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Table of Contents
                            1 Introduction
	1.1 Theory
	1.2 Experiment
	1.3 Overview of this work
2 QCD and chiral Symmetry
	2.1 Effective field theory
	2.2 Linear sigma model
	2.3 Chiral symmetry
	2.4 ChPT
		2.4.1 Lowest order
		2.4.2 L9 and L10
3 Hidden local symmetry model
4 +- and +- vertices
	4.1 High energy limit
5 Muon magnetic anomaly from light by light amplitude
	5.1 General
	5.2 Integration
		5.2.1 Pion exchange
		5.2.2 Bare pion loop
		5.2.3 HLS
		5.2.4 Full VMD
		5.2.5 L9 and L10
6 Relevant Momentum Regions for the pion Loop Contribution.
	6.1 Dependence on the photon cut–off
	6.2 Anatomy of the relevant momentum regions for the pion Loop Contribution.
7 Conclusions and Prospects
                        
Document Text Contents
Page 1

LU TP 12-26
20 June 2012

The Anatomy of the Pion Loop Hadronic Light by
Light Scattering Contribution to the Muon Magnetic

Anomaly

Mehran Zahiri Abyaneh

Thesis advisor : Johan Bijnens

Department of Theoretical Physics, Lund University

Sölvegatan 14A, S22362 Lund, Sweden

Abstract

This thesis investigates the Hadronic Light by Light (HLL) scattering contribution
to the muon g � 2, which is one of the most important low energy hadronic effects
and consists mainly of the quark loop, the pion pole and the charged pion and
kaon loops. In this work the charged pion loop has been investigated more closely.
After reviewing the subject a preliminary introduction to Chiral Perturbation Theory
(ChPT), Hidden Local Symmetry (HLS) model and the full Vector Meson Dominance
(VMD) model is given, and they are used to calculate the pion loop HLL scattering
contribution to the muon anomalous magnetic moment. The momentum regions
where the contributions of the bare pion loop, the VMD model, and the HLS come
from, have been studied, to understand why different models give very different
results. The effects of pion polarizability and charge radius on the HLL scattering,
which appear at order p4 in ChPT, from L9 and L10 Lagrangian terms and their
momentum regions have been studied.

Master of Science Thesis

a
rX

iv
:1

2
0
8
.2

5
5
4
v1


[h

e
p
-p

h
]
1

3
A

u
g
2

0
1
2

Page 2

Contents

1 Introduction 2
1.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Overview of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 QCD and chiral Symmetry 10
2.1 Effective field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Linear sigma model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Chiral symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 ChPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 Lowest order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.2 L9 and L10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Hidden local symmetry model 17

4 γπ+π− and γγπ+π− vertices 20
4.1 High energy limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Muon magnetic anomaly from light by light amplitude 22
5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.2.1 Pion exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.2.2 Bare pion loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2.3 HLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.2.4 Full VMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2.5 L9 and L10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6 Relevant Momentum Regions for the pion Loop Contribution. 36
6.1 Dependence on the photon cut–off Λ . . . . . . . . . . . . . . . . . . . . . 36
6.2 Anatomy of the relevant momentum regions for the pion Loop Contribution. 36

7 Conclusions and Prospects 44

1

Page 23

(a) (b)

Figure 14: The γγππ vertex.

It can be shown that the γγππ amplitude for two high energy photons with momenta
P1 ' −P2 ' P is proportional to 1/P 2. This is done by using the operator product
expansion for two vector currents and showing that the matrix element of the leading part
which is proportional to an axial current vanishes. Hence, when P → ∞ the amplitude
vanishes. The amplitude corresponding to the diagram 14 is

M = ie2
{(P1µ − 2K1µ) (P2ν − 2K2ν)

(P2 −K2)
2 −m2π

+
(P2ν − 2K1ν) (P1µ − 2K2µ)

(P1 −K2)
2 −m2π

− 2gµν
}
. (4.9)

For P1 ' −P2 ' P →∞ this reads

M =
(

2gµν − 2
PµPν
P 2

)
, (4.10)

which does not vanish fast enough. For the VMD case the above amplitude should be
multiplied with (4.6). The resulting amplitude vanishes at order P 0. Now let us examine
the HLS case. Then, the first and second term of (4.9) are multiplied with (4.3) and the
third is multiplied with (4.4). In the high energy limit the leading term is

M = 2
(
gµν −

PµPν
P 2

)
(1− a) . (4.11)

This is only satisfied for a = 1. However, the case HLS with a = 2 does not uphold this
condition. Hence, one can infer that something must be wrong with it.

