S

eminaire

P

oinar

e

2

(2002)

93

{

125

S

eminaire

P

oina

r

e

The

Anomalous

Magneti

Momen

ts

of

the

Eletron

and

the

Muon

Mar

Kne

h

t

Cen

tre

de

Ph

ysique

Th

eorique

CNRS-Lumin

y

,

Case

907

F-13288

Marseille

Cedex

9,

F

rane

1

In

tro

dution

In

F

ebruary

2001,

the

Muon

(g-2)

Collab

oration

of

the

E821

exp

erimen

t

at

the

Bro

okha

v

en

A

GS

released

a

new

v

alue

of

the

anomalous

magneti

momen

t

of

the

m

uon,

measured

with

an

unpree-

den

ted

auray

of

1.3

ppm.

This

announemen

t

has

aused

quite

some

exitemen

t

in

the

partile

ph

ysis

omm

unit

y

.

Indeed,

this

exp

erimen

tal

v

alue

w

as

laimed

to

sho

w

a

deviation

of

2.6

with

one

of

the

most

aurate

ev

aluation

of

the

anomalous

magneti

momen

t

of

the

m

uon

within

the

standard

mo

del.

It

w

as

subsequen

tly

sho

wn

that

a

sign

error

in

one

of

the

theoretial

on

tributions

w

as

resp

onsible

for

a

sizable

part

of

this

disrepany

,

whi

h

ev

en

tually

only

amoun

ted

to

1.6

.

Ho

w

ev

er,

this

ev

en

t

had

the

merit

to

dra

w

the

atten

tion

to

the

fat

that

lo

w

energy

but

high

preision

exp

erimen

ts

represen

t

real

p

oten

tialities,

omplemen

tary

to

the

high

energy

aelerator

programs,

for

evidening

p

ossible

new

degrees

of

freedom,

sup

ersymmetry

or

whatev

er

else,

b

ey

ond

those

desrib

ed

b

y

the

standard

mo

del

of

eletromagneti,

w

eak,

and

strong

in

terations.

Clearly

,

in

order

for

theory

to

mat

h

su

h

an

aurate

measuremen

t

[in

the

mean

time,

the

relativ

e

error

has

ev

en

b

een

further

redued,

to

0.7

ppm℄,

alulations

in

the

standard

mo

del

ha

v

e

to

b

e

pushed

to

their

v

ery

limits.

The

diÆult

y

is

not

only

one

of

ha

ving

to

ompute

higher

orders

in

p

erturbation

theory

,

but

also

to

orretly

tak

e

in

to

aoun

t

strong

in

teration

on

tributions

in

v

olving

lo

w-energy

sales,

where

non

p

erturbativ

e

eets

are

imp

ortan

t,

and

whi

h

therefore

represen

t

a

real

theoretial

hallenge.

The

purp

ose

of

this

aoun

t

is

to

giv

e

an

o

v

erview

of

the

main

features

of

the

theoretial

alulations

that

ha

v

e

b

een

done

in

order

to

obtain

aurate

preditions

for

the

anomalous

magneti

momen

ts

of

the

eletron

and

of

the

m

uon

within

the

standard

mo

del.

There

exist

sev

eral

exellen

t

reviews

of

the

sub

jet,

whi

h

the

in

terested

reader

ma

y

onsult.

As

far

as

the

situation

up

to

1990

is

onerned,

the

olletion

of

artiles

published

in

Ref.

[1℄

oers

a

w

ealth

of

information,

on

b

oth

theory

and

exp

erimen

t.

A

v

ery

useful

aoun

t

of

earlier

theoretial

w

ork

is

presen

ted

in

Ref.

[2℄.

Among

the

more

reen

t

reviews,

Refs.

[3

,

4,

5,

6

℄

are

most

informativ

e.

I

shall

not

tou

h

on

the

sub

jet

of

the

study

of

new

ph

ysis

senarios

whi

h

migh

t

oer

an

explanation

for

a

p

ossible

deviation

b

et

w

een

the

standard

mo

del

predition

of

the

magneti

momen

t

of

the

m

uon

and

its

exp

erimen

tal

v

alue.

F

or

this

asp

et,

I

refer

the

reader

to

[7

℄

and

to

the

artiles

quoted

therein,

or

to

[8

℄.

2

General

onsiderations

In

the

on

text

of

relativisti

quan

tum

me

hanis,

the

in

teration

of

a

p

oin

tlik

e

spin

one-half

partile

of

harge

e

`

and

mass

m

`

with

an

external

eletromagneti

eld

A

(x)

is

desrib

ed

b

y

the

Dira

equation

with

the

minimal

oupling

presription,

i

h

 

t

=

h

i

h

r

e

`

A

+

m

`

2

+

e

`

A

0

i

 

:

(2.1)

94

M.

Kne

h

t

S

eminaire

P

oinar

e

In

the

non

relativisti

limit,

this

redues

to

the

P

auli

equation

for

the

t

w

o-omp

onen

t

spinor

'

desribing

the

large

omp

onen

ts

of

the

Dira

spinor

 

,

i

h

'

t

=

(

i

h

r

(e

`

=)A)

2

2m

`

e

`

h

2m

`

B

+

e

`

A

0

'

:

(2.2)

As

is

w

ell

kno

wn,

this

equation

amoun

ts

to

asso

iate

with

the

partile's

spin

a

magneti

momen

t

M

s

=

g

`

e

`

2m

`

S

;

S

=

h

2

;

(2.3)

with

a

gyromagneti

ratio

predited

to

b

e

g

`

=

2.

In

the

on

text

of

quan

tum

eld

theory

,

the

resp

onse

to

an

external

eletromagneti

eld

is

desrib

ed

b

y

the

matrix

elemen

t

of

the

eletromagneti

urren

t

1

J

[spin

pro

jetions

and

Dira

indies

are

not

written

expliitly℄

h`

(p

0

)jJ

(0)j`

(p)i

=

u(p

0

)

(p

0

;

p)u(p)

;

(2.4)

with

[k

p

0

p

℄

(p

0

;

p)

=

F

1

(k

2

)

+

i

2m

`

F

2

(k

2

)

k

F

3

(k

2

)

5

k

:

(2.5)

This

expression

of

the

matrix

elemen

t

h`

(p

0

)jJ

(0)j`

(p)i

is

the

most

general

that

follo

ws

from

Loren

tz

in

v

ariane,

the

Dira

equation

for

the

t

w

o

spinors,

(6

p

m)u (p)

=

0,

u(p

0

)(6

p

0

m)

=

0,

and

the

onserv

ation

of

the

eletromagneti

urren

t,

( 

J

)

(x)

=

0.

