# Isovector and isoscalar tensor form factors of $N(1535)\rightarrow N$
transition in light-cone QCD

Ulaş Özdem<EMAIL_ADDRESS>Health Services Vocational School of Higher
Education, Istanbul Aydin University, Sefakoy-Kucukcekmece, 34295 Istanbul,
Turkey

###### Abstract

We have applied isovector and isoscalar tensor current to evaluate the tensor
form factors of the $N(1535)\rightarrow N$ transition with the help of the
light-cone QCD sum rule method. In numerical computations, have used the most
general forms of the interpolating current for the nucleon and the tensor
current together with two different sets of the input parameters in the DAs of
the $N(1535)$ state. We have obtained that the values of $N(1535)\rightarrow
N$ transition tensor form factors very sensitive to the input parameters of
the distribution amplitudes of the $N(1535)$ state. We have acquired that the
$Q^{2}$ dependence of $N(1535)\rightarrow$ transition tensor form factors is
well defined by a p-pole fit function.

Tensor form factors, Nucleon, N(1535), Light-cone QCD sum rule

##  I Introduction

The essential subject of QCD is to understand the internal structure of
hadrons and their features in terms of degrees of freedom of quark-gluons.
Hadron charges described as matrix elements of tensor, axial and vector
currents between hadron states include complete knowledge about internal
structure of the hadron. At the twist-two level, the corresponding charges are
characterized by the helicity distribution $g_{1}(x)$, transversity
distribution $h_{1}(x)$ and unpolarized distribution $f_{1}(x)$ function of
the quark. More generally, at the leading twist, eight generalized parton
distributions (GPDs) encompass full knowledge on the internal structure of
hadrons: four chiral-odd spin-dependent GPDs $H_{T}(x,\xi,t)$,
$E_{T}(x,\xi,t)$, $\tilde{H}_{T}(x,\xi,t)$, and $\tilde{E}_{T}(x,\xi,t)$; two
chiral-even spin-dependent GPDs $\tilde{H}(x,\xi,t)$, and $\tilde{E}(x,\xi,t)$
and; two chiral-even spin-independent GPDs $H(x,\xi,t)$ and $E(x,\xi,t)$,
where $\xi$ is the skewness and t = -$Q^{2}$ is the squared momentum transfer
Ji:1996ek ; Radyushkin:1997ki ; Hoodbhoy:1998vm ; Diehl:2001pm . These
observables include important knowledge about the internal structure of the
hadron. They characterize, e.g., how partons are distributed in the transverse
plane according to motion of the hadron or the contribution of quark orbital
angular momentum to total angular momentum of the hadron. The helicity and
unpolarized distribution functions can be extracted from inclusive deep-
inelastic scattering data because of their chiral-even nature. In the forward
limit, they are related to the electromagnetic, axial and pseudoscalar form
factors Goeke:2001tz . However, the transversity distribution function, which
is related to the tensor form factors in the forward limit, has chiral-odd
nature, so there is a big experimental problem to measure it. It can be
acquired Drell-Yan processes and semi-inclusive deep inelastic scattering, as
distributions of transversity do appear at leading twist in the cross-section.
Photo- and electro-production of mesons off the polarized nucleons and the
transversely polarized Drell-Yan process are recommended as suitable ways to
measure transversity distribution. In Ref. Anselmino:2007fs , transversity
distribution of the nucleon was extracted using the experimental data from
COMPASS Ageev:2006da , HERMES Airapetian:2004tw  and Belle Abe:2005zx
Collaborations. Afterward, in Ref. Cloet:2007em  the tensor charge of the
nucleon was extracted in the framework of the covariant quark-diquark model.
Moreover, tensor form factors of the nucleon have been investigated by the
help of QCD sum rule He:1994gz ; He:1996wy , axial vector meson dominance
model Gamberg:2001qc , quark model Schmidt:1997vm ; Pasquini:2005dk , chiral
quark soliton model Lorce:2007fa ; Ledwig:2010tu , light-cone QCD sum rule
Erkol:2011iw ; Aliev:2011ku , dihadron production Pisano:2015wnq , lattice QCD
Hagler:2007xi ; Gockeler:2005cj ; Bhattacharya:2015wna , relativistic confined
quark model Gutsche:2016xff , and skyrme model Olness:1992zb . Besides, the
tensor form factors of the octet hyperons are investigated in the framework of
the chiral quark soliton model Ledwig:2010tu  and light-cone QCD sum rule
kucukarslan:2016xhx .

