$\gamma^* N \rightarrow \Delta^{+}(1600)$ transition form factors in light-cone sum rules

The form factors of $\gamma^* N \rightarrow \Delta(1600)$ transition is calculated within the light-cone sum rules assuming that $\Delta^+(1600)$ is the first radial excitation of $\Delta(1232)$. The $Q^2$ dependence of the magnetic dipole $\tilde{G}_M(Q^2)$, electric quadrupole $\tilde{G}_E(Q^2)$, and Coulomb quadrupole $\tilde{G}_c(Q^2)$ form factors are investigated. Moreover, the $Q^2$ dependence of the ratios $R_{EM} = -\frac{\tilde{G}_E(Q^2)}{\tilde{G}_M{Q^2}}$ and $R_{SM} = - \frac{1}{4 m_{\Delta(1600)}^2} \sqrt{4 m_{\Delta(1600)}^2 Q^2 + (m_{\Delta(1600)}^2 - Q^2 - m_N^2)^2} \frac{\tilde{G}_c(Q^2)}{\tilde{G}_M(Q^2)}$ are studied. Finally, our predictions on $\tilde{G}_M(Q^2)$, $\tilde{G}_E(Q^2)$, and $\tilde{G}_C(Q^2)$ are compared with the results of other theoretical approaches.


I. INTRODUCTION
Advance technologies in accelerators enabled to search of high energy regions as well as improving the precision of the experiments by collecting data with high luminosity. This The electroproduction of ∆(1600) is studied within the quark model [1], and the effects of ∆(1600) in baryon-meson reactions is studied in [2,3]. However, the existing data [2,4] can be used for a more detailed analysis of this resonance. The γ * p → ∆ + (1232), (∆ + (1600)) transitions are computed using a diquark-quark picture and a covariant spectator constituent quark model in [5] and [6], respectively.
The form factors of the γ * N → ∆(1232) and γ * octet → decuplet baryon transitions within the same framework was studied in [7] and [8], respectively. And in this work, we study the transition form factors for the electroproduction of the ∆(1600) resonance within light-cone sum sum rules method.
The article is organized as follows. In section II, the sum rules for the transition formfactors of γ * N → ∆(1600) within the light-cone sum rules (LCSR) is derived. The numerical analysis of the obtained LCSRs is carried out in section III. This section also contains discussion and summary. mined in terms of the following form factors [9]: where i = 0, and i = 1 corresponds to the ground and first radial excitation of ∆ baryon, G i 1 , G i 2 , and G i 3 are the corresponding form-factors, and P α = 1 2 (p + p ) α = 1 2 (2p − q) α . However, the multipole form factors are more useful than the form factors G 1 , G 2 , and G 3 for the experimental point of view. The relations among the form factors G 1 (Q 2 ), G 2 (Q 2 ), and G 3 (Q 2 ) and multipole form factors (magnetic dipole G M , electric quadrupole G E , and Coulomb quadrupole G c ) form factors are given in [10]: After these preliminary remarks, we can proceed with the determination of these form factors for γ * N → ∆(1600) transitions within the light-cone QCD sum rules. For this purpose, we consider the following correlator function where η α is the interpolating current with the same quantum numbers of ∆(1232) and ∆(1600), and j el µ is the electromagnetic current. Since ∆ + (1232) and ∆ + (1600) states have the same quantum numbers, the interpolating current for these states is also same and it is given by the following expression where a, b, c are color indices, and C is the charge conjugation operator. According to the sum rules method approach, the correlation function should be calculated in two different regions. In one domain, the correlation function is saturated by the full tower of states carrying the quantum numbers of ∆ baryon in the region p 2 m 2 ∆ i . On the other hand, the correlation function is calculated in the deep Euclidean region where (p + q) 2 << 0 by using the operator product expansion (OPE) in terms of the nucleon distribution amplitudes with an increasing twist. The sum rules for the relevant physical quantities are obtained by matching these results of the representations of the correlation functions via the dispersion relation.
Following the mentioned prescription above and for the hadronic part of the correlation function after isolating the contributions of the ground ∆(1232), and its first radial excitation ∆(1600) state we get where i corresponds to the ground and first excited states. Parameterizing the matrix where u α (p ) is the Rarita-Schwinger spinor and p = p − q. Performing summation over the spins of Rarita-Schwinger spinors with the help of the formula we get the following result for the correlation function At this point, we would like to make the following remark. In general, the interpolating current, η α , interacts not only with spin-3/2 states, but also with the spin-1/2 ones. For the generic spin-1/2 states, the matrix element of the η µ current between the vacuum and spin 1/2 state is determined as In other words, the terms with ∼ γ α and p α contain the contributions of the spin-1/2 states. Comparing Eqs. (8) and (9), it follows that only the terms with ∼ g αβ contains the information of purely spin-3/2 states. Hence, for our problem the terms containing spin-1/2 contributions should be removed. Retaining the contributions of spin-3/2 states only, we in which λ 0 , m 0 (λ 1 , m 1 ) are the residue and mass of the ground state, ∆(1232), ∆(1600) baryons and G i ( G i ) are the form factors for γ * N → ∆(1232) and γ * N → ∆(1600) transitions, respectively. For simplicity, the mass of the ∆(1600) state we will be denote as m 1 from now on.
From Eq. (10), it follows that, for the description γ * N → ∆(1600) transition we have six form factors which should be determined. To determine the six form factors, we need six structures. It should be noted that all structures are not independent. To obtain the independent structures, the ordering procedure of the Dirac matrices is implemented. In this work, we choose the following order of Dirac matrices γ α/ p / qγ µ γ 5 . Taking into account this remark, the correlation function can be decomposed in terms of the following independent invariant functions as follows (see Eq. (8)): From Eqs. (9) and (10), the following six structures are found to determine the six form Solving these equations for the form factors we get From Eq. (13), it follows that to obtain the sum rules for the γ * N → ∆(1600) transition form factors, we need to know the invariant functions Π i . According to the sum rules methodology, the invariant functions Π i (i = 1 ÷ 6) are calculated at deep Euclidean domain with virtuality p 2 = (p − q) 2 << 0 in terms of the nucleon distribution amplitudes (DA's).
The nucleon DA's are the main non-perturbative ingredients of LCSR and they are calculated up to twist-6 in [11]. For completeness, definition of the nucleon's DA's and their expressions are presented in Appendix.
Using the expressions of the nucleon DA's and applying the quark-hadron duality ansatz, the invariant functions, Π i , can be written in the following form Matching the representations of the correlation functions and performing Borel transformations with respect to the variable −(p − q) 2 in order to suppress the contributions of higher states and continuum, the corresponding sum rules for the form-factors G 1 (Q 2 ), G 2 (Q 2 ) and The functions I i (M 2 , Q 2 , s 0 ) can be written in the form of a master formula (see [12] and [13]) , and x 0 is the solution of s 0 = s equation. The explicit forms of the functions, ρ n i are presented in the Appendix. From Eq. (13), we see that to determine the γ * N → ∆(1600) transition form factors, the residue of ∆(1600) is also needed. This value within QCD sum rules is already calculated in [14] and obtained as λ 1 = (0.057 ± 0.016) GeV 3 .
At the end of this section, we present the ratios R EM [10] and R SM [11] that are more suitable for the experimental point of view .
It should be noted that these ratios are identically zero in SU (6)  in [11] (see also [15,16]). In addition to these input parameters, the sum rules contain two Having specified all the input parameters and determined the working region of M 2 and s 0 , we are ready to perform the numerical calculations.
In Figures, 2,  By comparing the form factors of γ * N → ∆(1232) obtained in [7] and γ * N → ∆(1600) transitions, we infer the following results : • The electric quadrupole form factor is very small in magnitude compared to the form factors,G M (Q 2 ) andG c (Q 2 ) in both transitions.
Moreover, we also compared our results on the considered form factors with the predictions obtained by quark-diquark approximation to the Poincare-covariant three-body bound state problem in relativistic quantum field theory [5] and found out that our predictions on the form factors at the considered regions of Q 2 is around 2 times larger in magnitude than the one predicted in [5].
Furthermore in Figures 5 and 6, we present the Q 2 dependence of R EM (Q 2 ) and R SM at fixed values M 2 and s 0 considering their working regions. From these figures, we observe that while R SM (Q 2 ) is negative, R EM (Q 2 ) is positive at all values of Q 2 .
Comparing our predictions on R SM with the results obtained in [7], we observed similar qualitative behaviour considered in both works. However, behavior of R EM in our case is remarkably different than in γ * N → ∆(1232) transition, i.e. magnitude R EM is larger than the one in γ * N → ∆(1232) transition case. Finally, we compare our predictions on R SM and R EM with the results obtained within light-front relativistic quark model [17]. From Fig. 5, it follows that when Q 2 varies in the region Q 2 = 5 GeV 2 to 12 GeV 2 , R EM practically does not change and the value is around 0.22. However, the prediction of [17] on R EM grows from 0.06 to 0.1. In other words, our result on R EM is larger than the predicted in [17]. Besides, comparing our result on R SM , we deduce that with increasing Q 2 , R SM in our case grows mild, but Q 2 dependence is considerable [17]. For example, in our case, when Q 2 varies between 2 to 10 GeV 2 region, the R SM varies between (0.20 − 0.25), however it changes between (0.1 − 0.3) in [17].
Our final note is that the obtained results will shed light to the understanding the inner structures of the resonance ∆(1600), and can be checked in ongoing and planning experiments.

