Table Of Content

Deep exclusive charged π electroproduction above the resonance region Murat M. Kaskulov and Ulrich Mosel ∗ Institut fu¨r Theoretische Physik, Universita¨t Giessen, D-35392 Giessen, Germany (Dated: January 12, 2010) A description of exclusive charged pion electroproduction (e,e′π±) off nucleons at high energies isproposed. ThemodelcombinesaReggepoleapproachwithresidualeffectofnucleonresonances. Theexchangesofπ(140),vectorρ(770)andaxial-vectora (1260)andb (1235)Reggetrajectoriesare 1 1 considered. The contribution of nucleon resonances is described using a dual connection between the exclusive hadronic form factors and inclusive deep inelastic structure functions. The model describesthemeasuredlongitudinal,transverseandinterferencecrosssectionsatJLABandDESY. Thescaling behaviorof thecross sections is in agreement with JLAB and deeplyvirtual HERMES data. Theresultsforapolarizedbeam-spinazimuthalasymmetryin(~e,e′π±)arepresented. Model 0 predictions for JLAB at 12 GeV are given. 1 0 PACSnumbers: 12.39.Fe,13.40.Gp,13.60.Le,14.20.Dh 2 n a I. INTRODUCTION see Ref. [4] and references therein. Previous measure- J ments [1, 2, 11] at smaller and much higher values of Q2 2 At JLAB the exclusive reaction p(e,eπ+)n has been show a similar problem in the understanding of σT. Al- 1 ′ readyfromvaluesofQ2 >0.6GeV2 the meson-exchange studied for a wide range of photon virtualities Q2 at an and/or Regge pole models are not compatible with the invariant mass of the π+n system around the onset of ] measuredinterferenceσ andσ crosssectionsandthe h deep inelastic scattering (DIS) regime, W 2 GeV [1– TT LT p 5]. A separation of the cross section int≃o the trans- extraction of the pion form factor relies on the fit to the - longitudinal crosssection σ only [5]. A remarkably rich p verse σT, longitudinal σL and interference σTT and σLT L experimental data base obtained for N(e,eπ)N above e components has been performed. The CLAS data for ′ ′ the resonance region remains unexplained [3, 4]. On the h the polarizedbeamsingle-spinasymmetryinp(~e,eπ+)n [ are also available [6]. The HERMES data at DES′Y [7] other hand, a detailed knowledge of the p(e,e′π+)n re- 1 extend the kinematic region to much higher values of action above the resonances √s > 2 GeV is mandatory v W2 >10GeV2 towardthetrueDISregionQ2 1GeV2 fortheinterpretationofthecolortransparencysignalob- ≫ served in this reaction off nuclei [23, 24]. 2 and much higher values of t. The cross section for 5 p(e,eπ+)n has also been mea−sured above the resonance A possible description of σT at JLAB has been pro- ′ 9 posed in Ref. [25]. The approach followed there is to regionattheCambridgeElectronAccelerator(CEA)[8], 1 inp(e,eπ+)nandn(e,eπ )pattheWilsonSynchrotron complement the hadron-like interaction types in the t- . ′ ′ − channel,whichdominateinphotoproductionandlowQ2 1 Laboratory at Cornell [9–11] and DESY [12–16]. 0 electroproduction, with the direct interaction of virtual The longitudinal cross section σ is generally thought 0 L photonswithpartonsfollowedbystring(quark)fragmen- to be well understood in terms of the pion quasi-elastic 1 tation into π+n. Then σ can be readily explained and v: klonwockotut. mIfecthruane,ismthis[17m]abkeecsaiutsepoosfsitbhlee tpoiosntupdoylethaet both σL and σT can be Tdescribed from low up to high Xi charg−e form factor of the pion at momentum transfer values of Q2. In [25] the reaction p(e,e′π+)n is treated as an exclusive limit, z 1, of semi-inclusive DIS r much bigger than in the scattering of pions from atomic → a electrons [18]. On the other hand, the transverse cross p(e,e′π+)X z→1p(e,e′π+)n (1) sectionσ is predicted to be suppressedby 1/Q2 with −→ T respecttoσ forsufficientlyhighvaluesofQ2∼andW [19]. inthespiritofanexclusive-inclusiveconnection[26]. The L Onthe experimentalside,however,the JLAB datashow transverse cross section in n(e,e,π )p has been pre- ′ − that at forward angles σ is large. For instance, at dictedtobesmallerthaninp(e,e,π+)n. Themodel[27] T ′ Q2 =3.91GeV2 [3] σ is by abouta factor oftwo larger hasalsobeen appliedto valuesof(Q2,W) inthe DIS re- T than σ and at Q2 = 2.15 GeV2 it has same size as σ gionatHERMES[7]. In[27]σ inDISgetsmuchsmaller L L T in agreement with previous JLAB measurements [1]. intheforwardπ+ production,butstilldominatestheoff- There is a long standing issue concerning the reaction forward region. mechanisms contributing to deeply virtual π electropro- However,in [25, 27] the transversecrosssection σT it- duction above the resonance region [20–22]. The models self was modeled and the solution of the problem on the which describe (e,eπ ) in terms of hadronic degrees of amplitude level is still missing. Both the soft hadronic ′ ± freedomfailtoreproduceσ observedinthese reactions, and hard partonic parts