Chiral Perturbation Theory of Muonic-Hydrogen Lamb Shift: Polarizability Contribution

Eur. Phys. J. C (2014) 74:2852 DOI 10.1140/epjc/s10052-014-2852-0

Regular Article - Theoretical Physics

Chiral perturbation theory of muonic-hydrogen Lamb shift: polarizability contribution

Jose Manuel Alarcón1,a, Vadim Lensky2,3, Vladimir Pascalutsa1 1 Cluster of Excellence PRISMA Institut für Kernphysik, Johannes Gutenberg-Universität, Mainz 55099, Germany 2 Theoretical Physics Group, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK 3 Institute for Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya 25, 117218 Moscow, Russia

Received: 6 December 2013 / Accepted: 9 April 2014 / Published online: 29 April 2014 © The Author(s) 2014. This article is published with open access at Springerlink.com

Abstract The proton polarizability effect in the muonic- cf. Table 1. The ‘elastic’ and ‘inelastic’ 2γ contributions are hydrogen Lamb shift comes out as a prediction of baryon well constrained by the available empirical information on, chiral perturbation theory at leading order and our calcu- respectively, the proton form factors and unpolarized struc- (pol)( − ) = +3 µ lation yields E 2P 2S 8−1 eV. This result is ture functions. The ‘subtraction’ contribution must be mod- consistent with most of evaluations based on dispersive sum eled, and in principle one can make up a model where the rules, but it is about a factor of 2 smaller than the recent result effect is large enough to resolve the puzzle [8]. obtained in heavy-baryon chiral perturbation theory. We also In this work we observe that chiral perturbation theory find that the effect of (1232)-resonance excitation on the (χPT) contains definitive predictions for all of the above (α5 ) Lamb shift is suppressed, as is the entire contribution of the mentioned O em proton structure effects, hence no model- magnetic polarizability; the electric polarizability dominates. ing is needed, assuming of course that χPT is an adequate the- Our results reaffirm the point of view that the proton structure ory of the low-energy nucleon structure. Some of the effects effects, beyond the charge radius, are too small to resolve the were already assessed in the heavy-baryon variant of the the- ‘proton radius puzzle’. ory (HBχPT), namely: Nevado and Pineda [11] computed the polarizability effect to leading order (LO) [i.e., O(p3)], while Birse and McGovern [13] computed the ‘subtraction’ term 1 Introduction in O(p4) HBχPT (with the caveat explained in the end of Sect. 4). Here, on the other hand, we work in the framework of The eight standard-deviation (7.9σ) discrepancy in the value a manifestly Lorentz-invariant variant of χPT in the baryon of proton’s charge radius obtained from elastic electron– sector, referred to as BχPT [16–19]. At least the LO results proton scattering [1] and hydrogen spectroscopy [2] on one for nucleon polarizabilities are known to be very different hand and from the muonic-hydrogen (µH) spectroscopy in the two variants of the theory, e.g., the proton magnetic − [3,4] on the other, a.k.a. the proton charge radius puzzle [5,6], polarizability is (in units of 10 4 fm3): 1.2 in HBχPT [20] is yet to meet its fully agreeable solution. One way to solve vs. −1.8inBχPT [21–23]. Thus, the LO effect of the pion it is to find an effect that would raise the µH Lamb shift by cloud is paramagnetic in one case and diamagnetic in the about 310 µeV, and it has been suggested that proton struc- other (see [24,25] for more on HBχPT vs. BχPT). Due to (α5 ) ture could produce such an effect at O em ,e.g.[7,8]. Most these qualitative and quantitative differences it is interesting of the studies, however, derive an order of magnitude smaller to examine the BχPT predictions for the 2γ contributions to effect of proton structure beyond the charge radius [9–15]. the Lamb shift. Here we compute the polarizability effect at (α5 ) χ The O em effects of proton structure in the Lamb shift LO B PT and indeed find it significantly different from the are usually divided into the effect of (i) the 3rd Zemach LO HBχPT results of Nevado and Pineda [11]; see Table 1. moment, (ii) finite-size recoil, and (iii) polarizabilities. The Our result for the ‘subtraction’ and ‘inelastic’ contribu- first two are sometimes combined into (i) the ‘elastic’ 2γ tions differ from most of the previous works because we have contribution, while the polarizability effect is often split neglected the effect of the nucleon transition into its lowest between (ii) the ‘inelastic’ 2γ and (iii) a ‘subtraction’ term, excited state—the (1232). We argue, however (in Sect. 3), that the latter effect cancels out of the polarizability contri- a e-mail: [email protected] bution. Thus, even though the ‘subtraction’ and ‘inelastic’ 123 2852 Page 2 of 10 Eur. Phys. J. C (2014) 74:2852

Table 1 Summary of available calculations of the ‘subtraction’ (second row), ‘inelastic’ (third row), and their sum—polarizability (last row) effects on the 2S level of µH. The last column represents the χPT predictions obtained in this work; here the omitted effect of the (1232)-resonance excitation is missing in the ﬁrst two (‘subtraction’ and ‘inelastic’) numbers, but it does not affect the total polarizability contribution where it is to cancel out (µeV) Pachucki [9] Martynenko [10]Nevadoand Carlson and Birse and Gorchtein LO-BχPT Pineda [11] Vanderhaeghen [12] McGovern [13] et al. [14] [this work]

