Hamiltonian Light Front Field Theory:Recent Progress and Tantalizing Prospects

James P. VaryIowa State University

Light Cone 2011Dallas, Texas

May 23 - 27, 2011

Abstract

Fundamental theories, such as Quantum Chromodynamics (QCD) and Quantum Electrodynamics (QED) promise great predictive power spanning phenomena on all scales from the microscopic to cosmic scales. However, new non-perturbative tools are required to build bridges from one scale to the next. I will outline recent theoretical and computational progress to build these bridges and provide illustrative results for Hamiltonian Light Front Field Theory. One key area is our development of basis function approaches that cast the theory as a Hamiltonian matrix problem while preserving a maximal set of symmetries [1]. Regulating the theory with an external field that can be removed to obtain the continuum limit offers additional advantages as we showed recently in an application to the anomalous magnetic moment of the electron [2]. Recent progress capitalizes on algorithm and computer developments for setting up and solving very large sparse matrix eigenvalue problems. Matrices with dimensions of 10 billion basis states are now solved routinely in leadership-class computers for their low-lying eigenstates and eigenfunctions. This work was supported in part by US DOE Grant DE-FG02-87ER40371 [1] J. P. Vary, H. Honkanen, Jun Li, P. Maris, S. J. Brodsky, A. Harindranath, G. F. de Teramond, P. Sternberg, E. G. Ng, C. Yang, “Hamiltonian light-front field theory in a basis function approach”, Phys. Rev. C 81, 035205 (2010); arXiv nucl-th 0905.1411[2] H. Honkanen, P. Maris, J. P. Vary and S. J. Brodsky, “Electron in a transverse harmonic cavity”, Phys. Rev. Lett. 106, 061603 (2011); arXiv: 1008.0068

Ab initio nuclear physics - hierarchy of fundamental questions

Can hadron structures and their interactions be derived from QCD?

Can nuclei provide precision tests of the fundamental laws of nature?

What controls nuclear saturation - 3-nucleon interactions?

How does the nuclear shell model emerge from the underlying theory?

What are the properties of nuclei with extreme neutron/proton ratios?

Jaguar Franklin/Hopper Blue Gene/p Atlas

Nuclear Structure

QCD

Applications in astrophysics, defense, energy, and medicine

Bridging the nuclear physics scales

- D. Dean, JUSTIPEN Meeting, February 2009

http://extremecomputing.labworks.org/nuclearphysics/report.stm

Effective Nucleon Interaction

(Chiral Perturbation Theory)

R. Machleidt, D. R. Entem, nucl-th/0503025

Chiral perturbation theory (χPT) allows for controlled power series expansion

€

Expansion parameter : Q

Λχ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

υ

, Q − momentum transfer,

Λχ ≈1 GeV , χ - symmetry breaking scale

Within χPT 2π-NNN Low Energy Constants (LEC) are related to the NN-interaction

LECs {ci}.

Terms suggested within theChiral Perturbation Theory

Further renormalization is necessarysince momentum transfers still too high,reaching ~ 0.6 GeV/c

CD CE

• 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 or renormalized Heff, 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)]

• 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

€

Φn = [aα+ • • • aς

+]n 0

€

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

No Core Shell Model

A large sparse matrix eigenvalue problem

€

H = Trel + VNN + V3N + • • •

H Ψi = E i Ψi

Ψi = Ani

n= 0

∞

∑ Φn

Diagonalize Φm H Φn{ }

NOTE: No known limitations in principle on choice of

Hamiltonian

Basis space

Renormalization scheme

P. Maris, P. Navratil, J. P. Vary, to be published

Note additional predicted states!Shown as dashed lines

(CD= -0.2)

“Anomalous Long Lifetime of Carbon-14”

ImpactObjectives Solve the puzzle of the long but

useful lifetime of 14C

Determine the microscopic origin of the suppressed β-decay rate

Establishes a major role for strong 3-nucleon forces in nuclei Verifies accuracy of ab initio microscopic nuclear theory Provides foundation for guiding DOE-supported experiments

Dimension of matrix solved for 8 lowest states ~ 1x109

Solution takes ~ 6 hours on 215,000 cores on Cray XT5 Jaguar at ORNL

“Scaling of ab initio nuclear physics calculations on multicore computer architectures," P. Maris, M. Sosonkina, J. P. Vary, E. G. Ng and C. Yang, 2010 Intern. Conf. on Computer Science, Procedia Computer Science 1, 97 (2010)

3-nucleon forces suppress critical component

net decay rate Is very small

Descriptive Science

Predictive Science

“Proton-Dripping Fluorine-14”

ImpactObjectives Apply ab initio microscopic

nuclear theory’s predictive power to major test case

Deliver robust predictions important for improved energy sources Provide important guidance for DOE-supported experiments Compare with new experiment to improve theory of strong interactions

P. Maris, A. Shirokov and J.P. Vary, Phys. Rev. C 81 (2010) 021301(R)

V.Z. Goldberg et al., Phys. Lett. B 692, 307 (2010)

Experiment confirmsour publishedpredictions!

