$\gamma N \to N^*(1535)$ transition in soft-wall AdS/QCD

We present a study of the $N^*(1535)$ resonance electroexcitation in a soft-wall AdS/QCD model. Both the transverse $A_{1/2}^p$ and longitudinal $S_{1/2}^p$ helicity amplitudes are calculated resulting in good agreement with data and with the MAID parametrization.

coupling between the nucleon and the N * (1535) is possible for equal twists, which means that the leading minimal electromagnetic coupling between nucleon and N * (1535) occurs for τ N = τ N * (1535) = 4. Inclusion of this particular coupling helps to improve the description of both helicity amplitudes A 1/2 and S 1/2 at low Q 2 .
The paper is organized as follows. In Sec. II we briefly discuss our formalism. In Sec. III we present the analytical calculation and the numerical analysis of electromagnetic form factors and helicity amplitudes of the nucleon-N (1535) transition. Finally, Sec. IV contains our summary and conclusions.

II. FORMALISM
In this section we briefly review our approach [10,13,27,28]. We start with the definition of the conformal Poincaré metric where ǫ a M = δ a M /z is the vielbein, and we define g = |det(g MN )| = 1/z 10 as the magnitude of the determinant of g MN . The soft-wall AdS/QCD action S for the nucleon N = (p, n) and the N * (1535) resonance [in the following we use the notation N * = (N * p , N * n )] including photons is constructed in terms of the dual spin-1/2 fermion and vector fields. These fields have constrained (confined) dynamics in AdS space due to the presence of a background field -dilaton field ϕ(z) = κ 2 z 2 , where κ is its scale parameter.
The action S contains a free part S 0 , describing the confined dynamics of AdS fields, and an interaction part S int , describing the interactions of fermions with the vector field [below, for simplicity, we only display the coupling of N and N * (1535) to the vector field] L N , L N * , L V and L V N N * are the free and interaction Lagrangians, respectively, and are written as Here τ runs from 3. We thereby introduce the shortened notation where µ is the five-dimensional mass of the spin-1 2 AdS fermion with µ = 3/2+L (L is the orbital angular momentum); U F (z) = ϕ(z) is the dilaton potential; Q = diag(1, 0) is the nucleon (N * ) charge matrix; V MN = ∂ M V N − ∂ N V M is the stress tensor for the vector field; ω ab M = (δ a M δ b z − δ b M δ a z )/z is the spin connection term; σ MN = [Γ M , Γ N ] is the commutator of the Dirac matrices in AdS space, which are defined as Γ M = ǫ M a Γ a and Γ a = (γ µ , −iγ 5 ). The subscripts m and nm in the vector matrixV N * N ±,m/nm (x, z) refer to minimal (m) or nonminimal (nm) couplings, respectively. As was pointed out in the Introduction, due to gauge invariance the minimal coupling between the nucleon and the N * is only possible for the same twist, while for the nonminimal coupling it is not constrained and we include the coupling between nucleon with twist τ N = 3, 4, 5 and the N * (1535) resonance with the twist τ N * = τ N + 1 = 4, 5, 6.
The action (2) is constructed in terms of the 5D AdS fermion fields ψ N ±,τ (x, z), ψ N * ±,τ (x, z) and the vector field V M (x, z). Fermion fields are duals to the left-and right-handed chiral doublets of the nucleon and the N * (1535) These fields are in the fundamental representations of the chiral SU L (2) and SU R (2) subgroups and are holographic analogs of the nucleon N and N * (1535) resonance, respectively.
The 5D AdS fields ψ B ±,τ (x, z) are products of the left/right 4D spinor fields for the nucleon, and for the N * (1535) resonance with spin 1/2, and the bulk profiles with twist τ , which depend on the holographic (scale) variable z: where The nucleon is identified as the ground state with n = L = 0 while the N * (1535) resonance as the first orbitally excited state with n = 0 and L = 1. In the case of the vector field we work in the axial gauge V z = 0 and perform a Fourier transformation of the vector field V µ (x, z) with respect to the Minkowski coordinate We can then derive an equation of motion for the vector bulk-to-boundary propagator V (q, z) dual to the q 2dependent electromagnetic current The solution of