(subt) − a − . E2S 1.8 2.3 – 5.3 (1.9) 4.2 (1.0) 2.3 (4.6) 3 0 (inel) − . − . − . − . b − . − . E2S 13 9 13 8– 12 7(5) 12 7(5) 13 0(6) 5 2 (pol) − − . − . − . − . − . − . (+1.2) E2S 12 (2) 11 5 18 5 7 4 (2.4) 8 5 (1.1) 15 3 (5.6) 8 2 −2.5 a Adjusted value; the original value of Ref. [14], +3.3, is based on a different decomposition into the ‘elastic’ and ‘polarizability’ contributions b Taken from Ref. [12] values appear to be very different from the empirical values Upon the minimal inclusion of the electromagnetic field, due to neglect of the (1232) excitation, the polarizability the two Lagrangians give identical results for the O(p3) contribution is not affected by this neglect. Compton scattering amplitude and the isovector term pro- ( 2 − ) The details of our calculation and main results are pre- portional to gA 1 does not contribute. Working with the sented in the following section. Remarks on the role of the second Lagrangian, however, simplifies a lot the evaluation (1232) excitation are given in Sect. 3. The heavy-baryon of the two-loop graphs needed for the Lamb-shift calcula- expansion of our results is discussed in Sect. 4. An “effective- tion. The resulting Feynman diagrams, omitting crossed and ness” criterion is applied to the HBχPT and BχPT results in time-reversed ones, are shown in Fig. 1. (α2 ) Sect. 5. The conclusions are given in Sect. 6. Expressions for These graphs represent an O em correction to the the LO χPT forward doubly virtual proton Compton scat- Coulomb potential and can be treated in stationary pertur- 3/2 tering (VVCS) amplitude and pion electroproduction cross bation theory. Since the Coulomb wave function is O(αem ), sections are given in Appendices A and B, respectively. the first-order contribution of these graphs to the energy shift (α5 ) is O em as requested. As any energy transfer in the atomic system brings in extra powers of αem, we neglect it, and hence 2 Outline of the calculation and results consider strictly the zero-energy forward kinematics. In this case the Feynman amplitude M is a number in momentum M δ() We begin with the leading order chiral Lagrangian for the space, corresponding to a potential equal to r . Because δ pion and nucleon fields, as well as the minimally coupled of the -function only the S-levels are shifted: photons; see e.g. [16]. After a chiral rotation of the nucleon = φ2 M, field the Lagrangian resembles that of the chiral soliton EnS n (2) model; see [26] for details. As the result, the pseudovec- φ2 = 3α3 /(π 3) tor π NN interaction transforms into the pseudoscalar one, where n mr em n is the hydrogen wave function at while a new scalar–isoscalar ππNNinteraction is generated. the origin, for mr = m Mp/(m + Mp) the reduced mass The original and the redefined pion–nucleon Lagrangians, of the lepton–proton system, and m , Mp = MN the corre- expanded up to the second order in the pion field, take the sponding masses of the constituents. form It is customary for the 2γ contributions to be split into leptonic and hadronic parts, i.e., (1) / gA a / a Lπ = N i∂ − MN + τ ∂π γ5 2 4 N 2 fπ e d q 1 μν M = Lμν( , q) T (P, q), (3) 1 2m i(2π)4 q4 − τ aεabcπ b ∂π/ c N + O(π 3), (1a) 4 f 2 2 π where e = 4παem is the lepton charge squared, and L(1) = ∂/ − − gA τ aπaγ π N N i MN i MN 5 1 2 fπ Lμν = [q μ ν − (qμ ν + qν μ) · q 1 q4 − ( · q)2 2 ( 2 − ) 4 gA 2 gA 1 a abc b c 3 2 + M π + τ ε π ∂π/ N + O(π ), +gμν( · q) ] 2 N 2 (4) 2 fπ 4 fπ

(1b) is the leptonic tensor, with and q the 4-momenta of the lepton and the photons, respectively; gμν = diag(1, −1, −1, μν where N(x) and MN is the nucleon ﬁeld and mass, respec- −1) is the Minkowski metric tensor. The tensor T is the a tively, π (x) is the pion ﬁeld; gA 1.27, fπ 92.4MeV. unpolarized VVCS amplitude, which can be written in terms 123 Eur. Phys. J. C (2014) 74:2852 Page 3 of 10 2852

Fig. 1 The two-photon exchange diagrams of elastic lepton–nucleon scattering calculated in this work in the zero-energy (threshold) kinematics. Diagrams obtained (a) (b) (c) from these by crossing and time-reversal symmetry are included but not drawn

(d) (e) (f)

