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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: acomplex 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 observethat on-shell T-matrix elements of physical states $\vec{W}^\mu$,$\phi$ areindependent of global $SU(2)_{L}$ transformations, and the associated globalcurrent is exactly conserved for amplitudes of physical states. We identify twotowers of "1-soft-pion" global Ward-Takahashi Identities (WTI), which governthe $\phi$-sector, and represent a new global symmetry, $SU(2)_L\otimes$BRST, asymmetry not of the Lagrangian but of the physical states. The first givesrelations among 1-$\phi$-I (but one $W^{\mu},B^{\mu}$ reducible) off-shellGreen's functions, the second governs on-shell T-matrix elements, replacing theAdler self-consistency conditions. These WTI constrain the all-loop-ordersscalar-sector effective Lagrangian and guarantee IR finiteness of the theory.These on-shell WTI include a Lee-Stora-Symanzik (LSS) theorem, which enforcesthe condition $m_{\pi}^2=0$ (far stronger than $m_{\tilde{\pi}}^2=0$) andcauses all relevant-operator contributions to the effective Lagrangian tovanish exactly. The global $SU(2)_L$ and the BRST transformations commute in$R_\xi$ gauges. With the on-shell T-matrix constraints, the physics thereforehas more symmetry than does its BRST invariant Lagrangian. We also show thatthe statements made above hold for the electroweak sector of the Standard Modelbosons.

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