of the amplitude can in prin- T ciple interfere making non-additive contributions to σ L and to interference σ and σ cross sections. One TT LT might describe this transverse strength in the language ∗Electronicaddress: [email protected] of perturbative QCD by considering higher twist correc- 2 tions to a generalized parton distribution (GPD) based φ handbag diagram. This approach has been followed in Ref. [28] where p(γ ,π+)n is considered using the hand- π ∗ e’ bag approach with a π-pole contribution. Indeed, the data from JLAB demonstrate [3, 4] that the magnitude and sign of the interference cross sections are not compatible with the simple exchange of a pion trajectory in γ z the t-channel. Because,the contributions from exchange of heavy mesons are small [25] this would suggest the scattering plane N’ presence of a large transverse resonance or partonic in- e terfering backgroundto the meson-pole contributions. Inthis workwe attempta phenomenologicalapproach reaction plane to model the hard scattering or, using a duality argu- ment, the presence of nucleon resonances beyond the t-channel meson-pole amplitudes. The meson-exchange FIG. 1: Exclusive reaction N(e,e′π)N′ in the laboratory. φ processes dominate in high-energy photoproduction and standsfortheazimuthalanglebetweentheelectronscattering low Q2 electroproduction above the resonance region. (e,e′) plane and reaction N(γ∗,π)N′ plane. Onewaytodescribethis regionistoassumethattheco- herent sum of baryon resonance contributions would be also affects the longitudinal cross section making the π- expectedbydualityargumentstobeequivalenttoasum poledominanceinthe longitudinalresponseratherques- over t-channel Regge trajectories. However, in electro- tionable. Based on a quantitative description of electroproduction, with plausible assumptions concerning the production data achieved in this work in a large range coupling constants and transition form factors, the ex- of (Q2,W) from JLAB to DIS region at HERMES the changeofheavymesonsalonedoesnotexplainthetrans- present results may assist in the experimental analysis versecrosssectionandturnsouttobemarginal[25]. Itis and extraction of the pion charge form factor to mini- also a generic rule that single t-channel meson-exchange mize systematic uncertainties. Recall that it is essential processes vanish in the forward π+ direction. On the to use theoretical model input for the extraction of the otherhand,pionexchangedoesplayanimportantroleat pion form factor [5]. nearforwarddirectionsandmustbeincluded,asmustbe The outline of the present paper is as follows. In the the nucleon-polechargetermtosatisfygaugeinvariance. Section II we briefly recall the kinematics and definition The nucleon magnetic transitions vanish in the forward of the cross sections in exclusive (e,eπ ) electroproduc- ′ ± production and can be neglected [29]. For instance, in tion reaction. In Section III we discuss our treatment of photoproduction this suggests an extreme phenomeno- gauge invariant π-exchange in the Regge pole model. In logical scenario, known as an electric model, where the Section IV we consider the effect of nucleon resonances only relevant contributions to π production at forward ± andderivethetransitionformfactorsusinganexclusive- angles are the ones from π-exchange and the nucleon inclusive connection. In Section V we consider the con- Born term where the inclusion of the latter is manda- tribution of vector ρ(770) and axial-vector a (1260) and 1 tory to conserve gauge invariance. b (1235)Reggetrajectories. InSectionVIwebrieflydis- 1 By reggeization of the π-exchange one takes into ac- cuss the π photoproduction at forward angles. Then ± count higher mass and higher spin excitations. At for- the model is extended to electroproduction. In Sections ward angles considered here the momentum transfer t VII-IXtheresultsarecomparedtotheexperimentaldata − is small and the exchanged π-trajectory is close to its fromJLAB,DESYandCornell. InSectionXwecompare first materialization. However, the nucleons in the s(u)- our results with the HERMESdata. The Q2 behaviorof channel pole amplitudes are highly off-mass-shell and the crosssections is studied in SectionXI. The polarized with increasing values of (Q2,W) the effect of nucleon beam-spin asymmetry and the role played by the axial- resonances should become more and more important. vectormesonsin(~e,eπ )arediscussedinSectionXII.In ′ ± Thisisbecauseofthewellknownhardeningofthehigher Section XIII the model predictions for JLAB at 12 GeV mass resonance transition form factors which must re- are presented. The conclusions are summarized in Sec- spect the scaling properties of deep inelastic structure tion XIV. Some details of the calculations are relegated functions in inclusive scattering [30, 