# Study of $N(1440)$ structure via $\gamma^{*}p\to N(1440)$ transition

A. Kaewsnod<EMAIL_ADDRESS>K. Xu  Z. Zhao  School of Physics and Center
of Excellence in High Energy Physics and Astrophysics, Suranaree University of
Technology, Nakhon Ratchasima 30000, Thailand  X. Y. Liu  School of Physics
and Center of Excellence in High Energy Physics and Astrophysics, Suranaree
University of Technology, Nakhon Ratchasima 30000, Thailand  School of
Physics, Liaoning University, Shenyang 110036, China  S. Srisuphaphon
Department of Physics, Faculty of Science, Burapha University, Chonburi 20131,
Thailand  A. Limphirat  Y. Yan<EMAIL_ADDRESS>School of Physics and
Center of Excellence in High Energy Physics and Astrophysics, Suranaree
University of Technology, Nakhon Ratchasima 30000, Thailand

###### Abstract

We study the photoproduction of the $N(1440)$ resonance in $\gamma^{*}p\to
N^{*}$ process in quark models, where the $N(1440)$ takes different wave
functions: first radial excitation of the nucleon imported from low-lying
baryon mass spectrum calculations, a general radial excitation of the nucleon,
and a $q^{3}$ state with positive parity. The comparison between the
theoretical results and experimental data on the helicity amplitudes $A_{1/2}$
and $S_{1/2}$ and the analysis of the spatial wave function of the $N(1440)$
resonance reveal that the $N(1440)$ resonance is mainly the $q^{3}$ first
radial excitation.

##  I  Introduction

In recent years, the experiments on electro- and photoproductions of particles
have accumulated a large amount of data of helicity amplitudes over the four-
momentum transfer $Q^{2}$ of virtual photons Zyla et al. (2020); Aznauryan et
al. (2009); Mokeev et al. (2012, 2016). The most complete experimental data
has been obtained for the Roper resonance, $N(1440)$, up to 4.5 GeV2 Tiator et
al. (2011, 2009). Theoretical studies have been carried out to reveal the
properties and structures of nucleon resonance states by analyzing the
transverse transition amplitudes $A_{1/2}$ and $A_{3/2}$ and longitudinal
transition amplitude $S_{1/2}$ over $Q^{2}$ in a large range Aznauryan (2007);
Aznauryan et al. (2008); Capstick et al. (2007); Ramalho and Tsushima (2010);
Golli et al. (2009); Burkert and Roberts (2019); Li and Riska (2006);
Obukhovsky et al. (2011); Cano and Gonzalez (1998); Štajner et al. (2017).
However, the electro- and photoproduction data of the Roper resonance have not
been well described by the conventional quark models Aznauryan (2007);
Aznauryan et al. (2008); Capstick et al. (2007); Ramalho and Tsushima (2010);
Golli et al. (2009) as well as other approaches. The nature of the Roper
resonance is still an open question, which is also implied by its large decay
width Sarantsev et al. (2008) as well as the mass ordering with the lowest
negative-parity baryon resonance states $N(1520)$ and $N(1535)$ Capstick and
Roberts (2000). Detailed discussion about the Roper resonance may be found in
a good review paper Burkert and Roberts (2019). Among a large number of
theoretical works, the light-front relativistic quark model Aznauryan (2007);
Aznauryan et al. (2008); Capstick et al. (2007) and the covariant spectator
quark model Ramalho and Tsushima (2010), assuming that the $N(1440)$ is the
first radial excitation of the $q^{3}$ ground state, can give the right sign
of the transverse $\gamma^{*}p\to N(1440)$ amplitudes and reproduce the
experimental data of the transverse amplitudes at high $Q^{2}$, but fail to
describe the data of the $N(1440)$ helicity amplitude at low $Q^{2}$.

