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 − . . (1) 2 2Παk = 1π 1 1 z z√z +z3 1 3m2z2 2k22− 2 (1) k dz33π %1 1 &z z2 '2z4. e 4m+ z + k Π = ! " %+ 3 &2 2− 2 ' e (1) 3π! %" z &z 2z ' 4mez + k ! " 1
3π + 5 280 Singularity at k0 = e 2α 6 6.53 10 huge momenta = very short distances (2) 3π + 5 ≈280 × k0 = e 2α 6 6.53 10 (2) ≈ × S. Brodsky, P. J. Mohr, P. Indelicato Trento, October 2012 21 Extracting the proton charge radius from muonic hydrogen
All order numerical calculations for the largest contributions QED and Hyperfine energy F=2 2P3/2 F=1 • The two measured lines obey to: 2.03 THz 2P F=1 1/2 F=0
3 1 E 5 E 3 = ∆E + ∆E + ∆E (2p ) ∆E (2s) P3/2 − S1/2 LS FS 8 HFS 3/2 − 4 HFS 49.81 THz 5 3 E 3 E 1 = ∆E + ∆E ∆E (2p )+ ∆E (2s) ~ 6 µm P3/2 − S1/2 LS FS − 8 HFS 3/2 4 HFS (~708 nm)
finite size 0.96THz Lamb shift Fine structure
2p Hyperfine structure F=1 2s hyperfine structure 2S1/2 5.56 THz F=0
Trento, October 2012 23 Contributions included to all-order
All-order: the charge distribution is included exactly in the wavefunction and in the operator, when relevant. Higher order Vacuum Polarization contribution included by numerical solution of the Dirac equation
+ +...
a b
a b
c d
Trento, October 2012 24 Contributions included to all-order
+ +...
a b
a b
c d
Trento, October 2012 24 P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . . 5 P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . . 5
Table 2. Zemach radius values. “prot. elec. Scat.”: radius obtained from scattering data. “Hyd. HFS”: data obtained from hydrogen Table 2. Zemach radius values. “prot. elec. Scat.”: radius obtained fromhyperfine scattering structure. data. “Hyd. “Comb.”: HFS”: method data obtained combining from both hydrogen type of data. hyperfine structure. “Comb.”: method combining both type of data. Method Ref. RZ (fm) Rm Method Ref. RZ (fm) Rm prot. Elec. Scat. [18], this work 1.0466 0.8309 prot. Elec. Scat. [18], this work 1.0466 0.8309 Hyd. HFS [57] 1.045 0.016 ± Hyd. HFS [57] 1.045 0.016 Hyd.,Prot. Elec. [54] 1.016 0.016 ± prot. Elec. Scat. [58] 1.086 ± 0.012 Hyd.,Prot. Elec. [54] 1.016 0.016 ± prot. Elec. Scat. [58] 1.086 ± 0.012 prot. Elec. Scat. SC [19] 1.072 0.854 0.005 ± 0.007 ± prot. Elec. Scat. SC [19] 1.076 0.850 − prot. Elec. Scat. SC [19] 1.072 0.854 0.005 ± +0.002 Hyd.± HFS0.007 [59] 1.037 0.016 prot. Elec. Scat. SC [19] 1.076 0.850 +−0.002 ± Hyd. HFS [59] 1.037 0.016 ± ±
40 2 P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . . 40 30 P. Indelicato, P.J. Mohr:Experiment non-perturbative calculations in muonic. . . 5 Experiment 30 Ref. proton charge25 radius (fm) deuteron charge radius (fm) 30 Experiment GaussianExperiment Gaussian Hand et al. [1] 0.805 Table0.011 2. Zemach radius values. “prot. elec. Scat.”: radius obtained from scattering data. “Hyd. HFS”: data obtained from hydrogen 30 ± ± 25 Simon et a/ [2] 0.862 20 hyperfine0.012 structure. “Comb.”: method combining both type of data. Gaussian ± ± ExponentialGaussian Exponential Mergel et al. [11] 0.847 0.008 20 Method Ref. RZ (fm) Rm 20 15± Sick et al. [12] 2.130prot. Elec. Scat.0.010 [18], this work 1.0466 0.8309 Exponential ± ± FermiExponential Fermi Rosenfelder [13] 0.880 20 0.015 Hyd. HFS [57] 1.045 0.016 ± 15 10± Hyd.,Prot. Elec. [54] 1.016 0.016 Sick 2003 [14] 0.895 0.018 10 ± prot. Elec.± Scat. [58] 1.086 ± 0.012 Angeli [15]Fermi 0.8791 0.0088 2.1402 0.0091 Fermi ± 5 prot. Elec. Scat. SC [19] 1.072 0.854 0.005 10 ± ± ± 0.007 Kelly [16] 0.863 0.004 prot. Elec. Scat. SC [19] 1.076 0.850 − 10 ± ± +0.002 hammer et al. [17] 0.848 Hyd. HFS [59] 1.037 0.016 0 ± 0 5 CODATA 06 [10] 0.8768 0.0 0.00690.5 2.1394 1.0 0.0028 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Arington et al. [18] 0.850 ± ± 0 0 0.004 40 Belushkin et al. [19] SC approach 0.844 +−0.008 0.0 0.5 1.0 1.5 2.0 0.00.008 0.5 1.0 1.5 2.0 Belushkin et al. [19] pQCD app. 0.830 −1.2 30 1.2 +0.005 Experiment Experiment Babenko (a) [20] 2.124 0.006 Experiment Experiment 25 30 1.0 ± 1.0 Babenko (b) [20] 2.126 0.012Gaussian Gaussian 1.2 1.2 ± Gaussian Gaussian Pohl et al. [21]Experiment 0.84184 200.00067 Experiment Exponential Exponential Ref. [21] and [22]0.8± 2.12809 0.00031 20 0.8 1.0 1.0 15 ± Exponential Exponential Gaussian 2 2 Fermi Gaussian Fermi 0.6 rd rp values 0.6 2 2 2 − 10 0.8 r r (Fm0.8) Fermi 10 Fermi d − p de Beauvoir (1996)Exponential 3.827 0.4 0.0265 Exponential 0.4 ± 0.6 Udem et al. [6] 3.8212 0.6 0.0015 ± 0 0 Angeli [15]Fermi 3.808 0.2 0.0540.0 0.5 1.0 1.5 Fermi2.0 0.0 0.2 0.5 1.0 1.5 2.0 ± 0.4 CODATA 06 [10] 3.808 0.4 0.024 0.0± 0.0 Eides et al (2007) [23] 3.8203 1.20.0007 1.2 ± 0.0 0.5 1.0 Experiment1.5 2.0 0.0 0.5 Experiment1.0 1.5 2.0 0.2 Partney et al. (2010) [22] 3.82007 0.2 0.00065 ± 1.0 1.0 Fig. 1. Top: charge densities ρ(r), bottom: chargeGaussian densities r2ρ(r) Fig. 2. Top: magnetic momentGaussian densities ρ(r), bottom: charge 0.0 Table 1. Proton and deuteron charge radii. For the result0.0 corresponding0.8 to Ref. [18] see the analysis in Sec.2.1 0.8 2 0.0 0.5 1.0 1.5 2.0 for0.0 the experimental0.5 fits in Ref.1.0 [18], comparedExponential1.5 to Gaussian,2.0 densities r ρ(r) for the experimentalExponential fits in Ref. [18], compared Fermi and0.6 exponential models distributions. All models are0.6 cal- to Gaussian, Fermi and exponential models distributions. All Finite nuclear size effect byFermi direct numerical solution of Dirac EquationFermi 2 culated to have the same R = 0.850fm RMS radius as deduced models are calculated to have the same R = 0.831fm RMS radius Fig. 1. Top:In charge the present densities work,ρ(r), we bottom: use the charge latest densities versionr ofρ(r the) Fig.many 2. Top: years0.4 magnetic in the framework moment densities of the MultiConfigurationρ(r), bottom: charge0.4 for the experimental fits in Ref. [18], compared to Gaussian, densitiesfrom ther2 experimentalρ(r) for the experimental function. fits in Ref. [18], compared as deduced from the experimental function. MCDF code of Desclaux and Indelicato [44], which is de- Dirac-FockDirect numerical0.2 (MCDF) solution of method Dirac equation, to solve numerical the atomic grid, 10000 many- 0.2 Fermisigned and exponential to calculate models also properties distributions. of exotic All models atoms are [45], cal- to tobody Gaussian, problem Fermi [48–51]. and exponential models distributions. AllP. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . . 7 0.0 !"#%,%& 0.0 culated to have the same R = 0.850fm RMS radius as deduced modelspoints, are~4000 calculated inside the to proton have the same R = 0.831fm RMS4)560'&-5"'&70-&'(#)*-(0809(0950-:;-,*<,8-=!- radius evaluate exact contribution of the Uehling potential with The Dirac0.0 equation0.5 is written1.0 1.5 !"#%"%& 2.0 0.0 0.5 1.0 1.5 2.0 fromDirac the experimental wavefunction function. and finite nuclear size. We evaluate as deducedFor a point from the charge, experimental the Uehling¨ function. potential, which!"#%+%& repre- functions defined by !"#%*%&2 Fig. 1. Top: charge2 densities ρ(r), bottom: charge densities r ρ(r) Fig. 2. Top: magnetic./0&12&30&452678& moment densities ρ(r), bottom: charge !"#%)%& ./0&12&30&49:;18& sentscα thep leading+ βµrc + contributionVNuc(r) Φ toκµ the(r) vacuum= κµΦ polarization,κµ(r), (1) /0&12&30&4.<=>68& in the same way the Kall¨ en´ and Sabry contribution. Using n n n 2 ./0&12&30&4?@ABB8& for the experimental fits in Ref. [18], compared to Gaussian,!"#%(%& densities r ρ(r) forC1=6<&4D6;#8& the experimental fits∞ in Ref. [18], compared E@FGAFH6& 1 1 1 · E !"#$%&'(#)*+,-./01%2/+,3- C1=6<&4?@ABB8& xz √ 2 is expressed as [62] !"#$'%& the same code, we also evaluate the hyperfine structure. Fermi and exponential models distributions. All models are cal- to Gaussian, Fermi andχn exponential(x) = modelsdze− distributions.n + All3 z 1. (26) where αculatedand toβ haveare the the same DiracR = 0.850fm 4 4 RMS matrices, radius as deducedµr!"#$%%& the par-models are calculated to have the same1 R = 0.831fmz RMSz radius2z − The next largest contribution comes from the muon !"#$$%& '#*& '#(& %#%& %#-& %#"& %#*& %#(& $#%& $#-& functions defined by × 2m rz &'(#)*-.2/3- For a point charge, the Uehling¨ potential, which repre- ticle reducedfrom the experimental mass, V ∞ function.(r) the nuclear potential,e as deduced from the experimental function. ( ) Nuc n∆κEV11FNµ pn α Zα 2 1 e− Fig. 3. Dependence of as a function of R in meV/fm2 for self-energy and muon-loop vacuum polarization. We use 2 R2 The Uehling¨ potential for a spherically symmetric charge √ different charge distributionE models. sents the leading contribution to the vacuum polarization, theV atom11 (r) total= energy anddzΦ isz a one-electron1 2 + 4 Dirac four- the method developed by Mohr and Soff[46,47] to calcu- − 3π ∞ 1 xz 1 1− z1 z r distributionFig. 4. Feynman is diagramsexpressed obtained when the Uehling¨ as potential [60] is expressed as [62] √ 2 (25) is added to the nuclear potential in the Dirac equation. A double componentχn(x spinor) = : dze− n + 3 z 1. (26) line represents a bound electron wavefunction or propagator late the self-energy in all order in Zα with non perturba- For a point2α(Z charge,α) 1 the2 Uz ehling¨ z potential,2z which− repre-in the potential offunctions the Dirac equation defined (3) when solving by it and a wavy line a retarded photon propagator. The grey circles 1 numerically. This amounts to get the exact solution with correspond to the interaction with the nucleus ∞ = χ1 r any number of vacuum polarization insertions as shown 2α(Zα) 1 tive finite-nuclear size contribution. 2merz sents the leading contribution to the vacuum polarization, ∞ − 3π r 1 λP (r)χ (θ, φ) in Fig. 4. The numerical methods that we used areV de-11∞(r) = 1 1 1 dr rρ(r) pn α(Zα) 2 1 e− e nκ κµ scribed in Ref. [60,61]. Because of the Logarithmic de- xz 2 √ 2 The Uehling¨ is expressed potential asr [62] for a spherically symmetric charge χ (x) = dze− 12+π √z 1. (26) V (r)The= paper is organizeddz z as follow.1 + In Sec.2 we briefly re- Φnκµ( ) = pendence of the point(2) nucleus Uehling¨ n potential at the −n 3 r 0 11 2 4 r iQnκ(r)χ κµ(θ, φ) origin, we do not calculate the iterated vacuum polar-1 z z 2z − − 3π 1 − z z r distribution is expressed as [60] − 2m rz ization directly for point nucleus. We instead calculate call the technique we use to solve the Dirac equation and where me is the electronα( α) ∞ mass, λe is the electron e for Compton different mean square radii and charge distribution 2 2 (25) pn Z 2 1 e− 2 V (r) = dz √z2 1 + models, and fitThe the curves U withehling¨ f (R) = a potential+ bR . All 4 for a sphericallyχ2 symmetricr r chargeχ2 r + r . (27) the different2α(Z nuclearα) 1 model2 we have used. In Sec. 3 we wavelength11 and the function χ1 belongs2 4 to amodels family provides very ofsimilar values. The final value is with χκµ(θ, φ) the− two3π component1 − Pauliz z sphericalr 0.1510170 spinors0.0000003distribution meV. The value calculated is expressed in Ref. as [60] λ λ ∞ ± × e | − | − e | | = χ1 r 2α(Zα)1 [32](25) is 0.1509 meV and the one in Ref. [34] is 0.1510 meV Fig. 5. Feynman diagrams included in the Kallen¨ and Sabry in very good agreeement with ours. Our method provides evaluate the non-perturbative vacuum polarization con- [49], n the principal2α(Z quantumα) 1 2 number, κ the Dirac quan- V21(r) potential − 3π r λe V11(r) = dr rρ(r) in addition the proton size dependence for this correction, = χ r 22 α(Zα) 1 ∞ − 12π r 1 which is found to be -0.0000759 0.0000001 meV/fm , which tribution, with finite nuclear size, and examine the effect tum number, and− µ3πther eigenvalue0λe of Jz . Thiswas reduces, not calculated before.V The±11 uncertainties(r) = provided where dr rρ(r) here are the statistical average of the fit errors.− 12π r 0 where is the electron mass, λ is the electron Compton ∞ rt 2 8 ofm thee model used for the protone charge distribution. In for a spherically symmetric2 potential, to2 the differential L1(r) = dte− f (t) 5 where me is the electron mass, λe is the electron Compton 1 3t − 3t 2 2 χ2 r r χ2 r + r . (27) 2 4 wavelengthSec.5, we and evaluate the function the muonχ self-energy.belongs to The a next family section of χ2 r+ +r √t2 χ1 log2 8t t2 1 r + r . (27) 1 equation:wavelength and the function χ1 belongs to a family of 3 4 3 2 − − × λ | − | − λ | | × λe | −t t | − λe | (35)| e e 2 7 13 4.2 Reevaluation of the Kall¨ en` and Sabry potential + √t2 1 + + is devoted to the hyperfine structure. Section 7 contains − 9t6 108t4 54 t2 d κ 2 5 2 44 ( ) + + + + log √t2 1 + t , results that enable to predict the different hyperfine com- αVNuc r Pnκ(r) D Pnκ(r) 9t7 4t5 3t3 − 9t − dr r d κ − = αEnκµ Analytical expression for the evaluation of vacuum-polarization potentials in + αV (r) 2µ c Qnκ(r) TheQ Kall¨nenκ´ ( andr) Sabry potential [65], is a fourth order and ponents of transition between the 2s and 2p levels, and dr r Nuc r muonicpotential, correspondingatoms, S. toKlarsfeld. the diagrams inPhysics Fig. 5. The Letters 66B, 86-88 (1977). expression for this potential has also been derived on Ref. 2 √ 2 2 − ∞ 3x 1 log x 1 + x log 8x x 1 [66–69]. In the previous version of the mdfgme code, the f (t) = dx − − − . a detailed comparison with existing results, and Sec. 8 is (3) 2 Kall¨ en` and Sabry potential used was the one provided by t x (x 1) − √x2 1 − − Ref. [62], which is only accurate to 3 digits. The expression (36) our conclusion. where Pnκ(r) and Qnκ(r) respectively the largeof this and potential is the for a point charge The function f (t) can be calculated analytically in term of the log and dilogarithm functions. Blomqvist [70] has small radial components of the wavefunction, and shown that L1(r) can be expressed as Enκµα2(Zα) 2 V (r) = L ( r), (34) 2 21 2 1 L1(r) = g2(r) log (r) + g1(r)log(r) + g0(r), (37) π r λe theTrento, October 2012 binding energy. 25 2 Dirac equation and finite nuclear model To solve this equation numerically, we use a 5 point predictor-corrector method (order h7) [51,52] on a linear mesh defined as Techniques for the numerical solution of the Dirac equa- r tion in a Coulomb potential have been developed for t = ln + ar, (4) r0 P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . . 5
Table 2. Zemach radius values. “prot. elec. Scat.”: radius obtained from scattering data. “Hyd. HFS”: data obtained from hydrogen hyperfine structure. “Comb.”: method combining both type of data.
