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我们使用最近开发的相对论介子交换电流模型来计算2p-2h通道中12 C的(νμ,μ-)散射中子-质子和质子-质子的产率。 我们使用相对论的费米气体模型针对从中到高动量传递的不同运动学计算响应函数和横截面。 与质子-质子对相比,我们发现初始状态中的中子-质子构型有很大贡献。 在改变电荷的中微子散射的情况下,质子-质子发射(即初始状态下的np)的2p-2h横截面比中子-质子发射(即初始状态下的两个中子)大2p-2h截面。 (ω,q)依赖因子。 在我们的模型中,仅通过介子交换电流(主要是Δ等压线电流)产生了不同种类的核子对的不同发射概率。 我们还分析了其他影响,包括交换贡献以及轴向和矢量电流的影响。
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Physics Letters B 762 (2016) 124–130
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Emission of neutron–proton and proton–proton pairs in neutrino
scattering
I. Ruiz Simo
a
, J.E. Amaro
a,∗
, M.B. Barbaro
b,c
, A. De Pace
c
, J.A. Caballero
d
, G.D. Megias
d
,
T.W. Donnelly
e
a
Departamento de Física Atómica, Molecular y Nuclear, and Instituto de Física Teórica y Computacional Carlos I, Universidad de Granada, Granada 18071, Spain
b
Dipartimento di Fisica, Università di Torino, Via P. Giuria 1, 10125 Torino, Italy
c
INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy
d
Departamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla, Apdo. 1065, 41080 Sevilla, Spain
e
Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
a r t i c l e i n f o a b s t r a c t
Article history:
Received
29 July 2016
Received
in revised form 12 September
2016
Accepted
13 September 2016
Available
online 16 September 2016
Editor: W.
Haxton
We use a recently developed model of relativistic meson-exchange currents to compute the neutron–
proton
and proton–proton yields in (ν
μ
, μ
−
) scattering from
12
C in the 2p–2h channel. We compute the
response functions and cross sections with the relativistic Fermi gas model for different kinematics from
intermediate to high momentum transfers. We find a large contribution of neutron–proton configurations
in the initial state, as compared to proton–proton pairs. In the case of charge-changing neutrino scattering
the 2p–2h cross section of proton–proton emission (i.e., np in the initial state) is much larger than for
neutron–proton emission (i.e., two neutrons in the initial state) by a (ω, q)-dependent factor. The different
emission probabilities of distinct species of nucleon pairs are produced in our model only by meson-
exchange
currents, mainly by the isobar current. We also analyze other effects including exchange
contributions and the effect of the axial and vector currents.
© 2016 The Authors. Publishe d by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
The identification of nuclear effects in neutrino scattering is
essential for modern neutrino oscillation experiments [1–7]. In
particular the sensitivity of the neutrino energy reconstruction to
multi-nucleon events has been stressed in recent data analyses [8].
In the MINERvA neutrino experiment an enhanced population of
multi-proton states has been observed between the quasielastic
and peaks. On the other hand, observation of events with a pair
of energetic protons at the interaction vertex accompanying the
muon in
40
Ar(ν
μ
, μ
−
) reaction has been reported in the ArgoNeuT
experiment [9]. From these events several back-to-back nucleon
configurations have been identified and associated with nuclear
mechanisms involving short-range correlated (SRC) neutron–proton
(np) pairs in the nucleus [10]. However in [11] these “hammer
events” have been modeled by a simple pion production and reab-
sorption
model without nucleon–nucleon correlations, suggesting
that the distribution of pp pairs in the final state is less sensi-
*
Corresponding author.
E-mail
address: amaro@ugr.es (J.E. Amaro).
tive to details of the initial pair configuration. In the opinion of
the authors of [11], the events cannot teach us anything significant
about SRC. The NUWRO event generator supports that the excess
of back-to-back events in ArgoNeuT has a kinematic origin and is
not directly related to SRC [12].
