Directions toward the resolution of the proton charge radius puzzle

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1 Directions toward the resolution of the proton charge radius puzzle Krzysztof Pachucki Institute of Theoretical Physics, University of Warsaw ECT workshop on the proton radius puzzle, November 1, 2012

2 Proton charge radius puzzle global fit to H and D spectrum: r p = (77) fm (CODATA 2010) e p scattering: r p = (79) (Bernauer, 2010) from muonic hydrogen: r p = (39) fm (PSI, 2010, 2012) If all these measurements and Lamb shift calculations are correct, this discrepancy does not find explanation within the known description of electroweak and strong interactions.

3 Possible sources of the proton radius discrepancy: theory mistake in e H calculations: all corrections calculated independently by at least two groups, uncertainty in the two-loop correction enters at 1 khz level for 1S state, but this discrepancy corresponds to 100 khz mistake in µ H: QED theory is quite simple, dominated by nonrelativistic vacuum polarization, everything checked and verified missing QED corrections significant underestimation of the proton polarizability and of the related subtraction term in dispersion relations (not known from e-p inelastic scattering, (G. Paz and R.J. Hill, J.A. McGovern talk, G.A. Miller talk) new interactions between the muon and the proton: a scalar with 1 MeV mass is not completely ruled out, but requires fine tuning (I. Yavin talk)

4 Possible sources of the proton radius discrepancy: experiment µ H measurement is not verified by independent experiment the determination of r p from e p scattering data requires extrapolation to q 2 = 0, subject of systematic uncertainties and model dependence, main issue discussed during the conference 2S ns, D measurements (mostly from one laboratory, LKB Paris), not confirmed by independent and equally accurate measurements. Highly excited states of H are affected by various systematics, possible hints from E. Hessels talk. As a result the Rydberg constant might be not as accurate as claimed

5 Ways to go determine Ry by another accurate measurement in 2S-4P in H (Garching) 2S-nS,D in H (J. Flowers, NPL) 1S-3S (Garching,... ) transitions between Rydberg states of heavy H-like ions (NIST, N.D. Guise talk) 1S-2S and 1S hfs in e µ (A. Antonini, PSI) determine r p from 2S 2P transition in H: (E. Hessels talk) µ p elastic scattering (Arrington et al.) compare charge radii from electronic and muonic spectra of other atomic systems µd data are coming, r D from very accurate H-D isotope shift (Garching) r He charge radius from 1S-2S (two-photon) transition in He +, or 2 3 S 2 3 P in He

6 New interactions If discrepancy in r p is to be explained by a new type of interaction between the proton (neutron) and leptons, than we have two options long range λ e, not consistent with precise measurements of the Lamb shift in H- and Li-like heavy ions at GSI short range 1fm (or shorter), can be seen in µp scatt. Comparison of nuclear charge radii for H,D, 3 He and 4 He will give hints on the range of new interactions If it is local, than discrepancy for all these elements can be parametrized by δe = (Z δr 2 p + (A Z ) δr 2 n ) 2 δ l0 3 n 3 Z 3 α 4 µ 3 Determination of r N from muonic atoms spectra requires an accurate calculation of the nuclear polarizability correction (S. Schlesser talk), not necessarily easy task

7 Proposal to determine α charge radius from the atomic spectroscopy E(2 3 S 2 3 P, 4 He) centroid = (2.1) khz, Florence, 2004 finite size effect: E fs = khz since E fs is proportional to r 2 r r = 1 2 δe fs E fs = electron scattering gives r He = 1.681(4) fm, what corresponds to about relative accuracy can theoretical predictions be accurate enough 10 khz?

8 Theory of hydrogen energy levels energy according to Dirac equation ( (Z α) f (n, j) = [n+ (j+1/2) 2 (Z α) 2 j 1/2] 2 ) 1/2 total energy E = M + m + µ[f (n, j) 1] µ2 [f (n, j) 1]2 2 M + (Z α)4 µ 3 [ ] 1 2 n 3 M j + 1/2 1 (1 δ l0 ) + E L l + 1/2 E L (α) = E (5) + E (6) + E (7) + E (8) +... where E (n) α n E (n)

9 Contributions to the Lamb shift one-loop electron self-energy and vacuum polarization two-loops three-loops pure recoil correction radiative recoil correction finite nuclear size, and polarizability

10 One-loop contribution δe = α π (Z α)4 m F (Z α) analytic expansion F(Z α) = A 40 + A 41 ln(z α) 2 + (Z α) A 50 +(Z α) 2 [A 62 ln 2 (Z α) 2 + A 61 ln(z α) 2 + A 60 + O(Z α)] or direct numerical evaluation using the exact Coulomb-Dirac propagator