Form factors play a crucial role in our comprehension of the tomography of
baryons. The tensor form factors are missing part of the this tomography.
Recently, the measurements of exclusive electro-production of pseudoscalar
mesons ($\pi$ and $\eta$ mesons) has demonstrated that these processes are
responsive to chiral-odd GPDs Bedlinskiy:2012be ; Bedlinskiy:2014tvi ;
Defurne:2016eiy ; Bedlinskiy:2017yxe . Photo- and electro-production of
pseudoscalar mesons can be used to extract the tensor form factors of baryons
Bedlinskiy:2012be . In the short run, remarkably more accurate measurements of
the nucleon tensor form factors are expected at Jefferson Laboratory (JLab) by
the CLAS Collaboration. Besides, the experiments designed at CLAS
Collaboration, have been aimed to investigate features of electro-excitaton of
nucleon resonances in photo- and electro-production reactions Aznauryan:2012ba
. Inspired by the future experiments at JLab, we aim to investigate the
isovector and isoscalar tensor form factor $N(1535)\rightarrow N$ transition
up to a momentum transfer of $Q^{2}\leq$ 10 GeV2 with the help of the light-
cone QCD sum rule. To our knowledge, this is the first study in the literature
committed to the examination of the $N(1535)\rightarrow N$ transition tensor
form factors. In the light-cone QCD sum rule method, the hadronic observables
are described in connection with the properties of the vacuum and distribution
amplitudes (DAs) of the hadrons under the investigation Braun:1988qv ;
Balitsky:1989ry ; Chernyak:1990ag . Since the hadronic observables are
described in connection with the features of the QCD vacuum and the DAs, any
ambiguity in these variables reflects to the ambiguity of the predictions of
the hadronic observables. Note that the electromagnetic Anikin:2015ita , axial
Aliev:2019tmk  and gravitational ozdem:2019pkg  form factors for
$N(1535)\rightarrow N$ transition have been evaluated with the help of light-
cone QCD sum rule.

This article is organized in the following manner: In Sec. II we present the
details of our light-cone QCD sum rule calculations. In Sec. III we analyze
the obtained results and give our conclusions.

##  II Isovector and isoscalar tensor Form Factors of $N(1535)\rightarrow N$
transition

The matrix element of the isovector and isoscalar tensor current between
nucleon and $N(1535)$ baryons is defined by three dimensionless invariant form
factors as presented Hagler:2009ni ; Gockeler:2006zu

$\displaystyle\langle
N(p^{\prime})|J_{\mu\nu}|N(1535)(p)\rangle=\bar{u}(p^{\prime})\Bigg{[}i\sigma_{\mu\nu}H_{T}^{I=0,1}(Q^{2})+\frac{\gamma_{\mu}q_{\nu}-\gamma_{\nu}q_{\mu}}{2\bar{m}}E_{T}^{I=0,1}(Q^{2})+\frac{\tilde{P}_{\mu}q_{\nu}-\tilde{P}_{\nu}q_{\mu}}{2\bar{m}^{2}}\tilde{H}_{T}^{I=0,1}(Q^{2})\Bigg{]}\gamma_{5}u(p),$
(1)

where $\bar{m}$ =$(m_{N}+m_{N(1535)})/2$,
$\sigma_{\mu\nu}=\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]$, $q=p-p^{\prime}$,
$\tilde{P}=p^{\prime}+p$ and, $F^{I=1}=F^{u}-F^{d}$ and $F^{I=0}=F^{u}+F^{d}$
for any of the form factors, $F$= $E_{T}$, $H_{T}$ or $\tilde{H}_{T}$.

To derive the light-cone QCD sum rules for isovector and isoscalar tensor form
factors of $N(1535)\rightarrow N$ transition, we consider subsequent
correlator for our analysis

$\displaystyle\Pi_{\mu\nu}(p,q)=i\int d^{4}xe^{iqx}\langle
0|\mathcal{T}[J_{N}(0)J_{\mu\nu}(x)]|N(1535)(p)\rangle,$ (2)

where $J_{\mu\nu}(x)$ is the tensor current and $J_{N}(0)$ are interpolating
currents for nucleon states. The explicit forms of the $J_{N}(0)$ and
$J_{\mu\nu}(x)$ are given as

$\displaystyle J_{N}(0)$
$\displaystyle=2\epsilon^{abc}\sum_{\ell=1}^{2}(u^{aT}(x)CJ_{1}^{\ell}u^{b}(x))J_{2}^{\ell}d^{c}(x),$
$\displaystyle J_{\mu\nu}(x)$
$\displaystyle=\bar{u}^{d}(x)i\sigma_{\mu\nu}u^{d}(x)\pm\bar{d}^{e}(x)i\sigma_{\mu\nu}d^{e}(x),$
(3)

respectively, where $J_{1}^{1}=I$, $J_{1}^{2}=J_{2}^{1}=\gamma_{5}$,
$J_{2}^{2}=t$, which is an arbitrary parameter that fixes the mixing of two
local operators, and C denotes charge conjugation.

