Proton form-factor dependence of the finite-size correction to the Lamb shift in muonic hydrogen
aa r X i v : . [ phy s i c s . a t o m - ph ] A ug Proton form-factor dependence of the finite-size correction to the Lamb shift inmuonic hydrogen
J. D. Carroll ∗ and A. W. Thomas Centre for the Subatomic Structure of Matter (CSSM),Department of Physics, University of Adelaide, SA 5005, Australia † J. Rafelski
Departments of Physics, University of Arizona, Tucson, Arizona, 85721 USA
G. A. Miller
University of Washington, Seattle, WA 98195-1560 USA (Dated: October 15, 2018)The measurement of the 2 P F =23 / to 2 S F =11 / transition in muonic hydrogen by Pohl et al. [1] andsubsequent analysis has led to the conclusion that the rms radius of the proton differs from theaccepted (CODATA) [2] value by approximately 4%, corresponding to a 4.9 σ discrepancy. Weinvestigate the finite-size effects—in particular the dependence on the shape of the proton electricform-factor—relevant to this transition using bound-state QED with nonperturbative, relativisticDirac wave-functions for a wide range of idealised charge-distributions and a parameterization ofexperimental data in order to comment on the extent to which the perturbation-theory analysiswhich leads to the above conclusion can be confirmed. We find no statistically significant dependenceof this correction on the shape of the proton form-factor. PACS numbers: 36.10.Ee,31.30.jr,03.65.Pm,32.10.Fn
I. INTRODUCTION
The measurement and subsequent analysis of the2 P F =23 / to 2 S F =11 / transition by Pohl et al. [1] concludesthat the proton rms charge-radius is approximately 4%smaller than previously accepted (as per the 2006 CO-DATA value [2]). If accurate, this would indicate thateither QED is an incomplete description of the contri-butions to the transition, something has been missed, orsomething has been incorrectly calculated.Following the work of Refs. [3–5], we focus on one par-ticular contribution to this transition, namely the finite-size correction to the 2 P / –2 S / Lamb shift. We reporton the form-factor dependence of this correction as wellas the implications for the analysis leading to the protonrms charge-radius. Discussions of the form-factor depen-dence of finite-size effects have recently been reignited byde R´ujula [6] and this work should settle claims made inthat reference.
II. NUMERICAL METHOD
The numerical method used here has recently beensummarised in Carroll et al. [7] and will not be repeatedhere. The highly abridged description is that we usean effective Dirac equation for a muon (with a reduced ∗ Electronic address: [email protected] † Recently updated: http://physics.nist.gov/cgi-bin/cuu/Value?rp mass appropriate to the µ -p system). This is expectedto provide a precise approximation to the two-particleBethe-Saltpeter equation, yielding accurate muon wave-functions for the various potentials studied.The eigenvalues for each eigenstate can be calculatedby inserting the various potentials into the effective Diracequation and integrating iteratively to produce the con-verged wave-function. Accuracy is controlled by compar-ing the converged eigenvalues with those calculated usinga virial theorem and errors are conservatively found to be ± . µ eV. III. PROTON FINITE-SIZE CORRECTIONS
As a first non-perturbative approximation, the Lambshift in hydrogen is calculated using the point-Coulomband point vacuum polarization potentials V ( r ) = V C + V VP ( r )= − Zαr − Zαr α π Z ∞ e − m e qr q r − q (cid:18) q (cid:19) d ( q ) , (1)where here m e represents the electron mass which arisesas this is a consideration of the production of a vir-tual electron-positron pair. The momentum integrationvariable has been changed to d ( q ), perhaps disguisingthe fact that the lower cut-off of q = 4 corresponds to q = 2 m e —the energy required to produce the pair.These potentials can be modified to account for thefinite-size of the proton by convoluting the point po-tential with the proton charge-distribution. For exam-ple, the