# Superconducting Phases in Neutron Star Cores
## 2 The Ginzburg-Landau Formalism
### Entrainment and Local Phase Invariance  
The residual strong interaction between neutrons and protons in the core leads to a non-dissipative drag between their condensates known as entrainment [39; 45]. To illustrate the dynamical effect of entrainment, we temporarily neglect any coupling to the magnetic field by setting \(\mathbf{A}=\mathbf{0}\); the magnetic field will be reintroduced later by invoking gauge invariance. The hydrodynamical momenta of the condensates are then proportional to the gradient of the phases of the order parameters, i.e.,  
\[\hbar\mathbf{\
abla}\arg\psi_{\rm p}=2m_{\rm p}\mathbf{V}_{\rm p}\,,\qquad\hbar \mathbf{\
abla}\arg\psi_{\rm n}=2m_{\rm n}\mathbf{V}_{\rm n}\,, \tag{1}\]  
where \(\mathbf{V}_{\rm p}\) and \(\mathbf{V}_{\rm n}\) are the superfluid velocities. In the presence of entrainment, the velocity-dependent terms in the free-energy density, \(F_{\rm vel}\), of the condensates must take the form  
\[F_{\rm vel}=\tfrac{1}{2}\rho_{\rm p}|\mathbf{V}_{\rm p}|^{2}+\tfrac{1}{2}\rho_ {\rm n}|\mathbf{V}_{\rm n}|^{2}-\tfrac{1}{2}\rho^{\rm pn}|\mathbf{V}_{\rm p}- \mathbf{V}_{\rm n}|^{2}\,, \tag{2}\]  
where \(\rho_{\rm p}\) and \(\rho_{\rm n}\) are the true mass densities of the condensates and the coefficient \(\rho^{\rm pn}\), which determines the strength of entrainment, is generally negative [46]. This form of the free energy--with an interaction term that depends only on the relative velocity--is necessary to ensure Galilean invariance [31] (as well as the more general constraint of \"local phase invariance\" [30]). Equation (2) demonstrates that the primary effect of entrainment is to disfavor any relative flow between the two condensates by imposing an energetic penalty wherever \(\mathbf{V}_{\rm p}\
eq\mathbf{V}_{\rm n}\). A more subtle but equally important consequence is that the condensates' hydrodynamical momenta, given in Equation (1), are no longer proportional to their mass fluxes, defined as \(\partial F_{\rm vel}/\partial\mathbf{V}_{x}\) for \(x\in\{{\rm p},{\rm n}\}\). This has significant consequences for the structure of fluxtubes and vortices, and for their mutual interactions [39], but in the present work we are concerned only with fluxtubes, i.e., topological defects in the proton condensate. For this reason, we will neglect the star's rotation, meaning that no neutron vortices are present in the ground state. While neglecting rotation is certainly a simplification, it is justified when deriving a microscale model of the neutron star interior, in which the proton fluxtube density is many orders of magnitude larger than that of the neutron vortices, e.g., [47]. Within the Ginzburg-Landau mean-field framework, entrainment first enters the free-energy density at fourth order in the order parameters, and at second order in their derivatives [39]. The most general such term that satisfies global U(1) symmetry in each condensate is a linear combination of the quantities  
\[|\psi_{x}|^{2}|\mathbf{\
abla}\psi_{y}|^{2},\ \psi_{x}\psi_{y}\mathbf{\
abla}\psi_{x} ^{\star}\cdot\mathbf{\
abla}\psi_{y}^{\star},\ \psi_{x}\psi_{y}^{\star}\mathbf{\
abla}\psi_{x}^{\star}\cdot\mathbf{\
abla}\psi_{y},\ \psi_{x}^{\star}\mathbf{\
abla}\psi_{y}^{\star}\mathbf{\
abla}\psi_{x}\cdot\mathbf{\
abla} \psi_{y}, \tag{3}\]  
where \(x,y\in\{{\rm p},{\rm n}\}\) and \({}^{\star}\) indicates the complex conjugate.