Abstract
We obtain a model-independent prediction for the two-photon exchange contribution to the hyperfine splitting in muonic hydrogen. We use the relation of the Wilson coefficients of the spin-dependent dimension-six four-fermion operator of NRQED applied to the electron-proton and to the muon-proton sectors. Their difference can be reliably computed using chiral perturbation theory, whereas the Wilson coefficient of the electron-proton sector can be determined from the hyperfine splitting in hydrogen. This allows us to give a precise model-independent determination of the Wilson coefficient for the muon-proton sector, and consequently of the two-photon exchange contribution to the hyperfine splitting in muonic hydrogen, which reads \( \delta {\overline{E}}_{p\mu, \mathrm{H}\mathrm{F}}^{\mathrm{TPE}}(nS)=-\frac{1}{n^3}1.161(20) \) meV. Together with the associated QED analysis, we obtain a prediction for the hyperfine splitting in muonic hydrogen that reads E th pμ,HF (1S) = 182.623(27) meV and E th pμ,HF (2S) = 22.8123(33) meV. The error is dominated by the two-photon exchange contribution.
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References
H. Hellwig, R.F.C. Vessot, M.W. Levine et al., Measurement of the Unperturbed Hydrogen Hyperfine Transition Frequency, IEEE Trans. Instrum. Measur. 19 (1970) 200.
P.W. Zitzewitz, E.E. Uzgiris and N.F. Ramsey, Wall Shift of FEP Teflon in the Hydrogen Maser, Rev. Sci. Instr. 41 (1970) 81.
L. Essen, R.W. Donaldson, E.G. Hope and M.J. Bangham, Hydrogen Maser Work at the National Physical Laboratory, Metrologia 9 (1973) 128.
D. Morris, Hydrogen Maser Wall Shift Experiments at the National Research Council of Canada, Metrologia 7 (1971) 162.
V. S. Reinhard and J. Lavanceau, A comparison of the cesium and hydrogen hyperfine frequencies by means of Loran-C and portable clocks, in Proceedings of the 28th Annual Symposium on Frequency Control (Fort Mammouth, N.J. U.S.A., 1974), p. 379.
P. Petit, M. Desaintfuscien and C. Audoin, Temperature Dependence of the Hydrogen Maser Wall Shift in the Temperature Range 295-395 K, Metrologia 16 (1980) 7.
J. Vanier and R. Larouche, A Comparison of the Wall Shift of TFE and FEP Teflon Coatings in the Hydrogen Maser, Metrologia 14 (1976) 31.
Y.M. Cheng, Y.L. Hua, C.B. Chen, J.H. Gao and W. Shen, Hydrogen maser wall shift experiments and determination of the unperturbed hyperfine frequency of the ground state of the hydrogen atom, IEEE Trans. Instrum. Measur. 29 (1980) 316.
S.G. Karshenboim, Precision physics of simple atoms: QED tests, nuclear structure and fundamental constants, Phys. Rept. 422 (2005) 1 [hep-ph/0509010] [INSPIRE].
M.I. Eides, H. Grotch and V.A. Shelyuto, Theory of light hydrogen-like atoms, Phys. Rept. 342 (2001) 63 [hep-ph/0002158] [INSPIRE].
G.T. Bodwin and D.R. Yennie, Some Recoil Corrections to the Hydrogen Hyperfine Splitting, Phys. Rev. D 37 (1988) 498 [INSPIRE].
C. Peset and A. Pineda, The Lamb shift in muonic hydrogen and the proton radius from effective field theories, Eur. Phys. J. A 51 (2015) 156 [arXiv:1508.01948] [INSPIRE].
W.E. Caswell and G.P. Lepage, Effective Lagrangians for Bound State Problems in QED, QCD and Other Field Theories, Phys. Lett. 167B (1986) 437 [INSPIRE].
A. Pineda and J. Soto, Effective field theory for ultrasoft momenta in NRQCD and NRQED, Nucl. Phys. Proc. Suppl. 64 (1998) 428 [hep-ph/9707481] [INSPIRE].
A. Pineda and J. Soto, The Lamb shift in dimensional regularization, Phys. Lett. B 420 (1998) 391 [hep-ph/9711292] [INSPIRE].
A. Pineda and J. Soto, Potential NRQED: The Positronium case, Phys. Rev. D 59 (1999) 016005 [hep-ph/9805424] [INSPIRE].
A. Pineda, Leading chiral logs to the hyperfine splitting of the hydrogen and muonic hydrogen, Phys. Rev. C 67 (2003) 025201 [hep-ph/0210210] [INSPIRE].
A. Pineda, Learning about the chiral structure of the proton from the hyperfine splitting, hep-ph/0308193 [INSPIRE].
A. Pineda, The Chiral structure of the Lamb shift and the definition of the proton radius, Phys. Rev. C 71 (2005) 065205 [hep-ph/0412142] [INSPIRE].
S.D. Drell and J.D. Sullivan, Polarizability contribution to the hydrogen hyperfine structure, Phys. Rev. 154 (1967) 1477 [INSPIRE].
