# Coherent pion production in neutrino (anti-neutrino)-nucleus interaction

Hariom Sogarwal Prashant Shukla<EMAIL_ADDRESS>Nuclear Physics
Division, Bhabha Atomic Research Centre, Mumbai 400085, India  Homi Bhabha
National Institute, Anushakti Nagar, Mumbai 400094, India

###### Abstract

We present a study of coherent pion production in neutrino-nucleus
interactions using the formalism based on partially conserved axial current
theorem which connects the neutrino-nucleus cross section to the pion-nucleus
elastic scattering cross section. Pion-nucleus elastic scattering cross
section is calculated using Glauber model which takes three inputs, nuclear
densities, pion-nucleon cross section and $\alpha_{\pi N}$ which is the ratio
of real to imaginary part of $\pi N$ forward scattering amplitude, for which
the parametrizations are obtained from measured data. We calculate the
differential and integrated cross sections for charge and neutral current
coherent pion production in neutrino (anti-neutrino)-nucleus scattering for a
range of nuclear targets from light to heavy materials such as lithium,
carbon, hydrocarbon, oxygen, silicon, argon, iron and lead. The results of
these cross section calculations are compared with the measured data and with
the calculations from the Berger-Sehgal model and GENIE package. There is an
excellent agreement between the calculated and measured cross sections with
Glauber model. While GENIE and Berger-Sehgal model give a good description of
the data in the lower energy range the present calculations describe the data
in all energy ranges. Predictions are also made for upcoming experiments like
INO and DUNE in the coherent region of neutrino cross section.

###### keywords:

Coherent pion production, Neutrino-nucleus interactions, Glauber model, PCAC,
GENIE

##  1 Introduction

The neutrinos produced in upper atmosphere and in accelerators are used for
studying the phenomena of neutrino oscillations by many experiments worldwide
[1, 2, 3]. Most experiments focussing on muon neutrinos are designed to
measure recoil muons when neutrinos undergo charge current interaction in the
detector medium. Some examples of detector media are; iron in case of India-
based Neutrino Observatory (INO) [4] and argon in case of DUNE experiment [5].
Bulk of the interactions in the detector comes from intermediate energy
neutrinos having contribution from many processes which include quasi elastic
scattering, interaction via resonance pion production and deep inelastic
scattering [6, 7, 8]. One of the important processes in the resonance
production region is coherent pion production in which the nucleus interacts
as a whole with the neutrino with its quantum state remains unchanged. The
charge current (CC) and neutral current (NC) coherent pion production
processes are given as

$\displaystyle\nu_{\mu}+A$
$\displaystyle\rightarrow\mu^{-}+\pi^{+}+A.\,\,\,{\rm(CC)}$
$\displaystyle\nu_{\mu}+A$
$\displaystyle\rightarrow\nu_{\mu}+\pi^{0}+A~{}.\,\,\,{\rm(NC)}$

The four-momentum transfer $Q$ between the incoming neutrino and outgoing
lepton is given by $Q^{2}=-q^{2}={\bf q}^{2}-\nu^{2}$. Here, ${\bf q}$ is the
3-momentum transfer and $\nu~{}(=E_{\nu}-E_{\mu})$ is the energy difference
between the incident neutrino (with energy $E_{\nu}$) and outgoing lepton
(with energy $E_{\mu}$). For coherent pion production, the squared momentum
transfer to the nucleus from the lepton-pion system $|t|=|(q-p_{\pi})^{2}|$
remains small. Here $p_{\pi}$ is the 4-momentum of outgoing pion. Estimating
coherent pion production is important for the analysis of data of neutrino
oscillation experiments. The simplest theoretical approach for describing
coherent pion production is based on Adler’s Partially Conserved Axial Current
(PCAC) theorem which relates the neutrino induced coherent pion production to
the pion-nucleus elastic scattering [9, 10, 11, 12, 13, 14]. The PCAC model
has been successful in describing coherent pion production at high energy [9].
Work with the same assumption has been used at low energy in Ref. [12]. There
are several microscopic models as well for coherent scattering e.g. in Refs.
[15, 16, 17, 18]. The GENIE (Generates Events for Neutrino Interaction
Experiments) [19] uses PCAC theorem with Rein-Sehgal model [9] for the
coherent neutrino-nucleus scattering which results in the production of
forward going pions in both charge current and neutral current interactions.

