PHYSICAL REVIEW C 94, 024323 (2016)
Transverse isospin response function of asymmetric nuclear matter
from a local isospin density functional
Enrico Lipparini
*
and Francesco Pederiva
†
Dipartimento di Fisica, University of Trento, via Sommarive 14, I-38123 Povo, Trento, Italy
and INFN-TIFPA, Trento Institute for Fundamental Physics and Applications, Trento, Italy
(Received 4 June 2014; revised manuscript received 13 April 2016; published 17 August 2016)
The time dependent local isospin density approximation (TDLIDA) has been extended to the study of the
transverse isospin response function in nuclear matter with an arbitrary neutron-proton asymmetry parameter ξ .
The energy density functional has been chosen in order to fit existing accurate quantum Monte Carlo calculations
with a density dependent potential. The evolution of the response with ξ in the T
z
=±1 channels is quite
different. While the strength of the T
z
=+1 channel disappears rather quickly by increasing the asymmetry,
the T
z
=−1 channel develops a stronger and stronger collective mode that in the regime typical of neutron star
matter at β equilibrium almost completely exhausts the excitation spectrum of the system. The neutrino mean
free paths obtained from the TDLIDA responses are strongly dependent on ξ and on the presence of collective
modes, leading to a sizable difference with respect to the prediction of the Fermi gas model.
DOI: 10.1103/PhysRevC.94.024323
I. INTRODUCTION
Weak interaction processes are a key ingredient in a number
of phenomena both in terrestrial and astrophysical environ-
ments. In particular, in recent time there has been an increasing
interest in neutrino physics, not only related to detection
mechanisms, but also in connection to supernova explosions
and the cooling of neutron stars. In general, the energy range
of neutrinos of astroparticle interest is 0.1–50 MeV, implying
a direct connection with the excitation modes of nuclei and
nuclear matter [1]. The Weinberg-Salam model [2] allows for
deriving an expression of the neutrino cross section in terms
of the matrix elements of the weak current operators [3–6],
and of the corresponding response functions. The involved
excitation operators describe fluctuations in number, spin, and
isospin density. An accurate computation of the linear response
to these operators is fundamental to correctly analyze the
interaction of neutrinos with dense matter.
One of the open issues in this context is understanding
how the collective modes that are commonly observed in
nuclei (such as the giant dipole resonance) translate into
collective modes in the infinite matter, and how the energy and
the strength of such modes evolve with the density and with the
composition of the system [7,8]. Collective excitations tend
to suppress, for instance, the mean free path of neutrinos, and
could therefore play an important role in the cooling process
of neutron stars [5,6]. A thorough analysis of the response
function in nuclei and nuclear matter in the framework of
mean field theories was performed in Refs. [9–11]. In this case
particular attention was devoted to the study of the contribution
of tensor terms in the Skyrme-like functionals used. The
result is that many quantities related to the response function
(the occurrence and position of collective modes, mechanical
*
lipparin@science.unitn.it
†
pederiva@science.unitn.it
instability in nuclear matter, and the neutrino mean free path)
are particularly sensitive to the details of the interaction.
In general, we can conclude that all these problems can be
correctly addressed only if quantum correlations due to the
nucleon-nucleon interactions are taken into account.
The effect of the correlations induced by the strong
interaction on the response function has been widely studied
by Cowell and Pandhariphande [5,12] both for cold and
hot neutron matter. In that case the response function is
computed within a fully microscopic scheme. The correlations
induced in the wave function by the interaction, described by
a realistic Hamiltonian, are determined by the correlated basis
function (CBF) method. An effective interaction is then built,
and the response function is computed by diagonalizing the
matrix elements of the excitation operators on the space of
particle-hole states (correlated Tamm-Dancoff approximation
(CTD), and further extensions[13,14]). Calculations have been
carried out both at zero and finite temperature. This scheme has
been recently further extended with the inclusion of three-body
force effects by Lovato et al. [6], who studied the density and
spin density response in pure neutron matter.
The extension of these calculations to the case of nuclear
matter with an arbitrary value of the asymmetry parameter
ξ =
(N−Z)
A
is still difficult, and therefore it is important to
develop a scheme that can include as much as possible the
information coming from available fully microscopic results
and can be extended to arbitrary isospin polarizations. In a
previous work [15], we introduced a time dependent local
isospin density approximation (TDLIDA) that allowed us to
compute the longitudinal response function related to the
excitation operator
F
z
=
k
f (r
k
)τ
z
k
=
k
exp(i q · r
k
)τ
z
k
≡ τ
z
f
where q is the transferred momentum and r
k
are the coordi-
nates of the k-th particle with T
z
= 0, where T
z
=
∑
i
τ
z
i
and
τ
z
is the third component of the isospin operator (τ
z
|n〉=|n〉,
τ
z
|p〉 = −|p〉, and n,p stands for neutrons and protons,
2469-9985/2016/94(2)/024323(9) 024323-1 ©2016 American Physical Society