N.M. Kroll and F. Pollock, Second-Order Radiative Corrections to Hyperfine Structure, Phys. Rev. 86 (1952) 876 [INSPIRE].
C. Peset and A. Pineda, The two-photon exchange contribution to muonic hydrogen from chiral perturbation theory, Nucl. Phys. B 887 (2014) 69 [arXiv:1406.4524] [INSPIRE].
S.G. Karshenboim, Muonic vacuum polarization contribution to the energy levels of atomic hydrogen, J. Phys. B 28 (1995) L77.
A. Antognini, F. Kottmann, F. Biraben, P. Indelicato, F. Nez and R. Pohl, Theory of the 2S-2P Lamb shift and 2S hyperfine splitting in muonic hydrogen, Annals Phys. 331 (2013) 127 [arXiv:1208.2637] [INSPIRE].
P. Indelicato, Nonperturbative evaluation of some QED contributions to the muonic hydrogen N = 2 Lamb shift and hyperfine structure, Phys. Rev. A 87(2013) 022501 [arXiv:1210.5828] [INSPIRE].
K. Pachucki, Theory of the Lamb shift in muonic hydrogen, Phys. Rev. A 53 (1996) 2092 [INSPIRE].
A.P. Martynenko, 2S hyperfine splitting of muonic hydrogen, Phys. Rev. A 71 (2005) 022506 [hep-ph/0409107] [INSPIRE].
E. Borie, Lamb shift in light muonic atoms: Revisited, Annals Phys. 327 (2012) 733 [INSPIRE].
C.E. Carlson, V. Nazaryan and K. Griffioen, Proton structure corrections to electronic and muonic hydrogen hyperfine splitting, Phys. Rev. A 78 (2008) 022517 [arXiv:0805.2603] [INSPIRE].
A.P. Martynenko and R.N. Faustov, Hyperfine ground-state structure of muonic hydrogen, J. Exp. Theor. Phys. 98 (2004) 39 [Zh. Eksp. Teor. Fiz. 125 (2004) 48] [INSPIRE].
R. Pohl et al., The size of the proton, Nature 466 (2010) 213 [INSPIRE].
A. Antognini et al., Proton Structure from the Measurement of 2S − 2P Transition Frequencies of Muonic Hydrogen, Science 339 (2013) 417 [INSPIRE].
M. Sato et al., Laser Spectroscopy of Ground State Hyperfine Splitting Energy of Muonic Hydrogen, JPS Conf. Proc. 8 (2015) 025005 [INSPIRE].
FAMU collaboration, A. Adamczak et al., Steps towards the hyperfine splitting measurement of the muonic hydrogen ground state: pulsed muon beam and detection system characterization, 2016 JINST 11 P05007 [arXiv:1604.01572] [INSPIRE].
A. Antognini, private communication.
E.E. Jenkins and A.V. Manohar, Baryon chiral perturbation theory using a heavy fermion Lagrangian, Phys. Lett. B 255 (1991) 558 [INSPIRE].
Particle Data Group collaboration, C. Patrignani et al., Review of Particle Physics, Chin. Phys. C 40 (2016) 100001 [INSPIRE].
X.-D. Ji and J. Osborne, Generalized sum rules for spin dependent structure functions of the nucleon, J. Phys. G 27 (2001) 127 [hep-ph/9905410] [INSPIRE].
F. Hagelstein, Model-independent Calculations of Proton Structure Effects in Muonic Hydrogen talk given at the conference Hadronic Contributions to New Physics Searches (2016).
A.C. Zemach, Proton Structure and the Hyperfine Shift in Hydrogen, Phys. Rev. 104 (1956) 1771 [INSPIRE].
J. Gasser, M.E. Sainio and A. Svarc, Nucleons with Chiral Loops, Nucl. Phys. B 307 (1988) 779 [INSPIRE].
V. Bernard, N. Kaiser, J. Kambor and U.G. Meissner, Chiral structure of the nucleon, Nucl. Phys. B 388 (1992) 315 [INSPIRE].
V. Bernard, H.W. Fearing, T.R. Hemmert and U.G. Meissner, The form-factors of the nucleon at small momentum transfer, Nucl. Phys. A 635 (1998) 121 [Erratum ibid. A 642 (1998) 563] [hep-ph/9801297] [INSPIRE].
R.N. Faustov, E.V. Cherednikova and A.P. Martynenko, Proton polarizability contribution to the hyperfine splitting in muonic hydrogen, Nucl. Phys. A 703 (2002) 365 [hep-ph/0108044] [INSPIRE].
C.E. Carlson, V. Nazaryan and K. Griffioen, Proton structure corrections to hyperfine splitting in muonic hydrogen, Phys. Rev. A 83 (2011) 042509 [arXiv:1101.3239] [INSPIRE].
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Peset, C., Pineda, A. Model-independent determination of the two-photon exchange contribution to hyperfine splitting in muonic hydrogen. J. High Energ. Phys. 2017, 60 (2017). https://doi.org/10.1007/JHEP04(2017)060
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DOI: https://doi.org/10.1007/JHEP04(2017)060