# First Results on Nucleon Resonance Electroexcitation Amplitudes from $ep\to
e^{\prime}\pi^{+}\pi^{-}p^{\prime}$ Cross Sections at $W$ from 1.4-1.7 GeV and
$Q^{2}$ from 2.0-5.0 GeV2

V.I. Mokeev  Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606  P. Achenbach  Thomas Jefferson National Accelerator Facility,
Newport News, Virginia 23606  V.D. Burkert  Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606  D.S. Carman  Thomas
Jefferson National Accelerator Facility, Newport News, Virginia 23606  R.W.
Gothe  University of South Carolina, Columbia, South Carolina 29208  A.N.
Hiller Blin  Institute for Theoretical Physics, Tübingen University, D-72076
Tübingen, Germany  E.L. Isupov  Skobeltsyn Institute of Nuclear Physics and
Physics Department, Lomonosov Moscow State University, 119234 Moscow, Russia
K. Joo  University of Connecticut, Storrs, Connecticut 06269  K. Neupane
University of South Carolina, Columbia, South Carolina 29208  A. Trivedi
University of South Carolina, Columbia, South Carolina 29208

###### Abstract

The electroexcitation amplitudes or $\gamma_{v}pN^{*}$ electrocouplings of the
$N(1440)1/2^{+}$, $N(1520)3/2^{-}$, and $\Delta(1600)3/2^{+}$ resonances were
obtained for the first time from the $ep\to
e^{\prime}\pi^{+}\pi^{-}p^{\prime}$ differential cross sections measured with
the CLAS detector at Jefferson Lab within the range of invariant mass $W$ of
the final state hadrons from 1.4–1.7 GeV for photon virtualities $Q^{2}$ from
2.0–5.0 GeV2. A good description of the nine independent one-fold differential
$\gamma_{v}p\to\pi^{+}\pi^{-}p^{\prime}$ cross sections achieved within the
data-driven JM meson-baryon reaction model in each bin of ($W$,$Q^{2}$) allows
for separation of the resonant and non-resonant contributions. The
electrocouplings were determined in independent fits of the $\pi^{+}\pi^{-}p$
cross sections within three overlapping $W$ intervals with a substantial
contribution from each of the three resonances listed above. Consistent
results on the electrocouplings extracted from the data in these $W$ intervals
provide evidence for their reliable extraction. These studies extend
information on the electrocouplings of the $N(1440)1/2^{+}$ and
$N(1520)3/2^{-}$ available from this channel over a broader range of $Q^{2}$.
The electrocouplings of the $\Delta(1600)3/2^{+}$, which decays preferentially
into $\pi\pi N$ final states, have been determined for the first time.
Consistent results on the electrocouplings of the $N(1440)1/2^{+}$ and
$N(1520)3/2^{-}$ from the $\pi N$ and $\pi^{+}\pi^{-}p$ channels allows for
the determination of the uncertainties related to the reaction models employed
in the data fits. The reliable extraction of the electrocouplings for these
states is also supported by the description of the $\pi^{+}\pi^{-}p$
differential cross sections with $Q^{2}$-independent masses and total/partial
hadronic decay widths into the $\pi\Delta$ and $\rho p$ final states. Our
results provide further evidence for the structure of these resonances in
terms of an interplay between the inner core of three dressed quarks and an
external meson-baryon cloud.

Exclusive meson photo- and electroproduction, Charged double pion photo- and
electroproduction, Structure of Nucleon resonances

###### pacs:

75.25.-j, 13.60.-r, 13.88.+e, 24.85.+p

††preprint: APS/123-QED

##  I Introduction

Studies of exclusive $\pi^{+}\pi^{-}p$ photo- and electroproduction off
protons represent an effective tool for the exploration of the spectrum and
structure of nucleon resonances [1, 2, 3, 4, 5]. In this work we will use
$N^{*}$ to represent both excited nucleon states $N$(mass)$J^{P}$, as well as
Delta states $\Delta$(mass)$J^{P}$. The $\pi^{0}p$, $\pi^{+}n$, and
$\pi^{+}\pi^{-}p$ electroproduction channels account for the largest part of
the inclusive virtual photon-proton cross sections in the resonance excitation
region [1]. The data on $\pi N$ and $\pi^{+}\pi^{-}p$ electroproduction offer
complementary information on the nucleon resonance electroexcitation
amplitudes, the so-called $\gamma_{v}pN^{*}$ electrocouplings, and their
evolution with photon virtuality $Q^{2}$ ($Q^{2}=-q_{\mu}^{2}$), where
$q_{\mu}$ is the virtual photon four-momentum.

The low-lying $N^{*}$ states in the mass range below 1.6 GeV decay
preferentially into $\pi N$, making single pion electroproduction data the
driving source of information on the electrocouplings of these states [6, 7,
8]. On the other hand, the branching fraction ($BF$) of these resonance decays
into $\pi\pi N$ remains appreciable at the level of around 40%, allowing for
an independent determination of their electrocouplings from this channel [9,
10]. Consistent results on the electrocouplings from independent studies of
$\pi N$ and $\pi^{+}\pi^{-}p$ demonstrate the capability of the
$\pi^{+}\pi^{-}p$ reaction model to provide extraction of these quantities and
to evaluate the systematic uncertainties associated with their determination
[11, 10].

Several $N^{*}$s in the mass range above 1.6 GeV decay preferentially into the
$\pi\pi N$ final states with $BF$ around 70%, making studies of
$\pi^{+}\pi^{-}p$ electroproduction the major source of information on the
electrocouplings of these states [10]. At the same time, there are $N^{*}$s
with masses above 1.6 GeV that decay mostly to the $\pi N$ final states [12].
Therefore, studies of both $\pi N$ and $\pi^{+}\pi^{-}p$ electroproduction off
protons are of particular importance to get information on the
$Q^{2}$-evolution of the electrocouplings for most prominent nucleon
resonances.

Coupled-channel approaches are making progress towards the extraction of the
electrocouplings for low-lying $N^{*}$ states from the combined analyses of
meson photo-, electro-, and hadroproduction data. Recently, the $\pi N$ and
$\eta p$ electroproduction multipoles, which are directly related to the
$\gamma_{v}pN^{*}$ electrocouplings, were determined from CLAS data within a
multi-channel analysis [13, 14]. The first results on the electrocouplings of
the $\Delta(1232)3/2^{+}$ and $N(1440)1/2^{+}$ at their pole positions in the
complex energy plane have become available from the global multi-channel
analysis developed by the Argonne-Osaka Collaboration [15]. The contributions
from the $\pi\Delta$ and $\rho p$ channels deduced from the $\pi^{+}\pi^{-}p$
cross sections play an important role in the development of these approaches
[16, 10]. Experiments of the 6-GeV era with the CLAS detector [17] in Hall B
at Jefferson Lab have provided the first and still only available information
on the $Q^{2}$-evolution of the electrocouplings of most resonances in the
mass range up to 1.8 GeV for $Q^{2}<5$ GeV2 [1, 2].

Studies of $\pi^{+}\pi^{-}p$ photo- and electroproduction also provide a
promising avenue in the search for the so-called “missing” resonances.
Constituent quark models based on approximate symmetries of the strong
interaction that are relevant for the strongly coupled regime, when the QCD
running coupling $\alpha_{s}/\pi$ is comparable to unity, predict many more
$N^{*}$s than have been seen in experiments with both electromagnetic and
hadronic probes [18, 19, 20]. These expectations are supported by the results
on the $N^{*}$ spectrum obtained starting from the QCD Lagrangian both within
lattice and continuum QCD approaches [21, 22].

The combined analysis of the CLAS $\pi^{+}\pi^{-}p$ photo- and
electroproduction data [23, 24] carried out for $W$ from 1.6–1.8 GeV and
$Q^{2}$ from 0–1.5 GeV2 revealed the presence of a new
$N^{\prime}(1720)3/2^{+}$ baryon state [4]. Only after implementation of this
state with photo-/electrocouplings, mass, and decay widths fit to the CLAS
data, was a successful description of the $\pi^{+}\pi^{-}p$
photo-/electroproduction data achieved with $Q^{2}$-independent masses and
decay widths into the $\pi\Delta$ and $\rho p$ final states. The contributions
from the $N(1720)3/2^{+}$ and the new $N^{\prime}(1720)3/2^{+}$ are well
separated in the $\pi^{+}\pi^{-}p$ photo-/electroproduction data analyses
despite their close masses and same spin-parity due to their different
patterns for decay into intermediate $\pi\Delta$ and $\rho p$ states and the
different $Q^{2}$-evolution of their electrocouplings. These differences can
only be seen in the combined studies of $\pi^{+}\pi^{-}p$ photo- and
electroproduction, but they were elusive in previous studies of the two-body
meson-baryon channels. Therefore, such combined studies provide a promising
avenue in the quest to discover additional resonances.

The CLAS results on the electrocouplings make it possible for the first time
to determine the resonant contributions to the inclusive electron scattering
structure functions in the resonance region [25, 26, 27]. They also provide a
new opportunity to better understand the ground state nucleon parton
distribution functions (PDFs) at large values of the fractional parton
momentum $x$ within the resonance region.

Analyses of the CLAS results on the $Q^{2}$-evolution of the electrocouplings
within coupled-channel approaches [15, 28] and continuum Schwinger methods
(CSMs) [29], supported by the results from different quark models [30, 31, 32,
33], have revealed $N^{*}$ structure as an interplay between the inner core of
three dressed quarks and an external meson-baryon cloud [1, 3, 34, 15].
Studies of the electrocouplings for $Q^{2}\lesssim 2$ GeV2 provide important
information on the transition from confined dressed quarks to deconfined
mesons and baryons that give rise to the meson-baryon cloud.

Analyses of these electrocouplings suggest a gradual transition from the
convolution between the meson-baryon cloud and quark core in $N^{*}$ structure
towards quark core dominance with increasing $Q^{2}$ [29, 31, 32, 33]. Virtual
photons with $Q^{2}\gtrsim 1-2$ GeV2 penetrate the meson-baryon cloud and
interact mostly with the quark core. Consequently, studies of the
electrocouplings in this higher $Q^{2}$ regime provide a unique way to explore
the structure of dressed quarks and the evolution of their interactions at
distance scales from the strongly coupled to the perturbative (pQCD) regimes
[35, 36, 37, 38]. Therefore, the region of high $Q^{2}$ looks promising to
explore many facets of the strong interaction dynamics between three dressed
quarks apparent in the generation of various $N^{*}$s with different quantum
numbers and structure.

The description of the $\Delta(1232)3/2^{+}$ and $N(1440)1/2^{+}$
electrocouplings achieved within CSMs [39, 40] by employing the same momentum
dependence of the dressed quark mass deduced from the QCD Lagrangian [38, 36]
and used in the description of the pion and nucleon elastic electromagnetic
form factors [41, 42, 39] demonstrated the promising opportunity for gaining
insight into the emergence of more than 98% of the hadron mass from data on
the $Q^{2}$-evolution of the electrocouplings.

As of now, the electrocouplings are available from $\pi^{+}\pi^{-}p$
electroproduction cross sections within the range of $Q^{2}<1.5$ GeV2. In this
paper, we present an extension of the results on the electrocouplings of the
$N(1440)1/2^{+}$ and $N(1520)3/2^{-}$ determined from the $\pi^{+}\pi^{-}p$
cross sections at $W$ from 1.4–1.7 GeV for $Q^{2}$ from 2.0–5.0 GeV2 and
compare them with the available results from the studies of $\pi N$
electroproduction within the same kinematic domain. The $\Delta(1600)3/2^{+}$
recently was elevated to a four-star PDG status [12]. This state decays
preferentially into $\pi\pi N$. The electrocouplings of the
$\Delta(1600)3/2^{+}$ have become available for the first time from the
analysis of the $\pi^{+}\pi^{-}p$ electroproduction data presented here.

This paper is organized as follows. In Section II we present the kinematic
variables for the description of $\pi^{+}\pi^{-}p$ electroproduction and the
one-fold differential cross sections measured with CLAS used for the
extraction of the electrocouplings in the mass range below 1.7 GeV. Here, we
also discuss the updates to the Jefferson Lab-Moscow State University (JM)
reaction model relevant to the extraction of the electrocouplings for $Q^{2}$
from 2.0–5.0 GeV2. The procedures developed for the evaluation of the
electrocouplings from the cross section fits are presented in Section III. The
results on the electrocouplings and partial decay widths to $\pi\Delta$ and
$\rho p$ for the $N(1440)1/2^{+}$, $N(1520)3/2^{-}$, and $\Delta(1600)3/2^{+}$
are presented in Section IV, along with comparisons of the electrocouplings
from the $\pi^{+}\pi^{-}p$ data with those previously available from the
analyses of $\pi N$ data. The impact of these results on the understanding of
$N^{*}$ structure is presented in Section V. We conclude and highlight the
future prospects for resonance electrocoupling studies from exclusive meson
electroproduction data in Section VI.

##  II Cross Sections and Reaction Model for Electrocouplings

In this section we describe the $\pi^{+}\pi^{-}p$ differential cross sections
measured with CLAS for $W$ from 1.4–1.7 GeV and $Q^{2}$ from 2.0–5.0 GeV2 [43,
44] that were used for the extraction of the $N(1440)1/2^{+}$,
$N(1520)3/2^{-}$, and $\Delta(1600)3/2^{+}$ parameters. We also present the
basic features of the JM model relevant for the extraction of the
electrocouplings from the $\pi^{+}\pi^{-}p$ data and the most recent JM model
updates.