modification of the Coulomb potential gives theFourier transform of the Coulomb potential in atoms e V ( ~q ) = − Zα π G E ( ~q ) ~q , (2)where we note that as the energy transfer to the pro-ton is essentially negligible, ~q and the invariant, Q ,are functionally identical. The coordinate-space poten-tial can then be written in terms of a three-dimensionalFourier transform of G E ( ~q ): ρ ( r ) ≡ Z d q (2 π ) e − i~q · ~r G E ( ~q ) , (3)as V C ( r ) = − Zαr → − Zα Z ρ ( r ′ ) | ~r − ~r ′ | d r. (4)Since the potential of Eq. (4) involves the pro-ton charge-distribution ρ ( r )—itself a function of therms charge-radius—this leads to a radius-dependentquantity. The dependence on the choice of charge-distribution is investigated here. We enforce that acharge-distribution must satisfy Z ρ ( r ) d r = 1 , (5)and we can investigate the effect of using different formsfor ρ ( r ). If we use, say, an exponential form (correspond-ing to a dipole form-factor, see Eq. (21)) ρ ( r ) = Ae − Br , (6)then we can define the rms charge-radius via a ratio ofthe moments of the charge-distribution, as h r i = Z r ρ ( r ) d r Z ρ ( r ) d r , (7)for which the value of A in Eq. (6) is arbitrary, andwhich can be rearranged such that we arrive at B = q / h r p i (and A = B / π in order to correctly nor-malise the distribution). The normalized exponentialcharge-distribution is then given by ρ E ( r ) = η π e − ηr ; η = q / h r p i . (8)We can, however, perform the same procedure for al-ternative charge-distributions and determine the depen-dence on this choice. For a Gaussian charge-distribution —corresponding to a Gaussian form-factor —the nor-malised form is given by ρ G ( r ) = (cid:18) η ′ π (cid:19) / e − η ′ r ; η ′ = 3 / h r p i . (9)Similarly, for a Yukawa charge-distribution—corre-sponding to a monopole form-factor —the normalizedform is given by ρ Y ( r ) = (cid:18) η ′′ π (cid:19) e − η ′′ r /r ; η ′′ = q / h r p i . (10)To ensure that we are considering realistic distribu-tions, we also include in our analysis a charge distribu-tion extracted from a fit to experimental data of the Sachselectric form factor of the proton [11] G E ( Q ) given inthat reference by G E ( Q ) = 1 + q τ + q τ + q τ q τ + q τ + q τ + q τ + q τ , (11)for which the values of q i are given in Table I, andfor which τ = Q / M P . This parameterization is con-strained at O ( Q ) bylim Q → G E ( Q ) = 1 − Q h r p i O ( Q ) (12)to reproduce q h r p i = 0 .
878 fm. The charge distributionfor this form factor is calculated via a Fourier transformof G E . We will herein refer to this distribution as ‘ G E fitted’.We note several efforts [8, 9] to include an additional‘Darwin-Foldy’ (or similarly named) contribution to thedefinition of the charge radius beyond that determinedin Eq. (7) and used in Eq. (12). We note that theDarwin-Foldy term was explicitly calculated by Barkerand Glover [10] as part of the Breit potential. Togetherwith the other terms in the Breit potential this is al-ready included in the recoil correction to the Lamb shiftas calculated by previous authors (e.g. [3]) and as suchappears in the complete analysis determining r p (Line 17of Table 1 of Ref. [1] supplementary).The model charge-distributions used in our analysisare plotted for comparison in Fig. 1 and we note thestriking differences between the shapes below r = 0 . πr (as used in the normalization) and though thedifferences are reduced, they remain non-trivial.We can use these charge-distributions (calculated ateach of four selected rms charge-radii spanning 0.2 fmsurrounding the values given in Refs. [1] and [2]) to cal-culate the finite-size Coulomb potential of Eq. (4) —which is plotted in Fig. 3—and hence the convergedDirac wave-functions in response to this, in order to cal-culate the eigen-energies λ α of the 2 S / and 2 P / eigen-states for each charge-distribution. We can then calculate FIG. 1: Comparison of exponential, Yukawa, Gaussian, and G E -fitted charge-distributions, each normalized to unity asper Eq. (5), calculated for p h r p i = 0 .
