PHYSICAL REVIEW D 93, 114005 (2016) Single pion contribution to the hyperfine splitting in muonic hydrogen Nguyen Thu Huong,1 Emi Kou,2 and Bachir Moussallam3 Faculty of Physics, VNU University of Science, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam Laboratoire de l’Accélérateur Linéaire, Université Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, 91898 Orsay Cédex, France Groupe de physique théorique, IPN, Université Paris-Sud 11, 91406 Orsay, France (Received 28 December 2015; published June 2016) A detailed discussion of the long-range one-pion exchange (Yukawa potential) contribution to the 2S hyperfine splitting in muonic hydrogen, which had, until recently, been disregarded, is presented We evaluate the relevant vertex amplitudes, in particular ỵ , combining low energy chiral expansions together with experimental data on π and η decays into two leptons A value of EHFS ẳ 0.09 ặ 0.06Þ μeV is obtained for this contribution DOI: 10.1103/PhysRevD.93.114005 I MOTIVATION − The first accurate measurement of the Lamb shift transition in muonic hydrogen [1] has led, with the help of the currently accepted theoretical formulas (e.g., [2,3]), to a determination of the proton radius rE with a precision of 0.8 per mil The proton size puzzle arose from the discrepancy, by standard deviations, between this result and the CODATA-2010 value [4], which was based on ordinary hydrogen spectroscopy as well as ep scattering This has stimulated a number of new theoretical and experimental investigations (see, e.g., the review in [5]) In particular, Antognini et al [6] have measured both F¼0 F¼2 F¼1 the νt ≡ 2SF¼1 1=2 − 2P3=2 and the νs ≡ 2S1=2 − 2P3=2 transitions which has confirmed and refined the previous result on the Lamb shift (increasing the rE discrepancy to 7σ) and further provides an experimental value for the 2S hyperfine splitting1 2SF¼1 1=2 Eexp HFS ẳ 22.808951ị meVị: 2PFẳ2 1=2 1ị The hyperfine splitting (HFS) is interesting as it probes aspects of the proton structure somewhat differently from the Lamb shift While the influence of the proton radius rE is suppressed, the main structure dependent contribution is proportional to the Zemach radius rZ : EZHFS ẳ 0.162110ịrZ meV (with rZ in fm), as given in the review [8], and the next main structure dependent contribution is that associated with the forward proton polarizabilities It has been estimated in Ref [9] as ΔEpol HFS ẳ 8.0 ặ 2.6ị eV The 2S hyperfine splitting is extracted from the experimental 2P3=2 measurements through equation ΔE2S HFS ẳ hs ht ỵ EHFS where h is the Planck constant and the 2P hyperfine splitting 2P ΔEHFS3=2 and the 2P F ¼ mixing parameter δ are computed theoretically [3,7] (see also [10]) It is noteworthy that the value of rZ that one determines from the HFS measurement in muonic hydrogen, rZ ẳ 1.08237ị fm [6], is in agreement with the value computed in terms of the proton form factors GE , GM measured in ep scattering, rZ ẳ 1.08612ị fm [11], at the present level of accuracy A possible role in muonic hydrogen of light, exotic (universality violating) particles, with vector or axial-vector (J PC ẳ ; ẳ 1ỵỵ ) quantum numbers has been considered [12,13] Similarly, the influence of exchanging a light pseudoscalar particle (JPC ẳ 0ỵ ) was recently studied in Ref [14] In that case, the HFS splitting is affected but not the (appropriately defined) Lamb shift In this article, we point out that a light pseudoscalar particle exists within the standard model, the neutral pion, and we perform the exercise to estimate the influence of the one-pion exchange mechanism on ΔEHFS We will show that using chiral symmetry