# Superconducting Phases in Neutron Star Cores
## Appendix B Long-Range Interaction between Fluxtubes
The terms involving \(\delta f\) and \(\delta g\) can be evaluated by using the following property of the Bessel function \(K_{0}\):  
\[\
abla^{2}K_{0}(k|\mathbf{x}-\mathbf{x}_{i,0}|)=k^{2}K_{0}(k|\mathbf{x}- \mathbf{x}_{i,0}|)-2\pi\delta^{(2)}(\mathbf{x}-\mathbf{x}_{i,0}), \tag{112}\]  
where \(\delta^{(2)}\) is the two-dimensional delta function. Using Equations (44) and (45), we then find that all the terms cancel, apart from those involving delta functions, as expected in light of the comments below Equation (111). The only remaining terms are  
\[(\mathcal{F}-\mathcal{F}_{\infty})_{\delta f,\delta g}\simeq\] \[-2\pi\sum_{i\
eq 0}K_{0}\bigg{(}\frac{\sqrt{2}|\mathbf{x}_{i}|}{ \xi_{j}}\bigg{)}\sum_{j=1,2}\biggl{[}\bigg{(}1+\frac{h_{1}}{\epsilon}+h_{3} \bigg{)}f_{j}^{2}+2\frac{h_{2}}{\epsilon}f_{j}g_{j}+\frac{1}{\epsilon}\bigg{(} 1+h_{1}+\frac{h_{3}}{\epsilon}\bigg{)}g_{j}^{2}\bigg{]}\,. \tag{113}\]  
Again using Equations (44) and (45) to simplify this result, we obtain the second contribution in Equation (46). To evaluate the remaining terms in Equation (111), it is convenient to use the divergence theorem to rewrite the area integral over \(C_{0}\) as a contour integral over \(\partial C_{0}\) in order to avoid the singular behavior of the Bessel functions at \(r=0\). Remembering that the integrand of Equation (111) vanishes everywhere but at \(\mathbf{x}=0\), we can shrink the integration contour to a small circle of radius \(\epsilon\) centered around the origin. We then have \(\mathrm{d}\mathbf{S}=\epsilon\,\mathrm{d}\theta\,\mathbf{e}_{r}\), and so  
\[(\mathcal{F}-\mathcal{F}_{\infty})_{\delta\mathbf{V}}\simeq\kappa ^{2}\sum_{i\
eq 0}\int_{\partial C_{0}}\Bigl{[}\delta\mathbf{V}^{(i)}\times(\mathbf{ \
abla}\times\delta\mathbf{V}^{(0)})-\delta\mathbf{V}^{(0)}\times(\mathbf{\
abla} \times\delta\mathbf{V}^{(i)})\Bigr{]}\cdot\mathrm{d}\mathbf{S}\] \[=-\kappa^{2}V_{0}\sum_{i\
eq 0}\int_{0}^{2\pi}\epsilon\Bigl{[} \delta\mathbf{V}^{(i)}\big{|}_{\mathbf{x}=0}K_{0}\bigg{(}\frac{\epsilon}{ \lambda_{*}}\bigg{)}\frac{1}{\lambda_{*}}+K_{1}\bigg{(}\frac{\epsilon}{ \lambda_{*}}\bigg{)}(\mathbf{\
abla}\times\delta\mathbf{V}^{(i)})_{z}\big{|}_{ \mathbf{x}=0}\Bigr{]}\mathrm{d}\theta\,. \tag{114}\]Using the asymptotic behavior of the Bessel functions, i.e., \(K_{0}(r)\sim-\ln(r)\) and \(K_{1}(r)\sim 1/r\) as \(r\to 0\), we observe that in the limit \(\varepsilon\to 0\) the first term vanishes, while the second one remains finite. More precisely, we find  
\[(\mathcal{F}-\mathcal{F}_{\infty})_{\delta\!V} \simeq-\kappa^{2}V_{0}\sum_{i\
eq 0}\int_{0}^{2\pi}\lambda_{\star}( \boldsymbol{\
abla}\times\delta\mathbf{V}^{(i)})_{z}\big{|}_{\mathbf{x}=0}\, \mathrm{d}\theta\] \[=2\pi\kappa^{2}V_{0}^{2}\sum_{i\
eq 0}K_{0}\left(\frac{|\mathbf{x} _{i}|}{\lambda_{\star}}\right), \tag{100}\]  
the first contribution in Equation (46). In principle, if we can numerically compute the nonlinear solution for a _single_ fluxtube, from this we can determine the values of \(V_{0}\), \(f_{1}\) and \(f_{2}\), and then use Equation (46) to calculate the interaction energy for any lattice of our choosing. In practice, however, it is difficult to obtain both \(f_{1}\) and \(f_{2}\) to sufficiently high accuracy to achieve quantitatively reliable results. Moreover, the assumptions made in obtaining this result are only valid in the asymptotic limit of a widely-spaced lattice, so caution is needed when applying this result to a lattice with finite separation between fluxtubes. For these reasons, we would like to have a more robust method for estimating the interaction energy. [PERSON] and [PERSON] [PERSON] and [PERSON] (2014) have suggested a possible approach: they followed essentially the same steps leading to Equation (101), but chose to leave the result in the form of an integral over the boundary \(\partial C_{0}\). They then computed this integral numerically, approximating \(\delta\Psi^{(i)}\) using the solution obtained numerically for a single fluxtube. However, their approach has a number of shortcomings:  
1.