Comparing "gauge theory" and "(global) spontaneous symmetry breaking"

Question

To construct an effective field theory with spontaneously broken global symmetries, we need building blocks for the Lagrangian (such as covariant derivatives) that seem similar to gauge theory building blocks. Is it possible to view a "theory with spontaneously broken global symmetries" as a sort of "gauge theory", or vice versa?

My feeling is that, changing vacuum states somehow corresponds to performing gauge transformations. Alice may claim that there are no goldstone bosons present, while Bob may claim that there are goldstone bosons present, but they could be describing the same physical state, simply with different notions of "goldstone bosons". Following this line of thought, one would hope that the "gauge structure group" is the space of VEVs. However, the space of VEVs is the coset space $G/H$ which in general is not a group, so clearly something is still missing from this story.

Another evidence is that goldstone bosons $\pi^a(x)$ appear in the Lagrangian always with derivatives. This seems to suggest that $\partial_\mu\pi^a(x)$ should play a similar role to the gauge connection $A_\mu^a(x)$. [sidebar: I don't know whether this conflicts with the fact that $A_\mu(x)$ describes helicity-1 particles while $\partial_\mu\pi(x)$ describes helicity-0 particles. Maybe the paradox arises because "particle" is merely a perturbative notion.]

I imagine that to precisely formulate these theories, we need the machinery of fiber bundles. However, I'm still struggling to match the math definitions onto physical intuitions, so I have decided to leave the math definitions out of this post to minimize confusion. But please feel free to comment/answer with precise mathematics, preferably with reference to physics every now and then :)

Since the question is still at a rather primitive stage, I have left out the discussion of "spontaneously broken gauge symmetries" in this post. However, I do hope that a formalism (such as fiber bundles) exists that treats all these theories in a unified manner. Any comment in this respect is also welcome!

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[edit]: I would like to clarify the context of the question, in view of the comments by @flippiefanus and @CosmasZachos. The question arises when I'm reading Weinberg's QFT vol.2 chapter 19. Basically, I find his notations a bit cumbersome, with many indices. That's why I'm seeking a more abstract formulation, in terms of bundles and so on, to "get rid of the indices" and see the geometric picture.

Firstly, I just find it hard to interpret the "covariant derivatives" in, say chiral perturbation theory, as a connection; connection on which bundle?

Secondly, recent progress in scattering amplitudes show that gluon amplitudes and pion amplitudes have a lot in common. Gluon amplitudes vanish when replacing $\epsilon\to p$, while pion amplitudes vanish when $p\to0$, and both amplitudes (at tree level) are to some extent uniquely determined by such vanishing theorems. So there seems to be some deep connection between these theories.

I guess my goal is to make a theory with global SSB and a gauge theory look as similar as possible. At least, I would like to identify the gauge structure group in the appropriate sense in an SSB theory.

Answer 1

I'm abusing the answer format to avoid an interminable discussion in a chat room, past my last pessimistic comment. I'll do the trivial! U(1)_A nonlinear sigma model, distant analog of his O(4)/O(3) paradigm. Here, the analog of the coset space G/H is just this U(1), as there is no unbroken subgroup! There is only one projective coordinate, Goldstone boson ("goldston") θ(x) of the broken axial U(1). I let the order parameter v carry the dimension of the field, so θ(x) is dimensionless. I'm being cavalier with irrelevant normalizations.

Imagine having sent the mass of the σ to infinity, so it has decoupled and does not appear in the action at hand, $${\cal L}= v^2\partial_\mu \theta \partial^\mu \theta + \bar \psi \partial \!\!/ \psi + gv~\bar \psi e^{2i\gamma_5 \theta(x)}\psi. $$ This is invariant w.r.t. the broken U(1), $$ \psi \to e^{i\epsilon \gamma_5 } \psi ~~~~ \leadsto \overline \psi \to \overline{e^{i\epsilon \gamma_5} \psi} = \overline{\psi}e^{i\epsilon \gamma_5} \\ \theta \to \theta -\epsilon \implies \langle \delta \theta \rangle = -\epsilon, $$ the hallmark of a goldston.

