Nuclear structure corrections for $\mu$4He+ and $\mu$3He+ spectroscopy

The proton charge radius was recently determined from muonic hydrogen spectroscopy~[1] with tenfold improved precision but $7.9\sigma$ disagreement with the accepted value, leading to the ``proton radius puzzle''. To further investigate, and to obtain precise radii, these measurements were repeated in $\mu^4$He$^+$ and $\mu^3$He$^+$~[2]. This may also shed light on the discrepancy between isotope-shift measurements of the $^4$He--$^3$He radius difference. However, the precision of radii determined from the muonic experiments is limited by the uncertainties in the nuclear structure corrections~[2]. We present first {\it ab-initio} calculations of these corrections~[3,4] that reduced the uncertainties from $\sim$20\% to the few percent goal.\\ [1] R.~Pohl {\it et al.}, Nature {\bf 466}, 213 (2010); A.~Antognini {\it et al.}, Science {\bf 339}, 417 (2013). [2] M.~Diepold {\it et al.}, arXiv:1606.05231 [physics.atom-ph]. [3] C.~Ji {\it et al.}, Phys.\ Rev.\ Lett.\ {\bf 111}, 143402 (2013). [4] N.~Nevo Dinur {\it et al.}, Phys.\ Lett.\ B {\bf 755}, 380 (2016).