The Proton Charge Radius Unsolved Puzzle

The proton charge radius unsolved puzzle

Paul Indelicato

mardi 1 octobre 2013 CREMA: Muonic Hydrogen Collaboration

F.D Amaro, A. Antognini, F.Biraben, J.M.R. Cardoso, D.S. Covita, A. Dax, S. Dhawan, L.M.P. Fernandes, A. Giesen, T. Graf, T.W. Hänsch, P. Indelicato, L.Julien, C.-Y. Kao, P.E. Knowles, F. Kottmann, J.A.M. Lopes, E. Le Bigot, Y.-W. Liu, L. Ludhova, C.M.B. Monteiro, F. Mulhauser, T. Nebel, F. Nez, R. Pohl, P. Rabinowitz, J.M.F. dos Santos, L.A. Schaller, K. Schuhmann, C. Schwob, D. Taqqu, J.F.C.A. Veloso

http://muhy.web.psi.ch http://www.lkb.ens.fr/-Metrology-of-simple-sistems-and- Mainz MITP Oct. 2013 mardi 1 octobre 2013 • Description of the experiment • spectra for µp • specta for µd • Theory • Results for radii of p and d • explanations

Mainz MITP Oct. 2013 3 mardi 1 octobre 2013 8 juillet 2010

Mainz MITP Oct. 2013 4 mardi 1 octobre 2013 8 juillet 2010

The size of the proton, R. Pohl, A. Antognini, F. Nez et al. Nature 466, 213-216 (2010).

Mainz MITP Oct. 2013 4 mardi 1 octobre 2013 8 juillet 2010

The size of the proton, R. Pohl, A. Antognini, F. Nez et al. Nature 466, 213-216 (2010).

Proton Structure from the Measurement of 2S-2P Transition Frequencies of Muonic Hydrogen, A. Antognini, F. Nez, K. Schuhmann et al. Science 339, 417-420 (2013).

