  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.