allows one to evaluate the two vertex functions which are needed, represented by blobs in Fig 1, for small momentum transfer, based on experimental data The coupling of the π to a lepton pair proceeds (within the standard model) via two virtual photons The μp → μp 2470-0010=2016=93(11)=114005(7) FIG 114005-1 Single pion exchange in the μp → μp amplitude © 2016 American Physical Society HUONG, KOU, and MOUSSALLAM PHYSICAL REVIEW D 93, 114005 (2016) one-pion exchange amplitude can also be viewed as a two-photon exchange amplitude The pion pole in the Compton amplitude γp → γp contributes to the so-called proton backward spin polarizability γ π (e.g [15]) The corresponding contribution in muonic hydrogen is then expected to be suppressed by one power of α as compared to the forward proton polarizability contribution This explains why the simple mechanism of Fig does not seem to have been previously considered until very recently [16,17] Some enhancement might be expected from the fact that γ π is numerically large compared to the forward polarizabilities αp , βp and from the fact that the Yukawa potential has a relatively long range (on the scale of the proton size) which increases the overlap with the atomic wave functions As a final motivation, let us recall that the ỵ coupling plays a significant role among the hadronic contributions to the muon g − [18] and it is thus of interest to probe the level of sensitivity of muonic hydrogen to this coupling II PION COUPLING AMPLITUDES TO LEPTONS AND TO NUCLEONS A π0 -lepton coupling For low momentum transfer, the Plỵ l vertex amplitude, where P is a light neutral pseudoscalar meson (π or η) and lỈ is a light lepton ( or μỈ ), can be evaluated in the chiral expansion2 [19] At leading order, the amplitude is given from the two diagrams shown in Fig In the one-loop diagram, the Pγγ vertex is generated by the Wess-Zumino-Witten Lagrangian (see [20], Chap 22) LWZ ỵ p F F ẳ 8F 2ị with the sign corresponding to the convention ϵ0123 ¼ (we also use γ ¼ iγ γ γ γ ) This diagram accounts for the contributions of photons with low energy compared to GeV The higher energy contributions are parametrized through two chiral coupling constants χ , χ in the Lagrangian [19], LSLW ¼ 3iα2 μ lγ γ lðχ hQ2 U † Dμ U Q2 UD U i 32 ỵ hQU† QDμ U − QUQDμ U† iÞ; FIG Feynman graphs which generate the π -lepton vertex amplitude at leading order in the chiral expansion U ¼ exp iΦ ; F p ỵ p ỵ 2K ỵ p3 B p p C B η 2K C Φ ¼ B 2π − −π þ pffiffi3 C A @ pffiffiffi p ffiffi ffi − p23 2K 2K 4ị D U ẳ U iv ỵ a ịU ỵ iUv aμ Þ; ð5Þ and where vμ ðaμ Þ are external vector (axial-vector) sources (see [21]) and Q is the charge matrix, Q ẳ diag2=3; 1=3; 1=3ị The tree graph shown in Fig is computed from this Lagrangian The coupling constants χ , χ remove the ultraviolet divergence of the one-loop graph Assuming the leptons to be on their mass shell, the Plỵ l vertex amplitude can be expressed in terms of a single Dirac structure, iT Plỵ l ẳ rP ml Al p1 p2 Þ2 Þul ðp2 Þγ ul ðp1 Þ; 2π F 6ị p where rP ẳ 1; 1= if P ¼ π, η In practice, dimensional regularization brings in some scheme dependence because of the presence of the γ matrix For instance, the amplitudes computed in Refs [19] and [22] differ by a constant Some discussion of this point can be found in Ref [23] For definiteness, we will choose the convention of [22], which gives Al ðsÞ in the form m2 Al sị ẳ P ị ỵ log 2l ỵ Cl sị; 2 P ẳ ỵ χ Þ ð3Þ ð7Þ with where U is the chiral SUð3Þ matrix, We consider here the coupling mediated by the electromagnetic interaction The coupling mediated by the weak interaction is comparatively suppressed by