Mainz MITP Oct. 2013 4 mardi 1 octobre 2013 Form factor

A rapid definition

Mainz MITP Oct. 2013 5 mardi 1 octobre 2013 P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . . 3 where W is Lambert’s (or productLog) function. The wave- wich provides c = R . The gaussian charge distribution 2 √3 function and differential equation between 0 and r0 are and potentials are given by represented by a 10 terms series expansion. For a point 2 r 2 nucleus, the first point is usually given by r0 = 10− /Z and ( c ) 7 4e− h = 0.025. Here we used values down to r = 10 /Z and ρN(r) = Z , 0 − 3 h = 0.002 to obtain the best possible accuracy. For a finite − √πc charge distribution, r0 is obtained by fixing the nuclear r Erf c boundary at some arbitrary value of nNuc large enough VNuc(r) = Z , to get a good accuracy. The evaluation of mean values of − r 3c2 operators to obtain first-order contributions to the energy < r2 > = , calculated as 2 15c4 ∞ < r4 > = , ∆E = dr P(r)2 + Q(r)2 4 O P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . . 3 0 n+3 n r0 2Γ 2 c 2 2 < rn > = , (11) R = ( ) + ( ) where W is Lambert’s (or productLog) function. The wave- wich provides c = √ . The gaussian charge distribution dr P r Q r √π 2 3 0 function and differential equation between 0 and r0 are and potentials are given by represented by a 10 terms series expansion. For a point ∞ 2 2 dr 2 2 nucleus, the first point is usually given by = 10 / and r 2 r0 − Z ( c ) + dt P(r(t)) + Q(r(t)) (6) 7 4e− yieldingh = 0.025.c = Here3 weR. Erf(usedx values) is the down Error to r function.= 10 /Z and The Fermi ρN(r) = Z , r dt 0 − 3 0 = 0 002 to obtain the best possible accuracy. For a finite − √πc distributionh . is a two parameter distribution: charge distribution, r0 is obtained by fixing the nuclear Erf r using 8 and 14 points integration formulas due to Roothan. c boundary at some arbitrary value of nNuc large enough VNuc(r) = Z , Both integration formulas provide identical results within to get a good accuracy. The evaluationZ of mean values of − r ρN(r) = − . (12) 2 operators to obtain first-order contributionsr c to the energy 2 3c 9 decimal places. (4 ln(3) −t ) < r > = , calculated as 1 + e 2 4 4 15c Even though the relationship∞ 2 between2 c and R can be ex- < r > = , ∆E = dr P(r) + Q(r) 4 O 2.1 Charge distribution models pressed in term of0 the polylogarithm function, it is not n+3 n r 2Γ c 0 n 2 very useful, as= the potentialdr P(r)2 + Q itself(r)2 must be evaluated by < r > = , (11) For the proton charge distribution, mostly two models Proton form factor √π direct numerical integration,0 using standard techniques have been used. The first one correspond to a dipole pro- ∞ dr 2 quarks up (2/3 e) + 1 quark downfor (-1/3 the e) + strong numerical interaction +( evaluationgluons) dt P(r of(t)) Eq.2 + Q (9).(r(t)) The2 normalization(6) 2 dt u u yielding c = 3 R. Erf(x) is the Error function. The Fermi ton Dirac (charge) form factor, the second is a gaussian can be obtained by insuringr0 that the potential behave as Vertex EM interaction: Dirac and Pauli Form factors distribution is a two parameter distribution: model. Here we also use a uniform and Fermi charge Z/r at infinity. The Fermi distribution is more suitable (S, P: spin and 4-momentum of nucleon, f: usingquark flavor) 8 and 14 points integration formulasd due to Roothan. distribution. These distribution are parametrized so that for− heavyBoth integration nuclei, formulas but it provide behaves identical closely results to within the Gaussian Z ρN(r) = − . (12) 9 decimal places. 4 ln(3) r c they provide the same mean square radius R. We define distribution for the specific case where the thickness pa- 1 + e( −t ) the moment of the charge distribution, rameter t is set equal to c. We use this distributionEven as though a the relationship between c and R can be ex- check,2.1 as Charge it has distribution been implemented models in the MCDF codepressed since in term of the polylogarithm function, it is not n ∞ 2+n the origin and is well tested. very useful, as the potential itself must be evaluated by < r >= r ρ(r)dr, (7) For the proton charge distribution, mostly two models Physical charge density are derived fromIn the electron-nucleonSachs Form factors collision, the relevant quantitydirect numerical is integration, using standard techniques 0 have been used. The first one correspond to a dipole pro- for the numerical evaluation of Eq. (9). The normalization the Sach’ston Dirac form (charge) factor form defined factor, the as second is a gaussian can be obtained by insuring that the potential behave as where the nuclear charge distribution ρ(r) = ρN(r)/Z is model. Here we also use a uniform and Fermi charge Z/r at infinity. The Fermi distribution is more suitable distribution. These distribution are parametrized so that − normalized by 2 iq r ρN(r) for heavy nuclei, but it behaves closely to the Gaussian they provide theGN same(q ) mean= squaredre− radius· R. We, define distribution(13) for the specific case where the thickness pa- ∞ 2 4π ρ(r)r dr = 1. (8) the moment of the charge distribution, rameter t is set equal to c. We use this distribution as a 0 check, as it has been implemented in the MCDF code since ∞ Measure the moments of the chargewhich distribution: can be defined< rn > for= bothr2+n theρ(r)dr electric, charge(7) andthe mag- origin and is well tested. √ 2 In electron-nucleon collision, the relevant quantity is The mean square radius is R = < r >. netic moment distribution.0 For the exponential model this the Sach’s form factor defined as Mainz MITP Oct. 2013 6 The potential can be deduced from the charge density leadswhere to the nuclear charge distribution ρ(r) = ρN(r)/Z is using the well known expression: mardi 1 octobre 2013 normalized by 2 iq r ρN(r) 2 4 GN(q ) = dre− · , (13) 2 1 ∞ 2 R 2 R 4 4π r ρ(r)r dr = 1. (8) GN(q ) = 2 1 q + q + (14) 1 2 ∞ R20q2 ≈ − 6 48 ··· VNuc(r) = duρN(u)u + duuρN(u). (9) 1 + which can be defined for both the electric charge and mag- r The mean square radius12 is R = √< r2 >. netic moment distribution. For the exponential model this 0 r The potential can be deduced from the charge density leads to whileusing for the the well Gaussian known expression: model one gets The exponential charge distribution and potential are 2 4 2 1 R 2 R 4 r written GN(q ) = 2 1 q + q + (14) 1 2 2∞ 4 R2q2 ≈ − 6 48 ··· 1 VNuc(2r) = Rdu2qρ2 N(u)u +R duu2 ρNR(u).4 (9) 1 + 12 r GN(q ) =re−06 1 r q + q + (15) e− c 6 72 ρ (r) = Z , ≈ − ··· while for the Gaussian model one gets N 2 3 The exponential charge distribution and potential are − c written 2 2 r r The two models have an identical 6 slope in for 2 4 c c R / q 2 1 R2q2 R 2 R 4 1 e− e− r GN(q ) = e− 6 1 q + q + (15) 0 as expected (see, e.g,e− c [54]). VNuc(r) = Z − , q ρ (r) = Z , ≈ − 6 72 ··· − r − 2c → N 2 3 A recent analysis of− thec world’s data on elastic electron- 2 2 r r The two models have an identical R /6 slope in q for 2 2 1 e− c e− c < r > = 12c , proton scatteringVNuc and(r) = calculationsZ − of two-photon, exchangeq 0 as expected (see, e.g, [54]). n effects provides the electric− r form− 2 factors,c which have→A recent al- analysis of the world’s data on elastic electron- n (n + 2)!c < r > = (10) lowed us to obtain< r2 > a= new12c2, estimate of charge radiusproton [27]. scattering and calculations of two-photon exchange 2 (n + 2)!cn effects provides the electric form factors, which have al- < rn > = (10) 2 lowed us to obtain a new estimate of charge radius [27]. P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . . 3 P. Indelicato, P.J. Mohr: non-perturbative calculations in muonic. . . 3 where W is Lambert’s (or productLog) function. The wave- wich provides c = R . The gaussian charge distribution 2 √3 function and differential equation between 0 and r are where W is Lambert’s (or productLog) function. The0 wave-andwich potentials provides are givenc = R by. The gaussian charge distribution represented by a 10 terms series expansion. For a point 2 √3 function and differential equation between 0 and r are 2 0 and potentials are given by r 2 nucleus, the first point is usually given by r0 = 10− /Z and ( c ) represented by a 10 terms series expansion. For7 a point 4e− h = 0.025. Here we used values down to r0 = 10− /Z and ρN(r) = Z , 2 3 r 2 nucleus, the first point is usually given by r0 = 10− /Z and − √πc ( c ) h = 0.002 to obtain the best possible accuracy. For a finite7 4e− h = 0.025. Here we used values down to r0 = 10− /Z and ρN(r) = Z , charge distribution, r0 is obtained by fixing the nuclear Erf r 3 h = 0.002 to obtain the best possible accuracy. For a finite − c√πc boundary at some arbitrary value of nNuc large enough VNuc(r) = Z , r tocharge get a good distribution, accuracy. Ther0 is evaluation obtained by of mean fixing values the nuclear of − rErf 2 c operatorsboundary to obtain at some first-order arbitrary contributions value of nNuc to thelarge energy enough V2 Nuc(r3)c= Z , to get a good accuracy. The evaluation of mean values of < r > = ,− r calculated as 2 2 operators to obtain first-order contributions to the energy 2 43c 4< r >15=c , calculated as ∞ 2 2 < r > = ,2 ∆E = dr P(r) + Q(r) 4 4 O 0 15c ∞ < r4 > = n+3 ,n r0 2 2 2Γ 2 c ∆E = dr P2(r) + Q2(r) n 4 =O 0dr P(r) + Q(r) < r > = , (11) √π n+3 n 0 r0 2Γ c n 2 = dr P(r)2 + Q(r)2 < r > = , (11) ∞ dr 2 2 √π + 0 dt P(r(t)) + Q(r(t)) (6) yielding c = 2 R. Erf(x) is the Error function. The Fermi r dt 3 0 ∞ dr + dt P(r(t))2 + Q(r(t))2 (6)distribution is a two2 parameter distribution: using 8 and 14 points integrationdt formulas due to Roothan. yielding c = 3 R. Erf(x) is the Error function. The Fermi r0 Both integration formulas provide identical results within distribution is a two parameterZ distribution: ρN(r) = − . (12) 9 decimalusing 8 and places. 14 points integration formulas due to Roothan. 4 ln(3) r c 1 + e( −t ) Both integration formulas provide identical results within Z ρN(r) = − . (12) 9 decimal places. 4 ln(3) r c Even though the relationship1 between+ e( c and−t ) R can be ex- 2.1 Charge distribution models pressed in term of the polylogarithm function, it is not veryEven useful, though as the the potential relationship itself between must bec evaluatedand R can by be ex- For2.1 the Charge proton distribution charge distribution, models mostly two models directpressed numerical in term integration, of the polylogarithm using standard function, techniques it is not have been used. The first one correspond to a dipole pro- for thevery numerical useful, as evaluation the potential of Eq. itself (9). The must normalization be evaluated by tonFor Dirac the (charge) proton charge form factor, distribution, the second mostly is a twogaussian modelscandirect be obtained numerical by insuring integration, that the using potential standard behave techniques as model.have been Here used. we also The use first a one uniform correspond and Fermi to a dipole charge pro- Z/forr at the infinity. numerical The Fermi evaluation distribution of Eq. (9). is The more normalization suitable distribution.ton Dirac (charge) These distribution form factor, are the parametrized second is a so gaussian that for− can heavy be nuclei, obtained but by it insuring behaves that closely the to potential the Gaussian behave as theymodel. provide Here the we same also mean use square a uniform radius andR. Fermi We define chargedistributionZ/r at infinity. for the specific The Fermi case distribution where the thickness is more suitablepa- thedistribution. moment of the These charge distribution distribution, are parametrized so thatrameterfor− heavyt is set nuclei, equal but to c it. We behaves use this closely distribution to the Gaussian as a they provide the same mean square radius R. We definecheck,distribution as it has been for implementedthe specific case in the where MCDF the code thickness since pa- ∞ the moment of< thern charge>= distribution,r2+nρ(r)dr, (7) therameter origin andt is is set well equal tested. to c. We use this distribution as a In electron-nucleon collision, the relevant quantity is 0 check, as it has been implemented in the MCDF code since n ∞ 2+n thethe Sach’s origin form and factor is well defined tested. as where the nuclear charge< r >= distributionr ρ(r)ρdr(r,) = ρN(r)/Z is (7) 0 In electron-nucleon collision, the relevant quantity is normalized by 2 iq r ρN(r) the Sach’s formGN(q factor) = defineddre− · as , (13) where the nuclear charge∞ 2 distribution ρ(r) = ρN(r)/Z is 4π ρ(r)r dr = 1. (8) normalized by 0 2 iq r ρN(r) which can be definedGN for(q both) = thedr electrice− · charge, and mag- (13) ∞ 2 Electron-proton scattering 4π The mean square radius is Rρ(=r)r√dr< r=2 >1.. (8)netic moment distribution. For the exponential model this 0 - pf The potential can be deduced from the charge densitye leadswhich to can be defined for both the electric charge and mag- - θ usingThe the mean well square known radius expression: is R =e √< r2 >. θ netic moment distribution. For the exponential model this H 1 q =2Rp2 f sin(R4 2 ) The potential can be deduced from the2 charge density 2 2 4 r leadsGN(q to)q= = pf - pi 1 q + q + (14) 1 ∞ R2q2 2 6 48 using the well known expression:momentum2 p ≈ − ··· VNuc(r) = duρN(u)u + iduuρN(u). (9) 1 + 12 2 4 r 0 r 2 1 R 2 R 4 r energy Ei GN(q ) = 2 1 q + q + (14) 1 2 ∞ while for the Gaussian R model2q2 ≈ one− gets6 48 ··· The exponentialVNuc(r) = chargedu distributionρN(u)u + andduu potentialρN(u). are (9) 1 + 12 r 0 r written dσ(Ei,θ) dσRut.(Ei,θ) 2 2 4 = GE(q ) 2 1 R2q2 R 2 R 4 The exponential chargedω r distributiond andω potential are whileG forN(q the) = Gaussiane− 6 1 modelq one+ getsq + (15) e− c ≈ − 6 72 ··· written ρN(r) = Z , − 2c3 2 4 2 1 R2q2 R 2 2 R 4 2 r r r The two modelsGN(q have) = e− an6 identical1 Rq/6+ slopeq + in q for (15) 1 e−ec − c e− c ≈ − 6 72 ··· VNucρ(r)(=r) =Z Z − , , q 0 as expected (see, e.g, [54]). N − 2r3 − 2c → − c A recent analysis of the world’s data on elastic2 electron-2 r r The two models have an identical R /6 slope in q for 2 2 1 e− c e− c proton scattering and calculations of two-photon exchange (=r)12=c ,Z − , q 0 as expected (see, e.g, [54]). Nuc n 2 effects→ provides the electric form factors, which have al- n (n −+ 2)!c r − c A recent analysis of the world’s data on elastic electron- < r > = (10) lowed us to obtain a new estimate of charge radius [27]. < r2 > = 122c2, proton scattering and calculations of two-photon exchange n effects provides the electric form factors, which have al- n (n + 2)!c 4 < r > = (10) Fit function a0+ a1q²+a2q 2 lowed us to obtain a new estimate of charge radius [27].

