Dispersive analysis of low energy γ^* N\rightarrowπN process
AA Dispersive Analysis of Low Energy γ ∗ N → πN Process
Xiong-Hui Cao , Yao Ma , and Han-Qing Zheng , ∗ Department of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University, Beijing 100871, P. R. China Collaborative Innovation Center of Quantum Matter, Beijing, Peoples Republic of China
February 1, 2021
Abstract
We use a dispersion representation based on unitarity and analyticity to study the lowenergy γ ∗ N → πN process in the S channel. Final state interactions among the πN system are critical to this analysis. The left-hand part of the partial wave amplitude isimported from O ( p ) chiral perturbation theory result. On the right hand part, the finalstate interaction is calculated through Omn`es formula in S -wave. It is found that goodnumerical fit can be achieved with only one subtraction parameter, and the eletroproductionexperimental data of multipole amplitudes E , S in the energy region below ∆(1232)are well described when the photon virtuality Q ≤ . . The γ ∗ N coupling of thepossible sub-threshold resonance in S channel is also investigated. The electromagnetic interactions of nucleon have long been recognized as an important sourceof information for understanding strong interaction physics [1–7]. The investigation of pion pho-toproduction started in the 1950s with the seminal work of Chew, Goldberger, Low, and Nambu(CGLN) [1], where the formalism for pion photoproduction on nucleon target was developed, andfixed- t dispersion relations were used as a tool for the analyses of the reaction data. Postulatesunderlying the DR approach are analyticity, unitarity, and crossing symmetry of S -matrix. TheCGLN formalism was later extended to pion electroproduction [8, 9], and DR was used in theanalyses of the experimental data [9–12]. Based on the recent low energy experiments, partialwave analyses have been performed to study the underlying structure of the reaction amplitudesand describing the properties of the nucleon resonances [7, 13–15].Since 1980s, it has been successful to explore the electroproduction and relevant processesusing chiral perturbation theory ( χ PT) at low energies [16–20]. For the calculation of loopdiagrams, there are several renormalization schemes, which are e.g. heavy-baryon approach inRef. [16] and EOMS scheme adopted in Refs. [17,20], to solve the power-counting breaking (PCB)problems. However, χ PT only works well near the threshold and fails at slightly higher energies.So the unitary method is necessarily adopted in order to suppress the contributions from largeenergy and recast unitarity of the amplitude. ∗ Address after January 20: College of Physics, Sichuan University, Chengdu, Sichuan 610065, P. R. China a r X i v : . [ nu c l - t h ] J a n ome unitarity methods have already been explored (For a recent review, see Ref. [21]). Thecouple channel N/D method was used to unitarize χ PT amplitudes in Ref. [15], and J¨ulichmodel was adopted to study photoproduction and relevant process in Ref. [7]. In this paper, our γ ∗ N → πN amplitudes are obtained through the dispersive analysis [22], in the case we set upwith chiral O( p ) γ ∗ N → πN amplitudes and πN final state interaction estimated by the Omn`essolution [23] in single channel approximation. In order to achieve such a dispersive analysis,efforts have been made in understanding the complicated analytic structure of the amplitudes.Based on our dispersion representation, the multipole amplitudes ( S E and S S ) datafrom Refs. [5,13,24–27] below the ∆(1232) peak have been fitted. This work extends our previousanalysises on pion photoproduction [28], to the virtual-photon process with photon virtuality Q up to 0 . , and find a good description of the data when Q ≤ . , with only oneparameter. Besides, the comparison between the O ( p ) calculation in this paper and the one upto O ( p ) from Ref. [17] is performed and a discrepancy between the two results are noticed inhigher Q region.In Refs. [29–31], evidences are found on the possible existence of a sub-thresthod resonancenamed N ∗ (890) in the S channel using the method proposed in Refs. [32–36], assisted by chiralamplitudes obtained in Refs. [37–40]. In this paper, further results are provided on γ ∗ N couplingto this resonance for future reference.This paper is organized as follows. In section 2, a brief introduction to pion electroproductionis given. In section 3, we set up the dispersive formalism for γ ∗ N → πN process and make ananalysis about the singularities which appear in this process. In section 4, numerical resultsof multipoles are carried out. In section 5, we calculate, in particular, the virtual-photon decayamplitudes at the pole, as introduced in Refs. [41,42]. Finally we give our conclusions in section 6. In this section we provide a short introduction to the notations describing the electroductionof pions. Single pion electroproduction off the nucleon is the process described by e ( l ) + N ( p ) → e ( l ) + N ( p ) + π a ( q ) , (1)where a is the isospin index of the pion and l ( l ) , p ( p ) , q are incoming (outgoing) electron,incoming (outgoing) nucleon and pion momentum, respectively.Due to the fact that the interaction between electron and nucleon is pure electromagnetic,for every additional virtual photon exchange, there will be one more fine structure constant α = e / (4 π ) ≈ /
137 suppression factor. Hence we can only consider the lowest contribution orthe so-called one-photon-exchange approximation, see Fig. 1.2 ( l ) e ( l ) γ ∗ ( k ) N ( p ) N ( p ) π ( q )Figure 1: Pion electroproduction in the one-photon-exchange approximation. k = l − l repre-sents the momentum of the single exchanged virtual photon. The shaded circle represents thefull hadronic vertex.In this approximation, the invariant amplitude M is interpreted as the product of the po-larization vector (cid:15) µ of the virtual photon and the hadronic transition current matrix element M µ , M = (cid:15) µ M µ = e ¯ u ( l ) γ µ u ( l ) k M µ , (2)where M µ = − ie (cid:104) N ( p ) , π ( q ) | J µ (0) | N ( p ) (cid:105) , (3)with J µ the electromagnetic current operator. Since k µ (cid:15) µ = 0 in both photoproduction andeletroproduciton, it is possible to separate the pure electromagnetic part of the process from thehadronic part which is the process γ ∗ ( k ) + N ( p ) → N ( p ) + π ( q ) , (4)where γ ∗ refers to a (space-like) virtual photon, so we can define k = − Q <
0, and Q calledphoton virtuality. Mandelstam variables s, t and u are defined as s = ( p + k ) , t = ( p − p ) , u = ( p − q ) , (5)and satisfy s + t + u = 2 m N + m π − Q , where m N and m π denote the nucleon mass and thepion mass, respectively. In the center-of-mass (cm) frame, πN final state system , the energiesof the photon, k ∗ , the pion, E ∗ π and incoming (outgoing) nucleon, E ∗ ( E ∗ ) are given by k ∗ = W − Q − m N W , E ∗ π = W + m π − m N W ,E ∗ = W + m N + Q W , E ∗ = W + m N − m π W , (6)where W = √ s is the cm total energy. The values of the initial and final state momentum in the In this section, the superscript ∗ refers to the physical quantity in the cm frame.