However, the Yukawa term fails to exhibit decoupling at low momenta ("Adler zeros"). The elimination of the goldston non-derivative couplings involves a field redefinition gimmick of Weinberg's, $$ \tilde{\psi} \equiv e^{i\theta \gamma_5 } \psi ~~~~ \leadsto \overline{\tilde\psi} = \overline{e^{i\theta \gamma_5} \psi} = \overline{\psi}e^{i\theta \gamma_5}, $$ where the tilded nucleons are now U(1) invariant!

They now yield $$ {\cal L}= v^2\partial_\mu \theta \partial^\mu \theta + \overline{\tilde\psi} \Bigl (\partial \!\!/ +i\gamma_5\partial \!\!/ \theta + gv~\Bigr )\tilde\psi .$$ Here, all derivative couplings are manifest, whence the Adler zeros, and the nucleons have a mass, chirally invariant, to boot. This is the trivial analog of (19.5.49).

The first two terms in the big parenthesis comprise a remote/trivial analog of Weinberg's covariant $\mathcal{D}_\mu \gamma^\mu $ in (19.5.47). It is U(1) invariant acting on singlets of the broken group, and of course invariant in the null unbroken H! I might not call this a gauge theory, but if you wished to fibrate away in your project...

It is covariant to the extent that it does not succor complicated nonlinear terms in the unbroken group that is not there.

Belatedly, I realized the O(3)/O(2) might be more instructive as, then, you might have a nontrivial abelian (19.5.46), H being a U(1) there... But remember the covariant derivatives ensure terms involving them transform linearly w.r.t. the unbroken group H, albeit now with a goldstone-dependent parameter!

I hope this might shed some light on the trail of your contemplated construction.

Answer 2

I think the language of bundles is only going to confuse things more, all you need to do is compare Weinberg's discussion in Section 19.6 (the other answer roughly references Section 19.5 which is good to check all this) to the usual discussion of a local gauge theory.

In a gauge theory with (local) subgroup $H$, $\gamma(x) \in H$, where $\Psi'(x) = \gamma(x) \Psi(x)$, from $$\partial_{\mu} \Psi'(x) = \gamma(x) [\partial_{\mu} + \gamma^{-1}(x) \partial_{\mu} \gamma(x)]\Psi(x),$$ we see $\gamma^{-1}(x) \partial_{\mu} \gamma(x)$ lives in the Lie algebra $\mathfrak{h}$ of H, where $\gamma^{-1}(x) \partial_{\mu} \gamma(x) = i E_{\mu}^i t_i$ in the notation of (19.6.13), and we'd normally add a gauge field $A_{\mu} = A_{\mu}^i t_i$ also living in the Lie algebra $\mathfrak{h}$ whose transformation law is that of a connection so that it cancels out the inhomogeneous term $\gamma^{-1} \partial_{\mu} \gamma$, resulting in the usual covariant derivative.

In spontaneous symmetry breaking of a global group $G$ down to a global $H$, even though we're working with global groups, we actually begin by expressing/interpreting the wave functions of the theory (e.g. the wave functions in a Lagrangian that initially transform under some global $G$) using a parametrization of the fields $\psi(x)$ in terms of local group element $\gamma(x) \in G/H$ as in Weinberg's $$\psi(x) = \gamma(x) \tilde{\psi}(x),\tag{19.6.3}$$ where $\tilde{\psi}(x)$ transforms as a representation/'wave-function' of $H$, which transforms under a global $g \in G$ as $$\psi(x) \to \psi'(x) = g \psi(x) = g \gamma(x) \tilde{\psi}(x) = \gamma'(x) h(x) \tilde{\psi}(x) = \gamma'(x) \tilde{\psi}'(x) \tag{19.6.17-19}$$ Note that a global $g$ transformation is inducing a local $h(x)$ transformation on $\tilde{\psi}(x)$, thus we expect to find a connection or a covariant derivative associated to these local $H$ transformations somewhere in the formalism, despite spontaneously breaking only global symmetry transformations $g \in G$.

We now take a derivative of a field and express it in terms of the local group elements via $$\partial_{\mu} \psi(x) = \gamma(x)[\partial_{\mu} + \gamma^{-1} \partial_{\mu} \gamma]\tilde{\psi}(x) \tag{19.6.13}$$ which looks similar to the gauge theory case, however there are actually massive differences.

