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旨在直接检测暗物质的大型液体氙探测器将很快成为可行的工具,用于研究中微子物理学。 有关中微子-核散射中核结构影响的信息对于区分此类探测器中的中微子背景可能很重要。 我们对中性电流中微子散射最丰富的氙同位素的差分截面和总截面进行计算。 对于弹性散射,在核壳模型中进行了核结构计算;对于弹性散射和非弹性散射,在准粒子随机相近似(QRPA)和微观准粒子-声子模型(MQPM)中进行了核结构计算。 使用合适的中微子能量分布,我们计算8B太阳中微子和超新星中微子的总平均横截面估计。
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Research Article
Neutral-Current Neutrino-Nucleus Scattering off Xe Isotopes
P. Pirinen ,
1
J. Suhonen ,
1
and E. Ydrefors
2
1
University of Jyvaskyla, Department of Physics, P.O. Box 35, 40014, Finland
2
Instituto Tecnol
´
ogico de Aeron
´
autica, DCTA, 12228-900 S
˜
ao Jos
´
e dos Campos, Brazil
Correspondence should be addressed to P. Pirinen; pekka.a.pirinen@student.jyu.
Received 26 April 2018; Accepted 18 August 2018; Published 4 October 2018
Acad
emic Editor: Athanasios Hatzikoutelis
Copyright © P. Pirinen et al. is 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. e
publication of this article was funded by SCOAP
.
Large liquid xenon detectors aiming for dark matter direct detection will soon become viable tools also for investigating neutrino
physics. Information on the eects of nuclear structure in neutrino-nucleus scattering can be important in distinguishing neutrino
backgrounds in such detectors. We perform calculations for dierential and total cross sections of neutral-current neutrino
scattering o the most abundant xenon isotopes. e nuclear-structure calculations are made in the nuclear shell model for
elastic scattering and also in the quasiparticle random-phase approximation (QRPA) and microscopic quasiparticle-phonon model
(MQPM) for both elastic and inelastic scattering. Using suitable neutrino energy distributions, we compute estimates of total
averaged cross sections for
8
B solar neutrinos and supernova neutrinos.
1. Introduction
When the idea of neutrinos was rst suggested by Pauli
in,itwasthoughtthattheywouldneverbeobserved
experimentally. Only two decades later interaction of neu-
trinos with matter was detected in the famous Cowan-
Reines experiment []. More recently, detection and research
of neutrinos have become more and more of an everyday
commodity, and various more versatile ways to examine
interactions of the little neutral one have emerged and are
being tested in laboratories all over the world.
Coherent elastic neutrino-nucleus scattering (CE]NS) is
a process where the neutrino interacts with the target nucleus
as a whole instead of a single nucleon. Although CE]NS
has been predicted since the s [], it was discovered
only very recently by the COHERENT collaboration [].
Due to the coherent enhancement, this experiment had the
remarkable feature of detecting neutrinos with a compact
. kg detector instead of a massive detector volume which
is used in conventional neutrino experiments. Coherent
neutrino-nucleus scattering is on one hand an important
potential source of information for beyond-standard-model
physics [–], but on the other hand it may also hinder new
discoveries as it will start disturbing dark matter detectors in
the near future.
A great experimental eort has been put into directly
detecting dark matter in the past few decades (see [] for
a review). e next-generation detectors are expected to
be sensitive enough to probe cross sections low enough
to start observing CE]NS as an irreducible background
[, ]. Solar neutrinos, atmospheric neutrinos, and diuse
supernova background neutrinos provide a natural source
of background neutrinos, which for obvious reasons cannot
be shielded against. As there are uncertainties in the uxes
of each of the aforementioned neutrino types, the sensitivity
of WIMP (weakly interacting massive particle) detection is
basically limited to the magnitude of this uncertainty. To
make matters worse, it has been shown that for some specic
WIMP masses and cross sections the recoil spectra of CE]NS
very closely mimic that of scattering WIMPs [].
