# Light-cone sum rules for radial excitation of decuplet to octet baryons
electromagnetic transition form factors

T. M. Aliev<EMAIL_ADDRESS>Department of Physics, Middle East Technical
University, Ankara, 06800, Turkey  S. Bilmis<EMAIL_ADDRESS>Department
of Physics, Middle East Technical University, Ankara, 06800, Turkey  TUBITAK
ULAKBIM, Ankara, 06530, Turkey  M. Savci<EMAIL_ADDRESS>Department of
Physics, Middle East Technical University, Ankara, 06800, Turkey

###### Abstract

Magnetic dipole moment form factor, $G_{M}^{(2)}(Q^{2})$, describing the
radial excitation of decuplet baryons to octet baryons electromagnetic
transitions as well as the ratios,
$R_{SM}=-\frac{1}{4m_{2}^{2}}\sqrt{4m_{2}^{2}Q^{2}+(m_{2}^{2}-Q^{2}-m_{1}^{2})^{2}}\frac{G_{C}^{(2)}(Q^{2})}{G_{M}^{(2)}(Q^{2})}$
and $R_{EM}=-\frac{G_{E}^{(2)}(Q^{2})}{G_{M}^{(2)}(Q^{2})}$ are calculated in
the framework of light-cone sum rules. We also estimate the degree of the
violation of U-spin symmetry. The obtained results for the multipole form
factors can be useful in searching the properties of radially excited baryon
states.

##  I Introduction

The considerable progress in studying the spectrum of hadrons and obtaining
useful information about their inner structure using experimental data with
the electromagnetic interaction has essentially extended our knowledge about
the QCD dynamics at low energies. The properties of the excited states
essentially depend on the properties of QCD at low energy.

Many baryon states predicted by the quark model have been observed, and their
properties (masses, decay widths, quantum numbers) are carefully studied
Tanabashi  _et al._ (2018). However, part of the baryons predicted by the
quark model have not been discovered in experiments yet. These states are
called “missing-states” Koniuk and Isgur (1980). Searching for these baryons
constitutes one of the main motivations for future studies in the physics
program of Jefferson Lab, BATE, and MAMI facilities, namely investigation of
the spectrum of baryons containing strange quarks. Considering the potential
of these facilities, it is expected that these investigations will enrich our
knowledge on the application of QCD in this direction, and more information
will be acquired about the properties of hadrons around their first
excitations via electromagnetic probes.

In literature, there are many studies in this area. For instance, the
electroproduction of the first radial excitation of $\Delta$ baryon is
investigated in the framework of the light-front model Capstick and Keister
(1995), and the effect of $\Delta(1600)$ is studied in Burkert  _et al._
(2019). Moreover, the transition
$\gamma^{*}N\rightarrow\Delta(1232),~{}\Delta(1600)$ is analyzed within a
diquark-diquark and covariant constituent of the quark model in Lu  _et al._
(2019); Ramalho and Tsushima (2010), correspondingly. The transition form
factors $\gamma^{*}N\rightarrow\Delta(1600)$ within light-cone sum rules is
studied in Aliev  _et al._ (2020). In this study, the form factors of the
other members of the radial excitations decuplet baryons in reactions
$\gamma^{*}\Sigma\rightarrow\Sigma^{*}$ and $\gamma^{*}\Xi\rightarrow\Xi^{*}$
are calculated within the light-cone sum rules (LCSR). Here $\gamma^{*}$ means
virtual photon, and $B^{*}$ is the radial excitation of ground-state decuplet
baryons. Note that the form factors $\gamma^{*}N\rightarrow\Delta(1232)$ and
$\gamma^{*}\text{ octet}\rightarrow\text{ decuplet}$ baryon transition within
the same approach is studied in Braun  _et al._ (2006a) and Aliev  _et al._
(2013), respectively.

The paper is organized as follows. In section II, we derive the sum rules for
the multipole form factors responsible for $\gamma^{*}B\rightarrow B^{*}$
transition. In section III, the numerical analysis of the derived sum rules
for multipole form factors is performed. The summary and conclusion of the
study are presented in the last section.

##  II Light-cone sum rules for the $\gamma^{*}B\rightarrow B^{*}$ transition
form factors

In this section, the light-cone sum rules for the $\gamma^{*}B\rightarrow
B^{*}$ transition form factors is obtained. These form factors are defined by
the matrix element of the electromagnetic current
$j_{\mu}^{el}=e_{q_{1}}\bar{q}_{1}\gamma_{\mu}q_{1}+e_{q_{2}}\bar{q}_{2}\gamma_{\mu}q_{2}+e_{q_{3}}\bar{q}_{3}\gamma_{\mu}q_{3}$
between the initial ground octet and first radial excitations of decuplet
baryon states, i.e., $\langle B^{*}(p^{\prime})|j_{\mu}^{el}|B(p)\rangle$. In
general form, this matrix element can be written as

$\langle
B^{*}(p^{\prime})j_{\mu}^{el}|B(p)\rangle=\bar{u}_{\beta}(p^{\prime})\Gamma_{\beta\mu}u(p).$
(1)

The main issue is defining constraint-free and gauge-invariant form factors
from this matrix element. This problem is solved in Jones and Scadron (1973),
and the matrix element in Eq. (1) is determined as

$\begin{split}\langle
B^{*}(p^{\prime})|j_{\mu}^{el}|B(p)\rangle&=\sum_{i=1}^{2}\bar{u}_{\beta}(p^{\prime})\Bigl{\\{}G_{1}^{(i)}(Q^{2})(-q_{\beta}\gamma_{\mu}+\not{q}g_{\beta\mu})+G_{2}^{(i)}(Q^{2})(-q_{\beta}\mathcal{P}_{\mu}+(q\mathcal{P})g_{\beta\mu})\\\
&+G_{3}^{(i)}(q_{\beta}q_{\mu}-q^{2}g_{\beta\mu})\Bigr{\\}}\gamma_{5}u(p),\end{split}$
(2)

where $i=1(2)$ corresponds to the ground (first radial excitation) state
decuplet baryons, $q=p-p^{\prime}$,
$\mathcal{P}_{\alpha}=\frac{1}{2}(p+p^{\prime})_{\alpha}$, and
$u_{\beta}(p^{\prime})$ is the Rarita-Schwinger spinor for spin $3/2$ baryons.

From the experimental point of view, the multipole form factors are more
convenient to study. The relations among form factors $G_{i}(Q^{2})$ and
multipole magnetic dipole $G_{M}^{(2)}(Q^{2})$, electric quadrupole,
$G_{E}^{(2)}{(Q^{2})}$, and Coulomb quadrupole, $G_{C}^{(2)}(Q^{2})$ form
factors are obtained in Devenish  _et al._ (1976); Jones and Scadron (1973).

$\begin{split}G_{M}^{(i)}(Q^{2})&=\frac{m_{0}}{3(m_{0}+m_{i})}\big{[}(3(m_{i}+m_{0})(m_{i}+m_{0})+Q^{2})\frac{G_{1}^{(i)}(Q^{2})}{m_{i}}\\\
&+(m_{i}^{2}-m_{0}^{2})G_{2}^{(i)}(Q^{2})-2Q^{2}G_{3}^{(i)}(Q^{2})\big{]},\\\
G_{E}^{(i)}(Q^{2})&=\frac{m_{0}}{3(m_{0}+m_{i})}\big{[}\big{(}m_{i}-m_{0}^{2}-Q^{2})\frac{G_{1}^{(i)}(Q^{2})}{m_{i}}\\\
&+(m_{i}^{2}-m_{0}^{2})G_{2}^{(i)}-2Q^{2}G_{3}^{(i)}(Q^{2})\big{]},\\\
G_{C}^{(i)}(Q^{2})&=\frac{2m_{0}}{3(m_{0}+m_{i})}\big{[}2m_{i}G_{1}^{(i)}(Q^{2})+\frac{1}{2}(3m_{i}^{2}+m_{0}^{2}+Q^{2})G_{2}^{(i)}(Q^{2})\\\
&+(m_{i}^{2}-m_{0}^{2}-Q^{2})G_{3}^{(i)}(Q^{2})\big{]},\\\ \end{split}$ (3)

where $m_{0}$, $m_{1}$ and $m_{2}$ are the mass of the ground state $1/2$
baryons, ground and first radial excitations of decuplet baryons,
respectively. Instead of studying these multipole form factors, in the present
work, we analyze the magnetic dipole, $G_{M}^{(2)}(Q^{2})$, form factor as
well as the ratios of the multipole form factors $R_{SM}$ and $R_{EM}$ Braun
_et al._ (2006a), which are more convenient for analyzing the experimental
data