31]. We shall follow to the Appendix. thissuggestionandmodelthecontributionofnucleonres- onancesusingalocalBloom-Gilmanconnectionbetween the exclusive and inclusive processes. II. KINEMATICS AND DEFINITIONS Another question which we address here is a possible contribution of the resonance (or partonic) background We recallbriefly the kinematics in exclusive π electro- to the longitudinal cross section σL which is presently production used to get the information about the pion form factor. Indeed, the same resonance/partonic background e(l)+N(p) e(l )+π(k )+N (p), (2) ′ ′ ′ ′ ′ → 3 and specify the notations and definitions of variables. is the ratio of longitudinal to transverse polarization of The reaction (2) in the laboratory is shown in Figure 1 the virtual photon. The longitudinal/transverse (l/t) where the target nucleonis at rest, the z-axisis directed separatedvirtual-photonnucleoncrosssectionsaregiven along the three momentum ~q = (0,0, ν2+Q2) of the in Appendix A. The t-differential cross section for exchanged virtual photon γ∗ with q = l l′ = (ν,~q), N(γ∗,π)N′ integrated over φ is denoted here as p − Q2 = q2, ν = E E and l(l ) is the four momentum − e− e′ ′ of incoming (deflected) electrons. In Figure 1 φ stands dσ dσ dσ U T L for the azimuthal angle between the electron scattering = +ε . (6) dt dt dt (e,e) plane and γ N πN reaction plane. φ is zero ′ ∗ ′ → when the pion is closest to the outgoing electron [32]. In exclusive reaction(e,eπ) we shall deal with an un- The longitudinal beam single-spin asymmetry (SSA) ′ polarizedtargetand,bothunpolarizedandpolarizedlep- in (~e,eπ) scattering is defined so that ′ ton beams. The differential cross section is given by dσ (φ) dσ (φ) dσ = Φ dσT +εdσL ALU(φ)≡ dσ→(φ)+−dσ←(φ), (7) dQ2dνdtdφ 2π dt dt → ← (cid:20) dσ LT + 2ε(1+ε) cos(φ) where dσ refers to positive helicity h = +1 of the in- dt → comingelectron. Theazimuthalmomentassociatedwith pdσ TT + ε cos(2φ) the beam SSA is given by [32] dt dσLT′ + h 2ε(1 ε) sin(φ) , (3) − dt (cid:21) Asin(φ) = 2ε(1−ε)dσLT′. (8) p LU dσ +εdσ p T L wheredσ isthetransversecrosssection,dσ isthelongi- T L III. GAUGING THE PION EXCHANGE tudinalcrosssection,dσ isthecrosssectionoriginating TT fromtheinterferencebetweenthetransversecomponents of the virtual photon, dσ is the cross section arising The diagrams describing the π+ and/or π electro- LT − from the interference between the transverse and longi- production amplitudes in exclusive reactions p(e,eπ+)n ′ tudinal polarizations of the virtual photon and dσLT′ is and n(e,e′π−)p are shown in Figure 2. At high energies the beam-spin polarized cross section resulting from the the particles exchanged in the t-channel are understood interferencebetweenthetransverseandlongitudinalpho- as the Regge trajectories. In p(e,eπ+)n the s-channel ′ tons and helicity h= 1 of the incoming electron. nucleon-poleterm(ibdiagram)isaddedtothet-channel ± Thevirtualphotonfluxisconventionallydefinedas[33] π-pole exchange (ia diagram) to conserve the charge of thesystem. Similarly,thediagramiiaandtheu-channel π αe Ee ν 1 nucleon-pole diagram iib form a gauge invariant ampli- Φ= − K , (4) Ee(Ee−ν)(cid:18)2π2 Ee Q21−ε(cid:19) tude in n(e,e′π−)p. The last two diagrams (iii and iv) correspond to the exchange of vector V = ρ(770) and with α 1/137, =(W2 M2)/2M and axial-vector A=a (1260), b (1235) Regge trajectories. e ≃ K − N N 1 1 The π-exchange currents describing the reactions 1 ε= (5) p(γ ,π+)n and n(γ ,π )p take the form [24] 1+2 ν2+Q2 ∗ ∗ − 4Ee(Ee−ν)−Q2 (k+k )µ (p+q) γσγµ+M γµ −iJsµ(γ∗p→π+n)=√2gπNNu¯s′(p′)γ5 Fγππ(Q2,t)t m2 +′ i0+ +Fs(Q2,s,t) s σM2+i0+p (cid:20) − π − p (k k )µ +[ (Q2,t) (Q2,s,t)] − ′ u (p), (9) Fγππ −Fs Q2 s (cid:21) (k+k )µ γµ(p q) γσ+M γµ −iJuµ(γ∗n→π−p)=−√2gπNNu¯s′(p′) Fγππ(Q2,t)t m2 +′ i0+ −Fu(Q2,u,t) u′− Mσ2+i0+ p (cid:20) − π − p (k k )µ +[ (Q2,t) (Q2,u,t)] − ′ γ u (p), (10) Fγππ −Fu Q2 5 s (cid:21) where (Q2,t) denotes the transition form factor of the pion and (Q2,s(u),t) stands for the proton s(u)- γππ s(u) F F channel transition form factor. In Eqs. (9) and (10) g = 13.4 is the pseudoscalar πN coupling constant, t = k2, πNN s=W2, k=k q =p p and other notations are obvious. ′ ′ − − 4 e’ (k’) π e’ π e (q) γ π N’ e γ (k) (p’) N’ N N e’ e’ (p) π π Ia Ib e e γ V N’ γ A N’ e’ e’ π e e γ N N γ π N’ N’ III IV N N IIa π IIb FIG. 2: The diagrams describing the π+ and/or π− electroproduction amplitudes in exclusive reactions p(e,e′π+)n and n(e,e′π−)p. In p(e,e′π+)n the s-channel nucleon-pole term (ib diagram) is added to the t-channel π-pole exchange (ia diagram)toconservethechargeofthesystem. Similarly,thediagramiiaandtheu-channelnucleon-polediagramiibformgauge invariant amplitude in n(e,e′π−)p. The last two diagrams (iii and iv) correspond to the exchange of vector V = ρ(770) and axial-vector A=a (1260) and b (1235) Regge trajectories. The momentum flows are shown in the diagram ia. 1 1 The amplitudes are gaugeinvariantand the currentconservationcondition, q Jµ =0, is satisfiedin the presence µ s(u) ofdifferentformfactors, and , whichingeneralcandependonvaluesof(Q2,s(u),t). Eqs.