Method Ref. RZ (fm) Rm prot. Elec. Scat. [18], this work 1.0466 0.8309 Hyd. HFS [57] 1.045 0.016 Hyd.,Prot. Elec. [54] 1.016 ± 0.016 prot. Elec. Scat. [58] 1.086 ± 0.012 prot. Elec. Scat. SC [19] 1.072± 0.854 0.005 ± 0.007 prot. Elec. Scat. SC [19] 1.076 0.850 +−0.002 Hyd. HFS [59] 1.037 0.016 ± ±
40 2 P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . . 30 Experiment Experiment
Ref. proton charge25 radius (fm) deuteron charge radius (fm) 30 Hand et al. [1] 0.805 0.011 Gaussian Gaussian ± ± Simon et a/ [2] 0.862 20 0.012 ± ± Exponential Exponential Mergel et al. [11] 0.847 0.008 20 Sick et al. [12] 15± 2.130 0.010 Rosenfelder [13] 0.880 ± 0.015 ± Fermi Fermi 10± Sick 2003 [14] 0.895 0.018 10 Angeli [15] 0.8791 ± 0.0088 2.1402 ± 0.0091 Kelly [16] 0.863 ±5 0.004 ± hammer et al. [17] 0.848 ± 0 0 CODATA 06 [10] 0.8768 0.0 0.00690.5 2.1394 1.0 0.0028 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Arington et al. [18] 0.850 ± ± 0.004 Belushkin et al. [19] SC approach 0.844 +−0.008 1.20.008 1.2 Belushkin et al. [19] pQCD app. 0.830 +−0.005 Babenko (a) [20] 2.124 0.006 Experiment Experiment Babenko (b) [20]1.0 2.126 ± 0.012 1.0 Gaussian Gaussian Pohl et al. [21] 0.84184 0.00067 ± Ref. [21] and [22]0.8± 2.12809 0.00031 0.8 Exponential Exponential 2 2 ± 0.6 rd rp values 0.6 2 2 2 − r r (Fm ) Fermi Fermi d − p de Beauvoir (1996) 3.827 0.4 0.026 0.4 Udem et al. [6] 3.8212 ± 0.0015 ± Angeli [15] 3.808 0.2 0.054 0.2 CODATA 06 [10] 3.808 ± 0.024 Eides et al (2007) [23] 3.8203 0.0± 0.0007 0.0 Partney et al. (2010) [22] 3.82007 ± 0.0 0.00065 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 ± Table 1. Proton and deuteron charge radii. For the resultFig. corresponding 1. Top: charge to Ref. densities [18] seeρ(r the), bottom: analysis charge in Sec.2.1 densities r2ρ(r) Fig. 2. Top: magnetic moment densities ρ(r), bottom: charge for the experimental fits in Ref. [18], compared to Gaussian, densities r2ρ(r) for the experimental fits in Ref. [18], compared Fermi and exponential modelsRemark: distributions. scale All models of are QED cal- correctionsto Gaussian, Fermi and exponential models distributions. All culated to have the same R = 0.850fm RMS radius as deduced models are calculated to have the same R = 0.831fm RMS radius In the present work, we use the latest version of the manyfrom the years experimental in the framework function. of the MultiConfiguration as deduced from the experimental function. MCDF code of Desclaux and Indelicato [44], which is de- Dirac-Fock (MCDF) method to solve the atomic many- signed to calculate also properties of exotic atoms [45], to body problem [48–51]. evaluate exact contribution of the Uehling potential with The Dirac equation is written Dirac wavefunction and finite nuclear size. We evaluate For a point charge, the Uehling¨ potential, which repre- functions defined by 2 sentscα thep leading+ βµrc + contributionVNuc(r) Φn toκµ the(r) vacuum= nκµΦ polarization,nκµ(r), (1) in the same way the Kall¨ en´ and Sabry contribution. Using Bohr radius/particle∞ mass: · E xz 1 1 1 2 the same code, we also evaluate the hyperfine structure. is expressed as [62] χ (x) = dze− + √z 1. (26) α n zn z 2 3 − The next largest contribution comes from the muon where and β are the Dirac 4 4 matrices, µr the par- 1 z × 2merz ticle reducedα mass,(Zα) VNuc∞ (r) the nuclear2 1 potential,e nκµ self-energy and muon-loop vacuum polarization. We use Vpn(r) = dz √z2 1 + − E The Uehling¨ potential for a spherically symmetric charge the method developed by Mohr and Soff[46,47] to calcu- the atom11 total− energy3π and Φ is a− one-electronz2 z4 Diracr four- distribution is expressed as [60] component spinor : 1 (25) late the self-energy in all order in Zα with non perturba- 2α(Zα) 1 2 Electron Compton wavelength/2π=440 R tive finite-nuclear size contribution. = χ1 r 2α(Zα) 1 ∞ − 3π r 1 λPe nκ(r)χκµ(θ, φ) V11(r) = dr rρ(r) The paper is organized as follow. In Sec.2 we briefly re- Φnκµ(r) = n=1 in(2) hydrogen: a0=137λ−e=6034012π r R0 r iQnκ(r)χ κµ(θ, φ) call the technique we use to solve the Dirac equation and where me is the electron mass, λe is− the electron Compton 2 2 χ r r χ r + r . (27) wavelength and the function χ1 belongs to a familyn=2 in ofmuonic H: a=2.65λe 2 2 the different nuclear model we have used. In Sec. 3 we with χκµ(θ, φ) the two component Pauli spherical spinors × λe | − | − λe | | evaluate the non-perturbative vacuum polarization con- [49], n the principal quantum number, κ the Dirac quan- n=1 in h-like (Z=52) : a=2.65λe tribution, with finite nuclear size, and examine the effect tum number, and µ the eigenvalue of Jz . This reduces, of the model used for the proton charge distribution. In for a spherically symmetric potential, to the differential Sec.5, we evaluate the muon self-energy. The next section equation: is devoted to the hyperfine structure. Section 7 contains αV (r) d + κ P (r) P (r) results that enable to predict the different hyperfine com- Nuc dr r nκ = αED nκ d κ − Q (r) nκµ Q (r) ponents of transition between the 2s and 2p levels, and dr + r αVNuc(r) 2µrc nκ nκ a detailed comparison with existing results, and Sec. 8 is − (3) our conclusion. where Pnκ(r) and Qnκ(r) respectively the large and the small radial components of the wavefunction, and Enκµ theTrento, October 2012 binding energy. 26 2 Dirac equation and finite nuclear model To solve this equation numerically, we use a 5 point predictor-corrector method (order h7) [51,52] on a linear mesh defined as Techniques for the numerical solution of the Dirac equa- r tion in a Coulomb potential have been developed for t = ln + ar, (4) r0 Check point nucleus vacuum polarization
Non relativistic
Relativistic
Trento, October 2012 27 Numerical solution of the Dirac equation
Trento, October 2012 28 Numerical solution of the Dirac equation
Grid (point nucleus): Finite nucleus: 8000 points, 1801 in the nucleus
All calculations performed with the Dipole model Point Coulomb, finite size coulomb or Coulomb, finite size +Uehling
Trento, October 2012 29 Extraction of the size dependence
• Example: fit to the Coulomb contribution to 2s-2p1/2 separation
0
5
10 meV E
15
Point nucleus -1 20 2s1/2:-5098503.439147 cm -1 2p1/2:-5098503.439147 cm
25 0.5 1.0 1.5 2.0 proton size fm Trento, October 2012 30 Extraction of the size dependence
• Example: fit to the Coulomb contribution to 2s-2p1/2 separation
Trento, October 2012 31 Extraction of the size dependence
• Example: fit to the Coulomb contribution to 2s-2p1/2 separation
60
50
40 residues of 30 number
20
10
0 2. 106 1.5 106 1. 106 5. 107 0 5. 107 1. 106
Trento, October 2012 Re s id u e meV 32
Extraction of the size dependence
• Fit to the Coulomb+Vacuum polarization contribution to 2s-2p1/2 separation, plus higher order corrections
Trento, October 2012 33 P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . . 7
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∆EV11FN 2 Fig. 3. Dependence of R2 as a function of R in meV/fm for different charge distribution models.
