Natural Resolution of the Proton Size Puzzle
aa r X i v : . [ phy s i c s . a t o m - ph ] M a r Natural Resolution of the Proton Size Puzzle
G. A. Miller, A. W. Thomas , J. D. Carroll , and J. Rafelski Department of Physics, University of Washington, Seattle, WA 98195-1560, CSSM, School of Physics and Chemistry, University of Adelaide, Adelaide SA 5005, Australia Department of Physics, University of Arizona, Tucson, Arizona 85721, USA (Dated: January 21, 2011)We show that off-mass-shell effects arising from the internal structure of the proton provide a newproton polarization mechanism in the Lamb shift, proportional to the lepton mass to the fourthpower. This effect is capable of resolving the current puzzle regarding the difference in the protonradius extracted from muonic compared with electronic hydrogen experiments. These off-mass-shelleffects could be probed in several other experiments.
PACS numbers: 31.30.jn,14.20.Dh,24.10.Cn
The recent, extremely precise extraction of the protonradius [1] from the measured energy difference betweenthe 2 P F =23 / and 2 S F =11 / states of muonic hydrogen (H)has created considerable interest. Their analysis yields aproton radius that is smaller than the CODATA [2] value(extracted mainly from electronic H) by about 4% or 5.0standard deviations. This implies [1] that either the Ryd-berg constant has to be shifted by 4.9 standard deviationsor that the QED calculations for hydrogen are insuffi-cient. Since the Rydberg constant is extremely well mea-sured, and the QED calculations seem to be very exten-sive and highly accurate, the muonic H finding presentsa significant puzzle to the entire physics community.Our analysis is motivated by the fact that muonic hy-drogen is far smaller than electronic hydrogen and there-fore more sensitive to corrections arising from hadronstructure. In particular, we consider the lowest order cor-rection associated with off-shell behaviour at the photon-nucleon vertex, showing that it can very naturally ac-count for the difference reported by Pohl et al. . Sinceat the present state of development of hadronic physicsit is not possible to provide a precise value for this cor-rection, our result may be viewed as a phenomenologicalstudy of the sensitivity of muonic hydrogen to importantaspects of proton structure. It should spur further studyof processes which could be sensitive to off-shell changesin proton structure. In alternate language, the explana-tion which we present may be viewed as a new contribu-tion from proton polarization that is not constrained bydispersion relations but which can be studied in systemsother than the hydrogen atom.We begin with a brief discussion of the relevant phe-nomenology. Pohl et al. show that the energy differencebetween the 2 P F =23 / and 2 S F =11 / states, ∆ e E is given by∆ e E = 209 . − . r p + 0 . r p meV , (1)where r p is given in units of fm. Each of the three coef-ficients is obtained from extensive theoretical work [3–7], typically confirmed by several groups. Studies of therelevant atomic structure calculations and correspondingefforts to improve those have revealed no variations largeenough to significantly affect the above equation [8, 9]. Using this equation, we see that the difference betweenthe Pohl and CODATA values of the proton radius wouldbe entirely removed by an increase of the first term onthe rhs of Eq. (1) by just 0.31 meV=3 . × − MeV, butan effect of even half that much would be large enough todissipate the puzzle. Finding a new effect of about thatvalue resolves the puzzle provided that the correspond-ing effect in electronic H is no more than a few parts ina million (the current difference between theory and ex-periment [3]). An effect that gives a contribution to ∆ e E of the form α m M (with m the lepton mass and M theproton mass) could therefore resolve the proton radiuspuzzle and cause no disagreement in electronic H.The search to find such an effect has attracted consid-erable interest. New physics beyond the Standard Modelmust satisfy a variety of low-energy constraints and sofar no explanation of the proton radius puzzle has beenfound that satisfies these constraints [10–14]. Attentionhas been paid to the third term of Eq. (1) [15], with theresult that its current uncertainties are far too small toresolve the proton radius puzzle [16, 17].We therefore seek an explanation based on the factthat the proton is not an elementary Dirac particle, andthat many features of its interactions are still unknown.In particular, consider the electromagnetic vertex func-tion which must depend on all of the relevant invariants.For a proton of initial four-momentum p , the most gen-eral expression must include a