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I'm using the SOFTSUSY package to generate the sparticle spectrum at the EW scale. One of the input parameters is the ratio of the up and down-type Higgs vevs commonly known as $\tan\beta$. The $\mu$ parameter is computed as an output by constraining the Z-boson mass to be compatible with experiment. I was wondering if there a quick way to figure out the value of the $B\mu$ parameter from $\mu$ and $\tan\beta$. Even an approximate analytic expression would be nice to have.

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up vote 6 down vote accepted

At tree level, the conditions for the electroweak-symmetry breaking vacuum of the MSSM can be found in any of the standard review articles and are:

$$ \sin(2\beta) = \frac{2B_\mu}{2|\mu|^2 + m_{H_u}^2 + m_{H_d}^2} $$

$$ \frac{1}{2} m_Z^2 = -|\mu|^2 - \frac{m_{H_u}^2 \tan^2\beta - m_{H_d}^2}{\tan^2\beta - 1} $$

Now, the right logical way to think about this is that a top-down theory determines the values of $\mu$ and the soft-breaking parameters $m_{H_u}^2$, $m_{H_d}^2$, and $B_\mu$. These are really the inputs. On the other hand, we only want to consider the slice of parameter space on which EWSB is realized as in our world, with a particular value of $m_Z$. One can choose other useful coordinates on this slice, like $\tan \beta$, as software inputs to choose only those points on this slice of parameter space, rather than specifying the high-scale input, which in general will fail to realize correct EWSB.

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A vastly superior answer, I deleted mine. +1 – Luboš Motl Mar 9 '11 at 6:57
Thank you very much @Matt Reece ! I guess I could just take the SOFTSUSY output at the scale $M_S=\sqrt {m_{\tilde t_1}m_{\tilde t_2}}$ and use it to extract $B\mu$ at that scale but what I'm really interested in is the value of $B\mu$ at the unification scale. I'll see if I can find the corresponding RGE to estimate it. Thanks again! – stringpheno Mar 10 '11 at 1:22

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