Extracting the proton charge and magnetization radii from low-Q 2 polarized/unpolarized electron/muon scattering John Arrington, Argonne National Laboratory ECT* 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 Q 2 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: ]

New techniques: Polarization and A(e,e’N)  Mid ’90s brought measurements using improved techniques –High luminosity, highly polarized electron beams –Polarized targets ( 1 H, 2 H, 3 He) or recoil polarimeters –Large, efficient neutron detectors for 2 H, 3 He(e,e’n) Polarized 3 He target BLAST at MIT-Bates Focal plane polarimeter – Jefferson Lab Unpol:  G M 2 +  G E 2 Pol:  G E /G M

4 Two Photon Exchange  Proton form factor measurements –Comparison of precise Rosenbluth and Polarization measurements of G Ep /G Mp show clear discrepancy at high Q 2  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, (2003) I.A.Qattan, et al., PRL 94, (2005) P.A.M.Guichon and M.Vanderhaeghen, PRL 91, (2003) P. G. Blunden et al, PRC 72 (2005) A.V. Afanasev et al, PRD 72 (2005) D. Borisyuk, A. Kobushkin, PRC 78 (2008) C. Carlson, M. Vanderhaeghen, Ann. Rev. Nucl. Part. Sci. 57 (2007) 171 JA, P. Blunden, W. Melnitchouk, PPNP 66 (2011) several completed or ongoing experiments

Two Photon Exchange 5 Golden mode: e + -p vs. e - -p elastic scattering JA, PRC 69, (2004) Existing e+p/e-p data show some evidence for TPE TPE calculations largely resolve discrepancy Rosenbluth data with TPE correction Polarization transfer JA, W. Melnitchouk, and J. Tjon, PRC 76, (2007) IF THEN IF TPE corrections fully explain the discrepancy, THEN they are constrained well enough that they do not limit our extractions of the form factor Three new e+/e- experiments BINP Novosibirsk – internal target JLab – mixed e+/e- beam, CLAS DESY (OLYMPUS) - internal target

S. Boffi, et al. F. Cardarelli, et al. P. Chung, F. Coester F. 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 G E /G M –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 –G E (Q 2 ) ≠ G M (Q 2 )  different charge, magnetization distributions –Connection to GPDs: spin-space-momentum correlations A.Belitsky, X.Ji, F.Yuan, PRD69: (2004) G.Miller, PRC 68: (2003) x=0.7 x=0.4 x=0.1 1 fm Model-dependent extraction of charge, magnetization distribution of proton: J. Kelly, Phys. Rev. C 66, (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, (2007); G. Miller and JA, PRC 78:032201,2008   (b,x) = ∑ e q ∫ 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, (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, (2007); G. Miller and JA, PRC 78:032201,2008   (b,x) = ∑ e q ∫ 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, (2011)

Slide from G. Cates Q4F2q/Q4F2q/ Q 4 F 1 q

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, (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, (2010) Proton Charge Radius Extractions really  Directly related to strength of QCD in non-perturbative region (which would be really important if we actually knew how to extract “strength of QCD” in non-perturbative region)

Low Q 2 data:  JLab E and “LEDEX” polarization transfer data –1-2% uncertainty on G E /G M –Less sensitive to TPE  Updated global fit –Improves form factors over Q 2 range of the data –Constrain normalization of data sets over wider Q 2 range –Low Q 2 fit to extract radius; fix slopes for global (high-Q 2 ) fit Details of full (high-Q 2 ) fit: S.Venkat, JA, G.A.Miller, X. Zhan, PRC 83 (2011) X. Zhan, et al., PLB 705 (2011) 59; G. Ron, et al., PRC 84 (2011)

JLab radius extraction from ep scattering  Fit directly to cross sections and polarization ratios –Limit fit to low Q 2 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 G E and G M ) –Vary Q 2 cutoff (0.3, 0.4, 0.5, 1.0) P. G. Blunden, W. Melnitchouk, J. Tjon, PRC 72 (2005)

Some other issues Most older extractions dominated by Simon, et al., low Q 2 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 Radius Muonic hydrogen disagrees with atomic physics and electron scattering determinations of slope of G E at Q 2 = 0. #Extraction 2 [fm] 1Sick0.8950(180) 2Mainz0.8790(80) 2JLab0.8750(100) 4CODATA’ (69) 5 Combined (46) 6 Muonic Hydrogen (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 –G E dominates the cross section at low Q 2 Reduced sensitivity to G M High-Q 2 data can dominate fit when low-Q 2 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 R M is potentially sensitive to such effects, R E is much more robust

Difficulties in extracting the radius Want enough Q 2 range to constrain higher terms, but don’t want to be dominated by high Q 2 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) Q 2 [GeV 2 ] : Want enough Q 2 range to constrain higher terms, but don’t want to be dominated by high Q 2 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-G E (Q 2 )

Difficulties in extracting the radius (slope) I. Sick, PLB 576, 62 (2003) Q 2 [GeV 2 ] : Want enough Q 2 range to constrain higher terms, but don’t want to be dominated by high Q 2 data; Global fits almost always give poor estimates of the radii underestimate Note: linear fit will always give underestimate of radius for form factor that curves upwards 1-G E (Q 2 ) Linear fit error(stat) 4.7% 1.2% 0.5% 0.3% 0.2% Truncation Error (G Dip ) 0.8% 3.3% 7.5% 12% 19% Fits use ten 0.5% G E values for Q 2 from 0 to Q 2 max