$\begin{split}R_{SM}&=-\frac{1}{4m_{2}^{2}}\sqrt{4m_{2}^{2}Q^{2}+(m_{2}^{2}-Q^{2}-m_{N}^{2})^{2}}\frac{G_{C}^{(2)}(Q^{2})}{G_{M}^{(2)}(Q^{2})},\\\
R_{EM}&=-\frac{G_{E}^{(2)}(Q^{2})}{G_{M}^{(2)}{Q^{2}}}.\end{split}$ (4)

The main problem is the calculation of these ($G_{i}$ or $G_{M}$, $G_{E}$ and
$G_{C}$) form factors. For this purpose, we implement light cone QCD sum
rules. According to the LCSR methodology, we start our calculations by
considering the following correlation function

$\Pi_{\beta\mu}(p,q)=i\int d^{4}xe^{iqx}\langle
0|T\\{\eta_{\beta}(0)|j_{\mu}^{el}(x)\\}|B(p)\rangle,$ (5)

where $B(p)$ is the common notation for octet $\Sigma$ and $\Xi$ baryons,
$\eta_{\beta}$ is the interpolating current for ground or radial excitations
of the decuplet baryons, and
$j_{\mu}^{\text{el}}=\sum_{q}e_{q}\bar{q}\gamma_{\mu}q$ is the electromagnetic
current. Here $q$ denotes light quark, and $e_{q}$ is its electric charge. The
interpolating current for decuplet baryons in a general form can be written as

$\eta_{\beta}=N\epsilon^{abc}\big{[}({q_{1}^{a}}^{T}C\gamma_{\beta}q_{2}^{b})q_{3}^{c}+({q_{2}^{a}}^{T}C\gamma_{\beta}q_{3}^{b})q_{1}^{c}+({q_{3}^{a}}^{T}C\gamma_{\beta}q_{1}^{b})q_{2}^{c}\big{]},$
(6)

where $a,b,c$ are color indices, and $C$ is the charge conjugation operator.
The quark content of the considered decuplet baryons and the value of the
normalization factor is presented in Table 1.

| N | $q_{1}$ | $q_{2}$ | $q_{3}$
---|---|---|---|---
$\Sigma^{*+}$ | $\sqrt{\frac{1}{3}}$ | $u$ | $u$ | $s$
$\Sigma^{*0}$ | $\sqrt{\frac{2}{3}}$ | $u$ | $d$ | $s$
$\Sigma^{*-}$ | $\sqrt{\frac{1}{3}}$ | $d$ | $d$ | $s$
$\Xi^{*0}$ | $\sqrt{\frac{1}{3}}$ | $s$ | $s$ | $u$
$\Xi^{*-}$ | $\sqrt{\frac{1}{3}}$ | $s$ | $s$ | $d$
Table 1: The quark content and the normalization factor of the decuplet
baryons are presented.

To derive sum rules for the considered form factors, the correlation function
is calculated in two different kinematical domains. In a time like domain, its
expressions can be obtained by inserting the complete set of hadron states
which carry the same quantum numbers as the interpolating current, and then
the considered state is isolated. The phenomenological part of the correlation
function can also be calculated in the deep Euclidean region $Q^{2}<<0$ with
the operator product expansion (OPE). In LCSR, the OPE is performed over
twist. One can then obtain the relevant sum rules by matching these two
representations of the correlator functions via dispersion relation. Finally,
performing Borel transformation and subtracting the continuum contribution, we
get the result for the considered problem.

Let us first calculate the phenomenological part of the correlation function.
Inserting the hadronic states with quantum number $J^{P}=\frac{3}{2}^{+}$ and
isolating the contributions of the ground and first radial excitations of
decuplet baryons we get,

$\Pi_{\beta\mu}(p,q)=\sum_{i=1}^{2}\frac{\langle
0|\eta_{\beta}(0)|B_{i}^{*}\rangle\langle
B_{i}^{*}|j_{\mu}^{el}|B(p)\rangle}{m_{i}^{2}-p^{\prime^{2}}}+...,$ (7)

where $i=1(2)$ describe ground (first) radial excitation baryon and $...$
corresponds to the contributions of continuum and higher states. The matrix
element, $\langle 0|\eta_{\beta}|\beta_{i}^{*}\rangle$, is defined as;

$\langle 0|\eta_{\beta}|B_{i}^{*}\rangle=\lambda_{i}u_{\beta}(p^{\prime}),$
(8)

where $u_{\alpha}(p^{\prime})$ is the Rarita-Schwinger spinor, $\lambda_{i}$
is its residue and $p^{\prime}=p-q$. The second matrix element is given in Eq.
(2). Performing summation over the spins of Rarita-Schwinger spinors using the
formula,

$\sum_{s}u_{\alpha}^{(s)}(p^{\prime})\bar{u}_{\beta}^{(s)}(p^{\prime})=-(\not{p}^{\prime}+m_{i})\big{\\{}g_{\alpha\beta}-\frac{1}{3}\gamma_{\alpha}\gamma_{\beta}-\frac{2p_{\alpha}^{\prime}p_{\beta}^{\prime}}{3m_{i}^{2}}+\frac{p_{\alpha}^{\prime}\gamma_{\beta}-p_{\beta}^{\prime}\gamma_{\alpha}}{3m_{i}}\big{\\}}$
(9)

for the correlation function from hadronic part, we get

$\begin{split}\Pi_{\alpha\mu}=&-\sum_{i}\frac{\lambda_{i}}{m_{i}^{2}-p^{\prime
2}}(\not{p}^{\prime}+m_{i})\big{\\{}g_{\alpha\beta}-\frac{1}{3}\gamma_{\alpha}\gamma_{\beta}-\frac{2p_{\alpha}^{\prime}p_{\beta}^{\prime}}{3m_{i}^{2}}+\frac{p_{\alpha}^{\prime}\gamma_{\beta}-p_{\beta}^{\prime}\gamma_{\alpha}}{3m_{i}}\big{\\}}\\\
&\big{\\{}G_{1}^{i}(-q_{\beta}\gamma_{\mu}+g_{\beta\mu}\not{q})+G_{2}^{i}(-q_{\beta}\mathcal{P}_{\mu}+g_{\beta\mu}q\mathcal{P})+G_{3}^{i}(q_{\beta}q_{\mu}-g_{\beta\mu}q^{2})\big{\\}}\gamma_{5}u_{N}(p).\end{split}$
(10)

It should be noted that we face with the following drawbacks. The
interpolating current $\eta_{\alpha}$ couples not only to spin $3/2$ state but
also to spin $(1/2)^{-}$ one. The matrix element of the interpolating current
between vacuum and spin $(1/2)^{-}$ state in general is determined as

$\langle
0|\eta_{\alpha}|\frac{1}{2}(p^{\prime})\rangle=(A\gamma_{\alpha}+Bp^{\prime}_{\alpha})u(p^{\prime}).$
(11)

Multiplying both side to $\gamma_{\alpha}$ and using
$\gamma^{\alpha}\eta_{\alpha}=0$, we get

$\langle
0|\eta_{\alpha}|\frac{1}{2}(p^{\prime})\rangle=\frac{B}{4}(-m\gamma_{\alpha}+4p^{\prime}_{\alpha})u(p^{\prime}).$
(12)

From Eq. (12), we see that the structures proportional to $\gamma_{\alpha}$ or
$p_{\alpha}^{\prime}$ contain the contributions of $\frac{1}{2}^{-}$ states.
Hence, to take into account only the contributions of $\frac{3}{2}$ states,
these structures should be removed. Considering this fact, it follows from Eq.
(10) that contribution of the spin-$3/2$ term is solely due to the term
$g_{\alpha\beta}$.

There is another unpleasant situation, namely, the fact that all structures
being not independent. To obtain independent structures, we need an ordering
procedure. In the present work, we choose the ordering of Dirac matrices as
$\gamma_{\alpha}\not{p}^{\prime}\not{q}\gamma_{\mu}\gamma_{5}$. Taking into
account all the aforementioned circumstances, the hadronic part of the
correlation function for describing $3/2$ decuplet baryons (radial
excitations) to octet baryon transitions containing only the contributions of
spin $3/2$ states can be written as follows.