(g) (h) (j) of two scalar amplitudes: Focusing on the O(p3) corrections (i.e., the VVCS amplitude corresponding to the graphs in Fig. 1) we have explicitly ver- Pμ Pν T μν(P, q) =−gμν T (ν2, Q2) + T (ν2, Q2), iﬁed that the resulting NB amplitudes satisfy the dispersive 1 2 2 (5) Mp sum rules [28]: ( ) T NB (ν2, Q2) with P the proton 4-momentum, ν = P · q/M , Q2 =−q2, 1 p ∞ 2 = 2 2 2 P Mp. Note that the scalar amplitudes T1,2 are even ( ) 2ν σ (ν , Q ) = T NB (0, Q2) + dν T , (9a) functions of both the photon energy ν and the virtuality Q. 1 π ν2 − ν2 μ ν ν Terms proportional to q or q are omitted because they 0 vanish upon contraction with the lepton tensor. (NB)(ν2, 2) T2 Q Going back to the energy shift one obtains [12]: ∞ 2 2 2 2 2 ν Q σ (ν , Q ) + σ (ν , Q ) = dν T L , (9b) ∞ π ν2 + 2 ν2 − ν2 α φ2 1 Q em n 3 ν0 EnS = d q dν 4π 3m i 0 2 2 with ν0 = mπ + (mπ + Q )/(2Mp) the pion-production 2 2 2 2 2 2 2 2 (Q − 2ν ) T1(ν , Q ) − (Q + ν ) T2(ν , Q ) threshold, mπ the pion mass, and σ ( ) the tree-level cross × . (6) T L 4 4 2 2 Q [(Q /4m ) − ν ] section of pion production off the proton induced by trans- verse (longitudinal) virtual photons, cf. Appendix B. We In this work we calculate the functions T1 and T2 by hence establish that one is to calculate the ‘elastic’ con- extending the BχPT calculation of real Compton scatter- tribution from the Born part of the VVCS amplitudes and ing [26] to the case of virtual photons. We then split the the ‘polarizability’ contribution from the non-Born part, amplitudes into the Born (B) and non-Born (NB) pieces: in accordance with the procedure advocated by Birse and ( ) ( ) McGovern [13]. T = T B + T NB . (7) i i i Substituting the O(p3) NB amplitudes into Eq. (6)we The Born part is deﬁned in terms of the elastic nucleon form obtain the following value for the polarizability correction: factors as in, e.g. [13,27]: (pol) =− . µ . 4 2 2 2 E2S 8 16 eV (10) ( ) 4πα Q (F (Q )+F (Q )) T B = em D P −F2 (Q2) , 1 4 2 2 D (8a) Mp Q −4M ν χ p This is quite different from the corresponding HB PT result 2 2 for this effect obtained by Nevado and Pineda [11]: ( ) 16πα Mp Q Q T B = em F2 (Q2)+ F2 (Q2) . (8b) 2 Q4 − 4M2ν2 D 4M2 P p p (pol)( χ ) =− . µ . E2S LO-HB PT 18 45 eV (11) In our calculation the Born part was separated by subtract- ing the on-shell γ NN pion loop vertex in the one-particle- We postpone a detailed discussion of this difference till reducible VVCS graphs; see diagrams (b) and (c) in Fig. 1. Sect. 4. 123 2852 Page 4 of 10 Eur. Phys. J. C (2014) 74:2852

(NB)( , 2) π 2β , It is useful to observe that a much simpler formulas can be T1 0 Q 4 Q M1 (15a) obtained upon making the low-energy expansion (LEX) of (NB) 2 2 T (0, Q ) 4π Q (αE1 + βM1), (15b) the VVCS amplitude, assuming that the photon energy in the 2 atomic system is small compared to all other scales. To lead- where αE1 and βM1 are the electric and magnetic dipole ing order in LEX, we may neglect the ν dependence in the polarizabilities of the proton (hence the name “polarizabil- numerator of Eq. (6) and, after Wick-rotating q to Euclidean ity contribution”). Given the shape of the weighting func- hyperspherical coordinates [i.e., setting ν = iQcos χ, q = tion plotted in Fig. 2, the main contribution to the integral (Q sin χ sin θ cos ϕ, Q sin χ sin θ sin ϕ, Q sin χ cos θ)] and in Eq. (12) comes from low Q’s, and therefore βM1 can- angular integrations, find the following expression: cels out. The dominant polarizability effect in the Lamb shift thus comes from the electric polarizability αE1.TheBχPT ∞ α α physics of E1 is such that to obtain the empirical number (pol) = em φ2 dQ w(τ ) −4 3 π EnS n of about 11 (in units of 10 fm ), 7 comes from LO ( N π Q2 π 0 loops) and 4 from NLO ( loops), with uncertainty of about ( ) ( ) ±1fromtheO(p4) low-energy constant [26]. Since in the ×[T NB (0, Q2) − T NB (0, Q2)], (12) 1 2 present calculation we include only the LO π N loops, we expect our value to increase in magnitude when going to w(τ ) with the weighting function shown in Fig. 2 and given the next order (i.e., including the π loops). As the result, by we replace the usual uncertainty of 15 % ( mπ /GeV ) due to the higher-order effects by an uncertainty of 30 % √ Q2 ( − )/ w(τ ) = + τ − τ ,τ = . [ M Mp GeV] toward the magnitude increase, antic- 1 2 (13) 4m ipating in this way the effect of the π loops. The 15 % uncertainty remains toward the magnitude decrease. With χ (NB) the uncertainty thus defined, our result is Plugging in here the LO B PT expressions for Ti given in Appendix A, we obtain (pol)( χ ) =− . +1.2 µ . E2S LO-B PT 8 2−2.5 eV (16) ( ) E pol =−8.20 µeV, (14) 2S This is the number given in the third row of the last column in Table 1, where it can be compared to some previous results. i.e., nearly the same as before the LEX, cf. Eq. (10). This Most of them agree on the polarizability contribution. As for comparison shows that the LEX is applicable in this case, the ‘inelastic’ and ‘subtraction’ contributions, their meaning- ν i.e.: in the energy-shift formula of Eq. (6)the -dependence ful comparison can only be made together with discussing of the numerator can to an extremely good approxima- theroleofthe(1232)-resonance excitation. tion be neglected. As shown in Sect. 4, this approximation works well in the case of the HBχPT calculation too. 3 Remarks on the (1232) contribution To estimate the uncertainty of the LO result, we first and ‘subtraction’ observe that for low Q the VVCS amplitudes go as Presently the most common approach to calculate the polar- 1.0 izability effect relies on obtaining the VVCS amplitude from the sum rules of (9). Unfortunately, even a perfect knowledge 0.8 of the inclusive cross sections (or, equivalently, the unpolarized structure functions) determines the VVCS amplitude 0.6 (NB)( , 2) only up to the subtraction function T1 0 Q . The total result is therefore divided into the ‘inelastic’ part which is 0.4 determined by empirical cross sections, and the ‘subtrac- 0.2 tion’ term which stands for the contribution of the subtraction function. 0.0 We can also perform such a division and based on the 0.00 0.05 0.10 0.15 0.20 0.25 0.30 low-energy version of the sum rules [i.e., Eq. (12)] obtain 2 2 Q (GeV ) ∞ α (subt) em 2 dQ (NB) 2 n=2 2 E = φ w(τ ) T (0, Q ) =−3.0 µeV, Fig. 2 Plot of the Q behavior of the weighting function depending nS π n Q2 1 on the lepton mass. The blue dashed line is for the case of the electron, 0 w(τe), whereas the solid purple line is for the muon, w(τμ) (17a)

