David J' Dean

1
0n-bb decay and the nuclear many-body problem
David J. Dean ORNL
With input from Jon Engel (UNC)

• Outline
• Where we are today
• Potential ways to improve the calculations
• Increasing the Hilbert space
• Effective operators for small spaces
• Using Factorization schemes
• Using Coupled Cluster theory
• Conclusions

2
Nuclear structure landscapes

• Main theory goals
• Identify/investigate many-body
• methods that will extend to RIA
• Generate effective interactions
• Make reliable predictions
• Guide experimental efforts
• Use NN and 3N forces to build nuclei
• Various approaches to
• low-energy nuclear theory
• Coupled-Cluster theory
• Shell Model Monte Carlo
• DMRG/Factorization
• shell model diagonalization
• Continuum shell models
• HFB
• QRPA
• TDHF

3
One approach to the problem Green Function Monte
Carlo (ANL/LANL/UIUC)

• Since 1982
• algorithms
• Variational MC
• AV18
• Computing

• Indicate the need for
• 3 (and 4?) body interactions
• Future prospects
• A12 by 2003/2004
• triple alpha burning
• Reaction aspects
• NNN studies

For A10, 1.5 Tflop-hours/state For A12, 50
Tflop-hours/state
4
Two basic approaches have been applied to
bb-decay problem
5
Nuclear physics of the problem
6
Present published results
Kill outliers Factor of 3 in Cmm Assume T1/2
4E-27 years
7
What the calculations predict
Caurier et al
8
Microscopic nuclear structure theory
Begin with a bare NN (3N) Hamiltonian

• Solve the quantum many-body problem
• Easier said than done due to combinatorial
• growth of the problem as a function of
  particles.

Oscillator single-particle basis states
Many-body basis states
9
Choice of model space and the G-matrix
Q-Space
P-Space
ph intermediate states
10
Solving the quantum many-body problem in a basis
Many-body basis states
Reference Slater determinant

• Methods of solution
• Diagonalize Hab
• Determine the optimal (sometimes correlated)
  basis (Papenbrock)
• Reformulate problem as a path-integral (AFMC
  SMMC)
• Resum of quantum many-body perturbation theory
  diagrams

11
Diagonalization configuration-interaction,
interacting shell model
Yields eigenfunctions which are linear
combinations of particle-hole amplitudes
1p-1h
2p-2h

• Advantages
• Detailed spectroscopic information available
• Wave functions calculated and stored
• Disadvantages
• Dimension of problem increases dramatically with
  the
• number of active particles (combinatorial
  growth).
• disconnected diagrams enter if truncated

12
Efficient basis set selection (Other slide show)
Papenbrock Dean, PRC67,051303(R)
(2003) Papenbrock, Juodagalvis, Dean, PRC69,
024313 (2004)
13
Use Many-body perturbation theory to modify the
operator
The transformation operator
Effective operators
For initial application see Engel and Vogel
nucl-th/0311072
14
Another fascinating tool Coupled Cluster Theory

• Some interesting features of CCM
• Fully microscopic
• Size extensive
• only linked diagrams enter
• Size consistent
• the energy of two non-interacting
• fragments computed separately is the same as
  that
• computed for both fragments simultaneously
• Capable of systematic improvement
• Amenable to parallel computing

Computational chemistry 100s of publications in
2002 (Science Citation Index) for applications
and developments.
15
Coupled Cluster Theory
Correlation operator
Correlated Ground-State wave function
Reference Slater determinant
Energy

• With all Ts the spectrum of H is the
• same as the spectrum of the
• similarity transformed H formally valid
• In practice E closely approximates a
• variational theory when T is truncated

Amplitude equations
Dean Hjorth-Jensen, PRC submitted
2003 Kowalski, Dean, Hjorth-Jensen, Papenbrock,
Piecuch, PRL, in press 2004
16
Ground states of oxygen
Use realistic interactions, G-matrix
renormalization CCSD results
Dean Hjorth-Jensen (PRC, submitted)
17
Correcting the CCSD results by non-iterative
methods
Find a method that will yield the
complete diagonalization result in a given model
space How do we obtain the triples correction?
How do our results compare with exact results
in a given model space, for a given Hamiltonian?
Completely Renormalized Coupled Cluster
Theory P. Piecuch, K. Kowalski, P.-D. Fan,
I.S.O. Pimienta, and M.J. McGuire, Int. Rev.
Phys. Chem. 21, 527 (2002)
18
16O in four major oscillator shells

• Relative size of terms
• a) T1 and T2 of similar order
• b) T1T2 disconnected
• gtgt T3 connected triples
• c) diff between CISD and CISDT
• comes mainly from T1T2
• d) If T3 were large CCSD(T)
• would be far below CCSD
• 2) Size extensive nature of CC
• 3) CCSD CR-CCSD(T) bring
• T13T2, T1T22 T23 not in CISDTQ
• 4) Scaling

Kowalski, Dean, Hjorth-Jensen, Piecuch, PRL in
press (2004)
19
Conclusions and Perspectives

• Solutions to nuclear many-body problems requires
  extensive use of
• computational and mathematical tools.
  Numerical analysis becomes
• extremely important methods from other fields
  do help us.
• Nuclear theory progress is not presently tied to
  bb-decay it is tied to
• RIA and low-energy nuclear structure
  experiments.
• Big problems include correlations in Hilbert
  space, size of space,
• derivation of effective interactions and
  operators.
• bb-decay, near term plans.
• Do Ge76 by opening the f7/2 and including the
  d5/2 (others?)
• (with factorization method)
• limits (from small space, and larger space).
• Solve left wave function in CCSD Compute
• matrix element (Moores law).
• Reproduce/predict as much data as possible on a
  given nucleus
• (GT, 2nu-bb, M1,.)