$\begin{split}\Pi_{\alpha\mu}=&-\frac{\lambda_{1}}{m_{1}^{2}-{p^{\prime}}^{2}}(\not{p}^{\prime}+m_{1})\big{[}G_{1}^{(1)}(-q_{\alpha}\gamma_{\mu}+g_{\alpha\mu}\not{q})\\\
&+G_{2}^{(1)}[-q_{\alpha}(p^{\prime}+q/2)_{\mu}+q\cdot(p^{\prime}+q/2)g_{\alpha\mu}]+G_{3}^{(1)}[q_{\alpha}q_{\mu}-q^{2}g_{\alpha\mu}]\gamma_{5}u_{N}(p)\big{]}\\\
&-\frac{\lambda_{2}}{m_{2}^{2}-{p^{\prime}}^{2}}(\not{p}^{\prime}+m_{2})\big{[}G_{1}^{(2)}(-q_{\alpha}\gamma_{\mu}+g_{\alpha\mu}\not{q})\\\
&+G_{2}^{(2)}[-q_{\alpha}(p^{\prime}+q/2)_{\mu}+q\cdot(p^{\prime}+q/2)g_{\alpha\mu}]+G_{3}^{(2)}[q_{\alpha}q_{\mu}-q^{2}g_{\alpha\mu}]\gamma_{5}u_{N}(p)\big{]}.\end{split}$
(13)

where $m_{1}(m_{2})$, $G_{1}^{(i)}(G_{2}^{(i)})$ are the mass of ground
(radial excitation) $3/2$ state and form factors describing the transitions
$3/2~{}(\text{radial excitation})\rightarrow 1/2$, respectively.

This equation contains six form factors. Three of them correspond to the
ground, and the other three correspond to radial excited state decuplet
baryons to ground state octet baryon transition. To determine these six form
factors, we need six independent Lorentz structures.

From Eq. (13), we see that the correlation function can be written in terms of
the six independent structures in the following way.

$\begin{split}\Pi_{\alpha\mu}&=\Pi_{1}\not{p}^{\prime}\not{q}\gamma_{5}g_{\alpha\mu}+\Pi_{2}\not{q}\gamma_{5}g_{\alpha\mu}+\Pi_{3}\not{p}^{\prime}\gamma_{5}p^{\prime}_{\mu}q_{\alpha}+\Pi_{4}\gamma_{5}p^{\prime}_{\mu}q_{\alpha}+\Pi_{5}\not{p}^{\prime}\gamma_{5}q_{\alpha}q_{\mu}\\\
&+\Pi_{6}\gamma_{5}q_{\alpha}q_{\mu}+~{}\text{other structures}\end{split}$
(14)

Equating the coefficients of the relevant Lorentz structures from Eqs. (13)
and (14), we get,

$\begin{split}\Pi_{1}&=-\frac{\lambda_{1}G_{1}^{(1)}}{m_{1}^{2}-{p^{\prime}}^{2}}-\frac{\lambda_{2}G_{1}^{(2)}}{m_{2}^{2}-{p^{\prime}}^{2}},\\\
\Pi_{2}&=-\frac{\lambda_{1}m_{1}G_{1}^{(1)}}{m_{1}^{2}-{p^{\prime}}^{2}}-\frac{\lambda_{2}m_{2}G_{1}^{(2)}}{m_{2}^{2}-{p^{\prime}}^{2}},\\\
\Pi_{3}&=\frac{\lambda_{1}G_{2}^{(1)}}{m_{1}^{2}-{p^{\prime}}^{2}}+\frac{\lambda_{2}G_{2}^{(2)}}{m_{2}^{2}-{p^{\prime}}^{2}},\\\
\Pi_{4}&=\frac{\lambda_{1}m_{1}G_{2}^{(1)}}{m_{1}^{2}-{p^{\prime}}^{2}}+\frac{\lambda_{2}m_{2}G_{2}^{(2)}}{m_{2}^{2}-{p^{\prime}}^{2}},\\\
\Pi_{5}&=\frac{\lambda_{1}}{m_{1}^{2}-{p^{\prime}}^{2}}[\frac{G_{2}^{(1)}}{2}-G_{3}^{(1)}]+\frac{\lambda_{2}}{m_{2}^{2}-{p^{\prime}}^{2}}[\frac{G_{2}^{(2)}}{2}-G_{3}^{(2)}],\\\
\Pi_{6}&=\frac{\lambda_{1}m_{1}}{m_{1}^{2}-{p^{\prime}}^{2}}[\frac{G_{2}^{(1)}}{2}-G_{3}^{(1)}]+\frac{\lambda_{2}m_{2}}{m_{2}^{2}-{p^{\prime}}^{2}}[\frac{G_{2}^{(2)}}{2}-G_{3}^{(2)}].\end{split}$
(15)

Solving these equations for our goal to determine the form factors
$G_{1}^{(2)}$, $G_{2}^{(2)}$, and $G_{3}^{(2)}$, which describe the transition
of the radial excitation of decuplet baryons to the ground state octet
baryons, we get

$\begin{split}-m_{1}\Pi_{1}+\Pi_{2}&=-\frac{\lambda_{2}G_{1}^{(2)}}{m_{2}^{2}-{p^{\prime}}^{2}}(m_{2}-m_{1}),\\\
-m_{1}\Pi_{3}+\Pi_{4}&=\frac{\lambda_{2}G_{2}^{(2)}}{m_{2}^{2}-{p^{\prime}}^{2}}(m_{2}-m_{1}),\\\
-m_{1}\Pi_{5}+\Pi_{6}&=\frac{\lambda_{2}}{m_{2}^{2}-{p^{\prime}}^{2}}(m_{2}-m_{1})[\frac{G_{2}^{(2)}}{2}-G_{3}^{(2)}].\end{split}$
(16)

From this equation, it follows that to obtain the sum rules for the form
factors $G_{i}^{(2)}$, the expression of the correlation function from the OPE
part is needed. The OPE part of the correlation function can be obtained by
inserting the expression of the interpolating current given in Eq. (6) into
Eq. (5) and using the Wick theorem. As a result, one can get the expressions
of invariant functions in the deep Euclidean domain
${p^{\prime}}^{2}=(p-q)^{2}<<0$. In the LCSR method, the OPE part of the
correlation function for the considered problem is expressed in terms of octet
baryon distribution amplitudes (DA’s). The distribution amplitudes of the
octet baryons appear in matrix element of three quark non-local operator
between vacuum and members of the octet baryons

$\epsilon^{abc}\langle
0|q_{1\alpha}^{a}(a_{1}x)q_{2\beta}^{b}(a_{2}x)q_{3\gamma}^{c}(a_{3}x)|O(p)\rangle.$
(17)

Using the spin and parity of the baryons as well as the Lorentz covariance,
the general Lorentz composition of this matrix element can be written as

$\begin{split}4\epsilon^{abc}\langle
0|q_{1\alpha}^{a}(a_{1}x)q_{2\beta}^{b}(a_{2}x)q_{3\gamma}^{c}(a_{3}x)|O(p)\rangle=\sum_{i}\mathcal{F}_{i}\Gamma_{\alpha\beta}^{1i}(\Gamma^{2i}u(p))_{\gamma}\end{split}$
(18)

where $\Gamma^{1(2)i}$ are concrete Dirac matrices,
$\mathcal{F}_{i}=\mathcal{S},\mathcal{P}_{i},\mathcal{A}_{i},\mathcal{V}_{i}$
and $\mathcal{T}_{i}$ are the DA’s having no definite twists. The
decomposition of Eq. (12) in terms of $\mathcal{F}_{i}$ is given in Braun  _et
al._ (2006a). The DA’s with definite twists are defined as

$\begin{split}4\epsilon^{abc}\langle
0|q_{1\alpha}^{a}(a_{1}x)q_{2\beta}^{b}(a_{2}x)q_{3\gamma}^{c}(a_{3}x|O(p)\rangle=\sum_{i}F_{i}\Gamma_{\alpha\beta}^{1i}(\Gamma^{2i}u(p))_{\gamma}.\end{split}$
(19)

where $F_{i}=S,P_{i},A_{i},V_{i}$ and $T_{i}$. The relations among the two
sets of DA’s is given in Braun  _et al._ (2006a) and for completeness
presented in Appendix A. The DA’s, which are the main non-perturbative
ingredient, and up to twist-6 are calculated in Braun  _et al._ (2006a); Bali
_et al._ (2016, 2019); Wein and Schäfer (2015); Liu and Huang (2009a, b);
Braun  _et al._ (2006b).

