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A gauge theory with the gauge group $SU(2)_L$ is the simplest non-abelian spontaneous symmetry breaking theory. Its’ simplest bosonic representation is a complex scalar doublet in the linear representation with a scalar $h$, pseudoscalars $\vec{\pi}$ and vector gauge bosons $\vec{W}_{\mu}$. We observe that the on-shell T matrix elements of physical states are independent of global $SU(2)_L$ global transformations and the current corresponding to these global transformations is conserved exactly on the amplitudes of physical states. We identify two towers of 1-soft-pion Ward-Takahashi Identities which govern the scalar sector, and represent a symmetry which we call $SU(2)_L \otimes$BRST, a symmetry not of the Lagrangian but the physical states. The first tower gives relations among 1-$\phi$-I off-shell Green’s functions and the second tower governs on-shell T-matrix elements and replaces Adler self-consistency conditions with those for gauge theories. The T-matrix identities ensure IR finiteness of the theory despite zero Goldstone boson mass and include the LSS theorem which enforces the condition of masslessness on the pseudoscalars, a far stronger statement than the usual masslessness of the Goldstone bosons. The global $SU(2)_L$ and BRST transformations commute in $R_{\xi}$ gauges. With the on-shell constraints, the physics therefore has more symmetry than does its BRST invariant Lagrangian. In a previous work, some of us have shown that the above results hold for the Abelian Higgs Model.
References: arXiv: 1711.07349 (submitted to Phys. Rev. D)
Phys. Rev. D 96, 065006 (2017)
