively, the polar and azimuthal angles of the
rest frame of the $\pi(p_{\pi})N(p^{\prime})$ system.

Figure 6: Planes defining the scattering angles that characterize the
$e^{-}N\to e^{-}\gamma\pi N$ process. The angles $\Phi$ and $\phi^{*}_{\pi}$
are defined with respect to the $xz$-plane, which is the scattering plane of
the virtual photons with four-momenta $q$ and $q^{\prime}$. This figure is
taken from Ref. Semenov-Tian-Shansky:2023bsy .

Instead of the pion polar angle $\theta^{*}_{\pi}$, it is sometimes
instructive to consider the invariant mass of the $\pi\gamma$ system
$M_{\pi\gamma}^{2}=(p_{\pi}+q^{\prime})^{2}$ as independent kinematical
variable.

The sevenfold differential unpolarized cross section for the
$e^{-}N\rightarrow e^{-}\gamma\pi N$ reaction is expressed as

$\displaystyle\frac{d\sigma}{dQ^{2}dx_{B}dtd\Phi dM_{\pi
N}^{2}d\Omega_{\pi}^{*}}=\frac{1}{(2\pi)^{7}}\frac{x_{B}y^{2}}{32Q^{4}\sqrt{1+\frac{4M_{N}^{2}x_{B}^{2}}{Q^{2}}}}$
$\displaystyle\quad\times\frac{\left|\vec{p}_{\pi}^{*}\right|}{4M_{\pi
N}}\sum_{i}\sum_{f}\left|\mathcal{M}\left(e^{-}N\rightarrow e^{-}\gamma\pi
N\right)\right|^{2},$ (55)

where $y\equiv p\cdot q/p\cdot k$; $d\Omega_{\pi}^{*}\equiv
d\cos\theta_{\pi}^{*}d\phi_{\pi}^{*}$ is the solid angle of the produced pion;
and $\left|\vec{p}_{\pi}^{*}\right|$ is the value of the pion $3$-momentum in
the $\pi N$ rest frame. The squared invariant amplitude
$\left|\mathcal{M}\left(e^{-}N\rightarrow e^{-}\gamma\pi N\right)\right|^{2}$
is averaged (summed) over the initial (final) particle helicities.

Another characteristic observable of the reaction
(LABEL:ND_hard_electroproducton_of_gamma) is the beam-spin asymmetry (BSA)
defined as

$BSA=\frac{d\sigma^{+}-d\sigma^{-}}{d\sigma^{+}+d\sigma^{-}},$ (56)

where $d\sigma^{\pm}$ refer to the polarized cross sections with electron beam
helicity $\pm 1/2$.

Analogously to the usual hard exclusive electroproduction of photons off a
nucleon, the amplitude of the process
(LABEL:ND_hard_electroproducton_of_gamma) obtains contributions from the
Bethe-Heitler (BH) process, with the final-state real photon emitted off the
lepton lines; and the DVCS process, in which the photon is produced from the
hadron side. The BH contribution is a pure QED process, and the corresponding
amplitude can be calculated exactly provided an input on the nucleon-to-
resonance transition electromagnetic form factors.

Similarly to the diagonal DVCS case, for which factorization is proven
Ji:1998xh ; Collins:1998be , we assume the validity of the QCD collinear
factorization theorem for the non-diagonal DVCS in the generalized Bjorken
kinematics (large $Q^{2}$ and $W^{2}=(p+q)^{2}$; fixed $x_{B}$; $-t$ and
$W_{\pi N}^{2}$ of hadronic mass scale: $-t,\,W_{\pi N}^{2}\ll Q^{2}$).
Therefore, to the leading twist-$2$ accuracy, the reaction proceeds through
the reaction mechanism of Fig. 7; and the corresponding amplitude is
parameterized in terms of $N\to R$ transition GPDs.

Figure 7: Handbag mechanism for the $N\to\Delta,\,N^{*}$ DVCS process. The
diagram with the crossed real and virtual photon lines is not shown.

The interference between the BH and the non-diagonal DVCS leads to the
interference term in the cross section. The BH process is usually expected to
dominate the measured cross section for the kinematics accessible with current
lepton scattering experiments and has to be subtracted in order to extract the
non-diagonal DVCS signal.

As a first example we briefly review the description of the $e^{-}N\to
e^{-}\gamma\Delta(1232)\to e^{-}\gamma\pi N$ process. The corresponding
invariant amplitude entering the cross section of Eq. (55) is given by:

$\displaystyle\mathcal{M}(e^{-}N\to e^{-}\gamma\Delta\to e^{-}\gamma\pi
N)=\bar{C}_{iso}\frac{f_{\pi N\Delta}}{m_{\pi}}\left(-p_{\pi}\right)^{\alpha}$
$\displaystyle\times\,\bar{N}\left(p^{\prime},s^{\prime}_{N}\right)\frac{iP^{(3/2)}_{\alpha\beta}(p_{R})}{M_{\pi
N}^{2}-M_{R}^{2}+iM_{R}\Gamma_{\Delta}(M_{\pi N})}$
$\displaystyle\times\,\mathcal{M}_{R}^{\beta}(e^{-}N\to e^{-}\gamma\Delta),$
(57)

where $\mathcal{M}_{R}^{\beta}(e^{-}N\to e^{-}\gamma\Delta)$ is defined as the
sum of the BH and DVCS amplitudes; $f_{\pi N\Delta}\simeq 2.08$ is the $\pi
N\Delta$ coupling constant of the effective $\pi N\Delta$ Lagrangian (see Eq.
(45) of Ref. Semenov-Tian-Shansky:2023bsy ) determined from the $\Delta\to\pi
N$ decay width. The isospin factor $\bar{C}_{iso}$ takes on the values
$\bar{C}_{iso}=\sqrt{\frac{2}{3}}$ for $\Delta^{+}\to\pi^{0}p$; and
$\bar{C}_{iso}=-\sqrt{\frac{1}{3}}$ for $\Delta^{+}\to\pi^{+}n$. Furthermore,
in Eq. (57), $P^{(3/2)}_{\alpha\beta}(p_{R})$ denotes the spin-$\frac{3}{2}$
projector, cf. Ref. Pascalutsa:2006up .

The amplitude (57) results in the specific angular distribution for the decay
pion in the resonance rest frame:

$\displaystyle\mathcal{M}(e^{-}N\to e^{-}\gamma\Delta\to e^{-}\gamma\pi N)$
$\displaystyle=\bar{C}_{iso}\frac{f_{\pi
N\Delta}}{m_{\pi}}\sqrt{\frac{4\pi}{3}}\,\frac{i\,|\vec{p}^{\,*}_{\pi}|\,\left[(M_{\pi
N}+M_{N})^{2}-m_{\pi}^{2}\right]^{1/2}}{M_{\pi
N}^{2}-M_{R}^{2}+iM_{R}\Gamma_{\Delta}(M_{\pi N})}$
$\displaystyle\times\sum_{\lambda_{R}}\mathcal{M}(e^{-}N\to
e^{-}\gamma\,\Delta(M_{\pi N},\lambda_{R}))$
$\displaystyle\times\sum_{\lambda^{\prime}}\,\langle\,1\lambda^{\prime},\frac{1}{2}s_{N}^{\prime}\,|\,\frac{3}{2}s_{R}\,\rangle\,Y_{1\lambda^{\prime}}(\Omega^{\ast}_{\pi}),$
(58)

where $Y_{lm_{l}}(\Omega^{\ast}_{\pi})$ denote the spherical harmonic
functions and $\langle\,ss_{z},lm_{l}|JM\rangle$ are the Clebsch-Gordan
coefficients for the $\Delta\to\pi N$ decay.

