arXiv:0704.1991v1  [hep-ph]  16 Apr 2007
Preprint typeset in JHEP style - HYPER VERSION DO-TH 07/05
LPSC 07-29
SMU-HEP 07-07
Single pion electro– and neutrinoproduction on heavy
targets
E. A. Paschos
Theoretische Physik III, University of Dortmund, D-44221 Dortmund, Germany
E-mail: paschos@physik.uni-dortmund.de
I. Schienbein
Laboratoire de Physique Subatomique et de Cosmologie,
Université Joseph Fourier/CNRS-IN2P3,
53 Avenue des Martyrs, F-38026 Grenoble, France
E-mail: schien@lpsc.in2p3.fr
J.-Y. Yu
Southern Methodist University, Dallas, Texas 75275, USA
E-mail: yu@physics.smu.edu
Abstract: We present a calculation of single pion electroproduction cross sections on
heavy targets in the kinematic region of the ∆(1232) resonance. Final state interactions of
the pions are taken into account using the pion multiple scattering model of Adler, Nussinov
and Paschos (ANP model). For electroproduction and neutral current reactions we obtain
results for carbon, oxygen, argon and iron targets and find a significant reduction of the
W -spectra for π0 as compared to the free nucleon case. On the other hand, the charged
pion spectra are only little affected by final state interactions. Measurements of such cross
sections with the CLAS detector at JLAB could help to improve our understanding of pion
rescattering effects and serve as important/valuable input for calculations of single pion
neutrinoproduction on heavy targets relevant for current and future long baseline neutrino
experiments. Two ratios, in Eq. (3.8) and (3.10), will test important properties of the
model.
Keywords: single pion production, nuclear effects, long baseline experiments.
http://arxiv.org/abs/0704.1991v1
mailto:paschos@physik.uni-dortmund.de
mailto:schien@lpsc.in2p3.fr
mailto:yu@physics.smu.edu
http://jhep.sissa.it/stdsearch
Contents
1. Introduction 1
2. Free nucleon cross sections 3
3. Cross sections for heavy targets 5
3.1 Pion rescattering in the ANP model 5
3.2 Results for various targets 7
3.2.1 Neutrinoproduction 7
3.2.2 Electroproduction 7
4. Summary 9
A. Charge exchange matrices in the double averaging approximation 11
B. Charge exchange matrices for various amounts of pion absorption 11
C. Forward- and backward charge exchange matrices 12
1. Introduction
Neutrino interactions at low and medium energies are attracting attention because they
will be measured accurately in the new generation of experiments [1, 2]. One aim of the
experiments is to measure the precise form of the cross sections and their dependence
on the input parameters. This way we check their couplings and compare the functional
dependence of the form factors, where deviations from the dipole dependence have already
been established (see e.g. figure 1 in [3] and references therein). Deviation from the standard
model predictions can arise either from properties of the neutrinos or from new couplings
of the gauge bosons to the particles in the target. Another aim of the experiments is to
establish the properties of neutrinos including their masses, mixings and their fermionic
nature (Dirac or Majorana particles). This program requires a good understanding of the
cross sections, which motivated a new generation of calculations. Since the experiments use
nuclear targets, like C12, O16, Ar40, Fe56, ... it is necessary to understand the modifications
brought about by the targets.
The very old calculations for quasi-elastic scattering and resonance excitation on free
nucleons [4, 5] have been replaced by new results where couplings and form factors are
now better determined. For the vector couplings comparisons with electroproduction data
have been very useful [3, 6]. Axial couplings are frequently determined by PCAC. There
are already improvements and checks of the earlier quark models [7]. Comparisons with
– 1 –
experimental data are also available even though the experimental results are not always
consistent with each other [8, 9, 10] but there are plans for improvements that will resolve
the differences [1, 2].
For reactions on nuclear targets there are modifications brought about by the propaga-
tion of the produced particles in the nuclear medium. They involve absorption of particles,
restrictions from Pauli blocking, Fermi motion and charge-exchange rescatterings. One
group of papers uses nuclear potentials for the propagation of the particles [11]. Others use
a transport theory of the final particles including channels coupled to each other [12]. These
groups gained experience by analyzing reactions with electron beams (electroproduction)
and adopted their methods to neutrino reactions [12].
Our group investigated 1-π pion production on medium and heavy targets employing
the pion multiple scattering model by Adler, Nussinov and Paschos [13] that was developed
in order to understand neutral current neutrino interactions with nuclei. This model was
useful in the discovery of neutral currents and has been applied to predict neutrino-induced
single pion production on Oxygen, Argon and Iron targets [14, 15, 16] which are used in
long baseline(LBL) experiments. Among its characteristics is the importance of charge-
exchange reactions that modify the π+ : π0 : π− ratios of the original neutrino-nucleon
interaction through their scatterings within the nuclei. The presence of this effect has been
confirmed by experiments [17]. We note here that our results are valid for isoscalar targets.
For non-isoscalar targets like lead, used in the OPERA experiment, it is possible to extend
the ANP model [18], which can be done in the future.
In this article we take an inverse route and use our calculation in neutrino reactions to
go back to the electroproduction of pions on free nucleons and heavy nuclei. The plan of
the paper is as follows. In section 2 we summarize the neutrino production cross sections
on free nucleons and in the ∆ resonance region. This topic has been described by several
groups in the past few years. We present cross sections differential in several variables
Eπ, Q
2 and W . We pay special attention to the spectrum dσ/dEπ, where we correct an
error we found in our earlier calculation [14]. Then we obtain the electroproduction cross
section by setting the axial coupling equal to zero and rescaling, appropriately, the vector
current contribution.
The main content of the article appears in section 3 where we describe the salient
features and results of the ANP model. This model has the nice property that it can be
written in analytic form including charge exchange and absorption of pions. This way we
can trace the origin of the effects and formulate quantities which test specific terms and
parameters. As we mentioned above several features have been tested already, and we
wish to use electroproduction data in order to determine the accuracy of the predictions.
We present numerical results for different target materials, and study the quality of the
averaging approximation and uncertainties of the ANP model due to pion absorption ef-
fects. We discuss how the shape of the pion absorption cross section (per nucleon), an
important and almost unconstrained ingredient of the ANP model, can be delineated from
a measurement of the total fraction of absorbed pions. Finally, in Sec. 4 we summarize the
main results. Averaged rescattering matrices for carbon, oxygen, argon, and iron targets
and for different amounts of pion absorption have been collected in the appendices and are
– 2 –
useful for simple estimates of the rescattering effects.
2. Free nucleon cross sections
In the following sections, leptonic pion production on nuclear targets is regraded as a two
step process. In the first step, the pions are produced from constituent nucleons in the
target with free lepton-nucleon cross sections [13]. In the second step the produced pions
undergo a nuclear interaction described by a transport matrix. Of course, the resonances
themselves propagate in the nuclear medium before they decay, an effect that we will
investigate in the future.
The leptonic production of pions in the ∆-resonance region is theoretically available
and rather well understood as described in articles for both electro- and neutrino produc-
tion, where comparisons with available data are in good agreement [3, 6, 7, 19, 12].
The available data is described accurately with the proposed parameterizations. The
vector form factors are modified dipoles [3] which reproduce the helicity amplitudes mea-
sured in electroproduction experiments at Jefferson Laboratory [7]. The coupling in the
axial form factors are determined by PCAC and data. Their functional dependence in Q2
is determined by fitting the dσ
distributions. For the vector form factors the magnetic
dipole dominance for CV3 (q
2) and CV4 (q
2) gives an accurate description of the data. How-
ever, deviations with a non-zero CV5 (q
2) have also been established [7]. This way a small
(5%) isoscalar amplitude is reproduced.
For the propose of this article we shall use a scaling relation connecting neutrino- to
electroproduction. The weak vector current is in the same isospin multiplied with the
electromagnetic current and the two are related as follows:
< ∆++|V |p >=
3 < ∆+|JI=1em |p >=
3 < ∆0|JI=1em |n > .
Taking into account the isospin Clebsch-Gordan factors for the ∆ → Nπ branchings one
finds the following contributions of the ∆-resonance to the cross sections for ep → epπ0,
ep → enπ+, en → epπ− and en → enπ0
dσem,I=1
dQ2dW
dQ2dW


