Electromagnetic structure of nucleon and Roper in soft-wall AdS/QCD
aa r X i v : . [ h e p - ph ] D ec Electromagnetic structure of nucleon and Roper in soft-wall AdS/QCD
Thomas Gutsche, Valery E. Lyubovitskij,
1, 2, 3, 4 and Ivan Schmidt Institut f¨ur Theoretische Physik, Universit¨at T¨ubingen,Kepler Center for Astro and Particle Physics, Auf der Morgenstelle 14, D-72076 T¨ubingen, Germany Departamento de F´ısica y Centro Cient´ıfico Tecnol´ogico de Valpara´ıso-CCTVal,Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile Department of Physics, Tomsk State University, 634050 Tomsk, Russia Laboratory of Particle Physics, Tomsk Polytechnic University, 634050 Tomsk, Russia (Dated: December 27, 2017)We present an improved study of the electromagnetic form factors of the nucleon and of theRoper-nucleon transition using an extended version of the effective action of soft-wall AdS/QCD.We include novel contribution from additional non-minimal terms, which do not renormalize thecharge and do not change the normalization of the corresponding form factors, but the inclusion ofthese terms results in an important contribution to the momentum dependence of the form factorsand helicity amplitudes.
PACS numbers: 12.38.Lg, 13.40.Gp, 14.20.Dh, 14.20.GkKeywords: nucleons, Roper resonance, AdS/QCD, form factors
I. INTRODUCTION
Originally the soft-wall AdS/QCD action for the nucleon was proposed in Ref. [1]. It included a term describ-ing the nucleon confining dynamics and the electromagnetic field, and their minimal and non-minimal couplings Q N = diag(1 ,
0) (nucleon charge matrix) and η N = diag( η p , η n ) (nucleon matrix of anomalous magnetic moments),respectively. The use of the non-minimal couplings is essential to generate the Pauli spin-flip form factors. Later, inRef. [2], this action was used for the calculation of generalized parton distributions of the nucleon. In Ref. [3] it wasextended to take into account higher Fock states in the nucleon and additional couplings with the electromagneticfield in consistency with QCD constituent counting rules [4] for the power scaling of hadronic form factors at largevalues of the momentum transfer squared in the Euclidean region. In Ref. [5] soft-wall AdS/QCD was developed forthe description of baryons with adjustable quantum numbers n , J , L , and S . In another development, in Refs. [6]-[8],the nucleon properties have been analyzed using a Hamiltonian formalism. However, their calculation of the nucleonelectromagnetic properties ignored the contribution of the non-minimal coupling to the Dirac form factors, and there-fore, the analysis done in Refs. [6]-[8], is in our opinion not fully consistent. In Ref.[7] the ideas of Ref. [6] has beenextended by the inclusion of higher Fock states in the nucleon, in order to calculate nucleon electromagnetic formfactors in light-front holographic QCD. In this paper the Pauli form factor is again introduced by hand, using theoverlap of the L = 0 and L = 1 nucleon wave function. Additionally, the expression for the neutron Dirac form factorhas been multiplied by hand by a free parameter r .In a series of papers [9]-[11] we have developed a light-front quark-diquark approach for the nucleon structure,describing nucleon parton distributions and form factors from a unified point of view. In particular, in a recentpaper [11] we derived nucleon light-front wave functions, analytically matching the results of global fits to the quarkparton distributions in the nucleon at the initial scale µ ∼ q v ( x ) and δq v ( x ). In the context of the nucleon form factorsit is also important to mention a recent paper [13], where the γ ∗ → ρ transition form factor has been calculated insoft-wall AdS/QCD. Here it was shown that the form factor is consistent with quark counting rules for differentialcross sections with single and double vector meson production. It scales