(9)and(10)are γππ s(u) F F obtained using the requirement that the modified electromagnetic vertex functions entering the amplitude obey the same Ward-Takahashi identities as the bare ones [34–36]. Further aspects concerning the gauged electric amplitude are relegated to Appendix B. At highenergiesthe exchangeofhigh-spin andhigh-massparticles lying onthe π-Reggetrajectoryhas to be taken into account. Then, to continue the electric amplitude to high energies we define the half off-shell form factor (Q2,t)=F (Q2)(t m2) (α (t)). (11) Fγππ γππ − π R π In the π-pole term this procedure replaces the Feynman propagator by the Regge propagator suggested by the high energy limit of the amplitude D(t)= t m21+i0+ =⇒R(απ(t))= 1+e−2iπαπ(t) (−α′π)Γ[−απ(t)]eαπ(t)ln(α′πs), (12) − π (cid:20) (cid:21) where α (t) = α (t m2) is the Regge trajectory of π with a slope α = 0.74 GeV 2 and Γ function results from suppressπionofsin′πgul−aritiπesinthe physicalregion. Closeto the poleposi′πtiont m2 th−e Reggepropagatoris reduced → π to 1/(t m2) and we approachthe Feynman amplitude describing the first π(140) materialization of the trajectory. − π We further treat the nucleon-pole part as an indispensable part of the π-pole amplitude. At the real photon point gaugeinvariancerequiresforthenucleon-poletermthesamephaseandt-dependenceasintheπ-Reggeamplitude[29] (Q2,s(u),t)=F (Q2,s(u))(t m2) (α (t)). (13) Fs(u) s(u) − π R π Thisassumptionisjustifiedbytheobservationthatthereexistsagaugewheretheπ-exchangevanishesandtheπ-pole contribution is generated kinematically by the nucleon-pole term itself [37]. For the pion transition form factor F we use a highvalues of t [38]. In the forwardπ+ production the γππ − monopole parameterization momentum transfer t is rather small and the exchanged pion is close to its mass shell. In the fit to data we shall Fγππ(Q2)=[1+Q2/Λ2γππ]−1, (14) not allow large deviations from the VMD value. withacut-offΛ asafitparameter. Ingeneral,thecut- Since the π-pole contribution is replaced by an ex- γππ off can be a function of t, Λ =Λ (t), reflecting the change of reggeon-pion, the relation to the on-shell pion γππ γππ off-shellnessofthepioninthet-channelandtheunderly- form factor might be lost [39] and F (Q2) should be γππ ing space-time pattern of direct partonic interactions at understood as an effective transition form factor. 5 IV. EFFECT OF NUCLEON RESONANCES A resonance with mass M contributes to the structure i function at Bjorken x =Q2/(M2 M2+Q2). i i − p Similar argumentsshould apply to the transitionform Tobe inline withmeasurementsinthe DISregionthe factorinthes(u)-channelnucleon-poleterms. Asimplest resonance form factors F(Q2,Mi2) must fall with Q2 at choice would be to use in Eq. (13) least as fast as the nucleon dipole form factor. Futher- more, to be consistent with the scaling behavior of deep Fs(u)(Q2,s(u))=F1p(Q2), (15) inelasticstructurefunctionsthecut-offinthedipoletran- sition form factors must increase as the mass of the res- where Fp(Q2) is the protonDirac form factor. However, 1 onance is increases [30, 42]. Therefore we assume [41] since the nucleon is highly of-mass-shellthis assumption might be too naive [40]. Indeed, this prescription un- 2 derestimates the JLAB data for σ [25] and results in a 1 T F(Q2,M2)= , (20) wrong interference pattern between l/t components. i 1+ξQ2  A way to model an intermediate state which is highly Mi2   off-mass-shell is to increase the Fock space available for where the value ofξ is a commonaveragecut-off param- the virtual nucleon allowing the latter to excite into res- eter. Thisscenariosuggestsahardeningoftheresonance onances. Similar to the reggeized-exchange, these res- form factors with increasing value of M [42]. i onances with higher masses and spins may lie on the Athigh energiesthe leveldensity ofresonancesρ(M2) nucleon Regge trajectory or correspond to higher mass i is large and we can replace the sum in Eq. (18) over states with the same angular momentum as the nucleon. discrete spectrum of resonances by a continuous integral We replace the Born term in the s-channel for π+ production by a sum over all resonance excitations ∞ dM2ρ(M2). (21) sF−s(MQp22,+Mip0)+ → i r(Mi)c(Mi)sF−(MQ2i2,M+ii20)+, (16) Xi →MZp2 i i X where the sum runs from the nucleon-pole contribution, This is clearly a rough approximation in the resonance M is the ith resonance mass, r and c are the electro- region itself, but it makes no difference when we restrict i magnetic and strong couplings, respectively, relative