The

t

w

o

rst

form

fators,

F

1

(k

2

)

and

F

2

(k

2

),

are

kno

wn

as

the

Dira

(or

eletri)

form

fator

and

the

P

auli

(or

magneti)

form

fator,

resp

etiv

ely

.

Sine

the

eletri

harge

op

erator

Q

is

giv

en,

in

units

of

the

harge

e

`

,

b

y

Q

=

Z

dx

J

0

(x

0

;

x)

;

(2.6)

the

form

fator

F

1

(k

2

)

satises

the

normalization

ondition

F

1

(0)

=

1.

The

presene

of

the

form

fator

F

3

(k

2

)

requires

b

oth

parit

y

and

time

rev

ersal

in

v

ariane

to

b

e

brok

en.

It

is

therefore

absen

t

if

only

eletromagneti

in

terations

are

onsidered.

On

the

other

hand,

in

the

standard

mo

del,

the

w

eak

in

terations

violate

b

oth

parit

y

and

time

rev

ersal

symmetry

,

so

that

they

ma

y

indue

su

h

a

form

fator.

The

analyti

struture

of

these

form

fators

is

ditated

b

y

general

prop

erties

of

quan

tum

eld

theory

[ausalit

y

,

analytiit

y

,

and

rossing

symmetry℄.

They

are

real

funtions

of

k

2

in

the

spaelik

e

region

k

2

<

0.

In

the

timelik

e

region,

they

b

eome

omplex,

with

a

ut

starting

at

k

2

>

4m

2

`

.

A

t

k

2

=

0,

they

desrib

e

the

residue

of

the

s-

hannel

p

ole

in

the

S-matrix

elemen

t

for

elasti

`

+

`

sattering.

A

t

tree

lev

el,

i.e.

in

the

lassial

limit,

one

nds

F

tree

1

(k

2

)

=

1

;

F

tree

2

(k

2

)

=

0

;

F

tree

3

(k

2

)

=

0

:

(2.7)

In

order

to

obtain

non

zero

v

alues

for

F

2

(k

2

)

and

F

3

(k

2

)

already

at

tree

lev

el,

the

in

teration

of

the

Dira

eld

with

the

photon

eld

A

w

ould

ha

v

e

to

depart

from

the

minimal

oupling

presription.

F

or

instane,

the

mo

diation

[F

=

A

A

,

J

=

 

 

℄

Z

d

4

xL

in

t

=

e

`

Z

d

4

xJ

A

!

!

Z

d

4

x

b

L

in

t

=

e

`

Z

J

A

+

h

4m

`

a

`

 

 

F

+

h

2e

`

d

`

 

i

5

 

F

=

e

`

Z

d

4

x

b

J

A

;

(2.8)

1

In

the

standard

mo

del,

J

denotes

the

total

eletromagneti

urren

t,

with

the

on

tributions

of

all

the

harged

elemen

tary

elds

in

presene,

leptons,

quarks,

eletro

w

eak

gauge

b

osons,...

V

ol.

2,

2002

The

Anomalous

Magneti

Momen

ts

of

the

Eletron

and

the

Muon

95

with

2

b

J

=

J

h

2m

`

a

`

 

 

h

d

`

e

`

 

i

5

 

;

(2.9)

leads

to

b

F

tree

1

(k

2

)

=

1

;

b

F

tree

2

(k

2

)

=

a

`

;

b

F

tree

3

(k

2

)

=

d

`

=e

`

:

(2.10)

The

equation

satised

b

y

the

Dira

spinor

 

then

reads

i

h

 

t

=

i

hr

e

`

A

+

m

`

2

+

e

`

A

0

+

e

`

h

2m

`

a

`

(i

E

B)

hd

`

(

E

+

i

B)

 

;

(2.11)

and

the

orresp

onding

non

relativisti

limit

b

eomes

3

i

h

'

t

=

(

i

h

r

(e

`

=)A)

2

2m

`

e

`

h

2m

`

(1

+

a

`

)

B

E

+

e

`

A

0

+

'

:

(2.12)

Th

us

the

oupling

onstan

t

a

`

indues

a

shift

in

the

gyromagneti

fator,

g

`

=

2(1

+

a

`

),

while

d

`

giv

es

rise

to

an

eletri

dip

ole

momen

t.

The

mo

diation

(2.8)

of

the

in

teration

with

the

photon

eld

in

tro

dues

t

w

o

arbitrary

onstan

ts,

and

b

oth

terms

pro

dues

a

non

r

enormalizable

in

teration.

Non

onstan

t

v

alues

of

the

form

fators

ould

b

e

generated

at

tree

lev

el

up

on

in

tro

duing

[9

℄

additional

non

renormalizable

ouplings,

in

v

olving

deriv

ativ

es

of

the

external

eld

of

the

t

yp

e

2

n

A

,

whi

h

preserv

e

the

gauge

in

v

ariane

of

the

orresp

onding

eld

equation

satised

b

y

 

.

In

a

renormalizable

framew

ork,

lik

e

QED

or

the

standard

mo

del,

alulable

non

v

anishing

v

alues

for

F

2

(k

2

)

and

F

3

(k

2

)

are

generated

b

y

the

lo

op

orretions.

In

partiular,

the

latter

will

lik

ewise

indue

an

anomalous

magneti

moment

a

`

=

1

2

(g

`

2)

=

F

2

(0)

(2.13)

and

an

eletri

dip

ole

momen

t

d

`

=

e

`

F

3

(0).

If

w

e

onsider

only

the

eletromagneti

and

the

strong

in

terations,

the

urren

t

J

is

gauge

in

v

ari-

an

t,

and

the

t

w

o

form

fators

symmetry

F

1

(k

2

)

and

F

2

(k

2

)

do

not

dep

end

on

the

gauges

hosen

in

order

to

quan

tize

the

photon

and

the

gluon

gauge

elds.

This

is

no

longer

the

ase

if

the

w

eak

in

ter-

ations

are

inluded

as

w

ell,

sine

J

no

w

transforms

under

a

w

eak

gauge

transformation,

and

the

orresp

onding

form

fators

in

general

dep

end

on

the

gauge

hoies.