as 1 / p Q and, therefore, it contributes tothe electromagnetic form factors of the nucleons at subleading order.The Roper-nucleon transition form factors and helicity amplitudes can be also discussed within this same formalism.The Roper resonance was first considered in the context of AdS/QCD in Ref. [14], where the Dirac form factor forthe electromagnetic nucleon-Roper transition was calculated in light-front holographic QCD. Later, in Ref. [15], theformalism for the study of nucleon resonances in soft-wall AdS/QCD has been developed, and the first application fora detailed description of Roper-nucleon transition properties (form factors, helicity amplitudes and transition chargeradii) was performed. In Ref. [16, 17] the formalism proposed in [15] was used, with a different set of parameters.An overview of the application of other theoretical approaches can be found in Refs. [15, 18]. This includes recentnovel ideas about considering additional degrees of freedom for this state, such as a molecular nucleon-scalar σ mesoncomponent [19, 20], for a realistic description of current data on Roper electroproduction performed by the CLASCollaboration at JLab [21]-[23].In the present paper we include additional non-minimal couplings of the vector field (dual to the electromagneticfield) with fermions (dual to the nucleon and Roper). Such terms do not renormalize the charge, but gives animportant contribution to the momentum dependence of the nucleon and Roper-nucleon transition form factors(helicity amplitudes). The inclusion of these terms helps to improve the description of data. The paper is organizedas follows. In Sec. II we briefly discuss our formalism. In Sec. III we present the analytical calculation and thenumerical analysis of electromagnetic form factors and helicity amplitudes of the nucleon and the Roper. Finally,Sec. IV contains our summary and conclusions. II. FORMALISM
In this section we briefly review our approach. We start with the underlying action for the study of the nucleon N = ( p, n ) and Roper R = ( R p , R n ) resonance, extended by the inclusion of photons. It is constructed in terms ofthe 5D AdS fields ψ N ± ,τ ( x, z ) and ψ R± ,τ ( x, z ), which are duals to the left- and right-handed chiral doublets of nucleons(Roper resonances) O L = ( B L , B L ) T and O R = ( B R , B R ) T with B = p, R p and B = n, R n . These fields are in thefundamental representations of the chiral SU L (2) and SU R (2) subgroups and are holographic analogues of the nucleon N and Roper resonance R , respectively. The 5D AdS fields ψ B ± ,τ ( x, z ) are products of the left/right 4D spinor fields ψ L/Rn =0 , ( x ) = 1 ∓ γ ψ n =0 , ( x ) , (1)with spin 1 / F L/Rτ,n =0 , ( z ) = z f L/Rτ,n =0 , ( z ) with twist τ depending on the holographic (scale)variable z : ψ N ± ,τ ( x, z ) = 1 √ h ψ L ( x ) F L/Rτ, ( z ) ± ψ R ( x ) F R/Lτ, ( z ) i ,ψ R± ,τ ( x, z ) = 1 √ h ψ L ( x ) F L/Rτ, ( z ) ± ψ R ( x ) F R/Lτ, ( z ) i , (2)where f Lτ, = s τ ) κ τ z τ − / e − κ z / ,f Rτ, = s τ − κ τ − z τ − / e − κ z / ,f Lτ, = s τ + 1) κ τ z τ − / ( τ − κ z ) e − κ z / ,f Rτ, = s τ ) κ τ − z τ − / ( τ − − κ z ) e − κ z / . (3)Here the nucleon is identified as the ground state with n = 0 and the Roper resonance as the first radially excited statewith n = 1. We also include the vector field V M ( x, z ), dual to the electromagnetic 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 V µ ( x, z ) = Z d q (2 π ) e − iqx V µ ( q ) V ( q, z ) . (4)We derive an EOM for the vector bulk-to-boundary propagator V ( q, z ) dual to the q -dependent electromagneticcurrent ∂ z (cid:18) e − ϕ ( z ) z ∂ z V ( q, z ) (cid:19) + q e − ϕ ( z ) z V ( q, z ) = 0 , (5)where ϕ ( z ) = κ z is the dilaton field with its scale parameter κ , which is varied from 380 to 500 MeV in differentfits to hadron data.The solution of this equation in terms of the gamma Γ( n ) and Tricomi U ( a, b, z ) functions reads V ( q, z ) = Γ (cid:16) − q κ (cid:17) U (cid:16) − q κ , , κ z (cid:17) . (6)In the Euclidean region ( Q = − q >
