Uncertainty Quantification for Neutrino Opacities in Core-Collapse Supernovae and Neutron Star Mergers

We start from the Hartree-fock residual propagator $G_{0}^{\tau\tau^{\prime}}$ at finite temperature defined as Fetter and Walecka (1971):

$$G_{0}^{\tau\tau^{\prime}}(\vec{k},q_{0},\vec{q})=\dfrac{f^{\tau}(\vec{k})-f^{\tau^{\prime}}(\vec{k}+\vec{q})}{q_{0}+\epsilon^{\tau}(\vec{k})-\epsilon^{\tau^{\prime}}(\vec{k}+\vec{q})+i\eta},$$

(2)

where the neutrons and protons Fermi-Dirac distributions are

$$f^{\tau}(\vec{k})=\left\{1+\exp[(\epsilon^{\tau}(\vec{k})-\mu^{\tau})/T]\right\}^{-1}\,,$$

(3)

with $\tau$ and $\tau^{\prime}$ both refer to either neutrons or protons. The quantities $\epsilon^{\tau}$ are the mean-field single particle energies, see Eq. (6), and $\mu^{\tau}$ are the chemical potentials. Given the propagator, the imaginary part of the polarization function $\Pi_{0}$, in the mean field approximation Fetter and Walecka (1971); Reddy et al. (1998), is

$$\begin{split}\mathrm{Im}~{}\Pi_{0}&=\dfrac{2}{(2\pi)^{3}}\int\!\!d^{3}k\,G_{0}(\vec{k})\\ &=\dfrac{\left\{1-\exp[(-q_{0}-\mu^{\tau}+\mu^{\tau^{\prime}})/T]\right\}}{4\pi^{2}}\times\\ &\int\!\!d^{3}k\,\delta(\epsilon^{\tau}-\epsilon^{\tau^{\prime}}-q_{0})f^{\tau}(\vec{k})[1-f^{\tau^{\prime}}(\vec{k}+\vec{q})]\,,\end{split}$$

(4)

The detailed balance theorem and $1/(w+i\eta)=\mathcal{P}(1/w)-i\pi\delta(w)$ is used to derive the second line of Eq. (4). In the linear response theory, the dynamic structure factor $S_{0}(q_{0},q)$ at finite temperature is defined as Fetter and Walecka (1971); Burrows and Sawyer (1998); Roberts et al. (2012):

	$\displaystyle S_{0}(q_{0},q)$	$\displaystyle=$	$\displaystyle\dfrac{2~{}\mathrm{Im}\Pi_{0}}{1-\exp[(-q_{0}-\mu^{\tau}+\mu^{\tau^{\prime}})/T]}$		(5)
		$\displaystyle=$	$\displaystyle\dfrac{1}{2\pi^{2}}\int\!\!d^{3}k\,\delta(\epsilon^{\tau}-\epsilon^{\tau^{\prime}}-q_{0})f^{\tau}(\vec{k})\times$
			$\displaystyle\qquad\qquad[1-f^{\tau^{\prime}}(\vec{k}+\vec{q})]\,.$

In non-relativistic limit of the mean field approximation, the nucleon energy spectrum $\epsilon^{\tau}$ is given as the sum of an effective-kinetic and mean-field terms:

$$\epsilon^{\tau}(\vec{k})=\dfrac{k^{2}}{2m_{\tau}^{*}}+U_{\tau},$$

(6)

where $m^{*}_{\tau}$ is the Landau effective mass of nucleon with isospin $\tau$, and $U_{\tau}$ is the nucleon potential. Note that the energy delta function in $S(q_{0},q)$ can be written in terms of the angle $\theta$ between $\vec{q}$ and $\vec{k}$:

	$\displaystyle\delta(q_{0}+\epsilon_{\tau}-\epsilon_{\tau^{\prime}})$	$\displaystyle=$	$\displaystyle\dfrac{M^{*}_{\tau^{\prime}}}{kq}\,\delta(\cos\theta-\cos\theta_{0})\times$		(7)
			$\displaystyle\Theta(\epsilon_{K}^{\tau}-e_{-})\,\Theta(e_{+}-\epsilon_{K}^{\tau})\,,$

where

$$\cos\theta_{0}=\dfrac{M^{*}_{\tau^{\prime}}}{kq}\left(c-\dfrac{\chi k^{2}}{2M^{*}_{\tau^{\prime}}}\right),\hskip 14.22636pt\epsilon^{\tau}_{K}=\dfrac{k^{2}}{2m^{*}_{\tau}},$$

(8)

and

$$e_{\pm}=\dfrac{2q^{2}}{2\chi^{2}M^{*}_{\tau}}\left[\left(1+\dfrac{\chi M^{*}_{\tau^{\prime}}c}{q^{2}}\right)\pm\sqrt{1+\dfrac{2\chi M^{*}_{\tau^{\prime}}c}{q^{2}}}\right],$$