As

w

e

ha

v

e

already

men

tioned

ab

o

v

e,

the

zero

momen

tum

transfer

v

alues

F

i

(0),

i

=

1;

2;

3

desrib

e

a

ph

ysial

S-matrix

elemen

t.

T

o

the

exten

t

that

the

p

erturbativ

e

S-matrix

of

the

standard

mo

del

do

es

not

dep

end

on

the

gauge

xing

parameters

to

an

y

order

of

the

renormalized

p

erturbation

expansion,

the

quan

tities

F

i

(0)

should

dene

b

ona

de

gauge-xing

indep

enden

t

observ

ables.

The

omputation

of

(p

0

;

p)

is

often

a

tedious

task,

esp

eially

if

higher

lo

op

on

tributions

are

onsidered.

It

is

therefore

useful

to

onen

trate

the

eorts

on

omputing

the

form

fator

of

in

terest,

e.g.

F

2

(k

2

)

in

the

ase

of

the

anomalous

magneti

momen

t.

This

an

b

e

a

hiev

ed

up

on

pro

jeting

out

the

dieren

t

form

fators

[10

,

11

℄

using

the

follo

wing

general

expression

4

F

i

(k

2

)

=

tr

[

i

(p

0

;

p)(6

p

0

+

m

`

)

(p

0

;

p)(6

p

+

m

`

)℄

;

(2.14)

with

1

(p

0

;

p)

=

1

4

1

k

2

4m

2

`

+

3m

`

2

1

(k

2

4m

2

`

)

2

(p

0

+

p)

2

The

urren

t

b

J

is

still

a

onserv

ed

four-v

etor,

therefore

the

matrix

elemen

t

h`

(p

0

)j

b

J

(0)j`

(p)i

also

tak

es

the

form

(2.4),

(2.5),

with

appropriate

form

fators

b

F

i

(k

2

).

3

T

erms

in

v

olving

the

gradien

ts

of

the

external

elds

E

and

B

or

terms

nonlinear

in

these

elds

are

not

sho

wn.

4

F

rom

no

w

on,

I

most

of

the

time

use

the

system

of

units

where

h

=

1,

=

1.

96

M.

Kne

h

t

S

eminaire

P

oinar

e

2

(p

0

;

p)

=

m

2

`

k

2

1

k

2

4m

2

`

m

`

k

2

k

2

+

2m

2

`

(k

2

4m

2

`

)

2

(p

0

+

p)

r

ho

3

(p

0

;

p)

=

i

2k

2

1

k

2

4m

2

`

5

(p

0

+

p)

:

(2.15)

F

or

k

!

0,

one

has

2

(p;

p

0

)

=

1

4k

2

h

1

m

`

1

+

k

2

m

2

`

(p

+

1

2

k

)

+

i

;

(2.16)

and

(6

p

+

m

`

)

2

(p;

p

0

)(6

p

0

+

m

`

)

=

1

4

(6

p

+

m

`

)

h

k

k

2

+

(

p

m

`

)

6

k

k

2

+

i

:

(2.17)

The

last

expression

b

eha

v

es

as

1=k

as

the

external

photon

four

momen

tum

k

v

anishes,

so

that

one

ma

y

w

orry

ab

out

the

niteness

of

F

2

(0)

obtained

up

on

using

Eq.

(2.14).

This

problem

is

solv

ed

b

y

the

fat

that

(p

0

;

p)

satises

the

W

ard

iden

tit

y

(p

0

p)

(p

0

;

p)

=

0

;

(2.18)

follo

wing

from

the

onserv

ation

of

the

eletromagneti

urren

t.

Therefore,

the

iden

tit

y

(p

0

;

p)

=

k

k

(p

0

;

p)

(2.19)

pro

vides

the

additional

p

o

w

er

of

k

whi

h

ensures

a

nite

result

as

k

!

0.

The

presene

of

three

dieren

t

in

terations

in

the

standard

mo

del

naturally

leads

one

to

onsider

the

follo

wing

deomp

osition

of

the

anomalous

magneti

momen

t

a

`

:

a

`

=

a

QED

`

+

a

had

`

+

a

w

eak

`

:

(2.20)

By

a

QED

`

,

I

denote

all

the

on

tributions

whi

h

arise

from

lo

ops

in

v

olving

only

virtual

photons

and

leptons.

Among

these,

it

is

useful

to

distinguish

those

whi

h

in

v

olv

e

only

the

same

lepton

a

v

our

`

for

whi

h

w

e

wish

to

ompute

the

anomalous

magneti

momen

t,

and

those

whi

h

in

v

olv

e

lo

ops

with

leptons

of

dieren

t

a

v

ours,

denoted

olletiv

ely

as

`

0

[

e

2

=4

℄,

a

QED

`

=

X

n1

A

n

n

+

X

n2

B

n

(`;

`

0

)

n

:

(2.21)

The

seond

t

yp

e

of

on

tribution,

a

had

`

in

v

olv

es

also

quark

lo

ops.

Their

on

tribution

is

far

from

b

eing

limited

to

the

short

distane

sales,

and

a

had

`

is

an

in

trinsially

non

p

erturbativ

e

quan

tit

y

.

F

rom

a

theoretial

p

oin

t

of

view,

this

represen

ts

a

serious

diÆult

y

.

Finally

,

at

some

lev

el

of

preision,

the

w

eak

in

terations

an

no

longer

b

e

ignored,

and

on

tributions

of

virtual

Higgs

or

massiv

e

gauge

b

oson

degrees

of

freedom

indue

the

third

omp

onen

t

a

w

eak

`

.

Of

ourse,

starting

from

the

t

w

o

lo

op

lev

el,

a

hadroni

on

tribution

to

a

w

eak

`

will

also

b

e

presen

t.

The

remaining

of

this

presen

tation

is

dev

oted

to

a

detailed

disussion

of

these

v

arious

on

tributions.

Before

starting

this

guided

tour

of

the

anomalous

magneti

momen

ts

of

the

massiv

e

harged

leptons

of

the

standard

mo

del,

it

is

useful

to

k

eep

in

mind

a

few

simple

and

elemen

tary

onsiderations:

The

anomalous

magneti

momen

t

is

a

dimensionless

quan

tit

y

.

Therefore,

the

o

eÆien

ts

A

n

ab

o

v

e

are

universal,

i.e.

they

do

not

dep

end

on

the

a

v

our

of

the

lepton

whose

anomalous

magneti

momen

t

w

e

wish

to

ev

aluate.

V

ol.