5 Muon magnetic anomaly from light by light ampli-

tude

5.1 General

The response of a muon carrying momentum p to an external electromagnetic field Aµ with
momentum transferred p3 ≡ p− p′ is described by the matrix element

M≡ − | e | Aρū(p′)Γρ(ṕ, p)u(p) , (5.1)

22

Page 24

with

Γρ(p′, p) = F1(p
2
3)γ

ρ �
i

2ml
F2(p

2
3)σ

ρνp3ν � F3(p23)γ5σ
ρνp3ν +F4(p

2
3)[p

2


ρ � 2mlp
ρ
3]γ5 . (5.2)

The two first form factors are known as the Dirac and the Pauli form factor, respectively.
In fact [12], the magnetic moment of the fermion in magnetons is µ � 2(F1(0) + F2(0))
and in analogy with the classical limit, described in the introduction, one can define the
gyromagnetic ratio as g � 2µ and the anomalous magnetic moment as a � (g � 2)/2 =
F2(0) [12]. The form factor F3(p

2
3) can be different from zero provided parity and time

reversal invariance are broken and for F4(p
2
3) to be nonzero, parity invariance should be

broken. Therefore, both are absent in our survey. Since the task of computation of Γρ(p
′, p)

is very involved especially for higher order corrections, one can project out the form factor
of interest, F2(p

2
3) in our case, and then the general form of the contribution can be shown

to be [3]

alight−by−lightµ = �
1

48m
tr[(/p+m)Γ

λβ(0) (/p+m)[γλ, γβ]] . (5.3)

Defining the four point function Πρναλ as

Πρναλ(p1, p2, p3) = i
3


d4x1


d4x2


d4x3 exp i (p1 � x1 + p2 � x2 + p3 � x3)

� h 0 j Tjρ(0)jν(x1)jα(x2)jλ(x3) j 0i (5.4)

and using the Ward identity to rewrite it in the form

Πρναλ(p1, p2, p3) = � p3β
δΠρναβ(p1, p2, p3)

δp3λ
, (5.5)

the Γλβ(0) for the Figure 5 writes

Γλβ(p3) = jej6


d4p1
(2π)4


d4p2
(2π)4

1

q2 p21 p
2
2(p

2
4 � m2) (p25 � m2)


[
δΠρναβ(p1, p2, p3)

δp3λ

]
γα(/p4 +m)γν(/p5 +m)γρ . (5.6)

with p4 = p
′ � p2, p5 = p � q. The most formidable task ahead is then to build the

relevant four point functions and to calculate the integral (5.6). One should note that this
four-point function can be decomposed by using Lorentz covariance as follows

Πρναβ(p1, p2, p3) � Π1(p1, p2, p3)gρνgαβ + Π2(p1, p2, p3)gραgνβ

+ Π3(p1, p2, p3)g
ρβgνα

+ Π1jk(p1, p2, p3)g
ρνpαj p

β
k + Π

2jk(p1, p2, p3)g
ραpνjp

β
k

+ Π3jk(p1, p2, p3)g
ρβpνjp

α
k + Π

4jk(p1, p2, p3)g
ναp

ρ
jp
β
k

+ Π5jk(p1, p2, p3)g
νβp

ρ
jp
α
k + Π

6jk(p1, p2, p3)g
αβp

ρ
jp
ν
k

+ Πijkm(p1, p2, p3)p
ρ
i p
ν
jp
β
kp

α
m , (5.7)

23

Page 46

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