# The low energy inclusive $\nu_{\mu}(\nu_{e})$ \- 12C scattering revisited

M. Sajjad Athar<EMAIL_ADDRESS>Department of Physics, Aligarh Muslim
University, Aligarh-202002, India  S. K. Singh  Department of Physics, Aligarh
Muslim University, Aligarh-202002, India

###### Abstract

We have reviewed the current status of the inclusive neutrino scattering from
12C in the low energy region corresponding to the neutrino beams from the
pion, muon and kaon decaying at rest. The theoretical calculations of total
cross sections in various nuclear models with special emphasis on the recent
experiments with the monoenergetic neutrinos from KDAR Aguilar-Arevalo:2018ylq
along with the older experiments from KARMEN and LSND collaborations have been
discussed in the context of the recent works by Akbar et al. Akbar:2017dih
and Nikolakopoulos et al. Nikolakopoulos:2020alk . The inadequacy of the
various theoretical models used to explain the experimental results on the
inclusive neutrino scattering from nuclei at low energies has been highlighted
and the need for a better understanding of the nuclear medium effects beyond
the impulse approximation has been emphasized.

###### pacs:

12.15.-y, 13.15.+g

##  I Introduction

The inclusive neutrino scattering in 12C at low energies has been studied as a
background of the neutrino oscillation experiments performed with the liquid
scintillator detectors which contain 12C as the target. The first experiments
in the low energy region were done almost 40 years ago at the Brookhaven
National Laboratory with the muon neutrinos ($\nu_{\mu}$) as well as with the
electron neutrinos ($\nu_{e}$) Cortex . Since then quite a few experiments
have measured the inclusive cross sections with $\nu_{\mu}$ and $\nu_{e}$ in
12C nucleus in the low energy region i.e. $E_{\nu_{\mu}}\leq 280$ MeV and
$E_{\nu_{e}}\leq 52.8$ MeV Koetke:1992yk ; Albert:1994xs ;
Athanassopoulos:1997rn ; Auerbach:2002iy ; Krakauer:1991rf ; Bodmann:1994py ;
Kretschmer:2002iq ; Maschuw:1998qh ; Athanassopoulos:1997rm ; Auerbach:2001hz
. Theoretical calculations of these cross sections in 12C were done quite
early by many authors using various versions of the shell model and the Fermi
gas model to describe the structure of the 12C nucleus which are summarized in
excellent reviews of Llewellyn Smith LlewellynSmith:1971uhs , Walecka Walecka
, and Donnelly and Peccei Donnelly:1978tz . However, the recent topical
interest in studying the inclusive neutrino scattering in 12C at low energies
was regenerated again when the experimental results from the LSND
collaboration at LANL were found inconsistent with the existing theoretical
predictions specially in the case of $\nu_{\mu}$ scattering Koetke:1992yk ;
Albert:1994xs ; Athanassopoulos:1997rn ; Auerbach:2002iy . This led to many
new theoretical calculations for the neutrino scattering in 12C which were
done using various nuclear models, for example the various versions of
sophisticated microscopic models Nikolakopoulos:2020alk ; Kolbe:1994xb ;
Kolbe:1995af ; Kolbe:1999au ; Kolbe:2003ys ; Volpe:2000zn ; Auerbach:1997ay ;
Hayes:1999ew ; Auerbach:2002tw ; Paar:2008zza ; Krmpotic:2004gx ;
Pandey:2014tza ; Jachowicz:2002rr ; Gonzalez-Jimenez:2019qhq  and the
relativistic Fermi gas models Smith:1972xh ; Singh:1993rg ; Umino:1994wu ;
Umino:1996cz ; Gaisser:1986bv ; Kosmas:1996fh ; Singh:1998md ; Nieves:2004wx ;
SajjadAthar:2005ke ; Nieves:2017lij ; Valverde:2006zn ; Vagnoni:2017hll ;
Akbar:2017dih ; Ivanov:2018nlm  as well as some other models Mintz:1995ww ;
Kim:1985zs ; Frazier:1970rb ; Kim:1979 ; Kubodera:1993rk , to describe the
structure of 12C. Most of these calculations were done in the impulse
approximation(IA) and emphasized the need to go beyond the impulse
approximation by including the effects due to the nucleon-nucleon correlations
and meson exchange currents (MEC) in the nuclear medium. However, not much
work was done to study these effects except the work of Hayes and Towner
Hayes:1999ew  and Umino et al. Umino:1994wu ; Umino:1996cz  in the context of
the LSND experiment.

In recent times many experiments have measured the inclusive neutrino
scattering cross sections in 12C with $\nu_{\mu}$ in the intermediate energy
range of $E_{\nu_{\mu}}=0.7-1.2$ GeV and a theoretical understanding of these
experiments also emphasizes the need for including the effects due to the
nucleon-nucleon correlations and meson exchange currents Alvarez-Ruso:2017oui
; Alvarez-Ruso:2014bla ; Morfin:2012kn ; Formaggio:2013kya ; Gallagher:2011zza
; Katori:2016yel ; Benhar:2013bwa ; Benhar:2015wva . The muon neutrinos, in
these experiments, are produced by the decay of pions and kaons (created in
the scattering of the protons from nuclei in the proton accelerators) in
flight and arrive at the target-cum-detector with a continuous energy spectrum
having a peak around 0.6GeV. A precise determination of the energy spectrum of
$\nu_{\mu}$ used in these experiments is made difficult due to the
uncertainties in the knowledge of the hadronic interactions and the dynamics
of proton-nucleus scattering involved in the production of pions and kaons in
the energy region of a few GeV of protons. In most of these neutrino
experiments, the energy spectrum of the incident $\nu_{\mu}$ is reconstructed
using the dynamics of the two body quasielastic (QE) neutrino reactions in
nuclei. This QE dynamics is affected by the various nuclear medium effects
like the binding energy and the Fermi motion of the nucleons as well as the
nucleon-nucleon correlations which are needed to be taken into account. The
corrections due to these nuclear medium effects in the reconstruction of
incident neutrino spectrum are incorporated but they are model dependent and
lead to considerable uncertainties in determining the neutrino spectrum and
the flux averaged inclusive neutrino nucleus cross sections Alvarez-
Ruso:2017oui ; Alvarez-Ruso:2014bla ; Morfin:2012kn ; Katori:2016yel ;
Benhar:2015wva . In this context, attempts have been recently made to develop
the monoenergetic neutrino beams at the Fermilab Spitz:2012gp ; Spitz:2014hwa
; Axani:2015dha ; Acciarri:2016smi ; Antonello:2015lea ; Fava:2019fuz  and
JPARC Ajimura:2017fld ; Harada:2013yaa ; Park:2020uck ; Park:2020vxw
laboratories specially in the low energy region using kaon decaying at rest
(KDAR). The kaons decay through the $Kl_{2}$, $Kl_{3}$ and $Kl_{4}$ decay
modes ($l=e,~{}\mu$). The muon neutrinos from
$K^{+}\rightarrow\mu^{+}+\nu_{\mu}$ decay are monoenergetic with an energy
$E_{\nu_{\mu}}=236$ MeV accompanied by $\nu_{e}$ and $\nu_{\mu}$ from $K_{e3}$
and $K_{\mu 3}$ decays, respectively, with continuous energy spectra. The
MiniBooNE collaboration at the Fermilab Aguilar-Arevalo:2018ylq  has recently
reported the first measurement of the inclusive muon neutrino cross sections
with monoenergetic neutrinos of $E_{\nu_{\mu}}=236$ MeV. A comparison of the
experimental result with the recent theoretical calculations made for the
monoenergetic KDAR neutrinos Akbar:2017dih ; Nikolakopoulos:2020alk  shows a
clear discrepancy between the theoretical and the experimental results. This
discrepancy is similar to the discrepancy encountered almost 20 years ago in
the case of LSND neutrino experiment using muon neutrinos with a continuous
energy spectrum Koetke:1992yk ; Albert:1994xs ; Athanassopoulos:1997rn ;
Auerbach:2002iy  in the low energy region of $E_{\nu_{\mu}}\leq 280$ MeV. This
has motivated us to revisit the subject of the inclusive neutrino scattering
in 12C at low energies in the context of the new experiment with the
monoenergetic KDAR $\nu_{\mu}$ and emphasize the role of the nuclear medium
effects (NME).

