MSSM Higgs Mass

The MSSM Higgs Mass is a prediction of the Minimal Supersymmetric Standard Model.

The mass of the lightest Higgs boson is set by the Higgs quartic coupling. Quartic couplings are not soft supersymmetry-breaking parameters since they lead to a quadratic divergences to the Higgs mass. Furthermore, there are no supersymmetric parameters to make the Higgs mass a free parameter in the MSSM (though not in non-minimal extensions). This means that Higgs mass is a prediction of the MSSM. The LEP II and the four experiments placed a lower limit on the Higgs mass of 114.4 GeV. This lower limit is significantly above where the MSSM would typically predict it to be, and while it does not rule out the MSSM, the discovery of the Higgs with a mass of 125 GeV makes proponents of the MSSM nervous.

Formulas

The only susy preserving operator that create a quartic coupling for the Higgs in the MSSM arise for the D-terms of the SU(2) and U(1) gauge sector and the magnitude of the quartic coupling is set by the size of the gauge couplings. This leads to the prediction that the Standard Model-like Higgs mass (the scalar that couples approximately to the vev) is limited to be less than the Z mass

$m_{h^{0}}^{2}\leq m_{Z^{0}}^{2}\cos ^{2}2\beta$ .

Since supersymmetry is broken, there are radiative corrections to the quartic coupling that can increase the Higgs mass. These dominantly arise from the 'top sector'

$m_{h^{0}}^{2}\leq m_{Z^{0}}^{2}\cos ^{2}2\beta +{\frac {3}{\pi ^{2}}}{\frac {m_{t}^{4}\sin ^{4}\beta }{v^{2}}}\log {\frac {m_{\tilde {t}}}{m_{t}}}$

where $m_{t}$ is the top mass and $m_{\tilde {t}}$ is the mass of the top squark. This result can be interpreted as the RG running of the Higgs quartic coupling from the scale of supersymmetry to the top mass — however since the top squark mass should be relatively close to the top mass, this is usually a fairly modest contribution and increases the Higgs mass to roughly the LEP II bound of 114 GeV before the top squark becomes too heavy.

Finally there is a contribution from the top squark A-terms

${\mathcal {L}}=y_{t}\,m_{\tilde {t}}\,a\;h_{u}{\tilde {q}}_{3}{\tilde {u}}_{3}^{c}$

where $a$ is a dimensionless number. This contributes an additional term to the Higgs mass at loop level, but is not logarithmically enhanced

$m_{h^{0}}^{2}\leq m_{Z^{0}}^{2}\cos ^{2}2\beta +{\frac {3}{\pi ^{2}}}{\frac {m_{t}^{4}\sin ^{4}\beta }{v^{2}}}\left(\log {\frac {m_{\tilde {t}}}{m_{t}}}+a^{2}(1-a^{2}/12)\right)$

by pushing $a\rightarrow {\sqrt {6}}$ (known as 'maximal mixing') it is possible to push the Higgs mass to 125 GeV without decoupling the top squark or adding new dynamics to the MSSM.

As the Higgs was found at around 125 GeV (along with no other superparticles) at the LHC, this strongly hints at new dynamics beyond the MSSM such as the 'Next to Minimal Supersymmetric Standard Model' (NMSSM).

See also

• little hierarchy problem

References

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