Nucleon resonances with higher spins in soft-wall AdS/QCD

We present a study of electroexcitation of nucleon resonances with higher spins, in a soft-wall AdS/QCD model, comparing our results with existing data from the CLAS Collaboration at JLab, from MAMI, and other experiments.

In the past decade significant progress in the study of nucleon resonances has been achieved through the soft-wall AdS/QCD [47]- [53] formalism. For example, AdS/QCD is able to study the electromagnetic structure of nucleon and nucleon resonances in the whole region of Euclidean momentum squared Q 2 , and in particular, soft-wall AdS/QCD provides the correct power scaling description of form factors and helicity amplitudes of all hadrons at large Q 2 [55], while it is also able to give good agreement with data at low and intermediate Q 2 . In Refs. [47]- [53] soft-wall AdS/QCD was focused on the study of form factors and helicity amplitudes of electroexcitations of the Roper N (1440) (first radial excitation of the nucleon) and the negative-parity state N * (1535). In particular, in Refs. [48,[51][52][53][54] we proposed and developed a soft-wall AdS/QCD framework for the study of nucleon resonances with adjustable quantum numbers and successfully applied it to the unified description of electromagnetic structure of three states -nucleon, Roper, and N * (1535). In the present manuscript we apply this theoretical approach for the study of the electromagnetic structure of nucleon resonances with higher spins.
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 transitions between nucleon and nucleon resonances. Finally, Sec. IV contains our summary.

II. FORMALISM
In this section we discuss the basic principles of our approach [56]- [62] and focus on its application to nucleon resonances [48, 51-53, 56, 58] with higher spins. First, we define the conformal Poincaré metric, which is used in our formalism: where ǫ a M (z) = δ a M /z is the vielbein, g = |det(g MN )| = 1/z 10 . Next we discuss the construction of the effective action in terms of the 5D AdS fermion fields ψ i,τ , ψ MM1...M l−1 i,τ and the vector field V M (x, z), where τ = N + L is the twist, defined as the number of partons plus angular orbital momentum. The vector field is dual to the electromagnetic field, while the fermion fields are duals to the left-and right-handed chiral doublets of the nucleon and the nucleon resonances, with p and B 2 = n, N * n . These AdS 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 * resonance, respectively. They have constrained (confined) dynamics in AdS space, due to the presence of dilaton field ϕ(z) = κ 2 z 2 , where κ is its scale parameter. The action S for the description of electroexcitations of nucleon resonances contains a free part S 0 , describing the dynamics of AdS fields, and an interaction part S int , describing the interactions of fermions with the vector field dual to electromagnetic field where L N , L N * , L V , and L V N N * are the free and interaction Lagrangians, respectively, given by Here τ and τ * are the twists of both the nucleon and nucleon resonance, which run from their minimal possible value. We have introduced the following shortened notationŝ τ τ * and J where and Here 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 charge matrix corresponding to doublets of nucleon (nucleon resonances); 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; while Γ M = ǫ M a Γ a and Γ a = (γ µ , −iγ 5 ) are the Dirac matrices in AdS space, [Γ a , Γ b ] = Γ a Γ b − Γ b Γ a . Next we split 5D AdS fermion fields ψ ±,τ (x, z) and ψ MM1...M l−1 ±,τ (x, z) into left-and right-chirality components for the nucleon, and for the nucleon resonances with higher spins and perform the Kaluza-Klein expansion as: where n is the radial quantum number and Here are the bulk profiles with twist τ and radial quantum number n, which depend on the holographic variable z, where and L m n (x) are the generalized Laguerre polynomials. The bulk profiles f L/R τ (z) are normalized as The nucleon is identified as the ground state with n = L = 0, while the nucleon resonance have specific values of n and L. In Table I we display the quantum numbers (spin-parity J P , angular orbital moment L, radial quantum number n, mass) of the baryons considered in the present paper. The action describing transitions 1 2 + γ * → 1 2 ± has been derived and discussed in detail in Refs. [48,[51][52][53]. In Appendix A we briefly specify this action.
For the vector field V µ (x, z) we apply the axial gauge V z = 0 and perform a Fourier transformation with respect to the Minkowski coordinate where V (q, z) is the vector bulk-to-boundary (dual to the q 2 -dependent electromagnetic current) obeying the equation of motion with solution in terms of gamma Γ(n) and Tricomi U (a, b, z) functions In was shown in Ref. [63] that in the Euclidean region ( The sets of parameters c τ , c τ * , and g (iM) τ τ * induce mixing of the contributions of AdS fields with different twist dimensions. The parameters c τ and c τ * are constrained by the conditions τ c τ = 1 and τ c τ * = 1, to guarantee the correct normalization of the kinetic termsψ(x)i ∂ψ(x) of the four-dimensional spinor fields. This condition is also consistent with electromagnetic gauge invariance (see details in Refs. [48,58]). Therefore, the masses of the nucleon and nucleon resonance are identified by the expressions [48,58] where the leading twist from which the sums start in Eq. (20) is defined as τ = 3 + L, where L is the angular orbital moment specified for baryons in Table I. The baryon form factors are determined analytically using the bulk profiles of fermion fields and the bulk-toboundary propagator V (Q, z) of the vector field (for exact expressions see the next section). The calculational technique was already described in detail in Refs. [48,51,58]. The parameter κ = 383 MeV is universal and was fixed in previous studies (see, e.g., Refs. [48,58]), while the other parameters are fixed from a fit to the helicity amplitudes of the γ * N → N * transitions.

