Removal energies and final state interaction in lepton nucleus scattering

We investigate the binding energy parameters that should be used in modeling electron and neutrino scattering from nucleons bound in a nucleus within the framework of the impulse approximation. We discuss the relation between binding energy, missing energy, removal energy (ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document}), spectral functions and shell model energy levels and extract updated removal energy parameters from ee′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {ee}^{\prime }$$\end{document}p spectral function data. We address the difference in parameters for scattering from bound protons and neutrons. We also use inclusive e-A data to extract an empirical parameter UFSI((q3+k)2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{FSI}( (\varvec{q}_3+\varvec{k})^2)$$\end{document} to account for the interaction of final state nucleons (FSI) with the optical potential of the nucleus. Similarly we use Veff\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{eff}$$\end{document} to account for the Coulomb potential of the nucleus. With three parameters ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document}, UFSI((q3+k)2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{FSI}( (\varvec{q}_3+\varvec{k})^2)$$\end{document} and Veff\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{eff}$$\end{document} we can describe the energy of final state electrons for all available electron QE scattering data. The use of the updated parameters in neutrino Monte Carlo generators reduces the systematic uncertainty in the combined removal energy (with FSI corrections) from ± 20 to ± 5 MeV.

The RFG model does not describe the tails in the energy distribution of the final state lepton very well [17][18][19][20]. Improvements to the RFG model such as a better momentum distribution are usually made within the existing Monte Carlo a e-mail: bodek@pas.rochester.edu (MC) frameworks. All RFG-like models with one dimensional nucleon momentum distributions require in addition removal energy parameters ( P,N ) to account for the average removal energy of a proton or neutron from the nucleus. These parameters should be approximately the same for all one-dimensional momentum distributions.
Alternatively two dimensional spectral functions (as a function of nucleon momentum and missing energy) can be used. However, even in this case, MC generators currently used in neutrino oscillations experiments do not account for the final state interaction (FSI) of the final state lepton and nucleon in the optical and Coulomb potentials of the nucleus.
In this paper we extract empirical average removal energy parameters from spectral function measured in exclusive ee p electron scattering experiments on several nuclei. We use V e f f (see Appendix A) to account for the Coulomb potential of the nucleus, and extract empirical nucleon final state interaction parameter U F SI ((q 3 +k) 2 ) from all available inclusive e-A electron scattering data. With these three parameters , U F SI ((q 3 + k) 2 ) and V e f f we can describe the energy of final state electrons for all available electron QE scattering data. These parameters can be used to improve the predictions of current neutrino MC event generators such as genie and neut for the final state muon and nucleon energies in QE events.
A large amount of computer time has been used by various experiments to generate and reconstruct simulated neutrino interactions using MC generators such as genie 2. We show how approximate post-facto corrections could be applied to these existing MC samples to improve the modeling of the reconstructed muon, final state proton, and unobserved energy in quasielastic (QE) events.

Relevance to neutrino oscillations experiments
In a two neutrinos oscillations framework the oscillation parameters which are extracted from long baseline experi-ments are the mixing angle ϑ and the square of the difference in mass between the two neutrino mass eigenstates Δm 2 . A correct modeling of the reconstructed neutrino energy is very important in the measurement of Δm 2 . In general, the resolution in the measurement of energy in neutrino experiments is much worse than the resolution in electron scattering experiments. However, a precise determination of Δm 2 is possible if the MC prediction for average value of the experimentally reconstructed neutrino energy is unbiased. At present the uncertainty in the value of the removal energy parameters is a the largest source of systematic error in the extraction of the neutrino oscillation parameter Δm 2 (as shown below).
The location of the first oscillation maximum in neutrino energy (E 1st−min ν ) is when the term in brackets is equal to π/2. An estimate of the extracted value of Δm 2 is given by: For example, for the t2k experiment L = 295 K m, and E ν is peaked around 0.6 GeV. For the normal hierarchy the t2k experiment [21] reports a value of Using Eqs. 2 and 40 we estimate that a +20 MeV change in the removal energy used in the MC results in a change in Δm 2 32 of +0.03 × 10 −3 eV 2 , which is the largest contribution to the total systematic error in Δm 2 32 . The above estimate is consistent with the estimate of the t2k collaboration. The t2k collaboration reports [22,23] that "for the statistics of the 2018 data set, a shift of 20 MeV in the binding energy parameter introduces a bias of 20% for sin 2 θ 23 and 40% for Δm 2 32 with respect to the size of the systematics errors, assuming maximal sin 2 θ 23 ".
For the case of normal hierarchy a combined analysis [25] of the world's neutrino oscillations data in 2018 finds a best fit of which illustrates the importance of using a common definition of removal energy parameters and the importance in handling the correlations in the uncertainties between various experiments when performing a combined analysis.
For comparison, we find that a change of +20 MeV/c in the assumed value of the Fermi momentum k F yields a much smaller change of +0.005 × 10 −3 eV 2 in the extracted value of Δm 2 32 .