: pπ0
: nπ+
: pπ−
: nπ0
(2.1)
where dV
dQ2dW
denotes the cross section for the vector contribution alone to the reaction
νp → µ−pπ+. The free nucleon cross sections in Eq. (2.1) will be used in our numerical
analysis. We shall call this the reduced electromagnetic formula. Its accuracy was tested
in figure (5) of ref. [3]. Further comparisons can be found in [20].
For studies of the pion angular distributions (or what is the same of the pion energy
spectrum in the laboratory frame) we begin with the triple differential cross section for
– 3 –
neutrino production
dQ2dWd cos θ⋆π
16πM2
KiW̃i −
KiDi(3 cos
2 θ⋆π − 1)
(2.2)
with Ki being kinematic factors of W and Q
2 and the structure functions W̃i(Q
2,W ) and
2,W ) representing the dynamics for the process. All of them are found in ref. [5].
The angle θ⋆π is the polar angle of the pion in the CM frame with
cos θ⋆π =
−γECMSπ + Eπ
βγ|~p CMSπ |
(2.3)
where
|~p CMSπ | =
(ECMSπ )
2 −m2π with ECMSπ =
W 2 +m2π −M2N
(2.4)
and the rest of the variables defined as
W 2 +Q2 −M2N
, γ =
ν +MN
, βγ =
ν2 +Q2
. (2.5)
It is now straight-forward to convert the cross section differential in the solid angle to the
one differential in the laboratory energy of the pion, Eπ,
γβ|~p CMSπ |
d cos θ⋆π
. (2.6)
Having expressed all quantities in (2.2) and (2.5) in terms of W, Q2 and Eπ it is possible
to compute the pion energy spectrum
∫ Wmax
∫ Q2max
dQ2dWdEπ
θ(phys). (2.7)
The limits of integration are given as
Q2min = 0 , Q
max =
(S −W 2)(S −M2N )
Wmin = MN +mπ , Wmax ≃ 1.6 GeV (2.8)
where S = M2N + 2MNE1 is the center-of-mass energy squared with E1 the energy of the
incoming lepton in the LAB system. The θ-function takes care of the constraints from
the phase space. We integrated the cross section for Eν = 1 GeV and show the spectrum
in figures 1–3. In our earlier publication [14] the spectrum for Eπ was incorrect because
we did not impose the phase space constraints correctly. The pion spectrum for charged
current reactions is correctly reported in figure (4) in ref. [21]. The discrepancy in ref. [14]
has been pointed out for neutral currents in ref [12].
The neutrino–nucleon and electron–nucleon cross sections will be used in the rest of
this article in order to compute and test effects of nuclear corrections. We deduce the
electroproduction cross sections from neutrino production as in Eq. (2.1). For the triple
– 4 –
differential cross section we follow the same procedure by setting the axial form factors to
zero and using the relation
dσem,I=1
dQ2dWdEπ
dQ2dWdEπ


: ep → epπ0
: ep → enπ+
: en → epπ−
: en → enπ0
(2.9)
A small isoscalar part in the electromagnetic cross section is omitted since it does not
contribute to the ∆-resonance but only to the background, which for W < 1.3 GeV is
small and contributes for 1.3 GeV < W < 1.4 GeV.
3. Cross sections for heavy targets
In the following we will deal with single pion resonance production in the scattering of a
lepton l off a nuclear target T (6C
12, 8O
16, 18Ar
40, 26Fe
56), i.e., with the reactions
l + T → l′ + T ′ + π±,0 (3.1)
where l′ is the outgoing lepton and T ′ a final nuclear state. Furthermore, in our analysis of
nuclear rescattering effects we will restrict ourselves to the region of the ∆(1232) resonance,
1.1 GeV < W < 1.4 GeV, and to isoscalar targets with equal number of protons and
neutrons.
3.1 Pion rescattering in the ANP model
According to the ANP model [13, 22] the final cross sections for pions (π+, π0, π−)f can
be related to the initial cross sections (π+, π0, π−)i for a free nucleon target in the simple
form 

dσ(ZT
A;π+)
dQ2dW
dσ(ZT
A;π0)
dQ2dW
dσ(ZT
A;π−)
dQ2dW

= M [T ;Q2,W ]

dσ(NT ;π
dQ2dW
dσ(NT ;π
dQ2dW
dσ(NT ;π
dQ2dW

(3.2)
dσ(NT ;±0)
dQ2dW
dσ(p;±0)
dQ2dW
+ (A− Z)dσ(n;±0)
dQ2dW
(3.3)
where the free nucleon cross sections are averaged over the Fermi momentum of the nucle-
ons.1 For an isoscalar target the matrix M is described by three independent parameters
Ap, d, and c in the following form [13]
M = Ap
1− c− d d c
d 1− 2d d
c d 1− c− d
 , (3.4)
However, the Fermi motion has a very small effect on the W distribution and we neglect it in our
numerical analysis. On the other hand, effects of the Pauli exclusion principle have been absorbed into the
matrix M and are taken into account.
– 5 –
where Ap(Q
2,W ) = g(Q2,W ) × f(1,W ). Here, g(Q2,W ) is the Pauli suppression factor
and f(1,W ) is a transport function for equal populations of π+, π0, π− which depends on
the absorption cross section of pions in the nucleus. The parameters c and d describe the
charge exchange contribution. The final yields of π’s depend on the target material and
the final state kinematic variables, i.e., M = M [T ;Q2,W ].
In order to simplify the problem it is helpful to integrate the doubly differential cross
sections of Eq. (3.2) over W in the (3, 3) resonance region, say, mp +mπ ≤ W ≤ 1.4 GeV.
In this case Eq. (3.2) can be replaced by an equation of identical form

dσ(ZT
A;π+)
dσ(ZT
A;π0)
dσ(ZT
A;π−)

= M [T ;Q2]

dσ(NT ;π
dσ(NT ;π
dσ(NT ;π

(3.5)
where the matrix M [T ;Q2] can be obtained by averaging the matrix M [T ;Q2,W ] over W
with the leading W -dependence coming from the ∆ resonance contribution. Moreover, we
expect the matrix M to be a slowly varying function of Q2 (for Q2 & 0.3 GeV2). For this
reason we introduce a second averaging over Q2 and define the double averaged matrix
M [T ] which is particularly useful for giving a simple description of charge exchange effects
in different nuclear targets. In the double-averaging approximation (AV2) the final cross
sections including nuclear corrections are expressed as follows:

dσ(ZT
A;π+)
dQ2dW
dσ(ZT
A;π0)
dQ2dW
dσ(ZT
A;π−)
dQ2dW

= M [T ]