Using the expressions of the DAs of the octet baryons, the invariant functions
$\Pi_{i}$ is calculated straightforward. The general form of the expressions
of the invariant functions can be written as

$\Pi_{i}\big{(}(p-q)^{2},q^{2})=\sum_{n=1}^{3}\int_{0}^{1}\frac{\rho_{i}^{(n)}\big{(}x,q^{2},(p-q)^{2}\big{)}}{\big{(}(q-px)^{2}\big{)}^{n}}$
(20)

where $\rho_{i}^{n}$ are spectral densities. Explicit expressions for
$\rho_{i}^{n}$ for the considered transitions are presented in Appendix B.
Using the quark-hadron duality ansatz and performing Borel transformation from
both side of Eq. (16) with respect to $-{p^{\prime}}^{2}=-(p-q)^{2}$ to
enhance the contributions of first radial excitations as well as suppressing
the contributions of higher states and continuum, we arrive the desired sum
rules for the transition form factors of
$\gamma^{*}\text{octet}\rightarrow\text{first radial excitations of decuplet
baryons.}$

$\begin{split}-\lambda_{2}G_{1}^{(2)}(Q^{2})(m_{2}-m_{1})e^{-m_{2}^{{}^{2}}/M^{2}}&=-m_{1}I_{1}(M^{2},Q^{2},s_{0})+I_{2}(Q^{2},M^{2},s_{0}),\\\
\lambda_{2}G_{2}^{(2)}(Q^{2})(m_{2}-m_{1})e^{-m_{2}^{{}^{2}}/M^{2}}&=-m_{1}I_{3}(M^{2},Q^{2},s_{0})+I_{4}(Q^{2},M^{2},s_{0}),\\\
\lambda_{2}(\frac{G_{2}^{(2)}(Q^{2})}{2}-G_{3}^{(2)}(Q^{2}))(m_{2}-m_{1})e^{-m_{2}^{{}^{2}}/M^{2}}&=-m_{1}I_{5}(M^{2},Q^{2},s_{0})+I_{6}(Q^{2},M^{2},s_{0}),\end{split}$
(21)

where the functions $I_{i}(,Q^{2},M^{2},s_{0})$ are (see Gubernari  _et al._
(2019) and Aliev  _et al._ (2019))

$\begin{split}I_{i}=&\sum_{n=1}^{3}\int_{x_{0}}^{1}dxe^{-s/M^{2}}\frac{1}{(n-1)!}\frac{\rho_{i}^{(n)}}{x^{n}(M^{2})^{n-1}}\\\
&-e^{-s_{0}/M^{2}}\bigg{[}\frac{(-1)^{n-1}}{(n-1)!}\sum_{j=1}^{n-1}\frac{1}{(M^{2})^{n-j-1}}\frac{1}{s^{\prime}}\big{(}\frac{d}{dx}\frac{1}{s^{\prime}}\big{)}^{j-1}\frac{\rho_{i}^{(n)}}{x^{n}}\bigg{]}_{|_{x=x_{0}}},\end{split}$
(22)

where $s=\frac{m_{0}^{2}\bar{x}x+Q^{2}\bar{x}}{x}$, $\bar{x}=1-x$,
$s^{\prime}=\frac{ds}{dx}$, and $x_{0}$ is the solution of $s_{0}=s$ equation
in which $m_{0}$ is the ground state baryon mass. It follows from Eq. (21)
that to determine the transition form factors, the residues of the radial
excitations of decuplet baryons are needed. The mass and residues of these
baryons within QCD sum rules are estimated in Aliev  _et al._ (2017), and the
results are: $m_{2}=1.389~{}{GeV}$, $\lambda_{2}=0.045~{}{GeV^{3}}$ (for
$\Sigma^{*}$ baryon), $m_{2}=1.577~{}{GeV}$, $\lambda_{2}=0.045~{}{GeV^{3}}$
(for $\Xi^{*}$ baryon).

Once the form factors, $G_{i}^{(2)}$, are calculated from the sum rule
presented in Eq. (21) then using Eq. (2) we can calculate the multipole form
factors $G_{M}^{(2)}(Q^{2})$, $G_{E}^{(2)}(Q^{2})$, and $G_{C}^{(2)}(Q^{2})$.

##  III Numerical Analysis

In the present section, the numerical calculations are done for the multipole
form factors as well as $R_{EM}$ and $R_{SM}$. As we already noted, the main
non-perturbative inputs are DAs of the octet baryons. These DA’s are presented
in Bali  _et al._ (2019, 2016); Wein and Schäfer (2015); Liu and Huang (2009a,
b); Braun  _et al._ (2006b). In these works, the values of the parameters
entering the expressions of DA’s are also depicted. To predict the form
factors reliably, the working regions of two auxiliary parameters, the Borel
mass $M^{2}$ and the continuum threshold $s_{0}$, should be specified. The
working regions of these parameters are determined in a standard way. The
working window for $M^{2}$ is determined by demanding the following criteria:

  1. 1.

The lower bound of $M^{2}$ is obtained by requiring that, the higher states
and continuum’s contributions should be by less than say $40\%$ of the total
result.

  2. 2.

The upper limit is determined from the condition that the higher twist
conditions should be smaller than the leading twist one. Our numerical
analysis leads to the conclusion that these conditions are satisfied in the
regions shown below for the considered transitions.

$\begin{split}&1.5~{}GeV^{2}\leq M^{2}\leq~{}3.5~{}GeV^{2}\hskip
42.67912pt(\text{for }\gamma^{*}\Sigma\rightarrow\Sigma^{*})\\\
&1.5~{}GeV^{2}\leq M^{2}\leq~{}3.5~{}GeV^{2}\hskip 42.67912pt(\text{for
}\gamma^{*}\Xi\rightarrow\Xi^{*})\\\ &2.0~{}GeV^{2}\leq
M^{2}\leq~{}4.0~{}GeV^{2}\hskip 42.67912pt(\text{for
}\gamma^{*}\Xi\rightarrow\Xi^{*})\end{split}$ (23)

Besides, the continuum threshold $s_{0}$ is obtained by requiring that the
mass sum rules within $10\%$ accuracy reproduce the mass of
$\Sigma^{*}(1.727)$ and $\Xi^{*}(1.965)$ states. From these conditions, we get
$s_{0}=(2.6\pm 0.1)^{2}~{}\rm{GeV}^{2}$ and $(s_{0}=2.9\pm
0.1)^{2}~{}\rm{GeV}^{2}$ for $\gamma^{*}\Sigma\rightarrow\Sigma^{*}$ and
$\gamma^{*}\Xi\rightarrow\Xi^{*}$ transitions, respectively.

Having the values of all the input parameters, we can start performing
numerical calculations. In Figures 1, 2, 3, 4 and 5, we depict the dependency
of the magnetic dipole form factors $G_{M}(Q^{2})$ on $Q^{2}$ for the
considered transitions at the fixed values of $s_{0}$ and $M^{2}$. From these
figures, we can classify the results for the
$\gamma^{*}\Sigma\rightarrow\Sigma^{*}$ and $\gamma^{*}\Xi\rightarrow\Xi^{*}$
transitions using the magnitudes of the dipole form factors $G_{M}$ as

  * •

large: for $\gamma^{*}\Sigma^{+}\rightarrow\Sigma^{+*}$,
$\gamma\Sigma^{0}\rightarrow\Sigma^{0*}$ and
$\gamma\Xi^{0}\rightarrow\Sigma^{0*}$

  * •

small: for $\gamma^{*}\Sigma^{-}\rightarrow\Sigma^{-*}$ and
$\gamma^{*}\Xi^{-}\rightarrow\Xi^{-*}$.