Eq. (58) is instrumental in computing the resonance angular distributions of
the cross-section. It is particularly instructive to present the cross section
(55) integrated over the complete pion solid angle or integrated over the
invariant mass of the $\pi N$ system $M_{\pi N}^{2}$. It is worth mentioning
that, assuming the Breit-Wigner form of the spectral function
$\Gamma_{\Delta}(M_{\pi N})$ together with the narrow resonance limit
$\Gamma_{R}\ll M_{R}$, the unpolarized cross section (55) integrated both in
$\Omega_{\pi}^{*}$ and $M_{\pi N}^{2}$

$\displaystyle\int dM^{2}_{\pi N}\int
d\Omega^{\ast}_{\pi}\frac{d\sigma}{dQ^{2}dx_{B}dtd\Phi dM^{2}_{\pi
N}d\Omega^{*}_{\pi}}$
$\displaystyle\approx\frac{1}{(2\pi)^{4}}\frac{x_{B}\,y^{2}}{32\,Q^{4}\sqrt{1+\frac{4M_{N}^{2}x_{B}^{2}}{Q^{2}}}}$
$\displaystyle\times\overline{\sum_{i}}\sum_{f}\left|\mathcal{M}(e^{-}N\to
e^{-}\gamma\Delta(M_{R},\lambda_{R}))\right|^{2},$ (59)

where the sum runs over the final helicities of $e^{-}$, $\gamma$, and
$\Delta$, turns out to be fully consistent with the result for a stable
particle Goeke:2001tz .

A very instructive piece of information can be revealed through working out
the pion polar angular distribution integrated over the azimuthal angle
$\phi_{\pi}^{*}$ within the narrow resonance approximation:

$\displaystyle\int dM^{2}_{\pi N}\frac{d\sigma}{dQ^{2}dx_{B}dtd\Phi
dM^{2}_{\pi N}d\cos\theta^{*}_{\pi}}$
$\displaystyle\approx\frac{1}{(2\pi)^{4}}\frac{x_{B}\,y^{2}}{32\,Q^{4}\sqrt{1+\frac{4M_{N}^{2}x_{B}^{2}}{Q^{2}}}}\,\,\overline{\sum_{e,N}}\sum_{e,\gamma}(\bar{C}_{iso})^{2}$
$\displaystyle\times\bigg{\\{}\frac{1}{4}\left(1+3\cos^{2}\theta^{\ast}_{\pi}\right)\sum_{\lambda_{R}=\pm
1/2}\left|\mathcal{M}(e^{-}N\to e^{-}\gamma\Delta\right|^{2}$
$\displaystyle\quad+\frac{3}{4}\sin^{2}\theta^{\ast}_{\pi}\sum_{\lambda_{R}=\pm
3/2}\left|\mathcal{M}(e^{-}N\to e^{-}\gamma\Delta\right|^{2}\bigg{\\}},$ (60)

resulting in specific polar angular distributions for the $\Delta$ helicity
states $\lambda_{R}=\pm 1/2$ and $\lambda_{R}=\pm 3/2$. This gives rise to a
peculiar “autopolarization” effect providing access to information on
different polarization states of the produced $\Delta$-resonance. For a
sufficiently detailed experimental angular resolution, this can open
interesting options for a precise partial wave analysis of $\Delta$-resonance
production.

The cross-section estimates of $N\to\Delta(1232)$ in Refs. Guichon:2003ah ;
Guidal:2003ji  were based on a model for the $N\to\Delta$ transition GPDs
dominant in the large-$N_{c}$ limit within the framework developed in Ref.
Frankfurt:1999xe . The three dominant $N^{p}\to\Delta^{+}$ transition GPDs
$h_{M}$, $C_{1}$ and $C_{2}$ are related to the following combinations of
nucleon isovector GPDs Frankfurt:1999xe ; Goeke:2001tz :

$\displaystyle
h_{M}\left(x,\xi,\Delta^{2}\right)=\sqrt{2}\left[E^{u}\left(x,\xi,\Delta^{2}\right)-E^{d}\left(x,\xi,\Delta^{2}\right)\right];$
(61) $\displaystyle
C_{1}\left(x,\xi,\Delta^{2}\right)=\sqrt{3}\left[\tilde{H}^{u}\left(x,\xi,\Delta^{2}\right)-\tilde{H}^{d}\left(x,\xi,\Delta^{2}\right)\right];$
$\displaystyle
C_{2}\left(x,\xi,\Delta^{2}\right)=\frac{\sqrt{3}}{4}\left[\tilde{E}^{u}\left(x,\xi,\Delta^{2}\right)-\tilde{E}^{d}\left(x,\xi,\Delta^{2}\right)\right].$

The control of the accuracy of this approximation is performed through a
comparison of the first moments of (61) to values of the corresponding form
factors, see discussion in Ref. Pascalutsa:2006up .

In Ref. Semenov-Tian-Shansky:2023bsy  this model was employed to provide
estimates of the cross section for the kinematical setup accessible with
CLAS12@JLab. The phenomenological input included the nucleon GPD model of Sec.
2.7.2 of Ref. Pascalutsa:2006up . The GPD $\tilde{E}$ is determined by the
pion pole contribution, see Sec. 2.4.2 of Ref. Goeke:2001tz . For the relevant
transition electromagnetic form factors, the results of the MAID2007 analysis
reported in Drechsel:2007if ; Tiator:2011pw  were employed.

Figure 8: Dependence on the invariant mass of the $\pi^{+}n$ system ($M_{\pi
N}$) of the $e^{-}p\to e^{-}\gamma\Delta(1232)\to e^{-}\gamma\pi^{+}n$ cross
section (upper panels) integrated over the pion solid angle, with the cut
$M_{\pi\gamma}>1$ GeV, and corresponding beam-spin asymmetry (lower panels),
for three kinematical setups accessible with CLAS12@JLab. Blue dashed-dotted
curves: $p\to\Delta(1232)$ BH process; red dashed curves: $p\to\Delta(1232)$
DVCS process; black solid curves: BH + DVCS processes. The magenta dotted
curves show the $\pi^{0}$-pole contribution to the $p\to\Delta(1232)$ DVCS
process separately. This figure is taken from Ref. Semenov-Tian-
Shansky:2023bsy .