Our previous works Xu et al. (2019, 2020), where the mass ordering problem of
$N(1440)$, $N(1520)$, and $N(1535)$ is tackled by including ground state light
pentaquark components in the negative-parity nucleon resonances, point out
that the $N(1440)$ might be mainly the first radial excitation of the nucleon.
In this work, we push further by studying the photoproduction transitions
$\gamma^{*}p\to N^{*}$ in the three-quark picture. The paper is organized as
follows. In Sec. II, we briefly present the formalism of the proton electric
form factor, the longitudinal and transverse helicity amplitudes of the
$N(1440)$ resonance in the constituent quark model. The proton $G_{E}$ and the
$A_{1/2}$ and $S_{1/2}$ of the $N(1440)$ resonance are evaluated in Sec. III,
where the wave functions of the proton as the ground state of three quarks and
the Roper resonance as the first radial excitation of the nucleon are taken
from the study of the low-lying baryon mass spectra. In Sec. V, we study the
$A_{1/2}$ and $S_{1/2}$ of the $N(1440)$ resonance by assigning the roper
resonance as a general radial excitation of the nucleon as well as just a
$q^{3}$ state with positive parity. The quark distribution of the $N(1440)$ is
derived in Sec. V from the results in Sec. III and Sec. V. A summary is given
in Sec. VI.

##  II  Formalism of helicity amplitudes and electromagnetic form factor

We study the $N\gamma^{*}\rightarrow N^{*}$ electromagnetic transition in the
constituent quark model. The photoproduction diagram is shown in Fig. 1. Both
the nucleon and resonance states are composed of three quarks.

The proton electric form factor $G_{E}$, the time component of the
electromagnetic current, is derived in the Breit frame. The momenta of the
initial-state nucleon, final-state nucleon, and photon are respectively
defined as $P_{i}=(E_{N},0,0,-|\bm{k}/2|)$, $P_{f}=(E_{N},0,0,|\bm{k}/2|)$,
$k=(0,0,0,|\bm{k}|)$ in the Breit frame, with $E_{N}$ and $\bm{k}$ being the
energy and three-momentum of the photon. The photon energy is $\omega=0$ as
$\bm{P}_{i}+\bm{P}_{f}=0$ in the Breit frame. The square of the four-momentum
transfer is expressed by $Q^{2}=-k^{2}=|\bm{k}|^{2}$. The transverse helicity
amplitude $A_{1/2}$ and the longitudinal helicity amplitude $S_{1/2}$ are
usually defined in the $N^{*}$ rest frame. The momentum of the nucleon
$P_{i}$, the momentum of the resonance state $P_{f}$, and the photon momentum
$k=P_{f}-P_{i}$ are defined as $P_{i}=(E_{N},0,0,-|\bm{k}|)$,
$P_{f}=(M_{N^{*}},0,0,0)$, $k=(\omega,0,0,|\bm{k}|)$. One has

$\displaystyle E_{N}$ $\displaystyle=\
\frac{M_{N^{*}}^{2}+M_{N}^{2}+Q^{2}}{2M_{N^{*}}},$ $\displaystyle\omega$
$\displaystyle=\ \frac{M_{N^{*}}^{2}-M_{N}^{2}-Q^{2}}{2M_{N^{*}}},$
$\displaystyle|\bm{k}|$ $\displaystyle=\
\left[Q^{2}+\left(\frac{M_{N^{*}}^{2}-M_{N}^{2}-Q^{2}}{2M_{N^{*}}}\right)^{2}\right]^{\frac{1}{2}}.$
(1)

Figure 1: Diagram of photoproduction transition $N\gamma^{*}\rightarrow N^{*}$
where both the initial nucleon ($N$) and final nucleon resonance ($N^{*}$) are
in the three-quark picture.

