Precise Determination of the Electric and Magnetic Form Factors of the Proton

Precise Determination of the Electric and Magnetic Michael O. Distler for the A1 collaboration @ MAMI Institut für Kernphysik Johannes Gutenberg-Universität Mainz GDR Nucleon meeting at CEA/SPhN Saclay, France, November 25, 2010

Outline 1 Introduction I: The size of the proton from the Lamb shift in muonic hydrogen and electron scattering 2 Introduction II: Electric and magnetic form factors of the Proton 3 The Mainz high-precision p(e,e )p measurement 4 Results Design considerations Covered kinematical region Analysis technique Cross section results Checks: Rosenbluth and model dependence 5 Conclusion and Outlook 6 Discussion of the Lamb shift / electron scattering discrepancy

Introduction I: The size of the proton Nature 466, 213-216 (8 July 2010)

Cross section and form factors for elastic e-p scattering The cross section: ) with: ( dσ dω ( dσ dω ) Mott = 1 [ ( εge 2 Q 2) + τgm (Q 2 2)] ε (1 + τ) ( τ = Q2 4mp 2, ε = 1 + 2 (1 + τ) tan 2 θ ) 1 e 2 Fourier-transform of G E, G M spatial distribution (Breit frame) re 2 = 6 2 dg E dq 2 Q 2 =0 rm 2 = 6 2 d (G M/µ p ) dq 2 Q 2 =0

Overview of different proton charge-radius results Pohl et al. (2010) Bernauer et al. (2010) CODATA 06 (2008) CODATA 02 (2005) Melnikov et al. (2000) Udem et al. (1997) Blunden et al. (2005) Sick et al. (2003) Rosenfelder et al. (2000) Mergell et al. (1996) Wong et al. (1994) Eschrich et al. (2001) McCord et al. (1991) Simon et al. (1980) Borkowski et al. (1975) Akimov et al. (1972) Frerejacque et al. (1966) Hand et al. (1963) 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 <r E > [fm] Filled dots: Results from new measurements. Hollow dots: Reanalysis of existing data.

Introduction II: Original Motivation G EP /G dipole 1.6 1.4 1.2 1 0.8 total smooth Simon et al. Price et al. Berger et al. Hanson et al. polarisation data G MP /(µ p G dipole ) 1.6 1.4 1.2 1 0.8 total smooth Hoehler et al. Janssens et al. Berger et al. Bartel et al. Walker et al. Litt et al. Andivahis et al. Sill et al. 0.6 0.6 0.4 0 1 2 3 4 5 6 Q / (GeV/c) 0.4 0 1 2 3 4 5 6 Q / (GeV/c) G EP -G EP,phaen. 0.1 0.05 0 total-smooth Simon et al. Price et al. Berger et al. Hanson et al. polarisation data G MP -G MP,phaen. 0.1 0.05 0 total - smooth Hoehler et al. Janssens et al. Berger et al. Bartel et al. Walker et al. Litt et al. Andivahis et al. Sill et al. -0.05-0.05-0.1 0.001 0.01 0.1 1 10 Q 2 / (GeV/c) 2-0.1 0.001 0.01 0.1 1 10 Q 2 / (GeV/c) 2 (see J. Friedrich and Th. Walcher, Eur. Phys. J. A 17 (2003) 607)

Introduction II: Original Motivation G EP /G dipole 1.6 1.4 1.2 1 0.8 total smooth Simon et al. Price et al. Berger et al. Hanson et al. polarisation data G MP /(µ p G dipole ) 1.6 1.4 1.2 1 0.8 total smooth Hoehler et al. Janssens et al. Berger et al. Bartel et al. Walker et al. Litt et al. Andivahis et al. Sill et al. G EP -G EP,phaen. 0.6 0.4 0 1 2 3 4 5 6 Q / (GeV/c) -0.05 Discrepancy of existing values for proton electric radius: 0.6 0.4 0 1 2 3 4 5 6 Q / (GeV/c) 0.809(11) fm: standard dipole at HEPL (Hand et al. 1963) 0.1 0.1 total - smooth total-smooth Simon Price et et al. Hoehler et al. 0.862(12) fm: low Q 2 at Mainz (Simon et al. 1979) Janssens et al. al. Berger et al. Berger et al. 0.05 Hanson et al. 0.05 Bartel et al. Walker et al. 0.847(09) polarisation data fm: dispersion relation (Mergell et al. 1996) Litt et al. Andivahis et al. Sill et al. 0 0 0.890(14) fm: Hydrogen Lamb shift (Udem et al. 1997) G MP -G MP,phaen. -0.05-0.1 0.001 0.01 0.1 1 10 Q 2 / (GeV/c) 2-0.1 0.001 0.01 0.1 1 10 Q 2 / (GeV/c) 2 (see J. Friedrich and Th. Walcher, Eur. Phys. J. A 17 (2003) 607)

