Muonic Hydrogen: all orders calculations Paul Indelicato ECT* Workshop on the Proton Radius Puzzle, Trento, October 29-November 2, 2012 Theory • Peter Mohr, National Institute of Standards and Technology, Gaithersburg, Maryland • Paul Indelicato, Laboratoire Kastler Brossel, École Normale Supérieure, CNRS, Université Pierre et Marie Curie, Paris, France and Helmholtz Alliance HA216/ EMMI (Extreme Mater Institute) Trento, October 2012 2 Present status • Published result: – 49 881.88(76)GHz ➙rp= 0.84184(67) fm – CODATA 2006: 0.8768(69) fm (mostly from hydrogen and electron-proton scattering) – Difference: 5 σ • CODATA 2010 (on line since mid-june 2011) – 0.8775 (51) fm – Difference: 6.9 σ – Reasons (P.J. Mohr): • improved hydrogen calculation • new measurement of 1s-3s at LKB (F. Biraben group) • Mainz electron-proton scattering result • The mystery deepens... – Difference on the muonic hydrogen Lamb shift from the two radii: 0.3212 (51 µH) (465 CODATA) meV Trento, October 2012 3 Hydrogen QED corrections QED at order α and α2 Self Energy Vacuum Polarization H-like “One Photon” order α H-like “Two Photon” order α2 = + + +! Z α expansion; replace exact Coulomb propagator by expansion in number of interactions with the nucleus Trento, October 2012 5 Convergence of the Zα expansion Trento, October 2012 6 Two-loop self-energy (1s) V. A. Yerokhin, P. Indelicato, and V. M. Shabaev, Phys. rev. A 71, 040101(R) (2005). Trento, October 2012 7 Two-loop self-energy (1s) V. A. Yerokhin, P. Indelicato, and V. M. Shabaev, Phys. rev. A 71, 040101(R) (2005). V. A. Yerokhin, Physical Review A 80, 040501 (2009) Trento, October 2012 8 Evolution at low-Z All order numerical calculations V. A. Yerokhin, Phys. Rev. A 80, 040501 (2009) V. A. Yerokhin, P. Indelicato, and V. M. Shabaev, Phys. Rev. A 71, 040101(R) (2005). Analytic calculations K. Pachucki and U. D. Jentschura, Phys. Rev. Lett. 91, 113005 (4) (2003). U. D. Jentschura, A. Czarnecki, and K. Pachucki, Physical Review A 72, 062102 (2005). Trento, October 2012 9 Zoom sur l’hydrogène Proton size effect 1 Trento, October 2012 10 L’hydrogène 2 466 061 413 187 035 ±10 Hz 50 Hz 2 466 061 413 187 103±46Hz 2011 2000 1 Trento, October 2012 11 Why re-measure the proton charge radius? Trento, October 2012 12 The proton charge radius Using muonic hydrogen Trento, October 2012 14 P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. 21 Table 11. Contribution to the proton-size and Zemach’s radius Table 13. Contribution to the proton-size dependent part of the dependent part of the 2p1/2 level hyperfine structure energy 2p1/2–2p3/2 off-diagonal matrix element. Values are in meV. shift. Values are in meV. Contribution Value Dep. Contribution Value Dep. 6 Zemach’s correction -21.2237 10− RZ 6 × 6 Zemach’s correction -12.1857 10− RZ higher order struct. corrections 14.7080 10− R × 6 × 6 higher order structure 7.8967 10− R to HFS -7.2509 10− RRZ × × 6 2 corrections to HFS 14.4754 10− R 6 -3.7055 10− RRZ × 6 2 2.9124 10− RZ × 6 2 × 6 2 8.5130 10− R -10.9021 10− R RZ × 6 2 × 6 2 2 1.6684 10− RZ 3.8112 10− R R × 6 2 × Z -6.4939 10− R RZ Proton struct. correction × 6 2 2 10 2.1812 10− R RZ to all-order vac. pol. 60.9322 10− RRZ × 8 Effect of charge distribution -1.4500 10− R × 10 2 × -26.8178 10− R on ∆EHFS × 10 2 1γ,VP 34.5397 10− RZ 8 2 × 0.0859 10− R Value of the contributions for R = 0.850 and R = 1.047 × Z Proton structure correction Proton struct. correction to HFS -0.00000775Muonic hydrogen 8 to all order Vac. pol. -8.92903 10− R Proton struct. correction 0.00000001 × 8 -8.05897 10− RZ to all-order vac. pol. × 8 13.7173 10− RRZ Total correctionThe level scheme -0.00000774 is quite different (vacuum polarization × 8 2 17.8307 10− R × 8 2 dominates!) -12.1087 10− RZ × 8 2 -7.01228 10− R R Z 2p 3 5 × 2 P3 2 Value of the contributions for R = 0.850 and RZ = 1.047 2 Zemach’s correction -0.000045492 2p 1 2 1 Proton structure correction -0.000000095 2 P1 2 to all order-VP Effect of charge distribution -0.000000012 HFS on ∆E1γ,VP Total correction -0.000045599 Table 12. Contribution to the proton-size dependent part of the ELS 2p3/2 level hyperfine structure energy shift. Values are in meV. Contribution Value Dep. 8 Zemach’s correction -2.8841 10− RZ × 8 higher order struct. corrections 6.9207 10− R to HFS × 8 7.6327 10− RRZ × 8 2 -19.5901 10− R 2 3S × 8 2 2s 1 2 -5.4279 10− RZ EHFS × 8 2 2s 15.4073 10− R RZ 1 × 8 2 2 2 S12 -5.5858 10− R R × Z Proton struct. correction Fig. 19. The energy structure of the n = 2 level in muonic hy- 10 to all order vac. pol. 12.6258 10− RRZ drogen. The drawing is to scale. Details of the 2p level structure × 10 2 148.917 10− R are presented in Fig. 20. × 10 2 -8.29507 10− R × Z Value of the contributions for R = 0.850 and RZ = 1.047 Proton struct. correction to HFS -0.00000003 References Proton struct. correction 0.00000001 to all-order vac. pol. Trento, October 2012 15 Total correction -0.00000002 1. L.N. Hand, D.G. Miller, R. Wilson, Rev. Mod. Phys. 35(2), 335 (1963) 2. G.G. Simon, C. Schmitt, F. Borkowski, V.H.Walther, Nuclear Physics A 333(3), 381 (1980) 3. F. Nez, M.D. Plimmer, S. Bourzeix, L. Julien, F. Biraben, R. Felder, O. Acef, J.J. Zondy, P. Laurent, A. Clairon et al., Phys. Rev. Lett. 69(16), 2326 (1992) 4. M. Weitz, A. Huber, F. Schmidt-Kaler, D. Leibfried, W. Vassen, C. Zimmermann, K. Pachucki, T.W. Hansch,¨ L. Julien, F. Biraben, Phys. Rev. A 52(4), 2664 (1995) Muonic hydrogen The level scheme is quite different (vacuum polarization dominates!) Trento, October 2012 15 Muonic hydrogen The level scheme is quite different (vacuum polarization dominates!) Trento, October 2012 15 Muonic hydrogen The level scheme is quite different (vacuum polarization dominates!) 2 keV Trento, October 2012 15 New results • 2010 CODATA value uses improved theory for hydrogen and Mainz electron- proton scattering is now at 6.9σ mostly by a reduction of σ: – 0.8775 (59) fm 2010 – 0.8768 (69) fm 2006 • Experiment: a slightly reduced error bar for the first transition, an accurate value of a second transition which allows to – Get a measurement of the magnetic moment distribution mean radius – An improved charge radius Trento, October 2012 16 Possible origin for the discrepancy • QED? many checks performed • Proton polarization (strong discussion going on) • Electron-proton elastic scattering data analysis • Under-estimated systematic errors in some hydrogen measurements – possible, but many different kind of experiments (microwave, 1s-3s, 2s-ns and 2s-nd) • New physics – Constraints: • g-2 of the muon (3σ ), • g-2 of the electron (Harvard)+fine structure constant from atomic recoil (LKB) • Hydrogen • ... Trento, October 2012 17 Main QED contributions Källèn and Sabry All order VP Wichmann & Kroll Relativistic recoil Iterated Uehling Iterated Uehling on K&S Trento, October 2012 Self-energy 18 Main QED contributions Källèn and Sabry All order VP Wichmann & Kroll Relativistic recoil Iterated Uehling Iterated Uehling on K&S Trento, October 2012 Self-energy 18 Main QED contributions Källèn and Sabry All order VP Wichmann & Kroll Relativistic recoil Iterated Uehling Iterated Uehling on K&S Trento, October 2012 Self-energy 18 Main QED contributions Källèn and Sabry All order VP Wichmann & Kroll Relativistic recoil Iterated Uehling Iterated Uehling on K&S Trento, October 2012 Self-energy 18 Main QED contributions Källèn and Sabry All order VP Wichmann & Kroll Relativistic recoil Iterated Uehling Iterated Uehling on K&S Trento, October 2012 Self-energy 18 µP theory How well is it calculated? How dependent on nuclear model? QED QED QED QED Trento, October 2012 19 Proton size effect and VP potential Finite proton size Coulomb law Rp Trento, October 2012 20 Example of unsolved problems: Landau pole Dyson: the expansion in α of QED has zero convergence radius... e➝-e, no more bound states! Divergence of Perturbation Theory in Quantum Electrodynamics, F.J. Dyson. Physical Review 85, 631–632 (1952). Trento, October 2012 21 Example of unsolved problems: Landau pole Trento, October 2012 21 Formulas P. Indelicato1 1 FormulasFormulas Laboratoire Kastler Brossel, Ecole Normale Sup´erieure, Example of unsolved problems:CNRS, Universit´ePierre Landau pole1 et Marie Curie-Paris 6, P. IndelicatoP.1 Indelicato 1 Case 74, 4 place Jussieu, F-75005 Paris, France 1 Laboratoire Kastler Brossel, Ecole Normale Sup´erieure, Laboratoire Kastler Brossel, Ecole Normale Sup´erieure, CNRS, Universit´ePierre(Dated: et August, Marie Curie-Paris 23rd, 2010) 6, CNRS, Universit´ePierre et Marie Curie-Paris 6, Case 74, 4 place Jussieu, F-75005 Paris, France How toCase analyse 74, the 4 place DCS Jussieu, data, using F-75005 Perdro’s Paris, simulation France (Dated: August, 23rd, 2010) (Dated: August, 23rd, 2010) PACSHow numbers: to analyse the DCS data, using Perdro’s simulation How to analyse the DCS data, using Perdro’s simulation PACS numbers: + PACS numbers: + +... Vacuum Polarization re- summation V k2 1 k2 k2 1 k2 k2 1 k2 1 k2 ∆ VP∞ = 2 1+Π + Π 2 Π + Π 2 Π 2 Π + V k2 1k # k2 k2 1 kk2 k2 1 k2 k1 k2 k ···$ 2 ∆ VP1 ∞ ! " = 2k2 1+Π2 1 !+ Π"2 k!2 Π2" 1 +!Π 2 " 1 k2 Π! 2 " k2 Π ! "+ ! " VVP∞ k = 1+Π k +#Π k 2 Π k + Π k Π k Π k + ···$ k2 # ! " Π k !k2" ! " ! k2" ! k"2 ! " ···$! " ! " ! " k2! " ! " ! " ! " ! " 2 = Π 2 Π k = 1 !Π2(k" ) = 1 −!Π (k" ) 1 !Π (k"2) − − 2 2 ∞∞ 2 2 22 22ααkk 1 11 1 1 1 √z √1z 1 Π2kk ∞= dzdz + 2 − .