Neutrino near detectors
In general, neutrino oscillations experiments use data taken from a near detector to reduce the systematic error from uncertainties in the neutrino flux and in the modeling of neutrino interactions. However, near detector data cannot constrain the absolute energy scale of final state muons and protons, or account for the energy that goes into the undetected nuclear final state. These issues are addressed in this paper.

Simulation of QE events and reconstruction of neutrino energy
In order to simulate the reconstruction of neutrino QE events within the framework of the impulse approximation the experimental empirical parameters that are used should describe: 1. The momentum of the final state muon including the effect of Coulomb corrections [24]. 2. The mass, excitation energy, and recoil energy of the spectator nuclear state. 3. The effect of the interaction of the final state nucleon (FSI) with the optical and Coulomb potential of the spectator nucleus. Figure 1 shows a few models for the nucleon momentum distributions in the 12 6 C nucleus. The solid green line (labeled Global Fermi gas) is the nucleon momentum distribution for the Fermi gas [17] model which is currently implemented in all neutrino event generators and is related to global average density of nucleons. The solid black line is the projected momentum distribution of the Benhar-Fantoni [8][9][10][11] 2D spectral function as implemented in nuwro. The solid red line is the nucleon momentum distribution for the Local-Thomas-Fermi (LTF) gas which is is related to the local density of nucleons in the nucleus and is implemented in neut, nuwro and GiBUU.

Nucleon momentum distributions
A more sophisticated formalism is the ψ superscaling model [26,27], which is only valid for QE scattering. It can be used to predict the kinematic distribution of the final state muon but does not describe the details of the hadronic final state. Therefore, it has not been implemented in neutrino MC generators. However, the predictions of the ψ superscaling model can be approximated with an effective spectral function [28] which has been implemented in genie. The  6 C nucleus. The green curve (Global Fermi) is the momentum distribution for the relativistic Fermi gas (RFG) model. The red curve is the Local-Thomas-Fermi (LTM) gas distribution. The black curve is the projected momentum distribution of the Benhar-Fantoni two dimensional spectral function. The blue line is the momentum distribution for the effective spectral function model, which approximates the ψ superscaling prediction for the final state muon in quasielastic scattering momentum distribution of the effective spectral function for nucleons bound in 12 6 C is shown as the blue curve in Fig. 1. Although the nucleon momentum distributions are very different for the various models, the predictions for the normalized quasielastic neutrino cross section 1 σ dσ dν (Q 2 , ν) are similar as shown in Fig. 2. These predictions as a function of ν = E ν − E μ are calculated for 10 GeV neutrinos on 12 6 C at Q 2 = 0.5 GeV 2 . The prediction with the local Fermi gas distribution are similar to the prediction of the Benhar-Fantoni two dimensional spectral function as implemented in nuwro. Note that the prediction of the ψ superscaling model are based on fits to longitudinal QE differential cross sections. Subsequently, they includes 1p1h and some 2p2h processes (discussed in Sect. 2).
The following nuclear targets are (or were) used in neutrino experiments: Carbon (scintillator) used in the nova and minerνa experiments. Oxygen (water) used in t2k and in minerνa. Argon used in the argoneut and dune experiments. Calcium (marble) used in charm. Iron used in minerνa, minos, cdhs, nutev, and ccfr. Lead used in chorus and minerνa.  σ dσ dν (Q 2 , ν) at Q 2 = 0.5 GeV 2 for 10 GeV neutrinos on 12 6 C to the predictions with several momentum distribution (ν = E 0 − E ). Here the solid green curve labeled "Global Fermi" gas is the distribution for the Fermi gas model. The red line is the prediction for the local Thomas Fermi (LTF) gas, and the purple dots are the prediction using the two dimensional Benhar-Fantoni spectral function as implemented in nuwro nucleons in the nucleus. The on-shell recoil excited [A − 1] * spectator nucleus has a momentum p (A−1) * = −k and a mean excitation energy E P x . The off-shell energy of the interacting nucleon is We model the effect of FSI (strong and EM interactions) by setting E f = 2 ). For electron QE scattering on bound protons |V P e f f | = Z Z −1 |V e f f |, As discussed in Sect. 4 we model the effect of FSI (strong and EM interactions) by setting Table 1 shows the spin and parity of the initial state nucleus, and the spin parity of the ground state of the spectator nucleus when a bound proton or a bound neutron is removed via the 1p1h process.
The four-momentum transfer to the nuclear target is defined as q = (q 3 , ν). Here q 2 3 is the 3-momentum transfer, ν is the energy transfer, and Q 2 = −q 2 = q 2 − ν 2 is the square of the four-momentum transfer. For QE electron scattering on unbound protons (or neutrons) the energy transfer ν is equal to Q 2 /2M P,N where M P is mass of the proton and M N is the mass of the neutron, respectively.

