How to interpret vacuum instability of Higgs potential

Question

If the Higgs mass is in a certain range, the quartic self-coupling of the Higgs field becomes negative after renormalization group flow to a high energy scale, signalling an instability of the vacuum itself.

However, I don't know how to interpret the fact that the vacuum appears to be stable at a certain scale, and not stable at a higher scale. Does it mean the actual collapse of the vacuum will start from the higher momentum modes? In other words, when the vacuum collapses, is it true that the value of the scalar field averaged over large distances remain bounded for an extended period of time, but appears to be diverging quickly when averaged over short distances? Also, is it true that such a collapse only happens via the slow process of quantum tunneling under usual circumstances, but will take place quickly if a particle is produced at a high energy comparable to instability energy scale?

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Comments

Perhaps a condensed matter analogy is helpful. The standard Higgs mechanism is essentially an Ising magnetic, so the dynamics of the "collapse" can be thought analogously as the dynamics of coarsening as the magnetic spontaneously breaks the symmetry.

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Answer 1

The running coupling $\lambda(\mu)$, as a function of renormalization scale $\mu$, does run negative for large $\mu$ in the SM if the Higgs is not too heavy. But "renormalization scales" are not particularly physical things to talk about. A more physical quantity is the renormalization-group improved effective Higgs potential, $V(H)$. For large values of $H$, this is roughly just $\lambda(\left|H\right|) \left|H\right|^4$ (i.e., just evaluate the quartic at an RG scale equal to $H$). In other words, you can approximately equate the statement "$\lambda$ runs negative at large RG scales" with the statement "the Higgs potential turns over at large Higgs VEVs." The latter statement is clearly more connected to the existence of some kind of instanton giving rise to vacuum decay.

There's a long literature on this back to the 80s, if not earlier. You might start with hep-ph/0104016 by Isidori, Ridolfi, and Strumia and work your way in either direction through the literature....

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