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					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




                                         1
                  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
                                                              I
• Definition:                                                 jµ
                                                         q
                                                    p’            p
  t ≡ q 2 = (p′ − p)2 = −Q2

                                                     I
         ′      I               ′     I            F2 (t)
  N (p       )|jµ|N (p)    = u(p )
                             ¯       F1 (t)γµ   +i        σµν q ν u(p) ,   I = S, V
                                                    2m

                  S        V             S,V
• Normalization: F1 (0) = F1 (0) = 1/2, F2 (0) = (κp ± κn)/2

                                 t
• Sachs form factors: GE = F1 +     F2, GM = F1 + F2
                                4m2

• Radii: F (t) = F (0) 1 + t r 2 /6 + ...


                                                                                      4
           Starting Point: Spectral Decomposition

• Crossing:                            ¯ p
                    N (p′)|jµ|N (p) ←→ N (¯)N (p)|jµ|0
                            I                      I


• Spectral decomposition: (using microcausality, analyticity,...)
  (Federbush, Goldberger, Treiman, Chew,...)
     ¯ p         I
  Im N (¯)N (p)|jµ|0 ∼             ¯ p             I
                                   N (¯)N (p)|n n|jµ|0      =⇒     Im F
                               n

                                                 N

• On-shell intermediate states
  → imaginary part                                                          I
                                                                           jµ
  → relate physical matrix elements                       n n

                                                 N


• Intermediate States:        S                             ¯    ¯
                             jµ (isoscalar): 3π, 5π, ..., K K, K Kπ, ...
                              V
                             jµ (isovector): 2π, 4π, ...

                                                                                5
                         Dispersion Relations

• Form factors have multiple cuts in interval [tn, ∞[ (n = 0, 1, 2, ...)
                                                          Im t
• Dispersion relation for F :
  –Apply Cauchy’s formula in complex
   t-plane                                    spacelike   timelike
                                                                     Re t
 –Subtractions?
                                                 t        t0
             ∞
        1        ImFi(t′) ′
Fi(t) =            ′−t
                         dt ,    i = 1, 2
        π   t0    t


• 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)
                                        2
                            2         Mv                                          avi
 ImFi(t) =       πav δ(t − Mv ), av =
                   i              i       gvN N ⇒ Fi(t) =                         2
             v
                                      fv                                    v
                                                                                 Mv − t

                            Im F iS                           Im F iV
• EM Spectral Function:
                                  ω                 S’’                 ρ             ρ’

                                         ρπ                    ππ
                                               S’                               ρ’’    ρ’’’
                                                          t                                   t

                                      φ , KK




                                                                                                  7
                                     ¯
           Spectral Function: 2π-, K K-, ρπ-Continua
 • Continuum contributions:
   – 2π: analytic continuation of πN scattering data, Fπ (t)
         o
       (H¨hler, Pietarinen, ’75, Belushkin, HWH, Meißner, ’05)
   –      ¯
       K K: KN scattering data, FK (t) (HWH, Ramsey-Musolf, ’99)
   –                    u
       ρπ (→ 3π): J¨lich N N model (Meißner et al., ’97)


 • Evaluate with DR             1

                                                     F1       3                      F2
→ Form factor                 0.5

                                                              2
  contributions                 0

               ¯
   ..-..-..- K K
                                                              1

                              -0.5

   - - - - ρπ                                                 0

                                -1
   ——– 2π                         0   1    2
                                            2
                                                3
                                                 2
                                          Q [GeV ]
                                                      4   5       0   1    2
                                                                            2
                                                                                3
                                                                                 2
                                                                          Q [GeV ]
                                                                                      4   5




                                                                                              8
                         Asymptotic Behavior

• pQCD prediction (modulo logs): F1 ∼ 1/t2, F2 ∼ 1/t3 as t → ∞

• Various ways to implement asymptotic behavior from pQCD:
  – Superconvergence relations only
    → add broad resonance to generate imaginary part for t ≥ 4m2
                              2
     (I,broad)          aI (MI − t)
                         i
    Fi         (t)   =   2 − t)2 + Γ2 ,   i = 1, 2 ,   I = S, V ,
                       (MI          I

  – Explicit pQCD term in addition to SC relations
     (I,pQCD)             aI
                           i
    Fi         =      2t + b2 (−t)i+1 , i = 1, 2 , I = S, V ,
                 1 − ci      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 χ2 in interval