dσ(NT ;π
dQ2dW
dσ(NT ;π
dQ2dW
dσ(NT ;π
dQ2dW

. (3.6)
We note that the cross sections are differential in two variables while the matrix M [T ] is
the average over these variables.
The above discussion will be used for a phenomenological description of nuclear rescat-
tering effects. On the other hand, in Ref. [13] a dynamical model has been developed to
calculate the charge exchange matrix M . As an example, for oxygen the resulting matrix
in the double-averaging approximation is given by
16) = Ap
0.788 0.158 0.0537
0.158 0.684 0.158
0.0537 0.158 0.788
 . (3.7)
with Ap = 0.766, which contains the averaged Pauli suppression factor and absorption of
pions in the nucleus. There are various absorption models described in the original article.
Two of them are distinguished by the energy dependence of the absorption cross section
beyond the ∆ region. In model [A] the absorption increases as W increases while in [B]
– 6 –
it decreases for large W ’s (beyond the ∆ region). A comparison of the two absorption
models (A) and (B) can be found in [22]. Since the fraction of absorbed pions is still
rather uncertain we provide in the appendices ANP matrices for different amounts of
absorption. These matrices are useful to obtain an uncertainty band for the expected
nuclear corrections.
3.2 Results for various targets
In this section we present numerical results for 1-pion leptoproduction differential cross
sections including nuclear corrections using the ANP model outlined in the preceding sec-
tion.
3.2.1 Neutrinoproduction
We begin with a discussion of the nuclear corrections to the pion energy spectra in neutrino
scattering shown in Figs. 1–3, where the curves are neutral current reactions. The dotted
lines are the spectra for the free nucleon cross sections. The dashed lines include the effect
of the Pauli suppression (in step one of the two step process), whereas the solid line in
addition takes into account the pion multiple scattering. These curves correct Figs. 8–16
in Ref. [14]. Similar curves have been obtained recently by Leitner et al. [12] who also
noticed the error in [14]. Even though the models differ in the transport matrix, they
both include charge exchange effects. For example, they both find that for reactions where
the charge of the pions is the same with the charge of the current the pion yield shows a
substantial decrease.
3.2.2 Electroproduction
We now turn to the electroproduction. To be specific, our analysis will be done under the
conditions of the Cebaf Large Acceptance Spectrometer (CLAS) at Jefferson Lab (JLAB).
The CLAS detector [23] covers a large fraction of the full solid angle with efficient neutral
and charged particle detection. Therefore it is very well suited to perform a high statistics
measurement on various light and heavy nuclear targets and to test the ideas of pion
multiple scattering models. In the future these measurements can be compared with results
in neutrinoproduction from the Minerva experiment [1] using the high intensity Numi
neutrino beam. If not stated otherwise we use an electron energy Ee = 2.7 GeV in order
to come as close as possible to the relevant low energy range of the LBL experiments.
For the momentum transfer we take the values Q2 = 0.4, 0.8 GeV2 in order to avoid the
experimentally and theoretically more problematic region at very low Q2. Results for larger
Q2 and larger energies, say Ee = 10 GeV, are qualitatively very similar.
Figure 4 shows the double differential cross section dσ/dQ2dW for π+ and π0 produc-
tion versus W for an oxygen target. The solid lines have been obtained with help of Eq.
(3.2) including the nuclear corrections. The dashed lines show the result of the double-
averaging approximation according to Eq. (3.6) using the ANP matrix in Eq. (3.7). The
dotted line is the free cross section in Eq. (3.3). One sees, the double-averaging approxima-
tion and the exact calculation give very similar results such that the former is well-suited
for simple estimates to an accuracy of 10% of pion rescattering effects. We observe that
– 7 –
the cross sections for π0 production are largely reduced by about 40% due to the nuclear
corrections. This can be understood since the larger π0 cross sections are reduced by ab-
sorption effects and charge exchange effects. On the other hand, the π+ cross sections are
even slightly enlarged, because the reduction due to pion absorption is compensated by an
increase due to charge exchange. The compensation is substantial since the π0 yields are
dominant.
In Fig. 5 double differential cross sections per nucleon for different target materials
are presented. The electron energy and the momentum transfer have been chosen as
Ee = 2.7 GeV and Q
2 = 0.4 GeV2, respectively. The results for the pion rescattering
corrections have been obtained within the double-averaging approximation (3.6) which
allows for a simple comparison of the dependence on the target material in terms of the
matrices M [T ] which can be found in Eq. (3.7) and App. A. For comparison the free nucleon
cross section (3.3) (isoscalar
) is also shown. As expected, the nuclear corrections
become larger with increasing atomic number from carbon to iron.
One of the input quantities for calculating the transport function f(λ) in the ANP
model is the pion absorption cross section σabs(W ) describing the probability that the pion
is absorbed in a single rescattering process. For σabs(W ) the ANP article reported results
for two parameterizations, models A and B, taken from Refs. [24, 25] which have very
different W -dependence and normalization. However, the predictions of the ANP model
in the double-averaging approximation are primarily sensitive to the normalization of the
pion absorption cross section at W ≃ m∆ [22]. Using data by Merenyi et al. [26] for a
neon target it was found that about 25% ± 5% of pions are absorbed making possible the
determination of the normalization of σabs(W ) with a 20% accuracy.
In order to investigate the theoretical uncertainty due to pion absorption effects we
show in Fig. 6 double differential cross sections dσ/dQ2dW for π+ and π0 production vs
W for different amounts of pion absorption in oxygen: 25% (solid line), 20% (dashed line),
30% (dotted line). The π0 and π+ spectra have been calculated in the double-averaging
approximation (3.6) utilizing the matrices in App. B. The three curves represent the
theoretical uncertainty due to pion absorption effects. For comparison, the free nucleon
cross section (3.3) is shown as well.
Although the predictions of the ANP model are mainly sensitive to σabs(W ≃ m∆) it
would be interesting to obtain more information on the detailed W -shape. The fraction
of absorbed pions can be determined by measuring the inclusive pion production cross
sections for a nuclear target divided by the free nucleon cross sections,
Abs(Q2,W) = 1−
k=0,±
dσ(ZT
A;πk)
dQ2dW
j=0,±
dσ(NT;π
dQ2dW
= 1−Ap(Q2,W) , (3.8)
where Ap has been introduced in (3.4). This quantity is related to σabs(W ) as can be seen
by linearizing the transport function f(λ,W ) [16, 22]
Abs(Q2,W) ≃ 1
L̄ρ0 × σabs(W) . (3.9)
– 8 –
Here L̄ is the effective length of the nucleus averaged over impact parameters and ρ0 the
charge density in the center. As an example, for oxygen one finds L̄ ≃ 1.9R with radius
R ≃ 1.833 fm and ρ0 = 0.141 fm−3. Therefore, the W -dependence of σabs(W ) can be
reconstructed from the fraction of absorbed pions, i.e. Abs(Q2,W). Summing over the
three charged pions eliminates charge exchange effects.
In order to verify the linearized approximation in Eq. (3.9), we show in Fig. 7 the ANP
model prediction for Abs(Q2,W) for oxygen and iron targets with Q2 = 0.3 GeV2. This
prediction strongly depends on the shape of the cross section σabs(W ) for which we use
model B from Refs. [25]. σabs(W ) multiplied by a free normalization factors for oxygen
and iron, respectively, is depicted by the dashed lines. Obviously, Eq. (3.9) is quite well
satisfied for oxygen and still reasonably good for iron. Finally, the dotted line shows the
result of the averaging approximation. We conclude that σabs(W ) can be extracted with
help of Eqs. (3.8) and (3.9).
For completeness, we mention that the pion absorption in nuclei is reported in various
articles [27]. For comparisons one should be careful because the absorption cross sections in
pi-nucleus and in neutrino-nucleus reactions are different, in the former case it is a surface
effect while in the latter it occurs everywhere in the nucleus.
A useful test of charge exchange effects is provided by the double ratio
DR(Q2,W ) =
π+ + π−
π+ + π−
(3.10)
where (πi)A represents the doubly differential cross section dσ/dQ
2dW for the production
of a pion πi in eA scattering. This observable is expected to be rather robust with respect
to radiative corrections and acceptance differences between neutral and charged pions.2 In
Fig. 8 we show the double ratio for a carbon target in dependence of W for a fixed Q2 =
0.4 GeV2. The dependence on Q2 is weak and results for other values of Q2 are very similar.
The solid line shows the exact result, whereas the dotted lines have been obtained in the
double averaging approximation with minimal and maximal amounts of pion absorption.
As can be seen, the results are rather insensitive to the exact amount of pion absorption.
Without charge exchange effects (and assuming similar absorption of charged and neutral
pions) the double ratio would be close to unity. As can be seen, the ANP model predicts
a double ratio smaller than 0.6 in the region W ≃ 1.2 GeV. A confirmation of this
expectation would be a clear signal of pion charge exchange predominantly governed by
isospin symmetry. In this case it would be interesting to go a step further and to study
similar ratios for pion angular distributions.
4. Summary
Lepton induced reactions on medium and heavy nuclei include the rescattering of produced
pions inside the nuclei. This is especially noticeable in the ∆-resonance region, where the