2,

2002

The

Anomalous

Magneti

Momen

ts

of

the

Eletron

and

the

Muon

97

The

on

tributions

to

a

`

of

degrees

of

freedom

orresp

onding

to

a

t

ypial

sale

M

m

`

deouple

[12

℄,

i.e.

they

are

suppr

esse

d

b

y

p

o

w

ers

of

m

`

=

M

.

5

The

on

tributions

to

a

`

originating

from

ligh

t

degrees

of

freedom,

haraterized

b

y

a

t

ypial

sale

m

m

`

are

enhan

e

d

b

y

p

o

w

ers

of

ln(m

`

=m).

A

t

a

giv

en

order,

the

logarithmi

terms

that

do

not

v

anish

as

m

`

=m

!

0

an

often

b

e

omputed

from

the

kno

wledge

of

the

lesser

order

terms

and

of

the

funtion

through

the

renormalization

group

equations

[15

,

16

,

17

,

18

℄.

These

general

prop

erties

already

allo

w

to

dra

w

a

few

elemen

tary

onlusions.

The

eletron

b

eing

the

ligh

test

harged

lepton,

its

anomalous

magneti

momen

t

is

dominan

tly

determined

b

y

the

v

alues

of

the

o

eÆien

ts

A

n

.

The

rst

on

tribution

of

other

degrees

of

freedom

omes

from

graphs

in

v

olving,

sa

y

,

at

least

one

m

uon

lo

op,

whi

h

o

urs

rst

at

the

t

w

o-lo

op

lev

el,

and

is

of

the

order

of

(m

2

e

=m

)

2

(=

)

2

10

10

.

The

hadroni

eets,

i.e.

\quark

and

gluon

lo

ops",

haraterized

b

y

a

sale

of

1

GeV,

or

eets

of

degrees

of

freedom

b

ey

ond

the

standard

mo

del,

whi

h

ma

y

app

ear

at

some

high

sale

M

,

will

b

e

felt

more

strongly

,

b

y

a

onsiderable

fator

(m

=m

e

)

2

40

000,

in

a

than

in

a

e

.

Th

us,

a

e

is

w

ell

suited

for

testing

the

v

alidit

y

of

QED

at

higher

orders,

whereas

a

is

more

appropriate

for

deteting

new

ph

ysis.

If

w

e

follo

w

this

line

reasoning,

a

w

ould

ev

en

b

e

b

etter

suited

for

nding

evidene

of

degrees

of

freedom

b

ey

ond

the

standard

mo

del.

Unfortunately

,

the

v

ery

short

lifetime

of

the

lepton

[

3

10

13

s℄

mak

es

a

suÆien

tly

aurate

measuremen

t

of

a

imp

ossible

at

presen

t.

3

Brief

o

v

erview

of

the

exp

erimen

tal

situation

3.1

Measuremen

ts

of

the

magneti

momen

t

of

the

eletron

The

rst

indiation

that

the

gyromagneti

fator

of

the

eletron

is

dieren

t

from

the

v

alue

g

e

=

2

predited

b

y

the

Dira

theory

ame

from

the

preision

measuremen

t

of

h

yp

erne

splitting

in

h

ydrogen

and

deuterium

[19

℄.

The

rst

measuremen

t

of

the

gyromagneti

fator

of

free

eletrons

w

as

p

erformed

in

1958

[20

℄,

with

a

preision

of

3.6%.

The

situation

b

egan

to

impro

v

e

with

the

in

tro

dution

of

exp

erimen

tal

setups

based

on

the

P

enning

trap.

Some

of

the

suessiv

e

v

alues

obtained

o

v

er

a

p

erio

d

of

fort

y

y

ears

are

sho

wn

in

T

able

1.

T

e

hnial

impro

v

emen

ts,

ev

en

tually

allo

wing

for

the

trapping

of

a

single

eletron

or

p

ositron,

pro

dued,

in

the

ourse

of

time,

an

enormous

inrease

in

preision

whi

h,

starting

from

a

few

p

eren

ts,

w

en

t

through

the

ppm

[parts

p

er

million℄

lev

els,

b

efore

ulminating

at

4

ppb

[parts

p

er

billion℄

[21

℄

in

the

last

of

a

series

of

exp

erimen

ts

p

erformed

at

the

Univ

ersit

y

of

W

ashington

in

Seattle.

The

same

exp

erimen

t

has

also

pro

dued

a

measuremen

t

of

the

magneti

momen

t

of

the

p

ositron

with

the

same

auray

,

th

us

pro

viding

a

test

of

C

P

T

in

v

ariane

at

the

lev

el

of

10

12

,

g

e

=g

e

+

=

1

+

(0:5

2:1)

10

12

:

(3.1)

An

extensiv

e

surv

ey

of

the

literature

and

a

detailed

desription

of

the

v

arious

exp

erimen

tal

asp

ets

an

b

e

found

in

[22

℄.

The

earlier

exp

erimen

ts

are

review

ed

in

[23

℄.

3.2

Measuremen

ts

of

the

magneti

momen

t

of

the

m

uon

The

anomalous

magneti

momen

t

of

the

m

uon

has

also

b

een

the

sub

jet

of

quite

a

few

exp

erimen

ts.

The

v

ery

short

lifetime

of

the

m

uon,

=

(2:19703

0:00004

)

10

6

s,

mak

es

it

neessary

to

pro

eed

in

a

ompletely

dieren

t

w

a

y

in

order

to

attain

a

high

preision.

The

exp

erimen

ts

onduted

at

CERN

during

the

y

ears

1968-1977

used

a

m

uon

storage

ring

[for

details,

see

[31

℄

and

referenes

quoted

therein℄.

The

more

reen

t

exp

erimen

ts

at

the

A

GS

in

Bro

okha

v

en

are

based

on

the

same

5

In

the

presene

of

the

w

eak

in

terations,

this

statemen

t

has

to

b

e

reonsidered,

sine

the

neessit

y

for

the

anellation

of

the

S

U

(2)

U

(1)

gauge

anomalies

transforms

the

deoupling

of,

sa

y

,

a

single

hea

vy

fermion

in

a

giv

en

generation,

in

to

a

somewhat

subtle

issue

[13,

14 ℄.

98

M.

Kne

h

t

S

eminaire

P

oinar

e

T

able

1:

Some

exp

erimen

tal

determinations

of

the

eletron's

anomalous

magneti

momen

t

a

e

with

the

orresp

onding

relativ

e

preision.

