Advances in High Energy Physics

Volume 2018, Article ID 8963453, 11 pages

https://doi.org/10.1155/2018/8963453

## Neutrino Emission from Cooper Pairs at Finite Temperatures

Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation of the Russian Academy of Science (IZMIRAN), Troitsk, Moscow 108840, Russia

Correspondence should be addressed to Lev B. Leinson; ur.xednay@nosniel

Received 27 December 2017; Revised 5 March 2018; Accepted 22 March 2018; Published 10 May 2018

Academic Editor: Theocharis Kosmas

Copyright © 2018 Lev B. Leinson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

A brief review is given of the current state of the problem of neutrino pair emission through neutral weak currents caused by the Cooper pairs breaking and formation (PBF) in superfluid baryon matter at thermal equilibrium. The cases of singlet-state pairing with isotropic superfluid gap and spin-triplet pairing with an anisotropic gap are analyzed with allowance for the anomalous weak interactions caused by superfluidity. It is shown that taking into account the anomalous weak interactions in both the vector and axial channels is very important for a correct description of neutrino energy losses through the PBF processes. The anomalous contributions lead to an almost complete suppression of the PBF neutrino emission in spin-singlet superfluids and strong reduction of the PBF neutrino losses in the spin-triplet superfluid neutron matter, which considerably slows down the cooling rate of neutron stars with superfluid cores.

#### 1. Introduction

At the long cooling era, the evolution of a neutron star (NS) surface temperature crucially depends on the overall rate of neutrino emission out of the star. The cooling dynamics below the superfluid transition temperature is governed primarily by the superfluid component of nucleon matter. The superfluidity of nucleons in NSs strongly suppresses most mechanisms of neutrino emission operating in the nonsuperfluid nucleon matter (the bremsstrahlung at nucleon collisions, modified Urca processes, etc. [1, 2]) but simultaneously strongly reduces the heat capacity and triggers the emission of neutrino pairs through neutral weak currents caused by the nucleon Cooper pair breaking and formation (PBF) processes in thermal equilibrium. Neutrino emission from Cooper pairs is currently thought to be the dominant cooling mechanism of baryon matter, for some ranges of the temperature and/or matter density. The total energy and momentum of an escaping (massless) neutrino pair form a time-like four-momentum , so the process is kinematically allowed only because of the existence of a superfluid energy gap , which admits the nucleon transitions with and . (We use the Standard Model of weak interactions, the system of units , and the Boltzmann constant .)

The simplest case for baryon pairing corresponds to two particles correlated in the state with the total spin and orbital momentum . The neutrino emissivity due to the PBF processes in the spin-singlet superfluid nucleon matter was first suggested and calculated by Flowers et al. [3]. The result of this calculation was recovered later by other authors [4–6]. Similar mechanism for the neutrino energy losses due to spin-singlet pairing of hyperons was suggested in [7–9]. For more than three decades these ideas were a key ingredient in numerical simulations of NS evolution (e.g., [10–12]). However, after such a long period, it was unexpectedly found that the PBF emission of neutrino pairs is practically absent in a nonrelativistic spin-singlet superfluid liquid [13]. Later this result was confirmed in other calculations [14–16]. (Note also the controversial work [17].)

The importance of the suppression of the PBF neutrino emission from the superfluid was first understood in [18] in connection with the fact that the previous theory predicted a too rapid cooling of the NS’s crust, which dramatically contradicts the observed data of superbursts [19].

The neutron pairing in NS is essentially restricted to the crust. As a result, in the NS evolution, effects of the suppression are mostly observed during the thermal relaxation of the crust [20–22]. The significant revision of PBF neutrino emission from this relatively thin layer does not change substantially the total energy losses from the star. The most neutrino losses occur from the NS core, which occupies more than 90% of the star’s volume and contains the superfluid neutrons paired in the state with , , and [23, 24].

In the commonly used version of the minimal cooling paradigm, the emission of pairing was reduced by only about 30% due to the suppression of the vector channel of weak interactions [22, 25, 26]. This approach does not take into account the anomalous axial-vector weak interactions, existing due to spin fluctuations in the spin-triplet superfluid neutron matter [27]. Some simulations of the NS evolution accounting for the anomalous contributions predict a raising of its surface temperature and argue that a full exploration of this effect is necessary [28] (also see [29, 30]).

A correct description of the efficiency of neutrino emission in the PBF processes allows for a better understanding of observations [31–33]. This review is devoted to the current state of this problem. Since the complete calculations have been published repeatedly (e.g., [13, 27, 34]), I will briefly sketch the main steps of the derivation, referring the reader to the original papers for more detailed information.

#### 2. Preliminary Notes

The low-energy Hamiltonian of the weak interaction may be described in a point-like approximation. For interactions mediated by neutral weak currents, it can be written as follows (e.g., [1]): Here is the Fermi coupling constant, and the neutrino weak current is given by , where are Dirac matrices () and . The neutral weak current of the baryon, , represents the combination of the vector and axial-vector terms, and , respectively. Here represents the baryon field. The weak coupling constants and are determined by quark composition of the baryons. For the reactions with neutrons, one has and , while, for those with protons, and , where is the axial-vector constant. Notice that similar interaction Hamiltonian, but with other coupling constants, describes the neutrino weak interaction of hyperons in NS matter (e.g., [35]).

In the nonrelativistic nucleon system, the vector part of the weak current can be approximated by its temporal componentwhere . Throughout the text, a hat means a matrix in spin space and . The axial weak current is given dominantly by its space component where are Pauli spin matrices.

