Abstract

Without any mechanism to protect its mass, the self-energy of the Higgs boson diverges quadratically, leading to the hierarchy or fine-tuning problem. One bottom-up solution is to postulate some yet-to-be-discovered symmetry which forces the sum of the quadratic divergences to be zero, or almost negligible; this is known as the Veltman condition. Even if one assumes the existence of some new physics at a high scale, the fine-tuning problem is not eradicated, although it is softer than what it would have been with a Planck scale momentum cutoff. We study such divergences in an effective theory framework and construct the Veltman condition with dimension-six operators. We show that there are two classes of diagrams, the one-loop and the two-loop ones, that contribute to quadratic divergences, but the contribution of the latter is suppressed by a loop factor of $1/16{\ensuremath{\pi}}^{2}$. There are only six dimension-six operators that contribute to the one-loop category, and the Wilson coefficients of these operators play an important role towards softening the fine-tuning problem. We find the parameter space for the Wilson coefficients that satisfies the extended Veltman condition, and also discuss why one need not bother about the $d>6$ operators. The parameter space is consistent with the theoretical and experimental bounds of the Wilson coefficients and should act as a guide to the model builders.

Highlights

  • The title of this paper may appear to be an oxymoron for two reasons

  • We show that there are two classes of diagrams, the one-loop and the two-loop ones, that contribute to quadratic divergences, but the contribution of the latter is suppressed by a loop factor of 1=16π2

  • We have discussed the Veltman condition leading to the cancellation of the quadratic divergence of the Higgs self-energy in the context of an SM effective field theory (SMEFT) framework

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Summary

INTRODUCTION

The title of this paper may appear to be an oxymoron for two reasons. First, effective theories are known to be valid up to a certain energy scale, so why should one talk about the hierarchy problem, which essentially is a manifestation of extreme weakness of gravity, or the extremely high value of the Planck scale ∼1019 GeV? Second, any calculation of the scalar self-energy involves the evaluation of loop contributions to the self-energy, and how may one evaluate a loop in an effective theory with higherdimensional operators?. For d 1⁄4 2n, one needs operators of the generic form Φ†ΦðDpSÞ†ðDpSÞ, where p 1⁄4 n − 2 and S some scalar field, to throw a momentum dependence of k2p in the loop and make the one-loop diagram important as those coming from d 1⁄4 6 operators. All such operators can be reduced to harmless (i.e., not producing a oneloop Λ4 divergence) forms through equations of motion

VELTMAN CONDITION WITH DIMENSION-FOUR OPERATORS
VELTMAN CONDITION WITH DIMENSION-SIX OPERATORS
RESULT
CONCLUSION
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