To acquire the sum rules for isovector and isoscalar tensor form factors of
the $N(1535)\rightarrow N$ transition the correlator in Eq.(2) is obtained
from the subsequent three steps:
$\bullet$ The correlator is saturated by complete set of hadronic states,
which are have the same quantum numbers as interpolating currents (hadronic
representation),
$\bullet$ The correlator is obtained in connection with quark and gluon
degrees of freedom interacting with non-perturbative QCD vacuum (QCD
representation).
$\bullet$ Then match these two independent representations of the correlator
to one another employing the quark-hadron duality ansatz. To keep under
control undesirable contributions coming from the higher and excited states,
we perform a Borel transformation, in addition to continuum subtraction to
both representations of the obtained corresponding sum rules.

As we mentioned above in order to evaluate the correlator in connection with
hadron features, a complete hadronic set with the same quantum numbers as the
interpolation currents is inserted. After that, the correlation function
becomes

$\displaystyle\Pi_{\mu\nu}^{Had}(p,q)=$
$\displaystyle\sum_{s{{}^{\prime}}}\frac{\langle
0|J_{N}(0)|{N(p^{\prime},s^{\prime})}\rangle}{m^{2}_{N}-p^{\prime 2}}$
$\displaystyle\langle{N(p^{\prime},s^{\prime})}|J_{\mu\nu}(x)|N(1535)(p,s)\rangle+...,$
(4)

where

$\displaystyle\langle 0|J_{N}(0)|{N(p^{\prime},s^{\prime})}\rangle$
$\displaystyle=\lambda_{N}u_{N}(p^{\prime},s^{\prime}),$ (5)

with $\lambda_{N}$ and $u_{N}(p^{\prime},s^{\prime})$ being the residue and
Dirac spinor of nucleon, respectively. Summation over the spins of the nucleon
is performed as

$\displaystyle\sum_{s^{\prime}}u_{N}(p^{\prime},s^{\prime})\bar{u}_{N}(p^{\prime},s^{\prime})$
$\displaystyle=p\\!\\!\\!/^{\prime}+m_{N}.$ (6)

Substituting Eqs. (1), (5) and (6) into Eq. (II), we acquire the correlator in
the way of the hadronic features as

$\displaystyle\Pi_{\mu\nu}^{Had}(p,q)=\frac{\lambda_{N}}{m_{N}^{2}-p^{\prime
2}}(p\\!\\!\\!/^{\prime}+m_{N})\left[i\sigma_{\mu\nu}H_{T}^{I=0,1}(Q^{2})+\frac{\gamma_{\mu}q_{\nu}-\gamma_{\nu}q_{\mu}}{2\bar{m}}E_{T}^{I=0,1}(Q^{2})+\frac{\tilde{P}_{\mu}q_{\nu}-\tilde{P}_{\nu}q_{\mu}}{2\bar{m}^{2}}\tilde{H}_{T}^{I=0,1}(Q^{2})\right]\gamma_{5}\,u(p).$
(7)

The next step is to evaluate the correlator in Eq. (2) with respect to quarks
and gluon properties in deep Euclidean region. Employing the expression for
$J_{N}(0)$ and $J_{\mu\nu}(x)$ and Wick’s theorem, the QCD representation of
the correlator is obtained as

