Proton polarisability contribution to the Lamb shift in muonic hydrogen at fourth order in chiral perturbation theory*

The recent determination of the proton charge radius from the Lamb shift in muonic hydrogen [1] gives a value that differs by many standard deviations from the CODATA value [2] and from the results of recent electron scattering experiments [3]. In the theoretical calculations [4], the least-well-determined contribution is the ``proton polarisability'' contribution. This is the part of the two-photon exchange which involves proton excitations. The dominant effect can be determined via dispersion relations from the proton structure functions, but a subtraction term remains [5,6]. This subtraction term is the amplitude $T_1(0,Q^2)$ for forward, zero-energy, doubly-virtual Compton scattering, which we calculate in heavy-baryon chiral perturbation theory, to fourth order in the chiral expansion and with the leading contribution of the $\gamma N\Delta$ form factor. This provides a model-independent expression for the amplitude in the low-momentum region, which is the dominant one for its contribution to the Lamb shift, and allows us to significantly reduce the theoretical uncertainty in the latter [7].\\[4pt] [1] R. Pohl \textit{et al.}, Nature \textbf{466}, 213 (2010).\\[0pt] [2] P. J. Mohr, B. N. Taylor and D. B. Newell, Rev. Mod. Phys. \textbf{80}, 633 (2008) [arXiv:0801.0028].\\[0pt] [3] J. C. Bernauer \textit{et al.} (A1 Collaboration), Phys. Rev. Lett. \textbf{105}, 242001 (2010) [arXiv:1007.5076].\\[0pt] [4] U. D. Jentschura, Ann. Phys. \textbf{326}, 500 (2011) [arXiv:1011.5275]; E. Borie, Ann. Phys. \textbf{327}, 733 (2012) [arXiv:1103.1772].\\[0pt] [5] K. Pachucki, Phys. Rev. A \textbf{60}, 3593 (1999) [arXiv:physics/9906002].\\[0pt] [6] C. E. Carlson and M. Vanderhaeghen, Phys. Rev. A \textbf{84}, 020102 (2011) [arXiv:1101.5965]; also [arXiv:1109.3779].\\[0pt] [7] M. C. Birse and J. A. McGovern, Eur.\ Phys.\ J. A48, 120 (2012) [arXiv:1206.3030].