produced resonance decays into a nucleon and a pion. In the introduction and section
We are grateful to S. Manly for drawing our attention to the double ratio.
– 9 –
2 we reviewed the progress that has been made in the calculations of neutrino-induced
reactions on free protons and neutrons, because we needed them for following calculations.
For several resonances the vector form factors have been recently determined by using
electroproduction results in Jefferson Laboratory [7]. For the axial form factors modified
dipoles give an accurate description of the data. For the purposes of this article (studies of
nuclear corrections) it suffices to deduce the electroproduction cross sections through Eqs.
(2.1) and (2.9).
The main contribution of this article is contained in section 3, where we describe
important features of the ANP model and define single- and double averaged transport
matrices. Two important aspects of rescattering are emphasized: (i) the absorption of the
pions and (ii) charge exchange occurring in the multiple scattering, where we have shown
that special features of the data are attributed to each of them. Finally we propose specific
ratios of electroproduction reactions that are sensitive to the absorption cross section and
to charge exchange effects.
Using the model we calculate the transport matrix for various absorption cross sec-
tions and nuclei and present the results in appendix A. We also calculated the pion energy
spectra with and without nuclear corrections. The results appear in figures 1–3 and can be
compared with other calculations [12]. Comparison of the double averaged approximation
with the exact ANP calculation shows small differences (figure 4). As mentioned already,
electroproduction data are very useful in testing several aspects of the model and its pre-
dictions. For the absorption cross section we propose in Eq. (3.8) a ratio that depends only
on Ap(Q
2,W ) = g(Q2,W )f(1,W ). Since we consider isoscalar targets and sum over the
charges of the pions, charge exchange terms are eliminated. This leaves over the depen-
dence on charge independent effects, like the Pauli factor and the average absorption; this is
indeed the average absorption of pions and even includes the absorption of the ∆-resonance
itself.
Another ratio (DR(Q2,W )) is sensitive to charge exchange effects. In the double ratio
the dependence on Ap(Q
2,W ) drops out and the surviving terms are isospin dependent.
Our calculation shows that the ratio depends on W with the largest reduction occurring in
the region 1.1 < W < 1.25 GeV. Finally, the ∆(1232) is a sharply peaked resonance, where
the resonant interaction, takes place over small ranges of the kinematic variables, so that
averaging over them gives accurate approximations. This is analogous to a narrow width
approximation. Several comparisons in this article confirm the expectation that averaged
quantities give rather accurate approximations of more extensive calculations.
Acknowledgments
We wish to thank W. Brooks and S. Manly for many useful discussions, their interest and
encouragement. The work of J. Y. Yu is supported by the Deutsche Forschungsgemeinschaft
(DFG) through Grant No. YU 118/1-1.
– 10 –
Appendix
A. Charge exchange matrices in the double averaging approximation
Carbon:
12) = Ap
0.826 0.136 0.038
0.136 0.728 0.136
0.038 0.136 0.826
 (A.1)
with Ap = 0.791 .
Argon:
M(18Ar
40) = Ap
0.733 0.187 0.080
0.187 0.626 0.187
0.080 0.187 0.733
 (A.2)
with Ap = 0.657 .
Iron:
M (26Fe
56) = Ap
0.720 0.194 0.086
0.194 0.613 0.194
0.086 0.194 0.720
 (A.3)
with Ap = 0.631 .
B. Charge exchange matrices for various amounts of pion absorption
Carbon:
15% absorption
12) = Ap
0.817 0.141 0.041
0.141 0.718 0.141
0.041 0.141 0.817
 (B.1)
with Ap = 0.831 .
20% absorption
12) = Ap
0.829 0.134 0.037
0.134 0.731 0.134
0.037 0.134 0.829
 (B.2)
with Ap = 0.782 .
– 11 –
25% absorption
12) = Ap
0.840 0.127 0.032
0.127 0.745 0.127
0.032 0.127 0.840
 (B.3)
with Ap = 0.734 .
Oxygen:
15% absorption
16) = Ap
0.771 0.167 0.062
0.167 0.665 0.167
0.062 0.167 0.771
 (B.4)
with Ap = 0.833 .
20% absorption
16) = Ap
0.783 0.161 0.056
0.161 0.679 0.161
0.056 0.161 0.783
 (B.5)
with Ap = 0.784 .
25% absorption
16) = Ap
0.797 0.153 0.050
0.153 0.693 0.153
0.050 0.153 0.797
 (B.6)
with Ap = 0.735 .
30% absorption
16) = Ap
0.810 0.146 0.044
0.146 0.709 0.146
0.044 0.146 0.810
 (B.7)
with Ap = 0.687 .
C. Forward- and backward charge exchange matrices
Carbon:
15% absorption
M+(6C
12) = Ap+
0.870 0.100 0.029
0.100 0.799 0.100
0.029 0.100 0.870
 ,M−(6C12) = Ap−
0.675 0.251 0.074
0.251 0.498 0.251
0.074 0.251 0.675
 (C.1)
– 12 –
with Ap+ = 0.606 and Ap− = 0.225.
20% absorption
M+(6C
12) = Ap+
0.880 0.094 0.026
0.094 0.811 0.094
0.026 0.094 0.880
 ,M−(6C12) = Ap−
0.685 0.247 0.068
0.247 0.505 0.247
0.068 0.247 0.685
 (C.2)
with Ap+ = 0.578 and Ap− = 0.204.
25% absorption
M+(6C
12) = Ap+
0.889 0.088 0.022
0.088 0.823 0.088
0.022 0.088 0.889
 ,M−(6C12) = Ap−
0.695 0.243 0.062
0.243 0.513 0.243
0.062 0.243 0.695
 (C.3)
with Ap+ = 0.549 and Ap− = 0.184.
Oxygen:
15% absorption
M+(8O
16) = Ap+
0.829 0.125 0.046
0.125 0.750 0.125
0.046 0.125 0.829
 ,M−(8O16) = Ap−
0.635 0.265 0.100
0.265 0.470 0.265
0.100 0.265 0.635
 (C.4)
with Ap+ = 0.581 and Ap− = 0.252.
20% absorption
M+(8O
16) = Ap+
0.840 0.119 0.041
0.119 0.762 0.119
0.041 0.119 0.840
 ,M−(8O16) = Ap−
0.646 0.262 0.092
0.262 0.477 0.262
0.092 0.262 0.646
 (C.5)
with Ap+ = 0.554 and Ap− = 0.23.
25% absorption
M+(8O
16) = Ap+
0.852 0.112 0.036
0.112 0.776 0.112
0.036 0.112 0.852
 ,M−(8O16) = Ap−
0.657 0.258 0.085
0.258 0.485 0.257
0.085 0.258 0.657
 (C.6)
with Ap+ = 0.527 and Ap− = 0.208.
– 13 –
30% absorption
M+(8O
16) = Ap+
0.863 0.105 0.031
0.105 0.789 0.105
0.031 0.105 0.863
 ,M−(8O16) = Ap−
0.669 0.253 0.078
0.253 0.493 0.253
0.078 0.253 0.669
 (C.7)
with Ap+ = 0.499 and Ap− = 0.187.
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– 14 –
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hep-ph/0307223.
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C. H. Q. Ingram, Nucl. Phys. A684, 122 (2001).
R. D. Ransome, Nucl. Phys. Proc. Suppl. 139, 208 (2005), and references therein.
– 15 –
π+ ig
Eπ (GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
π0 ig
Eπ (GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
π- ig
Eπ (GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Figure 1: Differential cross section per nucleon for single pion spectra of π+, π0, π− for oxygen with
= 1 GeV in dependence of pion energy E
. The curves correspond to neutral current reactions.
– 16 –
π+ ig
Eπ (GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
π0 ig
Eπ (GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
π- ig
Eπ (GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Figure 2: The same as in fig. 1 for argon.
– 17 –
π+ ig
Eπ(GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
π0 ig
Eπ(GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
π- ig
Eπ(GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Figure 3: The same as in fig. 1 for iron.
– 18 –
Q2 = 0.4 GeV2
π0 f (exact)
π0 f (av. app.)
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Q2 = 0.4 GeV2
π+ f (exact)
π+ f (av. app.)
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Q2 = 0.8 GeV2
π0 f (exact)
π0 f (av. app.)
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Q2 = 0.8 GeV2
π+ f (exact)
π+ f (av. app.)
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Figure 4: Double differential cross sections for single-pion electroproduction for an oxygen target in
dependence of W . Spectra for π0 and π+ production are shown for Q2 = 0.4 GeV2 and Q2 = 0.8 GeV2
using an electron energy E
= 2.7 GeV. The solid and dotted lines have been obtained according to
(3.2) using the exact ANP matrix M(W,Q2) and (3.6) utilizing the double-averaged ANP matrix M in
(3.7), respectively. The dashed lines show the free nucleon cross section (3.3).
Q2 = 0.4 GeV2 π0 f (8O
π0 f (6C
π0 f (18Ar
π0 f (26Fe
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Q2 = 0.4 GeV2 π+ f (8O
π+ f (6C
π+ f (18Ar
π+ f (26Fe
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Figure 5: Double differential cross sections per nucleon for single-pion electroproduction for different
target materials. W -spectra for π0 and π+ production are shown for Q2 = 0.4 GeV2 using an electron
energy E
= 2.7 GeV. The pion rescattering corrections have been calculated in the double-averaging
approximation (3.6) using the ANP matrices in (3.7) and App. A. For comparison, the free nucleon
cross section (3.3) is shown.
– 19 –
Q2 = 0.8 GeV2
π0 f (Abs. 25%)
π0 f (Abs. 20%)
π0 f (Abs. 30%)
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Q2 = 0.8 GeV2
π+ f (Abs. 25%)
π+ f (Abs. 20%)
π+ f (Abs. 30%)
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Figure 6: Double differential cross sections per nucleon for single-pion electroproduction for oxygen with
20% (dashed line), 25% (solid line) and 30% (dotted line) pion absorption. Furthermore, Q2 = 0.8 GeV2
and E
= 2.7 GeV. The π0 and π+ spectra have been calculated in the double-averaging approximation
(3.6) utilizing the matrices in App. B. For comparison, the free nucleon cross section (3.3) is shown as
well.
exact
Av. Approx.
N * sigabs(W)
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45
Figure 7: The fraction of absorbed pions, Abs(Q2,W), in dependence of W for oxygen and iron
targets for Q2 = 0.3 GeV2. Also shown is the cross section σabs(W ) (model B) multiplied by free
normalization factors (dashed lines). The dotted lines are the result for Abs(Q2,W) in the averaging
approximation.
– 20 –
Q2 = 0.4 GeV2
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Figure 8: Double ratio of single pion electroproduction cross sections in dependence of W for fixed
Q2 = 0.4 GeV2 as defined in Eq. (3.10). The dotted lines show results in the double averaging
approximation with varying amounts of absorption.
– 21 –