$\displaystyle\Pi_{\mu\nu}^{QCD}(p,q)$ $\displaystyle=-\int
d^{4}xe^{iqx}\Bigg{[}\bigg{\\{}(\gamma_{5})_{\gamma\delta}\,C_{\alpha\beta}\,(i\sigma_{\mu\nu})_{\omega\rho}+t\,(I)_{\gamma\delta}\,(C\gamma_{5})_{\alpha\beta}\,(i\sigma_{\mu\nu})_{\omega\rho}\bigg{\\}}$
$\displaystyle\times\bigg{\\{}\langle
0|\epsilon^{abc}u_{\sigma}^{a}(0)u_{\theta}^{b}(x)d_{\phi}^{c}(0)|N(1535)(p)\rangle\Big{(}\delta_{\sigma}^{\alpha}\delta_{\theta}^{\rho}\delta_{\phi}^{\beta}S_{u}(-x)_{\delta\omega}+\,\delta_{\sigma}^{\delta}\delta_{\theta}^{\rho}\delta_{\phi}^{\beta}S_{u}(-x)_{\alpha\omega}\Big{)}$
$\displaystyle\pm\,\langle
0|\epsilon^{abc}u_{\sigma}^{a}(0)u_{\theta}^{b}(0)d_{\phi}^{c}(x)|N(1535)(p)\rangle\,\delta_{\sigma}^{\alpha}\delta_{\theta}^{\delta}\delta_{\phi}^{\rho}S_{d}(-x)_{\beta\omega}\bigg{\\}}\Bigg{]},$
(8)

where $S_{q}(x)$ is the light-quark propagator and it is given as ($m_{q}=0$)

$\displaystyle S_{q}(x)$
$\displaystyle=i\frac{x\\!\\!\\!/}{2\pi^{2}x^{4}}-\frac{\langle\bar{q}q\rangle}{12}-\frac{\langle\bar{q}\sigma.Gq\rangle}{192}x^{2}$
$\displaystyle-\frac{ig_{s}}{32\pi^{2}x^{2}}~{}G^{\mu\nu}(x)\bigg{[}\hbox
to0.0pt{/\hss}{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\hbox
to0.0pt{/\hss}{x}\bigg{]}.$ (9)

The $\langle
0|\epsilon^{abc}u_{\sigma}^{a}(x_{1})u_{\theta}^{b}(x_{2})d_{\phi}^{c}(x_{3})|N(1535)(p)\rangle$
matrix element in Eq. (II) is can be written in terms of the DAs of the
$N(1535)$ state and it is necessary for further computations. The
comprehensive expression of this matrix elements are presented in Ref. maxiphd
. After employing the explicit forms of the above matrix elements and the
light-quark propagator, we acquire expressions in x-space. Then we apply
Fourier transforms to transfer these expressions into the momentum space.

The desired light-cone sum rules are obtained by matching both representations
of the correlation function. In order to do this, we have to choose different
and independent Lorentz structures. For this purpose, we choose
$p_{\mu}q_{\nu}\gamma_{5}$, $p_{\mu}\gamma_{\nu}\gamma_{5}$ and
$p_{\mu}q_{\nu}q\\!\\!\\!/\gamma_{5}$ structures for $E_{T}^{I=0,1}(Q^{2})$,
$H_{T}^{I=0,1}(Q^{2})$ and $\tilde{H}_{T}^{I=0,1}(Q^{2})$ form factors,
respectively. As a result, we get the light-cone sum rules

$\displaystyle E_{T}^{I=1}(Q^{2})\frac{\lambda_{N}}{m_{N}^{2}-p^{\prime 2}}$
$\displaystyle=\bar{m}\,\Pi_{1}^{QCD},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}E_{T}^{I=0}(Q^{2})\frac{\lambda_{N}}{m_{N}^{2}-p^{\prime
2}}=\bar{m}\,\Pi_{2}^{QCD},$ (10) $\displaystyle
H_{T}^{I=1}(Q^{2})\frac{\lambda_{N}}{m_{N}^{2}-p^{\prime 2}}$
$\displaystyle=-\frac{1}{2}\,\Pi_{3}^{QCD},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}H_{T}^{I=0}(Q^{2})\frac{\lambda_{N}}{m_{N}^{2}-p^{\prime
2}}=-\frac{1}{2}\,\Pi_{4}^{QCD},$ (11)
$\displaystyle\tilde{H}_{T}^{I=1}(Q^{2})\frac{\lambda_{N}}{m_{N}^{2}-p^{\prime
2}}$
$\displaystyle=-\bar{m}^{2}\,\Pi_{5}^{QCD},~{}~{}~{}~{}~{}~{}~{}~{}~{}\tilde{H}_{T}^{I=0}(Q^{2})\frac{\lambda_{N}}{m_{N}^{2}-p^{\prime
2}}=-\bar{m}^{2}\,\Pi_{6}^{QCD}.,$ (12)

The $\Pi_{i}^{QCD}$ functions appearing in Eqs. (10)-(12) are quite long and
not illuminating. However, as an example, we give the result of the
$\Pi_{1}^{QCD}$. The remaining five of these functions have more or less
similar forms.

