Chiral Structure in Baryon Magnetic Moments

Chiral Structure in Baryon Magnetic 

Moments 

Ross Young, 

Derek Leinweber, Tony Thomas 

CSSM, University of Adelaide 

Chiral Structure in Baryon Magnetic Moments – p.1/18

Introduction 

Chiral extrapolation for lattice QCD 


Introduction 


Nontrivial problem due to nonanalytic behaviour 


Introduction 



Highlight nucleon mass extrapolation 

Reliable extrapolation with FRR χPT 


Introduction 



Highlight nucleon mass extrapolation 

Reliable extrapolation with FRR χPT 

Magnetic moments p and ∆ 

Observation of quenched chiral behaviour 


Expansion of Nucleon Mass 

General expansion about chiral limit (m q = 0) 

M N = {Analytic terms} + {Chiral loop corrections} 




M N = {Analytic terms} + {Chiral loop corrections} 

M N = {a 0 + a 2 m 2 π + a 4 m 4 π + a 6 m 6 π + ..} 

+{ χ π I π (m π ) + χ π∆ I π∆ (m π ) + ..} 




M N = {a 0 + a 2 m 2 π + a 4 m 4 π + a 6 m 6 π + ..} 

+{ χ π I π (m π ) + χ π∆ I π∆ (m π ) + ..} 

I π = 

I π∆ = 


Lattice QCD and Dim Reg χPT 

Lattice data: Ali Khan et al. PRD 65,054505 (2002) 

mN GeV 

2 

1.8 

1.6 

1.4 

1.2 

1 

0 0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 




mN GeV 

2 

1.8 

1.6 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

A: a 0 + a 2 m 2 π 




mN GeV 

2 

1.8 

1.6 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

B: a 0 + a 2 m 2 π + χ π I π 




mN GeV 

2 

1.8 

1.6 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

C: a 0 + a 2 m 2 π + a 4 m 4 π + χ π I π 




mN GeV 

2 

1.8 

1.6 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

D: a 0 + a 2 m 2 π + χ π I π + χ π∆ I π∆ 




mN GeV 

2 

1.8 

1.6 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

E: a 0 + a 2 m 2 π + a 4 m 4 π + χ π I π + χ π∆ I π∆ 




mN GeV 

2 

1.8 

1.6 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

F: a 0 + a 2 m 2 π + a 4 m 4 π + a 5 m 5 π + χ π I π + χ π∆ I π∆ 




mN GeV 

2 

1.8 

1.6 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

G: a 0 + a 2 m 2 π + a 4 m 4 π + a 6 m 6 π + χ π I π + χ π∆ I π∆ 


Convergence of Dim Reg χPT 

Extrapolated m N , different chiral expansions 

1.2 

1.0 

mN GeV 

0.8 

0.6 

0.4 

0.2 

A B C D E F G 


Lattice QCD and FRR χPT 

Dipole Regularisation 

mN GeV 

2 

1.8 

1.6 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

A: a 0 + a 2 m 2 π 




mN GeV 

2 

1.8 

1.6 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

B: a 0 + a 2 m 2 π + χ π I π 




mN GeV 

2 

1.8 

1.6 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

C: a 0 + a 2 m 2 π + a 4 m 4 π + χ π I π 




mN GeV 

2 

1.8 

1.6 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

D: a 0 + a 2 m 2 π + χ π I π + χ π∆ I π∆ 




mN GeV 

2 

1.8 

1.6 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

E: a 0 + a 2 m 2 π + a 4 m 4 π + χ π I π + χ π∆ I π∆ 




mN GeV 

2 

1.8 

1.6 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

F: a 0 + a 2 m 2 π + a 4 m 4 π + a 5 m 5 π + χ π I π + χ π∆ I π∆ 




mN GeV 

2 

1.8 

1.6 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

G: a 0 + a 2 m 2 π + a 4 m 4 π + a 6 m 6 π + χ π I π + χ π∆ I π∆ 


Convergence of FRR χPT (Dipole) 

Comparison: dipole vs. dim reg 

1.2 

1.0 

mN GeV 

0.8 

0.6 

0.4 

0.2 

A B C D E F G 


Regulator Dependence 

Best fit to lattice data, hep-lat/0302020 

1.8 

1.6 

mN GeV 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

Dipole 




1.8 

1.6 

mN GeV 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

Monopole 




1.8 

1.6 

mN GeV 

1.4 

1.2 

1 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 

Gaussian 




1.8 

1.6 

mN GeV 

1.4 

1.2 

1 

Sharp Cut-off 

0.2 0.4 0.6 0.8 1 

m Π 2 GeV 2 


LECs, Extrapolated 

Best fits to lattice data 

FRR a 0 c 0 a 2 c 2 

Sharp cutoff 1.06 0.892 1.06 3.16 

Monopole 1.38 0.914 0.950 2.63 

Dipole 1.15 0.910 0.998 2.72 

Gaussian 1.11 0.906 1.02 2.79 

Leinweber, Thomas & Young, hep-lat/0302020 


Statistical Errors 

Error bounds (dipole) 

