Many-Body Theory of the Electroweak Nuclear Response

Transcription

1 Many-Body Theory of the Electroweak Nuclear Response Omar Benhar INFN and Department of Physics Università La Sapienza, I Roma Collaborators N. Farina, D. Meloni, H. Nakamura, M. Sakuda, R. Seki References PRD 72 (2005) , hep-ph/ see also nucl-ex/ (with Donal Day & Ingo Sick, to appear in RMP) Workshop on Neutrino Response Functions - INT, August 16, 2006 p.1/36

2 Outline Motivation Theory of the electroweak nuclear response Results for electron scattering and comparison to data Results for neutrino scattering Conclusions & prospects Workshop on Neutrino Response Functions - INT, August 16, 2006 p.2/36

3 Motivation Neutrino experiments use nuclei as detectors Workshop on Neutrino Response Functions - INT, August 16, 2006 p.3/36

4 Motivation Neutrino experiments use nuclei as detectors Quantitative understanding of the weak nuclear response at GeV required for the analyses of many neutrino oscillation experiments Workshop on Neutrino Response Functions - INT, August 16, 2006 p.3/36

5 Motivation Neutrino experiments use nuclei as detectors Quantitative understanding of the weak nuclear response at GeV required for the analyses of many neutrino oscillation experiments Need to develop a theoretical framework applicable to a wide range of kinematical conditions and targets easily implementable in Monte Carlo simulations Workshop on Neutrino Response Functions - INT, August 16, 2006 p.3/36

6 Theory of the electroweak nuclear response Nuclear cross section lepton tensor, completely specified by kinematics Nuclear response tensor Workshop on Neutrino Response Functions - INT, August 16, 2006 p.4/36

7 Theory of the electroweak nuclear response Nuclear cross section lepton tensor, completely specified by kinematics Nuclear response tensor Need to develop a consistent theoretical description of Workshop on Neutrino Response Functions - INT, August 16, 2006 p.4/36

8 Non Relativistic Nuclear Many-Body Theory (NMBT) Bottom line: decouple the uncertainty associated with dynamical models from the approximations implied in many-body calculations Workshop on Neutrino Response Functions - INT, August 16, 2006 p.5/36

9 Non Relativistic Nuclear Many-Body Theory (NMBT) Bottom line: decouple the uncertainty associated with dynamical models from the approximations implied in many-body calculations Dynamics determined from the properties of two- and three-nucleon systems (exactly solvable) Workshop on Neutrino Response Functions - INT, August 16, 2006 p.5/36

10 Non Relativistic Nuclear Many-Body Theory (NMBT) Bottom line: decouple the uncertainty associated with dynamical models from the approximations implied in many-body calculations Dynamics determined from the properties of two- and three-nucleon systems (exactly solvable) Calculations of nuclear observables does not involve any additional adjustable parameters Workshop on Neutrino Response Functions - INT, August 16, 2006 p.5/36

11 Predictions of NMBT Ground and low-lying excited states of nuclei with approximations involved in the solution of the Schrödinger equation (Pieper & Wiringa). No -20 3/ He α+2n 6 He α+d 7 He 7/2 6 He+n 6 He+2n Energy (MeV) VMC GFMC Exp 6 Li 1/2 3/2 7 Li α+t 8 He 7 Li+n 8 Li α+α 8 Be Workshop on Neutrino Response Functions - INT, August 16, 2006 p.6/36

12 Energy dependence of the spectroscopic strengths of shell model states of (O.B. Fabrocini & Fantoni), measured in high resolution at NIKHEF Note: the shell model predicts Workshop on Neutrino Response Functions - INT, August 16, 2006 p.7/36

13 NMBT of the nuclear response Consider a scalar probe, for simplicity Exact calculations feasible for using integral transform techniques (Leidemann & Orlandini, Carlson & Schiavilla), for uniform nuclear matter can be obtained from Correlated Basis Function (CBF) perturbation theory (Fantoni & Pandharipande, Fabrocini & Fantoni) Workshop on Neutrino Response Functions - INT, August 16, 2006 p.8/36

14 Correlated states obtained from mean field (shell model or Fermi gas) states The two-nucleon correlation operator reflects the complexity of the nucleon-nucleon (NN) force (spin-isospin (ST) dependent, non central) Shape of state energy, determined from minimization of ground Workshop on Neutrino Response Functions - INT, August 16, 2006 p.9/36