3m frame are | k ∗ | = (cid:115)(cid:18) W − m N − Q W (cid:19) + Q , | q ∗ | = (cid:115)(cid:18) W − m N + m π W (cid:19) − m π , (7)The real photon equivalent energy in laboratory frame k lab is given by k lab = W − m N m N . (8)and k cm = ( m N /W ) k lab . The cm scattering angle θ ∗ between the pion three-momentum andthe z -axis, defined by the incoming photon direction, is depicted in Fig. 2 z axis N ( − q ∗ ) N (cid:48) ( − k ∗ ) π ( k ∗ ) γ ∗ ( q ∗ ) θ ∗ Figure 2: Scattering angle θ ∗ in the cm frame.The scattering amplitude of pion electroproduction can be parametrized in terms of the Ballamplitudes [10], which are defined in Lorentz-covariant form − ie (cid:104) N (cid:48) π | J µ (0) | N (cid:105) = ¯ u ( p ) (cid:32) (cid:88) i =1 B i V µi (cid:33) u ( p ) , (9)where u ( p ) and u ( p ) are the Dirac spinors of the nucleon in the initial and final states, respec-tively. Here we use the notation of [9, 16, 43], but slightly different from [2, 17]: V µ = γ γ µ /k , V µ = 2 γ P µ , V µ = 2 γ q µ , V µ = 2 γ k µ ,V µ = γ γ µ , V µ = γ P µ /k , V µ = γ k µ /k , V µ = γ q µ /k , (10)where P = ( p + p ) / /k = γ µ k µ . Using electromagnetic current conservation k µ M µ =0, only six independent amplitudes are required for the description of pion electroproduction.Furthermore, in pion photoproduction ( Q = 0) only four independent amplitudes survive.The parameterization of Ref. [16] takes care of current conservation already from the begin-ning which contains only six independent amplitudes A i , M µ = ¯ u ( p ) (cid:32) (cid:88) i =1 A i M µi (cid:33) u ( p ) (11)4ith M µ = 12 γ ( γ µ /k − /kγ µ ) ,M µ = 2 γ (cid:18) P µ k · (cid:18) q − k (cid:19) − (cid:18) q − k (cid:19) µ k · P (cid:19) ,M µ = γ ( γ µ k · q − /kq µ ) ,M µ = 2 γ ( γ µ k · P − /kP µ ) − m N M µ ,M µ = γ (cid:0) k µ k · q − k q µ (cid:1) ,M µ = γ (cid:0) k µ /k − k γ µ (cid:1) . (12)Each of them individually satisfies gauge invariance k µ M µi = 0. The scalar functions A i and B i can be linked through A = B − m N B , A = 2 m π − t B , A = − B , A = − B , A = B ,A = 2 s + u − m N (cid:18) B − s − u m π − t ) B + 2 B (cid:19) = 1 k (cid:18) s − ut − m π B − B (cid:19) . (13)The CGLN amplitudes F i are another common parameterization [1, 43], which plays animportant role in experiments and partial wave analyses. These amplitudes are defined in thecm frame via (cid:15) µ ¯ u ( p ) (cid:32) (cid:88) i =1 A i M µi (cid:33) u ( p ) = 4 πWm N χ † F χ , (14)where χ and χ denote initial and final Pauli spinors, respectively. Electromagnetic currentconservation allows us to work in the gauge where the polarization vector of virtual photon has avanishing longitudinal component. In terms of the polarization vector of Eq. (2) this is achievedby introducing the vector [16, 44, 45] b µ = (cid:15) µ − (cid:15) · ˆ k | k | k µ , (15)where b (cid:54) = 0, but b · ˆ k = 0 (ˆ k = k / | k | ). F may be written as ( σ = ( σ , σ , σ )) F = i σ · b F + σ · ˆ qσ · (ˆ k × b ) F + i σ · ˆ k ˆ q · b F + i σ · ˆ q ˆ q · b F − i σ · ˆ q b F − i σ · ˆ k b F . (16)We can connect A i and F i through algebraic calculations, and the results can be found inAppendix B. 5he CGLN amplitudes can be expanded into multipole amplitudes [43] F = ∞ (cid:88) l =0 (cid:8) [ lM l + + E l + ] P (cid:48) l +1 ( x ) + [( l + 1) M l − + E l − ] P (cid:48) l − ( x ) (cid:9) , F = ∞ (cid:88) l =1 { ( l + 1) M l + + lM l − } P (cid:48) l ( x ) , F = ∞ (cid:88) l =1 (cid:8) [ E l + − M l + ] P (cid:48)(cid:48) l +1 ( x ) + [ E l − + M l − ] P (cid:48)(cid:48) l − ( x ) (cid:9) , F = ∞ (cid:88) l =2 { M l + − E l + − M l − − E l − } P (cid:48)(cid:48) l ( x ) , F = ∞ (cid:88) l =1 [ lS l − − ( l + 1) S l + ] P (cid:48) l ( x ) = | k ∗ | k ∗ F , F = ∞ (cid:88) l =0 (cid:2) ( l + 1) S l + P (cid:48) l +1 ( x ) − lS l − P (cid:48) l − ( x ) (cid:3) = | k ∗ | k ∗ F , (17)with x = cos θ = ˆ q · ˆ k , P l ( x ) the Legendre polynomial of degree l , P (cid:48) l = d P l / d x and so on.Subscript l denotes the orbital angular momentum of the pion-nucleon system in the final state.The multipoles E l ± , M l ± , and S l ± are functions of the cm total energy W and the photonvirtuality Q , and refer to transversal electric, magnetic transitions and scalar transitions ,respectively. The subscript l ± denotes the total angular momentum j = l ± / E l + = (cid:90) − dx l + 1) (cid:20) P l F − P l +1 F + l l + 1 ( P l − − P l +1 ) F + l + 12 l + 3 ( P l − P l +2 ) F (cid:21) ,E l − = (cid:90) − dx l (cid:20) P l F − P l − F − l + 12 l + 1 ( P l − − P l +1 ) F + l l − P l − P l − ) F (cid:21) ,M l + = (cid:90) − dx l + 1) (cid:20) P l F − P l +1 F − l + 1 ( P l − − P l +1 ) F (cid:21) ,M l − = (cid:90) − dx l (cid:20) − P l F + P l − F + 12 l + 1 ( P l − − P l +1 ) F (cid:21) ,S l + = (cid:90) − dx l + 1) [ P l +1 F + P l F ] ,S l − = (cid:90) − dx l [ P l − F + P l F ] . (18)The isospin structure of the scattering amplitude can be written as A ( γ ∗ + N → π a + N (cid:48) ) = χ † (cid:26) δ a A (+) + i(cid:15) a b τ b A ( − ) + τ a A (0) (cid:27) χ , (19)where τ a ( a = 1 , ,