Nuclear density corrections to k P F and k N F
The values of the Fermi momentum k F that are currently used in neutrino Monte Carlo generators are usually taken from an analysis of e-A data by Moniz et al. [17][18][19]. The Moniz published values of k F were extracted using the RFG model under the assumption that the Fermi momenta for protons and neutrons are different and are related to k F via the relations What is actually measured is k P F , and what is published is k F . Moniz assumes that the nuclear density (nucleons per unit volume) is constant. Therefore, in the same nuclear radius R, k N F for neutrons is larger if N is greater than Z. Moniz used these expressions to extract the published value of k F from the measured value of k P F . We undo this correction and re-extract the measured values of k P F for nuclei which have a different number of neutrons and protons. In order to obtain the values of k N F from the measured values of k P F we use the fact that the Fermi momentum is proportional to the cube root of the nuclear density. Consequently k N For the proton and neutron radii, we use the fits for the half density radii of nuclei (in units of femtometer) given in Ref. [29].
We only these fits for nuclei which do not have an equal number of protons and neutrons. For nuclei which have an equal number of neutrons and protons we assume that k N F = k P F = k F (Moni z). However for the 208 82 Pb nucleus only we use k P F = 0.275 GeV which we obtain from our own fits to inclusive

Separation energy
The separation energy for a proton (S P ) or neutron S N is defined as follows: The energy to separate both a proton and neutron (S P+N ) is defined as follows: The proton and neutron separation energies S P and S N are available in nuclear data tables. The values of S P , S N and S N +P for various nuclei [30,31] are given in Table 1 2.4 Two nucleon correlations Figure 4 illustrates the 2p2h process originating from both long range and short range two nucleon correlations (src).
Here the scattering is from an off-shell bound proton of momentum p i = k. The momentum of the initial state offshell interacting nucleon is balanced by a single on-shell correlated recoil neutron which has momentum −k. The [A − 2] * spectator nucleus is left with two holes. Short range nucleon-proton correlations occur ≈ 20% of the time [32]. The off-shell energy of the interacting bound proton in a quasi-deuteron is ( where M D is the mass of the deuteron. For QE scattering there is an additional 2p2h transverse cross section from "Meson Exchange Currents" (mec) and "Isobar Excitation" (ie).
In this paper we only focus on the extraction of the average removal energy parameters for 1p1h processes. Processes leading to 2p2h final states (src, mec and ie) result in larger missing energy and should be modeled separately. Table 2 Summary of the relationships between excitation energy E P,N x (used in genie), separation energy S P,N , missing (missing) energy E P,N m (used in spectral function measurements), removal energy P,N (used in the reconstruction of neutrino energy from muon kinematics only), the Smith-Moniz removal energy P,N SM (that should be used in old-neut) and the parameters x and x ν,ν (q 3 + k) 2 ) which we use to include the effects of FSI in electron and neutrino/antineutrino scattering. For QE neutrino scattering on bound neutrons Missing energy used in spectral functions Q E−P , also used in effective spectral functions [28] (P,N ) SM We use x ν,ν to include the effects of FSI. Table 3 Average values of the proton kinetic energy T P SF and missing energy E P m SF for 1p1h final states (E P m < 80) extracted from published tests of the Koltun sum rule using spectral function (SF) measurements at Jefferson lab Hall A [37,38] and Saclay [39,40]. For a Fermi gas distribution T P = 3 5 (k P F ) 2 All energies are in MeV. The bolded numbers are the best estimates for each target