Optimizing the extractions Max. Q 2 [GeV 2 ] : Linear fit error (stat)4.7%1.2%0.5%0.3%0.2%0.1% Truncation error (G Dip ) 0.8%3.3%7.5%12%19%32% Quadratic fit error19%4.5%1.9%1.1%0.6%0.3% Truncation error: 00.1%0.6%1.4%3.1%7.5% Cubic fit error48%11.5%4.9%2.8%1.7%0.8% Truncation error: %0.2%0.5%1.7% Linear fit:Optimal Q 2 =0.024 GeV 2, dR=2.0%(stat), 2.0%(truncation) Quadratic fit:Optimal Q 2 = 0.13 GeV 2, dR=1.2%(stat), 1.2%(truncation) Cubic fit:Optimal Q 2 = 0.33 GeV 2, 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 Q 2 extractions “Tricks” may help further optimize: e.g. decrease data density at higher Q 2, exclude data with ‘large’ G M uncertainties

Difficulties in extracting the radius (slope) JA, W. Melnitchouk, J. Tjon, PRC 76, (2007) Very low Q 2 yields slope but sensitivity to radius is low Larger Q 2 values more sensitive, have corrections due to higher order terms in the expansion Want enough Q 2 range to constrain higher terms, but don’t want to be dominated by high Q 2 data; Global fits almost always give poor estimates of the radii More important for magnetic radius, where the precision on G M gets worse at low Q 2 values Very low Q 2 kinematics can have 1% cross sections yielding intercept (G M 2 ) known to 25%

Averaging of fits?  Limited precision on G M at low Q 2 means that more parameters are needed to reproduce low Q 2 data  Low N par fits may be less reliable  Statistics-weighted average of fits with different #/parameters  Emphasizes small N par  Expect fits with more parameters to be more reliable –Increase 2 by ~0.020 –Increase “statistical” uncertainty  No visible effect in 2 Weighted average: “By eye” average of high-N fits

Two-photon exchange corrections  Mainz analysis applied Q 2 =0 (point-proton) limit of “2 nd Born approximation” for Coulomb corrections  Applied 50% uncertainty in G E, G M fit (no uncertainty in radius) QED: straightforward to calculate  QED+QCD: depends on proton structure  Q 2 =0 Q 2 =0.1 Q 2 =0.3 Q 2 =1 Q 2 =0.03 JA, PRL 107, J.Bernauer, et al., PRL 107,

Impact of TPE Apply low-Q 2 TPE expansion, valid up to Q 2 =0.1 GeV 2 Small change, but still larger than total quoted uncertainty RADII: 1/2 goes from 0.879(8) to 0.876(8) fm [-0.3%] 1/2 goes from 0.777(17) to 0.803(17) fm [+3.0%] do not Note: these uncertainties do not include any contribution related to TPE: Change between default prescription and this suggests TPE uncertainty of approximately fm for r E, fm for r M Much (most?) of the effect associated with change in normalization factors of the different data subsets JA, PRL 107, ; J.Bernauer, et al., PRL 107, Borisyuk/Kobushkin, PRC 75, (2007)

Comparison of low Q 2 TPE calculations: Blunden, et al., hadronic calculation [PRC 72, (2005)] Borisyuk & Kobushkin: Low-Q 2 expansion, valid up to 0.1 GeV 2 [PRC 75, (2007)] B&K: Dispersion analysis (proton only) [PRC 78, (2008)] B&K: proton +  [arXiv: ] Typical uncertainties for radiative corrections are 1-1.5%; probably fair (or overestimate) after applying TPE calculations, at least for lower Q 2 Combining world’s data (or taking Mainz data set) yields enough data that it’s not sufficient to treat as uncorrelated or norm. uncertainties Full TPE calculations

Proton magnetic radius  Updated TPE yields  R M =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  R E value almost unchanged: 0.879(8)  0.876(8)  Higher-order Coulomb corrections?

Additional Coulomb Corrections? [JA, arXiv: ]  CC: 2 nd born approximation –Increases charge radius ~0.010 fm –[Rosenfelder PLB479(2000)381, Sick PLB576(2003)62]  + hard 2  corrections –Minimal impact (additional fm) –[Blunden and Sick, PRC72(2005)057601]  Low Q 2 : CC in 2 nd Born become small but non-zero  Very low energies, might expect large corrections (classical limit)  Could this have any impact on the radii extracted from data? 2 nd Born

Additional Coulomb Corrections? 2 nd Born EMA  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 Q 2  0.06 GeV 2 where scattering away from proton EMA 2 nd Born

Additional Coulomb Corrections?  Very little effect at high  ; no impact on charge radius  Large Q 2 dependence at low , especially at very low Q 2  Proton radius  slope  -600%/GeV 2  GeV 2 : CC slope  +100%/GeV 2  GeV 2 : slope  -8%/GeV 2  Higher  : up to ~15%/GeV 2  Couldimpact extraction of magnetic radius  Could impact extraction of magnetic radius –Need real calculation –Need to apply directly to real kinematics of the experiment EMA   = 0.02 EMA

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 G M at low Q 2 –Map out structure of G E at low Q 2 –Check TPE in both electron and muon scattering –Directly compare electron and muon scattering cross sections

Fin…

Impact of TPE RADII: 1/2 goes from 0.879(8) to 0.876(8) fm [-0.3%] 1/2 goes from 0.777(17) to 0.803(17) fm [+3.0%] do not Note: these uncertainties do not include any contribution related to TPE: Change between default prescription and this suggests TPE uncertainty of approximately fm for r E, fm for r M JA, PRL 107, ; J.Bernauer, et al., PRL 107, Borisyuk/Kobushkin, PRC 75, (2007) Apply low-Q 2 TPE expansion, valid up to Q 2 =0.1 GeV 2 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 ; Q 2 <0.5 (0.3, 0.4, 1.0) Fit (and normalization) uncertainties

 Quoted normalization uncertainties of ~2-4%; Fit yields %  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 Q 2 -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