15 CBF perturbation theory Split the hamiltonian according to If correlated states have large overlaps with the eigenstates of the hamiltonian, the matrix elements of are small and perturbation theory can be used to obtain, e.g., the ground state from Workshop on Neutrino Response Functions - INT, August 16, 2006 p.10/36

16 CBF calculation of the nuclear electroweak response Low : nonrelativistic approach expected to be OK. Ground and excited states from NMBT, expanded in powers of Calculations for uniform matter at equilibrium density carried out including intermediate contributions also estimated) correlated states ( The effect of short range correlations is large (Example: Fermi MeV) Workshop on Neutrino Response Functions - INT, August 16, 2006 p.11/36

17 RPA-like long range correlations are also important (indeed dominant at very low ), and need to be included Calculations are carried out using the mean field basis and an effective interaction defined through In CBF the link between the effective interaction and the bare NN interaction naturally emerges from the formalism Workshop on Neutrino Response Functions - INT, August 16, 2006 p.12/36

18 Large : nonrelativistic approximation breaks down. Hovewer, the impulse approximation (IA) regime sets in nuclear scattering incoherent sum of nucleon scattering processes Simplest implementation: neglect final state interactions (PWIA). : energy-momentum distribution (spectral function), obtained combining NMBT & data (O.B., Fabrocini & Fantoni) structure functions : nucleon tensor, written in terms of the measured nucleon Workshop on Neutrino Response Functions - INT, August 16, 2006 p.13/36

19 Local Density Approximation (LDA) for oxygen Workshop on Neutrino Response Functions - INT, August 16, 2006 p.14/36

20 Local Density Approximation (LDA) for oxygen from data Workshop on Neutrino Response Functions - INT, August 16, 2006 p.14/36

21 Local Density Approximation (LDA) for oxygen from data different densities from uniform nuclear matter calculation at Workshop on Neutrino Response Functions - INT, August 16, 2006 p.14/36

22 Local Density Approximation (LDA) for oxygen from data different densities from uniform nuclear matter calculation at Workshop on Neutrino Response Functions - INT, August 16, 2006 p.14/36

23 for oxygen Local Density Approximation (LDA) data from from uniform nuclear matter calculation at different densities Workshop on Neutrino Response Functions - INT, August 16, 2006 p.14/36

24 LDA spectral function of Oxygen can be obtained within the Local Density Approximation (LDA), combining data and theoretical calculations shell model states account for 80% of the strenght the remaining 20%, arising from NN correlations, is located at high momentum and large removal energy ( Workshop on Neutrino Response Functions - INT, August 16, 2006 p.15/36

25 Corrections to PWIA Final state interactions (FSI) are known to be large (e.g. from measurements of nuclear transparency) q,ν q,ν + Can be consistently included within NMBT using the eikonal approximation (reasonable at large ) Final state Workshop on Neutrino Response Functions - INT, August 16, 2006 p.16/36

26 Eikonal + Frozen Spectators approximation from NN scattering data (medium modifications of NN scattering tractable within NMBT) particle at position is the joint probability of finding the struck and spectators at positions Short range correlations strongly affect, leading to a complicated pattern of modifications of FSI, compared to mean field predictions Workshop on Neutrino Response Functions - INT, August 16, 2006 p.17/36

27 Medium modified NN cross-section results of NMBT calculation including Pauli blocking and dispersive corrections (Pandharipande & Pieper) Workshop on Neutrino Response Functions - INT, August 16, 2006 p.18/36

28 Correlation effect on the two-nucleon density Workshop on Neutrino Response Functions - INT, August 16, 2006 p.19/36

29 Corrections to PWIA (continued) Effects of Pauli blocking statistical FSI A rather crude prescription: modify the spectral function according to For beam energies in the few GeV region the effect of Pauli blocking is hardly visible in the energy spectrum of the outgoing lepton at fixed angle. However, it becomes dominant in the distribution for GeV. Workshop on Neutrino Response Functions - INT, August 16, 2006 p.20/36

30 Nuclear transparency (no FSI ) Nuclear transparency obtained from compared to MIT-Bates, SLAC and JLab data (O.B., Pandharipande & Pieper) Complicated pattern of correlation effects, leading to a sizable enhancement of the transparency Workshop on Neutrino Response Functions - INT, August 16, 2006 p.21/36