Here $\gamma(x)$ is an element of (a local form of) the quotient $G/H$ not $H$, so that it only depends on the Goldstone boson fields as in $$ \gamma(x) = \exp(i \xi^a(x) x_a) \tag{19.6.12}$$ while $\tilde{\psi}(x)$ transforms as a representation of $H$, and we're not looking at the $\partial_{\mu} \psi(x)$ on the LHS as some transformed version of the $\tilde{\psi}(x)$ on the RHS like we were above with $\partial_{\mu} \Psi'(x) = \gamma(x)[\partial_{\mu} + \gamma^{-1}\partial_{\mu} \gamma] \Psi(x)$, we're now basically ignoring the LHS and just looking to work with what it implies about representations $\tilde{\psi}$ of $H$ exclusively, meaning we care about working with $[\partial_{\mu} + \gamma^{-1} \partial_{\mu} \gamma]\tilde{\psi}(x)$ which (appears as if it) 'lives in $H$' directly, we're not trying to add something (like $A_{\mu}$) to modify it to eliminate the second term.

In other words, it seems like we're now treating $[\partial_{\mu} + \gamma^{-1}(x) \partial_{\mu} \gamma(x)]$ as though it were a covariant derivative of $H$ acting on $\tilde{\psi}(x)$, where $\gamma^{-1} \partial_{\mu} \gamma$ is interpreted directly as a connection, however the second term $\gamma^{-1} \partial_{\mu} \gamma$ now expands over the whole Lie algebra $\mathfrak{g}$, not just $\mathfrak{h}$, so there are now $x_a$ generators outside of $H$ as well as the $t_i$ generators, so we get his $$ \gamma^{-1} \partial_{\mu} \gamma = i D_{\mu} + i E_{\mu} = i \sum_a x_a D^a_{\mu}(x) + i \sum_i t_i E^i_{\mu}(x) \tag{19.6.14}$$ Thus, in working with $$[(\partial_{\mu} + iE_{\mu}) + iD_{\mu}]\tilde{\psi}(x),$$ we see $(\partial_{\mu} + iE_{\mu})$ seems to live in $H$ and act on $\tilde{\psi}$ in $H$ like a covariant derivative, while $D_{\mu}$ involves derivatives of the Goldstone boson fields.

By showing $(\partial_{\mu} + iE_{\mu})$ transforms as a connection under a global $g \in G$ symmetry transformation, as in (19.6.18), which he shows in his (19.6.25) --- (19.6.28), which amounts to showing $\partial_{\mu} \psi(x) \to \partial_{\mu} \psi'(x) = g \partial_{\mu} \psi(x)$, you see $$\mathcal{D}_{\mu} = (\partial_{\mu} + iE_{\mu}) \tag{19.6.30}$$ is a 'covariant derivative' acting on $\tilde{\psi}$ associated to the local $h(x)$ transformations that we predicted should have arisen despite starting from global $g$ transformations (which (19.6.29) just points out explicitly), while $D_{\mu}$ results in derivatives of the $\xi_a(x)$ Goldstone bosons that transforms covariantly (which (19.6.26) points out explicitly) under global $g$ transformations. It is worth noting that, since $g$ is global, our overall derivatives obviously must transform as $\partial_{\mu} \psi(x) \to \partial_{\mu} \psi'(x) = g \partial_{\mu} \psi(x)$, however expanding this out also involves derivatives of local $H$ transformations, thus of course the whole thing must implicitly involve an overall covariant derivative so that the overall result transforms like a global transformation, thus we could expect in advance that $\partial_{\mu} + E_{\mu}$ would end up being a covariant derivative in $H$. Thus, a Lagrangian involving $\tilde{\psi}$ that is invariant under $H$ and constructed from $D_{\mu}$ and $\mathcal{D}_{\mu} \tilde{\psi}$, will also be invariant under the larger $G$.

If we further made this $g$ local, $g = g(x)$, we'd then have to add a connection term to cancel out the additional $g^{-1}(x) \partial_{\mu} g(x)$-type term that would arise, as at the beginning of this post. Thus, although both begin from a $\gamma(x) \Psi(x)$ type construction where $\gamma(x)$ is an element of a (local representation of a) group ($H$ in gauge theory, $G/H$ in SSB, even though $G$ and $H$ are treated as global symmetry groups in the latter situation...), after this things slightly diverge.