0) it is convenient to use the integral representation for V ( Q, z ) [24] V ( Q, z ) = κ z Z dx (1 − x ) x a e − κ z x − x , (7)where x is the light-cone momentum fraction and a = Q / (4 κ ).The action contains a free part S , describing the confined dynamics of nucleon, Roper and the electromagneticfield in AdS space, and an electromagnetic interaction part S int with S = S + S int ,S = Z d xdz √ g e − ϕ ( z ) (cid:26) L N ( x, z ) + L R ( x, z ) + L V ( x, z ) (cid:27) ,S int = Z d xdz √ g e − ϕ ( z ) (cid:26) L V NN ( x, z ) + L V RR ( x, z ) + L V R N ( x, z ) (cid:27) . (8) L N , L R , L V ( x, z ) and L V NN ( x, z ), L V RR ( x, z ), L V R N ( x, z ) are the free and interaction Lagrangians, respectively,and are written as L B ( x, z ) = X i =+ , − ; τ c Bτ ¯ ψ Bi,τ ( x, z ) ˆ D i ( z ) ψ Bi,τ ( x, z ) , L V ( x, z ) = − V MN ( x, z ) V MN ( x, z ) , L V BB ( x, z ) = X i =+ , − ; τ c Bτ ¯ ψ Bi,τ ( x, z ) ˆ V Bi ( x, z ) ψ Bi,τ ( x, z ) , L V R N ( x, z ) = X i =+ , − ; τ c R Nτ ¯ ψ R i,τ ( x, z ) ˆ V R Ni ( x, z ) ψ Ni,τ ( x, z ) + H . c . , (9)where B = N, R andˆ D ± ( z ) = i M ↔ ∂ M − i M ω abM [Γ a , Γ b ] ∓ ( µ + U F ( z )) , ˆ V H ± ( x, z ) = Q Γ M V M ( x, z ) ± i η HV [Γ M , Γ N ] V MN ( x, z ) ± i λ HV z [Γ M , Γ N ] ∂ K ∂ K V MN ( x, z ) ± g HV Γ M i Γ z V M ( x, z ) + ζ HV z Γ M ∂ N V MN ( x, z ) ± ξ HV z Γ M i Γ z ∂ N V MN ( x, z ) , H = N, R , R N . (10)The set of parameters c Nτ , c R τ , and c R Nτ induce mixing of the contribution of AdS fields with different twist dimension.In Refs. [3, 15] we showed that the parameters c Bτ are constrained by the condition P τ c Bτ = 1 in order to getthe correct normalization of the kinetic term ¯ ψ n ( x ) i ∂ψ n ( x ) of the four-dimensional spinor field. This condition isalso consistent with electromagnetic gauge invariance. The couplings η HV = diag( η H V , η H V ), λ HV = diag( λ H V , λ H V ), g HV = diag( g H V , g H V ), ζ HV = diag( ζ H V , ζ H V ), and ξ HV = diag( ξ H V , ξ H V ), where H = p, R p , R p p and H = n, R n , R n n are fixed from the magnetic moments, slopes, and form factors of both the nucleon and Roper, while the couplings c R Nτ are fixed from the normalization of the Roper-nucleon helicity amplitudes. The terms proportional to the couplings λ HV , ζ HV , and ξ HV express novel nonminimal couplings of the fermions with the vector field. It does not renormalizethe charge and does not change the corresponding form factor normalizations, but gives an important contributionto the momentum dependence of the form factors and helicity amplitudes.We use the conformal metric g MN x M x N = ǫ aM ǫ bN η ab x M x N = ( dx µ dx µ − dz ) /z ; ǫ aM = δ aM /z is the vielbein; √ g = 1 /z . Here µ is the five-dimensional mass of the spin- AdS fermion µ = 3 / L , with L being the orbitalangular momentum; U F ( z ) = ϕ ( z ) is the dilaton potential; Q = diag(1 ,
0) is the nucleon (Roper) charge matrix; V MN = ∂ M V N − ∂ N V M is the stress tensor for the vector field; ω abM = ( δ aM δ bz − δ bM δ az ) /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 Ma Γ a andΓ a = ( γ µ , − iγ ).The nucleon and Roper masses are identified with the expressions [3, 15] M N = 2 κ X τ c Nτ √ τ − , M R = 2 κ X τ c R τ √ τ . (11)As we mentioned the set of mixing parameters c N, R τ is constrained by the correct normalization of the kinetic termof the four-dimensional spinor field and by charge conservation as (see detail in Ref. [3]): X τ c N, R τ = 1 . (12)Baryon form factors are calculated analytically using bulk profiles of fermion fields and the bulk-to-boundary prop-agator V ( Q, z ) of the vector field (see exact expressions in the next section). Calculation technique is discussed indetail in Refs. [3, 15].