The proton electric form factor, the transverse and longitudinal helicity
amplitudes of electromagnetic transitions of the $\gamma^{*}p\to N^{*}$ take
the form,

$\displaystyle G_{E}$ $\displaystyle=$ $\displaystyle\
\Braket{N,S^{\prime}_{z}=\frac{1}{2}}{q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime}}T_{B}(q_{1}q_{2}q_{3}\rightarrow
q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime})$
$\displaystyle\left.\times\Braket{q_{1}q_{2}q_{3}}{N,S_{z}=\frac{1}{2}}\right|_{\text{Breit
frame}},$ $\displaystyle A_{1/2}$ $\displaystyle=$ $\displaystyle\
\dfrac{1}{\sqrt{2K}}\Braket{N^{*},S^{\prime}_{z}=\frac{1}{2}}{q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime}}$
$\displaystyle\times T(q_{1}q_{2}q_{3}\rightarrow
q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime})\Braket{q_{1}q_{2}q_{3}}{N,S_{z}=-\frac{1}{2}},$
$\displaystyle S_{1/2}$ $\displaystyle=$ $\displaystyle\
\dfrac{1}{\sqrt{2K}}\dfrac{|\bm{k}|}{Q}\Braket{N^{*},S^{\prime}_{z}=\frac{1}{2}}{q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime}}$
$\displaystyle\times T(q_{1}q_{2}q_{3}\rightarrow
q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime})\Braket{q_{1}q_{2}q_{3}}{N,S_{z}=\frac{1}{2}},$
(2)

with

$\displaystyle K=\frac{M_{N^{*}}^{2}-M_{N}^{2}}{2M_{N^{*}}}$ (3)

where $K$ is the real-photon momentum in the $N^{*}$ rest frame, $M_{N^{*}}$
and $M_{N}$ are respectively the $N^{*}$ and $N$ masses, and
$\Braket{q_{1}q_{2}q_{3}}{N,S_{z}}$ and
$\Braket{q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime}}{N^{*},S^{\prime}_{z}}$
are the $N$ and $N^{*}$ wave functions in the three-quark picture.
$T(q_{1}q_{2}q_{3}\rightarrow q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime})$ in
Eq. (2) is the transition amplitude of the process $\gamma q\to q^{\prime}$
displayed in Fig. 1, which can be calculated in the standard language of
quantum field theory,

$\displaystyle T(q_{1}q_{2}q_{3}\rightarrow
q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime})=\ $ $\displaystyle
e_{3}\bar{u}_{s^{\prime}}(p^{\prime})\gamma^{\mu}u_{s}(p)\epsilon_{\mu}^{\lambda}(k)\Braket{q^{\prime}_{1}q^{\prime}_{2}}{q_{1}q_{2}}$
$\displaystyle=\ $ $\displaystyle
e_{3}T_{s^{\prime}s}^{\lambda}\Braket{q^{\prime}_{1}q^{\prime}_{2}}{q_{1}q_{2}},$
(4)

where $e_{3}$ and $u_{s(s^{\prime})}$ are the electric charge and the Dirac
spinners of the third quark with $s^{\prime},\,s$ being the single quark spin
projections (spin up $\uparrow$ and spin down $\downarrow$), and $\lambda$ is
the helicity of the photon. In the work, the mass of $u$ and $d$ quarks are
taken to be $m=5$ MeV. The photon polarization vectors
$\epsilon_{\mu}^{\lambda}(k)$ of the longitudinal ($\lambda=0$) and transverse
($\lambda=1$) helicity amplitudes in the Lorentz gauge are defined as
$\epsilon^{0}_{\mu}=\frac{1}{Q}(|\bm{k}|,0,0,\omega)$ and
$\epsilon^{+}_{\mu}=-\frac{1}{\sqrt{2}}(0,1,i,0)$. The matrix elements
$T_{s^{\prime}s}^{\lambda}$ are shown in Appendix A. In Eq. (2), one sums over
all possible quantum states of the intermediate three quarks and integrates
over the momenta of all quarks.