It is therefore of utmost importance to devise a way to go
through this neutrino oor. One potential way of achieving
this is having directional sensitivity in the detector [, ]. As
solar and atmospheric neutrinos have a distinct source within
the solar system, it is expected that their recoil direction
would be dierent to that of WIMPs, which are typically
Hindawi
Advances in High Energy Physics
Volume 2018, Article ID 9163586, 11 pages
https://doi.org/10.1155/2018/9163586
Advances in High Energy Physics
assumed to be gravitationally bound in a halo spanning the
galaxy. Also arising from the dierent origin of neutrinos
and WIMPs is the idea of using timing information to
discriminate between neutrino and WIMP induced events
in a detector []. Due to the motion of the Earth around
the Sun, it is expected that the solar neutrino ux peaks
around January, but the WIMP ux peaks in June when
the velocities of the Sun and Earth are the most in phase.
e recoil spectra of WIMPs and neutrinos could also be
distinguished if the WIMP-nucleus interaction happens via
a nonstandard operator emerging in the eective eld theory
framework [, ].
Some of the leading dark matter experiments use a
liquid xenon target [–], which allows for easy scalability
to larger detector volumes. It is expected that the xenon
detectors are the rst to hit the neutrino oor. In this article
we compute cross sections for elastic and inelastic neutrino-
nucleus scattering for the most abundant xenon isotopes.
For the coherent scattering we use the quasiparticle random-
phase approximation (QRPA) framework and the nuclear
shell model to model the nuclear structure and we compare
the results between the two models. e wave functions of the
states of odd-mass xenon isotopes are obtained by using the
microscopic quasiparticle-phonon model (MQPM) on top of
a QRPA calculation. Inelastic scattering is computed in the
QRPA/MQPM formalism. In our calculations we consider
8
B
solar neutrinos and supernova neutrinos.
A similar QRPA calculation has been made in []
for
136
Xe, where both charged-current and neutral-current
inelastic scattering was examined. Similar computations of
neutral-current neutrino-nucleus scattering cross sections
have been made before for the stable cadmium isotopes in
[] and for molybdenum isotopes in []. Both calcula-
tions used the QRPA/MQPM approach. To our knowledge
this article presents the rst calculation of neutral-current
neutrino-nucleus scattering within a complete microscopic
nuclear framework for Xe isotopes other than
136
Xe.
is article is organized as follows. In Section we outline
the formalism used to compute neutral-current neutrino-
nucleus scattering. In Section we summarize the nuclear-
structure calculations made for the target xenon isotopes.
In Section we discuss the results of our cross-section
calculations and in Section conclusions are drawn.
2. Neutral-Current
Neutrino-Nucleus Scattering
In this section we summarize the formalism used to compute
neutral-current neutrino-nucleus scattering processes. We
examine standard-model reactions mediated by the neutral
0
boson, namely, the processes
] +
(
,
)
→ ] +
(
,
)
,
()
] +
(
,
)
→ ] +
(
,
)
∗
,
()
i.e., the elastic and inelastic scattering of neutrinos o a
nucleus (with nucleons and protons), respectively. In
the elastic process the initial and nal states of the target
]
]
q
k
(
A, Z
)
(
A, Z
)
(∗)
Z
0
p
k
p
F : A diagram of the neutral-current scattering process. e
four momenta of the involved particles are labeled in the gure.
nucleus are the same, while in the inelastic process excitation
of the target nucleus takes place. e kinematics of the
scatteringprocessisillustratedinFigure.Welabelthefour
momenta of the incoming and outgoing neutrino as
and
, respectively. e momenta of the target nucleus before
and aer interacting with the neutrino are
and
.e
momentum transfer to the nucleus is referred to as
=
−
=
−
. e neutrino kinetic energy before and
aer scattering is
and
.
e neutral-current neutrino-nucleus scattering dieren-
tial cross section to an excited state of energy
ex
can be
written as []
2
ex
=
2
F
k
2
+1
≥0
CL
+
≥1
T
,
()
which comprises the Coulomb-longitudinal (
CL
)and trans-
verse (
T
)parts. ey are dened as
CL
=
(
1+cos
)
M
2
+1+cos −2sin
2
L
2
+
ex
(
1+cos
)
×2Re
M
∗
L
,
()
and
T
=1−cos +sin
2
⋅
T
mag
2
+
T
el
2
+
(
1−cos
)
×2
⋅Re
T
mag
T
el
∗
,
()
where the minus sign is taken for neutrino scattering and the
plus sign for antineutrino scattering.
and
are the initial
and nal state angular momenta of the nucleus. We use the
abbreviation
=
k
k
2
,
()
Advances in High Energy Physics
and is the magnitude of the three-momentum transfer.
e formalism and various dierent operators involved are
discussed in detail in [, ].