see S. Karshenboim in Can. J. Phys. 77, 241-266 (1999) andMainz MITP Oct. 2013 refs therein 7

mardi 1 octobre 2013 Metrology in hydrogen

Highest precision experiments

Mainz MITP Oct. 2013 8 mardi 1 octobre 2013 Hydrogen

Proton size effect

Mainz MITP Oct. 2013 9 mardi 1 octobre 2013 Hydrogen

2 466 061 413 187 035 ±10 Hz 2 466 061 413 187 103±46Hz 2011 2000

Mainz MITP Oct. 2013 10 mardi 1 octobre 2013 Rydberg constant

1 12 2006: R∞ = 10 973 731.568 525 ± 0.000 073m (ur = 6.6 x 10 ) is the most accurately determined fundamental constant.

Mainz MITP Oct. 2013 11 mardi 1 octobre 2013 Why re-measure the proton charge radius?

Mainz MITP Oct. 2013 12 mardi 1 octobre 2013 Hydrogen

QED corrections

Mainz MITP Oct. 2013 13 mardi 1 octobre 2013 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 Mainz MITP Oct. 2013 14 mardi 1 octobre 2013 Two-loop self-energy (1s)

V. A. Yerokhin, P. Indelicato, and V. M. Shabaev, Phys. rev. A 71, 040101(R) (2005). Mainz MITP Oct. 2013 15 mardi 1 octobre 2013 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)

Mainz MITP Oct. 2013 16 mardi 1 octobre 2013 Using muonic hydrogen

The exotic way...

Mainz MITP Oct. 2013 17 mardi 1 octobre 2013 Lamb shift

Self-energy: Vacuum Polarization: The heavier the particle, the smaller (in The closer the particle is, the stronger it is relative term) it is 2P (F=2) n=2 3/2 2S1/2 50 THz 1 GHz n=2 10-6s 2S1/2(F=1) 2P1/2 Muonic Hydrogen Hydrogen (electron) (muon 207 times heavier than the electron) Effect of R: 6x10-11 Effect of R: 1.7%

Mainz MITP Oct. 2013 18 mardi 1 octobre 2013 muonic hydrogen 2S Lamb shift determination of the “proton radius” µ− Exotic atom p

2P (8.5ps) Laser (6µm) Experiment 2S (1µs)

2keV

Challenges • production of muonic hydrogen in 2S • powerful triggerable 6µm laser • small signal analysis

Aim : better determination of proton radius rp Mainz MITP Oct. 2013 mardi 1 octobre 2013 The muonic hydrogen experiment

Getting up close and personal with the proton!

Mainz MITP Oct. 2013 20 mardi 1 octobre 2013 Principle of the experiment “prompt” (t~0) ppee + µ- → µ-p +… “delayed” (t~1µs) H n~14 2 2P laser 1% 2S

2P events 2S 2keV 99%

1S 1S

0.5 1 1.5 2 2.5 time(µs) laser - µ stop in H2 gas ⇒ µp* atoms formed (n~14) Fire laser (λ~6µm, ΔE~0.2eV) ⇒ induce µp(2S-2P) 99%: cascade to 1S emitting prompt Kα,Kβ,… ⇒ observe delayed Kα x-rays

1%: long lived 2S state (τ ~ 1µs at 1mbar) ⇒ normalize delayed K α x-rays Mainz MITP Oct. 2013 prompt Kα mardi 1 octobre 2013 Experimental set-up

Mainz MITP Oct. 2013 22 mardi 1 octobre 2013 muon beam apparatus laser hut π- µ-+ν below counting room µ concrete blocks

PSC solenoid,

Muon extraction channel

H2 target, laser cavity, detectors µp setMainz MITP Oct. 2013 up in πE5 23 mardi 1 octobre 2013 A muon’s Odyssey

