Basis Light-front Approach to Hadron Structure Xingbo Zhao - - PowerPoint PPT Presentation

Basis Light-front Approach to Hadron Structure

Xingbo Zhao Institute of Modern Physics Chinese Academy of Sciences Lanzhou, China

XVIII International Conference on Hadron Spectroscopy and Structure, Guilin China, 08/20/2019

Collaborators

• Institute of Modern Physics, Chinese Academy of Sciences

– Siqi Xu, Jiangshan Lan, Kaiyu Fu, Hengfei Zhao, Chandan Mondal, Sreeraj Nair

• Iowa State University

– Shaoyang Jia, Yang Li, Weijie Du, James Vary

• University of Maryland

– Henry Lamm

• Indian Institute of Technology, Kanpur

– Dipankar Chakrabarti

• Beihang University

– Li-Sheng Geng

• University of Washington

– Gerald Miller

2

Outline

• What is light-front and why?

– Relativistic bound states

• Basis Light-front Quantization

– Many body – Rotational symmetry

• Applications:

– QED: physical electron, positronium – QCD: nucleon, light meson, heavy quarkonium

3

Two Sides of Nuclear Physics

4 Nucleons & mesons Quarks & gluons Mass scale ∼ GeV Mass scale ∼ MeV

Low Energy High Energy

𝜌 Need frame-independent wave functions

Light-front Time

• We measure nucleon structure by virtual photon
• We "see” the world at fixed light-front time (𝑢 = 𝑦' + 𝑦()

5

equal-time dynamics vs light-front dynamics

Light-front vs Equal-time Quantization

6

i ∂ ∂x+ ϕ(x+) = 1 2 P− ϕ(x+) i ∂ ∂t ϕ(t) = H ϕ(t)

[Dirac 1949]

H = P0

P− = P0 − P3 t ≡ x0 t ≡ x+ = x0 + x3 𝑄*, 𝑄+, 𝐹* , 𝐹+, 𝐾/ 𝑄, ⃗ 𝐾 v.s.

Kinematic generators:

Why Go To Light-front?

• Boost invariant light-front wave functions
• Simple vacuum = free theory vacuum + zero modes
• Hamiltonian formalism for relativistic systems

7

⟩ |proton = 𝑏 ⟩ |𝑣𝑣𝑒 + 𝑐 ⟩ |𝑣𝑣𝑒𝑕 + c ⟩ |𝑣𝑣𝑒𝑕𝑕 + 𝑒 ⟩ |𝑣𝑣𝑒𝑟@ 𝑟 +. . . . ⟩ |pion = 𝑏 ⟩ |𝑟@ 𝑟 + 𝑐 ⟩ |𝑟@ 𝑟𝑕 + c ⟩ |𝑟@ 𝑟𝑕𝑕 + 𝑒 ⟩ |𝑟@ 𝑟𝑟@ 𝑟 +. . . . . . . .

Basis Light-Front Quantization

• Eigenvalue problem for Light-front Hamiltonian
• 𝑄B : light-front Hamiltonian
• 𝑄

C B : eigenvalue

hadron mass spectrum

• | ⟩

𝛾 : eigenvector light-front wave function

• Observables

[Vary et al., 2008]

O ≡ ⟨𝛾′| I 𝑃|𝛾⟩

8

Non-perturbative 𝑄B 𝛾 = 𝑄

C B|𝛾⟩

Basis Construction

• 1. Fock-space expansion:

e.g.

• 2. For each Fock particle:
• Transverse: 2D harmonic oscillator basis: ΦL,M

N

( ⃗ 𝑞*) labeled by radial (angular) quantum number n (m); scale parameter b e.g., n=4

• Longitudinal: plane-wave basis, labeled by k
• Helicity: labeled by 𝜇

e.g. with and

9

eγ = e ⊗ γ

e = {ne,me,ke,λ e}

γ = {nγ ,mγ ,kγ ,λγ }

⟩ |𝐐𝐭 = 𝑏 ⟩ |𝑓 ̅ 𝑓 + 𝑐 ⟩ |𝑓 ̅ 𝑓𝛿 + 𝑑 ⟩ |𝛿 + 𝑒 ⟩ |𝑓 ̅ 𝑓𝑓 ̅ 𝑓 +. . . . Y |𝒇𝒒 = 𝑏 ⟩ |𝑓 + 𝑐 ⟩ |𝑓𝛿 + c ⟩ |𝑓𝑓 ̅ 𝑓 + 𝑒 ⟩ |𝑓𝑓 ̅ 𝑓𝛿 +. . . .

