Three-Nucleon Forces and Triplet Pairing in Neutron Matter

Abstract

The existence of superfluidity of the neutron component in the core of a neutron star, associated specifically with triplet $P-$wave pairing, is currently an open question that is central to interpretation of the observed cooling curves and other neutron-star observables. Ab initio theoretical calculations aimed at resolving this issue face unique challenges in the relevant high-density domain, which reaches beyond the saturation density of symmetrical nuclear matter. These issues include uncertainties in the three-nucleon (3N) interaction and in the effects of strong short-range correlations—and more generally of in-medium modification of nucleonic self-energies and interactions. A survey of existing solutions of the gap equations in the triplet channel demonstrates that the net impact on the gap magnitude of 3N forces, coupled channels, and mass renormalization shows extreme variation dependent on specific theoretical inputs, in some cases even pointing to the absence of a triplet gap, thus motivating a detailed analysis of competing effects within a well-controlled model. In the present study, we track the effects of the 3N force and in-medium modifications in the representative case of the $^3P_2$ channel, based on the Argonne $v_{18}$ two-nucleon (2N) interaction supplemented by 3N interactions of the Urbana IX family. Sensitivity of the results to the input interaction is clearly demonstrated. We point out consistency issues with respect to the simultaneous treatment of 3N forces and in-medium effects, which warrant further investigation. We consider this pilot study as the first step toward a systematic and comprehensive exploration of coupled-channel $^3P\,F_2$ pairing using a broad range of 2N and 3N interactions from the current generation of refined semi-phenomenological models and models derived from chiral effective field theory.

Appendices

Appendix A

The present study is based on a generalization of the BCS-type gap equations describing triplet-P pairing by a 2N potential to the presence of a 3N interaction. The details of this generalization may be found in the Ph.D. thesis of Lingfeng Yuan [16]. As this thesis is not widely available, a pdf file will be sent upon request. In this appendix, we trace the most essential steps of the relevant derivation.

Yuan’s generalization is a direct extension of the pioneering work of Takatsuka and Tamagaki on nucleonic pairing [17,18,19,20,21,22, 24] (see also Ref. [23]). Their formulation, applicable to an arbitrary pairing channel of given spin, isospin, and total angular momentum, with specialization to the $T=1$ cases $^1S_0$, $^3P_2$, and $^3PF_2$, was carried out within the Bogoliubov canonical transformation method at a level commensurate with BCS theory. Here we indicate some key steps of the derivation leading to Eq. (8) and its approximation by Eq. (9), also providing an explicit justification of this simplification.

With the three-nucleon interaction present, the extended Bogoliubov analysis performed by Yuan yields the fundamental relation

$$\begin{aligned}&2(\varepsilon _k - \mu ) + \frac{1-2h_k}{\sqrt{(1-h_k)h_k}} \sum _{k'} \sqrt{(1-h_{k'})h_{k'}} \, \mathrm{Tr}[{G_L^{m_J}}^*(\hat{k'})\, G_{L'}^{{m_J}'}(\hat{k'})]\nonumber \\&\times \, \left( V_{2}(k,k') + \sum _{k''} 2h_{k''} \, V_3(k,k',k'')\,\mathrm{Tr}[{G_{L'}^{{m_J}'}}^*(\hat{k''})\,G_{L''}^{{m_J}''}(\hat{k''})] \right) \equiv 0 \end{aligned}$$

(17)

(written for the uncoupled case for transparency), a result equivalent to that obtained by applying the variational principle to the BCS trial ground state represented schematically by Eq. (1). The spin-angle matrices [18, 29] $G_L^{m_J}(k)$ contain all angular dependencies; the angular momentum notation is standard; and $h_k$ has its usual interpretation in terms of quasiparticle occupancy. Prefatory second-quantization analysis of the 3N interaction term in the Hamiltonian and disposition of the spin degrees of freedom is straightforward but tedious. In the end, expressions feature the 2N and 3N pairing interactions in terms of momentum-dependent functions $V_2(k,k')$ and $V_3(k,k',k'')$ alone.

