Eric Hessels and Marko Horbatsch York University Toronto, Canada

1. Determining the Proton Charge Radius from Electron-Proton Scattering and from Hydrogen Spectroscopy Eric Hessels and Marko Horbatsch York University Toronto, Canada

2. Determining the Proton Charge Radius from Electron-Proton Scattering and from Hydrogen Spectroscopy Three things I would like to talk about: A tabulation of the bound-state energies of hydrogen Evaluating the proton radius from ep scattering data Progress on our H 2S-2P Lamb-shift measurement (If I have time left)

3. A tabulation of the bound-state energies of hydrogen We need a correct description of the hydrogen energy levels to analyze our Lamb shift measurement (in progress) (not included in other tabulations) Our tabulation includes hyperfine structure with the anomalous moment corrections and mass corrections and corrections due to off-diagonal mixing of j states This work is the first appearance in the literature of the correct general formulas for these corrections

4. The tabulation of low-lying bound states includes all f states (hyperfine structure) (uses CREMA, 1S-2S, 1S,2S hfs expt. Input) has uncertainties of <1 kHz shows sensitivity to proton radius (CODATA vs muonic hydrogen) All states (not just S) depend on radius due to the necessary adjustment of Ry See poster for more details

5. Evaluating the proton radius from ep scattering data Determination of the rms charge radius RE from electron-proton elastic scattering data measured cross section corrected for TPE RE is directly obtainable from the derivative of the σred at Q2=0 since the magnetic moment is known precisely The most precise data for determining RE is the 2010 MAMI data 1422 measured cross sections with typical relative uncertainty of 0.35% 180 MeV ≤ E ≤ 855 MeV, 16° ≤ θ ≤ 135.5°, 0.0038 GeV2/c2 ≤ Q2 ≤ 1 GeV2/c2 34 data groups, 31 normalization constants

6. Evaluating the proton radius from ep scattering data Let me start with the punchline: We can get RE of anywhere from 0.84 fm to 0.89 fm (depending on our fit function) We fit all 1422 cross sections of the MAMI data We properly float the 31 normalization constants We only accept fits that give a reduced χ2 < 1.14 We properly deal with two-photon exchange Our fits give form factors that are well behaved at large Q2 Our fits do not give unphysical charge distributions Our fits do not allow for unphysical hooks in the form factors for Q2 < 0.004 (even Ingo should be happy) We conclude that any RE between 0.84 and 0.89 fm is consistent with the scattering data

7. Evaluating the proton radius from ep scattering data Pre-MAMI data have larger uncertainties, but I will look at these first to motivate our fit Pre-MAMI GE data

8. Evaluating the proton radius from ep scattering data Pre-MAMI GE data

9. Evaluating the proton radius from ep scattering data

10. Evaluating the proton radius from ep scattering data

11. Evaluating the proton radius from ep scattering data The square of the form factors is directly related to the cross sections We use the square of the form factors in our analysis

12. Evaluating the proton radius from ep scattering data GE2 0 as Q2 infinity, so linear trend cannot continue The effect of this nonlinearity back to Q2=0 (z=0) is the crux of the problem for determining the proton charge radius By how much should we let the tail wag the dog?

13. Evaluating the proton radius from ep scattering data GE2 0 as Q2 infinity, so linear trend cannot continue The effect of this nonlinearity back to Q2=0 (z=0) is the crux of the problem for determining the proton charge radius By how much should we let the tail wag the dog?

14. Evaluating the proton radius from ep scattering data GE2 0 as Q2 infinity, so linear trend cannot continue The effect of this nonlinearity back to Q2=0 (z=0) is the crux of the problem for determining the proton charge radius By how much should we let the tail wag the dog? dipole with RE = 0.84 fm These are not fits or determinations of the RE – just our inspiration for using fit functions for GE2(not GE) that are based on linear in z or dipole forms

15. Evaluating the proton radius from ep scattering data Need to fit to a model to GE2, GM2 to extrapolate the data to Q2=0 Fits must also determine normalization constants – leads to flexibility in the extrapolation Other best fits give a reduced χ2 of about 1.14, so we reject any fits that give χ2 > 1.14 As another warm-up exercise, fit the lowest Q2 MAMI data to see if it is approximately fit by two single-paramerter models (Later, include all Q2 to ensure that the fit used for the extrapolation is consistent with all of the MAMI data) One-parameter models for GE2, GM2: (1) Linear in z: (2) Dipole model:

16. Evaluating the proton radius from ep scattering data (1) Linear in z: Fit of Q2<0.1 GeV2 data 53% of measured MAMI cross sections Reduced χ2 of 1.11 Extrapolated slope gives: RE=0.888(1) fm This is still to show that a linear in z fit works reasonably well – our full fit gives a smaller χ2 contribution for this part of the data set

17. Evaluating the proton radius from ep scattering data (2) Dipole model: Fit of Q2<0.1 GeV2 data 53% of measured MAMI cross sections Reduced χ2 of 1.11 This is still to show that a dipole fit works reasonably well – our full fit gives a smaller χ2 contribution for this part of the data set Extrapolated slope gives: RE=0.842(2) fm

18. Evaluating the proton radius from ep scattering data Here is the deduced GE2 from the two fits to the low-Q2 MAMI data (along with the pre-MAMI data, for comparison) Different slopes at Q2=0 gives different radii RE=0.842 fm from one-parameter fit dipole model linear z model RE=0.888 fm from one-parameter fit

