Global $SU(2)_L \otimes$BRST symmetry and its LSS theorem: Ward-Takahashi identities governing Green's functions, on-shell T-Matrix elements, and $V_{eff}$, in the scalar-sector of certain spontaneously broken non-Abelian gauge theories

This work is dedicated to the memory of Raymond Stora (1930-2015). $SU(2)_L$ is the simplest spontaneous symmetry breaking (SSB) non-Abelian gauge theory: a complex scalar doublet $\phi=\frac{1}{\sqrt{2}}\begin{bmatrix}H+i\pi_3-\pi_2 +i\pi_1\end{bmatrix}\equiv\frac{1}{\sqrt{2}}\tilde{H}e^{2i\tilde{t}\cdot\tilde{\vec{\pi}}/<H>}\begin{bmatrix}10\end{bmatrix}$ and a vector $\vec{W}^\mu$. In Landau gauge, $\vec{W}^\mu$ is transverse, $\vec{\tilde{\pi}}$ are massless derivatively coupled Nambu-Goldstone bosons (NGB). A global shift symmetry enforces $m^{2}_{\tilde{\pi}}=0$. We observe that on-shell T-matrix elements of physical states $\vec{W}^\mu$,$\phi$ are independent of global $SU(2)_{L}$ transformations, and the associated global current is exactly conserved for amplitudes of physical states. We identify two towers of "1-soft-pion" global Ward-Takahashi Identities (WTI), which govern the $\phi$-sector, and represent a new global symmetry, $SU(2)_L\otimes$BRST, a symmetry not of the Lagrangian but of the physical states. The first gives relations among 1-$\phi$-I (but one $W^{\mu},B^{\mu}$ reducible) off-shell Green's functions, the second governs on-shell T-matrix elements, replacing the Adler self-consistency conditions. These WTI constrain the all-loop-orders scalar-sector effective Lagrangian and guarantee IR finiteness of the theory. These on-shell WTI include a Lee-Stora-Symanzik (LSS) theorem, which enforces the condition $m_{\pi}^2=0$ (far stronger than $m_{\tilde{\pi}}^2=0$) and causes all relevant-operator contributions to the effective Lagrangian to vanish exactly. The global $SU(2)_L$ and the BRST transformations commute in $R_\xi$ gauges. With the on-shell T-matrix constraints, the physics therefore has more symmetry than does its BRST invariant Lagrangian. We also show that the statements made above hold for the electroweak sector of the Standard Model bosons.