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 Low Energy High Energy 𝜌 Nucleons & mesons Quarks & gluons Mass scale ∼ GeV Mass scale ∼ MeV 4 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

Light-front vs Equal-time Quantization [Dirac 1949] v.s. equal-time dynamics vs light-front dynamics t ≡ x + = x 0 + x 3 t ≡ x 0 i ∂ i ∂ 2 P − ϕ ( x + ) ∂ x + ϕ ( x + ) = 1 ∂ t ϕ ( t ) = H ϕ ( t ) P − = P 0 − P 3 H = P 0 𝑄 * , 𝑄 + , 𝐹 * , 𝐹 + , 𝐾 / 𝑄 , ⃗ 𝐾 Kinematic generators: 6

Why Go To Light-front? • Boost invariant light-front wave functions • Simple vacuum = free theory vacuum + zero modes • Hamiltonian formalism for relativistic systems |proton = 𝑏 ⟩ |𝑣𝑣𝑒 + 𝑐 ⟩ |𝑣𝑣𝑒𝑕 + c ⟩ |𝑣𝑣𝑒𝑕𝑕 + 𝑒 ⟩ |𝑣𝑣𝑒𝑟@ 𝑟 + . . . . ⟩ ⟩ ⟩ ⟩ ⟩ ⟩ |pion = 𝑏 |𝑟@ 𝑟 + 𝑐 |𝑟@ 𝑟𝑕 + c |𝑟@ 𝑟𝑕𝑕 + 𝑒 |𝑟@ 𝑟𝑟@ 𝑟 + . . . . . . . . 7

Basis Light-Front Quantization [Vary et al. , 2008] • Eigenvalue problem for Light-front Hamiltonian 𝑄 B 𝛾 = 𝑄 B |𝛾⟩ Non-perturbative C - 𝑄 B : light-front Hamiltonian B : eigenvalue - 𝑄 hadron mass spectrum C - | ⟩ 𝛾 : eigenvector light-front wave function • Observables O ≡ ⟨𝛾′| I 𝑃|𝛾⟩ 8

Basis Construction 1. Fock-space expansion: e.g. |𝒇 𝒒 Y = 𝑏 |𝑓 + 𝑐 ⟩ |𝑓𝛿 + c ⟩ |𝑓𝑓 ̅ 𝑓 ⟩ + 𝑒 |𝑓𝑓 ̅ 𝑓𝛿 + . . . . ⟩ ⟩ ⟩ ⟩ ⟩ ⟩ |𝐐𝐭 = 𝑏 |𝑓 ̅ 𝑓 + 𝑐 |𝑓 ̅ 𝑓𝛿 + 𝑑 |𝛿 + 𝑒 |𝑓 ̅ 𝑓𝑓 ̅ 𝑓 + . . . . 2. For each Fock particle: N • Transverse: 2D harmonic oscillator basis: Φ L,M ( ⃗ 𝑞 * ) labeled by radial (angular) quantum number n (m); scale parameter b e.g., n=4 m=0 m=1 m=2 • Longitudinal: plane-wave basis, labeled by k • Helicity: labeled by 𝜇 γ = { n γ , m γ , k γ , λ γ } e.g. with and e γ = e ⊗ γ e = { n e , m e , k e , λ e } 9

Basis Truncation Scheme • Symmetries of Hamiltonian: ∑ - Net fermion number: = N f n f i i ∑ - Total angular momentum projection: + s i ) = J z ( m i i ∑ - Longitudinal momentum: = K k i i • Further truncation: - Fock-sector truncation 1, 2, 3…. bosons - Discretization in longitudinal direction 𝑙 j = 0.5, 1.5, 2.5 … fermions [ ] ≤ N max - “N max ” truncation in transverse directions ∑ 2 n i + | m i | + 1 i UV cutoff Λ~𝑐 𝑂 cde ; IR cutoff 𝜇~𝑐/ 𝑂 cde