Dimension of matrix solved for 14 lowest states ~ 2x109

Solution takes ~ 2.5 hours on 30,000 cores (Cray XT4 Jaguar at ORNL)

“Scaling of ab-initio nuclear physics calculations on multicore computer architectures," P. Maris, M. Sosonkina, J. P. Vary, E. G. Ng and C. Yang, 2010 Intern. Conf. on Computer Science, Procedia Computer Science 1, 97 (2010)

Recent noteworthy accomplishments of the ab initio no core shell model (NCSM) and no core full configuration (NCFC)

Described the anomaly of the nearly vanishing quadrupole moment of 6Li

Established need for NNN potentials to explain neutrino -12C cross sections

Explained quenching of Gamow-Teller transitions (beta-decays) in light nuclei

Obtained successful description of A=10-13 nuclei with chiral NN+NNN potentials

Explained ground state spin of 10B by including chiral NNN potentials

Developed/applied methods to extract phase shifts (J-matrix, external trap)

Successful prediction of low-lying 14F spectrum (resonances) before experiment

Explained the mystery of the anomalous long lifetime of 14C, useful for archeology

x0

x1

H=P0

P1

Light cone coordinates and generators

Equal time“Instant Form”€

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

Some perspectives on Hamiltonian applications to LFQ

1+1 dimensional theories DLCQ initiated many applications (Review: Brodsky, Pauli, Pinsky) Spontaneous symmetry breaking (Chakrabarti, Martinovic, Harindranath,…) Critical phenomena - e.g. kink condensation (Chakrabarti, …) Zero modes, boundary conditions, regulators, …(Bassetto, McCartor,…) QCD SU(3) color singlet structures (Hornbostel)

2+1 dimensional theories QCD - Bloch and SRG Heff treatments (Chakrabarti, Harindranath)

3+1 dimensional theories QED - LF wave equations (Hiller, Chabysheva, Brodsky, …) QCD - Transverse lattice (Dalley, van de Sande, Chakrabarti,….) SRG approach (Wilson, Glazek, Perry, … ) DIS - Q2 evolution (Zhang, Harindranath,…) “Near” LFQ (Franke, Prokhvatilov, Paston, Pirner, Naus, Lenz, Moniz, .…) Consistent quantization (D. Kulshreshtha, U. Kulshreshtha,…) Renormalization/Reg’n (Ji, Bakker, Karmanov, Mathiot, Smirnov, Grange’, …) DVCS (Brodsky, Mukherjee, Chakrabarti, …) BLFQ (this talk) QED - BLFQ approach (Zhao poster at this meeting)

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:

*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, PRC 81, 035205 (2010). ArXiv:0905:1411

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, PRC 81, 035205 (2010). ArXiv:0905:1411

• Enumerate Fock-space basis subject to symmetry constraints• Evaluate/renormalize/store H in that basis• Diagonalize (Lanczos)• Iterate previous two steps for sector-dep. renormalization• Evaluate observables using eigenvectors (LF amplitudes)• Repeat previous 4 steps for new regulator(s)• Extrapolate to infinite matrix limit – remove all regulators• Compare with experiment or predict new experimental results

Steps to implement BLFQ

Above now achieved for QED test case – electron in a trapH. Honkanen, P. Maris, J.P. Vary, S.J. Brodsky,Phys. Rev. Lett. 106, 061603 (2011)

Improvements: trap independence, (m,e) renormalization, . . .X. Zhao, P. Maris, J.P. Vary, S.J. Brodsky, poster at this meeting

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

€

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.

Λλ

“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, PRC 81, 035205 (2010). ArXiv:0905:1411

Comment based on Stan Glazek’s, Craig Roberts’ and Stan Brodsky’s presentations

Operator SRG (Glazek) appears to be a natural route to obtain the effective (Q-dependent) qluon mass invoked for the Dyson-Schwinger approach (Roberts)

If the “gluon condensate” indeed exists only within thehadron and is representable by the gluon mean field, as is consistent with LFQ approach,then the mean field of the glue could be defined (derived?)at a given Nmax ~ K ~ Q. We would then increase the number of gluons until convergence is reached atthat scale Q.