to ourselves to the experimental data above the resonance the lowest lying nucleon state, e.g. r(Mp)c(Mp) = 1. region. Performing the integration over Mi yields For π production we use a similar expansion over the − Fp(x ,Q2)=(s M2+Q2)r2(s)[F(Q2,s)]2ρ(s). (22) u-channel contributions 2 B − p Fu(Q2,Mp) r(M )c(M ) F(Q2,Mi2) . (17) The structure function F2p can be written in the form u−Mp2+i0+ →Xi i i u−Mi2+i0+ Bofloaopmo-lGynilommainalvianria1b−le1./Aωs′ Qwh2/eWre2ω′ = 1+orWω′2/Q21itshae →∞ → In the region of interest for the experiments to be dis- leading term yields the Drell-Yan-West behavior cussed later in this paper the invariant mass is W > 2 GeVandthusinaregionwheretheDISregimestarts∼. In F2p(ω′)∝(ω′−1)3, (23) order to make a connection to our earlier work in which whichshowsthatthepowerlawbehavioroftheformfac- we modeled the transverse cross section by a partonic torisrelatedtothesuppressionofthestructurefunctions subprocess [25, 27] we now invoke duality for the exclu- in the limit where one quark carries all of the hadron’s sive processes treated here. We start with the Bloom- momentum. The approximation (23) is supposed to be Gilman duality [31] in the local form reasonable down to x 0.2. Expanding the resonance B ≃ F2p(xB,Q2)= (Mi2−Mp2+Q2)W(Q2,Mi)δ(s−Mi2), form factors for ω′ →1 the leading term reads i X (18) F(Q2,s)= (ω′−1)2 + ((ω′ 1)3). (24) where x stands for a Bjorken scaling variable and the ξ2 O − B deepinelasticstructurefunction Fp(x ,Q2)is expressed 2 B Thedualityrelation,Eq.(22),canbewrittenintheform as a sum of resonances. In Eq. (18) the hadronic basis is used as a substitute for the quark basis. When Q2 is (ω 1)4 large the bulk structure of the resonances becomes less (ω′ 1)3 Q2 ′− r2(s)ρ(s). (25) − ∝ ξ4 and less important and we are justified when taking the zero-width approximation [41]. W(Q2,M ) defines the i This translates into ith resonance contribution to the γ p forwardscattering ∗ 1 1 amplitude; it is essentially the electromagnetic coupling r2(s)ρ(s) = . (26) constantr(Mi) times a resonanceformfactor F(Q2,Mi) ∝ Q2(ω′−1) s normalized to unity at Q2 =0 [42]: Since the level density grows with increasing s, for in- W(Q2,M )=r2(M )[F(Q2,M )]2, F(0,M )=1. (19) stance, ρ exp(const M ), the coupling strength to i i i i i ∝ × 6 resonances is decreasing, i.e, r(s ) (s ρ(s )) 1/2 where The s- and u-channel form factors read i i i − s = M2. This simple result has∝a remarkable conse- i i 2 qcnouannetncreciseb.udteeAcrtletoahsotehusegahsstar1un/cstiunwrfiietnhiftueinncctrtoeiwoanesrinthogfevraweleusoiegnhoatfnsoc.efsrecsaon- MZ∞2 dsis−ssi−i+β i0+ 1+1ξQsi2! A vanishing coupling of the higher spin (mass) reso- Fs(Q2,s) = p ,(31) nances to πN is expected from the chiral phenomenol- ∞ds s−i β ogy[43]. Thelatter claimis consistentwithourobserva- i s s +i0+ Z − i tionthatthemoreweexcitethenucleonthelessitdecays M2 p into the exclusive channel. Assuming for the strong cou- 2 pthlienginatesgimraitliaornfoirnmEcq(ss.i)(1∝6)(sa(in2dβ−(11)7ρ)(siis))s−u1p/e2rcwointhveβrg≥en1t MZ∞2 dsiu−ssi−i+β i0+ 1+1ξQsi2! and can be carried out analytically. Without an explicit F (Q2,u) = p ,(32) u assumption about the behavior of the level density we ∞ds s−i β get the following form for the product i u s +i0+ Z − i 1 Mp2 ρ(si)r(si)c(si)= λs−i β, (27) Because of the singularity at s = s+i0+ the s-channel i where λ is a normalization constant and β 1 accounts integrals develop an imaginary part which is missing in ≥ for the behaviour of coupling constants as well as a de- the u-channel contribution where the branch point sits viation of the level density contributing to the exclusive in the unphysical region. channelcomparedtothetotalinclusivedensityofstates. Concerning the terminology for regionslike the one at We now absorb all the effects of the higher lying reso- JLAB it would be appropriate to use the words resonances into the nucleon-pole term by setting nance effect. On the other hand, in the DIS region at HERMES it would be more natural to describe the ef- F(Q2,M2) r(M )c(M ) i = fect as of partonic origin. Since both descriptions are i i s M2+i0+ ⇒ i − i dual in the context of the present approach we shall in X line of [25, 27] refer to the terms derived above as the ∞ F(Q2,M2) dM2ρ(M2)r(M2)c(M2) i resonance/partonic (r/p) contributions. i i i i s M2+i0+ Z − i M2 p = ∞ds s−i β F(Q2,si) Fs(Q2,s) , (28) V. VECTOR AND AXIAL-VECTOR REGGE i λ s s +i0+ ≡ s M2+i0+ TRAJECTORIES Z − i − p M2 p The mesonic Regge trajectories can be characterized wherethesuminEq.(16)overdiscretespectrumofreso- by the signature and parity. The signature determines nances has been replacedagainby a continuous integral. whether the Regge polesin the scatteringamplitude will F (Q2,s)istheformfactoronther.h.s. ofEq.