                      [χ2 , χ2 + δχ2] ,
                        min  min                 δχ2 ≃ 1.04


→ keep form factor values with maximal deviation from best fit

 • Results: Belushkin, HWH, Meißner, Phys. Rev. C 75 (2007) 035202


                                                                     10
          Superconvergence Approach: Space-Like
                                  0.12
                                       0.1                                                             1.1
• 17 Parameters                   0.08
                                                                                                        1




                                                                                          GM /(µnGD)
                                                                                                       0.9
  – ω, φ



                          n
                          GE
                                  0.06                                                                 0.8




                                                                                         n
                                                                                                       0.7 1
  – 2 effective IS poles           0.04
                                                                                                       0.6
                                  0.02
  – 5 effective IV poles                 0
                                                                                                       0.5 0                      0.5
                                             0         0.5           1             1.5                   0.01          0.1               1        10
  – χ2/dof = 1.8                                                                                       1.2
                                                                                                       1.1
                                        1
• good description                                                                                      1




                                                                                         GM /(µpGD)
                              GE /GD
                                       0.8
  of data                                                                                              0.9
                          p




                                                                                         p
                                                 1
                                       0.6                                                             0.8 1

——– best fit                            0.4
                                            0                0.1         0.2
                                                                                                       0.7
                                                                                                             0         0.2         0.4
                                                                                                       0.6
                                        0.001        0.01        0.1           1                                 0.1            1            10
- - - - 1σ band                                               2
                                                             Q [GeV ]
                                                                     2                                                        2     2
                                                                                                                             Q [GeV ]




                                                                                                                                                   11
            Superconvergence Approach: Time-Like
 • Experimental extraction ambiguous (E/M separation)
 • Subthreshold resonance? (Antonelli et al., PRD 71 (’05) 054010)
    ¯
⇒ N N final-state interaction
   (Haidenbauer, HWH, Meißner, Sibirtsev, Phys. Lett. B 643 (2006) 29)


                                                  0.25                          0.8

                    0.5                            0.2                          0.7

                    0.4                                                         0.6
                                                  0.15
              p




                                             p




                                                                           n
               GM




                                             GM




                                                                           GM
                    0.3                                                         0.5
                                                   0.1
                    0.2                                                         0.4

                    0.1                           0.05                          0.3

                     0                              0                           0.2
                          4              5               5    10      15              3.6 3.8 4 4.2 4.4
                                    2                              2                             2
                              t [GeV ]                       t [GeV ]                      t [GeV ]


                                                                                                          12
                       ¯
                     N N Final-State Interaction
                                                           ¯
• Consider cross section/transition amplitude for e+e− → N N
  (→ source of timelike form factor data)




                           u       ¯
• Steep rise described by J¨lich N N model/Watson-Migdal treatment
  (Haidenbauer, HWH, Meißner, Sibirtsev, PLB 643 (’06) 29)

• Consistent with J/Ψ → γp¯, ωp¯, B → p¯K
                          p    p       p
  (Haidenbauer, Meißner, Sibirtsev, ’06,’08)

                                                                     13
            Explicit pQCD Approach: Space-Like
                                  0.12
• 14 Parameters                        0.1
                                                                                                       1.1
                                                                                                        1

  – ω, φ                          0.08




                                                                                          GM /(µnGD)
                                                                                                       0.9




                          n
                          GE
                                  0.06                                                                 0.8
  – 1 effective IS poles




                                                                                         n
                                  0.04                                                                 0.7 1

  – 3 effective IV poles           0.02
                                                                                                       0.6
                                                                                                       0.5 0                      0.5
  – explicit pQCD term                  0
                                             0         0.5           1             1.5                   0.01          0.1               1        10
                                                                                                       1.2
  – χ2/dof = 2.0                                                                                       1.1
                                        1
                                                                                                        1




                                                                                         GM /(µpGD)
• good description
                              GE /GD

                                       0.8
                                                                                                       0.9
                          p




  of data




                                                                                         p
                                                 1
                                       0.6                                                             0.8 1
                                                                                                       0.7
                                       0.4
——– best fit                                 0
                                        0.001        0.01
                                                             0.1
                                                                 0.1
                                                                         0.2
                                                                               1
                                                                                                       0.6
                                                                                                             0
                                                                                                                 0.1
                                                                                                                       0.2
                                                                                                                                1
                                                                                                                                   0.4
                                                                                                                                             10
                                                              2      2                                                        2    2

