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# Higher order proton structure corrections to the Lamb shift in muonic hydrogen

Carl E. Carlson Helmholtz Institut Mainz, Johannes Gutenberg-Universität, D-55099 Mainz, Germany Department of Physics, College of William and Mary, Williamsburg, VA 23187, USA    Marc Vanderhaeghen Institut für Kernphysik, Johannes Gutenberg-Universität, D-55099 Mainz, Germany
January 31, 2011
###### Abstract

The recent conundrum with the proton charge radius inspires reconsideration of the corrections that enter into determinations of the proton size. We study the two-photon proton-structure corrections, with special consideration of the non-pole subtraction term in the dispersion relation, and using fits to modern data to evaluate the energy contributions. We find that individual contributions change more than the total, and present results with error estimates.

preprint: MKPH-T-11-03

The recent measurement of the proton charge radius using the Lamb shift in muonic hydrogen Pohl et al. (2010) has given a value that is a startling 4%, or 5 of the previous standard deviations, lower than the values obtained from energy level shifts in electronic hydrogen Mohr et al. (2008) or from electron-proton scattering experiments Sick (2003); Bernauer et al. (2010). Specifically, the new muonic hydrogen measurement Pohl et al. (2010) gives

 RE=0.84184(67)fm, (1)

compared to the CODATA value Mohr et al. (2008)

 RE=0.8768(69)fm, (2)

or the latest electron scattering value Bernauer et al. (2010)

 RE=0.879(8) fm, (3)

where we have added in quadrature the several uncertainties given in Bernauer et al. (2010).

The promise of the muonic hydrogen measurement was that, because a muon would orbit closer to the proton than an electron, the effect of the proton structure on the energy level splittings would be enhanced and a more accurate proton radius could be obtained. Based on the quoted error limit, that promise has been achieved. However, the discrepancy from the previous results requires an explanation, and invites a reconsideration of the theoretical corrections that are involved in connecting the experimental energy shift to the proton charge radius Jentschura (2010a, b). In this note, we will focus on one of the corrections, namely the order proton size corrections to the Lamb shift.

The leading and proton structure contributions to the hydrogen Lamb shift are often given as

 ΔE=2πα3ϕn(0)2(R2E−12mrαR3(2)), (4)

where is the square of the -state wave function at the origin (which contains a factor ) and is the lepton-proton reduced mass. The quadratic term was obtained non-relativistically in Karplus et al. (1952), and one can verify from a relativistic calculation that is indeed the proton charge radius Eides et al. (2001).

The term was given by Friar Friar (1979) as,

 R3(2)=∫d3r1d3r2|→r1−→r2|3ρE(r1)ρE(r2). (5)

where is the charge density of the proton itself. Friar called the third Zemach moment, because it is reminiscent of an integral found by Zemach Zemach (1956) in the related context of hyperfine splitting.

In modern times, one should calculate the corrections field theoretically using the diagram shown in Fig. 1, as has been done by Pachucki Pachucki (1996, 1999) and by others Faustov and Martynenko (2000). We wish to reexamine the calculation here, for the purpose of better assessing the connection between the elastic and inelastic contributions, and to better evaluate the subtraction term needed in a dispersion relation that is part of the work.

The calculation of the elastic and inelastic contributions should be done together. Perhaps in the future a direct QCD calculation will be possible, and there is an exploration of the hadronic corrections to the Lamb shift using chiral perturbation theory Nevado and Pineda (2008), but for the present to obtain the required accuracy the calculation needs to be done dispersively, connecting the off-shell Compton scattering which is the hadronic side of the diagram to information obtained from electron-proton scattering. In particular, done that way the elastic contributions require no (non-existent) knowledge of form factors for situations where a proton is off-shell. It also means that certain non-pole contributions to the Compton amplitudes are not picked up by the dispersive calculation and in fact do not contribute.

We also analyze more concretely the subtraction function that appears because one of the dispersion relations does not converge if unsubtracted. The subtraction function depends on the photon four-momentum squared, , and its value at is given in terms of the proton magnetic polarizability. Its dependence can be estimated by calculating a two-pion loop contribution which couples to the nucleon as a scalar. One does not need to use a dependence assumed given by the nucleon electromagnetic form factor, as has been done previously.