This classification is similar to the ground state decuplet octet transitions,
with only one exception. In the last transitions, the magnitude of
$G_{M}(Q^{2})$ for $\gamma^{*}\Sigma^{0}\rightarrow\Sigma^{0*}$ transition is
moderate, but in our case, its magnitude is comparable with
$\gamma^{*}\Sigma^{+}\rightarrow\Sigma^{+*}$ and
$\gamma^{*}\Xi^{0}\rightarrow\Xi^{0*}$ transitions. This can be explained as a
consequence of U-spin symmetry Lipkin (1973).

Let us consider the following ratio
$T=|\frac{{G_{M}^{(2)}}^{\Sigma^{+}}}{{G_{M}^{(2)}}^{\Xi^{0}}}-1|$. In the
U-spin symmetry case, this factor must be equal to zero. However, from our
findings, we observe that the $U-spin$ symmetry is broken about $15\%$.
Similar results can be inferred for the $T$-factor of the
$\gamma^{*}\Sigma^{-}\rightarrow\Sigma^{-*}$ and
$\gamma^{*}\Xi^{-}\rightarrow\Xi^{-*}$ transitions.

Instead of studying the behavior of $G_{E}^{(2)}(Q^{2})$ and
$G_{C}^{(2)}(Q^{2})$ on $Q^{2}$, we study the dependencies of $R_{EM}$ and
$R_{SM}$ on $Q^{2}$ since these quantities are related with
$G_{E}^{(2)}(Q^{2})$ and $G_{C}^{(2)}(Q^{2})$ (see Eq. (4)). The dependencies
of these quantities on $Q^{2}$ are presented in Figs. 6 and 7, respectively.
From these figures, we get the following domains of variation of $R_{EM}$ and
$R_{SM}$:

| $R_{EM}(Q^{2})$ | $R_{SM}(Q^{2})$
---|---|---
$\gamma^{*}\Sigma^{0}\rightarrow\Sigma^{0*}$ | $0.28\div 0.55$ | $-0.1\div-0.5$
$\gamma^{*}\Sigma^{-}\rightarrow\Sigma^{-*}$ | $0.4\div 0.8$ | $0.15\div 0.30$
$\gamma^{*}\Sigma^{+}\rightarrow\Sigma^{+*}$ | $0.50\div 0.85$ | $-0.25\div 0.25$
$\gamma^{*}\Xi^{0}\rightarrow\Xi^{0*}$ | $-0.50\div-0.75$ | $-0.2\div 0.4$
$\gamma^{*}\Xi^{-}\rightarrow\Xi^{-*}$ | $1.3\div 1.4$ | $-0.2\div 0.5$
Table 2: The obtained results of $R_{EM}$ and $R_{SM}$ for the considered
transitions are shown with respect to the variation of $Q^{2}$.

For comparison, we also present the results of these quantities for the ground
state decuplet to ground state octet baryons in Table 3.

| $R_{EM}(Q^{2})$ | $R_{SM}(Q^{2})$
---|---|---
$\gamma^{*}\Sigma^{0}\rightarrow\Sigma^{0*}$ | $0.21\div 0.2$ | $-0.3\div-0.5$
$\gamma^{*}\Sigma^{-}\rightarrow\Sigma^{-*}$ | $-0.35\div 0.6$ | practically zero
$\gamma^{*}\Sigma^{+}\rightarrow\Sigma^{+*}$ | $0.1\div 0.2$ | $-0.25\div-0.3$
$\gamma^{*}\Xi^{0}\rightarrow\Xi^{0*}$ | $-0.1\div 0$ | $-0.35\div-0.4$
$\gamma^{*}\Xi^{-}\rightarrow\Xi^{-*}$ | $-0.5\div 0.8$ | $-0.1\div 0$
Table 3: The results for the ground state decuplet to ground state octet
baryons are shown Aliev  _et al._ (2013).

From Table 2, it follows that the values of $R_{EM}(Q^{2})$ for
$\gamma^{*}\Sigma^{\pm}\rightarrow\Sigma^{\pm*}$ and
$\gamma^{*}\Xi^{0}\rightarrow\Xi^{0*}$ in magnitude very close to each other
as a consequence of $U-spin$ symmetry. The large value of $R_{EM}$ for
$\gamma^{*}\Xi^{-}\rightarrow\Xi^{-*}$ channel is due to the large value of
$G_{E}^{(2)}(Q^{2})$. The magnitude of $R_{SM}(Q^{2})$ for the channels
$\gamma^{*}\Sigma^{-}\rightarrow\Sigma^{-*}$,
$\gamma^{*}\Sigma^{+}\rightarrow\Sigma^{+*}$ and
$\gamma^{*}\Xi^{0}\rightarrow\Xi^{0*}$ are very close to each other likewise
for the magnitudes.

Similar situation takes place for $\gamma^{*}\Sigma^{0}\rightarrow\Sigma^{0*}$
and $\gamma^{*}\Xi^{-}\rightarrow\Xi^{-*}$ channels too. Comparing the results
in Table 2 and 3, we observe that the values of $R_{SM}$ for the first radial
decuplet to octet baryons are close to the corresponding results for the
ground state decuplet to octet baryon transitions except the results for
$\gamma^{*}\Sigma^{-}\rightarrow\Sigma^{-*}$ and
$\gamma^{*}\Xi^{-}\rightarrow\Xi^{-*}$. On the other hand, for
$R_{EM}(Q^{2})$, we find that the results are close only for
$\gamma^{*}\Sigma^{-}\rightarrow\Sigma^{-*}$ channels.

##  IV Conclusion

In this work, we studied the electromagnetic transitions among the first
radial excitations of decuplet baryons to ground state baryons using the
light-cone QCD sum rules. The magnetic dipole form factors
$G_{M}^{(2)}(Q^{2})$ as well as the ratios
$R_{EM}=-\frac{G_{E}^{(2)}(Q^{2})}{G_{M}^{(2)}(Q^{2})}$ and
$R_{SM}=-\frac{1}{4m_{1}^{2}}\sqrt{4m_{1}^{2}Q^{2}+(m_{1}^{2}-Q^{2}-m_{N}^{2})^{2}}\frac{G_{C}^{(2)}(Q^{2})}{G_{M}^{(2)}(Q^{2})}$
are calculated when $Q^{2}$ varies in the domain $1~{}\rm{GeV^{2}}\leq
Q^{2}\leq 10~{}\rm{GeV^{2}}$. We observe that the results of the
${G_{M}^{(2)}}(Q^{2})$ for the transitions
$\gamma^{*}\Sigma^{+}\rightarrow\Sigma^{*+}$,
$\gamma^{*}\Xi^{-}\rightarrow\Xi^{-*}$ are very close to each other as a
result of the SU(3) symmetry. We also find that the results for
$R_{SM}(Q^{2})$ for the first radial decuplet excitation state to octet
baryons are close to each other except the results for the
$\gamma\Sigma^{-}\rightarrow\Sigma^{-*}$ and
$\gamma^{*}\Xi^{-}\rightarrow\Xi^{-*}$ transitions. For the quantity $R_{EM}$,
we see that the result only for $\gamma^{*}\Sigma^{-}\rightarrow\Sigma^{-*}$
and more or less for $\gamma^{*}\Xi^{-}\rightarrow\Xi^{-*}$ channels to
considered transitions are close to each other.

The results of the multipole form factors for the $\gamma^{*}B\rightarrow
B^{*}$ octet to the first radial excitation of decuplet baryons transition can
be very useful and play a critical role for studies of the properties of the
excited baryon spectrum and their properties in future experiments.

Figure 1: The dependency of the $G_{M}^{(2)}(Q^{2})$ on $Q^{2}$ at a fixed
values of $s_{0}$ and $M^{2}$ for $\gamma\Sigma^{+}\rightarrow\Sigma^{+*}$.
Figure 2: The same as in Fig. 1, but for
$\gamma\Sigma^{0}\rightarrow\Sigma^{0*}$. Figure 3: The same as in Fig. 1, but
for $\gamma\Sigma^{-}\rightarrow\Sigma^{-*}$. Figure 4: The same as in Fig. 1,
but for $\gamma\Xi^{0}\rightarrow\Xi^{0*}$ . Figure 5: The same as in Fig. 1,
but for $\gamma\Xi^{-}\rightarrow\Xi^{-*}$ . Figure 6: The dependency of the
$R_{EM}(Q^{2})$ on $Q^{2}$ at $s_{0}=5~{}\rm{GeV^{2}}$ and
$M^{2}=3~{}\rm{GeV^{2}}$ for $\gamma\Sigma^{+}\rightarrow\Sigma^{+*}$,
$\gamma\Sigma^{-}\rightarrow\Sigma^{-*}$, and
$\gamma\Sigma^{0}\rightarrow\Sigma^{0*}$ and $s_{0}=7~{}\rm{GeV^{2}}$ and
$M^{2}=4~{}\rm{GeV^{2}}$ for $\gamma\Xi^{0}\rightarrow\Xi^{0*}$ and
$\gamma\Xi^{-}\rightarrow\Xi^{-*}$ Figure 7: Same as in Fig. 6 but for
$R_{SM}$.