To obtain the elastic pion-nucleus scattering cross section, the Berger-Sehgal
(BS) model [13] is used in literature which basically follows Rein-Sehgal (RS)
method [9] with some improvement in the way parametrizations are done for
elementary cross sections. In this model, the elastic scattering pion nucleus
cross section is obtained from measured total and elastic pion nucleon cross
sections. Further an exponential function is assumed for $t$ dependence of
elastic scattering cross section and attenuation of pion is made dependent on
nuclear size. GENIE uses RS model and its difference with BS calculations can
arise due to input elementary cross sections and other input parameters. The
work presented in Ref. [20] calculates the pion-nucleus elastic scattering
cross section using the Glauber model in terms of measured nuclear densities
and measured pion nucleon cross sections. In view of the recent data at
various energies we aim to revisit this model and improve the treatment of
input parameters namely total pion nucleon cross sections, $\alpha_{\pi N}$
(the ratio of real to imaginary part of forward scattering amplitude) and
nuclear density function parameters. The parametrizations of the total pion
nucleon cross sections have been done and presented using the latest data. The
values of $\alpha_{\pi N}$ obtained with the elementary scattering amplitudes
may change when one applies the formalism to nuclear targets and are typically
around one [38]. In view of this, we use the pion nucleus elastic scattering
differential cross section data available for few incident energies of pions
on a broad range of nuclear targets to obtain $\alpha_{\pi N}$. We also
compare with the Berger-Sehgal model which gives reasonable description of
elastic scattering differential cross section at lower scattering angles only.
Recently, charge and neutral current coherent pion production has been
measured for several nuclei by many experiments such as MINER$\nu$A [21, 22],
CHARM-II [23], Aachen-Padova [24], Gargamelle [25], CHARM [26], SKAT [27],
NOMAD [28], 15’B.C. [29], SciBooNE [30], MINOS [31], NO$\nu$A [32], K2K [33],
T2K[34, 35] and MiniBooNE [36]. The comparison of Glauber approach has been
done with the calculations from Berger-Sehgal (BS) model and the GENIE package
(version v3_00_06a).

In this work, we calculate coherent pion production in neutrino-nucleus
interactions in the resonance region using the PCAC based formalism. Pion-
nucleus elastic scattering cross section is calculated with the Glauber model
using pion-nucleon cross section and $\alpha_{\pi N}$. The parametrizations of
latest measured total pion nucleon cross section has been presented. We
calculate the differential and integrated cross sections for charge and
neutral current coherent pion production in neutrino (anti-neutrino)-nucleus
scattering for a range of nuclear targets: lithium, carbon, hydrocarbon,
scintillator, oxygen, silicon, argon, iron, and lead using the model with
fixed parameters. The results of these cross section calculations are compared
with the measured data, BS model and GENIE package.

##  2 The formulation of the model

The differential cross section for the charge current coherent pion production
scattering process [10, 13] is

$\displaystyle\frac{d\sigma^{CC}}{dQ^{2}d\nu d|t|}$ $\displaystyle=$
$\displaystyle\frac{G^{2}_{F}\cos^{2}\theta_{C}f^{2}_{\pi}}{2\pi^{2}}\frac{uv}{|{\bf
q}|}~{}\Bigg{[}\Big{(}G_{A}-\frac{1}{2}\frac{Q^{2}_{m}}{(Q^{2}+m^{2}_{\pi})}\Big{)}^{2}$
(1)
$\displaystyle+\frac{\nu}{4E_{\nu}}(Q^{2}-Q^{2}_{m})\frac{Q^{2}_{m}}{(Q^{2}+m^{2}_{\pi})^{2}}\Bigg{]}\times\frac{d\sigma(\pi
A\rightarrow\pi A)}{d|t|}~{}.$

For the neutral current, the above expression is modified as

$\displaystyle\frac{d\sigma^{NC}}{dQ^{2}d\nu d|t|}$ $\displaystyle=$
$\displaystyle\frac{G^{2}_{F}f^{2}_{\pi}}{4\pi^{2}}\frac{G^{2}_{A}}{|{\bf
q}|}~{}u~{}v~{}\times\frac{d\sigma(\pi A\rightarrow\pi A)}{d|t|}~{}.$ (2)

