beta functions of a scalar theory coupled to gravity.
We study a scalar field theory coupled to gravity on a flat background, below Planck's energy. Einstein's theory is treated as an effective field theory. Within the context of Wilson's renormalization group, we compute gravitational corrections to the beta functions and the anomalous dimension of the scalar field, taking into account threshold effects.
The renormalization group, systems of units and the hierarchy problem
In the context of the renormalization group (RG) for gravity, I discuss the role of field rescalings and their relation to choices of units. I concentrate on a simple Higgs model coupled to gravity, where natural choices of units can be based on Newton's constant or on the Higgs mass. These quantities are not invariant under the RG, and the ratio between the units is scale-dependent. In the toy model, strong RG running occurs in the intermediate regime between the Higgs and the Planck scale, reproducing the results of the Randall–Sundrum I model. Possible connections with the problem of the mass hierarchy are pointed out.
Explicit solutions for N-dimensional Schrödinger equations with position-dependent mass
With the consideration of spherical symmetry for the potential and mass function, one-dimensional solutions of nonrelativistic Schrodinger equations with spatially varying effective mass are successfully extended to arbitrary dimensions within the frame of recently developed elegant nonperturbative technique, where the BenDaniel-Duke effective Hamiltonian in one dimension is assumed like the unperturbed piece, leading to well-known solutions, whereas the modification term due to possible use of other effective Hamiltonians in one dimension and, together with the corrections coming from the treatments in higher dimensions, are considered as an additional term like the perturbation. Application of the model and its generalization for the completeness are discussed.