Charge-Changing Neutrino Scattering from the Deuteron J. W. Van Orden ODU/Jlab Collaborators: T. W. Donnelly and Oscar Morino MIT W. P. Ford University of Tennessee Final State Nucleons for Neutrino-Nucleus Interactions, 5/15/2015 Introduction Detection of the final-state particles can be used to determine the initial neutrino energy. The deuteron is the simplest nucleus containing a neutron and can be calculated relatively easily. This reaction can then be used to measure the incident neutrino flux. This reaction can be used to study aspects of the weak interactions. Calculation for deuterium can be used as a testbed for adding relativistic and pionic contributions to calculations for heavier nuclei. Kinematics

hadron plane lepton plane Since the final state includes two protons, double counting can be avoided by always choosing p1 to be the proton with the largest momentum. The Model The model used for the calculations shown here is based on modifications to a model of deuteron electrodisintegration* developed for use at Q2 of several GeV2. The model is based on the Bethe-Salpeter equation and is manifestly covariant. Details of the electrodisintegration calculations can be found in the references below. *S. Jeschonnek and J. W. Van Orden, Phys. Rev. C 78, 014007 (2008); Phys. Rev. C 80, 054001 (2009); Phys. Rev. C 81, 014008 (2010). The Bethe-Salpeter Equation The Deuteron Vertex Function The Current Operator The Impulse Approximation The impulse approximation to CC scattering from the

deuteron is defined by the Feynman diagrams: Antisymmetric scattering matrix On Mass Shell We will show three approximations: 1. Plane Wave Impulse Approximation (PWIA): diagram (a) only. 2. Plane Wave Born Approximation (PWBA): diagrams (a)+(b). 3. Distorted Wave Born Approximation (DWBA): diagrams (a)+(b)+(c). Deuteron Vertex Function a b The invariant functions gi are given by where Charge conjugation matrix is the magnitude of the threemomentum in the deuteron rest frame.

The Single-Nucleon Current Operator where and The weak form factors used in the calculations shown here are: Final State Interactions There are no reliable meson-exchange models of NN scattering for the invariant masses where pions production channels are open. The scattering amplitudes must be obtained from data. The scattering amplitudes are represented by a parameterization in terms of five Fermi invariants. A complete description of on-shell NN scattering. Lorentz invariant. Has complete spin dependence. We use two approaches: 1. The invariant functions are constucted from the SAID helicity amplitudes. pp amplitudes are available for s<9.16 GeV2

2. We have recently performed a fit of the available NN data from s=5.4 GeV2 to s=4000 GeV2 based on a Regge model.* In the calculations shown here, only on-shell contributions from the pp amplitudes are used. * W. P. Ford and J. W. Van Orden, Phys. Rev. C 87, 014004 (2013). W. P. Ford, Ph.D. Dissertation, http://arxiv.org/abs/1310.0871 Elastic pp Scattering s= 5.913 GeV2 s= 5.934 GeV2 Regge SAID 1 10 0 10 0 10

20 30 40 50 cm (deg) 60 70 80 90 Integrated Cross Section 3 -3 d /(d'dk') (GeV ) 0.7 0.6

q (G eV ) For the purpose of initial study of this reaction, we assume kinematics such as are used in electron scattering. That is, the initial neutrino energy is assumed to be known and the muon and proton 1 are detected in the final state. 0.8 Maximum 0.5 0.4 0.3 10% of Maximum 0.2 0.1 0.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 (GeV) -10 4.0x10 -10 3.5x10 -10 3.0x10

PWIA PWBA DWBA Cross Section at Maximum -10 2.5x10 -10 2.0x10 -10 1.5x10 -10 1.0x10 -11 5.0x10

0 0.00 0.05 0.10 0.15 0.20 (GeV) 0.25 0.30 0.35 3.5 3.0 q (G eV ) 2.5 2.0 1.5 1.0

0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 (GeV) -8 3 -3 d /(d'dk') (G eV ) 10 -9

10 -10 10 -12 10 -13 10 0.0 -8 8x10 -9 6x10 -9 4x10

-9 2x10 -9 0 0.0 -11 10 1x10 1.0 0.2 0.3 (GeV) PWIA PWBA DWBA 0.5

0.1 1.5 (GeV) 2.0 2.5 3.0 Exclusive Cross Section where and for neutrino scattering for anti-neutrino scattering O. Moreno, T. W. Donnelly, W. P. Ford and J. W. Van Orden, Phys. Rev. D 90, 013014 (2014). Exclusive Cross Section Calculations -8 10

-9 10 -9 10 -10 -10 10 10 10 10 -11 -12 10 -13 -12

10 10 -13 10 -14 10 -15 10 0.0 -11 0.00 0.10 0.20 p2 (GeV)

0.30 PWIA PWBA DWBA SAID DWBA Regge 0.2 0.4 0.6 p2 (GeV) 0.8 1.0 Azimuthal Angular Dependence -8 10 -9 10

-10 10 10 =0 deg =36 deg =72 deg =108 deg =144 deg =180 deg -11 -12 -9 5x10 -9 4x10 -9 3x10 -9 2x10 -9 1x10 0

0.00 0.05 0.10 10 -13 10 -14 10 0.0 0.2 0.4 0.6 p2 (GeV) 0.8

1.0 Sensitivity to the Weak Form Factors -8 10 MA=1.03 GeV MA=1.3 GP=0 -9 10 -10 10 10 -11 -12 10 -13

10 -14 10 0.0 0.2 0.4 0.6 p2 (GeV) 0.8 1.0 Summary We have produced a working calculation of CC for the deuteron in the DWBA designed for use at large energies. We are just beginning to explore the results of this calculation. The existing calculations indicate that the bulk of the cross section is produced at forward lepton

scattering angles with small energy and momentum transfer. The exclusive calculations indicate that the cross sections are dominated by high-momentum protons close to q. The effects of final state interactions in this region are small but not necessarily neglegible. The exclusive cross sections are sensitive to the axial form factor GA(Q2). Inclusion of currents is underway. Comparison of Electron and Neutrino Scattering -3 10 -4 10 -5 10 -6 10 -7 10 -8 10 -9 10

-10 10 -11 10 -12 10 -13 10 -14 10 0.0 0.2 0.4 0.6 p2 (GeV) 0.8 1.0