acts as flavor-dependent field momentum. Meson and baryon fields formed from these $\Qt$ quark fields also acquire flavor-dependent momentum via the background $U(1)$ field namely, $$\label{eq:mmom}
\Dt_\mu \St = \partial_\mu \St + i [ B_\mu, \St ]
,$$ for the mesons [@Sachrajda:2004mi], and $$\label{eq:bmom}
[\cDt_\mu \cBt (x)]^{ijk} = \partial_\mu \cBt^{ijk} (x) + i (B_\mu^i + B_\mu^j + B_\mu^k) \cBt^{ijk}(x)
,$$ for the baryons [@Tiburzi:2005hg].

In analogy with QCD, we assume that the infinite volume theory described by Eq.  will undergo spontaneous symmetry breaking of the form $SU(4|2)_L \otimes SU(4|2)_R \to SU(4|2)_V$. The remaining vector symmetry is explicitly broken by the quark mass difference $m_j \neq m_u$ from $SU(4|2)_V \to SU(2|2)_V \otimes SU(2)_V$. The valence generators of the graded $SU(2|2)_V$ symmetry again lead to the operator relations in Eq. . Furthermore matrix element relations, e.g. that in Eq. , remain valid because the external states contain only valence quarks.

![Contraction encountered in the neutron-to-proton correlation function Eq. . The source and sink are denoted by filled circles, the operator insertion by an open circle. Single lines represent the propagators of twisted $d$-quark fields, while double lines represent the propagators of twisted $u$-quark fields. []{data-label="f:latcor"}](latcor.eps){width="25.00000%"}

Having related isovector matrix elements to isospin 