Resummation of Goldstone boson contributions to the MSSM effective potential

Nilanjana Kumar and Stephen P. Martin

Phys. Rev. D 94, 014013 – Published 12 July 2016

Abstract

We discuss the resummation of the Goldstone boson contributions to the effective potential of the minimal supersymmetric Standard Model. This eliminates the formal problems of spurious imaginary parts and logarithmic singularities in the minimization conditions when the tree-level Goldstone boson squared masses are negative or approach zero. The numerical impact of the resummation is shown to be almost always very small. We also show how to write the two-loop minimization conditions so that Goldstone boson squared masses do not appear at all, and so that they can be solved without iteration.

Received 12 May 2016

DOI:https://doi.org/10.1103/PhysRevD.94.014013

Physics Subject Headings (PhySH)

Electroweak radiative corrections Renormalization group Resummation methods Supersymmetric models Vacuum stability

Particles & Fields

Authors & Affiliations

Nilanjana Kumar and Stephen P. Martin

Department of Physics, Northern Illinois University, DeKalb, IL 60115, USA

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Issue

Vol. 94, Iss. 1 — 1 July 2016

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Images

Figure 1
The leading contribution at fixed loop order to the effective potential as $G^{0}, G^{\pm} \to 0$ comes from vacuum diagrams with chains of $ℓ - 1$ one-loop subdiagrams involving heavy particles connected by $ℓ - 1$ Goldstone boson propagators.
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Figure 2
Diagrams that give the most singular behavior as $h^{0} \to 0$ for the minimization condition of the effective potential, at two-loop order (a) and three-loop order (b).
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Figure 3
The dependences of tree-level masses on the renormalization scale, for the Goldstone bosons $G^{0}, G^{\pm}$ (solid blue line) and the lightest neutral Higgs bosons $h^{0}$ (long-dashed red line). The modified tree-level values ${\hat{h}}^{0}$ and ${\bar{h}}^{0}$ , defined by Eqs. (5.17) and (5.35), are visually indistinguishable from each other and are nearly constant, and are shown as the short-dashed green line. In each case, $\bar{sqrt} (m^{2})$ is plotted. The input parameters are defined by two-loop renormalization group running starting from Eqs. (7.1)–(7.11) at $Q_{0} = 2000 GeV$ .
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Figure 4
The dependences of $\tan β = v_{u} / v_{d}$ , the tree-level neutral pseudoscalar Higgs mixing parameter $\tan β_{0}$ , and the charged Higgs mixing parameter $\tan β_{\pm}$ , as a function of the renormalization scale $Q$ at which the two-loop effective potential is minimized. The input parameters are defined by running (7.1)–(7.11) starting from $Q_{0} = 2000 GeV$ .
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Figure 5
The dependence of the ratio of $μ_{\min} / μ_{run}$ (left panel) and $b_{\min} / b_{run}$ (right panel) on the renormalization scale $Q$ . Here “run” means obtained by two-loop renormalization group running starting from $Q_{0} = 2000 GeV$ with inputs from Eqs. (7.1)–(7.11), while “min” means obtained by applying the effective potential minimization conditions directly at $Q$ . The thinnest (green) lines are obtained with $V_{eff}$ found in Ref. [26]. The next thinnest (red) lines were obtained in the same way, but with $G^{0} = G^{\pm} = 0$ set by hand. The thicker (blue) lines were obtained with the resummed effective potential using Eqs. (5.3)–(5.8) in Eqs. (2.19)–(2.20). The thickest (black) lines were obtained by further reexpanding the minimization conditions to eliminate $μ$ and $b$ in the radiative correction part as described in Sec. 5c.
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Figure 6
As in Fig. 5, but for $m_{H_{u}}^{2}$ and $m_{H_{d}}^{2}$ . Here, the thickest (black) line is obtained by reexpanding the resummed effective potential to eliminate the dependence on $m_{H_{u}}^{2}$ and $m_{H_{d}}^{2}$ in the radiative correction part of the minimization conditions, as described in Sec. 5b.
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