Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary Magnetic Polarisability with the Background Field Method R. Bignell W. Kamleh D. Leinweber The Special Research Centre for the Subatomic Structure of Matter University of Adelaide Asia-Pacific Symposium for Lattice Field Theory August 6, 2020 R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 1 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary β p Status ◮ The magnetic polarisability is a fundamental property of a system of charged particles ◮ Describes the response to an external magnetic field ◮ Provides a description of hadron structure ◮ Experimentally measured in Compton scattering experiments ◮ What is the magnetic polarisability of the proton and neutron? R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 2 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary β p Status 7 Experiment: MacGibbon Experiment: Pasquini 6 Experiment: PDG Experiment: McGovern 5 Experiment: Beane Experiment: Blanpied 4 fm 3 ) Experiment: Olmos de León 4 p (10 3 2 1 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 m 2 ( GeV 2 ) R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 2 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary β p Status 7 Lattice: Chang: 1506.05518 Experiment 6 5 4 fm 3 ) 4 p (10 3 2 1 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 m 2 ( GeV 2 ) R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 2 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary Energy & Two-point Correlation Function ◮ Directly calculate hadron energies in an external magnetic field ◮ The energy of a baryon in an external magnetic field is B + | qe B | − 4 π 2 β B 2 + O � B 3 � µ · � E ( B ) = M + � 2 M ◮ Evaluate two point correlation functions G ( � α e − E α t p , t ) ∝ � x 0 Two point correlation function quark-flow diagram for a baryon R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 3 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary Background Field Method ◮ Introduce a background field on the lattice Lattice: U µ ( x ) → e i a qe A µ ( x ) U µ ( x ) Continuum: D µ → D QCD + iqe A µ µ ◮ Choose � A appropriately to generate constant magnetic field � B = + B ˆ z ◮ Periodic spatial boundary conditions impose a quantisation for a uniform field a 2 qe B 2 = 2 π k N x N y ◮ k d = 0 , 1 , 2 , . . . for the field strength experienced by the d quark R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 4 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary Wilson Term Mass Renormalisation ◮ Wilson term causes unphysical quark mass renormalisation in background magnetic field ◮ In free-field limit this change is m [ w ] ( B ) = m ( 0 ) + a 2 | qe B | ◮ Observe through investigation of QCD-free (connected) neutral pion energies ◮ First discussed by Bali et al. 1510.03899, 1707.05600 R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 5 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary Wilson Term Mass Renormalisation + 0.25 0 u 0.20 0 d aE(B) 0.15 0.10 0.00 0.02 0.04 0.06 eBa 2 R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 5 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary Clover term 0.250 + 0.225 0 0.200 u 0 0.175 d aE(B) 0.150 0.125 0.100 0.075 0.00 0.02 0.04 0.06 eBa 2 R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 6 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary Clover term ◮ Careful examination of the clover term � a c cl σ µν F µν µ<ν ◮ reveals it cancels the Landau shift induced by the Wilson term in the free-field limit ◮ This condition is modified by inclusion of QCD ◮ Allow QCD and electromagnetic field strengths to have different clover coefficients c cl → C SW F QCD + C EM F EM µν µν ◮ and set C EM such that Wilson Landau shift is cancelled C EM = C Tree EM ◮ 1910.14244 R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 6 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary Quark Operators ◮ Standard lattice QCD interpolators are inefficient at isolating energy eigenstates in a background magnetic field ◮ Quarks are charged! ◮ Quarks experience Landau type effects ◮ QCD causes quarks to hadronise for composite Landau energy ◮ Competing effects, introduce a quark projection operator that includes QCD and QED Figure: Left: Mode for the lowest quantised magnetic field strength k d = 1. Right: Two degenerate eigenmodes of second quantised field strength k d = 2. R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 7 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary SU ( 3 ) × U ( 1 ) Projection Operator ◮ Two-dimensional lattice Laplacian operator � x ′ + U † � � � � � � ∆ � x ′ = 4 δ � x ′ − U µ x δ � x − ˆ µ δ � x ′ , x ,� x ,� x +ˆ µ,� µ x − ˆ µ,� µ = 1 , 2 ◮ Use low-lying eigenmodes of the 2D Laplacian � � ψ i � to project the propagator B n � � � � x ′ , t ′ � � x ′ , t ′ � � � � x , t ; � � � � δ zz ′ δ tt ′ P n = x , t � ψ i ,� ψ i � � B B i = 1 ◮ Projected propagator is x , t ; � � x ′ , t ) S ( � x ′ , t ; � S n ( � P n ( � x , t ; � 0 , 0 ) = 0 , 0 ) x ′ � ◮ Also project hadronic level Landau effects for proton- using lattice Landau levels R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 8 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary Energy Shifts ◮ Recall the energy-field relation for an external magnetic field B + | qe B | − 4 π 2 β B 2 + O � B 3 � µ · � E ( B ) = M + � 2 M ◮ Construct energy difference E ( B ) − m for aligned and anti-aligned spin-field orientations and combine � G (+ s , + B ) + G ( − s , − B ) � � G (+ s , − B ) + G ( − s , + B ) � = e − ( 2 δ E ) t G (+ s , 0 ) + G ( − s , 0 ) G (+ s , 0 ) + G ( − s , 0 ) ◮ Extract effective energy shift in standard manner ◮ Hence determine β using δ E ( B , t ) = + | qe B | − 4 π 2 β B 2 + O � B 4 � 2 M R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 9 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary Proton Energy Shift 0.20 0.15 0.10 E ( B ) ( GeV ) 0.05 0.00 0.05 0.10 0.15 k B = 1, 2 dof = 1.0 len =6 k B = 3, 2 dof = 0.15 len =6 k B = 2, 2 dof = 0.43 len =6 0.20 16 18 20 22 24 26 28 30 32 t R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 10 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary Polarisability Fit ◮ Fit to these energy shifts δ E ( B , t ) δ E ( B , t ) − | qe B | = − 4 π β B 2 = c 2 k 2 2 M 2 ◮ where k is the field quanta from background magnetic field quantisation condition R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 11 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary Polarisability Fit 0.03 = 0.13727 0.02 0.01 ( GeV ) 0.00 0.01 E 0.02 0.03 0.04 q = 1 constrained c 2 k 2 , 2 dof = 1.099 0.05 0 1 2 3 4 k B R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 11 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary Ensemble Details κ ud m π (MeV) Number of Sources Number of configurations 0 . 13700 702 5 399 0 . 13727 570 4 400 0 . 13754 411 6 450 0 . 13770 296 7 400 ◮ Available through the International Lattice Data Grid and PACS-CS Collaboration: Phys. Rev. D79 (2009) 034503 ◮ Lattice Volume: 32 3 × 64 ◮ 2 + 1 flavour dynamical-fermion QCD ◮ Physical lattice spacing a = 0 . 0907 fm ◮ Electroquenched - “sea” quarks experience no background magnetic field R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 12 / 19
Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary Lattice Results 5 Lattice Experiment 4 4 fm 3 ) 3 p (10 2 1 0 0.0 0.1 0.2 0.3 0.4 m 2 ( GeV 2 ) R. Bignell (CSSM) Magnetic Polarisability with the Background Field Method APLAT 2020 13 / 19
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