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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.

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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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