Fig. 4. Feynman diagrams obtained when the Uehling¨ potential is added to the nuclear potential in the Dirac equation. A double line represents a bound electron wavefunction or propagator in the potential of the Dirac equation (3) when solving it and a wavy line a retarded photon propagator. The grey circles numerically. This amounts to get the exact solution with correspond to the interaction with the nucleus Källen and Sabry any number of vacuum polarization insertions as shown in Fig. 4. The numerical methods that we used are de- scribed in Ref. [60,61]. Because of the Logarithmic de- pendence of the point nucleus Uehling¨ potential at the origin, we do not calculate the iterated vacuum polar- P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . . 9 ization directly for point nucleus. We instead calculate for different mean square radii and charge distribution P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . . 9 models, and fit the curves with f (R) = a + bR2. All 4 models provides very similar values. The final value is Table 4. Finite size correction to the Kall¨ en´ and Sabry contribu- 0.1510170 0.0000003 meV. The value calculated in Ref. 2 3 4 ± tion. Coefficients corresponds to ∆E = aR + bR + cR , with ∆E [32] is 0.1509 meV and the one in Ref. [34] is 0.1510 meV Fig. 5. FeynmanTable 4.diagramsFinite included size correction in the Kallen¨ to and the Sabry Kall¨ en´ and Sabry contribu- in meV and R in fm. 2 3 4 in very good agreeement with ours. Our method provides V21(r) potentialtion. Coefficients corresponds to ∆E = aR + bR + cR , with ∆E in addition the proton size dependence for this correction, which is found to be -0.0000759 0.0000001 meV/fm2, which in meV and R in fm. was not calculated before. The± uncertainties provided where Model Uniform Exponential Fermi Gaussian here are the statistical average of the fit errors. Model∞ rt 2 Uniform8 a Exponential-0.0002145 -0.0002145 Fermi Gaussian -0.0002146 -0.0002145 L1(r) = dte− 5 f (t) a1 3t-0.0002145− 3t b -0.00021450.0000078 -0.0002146 0.0000086 -0.0002145 0.0000082 0.0000083 2 4 +b + √0.0000078t2 1c log 8t t2 1 0.0000086-0.0000008 0.0000082 -0.0000009 0.0000083 -0.0000008 -0.0000009 3 4 3 2 − − c t t -0.0000008 -0.0000009(35) -0.0000008 -0.0000009 2 7 13 4.2 Reevaluation of the Kall¨ en` and Sabry potential + √t2 1 + + − 9t6 108t4 54t2 2 5 2 44 + + + log √t2 1 + t , 9t7 4t5 3t3 − 9t − where Ei(r) is the exponential integral. where Ei(r) is the exponential integral. The Kall¨ en´ and Sabry potential [65], is a fourth order and Using these results we could evaluate the finite size potential, corresponding to the diagrams in Fig. 5. The Using these results we could evaluate the finite size expression for this potential has also been derived on Ref. 2 correction√ 2 to the2 Kall¨ en´ and Sabry contribution. The vari- correction∞ 3x 1 tolog thex Kall¨1 +en´x andlog Sabry8x x 1 contribution. The vari- [66–69]. In the previous version of the mdfgme code, the f (t) = dx − ation− as a function− of. the RMS radius is of the form ation as a function2 of the RMS2 radius is of the form Kall¨ en` and Sabry potential used was the one provided by t x (x 12) 3− √x4 1 2 3 −4 − Ref. [62], which is only accurate to 3 digits. The expression aR + bR + cRaR. We+ bR have+ fittedcR .(36) We such have a function fitted such of our a function of our of this potential is for a point charge TheTrento, October 2012 functionnumericalf (t) can be result, calculatednumerical for analytically values result, of in term 0. for3 valuesR 2.2fm. of 0.3 The re-R 2.2fm. The re- 34 of the log and dilogarithmsults functions. are Blomqvist presented [70] has in≤ Table≤ 4. ≤ ≤ shown thatsultsL1(r are) can presented be expressed as in Table 4. α2(Zα) 2 V (r) = L ( r), (34) 2 21 2 1 L1(r) = g2(r) log (r) + g1(r)log(r) + g0(r), (37) π r λe 4.3 Other higher-order4.3 Other U higher-orderehling¨ correction Uehling¨ correction
SInce we includeSInce the we vacuum include polarization the vacuum in the polarization Dirac in the Dirac equation potential,equation all energies potential, calculated all energies by perturbation calculated by perturbation using the numerical wavefunction contains the contribu- tion of higher-orderusing the diagrams numerical where wavefunction the external contains legs, the contribu- which representtion the of wavefunction, higher-order diagrams can be replaced where by the external legs, a wavefunctionwhich and represent a bound propagator the wavefunction, with one, can or be replaced by several vacuuma wavefunction polarization insertion. and a bound For example propagator the with one, or Kallen¨ and Sabryseveral correction vacuum calculated polarization in this insertion. way, con- For example the tains correctionKallen¨ of the and type Sabry presented correction in Fig. calculated 6. This cor- in this way, con- 2 rection is giventains by ∆ correctionE11 12 = 0.0021551 of the type0.0000012 presentedR with in Fig. 6. This cor- 7 × − 2 a 10− meV accuracy.rection is given by ∆E11 12 = 0.0021551 0.0000012R with 7 × − a 10− meV accuracy. Fig. 6. Lower order Feynman diagrams included in the Kallen¨ and Sabry V21(r) potential,Fig. 6. Lower when the order Uelhing¨ Feynman potential diagrams is in- included in the Kallen¨ 5 All-order 2s state self-energy cluded in the differential equation and Sabry V21(r) potential, when the Uelhing¨ potential is in- 5 All-order 2s state self-energy cluded in the differential equation 5.1 Muon self-energy and vacuum polarization 5.1 Muon self-energy and vacuum polarizationanomalous magnetic moment part and there is an extra The vacuum polarization due to the creation of virtual logarithmic correction: muon pairs for s states is given by [10] Eq. (27),[30] Eq. anomalous magnetic moment part and there is an extra (32) The vacuum polarization due to the creation of virtual logarithmic correction: 3 4 3 muon pairs4 for s states is given by [10] Eq. (27),[30] Eq. α (Zα) µr α(αZ) 4 µr ∆E = (F (Zα) E(32)µVP = mµ, (49) µSE,nlκ 3 nlκ 3 15 π n mµ πn − mµ 3 4 3 4 µ α (Zα) µr α(αZ) 4 r mµ ∆E1 δl,0 = 4 mµ (F (Zα) in which higher order termsEµ inVPZ=α have been neglected. mµ, (49) µ−SE,nlκ 3 nlκ 3 + +π δl,n0 ln mµ . πn −15 mµ µr 2κ(2l + 1) 3 µr For the 2s in muonic hydrogen it gives 0.01669 meV and is included in Refs. [32,34]. In the same work, the Self- mµ 1 δl,0(50) 4 mµ in which higher order terms in Zα have been neglected. + − + δ ln . energy is obtained from a formula valid only in the low- l,0 For the 2s in muonic hydrogen it gives 0.01669 meV and µr 2κ(2l + 1) 3 µr est order in Zα, and no finite nuclear size correction is is included in Refs. [32,34]. In the same work,Using thisthe with Self-F (α) = 10.546825185, F (α) = 0.12639637 (50) included. As it is a sizable contribution, and the muon 2s 2p1/2 − Compton wavelength,energy is which obtained represent from the a formula scale of valid QED onlyand inF2 thep3/2( low-α) = 0.12349856 [41] one gets the exact muon est order in Zα, and no finite nuclear sizeself-energy correction for is each state. For the 2s state, this gives 0.6752 meV corrections for muons is of the order of the finite nuclear Using this with F (α) = 10.546825185, F (α) = 0.12639637 size (1.9 fm), oneincluded. could expect As it a is non-negligible a sizable contribution, finite size andinstead the of muon 0.6754 meV. The finite size2s correction is given 2p1/2 and F (α) = 0.12349856 [41] one gets the exact− muon contribution.Compton The exact self-energywavelength, for which electronic represent atoms theby scale perturbation of QED theory [10]2p3/ Eq.2 (54) and point nucleuscorrections is known for muons from Ref.[41]. is of the It order requires of the finite nuclear self-energy for each state. For the 2s state, this gives 0.6752 meV instead of 0.6754 meV. The finite size correction is given correction forsize recoil, (1.9 as fm), described one could in [10]. expect The a global non-negligible de- finite size 23 pendence in the reduced mass has to be corrected in the ESE NSby= perturbation4 ln 2 α( theoryZα) NS [10], Eq. (54)(51) contribution. The exact self-energy for electronic atoms − − 4 E and point nucleus is known from Ref.[41]. It requires correction for recoil, as described in [10]. The global de- 23 pendence in the reduced mass has to be corrected in the ESE NS = 4 ln 2 α(Zα) NS, (51) − − 4 E Finite size correction on muon self-energy
All-orders calculations (P. Indelicato, P. Mohr)
Trento, October 2012 35 P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . . 9
Table 4. Finite size correction to the Kall¨ en´ and Sabry contribu- tion. Coefficients corresponds to ∆E = aR2 + bR3 + cR4, with ∆E in meV and R in fm.
Model Uniform Exponential Fermi Gaussian a -0.0002145 -0.0002145 -0.0002146 -0.0002145 b 0.0000078 0.0000086 0.0000082 0.0000083 c -0.0000008 -0.0000009 -0.0000008 -0.0000009
where Ei(r) is the exponential integral. Using these results we could evaluate the finite size correction to the Kall¨ en´ and Sabry contribution. The vari- ation as a function of the RMS radius is of the form aR2 + bR3 + cR4. We have fitted such a function of our numerical result, for values of 0.3 R 2.2fm. The re- sults are presented in Table 4. ≤ ≤
4.3 Other higher-order Uehling¨ correction
SInce we include the vacuum polarization in the Dirac equation potential, all energies calculated by perturbation using the numerical wavefunction contains the contribu- tion of higher-order diagrams where the external legs, which represent the wavefunction, can be replaced by a wavefunction and a bound propagator with one, or several vacuum polarization insertion. For example the Kallen¨ and Sabry correction calculated in this way, con- tains correction of the type presented in Fig. 6. This cor- 2 rection is given by ∆E11 12 = 0.0021551 0.0000012R with 7 × − a 10− meV accuracy. Fig. 6. Lower order Feynman diagrams included in the Kallen¨ 10 P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . . and Sabry V21(r) potential, when the Uelhing¨ potential is in- 5 All-order 2s state self-energy cluded in the differential equation !"#$%&'()&%*+,-.+$+/01%2$34+%'35+% !(#&% 5.1 Muon self-energy and vacuum polarization 10 P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . .
!(#$%anomalous magnetic moment part and there is an extra The vacuum polarization due to the creation of virtual logarithmic correction:!"#$%&'()&%*+,-.+$+/01%2$34+%'35+% !(#&% muon pairs for s states is given by [10] Eq. (27),[30] Eq. !'#&% ./01%23%42%56.78(#,$*9:5;<=>?<%@6AB&9% (32) CD%5(#,$*9% !(#$% 3 4 3 !'#$% 4 µ ./01%23%42%56.78&#'"9:5;<=>?<%6.7B&9% α(αZ) 4 µr α (Zα) r CD%5&#'"%E.9% ∆EµSE,nlκ = (Fnlκ(Z./01%23%42%56.78'#*(9:5;<=>?<%6.7B&9%α) EµVP = mµ, (49) !'#&% 3 3 π n mµ CD%5'#*(%E.9% πn −15 mµ !&#&% ./01%23%42%56.78(#,$*9:5;<=>?<%@6AB&9% ./01%23%42%56.78(#+((9:5;<=>?<%6.7B&9% CD%5(#,$*9% mµ 1 δ 0 4CD%5(#+((%E.9% mµ !'#$% l, ./01%23%42%56.78&#'"9:5;<=>?<%6.7B&9% ./01%23%42%56.78'#&*(9:5;<=>?<%6.7B&9% in which higher order terms in Zα have been neglected. !&#$% + − + δl,0 lnCD%5&#'"%E.9% . µ 2κ(2 + 1) 3CD%5'#&*(%E.9% µ For the 2s in muonic hydrogen it gives 0.01669 meV and r l ./01%23%42%56.78'#*(9:5;<=>?<%6.7B&9%r CD%5'#*(%E.9% !&#&% is included in Refs. [32,34]. In the same work, the Self- !"#&% ./01%23%42%56.78(#+((9:5;<=>?<%6.7B&9%(50) CD%5(#+((%E.9% energy is obtained from a formula valid only in the low- ./01%23%42%56.78'#&*(9:5;<=>?<%6.7B&9% !&#$% est order in Zα, and no finite nuclear size correction is !"#$% CD%5'#&*(%E.9% (% '(% &(% "(% )(% *(% +(% $(% ,(% -(% '((% included. As it is a sizable contribution, and the muon Using this with F2s(α) = 10.546825185, F2p1/2(α) = 0.12639637 !"#&% − and F2 3 2(α) = 0.12349856 [41] one gets the exact muon Compton wavelength, which represent the scale of QEDFig. 7. (F (Zpα/) F(Zα))/(Zα < r2 >) for 2s muon self-energy as self-energyFS − for each state. For the 2s state, this gives 0.6752 meV corrections for muons is of the order of the finite nucleara function!"#$% of Z for different nuclear size. size (1.9 fm), one could expect a non-negligible finite size instead(% of '(% 0.6754 &(% meV. "(% )(% The *(% finite +(% size $(% correction ,(% -(% '((% is given Finite size correctionby perturbation on muon theory self-energy [10] Eq. (54) contribution. The exact self-energy for electronic atoms 2 whereFig. ([10] 7. (F Eq.FS(Z (51))α) F(Zα))/(Zα < r >) for 2s muon self-energy as and point nucleus is known from Ref.[41]. It requires a function of Z for− different nuclear size. correction for recoil, as describedAll-orders in [10]. calculations The global (P. Indelicato, de- P. Mohr) 3 2 2 2 µr (Zα) 23Zα < r > pendence in the reduced mass has to be corrected in the NS = ESE NS = 4 ln 2mµ α(Zα) NS, , (52)(51)Fig. 8. Scaled finite nuclear size correction to 2s muon self-energy − 3 − 4 E whereE ([10] Eq.3 m (51))µ n C for different nuclear size. is the lowest-order finite nuclear3 2 size correction2 to the 2 µr (Zα) Zα < r > Coulomb energy. Here C is the muon Compton wave- Fig. 8. Scaled finite nuclear size correction to 2s muon self-energy NS = 3 mµ , (52) E 3 mµ n C 2 with length. Equation (52) provides NS = 5.19745 < r > for for different nuclear size. µ0 µ r E A(r) = × , (55) muonicis the hydrogen lowest-order in agreement finite nuclear with Refs. size correction [32,34]. The to the 4π r3 perturbativeCoulomb self-energy energy. Here finite-sizeC is the correction muon Compton to the 2 wave-s 2 2 wherewithµ is the nuclear magnetic moment and we have levellength. would Equation then be ESE (52)NS provides= 0.000824NS<=r5.>19745for muonic< r > for µ − − assumed a magnetic moment distribution0 µ r of a point par- hydrogen.muonic hydrogen in agreementE with Refs. [32,34]. The A(r) = × , (55) ticle for the nucleus. It is convenient4π tor3 express H using Theperturbative suitability self-energy of the perturbative finite-size approach correction has to been the 2s hfs shown to be questionable in heavy ions, where2 the wave- vectorwhere sphericalµ is the harmonics. nuclear On magnetic obtains moment [71–74] and we have level would then-0.00063 be ESE meVNS = (pert)0.000824 < r > for muonic function is contracted toward− the− nucleus[36]. As the assumed a magnetic moment distribution of a point par- hydrogen. -0.0008497(6) meV 1 1 reduction in size of the wavefunction is even larger in ticle for the nucleus.Hhfs It= isM convenientT , to express H (56)using The suitability(exact) of the perturbative approach has been · hfs muonicshown atoms, to be a questionable full calculation in heavy is needed ions, towhere check the the wave- vector spherical harmonics. On obtains [71–74] suitabilityfunction of the is contracted perturbative toward approach the in nucleus[36]. hydrogen. Us- As thewhere α Y1 (0)1(r) ing thereduction same technique in size of as the in wavefunction Ref [46,47], we is even have larger cal- in Hhfs8π= M 1Tq ,ˆ (56) culated the all-order in Zα finite size correction to the T 1 (r) = ie · · , (57) muonic atoms, a full calculation is needed to check the − 3 r2 self-energy for several nuclear size and 20 Z 90, as where suitability of the perturbative approach≤ in hydrogen.≤ Us- the code is not adapted to low-Z calculations. The plot and M 1 representing the magnetic moment(0) operator from ing the same technique as in Ref [46,47], we have cal- α Y1q (rˆ) Trento, October 2012 2 36 1 1 8π · of (FculatedFS(Zα) theF(Z all-orderα))/(Zα < inrZα>)finite is presented size correction on Fig. to 7 thethe nucleus. The operatorT (r) =T ieacts only on the bound, par- (57) − − 3 r2 for severalself-energy different for several nuclear nuclear radii. The sizeZ and0 20 limit is pre-90, as (0) → Z ticle coordinates. The vector spherical harmonic Y (rˆ) sented in Fig. 7 together with the perturbative result≤ from≤ 1 1q the code is not adapted to low-Z calculations. The plot and M representing2 the magnetic moment operator from (51). For muonic hydrogen one gets2 is an eigenfunction of J and Jz, defined as [75,72,76,77, of (FFS(Zα) F(Zα))/(Zα < r >) is presented on Fig. 7 the nucleus. The operator T 1 acts only on the bound par- − 74] for severalH different nuclear2 radii. The Z 3 0 limit is pre- (0) ESE NS = 0.00107 < r > +0.00035 < r→> ticle coordinates. The vector spherical harmonic Y1q (rˆ) sented− in Fig.− 7 together with the perturbative result from 4 (0) 2 0.00007 < > meV. (53) Y is( anrˆ) = eigenfunctionY11q (rˆ) = ofCJ1, 1and, 1; qJz, definedσ, σ, q Y1 as,q [75,72,76,77,σ (rˆ) ξσ (51). For muonic hydrogenr one gets 1q − − σ − H 74] For < r >= 0H.875, one gets E 2 = 0.00061 meV.3 This (58) ESE NS = 0.00107SE
All-orders calculations (P. Indelicato, P. Mohr)
-0.00063 meV (pert) -0.0008497(6) meV
Trento, October 2012 (exact) 37 Other contributions
Trento, October 2012 38 The role of the nuclear model
Dependence on the charge distribution P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . . 5
Table 2. Zemach radius values. “prot. elec. Scat.”: radius obtained from scattering data. “Hyd. HFS”: data obtained from hydrogen hyperfine structure. “Comb.”: method combining both type of data.