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 L = − 1 • QED Lagrangian 4 F µ ν F µ ν + ¯ Ψ ( i γ µ D µ − m e ) Ψ • Derived Light-front Hamiltonian ( ) A + = 0 Z − = − F µ + ∂ + A µ + i ¯ d 2 x ⊥ d x + ∂ + Ψ − L P Ψ γ Z + m 2 e + ( i ∂ ⊥ ) 2 − 1 Ψ + 1 Z − = ⊥ ) 2 A j + ¯ d 2 x 2 A j ( i ∂ ⊥ d x Ψ γ 2 i ∂ + kinetic energy terms j + ej µ A µ + e 2 + + e 2 1 γ + ¯ Ψ γ µ A µ 2 j + ( i ∂ + ) 2 j i ∂ + γ ν A ν Ψ 2 vertex instantaneous instantaneous interaction photon fermion interaction interaction 12

Light-front QCD Hamiltonian

Application to QED (I): Physical Electron Δ * 𝑸 + 𝒄 * | e phys i = a | e i + b | e γ i + c | e γγ i + d | ee ¯ e i + . . . . 15

Electron g-2 & GPD E(x, t) BLFQ vs Perturbation Theory X. Zhao, H. Honkanen, P. Maris, J. P. Vary, S. J. Brodsky, Phys. Letts. B737, 65 (2014) N max = K =18 0.115 0.002 Λ =1.5MeV m γ =0.085MeV Schwinger result = 0.1125395... N max = K =42 Λ =2.3MeV m γ =0.056MeV 0.11 E ( x, t → 0) N max = K =578 a e / e 2 0.0015 1 α = Λ =8.7MeV m γ =0.015MeV 137 . 036 0.105 0.001 � 0.1 even N max / 2 0.095 0.0005 odd N max / 2 even N max / 2 fit 0.09 odd N max / 2 fit 0 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 Nonperturbative x ( N max = K − 1 / 2 ) − 1 / 2 q 1 − iq 2 phys (  ∫ e ixP + y − /2 ψ (0) γ + ψ ( y ) e ↓ E ( x ,0, t = q 2 ) = e ↑ dy − • q ) phys (0) y + = 0, y ⊥ = 0 2 m e 1 ∫ • Anomalous magnetic moment a e = E ( x , t → 0) dx 0 • Less than 0.1% deviation from Schwinger result for a e 16 • Largest calculation with basis dim > 28 billion

Application to QED (II): Positronium e + γ e - 18

Energy spectrum and wavefunction See Kaiyu Fu’s talk Saturday (17) pm Nmax = 28,K = 29 0.03 0.02 binding energy ( MeV ) 𝐹 n (𝑁𝑓𝑊) 0.01 0.00 - 0.01 - 0.02 - 2 - 1 0 1 2 Mj lowest 8 states of Mj=0 : parity and charge conjugation parity agree with hydrogen atom. [Kaiyu Fu et al, in preparation] 19

Photon Distribution In Positronium Nmax=28, K=29 ▲ 5 ◆ ■ 1 1 S 0 ● 4 2 1 S 0 ■ 2 3 P 0 ◆ 3 ψ * ψ ● 2 3 P 2 ▲ 2 ▲ ◆ ■ ● 1 ▲ ● ■ ◆ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ▲ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ■ 0 ■ ▲ ▲ ■ ▲ ■ ▲ ■ ■ ▲ ■ ▲ ▲ ■ ▲ ■ ▲ ■ ■ ▲ ■ ▲ ■ ▲ ■ ▲ ▲ ■ ■ ▲ ▲ ■ ▲ ■ ■ ▲ ▲ ■ ▲ ■ ▲ ■ ▲ ■ ▲ ■ 0.2 0.4 0.6 0.8 x [Kaiyu Fu et al, in preparation] In excited states photons have larger probability at small-x region • Photon is massless, so peak is at small-x region • 20