It is important to notice that the vector weak current is conserved in the standard theory. The conservation law implies that the transition matrix element in the vector channel of the reaction obeys the relationThe transferred momentum enters into the medium response function through the quasiparticle energy, which for in a degenerate Fermi liquid takes the form . Thus, in the absence of external fields, the momentum transfer enters the response function of the medium only in combination with the Fermi velocity, which is small in the nonrelativistic system, . Therefore, for the PBF processes the relation is always satisfied. This allows one to evaluate the medium response function in the long-wave limit . Together with the conservation law (4) this immediately yields for , which means that the neutrino pair emission through the vector channel of weak interactions is strongly suppressed in the nonrelativistic system. This important fact was overlooked for a long time, since a direct calculation shows that the matrix element for the recombination of two Bogolons into the condensate does not vanish, which erroneously leads to a large neutrino emissivity through the vector channel.

First calculations of the PBF neutrino energy losses were performed using a vacuum-type weak interactions assuming that the medium effects can be taken into account by introducing effective masses of participating quasiparticles [3, 6]. This resulted in a substantial overestimate of the PBF neutrino energy losses from the superfluid core and inner crust of NSs. Only three decades later has it been understood that the calculation of neutrino radiation from a superfluid Fermi liquid requires a more delicate approach.

Within the Nambu-Gor’kov formalism the effective vertex of nucleon interactions with an external neutrino field represents a matrix in the particle-hole space. This matrix is diagonal for nucleons in the normal Fermi liquid but it gets the off-diagonal entries in superfluid systems [36–39]. The diagonal elements represent the ordinary (dressed) vertices of the field interaction with quasiparticles and holes, respectively, while the off-diagonal elements of the matrix represent the effective vertices for a virtual breaking and formation of Cooper pairs in the external field. In other words, the off-diagonal components of the vertex matrix describe a coupling of the external field with fluctuations of the order parameter in the superfluid Fermi liquid. These so-called “anomalous weak interactions” should be necessarily taken into account when calculating the neutrino energy losses from superfluid cores of NSs.

In particular, the anomalous weak interactions are crucial for the neutrino emission caused by the PBF processes. For example, in nonrelativistic systems, the ordinary and anomalous contributions into the matrix element of the weak vector transition current are mutually cancelled in the long-wave limit, leading to a strong suppression of the PBF neutrino emission [13]. The more accurate calculation [14, 16] has shown that the neutrino pair emission owing to the density fluctuations is suppressed proportionally to . This reflects the well known fact that the dipole radiation is not possible in the vector channel in the collision of two identical particles. Thus, exactly due to the anomalous contributions, the PBF neutrino emission in the vector channel of weak interactions is practically absent.

In the case of pairing this has far-reaching consequences. The total spin of the nonrelativistic Cooper pair is conserved. Therefore the neutrino emission through the axial-vector channel of weak interactions could arise only due to small relativistic effects and is proportional to [3, 15]. Thus the PBF neutrino energy losses due to singlet-state pairing of baryons can, in practice, be neglected in simulations of NS cooling. This makes the neutrino radiation from pairing of protons or hyperons unimportant.

The minimal cooling paradigm [22] suggests that, below the critical temperature for a triplet pairing of neutrons, the dominant neutrino energy losses occur from the superfluid neutron liquid in the inner core of a NS. It is commonly believed [23, 24, 40–42] that, in this case, the pairing (with a small admixture of state) takes place with a preferred magnetic quantum number . Since the spin of a Cooper pair in the state is the spin fluctuations are possible and the PBF neutrino energy losses from the neutron superfluid occur through the axial channel of weak interactions.

The pairing interaction, in the most attractive channel, can be written as [23] where is the corresponding interaction amplitude; and are the Fermi momentum and the neutron effective mass, respectively, so that is the density of states near the Fermi surface. The angular dependence of the interaction is represented by Cartesian components of the unit vector which involves the polar angles on the Fermi surface, Further, is a real vector in the spin space, normalizable by conditionHereafter we use the angle brackets to denote angle averages,

For spin-triplet pairing, the order parameter is a symmetric matrix in the spin space, which near the Fermi surface can be written as follows (see, e.g., [43]): where the temperature-dependent gap amplitude is a real constant.

The vector defines the angle anisotropy of energy gap which depends on the phase state of the superfluid condensate. In general, this vector can be written in the form , where is a matrix. In the case of a unitary condensate the matrix must be a real symmetric traceless tensor. It may be specified by giving the orientation of its principal axes and its two independent diagonal elements in its principal-axis coordinate system. Within the preferred coordinate system, the ground state with is described by the matrix and .

#### 3. General Approach to Neutrino Energy Losses

Thermal fluctuations of the neutral weak currents in nucleon matter are closely related to the imaginary, dissipative part of the response function of the medium onto the external neutrino field. According to the fluctuation-dissipation theorem, the total energy loss per unit volume and time caused by thermal fluctuations of the neutral weak current in the nucleon matter is given by the following formula: where is the imaginary part of the retarded weak polarization tensor. The integration goes over the phase volume of neutrinos and antineutrinos of total energy and total momentum . The symbol indicates a summation over three neutrino flavors. The factor occurs as a result of averaging over the Gibbs distribution, which must be performed at finite ambient temperatures.

By inserting in this equation and making use of the Lenard’s integral where , , is the Heaviside step function, and is the signature tensor, we can write where is the number of neutrino flavors.

In general, the weak polarization tensor of the medium is a sum of the vector-vector, axial-axial, and mixed terms. The mixed vector-axial polarization has to be an antisymmetric tensor, and its contraction in (13) with the symmetric tensor vanishes. Thus only the pure-vector and pure-axial polarization should be taken into account. We then obtain where and are vector and axial-vector weak coupling constants of a neutron, respectively.

#### 4. Weak Interactions in Superfluid Fermi Liquids

Physically, the polarization tensor represents a correction to the Z-boson self-energy in the medium. Making use of the adopted graphical notation for the ordinary and anomalous propagators, =