Two-Loop Correction to Bhabha Scattering

hep-ph/0010075 SLAC{PUB{8655

UCLA/00/TEP/26

Octob er, 2000

Two-Lo op Correction to Bhabha Scattering

Z. Bern

Department of Physics and Astronomy

UCLA, Los Angeles, CA 90095-1547

L. Dixon

Stanford Linear Accelerator Center

Stanford University

Stanford, CA 94309

and

A. Ghinculov

Department of Physics and Astronomy

UCLA, Los Angeles, CA 90095-1547

Abstract

+ +

Wepresentthetwo-lo op virtual QED corrections to e e ! and Bhabha scattering

in dimensional regularization. The results are expressed in terms of p olylogarithms. The form of

the infrared divergences agrees with previous exp ectations. These results are a crucial ingredient

in the complete next-to-next-to-leading order QED corrections to these pro cesses. A future

application will b e to reduce theoretical uncertainties asso ciated with luminosity measurements

at e e colliders. The calculation also tests metho ds that may b e applied to analogous QCD

pro cesses.

Submitted to Physical Review D

Research supp orted by the US Department of Energy under grantDE-FG03-91ER40662.

Research supp orted by the US Department of Energy under grant DE-AC03-76SF00515.

1 Intro duction

Bhabha scattering is an imp ortant pro cess for extracting physics from exp eriments at electron-

p ositron colliders primarily b ecause it provides an e ective means for determining luminosity. These

measurements dep end on having precise theoretical predictions for the Bhabha scattering cross

sections. As yet, the complete next-to-next-to-leading order (NNLO) QED corrections needed for

reducing theoretical uncertainties have not b een computed. In this pap er we present the complete

two-lo op matrix elements that would enter into such a computation. This calculation also provides

a means for validating techniques that can b e applied to physically imp ortant but more intricate

QCD calculations. It also provides an additional explicit veri cation of a general formula due

to Catani [1 ] for the structure of two-lo op infrared divergences, and allows us to determine the

pro cess-dep endent terms for the pro cesses at hand.

In Bhabha scattering there are two distinct kinematic regions: small angle Bhabha scattering

(SABS), and large angle (LABS). In the LEP/SLC energy range, SABS is used to measure the

machine luminosity via a dedicated small angle luminosity detector. SABS has a large cross section

| ab out four times larger than Z decay in the 1 3 window | making it particularly e ectiveas

a luminosity monitor. At the same time, SABS is calculable theoretically with high accuracy from

known physics (mainly QED), apart from hadronic vacuum p olarization corrections that rely up on

the exp erimental data for e e annihilation into hadrons at low energy [2 , 3]. Therefore, SABS is

an imp ortant ingredient in measuring any absolute cross section. For instance, the measurementof

the hadronic cross section at the Z p eak, ,whichenters several precision observables, is esp ecially

dep endent on an accurate theoretical understanding of Bhabha scattering.

+ +

At LEP/SLC, large angle Bhabha scattering interferes with e e ! Z ! e e and so it is

needed to disentangle imp ortant parameters such as the electroweak mixing angle. It is also useful

for measuring the luminosityat avor factories such as BABAR, BELLE, DANE, VEPP-2M, and

BEPC/BES [4 ]. A p eculiarity of future electron linear colliders is that the luminosity sp ectrum

is not mono chromatic b ecause of the b eam-b eam e ect. Because of this, measuring the total

small angle cross section of Bhabha scattering alone is not sucient, and therefore the angular

distribution of LABS was prop osed for disentangling the luminosity sp ectrum [5].

Due to the exp erimental imp ortance of this pro cess, signi cant e ort has been devoted to

developing Monte Carlo event generators | see for instance ref. [6 ] for an overview. In order to

match the impressive exp erimental precision, a complete inclusion of NNLO QED quantum e ects

has b ecome necessary. On the theoretical side, however, the calculation of two-lo op four-p oint

amplitudes has b een a roadblo ck to further progress.

In this article we present the two-lo op virtual QED corrections to the di erential cross section for

Bhabha scattering, i.e., the two-lo op amplitude interfered with the tree amplitude and summed over

all spins. We neglect the small electron mass in comparison to all other kinematic invariants, and use

dimensional regularization to handle the ensuing infrared divergences. Besides these contributions,

anumb er of other virtual and real emission contributions (discussed in the conclusions) still need 1

to be obtained b efore a full Monte Carlo program for the Bhabha scattering cross section can be

constructed.