Mainz MITP Oct. 2013 24 mardi 1 octobre 2013 A muon’s Odyssey

Mainz MITP Oct. 2013 24 mardi 1 octobre 2013 The laser trigger signal

Mainz MITP Oct. 2013 25 mardi 1 octobre 2013 Laser chain Laser chain • Each single muon triggers the laser system (random trigger) • 2S lifetime ~1µs → short laser delay (disk laser) • 6 µm tunable laser pulse (0.2mJ)

µ- trigger

Thin disk laser (1030 nm) + LBO (515 nm)

pulsed TiSa oscillator + amplifier (708 nm determined by cw TiSa seeding)

Raman cell 6.02 µm

Multipass cavity at 6μm surrounding the H2 target

Mainz MITP Oct. 2013 mardi 1 octobre 2013 Laser chain Laser chain

Thin disk laser

•Large pulse energy: 85 (160) mJ •Short trigger-to-pulse delay: ≤ 400 ns •Random trigger •Pulse-to-pulse delays down to 2 ms (rep. rate ≥ 500 Hz)

Mainz MITP Oct. 2013 mardi 1 octobre 2013 Laser chain Laser chain MOPA TiSa laser •cw laser, frequency stabilized •referenced to a stable FP cavity •FP cavity calibrated with I2, Rb, Cs lines •FP = N . FSR (free spectral range) •FSR = 1497.344(6) MHz •cw TiSa frequency absolutely known to 30 MHz •Γ2P2S = 18.6 GHz •Seeded oscillator •TiSa = cw → pulsed TiSa (frequency chirp ≤ 100 MHz) •Multipass amplifier (2f- configuration) •gain=10

Mainz MITP Oct. 2013 mardi 1 octobre 2013 Laser chain : Raman cell

708 nm 6.02 µm Threshold H2 15.5 bars but reliable 12 mJ 0.2 mJ

1st Stokes

2nd Stokes 6 µm energy (mJ) 3rd Stokes 708 nm 1.00 µm 1.72 µm v=1 6.02 µm 708 nm pump energy (mJ) 4155,2 cm-1 H2 v=0 6 µm frequency calibration : H20 lines absorption@1662cm-1 in cell (37cm)

Mainz MITP Oct. 2013

mardi 1 octobre 2013 Laser chain : Raman cell

708 nm 6.02 µm H2 15.5 bars 12 mJ 0.2 mJ

1st Stokes 2nd Stokes 3rd Stokes 708 nm 1.00 µm 1.72 µm v=1 6.02 µm 4155,2 cm-1 H2 v=0

•Vacuum tube for 6µm laser beam transport •Direct frequency calibration at 6µm

Mainz MITP Oct. 2013 •Well known lines

mardi 1 octobre 2013 Laser chain : frequency calibration

FSR measured/controlled in cw with I2 (1 ph abs), Cs (2 ph fluo), Rb (2 ph fluo), lines

Cs 2-ph I2 Line 1 ……..I2 Line xx Cw-TiSa frequency

FP/I2 FSR FP/H20 FP/H 0 2 FP frequency N N’ µp 2S-2P H 0 Line 2 H20 Line 1 2 in air in a cell 6µm pulsed frequency

absorption@1662cm-1 in cell (37cm)

FP absolute frequency ν(µp:2S-2P) = ν(H20 Line 2 ) + (N-N’) FSR calibration @ 6µm with H20 lines Mainz MITP Oct. 2013 mardi 1 octobre 2013 The laser hut

Mainz MITP Oct. 2013 32 mardi 1 octobre 2013 Ti:Sa and raman cell

Mainz MITP Oct. 2013 33 mardi 1 octobre 2013 Laser chain : multipass cavity → illuminate at 6 µm all the muon stopping volume (5×15×190 mm3)

• coupling through a 0.63mm diameter hole • R=99.90% at 6 µm 6 µm mirrors • 1000 reflections LAAPD (14´14mm²) • 0.15mJ injected → 2S-2P saturated muons

laser pulse

Mainz MITP Oct. 2013 mardi 1 octobre 2013 X-rays analysis → event gate sorting → noise rejection

Example : FP 900 - 11 hrs meas.

1.56 million detector events

expected 2-3 laser induced events/hour !

time signature in LAAPD energy signature in LAAPD • photon < 10keV → 1 shot in the LAAPD • E > 8keV ⇔ electron • e- in B = 5T → many counts in detectors • 1keV < E < 8keV ⇔ X ray • E<1keV ⇔ neutron e- in B field L.Ludhova µp K phd thesis rest Kα n=3 n=2

µp Kβ µp Kα K rest n=1 Kβ E(keV) targetMainz MITP Oct. 2013 in the solenoid µO µN mardi 1 octobre 2013 X-rays analysis → noise rejection Example : FP 900 - 11 hrs meas. Ø 400 µ-/s Ø 240 laser shot/s Ø 860 000 laser shot/hour Ø 1.56 million detector clicks Ø 19600 clicks in the laser region Ø expected 2-3 laser induced events/hour !

µ→e νµ νe x rays multiplicity 1 LAAPD energy resolution

Mainz MITP Oct. 2013 mardi 1 octobre 2013 Time spectra

Mainz MITP Oct. 2013 37 mardi 1 octobre 2013 New analysis: tacking into account more events

1.9 keV Ka x-ray must followed by the detection of an MeV-energy electron, but there are several detectors Proton Structure from the Measurement of 2S-2P Transition Frequencies of Muonic Hydrogen, A. Antognini, F. Nez, K. Schuhmann et al. Science 339, 417-420 (2013). Mainz MITP Oct. 2013 38 mardi 1 octobre 2013 2 2 muonic hydrogen : S1/2(F=1) - P3/2(F=2) F=2 • 550 events measured 2P3/2 F=1 • 155 backgrounds 2.03 THz • 31 FP fringes 2P F=1 1/2 F=0 • 250 hours

49.81 THz ~ 6 µm (~708 nm)

finite size 0.96THz

F=1 2S1/2 5.56 THz F=0

R. Pohl, A. Antognini, F. Nez, et al., Nature 466, 213 (2010). Mainz MITP Oct. 2013 → proton charge radius (~0.1%) mardi 1 octobre 2013 2 2 muonic hydrogen : S1/2(F=1) - P3/2(F=2) F=2 2P3/2 F=1 2.03 THz 2P F=1 1/2 F=0

49.81 THz ~ 6 µm (~708 nm)

finite size 0.96THz

F=1 2S1/2 5.56 THz F=0

49.81 THz ~ 6 µm (~708 nm)

finite size 0.96THz

F=1 2S1/2 5.56 THz F=0

49.81 THz ~ 6 µm (~708 nm)

finite size 0.96THz

F=1 2S1/2 5.56 THz F=0

49.81 THz ~ 6 µm (~708 nm)

finite size 0.96THz

F=1 2S1/2 5.56 THz F=0

49.81 THz ~ 6 µm (~708 nm)

finite size 0.96THz

F=1 2S1/2 5.56 THz F=0

49.81 THz ~ 6 µm (~708 nm)

finite size 0.96THz

F=1 2S1/2 5.56 THz F=0

49.81 THz 550 events measured on resonance ~ 6 µm where 155 bgr events are expected (~708 nm) fit Lorentz + flat bgr⇒χ2/dof = 28.1/28

finite size width agrees with expectation 0.96THz bgr agrees with laser OFF data χ2/dof = 283/31 for flat line → 16σ F=1 2S1/2 5.56 THz F=0

