Reduced cross sections of electron and neutrino charged current quasielastic scattering on nuclei

The semi-exclusive averaged reduced cross sections for (anti)neutrino charged current quasi-elastic scattering on carbon, oxygen, and argon are analyzed within the relativistic distorted wave impulse approximation. We found that these cross sections as functions of missing nucleon energy are similar to those of electron scattering and are in agreement with electron scattering data for three nuclei. The difference between the electron and neutrino cross sections can be attributed to Coulomb distortion on the electron wave function. The averaged reduced cross sections depend slowly upon incoming lepton energy. The approach presented in this paper provide novel constraints on nuclear models of quasi-elastic neutrino-nucleus scattering and can be easily applied to test spectral functions and final state interactions, employed in neutrino event generators.


I. INTRODUCTION
For current [1,2] and future [3][4][5] accelerator-based neutrino experiments the primary physics goals are measuring the lepton CP violation phase, determing neutrino mass ordering and testing the three flavor paradigm.In these experiments to evaluate the oscillation parameters, the probabilities of neutrino oscillations as functions of neutrino energy are measured.The neutrino beams are not monoenergetic and have broad distributions that range from tens of MeVs to a few GeVs.The accuracy to which neutrino oscillation parameters can be extracted depends on the ability of experiments to determine the individual energy of detected neutrino.
Measurements at neutrino energy 1 GeV are critical for the T2K [2] and HK [4] programs, which are carbon and water (oxygen) detectors as well as for the SBN (argon) [5] program.Measurements from 1 to 2 GeV are important for the NOvA (carbon, chlorine) [1] experiment, and measurements spanning from 1 to 10 GeV critical for the DUNE (argon) [3] program.At the GeV-scale neutrino energies the neutrino can interact with a nucleus through a wide range of reaction channels.These include the charged-current (CC) quasielastic (QE) scattering, two-body meson exchange current (MEC) channels, resonance production and deep inelastic scattering.
The incident neutrino energy is reconstructed using kinematic or calorimetric methods.
At energy about 1 GeV, where the CCQE scattering is dominant, the incoming neutrino energy can be derived from lepton kinematics alone.The calorimetric method relies not only on the visible energy measured in the detector, but also on the models of the neutrinonucleus interactions that are implemented in neutrino event generators.In addition the neutrino-nucleus scattering model is critical for obtain background estimates, and for correct extrapolations of the near detector constraints to the far detector in analyses aimed at determing the neutrino oscillation parameters.
Unfortunately, due to wide range of neutrino energy beams and poor statistics available from current experiments, it is very difficult to measure differential neutrino-nucleus cross sections for specific energies and to test beam energy reconstruction techniques.On the theoretical side, many studies have been presented aiming at improving our knowledge on lepton-nucleus interaction .However, it is extremely challenging to provide reliable and consistent predictions for the diversity of processes that can take place in the energy range covered by the neutrino beams.Various contributions to the cross sections can significantly overlap with each other making it difficult to identify, diagnose and remedy shortcoming of nuclear models.
While electron and neutrino interactions are different at the primary vertex, many underlying physics process in the nucleus are the same, and electron scattering data collected with precisely controlled kinematics (initial and final energies and scattering angles) and large statistic allows validation and improvement of the description of nuclear effects.There are a large body of electron-scattering data on carbon and calcium and only a few data sets available for scattering on argon.
All of the above reaction mechanisms are very similar for electrons and for neutrinos.
From the nuclear point of view the influence of nuclear medium effects such as the nuclear ground state and interaction of the outgoing nucleon with the residual nucleus can be expected to be largely the same for electron as for neutrino-induced processes.We can exploit this similarity and use electron scattering data with known beam energies to test the neutrino energy reconstruction methods [28] and interaction models.The vector part of the electroweak interaction can be inferred directly from the electron scattering data.Because electron and neutrino scattering are strongly linked in theory, any model of neutrino interactions (vector+axial) should also be able to reproduce electron (vector) interactions.A model unable to reproduce electron measurements cannot be expected to provide accurate prediction for neutrino cross sections.
It is therefore unsurprising that recent years have seen a plethora of analyses of electronscattering data to test the vector current part of the lepton-nucleus interaction against existing inclusive electron scattering cross sections for different target nuclei at several incident beam energies and scattering electron angles.The relativistic distorted wave impulse approximation (RDWIA), initially designed for description of exclusive (e, e ′ p) data [29,31,36] and then adopted for neutrino reactions was successfully tested against of inclusive (e, e ′ ) data [23,24].The SuSAv2 model exploits the similarities between both interaction types to guide the description of weak scattering process [16,17].The utility of validating neutrino events generators against inclusive electron scattering data that they had not been tuned to was demonstrated in Refs.[32][33][34][35].
Such inclusive reactions involve total hadronic cross sections and typically are relatively insensitive to the details of the final nuclear states.Rather simple models may yield cross sections that are not very different from those found in the most sophisticated models.Typically, the inclusive predictions using different models are rather similar and agree to about 10-20%, but they cannot make predictions on both leptons and hadrons in final states.The semi-exclusive (l, l ′ p) lepton scattering process involves not the total cross sections, but the specific asymptotic states and allows to test more in detail the nuclear model.Microscopic and unfactorized models like the RDWIA can be used to model both lepton-boson and boson-nucleus vertexes in the same detail and compare the results to semi-exclusive observables.The comparison of the results of the RDWIA approach and cascade models employed in the neutrino event generators provides constraints on cascade models from proton-nucleus scattering [36].
The reduced cross section, obtained from the measured differential semi-exclusive electron scattering cross section dividing on the kinematic factor and the off-shell electron-proton cross section, can be identified with the distorted spectral function.Final state interactions between the ejected nucleon and the residual nucleus make the reduced cross sections depend upon the initial and ejectile nucleon's momenta and angle between them (depends upon momentum transfer).Thus, irrespective of the type of interaction (electromagnetic or weak) the distorted spectral function is determined mainly by the intrinsic properties of the target and the ejected nucleon interaction with residual nucleus.
The purpose of the present work is calculation of the CCQE neutrino scattering reduced cross sections averaged over phase space as functions of the missing nucleon momentum and incoming neutrino energy, and comparison of them with ones obtained from measurements of (e, e ′ p) scattering on carbon, oxygen and argon targets.The direct comparison of the spectral functions used in the factorized approach in neutrino event generators to the measured reduced cross sections of the electron-nucleus scattering can provide an additional test of the nuclear models employed in these generators.
The outline of this paper is the following.In Sec.II we introduce the formalism needed to describe the semi-exclusive lepton-nucleus CCQE scattering process.The RDWIA model is briefly introduced in Sec.III.Results of the calculations are presented in Sec.IV.Our conclusions are summarized in Sec.V.