Figure 1: Kinematic variables for the description of the reaction $\gamma_{v}p\to\pi^{+}\pi^{-}p^{\prime}$ in the CM frame of the final state hadrons corresponding to the $d^{5}\tau$ differential assignment given in Section II.1. Panel (a) shows the $\pi^{-}$ polar and azimuthal angles $\theta_{\pi^{-}}$ and $\phi_{\pi^{-}}$. Plane C represents the electron scattering plane. The $z$-axis is directed along the $\gamma_{v}$ three-momentum, while the $x$-axis is located in the electron scattering plane C and the $y$-axis forms a right-handed coordinate system. Plane A is defined by the three-momenta of the initial state proton and the final state $\pi^{-}$. Panel (b) shows the angle $\alpha_{[\pi^{-}p][\pi^{+}p^{\prime}]}$ between the two hadronic planes A and B or the plane B rotation angle around the axis aligned along the three-momentum of the final state $\pi^{-}$. Plane B is defined by the three-momenta of the final state $\pi^{+}$ and $p^{\prime}$. | 2.0-2.4
---|---
| 2.2 for computed cross sections
| 2.4-3.0
| 2.6 for computed cross sections
| 3.0-3.5
$Q^{2}$ Interval, GeV2 | 3.2 for computed cross sections
| 3.5-4.2
| 3.6 for computed cross sections
| 4.2-5.0
| 4.4 for computed cross sections
$W$ Interval, GeV | 1.41-1.66
covered in each $Q^{2}$ bin | 11 bins
Table 1: Kinematic area covered in the fit of the CLAS $\pi^{+}\pi^{-}p$ electroproduction cross sections for the extraction of the resonance parameters [43, 44]. One-Fold Differential | Interval | Number of
---|---|---
Cross Section | Covered | Bins
$\frac{d\sigma}{dM_{\pi^{+}p}}$ ($\mu$b/GeV) |  $M_{\pi^{+}p}^{min}$-$M_{\pi^{+}p}^{max}$ | 14
$\frac{d\sigma}{dM_{\pi^{+}\pi^{-}}}$ ($\mu$b/GeV) |  $M_{\pi^{+}\pi^{-}}^{min}$-$M_{\pi^{+}\pi^{-}}^{max}$ | 14
$\frac{d\sigma}{dM_{\pi^{-}p}}$ ($\mu$b/GeV) |  $M_{\pi^{-}p}^{min}$-$M_{\pi^{-}p}^{max}$ | 14
$\frac{d\sigma}{\sin\theta_{\pi^{-}}d\theta_{\pi^{-}}}$ ($\mu$b/rad) | 0-180∘ | 10
$\frac{d\sigma}{\sin\theta_{\pi^{+}}d\theta_{\pi^{+}}}$ ($\mu$b/rad) | 0-180∘ | 10
$\frac{d\sigma}{\sin\theta_{p^{\prime}}d\theta_{p^{\prime}}}$ ($\mu$b/rad) | 0-180∘ | 10
$d\sigma/d\alpha_{[\pi^{-}p][\pi^{+}p^{\prime}]}$ ($\mu$b/rad) | 0-360∘ | 10
$d\sigma/d\alpha_{[\pi^{+}p][\pi^{-}p^{\prime}]}$ ($\mu$b/rad) | 0-360∘ | 10
$d\sigma/d\alpha_{[\pi^{+}\pi^{-}][pp^{\prime}]}$ ($\mu$b/rad) | 0-360∘ | 10
Table 2: List of the one-fold differential cross sections measured with CLAS
[43, 44] and the binning over the kinematic variables.
$M_{i,j}^{min}=M_{i}+M_{j}$ and $M_{i,j}^{max}=W-M_{k}$, where $M_{i,j}$ and
$M_{k}$ are the invariant masses of the final state hadron pair $(i,j)$, and
the mass of the third final state hadron $k$, respectively.

###  II.1 Kinematic Variables and $\pi^{+}\pi^{-}p$ Electroproduction Cross
Sections

For the $\gamma_{v}p\to\pi^{+}\pi^{-}p^{\prime}$ reaction, the invariant mass
of the final state hadrons $W$ and photon virtuality $Q^{2}$ unambiguously
determine the initial state virtual photon and proton four-momenta in their
center-of-mass (CM) frame with the $z$-axis directed along the $\gamma_{v}$
three-momentum as shown in Fig. 1. The final $\pi^{+}\pi^{-}p$ state is
described by the four-momenta of the three final state hadrons by twelve
variables. Energy-momentum conservation reduces the number of variables down
to eight. Since the three final state hadrons are on-shell, three additional
relations between the final state hadron energies and absolute momentum values
reduce the number of independent variables down to five. Hence, at a given $W$
and $Q^{2}$, the reaction can be fully described by the five-fold differential
cross section $d^{5}\sigma/d^{5}\tau$, where $d^{5}\tau$ is differential in
the five independent variables that determine the final state hadron four-
momenta. There are many choices for these five variables [45]. After defining
$M_{\pi^{+}p}$, $M_{\pi^{-}p}$, and $M_{\pi^{+}\pi^{-}}$ as the invariant
masses of the three possible two-hadron pairs in the final state, we adopt
here the following assignment for the computation of the five-fold
differential cross section:
$d^{5}\tau=dM_{\pi^{+}p}dM_{\pi^{+}\pi^{-}}d\Omega_{\pi^{-}}d\alpha_{[\pi^{-}p][\pi^{+}p^{\prime}]}$,
where $\Omega_{\pi^{-}}$ is the final state $\pi^{-}$ solid angle defined by
the polar ($\theta_{\pi^{-}}$) and azimuthal ($\phi_{\pi^{-}}$) angles shown
in Fig. 1(a), and $\alpha_{[\pi^{-}p][\pi^{+}p^{\prime}]}$ is the rotation
angle of plane B defined by the momenta of the final state $\pi^{+}$ and
$p^{\prime}$ around the axis defined by the final state $\pi^{-}$ momentum,
see Fig. 1(b). This $d^{5}\tau$ differential is used in the computation of the
$\pi^{+}\pi^{-}p$ cross sections within the JM model for comparison with the
experimental data [46, 47]. All frame-dependent variables are defined in the
final state hadron CM frame.

The $\pi^{+}\pi^{-}p$ electroproduction data have been collected in the bins
of a seven-dimensional space, since for the description of the initial state
kinematics, $W$ and $Q^{2}$ are required. The number of bins in the seven-
dimensional reaction phase space and the kinematic area covered by the data
for extraction of the differential cross sections are detailed in Tables 1 and
2. The huge number of seven-dimensional bins over the reaction phase space
($\approx 1\times 10^{7}$ bins) does not allow us to use the correlated multi-
fold differential cross sections in the analysis of the data. More than half
of the five-dimensional phase-space bins of the final state hadrons at any
given $W$ and $Q^{2}$ remain unpopulated due to statistical limitations.
Therefore, we use the following one-fold differential cross sections in each
bin of $W$ and $Q^{2}$ covered by the data:

  * •

invariant mass distributions for the three pairs of the final state particles
$d\sigma/dM_{\pi^{+}\pi^{-}}$, $d\sigma/dM_{\pi^{+}p}$, and
$d\sigma/dM_{\pi^{-}p}$;

  * •

distributions for the CM polar angles of the three final state particles
$d\sigma/(\sin\theta_{\pi^{-}}d\theta_{\pi^{-}})$,
$d\sigma/(\sin\theta_{\pi^{+}}d\theta_{\pi^{+}})$, and
$d\sigma/(\sin\theta_{p^{\prime}}d\theta_{p^{\prime}})$;

  * •

distributions for the three $\alpha$-angles determined in the CM frame:
$d\sigma/d\alpha_{[\pi^{-}p][\pi^{+}p^{\prime}]}$,
$d\sigma/d\alpha_{[\pi^{+}p][\pi^{-}p^{\prime}]}$, and
$d\sigma/d\alpha_{[\pi^{+}\pi^{-}][pp^{\prime}]}$, where
$d\sigma/d\alpha_{[\pi^{+}p][\pi^{-}p^{\prime}]}$ and
$d\sigma/d\alpha_{[\pi^{+}\pi^{-}][pp^{\prime}]}$ are defined analogously to
$d\sigma/d\alpha_{[\pi^{-}p][\pi^{+}p^{\prime}]}$ described above.

The one-fold differential cross sections were obtained by integrating the
five-fold differential cross sections over the other four kinematic variables
of $d^{5}\tau$. However, the angular distributions for the polar angles of the
final state $\pi^{+}$ and $p$, as well as for the rotation angles around the
axes along the momenta of these final state hadrons, cannot be obtained from
$d^{5}\tau$ described above, since this differential does not depend on these
variables. Two other sets of differentials $d^{5}\tau^{\prime}$ and
$d^{5}\tau^{\prime\prime}$ are required, which contain
$d\Omega_{\pi^{+}}d\alpha_{[\pi^{+}p][\pi^{-}p^{\prime}]}$ and
$d\Omega_{p^{\prime}}d\alpha_{[pp^{\prime}][\pi^{+}\pi^{-}]}$, respectively,
as described in Refs. [47, 9]. The five-fold differential cross sections
evaluated over the other two differentials were computed from the five-fold
differential cross section over the $d^{5}\tau$ differential by means of cross
section interpolation. For each kinematic point in the five-dimensional phase
space determined by the variables of the $d^{5}\tau^{\prime}$ and
$d^{5}\tau^{\prime\prime}$ differentials, the four-momenta of the three final
state hadrons were computed, and from these values, the five variables of the
$d^{5}\tau$ differential were determined. The $d^{5}\sigma/d^{5}\tau$ cross
sections were interpolated into this five-dimensional kinematic point.

###  II.2 Reaction Model for Extraction of Electrocouplings

The $N(1440)1/2^{+}$, $N(1520)3/2^{-}$, and $\Delta(1600)3/2^{+}$
electrocouplings have been extracted for $Q^{2}$ from 2.0–5.0 GeV2 from the
data on the $\pi^{+}\pi^{-}p$ differential cross sections by fitting them
within the framework of the data-driven JM reaction model detailed in Refs.
[16, 9, 10], referred to as JM17, which was used for the extraction of the
electrocouplings for $Q^{2}<1.5$ GeV2 and $W<1.8$ GeV. Within this approach,
the $\pi^{+}\pi^{-}p$ electroproduction mechanisms seen through their
manifestation in the observables as peaks in the invariant mass distributions
for the final state hadrons and with pronounced dependencies in the CM angular
distributions for the final state hadrons were incorporated. The remaining
mechanisms without pronounced kinematic dependencies were accounted for by
exploring the correlations between the shapes of their contributions into the
nine independent one-fold differential cross sections.

Figure 2: The $\gamma_{v}p\to\pi^{+}\pi^{-}p^{\prime}$ electroproduction
mechanisms incorporated at the amplitude level into the JM17 model [16, 9,
10]: a) full amplitude; b) $\pi^{-}\Delta^{++}$ and
$\pi^{-}\Delta^{++}(1600)3/2^{+}$ sub-channels; c) $\pi^{+}\Delta^{0}$,
$\pi^{+}N^{0}(1520)3/2^{-}$, and $\pi^{+}N^{0}(1680)5/2^{+}$ sub-channels; d)
$\rho p$ sub-channel; e) direct $2\pi$ mechanisms.

The mechanisms incorporated into the JM model are shown in Fig. 2. The
amplitudes of the $\gamma_{v}p\to\pi^{+}\pi^{-}p^{\prime}$ reaction are
described as a superposition of the $\pi^{-}\Delta^{++}$, $\pi^{+}\Delta^{0}$,
$\rho p$, $\pi^{+}N^{0}(1520)$, and $\pi^{+}N^{0}(1680)$ sub-channels with
subsequent decays of the unstable hadrons to the final state $\pi^{+}\pi^{-}p$
state as detailed in Appendix III of Ref. [16]. In addition, direct $2\pi$
production mechanisms, where the final $\pi^{+}\pi^{-}p$ comes about without
going through the intermediate process of forming unstable hadron states are
included. Evidence for these contributions is seen in analyses of the final
state hadron angular distributions with the phenomenological amplitudes
described in Ref. [16].

$N^{*}$ States | Mass, | Total Decay | Refs.
---|---|---|---
Incorporated | GeV | Width |
in Data Fit |  |  $\Gamma_{tot}$, GeV |
$N(1440)1/2^{+}$ | 1.43-1.48 | 0.25-0.40 | [8]
$N(1520)3/2^{-}$ | 1.51-1.53 | 0.12-0.13 | [8]
$N(1535)1/2^{-}$ | 1.51-1.55 | 0.12-0.18 | [8]
$N(1650)1/2^{-}$ | 1.64-1.67 | 0.15-0.16 | [27]
$N(1680)5/2^{+}$ | 1.68-1.69 | 0.11-0.13 | [7]
$N(1700)3/2^{-}$ | 1.65-1.75 | 0.16-0.18 | [7]
$N^{\prime}(1720)3/2^{+}$ | 1.71-1.74 | 0.11-0.13 | [4]
$N(1720)3/2^{+}$ | 1.73-1.76 | 0.11-0.13 | [27]
$\Delta(1600)3/2^{+}$ | 1.50-1.64 | 0.20-0.30 | [48]
$\Delta(1620)1/2^{-}$ | 1.60-1.66 | 0.11-0.15 | [27]
$\Delta(1700)3/2^{-}$ | 1.67-1.73 | 0.23-0.32 | [27]
Table 3: List of resonances included in the fit of the $\pi^{+}\pi^{-}p$
differential cross sections within the JM23 model and their parameters:
masses, total decay widths $\Gamma_{tot}$, and ranges of their variation. The
JM17 model contains all listed resonances, except for the
$\Delta(1600)3/2^{+}$. The starting values for the resonance electrocouplings
were taken from the references given in the last column. The electrocouplings
of the $N(1650)1/2^{-}$, $N(1720)3/2^{+}$, $N^{\prime}(1720)3/2^{+}$, and
$\Delta(1620)1/2^{-}$ were obtained for $Q^{2}<1.5$ GeV2 and extrapolated to
the $Q^{2}$ area covered by the CLAS data [43, 44] as described in Ref. [27].
The predictions of Ref. [48] were used as the starting values for the
$\Delta(1600)3/2^{+}$ electrocouplings.

Within the JM17 model, only the $\pi^{-}\Delta^{++}$, $\pi^{+}\Delta^{0}$, and
$\rho p$ channels contain contributions from $N^{*}$s excited in the
$s$-channel for the $\gamma_{v}p$ interaction. The JM17 model incorporates
contributions from all well-established $N^{*}$ states listed in Table 3,
except for the $\Delta(1600)3/2^{+}$ that was not included. Note that the
four-star $N(1675)5/2^{-}$ and $N(1710)1/2^{+}$ states were not included. The
amplitudes for electroexcitation of the $N(1675)5/2^{-}$ off protons are
suppressed in comparison with the electrocouplings of other resonances in the
third resonance region. Furthermore, in this work we have only determined the
$\gamma_{v}pN^{*}$ electrocouplings for resonances in the mass range below 1.6
GeV. In this case, only the tail from the weakly excited $N(1675)5/2^{-}$ can
contribute. As well, studies of $\pi^{+}\pi^{-}p$ electroproduction in the
third resonance region have revealed no evidence for contributions from the
$N(1710)1/2^{+}$[4, 23].

