Extracting the proton charge

and magnetization

radiifrom low-Q2

polarized/unpolarized electron/muon

scattering

John Arrington, Argonne National LaboratoryECT* Workshop on the Proton Radius Puzzle

Graphic by Joshua Rubin, ANL

Outline

JLab form factor measurements– Polarization technique– Two-photon exchange– Proton structure

JLab low Q2 data, proton radius analysis [X. Zhan, et al., PLB 705 (2011) 59]

General considerations in extracting radius from scattering data

Corrections beyond two-photon exchange?? [JA, arXiv:1210.2667]

New techniques: Polarization and A(e,e’N) Mid ’90s brought measurements using improved techniques

– High luminosity, highly polarized electron beams– Polarized targets (1H, 2H, 3He) or recoil polarimeters– Large, efficient neutron detectors for 2H, 3He(e,e’n)

Polarized 3He target

BLAST at MIT-Bates

Focal plane polarimeter – Jefferson Lab

Unpol:GM2+GE

2

Pol:GE/GM

4

Two Photon Exchange Proton form factor measurements

– Comparison of precise Rosenbluth and Polarization measurements of GEp/GMp show clear discrepancy at high Q2

Two-photon exchange corrections believed to explain the discrepancy– Minimal impact on polarization data

Have only limited direct evidence of effect on cross section

– Active program to fully understand TPE

M.K.Jones, et al., PRL 84, 1398 (2000)O.Gayou, et al., PRL 88, 092301 (2003)

I.A.Qattan, et al., PRL 94, 142301 (2005)

P.A.M.Guichon and M.Vanderhaeghen, PRL 91, 142303 (2003)

P. G. Blunden et al, PRC 72 (2005) 034612A.V. Afanasev et al, PRD 72 (2005) 013008D. Borisyuk, A. Kobushkin, PRC 78 (2008) 025208C. Carlson, M. Vanderhaeghen, Ann. Rev. Nucl. Part. Sci. 57 (2007) 171JA, P. Blunden, W. Melnitchouk, PPNP 66 (2011) 782+ several completed or ongoing experiments

S. Boffi, et al.

F. Cardarelli, et al.

P. Chung, F. CoesterF. Gross, P. Agbakpe

G.A. Miller, M. Frank

Quark Orbital Angular Momentum

C. Perdrisat, V. Punjabi, and M. Vanderhaeghen, PPNP 59 (2007)

Many calculations reproduce recently observed falloff in GE/GM

– Descriptions differ in details, but nearly all were directly or indirectly related to quark angular momentum

Insight from Recent Measurements New information on proton structure

– GE(Q2) ≠ GM(Q2) different charge, magnetization distributions– Connection to GPDs: spin-space-momentum correlations

A.Belitsky, X.Ji, F.Yuan, PRD69:074014 (2004)

G.Miller, PRC 68:022201 (2003)

x=0.7x=0.4x=0.1

1 fm

Model-dependent extraction of charge, magnetization distribution of proton:

J. Kelly, Phys. Rev. C 66, 065203 (2002)

Transverse Spatial Distributions

Simple picture: Fourier transform of the spatial distribution– Relativistic case: model dependent “boost” corrections

Model-independent relation found between form factors and transverse spatial distribution

G. Miller, PRL 99, 112001 (2007); G. Miller and JA, PRC 78:032201,2008

(b,x) = ∑ eq ∫ dx q(x,b) = transverse density distribution in infinite momentum frame (IMF) for quarks with momentum x

Natural connection to GPD picture

Evaluated for proton, with experimental and truncation uncertainties

PROTON

NEUTRON

S.Venkat, JA, G.A.Miller, X.Zhan, PRC83, 015203 (2011)

Transverse Spatial Distributions

Simple picture: Fourier transform of the spatial distribution– Relativistic case: model dependent “boost” corrections

Model-independent relation found between form factors and transverse spatial distribution

G. Miller, PRL 99, 112001 (2007); G. Miller and JA, PRC 78:032201,2008

(b,x) = ∑ eq ∫ dx q(x,b) = transverse density distribution in infinite momentum frame (IMF) for quarks with momentum x

Natural connection to GPD picture

Evaluated for proton, with experimental and truncation uncertainties

(b,x): neutron Sea quarks

(x<0.1)

Valence quarks

Intermediate x region

S.Venkat, JA, G.A.Miller, X.Zhan, PRC83, 015203 (2011)

Slide from G. Cates

Q4F2q/

Q4 F1q

Slide from G. Cates

Lamb shift: largest ‘uncertainty’ is correction for size of proton

Precise measurement of Lamb shift measure proton RMS radius

Muonic Hydrogen: Radius 4% below previous best value

Proton 13% smaller, 13% denser than previously believed

Pohl, R. et al. Nature 466, 213-217 (2010)

