# Hadron Form Factors : theory

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Hadron Form Factors : theory Marc VanderhaeghenCollege of William & Mary / JLab

EINN 2005, Milos (Greece), September 20-24, 2005

Nucleon electromagnetic form factors : theoretical approaches

N form factors

two-photon exchange effects nucleon FF : Rosenbluth vs polarization data extension to N -> FF For weak form factors/parity violation : see talks -> D. LHuillier, K. PaschkeOutlineRecent review on electromagnetic form factors of the nucleon and Compton scattering Ch. Hyde-Wright and K. de Jager : Ann. Rev. Nucl. Part. Sci. 2004, 54

Nucleon electromagnetic form factors : theoretical approaches

i) Dispersion theoryii) Mapping out pion cloud, chiral perturbation theory iii) lattice QCD : recent results & chiral extrapolationiv) link to Generalized Parton Distributions : nucleon tomography v) other

disclaimer : v) will not be discussed in this talk

nucleon FF : dispersion theory * N N V q Hoehler et al. (1976)Mergell, Meiner, Drechsel (1995)Hammer, Meiner, Drechsel (1996)Hammer, Meiner (2004)general principles : analyticity in q2 , unitarityFF -> dispersion relation in q2 branch cuts for q2 > 4 m2 : vector meson poles + continua (, ) q2 > 0 : timelike q2 = - Q2 < 0 : spacelike basic dipole behavior : explained by 2 nearby poles with residua of equal size but opposite signanalysis of Hammer & Meiner (2004)isovector channel : 2 continuum + 4 poles : , (1050), (1465), (1700)isoscalar channel : 4 poles : , (1019), S(1650) S(1680)masses & 16 residua (V, T) fitted + PQCD scaling behavior parametrized

nucleon FF : dispersion theoryHammer, Meiner (2004)Hammer, Meiner, Drechsel (1996)phenomenological fit by Friedrich, Walcher (2003) DR : good description, except for GEp / GMp

nucleon form factors : pion cloudFriedrich, Walcher (2003)phenomenological fit :smooth part (sum of 2 dipoles) + bump(gaussian)6 parameter fit for each FFpronounced structure in all FF around Q 0.5 GeV/cpion cloud extending out to 2 fm

nucleon FF : Chiral Perturbation TheoryKubis, Meiner (2001)SU(3)SU(2) : NDR Goldstone boson -Baryon loops (relativistic ChPT, 4th order, IR reg.)+ vector mesonssee also Schindler et al. (2005)(EOM renorm. scheme)

nucleon FF : lattice QCDQCDSF Coll. : Goeckeler et al. (2003)Lattice results fitted by dipoles -> for isovector channel : masses MeV , MmV latticeExpt.Expt.latticeExpt.latticequenched approximation : qq loops neglectedlinear extrapolation in mreasonable good description of GEp / GMp at larger Q2 (where role of pion cloud is diminished)

nucleon FF : lattice & chiral extrapolationLeinweber, Lu, Thomas (1999)Hemmert and Weise (2002)+ 4 LEC fit3 LEC fitlattice : QCDSFlattice : QCDSFlattice : QCDSFHemmert : chiral extrapolation using SSE at O(3) -> fit LEC to available lattice points V(r1V )2(r2V )2qualitative description obtained, not clear for (r1V )2

Relativistic chiral loops (SR) give smoother behavior than the heavy-baryon expansion (HB) or Infrared-Regularized ChPT (IR) red curve is the 2-parameter fit to lattice data based on sum rule (SR) resultPascalutsa, Holstein, Vdh (2004)nucleon FF : lattice & chiral extrapolationFor : resummation of higher order terms by using a new sum rule (SR) (linearized version of GDH) -> analyticity is built in chiral loopslattice : Adelaide group (Zanotti)

nucleon FF : lattice prospectsLHP Collaboration (R. Edwards)state of art : employ full QCD lattices (e.g. MILC Coll.) using staggered fermions for sea quarks employ domain wall fermions for valence quarksPion masses down to less than 300 MeV As the pion mass approaches the physical value, the calculated nucleon size approaches the correct value next step : fully consistent treatment of chiral symmetry for both valence & sea quarks F1V(r2)1V

FF : link to Generalized Parton Distributionsx + x - P - /2P + /2*Q2 larget = 2 low t process : -t Diehl, Camacho, Hadjidakis

known information on GPDsfirst moments : nucleon electroweak form factors independence : Lorentz invariancePauliDiracaxialpseudo-scalarforward limit : ordinary parton distributions unpolarized quark distrpolarized quark distr: do NOT appear in DIS new information

GPDs : 3D quark/gluon imaging of nucleonFourier transform of GPDs :simultaneous distributions of quarks w.r.t. longitudinal momentum x P and transverse position btheoretical parametrization needed

modified Regge parametrization : Guidal, Polyakov, Radyushkin, Vdh (2004)Input : forward parton distributions at m2 = 1 GeV2 (MRST2002 NNLO) regge slopes : 1 = 2 determined from rms radii determined from F2 / F1 at large -t Drell-Yan-West relation : exp(- t ) -> exp(- (1 x) t) : Burkardt (2001) parameters : future constraints : moments from lattice QCD GPDs : t dependence

electromagnetic form factorsmodified Regge parametrizationRegge parametrizationPROTONNEUTRON