That is, we adopt the scale Q as defining the border between dynamical gluons and high momentum gluons that define the gluon mean field.

For such calculations we initially retain one parameter for the strength of the SRG mean field interaction Vmf between pairs of constituent quarks and gluons.

QED & QCD

QCD

Elementary vertices in LF gauge

Initial application to QED*

**

*H. Honkanen, P. Maris, J.P. Vary, S.J. Brodsky, Phys. Rev. Lett. 106, 061603 (2011); X. Zhao, H. Honkanen, P. Maris, J.P. Vary, S.J. Brodsky, poster at this meeting** T. Heinzl, A. Ilderton and H. Marklund, Phys. Lett. B692, 250(2010); arXiv:1002.4018

*H. Honkanen, P. Maris, J.P. Vary, S.J. Brodsky, Phys. Rev. Lett. 106, 061603 (2011);X. Zhao, P. Maris, J.P. Vary, S.J. Brodsky, poster at this meeting

Invariant M2 spectra

Initial QED problemElectron in a transverse harmonic trap*

Extended and Improved QED Calculations Xingbo Zhao, Heli Honkanen, Pieter Maris, James P. Vary and Stanley J. Brodsky – Poster Session

1.Corrects some implementation errors2.Implements sector-dependent renormalization of mass and charge3.Decouples trap frequency from basis frequency to improve convergence4.Preliminary results lead to a puzzle in the electron anomalous magnetic moment => Spurious CM motion?5. Working now on alternative setup with external field removed completely to verify Schwinger moment

Stay tuned for the paper!

• δμ=(g-2)/2 vs. basis ω (at Ω=0.5MeV):

• δμ=(g-2)/2 vs. trap potential Ω:

38

g-2 for Electron in Harmonic Trap

Convergence reached at large ω or Nmax

δμ should be independent of the choice of basis in the limit of Nmax->∞

• Each point is the extrapolated result at Nmax->∞• δμ decreases with the strength of trap potential• Extrapolated value at Ω=0 larger than the result from perturbation theory• Contribution from center-of-mass motion in truncated HO basis? • Remove center-of-mass KE from Hamiltonian!

?

Additional recent progress

Derivation of all HQCD vertices in momentumrepresentation and HO basis spaces (Harindranath, Honkanen, Zhao, Wiecki, Li)

Comprehensive notes under development (All)

Jun Li’s color singlet code transferred to Wiecki and is undergoing verification tests.

Programming of additional QCD vertices under development(Zhao, Wiecki, Li)

Commencing initial applications to quarkonia

Applications of LF amplitudes to experiment - DVCS S.J. Brodsky, D. Chakrabarti, A. Harindranath, A. Mukherjee, and J.P. Vary, Phys. Letts B641, 440 (2006); Phys. Rev. D75, 14003 (2007)

Key to graphsx Bjorken variable invariant longitudinal impact parameter invariant conjugate longitudinal momentumM++ Helicity non-flip DVCS amplitudeFS Fourier SpectrumF2 DIS structure function

HadronOptics!

Applications of LF AdS/CFT amplitudes to experiment - DVCS S.J. Brodsky, D. Chakrabarti, A. Harindranath, A. Mukherjee, and J.P. Vary, Phys. Letts B641, 440 (2006); Phys. Rev. D75, 14003 (2007)

Observation

Ab initio approaches maximize predictive power& represent a theoretical and computational physics challenge

Key issue

How to achieve the full physics potential of ab initio theory

Conclusions

We have entered an era of first principles, high precision,many-body and quantum field theory for strongly interacting systems

Linking hadronic physics and the cosmosthrough the Standard Model (and beyond) is well underway

and LFQ could play a leading role

Avaroth Harindranath, Saha Institute, KolkotaDipankar Chakarbarti, IIT, KanpurAsmita Mukherjee, IIT, MumbaiStan Brodsky, SLACGuy de Teramond, Costa RicaUsha Kulshreshtha, Daya Kulshreshtha, University of DelhiXingbo Zhao, Pieter Maris, Jun Li, Paul Wiecki, Young LiHeli Honkanen, University of JyvaskylaEsmond Ng, Chou Yang, Metin Aktulga, Philip Sternberg, Lawrence Berkeley Laboratory

Collaborators on BLFQ

Thank You!