R. Pohl, A. Antognini, F. Nez, et al., Nature 466, 213 (2010). Mainz MITP Oct. 2013 → proton charge radius (~0.1%) mardi 1 octobre 2013 Reanalysis 2012

Mainz MITP Oct. 2013 mardi 1 octobre 2013 New resonance 2012

Mainz MITP Oct. 2013 mardi 1 octobre 2013 2 2 µP : S1/2(F=1) - P3/2(F=2) uncertainty budget Statistics • uncertainty on position (fit) 541 MHz (~ 3 % of Γnat)

Δνexperimental= 20 (1) GHz ( Γnat = 18.6 GHz )

Sources : • Laser frequency (H20 calibration, lines known to ~1 MHz) 300 MHz • AC and DC stark shift < 1 MHz • Zeeman shift ( 5 Telsa) < 30 MHz • Doppler shift < 1 MHz • Collisional shift 2 MHz

TOTAL UNCERTAINTY ON FREQUENCY 618 MHz

Broadening : • 6 µm laser line width ~ 2 GHz • Doppler Broadening < 1 GHz • Collisional broadening 2.4 MHz

Updated: ν (µp : 2S1/2(F=1)- 2P3/2(F=2)) <1σ (12.5 ppm) Nature: ν (µp : 2S1/2(F=1)- 2P3/2(F=2)) = 49 881.88 (76) GHz (16 ppm) Mainz MITP Oct. 2013 mardi 1 octobre 2013 2 2 µP : S1/2(F=0) - P3/2(F=1) uncertainty budget

Statistics • uncertainty on position (fit) 960 MHz

Sources : • Laser frequency (H20 calibration) 300 MHz • AC and DC stark shift < 1 MHz • Zeeman shift ( 5 Telsa) < 30 MHz • Doppler shift < 1 MHz • Collisional shift 2 MHz

TOTAL UNCERTAINTY ON FREQUENCY 1006 MHz

Broadening : • 6 µm laser line width ~ 2 GHz • Doppler Broadening < 1 GHz • Collisional broadening 2.4 MHz

ν (µp : 2S1/2(F=0)- 2P3/2(F=1)) good agreement with the other (18.5ppm)

Mainz MITP Oct. 2013 mardi 1 octobre 2013 2 2 muonic deuterium : S1/2(F=3/2) - P3/2(F=5/2)

50815.491±0.815GHz

Mainz MITP Oct. 2013 mardi 1 octobre 2013 2 2 muonic deuterium : S1/2(F=3/2) - P3/2(F=5/2)

50815.491±0.815GHz

x10 -4 10

8 PRELIMINARY

6 delayed / prompt events

0 460 480 500 520 540 560 580 laser frequency (FP fringes)

Mainz MITP Oct. 2013 mardi 1 octobre 2013 2 2 2 2 muonic deuterium : S1/2(F=1/2)- P3/2(F=3/2) and S1/2(F=1/2)- P3/2(F=1/2)

52061.0±1.6GHz preliminary 52154.4±3.0GHz preliminary -4 ×10

PRELIMINARY 5

4 delayed / prompt events

0 280 300 320 340 360 380 400 420 laser frequency (FP fringes)

Mainz MITP Oct. 2013 mardi 1 octobre 2013 2 2 2 2 muonic deuterium : S1/2(F=1/2)- P3/2(F=3/2) and S1/2(F=1/2)- P3/2(F=1/2)

52061.0±1.6GHz preliminary 52154.4±3.0GHz preliminary -4 ×10

7 Still to be investigated: possible shift due to proximity of both 6 lines: PRELIMINARY 5 [1] Shifts due to distant neighboring resonances for laser 3 3 measurements of 2 S1-2 PJ transitions of helium, A. Marsman, 4

delayed / prompt events M. Horbatsch et E.A. Hessels. Phys. Rev. A 86, 040501 (2012).

3 [2] Shifts from a distant neighboring resonance for a four- level atom, M. Horbatsch et E.A. Hessels. Phys. Rev. A 84, 2 032508 (2011).

1 [3] Shifts from a distant neighboring resonance, M. Horbatsch et E.A. Hessels. Phys. Rev. A 82, 052519 (2010). 0 280 300 320 340 360 380 400 420 laser frequency (FP fringes)

Mainz MITP Oct. 2013 mardi 1 octobre 2013 Extraction of the radii

Charge, magnetic and Zemach’s radii

Mainz MITP Oct. 2013 48 mardi 1 octobre 2013 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 • We have analyzed in details the second transition that was observed, using an improved algorithm that correct for the variation of the laser pulse energy from shot to shot • We take into account more events • We have reanalyzed the first observed line using the improved method • This lead to 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

Mainz MITP Oct. 2013 49 mardi 1 octobre 2013 µP theory

Discrepancy: 0.31 meV Th. Uncertainty 0.0025 meV How well is it calculated? That’s 120 times smaller! How dependent on nuclear model? QED QED QED QED

Mainz MITP Oct. 2013 50 mardi 1 octobre 2013 µP theory Discrepancy: 0.31 meV Th. Uncertainty 0.0025 meV That’s 120 times smaller!

Mainz MITP Oct. 2013 51 mardi 1 octobre 2013 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

Mainz MITP Oct. 2013 52 mardi 1 octobre 2013 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

Nonperturbative evaluation of some QED contributions toc the muonicd hydrogen n=2

Mainz MITP Oct. 2013 Lamb shift and hyperfine structure, P. Indelicato. Phys. Rev. A 87, 022501 (2013). 53 mardi 1 octobre 2013 Contributions included to all-order

+ +...

a b

Nonperturbative evaluation of some QED contributions toc the muonicd hydrogen n=2

Mainz MITP Oct. 2013 Lamb shift and hyperfine structure, P. Indelicato. Phys. Rev. A 87, 022501 (2013). 53 mardi 1 octobre 2013 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 theMainz MITP Oct. 2013 binding energy. 54 2 Dirac equation and finite nuclear model To solve this equation numerically, we use a 5 point mardipredictor-corrector 1 octobre 2013 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 Extraction of the size dependence

• Fit to the Coulomb+Vacuum polarization contribution to 2s-2p1/2 separation, plus higher order corrections using Friar functional form

Nonperturbative evaluation of some QED contributions to the muonic hydrogen n=2 Lamb shift and hyperfine structure, P. Indelicato. Phys. Rev. A 87, 022501 (2013).