In the upper panels of Fig. 8 we present the dependence of the cross-section
(55) $e^{-}p\to e^{-}\gamma\Delta(1232)\to e^{-}\gamma\pi^{+}n$ integrated
over the pion solid angle $d\Omega_{\pi}^{*}$ on the invariant mass of the
$\pi^{+}n$ system ($M_{\pi N}$). The kinematic cut $M_{\pi\gamma}>1$ GeV is
imposed to reduce the resonating background from the $\gamma\pi$ subsystem
arising from the $e^{-}p\to e^{-}\rho^{+}n\to e^{-}\gamma\pi^{+}n$ process.
The angle between the leptonic and production planes $\Phi$ is set to
$90^{\circ}$ to maximize the BSA. We note that the non-diagonal $N\to\pi N$
DVCS process in the $\Delta(1232)$ region is dominated by the $\pi^{0}$-pole
contribution into the $N\to\Delta$ polarized transition GPD $C_{2}$. It is
interesting that in the $\Delta(1232)$ resonance region, for $Q^{2}=2.3$ GeV2,
the model estimates show the change from being dominated by the BH
contribution for $-t=0.5$ GeV2 to being DVCS-dominated for $-t=1.0$ GeV2. Such
behavior is unlike the DVCS process on a nucleon in similar valence region
kinematics, for which the BH dominates in both cases. This can, probably, be
attributed to an effect of kinematic power correction proportional to
$t/Q^{2}$ and $M_{N}^{2}/Q^{2}$ Braun:2014sta  that stays large for
$Q^{2}=2.3$ GeV2. For larger $Q^{2}=5$ GeV2 the dominance of BH over the DVCS
seems to be restored. The lower panels of Fig. 8 present the corresponding BSA
(56), as a function of $M_{\pi N}$. It originates from the BH and DVCS
interference, and in the lower range of $-t$ is estimated to be in the range
of $10\%$ with a tendency to slightly decrease with growth of $-t$.

Ref. Semenov-Tian-Shansky:2023bsy  also presented a generalization of the
formalism for the second resonance region, including the contribution of
$P_{11}(1440)$, $D_{13}(1520)$ and $S_{11}(1535)$ resonances with
isospin-$\frac{1}{2}$. This required the construction of phenomenological
models for the corresponding twist-$2$ unpolarized and polarized transition
GPDs, see Sec. 3.2: ($2$ of each for the spin-$\frac{1}{2}$ resonances and $4$
of each for the spin-$\frac{3}{2}$ resonance). Let us briefly summarize the
key ingredients and the phenomenological input.

  * •

The unpolarized transition GPDs were constructed relying on the constraints
for the corresponding first Mellin moments in terms of the transition
electromagnetic form factors $F_{1,2}^{NP_{11}}$, $F_{1,2,3}^{ND_{13}}$, and
$F_{1,2}^{NS_{11}}$ available from the MAID2007 and MAID2008 analyses detailed
in Ref. Tiator:2011pw . The $x$ and $\xi$ dependence of GPDs was provided by
two types of models:

    * –

I: $\xi$-independent valence PDF type parametrization;

    * –

II: the Radyushkin double distribution Ansatz with a usual $b$-dependent
profile with $b=1$ and the same valence PDF used as input.

In fact, model I turns out to be the $b=\infty$ limit of model II. The
comparison of the results of the two models allows us to roughly quantify the
sensitivity of the non-diagonal DVCS observables to the quark momentum
fraction and skewness dependence of GPDs.

  * •

The polarized transition GPDs $\tilde{H}_{1}^{pP_{11}}$,
$\tilde{H}_{1}^{pD_{13}}$, $\tilde{H}_{1}^{pS_{11}}$ were modeled relying on
the normalization provided by the PCAC through the generalized Goldberger-
Treiman relations for the dominant axial transition FFs and adopting the
dipole form for the $t$-dependence with the $x$ and $\xi$ dependence from the
two types of models (I and II) described above.

  * •

Finally, the GPDs $\tilde{H}_{2}^{pP_{11}}$, $\tilde{H}_{2}^{pD_{13}}$,
$\tilde{H}_{2}^{pS_{11}}$ were modeled assuming the pion pole dominance with
the asymptotic form of the pion distribution amplitude.

  * •

GPDs $H_{4}^{pD_{13}}$, $\tilde{H}_{3}^{pD_{13}}$, $\tilde{H}_{4}^{pD_{13}}$,
for which no phenomenological constraints are available, were neglected.

Fig. 9 presents the $M_{\pi N}$ invariant mass dependence in the first and
second nucleon resonance regions of the $e^{-}p\to e^{-}\gamma\pi^{+}n$ cross
section and corresponding BSA with the cut $M_{\pi\gamma}>1$ GeV to minimize
the possible contamination from the $\rho^{+}$ production channel.

We first notice from the cross section behavior that, with increasing values
of $-t$, the second nucleon resonance region becomes more important relative
to the $\Delta(1232)$ resonance region. This can be explained due to the
$\gamma^{\ast}N\Delta$ transition form factors dropping faster with increasing
$-t$ value in comparison with the corresponding ones for the $D_{13}(1520)$
and $S_{11}(1535)$ resonances. Moreover, in the second nucleon resonance
region, the $D_{13}(1520)$ excitation provides the largest contribution,
followed by the $S_{11}(1535)$ resonance. On the other hand, the contribution
of the $P_{11}(1440)$ excitation to the unpolarized cross section is only very
small. For the BSA, one notices that at $-t=0.5$ GeV2 it reaches a value of
around 10% in the $\Delta$-resonance region and shows a sharp drop in the
second resonance region. Fig. 9 also shows that, with an increasing value of
$-t$, the BSA for the $D_{13}(1520)$ displays a sign change, resulting in a
positive BSA in the second nucleon resonance region at $-t=1$ GeV2.

Figure 9: Dependence on the invariant mass of the $\pi^{+}n$ system ($M_{\pi
N}$) of the $e^{-}p\to e^{-}\gamma R\to e^{-}\gamma\pi^{+}n$ cross section
(upper panels) and corresponding beam-spin asymmetry (lower panels),
integrated over the decay pion solid angle, with the cut $M_{\pi\gamma}>1$
GeV, for three kinematics accessible at CLAS12@JLab. Magenta dotted curves: BH
+ DVCS process for $R=\Delta(1232)$; red dashed curves and red bands: BH +
DVCS process for $R=D_{13}(1520)$; blue dashed-dotted curves and blue bands:
BH + DVCS process for $R=P_{11}(1440)+D_{13}(1520)+S_{11}(1535)$; black solid
curves and grey bands: BH + DVCS process for
$R=\Delta(1232)+P_{11}(1440)+D_{13}(1520)+S_{11}(1535)$. The thin (thick)
curves represent the result of model I (model II) for $x$ and $\xi$ dependence
of $N\to R$ GPDs, see a description in the text. The bands indicate the
corresponding variation due to the modeling. This figure is taken from Ref.
Semenov-Tian-Shansky:2023bsy .

###  6.2 $N\to N^{*}$ DVMP and transition GPDs

While the $N\to N^{*}$ DVCS process is only sensitive to the twist-2
transition GPDs, the $N\to N^{*}$ DVMP process

$\gamma^{*}~{}N\rightarrow\text{meson}~{}N^{*}\rightarrow~{}\text{meson}~{}[~{}N~{}\text{meson}~{}],$
(62)

which is shown schematically in Fig. 10, can be used to access all 16
transition GPDs, including the transversity transition GPDs.

Figure 10: Schematic drawing of the $N\to N^{*}$ DVMP process.

Due to the possible different mesons that can be produced and due to the
different decay channels of the resonance, a whole series of $N\to N^{*}$ DVMP
processes can be studied, for example:

  1. 1.

$\gamma^{*}~{}p~{}~{}\rightarrow~{}~{}\pi^{-}~{}N^{*++}$

  2. 2.

$\gamma^{*}~{}p~{}~{}\rightarrow~{}~{}\pi^{0}~{}N^{*+}$

  3. 3.

