Spontaneous Chiral Symmetry Breaking in Schwartz

Question

I am reading M. Schwartz's book on QFT, equation (28.24)/(28.22). They say that a set scalar fields will transform as, where $g_L$ belongs to $SU(2)_{L}$ and $g_R$ belongs to $SU(2)_R$: $$\Sigma\rightarrow g_L\Sigma\ g_R^{\dagger}.\tag{28.22}$$ The Chiral Lagrangian is (spontaneous symmetry broken) $$L=tr[(\partial_\mu\Sigma)(\partial_\mu\Sigma^{\dagger})]+m^2tr[\Sigma\Sigma^{\dagger}]-\frac{\lambda}{4}tr[\Sigma\Sigma^{\dagger}\Sigma\Sigma^{\dagger}].$$ After Spontaneous symmetry breaking: $$\Sigma(x)=\frac{v+\sigma(x)}{\sqrt{2}}\exp\left(2i\frac{\pi^a(x)\tau^a}{F_\pi}\right),\tag{28.24}$$ where $\tau^a$ are Pauli matrices.

My question is that, were there any constraint on $\Sigma$? Because if it was four unconstrained complex scalar field, then there would be altogether 8 degrees of freedom. Why after symmetry breaking, there are only 4 left?

Answer 1

Symmetry breaking has little to do with it: it dictates the renaming/rearrangement of the fields so as to expand around the peculiar vacuum involved; it amounts to a writing of variables, here $\sigma\mapsto \sigma + v$. It does not eat up fields represented in the kinetic term!

The text you are reading is notorious for pedagogical complacency and omissions. What Matt neglected to specify, deeming it obvious, is that he is stretching the celebrated Gürsey chiral model to variables $\Sigma\equiv \frac{\sigma}{\sqrt{2}} U$, where U is a unitary unimodular matrix, an element of SU(2). Upon the SSB shift, it becomes the matrix you have.

The 4 missing degrees of freedom you are worrying about were never there to consider.

Answer 2

1. OP has a point. If the matrix-valued field $\Sigma\in{\rm Mat}_{2\times 2}(\mathbb{C})$ in eq. (28.22) has originally 4 complex DOF, then $\sigma(x),\pi^a(x)\in\mathbb{C}$ in eq. (28.24) are in principle 4 complex fields. Then after SSB $$G~=~SU(2)_L\times SU(2)_R\quad\longrightarrow\quad H~=~SU(2)_D,$$ the 3 real parts ${\rm Re}\pi^a(x)$ becomes massless Goldstone modes.

2. Note however that to make sense of the text after eq. (28.26) it seems that the matrix-valued field $\Sigma\in \mathbb{R}_+\times SU(2)$ in eq. (28.22) has originally 4 real DOF. Then $\sigma(x),\pi^a(x)\in\mathbb{R}$ in eq. (28.24) are 4 real fields, so that the counting of DOF is OK.