ab initio Hamiltonian approach to light nuclei
and to quantum field theory
James P. Vary
Iowa State University
International Symposium on Nuclear Physics
BARC and TIFR
Mumbai, India
December 8-12, 2009
Ab initio nuclear physics - fundamental questions
 Can we develop a predictive theory from QCD to nuclear reactions
 What controls nuclear saturation?
 How does the nuclear shell model emerge from the underlying theory?
 What are the properties of nuclei with extreme neutron/proton ratios?
 Can nuclei provide precision tests of the fundamental laws of nature?
Jaguar
Franklin
Blue Gene/p
Atlas
QCD
Theory of strong interactions
χEFT
Chiral Effective Field Theory
Big Bang
Nucleosynthesis
& Stellar Reactions
r,s processes
& Supernovae
www.unedf.org
http://extremecomputing.labworks.org/nuclearphysics/report.stm
Fundamental Challenges for a Successful Theory
 What is the Hamiltonian
 How to renormalize in a Hamiltonian framework
 How to solve for non-perturbative observables
 How to take the continuum limit (IR -> 0, UV-> !)
Focii of the both the Nuclear Many-Body
and Light-Front QCD communities!
Realistic NN & NNN interactions
High quality fits to 2- & 3- body data
Meson-exchange
NN: AV18, CD-Bonn, Nijmegen, . . .
NNN: Tucson-Melbourne, UIX, IL7, . . .
Need
Consistent
EW operators
Need
Improved NNN
Need
Chiral EFT (Idaho)
Fully derived/coded
NN: N3LO
N3LO
NNN: N2LO
4N: predicted & needed for consistent N3LO
Inverse Scattering
NN: JISP16
Need
JISP40
Consistent NNN
The Nuclear Many-Body Problem
The many-body Schroedinger equation for bound states consists
A
of 2( Z ) coupled second-order differential equations in 3A coordinates
using strong (NN & NNN) and electromagnetic interactions.
Successful Ab initio quantum many-body approaches
Stochastic approach in coordinate space
Greens Function Monte Carlo (GFMC)
Hamiltonian matrix in basis function space
No Core Shell Model (NCSM)
Cluster hierarchy in basis function space
Coupled Cluster (CC)
Comments
All work to preserve and exploit symmetries
Extensions of each to scattering/reactions are well-underway
They have different advantages and limitations
No Core Shell Model
A large sparse matrix eigenvalue problem
H = Trel + VNN + V3N + •••
H !i = E i !i
"
!i = # Ani $n
n= 0
Diagonalize { $m H $n
•
•
•
•
}
Adopt realistic NN (and NNN) interaction(s) & renormalize as needed - retain induced
many-body interactions: Chiral EFT interactions and JISP16
Adopt the 3-D Harmonic Oscillator (HO) for the single-nucleon basis states, α, β,…
Evaluate the nuclear Hamiltonian, H, in basis space of HO (Slater) determinants
(manages the bookkeepping of anti-symmetrization)
Diagonalize this sparse many-body H in its “m-scheme” basis where [α =(n,l,j,mj,τz)]
!n = [a"+ ••• a#+ ]n 0
n = 1,2,...,1010 or more!
•
Evaluate observables and compare with experiment
Comments
• Straightforward but computationally demanding => new algorithms/computers
• Requires convergence assessments and extrapolation tools
• Achievable for nuclei up to A=16 (40) today with largest computers available
Experiment-Theory comparison
RMS(Total E) 0.739 MeV (2%)
RMS(Excit’n E) 0.336 MeV (1%)
GTexp 2.161 vs GTthy 2.198(7) (2%)
HH+EFT*: Vaintraub, Barnea & Gazit,
PRC79,065501(2009);arXiv0903.1048
Solid - JISP16 (bare)
Dotted - Extrap. B
2,1
1,0
2,0
0,1
3,0
1,0
P. Maris, A. Shirokov and J.P. Vary, ArXiv 0911.2281
How good is ab initio theory
for predicting large scale collective motion?
Quantum rotator
25
Experiment = 3.17
Theory(N max = 10) = 3.54
20
15
E (MeV)
0+; 0
Ĵ 2 J(J + 1)! 2
EJ =
=
2I
2I
E4 20
=
= 3.33
E2
6
3.535
3.173
3.470
3.333
3.129
2.927
2.994
12C
E /E
4+
4+; 0
!! = 20MeV
4+; 0
E4
10
2+; 0
5
2+; 0
E2
0
0+; 0
0+; 0
Exp
N
=10
max
N
=8
max
N
=6
max
Dimension = 8x109
N
=4
max
N
=2
max
N
=0
max
2+
12C
- At the heart of matter
The first excited 0+ state of 12C, the “Hoyle
state”, is the key state of 12C formation in the
triple-alpha fusion process that occurs in stars.
Due to its role in astrophysics and the fact that
carbon is central to life, some refer to this as
one of the “holy grails” of nuclear theory.