m=1 m=2 m=0

Basis Truncation Scheme

• Symmetries of Hamiltonian:
• Further truncation:
• Fock-sector truncation
• Net fermion number:
• Total angular momentum projection:
• Longitudinal momentum:
• Discretization in longitudinal direction

2ni+ | mi | +1

[ ] ≤ Nmax

i

∑

• “Nmax” truncation in transverse directions

n f

i i

∑

= N f (mi

i

∑

+ si) = Jz ki

i

∑

= K UV cutoff Λ~𝑐 𝑂cde; IR cutoff 𝜇~𝑐/ 𝑂cde 𝑙j = 1, 2, 3…. bosons 0.5, 1.5, 2.5 … fermions

Features of BLFQ

• Basis respects (transverse) rotational symmetry
• Basis states are eigenstates of 𝐾/
• Single-particle basis for many-body system
• (Anti-)symmetrization of identical particles
• Exact factorization of intrinsic and c.m. motion
• Harmonic oscillator basis with Nmax truncation
• Harmonic oscillator basis suitable for bound states

11

Applications to QED

• QED Lagrangian
• Derived Light-front Hamiltonian

12

L = −1 4FµνF µν + ¯ Ψ(iγµDµ − me)Ψ P

− =

Z d2x

⊥dx − F µ+∂+Aµ + i¯

Ψγ

+∂+Ψ − L

Z

− =

Z d2x

⊥dx − 1

2 ¯ Ψγ

+ m2

e + (i∂⊥)2

i∂+ Ψ + 1 2Aj(i∂

⊥)2Aj +

j + ejµAµ + e2

2 j

+

1 (i∂+)2 j

+ + e2

2 ¯ ΨγµAµ γ+ i∂+ γνAνΨ kinetic energy terms vertex interaction instantaneous photon interaction instantaneous fermion interaction

A+ = 0

( )

Light-front QCD Hamiltonian

Application to QED (I): Physical Electron

|ephysi = a|ei + b|eγi + c|eγγi + d|ee¯ ei + . . . .

𝑸+

Δ* 𝒄*

15

0.09 0.095 0.1 0.105 0.11 0.115 0.05 0.1 0.15 0.2 (Nmax = K−1/2)−1/2 α =

1 137.036

Schwinger result = 0.1125395... even Nmax/2

• dd Nmax/2

even Nmax/2 fit

• dd Nmax/2 fit

Electron g-2 & GPD E(x, t) BLFQ vs Perturbation Theory

q1 − iq2 2me E(x,0,t = q2) = e↑

phys(

q) dy−

∫

eixP+y−/2ψ (0)γ +ψ (y) e↓

phys(0) y+=0,y⊥=0

• Anomalous magnetic moment
• Less than 0.1% deviation from Schwinger result for
• Largest calculation with basis dim > 28 billion
• ae =

E(x,t → 0)dx

1

∫

ae

• X. Zhao, H. Honkanen, P. Maris, J. P. Vary, S. J. Brodsky, Phys. Letts. B737, 65 (2014)

E(x, t → 0)

• ae/e2

0.0005 0.001 0.0015 0.002 0.2 0.4 0.6 0.8 1

Nmax=K=18 Λ=1.5MeV mγ=0.085MeV Nmax=K=42 Λ=2.3MeV mγ=0.056MeV Nmax=K=578 Λ=8.7MeV mγ=0.015MeV

x

Nonperturbative

16

Application to QED (II): Positronium

18

e+ e- γ

[Kaiyu Fu et al, in preparation] lowest 8 states of Mj=0 : parity and charge conjugation parity agree with hydrogen atom.