III. ELECTROMAGNETIC FORM FACTORS AND HELICITY AMPLITUDES OF THE
Due to Lorenz covariance and gauge invariance, the matrix elements of the electromagnetic γ * N → N * transitions can be expressed in terms of their general Lorenz structures as and the relativistic form factors G i (Q 2 ), i = 1, 2, 3 as [1-3, 66, 67] Here u N (p 2 λ 2 ) and u νν1...ν l−1 N * (p 1 λ 1 ) are spin-1 2 and higher spin (Rarita-Schwinger) spinors, respectively. The Rarita-Schwinger spinor satisfies the conditions 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 In the rest frame of the N * 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 convenient to introduce the helicity amplitudes A 1/2 , A 3/2 , and S 1/2 responsible for the helicity transitions where where α = 1/137.036 is the fine-structure constant. The relations expressing the relativistic form factors G i through the set of form factors h i and helicity amplitudes read: The structure of the 1 2 + γ * → 1 2 ± is simpler and is given by the form where γ µ ⊥ = γ µ − q µ q/q 2 . The helicity amplitudes defining the 1 2 + → 1 2 + and 1 2 + → 1 2 − transitions in terms of form factors are defined, respectively, as: In the case of the high-spin resonances, the set of helicity amplitudes (A 1/2 , A 3/2 , S 1/2 ) is related with the set of the charge (G E ), magnetic (G M ), and Coulomb (G Q ) form factors [1][2][3]: for normal parity transitions In terms of the relativistic form factor h i , the charge, magnetic, and Coulomb form factors are expressed as for abnormal parity transitions and for normal parity transitions. Here The form factors G (n) i (Q 2 ) (here n is the radial quantum number), defining the abnormal parity transitions, are given by where F i±τ τ * (Q 2 ) are written as and For n = 0 one gets L(a, τ * , τ ) = Γ τ * + τ + 2 2 B a + 1, M (a, τ * , τ ) = Γ τ * + τ + 1 2 B a + 1, τ * + τ + 1 2 , N (a, τ * , τ ) = K(a, τ * , τ ) − 2K(a, τ * + 2, τ ) . Our results are shown with a variation of the parameters of our approach (shaded band), and comparing with data taken from the CLAS Collaboration [68]. Here and in the following superscript (p) in the notation of the helicity amplitudes means the proton channel. Our results are shown with a variation of the parameters of our approach (shaded band), comparing with data taken from the CLAS Collaboration [69,70] and compilation of the world analyses of the N π electroproduction data [81]. Our results (shaded band) are compared with data taken from the CLAS Collaboration [6,69], a compilation of the world analyses of N π electroproduction data [81] and Particle Data Group (PDG) [64]. , and S p 1/2 (Q 2 ) (centered lower panel), for N γ * → N (1520) transition up to Q 2 = 10 GeV 2 . Our results (shaded band) are compared with data taken from the CLAS Collaboration [4,6,71,72], a compilation of data [81], and PDG [64]. , and S p 1/2 (Q 2 ) (centered lower panel), for N γ * → ∆(1700) transition up to Q 2 = 10 GeV 2 . Our results (shaded band) are compared with data taken from the CLAS Collaboration [70,71], a compilation of data [81], and PDG [64]. Our results (shaded band) are compared with data taken from the CLAS Collaboration [68,71] and PDG [64]. , and S p 1/2 (Q 2 ) (centered lower panel), for N γ * → N (1720) transition up to Q 2 = 10 GeV 2 . Our results (shaded band) are compared with data taken from the CLAS Collaboration [65] and PDG [64]. , and S p 1/2 (Q 2 ) (centered lower panel), for N γ * → N ′ (1720) transition up to Q 2 = 10 GeV 2 . Our results (shaded