In sections IIA and IIB, we present a review of the experimental and
theoretical works in the low energy inclusive neutrino scattering in 12C in
the context of KARMEN and LSND experiments and discuss in section II.3, the
results of the recent MiniBooNE experiment in the intermediate energy range of
0.6-1.2GeV and the latest theoretical developments in understanding the
nuclear medium effects in the QE neutrino-nucleus scattering. In section III,
we have discussed the experimental as well as theoretical results of inclusive
cross sections with monoenergetic KDAR neutrinos with $E_{\nu_{\mu}}$=236MeV
in the impulse approximation as well as some effects beyond the impulse
approximation. We conclude in section IV with some comments emphasizing the
need for further theoretical work to explain the earlier and recent results in
the low energy inclusive neutrino scattering in 12C which could help to
analyze the results expected from the future experiments planned at the
Fermilab and at the JPARC in 12C and 40Ar nuclear targets.

##  II Inclusive weak nuclear processes at low energies

###  II.1 Experimental results

In the low energy region, the following weak inclusive processes have been
studied in 12C using $\mu^{-}$, $\nu_{\mu}$ and $\nu_{e}$ beams:

$\displaystyle\mu^{-}+^{12}C$ $\displaystyle\longrightarrow$
$\displaystyle\nu_{\mu}+X,$ (1) $\displaystyle\nu_{\mu}+^{12}C$
$\displaystyle\longrightarrow$ $\displaystyle\mu^{-}+X,$ (2)
$\displaystyle\nu_{e}+^{12}C$ $\displaystyle\longrightarrow$ $\displaystyle
e^{-}+X.$ (3)

The study of the low energy weak inclusive processes in nuclei started with
the attempts made to understand the early experimental results in the
inclusive nuclear muon capture in various nuclei including 12C. Accordingly
many microscopic calculations were done for the inclusive muon capture in 12C
employing quite elaborate nuclear models to obtain the nuclear wave functions
of the initial and final nuclei. The transition rates to the ground state as
well as to the various excited states in the final nucleus were calculated and
summed over to obtain the inclusive muon capture rate. In addition to these
explicit microscopic calculations, studies were also made by applying the
closure approximation over the final nuclear states using many particle shell
model wave functions of the initial state. Furthermore, similar calculations
in various verminous versions of the relativistic Fermi gas model and the
other models were also done. A comparison of the theoretical results on
inclusive muon capture with the available experimental results emphasized the
need for understanding various nuclear medium effects beyond the impulse
approximation like the effects due to the nucleon-nucleon correlations,
residual interaction and pairing between the nucleons as well as the meson
exchange currents in nuclei. The earlier works in the inclusive muon capture
in nuclei using these methods are summarized in excellent reviews by
Mukhopadhyay Mukhopadhyay:1976hu , Measday Measday:2001yr , Auerbach and Klein
Auerbach:1984ph  and Suzuki et al. Suzuki:1987jf . These effects were also
found to be quite important in the other low energy weak process like
$p+p\rightarrow d+e^{+}+\nu_{e}$ and leading to the renormalization of the
axial coupling in the nuclear beta decays Dautry:1975xq ; Ohta:1974fa ;
Ericson:1973vj ; Riska:1970jxh ; Nieves:2011yp ; Delorme:1985ps ;
Muller:1981dc .

As mentioned in the introduction, the importance of these nuclear medium
effects was reiterated again when the inclusive neutrino scattering cross
sections at low energies were measured by the KARMEN experiment at the ISIS
facility at the Rutherford Appleton Laboratory using $\nu_{e}$ beams
Bodmann:1994py ; Kretschmer:2002iq ; Maschuw:1998qh  and by the LSND
experiment at the LAMPF Krakauer:1991rf ; Athanassopoulos:1997rn  with
$\nu_{e}$ and $\nu_{\mu}$ beams.

In the case of $\nu_{e}$ beams, the inclusive cross sections were measured in
both experiments and the results are reported in many publications
Krakauer:1991rf ; Bodmann:1994py ; Kretschmer:2002iq ; Maschuw:1998qh ;
Athanassopoulos:1997rm . The $\nu_{e}$ and $\bar{\nu}_{\mu}$ beams are
obtained from the muons decaying at rest ($\mu$DAR) with $E_{\nu_{e}}\leq
52.8$ MeV and the neutrino energy spectrum described by the Michel spectrum,
for $\nu_{e}$ is given by

$\displaystyle f(E_{\nu_{e}})$ $\displaystyle=$
$\displaystyle\frac{96{E_{\nu}}^{2}}{{m_{\mu}}^{4}}(m_{\mu}-2E_{\nu}),$ (4)

and for $\bar{\nu}_{\mu}$ by

$\displaystyle f(E_{\bar{\nu}_{\mu}})$ $\displaystyle=$
$\displaystyle\frac{16{E_{\nu}}^{2}}{{m_{\mu}}^{4}}(3m_{\mu}-4E_{\nu}),$ (5)

as shown in Fig.1 Divari:2012cj . It should be emphasized here that the
$\nu_{e}$ spectrum resembles, to a great extent, with the various simulated
spectra for supernova neutrinos available in literature Totani:1997vj ;
Duan:2010bf ; Gava:2009pj . This makes the experimental information about the
inclusive cross sections with the $\mu$DAR neutrinos in this energy range very
useful for studying the $\nu_{e}$ induced nuclear reactions in 12C and other
nuclei for the detection of supernova neutrinos Scholberg:2012id .