GPDs : transverse image of the nucleon (tomography) Hu(x, b? ) b? (GeV-1)x

proton Dirac & Pauli form factorsmodified Regge modelRegge modelPQCDBelitsky, Ji, Yuan (2003)

timelike proton FF : GM = F1 + F2PQCDanalytic function in q2(Phragmen-Lindelf theorem)around |q2| = 10 GeV2 timelike FF twice as large as spacelike FFFermilab p p -> e+ e- timelike (q2 > 0) spacelike (q2 < 0) HESR@GSI can measure timelike FF up to q2 25 GeV2 q2

timelike proton FF : F2 / F1PQCDVMDBelitsky, Ji, Yuan (2003)Iachello et al. (1973, 2004)q2REAL partIMAG partVMDREAL partIMAG partPQCD

measurement of timelike F2 / F1Polarization Py normal to elastic scattering plane(polarized beam OR target) VMDPQCDBrodsky et al. (2003)

N -> transition form factorsmodified Regge modelRegge model in large Nc limit

- electromagnetic N -> (1232) transition in chiral effective field theoryRole of quark core (quark spin flip) versus pion cloud non-zero values for E2 and C2 : measure of non-spherical distribution of charges spin 3/2J P=3/2+ (P33), M ' 1232 MeV, ' 115 MeVN ! transition: N ! (99%), N ! (
Effective field theory calculation of the e p -> e p 0 process in (1232) regioncalculation to NLO in expansion (powers of ) Power counting : in region, treat parameters = (M MN)/MN and m on different footing ( m ~ 2 ) in threshold region : momentum p ~ m / in region : p ~ M - MNPascalutsa, Vdh ( hep-ph/0508060 ) LOvertex corrections : unitarity & gauge invariance exactly preserved to NLO

data : MIT-BATES (2001, 2003, 2005)e p -> e p 0 in (1232) region : observablesEFT calculation error bands due to NNLO, estimated as : ~ || 2 W = 1.232 GeV , Q2 = 0.127 GeV2

Q2 dependence of E2/M1 and C2/M1 ratiosEFT calculation predicts the Q2 dependence data points : MIT-Bates (2005)see talk -> Sparveris MAMI : REM (Beck et al., 2000) RSM (Pospischil et al., 2001; Elsner et al., 2005)EFT calculation error bands due to NNLO, estimated as : R ~ |R| 2 + |Rav| Q2/MN2 REM = E2/M1RSM = C2/M1

m dependence of E2/M1 and C2/M1 ratiosquenched lattice QCD results :at m = 0.37, 0.45, 0.51 GeVlinear extrapolation in mq ~ m2discrepancy with lattice explained by chiral loops (pion cloud) !Alexandrou et al., (2005)data points : MAMI, MIT-BatesEFT calculationPascalutsa, Vdh (2005)Q2 = 0.1 GeV2see also talk -> Gail see talk -> Tsapalis

Rosenbluth vs polarization transfer measurements of GE/GM of proton Jlab/Hall A Polarization dataJones et al. (2000)Gayou et al. (2002)SLAC, Jlab Rosenbluth dataTwo methods, two different results !Two-photon exchange effects

Observables including two-photon exchangeReal parts of two-photon amplitudes

Phenomenological analysisGuichon, Vdh (2003)2-photon exchange corrections can become large on the Rosenbluth extraction,and are of different size for both observables relevance when extracting form factors at large Q2

Two-photon exchange calculation : elastic contributionBlunden, Tjon, Melnitchouk (2003, 2005)Nworld Rosenbluth dataPolarization Transfer

hard scattering amplitudeTwo-photon exchange : partonic calculationGPD integralsmagnetic GPDelectric GPDaxial GPD

Two-photon exchange : partonic calculationGPDsChen, Afanasev, Brodsky, Carlson, Vdh (2004)

1 result1 + 2 resultTwo-photon exchange in N -> transition General formalism for eN -> e has been worked outModel calculation for large Q2 in terms of N -> GPDsPascalutsa, Carlson, Vdh ( hep-ph/0509055 ) N REM little affected < 1 %RSM mainly affected when extracted through Rosenbluth method

Nucleon electromagnetic form factors : -> dispersion theory, chiral EFT : map out pion cloud of nucleon -> lattice QCD : state-of-art calculations go down to m ~ 300 MeV, into the regime where chiral effects are important / ChPT regime -> link with GPD : provide a tomographic view of nucleon

N form factors : -> chiral EFT ( -expansion) is used in dual role : describe both observables and use in lattice extrapolations, -> resolve a standing discrepancy : strong non-analytic behavior in quark mass due to opening of N decay channel

difference Rosenbluth vs polarization data -> GEp /GMp : understood as due to two-photon exchange effects -> precision test : new expt. planned -> N transition : effect on RSM when using Rosenbluth method Summary

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