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].