# Generalized Mie theory for full-wave numerical calculations of scattering
near-field optical microscopy with arbitrary geometries

###### Abstract

Scattering-type scanning near-field optical microscopy is becoming a premier
method for the nanoscale optical investigation of materials well beyond the
diffraction limit. A number of popular numerical methods exist to predict the
near-field contrast for axisymmetric configurations of scatterers on a surface
in the quasi-electrostatic approximation. Here, a fully electrodynamic
approach is given for the calculation of near-field contrast of several
scatterers in arbitrary configuration, based on the generalized Mie scattering
method. Examples for the potential of this new approach are given by showing
the coupling of hyperbolic phonon polaritons in hexagonal boron nitride layers
and showing enhanced scattering in core-shell systems. In general, this method
enables the numerical calculation of the near-field contrast in a variety of
strongly resonant scatterers and is able to accurately recreate spatial near-
field maps.

###### keywords:

Mie, near-field, numerical model, scattering

Dániel Datz* Gergely Németh László Rátkai Áron Pekker Katalin Kamarás

Dániel Datz, Dr. Gergely Németh, Dr. Áron Pekker, Dr. László Rátkai, Prof.
Katalin Kamarás
Institute for Solid State Physics and Optics, Wigner Research Centre for
Physics, Konkoly-Thege Miklós út 29-33, 1121, P.O. Box 49, H-1525 Budapest,
Hungary Email Address<EMAIL_ADDRESS>
Prof. Katalin Kamarás Institute of Technical Physics and Materials Science,
Centre for Energy Research P.O. Box 49, H-1525 Budapest, Hungary

Dr. Gergely Németh SOLEIL Synchrotron, L’Orme des Merisiers, RD128, Saint
Aubin, France 91190

##  1 Introduction

Scattering-type scanning near-field optical microscopy (s-SNOM) has become one
of the leading methods for determining local optical information of materials
with spatial resolution well below the diffraction limit. This method is
especially effective in visualizing and otherwise investigating exotic optical
phenomena, such as plasmon and phonon polaritons in 2D van der Waals crystals
[1, 2] and detecting strong coupling between hexagonal boron nitride (hBN) and
nanotube plasmons [3] or molecular vibrations [4, 5, 6].

The extreme spatial focusing and amplification of the illuminating light
happens by approaching the (often metallized) probing tip of an atomic force
microscope (AFM) to close proximity of the investigated sample. In this
electromagnetic environment the AFM tip acts as an antenna and scatters light
in every direction. The complex scattering processes between the
sample/substrate and the tip slightly modify the scattering character of the
probe. This small change in the amplitude and the phase of the scattered light
can be detected by interferometric techniques, such as the heterodyne or the
pseudo-heterodyne method [7].

The exact nuances of the tip-sample interaction are still not entirely
understood. A number of methods exist, with different complexity levels, that
try to capture the essential details of the scattering process. Full-wave
calculations with large program packages, such as finite element modeling
(FEM) with COMSOL [8, 9, 10] or finite difference time domain (FDTD) [11]
include all the complexities in exchange for largely increased computational
time and effort. Simpler models, such as the point dipole model (PDM) [12],
the finite dipole model (FDM) [13] or the the extended finite dipole model
(EFDM) [14] provide a quasi-static approximation to the solution of the
scattering problem. In the PDM, the AFM tip is approximated by a point dipole,
or a sphere with much smaller effective radius than the exciting laser
wavelength. In the FDM and EFDM spheroidal scatterers are included for more
realistic tip shape approximation. Lately, full-wave finite element
calculations were combined with PDM and FDM formulations to achieve efficient
extraction of the near-field contrast, albeit still in the quasi-static limit
[9, 10].

More involved quasi-static calculations include the polarizability of
spheroidal and more complicated tip shapes and multilayer structures [15].
These models all severely underestimate the penetration depth which is
important in the examination of multilayered thin films. A common drawback of
these approximations is the use of fitting parameters with questionable
physical interpretation to adjust the calculated results to measured ones.

More sophisticated numerical methods without ad hoc fitting parameters have
also been developed for the calculation of the scattered signal in the fully
electrodynamical limit. The so-called ”lightning rod” model [16] does give a
full description of electrodynamic effects, such as retardation effects, by
using a method similar to the one presented in this paper.

Reference [17] uses the generalized spectral method for the description of the
near-field contrast in case of highly resonant samples and realistic tip
shapes.

While these models are able to calculate the near-field contrast for realistic
tip shapes in the fully electrodynamic limit, they require translational
symmetry in the plane (not accounting for the tip), which limits their
applications regarding multiple scatterers in arbitrary configuration.

In this paper, we present a full-wave, tractable, relatively time-efficient
method of calculating complex far-field scattering amplitudes and phases
related to s-SNOM measurements, without spurious fitting parameters, using the
”generalized Mie scattering” or multipole reflection theory (MRT) method.

##  2 Methods

Mie’s theory yields the scattering properties of a sphere suspended in a
homogeneous, non-absorbing medium by expanding the electromagnetic field in
terms of spherical vector functions. The generalization of the classical Mie
theory involves the inclusion of multiple, possibly layered and non-spherical
scatterers in the vicinity of a plane interface representing the surface of a
possibly layered substrate.

Following the MRT method [18, 19, 20, 21, 22, 23], the fields can be expanded
in terms of the spherical multipoles $\textbf{J}^{(p)}_{lm}$ and
$\textbf{H}^{(p)}_{lm}$. The extension of the scattered electric field takes
the form

$\mathbf{E}_{sca}=\sum_{\eta}E_{sca,\eta}\sum_{plm}\mathbf{H}^{(p)}_{lm}(\mathbf{r}^{\prime})A^{(p)}_{\eta
lm},$ (1)

where $\eta$ is the index for the different polarizations in a given
polarization basis, and $A^{(p)}_{\eta lm}$ are the expansion coefficients.
The index p is either 1 or 2 and distinguishes between electric and magnetic
type of vector functions. The indices l and m are the azimuthal and magnetic
indices, respectively.

In the presence of multiple scatterers, the usual boundary conditions on the
surface of each scatterer (the continuity of the electric and the magnetic
field) result in a system of linear equations for the expansion coefficients:

$\sum_{p^{\prime}l^{\prime}m^{\prime}}M^{(pp^{\prime})}_{lm,l^{\prime}m^{\prime}}A^{(p^{\prime})}_{\eta
l^{\prime}m^{\prime}}=-\mathcal{W}^{(p)}_{\eta lm},$ (2)

where the matrix M is the matrix that describes the boundary conditions, while
the coefficients $\mathcal{W}^{(p)}_{\eta lm}$ are the expansion coefficients
of the exciting field.

In the presence of a plane interface, the reflected spherical multipoles are
described by the formulation of Bobbert and Vlieger [24]. Solving Equation 2
for the expansion coefficients of the scattered field allows the formulation
of the scattering amplitude matrix that connects the incident electric field
to the observed electric field scattered in a given direction

$\mathbf{E}_{obs}(\theta,\phi)=\mathbf{f}(\theta,\phi)\cdot\mathbf{E}_{inc}.$
(3)

This equation is analogous to the commonly cited relation between the near
field and the far field in SNOM literature ($E_{N}=\sigma E_{I}$) and makes
the scattering amplitude matrix the final derived quantity of this
formulation.

The scattering parameters of objects of varying shapes can be given by the
extended boundary condition method (EBCM). This method does not limit the
shape of the particle as long as the surface is parametrizable [25, 26].

Details about the calculations can be found in the supporting information.

##  3 Results

###  3.1 Phonon polariton coupling in hexagonal boron nitride

Hexagonal boron nitride is a naturally occurring hyperbolic material [27].
This type of anisotropy causes the bulk phonon polaritons in the crystal to
propagate on the surface of a cone. For the near-field detection of these
polaritons, the phonon polaritons have to be coupled into the crystal by sharp
features of the surface, such as the SNOM tip itself [27], the crystal edge or
other objects on the surface [28] that are able to provide the missing
momentum components for the coupling. The s-SNOM method is then able to
visualize the phonon polaritons by mapping the fringe pattern arising from the
interference of the polaritons coupled in by the tip of the SNOM device and
the sharp feature.

The main strength of the MRT formulation is the ability to handle additional
particles besides the probing tip. To illustrate the power of this method, we
calculated the phonon polariton interference fringes in a thin layer of
hexagonal boron nitride (hBN). A gold nanosphere is placed on the surface of
the hBN layer to couple phonon polaritons into the layer that can interfere
with the tip-launched polaritons.

The layered structure consists of a 40 nm thick hBN layer on top of a 4 nm
SiO2 layer on a semi-infinite silicon substrate (see Figure 1a). The
refractive index of the 3 nm radius gold sphere is extrapolated from measured
data of Johnson and Cristi [29]. The SNOM tip is modeled as a platinum
spheroid of 600 nm length and 20 nm tip-inscribed sphere radius. The
refractive index of platinum is also extrapolated from measured data [30]. The
illumination is a p-polarized plane wave with 60∘ angle of incidence. The
calculations were conducted at 1540 cm-1 excitation. A measurement value
(amplitude and phase) is obtained by imitating the vibration of the tip by
calculating at different separations between the tip and the sample. All the
steps of the pseudo-heterodyne detection technique are replicated in the
background-free extraction of the near-field amplitude and phase.

Figure 1: (a) Modeled geometry of a gold sphere and platinum tip above the
multilayered Si/SiO2/hBN surface. (b) Calculated near-field amplitude (third
harmonic) acquired along the dashed line in (a).

The near-field amplitude, demodulated at the third harmonic, is shown in
Figure 1b. It clearly shows a high contrast peak at the position of the gold
nanosphere caused by the high refractive index of gold. On either side of the
sphere peak, the amplitude shows a characteristic ripple pattern. The
approximate wavelength of this pattern (120-130 nm) agrees reasonably well
with measured data [31]. This shows the ability of the generalized Mie
scattering model to capture the propagation of phonon polaritons in
multilayered surfaces containing hBN.

A feature of the calculated linescan in Figure 1 is the enhanced forward
scattering of phonon polaritons marked by the larger amplitude in the forward
scattering direction. This can be understood by analyzing the electric field
on the surface of the multilayer structure. In Figure 2a, the in-plane
components of the electric field are visualised on the surface of the
substrate in the presence of only the SNOM tip.

Figure 2: (a) Magnitude of the in-plane Fourier components of the near field
above the surface of the multilayered structure in the presence of the
platinum tip. Black and red dashed lines indicate the forward and backward
scattering directions respectively. (b) Amplitude of the z component of the
electric field (logarithmic scale), showing the nanofocus and the asymmetry of
the field distribution. (c) (Below) Imaginary part of the reflection
coefficient of the multilayered structure. (Above) The Fourier components in
the forward (black) and backward (red) scattering directions.

The asymmetry of the in-plane field distribution in the forward and backward
scattering direction (see Figure 2b for the z component) results in
significantly different amplitude Fourier components for the propagating modes
in the multilayer pattern. In Figure 2c the available Fourier amplitudes are
depicted for 1540 cm-1 excitation, showing the larger in-plane amplitude in
the forward scattering direction.

###  3.2 Enhanced scattering from encapsulated molecules in boron nitride
nanotubes

The near-field optical properties of boron nitride nanotubes (BNNTs) are
exciting due to the nanotubes’ ability to enhance the scattering of the
encapsulated material inside its cavity [32]. Using the generalized Mie
scattering model, the approximate numerical modeling of BNNT scattering is
possible. For simplicity, the BNNT is modeled as a hollow sphere (3 nm outer,
2 nm inner radius) with its refractive index set to that of hBN’s out-of-plane
refractive index (see Figure 3a). The substrate is 4 nm SiO2 on top of a semi-
infinite silicon half-space. The calculation parameters otherwise coincide
with the ones mentioned in the previous section.