(9)

with $\chi=1-M^{*}_{\tau^{\prime}}/M^{*}_{\tau}$ and $c=q_{0}+U_{\tau}-U_{\tau^{\prime}}-q^{2}/(2M^{*}_{\tau^{\prime}}).$

For charged current (CC) reactions, $\tau\neq\tau^{\prime}$ and we focus in this work on $\nu_{e}+n\rightarrow p+e^{-}$, with $\tau=n$ and $\tau^{\prime}=p$. By using the delta function (7), the $S(q_{0},q)$ in the CC channel becomes

$$S^{CC}(q_{0},q)=\dfrac{M^{*}_{\tau}M^{*}_{\tau^{\prime}}T}{\pi q}\dfrac{\xi_{-}-\xi_{+}}{1-\exp[-(q_{0}-\mu_{\tau}+\mu_{\tau^{\prime}})/T]},$$

(10)

where

$$\xi_{\pm}=\ln\left\{\dfrac{1+\exp[(e_{\pm}-\mu_{\tau}+U_{\tau})/T]}{1+\exp[(e_{\pm}+q_{0}-\mu_{\tau^{\prime}}+U_{\tau})/T]}\right\}.$$

(11)

For neutral current (NC) reactions $\tau=\tau^{\prime}$, the NC $S(q_{0},q)$ reduces to

$$S^{NC}(q_{0},q)=\dfrac{M^{*2}_{\tau}T}{\pi q}\left[\dfrac{q_{0}/T}{1-\exp(-q_{0}/T)}\left(1+\dfrac{T\xi_{-}}{q_{0}}\right)\right].$$

(12)

Note that in NC reactions $\chi\rightarrow 0$. By performing a Taylor expansion of the second term in eq. 9, $e_{-}$ reduces to Reddy et al. (1998)

$$e_{-}=\dfrac{M^{*}_{\tau}}{2q^{2}}\left(q_{0}-\dfrac{q^{2}}{2M^{*}_{\tau}}\right)^{2}.$$

(13)

To go beyond the Hartree-Fock approximation by including the long-range correlations, we calculate the HF+RPA residual propagator and solve the Bethe-Salpeter integral equation Fetter and Walecka (1971); García-Recio et al. (1992); Hernánhez et al. (1997); Pastore et al. (2014); Dzhioev and Martínez-Pinedo (2018):

$$\begin{split}G_{\mathrm{RPA}}^{\alpha}(\vec{k},q_{0},\vec{q})&=G_{0}(\vec{k},q_{0},\vec{q})+G_{0}(\vec{k},q_{0},\vec{q})\\ &\times\sum_{\alpha^{\prime}}\int\dfrac{d^{3}k^{\prime}}{(2\pi)^{3}}V_{ph}^{\alpha,\alpha^{\prime}}(\vec{k},\vec{k}^{\prime},q)G_{\mathrm{RPA}}^{\alpha^{\prime}}(k^{\prime},q_{0},q)\,.\end{split}$$

(14)

In this equation, $\alpha$ and $\alpha^{\prime}$ are quantum numbers of residual pairs, e.g., in spin-isospin channels $\alpha=(S,T)$, $\vec{k}$ and $\vec{k}^{\prime}$ are hole momenta, and $V_{ph}^{\alpha,\alpha^{\prime}}(\vec{k},\vec{k}^{\prime},q)$ is the residual interaction matrix element which describes the RPA collective excitations of the system built on a mean field (Hartree-Fock) ground state.

Neglecting the possible mixing between spin/isospin channels, $V^{\alpha,\alpha^{\prime}}_{ph}=\delta(\alpha,\alpha^{\prime})V^{\alpha}_{ph}$, and we get the RPA polarization function as

$$\Pi_{\mathrm{RPA}}^{\alpha}=\dfrac{\Pi_{0}}{1-V_{ph}^{\alpha}\Pi_{0}}.$$

(15)

In the next step, we discuss the residual interactions relevant more specifically to NC and CC neutrino-nucleon reactions.

In the monopolar Laudau approximation, where only the low-energy $\ell=0$ interaction is considered and the momenta are taken at the Fermi surface, $V_{ph}^{\alpha}(\vec{k},\vec{k}^{\prime},q)=W_{1}^{\alpha}+W_{1R}^{\alpha}+W^{\alpha}_{2}(\vec{k}_{F}^{2}+\vec{k}_{F}^{\prime 2})$, where $W_{1(2)}^{\alpha}$ and $W_{1R}^{\alpha}$ are the strength functions of the residual interactions. Note that $W_{1R}^{\alpha,\alpha^{\prime}}$ is the contribution from the rearrangement term, i.e., the term which derives from the density dependence of the effective nuclear interaction. For Skyrme force, since the density dependent term implies only the isoscalar density $\rho$, the rearrangement term does not contribute to the spin-density channel. The $W_{1}^{\alpha}$, $W_{1R}^{\alpha}$ and $W_{2}^{\alpha}$ are strength functions, see Appendix B, which can be written in terms of the Skyrme parameters as well as of the isoscalar density $\rho$ Hernánhez et al. (1997). The density and spin-density dependent residual interactions in $(pp^{-1},pp^{-1})$, $(nn^{-1},nn^{-1})$, $(pp^{-1},nn^{-1})$ and $(nn^{-1},pp^{-1})$ transitions are then given by

