Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary Computing the magnetic field response of the proton R. Bignell W. Kamleh D. Leinweber The Special Research Centre for the Subatomic Structure of Matter University of Adelaide Computing in High Energy & Nuclear Physics November 5, 2019 R. Bignell (CSSM) Computing the magnetic field response of the proton CHEP 2019 1 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results 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 proton’s magnetic polarisability? R. Bignell (CSSM) Computing the magnetic field response of the proton CHEP 2019 2 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results 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) Computing the magnetic field response of the proton CHEP 2019 2 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary β p Status 7 Lattice 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) Computing the magnetic field response of the proton CHEP 2019 2 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary Energy & Two-point Correlation Function ◮ Baryon energy in 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) Computing the magnetic field response of the proton CHEP 2019 3 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary Computing Resources ◮ Calculating two point correlation functions is expensive! ◮ Requires quark propagators - large-multidimensional matrix inversions ◮ Work performed on University of Adelaide Phoenix supercomputer and NCI Raijin ◮ Phoenix calculations utilising CUDA ◮ Raijin uses Fortran MPI R. Bignell (CSSM) Computing the magnetic field response of the proton CHEP 2019 4 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary Two-point Correlation Function & Background Field Method t →∞ e − E 0 t G ( � p , t ) ∝ ◮ Define effective energy � � E ( t ) = 1 G ( t ) δ t log G ( t + δ t ) R. Bignell (CSSM) Computing the magnetic field response of the proton CHEP 2019 5 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary Two-point Correlation Function & Background Field Method 0.45 BF0 0.40 0.35 ( GeV ) 0.30 E 0.25 0.20 0.15 20 25 30 35 40 t R. Bignell (CSSM) Computing the magnetic field response of the proton CHEP 2019 5 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary Two-point Correlation Function & Background Field Method t →∞ e − E 0 t G ( � p , t ) ∝ ◮ Define effective energy E ( t ) = 1 � G ( t ) � δ t log G ( t + δ t ) ◮ Background field is introduced by modification of gluon-field gauge links U µ ( x ) → U µ ( x ) e ( i a qe A µ ( x )) ◮ Periodic boundary conditions in the x − y plane introduce a quantisation condition qe B a 2 = 2 π k N x N y . ◮ Often refer to field strength B in terms of field quanta k R. Bignell (CSSM) Computing the magnetic field response of the proton CHEP 2019 5 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results 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) Computing the magnetic field response of the proton CHEP 2019 6 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results 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 Laplacian to project the propagator n � P n = | ψ i � � ψ i | i = 1 ◮ Projected propagator is x , t ; � � x ′ , t ; � x ′ ) S ( � S n ( � P n ( � x ,� 0 , 0 ) = 0 , 0 ) � x ′ ◮ Also project hadronic level Landau effects - using lattice Landau levels R. Bignell (CSSM) Computing the magnetic field response of the proton CHEP 2019 7 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results 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 ratio of spin-up (+s) and spin-down (-s) relative to magnetic field orientation. � 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) Computing the magnetic field response of the proton CHEP 2019 8 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results 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, dof = 1.0 len =6 2 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) Computing the magnetic field response of the proton CHEP 2019 9 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results 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) Computing the magnetic field response of the proton CHEP 2019 10 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results 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) Computing the magnetic field response of the proton CHEP 2019 10 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results 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) Computing the magnetic field response of the proton CHEP 2019 11 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary Making Contact With Experiment ◮ Use a chiral effective field theory analysis to 1. Account for finite volume effects 2. Incorporate Sea-quark-loop contributions to β R. Bignell (CSSM) Computing the magnetic field response of the proton CHEP 2019 12 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary Making Contact With Experiment 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) Computing the magnetic field response of the proton CHEP 2019 12 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary Making Contact With Experiment 5 Full-QCD Infinite Volume 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) Computing the magnetic field response of the proton CHEP 2019 12 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary Making Contact With Experiment ◮ Use a chiral effective field theory analysis to 1. Account for finite volume effects 2. Incorporate Sea-quark-loop contributions to β 3. Perform a chiral extrapolation to the physical point ◮ Use the techniques of ◮ J. M. M. Hall, D. B. Leinweber, and R. D. Young, Phys. Rev. D89, 054511 (2014), arXiv:1312.5781 [hep-lat] ◮ R. Bignell, J. Hall, W. Kamleh, D. Leinweber, and M. Burkardt, Phys. Rev. D98, 034504 (2018), arXiv:1804.06574 [hep-lat] R. Bignell (CSSM) Computing the magnetic field response of the proton CHEP 2019 12 / 13
Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary Making Contact With Experiment 5 Full-QCD Infinite Volume EFT Extrapolation 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) Computing the magnetic field response of the proton CHEP 2019 12 / 13
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