Figure 3: MRT calculations of the layered sphere representing a BNNT (a)
Schematic geometry of the system. (b) Calculated spectrum with peaks at 1380
cm-1 and 1500 cm-1. The 1380 cm-1 peak comes from a separate calculation of an
hBN layer of the same thickness as the BNNT wall. (c) Extracted nano-FTIR
spectrum of 20 nm thick BNNT from [33]. The peak positions match well with the
calculated results. The observed broadening of the measured peaks can be
caused by the significantly larger diameter of the BNNTs.

The calculated near-field amplitude and phase (third harmonic) are shown in
Figure 3b. Since the scattering properties of anisotropic spheres are hard to
calculate numerically [34], the particle under the SNOM tip is practically a
hollow, metallic shell. Since the average diameter of BNNTs is rather large
(5-20 nm diameter), the TO phonon polariton peak at 1370 cm-1 coincides well
with the same peak in hBN. The results in Figure 3b are thus the sum of the
results from the hollow sphere and from an hBN layer of the same thickness as
the BNNT wall thickness. These results are compared to nano-FTIR measurements
reported in Reference [33]. The calculated spectra show reasonable agreement
for the position and the symmetry of the peaks. The difference of the peak
widths can be attributed to the large difference in diameter between the
measured nanotubes and the one used for the calculation.

An important conclusion drawn from the calculated results is that the BNNT
peak at around 1500 cm-1 is caused by the characteristic Mie scattering of
hollow nanospheres and not by any additional polaritonic excitation inherent
to only boron nitride materials. This example shows that the generalized Mie
scattering method is able to reproduce the near-field spectral features of
more complicated geometries and give meaningful information about the origin
of the peaks.

Figure 4: MRT calculations of layered spheres representing filled BNNTs. (a)
The resonance frequency of the dielectric function of the inner sphere is set
to 1428 cm-1 (see inset). A small peak is visible at this wavenumber. (b) The
resonance frequency of the dielectric function is set to 1358 cm-1. The
additional peak from the inner sphere disappears.

In Reference [32], we showed that the signal of weakly absorbing C60 molecules
can be detected inside BNNTs due to the enhancing effect of the highly
confined electromagnetic fields inside the walls of the nanotube. This
configuration can also be handled with the MRT model, using core-shell layered
spheres representing the nanotubes. The inner sphere’s refractive index is
given by a single Lorentzian with parameters describing a weak oscillator. In
Figure 4a, the resonance frequency of the inner sphere is set to 1428 cm-1
which coincides with a $T_{1u}$ vibrational mode of C60 [35, 36, 37]. Using
the MRT method, a small peak in both the near-field amplitude and phase is
present, the shape of which is typical for weak vibrations. On the other hand,
if the resonance frequency of the Lorentzian of the inner sphere is shifted to
1358 cm-1 (see Figure 4)b) which approximately coincides with a $(C_{60})_{3}$
trimer vibration mode [38] and is outside of the Reststrahlen band, this
effect disappears. The results show that the field enhancement in the inner
sphere predicted by Mie’s theory of a core-shell spherical system [39] is
enough for a significant enhancement of the near-field signal. In a more
realistic configuration, the additional field confinement due to the
hyperbolic nature of the nanotube can result in even higher electric field
amplitude which can explain the amplitude of the detected peaks in Reference
[32].

##  4 Conclusion

In this paper, we introduced a novel approach to calculate near-field contrast
in s-SNOM measurements using a generalized Mie theory approach, the multipole
reflection theory. This model is able to calculate the near-field contrasts of
several, complex scatterers in arbitrary configuration in a fully
electrodynamic treatment. The scattering parameters of all of the interacting
particles are calculated using Mie’s theory, therefore the use of arbitrary
fitting parameters is unnecessary.

Using the MRT model, we showed that it is suitable to calculate hyperbolic
polariton interference fringes coupled into a thin hBN layer by a nanoparticle
on its surface. Furthermore, the polaritonic enhancement of scattering from
the inner sphere of core-shell nanoparticles can also be calculated that
provides valuable insight into the enhancement of molecular vibrations inside
BNNTs.

Further improvements to the presented numerical model are still possible. The
scattering properties of disks ([40]) and cylinders ([41]) are possible to
calculate numerically and can be included into the model without a great
increase in complexity. The illumination is also better described by a
Gaussian beam [42] rather than a plane wave. The results presented in this
paper are the first steps in accurate numerical modeling of non-symmetric
geometries with multiple particles using the scattering theory approach.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the
author.

Acknowledgements

We gratefully acknowledge the support from the National Research, Development
and Innovation Office - NKFIH FK-138411 and K-143153. Research infrastructure
was provided by the Hungarian Academy of Sciences (MTA).

## References

  * [1] S. Dai, W. Fang, N. Rivera, Y. Stehle, B.-Y. Jiang, J. Shen, R. Y. Tay, C. J. Ciccarino, Q. Ma, D. Rodan-Legrain, P. Jarillo-Herrero, E. Hang Tong Teo, M. M. Fogler, P. Narang, J. Kong, D. N. Basov,  _Adv Mater_ 2019, _31_ , 37 1806603.
  * [2] V. E. Babicheva, S. Gamage, L. Zhen, S. B. Cronin, V. S. Yakovlev, Y. Abate,  _ACS Photonics_ 2018, _5_ , 6 2106.
  * [3] Németh, Gergely and Otsuka, Keigo and Datz, Dániel and Pekker, Áron and Maruyama, Shigeo and Borondics, Ferenc and Kamarás, Katalin,  _Nano Lett_ 2022, _22_ , 8 3495.
  * [4] M. Autore, P. Li, I. Dolado, F. J. Alfaro-Mozaz, R. Esteban, A. Atxabal, F. Casanova, L. E. Hueso, P. Alonso-González, J. Aizpurua, A. Y. Nikitin, S. Velez, R. Hillenbrand,  _Light: Sci Appl_ 2018, _7_ , 4 17172.
  * [5] A. Bylinkin, M. Schnell, M. Autore, F. Calavalle, P. Li, J. Taboada-Gutierrez, S. Liu, J. H. Edgar, F. Casanova, L. E. Hueso, P. Alonso-Gonzales, A. Y. Nikitin, R. Hillenbrand,  _Nat Photonics_ 2021, _15_ , 3 197.
  * [6] I. Dolado, C. Maciel-Escudero, E. Nikulina, E. Modin, F. Calavalle, S. Chen, A. Bylinkin, F. J. Alfaro-Mozaz, J. Li, J. H. Edgar, F. Casanova, S. Vélez, L. E. Hueso, R. Esteban, J. Aizupura, R. Hillenbrand,  _Nat Commun_ 2022, _13_ , 1 6850.
  * [7] N. Ocelic,  Ph.D. thesis, Technische Universität München, 2007.
  * [8] L. Mester, A. A. Govyadinov, S. Chen, M. Goikoetxea, R. Hillenbrand,  _Nat Commun_ 2020, _11_ , 1 3359.
  * [9] X. Chen, Z. Yao, S. G. Stanciu, D. Basov, R. Hillenbrand, M. Liu,  _Opt Express_ 2021, _29_ , 24 39648.
  * [10] X. Chen, Z. Yao, Z. Sun, S. G. Stanciu, D. Basov, R. Hillenbrand, M. Liu,  _Opt Express_ 2022, _30_ , 7 11228.
  * [11] V. V. Kotlyar, S. S. Stafeev, A. G. Nalimov, L. O’Faolain, M. V. Kotlyar,  _Photonics Nanostruct_ 2021, _43_ 100898\.
  * [12] M. B. Raschke, C. Lienau,  _Appl Phys Lett_ 2003, _83_ , 24 5089.
  * [13] A. Cvitkovic, N. Ocelic, R. Hillenbrand,  _Opt Express_ 2007, _15_ , 14 8550.
  * [14] A. Cvitkovic,  Ph.D. thesis, Technische Universität München, 2009.
  * [15] B. Hauer, A. P. Engelhardt, T. Taubner,  _Opt Express_ 2012, _20_ , 12 13173.
  * [16] A. S. McLeod, P. Kelly, M. D. Goldflam, Z. Gainsforth, A. J. Westphal, G. Dominguez, M. H. Thiemens, M. M. Fogler, D. N. Basov,  _Phys Rev B_ 2014, _90_ , 8 085136.
  * [17] B.-Y. Jiang, L. M. Zhang, A. H. Castro Neto, D. N. Basov, M. M. Fogler,  _J Appl Phys_ 2016, _119_ , 5 054305.
  * [18] F. Borghese, P. Denti, R. Saija, G. Toscano, O. Sindoni,  _Aerosol Sci Tech_ 1984, _3_ , 2 227.
  * [19] E. Fucile, F. Borghese, P. Denti, R. Saija, O. Sindoni,  _IEEE T Antenn Propag_ 1997, _45_ , 5 868.
  * [20] E. Fucile, P. Denti, F. Borghese, R. Saija, O. Sindoni,  _J Opt Soc Am A_ 1997, _14_ , 7 1505.
  * [21] P. Denti, F. Borghese, R. Saija, E. Fucile, O. I. Sindoni,  _J Opt Soc Am A_ 1999, _16_ , 1 167.
  * [22] F. Borghese, P. Denti, R. Saija,  _J Quant Spectrosc Ra_ 2013, _129_ 69\.
  * [23] F. Borghese, P. Denti, R. Saija,  _Scattering from model nonspherical particles: theory and applications to environmental physics_ ,  Springer Science & Business Media, 2007.
  * [24] P. A. Bobbert, J. Vlieger,  _Physica A_ 1986, _137_ , 1-2 209.
  * [25] M. I. Mishchenko, J. W. Hovenier, L. D. Travis,  _Light scattering by nonspherical particles: theory, measurements, and applications_ ,  Elsevier, 1999.
  * [26] M. I. Mishchenko,  _J Quant Spectrosc Ra_ 2009, _110_ , 11 808.
  * [27] S. Dai, Z. Fei, Q. Ma, A. Rodin, M. Wagner, A. S. McLeod, M. K. Liu, W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M. Thiemens, G. Dominguez, A. H. Castro Neto, A. Zettl, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, D. N. Basov,  _Science_ 2014, _343_ , 6175 1125.
  * [28] S. Dai, Q. Ma, Y. Yang, J. Rosenfeld, M. D. Goldflam, A. McLeod, Z. Sun, T. I. Andersen, Z. Fei, M. Liu, Y. Shao, K. Watanabe, T. Taniguchi, M. Thiemens, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, D. N. Basov,  _Nano Lett_ 2017, _17_ , 9 5285.
  * [29] P. B. Johnson, R.-W. Christy,  _Phys Rev B_ 1972, _6_ , 12 4370.
  * [30] A. D. Rakić, A. B. Djurišić, J. M. Elazar, M. L. Majewski,  _Appl Optics_ 1998, _37_ , 22 5271.
  * [31] S. G. Menabde, S. Boroviks, J. Ahn, J. T. Heiden, K. Watanabe, T. Taniguchi, T. Low, D. K. Hwang, N. A. Mortensen, M. S. Jang,  _Science Adv_ 2022, _8_ , 28 eabn0627.
  * [32] D. Datz, G. Németh, K. E. Walker, G. A. Rance, Á. Pekker, A. N. Khlobystov, K. Kamarás,  _ACS Appl Nano Mater_ 2021, _4_ , 5 4335.
  * [33] C. A. Phillips,  Ph.D. thesis, University of Toronto (Canada), 2022.
  * [34] J. Roth, M. J. Dignam,  _J Opt Soc Am_ 1973, _63_ , 3 308.
  * [35] T. Pusztai, G. Oszlányi, G. Faigel, K. Kamarás, L. Gránásy, S. Pekker,  _Solid State Commun_ 1999, _111_ , 11 595.
  * [36] É. Kováts, G. Oszlányi, S. Pekker,  _J Phys Chem B_ 2005, _109_ , 24 11913.
  * [37] G. Klupp, F. Borondics, É. Kováts, Á. Pekker, G. Bényei, I. Jalsovszky, R. Hackl, S. Pekker, K. Kamarás,  _J Phys Chem B_ 2007, _111_ , 43 12375.
  * [38] S. G. Stepanian, V. A. Karachevtsev, A. M. Plokhotnichenko, L. Adamowicz, A. M. Rao,  _J Phys Chem B_ 2006, _110_ , 32 15769.
  * [39] H.-Y. Xie, M. Chen, Y.-C. Chang, R. S. Moirangthem,  _Plasmonics_ 2015, _10_ 847\.
  * [40] G. Kristensson, P. C. Waterman,  _J Acoust Soc Am_ 1982, _72_ , 5 1612.
  * [41] W.-Z. Yan, Y. Du, H. Wu, D. Liu, B.-I. Wu,  _Prog Electromagn Res_ 2008, _85_ 39\.
  * [42] P. B. Bareil, Y. Sheng,  _J Opt Soc Am A_ 2013, _30_ , 1 1.