- - - - 1σ band                                              Q [GeV ]                                                        Q [GeV ]




                                                                                                                                                   14
             Explicit pQCD Approach: Time-Like

• Timelike neutron data is not included in the fit (also in SC)

• No predictive power at threshold

                                                0.25                          0.8

                  0.5                            0.2                          0.7

                  0.4                                                         0.6
                                                0.15
            p




                                           p




                                                                         n
             GM




                                           GM




                                                                         GM
                  0.3                                                         0.5
                                                 0.1
                  0.2                                                         0.4

                  0.1                           0.05                          0.3

                   0                              0                           0.2
                        4              5               5    10      15              3.6 3.8 4 4.2 4.4
                                  2                              2                             2
                            t [GeV ]                       t [GeV ]                      t [GeV ]




                                                                                                        15
                                  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.
          p
         rE [fm]      0.84...0.85       0.82...0.84    0.886(15) [1,2,3]
          p
         rM [fm]      0.85...0.86       0.84...0.85     0.855(35) [2,4]
       (rE )2 [fm2] −0.11...−0.13 −0.11...−0.13
         n
                                                         −0.115(4) [5]
          n
         rM [fm]      0.85...0.87       0.86...0.87      0.873(11) [6]
  [1]   Rosenfelder, Phys. Lett. B 479 (’00) 381
  [2]   Sick, Phys. Lett. B 576 (’03) 62
  [3]   Melnikov, van Ritbergen, Phys. Rev. Lett. 84 (’00) 1673
  [4]   Sick, private communication
  [5]   Kopecky et al., Phys. Rev. C 56 (’97) 2229
  [6]   Kubon et al., Phys. Lett. B 524 (’02) 26



                                                                          16
                         CLAS data: Space-Like

cf. CLAS collaboration, arXiv:0811.1716v1; W. Brooks, private communication
                                       0.12
• 14 Parameters                             0.1                                                             1.1
                                                                                                             1
  – ω                                  0.08




                                                                                               GM /(µnGD)
                                                                                                            0.9




                               n
                                GE
                                       0.06                                                                 0.8
  – 2 effective IS poles




                                                                                              n
                                       0.04                                                                 0.7 1

  – 3 effective IV poles                0.02                                                                 0.6
                                                                                                            0.5 0                      0.5
  – explicit pQCD term                       0
                                                  0         0.5           1             1.5                   0.01          0.1               1        10
                                                                                                            1.2
  – χ2/dof = 2.2                                                                                            1.1
                                             1
                                                                                                             1




                                                                                              GM /(µpGD)
• good description
                                   GE /GD




                                            0.8
                                                                                                            0.9
                                p




  of data




                                                                                              p
                                                      1
                                            0.6                                                             0.8 1
                                                                                                            0.7
                                            0.4
——– best fit                                      0
                                             0.001        0.01
                                                                  0.1
                                                                      0.1
                                                                              0.2
                                                                                    1
                                                                                                            0.6
                                                                                                                  0
                                                                                                                      0.1
                                                                                                                            0.2
                                                                                                                                     1
                                                                                                                                        0.4
                                                                                                                                                  10
                                                                   2      2                                                        2     2

- - - - 1σ band                                                   Q [GeV ]                                                        Q [GeV ]




                                                                                                                                                        17
                       Two-Photon Corrections

• Discrepancy between Rosenbluth and polarization transfer (PT) data
  ⇒ Two-Photon Exchange Effects

• Direct model calculations
  (Blunden, Melnichouk, Tjon, 2003, 2005; Afanasev et al., 2004, 2005, ...)
  ⇒ 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)

                                                                              18
                       Two-Photon Corrections

• Hybrid analysis: FF data for neutron, cross sections for proton
  (Belushkin, HWH, Meißner, Phys. Lett. B 658 (2008) 138)

• Easiest to compare at cross section level
  ⇒ reconstruct “PT cross section” from FF data
                            dσ           2γ        dσ
                                   1+δ        =
                            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 70 (2004) 028203)


                                                                    19
                                                 Cross Section Analysis

• Example: cross section analysis in SC approach
                                        1.02                                                                    1.05
              (dσ/dΩ) / (dσ/dΩ)dipole