To compute the averaged cross section , we need to
fold the computed cross sections with the energy distribution
of the incoming neutrinos. We take the supernova neutrino
spectrum to be of a two-parameter Fermi-Dirac character
FD
=
1
2
]
]
/
]
2
1+
𝑘
/(
]
−
]
)
,
()
where
]
is the so-called pinching parameter and
]
is the
neutrino temperature. e normalization factor
2
(
]
) is
dened by the formula
]
=
1+
−
]
,
()
and the temperature and mean energy of neutrinos are related
by
]
]
=
3
]
2
]
. ()
We also examine solar neutrinos from
8
B beta decay. We use
an
8
B neutrino energy spectrum from [].
3. Nuclear Structure of the Target Nuclei
In this section we outline the nuclear-structure
calculations performed for the investigated nuclei
128,129,130,131,132,134,136
Xe. We have performed computations
in the quasiparticle random-phase approximation (QRPA),
microscopic quasiparticle-phonon model (MQPM), and the
nuclear shell model.
3.1. QRPA/MQPM Calculations. e nuclear structure of
even-even Xe isotopes was computed by using the charge-
conserving QRPA framework. e QRPA is based on a
BCS calculation [], where quasiparticle creation and anni-
hilation operators are dened via the Bogoliubov-Valatin
transformation as
†
=
†
+V
,
=
−V
†
,
()
with the regular particle creation and annihilation operators
†
and
dened in []. Here contains the quantum
numbers (,
)with =(
,
,
). e excited states with
respect to the QRPA vacuum are created with the phonon
creation operator
†
=
N
†
†
𝜔
𝜔
+
𝜔
𝜔
()
for an excited state =(
,
,
,
),where
is a
number labeling the excited states of given
.Intheabove
equation
N
=
1+
(
−1
)
𝜔
1+
,
()
and
and
are amplitudes describing the wave function
that are solved from the QRPA equation
AB
−B
∗
−A
∗
X
Y
=
X
Y
, ()
where the matrix A is the basic Tamm-Danko matrix and B
is the so-called correlation matrix, both dened in detail in
[].
We perform the QRPA calculations using large model
spaces consisting of the entire –,––,––,
and – major shells, adding also the 0
13/2
and 0
11/2
orbitals. e single-particle bases are constructed by solving
the Schr
¨
odinger equation for a Coulomb-corrected Woods-
Saxon potential. We use the Woods-Saxon parameters given
in []. We make an exception for
136
Xe, adopting the set of
adjusted values of single-particle energies from []. Due to
the neutron-magic nature of
136
Xe, adjusted single-particle
energies are necessary to get agreement with experimental
energy levels. e Bonn one-boson exchange potential []
was used to estimate the residual two-body interaction.
e QRPA formalism involves several parameters that
have to be xed by tting observables to experimental data.
In the BCS calculation we t the proton and neutron pairing
strengths
p
pair
and
n
pair
so that the lowest quasiparticle
energy matches the empirical pairing gap given by the three-
point formula []:
p
(
,
)
=
1
4
(
−1
)
+1
p
(
+1,+1
)
−2
p
(
,
)
+
p
(
−1,−1
)
,
n
(
,
)
=
1
4
(
−1
)
−+1
n
(
+1,
)
−2
n
(
,
)
+
n
(
−1,
)
.
()
It should be noted that for the neutron-magic
136
Xe this
procedure cannot be done for the neutron pairing strength.
We have instead used a bare value of
pair
=1.0for
136
Xe.
e particle-particle and particle-hole terms of the two-
body matrix elements are scaled by strength parameters
pp
and
ph
, respectively. e energies of the computed QRPA
states are quite sensitive to these model parameters. We t
the lowest excited states of each
separately to experimental
values from [] by altering the values of
pp
and
ph
.e
values used for the model parameters are given in Table .
e QRPA process is known to produce states that are
spurious, namely, the rst excited 0
+
state and the rst 1
−
state. e rst 0
+
state has been deemed spurious in [, ].
e rst 1
−
stateisspuriousduetocenter-of-massmotion
as described in []. We have tted the energies of these
states to zero, if possible, by using the model parameters
pp
and
ph
, and subsequently the states have been omitted
from calculations for the even-mass isotopes and also from
the MQPM calculations for the odd-mass isotopes. e
contributions of these spurious states to the total neutrino-
nucleus scattering cross section would be tiny in any case.
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