$\gamma^{*}~{}p~{}~{}\rightarrow~{}~{}\rho^{0}~{}N^{*+}$

  4. 4.

$\gamma^{*}~{}p~{}~{}\rightarrow~{}~{}\pi^{+}~{}N^{*0}$

  5. 5.

$\gamma^{*}~{}n~{}~{}\rightarrow~{}~{}\pi^{+}~{}N^{*-}$

with a nucleon or a delta resonance $N^{*}$. Furthermore, deeply virtual Kaon
production in relation to hyperons like the $\Lambda$ is possible and enables
the study of transitions related to the strangeness sector. Flavor symmetry
relates the proton-hyperon GPDs to the proton-proton ones Frankfurt:1999fp ,
e.g.

$K_{i}(p-\Lambda)=-\frac{1}{\sqrt{6}}\Big{(}2K_{i}^{u}-K_{i}^{d}-K_{i}^{s}\Big{)}$
(63)

where $K_{i}$ is a GPD. Predictions for the kaon-hyperon channels can be made
analogously to those for pion production Kroll:2019wug .

For DVMP reactions, the cross-section is given by Arens:1996xw :

$\displaystyle\frac{d^{4}\sigma}{dsdQ^{2}d\phi
dt}\propto\bigg{[}\frac{d\sigma_{T}}{dt}+\varepsilon\frac{d\sigma_{L}}{dt}+\varepsilon\cos
2\phi\frac{d\sigma_{TT}}{dt}$
$\displaystyle+\sqrt{2\varepsilon(1+\varepsilon)}\cos\phi\frac{d\sigma_{LT}}{dt}+h\sqrt{2\varepsilon(1-\varepsilon)}\frac{d\sigma_{LT^{\prime}}}{dt}\sin\phi\bigg{]}.$
(64)

Here the terms
$\sigma_{L},\sigma_{T},\sigma_{LT},\sigma_{LT^{\prime}},\sigma_{TT}$
correspond to the interaction of longitudinally ($L$) and transversely ($T$)
polarized virtual photons, $\phi$ is the angle between the leptonic and
hadronic planes, and $\varepsilon$ is the polarization of the virtual photon,
representing the ratio of longitudinal to transverse photon flux.

As a first reaction, hard exclusive $\pi^{-}\Delta^{++}$ electroproduction:

$\gamma^{*}~{}p\rightarrow\pi^{-}\Delta^{++}\rightarrow~{}\pi^{-}~{}[~{}p~{}\pi^{+}~{}],$
(65)

has been theoretically described based on transition GPDs in Ref.
Kroll:2022roq . The process is expected, but not proven yet, to factorize
under the same conditions as the $N\to N^{*}$ DVCS process. At present,
nothing is known about the $p-\Delta$ GPDs neither from experiment nor from
theory leaving aside a few relations between the $p-p$ and the $p-\Delta$ GPDs
obtained in the large-$N_{c}$ limit Belitsky:2005qn ; Kroll:2022roq ;
Frankfurt:1999xe . One may expect, however, that the process
$\gamma^{*}p\to\pi^{-}\Delta^{++}$ is under control of similar dynamics as
$\gamma^{*}N\to\pi N^{\prime}$. From HERMES and the JLab experiments, we
learned that the transverse cross section is much larger than the longitudinal
one for the $\pi^{0}p$ channel. For charged pions, there is a strong
contribution from the pion pole to the longitudinal amplitudes. The pole is to
be treated as a one-boson-exchange contribution in the experimentally
accessible range of kinematics since its contribution evaluated from the GPD
$\tilde{E}$ is too small Goloskokov:2009ia . Based on this expectation and
with the help of large-$N_{c}$ results first, admittedly rough, estimates of
the $\pi^{-}\Delta^{++}$ cross sections have been given in Kroll:2022roq .

The helicity amplitudes for longitudinally polarized photons, which are fed by
twist-2 contributions, reads ($e_{0}$ is the positron charge)

${\cal M}^{tw2}_{0\lambda_{R},0\lambda_{N}}=e_{0}\int_{-1}^{1}dx{\cal
H}^{\pi}_{0+0+}\Big{[}A^{\Delta}_{\lambda_{R}+,\lambda_{N}+}-A^{\Delta}_{\lambda_{R}-,\lambda_{N}-}\Big{]},$
(66)

where ${\cal H}^{\pi}_{0\lambda,0\lambda}$ is the leading-twist amplitude for
the hard partonic subprocess $\gamma^{*}q(\lambda)\to\pi q(\lambda)$
($\lambda=\pm\frac{1}{2}$) and

$\displaystyle
A^{\Delta}_{\lambda_{R}\lambda,\lambda_{N}\lambda}=\int\frac{dz^{-}}{2\pi}e^{ixP^{+}z^{-}}\langle\Delta^{++}(p_{R},\lambda_{R})\mid$
$\displaystyle\times\bar{u}(-z/2)\frac{1}{4}\gamma^{+}(1+2\lambda)\gamma_{5})d(z/2)\mid
p(p,\lambda_{N})\rangle_{z^{+}=0,z_{\perp}=0}.$ (67)

To the difference of the $A^{\Delta}$s in Eq. (66) only the odd-parity GPDs
$\tilde{G}_{i}$ (2) contribute.

The amplitudes for transversally polarized photons ($\mu=\pm 1$) read

$\displaystyle{\cal M}^{tw3}_{0\lambda_{R},\mu\lambda_{N}}$
$\displaystyle=e_{0}\int_{-1}^{1}dx\Big{[}{\cal
H}^{\pi}_{0-,\mu+}A^{\Delta}_{\lambda_{R}-,\lambda_{N}+}+{\cal
H}^{\pi}_{0+,\mu-}A^{\Delta}_{\lambda_{R}+,\lambda_{N}-}\Big{]},$ (68)

where

$\displaystyle
A^{\Delta}_{\lambda_{R}-\lambda,\lambda_{N}\lambda}=\int\frac{dz^{-}}{2\pi}e^{ixP^{+}z^{-}}\langle\Delta^{++}(p_{R},\lambda_{R})\mid$
$\displaystyle\times-\lambda\frac{i}{2}\bar{u}(-z/2)(\sigma^{+1}-2\lambda\sigma^{+2})d(z/2)\mid
p(p,\lambda_{N})\rangle_{z^{+}=0,z_{\perp}=0}$

to which the transversity GPDs (6) contribute. The pion wave function
appearing in ${\cal H}^{\pi}_{0-\lambda,\mu\lambda}$, is of twist-3 nature.
The matrix elements $A^{\Delta}$ as well as the helicity amplitudes are
explicitly given in terms of the transition GPDs in Kroll:2022roq . The
subprocess amplitudes, ${\cal H}^{\pi}$, can be found in Goloskokov:2009ia .
They have been calculated within the so-called modified perturbative approach
in which the quark transverse momenta and the Sudakov suppressions are taken
into account.