The corresponding gap equation in the presence of a 3N interaction is implicit in relation (17). Appealing to the definition

$$\begin{aligned} 2(\varepsilon _{k} -\mu )\equiv \frac{1-2h_{k}}{\sqrt{(1-h_{k})h_{k}}} \Delta _{L}^{m_J}(k) \end{aligned}$$

(18)

this relation yields

$$\begin{aligned} \Delta _{L}^{m_J}(k) = - {\sum _{k'}}^{} \sqrt{(1-h_{k'})h_{k'}} \, V(k,k') \, Tr[{G_{L}^{m_J}}^*(\hat{k'})\,G_{L'}^{{m_J}'}(\hat{k'})], \end{aligned}$$

(19)

for the gap components in the space of the G matrices. This result has exactly the same form as that for a pure 2N interaction, but now with

$$\begin{aligned} V(k,k') = V_{2}(k,k') + \sum _{k''} 2h_{k''} \, V_3(k,k',k'') \, Tr[{G_{L'}^{{m_J}'}}^*(\hat{k''})\,G_{L''}^{{m_J}''} (\hat{k''})]. \end{aligned}$$

(20)

Solving Eq. (18) for $h_{k}$, we also obtain the usual expression

$$\begin{aligned} h_{k} \equiv \frac{1}{2} \biggl [ 1-\frac{\varepsilon _{k} - \mu }{\sqrt{(\varepsilon _{k} - \mu )^2+ D^2(k)}} \biggr ] \end{aligned}$$

(21)

with

$$\begin{aligned} D^2(k)=\frac{1}{2}\sum _{LL'}\sum _{{m_J}{m_J}'} {\Delta _{L}^{m_J}}^*(k) \, \Delta _{L'}^{{m_J}'}(k) \, \mathrm{Tr}[ {G_{L}^{m_J}}^*(\hat{k})\, G_{L'}^{{m_J}'}(\hat{k})] \end{aligned}$$

(22)

having the role of a generalized energy gap.

Combining Eq. (21) with Eq. (19), we arrive at the single-channel gap equation in its familiar form

$$\begin{aligned} \Delta _{L}^{m_J}(k) = -{\sum _{k'}}^{} \frac{\Delta _{L}^{m_J}(k') \langle k|V|k' \rangle }{2\sqrt{(\varepsilon _{k'} - \mu )^{2}+D^2(k')}} Tr[{G_{L}^{m_J}}^*(\hat{k'})\,G_{L'}^{{m_J}'}(\hat{k'})]. \end{aligned}$$

(23)

That the contribution of the 3N potential enters in the simple additive manner (20) stems from the fact that only two of the three nucleons can form a pair, while the third nucleon plays the role of a spectator. At this BCS stage of analysis, the spectator is a plane-wave state; in the subsequent approximation introducing some non-BCS correlations, it is considered to interact only in the $^1S_0$ state.

Specialization to the case of $^3P_2$ pairing and application of angle-averaging leads to Eq. (8), and evaluation of $V_{3}(k,k',k'')$ for the Urbana IX form to Eq. (10). (In particular, the angle-averaging was performed for the type 2 solution in the classification of Takatsuka and Tamagaki [18]—the simplest, noting that the five solutions are nearly degenerate.)

The remaining angular integral affords the analytical result

$$\begin{aligned} \int _{-1}^1 \text {d}x \left[ 1 - \frac{e}{\sqrt{e^2 + \frac{1+3x^2}{16\pi }\Delta ^2} } \right] = 2 + \frac{4\sqrt{3\pi }}{3} \frac{e}{\Delta } \log \frac{-\sqrt{3}+ 2\sqrt{1 + 4\pi \left( \frac{e}{\Delta }\right) ^2}}{\sqrt{3} + 2\sqrt{1 + 4\pi \left( \frac{e}{\Delta }\right) ^2} } \nonumber \\ \end{aligned}$$

(24)

where, for brevity, e stands for $\varepsilon _{k''}-\mu$ and $\Delta$ for $\Delta (k'')$. This expression is plotted as a function of $\frac{e}{\Delta }$ in Fig. 7. Clearly to a very good approximation, and especially for $|e|>\Delta$, the function equals $4\theta (-e/\Delta )$, i.e., four times a step function, from which Eq. (9) follows.

