# Superconducting Phases in Neutron Star Cores
## 3 Superconducting Ground States  
### The Free Energy Density  
The total free-energy density in our model is obtained by adding the entrainment terms (4) to the usual free energy of a two-component superfluid, and introducing the magnetic vector potential \(\mathbf{A}\) by minimal coupling. To reduce the number of parameters in the model, in what follows we assume that the protons and neutrons have equal masses, \(m_{\rm p}=m_{\rm n}=m_{\rm u}\), and that \(h_{3}=h_{4}\), which means that the entrainment interaction (4) is symmetric in the condensates. Using Gaussian c.g.s. units, the free energy density in its most compact form can then be expressed as  
\[F[\psi_{\rm p},\psi_{\rm n},\mathbf{A}]=\frac{g_{\rm pp}}{2} \Big{(}|\psi_{\rm p}|^{2}-\frac{n_{\rm p}}{2}\Big{)}^{2}+\frac{g_{\rm nn}}{2} \Big{(}|\psi_{\rm n}|^{2}-\frac{n_{\rm n}}{2}\Big{)}^{2}\] \[+g_{\rm pn}\Big{(}|\psi_{\rm p}|^{2}-\frac{n_{\rm p}}{2}\Big{)} \Big{(}|\psi_{\rm n}|^{2}-\frac{n_{\rm n}}{2}\Big{)}+\frac{1}{8\pi}|\mathbf{ \
abla}\times\mathbf{A}|^{2}\] \[+\frac{\hbar^{2}}{4m_{\rm u}}\bigg{|}\bigg{(}\mathbf{\
abla}-\frac{2 \mathrm{i}e}{\hbar c}\mathbf{A}\bigg{)}\psi_{\rm p}\bigg{|}^{2}+\frac{\hbar^{2 }}{4m_{\rm u}}|\mathbf{\
abla}\psi_{\rm n}|^{2}\] \[+h_{1}\bigg{|}\bigg{(}\mathbf{\
abla}-\frac{2\mathrm{i}e}{\hbar c} \mathbf{A}\bigg{)}(\psi_{\rm n}^{*}\psi_{\rm p})\bigg{|}^{2}+\frac{h_{2}-h_{1 }}{2}\mathbf{\
abla}(|\psi_{\rm p}|^{2})\cdot\mathbf{\
abla}(|\psi_{\rm n}|^{2})\] \[+\frac{h_{3}}{4}\bigg{(}\Big{|}\mathbf{\
abla}(|\psi_{\rm p}|^{2}) \Big{|}^{2}+\Big{|}\mathbf{\
abla}(|\psi_{\rm n}|^{2})\Big{|}^{2}\bigg{)}\,, \tag{9}\]  
where we have assumed that the proton Cooper pairs have charge \(2e\). The coefficients \(g_{\rm pp}\) and \(g_{\rm nn}\) measure the self-repulsion of the condensates, and \(g_{\rm pn}\) measures their mutual repulsion or attraction. In the absence of magnetic field (i.e., if \(\mathbf{B}\equiv\mathbf{\
abla}\times\mathbf{A}=\mathbf{0}\)) we expect the ground state to be a uniform mixture of proton and neutron condensates, with position-independent densities \(|\psi_{\rm p}|^{2}=n_{\rm p}/2\) and \(|\psi_{\rm n}|^{2}=n_{\rm n}/2\). Therefore the parameters \(n_{\rm p}\) and \(n_{\rm n}\) represent the expected number density of superfluid protons and neutrons, respectively, in the absence of magnetic field. For reasons explained earlier, in a mature neutron star these will have values very close to the total (i.e., normal plus condensate) particle number densities. We will hereafter refer to this state, with uniform condensate densities and vanishing magnetic field, as the Meissner state, and we note that it has \(F=0\), according to Equation (3.1). However, if the mutual attraction/repulsion between the condensates is too strong, such that \(g_{\rm pn}^{2}>g_{\rm pp}g_{\rm nn}\), then this Meissner state becomes unstable. Such behavior is not expected in the neutron star core, where the two condensates are believed to be only weakly attractive [6], and so in what follows we will always assume that \(g_{\rm pn}^{2}<g_{\rm pp}g_{\rm nn}\). Moreover, we expect a neutron condensate to be present even in non-superconducting regions, where \(\psi_{\rm p}=0\), and this implies the further restriction \(g_{\rm pn}>-g_{\rm nn}n_{\rm n}/n_{\rm p}\). The consequences of violating these restrictions have been discussed in detail by [PERSON] and [PERSON] [8], for instance. In most studies of two-component condensates, the coefficient \(g_{\rm pn}\) represents the principle interaction between the two components, e.g., [48; 49; 50; 51], and its effect on superconductivity in the neutron star core has been studied extensively [7; 8; 37]. In the present work, however, our main focus is on the effect of entrainment and other higher-order coupling terms. In the numerical results we present later, we therefore take \(g_{\rm pn}=0\), and instead study the effect of the \(h_{i}\) parameters on the superconductor. For completeness, and to facilitate comparison with earlier studies, we retain \(g_{\rm pn}\) in all of our analytical results.