$$f_{0}^{\tau\tau}=\dfrac{1}{2}(W_{1}^{\tau\tau,0}+W_{1R}^{\tau\tau,0})+W_{2}^{\tau\tau,0}k_{F}(\tau)^{2},$$

(16)

$$f_{0}^{\tau-\tau}=\dfrac{1}{2}(W_{1}^{\tau-\tau,0}+W_{1R}^{\tau-\tau,0})+\dfrac{1}{2}W_{2}^{\tau-\tau,0}\left[k_{F}^{2}(\tau)+k_{F}^{2}(-\tau)\right],$$

(17)

$$g_{0}^{\tau\tau}=\dfrac{1}{2}W_{1}^{\tau\tau,1}+W_{2}^{\tau\tau,1}k_{F}(\tau)^{2},$$

(18)

$$g_{0}^{\tau-\tau}=\dfrac{1}{2}W_{1}^{\tau-\tau,1}+\dfrac{1}{2}W_{2}^{\tau-\tau,1}\left[k_{F}^{2}(\tau)+k_{F}^{2}(-\tau)\right]\,.$$

(19)

The strength functions $W_{i}^{\alpha}$ ($i=1$, $1R$, $2$) in symmetric nuclear matter (SNM) can be obtained from the general expression in $(\tau,\tau^{\prime})$ channel $W_{i}^{\tau\tau^{\prime},S}$ as Hernandez et al. (1999),

$$W_{i}^{0,S}=W_{i}^{\tau\tau,S}+W_{i}^{\tau-\tau,S},$$

(20)

$$W_{i}^{1,S}=W_{i}^{\tau\tau,S}-W_{i}^{\tau-\tau,S}.$$

(21)

from which the Landau parameters in SNM are obtained:

$$f_{0}=\dfrac{1}{2}\left(f^{\tau\tau}_{0}+f_{0}^{\tau-\tau}\right),\hskip 14.22636ptf_{0}^{\prime}=\dfrac{1}{2}\left(f^{\tau\tau}_{0}-f_{0}^{\tau-\tau}\right)$$

(22)

$$g_{0}=\dfrac{1}{2}\left(g^{\tau\tau}_{0}+g_{0}^{\tau-\tau}\right),\hskip 14.22636ptg_{0}^{\prime}=\dfrac{1}{2}\left(g^{\tau\tau}_{0}-g_{0}^{\tau-\tau}\right)$$

(23)

We now focus on the NC processes, e.g., $\nu+n\rightarrow\nu+n$, where the residual interaction $V_{ph}$ can be expressed in terms of the Landau parameters (16)-(19). The polarization functions for Fermi (Vector) and Gamow-Teller (Axial) operators are

$$\Pi^{NC}_{V}=\dfrac{\Pi^{NC}_{0}}{1-f^{nn}_{0}\Pi^{NC}_{0}}\,,$$

(24)

and

$$\Pi^{NC}_{A}=\dfrac{\Pi^{NC}_{0}}{1-g^{nn}_{0}\Pi^{NC}_{0}}\,.$$

(25)

In $(n.p.e)$ beta equilibrium condition, residual interactions corresponding to $(nn^{-1},pp^{-1})$ are also involved for the NC RPA calculation. The vector and the axial vector residual interactions in matrix form are written as:

$$V_{ph}^{V}=\begin{bmatrix}f_{nn}&f_{np}\\ f_{pn}&f_{pp}\end{bmatrix}\,,\hskip 14.22636ptV_{ph}^{A}=\begin{bmatrix}g_{nn}&g_{np}\\ g_{pn}&g_{pp}\end{bmatrix}\,.$$

(26)

Furthermore, the Hartree-Fork polarization function may be written as a $2\times 2$ diagonal matrix. We then get Reddy et al. (1999)

$$\begin{split}\begin{bmatrix}\Pi_{\mathrm{V(A)}}^{nn}&\Pi_{\mathrm{V(A)}}^{np}\\ \Pi_{\mathrm{V(A)}}^{pn}&\Pi_{\mathrm{V(A)}}^{pp}\end{bmatrix}&=\begin{bmatrix}\Pi_{0}^{n}&0\\ 0&\Pi_{0}^{p}\end{bmatrix}\\ &+\begin{bmatrix}\Pi_{\mathrm{V(A)}}^{nn}&\Pi_{\mathrm{V(A)}}^{np}\\ \Pi_{\mathrm{V(A)}}^{pn}&\Pi_{\mathrm{V(A)}}^{pp}\end{bmatrix}V_{ph}^{V(A)}\begin{bmatrix}\Pi_{0}^{n}&0\\ 0&\Pi_{0}^{p}\end{bmatrix}.\end{split}$$