                                                                                    (dσ/dΩ) / (dσ/dΩ)dipole
                                                                                                                                                             o
                                        1.01                     E0 = 0.1494 GeV                                                                      Θ = 60
                                                                                                                        1
                                           1
                                                                                                                0.95
                                        0.99

                                        0.98                                                                           0.9
                                        0.97
                                            20   30   40   50        60     70     80                                    0     0.2       0.4    0.6    0.8           1
                                                                                                                        3
                                        1.08                                                                                                                 o
              (dσ/dΩ) / (dσ/dΩ)dipole




                                                                                             (dσ/dΩ) / (dσ/dΩ)dipole
                                                                 E0=0.900 GeV                                                                         Θ = 90
                                                                                                                       2.5
                                        1.05
                                        1.02                                                                            2

                                        0.99                                                                           1.5
                                        0.96
                                                                                                                        1
                                        0.93
                                                                                                                       0.5
                                           40         60             80            100                                    0   0.5    1     1.5    2    2.5       3
                                                             o
                                                           Θ[ ]                                                                           E0 [GeV]




                                                                                                                                                                         20
                       Two-Photon Corrections

• Extracted two-photon correction term


• ∆2γ = δ 2γ + δ C                0.02
                                                                              0.03
                                                                                 0
                                     0
                             2γ
                                                                              -0.03
                             ∆
                                                                                                    2         2
                                  -0.02                   2         2                             Q =1 GeV
                                                      Q =0.5 GeV              -0.06               Ref. [1]
                                  -0.04
                                                                              -0.09
• ——– SC                          0.05                                                                                  cf.
                                                      2
                                                  Q =2 GeV
                                                               2
                                                                              0.05                  2
                                                                                                  Q =3 GeV
                                                                                                              2

                                                  Ref. [1]                                        Ref. [1]
                                     0                                           0
                             2γ
                             ∆




                                  -0.05                                       -0.05
• .-.-.-.- Blunden et al.          -0.1                                        -0.1

                                      0.1   0.3   0.5         0.7       0.9       0.1   0.3   0.5       0.7       0.9
                                                  ε                                           ε
Blunden, Melnitchouk, Tjon, Phys. Rev. C 72 (2005) 034612


                                                                                                                        21
            Comparison with Form Factor Ratios
• Form factor ratio from cross section analysis (SC)
                                        2
                                                Rosenbluth
                                                Polarisation transfer

                                       1.5




                       µpGE /GM (SC)
                       p
                                        1
                       p




                                       0.5



                                        0
                                            0   1           2        3     4   5
                                                                 2    2
                                                                Q [GeV ]




• 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
                                                 p
• Radii: agreement with other analyses except rE
  → not likely explained by 2γ physics (Blunden, Sick, ’05)

• 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:0811.1716v1; W. Brooks, private communication

• 14 Parameters
                                             1.2
  – ω
  – 2 effective IS poles
                                             1.1

                                GM /(µnGD)
  – 3 effective IV poles
  – explicit pQCD term
  – χ2/dof = 2.2                               1
                               n



• good description
                                             0.9
  of data
——– best fit                                  0.8
                                                0   1   2        3       4    5
- - - - 1σ band


                                                                              25
                      Interpretation of Radii

• Nucleon FF’s in space-like region:           p=(E, p)
                                                                 q=(0,−2p)
  can always find reference frame
  where no energy is transferred
  (Breit frame)                                p’=(E,− p)



                     F (q 2) = F (−q2) =        d3reiq·rρ(r)

                        4π
           −→ r 2 =    F (0)   dr r 4 ρ(r) ,      F (0) = 4π   dr r 2 ρ(r)
• Interpretation:
  – GE (GM ): FT of charge (magnetization) distribution (q2 ≪ m2)
  – F1, F2: only formal definition

                                                                             26
         Isovector Spectral Function: 2π-Continuum

• 2π-contribution to spectral functions: (Frazer, Fulco, ’59 → predicted ρ)
              3
             q√        1
  ImGV (t)
     E    = t Fπ (t)∗f+(t)
            m t                                   Fπ*
              3
             qt        1
  ImGV (t) = √ Fπ (t)∗f−(t)                                      f+1
                                                                  _
     M
              2t
         where qt = t/4 − Mπ2

       ¯             1
• ππ → N N P-waves: f±(t)
  −→ from analytic continuation of πN data (H¨hler, Pietarinen, ’75)
                                             o
                                            2
• Singularity close to threshold: tc ≈ 3.98Mπ → isovector radii
• Pion EM form factor Fπ (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, ’06)
          50                                                        0.06