Spectral functions and ee p experiments
In ee p experiments the following process is investigated: Here, an electron beam is incident on a nuclear target of mass M A . The hadronic final state consists of a proton of four momentum p f ≡ (E f , p f ) and an undetected nuclear remnant (A − 1) . Both the final state electron and the final state proton are measured. The (A − 1) nuclear remnant can be a (A − 1, Z − 1) spectator nucleus with excitation E P x , or a nuclear remnant with additional unbound nucleons.
At high energies, within the plane wave impulse approximation (PWIA) the initial momentum k of the initial state off-shell interacting nucleon can be identified approximately with the missing momentum p m . Here we define p m = | p m | and k = |k| The missing energy E m is defined by the following relativistic energy conservation expression, The missing energy E m can be expressed in term of the excitation energy (E x ) of the spectator (A-1) nucleus and the separation energy of the proton S P (or neutron S N ).
The probability distribution of finding a nucleon with initial state momentum p m ≈ k and missing energy E m from the target nucleus is described by the spectral function, defined as P SF ( p m , E m ). Note that for spectral functions both P( p m , E m ) and S( p m , E m ) notation are used in some publications. The spectral functions P P SF ( p m , E P m ) and P N SF ( p m , E N m ) for protons and neutrons are two dimensional distributions which can be measured (or calculated theoretically). Corrections for final state interactions of the outgoing nucleon are required in the extraction of P P SF ( p m , E P m ) from ee p data. The kinematical region corresponding to low missing momentum and energy is where shell model [33] states dominate [34]. In practice, only the spectral function for protons can be measured reliably.
In addition to the 1p1h contribution in which the residual nucleus is left in the ground or excited bound state, the measured spectral function includes contributions from nucleonnucleon correlations in the initial state (2p2h) where there is one or more additional spectator nucleons. Spectral function measurements cannot differentiate between a spectator (A-1) nucleus and a spectator (A-2) nucleus from src because the 2nd final state src spectator nucleon is not detected.
Here, we focus on the spectral function for the 1p1h process, which dominates for E m less than 80 MeV, and ignore the spectral function for the 2p2h process which dominates at higher values of E m . We use shell model calculations to obtain the difference in the binding energy parameters for neutrons and protons.

Effects of the optical and Coulomb potentials (FSI)
We use empirical parameter U F SI ((q 3 + k) 2 ) to approximate the effect of the interaction of the final state proton with the optical potential of the spectator nucleus. This is important at low values of q 2 3 . In addition, we include the effect of the interaction of the final state proton with the Coulomb field of the nucleus (V P e f f ). In QE scattering of electrons a three momentum transfer q 3 to a bound proton with initial momentum k results in the following energy E P f of the final state proton: We define the average removal energy P,N in terms of the average momentum k 2 P,N of the bound nucleon as follows: In order to properly simulate neutrino interactions we extract values of the average missing energy (or equivalently the average excitation energy) from spectral functions measured in ee p experiments. We then use these values and extract U F SI ((q 3 + k) 2 ) from inclusive e-A as discussed in Sect. 9.

Smith-Moniz formalism
The Smith-Moniz [20] formalism uses on-shell description of the initial state. In the on-shell formalism, the energy conserving expression is Table 4 Results of a DPWA analysis of the "level missing energies" for different shell-model levels done by the Saclay [39,40] and Tokyo [41][42][43] ee p experiments on 12 6 C, 28 14 Si and 58 28 Ni.  Table 5 Results of a DPWA analysis of the "level missing energies" for different shell-model levels done by the Saclay [39,40] and Tokyo [41][42][43] ee p experiments on on 6 3 Li, 27 13 Al, 40 20 [44,45] NIKHEF high resolution spectral function for protons in the 1p level of 12 6 C as a function of the spectator nucleus excitation energy E P x for p m = k = 172 MeV/c. Bottom panel: The Jlab measurement [37,38] of the one-dimensional spectral function for 12 6 C as a function of missing energy E P m for Q 2 = 0.64 GeV 2 where A summary of the relationships between excitation energy E P,N x used in genie (which incorporates the Bodek-Ritchie [35] off-shell formalism), separation energy S P,N , missing  Table 2.

Extraction of average missing energy E m
We extract the average missing energy E P m and excitation energy E P x for the 1p1h process from ee P electron scattering data using two methods.
1. E P m SF : From direct measurements of the average missing energy E P m and average proton kinetic energy T P . These quantities have been extracted from spectral functions measured in ee p experiments for tests of the Koltun sum rule [36]. The contribution of two nucleon corrections is minimized by restricting the analysis to E P m < 80 MeV. This is the most reliable determination of E P m . We refer to this average as E P m SF .
2. E P m levels : By taking the average (weighted by shell model number of nucleons) of the nucleon "level missing energies" of all shell model levels which are extracted from spectral functions measured in ee p experiments. We refer to this average as E P m levels .
There could be bias in method 2 originating from the fact that a fraction of the nucleons (≈ 20%) in each level are in a correlated state with other nucleons (leading to 2p2h final states). The fraction of correlated nucleons is not necessarily the same for all shell-model levels. As discussed in Sect. 5.4 (and shown Fig. 6)   and protons can be approximated by the difference in the weighted average of the single particle binding energies of all shell-model levels. We obtain N-P differences of 2.6 and 2. 2.9 ± 1 the nuclear shell model to estimate difference between the missing energies for neutrons and protons.

Direct measurements of E P m SF and T P SF
The best estimates of the average missing energy E P m and average nucleon kinetic energy T P are those that are directly extracted from spectral function measurements in analyses that test the Koltun sum rule [36]. The Koltun's sum rule states that where E B /A is the nuclear binding energy per particle obtained from nuclear masses and includes a (small) correction for the Coulomb energy, and For precise tests of the Koltun sum rule a small contribution from three-nucleon processes should be taken into account.
Values of E P m SF and T P SF for the 1p1h process (E P m <80 MeV) published by Jlab Hall C experiments [37,38] and by the Saclay group [39,40] are given in Table 3.