123 Eur. Phys. J. C (2014) 74:2852 Page 5 of 10 2852

invariant formulation of χPT in the baryon sector [16]. How- ever, as pointed out by Gegelia et al. [30,31], the “power- counting violating terms” are renormalization scheme dependent and as such do not alter physical quantities. Furthermore, in HBχPT they are absent only in dimensional regularization. Fig. 3 The (1232)-excitation mechanism. Double line represents the If a cutoff regularization is used the terms which superficially propagator of the violate power counting arise in HBχPT as well, and must be handled in the same way as they are handled nowadays in ∞ BχPT—by renormalization. α (inel) em 2 dQ (NB) 2 n=2 E =− φ w(τ ) T (0, Q ) =−5.2 µeV. In this work for example, all such (superficially power- nS π n Q2 2 0 counting-violating) terms, together with ultraviolet divergen- (17b) cies, are removed in the course of renormalization of the proton field, charge, anomalous magnetic moment, and mass. This looks very different from the dispersive calculation, We use the physical values for these parameters and hence cf. Table 1. The main reason for this is the (1232)- the on-mass-shell (OMS) scheme. This is different from the resonance excitation mechanism shown by the graph in extended on-mass-shell scheme (EOMS) [17], where one Fig. 3. starts with the parameters in the chiral limit. The physical We have checked that the dominant, magnetic-dipole observables, such as the Lamb shift in this case, would of (M1), part of electromagnetic nucleon-to- transition is course come out exactly the same in both schemes, pro- strongly suppressed here, as is the entire magnetic polar- vided the parameters in the EOMS calculation are cho- izability (βM1) contribution, cf. discussion below Eq. (15). It sen to yield the physical proton mass at the physical pion is not suppressed in the ‘inelastic’ and ‘subtraction’ contri- mass. bution separately, but it cancels out in the total. Thus, even Coming back to HBχPT. Despite the above-mentioned though it is well justified to neglect the graph in Fig. 3 at developments the HBχPT is still often in use. The two EFT the current level of precision, the split into ‘inelastic’ and studies of proton structure corrections done until now [11,13] ‘subtraction’ looks unfair without it. are done in fact within HBχPT. Wenext examine these results In most of the dispersive calculations the cancelation of from the BχPT perspective. the excitation, as well as of the entire contribution of One of the advantages of having worked out a BχPT result βM1, occurs too, because the subtraction function is at low is that the one of HBχPT can easily be recovered. We do it by β Q expressed though the empirical value for M1.Eventhe expanding the expressions of Appendix A in μ = mπ /MN , χ 2 2 HB PT-inspired calculation of the subtraction function [13], while keeping the ratio of light scales τπ = Q /4mπ fixed. which does not include the (1232) explicitly, is not an For the leading term the Feynman-parameter integrations are 4 exception, as a low-energy constant from O(p ) is cho- elementary and we thus obtain the following heavy-baryon 3 sen to achieve the empirical value for βM1.EvenatO(p ) expressions: HBχPT, the chiral-loop contribution to β is—somewhat M1 α 2 √ (NB) 2 HB emgA 1 counterintuitively—paramagnetic and not too far from the T ( , Q ) = mπ − √ τπ , 1 0 2 1 arctan empirical value, leading to a reasonable result for the ‘sub- 4 fπ τπ traction’ contribution. We take a closer look at the HBχPT (18a) prediction for the various Lamb-shift contributions in the fol- α 2 + τ √ (NB) 2 HB emgA 1 4 π lowing section. T ( , Q ) =− mπ − √ τπ . 2 0 2 1 arctan The central value for the ‘subtraction’ contribution obtained 4 fπ τπ by Gorchtein et al. [14] is negative, even though the - (18b) excitation is included in their ‘inelastic’ piece. The quoted The first expression reproduces the result of Birse and uncertainty of their subtraction value, however, is too large ( ) 3 1 to point out any contradiction of this result with the other McGovern (cf. T 1 in the appendix of [13] ). We have studies. also verified that these amplitudes correspond to the ones

1 At subleading order in the heavy-baryon expansion, we obtain NB (4) α g2 τ ( +τ )− √ HB= em A 2 − τ + 48√π 1 π 3 τ 4 Heavy-baryon expansion T 1 π 2 mπ 3 50 π τ ( +τ ) arcsinh π 12 fπ MN π 1π mπ +18τπ 7 + 4log . The heavy-baryon expansion, or HBχPT [20,29], was called MN ( ) to salvage “consistent power counting” which seemed to be 2 4 This expression reproduces the gA terms of T 1 in the appendix of lost in BχPT, i.e. the straightforward, manifestly Lorentz- Ref. [13], apart from the terms inside the square brackets. These terms 123 2852 Page 6 of 10 Eur. Phys. J. C (2014) 74:2852 of Nevado and Pineda [11] at zero energy (ν = 0), up to a ing. Birse and McGovern [13] computed the VVCS ampli- 2 4 convention for an overall normalization of the amplitudes. tude T1(0, Q ) to order O(p ), but they evaded the consider- We have also reproduced their expressions for T1 and T2 (cf. ation of the lepton–nucleon terms by introducing a “physical Eq. (3.2) and (3.5) in Ref. [11]) for all ν and Q2. cutoff” in Q. Hence, their resulting calculation of the subtrac- Substituting these expressions into (12), we obtain the tion term is strongly cutoff dependent and lies, strictly speak- following value for the polarizability contribution to the 2S- ing, outside the χPT framework; we refer to it as “HBχPT- level shift in µH: inspired” calculation.