Figure 1: Michel spectra, showing the $\nu_{e}$ (dotted line) and
$\bar{\nu}_{\mu}$ (solid line) distributions originating from muon decay at
rest ($\mu$-DAR). Figure 2: The solid line shows $\nu_{\mu}$ flux from
$\pi^{+}$ decay in flight and the dotted line shows the flux of
$\bar{\nu}_{\mu}$ from $\pi^{-}$ decay in flight for the same integrated
proton beam.

In the case of $\nu_{\mu}$, the inclusive reaction cross sections have been
measured by the LSND collaboration at LAMPF during the years 1992–2002 and the
results are reported in many publications Koetke:1992yk ; Albert:1994xs ;
Athanassopoulos:1997rn ; Auerbach:2002iy . These experiments were done with
$\nu_{\mu}$ coming from pions decaying in flight ($\pi$DIF) in which the pions
are produced by the Los Alamos Neutrino Scattering Center (LANSCE) at LAMPF
using proton beam of energy $E_{P}\approx$ 800 MeV. The muon neutrino spectrum
is generated by a Monte-Carlo simulation which reproduces the exact spectrum
of neutrinos from pions and muons decay at rest (DAR) quite satisfactorily.
This Monte Carlo simulation program was then used to simulate the $\nu_{\mu}$
flux from the pions decay in flight. The $\nu_{\mu}$ fluxes for $\pi$DIF
neutrinos have an uncertainty of about $15\%$ due to the uncertainty in the
knowledge of the pion production cross section in this energy region. The
simulated flux of the muon neutrinos from pions decay is shown in Fig.2, where
the vertical line shows the threshold energy for the muon production. The
average neutrino energy above the muon production threshold
($E_{\text{th}}=123.1$ MeV) is around $E_{\nu}$=170MeV.

Very recently, the inclusive $\nu_{\mu}$ cross section in 12C using
monoenergetic KDAR neutrinos with $E_{\nu_{\mu}}$=236 MeV has also been
measured by the MiniBooNE collaboration Aguilar-Arevalo:2018ylq . In Table-1,
we present all the experimental results of the low energy inclusive neutrino
cross sections for $\nu_{\mu}$ and $\nu_{e}$ scattering from 12C reported in
the literature along with the respective energy of the neutrinos. It can be
seen from Table-1 that all the experimental results from KARMEN Bodmann:1994py
; Kretschmer:2002iq ; Maschuw:1998qh  and LSND Albert:1994xs ;
Athanassopoulos:1997rn ; Auerbach:2002iy  collaborations for $\nu_{e}$ are
compatible with each other. In the case of $\nu_{\mu}$, all the experimental
results are also compatible with each other except for an earlier result
reported from Koetke et al. Koetke:1992yk  which reportedly had a different
$\nu_{\mu}$ flux. In Table-1, we also show the recent results for the
inclusive neutrino cross sections in the case of the monoenergetic $\nu_{\mu}$
from KDAR(row-3).

The other low energy inclusive weak process is the ordinary muon capture in
12C, which has been experimentally studied for a long time and the total
capture rate is reported from almost a dozen experiments with varying
precision Suzuki:1987jf . We quote in Table-1, some of the most referred
experimental results reported by Martino Martino:1986gq , Eckhause et al.
Eckhause:1966 , Suzuki et al. Suzuki:1987jf , and Ishida et al. Ishida:1986jz
.

###  II.2 Theoretical results and comparison with experiments

The weak processes of inclusive muon capture and neutrino scattering from
nuclei at low energies have been theoretically studied for a long time. The
various theoretical calculations for these processes are done using:

  * (i)

the microscopic nuclear models to describe the nuclear structure of the
initial and final nuclei and to calculate the transition rate leading to the
ground state and excited state of the final nucleus. These transition rates
are then summed over all the states to obtain the inclusive cross sections.
Subsequently, these nuclear models were improved by enlarging the basis space
in the shell model(SM) calculations and incorporating the nucleon-nucleon
correlations and pairing. These microscopic nuclear models use various
parametric forms of the nucleon-nucleon potential to calculate the nuclear
states in the initial and final nuclei. These nuclear states are then used to
calculate the transitions to the ground state as well as to the various
excited states in the final nucleus using multipole expansion of the
electroweak transition operator Walecka ; Donnelly:1973enn ; Donnelly:1978tz .
In the case of inclusive reactions, these methods have limited applicability
where the transitions to only a small number of the low lying excited states
in the final nucleus can be calculated in a consistent and reliable way. As
the initial neutrino energy($E_{\nu}$) increases, the number of excited states
become large and reliable calculation of the wave function of the higher
excited states and the transition rates to all of these excited states in the
final nucleus becomes very difficult and intractable and some approximate
methods described below have been used.

  * (ii)

non-relativistic global and local Fermi gas models and their relativistic
versions have been used to describe the nucleus. In some of these models, the
experimentally determined spectral function of the nucleons is used to
describe their momentum distribution in the nucleus. These theoretical models
were later improved by incorporating the long-range nucleon-nucleon
correlation effect using ladder approximation in the calculation of particle-
hole excitations Smith:1972xh ; Gaisser:1986bv ; Singh:1993rg ; Umino:1994wu ;
Umino:1996cz ; Kosmas:1996fh ; Singh:1998md ; Nieves:2004wx ; Nieves:2017lij ;
Valverde:2006zn ; Vagnoni:2017hll ; Akbar:2017dih ; Ivanov:2018nlm .

  * (iii)

other nuclear models using the closure approximation for summing over the
contributions from various final states in the case of medium and heavy nuclei
or using the elementary particle models specially in case of some light nuclei
like ${}^{3}He,^{6}Li$ and 12C etc. Mintz:1995ww ; Kim:1985zs ; Frazier:1970rb
; Kim:1979 ; Kubodera:1993rk  were also used.