II. FORMALISM OF QUASI-ELASTIC SCATTERING
We consider the formalism used to describe electron and neutrino quasi-elastic exclusive scattering off nuclei in the one-photon (W-boson) exchange approximation.Here l labels the incident lepton [electron or muon (anti)neutrino], and l ′ represents the scattered lepton (electron or muon), k i = (ε i , k i ) and k f = (ε f , k f ) are the initial and final lepton momenta, p A = (ε A , p A ), and p B = (ε B , p B ) are the initial and final target momenta, p x = (ε x , p x ) is the ejectile nucleon momentum, q = (ω, q) is the momentum transfer carried by the virtual photon (W-boson), and Q 2 = −q 2 = q 2 − ω 2 is the photon (W-boson) virtuality.

A. CCQE lepton-nucleus cross sections
In the laboratory frame the differential cross section for exclusive electron (σ el ) and (anti)neutrino (σ cc ) CC scattering can be written as where Ω f is the solid angle for the lepton momentum, Ω x is the solid angle for the ejectile and µν are correspondingly the electromagnetic and weak CC nuclear tensors.For exclusive reactions in which only a single discrete state or a narrow resonance of the target is excited, it is possible to integrate over the peak in missing energy and obtain a fivefold differential cross section of the form where R is a recoil factor εx is solution to equation ε m A and m B are masses of the target and recoil nucleus, respectively.Note, that missing momentum is p m = p x − q and missing energy ε m is defined by All information about the nuclear structure and effects of final-state interaction (FSI) between the ejectile nucleon and residual nucleus is contained in the electromagnetic and weak CC hadronic tensors, W where the sum is taken over undetected states.
In the exclusive reaction (1) the outgoing lepton and proton are detected and the exclusive lepton scattering cross sections (3a) and (3b) in terms of response functions can be written as where is the Mott cross section and h is +1 for positive lepton helicity and -1 for negative lepton helicity.The coupling coefficient V k and v k , the expression of which are given in Ref. [6] are kinematic factors depending on the lepton's kinematics.The response functions R i are given in terms of components of the exclusive hadronic tensors [6] and depend on the variables It is also useful define a reduced cross section where K el = Rp x ε x /(2π) 3 and K cc = Rp x ε x /(2π) 5 are phase-space factors for electron and neutrino scattering and σ lN is the corresponding elementary cross section for the lepton scattering from the moving free nucleon normalized to unit flux.The reduced cross section is an interesting quantity that can be regarded as the nucleon momentum distribution modified by FSI, i.e. as the distorted spectral function.Final-state interactions make the reduced cross sections σ red (p m , p x ) depend upon ejectile momentum p x , angle between the initial and final nucleon momentum and upon incident lepton energy.These cross sections for (anti)neutrino scattering off nuclei are similar to the electron scattering apart from small differences at low beam energy due to effects of Coulomb distortion of the incoming electron wave function as shown in Refs.[6,7,12] The factorization approximation to the knockout cross section stipulates that This factorization implies that the initial nuclear sate and FSI effects are decoupled from leptonic vertex with preserved the correlations between the final lepton and nucleon.
The reduced cross section as a function of missing momentum p m , averaged over phase volume in (ω, Ω f , φ) coordinates, where Ω x = (cos θ pq , φ), can be written as where Precise electron reduced cross sections data can be used to validate the neutrino reduced cross sections (spectral functions) that are implemented in neutrino generators.

B. Nuclear current
Obviously, the determination of the response tensor W µν requires the knowledge of the nuclear current matrix elements in Eq.( 5).We describe the lepton-nucleon scattering in the impulse approximation, assuming that the incoming lepton interacts with only one nucleon, which is subsequently emitted.The nuclear current is written as the sum of single-nucleon currents.Then, the nuclear matrix element in Eq.( 5) takes the form where Γ µ is the vertex function, t = ε B q/W is the recoil-corrected momentum transfer, W = (m A + ω) 2 − q 2 is the invariant mass, Φ and Ψ (−) are relativistic bound-state and outgoing wave functions.For electron scattering, most calculations use the CC2 electromagnetic vertex function for a free nucleon [37] Γ µ = F (el) where and F (el) M are the Dirac and Pauli nucleon form factors.Because the bound nucleons are off shell, the vertex Γ µ in Eq.( 13) should be taken for an off-shell nucleon.We employ the de Forest prescription for the off-shell vertex [37] Γµ = F (el) where q = (ε x − Ẽ, q) and the nucleon energy Ẽ = m 2 + (p x − q) 2 is placed on shell.We use the approximation of [38] on the nucleon form factors.The Coulomb gauge is assumed for the single-nucleon current.Although the experimental analysis usually employ the de Forest CC1 prescription for σ lN , consistency requires that calculation of σ red to employ the σ lN that corresponds to the current operator used in the RDWIA calculations.
The single-nucleon charged current has and the axial current vertex function Weak vector form factors F V and F M are related to corresponding electromagnetic ones for proton F (el) i,p and neutron F (el) i,n by the hypothesis of conserved vector current (CVC) The axial F A and psevdoscalar F P form factors in the dipole approximation are parameterized as where F A (0) = 1.2724, m π is the pion mass, and M A is the axial mass.We use de Forest prescription for off-shell extrapolation of Γ µ(cc) .Similar to electromagnetic current, the Coulomb gauge is applied for the vector current J V .

