Veltman Criteria in Beyond Standard Model Effective Field Theory of Complex Scalar Triplet

TL;DR

This work addresses the Higgs mass naturalness problem by examining the Veltman condition within a Beyond Standard Model Effective Field Theory framework. It shows that a heavy complex scalar triplet with hypercharge $Y=1$ can generate all four SMEFT operators that contribute to the Veltman cancellation at one loop, enabling exact $\delta m_h^2$ cancellation even for high cutoff scales $\Lambda$; the cancellations arise from explicit relations among Wilson coefficients and model parameters, rather than fine-tuning. The analysis demonstrates how the Wilson coefficients depend on $\mu_\Delta$, $M$, $\lambda_1$, and $\lambda_4$, and finds allowed regions in the $ (\lambda_1, \lambda_4) $ plane that satisfy the Veltman condition while remaining consistent with stability and unitarity. The results persist under RG running to lower scales and stay within current experimental bounds, highlighting the viability of Veltman cancellations in SMEFT with a high-scale complex triplet and suggesting connections to type-II seesaw phenomenology and high-scale Leptogenesis.

Abstract

The Higgs mass is not protected by any symmetry in the Standard Model. Hence, the self-energy corrections to the Higgs mass become large due to the quadratic divergence terms. Veltman condition (V.C.) ensures that the coefficient of the quadratic divergent term either vanishes or becomes negligible. The non-observation of new physics has pushed the new physics scale to be larger than 1 TeV, making it impossible to satisfy the Veltman condition in the Standard Model without very large fine-tuning. Many attempts are made to satisfy the V.C. in Beyond Standard Model theories, but the V.C. is hard to achieve at a very large scale ($Λ$). Alternatively, it is possible that the new physics appears much above the Electroweak scale, and the effect of the new physics is observed in terms of the Wilson coefficients of the Standard Model Effective Field Theory (SMEFT) operators. The V.C. can be addressed in the SMEFT framework. In this paper, some specific new physics scenarios are considered at a very large scale. Below that scale, the effect of the new physics is observed as Beyond Standard Model Effective Field Theory (BSM-EFT). We particularly study the type-II seesaw model with the complex scalar triplet ($Y=1$) in the context of V.C. We found that this particular model is the minimal model to generate all SMEFT operators that appear in V.C. and satisfies V.C. We also examine the model parameter dependence of the Wilson coefficients in detail and show how the cancellation of the Wilson coefficients is highly dependent on some specific values of the model parameters.

Figures (3)

• Figure 1: Variation of $\lambda_1$ and $\lambda_4$ with $\mu_{\Delta}$ at two benchmark values of $\Lambda$. V.C. is satisfied over the lines. The blue points show the allowed region when the full theory with Triplet is bounded from below and satisfies perturbative unitarity conditions.
• Figure 2: Variation of the Wilson coefficients with model parameter $\lambda_4$ when V.C. is satisfied.
• Figure 3: Running of the Wilson coefficients from $\Lambda=10^6$ TeV to the cut-off scale for one set of model parameters. We have kept the value of $\mu_\Delta$ to be fixed at $1$ TeV.