Here $G_{F}$ (=1.16639 $\times 10^{-5}$ GeV-2) is the Fermi coupling constant
and $\cos\theta_{C}~{}(=0.9725)$. The kinematic factors $u$ and $v$ are given
by : $u,v=\Big{(}E_{\nu}+E_{\mu}~{}\pm~{}|{\bf q}|\Big{)}/(2~{}E_{\nu})$. The
pion decay constant is $f_{\pi}~{}(=0.93~{}m_{\pi})$ and ${d\sigma(\pi
A\rightarrow\pi A)}/{d|t|}$ is the pion-nucleus differential elastic cross
section calculated assuming $\nu$ as the total incident energy of pion in
laboratory frame. The axial vector form factor can be defined as
$G_{A}=m^{2}_{A}/(Q^{2}+m^{2}_{A})$ [13] with the axial vector meson mass
$m_{A}$ (= 1.05 GeV/c2) [7]. The high energy approximation to the true minimal
$Q^{2}$ is given by $Q^{2}_{m}=m^{2}_{l}~{}\nu/(E_{\nu}-\nu)$, where $m_{l}$
is the mass of outgoing lepton which will be muon mass in case of CC. The
expression for neutral current (NC) (Eq. 2) is obtained from Eq. 1 by putting
$m_{l}$=0, $\theta_{C}$=0 and divide the right hand side of the Eq. 1 by 2,
because $f_{\pi^{0}}=f_{\pi}/\sqrt{2}$.

The kinematic limits are guided by the work in Ref. [12]. The upper limit of
$Q^{2}$ is taken as 1.0 GeV2. The $\nu$ integration should be done in the
range $\rm max(\xi\sqrt{Q^{2}},\nu_{min})<\nu<\nu_{max}$. In the previous work
[20] the calculations were performed at two values of $\xi$ (=1,2). In the
present work, we include this variation as uncertainty in both Glauber and BS
calculations.

Pion-nucleus differential cross section

To obtain the elastic pion-nucleus scattering cross section, the Berger-Sehgal
(BS) model [13] is used in literature which basically follows Rein-Sehgal (RS)
method [9]. In their model, the pion-nucleus elastic scattering cross section
is obtained from measured total pion nucleon cross sections. Further an
exponential function is assumed for $t$ dependence and the attenuation of
pions is made dependent on nuclear radius. We use scattering theory to obtain
the pion-nucleus differential elastic cross section given as

$\frac{d\sigma_{el}}{d|t|}=\frac{\pi}{k^{2}}|f(t)|^{2},$ (3)

where $f(t)$ is given by

$\displaystyle
f(t)=\frac{1}{2ik}\sum\limits_{l=0}^{\infty}\Big{(}2l+1\Big{)}\Big{(}S_{l}-1\Big{)}P_{l}\Big{(}\cos\theta\Big{)}.$
(4)

Here $t=-4k^{2}\,\sin^{2}\theta/2$ and $k$ is the momentum of pion and
$\theta$ is the scattering angle of pion in center of mass frame.

We use the Glauber model to obtain the scattering matrix $S_{l}$ in terms of
pion-nucleus impact parameter $b$ by [37]

$S_{l}=\exp(i\chi(b)),\quad bk=\Big{(}l+\frac{1}{2}\Big{)},~{}$ (5)

The Glauber phase shift $\chi(b)$ is given by

$\chi(b)=\frac{1}{2}\sigma_{\pi N}\Big{(}\alpha_{\pi N}+i\Big{)}AT(b)$ (6)

Here $\sigma_{\pi N}$ is the average total pion-nucleon cross section and
$\alpha_{\pi N}$ is the ratio of real to imaginary part of the $\pi N$ forward
scattering amplitude. $T(b)$ is the overlap function obtained by the Glauber
model, the full details of which are given in Ref. [20]. The pion-nucleon
cross sections are taken from the Particle Data Group [3].

Figure 1 (a) and (b) show the measured total cross sections for $\pi^{+}$p and
$\pi^{-}$p collisions respectively as a function of pion momentum in
laboratory frame. The data is fitted with Breit-Wigners and Regge function for
pion momentum up to 4 GeV/$c$ and a pure Regge function above 4 GeV/$c$. The
Regge term above 4 GeV/$c$ is given by
$25.06-15.16/\sqrt{p_{\pi}}+41.07\sqrt{p_{\pi}}$ for $\pi^{+}$p case and
$24.73-6.91/\sqrt{p_{\pi}}+37.37\sqrt{p_{\pi}}$ for $\pi^{-}$p case.

Figure 1: Measured total cross sections for (a) $\pi^{+}$p and (b) $\pi^{-}$p
collisions as a function of pion momentum along with the fit function. Figure
2: Pion-12C elastic scattering differential cross section as function of pion
scattering angle at three pion kinetic energies of 400 MeV, 500 MeV and 675.5
MeV scaled respectively by factors 1000, 10 and 0.1 [39, 40]. The blue bands
are obtained using the Glauber model for $\alpha_{\pi N}$ = 0.9-1.4 and dashed
lines are the calculations of the Berger-Sehgal (BS) model [13]. Figure 3:
Pion-40Ca elastic scattering differential cross section as function of pion
scattering angle at three pion kinetic energies of 400 MeV, 500 MeV and 675.5
MeV scaled respectively by factors 1000, 10 and 0.1 [39, 40]. The blue bands
are obtained using the Glauber model for $\alpha_{\pi N}$ = 0.9-1.4 and dashed
lines are the calculations of the Berger-Sehgal model [13]. Figure 4:
Pion-208Pb elastic scattering differential cross section as function of pion
scattering angle at three pion kinetic energies of 400 MeV, 500 MeV and 662.7
MeV scaled respectively by factors 1000, 10 and 0.1 [39, 40]. The blue bands
are obtained using the Glauber model for $\alpha_{\pi N}$ = 0.9-1.4 and dashed
lines are the calculations of the Berger-Sehgal model [13].