SRC
with back-to-back configurations have been also identified
in two-nucleon knock-out electron scattering experiments on
12
C
for high momentum transfer and missing momentum [13,14]. In
this case one expects an excess of np pairs over pp pairs [15,
16].
The experiment reported a number of np pairs 18 times
larger than their pp counterparts. The analysis of these experi-
ments
is compatible with theoretical single-nucleon and nucleon
pair momentum distributions in variational Monte Carlo calcula-
tions,
where the importance of the tensor forces in the ground-
state
correlations of nuclei has been emphasized [17,18]. While the
kinematics of the experiments have been selected to minimize the
contribution from other mechanisms that can induce two-particle
emission, such as meson-exchange currents (MEC) and isobar exci-
tations
[13], the contribution of MEC cannot be ruled out a priori
[19].
In
this work we investigate the relative effect of MEC on the
separate pp and np emission channels in the inclusive 2p–2h neu-
http://dx.doi.org/10.1016/j.physletb.2016.09.021
0370-2693/
© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
I. Ruiz Simo et al. / Physics Letters B 762 (2016) 124–130 125
trino cross section. Through this work we use the convention
that, unless specified otherwise, the charge label xy for the pair
concerns the final state pair. It has been emphasized that the
separate charge distributions of 2p–2h events are useful. One of
the reasons is for their use in Monte Carlo event generators [12,
20].
For instance in NUWRO configurations the MEC 2p–2h exci-
tations
are assumed to occur 95% of the time for events where
the interaction occurs in initial np pairs [12,21] (or final pp pairs
for charged current neutrino scattering). This value was estimated
based on the assumption, claimed also in [22], that neutrinos inter-
act
mostly with correlated np pairs. From a naive calculation this
value agrees with a factor 18/19, corresponding to the extracted
value of np/(np + pp) in the
12
C(e, e
Np) experiment of [13]. How-
ever
this neutrino generator uses a 2p–2h model that does not
give separate pp and np contributions, and therefore this choice
is not fully consistent from the theoretical point of view [12]. On
the other hand it is expected that the ratio between np and pp in-
teractions
should be kinematics dependent and not only a global
factor. Thus a theoretical quantification of the np/pp ratio and its
dependence on the typical kinematics would be desired for each
implementation of 2p–2h cross sections. Results for the separate
pp and np contributions due to short-range correlations have been
presented in [23], for the R
T
and R
CC
response functions, and for
q = 400 MeV/c, but not for the differential cross section. The con-
tribution
of initial np pairs to the T response found in [23] is about
twice that of the initial nn pairs.
We
have recently developed a fully relativistic model of meson-
exchange
currents in the 2p–2h channel for electron and neutrino
scattering [24]. This model is an extension of the relativistic MEC
model of [25] to the weak sector. It has been recently validated by
comparing to the
12
C(e, e
) inclusive cross section data for a wide
kinematic range within the SuperScaling approach (SuSA) [26]. This
model describes jointly the quasielastic and inelastic regions using
two scaling functions fitted to reproduce the data, while the 2p–2h
MEC contribution properly fills the dip region in between, result-
ing
in excellent global agreement with the data. The model has
been recently extended to the description of neutrino scattering
reactions for a variety of experiments providing an excellent agree-
ment
with data [27]. With this benchmark model we are able to
study the separate np and pp channels in the response functions
and cross section for the three (e, e
), (ν
l
, l
−
) and (
¯
ν
l
, l
+
) reactions.
While this analysis is performed in [19] for electron scattering,
in this work we consider neutrino reactions. Our model includes
the contributions of pion-in-flight, seagull, pion-pole and (1232)
excitation diagrams of the MEC. The two-body matrix elements be-
tween
relativistic spinors were presented in our recent work [24],
where they have been deduced from the weak pion production
amplitudes of [28].
