Bound states and fermiophobic Unparticle oblique corrections to the photon

Abstract

We study the effects of fermiophobic scalar/pseudo-scalar oblique corrections on bound state energy levels in muonic atoms. To make the treatment sufficiently general we will consider Unparticle scalars/pseudo-scalars which couple only to photons. We derive the relevant vacuum polarization functions and comment on the functional forms of the Unparticle Uehling potentials for various scaling dimensions in the point nucleus and finite nucleus approximations. It is estimated that for an infra-red fixed point near the scale of electroweak symmetry breaking, in the low TeV range, the energy shifts in the low-lying muonic lead transitions could typically be of the order of a few times 0.1 eV to a few times 0.01 eV. It is shown that this conclusion is not changed even when scale invariance is broken and is in fact relatively insensitive to the scale at which it is broken. It is pointed out nevertheless that the estimated magnitude of the Unparticle Uehling shift is a factor of $10^{3}-10^{4}$ below the discrepancy in QED/nuclear theory and precision muonic lead spectroscopy from about two decades ago. A minimum value for the conformal invariance breaking scale is also estimated for the fermiophobic scalar/pseudo-scalar sector to ensure consistency with astrophysics and cosmology. The energy level structure of the Unparticle Uehling shifts are inferred using general methods for the scalar and pseudo-scalar cases and it is shown that the two cases contribute to the energy shifts with the same sign.