Schwinger-Keldysh approach to tunneling transport at a hadron-quark interface

TL;DR

This work develops a field-theoretical framework based on the Schwinger-Keldysh formalism to study tunneling transport and friction at a hadron-quark interface in dense neutron-star matter. By modeling a hybrid hadron-quark system with a tunneling Hamiltonian that converts a baryon into three quarks, it derives perturbative expressions for the interface current and friction, and demonstrates a DC Josephson current at the hadron-quark superfluid interface. The key finding is that the DC Josephson current scales as $I_{DC} \propto |\Delta_B|\,|\Delta_Q|^3\sin(\Delta\phi)$, where $\Delta\phi = \phi_B - 3\phi_Q$, highlighting the role of the phase difference between baryon and quark pairing. The approach offers a versatile tool for exploring transport at interfaces in astrophysical contexts and motivates future studies of differential rotation, vortices, and thermal transport in neutron-star interiors.

Abstract

We theoretically discuss quantum tunneling transport and frictions at a hadron-quark matter interface based on the Schwinger-Keldysh approach combined with the tunneling Hamiltonian, which has been developed in the context of condensed matter physics. In the inner core of massive neutron stars, it is expected that cold quark matter appears at sufficiently high densities and hence exhibits color superconductivity, surrounded by nucleon superfluids at lower densities. The perturbative expressions of the tunneling current and the friction at the interface are obtained in terms of the non-equilibrium Green's functions. We demonstrate the DC Josephson current that occurs at the hadron-quark superfluid interface in the present scheme. Our framework can be applied to various conflagrations involving the interfaces relevant to astrophysical phenomena.

Figures (4)

• Figure 1: Schematics of the hadron-quark interface in a neutron star consisting of baryon matter (BM) and quark matter (QM). We consider the tunneling transport and the friction driven by the thermodynamic bias (e.g., difference between chemical potentials $\mu_{B/Q}$ or pairing gaps $\Delta_{B/Q}$ in BM and QM) and the BM velocity $\bm{v}_{B}$ in the rest frame of QM (i.e., the frame with vanishing QM velocity $\bm{v}_{Q}=\bm{0}$). In this sense, $\bm{v}_{B}$ can be regarded as the relative velocity between BM and QM. In the local area of the hadron-quark interface, the global differential rotation is considered as the linear relative motion parallel to the interface.
• Figure 2: Keldysh contour $C=C_-+C_+$ along the $t"$ axis in Eq. \ref{['eq:3.14']}, consisting of a forward branch $C_-$ and a backward one $C_{+}$. Two time parameters $t$ and $t'$ in $\langle\hat{I}(t,t')\rangle$ and $\langle\hat{\bm{F}}(t,t')\rangle$ locate on $C_+$ and $C_-$, respectively.
• Figure 3: Feynman diagrams for the Josephson tunneling current $I_{\rm J}$ and the quasiparticle tunneling current $I_{\rm qp}$. The single- and triple-solid lines show baryon and three-quark propagators, respectively, within the Nambu-Gor'kov representation. The circle represents the tunneling coupling $\mathcal{T}$.
• Figure 4: The DC Josephson current $I_{\rm DC}$ at the hadron-quark interface for different momentum cutoff $\Lambda$ normalized by the quark Fermi momentum $k_{\rm F}$. $X =N_{c}N_{s}N_f|\Delta_B|\mathcal{T}^2k_{\rm F}^{12}\sin(\Delta\phi)/(2^7\pi^{11}\epsilon^5_{\rm F})$ is the normalization factor with color, spin and flavor degrees of freedom denoted by $N_c$, $N_{s}$, and $N_f$, respectively. The diquark pairing gap $|\Delta_Q|$ is taken such that it is comparable with that employed in Ref. PhysRevD.74.074020. Since $|\Delta_B|$ is sufficiently small compared to the baryon Fermi energy around the core region, we only keep the leading-order expression where $|\Delta_B|$ is absorbed into $X$ and $|\Delta_B|\rightarrow 0$ is taken in the integrand of Eq. \ref{['eq:IDC']}.