Energy spectrum and wavefunction

19

𝐹n(𝑁𝑓𝑊)

• 2
• 1

1 2

• 0.02
• 0.01

0.00 0.01 0.02 0.03 Mj binding energy(MeV)

Nmax=28,K=29

See Kaiyu Fu’s talk Saturday (17) pm

Photon Distribution In Positronium

• In excited states photons have larger probability at small-x region
• Photon is massless, so peak is at small-x region

[Kaiyu Fu et al, in preparation]

20

• ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲

• 11S0

■

21S0

◆

23P0

▲

23P2 0.2 0.4 0.6 0.8 1 2 3 4 5 x ψ*ψ

Nmax=28, K=29

Application to Heavy Quarkonium

21

Hamiltonian

𝑄B = 2𝑕 r s 𝑒(𝑦 @ 𝜔 𝑦 𝛿u𝑈w𝜔 𝑦 𝐵uy

z{|'

𝑄B = 𝑕} r s 𝑒(𝑦 @ 𝜔𝛿+𝑈w𝜔 1 (𝑗𝜖+)} @ 𝜔𝛿+𝑈w𝜔

z{|'

• 2. Vector coupling vertex
• 3. Vector coupling with instantaneous gluon

𝐼

‚ƒ„„ + 𝐼…†L‚ = ⃗

𝑟*

} + 𝜆ˆ ⃗

𝜊*

} + 𝑛‹ }

𝑦 + 𝑛@

‹ }

1 − 𝑦 − 𝜆ˆ 𝑛‹ + 𝑛@

‹ } 𝜖z(𝑦(1 − 𝑦)𝜖z)

• 1. Kinetic Hamiltonian and confining potentials

𝑟• 𝑟} g 𝑟• 𝑟} @ 𝑟( @ 𝑟ˆ

𝑄B = 𝐼

‚ƒ„„ + 𝐼…†L‚ + 𝐼‹Ž†‹• + 𝐼‹‹Ž†‹‹

22

Energy Spectrum

Kt=11 Nmax=10 Mj=0 me=1.5GeV b=1.64GeV binst=3.2GeV k2l=0.3 k2t=0.1

OGE : Yang Li,Maris & Vary ,PRD 17

[Hengfei Zhao, In progress]

23

Wave function

State 𝐾/𝜔(1𝑇) OGE BLFQ

OGE; Li, Maris & Vary ,PRD 17

𝜃…(1𝑇) [Hengfei Zhao, In progress]

24

Wave function

State OGE BLFQ 𝜓…'(1𝑄) 𝜓…•(1𝑄) [Hengfei Zhao, In progress]

OGE; Li, Maris & Vary ,PRD 17

25

Wave function

stat e OGE BLFQ ℎ…(1𝑄) 𝜓…}(1𝑄) [Hengfei Zhao, In progress]

OGE; Li, Maris & Vary ,PRD 17

26

Wave function

State 𝜔(2𝑇) OGE BLFQ 𝜃…(2𝑇) [Hengfei Zhao, In progress]

OGE; Li, Maris & Vary ,PRD 17

27

Decay constants

Wave function at the origin – probe short-distance physics LFWF representation 𝑔

• ,–

2 2𝑂… = s

'

• 𝑒𝑦

2 𝑦(1 − 𝑦) s 𝑒}𝑙* 2𝜌 ( 𝜒↑↓∓↓↑

›|' (𝑦, 𝑙* )

Decay constants (GeV) [Hengfei Zhao, In progress]

28

PDA

Generalized Van Royen-Weisskopf formula: 𝑔

• ,–

2 2𝑂… 𝜚•,– 𝑦; 𝜈 = 1 4𝜌 𝜔↑↓∓↓↑

›|' (𝑦, 𝑐* = 0)

OGE; Li, Maris & Vary ,PRD 17

29

Gluon PDF

[Hengfei Zhao, In progress]

30

Application to Pion

31

PDF with QCD Evolution

[Lan, Mondal, Jia, Zhao, Vary, PRL122, 172001(2019)]

large x: (1-x)1.44

HLF=

PDF for the valence quark result from the light front wave functions

• btain by diagonalizing the effective Hamiltonian.

Pion PDF Valence u PDF Kaon/Pion

See Jiangshan Lan’s talk Sunday (18) pm See Shaoyang Jia’s talk Tuesday (20) pm

The moments of pion valence quark PDF:

[Lan, Mondal, Jia, Zhao, Vary, arxiv: 1907.01509]

<x> @4 GeV2 Valence Gluon Sea BLFQ-NJL 0.489 0.398 0.113 [Aguilar et. al., Pion and Kaon Structure at the Electron-Ion Collider] 0.48(3) 0.41(2) 0.11(2)

PDF with QCD Evolution

Drell-Yan cross section

[S. D. Drell and T.-M. Yan, PRL (1970)] [McGaughey et al, Drell-Yan experiment FNAL-E-0772, PRD (1994)]

Pt W C W W

Agree with experimental data (FNAL E615, 326, 444, & CERN NA3, WA-039).