(27)

A compact form of RPA polarization functions including coupling constants is

	$\displaystyle C^{T}\Pi_{\mathrm{V(A)}}^{NC}C$	$\displaystyle=$	$\displaystyle c^{2}_{n,V(A)}\Pi^{nn,V(A)}_{\mathrm{V(A)}}+c^{2}_{p,V(A)}\Pi^{pp,V(A)}_{\mathrm{V(A)}}$		(28)
			$\displaystyle+2c_{n,V(A)}c_{p,V(A)}\Pi^{np,V(A)}_{\mathrm{V(A)}},$

where

$$C=\begin{bmatrix}c_{n,V(A)}\\ c_{p,V(A)}\end{bmatrix}$$

(29)

are the coupling constants. When calculating NC vector current neutrino-proton interactions, we drop the second and the third term in the right handed side of Eq. (28), which are proportional to the vector current neutrino-proton coupling constant $c_{p,V}\approx 0$. The RPA polarization function in NC vector current channel, is

$$\begin{split}\Pi_{\mathrm{V}}^{NC}=\dfrac{(1-f_{pp}\Pi_{0}^{p})\Pi_{0}^{n}}{\Delta_{V}}\,.\end{split}$$

(30)

For the neutral current vector polarization functions, we have

$$\Delta_{V}=1-f_{nn}\Pi_{0}^{n}-f_{pp}\Pi_{0}^{p}+f_{pp}\Pi_{0}^{p}f_{nn}\Pi_{0}^{n}-f_{np}^{2}\Pi_{0}^{p}\Pi_{0}^{n}\,.$$

(31)

Note that the effect of Coulomb force in the NC vector polarization functions has been considered, by replacing $f_{pp}$ with $f_{pp}+4\pi e^{2}(q^{2}+q_{TF}^{2})^{-1}$, where $q_{TF}=4e^{2}\pi^{1/3}(3n_{p})^{2/3}$.

The axial vector polarization function is given by

$$\Pi_{\mathrm{A}}^{NC}=\dfrac{(1-g_{nn}\Pi_{0}^{n})\Pi_{0}^{p}+(1-g_{pp}\Pi_{0}^{p})\Pi_{0}^{n}-2g_{np}\Pi_{0}^{n}\Pi_{0}^{p}}{\Delta_{A}},$$

(32)

where

$$\Delta_{A}=1-g_{nn}\Pi_{0}^{n}-g_{pp}\Pi_{0}^{p}+g_{pp}\Pi_{0}^{p}g_{nn}\Pi_{0}^{n}-g_{np}^{2}\Pi_{0}^{p}\Pi_{0}^{n}.$$

(33)

The NC polarization functions have the form similar to those derived in Ref. Reddy et al. (1999). Additionally, note that in SNM, we have $f_{nn}=f_{pp}=f_{0}+f_{0}^{\prime}$ and $g_{nn}=g_{pp}=g_{0}+g_{0}^{\prime}$. By replacing $f_{nn}$ and $f_{pp}$ with $f_{0}+f_{0}^{\prime}$, Eq. (30) reproduces the vector RPA polarization function in Burrows and Sawyer (1998). And by assuming $g_{nn}=g_{pp}=-g_{np}$, Eq. (32) reproduces the axial RPA polarization function in Ref. Burrows and Sawyer (1998).

For CC, the residual interactions for charge exchange (CE) process are given by Hernandez et al. (1999):

$$V_{ph}^{CE;S}=\sum_{S}W_{i}^{CE;S}P^{S},$$

(34)

and we get the residual interactions in vector and axial vector channel

$$V_{\mathrm{f}}=\dfrac{1}{2}W_{1}^{CE;0}+W_{2}^{CE;0}k_{F}^{2},$$

(35)

$$V_{\mathrm{gt}}=\dfrac{1}{2}W_{1}^{CE;1}+W_{2}^{CE;1}k_{F}^{2}$$

(36)

corresponding to $(pn^{-1},pn^{-1})$ transitions, where $k_{F}$ is the Fermi momentum of holes states. The functions $W^{CE;S}$ for $S=0$ and 1 are given in terms of the Skyrme parameters:

$$W_{1}^{CE;0}(q=0)=-t_{0}(1+2x_{0})-\dfrac{1}{6}t_{3}\rho^{\gamma}(1+2x_{3}),$$

(37)

$$W_{2}^{CE;0}(q=0)=-\dfrac{1}{4}[t_{1}(1+2x_{1})-t_{2}(1+2x_{2})],$$

(38)

$$W_{1}^{CE;1}(q=0)=-t_{0}-\dfrac{1}{6}t_{3}\rho^{\gamma},$$

(39)

$$W_{2}^{CE;1}(q=0)=-\dfrac{1}{4}(t_{1}-t_{2}).$$

(40)