For the proton electric form factor $G_{E}$, the matrix elements of
electromagnetic transition in the Breit frame are represented as

$\displaystyle T_{B}(q_{1}q_{2}q_{3}\rightarrow
q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime})=\ $ $\displaystyle
e_{3}\bar{u}_{s^{\prime}}(p^{\prime})\gamma^{\mu}u_{s}(p)\epsilon_{\mu}(k)\Braket{q^{\prime}_{1}q^{\prime}_{2}}{q_{1}q_{2}}$
$\displaystyle=\ $ $\displaystyle
e_{3}T_{s^{\prime}s}^{B}\Braket{q^{\prime}_{1}q^{\prime}_{2}}{q_{1}q_{2}}.$
(5)

The polarization vector of the photon in Eq. II is defined as
$\epsilon_{\mu}=(1,0,0,0)$. The matrix elements $T_{s^{\prime}s}^{B}$ are
shown in Appendix B.

The proton charge form factor is calculated in the $q^{3}$ picture, and the
proton wave function is nailed down by comparing the theoretical results with
experimental data. The helicity amplitudes $A_{1/2}$ and $S_{1/2}$ of the
$N(1440)$ resonance will be evaluated in the next sections with different
$N(1440)$ wave functions: first radial excitation of the nucleon imported from
low-lying baryon mass spectrum calculations, a general radial excitation of
the nucleon, and a $q^{3}$ state with positive parity.

##  III Helicity amplitudes with $N(1440)$ wave function from mass spectrum
study

In this section, we evaluate the proton electric form factor and the helicity
amplitudes with the spatial wave functions of the proton and the Roper
resonance derived in the study of low-lying baryon mass spectra Xu et al.
(2020), where the proton is assumed to be the three-quark ground state and the
$N(1440)$ to be the first radial excitation of the nucleon. There is no free
parameter in the calculations.

Figure 2: Proton electric form factor $G_{E}$ compared to experimental data.
The solid curve is the fitting result that leads to the proton spatial wave
function to be used in the helicity amplitude calculations, and the short
dashed curve is derived with the proton spatial wave function from low-lying
baryon mass spectrum calculations. The experimental data are taken from
Janssens et al. (1966); Berger et al. (1971); Price et al. (1971); Hanson et
al. (1973); Murphy et al. (1974); Höhler et al. (1976); Simon et al. (1980);
Walker et al. (1994).

The calculated proton electric form factor $G_{E}$ with the proton spatial
wave function imported from the mass spectra fitting is shown as the dotted
curve in Fig. 2 while the theoretical results of transverse and longitudinal
helicity amplitudes of the $N(1440)$ resonance are shown as the dotted curve
in Figs. 3 and 4, respectively. It is found in Fig. 2 that the theoretical
results for the proton electric form factor $G_{E}$ are consistent with the
experimental data. It is a surprise that the proton wave function derived from
the baryon mass spectrum calculation leads to perfect results for the electric
form factor at lower $Q^{2}$. One may believe that meson cloud contributions
may account for a considerable part of the proton electric form factor at
lower $Q^{2}$. However, this work reveals that the three-quark core dominantly
contributes to the proton electric form factor, even for very low $Q^{2}$, and
meson cloud contributions may be negligible.

For both the $A_{1/2}$ and $S_{1/2}$, the theoretical results can fairly
describe the experimental data at large $Q^{2}$ region ($1.5\leq Q^{2}\leq
4.5$ GeV2). This is consistent with the theoretical results in the light-front
relativistic quark model Aznauryan (2007) and covariant spectator quark model
Ramalho and Tsushima (2010). The theoretical results of the helicity amplitude
$A_{1/2}$ are in the right tendency of the $A_{1/2}$ data and can give the
right sign at the real photon point though the discrepancy between the
theoretical results and the experimental data is obvious. Considering that the
$N(1440)$ wave function employed in the calculation is imported, without any
modification, from the baryon mass spectrum calculation, one may have a good
belief that the $N(1440)$ is mainly a three-quark state, the first radial
excitation of the proton. The theoretical results of the $S_{1/2}$ at low
$Q^{2}$ region are largely inconsistent with the experimental data, giving a
much bigger value at the real photon point, which may indicate that the
$N(1440)$ might have some other components rather than the first radial
excitation of the nucleon in the three quark picture. However, it is not
necessary to jump to the point that the meson cloud is important among the
additional components. Note that the three quark ground state of the proton
derived in the baryon mass spectrum calculations reproduces well the
experimental data of the proton charge form factor at small $Q^{2}$.

