1. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 1 Eric Hessels and Marko Horbatsch York University Toronto, Canada Determining the Proton Charge Radius from Electron-Proton Scattering and from Hydrogen Spectroscopy

2. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 2 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) Determining the Proton Charge Radius from Electron-Proton Scattering and from Hydrogen Spectroscopy

3. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 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) 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 Our tabulation includes hyperfine structure (not included in other tabulations)

4. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 4 The tabulation of low-lying bound states includes all f states (hyperfine structure) 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 (uses CREMA, 1S-2S, 1S,2S hfs expt. Input)

5. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 5 180 MeV ≤ E ≤ 855 MeV, 16° ≤ θ ≤ 135.5°, 0.0038 GeV 2 /c 2 ≤ Q 2 ≤ 1 GeV 2 /c 2 Evaluating the proton radius from ep scattering data The most precise data for determining R E is the 2010 MAMI data 1422 measured cross sections with typical relative uncertainty of 0.35% 34 data groups, 31 normalization constants Determination of the rms charge radius R E from electron-proton elastic scattering data R E is directly obtainable from the derivative of the σ red at Q 2 =0 since the magnetic moment is known precisely measured cross section corrected for TPE

6. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 6 Evaluating the proton radius from ep scattering data Let me start with the punchline: We can get R E 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 χ 2 < 1.14 We properly deal with two-photon exchange Our fits give form factors that are well behaved at large Q 2 Our fits do not give unphysical charge distributions Our fits do not allow for unphysical hooks in the form factors for Q 2 < 0.004 (even Ingo should be happy) We conclude that any R E between 0.84 and 0.89 fm is consistent with the scattering data

7. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 7 Pre-MAMI G E data 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

8. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 8 Evaluating the proton radius from ep scattering data Pre-MAMI G E data

9. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 9 Evaluating the proton radius from ep scattering data

10. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 10 Evaluating the proton radius from ep scattering data

11. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 11 The square of the form factors is directly related to the cross sections Evaluating the proton radius from ep scattering data We use the square of the form factors in our analysis

12. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 12 G E 2  0 as Q 2  infinity, so linear trend cannot continue The effect of this nonlinearity back to Q 2 =0 (z=0) is the crux of the problem for determining the proton charge radius Evaluating the proton radius from ep scattering data By how much should we let the tail wag the dog?

13. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 13 G E 2  0 as Q 2  infinity, so linear trend cannot continue The effect of this nonlinearity back to Q 2 =0 (z=0) is the crux of the problem for determining the proton charge radius Evaluating the proton radius from ep scattering data By how much should we let the tail wag the dog?

14. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 14 Evaluating the proton radius from ep scattering data dipole with R E = 0.84 fm G E 2  0 as Q 2  infinity, so linear trend cannot continue The effect of this nonlinearity back to Q 2 =0 (z=0) is the crux of the problem for determining the proton charge radius These are not fits or determinations of the R E – just our inspiration for using fit functions for G E 2 (not G E ) that are based on linear in z or dipole forms By how much should we let the tail wag the dog?

15. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 15 As another warm-up exercise, fit the lowest Q 2 MAMI data to see if it is approximately fit by two single-paramerter models (Later, include all Q 2 to ensure that the fit used for the extrapolation is consistent with all of the MAMI data) One-parameter models for G E 2, G M 2 : (1) Linear in z: (2) Dipole model: Need to fit to a model to G E 2, G M 2 to extrapolate the data to Q 2 =0 Fits must also determine normalization constants – leads to flexibility in the extrapolation Other best fits give a reduced χ 2 of about 1.14, so we reject any fits that give χ 2 > 1.14 Evaluating the proton radius from ep scattering data

16. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 16 ( 1) Linear in z: Fit of Q 2 <0.1 GeV 2 data 53% of measured MAMI cross sections Reduced χ 2 of 1.11 Extrapolated slope gives: R E =0.888(1) fm Evaluating the proton radius from ep scattering data This is still to show that a linear in z fit works reasonably well – our full fit gives a smaller χ 2 contribution for this part of the data set

17. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 17 ( 2) Dipole model: Fit of Q 2 <0.1 GeV 2 data 53% of measured MAMI cross sections Reduced χ 2 of 1.11 Extrapolated slope gives: R E =0.842(2) fm Evaluating the proton radius from ep scattering data This is still to show that a dipole fit works reasonably well – our full fit gives a smaller χ 2 contribution for this part of the data set

18. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 18 dipole model linear z model Here is the deduced G E 2 from the two fits to the low-Q 2 MAMI data (along with the pre-MAMI data, for comparison) Different slopes at Q 2 =0 gives different radii R E =0.842 fm from one-parameter fit R E =0.888 fm from one- parameter fit Evaluating the proton radius from ep scattering data

19. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 19 The fits are extended to higher Q 2 by replacing the single parameter for each form factor with a cubic spline The constants b E and b M are replaced by cubic splines (continuous function and derivative) for z > 0.1 Only fits with a reduced χ 2 < 1.14 are included – and all fits give 0.84 fm < R E < 0.85 fm 10-knot spline (11-parameter G E ) of all of the MAMI data still gives R E ~0.84 fm, χ 2 < 1.14 Evaluating the proton radius from ep scattering data constant b E, b M

20. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 20 linear fit Evaluating the proton radius from ep scattering data The constants c E and c M are replaced by cubic splines for z > 0.1 denominator forces G  0 as Q 2  ∞ P = 4 to 14 works, give similar results Both of these fits include all of the data, have χ 2 < 1.14 We conclude that MAMI data is consistent with any R E within the range of 0.84 to 0.89 fm “ … ”

21. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 21 linear fit Evaluating the proton radius from ep scattering data The constants c E and c M are replaced by a uniform cubic splines for z > 0.1 We can produce families of curves that give good fits and give all intermediate values of R E Can do this by any one of the following: (1)varying t c away from 4m  2 in definition of z (2)varying the expansion point z 0 for definition of z (3)varying an added fixed quadratic terms in numerator

22. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 22 Evaluating the proton radius from ep scattering data Sensitivity to two-photon exchange corrections dipole fits We calculate TPE from Borisyuk & Kobushkin PRC 86 055204; arXiv:1209. 2746 (2012) To check sensitivity to TPE, we reanalyzed using low-Q 2 approximation (Borisyuk & Kobushkin PRC 75 038202 (2007)) – dashed lines show negligible change in R E Even if Feshbach correction is used instead of correct TPE, R E changes by only -0.003 fm (solid curves) – not so sensitive to TPE

23. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 23 Q2Q2 dG E dQ 2 Dipole R E =.84 fm Bernauer et al 10 th -order polynomial Evaluating the proton radius from ep scattering data Both give good fits to the MAMI data Difference of behavior of derivatives at low Q 2 leads to difference in radius Difference of behavior of derivatives at low Q 2 also leads to dependence of fit on Q 2 max when fitting to one of these functions while using pseudo-data generated from the other

24. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 24 Evaluating the proton radius from ep scattering data I haven’t mentioned it, but, of course, our fits also give values for the R M. Dipole fit gives R E 2 +R M 2 =1.349(4) Linear fit gives R E 2 +R M 2 =1.553(4) Both compare reasonably well with 1.39(10) from hydrogen and muonic hydrogen hfs We conclude that any R E between 0.84 and 0.89 fm is possible from scattering data

25. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 25 A quick update on our Lamb-shift measurement We are measuring the atomic hydrogen 2S 1/2 to 2P 1/2 interval 2P 3/2 m F  -1 mF0mF0mF1mF1 F2F2 F1F1 2P 1/2 2S 1/2 F1F1 F1F1 F0F0 F0F0 1S 1/2 F1F1 F0F0 Using microwave spectroscopy

26. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 26 Stable ions source with 10  A of 50-keV protons Lamb shift

27. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 27 protons charge exchange with H 2 gas 10  A 50-keV protons Lamb shift

28. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 28 Charge exchange 2% H(2S) We empty the 2S 1/2 F=1 states using 2 rf cavities that drive them down to short-lived 2P 1/2 states 2P 3/2 m F  -1 mF0mF0mF1mF1 F2F2 F1F1 2P 1/2 2S 1/2 F1F1 F1F1 F0F0 F0F0 1S 1/2 F1F1 F0F0 With F=1 states empty, can make a measurment of the isolated transition from the 2S 1/2 F=0 transition 10  A 50-keV protons Lamb shift

29. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 29 10  A 50-keV to 100-keV protons 2S(F=1) quench low-Q microwave cavities to create standing waves which drive the main SOF fields Critical parameter for the SOF measurement is the relative phase of the microwaves in the two cavities Any unanticipated error in relative phase is reversed by rotating entire microwave system by 180 O – all in situ Lamb shift

30. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 30 2S(F=1) quench We detect the 2S atoms that remain by mixing 2S with 2P with a DC electric field and resulting Ly-  is detected by ionizing gas – almost 4  ~50% Ly-  detection efficiency 10  A 50-keV protons Lamb shift

31. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 31 f f+  f  f | =~100 Hz rf in two SOF regions oscillate between being in phase and out of phase at  f offset frequency We are using a new Frequency-offset SOF technique (FOSOF) (AC Vutha and EA Hessels Phys. Rev. A052504 (2015)) Lamb shift

32. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 32 f f+  f  f | =~100 Hz rf in two SOF regions oscillate between being in phase and out of phase at  f beat frequency SOF signal  f =100 Hz Diffference in phase between mixer signal and SOF signal is zero if rf frequency (f) is in resonance with the atomic transition. If there is a phase difference, it predicts the difference between the applied frequency f and the atomic resonance frequency. PRELIMINARY We are using a new Frequency-offset SOF technique (FOSOF) (AC Vutha and EA Hessels Phys. Rev. A October 2015) Lamb shift

33. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 33 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 Hydrogen Lamb shift See poster for more details

34. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 34 Summary A tabulation of the bound-state energies of hydrogen Evaluating the proton radius from ep scattering data We conclude that any R E 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. ECT* Proton Radius Workshop Trento 2016 Eric Hessels & Marko Horbatsch York University Toronto Canada 35 Summary A tabulation of the bound-state energies of hydrogen Evaluating the proton radius from ep scattering data We conclude that any R E 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