Many important unsolved problems of the Hoyle state:
Microscopic origins of the triple-alpha structure are unsolved
Breathing mode puzzle - experiments disagree on sum rule fraction
Laboratory experiments to measure the formation rate are very
difficult - resulting uncertainties are too large for predicting
the 12C formation rate through this state that dictates the size
of the iron core in pre-supernova stars
Conclusion: Need ab initio solutions of the Hoyle state with no-core
method that accurately predicts the ground state binding energy
==> parameter free predictions for the Hoyle state
achievable with petascale within 1-2 years
P. Maris, J.P. Vary and A. Shirokov, Phys. Rev. C. 79, 014308(2009), ArXiv:0808.3420; and to be published
•
•
ab initio NCSM with χEFT Interactions
Only method capable to apply the χEFT NN+NNN interactions to all p-shell nuclei
Importance of NNN interactions for describing nuclear structure and transition rates
P. Navratil, V.G. Gueorguiev,
J. P. Vary, W. E. Ormand
and A. Nogga,
PRL 99, 042501(2007);
ArXiV: nucl-th 0701038.
Extensions and work in progress
•
•
•
•
•
•
Better determination of the NNN force itself, feedback to χEFT (LLNL, OSU, MSU, TRIUMF)
Implement Vlowk & SRG renormalizations (Bogner, Furnstahl, Maris, Perry, Schwenk & Vary, NPA 801,
21(2008); ArXiv 0708.3754)
Response to external fields - bridges to DFT/DME/EDF (SciDAC/UNEDF)
- Axially symmetric quadratic external fields - in progress
- Triaxial and spin-dependent external fields - planning process
Cold trapped atoms (Stetcu, Barrett, van Kolck & Vary, PRA 76, 063613(2007); ArXiv 0706.4123) and
applications to other fields of physics (e.g. quantum field theory)
Effective interactions with a core (Lisetsky, Barrett, Navratil, Stetcu, Vary)
Nuclear reactions & scattering (Forssen, Navratil, Quaglioni, Shirokov, Mazur, Vary)
Jaguar PF award of 30,000,000 cpu hours
Transition
Experimental
B(GT) : Be! B
10
10
B(GT) : 14 N!14 C(" 2 + )
Calculated
0.08
0.06 (3 - body : 0.066)
0.92(33)
2.61 (CD - Bonn)
CD-Bonn
1.62 (JISP16)
1.62 JISP16
Collaborators: David Dean, Pieter Maris, Hai Ah Nam, Petr Navratil, Erich Ormand
GT(T=0 T=1 states)
Nmax = 8, NN+NNN
Results under analysis
Jaguar PF - 110,000 cores
“Early Science”
Preliminary
Measured = 2.4(1)E-06
Calculated = 3.7(6)E-02
Assuming isospin purity
Results with JISP16
Iowa State - ORNL - LLNL collaboration
RMS Eabs (45 states) = 1.5 MeV
RMS Eex (32 states) = 0.7 MeV
P. Maris, J.P. Vary and A. Shirokov, Phys. Rev. C. 79, 014308(2009), ArXiv:0808.3420
Descriptive Science
Predictive Science
Proton-Dripping Fluorine-14
First principles quantum solution for yet-to-be-measured unstable nucleus 14F
 Apply ab initio microscopic nuclear theory’s predictive power to major test case
 Robust predictions important for improved energy sources
 Providing important guidance for DOE-supported experiments
 Comparison with new experiment will improve theory of strong interactions
 Dimension of matrix solved for 14 lowest states ~ 2x109
 Solution takes ~ 2.5 hours on 30,000 cores (Cray XT4 Jaguar at ORNL)
Predictions:
Binding energy: 72 ± 4 MeV indicating
that Fluorine-14 will emit (drip) one
proton to produce more stable Oxygen-13.
Predicted spectrum (Extrapolation B)
for Fluorine-14 which is nearly identical
with predicted spectrum of its “mirror”
nucleus Boron-14. Experimental data
exist only for Boron-14 (far right column).
Ab initio Nuclear Structure
Ab initio Nuclear Reactions
*Navratil*
Phase shifts in PRL101, 092501 (2008)
and PRC79, 044606 (2009); arXiv0901.0950;
Cross sections and polarizations to be published
Ab initio Nuclear Structure
Ab initio Quantum Field Theory
Light cone coordinates and generators
+
!
M = P P0 ! P P1 = (P ! P )(P0 + P1 ) = P P = KE
2
0
1
Equal time
x0
H=P0
x1
P1
0
1
Discretized Light Cone Quantization (c1985)
Basis Light Front Quantization
!
! +
* !
! ( x ) = # [ f" ( x ) a" + f " ( x ) a" ]
"
where {a" } satisfy usual (anti-) commutation rules.
!
Furthermore, f " ( x ) are arbitrary except for conditions :
! * ! 3
Orthonormal: $ f " ( x ) f " ' ( x )d x = %"" '
! * !
!
3 !
f
x
f
x
'
=
%
x
&
x
# " ( ) " ( ) ( ')
Complete:
"
!