# Nucleon resonances and transition form factors

V. D. Burkert Thomas Jefferson National Accelerator Facility, Newport News,
VA, USA

##  .1 Abstract

This is a contribution to the review ”50 Years of Quantum Chromodynamics”
edited by F. Gross and E. Klempt, to be published in Journal EPJC. This
contribution reviews the nucleon resonance transition form factors determined
from meson electro-production experiments at electron accelerator facilities,
i.e. this contribution focuses on ”space-like” transition form factors and
amplitudes. Comparisons are made when available to LQCD and to approaches with
traceable links to strong QCD and to advanced quark model calculations.

##  .2 Introduction and Formalism

Meson photoproduction has become an essential tool in the search for new
excited light-quark baryon states. As discussed in the previous section, many
new excited states have been discovered thanks to high precision
photoproduction data in different final states Anisovich  _et al._ (2017), and
are now included in recent editions of the Review of Particle Physics (RPP)
Workman (2022). The exploration of the internal structure of excited states
and the effective degrees of freedom contributing to s-channel resonance
excitation requires the use of electron beams, which is the subject of this
contribution, where the virtuality ($Q^{2}$) of the exchanged photon can be
varied to pierce through the peripheral meson cloud and probe the quark core
and its spatial structure. Electroproduction can thus say something about if a
resonance is generated through short distance photon interaction with the
small quark core, or through interaction with a more extended hadronic system.

The experimental exploration of resonance transition form factors reaches over
60 years with many review articles describing this history. Here we refer to a
few recent ones Stoler (1993); Burkert and Lee (2004); Aznauryan and Burkert
(2012a); Aznauryan  _et al._ (2013). A review of recent electroproduction
experiments in hadron physics and their interpretation within modern
approaches of strong interaction physics can be found in Ref. Brodsky  _et
al._ (2020).

Electroproduction of final states with pseudoscalar mesons (e.g. $N\pi$,
$p\eta$, $K\Lambda$) have been employed at Jefferson Laboratory mostly with
the CEBAF Large Acceptance Spectrometer (CLAS) operating at an instantaneous
luminosity of $10^{34}$sec-1cm-2. In Hall A and Hall C, pairs of individual
well-shielded focusing magnetic spectrometers are employed with more
specialized aims and limited acceptance, but operating at much higher
luminosity. This experimental program led to new insights into the scale
dependence of effective degrees of freedom, e.g. meson-baryon, constituent
quarks, and dressed quark contributions. Several excited states, shown in Fig.
1 assigned to their primary $SU(6)\otimes O(3)$ supermultiplets, have been
studied this way, mostly with CLAS in Hall B.

Figure 1: Excited states of the proton that have been studied in
electroproduction to determine their resonance transition amplitudes or form
factors. States highlighted in red are discussed in this subsection. Graphics
from Ref. Burkert (2020).

Most of the resonance couplings have been extracted from single pseudoscalar
meson production. In electroproduction, there are 6 complex helicity
amplitudes, requiring a minimum of 11 independent measurements for a complete
111With the exception of an overall phase that cannot be determined model-
independent determination of the amplitudes. In addition, measurements of
isospin amplitudes require additional measurements. Following this, the
complex amplitudes would need to be subjected to analyses of their phase
motions to determine resonance masses on the (real) energy axis, or poles in
the (complex) energy plane. Fortunately, in the lower mass range a variety of
constraints can be applied to limit the number of unknowns when fitting the
cross section data. These include the masses of states quite well known from
hadronic processes or from meson photoproduction. Also, the number of possible
angular momenta is limited to $l_{\pi}\leq 2$ in the examples discussed in the
following. Additional constraints come from the Watson theorem Watson (1954)
that relates the electromagnetic phases to the hadronic ones, and the use of
dispersion relations, assuming the imaginary parts of the amplitude are given
by the resonance contribution, and the real parts determined through
dispersion integrals and additional pole terms. Other approaches use unitary
isobar models that parameterize all known resonances and background terms, and
unitarize the full amplitudes in a K-matrix procedure. In the following, we
show results based on both approaches, where additional systematic
uncertainties have been derived from the differences in the two procedures.

The availability of electron accelerators with the possibility of generating
high beam currents at CEBAF at Jefferson Lab in the US and MAMI at Mainz in
Germany, has enabled precise studies of the internal structure of excited
states in the $N^{*}$ and the $\Delta^{*}$ sectors employing s-channel
resonance excitations in large ranges of photon virtuality $Q^{2}$. This has
enabled probing the degrees of freedom relevant in the resonance excitation as
a function of the distance scale probed. This is the topic we will elucidate
in the following sections and the relevance to (approximations to) QCD. First
we briefly discuss the formalism needed for a quantitative analysis of the
single pseudoscalar meson electroproduction.

Figure 2: The kinematics of single $\pi^{+}$ electro-production off protons in
the laboratory system.

###  .2.1 Formalism in the analysis of electroproduction of single
pseudoscalar mesons

The simplest process used in the study of resonance transition amplitudes is
single pion or kaon production, e.g. $ep\to e\pi^{+}n$. Single $\pi^{+}$ and
$\pi^{0}$ production are most suitable for the study of the lower-mass excited
states as they couple dominantly to the excited states with masses up to
$\approx 1.7$ GeV. It may then be useful to describe in more detail the
analysis techniques and the formalism used. The unpolarized differential cross
section for single pseudoscalar meson production can be written in the one-
photon exchange approximation as:

$\displaystyle\frac{d\sigma}{dE_{f}d\Omega_{e}d\Omega_{\pi}}$ $\displaystyle=$
$\displaystyle\Gamma{\frac{d\sigma}{d\Omega_{\pi}}}\,,$ (1)

where $\Gamma$ is the virtual photon flux,

$\displaystyle\Gamma=\frac{\alpha_{em}}{2\pi^{2}Q^{2}}\frac{(W^{2}-M^{2})E_{f}}{2ME_{i}}\frac{1}{1-\epsilon},$
(2)

where $M$ is the proton mass, $W$ the mass of the hadronic final state,
$\epsilon$ is the photon polarization parameter, $Q^{2}$ the photon
virtuality, $E_{i}$ and $E_{f}$ represent the initial and the final electron
energies, respectively. Moreover,

$\displaystyle\epsilon=\left[1+2\left(1+\frac{\nu^{2}}{Q^{2}}\right)\tan^{2}\frac{\theta_{e}}{2}\right]^{-1}$
(3)

and

$\displaystyle\frac{d\sigma}{d\Omega_{\pi}}$ $\displaystyle=$
$\displaystyle\sigma_{T}+\epsilon\sigma_{L}+\epsilon\sigma_{TT}\cos
2\phi_{\pi}$
$\displaystyle\qquad\qquad\qquad+\sqrt{2\epsilon(1+\epsilon)}\sigma_{LT}\cos{\phi}_{\pi}\,.$

The kinematics for single $\pi^{+}$ production is shown in Fig. 2.

The observables of the process $\gamma_{v}p\to\pi N^{\prime}$ can be expressed
in terms of six parity-conserving helicity amplitudes Aznauryan and Burkert
(2012a); Walker (1969); Berends  _et al._ (1967) :

$\displaystyle{H_{i}=\left<\lambda_{\pi};\lambda_{N}|T|\lambda_{\gamma_{\nu}};\lambda_{p}\right>,}$
(4)

where $\lambda$ denotes the helicity of the respective particle,
$\lambda_{\pi}=0$, $\lambda_{N}=\pm\frac{1}{2}$, $\lambda_{\gamma_{v}}=\pm
1,0$, and $\lambda_{p}=\pm\frac{1}{2}$, and $H_{i}$ are complex functions of
$Q^{2}$, $W$, and $\theta^{*}_{\pi}$.

###  .2.2 Multipoles and partial wave decompositions

The response functions in (1) are given by:

$\displaystyle\sigma_{T}$ $\displaystyle=$ $\displaystyle\frac{\vec{\it
p}_{\pi}W}{2KM}(|H_{1}|^{2}+|H_{2}|^{2}+|H_{3}|^{2}+|H_{4}|^{2}),$ (5)
$\displaystyle\sigma_{L}$ $\displaystyle=$ $\displaystyle\frac{\vec{\it
p}_{\pi}W}{2KM}(|H_{5}|^{2}+|H_{6}|^{2}),$ (6) $\displaystyle\sigma_{TT}$
$\displaystyle=$ $\displaystyle\frac{\vec{\it
p}_{\pi}W}{2KM}Re(H_{2}H_{3}^{*}-H_{1}H_{4}^{*}),$ (7)
$\displaystyle\sigma_{LT}$ $\displaystyle=$ $\displaystyle\frac{\vec{\it
p}_{\pi}W}{2KM}Re[H_{5}^{*}(H_{1}-H_{4})+H_{6}^{*}(H_{2}+H_{3})]\,,$ (8)

where $\vec{\it p}_{\pi}$ is the pion 3-momentum in the hadronic center-of-
mass system, and $K$ is the equivalent real photon lab energy needed to
generate a state with mass $W$:

$\displaystyle K=\frac{W^{2}-M^{2}}{2M}\,.$ (9)

The helicity amplitudes $H_{i},i=1$–6, can be expanded into Legendre
polynomials:

$\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\sin\theta\cos{\frac{\theta}{2}}\sum_{l=1}^{\infty}(B_{l+}-B_{(l+1)-})(P^{\prime\prime}_{l}-P^{\prime\prime}_{l+1})$
$\displaystyle H_{2}$ $\displaystyle=$
$\displaystyle\sqrt{2}\cos{\frac{\theta}{2}}\sum_{l=1}^{\infty}(A_{l+}-A_{(l+1)-})(P^{\prime}_{l}-P^{\prime}_{l+1})$
$\displaystyle H_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\sin\theta\sin{\frac{\theta}{2}}\sum_{l=1}^{\infty}(B_{l+}+B_{(l+1)-})(P^{\prime\prime}_{l}+P^{\prime\prime}_{l+1})$
$\displaystyle H_{4}$ $\displaystyle=$
$\displaystyle\sqrt{2}\sin{\frac{\theta}{2}}\sum_{l=1}^{\infty}(A_{l+}+A_{(l+1)-})(P^{\prime}_{l}+P^{\prime}_{l+1})$
$\displaystyle H_{5}$ $\displaystyle=$
$\displaystyle\sqrt{2}\cos{\frac{\theta}{2}}\sum_{l=1}^{\infty}(C_{l+}-C_{(l+1)-})(P^{\prime}_{l}-P^{\prime}_{l+1})$
$\displaystyle H_{6}$ $\displaystyle=$
$\displaystyle\sqrt{2}\sin{\frac{\theta}{2}}\sum_{l=1}^{\infty}(C_{l+}+C_{(l+1)-})(P^{\prime}_{l}+P^{\prime}_{l+1})\,,$
(10)

where the $A_{l+}$ and $B_{l+}$ etc., are the transverse partial wave helicity
elements for $\lambda_{\gamma p}=\frac{1}{2}$ and $\lambda_{\gamma
p}=\frac{3}{2}$, and $C_{\pm}$ the longitudinal partial wave helicity
elements. In the subscript, $l+$ and $(l+1)-$ define the $\pi$ orbital angular
momenta, and the sign $\pm$ is related to the total angular momentum
$J=l_{\pi}\pm\frac{1}{2}$. In the analysis of data on the $N\Delta(1232)$
transition, linear combinations of partial wave helicity elements are
expressed in terms of electromagnetic multipoles:

