The parameter space for tree-level hybrid inflation

In a large region of parameter space, the tree-level hybrid inflation model is likely to be be invalidated by loop corrections. In particular, this is likely to be the case if both quartic couplings are of order unity, as is often supposed. It is likely also to be the case if there is an ultra-violet cutoff far below the Planck scale, (ΛUV ∼ < 10 GeV(V 1/4 0 /1MeV) 2/5, where V0 is the height of the potential) unless one allows field values bigger than the cutoff. 1. The original tree-level hybrid inflation model [1] involves the mass of the inflaton field φ, the coupling λφψ of the inflaton to another field ψ, and the self-coupling λψ. It was immediately noted that the model can give a density perturbation of the required magnitude with the natural choice λ ∼ λ ∼ 1. This feature, desirable in itself, yields the bonus [2] that all relevant field values are far below the Planck scale, making it feasible to justify the neglect of non-renormalizable terms in the potential [3, 4]. Supersymmetric realizations of the tree-level model were soon proposed [2, 3], which seemed to confirm its theoretical viability. Unfortunately, with supersymmetry in place it became sensible to evaluate the 1-loop correction to the potential. In both models, the loop corrections was found [5, 6] to be large, necessarily dominating the tree-level term if both couplings are of order 1. In this note I argue that the loop correction in a generic model will be at least as big as the one found in these particular cases, unless there are unforseen cancellations between unrelated parameters. On this basis, I describe the maximal region of parameter space in which the tree-level model can be valid. As already indicated, it does not include the regime of unsuppressed couplings, which unfortunately is still in common use. I shall One of the models had the F term dominating [2, 3] while the other had the D term dominating [3]. The models actually made the potential completely flat during inflation, but supergravity corrections generate [2, 3] a non-zero (in fact generically rather too large) curvature for the potential. The loop correction was actually first evaluated [5, 6] for slight variants of the original models. The two most recent examples are [7] which invokes this regime to construct a model of inflation with TeV-scale quantum gravity, and [8] which uses it to construct a model of baryogenesis.