ab initio Hamiltonian approach to nuclear physics and to light-front field theory

James P. VaryIowa State University

High Energy Physics in the LHC EraHEP-2010

Valparaiso, ChileJanuary 4-8, 2010

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

DOE investments:~60 cpu-centuriesduring calendar ‘09

www.unedf.org

QCDTheory of strong interactions

EFTChiral Effective Field Theory

Big Bang Nucleosynthesis

& Stellar Reactions

r,s processes& Supernovae

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 interactionsHigh quality fits to 2- & 3- body data

Meson-exchange NN: AV18, CD-Bonn, Nijmegen, . . . NNN: Tucson-Melbourne, UIX, IL7, . . .

Chiral EFT (Idaho) NN: N3LO NNN: N2LO 4N: predicted & needed for consistent N3LO

Inverse Scattering NN: JISP16

NeedImproved NNN

NeedFully derived/coded

N3LO

NeedJISP40

Consistent NNN

NeedConsistent

EW operators

The Nuclear Many-Body Problem

The many-body Schroedinger equation for bound states consistsof 2( ) 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 spaceGreens Function Monte Carlo (GFMC)

Hamiltonian matrix in basis function spaceNo Core Shell Model (NCSM)

Cluster hierarchy in basis function spaceCoupled Cluster (CC)

CommentsAll work to preserve and exploit symmetries

Extensions of each to scattering/reactions are well-underwayThey have different advantages and limitations

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A

Z

• 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, , ,…• 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 [ =(n,l,j,mj,z)]

• 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

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Φn = [aα+ • • • aς

+]n 0

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n =1,2,...,1010 or more!

No Core Shell Model

A large sparse matrix eigenvalue problem

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H = Trel + VNN + V3N + • • •

H Ψi = E i Ψi

Ψi = Ani

n= 0

∞

∑ Φn

Diagonalize Φm H Φn{ }

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

P. Maris, A. Shirokov and J.P. Vary, ArXiv 0911.2281

1,0

3,0

0,12,0

2,11,0

How good is ab initio theory for predicting large scale collective motion?

Quantum rotator

EJ =J 2

2I=

J (J +1)h2

2IE4

E2

=206

=3.33

Experiment=3.17Theory(Nmax =10) =3.54

0+; 00+; 0

0

5

10

15

20

25

E

(MeV)4+; 0

0+; 0

2+; 0

4+; 0

0+; 0

2+; 0

Exp Nmax

=10 Nmax

=6 Nmax

=4 Nmax

=0Nmax

=2Nmax

=8

3.173 3.535 3.333 3.129 2.9942.9273.470 E4+

/E2+

12C hΩ =20MeV

Dimension = 8x109

E4

E2

ab initio NCSM with EFT Interactions• Only method capable to apply the EFT NN+NNN interactions to all p-shell nuclei • Importance of NNN interactions for describing nuclear structure and transition rates

• Better determination of the NNN force itself, feedback to EFT (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)

Extensions and work in progress

P. Navratil, V.G. Gueorguiev, J. P. Vary, W. E. Ormand and A. Nogga, PRL 99, 042501(2007);ArXiV: nucl-th 0701038.

P. Maris, J.P. Vary and A. Shirokov, Phys. Rev. C. 79, 014308(2009), ArXiv:0808.3420

RMS Eabs (45 states) = 1.5 MeVRMS Eex (32 states) = 0.7 MeV

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 indicatingthat Fluorine-14 will emit (drip) oneproton 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).

P. Maris, A. M. Shirokov and J. P. Vary, PRC, Rapid Comm., accepted, nucl-th 0911.2281

Ab initio Nuclear Structure

Ab initio Quantum Field Theory

x0

x1

H=P0

P1

Light cone coordinates and generators

Equal time€

M 2 = P 0P0 − P1P1 = (P 0 − P1)(P0 + P1) = P+P− = KE

Discretized Light Cone Quantization (c1985)

Basis Light Front Quantization

€

φ r

x ( ) = fα

r x ( )aα

+ + fα* r

x ( )aα[ ]α

∑

where aα{ } satisfy usual (anti-) commutation rules.