In the large-$N_{c}$ limit, the following relations between the
$p-\Delta^{++}$ transition GPDs and the proton-proton ones are found
Belitsky:2005qn ; Frankfurt:1999xe

$\tilde{G}_{3}=\frac{3}{2}\Big{(}\tilde{H}^{u}-\tilde{H}^{d}\Big{)}\qquad\tilde{G}_{4}=\frac{3}{8}\Big{(}\tilde{E}^{u}-\tilde{E}^{d}\Big{)}$
(70)

and Kroll:2022roq

$G_{T5}+\frac{1}{2}G_{T7}=-\frac{3}{2}\Big{(}H_{T}^{u}-H_{T}^{d}\Big{)}\,.$
(71)

Neglecting either $G_{T5}$ (scenario I) or $G_{T7}$ (scenario II) as well as
all other transversity and odd-parity GPDs except of $\tilde{G}_{3}$ and
$\tilde{G}_{4}$ and making use of the parametrization of the proton-proton
GPDs advocated for in Goloskokov:2007nt ; Goloskokov:2011rd , one is in the
position to evaluate the helicity amplitudes (66) and (68). In addition to the
GPD contributions one has to take into account also the pion-pole contribution
which reads Kroll:2022roq

$\displaystyle{\cal M}^{pole}_{0\lambda_{R},\mu\lambda_{N}}$
$\displaystyle=e_{0}\frac{\varrho_{\pi\Delta}}{t-m_{\pi}^{2}}\bar{\cal
U}_{\delta}(p_{R},\lambda_{R})\frac{\Delta^{\delta}}{M_{N}}u(p,\lambda_{N})(q-2q^{\prime})_{\rho}\epsilon^{\rho}(q,\mu),$

where

$\varrho_{\pi\Delta}=\sqrt{2}g_{\pi\Delta^{++}p}F_{\pi\Delta
p}(t)F_{\pi}(Q^{2})\,.$ (73)

The coupling of the pion to the proton and the $\Delta^{++}$ is described by a
coupling constant, $g_{\pi\Delta^{++}p}$, and a $t$-dependent form factor
$F_{\pi\Delta p}$. The last item in Eq. (73), $F_{\pi}(Q^{2})$, is the
electromagnetic form factor of the pion. The photon polarization vector is
denoted by $\epsilon$. The photon and meson momenta are denoted by $q$ and
$q^{\prime}$, respectively. The longitudinal pole contribution dominates over
the transversal ones. The transversal/longitudinal ratio of the pole
contributions is proportional to $1/Q$ as is the case for the corresponding
ratio of the amplitudes (68) and (66).

With the above described model for the $p-\Delta$ GPDs and the pion-pole
contribution, the partial cross sections

$\displaystyle\frac{d\sigma_{L}}{dt}$ $\displaystyle=$
$\displaystyle\frac{\sum_{\lambda_{R}}|{\cal
M}_{0\lambda_{R},0+}|^{2}}{16\pi(W^{2}-M_{N}^{2})\sqrt{\Lambda(W^{2},-Q^{2},M_{N}^{2})}}\,,$
$\displaystyle\frac{d\sigma_{T}}{dt}$ $\displaystyle=$
$\displaystyle\frac{\sum_{\lambda_{R}}\Big{[}|{\cal
M}_{0\lambda_{R},++}|^{2}+|{\cal
M}_{0\lambda_{R},-+}|^{2}\Big{]}}{32\pi(W^{2}-M_{N}^{2})\sqrt{\Lambda(W^{2},-Q^{2},M_{N}^{2})}}\,,$
$\displaystyle\frac{d\sigma_{TT}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{\sum_{\lambda_{R}}{\rm Re}\Big{[}{\cal
M}_{0\lambda_{R},++}^{*}{\cal
M}_{0\lambda_{R},-+}\Big{]}}{16\pi(W^{2}-M_{N}^{2})\sqrt{\Lambda(W^{2},-Q^{2},M_{N}^{2})}}\,,$
$\displaystyle\frac{d\sigma_{LT}}{dt}$ $\displaystyle=$
$\displaystyle-\sqrt{2}\frac{\sum_{\lambda_{R}}{\rm Re}\Big{[}{\cal
M}_{0\lambda_{R},0+}^{*}\big{(}{\cal M}_{0\lambda_{R},++}-{\cal
M}_{0\lambda_{R},-+}\big{)}\Big{]}}{32\pi(W^{2}-M_{N}^{2})\sqrt{\Lambda(W^{2},-Q^{2},M_{N}^{2})}}\,.$

for the $\pi^{-}\Delta^{++}$ channel are evaluated. The results at
$Q^{2}=2.48\,{\rm GeV}^{2}$ and a Bjorken-$x$ of $x_{B}=0.27$ are shown in
Figs. 11, 12 as a function of $t^{\prime}=t-t_{0}$ (see Eq. (5)) and compared
to the cross sections for the $\pi^{+}n$ channel. Note that the $t_{0}$-values
are quite different in both processes ($t_{0}=-0.323\,(-0.088)\,{\rm GeV}^{2}$
for the $\pi^{-}\Delta^{++}$ ($\pi^{+}n$) channel at the quoted kinematics).
The amplitudes in (LABEL:eq:partial-cross-sections) are the sum of the GPD and
the pole contributions. The Mandelstam function is

$\displaystyle\Lambda(W^{2},Q^{2},M_{N}^{2})=$ $\displaystyle
W^{4}+Q^{4}+M_{N}^{4}+2W^{2}Q^{2}$ (75)
$\displaystyle-2W^{2}M_{N}^{2}+2Q^{2}M_{N}^{2}\,.$

Figure 11: The longitudinal (left) and the transverse (right) cross sections
of $\gamma^{*}p\to\pi^{-}\Delta^{++}$ versus $-t^{\prime}=-t+t_{0}$. The solid
(dashed) lines represent the predictions obtained for scenario I (II). For
comparison, the dotted lines are the results for $\gamma^{*}p\to\pi^{+}n$
obtained with the same GPDs. The figure is taken from Ref. Kroll:2022roq .

As is to be seen from Fig. 11 the longitudinal cross section is large. The
reason for this feature is a strong contribution from the pion pole. For the
other partial cross sections there are substantial differences between the two
scenarios. Particularly interesting is the opposite sign of $d\sigma_{TT}$ for
the $\pi^{-}\Delta^{++}$ and the $\pi^{+}n$ channels. The dominant
contribution to $d\sigma_{TT}$ comes from an interference of a twist-3
helicity non-flip amplitude being proportional to the convolution $\langle
H_{T}\rangle$ (in the large-$N_{c}$ limit for the $\pi^{-}\Delta^{++}$
channel), and a pion-pole contribution. The $Q^{2}$ and $x_{B}$ dependencies
of the $\pi^{-}\Delta^{++}$ partial cross sections are similar to those of the
$\pi^{+}n$ ones. It is to be stressed that the predictions are to be
understood as rough estimates, their uncertainties are very large.

Figure 12: The longitudinal-transverse (left) and the transverse-transverse
(right) interference cross sections of $\gamma^{*}p\to\pi^{-}\Delta^{++}$. For
other notations see Fig. 11. The figure is taken from Ref. Kroll:2022roq .

