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 (), spectral functions and shell model energy levels and extract updated removal energy parameters from eep 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 to account for the interaction of final state nucleons (FSI) with the optical potential of the nucleus. Similarly we use to account for the Coulomb potential of the nucleus. With three parameters , and 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.
pacs:13.60.HbTotal and inclusive cross sections (including deep-inelastic processes) and 13.15.+g Neutrino interactions and 13.60.-rPhoton and charged-lepton interactions with hadrons
The modeling of neutrino cross sections on nuclear targets is of great interest to neutrino oscillations experiments. Neutrino Monte Carlo (MC) generators include geniegenie (), neugenneugen (), neutneut (), nuwronuwro () and GiBUUgibuu ().
Although more sophisticated models are availablebenhar (); spectral-theory2 (); Sakuda (); optical (); optical1 (), calculations using a one-dimensional momentum distribution and an average removal energy parameter are still widely used. One example is the simple relativistic Fermi gas (RFG) model.
The RFG model does not describe the tails in the energy distribution of the final state lepton very wellelectron (); neutrino (). Improvements to the RFG model such as a better momentum distribution are usually made within the existing Monte Carlo (MC) frameworks. All RFG-like models with one dimenssional nucleon momentum distributions require in addition binding energy parameters () 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 eep electron scattering experiments on several nuclei. We use (see Appendix A) to account for the Coulomb potential of the nucleus, and extract empirical final state interaction parameters from all available inclusive e-A electron scattering data. With these three parameters , and 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.
1.1 Relevance to neutrino oscillations experiments
In a two neutrinos oscillations framework the oscillation parameters which are extracted from long baseline experiments are the mixing angle and the square of the difference in mass between the two neutrino mass eigenstates . A correct modeling of the reconstructed neutrino energy is very important in the measurement of . 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 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 (as shown below).
The two-neutrino transition probability can be written as
Here, L (in km) is the distance between the neutrino source and the detector and is in .
The location of the first oscillation maximum in neutrino energy () is when the term in brackets is equal to . An estimate of the extracted value of is given by:
For example, for the t2k experiment , and is peaked around 0.6 GeV. For the normal hierarchy the t2k experimentT2K () reports a value of
The above estimate is consistent with the estimate of the t2k collaboration. The t2k collaboration reportst2k-impact () 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 and 40% for with respect to the size of the systematics errors, assuming maximal ”.
For the case of normal hierarchy a combined analysiscombined () 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 yields a smaller change of in the extracted value of .
1.2 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.
1.3 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:
The momentum of the final state muon including the effect of Coulomb correctionsgueye ().
The mass, excitation energy, and recoil energy of the spectator nuclear state.
The effect of the interaction of the final state nucleon (FSI) with the optical and Coulomb potential of the spectator nucleus.
1.4 Nucleon momentum distributions
Fig. 1 shows a few models for the nucleon momentum distributions in the C nucleus. The solid green line (labeled Global Fermi gas) is the nucleon momentum distribution for the Fermi gaselectron () 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-Fantonibenhar () 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.
A more sophisticated formalism is the superscaling modelDonnelly (), 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 superscaling model can be approximated with an effective spectral functioneffective () which has been implemented in genie. The momentum distribution of the effective spectral function for nucleons bound in 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 are similar as shown in Fig. 2. These predictions as a function of are calculated for 10 GeV neutrinos on C at =0.5 GeV. 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 superscaling model are based on fits to longitudinal QE differential cross sections. Subsequently, they includes 1p1h and some 2p2h processes (discussed in section 2).
The following nuclear targets are (or were) used in neutrino experiments: Carbon (scintillator) used in the nova and minera experiments. Oxygen (water) used in t2k and in minera. Argon used in the argoneut and dune experiments. Calcium (marble) used in charm. Iron used in minera, minos, cdhs, nutev, and ccfr. Lead used in chorus and minera.