III. ELECTROMAGNETIC FORM FACTORS OF NUCLEON, ROPER AND ROPER-NUCLEONTRANSITIONS
The electromagnetic form factors of the nucleon, Roper and Roper-nucleon transitions are defined by the followingmatrix elements, due to Lorentz and gauge invariance, N → N : M µ ( p , λ ; p , λ ) = ¯ u N ( p , λ ) (cid:20) γ µ F N ( q ) − iσ µν q ν M N F N ( q ) (cid:21) u N ( p , λ ) , R → R : M µ ( p , λ ; p , λ ) = ¯ u R ( p , λ ) (cid:20) γ µ F R ( q ) − iσ µν q ν M R F R ( q ) (cid:21) u R ( p , λ ) , (13) R → N : M µ ( p , λ ; p , λ ) = ¯ u N ( p , λ ) (cid:20) γ µ ⊥ F R N ( q ) − iσ µν q ν M R + M N F R N ( q ) (cid:21) u R ( p , λ ) , where γ µ ⊥ = γ µ − q µ q/q , q = p − p , and λ , λ , and λ are the helicities of the initial, final baryon and photon,obeying the relation λ = λ − λ .We recall the definitions of the nucleon Sachs form factors G NE/M ( Q ) and the electromagnetic radii h r E/M i N interms of the Dirac F N ( Q ) and Pauli F N ( Q ) form factors G NE ( Q ) = F N ( Q ) − Q M N F N ( Q ) ,G NM ( Q ) = F N ( Q ) + F N ( Q ) , h r E i N = − dG NE ( Q ) dQ (cid:12)(cid:12)(cid:12)(cid:12) Q =0 , h r M i N = − G NM (0) dG NM ( Q ) dQ (cid:12)(cid:12)(cid:12)(cid:12) Q =0 , (14)where G NM (0) ≡ µ N is the nucleon magnetic moment.Now we introduce the helicity amplitudes H λ λ , which in turn can be related to the invariant form factors F R Ni (see details in Refs. [25–28]. The pertinent relation is H λ λ = M µ ( p , λ ; p , λ ) ǫ ∗ µ ( q, λ ) , (15)where ǫ ∗ µ ( q, λ ) is the polarization vector of the outgoing photon. A straightforward calculation gives [25–28] H ± = s Q − Q (cid:18) F R N M + − F R N Q M (cid:19) , H ± ± = − p Q − (cid:18) F R N + F R N M + M (cid:19) , (16)where M ± = M ± M , Q ± = M ± + Q .In the case of the Roper-nucleon transition there exists the alternative set of helicity amplitudes ( A / , S / ) relatedto the set ( H , H ) by [29–33] A / = − b H , S / = b | p | p Q H , (17)where | p | = p Q + Q − M R , b = r παM + M − M N (18)and α = 1 / .
036 is the fine-structure constant.Expressions for the electromagnetic form factors of the nucleons, Roper, and Roper-nucleon transitions are givenas follows:nucleon-nucleon transition, F p ( Q ) = C ( Q ) + g pV C ( Q ) + η pV C ( Q ) + λ pV C ( Q ) + ζ pV C ( Q ) + ξ pV C ( Q ) ,F n ( Q ) = g nV C ( Q ) + η nV C ( Q ) + λ nV C ( Q ) + ζ nV C ( Q ) + ξ nV C ( Q ) ,F p ( Q ) = η pV C ( Q ) + λ pV C ( Q ) ,F n ( Q ) = η nV C ( Q ) + λ nV C ( Q ) . (19)Roper-nucleon transition, F R p p ( Q ) = D ( Q ) + g R p pV D ( Q ) + η R p pV D ( Q ) + λ R p pV D ( Q ) + ζ R p pV D ( Q ) + ξ R p pV D ( Q ) ,F R n n ( Q ) = g R n nV D ( Q ) + η R n nV D ( Q ) + λ R n nV D ( Q ) + ζ R n nV D ( Q ) + ξ R n nV D ( Q ) ,F R p p ( Q ) = η R p pV D ( Q ) + λ R p pV D ( Q ) ,F R n n ( Q ) = η R n nV D ( Q ) + λ R n nV D ( Q ) . (20)Roper-Roper transition, F R p ( Q ) = E ( Q ) + g R p V E ( Q ) + η R p V E ( Q ) + λ R p V E ( Q ) + ζ R p V E ( Q ) + ξ R p V E ( Q ) ,F R n ( Q ) = g R n V E ( Q ) + η R n V E ( Q ) + λ R n V E ( Q ) + ζ R n V E ( Q ) + ξ R n V E ( Q ) ,F R p ( Q ) = η R p V E ( Q ) + λ R p V E ( Q ) ,F R n ( Q ) = η R n V E ( Q ) + λ R n V E ( Q ) . (21)The structure integrals C i ( Q ), D i ( Q ), and E i ( Q ) are given by the analytical expressions (see in Appendix). Allcalculated form factors are consistent with QCD constituent counting rules [4] for the power scaling of hadronic formfactors at large values of the momentum transfer squared in the Euclidean region.The parameters, which will be used in the numerical evaluations, are fixed as follows: we use the universal dilatonparameter of κ = 383 MeV, the sets of twist mixing parameters are fixed from data on masses of nucleon ( c N = 1 .
800 , c N = − .
042 , c N = 0 . c R = 0 .
820 , c R = − .
242 , c R = 0 . κ = 383 MeV and baryonmasses M N = 938 .
27 MeV and M R = 1440 MeV only two parameters from the set of six twist mixing parameters arefree. E.g., parameters c N, R and c N, R can be fixed through the parameters c N, R and ratios M N, R using the matchingconditions (11) and (12). The parameters η pV = 0 . η nV = − . η pV = (cid:16) κM N (cid:17) ( µ p − , η nV = (cid:16) κM N (cid:17) µ n , (22)where µ p = 2 .
793 n.m. and µ n = − .
913 n.m. [34].The set on the nucleon parameters g pV = − .
001 , g nV = 1 .
731 , ζ pV = − .
109 , ζ nV = 0 .
101 , ξ pV = − .
