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The Proton Charge Radius Unsolved Puzzle

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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].
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  • Physics Has a Core Problem

    Physics Has a Core Problem

    Physics Has a Core Problem Physicists can solve many puzzles by taking more accurate and careful measurements. Randolf Pohl and his colleagues at the Max Planck Institute of Quantum Optics in Garching, however, actually created a new problem with their precise measurements of the proton radius, because the value they measured differs significantly from the value previously considered to be valid. The difference could point to gaps in physicists’ picture of matter. Measuring with a light ruler: Randolf Pohl and his team used laser spectroscopy to determine the proton radius – and got a surprising result. PHYSICS & ASTRONOMY_Proton Radius TEXT PETER HERGERSBERG he atmosphere at the scien- tific conferences that Ran- dolf Pohl has attended in the past three years has been very lively. And the T physicist from the Max Planck Institute of Quantum Optics is a good part of the reason for this liveliness: the commu- nity of experts that gathers there is working together to solve a puzzle that Pohl and his team created with its mea- surements of the proton radius. Time and time again, speakers present possible solutions and substantiate them with mathematically formulated arguments. In the process, they some- times also cast doubt on theories that for decades have been considered veri- fied. Other speakers try to find weak spots in their fellow scientists’ explana- tions, and present their own calcula- tions to refute others’ hypotheses. In the end, everyone goes back to their desks and their labs to come up with subtle new deliberations to fuel the de- bate at the next meeting. In 2010, the Garching-based physi- cists, in collaboration with an interna- tional team, published a new value for the charge radius of the proton – that is, the nucleus at the core of a hydro- gen atom.