term, dependent on theproton virtuality, that is proportional to p − M and/or p · γ N − M , where the subscript N denotes acting on anucleon, and M is the nucleon mass. Such terms havebeen discussed for a very long time in atomic [6, 7] andnuclear physics [18]-[32]. They have been of special con-cern in relation to the difference between free and bounddeep inelastic structure functions measured in the EMCeffect [18]-[22], nucleon-nucleon scattering [23] and elec-tromagnetic interactions involving nucleons [24, 25], no-tably quasi-elastic scattering [26]-[32].Many possible forms [24, 25] include the effects of pro-ton virtuality; we consider three that could be significantfor the Lamb shift. We write the Dirac part of the vertexfunction for a proton of momentum p to absorb a photonof momentum q = p ′ − p .as:Γ µ ( p ′ , p ) = γ µN F ( − q ) + F ( − q ) F ( − q ) O µa,b,c (2) O µa = ( p + p ′ ) µ M [Λ + ( p ′ ) ( p · γ N − M ) M + ( p ′ · γ N − M ) M Λ + ( p )] O µb = (( p − M ) /M + ( p ′ − M ) /M ) γ µN O µc = Λ + ( p ′ ) γ µN ( p · γ N − M ) M + ( p ′ · γ N − M ) M γ µN Λ + ( p ) , where three possible forms are displayed. Other terms ofthe vertex function needed to satisfy the WT identity donot contribute significantly to the Lamb shift and are notshown explicitly. The proton Dirac form factor, F ( − q )is empirically well represented as a dipole F ( − q ) = (1 − q / Λ ) − , (Λ = 840 MeV) for the values of − q ≡ Q > needed here. F ( − q ) is an off-shell form factor, and Λ + ( p ) = ( p · γ N + M ) / (2 M ) is anoperator that projects on the on-mass-shell proton state.We use O a unless otherwise stated.We take the off-shell form factor F ( − q ) to vanish at q = 0. This means that the charge of the off-shell protonwill be the same as the charge of a free proton, and isdemanded by current conservation as expressed throughthe Ward-Takahashi identity [24, 25]. We assume F ( − q ) = − λq /b (1 − q / e Λ ) ξ . (3)This purely phenomenological form is simple and clearlynot unique. The parameter b is expected to be of theorder of the pion mass, because these longest range com-ponents of the nucleon are least bound and more suscep-tible to the external perturbations putting the nucleonoff its mass shell. At large values of | q | , F has the samefall-off as F , if ξ = 0. We take e Λ = Λ here.We briefly discuss the expected influence of usingEq. (2). The ratio, R , of off-shell effects to on-shell ef-fects, R ∼ ( p · γ N − M ) M λ q b , ( | q | ≪ Λ ) is constrained bya variety of nuclear phenomena such as the EMC effect(10-15%), uncertainties in quasi-elastic electron-nuclearscattering [26], and deviations from the Coulomb sumrule [27]. For a nucleon experiencing a 50 MeV centralpotential, ( p · γ N − M ) /M ∼ .
05, so λq /b is of or-der 2. The nucleon wave functions of light-front quark-models [33] contain a propagator depending on M .Thus the effect of nucleon virtuality is proportional tothe derivative of the propagator with respect to M , or ofthe order of the wave function divided by difference be-tween quark kinetic energy and M . This is about threetimes the average momentum of a quark ( ∼
200 MeV/c)divided by the nucleon radius or roughly M/
2. Thus R ∼ ( p · γ N − M )2 /M , and the natural value of λq /b is of order 2.The lowest order term in which the nucleon is suffi-ciently off-shell in a muonic atom for this correction toproduce a significant effect is the two-photon exchangediagram of Fig. 1 and its crossed partner, including an ℓP ℓ − k Pℓ FIG. 1: Direct two-photon exchange graph corresponding tothe hitherto neglected term. The dashed line denotes thelepton; the solid line, the nucleon; the wavy lines photons;and the ellipse the off-shell nucleon. interference between one on-shell and one off-shell partof the vertex function. The change in the invariant am-plitude, M Off , due to using Eq. (2) along with O µa , to beevaluated between fermion spinors, is given in the restframe by M Off = e M Z d k (2 π ) F ( − k ) F ( − k )( k + iǫ ) (4) × ( γ µN (2 p + k ) ν + γ νN (2 p + k ) µ ) × (cid:20) γ µ ( l · γ − k · γ + m ) k − l · k + iǫ γ ν + γ ν ( l · γ + k · γ + m ) k + 2 l · k + iǫ γ µ (cid:21) , where the lepton momentum is l = ( m, , , k and the nucleon momentum p = ( M, , , γ µN γ νN corresponds tothe T term of conventional notation [35], [36]. Eq. (4)is gauge-invariant; not changed by adding a term of theform k µ k ν /k to the photon propagator.Evaluation proceeds in a standard way by taking thesum over Dirac indices, performing the integral over k by contour rotation, k → − ik , and integrating over theangular variables. The matrix element M is well approx-imated by a constant in momentum space, for momentatypical of a muonic