The Feynman diagram for the two-photon proton-structure correction to the Lamb shift is shown in Fig. 1. To the level of accuracy needed here, all external lines have zero three-momentum. The blob corresponds to off-shell forward Compton scattering, given in terms of the Compton tensor

 Tμν(p,q)=i8πM∫d4xeiqx⟨p|Tjμ(x)jν(0)|p⟩ =(−gμν+qμqνq2)T1(ν,Q2) +1M2(pμ−p⋅qq2qμ)(pν−p⋅qq2qν)T2(ν,Q2), (6)

where , , and is the nucleon mass. A spin average is implied and the state normalization is . The functions are each even in and their imaginary parts are related to the structure functions measured in electron or muon scattering by

 ImT1(ν,Q2) =14MF1(ν,Q2), ImT2(ν,Q2) =14νF2(ν,Q2), (7)

with and where are standard Nakamura et al. (2010).

After doing a Wick rotation, where and , one obtains the energy shift as

 ΔE =8α2mπϕ2n(0)∫d4Q ×(Q2+2Q20)T1(iQ0,Q2)−(Q2−Q20)T2(iQ0,Q2)Q4(Q4+4m2Q20), (8)

where is the lepton mass, and with .

The are obtained using dispersion relations. Regge arguments Abarbanel and Nussinov (1967) suggest that satisfies an unsubtracted dispersion relation in at fixed , but that will require one subtraction. Before proceeding, we will note that the Born terms, obtained from the elastic box and crossed box of Fig. 2 and the vertex function for an incoming photon, are

 TB1(q0,Q2) =14πM{Q4G2M(Q2)(Q2−iε)2−4M2q20−F21(Q2)}, TB2(q0,Q2) =MQ2π(1+τp)G2E(Q2)+τpG2M(Q2)(Q2−iε)2−4M2q20, (9)

where , and the electric and magnetic form factors are

 GE(Q2) =F1(Q2)−τpF2(Q2), GM(Q2) =F1(Q2)+F2(Q2). (10)

The Born terms are reliable for obtaining the imaginary parts of the nucleon pole terms, but not reliable in general, since the given vertex assumes the incoming and outgoing nucleons are both on shell.

Calling the first term in the pole term, one can split the whole of into pole term and non-pole terms,

 T1(q0,Q2)=Tpole1+¯¯¯¯T1. (11)

The pole term alone evidently allows an unsubtracted dispersion relation, and this term calculated from the dispersion relation simply reproduces itself. With a once subtracted dispersion relation for , one has

 T1(q0,Q2) =Tpole1(q0,Q2)+¯¯¯¯T1(0,Q2) +q202πM∫∞νthdνF1(ν,Q2)ν(ν2−q20). (12)

The nucleon pole is isolated in and the integral begins at the inelastic threshold . Similarly, as contains only a pole term,

 T2(q0,Q2) =TB2(q0,Q2)+12π∫∞νthdνF2(ν,Q2)ν2−q20. (13)

With

 ΔE=ΔEsubt+ΔEinel+ΔEel, (14)

we obtain

 ΔEsubt=4πα2mϕ2n(0)∫∞0dQ2Q2γ1(τℓ)√τℓ¯¯¯¯T1(0,Q2), (15)
 ΔEinel=−2α2mMϕ2n(0)∫∞0dQ2Q2 ×∫∞νthdν[˜γ1(τ,τℓ)F1(ν,Q2)ν+˜γ2(τ,τℓ)F2(ν,Q2)Q2/M], (16) ΔEel=−α2mM(M2−m2)ϕ2n(0)∫∞0dQ2Q2 ×{(γ2(τp)√τp−γ2(τℓ)√τℓ)G2E+τpG2Mτp(1+τp) −(γ1(τp)√τp−γ1(τℓ)√τℓ)G2M}, (17)

where and . The auxiliary functions are

 γ1(τ) =(1−2τ)((1+τ)1/2−τ1/2)+τ1/2, γ2(τ) =(1+τ)3/2−τ3/2−32τ1/2. (18)

Both are monotonically falling functions, reducing to 1 at and falling like at large . Also

 ˜γ1(τ,τℓ) =1τℓ−τ(√τℓγ1(τℓ)−√τγ1(τ)), ˜γ2(τ,τℓ) =1τℓ−τ(γ2(τ)√τ−γ2(τℓ)√τℓ). (19)

The subtraction function has unphysical arguments, excepting the point . It comes from the excitation of the proton, and can at low (and low , in general) be described using the electric () and magnetic () polarizabilities and the effective Hamiltonian

 H=−124παE→E2−124πβM→B2. (20)

For small and , this gives

 limν2,Q2→0¯¯¯¯T1(ν,Q2)=ν2e2(αE+βM)+Q2e2βM. (21)

The term is shown to connect to known results in another context Drechsel et al. (2003), and the term was obtained by Pachucki Pachucki (1996). With the above result, the integral over converges at the lower limit.

