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

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.

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

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

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

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

where

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

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

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

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

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

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.

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 :

where

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 :

where

and

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 .

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

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:

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.

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.

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