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Computing the magnetic field response of the proton R. Bignell W. - - PowerPoint PPT Presentation

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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

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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

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Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary

βp 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

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Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary

βp Status

0.0 0.1 0.2 0.3 0.4 0.5 0.6 m2 ( GeV2) 1 2 3 4 5 6 7

p (10 4 fm3)

Experiment: MacGibbon Experiment: Pasquini Experiment: PDG Experiment: McGovern Experiment: Beane Experiment: Blanpied Experiment: Olmos de León

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 2 / 13

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Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary

βp Status

0.0 0.1 0.2 0.3 0.4 0.5 0.6 m2 ( GeV2) 1 2 3 4 5 6 7

p (10 4 fm3)

Lattice Experiment

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 2 / 13

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Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary

Energy & Two-point Correlation Function

◮ Baryon energy in external magnetic field is

E(B) = M + µ · B + |qe B| 2 M − 4 π 2 β B2 + O

  • B3

◮ Evaluate two point correlation functions G(

p, t) ∝

α e−Eα t

x 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

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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

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Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary

Two-point Correlation Function & Background Field Method

G( p, t)

t→∞

∝ e−E0 t

◮ Define effective energy

E(t) = 1 δ t log

  • G(t)

G(t + δ t)

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 5 / 13

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Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary

Two-point Correlation Function & Background Field Method

20 25 30 35 40

t

0.15 0.20 0.25 0.30 0.35 0.40 0.45

E (GeV)

BF0

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 5 / 13

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Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary

Two-point Correlation Function & Background Field Method

G( p, t)

t→∞

∝ e−E0 t

◮ Define effective energy

E(t) = 1 δ t log

  • G(t)

G(t + δ t)

  • ◮ Background field is introduced by modification of gluon-field gauge links

Uµ(x) → Uµ(x) e(i a qe Aµ(x))

◮ Periodic boundary conditions in the x − y plane introduce a quantisation condition

qe B a 2 = 2 π k Nx Ny .

◮ 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

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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 kd = 1. Right: Two degenerate eigenmodes of second quantised field strength kd = 2.

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 6 / 13

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Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary

SU(3) × U(1) Projection Operator

◮ Two-dimensional lattice Laplacian operator

∆

x, x′ = 4 δ x, x′ −

  • µ=1,2

Uµ

  • x
  • δ

x+ˆ µ, x′ + U† µ

  • x − ˆ

µ

  • δ

x−ˆ µ, x′,

◮ Use low-lying eigenmodes of the Laplacian to project the propagator

Pn =

n

  • i=1

|ψi ψi|

◮ Projected propagator is

Sn( x, t; 0, 0) =

  • x′

Pn( x, x′) S( x′, t; 0, 0)

◮ 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

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Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary

Energy Shifts

◮ Recall the energy-field relation for an external magnetic field

E(B) = M + µ · B + |qe B| 2 M − 4 π 2 β B2 + O

  • B3

◮ Construct ratio of spin-up (+s) and spin-down (-s) relative to magnetic field orientation.

G(+s, +B) + G(−s, −B) G(+s, 0) + G(−s, 0) G(+s, −B) + G(−s, +B) G(+s, 0) + G(−s, 0)

  • = e−(2 δ E) t

◮ Extract effective energy shift in standard manner ◮ Hence determine β using

δ E(B, t) = +|qe B| 2 M − 4 π 2 β B2 + O

  • B4
  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 8 / 13

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Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary

Proton Energy Shift

16 18 20 22 24 26 28 30 32

t

0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20

E(B) (GeV)

kB = 1,

2 dof = 1.0 len =6

kB = 2,

2 dof = 0.43 len =6

kB = 3,

2 dof = 0.15 len =6

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 9 / 13

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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| 2 M = −4 π 2 β B2 = c2 k2

◮ 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

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Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary

Polarisability Fit

1 2 3 4

kB

0.05 0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03

E (GeV)

= 0.13727 q = 1 constrained c2 k2,

2 dof = 1.099

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 10 / 13

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Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary

Lattice Results

0.0 0.1 0.2 0.3 0.4 m2 ( GeV2) 1 2 3 4 5

p (10 4 fm3)

Lattice Experiment

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 11 / 13

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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

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Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary

Making Contact With Experiment

0.0 0.1 0.2 0.3 0.4 m2 ( GeV2) 1 2 3 4 5

p (10 4 fm3)

Lattice Experiment

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 12 / 13

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Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary

Making Contact With Experiment

0.0 0.1 0.2 0.3 0.4 m2 ( GeV2) 1 2 3 4 5

p (10 4 fm3)

Full-QCD Infinite Volume Experiment

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 12 / 13

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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

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Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary

Making Contact With Experiment

0.0 0.1 0.2 0.3 0.4 m2 ( GeV2) 1 2 3 4 5

p (10 4 fm3)

Full-QCD Infinite Volume EFT Extrapolation Experiment

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 12 / 13

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Introduction Lattice QCD Quark Operators Magnetic Polarisability Results Summary

Summary

◮ Calculated the magnetic polarisability of the proton using lattice QCD and background field

method

◮ Specialised projection techniques have been used to account Landau effects ◮ Enabling energy shift plateaus ◮ Chiral effective field theory analysis has been performed to connect lattice results to

experiment.

◮ Techniques are applicable to further elements of the hadronic spectrum

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 13 / 13

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BONUS SLIDES

Neutron βn

0.0 0.1 0.2 0.3 0.4 m2 ( GeV2) 1.0 1.5 2.0 2.5 3.0 3.5

n (10 4 fm3)

  • inf. volume extrapolation

Finite volume 3.0 fm Finite volume 4.0 fm Finite volume 5.0 fm Finite volume 6.0 fm Finite volume 7.0 fm Full-QCD Infinite Volume Results Lattice Results

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 1 / 3

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BONUS SLIDES

Neutron βn

0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 m2 ( GeV2) 1 2 3 4 5 6 7

n (10 4 fm3)

  • inf. volume extrapolation

Experiment: Kossert Prediction at the physical point Experiment: PDG Experiment: Myers Experiment: Griesshammer Full-QCD Infinite Volume Results

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 1 / 3

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BONUS SLIDES

Results

Table: Magnetic polarisability values for the neutron and proton at each quark mass. The numbers in parantheses describe statistical (systematic) uncertainties. The value at mπ = 0.140 GeV is the result of our chiral extrapolation.

mπ (GeV) βn fm3 × 10−4

nχ2 dof

βp fm3 × 10−4

pχ2 dof

0.702 1.91(12) 0.85 1.90(19) 0.96 0.570 1.66(10) 0.88 1.87(18) 1.10 0.411 1.53(29) 0.74 1.98(21) 0.67 0.296 1.27(37) 0.91 1.93(22) 0.33 0.140 2.05(26)(19) 2.79(22)(17)

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 2 / 3

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BONUS SLIDES

Magnetic Moment

◮ Considerably easier than magnetic polarisability ◮ Take a different ratio

R(B, t) = G(+s, +B) + G(−s, −B) G(+s, −B) + G(−s, +B)

  • ◮ to get an energy shift of

δEµ(B) = −µ B + O

  • B3

◮ µn = −1.49(7) µN

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 3 / 3

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BONUS SLIDES

Magnetic Moment

20 25 30 35 t 0.10 0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 δE(B) (GeV)

Plot of fit to BF3 deltaEmu showing the χ2

dof of the fit

kB = 1, χ2

dof = 0.24 len =2

kB = 2, χ2

dof = 0.27 len =2

kB = 3, χ2

dof = 0.51 len =2

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 3 / 3

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BONUS SLIDES

Magnetic Moment

1 2 3 4 kB 0.00 0.05 0.10 0.15 0.20 0.25 δEµ (GeV)

Fits to energy shifts; t = [22, 24] c1 k, χ2

dof = 18.65

c1 k + c0, χ2

dof = 8.53

c3 k 3 + c1 k, χ2

dof = 0.907

  • R. Bignell (CSSM)

Computing the magnetic field response of the proton CHEP 2019 3 / 3