Mainz MITP Oct. 2013 55 mardi 1 octobre 2013 Other contributions for µP

Mainz MITP Oct. 2013 56 mardi 1 octobre 2013 The role of the nuclear model

Dependence on the charge distribution

mardi 1 octobre 2013 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

Mainz MITP Oct. 2013 58 mardi 1 octobre 2013 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 -1.62 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 freedomMainz MITP Oct. 2013 of the proton. The other symbols bution2 to the Lamb shift, but is much larger than the59 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 mardi 1 octobre 2013 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 hyperfine 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

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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, compared 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) contribution 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- • Several calculations 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 polarization [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 − – Proton polarisability contribution to the Lamb shiftare explained in muonic in Fig. hydrogen 5. at fourth order in where the error is set to encompass the error bars of all calculations. This is the value we retain for our ﬁnal table. chiral perturbation theory, M.C. Birse et J.A. McGovern. Eur. Phys. J. A 48, 1-9 (2012). 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) We calculate the amplitude T1 for forward doubly virtual Compton2s − nscattering3 −in heavy-baryon 7.4 Hyperfine structure chiral perturbation theory, to fourth order in withoutthe chiral giving expansion error bars. and More with recently the leading Martynenko [99] In order to compare with experiment, one has to combine contribution of the NΔ form factor. This providesreevaluated a model-independent the proton polarization, expression given for as thethe sum of the calculations of the Lamb shift with the fine struc- γ an inelastic and subtraction terms. He obtained ture (provided in the next section) and 2s and 2p hyperfine structure (HFS). The line measured in Ref. [12] is amplitude in the low-momentum region, which is the dominantp.pol one forsubt its. contributioninel. to the ∆E2s = ∆E + ∆E described by Lamb shift. It allows us to significantly reduce the theoretical uncertainty= 0.0023 in0.01613 the proton meV − 3 2p3/2 1 2s E 5 E 3 = ∆E + ∆E + ∆E ∆E , (97) polarisability contributions to the Lamb shift in muonic hydrogen.= We0. 0138(29)also stress meV ,the (94) P3/2 S1/2 LS FS HFS HFS − − 8 − 4 importance of consistency between the definitions of the Born and structure parts of the amplitude. Our result leaves no room for any effect large enough to explain the discrepancy between proton charge radii as determined from muonic and normal hydrogen.

Mainz MITP Oct. 2013 60 mardi 1 octobre 2013 P. Indelicato, P.J. Mohr: Non-perturbative evaluation of muonic H Lamb-shift 15

"#"""& -1.2

!"#""%& -1.4

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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, compared 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) contribution 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 corresponds 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 polarization [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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Mainz MITP Oct. 2013 61 mardi 1 octobre 2013 P. Indelicato, P.J. Mohr: Non-perturbative evaluation of muonic H Lamb-shift 15

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!"#""%& -1.4

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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, compared 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) contribution 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 corresponds 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 polarization [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 No radial excitations in low energy QCD. I. Diquarks and classification of mesons∆E, T. Friedmann.= meV The= European0.0115 meV Physical, (93)Journal !"#"'"& 2s (:3<36&− n3 − 7.4 Hyperfine structure C 73, 2298 (2013) C38#& without giving error bars. More recently Martynenko [99] In order to compare with experiment, one has to combine D5.E+<59& No radial excitations in low energy QCD. II. The shrinking radius of hadronsreevaluated, T. Friedmann. the proton The polarization, European givenPhysical as the Journal sum of C 73,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- 2299 (2013). !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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Mainz MITP Oct. 2013 61

mardi 1 octobre 2013 P. Indelicato, P.J. Mohr: Non-perturbative evaluation of muonic H Lamb-shift 15

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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, compared 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. Deuteron polarization6 (Friar, Pachucki) tivistic recoil. It is in fact the Zemach moment (24) contribution 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 corresponds 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 polarization [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 − 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 hyperfine 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

Mainz MITP Oct. 2013 62 mardi 1 octobre 2013 P. Indelicato, P.J. Mohr: Non-perturbative evaluation of muonic H Lamb-shift 15

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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, compared 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. Deuteron polarization6 tivistic recoil. It is in fact the Zemach moment (24) contribution 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 corresponds 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- Chen Ji, Nir Nevo Dinur, Sonia Bacca, Nir Barneaization (arXiv:1307.6577) [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.

Mainz MITP Oct. 2013 63 mardi 1 octobre 2013 Hyperfine structure

Beyond Zemach

Mainz MITP Oct. 2013 64 mardi 1 octobre 2013 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 Mainz MITP Oct. 2013HFS on the chargenucleusand magnetic finiteZ65 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 mardinuclear 1 octobre chargeE 2013F 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 calculatedMainz MITP Oct. 2013 the cor- 66 rection EHFS (RZ, R) for several value of RZ betweenmardi 1 octobre 0.4 fm 2013 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 Fullfunction calculation for which beyond I find Zemach 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 Nonperturbative evaluation of some QED + 0.000063748539R RZ 0.002595 meV, to be compared to 0.0026 meV in Ref. [40] 2 0.00020892783R contributions to the muonic hydrogen n=2 − Z (Table II, line 3) and 0.00258 meV in Ref. [70]. Mar- 2 4 3 Lamb shift and hyperfine structure, P. 0.00032967277RRZ 2 (Zα) µr − s Indelicato.tynenko Phys. [40] evaluatesRev. A 87, 022501 this correction, (2013). 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 Mainz MITP Oct. 2013 67 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. mardi 1 goodoctobre agreement 2013 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

Mainz MITP Oct. 2013 68 mardi 1 octobre 2013 Results HFS

Mainz MITP Oct. 2013 69 mardi 1 octobre 2013 Results • 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

Mainz MITP Oct. 2013 70 mardi 1 octobre 2013 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

A magnetic radius larger than the charge radius leads to large discrepancies when applied to electron proton scattering data

Mainz MITP Oct. 2013 71 mardi 1 octobre 2013 Summary of results: charge radius

A. Antognini, F. Nez, K. Schuhmann, et al., Science 339, 417 (2013).

A. Antognini, F. Kottmann, F. Biraben, et al., Annals of Physics 331, 127 (2013).

Mainz MITP Oct. 2013 72 mardi 1 octobre 2013 Summary of results: Zemach radius

A. Antognini, F. Kottmann, F. Biraben, et al., Annals of Physics 331, 127 (2013).

Mainz MITP Oct. 2013 73 mardi 1 octobre 2013 Summary of present status

Mainz

Jefferson Lab

High-precision measurement of the proton elastic form factor ratio at low, X. Zhan, et al Physics Letters B 705, 59-64 (2011).

Mainz MITP Oct. 2013 74 mardi 1 octobre 2013 New 2013: deuterium charge radius

CODATA-2010

CODATA D + e-d

e-d scatt.

n-p scatt. 2.11 2.115 2.12 2.125 2.13 2.135 2.14 2.145 Deuteron charge radius [fm]

Mainz MITP Oct. 2013 75 mardi 1 octobre 2013 New 2013: deuterium charge radius

µH + iso H/D(1S-2S) CODATA-2010

CODATA D + e-d

e-d scatt.

n-p scatt. 2.11 2.115 2.12 2.125 2.13 2.135 2.14 2.145 Deuteron charge radius [fm]

Mainz MITP Oct. 2013 76 mardi 1 octobre 2013 New 2013: deuterium charge radius

µD 2013 PRELIMINARY µH + iso H/D(1S-2S) CODATA-2010

CODATA D + e-d

e-d scatt.

n-p scatt. 2.11 2.115 2.12 2.125 2.13 2.135 2.14 2.145 Deuteron charge radius [fm]

Mainz MITP Oct. 2013 77 mardi 1 octobre 2013 New 2013: deuterium charge radius

µD 2013 PRELIMINARY µH + iso H/D(1S-2S) CODATA-2010

CODATA D + e-d

e-d scatt.

n-p scatt. 2.11 2.115 2.12 2.125 2.13 2.135 2.14 2.145 It is not possible to extract the magnetic moment distribution Deuteron charge radius [fm] radius: uncalculated µD 2S hyperfine structure polarization correction.