The values of $\alpha_{\pi N}$ obtained with the elementary scattering
amplitudes may change when one applies the formalism to nuclear targets and
are typically around one [38]. In view of this, we use the pion nucleus
elastic scattering data available for few incident energies of pions on a
broad range of nuclear targets to obtain $\alpha_{\pi N}$.

Figure 2 shows pion-12C elastic scattering differential cross section as
function of pion scattering angle at three pion kinetic energies of 400 MeV,
500 MeV and 675.5 MeV scaled respectively by factors 1000, 10 and 0.1 [39,
40]. The blue bands are obtained using the Glauber model for $\alpha_{\pi N}$
= 0.9-1.4 and dashed lines are the calculations of the Berger-Sehgal model
[13].

Figure 3 shows pion-40Ca elastic scattering differential cross section as
function of pion scattering angle at three pion kinetic energies of 400 MeV,
500 MeV and 675.5 MeV scaled respectively by factors 1000, 10 and 0.1 [39,
40]. The blue bands are obtained using the Glauber model for $\alpha_{\pi N}$
= 0.9-1.4 and dashed lines are the calculations of the Berger-Sehgal model
[13].

Figure 4 shows pion-208Pb elastic scattering differential cross section as
function of pion scattering angle at three pion kinetic energies of 400 MeV,
500 MeV and 662.7 MeV scaled respectively by factors 1000, 10 and 0.1 [39,
40]. The blue bands are obtained using the Glauber model for $\alpha_{\pi N}$
= 0.9-1.4 and dashed lines are the calculations of the Berger-Sehgal model
[13].

It is shown that the values of $\alpha_{\pi N}$ in the range 0.9 to 1.4 give a
very good description of all the available data. We also compare the
calculations with the Berger-Sehgal model which gives reasonable description
at lower scattering angles only. These values of $\alpha_{\pi N}$ are then
used to calculate the band of pion production cross section in neutrino
nucleus interactions.

The nuclear density function for lighter nuclei with nuclear mass upto 16O are
taken as the harmonic oscillator type as given by

$\displaystyle\rho(r)$ $\displaystyle=$
$\displaystyle\rho_{0}\Big{(}1+\alpha\,\frac{r^{2}}{a^{2}}\Big{)}\exp\Big{(}-\frac{r^{2}}{a^{2}}\Big{)},\quad\rho_{0}=\frac{1}{(1+1.5\alpha)(\sqrt{\pi}\,a)^{3}}~{}.$
(7)

where $a$ and $\alpha$ are given in Table 1. For light composite material
13.8CH, we use harmonic oscillator type density with the value of $a$ = 1.729
fm (Same as that for 14N) and $\alpha$= 1.024 calculated using formula given
in Ref. [41].

For nuclei heavier than 16O, the nuclear density function is taken as two-
parameter Fermi (2pF) type function as given by

$\displaystyle\rho(r)$ $\displaystyle=$ $\displaystyle\frac{\rho_{0}}{1+{\rm
exp}(\frac{r-c}{d})},\quad\rho_{0}=\frac{3}{4\pi
c^{3}(1+\frac{\pi^{2}d^{2}}{c^{2}})}~{}.$ (8)

Here $d$ is the diffuseness and $c$ is the half value radius in terms of rms
radius $R_{\rm rms}$ as $c=\sqrt{(5R_{\rm rms}^{2}-7\pi^{2}d^{2})/3}$ using
the given average mass number $A$. For the composite targets with $A_{\rm
eff}>16$, the $R_{\rm rms}$ is calculated by $R_{\rm
rms}=0.891A^{1/3}(1+1.565A^{-2/3}-1.043A^{-4/3})$ [42] with $d=0.537$ fm.
Table 1 shows the nuclear density function parameters for Harmonic oscillator
[41] and 2pF [43] for various nuclei used in the present work.

