### Beyond the MSSM

One of the talks at Strings 2007 that I surely would have blogged about, but for my laptop’s ailments was Nati Seiberg’s report on his most recent paper with Dine and Thomas.

As has become painfully familiar, the MSSM has a wee bit of a problem. At tree level, the mass of the lightest Higgs is lighter than that of the $Z$, $m_h \lt m_Z \cos(2\beta)$ Radiative corrections, due to stop and top loops, push the mass up. But, to get a heavy-enough Higgs compatible with the LEP-II bounds, requires a very heavy stop. That, in turn, makes the MSSM rather finely-tuned.

There are lots of ways to get around this. But, surprisingly, no one had attempted a general effective field theory analysis. Imagine there’s new physics at some scale, $M$. What new operators does this induce in the effective Lagrangian? And how do they affect the Higgs mass?

More broadly, the MSSM predicts tree-level relations among the masses of the five Higgses:

where $\begin{aligned} |\langle H_u\rangle | &= v \sin\beta\\ |\langle H_d\rangle | &= v \cos\beta \end{aligned}$

These relations are modified by (known) radiative corrections and by new operators in the effective Lagrangian, induced by new physics at some scale $M$.

Surprisingly, at order $(1/M)$, there are only two new operators. There’s a supersymmetry-preserving quartic term in the superpotential, $W_{\text{Higgs}} = \int d^2\theta\, \mu H_u H_d + \frac{\lambda_1}{M} {(H_u H_d)}^2$ and there’s a supersymmetry-breaking correction to the scalar potential, which can be written as $\int d^2\theta \mathcal{Z} \frac{\lambda_2}{M} {(H_u H_d)}^2$ where $\mathcal{Z}= \theta^2 m_{\text{SUSY}}$ is a spurion field. Corrections to the Kähler potential can, to this order, be absorbed in field redefinitions.

These two terms yield corrections to the quartic term in the Higgs scalar potential

where $\epsilon_1 = \frac{\overline{\mu}\lambda_1}{M},\quad \epsilon_2 = - \frac{m_{\text{SUSY}}\lambda_2}{M}$

It’s useful to study the problem in the limit of small $\eta \equiv \cot(\beta)$. This is the limit in which $h$ is predominantly $H_u$ and its tree-level mass in the MSSM is a large as can be. Seiberg *et al* hold the mass of the CP-odd state, $m_A$, fixed as they take the limit of large $\tan(\beta)$.

In this limit, we can expand (1) and include the corrections due to (2) $\begin{aligned} m^2_h &= m_Z^2 -\frac{4 m_Z^2 m_A^2}{m_A^2-m_Z^2}\eta^2 + \frac{16 m_A^2}{m_A^2-m_Z^2} v^2 \eta Re(\epsilon_1) + \mathcal{O}(\eta^2\epsilon,\epsilon^2)\\ m^2_H &= m_A^2 +\frac{4 m_Z^2 m_A^2}{m_A^2-m_Z^2}\eta^2 +4 v^2 Re(\epsilon_2)- \frac{16 m_Z^2}{m_A^2-m_Z^2} v^2 \eta Re(\epsilon_1)+\mathcal{O}(\eta^2\epsilon,\epsilon^2)\\ m^2_{H^\pm} &= m_A^2 +m_W^2 +2 v^2 Re(\epsilon_2) \end{aligned}$

If $\eta$ is small, but still larger than the $\epsilon_i$, these are the dominant corrections to the MSSM expressions for the Higgs masses. At $\mathcal{O}(\epsilon^2)$, we need to include the effect of further operators, suppressed by $1/M^2$. If we’re only interested in the contributions to $m_h^2$, there are a half-dozen of these, all corrections to the Kähler potential.

Even ignoring these higher corrections, it’s quite easy to reconcile the LEP II bounds with new physics in the 1-5 TeV range. In particular, a light top squark, or violations of the (loop-corrected) mass relations (1) — e.g. $m_h^2 +m_H^2 = m_Z^2 +m_A^2,\quad m_{H^\pm}^2 = m_W^2 + m_A^2$ which are $\tan(\beta)$-independent — will provide a direct window into new beyond-the-MSSM physics.

What’s really surprising about their result is that it is so simple (only two operators at leading order) and that it wasn’t found 25 years ago.

## Re: Beyond the MSSM

“But, surprisingly, no one had attempted a general effective field theory analysis. Imagine there’s new physics at some scale, M. What new operators does this induce in the effective Lagrangian?And how do they affect the Higgs mass?”

Except unsurprisingly, they had. e.g. see hep-ph/0301121 and related references.

(I have no stake in this myself, but correct assignment of credit is important)