orders of magnitude βl ðsÞ − π 2 l sị ỵ Cl sị ẳ Li2 ỵ þ log ; βl ðsÞ βl ðsÞ þ l sị q l sị ẳ − 4m2l =s: ð8Þ 114005-2 SINGLE PION CONTRIBUTION TO THE HYPERFINE … Using MS renormalization, the coupling constant combination χ P becomes scale dependent with d=d P ị ẳ 3=, which ensures that Al is scale independent The value of χ P ðΛÞ must be determined from experiment For this purpose, we can use either eỵ e which was measured recently by the KTeV Collaboration [24] or ỵ (see [25]) It is convenient to consider the ratio RP ẳ P lỵ l ị=P → γγÞ which should be less sensitive to higher order chiral corrections than the individual modes It is expressed as follows, in terms of the amplitude Al : RP ¼ 2α2 m2l βl ðm2P ÞjAl ðm2P Þj2 : π m2P ð10Þ There are two solutions for χ P which correspond to this experimental result, ðaÞ χ P ðmρ Þ ẳ 4.51 ặ 0.97; bị P m ị ẳ 19.41 ặ 0.97 11ị (in which the scale was set to Λ ¼ mρ ¼ 0.774 GeV) To decide which solution to choose, we can compare with the model proposed in Ref [22] It is based on a rigorous sum rule which holds in the large N c limit of QCD and the approximation of retaining only the lightest resonance in the sum This model gives LMD ị ẳ P m2ρ 4π F2π 11 − log − ; Λ m2ρ ð13Þ This result lies within one sigma of solution ðaÞ and is not compatible with solution ðbÞ This argument suggests that solution ðaÞ is more likely to be the physically correct one Alternatively, we can determine the coupling constant χ P from the decay mode of the meson, ỵ for which the experimental branching fraction is (see [25]) BFðη → ỵ ị ẳ 5.8 ặ 0.8ị ì 106 leading to Rexp ẳ 1.47 ặ 0.20ị ì 10 : a0 ị P m ị ẳ 1.69 ặ 0.87; b0 ị P m ị ẳ 7.96 ặ 0.87: ð14Þ ð15Þ None of these solutions is compatible with ðbÞ of Eq (11): one can therefore safely conclude that solution ðbÞ must be eliminated We can also eliminate ðb0 Þ which is not compatible with the model estimate (12) while ða0 Þ is It seems reasonable, for our purposes, to perform an average of the ðaÞ and ða0 Þ values and thus use P m ị ẳ 3.10 Æ 1.50; ð16Þ where we have slightly rescaled the error such that the two central values of ðaÞ and ða0 Þ lie within the error B π0 -proton coupling At leading order in the chiral expansion, the pionnucleon coupling is given, at tree level, from the chiral Lagrangian [27] gA μ † μ † LπNN ¼ ψ i mN ỵ i u D Uu ψ; ð17Þ pffiffiffiffi where U is the SUð2Þ chiral matrix here, u ¼ U , and Δμ ¼ ỵ ; 1 ẳ ẵu ; u iu v ỵ a Þu − iuðvμ − aμ Þu† ; 2 ð18Þ vμ (aμ ) being external vector (axial-vector) sources and ψ is an isospin spinor containing the proton and the neutron, ẳ 12ị and the uncertainty was estimated in Ref [22] to be of the order of 40% Thus, one has LMD m ị 2.2 ặ 0.8: P There are again two solutions for χ P corresponding to this experimental result, ð9Þ In the case of the π , the quantity measured experimentally is the branching ratio for the decay mode eỵ e ị, including photons in the final state such that seỵ e− ≥ 0.95m2π0 The ratio which interest us, Rπ0 , can be deduced from this result by removing the bremsstrahlung and the associated radiative corrections These have been revised recently in Ref [26] Using the results of that work, one deduces Rexp ẳ 6.96 ặ 0.36ị ì 108 : π0 PHYSICAL REVIEW D 93, 114005 (2016) ψp ψn : ð19Þ The coupling constant gA in the Lagrangian (17) is easily identified as the axial charge of the proton and also controls the neutron-proton matrix element of the charged axial current, lim hpq0 ịju djnqịi ẳ gA up qị un qị: q0 ẳq ð20Þ It is determined from neutron beta decay experiments to have the following positive3 value [25]: The absolute value of gA is obtained from