Can someone explain with a simple example what is meant by vacuum alignment in a field theory? Recently I have heard this term in a seminar and when I asked the speaker I got an unsatisfactory answer:

vacuum expectation values cannot be arbitrarily chosen.

But I have often seen in models beyond standard models that additional scalars are assigned zero vacuum expectation value (For example, in two-Higgs inert doublet model, he said that there is no vacuum alignment problem and therefore, one can choose the additional Higgs doublet $H_2$ to have zero vacuum expectation value (vev) i.e., $\langle H_2\rangle=0$). Click here for a review of two-Higgs inert doublet model.

When I searched the internet, I found extremely technical articles related to supersymmetry, technicolor theory etc that I'm not familiar with.


Vacuum alignment is the lifting of the vacuum degeneracy present in SSB, such as chiral symmetry breaking, technicolor, etc, in comportance with a small external perturbation ΔΗ which explicitly breaks the symmetry, simultaneously, Dashen 1971, Phys. Rev. D 3, 1879 .

SSB only “hides” the symmetry. Recall Goldstone’s 1961 celebrated U(1) sombrero potential, enter image description here $$ {\cal L}= \partial \phi ^* \partial \phi -\lambda (\phi^* \phi -v^2/2)^2 = \tfrac{1}{2}\left ( \partial R \partial R +\frac{R^2}{v^2}\partial \Theta \partial \Theta \right ) -\frac{\lambda}{4}(R^2-v^2)^2, $$ where we deploy radial variables, $\phi\equiv R e^{i\Theta/v}/\sqrt 2$.

In these variables, the U(1) symmetry amounts to shifting $\Theta$ and leaving R alone. So the sombrero potential is independent of $\Theta$, and so is its vacuum. Whereas the “Higgs” R must be at the bottom, $\langle R \rangle =v$, absolutely any v.e.v. for the manifestly massless Goldstone mode $\Theta$ will give the same energy, since, redefining $R\equiv \rho + v$, $$ {\cal L}= \tfrac{1}{2}( \partial \rho \partial \rho +\partial \Theta \partial \Theta )+ \left (\frac{\rho^2}{2v^2} +\frac{\rho}{v} \right )\partial \Theta \partial \Theta -\lambda v^2 \rho^2 -\frac{\lambda}{4}\rho^4 - \lambda v \rho^3, $$ so all vacua parameterized by the v.e.v. of $\Theta$ are degenerate, as required.

When, however, one introduces by hand/fiat a small explicit symmetry breaking mass term $-\Delta H=-m_\Theta ^2\Theta^2/2$ for the goldston, $m_\Theta \ll m_\rho=v\sqrt{2 \lambda}$, the U(1) shift symmetry is lost.

This amounts to tilting by a small bit the sombrero off its flat position, with a lowest point at a real minimum of $\Theta=0$, so $\langle \Theta \rangle$ is not arbitrarily chosen anymore, and must be chosen at 0: the vacuum is now unique (non-degenerate, aligned) and the goldstone has turned into a massive pseudogoldston—like the pions of QCD in real life.

The glorious SSB structure is still around in the infinitesimal $m_\Theta$ limit, but for this “inherent vice” aligning it. A marble at $\Theta=0$ is itching to easily oscillate around 0 at the bottom of the hat, clearly much more easily than up the walls of the hat (corresponding to the $\rho$ excitation—the Higgs).

Dashen does the heavy lifting of working this out for chiral symmetry breaking in SU(3)×SU(3) in the strong interactions and thereby derives his celebrated eponymous formula for the pseudo-goldstone pseudoscalar masses which outranges our discussion, ($m_\pi^2 f_\pi^2=-\langle 0|[Q_5,[Q_5,\Delta H]]|0\rangle=-\langle 0| \bar{u} u + \bar{d} d|0\rangle (m_u+m_d)/2 ~$ ).

(A similar formula obtains for pseudogoldstone fermions when susy is broken, instead, of course.)


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