0.001

19(5)

4.2%

[24

℄

0.001

165(11)

1%

[25

℄

0.001

116(40)

3.6%

[20

℄

0.001

160

9(2

4)

2

100

ppm

[26

℄

0.001

159

622(27)

23

ppm

[27

℄

0.001

159

660(300)

258

ppm

[28

℄

0.001

159

657

7(3

5)

3

ppm

[29

℄

0.001

159

652

41(20)

172

ppb

[30

℄

0.001

159

652

188

4(4

3)

4

ppb

[21

℄

onept.

Pions

are

pro

dued

b

y

sending

a

proton

b

eam

on

a

target.

The

pions

subsequen

tly

dea

y

in

to

longitudinally

p

olarized

m

uons,

whi

h

are

aptured

inside

a

storage

ring,

where

they

follo

w

a

irular

orbit

in

the

presene

of

b

oth

a

uniform

magneti

eld

and

a

quadrup

ole

eletri

eld,

the

latter

serving

the

purp

ose

of

fo

using

the

m

uon

b

eam.

The

dierene

b

et

w

een

the

spin

preession

frequeny

and

the

orbit,

or

syn

hrotron,

frequeny

is

giv

en

b

y

!

s

!

=

e

m

a

B

a

1

1

2

^

E

:

(3.2)

Therefore,

if

the

Loren

tz

fator

is

tuned

to

its

\magi"

v

alue

=

p

1

+

1=a

=

29:3,

the

measuremen

t

of

!

s

!

and

of

the

magneti

eld

B

allo

ws

to

determine

a

.

The

spin

diretion

of

the

m

uon

is

determined

b

y

deteting

the

eletrons

or

p

ositrons

pro

dued

in

the

dea

y

of

the

m

uons

with

an

energy

greater

than

some

threshold

energy

E

t

.

The

n

um

b

er

of

eletrons

deteted

dereases

exp

onen

tially

in

time,

with

a

time

onstan

t

set

b

y

the

m

uon's

lifetime,

and

is

mo

dulated

b

y

the

frequeny

!

s

!

,

N

e

(t)

=

N

0

e

t=

f1

+

A

os

[(!

s

!

)t

+

℄g

:

(3.3)

T

able

2:

Determinations

of

the

anomalous

magneti

momen

t

of

the

p

ositiv

ely

harged

m

uon

from

the

storage

ring

exp

erimen

ts

onduted

at

the

CERN

PS

and

at

the

BNL

A

GS.

0.001

166

16(31)

265

ppm

[32

℄

0.001

165

895(27)

23

ppm

[33

℄

0.001

165

911(11)

10

ppm

[34

℄

0.001

165

925(15)

13

ppm

[35

℄

0.001

165

9191(59)

5

ppm

[36

℄

0.001

165

920

2(16)

1.3

ppm

[37

℄

0.001

165

920

3(8)

0.7

ppm

[38

℄

Sev

eral

exp

erimen

tal

results

for

the

anomalous

magneti

momen

t

of

the

p

ositiv

ely

harged

m

uon,

obtained

at

the

CERN

PS

or,

more

reen

tly

,

at

the

BNL

A

GS,

are

reorded

in

T

able

2.

Notie

that

the

relativ

e

errors

are

measured

in

ppm

units,

to

b

e

on

trasted

with

the

ppb

lev

el

of

auray

a

hiev

ed

in

the

eletron

ase.

The

four

last

v

alues

in

T

able

2

w

ere

obtained

b

y

the

E821

exp

erimen

t

at

BNL.

They

sho

w

a

remark

able

stabilit

y

and

a

steady

inrease

in

preision,

and

no

w

ompletely

V

ol.

2,

2002

The

Anomalous

Magneti

Momen

ts

of

the

Eletron

and

the

Muon

99

dominate

the

w

orld

a

v

erage

v

alue.

F

urther

data,

for

negativ

ely

harged

m

uons

6

are

presen

tly

b

eing

analyzed.

The

aim

of

the

Bro

okha

v

en

Muon

(g

-

2)

Collab

oration

is

to

rea

h

a

preision

of

0.35

ppm,

but

this

will

dep

end

on

whether

the

exp

erimen

t

will

reeiv

e

nanial

supp

ort

to

ollet

more

data

or

not.

3.3

Exp

erimen

tal

b

ounds

on

the

anomalous

magneti

momen

t

of

the

lepton

As

already

men

tioned,

the

v

ery

short

lifetime

of

the

preludes

a

measuremen

t

of

its

anomalous

magneti

momen

t

follo

wing

an

y

of

the

te

hniques

desrib

ed

ab

o

v

e.

Indiret

aess

to

a

is

pro

vided

b

y

the

reation

e

+

e

!

+

.

The

results

obtained

b

y

OP

AL

[39

℄

and

L3

[40

℄

at

LEP

only

pro

vide

v

ery

lo

ose

b

ounds,

0:052

<

a

<

0:058

(95%C :L:)

0:068

<

a

<

0:065

(95%C :L:)

;

(3.4)

resp

etiv

ely

.

W

e

shall

no

w

turn

to

w

ards

theory

,

in

order

to

see

ho

w

the

standard

mo

del

preditions

ompare

with

these

exp

erimen

tal

v

alues.

Only

the

ases

of

the

eletron

and

of

the

m

uon

will

b

e

treated

in

some

detail.

The

theoretial

asp

ets

as

far

as

the

anomalous

magneti

momen

t

of

the

are

onerned

are

disussed

in

[41

℄.

4

The

anomalous

magneti

momen

t

of

the

eletron

W

e

start

with

the

anomalous

magneti

momen

t

of

the

ligh

test

harged

lepton,

the

eletron.

Sine

the

eletron

mass

m

e

is

m

u

h

smaller

than

an

y

other

mass

sale

presen

t

in

the

standard

mo

del,

the

mass

indep

enden

t

part

of

a

QED

e

dominates

its

v

alue.

As

men

tioned

b

efore,

non

v

anishing

on

tributions

app

ear

at

the

lev

el

of

the

lo

op

diagrams

sho

wn

in

Fig.

1.

=

+

+

+ ...

QED

Figure

1:

The

p

erturbativ

e

expansion

of

(p

0

;

p)

in

single

a

v

our

QED.

The

tree

graph

giv

es

F

1

=

1,

F

2

=

F

3

=

0.

The

one

lo

op

v

ertex

orretion

graph

giv

es

the

o

eÆien

t

A

1

in

Eq.