IV. CONCLUSION
In this article, we studied the LCSR to evaluate the magnetic dipoleG M (Q 2 ) electric quadrupoleG E (Q 2 ) and Coulomb quadrupoleG * (Q 2 ) form factors as well as the ratios on Q 2 when Q 2 varies in the region 1 GeV 2 ≤ Q 2 ≤ 10 GeV 2 . This domain may be covered in the incoming CLAS-12 at the Jefferson Lab. Appearance of experimental information would be very useful to establish the nature of ∆ + (1600) resonance by assuming it as radial excitation of ∆(1232) in the γ * N → ∆(1600) transition. We also compared our predictions on the form factors      In this Appendix we present the explicit expressions of the functions ρ n i entering to the sum rules for the form factors G 1 (Q 2 ), G 2 (Q 2 ) and G 3 (Q 2 ) for the γ * N → ∆(1600) transition. (1) (2) 6 (x) = −8e q 2 m 2 N B 6 (x) − 16e q 3 m 2 N (1 − x) C 6 (x) where q 1 = u, q 2 = u, and q 3 = d, respectively.
In the above expressions for ρ 2 , ρ 4 , and ρ 6 the functions F(x i ) are defined in the following way:F (x 1 ) = Definitions of the functions B i , C i , D i , E 1 and H 1 that appear in the expressions for ρ i (x) are given as follows: The forms of these functions can be found in [11].