The Mainz Microtron MAMI

The Mainz Microtron MAMI MAMI-A: 180 MeV fixed

The Mainz Microtron MAMI MAMI-A: 180 MeV fixed MAMI-B: 855 MeV,15 MeV steps

The Mainz Microtron MAMI MAMI-A: 180 MeV fixed MAMI-B: 855 MeV,15 MeV steps MAMI-C: 1.5 GeV,15 MeV steps

The Mainz high-precision p(e,e )p measurement: Three spectrometer facility of the A1 collaboration

Design goal: High precision Statistical precision: 20 min beam time for <0.1%

Design goal: High precision through redundancy Statistical precision: 20 min beam time for <0.1% Control of luminosity and systematic errors: Measure all quantities in as many ways as possible:

Design goal: High precision through redundancy Statistical precision: 20 min beam time for <0.1% Control of luminosity and systematic errors: Measure all quantities in as many ways as possible: Beam current: Foerster probe (usual way) pa-meter measures down to extremely low currents for small θ Luminosity: current density target length third magnetic spectrometer as monitor

Measured settings and future (high Q 2 ) expansion dσ dω = ( ) dσ dω Mott 1 [ ( εge 2 Q 2) + τgm (Q 2 2)] ε (1 + τ) Q [GeV/c] 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 spectrometer A limit spectrometer B limit ε MAMI min. E= 180 MeV MAMI-C max. E=1.53 GeV MAMI-B max. E= 855 MeV 2.56 1.96 1.44 1 0.64 0.36 0.16 0.04 Q 2 [(GeV/c) 2 ]

Measured settings and future (high Q 2 ) expansion dσ dω = ( ) dσ dω Mott 1 [ ( εge 2 Q 2) + τgm (Q 2 2)] ε (1 + τ) 1.6 2.56 1.4 1.96 Q [GeV/c] 1.2 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 spectrometer A spectrometer B spectrometer C ε 1.44 1 0.64 0.36 0.16 0.04 Q 2 [(GeV/c) 2 ] 90 GByte on disc 1400 Settings > 10 9 events.

Background 48 mm 20 mm 10 mm

Background 48 mm 20 mm 10 mm Liquid Hydrogen -20-10 0 10 20 de [MeV]

Background 48 mm 20 mm 10 mm Liquid Hydrogen Havar foil -20-10 0 10 20 de [MeV] -20-10 0 10 20 de [MeV] -20-10 0 10 20 de [MeV]

Data Simulation matching Simulation: Model for energy loss and small angle scattering Input: momentum-, angular-, vertex resolution 500 400 300 200 100 6000 5000 4000 3000 2000 1000 0-20 -10 0 10 20 data simulated background residue 0-100 -20-10 0 10 20 de [MeV]

Feynman graphs of leading and next to leading order for elastic scattering All graphs are taken into account: vacuum polarization (v1): e, (µ, τ) Maximon/Tjon (2000) and Vanderhaeghen et al. (2000) electron vertex correction Coulomb distortion (two photon exchange) real photon emission

Description of the radiative tail 10000 1000 counts 100 10 1 0 10 20 30 40 50 60 70 80 E [MeV] integrated tail exp./sim. [arb. units] 1.005 1.004 1.003 1.002 1.001 1 0.999 0.998 0.997 0.996 0.995 10 20 30 40 50 60 70 E [MeV]

Cross sections 10000 standard dipole 1000 cross section dσ/dω / µbarn/sr 100 10 1 0.1 0.01 0.001 0.0001 20 40 60 80 100 120 140 scattering angle

Cross sections / standard dipole 1.7 +0.5, 855 MeV 1.6 σ exp /σ std. dipole 1.5 1.4 1.3 1.2 1.1 +0.4, 720 MeV +0.3, 585 MeV +0.2, 450 MeV +0.1, 315 MeV 1 0.9 20 40 60 80 100 120 scattering angle 180 MeV

How to extract the form factors? Two methods: 1 Classical Rosenbluth separation

How to extract the form factors? Two methods: 1 Classical Rosenbluth separation 2 Super-Rosenbluth separation : Fit of form factor models directly to the measured cross sections Feasible due to fast computers. All data at all Q 2 and ε values contribute to the fit, i.e. full kinematical region used, no projection (to specific Q 2 ) needed. Easy fixing of normalization.