this equation in terms of the gamma Γ(n) and Tricomi U (a, b, z) functions reads In the Euclidean region (Q 2 = −q 2 > 0) it is convenient to use the integral representation for V (Q, z) [29] V (Q, z) = κ 2 z 2 where x is the light-cone momentum fraction and a = Q 2 /(4κ 2 ). The set of parameters c N τ , c N * τ +1 , c N * N τ +1 , and d N * N τ induces mixing of the contributions of AdS fields with different twist dimensions. In Refs. [10,28] we showed that the parameters c N τ and c N * τ +1 are constrained by the conditions τ c N τ = 1 and τ c N * τ +1 = 1 in order to get the correct normalization of the kinetic termψ(x)i ∂ψ(x) of the four-dimensional spinor field. This condition is also consistent with electromagnetic gauge invariance. Therefore, the nucleon and N * (1535) masses can be identified with following the expressions [10,28] As in the previous case of the nucleon and the Roper resonance we restrict our calculation to the three leading twist contributions to the N * The electromagnetic form factors of the γN → N * (1535) transition are defined, due to Lorentz and gauge invariance, by the following matrix element We have u N * (p 1 λ 1 ) and u N (p 2 λ 2 ) which are the usual spin-1 2 Dirac spinors describing the N * (1535) resonance and nucleon, M ± = M N * ± M N , γ µ ⊥ = γ µ − q µ q/q 2 , q = p 1 − p 2 , and λ 1 , λ 2 , and λ are the helicities of the final, initial baryon and photon, respectively, with the relation λ 2 = λ 1 − λ. In the rest frame of the N * (1535) the four momenta of N * , N , photon and the polarization vector of photon are specified as: is the absolute value of the three-momentum of the nucleon or the photon. It is important to point out that the matrix element (16) is manifestly gauge invariant. The form factor F N * N 1 (Q 2 ) vanishes at Q 2 = 0. In more detail the contribution to F N * N 1 (Q 2 ) from nonminimal terms of the action (2) includes the z-derivative acting on the vector bulk-to-boundary propagator ∂ z V (Q, z), which is zero at Q 2 = 0 because of V (0, z) ≡ 1. In the case of the minimal term its contribution to the F N * N 1 (Q 2 ) reads: where a = Q 2 /(4κ 2 ), B(x, y) = Γ(x)Γ(y)/Γ(x + y) and Γ(x) are the beta and gamma functions. It should be clear that F N * N 1,m (0) = 0 at Q 2 = 0. Next we introduce the helicity amplitudes H λ1λ which are related to the invariant form factors F RN i as (see details in Refs. [31,32]) A straightforward evaluation gives [1,2,4,31,32] where Q ± = M 2 ± +Q 2 . In the case of the Roper-nucleon transition we also have an additional set of helicity amplitudes (A 1/2 , S 1/2 ) related to the (H 1 2 0 , H 1 2 1 ) by and α = 1/137.036 is the fine-structure constant. At Q 2 = 0 our predictions for the last set of helicity amplitudes (proton channel) in the N − N * (1535) transition are Our results for the Q 2 dependence of the helicity amplitudes in the N − N * (1535) transition (proton channel) are fully displayed in the left panels in Figs. 1 and 2. We compare them to experimental results of the CLAS Collaboration (JLab) [3][4][5] and to the MAID parametrization [18] We further display a parametrization proposed by us: where A p 1/2 (0) = 0.090 GeV −1/2 , S p 1/2 (0) = −0.002 GeV −1/2 , We also present an analysis of the error of our approach due to a variation of the parameters (up to 15%) for both helicity amplitudes in the right panels of Figs. 1 and 2.

IV. SUMMARY
We extended our formalism based on a soft-wall AdS/QCD approach to the description of the γN → N * (1535) transition. We showed that inclusion of the minimal electromagnetic coupling of the nucleon and the N * (1535) resonance, based on the coupling of two fermion AdS fields with the same twist-dimension, is manifestly gauge invariant. It then results in a satisfactory description of data on the helicity amplitudes even at small Q 2 . The failure to reproduce the low-Q 2 behavior of these helicity amplitudes was a long-standing problem of most theory descriptions, the present soft-wall AdS/QCD approach can offer a solution. In the future we plan to apply our formalism to the calculation of electromagnetic transitions between the nucleon and further high-spin resonances.