##  IV  Helicity amplitudes with $N(1440)$ as three quark states

In this section, we calculate the helicity amplitudes of the $N(1440)$ which
is allowed to include higher radial excitations beside the first radial
excitation of the nucleon as well as just a three quark state of positive
parity. As shown in the previous section, the proton wave function derived
from the baryon mass spectrum calculation leads to a reasonable, but not
perfect, proton charge form factor. Here we first extract the proton spatial
wave function in the three-quark picture by fitting the theoretical result of
the proton electric form factor to the experimental data. This extracted
proton spatial wave function will be employed to study the $N(1440)$. The
proton spatial wave function is extracted from the form factor shown as the
solid curve in Fig. 2.

In this calculation we assume that the $N(1440)$ is in the three-quark
configuration, but may contain higher radial excitations of the nucleon or may
be just a positive-parity state. We let the $N(1440)$ spatial wave function be
free for fitting the experimental data of the helicity amplitudes of the
$N(1440)$ resonance. We do two calculations:
Fitting I: the proton and the $N(1440)$ spatial wave functions are orthogonal
each other, $\Braket{\psi^{O}_{p}}{\psi^{O}_{N(1440)}}=0$ which constrains the
$N(1440)$ spatial wave function to be the radial excitations of the nucleon;
Fitting II: the proton and the $N(1440)$ spatial wave functions may not be
orthogonal each other, which allows the $N(1440)$ spatial wave function
composed of the radial excitation of the nucleon as well as others.

Figure 3: Transverse helicity amplitude $A_{1/2}$ of the
$p\gamma^{*}\rightarrow N(1440)$ transition compared to the measurements. The
short dashed, dash-dotted, and solid curves are the results with the $N(1440)$
spatial wave function from the low-lying baryon mass spectrum calculations, in
Fitting I, and in Fitting II, respectively. The experimental data are taken
from Aznauryan et al. (2009); Mokeev et al. (2012, 2016); Zyla et al. (2020).

The theoretical results of the helicity amplitudes of the $N(1440)$ resonance
are shown in Figs. 3 and 4 with the dash-dotted lines for Fitting I and the
solid lines for Fitting II. It is found that the theoretical results of the
helicity amplitude $A_{1/2}$ in Fitting I improve more or less the results in
Section III, especially for small $Q^{2}$, but for the amplitude $S_{1/2}$ the
results here are rather similar to the ones in Section III without any
improvement. Therefore, one may conclude that the results in Fitting I rule
out that the $N(1440)$ may be composed of a sizeable component of higher
radial excitations of the nucleon.

As shown in Figs. 3 and 4, Fitting II results are in a perfect fit to the
amplitude $A_{1/2}$ and a very reasonable fit to the amplitude $S_{1/2}$,
particularly to the two values at the real photon point. The results imply
that the $N(1440)$ may consist of components other than radial excitations of
three quarks.

##  V  Extraction of quark distribution of $N(1440)$

To see the inner structure of the $N(1440)$ closely, we extract the spatial
wave functions from the calculations in Section III and IV. Shown in Fig. 5
are the $N(1440)$ spatial wave functions with respect to the relative distance
between the quark to the center of mass, where the solid and dashed curves are
for the $N(1440)$ in Fitting I and Fitting II, respectively. Considering the
full permutation symmetry of the three-quark baryon state, we set
$|\bm{r_{\lambda}}|=|\bm{r_{\rho}}|$, where
$\bm{r_{\rho}}=\frac{1}{\sqrt{2}}(\bm{r}_{1}-\bm{r}_{2})$ and
$\bm{r_{\lambda}}=\frac{1}{\sqrt{6}}(\bm{r}_{1}+\bm{r}_{2}-2\bm{r}_{3})$ to
project the three-dimensional spatial wave functions to a 2D plot. The non-
orthogonality between the proton spatial wave function and the $N(1440)$ one
derived in Fitting II is
$|\braket{\psi^{O}_{p}}{\psi^{O}_{N(1440)}}|^{2}=0.03$, indicting that the
$N(1440)$ may be composed of components rather than the radial excitations of
three quarks.

