Fig. \[f:moreNvecff\]. Quite interestingly these diagrams are only non-vanishing in a finite volume with twisted boundary conditions. When the twisting parameters vanish, so too does this finite volume effect. This can easily be explained. In infinite volume the diagrams in Fig. \[f:moreNvecff\] vanish due to $SO(4)$ rotational invariance, while in a periodic finite volume they vanish due to invariance under lattice rotations. Lattice rotational invariance is broken in the direction of the twisted boundary conditions, hence the diagrams make non-vanishing contributions. Notice that the hairpin diagrams each vanish because the flavor-neutral mesons are additionally neutral under $B_\mu$.

![Additional one-loop contributions to the isospin transition matrix elements. Diagram elements are the same as in Fig. \[f:Nvecff\]. These diagrams vanish for periodic fields in a finite volume. []{data-label="f:moreNvecff"}](moreNvecff.eps){width="65.00000%"}

Combining the infinite volume and finite volume results with a twisted valence $u$-quark and specifying the case $m_j = m_u$, we arrive at $$\begin{aligned}
 \label{eq:twistedanswer}
\langle p(\bm{0}) \downarrow | J_3^+  | n(\bm{0}) \uparrow \rangle 
&=& 
\frac{- i B }{2 M} \Bigg(
F_2(B^2)
-
\frac{g_A^2 M}{ 4 \pi^2 f^2 B} \cK_2(m_\pi, B \hat{y}, 0)
-
\frac{g_{\D N}^2 M}{36 \pi^2 f^2 B}
\cK_2(m_\pi, B \hat{y}, \D)
\notag\\
&&
+ 
\frac{3 M }{4 \pi^2 f^2}
\int_0^1 dx 