In SNM, $f_{0}^{\mathrm{CE}}=2f_{0}^{\prime}$ and $g_{0}^{\mathrm{CE}}=2g_{0}^{\prime}$, which is consistent with the CC residual interactions used in Reddy et al. (1999). The relationship between CC and NC residual interactions is discussed in more details in Appendix A. The Landau parameters in Eqs. (35)-(36) are relevant to $V_{ph}$ in CC process. Indeed, in $\nu_{e}+n\rightarrow e^{-}+p$, the polarization function for Fermi and Gammow-Teller operators are:

$$\Pi^{CC}_{V}=\dfrac{\Pi^{CC}_{0}}{1-V_{\mathrm{f}}\Pi^{CC}_{0}},$$

(41)

and

$$\Pi^{CC}_{A}=\dfrac{\Pi^{CC}_{0}}{1-V_{\mathrm{gt}}\Pi^{CC}_{0}}.$$

(42)

In low density region where the nucleon gas is hot and dilute, we use the virial EoS Horowitz et al. (2017) to deduce the residual interactions. In the Virial EoS, we have nucleon densities and pressures written in terms of spin and isospin dependent fugacities $z_{i}=e^{\mu_{i}/T}$, where $z_{i}$ could be $z_{n\uparrow}$, $z_{n\downarrow}$, $z_{p\uparrow}$ or $z_{p\downarrow}$. The thermal wavelength $\lambda=(2\pi/(mT))^{1/2}$, the spin-like virial coefficients are $b_{1}$, and the spin-opposite virial coefficients are $b_{0}$. The pressure in virial expansion reads:

$$\begin{split}P&=\dfrac{T}{\lambda^{3}}(z_{n\uparrow}+z_{n\downarrow}+z_{p\uparrow}+z_{p\downarrow}\\ &+b_{n,1}(z_{n\uparrow}^{2}+z_{n\downarrow}^{2}+z_{p\uparrow}^{2}+z_{p\downarrow}^{2})+2b_{n,0}(z_{n\uparrow}z_{n\downarrow}+z_{p\uparrow}z_{p\downarrow})\\ &+2b_{pn,1}(z_{n\uparrow}z_{p\uparrow}+z_{n\downarrow}z_{p\downarrow})+2b_{pn,0}(z_{n\uparrow}z_{p\downarrow}+z_{n\downarrow}z_{p\uparrow})).\end{split}$$

(43)

Given the pressure, the nucleon density of specified spin and isospin can be derived using

$$n_{i}=\dfrac{1}{T}\dfrac{\partial P}{\partial z_{i}}z_{i}.$$

(44)

We further write down the free energy in terms of the virial coefficients $b_{i}$ and the fugacities $z_{i}$:

$$\begin{split}f&=n_{n}\mu_{n}+n_{p}\mu_{p}-P\\ &=\dfrac{T}{2}(n_{n\uparrow}\ln z_{n\uparrow}+n_{n\downarrow}\ln z_{n\downarrow}+n_{n\uparrow}\ln z_{n\downarrow}+n_{n\downarrow}\ln z_{n\uparrow}\\ &+n_{p\uparrow}\ln z_{p\uparrow}+n_{p\downarrow}\ln z_{p\downarrow}+n_{p\uparrow}\ln z_{p\downarrow}+n_{p\downarrow}\ln z_{p\uparrow})\\ &-P,\end{split}$$

(45)

where the second line was derived by using the fact that $n_{n(p)}=n_{n\uparrow(p\uparrow)}+n_{n\uparrow(p\uparrow)}$ and $z_{n(p)}=\sqrt{z_{n\uparrow(p\uparrow)}z_{n\downarrow(p\downarrow)}}$. Note that the free energy in non-interacting nucleon gas $f^{0}$ in virial expansion can be easily calculated by replacing the virial coefficients: $b_{n,1}\rightarrow b_{n,1}^{\mathrm{free}}=-1/4\sqrt{2}$, $b_{n,0}\rightarrow b_{n,0}^{\mathrm{free}}=0$, $b_{pn,1}\rightarrow b_{pn,1}^{\mathrm{free}}=0$ and $b_{pn,0}\rightarrow b_{pn,0}^{\mathrm{free}}=0$. Finally, we have the nucleon potential energy $U_{i}$ in virial expansion:

$$U_{i}=\dfrac{\partial(f-f^{0})}{\partial n_{i}},$$

(46)

which is defined similarly as in Ref. Horowitz et al. (2012). Given the potential density $E=f-f_{0}$ and the single nucleon potential $U_{i}$, in principle we can derive the spin and isospin dependent residual interactions by finding the double density functional derivative of potential energy density $E$:

$$v_{ph}(i,j)=\dfrac{\delta^{2}E}{\delta n_{i}\delta n_{j}}=\dfrac{\delta U_{i}}{\delta n_{j}},$$