Most of the theoretical calculations mentioned above were done within the
impulse approximation and no serious attempts were made to include the nuclear
medium effects(NME) beyond the impulse approximation. Recently the importance
of including these NME effects beyond the impulse approximation in the
inclusive neutrino scattering in ${}^{12}C$ was highlighted again when the
discrepancy between the experimental results and the theoretical predictions
in the inclusive neutrino cross sections in the intermediate energy region of
500 MeV to 1 GeV were reported AguilarArevalo:2010zc ; Katori:2009du  which
led to the axial dipole mass($M_{A}$) anomaly and attempts were made to
understand this anomaly Martini:2009uj ; Martini:2010ex ; Martini:2011wp ;
Nieves:2011pp ; Nieves:2012yz .

Before we discuss the nuclear medium effects in the context of understanding
$M_{A}$ anomaly in the next section, we summarize in this section, the
representative or all the experimental results for the low energy inclusive
$(\nu_{e},e^{-}),(\nu_{\mu},\mu^{-})$ scattering cross sections from the
KARMEN, LSND and MiniBooNE collaborations and the inclusive muon capture rates
in 12C, presented in Table-2, along with the theoretical predictions for these
processes in various nuclear models. We have included theoretical results only
from those calculations which quote results for the relevant physical
observable for all the three processes i.e. the total cross section for the
$(\nu_{e},e^{-})$ and $(\nu_{\mu},\mu^{-})$ inclusive scattering as well as
the total rates for muon capture in 12C using the same nuclear model. We show
in Table-2, the physical processes (column-1), the most recent experimental
results (column-2) and the theoretical predictions for the inclusive cross
sections/capture rates in the various microscopic nuclear models (column 3)
and the Fermi gas models (column 4).

Figure 3: $\nu_{\mu}$ scattering cross section in ${}^{12}C$ per neutron
target. Clockwise from the top left: (1) Nieves et al. Nieves:2011pp  in the
relativistic Fermi gas model with RPA effect and without 2p-2h
contribution(solid line) and only the 2p-2h contribution(dashed line); (2)
Martini et al. Martini:2010ex ; Martini:2011wp  relativistic Fermi gas model
with RPA effect and without np-nh contribution (solid line) and only the np-nh
contribution(dashed line); (3) Ivanov et al. Ivanov:2018nlm  using realistic
spectral function with nucleon-nucleon correlations and without MEC
contribution (solid line) and only the MEC contribution (dashed line), and (4)
Rocco et al. Rocco:2018mwt  using impulse approximation with spectral function
and without 2p-2h contribution, and only the 2p-2h contribution(dashed line).
The results of Akbar et al. Akbar:2017dih  has been shown by dash-dotted line
using the relativistic Fermi gas model with RPA effect.

The microscopic nuclear models referred in the column 3 of Table-2, in general
use the shell model with large varying model spaces including states from 1
$\hbar\omega$ to 4 $\hbar\omega$ excitations to calculate the ground state of
the initial and the final nuclei and transitions between them. The various
models use different forms of the phenomenological nucleon-nucleon potential
like the Bonn potential, Landau-Migdal potential or the Skymre model potential
to calculate the nuclear states. Moreover, the residual interaction between
the nucleons which leads to the pairing and also to the quasi-particle
excitations of nucleons in the nuclei are also included in some nuclear models
to calculate the nuclear wave functions of the higher excited states and the
continuum state in the final nucleus. Depending upon the various assumptions
and approximations, many nuclear models have been developed to calculate the
nuclear wave functions of the initial and final nuclei and various transitions
between them. For example, the models using RPA and CRPA by Kolbe et al.
Kolbe:1994xb ; Kolbe:1995af ; Kolbe:1999au  and Jachowicz et al.
Jachowicz:2002rr , RPA with Hartree-Fock wave functions by Auerbach et al.
Auerbach:1997ay , QRPA by Volpe et al. Volpe:2000zn , PQRPA by Paar et al.
Paar:2008zza  and RQPRPA by Krmpotic et al. Krmpotic:2004gx  as well as the
large basis elaborate multiparticle shell model by Hayes and Towner
Hayes:1999ew  and Auerbach and Brown Auerbach:2002tw  have been used to
calculate the weak reactions given in Eq. (1) in 12C nucleus. The results for
the inclusive cross sections from these calculations are shown in Table-2,
column-3.

In the case of Fermi gas models, the non-relativistic global Fermi gas models
were historically the first ones to be used to calculate these reactions
LlewellynSmith:1971uhs . Later various versions like the relativistic Fermi
gas model(RFG) Smith:1972xh ; Gaisser:1986bv , non-relativistic and
relativistic version of the local Fermi gas models (LFG) Gaisser:1986bv ;
Singh:1993rg ; Umino:1994wu ; Umino:1996cz ; Kosmas:1996fh ; Singh:1998md ;
Nieves:2004wx ; Valverde:2006zn ; Nieves:2017lij ; Vagnoni:2017hll  have been
used to calculate these reactions in 12C. Further improvements in the local
Fermi gas models have been incorporated by taking into account the long range
nucleon-nucleon correlation using the Random Phase Approximation (RPA) as well
as the spectral function (SF) of the nucleons in the nucleus to describe the
momentum distribution of the nucleon to calculate the total cross section in
the low energy inclusive scattering of neutrinos and inclusive muon capture in
12C Nieves:2017lij ; Vagnoni:2017hll ; Ivanov:2018nlm . The results from these
models are shown in column-4 of Table-2.

The following general observations about the theoretical results and their
comparisons with the experimental results shown here in Table-2, can be made:

  * (i)

None of the microscopic nuclear model calculations is able to explain all the
three weak processes in a given version of the model. For example, the models
of Kolbe et al. Kolbe:1994xb ; Kolbe:1995af ; Kolbe:1999au  using RPA and CRPA
explain the $(\nu_{e},e^{-})$ reactions but overestimate the
$(\nu_{\mu},\mu^{-})$ and underestimate the $(\mu^{-},\nu_{\mu})$ reactions.
The RPA and QRPA models of Volpe et al. Volpe:2000zn  overestimates all the
processes and PQPRPA model of Krmpotic et al. Krmpotic:2004gx  explains
$(\nu_{\mu},\mu^{-})$ reaction but overestimate the other two reactions. The
HF+RPA nuclear model of Auerbach et al. Auerbach:1997ay  could explain the
$(\nu_{e},e^{-})$ and $(\nu_{\mu},\mu^{-})$ reactions but underestimate the
($\mu^{-},\nu_{\mu}$) reactions. On the other hand, the elaborate shell model
calculations with large basis space done by Hayes and Towner Hayes:1999ew  and
Auerbach and Brown Auerbach:2002tw  do better in explaining all the reactions
qualitatively well but fail to reproduce quantitatively the cross
section(capture rate) of $\nu_{e}$ and $\nu_{\mu}$ reactions ($\mu$-capture).