The two-lo op QED four-fermion amplitudes are also a useful testing ground for two-lo op QCD

calculations containing more than one kinematic invariant, which are required for higher-order

jet cross sections and other asp ects of collider physics. For pro cesses that dep end on a single

momentum invariant, a number of imp ortant quantities have been calculated up to four lo ops,

such as the total cross section for e e annihilation into hadrons and the QCD -function [7]. In

contrast, the only complete two-lo op four-p oint scattering amplitudes presently known for generic

kinematics in massless gauge theory are the N = 4 sup er-Yang-Mills amplitudes [8, 9], and gg ! gg

in a single helicity con guration in pure gauge theory [10 ]. The two-lo op amplitudes required for

NNLO computations of jet pro duction in hadron colliders, or for NNLO three-jet rates and other

eventshapevariables at e e colliders, remain uncalculated. We note in passing that partial results

for the leading-color part of two-lo op contributions to quark-quark scattering have very recently

app eared [11 ].

Two imp ortant technical breakthroughs are the calculations of the dimensionally regularized

scalar double box integrals with planar [12 ] and non-planar [13 ] top ologies and all external legs

massless, and the development of reduction algorithms for the same typ es of integrals with lo op

momenta in the numerator (tensor integrals) [14 , 15 , 16 , 17 , 11 ]. Related integrals, which also arise

in the reduction pro cedure, have b een computed in refs. [18 , 19 ]. Taken together, these results are

sucient to compute all lo op integrals required for 2 ! 2 massless scattering amplitudes at two

lo ops, thus removing a ma jor obstacle to several typ es of NNLO calculations. In this pap er weuse

these techniques to evaluate the integrals encountered in the Bhabha calculation. An even more

recent result concerning two-lo op planar double box integrals with one massive external leg [20 ]

holds promise for the NNLO computation of three-jet rates at e e colliders.

There has also been signi cant progress in developing general formalisms for other asp ects of

NNLO computations involving massless particles. The motivation has typically b een infrared-safe

observables in QCD, but many of the developments can b e applied to the Bhabha pro cess as well.

The developments include an understanding of the intricate structure of the infrared singularities

that arise when more than one particle is unresolved (i.e., is soft or collinear with another parti-

cle) [21 , 22 , 23 ]. Improved approximations to the NNLO correction to splitting functions have b een

constructed recently as well [24 ].

Infrared divergences are a signi cant complication in all the QCD and QED computations men-

tioned ab ove. In any suitably \infrared-safe" observable all nal-state divergences will cancel [25 ].

However, divergences o ccur in individual amplitudes for xed particle number, and it is very useful

to have a general description of such divergences. Catani has presented a general formula for the in-

frared divergence app earing in anytwo-lo op QCD amplitude [1 ]. By appropriately adjusting group

theory factors, it is straightforwardtoconvert Catani's QCD formula to a QED formula, allowing

us to directly verify it. Moreover, we extract the exact form of a pro cess-dep endent term in the

formula, for the case of QED scattering of four charged fermions. Previously, the only pro cess for 2

which this term had b een extracted [1] was the quark form factor whichenters Drell-Yan pro duc-

tion [26 ]. (It should also now be p ossible to extract it for gg ! Higgs using the recent two-lo op

computation [27 ].) Interestingly, a simple generalization of the quark form factor term (converted

+ +

to QED) correctly predicts the pro cess-dep endent term for the e e ! and Bhabha am-

plitudes. We also use Catani's formula to conveniently organize the infrared divergences and to

absorb some of the nite terms.

The previously computed non-ab elian gauge theory amplitudes [8 , 9, 10 ] were obtained via

+ +

cutting metho ds. The low multiplicity and relative simplicity of the e e ! and Bhabha

scattering Feynman diagrams makes it relatively easy to directly compute the diagrams, as wedo

here. We include here only the pure QED diagrams, neglecting for example the contributions of

Z exchange, and hadronic vacuum p olarization e ects. The former are negligible at this order in

SABS and in LABS at avor factories. The hadronic contributions are imp ortant, but much of

their e ect is straightforward to include byintro ducing a running coupling.

We perform the calculation in dimensional regularization [28 ] with d = 4 2 and set the

small electron mass to zero, since it is the only form in which the required two-lo op momentum

integrals are known. Moreover, it provides a p owerful metho d for simultaneously dealing with b oth

the infrared and ultraviolet divergences encountered in gauge theories. Traditionally, dimensional

regularization is not used for QED, in part b ecause the infrared divergences are relatively tame

compared to non-ab elian gauge theories, so photon and electron masses are sucient for cutting o

the theory. Another imp ortant reason for using dimensional regularization is to validate techniques

that can also b e applied to the more complicated case of QCD. In QCD, dimensional regularization

is the universally utilized metho d for dealing with divergences.