The resonant amplitudes are described by a unitarized Breit-Wigner ansatz [9],
which accounts for the transition between the same and different resonances in
the dressed resonance propagator, which makes the resonant amplitudes
consistent with restrictions imposed by a general unitarity condition [49,
50]. Quantum number conservation in the strong interaction allows for
transitions between the pairs of $N^{*}$ states,
$N(1520)3/2^{-}\leftrightarrow N(1700)3/2^{-}$, $N(1535)1/2^{-}\leftrightarrow
N(1650)1/2^{-}$, and $N(1720)3/2^{+}\leftrightarrow N^{\prime}(1720)3/2^{+}$,
which are incorporated into the JM17 model.

The non-resonant amplitudes in the $\pi\Delta$ sub-channels are described by
the minimal set of current conserving Reggeized Born terms detailed in Ref.
[16]. They include the contact, Reggeized $t$-channel $\pi$-in-flight,
$s$-channel nucleon, and $u$-channel $\Delta$-in-flight terms. Note, as $W$ is
going to threshold, the Reggeized $t$-channel term gradually transforms into
the $\pi$-pole term, allowing use of the Reggeized Born terms at low $W$.

Figure 3: Description of the nine one-fold differential $\pi^{+}\pi^{-}p$
electroproduction cross sections measured with CLAS [43, 44] (black points)
achieved within the JM17 reaction model (left) and after improvements within
the updated version JM23 (right) at $W$ from 1.45–1.48 GeV and $Q^{2}$ from
3.50–4.20 GeV2. $\theta_{h}$ represents the CM emission angles of the final
state hadrons ($h=\pi^{-},\pi^{+},p$). The computed differential cross
sections are shown by the black solid lines, while the contributions from the
$\pi^{-}\Delta^{++}$ and $\pi^{+}\Delta^{0}$ channels are shown by the red
dashed and blue dotted lines, respectively. The contributions from direct
$2\pi$ production are shown by the magenta long-dashed lines.

Figure 4: Description of the nine one-fold differential $\pi^{+}\pi^{-}p$
electroproduction cross sections measured with CLAS [43, 44] (in red) achieved
within the JM17 reaction model (left) and after improvements within the
updated version JM23 (right) at $W$ from 1.50–1.53 GeV and $Q^{2}$ from
3.50–4.20 GeV2. The legend for the curves is the same as in Fig. 3.

Figure 5: Description of the nine one-fold differential $\pi^{+}\pi^{-}p$
electroproduction cross sections measured with CLAS [43, 44] (in red) achieved
within the JM17 reaction model (left) and after improvements within the
updated version JM23 (right) at $W$ from 1.55–1.58 GeV and $Q^{2}$ from
3.50–4.20 GeV2. The legend for the curves is the same as in Fig. 3.

The $\pi^{+}N^{0}(1520)3/2^{-}$ and $\pi^{+}N^{0}(1680)5/2^{+}$ channels are
described in the JM17 model by non-resonant contributions only. The amplitudes
of the $\pi^{+}N^{0}(1520)3/2^{-}$ sub-channel were derived from the non-
resonant Born terms in the $\pi\Delta$ sub-channels by implementing an
additional $\gamma_{5}$-matrix that accounts for the opposite parities of the
$\Delta(1232)3/2^{+}$ and $N(1520)3/2^{-}$ [16]. The magnitudes of the
$\pi^{+}N^{0}(1520)3/2^{-}$ production amplitudes were independently fit to
the data for each bin in $W$ and $Q^{2}$. The contributions from the
$\pi^{+}N^{0}(1520)3/2^{-}$ sub-channel should be taken into account for
$W>1.5$ GeV. The $\pi^{+}N^{0}(1680)5/2^{+}$ contributions are seen in the
data at $W>1.6$ GeV. These contributions are almost negligible at smaller $W$.
Effective $t$-channel exchange terms were employed in the JM17 model for
parameterization of the amplitudes of this sub-channel [16]. The magnitudes of
the $\pi^{+}N^{0}(1680)5/2^{+}$ amplitudes were fit to the data for each bin
in $W$ and $Q^{2}$.

In general, unitarity requires the presence of direct $2\pi$ production
mechanisms in the $\pi^{+}\pi^{-}p$ electroproduction amplitudes, where the
final state is created without going through the intermediate step of forming
unstable hadron states [51, 52]. These $2\pi$ processes are beyond the
aforementioned contributions from the two-body sub-channels and are
implemented into the JM17 model. These mechanisms are incorporated by a
sequence of two exchanges in the $t$\- and/or $u$-channel by unspecified
particles that belong to two Regge trajectories. The amplitudes of the $2\pi$
mechanisms are parameterized by a Lorentz-invariant contraction between the
spin-tensors of the initial and final state particles, while two exponential
propagators describe the exchanges by unspecified particles. All details on
the parameterization of the $2\pi$ mechanisms are available in Refs. [16, 10].
The magnitudes of these amplitudes are fit to the data for each bin in $W$ and
$Q^{2}$.

The studies of the final state hadron angular distributions over $\alpha_{i}$
($i=[\pi^{-}p][\pi^{+}p^{\prime}]$, $[\pi^{+}p][\pi^{-}p^{\prime}]$,
$[\pi^{+}\pi^{-}][pp^{\prime}]$) in Ref. [10] demonstrated the need to
implement relative phases for all $2\pi$ mechanisms included in the JM17 model
determined in the data fit. The contributions from these mechanisms are
maximal and substantial ($\approx$30%) for $W<1.5$ GeV and they decrease with
increasing $W$, contributing less than 10% for $W>1.6$ GeV. However, even in
this kinematic regime, these mechanisms can be seen in the $\pi^{+}\pi^{-}p$
cross sections due to an interference of the amplitudes with the two-body sub-
channels.

Representative examples of the description of the $\pi^{+}\pi^{-}p$ cross
sections achieved within the JM17 model in several bins of $W<1.6$ GeV for
$Q^{2}$ from 3.50–4.20 GeV2 are shown in Figs. 3, 4, and 5 (left). The chosen
$W$ intervals are closest to the Breit-Wigner masses of the $N(1440)1/2^{+}$,
$N(1520)3/2^{-}$, and $\Delta(1600)3/2^{+}$. The following discrepancies have
been observed in the description of these data:

  * •

The JM17 model overestimates the $\pi^{-}$ and $p$ polar angular distributions
for forward CM angles.

  * •

These discrepancies are correlated with overestimated $\pi^{+}$ angular
distributions at backward CM polar angles.

  * •

Substantial deviations between the computed and measured
$d\sigma/d\alpha_{[\pi^{+}\pi^{-}][pp^{\prime}]}$ differential cross sections
become evident.

  * •

The JM17 model cannot reproduce the shape of the $\pi^{+}\pi^{-}$ invariant
mass distributions for $W<1.55$ GeV and of the $\pi^{-}p$ invariant mass
distributions at $W<1.50$ GeV for $Q^{2}$ from 2.0–5.0 GeV2.

Comparisons of Figs. 3, 4, and 5 (left) demonstrate that these discrepancies
are related mostly to the limitations of the JM17 model for the description of
the $\pi\Delta$ channels for $Q^{2}$ from 2.0–5.0 GeV2. The phenomenological
extra contact terms employed for the description of the contributions into the
$\pi\Delta$ amplitudes beyond the Born terms have been further modified to
achieve the quality of the data description needed for the extraction of the
electrocouplings.

These modifications are achieved by multiplying the extra contact term
amplitudes $T^{0}_{e.c.t.\,\pi\Delta}$ employed in the JM17 model (detailed in
Ref. [16], Appendix II) with (a) the four factors
$F_{1}(t^{\prime}_{pp^{\prime}})$, $F_{2}(t^{\prime}_{\gamma\pi^{-}})$,
$F_{3}(t^{\prime}_{\gamma\pi^{+}})$, and $F_{4}(t^{\prime}_{\gamma
p^{\prime}})$, which allow for a better description of the CM $\theta_{i}$
($i=\pi^{+},\pi^{-},p^{\prime}$) angular distributions of the final state
hadrons and (b) the factor $M_{1}(M_{\pi^{+}\pi^{-}})$ needed in order to
improve the description of $\pi^{+}\pi^{-}$ invariant mass distributions. The
updated extra contact term amplitudes $T_{e.c.t.\pi\Delta}$ are parameterized
as:

$\displaystyle T_{e.c.t.\,\pi\Delta}=T^{0}_{e.c.t.\,\pi\Delta}\cdot
F_{1}(t^{\prime}_{pp^{\prime}})\cdot F_{2}(t^{\prime}_{\gamma\pi^{-}})$
$\displaystyle~{}~{}\cdot F_{3}(t^{\prime}_{\gamma\pi^{+}})\cdot
F_{4}(t^{\prime}_{\gamma p^{\prime}})\cdot M_{1}(M_{\pi^{+}\pi^{-}}).$ (1)

Here $t^{\prime}_{pp^{\prime}}$, $t^{\prime}_{\gamma\pi^{-}}$,
$t^{\prime}_{\gamma\pi^{+}}$, and $t^{\prime}_{\gamma p^{\prime}}$ are the
squared four-momentum transfers defined by the difference between the initial
state $\gamma_{v}$ and $p$ and one of the final state hadrons, and their
maximum values in the respective physics regions. They are defined by:

$\displaystyle
t_{pp^{\prime}}=(p_{p}-p_{p^{\prime}})^{2},~{}~{}t^{\prime}_{pp^{\prime}}=t_{pp^{\prime}}-t_{pp^{\prime}}^{max},$
$\displaystyle
t_{pp^{\prime}}^{max}=2m^{2}_{p}-2E_{p}E_{p^{\prime}}+2\left|p_{p}\right|\left|p_{p^{\prime}}\right|$
(2)

$\displaystyle
t_{\gamma\pi^{-}}=(q_{\gamma}-p_{\pi^{-}})^{2},~{}~{}t^{\prime}_{\gamma\pi^{-}}=t_{\gamma\pi^{-}}-t_{\gamma\pi^{-}}^{max},$
$\displaystyle
t_{\gamma\pi^{-}}^{max}=-Q^{2}+m^{2}_{\pi}-2E_{\gamma}E_{\pi^{-}}+2\left|q_{\gamma}\right|\left|p_{\pi^{-}}\right|,$
$\displaystyle Q^{2}=-q_{\gamma}^{2}$ (3) $\displaystyle
t_{\gamma\pi^{+}}=(q_{\gamma}-p_{\pi^{+}})^{2},~{}~{}t^{\prime}_{\gamma\pi^{+}}=t_{\gamma\pi^{+}}-t_{\gamma\pi^{+}}^{max},$
$\displaystyle
t_{\gamma\pi^{+}}^{max}=-Q^{2}+m^{2}_{\pi}-2E_{\gamma}E_{\pi^{+}}+2\left|q_{\gamma}\right|\left|p_{\pi^{+}}\right|$
(4) $\displaystyle t_{\gamma
p^{\prime}}=(q_{\gamma}-p_{p^{\prime}})^{2},~{}~{}t^{\prime}_{\gamma
p^{\prime}}=t_{\gamma p^{\prime}}-t_{\gamma p^{\prime}}^{max},$ $\displaystyle
t_{\gamma
p^{\prime}}^{max}=-Q^{2}+m^{2}_{p}-2E_{\gamma}E_{p^{\prime}}+2\left|q_{\gamma}\right|\left|p_{p^{\prime}}\right|.$
(5)

Here $q_{\gamma}$, $p_{\pi^{+}}$, $p_{\pi^{-}}$, and $p_{p^{\prime}}$ are the
four-momenta of the initial state photon and the final state $\pi^{+}$,
$\pi^{-}$, and $p$, respectively, and $p_{p}$ is the four-momentum of the
initial state proton. $q^{2}$, $E_{p}$, and $m_{p}$ represent the virtual
photon four-momentum squared, the initial state proton CM energy, and the
proton mass. $E_{\pi^{+}}$, $E_{\pi^{-}}$, $E_{p^{\prime}}$, and
$\left|p_{\pi^{-}}\right|$, $\left|p_{\pi^{+}}\right|$,
$\left|p_{p^{\prime}}\right|$ are the CM energies and absolute values of the
three-momenta for the final state $\pi^{+}$, $\pi^{-}$, and $p$, respectively,
while $\left|q_{\gamma}\right|$ is the absolute value of the virtual photon
three-momentum in the CM frame.

The factors $F_{j}$ in Eq.(1) are parameterized as:

$F_{j}=\begin{cases}-t^{\prime}_{j},&\mbox{if}\,\,\,t_{j}>\Lambda_{j}(W,Q^{2})\\\
\Lambda_{j}(W)-t_{j}^{max}&\mbox{if}\,\,\,t_{j}<\Lambda_{j}(W,Q^{2}),\end{cases}$
(6)

where $j$=$pp^{\prime}$, $\gamma\pi^{-}$, $\gamma\pi^{+}$, $\gamma p^{\prime}$
and the parameters $\Lambda_{j}(W,Q^{2})$ are adjusted to reproduce the data
on the CM $\pi^{+}$, $\pi^{-}$, and $p$ angular distributions in each bin of
$W$ and $Q^{2}$ independently.

Figure 6: Description of the nine one-fold differential $\pi^{+}\pi^{-}p$
electroproduction cross sections in terms of $\chi^{2}/d.p.$ determined from
the point-by-point comparison between the experimental data (only statistical
uncertainties taken into account) and the computed values within the JM23
model as a function of $W$ for $Q^{2}$ as indicated on the plots.
$\chi^{2}/d.p.$ with/without implementation of the $\Delta(1600)3/2^{+}$ is
shown by the blue dashed/black solid lines, respectively. The Breit-Wigner
mass of the $\Delta(1600)3/2^{+}$ is indicated by the middle vertical line and
the outer vertical lines show the total decay width.

The $F_{i}$ factors make the extra contact term contributions equal to zero at
the maximum accessible values of the respective squared four-momentum
transfers. They cause the amplitudes to increase with $-t^{\prime}_{j}$ (or
with absolute $t^{\prime}_{j}$ values) in the range of
$t^{\prime}_{j}>\Lambda_{j}$, making the final state hadron CM angular
distributions closer to those measured with CLAS.