166 , ξ nV = 0 .
174 , λ pV = − . λ nV = 0 . c R N = 0 .
142 , c R N = − .
942 , c R N = 3 .
449 , g R p pV = − .
095 , η R p pV = − .
551 , ζ R p pV = 0 .
020 , and ξ R p pV = − .
770 is fixed from data on Roper-nucleon transition data. For simplicity we put λ R p pV = 0. Our results for quark and nucleon electromagnetic form factors are shown in Figs. 1-9. We compareour results with data [35]-[84] and the dipole fit G D ( Q ) = 1 / (1 + Q / Λ ) . As scale parameter Λ we use twovalues Λ = √ .
71 GeV and Λ = √ .
66 GeV, corresponding to the root-mean-square (rms) radius r p = 0 . r p = 0 .
84 fm, respectively. In particular, in Fig. 1 and 2 we present our results for the Dirac and Pauli u (left panel)and d (right panel) quark form factors. Here data are taken from Refs. [35, 36].In Fig. 3 we display the Dirac proton form factor multiplied by Q (left panel) and the ratio Q F p ( Q ) /F p ( Q )(right panel). Results for the Dirac neutron form factor multiplied by Q (left panel) and ratio µ p G pE ( Q ) /G pM ( Q )in comparison with global Fit I and Fit II (right panel) are shown in Fig. 4. We take the central values of the resultsfor a global fit of the charge and magnetic proton form factors from Ref. [37]:Fit I: G pE ( Q ) = 1 + a E τ b E τ + c E τ + d E τ ,G pM ( Q ) = 1 + a M τ b M τ + c M τ + d M τ , (23)where a E = − . , b E = 12 . , c E = 12 . , d E = 23 . ,a M = 0 . , b M = 10 . , c M = 19 . , d M = 4 . , (24)Fit II: G pE ( Q ) = 1 + a E τ b E τ + c E τ + d E τ ,G pM ( Q ) = 1 + a M τ b M τ + c M τ + d M τ . (25)where a E = − . , b E = 12 . , c E = 9 . , d E = 37 . ,a M = 0 . , b M = 11 . , c M = 19 . , d M = 5 . . (26)Here τ = Q / (4 M N ).In Figs. 5 and 6 we present the ratios G pE ( Q ) /G D ( Q ) and G pM ( Q ) / ( µ p G D ( Q )) in comparison with the globalFit I and Fit II for the dipole scale parameter Λ = 0 .
71 GeV (left panel) and Λ = 0 .
66 GeV (right panel). Adetailed comparison of different ratios of the nucleon Sachs form factors is shown in Fig. 7-9. Here we use the dipolefunction G D ( Q ) with Λ = 0 .
71 GeV . The Roper-nucleon transition form factors and helicity amplitudes are shownin Figs. 10 and 11. Our predictions for the Roper-nucleon helicity amplitudes are compared with experimental dataof the CLAS (JLab) [23] and A1 (MAMI) [85] Collaborations, and with the MAID parametrization [86] A p / ( Q ) = − . − / (1 − .
22 GeV − Q − .
55 GeV − Q ) exp[ − .
51 GeV − Q ] ,S p / ( Q ) = 0 . − / (1 + 40 GeV − Q + 1 . − Q ) exp[ − .
75 GeV − Q ] , (27)and with the parametrization proposed by us. We find that the present data on helicity amplitudes can be fitted withthe use of the formulas A p / ( Q ) = A p / (0) 1 + a Q a Q + a Q + a Q ,S p / ( Q ) = S p / (0) 1 + s Q s Q + s Q + s Q , (28)where A p / (0) = − .
064 GeV − / , S p / (0) = 0 .
010 GeV − / , (29)and a = − . − , a = 1 . − , a = − . − , a = 0 . − ,s = 16 . − , s = 1 . − , s = 3 . − , s = − . − . (30)Our results for magnetic moments, slope radii and Roper-nucleon transition helicity amplitudes at q = 0 are sum-marized in Table I. TABLE I: Electromagnetic properties of nucleons and RoperQuantity Our results Data [34] µ p (in n.m.) 2.793 2.793 µ n (in n.m.) -1.913 -1.913 r pE (fm) 0.832 0.84087 ± ± h r E i n (fm ) -0.116 -0.1161 ± r pM (fm) 0.793 0.78 ± r nM (fm) 0.813 0.864 +0 . − . A p / (0) (GeV − / ) -0.061 -0.060 ± S p / (0) (GeV − / ) 0.008 — IV. SUMMARY
In the present paper we significantly improved the description of both the nucleon and the Roper structure usinga soft-wall AdS/QCD approach. We included novel contributions to the AdS/QCD action from additional non-minimal terms, which do not renormalize the charge and do not change the normalization of the corresponding formfactors. They give important contributions to the momentum dependence of the form factors and helicity amplitudesin reasonable agreement with data. In the future we plan to extend our formalism to the study of other nucleonresonances.