Method Ref. RZ (fm) Rm prot. Elec. Scat. [18], this work 1.0466 0.8309 Hyd. HFS [57] 1.045 0.016 Hyd.,Prot. Elec. [54] 1.016 ± 0.016 prot. Elec. Scat. [58] 1.086 ± 0.012 prot. Elec. Scat. SC [19] 1.072± 0.854 0.005 ± 0.007 prot. Elec. Scat. SC [19] 1.076 0.850 +−0.002 Hyd. HFS [59] 1.037 0.016 ± Nuclear Models and ±experiment
40
30 Experiment Experiment
25 30 Gaussian Gaussian 20 Exponential Exponential 20 15 Fermi Fermi 10 10
5
0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
1.2 1.2 Experiment Experiment 1.0 1.0 Gaussian Gaussian 0.8 0.8 Exponential Exponential 0.6 0.6 Fermi Fermi 0.4 0.4
0.2 0.2
0.0 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
Fig. 1. Top: charge densities ρ(r), bottom: charge densities r2ρ(r) Fig. 2. Top: magnetic moment densities ρ(r), bottom: charge Global analysisfor of proton the experimental elastic form factor fits in data Ref. with [18], two-photon compared exchange to Gaussian, correctionsdensities, J. Arrington,r2ρ(r) for W. Melnitchouk the experimental et J.A. Tjon. fits in Phys. Ref. Rev. [18], C compared76, 035205 (2007). Fermi and exponential models distributions. All models are cal- to Gaussian, Fermi and exponential models distributions. All culated to have the same R = 0.850fm RMS radius as deduced models are calculated to have the same R = 0.831fm RMS radius Rp = 0.85035from fm the Rm experimental = 0.831 fm function. Rz = 1.0466 fm as deduced from the experimental function. Trento, October 2012 40
For a point charge, the Uehling¨ potential, which repre- functions defined by sents the leading contribution to the vacuum polarization, ∞ 1 1 1 is expressed as [62] xz √ 2 χn(x) = dze− n + 3 z 1. (26) 1 z z 2z − 2merz α(Zα) ∞ 2 1 e Vpn(r) = dz √z2 1 + − The Uehling¨ potential for a spherically symmetric charge 11 − 3π − z2 z4 r distribution is expressed as [60] 1 (25) 2α(Zα) 1 2 = χ1 r 2α(Zα) 1 ∞ − 3π r λ V11(r) = dr rρ(r) e − 12π r 0 where me is the electron mass, λe is the electron Compton 2 2 χ r r χ r + r . (27) wavelength and the function χ1 belongs to a family of 2 2 × λe | − | − λe | | Comparison of Zemach’s 3rd moments
3 2 3/2 •
[1] QED is not endangered by the proton's size, A. De Rújula. Physics Letters B 693, 555-558 (2010) [2] QED confronts the radius of the proton, A. De Rújula. Physics Letters B 697, 26-31 (2011) [3] The RMS charge radius of the proton and Zemach moments, M.O. Distler, J.C. Bernauer et T. Walcher. Physics Letters B 696, 343-347 (2011). . Trento, October 2012 41 Nuclear Models and experiment • Using the electronic density from Arrington et al. I get – Rp = 0.85035 fm Rm = 0.831 fm Rz = 1.0466 fm • Dirac + Uehling vacuum polarization with this density: – E=201.2789 meV • Dirac + Uehling vacuum polarization with same radius and other models – Gauss: E=201.2680 (-0.0109) meV – Dipole: E=201.2700 (-0.0089) meV – Uniform: E=201.2669 (-0.0120) meV – Fermi: E=201.2686 (-0.0102)
• Solving EDipole(R)=201.2789 meV gives: – R=0.84934 (-0.00101) fm
Trento, October 2012 42 Hyperfine structure
Beyond Zemach
Trento, October 2012 43 12
reduction to radial and angular integrals is presented This means that µBR(r) = µ(r)/(4π). Evaluation of the in various works [84–86]. In heavy atoms, the hyper- Wigner 3J and 6J symbols in (89) give the same angular fine structure correction due to the magnetic moment factor than in Eq. (88). contribution is usually calculated for a finite charge dis- The equivalence of the two formalism can be easily tribution, but a point magnetic dipole moment (see, e.g., checked: starting from (92) and droping the angular fac- [84, 85]). When matrix elements non-diagonal in J are tors, we get, doing an integration by part needed, one can use [90] for a one-particle atom
gα ∞ P (r)Q (r) + P (r)Q (r) ∆EHFS = A dr 1 2 2 1 , (88) r M1 2 ∞ P1(r)Q2(r) 2 2Mp 0 r dr dr r µ(r ) r2 n n n 0 0 where g = µp/2 = 2.792847356 for the proton, is the r r P1(t)Q2(t) ∞ anomalous magnetic moment, A is an angular coefficient = dr r2µ(r ) dt n n n t2 0 0 0 r ∞ n ( ) ( ) Ij1 F 2 P1 r Q2 r drnrnµ(rn) dr 2 j2 Ik − 0 0 r A = ( 1)I+j1+F − I 1 I ∞ P1(r)Q2(r) = dr 2 0 0 r I I − rn ∞ P1(r)Q2(r) 1 2 J1 j1 1 j2 drnr µ(rn) dr , (94) ( 1) − 2 (2J + 1)(2J + 1) π (l , k, l ) , n 2 × − 1 2 1 0 1 1 2 − 0 0 r 2 − 2 (89) 13 where we have used (91). We thus find that the for- where π (l1, k, l2) = 0 if l1 + l2 + 1 is odd and 1 otherwise. mula in Borie and Rinker represents12 the full hyperfine The ji are the total angular momentum of the i state for Ref. [95] corrected for a misprint):structure correction, including the Bohr-Weisskopffunction of part.R and RZ, which gives: thereduction bound to particle, radial andli are angular orbital integrals angular is presented momentum,ThisI means that µBR(r) = µ(r)/(4π). Evaluation of the is thein various nuclear works spin, [84–86].k the In multipole heavy atoms, order the ( hyper-k = 1 forWigner the 3J andIn 6J 1956, symbols Zemach in (89) give [95] the calculated same angular the fine structure en- fine structure correction due to the magnetic moment factor than in Eq. (88). 2s magnetic dipole contribution described in Eq. (88)) and F ergy of hydrogen, including recoil effects.E He(R showedZ, R) = 22.807995 contribution is usually calculated for a finite charge dis- HFS The equivalencethat in of first the ordertwo formalism in the can finite be easily size, the HFS depends thetribution, total angular but a point momentum magnetic dipole ofHyperfine the moment atom. (see, Thestructure e.g., difference 2 3 HFS checked: starting from (92) and droping the angular fac- 0.0022324349R + 0.00072910794R between[84, 85]).∆ WhenE values matrix elements calculated non-diagonal with a finitein J are or2 point on the charge2 and magnetic distribution moments only Z tors, we get, doing an integration by part − nuclearneeded, charge one can contribution use [90] for a one-particle is called∆E the atomHFS Breit-Rosenthal= Sp Sµ throughφC(0) the Zemacs’s form factor defined in Eq. (15). 4 −3 · 0.000065912957R 0.16034434RZ correction [91]. The proton is assumed to be at the origin of coordinates. gα ∞ P (r)Q (r) + P (r)Q (r) − − ∆EHFS = A dr 1 2 2 1 , (88) Its charge andr magnetic moment distribution are given 0.00057179529 To considerM1 a finite magnetic moment2 distribution, one ∞ P1(r)Q2(r) 2 RRZ 2Mp 0 r 1 2αmµdr ρ(u) udr rrµ(rµ)(r)dudr , in terms2 of chargen n n distribution ρ( ) and magnetic mo-− uses the Bohr-Weisskopf correction [92]. The correction× − 0 r | 0 − | r 2 r r 0.00069518048R RZ where g = µp/2 = 2.792847356 for the proton, is the ment distributions µ(r).∞ Zemach calculate the correction can be written [93] 2 P1(t)Q2(t) − anomalous magnetic moment, A is an angular coefficient = drnrnµ(rn) dt 3 in first order to thet2 hyperfine energy of s-states of hy- = EF 1 2αmµ0 ρ(u)0 u r µ(r0 )dudr , 0.00018463878R RZ gα ∞ drogen duer ton the electric charge distribution. The HFS BW 2 − ∞ | −P (r)Q| (r) − (100) ∆E = AIj1 F drnrnµ(rn) dr r2µ(r ) dr 1 2 2 2 energyn n isn written as2 13 + 0.0010566454R −j2 IkMp 0 − 0 0 r Z I+j1+F (90) (97) A = ( 1) rn 2 − P1(r)Q2(r) + P2(r)Q1(r) ∞ P1(r)Q2(r) Ref. [95] correctedI for1 aI misprint): function of R=and RZdr, which gives: + 0.00096830453RR dr , 2 Z 0 2 0 r × I I r − 0 2s Z 2 rn 2 2 2 EHFS (RZ, R) = 22.807995 ∆E∞ =2 S SP1(r)Q2(r) φ(r) µ(r)dr (95)+ 0.00037883473R R 1 j1 1 j2 HFS p µ Z J1 2 drnrnµ(rn) dr , (94) ( 1) − (2J1 + 1)(2J2 + 1) 1 1 π (l1, k, l2) , 2 −3 ·3 2 | | where the magnetic2 moment2 density0 µ(rn) is normalized 0.0022324349− 0R + 0.000729107940 R r 3 ∆×EZ− = S S φ (0) 2 2 − HFS p µ C where − is the well known HFS Fermi4 energy. Trans- 0.00048210961R as −3 · EF (89) 0.000065912957R 0.16034434RZ Z − − − forming Eq. (97) using r 0.00057179529r + u,RR etc.Z Zemach obtains 3 1 2αmµ ρ(u) u r µ(r)dudr , where− weφ havethe used non-relativistic (91). We thus electron find that the wavefunction for- and S are 0.00041573690RR where π∞(l , k, l )×= 0− if l + l + 1 is| odd− and| 1 otherwise. 2 x Z 1 22 1 2 mula in→0 Borie.00069518048 and RinkerR RZ represents the full hyperfine drnrnµ(rn) = 1. (91) − the spin operators of the electron and proton. If the− The ji are the total angular momentum of the i state for structure correction, including3 the Bohr-Weisskopf part. 4 0 = EF 1 2αmµ ρ(u) u r µ(r)dudr , 0.00018463878R RZ + 0.00018238754R meV. the bound particle, l−i are orbital angular| − | momentum, I − magnetic moment distribution(100) is taken to be the one of a Z In 1956,+ 0.0010566454 Zemach [95]R2 calculated the fine structure en- 2 is the nuclear spin, k the multipole order (k = 1(97) for the point charge,Z µ(r) = δ(r), the integral reduces to φ(0) . Borie and Rinker [94], write the totalZ diagonal hyper-ergy of hydrogen, including2 recoil effects. He showed magnetic dipole contribution described in Eq. (88)) and F + 0.00096830453RRZ | | fine energy correction for a muonic∆ atomEHFS as= EF 1that2 inαm firstµThe orderrZ first in, order the finite correction size, the HFS to the depends(98) wavefunctionThe constant due to the term should be close to the sum of the Fermi the total angular momentum of the atom. The difference − + 0.00037883473 R2R2 Trento, October 2012HFS on the chargenucleusand magnetic finiteZ44 charge distribution distribution moments only is given by between ∆E values calculated with a finite or point 3 energy 22.80541 meV and of the Breit term [96]. the HFS where is the well known HFS Fermi energy. Trans- 0.00048210961R nuclear chargeEF 4 contributionπκ F(F + is1) calledI(I the+ 1) Breit-Rosenthalj(j + 1) gαthrough− the Zemacs’s Z form factor defined in Eq. (15). forming∆Ei,j Eq.= (97) using r r + u−, etc. Zemach− obtains 3 correction calculated with a point-nucleus Dirac wave- correction [91]. 1 The proton0.00041573690 is assumedRR toZ be at the origin of coordinates. → κ2 2Mp − To consider a finite magnetic moment4 distribution, one Its charge+ 0.00018238754 and magneticR4 meV. moment distribution are given function for which I find 22.807995 meV. When setting with− rZ given in Eq. (16). The 2s Zstate Fermi energy is uses the Bohr-Weisskopf correction [92]. Ther correction in terms of charge distribution ρ(r) and magnetic mo- Z ∞ P1(r)Q2(r) 2 φ(r) = φC(0) 1 αmµ ρ(u) u ther du speed, of(96) light to infinity in the program I recover ∆E = EF 1 2αmµ rZ , given by (98) The constantment distributions term should beµ close(r). Zemach to the sum calculate of the Fermi the correction can be writtenHFS [93] − dr 2 drnrnµBR(rn). (92) − | − | × 0 r 0 energyin 22 first.80541 order meV to and the of thehyperfine Breit term energy [96]. the of HFS s-states of hy- exactly the Fermi energy. The Breit contribution is thus gα ∞ correction calculated with a point-nucleus Dirac wave- BW 2 drogen due to the electric charge distribution. The HFS 0.002595 meV, to be compared to 0.0026 meV in Ref. [40] wherewith the∆Er normalizationgiven= A in Eq. (16). Thedr isnr di 2nsµffstate(rerent:n) Fermi energy is function for which I find 22.807995 meV. When setting Z − 2Mp 0 energy is written as given by (90)the speed of lightwhere to infinityφC in(0) the is program the unperturbed I recover Coulomb wavefunction rn (Table II, line 3) and 0.00258 meV in Ref. [70]. Mar- ∞ P1(r)Q2(r)∞+ P2(r)Q1(r) exactly4 the3 Fermi energy. The Breit contribution is thus 3 dr 2 , (Zα) µ at the origin for a point nucleus. Replacing into Eq. (95) d rnµBR(rn) = 4π 2 drnr µ2BRs (rn) = 1.0.002595(93) meV,r to be compared to 0.0026 meV in Ref. [40] tynenko [40] evaluates this correction, which he names × 0 r nE = g . 2 (99) 0 0 F (Tablep II, line 3)Z and 0.00258 keeping meV in only Ref. first [70].2 Mar-order terms, we get (Eq. 2.8 of 5 6 4 3 3 ∆EHFS = Sp Sµ φ(r) µ(r)dr (95) 2s (Zα) µr tynenkomp [40]mµ evaluates− this3 correction,· which| he| names “Proton structure corrections of order α and α ”, to be where theE magnetic= gp moment. density µ(r ) is normalized(99) F 3 m m n 5 6 as p µ “Proton structure corrections of order α and α ”, to be 0.1535 meV, following [35]. He finds the coefficient 0.1535 meV, following [35]. He finds the coefficient − 1 − 1 for theφ Zemach’sthe non-relativistic radius to be electron0.16018 meVf wavefunction− , in very and Sx are for the Zemach’s radius to be 0.16018 meVf− , in very ∞ 2 − drnrnµ(rn) = 1. (91)goodthe agreement spin operators with the present of the all-order electron calculation and proton. If the − 1 1 good agreement with the present all-order calculation 0 0.16034 meVf− . Borie’s value [70] 0.16037 meVf− is − magnetic moment distribution− is taken to be the one of a even closer. The difference between Borie’s value and 2 1 1 Borie and Rinker [94], write the total diagonal hyper- point charge, µ(r) = δ(r), the integral reduces to φ(0) . 