(pol)( χ ) =− . µ . E2S LO-HB PT 17 85 eV (19) 5 “Effectiveness” of HBχPT vs. BχPT This is slightly different from the result of Ref. [11] that we quote in Eq. (11), which is because of the neglected energy Although at high enough orders HBχPT and BχPT are dependence, i.e., the use of the LEX in deriving Eq. (12)from bound to yield the same results, at low orders this is not (6). Still, the difference between the exact and LEX result is necessarily so and practice shows that especially at ‘predic- well within the expected 15 % uncertainty of such calculation tive’ orders, where there are no free low-energy constants and hence we conclude that the LEX approximation works to absorb the differences, HBχPT and BχPT results differ well in this case too. substantially, sometimes even in the sign of the total effect Substitution to Eq. (17) yields the HBχPT predictions for (cf. the order p3 result for the magnetic polarizability of the the ‘inelastic’ and ‘subtraction’ contributions: nucleon [24,26]). The proton polarizability contribution to ( ) the Lamb shift is apparently such a case as well. So, hav- E subt (LO-HBχPT) = 1.3 µeV, (20a) 2S ing found the substantial differences between the HBχPT (inel)( χ ) =− . µ . E2S LO-HB PT 19 1 eV (20b) and BχPT predictions the obvious question is: which one is more reliable, if any? Neglecting for a moment the difference between τπ and τμ, χ we obtain very simple closed expressions for the Lamb-shift A rather common point of view is that, since HB PT contributions: neglects only the effects of “higher order”, any substantial disagreement only signals the importance of higher-order (pol)( χ ) E2S LO-HB PT effects and hence neither of the calculations should be trusted α5 3 2 at this order. On the other hand, it is plausible that not all the emmr gA mμ ≈ (1−10G+6log2)=−16.1 µeV, (21a) higher-order effects are large, but only the ones present in 4(4π fπ )2 mπ the BχPT calculation and dismissed in the one of HBχPT. (subt)( χ ) E2S LO-HB PT In support of the latter scenario is the physical principle 2 α5 m3g mμ of analyticity—consequence of (micro-)causality, which in ≈ em r A (1 − 2G + 2log2) = 1.1 µeV, (21b) 8(4π fπ )2 mπ BχPT is obeyed exactly, while in HBχPT it is obeyed only (inel) approximately, albeit improvable order by order. E (LO-HBχPT) 2S Another, perhaps more quantitative criterion is the one put 2 α5 m3g mμ ≈ em r A (1 − 18G+10 log 2) =−17.2 µeV, forward by Strikman and Weiss [32]. In the interpretation of 8(4π fπ )2 mπ Ref. [24], it requires that the high-momentum contribution (21c) of finite (renormalized) loop integrals over quantities which where G 0.9160 is the Catalan constant. This should pro- are invariant under redefinitions of hadron fields should not vide an impression of the parametric dependencies arising exceed the expected uncertainty of the given-order calcula- in χPT for this effect. The resulting numbers are within the tion. In other words, the contribution from beyond the scales expected uncertainty for HBχPT result, and they can in prin- at which the effective theory is applicable should not exceed ciple be easily improved in a perturbative treatment of the a natural estimate of missing higher-order effects. pion–muon mass difference. In our case the VVCS amplitudes are such quantities So far we have been discussing the O(p3) result. At higher invariant under redefinitions of pion and nucleon fields and orders one in addition to the VVCS calculation needs to con- hence it makes sense to examine Fig. 4, where the polariz- sider the appropriate operators from the effective lepton– ability effect is plotted as a function of the ultraviolet cutoff Q nucleon Lagrangian with corresponding low-energy con- max imposed on the momentum integration in (12). stants fixed to, e.g., the low-energy lepton–nucleon scatter- The figure clearly shows that the relative size of the high- momentum contribution in the HBχPT case is substantially larger than in BχPT. Footnote 1 continued χ come from the expansion of the leading pion loop contribution to the Assuming the breakdown scale for PT is of order of the 2 term βM1 Q in powers of mπ and hence are part of δβ in that reference. ρ-meson mass, mρ = 777 MeV, we can make a more quanti- 123 Eur. Phys. J. C (2014) 74:2852 Page 7 of 10 2852