19. Evaluating the proton radius from ep scattering data The fits are extended to higher Q2 by replacing the single parameter for each form factor with a cubic spline The constants bE and bM are replaced by cubic splines (continuous function and derivative) for z > 0.1 constant bE, bM Only fits with a reduced χ2 < 1.14 are included – and all fits give 0.84 fm < RE < 0.85 fm 10-knot spline (11-parameter GE) of all of the MAMI data still gives RE~0.84 fm, χ2 < 1.14

20. Evaluating the proton radius from ep scattering data denominator forces G  0 as Q2∞ P = 4 to 14 works, give similar results The constants cE and cM are replaced by cubic splines for z > 0.1 linear fit Both of these fits include all of the data, have χ2 < 1.14 We conclude that MAMI data is consistent with any RE within the range of 0.84 to 0.89 fm “ … ”

21. Evaluating the proton radius from ep scattering data The constants cE and cM are replaced by a uniform cubic splines for z > 0.1 linear fit • We can produce families of curves that give good fits and give all intermediate values of RE • Can do this by any one of the following: • varying tc away from 4mp2 in definition of z • varying the expansion point z0 for definition of z • varying an added fixed quadratic terms in numerator

22. Evaluating the proton radius from ep scattering data Sensitivity to two-photon exchange corrections We calculate TPE from Borisyuk & Kobushkin PRC 86 055204; arXiv:1209. 2746 (2012) dipole fits To check sensitivity to TPE, we reanalyzed using low-Q2 approximation (Borisyuk & Kobushkin PRC 75 038202 (2007)) – dashed lines show negligible change in RE Even if Feshbach correction is used instead of correct TPE, RE changes by only -0.003 fm (solid curves) – not so sensitive to TPE

23. Evaluating the proton radius from ep scattering data Both give good fits to the MAMI data Difference of behavior of derivatives at low Q2 leads to difference in radius Dipole RE=.84 fm Difference of behavior of derivatives at low Q2 also leads to dependence of fit on Q2max when fitting to one of these functions while using pseudo-data generated from the other dGE dQ2 Bernauer et al 10th-order polynomial Q2

24. Evaluating the proton radius from ep scattering data I haven’t mentioned it, but, of course, our fits also give values for the RM. Dipole fit givesRE2+RM2=1.349(4) Linear fit gives RE2+RM2=1.553(4) Both compare reasonably well with 1.39(10) from hydrogen and muonic hydrogen hfs We conclude that any RE between 0.84 and 0.89 fm is possible from scattering data

25. A quick update on our Lamb-shift measurement We are measuring the atomic hydrogen 2S1/2 to 2P1/2 interval mF=0 mF=1 mF=-1 Using microwave spectroscopy F=2 2P3/2 F=1 F=1 2S1/2 F=0 F=1 2P1/2 F=0 1S1/2 F=1 F=0

26. Lamb shift Stable ions source with 10 mA of 50-keV protons

27. Lamb shift 10 mA 50-keV protons protons charge exchange with H2 gas

28. Lamb shift We empty the 2S1/2 F=1 states using 2 rf cavities that drive them down to short-lived 2P1/2 states Charge exchange 2% H(2S) With F=1 states empty, can make a measurment of the isolated transition from the 2S1/2 F=0 transition 10 mA 50-keV protons mF=0 mF=1 mF=-1 F=2 2P3/2 F=1 F=1 2S1/2 F=0 F=1 2P1/2 F=0 1S1/2 F=1 F=0

29. Lamb shift low-Q microwave cavities to create standing waves which drive the main SOF fields 2S(F=1) quench Critical parameter for the SOF measurement is the relative phase of the microwaves in the two cavities 10 mA 50-keV to 100-keV protons Any unanticipated error in relative phase is reversed by rotating entire microwave system by 180O – all in situ

30. Lamb shift We detect the 2S atoms that remain by mixing 2S with 2P with a DC electric field and resulting Ly-a is detected by ionizing gas – almost 4p ~50% Ly-a detection efficiency 2S(F=1) quench 10 mA 50-keV protons

31. Lamb shift We are using a new Frequency-offset SOF technique (FOSOF) (AC Vutha and EA Hessels Phys. Rev. A052504 (2015)) |df | =~100 Hz rf in two SOF regions oscillate between being in phase and out of phase at df offset frequency f f+df

32. Lamb shift We are using a new Frequency-offset SOF technique (FOSOF) (AC Vutha and EA Hessels Phys. Rev. A October 2015) Diffference in phase between mixer signal and SOF signal is zero if rf frequency (f) is in resonance with the atomic transition. |df | =~100 Hz rf in two SOF regions oscillate between being in phase and out of phase at df beat frequency f f+df If there is a phase difference, it predicts the difference between the applied frequency f and the atomic resonance frequency. SOF signal df =100 Hz PRELIMINARY

33. Lamb shift We will spend the next year studying systematic effects – we can see that several are affecting our measurement Our aim is an accuracy of 2 kHz which would provide a new measurements of the proton radius with uncertainty indicated See poster for more details Hydrogen

34. Summary A tabulation of the bound-state energies of hydrogen Evaluating the proton radius from ep scattering data We conclude that any RE between 0.84 and 0.89 fm is possible from scattering data We report progress on our H 2S-2P Lamb-shift measurement We hope that we might have a measurement by the end of 2017

35. Summary A tabulation of the bound-state energies of hydrogen Evaluating the proton radius from ep scattering data We conclude that any RE between 0.84 and 0.89 fm is possible from scattering data We report progress on our H 2S-2P Lamb-shift measurement We hope that we might have a measurement by the end of 2017