In the high-energy Bhabha pro cess, even with an \infrared-safe" (calorimetric) nal-state def-

2 2

inition, the electron mass will still app ear in large logarithms of the form L ln(Q =m ) due to

initial-state radiation. However, in the dimensionally regulated amplitudes these singularities (like

all others) app ear as poles in . It may therefore be most convenient to handle the initial-state

singularities using an electron structure function metho d [29 ] implemented in the MS collinear

factorization scheme.

In the next section we brie y describ e our metho d for computing the two-lo op amplitudes.

Then we describ e Catani's formula for the divergence structure of the amplitudes, followed by a

0 + +

presentation of the nite (O ( )) terms for b oth e e ! and Bhabha scattering. In the

nal section we give our conclusions, including some discussion of the remaining ingredients still

required for construction of a numerical program for Bhabha scattering at this order.

2 The Two-Lo op Amplitudes

The 16 indep endent Feynman diagram top ologies describing the two-lo op QED corrections to

+ +

e e ! and Bhabha scattering are enumerated in g. 1. In this gure wehave suppressed

the fermion arrows. After including the fermion arrows and distinct lab els for the external legs, 3

there are a total of 47 Feynman diagrams; however, many of these diagrams generate identical re-

sults. Of the 47 diagrams, 35 contain no fermion lo op, 11 contain one fermion lo op, and 1 contains

+ +

two fermion lo ops. The Bhabha amplitude may be obtained from the e e ! amplitude

by adding to it the same set of diagrams, but with an exchange of one pair of external legs. The

+ +

e ! e and e ! e amplitudes may, of course, b e obtained by crossing.

Figure 1: The indep endent diagrammatic top ologies for two-lo op four-fermion scattering in QED.

We have evaluated these diagrams interfered with the tree amplitudes and summed over spins

in the conventional dimensional regularization (CDR) scheme. This interference gives directly the

two-lo op virtual correction to the 2 ! 2 di erential cross section. The rules for implementing CDR

are straightforward b ecause all particle are treated uniformly in all parts of the calculation. In this

scheme, all momenta and all Lorentz indices are taken to b e D =4 2 dimensional vectors. (The

-matrices remain as 4 4 matrices; i.e., Tr[1] = 4.)

After p erforming all -matrix algebra present in the two-lo op Feynman diagrams, we use the

conservationofmomenta owing on the internal lines to express the tensor structure of the diagrams

in terms of inverse scalar propagators and a small numb er of additional scalar invariants containing

lo op momenta. The inverse scalar propagators cancel propagators in the denominator to generate

simpler \b oundary" integrals. To handle the integrals containing scalar invariants, we intro duce

Feynman parameters and interpret the resulting integrals in terms of scalar integrals with multiple

propagators, which are then reduced to a set of master integrals with the help of equations in

refs. [14 , 15 , 19 ].

Pro ceeding in this way,we obtain an expression for the amplitude in terms of master integrals

(of the typ e listed in ref. [15 ], plus a few more for the planar double b ox top ology) multiplied by 4

co ecient functions. This expression is in principle valid to an arbitrary order in , assuming that

the master integrals could b e evaluated to such an order. However, it is a bit to o lengthy to present

here, and for NNLO computations only the series expansion in through O ( ) is required. To carry

out this expansion, we use expansions of the master integrals presented in refs. [12 , 13 , 14 , 15, 18 , 19 ].

As noted in ref. [11 ], there is a slight problem with the original choice of basis [14 ] for the two master

planar double b ox integrals. In that basis, the co ecients for generic tensor integrals contain 1=

p oles, necessitating an O () evaluation of the master integrals. Several solutions to this problem

have b een presented [17 , 11 ]. We have used a slightly di erent solution, which is simply to use

the original pair of master integrals de ned in ref. [14 ], except evaluated in d =6 2 instead of

d =4 2. In d =6 2 the integrals have neither ultraviolet nor infrared divergences, making

them simpler to evaluate through O ( ) than the d =4 2 integrals.