                                        spectral function [1/mπ ]
                                        4
                                                                                                         2
                                                                                               2ImF1/t
          40                                                                                   -2τImF2/t
                                                                                                             2

                                                                    0.04                       2ImGE/t
                                                                                                         2
          30
  2




                                                                                                         2
   |Fπ|




                                                                                               2ImGM/t

          20
                                                                    0.02
          10

           0                                                          0
               0.4   0.6      0.8   1                                  0   0.2   0.4 0.6     0.8             1
                           2                                                            2
                     t [GeV ]                                                     t [GeV ]


• Pronounced ρ peak with strong ρ − ω mixing
• Contains information on long-range pion cloud (cf. Drechsel et al., ’04)

                                                                                                                 28
                On the Pion Cloud of the Nucleon
                                                                  1.2
• FW find very long-ranged pion                                                                        V
                                                                                                 GM
  cloud contribution: r ≃ 2 fm




                                                4πr ρ(r) [1/fm]
                                                                                                     V
                                                                  0.8                            GE
   Friedrich, Walcher, EPJA 17 (’03) 607




                                                2
                                                                  0.4
• Long-range pion contribution given
  by 2π-continuum
                                                                  0.0
HWH, Drechsel, Meißner, PLB 586 (’04) 291                           0.0       0.5       1.0    1.5        2.0
                                                                                      r [fm]
                                                                     √
                              40Mπ2                                −r t
                        1
            ρV (r) =
             i         4π2   4Mπ2     dt Im GV (t) e
                                             i                      r     ,         i = E, M

• Maxima around rmax ≈ 0.3 fm ←→ FW: rmax ≈ 1.5 fm
• Smaller pion cloud contribution beyond r ∼ 1 fm compared to FW
                                          2
• Independent of contribution from t > 40Mπ

                                                                                                                29
                    Bump-Dip Structure in Gn
                                           E

                                                    n     2                           BHM best fit
• Can structure be generated in         0.1
                                              GE (Q )                                 BHM light mass poles
                                                                                      FW pheno fit
  dispersive approach?                                                                BHM 1 sigma

 → low mass strength required!
                                       0.05
 → e.g. low-mass poles:
       2
     MS = 0.7 GeV2
       2
     MV = 0.1 GeV2                       0
                                          0   0.2       0.4   0.6       0.8       1     1.2   1.4   1.6
                                                                    2         2
                                                                Q [GeV ]
• No known vector mesons in this region
• Vector meson dominance applicable for t ≤ 1 GeV2
• Higher mass continua? (|3π : tth ≈ 0.17 GeV2, |4π : tth ≈ 0.31 GeV2)
  ⇔ |3π small in ChPT (Bernard, Kaiser, Meißner, Nucl. Phys. A 611 (’96) 429)

                                                                                                             30
             Bump-Dip Structure: Space-Like Only

• 22 Parameters                   0.12
                                                                                                       1.1
                                       0.1
                                                                                                        1
  – ω                             0.08




                                                                                          GM /(µnGD)
                                                                                                       0.9
  – 3 effective IS poles


                          n
                          GE
                                  0.06                                                                 0.8




                                                                                         n
                                                                                                       0.7 1
  – 4 effective IV poles           0.04
                                                                                                       0.6
                                  0.02
  – explicit pQCD term                  0
                                                                                                       0.5 0                 0.5
                                             0         0.5           1             1.5                   0.01        0.1           1       10
  – χ2/dof = 0.9                                                                                       1.2
                                                                                                       1.1
     (only spacelike)                   1
                                                                                                        1




                                                                                         GM /(µpGD)
                              GE /GD


                                       0.8
• good description                                                                                     0.9
                          p




                                                                                         p
                                                 1
                                       0.6                                                             0.8 1
  of data                                                                                              0.7
                                       0.4
                                            0                0.1         0.2                                 0    0.1 0.2 0.3 0.4 0.5
                                                                                                       0.6
——– best fit                             0.001        0.01
                                                              2
                                                                 0.1
                                                             Q [GeV ]
                                                                     2
                                                                               1                                 0.1
                                                                                                                          2
                                                                                                                            1
                                                                                                                               2
                                                                                                                         Q [GeV ]
                                                                                                                                      10


- - - - 1σ band


                                                                                                                                            31

				
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posted:12/19/2011
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