$\displaystyle M_{l+}$ $\displaystyle=$
$\displaystyle\frac{1}{2(l+1)}[2A_{l+}-(l+2)B_{l+}]$ (11) $\displaystyle
E_{l+}$ $\displaystyle=$ $\displaystyle\frac{1}{2(l+1)}(2A_{l+}+lB_{l+})$ (12)
$\displaystyle M_{l+1,-}$ $\displaystyle=$
$\displaystyle\frac{1}{2(l+1)}(2A_{l+1,-}-lB_{l+1,-})$ (13) $\displaystyle
E_{l+1,-}$ $\displaystyle=$
$\displaystyle\frac{1}{2(l+1)}[-2A_{l+1,-}+(l+2)B_{l+1,-}]$ (14)
$\displaystyle S_{l+}$ $\displaystyle=$
$\displaystyle\frac{1}{l+1}\sqrt{\frac{{\vec{\it Q^{*}}^{2}}}{Q^{2}}}C_{l+}$
(15) $\displaystyle S_{l+1,-}$ $\displaystyle=$
$\displaystyle\frac{1}{l+1}\sqrt{\frac{{\vec{\it
Q^{*}}^{2}}}{Q^{2}}}C_{l+1,-}\,,$ (16)

where $\vec{\it Q}^{*}$ is the photon 3-momentum in the hadronic rest frame.
The electromagnetic multipoles are often used to describe the transition from
the nucleon ground state to the $\Delta(1232)$, which is dominantly described
as a magnetic dipole transition $M_{1+}$. The electromagnetic multipoles as
well as the partial wave helicity elements are complex quantities and contain
both non-resonant and resonant contributions. In order to compare the results
to model predictions and LQCD, an additional analysis must be performed to
separate the resonant parts $\hat{\rm A}_{\pm}$, $\hat{\rm B}_{\pm}$, etc.,
from the non-resonant parts of the amplitudes. In a final step, the known
hadronic properties of a given resonance can be used to determine
photocoupling helicity amplitudes that characterize the electromagnetic
vertex:

$\displaystyle\hat{A}_{l\pm}$ $\displaystyle=$ $\displaystyle\mp FC^{I}_{\pi
N}A_{1/2},$ (17) $\displaystyle\hat{B}_{l\pm}$ $\displaystyle=$
$\displaystyle\pm F\sqrt{\frac{16}{(2j-1)(2j+3)}}C^{I}_{\pi N}A_{3/2},$ (18)
$\displaystyle\hat{S}_{l\pm}$ $\displaystyle=$
$\displaystyle-F\frac{2\sqrt{2}}{2J+1}C^{I}_{\pi N}S_{1/2},$ (19)
$\displaystyle F$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{(2j+1)}\pi}\frac{K}{p_{\pi}}\frac{\Gamma_{\pi}}{\Gamma^{2}}$

where the $C^{I}_{\pi N}$ are isospin coefficients. The total transverse
absorption cross section for the transition into a specific resonance is given
by:

$\displaystyle\sigma_{T}=\frac{2M}{W_{R}\Gamma}(A^{2}_{1/2}+A^{2}_{3/2}).$
(20)

Experiments in the region of the $\Delta(1232)\frac{3}{2}^{+}$ resonance often
determine the electric quadrupole ratio $R_{EM}$

$R_{EM}=\frac{Im(E_{1+})}{Im(M_{1+})}$ (21)

and the scalar quadrupole ratio $R_{SM}$

$R_{SM}=\frac{Im(S_{1+})}{Im(M_{1+})}$ (22)

where $E_{1+}$, $S_{1+}$, and $M_{1+}$ are the electromagnetic transition
multipoles at the mass of the $\Delta(1232)\frac{3}{2}^{+}$ resonance.

It is worth noting that the electric and the scalar transition amplitudes are
connected at the so-called pseudo-threshold $Q^{2}_{pt}=-(W-M)^{2}$ through
the Siegert theorem Siegert (1937), whose impact on the extraction of nucleon
resonance transition amplitudes from pion electro-production is discussed in
Tiator (2016); Ramalho (2016).

###  .2.3 Resonance analysis tools

A model-independent determination of the amplitudes contributing to the
electro-excitation of resonances in single pseudoscalar pion production $ep\to
e^{\prime}N\pi$ (see kinematics of single pion production in Fig. 2) requires
a large number of independent measurements at each value of the electron
kinematics $W$, $Q^{2}$, the hadronic cms angle $\cos{\theta^{\pi}}$, and the
azimuthal angle $\phi^{\pi}$ describing the angle between the electron
scattering plane and the hadronic decay plane. Such a measurement requires
full exclusivity of the final state and employing both polarized electron
beams and the measurements of the nucleon recoil polarization.

Figure 3: JLab/Hall A data for $\vec{e}p\to e\vec{p}\pi^{0}$ response
functions at W=1.232 GeV and $Q^{2}=1.0$ Kelly  _et al._ (2007). Notations
refer to transverse (t), normal (n) and longitudinal (l) components of the
proton recoil polarization. The curves correspond to results obtained using
SAID (short dashed), MAID (dashed-dotted), and the dynamical models DMT
Kamalov  _et al._ (2001) (dotted), and SL Sato and Lee (2001) (long-
dashed/green). The other curves correspond to Legendre and multipole fits
performed by the authors.

Such measurements would in general require full $4\pi$ coverage for the
hadronic final state. The only measurement that could claim to be complete was
carried out at Jefferson Lab in Hall A Kelly  _et al._ (2007) employing a
limited kinematics centered at resonance for $\vec{e}p\to
e^{\prime}\vec{p}\pi^{0}$ at $W=1.232$ GeV, and $Q^{2}\approx 1$ GeV2. Figure
5 shows the 16 response functions extracted from this measurement. The results
of this measurement in terms of the magnetic $N\Delta$ transition form factor
and the quadrupole ratios are included in Fig. 4 among other data. They
coincide very well with results of other experiments Aznauryan  _et al._
(2009); Ungaro  _et al._ (2006); Joo  _et al._ (2002); Frolov  _et al._ (1999)
using different analysis techniques that may be also applied to broader
kinematic conditions, especially higher mass resonances. Details of the latter
are discussed in Aznauryan and Burkert (2012a); Tiator  _et al._ (2011). We
briefly summarize them here:

|  |
---|---|---
Figure 4: The $N\Delta(1232)$ transition amplitudes. Left: The magnetic
N$\Delta$ transition form factor normalized to the dipole form factor and
compared with the Light-Front Relativistic Quark Model (LFRQM)Aznauryan and
Burkert (2012b, 2016) with running quark mass, and with results using the
Dyson-Schwinger Equation Segovia  _et al._ (2014). Both predictions are close
to the data at high $Q^{2}$. At $Q^{2}<3$GeV2 meson-baryon contributions are
significant. Middle: The electric (top) and scalar (bottom)
quadrupole/magnetic-dipole ratios $R_{EM}$ and $R_{SM}$. Right: $R_{EM}$ and
$R_{SM}$ from Lattice QCD Alexandrou  _et al._ (2008); Behrndt and Cvetic
(2005) compared to data in the low $Q^{2}$ domain.

  * •

Dispersion Relations have been employed in two ways: One is based on fixed-t
dispersion relations for the invariant amplitudes and was successfully used
throughout the nucleon resonance region. Another way is based on DR for the
mulipole amplitudes of the $\Delta(1232)$ resonance, and allows getting
functional forms of these amplitudes with one free parameter for each of them.
It was employed for the analysis of the more recent data.

  * •

The Unitary Isobar Model (UIM)  was developed in Drechsel  _et al._ (1999)
from the effective Lagrangian approach for pion photoproduction Peccei (1969).
Background contributions from t-channel $\rho$ and $\omega$ exchanges are
introduced and the overall amplitude is unitarized in a K-matrix
approximation.

  * •

Dynamical Models have been developed, as SAID from pion photoproduction data
Arndt  _et al._ (2002), the Sato-Lee model was developed in Sato and Lee
(1996). Its essential feature is the consistent description of $\pi N$
scattering and the pion electroproduction from nucleons. It was utilized in
the study of $\Delta(1232)$ excitations in the $ep\to ep\pi^{0}$ channel Sato
and Lee (2001). The Dubna-Mainz-Taipei model Kamalov and Yang (1999) builds
unitarity via direct inclusion of the $\pi N$ final state in the T-matrix of
photo- and electroproduction.

##  .3 Light-quark baryon resonance electroproduction

In order to learn from the meson electroproduction data about the internal
spin and spatial electromagnetic structure, it is essential to have advanced
models available with links to the fundamentals of $QCD$.

Figure 5: Sample of results of an analysis by the JLab group of the Legendre
moments of $\vec{e}p\to e\pi^{+}n$ structure functions in comparison with
experimental data Park  _et al._ (2008) at $Q^{2}=2.44$ GeV2. The solid
(dashed) curves correspond to results obtained using the DR (UIM) approach.

While most of the analyses have focused on single pseudoscalar meson
production, such as $\gamma_{v}p\to N\pi,p\eta,K\Lambda$, $K\Sigma$, more
recent work included the $p\pi^{+}\pi^{-}$ final state both in real
photoproduction Golovatch  _et al._ (2019) as well as in electroproduction
Mokeev  _et al._ (2020). The 2-pion final state has more sensitivity to
excited $N^{*}$ and $\Delta^{*}$ states in the mass range above 1.6 GeV, with
several states dominantly coupling to $N\pi\pi$ final states, enabling the
study of their electromagnetic transition form factors in the future.

###  .3.1 The $N\Delta(1232)\frac{3}{2}^{+}$ transition

The $\Delta^{++}$ isobar was first observed 70 years ago in Enrico Fermi’s
experiment that used a $\pi^{+}$ meson beam scattered off the protons in a
hydrogen target Anderson  _et al._ (1952). The cross section showed a sharp
rise above threshold towards a mass near 1200 MeV. While the energy of the
pion beam was not high enough to see the maximum and the fall-off following
the peak, a strong indication of the first baryon resonance was observed. It
took 12 more years and the development of the underlying symmetry in the quark
model before a microscopic explanation of this observation could emerge. There
was, however, a problem; while the existence of the $\Delta^{+,0,-}$ could be
explained within the model, the existence of the $\Delta(1232)^{++}$, which
within the quark model would correspond to a state
$|u{\uparrow}u{\uparrow}u{\uparrow}\rangle$, was forbidden as it would have an
overall symmetric wave function. It took the introduction of para Fermi
statistics Greenberg (1964) what later became ”color”, to make the overall
wave function anti-symmetric. In this way the $\Delta^{++}(1232)$ resonance
may be considered a harbinger of the development of QCD.