Many of the master integral expansions quoted in refs. [12 , 13 , 14 , 15 , 18 , 19 ] are in terms of

Nielsen functions [30 ],

n+p1

(1) dt

n1 p

S (x)= ln t ln (1 xt) ; (2:1)

n;p

(n 1)! p! t

with n + p 4. Wehave found it useful to express the results instead in terms of a minimal set of

p olylogarithms [31 ],

dt x

Li (x)= = Li (t);

n n1

i t

i=1

(2:2)

Li (x)= ln(1 t) ;

with n =2; 3; 4, using relations suchas

1 1

2 3

ln (1 x) Li (x) + ln (1 x)lnx + ; S (x)= Li (1 x)+ln(1 x)Li (1 x)+

2 2 4 13 4 3

2 3

ln(1 x) Li (x) S (x)=Li (x) Li (1 x)+Li

3 3 22 4 4 4

1 x

1 1 1

4 3 2

+ ln (1 x) ln (1 x)lnx + ln (1 x)+ ;

2 4

24 6 2

for 0

(2:3)

Here

2 4

n ; = ; =1:202057 ::: ; = : (2:4)

s 2 3 4

6 90

n=1

The analytic prop erties of the non-planar double b oxintegrals are somewhat intricate [13 ], since

they are not real in any of the three kinematic channels for the 2 ! 2 pro cess,

s-channel : s>0; t; u < 0 ;

t-channel : t > 0; s; u < 0 ;

(2:5)

u-channel : u>0; s; t<0 ; 5

2 2 2

where s = (k + k ) , t = (k k ) , and u = (k k ) . Therefore we shall present explicit

1 2 1 4 1 3

formulae for the nite terms in the amplitude in b oth the s-andu-channels; those in the t-channel

will b e related by symmetries.

2.1 General Structure of Divergences

Dimensionally regulated two-lo op amplitudes for four massless fermions contain p oles in =(4

d)=2upto1= . The structure of most of these singularities has already b een exp osed by Catani [1 ],

who describ ed the infrared b ehavior of general two-lo op QCD pro cesses. We shall therefore adopt

his notation in presenting our results.

MS running coupling for We work with ultraviolet renormalized amplitudes, and employ the

2 2

QED, ( ). Of course this scheme can always be converted to another one, for example (Q )

de ned via the photon propagator at momentum transfer Q, by a nite renormalization. The

u 2

relation b etween the bare coupling and ( ) through two-lo op order can b e expressed as [1 ]

" #

0 1

2 2 2 u 2 2 2 3 2

S = ( ) 1 ( ) + ( ) + O ( ( )) ; (2:6)

where S = exp[(ln 4 + (1))] and = (1) = 0:5772 ::: is Euler's constant. The rst two

co ecients of the QED b eta function are

N N

f f

= ; = ; (2:7)

0 1

3 4

where N is the number of light (massless) charge 1 fermions.

The renormalized four-fermion amplitude is expanded as

( )

(1) (0)

2 2 2 2 2

M ( ; fpg) M ( ; fpg)+ M ( ( ); ; fpg)=4 ( )

4 4

(2:8)

( )

(2)

2 3 2

( ; fpg)+O ( ( )) : M +

The infrared divergences of a renormalized two-lo op amplitude in QCD or QED are [1 ],

(2) 2 (1) 2 (1) 2

jM ( ; fpg)i = I (; ; fpg) jM ( ; fpg)i

R:S: R:S:

n n

(2:9)

(2)

2 (0) 2 (2) n 2

+ I (; ; fpg) jM ( ; fpg)i + jM ( ; fpg)i ;

R:S: R:S:

R:S:

n n

(L)

where jM ( ; fpg)i is a color space vector representing the renormalized L lo op amplitude.

R:S:

The subscript R:S: stands for the choice of renormalization scheme, and is the renormalization

scale. These color space vectors give the amplitudes via,

a a

M (1 ;::: ;n ) ha ;::: ;a jM (p ;::: ;p )i ; (2:10 )

n 1 n n 1 n

(1)

where the a are color indices. The divergences of M are enco ded in the color op erators I (; ; fpg)

i n

(L)

(2)

and and I (; ; fpg). In the QED case, the color space language is clearly unnecessary; M

(L)

I are just numbers. 6

(1)

In QCD, the op erator I (; ; fpg) is given by

n n

2 i (1)

X X

1 e 1 e 1

(1) 2

I (; ; fpg)= + ; (2:11 ) T T

i j

2 (1 ) 2p p

i j

i=1

j 6=i

where =+1if i and j are b oth incoming or outgoing partons and = 0 otherwise. The color

ij ij

charge T = fT g is a vector with resp ect to the generator lab el a, and an SU (N ) matrix with

i c

resp ect to the color indices of the outgoing parton i. The values required for QCD are

2 2 2

T = T = C ; T = C = N ;

F A c

q q g

(2:12 )

11 2 3

C ; = C T N : = =

F g A R q q

2 6 3

For QED we let C ! 0, C ! 1, T ! 1 and T T ! e e = 1, where the e are the

A F R i j i j i

electric charges, to obtain

" #

2 2 2 (1)