Proton Charge Radius Extractions

Directly related to strength of QCD in non-perturbative region

Lamb shift: largest ‘uncertainty’ is correction for size of proton

Precise measurement of Lamb shift measure proton RMS radius

Muonic Hydrogen: Radius 4% below previous best value

Proton 13% smaller, 13% denser than previously believed

Pohl, R. et al. Nature 466, 213-217 (2010)

Proton Charge Radius Extractions

Directly related to strength of QCD in non-perturbative region (which would be reallyreally important if we actually knew how to extract “strength of QCD” in non-perturbative region)

Low Q2 data:

JLab E08-007 and “LEDEX” polarization transfer data– 1-2% uncertainty on GE/GM – Less sensitive to TPE

Updated global fit– Improves form factors over Q2

range of the data– Constrain normalization of data

sets over wider Q2 range– Low Q2 fit to extract radius; fix

slopes for global (high-Q2) fitDetails of full (high-Q2) fit: S.Venkat, JA,

G.A.Miller, X. Zhan, PRC 83 (2011) 015203

X. Zhan, et al., PLB 705 (2011) 59; G. Ron, et al., PRC 84 (2011) 055204

JLab radius extraction from ep scattering

Fit directly to cross sections and polarization ratios– Limit fit to low Q2 data– Two-photon exchange corrections (hadronic) applied to cross sections

Estimate model uncertainty by varying fit function, cutoffs– Different parameterizations (continued fraction, inverse polynomial)– Vary number of parameters (2-5 each for GE and GM )– Vary Q2 cutoff (0.3, 0.4, 0.5, 1.0)

11

1

1)(

21

20

2

QbQb

QGCF...1

1)(

62

41

20

2

QbQbQbQGpoly

P. G. Blunden, W. Melnitchouk, J. Tjon, PRC 72 (2005) 034612

Some other issues

Most older extractions dominated by Simon, et al., low Q2 data - 0.5% pt-to-pt and norm. systematics - Neglects uncertainty in Radiative Corr.

We apply TPE uncertainty consistent with other data sets

Relative normalization of experiments:

- Typical approach: fit normalizations and then neglect uncertainty (wrong)

- Ingo Sick’s approach: do not fit normalizations; vary based on quoted uncertainties to evaluate uncertainties (correct - conservative)

- Our approach: Fit normalization factors, vary based on remaining uncertainty from fit

- Systematics hard to tell how well we can REALLY determine normalization

- We set minimum uncertainty to 0.5%

Proton RMS Charge RadiusMuonic hydrogen disagrees with atomic physics and electron scattering determinations of slope of GE at Q2 = 0.

#Extractio

n<RE>2 [fm]

1 Sick 0.8950(180)

2 Mainz 0.8790(80)

2 JLab 0.8750(100)

4CODATA’

060.8768(69)

5Combine

d 2-40.8772(46)

6Muonic

Hydrogen0.8418(7)

JLab

CODATA 10 9 between electron average and muonic hydrogen

Proton magnetic radius

Significant (3.4) difference between Mainz and JLab results

– 0.777(17) fm– 0.867(20) fm

Need to fully understand this before we can reliably combine the electron scattering values?

Robustness of the results

Magnetic form factor, radius much more difficult to extract– GE dominates the cross section at low Q2

• Reduced sensitivity to GM

• High-Q2 data can dominate fit when low-Q2 data is less precise

– Extrapolation to =0 very sensitive to -dependent corrections• Two-photon exchange• Experimental systematics

– Cross section, electron momentum, radiative corrections all vary rapidly with scattering angle

– Relative normalization between data sets with different ranges

From here on, I take liberties with the Mainz data to demonstrate that while RM is potentially

sensitive to such effects, RE is much more robust

Difficulties in extracting the radius

Want enough Q2 range to constrain higher terms, but don’t want to be dominated by high Q2 data; Global fits almost always give poor estimates of the radii

Note: linear fit will always give underestimate of radius for form factor that curves upwards

Dipole

Linear fit

Difficulties in extracting the radius (slope)

I. Sick, PLB 576, 62 (2003)

Q2 [GeV2] : 0 0.01 0.04 0.09 0.15 0.23

Want enough Q2 range to constrain higher terms, but don’t want to be dominated by high Q2 data; Global fits almost always give poor estimates of the radii

Note: linear fit will always give underestimate of radius for form factor that curves upwards

1-GE(Q2)

Difficulties in extracting the radius (slope)

