Electron- Vs Neutrino-Nucleus Scattering

Electron- vs Neutrino-Nucleus Scattering

Omar Benhar

INFN and Department of Physics Universita` “La Sapienza” I-00185 Roma, Italy

NUFACT11 University of Geneva, August 2nd, 2011 Neutrino-nucleus scattering . impact of the ﬂux average . ﬂux-averaged electron scattering x-section: a numerical experiment . contributions of reaction mechanisms other than quasi elastic single nucleon knock out to the MiniBooNE CCQE data sample Summary & Outlook

Outline

Electron scattering . standard data representation . theoretical description: the impulse approximation

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 2 / 24 Summary & Outlook

Outline

Electron scattering . standard data representation . theoretical description: the impulse approximation Neutrino-nucleus scattering . impact of the ﬂux average . ﬂux-averaged electron scattering x-section: a numerical experiment . contributions of reaction mechanisms other than quasi elastic single nucleon knock out to the MiniBooNE CCQE data sample

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 2 / 24 Outline

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 2 / 24 The inclusive electron-nucleus x-section The x-section of the process e + A → e0 + X is generally analyzed at fixed beam energy and electron scattering angle, as a function of the electron energy loss ω = Ee − Ee0 Different rection mechanisms can be readily identified

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 3 / 24 Kinematical range covered by the available data

Carbon target 2 2 2 θe Q Q = 4E E 0 sin , x = e e 2 2Mω

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 4 / 24 Theory of electron-nucleus scattering

Double diﬀerential electron-nucleus cross section dσ α2 E0 eA = e L Wµν 4 µν A dΩe0 dEe0 Q Ee

. Lµν is fully speciﬁed by the measured electron kinematical variables . the determination of the target response tensor

µν X µ ν (4) WA = h0|JA|nih|JA|0iδ (P0 + ke − Pn − ke0 ) n requires a consistent description of the target internal dynamics and electromagnetic current, unavoidably implying approximations Impulse approximation (IA): at large momentum transfer electron-nucleus scattering reduces to the incoherent sum of scattering processes involving individual nucleons

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 5 / 24 The impulse approximation

The IA amounts to the replacement

22 q,ω q,ω x i

Σi

µ X µ JA = ji , |Xi → |x, pxi ⊗ |R, pRi . i Nuclear dynamics and electromagnetic interactions are decoupled Z 3 dσA = d kdE dσN P(k, E)

. The electron-nucleon cross section dσN can be extracted from electron-proton and electron-deuteron scattering data . The spectral function P(k, E) , momentum and energy distribution of the knocked out nucleon, can be obtained from ab initio many-body calculations

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 6 / 24 Theory vs data

Memento: theoretical calculations involve no adjustable parameters

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 7 / 24 The quasi elastic (QE) sector

Elementary interaction vertex described in terms of the vector form (p,n) (p,n) 2 factors, F1 and F2 , precisely measured over a broad range of Q

Position and width of the peak are determined by P(k, E) The tail extending to the region of high energy loss is due to nucleon-nucleon correlations in the initial state, leading to the occurrence of two particle-two hole (2p2h) ﬁnal states

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 8 / 24 QE neutrino nucleus scattering

Consider the data sample of charged current QE data collected by the MiniBooNE Collaboration using a CH2 target The measured double diﬀerential cross section is averaged over the energy of the incoming neutrino, distributed according to the ﬂux Φ Z dσA 1 dσA = dEνΦ(Eν) dTµd cos θµ NΦ dEνdTµd cos θµ

In addition to F1 and F2 , the QE electron-nucleon cross section is determined by the axial form factor FA, assumed to be of dipole form and parametrized in terms of the axial mass MA According to the paradigm succesfully employed to describe electron scattering data, in order to minimize the bias associated with nuclear eﬀetcs, MA must be determined from measurement carried out using a deuterium target. The resulting value is MA = 1.03 GeV

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 9 / 24 The electron scattering paradigm fails to explain the ﬂux-averaged neutrino scattering x-section

Analysis of CCQE data

. MiniBooNe CCQE data

. MiniBooNE ﬂux

Theoretical calculations carried out setting MA = 1.03

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 10 / 24 Analysis of CCQE data

. MiniBooNe CCQE data

. MiniBooNE ﬂux

Theoretical calculations carried out setting MA = 1.03 The electron scattering paradigm fails to explain the ﬂux-averaged neutrino scattering x-section Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 10 / 24 Contribution of diﬀerent reaction mechanisms

In neutrino interactions the lepton kinematics is not determined. The flux-averaged cross sections at fixed Tµ and cos θµ picks up contributions at different beam energies, corresponding to a variety of kinematical regimes in which different reaction mechanisms dominate

. x = 1 → Eν 0.788 GeV , x = 0.5 → Eν 0.975 GeV . Φ(0.975)/Φ(0.788) = 0.83

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 11 / 24 “Flux averaged” electron-nucleus x-section