=> Wide range of choices for f a ( x ) and our initial choice is
!
f! ( x ) = Ne ik x #n,m ($,% ) = Ne ik
+ "
+ "
x
f n,m ($) & m (% )
Set of transverse 2D HO modes for n=0
m=0
m=1
m=3
m=2
m=4
J.P. Vary, H. Honkanen, J. Li, P. Maris, S.J. Brodsky, A. Harindranath,
G.F. de Teramond, P. Sternberg, E.G. Ng and C. Yang, ArXiv:0905:1411
Symmetries & Constraints
! bi = B
i
! (m i + s i ) = J z
i
! ki = K
i
! [2ni + | m i | +1] " N max
Finite basis regulators
i
Global Color Singlets (QCD)
Light Front Gauge
Optional - Fock space cutoffs
Hamiltonian for “cavity mode” QCD in the chiral limit
Why interesting - cavity modes of AdS/QCD
H = H 0 + H int
Massless partons in a 2D harmonic trap solved in basis functions
commensurate with the trap :
Λλ
2M 0 # 1
$ [2ni + | m i | +1]
H 0 ! 2M 0 PC" !
K i xi
with %& defining the confining scale as well as the basis function scale.
Initially, we study this toy model of harmonically trapped partons in the
chiral limit on the light front. Note Kx i = k i and BC's will be specified.
Nucleon radial excitations
j
M
M
EXP
j
EXP
0
M BLFQ
j
M BLFQ
0
1
1.53
2 = 1.41
2
1.82
3 = 1.73
Quantum statistical mechanics of trapped systems in BLFQ:
Microcanonical Ensemble (MCE)
Develop along the following path:
Select the trap shape (transverse 2D HO)
Select the basis functions (BLFQ)
Enumerate the many-parton basis in unperturbed
energy order dictated by the trap - obeying all symmetries
Count the number of states in each energy interval
that corresponds to the experimental resolution = > state density
Evaluate Entropy, Temperature, Pressure, Heat Capacity,
Gibbs Free Energy, Helmholtz Free Energy, . . .
Note: With interactions, we will remove the trap and examine
mass spectra and other observables.
Microcanonical Ensemble (MCE) for Trapped Partons
Solve the finite many - body problem :
H !i = E i !i
and form the density matrix :
"(E) #
&!
i
!i
i$ E i = E ±%
Statistical Mechanical Observables :
O =
Tr( "O)
Tr( ")
Tr( ") # '(E) = Total number of states in MCE at E
'(E) # ( (E)%
( (E) # Density of states at E
S(E,V ) # k ln('(E))
* )S * )E 1 )S
#
; P # T, / ; CV # , /
+ )V . E
+ )T .
T )E
E =!
i
1
[2ni + | m i | +1]
xi
J.P. Vary, H. Honkanen, J. Li, P. Maris, S.J. Brodsky, A. Harindranath,
G.F. de Teramond, P. Sternberg, E.G. Ng and C. Yang, ArXiv:0905:1411
Non-interacting QED cavity mode with zero net charge
Photon distribution functions
Labels: Nmax = Kmax ~ Q
“Weak” coupling:
Equal weight to low-lying states
“Strong” coupling:
Equal weight to all states
J.P. Vary, H. Honkanen, J. Li, P. Maris, S.J. Brodsky, A. Harindranath,
G.F. de Teramond, P. Sternberg, E.G. Ng and C. Yang, ArXiv:0905:1411
J.P. Vary, H. Honkanen, J. Li, P. Maris, S.J. Brodsky, A. Harindranath,
G.F. de Teramond, P. Sternberg, E.G. Ng and C. Yang, ArXiv:0905:1411
without color
with color but no restriction
with color and space-spin
degeneracy
Jun Li, PhD Thesis 2009, Iowa State University
Elementary vertices in LF gauge
QED & QCD
QCD
Cavity mode QED
M0=me=0.511
Mj=1/2
gQED= [4πα]1/2
lepton & lepton-photon Fock space only
k photon ! " ! m e
Preliminary
Next steps
 Increase basis space size
 Remove cavity
H. Honkanen, et al., to be published
QFT Application - Status
 Progress in line with Ken Wilson’s advice = adopt MBT advances
 Exact treatment of all symmetries is challenging but doable
 Important progress in managing IR and UV cutoff dependences
 Advances in algorithms and computer technology crucial
 First results with interaction terms in QED - anomalous moment
 Community effort welcome to advance the field dramatically
Collaborators in India
Avaroth Harindranath, Saha Institute, Kolkota
Dipankar Chakarbarti, IIT, Kanpur
Asmita Mukherjee, IIT, Mumbai
Daya Kulshreshtha, Univ. of Delhi
Usha Kulshreshtha, Univ. of Delhi
Summary
ab initio nuclear theory - building bridges
Standard Model (QCD + Electroweak)
NN + NNN interactions & effective EW operators
quantum many-body theory
describe/predict experimental data
U
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D
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