Models: Dipols Dipole (different b for G E and G M ): ) G D (Q 2, b = ( 1 1 + Q2 b ) 2 Double Dipole (as in Friedrich/Walcher phenomenological fit [Eur. Phys. J. A 17 (2003) 607]): G DD ( Q 2, a, b 1, b 2 ) = ag D ( Q 2, b 1 ) + (1 a) G D ( Q 2, b 2 )

Models: Polynomial Polynomial G P ( Q 2, a 1,..., a n ) = 1 + Polynomial + standard Dipole n a i Q 2 i i=1 ( ) ) G PAD Q 2, a 1,..., a n = G D (Q 2, 0.71 + Polynomial standard Dipole ( ( ) ) G PMD Q 2, a 1,..., a n = G D (Q 2, 0.71 1 + n a i Q 2 i i=1 ) n a i Q 2 i i=1

Models: Splines Uniform cubic splines ) spline (Q 2, a 1,..., a n Spline: 4 3.5 3 2.5 2 1.5 1 0.5 0-3 -2-1 0 1 Knots λ i ( ) G Spline Q 2, a 1,..., a n = 1 + Q 2 spline (Q 2) Spline standard Dipole ( ) ) ( G SMD Q 2, a 1,..., a n = G D (Q 2, 0.71 1 + Q 2 spline (Q 2))

Models: Misc Also: Friedrich / Walcher phenomenological ansatz extended Gari-Krümpelmann (VMD), Lomon et al. Arrington type: P N P N+2

Cross sections + spline fit 1.7 +0.5, 855 MeV 1.6 σ exp /σ std. dipole 1.5 1.4 1.3 1.2 1.1 +0.4, 720 MeV +0.3, 585 MeV +0.2, 450 MeV +0.1, 315 MeV 1 0.9 20 40 60 80 100 120 scattering angle 180 MeV

Cross sections: 180 MeV σ exp /σ std. dipole 1.02 1.01 1 0.99 0.98 Polynomial Poly. + dip. Poly. dip. Inv. poly. Spline Spline dip. Friedrich-Walcher Double Dipole Extended G.K. 0.97 0.96 0 0.02 0.04 0.06 0.08 0.1 Q 2 [(GeV/c) 2 ]

Form factor results G E /G std. dipole 1.05 1 0.95 0.9 0.85 0.8 Christy et al. Simon et al. Price et al. Berger et al. Hanson et al. Borkowski et al. Janssens et al. Murphy et al. 0 0.2 0.4 0.6 0.8 1 G M /(µ p G std. dipole ) 1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 Christy et al. Price et al. Berger et al. Hanson et al. Borkowski et al. Janssens et al. Bosted et al. Bartel et al. 0.94 0 0.2 0.4 0.6 0.8 1 G E /G std. dipole 1.04 1.03 1.02 1.01 1 0.99 0.98 0.97 0.96 0.95 Arrington et al. Friedrich/Walcher Simon et al. Price et al. Borkowski et al. Janssens et al. Murphy et al. 0 0.05 0.1 0.15 0.2 µ p G E /G M 1.1 1.05 1 0.95 0.9 Arrington w/o TPE Jones et al. 0.85 Arrington w/ TPE Crawford et al. Pospischil et al. Dieterich et al. 0.8 Gayou et al. Milbrath et al. Ron et al. Punjabi et al. 0.75 0 0.2 0.4 0.6 Q 2 / (GeV/c) 2 0.8 1 Jan C. Bernauer et al., High-precision determination of the electric and magnetic form factors of the proton, arxiv:1007.5076

The electric rms radius - extracted by different models Spline 3 Ord. 8 par. Spline 3 Ord. 9 par. Spline 3 Ord. 10 par. Spline 3 Ord. 11 par. Spline 4 Ord. 8 par. Spline 4 Ord. 9 par. Spline 4 Ord. 10 par. Spline 4 Ord. 11 par. Spline 5 Ord. 8 par. Spline 5 Ord. 9 par. Spline 5 Ord. 10 par. Spline 5 Ord. 11 par. Spline dipole 7 par. Spline dipole 8 par. Spline dipole 9 par. Spline dipole 10 par. Spline dipole 11 par. Inv. Poly. 6 par. Inv. Poly. 7 par. Inv. Poly. 8 par. Inv. Poly. 9 par. Poly. 9 par. Poly. 10 par. Poly. 11 par. Poly. 12 par. Poly. + dipole 9 par. Poly. + dipole 10 par. Poly. + dipole 11 par. Poly. + dipole 12 par. Poly. dipole 7 par. Poly. dipole 8 par. Poly. dipole 9 par. Poly. dipole 10 par. Friedrich-Walcher 0.86 0.87 0.88 0.89 0.9 0.91 <r E > [fm]