Mainz MITP Oct. 2013 77 mardi 1 octobre 2013 Other measurements

How to get the radius from hydrogen and electron-proton scattering

Mainz MITP Oct. 2013 78 mardi 1 octobre 2013 Hydrogen spectroscopy

ns,nd 3s

2 466 061 413 187 103±46Hz Two lines needed: Rydberg constant Proton radius

Mainz MITP Oct. 2013 79 mardi 1 octobre 2013 Hydrogen+Deuterium

'"!%$

!"&%$

!"&$ KF$ A26$ !"#%$ Analysis by F. Biraben !"#$ Analysis by F. Biraben D>;/E$.3F$ GH6IJI$0!!8$ GH6IJI$0!'!$ =.;-(:.*$>?.+>-.$@AB$ ()*+,-./$01'20351'20$ ()*+,-./$01'20381'20$ ()*+,-./$01'203#1'20$ ()*+,-./$'1'20371'20$ ()*+,-./$01'20304'20$ ()*+,-./$01'20304'20$ ()*+,-./$01'20304'20$ ()*+,-./$01'20354'20$ ()*+,-./$01'20354720$ ()*+,-./$01'20356%20$ ()*+,-./$01'20386%20$ ()*+,-./$01'203#6720$ ()*+,-./$01'203#6%20$ *.9:.+;9<$01'203#1'20$ *.9:.+;9<$01'203#6720$ *.9:.+;9<$01'203#6%20$ ()*+,-./$01'203'06720$ ()*+,-./$01'203'06%20$ =.;-(:.*$>?.+>-.$@AC6B$ *.9:.+;9<$01'203'06720$ *.9:.+;9<$01'203'06%20$ Mainz MITP Oct. 2013 80 mardi 1 octobre 2013 Hydrogen+Deuterium • Needed: new measurements with independent systematic errors and get an independent Rydberg constant value: – 2S-4P in H (Garching) – 2S-nS,D in H (J. Flowers, NPL) – 1S-3S (Garching, Paris) – transitions between Rydberg states of heavy H-like ion (NIST) – 1S-2S and 1S hfs in µe (A. Antognini, PSI)

Mainz MITP Oct. 2013 81 mardi 1 octobre 2013 Electron-Proton scattering

M.O. Distler, Trento, 2012 Normalization! Physical model?

Mainz MITP Oct. 2013 82 mardi 1 octobre 2013 Electron-Proton scattering

M.O. Distler, Trento, 2012

Mainz MITP Oct. 2013 83 mardi 1 octobre 2013 Possible origin of the discrepancy

Systematic errors or new physics?

Mainz MITP Oct. 2013 84 mardi 1 octobre 2013 Possible sources of discrepancy (µP) • Frequency shift: unlikely - several redundant measurements at 708 nm (Fabry- Perot, two-photon transition in Rb) and 6µm (water lines) • µ e- p molecules or p p µ molecules. Not possible - Why Three-Body Physics Does Not Solve the Proton-Radius Puzzle, J.-P. Karr and L. Hilico. Phys. Rev. Lett. 109, 103401 (2012). • Experimental problems, e.g., a small air leak in the hydrogen target: we see characteristic µN and µO x-rays

– Less than 1% of all created µP atoms see any N2 molecules

– Less than 0.1% of all µP in 2S state see any N2 molecule during laser time • µP theory: many checks, no effect seems large enough to explain a 0.3 meV energy shift, probably not even proton polarization (30 times too small)

Mainz MITP Oct. 2013 85 mardi 1 octobre 2013 Possible origin for the discrepancy • 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) • Proton structure • New physics – Constraints: • g-2 of the muon (3σ ), • g-2 of the electron (Harvard)+fine structure constant from atomic recoil (LKB) • Hydrogen • Precision highly charged ions experiments at GSI (if long range interaction) • ...

Mainz MITP Oct. 2013 86 mardi 1 octobre 2013 Unsolved problems in the muon corner...

Example: muon g-2, discrepancy not solved after improved QED calculation

For electrons:

Reevaluation of the hadronic contributions to the muon g-2 and to $\alpha (M^{2}_{Z})$", M. Davier, A. Hoecker, B. Malaescu and Z. Zhang. The European Physical Journal C - Particles and Fields 71, 1-13 (2011). Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant, T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio. Phys. Rev. Lett. 109, 111807 (2012). Complete Tenth-Order QED Contribution to the Muon g-2, T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio. Phys. Rev. Lett. 109, 111808 (2012).

Mainz MITP Oct. 2013 87 mardi 1 octobre 2013 A new polarization term ?

Toward a resolution of the proton size puzzle, G.A. Miller, A.W. Thomas, J.D. Carroll and J. Rafelski. Phys. Rev. A 84, 020101 (2011).

0.31 meV for µH and 9Hz for hydrogen (with model dependent parametrization)

Mainz MITP Oct. 2013 No other work supports such a large effect 88 mardi 1 octobre 2013 Where could it come from? • Proton Size Anomaly, V. Barger, C.-W. Chiang, W.-Y. Keung et al. Phys. Rev. Lett. 106, 153001 (2011): A measurement of the Lamb shift in muonic hydrogen yields a charge radius of the proton that is smaller than the CODATA value by about 5 standard deviations. We explore the possibility that new scalar, pseudoscalar, vector, and tensor flavor-conserving nonuniversal interactions may be responsible for the discrepancy. We consider exotic particles that, among leptons, couple preferentially to muons and mediate an attractive nucleon-muon interaction. We find that the many constraints from low energy data disfavor new spin-0, spin-1, and spin-2 particles as an explanation. • Lamb shift in muonic hydrogen--II. Analysis of the discrepancy of theory and experiment, U.D. Jentschura. Annals of Physics 326, 516-533 (2011). No unstable vector boson, no millicharged particles,

Mainz MITP Oct. 2013 89 mardi 1 octobre 2013 Where could it come from? • New physics and the proton radius problem, C.E. Carlson and B.C. Rislow. Physical Review D 86, 035013 (2012): – Particles that couple to muons and hadrons but not electrons – For the scalar-pseudoscalar model, masses between 100 to 200 MeV are not allowed. – For the vector model, masses below about 200 MeV are not allowed. The strength of the couplings for both models approach that of electrodynamics for particle masses of about 2 GeV. – New physics with fine-tuned couplings may be entertained as a possible explanation for the Lamb shift discrepancy. • New Parity-Violating Muonic Forces and the Proton Charge Radius, B. Batell, D. McKeen and M. Pospelov. Phys. Rev. Lett. 107, 011803 (2011). – We identify a class of models with gauged right-handed muon number, which contains new vector and scalar force carriers at the100 MeV scale or lighter, that is consistent with observations. – Such forces would lead to an enhancement by several orders-of-magnitude of the parity- violating asymmetries in the scattering of low-energy muons on nuclei.

Mainz MITP Oct. 2013 90 mardi 1 octobre 2013 Where could it come from?