  We present a calculation of single pion electroproduction cross sections on
heavy targets in the kinematic region of the Delta(1232) resonance. Final state
interactions of the pions are taken into account using the pion multiple
scattering model of Adler, Nussinov and Paschos (ANP model). For
electroproduction and neutral current reactions we obtain results for carbon,
oxygen, argon and iron targets and find a significant reduction of the
W-spectra for pi^0 as compared to the free nucleon case. On the other hand, the
charged pion spectra are only little affected by final state interactions.
Measurements of such cross sections with the CLAS detector at JLAB could help
to improve our understanding of pion rescattering effects and serve as
important/valuable input for calculations of single pion neutrinoproduction on
heavy targets relevant for current and future long baseline neutrino
experiments. Two ratios, in Eq. (3.8) and (3.10), will test important
properties of the model.

Introduction 1
2. Free nucleon cross sections 3
3. Cross sections for heavy targets 5
3.1 Pion rescattering in the ANP model 5
3.2 Results for various targets 7
3.2.1 Neutrinoproduction 7
3.2.2 Electroproduction 7
4. Summary 9
A. Charge exchange matrices in the double averaging approximation 11
B. Charge exchange matrices for various amounts of pion absorption 11
C. Forward- and backward charge exchange matrices 12
1. Introduction
Neutrino interactions at low and medium energies are attracting attention because they
will be measured accurately in the new generation of experiments [1, 2]. One aim of the
experiments is to measure the precise form of the cross sections and their dependence
on the input parameters. This way we check their couplings and compare the functional
dependence of the form factors, where deviations from the dipole dependence have already
been established (see e.g. figure 1 in [3] and references therein). Deviation from the standard
model predictions can arise either from properties of the neutrinos or from new couplings
of the gauge bosons to the particles in the target. Another aim of the experiments is to
establish the properties of neutrinos including their masses, mixings and their fermionic
nature (Dirac or Majorana particles). This program requires a good understanding of the
cross sections, which motivated a new generation of calculations. Since the experiments use
nuclear targets, like C12, O16, Ar40, Fe56, ... it is necessary to understand the modifications
brought about by the targets.
The very old calculations for quasi-elastic scattering and resonance excitation on free
nucleons [4, 5] have been replaced by new results where couplings and form factors are
now better determined. For the vector couplings comparisons with electroproduction data
have been very useful [3, 6]. Axial couplings are frequently determined by PCAC. There
are already improvements and checks of the earlier quark models [7]. Comparisons with
– 1 –
experimental data are also available even though the experimental results are not always
consistent with each other [8, 9, 10] but there are plans for improvements that will resolve
the differences [1, 2].
For reactions on nuclear targets there are modifications brought about by the propaga-
tion of the produced particles in the nuclear medium. They involve absorption of particles,
restrictions from Pauli blocking, Fermi motion and charge-exchange rescatterings. One
group of papers uses nuclear potentials for the propagation of the particles [11]. Others use
a transport theory of the final particles including channels coupled to each other [12]. These
groups gained experience by analyzing reactions with electron beams (electroproduction)
and adopted their methods to neutrino reactions [12].
Our group investigated 1-π pion production on medium and heavy targets employing
the pion multiple scattering model by Adler, Nussinov and Paschos [13] that was developed
in order to understand neutral current neutrino interactions with nuclei. This model was
useful in the discovery of neutral currents and has been applied to predict neutrino-induced
single pion production on Oxygen, Argon and Iron targets [14, 15, 16] which are used in
long baseline(LBL) experiments. Among its characteristics is the importance of charge-
exchange reactions that modify the π+ : π0 : π− ratios of the original neutrino-nucleon
interaction through their scatterings within the nuclei. The presence of this effect has been
confirmed by experiments [17]. We note here that our results are valid for isoscalar targets.
For non-isoscalar targets like lead, used in the OPERA experiment, it is possible to extend
the ANP model [18], which can be done in the future.
In this article we take an inverse route and use our calculation in neutrino reactions to
go back to the electroproduction of pions on free nucleons and heavy nuclei. The plan of
the paper is as follows. In section 2 we summarize the neutrino production cross sections
on free nucleons and in the ∆ resonance region. This topic has been described by several
groups in the past few years. We present cross sections differential in several variables
Eπ, Q
2 and W . We pay special attention to the spectrum dσ/dEπ, where we correct an
error we found in our earlier calculation [14]. Then we obtain the electroproduction cross
section by setting the axial coupling equal to zero and rescaling, appropriately, the vector
current contribution.
The main content of the article appears in section 3 where we describe the salient
features and results of the ANP model. This model has the nice property that it can be
written in analytic form including charge exchange and absorption of pions. This way we
can trace the origin of the effects and formulate quantities which test specific terms and
parameters. As we mentioned above several features have been tested already, and we
wish to use electroproduction data in order to determine the accuracy of the predictions.
We present numerical results for different target materials, and study the quality of the
averaging approximation and uncertainties of the ANP model due to pion absorption ef-
fects. We discuss how the shape of the pion absorption cross section (per nucleon), an
important and almost unconstrained ingredient of the ANP model, can be delineated from
a measurement of the total fraction of absorbed pions. Finally, in Sec. 4 we summarize the
main results. Averaged rescattering matrices for carbon, oxygen, argon, and iron targets
and for different amounts of pion absorption have been collected in the appendices and are
– 2 –
useful for simple estimates of the rescattering effects.
2. Free nucleon cross sections
In the following sections, leptonic pion production on nuclear targets is regraded as a two
step process. In the first step, the pions are produced from constituent nucleons in the
target with free lepton-nucleon cross sections [13]. In the second step the produced pions
undergo a nuclear interaction described by a transport matrix. Of course, the resonances
themselves propagate in the nuclear medium before they decay, an effect that we will
investigate in the future.
The leptonic production of pions in the ∆-resonance region is theoretically available
and rather well understood as described in articles for both electro- and neutrino produc-
tion, where comparisons with available data are in good agreement [3, 6, 7, 19, 12].
The available data is described accurately with the proposed parameterizations. The
vector form factors are modified dipoles [3] which reproduce the helicity amplitudes mea-
sured in electroproduction experiments at Jefferson Laboratory [7]. The coupling in the
axial form factors are determined by PCAC and data. Their functional dependence in Q2
is determined by fitting the dσ
distributions. For the vector form factors the magnetic
dipole dominance for CV3 (q
2) and CV4 (q
2) gives an accurate description of the data. How-
ever, deviations with a non-zero CV5 (q
2) have also been established [7]. This way a small
(5%) isoscalar amplitude is reproduced.
For the propose of this article we shall use a scaling relation connecting neutrino- to
electroproduction. The weak vector current is in the same isospin multiplied with the
electromagnetic current and the two are related as follows:
< ∆++|V |p >=
3 < ∆+|JI=1em |p >=
3 < ∆0|JI=1em |n > .
Taking into account the isospin Clebsch-Gordan factors for the ∆ → Nπ branchings one
finds the following contributions of the ∆-resonance to the cross sections for ep → epπ0,
ep → enπ+, en → epπ− and en → enπ0
dσem,I=1
dQ2dW
dQ2dW