The nucleon to $\Delta(1232)\frac{3}{2}^{+}$ transition is now well measured
in a large range of $Q^{2}$ Ungaro  _et al._ (2006); Joo  _et al._ (2002);
Frolov  _et al._ (1999). At the real photon point, it is explained by a
dominant magnetic transition from the nucleon ground state to the
$\Delta(1232)$ excited state. Additional contributions are related to small
D-wave components in both the nucleon and the $\Delta(1232)$ wave functions
leading to electric quadrupole and scalar quadrupole transitions. These remain
in the few % ranges at small $Q^{2}$. The magnetic transition is to $\approx
65\%$ given by a simple spin flip of one of the valence quarks as seen in Fig.
4. The remaining 35% of the magnetic dipole strength is attributed to meson-
baryon contributions. There exist extensive theoretical reviews of the
$N\Delta(1232)$ transition in the lower $Q^{2}$ range are available Pascalutsa
and Vanderhaeghen (2006), and more recent reviews that cover the full $Q^{2}$
covered by data Aznauryan and Burkert (2012a); Tiator  _et al._ (2011).

The electric quadrupole ratio $R_{EM}$ was found as:

$\displaystyle R_{EM}\approx-0.02.$ (23)

There has been a longstanding prediction of asymptotic pQCD, that
$R_{EM}\to+1$ at $Q^{2}\to\infty$. Results on the magnetic transition form
factor $\rm G_{Mn,Ash}$, defined in the Ash convention Ash  _et al._ (1967),
and on the quadrupole transition ratios are shown in Fig. 4.

---
Figure 6: Helicity transition amplitudes for the proton to Roper
$N(1440)\frac{1}{2}^{+}$ excitation in units of $10^{-3}\rm{GeV}^{-1/2}$
compared to various model calculations; see text. Left: Transverse $A_{1/2}$
amplitude. Middle: Scalar $S_{1/2}$ amplitude. Right: Helicity amplitudes of
the Roper resonance at low $Q^{2}$. Data are compared to calculations within
Effective Field Theory Bauer  _et al._ (2014) shown in solid black lines. The
other broken lines are parts of the full calculations. The data are from
Aznauryan  _et al._ (2009); Mokeev  _et al._ (2012); Drechsel  _et al._
(2007). The open red circle at $Q^{2}\approx 0.1$GeV2 is the result of an
analysis of $ep\to ep\pi^{0}$ data from Štajner  _et al._ (2017).

$\rm G_{Mn,Ash}$ is shown normalized to the dipole form factor, but indicates
a much faster $Q^{2}$ fall-off compared to that. In comparison to the advanced
LFRQM with momentum-dependent constituent quark mass, and with the Dyson-
Schwinger Equation (DSE-QCD) results, there is good agreement at the
high-$Q^{2}$ end of the data. The discrepancy at small $Q^{2}=0$ is likely due
to the meson-baryon contributions at small $Q^{2}$, which are not modeled in
either of the calculations.

The quadrupole ratio $R_{EM}$ shows no sign of departing significantly from
its value at $Q^{2}=0$, even at the highest $Q^{2}\approx 6.5$ GeV2. Both
calculations barely depart from $R_{EM}=0$, and remain near zero at all
$Q^{2}>2$ GeV2. This indicates that the negative constant value shown by the
data is likely due to meson-baryon contributions that are not included in the
theoretical models. For the scalar quadrupole ratio $R_{SM}$ the asymptotic
prediction in holographic QCD (hQCD) Grigoryan  _et al._ (2009) is:

$\displaystyle R_{SM}=\frac{ImS_{1+}}{ImM_{1+}}\to-1,~{}~{}{\rm
at~{}Q^{2}\to\infty},$ (24)

while $R_{EM}$ in hQCD is predicted to approach +1 asymptotically. The
$R_{SM}$ data show indeed a strong trend towards increasing negative values at
larger $Q^{2}$, semi-quantitatively described by both calculations at
$Q^{2}<4$ GeV2. The Dyson-Schwinger equation approach predicts a flattening of
$R_{SM}$ at $Q^{2}>4$ GeV2, while the Light Front relativistic Quark Model
predicts a near constant negative slope of $R_{SM}(Q^{2})$ also at higher
$Q^{2}$.

###  .3.2 The Roper resonance $N(1440)\frac{1}{2}^{+}$

The Roper resonance, discovered in 1964 Roper (1964) in a phase shift analysis
of elastic $\pi N$ scattering data, has been differently interpreted for half
a century. In the non-relativistic quark model (nrQM), the state is the first
radial excitation of the nucleon ground state with a mass expected around 1750
MeV, much higher than the measured Breit-Wigner mass of $\approx 1440$ MeV.
This discrepancy is now understood as the consequence of a dynamical coupled
channel effect that shifts the mass below the mass of the $N(1535){1/2}^{-}$
state, the negative-parity partner of the nucleon Suzuki  _et al._ (2010).
Another problem with the quark model was the sign of the transition form
factor $A_{1/2}(Q^{2}=0)$, predicted in the nrQM as large and positive, while
experimental analyses showed a negative value.

These discrepancies resulted in different interpretations of the state that
could only be resolved with electroproduction data from CLAS at Jefferson Lab,
the development of continuous QCD approximations in the Dyson-Schwinger
equation approach Segovia  _et al._ (2015) and Light Front Relativistic QM
with momentum-dependent quark masses Aznauryan and Burkert (2012b) shown in
Fig. 6, and Lattice data Mathur  _et al._ (2005); Lin and Cohen (2012). A
recent review of the history and current status of the Roper resonance, is
presented in a colloquium-style article published in Review of Modern Physics
Burkert and Roberts (2019).

Descriptions of the baryon resonance transitions form factors, including the
Roper resonance $N(1440)\frac{1}{2}^{+}$, have also been carried out within
holographic models de Teramond and Brodsky (2012); Ramalho and Melnikov
(2018). In the range $Q^{2}<0.6$ GeV2, calculations based on meson-baryon
degrees of freedom and effective field theory Bauer  _et al._ (2014) have been
successfully performed, as may be seen in Fig. 6. Earlier model descriptions,
such as the Isgur-Karl model that describe the nucleon as a system of 3
constituent quarks in a confining potential and a one-gluon exchange
contribution leading to a magnetic hyperfine splitting of states Isgur and
Karl (1978, 1977), and the relativized version of Capstick Capstick and Isgur
(1986) have popularized the model that became the basis for many further
developments and variations, e.g. the light front relativistic quark model,
and the hypercentral quark model Giannini  _et al._ (2003). Other models were
developed in parallel. The cloudy bag model Thomas  _et al._ (1981) describes
the nucleon as a bag of 3 constituent quarks surrounded by a cloud of pions.
It has been mostly applied to nucleon resonance excitations in real
photoproduction, $Q^{2}=0$ Bermuth  _et al._ (1988); Thomas  _et al._ (1981),
with some success in the description of the $\Delta(1232)\frac{3}{2}^{+}$ and
the Roper resonance transitions.

There is agreement with the data at $Q^{2}>1.5$ GeV2 for these two states,
while the meson-baryon contributions for the $\Delta(1232)$ are more extended,
and agreement with the quark based calculations is reached at $Q^{2}>4$ GeV2.
The calculations deviate significantly from the data at lower $Q^{2}$, which
indicates the presence of non-quark core effects. For the Roper resonance such
contributions have been described successfully in dynamical meson-baryon
models Obukhovsky  _et al._ (2011) and in effective field theory Bauer  _et
al._ (2014). Calculations on the Lattice for the N-Roper transition form
factors $F_{1}^{pR}$ and $F_{2}^{pR}$, which are combinations of the
transition amplitudes $A_{1/2}$ and $S_{1/2}$, have been carried out with
dynamical quarks Lin and Cohen (2012). The results agree well with the data in
the range $Q^{2}<1.0$ GeV2, where data and calculations overlap Fig. 7.

New electroproduction data on the Roper Štajner  _et al._ (2017) and on
several higher mass states have been obtained in the 2-pion channel,
specifically in $ep\to e^{\prime}p\pi^{+}\pi^{-}$ Mokeev  _et al._ (2016).

---
Figure 7: Dirac and Pauli transition form factors $F_{1}$ and $F_{2}$ for the
proton to $N(1440)1/2^{+}$ transition compared to Lattice QCD calculations Lin
and Cohen (2012) with pion masses (in GeV): 0.39 (red squares), 0.45 (orange
triangles), and 0.875 (green circles) on the $N_{f}=2+1$ anisotropic lattices,
compared to CLAS results (black circles). The $F_{1}$ and $F_{2}$ form factors
are linear combinations of the $A_{1/2}$ and $S_{1/2}$ amplitudes.

The mass of the Roper state has been computed on the Lattice and extrapolated
to the physical pion mass, showing good agreement with the physical value
measured with a Breit-Wigner parametrization. It should be noted that the
Roper mass measured at the pole in the complex plane is significantly
different from the value obtained using the BW ansatz.

Figure 8: The transverse helicity transition amplitudes of
$N(1520)\frac{3}{2}^{-}$ versus $Q^{2}$, compared to the LFRQM, $A_{1/2}$
(left), $A_{3/2}$ (middle). The shaded area indicates the contribution from
non-quark contributions as estimated from the difference of the measured data
points and the LF RQM contribution, likely due to hadronic contributions.
Right: Helicity asymmetry $A_{hel}$, as defined in Eq. 26. Graphics from Ref.
Aznauryan and Burkert (2012a)

Supported by an extensive amount of single pion electroproduction data,
covering the full phase space in the pion polar and azimuthal center-of-mass
angles, and accompanied by several theoretical modeling, we can summarize our
current understanding of the $N(1440)\frac{1}{2}^{+}$ state as follows:

  * •

The Roper resonance is, at heart, the first radial excitation of the nucleon.

  * •

It consists of a well-defined dressed-quark core, which plays a role in
determining the system’s properties at all length scales, but exerts a
dominant influence on probes with $Q^{2}>m_{N}^{2}$, where $m_{N}$ is the
nucleon mass.

  * •

The core is augmented by a meson cloud, which both reduces the Roper’s core
mass by $\approx 20\%$, thereby solving the mass problem that was such a
puzzle in constituent quark model treatments, and, at low $Q^{2}$, contributes
an amount to the electroproduction transition form factors that is comparable
in magnitude with that of the dressed quark core, but vanishes rapidly as
$Q^{2}$ is increased beyond $m_{N}^{2}$.

As stated in the conclusions of Burkert and Roberts (2019): ”The fifty years
of experience with the Roper resonance have delivered lessons that cannot be
emphasized too strongly. Namely, in attempting to predict and explain the QCD
spectrum, one must fully consider the impact of meson-baryon final state
interactions and the coupling between channels and states that they generate,
and look beyond merely locating the poles in the S-matrix, which themselves
reveal little structural information, to also consider the $Q^{2}$
dependencies of the residues, which serve as a penetrating scale-dependent
probe of resonance composition.”

###  .3.3 The helicity structure of the $N(1520)\frac{3}{2}^{-}$

The $N(1520)\frac{3}{2}^{-}$ state corresponds to the lowest excited nucleon
resonance with $J^{P}=\frac{3}{2}^{-}$. Its helicity structure is defined by
the $Q^{2}$ dependence of the two transverse transition amplitudes $A_{1/2}$
and $A_{3/2}$. They are both shown in Fig. 8. A particularly interesting
feature of this state is that at the real photon point, $A_{3/2}$ is strongly
dominant, while $A_{1/2}$ is very small. However, at high $Q^{2}$, $A_{1/2}$
is becoming dominant, while $A_{3/2}$ drops rapidly. This behavior is
qualitatively consistent with the expectation of asymptotic QCD, which
predicts the transition helicity amplitudes to behave like:

$\displaystyle\hskip
71.13188ptA_{1/2}\propto\frac{a}{Q^{3}},A_{3/2}\propto\frac{b}{Q^{5}}\,.$ (25)

The helicity asymmetry

$\displaystyle\hskip
71.13188ptA_{hel}=\frac{A_{1/2}^{2}-A_{3/2}^{2}}{A_{1/2}^{2}+A_{3/2}^{2}},$
(26)

shown in Fig. 8, illustrates this rapid change in the helicity structure of
the $\gamma_{v}pN(1520){3/2}^{-}$ transition. At $Q^{2}>2~{}$GeV2, $A_{1/2}$
fully dominates the process.