band) are compared with data taken from the CLAS Collaboration [65] and PDG [64]. Our results (shaded band) are compared with data taken from the CLAS Collaboration [68,70,71], data analisys [81] and PDG [64]. , and S p 1/2 (Q 2 ) (centered lower panel), for N γ * → ∆(1232) transition up to Q 2 = 10 GeV 2 . Our results (shaded band) are compared with data taken from the CLAS Collaboration [4] and PDG [64]. Our results are compared with data taken from PDG [64] and Refs. [77][78][79]. Our results are compared with data taken from PDG [64] and Refs. [77][78][79].
It is important to stress that at large values of Q 2 the form factors and helicity amplitudes for the electroexcitation of nucleon resonances are consistent with quark counting rules [55]. In particular, the sets of the form factors h i , G i , (G E , G M , G C ) and helicity amplitudes (A 1/2 , A 3/2 , S 1/2 ) scale as Model parameters (central values) used for each γ * N → N * transition are shown in Tables II and III. As in previous calculations we include the contributions of three leading twists. Also, to reduce a number of free parameters we drop the contribution to the form factors induced by the couplings g 3M τ τ * and g 4M τ τ * .   Our results for the Q 2 dependence of the helicity amplitudes in the γ * N → N * transitions including a variation of the parameters (up to 20%) are fully displayed in Figs. 1-11. In Figs. 1-3 we present the results for the modes with nucleon resonances having spin 1 2 , which were not considered by us before and in addition in Figs. 4-14 we display the results for the nucleon resonances with higher spins 3 2 and 5 2 . We compare our reuslts to data from the CLAS Collaboration (JLab) [4][5][6][68][69][70][71][72], other experiments [73][74][75][76][77][78][79][80], and world data analyses [81][82][83]. Also we consider in detail the observables of the γ * N → ∆(1232) transitions: helicity amplitudes (Fig. 11), the Q 2 dependence of the magnetic form factor G * M (Q 2 ) divided by the dipole form factor 3D(Q 2 ) (Fig. 12), where D(Q 2 ) = 1/(1 + Q 2 /0.71 GeV 2 ) 2 , the Q 2 dependence of the R EM = E/M and R SM = S/M ratios (Fig. 13 up to 5 GeV 2 and Fig. 14 up to 10 GeV 2 ) magnetic dipole µ N ∆ and electric quadrupole Q N ∆ moments: Note the magnetic G * M (Q 2 ) and electric G * E (Q 2 ) form factors are normalized as [82]: where In Table IV our results for R EM (0) are compared with existing data (PDG [64], MAMI experiment [77], LEGS Collaboration [80]) and some theoretical approaches [model bases on partial-Wave analysis (SAID) [84], approach based on dispersion relations and unitarity (DR) [85] and relativistic quark model (RQM) [22]]. For R SM (0) we get −5.5 ± 0.5. Our predictions for the moments µ N ∆ and Q N ∆ µ N ∆ = 3.7 ± 0.4 , Q N ∆ = −(0.09 ± 0.01) fm 2 .

IV. SUMMARY
We extended our formalism based on a soft-wall AdS/QCD approach to the description of the electro-couplings of nucleons with nucleon resonances with high spins. All form factors and helicity amplitudes characterizing the electromagnetic transitions between nucleons and nucleon resonances are consistent with quark counting rules [55]. We fix free parameters in our approach using data from the CLAS Collaboration [4][5][6][68][69][70][71][72] and a compilation of the world analyses of the N π electroproduction data [81]. In our calculations we adopt a variation of free parameters up to 20%. The main success of our approach is based on analytical implementation of quark counting rules [55].