For higher , the subtraction function comes from non-nucleon-pole contributions, and the forward amplitude is dominated by low mass intermediate states. With the limit fixed in terms of , we estimate the dependence from pion loop contributions where the two-pion state has a scalar coupling to the nucleon, as illustrated in Fig. 3. Figure 3: Diagrams used for estimating the Q2 dependence of the non-pole part of the subtraction term.

With standard Feynman rules for scalar QED and an effective coupling for the lower vertex, one obtains from these terms

 ¯¯¯¯Tμν=gS192π3m2π(q2gμν−qμqν)Floop(Q2), (22)

where, with ,

 Floop(Q2)=3λ2{√1+λln√1+λ+1√1+λ−1−2} =⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩1−Q210m2π+O(Q4m2π),Q→06m2πQ2(lnQ2m2π−2)+O(m4πQ4),Q→∞. (23)

We can identify and obtain

 ¯¯¯¯T1(0,Q2)=βM4παQ2Floop(Q2). (24)

The Particle Data Group gives Nakamura et al. (2010),

 βM=(1.9±0.5)×10−4 fm3. (25)

However, according to some recent analyses,

 βM={(4.0±0.7)×10−4 fm3(3.4±1.2)×10−4 fm3[19,20]. (26)

Using the subtraction function from Eq. (24), we find

 ΔEsubt=5.3 μeV×βM(3.4×10−4fm3). (27)

Much of the support for the integral is at low , being controlled by as well as by the dependence from the pion loop, and half the contributions to come from GeV, albeit with a long tail.

Refs. Pachucki (1999) and Martynenko (2006) found to be and eV, using and fm, respectively, and using a falloff related to the nucleon electromagnetic form factor. For the same , our results are about 30% larger due to having flatter falloff.

One can also consider inserting a form factor for each incoming photon coupling to pions, modifying the subtraction function of Eq. (24) by multiplying it with . Obtaining from the fit of Huber et al. (2008), we find eV.

The inelastic contributions depend on , and good data in the low- and resonance region is available from Jefferson Lab. Analytic representations of this data are given by Christy and Bosted Christy and Bosted (2010), in a fit valid for GeV and from threshold to GeV, where is the final hadronic mass for inelastic scattering, . From the Bosted-Christy region, we obtain a eV contribution to . We also use the fit of Capella et al. Capella et al. (1994), valid for data at low and intermediate above the resonance region, specifically GeV and GeV. This gives a eV contribution using Capella et al. (1994) for GeV in the allowed region. Contributions from higher are quite small (on the order of 0.002 eV from GeV and GeV). We thus have

 ΔEinel=−12.7 μeV. (28)

Refs. Pachucki (1999) and Martynenko (2006) quoted and eV for this contribution.

The elastic contribution depends on the nucleon form factors, for a selection of modern form factors we get

 ΔEel=⎧⎪⎨⎪⎩−27.8 μeVKelly −29.5 μeVAMT −30.8 μeVMainz 2010 [4,27]. (29)

Ref. Pachucki (1996) quoted eV using the Simon et al. form factors Simon et al. (1980) from 1980. However, the main difference between our results is not due to the newness of the form factors, but rather to our exclusion of the non-pole contributions from the elastic contributions. The non-pole contributions would be a positive (Kelly) or eV (AMT or Mainz 2010) contribution were they included.

Table 1 summarizes our numerical results (selecting the AMT form factors Arrington et al. (2007) for the elastic terms) and compares them to earlier work. (Ref. Martynenko (2006) did not calculate the elastic term, so we carried over the result from Pachucki (1996).)

Regarding the uncertainties, for the subtraction term energy, we took from Beane et al. (2005), and propagated their error limits, which are large enough to accommodate the other two values. The inelastic energy, comes mainly from Christy and Bosted (2010), which states that most of the data points are fit to within 3%. The data itself typically had 3% error limits, and we added these two errors in quadrature. For the elastic term, we estimated the error from the spread between the two newer form factor fits that we used. We added the errors in quadrature to obtain the total error.

Our results are similar to previous results in aggregate. This seems to be happenstance, since changes in the individual contributions are larger than the change in the total. The main changes occurred because we feel use of a larger magnetic polarizability is justified and because using a dispersive treatment throughout does not allow keeping the elastic non-pole contributions.

###### Acknowledgements.
CEC thanks the National Science Foundation for support under Grant PHY-0855618 and thanks the Helmholtz Gemeinschaft in Mainz and the Helsinki Institute for Physics for their hospitality. We thank Vladimir Pascalutsa for useful comments.
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