(13). Sim- s ilarlyweproceedforthetransitionformfactorF (Q2,u) u in the u-channel, Eq. (17). The integration covers the 6 full region from the nucleon pole Mp up to . Further- ρ− − more, the normalization constants λ are de∞termined by 5 5 the chargeconservationatthe realphotonpointQ2 =0, i.e. 4 a4+ + π4− + λ(cid:12)(cid:12)s−channel =(s−Mp2)MZ∞p2 dsi ρ(ss−i)sri(s+i)ic0(+si), (29) α (t)23 a2+ + ρ3− − π− + b3+ - a3+ + (cid:12) 2 λ(cid:12)(cid:12)(cid:12)u−channel =(u−Mp2)MZ∞p2 dsi ρu(s−i)sr2i(s+i)ci0(s+i), (30) 01 ρ1π− −− + b1+ -a1+ + forthes-andu-channels,respectively. Thismerelyguar- 0 antees that the effective form factors are normalized to 0 1 2 3 4 5 6 unity, e.g. F (0,s(u)) = 1. With this prescription we t [GeV2] s(u) demand that the contributions of resonances show up in the modified off-mass-shellbehavior of the nucleontran- FIG. 3: (Color online) ρ(770)/a (1320) (solid), π/b (1235) 2 1 sition form factors. (dashed) and a (1260) (dash-dotted) Regge trajectories. 1 7 occur for even or odd positive integer value of the tra- photons polarizedparallel to the reactionplane, one can jectory α(t) = J (the spin). The leading mesons con- directly access the difference between recoil and the po- tributing to (e,eπ ) are the natural P = ( 1)J parity larized target asymmetries which is proportional to the ′ ± vector ρ(770) and the unnatural P = ( 1−)J+1 parity exchangeofthea (1260)-trajectory[44]. However,being 1 axial-vector mesons a1(1260) and b1(1235−). The Regge suppressed by the Regge factor e−α′a1ln(α′a1s)m2a1 at trajectories α(t) considered here are shown in Figure 3 t = 0, its contribution to the forw∼ard unpolarized cross The absolute contribution of the reggeizedρ-exchange section turns out to be small. With our choice of the amplitude to (e,e′π±) turns out to be small, but by b NN tensor coupling the contribution of b (1235) ex- 1 1 its interference with the s- and u-channel terms consid- change is even smaller. On the other hand, as we shall eredabove,itisresponsiblefortheπ−/π+ asymmetryin see, it is absolutely essential to consider the exchange photoproduction and gives a sizable contribution to the of a (1260) Regge trajectory in the polarization (~e,eπ) 1 ′ π−/π+ ratio in electroproduction. observableslikethebeamspinazimuthalasymmetrycon- In the axial-vector sector, the experimental isolation sidered in the following. Other aspects related to a pos- of the amplitudes with axial-vector quantum numbers sible role of a (1260) in (e,eπ) are discussed in [45]. 1 ′ would be of great interest. For instance, using a proton targetpolarizedperpendiculartothereactionplane,and A. Vector-isovector IG(JPC)=1+(1−−) exchange currents The currents Jµ describing the exchange of the natural parity ρ(770)-meson Regge trajectory are given by ρ iJµ(γ p π+n) − ρ ∗ → = i √2GργπGρNNFργπ(Q2)εµναβqνkαu¯s′(p′) (1+κρ)γβ κρ (p+p′)β us(p) −iJρµ(γ∗n→π−p) − (cid:20) − 2Mp (cid:21)   × 1−e−2iπαρ(t) −α′ρ Γ[1−αρ(t)]eln(α′ρs)(αρ(t)−1) (33) (cid:20) (cid:21) (cid:0) (cid:1) where G =3.4 and κ =6.1 are the standard vector and anomalous tensor coupling constants, respectively. The ρNN ρ ρ-trajectoryadoptedherereadsα (t)=0.53+α twithaslopeα =0.85GeV 2. TheΓfunctionin(33)containsthe ρ ′ρ ′ρ − pole propagator 1/sin(παρ(t)) but no zeroes and the amplitude zeroes only occur through the factor 1 e−iπαρ(t). ∼ − The ργπ coupling constants G can be deduced from the radiative γπ decay widths of ρ ργπ Γ(ρ γπ )= αeG2ργπ m2 m2 3. (34) ± → ± 24 m3 ρ− π ρ (cid:0) (cid:1) FTohretmheeatsruanresditiwonidftohrm[46fa]:ctΓoρr±F→γπ±(Q=2)(w68e±us7e)akVeVM,DwhmeoredetlhFe cen(tQra2l)v=al(u1e+coQrr2e/sΛp2ond)s1towGithργΛπ = 0=.72m8 GeV−. 1. ργπ ργπ ργπ − ργπ ω(782) B. Axial-vector IG(JPC)=1−(1++) exchange currents Theaxial-vectora (1260)mesonwithIG(JPC)=1 (1++)hasalargewidthintothea (1260) γπ channel[46]. 1 − 1 ± ± A conversionof a into γπ is described by the a γπ vertex = ieG Fµν Q[A ,ϕ] wh→ere Fµν denotes the 1 1 La1γπ −4 a1γπ h µν i field tensor of photons, A stands for the field tensor of the axial-vector meson with A =∂ A ∂ A and µν µν µ ν ν µ − a0 √2a+ A = 1 1 . (35) µ (cid:18)√2a−1 −a01 (cid:19)µ ϕ is a standard SU(2) pion matrix, Q = diag(2/3, 1/3) is a quark charge matrix, .. and [..] denote a trace and a − h i commutator of fields. The hadronic a NN interaction is described by 1 =G ψ¯γµγ A ψ, (36) La1NN a1NN 5 µ where ψ =(p,n)T. Because of G-parity conservation in the vertex there is no tensor coupling of a to nucleons. 