The factor $M_{1}(M_{\pi^{+}\pi^{-}})$ in Eq.(1) is essential to improve the
description of the $\pi^{+}\pi^{-}$ invariant mass distributions. It is given
by:

$M_{1}(M_{\pi^{+}\pi^{-}})=\begin{cases}\frac{M^{2}_{\pi^{+}\pi^{-}}-a_{\pi^{+}\pi^{-}}(W,Q^{2})m^{2}_{\pi}}{(W-b_{\pi^{+}\pi^{-}}(W,Q^{2})M_{p})^{2}-M^{2}_{\pi^{+}\pi^{-}}}&\mbox{if}\,\,\,W<1.6\,\,\mbox{GeV}\\\
1&\mbox{if}\,\,\,W>1.6\,\,\mbox{GeV,}\end{cases}$ (7)

where the parameters $a_{\pi^{+}\pi^{-}}(W,Q^{2})$ ($a_{\pi^{+}\pi^{-}}>4.0$)
and $b_{\pi^{+}\pi^{-}}(W,Q^{2})$ ($0.0<b_{\pi^{+}\pi^{-}}<1.0$) are adjusted
to the data in each bin of $W$ and $Q^{2}$ independently. At values of
$b_{\pi^{+}\pi^{-}}(W,Q^{2})$ equal to unity, the factor
$M_{1}(M_{\pi^{+}\pi^{-}})$ develops a pole at the maximum kinematically
allowed invariant masses $M_{\pi^{+}\pi^{-}}$. Hence, the denominator in
Eq.(7) defines the shape of the $M_{\pi^{+}\pi^{-}}$ invariant mass
distributions at their largest kinematically allowed values, while the
numerator in Eq.(7) regulates the slope of the $M_{\pi^{+}\pi^{-}}$ mass
distributions.

The modifications of the JM17 model extra contact term described above and the
implementation of the $\Delta(1600)3/2^{+}$ led to the updated model version
referred to as JM23. With the JM23 model, a reasonable description of the
experimental data on the nine independent one-fold differential
$\pi^{+}\pi^{-}p$ electroproduction cross sections has been achieved in the
extended $Q^{2}$ range as exemplified in Figs. 3, 4, and 5 (right). The
$\chi^{2}/d.p.$ ($d.p.$ = data point) values computed within the range of
$W<1.7$ GeV for $Q^{2}$ from 2.0–5.0 GeV2 from the point-by-point comparison
between the measured $\pi^{+}\pi^{-}p$ differential cross sections and those
evaluated within JM23 are shown in Fig. 6 as a function of $W$ in each $Q^{2}$
interval covered by the analyzed CLAS data. Only the statistical data
uncertainties were included in the evaluation of $\chi^{2}/d.p.$ to enhance
the sensitivity of these quantities to the parameterization of non-resonant
contributions. In each bin the $\chi^{2}/d.p.$ values are comparable with
those achieved in previous analyses of $\pi^{+}\pi^{-}p$ electroproduction
data [9, 10] where the electrocouplings were deduced from the data fits within
the previous versions of the JM model. From 6 to 8 parameters of the non-
resonant amplitudes were varied in the extraction of the $\gamma_{v}pN^{*}$
electrocouplings, which were fit to 102 data points in each bin of
$(W,Q^{2})$.

We also explored the impact of the $\Delta(1600)3/2^{+}$ on the description of
the $\pi^{+}\pi^{-}p$ differential cross sections. The four-star status for
this state was assigned after 2017, hence, it was not included in the JM17
model. The $\Delta(1600)3/2^{+}$ was incorporated into the JM23 version with
starting hadronic decay parameters taken from the PDG [12] and with the
initial $\gamma_{v}pN^{*}$ electrocouplings from the CSM predictions [48]. The
results in Fig. 6 demonstrate improvements in the description of the
$\pi^{+}\pi^{-}p$ differential cross sections after implementation of this
state. The improvements become more pronounced with increasing $Q^{2}$,
suggesting a relative increase of the signal from the $\Delta(1600)3/2^{+}$
with $Q^{2}$. For the highest $Q^{2}$ bin, because of the increase of the data
uncertainties, the improvement of the data description is less pronounced. It
is also worth noting that the implementation of the $\Delta(1600)3/2^{+}$ into
JM23 has allowed for the reduction of the magnitudes of the direct $2\pi$
production amplitudes by 20–50% and for the extra contact term amplitudes in
the $\pi\Delta$ sub-channels by 10–30%, which represent the contributions
described within the entirely phenomenological parameterization. These
observations have confirmed the impact of the $\Delta(1600)3/2^{+}$
contribution in $\pi^{+}\pi^{-}p$ electroproduction.

Resonance |  $W$ Interval, GeV
---|---
| 1.41-1.51
$N(1440)1/2^{+}$ | 1.46-1.56
| 1.51-1.61
| 1.41-1.51
$N(1520)3/2^{-}$ | 1.46-1.56
| 1.51-1.61
| 1.46-1.56
$\Delta(1600)3/2^{+}$ | 1.51-1.61
| 1.56-1.66
Table 4: $W$ intervals where the nine independent one-fold differential
$\pi^{+}\pi^{-}p$ electroproduction cross sections were fit independently for
$Q^{2}$ from 2.0–5.0 GeV2 to extract the $N(1440)1/2^{+}$, $N(1520)3/2^{-}$,
and $\Delta(1600)3/2^{+}$ electrocouplings and their $BF$ for decays into the
$\pi\Delta$ and $\rho p$ intermediate states with JM23.

##  III Resonance Parameter Extraction From Cross Section Fits

The electrocouplings of the $N(1440)1/2^{+}$, $N(1520)3/2^{-}$, and
$\Delta(1600)3/2^{+}$, as well as their branching fractions for decays into
$\pi\Delta$ and $\rho p$, were extracted from fits of the nine independent
one-fold differential $\pi^{+}\pi^{-}p$ cross sections as described in Section
II.1. A good description of the $\pi^{+}\pi^{-}p$ electroproduction data
achieved with the JM23 model at $W<1.7$ GeV for $Q^{2}$ from 2.0–5.0 GeV2
allows for the determination of the resonance parameters from the data fit for
$W$ from 1.40–1.66 GeV and $Q^{2}$ from 2.0–5.0 GeV2. The electrocouplings and
hadronic decay parameters were obtained from independent fits within the three
overlapping $W$ intervals given in Table 4. The non-resonant contributions in
these $W$ intervals are different, while the electrocouplings determined from
the data fits should be the same within their uncertainties, validating
extraction of these quantities.

In the data fit we simultaneously varied the electrocouplings, the resonance
partial decay widths into $\pi\Delta$ and $\rho p$, and the Breit-Wigner
masses for all $N^{*}$ states listed in Table 3. The starting values for the
resonance decay amplitudes into $\pi\Delta$ and $\rho p$ of orbital angular
momentum $L$ and total spin $S$, are defined by

$\sqrt{\Gamma^{i}_{LS}}=\sqrt{\Gamma_{tot}\cdot BF^{i}_{LS}},$ (8)

where the resonance total decay widths $\Gamma_{tot}$ were taken from Ref.
[12], and the branching fractions $BF^{i}_{LS}$ ($i=\pi\Delta,\rho p$) for the
$N(1440)1/2^{+}$ and $N(1520)3/2^{-}$ were taken from the previous studies of
CLAS $\pi N$ and $\pi^{+}\pi^{-}p$ electroproduction data [8, 9, 10]. For
other excited states, the outcome of the analyses of Refs. [53, 54] were used
for the $BF^{i}_{LS}$ starting values. For each resonance, the total decay
width was computed as the sum of all partial decay widths. The floating of the
resonance masses and total decay widths $\Gamma_{tot}$ caused by variation of
the partial hadronic decay widths into $\pi\Delta$ and $\rho p$ were limited
by the intervals given in Ref. [12]. In this way, we imposed restrictions for
the variation of the $N^{*}$ partial hadronic decay widths $\Gamma^{i}_{LS}$
($i=\pi\Delta,\rho p$)into $\pi\Delta$ and $\rho p$ decomposed over the
$LS$-partial waves.

The starting values for the $N(1440)1/2^{+}$ and $N(1520)3/2^{-}$
electrocouplings used in the fit of the $\pi^{+}\pi^{-}p$ electroproduction
cross sections were taken from the analysis of $\pi N$ electroproduction [8].
The transverse electrocouplings $A_{1/2}$ and $A_{3/2}$ of these states were
varied by employing normal distributions with the $\sigma$ parameters equal to
30% of their starting values. There were no restrictions on the minimum or
maximum trial electrocoupling values, allowing us to explore the area of
$\approx\pm 3\sigma$ around the starting values. The longitudinal $S_{1/2}$
electrocouplings of smaller absolute values were varied within a broader range
so that their absolute values overlapped with the absolute values of the
transverse electrocouplings.

The starting values for the electrocouplings of the $\Delta(1600)3/2^{+}$ were
based on the CSM predictions [48]. Currently, CSM takes into account only the
contributions from the quark core that gradually dominate as $Q^{2}$
increases, typically for $Q^{2}\gtrsim 2$ GeV2, which is the range covered in
this analysis. Therefore, the actual starting values of the
$\Delta(1600)3/2^{+}$ electrocouplings were taken from the values of Ref. [48]
multiplied by a common factor of 0.6 applied over the entire range of $Q^{2}$
for all three electrocouplings $A_{1/2}$, $A_{3/2}$, and $S_{1/2}$. This
factor accounts for the fact that the wave function of the
$\Delta(1600)3/2^{+}$ represents a superposition of the contributions from the
quark core and meson-baryon cloud. In the data fit, implementation of this
factor was needed to reproduce the results on the final state proton CM
angular distributions in the forward hemisphere. The $\Delta(1600)3/2^{+}$
electrocouplings were varied employing normal distributions with the $\sigma$
parameters equal to 50% of their starting values.

In the fits we simultaneously varied the electrocouplings of all $N^{*}$s of
four-star status (other than the $N(1440)1/2^{+}$, $N(1520)3/2^{-}$, and
$\Delta(1600)3/2^{+}$) as described above within the mass range below 1.7 GeV.
This variation employed a normal distribution with the $\sigma$ parameters
equal to 20% of their starting values, which were taken from the analyses of
the CLAS exclusive meson electroproduction data [1] with the numerical results
available in Ref. [27].

Figure 7: Fit of the nine one-fold differential $\pi^{+}\pi^{-}p$
electroproduction cross sections measured with CLAS [44] (in black) achieved
within the JM23 model for $W$ from 1.450–1.475 GeV (top left), 1.500–1.525
GeV, (top right) and 1.550–1.575 GeV (bottom) for $Q^{2}$ from 3.0-3.5 GeV2.
The data point uncertainties are evaluated as a quadratic sum of statistical
and relevant systematic uncertainties. The groups of red curves represent the
JM23 fits closest to the data. The resonant contributions are shown in blue.

In the data fit we also varied the following parameters of the non-resonant
mechanisms employed in the JM23 model:

  * •

the magnitudes of the additional contact-term amplitudes in the
$\pi^{-}\Delta^{++}$ and $\pi^{+}\Delta^{0}$ channels (1 parameter per
$Q^{2}$-bin);

  * •

the magnitudes of the $\pi^{+}N^{0}(1520)3/2^{-}$ channel (1 parameter per
$Q^{2}$-bin);

  * •

the magnitudes of all direct $2\pi$ production amplitudes (up to 6 parameters
per $Q^{2}$-bin).

The starting values for these parameters were determined in their initial
adjustment to the nine independent one-fold $\pi^{+}\pi^{-}p$ differential
cross sections. We applied $W$-independent multiplicative factors to the
magnitudes of the non-resonant amplitudes listed above. They remained the same
in the entire $W$ interval covered by the fit within any $Q^{2}$-bin, but they
depended on $Q^{2}$ and were fit to the data in each $Q^{2}$-bin
independently. The multiplicative factors were varied around unity, employing
normal distributions with $\sigma$ values in the range of 20%. In this way, we
retained a smooth $W$-dependence of the non-resonant contributions established
in the adjustment to the data and explored the possibility of improving the
data description in the simultaneous variation of the resonant and non-
resonant parameters.

The special data fit procedure described in Ref. [9] was employed for the
extraction of the resonance parameters. It allowed us to obtain not only the
best fit but also to establish bands of the computed cross sections that were
compatible with the data within their uncertainties. For each trial set of
JM23 resonant and non-resonant parameters, we computed the nine one-fold
differential $\pi^{+}\pi^{-}p$ cross sections and $\chi^{2}$/$d.p.$ values.
The latter were estimated in point-by-point comparisons between the measured
and computed cross sections in all bins of $W$ from 1.41–1.66 GeV for $Q^{2}$
from 2.0–5.0 GeV2 covered by the CLAS data. The data uncertainties account for
both the statistical and that part of the systematic uncertainty dependent on
the final state hadron kinematics, which were added in quadrature. In the fit,
we selected the computed one-fold differential cross sections closest to the
data with $\chi^{2}/d.p.$ less than a predetermined maximum value. These
values of $\chi^{2}_{max}/d.p.$ were obtained by requiring that the computed
cross sections with smaller $\chi^{2}/d.p.$ be within the data uncertainties
for the majority of the data points. In this fit procedure, we obtained the
$\chi^{2}/d.p.$ intervals within which the computed cross sections described
the data equally well within the data uncertainties. The ranges of
$\chi^{2}/d.p.$ listed in Table 5 demonstrate that the cross sections selected
in the data fit are indeed distributed within the data uncertainties over the
entire area of $(W,Q^{2})$ covered in the data analysis.

$W$ Interval, |  |  |  |
---|---|---|---|---
GeV | 1.41-1.51 | 1.46-1.56 | 1.51-1.61 | 1.56-1.66
$\chi^{2}/d.p.$ |  |  |  |
Ranges | 0.51-0.57 | 0.52-0.67 | 0.52-0.69 | 0.69-0.76
Table 5: The ranges of $\chi^{2}/d.p.$ for the nine one-fold differential
$\pi^{+}\pi^{-}p$ electroproduction cross sections selected in the data fit
computed within JM23 in overlapping $W$ intervals for $Q^{2}$ from 2.0–5.0
GeV2. The uncertainties for the measured data are given by the quadratic sum
of the statistical and that part of the systematic uncertainty dependent on
the final state hadron kinematics.

Representative examples for the fit quality of the cross sections within the
$W$ intervals closest to the Breit-Wigner masses of the $N(1440)1/2^{+}$,
$N(1520)3/2^{-}$, and $\Delta(1600)3/2^{+}$ for $Q^{2}$ from 3.0–3.5 GeV2 are
shown in Fig. 7. The computed cross sections selected in the data fit are
shown by the red curves. The resonant contributions were obtained from the
differential cross sections computed from only the resonance amplitudes and
are shown by the blue bars. The deduced uncertainties for the resonance
contributions are comparable both with the uncertainties of the measured
differential cross sections and with the spread of the fits computed within
the JM23 model. Pronounced differences are evident in the shapes of the
computed differential cross sections selected in the data fit and the
respective resonant contributions, in particular, for all angular
distributions. The shapes of the resonant contributions are different in each
of the nine one-fold differential cross sections but they are highly
correlated by the reaction dynamics that underlie the resonance excitations in
the $s$-channel and their subsequent decays into either the $\pi\Delta$ or
$\rho p$ intermediate states. The resonant/non-resonant contribution
differences allow for isolation of the resonant contributions. The
electrocouplings and decay widths into $\pi\Delta$ and $\rho p$ have been
determined from the resonant contributions by fitting them within the
unitarized Breit-Wigner ansatz [9, 49], taking into account the constraints
imposed on the resonant amplitudes by a general unitarity condition.

