Top Quark and Higgs Boson Masses: Interplay between Infrared and Ultraviolet Physics

TL;DR

The paper investigates how quantum effects encoded in renormalization group equations relate ultraviolet boundary conditions to infrared masses of heavy particles in the Standard Model and the MSSM. It develops and analyzes a hierarchy of infrared attractive structures—points, lines, and surfaces—in increasingly complex coupling spaces, showing that MSSM flows are more strongly attracted to these manifolds than SM flows. It derives mass relations and bounds from IR dynamics, including triviality and vacuum stability in the SM and the MSSM’s upper bound on the lightest Higgs, and highlights Yukawa-unification-driven IR fixed lines that constrain $m_t$ and $\tan\beta$. The work also discusses the reduction of parameters and the role of grand unification, arguing that IR RG structure can largely determine heavy-mass predictions regardless of UV details, with practical implications for collider phenomenology.

Abstract

We review recent efforts to explore the information on masses of heavy matter particles, notably of the top quark and the Higgs boson, as encoded at the quantum level in the renormalization group (RG) equations. The Standard Model (SM) and the Minimal Supersymmetric Standard Model (MSSM) are considered in parallel. First, the question is addressed to which extent the infrared (IR) physics of the ``top-down'' RG flow is independent of the ultraviolet (UV) physics. The central issues are i) IR attractive fixed point values for the top and the Higgs mass, the most outstanding one being m_t=O(190 GeV)sin(beta) in the MSSM, ii) IR attractive relations between parameters, the most prominent ones being an IR fixed top-Higgs mass relation in the SM, leading to m_H=O(156) GeV for the experimental top mass, and an IR fixed relation between the top mass and tan(beta) in the MSSM, and iii) an analytical assessment of their respective strengths of attraction. The triviality and vacuum stability bounds on the Higgs and top masses in the SM and the upper bound on the lightest Higgs boson mass in the MSSM are reviewed. The mathematical backbone, the rich structure of IR attractive fixed points, lines, surfaces,... in the multiparameter space, is made transparent. Interesting hierarchies emerge, most remarkably: IR attraction in the MSSM is systematically stronger than in the SM. Tau-bottom-(top) Yukawa coupling unification in supersymmetric grand unified theories and its power to focus the ``top-down'' RG flow into the IR top mass fixed point resp. onto the IR fixed line in the m_t-tan(beta) plane is reviewed. The program of reduction of parameters, a search for RG invariant relations between couplings, guided by the requirement of asymptotically free couplings in the UV limit,is summarized; its interrelations with the search for

Figures (13)

• Figure 1: Radiative corrections, applied to the $\overline{\rm MS}$ Higgs-top mass pairs, denoted by open squares, leading to the corresponding physical pole mass pairs at the tips of the arrows.
• Figure 2: Upper bound on the Higgs boson mass as a function of $\Lambda$/$m_H$, where $\Lambda$ is the scale of new physics. The dotted curve is obtained by identifying $\Lambda$ with the Landau pole of $\lambda$ and is given by Eq. (\ref{['tribo']}). The solid curve marc is the renormalization group improved unitarity bound (\ref{['marval']}). The dashed curve is the result of a lattice calculation of Ref. lues. The figure was taken from Ref. marc.
• Figure 3: The IR attractive fixed line (fat line) and the IR attractive fixed point (symbol $\Diamond$) in the $\rho_H$-$\rho_t$-plane; solutions (thin lines) representative for the "top-down" RG flow are shown, which demonstrate the strong IR attraction of the fixed line. This figure is an update of a figure in Ref. sch1.
• Figure 4: The IR fixed line (fat line) and fixed point (symbol $\Diamond$) in relation to the $\Lambda$ dependent triviality bounds and vacuum stability bounds (thin lines) for UV scale $\hbox{$\Lambda$}=10^4,\;10^6,\;10^{10},\;10^{15}\hbox{$\,{{\rm GeV}}$}$, bounding wedge-formed allowed regions. The IR attraction of the bounds towards the IR fixed line is demonstrated. The tip of the wedge, the absolute upper bound on $\rho_t$ and $\rho_H$ slides down the IR fixed line towards the IR fixed point for increasing values of $\Lambda$.
• Figure 5: The IR attractive surface in the $\rho_H$-$\rho_b$-$\rho_b$-space containing the more attractive IR fixed line (fat line 1), the less attractive one (fat line 2) and the IR fixed point (symbol $\Diamond$) at their intersection. The "top-down" RG flow is first drawn towards the surface (not shown), then within the surface towards the IR fixed line 1, then along or close to this line towards the IR fixed point, demonstrated by representative solutions (thin lines). The figure was taken from Ref. schwi.