3) are Pauli matrices. We can define the isospin transition amplitudes by A I,I ( A , ± , A , ± ), where { I, I } denote isospin of the final πN system. In the notation | I, I (cid:105) Sometimes the longitudinal multipoles are used instead of the scalar multipoles, they satisfies a relationship L l ± = ( k / | k | ) S l ± | p (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12) , + 12 (cid:29) , | n (cid:105) = (cid:12)(cid:12)(cid:12)(cid:12) , − (cid:29) , (20) (cid:12)(cid:12) π + (cid:11) = −| , +1 (cid:105) , (cid:12)(cid:12) π (cid:11) = | , (cid:105) , (cid:12)(cid:12) π − (cid:11) = | , − (cid:105) . (21)So isospin transition amplitudes can be obtained from A ( ± ) and A (0) via A , = A , − = (cid:114) (cid:16) A (+) − A ( − ) (cid:17) , (22) A , = − (cid:114) (cid:16) A (+) + 2 A ( − ) + 3 A (0) (cid:17) , (23) A , − = (cid:114) (cid:16) A (+) + 2 A ( − ) − A (0) (cid:17) . (24)In the one-photon-exchange approximation, the differential cross section can be factorizedas [3, 4] d σ d E dΩ l dΩ ∗ π = α π E E Q k lab − (cid:15) d σ v dΩ ∗ π ≡ Γ d σ v dΩ ∗ π , (25)where Γ is the flux of the virtual photon, E , denote the energy of the initial and final electronsin the laboratory frame, respectively. The parameter (cid:15) expresses the transverse polarization ofthe virtual photon in the laboratory frame and it’s an invariant under collinear transformations.In terms of laboratory electron variables, it is given by [3] (cid:15) = (cid:18) k Q tan (cid:18) θ l (cid:19)(cid:19) − , (26)where θ l is the scattering angle of the electron in the laboratory frame. The virtual photondifferential cross section, d σ v / dΩ ∗ π , for an unpolarized target without recoil polarization can bewritten in the form [4]:d σ v dΩ ∗ π = d σ T dΩ ∗ π + (cid:15) d σ L dΩ ∗ π + (cid:112) (cid:15) (1 + (cid:15) ) d σ LT dΩ ∗ π cos φ ∗ π + (cid:15) d σ T T dΩ ∗ π cos 2 φ ∗ π + h (cid:112) (cid:15) (1 − (cid:15) ) d σ LT (cid:48) dΩ ∗ π sin φ ∗ π + h (cid:112) − (cid:15) d σ T T (cid:48) dΩ ∗ π , (27)in which φ ∗ π is the azimuthal angle of pion and h is the helicity of the incoming electron. Forfurther details about Eq. (27), especially concerning polarization observables, we refer to Ref. [4].If we integrate the dependence of azimuthal angle, at the end we will get σ v = σ T + (cid:15)σ L . (28)In the following chapter, we will introduce χ PT as an effective field theory which allow usto calculate pion production. The upper limit for the cm total energy W , restricted by the factthat we only consider pion and nucleon degrees of freedom, is below the ∆(1232) resonance peak.Furthermore, through the experience gained by studying EM form factors [46, 47], the estimateof the upper limit of momentum transfers is Q (cid:39) . in χ PT [17, 48].7
Partial wave amplitudes χ PT amplitudes and unitarity method
We recalculated the pion electroproduction process close to threshold using χ PT up to O ( p )and confirm the results of [16]. The invariant scalar functions can be extracted from full ampli-tudes. The results are listed in the Appendix A. for higher order O ( p ) contributions and theinfluence of ∆(1232) resonance, readers can refer to Ref. [20].In the following part, superscripts and subscripts I, J (isospin, total angular momentum)are ignored for brevity. Considering final-state theorem [49] and using the dispersion relation,the unitarized S wave amplitude can be written as [22, 28, 50–56] M ( s ) = M L ( s ) + Ω( s ) (cid:32) − sπ (cid:90) ∞ ( m π + m N ) (cid:0) Im Ω( s ) − (cid:1) M L ( s ) s (cid:48) ( s (cid:48) − s ) d s (cid:48) + P ( s ) (cid:33) , (29)where P ( s ) is subtraction polynomial. The amplitude M L only contains left hand cut singularity.Thus, the pion electroproduction amplitude M ( s ) is determined up to a polynomial. Ω( s ) is theso-called Omn`es function [23]:Ω IJ ( s ) = ˜ P ( s ) exp (cid:20) sπ (cid:90) ∞ ( m π + m N ) δ IJ ( s (cid:48) ) s (cid:48) ( s (cid:48) − s ) d s (cid:48) (cid:21) (30)with ˜ P representing a polynomial, reflecting the zeros of Ω( s ) in the complex plane and δ IJ ( s )being the elastic πN partial wave phase shift.For our calculation, we use the χ PT result to esimate M L , so as long as function Ω( s ) isknown, we can get the amplitude with correct unitarity and analyticity property. Applicability of the Omn`es method to the amplitudes of interest relies on the ability toseparate the amplitude into a piece having only a left-hand cut and a piece having only a righthand one. This, a priori, is not the case if the left-hand cuts overlapped with the unitary cut.So we review the analytic structures arising in our calculation and find that the singularitiesin this virtual process are rather complicated than real photoproduction. There will be someadditional cuts in the complex s plane, compared with the