Nucleon EM Form Factors in Dispersion Theory H.-W. Hammer, University of Bonn supported by DFG, EU and the Virtual Institute on Spin and strong QCD Collaborators: M. Belushkin, U.-G. Meißner
Agenda Introduction Dispersion Analysis Results Two-Photon-Echange Corrections Conclusions & Outlook
Why Nucleon Form Factors? Nucleons are basic constituents of matter Detailed understanding of nucleon structure nucleon radii, vector meson coupling constants,... perturbative and nonperturbative aspects of QCD input for wide variety of experiments, e.g. Lamb shift measurements in atomic hydrogen Sea quark dynamics in strange form factors Theoretical tools Effective Field Theory ChPT (low momentum transfer) Dispersion theory (all momentum transfers, space- & time-like) Lattice QCD (this workshop) 2
Why Dispersion Theory? Based on fundamental principles: unitarity, analyticity (Largely) model independent Connects data over full range of momentum transfers: time-like and space-like data Connects to data from different processes (πn scattering,...) Simultaneous analysis of all four form factors Spectral functions encode perturbative and non-perturbative physics: vector meson couplings, multi-meson continua, pion cloud,... Constraints from ChPT, pqcd Extraction of nucleon radii 3
Electromagnetic Form Factors Definition: t q 2 = (p p) 2 = Q 2 p q I jµ p N(p ) j I µ N(p) = ū(p ) [ F I (t)γ µ + i F 2 I ] (t) 2m σ µνq ν u(p), I = S, V Normalization: F S () = F V () = /2, F S,V 2 () = (κ p ± κ n )/2 Sachs form factors: G E = F + Radii: F(t) = F() [ + t r 2 /6 +... ] t 4m 2F 2, G M = F + F 2 4
Starting Point: Spectral Decomposition Crossing: N(p ) j I µ N(p) N( p)n(p) j I µ Spectral decomposition: (using microcausality, analyticity,...) (Federbush, Goldberger, Treiman, Chew,...) Im N( p)n(p) j I µ n N( p)n(p) n n j I µ = ImF N On-shell intermediate states imaginary part relate physical matrix elements n n I j µ N Intermediate States: j S µ (isoscalar): 3π, 5π,...,K K,K Kπ,... j V µ (isovector): 2π,4π,... 5
Dispersion Relations Form factors have multiple cuts in interval [t n, [ (n =,, 2,...) Dispersion relation for F: Apply Cauchy s formula in complex t-plane Subtractions? spacelike t Im t timelike t Re t F i (t) = π ImF i (t ) t t t dt, i =, 2 Higher mass intermediate states are suppressed Spectral functions are central objects from where to take? 6
EM Spectral Functions Include 2π and K K, ρπ ( 3π) continua as independent input Higher mass states: dominated/parametrized by vector meson poles (widths can be included) ImF i (t) = v πa v iδ(t M 2 v), a v i = M2 v f v g vnn F i (t) = v a v i M 2 v t EM Spectral Function: Im F i S ω S Im F V i ρ ρ ρπ S t ππ ρ ρ t φ, KK 7
Spectral Function: 2π-, K K-, ρπ-continua Continuum contributions: 2π: analytic continuation of πn scattering data, F π (t) (Höhler, Pietarinen, 75, Belushkin, HWH, Meißner, 5) K K: KN scattering data, F K (t) (HWH, Ramsey-Musolf, 99) ρπ ( 3π): Jülich N N model (Meißner et al., 97) Evaluate with DR Form factor contributions..-..-..- K K - - - - ρπ - 2 3 4 5 2π Q 2 [GeV 2 ].5 -.5 F 3 F 2 2 2 3 4 5 Q 2 [GeV 2 ] 8
Asymptotic Behavior pqcd prediction (modulo logs): F /t 2, F 2 /t 3 as t Various ways to implement asymptotic behavior from pqcd: Superconvergence relations only add broad resonance to generate imaginary part for t 4m 2 F (I,broad) i (t) = ai i (M2 I t) (M 2 I t)2 + Γ 2 I, i =, 2, I = S,V, Explicit pqcd term in addition to SC relations F (I,pQCD) i = a I i c 2 i t +, i =,2, I = S,V, b2 i ( t)i+... 9
Error Bands General Problem: What are the errors in dispersion analyses? Generate theory error bands for best fits Difficult since problem is highly non-linear Perform Monte Carlo sampling of fits with χ 2 in interval [χ 2 min, χ 2 min + δχ 2 ], δχ 2.4 keep form factor values with maximal deviation from best fit Results: Belushkin, HWH, Meißner, Phys. Rev. C 75 (27) 3522