In Kroll:2022roq  also the asymmetry $A_{LL}$ has been calculated, which can
be measured with longitudinally polarized beam and target and is to be
integrated upon the electroproduction cross section

$A_{LL}=\sqrt{1-\varepsilon^{2}}\,\frac{1}{2\sigma_{0}}\,\sum_{\lambda_{R}}\Big{[}|{\cal
M}_{0\lambda_{R},++}|^{2}-|{\cal M}_{0\lambda_{R},-+}|^{2}\Big{]}$ (76)

where

$\sigma_{0}=\sum_{\lambda_{R}}\Big{[}|{\cal M}_{0\lambda_{R},++}|^{2}+|{\cal
M}_{0\lambda_{R},-+}|^{2}+\varepsilon|{\cal
M}_{0\lambda_{R},0+}|^{2}\big{]}\,.$ (77)

The magnitude of the predictions are close to that one for the $\pi^{+}n$
channel but the two scenarios lead to opposite signs of $A_{LL}$. This result
is understandable: for scenario I the amplitudes ${\cal M}_{0\lambda_{R},++}$
are large, whereas for scenario II the amplitudes ${\cal M}_{0\lambda_{R},-+}$
dominate. This feature provides an opportunity to discriminate between
dominant $G_{T5}$ and $G_{T7}$.

There are many other asymmetries, like the beam-spin one

$A_{LU}^{\sin{(\phi)}}\sim\frac{1}{\sigma_{0}}\sum_{\lambda_{R}}{\rm
Im}\Big{[}({\cal M}_{0\lambda_{R},++}-{\cal M}_{0\lambda_{R},-+})^{*}{\cal
M}_{0\lambda_{R},0+}\Big{]}\,.$ (78)

Most of these asymmetries depend on relative phases between different helicity
amplitudes. At present, it is not possible to estimate these asymmetries
reliably.

For vector-meson channels, as for instance $\rho^{0}\Delta^{+}$, the same
$A^{\Delta}$s hold. Only a plus sign appears between the $A^{\Delta}$s in Eq.
(66) instead of the minus sign. This projects out the parity-even GPDs $G_{i}$
(1).

###  6.3 Near-threshold pion production

Pion photo- and electroproduction on the nucleon $\gamma N\to\pi N$,
$\gamma^{\ast}N\to\pi N$ close to threshold has been studied extensively since
the 1950s both experimentally and theoretically. The celebrated low-energy
theorems (LET) Kroll:1953vq ; Nambu:1962ilv ; Nambu:1962lbq  relate the S-wave
transverse $E_{0+}$ and longitudinal $L_{0+}$ multipoles at threshold to the
nucleon electromagnetic and axial form factors in the chiral limit
$m_{\pi}=0$. The new insight gained from CHPT calculations Bernard:1992ys ;
Bernard:1995dp  is that the expansion at small $Q^{2}$ has to be done with
care as the limits $m_{\pi}\to 0$ and $Q^{2}\to 0$ do not commute, in general.
For large momentum transfers, the power counting of CHPT cannot be applied.
However, the traditional derivation of LET using PCAC and current algebra is
not affected as long as the emitted pion is “soft” with respect to the initial
and final state nucleons simultaneously. The corresponding condition is,
parametrically, $Q^{2}\ll\Lambda^{3}/m_{\pi}$ Vainshtein:1972ih ;
Pobylitsa:2001cz  where $\Lambda$ is some hadronic scale, and might be
violated at the highest $Q^{2}$ available in future experiments at JLab and
EIC.

Such an interplay of chiral dynamics with “hard” QCD factorization picture of
large-$Q^{2}$ reactions is potentially very interesting since the generalized
pion-nucleon LCDAs in the chiral limit can be expressed in terms of the
“chirally-rotated” nucleon LCDAs Pobylitsa:2001cz ; Braun:2006td , thus
providing one with an additional handle to separate the different components
in the nucleon wave function. In addition, if the structure of the corrections
beyond LET can be quantified, pion electroproduction would allow one to
determine the nucleon axial form factor at large momentum transfers, which is
difficult to access by direct measurements.

The difficulty is that the onset of the pQCD regime in the form factors is
generally believed to be postponed to very large momentum transfers, and at
intermediate $Q^{2}\sim 1-10\,\text{GeV}^{2}$ one has to take into account
non-factorizable “soft” or “end-point” contributions. The light-cone sum rule
(LCSR) technique Balitsky:1989ry  allows one to estimate such contributions in
terms of the same nucleon LCDAs that enter pQCD calculation using dispersion
relations and duality. It can be generalized and was applied to the pion
electroproduction at threshold in Ref. Braun:2007pz . The LCSR calculation (so
far available in leading order only) suggests that the leading transverse
$E_{0+}$ multipole receives moderate corrections w.r.t. the LET predictions in
the whole studied $Q^{2}$ range, whereas the longitudinal $L_{0+}$ multipole
receives large corrections and tends to change sign at large $Q^{2}$. These
conclusions are, qualitatively, in agreement with the existing CLAS
measurements for charged CLAS:2012ich  and neutral CLAS:2012zia  pion
production. More studies, both theoretical and experimental, are needed to
make this comparison fully quantitative.

##  7 First experimental results from JLab 12 GeV

Except for the information obtained from the relation to the transition form
factors and the relation to the ordinary GPDs in the large $N_{c}$ limit,
little is known about the transition GPDs so far. On the experimental side,
recently, the first beam spin asymmetries for hard exclusive
$\pi^{-}\Delta^{++}$ electro-production were published by the CLAS
collaboration at JLab and further measurements on $N\to\Delta^{+}$ DVCS with
CLAS and $\pi^{+}\Delta^{0}$ DVMP based on data from hall C at JLab are
ongoing. This section will summarize the published and ongoing studies and
their impact on constraining transition GPDs.

###  7.1 $\pi^{-}\Delta^{++}$ electroproduction with CLAS12

The first observable sensitive to $N\to\Delta$ transversity transition GPDs
and $N\to\Delta$ transition GPDs in general was provided by the CLAS
collaboration with hard exclusive
($\gamma^{*}N\to\pi^{-}\Delta^{++}\to\pi^{-}[p\pi^{+}]$) electro-production
beam-spin asymmetries, based on the scattering of longitudinally polarized
10.6 GeV electrons on an unpolarized hydrogen target in the CLAS12 detector
Burkert:2020akg , published in 2023 CLAS:2023akb .

For the study, the $ep\pi^{-}X$ final state was detected, and the $\pi+$ was
reconstructed via a cut on the missing mass. Kinematic cuts on W $>$ 2 GeV,
$Q^{2}$ $>$ 1.5 GeV2, y $<$ 0.75, and $-t$ $<$ 1.5 GeV2 have been applied.
Furthermore, a cut on $M(\pi^{+}\pi^{-})$ $>$ 1.1 GeV is applied to reject the
dominant background from
$\gamma^{*}~{}p~{}\rightarrow~{}p~{}\rho^{0}~{}\rightarrow~{}p~{}\pi^{+}~{}\pi^{-}$.
Fig. 13 shows the $\Delta^{++}$ peak in the resonance spectrum of the
$p\pi^{+}$ invariant mass and compares it to the background distribution from
a full semi-inclusive deep inelastic scattering Monte Carlo simulation. In
addition, the background-subtracted data is compared to a Monte Carlo
simulation of the exclusive process.

Figure 13: Upper row: $\Delta^{++}$ peak in the $p\pi^{+}$ invariant mass of
the experimental data (blue, solid) in comparison to the non-resonant
background obtained with the semi-inclusive deep inelastic scattering MC
(black) for selected bins of $-t$ in the forward region ($Q^{2}$ = 2.48 GeV2,
$x_{B}$ = 0.27) after a cut on $M_{\pi^{+}\pi^{-}}>1.1$ GeV. Lower row:
$\Delta^{++}$ peak in the same bins after the subtraction of the background
(blue, solid) in comparison to the result from the exclusive MC (green).
Figure in modified form taken from Ref. CLAS:2023akb .