◦ The electron scattering x-section oﬀ Carbon at θe= 37 has been measured for a number of beam energies

MIT-Bates data compared to theoretical calculations including QE scattering only

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 12 / 24 Compute the flux averaged cross section using experimental data and assuming that the electron beam energies be distributed according to the MiniBooNE neutrino flux (Σexp) Compute the flux averaged cross section using the results of theoretical calculations including QE scattering only (Σth) The above procedure yields

Σexp >∼ 1.20 Σth

The electron scattering paradigm fails to explain the ﬂux averaged electron scattering x-section

A numerical experiment

◦ Consider the cross section at θ=37 and 550 ≤ Te0 ≤ 650 MeV (corresponding to the QE peak at 730 MeV)

Data are available at Ee = 730, 961, 1108, 1299 and 1501 MeV

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 13 / 24 The above procedure yields

Σexp >∼ 1.20 Σth

The electron scattering paradigm fails to explain the ﬂux averaged electron scattering x-section

A numerical experiment

◦ Consider the cross section at θ=37 and 550 ≤ Te0 ≤ 650 MeV (corresponding to the QE peak at 730 MeV)

Data are available at Ee = 730, 961, 1108, 1299 and 1501 MeV Compute the flux averaged cross section using experimental data and assuming that the electron beam energies be distributed according to the MiniBooNE neutrino flux (Σexp) Compute the flux averaged cross section using the results of theoretical calculations including QE scattering only (Σth)

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 13 / 24 The electron scattering paradigm fails to explain the ﬂux averaged electron scattering x-section

A numerical experiment

◦ Consider the cross section at θ=37 and 550 ≤ Te0 ≤ 650 MeV (corresponding to the QE peak at 730 MeV)

Σexp >∼ 1.20 Σth

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 13 / 24 A numerical experiment

◦ Consider the cross section at θ=37 and 550 ≤ Te0 ≤ 650 MeV (corresponding to the QE peak at 730 MeV)

Σexp >∼ 1.20 Σth

The electron scattering paradigm fails to explain the ﬂux averaged electron scattering x-section

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 13 / 24 Where does the additional strength come from?

It has been suggested that 2p2h final states provide a large contribution to the measured neutrino cross section Two particle-two hole final states may be produced through different mechanisms . Initial state correlations : lead to the tail extending to large energy loss, clearly visible in the calculated QE cross section. The corresponding strength is consistent with the measurements of the coincidence( e, e0p) x-section carried out by the JLAB E97-006 Collaboration. . Final state interactions : lead to a redistribution of the inclusive strength, mainly affecting the region of x > 1 , corresponding to low energy loss . Coupling to the two-body current : leads to the appearance of strength at x < 1 , in the dip region between the QE and ∆-excitation peaks

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 14 / 24 Measured correlation strength

the correlation strength in the 2p2h sector has been measured by the JLAB E97-006 Collaboration using a carbon target A−2 k2 strong energy-momentum correlation: E ∼ Ethr + A−1 2m

-1 10 ] -1

sr -2 3 10 ) [fm

m -3 10 n(p 0.2 0.3 0.4 0.5 0.6 pm [GeV/c] Measured correlation strength 0.61 ± 0.06, to be compared with the theoretical predictions 0.64 (CBF) and 0.56 (G-Matrix) Correlated nucleons (most of the times a proton and a neutron) have momenta > 250 MeV, pointing in opposite directions

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 15 / 24 Two-body corrent

The nuclear electromagnetic current involves two-nucleon contributions, arising from nucleon-nucleon interactions and nucleon excitations

A A µ X µ X µ JA = j1 (i) + j2 (i, j) i=1 j>1=1

Meson exchange current (MEC) arising from processes involving pions

In electron scattering MEC produce a signiﬁcant enhancement of the 3He and 4He inclusive x-section in the transverse channel

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 16 / 24 MEC eﬀect on the transverse electromagnetic response

L/T separation of the the inclusive electron-nucleus x-section ! " ! # d2σ dσ Q4 1 Q2 θ = R (|q|, ω) + + tan2 R (|q|, ω) 4 L 2 T dΩe0 dEe0 dΩe0 M |q| 2 |q| 2

Calculations by M.J. Dekker et al (Fermi gas model, AD 1991)

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 17 / 24 Dynamically Consistent Treatment of MEC

The nuclear electromagnetic current must satisfy the continuity equation

µ 0 ∂µJ = ∇ · J + i[H, J ] = 0 , where H is the nuclear hamiltonian The two-body current can be separated into a model-independent term, fully determined from the interaction, and a model-dependent one associated with the ρπγ and ωπγ electromagnetic couplings A fully consistent treatment of the contribution of 2p2h ﬁnal states requires the inclusion of initial state nucleon-nucleon correlations, ﬁnal state interactions and MEC ab initio calculations based on realistic nuclear hamiltonians and wave functions provide a very good description of the electron scattering x-sections of light nuclei

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 18 / 24 May inclusion of MEC contributions explain the MiniBooNE data?