: pπ0
: nπ+
: pπ−
: nπ0
(2.1)
where dV
dQ2dW
denotes the cross section for the vector contribution alone to the reaction
νp → µ−pπ+. The free nucleon cross sections in Eq. (2.1) will be used in our numerical
analysis. We shall call this the reduced electromagnetic formula. Its accuracy was tested
in figure (5) of ref. [3]. Further comparisons can be found in [20].
For studies of the pion angular distributions (or what is the same of the pion energy
spectrum in the laboratory frame) we begin with the triple differential cross section for
– 3 –
neutrino production
dQ2dWd cos θ⋆π
16πM2
KiW̃i −
KiDi(3 cos
2 θ⋆π − 1)
(2.2)
with Ki being kinematic factors of W and Q
2 and the structure functions W̃i(Q
2,W ) and
2,W ) representing the dynamics for the process. All of them are found in ref. [5].
The angle θ⋆π is the polar angle of the pion in the CM frame with
cos θ⋆π =
−γECMSπ + Eπ
βγ|~p CMSπ |
(2.3)
where
|~p CMSπ | =
(ECMSπ )
2 −m2π with ECMSπ =
W 2 +m2π −M2N
(2.4)
and the rest of the variables defined as
W 2 +Q2 −M2N
, γ =
ν +MN
, βγ =
ν2 +Q2
. (2.5)
It is now straight-forward to convert the cross section differential in the solid angle to the
one differential in the laboratory energy of the pion, Eπ,
γβ|~p CMSπ |
d cos θ⋆π
. (2.6)
Having expressed all quantities in (2.2) and (2.5) in terms of W, Q2 and Eπ it is possible
to compute the pion energy spectrum
∫ Wmax
∫ Q2max
dQ2dWdEπ
θ(phys). (2.7)
The limits of integration are given as
Q2min = 0 , Q
max =
(S −W 2)(S −M2N )
Wmin = MN +mπ , Wmax ≃ 1.6 GeV (2.8)
where S = M2N + 2MNE1 is the center-of-mass energy squared with E1 the energy of the
incoming lepton in the LAB system. The θ-function takes care of the constraints from
the phase space. We integrated the cross section for Eν = 1 GeV and show the spectrum
in figures 1–3. In our earlier publication [14] the spectrum for Eπ was incorrect because
we did not impose the phase space constraints correctly. The pion spectrum for charged
current reactions is correctly reported in figure (4) in ref. [21]. The discrepancy in ref. [14]
has been pointed out for neutral currents in ref [12].
The neutrino–nucleon and electron–nucleon cross sections will be used in the rest of
this article in order to compute and test effects of nuclear corrections. We deduce the
electroproduction cross sections from neutrino production as in Eq. (2.1). For the triple
– 4 –
differential cross section we follow the same procedure by setting the axial form factors to
zero and using the relation
dσem,I=1
dQ2dWdEπ
dQ2dWdEπ


: ep → epπ0
: ep → enπ+
: en → epπ−
: en → enπ0
(2.9)
A small isoscalar part in the electromagnetic cross section is omitted since it does not
contribute to the ∆-resonance but only to the background, which for W < 1.3 GeV is
small and contributes for 1.3 GeV < W < 1.4 GeV.
3. Cross sections for heavy targets
In the following we will deal with single pion resonance production in the scattering of a
lepton l off a nuclear target T (6C
12, 8O
16, 18Ar
40, 26Fe
56), i.e., with the reactions
l + T → l′ + T ′ + π±,0 (3.1)
where l′ is the outgoing lepton and T ′ a final nuclear state. Furthermore, in our analysis of
nuclear rescattering effects we will restrict ourselves to the region of the ∆(1232) resonance,
1.1 GeV < W < 1.4 GeV, and to isoscalar targets with equal number of protons and
neutrons.
3.1 Pion rescattering in the ANP model
According to the ANP model [13, 22] the final cross sections for pions (π+, π0, π−)f can
be related to the initial cross sections (π+, π0, π−)i for a free nucleon target in the simple
form 

dσ(ZT
A;π+)
dQ2dW
dσ(ZT
A;π0)
dQ2dW
dσ(ZT
A;π−)
dQ2dW

= M [T ;Q2,W ]

dσ(NT ;π
dQ2dW
dσ(NT ;π
dQ2dW
dσ(NT ;π
dQ2dW

(3.2)
dσ(NT ;±0)
dQ2dW
dσ(p;±0)
dQ2dW
+ (A− Z)dσ(n;±0)
dQ2dW
(3.3)
where the free nucleon cross sections are averaged over the Fermi momentum of the nucle-
ons.1 For an isoscalar target the matrix M is described by three independent parameters
Ap, d, and c in the following form [13]
M = Ap
1− c− d d c
d 1− 2d d
c d 1− c− d
 , (3.4)
However, the Fermi motion has a very small effect on the W distribution and we neglect it in our
numerical analysis. On the other hand, effects of the Pauli exclusion principle have been absorbed into the
matrix M and are taken into account.
– 5 –
where Ap(Q
2,W ) = g(Q2,W ) × f(1,W ). Here, g(Q2,W ) is the Pauli suppression factor
and f(1,W ) is a transport function for equal populations of π+, π0, π− which depends on
the absorption cross section of pions in the nucleus. The parameters c and d describe the
charge exchange contribution. The final yields of π’s depend on the target material and
the final state kinematic variables, i.e., M = M [T ;Q2,W ].
In order to simplify the problem it is helpful to integrate the doubly differential cross
sections of Eq. (3.2) over W in the (3, 3) resonance region, say, mp +mπ ≤ W ≤ 1.4 GeV.
In this case Eq. (3.2) can be replaced by an equation of identical form

dσ(ZT
A;π+)
dσ(ZT
A;π0)
dσ(ZT
A;π−)

= M [T ;Q2]

dσ(NT ;π
dσ(NT ;π
dσ(NT ;π

(3.5)
where the matrix M [T ;Q2] can be obtained by averaging the matrix M [T ;Q2,W ] over W
with the leading W -dependence coming from the ∆ resonance contribution. Moreover, we
expect the matrix M to be a slowly varying function of Q2 (for Q2 & 0.3 GeV2). For this
reason we introduce a second averaging over Q2 and define the double averaged matrix
M [T ] which is particularly useful for giving a simple description of charge exchange effects
in different nuclear targets. In the double-averaging approximation (AV2) the final cross
sections including nuclear corrections are expressed as follows:

dσ(ZT
A;π+)
dQ2dW
dσ(ZT
A;π0)
dQ2dW
dσ(ZT
A;π−)
dQ2dW

= M [T ]