It can be observed that the background contribution increases with increasing
values of $-t$. However, the good agreement of the falling tail of the
$\Delta^{++}$ peak after the background subtraction with the exclusive Monte
Carlo shows that the background is well understood and under control. To
select the $\pi^{-}\Delta^{++}$ signal region, a cut on $M(p\pi^{+})$ $<$ 1.3
GeV is applied. For these events, the beam spin asymmetries are determined for
3 $Q^{2}-x_{B}$ bins and several $-t$ and $\phi$ bins. For each $-t$ bin, the
structure function ratio $\sigma_{LT^{\prime}}/\sigma_{0}$ is extracted based
on a fit of the $\phi$ dependence of the beam spin asymmetry, and a background
subtraction is performed for each kinematic bin. More details on the analysis
can be found in Ref. CLAS:2023akb .

Fig. 14, shows a comparison of the obtained structure function ratio
$\sigma_{LT^{\prime}}/\sigma_{0}$ for $\pi^{-}\Delta^{++}$ with results from
$\pi^{+}n$ CLAS:2022iqy  and $\pi^{0}p$ CLAS:2023wda .

Figure 14: $\sigma_{LT^{\prime}}/\sigma_{0}$ and its statistical uncertainty
for $\pi^{-}\Delta^{++}$ (black squares, Ref. CLAS:2023akb ) as a function of
$-t^{\prime}=(|t|-|t_{min}|)$ and its systematic uncertainty (gray band),
measured with CLAS12. For comparison, the CLAS12 results from the hard
exclusive $\pi^{+}n$ (red triangles, Ref. CLAS:2022iqy ) and $\pi^{0}p$ (blue
circles, Ref. CLAS:2023wda ) electro-production with similar kinematics are
shown. The sub-figures correspond to the results for the different $Q^{2}$ and
$x_{B}$ bins defined in the upper right insert. The mean kinematics are shown
on top of each sub-figure. Figures in modified form taken from Ref.
CLAS:2023akb .

The figure shows that the structure function ratio of $\pi^{-}\Delta^{++}$
shows an opposite sign than for $\pi^{+}n$ and an approximately doubled
magnitude compared to $\pi^{+}n$. The different sign of the asymmetry can be
related to an interaction of the virtual photon with different quarks ($u$,
$d$) for the two reactions and the opposite sign of the quark polarization.
However, the large absolute magnitude of $\sigma_{LT^{\prime}}/\sigma_{0}$ for
$\pi^{-}\Delta^{++}$, compared to $\pi^{+}n$ can be seen as a clear effect of
the excitation process, which is encoded in the transition GPDs CLAS:2023akb .

The studied polarized structure function $\sigma_{LT^{\prime}}$ is given by
products of convolutions of transversity and helicity non-flip transition GPDs
with sub-process amplitudes and shows the following relation to the two
dominant transversity transition GPDs CLAS:2023akb :

$\sigma_{LT^{\prime}}\sim\sqrt{-t^{\prime}}~{}~{}{\rm
Im}\left[G^{3}_{T_{5}}\cdot A+c~{}G^{3}_{T_{7}}\cdot A^{\prime}\right],$ (79)

with an unknown kinematic factor $c$ and helicity amplitudes for
longitudinally polarized virtual photons $A$ and $A^{\prime}$, which are
determined by the helicity non-flip transition GPDs $\widetilde{G}_{3}$ and
$\widetilde{G}_{4}$ within the large $N_{c}$ limit. The transversity
transition GPDs $G^{3}_{T_{5}}$ and $G^{3}_{T_{7}}$ can be related to the
ground state transversity GPD $H_{T}$ in the large $N_{c}$ limit, as shown in
Eq. 71. Therefore, they are expected to be sensitive to the tensor charge of
the resonance following Eq. 48.

Based on CLAS12 data with a longitudinally polarized $NH_{3}$ target, it is
planned to extend the study to target polarization observables like $A_{UL}$
and $A_{LL}$. Furthermore, the extraction of absolute cross sections
($\sigma_{0}=\sigma_{L}+\sigma_{T}$) and the $\cos(\phi)$ ($\sigma_{LT}$) and
$\cos(2\phi)$ ($\sigma_{TT}$) modulations of the cross section is planed based
on data from an unpolarized proton target.

###  7.2 Further $N\to N^{*}$ DVMP channels with CLAS12

As shown in section 6.2, different $N\to N^{*}$ DVMP channels can be used to
study transition GPDs, and a combination of data from different mesons in the
final state will allow a flavor decomposition of the transition GPDs.

Currently, the feasibility of the study of different channels, like

  * •

$ep\to e^{\prime}\pi^{+}\Delta^{0}\to e^{\prime}\pi^{+}p\pi^{-}$,

  * •

$ep\to e^{\prime}\pi^{+}\Delta^{0}\to e^{\prime}\pi^{+}n\pi^{0}$,

  * •

$ep\to e^{\prime}\pi^{0}\Delta^{+}\to e^{\prime}\pi^{0}n\pi^{+}$,

is under investigation. Since the electromagnetic calorimter of CLAS12 is
limited to a polar angle below 35∘, only resonance decays with a charged pion
can be efficiently detected. However, if the DVMP meson is a charged pion, the
resolution of CLAS12 is sufficient to consider the application of the missing
mass technique to reconstruct the pion from the resonance decay.

As an example, Fig. 15 shows the $p\pi^{-}$ invariant mass for the
$e^{\prime}\pi^{+}\Delta^{0}\to e^{\prime}\pi^{+}p\pi^{-}$ DVMP process.

Figure 15: $p\pi^{-}$ invariant mass for the $e^{\prime}\pi^{+}\Delta^{0}\to
e^{\prime}\pi^{+}p\pi^{-}$ DVMP process for $-t<$ 1 GeV2, after a cut on
$M(\pi^{+}\pi^{-})$ $>$ 1.1 GeV to reject the exclusive $\rho$ events,
measured with CLAS12.

It can be clearly seen that, in contrast to the $\pi^{-}[p\pi^{+}]$ final
state, which was discussed in the last section and which can only be populated
by $\Delta$ resonances, the $\pi^{+}[p\pi^{-}]$ final state shows the complete
resonance spectrum with nucleon and $\Delta$ resonances. Besides the
$\Delta(1232)$ peak, the second and third resonance regions are clearly
visible in Fig. 15. All other $N\to N^{*}$ DVMP channels show a similar
picture. Therefore, a clean separation of the $\Delta(1232)$ events is more
difficult for these channels, and either a full fit of the spectrum, including
non-resonant background, or a partial wave decomposition have to be performed
to access the $\Delta(1232)$ events, and especially the higher resonances with
a higher mass.