Figure 9: Right: The transverse transition helicity amplitude $A_{1/2}$ versus
$Q^{2}$. At $Q^{2}>2$GeV2 the data are well described by the light-cone sum
rules LCSR Braun  _et al._ (2009). The light front relativistic quark model
(LFRQM) Aznauryan and Burkert (2017) describes that data at $Q^{2}>1$GeV2.

###  .3.4 Transition Form Factors of $N(1535)\frac{1}{2}^{-}$ \- A state with
a hard quark core.

This state is the parity partner state to the ground state nucleon, with the
same spin 1/2 but with opposite parity, its quark content requires an orbital
L=1 excitation in the transition from the proton. In the $SU(6)\otimes O(3)$
symmetry scheme, the state is a member of the $[70,1^{-}]$ super multiplet.
This state couples equally to $N\pi$ and to $N\eta$ final state. It has
therefore be probed using both decay channels $ep\to ep\eta$ and $ep\to
eN\pi^{+,0}$. Because of isospin $I=1/2$ for nucleon states, the coupling to
the charged $\pi^{+}n$ channel is preferred over $\pi^{0}p$ owing to the
Clebsch-Gordon coefficients.

The $A_{1/2}$ helicity amplitude for the $\gamma pN(1535)\frac{1}{2}^{-}$
resonance excitation shown in Fig. 9 represents the largest range in $Q^{2}$
of all nucleon states for which resonance transition form factors have been
measured as part of the broad experimental program at JLab.

For this state, as well as for the $N(1440)\frac{1}{2}^{+}$ state, advanced
relativistic quark model calculations Aznauryan and Burkert (2015a), DSE-QCD
calculations Segovia  _et al._ (2015) and Light Cone sum rule results Anikin
_et al._ (2015) are available, employing QCD-based modeling of the excitation
of the quark core for the first time.

---
Figure 10: The transverse amplitudes of the proton to $N(1675)\frac{5}{2}^{-}$
transition compared to the LF RQM Aznauryan and Burkert (2015b), hypercentral
QM Santopinto and Giannini (2012), and contributions from meson-baryon (MB)
coupled channel dynamics Julia-Diaz  _et al._ (2008). Left: $A_{1/2}$,
Middle:$A_{3/2}$. Both quark models predict very small amplitudes for the
proton, while the meson-baryon contributions estimate is large and is close to
the data. Right: $A_{1/2}$ for neutron target (only photoproduction data
available) compared to the LFRQM and hCQM. Both quark models predict large
amplitudes for neutrons, more than factor 10 compared to protons at $Q^{2}=0$.
Assuming similar meson-baryon contributions as in the proton case with
opposite sign could quantitatively explain the single measured value at the
photon point.

The transverse transition amplitude $A_{1/2}$ of $N(1535)\frac{1}{2}^{-}$ is a
prime example of the power of meson electroproduction to unravel the internal
structure of the resonance transition. In another section of the Volume ”50
Years of Quantum Chromodynamics” Burkert  _et al._ (2022), the nature of this
state is discussed as a possible example of a dynamically generated resonance.
The electroproduction data shown here reveal structural aspects of the state
and its nature that require a different interpretation. The transition form
factor $A_{1/2}$ of the state, shown in Fig. 9, is quantitatively reproduced
over a large range in $Q^{2}$ by two alternative approaches, the LFRQM and the
LCSR. Both calculations are based on the assumptions of the presence of a
3-quark core. Notice that there is deviation from the quark calculations at
$Q^{2}<1-2$ GeV2, highlighted as the shaded area in Fig. 9, which may be
assigned to the presence of non-quark contributions. Attempts to compute the
transition form factors within strictly dynamical models have not succeeded in
explaining the available data Jido  _et al._ (2008). The discrepancy could be
resolved if the character of the probe, meson (pion) in the case of hadron
interaction and short range photon interaction in the case of
electroproduction, probe different parts of the resonance’s spatial structure:
peripheral in case of meson scattering and short distance behavior in electro-
production. The peripheral meson scattering and low $Q^{2}$ meson photo-
production reveal the dynamical features of the state, whereas high $Q^{2}$
electroproduction reveals the structure of the quark core.

###  .3.5 The $N(1675)\frac{5}{2}^{-}$ resonance reveals the meson-baryon
contributions

In previous discussions we have concluded that meson-baryon degrees of freedom
provide significant strength to the resonance excitation in the low-$Q^{2}$
domain where quark based approaches LF RQM, DSE/QCD, and LC SR calculations
fail to reproduce the transition amplitudes quantitatively. Our conclusion
rests, in part, with this assumption. But, how can we be certain of the
validity of this conclusion?

The $N(1675)\frac{5}{2}^{-}$ resonance allows us to test this assumption,
quantitatively. Figure 10 shows our current knowledge of the transverse
helicity amplitudes $A_{1/2}(Q^{2})$ and $A_{3/2}(Q^{2})$, for proton target
compared to RQM Aznauryan and Burkert (2017) and hypercentral CQM Santopinto
and Giannini (2012) calculations. The specific quark transition for a
$J^{P}=\frac{5}{2}^{-}$ state belonging to the $[SU(6)\otimes
O(3)]=[70,1^{-}]$ supermultiplet configuration, in non-relativistic
approximation prohibits the transition from the proton in a single quark
transition. This suppression, known as the Moorhouse selection rule Moorhouse
(1966), is valid for the transverse transition amplitudes $A_{1/2}$ and
$A_{3/2}$ at all $Q^{2}$. It should be noted that this selection rule does
apply to the transition from a proton target, it does not apply to the
transition from the neutron, which is consistent with the data. Modern quark
models that go beyond single quark transitions, confirm quantitatively the
suppression resulting in very small amplitudes from protons but large ones
from neutrons. Furthermore, a direct computation of the hadronic contribution
to the transition from protons confirms this (Fig. 10). The measured helicity
amplitudes off the protons are almost exclusively due to meson-baryon
contributions as the dynamical coupled channel (DCC) calculation indicates
(dashed line). The close correlation of the DCC calculation and the measured
data for the case when quark contributions are nearly absent, supports the
phenomenological description of the helicity amplitudes in terms of a 3-quark
core that dominate at high $Q^{2}$ and meson-baryon contributions that can
make important contributions at lower $Q^{2}$.

###  .3.6 Resonance lightfront transition charge densities.

Knowledge of the helicity amplitudes in a large $Q^{2}$ allows for the
determination of the transition charge densities on the light cone in
transverse impact parameter space ($b_{x},b_{y}$) Tiator and Vanderhaeghen
(2009). The relations between the helicity transition amplitudes and the Dirac
and Pauli resonance transition form factors are given by:

$\displaystyle
A_{1/2}=e\frac{Q_{-}}{\sqrt{K}(4M_{N}M^{*})^{1/2}}\\{F_{1}^{NN^{*}}+F_{2}^{NN^{*}}\\}~{}~{}~{}~{}~{}$
(27) $\displaystyle
S_{1/2}=e\frac{Q_{-}}{\sqrt{K}(4M_{N}M^{*})^{1/2}}\left(\frac{Q_{+}Q_{-}}{2M^{*}}\right)\frac{(M^{*}+M_{N})}{Q^{2}}$
$\displaystyle\times\\{F_{1}^{NN^{*}}-\frac{Q^{2}}{(M^{*}+M_{N})^{2}}F_{2}^{NN^{*}}\\}\,,~{}~{}~{}~{}~{}$
(28)

where $M^{*}$ is the mass of the excited state $N^{*}$,
$K=\frac{{M^{*}}^{2}-M^{2}_{N}}{2M^{*}}$ is the equivalent photon energy,
$Q_{+}$ and $Q_{-}$ are short hands for $Q_{\pm}\equiv\sqrt{M^{*}\pm
M_{N})^{2}+Q^{2}}$. The charge and magnetic lightfront transition densities
$\rho_{0}^{NN^{*}}$ and $\rho_{T}^{NN^{*}}$, respectively, are given as:

$\displaystyle\rho_{0}^{NN^{*}}(\vec{b})=\int_{0}^{\infty}\frac{dQ}{2\pi}J_{0}(bQ)F_{1}^{NN^{*}}(Q^{2})$
(29)
$\displaystyle\rho_{T}^{NN^{*}}(\vec{b})=\rho_{0}^{NN^{*}}(\vec{b})+\sin(\phi_{b}-\phi_{s})\times$
$\displaystyle\int_{0}^{\infty}\frac{dQ}{2\pi}\frac{Q^{2}}{(M^{*}+M_{N})}J_{1}(bQ)F_{2}^{NN^{*}}(Q^{2})\,.$
(30)

Figure 11: Left panels: $N(1440)$, top: projection of charge densities on
$b_{y}$, bottom: transition charge densities when the proton is spin polarized
along $b_{x}$. Right panels: same for $N(1535)$. Note that the densities are
scaled with $b^{2}$ to emphasize the outer wings. Color code:negative charge
is blue, positive charge is red. Note that all scales are the same for ease of
comparison Burkert (2018b). Graphics credit: F.X. Girod.

Similar transverse charge transition densities can be defined for
$J^{P}=\frac{3}{2}^{+}$ states such as the $\Delta(1232)\frac{3}{2}^{+}$. This
has been studied in Carlson and Vanderhaeghen (2008) and is shown in Fig. 12.

A comparison of $N(1440)\frac{1}{2}^{+}$ and $N(1535)\frac{1}{2}^{-}$ is shown
in Figure 11. There are clear differences in the charge transition densities
between the two states. The Roper state has a softer positive core and a wider
negative outer cloud than $N(1535)\frac{1}{2}^{-}$ and develops a larger shift
in $b_{y}$ when the proton is polarized along the $b_{x}$ axis.

###  .3.7 Single Quark Transition Model

Many of the exited states for which there is information about the transition
form factors available have been assigned as members of the $[SU(6),L^{P}]$ =
$[70,1^{-}]$ super multiplet of the $[SU(6)\otimes O(3)]$ symmetry group. In a
model, where only single quark transitions to the excited states are
considered Hey and Weyers (1974); Cottingham and Dunbar (1979); Burkert  _et
al._ (2003), only 3 of the amplitudes need to be known to determine the
remaining 16 transverse helicity amplitudes for all states in $[70,1^{-}]$
including on neutrons. However, the picture is now more complicated due to the
strong admixture of meson-baryon components to the single quark transition
especially in the lower $Q^{2}$ range. This requires a model to separate the
single quark contributions from the hadronic part before projections for other
states can be made Ramalho (2014).

##  .4 Higher mass baryons and hybrid baryons

The existence of baryons containing significant active gluonic components in
the wave function has been predicted some decade ago Dudek and Edwards (2012)
employing Lattice QCD simulations. The lowest such ”hybrid” state is expected
to be a $J^{P}=\frac{1}{2}^{+}$ nucleon state. LQCD projects a mass of 1.3 GeV
above the nucleon mass, i.e. approximately 2.2-2.3 GeV, and several other
states should appear close by in $J^{P}$ = $\frac{1}{2}^{+}$ and $J^{P}$ =
$\frac{3}{2}^{+}$, as seen in Fig. 13.

How do we identify these states? Hybrid baryons have same spin-parity as other
ordinary baryons. In contrast to hybrid mesons, there are no hybrid baryons
with ”exotic” quantum numbers. One possibility is to search for more states
than the quark model predicts in some mass range. The other possibility is to
study the transition form factors of excited states. Hybrid states may be
identified as states with a different $Q^{2}$ behavior than what is expected
from a 3-quark state. The sensitivity Li  _et al._ (1992) is demonstrated for
the Roper resonance that projected a very rapid drop of the $A_{1/2}(Q^{2})$
with $Q^{2}$, and $S_{1/2}(Q^{2})\sim 0$ prediction. Both are incompatible
with what we know today about the Roper resonance. Precision electroproduction
data in the mass range above 2 GeV will be needed to test high mass states for
their potential hybrid character, e.g. from experiments at CLAS12 Lanza and
D’Angelo (2021).