Fig. 5 shows that the quark density about the center of mass in Fitting II is
lower than the one in Fitting I, and that the $N(1440)$ spatial wave functions
are almost the same for larger distances starting from the node. The results
may tell that it is not the skirt but the very inner part of the $N(1440)$
which accounts for the helicity amplitudes at very small $Q^{2}$.

Figure 4: Longitudinal helicity amplitude $S_{1/2}$ of $p\gamma^{*}\to
N(1440)$ transition in comparison with experimental data Štajner et al.
(2017); Aznauryan et al. (2009). The short dashed, dash-dotted, and solid
curves are the results with the $N(1440)$ spatial wave function from the low-
lying baryon mass spectrum calculations, in Fitting I, and in Fitting II,
respectively. Figure 5: $N(1440)$ spatial wave functions with respect to the
relative distance between the quark to the center of mass: solid curve from
Fitting I and dashed curve from Fitting II.

##  VI  Summary

In this work, we have studied the proton electric form factor and the helicity
amplitudes of the $N(1440)$ resonance via the process $p\gamma^{*}\to N^{*}$.
The spatial wave functions of the proton and $N(1440)$ are extracted by
fitting the theoretical results to the experimental data of the proton
electric form factor and helicity amplitudes.

It is found that the proton wave function derived in the baryon mass spectrum
study in the three-quark picture leads to an almost perfect proton electric
form factor, particularly at low $Q^{2}$. One may conclude that the meson
cloud contribution to the proton electric form factor is negligible.

The work supports the argument that the $N(1440)$ resonance is mainly the
first radial excited state of the nucleon. However, the Roper may be composed
of small components rather than radial excitations of three quarks, such as
$D$-wave three quark states, hadronic molecule states, and pentaquarks.
Further studies including these components into the N1440 are under way.

## ACKNOWLEDGMENTS

A.K. and Y.Y. would like to thank for the support of the Thailand Science
Research and Innovation (TSRI) for their support through the Royal Golden
Jubilee Ph.D. (RGJ-PHD) Program (Grant No. PHD/0242/2558) and A.K. would like
to thank for the support of Suranaree University of Technology under SUT-OROG
scholarship (contract no. 46/2558).

##  Appendix A Matrix elements of $\gamma q\to q^{\prime}$ for helicity
amplitudes

The matrix elements of the electromagnetic transition $\gamma q\to q^{\prime}$
for the helicity $\lambda=0,1$ are derived in detail,