##  IV $N(1440)1/2^{+}$, $N(1520)3/2^{-}$, and $\Delta(1600)3/2^{+}$
Electrocouplings and Hadronic Decay Widths to $\pi\Delta$ and $\rho p$

The resonance parameters determined from the data fit include the
electrocouplings, the partial decay widths into $\pi\Delta$ and $\rho p$, and
the total resonance decay widths. They are averaged from the group of fits
selected by the $\chi^{2}/d.p.$ limits for each bin and their mean values are
taken as the resonance parameters extracted from the data. The r.m.s
dispersions in these parameters are taken as the uncertainties. The
electrocoupling uncertainties obtained in this manner take into account both
the statistical and systematic uncertainties in the data, as well as the
systematic uncertainties associated with the JM23 model. Furthermore, we
consistently account for the correlations between the variations of the
resonant and non-resonant contributions when extracting the resonance
parameters. In the cases where the ranges of the extracted electrocouplings
covered more than 90% of the intervals for the electrocoupling variation
(starting values $\pm\sigma$) employed in the data fit, we further increased
the ranges of the variation and repeated the data fit, so that eventually the
electrocouplings extracted from the data were inside the intervals of the
variations employed in the data fit. In this way, we made sure that the
employed ranges were sufficient to determine both the mean values of the
resonance parameters and their uncertainties.

Figure 8: Electrocouplings of the $N(1440)1/2^{+}$ determined from independent fits of the $\pi^{+}\pi^{-}p$ electroproduction cross sections in three $W$ intervals, 1.41–1.51 GeV (blue squares), 1.46–1.56 GeV (red circles), and 1.51–1.61 GeV (black triangles), for $Q^{2}$ from 2.0–5.0 GeV2 within the JM23 model. Figure 9: Electrocouplings of the $N(1520)3/2^{-}$ determined from independent fits of the $\pi^{+}\pi^{-}p$ electroproduction cross sections in three $W$ intervals, 1.41–1.51 GeV (blue squares), 1.46–1.56 GeV (red circles), and 1.51–1.61 GeV (black triangles), for $Q^{2}$ from 2.0–5.0 GeV2 within the JM23 model. $Q^{2}$ Inter- | Mass, |  $\Gamma_{tot}$, | $\Gamma_{\pi\Delta},$ |  $BF_{\pi\Delta}$, |  $\Gamma_{\rho p}$, |  $BF_{\rho p}$,
---|---|---|---|---|---|---
val, GeV2 | GeV | MeV | MeV | % | MeV | %
0.25-0.60 | 1.458$\pm$0.012 | 363$\pm$39 | 142$\pm$48 | 23-58 | 6$\pm$4 |  $<$2
0.5-1.5 | 1.450$\pm$0.011 | 352$\pm$37 | 120$\pm$41 | 20-52 | 5$\pm$2 |  $<$2
2.0-3.5 | 1.457$\pm$0.008 | 331$\pm$54 | 129$\pm$52 | 20-65 | 6$\pm$2 | 1.1-2.6
3.0-5.0 | 1.446$\pm$0.013 | 352$\pm$33 | 151$\pm$32 | 31-57 | 5$\pm$1 | 1.2-2.0
Table 6: Masses and total/partial hadronic decay widths of the $N(1440)1/2^{+}$ into $\pi\Delta$ and $\rho p$ determined from fits of the $\pi^{+}\pi^{-}p$ electroproduction cross sections carried out independently within different $Q^{2}$ intervals. The new results from this work are given in the last two rows. The results in the upper rows are available from previous studies [9, 10] of the $\pi^{+}\pi^{-}p$ electroproduction cross sections. $Q^{2}$ Inter- | Mass, |  $\Gamma_{tot}$, | $\Gamma_{\pi\Delta},$ |  $BF_{\pi\Delta}$, |  $\Gamma_{\rho p}$, |  $BF_{\rho p}$,
---|---|---|---|---|---|---
val, GeV2 | GeV | MeV | MeV | % | MeV | %
0.25-0.60 | 1.521$\pm$0.004 | 127$\pm$4 | 35$\pm$4 | 24-32 | 16$\pm$5 | 8-17
0.5-1.5 | 1.520$\pm$0.001 | 125$\pm$4 | 36$\pm$5 | 25-34 | 13$\pm$6 | 5-16
2.0-3.5 | 1.518$\pm$0.003 | 122$\pm$7 | 29$\pm$5 | 19-30 | 15$\pm$6 | 7-18
3.0-5.0 | 1.522$\pm$0.003 | 121$\pm$7 | 30$\pm$5 | 20-30 | 13$\pm$5 | 6-16
Table 7: Masses and total/partial hadronic decay widths of the
$N(1520)3/2^{-}$ into $\pi\Delta$ and $\rho p$ determined from fits of the
$\pi^{+}\pi^{-}p$ electroproduction cross sections carried out independently
within different $Q^{2}$ intervals. The new results from this work are given
in the last two rows. The results in the upper rows are available from
previous studies [9, 10] of the $\pi^{+}\pi^{-}p$ electroproduction cross
sections.

###  IV.1 Parameters for the $N(1440)1/2^{+}$ and $N(1520)3/2^{-}$

Figure 10: (Left) $N(1440)1/2^{+}$ electrocouplings determined from the $\pi
N$ differential cross sections, beam, target, and beam-target asymmetries [8]
(red triangles) and from the $\pi^{+}\pi^{-}p$ differential cross sections
(blue squares) for $Q^{2}$ from 2.0–5.0 GeV2 presented in this work. The
electrocouplings from the $\pi N$ data after interpolation over $Q^{2}$ are
compared with the results from the $\pi^{+}\pi^{-}p$ data. (Right)
$N(1440)1/2^{+}$ electrocouplings from the $\pi N$ and $\pi^{+}\pi^{-}p$ data
for $Q^{2}$ from 0.25–5.0 GeV2. The results from $\pi N$ electroproduction [8]
are shown by the red circles. The electrocouplings from the $\pi^{+}\pi^{-}p$
differential cross sections measured with CLAS for $Q^{2}$ from 0.25–1.5 GeV2
[9, 10] are shown by the green triangles. The electrocouplings determined
within the JM23 model are shown by the magenta diamonds. The photocouplings
from the PDG [12] and from the CLAS $\pi N$ photoproduction data [55] are
shown by the blue triangle and square, respectively.

The hadronic decay widths of the $N(1440)1/2^{+}$ and $N(1520)3/2^{-}$ were
deduced from the fits of the $\pi^{+}\pi^{-}p$ differential cross sections
using the procedures described in Section III. The results are presented in
Table 6 for the $N(1440)1/2^{+}$ and in Table 7 for the $N(1520)3/2^{-}$. For
both resonances, their masses and total/partial hadronic decay widths into
$\pi\Delta$ and $\rho p$ are self-consistent in the four $Q^{2}$ intervals
covered by the CLAS $\pi^{+}\pi^{-}p$ electroproduction cross sections from
the previous studies [9, 10] and those reported in this work. The successful
fit of the data achieved within a broad range of $Q^{2}$ from 0.25–5.0 GeV2
with $Q^{2}$-independent resonance masses and total/partial hadronic decay
widths given the pronounced evolution of the non-resonant mechanisms with
$Q^{2}$, demonstrates that both the $N(1440)1/2^{+}$ and $N(1520)3/2^{-}$ are
excited states of the proton produced in the $s$-channel for the $\gamma_{v}p$
interaction.

The electrocouplings for the $N(1440)1/2^{+}$ and $N(1520)3/2^{-}$ determined
from the fits of the $\pi^{+}\pi^{-}p$ cross sections carried out
independently in three overlapping $W$ intervals for $Q^{2}$ from 2.0–5.0 GeV2
are shown in Figs. 8 and 9, respectively. The non-resonant contributions in
these three $W$ intervals are different, however the extracted
electrocouplings are consistent within the uncertainties, suggesting credible
extraction of these quantities.

Figure 11: (Left) $N(1520)3/2^{-}$ electrocouplings determined from the $\pi
N$ differential cross sections, beam, target, and beam-target asymmetries [8]
(red triangles) and from the $\pi^{+}\pi^{-}p$ differential cross sections
(blue squares) for $Q^{2}$ from 2.0–5.0 GeV2 presented in this work. The
electrocouplings from the $\pi N$ data after interpolation over $Q^{2}$ are
compared with the results from the $\pi^{+}\pi^{-}p$ data. (Right)
$N(1520)3/2^{-}$ electrocouplings from the $\pi N$ and $\pi^{+}\pi^{-}p$ data
for $Q^{2}$ from 0.25–5.0 GeV2. The results from $\pi N$ electroproduction [8]
are shown by the red circles. The electrocouplings from the $\pi^{+}\pi^{-}p$
differential cross sections measured with CLAS for $Q^{2}$ from 0.25–1.5 GeV2
[9, 10] are shown by the green triangles. The electrocouplings determined
within the JM23 model are shown by the magenta diamonds. The photocouplings
from the PDG [12] and from the CLAS $\pi N$ photoproduction data [55] are
shown by the blue triangles and squares, respectively.

In order to compare results for the $N(1440)1/2^{+}$ and $N(1520)3/2^{-}$
electrocouplings in the $\pi^{+}\pi^{-}p$ electroproduction channel with the
values from the analysis of $\pi N$ electroproduction, we must use common
decay branching fractions to these final states for each resonance. Within the
JM23 model, the sum of the branching fractions into $\pi N$ and $\pi\pi N$
accounts for almost 100% of the total decay widths of the $N(1440)1/2^{+}$ and
$N(1520)3/2^{-}$. Since the $\pi N$ exclusive electroproduction channels are
the most sensitive to contributions from the $N(1440)1/2^{+}$ and
$N(1520)3/2^{-}$, we re-evaluated the branching fraction for the decay to the
$\pi\pi N$ final states $BF(\pi\pi N)_{corr}$ as

$BF(\pi\pi N)_{corr}=1-BF(\pi N).$ (9)

For these resonance decays to $\pi\pi N$, it turns out that the estimated
branching fractions $BF(\pi\pi N)_{corr}$ are slightly ($<$10%) different with
respect to those obtained from the $\pi^{+}\pi^{-}p$ fit ($BF(\pi\pi N)_{0}$).
Therefore, we multiplied the $\pi\Delta$ and $\rho p$ hadronic decay widths of
the $N(1440)1/2^{+}$ and $N(1520)3/2^{-}$ from the $\pi^{+}\pi^{-}p$ fit by
the ratio $\frac{BF(\pi\pi N)_{corr}}{BF(\pi\pi N)_{0}}$. The electrocouplings
obtained were then multiplied by the correction factors

$C_{hd}=\sqrt{\frac{BF(\pi\pi N)_{corr}}{BF(\pi\pi N)_{0}}}$ (10)

in order to keep the resonant parts and the computed $\pi^{+}\pi^{-}p$
differential cross sections unchanged under the re-scaling of the resonance
hadronic decay parameters described above.

A special procedure was developed for the evaluation of the transverse $A_{i}$
($i=1/2$, $3/2$) and longitudinal $S_{i}$ ($i=1/2$) electrocouplings analyzing
the results from independent fits of the electroproduction cross sections in
the three $W$ intervals with electrocouplings and uncertainties
$A_{i,j}\pm\delta A_{i,j}$ and $S_{i,j}\pm\delta S_{i,j}$, where the index
$j=1\to 3$ is the $W$ interval. First, we found the overlap range for the
electrocouplings [$A_{i}^{min}-A_{i}^{max}$] ($i=1/2$, $3/2$) and
[$S_{i}^{min}-S_{i}^{max}$] ($i=1/2$) from the data fit in the three $W$
intervals

$\displaystyle A_{i}^{min}=min[A_{i,j}+\delta A_{i,j}]$ (11) $\displaystyle
S_{i}^{min}=min[S_{i,j}+\delta S_{i,j}]$ $\displaystyle
A_{i}^{max}=max[A_{i,j}-\delta A_{i,j}]$ $\displaystyle
S_{i}^{max}=max[S_{i,j}-\delta S_{i,j}].$

Within these ranges the best data description was achieved in all $W$
intervals. Consequently, the mean values for $A_{i}$ and $S_{i}$ were
redefined as:

$\displaystyle A_{i}=\frac{A_{i}^{min}+A_{i}^{max}}{2},~{}(i=1/2,3/2)$ (12)
$\displaystyle S_{i}=\frac{S_{i}^{min}+S_{i}^{max}}{2},~{}(i=1/2).$

There are three sources of uncertainties in the evaluation of $A_{i}$ and
$S_{i}$ in each of the three $W$ intervals: a) the range of overlap between
the electrocouplings determined from the data fit defined in Eq.(11), b) the
root mean square (RMS) for the mean values of the determined electrocouplings,
i.e. RMS [$A_{i,j}$] and RMS [$S_{i,j}$], and c) the differences between the
redefined electrocouplings according to Eq.(12) and their average values
obtained from the data fit. The total uncertainties were obtained as the
quadrature sum of the contributions a) to c)

$\displaystyle\delta
A_{i}=\sqrt{\frac{(A_{i}^{max}-A_{i}^{min})^{2}}{4}+(RMS[A_{i,j}])^{2}+\Delta
A_{i}^{2}}\,$ (13) $\displaystyle\Delta
A_{i}=\frac{A_{i}^{min}+A_{i}^{max}}{2}-\frac{\sum_{j=1,2,3}A_{i,j}}{3}$
$\displaystyle\delta
S_{i}=\sqrt{\frac{(S_{i}^{max}-S_{i}^{min})^{2}}{4}+(RMS[S_{i,j}])^{2}+\Delta
S_{i}^{2}}\,$ $\displaystyle\Delta
S_{i}=\frac{S_{i}^{min}+S_{i}^{max}}{2}-\frac{\sum_{j=1,2,3}S_{i,j}}{3}.$

The electrocouplings determined for the $N(1440)1/2^{+}$ and $N(1520)3/2^{-}$
are listed in Tables 8 and 9, respectively. We consider these results as the
final electrocouplings from the analysis of the $\pi^{+}\pi^{-}p$
electroproduction cross sections within the JM23 model.