Spectral function "level missing energies"
Measured 2D spectral functions can be analyzed within the distorted plane wave approximation (DPWA) to extract the peak and width of the missing energy distribution E P m for protons for each shell model level. We refer to it as the "level missing energy". In some publications it is referred to as the "shell separation energy". The energies and widths of the "level missing energies" for 6 3 Li, 12 6 C, 17 3 Al, 40 20 Ca, 50 23 V, extracted from data published by the Tokyo group [41][42][43] are shown in Tables 4 and 5. Also shown are the "level missing energies" for 12 6 C, 28 14 Si, 40 20 Ca, and 58.7 28 Ni, extracted from the data published by the Saclay [39,40] group.
We obtain an estimate of the average missing energy E P m levels for the 1p1h process by taking the average (weighted by the number of nucleons) of the "level missing energies" of all shell model levels with E P m <80 MeV. The results of our analysis of the Saclay and Tokyo data are given in Tables 4 and 5. As shown in Tables 4 and 5 for the deeply bound 1s and 1p levels in heavy nuclei the averages and widths of the missing energy distributions are large. 5.3 12 6 C spectral function The measured [44][45][46][47] NIKHEF high resolution spectral function for the missing energy of a bound proton in the 1p level of 12 6 C as a function of the spectator nucleus excitation energy E P x for p m = k = 172 MeV/c is shown in the top panel of Fig. 5. The Jlab measurement [37,38] of the one-dimensional spectral function for the missing energy for bound proton from 12 6 C as a function of E P m for Q 2 = 0.64 GeV 2 is shown in the bottom panel of of Fig. 5. The second peak at an average value of E P m ≈ 42.6±5 MeV is for protons in the 1s level. Combining the two results (weighted by the number of nucleons in each level) we obtain E P m levels =25.7 ±2 MeV for 12 6 C. Additional details are given in Table 4.

Comparison of the two methods
Tests of the Koltun sum rule as a function of Q 2 were done by ee p experiments at Jlab Hall C [37,38] for 12 6 C, 56 26 Fe, and 197 79 Au. Tests of the Koltun sum rule were also reported by the Saclay [39,40] group for 12 6 C, 28 14 Si, 40 20 Ca, and 59 28 Ni. For both groups values of E P m SF and T P SF were extracted from the measured spectral functions. The results from both groups are summarized in Table 3. We take the RMS variation with Q 2 of the Jefferson Lab Hall C data shown in  Fig. 6. For example, for the Saclay data shown in Table 3 the average of the difference between E P m levels and E P m SF for 12 6 C, 28 14 Si, 40 20 Ca, and 59 28 Ni is 0.9±1.0 MeV.

Spectral function measurement of 16 8 O
The spectral function for 16  Using T A−1 = 1.1 MeV for 16 8 O we also obtain a average P levels = 24.1 ± 2 MeV. 5.6 Difference between neutrons and protons for 12 8  ). Since E P,N m = S P,N + E P,N x the difference in the average missing energies for neutrons and protons is approximately equal to the difference in separation energies S N − S P . By definition, the single particle binding energy of the least bound state is equal to the separation energy. The differences in the separation energies between neutrons and protons (S N − S P ) bound in 12 8 C and 16 8 O are of 2.7 MeV and 3.6 MeV, respectively.
More generally, a better estimate of the difference between the average missing energies for neutrons and protons can be obtained from the nuclear shell model. The single nucleon missing energy (E P,N m ) shell-model-level for a nucleon in a given shell-model level is close (somewhat larger) to the single nucleon binding energy for that level. Consequently, the difference in the average missing energies for neutrons and protons for a nucleus E P m − E N m is also approximately equal to the difference in the average binding energies.
The binding energies of different shell-model levels [51] for 12 8 C and 16 8 O are shown in Table 6. When available, the experimental values shown in italics are used.
The differences between the averages of the nucleon binding energies in all shell-model levels for neutrons and protons is 2.6 and 2.9 MeV for 12 8 C and 16 8 O, respectively. As expected these values are similar (within 1 MeV) to the differences in the separation energies for neutrons and protons (S N − S P ) bound in 12 8 C and 16 8 O of of 2.7 and 3.6 MeV, respectively.

Inclusive e-A electron scattering
For QE electron scattering at low (k+q 3 ) 2 we use an empirical parameter U F SI ((k + q 3 ) 2 ) to account for the effect of final state interactions. The off-shell Bodek-Ritchie formalism (used by genie) for the case of QE electron scattering from a bound pr oton, should be implemented as follows: and U F SI = U F SI ((q 3 + k) 2 ). For electron scattering from a bound pr oton |V P e f f | = Z −1 Z |V e f f |, where (Z-1) is the number of protons in the spectator final state nucleus.