pol µ the calculation. The BχPT result is more “effective” in this E2 S eV 5 respect, as the high-momentum contribution therein is well within the expected uncertainty. B PT 8.2 Within the BχPT calculation, we have verified the dis- 10 persive sum rules given in (9) and confirmed the statement of Ref. [13] that the split between the ‘elastic’ and ‘inelastic’ 2γ contributions corresponds unambiguously to the split 15 HB PT between the Born and non-Born parts of the VVCS ampli- 17.9 tude, rather than between the pole and non-pole parts. 0 0.2 0.4 0.6 0.8 1 We have observed that the (1232)-excitation mechanism 2 2 Qmax GeV showninFig.3 does not impact the Lamb shift in a significant way because the dominant magnetic-dipole (M1) transition Fig. 4 The polarizability effect on the 2S-level shift in µH computed is suppressed, as is the entire magnetic polarizability effect. χ χ Q in HB PT and B PT as a function of the ultraviolet cutoff max.The The (1232)-excitation effect is, however, important for the arrows on the right indicate the asymptotic (Qmax →∞)values dispersive calculation because it is prominent in the proton structure functions and hence must be included in the ‘sub- tative statement. In the present HBχPT calculation the contri- traction’ contribution to achieve a consistent cancelation of bution from Q > mρ is at least 25 % of the total result, hence the M1 (1232) excitation. In most of the models this is exceeding the natural expectation of uncertainty of such cal- roughly achieved by using an empirical value for the mag- culations. In the BχPT case, the contribution from momenta netic polarizability which includes the large paramagnetic above mρ is less than 15 %, well within the expected uncer- effect of the M1 (1232) excitation. In the HBχPT-inspired tainty. calculation of the ‘subtraction’ term [13]the-excitation is not included; however, the situation is ameliorated by the low-energy constant from O(p4), which is chosen to 6 Conclusion and outlook reproduce the empirical value of the magnetic polarizability. Is the proton polarizability effect different in muonic versus Naive dimensional analysis shows that χPT at leading electronic hydrogen so as to affect the charge radius extrac- order is capable of yielding predictions for the entire two- tion? The answer is ‘yes’. From the LEX formula in Eq. (12), photon correction to the Lamb shift. The polarizability part one sees that the polarizability contribution not only affects of that correction has been considered in this work. The last the charge radius extraction from the Lamb shift but also that row of the last column of Table 1 contains the O(p3) BχPT this effect is about mμ/me ≈ 200 times stronger in µH than prediction for the proton polarizability effect on the 2S-level in eH. Indeed, the weighting function plotted in Fig. 2 for of µH. One needs to add to it the ‘elastic’ contribution (or, the two cases is much larger in the muon case. The lepton alternatively, the third Zemach moment together with ‘finite- χ (α5 ) mass acts, in fact, as a cutoff scale. Nonetheless, the B PT size recoil’), to obtain the full O em effect of the proton result obtained hereby demonstrates that the magnitude of structure in µH Lamb shift. Using an empirical value for the this effect is not nearly enough to explain the ‘proton radius ‘elastic’ contribution from Ref. [13] [i.e., −24.7(1.6) µeV], puzzle’, which amounts to a discrepancy of about 300 µeV. our result for the full 2γ contribution to the 2P –2S Lamb As seen from Table 1, our BχPT result for the polariz- shift is in nearly perfect agreement with the presently favored ability effect agrees with the previous evaluations based on value [5,13]of33(2) µeV. dispersive sum rules, but it is substantially smaller in magni- While the leading-order χPT calculation gives a reliable tude than the HBχPT result of Nevado and Pineda [11]. This prediction for the polarizability contribution, the splitting of is of course not the first case when the BχPT and HBχPT it into ‘inelastic’ and ‘subtraction’ works less well, because of results differ significantly—the polarizabilities themselves the missing (1232)-excitation effect, which will only enter provide such an example. at the (future) next-to-leading order calculation. Indeed, χPT The differences between HBχPT and BχPT results are is capable of providing results for the Lamb-shift contribu- often interpreted as the uncertainty of χPT calculations. This tion beyond O(p3). The main difficulty then is to include all interpretation is too naive as there are physical effects that the appropriate operators from the effective lepton–nucleon distinguish the two. For example, the BχPT calculations Lagrangian, with corresponding low-energy constants fixed obey analyticity exactly while the HBχPT ones only approx- to the two-photon exchange component of the low-energy imately. Furthermore, we have checked that in HBχPT the lepton–nucleon scattering. It will therefore be interesting but contribution from momenta beyond the χPT applicability very difficult to carry out any beyond-the-leading-order cal- domain is somewhat bigger than the expected uncertainty of culation in a systematic way. 123 2852 Page 8 of 10 Eur. Phys. J. C (2014) 74:2852