$Q^{2}$ Interval, |  $A_{1/2}\times 1000$, |  $S_{1/2}\times 1000$,
---|---|---
GeV2 | GeV-1/2 | GeV-1/2
2.0-2.4 | 68.7 $\pm$ 6.6 | 7.6 $\pm$ 4.1
2.4-3.0 | 60.2 $\pm$ 4.9 | 11.8 $\pm$ 4.2
3.0-3.5 | 32.3 $\pm$ 7.5 | 8.7 $\pm$ 5.3
3.5-4.2 | 24.6 $\pm$ 3.6 | 11.6 $\pm$ 3.0
4.2-5.0 | 22.8 $\pm$ 4.3 | 3.3 $\pm$ 2.0
Table 8: $N(1440)1/2^{+}$ electrocouplings determined from the $\pi^{+}\pi^{-}p$ differential cross sections measured with the CLAS detector [43, 44] in three $W$ intervals, 1.41–1.51 GeV, 1.46–1.56 GeV, and 1.51–1.61 GeV, for $Q^{2}$ from 2.0–5.0 GeV2 evaluated according to Eqs.(12,13). $Q^{2}$ Interval, |  $A_{1/2}\times 1000$, |  $S_{1/2}\times 1000$, |  $A_{3/2}\times 1000$,
---|---|---|---
GeV2 | GeV-1/2 | GeV-1/2 | GeV-1/2
2.0-2.4 | -35.8 $\pm$ 4.8 | -3.8 $\pm$ 5.4 | 16.8 $\pm$ 2.2
2.4-3.0 | -37.5 $\pm$ 5.2 | -4.1 $\pm$ 5.6 | 15.8 $\pm$ 3.2
3.0-3.5 | -25.7 $\pm$ 3.5 | -0.1 $\pm$ 2.4 | 9.0 $\pm$ 3.5
3.5-4.2 | -16.8 $\pm$ 5.2 | -3.3 $\pm$ 1.4 | 11.2 $\pm$ 4.3
4.2-5.0 | -15.6 $\pm$ 6.7 | -0.5 $\pm$ 3.0 | 7.0 $\pm$ 4.1
Table 9: $N(1520)3/2^{-}$ electrocouplings determined from the
$\pi^{+}\pi^{-}p$ differential cross sections measured with the CLAS detector
[43, 44] in three $W$ intervals, 1.41–1.51 GeV, 1.46–1.56 GeV, and 1.51–1.61
GeV, for $Q^{2}$ from 2.0–5.0 GeV2 evaluated according to Eqs.(12,13).

In Figs. 10 (left) and 11 (left) we compare the electrocouplings of the
$N(1440)1/2^{+}$ and $N(1520)3/2^{-}$ obtained from analyses of $\pi N$
electroproduction cross sections, beam, target, and beam-target asymmetries
within the unitary isobar model and dispersion relation approach [8, 56] with
the results available from analysis of the nine independent one-fold
differential $\pi^{+}\pi^{-}p$ electroproduction cross sections within JM23.
The analyses of the $\pi N$ and $\pi^{+}\pi^{-}p$ data were carried out with
different $Q^{2}$-binning. For direct comparison, the electrocouplings
obtained from $\pi N$ were interpolated over $Q^{2}$, so that the
electrocouplings could be compared at the same $Q^{2}$-values. The comparison
between all currently available electrocouplings for the $N(1440)1/2^{+}$ and
$N(1520)3/2^{-}$ from CLAS $\pi N$ and $\pi^{+}\pi^{-}p$ electroproduction is
shown in Figs. 10 (right) and 11 (right). Overall, good agreement has been
achieved between the electrocouplings of both the $N(1440)1/2^{+}$ and
$N(1520)3/2^{-}$ determined from independent analyses of the $\pi N$ and
$\pi^{+}\pi^{-}p$ data.

The $\pi N$ and $\pi^{+}\pi^{-}p$ electroproduction channels account for the
largest part of the total meson electroproduction cross sections in the
resonance region for $W<2$ GeV. Their non-resonant amplitudes are different,
however, the $N^{*}$ electrocouplings obtained from independent studies of
these channels should be the same at each $Q^{2}$, since the resonance
electroexcitation and hadronic decay amplitudes into the different final
states should be independent. Hence, consistent results on the
electrocouplings of the $N(1440)1/2^{+}$ and $N(1520)3/2^{-}$ deduced from
$\pi N$ and $\pi^{+}\pi^{-}p$ observed within a broad range of $Q^{2}$ from
the photon point to 5.0 GeV2 validates the extraction of the
$\gamma_{v}pN^{*}$ electrocouplings from the $\pi^{+}\pi^{-}p$
electroproduction data.

Figure 12: (Left) $\Delta(1600)3/2^{+}$ electrocouplings deduced from
independent fits of the $\pi^{+}\pi^{-}p$ differential cross sections carried
out within three $W$ intervals, 1.46–1.56 GeV (blue squares), 1.51–1.61 GeV
(red circles), and 1.56–1.66 GeV (black triangles), for $Q^{2}$ from 2.0–5.0
GeV2. (Right) $\Delta(1600)3/2^{+}$ electrocouplings evaluated by combining
the results from the three $W$ intervals as described in Section IV.1.

###  IV.2 Parameters for the $\Delta(1600)3/2^{+}$

After the discovery of several new excited states of the proton from global
multichannel analysis of exclusive meson photo- and hadroproduction data [57],
the status of the $\Delta(1600)3/2^{+}$ was elevated to a four-star firmly
established state [12]. This resonance decays preferentially into $\pi\Delta$
and has been included in the JM23 model.

$W$ Inter- |  $Q^{2}$ Inter- | Mass, |  $\Gamma_{tot}$, |  $\Gamma_{\pi\Delta}$, |  $BF_{\pi\Delta}$,
---|---|---|---|---|---
val, GeV | val, GeV2 | GeV | GeV | GeV | %
1.46-1.56 | 2.0-3.5 | 1.55$\pm$0.014 | 244$\pm$21 | 154$\pm$21 | 50-78
1.51-1.61 | 2.0-3.5 | 1.57$\pm$0.018 | 259$\pm$21 | 169$\pm$22 | 52-81
1.56-1.66 | 2.0-3.5 | 1.57$\pm$0.042 | 256$\pm$33 | 166$\pm$34 | 46-90
1.46-1.56 | 3.0-5.0 | 1.56$\pm$0.030 | 249$\pm$37 | 158$\pm$37 | 42-92
1.51-1.61 | 3.0-5.0 | 1.56$\pm$0.030 | 249$\pm$34 | 158$\pm$34 | 44-89
1.56-1.66 | 3.0-5.0 | 1.58$\pm$0.039 | 263$\pm$29 | 172$\pm$29 | 49-86
PDG | PDG | 1.50-1.64 | 200-300 | 172$\pm$29 | 73-83
Table 10: Mass and total/partial decay widths of the $\Delta(1600)3/2^{+}$
into $\pi\Delta$ determined from the fit of $\pi^{+}\pi^{-}p$
electroproduction cross sections carried out independently within two
intervals in $Q^{2}$ and within three overlapping intervals in $W$. The PDG
parameters are listed in the bottom row.

The electrocouplings, mass, and total/partial hadronic decay widths of the
$\Delta(1600)3/2^{+}$ have been determined for the first time from independent
analysis of the $\pi^{+}\pi^{-}p$ differential cross sections within three
overlapping $W$ intervals, 1.46–1.56 GeV, 1.51–1.61 GeV, and 1.56–1.66 GeV,
for $Q^{2}$ from 2.0–3.5 GeV2 and from 3.0–5.0 GeV2. The $\Delta(1600)3/2^{+}$
contributes substantially to each of these intervals.

The mass and hadronic decay parameters of the $\Delta(1600)3/2^{+}$ determined
from independent fits within the six overlapping ($W$,$Q^{2}$) bins listed in
Table 10 are consistent within their uncertainties. They are also in good
agreement with reported PDG values [12]. The masses and hadronic decay widths
of $N^{*}$s excited in the $s$-channel for $\gamma_{v}p$ interactions should
be $Q^{2}$-independent, since the resonance electroexcitation and hadronic
decay amplitudes are independent. The resonance hadronic decay widths obtained
from the data fit within the three $W$ intervals should also be the same since
the corresponding hadronic decay amplitudes are defined at the resonant point
$W=M_{N^{*}}$. Therefore, the results in Table 10 provide evidence for the
manifestation of the $\Delta(1600)3/2^{+}$ as an $s$-channel resonance seen in
$\pi^{+}\pi^{-}p$ electroproduction. Furthermore, the pronounced
$Q^{2}$-evolution observed in the non-resonant contributions makes
interpretation of the $\Delta(1600)3/2^{+}$ as a singularity of the non-
resonant amplitudes or a dynamically generated resonance unlikely. The
successful description of the $\pi^{+}\pi^{-}p$ electroproduction data within
the ($W$,$Q^{2}$) bins listed in Table 10 achieved with $W$\- and
$Q^{2}$-independent $\Delta(1600)3/2^{+}$ mass and total/partial hadronic
decay widths has also demonstrated the capability of the JM23 model for the
evaluation of the resonant contributions from this state.

$Q^{2}$ Interval, |  $A_{1/2}\times 1000$, |  $S_{1/2}\times 1000$, |  $A_{3/2}\times 1000$,
---|---|---|---
GeV2 | GeV-1/2 | GeV-1/2 | GeV-1/2
2.0-2.4 | -11.7 $\pm$ 2.5 | 7.5 $\pm$ 2.2 | -22.1 $\pm$ 4.0
2.4-3.0 | -10.6 $\pm$ 3.5 | 9.0 $\pm$ 1.8 | -16.4 $\pm$ 3.7
3.0-3.5 | -8.4 $\pm$ 1.0 | 8.0 $\pm$ 2.1 | -14.5 $\pm$ 3.8
3.5-4.2 | -7.0 $\pm$ 1.7 | 4.2 $\pm$ 1.5 | -11.3 $\pm$ 4.4
4.2-5.0 | -4.9 $\pm$ 1.1 | 5.1 $\pm$ 2.1 | -9.2 $\pm$ 2.7
Table 11: $\Delta(1600)3/2^{+}$ electrocouplings determined from the
$\pi^{+}\pi^{-}p$ differential cross sections measured with the CLAS detector
in three $W$ intervals, 1.46–1.56 GeV, 1.51–1.61 GeV, and 1.56–1.66 GeV, for
$Q^{2}$ from 2.0–5.0 GeV2 evaluated as described in Section IV.1.

The procedure for the extraction of the $\Delta(1600)3/2^{+}$ electrocouplings
is similar to that used for the $N(1440)1/2^{+}$ and $N(1520)3/2^{-}$
described in Section III. As starting values for the $\Delta(1600)3/2^{+}$
electrocouplings, we explored a $\pm$50% range around the values predicted by
CSM [48]. Under this variation, the $\pi^{+}\pi^{-}p$ differential cross
sections computed within JM23 are spread within a range that overlaps the
measured differential cross sections for the dominant part of the CLAS data
points [43, 44]. The extracted $\Delta(1600)3/2^{+}$ electrocouplings within
each of the three $W$ intervals are shown in Fig. 12 (left). The non-resonant
amplitudes in the three $W$ intervals are different, however, the determined
$\Delta(1600)3/2^{+}$ electrocouplings are the same within their
uncertainties. This success solidifies the evidence for the extraction of the
$\Delta(1600)3/2^{+}$ electrocouplings. The final results for the
$\Delta(1600)3/2^{+}$ electrocouplings were determined by combining the
results obtained from the data fits in the three $W$ intervals using the
procedure described in Section IV.1. They are listed in Table 11 and shown in
Fig. 12 (right). Currently, the CLAS $\pi^{+}\pi^{-}p$ electroproduction cross
sections are the only data from which electrocouplings of this state have
become available. Extraction of the $\Delta(1600)3/2^{+}$ electrocouplings
from $\pi^{0}p$ electroproduction data [58, 59] will be the next important
step in the exploration of the structure of this state.

Figure 13: (Left) $N(1440)1/2^{+}$ $A_{1/2}$ electrocouplings determined from
studies of $\pi N$ electroproduction (red circles) [8] and from
$\pi^{+}\pi^{-}p$ electroproduction for $Q^{2}<1.5$ GeV2 (green triangles) [9,
10] and for $Q^{2}$ from 2.0–5.0 GeV2 available from this work (magenta
diamonds). The electrocoupling description within CSM [40], employing a
momentum dependent dressed quark mass deduced from the QCD Lagrangian, is
shown by the black solid line. The descriptions achieved within light-front
quark models are shown that a) implement a phenomenological momentum-dependent
dressed quark mass [31] (black dashed line) and b) account for both an inner
core of three constituent quarks and an external meson-baryon cloud [60] (blue
dashed line). (Right) The evolution of the $N(1440)1/2^{+}$ complex pole mass
available from the analysis of meson photo- and hadroproduction data within
the Argonne-Osaka coupled-channel approach [61, 62] for running values of the
meson-baryon couplings from zero (corresponding to the bare quark-core mass on
the real energy axis shown by the shaded gray circle) to the finite values
determined from the data. The mass of the observed $N(1440)1/2^{+}$ is
determined by the two poles in the complex energy plane labeled on the graph
as A and B. The colored lines show the pole movement and splitting as the
meson-baryon couplings increase. The horizontal dashed lines show the cuts
owing to the opening of the quasi-two-body channels with unstable hadrons.
Figure 14: Faddeev equation for computation of the masses and wave functions
of the quark core of the ground and excited states of the nucleon. The kernel
for the matrix-valued integral equations is represented by the blue area.

##  V Insight into Nucleon Resonance Structure

The new results on the $\gamma_{v}pN^{*}$ electrocouplings available from the
analysis of the CLAS $\pi^{+}\pi^{-}p$ electroproduction data for $Q^{2}$ from
2.0–5.0 GeV2, together with the previously available results from $\pi N$ and
$\pi^{+}\pi^{-}p$ electroproduction off protons for $Q^{2}<1.5$ GeV2, provide
important input needed to check theory predictions on the structure of $N^{*}$
states and their emergence from QCD [3, 34, 62, 6]. In this Section, we
describe new opportunities for the exploration of the structure of the
$N(1440)1/2^{+}$, $N(1520)3/2^{-}$, and $\Delta(1600)3/2^{+}$ provided by the
results on their electrocouplings.

###  V.1 $N(1440)1/2^{+}$ Resonance Structure

The CLAS results on the electrocouplings of the $N(1440)1/2^{+}$ have been
described for $Q^{2}$ from 2.0–5.0 GeV2 within different approaches. The
relativistic light-front quark model [31] and the CSM [40] are shown as
representative examples in Fig. 13 (left) for the description of its $A_{1/2}$
electrocoupling.