1 8 In the reactions p(γ ,π+)n and n(γ ,π )p the currents describing the exchange of a (1260) trajectory read ∗ ∗ − 1 iJµ(γ p π+n) + −iJaµ1(γ∗n→ π p)= √2Ga1NNGa1γπFa1γπ(Q) kµqν −(qk)gµν u¯s′(p′)γνγ5us(p) − a1 ∗ → − − h i     × 1−e−2iπαa1(t) −α′a1 Γ[1−αa1(t)]eln(α′a1s)(αa1(t)−1), (37) (cid:20) (cid:21) (cid:0) (cid:1) Thea Reggetrajectoryadoptedhereisα (t)=α (t) 1whereα (t)isthetrajectoryofρ. Theγ a π transition is isov1ector and contributes with oppositeas1igns toργ p− π+n andργ n π p reactions. ∗ → 1 ∗ ∗ − → → Toestimatethea -nucleoncouplingconstantG onecanrelatesayG totheobservedaxial-vectorcoupling 1 a1NN a1pp constant using axial-vector dominance [47, 48] gA = √2fa1Ga1pp, where the weak decay constant f is deduced gV m2a1 a1 from τ decay: τ a + ν . With g /g = 1.267, f = (0.19 0.03) GeV2 one gets the following estimate → 1 τ A V a1 ± G =G =7.1 1.0. a1pp a1NN ± The radiative decay width a γπ is given by 1 → α G2 Γ = e a1γπ(m2 m2)3, (38) a+1→γπ+ 24 m3a1 a1 − π The empirical width a+ γπ+ is Γ (640 246) keV [46]. The coupling constant G 1.1 GeV 1 correspondstothe cent1ral→value. InaaV+1M→Dγπ+pic≃tureac±onversionofγ toρwithsubsequenta ρπ intae1rγaπct≃iongenerat−es 1 the monopole form factor F (Q2) = (1+Q2/Λ2 ) 1 with Λ = m . This form is used to model the Q2 a1γπ a1γπ − a1γπ ρ(770) dependence of the a γ π vertex. 1 ∗ C. Axial-vector IG(JPC)=1+(1+−) exchange currents We consider the exchange of b (1235) axial-vector meson with IG(JPC)=1+(1+ ). The conversion of b γπ is 1 − 1 → described by the vertex = eGb1γπFµν Q B ,ϕ , where .. anti-commutes and B =∂ B ∂ B with Lb1γπ 4 h { µν }i { } µν µ ν − ν µ b0 √2b+ B = 1 1 . (39) µ (cid:18)√2b−1 −b01 (cid:19)µ The b (1235) coupling to nucleons takes the form of axial-tensor interaction 1 G = i b1NN ψ¯σµνγ B ψ. (40) Lb1NN 4M 5 µν N where σµν = i[γµ,γν]. The hadronic currents iJµ describing the exchange of b (1235) Regge trajectory read 2 − b1 1 iJµ(γ p π+n) −iJbµ1(γ∗n→ π p) = √32 G2bM1NNNGb1γπFb1γπ(Q) kµqν −(qk)gµν (p+p′)νu¯s′(p′)γ5us(p) − b1 ∗ → − h i   × 1−e−2iπαb1(t) −α′b1 Γ[1−αb1(t)]eln(α′b1s)(αb1(t)−1). (41) (cid:20) (cid:21) (cid:0) (cid:1) ETqh.e(r3a8d)iaotniveegdeetcsaGy widt/h3o=f b0±1.64→7 γGπe±Vis1.ΓbT±1h→eγππ±a=nd(2b30(1±23650))RkeeVgge[4t6r]a.jMecatokriinegs uasree onfeaarnlyexdpergeesnseiorantesim(dialasrhetdo b1γπ − 1 curveifFigure3). Thereforeweassumeα (t)=α (t). Intheb γ π vertexweusetheVMDformfactorF (Q2)= b1 π 1 ∗ b1γπ (1+Q2/Λ2 ) 1 with Λ =m . b1γπ − b1γπ ω(782) It was proposed [29] that the polarized photon asym- strong coupling constants. In [29] the b0(1235) is cou- 1 metry in p(γ,π0)p reaction at high energies can be used pled to the axial-vector current and the axial-tensor in- to estimate the product of the b electromagnetic and teraction has been neglected. However, the axial-vector 1 9 8 20 7 JLAB JLAB DESY DESY 6 15 ] ] 2V 5 2V e e G G b/ b/ µ 4 µ10 [ [ dt dt σ/T 3 σ/L d d 2 5 1 0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 2 2 -t [GeV ] -t [GeV ] FIG. 4: (Color online) t dependence of the transverse dσ /dt (left panel) and longitudinal dσ /dt (right panel) differential T L cross sections in exclusi−vereaction p(γ∗,π+)n. The compilation of JLAB and old DESY data are from Ref. [4] and havebeen scaled to common values of W = 2.19 GeV and Q2 = 0.7 GeV2. The solid curves describe the model results which include the effect of resonances and exchange of π/b (1235), ρ(770)/a (1320) and a (1260) Regge trajectories. The same is true for 1 2 1 the dash-dash-dotted curves but without the DIS slope of Eq. (45). The dashed curves correspond to the contribution of the π-reggeon exchange only. The dash-dotted curvesdescribe the model results which includethe exchange of Regge trajectories and on-mass-shell parameterization of theproton Dirac form factor. γµγ vertex is C-parity even and can not couple to the have to be made. In the gauged π-Regge amplitudes, 5 b0(1235)-mesonwhich has negative C-parity. This is op- Eqs. (9) and (10), an assumption concerning an exact 1 posed to the a (1260) which couples to γµγ . By C(G)- degeneracy of π and axial-vector b (1235) Regge trajec- 1 5 1 parityonlyatensorb NN coupling,Eq.(40),ispossible. torieswithachoiceofrotatingphaseinπ+andaconstant 1 We have checked and found that with the proper b NN phaseinπ photoproductionyields aremarkableconsis- 1 − vertexthisextractionisnotobvious. Withthetensorin- tencywithdata[29]. Thedegeneracyofρ(770)/a (1320) 2 teraction the exchange of the b (1235) trajectory is neg- ReggetrajectoriesandG-parityargumentsresultinaro- 1 ligibly small and can be readily neglected. For instance, tatingphaseinπ+andaconstantphaseinπ production − using G = G or even increasing considerably described by the ρ-exchange amplitude, Eq. (33). Here, b1NN a1NN the latter value does not produce any noticeable effects to be consistent with the real photon limit, we follow on the observables considered here. these assumptions [29]. However, in the high-Q2 electroproduction a particular choice of phases in the Regge amplitudes is of minor importance. The virtual photon VI. ELECTROPRODUCTION ABOVE THE (γ ,π ) results presented here can be well reproduced ∗ ± RESONANCE REGION with the standard Regge propagators. From the meson spectrum there is no conclusive evidence that a leading In this sectionwe demonstrate the resonanceinterpre- Regge trajectory for an unnatural parity ρ state exists. 