1 1 2 Acknowledgments It is a pleasure to thank M. Birse, C. E. Carl- ( ) αem g Mp T NB ( , Q2)=− A x y 2 0 2 d d son, M. Gorchtein, R. J. Hill, S. Karshenboim, N. Kivel, J. McGovern, π fπ 0 0 G. A. Miller, A. Pineda, M. Vanderhaeghen, and T. Walcher for insight- ( − )2 ( − )[( − )(− 2 + − ) + μ2 ] ful, often inspiring, discussions and communications. We furthermore × x 1 x y 1 x 1 Q y 2x 2 x 2 2 2 2 2 thank A. Antognini, M. Birse, C. E. Carlson, M. Gorchtein, J. McGov- [(x − 1)(Q (x − 1)y + Q y − x + 1) − μ x] ( − ) 2 [ 2( 2( − ) + ) − (μ2 + ) + ] ern, R. Pohl, M. Vanderhaeghen for helpful remarks on the manuscript. + 4 x 1 x y x Q y 1 y 1 2 x 1 This work was partially supported by the Deutsche Forschungsgemein- [x(−μ2 +x(Q2(y−1)y−1)+2) − 1][x2(Q2 y2 −1) − x(μ2 +Q2 y−2) − 1] schaft (DFG) through the Collaborative Research Center “The Low- 4x + [log(Q2 xy(1 − xy) + μ2 x + (x − 1)2) Energy Frontier of the Standard Model” (SFB 1044), by the Cluster of Q2 Excellence “Precision Physics, Fundamental Interactions and Structure − log(x(μ2 + x(1 − Q2(y − 1)y) − 2)+1)] of Matter” (PRISMA), and by the UK Science and Technology Facil- 4(x − 1)x3(y − 1)[Q2 y(xy − 1) − μ2] ities Council through the grant ST/J000159/1. V. L. thanks the Institut + [x(μ2 + Q2(−x)y2 + Q2 y + x − 2) + 1]2 für Kernphysik at the Johannes-Gutenberg-Universität Mainz for their ( − ) ( − )2( 2( − ) + ) kind hospitality. + 2 x 1 x 1 Q y 1 y 1 Q2 [(x − 1)2(Q2(y − 1)y − 1) − μ2 x] ( − )2( 2( − ) + ) Open Access This article is distributed under the terms of the Creative − x 1 Q y 1 y 1 Commons Attribution License which permits any use, distribution, and [(x − 1)(Q2(x − 1)y2 + Q2 y − x + 1) − μ2 x] reproduction in any medium, provided the original author(s) and the − log[Q2(1 − x)y((x − 1)y + 1) + μ2 x + (x − 1)2] source are credited. Funded by SCOAP3 /LicenseVersionCCBY4.0. + log[μ2 x − (x − 1)2(Q2(y − 1)y − 1)] 2 ( − )2 − 3 − 2Q x x 1 4 (μ2 + − ) + Appendix A: Non-Born amplitudes of zero-energy VVCS Q x x 2 1 −[(Q2 − 2)x + 2] log[x(μ2 + x − 2) + 1] Here we specify the VVCS amplitudes at ν = 0. The expres- +[(Q2 − 2)x + 2] log[x(μ2 + Q2(1 − x) + x − 2) + 1] . (23) sions are given in terms of dimensionless variables: the pion– proton mass ratio μ = mπ /Mp and the momentum-transfer Q expressed in the proton mass units. The pre-factor contains the ﬁne-structure constant αem 1/137.036, the pro- Appendix B: Tree-level electroproduction cross sections ton mass Mp 938.3 MeV, the nucleon axial coupling gA 1.27 and the pion decay constant fπ 92.4MeV.We Here we present our results for the electroproduction cross neglect the isospin breaking effects, such as differences in the sections corresponding to diagrams in Fig. 5. We give them nucleon or pion masses. For the latter we assume mπ 139 in terms of the following dimensionless variables: MeV. O(p3) χ √ + 2 + 2 The B PT expressions are given by N s Mp Q αγ = (E ) / s = , (24a) i cm 2s √ + 2 − 2 1 s Mp mπ α g2 M α = ( N ) / = , (NB)( , 2) =− em A p π E f cm s (24b) T1 0 Q 2 dx 2π fπ 2s 0 − 2 − 2 γ √ s Mp Q ⎧ βγ = / = , 1 ⎨ Ecm s (24c) 4μ2 (4μ2/Q2) + 1 + 1 2s × dy + 1log ⎩ Q2 ( μ2/ 2) + − − 2 + 2 4 Q 1 1 π √ s Mp mπ 0 βπ = E / s = , (24d) cm 2s 3(x − 1) + [log(Q2(−(x − 1))x + μ2 x + (x − 1)2) Q2 ( − 2 − 2)2 + 2 √ s Mp Q 4sQ λγ =|| / = , − ( 2 + (μ2 − ) + )] qi cm s (24e) log x 2 x 1 2s ( − )2 [( − )2( 2 2 − ) − μ2 ] (s − M2 + m2 )2 − sm2 − 2 x 1 x x 1 Q y 1 x √ p π 4 π [(x − 1)2(Q2(y − 1)y − 1) − μ2 x][(x − 1)(Q2(x−1)y2+Q2 y−x+1)−μ2 x] λπ =|q f | / s = , (24f) cm 2s ( − )2( − )[( − )( 2( − ) 2 − 2( − ) + − ) − μ2 2] + x 1 y 1 x 1 Q x 1 y Q x 2 y x 1 x [(x−1)(Q2(x−1)y2+Q2 y − x+1) − μ2 x]2 N where (E )cm is the energy of the incoming nucleon, i γ 2 ( N ) 4x (x − 1)(y − 1) E f cm is the energy of the outgoing nucleon, Ecm the energy − π x2(Q2 y2 − 1) − x(μ2 + Q2 y − 2) − 1 of the incoming photon, Ecm the energy of the outgoing pion, ( − )2 |qi |cm the relative three-momentum of the incoming particles − 4x x 1 x2[Q2(y − 1)y − 1]−(μ2 − 2)x − 1 and |q f |cm the relative three-momentum of the outgoing par- ⎫ ticles, all in the center-of-mass frame (CM). ⎬ 2x(x − 1) + − 2 , (22) We show below the results obtained for the pion electro- 2 + μ2 − + ⎭ x 2 x 1 production cross sections for the different channels. They 123 Eur. Phys. J. C (2014) 74:2852 Page 9 of 10 2852

1 (1) (2) (3)

Fig. 5 Graphs for pion electroproduction amplitude at leading order. The π NN couplings are pseudoscalar as derived from the transformed Lagrangian Eq. (1b)