Acknowledgments
This work was supported by the German Bundesministerium f¨ur Bildung und Forschung (BMBF) under Project05P2015 - ALICE at High Rate (BMBF-FSP 202): “Jet- and fragmentation processes at ALICE and the partonstructure of nuclei and structure of heavy hadrons”, by CONICYT (Chile) Research Project No. 80140097 and underGrants No. 7912010025, 1140390 and PIA/Basal FB0821, by Tomsk State University Competitiveness ImprovementProgram and the Russian Federation program “Nauka” (Contract No. 0.1764.GZB.2017), and by Tomsk PolytechnicUniversity Competitiveness Enhancement Program (Grant No. VIU-FTI-72/2017).
Appendix A: The structure integrals C i ( Q ) , D i ( Q ) , and E i ( Q ) Functions C i ( Q ), D i ( Q ), and E i ( Q ) are given by the analytical expressions C i ( Q ) = X τ c Nτ C τi ( Q ) ,C τ ( Q ) = B ( a + 1 , τ ) (cid:16) τ + a (cid:17) ,C τ ( Q ) = a B ( a + 1 , τ ) ,C τ ( Q ) = a B ( a + 1 , τ + 1) a ( τ − − τ ,C τ ( Q ) = 2 a (cid:20) ( τ − B ( a + 1 , τ ) − τ − B ( a + 1 , τ + 1) + 3( τ + 1) B ( a + 1 , τ + 2)+ 2 ( τ − B ( a + 2 , τ + 1) − τ + 1)( τ + 2) B ( a + 2 , τ + 2) (cid:21) ,C τ ( Q ) = − a (cid:20) ( τ − B ( a + 1 , τ ) + τ (2 τ − B ( a + 1 , τ + 1) + 2 τ ( τ + 1) B ( a + 1 , τ + 2) (cid:21) ,C τ ( Q ) = − a (cid:20) ( τ − B ( a + 1 , τ ) + τ (2 τ − B ( a + 1 , τ + 1) − τ ( τ + 1) B ( a + 1 , τ + 2) (cid:21) ,C τ ( Q ) = 2 M N κ ( a + 1 + τ ) √ τ − B ( a + 1 , τ + 1) ,C τ ( Q ) = 4 M N κ a τ √ τ − (cid:20) B ( a + 1 , τ + 1) + 2( τ + 1) B ( a + 1 , τ + 2) (cid:21) , (A1) D i ( Q ) = X τ c R Nτ D τi ( Q ) ,D τ ( Q ) = a B ( a + 1 , τ + 1) (cid:20) √ τ − (cid:18) a + 1 τ (cid:19) + √ τ (cid:21) ,D τ ( Q ) = a B ( a + 1 , τ + 1) (cid:18) √ τ − (cid:18) a + 1 τ (cid:19) − √ τ (cid:19) ,D τ ( Q ) = a (cid:20) ( τ − / B ( a + 1 , τ ) − τ ( √ τ + √ τ − B ( a + 1 , τ + 1) + ( τ + 1) √ τ B ( a + 1 , τ + 2) (cid:21) ,D τ ( Q ) = 2 a (cid:20) τ ( τ − / B ( a + 1 , τ + 1) + τ / (( τ − p τ ( τ − − τ − B ( a + 1 , τ + 2) − ( τ + 1) p τ ( τ −
1) (2 ( τ + 1) √ τ − τ − √ τ ) B ( a + 1 , τ + 3)+ ( τ + 1)( τ + 2) √ τ (3 + 4 τ + 2 p τ ( τ − B ( a + 1 , τ + 4) − τ + 1)( τ + 2)( τ + 3) √ τ B ( a + 1 , τ + 5) (cid:21) ,D τ ( Q ) = − a (cid:20) ( τ − / B ( a + 1 , τ ) + τ ( √ τ + √ τ − τ − B ( a + 1 , τ + 1)+ √ τ ( √ τ − √ τ − ( τ + 1) B ( a + 1 , τ + 2) − √ τ ( τ + 1)( τ + 2) B ( a + 1 , τ + 3) (cid:21) ,D τ ( Q ) = − a (cid:20) ( τ − / B ( a + 1 , τ ) − τ ( √ τ − √ τ − τ − B ( a + 1 , τ + 1) − √ τ ( √ τ + √ τ − ( τ + 1) B ( a + 1 , τ + 2) + 2 √ τ ( τ + 1)( τ + 2) B ( a + 1 , τ + 3) (cid:21) ,D τ ( Q ) = M N + M R κ B ( a + 1 , τ + 1) (cid:20) a ( τ − − τ − a p τ ( τ − (cid:21) ,D τ ( Q ) = M N + M R κ a √ τ ( √ τ + √ τ − (cid:20)p τ ( τ − B ( a + 1 , τ + 1)+ ( τ + 1) (2 p τ ( τ − − B ( a + 1 , τ + 2) − τ + 1)( τ + 2) B ( a + 1 , τ + 3) (cid:21) , (A2)0and E i ( Q ) = X τ c R τ E τi ( Q ) ,E τ ( Q ) = 12 (cid:20) ( τ − B ( a + 1 , τ −
