What can drive the higgs mass in mSUGRA up?

Question

As a test to familiarize myself with the program softsusy, I generated a spectrum for the following (already excluded, but once considered) mSUGRA point: $m_0 = 170\,\mathrm{GeV}$, $m_{1/2} = 190\,\mathrm{GeV}$, $\tan\beta = 3$, $\mathop{\mathrm{sign}}{\mu} = +1$ and $A_0 = 0$. The result is displayed in the picture below. While the mass hierarchy looks OK, I'm worried that the lightest scalar Higgs is around 100 GeV.

Now, as we all know ;-), the Higgs lives around 125 GeV. Also, I understand that in leading order the lightest higgs should be around the Z mass, and that it gets a large correction from the top quark that pushes it upwards. What I'm wondering is now:

1. Is is really a feature of mSUGRA that the higgs is so light? Why have people been studying points like this, when this would have been ruled out by LEP? The $h_0$ should be sufficently similar to a "SM" Higgs, right?

2. If this is not a genuine feature, have I made a mistake? Is there a way to persuade softsusy to add higher order corrections that drive the mass up? (I'm already using the "Include 2-loop scalar mass squared/trilinear RGEs" option.)

Answer 1

I think it simply means that you didn't choose the point in the parameter space that gives the Higgs mass you want. Try and increase $\tan \beta$ and/or $m_0, m_{1/2}, A$ and see how this affects the mass of the lightest Higgs. And yes, this little $h_0$ is a SM-like Higgs.

To answer the question in your title, the mass of the lightest Higgs in the MSSM is given by $$ m_h^2 \approx M_z^2 cos^22\beta + Loops, $$ and $$ Loops = \frac{3m_t^4}{2\pi^2 v^2} ln(\frac{m_{\tilde{t}_1}m_{\tilde{t}_1} }{m_t^1}) + Mixing $$ where the mixing term depends on $A_t$ and other parameters.

So if you want to increase the mass, you have to increase $\tan \beta$ as well as the mSUGRA input parameters, because they determine the average mass of the stop (and the mixing)