III. MODEL
The semi-exclusive differential and reduced cross sections for neutrino scattering were studied in Refs.[6,7,12,25,26], using the relativistic shell model approach and taking into account the FSI effects.A formalism for the A(e, e ′ N)B reaction that describes the channel coupling in the FSI of N + B system was developed in Ref. [31].
In this work the independent particle shell model (IPSM) is assumed for the nuclear structure.The model space for 12 C(l, l ′ N) consists of 1s  39 Ar nuclei.All states in these nuclei are regarded as discrete states even though their spreading widths are actually appreciable.
In the independent particle shell model the relativistic bound-state function Φ in Eq.( 12) is obtained as the self-consistent solutions of a Dirac equation, derived within a relativistic mean-field approach, from a Lagrangian containing σ, ω, and ρ mesons [39].The nucleon bound-state functions were calculated by the TIMORA code [40] with the normalization factors S(α) relative to full occupancy of the IPSM orbitals.According the RDWIA analysis of the JLab 12 C(e, e ′ p) data [41,42] S(1p 3/2 ) = 84%, S(1s 1/2 ) = 100% and average factor about ≈ 89%.We use also the following values of normalization factors of 16 O: S(1p 3/2 ) = 66%, S(1p 1/2 ) = 70%, and S(1s 1/2 ) = 100%, that were obtained in the RDWIA analysis of the JLab data [43].From the RDWIA analysis [12] of NIKHEF data [44][45][46] follows that the occupancy of the orbitals of 40 Ca and 40 Ar are approximately 87% on average.Proton and neutron binding energies and the occupancy's of the orbitals in 40 Ar are given in Table II of Ref. [12].In this work we assume that the missing strength can be attributed to the shortrange nucleon-nucleon (NN) correlations, leading to the appearance of the high-momentum and high-energy nucleon distribution in the target.
Figures 1 and 2 show the proton momentum distributions for occupied orbitals in 12 C and 16 O, calculated within the mean-field approach.The neutron momentum distributions in these nuclei are almost identical to proton ones.The total proton and neutron momentum distributions in 40 Ar are presented in Fig. 3.These distributions are normalized to the total number of protons/neutrons on the IPSM shells.
For an outgoing nucleon, the simplest chose is to use plane-wave function Ψ in Eq.( 12) that is, no interactions are between the ejected nucleon N and the residual nucleus B, i.e. to use the so-called plane-wave impulse approximation (PWIA).For a more realistic description, final state interaction effects should be taken into account.In the RDWIA the distortedwave function of the knocked out nucleon Ψ is evaluated as a solution of a Dirac equations containing a phenomenological relativistic optical potential [43].This potential consists of a real part, which describes the rescattering of the ejected nucleon and an imaginary part for the absorption of it into unobserved channels.We use the LEA program [47] for numerical calculation of the distorted-wave function with the EDAD1 parameterization [48] of the relativistic optical potential for carbon, oxygen and calcium.