(47)

where the index $i$ indicate the spin and isospin of the nucleon states. In SNM, the monopolar Laudau parameters are defined as Sawyer (1989):

$$f_{0}=\dfrac{\partial^{2}E}{\partial n\partial n},$$

(48)

$$f^{\prime}_{0}=\dfrac{\partial^{2}E}{\partial n_{3,0}\partial n_{3,0}},$$

(49)

$$g_{0}=\dfrac{\partial^{2}E}{\partial n_{0,3}\partial n_{0,3}},$$

(50)

and

$$g^{\prime}_{0}=\dfrac{\partial^{2}E}{\partial n_{3,3}\partial n_{3,3}},$$

(51)

where $n_{3,0}=n_{p}-n_{n}$, $n_{0,3}=n_{p\uparrow}-n_{p\downarrow}+n_{n\uparrow}-n_{n\downarrow}$ and $n_{3,3}=n_{p\uparrow}-n_{p\downarrow}-n_{n\uparrow}+n_{n\downarrow}$. Following Ref. Horowitz et al. (2012), we invert Eq. (44) to second order in densities. In this way, the free energy is a function of densities up to the second order:

$$\begin{split}f&=-T(n_{n\uparrow}+n_{n\downarrow}+n_{p\uparrow}+n_{p\downarrow})-2b_{pn,0}T\lambda^{3}(n_{p\uparrow}n_{n\downarrow}\\ &+n_{p\downarrow}n_{n\uparrow})-2b_{pn,1}T\lambda^{3}(n_{p\uparrow}n_{n\uparrow}+n_{p\downarrow}n_{n\downarrow})-2b_{n,0}T\lambda^{3}\\ &(n_{n\uparrow}n_{n\downarrow}+n_{p\uparrow}n_{p\downarrow})-b_{n,1}T\lambda^{3}(n_{n\uparrow}^{2}+n_{p\uparrow}^{2}+n_{n\downarrow}^{2}+n_{p\downarrow}^{2})+\\ &\dfrac{T}{2}(n_{n\uparrow}\ln[\lambda^{3}n_{n\uparrow}]+n_{n\downarrow}\ln[\lambda^{3}n_{n\downarrow}]+n_{n\uparrow}\ln[\lambda^{3}n_{n\downarrow}]+\\ &n_{n\downarrow}\ln[\lambda^{3}n_{n\uparrow}]+n_{p\uparrow}\ln[\lambda^{3}n_{p\uparrow}]+n_{p\downarrow}\ln[\lambda^{3}n_{p\downarrow}]\\ &+n_{p\uparrow}\ln[\lambda^{3}n_{p\downarrow}]+n_{p\downarrow}\ln[\lambda^{3}n_{p\uparrow}]).\end{split}$$

(52)

Note that the nucleon potential $U_{i}$ in Eq. (46) reproduces the nucleon potentials in spin-symmetric matter in Horowitz et al. (2012). Since the virial EoS include virial coefficients up to the 2nd order, $f$ is a function of $n_{i}$ up to $\mathcal{O}(n^{2})$. Consequently, the residual interactions based on low-density virial expansion are density and isospin-density independent. They are given by:

$$\begin{split}f_{0,\mathrm{virial}}&=-\dfrac{\lambda^{3}T}{2}(b_{n,0}-b_{n,0}^{\mathrm{free}}+b_{n,1}\\ &-b_{n,1}^{\mathrm{free}}+b_{pn,0}-b_{pn,0}^{\mathrm{free}}+b_{pn,1}-b_{pn,1}^{\mathrm{free}}),\end{split}$$

(53)

$$\begin{split}f_{0,\mathrm{virial}}^{\prime}&=-\dfrac{\lambda^{3}T}{2}(b_{n,0}-b_{n,0}^{\mathrm{free}}+b_{n,1}\\ &-b_{n,1}^{\mathrm{free}}-b_{pn,0}+b_{pn,0}^{\mathrm{free}}-b_{pn,1}+b_{pn,1}^{\mathrm{free}}),\end{split}$$

(54)

$$\begin{split}g_{0,\mathrm{virial}}&=\dfrac{\lambda^{3}T}{2}(b_{n,0}-b_{n,0}^{\mathrm{free}}-b_{n,1}+b_{n,1}^{\mathrm{free}}+b_{pn,0}\\ &-b_{pn,0}^{\mathrm{free}}-b_{pn,1}+b_{pn,1}^{\mathrm{free}}),\end{split}$$