Smith-Moniz on-shell formalism
For QE electron scattering on a bound proton in the Smith-Moniz on-shell formalism (used by old-neut) the following equations should be used: and U F SI = U F SI ((q 3 + k) 2 ).

Extraction of U F SI from in inclusive e-A QE data
We define k z as the component of k along the direction of the of the 3-momentum transfer q 3 .
where in the calculation of q 2 3 we have applied Coulomb corrections to the initial and final electron energies as described in Appendix A.
In the peak region of the QE distribution k z ≈ 0. Therefore, from the location of the peak in ν we extract U F SI ((q 3 + k) 2 ) peak for where we have used Eq. 33 for the Fermi gas distribution. If simplicity is needed then (q 3 + k) 2 ≈ q 2 3 is a good approximation. Fig. 8 Examples of fits for three out of four 6 3 Li (k P F = 0.169 GeV) QE differential cross sections. The solid blue curve is the RFG fit with the best value of U F SI . The black dashed curve is the simple parabolic fit used to estimate the systematic error. The red dashed curve is the RFG model with U F SI = V ef f = 0. Above each panel we show: The element, E 0 in GeV, θ in degrees, Q 2 in GeV 2 , First Author, and year of publication Fig. 9 Same as Fig. 8 for three out of 33 12 6 C (k P F = 0.221 GeV) QE differential cross sections We fit a large number of electron scattering QE differential cross sections for various nuclei and extract the values of U F SI ((q 3 + k) 2 peak ). The data samples include: four 6 3 Li spectra, 33 12 6 C spectra, five 16 8 O spectra, seven 27 18 Al spectra, 29 40 20 Ca spectra, two 40 18 Ar spectra, 30 56 26 Fe spectra, 23 208 82 Pb spectra and one 197 79 Au) spectra. Most (but not all) of the QE differential cross sections given in references ) are available on the QE electron scattering archive [53,54]. Figures 8,9,10,11,12,13,14,15 show examples of these fits to QE differential cross sections for all these elements. The solid blue curve is the RFG fit with the best value of U F SI . The black dashed curve is a simple parabolic fit used to estimate the systematic error. The red dashed curve is the RFG model with U F SI = V e f f = 0.
The extracted values of U F SI versus (q 3 +k) 2 for Lithium, Carbon+Oxygen, Aluminum, Calcium +Argon, iron, and Lead+Gold are shown in Figs. 16,17,18,19,20, and 21, respectively. Here (q 3 +k) 2 is evaluated at the peak of the QE distribution. We fit the extracted values of U F SI ((q 3 + k) 2 ) versus (q 3 + k) 2 for (q 3 + k) 2 > 0.1 GeV 2 to a linear function. The intercepts at (q 3 + k) 2 = 0 and the slopes of U F SI ((q 3 + k) 2 ) are given in Table 9 for various nuclei.
For the Relativistic Fermi Gas (rfg) the probability distribution P r f g (k z ) and the average k 2 (k z ) r f g are given in Appendix B. We compare the e-A QE cross sections versus ν to the rfg model for QE scattering. We account for the nucleon Q 2 dependent form factors and for Pauli suppression (discussed in Appendix B.2) at low q 2 3 . We only fit to the data in the top 1/3 of the QE distribution to extract the best value of U F SI for (q 3 + k) 2 at the peak. In the fit we let the normalization of the QE peak float to agree with data. For the estimate of the systematic error we also fit the QE differential cross section versus ν near the peak region to a simple parabola and extract the value of ν parabola peak . We use the difference between ν parabola peak and ν r f g peak as a systematic error in our extraction of U F SI ((q 3 + k) 2 ).

6.3
The Δ(1232) resonance shown in Figs. 8, 9, 10, 11, 12, 13,14 and 15 A simple calculation of the cross section for the production of Δ(1232) resonance is shown in Figs. 8, 9, 10, 11, 12, 13,14 and 15. The calculation uses Jlab fits to the structure functions in the resonance region for protons and neutrons. These structure functions were extracted from hydrogen and deuterium data. The proton and neutron structure functions in the resonance region were used as input to a simple Fermi Gas  Fig. 8 for three out of four 16 8 O (k P F = 0.225 GeV) QE differential cross sections Fig. 11 Same as Fig. 8 for three out of seven 27 13 Al (k P F = 0.238 GeV) QE differential cross sections  40 20 Ca (k P F = 0.251 GeV) QE differential cross sections Fig. 13 Same as Fig. 8 for two 40 18 Ar (k P F = 0.251 GeV) QE differential cross sections  Fig. 8 for three of 30 56 20 Fe (k P F = 0.254 GeV) QE differential cross sections  2 for four Lithium ( 6 3 Li) spectra smearing model. In the calculation, U F SI for the Δ(1232) resonance is assumed to be the same as U F SI for QE scattering. The curves shown Figs. 8, 9, 10, 11, 12, 13, 14 and 15 do not include the contributions of 2p2h final states from meson exchange currents (MEC) and isobar excitation. These 2p2h contributions yield additional cross section in the region between the QE peak and the Δ(1232) resonance. The 2p2h contributions are primarily transverse and therefore are more significant for electron scattering at larger angles than at small angles (as observed in the figures). The investigations of MEC (which is model dependent) and the values of U F SI for a Δ(1232) resonance in the final state are the subject of a future investigation.

