Next-to-Minimal Supersymmetric Standard Model

In particle physics, NMSSM is an acronym for Next-to-Minimal Supersymmetric Standard Model. [1][2][3][4][5] It is a supersymmetric extension to the Standard Model that adds an additional singlet chiral superfield to the MSSM and can be used to dynamically generate the ${\displaystyle \mu }$ term, solving the ${\displaystyle \mu }$-problem. Articles about the NMSSM are available for review.[6][7]

The Minimal Supersymmetric Standard Model does not explain why the ${\displaystyle \mu }$ parameter in the superpotential term ${\displaystyle \mu H_{u}H_{d}}$ is at the electroweak scale. The idea behind the Next-to-Minimal Supersymmetric Standard Model is to promote the ${\displaystyle \mu }$ term to a gauge singlet, chiral superfield ${\displaystyle S}$. Note that the scalar superpartner of the singlino ${\displaystyle S}$ is denoted by ${\displaystyle {\hat {S}}}$ and the spin-1/2 singlino superpartner by ${\displaystyle {\tilde {S}}}$ in the following. The superpotential for the NMSSM is given by

${\displaystyle W_{\text{NMSSM}}=W_{\text{Yuk}}+\lambda SH_{u}H_{d}+{\frac {\kappa }{3}}S^{3}}$

where ${\displaystyle W_{\text{Yuk}}}$ gives the Yukawa couplings for the Standard Model fermions. Since the superpotential has a mass dimension of 3, the couplings ${\displaystyle \lambda }$ and ${\displaystyle \kappa }$ are dimensionless; hence the ${\displaystyle \mu }$-problem of the MSSM is solved in the NMSSM, the superpotential of the NMSSM being scale-invariant. The role of the ${\displaystyle \lambda }$ term is to generate an effective ${\displaystyle \mu }$ term. This is done with the scalar component of the singlet ${\displaystyle {\hat {S}}}$ getting a vacuum-expectation value of ${\displaystyle \langle {\hat {S}}\rangle }$; that is, we have

${\displaystyle \mu _{\text{eff}}=\lambda \langle {\hat {S}}\rangle }$

Without the ${\displaystyle \kappa }$ term the superpotential would have a U(1)' symmetry, so-called Peccei–Quinn symmetry; see Peccei–Quinn theory. This additional symmetry would alter the phenomenology completely. The role of the ${\displaystyle \kappa }$ term is to break this U(1)' symmetry. The ${\displaystyle \kappa }$ term is introduced trilinearly such that ${\displaystyle \kappa }$ is dimensionless. However, there remains a discrete ${\displaystyle \mathbb {Z} _{3}}$ symmetry, which is moreover broken spontaneously.[8] In principle this leads to the domain wall problem. Introducing additional but suppressed terms, the ${\displaystyle \mathbb {Z} _{3}}$ symmetry can be broken without changing phenomenology at the electroweak scale.[9] It is assumed that the domain wall problem is circumvented in this way without any modifications except far beyond the electroweak scale.

Other models have been proposed which solve the ${\displaystyle \mu }$-problem of the MSSM. One idea is to keep the ${\displaystyle \kappa }$ term in the superpotential and take the U(1)' symmetry into account. Assuming this symmetry to be local, an additional, ${\displaystyle Z'}$ gauge boson is predicted in this model, called the UMSSM.