Musings

Fine-tuned

Corrections to quartic Higgs self-coupling from stop and top loops.

We all learned on our grandfather’s knee that supersymmetry required a light Higgs. Back then, this was a cheering thought, for it meant that we would not have to wait too long for the Higgs to be discovered. The years passed, and the experimental lower bound on the mass of the Higgs crept slowly upwards. We now know that it must be heavier than 114 GeV or so.

Scott Thomas was in town the other week, and gave a very nice colloquium, explaining how serious the situation has become for the MSSM.

At tree level, $$m_h \lt m_Z \cos(2\beta)$$ where $\tan(\beta)= \langle H\rangle/\langle\tilde{H}\rangle$, and $H$ & $\tilde{H}$ give masses, respectively, to the up and down type quarks. The inequality becomes an equality in the limit that the mass of one of the other neutral scalars in the Higgs sector, $m_A\to\infty$.

With $m_Z = 91$ GeV, and $m_h\gt 114$ GeV, this bound is clearly violated. Fortunately, the one-loop corrections to the quartic self-coupling, depicted above tend to push this number up. $$m_h^2 = m_Z^2 \cos^2(2\beta)+\frac{6|\lambda_t|^2 m_t^2}{4\pi^2}\log(m_{\tilde{t}}/m_t)$$ Note that the supersymmetric cancellation between the two diagrams means that the result depends only logarithmically on the stop mass. To fit the current lower bound on $m_h$, the stop must be heavy $$m_{\tilde{t}} \gt 850\, \text{GeV}$$ And each time we push up the lower bound on the Higgs mass, the lower bound on the stop mass goes up exponentially.

Corrections to the quadratic Higgs self-coupling dominated by stop loops.

While the corrections to the quartic terms in the Higgs potential depend only logarithmically on the stop mass, the corrections to the quadratic terms are proportional to $m_{\tilde{t}}^2$. $$m^2 \sim |\mu|^2 - \frac{3 |\lambda_t|^2 m_{\tilde{t}}^2}{8\pi^2} \log (m_{\tilde{t}}/M)$$ where $M$ is a messenger mass, at which the loop-momentum integral is effectively cut off. (It’s precisely these radiative corrections that drive this term negative, and lead to the electroweak symmetry-breaking.)

To end up with an electroweak symmetry-breaking scale around $(100\, \text{GeV})^2$, one needs the $\mu$ parameter (the coefficient of $H\tilde{H}$ in the superpotential) to be in the TeV range, and its value must be tuned to within a few percent.

Personally, I can live with a fine-tuning in the 1% range. But you would not have to push the Higgs mass up too much further to make even me nervous.