photoproduction one. we followRef. [57] which relying on the Mandelstam double spectral representation to illustrate the analyticstructure of the partial wave amplitudes. According to crossing symmetry, one amplitude cansimultaneously describe the three channels of s, t, u : s : γ ∗ + N → π + N (cid:48) , σ = M , ρ = ( m + M ) ; t : γ ∗ + π → N + N (cid:48) , σ = m , ρ = 4 m ; u : γ ∗ + N (cid:48) → π + N , σ = M , ρ = ( m + M ) . (31)Here, for brevity, we define m = m π , M = m N , σ i represent the the mass squares of stronglyinteracting intermediate bound states, and the continuous spectra will begin at ρ i which is thethreshold of two particle intermediate states. Note here that the Mandelstam variable t definedin the s plane is related to z s = cos θ via t = − Q + m − (cid:0) s − Q − M (cid:1) (cid:0) s + m − M (cid:1) s + (cid:110)(cid:104) s + 2 (cid:0) Q − M (cid:1) s + (cid:0) Q + M (cid:1) (cid:105) ( s − s L ) ( s − s R ) (cid:111) z s s . (32)8ith regard to dynamical cut positions of the partial wave T matrix, we first take the t channel as an illuastration. The full amplitude can be written as a dispersion integral: T ( s, t ) = (cid:90) ∞ σ F ( s, t (cid:48) ) t (cid:48) − t dt (cid:48) , (33)where F is a spectral function. The partial wave amplitude is the projection of the full amplitudeonto rotation function d J , T J ( s ) = (cid:90) − d z s d J ( z s ) (cid:90) ∞ σ d t (cid:48) F ( s, t (cid:48) ) t (cid:48) − t ( s, z s )= (cid:90) ∞ σ d t (cid:48) F ( s, t (cid:48) ) (cid:90) − d z s d J ( z s ) α ( t (cid:48) , s ) − β ( s ) z s , (34)where integration (cid:82) ∞ σ denotes the sum of the value at pole t (cid:48) = σ and (cid:82) ∞ ρ . It can be provedthat the final singularity only comes from the form in logarithmic functionln( α + β ) − ln( α − β ) . (35)We classify all cuts as follows: • unitary cut: s ∈ [ s R , ∞ ) on account of the s -channel continuous spectrum; • t -channel cut: the arc, on the left of s = s c , stems from t -channel continuous spectrum for4 m ≤ t ≤ M ; • u -channel cut: s ∈ ( −∞ , s u ] with s u = M − m M − m ( M + Q ) m + M due to the u -channel continu-ous spectrum for u ≥ ( m + M ) ; • t -channel cut from pion pole: due to t channel single pion exchange, and the branch pointslocate at 0 , C t , C † t ; • u -channel cut from nucleon pole: due to t channel single nucleon exchange, and the branchpoints located at 0 , C u , C † u .where the branch points in the complex plane are (the other three cases are symmetric aboutthe real axis) C t = M − Q i (cid:114) M Q − m Q − Q M m Q , (36) C u = M − m M Q + i (cid:115) m Q − m M m Q + m M Q − (cid:18) m M (cid:19) Q . (37)The singularities caused by the pole exchanges of t, u channels are complicated but definitelyseparated from the unitarity cut.Aside from the above dynamical singularities, there exist additional kinematical singularitiesfrom relativistic kinematics and polarization spinor of fermions, especially in an inelastic scat-tering process. These inelastic ones will naturally introduce some square-root functions in thepartial wave amplitudes (or multipole amplitudes) which will cause the kinematical singularities.They provide some of the most obvious characteristics in the case of relativistic theory. Herekinematical cuts are introduced when the arguments of the square-root functions from Eq. (32)9re negative. All the involved arguments together with their corresponding domains with theirvariable less than zero are listed in Table 1.Table 1: Arguments causing singularities Arguments Domain s ( −∞ , s − s R ( −∞ , s R ) s − s L ( −∞ , s L ) s + 2 (cid:0) Q − M (cid:1) s + (cid:0) Q + M (cid:1) ( M − Q ± iMQ, M − Q ± i ∞ ) There are some arbitrariness when fixing the cut position [58, 59]. For example, compare (cid:112) ( s − s L ) ( s − s R ) and √ s − s L √ s − s R , they may correspond to different cut structure. Theformer will have an extra cut, which is perpendicular to the real axis and passes the midpointof s L and s R . So we choose the latter one to make sure that left cuts are lying on the real axis.In addition, there is a pole like singularity at M comes from the fact that Eq. (17) has the1 / ( s − M ) term (See Appendix B), which will appear in partial wave amplitudes. Finally, theremay be a pole derived from the gauge invariant amplitudes. Relations (13) has introduced the1 / ( t − m ) pole singularity, If we consider the partial wave integral, e.g., (cid:90) d z z ( t ( z ) − m ) ( u ( z ) − M ) ∝ (cid:90) d z z ( a + bz )( c − bz ) ∝ a + c ∝ s − M + Q , (38)where a = − m M + M − m Q + M Q + m s − M s + Q s + s , c = m M − M + m Q − M Q − m s + Q s + s , b = √ s − s L √ s − s R (cid:113) s + 2 ( Q − M ) s + ( Q + M ) . Sothe possible additional singularities in our partial wave analysis are displayed in Fig. 3. s R s L m N m N − Q ∆∆ † Figure 3: Kinematical singularities. The red dot represents the nucleon pole, and the two verticalsolid rays represent