$\displaystyle\Pi_{1}^{QCD}$
$\displaystyle=2\,m^{2}_{N(1535)}\Bigg{\\{}\int_{0}^{1}d\alpha\frac{\alpha}{(q-p\alpha)^{4}}\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}\Bigg{[}(1-t)[2A_{1}-2A_{2}-A_{3}+3A_{4}+2V_{1}+2V_{2}-4V_{3}$
$\displaystyle+2V_{4}-4V_{5}]+(1+t)[2P_{1}-2P_{2}+2S_{1}-2S_{2}+2T_{2}+4T_{3}-6T_{5}-4T_{7}]\Bigg{]}(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle+2\,\int_{0}^{1}d\alpha\frac{\alpha}{(q-p\alpha)^{4}}\int_{\alpha}^{1}dx_{3}\int_{0}^{1-x_{3}}dx_{1}\Bigg{[}(1+t)[-P_{1}+P_{2}-S_{1}+S_{2}-T_{1}+T_{5}+T_{7}+T_{8}]\Bigg{]}$
$\displaystyle\times(x_{1},1-x_{1}-x_{3},x_{3})$
$\displaystyle+2\,\int_{0}^{1}d\beta\int_{\beta}^{1}d\alpha\frac{1}{(q-p\beta)^{4}}\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}\Bigg{[}(1+t)[-T_{2}+T_{3}+T_{4}-T_{5}-T_{7}-T_{8}]\Bigg{]}$
$\displaystyle\times(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle-2\,\int_{0}^{1}d\beta\int_{\beta}^{1}d\alpha\frac{1}{(q-p\beta)^{4}}\int_{\alpha}^{1}dx_{3}\int_{0}^{1-x_{3}}dx_{1}\Bigg{[}(1+t)[T_{2}-T_{3}-T_{4}+T_{5}+T_{7}+T_{8}]\Bigg{]}$
$\displaystyle\times(x_{1},1-x_{1}-x_{3},x_{3})$
$\displaystyle+8\,m^{2}_{N(1535)}\int_{0}^{1}d\beta\int_{\beta}^{1}d\alpha\frac{\beta^{2}}{(q-p\beta)^{6}}\int_{\alpha}^{1}dx_{2}\int_{0}^{1-x_{2}}dx_{1}\Bigg{[}(1+t)[-T_{2}+T_{3}+T_{4}-T_{5}-T_{7}-T_{8}]\Bigg{]}$
$\displaystyle\times(x_{1},x_{2},1-x_{1}-x_{2})$
$\displaystyle-8\,m^{2}_{N(1535)}\int_{0}^{1}d\beta\int_{\beta}^{1}d\alpha\frac{\beta^{2}}{(q-p\beta)^{6}}\int_{\alpha}^{1}dx_{3}\int_{0}^{1-x_{3}}dx_{1}\Bigg{[}(1+t)[T_{2}-T_{3}-T_{4}+T_{5}+T_{7}+T_{8}]\Bigg{]}$
$\displaystyle\times(x_{1},1-x_{1}-x_{3},x_{3})\Bigg{\\}},$ (13)

where, $A_{i}$, $P_{i}$, $V_{i}$, $S_{i}$ and $T_{i}$ are distribution
amplitudes of different twists. They have been expressed with respect to
$N(1535)$ state wavefunctions. The explicit forms of these wavefunctions are
presented in Ref. maxiphd . To eliminate contributions coming from the excited
and continuum states the Borel transformation and continuum subtraction are
performed. The suppression of the excited and continuum states can be
accomplished by means of the subsequent subtraction rules Braun:2006hz :