0.16034 meVf− . Borie’s value [70] 0.16037 meVf− is Eq. (100)The firstis represented order correction in Fig. 11 to as the a function wavefunction of the due| to the| − − fine energyA. correction Hyperfine for structure a muonic of the atom2s level as charge and Zemach radii. The maximum difference is even closer. The difference between Borie’s value and aroundnucleus 1 µeV. finite charge distribution is given by 4πκ F(F + 1) I(I + 1) j(j + 1) gα Eq. (100) is represented in Fig. 11 as a function of the − − In Ref. [35], the charge and magnetic moment dis- ∆Ei,j = 1 In order to check the dependenceκ2 of the hyperfine2Mp tributions are written down in the dipole2 form, which 4 A. Hyperfine structure of the s level charge and Zemach radii. The maximum difference is structure on the Zemach radius− andr on the proton fi- corresponds to (13), nite size, I have performed∞ P1(r)Q a2 series(r) of calculations2 for φ(r) = φC(0) 1 αmµ ρ(u) u r du , (96) around 1 µeV. dr drnrnµBR(rn). (92) − | − | a dipolar distribution× for both2 the charge and magnetic 2 0 r 0 GM q 4 moment distribution. We can then study the dependence 2 Λ In Ref. [35], the charge and magnetic moment dis- GE q = = 2 , (101) whereof the the HFS normalization beyond the first is di orderfferent: corresponding to the 1 +κp Λ2 + 2 In order to checkwhere the dependence(0) is the unperturbedq of Coulomb the hyperfine wavefunction tributions are written down in the dipole form, which Zemach correction. I calculated the hyperfine energy φC ∞ 3 ∞ structureBW2 on thewith ZemachatΛ the= 848 origin.5MeV. radius for This a point leads and nucleus. to R on= 0. the Replacing806 fm proton as in into Eq. fi- (95) corresponds to (13), splitting d∆ErHFSµ (R(Zr, R) )==4EπHFS(R)dr+ ErHFSµ (R(,rR)M=) numer-1. (93) n BR n n n BR n and keeping only first order terms, we get (Eq. 2.8 of ically.0 I also evaluate with and0 withoutnite size, self-consistent I haveRef. performed [2] and RZ = a1.017 series fm using of this calculations definition for for inclusion of the Uëhling potential in the calculation, to the form factor in Eq. (17). Moreover there are re- obtain all-order Uëhling contributiona dipolar to the HFS energy. distributioncoil corrections for both included. the Pachucki charge [35] finds and that magnetic the G q2 4 We calculated the correction ∆E (R , R) for several pure Zemach contribution (in the limit mp ) is M Λ HFSmomentZ distribution. We can then study the dependence→∞ 2 value of RZ between 0.8 fm and 1.15 fm, and proton sizes 0.183 meV. In Ref. [97], the Zemach corrections is given GE q = = , (101) − 4 2 ranging from 0.3 fm to 1.2 fm, by steps of 0.05 fm, which as δ(Zemach) EF = 71.80 10− EF, for RZ = 1.022 fm. 1 + κp 2 2 of the HFS beyond the first× order− × corresponding1 to the Λ + q represents 285 values. The results show that the cor- This leads to a coefficient 0.1602 meVfm− , in excellent − 1 rection to the HFS energy due toZemach charge and magnetic correction.agreement I with calculated our value 0.16036 the meVfm hyperfine− . energy − moment distribution is not quite independent of R as The effect of the vacuum polarizationBW on the 2s hyper- with Λ = 848.5MeV. This leads to R = 0.806 fm as in one would expect from Eq. (98), insplitting which the finite∆E sizeHFS (fineRZ structure, R) = energyEHFS shift(R) as+ a functionEHFS ( ofR the, R ZemachM) numer- contribution depends only on RZ.ically. We fitted the I also hyper- evaluateand charge with radius have and been without calculated for self-consistent the same set Ref. [2] and RZ = 1.017 fm using this definition for fine structure splitting of the 2s level, E2s (R , R) by a of values as the main contribution. The data can be de- inclusionHFS Z of the Uëhling potential in the calculation, to the form factor in Eq. (17). Moreover there are re- obtain all-order Uëhling contribution to the HFS energy. coil corrections included. Pachucki [35] finds that the We calculated the correction ∆E (R , R) for several pure Zemach contribution (in the limit mp ) is HFS Z →∞ value of RZ between 0.8 fm and 1.15 fm, and proton sizes 0.183 meV. In Ref. [97], the Zemach corrections is given − 4 ranging from 0.3 fm to 1.2 fm, by steps of 0.05 fm, which as δ(Zemach) EF = 71.80 10− EF, for RZ = 1.022 fm. × − × 1 represents 285 values. The results show that the cor- This leads to a coefficient 0.1602 meVfm− , in excellent − 1 rection to the HFS energy due to charge and magnetic agreement with our value 0.16036 meVfm− . − moment distribution is not quite independent of R as The effect of the vacuum polarization on the 2s hyper- one would expect from Eq. (98), in which the finite size fine structure energy shift as a function of the Zemach contribution depends only on RZ. We fitted the hyper- and charge radius have been calculated for the same set 2s fine structure splitting of the 2s level, EHFS (RZ, R) by a of values as the main contribution. The data can be de- 14 P. Indelicato, P.J. Mohr: non-perturbative calculations inHyperfine muonic. . . structure where Ξn is a new function defined by !"#$)*' -./'0122'3$#"+)4'5678' -./'0122'3$#"+)4'9:;<<8' ∞ 1 1 1 -./'0122'3"#%4'5678' √ 2 !"#$)*)' Ξn (x) = dzΓ (0, xz) n + 3 z 1 (89) -./'0122'3"#=4'5678' 1 z z 2z − -./'0122'3"#,4'5678' !"#$%' -./'0122'3"#*4'5678' -./'0122'3$#(4'5678' Some properties of Ξn are given in Appendix A. (>'?@'3$#"+)4'5678' !"#$%")' (>'?@'3"#%4'5678' (>'?@'3"#=4'5678' !"#$%$' (>'?@'3"#,4'5678' 2 (>'?@'3"#*4'5678' 6.2 Hyperfine structure of the s level (>'?@'3$#(4'5678' !"#$%$)' !"#$%&&'$(%)*+,-#./'01# In order to check the dependence of the hyperfine struc- !"#$%(' ture in the Zemach radius, we have performed a series of calculations for the different nuclear models we have !"#$%()' presented in Sec. 2. The calculation are done for different !"#$%&' pairs of R and RM, such that RZ remains constant. We can "#"""""""' "#(""""""' "#+""""""' "#%""""""' "#,""""""' $#"""""""' $#(""""""' $#+""""""' $#%""""""' then study the dependence of the HFS beyond the first or- +# der corresponding to the Zemach correction. We calculate Fig. 10. Finite charge and magnetic moment distribution energy E (R , R) = E (R)+EBW (R ) E (0). We also eval- shift for the 2s state as a function of the charge R and Zemach HFS Z HFS HFS M − HFS uate with and without the Uehling¨ potential to obtain all- radius RZ for the Gaussian ( RZ = 1.045fm) and exponential order Uehling¨ contribution to the HFS energy. Values for model, divided by RZ. The lines correspond to the function in the 2s level and RZ = 1.045fm for two different type of dis- Eq. (90). tributions are presented in Table 5. We calculatedTrento, October 2012 the cor- 45 rection EHFS (RZ, R) for several value of RZ between 0.4 fm and 1.2 fm. The correction, divided by RZ, is plotted on The effect of the vacuum polarization on the 2s hy- Fig. 10. The figure shows clearly that the correction to the perfine structure energy shift as a function of the Zemach HFS energy due to charge and magnetic moment distri- and charge radius have been calculated for the same set bution is not quite independent of R as one would expect of values as the main contribution. A set of value can be from Eq. 69, in which the finite size contribution depends found in the last column of Table 5 for RZ = 1.045fm. The only on RZ. For each value of RZ, the EHFS (RZ, R) can be data can be described as a function of RZ and R as 2 approximated by RZ (a (RZ) + b (RZ)) R . We checked that a (RZ) is a linear function of Z and RZ a quadratic function 2s,VP 2 2 EHFS (RZ, R) = 0.000025339R + 0.000154707RZ of RZ. We thus fitted EHFS (RZ, R) /RZ by a function of R − (92) 0.00203434RZ + 0.0744207meV. and RZ, which gives: − 2s ( , ) It correspond to the diagrams presented in Fig. 12. The EHFS RZ R 2 2 2 = 0.00168747R RZ + 0.00483539R RZ size-independent term 0.07442 meV corresponds to the RZ − 2 sum of the two contributions represented by the two top 0.00464802R + 0.000899461RRZ (90) HFS − diagrams in Fig. 12) and correspond to ∆E1loop-after-loop VP = 2 0.00166526R 0.000562326RZ HFS − − 0.0746 meV in Ref. [35]. The term ∆E1γ,VP = 0.0481 meV 1 + 0.00160609RZ 0.16045 meV fm− . correspond to a vacuum polarization loop in the HFS po- − tential [82,30,35]. Martynenko [35] evaluate this correction “Proton struc- A summary of all known contributions from previous ture corrections of order α5 and α6 ” to be 0.1535 meV, − work and the present one as presented in Table 6 for following [30]. In Ref. [30], The charge and magnetic mo- the part independent of charge and magnetic moment ment distributions are written down in the dipole form, distribution, and in Table for the part depending on the which corresponds to (14), charge and Zemach’s radiii.
2 GM q 4 2 Λ GE q = = 2 , (91) 1 +κp Λ2 + q2 6.3 Hyperfine structure of the 2p level j with Λ = 848.5MeV. This leads toR = 0.806 fm as in Ref. [2] and RZ = 1.017 fm using this definition for the form Calculations of the 2p level hyperfine structure for muonic factor in Eq. (18). Moreover there are recoil corrections hydrogen have been provided in [30,34]. More recently included. Pachucki [30] finds that the pure Zemach con- Martynenko has made a detailed calculation [84] to the tribution (in the limit mp ) is 0.183 meV. In Ref. same level of accuracy as in Ref. [35] for the 2s level. →∞ − [86], the Zemach corrections is given as δ(Zemach) EF = We have used the same technique as in the previous 4 × 71.80 10− EF, for RZ = 1.022 fm. This leads to a coeffi- section, to evaluate the effect of the charge and magnetic − × 1 cient 0.1602 meVfm− , in excellent agreement with our moment distribution on the 2p1/2 and 2p3/2 states. The − 1 value 0.16045 meVfm− . results for the former are presented in Fig. 13. The fitted − 13 Hyperfine structure FS corrections Ref. [95] corrected for a misprint): function of R and RZ, which gives:
2s EHFS (RZ, R) = 22.807995 0.0022324349R2 + 0.00072910794R3 Z 2 2 − ∆EHFS = Sp Sµ φC(0) 4 −3 · 0.000065912957R 0.16034434RZ − − 0.00057179529RRZ 1 2αmµ ρ(u) u r µ(r)dudr , − × − | − | 0.00069518048R2R − Z 3 = EF 1 2αm ρ(u) u r µ(r)dudr , 0.00018463878R RZ − µ | − | − (100) + 0.0010566454R2 (97) Z 2 + 0.00096830453RRZ + 0.00037883473R2R2 Z 16 3 where EF is the well known HFS Fermi energy. Trans- 0.00048210961RZ − Using the results presented above we get forming Eq. (97) using r r + u, etc. Zemach obtains 0.00041573690RR3 → Z F=2 F=1 2 − E E (RZ, R) = 209.9465616 5.2273353R + 0.00018238754R4 meV. 2p3/2 − 2s1/2 − Z + 0.036783284R3 Z 0.0011425674R4 ∆EHFS = EF 1 2αmµ rZ , (98) The constant term should be close to the sum of the Fermi − − + 0.00044096683R5 energy 22.80541 meV and of the Breit term [96]. the HFS 0.000066623155R6 correction calculated with a point-nucleus Dirac wave- − + 0.040533092R function for which I find 22.807995 meV. When setting Z with rZ given in Eq. (16). The 2s state Fermi energy is + 0.00018596008RRZ the speed of light to infinity in the program I recover 2 given by + 0.00026754376R RZ (115) exactly the Fermi energy. The Breit contribution is thus 3 + 0.000063748539R RZ 0.002595 meV, to be compared to 0.0026 meV in Ref. [40] 2 0.00020892783R (Table II, line 3) and 0.00258 meV in Ref. [70]. Mar- − Z 4 3 0.00032967277RR2 (Zα) µ − Z 2s r tynenko [40] evaluates this correction, which he names 2 2 EF = gp . (99) 5 6 0.00014609447R RZ 3 mpmµ “Proton structure corrections of order α and α ”, to be − + 0.000057775798R3 Trento, October 2012 46 Z 0.1535 meV, following [35]. He finds the coefficient 3 − 1 FIG. 12. Feynman diagrams corresponding to the evaluation + 0.00014693531RRZ for the Zemach’s radius to be 0.16018 meVf− , inof the very hyperfine structure using wavefunctions obtained with 4 − 0.000030280142RZ meV. good agreement with the present all-order calculationthe Uehling potential in the Dirac equation. The grey squares − 1 correspond1 to the hyperfine interaction. 0.16034 meVf− . Borie’s value [70] 0.16037 meVf− is This can be compared with the result from even− closer. The difference between− Borie’s value and U. Jentschura [101] Eq. (100) is represented in Fig. 11 as a functionand of Carroll the et al. [100] ∆EJents.,F=2 ∆EJents.,F=1 = 209.9974(48) 2p3/2 − 2s1/2 (116) A. Hyperfine structure of the 2s level charge and Zemach radii. The maximum difference is 5.2262R2 meV, ECar.,fs ECar.,fs(R) = 206.0604 5.2794R2 − around 1 µeV. 2p1/2 − 2s1/2 − (111) + 0.0546R3 meV. using the 2s hyperfine structure of Ref. [40]. In Ref. [35], the charge and magnetic moment dis- For the other transition I obtain
In order to check the dependence of the hyperfine tributions are written down in the dipole form, which F=1 F=0 2 E E (RZ, R) = 229.6634776 5.2294935R structure on the Zemach radius and on the proton fi- corresponds to (13), 2p3/2 − 2s1/2 − + 0.037645166R3 nite size, I have performed a series of calculations for B. Transitions between hyperfine sublevels 0.0012165791R4 a dipolar distribution for both the charge and magnetic 2 − GM q 4 5 2 Λ + 0.00044096683R moment distribution. We can then study the dependence GE q = = , The(101) energies of the two transitions observed experi- 1 + κ 2 2 2 mentally in muonic hydrogen are given by 0.000066623155R6 of the HFS beyond the first order corresponding to the p Λ + q − 0.12159928RZ Zemach correction. I calculated the hyperfine energy F=2 F=1 − E2p E2s = E2p1/2 E2s1/2 + E2p3/2 E2p1/2 0.00055788025RR splitting ∆ ( ) = ( ) + BW ( ) numer- with Λ = 848.5MeV. This leads to R = 0.806 fm as in3/2 − 1/2 − − − Z EHFS RZ, R EHFS R EHFS R, RM (112) 2 3 2p3/2 1 2s 0.00080263129R RZ (117) Ref. [2] and RZ = 1.017 fm using this definition for + EHFS EHFS, − ically. I also evaluate with and without self-consistent 8 4 3 − 0.00019124562R R inclusion of the Uëhling potential in the calculation, to the form factor in Eq. (17). Moreover there are re- − Z + 0.00062678350R2 obtain all-order Uëhling contribution to the HFS energy. coil corrections included. Pachucki [35] finds thatand the Z 2 F=1 F=0 + 0.00098901832RRZ We calculated the correction ∆EHFS (RZ, R) for several pure Zemach contribution (in the limit mp )E is E = E2p E2s + E2p E2p →∞ 2p3/2 − 2s1/2 1/2 − 1/2 3/2 − 1/2 + 0.00043828342R2R2 value of RZ between 0.8 fm and 1.15 fm, and proton sizes 0.183 meV. In Ref. [97], the Zemach corrections is given 5 3 (113) Z 2p3/2 2s F=1 3 − 4 E + EHFS + δEHFS. 0.00017332740R ranging from 0.3 fm to 1.2 fm, by steps of 0.05 fm, which as δ(Zemach) EF = 71.80 10− EF, for RZ = 1.022 fm. − 8 HFS 4 − Z × − × 1 0.00044080593 3 This leads to a coefficient 0.1602 meVfm , in excellent RRZ represents 285 values. The results show that the cor- − Here we use the results from [99] for the 2p states: − − 1 + 0.000090840426R4 meV. rection to the HFS energy due to charge and magnetic agreement with our value 0.16036 meVfm− . Z
− 2p1/2 moment distribution is not quite independent of R as The effect of the vacuum polarization on the 2s hyper-EHFS = 7.964364 meV Using Eq. (115), the Zemach radius from Ref. [40] 2 and the transition energy from Ref. [12], I obtain a one would expect from Eq. (98), in which the finite size fine structure energy shift as a function of the ZemachE p3/2 = 3.392588 meV (114) HFS charge radius for the proton of 0.85248(45) fm in place contribution depends only on RZ. We fitted the hyper- and charge radius have been calculated for the same setδEF=1 = 0.14456 meV. of 0.84184(69) fm in Ref. [12] and 0.8775(51) fm in the 2s HFS fine structure splitting of the 2s level, EHFS (RZ, R) by a of values as the main contribution. The data can be de- Hyperfine structure FS corrections
Would need to be evaluated as well
Trento, October 2012 47 Hyperfine structure •Difference between Zemach’s calculation and all-order inclusion of charge and magnetic moment distribution (meV) 1.4 0.001 1.2 0.001
1.0 fm 0.8 0.0005 radius
0.6 0.0005 emch Zeam 0.4
0.2
0 0.0015 0.0 0.0 0.2 0.4 0.6 0.8 1.0 charge radius fm
Trento, October 2012 48 Zemach radius
Borie 22.8140
Hydrogen (Volotka et al. 2005) 1.045 (16)
Trento, October 2012 49 Results HFS
Trento, October 2012 50 Proton polarization 14 P. Indelicato, P.J. Mohr: Non-perturbative evaluation of muonic H Lamb-shift
14 P. Indelicato, P.J. Mohr: Non-perturbative evaluation of muonic H Lamb-shift Table 4. Finite size effect on the muon self-energy. Values of FNS(R, Zα) together with numerical accuracy.