dσ(NT ;π
dQ2dW
dσ(NT ;π
dQ2dW
dσ(NT ;π
dQ2dW

. (3.6)
We note that the cross sections are differential in two variables while the matrix M [T ] is
the average over these variables.
The above discussion will be used for a phenomenological description of nuclear rescat-
tering effects. On the other hand, in Ref. [13] a dynamical model has been developed to
calculate the charge exchange matrix M . As an example, for oxygen the resulting matrix
in the double-averaging approximation is given by
16) = Ap
0.788 0.158 0.0537
0.158 0.684 0.158
0.0537 0.158 0.788
 . (3.7)
with Ap = 0.766, which contains the averaged Pauli suppression factor and absorption of
pions in the nucleus. There are various absorption models described in the original article.
Two of them are distinguished by the energy dependence of the absorption cross section
beyond the ∆ region. In model [A] the absorption increases as W increases while in [B]
– 6 –
it decreases for large W ’s (beyond the ∆ region). A comparison of the two absorption
models (A) and (B) can be found in [22]. Since the fraction of absorbed pions is still
rather uncertain we provide in the appendices ANP matrices for different amounts of
absorption. These matrices are useful to obtain an uncertainty band for the expected
nuclear corrections.
3.2 Results for various targets
In this section we present numerical results for 1-pion leptoproduction differential cross
sections including nuclear corrections using the ANP model outlined in the preceding sec-
tion.
3.2.1 Neutrinoproduction
We begin with a discussion of the nuclear corrections to the pion energy spectra in neutrino
scattering shown in Figs. 1–3, where the curves are neutral current reactions. The dotted
lines are the spectra for the free nucleon cross sections. The dashed lines include the effect
of the Pauli suppression (in step one of the two step process), whereas the solid line in
addition takes into account the pion multiple scattering. These curves correct Figs. 8–16
in Ref. [14]. Similar curves have been obtained recently by Leitner et al. [12] who also
noticed the error in [14]. Even though the models differ in the transport matrix, they
both include charge exchange effects. For example, they both find that for reactions where
the charge of the pions is the same with the charge of the current the pion yield shows a
substantial decrease.
3.2.2 Electroproduction
We now turn to the electroproduction. To be specific, our analysis will be done under the
conditions of the Cebaf Large Acceptance Spectrometer (CLAS) at Jefferson Lab (JLAB).
The CLAS detector [23] covers a large fraction of the full solid angle with efficient neutral
and charged particle detection. Therefore it is very well suited to perform a high statistics
measurement on various light and heavy nuclear targets and to test the ideas of pion
multiple scattering models. In the future these measurements can be compared with results
in neutrinoproduction from the Minerva experiment [1] using the high intensity Numi
neutrino beam. If not stated otherwise we use an electron energy Ee = 2.7 GeV in order
to come as close as possible to the relevant low energy range of the LBL experiments.
For the momentum transfer we take the values Q2 = 0.4, 0.8 GeV2 in order to avoid the
experimentally and theoretically more problematic region at very low Q2. Results for larger
Q2 and larger energies, say Ee = 10 GeV, are qualitatively very similar.
Figure 4 shows the double differential cross section dσ/dQ2dW for π+ and π0 produc-
tion versus W for an oxygen target. The solid lines have been obtained with help of Eq.
(3.2) including the nuclear corrections. The dashed lines show the result of the double-
averaging approximation according to Eq. (3.6) using the ANP matrix in Eq. (3.7). The
dotted line is the free cross section in Eq. (3.3). One sees, the double-averaging approxima-
tion and the exact calculation give very similar results such that the former is well-suited
for simple estimates to an accuracy of 10% of pion rescattering effects. We observe that
– 7 –
the cross sections for π0 production are largely reduced by about 40% due to the nuclear
corrections. This can be understood since the larger π0 cross sections are reduced by ab-
sorption effects and charge exchange effects. On the other hand, the π+ cross sections are
even slightly enlarged, because the reduction due to pion absorption is compensated by an
increase due to charge exchange. The compensation is substantial since the π0 yields are
dominant.
In Fig. 5 double differential cross sections per nucleon for different target materials
are presented. The electron energy and the momentum transfer have been chosen as
Ee = 2.7 GeV and Q
2 = 0.4 GeV2, respectively. The results for the pion rescattering
corrections have been obtained within the double-averaging approximation (3.6) which
allows for a simple comparison of the dependence on the target material in terms of the
matrices M [T ] which can be found in Eq. (3.7) and App. A. For comparison the free nucleon
cross section (3.3) (isoscalar
) is also shown. As expected, the nuclear corrections
become larger with increasing atomic number from carbon to iron.
One of the input quantities for calculating the transport function f(λ) in the ANP
model is the pion absorption cross section σabs(W ) describing the probability that the pion
is absorbed in a single rescattering process. For σabs(W ) the ANP article reported results
for two parameterizations, models A and B, taken from Refs. [24, 25] which have very
different W -dependence and normalization. However, the predictions of the ANP model
in the double-averaging approximation are primarily sensitive to the normalization of the
pion absorption cross section at W ≃ m∆ [22]. Using data by Merenyi et al. [26] for a
neon target it was found that about 25% ± 5% of pions are absorbed making possible the
determination of the normalization of σabs(W ) with a 20% accuracy.
In order to investigate the theoretical uncertainty due to pion absorption effects we
show in Fig. 6 double differential cross sections dσ/dQ2dW for π+ and π0 production vs
W for different amounts of pion absorption in oxygen: 25% (solid line), 20% (dashed line),
30% (dotted line). The π0 and π+ spectra have been calculated in the double-averaging
approximation (3.6) utilizing the matrices in App. B. The three curves represent the
theoretical uncertainty due to pion absorption effects. For comparison, the free nucleon
cross section (3.3) is shown as well.
Although the predictions of the ANP model are mainly sensitive to σabs(W ≃ m∆) it
would be interesting to obtain more information on the detailed W -shape. The fraction
of absorbed pions can be determined by measuring the inclusive pion production cross
sections for a nuclear target divided by the free nucleon cross sections,
Abs(Q2,W) = 1−
k=0,±
dσ(ZT
A;πk)
dQ2dW
j=0,±
dσ(NT;π
dQ2dW
= 1−Ap(Q2,W) , (3.8)
where Ap has been introduced in (3.4). This quantity is related to σabs(W ) as can be seen
by linearizing the transport function f(λ,W ) [16, 22]
Abs(Q2,W) ≃ 1
L̄ρ0 × σabs(W) . (3.9)
– 8 –
Here L̄ is the effective length of the nucleus averaged over impact parameters and ρ0 the
charge density in the center. As an example, for oxygen one finds L̄ ≃ 1.9R with radius
R ≃ 1.833 fm and ρ0 = 0.141 fm−3. Therefore, the W -dependence of σabs(W ) can be
reconstructed from the fraction of absorbed pions, i.e. Abs(Q2,W). Summing over the
three charged pions eliminates charge exchange effects.
In order to verify the linearized approximation in Eq. (3.9), we show in Fig. 7 the ANP
model prediction for Abs(Q2,W) for oxygen and iron targets with Q2 = 0.3 GeV2. This
prediction strongly depends on the shape of the cross section σabs(W ) for which we use
model B from Refs. [25]. σabs(W ) multiplied by a free normalization factors for oxygen
and iron, respectively, is depicted by the dashed lines. Obviously, Eq. (3.9) is quite well
satisfied for oxygen and still reasonably good for iron. Finally, the dotted line shows the
result of the averaging approximation. We conclude that σabs(W ) can be extracted with
help of Eqs. (3.8) and (3.9).
For completeness, we mention that the pion absorption in nuclei is reported in various
articles [27]. For comparisons one should be careful because the absorption cross sections in
pi-nucleus and in neutrino-nucleus reactions are different, in the former case it is a surface
effect while in the latter it occurs everywhere in the nucleus.
A useful test of charge exchange effects is provided by the double ratio
DR(Q2,W ) =
π+ + π−
π+ + π−
(3.10)
where (πi)A represents the doubly differential cross section dσ/dQ
2dW for the production
of a pion πi in eA scattering. This observable is expected to be rather robust with respect
to radiative corrections and acceptance differences between neutral and charged pions.2 In
Fig. 8 we show the double ratio for a carbon target in dependence of W for a fixed Q2 =
0.4 GeV2. The dependence on Q2 is weak and results for other values of Q2 are very similar.
The solid line shows the exact result, whereas the dotted lines have been obtained in the
double averaging approximation with minimal and maximal amounts of pion absorption.
As can be seen, the results are rather insensitive to the exact amount of pion absorption.
Without charge exchange effects (and assuming similar absorption of charged and neutral
pions) the double ratio would be close to unity. As can be seen, the ANP model predicts
a double ratio smaller than 0.6 in the region W ≃ 1.2 GeV. A confirmation of this
expectation would be a clear signal of pion charge exchange predominantly governed by
isospin symmetry. In this case it would be interesting to go a step further and to study
similar ratios for pion angular distributions.
4. Summary