2 2 2 have been calculated by using the energy of the incoming ×[2λγ (−Q (1 − s)(βγ (Q virtual photons in the laboratory frame as the ﬂux factor of 2 + 2sβγ βπ ) + (1 + s(−1 + 2βγ (−1 + 2απ + βπ )))λγ ) the incoming particles. We have checked that they reproduce 2 2 2 2 2 2 2 the result at the real photon point shown in Refs. [26]. As × ((Q + 2sβγ βπ ) − 4s λπ λγ ) + μ (−2(s − 1) 2 2 2 8 6 in Appendix A, Q and s are in the units of proton mass. We × βγ (Q + 2sβγ βπ ) + (Q + 4Q sβγ βπ have 2 − 4Q (1 − s)sβγ ((1 − s)(−1 + απ ) + 4sβγ βπ ) 2 4 2 (π+ ) αemg λπ 2sλγ + 4Q sβγ (s − 1 + sβγ β ) σ n = A [2μ2((s − 1)2 π T 4 f 2s2(s − 1 + Q2)λ3 (s − 1)2 2 2 π γ + 4s (s − 1)βγ (2βπ ((1 − s)(−1 + απ ) 2 2 4 2 −Q sλ ) + (1 − s)(Q + 2Q sβγ βπ 2 2 γ + 2sβγ βπ ) + (s − 1)λπ ))λγ + ( − + β β )λ2 )] 2 2 2 2 2 2s 1 s 2s γ π γ − 4s ((1 + s )(απ − 1) + (Q + 2sβγ βπ ) 1 2 2 4 2 2 2 2 4 + [2μ (1 − s)(Q + 2sβγ βπ ) + (Q + 4s βγ )λπ − 2s((απ − 1) + 2βγ λπ ))λγ (s − 1)λπ 1 − s 2 2 2 4 2 6 2 + Q ((Q + 2sβγ βπ ) + 16s λπ λγ ))]+ [βγ (Q + 2sβγ βπ ) sλπ λ λ 2 2 2 2s π γ + (α − )λ2 ][β ( 4 + μ2( − ) + 2 β β ) −4s λπ λγ )] arctanh , (25) 2s π 1 π γ Q 2 1 s 2Q s γ π Q2 + 2sβγ βπ λ λ 2 2 2 2s π γ 2 +2s(2μ + Q απ )λγ ] arctanh , 0 α g λπ 2 σ (π p) = em A Q + 2sβγ βπ T 2 f 2(s − 1 + Q2)(s − 1)2 π (27) 2 1 0 α g λπ × σ (π p) = em A L 2 2 2 2 3 −2s(1 + s(−1 + 2βγ βπ ))2λ2 + 8s3λ2 λ4 4 fπ Q (s − 1 + Q )(s − 1) λγ γ π γ 2 ×[(1 − s)(Q (s − 1 − 2sβγ βπ ) 1 2 2 × [ μ λγ (−( − s) 2 2 2 2 4 1 2 2 (1 + s(−1 + 2βγ βπ )) − 4s λπ λγ − 2s(s − 1 + 2sβγ βπ )λγ )((1 + s(−1 + 2βγ βπ )) 2 2 2 2 2 2 2 × βγ (1 + s(−1 + 2βγ βπ )) + (−2s(1 + απ )βγ − 4s λπ λγ ) + 2μ (−(s − 1) (1 4 2 2 2 2 + (Q + 2s βγ (3 + 3απ − 4βγ βπ − 2απ βγ βπ + βγ λπ )) + s(−1 + 2βγ βπ )) + 2s(Q (1 + 2s(−1 + βγ βπ ) 2 4 2 2 2 2 2 + s (Q (1 + 2βγ βπ (−1 + βγ βπ )) + 2s βγ (απ − 2απ βγ βπ + s (1 + 2βγ βπ (−1 + βγ βπ ))) + 2(s − 1) sλπ )λγ 2 2 4 + (1 − 2βγ βπ ) + βγ λπ )) − 2s(Q (1 − βγ βπ ) 2 3 2 4 1 − 4Q s λ λ )]+ (1 − s) 2 π γ 2 3 + βγ ( + απ ( − βγ βπ ) + βγ ( βπ (βγ βπ − ) 4s λπ λγ s 3 3 4 2 2 2 + λ2 ))))λ2 − 2(( + α2 ) + 4λ2 − ( + α2 ×[−((2μ2 + Q2)(1 − s) π γ 2s 1 π Q π 2s 1 π 2 2 2 2 2 4 2 − 2βγ βπ + 2βγ λ ) + s (α + (1 − 2βγ βπ ) + 4βγ λ ))λ + 2Q sβγ βπ )(1 + s(−1 + 2βγ βπ )) π π π γ 4 2 6 2 2 2 2 2 2 2 + 8s λπ λγ ) − 2Q (1 − s)λγ (sβγ (−1 + 2βγ βπ ) + 2s(1 − 2s + s + 2Q (μ + sλπ ))λγ ] 2 2 2 + s(1 + 2βγ (−2 + 2απ + βπ ))λγ + βγ − λγ ) 2sλπ λγ × , arctanh (26) × (1 + s(−1 + 2βγ βπ − 2λπ λγ )) 1 + s(−1 + 2βγ βπ ) 2 × (1 + s(−1 + 2βγ βπ + 2λπ λγ ))] (π+ ) α g λπ σ n = em A − L 2 2 2 2 3 1 s 2 4 2 2 2 fπ Q (s − 1 + Q )(s − 1) λγ + [2μ (−Q λγ + ((1 − s)βγ + 2sλγ )(βγ + sβγ sλπ 1 2 2 × × (−1 + 2βγ βπ ) + 2sαπ λγ )) + Q ((βγ − λγ ) 2 2 2 2 2 (Q + 2sβγ βπ ) − 4s λπ λγ 2 + s(βγ (−1 + 2βγ βπ ) + λγ + 2(απ − 1)λγ ))((βγ + λγ ) 123 2852 Page 10 of 10 Eur. Phys. J. C (2014) 74:2852

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