  * (ii)

In the case of Fermi gas models also the earlier non-relativistic calculations
by Singh and Oset Singh:1993rg , Kosmas and Oset Kosmas:1996fh , and Nieves et
al. Nieves:2004wx ; Valverde:2006zn  in the local Fermi gas model including
RPA reproduce ($\nu_{e},e^{-})$ but overestimate the ($\nu_{\mu},\mu^{-}$)
reaction cross section and underestimate $(\mu^{-},\nu_{\mu})$ capture rate.
The local Fermi gas model has been considerably improved in the latest
calculations by Nieves and Sobczyk Nieves:2017lij , Vagnoni et al.
Vagnoni:2017hll  and Ivanov et al. Ivanov:2018nlm , where relativistic
transition operators and the spectral function of nucleons are used to
calculate all the three processes. These models reproduce ($\nu_{e},e^{-}$)
and ($\nu_{\mu},\mu^{-}$) inclusive cross sections quite well but
underestimate the inclusive ($\mu^{-},\nu_{\mu}$) capture rates.

It can be concluded from the results shown in Table-2 and the discussions
given above that there exists real discrepancy between the theoretical and
experimental results of the low energy weak inclusive processes of
($\nu_{e},e^{-}$), ($\nu_{\mu},\mu^{-}$) and ($\mu^{-},\nu_{\mu}$) in 12C. It
should be noted that a discrepancy of about $15\%$ in the experimental cross
section ($\nu_{\mu},\mu^{-}$) is expected from the uncertainty due to the lack
of the exact knowledge of the $\nu_{\mu}$ flux in the LSND experiment. On the
theoretical side, it has been suggested that nuclear medium effects beyond the
impulse approximation like the meson exchange currents and multinucleon
effects due to the ph-$\Delta$h and np-nh excitations may be important even in
these low energy processes. However, some early calculation of these effects
in the ($\nu_{\mu},\mu^{-}$) reaction were not able to remove this discrepancy
Hayes:1999ew ; Umino:1994wu ; Umino:1996cz .

The important role of these nuclear medium effects beyond the impulse
approximation was emphasized again almost a decade later when the theoretical
attempts were made to understand the MiniBooNE results AguilarArevalo:2010zc ;
Katori:2009du ; Sanchez:2009zz ; Singh:2011zzp  on the inclusive muon neutrino
scattering cross sections in ($\nu_{\mu},\mu^{-}$) reaction in 12C showed that
their contributions is quite large and could explain the experimental results
using the world average value of the axial dipole mass $M_{A}$ in the
intermediate energy region of $E_{\nu_{\mu}}$=0.5 to 1GeV Martini:2009uj ;
Martini:2010ex ; Martini:2011wp ; Nieves:2011pp . The impulse approximation
calculations done to explain the experimentally observed cross sections
required $M_{A}$ to be much higher than the world average value. This led to
the MiniBooNE axial dipole mass ($M_{A}$) anomaly(section IIC). However, the
calculations of the nuclear medium effects due to the multi-nucleon
correlations and MEC effects in $\nu_{\mu}-^{12}$C scattering showed that
their contribution is quite large and could explain the experimental results
using the world average value of $M_{A}$. Such calculations are required to be
done at the lower neutrino energies relevant to the LSND experiments with muon
neutrinos.

Therefore, in the case of LSND experiments, this discrepancy could be a
manifestation of the theoretical problem of not able to describe correctly
enough the nuclear medium effect in 12C or an experimental problem of not
doing a precise enough measurement of the inclusive cross-section specially in
$(\nu_{\mu}-^{12}C)$ scattering in the energy region of muon neutrino from
pion DIF.

###  II.3 MiniBooNE Axial Dipole Mass ($M_{A}$) Anomaly and Nuclear Medium
Effects

In the V-A theory of the charged weak interactions in the standard model, the
inclusive neutrino scattering cross sections receive direct contribution from
the vector currents and axial vector currents as well as from the interference
between the vector and the axial vector currents. The direct contribution from
the vector currents is expressed in terms of the weak vector form factors
determined by the electric and magnetic Sachs’ form factors $G_{E}(q^{2})$ and
$G_{M}(q^{2})$ Punjabi:2015bba , respectively. On the other hand, the direct
contribution from the axial currents and the contribution from the
interference term involve the axial vector form factor $g_{A}(q^{2})$, which
is parameterized as:

$g_{A}(q^{2})=\frac{g_{A}(0)}{\left(1-\frac{q^{2}}{M_{A}^{2}}\right)^{2}}$ (6)

where $g_{A}(0)$ is the axial charge determined from the neutron decay to be
$g_{A}(0)=-1.26$ and $M_{A}$ is the axial dipole mass, generally taken to be
the world average value of $M_{A}=1.026\pm 0.016$GeV, determined from the
quasielastic scattering or $M_{A}=1.014\pm 0.016$ GeV determined from
threshold pion electroproduction from proton/deuteron Bernard:2001rs ;
Bodek:2007ym . However, using these values of $M_{A}$, the inclusive total
cross sections and the differential cross sections, obtained from the high
statistics experiment in 12C, performed by the MiniBooNE collaborations, were
underestimated AguilarArevalo:2010zc ; Katori:2009du ; Sanchez:2009zz . The
MiniBooNE results were analyzed using the relativistic Fermi gas model and the
microscopic nuclear models, which failed to explain the observed cross
sections using the world average value of $M_{A}$ Singh:2011zzp . It was also
reported that a higher value of $M_{A}=1.35\pm 0.17$ Benhar:2006nr ;
Benhar:2013bwa  can explain both the total cross sections and the differential
cross sections. This value of $M_{A}$ is considerably larger than the world
value of $M_{A}$ determined from earlier experiments. The higher value of
$M_{A}$ is also in disagreement with the results of another high statistics
experiment recently performed by the NOMAD collaboration in ${}^{56}Fe$, which
reported a value of $M_{A}=1.05\pm 0.02\pm 0.06$ GeV Lyubushkin:2008pe ,
consistent with the world average value of $M_{A}$. This is known as the
MiniBooNE axial dipole mass anomaly.

Assuming that the uncertainties in the neutrino flux at the MiniBooNE detector
were well estimated and are reflected in the uncertainties quoted in the cross
section measurements, there was a general consensus that the nuclear medium
effects are not adequately taken into account. This may be because:

  * 1.

The effect of nuclear medium beyond the impulse approximation like the meson
exchange currents (MEC) and the $np-nh$ and the $ph-\Delta h$ excitations are
not included adequately in these models, despite indications that they are
quite important in the region of the low and intermediate energy neutrino
reactions.

  * 2.