$\displaystyle T^{0}_{\uparrow\uparrow}$ $\displaystyle=$ $\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left[\dfrac{|\bm{k}|}{Q}\left(1+\dfrac{p_{z}^{\prime}p_{z}+2p_{-}^{\prime}p_{+}}{(E^{\prime}+m)(E+m)}\right)\right.$
$\displaystyle\
\left.-\dfrac{\omega}{Q}\left(\dfrac{p_{z}}{E+m}+\dfrac{p_{z}^{\prime}}{E^{\prime}+m}\right)\right],$
$\displaystyle T^{0}_{\uparrow\downarrow}$ $\displaystyle=$ $\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left[\dfrac{|\bm{k}|}{Q}\left(\dfrac{\sqrt{2}(p_{z}^{\prime}p_{-}-p_{-}^{\prime}p_{z})}{(E^{\prime}+m)(E+m)}\right)\right.$
$\displaystyle\
\left.-\dfrac{\omega}{Q}\left(\dfrac{\sqrt{2}p_{-}}{E+m}-\dfrac{\sqrt{2}p_{-}^{\prime}}{E^{\prime}+m}\right)\right],$
$\displaystyle T^{0}_{\downarrow\uparrow}$ $\displaystyle=$ $\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left[\dfrac{|\bm{k}|}{Q}\left(\dfrac{\sqrt{2}(-p_{z}^{\prime}p_{+}+p_{+}^{\prime}p_{z})}{(E^{\prime}+m)(E+m)}\right)\right.$
$\displaystyle\
\left.-\dfrac{\omega}{Q}\left(\dfrac{-\sqrt{2}p_{+}}{E+m}+\dfrac{\sqrt{2}p_{+}^{\prime}}{E^{\prime}+m})\right)\right],$
$\displaystyle T^{0}_{\downarrow\downarrow}$ $\displaystyle=$ $\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left[\dfrac{|\bm{k}|}{Q}\left(1+\dfrac{p_{z}^{\prime}p_{z}+2p_{+}^{\prime}p_{-}}{(E^{\prime}+m)(E+m)}\right)\right.$
$\displaystyle\
\left.-\dfrac{\omega}{Q}\left(\dfrac{p_{z}}{E+m}+\dfrac{p_{z}^{\prime}}{E^{\prime}+m}\right)\right],$
$\displaystyle T^{+}_{\uparrow\uparrow}$ $\displaystyle=$ $\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left[\dfrac{2p_{+}}{E+m}\right],$
$\displaystyle T^{+}_{\uparrow\downarrow}$ $\displaystyle=$ $\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left[-\dfrac{\sqrt{2}p_{z}}{E+m}+\dfrac{\sqrt{2}p_{z}^{\prime}}{E^{\prime}+m}\right],$
$\displaystyle T^{+}_{\downarrow\uparrow}$ $\displaystyle=$ $\displaystyle\
0,$ $\displaystyle T^{+}_{\downarrow\downarrow}$ $\displaystyle=$
$\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left[\dfrac{2p_{+}^{\prime}}{E^{\prime}+m}\right],$
(6)

where $E$ and $E^{\prime}$ are respectively the energies of the initial and
final interacting quarks with the dynamical quark mass of $u$ and $d$ quarks
as $m$, and $p_{\pm}=\frac{1}{\sqrt{2}}(p_{x}\pm ip_{y})$.

##  Appendix B Matrix elements of $\gamma q\to q^{\prime}$ for proton electric
form factor

The matrix elements of the electromagnetic transition $\gamma q\to q^{\prime}$
for the proton electric form factor are derived in detail,

$\displaystyle T^{B}_{\uparrow\uparrow}$ $\displaystyle=$ $\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left(1+\dfrac{p_{z}^{\prime}p_{z}+2p_{-}^{\prime}p_{+}}{(E^{\prime}+m)(E+m)}\right),$
$\displaystyle T^{B}_{\uparrow\downarrow}$ $\displaystyle=$ $\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left(\dfrac{\sqrt{2}(p_{z}^{\prime}p_{-}-p_{-}^{\prime}p_{z})}{(E^{\prime}+m)(E+m)}\right),$
$\displaystyle T^{B}_{\downarrow\uparrow}$ $\displaystyle=$ $\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left(\dfrac{\sqrt{2}(-p_{z}^{\prime}p_{+}+p_{+}^{\prime}p_{z})}{(E^{\prime}+m)(E+m)}\right),$
$\displaystyle T^{B}_{\downarrow\downarrow}$ $\displaystyle=$ $\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left(1+\dfrac{p_{z}^{\prime}p_{z}+2p_{+}^{\prime}p_{-}}{(E^{\prime}+m)(E+m)}\right),$
(7)

where $E$ and $E^{\prime}$ respectively are the energies of the initial and
final interacting quarks with the dynamical quark mass of $u$ and $d$ quarks
as $m$. The momentum $p_{\pm}$ are defined by
$p_{\pm}=\frac{1}{\sqrt{2}}(p_{x}\pm ip_{y})$.

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