[Lan, Mondal, Jia, Zhao, Vary, arxiv: 1907.01509]

[nCTEQ 2015]

Application to Proton

35

Lig Light-Fr Front Hamiltonian

𝑸B = 𝑰𝑳.𝑭. + 𝑰𝒖𝒔𝒃𝒐𝒕 + 𝑰𝒎𝒑𝒐𝒉𝒋 + 𝑰𝑷𝑯𝑭

𝑰𝒖𝒔𝒃𝒐𝒕 ~ 𝝀𝑼

𝟓𝒔𝟑

𝑰𝒎𝒑𝒐𝒉𝒋 ~ − µ

𝒋𝒌

𝝀𝑴

𝟓𝝐𝒚𝒋 𝒚𝒋𝒚𝒌𝝐𝒚𝒌

𝑰𝑳.𝑭. = µ

𝒋

𝒒𝒋

𝟑 + 𝒏𝒓 𝟑

𝒒𝒋

+

𝑰𝑷𝑯𝑭 = − 𝑫𝑮𝟓𝝆𝜷𝒕 𝑹𝟑 µ

𝒋,𝒌(𝒋Á𝒌)

 𝒗𝒕𝒋

Ä 𝒍𝒋

Æ 𝜹𝝂𝒗𝒕𝒋 𝒍𝒋 Â

𝒗𝒕𝒌

Ä 𝒍𝒌

Æ 𝜹𝝂𝒗𝒕𝒌(𝒍𝒌)

• --Y Li, X Zhao , P Maris , J Vary, PLB 758(2016)

Color factor : 𝐷Ê = − }

(

• - Brodsky, Teramond arXiv: 1203.4025

| Y 𝑄NyƒË†L = | ⟩ 𝑟𝑟𝑟 + 𝑟𝑟𝑟𝑕 + 𝑟𝑟𝑟 𝑟@ 𝑟 + ⋅⋅⋅⋅⋅⋅

Three active-quark approach

36

See Siqi Xu’s talk Tuesday (20) pm

αOGE=0.6,mqk=0.3/0.306GeV, mqOGE=0.2/0.2GEV,Nmax&Kmax=6&10 αOGE=0.8,mqk=0.3/0.306GeV, mqOGE=0.2/0.2GEV,Nmax&Kmax=6&16 Herberg 99 Glazier 05 Plaster 06 n Jlab Hall A Bermuth 03 warren 04 Passchier 99 Blast 05 Zhu 01 Plaster 06, n(e,e',n 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.00 0.02 0.04 0.06 0.08 Q2 GEN

Kmax=10&16,kappat=0.283GeV,kappal=0.424GeV,b=0.6GeV,mg=0.05GeV

αOGE=0.6,mqk=0.3/0.306GeV, mqOGE=0.2/0.2GEV,Nmax&Kmax=6&10 αOGE=0.8,mqk=0.3/0.306GeV,mqOGE=0.2/0.2GEV, Nmax&Kmax=6&16 PRC79, 035205 Arrington 07 Milbrath 99, PRL 82, 2221 Pospischil 01, EPJ A12, 125 Jones 2000 PRL 84, 1398 Gayou 01, PRC, 038202 Gayou 02, PRL 88, 092301 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.2 0.4 0.6 0.8 1.0 Q2 GEP

Kmax=10&16,kappat=0.283GeV,kappal=0.424GeV,b=0.6GeV,mg=0.05GeV

Sa Sach ch Fo Form Factor

[Work in progress, C. Mondal, Siqi Xu, et.al ]

αOGE=0.6,mqk=0.3/0.306GeV, mqOGE=0.2/0.2GEV,Nmax&Kmax=6&10 αOGE=0.8,mqk=0.3/0.306GeV, mqOGE=0.2/0.2GEV,Nmax&Kmax=6&16 Arrington 05 Arrington 07 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Q2 GMP