Lepton induced reactions on medium and heavy nuclei include the rescattering of produced
pions inside the nuclei. This is especially noticeable in the ∆-resonance region, where the
produced resonance decays into a nucleon and a pion. In the introduction and section
We are grateful to S. Manly for drawing our attention to the double ratio.
– 9 –
2 we reviewed the progress that has been made in the calculations of neutrino-induced
reactions on free protons and neutrons, because we needed them for following calculations.
For several resonances the vector form factors have been recently determined by using
electroproduction results in Jefferson Laboratory [7]. For the axial form factors modified
dipoles give an accurate description of the data. For the purposes of this article (studies of
nuclear corrections) it suffices to deduce the electroproduction cross sections through Eqs.
(2.1) and (2.9).
The main contribution of this article is contained in section 3, where we describe
important features of the ANP model and define single- and double averaged transport
matrices. Two important aspects of rescattering are emphasized: (i) the absorption of the
pions and (ii) charge exchange occurring in the multiple scattering, where we have shown
that special features of the data are attributed to each of them. Finally we propose specific
ratios of electroproduction reactions that are sensitive to the absorption cross section and
to charge exchange effects.
Using the model we calculate the transport matrix for various absorption cross sec-
tions and nuclei and present the results in appendix A. We also calculated the pion energy
spectra with and without nuclear corrections. The results appear in figures 1–3 and can be
compared with other calculations [12]. Comparison of the double averaged approximation
with the exact ANP calculation shows small differences (figure 4). As mentioned already,
electroproduction data are very useful in testing several aspects of the model and its pre-
dictions. For the absorption cross section we propose in Eq. (3.8) a ratio that depends only
on Ap(Q
2,W ) = g(Q2,W )f(1,W ). Since we consider isoscalar targets and sum over the
charges of the pions, charge exchange terms are eliminated. This leaves over the depen-
dence on charge independent effects, like the Pauli factor and the average absorption; this is
indeed the average absorption of pions and even includes the absorption of the ∆-resonance
itself.
Another ratio (DR(Q2,W )) is sensitive to charge exchange effects. In the double ratio
the dependence on Ap(Q
2,W ) drops out and the surviving terms are isospin dependent.
Our calculation shows that the ratio depends on W with the largest reduction occurring in
the region 1.1 < W < 1.25 GeV. Finally, the ∆(1232) is a sharply peaked resonance, where
the resonant interaction, takes place over small ranges of the kinematic variables, so that
averaging over them gives accurate approximations. This is analogous to a narrow width
approximation. Several comparisons in this article confirm the expectation that averaged
quantities give rather accurate approximations of more extensive calculations.
Acknowledgments
We wish to thank W. Brooks and S. Manly for many useful discussions, their interest and
encouragement. The work of J. Y. Yu is supported by the Deutsche Forschungsgemeinschaft
(DFG) through Grant No. YU 118/1-1.
– 10 –
Appendix
A. Charge exchange matrices in the double averaging approximation
Carbon:
12) = Ap
0.826 0.136 0.038
0.136 0.728 0.136
0.038 0.136 0.826
 (A.1)
with Ap = 0.791 .
Argon:
M(18Ar
40) = Ap
0.733 0.187 0.080
0.187 0.626 0.187
0.080 0.187 0.733
 (A.2)
with Ap = 0.657 .
Iron:
M (26Fe
56) = Ap
0.720 0.194 0.086
0.194 0.613 0.194
0.086 0.194 0.720
 (A.3)
with Ap = 0.631 .
B. Charge exchange matrices for various amounts of pion absorption
Carbon:
15% absorption
12) = Ap
0.817 0.141 0.041
0.141 0.718 0.141
0.041 0.141 0.817
 (B.1)
with Ap = 0.831 .
20% absorption
12) = Ap
0.829 0.134 0.037
0.134 0.731 0.134
0.037 0.134 0.829
 (B.2)
with Ap = 0.782 .
– 11 –
25% absorption
12) = Ap
0.840 0.127 0.032
0.127 0.745 0.127
0.032 0.127 0.840
 (B.3)
with Ap = 0.734 .
Oxygen:
15% absorption
16) = Ap
0.771 0.167 0.062
0.167 0.665 0.167
0.062 0.167 0.771
 (B.4)
with Ap = 0.833 .
20% absorption
16) = Ap
0.783 0.161 0.056
0.161 0.679 0.161
0.056 0.161 0.783
 (B.5)
with Ap = 0.784 .
25% absorption
16) = Ap
0.797 0.153 0.050
0.153 0.693 0.153
0.050 0.153 0.797
 (B.6)
with Ap = 0.735 .
30% absorption
16) = Ap
0.810 0.146 0.044
0.146 0.709 0.146
0.044 0.146 0.810
 (B.7)
with Ap = 0.687 .
C. Forward- and backward charge exchange matrices
Carbon:
15% absorption
M+(6C
12) = Ap+
0.870 0.100 0.029
0.100 0.799 0.100
0.029 0.100 0.870
 ,M−(6C12) = Ap−
0.675 0.251 0.074
0.251 0.498 0.251
0.074 0.251 0.675
 (C.1)
– 12 –
with Ap+ = 0.606 and Ap− = 0.225.
20% absorption
M+(6C
12) = Ap+
0.880 0.094 0.026
0.094 0.811 0.094
0.026 0.094 0.880
 ,M−(6C12) = Ap−
0.685 0.247 0.068
0.247 0.505 0.247
0.068 0.247 0.685
 (C.2)
with Ap+ = 0.578 and Ap− = 0.204.
25% absorption
M+(6C
12) = Ap+
0.889 0.088 0.022
0.088 0.823 0.088
0.022 0.088 0.889
 ,M−(6C12) = Ap−
0.695 0.243 0.062
0.243 0.513 0.243
0.062 0.243 0.695
 (C.3)
with Ap+ = 0.549 and Ap− = 0.184.
Oxygen:
15% absorption
M+(8O
16) = Ap+
0.829 0.125 0.046
0.125 0.750 0.125
0.046 0.125 0.829
 ,M−(8O16) = Ap−
0.635 0.265 0.100
0.265 0.470 0.265
0.100 0.265 0.635
 (C.4)
with Ap+ = 0.581 and Ap− = 0.252.
20% absorption
M+(8O
16) = Ap+
0.840 0.119 0.041
0.119 0.762 0.119
0.041 0.119 0.840
 ,M−(8O16) = Ap−
0.646 0.262 0.092
0.262 0.477 0.262
0.092 0.262 0.646
 (C.5)
with Ap+ = 0.554 and Ap− = 0.23.
25% absorption
M+(8O
16) = Ap+
0.852 0.112 0.036
0.112 0.776 0.112
0.036 0.112 0.852
 ,M−(8O16) = Ap−
0.657 0.258 0.085
0.258 0.485 0.257
0.085 0.258 0.657
 (C.6)
with Ap+ = 0.527 and Ap− = 0.208.
– 13 –
30% absorption
M+(8O
16) = Ap+
0.863 0.105 0.031
0.105 0.789 0.105
0.031 0.105 0.863
 ,M−(8O16) = Ap−
0.669 0.253 0.078
0.253 0.493 0.253
0.078 0.253 0.669
 (C.7)
with Ap+ = 0.499 and Ap− = 0.187.
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– 15 –
π+ ig
Eπ (GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
π0 ig
Eπ (GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
π- ig
Eπ (GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Figure 1: Differential cross section per nucleon for single pion spectra of π+, π0, π− for oxygen with
= 1 GeV in dependence of pion energy E
. The curves correspond to neutral current reactions.
– 16 –
π+ ig
Eπ (GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
π0 ig
Eπ (GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
π- ig
Eπ (GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Figure 2: The same as in fig. 1 for argon.
– 17 –
π+ ig
Eπ(GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
π0 ig
Eπ(GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
π- ig
Eπ(GeV)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Figure 3: The same as in fig. 1 for iron.
– 18 –
Q2 = 0.4 GeV2
π0 f (exact)
π0 f (av. app.)
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Q2 = 0.4 GeV2
π+ f (exact)
π+ f (av. app.)
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Q2 = 0.8 GeV2
π0 f (exact)
π0 f (av. app.)
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Q2 = 0.8 GeV2
π+ f (exact)
π+ f (av. app.)
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Figure 4: Double differential cross sections for single-pion electroproduction for an oxygen target in
dependence of W . Spectra for π0 and π+ production are shown for Q2 = 0.4 GeV2 and Q2 = 0.8 GeV2
using an electron energy E
= 2.7 GeV. The solid and dotted lines have been obtained according to
(3.2) using the exact ANP matrix M(W,Q2) and (3.6) utilizing the double-averaged ANP matrix M in
(3.7), respectively. The dashed lines show the free nucleon cross section (3.3).
Q2 = 0.4 GeV2 π0 f (8O
π0 f (6C
π0 f (18Ar
π0 f (26Fe
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Q2 = 0.4 GeV2 π+ f (8O
π+ f (6C
π+ f (18Ar
π+ f (26Fe
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Figure 5: Double differential cross sections per nucleon for single-pion electroproduction for different
target materials. W -spectra for π0 and π+ production are shown for Q2 = 0.4 GeV2 using an electron
energy E
= 2.7 GeV. The pion rescattering corrections have been calculated in the double-averaging
approximation (3.6) using the ANP matrices in (3.7) and App. A. For comparison, the free nucleon
cross section (3.3) is shown.
– 19 –
Q2 = 0.8 GeV2
π0 f (Abs. 25%)
π0 f (Abs. 20%)
π0 f (Abs. 30%)
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Q2 = 0.8 GeV2
π+ f (Abs. 25%)
π+ f (Abs. 20%)
π+ f (Abs. 30%)
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Figure 6: Double differential cross sections per nucleon for single-pion electroproduction for oxygen with
20% (dashed line), 25% (solid line) and 30% (dotted line) pion absorption. Furthermore, Q2 = 0.8 GeV2
and E
= 2.7 GeV. The π0 and π+ spectra have been calculated in the double-averaging approximation
(3.6) utilizing the matrices in App. B. For comparison, the free nucleon cross section (3.3) is shown as
well.
exact
Av. Approx.
N * sigabs(W)
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45
Figure 7: The fraction of absorbed pions, Abs(Q2,W), in dependence of W for oxygen and iron
targets for Q2 = 0.3 GeV2. Also shown is the cross section σabs(W ) (model B) multiplied by free
normalization factors (dashed lines). The dotted lines are the result for Abs(Q2,W) in the averaging
approximation.
– 20 –
Q2 = 0.4 GeV2
W(GeV)
1.1 1.15 1.2 1.25 1.3 1.35 1.4
Figure 8: Double ratio of single pion electroproduction cross sections in dependence of W for fixed
Q2 = 0.4 GeV2 as defined in Eq. (3.10). The dotted lines show results in the double averaging
approximation with varying amounts of absorption.
– 21 –
