In the case of discretely degenerate vacua, for example in the double well potential, there are instantonic corrections to the energies. The degeneracy is lifted, and the true vacuum becomes a spatially even linear combination of the left- and right-vacua. A great introduction/reference is Coleman's Aspects of Symmetry, in Chapter 7, ``The uses of instantons''.

Are there similar corrections to the energies when the degeneracy is continuous, for example in the case of a Mexican-Hat potential? If so, please provide a reference.


1 Answer 1


Short answer: No, there are no tunneling solutions if the potential is flat.

1) In QFT continously degenerate vacua are either resolved by spontaenous symmetry breaking, and there is a physical (Goldstone) mode that connects the vacua, or the groundstate really has flat directions that are labeled by moduli (vacuum expectation values). These flat directions are sometimes lifted by quantum effects.

2) In QM flat directions typically imply that the states are superpositions, but there is a unique ground state. The standard example is Bloch waves. For example, a particle on a circle (no potential) has states $\psi\sim\exp(il\phi)$. There can be topological terms $\cal{L}\sim \dot\phi$, but these are related to instantons only if there is a periodic potential.

3) Flat directions in QFT can be lifted by instanton effects (the standard example is the Affleck-Dine-Seiberg potential in ${\cal N}=1$ SUSY YM), but the instantons are tunneling events between vacua with different winding number, as in QCD.

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    $\begingroup$ I'm confused because you say that the short answer is "No", but then in points 1) and 3) you mention quantum or instanton effects, which is precisely what I am asking about. So is it the case that there are instantonic effects, but that these do not correct the energy? $\endgroup$ Aug 22, 2016 at 17:04
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    $\begingroup$ What happens in N=1 SUSY YM is that the effective potential for the Higgs vev has flat directions that remain flat in perturbation theory, but that are lifted by non-perturbative (instanton) effects. However, these instantons are not tunneling events between vacua with different Higgs vev, they are tunneling events between different winding number sectors (separated by a barrier) for a fixed Higgs vev. The resulting shift in teh ground state energy depends on the vev, so we get a non-trivial effective potential. $\endgroup$
    – Thomas
    Aug 22, 2016 at 19:41

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