the neutron lifetime, and its sign, we remind, is unambiguously determined from the asymmetry parameter of the neutron beta decay which, using Eq (20), is given by A ẳ 2gA g2A ị=1 ỵ 3g2A Þ The experimental value is [25] A ¼ −0.1184ð10Þ 114005-3 HUONG, KOU, and MOUSSALLAM PHYSICAL REVIEW D 93, 114005 (2016) gA ẳ 1.2723 ặ 0.0023: expm rị δ ð~rÞ; r mπ 3 expð−mπ rị V T rị ẳ ỵ : ỵ 2 mπ r mπ r r ð21Þ V SS ð~rÞ ¼ The pion-proton vertex amplitude is then deduced from the Lagrangian (17) to be iT πpp ¼ −gπpp up ðq2 Þγ up ðq1 Þ; gπpp ¼ gA mp : ð22Þ Fπ The expression of the coupling constant gπpp at leading chiral order, in terms of gA , mp , and Fπ , as it appears in the above expression is, of course, the content of the NambuGoldberger-Treiman relation (e.g., [20], Chap 19) It is known that the higher order chiral corrections to this relation not exceed a few percent III ENERGY SHIFTS IN MUONIC HYDROGEN A q2 = approximation Making use of the average result (16) for χ P, one obtains the following values for Aμ ð0Þ and for the overall coupling λ~ in muonic hydrogen4 Aμ 0ị ẳ 5.37 ặ 1.5; T 23ị For our purposes, we can consider that both the muon and the proton are nonrelativistic; therefore ~ Þ2 ≡ −q2 : ðp1 p2 ị2 ẳ q1 q2 ị2 ~ p1 − p We can now compute the energy shifts of muonic hydrogen caused by the one-pion exchange amplitude We will consider both the 2S and 2P energy shifts for completeness, and the relevant radial Coulomb wave functions are V p ~qị ẳ ẳ T p ~ ã q~ ~ p ã q~ ẳ A 0ị ; 4m mp q ỵ m2 gA : F2π ~ σ μ · σ~ p V SS ð~rÞ ỵ S12 V T rị; V p ~rị ẳ ẵ~ 26ị where S12 ẳ 3~ ã r ~ p · rˆ − σ~ μ · σ~ p is the so-called tensor operator, and ð29Þ where μ is the muon-proton reduced mass 1= ẳ 1=m ỵ 1=mp From these, one computes the expectation values of the components V SS and V T of the Yukawa potential For the S wave, first, one has h2SjV SS j2Si ≡ Y S m ị ẳ ~ ẳ : m ị4 ỵ 11~ ỵ 8~ ỵ 2~ ; ~ m3 41 ỵ ị 30ị When computing the expectation value in the 2S state, the contribution from the delta function in the potential V SS cancels the leading term in α from the contribution of the first piece As a result of this cancellation, Y S scales as α4 and has a negative sign For the 2P states, one has h2PjV SS j2Pi ≡ Y P ¼ mπ α~ ð25Þ The contributions to the atomic energy shifts are most easily performed by Fourier transforming to configuration space, m2 ~ ẳ A 0ị ; 12 r r 2S rị ẳ p exp − 1− ; 2 μαr 2P rị ẳ p exp r; 2 ð24Þ At first, let us make the approximation to set q2 ¼ in the vertex function Aμ We then obtain the nonrelativistic Yukawa potential in momentum space, ~ ẳ 2.61 ặ 0.49ị ì 107 : 28ị Having determined the π μμ vertex [Eq (6)] and the π pp vertex [Eq (22)], it is straightforward to derive the muon-proton scattering amplitude, μðp1 Þpðq1 Þ → μðp2 Þpðq2 Þ associated with one-pion exchange (Fig 1), 4mμ mp gA A p1 p2 ị2 ị p ẳ − 8π F2π uμ ðp2 Þγ uμ ðp1 Þup ðq2 Þγ up ðq1 Þ × : ðp1 − p2 Þ2 − m2π ð27Þ h2PjV T j2Pi ≡ T P ¼ mπ α~ ; ~ 41 ỵ ị ỵ 4~ ỵ ~ : ~ 81 ỵ ị 31ị Table I lists the expressions for the shifts in the 2P and the 2S states of muonic hydrogen in terms of the integrals Y S , Y P , T P and the overall coupling λ~ [given in Eqs (26) and (28)] as well as the central numerical values The contributions to the 2P3=2 states are particularly suppressed We also use F ẳ 92.2114ị MeV and m ẳ m ẳ 134.97666ị 114005-4 SINGLE PION CONTRIBUTION TO THE HYPERFINE … TABLE I Contributions from the single pion exchange amplitude to the 2S and the 2P energy levels in muonic hydrogen where Y S ð≡Y S ðmπ ÞÞ, Y P and T P are expectation values given in Eqs (30) and (31), and λ~ is given by Eq (26) State Expression Value (μeV) 2PF¼2 3=2 ~ P TPị Y 1.3 ì 10−7 2PF¼1 3=2 ~ P − 53 λðY 2PF¼1 1=2 ~ P − 4T P Þ − 13 λðY 0.9 ì 104 2PFẳ0 1=2 ~ P 4T P ị Y 2.8 ì 104 2SFẳ1 1=2 ~ S Y ~ S 3Y 0.049 2SFẳ0 1=2 TPị 2.1 × 10−7 0.146 because the leading terms in α cancel in the combination Y P − 25 T P Finally, in the q2 ¼ approximation, the contribution from the single pion exchange to the 2S hyperfine splitting in muonic hydrogen is ~ S m ị ẳ 0.19 ặ 0.05ị eV; EHFS ẳ 4Y 32ị which is small but not irrelevant In contrast, the contributions to the HFS in the 2P states, as can be deduced from Table I are too small to be of physical relevance Our result (32) disagrees with the one quoted in Ref [16] which uses the same approximation We could trace the origin of the discrepancy, essentially, to an incorrect coefficient for the delta function in the Yukawa potential B Influence of the vertex functions momentum dependence PHYSICAL REVIEW D 93, 114005 (2016) variation of Aμ This is easily done by using the dispersion relation representation of the function Aμ ð−q2 Þ, q2 A q ị ẳ A 0ị ImA s0 ị : s0 s0 ỵ q2 ị 33ị 34ị [which is easily verified to be reproduced by the explicit expressions (7) and (8) of Al ] Beyond the low q2 region, estimates of the behavior of Aμ may be obtained based on modelings of the π γ Ã γ Ã form factor (e.g., [28,29] for recent work; see also [30] where a list of references to earlier work can be found) We will not consider these in detail here and content ourselves with a simple estimate of the role of the q2 ≳ GeV2 region, taking into account the q2 dependence attached to the πpp vertex In this case, a weak cusp is expected from the three pions threshold at q2 ¼ −9m2π , and the q2 dependence is expected to be smooth in the q2 > region Models of the nucleon-nucleon interaction suggest a simple approximation for the behavior in this region [31], gπpp ð−q2 ị gpp 0ị ỵ q2 35ị with Λπ ≃ 1.3 GeV We can now write the μp potential, taking into account a more complete picture of the momentum dependence, as Λ2π Aμ ð−q2 Þ~σ μ · q~ ~ p ã q~ q2 ỵ m2 ỵ q2 ð36Þ [where λ is given in Eq (25)] From this, it is not difficult to compute the Fourier transform, using the representation (33) for Aμ ð−q2 Þ, and then the expectation values using the formulas of the preceding section The result for the 2S states can be written in the form -1 -2 -3 -4 h2SjV μp j2Si ẳ h ã p iA 0ị ỵ A1 þ δA2 Þ -5 -6 -0.2 -0.1 FIG ds0 q arctan 4m2l =s0 ImAl s0 ị ẳ −π qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs0 ≤ 4m2l Þ; 4m2l =s0 − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arctanh − 4m2l =s0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ImAl ðs0 Þ ¼ −π ðs0 ≥ 4m2l Þ − 4ml =s V p q2 ị ẳ ∞ For small values of q2 (compared to GeV2 ) we can use the leading order chiral approximation which gives, for the imaginary part [19], The results quoted above were obtained setting q ¼ in the vertex function Aμ It was pointed out in Ref [17] that this is not a good approximation Plotting Aμ ð−q2 Þ (see Fig 3) shows indeed that the vertex function has a strong cusp at q2 ¼ which induces a rapid variation In the following we evaluate the corrections induced by the q2 Z ×λ 0.1 0.2 0.3 0.4 0.5 0.6 Vertex function Aμ as a function of q m2π Y ðm Þ; 12π S π ð37Þ where the two corrective terms δA1 and δA2 have the following expressions: 114005-5 HUONG, KOU, and MOUSSALLAM PHYSICAL REVIEW D 93, 114005 (2016) these corrective terms We thus arrive at the following final estimate for the 2S hyperfine splitting induced by the exchange of one pion in muonic hydrogen, -1 EHFS ẳ 0.09 ặ 0.06ị eV; -2 -3 10 Integrand of the corrective term δA2 given in Eq (39) FIG