(55)

$$\begin{split}g_{0,\mathrm{virial}}^{\prime}&=\dfrac{\lambda^{3}T}{2}(b_{n,0}-b_{n,0}^{\mathrm{free}}-b_{n,1}+b_{n,1}^{\mathrm{free}}-b_{pn,0}\\ &+b_{pn,0}^{\mathrm{free}}+b_{pn,1}-b_{pn,1}^{\mathrm{free}}).\end{split}$$

(56)

Here the $b_{0,1}$ are virial coefficients for spin-opposite and spin-like particles, $b^{\mathrm{free}}$ are virial coefficients for non-interacting nucleon gas, and the length parameter $\lambda=(2\pi/(MT))^{1/2}$. The residual interactions based on virial coefficients at $\mathrm{T}=10~{}\mathrm{MeV}$ are

	$\displaystyle f_{0,\mathrm{virial}}$	$\displaystyle=$	$\displaystyle-2.03\times 10^{-4}\;\hbox{MeV}^{-2}\,,$		(57)
	$\displaystyle f_{0,\mathrm{virial}}^{\prime}$	$\displaystyle=$	$\displaystyle 1.19\times 10^{-4}\;\hbox{MeV}^{-2}\,,$		(58)
	$\displaystyle g_{0,\mathrm{virial}}$	$\displaystyle=$	$\displaystyle 2.88\times 10^{-6}\;\hbox{MeV}^{-2}\,,$		(59)
	$\displaystyle g_{0,\mathrm{virial}}^{\prime}$	$\displaystyle=$	$\displaystyle 7.31\times 10^{-5}\;\hbox{MeV}^{-2}\,.$		(60)

Correspondingly, the virial coefficients at $\mathrm{T}=10~{}\mathrm{MeV}$ are: $b_{n,0}=0.463$, $b_{n,1}=-0.152$, $b_{pn,0}=0.725$, $b_{pn,1}=1.130$, $b_{n,0}^{\mathrm{free}}=0$,$b_{n,1}^{\mathrm{free}}=-1/(4\sqrt{2})$, $b_{pn,0}^{\mathrm{free}}=b_{pn,1}^{\mathrm{free}}=0$. Because the residual interactions are density and isospin-density independent in low density regimes where the viral approximation applies, Eq. (22)-(23) are inverted to find $f_{nn}$,$f_{np}$, $g_{nn}$ and $g_{np}$ in NC vector and axial vector RPA polarization functions in asymmetric matter here.

Furthermore,

$$V_{\mathrm{f},\mathrm{virial}}=2f_{0,\mathrm{virial}}^{\prime}$$

(61)

and

$$V_{\mathrm{gt},\mathrm{virial}}=2g_{0,\mathrm{virial}}^{\prime}$$

(62)

are residual interactions in CC vector and axial vector RPA polarization functions in low density region where virial EoS is valid. Similarly, these residual interactions follow the form of $V_{\mathrm{f}}$ and $V_{\mathrm{gt}}$ in SNM, since the residual interactions are approximately density and isospin-density independent in low density regimes where the viral approximation applies.

Finally, following Ref. Du et al. (2019), a thermodynamically consistent approach is employed to connect the residual interactions calculated by the virial approach with the ones calculated using Skyrme models. We discuss this method in more details in Appendix A.

Concerning the EoS, we use a similar approach to that used in Ref. Du et al. (2019, 2022). In particular, the benefit of this EoS is that it reproduces exactly the virial approximation in the nondegenerate limit, and in the high-density limit, it reproduces the Skyrme interaction. The low-density behavior is important for the neutrinosphere in core-collapse supernovae Horowitz et al. (2012) where the Skyrme interaction fails to accurately describe the EoS. Our free energy is defined as

	$\displaystyle f(n_{B},x_{p},T)$	$\displaystyle=$	$\displaystyle f_{\mathrm{virial}}(n_{B},x_{p},T)\eta$		(63)
			$\displaystyle+f_{\mathrm{Sk}}(n_{B},x_{p},T)(1-\eta)\,.$

where the function $\eta(z_{n},z_{p})$ is given by

$$\eta(z_{n},z_{p})=1/\big{(}1+10(z_{n}^{2}+z_{p}^{2})\big{)}\,,$$

(64)

