Revisiting the Naturalness Problem -- Who is afraid of quadratic divergences? --

TL;DR

The paper reexamines naturalness through the Wilsonian RG, arguing that quadratic divergences simply shift the critical surface $m^2_c(\lambda)$ and can be subtracted, leaving the RG flow around the critical surface governed by logarithmic divergences. This reframes naturalness as a question of how radiative logarithmic mixings couple multiple scales, rather than the presence of quadratic sensitivities. Subtractive renormalization is shown to correspond to a coordinate transformation in theory space, with the physically relevant tuning being the distance from the bare parameters to the critical surface. Consequently, scale-invariant SM dynamics (except soft terms) and broader beyond-SM constructions become natural possibilities, provided one controls logarithmic runnings and inter-scale mixings.

Abstract

It is widely believed that quadratic divergences severely restrict natural constructions of particle physics models beyond the standard model (SM). Supersymmetry provides a beautiful solution, but the recent LHC experiments have excluded large parameter regions of supersymmetric extensions of the SM. It will now be important to reconsider whether we have been misinterpreting the quadratic divergences in field theories. In this paper, we revisit the problem from the viewpoint of the Wilsonian renormalization group and argue that quadratic divergences, which can always be absorbed into a position of the critical surface, should be simply subtracted in model constructions. Such a picture gives another justification to the argument that the scale invariance of the SM, except for the soft-breaking terms, is an alternative solution to the naturalness problem. It also largely broadens possibilities of model constructions beyond the SM since we just need to take care of logarithmic divergences, which cause mixings of various physical scales and runnings of couplings.

Figures (6)

• Figure 1: (a) Feynman diagrams that contribute to the mass renormalization transformation (\ref{['RGtr1mass']}). The cross represents a mass insertion. (b) A diagram in higher order in $m^2$, which does not contribute to (\ref{['RGtr1mass']}) in the limit (\ref{['molll']}).
• Figure 2: (a) A Feynman diagram that contributes to the coupling renormalization transformation (\ref{['RGtr1lambda']}). (b) A diagram in higher order in $m^2$, which does not contribute to (\ref{['RGtr1lambda']}) in the limit (\ref{['molll']}).
• Figure 3: RG flow for $d=4$. The blob indicates the Gaussian fixed point, and the thick line $m^2=m^2_c(\lambda)$ corresponds to the critical line.
• Figure 4: The Feynman diagrams that contribute to $f\Lambda^2$ up to order $\lambda^2$. The dot represents an insertion of $m_c^2$. The first two diagrams give order $\lambda^1$ contributions, while the last four give order $\lambda^2$. Note that the second diagram gives both order $\lambda^1$ and $\lambda^2$ contributions.
• Figure 5: The Feynman diagrams that contribute to $g$ up to order $\lambda^2$. The cross and the dot represent an insertion of $m^2-m_c^2$ and $m_c^2$, respectively. The first diagram gives an order $\lambda^0$ contribution, the second $\lambda^1$, and the last four $\lambda^2$.