Electroweak Interactions

1
Electroweak Interactions A Correlated Basis
Theory Approach Progress Report
Shannon T. Cowell Los Alamos National Laboratory
Collaborators J Carlson (LANL) A
Hayes-Sterbenz VR Pandharipande (UIUC) R
Schiavilla (JLAB/ODU)
Advanced Many-Body Methods for Nuclear
Structure ECT July, 2007
2
Outline

• Introduction
• Correlated Basis Theory - 101
• Effective Interaction Operators
• One-body Response
• Two-body Response
• Conclusion Outlook

3
Motivations
Modern simulations of astrophysical environments
(core-collapse supernovae, neutron star cooling)
use simple approximations of electroweak
interaction.
Calculations predict an enhancement factors of
2-3 in n mean free paths Hot Asymmetric Matter
Skyrme-like effective interactions and RPA
(Reddy, et.al.) Hot Neutron Matter Fermi Liquid
Theory (Iwamoto and Pethick)
Need reliable predictions over a range of
densities, temperatures and proton fractions
that are consistent with the underlying EOS
4
Motivations
Modern simulations of astrophysical environments
(core-collapse supernovae, neutron star cooling)
use simple approximations of electroweak
interaction. Need reliable predictions over a
range of densities, temperatures and proton
fractions that are consistent with the underlying
EOS
Modern n experiments employ nuclei as
detectors n-nucleus reaction must be well
understood to extract reliable information about
n properties.
5
Microscopic (Quantum Monte Carlo) Calculations
with BARE realistic interactions and BARE
operators AGREE with experiment (Schiavilla and
Wiringa (2002))
Reduced GT Matrix elements
But Quantum Monte Carlo calculations are possible
only for A 12
6
n nucleus Approximation Methods
n12C (LSND)
Hayes Towner PRC 61
pf shell model GT predictions incorporate a
factor of 0.6 to quench the strength
Improvements are needed to calculate interaction
rates from bare realistic interactions in
larger nuclei and nucleon matter.
7
Motivations
Modern simulations of astrophysical environments
(core-collapse supernovae, neutron star cooling)
use simple approximations of electroweak
interaction. Need reliable predictions over a
range of densities, temperatures and proton
fractions that are consistent with the underlying
EOS
Modern n experiments employ nuclei as
detectors n-nucleus reaction must be well
understood to extract reliable information about
n properties.
Improvements are needed to calculate interaction
rates from bare realistic interactions in larger
nuclei and nucleon matter.
Electron Scattering plenty of data against
which to test theory Quasi-elastic scattering on
nuclei probes energy and momentum distributions
of nucleons large momentum transfers mean short
range correlations (and MEC) MUST be included.
8
Why dont approximations do as well?
Nuclear wave functions are highly correlated
difficult to calculate nuclear matrix elements
Traditionally, correlations included using an
effective interaction
basis states
model states
HOWEVER
Correlations must be included consistently in ALL
operators.
Correlated Basis Theory (CBT) allows for a
consistent and systematic treatment of BOTH the
effective interactions and effective operators.
9
Needed

• Effective Interaction
• (preferably obtained using realistic bare
  interactions)
• Effective electro-weak operators
• (consistent with effective interaction)
• Manageable model space

10
Correlated Basis Theory 101
Assumption
Model Space Uncorrelated Fermi gas states (Shell
Model)
Pair Correlation Operator
pair correlations function,
,obtained by minimizing the energy of SNM using
AV18 UIX with FHNC-SOC methods. (Morales,
Ravenhall, Pandharipande PRC 66 (2002))
Original FHNC calculation included spin-orbit
terms ignored here.
11
Correlation Functions
central
12
I Effective Interaction
Momentum dependent
Two-body cluster level
Use static part of v8
13
Effective Two-body Interaction
E/A for SNM and PNM
We ignore momentum dependent term
Minimum at r0.16 not obtained. Requires 3-body
terms.
Add r dependent delta function interaction to
correct.
14
II Effective Operators
One-body weak operators
Two-Body cluster approximation
15
II Effective Operators One-Body Weak