###  7.3 $\pi^{+}\Delta^{0}$ electroproduction in JLab Hall C

Hall C at JLab blok2008charged  is a unique facility that gives access to high
precision DVMP reaction measurements over a wide range of kinematics. The hall
has a fixed target geometry with two focusing magnetic spectrometers, which
detect the scattered electron and final state meson in coincidence. The final
state hadron is reconstructed to high resolution through missing mass. Hall C
is also the only facility worldwide with the capability to perform high
$Q^{2}$ Longitudinal/Transverse (L/T) separations of the reaction cross
section via the Rosenbluth separation process. This is done by taking data at
two values of $\epsilon$ (virtual photon polarization factor) for each
kinematic point, where

$\centering{\epsilon}=1+2\frac{(E_{e}-E_{e^{{}^{\prime}}})^{2}+Q^{2}}{Q^{2}}\tan^{2}\frac{\theta_{e^{{}^{\prime}}}}{2}\@add@centering$
(80)

Figure 16: Missing mass distribution for exclusive pion electroproduction
reaction from Kaon-LT data ($Q^{2}=2.1$ GeV2, $W=2.95$ GeV) from JLab hall C.
All the curves except the data are from Monte Carlo simulations. The fits of
the simulations to the data are preliminary.

The Kaon-LT experiment (E12-09-011) JLABPAC:KaonLT  is the first L/T
separation experiment after the 12 GeV upgrade to study exclusive $K^{+}$
electroproduction reaction $p(e,e^{\prime}K^{+})\Lambda$. The experiment also
gathered high-statistics exclusive $\pi^{+}$ electroproduction data, as it is
one of the background processes for the exclusive $K^{+}$ channel. The
exclusive $\pi^{+}$ data include both the $p(e,e^{\prime}\pi^{+})n$ and
$p(e,e^{{}^{\prime}}\pi^{+})\Delta^{0}$ reactions. Table 3 lists all settings
collected to date during the Kaon-LT experiment. The BSA analysis is ongoing
for the 10.6 GeV kinematic settings listed. In this experiment, the meson
spectrometer is rotated 3 degrees on either side of the $\vec{q}$-vector to
get full $\phi$ coverage at fixed t, needed for the BSA measurement.

$E_{b}$ (GeV) |  $Q^{2}$ ($GeV^{2}$/$c^{2}$) | W (GeV) |  $\epsilon_{high}$/$\epsilon_{low}$
---|---|---|---
10.591/8.213 | 5.5 | 3.02 | 0.53/0.18
10.591/8.213 | 4.4 | 2.74 | 0..72/0.48
10.591/8.213 | 3.0 | 3.14 | 0.67/0.39
10.591/6.187 | 3.0 | 2.32 | 0.88/0.57
10.590/6.187 | 2.115 | 2.95 | 0.79/0.25
4.930/3.834 | 0.5 | 2.40 | 0.70/0.45
Table 3: The kinematics settings acquired to date in the KaonLT experiment
JLABPAC:KaonLT .

The Hall C spectrometers give access to very good missing mass resolution.
Figure 16 shows the missing mass distribution for $p(e,e^{\prime}\pi^{+})X$
including experimental data and different simulated processes. The plot on the
left gives the full range of missing mass, which clearly shows a clean
$p(e,e^{\prime}\pi^{+})n$ peak. The figure on the right is a zoomed-in version
to focus on the $\Delta^{0}$ region. The $p(e,e^{\prime}\pi^{+})n$ sample has
very little background underneath; therefore, it can be selected by using a
missing mass cut around the peak. On the other hand, the
$p(e,e^{\prime}\pi^{+})\Delta^{0}$ reaction has a Breit-Wigner distribution
that sits atop other multi-pion processes. This requires a background shape
study to cleanly isolate the $p(e,e^{\prime}\pi^{+})\Delta^{0}$ reaction. A
shape study was performed for each $\phi$ bin for each beam helicity state by
fitting the $\Delta^{0}$ Monte Carlo to the background-subtracted experimental
data, and the BSA was calculated. Figure 17 shows preliminary
$p(e,e^{\prime}\pi^{+})\Delta^{0}$ BSA results for one kinematic setting. A
parallel analysis of $p(e,e^{{}^{\prime}}\pi^{+})n$ BSA is also in progress
for the same kinematics. The $\Delta^{0}$ BSA paper will include the
$\sigma_{LT^{\prime}}$ ratio for both $n$ and $\Delta^{0}$ final states.

Figure 17: Preliminary Beam Spin Asymmetry results for the
$p(e,e^{\prime}\pi^{+})\Delta^{0}$ reaction from the KaonLT experiment
($Q^{2}=2.1$ GeV2, $W=2.95$ GeV) in JLab hall C. Only statistical errors are
shown. The final results will include three $\times$ more data after including
all three meson spectrometer settings.

The future plan is to perform an L/T separation for all kinematics listed in
Table 3 for both $p(e,e^{\prime}\pi^{+})\Delta^{0}$,
$p(e,e^{{}^{\prime}}\pi^{+})n$ reactions. This involves a more detailed
understanding of all the experimental systematics, as well as fine-tuning the
simulation models. The L/T separated cross-section ratio for
$p(e,e^{\prime}\pi^{+})\Delta^{0}$ and $p(e,e^{\prime}\pi^{+})n$ can give
access to $N\rightarrow\Delta^{0}$ transition GPDs. The KaonLT experiment is
anticipated to collect more data to complete the proposed kinematics. There is
also the possibility of increasing the kinematic coverage with a future
experiment using the Hall C experimental setup. An energy upgrade at Jefferson
Lab will allow a further increase in kinematic coverage for the
$p(e,e^{\prime}\pi^{+})\Delta^{0}$ L/T separation experiment.

###  7.4 $N\to\Delta^{+}$ DVCS with CLAS12

First feasibility studies for the non-diagonal DVCS process with limited
statistics and kinematic coverage, not allowing a real background separation,
have been performed based on CLAS data from the 6 GeV era in 2009
transGPDMor09 . Studies of associated electroproduction of real photons,
$e+p\rightarrow e^{\prime}\gamma\pi N$, in the $\Delta(1232)$-resonance region
were also performed with HERMES Duren:2014ola , but similarly to the first
CLAS studies, they also suffered from limited statistics and a missing
background separation. The 12 GeV upgrade of JLab finally allowed initial
measurements of the $N\to N^{*}$ DVCS and DVMP processes at reasonably high
$Q^{2}$ values and with sufficient statistics for the first time. For this
process, the detection of all final state particles is important for a proper
rejection of potential backgrounds. For
$\gamma^{*}~{}p\rightarrow~{}~{}\gamma~{}\pi^{+}~{}n$ $N\to N^{*}$ DVCS, two
final states are possible based on the $N^{*+}$ decay:

  1. 1.

$\gamma^{*}~{}p~{}~{}\rightarrow~{}~{}\gamma~{}N^{*+}~{}~{}\rightarrow~{}~{}\gamma~{}n~{}\pi^{+}$

  2. 2.

$\gamma^{*}~{}p~{}~{}\rightarrow~{}~{}\gamma~{}N^{*+}~{}~{}\rightarrow~{}~{}\gamma~{}p~{}\pi^{0}$

With a deuterium (neutron) target, also the following reactions can be
studied:

  1. 1.

$\gamma^{*}~{}n~{}~{}\rightarrow~{}~{}\gamma~{}N^{*0}~{}~{}\rightarrow~{}~{}\gamma~{}p~{}\pi^{-}$

  2. 2.

$\gamma^{*}~{}n~{}~{}\rightarrow~{}~{}\gamma~{}N^{*0}~{}~{}\rightarrow~{}~{}\gamma~{}n~{}\pi^{0}$

In all cases, the $N^{*}$ can be a nucleon or a $\Delta$ resonance. For the
very forward regime of the DVCS photon (low $-t$), the $N^{*}$ and its decay
products are expected to be detected under central to backward lab angles.