|Li 1+||He -||4.4||Li -||5.7||4.0|
|C 0+||B -||16.0||C -||18.7||27.4|
|O 0+||N -||12.1||O -||15.7||23.0|
|Mg 0+||+||11.7||Mg +||16.5||24.1|
|Si 0+||Al+||11.6||Si +||17.2||24.7|
|Ar+||CL +||12.5||Ar -||9.9||20.6|
|Ca 0+||K +||8.3||Ca +||15.6||21.4|
|V -||Ti 0+||8.1||V 6+||11.1||19.0|
|Fe 0+||Mn -||10.2||Fe -||11.2||20.4|
|Ni -||Co 2+||8.2||Ni 0+||12.2||19.5|
|Y -||Sr -||7.1||Y 4-||11.5||18.2|
|Zr 0+||Y -||8.4||Zr+||12.0||17.8|
|Sn 0+||In +||10.1||Sn +||8.5||17.3|
|Ta -||Hf 0+||5.9||Ta 1+||7.6||13.5|
|Au+||Pt 0+||5.8||Au 2-||8.1||13.7|
|Pb 0+||TI +||8.0||Pb -||7.4||14.9|
2 The Impulse Approximation
2.1 1p1h process
Fig. 3 is a descriptive diagram for QE electron scattering on an off-shell proton which is bound in a nucleus of mass , and is moving in the mean field (MF) of all other nucleons in the nucleus. The on-shell recoil excited spectator nucleus has a momentum and a mean excitation energy . The off-shell energy of the interacting nucleon is , where . As discussed in section 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 . Here is the 3-momentum transfer, is the energy transfer, and is the square of the four-momentum transfer. For QE electron scattering on unbound protons (or neutrons) the energy transfer is equal to where is mass of the proton and is the mass of the neutron, respectively.
2.2 Nuclear Density corrections to and
The values of the Fermi momentum that are currently used in neutrino Monte Carlo generators are usually taken from an analysis of e-A data by Moniz et al.electron (). The Moniz published values of were extracted using the RFG model under the assumption that the Fermi momenta for protons and neutrons are different and are related to via the relations and , respectively. What is actually measured is , and what is published is . Moniz assumes that the nuclear density (nucleons per unit volume) is constant. Therefore, in the same nuclear radius R, for neutrons is larger if N is greater than Z. Moniz used these expressions to extract the published value of from the measured value of .
We undo this correction and re-extract the measured values of for nuclei which have a different number of neutrons and protons. In order to obtain the values of from the measured values of we use the fact that the Fermi momentum is proportional to the cube root of the nuclear density. Consequently , and , and . For the proton and neutron radii, we use the fits for the half density radii of nuclei (in units of femtometer) given in ref.radii ().
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 =.
However for the Pb nucleus only we use =0.275 GeV which we obtain from our own fits to inclusive e-A scattering data. For all other nuclei, our values are consistent with the values extracted by Moniz et. al.
2.3 Separation energy
The separation energy for a proton () or neutron is defined as follows:
The energy to separate both a proton and neutron () is defined as follows:
2.4 Two nucleon correlations
Fig. 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 =. The momentum of the initial state off-shell interacting nucleon is balanced by a single on-shell correlated recoil neutron which has momentum . The spectator nucleus is left with two holes. Short range nucleon-proton correlations occur % of the timemiller (). The off-shell energy of the interacting bound proton in a quasi-deuteron is , where 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.
|Spectator Nucleus Excitation|
|Used in spectral functions|
|implemented in geniegenie ()|
|=||Nuclear Data Tables|
|(measured) TUNL (); nuclear-data ()|
|=+||used in spectral functions|
|removal energy is|
|used in , ,|
|and , also used in|
|effective spectral functionseffective ()|
|=||is Smith Monizneutrino ()|
|used in old-neutneut ()|
|=||we use to include|
|the effects of FSI.|
|Jlab Hall C C12 ()||1.2||16.3||25.8|
|Jlab Hall C C12 ()||1.2||18.1||29.4|
|Jlab Hall C C12 ()||1.2||18.4||25.7|
3 Spectral functions and experiments
In experiments the following process is investigated:
Here, an electron beam is incident on a nuclear target of mass . The hadronic final state consists of a proton of four momentum and an undetected nuclear remnant . Both the final state electron and the final state proton are measured. The nuclear remnant can be a spectator nucleus with excitation , or a nuclear remnant with additional unbound nucleons.