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## Appendix

Relations between two sets of octet baryon distribution amplitudes are given
below (see Braun  _et al._ (2006b)).

$\displaystyle\begin{array}[]{ll}{\cal
S}_{1}=S_{1}~{},&(2P\\!\cdot\\!x)\,{\cal S}_{2}=S_{1}-S_{2}~{},\\\ {\cal
P}_{1}=P_{1}~{},&(2P\\!\cdot\\!x)\,{\cal P}_{2}=P_{2}-P_{1}~{},\\\ {\cal
V}_{1}=V_{1}~{},&(2P\\!\cdot\\!x)\,{\cal V}_{2}=V_{1}-V_{2}-V_{3}~{},\\\
2{\cal V}_{3}=V_{3}~{},&(4P\\!\cdot\\!x)\,{\cal
V}_{4}=-2V_{1}+V_{3}+V_{4}+2V_{5}~{},\\\ (4P\\!\cdot\\!x)\,{\cal
V}_{5}=V_{4}-V_{3}~{},&(2P\\!\cdot\\!x)^{2}\,{\cal
V}_{6}=-V_{1}+V_{2}+V_{3}+V_{4}+V_{5}-V_{6}~{},\\\ {\cal
A}_{1}=A_{1}~{},&(2P\\!\cdot\\!x)\,{\cal A}_{2}=-A_{1}+A_{2}-A_{3}~{},\\\
2{\cal A}_{3}=A_{3}~{},&(4P\\!\cdot\\!x)\,{\cal
A}_{4}=-2A_{1}-A_{3}-A_{4}+2A_{5}~{},\\\ (4P\\!\cdot\\!x)\,{\cal
A}_{5}=A_{3}-A_{4}~{},&(2P\\!\cdot\\!x)^{2}\,{\cal
A}_{6}=A_{1}-A_{2}+A_{3}+A_{4}-A_{5}+A_{6}~{},\\\ {\cal
T}_{1}=T_{1}~{},&(2P\\!\cdot\\!x)\,{\cal T}_{2}=T_{1}+T_{2}-2T_{3}~{},\\\
2{\cal T}_{3}=T_{7}~{},&(2P\\!\cdot\\!x)\,{\cal
T}_{4}=T_{1}-T_{2}-2T_{7}~{},\\\ (2P\\!\cdot\\!x)\,{\cal
T}_{5}=-T_{1}+T_{5}+2T_{8}~{},&(2P\\!\cdot\\!x)^{2}\,{\cal
T}_{6}=2T_{2}-2T_{3}-2T_{4}+2T_{5}+2T_{7}+2T_{8}~{},\\\
(4P\\!\cdot\\!x)\,{\cal T}_{7}=T_{7}-T_{8}~{},&(2P\\!\cdot\\!x)^{2}\,{\cal
T}_{8}=-T_{1}+T_{2}+T_{5}-T_{6}+2T_{7}+2T_{8}~{}.\end{array}$

Explicit expressions of DAs ${\cal S}_{i}$, ${\cal P}_{i}$, ${\cal A}_{i}$,
${\cal V}_{i}$ and ${\cal T}_{i}$ at the leading order of conformal spin
expansion can be found in Braun  _et al._ (2006a); Bali  _et al._ (2016,
2019); Wein and Schäfer (2015); Liu and Huang (2009a, b); Braun  _et al._
(2006b).

## Appendix B

In this appendix, we present the expressions for the functions
$\rho_{1}^{(i)}$, $\rho_{2}^{(i)}$ and $\rho_{3}^{(i)}$ which appear in the
sum rules for $G_{1}^{(2)}(Q^{2})$, $G_{2}^{(2)}(Q^{2})$, and
$\displaystyle{G_{2}^{(2)}(Q^{2})\over 2}-G_{3}^{(2)}(Q^{2})$, for the
$\gamma^{\ast}\Sigma^{+}\to\Sigma^{\ast+}$ transition.

## Functions $\rho_{i}^{(n)}$ for the form factor $G_{1}^{(2)}$

$\begin{split}\rho_{1}^{(3)}(x)&=0\\\
\rho_{2}^{(3)}(x)&=\frac{8(1-x)}{x}e_{q_{2}}m_{0}^{2}m_{q_{2}}(x^{2}m_{0}^{2}+Q^{2})\,\widetilde{\\!\widetilde{B}}_{6}\\\
\rho_{1}^{(2)}(x)&=-4e_{q_{3}}m_{0}(m_{0}\widehat{\\!\widehat{B}}_{6}-2m_{q_{3}}\widehat{B}_{4})+8e_{q_{2}}m_{0}m_{q_{2}}\widetilde{B}_{2}\\\
&-8e_{q_{2}}m_{0}^{2}\int_{0}^{\bar{x}}dx_{1}({T_{1}}^{M}-{A_{1}}^{M})(x_{1},x,1-x_{1}-x)\\\
&+8e_{q_{3}}m_{0}^{2}\int_{0}^{\bar{x}}dx_{1}{T_{1}}^{M}(x_{1},1-x_{1}-x,x)\\\
\rho_{2}^{(2)}(x)&=-\frac{4m_{0}}{x}\Big{\\{}-e_{q_{1}}\Big{[}(x-1)(x^{2}m_{0}^{2}+Q^{2})\check{C}_{2}+2x(x+1)\check{D}_{2}\Big{]}\\\
&+e_{q_{2}}\Big{[}x^{3}m_{0}^{2}\widetilde{B}_{2}+x^{3}m_{0}^{2}\widetilde{B}_{4}+(x-1)(x^{2}m_{0}^{2}+Q^{2})\widetilde{D}_{2}-(x-1)(x^{2}m_{0}^{2}+Q^{2})\widetilde{C}_{2}\\\
&-x^{2}m_{0}^{2}\widetilde{B}_{2}-x^{2}m_{0}^{2}\widetilde{B}_{4}-2x^{2}m_{0}m_{q_{2}}\widetilde{H}_{1}+2x^{2}m_{0}m_{q_{2}}\widetilde{B}_{5}+2x^{2}m_{0}m_{q_{2}}\widetilde{B}_{7}\\\
&+2(x-1)xm_{0}^{2}\,\widetilde{\\!\widetilde{B}}_{8}+2xm_{0}m_{q_{2}}\widetilde{H}_{1}-2xm_{0}m_{q_{2}}\widetilde{B}_{5}-2xm_{0}m_{q_{2}}\widetilde{B}_{7}-xm_{0}m_{q_{2}}\,\widetilde{\\!\widetilde{B}}_{6}\\\
&+xQ^{2}\widetilde{B}_{2}+xQ^{2}\widetilde{B}_{4}-2m_{0}m_{q_{2}}\,\widetilde{\\!\widetilde{B}}_{6}-Q^{2}\widetilde{B}_{2}-Q^{2}\widetilde{B}_{4}\Big{]}\\\
&+xe_{q_{3}}m_{0}\Big{[}(x-1)\Big{(}m_{0}(\,\widehat{\\!\widehat{D}}_{6}-2\;\widehat{\\!\widehat{C}}_{6})+m_{q_{3}}(\widehat{D}_{5}-2\widehat{C}_{5}+2\widehat{B}_{5}+4\widehat{B}_{7})\Big{)}+m_{q_{3}}\,\widehat{\\!\widehat{B}}_{6}\Big{]}\Big{\\}}\\\
\rho_{1}^{(1)}(x)&=-e_{q_{2}}\int_{0}^{\bar{x}}dx_{1}(8{B_{1}}-8{D_{1}})(x_{1},x,1-x_{1}-x)+8e_{q_{3}}\int_{0}^{\bar{x}}dx_{1}{B_{1}}(x_{1},1-x_{1}-x,x)\\\
\rho_{2}^{(1)}(x)&=\frac{4m_{0}}{x}\Big{[}e_{q_{2}}(\widetilde{D}_{2}-\widetilde{C}_{2}+\widetilde{B}_{2}+\widetilde{B}_{4})-e_{q_{1}}(x\check{D}_{2}+\check{C}_{2})\Big{]}\\\
&+4(x-1)e_{q_{1}}m_{0}\int_{0}^{\bar{x}}dx_{3}({C_{3}}-{D_{3}})(x,1-x-x_{3},x_{3})\\\
&-4e_{q_{2}}(x-1)m_{0}\int_{0}^{\bar{x}}dx_{1}\Big{[}({D_{3}}-{C_{3}}+2{P_{1}}-2{S_{1}})-8m_{q_{2}}{B_{1}}\Big{]}(x_{1},x,1-x_{1}-x)\\\
&+4e_{q_{3}}(x-1)m_{0}\int_{0}^{\bar{x}}dx_{1}\Big{[}({D_{3}}-2{C_{3}})-8m_{q_{3}}{B_{1}}\Big{]}(x_{1},1-x_{1}-x,x)\end{split}$
(1)