In the intermediate energy region of the MiniBooNE experiment, where the
$\nu_{\mu}$ spectrum peaks around 750 MeV, the real pions would be produced
which could be re-absorbed in the nuclear medium mimicking the genuine
quasielastic inclusive events leading to an enhancement in the observed
inclusive cross sections for quasielastic reactions. The effect of these
events called the quasielastic like(QE-like) events were not included
adequately in the theoretical calculations.

  * 3.

In most of the neutrino reactions, the energy of the initial neutrinos is
reconstructed using free particle quasielastic kinematics of neutrino-nucleon
reactions in the nuclear medium. This kinematics is affected by the
entanglement of the kinematics of the quasielastic-like events due to the
inelastic processes i.e. $\nu_{\mu}NN\rightarrow\mu\Delta N\rightarrow\mu NN$
or scattering from the correlated pair $\nu_{\mu}NN\rightarrow\mu NN$ in the
nucleus with the genuine lepton events produced in the real quasielastic
$\nu_{\mu}N\rightarrow\mu N$ scattering. The effect of this entanglement was
not included in reconstructing the neutrino energy leading to underestimate
the flux averaged cross sections.

A careful investigation of the above nuclear medium effects beyond impulse
approximation was undertaken in view of the MiniBooNE axial dipole mass
anomaly. The earlier calculations of Marteau et al. Marteau:1999kt , Singh and
Oset Singh:1993rg , and Nieves et al. Nieves:2004wx ; Valverde:2006zn  were,
respectively, improved by Martini et al. Martini:2010ex ; Martini:2011wp  and
Nieves et al. Nieves:2011pp  in which the $2p-2h$, $ph-\Delta h$, MEC effects
as well as the pion reabsorption effects were taken into account. It was shown
that the contribution of these effects is quite substantial in the energy
region of MiniBooNE experiment and the observed cross sections are reproduced
quite well when the above mentioned nuclear medium effects are taken into
account using the world average value of $M_{A}$ and can explain the axial
dipole mass anomaly. These results were further improved by the calculations
of Rocco et al. Rocco:2018mwt  and Ivanov et al. Ivanov:2018nlm  using the
nucleon spectral functions to describe the momentum distribution of the
nucleons in the nucleus.

##  III Inclusive Cross sections with monoenergetic KDAR neutrinos with
$E_{\nu_{\mu}}$=236MeV

###  III.1 Experimental and Theoretical results in the Impulse Approximation

The monoenergetic muon neutrinos from KDAR are identified as an ideal probe to
study the neutrino nucleus cross sections in the low energy region in order to
benchmark the nuclear medium effects in the exclusive as well as in the
inclusive reactions in this energy region. An experiment for measuring
neutrino cross section $\sigma(E_{\nu_{\mu}})$ with monoenergetic neutrinos
would be free from the uncertainties arising from the reconstruction procedure
of the initial neutrino energy present in most of the present experiments
using beams of continuous energy of the muon neutrinos from accelerators. Some
new experiments have been planned to measure inclusive cross sections in 12C
and 40Ar using the monoenergetic neutrinos from kaons decaying at rest. The
monoenergetic neutrinos from kaons decaying at rest are copiously produced
with an energy $E_{\nu_{\mu}}=236$ MeV from the
$K^{+}\rightarrow\mu^{+}\nu_{\mu}$ decays along with a continuous energy
spectrum of $\nu_{e}$ and $\nu_{\mu}$ from $Kl_{3}$ decays like
$K^{+}\rightarrow\pi^{0}e^{+}\nu_{e}$ and
$K^{+}\rightarrow\pi^{0}\mu^{+}\nu_{\mu}$.

The first measurements of the inclusive cross section in 12C nucleus with the
monoenergetic KDAR muon neutrinos has been recently reported by the MiniBooNE
collaboration to be $(2.7\pm 0.9\pm 0.8)\times 10^{-39}$ cm2/neutron Aguilar-
Arevalo:2018ylq . Adding the errors in quadrature, the collaboration has also
quoted a value of $\sigma=(2.7\pm 1.2)\times 10^{-39}cm^{2}$. Theoretically,
this reaction in 12C in the low energy region of few hundreds of MeV, has been
studied by many authors as discussed in section-II, but specific calculations
and discussions of the inclusive cross section for $E_{\nu_{\mu}}=236$ MeV
have been done recently by Akbar et al. Akbar:2017dih  in the relativistic
Fermi gas model with RPA to include the effect of correlations and
Nikolakopoulos et al. Nikolakopoulos:2020alk  in a microscopic model using
CRPA to include the effect of nucleon -nucleon correlations. Nikolakopoulos et
al. Nikolakopoulos:2020alk  have also extrapolated the results of some earlier
calculations to predict the inclusive neutrino cross sections at
$E_{\nu_{\mu}}$=236MeV and presented a comparative study of the theoretical
and experimental results. In Table-3, we present a list of the theoretical
results for the inclusive cross sections at $E_{\nu_{\mu}}$=236MeV in the
process $\nu_{\mu}+^{12}C\rightarrow\mu^{-}+X$ obtained in various theoretical
calculations along with the experimental result from the MiniBooNE experiment
Aguilar-Arevalo:2018ylq  for comparison.

