EPJ manuscript No. (will be inserted by the editor) Removal and Binding Energies in Lepton Nucleus Scattering A. Bodek Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627-0171 Received: date / Revised version: July 8 , 2018 Abstract. 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, removal energy, interaction energy, spectral functions and shell model energy levels and extract updated interaction energy parameters from modern ee0p spectral function data. We address the difference in parameters for scattering from bound protons and neutrons, and the difference between the parameters for modeling the peak (most probable value) versus the mean of distributions. We show that different MC generators use different definitions of what is referred to as 16 nuclear interaction energy parameters. For example, for neutrino scattering from neutrons bound in 8 O 0N the Smith-Moniz interaction energy hSM i = 43.0±3 MeV should be used in neut, the excitation energy N N hEx i= 10.2±3 MeV should be used in genie, and the interaction energy hR i = 27.0±3 should be used to calculate the neutrino energy from muon variables only. At present the uncertainty in the value of the interaction energy (±15 MeV) results in the largest systematic uncertainty (± 0.033 eV 2) in the extraction of the neutrino oscillation parameter ∆m2. We reduce this uncertainty by a factor of 5. PACS. 13.60.Hb Total and inclusive cross sections (including deep-inelastic processes) { 13.15.+g Neu- trino interactions { 13.60.-r Photon and charged-lepton interactions with hadrons 1 Introduction 1.1 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 # and the square of the differ- The modeling of neutrino cross sections on nuclear tar- ence in mass between the two neutrino mass eigenstates gets is of great interest to neutrino oscillations exper- ∆m2. A correct modeling of the reconstructed neutrino iments. Neutrino Monte Carlo (MC) generators include energy is very important in the measurement of ∆m2. In genie[1], neugen[2], neut[3], nuance[4], nuwro[5] and general, the resolution in the measurement of energy in GiBUU[6]. neutrino experiments is much worse than the resolution in Although more sophisticated models are available[7{9], electron scattering experiments. However, a precise deter- calculations using a one-dimensional momentum distribu- mination of ∆m2 is possible if the MC prediction for mean tion and an average binding energy parameter are still value of the experimentally reconstructed neutrino energy widely used. One example is the simple relativistic Fermi is unbiased. At present the uncertainty in the value of the gas (RFG) model. interaction energy is a significant source of systematic er- ror in the extraction of the neutrino oscillation parameter The RFG model does not describe the tails in the en- ∆m2 (as shown below). ergy distribution of the final state lepton very well[10,11]. The two-neutrino transition probability can be written Improvements to the RFG model such as a better mo- as mentum distribution are usually made within the existing Monte Carlo (MC) frameworks. All RFG-like models with 2 2 ! 2 2 ∆m =eV (L=km) different nucleon momentum distributions require in addi- Pνα!νβ (L) = sin 2# sin 1:27 : tion an average binding energy parameter (interaction en- (Eν =GeV) ergy) to account for the removal energy of nucleons from (1) the nucleus. The interaction energy should be the same Here, L (in km) is the distance between the neutrino source for all one-dimensional momentum distributions. In more and the detector and ∆m2 is in eV2. sophisticated impulse approximation models[7{9] two di- The location of the first oscillation maximum in neu- 1st−min mensional spectral functions (as a function of nucleon mo- trino energy (Eν ) is when the term in brackets is mentum and removal energy) are used. equal to π=2. An estimate of the extracted value of ∆m2 2 A. Bodek: Removal and Binding Energies in Lepton Nucleus Scattering is given by: Momentum"DistribuGons" 2E1st−min 14" ∆m2 = ν : (2) NUWRO"Spectral"(BenharQ Fantoni)" 1:27πL 12" EffecGve"Spectral"FuncGon" For example, for the T2K experiment[12] L = 295 Km, 10" Global"Fermi" and Eν is peaked around 0.6 GeV. The T2K experiment[12] reports a value of Local"Thomas"Fermi" 8" 2 −3 2 ∆m32 = (2:434 ± 0:064) × 10 eV : 6" In the Monte Carlo generator used by T2K (NEUT) a PROBABILITY/GeV" 16 4" value of the interaction energy of 27 MeV for 8 O has been used. However, as we show in section 7 a value of 43 