Table 1: Nuclear density function parameters for Harmonic oscillator [41] and
2pF [43] density functions for various nuclei. Nuclear density parameters
---
Harmonic oscillator | 2pF
Nucleus |  $a$ (fm) | $\alpha$ | Nucleus |  $c$ (fm) |  $d$ (fm)
7Li | 1.77 | 0.329 |  28Si | 3.14 | 0.537
12C | 1.687 | 1.029 |  40Ar | 3.53 | 0.542
16O | 1.805 | 1.446 |  56Fe | 4.106 | 0.519
|  |  |  207Pb | 6.62 | 0.546

##  3 Results and discussions

In this section, we present the results of cross sections calculated using Eq.
1 for both neutrino and anti-neutrino interactions with nucleus. The
differential cross section corresponding to an experiment is obtained by
averaging cross section over all energies weighted by the neutrino (anti-
neutrino) energy spectrum for the experiment.

$<\frac{d\sigma}{dQ^{2}}>=\frac{\int^{E_{\rm{max}}}_{E_{\rm{min}}}\frac{d\sigma}{dQ^{2}}(E)~{}\phi(E)~{}dE}{\int^{E_{\rm{max}}}_{E_{\rm{min}}}\phi(E)~{}dE}.$
(9)

Figure 5: Differential cross section $d\sigma/dQ^{2}$ (averaged over neutrino
and anti-neutrino flux) for the charge current coherent pion production in
$\nu_{\mu}-^{12}$C interaction as a function of the square of four-momentum
transfer $Q^{2}$ obtained using the Glauber model based present approach and
BS approach for (a) neutrinos with $<E_{\nu}>$ = 4.5 GeV and (b) anti-
neutrinos with $<E_{\bar{\nu}}>$ = 4 GeV in comparison with MINER$\nu$A data
[21].

Figure 5 (a) shows the differential cross section $d\sigma/dQ^{2}$ (averaged
over neutrino flux) for the charge current coherent pion production in
neutrino-carbon interaction as a function of the square of four-momentum
transfer $Q^{2}$ and Figure 5 (b) shows the same for anti-neutrino-carbon
interaction using the Glauber model based present approach and BS approach.
The band in the Glauber model corresponds to the maximum difference due to
variation of both $\alpha_{\pi N}$ (in range 0.9-1.4) and $\xi$ (in range 1-2)
while for BS model the band includes variation of $\xi$ only. The calculations
correspond to the average energy 4.5 GeV for neutrinos and 4 GeV for anti-
neutrinos for the case of MINER$\nu$A experiment [21]. The Glauber approach
gives a better agreement (as compared to BS) with the data especially at high
$Q^{2}$.

Figure 6: Total cross section $\sigma$ for the charge current coherent pion
production in $\nu_{\mu}-^{12}$C interaction as a function of neutrino energy
$E_{\nu}$ obtained using the Glauber model based present approach and BS
approach for (a) neutrinos and (b) anti-neutrinos. The calculations are
compared with the MINER$\nu$A data [21] and GENIE.

Figure 6 (a) shows the total cross section for the charge current coherent
pion production in neutrino-carbon interaction as a function of neutrino
energy and Figure 6 (b) shows the same for anti-neutrino obtained using the
Glauber model based present approach and BS approach. The band in the Glauber
model corresponds to the maximum difference due to variation of both
$\alpha_{\pi N}$ (in range 0.9-1.4) and $\xi$ (in range 1-2) while for BS
model, the band includes variation of $\xi$ only. The calculations are
compared with the measured data of MINER$\nu$A experiment [21] and GENIE
calculations. The present calculations and GENIE give a good description of
the data at low energies while at high energies, the present calculation is in
better agreement with data as compared to BS approach and GENIE.

The GENIE package uses Rein sehgal (RS) model. The BS model is actually RS
model with improvement in the parametrization of the pion nucleon data. The
difference between BS approach and GENIE can arise due to the differences
between the treatment of pion nucleon data. We have shown the comparison
between Berger Sehgal (BS) model and Glauber model in Figs. 2, 3 and 4. The
pion nucleus cross section goes as input in PCAC calculations and creates the
differences amoung the neutrino nucleus cross section. The same parameters and
input pion nucleon cross sections used for Glauber model are used in BS
calculations here and thus the difference between the two is only due to
method of the modeling. It looks that the BS model underestimates the pion
nucleus cross section at higher energies. The Glauber model has assumptions
which work better at higher collision energies.

Figure 7: Total cross section $\sigma$ for the charge current coherent pion
production in $\nu_{\mu}$ or $\bar{\nu_{\mu}}-^{12}$C interaction as a
function of neutrino energy $E_{\nu}$ obtained using our calculations
($\alpha_{\pi N}$ =1.1 and $\xi$=1) for four different values of axial mass
parameter $m_{A}$ in comparison with MINER$\nu$A experimental data [21].