2 tation proposed in this work and fix the model parame- Therefore, we do not make any assumptions on a degen- ters using the JLAB data. At first, we briefly consider eracy pattern of a (1260). 1 the real photon limit of the Regge amplitudes at high The resulting Regge model based on reggeized gauge energies. Inπ+ andπ photoproductionatveryforward invariantFeynman amplitudes describes the high energy − anglesthereggeizedelectricamplitudes,seeEqs.(9)and π photoproductiondatarelevantforthepresentstudies ± (10), are supposed to be dominant. Since the vector and reasonably well. These include the differential cross sec- axial-vectormeson-exchangecontributions vanish at for- tions abovethe resonanceregion,the π /π+ ratioofthe − ward angles, Eqs. (9) and (10) are parameter free, pro- n(γ,π )p and p(γ,π+)n differential cross sections and − videdtheinterceptoftheπ-trajectoryandtheg cou- polarized photon asymmetries. Our description of pho- πNN pling constant are fixed. However, further assumptions toproductiondata is quantitatively similar to the results concerningachoiceofthephasesintheReggeamplitudes of Ref. [29] and we do not repeat this comparison with 10 1 Thedash-dottedcurvescorrespondtothegaugedelectric model with the on-shell Dirac form factor, see Eq. (15), 0.8 Gp(Q2)/Gp (Q2)+Q2/4M2 Fp(Q2)= E M pGp (Q2), (42) 2, s)| 0.6 1 1+Q2/4Mp2 M Q F(s0.4 and exchange of ρ(770)/a2(1320) and a1(1260) Regge | trajectories. In Eq. (42) the electric form factor Gp E decreases linearly as a function of Q2 with respect to 0.2 the magnetic form factor Gp with a node around Q2 8 GeV2 [49] provided µ GpM/Gp = 1 Q2/Q2. On0th≃e 0 otherhandupto 5GepV2EtheMmagnet−icform0factorisa 0 1 2 3 4 5 6 7 8 9 10 dipole Gp = µ /≃(1+Q2/0.71GeV2)2 where µ = 2.793 2 2 M p p Q [GeV ] is the magnetic moment of the proton. As one can see, this model (dash-dotted curves) FIG. 5: (Color online) The Q2 dependence of the absolute with the nucleon-pole (gauge invariance), vector and valueofthetransitionformfactor Fs(Q2,s) (dashedcurve), axial-vectormeson-exchangeReggetrajectoriesdescribes | | Eq. (43), at √s = 2.2 GeV. The solid curve describes the dσ /dt well and grossly underestimates dσ /dt. Varia- L T protonDiracform factor, Eq(42),incomparison withdata. tions of the cut-offs inthe vectorand axial-vectormeson transition form factors do not improve the description. experimental data here. Thispreliminarycomparisonwithdatashowsthatbeing An extension of the model to electroproduction is consistentwithphotoproductiontheabovesimpleexten- straightforward provided the Q2-dependent transition sion of the Regge model to electroproduction is not able form factors are defined. In Figure 4 we plot the trans- to describe the data already at values of Q2 as low as verse dσ /dt (left panel) and longitudinal dσ /dt (right Q2 1GeV2. Thediscrepancieswithdataincreasewith panel)π+T electroproductiondatafromJLABLandDESY incr≃easing value of Q2 [4]. (old data) scaled to the same values of Q2 = 0.7 GeV2 Next, consider the resonance contributions using the and W = 2.19 GeV [4]. The value of the momentum transition form factors as defined in Eq. (31). We have cut-off Λ in the pion form factor, Eq. (14), is largely twoparametersathand: theparameterβwhichisrelated γππ constrained by the magnitude of dσ /dt at forward an- to the level density of states and a parameter ξ describ- L gles. InthefollowingtheJLABdataareconsideredtobe ing the average cut-off in the resonance transition form a guideline for fixing the model parameters. The dashed factors. Alltheexclusivep(γ ,π+)nandn(γ ,π )pelec- ∗ ∗ − curves correspond to Λ2 = 0.46 GeV2 and describe troproduction data considered in this work from JLAB, γππ the contribution of the π-reggeon exchange only. The DESY and Cornell to DIS region at HERMES can be exchange of π dominates in dσ /dt at forward angles. well described by the choice β = 3 and ξ = 0.4. The L However, in dσ /dt the π-exchange is not compatible formulaeforthetransitionformfactorsgivenin(31)and T with data and also vanishes in the forward direction. (32) can be integrated and yield (for β =3) ξQ2 (2ξQ2+s) s(ξQ2+s) s M2 sln +1 +ln − p iπ F (Q2,s)= (cid:20)Mp2 (cid:21) (ξQ2)2 − ξQ2(ξQ2+Mp2) " Mp2 #− , (43) s ξQ2 2 s2+2sM2 s M2 +1 p +ln − p iπ (cid:18) s (cid:19) 2Mp4 " Mp2 #− ! ξQ2 (2ξQ2+u) u(ξQ2+u) M2 u uln +1 +ln p − F (Q2,u)= (cid:20)Mp2 (cid:21) (ξQ2)2 − ξQ2(ξQ2+Mp2) " Mp2 #. (44) u ξQ2 2 u2+2uM2 M2 u +1 p +ln p − u 2M4 M2 (cid:18) (cid:19) p " p #! In Figure 5 we plot the Q2 dependence of the absolute value of the transition form factor F (Q2,s) (dashed s | |