## Functions $\rho_{i}$ for the form factor $G_{2}^{(2)}$

$\begin{split}\rho_{3}^{(3)}(x)&=64(x-1)x^{2}e_{q_{1}}m_{0}^{3}\,\check{\\!\check{C}}_{6}+16xe_{q_{2}}m_{0}^{2}\Big{[}4(x-1)xm_{0}(\,\widetilde{\\!\widetilde{C}}_{6}-2\,\widetilde{\\!\widetilde{B}}_{8})-m_{q_{2}}\,\widetilde{\\!\widetilde{B}}_{6}\Big{]}\\\
&+16xe_{q_{3}}m_{0}^{2}\Big{[}m_{q_{3}}\,\widehat{\\!\widehat{B}}_{6}-2(x-1)xm_{0}(\,\widehat{\\!\widehat{D}}_{6}-2\;\widehat{\\!\widehat{C}}_{6}+2\,\widehat{\\!\widehat{B}}_{8})\Big{]}\\\
\rho_{4}^{(3)}(x)&=32(x-1)e_{q_{2}}m_{0}^{2}\Big{[}(x^{2}m_{0}^{2}+2xQ^{2}-Q^{2})\,\widetilde{\\!\widetilde{B}}_{6}+xm_{0}m_{q_{2}}\,\widetilde{\\!\widetilde{B}}_{8}\Big{]}\\\
&+16(x-1)e_{q_{3}}m_{0}^{2}\Big{[}2x(xm_{0}^{2}+Q^{2})\,\widehat{\\!\widehat{B}}_{6}-xm_{0}m_{q_{3}}(\,\widehat{\\!\widehat{D}}_{6}+2\;\widehat{\\!\widehat{C}}_{6})\Big{]}\\\
\rho_{3}^{(2)}(x)&=-8(1-2x)xe_{q_{1}}m_{0}\check{C}_{2}+8xe_{q_{2}}m_{0}\Big{[}(2x-1)\widetilde{C}_{2}+2(1-2x)\widetilde{B}_{4}-\widetilde{D}_{2}-2\widetilde{B}_{2}\Big{]}\\\
&-8xe_{q_{3}}m_{0}\Big{[}x\widehat{D}_{2}-2x\widehat{C}_{2}+2(x-1)\widehat{B}_{4}\Big{]}\\\
\rho_{4}^{(2)}(x)&=-8(x-1)xe_{q_{1}}m_{0}^{2}(\check{D}_{5}-\check{C}_{4})-8e_{q_{2}}m_{0}\Big{\\{}(x-1)xm_{0}\Big{[}\widetilde{D}_{5}-\widetilde{C}_{4}-2(\widetilde{H}_{1}+\widetilde{E}_{1}-\widetilde{B}_{5})\Big{]}\\\
&+(4x-3)m_{0}\,\widetilde{\\!\widetilde{B}}_{6}+2xm_{q_{2}}\widetilde{B}_{4}+m_{q_{2}}(\widetilde{B}_{2}-\widetilde{B}_{4})\Big{\\}}+8xe_{q_{3}}m_{0}\Big{[}(x-1)m_{0}(\widehat{D}_{5}+2\widehat{C}_{5}-2\widehat{B}_{5})\\\
&+m_{q_{3}}(\widehat{D}_{2}+2\widehat{C}_{2})\Big{]}+8(2x-1)e_{q_{1}}m_{0}^{2}\int_{0}^{\bar{x}}dx_{3}{V_{1}}^{M}(x,1-x-x_{3},x_{3})\\\
&-8e_{q_{2}}m_{0}^{2}\int_{0}^{\bar{x}}dx_{1}\Big{[}{A_{1}}^{M}+(1-2x){V_{1}}^{M}+2(x-1){T_{1}}^{M}\Big{]}(x_{1},x,1-x_{1}-x)\\\
&-16xe_{q_{3}}m_{0}^{2}\int_{0}^{\bar{x}}dx_{1}{T_{1}}^{M}(x_{1},1-x_{1}-x,x)\\\
\rho_{3}^{(1)}(x)&=0\\\
\rho_{4}^{(1)}(x)&=8(2x-1)e_{q_{1}}\int_{0}^{\bar{x}}dx_{3}{C_{1}}(x,1-x-x_{3},x_{3})\\\
&-8e_{q_{2}}\int_{0}^{\bar{x}}dx_{1}\Big{[}{D_{1}}-(2x-1){C_{1}}+2(x-1){B_{1}}\Big{]}(x_{1},x,1-x_{1}-x)\\\
&-16xe_{q_{3}}\int_{0}^{\bar{x}}dx_{1}{B_{1}}(x_{1},1-x_{1}-x,x)\end{split}$
(2)

## Functions $\rho_{i}$ for the form factor
$\frac{G_{2}^{(2)}}{2}-G_{3}^{(2)}$

$\begin{split}\rho_{5}^{(3)}(x)&=-64(x-1)^{2}xe_{q_{1}}m_{0}^{3}\,\check{\\!\check{C}}_{6}-16(x-1)e_{q_{2}}m_{0}^{2}\Big{[}4(x-1)xm_{0}(\,\widetilde{\\!\widetilde{C}}_{6}-2\,\widetilde{\\!\widetilde{B}}_{8})-2m_{q_{2}}\,\widetilde{\\!\widetilde{B}}_{6}\Big{]}\\\
&-16(x-1)e_{q_{3}}m_{0}^{2}\Big{[}2(x-1)xm_{0}(2{C_{6}}-\widehat{\\!\widehat{D}}_{6}-2\,\widehat{\\!\widehat{B}}_{8})+m_{q_{3}}\,\widehat{\\!\widehat{B}}_{6}\Big{]}\\\
\rho_{6}^{(3)}(x)&=\frac{-32(x-1)^{2}}{x}e_{q_{2}}m_{0}^{2}\Big{[}(x^{2}m_{0}^{2}+2xQ^{2}-Q^{2})\,\widetilde{\\!\widetilde{B}}_{6}+xm_{0}m_{q_{2}}\,\widetilde{\\!\widetilde{B}}_{8}\Big{]}\\\
&-\frac{16(x-1)^{2}}{x}e_{q_{3}}m_{0}^{2}\Big{[}2x(xm_{0}^{2}+Q^{2})\,\widehat{\\!\widehat{B}}_{6}-xm_{0}m_{q_{3}}(2{C_{6}}+\widehat{\\!\widehat{D}}_{6})\Big{]}\\\
\rho_{5}^{(2)}(x)&=-16(x-1)xe_{q_{1}}m_{0}\check{C}_{2}+16(x-1)e_{q_{2}}m_{0}(-x\widetilde{C}_{2}+2x\widetilde{B}_{4}+\widetilde{D}_{2}+\widetilde{B}_{2})\\\
&+8(x-1)e_{q_{3}}m_{0}\Big{[}x\widehat{D}_{2}-2x\widehat{C}_{2}+2(x-1)\widehat{B}_{4}\Big{]}\\\
\rho_{6}^{(2)}(x)&=8(x-1)^{2}e_{q_{1}}m_{0}^{2}(\check{D}_{5}-\check{C}_{4})+\frac{8(x-1)}{x}e_{q_{2}}m_{0}\Big{\\{}(x-1)xm_{0}\Big{[}\widetilde{D}_{5}-\widetilde{C}_{4}-2(\widetilde{H}_{1}+\widetilde{E}_{1}-\widetilde{B}_{5})\Big{]}\\\
&+4(x-1)m_{0}\,\widetilde{\\!\widetilde{B}}_{6}+2xm_{q_{2}}\widetilde{B}_{4}\Big{\\}}\\\
&-8(x-1)e_{q_{3}}m_{0}\Big{[}(x-1)m_{0}(\widehat{D}_{5}+2\widehat{C}_{5}-2\widehat{B}_{5})+m_{q_{3}}(\widehat{D}_{2}+2\widehat{C}_{2})\Big{]}\\\
&-16(x-1)e_{q_{1}}m_{0}^{2}\int_{0}^{\bar{x}}dx_{3}{V_{1}}^{M}(x,1-x-x_{3},x_{3})\\\
&-16(x-1)e_{q_{2}}m_{0}^{2}\int_{0}^{\bar{x}}dx_{1}({V_{1}}^{M}-{T_{1}}^{M})(x_{1},x,1-x_{1}-x)\\\
&+16(x-1)e_{q_{3}}m_{0}^{2}\int_{0}^{\bar{x}}dx_{T_{1}}^{M}(x_{1},1-x_{1}-x,x)\\\
\rho_{5}^{(1)}(x)&=0\\\
\rho_{6}^{(1)}(x)&=-16(x-1)e_{q_{1}}\int_{0}^{\bar{x}}dx_{3}{C_{1}}(x,1-x-x_{3},x_{3})\\\
&-16(x-1)e_{q_{2}}\int_{0}^{\bar{x}}dx_{1}({C_{1}}-{B_{1}})(x_{1},x,1-x_{1}-x)\\\
&+16(x-1)e_{q_{3}}\int_{0}^{\bar{x}}dx_{1}{B_{1}}(x_{1},1-x_{1}-x,x)\end{split}$
(3)