the kinematic cuts from (cid:113) s + 2 ( Q − M ) s + ( Q + M ) .For a certain channel we are considering, these singularities may not all appear due to can-cellation from linear combinations. Therefore, it must be analyzed in detail when it is used. We are now in the position to compare the unitary representation of virtual photoproductionamplitude given in Eq. (29) with experimental multipole amplitude data in S channel. Herewe use MAID2007 [13, 24] and DMT2001 [5, 25–27] results for fitting. These model providea good description to multipole amplitudes, differential cross sections as well as polarizationobservables. They can be used as the basis for the prediction and the analysis of meson photo-and electroproduction data on proton and neutron targets.10 .1 Fitting procedure In the fit, unknown parameters include the LECs appeared in M χP T ( s ), the subtraction con-stants in the auxiliary function Ω( s ) and the ones in the subtraction polynomial P ( s ). However,the parameters in chiral lagrangian appearing in M χP T ( s ) up to O ( p ) are well fixed. Theyare m N = 0 . m π = 0 . e = √ πα = 0 . g A = 1 . F π = 0 . c = 3 . / (4 m N ) and c = − . / (2 m N ) [60] . Hence, M χP T ( s ) is parameter free. Futher, weset ˜ P ( s ) = 1, and compute Ω( s ) by using the partial wave phase shift extracted from the πNS matrix given in Ref. [61]. Note that it should be a good approximation for a single channelcase that the integrations in Eqs. (30) and (29) are performed up to 2 . (below the ηN threshold). Lastly, the subtraction polynomial P is taken to be a constant, P ( s ) = a , i.e., herewe only consider once subtraction. The above fit method is simultaneously performed on thedata from the MAID and DMT models.In the numerical analyses, we fit the multipole amplitudes with S channel from πN thresholdto 1 .
440 GeV just below the resonance ∆(1232). Since no error bars are given, we assign themaccording to Refs. [30, 37] err( M Il ) = (cid:114)(cid:16) e R,Is (cid:17) + (cid:16) e R,Ir (cid:17) (cid:0) M Il (cid:1) . (39)Here the superscripts R, I represent the real and imaginary parts of the amplitude. We choose e R,Is = 0 . , . − /m π ] , e R,Ir = 10%. We take into account the errors caused by the modeldependence of the partial wave data as much as possible [5,13]. The fit results to MAID2007 andDMT2001 data are displayed in Figs. 4, 5 and 6, 7, respectively. For comparison, we also showthe O ( p ) chiral results of multipole amplitudes. As expected, the chiral results only describethe data at low energies close to threshold and in low Q . The values of the fit parameters arecollected in Table 2. Neglecting χ PT correction beyond tree level, the two LECs c and c can be related to the anomalousmagnetic moments of the nucleon via c = ( k p + k n ) / m N , c = ( k p − k n ) / m N , with k p and k n beinganomalous magnetic moments of proton and neutron, respectively. Since k p and k n are precisely determined byexperiments, one can infer the uncertainties of c and c must be negligible and shall hardly change our results. The influence of twice subtractions P ( s ) = a + b s is also examined to test the fit result. It is found that themajor physical outputs are almost inert. pE : the two left columns correspond to the results of MAID, the others are fromDMT. Moreover, the blue lines represent our fit result. For comparison, the chiral result is alsoshown (the black line). 12igure 5: nE : description the same as in Fig 413igure 6: pS : description the same as in Fig 414igure 7: nS : description the same as in Fig 415able 2: Fit results of once subtraction ( P = a ). The parameter a is given in unit of [10 − /m π ]. Multipole Target Case Value χ /d.o.fE p MAID 0 . ± . . . ± . .
32n MAID 1 . ± . . . ± . . S p MAID − . ± . . − . ± . .
36n MAID 1 . ± . . . ± . . In Figs. 4, 5 and 6, 7, we fit amplitudes from Q = 0 to Q = 0 . in the increments of0 . . We also draw the result where Q = 0 . . It can be seen that, except that thefit to pS is rather good, the other fit results do not improve much when Q = 0 . . Thisis within expectation that we did not consider corrections of vector meson exchanges. [46,47]. Ingeneral, as we can see in the Table 2, our results are in good agreement with the experimentaldata in the sense that the averaged χ are close to one. Meanwhile, the central value of a is verysmall in any case. That can be understood by the fact that multipoles calculated from χ PT andunitarity method can already well describe the experimental data. We also do the fit which usesa twice subtraction polynomial, i.e., P = a + bs . However, the fit parameters a and b are foundto be highly negative correlated. Thus, once subtraction is more advisable.It is very convincing that no matter what data are used, the fit results of multipole amplitudesare very similar. The results can illustrate that our unitarity method is very powerful and effectivein low energy regions and low Q