Next-to-Minimal Supersymmetric Standard Model with the Gauge Mediation of Supersymmetry Breaking

TL;DR

This work critically tests whether the Next-to-Minimal Supersymmetric Standard Model (NMSSM) can resolve the μ-problem within gauge-mediated SUSY breaking (GMSB). It first surveys proposed μ-generation mechanisms in GMSB and highlights persistent issues, then analyzes MSSM and NMSSM realizations under both low- and high-scale GMSB. The key finding is that the NMSSM fails to yield viable electroweak symmetry breaking in either regime, and viable modifications (e.g., adding vector-like quarks) only succeed at the cost of percent-level parameter tuning. The results underscore substantial challenges for naturalizing μ in GMSB and motivate exploring more intricate model-building avenues. Overall, the paper delineates the constraints and tuning requirements for NMSSM-based approaches to the μ-problem in gauge-mediated SUSY breaking and points to directions for future work.

Abstract

We study the Next-to-Minimal Supersymmetric Standard Model (NMSSM) as the simplest candidate solution to the $μ$-problem in the context of the gauge mediation of supersymmetry breaking (GMSB). We first review various proposals to solve the $μ$-problem in models with the GMSB. We find none of them entirely satisfactory and point out that many of the scenarios still lack quantitative studies, and motivate the NMSSM as the simplest possible solution. We then study the situation in the Minimal Supersymmetric Standard Model (MSSM) with the GMSB and find that an order 10% cancellation is necessary between the $μ$-parameter and the soft SUSY-breaking parameters to correctly reproduce $M_Z$. Unfortunately, the NMSSM does not to give a phenomenologically viable solution to the $μ$-problem. We present quantitative arguments which apply both for the low-energy and high-energy GMSB and prove that the NMSSM does not work for either case. Possible modifications to the NMSSM are then discussed. The NMSSM with additional vector-like quarks works phenomenologically, but requires an order a few percent cancellation among parameters. We point out that this cancellation has the same origin as the cancellation required in the MSSM.

Figures (5)

• Figure 1: Lower bounds on $\mu$ in models with the GMSB subject to the constraint $M_{Z}=91$ GeV and to the lower bounds on superparticle masses (see text), (a) as a function of the messenger scale, for $\tan\beta=$ 2, 10, and 30, and (b) as a function of $\tan\beta$ for a fixed messenger scale of $10^8$ GeV.
• Figure 2: (a) Lower bounds on $|m_{H_u}^2|^{1/2}$ and $|m_{H_d}^2|^{1/2}$ as a function of the messenger scale $\Lambda$ from the selectron mass constraint $m_{\tilde{e}}>80$ GeV. Here $n=1$, $h_{t}=1.07$, $k=0.3$ and $\lambda=0.29$ at the weak scale. These bounds do not change for different values of $k$ or $\lambda$. The other plots show typical values of (b) $A_\lambda$, (c) $A_k$, and (d) $m_N^2$, for the same choice of parameters that yielded (a). The values of these parameters do not change significantly for different values of $k$ or $\lambda$.
• Figure 3: The value of $v\equiv \sqrt{v_d^2+v_u^2}$ as a function of $\lambda$ and $k$. The inputs are $n=1$, $m_N^2=-(190~\hbox{GeV})^2$, $B=50$ TeV, $\Lambda=100$ TeV, $h_t=0.99$.
• Figure 4: The dependence of $v$ on the value of $\lambda$ for the high- and low-energy GMSB. The other input parameters are the same as in Fig. \ref{['fig:3dplot']}.
• Figure 5: The probability densities of finding specific values of $v$ in the NMSSM with extra vector-like quarks upon random choices of $\lambda$. All other parameters are the same as in Fig. \ref{['fig:2plots']}. The probability densities are normalized so that $P(v=174~\rm{GeV})=1$.