Furthermore, fα

r x ( ) are arbitrary except for conditions :

fα

r x ( ) fα '

* r x ( )d3x∫ = δαα '

fα

r x ( ) fα

* r x '( )

α

∑ = δ 3 r x −

r x '( )

=> Wide range of choices for and our initial choice is

€

fa

r x ( )

€

fα

r x ( ) = Ne ik +x −

Ψn ,m (ρ,ϕ ) = Ne ik +x −

fn ,m (ρ )χ m (ϕ )

Orthonormal:

Complete:

Set of transverse 2D HO modes for n=0

m=0 m=1 m=2

m=3 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

€

ρ

€

ϕ

€

x-

€

k = 12, n =1, m = 0

€

Ψknm = ψ k x−( ) fnm ρ( )χ φ( )

APBC : − L ≤ x− ≤ L

ψ k x−( ) =

1

2Le

iπ

Lkx −

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

€

bii

∑ = B

(m ii

∑ + s i ) = Jz

k ii

∑ = K

2ni + | m i | +1[ ] ≤ N maxi

∑

Global Color Singlets (QCD)

Light Front Gauge

Optional - Fock space cutoffs

Finite basis regulators

Hamiltonian for “cavity mode” QCD in the chiral limit

Why interesting - cavity modes of AdS/QCD

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H = H 0 + H int

Massless partons in a 2D harmonic trap solved in basis functions

commensurate with the trap :

H 0 ≡ 2M 0PC

− ≡2M 0Ω

K

1

xii

∑ 2ni + | m i | +1[ ]

with Λλ 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.

j

1 1.53

2 1.82

Nucleon radial excitations

€

M jEXP

M0EXP

€

M j

BLFQ

M 0

BLFQ

€

2 =1.41

€

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 symmetriesCount the number of states in each energy interval that corresponds to the experimental resolution = > state densityEvaluate 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

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Solve the finite many - body problem:

H Ψi = E i Ψi

and form the density matrix :

ρ(E) ≡ Ψi Ψi

i∋E i = E ±Δ

∑

Statistical Mechanical Observables :

O =Tr ρO( )Tr ρ( )

Tr ρ( ) ≡ Γ(E) = Total number of states in MCE at E

Γ(E) ≡ ω(E)Δ

ω(E) ≡ Density of states at E

S(E,V ) ≡ k ln(Γ(E))

1

T≡

∂S

∂E; P ≡ T

∂S

∂V

⎛

⎝ ⎜

⎞

⎠ ⎟E

; CV ≡∂E

∂T

⎛

⎝ ⎜

⎞

⎠ ⎟

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

€

E =1

xii

∑ 2ni + | m i | +1[ ]

Cavity mode QED with no net charge & K = Nmax

Distribution of multi-parton states by Fock-space sector

K=Nmax81012

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

“Weak” coupling:Equal weight to low-lying states

“Strong” coupling:Equal weight to all states

Non-interacting QED cavity mode with zero net charge Photon distribution functions

Labels: Nmax = Kmax ~ Q

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 colorwith color but no restrictionwith color and space-spin degeneracy

Jun Li, PhD Thesis 2009, Iowa State University

QED & QCD

QCD

Elementary vertices in LF gauge

Renormalization in BLFQ => Analyze divergences

Are matrix elements finite - No => counterterms Are eigenstates convergent as regulators removed?

Examine behavior of off-diagonal matrix elements of the vertex for the spin-flip case:As a function of the 2D HO principal quantum number, n.Second order perturbation theory gives log divergence if sucha matrix element goes as 1/Sqrt(n+1)

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

Next steps

Increase basis space size Remove cavity Evaluate form factors

H. Honkanen, et al., to be published Cavity mode QEDM0=me=0.511Mj=1/2gQED= [412

lepton & lepton-photon Fock space only

€

k photon ≈ ω ≈ me

PreliminarySchwinger perturbative result

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 Connections with results of AdS/QCD assist intuition Advances in algorithms and computer technology crucial First results with interaction terms in QED - anomalous moments Community effort welcome to advance the field dramatically

Collaborations - See Individual Slides

Avaroth Harindranath, Saha Institute, KolkotaDipankar Chakarbarti, IIT, KanpurAsmita Mukherjee, IIT, MumbaiStan Brodsky, SLACGuy de Teramond, Costa RicaUsha Kulshreshtha, Daya Kulshreshtha, University of DelhiPieter Maris, Jun Li, Heli Honkanen, Iowa State UniversityEsmond Ng, Chou Yang, Philip Sternberg, Lawrence Berkeley National Laboratory

Collaborators on BLFQ

Thank You!