regions. But our results do not fit well in the case of high Q .In our future work, we will consider using resonance χ PT to improve the description ability ofthis method in high Q . Here we find that the O ( p ) results [17] and our amplitudes of multipole S pE differentin higher Q regions. Furthemore, S nE , S pS and S nS are even different from ourcalculations at lower Q . For comparison, we list UIM , DMT model, our results and that of [17].16igure 8: The black solid curves show our calculations at O ( p ) and the blue long-dashed curvesare the outputs of the O ( p ) results [17]. The red dot-dashed and green short-dashed curves arethe predictions of the UIM and the DMT model, respectively. In the above subsection, all the involved parameters in the partial wave virtual photopro-duction amplitude M l ( s ) have been determined. Since N ∗ (890), as a subthreshold resonance, islocated on the second Riemann sheet (RS), one needs to perform analytic continuation in orderto extract its couplings to the γ ∗ N and πN systems.It can be proved that the residue can be extracted from [28] g γN g πN (cid:39) M l ( s p ) S (cid:48) l ( s p ) , (40)where S l ( s ) corresponds to partial wave S matrix of elastic πN scattering. Residues g γN and g πN denote the γN and πN couplings, respectively. The πN coupling can also be extracted fromelastic πN scattering, i.e., g πN (cid:39) T l ( s p ) / S (cid:48) l ( s p ), where T l is the corresponding partial wave πN scattering amplitude. In order to compare the results with Refs. [41, 42], which are extracted17irectly from multipole amplitudes parameterized in the √ s plane, so we can write E II0+ ( s → s p ) (cid:39) g EγN g πN s − s p (cid:39) g EγN g πN √ s p ( √ s − √ s p ) = (cid:0) g EγN g πN / W p (cid:1) W − W p , (41) S II0+ ( s → s p ) (cid:39) g SγN g πN s − s p (cid:39) g SγN g πN √ s p ( √ s − √ s p ) = (cid:0) g SγN g πN / W p (cid:1) W − W p , (42)where subscript p stands for pole parameters.Using the above formulas, we can calculate the virtual-photon decay amplitudes A pole h , S pole1 / at the S N ∗ (890) pole position, which is Refs. [41, 42, 62]: A pole h = C (cid:115) | q p | k cm p π (2 J + 1) W p m N Res T πN Res A hα , (43) S pole1 / = C (cid:115) | q p | k cm p π (2 J + 1) W p m N Res T πN Res S / α . (44)Refer to appendix C for the definition of A hα , S / α , here we use A / = − E , S / = − (1 / √ S . Intuitively, A hα , S / α characterize the power of electromagnetic couplings andthe amplitudes of the “decay” process N ∗ → γ ∗ N . | q p | is the pion momenta at the pole. Thefactor C is (cid:112) / / −√ /
2. So if we focus only on S channel,the corresponding virtual-photon decay amplitudes are given by A pole1 / ( Q ) = g EγN (cid:115) πW p m N k cm p , S pole1 / ( Q ) = g SγN (cid:115) πW p m N k cm p . (45)It can be seen that the amplitudes, A / α and S / α , as well as the residues, A pole1 / and S pole1 / , arefunctions of the photon virtuality Q . According to Eqs. (40), N ∗ (890) residues or couplings, g γN g πN , can be extracted from multi-pole amplitudes E , S . In the meantime, g πN can be computed by using g πN (cid:39) T l ( s p ) / S (cid:48) l ( s p ),which was already obtained in Ref. [29]. We employed the MAID solution of the fit (The resultof DMT is similar), and chose central value √ s = 0 . − . i for pole position to extract poleresidues. T ( s p ) can be obtained through S ( s p ) = 1 + 2 iρ πN ( s p ) T ( s p ) = 0 and S (cid:48) ( s p ) is just theresidue of S II as explained Ref. [29].The values of the decay amplitudes A / and S / at the pole position are shown in Figs. 9.18igure 9: The blue solid line and the red solid line represent real and imaginary parts of virtual-photon decay amplitudes at pole position, respectively. In this paper, we have performed a careful dispersive analysis about the process of single pionelectroproduction off the nucleon in the S channel of the final πN system. In the dispersiverepresentation, the right-hand cut contribution can be related to an Omn`es solution, whichtakes the elastic πN phase shifts as inputs. At the same time, we estimate the left-hand cutcontribution by making use of the O ( p ) amplitudes taken from χ PT. A detailed discussionon virtual photoproduction amplitude at the level of multipoles is presented. Different fromRef. [17, 20], here we go beyond pure χ PT calculations by applying the final state interactiontheorem to partial wave amplitudes. To pin down the free parameters in the dispersive amplitude,we perform fits to experimental data of multipole amplitudes E and S for the energiesranging from πN threshold to 1 .
