A Appendix

A.1 Notation

In this Appendix, we summarize the notation that are used throughout thetext. We use e to denote the muon electric charge, e = −|e| and m to denotethe muon mass.

The Lorentz indices are denoted by Greek letters, µ, ν, ... = 0, 1, 2, 3; thethree-dimensional indices by Latin letters, i, j, k, ... = 1, 2, 3. The spatial vec-tors refer to upper Lorentz indices, e.g. xµ = (t,x). The metric tensor isgµν = Diag{1,−1,−1,−1}, and the totally antisymmetric tensor εµναβ is de-fined through ε0123 = 1. The product of two Lorentz vectors is denoted byab = aµbµ = a0b0 − ab.

The Dirac matrices are denoted by γµ and

γ5 = −iγ0γ1γ2γ3 = − i

4!εµναβγµγνγαγβ .

We use the short-hand notation a = aµγµ. The Pauli matrices are denotedby σi, i = 1, 2, 3.

Unless stated explicitly, we use the system of units where the Planckconstant � and the speed of light c are equal to 1.

A.2 Gounaris-Sakurai Parametrizationof the Pion Form Factor

In this Appendix, we summarize the formulas related to Gounaris-Sakurai(GS) parametrization of the pion form factor. The form factor is written as

Fπ(s) =1

1+β+γ

[BWGS

ρ (s)(1+δ

s

m2ω

BWω(s))+βBWGS

ρ′ (s)+γBWGSρ′′ (s)

],

(A.1)where

BWGSρ (s) =

m2ρ

m2ρ − s + fρ(s) − i

√sΓρ(s)

(1 +

d

mρΓρ

);

BWω(s) =m2

ω

m2ω − s + iΓωmω

, (A.2)

K. Melnikov and A. Vainshtein: Theory of the Muon Anomalous Magnetic MomentSTMP 216, 173–174 (2006)c© Springer-Verlag Berlin Heidelberg 2006

174 A Appendix

with

fρ(s) = Γρ

m2ρ

p0

(p(s)2(h(s) − h(m2

ρ)) + (m2ρ − s)p2

0

dh

ds

∣∣∣∣s=m2

ρ

),

h(s) =2p(s)π√

sln

√s + 2p(s)2mπ

, (A.3)

d =3π

m2π

p20

lnmρ + 2p0

2mπ+

mρ

2πp0− m2

πmρ

πp30

,

We also use p(s) =√

1 − 4m2π/s, p0 = p(m2

ρ) and

Γρ(s) = Γρ

(p(s)p0

)3 (m2ρ

s

)1/2

. (A.4)

All the parameters involved in the Gounaris-Sakurai parameterization ofthe pion form factor are extracted from the fit to the experimental data. Fornumerical computations, we use [1],

mρ = 773.1 ± 0.5 MeV, Γρ = 148.0 ± 0.9 MeV ,

mρ− = 775.5 ± 0.6 MeV, Γρ− = 148.2 ± 0.8 MeV ,

|δ| = (2.03 ± 0.1) × 10−3, arg δ = 13.00 ± 2.30 ,

mρ′ = 1409 ± 12 MeV, Γρ′ = 501 ± 37 MeV ,

mρ′′ = 1740 ± 21 MeV, Γρ′′ = 235 MeV ,

as well as β = −0.167 ± 0.006 and γ = 0.071 ± 0.006. For the mass and thewidth of ω, we adopt mω = 782.71 ± 0.08 MeV and Γω = 8.68 MeV.

References

1. M. Davier, Nucl. Phys. Proc. Suppl. 131, 123 (2004); for an update, see S. Schaelet al. [ALEPH Collaboration], Phys. Rept. 421, 191 (2005).