where $q_{1}=u$, $q_{2}=u$, and $q_{3}=d$, respectively.

In the above expressions for $\rho_{2}$, $\rho_{4}$, and $\rho_{6}$ the
functions ${\cal F}(x_{i})$ are defined in the following way:

$\displaystyle\check{\cal F}(x_{1})\\!\\!\\!$ $\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!\int_{1}^{x_{1}}\\!\\!dx_{1}^{{}^{\prime}}\int_{0}^{1-x^{{}^{\prime}}_{1}}\\!\\!dx_{3}\,{\cal
F}(x_{1}^{{}^{\prime}},1-x_{1}^{{}^{\prime}}-x_{3},x_{3})~{},$
$\displaystyle\check{\\!\\!\\!\;\check{{\cal F}}}(x_{1})\\!\\!\\!$
$\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!\int_{1}^{x_{1}}\\!\\!dx_{1}^{{}^{\prime}}\int_{1}^{x^{{}^{\prime}}_{1}}\\!\\!dx_{1}^{{}^{\prime\prime}}\int_{0}^{1-x^{{}^{\prime\prime}}_{1}}\\!\\!dx_{3}\,{\cal
F}(x_{1}^{{}^{\prime\prime}},1-x_{1}^{{}^{\prime\prime}}-x_{3},x_{3})~{},$
$\displaystyle\widetilde{\cal F}(x_{2})\\!\\!\\!$ $\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!\int_{1}^{x_{2}}\\!\\!dx_{2}^{{}^{\prime}}\int_{0}^{1-x^{{}^{\prime}}_{2}}\\!\\!dx_{1}\,{\cal
F}(x_{1},x_{2}^{{}^{\prime}},1-x_{1}-x_{2}^{{}^{\prime}})~{},$
$\displaystyle\widetilde{\\!\widetilde{\cal F}}(x_{2})\\!\\!\\!$
$\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!\int_{1}^{x_{2}}\\!\\!dx_{2}^{{}^{\prime}}\int_{1}^{x^{{}^{\prime}}_{2}}\\!\\!dx_{2}^{{}^{\prime\prime}}\int_{0}^{1-x^{{}^{\prime\prime}}_{2}}\\!\\!dx_{1}\,{\cal
F}(x_{1},x_{2}^{{}^{\prime\prime}},1-x_{1}-x_{2}^{{}^{\prime\prime}})~{},$
$\displaystyle\widehat{\cal F}(x_{3})\\!\\!\\!$ $\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!\int_{1}^{x_{3}}\\!\\!dx_{3}^{{}^{\prime}}\int_{0}^{1-x^{{}^{\prime}}_{3}}\\!\\!dx_{1}\,{\cal
F}(x_{1},1-x_{1}-x_{3}^{{}^{\prime}},x_{3}^{{}^{\prime}})~{},$
$\displaystyle\widehat{\\!\widehat{\cal F}}(x_{3})\\!\\!\\!$
$\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!\int_{1}^{x_{3}}\\!\\!dx_{3}^{{}^{\prime}}\int_{1}^{x^{{}^{\prime}}_{3}}\\!\\!dx_{3}^{{}^{\prime\prime}}\int_{0}^{1-x^{{}^{\prime\prime}}_{3}}\\!\\!dx_{1}\,{\cal
F}(x_{1},1-x_{1}-x_{3}^{{}^{\prime\prime}},x_{3}^{{}^{\prime\prime}})~{}.$

Definitions of the functions $B_{i}$, $C_{i}$, $D_{i}$, $E_{1}$ and $H_{1}$
that appear in the expressions for $\rho_{i}^{(n)}(x)$ are given as follows:

$\displaystyle B_{2}\\!\\!\\!$ $\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!T_{1}+T_{2}-2T_{3}~{},$ $\displaystyle B_{4}\\!\\!\\!$
$\displaystyle~{}=~{}$ $\displaystyle\\!\\!\\!T_{1}-T_{2}-2T_{7}~{},$
$\displaystyle B_{5}\\!\\!\\!$ $\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!-T_{1}+T_{5}+2T_{8}~{},$ $\displaystyle B_{6}\\!\\!\\!$
$\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!2T_{1}-2T_{3}-2T_{4}+2T_{5}+2T_{7}+2T_{8}~{},$
$\displaystyle B_{7}\\!\\!\\!$ $\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!T_{7}-T_{8}~{},$ $\displaystyle B_{8}\\!\\!\\!$
$\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!-T_{1}+T_{2}+T_{5}-T_{6}+2T_{7}+2T_{8}~{},$
$\displaystyle C_{2}\\!\\!\\!$ $\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!V_{1}-V_{2}-V_{3}~{},$ $\displaystyle C_{4}\\!\\!\\!$
$\displaystyle~{}=~{}$ $\displaystyle\\!\\!\\!-2V_{1}+V_{3}+V_{4}+2V_{5}~{},$
$\displaystyle C_{5}\\!\\!\\!$ $\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!V_{4}-V_{3}~{},$ $\displaystyle C_{6}\\!\\!\\!$
$\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!-V_{1}+V_{2}+V_{3}+V_{4}+V_{5}-V_{6}~{},$
$\displaystyle D_{2}\\!\\!\\!$ $\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!-A_{1}+A_{2}-A_{3}~{},$ $\displaystyle D_{4}\\!\\!\\!$
$\displaystyle~{}=~{}$ $\displaystyle\\!\\!\\!-2A_{1}-A_{3}-A_{4}+2A_{5}~{},$
$\displaystyle D_{5}\\!\\!\\!$ $\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!A_{3}-A_{4}~{},$ $\displaystyle D_{6}\\!\\!\\!$
$\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!A_{1}-A_{2}+A_{3}+A_{4}-A_{5}+A_{6}~{},$ $\displaystyle
E_{1}\\!\\!\\!$ $\displaystyle~{}=~{}$ $\displaystyle\\!\\!\\!S_{1}-S_{2}~{},$
$\displaystyle H_{1}\\!\\!\\!$ $\displaystyle~{}=~{}$
$\displaystyle\\!\\!\\!P_{2}-P_{1}~{}.$

The expressions of the functions $V_{i}$, $A_{i}$, $T_{i}$, $S_{i}$ and
$P_{i}$ are presented in Appendix A.