$\sigma(\nu_{\mu})$ |  123.7$<E_{\nu_{\mu}}<$280MeV |  159$\pm$26$\pm$37 Koetke:1992yk  |  8.3$\pm$0.7$\pm$1.6 Albert:1994xs  |  11.2$\pm$0.3$\pm$1.8 Athanassopoulos:1997rn
---|---|---|---|---
$(\times 10^{-40}cm^{2})$ |  |  |  12.4$\pm$0.3$\pm$1.8 Athanassopoulos:1997pv  |  10.6$\pm$ 0.3$\pm$1.8 Auerbach:2002iy
$\sigma(\nu_{e})$ |  $E_{\nu_{e}}<52.8$ MeV |  14.1$\pm$2.3 Krakauer:1991rf  |  15.2$\pm$1.0$\pm$1.3 Bodmann:1994py  |  14.8$\pm$0.7 $\pm$1.1 Athanassopoulos:1997rm
$(\times 10^{-42}cm^{2})$ |  |  |  |  16.8$\pm$1.7 Kleinfelleret:1993
$\sigma(\nu_{\mu})$ |  |  2.7$\pm$0.9$\pm$0.8 Aguilar-Arevalo:2018ylq  |  |
$(\times 10^{-39}cm^{2}/neutrons)$ |  $E_{\nu_{\mu}}=$ 236 MeV |  |  |
$\Gamma(\mu^{-})$ |  $m_{\mu}$=105.6 MeV |  3.88 $\pm$ 0.05 Suzuki:1987jf  |  |  3.71$\pm$0.11 Eckhause:1966
($10^{4}s^{-1}$) |  |  |  |  3.77$\pm$0.07 Ishida:1986jz
|  |  |  |  3.76$\pm$0.04 Martino:1982
Table 1: A summary of the experimental results for the flux averaged inclusive $\nu_{e}$ and $\nu_{\mu}$ scattering cross sections and muon capture rates in 12C. Process | Experiments | Microscopic Theories | Fermi Gas
---|---|---|---
|  |  | Model
$\sigma(\nu_{\mu})$ |  | 28.8,22.4,14.5,15.2 Auerbach:1997ay  | 16.65$\pm$1.37 Singh:1998md
$(\times 10^{-40})$ |  | 27.0,21.1,13.5,14.3 Auerbach:1997ay  | 19 Kosmas:1996fh
$cm^{2}$ | 12.4$\pm$0.3$\pm$1.8 Athanassopoulos:1997rn ; Kolbe:1999au  | 19.25 Kolbe:1995af ,19.59 Paar:2008zza  | 13.2$\pm$0.7,9.7$\pm$0.3,12.2 Nieves:2017lij
$\nu_{\mu}(^{12}C,X)\mu^{-}$ |  | 18.18 Kolbe:1999au  | 22.7-24.1 Umino:1996cz ; Umino:1994wu
|  | 15.6,13.2,17.0,31.3,19.1 Hayes:1999ew  | 25 Singh:1993rg
|  | 15.18,17.80,19.23,20.29,21.08 Volpe:2000zn  | 22.7-24.1 Umino:1994wu ; Umino:1996cz
|  | 30.0, 19.2 Auerbach:2002tw  | 11.9 Nieves:2004wx
$\sigma(\nu_{e})$ |  | 15.2,15.6 Kolbe:1994xb  | 14 Nieves:2004wx ,15.48$\pm$1.13 Singh:1998md
$(\times 10^{-42})$ |  | 16.42,16.70,55.1,52.0 Volpe:2000zn  | 14 Kosmas:1996fh
$cm^{2}$ |  | 19.28, Kolbe:1999au  | 13.8$\pm$0.4,14.3$\pm$0.6,8.6 Nieves:2017lij
|  | 6.9,3.5,4.1,5.4,3.1 Hayes:1999ew  |
$\nu_{e}(^{12}C,X)e^{-}$ | 14.8$\pm$0.7$\pm$1.4 | 23.7, 15.1 Auerbach:2002tw ,15 Kolbe:2003ys ,12.14 Paar:2008zza  | 15.3 Singh:1993rg
|  | 14.6 Donnelly:1978tz ,13.1 Donnelly:1973enn  | 13.6 SajjadAthar:2005ke
|  | 90.6,63.2,12.9,17.6 Auerbach:1997ay  |
|  | 114.4,76.3,16.5,22.7 Auerbach:1997ay  |
$\Gamma(\mu^{-})$ | 3.88 $\pm$ 0.05 Suzuki:1987jf  | 4.82,4.26,4.07,4.47 Krmpotic:2004gx  | 3.3 Singh:1993rg
$(\times 10^{4})$ |  |  | 3.37$\pm$0.16,3.22,3.19$\pm$0.06 Nieves:2017lij
$sec^{-1}$ |  | 2.98,2.99,3.17,3.40 Hayes:1999ew  | 3.60$\pm$0.22 Singh:1998md
$\mu^{-}(^{12}C,X)\nu_{\mu}$ |  | 3.56,4.53 Hayes:1999ew  |
|  |  | 3.21 Nieves:2004wx
|  | 5.24,3.35 Auerbach:2002tw  |
|  | 8.0,6.87,3.09,3.48 Auerbach:1997ay  |
|  | 8.4,7.22,3.23,3.64 Auerbach:1997ay  |
|  | 3.32,4.06,5.12,5.79 Volpe:2000zn  |
Table 2: Latest experimental results and various theoretical results in different nuclear models for inclusive cross section for $\nu_{e}$ and $\nu_{\mu}$ scattering and muon capture rates in12C. Experimental and Theoretical Models | Cross section
---|---
MiniBooNE Exp. Aguilar-Arevalo:2018ylq  | 2.7$\pm$1.2
Akbar et al. Akbar:2017dih  | 0.91
Martini et al. Martini:2011wp ; Auerbach:2001hz  | 1.3+0.2(np-nh)
NuWro Juszczak:2009qa ; Golan:2012wx  | 1.3+0.4(np-nh)
GENIE Andreopoulos:2009rq  | 1.75
NUANCE Casper:2002sd  | 1.4
CRPA Gonzalez-Jimenez:2019qhq  | 1.58
RMF Gonzalez-Jimenez:2019qhq  | 1.56
RFG Nikolakopoulos:2020alk  | 1.66
RFG 34 Nikolakopoulos:2020alk  | 1.38
Table 3: Experimental and theoretical results for the inclusive cross section
for KDAR neutrinos. The cross sections are in units of $10^{-39}cm^{2}$.

We see that all the theoretical predictions for the theoretical cross sections
lie in the wide range of $(0.91~{}\text{to}~{}1.66)\times
10^{-39}cm^{2}/neutron$ and underestimate the experimental results for the
inclusive cross sections at $E_{\nu_{\mu}}$=236MeV. A comparison of the
theoretical results of the inclusive neutrino cross sections in 12C in the
case of monoenergetic neutrinos at $E_{\nu_{\mu}}$=236MeV and the earlier
theoretical results in the energy region of the LSND experiments i.e.
$E_{\nu_{\mu}}<$ 280MeV with the experimental data, discussed in section-2,
shows that :

  * (i)

The theoretical predictions using various nuclear models for the inclusive
cross sections for the reaction $\nu_{\mu}+^{12}C\rightarrow\mu^{-}+X$ have a
large range of variation. This is surprising in the case of 12C which is one
of the theoretically better studied nucleus.

  * (ii)

Most of the theoretical predictions for the inclusive cross section for this
reaction overestimate the experimental results for the LSND experiment with
neutrino energy $E_{\nu_{\mu}}$ in the range of $120MeV<E_{\nu_{\mu}}<280$
MeV, while the theoretical predictions in the case of the KDAR neutrino with
$E_{\nu_{\mu}}=236$ MeV underestimate the experimental result.

  * (iii)

The latest experimental as well as theoretical results for the inclusive cross
sections using the monoenergetic KDAR muon neutrinos together with the results
obtained in the case of LSND and KARMEN experiments with electron and muon
neutrinos along with the capture rate of $(\mu^{-},\nu_{\mu})$ process in 12C
show that the theoretical calculations done in the impulse approximation for
all the weak nuclear processes of $(\mu^{-},\nu_{\mu}),~{}(\nu_{e},e^{-})$ and
$(\nu_{\mu},\mu^{-})$ in the low energy region are not theoretically
understood satisfactorily, with a given nuclear model used to describe the
structure of 12C nucleus.