Superconvergence Approach: Space-Like 7 Parameters ω, φ 2 effective IS poles 5 effective IV poles χ 2 /dof =.8 good description of data best fit - - - - σ band G E n G E p /GD.2..8.6.4.2.5.5.8.6.4..2... Q 2 [GeV 2 ] G M n /(µn G D ) G M p /(µp G D )..9.8.7.6.5.5...2..9.8.7.6.2.4. Q 2 [GeV 2 ]
Superconvergence Approach: Time-Like Experimental extraction ambiguous (E/M separation) Subthreshold resonance? (Antonelli et al., PRD 7 ( 5) 54) N N final-state interaction (Haidenbauer, HWH, Meißner, Sibirtsev, Phys. Lett. B 643 (26) 29).25.8.5.2.7 G M p.4.3.2 G M p.5. G M n.6.5.4..5.3 4 5 t [GeV 2 ] 5 5 t [GeV 2 ].2 3.6 3.8 4 4.2 t [GeV 2 ] 4.4 2
N N Final-State Interaction Consider cross section/transition amplitude for e + e N N ( source of timelike form factor data) Steep rise described by Jülich N N model/watson-migdal treatment (Haidenbauer, HWH, Meißner, Sibirtsev, PLB 643 ( 6) 29) Consistent with J/Ψ γp p, ωp p, B p pk (Haidenbauer, Meißner, Sibirtsev, 6, 8) 3
Explicit pqcd Approach: Space-Like 4 Parameters ω, φ effective IS poles 3 effective IV poles explicit pqcd term χ 2 /dof = 2. good description of data best fit - - - - σ band G E n G E p /GD.2..8.6.4.2.5.5.8.6.4..2... Q 2 [GeV 2 ] G M n /(µn G D ) G M p /(µp G D )..9.8.7.6.5.5...2..9.8.7.6.2.4. Q 2 [GeV 2 ] 4
Explicit pqcd Approach: Time-Like Timelike neutron data is not included in the fit (also in SC) No predictive power at threshold.25.8.5.2.7 G M p.4.3.2 G M p.5. G M n.6.5.4..5.3 4 5 t [GeV 2 ] 5 5 t [GeV 2 ].2 3.6 3.8 4 4.2 t [GeV 2 ] 4.4 5
General Comments/Radii Successive improvement by reduction of number of poles (stability constraint) Theoretical/systematic uncertainties? (2γ physics, consistency of data,..) Extraction of radii SC ex. pqcd recent determ. r p E [fm].84...85.82...84.886(5) [,2,3] r p M [fm].85...86.84...85.855(35) [2,4] (re n)2 [fm 2 ]....3....3.5(4) [5] rm n [fm].85...87.86...87.873() [6] [] Rosenfelder, Phys. Lett. B 479 ( ) 38 [2] Sick, Phys. Lett. B 576 ( 3) 62 [3] Melnikov, van Ritbergen, Phys. Rev. Lett. 84 ( ) 673 [4] Sick, private communication [5] Kopecky et al., Phys. Rev. C 56 ( 97) 2229 [6] Kubon et al., Phys. Lett. B 524 ( 2) 26 6
CLAS data: Space-Like cf. CLAS collaboration, arxiv:8.76v; W. Brooks, private communication 4 Parameters ω 2 effective IS poles 3 effective IV poles explicit pqcd term χ 2 /dof = 2.2 good description of data best fit - - - - σ band G E n G E p /GD.2..8.6.4.2.5.5.8.6.4..2... Q 2 [GeV 2 ] G M n /(µn G D ) G M p /(µp G D )..9.8.7.6.5.5...2..9.8.7.6.2.4. Q 2 [GeV 2 ] 7
Two-Photon Corrections Discrepancy between Rosenbluth and polarization transfer (PT) data Two-Photon Exchange Effects Direct model calculations (Blunden, Melnichouk, Tjon, 23, 25; Afanasev et al., 24, 25,...) right direction, effect too small Model-independent extraction from data? Assumption: no significant two-photon effects in PT data Estimate hard 2γ corrections from comparison of our previous analysis (mainly PT data) and direct analysis of Rosenbluth cross sections for proton (including Coulomb corrections soft 2γ corrections) 8
Two-Photon Corrections Hybrid analysis: FF data for neutron, cross sections for proton (Belushkin, HWH, Meißner, Phys. Lett. B 658 (28) 38) Easiest to compare at cross section level reconstruct PT cross section from FF data ( ) dσ ( ) ( ) + δ 2γ dσ = dω dω Ros Comparison with direct calculation (Blunden et al.) add in Coulomb correction 2γ = δ 2γ + δ C (cf. Arrington, Sick, Phys. Rev. C 7 (24) 2823) 9
Cross Section Analysis Example: cross section analysis in SC approach.2.5 (dσ/dω) / (dσ/dω) dipole..99.98 E =.494 GeV (dσ/dω) / (dσ/dω) dipole.95.9 Θ = 6 o (dσ/dω) / (dσ/dω) dipole.97 2 3 4 5 6 7 8.8.5.2.99.96.93 E =.9 GeV 4 6 8 Θ [ o ] (dσ/dω) / (dσ/dω) dipole.2.4.6.8 3 2.5 2.5 Θ = 9 o.5.5.5 2 2.5 3 E [GeV] 2