I. Sick, PLB 576, 62 (2003)

Q2 [GeV2] : 0 0.01 0.04 0.09 0.15 0.23

Want enough Q2 range to constrain higher terms, but don’t want to be dominated by high Q2 data; Global fits almost always give poor estimates of the radii

Note: linear fit will always give underestimateunderestimate of radius for form factor that curves upwards

1-GE(Q2)

Linear fit error(stat) 4.7% 1.2% 0.5% 0.3% 0.2%

Truncation Error (GDip) 0.8% 3.3% 7.5% 12% 19%

Fits use ten 0.5% GE values for Q2 from 0 to Q2

max

Optimizing the extractions

Max. Q2 [GeV2] : 0.01 0.04 0.09 0.15 0.230.4

Linear fit error (stat) 4.7% 1.2% 0.5% 0.3% 0.2% 0.1%

Truncation error (GDip) 0.8% 3.3% 7.5% 12% 19% 32%

Quadratic fit error 19% 4.5% 1.9% 1.1% 0.6%0.3%

Truncation error: 0 0.1% 0.6% 1.4% 3.1%7.5%

Cubic fit error 48% 11.5% 4.9% 2.8% 1.7% 0.8%

Truncation error: 0 0 0.1% 0.2% 0.5%1.7%

Linear fit: Optimal Q2=0.024 GeV2, dR=2.0%(stat), 2.0%(truncation)

Quadratic fit: Optimal Q2 = 0.13 GeV2, dR=1.2%(stat), 1.2%(truncation)

Cubic fit: Optimal Q2 = 0.33 GeV2, dR=1.1%(stat), 1.1%(truncation)

Note: Brute force (more data points, more precision) can reduce stat. error

Improved fit functions (e.g. z-pole, CF form) can reduce truncation error, especially for low Q2 extractions

“Tricks” may help further optimize: e.g. decrease data density at higher Q2, exclude data with ‘large’ GM uncertainties

Difficulties in extracting the radius (slope)

JA, W. Melnitchouk, J. Tjon, PRC 76, 035205 (2007)Very low Q2 yields slope but sensitivity to radius is low

Larger Q2 values more sensitive, have corrections due to higher order terms in the expansion

Want enough Q2 range to constrain higher terms, but don’t want to be dominated by high Q2 data; Global fits almost always give poor estimates of the radii

More important for magnetic radius, where the precision on GM gets worse at low Q2 values

Very low Q2 kinematics can have 1% cross sections yielding intercept (GM2) known to

25%

Averaging of fits?

Limited precision on GM at low Q2 means that more parameters are needed to reproduce low Q2 data Low Npar fits may be less reliable

Statistics-weighted average of fits with different #/parameters Emphasizes small Npar

Expect fits with more parameters to be more reliable

–Increase <rM>2 by ~0.020–Increase “statistical” uncertainty

No visible effect in <RE>2

Weighted average: 0.777

“By eye” average of high-N fits

Two-photon exchange corrections

Mainz analysis applied Q2=0 (point-proton) limit of “2nd Born approximation” for Coulomb corrections

Applied 50% uncertainty in GE, GM fit (no uncertainty in radius)

QED: straightforward to calculate

QED+QCD: depends on proton structure

Q2=0

Q2=0.1

Q2=0.3

Q2=1

Q2=0.03

JA , PRL 107, 119101

J.Bernauer, et al., PRL 107, 119102

Impact of TPE

Apply low-Q2 TPE expansion, valid up to Q2=0.1 GeV2

Small change, but still larger than total quoted uncertainty

RADII: <rE2>1/2 goes from 0.879(8) to 0.876(8) fm [-0.3%]

<rM2>1/2 goes from 0.777(17) to 0.803(17) fm [+3.0%]

Note: these uncertainties do notdo not include any contribution related to TPE: Change between default prescription and this suggests TPE uncertainty of approximately 0.003 fm for rE, 0.026 fm for rM

Much (most?) of the effect associated with change in normalization factors of the different data subsets

JA , PRL 107, 119101; J.Bernauer, et al., PRL 107, 119102

Borisyuk/Kobushkin, PRC 75, 028203 (2007)

Comparison of low Q2 TPE calculations:

Blunden, et al., hadronic calculation [PRC 72, 034612 (2005)]

Borisyuk & Kobushkin: Low-Q2 expansion, valid up to 0.1 GeV2 [PRC 75, 038202 (2007)]

B&K: Dispersion analysis (proton only) [PRC 78, 025208 (2008)]

B&K: proton + [arXiv:1206.0155]

Typical uncertainties for radiativecorrections are 1-1.5%; probably fair(or overestimate) after applying TPEcalculations, at least for lower Q2

Combining world’s data (or takingMainz data set) yields enough datathat it’s not sufficient to treat as uncorrelated or norm. uncertainties

Full TPE Full TPE calculationscalculations

Proton magnetic radius

Updated TPE yields RM=0.026 fm

0.777(17) 0.803(17)

Remove fits that may not have sufficient flexibility: R≈0.02 fm?