2. Formalism for neutrino scattering
The formalism of 2p–2h cross section including MEC in the rel-
ativistic
Fermi gas was given in [24]. We write the charged current
(CC) cross section as
dσ
d
d
= σ
0
V
CC
R
CC
+ 2
V
CL
R
CL
+
V
LL
R
LL
+
V
T
R
T
± 2
V
T
R
T
,
(1)
where σ
0
is a kinematic factor including the weak couplings de-
fined
in [29,30]. Note that there is a linear combination of five re-
sponse
functions, labeled as CC, CL, LL, T and T
. The T
response
function contributes differently for neutrinos (plus sign) than for
antineutrinos (minus sign). The
V
K
factors are kinematic functions
that were defined in [29,30].
The
weak response functions R
K
(ω, q)—not to be confused with
the electromagnetic response functions used in previous works—
depend
on the energy and momentum transfer. They are computed
here in a relativistic Fermi gas (RFG) model, with Fermi momen-
tum
k
F
, where they can be expanded as the sum of one-particle
one-hole (1p–1h), two-particle two-hole (2p–2h), plus additional
channels. Here we are interested in the 2p–2h channel, where two
nucleons with momenta p
1
and p
2
are ejected out of the Fermi
sea, p
i
> k
F
, leaving two hole states in the daughter nucleus, with
momenta h
1
and h
2
(with h
i
< k
F
).
The
2p–2h response functions are computed as
R
K
2p
−2h
=
V
(2π)
9
d
3
p
1
d
3
h
1
d
3
h
2
m
4
N
E
1
E
2
E
1
E
2
r
K
(p
1
, p
2
, h
1
, h
2
)δ(E
1
+ E
2
− E
1
− E
2
− ω)
× θ(
p
2
− k
F
)θ(p
1
− k
F
)
× θ(
k
F
− h
1
)θ(k
F
− h
2
), (2)
where the momentum of the second nucleon is fixed by momen-
tum
conservation inside the integral sign, p
2
= h
1
+ h
2
+ q − p
1
,
V is the volume of the system, m
N
is the nucleon mass, while E
i
and E
i
are the energies of the holes and particles, respectively .
Using
energy conservation, the calculation of the inclusive
2p–2h responses of Eq. (2) for given energy and momentum trans-
fer
(ω, q), is reduced to a seven-dimensional integral that is com-
puted
numerically following the methods developed in [31,32]. The
main ingredient of the calculation is the set of five response func-
tions
r
K
(p
1
, p
2
, h
1
, h
2
), for the elementary 2p–2h transition. These
elementary response functions are written in terms of the two-
body
MEC antisymmetrized matrix elements, summed over spin.
We separate the contributions of the different charge channels to
the response functions. These can be (np, pp) for neutrinos, and
(np, nn) for antineutrinos. In [24] we derived general formulae for
the separate np and pp response functions.
The
total CC MEC for neutrino scattering can be written as
j
μ
MEC
= τ
+
(1) J
μ
1
(1
2
; 12) + τ
+
(2) J
μ
2
(1
2
; 12)
+
(
I
V
)
+
J
μ
3
(1
2
; 12), (3)
where τ
+
= τ
x
+ iτ
y
and we have defined the isospin operators
(
I
V
)
±
= (I
V
)
x
± i(I
V
)
y
(4)
that stands for the ±-component of the two-body isovector opera-
tor
I
V
= i
[
τ (1) × τ (2)
]
. (5)
The isospin-independent two-body currents J
μ
1
, J
μ
2
, and J
μ
3
, fol-
low
from the amplitudes of weak pion production model of [28]
and
are written in [24].
In
other models of neutrino scattering [22,33], only the direct
diagrams (a, b) of Fig. 1 are included, while the direct-exchange
contribution corresponding to the diagrams (c, d) are disregarded.
In our model, on the contrary, both contributions are considered.
The elementary 2p–2h transverse response function is given for pp
emission by
r
T
pp
= 4
2
μ=1
s
1
s
2
s
1
s
2
J
μ
pp
(1
2
; 12)
2
− Re J
μ
pp
(1
2
; 12)
∗
J
μ
pp
(2
1
; 12)
, (6)
where J
μ
pp
(1
2
; 12) is the effective two-body current for pp emis-
sion
with neutrinos given by
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