Planck Scale Boundary Conditions and the Higgs Mass

TL;DR

The paper investigates whether Planck-scale boundary conditions, potentially arising from quantum gravity, can fix the Higgs mass within the Standard Model if no new physics appears between the electroweak and Planck scales. By imposing conditions such as λ(M_pl)=0, β_λ(M_pl)=0, Str M^2=0, and γ_m(M_pl)=0 and evolving the SM couplings via RGEs with careful matching, it predicts MH values at the electroweak scale. The key result is that λ(M_pl)=0 yields MH≈127 GeV (for Mt≈173 GeV), while other boundary conditions also produce MH in the 127–145 GeV range; random high-scale λ values tend to give MH>150 GeV, which is excluded by data. The findings suggest that a light Higgs could be a signal of Planck-scale boundary conditions and highlight the need for higher-loop calculations and precise top-quark mass measurements to distinguish among possible high-scale scenarios.

Abstract

If the LHC does only find a Higgs boson in the low mass region and no other new physics, then one should reconsider scenarios where the Standard Model with three right-handed neutrinos is valid up to Planck scale. We assume in this spirit that the Standard Model couplings are remnants of quantum gravity which implies certain generic boundary conditions for the Higgs quartic coupling at Planck scale. This leads to Higgs mass predictions at the electroweak scale via renormalization group equations. We find that several physically well motivated conditions yield a range of Higgs masses from 127-142 GeV. We also argue that a random quartic Higgs coupling at the Planck scale favors M_H > 150 GeV, which is clearly excluded. We discuss also the prospects for differentiating different boundary conditions imposed for λ(M_{pl}) at the LHC. A striking example is M_H = 127\pm 5 GeV corresponding to λ(M_{pl})=0, which would imply that the quartic Higgs coupling at the electroweak scale is entirely radiatively generated.

Figures (5)

• Figure 1: The SM Higgs mass could be determined and fixed by unknown physics connected to quantum gravity, which should be based on new concepts other than conventional QFT. The running of the Higgs mass from Planck scale down to electroweak scale is fully dictated by the SM as a QFT.
• Figure 2: Higgs and top (pole) mass determinations for different boundary conditions at the Planck scale. The coloured bands correspond to the conditions discussed in the text and which are also labelled in the insert. The middle of each band is the best value, while the width of the band is a "RGE error band" inferred from assuming that all omitted higher orders in the beta functions beyond two loops are limited by the difference between the one and two loop results. Note that the Veltman condition is truncated at the point where its Higgs mass prediction violates the vacuum stability bound (both at two-loops). The gray-hatched line at the bottom is the lower direct Higgs mass bound from LEP. Similarly the purple (brown) lines indicate the LHC Higgs searches at 95% (90%) CL from the 2010 data. The black dashed lines show the electroweak precision fit from GFitter Baak:2011zeBardin:1999yd*Arbuzov:2005ma for 68%, 95% and 99% confidence intervals (which include limits from radiative corrections and also the direct searches).
• Figure 3: Running of $\lambda$ from the Planck scale to the Fermi scale. Different values of $\lambda(M_{pl})$ and it can be seen that a large parameter space of $\lambda(M_{pl})$ tends to induce $\lambda(M_t=\unit[173]{GeV})\gtrsim\, 0.186$ which is equivalent to Higgs mass greater than $\unit[150]{GeV}$.
• Figure 4: Scatter plot of Higgs mass at the Fermi scale determined by random $\lambda$ at the Planck scale with random top mass constrained to the interval $\left[170,175\right]$ GeV.
• Figure 5: A blow up of the vacuum stability bound in the interesting Higgs and top mass region. The blue line in the center represents the vacuum stability bound obtained via two-loop beta functions, which has been thoroughly discussed in main text. The yellow band represents the uncertainties of the Higgs mass obtained via two-loop RGEs due to $\alpha_s$ uncertainties. The outer blue band is identical to the blue band in Fig. (\ref{['fig:pd1']}) and it represents the full "RGE error band" estimated from difference between one- and two-loop RGEs. With the best world average top pole mass $\unit[173.2]{GeV}$ the inferred Higgs mass from the vacuum stability condition $\lambda(M_{pl})=0$ is $\unit[127 \pm 5]{GeV}$.