$\displaystyle\int dz\frac{\rho(z)}{(q-zp)^{2}}\rightarrow$
$\displaystyle-\int_{x_{0}}^{1}\frac{dz}{z}\rho(z)e^{-s(z)/M^{2}},$
$\displaystyle\int dz\frac{\rho(z)}{(q-zp)^{4}}\rightarrow$
$\displaystyle\frac{1}{M^{2}}\int_{x_{0}}^{1}\frac{dz}{z^{2}}\rho(z)e^{-s(z)/M^{2}}+\frac{\rho(x_{0})}{Q^{2}+x_{0}^{2}m^{2}_{N}}e^{-s_{0}/M^{2}},$
$\displaystyle\int dz\frac{\rho(z)}{(q-zp)^{6}}\rightarrow$
$\displaystyle-\frac{1}{2M^{4}}\int_{x_{0}}^{1}\frac{dz}{z^{3}}\rho(z)e^{-s(z)/M^{2}}-\frac{1}{2M^{2}}\frac{\rho(x_{0})}{x_{0}(Q^{2}+x_{0}^{2}m_{N}^{2})}e^{-s_{0}/M^{2}}$
$\displaystyle+\frac{1}{2}\frac{x_{0}^{2}\,e^{-s_{0}/M^{2}}}{Q^{2}+x_{0}^{2}m_{N}^{2}}\bigg{[}\frac{d}{dx_{0}}\frac{\rho(x_{0})}{x_{0}(Q^{2}+x_{0}^{2}m_{N}^{2})}\bigg{]},$
(14)

where

$\displaystyle s(z)=$ $\displaystyle(1-z)m^{2}_{N}+\frac{1-z}{z}Q^{2},$
$\displaystyle x_{0}=$
$\displaystyle\frac{\sqrt{(Q^{2}+s_{0}-m^{2}_{N})^{2}+4m^{2}_{N}Q^{2}}-(Q^{2}+s_{0}-m^{2}_{N})}{2m^{2}_{N}}.$
(15)

The residue of the nucleon, $\lambda_{N}$, is needed for the numerical
computation of $N(1535)\rightarrow N$ transition tensor form factors. The
$\lambda_{N}$ is specified from two-point QCD sum rules Aliev:2011ku :

$\displaystyle\lambda_{N}$
$\displaystyle=\Bigg{[}e^{\frac{m_{N}^{2}}{M^{2}}}\bigg{\\{}\frac{M^{6}}{256\pi^{4}}(5+2t+t^{2})E_{2}(y)$
$\displaystyle-\frac{\langle\bar{q}q\rangle^{2}}{6}\bigg{(}6(1-t^{2})-(1-t)^{2}$
$\displaystyle-\frac{m_{0}^{2}}{4M^{2}}\Big{[}12(1-t^{2})-(1-t)^{2}\Big{]}\bigg{)}\Bigg{\\}}\Bigg{]}^{1/2},$
(16)

where

$\displaystyle y$ $\displaystyle=$ $\displaystyle s_{0}/M^{2},$

and

$\displaystyle E_{n}(y)$ $\displaystyle=$ $\displaystyle
1-e^{-y}\sum_{i=0}^{n}\frac{y^{i}}{i!}~{}.$

##  III Numerical analysis and conclusion

In this section, we have obtained numerical analysis of $N(1535)\rightarrow N$
transition tensor form factors. The DAs of $N(1535)$ state have been evaluated
by means of the light-cone QCD sum rule in Ref. maxiphd . The numerical values
of the input parameters inside the DAs of the $N(1535)$ state are given in
Table 1, which are obtained at renormalization scale
$\mu^{2}=2.0~{}\mathrm{GeV}^{2}$. Furthermore, we use
$\lambda_{1}^{N}m_{N}=-3.88(2)(19)\times 10^{-2}$ GeV3 and
$\lambda_{2}^{N(1535)}m_{N(1535)}=8.97(45)\times 10^{-2}$ GeV3, given in Ref.
Braun:2014wpa  at renormalization $\mu^{2}=4.0$ GeV2, by rescaling to
$\mu^{2}=2.0~{}\mathrm{GeV}^{2}$. Beside these values, we use
$m_{N(1535)}=1.51\pm 0.01$ GeV, $m_{N}=0.94$ GeV Tanabashi:2018oca ,
$m_{0}^{2}=0.8\pm 0.1$ GeV2 and $\langle\bar{q}q\rangle=(-0.24\pm 0.01)^{3}$
GeV3 Ioffe:2005ym .