3 2 e

(1) 2

+ + ; (2:13 ) I (; ; fpg)=

(1 ) s t u

for the four-fermion amplitude

+ +

e (k ) e (k ) ! (k ) (k ) : (2:14 )

1 2 4 3

(Note that the charges of incoming states should b e reversed in computing T T .)

i j

(2)

is given in QCD by[1] The op erator I

R:S:

1 4

(2)

2 (1) 2 (1) 2

(; ; fpg)= I I (; ; fpg) I (; ; fpg)+

R:S:

+ (1)

2 e (1 2)

(2:15 )

(1) 2

+ K I (2; ; fpg) +

(1 )

(2)

+ H (; ; fpg) ;

R:S:

where the co ecient K is:

67 10

K = C T N : (2:16 )

A R

18 6 9

(1)

For the QED pro cess (2.14), we insert I from eq. (2.13), take and from eq. (2.7), and let

0 1

K !10N =9.

(2)

The function H is pro cess-dep endent but has only single p oles:

R:S:

(2)

H (; ; fpg) = O (1=) : (2:17 )

R:S:

(2)

Ref. [1 ] do es not give an expression for H for a general amplitude, but only for the case of a

R:S:

q q pair, i.e. asinglecharged fermion pair. The result, which is extracted from the two-lo op QCD

computation of the electromagnetic form factor of the quark [26 ], is

(1) 2 i

e 1 56 e 1

(2)

+3C K +5 C C H (; ; fpg)=

F 2 0 F 0 F

(1)

q q; CDR

4 (1 ) 2p p 4 9

1 2

(2:18 )

; C C 7

F A 3

9 7

where

17 88 4 32

= (3+24 48 )C + C C + C T N : (2:19 ) +24 +

2 3 F A F R 2 3 2

(1)

3 3 3 3

Performing the usual conversion to QED yields a result applicable to the electromagnetic form

factor of the electron,

(1) 2 i

e e 1 25 3

(2)

H (; ; fpg)= +6 12 + + N : (2:20 )

2 3 2 +

e e ;CDR

4 (1 ) 2p p 4 27

1 2

Using our two-lo op computation, and an all-orders-in- computation of the one-lo op amplitude

+ + + +

for e e ! (see sect. 2.2), wehaveveri ed that the singular b ehavior of the e e !

amplitude in CDR agrees precisely with that predicted by eq. (2.9) in all three kinematic channels.

(2)

In addition, we have extracted the function H controlling the 1= p oles in eq. (2.9).

+ +

e e ;CDR

Weobtain

" #

2 2 2

(1) 2 2 2

1 e

(2)

H (; ; fpg)= 2 +

+ +

e e ;CDR

4 (1 ) s t u

(2:21 )

3 25

+6 12 + + N :

2 3 2

4 27

This result agrees with a \naive" generalization from the form factor case, in which one sums

eq. (2.20) over the six pairs of charged legs in the four-p oint amplitude, weighted by the sign of

2 2

the charge pro duct e e . (Note that the factors of ( =(s )) are purely conventional here, since

i j ij

their deviation from unity only contributes at the level of nite parts, O ( ). However, the overall

normalization is predicted correctly bythesumover the six pairs.)

+ +

2.2 e e ! at One Lo op to All Orders in

In order to verify the structure of the infrared singularities, and to extract the nite remainder

+ +

of the two-lo op amplitude presented below, we computed the one-lo op e e ! amplitude

(interfered with the tree amplitude) to all orders in . The result is

h i

X X

(0) (0) y (1) (0) y

(1) (1)

M M + A + S A ; (2:22 ) M M =

4 4 4 4

spins spins

where the rst term is the MS counterterm, expressed in terms of the tree-level interference

2 2

t + u

(0) (0) y

M ; (2:23 ) M =8

4 4

spins

Strictly sp eaking, wehave computed the interference of the two-lo op amplitude with the tree-amplitude, summed

over intermediate fermion spins, so in our veri cation eq. (2.9) should b e similarly understo o d to b e interfered with

the tree amplitude. 8

and

h i

(1) 2 2 (6)

A =4 (1 2) (2 3)u 6tu + 3(2 )t Box (s; t)

i h

2 2 2 2

Tri(t) 4 (4 12 +7 )t 6(1 2)tu +(4 10 +5 )u

1 2 s

(2:24 )

8 1

2(1 ) t((1 )t u) N (3 2)(2 +2 )tu

(1 2)(3 2) s

2 2

+(1 )(3 2)(2 (1 ) +2 )t Tri(s) :

The symmetry op eration S acts as

S : t $ u; $: (2:25 )

After carrying out the op eration of S , one should then set =1. (Basically, allows us to separate

diagrams based on whether they have an even or odd number of photons attached to the muon

line. Because photons haveC=1, this criterion governs the t $ u symmetry prop erties.)