Two-Photon Corrections Extracted two-photon correction term 2γ = δ 2γ + δ C SC.-.-.-.- Blunden et al. 2γ 2γ.2 -.2 -.4.5 -.5 -. Q 2 =.5 GeV 2 Q 2 =2 GeV 2 Ref. []..3.5.7.9 ε Blunden, Melnitchouk, Tjon, Phys. Rev. C 72 (25) 3462.3 -.3 -.6 -.9.5 Q 2 =3 GeV 2 -.5 -. Q 2 = GeV 2 Ref. [] Ref. []..3.5.7.9 ε cf. 2
Comparison with Form Factor Ratios Form factor ratio from cross section analysis (SC) 2 Rosenbluth Polarisation transfer.5 µ p G E p /GM p (SC).5 2 3 4 5 Q 2 [GeV 2 ] Consistent within error bars Form factor data not included in analysis 22
Summary Dispersion analysis of nucleon EM form factors Improved spectral functions: 2π, K K, ρπ continua Consistent description of EM FF data Radii: agreement with other analyses except r p E not likely explained by 2γ physics (Blunden, Sick, 5) Model independent extraction of hard two-photon effects no discrepancy between Ros and PT data within errors Currently investigating subtracted dispesion relations 23
Additional Transparencies 24
CLAS data: Space-Like II cf. CLAS collaboration, arxiv:8.76v; W. Brooks, private communication 4 Parameters ω 2 effective IS poles 3 effective IV poles explicit pqcd term χ 2 /dof = 2.2 good description of data best fit - - - - σ band G M n /(µn G D ).2..9.8 2 3 4 5 25
Interpretation of Radii Nucleon FF s in space-like region: can always find reference frame where no energy is transferred (Breit frame) F(q 2 ) = F( q 2 ) = p=(e, p) p =(E, p) d 3 re iq r ρ(r) q=(, 2p) Interpretation: r 2 = 4π F() dr r 4 ρ(r), F() = 4π dr r 2 ρ(r) G E (G M ): FT of charge (magnetization) distribution (q 2 m 2 ) F, F 2 : only formal definition 26
Isovector Spectral Function: 2π-Continuum 2π-contribution to spectral functions: (Frazer, Fulco, 59 predicted ρ) ImG V E (t) = q3 t m t F π(t) f +(t) ImG V M (t) = q3 t F π (t) f (t) 2t F π * f_+ where q t = t/4 M 2 π ππ NN P-waves: f ±(t) from analytic continuation of πn data (Höhler, Pietarinen, 75) Singularity close to threshold: t c 3.98M 2 π isovector radii Pion EM form factor F π (t): from e + e π + π 27
Isovector Spectral Function: 2π-Continuum New data for pion form factor (CMD, KLOE, SMD) New determination of 2π-continuum (Belushkin, HWH, Meißner, 6) F π 2 5 4 3 2.4.6.8 t [GeV 2 ].6.4.2 Pronounced ρ peak with strong ρ ω mixing spectral function [/m π 4 ] 2ImF /t 2-2τImF 2 /t 2 2ImG E /t 2 2ImG M /t 2.2.4.6.8 t [GeV 2 ] Contains information on long-range pion cloud (cf. Drechsel et al., 4) 28
On the Pion Cloud of the Nucleon FW find very long-ranged pion cloud contribution: r 2 fm Friedrich, Walcher, EPJA 7 ( 3) 67 Long-range pion contribution given by 2π-continuum HWH, Drechsel, Meißner, PLB 586 ( 4) 29..5..5 2. r [fm] 4πr 2 ρ(r) [/fm].2.8.4. G M V G E V ρ V i (r) = 4M 2 4π 2 π 4M 2 π dtim GV i t e r (t) r, i = E,M Maxima around r max.3 fm FW: r max.5 fm Smaller pion cloud contribution beyond r fm compared to FW Independent of contribution from t > 4M 2 π 29
Bump-Dip Structure in G n E Can structure be generated in dispersive approach? low mass strength required! e.g. low-mass poles: M 2 S =.7 GeV2. G E n (Q 2 ) BHM best fit BHM light mass poles FW pheno fit BHM sigma.5 MV 2 =. GeV2.2.4.6.8.2.4.6 No known vector mesons in this region Q 2 [GeV 2 ] Vector meson dominance applicable for t GeV 2 Higher mass continua? ( 3π : t th.7 GeV 2, 4π : t th.3 GeV 2 ) 3π small in ChPT (Bernard, Kaiser, Meißner, Nucl. Phys. A 6 ( 96) 429) 3
Bump-Dip Structure: Space-Like Only 22 Parameters ω 3 effective IS poles 4 effective IV poles explicit pqcd term χ 2 /dof =.9 (only spacelike) good description of data best fit - - - - σ band G E n G E p /GD.2..8.6.4.2.5.5.8.6.4..2... Q 2 [GeV 2 ] G M n /(µn G D ) G M p /(µp G D )..9.8.7.6.5.5...2..9.8.7.6..2.3.4.5. Q 2 [GeV 2 ] 3