Conclusion Part I High precision e-p scattering data from MAMI. e-print: arxiv:1007.5076 accepted by PRL Q 2 range from 0.003 to 1 (GeV/c) 2. Consistent data set. Super-Rosenbluth fit to determine form factors and radii. The charge and magnetic rms radii are determined as r e = 0.879 ± 0.005 stat. ± 0.004 syst. ± 0.002 model ± 0.004 group fm, r m = 0.777 ± 0.013 stat. ± 0.009 syst. ± 0.005 model ± 0.002 group fm. Supported by the Deutsche Forschungsgemeinschaft (DFG) with a Sonderforschungsbereich (SFB443).

Discussion of the Lamb shift / electron scattering discrepancy The following tables are taken from the QED supplement published in Nature 466, 213-216 (8 July 2010). All known radius-independent contributions and all relevant radius-dependent contributions to the Lamb shift in µp from different authors are listed.

2S 2P splitting in muonic hydrogen

Discussion of the Lamb shift / electron scattering discrepancy

Discussion of the Lamb shift / electron scattering discrepancy E = 209.9779(49) 5.2262 r 2 p + 0.0347 r 3 p Values are in mev and radii in fm.

Discussion of the Lamb shift / electron scattering discrepancy Zemach-Moments: A. C. Zemach, Proton Structure and the Hyperfine Shift in Hydrogen, Phys. Rev. 104, 1771 (1956). < r 3 > (2) = 0 dq ( ) q 4 GE 2 (q2 ) 1 + q 2 < r 2 > p /3 < r 3 > (2) = 2.27 fm 3 r p = 0.84184(67) fm < r 3 > (2) = 2.85(8) fm 3 r p = 0.84245(67) fm M.O.D., J.C. Bernauer, and Th. Walcher, The RMS Charge Radius of the Proton and Zemach Moments, arxiv:1011.1861.

2S 2P splitting in muonic hydrogen

De Rújula s toy model A. De Rújula, QED is not endangered by the proton s size, Phys. Lett. B693, 555 (2010). Sum of single pole and dipole [ ] ρ Proton (r) = 1 M 4 e Mr cos 2 (θ) + m5 e mr sin 2 (θ) D 4πr 8π D M 2 cos 2 (θ) + m 2 sin 2 (θ) using M = 0.750 GeV/c 2, m = 0.020 GeV/c 2, and sin 2 (θ) = 0.3 and ρ (2) (r) = d 3 r 2 ρ charge ( r r 2 ) ρ charge (r 2 ) we get the third Zemach moment: r 3 (2) = d 3 r r 3 ρ (2) (r) = 36.2 fm 3

De Rújula s toy model... We put r 3 (2) = 36.2 fm 3 in the Lamb shift formular: L 5th [ r 2, r 3 (2) ] = ( 209.9779(49) 5.2262 r 2 fm 2 + 0.00913 r ) 3 (2) fm 3 mev and get r p = 0.878 fm

De Rújula s toy model... We put r 3 (2) = 36.2 fm 3 in the Lamb shift formular: L 5th [ r 2, r 3 (2) ] = ( 209.9779(49) 5.2262 r 2 fm 2 + 0.00913 r ) 3 (2) fm 3 mev and get r p = 0.878 fm problem solved

De Rújula s toy model is excluded by experiment charge distribution form factor 4 Π r 2 Ρ r fm 1.2 1.0 0.8 0.6 0.4 0.2 G E,p Q 0.1 0.01 0.001 10 4 0.0 0 1 2 3 4 0 2 4 6 8 10 De Rújula s toy model standard dipole Bernauer-Arrington fit assembly

Conclusion Part II High precision form factors from MAMI provide constraints for the charge distribution of the proton. Standard dipole approximation is not sufficient for correction of the muonic hydrogen Lamb shift. M.O.D., J.C. Bernauer, and Th. Walcher, The RMS Charge Radius of the Proton and Zemach Moments, arxiv:1011.1861. Supported by the Deutsche Forschungsgemeinschaft (DFG) with a Sonderforschungsbereich (SFB443).