and $z_{n}$ and $z_{p}$ are the neutron and proton fugacities in the virial expansion. Note that, in Ref. Du et al. (2019), slightly different coefficients were used in the denominator of $\eta$ but these have been modified to ensure the entropy is positive everywhere. The Skyrme free energy, $f_{\mathrm{Sk}}$, is taken from the UNEDF posterior distribution from Refs. Kortelainen et al. (2014); McDonnell et al. (2015). The posterior distribution from Ref. McDonnell et al. (2015) takes the form of a table of 1000 Skyrme models, each selected from a likelihood function based on nuclear masses, charge radii, fision barriers, and other nuclear data. In Ref. Du et al. (2019) and in this work, we randomly select from that table of 1000 Skyrme models in order to obtain our results. The Skyrme parameters which we select are thus not uncorrelated, they retain the correlations which were obtained in Ref. McDonnell et al. (2015) as a result of the matching to the experimental data. These correlations are discussed below and displayed in Fig. 8. In some cases, we replace this Skyrme model with an alternate parameterization to test the variation beyond that obtained in this posterior. We use NRAPR Steiner et al. (2005) because it has been shown to be a good model for high-density matter Dutra et al. (2012) and is often used in EoS for core-collapse supernovae Schneider et al. (2019). We use SGII because it was explicitly constructed to fix the spin instability encountered in Skyrme models Van Giai and Sagawa (1981). We also use the UNEDF0 Kortelainen et al. (2010) and UNEDF2 Kortelainen et al. (2014) EoS to compare with the original posterior distribution used in Refs. Du et al. (2022, 2019). As in Ref. Du et al. (2019), we also randomly select Skyrme models from the posterior distribution generated in Ref. McDonnell et al. (2015). All of these models give a reasonable description of the binding energies, charge radii, and other experimental nuclear properties. The symmetry energy $S$ and its derivative $L$ are not taken from the posterior but used as additional parameters. Results obtained from EoS selected in this way are referred to as “MC” hereafter. In order to focus on the uncertainty of the neutrino opacities, we approximate the electron fraction in beta equilibrium to be the same for all models, $Y_{e}\approx 0.05+0.28\exp[-126-31.49\log_{10}(n_{B})]$

Given an EoS, the residual interactions, the nucleon chemical potentials and the nucleon effective mass were derived and then were applied in the neutrino opacity calculations. Note that, the effective mass is defined differently in relativistic (the “Dirac mass”) and in non-relativistic models (the “Landau mass”). Consequently, one cannot directly use the non-relativistic type effective mass derived from Skyrme EoSs in relativistic neutrino opacity calculations.

In the present study, the uncertainties in the neutrino opacities are directly resulting from the uncertainties in the EoS. The aforementioned Skyrme-type EoSs are different to each other mainly because they are constrained by different experimental measurements/astronomical observations. In this way, we evaluate the impact on the neutrino opacities due to changing the Skyrme interaction, e.g., NRAPR, SGII, UNEDF0, UNEDF2 and SVmin. Similarly, the impact of the MC EoSs on the neutrino opacities are also evaluated. Since the MC EoSs are constrained differently from the other Skyrme interactions previously mentioned, one could estimate the contribution of the nuclear and astrophysical uncertainties on the prediction of the neutrino opacities. This will be done in the discussion of our results in the following.

In the following we discuss uncertainties of the neutrino IMFPs, in the framework of HF+RPA. As shown in Eqs. (42), (41), (25) and Eq. (24), the input for the calculation of IMFPs based on HF+RPA are EoS-based quantities such as $M^{*}_{\tau}$, $M^{*}_{\tau^{\prime}}$, $\mu_{\tau}$, $\mu_{\tau^{\prime}}$, and $V_{ph}$. We first investigate the Pearson correlations between (1) two different EoS-based quantities; (2) EoS-based quantities and IMFPs and (3) two IMFPs at different densities. The Pearson coefficient is given by:

$$\rho(A,B)=\dfrac{\sum_{M=1}^{N}(A_{M}-\bar{A})(B_{M}-\bar{B})}{\sqrt{\sum_{M=1}^{N}(A_{M}-\bar{A})^{2}}\sqrt{\sum_{M=1}^{N}(B_{M}-\bar{B})^{2}}},$$

(65)

where $A$ and $B$ are (1) EoS-based quantities and EoS-based quantities; (2) EoS-based quantities and IMFPs and (3) IMFPs and IMFPs of a specific model $M$. In this work the EoS-based quantities are from $N$ hybrid EoSs, which were introduced with more details in Ref. Du et al. (2022). Given EoS-based quantities from $N$ models we define the covariance matrix $C_{ij}$ Fattoyev and Piekarewicz (2011):

$$C_{ij}=\dfrac{1}{N}\sum_{M=1}^{N}x_{i,M}x_{j,M},$$

(66)

where $x_{i,M}=(P_{i,M}-\bar{P}_{i})/\bar{P}_{i}$ and $P_{i,M}$ is the $i$th EoS-based quantity (e.g. nucleon effective mass, residual interaction, etc) predicted in EoS model $M$. The variance of IMFP is:

$$\sigma_{A}^{2}=\sum_{i,j=1}^{F}\dfrac{\partial A}{\partial x_{i}}C_{ij}\dfrac{\partial A}{\partial x_{j}},$$

(67)

where $A$ is the IMFP.