Muonic hydrogen and MeV forces, D. Tucker-Smith et I. Yavin. Physical Review D 83, 101702 (2011).

We explore the possibility that a new interaction between muons and protons is responsible for the discrepancy between the CODATA value of the proton-radius and the value deduced from the measurement of the Lamb shift in muonic hydrogen. We show that a new force carrier with roughly MeV-mass can account for the observed energy- shift as well as the discrepancy in the muon anomalous magnetic moment. However, measurements in other systems constrain the couplings to electrons and neutrons to be suppressed relative to the couplings to muons and protons, which seems challenging from a theoretical point of view. One can nevertheless make predictions for energy shifts in muonic deuterium, muonic helium, and true muonium under the assumption that the new particle couples dominantly to muons and protons.

Mainz MITP Oct. 2013 91 mardi 1 octobre 2013 Where could it come from?

Muonic hydrogen and MeV forces, D. Tucker-Smith et I. Yavin. Physical Review D 83, 101702 (2011).

Compatible with muon g-2

Mainz MITP Oct. 2013 92 mardi 1 octobre 2013 Where could it come from?

Nonidentical protons, T. Mart et A. Sulaksono. Phys. Rev. C 87, 025807 (2013).

We have calculated the proton charge radius by assuming that the real proton radius is not unique and the radii are randomly distributed in a certain range. This is performed by averaging the elastic electron-proton differential cross section over the form factor cutoff. By using a dipole form factor and fitting the middle value of the cutoff to the low-Q2 Mainz data, we found the lowest χ2/N for a cutoff = 0.8203 ± 0.0003 GeV, which corresponds to a proton charge radius rE= 0.8333± 0.0004 fm. The result is compatible with the recent precision measurement of the Lamb shift in muonic hydrogen as well as recent calculations using more sophisticated techniques. Our result indicates that the relative variation of the form factor cutoff should be around 21.5%. Based on this result we have investigated effects of the nucleon radius variation on the symmetric nuclear matter (SNM) and the neutron star matter (NSM) by considering the excluded volume effect in our calculation. The mass-radius relation of a neutron star is found to be sensitive to this variation. The nucleon effective mass in the SNM and the equation of state of both the SNM and the NSM exhibit a similar sensitivity.

Mainz MITP Oct. 2013 93 mardi 1 octobre 2013 Where could it come from?

No radial excitations in low energy QCD. II. The shrinking radius of hadrons, T. Friedmann. The European Physical Journal C 73, 2299 (2013). We discuss the implications of our prior results obtained in our companion paper (Eur. Phys. J. C (2013 ) ). Inescapably, they lead to three laws governing the size of hadrons, including in particular protons and neutrons that make up the bulk of ordinary matter: (a) there are no radial excitations in low-energy QCD; (b) the size of a hadron is largest in its ground state; (c) the hadron’s size shrinks when its orbital excitation increases. The second and third laws follow from the first law. It follows that the path from confinement to asymptotic freedom is a Regge trajectory. It also follows that the top quark is a free, albeit short-lived, quark.

Note added Nine months after this paper was originally posted to arXiv [32, 33], an experiment studying muonic hydrogen [34], repeated more recently [35], observed a smaller size of the proton than previously expected, consistent with our predictions. It is possible that this is a manifestation of our three laws, and may be a QCD, rather than QED, effect.

Mainz MITP Oct. 2013 94 mardi 1 octobre 2013 Where could it come from?

•PROTON RADIUS PUZZLE AND LARGE EXTRA DIMENSIONS, L.B. Wang et W.T. Ni. Modern Physics Letters A 28, 1350094 (2013).

We propose a theoretical scenario to solve the proton radius puzzle which recently arises from the muonic hydrogen experiment. In this framework, 4 + n dimensional theory is incorporated with modified gravity. The extra gravitational interaction between the proton and muon at very short range provides an energy shift which accounts for the discrepancy between spectroscopic results from muonic and electronic hydrogen experiments. Assuming the modified gravity is a small perturbation to the existing electromagnetic interaction, we find the puzzle can be solved with stringent constraint on the range of the new force. Our result not only provides a possible solution to the proton radius puzzle but also suggest a direction to test new physics at very small length scale.

Mainz MITP Oct. 2013 95 mardi 1 octobre 2013 Where could it come from?

Can Large Extra Dimensions Solve the Proton Radius Puzzle? Zhigang Li, Xuelei Chen (http://arxiv.org/abs/1303.5146v1)

The proton charge radius extracted from the recent muonic hydrogen spectroscopy [Antognini et al. 2013; Pohl et al. 2010] differs from the CODATA 2010 recommended value [Mohr et al. 2012] by more than 4%. This discrepancy, dubbed as the "Proton Radius Puzzle", is a big challenge to the Standard Model of particle physics, and has triggered a number of works on the quantum electrodynamic calculations recently. The proton radius puzzle may indicate the presence of an extra correction which enlarges the 2S-2P energy gap in muonic hydrogen. Here we explore the possibility of large extra dimensions which could modify the Newtonian gravity at small scales and lower the 2S state energy while leaving the 2P state nearly unchanged. We find that such effect could be produced by four or more large extra dimensions which are allowed by the current constraints from low energy physics.

Mainz MITP Oct. 2013 96 mardi 1 octobre 2013 What’s next

Muonic Helium: experiment set up October 8th, 2013

Mainz MITP Oct. 2013 97 mardi 1 octobre 2013 Muonic He spectroscopy 812 nm pHe = 40 bars 2011-2013→ muonic helium spectroscopy (4 mbar)

µ4He+ µ3He+

F=1 2P3/2 2P3/2 F=2 35 THz 35 THz 2P 2P F=1 1/2 1/2 F=0 898 nm 812 849 nm 863 nm nm 958 nm finite size 964 nm Nuclear Physics A278 (1977) p. 381 70 THz 923 nm but signal never F=1 2S1/2 2S reproduced 1/2 40 THz (10 bars, 40 bars) finite size F=0 96 THz

• µHe+ spectroscopy + He+ spectroscopy → QED test (Zα)

• improve He spectroscopy Mainz MITP Oct. 2013 98 mardi 1 octobre 2013 Muonic He spectroscopy

Mainz MITP Oct. 2013 99 mardi 1 octobre 2013 Improve statistics...

16% more from the beam time: reliquefying He Moving in on October 3rd Better disk laser beam time until December 19th Much more intensity (no Raman shift) Mainz MITP Oct. 2013 100 mardi 1 octobre 2013 Conclusions • We have performed a 12.5 ppm measurement of the Lamb-shift in muonic hydrogen • The deduced proton radius using a Dipole model is 6.9 standard deviations away from the hydrogen and electron-proton elastic scattering data • Better modeling of the proton form-factor and polarization required to confirm or reduce the disagreement • Experiment confirmed with 2nd µH line • 3 µD lines observed and being analyzed • No explanation of the discrepancy yet, but possibilities – QCD – Problems with hydrogen experiments – New physics • Muonic He in 2013 (check of theory, different laser wavelength-in the red) predictions of measurable effects from new physics!!

Mainz MITP Oct. 2013 101 mardi 1 octobre 2013 CREMA 2011

Mainz MITP Oct. 2013 102 mardi 1 octobre 2013