At high energies, within the plane wave impulse approximation (PWIA) the initial momentum of the initial state off-shell interacting nucleon can be identified approximately with the missing momentum . Here we define = and
The missing energy is defined by the following relativistic energy conservation expression,
The missing energy can be expressed in term of the excitation energy () of the spectator (A-1) nucleus and the separation energy of the proton (or neutron ).
The probability distribution of finding a nucleon with initial state momentum and missing energy from the target nucleus is described by the spectral function, defined as . Note that for spectral functions both and notation are used in some publications. The spectral functions and 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 from data. The kinematical region corresponding to low missing momentum and energy is where shell modelbrown () states dominatereview (). 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 nucleon-nucleon 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 less than 80 MeV, and ignore the spectral function for the 2p2h process which dominates at higher values of . We use shell model calculations to obtain the difference in the binding energy parameters for neutrons and protons.
4 Effects of the optical and Coulomb potentials (FSI)
We use empirical parameter 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 . In addition, we include the effect of the interaction of the final state proton with the Coulomb field of the nucleus ().
In QE electron scattering A three momentum transfer to a bound proton with initial momentum results in the following energy of the final state proton:
where . The Coulomb correction is discussed in Appendix A.
We define the average removal energy in terms of the average momentum 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 eep experiments. We then use these values and extract from inclusive e-A as discussed in section 9.
4.1 Smith-Moniz formalism
The Smith-Monizneutrino () formalism uses on-shell description of the initial state. In the on-shell formalism, the energy conserving expression is
is replaced with
A summary of the relationships between excitation energy used in genie (which incorporates the Bodek-Ritchie BodekRitchie () off-shell formalism), separation energy , missing missing energy (used in spectral function measurements), removal energy (used in the reconstruction of neutrino energy from muon kinematics only) and the Smith-Moniz removal energy (that should be used in old-neut) is given in Table 2.
5 Extraction of average missing energy
We extract the average missing energy and excitation energy for the 1p1h process from electron scattering data using two methods.
: From direct measurements of the average missing energy and average proton kinetic energy . These quantities have been extracted from spectral functions measured in eep experiments for tests of the Koltun sum ruleKoltun (). The contribution of two nucleon corrections is minimized by restricting the analysis to 80 MeV. This is the most reliable determination of . We refer to this average as
: 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 eep experiments. We refer to this average as
There could be bias in method 2 originating from the fact that a fraction of the nucleons () 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 section 5.4 (and shown Fig. 6)we find that the values of are consistent with for nuclei for which both are available.
When available, we extract the removal energy parameters using from method 1. Otherwise we use from method 2. For each of the two methods, we also use the nuclear shell model to estimate difference between the missing energies for neutrons and protons.
|Nucl.||5||ref.Donnelly ()||ref.Donnelly ()||ref.gueye ()|
|eep TokyoTokyo1 (); Tokyo2 (); Tokyo3 ()||169,169||165||15.1||1.4||*18.43|
|eep TokyoTokyo1 (); Tokyo2 (); Tokyo3 ()||221,221||228||20.0||3.10.25||24.03|
|eep NIKHEFNIKHEFC12 ()||27.13|
|eep SaclaySaclay ()||25.83|
|Shell Model binding E||24.9 5|
|eep Jlab Hall C C12 ()||*27.53|
|eep Jlab Hall A O16 ()||225,225||3.4||*24.13|
|Shell Model binding E||23.55|
|eep TokyoTokyo1 (); Tokyo2 (); Tokyo3 ()||238,241||236||18.0||5.1||30.6 3|
|eep SaclaySaclay ()||239,241||5.5||28.32|
|eep TokyoTokyo1 (); Tokyo2 (); Tokyo3 ()||251,251||241||28.8||7.40.6||26.33|
|eep SaclaySaclay ()||27.03|
|Shell-model binding E||23.65|
|eep TokyoTokyo1 (); Tokyo2 (); Tokyo3 ()||253,266||8.1||*25.63|
|eep Jlab hall C C12 ()||254,268||241||23.0||8.90.7||*29.63|
|eep SaclaySaclay ()||257,269||245||30.0||9.8||25.73|
|Shell-model binding E||11.90.9|