Outlook Jan Bernauer joined the OLYMPUS Experiment @ DESY/Hamburg (Spokesperson: Richard Milner, MIT) OLYMPUS: Determine the effect of two-photon exchange in elastic lepton-proton scattering by precisely measuring the ratio of positron-proton to electron-proton elastic unpolarized cross sections. low Q 2 extension: ISR @ MAMI

Outlook: Initial state radiation 10 4 10 3 e e e e dσ/(de dω ) LAB (dω γ ) CM [µb/(gev sr 2 )] 10 2 10 1 10 1 10 2 10 3 10 4 10 5 e e γ* p p γ* γ* p p p p e e γ* p p 10 6 180 o 90 o 0 o 90 o 180 o ϑ γγ

Outlook: Initial state radiation 10-1 10 8 Reconstructed Q 2 (E,θ ) [GeV 2 /c 2 ] 10-2 10-3 10-4 10-5 10-5 10-4 10-3 10-2 10-1 Q 2 at Vertex (Simulation) [GeV 2 /c 2 ] 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 Events / (µa h)

Backup

Discussion of the Lamb shift / electron scattering discrepancy

Discussion of the Lamb shift / electron scattering discrepancy E = 209.9779(49) 5.2262 r 2 p + 0.0347 r 3 p Values are in mev and radii in fm.

Data taking periods August 2006 November 2006 May 2007 Duration 10 days 11 days 17 days without setup/calibration 8 days 9 days 11 days Energies 575, 850 MeV 180, 720 MeV 315, 450 MeV Setting changes 152 173 202 data taking time 3.3 days 4.3 days 5.3 days Average time per setting 31 min 35 min 38 min Overhead per setting 44 min 40 min 40 min Overhead includes angle changes, momentum changes and down times. Average time for angle only setting changes: 10 min.

Form factor results G E /G std. dip 0.95 0.9 0.85 0.8 [13] [2] Christy et al. Simon et al. Price et al. Berger et al. Hanson et al. Borkowski et al. [15] Janssens et al. Murphy et al. [16] 0 0.2 0.4 0.6 0.8 1 G E /G std. dipole G E /G std. dipole 1.04 1.03 1.02 1.01 1 0.99 0.98 0.97 0.96 0.95 1.05 1 0.95 0.9 0.85 0.8 [13] [2] Simon et al. Price et al. Borkowski et al. [15] Janssens et al. Murphy et al. [16] 0 0.05 0.1 0.15 0.2 [13] [2] Christy et al. Simon et al. Price et al. Berger et al. Hanson et al. Borkowski et al. [15] Janssens et al. Murphy et al. [16] 0 0.2 0.4 0.6 0.8 1 G M /(µ p G std. dipole ) µ p G E /G M 1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 [13] [2] Christy et al. Price et al. Berger et al. Hanson et al. Borkowski et al. [15] Janssens et al. Bosted et al. Bartel et al. 0.94 0 0.2 0.4 0.6 0.8 1 1.1 1.05 1 0.95 0.9 0.85 [13] w/o TPE Gayou et al. [13] w/ TPE Milbrath et al. Pospischil et al. 0.8 [2] Punjabi et al. Dieterich et al. Crawford et al. Jones et al. Ron et al. [17] 0.75 0 0.2 0.4 0.6 0.8 1 Q 2 / (GeV/c) 2 M /(µ p G std. dipole ) 1.1 1.08 1.06 1.04 1.02 1 [13] [2] Christy et al. Price et al. Berger et al. Hanson et al. Borkowski et al. [15] Janssens et al. Bosted et al. Bartel et al.

Backup 1.08 1.06 1.04 Polynomial Poly. + Dip. Poly. Dip. Inv. Poly. Spline Spline Dip. Friedrich/Walcher σ exp /σ std. dipole 1.02 1 0.98 0.96 PRELIMINARY 0.94 0 20 40 60 80 100 120 140 Central angle of spectrometer

Cross sections: 315 MeV σ exp /σ std. dipole 1.02 1.01 1 0.99 0.98 Polynomial Poly. + dip. Poly. dip. Inv. poly. Spline Spline dip. Friedrich-Walcher Double Dipole Extended G.K. 0.97 0.96 0.95 0 0.05 0.1 0.15 0.2 0.25 Q 2 [(GeV/c) 2 ]