11 Numerical evaluation of the one-loop self-energy [U. Jentschura, P.J. Mohr, and G. Soff, Phys. Rev. Lett. 82, 53 (1999)] δe = α π (Z α)4 m F (Z α) F(Z α) = A 40 + A 41 ln(z α) 2 + (Z α) A 50 +(Z α) 2 [A 62 ln 2 (Z α) 2 + A 61 ln(z α) 2 + G 60 ]

12 Two-loop electron self-energy correction Electron propagators include external Coulomb field, external legs are bound state wave functions. The expansion of the energy shift in powers of Z α ( α ) 2 δ (2) E = m F(Z α) π { F(Z α) = B 40 + (Z α) B 50 + (Z α) 2 [ln(z α) 2 ] 3 B 63 } +[ln(z α) 2 ] 2 B 62 + ln(z α) 2 B 61 + G 60 (Z α)

13 Direct numerical calculation versus analytical expansion G h.o. (Z) G 60 (1) 86(15) (Yerokhin, 2009), uncertainty δe(1s) = ±1.5 khz B 60 = 61.6(9.2) (K.P., U.J., 2003) uncertainty due to the unknown high energy contribution from the class of about 80 diagrams discrepancy in the proton charge radius δe(1s) 100 khz

14 Pure recoil corrections finite nuclear mass effects, beyond the Dirac equation leading O(α 5 ) terms are known for an arbitrary mass ratio δe (5) = µ3 (Z α) 5 { 1 m M π n 3 3 δ l0 ln(z α) ln k 0(n, l) 1 9 δ l0 7 3 [ 2 M 2 m 2 δ l0 M 2 ln m µ m2 ln M ]} µ where a n = 2 [ln 2n ( ) ] 1 δ l0 δ l0 + n 2 n l (l + 1) (2 l + 1) ( ) δe (6) = m2 M 4 ln 2 7 2

15 Experimental results for hydrogen and r p Some accurate results ν exp. (2P 1/2 2S 1/2 ) = (9.0) khz, [Lundeen, Pipkin, 1994] ν exp (1S 1/2 2S 1/2 ) = (10) khz, [MPQ, 2011] ν exp (2S 1/2 8D 5/2 ) = (5.9) khz, [Paris, 2001] global fit to the hydrogen data r p = (77) fm

16 Helium energy levels

17 Expansion of energy in α and µ/m Expansion of energy as a function of the fine structure constant α and the electron-nucleus mass ratio: E(α, η) = m α 2 [ E (2,0) + η E (2,1) + η 2 E (2,2)] +m α 4 [ E (4,0) + η E (4,1)] +m α 5 [ E (5,0) + η E (5,1)] +m α 6 [ E (6,0) + η E (6,1)] +m α 7 E (7,0) with η = µ/m, E (6,1) and E (7,0) are not yet known

18 Nonrelativistic wave function φ( r 1, r 2 ) = N i=1 c [ i exp( αi r 1 β i r 2 γ i r 12 ) + (1 2) ] parameters α i, β i, and γ i are chosen quasi randomly α i [A 1, A 2 ] β i [B 1, B 2 ] γ i [C 1, C 2 ] master integral 1 16 π 2 d 3 r 1 d 3 r 2 e αr 1 βr 2 γr 12 r 1 r 2 r 12 = 1 (α + β)(β + γ)(γ + α) E (2) (1 1 S) = more accurate results by Drake, Korobov, Schwartz, Nakatsuji

19 QED theory of He ionization energies ν(2 1 S)[MHz] ν(2 1 P)[MHz] ν(2 3 S)[MHz] ν(2 3 P)[MHz] µ α µ 2 /M α µ 3 /M 2 α E fs m α m 2 /M α m α m 2 /M α m α m α (1.90) 0.16(16) 5.31(1.00) 1.93(40) E the (1.90) (16) (1.00) (40) E exp (15) (3.0) (0.06) (6)

20 3 He - 4 He isotope shift of centroid energies in khz Contribution 2 3 S 2 3 P 2 3 S 2 1 S m r α m r α 2 (m r /M) m r α 2 (m r /M) m r α 2 (m r /M) m r α m r α 4 (m r /M) m r α 4 (m r /M) m α 5 (m/m) m α 6 (m/m) 15.5 (3.9) 2.75 (69) Nuclear polarizability 1.1 (1) 0.20 (2) HFS mixing Total (3.9) (69) Results obained in collaboration with V. Yerokhin

21 3 He - 4 He charge radii difference δr 2 (Florence 2012) = (3) fm 2, δr 2 (Shiner 1995) = (4) fm 2, δr 2 (Amsterdam 2011) = (11) fm 2 4 σ discrepancy

22 Higher order corrections to 2 3 S 2 3 P energy nuclear polarizability E pol (2 3 S 2 3 P) = 1.6(2) khz nuclear recoil corrections µ2 M α6, known only for hydrogen, calculations for He 3 X states is straightforward higher order QED µ α 7, calculations are challenging, similar to that in H + 2, Bethe logs