Z Table 4. FiniteR = size0.6 fm effect on the muon self-energy.R = 0 Values.875 fm of FNS(R, Zα) together withR = numerical1.25 fm accuracy. 20 3.4200 0.0002 3.3424 0.0001 3.2160P. Indelicato, P.J. 0.0001 Mohr: Non-perturbative evaluation of muonic H Lamb-shift 15 25Z 3.0137R = 0.6 fm 0.0001 2.9201R = 0.875 fm 0.0001 2.7727R = 1.25 fm 0.0001 3020 2.7056 3.4200 0.0001 0.0002 2.59683.3424 0.0001 0.0001 2.43153.2160 0.0001 0.0001 "#"""& 3525 2.4631 3.0137 0.0001 0.0001 2.33992.9201 0.0001-1.2 0.0001 2.15922.7727 0.0001 0.0001 4030 2.26717 2.7056 0.00006 0.0001 2.130032.5968 0.00005 0.0001 1.936162.4315 0.00004 0.0001 4535 2.10558 2.4631 0.00003 0.0001 1.954982.3399 0.00003 0.0001 1.749862.1592 0.00002 0.0001 !"#""%& 5040 1.97006 2.26717 0.00004 0.00006 1.806402.13003 0.00004-1.4 0.00005 1.591701.93616 0.00004 0.00004 5545 1.85474 2.10558 0.00003 0.00003 1.678421.95498 0.00003 0.00003 1.455651.74986 0.00003 0.00002 Z=1 !"#"'"& 6050 1.75537 1.97006 0.00001 0.00004 1.566781.80640 0.00001 0.00004 1.337321.59170 0.00001 0.00004 LO 55 1.85474 0.00003 1.67842 0.00003 1.45565 0.00003 65 1.66873 0.00002 1.46832 0.00001-1.6 1.23342 0.00001 (:3<36& 7060 1.59235 1.75537 0.00002 0.00001 1.380611.56678 0.00002 0.00001 1.141441.33732 0.00002 0.00001 !"#"'%& @38#& Z=1 7565 1.52430 1.66873 0.00002 0.00002 1.301791.46832 0.00002 0.00001 1.059431.23342 0.00002 0.00001 !"#$#%&'#()&*+,-.& A5.B+<59& CD5:#& 70 1.59235 0.00002 1.38061-1.8 0.00002 1.14144 0.00002 80 1.46301 0.00001 1.23037 0.00001 0.98584 0.00001 !E& 8575 1.407206 1.52430 0.000006 0.00002 1.1651721.30179 0.000006 0.00002 0.9194461.05943 0.000007 0.00002 !"#"$"& 9080 1.355868 1.46301 0.000002 0.00001 1.1052621.23037 0.000002 0.00001 0.8592360.98584 0.000003 0.00001 85 1.407206 0.000006 1.165172-2.0 0.000006 0.919446 0.000007 Z R = 1.5 fm R = 2.130 fm R = 0 fm !"#"$%& 2090 3.12370 1.355868 0.00004 0.000002 2.879761.105262 0.00016 0.000002 3.5066480.859236 0.000003 ()*+,*-.&/'0001& 2345675895:&/'0001&;)<:=656-3&/$"">1&?):8436&/$"''1& 25Z 2.66826R = 1.5 fm 0.00007 2.40251R = 2.130 fm 0.00007 R3.122959= 0 fm 3020 2.31779 3.12370 0.00009 0.00004 2.039032.87976 0.00009-2.2 0.00016 2.8388393.506648 0.50 1.0 1.5 2.0 2.5 3525 2.03861 2.66826 0.00009 0.00007 1.753292.40251 0.00012 0.00007 2.6223363.122959 4030 1.81063 2.31779 0.00004 0.00009 1.523482.03903 0.00005 0.00009 2.4548292.838839R Fig. 20. Plot of the proton polarization for the 2s level [Eqs. (91) 35 2.03861 0.00009 1.75329 0.00012 2.622336 45 1.62090 0.00002 1.33529Fig. 17. 0.00004(F (R, α) F(α2.))324690/(αR2) as a function of nuclear size, to (95)] 40 1.81063 0.00004 1.52348 0.00005NS 2.454829 50 1.46059 0.00004 1.17897obtained 0.00005 by extrapolation− 2.224337 of the results plotted in Fig. 15, com- 5545 1.32344 1.62090 0.00003 0.00002 1.047561.33529 0.00004 0.00004 2.1487272.324690lo 2 pared to the low-order result FNS(R, α)/(αR ) (86) 6050 1.20487 1.46059 0.00001 0.00004 0.935961.17897 0.00002 0.00005 2.0945182.224337 P. Indelicato, P.J.in Mohr: reasonable Non-perturbative agreement evaluation of muonic with H earlierLamb-shift work cited above. 15 6555 1.101443 1.32344 0.000004 0.00003 0.8403361.04756 0.000006 0.00004 2.0596112.148727 Carlson and Vanderhaeghen [100] provide a new value, "#"""& 7060 1.01053 1.20487 0.00001 0.00001 0.7577710.93596 0.000004 0.00002 2.0428912.094518 -1.2 which is the sum of an elastic, inelastic and subtraction 7565 0.93008 1.101443 0.00001 0.000004 0.6859930.840336 0.000008 0.000006 2.0441152.059611 terms 70 1.01053 0.00001 0.757771 0.000004 2.042891 !"#""%& 80 0.85847 0.00001 0.623211 0.000007 2.063906 -1.4 p.pol 75 0.93008 0.00001 0.685993 0.000008 2.044115 ∆E = ∆Esubt. + ∆Einel. + ∆Eel. 85 0.794378 0.000008 0.567998 0.000005 2.103876 2s Z=1 !"#"'"& LO 9080 0.736752 0.85847 0.000003 0.00001 0.5192340.623211 0.000094 0.000007 2.1668832.063906 -1.6 = 0.0053(19) 0.0127(5) 0.0295(13) meV(:3<36& 85 0.794378 0.000008 0.567998 0.000005 2.103876 !"#"'%& − − @38#& Z=1 = 0.!"#$#%&'#()&*+,-.& 0074(20) 0.0295(13) meV. A5.B+<59&(95) CD5:#& 90 0.736752 0.000003 0.519234 0.000094 2.166883 -1.8 − − !E& !"#"$"& P. Indelicato, P.J. Mohr: Non-perturbative evaluation of muonic H Lamb-shiftThe elastic 15 contribution corresponds to the contribution -2.0 as a cut-off parameter: Martynenko and Faustov [95,96,97], using experimental to the diagrams in!"#"$%& Fig. 18 that is identical to the one from Fig. 18. Feynman+ diagrams corresponding to the proton self- ()*+,*-.&/'0001& 2345675895:&/'0001&;)<:=656-3&/$"">1&?):8436&/$"''1& data from e + e− hadrons collisions. In Ref. [92], the the relativistic recoil in Fig. 5 and Eq. (44), but which as a cut-off parameter:2 4 3 2 Martynenkoenergy (88). The and→ heavy Faustovan inelastic double [95,96,97], line and represents subtraction using the terms. experimental proton He wave obtained prot. 4Z α(Zα) µr c 1 -1.2 Λ 11 resulting correction is given for the 2s state-2.2 as contains proton structure terms not present in the rela- function or propagator.+ The other symbols are0.50 explained1.0 in1.5 Fig. 2.0 2.5 ∆ESE (n, l) = 3 2 ln 2 + data from e + e− hadrons collisions.p.pol Insubt. Ref. [92],inel. the ∆ = ∆ + ∆ R tivistic recoil. It is in fact the Zemach moment (24) contri- πn2 M4 p 3 23 µr(Zα) 72 6 Hadronic→ E2sµ E E Fig. 20. Plot of the proton polarization for the 2s level [Eqs. (91) prot. 4Z α(Zα) µrc 1 Λ P. Indelicato,11 P.J. Mohr:resulting Non-perturbative correction is evaluation given for of the muonic 2s state H Lamb-shiftas 15 E = 0.671(15)E . (90)2 bution to the Lambto (95)] shift, but is much larger than the value 2 VP,2s VPFig.,2= 17.s 0(.F0023NS(R, α) 0F.(01613α))/(αR meV) as a function of nuclear size, ∆ESE (n, l) = 3 22 -1.4ln 2 2 + − 1πn 7π ΛM 23 Λµr(Zα) 72 obtained by extrapolation− of the results plotted0 0232(10) in Fig. 15, com- meV provided in Ref. [34] Eq. (25), using the p Hadronic µ lo 2 . Z=1 pared= to the0.0138(29) low-order result meVF, (R, α)/(αR ) (86) (94) + + + δl,0 E = 0.671(15)E . (90)NS − −24 − 32 2 3 2 2··· Combined withLO the muonicVP,2s "#"""& vacuum polarizationVP−,2s (79), this proton form factorin reasonable parametrization agreement with in earlier Ref. work [2], cited which above. cor- 4M-1.2p 2 4Mp 2 Carlson and Vanderhaeghen [100] provide a new value, 1 7π Λ -1.62 Λ gives 0.01121(25) meV,in while reasonable it is calculated agreement as with 0.01077(38) earlier work meV citedresponds above. to a proton size of 0.862 fm. The Carlson and + + + δl,0 which is the sum of an elastic, inelastic and subtraction 1 24 32 2 3 2 in Ref.Combined [97]. Here with we the takeCarlson muonic then and vacuum the Vanderhaeghen value polarization of Ref. [101] [92] (79), with provide this a new value, terms − − 4Z=1 M 4M ··· Vanderhaeghen proton polarization value is somewhat ln ( , ) . p p (89) !"#""%& k0 n l angives enlarged 0.01121(25) error bar. meV,which while is the it sum is calculated of an elastic, as 0. inelastic01077(38) and meV subtraction p.pol subt. inel. el. −3 -1.8 lower than previous∆E2 works= ∆E had+ ∆E provided.+ ∆E All the results -1.4 in Ref. [97]. Here weterms take then the value of Ref. [92] with s 1 with provided uncertainties= 0.0053(19) are0 plotted.0127(5) 0 in.0295(13) Fig. meV 20. The ln k0(n, l) . (89) − − an enlarged errorZ=1 bar. !"#"'"&p.pol subt. inel. el. −3 weighted average = 0.0074(20) 0.0295(13) meV. (95) For the 2s level, for which the two expressions differ, Eq. LO ∆E2s = ∆E + ∆E + ∆E − − prot. -2.0 7.3Fig. Proton 19. Feynman polarization diagrams corresponding to the proton polar- -1.6 = 0.0053(19) 0.0127(5) 0.0295(13) meV The elastic contribution corresponds to the contribution (88) gives ∆ESE (n, l) = 0.0108 meV, while Eq. (89) gives (:3<36& to thep. diagramspol in Fig. 18 that is identical to the one from For the 2s level, for which the two expressions differ, Eq. ization [Eqs. (91) to (95)]. The blob representsFig. 18. Feynman− the diagrams excitation− corresponding of to the proton self- ∆E = 0.0129(36) meV (96) prot. !"#"'%& = 0.0074(20) 0.0295(13) meV. (95) @38#& the relativistic2s recoil in Fig. 5 and Eq. (44), but which ∆E (n, l) = 0.0098prot. meV.Z=1 We retain the first expression 7.3 Proton polarization energy (88). The heavy double line represents the proton wave − SE Theinternal proton degrees polarization of freedom!"#$#%&'#()&*+,-.& correction of the− proton. to the LambThe− other shift symbols in A5.B+<59&contains proton structure terms not present in the rela- (88) gives ∆ESE (n, l) = 0.0108 meV,-2.2 while Eq. (89) gives function or propagator. The other symbols are explained in Fig. and assign the difference as uncertainty.0.50 1.0 1.5 2.0 2.5 where theCD5:#& errortivistic is set recoil. to It encompass is in fact the Zemach the moment error bars(24) contri- of all prot. -1.8 muonicare explained hydrogen in Fig. hasProtonThe 5. been elastic calculated contributionpolarization6 by several corresponds authors to the contribution !E& bution to the Lamb shift, but is much larger than the value ∆ESE (n, l) = 0.0098 meV. We retain the first expression The proton polarization!"#"$"& correction to the Lamb shift in calculations. This is the value we retain for our final table. R [32,98,34,96,97,99]. Itto is the represented diagrams in by Fig. the 18 Feyman that is identical dia- to the one from 0.0232(10) meV provided in Ref. [34] Eq. (25), using the and assign the difference as uncertainty. muonic hydrogen has been calculated by several authors − 2 grams in Fig. 19. Rosenfelderthe relativistic [98] Eq. recoil (16) in Fig. finds 5 and Eq. (44), but whichHigher con- ordersproton polarization form factor parametrization corrections in provided Ref. [2], which in cor- [97] Fig. 17. (FNS(R, α) F(α))•/(αRSeveral) as a function calculations of nuclear size, 7.2 Hadronic vacuum polarization-2.0 − [32,98,34,96,97,99].while Martynenkotains It and is proton Faustovrepresented structure [96] by provides terms the not Feyman present dia- in the relativisticare negligible. responds to a proton size of 0.862 fm. The Carlson and obtained by extrapolation of the results plotted in Fig. 15, com- !"#"$%& Vanderhaeghen proton polarization value is somewhat lo grams2 p.pol in Fig.136 19. Rosenfelder30recoil. It is in [98] fact()*+,*-.