(2.21).

The

ross

denotes

the

insertion

of

the

external

eld.

4.1

The

lo

w

est

order

on

tribution

The

one

lo

op

diagram

giv

es

(p

0

;

p)

1

lo

op

=

(

ie)

2

Z

d

4

q

(2

)

4

(6

p

0

+

6

q

+

m

e

)

(6

p+

6

q

+

m

e

)

i

(p

0

+

q

)

2

m

2

e

i

(p

+

q

)

2

m

2

e

(

i)

q

2

:

(4.1)

6

The

CERN

exp

erimen

t

had

also

measured

a

=

0:001

165

937(12)

with

a

10

ppm

auray

,

giving

the

a

v

erage

v

alue

a

=

0:001

165

924(8:5),

with

an

auray

of

7

ppm.

100

M.

Kne

h

t

S

eminaire

P

oinar

e

The

form

fator

F

2

(k

2

)

is

obtained

b

y

using

Eqs.

(2.14)

and

(2.15)

and,

up

on

ev

aluating

the

orresp

onding

trae

of

Dira

matries,

one

nds

F

2

(k

2

)

1

lo

op

=

ie

2

32m

2

e

k

2

(k

2

4m

2

e

)

2

Z

d

4

q

(2

)

4

1

(p

0

+

q

)

2

m

2

e

1

(p

+

q

)

2

m

2

e

1

q

2

3k

2

(p

q

)

2

+

2k

2

m

2

e

(p

q

)

+

k

2

m

2

e

q

2

m

2

e

(k

q

)

2

:

(4.2)

Then

follo

w

the

usual

steps

of

in

tro

duing

t

w

o

F

eynman

parameters,

of

p

erforming

a

trivial

hange

of

v

ariables

and

a

symmetri

in

tegration

o

v

er

the

lo

op

momen

tum

q

,

so

that

one

arriv

es

at

F

2

(k

2

)

1

lo

op

=

ie

2

64m

2

e

(k

2

4m

2

e

)

2

Z

1

0

dxx

Z

1

0

dy

Z

d

4

q

(2

)

4

1

(q

2

R

2

)

3

2x(1

x)m

4

e

3

4

x

2

y

2

(k

2

)

2

+

m

2

e

k

2

x

3xy

y

+

1

2

x

 

=

e

2

2

2m

2

e

(k

2

4m

2

e

)

2

Z

1

0

dxx

Z

1

0

dy

1

R

2

2x(1

x)m

4

e

3

4

x

2

y

2

(k

2

)

2

+

m

2

e

k

2

x

3xy

y

+

1

2

x

 

;

(4.3)

with

R

2

=

x

2

y

(1

y

)(2m

2

e

k

2

)

+

x

2

y

2

m

2

e

+

x

2

(1

y

)

2

m

2

e

:

(4.4)

As

exp

eted,

the

limit

k

2

!

0

an

b

e

tak

en

without

problem,

and

giv

es

a

e

j

1

lo

op

F

2

(0)

1

lo

op

=

1

2

:

(4.5)

Let

us

stress

that

although

the

in

tegral

(4.1)

div

erges,

w

e

ha

v

e

obtained

a

nite

result

for

F

2

(k

2

),

and

hene

for

a

e

,

without

in

tro

duing

an

y

regularization.

This

is

of

ourse

exp

eted,

sine

a

di-

v

ergene

in,

sa

y

,

F

2

(0)

w

ould

require

that

a

oun

terterm

of

the

form

giv

en

b

y

the

seond

term

in

b

L

in

t

,

see

Eq.

(2.8),

b

e

in

tro

dued.

This

w

ould

in

turn

sp

oil

the

renormalizabilit

y

of

the

theory

.

In

fat,

as

is

w

ell

kno

wn,

the

div

ergene

lies

in

F

1

(0),

and

is

absorb

ed

in

to

the

renormalization

of

the

eletron's

harge.

+

sym

+

sym

Figure

2:

The

F

eynman

diagrams

whi

h

on

tribute

to

the

o

eÆien

t

A

2

in

Eq.

(2.21).

4.2

Higher

order

mass

indep

enden

t

orretions

The

previous

alulation

is

rather

straigh

tforw

ard

and

amoun

ts

to

the

result

A

1

=

1

2

(4.6)

rst

obtained

b

y

S

h

winger

[42

℄.

S

h

winger's

alulation

w

as

so

on

follo

w

ed

b

y

a

omputation

of

A

2

[43

℄,

whi

h

requires

the

ev

aluation

of

7

graphs,

represen

ting

v

e

distint

top

ologies,

and

sho

wn

in

Fig.

2.

Historially

,

the

result

of

Ref.

[43

℄

w

as

imp

ortan

t,

b

eause

it

pro

vided

the

rst

expliit

example

of

the

realization

of

the

renormalization

program

of

QED

at

t

w

o

lo

ops.

Ho

w

ev

er,

the

V

ol.

2,

2002

The

Anomalous

Magneti

Momen

ts

of

the

Eletron

and

the

Muon

101

v

alue

for

A

2

w

as

not

giv

en

orretly

.

The

orret

expression

of

the

seond

order

mass

indep

enden

t

on

tribution

w

as

deriv

ed

in

[44

,

45

,

46

℄

(see

also

[47

,

48

℄)

and

reads

7

A

2

=

197

144

+

1

2

3

ln

2

(2)

+

3

4

(3)

=

0:328

478

965:::

(4.7)

with

(p)

=

1

X

n=1

1=n

p

,

(2)

=

2

=6.

The

o

urrene

of

transenden

tal

n

um

b

ers

lik

e

zeta

funtions

or

p

olylogarithms

is

a

general

feature

of

higher

order

alulations

in

p

erturbativ

e

quan

tum

eld

theory

.

The

pattern

of

these

transenden

tals

in

p

erturbation

theory

has

also

b

een

put

in

relationship

with

other

mathematial

strutures,

lik

e

knot

theory

.

The

analyti

ev

aluation

of

the

three-lo

op

mass

indep

enden

t

on

tribution

to

the

anomalous

magneti

momen

t

required

quite

some

time,

and

is

mainly

due

to

the

dediation

of

E.

Remiddi

and

his

o

w

ork

ers

during

the

p

erio

d

1969-1996.

There

are

no

w

72

diagrams

to

onsider,

in

v

olving

man

y

dieren

t

top

ologies,

see

Fig.