The CSM approach provides a good description in the range of $Q^{2}$ from
2.0–5.0 GeV2 of the electrocouplings of the $N(1440)1/2^{+}$ as a bound
quark$+$diquark system in its first radial excitation [40]. Within this
approach, the momentum dependence of the dressed quark and gluon masses has
been evaluated from the solution of the QCD equations of motion for the quark
and gluon fields. The gluon self-interaction encoded in the QCD Lagrangian
underpins the emergence of the dynamically generated gluon mass, which at
distances on the order of the hadron size, approaches the mass scale of
$\approx$0.4 GeV. This process is responsible for the sharp increase of the
QCD running coupling $\alpha_{s}/\pi$ at distances where perturbative QCD
evolves into the strongly coupled QCD (sQCD) regime [36, 38]. At quark momenta
below 2 GeV, where $\alpha_{s}/\pi$ increases rapidly and becomes comparable
with unity, the energy stored in the gluon field is transformed into the
momentum dependence of the dynamically generated dressed quark mass, which
increases rapidly with increasing distance (or inverse quark momentum) and
approaches the mass scale of $\approx$0.4 GeV at quark momenta $<$ 0.5 GeV.
The dressed quarks with momentum-dependent masses deduced from QCD are treated
as the building blocks for the quark core of the ground and excited state
nucleons. Their masses and wave functions are obtained from the solution of
the Faddeev equations for three dressed quarks in the approximation of a
quark$+$diquark kernel [39, 40, 34].

The diquark correlations employed in CSM are different in comparison with the
rigid diquarks of constituent quark models. In the CSM approach, diquarks
represent correlated quark pairs, whose correlation amplitudes are computed as
the solution of the Bethe-Salpeter equation with the kernel for the two-
dressed-quark interaction mediated by dressed gluon exchange starting from the
QCD Lagrangian. The CSM diquark is a dynamical object that interacts with the
corresponding third quark, forming a new correlated diquark pair, as shown in
Fig. 14. The masses of the ground and excited states of the nucleon of a given
spin-parity $J^{P}$ have been obtained as poles in the respective $J^{P}$
partial waves of the Faddeev amplitude of the three dressed quarks from the
solution of the Faddeev equations depicted in Fig. 14. The wave functions for
the ground and excited states of the nucleon were obtained from the Faddeev
amplitude residues at the pole positions. The electrocouplings are evaluated
considering the virtual photon interaction with the electromagnetic currents
of the dressed quark and diquark system for the transitions between diquarks
of the same or different spin-parities, and account for the virtual photon
interaction at the vertex describing diquark decay/recombination to/from the
pair of uncorrelated quarks shown by the blue shadowed area in Fig. 14 [39].

Within the light-front quark model [63, 31], the $N(1440)1/2^{+}$ is treated
as a bound system of three constituent quarks in their first radial
excitation. The momentum dependence of the constituent quark mass has been
employed in order to reproduce the experimental results on the nucleon elastic
form factors. With the same momentum dependence of the constituent quark mass
the model has succeeded in providing a reasonable description of the
electrocouplings of all $N^{*}$ states in the mass range up to 1.6 GeV.

As shown in Fig. 13 (left), both CSM and the light-front quark model describe
the $N(1440)1/2^{+}$ electrocouplings for $Q^{2}$ from 2.0–5.0 GeV2 by
employing a momentum-dependent quark mass with virtually coincident
predictions. This success demonstrates strong support for quarks with running
mass as active structural components in the ground and excited states of the
nucleon at distances where the contributions from the quark core become the
largest, which occurs at $Q^{2}\gtrsim 2$ GeV2 for the $N(1440)1/2^{+}$.
However, both the CSM [40] and the light-front quark model of Ref. [31] fail
to reproduce the $N(1440)1/2^{+}$ electrocouplings for $Q^{2}<1$ GeV2. This
failure points to additional contributions to the structure relevant at
distances on the order of the baryon size. These contributions arise from the
meson-baryon cloud. The light-front quark model of Ref. [60], which takes into
account the contributions from both the meson-baryon cloud and the quark core
to the structure of the $N(1440)1/2^{+}$, provides a much better description
of the data at low $Q^{2}$, while retaining a reasonable description at higher
$Q^{2}$. This feature explains the success of the models that account for only
the meson-baryon degrees of freedom in describing the electrocouplings of the
$N(1440)1/2^{+}$ for $Q^{2}<1$ GeV2 [64, 65, 66].

Taking into account the contributions from the quark core and meson-baryon
cloud allowed for the resolution of the long-standing puzzle on the ordering
of the masses of the radial and orbital excitations in the $N^{*}$ spectrum.
Most quark models with quark interactions mediated by gluon exchange predict
the mass of the first radial excitation of three quarks above the mass of the
first orbital excitation. The experimental results, in contrast with these
quark model expectations, revealed that the mass of the $N(1440)1/2^{+}$
(1.440 GeV), a state that represents the first radial excitation of three
quarks, is below that of the states belonging to the [70,1-] SU(6) spin-flavor
super-multiplet (1.520 GeV and 1.535 GeV for the lightest states) and thus
should be expected as the first orbital excitation of three quarks. Coupled-
channel analyses of the meson photo- and hadroproduction data carried out by
the Argonne-Osaka group [61, 62] (see Fig. 13 (right)) revealed that in the
limit of zero meson-baryon coupling corresponding to the contribution from
just the bare quark core, the mass of 1.76 GeV of the $N(1440)1/2^{+}$ radial
excitation would be above the masses of the lightest $N^{*}$ in the [70,1-]
super-multiplet, and hence in agreement with the quark model expectations. As
the meson-baryon couplings increase toward the values established in the data
analysis, the single pole on the real energy axis moves into the complex
energy plane and eventually splits into two poles A and B related to the
$N(1440)1/2^{+}$ properties, while the third pole C moves toward the mass
range above 1.7 GeV as shown in Fig. 13 (right). Therefore, the meson-baryon
dressing is responsible for the shift of the bare quark core mass down to the
value of the measured mass, below the lightest states in the [70,1-] super-
multiplet.

The CLAS results on the $N(1440)1/2^{+}$ electrocouplings available over a
broad range of $Q^{2}<5.0$ GeV2 (see Fig. 13) have revealed its structure as
an interplay between the inner core of three dressed quarks in their first
radial excitation augmented by the external meson-baryon cloud. The meson-
baryon degrees of freedom are most relevant for $Q^{2}\lesssim 1$ GeV2, while
photons of virtualities $Q^{2}\gtrsim 2$ GeV2 interact mostly with the quark
core. Therefore, higher $Q^{2}$ studies are preferential for the exploration
of the quark degrees of freedom in the structure of this state. Thus studies
of the $N^{*}$ electrocouplings over a broad $Q^{2}$ range are necessary to
establish the relevant degrees of freedom in their structure and to shed light
on their distance-dependent evolution.

The CSM analysis has provided predictions for the quark core wave function of
the $N(1440)1/2^{+}$ in terms of quark$+$diquark configurations of certain
values of the orbital angular momentum of the quark relative to the diquark,
as well as for certain isospin and spin-parity values for the diquarks [67].
The Faddeev equations are Poincaré covariant and their solution is available
in any reference frame. For comparison with the quark model expectations, the
quark$+$diquark content in the analysis of Ref. [67] was evaluated in the
resonance rest frame.

Figure 15: $N(1520)3/2^{-}$ electrocouplings determined from studies of $\pi
N$ electroproduction (red circles) [8], and from $\pi^{+}\pi^{-}p$
electroproduction for $Q^{2}<1.5$ GeV2(green triangles) [9, 10] and for
$Q^{2}$ from 2.0–5.0 GeV2 available from this work (magenta diamonds):
$A_{1/2}(Q^{2})$ (top), $S_{1/2}(Q^{2})$ (middle), and $A_{3/2}(Q^{2})$
(bottom). The electrocoupling descriptions achieved within the light-front
quark model that includes only the three-quark configuration expected for the
SU(6) assignment of the $N(1520)3/2^{-}$ [68] are shown by the dotted red
lines. The results of the light-front quark model [31] that employs three
quark configuration mixing and a phenomenological parameterization for the
running quark mass and from the hypercentral constituent quark model [19] are
shown by the solid blue and dashed black curves, respectively.

Evaluations of the contributions from quark$+$diquark configurations into the
resonance mass within CSM showed that the dominant part of the bare quark core
mass of the $N(1440)1/2^{+}$ is created by a configuration with a scalar
diquark of spin-parity $J^{P}=0^{+}$ and relative orbital angular momentum
$L=0$ of the quark. This fully relativistic finding based on the QCD
Lagrangian is in good agreement with expectations from the majority of
constituent quark models [19, 18, 32, 62, 63]. However, the evaluation of the
contributions from the quark$+$diquark configurations in the $N(1440)1/2^{+}$
wave function has revealed a more complex pattern: a) the contributions from
scalar $J^{P}=0^{+}$ and axial-vector diquarks become comparable and b) higher
orbital angular momenta of the quark also contribute to the wave function.
Therefore, studies of only the spectrum of $N^{*}$ states have limited
sensitivity to their structure. Any statement on $N^{*}$ structure based
solely on analysis of the $N^{*}$ spectrum is tenuous as such studies only
account for the wave function behavior at distances comparable with the hadron
size. The full complexity of the $N^{*}$ wave function can be mapped out from
the results on the $Q^{2}$-evolution of the electrocouplings as they are
sensitive to all configurations contributing to the wave function and their
distance-dependent evolution.

Lowest-order Chebyshev projections for all configurations contributing to the
$N(1440)1/2^{+}$ quark core wave function evaluated within CSM [67] revealed
zeros in their dependencies on the relative quark$+$diquark momentum. This
serves as evidence for a radial excitation in the three quark system. Again,
this finding, obtained under direct connection to QCD, explains the success of
the quark model results [19, 18, 32, 62, 63] for the quark core of the
$N(1440)1/2^{+}$ as a radial excitation of the three quark system. The good
description of the $N(1440)1/2^{+}$ electrocouplings obtained from the
$\pi^{+}\pi^{-}p$ electroproduction data of this work and from $\pi N$
electroproduction for $Q^{2}$ from 2.0–5.0 GeV2 achieved with CSM supports
these predictions on its quark core wave function.

###  V.2 $N(1520)3/2^{-}$ Resonance Structure

The detailed and consistent information on the electrocouplings of the
$N(1520)3/2^{-}$ available for $Q^{2}<5.0$ GeV2 from the studies of $\pi N$
and $\pi^{+}\pi^{-}p$ electroproduction sheds light on the relevant degrees of
freedom in its structure. From its SU(6) assignment, the quark core of the
$N(1520)3/2^{-}$ can be described as three quarks in the octet flavor
SU(3)-multiplet of mixed permutation symmetry with spin $S=1/2$ and orbital
angular momentum $L=1$ [19, 68]. The predicted $N(1520)3/2^{-}$
electrocouplings accounting for only this three-quark configuration have
become available from the relativistic quark model [68] and are shown in Fig.
15 by the dotted red lines. These predictions are in strong disagreement with
the experimental results. These significant discrepancies suggest that the
structure of the $N(1520)3/2^{-}$ quark core is more complex than expected
from its SU(6) assignment.

Figure 16: (Top) Color map for the contributions to the $N(1520)3/2^{-}$ wave
function from quark$+$diquark configurations with $L$ and $L^{\prime}$ orbital
angular momenta in the canonical normalization constant of the $N^{*}$ wave
functions accounting for $S$, $P$, $D$, and $F$ quark orbital angular momenta.
Here, $L$ and $L^{\prime}$ represent the quark orbital angular momenta in the
resonance wave function and its conjugate, respectively. The axis labels
represent the parts of the Faddeev amplitude that contain this information
[69]. (Bottom) The contributions from the pairs of quark$+$diquark
configurations to the canonical normalization constant of the $N(1520)3/2^{-}$
quark core wave function computed within CSM [69] in the resonance rest frame.
The color code is shown in the top figure.

A reasonable description of the $N(1520)3/2^{-}$ electrocouplings has been
achieved within the framework of the hypercentral constituent quark model
(hCQM) [19]. Within this approach, the space part of the $N(1520)3/2^{-}$
quark core wave function in the resonance rest frame is described by six
coordinates consisting of two solid angles $\Omega_{\rho}$,
$\Omega_{\lambda}$, hyperangle $\xi$, and hyperradius $x$. Relations between
$\xi$, $x$, and the $\rho$ and $\lambda$ vectors that describe the three quark
systems can be found in Ref. [19]. Both $\xi$ and $x$ are determined by the
coordinates of the three quarks. The wave function of the three-quark system
expressed in terms of the hypercentral coordinates effectively accounts for
the interactions between the three quarks, resulting in their binding into the
quark core. The confining potential employed in the hCQM is a function of $x$
and consists of two parts: a Coulomb term relevant for distances close to the
pQCD regime $\propto 1/x$ and a linear confining part $\propto x$. The
SU(6)-violating part of the $qq$-interaction consists of the hyperfine term
stemming from vector particle exchange between the two quarks and the quark-
flavor mixing term mediated by pseudoscalar meson exchange. Both terms are
supported qualitatively by the CSM approach [39, 22]. The dressed gluon
exchange represents the vector particle exchange between two quarks, while
pseudoscalar meson exchange could be traced back to dynamical chiral symmetry
breaking at distances comparable with the baryon size scale. Both SU(6)
violating terms and the linear confining term underlie the three-quark
configuration mixing. The electrocouplings of most $N^{*}$s have been computed
within the hCQM for $Q^{2}<5.0$ GeV2, keeping all model parameters at the
values that fit the $N^{*}$ spectrum. The hCQM results on the
$Q^{2}$-evolution of the $N(1520)3/2^{-}$ electrocouplings [19] are shown in
Fig. 15 by the dashed black lines, and a reasonable description has been
achieved for $Q^{2}>2.5$ GeV2.