ð41Þ which is negative and differs from zero within the error The error is obtained by adding in quadrature the error associated with Aμ ð0Þ [see Eq (28)] and that arising in the integral giving δA2 as discussed above Our result agrees in magnitude with that of Ref [17] The difference arises mainly from taking into account hadronic form factor effects in the momentum integral for δA2 IV CONCLUSIONS mπ δA1 ¼ A 0ị ỵ m 38ị and A2 ẳ ImAμ ðm2π x2 Þ x4 Y S ðmπ xÞ 2 Y ðm Þ xðx − 1Þ − R x S π π x2 − Y S ị ỵ1 R ð1 − Rπ x2 ÞRπ Y S ðmπ Þ Z dx 39ị with R ẳ m2 =2 This expression agrees with the result of Ref [17] in the limit Λπ → ∞ and using the leading order approximation in α of the function Y S (which is valid except when x is very close to zero) Figure shows that the integrand in Eq (39) is peaked at x ¼ The effect of Λπ is essentially to cut off the integration region x > Λπ =mπ which reduces the size of δA2 by 30% approximately Using the numerical result (28) for Aμ ð0Þ we find, for the two corrective terms induced by the q2 dependence of the vertices, δA1 ≃ −0.52; δA2 ≃ −2.30; ð40Þ which reduce the result based on Aμ ð0Þ by roughly 50% It seems reasonable to affect an uncertainty of ≃30% to [1] [2] [3] [4] The recent measurement of the 2S HFS in muonic hydrogen [6] incites one to try to improve the theoretical evaluations of the strong interaction effects, in order to reduce the error in the determination of the Zemach radius rZ In this context, we have considered here the “simple” one-pion exchange (Yukawa) contribution We have indicated how to compute this contribution based on experimental results on eỵ e , ỵ μ− , and the associated low energy chiral expansion as developed, in this sector, in Ref [19] The use of chiral symmetry is important in order to properly fix the signs of the relevant πll and πNN coupling constants and is also necessary in order to perform low-momentum expansions at the vertices The final result for the contribution of one-pion exchange to the HFS is given in Eq (41) It has a magnitude comparable to the smallest contributions which are already taken into account in the theoretical evaluation of the HFS (see the list of 28 contributions collected in Table of Ref [8]) At present, however, the main source of uncertainty affecting the strong interaction effects in the 2S HFS is that attached to the proton forward polarizabilities R Pohl et al., Nature (London) 466, 213 (2010) K Pachucki, Phys Rev A 53, 2092 (1996) E Borie, Phys Rev A 71, 032508 (2005) P J Mohr, B N Taylor, and D B Newell, Rev Mod Phys 84, 1527 (2012) [5] C E Carlson, Prog Part Nucl Phys 82, 59 (2015) [6] A Antognini et al., Science 339, 417 (2013) [7] A P Martynenko, Phys At Nucl 71, 125 (2008) ACKNOWLEDGMENTS We thank Vladimir Pascalutsa, Franziska Hagelstein, and Hai-Qing Zhou for clarifying correspondence [8] A Antognini, F Kottmann, F Biraben, P Indelicato, F Nez, and R Pohl, Ann Phys (Amsterdam) 331, 127 (2013) [9] C E Carlson, V Nazaryan, and K Griffioen, Phys Rev A 78, 022517 (2008) [10] R N Faustov and A P Martynenko, Eur Phys J C 24, 281 (2002) [11] V Nazaryan, C E Carlson, and K A Griffioen, Phys Rev Lett 96, 163001 (2006) 114005-6 SINGLE PION CONTRIBUTION TO THE HYPERFINE … [12] V Barger, C W Chiang, W Y Keung, and D Marfatia, Phys Rev Lett 106, 153001 (2011) [13] S G Karshenboim, D McKeen, and M Pospelov, Phys Rev D 90, 073004 (2014); 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REVIEW D 93, 114005 (2016) these corrective terms We thus arrive at the following final estimate for the 2S hyperfine splitting induced by the exchange of one pion in muonic hydrogen, -1 EHFS ẳ 0.09