Table 1: Input parameters of the $N(1535)$ state DAs for the two different
sets.
$\begin{array}[]{|c|c|c|c|c|c|c|c|c|c|}\hline\cr\hline\cr\mbox{Model}&\mid\lambda_{1}^{N(1535)}/\lambda_{1}^{N}\mid&f_{N(1535)}/\lambda_{1}^{N(1535)}&\varphi_{10}&\varphi_{11}&\varphi_{20}&\varphi_{21}&\varphi_{22}&\eta_{10}&\eta_{11}\\\
\hline\cr\mbox{ LCSR-I}&0.633&0.027&0.36&-0.95&0&0&0&0&0.94\\\ \mbox{LCSR-
II}&0.633&0.027&0.37&-0.96&0&0&0&-0.29&0.23\\\ \hline\cr\hline\cr\end{array}$

The predictions for the isovector and isoscalar tensor form factors depend on
three auxiliary parameters; the Borel mass parameter $M^{2}$, arbitrary mixing
parameter $t$ and continuum threshold $s_{0}$. For the quality of the
numerical values of the physical observables, we should minimize the
dependence of the results on these parameters. The $M^{2}$ can change in the
interval that the results relatively weakly depend on it with respect to the
standard definition. The upper limit of it is acquired by demanding the
maximum pole contributions and its lower limit is acquired from the
convergence of the operator product expansion and exceeding of the
perturbative part over nonperturbative contributions. The $t$ is chosen such
that, the estimations of the isovector and isoscalar tensor form factors are
reasonably insensitive of the values of $t$. The working region for the
$s_{0}$ is chosen such that the maximum pole contribution is obtained and the
results relatively weakly depend on its choices. These constraints lead to the
working intervals for auxiliary parameters as

$\displaystyle 2.50~{}\mathrm{GeV}^{2}\leq M^{2}\leq 3.50~{}\mathrm{GeV}^{2},$
$\displaystyle 2.50~{}\mathrm{GeV}^{2}\leq s_{0}\leq 3.00~{}\mathrm{GeV}^{2},$
$\displaystyle~{}~{}~{}-3.00~{}\leq~{}t~{}\leq~{}-5.00.$

In Figs. 1 and 2, we show dependency of isovector and isoscalar tensor form
factors with respect to the Borel mass parameter at three fixed values of the
continuum threshold and two fixed values of the arbitrary mixing parameter in
their working interval. The results show good stability against the variations
of the Borel mass parameters, as desired. In Figs. 3 and 4, we plot the
dependence of the isovector and isoscalar tensor form factors on $Q^{2}$ for
various values of $s_{0}$ and $t$ in their working regions and at the fixed
values of $M^{2}$ = 3.00 GeV2 for LCSR-I and LCSR-II values of input
parameters entering the DAs. All the form factors taken into account show a
similar dependence on $Q^{2}$ for LCSR-I and LCSR-II except the form factor
$\tilde{H}_{T}^{I=0}$($Q^{2}$). This form factor changes its sign in the
region under consideration, so its results are not given in the text. We
should note here that the light-cone QCD sum rule approach is trustworthy only
for $Q^{2}>1.0$ GeV2. On the other hand, the baryon mass corrections of the
DAs $\sim m^{2}/Q^{2}$ become very large for $Q^{2}<2.0$ GeV2, in other words
the light-cone QCD sum rules turn out to be untrustworthy. Thus, we expect the
light-cone QCD sum rule to be more reliable and effective in the region of 2.0
GeV ${}^{2}\leq Q^{2}\leq$ 10.0 GeV2.

As we mentioned above our sum rules work only for $Q^{2}\geq$ 2.0 GeV2.
However, we want to extend our analysis to the region $0\leq Q^{2}<2$. To do
this, some fit parameters need to be used. Our numerical investigations
indicate that the isovector and isoscalar tensor form factors of
$N(1535)\rightarrow N$ transition are nicely defined by employing the p-pole
fit function:

$\displaystyle{\cal F}(Q^{2})=\frac{{\cal
F}(0)}{\Big{(}1+Q^{2}/(p\,m^{2}_{p})\Big{)}^{p}}.$ (17)

The numerical results obtained for $N(1535)\rightarrow N$ transition isovector
and isoscalar tensor form factors are given in Table 2. The results obtained
by employing LCSR-I and LCSR-II parameters were found to be quite different
from each other. The numerical values of the form factors
$E_{T}^{I=0,1}$($Q^{2}=0$), $H_{T}^{I=0,1}$($Q^{2}=0$) and
$\tilde{H}_{T}^{I=0,1}$($Q^{2}=0$) for the LCSR-II numerical values are
smaller than those for the LCSR-I parameters. As one can see from Table 1, the
essential difference between input parameters of the DAs is the numerical
values for the $\eta_{10}$ and $\eta_{11}$, which are related to the p-wave
three-quark wave functions of the $N(1535)$ state and, therefore to the
distribution of orbital angular momentum. This means these form factors are
very sensitive to the shape parameters of the DAs of the $N(1535)$ state that
parametrize relative orbital angular momentum of the quarks.