Figure 12: Quark transverse transition charge density corresponding to the
$p\to\Delta^{+}$ transition. Light color indicates positive charge, dark color
indicates negative charge. Top: p and $\Delta$ are unpolarized. Middle: p and
$\Delta$ are polarized along $b_{x}$ axis generating an electric dipole along
the $b_{y}$ axis. Bottom: Quadrupole contribution to transition density.
Graphics adapted from Carlson and Vanderhaeghen (2008).

##  .5 Conclusions and Outlook

In this contribution we have focused on more recent results of nucleon
resonance transition amplitudes and their interpretation within LQCD and
within most advanced approaches, e.g. in light front relativistic quark models
and approaches with traceable links to first principle QCD such as Dyson-
Schwinger Equations Roberts (2008) and light cone sum rules Braun  _et al._
(2009). These calculations describe the transition form factors at $Q^{2}\geq
2$ GeV2, while at lower $Q^{2}$ values hadronic degrees of freedom must be
included and could even dominate contributions of the quark core.

For the lowest mass states, $\Delta(1232)\frac{3}{2}^{+}$ and the Roper
$N(1440)\frac{1}{2}^{+}$, LQCD calculations have been carried out that are
consistent with the data within large uncertainties. These calculations are
about one decade old, and new data, with higher precision in more extended
kinematic range have been added to the database that warrant new Lattice
calculations at the physical pion mass to be carried out.

Over the past decade, eight baryon states in the mass range from 1.85 to 2.15
GeV have been either discovered or evidence for the existence of states has
been significantly strengthened. Some of these states are in the mass range
and have $J^{PC}$ quantum numbers that could have significant contributions of
gluonic components. Such ”hybrid” states are in fact predicted in LQCD Dudek
and Edwards (2012). These states appear with the same quantum numbers as
ordinary quark excitations, and can only be isolated from ordinary states due
to the $Q^{2}$ dependence of their helicity amplitudes Li  _et al._ (1992),
which is expected to be quite different from ordinary 3-quark excitation. The
study of hybrid baryon excitations then requires new electroproduction data
especially at low $Q^{2}$ Lanza and D’Angelo (2021) with different final
states and with masses above 2 GeV.

Figure 13: Projections of excited baryons with dominant gluonic components
(marked in blue shades) in LQCD with 400MeV pions. The lowest hybrid baryon is
projected with mass 1.3 GeV above the nucleon mass. The $1/2^{+}$ and
$3/2^{+}$ states are clustered in a narrow mass range of about 200 MeV.

Despite the very significant progress made in recent years to further
establish the light-quark baryon spectrum and explore the internal structure
of excited states and the relationship to QCD Carman  _et al._ (2020); Brodsky
_et al._ (2020), much remains to be done. A vast amount of precision data
already collected needs to be included in the multi-channel analysis
frameworks, and polarization data are still to be analyzed. There are approved
proposals to study resonance excitation at much higher $Q^{2}$ and with higher
precision at Jefferson Lab with CLAS12 Tian  _et al._ (2022); Burkert (2018a),
which may begin to reveal the transition to the bare quark core contributions
at short distances.

A new avenue of experimental research has recently been opened up with the
first data-based extraction of a gravitational property of the proton, its
internal pressure distribution, which is represented by the gravitational form
factor $D^{q}(t)$. It is one of the form factors of the QCD matrix element of
the energy-momentum tensor, its internal pressure and shear stress
distribution in space Burkert  _et al._ (2018, 2021). These properties, as
well as the distribution of mass and angular momentum, and are accessible
directly in gravitational interaction, which is highly impractical. However
the relevant gravitational form factor $D^{q}(t)$ for the ground state nucleon
can be accessed indirectly through the process of deeply virtual Compton
scattering and in time-like Compton scattering Ji (1997); Chatagnon  _et al._
(2021). Both processes, having a $J=1$ photon in the initial state as well as
in the final state, contain components of $J=2$ that couple to the proton
through a tensor interaction, as gravity does Polyakov and Schweitzer (2018).

Mechanical properties of resonance transitions have recently been explored for
the $N(1535)\frac{1}{2}^{-}\to N(938)$ gravitational transition form factors
calculations in Özdem and Azizi (2020) and in Polyakov and Tandogan (2020). In
order to access these novel gravitational transition form factors
experimentally, the nucleon to resonance transition generalized parton
distributions must be studied. The framework for studying the $N\to N(1535)$
transition GPDs, which would enable experimental access to these mechanical
properties, remains to be developed. The required effort is quite worthwhile
as a new avenue of hadron physics has opened up that remains to be fully
explored.

