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].