Positronium in Basis Light-front Quantization

Transcription

1 Positronium in Basis Light-front Quantization Xingbo Zhao With Yang Li, Pieter Maris, James P. Vary Institute of Modern Physics Chinese Academy of Sciences Lanzhou, China Lightcone 2016, Lisbon, Portugal, Sept. 6, 2016

2 Key Questions What does the bound state light-front wavefunction look like beyond valence Fock sector What does positronium look like? The probability of finding a photon in it? How to solve for the bound state light-front wavefunction in an ab initio approach in Hamiltonian formalism Truncations, renormalization, divergences? How far can we go 2

3 Outline Introdction to Basis Light-Front Quantization (BLFQ) Electron Renormalization in BLFQ Positronium Embedding in BLFQ 3

4 Basis Light-front Quantization Based on quantum field theory For both first-principles and effective theory Non-perturbative Hamiltonian formalism Light-front dynamics 4

5 Basis Light-front Quantization Solve quantum field theory through eigenvalue problem of light-front Hamiltonian P - b = P b- b - P : light-front Hamiltonian - ȁ β : light-front amplitude for mass eigenstates - P β : eigenvalue (light-front energy) for eigenstate b Evaluate observables for eigenstate b O º b Ô b 5

6 General Procedure for BLFQ 1. Derive LF-Hamiltonian P from Lagrangian 2. Construct basis states 3. Calculate Hamiltonian matrix elements 4. Diagonalize P - (solve P - b = P b- b ) and obtain its eigenspectrum 5. Evaluate observables a O º b Ô b a ' P - a 6

7 General Procedure for BLFQ 1. Derive LF-Hamiltonian P from Lagrangian 2. Construct basis states 3. Calculate Hamiltonian matrix elements 4. Diagonalize P - (solve P - b = P b- b ) and obtain its eigenspectrum 5. Evaluate observables a O º b Ô b a ' P - a 7

8 Example: Obtain LF QED Hamiltonian QED Lagrangian Derived Light-front Hamiltonian ( A + = 0) kinetic energy terms vertex interaction instantaneous photon interaction instantaneous fermion interaction 8

9 General Procedure for BLFQ 1. Derive LF-Hamiltonian from Lagrangian 2. Construct basis states 3. Calculate Hamiltonian matrix elements 4. Diagonalize P - (solve P - b = P b- b ) and obtain its eigenspectrum 5. Evaluate observables a O º b Ô b a ' P - a 11

10 General Procedure for BLFQ 1. Derive LF-Hamiltonian from Lagrangian 2. Construct basis states 3. Calculate Hamiltonian matrix elements 4. Diagonalize P - (solve P - b = P b- b ) and obtain its eigenspectrum 5. Evaluate observables a O º b Ô b a ' P - a Fock-sector expansion + 2D Harmonic Oscillator + 1D Planewave 12

11 Basis Construction Example: single physical electron in QED 1. Fock-space expansion e.g. 2. For each Fock particle: transverse: 2D-HO basis ( F n,m (r,j)), labeled by n,m quantum number (HO basis parameter b = MW) longitudinal: plane-wave basis, labeled by k e.g. with eg = e Ä g and e = {n e,m e,k e,l e } g = {n g,m g,k g,l g } 13

12 Set of Transverse 2D HO Modes for n=4 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, (2010). 14 ArXiv:0905:1411

13 Basis Truncation Fock sector truncation Longitudinal periodic boundary condition (integer or half integer k i ) å k = K i i N max truncation in the transverse directions å i [ 2n + m +1] N i i max

14 General Procedure for BLFQ 1. Derive LF-Hamiltonian from Lagrangian 2. Construct basis states 3. Calculate Hamiltonian matrix elements 4. Diagonalize P - (solve P - b = P b- b ) and obtain its eigenspectrum 5. Evaluate observables a O º b Ô b a ' P - a 16

15 General Procedure for BLFQ 1. Derive LF-Hamiltonian from Lagrangian 2. Construct basis states 3. Calculate Hamiltonian matrix elements 4. Diagonalize P - (solve P - b = P b- b ) and obtain its eigenspectrum 5. Evaluate observables a O º b Ô b a ' P - a 22

16 Single Physical Electron Fock-sector truncation: XZ, H. Honkanen, P. Maris, J. P. Vary, S. J. Brodsky, Phys. Letts. B737, 65 (2014) Interaction part of the Hamiltonian Ground state is identified as the physical electron

17 Renormalization In quantum field theory, bare electron mass in the Hamiltonian is not the physical mass. Bare electron mass is found by matching the physical electron mass to the experimental value 27

18 Structure of Electron Wavefunction In electron, the probability of finding ȁe is almost zero. 28

19 Comparison of GPD E(x,q 2 ) obtained from BLFQ and Light Cone perturbation theory, for selected q 2. The integral of GPD E(x,q 2 ) over the momentum fraction x contributes F 2. Top panel: q 2 = 0, bottom panel: q 2 = 5MeV 2. Form factors as functions of q 2 obtained from BLFQ and Light Cone perturbation theory (Stanley J. Brodsky, et al.) Top panel: Pauli form factor F 2 (q 2 ), bottom panel: gravitomagnetic form factor B(q 2 ) = B f + B b In the perturbative calculation, UV and IR cutoffs are imposed on the transverse momenta to match the BLFQ basis truncation. UV cutoff, IR cutoff.

20 Positronium α = 0.31 LF-QED Interaction: + ( eeҧ sector only) 30

21 Mass Renormalization Embedding Mass counterterms (bare mass) should be evaluated on the level of single electron Construct a series of parallel single electron problem with matching kinematics Embed the physical electron in the positronium basis, allowing only the electron to be coupled with the photon: the positron being spectator Counterterms are not only Fock-sector dependent but also basis-state dependent 31

22 State-dependent Mass Couterterms With state-dependent mass counterterms applied, the ground state binding energy convergence is much improved 32

23 Divergence in Longitudinal Direction Nmax=6 1 k + 2 This divergence is due to violation of current conservation introduced by basis truncation 33

24 Longitudinal regulator Introduce a regulator in the longitudinal direction Applied to both embedded single electron system and positronium system, both the vertex interaction and the instantaneous interaction Results depend on x c, still a problem 34

25 Longitudinal Regulator NR binding value: α 2 4 m e Nmax=6 x c =0.1 seems reasonable since hard photons are not expected to be responsible for binding 35

26 Structure of Wavefunction Probability of finding positronium in sector ȁeeҧ Fock Seems that ȁeeγ ҧ will eventually dominate, from constituent electron? 36

27 Conclusion Mass renormalization is needed when BLFQ is dealing with more than a single Fock sector Embedding approach with basis statedependent renormalization seems working Multiple divergences encountered Higher Fock sector seems dominating positronium 37

28 Next Explore parameter space further Better treatment of longitudinal divergence Mass spectrum Observables QCD counterpart 38

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