878 fm. Note the strik-ing differences between the shapes below r = 0 . G E -fitted distribution, asexpected. the deviation of the Lamb shifts ( δ = λ S − λ P ) calcu-lated in the point-Coulomb and finite-Coulomb cases todetermine the magnitude of the correction∆ E finite = δ finite − δ point . (13)We calculate the proton finite-size correction to theLamb shift using the aforementioned effective Dirac equa-tion method for several choices of charge-distribution TABLE I: Coefficients of polynomial fit to Sachs electric formfactor data for the proton taken from [11] as used in Eq. (12). i q i . . . . − . . . . G E -fitted charge-distributions weighted appropriately as theycontribute to the Lamb shift, each normalized to unity as perEq. (5), calculated for p h r p i = 0 .
878 fm. Here the differencesare perhaps not as striking, but still noticeably different. (viz exponential, Gaussian, and Yukawa) at severalseparated values of the proton rms charge-radius (viz q h r p i = 0.7 fm, 0.84184 fm, 0.8768 fm, 0.9 fm). Withthis information, we calculate a polynomial fit to the dataof the form ∆ E finite = a h r i + b h r i / , (14)in order to compare with other published data. A dis-cussion of the role of finite proton size in vacuum po-larization potential is given in Ref. [7], Table I.. Therelevant parameters of our fits are shown in Table II, andwe find no significant dependence on the shape of theproton charge distribution.Moreover, we are able to make a comparison to the per-turbative finite-size correction (due to the finite-Coulombpotential) to the Lamb shift as referenced in Ref. [1] andderived in full in Ref. [14] as∆ E finite = − π Zα ( Zαµ ) π (cid:20) h r p i − Zαµ h r p i + ( Zα ) (cid:0) F Rel + µ F NR (cid:1)i , (15)with the caveat that this expression was derived for an ex-ponential charge distribution (corresponding to a dipoleform-factor) and does not equally apply for other distri-butions. The derivation of Eq. (15) assumes that theSchr¨odinger wave-function is appropriate, in that thevalue at the origin | φ n (0) | = ( Zαµ ) n π (16)appears, and as such this expression requires a relativis-tic correction F Rel . Alternatively, by numerically cal-culating the converged Dirac wave-functions we requireno such perturbative correction, and a comparison withthat given in Eq. (15) is consistent. We find agreementbetween our Dirac calculation with an exponential dis-tribution and various evaluations of Eq. (15) to within0 . r p we cannot determine apolynomial dependence on this quantity. We can howeverinterpolate the shifts from our three models to compareat a single value of r p . The result of such a comparisonis that at r p = 0 .
878 fm, the contribution to the Lambshift due to the finite size of the proton is given by∆ E Exponentialfinite (0 .
878 fm) = − . , (17)∆ E Yukawafinite (0 .
878 fm) = − . , (18)∆ E Gaussianfinite (0 .
878 fm) = − . , (19)∆ E G E fittedfinite (0 .
878 fm) = − . , (20)in keeping with our conclusion that the form-factor shapeis of negligible influence.We note Ref. [12] in which a choice of electric form-factor parameterization is compared to an idealizeddipole form-factor G D ( Q ) = (cid:0) Q / Λ (cid:1) − , (21)and for which the ratio of the two tends to unity at low Q . The ratio remains close to unity up to approximately1 (GeV/c) , re-enforcing that a dipole is a suitable pa-rameterization of the electric form-factor for the purposesof this analysis. IV. IMPLICATIONS FOR THE PROTONRADIUS
With the parameterizations of the finite-size contribu-tion to the Lamb shift determined, it is possible to infera proton rms charge radius h r p i / by re-analyzing themeasured transition in muonic hydrogen of Ref. [1]. Ifwe take all other contributions to the transition primafacie (we note that because of the unknown magnitudeof off-shell corrections to the photon-nucleon vertex [15]such an analysis is physically inappropriate) we can solvethe cubic equation L measured = L r − indep + a ′ h r i + b ′ h r i / , (22) FIG. 3: Comparison of exponential, Yukawa, Gaussian,and G E -fitted finite-size Coulomb potentials calculated for p h r p i = 0 .
