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I need a little bit of help in understanding the mu problem in SUSY. As I understand it, the gist of the problem is that $\mu$ must be on the same order of magnitude as the electroweak scale (246 GeV). Thus, $\mu$ is much smaller than the Planck scale and we run into problems of naturalness. Correct me if I am wrong, but isn't this basically the same as the hierarchy problem in the SM that SUSY is supposed to help fix? So in SUSY, are we just replacing one hierarchy problem with another?

Furthermore, why is $\mu$ constrained to be ~246 GeV in SUSY? Mathematically, or physically, why can't $\mu$ be very large?

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The difference between the $\mu$-problem and the hierarchy problem is that loop corrections to the value of $\mu$ in MSSM are small and convergent, because of supersymmetry, while the loop corrections to $m_h^2$ in the SM are divergent. So to explain why $\mu$ is small, it is enough to explain why its approximate – tree-level – value is small. (Well, the superpotential doesn't receive any perturbative corrections whatsoever, but I don't want to get to the interesting realm of stronger statements here.) Once you explain or assume the value of the tree value, the theory is OK.

On the other hand, to explain why $m_h^2$ is small, one always needs to work with the full value including all the loop corrections which are large. So in the SM case, it's much easier to convince oneself that the almost exact cancellation of the Higgs mass is a coincidence that doesn't happen naturally. In the MSSM case, a low value of $\mu$ is a feature that may occur naturally.

The coefficient $\mu$ in $\mu H_u H_d$ term in the superpotential shouldn't be much heavier than the electroweak scale because it determines the masses of higgsinos; and it contributes to the Higgs bosons' self-interactions. Higgsinos that are vastly heavier than the actual Higgs mass would mean that the SUSY breaking (via higgs-higgsino mass difference) is extremely big and a big part of the hierarchy problem of the SM would re-emerge in the MSSM. Too strong interactions of the Higgs bosons could also be a problem.

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I'm confused by your last paragraph. $\mu$ is in the superpotential - it respects SUSY. If $\mu$ is large, we need large soft-breaking masses for the correct $m_Z$, and the hierarchy problem re-emerges as a naturalness problem with $m_Z$. Because $\mu$ respects all (super)symmetries, it knows nothing of the EWSB or SUSY scale, so why should be close to these scales? We could set it and forget it, because it won't have quadratic corrections, or we could generate $\mu$ via SSB with a new scalar, i.e. NMSSM. Is my understanding correct? – innisfree Apr 23 '13 at 21:57

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