Fig. 7

The expression (24) compared with a step function (Color figure online)

The development sketched above applies to any uncoupled partial wave. Its extension to coupled channels is in principle straightforward, but much more demanding in detail and computationally intensive. The $^3P\,F_2$ case will entail three additional expressions analogous to Eq. (10), but considerably more complicated.

Appendix B

This second appendix collects the essential theoretical background that motivates our treatment of the 3N interaction in constructing the net pairing interaction, which does not involve its simulation by a density-dependent effective 2N interaction.

With various exceptions and qualifications, all past and current studies of the triplet-pairing problem in neutron or neutron-star matter have been carried out at the BCS level of description, with the pairing matrix elements calculated for bare 2N and 3N interactions. The exceptions and qualifications are related primarily to the fact that a strict application of BCS theory (for both singlet and triplet cases) is problematic for bare interactions that are strongly repulsive at short range. This becomes apparent when one considers what to do about the single-particle (sp) energies $\varepsilon (k)$ entering the gap Eq. (2). Bare-BCS implies that these should be Hartree-Fock sp energies. However, this choice is clearly wrong, if the 2N interaction features a strong repulsive core, producing divergent sp energies in the extreme case of a hard core. So, without performing the necessary self-consistent ladder and bubble sums (i.e., proper treatment of pp- and ph-reducible vertex and self-energy parts), it is customary to resort to BHF sp energies, or variational/CBF sp energies, or simply an effective-mass approximation—which almost always entails a sacrifice of self-consistency. An essentially parallel situation arises in dealing with the contribution to the (two-body) pairing interaction implied by a 3N interaction, which, in the gap equation, is naturally expressed in the momentum representation. In this situation, the extra, unpaired nucleon plays a role analogous to that of the single particle involved in the denominator of the gap Eq. (2).

In somewhat ironic contrast, strict BCS theory is capable of dealing with the inner repulsion of a bare 2N pairing interaction, because BCS is designed to introduce the coherent effects of two-body correlations (specifically, pairing correlations) acting near the Fermi surface in momentum space. Thus, nothing obviously goes wrong for that aspect of the problem, unless the interaction involves a hard core, and even that can be handled if the gap equation is rewritten and solved in coordinate space [51]. However, it should be recognized that some poorly controlled compromise may be occurring in simultaneous description of pairing correlations for momenta near $k_F$ and strong geometrical correlations at small r, within the strict BCS framework. But that is beside the point, for the model study we have carried out.

Instead, what is important for our treatment of the effect of 3N forces on triplet pairing is that, motivated by the above considerations, we have chosen to avoid replacing the 3N interaction by an effective 2N interaction in constructing the pairing interaction of the BCS problem, thereby preserving its three-body character without any approximation, within the strict BCS context. No phenomenological averaging has been performed in the sense involved in the usual practice of approximating the 3N interaction by a density-dependent effective 2N interaction. True, an integration is performed over the momentum of the unpaired nucleon, but that is exactly what the BCS/Bogoliubov formulations require. It is also true that in practice we do make the approximation of replacing the momentum distribution by a Fermi step, but that is quite different, and it is well justified. Indeed, this extra step coincides with the so-called decoupling approximation normally applied in eliminating the dependence of the sp energies on the Bogoliubov amplitude $v_k$ or on the BCS amplitude $h_k$. However, this step is not strictly necessary, although it is expedient from a practical standpoint in trivializing one momentum integration.

To some extent, contact can be made with calculations based on use of a density-dependent effective 2N interaction in place of the 3N force, by introduction of a defect function to screen the repulsion of the UIX interaction. This is essentially equivalent to introduction of a Jastrow two-body correlation factor and calculating in lowest cluster order [14, 49]. It would appear that common ground may be reached for the two approaches, depending on details of (i) averaging over the third particle coordinate on the one hand and (ii) the choice of defect function on the other, but explicit comparison at this stage would not be very useful. What would really be needed is a consistent treatment of nuclear interactions within a comprehensive theoretical framework for superfluid states, as can be formulated within the Gorkov formalism, correlated BCS theory, or some other generic many-body theory. Although in this sense the strategy we have adopted is far from ideal, it is preferable for our study, since the effective-interaction treatment, in suppressing the 3-body character of the assumed 3N interaction, introduces an extra degree of descriptive uncertainty.