1.8 

1.6 

MN GeV 

1.4 

1.2 

1.0 

0.8 

0.0 0.2 0.4 0.6 0.8 1.0 

m 2 Π GeV 2 


Magnetic Moments 

Consider extrapolation of nucleon magnetic moments 

from lattice QCD 





Many one-loop graphs to be considered 





Many one-loop graphs to be considered 

Additional lattice complication 

→ The quenched approximation 


Chiral Expansion 

Mass expansion 

M N = {a 0 + a 2 m 2 π + a 4 m 4 π + a 6 m 6 π + ..} 

+{ χ π I π (m π ) + χ π∆ I π∆ (m π ) + ..} 


Chiral Expansion 

Mass expansion 

M N = {a 0 + a 2 m 2 π + a 4 m 4 π + a 6 m 6 π + ..} 

+{ χ π I π (m π ) + χ π∆ I π∆ (m π ) + ..} 

Similarly, the magnetic moment expansion 

µ p = β 0 + β 2 m 2 π + β 4 m 4 π + β 6 m 6 π + ... 

+ ∑ α 

χ α I µ α 


One-loop Contributions 


Nonanalytic Structure 

Leading nonanalytic contribution 

(a) 

(b) 

(c) 

m π 

m 2 π ln m π 

µ 


µ 

(d) m 4 π ln m π 

µ 

(e) m 3 π + 

4π∆ 3 π ln m π 

µ 

(f) m 3 π 

(g) m 4 π ln m π 

µ 

(h) 

(i) 


µ 


µ 


Quenched Approximation 

Virtual q¯q-pair creation in vacuum neglected — pure 

glue 




glue 

Chiral loops modified in absence of sea quarks 

Different nonanalytic contributions alter extrapolation 




glue 

Chiral loops modified in absence of sea quarks 

Different nonanalytic contributions alter extrapolation 

The η ′ a new light degree of freedom 


Quenched Diagrams 


Quenched QCD 

Proton magnetic moment 

3 

2.5 

Μ p ΜN 

2 

1.5 

1 

0 0.2 0.4 0.6 0.8 

m 2 Π GeV 2 

Lattice data: Preliminary CSSM results Zanotti et al. 


Quenched QCD 

Proton magnetic moment 

3 

2.5 

Μ p ΜN 

2 

1.5 

1 

0 0.2 0.4 0.6 0.8 

m 2 Π GeV 2 

Quenched fit: β 0 + β 2 m 2 π + Loops 


Quenched QCD 

∆ ++ magnetic moment 

4 

3.5 

Μ 

ΜN 

3 

2.5 

2 

0 0.2 0.4 0.6 0.8 

m 2 Π GeV 2 

Quenched simulation 


Quenched QCD 


4 

3.5 

Μ 

ΜN 

3 

2.5 

2 

0 0.2 0.4 0.6 0.8 

m 2 Π GeV 2 

Quenched fit: β 0 + β 2 m 2 π + Loops 


Quenched QCD 


4 

3.5 

Μ 

ΜN 

3 

2.5 

2 

0 0.2 0.4 0.6 0.8 

m 2 Π GeV 2 

Flip sign on chiral loop 


Quenched QCD 

∆ + -Proton magnetic moment 

2.4 

2.2 

2 

Μ B ΜN 

1.8 

1.6 

1.4 

1.2 

1 

0 0.2 0.4 0.6 0.8 

m 2 Π GeV 2 

Quenched simulation 


Quenched QCD 

∆ + -Proton magnetic moment 

2.4 

2.2 

2 

Μ B ΜN 

1.8 

1.6 

1.4 

1.2 

1 

0 0.2 0.4 0.6 0.8 

m 2 Π GeV 2 

Quenched fits 


Conclusions 

Finite-range regulator (FRR) improves the 

convergence properties of χ’al EFT 


Conclusions 



Provides accurate chiral extrapolation for lattice 

simulations 


Conclusions 



Provides accurate chiral extrapolation for lattice 

simulations 

Application of FRR to magnetic moments 

Quenched FRR-χPT to assist understanding 

simulation results

Chiral Structure in Baryon Magnetic Moments

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