Figure 7 shows the total cross section for the charge current coherent pion
production neutrino (anti-neutrino)-carbon interaction as a function of
neutrino energy obtained using the Glauber model based present approach with
$\alpha_{\pi N}$ =1.1 and $\xi$=1. The calculations are done for four
different values of axial mass parameter $m_{A}$ and are compared with
MINER$\nu$A experimental data [21]. Different values of $m_{A}$ make
difference only at high energies where there are large uncertainties of the
measurements. We have chosen $m_{A}=1.05$ GeV/$c^{2}$ in all our calculations.

Figure 8: Charge current coherent pion production in $\nu_{\mu}-^{13.8}$CH
interaction obtained using the Glauber model based present approach and BS
approach (a) Total cross section $\sigma$ as a function of neutrino energy
$E_{\nu}$ compared with the preliminary MINER$\nu$A data [22] and GENIE and
(b) Differential cross section (averaged over neutrino flux [22]) as a
function of the square of four-momentum transfer $Q^{2}$ compared with the
preliminary MINER$\nu$A data [22].

Figure 8 (a) shows the total cross section for the charge current coherent
pion production in neutrino-hydrocarbon interaction as a function of neutrino
energy obtained using the Glauber model based present approach and BS approach
compared with the preliminary data of MINER$\nu$A experiment [22] and GENIE
calculations. Figure 8 (b) shows differential cross section (averaged over
neutrino flux [22]) as a function of the square of four-momentum transfer
$Q^{2}$ obtained using the Glauber model based present approach and BS
approach compared with the preliminary data of MINER$\nu$A experiment [22].
The present calculations and GENIE give good description of the total cross
section data at low energies while at high energies our calculation is in
better agreement with data as compared to GENIE and BS approach. The present
calculations of the differential cross section match much better with the data
over whole $Q^{2}$ range as compared to BS approach. The cross section by
GENIE shown in the figure is obtained by scaling their result on carbon with
$(13.8/12)^{\frac{2}{3}}$.

Figure 9: Total cross section $\sigma$ for the charge current coherent pion
production in neutrino-20.1 Glass+Scintillator interaction as a function of
neutrino energy $E_{\nu}$ obtained using the Glauber model based present
approach and BS approach for (a) neutrino and (b) anti-neutrino. The
calculations are compared with the CHARM-II data [23] and GENIE.

Figure 9 (a) shows the total cross section for the charge current coherent
pion production in neutrino-glass+scintillator interaction as a function of
neutrino energy obtained using the Glauber model based present approach and BS
approach. Figure 9 (b) shows the same for anti-neutrino. The calculations are
compared with the measured data of CHARM-II experiment [23] and GENIE. The
present calculations are in better agreement with data within experimental
error as compared to GENIE and BS approach.

Figure 10: Charge current coherent pion production in $\nu_{\mu}-^{207}$Pb
interaction obtained using the Glauber model based present approach and BS
approach compared with the MINER$\nu$A data [22] and GENIE. (a) Total cross
section $\sigma$ as a function of neutrino energy and (b) Differential cross
section (averaged over neutrino flux) as a function of the square of four-
momentum transfer $Q^{2}$ [22] compared with the MINER$\nu$A data [22].

Figure 10 (a) shows the total cross section for the charge current coherent
pion production in neutrino-lead interaction as a function of neutrino energy
obtained using the Glauber model based present approach and BS approach
compared with the measured data of MINER$\nu$A experiment [22] and GENIE
calculations. Figure 10 (b) shows differential cross section (averaged over
neutrino flux [22]) as a function of the square of four-momentum transfer
$Q^{2}$ compared with the measured data of MINER$\nu$A experiment [22]. The
present calculations and GENIE give a good description of the total cross
section at low energies while at high energies our calculation is in better
agreement with data as compared to GENIE and BS model. The present
calculations of the differential cross section match much better with the data
over whole $Q^{2}$ range as compared to BS approach.

Figure 11: Charge current coherent pion production in $\nu_{\mu}-^{56}$Fe
interaction obtained using the Glauber model based present approach and BS
approach (a) Total cross section $\sigma$ as a function of neutrino energy are
compared with MINER$\nu$A data [22] and GENIE. (b) Differential cross section
(averaged over neutrino flux) as a function of the square of four-momentum
transfer $Q^{2}$ are compared with the MINER$\nu$A data [22].