(6)

In eq. (2.24), Box (s; t)andTri(s) are one-lo op b ox and triangle integrals, the former evaluated

in an expansion around d =6 2. For the divergence formula (2.9), we need their series expansions

in through O ( ). In the u-channel, where the functions are manifestly real, their expansions are

given by

1 2 2

u 1 1

(6) 2 2 2 3

Box (s; t)= 1 (V W ) + +2 Li (v ) V Li (v ) V V

3 2

2(1 2) 12 2 3 2

1 1 1 1

2 2 4 3 2 2

2 Li (v )+W Li (v ) V Li (v ) V V W + V W

4 3 2

2 8 6 4

2 2

2 3

V VW 2 + O ( ); + (s $ t)

4 3

2 1

7 47 (s)

2 3 4 3

1 + O ( ) ; Tri(s)=

3 4

12 3 16

(2:26 )

where

s t s t

v = ; w = ; V =ln ; W =ln : (2:27 )

u u u u

The expansions in the s- and t-channels can be found using analytic continuation formulae such

as [19 ]

ln(1 x + i")=ln(x 1) + i ;

1 1

ln x + + i ln x; Li (x + i")=Li

2 2

x 2 3

1 1

3 2

(2:28 )

ln x + ln x + i ln x; Li (x + i")=Li

3 3

x 6 3 2

1 1

4 2 3

Li (x + i")=Li ln x + ln x +2 + i ln x;

4 4 4

x 24 6 6

x>1 ; 9

where i" is an imaginary in nitesimal added to s; t or u b efore continuing.

We have veri ed that through O ( ) our result for the one-lo op amplitude agrees with a pre-

vious calculation [32 ], up to terms whichcanbeidenti ed as b eing due to the conversion b etween

dimensional regularization and a photon mass regularization.

2.3 Mo di cations for Bhabha Scattering

+ +

In comparison with the pro cess e e ! described above, the Bhabha scattering pro cess

+ +

e (k ) e (k ) ! e (k ) e (k ) ; (2:29 )

1 2 4 3

has additional exchange diagrams. In general, the interference required for Bhabha scattering is

given by

X X X

(L ) (L ) y (L ) (L ) y (L ) (L ) y

1 2 1 2 1 2

M M = M M + M M

4 4 4 4 4 4

Bhabha

spins spins spins

" #

(2:30 )

X X

(L ) y (L ) (L ) (L ) y

2 1 1 2

+ U M + M M ; M

4 4 4 4

spins spins

where the symmetry U acts as

U : s $ t; (2:31 )

(L) (L)

+ +

is the L-lo op amplitude for e e ! , and M is the same L-lo op amplitude but with M

4 4

legs1and3interchanged (taking into accounttheFermi statistics minus sign).

2.4 Bhabha Scattering at One Lo op to All Orders in

In the CDR scheme, the tree-level exchange contribution required for Bhabha scattering in eq. (2.30)

(0) (0) y

+ : (2:32 ) M =8(1 ) M

4 4

spins

Theone-loopexchange contribution, evaluated to all orders in ,isgiven by

X X

2 N

(1) (0) y (0) (0) y

(1)

~ ~ ~

M M = M M + A ; (2:33 )

4 4 4 4

spins spins 10

where

(1) 2 2 2 2 (6)

A = 8(1 2) (1 4 + )t 2(2 )tu +(1 ) u Box (s; t)

2 2 2 2 3 2 (6)

(2 3 )t +2(1 3 )tu (2 2 +3 + )u Box (s; u) + 8(1 2)

8(1 ) 1

2(1 )(u + st)N

(1 2)(3 2) t

2 2 2 2 2

Tri(s) (3 2) 2(1 + )t + (3 + 2 )tu 2(1 + )u

8 1

2 3 2 2 3 2 2

+ (2 5 +2 )t + (1 3 + )tu (1 )(2 3 )u Tri(t)

1 2 s

u 8

2 3 2 3 2 2

(2 4 + )t + (2 3 )tu (1 )(2 4 )u Tri(u) :

1 2 st

(2:34 )

Using these results, and the computation of the two-lo op exchange terms, we again nd that the

additional singular terms in Bhabha scattering are describ ed by eq. (2.9), where (not surprisingly)

(2)

+ +

H is given by precisely the same expression (2.21) that we found for e e ! .

