summary in Sect. \[summy\].

Vector Current Conservation {#s:SU(2)_V}
===========================

The Lagrangian of two flavor QCD,[^2] the quark part of which is $$\mathcal{L} 
= \sum_{j,k=1}^2 \ol{Q}{}^{\hskip 0.2em j} \left(
 \Dslash + m_Q \right)_j^{\hskip 0.3em k} Q_k
,$$ with $Q = ( u, d )^T$ and $m = \diag (m_u, m_d)$, has an exact $SU(2)_V$ symmetry in the isospin limit, $m_u = m_d$, that cannot be spontaneously broken. We shall work exclusively in the isospin limit. Let us denote the generators of $SU(2)_V$ by $T^a$. Noether’s theorem yields the current $$J^{a}_\mu(x) = \ol Q (x) T^a \gamma_\mu Q(x) 
,$$ with conserved charges $$\cQ^a = \int d\bm{x} \, J^{a0} (x)
,$$ that are the generators of isospin rotations.

Now consider a quark bilinear operator $\mathcal{O}^a$ of the form $$\label{eq:bilinear}
\mathcal{O}^a (x) = \ol Q(x) \, T^a \, \Gamma \, Q(x)
.$$ The $\Gamma$ represents any Dirac matrix and any Lorentz tensor. For example, the twist-two operator $$A^a_{\mu \mu_1 \ldots \mu_n} (x) = \ol Q (x) \, T^a \, \gamma_5 \gamma_{\{ \mu}
D_{\mu_1} \cdots D_{\mu_n \}} Q (x),$$ is such an $\mathcal{O}^a(x)$, *etc*. We leave $\Gamma$ unspecified because only the flavor structure is relevant for our discussion. It is straightforward to show that $$[ 