878 fm. (where we note that a ′ and b ′ may also account for finite-size effects in the 2 S hyperfine splitting [7], not includedhere, and thus a ′ = a , b ′ = b ) in which the measured tran-sition energy is L measured = 206 . ± . L r − indep = 209 . ± . TABLE II: Coefficients of polynomial fits to the finite-sizecorrection to the Lamb shift ∆ E Lambfinite (refer to Eq. (14)) inmuonic hydrogen for several choices of charge-distribution(calculated using the finite-Coulomb potential), and selectedpublished values. All values in this table include a radiativecorrection of − . h r p i as per Ref. [1]. We note that thefinite-size effects of the vacuum polarization alter these val-ues further, and a discussion of this matter can be found inRef. [7].Name ρ ( r ) a [ ∝ h r i ] b [ ∝ h r i / ]exponential Ae − Br − . . Ae − Br /r − . . Ae − Br − . . − . . − . . − . . TABLE III: Proton rms charge-radius p h r p i calculated us-ing various charge-distributions. In these calculations, theremaining analysis of Ref. [1] is taken prima facie , includingthe radiative (and other) corrections to the finite-size effect.The errors in the radii calculated here are dominated by theexperimental error in L measured . Also shown are the values ob-tained in Ref. [1] and the previously accepted 2006 CODATAvalue [2].Name ρ ( r ) p h r p i [fm]exponential Ae − Br . Ae − Br /r . Ae − Br . . . and the remaining coefficients are taken from Table II.Of the three solutions to Eq. (22), only one is physicallymeaningful. The physically meaningful value of the pro-ton rms charge-radius calculated for each of the choices ofcharge-distribution are given in Table III and comparedin Fig. 4.We do not calculate a prediction for the proton rmscharge radius based on the G E fitted charge distribu-tion as the above analysis requires knowledge of thepolynomial dependence on this quantity, which is absentfrom this aspect of our investigation. Nonetheless, thesimilarity between predictions of the finite-size effect at r p = 0 .
878 fm as given in Eqs. (17–20) suggest that nosignificant changes would be found.
V. CONCLUSIONS
The dependence of the proton rms charge-radiusextracted from an analysis of the measured transi-tion in muonic hydrogen on the choice of protoncharge-distribution (and thus form-factor) investigatedhere is shown to be of negligible importance, despite the wide range of higher-order moments (given that h r i = 5 h r i / h r i / h r i / Acknowledgments
This research was supported in part by the UnitedStates Department of Energy (under which Jeffer-son Science Associates, LLC, operates Jefferson Lab)via contract DE-AC05-06OR23177 (JDC, in part);grant FG02-97ER41014 (GAM); and grant DE-FG02-04ER41318 (JR), and by the Australian Research Coun-cil, FL0992247, and the University of Adelaide (JDC,AWT). GAM and JR gratefully acknowledge the sup-port and hospitality of the University of Adelaide whilethe project was undertaken. [1] R. Pohl, A. Antognini, F. Nez, F. D. Amaro, F. Biraben,et al., Nature (and Supplementary Material) , 213(2010).[2] P. J. Mohr, B. N. Taylor, and D. B. Newell, Rev. Mod.Phys. , 633 (2008), 0801.0028.[3] E. Borie, Phys. Rev. A , 032508 (2005),physics/0410051.[4] A. Martynenko, Phys.Atom.Nucl. , 125 (2008), hep-ph/0610226.[5] A. Martynenko, Phys.Rev. A71 , 022506 (2005), hep-ph/0409107.[6] A. De Rujula, Phys. Lett.
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