Figure 11 (a) shows the total cross section for the charge current coherent
pion production in neutrino-iron interaction as a function of neutrino energy
obtained using the Glauber model based present approach and BS approach
compared with the data of MINER$\nu$A experiment [22] and GENIE. This result
is relevant for INO as well. Figure 11 (b) shows differential cross section
(averaged over neutrino flux) as a function of the square of four-momentum
transfer $Q^{2}$ [22] compared with the data of MINER$\nu$A experiment [22].
The GENIE and BS give a good description of the data at lower energies while
our calculations give reasonable description in the whole energy range.

Figure 12: Neutral current coherent pion production in $\nu_{\mu}-^{56}$Fe
interaction obtained using the Glauber model based present approach and BS
approach (a) Total cross section $\sigma$ as a function of neutrino energy
compared with GENIE and (b) Differential cross section (averaged over neutrino
flux at INO-site [4] for solar minimum [44]) as a function of the square of
four-momentum transfer $Q^{2}$.  Figure 13: Charge current coherent pion
production in $\nu_{\mu}-^{56}$Fe interaction obtained using the Glauber model
based present approach and BS approach for differential cross section
(averaged over neutrino flux at INO-site for solar minimum [44]) as a function
of the square of four-momentum transfer $Q^{2}$.

Figure 12 (a) shows the total cross section prediction for the neutral current
coherent pion production in neutrino-iron interaction relevant for INO as a
function of neutrino energy obtained using the Glauber model based present
approach and BS approach. The calculations are compared with GENIE. Figure 12
(b) shows differential cross section (averaged over neutrino flux at INO-site
[4] for solar minimum from 1 GeV to 20 GeV energy range [44]) as a function of
the square of four-momentum transfer $Q^{2}$ calculated using Glauber based
approach and BS approach.

Figure 13 shows differential cross section for charge current coherent pion
production in neutrino-iron interaction as a function of the square of four-
momentum transfer $Q^{2}$ (averaged over neutrino flux at INO-site for solar
minimum from 1 GeV to 20 GeV energy range [44]) calculated using Glauber based
approach and BS approach.

Figure 14: Charge current coherent pion production in $\nu_{\mu}-^{40}$Ar
interaction obtained using the Glauber model based present approach and BS
approach (a) Total cross section predictions as a function of neutrino energy
compared with GENIE (b) Differential cross section (averaged over neutrino
flux corresponding to the DUNE experiment [5]) as a function of the square of
four-momentum transfer $Q^{2}$.

Figure 14 (a) shows the total cross section for the charge current coherent
pion production in neutrino-argon interaction as a function of neutrino energy
obtained using the Glauber model based present approach and BS approach as
prediction for the DUNE experiment. The calculations are compared with GENIE.
Figure 14 (b) shows differential cross section(averaged over neutrino flux
corresponding to the DUNE experiment [5]) as a function of the square of four-
momentum transfer $Q^{2}$ calculated using Glauber based approach and BS
approach.

Figure 15: Neutral current coherent pion production in $\nu_{\mu}-^{40}$Ar
interaction obtained using the Glauber model based present approach and BS
approach (a) Total cross section $\sigma$ as a function of neutrino energy are
compared with GENIE (b) Differential cross section (averaged over neutrino
flux corresponding to the DUNE experiment [5]) as a function of the square of
four-momentum transfer $Q^{2}$.

Figure 15 (a) shows the total cross section for the neutral current coherent
pion production in neutrino-argon interaction as a function of neutrino energy
obtained using the Glauber model based present approach and BS approach as
predictions for the DUNE experiment. The calculations are compared with GENIE.
Figure 15 (b) shows differential cross section (averaged over neutrino flux
corresponding to the DUNE experiment [5]) as a function of the square of four-
momentum transfer $Q^{2}$ calculated using the Glauber model based present
approach and BS approach.

Figure 16: Total cross section $\sigma$ for the neutral current coherent pion
production in $\nu_{\mu}$-nucleus interaction as a function of nuclear mass
$A$ at 4.9 GeV neutrino energy obtained using the Glauber model based present
approach. The measurements of experiments are scaled to $E_{\nu}$ = 4.9 GeV
[31] using the Berger-Sehgal [13] approach. The results obtained using our
calculations are compared with BS approach, GENIE and experiments: Aachen-
Padova [24], Gargamelle [25], CHARM [26], SKAT [27], NOMAD [28], 15’B.C. [29]
and SciBooNE [30]

Figure 16 shows the total cross section for the neutral current coherent pion
production in neutrino-nucleus interaction as a function of target nucleus
mass (7Li, 12C, 16O, 28Si, 40Ar and 56Fe) obtained using the Glauber model
based present approach. The calculations are compared with BS approach, GENIE
and experiments: Aachen-Padova [24], Gargamelle [25], CHARM [26], SKAT [27],
NOMAD [28], 15’B.C. [29] and SciBooNE [30]. The calculations give a very good
description for several experimental data for a wide range of targets and are
much better than GENIE. The measurements of experiments are scaled to
$E_{\nu}$ = 4.9 GeV [31] using the Berger-Sehgal [13] approach.