A circumstantial evidence: the disagreement between data and theoretical calculations not including MEC turns out to be less pronounced at small θµ

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 19 / 24 Summary and outlook

In electron-nucleus scattering, the energy loss spectra at fixed beam energy allow for a clearcut identification of different reaction mechanisms This feature has been exploited to carry out theoretical calculations of the inclusive cross section, providing a quantitative description of the large body of data over a broad kinematical range Due to flux average, the measured “quasi elastic” neutrino-nucleus cross section picks up contributions from different kinematical regions, where reaction mechanisms other than single nucleon knock out are dominant Systematic studies of the flux-averaged electron scattering cross section may prove very useful to pin down the relevant reaction mechanisms The development of a theoretical description of the neutrino x-section consistenttly taking into account all the relevant mechanisms will shed light on the controversial issue of medium modifications of the axial form factor.

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 20 / 24 Background slides

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 21 / 24 Local Density Approximation (LDA) P(k, E) for oxygen

P(k, E) = PMF(k, E) + Pcorr(k, E)

0 PMF(k, E) → from (e, e p) data

Pcorr(p, E) → from uniform nuclear matter calculations at diﬀerent densities:

X 2 PMF(k, E) = Zn|φn(k)| Fn(E − En) n

Z 3 NM Pcorr(k, E) = d r ρA(r) Pcorr(k, E; ρ = ρA(r))

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 22 / 24 BENHAR, FARINA, NAKAMURA, SAKUDA, AND SEKI PHYSICAL REVIEW D 72, 053005 (2005) spectral functions of the few-nucleon systems, having A Strong dynamical NN correlations give rise to virtual 3 [29–31] and 4 [17,32,33], as well as of nuclear matter, scattering processes leading to the excitation of the partic- i.e. in the limit A with Z A=2 [34,35]. Calculations ipating nucleons to states of energy larger than the Fermi based on G-matrix!1 perturbation theory have also been energy, thus depleting the shell model states within the carried out for oxygen [36,37]. Fermi sea. As a consequence, the spectral function asso- The spectral functions of different nuclei, ranging from ciated with nucleons belonging to correlated pairs extends carbon to gold, have been modeled using the local density to the region of p pF and E eF, where pF and eF approximation (LDA) [18], in which the experimental denote the Fermij momentumj and the Fermi energy, typi- information obtained from nucleon knockout measure- cally & 250 and & 30 MeV, respectively. ments is combined with the results of theoretical calcula- The correlation contribution to P p;E of uniform nu- tions of the nuclear matter P p;E at different densities clear matter has been calculated by Benhar et al. for a wide [18]. range of density values [18]. Within the LDA scheme, the Nucleon removal from shell model states has been ex- results of Ref. [18] can be used to obtain the corresponding tensively studied by coincidence e; e0p experiments (see, quantity for a ﬁnite nucleus of mass number A from e.g., Ref. [38]). The corresponding measured spectral function is usually parametrized in the factorized form P p;E d3r r PNM p;E; r ; (22) corr A corr A 2 Z PMF p;E Zn n p Fn E En ; (21) n j j ÿ where A r is the nuclear density distribution and X PNM p;E; is the correlation component of the spectral corr where n p is the momentum-space wave function of the function of uniform nuclear matter at density . single particle shell mode state n (e.g. Woods-Saxon wave Finally, the full LDA nuclear spectral function can be functions), whose energy width is described by the func- written tion Fn E En (e.g. a Lorentzian). The normalization of the nth stateÿ is given by the so called spectroscopic factor P p;E P p;E P p;E ; (23) LDA MF corr Zn < 1, and the sum in Eq. (21) is extended to all occupied states. Typically, PMF p;E vanishes at E larger than the spectroscopic factors Zn of Eq. (21) being constrained 30 MeV and pSpectrallarger than 250 MeV function. Note that in andby the normalization momentum requirement distribution of Oxygen the absence of NNj j correlations the full spectral function could be written as in Eq. (21), with F E E E 3 n ÿ n ÿ d pdEPLDA p;E 1: (24) En and Zn 1. Z

FIG. 2 (color online). Three-dimensional plot (left panel) and scatter plot (right panel) of the oxygen spectral functionp obtained2 using 2 the LDA approximation described• FG in the model: text. Ph(k, E) ∝ θ(kF − |k|) δ(E − |k| + m + ) • shell model states account for ∼ 80% of the strenght 053005-4 • the remaining ∼ 20%, arising from NN correlations, is located at high momentum and large removal energy (k kF, E )

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 23 / 24 MEC model of Dekker et al (A.D. 1991)

Nuclear structure described according to the Relativistic Fermi Gas model

Omar Benhar (INFN, Roma) NUFACT11 Geneva 02/08/2011 24 / 24