&/'0001& Eq. the (16) Zemach2345675895:&/'0001& finds moment;)<:=656-3&/$"">1& (24) contribution?):8436&/$"''1& pared to the low-order result FNS– (RRosenfelder, α)/(αR ) (86) (1999) lower than previous works had provided. All the results The7.2 hadronic Hadronic contributions vacuum polarization comes from both vacuum po- ∆E2s = p.pol±3 toµ the0eV.092 Lamb= 0. shift,017 but0.004 is somewhat meV. larger(91) than the value with provided uncertainties are plotted in Fig. 20. The −∆E n = meV− = ±0.0115 meV, (93) p.pol 2s136 030.0232(10)3 meV provided in Ref. [34] Eq. (25), using the weighted average larization from virtual pions-2.2 and other resonances like −− n Fig.− 19. Feynman diagrams corresponding7.4 to the Hyperfine proton polar- structure 0.50 1.0 1.5 2.0 ∆E 2.5 = ±protonµeV form= factor0.017 parametrization0.004 meV. in Ref.(91) [2], which cor- The hadronic contributions comes from both vacuum po-Pachucki2s [34] provides3 an independentization value [Eqs. (91) to (95)]. The blob represents the excitation of p.pol ω and ρ. The hadronic polarization correction has been − n − ± ∆E2s = 0.0129(36) meV (96) larization from virtual pions and other resonances like without giving errorresponds bars. More to a proton recentlyinternal size degrees Martynenko of of 0. freedom862 fm. of [99]The the proton. CarlsonIn The order other and symbols to compare with experiment,− one has to combine evaluated for hydrogen [91,92], for muonic hydrogenR by where the error is set to encompass the error bars of all p.pol VanderhaeghenFig. 20. Plot of the protonare explained proton polarization in polarization Fig. 5. value for is the somewhat 2s level [Eqs. (91) ω and ρ. The hadronic polarization correction has– beenPachuckiPachuckireevaluated (1999) [34] the provides proton an polarization, independent given value as the sum of the calculationscalculations. of the This Lamb is the shiftvalue we with retain for the our fine final table. struc- Borie [93,94] and more recently by Friar and coll. [92]2 and ∆E2s =lessto0 (95)] negative.012 0 than.002 meV previous, works had(92) provided. All the evaluated for hydrogenFig. 17. [91,92],(FNS(R for, α) muonicF(α))/ hydrogen(αR ) as a function by an of inelastic nuclear and size, subtraction− ± terms. He obtained ture (provided inHigher the orders next polarization section) corrections and 2s and provided 2p inhyper- [97] − p.polresults with providedwhile uncertaintiesMartynenko and areFaustov plotted [96] provides in Fig. 20. are negligible. Borie [93,94] and moreobtained recently by extrapolation by Friar and of coll. the [92] results and plotted in Fig. 15, com-∆E = 0.012 0.002 meV, (92) fine structure (HFS). The line measured in Ref. [12] is lo 2 p2.spolThe weightedsubt. averageinel. pared to the low-order result F (R, α)/(αR ) (86) − ± p.pol 0.092 NS – Martynenko (2006) ∆E2s in= reasonable∆E + ∆E agreement∆E = with earliermeV = work0.0115described meV cited, above.by(93) p.pol 2s − n3 − 7.4 Hyperfine structure Carlson= 0.0023 and0∆ Vanderhaeghen.01613E2s = meV0.0129(36) [100] meV provide a (96) new value, − without giving− error bars. More recently Martynenko [99] In order to compare with experiment,3 2p3/2 one1 has to2 combines Fig. 18. Feynman diagrams corresponding to the proton self- which is the sum of an elastic, inelastic andE 5 subtractionE 3 = ∆E + ∆E + ∆E ∆E , (97) where= 0 the.0138(29) error isreevaluated meV set to, encompass the proton polarization, the(94) error given bars asP of3/ the2 all sumS of1/2 the calculationsLS ofFS the Lamb shiftHFS with the fineHFS struc- energy (88). The heavy double line represents the proton wave terms− an inelastic and subtraction terms. He obtained − ture (provided in the next8 section) and− 24s and 2p hyper- calculations. In Ref. [91], it is argued that the two-photon fine structure (HFS). The line measured in Ref. [12] is function or propagator. The other symbols are explained in Fig. p.pol correctionp.pol not only includes∆E the elastic= ∆Esubt. relativistic+ ∆Einel. recoil 6 – Carlson and Vanderhaeghen (2011) ∆E = ∆Esubt. + ∆E2sinel. + ∆Eel. described by term ∆Eel2.s and the polarization term= 0.0023 but0 also.01613 the meV finite − 3 2p3/2 1 2s E 5 E 3 = ∆E + ∆E + ∆E ∆E , (97) size correction= to0 the.0053(19) relativistic0.0127(5) recoil.= 0.0138(29) The0 correctionmeV.0295(13), is meV(94) P3/2 S1/2 LS FS HFS HFS − − − − 8 − 4 written as = 0.0074(20) 0.0295(13) meV. (95) p.pol − − ∆E = ∆Esubt. + ∆Einel. – Hill and Paz (2011+DPF 2011) The elastic2s contribution corresponds to the contribution 2 to the diagrams= δEW1 in(0,Q Fig.) + δ 18Eproton that pole is identical+ δEcontinuum to the one from Fig. 18. Feynman diagrams corresponding to the proton self- the relativistic recoil2 in Fig. 5 and Eq. (44), but which energy (88). The heavy double lineCould represents be wrong by the 0.04 proton meV wave W1(0,Q ) contains proton= δE structure+ 0016 terms0.0127(5) not present meV, (97) in the rela- functionFig. or 19. propagator.Feynman diagrams The other corresponding symbols are to explained the proton in polar- Fig. − tivistic recoil. It is in fact the Zemach moment (24) contri- 6 ization [Eqs. (91) to (95)]. The blob represents the excitation of where, using definition from previous authors ∆Esubt. = internal degrees of freedomEMMI days 2012 of the proton. The other symbols bution2 to the Lamb shift, but is much larger than the52 value W1(0,Q ) proton pole inel. continuum are explained in Fig. 5. δE 0.0232(10)+ δE meV providedand ∆E in Ref.= δE [34] Eq.. (25), Using using the the− same model of form factor than Ref. [32], Hill and Paz proton form factor2 parametrization in Ref. [2], which cor- obtains δEW1(0,Q ) = 0.034 meV, reproducing ∆Esubt. = responds to a proton− size of 0.862 fm. The Carlson and Pachucki [34] provides an independent value 0.018 meV from [32]. However they claim that the model −Vanderhaeghen proton polarization2 value is somewhat used for the subtraction function W1 0, Q does not have p.pol lower than previous works had provided. All the results ∆E = 0.012 0.002 meV, (92) 2 2s the correct behavior at small and large Q , that the values − ± with provided uncertainties2 are plotted in Fig. 20. The forweighted intermediate averageQ are not constrained by experiment, while Martynenko and Faustov [97] provides and that an error bar as large as 0.04 meV should be used, Fig. 19. Feynman diagrams corresponding to the proton polar- ten times larger than the dispersion between the different ization [Eqs. (91) to (95)]. The blob represents the excitation of p.pol p.pol 0.092 calculations. We thus∆E2 retains = the0. value0129(36) from meV Eq. (96) for (96) internal degrees∆ ofE freedom= of themeV proton.= 0.0115 The meV other, symbols(93) − 2s − n3 − our final table, with an error bar increased to 0.04 meV. are explained in Fig. 5. Thiswhere correction the error represents is set to then encompass overwhelmingly the error domi- bars of all without giving error bars. More recently Martynenko [100] nantcalculations. source of uncertainty. This is the valueHigher we orders retain polarization for our final table. reevaluated the proton polarization, given as the sum of correctionsHigher provided orders polarization in [98] are negligible. corrections provided in [97] while Martynenko and Faustov [96] provides are negligible. p.pol 0.092 ∆E = meV = 0.0115 meV, (93) 2s − n3 − 7.4 Hyperfine structure without giving error bars. More recently Martynenko [99] In order to compare with experiment, one has to combine reevaluated the proton polarization, given as the sum of the calculations of the Lamb shift with the fine struc- an inelastic and subtraction terms. He obtained ture (provided in the next section) and 2s and 2p hyper- fine structure (HFS). The line measured in Ref. [12] is p.pol subt. inel. ∆E2s = ∆E + ∆E described by = 0.0023 0.01613 meV − 3 2p3/2 1 2s 5 3 = 0.0138(29) meV, (94) E P3/2 E S1/2 = ∆ELS + ∆EFS + ∆E ∆EHFS, (97) − − 8 HFS − 4 P. Indelicato, P.J. Mohr: Non-perturbative evaluation of muonic H Lamb-shift 15
"#"""& -1.2
!"#""%& -1.4
Z=1 !"#"'"& LO
-1.6 (:3<36& !"#"'%& @38#& Z=1
!"#$#%&'#()&*+,-.& A5.B+<59& CD5:#& -1.8 !E& !"#"$"&
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-2.2 0.50 1.0 1.5 2.0 2.5
R Fig. 20. Plot of the proton polarization for the 2s level [Eqs. (91) 2 to (95)] Fig. 17. (FNS(R, α) F(α))/(αR ) as a function of nuclear size, obtained by extrapolation− of the results plotted in Fig. 15, com- pared to the low-order result Flo (R, α)/(αR2) (86) NS in reasonable agreement with earlier work cited above. Carlson and Vanderhaeghen [100] provide a new value, which is the sum of an elastic, inelastic and subtraction terms p.pol subt. inel. el. ∆E2s = ∆E + ∆E + ∆E = 0.0053(19) 0.0127(5) 0.0295(13) meV − − = 0.0074(20) 0.0295(13) meV. (95) − − The elastic contribution corresponds to the contribution to the diagrams in Fig. 18 that is identical to the one from Fig. 18. Feynman diagrams corresponding to the proton self- the relativistic recoil in Fig. 5 and Eq. (44), but which energy (88). The heavy double line represents the proton wave contains proton structure terms not present in the rela- function or propagator. The other symbols are explained in Fig. Proton polarization6 tivistic recoil. It is in fact the Zemach moment (24) contri- bution to the Lamb shift, but is much larger than the value 0.0232(10) meV provided in Ref. [34] Eq. (25), using the proton− form factor parametrization in Ref. [2], which cor- responds to a proton size of 0.862 fm. The Carlson and Vanderhaeghen proton polarization value is somewhat lower than previous works had provided. All the results with provided uncertainties are plotted in Fig. 20. The weighted average Fig. 19. Feynman diagrams corresponding to the proton polar- ization [Eqs. (91) to (95)]. The blob represents the excitation of p.pol "#"""& ∆E2s = 0.0129(36) meV (96) internal degrees of freedom of the proton. The other symbols − are explained in Fig. 5. where the error is set to encompass the error bars of all calculations. This is the value we retain for our final table. !"#""%& Higher orders polarization corrections provided in [97] while Martynenko and Faustov [96] provides are negligible.