MeV should be used. Using equation 2 in conjunc- 2" tion with equation 40 of Appendix D we estimate that a +15 MeV change in h0N i results in a change in ∆m2 0" SM 32 0" 0.1" 0.2" 0.3" 0.4" 0.5" 0.6" 0.7" −3 2 73 of +0:033 × 10 eV , which is the largest contribution Momentum"(k)""GeV" to the total systematic error in ∆m2 . As we show in this 32 Fig. 1. One-dimensional nucleon momentum distributions in communication, this contribution can be reduced by a fac- 12 a 6 C nucleus. The green curve (Global Fermi) is the momen- tor of 5. tum distribution for the relativistic Fermi gas (RFG) model For comparison, a change of +15 MeV/c in the as- which is related to global average density of nucleons. The red sumed value of the Fermi momentum KF yields a smaller curve is the local Fermi gas distribution which is related to the −3 2 change of +0:003 × 10 eV in the extracted value of local density of nucleons in the nucleus. The black curve is the 2 ∆m32. projected momentum distribution of the Benhar-Fantoni two dimensional spectral function as implemented in nuwro. The blue line is the momentum distribution for the effective spec- 1.2 Nucleon momentum distributions tral function model, which approximates the 0 superscaling prediction for the final state muon in quasielastic scattering. Fig. 1 shows a few models for the nucleon momentum dis- The effective spectral function is only valid for QE scattering 12 0 tributions in the 6 C nucleus. The solid green line (labeled because the superscaling model is only valid for QE scat- Global Fermi gas) is the nucleon momentum distribution tering. for the Fermi gas[10] 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 superscaling model are based on fits to electron QE scat- momentum distribution of the Benhar-Fantoni[7] 2D spec- tering data and therefore includes both the 1p1h and 2p2h tral function as implemented in nuwro. The solid red processes (discussed in section 3). line is the nucleon momentum distribution for the Local- The following nuclear targets are (or were) used in neu- Thomas-Fermi (LTF) gas which is is related to the local trino experiments: Carbon (scintillator) used in the nova density of nucleons in the nucleus and is implemented in and minerνaexperiments. Oxygen (water) used in T2K, , and GiBUU. neut nuwro and in minerνa. Argon used in the argoneut and dune For QE scattering, another more sophisticated formal- experiments. Calcium (marble) used in charm. Iron used ism is the 0 superscaling model[13] discussed in Appendix in minerνa, minos, cdhs, nutev, and ccfr. Lead used E. This model is only valid for QE scattering. It can be in chorus and minerνa. used to predict the kinematic distribution of the final state muon but does not describe the details of the hadronic fi- nal state. Therefore, it has not been implemented in neu- trino MC generators. However, the predictions of the 0 2 Deficiencies in current MC generators superscaling model can be approximated with an effec- Deficiencies in the current implementations in neutrino tive spectral function[8] which has been implemented in MC generators originate from several sources. genie. The momentum distribution of the effective spec- 12 tral function for nucleons bound in 6 C is shown as the 1. Using the published values of parameters extracted by blue curve in Fig. 1. Moniz et. al.[10] and not correcting for the approxima- Although the nucleon momentum distributions are very tions used in that analysis. different for the various model, the predictions for the nor- 2. Using the same values of the interaction energy in dif- 1 dσ 2 malized quasielastic neutrino cross section σ dν (Q ; ν) are ferent MC generators not accounting for the fact that similar as shown in Fig. 2. These predictions as a func- they are defined differently in each generator. tion of ν = Eν − Eµ are calculated for 10 GeV neutrinos 3. Not accounting for nuclear Coulomb corrections[14] to 12 2 2 on 6 C at Q =0.5 GeV . The prediction with the local the energies of charged leptons in the initial and final Fermi gas distribution are similar to the prediction of the state. Benhar-Fantoni two dimensional spectral function as im- 4. The interaction energies extracted by Moniz et. al. plemented in nuwro. Note that the prediction of the 0 from high resolution electron scattering experiments A. Bodek: Removal and Binding Energies in Lepton Nucleus Scattering 3 Carbon#Q2=0.5##GeV2# As discussed in section 6, method 1 is the most reliable, 6.0# followed by method 2.