440 GeV . It is found that the experimental data can be welldescribed by the dispersive amplitude with only one free subtraction parameter, when Q ≤ . . As Q further increases to 0 . , the fit fails, similar to what happened in otherliterature [17, 46, 47].We also extend our analyses to study the property of S N ∗ (890) as proposed recently inthe literature. We analytically continue the dispersive amplitude to the second RS to extractthe couplings of N ∗ to the γ ∗ N and πN systems, and calculated the Q dependence of them.Particularly, the decay amplitudes A / ( Q ) and S / ( Q ) at the N ∗ (890) pole position areobtained. Our results serve providing further motivation for a new study of electromagneticproperties of N ∗ (890). 19 cknowledgments The authors would like to thank De-Liang Yao, Yu-Fei Wang and Wen-Qi Niu for helpfuldiscussions. This work is supported in part by National Nature Science Foundations of China(NSFC) under contract numbers 11975028 and 10925522.20 ppendices
A Invariant amplitudes A (+)1 = − eg A m N F (cid:18) s − m N + 1 u − m N (cid:19) − eg A c F ,A (+)2 = − eg A m N F t − m π (cid:18) s − m N + 1 u − m N (cid:19) ,A (+)3 = eg A m N c F (cid:18) s − m N − u − m N (cid:19) ,A (+)4 = eg A m N c F (cid:18) s − m N + 1 u − m N (cid:19) ,A (+)5 = − eg A m N F t − m π (cid:18) s − m N − u − m N (cid:19) ,A (+)6 = 0 ,A ( − )1 = − eg A m N F (cid:18) s − m N − u − m N (cid:19) ,A ( − )2 = − eg A m N F t − m π (cid:18) s − m N − u − m N (cid:19) ,A ( − )3 = eg A m N c F (cid:18) s − m N + 1 u − m N (cid:19) ,A ( − )4 = eg A m N c F (cid:18) s − m N − u − m N (cid:19) ,A ( − )5 = − eg A m N F t − m π (cid:18) s − m N + 1 u − m N (cid:19) ,A ( − )6 = 0 ,A (0)1 = − eg A m N F (cid:18) s − m N + 1 u − m N (cid:19) − eg A c F ,A (0)2 = − eg A m N F t − m π (cid:18) s − m N + 1 u − m N (cid:19) ,A (0)3 = eg A m N c F (cid:18) s − m N − u − m N (cid:19) ,A (0)4 = eg A m N c F (cid:18) s − m N + 1 u − m N (cid:19) ,A (0)5 = − eg A m N F t − m π (cid:18) s − m N − u − m N (cid:19) ,A (0)6 = 0 , (46)where the two LECs F and g A denote the chiral limit of pion decay constant and the axial-vectorcoupling constant, respectively. Here c and c are LECs of the O ( p ) chiral lagrangian.21 The relations between CGLN amplitudes and invariantamplitudes
The functions A i and F i are connected with each other as the following relation [45, 63]: F = (cid:0) √ s − m N (cid:1) N N π √ s × (cid:20) A + k · q √ s − m N A + (cid:18) √ s − m N − k · q √ s − m N (cid:19) A − k √ s − m N A (cid:21) , (47) F = (cid:0) √ s + m N (cid:1) N N π √ s | q || k | ( E + m N ) ( E + m N ) × (cid:20) − A + k · q √ s + m N A + (cid:18) √ s + m N − k · q √ s + m N (cid:19) A − k √ s + m N A (cid:21) , (48) F = (cid:0) √ s + m N (cid:1) N N π √ s | q || k | E + m N (cid:20) m N − s + k √ s + m N A + A − A − k √ s + m N A (cid:21) , (49) F = (cid:0) √ s − m N (cid:1) N N π √ s | q | E + m N (cid:20) s − m N − k √ s − m N A + A − A + k √ s − m N A (cid:21) , (50) F = N N π √ s | q | E + m N (cid:34) ( m N − E ) A − (cid:18) | k | k (cid:0) k √ s − k · q (cid:1) − q · k k (cid:0) s − m N − k (cid:1)(cid:19) A + (cid:0) q (cid:0) √ s − m N (cid:1) − k · q (cid:1) A + (cid:0) k · q − q (cid:0) √ s − m N (cid:1) + ( E − m N ) (cid:0) √ s + m N (cid:1)(cid:1) A + (cid:0) q k − k k · q (cid:1) A − ( E − m N ) (cid:0) √ s + m N (cid:1) A (cid:35) , (51) F = N N π √ s | k | E + m N (cid:34) ( m N + E ) A + (cid:18) | k | k (cid:0) k √ s − k · q (cid:1) − q · k k (cid:0) s − m N − k (cid:1)(cid:19) A + (cid:0) q (cid:0) √ s + m N (cid:1) − k · q (cid:1) A + (cid:0) k · q − q (cid:0) √ s + m N (cid:1) + ( E + m N ) (cid:0) √ s − m N (cid:1)(cid:1) A + (cid:0) q k − k k · q (cid:1) A − ( E + m N ) (cid:0) √ s − m N (cid:1) A (cid:35) , (52)with N i = (cid:112) E i + m N , E i = (cid:113) p i + m N , i = 1 , . (53) C Partial wave helicity amplitudes
In the following part, we introduce the partial wave helicity amplitude method of pion photo-and electroproduction [64].It is convenient to perform partial wave projection using the helicity formalism proposed inRef. [64, 65]. Here we define λ i ( i = 1 , , ,
4) which stand for the helicity of photon, initialnucleon, pion and final nucleon. For each set of helicity quantum numbers, denoted by H s ≡{ λ λ λ λ } , there is a helicity amplitude A H s , which can be expanded as A H s ( s, t ( θ )) = 16 π ∞ (cid:88) J = M (2 J + 1) A JH s ( s ) d Jλµ ( θ ) , (54)22able 3: Helicity amplitudes { A µλ ( θ ) } = { A H s } µ λ λ = +1 λ = − λ = 0
32 12 − −
32 12 − H H H − H H H − H H − H H H − H where M = max {| λ | , | µ |} , λ ≡ λ − λ and µ ≡ λ − λ = − λ , and d J ( θ ) is the standardWigner function. By imposing the orthogonal properties of the d functions, the partial wavehelicity amplitudes A JH s ( s ) in the above equation may be projected, i.e. A JH s ( s ) = 132 π (cid:90) − d cos θA H s ( s, t ) d Jλ,λ (cid:48) ( θ ) . (55)In particular, we use H i ( i = 1 ∼