31 Comparison to Oxygen GeV (O.B., Farina, Nakamura, Sakuda & Seki) Workshop on Neutrino Response Functions - INT, August 16, 2006 p.22/36

32 Theoretical results OK in the region of quasi elastic peak data significantly underestimated above threshold production deficiencies of the SF unlikely, as calculation of quasi elastic and -production x-sections involve integration regions almost exactly overlapping one another Workshop on Neutrino Response Functions - INT, August 16, 2006 p.23/36

33 Comparison to data at larger : extrapolation of SLAC 2 GeV problems likely to be ascribable to the description of and in the production region at low Workshop on Neutrino Response Functions - INT, August 16, 2006 p.24/36

34 Digression: how well do we know production at low? Oxygen results obtained using Bodek-Ritchie (BR) model of the nucleon structure functions, resulting from a global fit to SLAC data Main features of the BR model provides for both proton and neutron neutron structure function are obtained combining proton and deuteron data and unfolding nuclear effetcs extends over a broad kinematical region, including both resonance production and deep inelastic scattering only includes data with involves an extrapolation GeV. Use at lower Workshop on Neutrino Response Functions - INT, August 16, 2006 p.25/36

35 JLab and data at low Fits and dynamical models provide a good account of proton data The BR model fails to reproduce the deuteron data at GeV Problems with the neutron structure functions! Workshop on Neutrino Response Functions - INT, August 16, 2006 p.26/36

36 Systematic uncertainty in the BR treatment of nuclear effect (deuteron wf, relativistic normalization of, FSI) small Estimate à la BR from JLab GeV write the deuteron cross section as where are the smeared and x-sections. define the smearing ratio Assuming Workshop on Neutrino Response Functions - INT, August 16, 2006 p.27/36

37 from GeV and Workshop on Neutrino Response Functions - INT, August 16, 2006 p.28/36

38 Nuclear cross sections SLAC data (Sealock et al (1989)) LNF data (Anghinolfi et al (1996)) Workshop on Neutrino Response Functions - INT, August 16, 2006 p.29/36

39 Results for scattering dσ/de µ [10-14 fm 2 /MeV] ν µ + 16 O µ + X E ν = 0.8 GeV dσ/de µ [10-14 fm 2 /MeV] ν µ + 16 O µ + X E ν = 2 GeV E µ [GeV] E µ [GeV] Workshop on Neutrino Response Functions - INT, August 16, 2006 p.30/36

40 Total CC cross section for -beams of and (QE only) more results in Davide Meloni s talk at NUFACT Flux (Arbitarary Unit) γ=250 for ν e γ=150 for anti ν e E ν (MeV) Workshop on Neutrino Response Functions - INT, August 16, 2006 p.31/36

41 Monte Carlo simulation of Workshop on Neutrino Response Functions - INT, August 16, 2006 p.32/36

42 Semi-inclusive Theory: Schiavilla, O.B., Kievsky, Marcucci & Viviani Data: Jlab E PWIA GLB GLB+MEC σ(fm 2 /MeV sr 2 ) p m (MeV/c) Workshop on Neutrino Response Functions - INT, August 16, 2006 p.33/36

43 Semi-inclusive (continued) Theory: Schiavilla, O.B., Kievsky, Marcucci & Viviani Data: Jlab E PWIA GLB GLB+MEC σ(fm 2 /MeV sr 2 ) p m (MeV/c) Workshop on Neutrino Response Functions - INT, August 16, 2006 p.34/36

44 Implementig NMBT in Monte Carlo simulations distribution of the primary vertex and two-nucleon density obtained sampling the probability distributions associated with the wave function of Pieper et al Workshop on Neutrino Response Functions - INT, August 16, 2006 p.35/36

45 Conclusions & prospects Many-body theory provides quantitative parameter free predictions of the nuclear response for a broad range of targets and kinematical conditions Nuclear dynamics can be included in both the initial and final state in a consistent fashion Spectral functions and FSI in eikonal approximation are easily inplementable in Monte Carlo simulation (application to semi-inclusive processes is under way) Workshop on Neutrino Response Functions - INT, August 16, 2006 p.36/36