Figure 17: Total cross section $\sigma$ for the neutral current coherent pion
production in $\nu_{\mu}-^{12}$C interaction as a function of neutrino energy.
The cross sections obtained using the Glauber model based present approach and
BS approach are compared with GENIE and experiments: Aachen-Padova [24],
Gargamelle [25], CHARM [26], SKAT [27], NOMAD [28], MINOS [31] and NO$\nu$A
[32]. The cross section measurements of experiments with different nuclei of
mass number $A$ are scaled to carbon [31] using $(12/A)^{\frac{2}{3}}$.

Figure 17 shows the total cross section for the neutral current coherent pion
production in neutrino-carbon interaction as a function of neutrino energy
obtained using the Glauber model based present approach and BS approach. The
calculations are compared with GENIE and experiments: Aachen-Padova [24],
Gargamelle [25], CHARM [26], SKAT [27], NOMAD [28], MINOS [31] and NO$\nu$A
[32]. The Glauber based calculations give a very good description for
experimental data in a wide range of energy. GENIE and BS approach give a good
description of data only at low neutrino energies. The cross section
measurements of experiments with different nuclei of mass number $A$ are
scaled to carbon [31] using $(12/A)^{\frac{2}{3}}$.

Table 2: Comparison of the calculated total cross sections $\sigma$ for coherent pion production using Glauber and BS approach with the measurements by different experiments along with the average energy of neutrinos. Exps. | $<A>$ | Current, | $\left\langle E\right\rangle$ |  $\sigma$ ($10^{-39}cm^{2}$)
---|---|---|---|---
|  | particle | GeV | Glauber | BS | Exp.
SKAT [27] | 30 | CC, $\nu$ | 7 | 8.8-13.3 | 7.7-9.2 | 10.6$\pm$1.6
SKAT [27] | 30 | CC, $\bar{\nu}$ | 7 | 8.8-13.3 | 7.7-9.2 | 11.3$\pm$3.5
SKAT [27] | 30 | NC, $\nu$ | 7 | 5.0-7.5 | 4.3-5.1 | 5.2$\pm$1.9
MINOS [31] | 48 | NC, $\nu$ | 4.9 | 5.1-7.3 | 4.9-5.7 | 3.26$\pm$0.21
NO$\nu$A [32] | 13.8 | NC, $\nu$ | 2.7 | 1.0-1.8 | 1.0-1.5 | 1.4$\pm$0.2

Table 2 shows a comparison of the calculated total cross sections $\sigma$ for
coherent pion production using Glauber and BS approach with the measurements
by different experiments SKAT [27], MINOS [31] and NO$\nu$A [32] along with
the average energy of neutrinos. In the SKAT experiment, target material is
Heavy freon ($A$ = 30) and the calculation is done on the average energy of
neutrinos in the experiment. In the MINOS experiment, target material is the
composition of 80$\%$ iron and 20$\%$ carbon ($A$ = 48). The cross section is
averaged over the flux. In the NO$\nu$A experiment target material is mainly
the composition of 66.7$\%$ carbon, 16.1$\%$ chlorine and 10.8$\%$ hydrogen
and other nuclei ($A$ = 13.8). The table shows excellent agreement between the
data and the calculations, especially at higher energy.

##  4 Conclusions

In this work, we presented a comprehensive study of coherent pion production
in neutrino-nucleus interactions in the resonance region using the formalism
based on PCAC theorem. Pion-nucleus elastic scattering cross section is
calculated using the Glauber model which takes three inputs, nuclear
densities, pion-nucleon cross section and $\alpha_{\pi N}$ for which the
parametrizations are obtained from measured data. We obtain the differential
and integrated cross sections for charge and neutral current coherent pion
production in neutrino (anti-neutrino)-nucleus scattering for a range of
targets such as lithium, carbon, hydrocarbon, scintillator, oxygen, silicon,
argon, iron and lead. The results of these cross section calculations are
compared with the measured data, BS model and GENIE package. There is an
excellent agreement between the calculated cross section fom Glauber model and
measured cross sections. Predictions are also made for upcoming experiments
like INO and DUNE in the coherent pion production region of the neutrino cross
section.

Acknowledgements We thank Dr. Kapil Saraswat for many fruitful discussions
during the course of this work.

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