3.

6

20

12

24

4

6

Figure

3:

The

72

F

eynman

diagrams

whi

h

mak

e

up

the

o

eÆien

t

A

3

in

Eq.

(2.21).

The

alulation

w

as

ompleted

[49

℄

in

1996,

with

the

analytial

ev

aluation

of

a

last

lass

of

diagrams,

the

non

planar

\triple

ross"

top

ologies.

The

result

reads

8

A

3

=

87

72

2

(3)

215

24

(5)

+

100

3

a

4

+

1

24

ln

4

2

1

24

2

ln

2

2

239

2160

4

+

139

18

(3)

298

9

2

ln

2

+

17101

810

2

+

28259

5184

=

1:181

241

456:::

(4.8)

7

Atually

,

the

exp

erimen

tal

result

of

Ref.

[25℄

disagreed

with

the

v

alue

A

2

=

2:973

obtained

in

[43℄,

and

prompted

theoretiians

to

reonsider

the

alulation.

The

result

obtained

b

y

the

authors

of

Refs.

[44,

45,

46 ℄

reoniled

theory

with

exp

erimen

t.

8

The

ompletion

of

this

three-lo

op

program

an

b

e

follo

w

ed

through

Refs.

[50℄-[55℄

and

[49 ℄.

A

desription

of

the

te

hnial

asp

ets

related

to

this

w

ork

and

an

aoun

t

of

its

status

up

to

1990,

with

referenes

to

the

orresp

onding

literature,

are

giv

en

in

Ref.

[56℄.

102

M.

Kne

h

t

S

eminaire

P

oinar

e

where

9

a

p

=

1

X

n=1

1

2

n

n

p

.

The

n

umerial

v

alue

extrated

from

the

exat

analytial

expression

giv

en

ab

o

v

e

an

b

e

impro

v

ed

to

an

y

desired

order

of

preision.

In

parallel

to

these

analytial

alulations,

n

umerial

metho

ds

for

the

ev

aluation

of

the

higher

order

on

tributions

w

ere

also

dev

elop

ed,

in

partiular

b

y

Kinoshita

and

his

ollab

orators

(for

details,

see

[57

℄).

The

n

umerial

ev

aluation

of

the

full

set

of

three

lo

op

diagrams

w

as

a

hiev

ed

in

sev

eral

steps

[58

℄-[64

℄.

The

v

alue

quoted

in

[64

℄

is

A

3

=

1:195(26),

where

the

error

omes

from

the

n

umerial

pro

edure.

In

omparison,

let

us

quote

the

v

alue

[65

,

57

℄

A

3

=

1:176

11

(42)

obtained

if

only

a

subset

of

21

three

lo

op

diagrams

out

of

the

original

set

of

72

is

ev

aluated

n

umerially

,

relying

on

the

analytial

results

for

the

remaining

51

ones,

and

reall

the

v

alue

A

3

=

1:181

241

456:::

obtained

from

the

full

analytial

ev

aluation.

The

error

indued

on

a

e

due

to

the

n

umerial

unertain

t

y

in

the

seond,

more

aurate,

v

alue

is

still

(a

e

)

=

5:3

10

12

,

whereas

the

exp

erimen

tal

error

is

only

(a

e

)j

exp

=

4:3

10

12

.

This

disussion

sho

ws

that

the

analytial

ev

aluations

of

higher

lo

op

on

tributions

to

the

anomalous

magneti

momen

t

of

the

eletron

ha

v

e

a

strong

pratial

in

terest

as

far

as

the

preision

of

the

theoretial

predition

is

onerned,

and

whi

h

go

es

w

ell

b

ey

ond

the

mere

in

telletual

satisfation

and

te

hnial

skills

in

v

olv

ed

in

these

alulations.

10

A

t

the

four

lo

op

lev

el,

there

are

891

diagrams

to

onsider.

Clearly

,

only

a

few

of

them

ha

v

e

b

een

ev

aluated

analytially

[66

,

67

℄.

The

omplete

n

umerial

ev

aluation

of

the

whole

set

ga

v

e

[65

℄

A

4

=

1:434(138).

The

dev

elopmen

t

of

omputers

allo

w

ed

subsequen

t

reanalyzes

to

b

e

more

aurate,

i.e.

A

4

=

1:557(70)

[68

℄,

while

the

\latest

of

[these℄

onstan

tly

impro

ving

v

alues"

is

[4℄

A

4

=

1:509

8(38

4)

:

(4.9)

Needless

to

sa

y

,

so

far

the

v

e

lo

op

on

tribution

A

5

is

unkno

wn

territory

.

On

the

other

hand,

(=

)

5

7

10

14

,

so

that

one

ma

y

reasonably

exp

et

that,

in

view

of

the

presen

t

exp

erimen

tal

situation,

its

kno

wledge

is

not

y

et

required.

4.3

Mass

dep

enden

t

QED

orretions

W

e

no

w

turn

to

the

QED

lo

op

on

tributions

to

the

eletron's

anomalous

magneti

momen

t

in-

v

olving

the

hea

vier

leptons,

and

.

The

lo

w

est

order

on

tribution

of

this

t

yp

e

o

urs

at

the

t

w

o

lo

op

lev

el,

O

(

2

),

and

orresp

onds

to

a

hea

vy

lepton

v

auum

p

olarization

insertion

in

the

one

lo

op

v

ertex

graph,

f.

Fig.

4.

Quite

generally

,

the

on

tribution

to

a

`

arising

from

the

insertion,

in

to

the

one

lo

op

v

ertex

orretion,

of

a

v

auum

p

olarization

graph

due

to

a

lo

op

of

lepton

`

0

,

reads

[69

,

70

℄

11

B

2

(`;

`

0

)

=

1

3

Z

1

4m

2

`

0

dt

r

1

4m

2

`

0

t

t

+

2m

2

`

0

t

2

Z

1

0

dx

x

2

(1

x)

x

2

+

(1

x)

t

m

2

`

:

(4.10)

If

m

`

0

m

`

,

the

seond

in

tegrand

an

b

e

appro

ximated

b

y

x

2

m

2

`

=t,

and

one

obtains

[72

℄

B

2

(`;

`

0

)

=

1

45

m

`

m

`

0

2

+

O

"

m

`

m

`

0

3

#

;

m

`

0

m

`

:

(4.11)

9

The

rst

three

v

alues

are

kno

wn

to

b

e

a

1

=

ln

2,

a

2

=

Li

2

(1=2)

=

(

(2)

ln

2

2)=2,

a

3

=

7

8

(3)

1

2

(2)

ln

2

+

1

6

ln

3

2

[56℄.