1) + τ (2 − τ ) B ( a + 1 , τ ) − τ ( τ + 1) B ( a + 1 , τ + 1)+ ( τ + 1)( τ + 2) B ( a + 1 , τ + 2) (cid:21) ,E τ ( Q ) = 12 (cid:20) ( τ − B ( a + 1 , τ −
1) + τ (2 − τ ) B ( a + 1 , τ )+ 3 τ ( τ + 1) B ( a + 1 , τ + 1) − ( τ + 1)( τ + 2) B ( a + 1 , τ + 2) (cid:21) ,E τ ( Q ) = a (cid:20) ( τ − B ( a + 1 , τ ) + τ (2 − τ ) B ( a + 1 , τ + 1)+ 3 τ ( τ + 1) B ( a + 1 , τ + 2) − ( τ + 1)( τ + 2) B ( a + 1 , τ + 3) (cid:21) ,E τ ( Q ) = 2 a ( τ + 1) (cid:20) τ ( τ − B ( a + 1 , τ + 2) + τ (2 τ − τ − τ + 5) B ( a + 1 , τ + 3) − τ ( τ + 2) (8 τ − τ − B ( a + 1 , τ + 4) + ( τ + 2)( τ + 3) (12 τ + 10 τ − B ( a + 1 , τ + 5) − (8 τ + 7)( τ + 2)( τ + 3)( τ + 4) B ( a + 1 , τ + 6) + 2( τ + 2)( τ + 3)( τ + 4)( τ + 5) B ( a + 1 , τ + 7) (cid:21) ,E τ ( Q ) = − a (cid:20) ( τ − B ( a + 1 , τ ) + 2 τ ( τ − B ( a + 1 , τ + 1) − ( τ + 1) B ( a + 1 , τ + 2) − τ + 1)( τ + 2) B ( a + 1 , τ + 3) (cid:21) ,E τ ( Q ) = − a (cid:20) ( τ − B ( a + 1 , τ ) + 2 τ ( τ − B ( a + 1 , τ + 1) − ( τ + 1)(4 τ − B ( a + 1 , τ + 2) + 2( τ + 1)( τ + 2) B ( a + 1 , τ + 3) (cid:21) ,E τ ( Q ) = 2 M R κ √ τ (cid:20) τ ( τ − B ( a + 1 , τ ) − ( τ + 1)(2 τ − B ( a + 1 , τ + 1)+ ( τ + 1)( τ + 2) B ( a + 1 , τ + 2) (cid:21) ,E τ ( Q ) = 4 M R κ a √ τ ( τ + 1) (cid:20) τ ( τ − B ( a + 1 , τ + 2) + (2 τ − τ + 1)( τ + 2) B ( a + 1 , τ + 3)+ (3 − τ )( τ + 2)( τ + 3) B ( a + 1 , τ + 4) + 2 ( τ + 2)( τ + 3)( τ + 4) B ( a + 1 , τ + 5) (cid:21) , (A3)where B ( m, n ) = Γ( m )Γ( n )Γ( m + n ) (A4)is the Beta function. 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Q H GeV L Q F p H Q LH G e V L Q H GeV L Q F p H Q L (cid:144) F p H Q LH G e V L FIG. 3: Dirac proton form factor multiplied by Q and ratio Q F p ( Q ) /F p ( Q ). - - - - - Q H GeV L Q F n H Q LH G e V L OurFit IFit II0 2 4 6 8 100.00.20.40.60.81.01.2 Q H GeV L Μ P G E p H Q L (cid:144) G M p H Q L FIG. 4: Dirac neutron form factor multiplied by Q and ratio µ p G pE ( Q ) /G pM ( Q ) in comparison with global Fit I and Fit II. OurFit IFit II0 2 4 6 8 100.20.40.60.81.0 Q H GeV L G E p H Q L (cid:144) G D H Q L OurFit IFit II0 2 4 6 8 100.20.40.60.81.0 Q H GeV L G E p H Q L (cid:144) G D H Q L FIG. 5: Ratio G pE ( Q ) /G D ( Q ) in comparison with global Fit I and Fit II for dipole scale parameter Λ = 0 .