Cross sections: 450 MeV σ exp /σ std. dipole 1.03 1.02 1.01 1 0.99 0.98 0.97 0.96 Polynomial Poly. + dip. Poly. dip. Inv. poly. Spline Spline dip. Friedrich-Walcher Double Dipole Extended G.K. 0.95 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Q 2 [(GeV/c) 2 ]

Cross sections: 585 MeV σ exp /σ std. dipole 1.08 1.06 1.04 1.02 1 0.98 Polynomial Poly. + dip. Poly. dip. Inv. poly. Spline Spline dip. Friedrich-Walcher Double Dipole Extended G.K. 0.96 0.94 0 0.1 0.2 0.3 0.4 0.5 0.6 Q 2 [(GeV/c) 2 ]

Cross sections: 720 MeV σ exp /σ std. dipole 1.15 1.1 1.05 1 Polynomial Poly. + dip. Poly. dip. Inv. poly. Spline Spline dip. Friedrich-Walcher Double Dipole Extended G.K. 0.95 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Q 2 [(GeV/c) 2 ]

Cross sections: 855 MeV σ exp /σ std. dipole 1.2 1.15 1.1 1.05 1 Polynomial Poly. + dip. Poly. dip. Inv. poly. Spline Spline dip. Friedrich-Walcher Double Dipole Extended G.K. 0.95 0 0.2 0.4 0.6 0.8 1 Q 2 [(GeV/c) 2 ]

Rosenbluth separation Q 2 = 0.15 (GeV/c) 2 0.55 0.5 ε (1+τ) σ exp /σ Mott 0.45 0.4 0.35 0.3 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ε

... more...

... and more...

... and even more

Comparison: Rosenbluth vs. Spline fit 1.01 1 0.99 G E /G std. dipole 0.98 0.97 0.96 0.95 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Q 2 [(GeV/c) 2 ] Spline fit Rosenbluth

Model dependence of radius extraction Check extraction of radii with Monte-Carlo data: Monte-Carlo data from given parametrization (known radii!) Error distribution of this simulated data according to errors from real data Fit with different models Assumption: ±5% normalization error (per spectrometer/energy)

Model dependence: Charge Radius Analysis of simulated pseudo data Input Analysis Dipole Dbl-D. Poly. P.+D. P. D. Spline S. D. F./W. Std. dipole 811 0±1 0±1 0±3 0±3 0±4 0±5 0±7 0±1 Arrington 07 878 18±1 3±3 3±3 2±3 1±4 4±5 1±6 2±3 Arr. 03 (P) 829 29±1 10±1 1±3 1±3 0±4 1±5 0±6 2±6 Arr. 03 (R) 868 9±1 0±2 0±3 0±3 0±4 3±5 0±6 1±3 FW 860 4±1 31±14 1±3 1±3 1±4 0±5 0±6 0±3

Spline fit + error band 1.01 1 0.99 G E /G std. dipole 0.98 0.97 0.96 0.95 0.94 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Q 2 [(GeV/c) 2 ] Spline fit stat. errors exp. syst. errors theo. syst. errors Simon et al. Price et al. Berger et al. Hanson et al. Borkowski et al. Janssens et al. Murphy et al.

Spline fit + error band 1.02 1.01 µ p G M /G std. dipole 1 0.99 0.98 0.97 0.96 0.95 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Q 2 [(GeV/c) 2 ] Spline fit stat. errors exp. syst. errors theo. syst. errors Price et al. Berger et al. Hanson et al. Borkowski et al. Janssens et al.

The cryogenic target system 11.5 mm 49.5 mm Electronbeam liquid hydrogen 10 µm Havar Target cell Ventilator Heat exchanger "Basel-Loop" Scattering chamber

Model dependence: Magnetic Radius Analysis of simulated pseudo data Input Analysis Dipole Dbl-D. Poly. P.+D. P. D. Spline S. D. F./W. Std. dipole 811 0±1 0±1 1±7 0±7 0±10 2±14 1±18 0±1 Arrington 07 858 55±1 4±4 5±6 4±6 1±9 2±13 0±17 10±4 Arr. 03 (P) 837 33±1 12±3 1±7 0±7 0±9 2±13 0±19 5±5 Arr. 03 (R) 863 52±1 2±4 4±6 3±6 0±9 3±13 0±17 8±4 FW 805 4±1 49±2 0±7 1±7 1±10 1±13 1±18 1±4