6) as symbols to define the helicity amplitude. The relationsbetween A µλ and H i are listed in Table. 3 [64].The differential scattering cross section can be written asd σ dΩ = 12 | q | k cm (cid:88) λ i | A µλ | . (56)From Eq. (56) and Table. 3, we can integrate the angle dependence σ = 2 π | q | k cm (cid:88) J (cid:88) i =1 (2 j + 1) (cid:12)(cid:12) H Ji (cid:12)(cid:12) . (57) A Jµλ ( H Ji ) has definite angular momentum but cannot be determined in parity. Therefore,we can add the final state with the opposite helicity µ, − µ to obtain the so-called partial wavehelicity parity eigenstates A l + = − √ (cid:16) A J , ( λ =1) + A J − , ( λ =1) (cid:17) ,A ( l +1) − = 1 √ (cid:16) A J , ( λ =1) − A J − , ( λ =1) (cid:17) ,B l + = (cid:115) l ( l + 2) (cid:16) A J , + A J − , (cid:17) (cid:96) ≥ ,B ( l +1) − = − (cid:115) l ( l + 2) (cid:16) A J , − A J − , (cid:17) (cid:96) ≥ ,S l + = − Q | k | ( l + 1) (cid:16) A J , ( λ =0) + A J − , ( λ =0) (cid:17) ,S ( l +1) − = − Q | k | ( l + 1) (cid:16) A J , ( λ =0) − A J − , ( λ =0) (cid:17) . (58)Notice that the normalization coefficients we use here are different from those in Refs. [66,67], and J = l + 1 / J = l − / − ’ amplitudes. A, B and S represent amplitudes with initial helicity of 1 / , / , /
2, respectively, so it can also be writtenas A / , A / and S / up to some normalization factors, see Eq. (62).23urthermore, with the definitions of Eqs. (54) and (55), then we can obtain H = 1 √ θ cos θ (cid:88) (cid:0) B l + − B ( l +1) − (cid:1) (cid:0) P (cid:48)(cid:48) l − P (cid:48)(cid:48) l +1 (cid:1) ,H = √ θ (cid:88) (cid:0) A l + − A ( l +1) − (cid:1) (cid:0) P (cid:48) l − P (cid:48) l +1 (cid:1) ,H = 1 √ θ sin θ (cid:88) (cid:0) B l + + B ( l +1) − (cid:1) (cid:0) P (cid:48)(cid:48) l + P (cid:48)(cid:48) l +1 (cid:1) ,H = √ θ (cid:88) (cid:0) A l + + A ( l +1) − (cid:1) (cid:0) P (cid:48) l + P (cid:48) l +1 (cid:1) ,H = Q | k | cos θ (cid:88) ( l + 1) (cid:0) S l + + S ( l +1) − (cid:1) (cid:0) P (cid:48) l − P (cid:48) l +1 (cid:1) ,H = Q | k | sin θ (cid:88) ( l + 1) (cid:0) S l + − S ( l +1) − (cid:1) (cid:0) P (cid:48) l + P (cid:48) l +1 (cid:1) . (59)According to the expansion method of CGLN [1], the relationship between helicity amplitudesand CGLN multipole amplitudes can also be obtained [1, 68]: H = − √ θ cos θ F + F ) ,H = √ θ (cid:20) ( F − F ) + 12 (1 − cos θ )( F − F ) (cid:21) ,H = 1 √ θ sin θ F − F ) ,H = √ θ (cid:20) ( F + F ) + 12 (1 + cos θ )( F + F ) (cid:21) ,H = cos θ F + F ) ,H = − sin θ F − F ) . (60)Compare Eqs. (59) and (60) with the CGLN expansion, we have A l + = 12 [( l + 2) E l + + lM l + ] ,B l + = E l + − M l + ,A ( l +1) − = − (cid:2) lE ( l +1) − − ( l + 2) M ( l +1) − (cid:3) ,B ( l +1) − = E ( l +1) − + M ( l +1) − . (61) A h , S / can be related to the resonant part of the corresponding multipole amplitudes at the24ole position in the following way: A / l + = −
12 [( l + 2) E l + + lM l + ] , A / l + = 12 (cid:112) l ( l + 2) ( E l + − M l + ) , S / l + = − l + 1 √ S l + , A / l +1) − = − (cid:2) lE ( l +1) − − ( l + 2) M ( l +1) − (cid:3) , A / l +1) − = − (cid:112) l ( l + 2) (cid:0) E ( l +1) − + M ( l +1) − (cid:1) , S / l +1) − = − l + 1 √ S ( l +1) − . (62)The scattering cross section is written in terms of A hα as σ T = (cid:16) σ / T + σ / T (cid:17) + (cid:15)σ L ,σ hT = 2 π | q | k cm (cid:88) α ( (cid:96),J ) (2 J + 1) (cid:12)(cid:12) A hα (cid:12)(cid:12) ,σ L = 2 π | q | k cm Q k (cid:88) α ( (cid:96),J ) (2 J + 1) (cid:12)(cid:12)(cid:12) S / α (cid:12)(cid:12)(cid:12) , (63)where superscript h stands for helicity. Expand the above formula, it can be obtained: σ / T = 2 π | q | k cm (cid:88) l + 1) (cid:104) | A l + | + (cid:12)(cid:12) A (1+1) − (cid:12)(cid:12) (cid:105) ,σ / T = 2 π | q | k cm (cid:88) l l + 1)( l + 2) (cid:104) | B l + | + (cid:12)(cid:12) B ( l +1) − (cid:12)(cid:12) (cid:105) ,σ L = 4 π | q | k cm (cid:88) Q k ( l + 1) (cid:104) | C l + | + (cid:12)(cid:12) C ( l +1) − (cid:12)(cid:12) (cid:105) . (64)25 eferences [1] G. F. Chew, M. L. Goldberger, F. E. Low, and Y. Nambu, Phys. Rev. , 1345 (1957).[2] S. L. Adler, Annals Phys. , 189 (1968).[3] E. Amaldi, S. Fubini, and G. Furlan, Springer Tracts Mod. Phys. , 1 (1979).[4] D. Drechsel and L. Tiator, J. Phys. G , 449 (1992).[5] V. Pascalutsa, M. Vanderhaeghen, and S. N. Yang, Phys. Rept. , 125 (2007).[6] I. Aznauryan and V. Burkert, Prog. Part. Nucl. Phys. , 1 (2012).[7] D. R¨onchen et al. , Eur. Phys. J. A , 101 (2014), [Erratum: Eur.Phys.J.A 51, 63 (2015)].[8] S. Fubini, Y. Nambu, and V. Wataghin, Phys. Rev. , 329 (1958).[9] F. A. Berends, A. Donnachie, and D. L. Weaver, Nucl. Phys. B4 , 1 (1967).[10] J. S. Ball, Phys. Rev. , 2014 (1961).[11] R. Devenish and D. Lyth, Phys. Rev. D , 47 (1972), [Erratum: Phys.Rev.D 6, 2067 (1972)].[12] R. Crawford and W. Morton, Nucl. Phys. B , 1 (1983).[13] D. Drechsel, S. S. Kamalov, and L. Tiator, Eur. Phys. J. A34 , 69 (2007).[14] M. Doring and K. Nakayama, Eur. Phys. J. A , 83 (2010).[15] A. Gasparyan and M. Lutz, Nucl. Phys. A848 , 126 (2010).[16] V. Bernard, N. Kaiser, T. S. H. Lee, and U.-G. Meissner, Phys. Rept. , 315 (1994).[17] M. Hilt, B. C. Lehnhart, S. Scherer, and L. Tiator, Phys. Rev.
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