Hyperon–nucleon three-body forces and strangeness in neutron stars

Abstract

Three-body forces acting on a $\varLambda$ hyperon in a nuclear medium are investigated, with special focus on the so-called hyperon puzzle in neutron stars. The hyperon–nucleon two-body interaction deduced from SU(3) chiral effective field theory is employed at next-to-leading order. Hyperon–nucleon three-body forces are approximated using saturation by decuplet baryons and are transcribed to density-dependent effective two-body interactions. These together are taken as input in a Brueckner–Bethe–Goldstone equation with explicit treatment of the $\varLambda N\leftrightarrow \varSigma N$ and $\varLambda NN\leftrightarrow \varSigma NN$ coupled channels. Single-particle potentials of a $\varLambda$ hyperon in symmetric nuclear matter and neutron matter are calculated. With parameters of the $\varLambda NN$ three-body force constrained by hypernuclear phenomenology, extrapolations to high baryon density are performed. By comparison of the $\varLambda$ and neutron chemical potentials at densities characteristic of the core of neutron stars it is found that the combined repulsive effects of two- and three-body correlations can make the appearance of $\varLambda$ hyperons in neutron stars energetically unfavourable, thus potentially offering a possible answer to a longstanding query.

Appendices

Appendix A: Angular average in the Bethe–Goldstone equation

Here, we present details of the angular averaging procedure that precedes the derivation of Eq. (15). Total and relative momenta for two baryons $B_1$ and $B_2$ are introduced by

$$\begin{aligned} \mathbf {P} = \mathbf {p}_1 + \mathbf {p}_2,\quad \mathbf {k} = \frac{\xi _{12}\mathbf {p}_1-\mathbf {p}_2}{1+\xi _{12}},\quad \xi _{12} = \frac{M_{2}}{M_{1}}. \end{aligned}$$

(24)

A standard approximation is applied by replacing Q/e by the ratio of angle-averaged quantities, $\bar{Q}/\bar{e}$. The Pauli operator, averaged over the angle $\theta$ between $\mathbf {P}$ and $\mathbf {k}$, is given by

$$\begin{aligned} \bar{Q}_{\nu }(P,k)&= \frac{1}{2}\int _{-1}^1\!\mathrm d\cos \theta \ \varTheta \big (|\mathbf {p}_1| - k_F^{(1)}\big )\varTheta \big (|\mathbf {p}_2| - k_F^{(2)}\big )\nonumber \\&= [0|\frac{[-1|z_1|1]+[-1|z_2|1]}{2}|1] , \end{aligned}$$

(25)

with

$$\begin{aligned} z_1&= \frac{1+\xi _{12}}{2kP}\bigg \{\left( \frac{1}{1+\xi _{12}}P\right) ^2+k^2-(k_F^{(1)})^2\bigg \}, \nonumber \\ z_2&= \frac{1+1/\xi _{12}}{2kP} \bigg \{\left( \frac{\xi _{12}}{1+\xi _{12}}P\right) ^2 + k^2-(k_F^{(2)})^2\bigg \}, \end{aligned}$$

(26)

and the notation $[a|b|c]\equiv {\text {max}}(a,{\text {min}}(b,c))$ introduced in Ref. [44]. The angle-averaged energy denominator is given by

$$\begin{aligned} \bar{e}_{\nu }(P,k;\omega ) =&\omega - \frac{P^2}{2M_\nu }-\frac{k^2}{2\mu _\nu } -M_\nu \nonumber \\&- {\text {Re}}U_{B_1}(\bar{p}_1)- {\text {Re}}U_{B_2}(\bar{p}_2) , \end{aligned}$$

(27)

with $M_{\nu }=M_1+M_2$ and $\mu _{\nu }=M_1 M_2/(M_1+M_2)$. The angle-average is done for the arguments of the single particle potentials $U_{B_i}$ of the intermediate baryons:

$$\begin{aligned} \bar{p}_1&= \left( \tfrac{1}{(1+\xi _{12})^2} P^2 + k^2 + 2\tfrac{1}{1+\xi _{12}} P k\, \overline{\cos \theta }\right) ^{1/2} , \nonumber \\ \bar{p}_2&= \left( \tfrac{\xi ^2_{12}}{(1+\xi _{12})^2} P^2 + k^2 - 2\tfrac{\xi _{12}}{1+\xi _{12}} P k\, \overline{\cos \theta }\right) ^{1/2} , \end{aligned}$$

(28)

with

$$\begin{aligned} \overline{\cos \theta }&= \frac{ \int _{-1}^1\!\mathrm d\cos \theta \, \cos \theta \, Q(\mathbf {P},\mathbf {k}) }{ \int _{-1}^1\!\mathrm d\cos \theta \, Q(\mathbf {P},\mathbf {k}) } \nonumber \\&= \frac{1}{2}\big ([-1|z_2|1]-[-1|z_1|1]\big ), \end{aligned}$$

(29)

where $Q(\mathbf {P},\mathbf {k})$ is the exact Pauli blocking operator.

