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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

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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

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron 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 magnetic polarisability of the proton and neutron?

  • R. Bignell (CSSM)

Magnetic Polarisability with the Background Field Method APLAT 2020 2 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron 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)

Magnetic Polarisability with the Background Field Method APLAT 2020 2 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron 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: Chang: 1506.05518 Experiment

  • R. Bignell (CSSM)

Magnetic Polarisability with the Background Field Method APLAT 2020 2 / 19

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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

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)

Magnetic Polarisability with the Background Field Method APLAT 2020 3 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary

Background Field Method

◮ Introduce a background field on the lattice

Continuum:Dµ → DQCD

µ

+ iqe Aµ Lattice:Uµ (x) → ei a qe Aµ(x) Uµ (x)

◮ Choose

A appropriately to generate constant magnetic field B = +B ˆ z

◮ Periodic spatial boundary conditions impose a quantisation for a uniform field

a2 qe B2 = 2 πk Nx Ny

◮ kd = 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

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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

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary

Wilson Term Mass Renormalisation

0.00 0.02 0.04 0.06

eBa2

0.10 0.15 0.20 0.25

aE(B)

+ u d

  • R. Bignell (CSSM)

Magnetic Polarisability with the Background Field Method APLAT 2020 5 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary

Clover term

0.00 0.02 0.04 0.06

eBa2

0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250

aE(B)

+ u d

  • R. Bignell (CSSM)

Magnetic Polarisability with the Background Field Method APLAT 2020 6 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary

Clover term

◮ Careful examination of the clover term

a ccl

  • µ<ν

σµν Fµν

◮ 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

ccl → CSW F QCD

µν

+ CEM F EM

µν

◮ and set CEM such that Wilson Landau shift is cancelled

CEM = CTree

EM

◮ 1910.14244

  • R. Bignell (CSSM)

Magnetic Polarisability with the Background Field Method APLAT 2020 6 / 19

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

  • R. Bignell (CSSM)

Magnetic Polarisability with the Background Field Method APLAT 2020 7 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron 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 2D Laplacian

  • ψi

B

  • to project the propagator

Pn

  • x, t;

x′, t′ =

n

  • i=1
  • x, t
  • ψi,

B

ψi

B

  • x′, t′

δzz′ δtt′

◮ Projected propagator is

Sn( x, t; 0, 0) =

  • x′

Pn( x, t; x′, t) S( x′, t; 0, 0)

◮ 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

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron 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 energy difference E(B) − m for aligned and anti-aligned spin-field orientations and

combine 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)

Magnetic Polarisability with the Background Field Method APLAT 2020 9 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron 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)

Magnetic Polarisability with the Background Field Method APLAT 2020 10 / 19

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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| 2 M = −4 π 2 β B2 = c2 k2

◮ 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

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron 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)

Magnetic Polarisability with the Background Field Method APLAT 2020 11 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary

Ensemble Details

κud mπ (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: 323 × 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

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron 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)

Magnetic Polarisability with the Background Field Method APLAT 2020 13 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary

Making Contact With Experiment

◮ Use a chiral effective field theory analysis to

  • 1. Account for finite volume effects
  • 2. Model sea-quark-loop contributions to β using techniques of partially quenched χPT

p p n π+

u d d d u u u u d u d d u u d d u u

  • R. Bignell (CSSM)

Magnetic Polarisability with the Background Field Method APLAT 2020 14 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron 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)

Magnetic Polarisability with the Background Field Method APLAT 2020 14 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron 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)

Magnetic Polarisability with the Background Field Method APLAT 2020 14 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary

Making Contact With Experiment

◮ Use a chiral effective field theory analysis to

  • 1. Account for finite volume effects
  • 2. Model sea-quark-loop contributions to β using techniques of partially quenched χPT
  • 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)

Magnetic Polarisability with the Background Field Method APLAT 2020 14 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron 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)

Magnetic Polarisability with the Background Field Method APLAT 2020 14 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary

Neutron Energy Shift

◮ Identical process for neutron correlation functions ◮ A different fit function used

δ E(B, t) = −4 π 2 β B2 = c2 k2

◮ No Hadronic U1 Landau wavefunction projection

  • R. Bignell (CSSM)

Magnetic Polarisability with the Background Field Method APLAT 2020 15 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary

Neutron Energy Shift

20 25 30 35

t

0.20 0.15 0.10 0.05 0.00 0.05 0.10

E(B) (GeV)

kd = 1,

2 dof =1.0, len =6

kd = 2,

2 dof =0.42, len =5

kd = 3,

2 dof =0.59, len =4

  • R. Bignell (CSSM)

Magnetic Polarisability with the Background Field Method APLAT 2020 15 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary

Polarisability Fit

1 2 3 4

kd

0.12 0.10 0.08 0.06 0.04 0.02 0.00

E (GeV)

= 0.13727 c2 k2,

2 dof = 0.8754

  • R. Bignell (CSSM)

Magnetic Polarisability with the Background Field Method APLAT 2020 16 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary

Neutron Extrapolation

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

m2 ( GeV2 )

1 2 3 4 5 6 7

n (10 4 fm) Full-QCD Infinite Volume EFT Extrapolation Experiment

  • R. Bignell (CSSM)

Magnetic Polarisability with the Background Field Method APLAT 2020 17 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary

βp − βn = 0.80 (28) (4) × 10−4 fm3

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

m2 ( GeV2 )

2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5

p n (10 4 fm)

  • Inf. Vol.

FV Corr. This work Experiment: PDG Reggeon Dom. : GHLR 2015

  • R. Bignell (CSSM)

Magnetic Polarisability with the Background Field Method APLAT 2020 18 / 19

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Introduction Lattice QCD BFCC Action Quark Operators Magnetic Polarisability Proton Neutron Summary

Summary

◮ Removed the additive mass renormalisation due to the Wilson term in a background magnetic

field

◮ Calculated the magnetic polarisability of the proton and neutron 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)

Magnetic Polarisability with the Background Field Method APLAT 2020 19 / 19

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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)

Magnetic Polarisability with the Background Field Method APLAT 2020 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 1.91(19) 0.96 0.570 1.68(10) 0.88 1.89(18) 1.10 0.411 1.58(29) 0.74 2.03(21) 0.67 0.296 1.42(37) 0.91 2.08(22) 0.33 0.140 2.06(26)(20) 2.79(22)(18)

  • R. Bignell (CSSM)

Magnetic Polarisability with the Background Field Method APLAT 2020 2 / 3

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BONUS SLIDES

Review of Nucleon Polarisabilities

Figure: Magnetic dipole polarizability βM1 of the nucleon. Figure from 2006.16124

  • R. Bignell (CSSM)

Magnetic Polarisability with the Background Field Method APLAT 2020 3 / 3