Table 2:  The obtained numerical values for the parameters of the isovector and isoscalar tensor form factors by employing the p-pole fit functions.  | LCSR-I | LCSR-II
---|---|---
| Form factors
---
| ${\cal F}(0)$
---
|  $m_{p}$(GeV)
---
| p
---
${\cal F}(0)$ |  $m_{p}$(GeV) | p
$E_{T}^{I=1}$($Q^{2}$) | $7.54\pm 1.26$ | $1.10\pm 0.05$ | $3.6-4.0$ | $3.48\pm 0.84$ | $1.07\pm 0.07$ | $3.8-4.2$
$E_{T}^{I=0}$($Q^{2}$) | $5.05\pm 1.01$ | $1.13\pm 0.08$ | $3.6-4.0$ | $3.00\pm 0.66$ | $1.14\pm 0.10$ | $3.6-4.0$
$H_{T}^{I=1}$($Q^{2}$) | $5.22\pm 0.27$ | $1.30\pm 0.10$ | $3.0-3.4$ | $1.51\pm 0.20$ | $1.28\pm 0.10$ | $3.0-3.4$
$H_{T}^{I=0}$($Q^{2}$) | $3.37\pm 0.47$ | $1.28\pm 0.11$ | $3.0-3.4$ | $1.10\pm 0.20$ | $1.32\pm 0.10$ | $3.0-3.4$
$\bar{H}_{T}^{I=1}$($Q^{2}$) | $14.51\pm 4.43$ | $1.02\pm 0.10$ | $3.6-4.0$ | $1.30\pm 0.39$ | $1.18\pm 0.14$ | $3.6-4.0$
$\bar{H}_{T}^{I=0}$($Q^{2}$) | $-$ | $-$ | $-$ | $-$ | $-$ | $-$

In summary, we have applied isovector and isoscalar tensor current to evaluate
the tensor form factors of the $N(1535)\rightarrow N$ transition with the help
of the light-cone QCD sum rule method. In numerical computations, we have used
the most general forms of the interpolating current for the nucleon and the
tensor current together with two different sets of the input parameters in the
DAs of the $N(1535)$ state. We have obtained that the values of
$N(1535)\rightarrow N$ transition tensor form factors are very sensitive to
the input parameters of the DAs of the $N(1535)$ state. We have acquired that
the $Q^{2}$ dependence of $N(1535)\rightarrow$ transition tensor form factors
are well defined by a p-pole fit function. To our knowledge, this is the first
study in the literature committed to the examination of the
$N(1535)\rightarrow N$ transition tensor form factors. Thus, experimental data
or theoretical predictions are not yet available to compare our numerical
results with them. A comparison of the results acquired with the estimations
of other theoretical approximations, such as the quark model, chiral
perturbation theory, lattice QCD, etc., would also be interesting.

##  IV Acknowledgments

The author is thankful to K. Azizi for helpful remarks, comments and
discussions.

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(a)  (b)

(c)  (d)

(e)  (f)

Figure 1: The dependence of the isovector tensor form factors of the
$N(1535)\rightarrow N$ transition on $M^{2}$ at $Q^{2}$ = 2.0 GeV2 and
different values of $s_{0}$ and $t$ at their working windows. (a), (c) and (e)
for LCSR-I, and; (b), (d) and (f) for LCSR-II.

(a)  (b)

(c)  (d)

Figure 2: The dependence of the isoscalar tensor form factors of the
$N(1535)\rightarrow N$ transition on $M^{2}$ at $Q^{2}$ = 2.0 GeV2 and
different values of $s_{0}$ and $t$ at their working windows. (a) and (c) for
LCSR-I, and; (b) and (d) for LCSR-II.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 3: The dependence of the isovector tensor form factors of the
$N(1535)\rightarrow N$ transition on $Q^{2}$ at $M^{2}$ = 3.00 GeV2 and
different values of $s_{0}$ and $t$ at their working windows. (a), (c) and (e)
for LCSR-I, and; (b), (d) and (f) for LCSR-II.

(a)  (b)

(c)  (d)

Figure 4: The dependence of the isoscalar tensor form factors of the
$N(1535)\rightarrow N$ transition on $Q^{2}$ at $M^{2}$ = 3.00 GeV2 and
different values of $s_{0}$ and $t$ at their working windows. (a) and (c) for
LCSR-I, and; (b) and (d) for LCSR-II.