It is common practice to introduce a further simplification, replacing the squared momenta $P^2=P^2(\mathbf {p}_1, \mathbf {k})$ and $p^2_2=p^2_2(\mathbf {p}_1, \mathbf {k})$ entering the Bethe–Goldstone equation by their angle averages:

$$\begin{aligned} \bar{P}^2(p_1,k)&= \frac{\int _{|\mathbf {p}_2|\le k_F^{(2)}}\!\mathrm d\cos \vartheta \, P^2(p_1,k,\cos \vartheta )}{\int _{|\mathbf {p}_2|\le k_F^{(2)}}\!\mathrm d\cos \vartheta } \nonumber \\&= (1+\xi _{12})^2\left[ p_1^2+k^2-p_1k(1+[-1|x_0|1])\right] , \nonumber \\ \bar{p}_2^2(p_1,k)&= \frac{\xi _{12}}{1+\xi _{12}}\bar{P}^2(p_1,k) + (1+\xi _{12})k^2 -\xi _{12}\,p_1^2 , \end{aligned}$$

(30)

where $\vartheta$ is the angle between $\mathbf {p}_1$ and $\mathbf {k}$, and one finds

$$\begin{aligned} x_0 = \frac{\xi _{12}^2p_1^2+(1+\xi _{12})^2k^2-(k_F^{(2)})^2}{2\xi _{12}(1+\xi _{12})p_1k} . \end{aligned}$$

(31)

Note that baryon $B_2$ in the initial state is within its Fermi sea.

The weight function W(p, k) that appears in the calculation of the single-particle potential $U_B(p)$ via Eq. (17) is given by

$$\begin{aligned} W(p,k)&= \frac{1}{4\pi }\int _{|\mathbf {p}_2|\le k_F^{(2)}}\!\mathrm d\varOmega _k = \frac{1}{2}(1-[-1|x_0|1]) . \end{aligned}$$

(32)

The integration boundaries, $k_\mathrm {min}$ and $k_\mathrm {max}$, of the relative momentum are determined by the condition $W(k_1,k) = 0$, which leads to

$$\begin{aligned} k_\mathrm {min}&= {\text {max}}\left( 0, \frac{-k_F^{(2)}+\xi _{12} p}{1+\xi _{12}} \right) ,\ k_\mathrm {max} = \frac{k_F^{(2)}+\xi _{12} p}{1+\xi _{12}} . \end{aligned}$$

(33)

Appendix B: Some technical details

Equations (15) and (17) are solved numerically by alternately iterating both equations until the potentials $U_B$ converge. For certain kinematical conditions, e.g. involving large momenta in the initial state, the energy denominator in Eq. (15) vanishes, giving rise to a pole in the Bethe–Goldstone equation. To make the integral numerically manageable, the principal value prescription is utilized:

$$\begin{aligned} -\!\!\!\!\!\!\int _0^\infty \!dk \frac{N(k)}{D(k)} = \int _0^\infty \!dk \left( \frac{N(k)}{D(k)}-\frac{N(k_0)}{D^\prime (k_0)} \frac{2k_0}{k^2-k_0^2}\right) , \end{aligned}$$

(34)

for a simple pole at $k_0$, i.e. $D(k_0)=0$ and $D'(k_0)\ne 0$. This prescription eventually meets its limits at densities beyond $3.5\,\rho _0$, once second-order poles begin to show up.

For densities $\rho > 2\rho _0$ in pure neutron matter, slow convergence of $U_B$ occurs because of increasingly strong input potentials. Convergence is improved by averaging $U_B(k)$ over subsequent iterations. High-density calculations occasionally require intervention by estimating an appropriate starting point for iterations from lower-density results. However, numerical stability is generally not guaranteed any more for $\rho > \rho _c \simeq 3.5\,\rho _0$. For densities $\rho > \rho _c$ we use the power series extrapolation in Eq. (19), fitted to numerically stable results at $\rho \lesssim \rho _c$ as described in the text.

As discussed in Ref. [37], the cutoff $\lambda$ in the chiral input potential causes numerical oscillations of $U_B(k)$ for momenta $k > \lambda$. These oscillations tend to slow down the convergence of $U_B$ and are of no physical relevance. They are suppressed by the additional cutoff factor $\exp [-(k/\lambda '_{spp})^6]$ attached to the single-particle potentials as described in Sect. 4.1.