## References

  * Alexandrou  _et al._ (2008) Alexandrou, C, G. Koutsou, H. Neff, John W. Negele, W. Schroers, and A. Tsapalis (2008), Phys. Rev. D 77, 085012.
  * Anderson  _et al._ (1952) Anderson, H L, E. Fermi, E. A. Long, and D. E. Nagle (1952), Phys. Rev. 85, 936.
  * Anikin  _et al._ (2015) Anikin, I V, V. M. Braun, and N. Offen (2015), Phys. Rev. D 92 (1), 014018.
  * Anisovich  _et al._ (2017) Anisovich, A V, _et al._ (2017), “Strong evidence for nucleon resonances near 1900\,MeV,” Phys. Rev. Lett. 119 (6), 062004, arXiv:1712.07549 [nucl-ex] .
  * Arndt  _et al._ (2002) Arndt, R A, W. J. Briscoe, I. I. Strakovsky, and R. L. Workman (2002), Phys. Rev. C 66, 055213.
  * Ash  _et al._ (1967) Ash, W W, K. Berkelman, C. A. Lichtenstein, A. Ramanauskas, and R. H. Siemann (1967), Phys. Lett. B 24, 165–168.
  * Aznauryan and Burkert (2012a) Aznauryan, I G, and V. D. Burkert (2012a), Prog. Part. Nucl. Phys. 67, 1–54.
  * Aznauryan and Burkert (2012b) Aznauryan, I G, and V. D. Burkert (2012b), Phys. Rev. C 85, 055202.
  * Aznauryan and Burkert (2015a) Aznauryan, I G, and V. D. Burkert (2015a), Phys. Rev. C 92 (3), 035211.
  * Aznauryan and Burkert (2015b) Aznauryan, I G, and V. D. Burkert (2015b), “Extracting meson-baryon contributions to the electroexcitation of the $N(1675){\frac{5}{2}}^{-}$ nucleon resonance,” Phys. Rev. C 92 (1), 015203, arXiv:1412.1296 [nucl-th] .
  * Aznauryan and Burkert (2016) Aznauryan, I G, and V. D. Burkert (2016), “Configuration mixings and light-front relativistic quark model predictions for the electroexcitation of the $\Delta(1232)3/2^{+}$, $N(1440)1/2^{+}$, and $\Delta(1600)3/2^{+}$,” arXiv:1603.06692 [hep-ph] .
  * Aznauryan  _et al._ (2009) Aznauryan, I G, _et al._ (CLAS) (2009), Phys. Rev. C 80, 055203.
  * Aznauryan  _et al._ (2013) Aznauryan, I G, _et al._ (2013), Int. J. Mod. Phys. E 22, 1330015.
  * Aznauryan and Burkert (2017) Aznauryan, Inna G, and Volker Burkert (2017), Phys. Rev. C 95 (6), 065207.
  * Bauer  _et al._ (2014) Bauer, T, S. Scherer, and L. Tiator (2014), “Electromagnetic transition form factors of the Roper resonance in effective field theory,” Phys. Rev. C 90 (1), 015201, arXiv:1402.0741 [nucl-th] .
  * Behrndt and Cvetic (2005) Behrndt, Klaus, and Mirjam Cvetic (2005), Phys. Rev. Lett. 95, 021601.
  * Berends  _et al._ (1967) Berends, Frits A, A. Donnachie, and D. L. Weaver (1967), Nucl. Phys. B 4, 1–53.
  * Bermuth  _et al._ (1988) Bermuth, K, D. Drechsel, L. Tiator, and J. B. Seaborn (1988), “Photoproduction of $\Delta$ and Roper Resonances in the Cloudy Bag Model,” Phys. Rev. D 37, 89–100.
  * Braun  _et al._ (2009) Braun, V M, _et al._ (2009), “Electroproduction of the N*(1535) resonance at large momentum transfer,” Phys. Rev. Lett. 103, 072001, arXiv:0902.3087 [hep-ph] .
  * Brodsky  _et al._ (2020) Brodsky, S J, _et al._ (2020), “Strong QCD from Hadron Structure Experiments: Newport News, VA, USA, November 4-8, 2019,” Int. J. Mod. Phys. E 29 (08), 2030006, arXiv:2006.06802 [hep-ph] .
  * Burkert  _et al._ (2022) Burkert, V, _et al._ (2022), “Precision Studies of QCD in the Low Energy Domain of the EIC,” arXiv:2211.15746 [nucl-ex] .
  * Burkert  _et al._ (2003) Burkert, V D, R. De Vita, M. Battaglieri, M. Ripani, and V. Mokeev (2003), “Single quark transition model analysis of electromagnetic nucleon resonance transitions in the [70,1-] supermultiplet,” Phys. Rev. C 67, 035204, arXiv:hep-ph/0212108 .
  * Burkert  _et al._ (2018) Burkert, V D, L. Elouadrhiri, and F. X. Girod (2018), Nature 557 (7705), 396–399.
  * Burkert  _et al._ (2021) Burkert, V D, L. Elouadrhiri, and F. X. Girod (2021), “Determination of shear forces inside the proton,” arXiv:2104.02031 [nucl-ex] .
  * Burkert and Lee (2004) Burkert, V D, and T. S. H. Lee (2004), Int. J. Mod. Phys. E 13, 1035–1112.
  * Burkert (2020) Burkert, VD (2020), “$N^{*}$ Experiments and what they tell us about Strong QCD Physics,” EPJ Web Conf. 241, 01004, arXiv:1912.11400 [nucl-ex] .
  * Burkert (2018a) Burkert, Volker D (2018a), Ann. Rev. Nucl. Part. Sci. 68, 405–428.
  * Burkert (2018b) Burkert, Volker D (2018b), “N∗ Experiments and Their Impact on Strong QCD Physics,” Few Body Syst. 59 (4), 57, arXiv:1801.10480 [nucl-ex] .
  * Burkert and Roberts (2019) Burkert, Volker D, and Craig D. Roberts (2019), “Colloquium : Roper resonance: Toward a solution to the fifty year puzzle,” Rev. Mod. Phys. 91 (1), 011003.
  * Capstick and Isgur (1986) Capstick, Simon, and Nathan Isgur (1986), “Baryons in a relativized quark model with chromodynamics,” Phys. Rev. D 34 (9), 2809–2835.
  * Carlson and Vanderhaeghen (2008) Carlson, Carl E, and Marc Vanderhaeghen (2008), “Empirical transverse charge densities in the nucleon and the nucleon-to-Delta transition,” Phys. Rev. Lett. 100, 032004, arXiv:0710.0835 [hep-ph] .
  * Carman  _et al._ (2020) Carman, D S, K. Joo, and V. I. Mokeev (2020), “Strong QCD Insights from Excited Nucleon Structure Studies with CLAS and CLAS12,” Few Body Syst. 61 (3), 29, arXiv:2006.15566 [nucl-ex] .
  * Chatagnon  _et al._ (2021) Chatagnon, P, _et al._ (CLAS) (2021), Phys. Rev. Lett. 127 (26), 262501.
  * Cottingham and Dunbar (1979) Cottingham, W N, and I. H. Dunbar (1979), “Baryon Multipole Moments in the Single Quark Transition Model,” Z. Phys. C 2, 41.
  * Drechsel  _et al._ (1999) Drechsel, D, O. Hanstein, S. S. Kamalov, and L. Tiator (1999), Nucl. Phys. A 645, 145–174.
  * Drechsel  _et al._ (2007) Drechsel, D, S. S. Kamalov, and L. Tiator (2007), “Unitary Isobar Model - MAID2007,” Eur. Phys. J. A 34, 69–97, arXiv:0710.0306 [nucl-th] .
  * Dudek and Edwards (2012) Dudek, Jozef J, and Robert G. Edwards (2012), Phys. Rev. D 85, 054016.
  * Frolov  _et al._ (1999) Frolov, V V, _et al._ (1999), Phys. Rev. Lett. 82, 45–48.
  * Giannini  _et al._ (2003) Giannini, M M, E. Santopinto, and A. Vassallo (2003), “An Overview of the hypercentral constituent quark model,” Prog. Part. Nucl. Phys. 50, 263–272, arXiv:nucl-th/0301017 .
  * Golovatch  _et al._ (2019) Golovatch, E, _et al._ (CLAS) (2019), “First results on nucleon resonance photocouplings from the $\gamma p\to\pi^{+}\pi^{-}p$ reaction,” Phys. Lett. B 788, 371–379, arXiv:1806.01767 [nucl-ex] .
  * Greenberg (1964) Greenberg, O W (1964), Phys. Rev. Lett. 13, 598–602.
  * Grigoryan  _et al._ (2009) Grigoryan, Hovhannes R, T. S. H. Lee, and Ho-Ung Yee (2009), Phys. Rev. D 80, 055006.
  * Hey and Weyers (1974) Hey, A J G, and J. Weyers (1974), “Quarks and the helicity structure of photoproduction amplitudes,” Phys. Lett. B 48, 69–72.
  * Isgur and Karl (1977) Isgur, Nathan, and Gabriel Karl (1977), “Hyperfine Interactions in Negative Parity Baryons,” Phys. Lett. B 72, 109.
  * Isgur and Karl (1978) Isgur, Nathan, and Gabriel Karl (1978), “P Wave Baryons in the Quark Model,” Phys. Rev. D 18, 4187.
  * Ji (1997) Ji, Xiang-Dong (1997), Phys. Rev. D 55, 7114–7125.
  * Jido  _et al._ (2008) Jido, D, M. Doering, and E. Oset (2008), “Transition form factors of the N*(1535) as a dynamically generated resonance,” Phys. Rev. C 77, 065207, arXiv:0712.0038 [nucl-th] .
  * Joo  _et al._ (2002) Joo, K, _et al._ (CLAS) (2002), Phys. Rev. Lett. 88, 122001.
  * Julia-Diaz  _et al._ (2008) Julia-Diaz, B, T. S. H. Lee, A. Matsuyama, T. Sato, and L. C. Smith (2008), “Dynamical coupled-channels effects on pion photoproduction,” Phys. Rev. C 77, 045205, arXiv:0712.2283 [nucl-th] .
  * Kamalov and Yang (1999) Kamalov, S S, and Shin Nan Yang (1999), Phys. Rev. Lett. 83, 4494–4497.
  * Kamalov  _et al._ (2001) Kamalov, Sabit S, Shin Nan Yang, Dieter Drechsel, Olaf Hanstein, and Lothar Tiator (2001), “Gamma* N —$>$ Delta transition form-factors: A New analysis of the JLab data on p (e, e-prime p) pi0 at Q**2=(2.8-(GeV/c)**2 and 4.0-(GeV/c)**2),” Phys. Rev. C 64, 032201, arXiv:nucl-th/0006068 .
  * Kelly  _et al._ (2007) Kelly, J J, _et al._ (2007), Phys. Rev. C 75, 025201.
  * Lanza and D’Angelo (2021) Lanza, L, and A. D’Angelo (CLAS) (2021), Nuovo Cim. C 44 (2-3), 51.
  * Li  _et al._ (1992) Li, Zhen-ping, Volker Burkert, and Zhu-jun Li (1992), Phys. Rev. D 46, 70–74.
  * Lin and Cohen (2012) Lin, Huey-Wen, and Saul D. Cohen (2012), AIP Conf. Proc. 1432 (1), 305–308.
  * Mathur  _et al._ (2005) Mathur, N, Y. Chen, S. J. Dong, Terrence Draper, I. Horvath, F. X. Lee, K. F. Liu, and J. B. Zhang (2005), Phys. Lett. B 605, 137–143.
  * Mokeev  _et al._ (2012) Mokeev, V I, _et al._ (CLAS) (2012), “Experimental Study of the $P_{11}(1440)$ and $D_{13}(1520)$ resonances from CLAS data on $ep\rightarrow e^{\prime}\pi^{+}\pi^{-}p^{\prime}$,” Phys. Rev. C 86, 035203, arXiv:1205.3948 [nucl-ex] .
  * Mokeev  _et al._ (2016) Mokeev, V I, _et al._ (2016), Phys. Rev. C 93 (2), 025206.
  * Mokeev  _et al._ (2020) Mokeev, V I, _et al._ (2020), “Evidence for the $N^{\prime}(1720)3/2^{+}$ Nucleon Resonance from Combined Studies of CLAS $\pi^{+}\pi^{-}p$ Photo- and Electroproduction Data,” Phys. Lett. B 805, 135457, arXiv:2004.13531 [nucl-ex] .
  * Moorhouse (1966) Moorhouse, R G (1966), “PHOTOPRODUCTION OF N* RESONANCES IN THE QUARK MODEL,” Phys. Rev. Lett. 16, 772–774.
  * Obukhovsky  _et al._ (2011) Obukhovsky, I T, Amand Faessler, D. K. Fedorov, Thomas Gutsche, and Valery E. Lyubovitskij (2011), Phys. Rev. D 84, 014004.
  * Özdem and Azizi (2020) Özdem, U, and K. Azizi (2020), Phys. Rev. D 101 (5), 054031.
  * Park  _et al._ (2008) Park, K, _et al._ (CLAS) (2008), “Cross sections and beam asymmetries for $\vec{e}p\to n\pi^{+}$ in the nucleon resonance region for $1.7<Q^{2}\leq 4.5$ (GeV)2,” Phys. Rev. C 77, 015208, arXiv:0709.1946 [nucl-ex] .
  * Pascalutsa and Vanderhaeghen (2006) Pascalutsa, Vladimir, and Marc Vanderhaeghen (2006), “Chiral effective-field theory in the Delta(1232) region: I. Pion electroproduction on the nucleon,” Phys. Rev. D 73, 034003, arXiv:hep-ph/0512244 .
  * Peccei (1969) Peccei, R D (1969), Phys. Rev. 181, 1902–1904.
  * Polyakov and Schweitzer (2018) Polyakov, Maxim V, and Peter Schweitzer (2018), Int. J. Mod. Phys. A 33 (26), 1830025.
  * Polyakov and Tandogan (2020) Polyakov, Maxim V, and Asli Tandogan (2020), “Comment on “Gravitational transition form factors of $N(1535)\to N$”,” Phys. Rev. D 101 (11), 118501.
  * Ramalho (2014) Ramalho, G (2014), “Using the Single Quark Transition Model to predict nucleon resonance amplitudes,” Phys. Rev. D 90 (3), 033010, arXiv:1407.0649 [hep-ph] .
  * Ramalho (2016) Ramalho, G (2016), “Improved empirical parametrizations of the $\gamma^{\ast}N\to\Delta(1232)$ and $\gamma^{\ast}N\to N(1520)$ helicity amplitudes and the Siegert’s theorem,” Phys. Rev. D 93 (11), 113012, arXiv:1602.03832 [hep-ph] .
  * Ramalho and Melnikov (2018) Ramalho, G, and D. Melnikov (2018), “Valence quark contributions for the $\gamma^{\ast}N\to N(1440)$ form factors from light-front holography,” Phys. Rev. D 97 (3), 034037, arXiv:1703.03819 [hep-ph] .
  * Roberts (2008) Roberts, C D (2008), “Hadron Properties and Dyson-Schwinger Equations,” Prog. Part. Nucl. Phys. 61, 50–65, arXiv:0712.0633 [nucl-th] .
  * Roper (1964) Roper, L David (1964), Phys. Rev. Lett. 12, 340–342.
  * Santopinto and Giannini (2012) Santopinto, E, and M. M. Giannini (2012), Phys. Rev. C 86, 065202.
  * Sato and Lee (1996) Sato, T, and T-. S. H. Lee (1996), Phys. Rev. C 54, 2660–2684.
  * Sato and Lee (2001) Sato, T, and T. S. H. Lee (2001), Phys. Rev. C 63, 055201.
  * Segovia  _et al._ (2014) Segovia, Jorge, Ian C. Cloet, Craig D. Roberts, and Sebastian M. Schmidt (2014), “Nucleon and $\Delta$ elastic and transition form factors,” Few Body Syst. 55, 1185–1222, arXiv:1408.2919 [nucl-th] .
  * Segovia  _et al._ (2015) Segovia, Jorge, Bruno El-Bennich, Eduardo Rojas, Ian C. Cloet, Craig D. Roberts, Shu-Sheng Xu, and Hong-Shi Zong (2015), Phys. Rev. Lett. 115 (17), 171801.
  * Siegert (1937) Siegert, A J F (1937), “Note on the interaction between nuclei and electromagnetic radiation,” Phys. Rev. 52, 787–789.
  * Stoler (1993) Stoler, P (1993), Phys. Rept. 226, 103–171.
  * Suzuki  _et al._ (2010) Suzuki, N, B. Julia-Diaz, H. Kamano, T. S. H. Lee, A. Matsuyama, and T. Sato (2010), Phys. Rev. Lett. 104, 042302.
  * de Teramond and Brodsky (2012) de Teramond, Guy F, and Stanley J. Brodsky (2012), “Excited Baryons in Holographic QCD,” AIP Conf. Proc. 1432 (1), 168–175, arXiv:1108.0965 [hep-ph] .
  * Thomas  _et al._ (1981) Thomas, Anthony William, S. Theberge, and Gerald A. Miller (1981), “The Cloudy Bag Model of the Nucleon,” Phys. Rev. D 24, 216.
  * Tian  _et al._ (2022) Tian, Y, _et al._ (CLAS) (2022), “Exclusive $\pi^{-}$ Electroproduction off the Neutron in Deuterium in the Resonance Region,” arXiv:2203.16785 [nucl-ex] .
  * Tiator  _et al._ (2011) Tiator, L, D. Drechsel, S. S. Kamalov, and M. Vanderhaeghen (2011), “Electromagnetic Excitation of Nucleon Resonances,” Eur. Phys. J. ST 198, 141–170, arXiv:1109.6745 [nucl-th] .
  * Tiator (2016) Tiator, Lothar (2016), “Pion Electroproduction and Siegert’s Theorem,” Few Body Syst. 57 (11), 1087–1093.
  * Tiator and Vanderhaeghen (2009) Tiator, Lothar, and Marc Vanderhaeghen (2009), Phys. Lett. B 672, 344–348.
  * Ungaro  _et al._ (2006) Ungaro, M, _et al._ (CLAS) (2006), Phys. Rev. Lett. 97, 112003.
  * Štajner  _et al._ (2017) Štajner, S, _et al._ (2017), “Beam-Recoil Polarization Measurement of $\pi^{0}$ Electroproduction on the Proton in the Region of the Roper Resonance,” Phys. Rev. Lett. 119 (2), 022001.
  * Walker (1969) Walker, R L (1969), Phys. Rev. 182, 1729–1748.
  * Watson (1954) Watson, Kenneth M (1954), “Some general relations between the photoproduction and scattering of pi mesons,” Phys. Rev. 95, 228–236.
  * Workman (2022) Workman, R L (Particle Data Group) (2022), PTEP 2022, 083C01.