IV. RESULTS AND ANALYSIS
The reduced cross sections of 12 C(e, e ′ p) reaction in the range of missing energy, that corresponds to knockout of 1s and 1p-shells protons were measured at Tokyo [49], Saclay [50,51], NIKHEF [53], SLAC [52], and JLab [41].The knockout of 1p-shell protons in 16 O(e, e ′ p) was studied at Saclay [50,54], NIKHEF [55,56], Mainz [57], and JLab [43].In these experiments, cross sections data for the lowest-lying fragments of each shell were measured as functions of p m , and normalization factors (relating how much the measured cross section data were less than predicted in IPSM) were extracted.The E12-14-012 experiment [58], performed in JLab has measured the (e, e ′ p) reduced cross sections using 40 Ar [59] and The distorted spectral function depends upon initial momentum p m , ejectile momentum p x and angle between the initial and final nucleon momenta.Thus it depends upon kinematical conditions and is different for parallel and perpendicular kinematics.Furthermore, σ red depends upon initial electron energy due to Coulomb distortion.The RDWIA approach with LEA code was successfully tested against measured 12 C(e, e ′ p) [42], 16 O(e, e ′ p) [43], and 40 Ca(e, e ′ p) [12] differential and reduced cross sections, and the normalization factors S(α) for the IPSM orbitals were derived.
In Refs.[6,7,12] electron and CCQE (anti)neutrino scattering on oxygen, carbon, cal-FIG.5: The RDWIA calculation of neutrino (solid line) and antineutrino (dashed-dotted line) averaged reduced cross sections compared with measured exclusive cross section data for the removal of nucleons from 1p and 1s + 1p shells of 12 C as functions of the missing momentum.The data are from Saclay [50] for 1p and the beam energy E beam = 500 MeV, SLAC [52] for 1s + 1p shells and E beam = 2015 MeV, and from JLab [41] for 1s + 1p shells and E beam =2455 MeV cium, and argon targets were studied.It was found that the reduced cross sections for (anti)neutrino scattering are similar to those of electron scattering, and the latter are in good agreement with electron data.The difference between the electron and (anti)neutrino reduced cross sections calculated for Saclay kinematics is less than 10%.This can be attributed to Coulomb distortion upon electron wave function which is usually described as the effective momentum approximation (EMA) [61].In the EMA, the electron Coulomb wave function is replaced by a plane wave function with effective momentum whose value is larger than the value of electron momentum at infinity, because of Coulomb attraction.
The flux is also increased in the interaction zone by focusing of electron wave.This effect is proportional to charge of the target and weakens as the beam energy increases.The small difference between neutrino and antineutrino reduced cross sections is due to the difference in the FSI of the proton and neutron with the residual nucleus.
In this section we present the results of the RDWIA calculations of the averaged reduced cross sections Eq.( 10) for (anti)neutrino scattering off carbon, oxygen, and argon as functions of the missing momentum p m and compare them with the measured (e, e ′ p) reduced cross sections.In Ref. [7] electron, neutrino, and antineutrino cross sections for the removal protons from the 1s, 1p, and 1s + 1p shells of 120 C as functions of missing momentum p m were calculated and compared with JLab data [41].For illustration, Fig. 4 shows measured the removal cross sections as compared with the LEA code calculations [7].It should be note that negative value of p m corresponds to φ = π and positive to φ = 0, where φ is the angle between the scattering (k i , k f ) and reaction (p x , p B ) planes.The data for beam energy E beam = 2.445 GeV and Q 2 = 0.64 (GeV/c) 2 were measured in the quasi-perpendicular kinematics with constant (ω, q).The electron and neutrino scattering off the nuclei are closely interrelated and one can treat both processes within the same formalism.There is an overall good agreement between the cross sections calculated in the RDWIA and data.
The averaged reduced cross sections for removal of nucleons from the 1p, and 1s + 1p FIG.7: Comparison of the RDWIA calculations for neutrino (dashed line) and antineutrino (dashed-dotted line) averaged reduced cross sections for the removal of nucleons from 1p shell of 16 O with Saclay [50] and NIKHEF [55] data as functions of p m .Also shown are the RDWIA calculations of the reduced cross section for electron scattering (solid line) from Ref. [7].