atom, and the corresponding poten-tial V = i M has the form V ( r ) = V δ ( r ) in coordinatespace. This is the “scattering approximation” [3]. Thenthe relevant matrix elements have the form V | Ψ S (0) | ,where Ψ S is the muonic hydrogen wave function of thestate relevant to the experiment of Pohl et al. We use | Ψ S (0) | = ( αm r ) / (8 π ), with the lepton-proton re-duced mass, m r . The result h S | V | S i = − α m r M π λ mMb F L ( m ) ,F L ( m ) ≡ β Z ∞ p ( x + β ) /x − x ) ξ xdx , (5)(where β ≡ m / Λ ) shows a new contribution to theLamb shift, proportional to m and therefore negligiblefor electronic hydrogen. Using O µa leads to a vanishinghyperfine HFS splitting because the operator γ µN is oddunless µ = 0.We next seek values of the model parameters λ, b, ξ of Eq. (3), chosen to reproduce the value of the neededenergy shift of 0.31 meV with a value of λ of order unity.Numerical evaluation, using ξ = 0 , e Λ = Λ, shows that λb = 2(79MeV) (6)leads to 0.31 meV. If ξ is changed substantially from 0to 1 the required value of λ would be increased by about10%. If our mechanism increases the muonic Lamb shiftby 0.31 meV, the change in the electronic H Lamb shiftfor the 2S-state is about 9 Hz, significantly below thecurrent uncertainty in both theory and experiment [3].Should some other effect account for part of the protonradius puzzle, the value of λ/b would be decreased. Wealso caution that other systems in which one might aimto test this effect could show sensitivity to the value of ξ or ˜Λ in Eq. (3).The other operators appearing in Eq. (2) yield similarresults when used to evaluate M off . Using O b , gives aterm of the T form with a Lamb shift twice that of O a ,and also a HFS term that is about -1/6 of its Lamb shift,so the value of λ/b would be decreased by 3/5. The useof O c , gives a term of the T form and the same Lambshift as O a , as well as a HFS term that is -1.7 times itsLamb shift. In this case, the value of λ/b would be about- 3/2 times that of Eq. (6). The HFSs are small enoughto be well within current experimental and theoreticallimits for electronic hydrogen. Thus each operator leadsto a reasonable explanation of the proton radius puzzle.It is necessary to comment on the difference betweenour approach, which yields a relevant proton polarizationeffect, and others [36], [37] which do not. The latter usea current-conserving representation of the virtual-photonproton scattering amplitude in terms of two unmeasur-able scalar functions, T , . Dispersion relations are usedto relate T , to their measured imaginary parts. How-ever, terms with intermediate nucleon states are treated by evaluating Feynman diagrams. This allows the re-moval of an infrared divergence by subtracting the firstiteration of the effective potential that appears in thewave function. But the Feynman diagrams involve off-shell nucleons, so that their evaluation for composite par-ticles must be ambiguous. For example, using two differ-ent forms of the on-shell electromagnetic vertex function,related by using the Gordon identity, leads to results thatdiffer. This ambiguity in obtaining T , is removed in ourapproach by postulating Eq. (2) and evaluating its con-sequences. Note also that in order to evaluate the terminvolving T using a dispersion relation one must intro-duce a subtraction function, T (0 , q ). This is uncon-strained by prior data [35] because the value of σ L /σ T at ν = ∞ is not determined [38]. Pachucki [36], in Eq.(31),assumes a form proportional to q (see our Eq. (3)) timesthe very small proton magnetic polarizability. Howeverwe are aware of no published derivation of this result.In conclusion, we have shown that a simple off-shellcorrection to the photon-proton vertex, which arises nat-urally in quantum field theory and is of natural size andconsistent with gauge invariance, is capable of resolv-ing the discrepancy between the extraction of the protoncharge radius from Lamb shift measurements in muonicand electronic hydrogen. Off-shell effects of the protonform factor were an explicit concern of both Zemach [6]and Grotch & Yennie [7]. However, it is only with the re-markable improvement in experimental precision recentlyachieved [1] that it has become of practical importance.Within the field of nuclear physics there is great inter-est in the role that the modification of nucleon structurein-medium may play in nuclear structure [39, 40]. Westress that the effect postulated here can be investigatedin lepton-nucleus scattering via the binding effects of thenucleon, as well as by lepton-proton scattering in arenaswhere two photon (or γ, Z ) effects are relevant. Acknowledgments:
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