R:S:

2.5 Finite Contributions to the Amplitudes

+ +

2.5.1 e e !

Finally wegive the real (disp ersive) part of the nite remainder in eq. (2.9), interfered with the tree

+ +

amplitude in the CDR scheme. First we treat the e e ! pro cess (2.14). It is convenient

to decomp ose the nite part according to the numb er of light avors, N ,

h i h i

(2) n (0) y

(0) (1) 2 (2)

Re M M = 8 F + N F + N F : (2:35 )

4 4

spins

(i)

In the s-channel, the functions F are given by

i h

(i) (i) (i)

+ S F ; (2:36 ) F = F

s s

=1 11

where

x 1 4

(0) 2 2 2 2 2

F = 2 X +(x + y ) 4(2 ) Li (x) X Li (x)+ X Li (x) + Li (x)

4 3 2 2

y 2 3

2 1 1

3 2 2

(3 + )X 3Y + (1 + )X + (2 )X Y XY 3(X Y )X 2

6 3 2 3

1 9 93 43 15 29 511

2 2

+ (11 16 )X Y + 12 + + X + +

3 4 3

2 2 4 2 2 24 32

+(x y ) 8Li (x=y ) + 6(2 + )Li (x)+(4(1 )X 12Y )Li (x)

4 4 3

2 2

Li (x) (2 + ) Li (x) X Li (x) (2 )X 4XY +

2 3 2

1 2 1 1 1

3 2 2 2

+ (6 + )X + X Y + (1 + 4 ) X + (10 + )X (2 )XY

12 3 6 6 2

2 2

+ (1 + 6 )X 4(2 ) (1 + 4 ) 6 X + 6 2 +2

3 4 3

2 6 3

1 1

(5 3 )X (3 )XY (1 4 ) X +(2 ) Li (x) X Li (x) +

3 2

6 2 6

1 1

+ (1 + 6 )X Y 6 X 4 + ;

2 2 3

(2:37 )

(

(1) 2 2 2 2 2

F (x + y ) 12 Li (x) X Li (x) + 4X 6X (Y +2ln( =s)) + 20 =

3 2

35 7 685 87

2 2 2

ln( =s)+ + + 29X + 36 ln( =s)+33 X +

2 2 4 9

2 2 2 2

2 (x y ) X 3(X 1) ln( =s) X +4 +8 X 2

)

(3X +6ln( =s) + 16)X ;

(2:38 )

4 10 25

(2) 2 2 2 2 2

= F ln ( =s)+ ; (2:39 ) x ln( =s) +

9 3 9

with

u t u t

; y = ; X =ln ; Y =ln ; (2:40 ) x =

s s s s

and the symmetry op eration S is given in eq. (2.25).

(i)

In the u-channel, the F are given by

(i) (i)

; (2:41 ) F = F

=1 12

where

x y 1

(0) 2 2 2

2(x y ) F (V W ) + = 2 3 V

y x

2 2 2

V Li (v ) +(x + y ) 4 (2 + ) Li (v ) V Li (v )+

2 4 3

+(2 ) Li (v=w)+(V W ) Li (v )+Li (w ) W Li (w )

4 3 3 2

(V + W )Li (v ) + 2(6 + ) Li (v )

2 2

2 3

4 2 1

3 2 2 3 4

V W +(4 )V W (7 2 )VW (1 2 )W

3 3 6

1 93

2 3 2

+6VW 3W +16V W + (9 16 )W + W

2 4

2 1

2 2 2

2V + + (3 + )VW (3 )W 6V (3 )W

3 3

511

+ +4 (2 )V 2(1 + )W +34 15 (25 + 96 )

3 4 3

12 16

6 (2 + )Li (v=w) (2 )Li (v ) 16Li (w ) (x y )

4 4 4

+4 (1 )W +(2+ )V Li (w )+4 4V (2 + )W Li (v )

3 3

(2 + ) Li (w )+W Li (v ) 4 Li (v ) V Li (v )

3 2 3 2

2 2

+ 4V + 2(2 + )VW +(2 )W (10 + 3 ) Li (v )

1 2 5 1 1

4 3 2 2 3 4

+ V (2 + )V W + (2 + )V W (4 + 7 )VW + (2 + )W

3 3 2 3 3

2 1 1

3 2 2 3 2

+ V (2 + )V W +2VW (10 + )W +(5+6 )VW (5 + 6 )W

3 6 2

2 2

2(6 )V 2(13 + 4 )VW +(5+3 )W +2(12 )V (3 2 )W +

4 (2 )V W 6 (2V W ) + (121 + 3 ) +(2+5 ) (15 + 26 )