71 GeV (leftpanel) and Λ = 0 .
66 GeV (right panel). OurFit I Fit II0 5 10 15 20 25 30 350.70.80.91.0 Q H GeV L G M p H Q L (cid:144) H Μ p G D H Q LL OurFit I Fit II0 5 10 15 20 25 30 350.80.91.01.11.2 Q H GeV L G M p H Q L (cid:144) H Μ p G D H Q LL FIG. 6: Ratio G pM ( Q ) / ( µ p G D ( Q )) in comparison with global Fit I and Fit II for dipole scale parameter Λ = 0 .
71 GeV (leftpanel) and Λ = 0 .
66 GeV (right panel). ▽▽▽▽▽▽△▽ ×× △▽ ⋆ ▽ ×× ▽▽△▽▽▽△▽▽△▽ ▽△ Q (GeV ) G pE ( Q ) /G D ( Q ) G p E ( Q ) / G D ( Q ) Berger et al. [38] ▽ Price et al. [39] △ Hanson et al. [40]Simon et al. [41] × Milbrath et al. [42]Jones et al. [43] ⋆ Dieterich et al. [44] △▽ ⋄ ⋆ × N N N ▽ Eden et al. [45]Herberg et al. [46] ⋄ Ostrick et al. [47] △ Passchier et al. [48]Rohe et al. [49] ⋆ Golak et al. [50]Schiavilla & Sick [51] × Zhu et al. [52]Madey et al. [53] N Riordan et al. [54] Q (GeV ) G nE ( Q ) G n E ( Q ) FIG. 7: Ratio G pE ( Q ) /G D ( Q ) and charge neutron form factor G nE ( Q ) in comparison with data. NN NNNNN N N N ▽ ▽ ⋄⋄⋄⋄⋄ △ △ △ △ N Punjabi et al. [55]Gayou et al. [56]Ron et al. [57]Puckett et al. [37] △ Puckett et al. [58] ⋄ Zhan et al. [59]Paolone et al. [60] Q (GeV ) µ p G pE ( Q ) /G pM ( Q ) µ p G p E ( Q ) / G p M ( Q ) NNN N ⋆ ⋄× ▽ △ △ △⊗ Q (GeV ) µ n G nE ( Q ) /G nM ( Q ) µ n G n E ( Q ) / G n M ( Q ) ▽ Eden et al. [30]Herberg et al. [31] × Passchier et al. [33] ⋄ Zhu et al. [37] △ Riordan et al. [39] ⋆ Bermuth et al. [63]Warren et al. [64]Glazier et al. [65]Plaster et al. [66] N Geis et al. [67] ⊗ Schlimme et al. [68]
FIG. 8: Ratios µ p G pE ( Q ) /G pM ( Q ) and µ n G nE ( Q ) /G nM ( Q ) in comparison with data. ×××× △ ×××× ▽ × △ ××××× ▽△ ×× ⋆ ××× ▽△ × ▽ ⋆ ⋄ ▽ ⋆ △▽ ⋆ ⋄ ▽ ⋄ ⋆ △ ⋆ ⋄⋄ ⋆ ⋄ Q (GeV ) G pM ( Q ) / ( µ p G D ( Q )) G p M ( Q ) / ( µ p G D ( Q )) ▽ Berger et al. [38] △ Hanson et al. [40] × Janssens et al. [67]Litt et al. [68] ⋆ Bartel et al. [69] Hoehler et al. [70]Sill et al. [71]Andivahis et al. [72] ⋄ Walker et al. [73] A △▽△ ⋆⋆⋆⋆⋆⋆⋆ ⋄⋄⋄⋄ ⊗ NNNNNNNN NNNNNNNNNNNNNNNN Q (GeV ) G nM ( Q ) / ( µ n G D ( Q )) G n M ( Q ) / ( µ n G D ( Q )) Rock et al. [52]Lung et al. [53] ⋆ Markowitz et al. [54] ▽ Anklin et al. [55] ⊗ Gao et al. [56] ⋄ Bruins et al. [57] × Anklin et al. [58] △ Xu et al. [59]Kubon et al. [60]Xu et al. [61] N Lachniet et al. [62]
FIG. 9: Ratios G pM ( Q ) / ( µ p G D ( Q )). and G nM ( Q ) / ( µ n G D ( Q )) in comparison with data. Q F R pp H Q L - - - - - Q F R pp H Q L FIG. 10: Roper-nucleon transition form factors F R p p ( Q ) and F R p p ( Q ) up to 10 GeV . Our Fit MAID0 2 4 6 8 10 - - - Q A p (cid:144) H Q LH - G e V - (cid:144) L OurFitMAID0 2 4 6 8 10010203040 Q S p (cid:144) H Q LH - G e V - (cid:144) L FIG. 11: Helicity amplitudes A p / ( Q ) and S p / ( Q ) up to 10 GeV2