shells of 12 C(ν µ , µp) and 12 C(ν µ , µn) reactions are shown in Fig. 5 as functions of positive p m values together with Saclay [51], SLAC [42], and JLab [41] data.The data for beam energies Saclay [50] and NIKHEF [55] data.There is an overall agreement between calculated cross sections and data, but the values of the calculated cross sections at maximum is systematically higher (about 15-20%) than measured ones for NIKHEF kinematics.Unfortunately, there are no data for removal protons from the 1s and 1s + 1p shells of 16 O.Therefore the models of lepton-nucleus interaction that don't take into account the shell structure of nucleus can not be tested against the available reduced cross sections measured in 16 O(e, e ′ p) reaction.The RDWIA averaged reduced cross section for removal nucleon from 1s+1p shells in 16 O(ν µ , µp) is shown in Fig. 8 as a function of incoming neutrino energy and missing momentum p m .As can be seen from figures 6 and 8 the dependence of these cross sections and antineutrino (dashed-dotted line) reduced cross sections for the removal of nucleons from 1d 3/2 , 2s 1/2 , and 1d 5/2 shells of 40 Ca with NIKHEF data [46].The cross sections are presented as functions of missing momentum p m .The figure taken from Ref. [12].
upon neutrino energy and p m are almost similar.
The structure of calcium and argon nuclei is similar, although unlike 40  18 Ar, 40  20 Ca is a symmetric and closed-shell nucleus.In Ref. [12] the (e, e ′ p) reduced cross sections for removal of the proton from 1d 3/2 shell, for transition to the 1/2 + exited state of the 39 K nucleus at excitation energy E x = 2.522 MeV, and for the transitions to the 5/2 + excited states at E x =5.258 MeV and E x =6.328 MeV, obtained by knocking out protons from the 2s 1/2 and 1d 5/2 orbitals, correspondingly were calculated.The calculated reduced cross sections are shown in Fig. 9 with the NIKHEF data [44,45] and provide a good description of the shape and magnitude of the measured distribution.Neutrino and antineutrino calculated reduced cross sections of 40 Ca(ν, µ − p) 39 Ca and 40 Ca(ν, µ + n) 39 K reactions also shown are in Fig. 9.There is an overall good agreement between calculated cross sections, but the values of the electron cross sections at the maximum is systematically higher than those for (anti)neutrino.This can be attributed to Coulomb distortion upon the incident electron wave function.
The JLab experiment [58] has measured the (e, e ′ p) cross sections using argon and titanium targets [59,60].The reduced cross sections were obtained in the missing momentum 15 ≤ p m ≤ 300 MeV/c and missing energy range 12 ≤ p m ≤ 80 MeV.The procedure to obtained information on neutron distribution in argon is based on the observation that neutron spectrum of 40  18 Ar is mirrored by proton spectrum of the nucleus of titanium, having charge Z = 22.Therefor one can expect that the proton spectral function obtained from Ti(e, e ′ p) data provides information on the neutron spectral function of argon.
The 40  18 Ar and 48  22 Ti data were analyzed to obtain the spectral functions, describing the energy and momentum distributions of protons in the argon and titanium ground states.The effect of FSI, which are known to be significant in (e, e ′ p) reactions, was taken into account within the distorted-wave impulse approximation approach.Figure 10 [12].There is an overall agreement between the RPWIA calculations and data within sizable uncertainties of the measured proton momentum distributions.A more accurate determination of the distorted spectral functions for different shells of 40 Ar and 48 Ti will improve the testing of models using for description of neutrino interaction with these nucleus.
The averaged reduced cross section of 40 Ar(ν µ , µp) reaction calculated in the RDWIA approach is shown in Fig. 12 as a function of neutrino energy and missing momentum p m .It has to be pointed out that unlike 12 C and 16 O, in 40 Ar the maximum of these cross sections is shifted to the range of lower missing momentum p m ≈ 15 MeV/c.The cross section increases very slowly with neutrino energy.
Neutrino event generators employ the factorization approach to make predictions about the lepton and also the outgoing nucleon kinematics from inclusive models.These models are aimed to describe inclusive cross section that is only as function of the final lepton kinematics.This factorization uses the spectral functions, which are generated from different nucleon distributions in the initial nuclear state (local Fermi gas, shell model, etc).While the behavior of the cross section against the lepton kinematics may be described correctly, there is no guarantee that the correlations between the final lepton and nucleon for a given event are preserved.The comparison of the employed spectral function with the measured reduced cross sections allows the estimation of the accuracy of the nuclear effects calculations.On the other hand, the effective spectral functions can be obtained within the microscopic and µν , which are given by the bilinear products of the transition matrix elements of the nuclear electromagnetic or CC operator J (el)(cc) µ between the initial nucleus state |A and the final state |B f as

