Goofy transformations and the hierarchy problem

Goofy transformations of the Standard Model (SM) Higgs field generally prohibit its bare mass term. This opens up an entirely new class of solutions to the electroweak (EW) hierarchy problem. We argue that these can be intrinsically linked to the flavor structure and origin of CP violation.

💡 Research Summary

The paper introduces a novel class of global transformations, dubbed “Goofy transformations,” which act non‑trivially on gauge‑kinetic terms of quantum fields. Unlike ordinary symmetries that leave kinetic terms invariant, a Goofy transformation flips the sign of both the Higgs kinetic term ((D_\mu H)^\dagger D^\mu H) and the bare scalar mass term (\mu^2 H^\dagger H). Consequently, the bare Higgs mass term is forbidden by symmetry, providing a fresh avenue to address the electroweak hierarchy problem.

The authors first situate Goofy transformations among known hierarchy‑solving mechanisms: supersymmetry (linking scalar masses to chiral fermion masses), composite Higgs models (pseudo‑Nambu‑Goldstone Higgs with shift symmetry), and conformal symmetry (where all dimensionful parameters are broken homogeneously). They argue that Goofy transformations occupy a distinct niche: they protect the Higgs quadratic term without invoking supersymmetry or compositeness, yet they can be broken softly so that a realistic electroweak scale can still be generated.

Two families of Goofy transformations are defined: a “flavor‑type” (Eq. 2) and a “CP‑type” (Eq. 3). Both involve a 2×2 matrix acting on the Higgs doublet (H) and its conjugate (\tilde H). The phases (\alpha) (flavor) and (\beta) (CP) can be absorbed by a hypercharge rotation, and the authors often set them to zero for simplicity. Under either transformation the Higgs kinetic term and the mass term acquire an overall minus sign, explicitly breaking the symmetry.

To retain the Standard Model Yukawa couplings, the fermion sector must transform in a compatible way. The paper systematically explores possible unitary transformations on the left‑handed quark doublet (Q_L) and the right‑handed singlets (u_R, d_R). For the flavor‑type case, a simple choice is to flip the sign of (u_R) and (d_R) while leaving (Q_L) inert (Eq. 10). This leaves the Yukawa matrices (Y_u) and (Y_d) invariant, thereby preserving the observed flavor structure while still forbidding the Higgs mass term. For the CP‑type case, all fermions must be conjugated; the authors show that this forces the Yukawa matrices to be purely imaginary (Eq. 11), linking the prohibition of the Higgs mass term to a specific pattern of CP violation.

The paper also discusses the role of additional exotic space‑time transformations (\partial_\mu\to -i\partial_\mu) and (A_\mu\to -iA_\mu) (Eq. 1). When combined with the Higgs Goofy transformation, these render the full effective potential invariant only if the UV cutoff (or (\overline{\text{MS}}) scale) also flips sign, (\Lambda^2\to -\Lambda^2). This observation is presented as a possible resolution of the “intrinsic” hierarchy problem, where the cutoff itself transforms under the symmetry.

A central technical result is the existence of all‑order renormalization‑group (RG) fixed points for the Goofy‑protected parameter relations. The authors argue that the beta‑functions of the Higgs mass parameter (\mu_H) can be made to vanish to all loop orders because the symmetry of the beta‑functions can be larger than the symmetry of the action itself (an extension of ’t Hooft’s technical naturalness). Explicit two‑loop calculations in a two‑Higgs‑doublet model (2HDM) confirm the presence of such fixed points, even when the canonical gauge‑kinetic terms break the Goofy symmetry softly.

To illustrate how a non‑zero Higgs mass can re‑appear, the authors employ a spurion analysis. If a new high scale (M) transforms oddly under the Goofy symmetry, it cannot contribute to the (\beta_{\mu_H}) function, leaving (\mu_H) insensitive to that scale. Conversely, if a new scalar field is odd under the symmetry and acquires a vacuum expectation value, it can break the Goofy symmetry spontaneously, generating a calculable contribution to the Higgs mass. This mechanism mirrors soft supersymmetry breaking: the symmetry must be broken in a hidden sector and then mediated to the Higgs sector.

The paper concludes that Goofy transformations provide a robust, symmetry‑based protection of the Higgs quadratic term, naturally intertwining the hierarchy problem with flavor structure and CP violation. While the symmetry must ultimately be broken to generate the electroweak scale, the breaking can be soft or spontaneous, preserving the all‑order RG stability of the Higgs mass. The authors acknowledge that the model‑building landscape is vast—comparable to supersymmetric constructions—and that a full phenomenological exploration (including collider signatures) lies beyond the scope of the present letter. Nonetheless, the work opens a new theoretical direction for addressing the hierarchy problem through unconventional global symmetries.