The $N(1520)3/2^{-}$ electrocouplings have also been computed within the
light-front quark model [31] and are shown in Fig. 15 by the solid blue lines.
The major features of this model were highlighted in Section V.1. The
electrocouplings of most $N^{*}$s in the mass range below 1.6 GeV have been
evaluated within this model by employing the same momentum-dependent quark
mass deduced from the fit of the electromagnetic nucleon elastic form factors.
A reasonable description of the CLAS results for the $N(1520)3/2^{-}$
electrocouplings has been achieved for $Q^{2}$ from 1.5–5.0 GeV2. The analysis
of the CLAS results on the $N(1440)1/2^{+}$ and $N(1520)3/2^{-}$
electrocouplings within the light-front quark model [31] suggests that quarks
with momentum-dependent mass represent the active degrees of freedom for the
quark core structure of these states. Studies of the $N(1520)3/2^{-}$
electrocouplings within these models [19, 31] have also demonstrated the
important role of three-quark configuration mixing in the generation of this
state. However, both of these models fail to describe the $N(1520)3/2^{-}$
electrocouplings for $Q^{2}\lesssim 1$ GeV2. These discrepancies are
suggestive of substantial meson-baryon cloud contributions relevant at
distances comparable with the baryon size. The CLAS results on the
$N(1520)3/2^{-}$ electrocouplings show that its structure arises as a
convolution between the external meson-baryon cloud and the inner core of
three dressed quarks.

The contributions from different quark$+$diquark configurations to the quark
core of the $N(1520)3/2^{-}$ have been evaluated within the CSM approach [69].
These studies demonstrated that 92% of its quark core mass is generated by
configurations with axial-vector diquarks of $J^{P}=1^{+}$, while the combined
contributions of the configurations with scalar $J^{P}=0^{+}$ and axial-vector
diquarks account for almost 100% of the quark core mass, and with respect to
the relative angular momentum the $P$-wave quark in the quark$+$diquark rest
frame generates 98% of the $N(1520)3/2^{-}$ quark core mass. These CSM
evaluations demonstrate that the dominant part of the quark core mass of this
state is created by quark$+$diquark configurations consistent with its SU(6)
assignment as the first orbital excitation of three quarks of $L=1$ and total
quark spin $S=1/2$.

However, analysis of the contributions from quark$+$diquark configurations to
the Faddeev amplitude or wave function of the $N(1520)3/2^{-}$ reveals a more
complex pattern that becomes evident in the studies of the contributions from
the pairs of quark$+$diquark configurations with orbital angular momenta $L$
and $L^{\prime}$ in the canonical normalization constant for the resonance
wave function evaluated in its rest frame. Here, $L$ and $L^{\prime}$
represent the quark orbital angular momenta in the resonance wave function and
its conjugate, respectively. The results are shown in Fig. 16. The major part
of the $N(1520)3/2^{-}$ quark core wave function is determined by interference
between $P$\- and $D$-waves and by the negative contribution of the $D$-wave.
The contribution from the $P$-wave is smaller but non-negligible. These
results qualitatively support the quark model findings on the substantial role
of quark configuration mixing in the $N(1520)3/2^{-}$ quark core wave
function. The CSM studies demonstrated that the spectroscopic $N(1520)3/2^{-}$
mass is determined by just a $P$-wave quark$+$diquark configuration consistent
with the SU(6) assignment for this state. Therefore, studies of only the
$N^{*}$ spectrum do not have sufficient sensitivity to elucidate the
complexity of its structure. Indeed, the resonance masses are determined by
just the long wavelength part of the wave function. Instead, the
electrocouplings are sensitive to the contributions from different
quark$+$diquark configurations, shedding light on the full complexity of their
structure.

Consequently, a comparison between the predicted CSM results on the
$Q^{2}$-evolution of the $N(1520)3/2^{-}$ electrocouplings with the results
determined deduced from the $\pi N$ and $\pi^{+}\pi^{-}p$ electroproduction
data measured with CLAS provides a sensitive tool for validation of the CSM
expectations on the complexity of quark$+$diquark configuration mixing in the
$N(1520)3/2^{-}$ wave function obtained under a traceable connection to the
QCD Lagrangian. The evaluations of the $N(1520)3/2^{-}$ electrocouplings
within CSM are currently in progress.

###  V.3 $\Delta(1600)3/2^{+}$ Resonance Structure

The predictions of the $Q^{2}$-evolution of the $\Delta(1600)3/2^{+}$
electrocouplings became available in 2019 within the CSM [48]. These
evaluations employed the same momentum-dependent dressed quark mass deduced
from the QCD Lagrangian and the same framework was used for solving the
Faddeev equation to obtain the $\Delta(1600)3/2^{+}$ wave function as used
previously in the description of the $N(1440)1/2^{+}$ electrocouplings [40].
There are no additional free parameters in the computation of the
$\Delta(1600)3/2^{+}$ electrocouplings. Furthermore, in 2019 no experimental
results for the $\Delta(1600)3/2^{+}$ electrocouplings were available. These
have been obtained for the first time from this work. These CSM predictions
are shown in Fig. 17 by the solid lines in comparison with the results
determined from the data.

Figure 17: $\Delta(1600)3/2^{+}$ electrocouplings obtained in this work:
$A_{1/2}$ (top), $S_{1/2}$ (middle), and $A_{3/2}$ (bottom). For comparison
with the CSM predictions [48] (solid black lines), the electrocouplings
determined from the $\pi^{+}\pi^{-}p$ data analysis have been divided by a
factor of 0.6 to account for the missing contributions from the meson-baryon
cloud in the CSM (see Section V.3 for details).

The studies of the electrocouplings have demonstrated that the $N^{*}$ wave
functions are determined by the combined contributions from the quark core and
meson-baryon cloud. The CSM evaluation of the $\Delta(1600)3/2^{+}$
electrocouplings accounts for the contribution from only the quark core.
Therefore, for comparison with the CSM expectations, the values deduced in
this work should be divided by a factor to account for the contribution from
only the three quark core component to the full resonance wave function
normalization. This factor should be the same for all three
$\Delta(1600)3/2^{+}$ electrocouplings and $Q^{2}$-independent. We obtained
this factor of 0.6 from the best description of the nine one-fold differential
cross sections for $W$ from 1.46–1.66 GeV and $Q^{2}$ from 2.0–5.0 GeV2 by
varying the common multiplicative factor for the three $\Delta(1600)3/2^{+}$
electrocouplings in the range from 0 to 1 with the other parameters of the
JM23 model that had been adjusted to fit the $\pi^{+}\pi^{-}p$
electroproduction cross section data. Figure 17 shows the electrocouplings
from the right column of Fig. 12 divided by a factor of 0.6 and compared with
the CSM expectations. The results on the $Q^{2}$-evolution of the
$\Delta(1600)3/2^{+}$ electrocouplings deduced from the $\pi^{+}\pi^{-}p$
electroproduction data in this work have confirmed the CSM predictions.

Figure 18: The contributions of quark$+$diquark configurations with orbital
angular momenta $L$ and $L^{\prime}$ in the canonical normalization constant
of the $\Delta(1600)3/2^{+}$ quark core wave function computed within CSM in
the resonance rest frame [70]. The color code is shown in Fig. 16.

The quark core structure of the $\Delta(1600)3/2^{+}$ in terms of the
contributing quark$+$diquark configurations has been computed in Ref. [70]. It
was found that nearly 100% of the mass of this state is generated by the
quark$+$diquark configuration with an axial-vector diquark of $J^{P}=1^{+}$
and a quark in a relative $S$-wave in the resonance rest frame. The Chebyshev
moment of this dominant configuration shows a clear zero crossing, providing
evidence for a radial excitation. Hence, the underlying mass generation
configuration for the $\Delta(1600)3/2^{+}$ quark core evaluated within the
CSM is consistent with the SU(6) assignment of this state as the first radial
excitation of three quarks in an $S$-wave coupled to isospin 3/2.

Studies of the contributions of quark$+$diquark configurations with orbital
angular momenta $L$ and $L^{\prime}$ in the canonical normalization constant
for the resonance wave function in its rest frame again revealed a much more
complex structure of this state as shown in Fig. 18. The leading contribution
arises from interference between $S$\- and $F$-waves with sub-leading
contributions from the $D$-wave and interference between $P$\- and $D$-waves.
As in the case of the $N(1520)3/2^{-}$, the mass of the $\Delta(1600)3/2^{+}$
does not have enough sensitivity to unravel the full complexity of the wave
function that can only be revealed in the studies of the $Q^{2}$-evolution of
its electrocouplings. Confirmation of the CSM predictions on the
$Q^{2}$-evolution of the $\Delta(1600)3/2^{+}$ electrocouplings by the
experimental results obtained in this work validates the structure of its
quark core evaluated within CSM under connection to the QCD Lagrangian.

##  VI Conclusions and Outlook

A good description of the nine independent one-fold differential
$\pi^{+}\pi^{-}p$ electroproduction cross sections off the proton has been
achieved within the data-driven JM23 reaction model at $W$ from 1.41–1.66 GeV
for $Q^{2}$ from 2.0–5.0 GeV2. Comparable uncertainties have been achieved for
the extracted resonant contributions in the model fits and the measured cross
sections. The resonance electrocouplings have been determined from the
resonant contributions by employing a unitarized Breit-Wigner ansatz, allowing
the restrictions imposed by a general unitarity condition on the resonant
amplitudes in exclusive $\pi^{+}\pi^{-}p$ electroproduction to be taken into
account [9, 49]. The $\gamma_{v}pN^{*}$ electrocouplings of the
$N(1440)1/2^{+}$, $N(1520)3/2^{-}$, and $\Delta(1600)3/2^{+}$ resonances have
been determined from the $\pi^{+}\pi^{-}p$ electroproduction data for the
first time for $Q^{2}$ from 2.0–5.0 GeV2.

The electrocouplings of the $N(1440)1/2^{+}$ and $N(1520)3/2^{-}$ determined
in this work are in good agreement with the results determined independently
from the $\pi^{+}n$ and $\pi^{0}p$ electroproduction channels [8]. The
consistency of the results from independent studies of $\pi N$ and
$\pi^{+}\pi^{-}p$ with completely different non-resonant contributions,
supports the capabilities of the JM23 reaction model for the extraction of the
resonance electrocouplings from $\pi^{+}\pi^{-}p$ electroproduction data.
Furthermore, consistent results on the electrocouplings of the
$N(1440)1/2^{+}$, $N(1520)3/2^{-}$, and $\Delta(1600)3/2^{+}$ for $Q^{2}$ from
2.0–5.0 GeV2 available from the independent fits of the $\pi^{+}\pi^{-}p$
electroproduction cross sections in three overlapping $W$ intervals with the
contribution from the same resonance and different non-resonant amplitudes,
solidifies the capability of the JM23 model to provide information on the
resonance electrocouplings and their partial hadronic decay widths into the
$\pi\Delta$ and $\rho p$ final states.

A successful description of the $\pi^{+}\pi^{-}p$ electroproduction cross
sections achieved for $Q^{2}$ from 0.2–5.0 GeV2 [16, 9, 10] with the same
$Q^{2}$-independent masses and total/partial hadronic decay widths for the
$N(1440)1/2^{+}$, $N(1520)3/2^{+}$, and $\Delta(1600)3/2^{+}$, suggests that
these nucleon excited states are produced in the $s$-channel of the
$\gamma_{v}p$ interaction.

The new results on the electrocouplings presented in this work extend the
available information on the structure of $N^{*}$ states in the mass range up
to 1.6 GeV. Analyses of these results within the Argonne-Osaka coupled-channel
approach [61, 28, 29], quark models [60, 32, 31], and within CSM under
connection to the QCD Lagrangian [40, 36], have revealed the structure of
these states as a complex interplay between an inner core of three dressed
quarks and an external meson-baryon cloud. Studies of the electrocouplings
over a broad range of $Q^{2}$ are critical to reveal the structure of these
states. At $Q^{2}\lesssim 1$ GeV2, the contribution from the meson-baryon
cloud to the interaction with virtual photons at large distances is maximal.
With increasing $Q^{2}$, photons of high virtuality penetrate the external
meson-baryon cloud and interact mostly with the core of three dressed quarks,
elucidating the three-quark component in the structure of the $N^{*}$ states.

Continuum Schwinger methods [40, 35] have provided a successful description of
the $N(1440)1/2^{+}$ electrocouplings for $Q^{2}$ from 2.0–5.0 GeV2 by
employing the momentum dependence of the dressed quark mass deduced from the
QCD Lagrangian. The CSM results are virtually the same as those obtained
within the relativistic light-front quark model [63, 31], which employed a
phenomenological momentum-dependent dressed quark mass fit to the results on
the nucleon electromagnetic elastic form factors. The predictions of the
$Q^{2}$-evolution of the $\Delta(1600)3/2^{+}$ electrocouplings made by CSM in
2019 [48] with the same dressed quark mass function as described above have
been confirmed by the results obtained in this work from the $\pi^{+}\pi^{-}p$
electroproduction data. All of these studies demonstrate the relevance of
dressed quarks with momentum-dependent running mass as the active degrees of
freedom in $N^{*}$ structure.

The results of this work on the electrocouplings of the $N(1520)3/2^{-}$, as
well as the previously available results from the studies of $\pi N$
electroproduction [8], provide a promising opportunity for CSM and other QCD-
based approaches as they become available to explore the relevance of dressed
quarks to $N^{*}$s of different structure. In constituent quark models, this
state is treated as three quarks with orbital angular momentum $L=1$ and
belonging to the [70,1-] SU(6) spin-flavor multiplet. CSM evaluations of the
electrocouplings of this state are currently in progress.

In the near future we are planning to determine from the CLAS
$\pi^{+}\pi^{-}p$ electroproduction data the electrocouplings of the most
prominent nucleon excited states in the mass range of $W$ from 1.6–2.0 GeV for
$Q^{2}$ from 2.0–5.0 GeV2. Analyses of these results provide additional
promising opportunities for hadron structure theory to shed light on many
facets of the dynamics in the strong QCD regime as more excited states of the
nucleon with different structural features emerge from QCD. These studies also
motivate the potential increase of the Jefferson Lab electron beam energy up
to 22 GeV, which will offer a unique way to explore the full range of
distances where $N^{*}$ structure emerges in the transition from the
perturbative to the strongly coupled QCD regime [71].

###### Acknowledgements.

This work was supported in part by the U.S. Department of Energy (DOE) under
Contract No. DE-AC05-06OR23177, the National Science Foundation (NSF) under
Grant PHY 1812382, the Physics Department of the University of South Carolina
(USC) under NSF Grant PHY 10011349, Jefferson Science Associates (JSA), and
the Skobeltsyn Nuclear Physics Institute and Physics Department at Lomonosov
Moscow State University. ANHB is supported by the DFG through the Research
Unit FOR 2926 (project number 409651613). We are grateful for theory support
and constructive comments on the CSM approach from C. Chen, Z.-F. Cui, L. Liu,
Y. Lu, C.D. Roberts, and J. Segovia, as well as on the quark models from I.G.
Aznauryan, V.E. Lyubovitskij, I.T. Obukhovsky, and E. Santopinto.

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