FIG. 1 :FIG. 2 :FIG. 3 :
FIG.1: Proton momentum distributions for the different single particle states in12 C nucleus.Also shown is the total proton momentum distribution (solid line).

FIG. 4 :
FIG.4: Comparison of the RDWIA calculations for electron, neutrino and antineutrino reduced cross sections for the removal of nucleons from 1s and 1p shells of12 C as functions of the missing momentum.JLab data[41] for beam energy E beam =2.455 GeV, proton kinetic energy T p =350 MeV, and Q 2 =0.64 (GeV/c) 2 .The RDWIA calculations are shown for electron scattering (dashed line) and neutrino (solid line) and antineutrino (dashed-dotted line) scattering.This figure taken from Ref.[7].

FIG. 6 :
FIG. 6:The RDWIA neutrino averaged reduced cross section for removal of nucleons from the 1s + 1p shells of12 C as a function of neutrino energy and missing momentum p m .

FIG. 10 :
FIG. 10: Missing momentum distribution in argon obtained by integrating over the missing energy range of 0 -30 MeV (left panel) and 30 -54 MeV (right panel), presented with the geometrical factor of 4πp 2 m .The gray band shows the measured spectral function including the full error.

FIG. 11 :
FIG. 11: Missing momentum distribution in titanium obtained by integrating over the missing energy range of 0 -30 MeV, presented with the geometrical factor of 4πp 2 m .The gray band shows the measured spectral function including the full error.
shows the missing momentum distributions of protons in argon obtained by integrating the data over the missing energy ranges 0 − 30 MeV and 30 − 50 MeV.The proton missing momentum distribution in titanium was obtained by integrating the data over the missing energy range 0 − 30 MeV is shown in Fig. 11.Also shown in Figs. 10 and 11 are the results obtained without FSI effects in the relativistic plane wave impulse approximation (RPWIA), with normalization factors S α from Ref.