Application to Heavy Quarkonium 21

Hamiltonian 𝑄 B = 𝐼 ‚ƒ„„ + 𝐼 …†L‚ + 𝐼 ‹Ž†‹• + 𝐼 ‹‹Ž†‹‹ 1. Kinetic Hamiltonian and confining potentials } } 𝜆 ˆ } + 𝑛 ‹ 𝑛 @ } + 𝜆 ˆ ⃗ ‹ 𝐼 ‚ƒ„„ + 𝐼 …†L‚ = ⃗ 𝑟 * 𝜊 * 𝑦 + 1 − 𝑦 − } 𝜖 z (𝑦(1 − 𝑦)𝜖 z ) 𝑛 ‹ + 𝑛 @ ‹ 2. Vector coupling vertex 𝑟 } 𝑄 B = 2𝑕 s 𝑒 ( 𝑦 @ 𝜔 𝑦 𝛿 u 𝑈 w 𝜔 𝑦 𝐵 uy r 𝑟 • g z { |' 3. Vector coupling with instantaneous gluon 𝑟 • 𝑟 } 1 𝑄 B = 𝑕 } s 𝑒 ( 𝑦 @ (𝑗𝜖 + ) } @ 𝜔𝛿 + 𝑈 w 𝜔 𝜔𝛿 + 𝑈 w 𝜔 r z { |' 𝑟 ( @ 𝑟 ˆ @ 22

Energy Spectrum [Hengfei Zhao, In progress] OGE : Yang Li,Maris & Vary ,PRD 17 Kt=11 Nmax=10 Mj=0 me=1.5GeV b=1.64GeV binst=3.2GeV k2l=0.3 k2t=0.1 23

Wave function [Hengfei Zhao, In progress] 𝜃 … (1𝑇) 𝐾/𝜔(1𝑇) State BLFQ OGE 24 OGE; Li, Maris & Vary ,PRD 17

Wave function [Hengfei Zhao, In progress] 𝜓 …• (1𝑄) State 𝜓 …' (1𝑄) BLFQ OGE 25 OGE; Li, Maris & Vary ,PRD 17

Wave function [Hengfei Zhao, In progress] ℎ … (1𝑄) 𝜓 …} (1𝑄) stat e BLFQ OGE 26 OGE; Li, Maris & Vary ,PRD 17

Wave function [Hengfei Zhao, In progress] 𝜃 … (2𝑇) 𝜔(2𝑇) State BLFQ OGE 27 OGE; Li, Maris & Vary ,PRD 17

Decay constants [Hengfei Zhao, In progress] Wave function at the origin – probe short-distance physics LFWF representation • s 𝑒 } 𝑙 * 𝑔 𝑒𝑦 ›|' (𝑦, 𝑙 * ) •,– = s 2𝜌 ( 𝜒 ↑↓∓↓↑ 2 2𝑂 … 2 𝑦(1 − 𝑦) ' Decay constants (GeV) 28

PDA OGE; Li, Maris & Vary ,PRD 17 𝑔 𝜚 •,– 𝑦; 𝜈 = 1 ›|' (𝑦, 𝑐 * = 0) •,– 𝜔 ↑↓∓↓↑ Generalized Van Royen-Weisskopf formula: 4𝜌 2 2𝑂 … 29

Gluon PDF [Hengfei Zhao, In progress] 30

Application to Pion 31

See Jiangshan Lan’s talk See Shaoyang Jia’s talk PDF with QCD Evolution Sunday (18) pm Tuesday (20) pm H LF = PDF for the valence quark result from the light front wave functions obtain by diagonalizing the effective Hamiltonian. Pion PDF Valence u PDF Kaon/Pion large x: (1-x) 1.44 [Lan, Mondal, Jia, Zhao, Vary, PRL122, 172001(2019)]

PDF with QCD Evolution [Lan, Mondal, Jia, Zhao, Vary, arxiv: 1907.01509] The moments of pion valence quark PDF: <x> @4 GeV 2 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)