p.pol 0.092 ∆E = meV = 0.0115 meV, (93) !"#"'"& 2s (:3<36&− n3 − 7.4 Hyperfine structure C38#& without giving error bars. More recently Martynenko [99] In order to compare with experiment, one has to combine D5.E+<59& reevaluated the proton polarization, given as the sum of the calculations of the Lamb shift with the fine struc- !"#"'%& an inelastic andF@5:#& subtraction terms. He obtained ture (provided in the next section) and 2s and 2p hyper- !G& !"#$#%&'#()&*+,-.& fine structure (HFS). The line measured in Ref. [12] is p.pol subt. inel. ∆E2s = ∆E + ∆E described by HG& = 0.0023 0.01613 meV !"#"$"& − 3 2p3/2 1 2s 5 3 = 0.0138(29) meV, (94) E P3/2 E S1/2 = ∆ELS + ∆EFS + ∆E ∆EHFS, (97) − − 8 HFS − 4
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Trento, October 2012 53 15
TABLE II. Comparison of contributions to the 2s hyperfine structure from Refs. [40, 70] and the present work (meV) for R = 0.84184 fm and RZ = 0.9477 fm
Ref. [40] Ref. [70] This work Fermi energy 22.8054 22.8054 Dirac energy (includes Breit corr.) Eq. 22.807995 Vacuum polarization corrections of orders α5, α6 in the second-order perturbation theory 0.0746 0.07443 All order vacuum polarization correction (102) 0.07261 finite extent of magnetization density correction to the above 0.00114 Proton structure corr. of order α5 0.1518 −0.15198 0.15280 Proton structure corrections of order α6 −0.0017 − − Electron vacuum polarization contribution+ − proton structure corrections of order α6 0.0026 Two-loop electron vacuum polarization − contribution of 1γ interaction of order α6 0.0003 0.00037 0.00037 VP2EF (* neglected by Martynenko (ref63)) 0.00056 0.00056 muonVP (part corresponding to epsVP2 neglected by Martynenko (ref63)) 0.00091 0.00091 Hadron VP 0.0006 0.0006 Vertex(order α5) 0.00311 0.00311 Vertex(order α6), only part with powers of ln(α) - see ref 55 −0.00017 −0.00017 Breit correction −0.00258 − Muon anomalous magnetic moment correction of order α5, α6 0.0266 0.02659 0.02659 Relativistic and radiative recoil corrections 0.0026 Relativistic and radiative recoil corrections with the account proton AMM of order α6 0.0018 One-loop electron vacuum polarization contribution of 1γ interaction of orders α5, α6 0.0482 0.04818 0.04818 finite extent of magnetization density correction to the above 0.00114 One-loop muon vacuum polarization contribution of 1γ interaction of order α6 0.0004− 0.00037 0.00037 Muon self energy+proton structure correction of order α6 0.001 6 Vertex corrections+proton structure corrections of order α 0.0018 16 ?Jellyfish? diagram correction+ proton structure corrections of order α6 −0.0005 HVP contribution of order α6 0.0005 Proton polarizability contribution of order α5 0.0105 Using the results presented above we get Proton polarization (ref66) 0.00801 0.00801 Weak interaction contribution 0.0003 0.00027 0.00027 EF=2 EF=1 (R , R) = 209.9465616 5.2273353R2 2p3/2 2s1/2 Z Total 22.8148 22.8107 22.8104 − − + 0.036783284R3 0.0011425674R4 − + 0.00044096683R5 In the same way, Eqs. (43), (51), (54), 57, (59), (63), (72) contained in Table III, I obtain the final 2s 2p1/2 energy: and (82) lead to the fine structure interval − 0.000066623155R6 − Tot,fs Tot,fs + 0.040533092RZ E2p E2s (R) = 206.063525482 ETot,fs ETot,fs(R) = 8.3520516 1/2 − 1/2 Summary Lamb shift + 0.00018596008RRZ 2p3/2 2p1/2 2 − 5.2278228R 2 2 − + 0.00026754376R RZ (115) 0.00005206070R 3 − + 0.036998584R 3 7 3 (109) + 0.000063748539R RZ + 1.70858 10− R 4 × (108) 0.0011610273R 2 8 4 0.00020892783R 4.30422 10− R − Z − × + 0.00044095136R5 − 8 5 2 + 1.54724 10− R P. Indelicato, arXiv:1210.5828 0.00032967277RRZ × 0.000066621019R6 meV. − 9 6 − 0.00014609447R2R2 2.13593 10− R meV. − Z − × 3 + 0.000057775798RZ This can be compared with the result from from E. Borie Tot,fs Tot,fs 3 Martynenko [99] finds E E = 8.352082 meV for FIG.[70] 12. Feynman diagrams corresponding to the evaluation + 0.00014693531RRZ 2p3/2 − 2p1/2 the fine structure. of the hyperfine structure using wavefunctions obtained with 0.000030280142R4 meV. the Uehling potential in the Dirac equation. The grey squares − Z A number of terms not included in Eqs. (107) and correspondBorie, tofs theBorie hyperfine,fs interaction. 2 E E (R) = 206.0579(60) 5.22713R This can be compared with the result from (108), are presented in Table III together with the rele- 2p1/2 − 2s1/2 − (110) 3 Lamb shift in light muonicU. atoms Jentschura — Revisited, [101]E. Borie. Annals of vant references. Combining Eq. (107) with the values + 0.0365(18)R meV, Physics 327, 733-763 (2012). and Carroll et al. [100] ∆EJents.,F=2 ∆EJents.,F=1 = 209.9974(48) 2p3/2 − 2s1/2 (116) 5.2262R2 meV, ECar.,fs ECar.,fs(R) = 206.0604 5.2794R2 − 2p1/2 − 2s1/2 − (111) + 0.0546R3 meV. using the 2s hyperfine structure of Ref. [40]. Nonperturbative relativistic calculation of the muonic hydrogenFor spectrum the other, J.D. Carroll, transition I obtain A.W. Thomas, J. Rafelski et al. Phys. Rev. A 84, 012506 (2011). Trento, October 2012 F=1 F=0 54 2 E E (RZ, R) = 229.6634776 5.2294935R 2p3/2 − 2s1/2 − + 0.037645166R3 B. Transitions between hyperfine sublevels 0.0012165791R4 − + 0.00044096683R5 The energies of the two transitions observed experi- mentally in muonic hydrogen are given by 0.000066623155R6 − 0.12159928RZ F=2 F=1 − E2p E2s = E2p1/2 E2s1/2 + E2p3/2 E2p1/2 0.00055788025RR 3/2 − 1/2 − − − Z (112) 2 3 2p3/2 1 2s 0.00080263129R RZ (117) + EHFS EHFS, − 8 4 3 − 0.00019124562R R − Z 2 and + 0.00062678350RZ 2 F=1 F=0 + 0.00098901832RRZ E E = E2p E2s + E2p E2p 2p3/2 − 2s1/2 1/2 − 1/2 3/2 − 1/2 + 0.00043828342R2R2 5 3 (113) Z 2p3/2 2s F=1 3 E + EHFS + δEHFS. 0.00017332740R − 8 HFS 4 − Z 0.00044080593RR3 − Z Here we use the results from [99] for the 2p states: 4 + 0.000090840426RZ meV.
2p1/2 EHFS = 7.964364 meV Using Eq. (115), the Zemach radius from Ref. [40] 2 and the transition energy from Ref. [12], I obtain a E p3/2 = 3.392588 meV (114) HFS charge radius for the proton of 0.85248(45) fm in place F=1 δEHFS = 0.14456 meV. of 0.84184(69) fm in Ref. [12] and 0.8775(51) fm in the 15
TABLE II. Comparison of contributions to the 2s hyperfine structure from Refs. [40, 70] and the present work (meV) for R = 0.84184 fm and RZ = 0.9477 fm
Ref. [40] Ref. [70] This work Fermi energy 22.8054 22.8054 Dirac energy (includes Breit corr.) Eq. 22.807995 Vacuum polarization corrections of orders α5, α6 in the second-order perturbation theory 0.0746 0.07443 All order vacuum polarization correction (102) 0.07261 finite extent of magnetization density correction to the above 0.00114 Proton structure corr. of order α5 0.1518 −0.15198 0.15280 Proton structure corrections of order α6 −0.0017 − − Electron vacuum polarization contribution+ − proton structure corrections of order α6 0.0026 Two-loop electron vacuum polarization − contribution of 1γ interaction of order α6 0.0003 0.00037 0.00037 VP2EF (* neglected by Martynenko (ref63)) 0.00056 0.00056 muonVP (part corresponding to epsVP2 neglected by Martynenko (ref63)) 0.00091 0.00091 Hadron VP 0.0006 0.0006 Vertex(order α5) 0.00311 0.00311 Vertex(order α6), only part with powers of ln(α) - see ref 55 −0.00017 −0.00017 Breit correction −0.00258 − Muon anomalous magnetic moment correction of order α5, α6 0.0266 0.02659 0.02659 Relativistic and radiative recoil corrections 0.0026 Relativistic and radiative recoil corrections with the account proton AMM of order α6 0.0018 One-loop electron vacuum polarization contribution of 1γ interaction of orders α5, α6 0.0482 0.04818 0.04818 finite extent of magnetization density correction to the above 0.00114 One-loop muon vacuum polarization contribution of 1γ interaction of order α6 0.0004− 0.00037 0.00037 Muon self energy+proton structure correction of order α6 0.001 Vertex corrections+proton structure corrections of order α6 0.0018 ?Jellyfish? diagram correction+ proton structure corrections of order α6 −0.0005 HVP contribution of order α6 0.0005 Proton polarizability contribution of order α5 0.0105 Proton polarization (ref66) 0.00801 0.00801 Weak interaction contribution 0.0003 0.00027 0.00027 Total 22.8148 22.8107 22.8104
Summary Fine structure
In the same way, Eqs. (43), (51), (54), 57, (59), (63), (72) contained in Table III, I obtain the final 2s 2p1/2 energy: and (82) lead to the fine structure interval −
ETotP.,fs Indelicato,ETot arXiv:1210.5828,fs(R) = 206.063525482 Tot,fs Tot,fs 2p1/2 − 2s1/2 E2p E2p (R) = 8.3520516 3/2 − 1/2 5.2278228R2 0.00005206070R2 − − + 0.036998584R3 7 3 + 1.70858 10− R (109) × (108) 0.0011610273R4 8 4 4.30422 10− R − − × + 0.00044095136R5 8 5 + 1.54724 10− R × 0.000066621019R6 meV. 9 6 2.13593 10− R meV. − − × This can be compared with the result from from E. Borie Tot,fs Tot,fs Martynenko [99] finds E E = 8.352082 meV for [70] 2p3/2 − 2p1/2 the fine structure. Fine and hyperfine structure of P-wave levels in muonic hydrogen, A.P. Martynenko. Physics of Atomic Nuclei 71, 125-135 (2008). A number of terms not included in Eqs. (107) and Borie,fs Borie,fs 2 E2p E2s (R) = 206.0579(60) 5.22713R (108), are presented inTrento, October 2012 Table III together with the rele- 1/2 − 1/2 55 − (110) vant references. Combining Eq. (107) with the values + 0.0365(18)R3 meV, 16
Using the results presented above we get
F=2 F=1 2 E E (RZ, R) = 209.9465616 5.2273353R 2p3/2 − 2s1/2 − + 0.036783284R3 0.0011425674R4 − + 0.00044096683R5 0.000066623155R6 − + 0.040533092RZ
+ 0.00018596008RRZ 2 + 0.00026754376R RZ (115) 3 + 0.000063748539R RZ 0.00020892783R2 − Z 0.00032967277RR2 − Z 0.00014609447R2R2 − Z 3 + 0.000057775798RZ 3 FIG. 12. Feynman diagrams corresponding to the evaluation + 0.00014693531RRZ of the hyperfine structure using wavefunctions obtained with 0.000030280142R4 meV. the Uehling potential in the Dirac equation. The grey squares − Z correspond to the hyperfine interaction. ResultsThis can be compared with the result from U. Jentschura [101] and Carroll et al. [100] ∆EJents.,F=2 ∆EJents.,F=1 = 209.9974(48) 2p3/2 − 2s1/2 (116) 5.2262R2 meV, ECar.,fs ECar.,fs(R) = 206.0604 5.2794R2 − 2p1/2 − 2s1/2 − (111) + 0.0546R3 meV. using the 2s hyperfine structure of Ref. [40]. Lamb shiftFor in muonic the other hydrogen--I. transition Verification I obtain and update of theoretical predictions, U.D. Jentschura. Annals of Physics 326, 500-515 (2011). F=1 F=0 2 E E (RZ, R) = 229.6634776 5.2294935R 2p3/2 − 2s1/2 − + 0.037645166R3 B. Transitions between hyperfine sublevels 0.0012165791R4 − + 0.00044096683R5 The energies of the two transitions observed experi- mentally in muonic hydrogen are given by 0.000066623155R6 − 0.12159928RZ F=2 F=1 − E2p E2s = E2p1/2 E2s1/2 + E2p3/2 E2p1/2 0.00055788025RR 3/2 − 1/2 − − − Z (112) 2 3 2p3/2 1 2s 0.00080263129R RZ (117) + EHFS EHFS, − 8 4 3 − 0.00019124562R R − Z 2 and + 0.00062678350RZ 2 F=1 F=0 + 0.00098901832RRZ E E = E2p E2s + E2p E2p 2p3/2 − 2s1/2 1/2 − 1/2 3/2 − 1/2 + 0.00043828342R2R2 5 3 (113) Z 2p3/2 2s F=1 3 E + EHFS + δEHFS. 0.00017332740R − 8 HFS 4 − Z 0.00044080593RR3 Trento, October 2012 − Z 56 Here we use the results from [99] for the 2p states: 4 + 0.000090840426RZ meV.
2p1/2 EHFS = 7.964364 meV Using Eq. (115), the Zemach radius from Ref. [40] 2 and the transition energy from Ref. [12], I obtain a E p3/2 = 3.392588 meV (114) HFS charge radius for the proton of 0.85248(45) fm in place F=1 δEHFS = 0.14456 meV. of 0.84184(69) fm in Ref. [12] and 0.8775(51) fm in the Results 1
Trento, October 2012 57 Results 2 • Using the second line measured during the experiment we can improve the charge radius and get a value for the Zemach’s radius.
Simultaneous solution of these two equations with the two line energies Rc = 0.84100(63) fm and Rz=1.086(40) fm Assuming a dipole model, this gives Rm: 0.879(50) fm Mainz results: Rc = 0.879 fm, Rm = 0.777 fm and Rz = 1.047 fm
Trento, October 2012 58 Extracting the magnetic radius • We can extract the magnetic radius by using the dipole model to be consistent withe the calculations.
Simultaneous solution of the two equations with the two line energies Rc = 0.84100(63) fm and Rz=1.086(40) fm Assuming a dipole model, this gives Rm: 0.879(50) fm Mainz results: Rc = 0.879 fm, Rm = 0.777 fm and Rz = 1.047 fm
Trento, October 2012 59 Results
0.91!
0.89! !
0.87!
electron scat.! 0.85! CODATA! µH LS! Lattice QCD! 0.83! proton RMS charge radius RMS(fm) charge proton
0.81! Chiral extrapolation of octet-baryon charge radii, P. Wang, D.B. Leinweber, A.W. Thomas et al. Phys. Rev. D 79, 094001 (2009). 0.79! 1960! 1970! 1980! 1990! 2000! 2010! 2020! Year!
Trento, October 2012 60 Conclusions
• We have performed very accurate numerical calculations of the muonic hydrogen Lamb shift • We have evaluated the dependence in the charge radius to high orders (log terms) • We have evaluated crossed Zemach and charge radius terms for the HFS • Vacuum polarization to all orders has been evaluated, enabling to calculate several contributions • Results are consistent with, but more accurate than NRQED calculations
Trento, October 2012 61