Kmax=10&16,kappat=0.283GeV,kappal=0.424GeV,b=0.6GeV,mg=0.05GeV

αOGE=0.6,mqk=0.3/0.306GeV, mqOGE=0.2/0.2GEV,Nmax&Kmax=6&10 αOGE=0.8,mqk=0.3/0.306GeV, mqOGE=0.2/0.2GEV,Nmax&Kmax=6&16 Anikin 98 Kubon 02 Xu 03 Anderson 07 Lachniet 09 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

• 1.5
• 1.0
• 0.5

0.0 Q2 GMN

Kmax=10&16,kappat=0.283GeV,kappal=0.424GeV,b=0.6GeV,mg=0.05GeV

With increasing the

basis size, the magnetic

and electric form factors

approach the experiment data

𝐻Î(𝑅}) = ∑‹ 𝑓‹𝐺

• ‹(𝑅}) − ÒÓ

ˆÔÓ ∑‹ 𝑓‹𝐺 } ‹(𝑅}) ,

𝐻Ô(𝑅}) = ∑‹ 𝑓‹𝐺

• ‹(𝑅}) + ∑‹ 𝑓‹𝐺

} ‹(𝑅}).

Anomalous magnetic moment

Proton : 𝜈Õ = 2.44 (Exp. : 2.79) Neutron : 𝜈Õ = −1.41 (Exp. : −1.91) neutron neutron proton proton

37

Use the NNLO DGLAP to evolve the PDF. Qualitative behavior is consistent with the CTEQ 15 PDF.

Pa Parton Distribution Functions (PDF)

x

4

• 10

3

• 10

2

• 10

1

• 10

1

x f(x)

0.2 0.4 0.6 0.8

2

= 0.195 GeV

2

µ

2

= 10 GeV

2

µ dashed: CTEQ15

u d

[Work in progress, C. Mondal, Siqi Xu, Jiangshan Lan, et.al ] 38

Gen Gener eraliz alized ed Par arton Dis istrib ibutio ion Functio ions (GP (GPD)

𝑢 = Δ}, 𝑦 =

𝑙+ 𝑄+ , 𝜼 = 𝚬+ 𝑸+ = 𝟏 With increasing momentum transfer

(𝒖), the peaks of distributions shift to larger 𝒚;

[Work in progress, C. Mondal, Siqi Xu, et.al ]

39

, U2 = 1 + i5~ ⌧ · ~ ⇡/(2f) 1 i5~ ⌧ · ~ ⇡/(2f)

Relativistic 𝑂𝜌 chiral Lagrangian density

Chiral model of nucleon and pion

𝑔 = 93 MeV: pion decay constant 𝑁Û = 137 MeV: pion mass 𝑁Õ = 938 MeV: nucleon mass

[Miller, 1997] 40

The idea: cloudy-bag model

41

Solve the proton wave function in terms of the nucleon and pion ⟩ |proton = 𝑏 ⟩ |𝑂 + 𝑐 ⟩ |𝑂𝜌 + c ⟩ |𝑂𝜌𝜌 + 𝑒 ⟩ |𝑂 Â 𝑂𝑂 +. . . . ⟩ |proton = 𝑏 ⟩ |𝑣𝑣𝑒 + 𝑐 ⟩ |𝑣𝑣𝑒𝑕 + c ⟩ |𝑣𝑣𝑒𝑕𝑕 + 𝑒 ⟩ |𝑣𝑣𝑒𝑟@ 𝑟 +. . . .

Step: 1 Step: 2

Attach the fundamental degree of freedom: quarks and gluons

N

π π

Mass spectrum of the 𝑂𝜌 system

[Du et al., in preparation]

§ Fock sector-dependent renormalization applied § Mass counterterm applied to |𝑂⟩ sector only

[Karmanov et al, 2008, 2012] 42

the physical proton 938 MeV 𝑂Myz(= 𝐿Myz − 1/2 )

Proton parton distribution function

43

Preliminary

𝑔

Û(𝑦Û)

Proton Dirac form factor

Preliminary

Proton Dipole form with constituents’ internal structures

Summary and Outlook

• Nonperturbative approach for relativistic bound states
• Access to (frame-independent) many-body wave functions
• Straightforward to calculate lightcone observables eg. PDFs
• Can be applied to effective/first-principles interactions
• Systematically improvable by including higher Fock sectors

45

• Apply to different systems: excited/exotic hadron states
• Apply to different interactions: NJL, Chiral EFT...
• More observables: GPD, TMD, GTMD…
• Apply to first-principles QCD interactions