###  III.2 Beyond the Impulse Approximation

We have discussed the importance of the nuclear medium effects like the 2p-2h,
ph-$\Delta$h and meson exchange currents beyond the impulse approximation in
the case of inclusive neutrino scattering in 12C in the intermediate energy
region of several hundreds of MeV. These effects were also shown earlier to be
important in the very low energy region of the nuclear beta decays
Dautry:1975xq ; Ohta:1974fa ; Ericson:1973vj ; Riska:1970jxh ; Nieves:2011yp ;
Delorme:1985ps ; Muller:1981dc , solar neutrino reaction Gari  and muon
capture Suzuki:1987jf ; Mukhopadhyay:1998me . However, very few studies of
these effects have been made in the case of inclusive neutrino reactions in
12C relevant to the neutrino energies of $\pi$-DIF and KDAR neutrinos, for
example, the early works of Hayes and Towner Hayes:1999ew  and Umino et al.
Umino:1994wu ; Umino:1996cz . in the case of $\pi$-DIF neutrinos and the
recent work of Nikolakopoulos et al. Nikolakopoulos:2020alk  in the case of
KDAR neutrinos.

In the case of $\pi$-DIF neutrinos, the calculations of Hayes and Towner
Hayes:1999ew  were performed in a microscopic nuclear model using a
multiparticle shell model with large basis space, while the calculations of
Umino et al. Umino:1994wu ; Umino:1996cz  were done using the relativistic
Fermi gas model. Both the calculations find a reduction of about 20$\%$ in the
inclusive cross sections while in the case of KDAR neutrinos, the work of
Nikolakopoulos et al. Nikolakopoulos:2020alk  finds an increase of about
20-25$\%$ obtained from extrapolating the works of NuWro Juszczak:2009qa ;
Golan:2012wx  and Martini et al. Martini:2010ex ; Martini:2011wp .

Recently an ab initio calculation of the inclusive $\nu_{\mu}$ cross section
in 12C has been done by Rocco et al. Rocco:2018mwt  including the contribution
of some two body effects. Moreover, Ivanov et al. Ivanov:2018nlm  have made an
improvement over the calculations of Nieves et al. Nieves:2011pp  by using an
spectral function $S(p,E)$ for the nucleon momentum distribution to calculate
the inclusive cross sections in the relativistic Fermi gas model. We show in
Fig.3, the results for the inclusive cross section $\sigma(E_{\nu_{\mu}})$ as
a function of the neutrino energy E${}_{\nu_{\mu}}$ in the energy range
$0<E_{\nu_{\mu}}<500$MeV in various models. It is clear from Fig.3 that the
calculations by Rocco et al. Rocco:2018mwt  and Ivanov et al. Ivanov:2018nlm
show an enhancement in the inclusive cross section at $E_{\nu_{\mu}}$=236MeV
which are quantitatively small as compared to the results quoted by
Nikolakopoulos et al. Nikolakopoulos:2020alk .

We observe from the results shown in Table-3, that:

  * (i)

The contribution of the nuclear medium effects beyond the impulse
approximation is to increase the inclusive cross sections but the increase is
not sufficient enough to explain the results of the KDAR neutrinos.

  * (ii)

Such an increase in the inclusive cross section in the theoretical predictions
due to the nuclear medium effects, in the similar energy region of the
$\pi$-DIF neutrinos would further enhance the disagreement between the
theoretical and the experimental results in the case of LSND experiment.

  * (iii)

Moreover, this would also be in contradiction with earlier results of such
effects calculated in the work of Hayes and Towner Hayes:1999ew  in
microscopic models and Umino et al. Umino:1994wu ; Umino:1996cz  in the case
of Fermi gas models.

It is clear that present status of the theoretical calculations for the
inclusive cross section in the process $\nu_{\mu}+^{12}C\rightarrow\mu^{-}+X$
in the low energy region of few hundreds of MeV is not satisfactory even with
the inclusion of nuclear medium effects beyond the impulse approximation
calculated in various models available in the literature.

##  IV Summary and Conclusions

The experimental and the theoretical status of the low energy inclusive
neutrino scattering cross sections and inclusive muon capture from 12C nucleus
has been reviewed in the context of new measurements reported by the MiniBooNE
collaboration with monoenergetic muon neutrinos from KDAR with energy
$E_{\nu_{\mu}}=236$ MeV. To summarize, we find that

  * (1)

The various theoretical predictions for the inclusive cross sections in the
case of KDAR neutrinos underestimate the experimentally observed cross
sections showing a discrepancy between the theory and experiment. This reminds
us of the discrepancy observed between the experimental and theoretical
results in the case of the low energy measurements of the inclusive cross
section reported about 20 years ago by the LSND experiments using muon
neutrinos from $\pi$DIF, in the energy range of $120<E_{\nu_{\mu}}<280$ MeV,
with the difference that in the case of LSND experiment the results were
overestimated by the theoretical predictions.

  * (2)

There is no theoretical model which succeeds in reproducing the inclusive
scattering cross sections and the inclusive capture rates, respectively, in
all the three weak nuclear processes at low energies induced by the muon
neutrinos, electron neutrinos and muons in 12C. This is indeed surprising in
the case of 12C nucleus, which is considered to be one of the best known
nucleus regarding its structure. This calls for a better understanding of the
nuclear medium effects in the low energy weak interaction processes induced by
neutrinos and muons in 12C as well as in other nuclei.

  * (3)

The recent theoretical calculations of the inclusive neutrino cross sections
in 12C in the region of intermediate energy of neutrinos around 1 GeV show
that the role of meson exchange currents, $2p-2h$ and $\Delta h$ excitations
etc., is quite important in explaining the inclusive neutrino cross sections
in 12C observed in the MiniBooNE experiment to explain the axial dipole mass
anomaly. These calculations when extended to the low energy region of
neutrinos from pion decays in flight corresponding to the LSND experiments and
the neutrinos from the kaon decays at rest, do not help to remove the
discrepancy between the theoretical and experimental results.

  * (4)

In view of the LSND experiments being completed almost 20 years ago, it is
desirable that future experiments planned at the Fermilab and JPARC
laboratories with the low energy monoenergetic KDAR neutrinos are performed
with enhanced precision and the theoretical calculations are done using the
advanced state of the art methods to compute the nuclear medium effects
including the many body effects for the low energy inclusive reactions in 12C
and 40Ar.

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