10

It

is

only

fair

to

p

oin

t

out

that

the

n

umerial

v

alues

that

are

quoted

here

orresp

ond

to

those

giv

en

in

the

original

referenes.

It

is

to

b

e

exp

eted

that

they

w

ould

impro

v

e

if

to

da

y's

n

umerial

p

ossibilities

w

ere

used.

11

A

trivial

hange

of

v

ariable

on

t

brings

the

expression

(4.10)

in

to

the

form

giv

en

in

[69,

70℄.

F

urthermore,

the

analytial

result

obtained

up

on

p

erforming

the

double

in

tegration

is

a

v

ailable

in

[71 ℄.

V

ol.

2,

2002

The

Anomalous

Magneti

Momen

ts

of

the

Eletron

and

the

Muon

103

e

e

µ

e

µ

µ

Figure

4:

The

insertion

of

a

m

uon

v

auum

p

olarization

lo

op

in

to

the

eletron

v

ertex

orretion

(left)

or

of

an

eletron

v

auum

p

olarization

lo

op

in

to

the

m

uon

v

ertex

orretion

(righ

t).

Numerially

,

this

translates

in

to

[3

℄

[m

e

=

0:51099907(15)

MeV,

m

=m

e

=

206:768273(24),

m

=

1

777:05(26)℄

B

2

(e;

)

=

5:197

10

7

B

2

(e;

)

=

1:838

10

9

:

(4.12)

F

or

later

use,

it

is

in

teresting

to

briey

disuss

the

struture

of

Eq.

(4.10).

The

quan

tit

y

whi

h

app

ears

under

the

in

tegral

is

related

to

the

ross

setion

for

the

sattering

of

a

`

+

`

pair

in

to

a

pair

(`

0

)

+

(`

0

)

at

lowest

or

der

in

QED,

(`

+

`

!(`

0

)

+

(`

0

)

)

QE

D

(s)

=

4

2

3s

2

r

1

4m

2

`

0

s

(s

+

2m

2

`

0

)

;

(4.13)

so

that

B

2

(`;

`

0

)

=

1

3

Z

1

4m

2

`

0

dtK

(t)R

(`

0

)

(t)

;

(4.14)

where

K

(t)

=

Z

1

0

dx

x

2

(1

x)

x

2

+

(1

x)

t

m

2

`

;

(4.15)

and

R

(`

0

)

(t)

is

the

lowest

or

der

QED

ross

setion

(`

+

`

!(`

0

)

+

(`

0

)

)

Q E

D

(s)

divided

b

y

the

asymptoti

form

of

the

ross

setion

of

the

reation

e

+

e

!

+

for

s

m

2

,

(e

+

e

!

+

)

1

(s)

=

4

2

3s

.

The

three

lo

op

on

tributions

with

dieren

t

lepton

a

v

ours

in

the

lo

ops

are

also

kno

wn

analytially

[73

,

74

℄.

It

is

on

v

enien

t

to

distinguish

three

lasses

of

diagrams.

The

rst

group

on

tains

all

the

diagrams

with

one

or

t

w

o

v

auum

p

olarization

insertion

in

v

olving

the

same

lepton,

or

,

of

the

t

yp

e

sho

wn

in

Fig.

5.

The

seond

group

onsists

of

the

leptoni

ligh

t-b

y-ligh

t

sattering

insertion

diagrams,

Fig.

6.

Finally

,

sine

there

are

three

a

v

ours

of

massiv

e

leptons

in

the

standard

mo

del,

one

has

also

the

p

ossibilit

y

of

ha

ving

graphs

with

t

w

o

hea

vy

lepton

v

auum

p

olarization

insertions,

one

made

of

a

m

uon

lo

op,

the

other

of

a

lo

op.

This

giv

es

B

3

(e;

`)

=

B

(v.p.

)

3

(e;

)

+

B

(v.p.)

3

(e;

)

+

B

(LL)

3

(e;

)

+

B

(LL)

3

(e;

)

+

B

(v.p.)

3

(e;

;

)

:

(4.16)

The

analytial

expression

for

B

(v.p.)

3

(e;

)

an

b

e

found

in

Ref.

[73

℄,

whereas

[74

℄

giv

es

the

orre-

sp

onding

result

for

B

(LL)

3

(e;

).

F

or

pratial

purp

oses,

it

is

b

oth

suÆien

t

and

more

on

v

enien

t

to

use

their

expansions

in

p

o

w

ers

of

m

e

=m

,

B

(v.p.)

3

(e;

)

=

m

e

m

2

23

135

ln

m

m

e

2

45

2

+

10117

24300

104

M.

Kne

h

t

S

eminaire

P

oinar

e

+

m

e

m

4

19

2520

ln

2

m

m

e

14233

132300

ln

m

m

e

+

49

768

(3)

11

945

2

+

2976691

296352000

+

O

"

m

e

m

6

#

=

0:000

021

768:::

(4.17)

12

6

Figure

5:

Three

lo

op

QED

orretions

with

insertion

of

a

hea

vy

lepton

v

auum

p

olarization

whi

h

mak

e

up

the

o

eÆien

t

B

(v.p.)

3

(e;

).

and

[74℄

B

(LL)

3

(e;

)

=

m

e

m

2

3

2

(3)

19

16

+

m

e

m

4

161

810

ln

2

m

m

e

16189

48600

ln

m

m

e

+

13

18

(3)

161

9720

2

831931

972000

+

O

"

m

e

m

6

#

=

0:000

014

394

5:::

(4.18)

6

Figure

6:

The

three

lo

op

QED

orretion

with

the

insertion

of

a

hea

vy

lepton

ligh

t-b

y-ligh

t

sat-

tering

subgraph,

orresp

onding

to

the

o

eÆien

t

B

(LL)

3

(e;

).

The

expressions

for

B

(v.p.)

3

(e;

)

and

B

(LL)

3

(e;

)

follo

w

up

on

replaing

the

m

uon

mass

m

b

y

m

.

This

again

giv

es

a

suppression

fator

(m

=m

)

2

,

whi

h

mak

es

these

on

tributions

negligible

at

the

presen

t

lev

el

of

preision.

F

or

the

same

reason,

B

(v.p.)

3

(e;

;

)

an

also

b

e

disarded.