Effective range expansions and relativistic deuteron wave functions
Charge-Changing Neutrino Scattering from the Deuteron J. W. Van Orden ODU/Jlab Collaborators: T. W....
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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
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Transcript of Charge-Changing Neutrino Scattering from the Deuteron J. W. Van Orden ODU/Jlab Collaborators: T. W....
Charge-Changing Neutrino Scattering from the Deuteron
J. W. Van OrdenODU/Jlab
Collaborators:
T. W. Donnelly and Oscar MorinoMIT
W. P. FordUniversity 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
lepton plane
hadron 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 ApproximationThe impulse approximation to CCν scattering from the deuteron is defined by the Feynman diagrams:
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).
On Mass Shell
Antisymmetric scattering matrix
Deuteron Vertex Function
The invariant functions gi are given by
where
a b
is the magnitude of the three-momentum in the deuteron rest frame.
Charge conjugation matrix
The Single-Nucleon Current Operator
where
and
The weak form factors used in the calculations shown here are:
Final State InteractionsThere 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
cm (deg)
0 10 20 30 40 50 60 70 80 90
100
101
s= 5.913 GeV2s= 5.934 GeV2ReggeSAID
Elastic pp Scattering
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.
(GeV)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
q (
GeV
)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(GeV)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
d3 /(d
'd
k')
(G
eV-3
)
0
5.0x10-11
1.0x10-10
1.5x10-10
2.0x10-10
2.5x10-10
3.0x10-10
3.5x10-10
4.0x10-10
PWIAPWBADWBA
Integrated Cross Section
Maximum
10% of Maximum
Cross Section at Maximum
(GeV)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
q (
GeV
)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
(GeV)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
d3 /(d'
dk
') (
GeV
-3)
10-13
10-12
10-11
10-10
10-9
10-8
PWIAPWBADWBA
(GeV)0.0 0.1 0.2 0.3
0
2x10-9
4x10-9
6x10-9
8x10-9
1x10-8
Exclusive Cross Section
where
and
O. Moreno, T. W. Donnelly, W. P. Ford and J. W. Van Orden, Phys. Rev. D 90, 013014 (2014).
for neutrino scatteringfor anti-neutrino scattering
p2 (GeV) 0.0 0.2 0.4 0.6 0.8 1.0
10-15
10-14
10-13
10-12
10-11
10-10
10-9
PWIAPWBADWBA SAIDDWBA Regge
p2 (GeV)0.00 0.10 0.20 0.30
10-13
10-12
10-11
10-10
10-9
10-8
Exclusive Cross Section Calculations
0.00 0.05 0.100
1x10-92x10-93x10-94x10-95x10-9
p2 (GeV)0.0 0.2 0.4 0.6 0.8 1.0
10-14
10-13
10-12
10-11
10-10
10-9
10-8=0 deg=36 deg=72 deg=108 deg=144 deg=180 deg
Azimuthal Angular Dependence
p2 (GeV)0.0 0.2 0.4 0.6 0.8 1.0
10-14
10-13
10-12
10-11
10-10
10-9
10-8
MA=1.03 GeVMA=1.3GP=0
Sensitivity to the Weak Form Factors
Summary• We have produced a working calculation of CCν 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 Δ currents is underway.
p2 (GeV) 0.0 0.2 0.4 0.6 0.8 1.0
10-1410-1310-1210
-1110-1010-910-810
-710-610-510-410
-3
Comparison of Electron and Neutrino Scattering