Implementation for neutrino experiments
For QE scattering of neutrinos (antineutrinos) on bound neutrons (protons) the final state nucleon is a proton (neutron). The following equations should be used in neutrino/antineutrino MC generators: For neutrino QE scattering on bound neutrons: where |V P e f f | = |V e f f |. For antineutrino QE scattering on bound protons: Where Rearranging, we have where for neutrinos and antineutrinos we have:   Table 9 The intercepts (GeV) at q 2 3 = 0 and slopes (GeV/GeV 2 ) of fits to U F SI versus (q 3 + k) 2 For both neutrinos and antineutrinos P,N is the unobserved removal energy (Fig. 22).
In neutrino experiments in which both the final state lepton and final state proton (or neutron) are measured (e.g. nova, minerνa, dune) the neutrino energy can be calculated as follows:

Corrections to GENIE version 2
The generation of events in genie 2 as currently done is equivalent to using Eqs. 19 and 20, but with V e f f = 0, U F SI = 0, and E P,N x = 0. In addition, an amount Δ nucleon genie (25 MeV for 12 6 C) is subtracted from the energy of the final state nucleon (or quark for inelastic events) to account for "binding energy" in genie 2. For neutrino QE scattering on bound neutrons events are generated in genie 2 using the following equations: Where ν,ν genie is the removal energy for neutrino (antineutrino) assumed in genie . Therefore the difference between the correct muon energy and the muon energy generated by genie is approximately equal to Δx ν Where Δ nucleon genie = 25 MeV is used in genie. As given in the Tables 7 and 8 (Table 9).
Tables 10 and 11 show the differences between the correctly simulated muon, final state nucleon and removal (unobserved) energies and those generated by genie 2 for QE events in carbon, and oxygen respectively. These differences are shown for the case of q 2 3 = 0.2 GeV 2 (|U F SI | = 20 MeV) and for q 2 3 = 0.8 GeV 2 (|U F SI | = 0).

Conclusion
We investigate the binding energy parameters that should be used in modeling electron and neutrino scattering from nucleons bound in a nucleus within the framework of the impulse approximation. We discuss the relation between binding energy, missing energy, removal energy ( ), spectral functions and shell model energy levels and extract updated removal energy parameters from ee p spectral function data. We address the difference in parameters for scattering from bound protons and neutrons. We also use inclusive e-A data to extract an empirical parameter U F SI ((q 3 + k) 2 ) to account for the interaction of final state nucleons (FSI) with the optical potential of the nucleus. Similarly we use V e f f to account for the Coulomb potential of the nucleus.
With three parameters , U F SI ((q 3 + k) 2 )and V e f f we can describe the energy of final state electrons for all available electron QE scattering data. The use of the updated parameters in neutrino Monte Carlo generators reduces the systematic uncertainty in the combined removal energy (with FSI corrections) from ± 20 MeV to ± 5 MeV.

A Appendix: Coulomb corrections
For targets with atomic number Z greater than one we should take into account the effect of the electric field of the nucleus on the incident and scattered electrons (and also on the final state proton in QE events). These corrections are called Coulomb corrections. For atomic weight A and atomic number Z the protons create an electrostatic potential V(r). In the effective momentum approximation (EMA), the effective potential for an incident electron is V e f f , which can be calculated as follows: The values for |V e f f | calculated from Eq. 29 agree (within errors) with values extracted from a comparison of the peak positions and cross sections of positron and electron QE scattering [24]. For our estimates of |V e f f | shown in Table 7 we use the experimental values for the nuclei that were measured in Ref. [24]. We use Eq. 29 to interpolate to other nuclei. For completeness, though not relevant in this analysis, there is also a focusing factor F f oc = ( ) 2 that enhances the cross section for electrons and reduces the cross section for positrons. The focussing factor cancels the 1/E 2 0 factor in the Mott cross section. Therefore, the Coulomb correction should only be applied to the structure functions W 1 and W 2 .