Mainz/JLab difference goes from 3.4 to 1.7, less if include TPE uncertainty

RE value almost unchanged: 0.879(8) 0.876(8)

Higher-order Coulomb corrections?

Additional Coulomb Corrections? [JA, arXiv:1210.2677]

CC: 2nd born approximation– Increases charge radius ~0.010 fm– [Rosenfelder PLB479(2000)381, Sick PLB576(2003)62]

+ hard 2 corrections– Minimal impact (additional 0.002 fm)– [Blunden and Sick, PRC72(2005)057601]

Low Q2: CC in 2nd Born become small but non-zeroVery low energies, might expect large corrections (classical limit)

Could this have any impact on the radii extracted from data?

22ndnd Born Born

Additional Coulomb Corrections?

22ndnd Born Born

EMAEMA

Effective Momentum Approximation– Coulomb potential boosts energy at

scattering vertex– Flux factor enhancement

– Used in QE scattering (Coulomb field of nucleus)

Key parameter: average e-p separation at the scattering

– ~1.6 MeV at surface of proton– Decreases as 1/R outside proton

Additional Coulomb Corrections?

Effective Momentum Approximation– Coulomb potential boosts energy at

scattering vertex– Flux factor enhancement

– Used in QE scattering (Coulomb field of nucleus)

Key parameter: average e-p separation at the scattering

– ~1.6 MeV at surface of proton– Decreases as 1/R outside proton

Assume scattering occurs at R = 1/q– Limits correction below Q20.06 GeV2

where scattering away from proton

EMAEMA

22ndnd Born Born

Additional Coulomb Corrections?

Very little effect at high ; no impact on charge radius

Large Q2 dependence at low , especially at very low Q2

Proton radius slope -600%/GeV2

0-0.02 GeV2: CC slope +100%/GeV2

0.05-0.2 GeV2: slope -8%/GeV2

Higher : up to ~15%/GeV2

CouldCould impact extraction of magnetic impact extraction of magnetic radiusradius

– Need real calculation– Need to apply directly to real

kinematics of the experiment

EMAEMA

= 0.02= 0.02

EMAEMA

How many parameters is enough? Too many?

Simulated data World’s data (w/o Bernauer)

Black points: Total chi-squared for fit to “Fit” data vs. N = # of param.

Red points: Comparing result of fit to independent “Reference” data set (generated according to same distribution as “Fit” data)

Summary

Inconsistency between muonic hydrogen and electron-based extractions

Fits from scattering data must take care to avoid underestimating uncertainties, but charge radius is significantly more robust

Future experiments planned– Better constrain GM at low Q2

– Map out structure of GE at low Q2

– Check TPE in both electron and muon scattering– Directly compare electron and muon scattering cross sections

Fin…

Impact of TPE

RADII: <rE2>1/2 goes from 0.879(8) to 0.876(8) fm [-0.3%]

<rM2>1/2 goes from 0.777(17) to 0.803(17) fm [+3.0%]

Note: these uncertainties do notdo not include any contribution related to TPE: Change between default prescription and this suggests TPE uncertainty of approximately 0.003 fm for rE, 0.026 fm for rM

JA , PRL 107, 119101; J.Bernauer, et al., PRL 107, 119102

Borisyuk/Kobushkin, PRC 75, 028203 (2007)

Apply low-Q2 TPE expansion, valid up to Q2=0.1 GeV2

Small change, but still larger than total quoted uncertainty

Best fit starting radius, normalization factors Vary radius parameter and refit, determine 2 vs. radius (allowing everything including normalizations to vary) Different functional forms & data range to check systematic:

- CF, Polynomial fits ; N = 3, 4, 5 ; Q2 <0.5 (0.3, 0.4, 1.0)

Fit (and normalization) uncertainties

Quoted normalization uncertainties of ~2-4%; Fit yields 0.2-1.0%

Polarization data very helpful in linking low and high data; less room to trade off between slope in reduced cross section and normalization factors

These analyses neglect correlated uncertainties: a Q2-dependent or -dependent systematic can yield incorrect normalization

Hard to be sure that normalization is known to much better than 0.5%

How well can we determine the normalizations?

Assumed minimum uncertainty