• Quenching depends upon
• density
• momenta
• proton fraction

Dominant contribution reduced probability that
the quasinucleon is neutron in initial state
proton in final state (spin isospin for GT)
16
II Effective Operators
One-body weak operators
Two-Body cluster approximation
17
Electroweak Operators
Charge Component
Pion charge contribution is small (ignored)
Current Component
Neutral Weak Currents wont be shown here
18
III Model Space
Model the system as non-interacting Fermi gas in
a periodic box
Single particle states
Methods developed for T0 Symmetric Nuclear
Matter (SNM) box filled up to Fermi
momentum Finite Temperature Asymmetric Matter
box filled using Fermi-Dirac statistics
19
Response Functions
Recall
Many-body effective operator
1-body bare operator
We start with the 1p-1h contribution
Final State 1p-1h excitations
Have employed two approximations Correlated
Hartree-Fock 1p-1h excitations do not
interact. Correlated Tamm-Dancoff
Approximation 1p-1h excitations
interact Always compared to non-interacting FG
20
SNM Weak Response FunctionsTamm-Dancoff
Approximation
25 reduction
Strength Shifted
21
SNM Weak Response FunctionsTamm-Dancoff
Approximation
22
Weak Operators electron capture Correlated
Tamm-Dancoff Approximation
r0.16
Strength of response shifted to larger energy
transfers
CHF CTDA very similar for large q
23
Neutrino Mean Free Path in SNM
Factor of 2-3 ½ enhancement
Sensitive to low w,q response
24
Sum Rules Gamow-Teller
Variational
Fermi Gas
Many p-h contribution large (small for charge)
1p-1h
Many p-h contribution at high w
Variational
1p-1h
Fermi Gas
25
How do we compare? Neutrino Mean Free Paths (T0
SNM)
Skyrme-like force fit to E/A and
susceptibilities.
m .64m
m with Obare
Previous approximations too small
Depends upon low w response which agrees well.
26
How do we compare? Electron Capture Response
(T10xp0.4)
q 2.0 fm-1
r 0.16 fm-3
m .64m
CTDA
So different we didnt compare
27
Electromagnetic Charge Response Correlated
Hartree-Fock
FG
CBT Obare

20 reduction
CBT
Strength shifted to higher w
q2.0 fm-1
28
Electromagnetic Current Operator Correlated
Hartree-Fock
FG
Full
q1.5 fm-1
r0.08 fm-3
1-body
1000
1-body (80) MEC (20)
29
Have a good handle on lowest order 1p-1h
calculations
BUT they do not saturate the sum rules!
What if we include the two-body contribution?
Methods developed can be used to address neutron
star cooling
Final State 2p-2h excitations
The size of the 2p-2h basis introduces
complications
30
Complications Shear size of it all!

• Approximation schemes limited
• 1p-1h states could be orthogonalized w.r.t the
  Hamiltonian (TDA)
• Results differ significantly for low momentum
  transfers
• 2p-2h basis too large for matrix manipulations
• Only Hartree-Fock currently possible
• Appropriate only for large momentum transfers
  (QE)

MEC currents involve 3-dimensional
integration limits the size of the system to 28
in many cases poor resolution
31
Complications Orthogonality
Orthogonalization procedure use a combination of
Schmidt and Löwdin orthogonalizations. (Fantoni
Pandharipande prescription)
Only consider 1p-1h contributions at the
two-body cluster level, orthogonization of 2p-2h
states is negligible (1/V) Even with this
simplification, the size of the 2p2h basis makes
orthogonalization difficult Cannot currently
treat all MEC
32
PRELIMINARY
2p-2h Gamow-Teller Response
OC most important at high w response
r0.16 fm-3
q2.87 fm-1
Nn,p54
1p-1h corrections 15-20
33
PRELIMINARY
2p-2h Electromagnetic Charge Response
q2.25 fm-1
r0.16 fm-3
28 particles
34
1p-1h 2p-2h GT Sum Rule
PRELIMINARY
THINK the bugs are out.


OC 10-20

OC 15-30



OC corrections to 1p-1h component included but lt
5
35
The miles ahead
These are not quantitative results

• This collaborations
• Finish MEC contributions to 1p-1h (weak
  interactions)
• Extend 2p-2h calculation to include current MEC
• (computational obstacles)
• Add 3-body terms (interactions and clusters)
• Make more approachable for use in simulations
• Extend to finite nuclei - combine CBT with shell
  model methods
• Someone else
• Extend variational calculations to higher/lower
  densities
• More accurate temperature dependence
• Proton fraction dependence

Thank you!
36
Most Modern Calculations Use
Effective N-N Interactions ( )
With bare operators
RPA, Shell Model, etc.
Overestimates at high q
Electron Scattering on 12C
0 2 Form Factor
Underestimates at low q
0hw Shell Model Using
37
(No Transcript)
38
(No Transcript)