B Appendix: Relativistic Fermi Gas (RFG)
For the Fermi gas model the momentum distribution is zero for k > k F , and for k < k F it is given by and

B.1 Distributions and parameters of RFG versus k z
Here we do the calculation in cylindrical coordinates the probability distribution of the Z component of the momentum k z P(k z ) dk z , and average square of the transverse momentum k 2 T (k Z ) as a function of k z are given below.

B.2 Pauli blocking
We multiply the QE differetial cross sections by a Pauli blocking factor K nuclei Pauli (q 2 3 ) which reduces the predicted cross sections at low q 2 3 . The Pauli suppression factor shown below is from Eq. B54 of Ref. [79].
For |q 3 | < 2k F , otherwise no Pauli suppression correction is made. Here |q 3 | = Q 2 + ν 2 is the absolute magnitude of the 3-momentum transfer to the target nucleus,

C Reconstruction of E Q E-μ ν
, Q 2 Q E-μ and Q 2

Q E-P
In this section we update the expressions for the mean reconstructed neutrino energy E Q E-μ ν and square of the fourmomentum transfer Q 2 Q E-μ extracted only from the kinematics of final state muons in QE events. In addition we can also reconstruct the four momentum transfer Q 2 Q-(P,N ) from the kinematics of the final state recoil proton or neutron in QE events.
The expressions are updated to include: 1. The contribution of final state interaction |U F SI |.
2. The contribution of Coulomb corrections V e f f .
3. The contribution of the proton and neutron transverse momentum k T at the location of the QE peak.
In the derivation of the expressions we use relativistic kinematics. The "primed" energies and momenta are at the vertex before FSI with the nuclear and Coulomb field.
For neutrino scattering on bound neutrons |V P e f f | = |V e f f |. We define M N , M P , m μ as the neutron, proton and muon masses. At the peak location of the QE distribution the bound neutron momentum is perpendicular to q (i.e. k z =0). In this case, the average of the square of transverse momenta of the neutron (proton) for a Fermi gas momentum distribution (and also for a Gaussian distribution) is k 2 T -N = (k N F ) 2 2 for a bound neutron in the initial state and k 2 T -P = (k P F ) 2 2 ) for bound proton in the initial state.
C.1 Using only the kinematics of the μ − For neutrino QE events we define E μ-= T μ-+ m μ + |V e f f | as the total Coulomb corrected muon energy. We define (M P ) 2 = M 2 P + k 2 T -N to account for the fact that the final state pr oton has the same average transverse momentum as that of the initial state neutron k 2 T -N with respect to the neutrino-muon scattering plane. From energy-momentum conservation we get: E ν = p μ-cos θ μ-+ P P cos θ P p μ-sin θ μ = P P sin θ P Here. for neutrino scattering on bound neutrons |V P e f f | = |V e f f |, |U F SI = |U F SI (q 2 3 + 1 2 k 2 F )|, p μ-= (E μ-) 2 − m 2 μ , E μ-= E μ-− |V e f f |, P P is the momentum of the final state pr oton (before FSI) in the neutrino − muon plane, and θ P is the angle of the proton in the neutrino − muon scattering plane. From Eq. 39 we obtain the following expressions.
Note that because |U F SI (q 2 3 + 1 2 k 2 F )| is q 2 3 dependent, the above expressions should be solved iteratively, or an average value corresponding to the mean q 2 3 should be used.
C.2 Using only the kinematics of the μ + For antineutrino QE events we define E μ+ = T μ+ + m μ − |V e f f | as the total Coulomb corrected muon energy. We define (M N ) 2 = M 2 N + k 2 T -P to account for the fact that the final state neutron has the same average transverse momentum as that of the initial state pr oton k 2 T -P with respect to the antineutrino − muon scattering plane. From energymomentum conservation we get: Here, |U F SI = |U F SI (q 2 3 + 1 2 k 2 F )|, p μ+ = (E 2 μ+ − m 2 μ , E μ+ = E μ+ + |V e f f |, P N is the momentum of the final state neutron (before FSI) in the antineutrino−muon scattering plane, and θ N is the angle of the neutron in the neutrino-muon plane. From Eq. 40 we obtain the following expressions.
Note that because |U F SI (q 2 3 ) + 1 2 k 2 F )| is q 2 3 dependent, the above expressions should be solved iteratively, or an average value corresponding to the mean q 2 3 should be used.

C.2.1 Using only the kinematics of the final state nucleon
For neutrino QE events the average reconstructed Q 2 Q E-P can be extracted from final state pr oton variables only by using following expression: For antineutrino QE events the average reconstructed Q 2 Q E-N can be extracted from final state neutron variables only by using following expression:

C.3 Comparison to previous analyses
If we set k 2 T = 0, U F SI = 0, and |V e f f | = 0, the above equations are reduced to the equations used in previous analyses except that x ν and xν (equation 24) are used.