Which SUSY models are affected by the recent LHCb result?

Question

The LHCb has recently published the observation of $B_s \rightarrow \mu^+ \mu^-$ with a branching ratio that agrees with the Standard Model (SM). There are many blog posts about it (See: Of Particular Significance). The result initiated a debate over its implication on Supersymmetry in general, which I don't want to discuss here. But it certainly affects some Supersymmetric models that may have a different prediction than that of the SM.

My questions are,

1. Which models in particular are affected (or variants of models)? What is the MSSM status? The NMSSM?
2. Does the result affect some of the parameter space of a particular model, or does it rule out an entire model/or variant of a model?

Answer 1

In both the MSSM and SM, $ B^0_s\to\mu^+\mu^- $ is a rare, loop-induced decay. The contributions to this from the MSSM arise from flavor-changing current loops to convert one of the initial valence quarks to be the anti-quark of the other. This will then lead to a Yukawa vertex with a neutral Higgs boson, of which there are three.

To be more clear, the $B^0_s$ meson is a bound state of the $b$ and $\bar{s}$ valence quarks, so those are the initial particles. One of the these quarks, e.g., the $b$ will go through a supersymmetric flavor-changing loop to turn it into an $s$ quark. Now the $s$ and $\bar{s}$ meet at a Yukawa vertex, so that the Higgs mediates the decay into muons. Here is a diagram to illustrate the process as I have described it:

Now, in the large $\tan\beta$ limit, the branching ratio for this decay is given in the MSSM as

\begin{multline*} \mathcal{B}r\left(B^0_s\to\mu^+\mu^-\right) \simeq 3.5\times10^{-5} \left(\frac{\tau_{B_s}}{1.5\,\mathrm{ps}}\right) \left(\frac{f_{B_s}}{230\,\mathrm{MeV}}\right)^2 \left(\frac{\left|V_{ts}^\mathrm{eff}\right|}{0.040}\right)^2 \\ \times \left(\frac{\tan\beta}{50}\right)^6 \left(\frac{m_t}{m_A}\right)^4 \frac{(16\pi^2)^2\epsilon_Y^2}{(1+(\epsilon_0 + \epsilon_Y y^2_t)\tan\beta)^2(1+\epsilon_0\tan\beta)^2} \end{multline*} where $\tau_{B_s}$ is the mean lifetime, $f_{B_s}$ is the decay constant, and $V^\mathrm{eff}_{ts}$ is the eective CKM matrix element. The loop factors $\epsilon_0$ and $\epsilon_Y$ are given in terms of the 3rd generation squark soft masses.

The important thing to see is that large corrections arise if $\tan\beta\gtrsim50$, and if $m_A\lesssim m_t \simeq 173\,\mathrm{GeV}$. Typically in the MSSM, $m_A \simeq m_H$, so that factor accounts for both additional propagators in the MSSM case.

Thus in low energy MSSM theories, it is fair to say that the $ \mathcal{B}r\left(B^0_s\to\mu^+\mu^-\right) $ measurement requires one to have small $\tan\beta$ and/or heavy Higgs bosons. In fact, this is already necessary given that $m_h$ has been measured to be $\sim126\,\mathrm{GeV}$, but that is another story.

References

1. S. Akula, P. Nath, Phys. Rev. D 87, 115022 (2013)
2. A. J. Buras, P. H. Chankowski, J. Rosiek and L. Slawianowska, Phys. Lett. B 546, 96 (2002).