3 4 3

+2 Li (v ) V Li (v ) V W (2 ) Li (w ) W Li (w )

3 2 3 2

1 1

3 2 2

(5 3 )W +6V W + (1 6 )W 6W (1 )VW +

6 2

2 2

+ 2(4 + 3 )V +(8 5 )W (6 + ) +(1 18 ) ;

6 6

(2:42 ) 13

( "

2 2 (1)

(x + y ) 12 Li (w ) W Li (w ) + 2(Li (v ) V Li (v )) F =

3 2 3 2

3 1 1

2 2 3 2 2 2

V W + VW W +(W 2VW 3W + )ln( =(s))

2 2 3

+29(W 2V )W 33W +2 (3V 4W )

1370

+ +87ln( =(s)) + (35 + 12 ) +(7 58 )

2 9

3 3 2

+ (x y ) 2V +2(V W ) + 13(2(V W )V + W )

2 2 2

ln( =(s)) + 16(2V W ) +6 2(V W )V + W +2V W +

2 2

2 (V + W )+9

)

2 2

3W 6V +6ln( =(s)) + 16 W +3 ;

(2:43 )

4 10 25

(2) 2 2 2 2 2

F = (x + y ) ln ( =(s)) + ln( =(s)) + : (2:44 )

9 3 9

Here x, y are de ned in eq. (2.40), whereas v , w , V , W are de ned in eq. (2.27).

(i)

In the t-channel, the functions F are given by the action of the symmetry S of eq. (2.25) on

the u-channel results,

h i

(i) (i)

F = S F : (2:45 )

+ +

The two-lo op virtual contribution to the e e ! unp olarized cross section, restoring

overall factors and averaging over initial spins, is given by

i 2 h (2)

1 (4 ) d

(0) y (2)

M : (2:46 ) = 2 Re M

4 4

dt 16s 4 2

spins

2.5.2 Bhabha Scattering

For the nite two-lo op remainder for the Bhabha scattering pro cess (2.29), we quote only the

(s $ t) symmetric sum of the twoexchange terms required by eq. (2.30). Again we decomp ose the

answer according to N ,

h i h i h i

(2) n (0) y (2) n (0) y

(0) (1) 2 (2)

~ ~ ~ ~ ~

Re M M + U Re M M = 8 F + N F + N F : (2:47 )

4 4 4 4

spins

(i)

In the s-channel, the functions F are given by

(i) (i)

~ ~

; (2:48 ) F = F

s 14

where

2 2 2 2 (0)

(X Y ) + Y 2x F = 2

+ 4 Li (x=y ) Li (y )+X Li (y ) +2(4Y 3X 1)Li (x)

4 4 3 3

1 4 2 1 1

2 4 3 2 2 3 4 2

Li (x)+ X + X + X Y 4X Y + XY Y +4 X 2XY +

2 8 3 3 6

23 3 93

3 2 2 3 2 2

X + X Y +9XY 6Y 5X 21XY +23Y + (X 2Y )

12 2 4

2 2

( 17X +32XY 18Y 17X 26Y ) +

511

2 (3X 8Y )+15 38 +47 +

3 4 3

6 8

y (1 x) 1 13

2 4 3

+ 10Li (x)+6X Li (x) X +2 Li (x)+ X X

4 3 2

x 3 24 12

5 1 5

2 2 2 2 2

+ X ((X Y ) + )+ Y + 6 + +12 X +20

3 4

3 2 2 2

+16 Li (x=y ) Li (y )+X Li (y ) + 4(3X 2Y 2)Li (x)

4 4 3 3

4 8 2 5

4 3 2 2 3 4 2 2

X X Y +8X Y XY + Y 4(X 2XY 2X + )Li (x)

12 3 3 3

3 2 2 2 2 2

+ X + X Y +18X 4XY +2Y + (11X 20XY +4Y +9X )

6 3

4 (3X 2Y )+88 +8 +2 ;

3 4 3

(2:49 )

(

y 1 1

2 (1)

36 Li (x) X Li (x) (X 4)X ln( =s) F =

3 2

9 x 6

2 2 3 2

+ (Y 3)Y ln( =s)+ln( =(t)) +2X 24 X +ln( =(t)) XY

2 2 3 2 2

18X 28Y +12ln( =(t)) +12XY 8Y 19X 58(X Y )Y +

2740

2 2 2

+ X 66Y +87 ln( =s)+ln( =(t)) +58 +44 +

2 2 2 2

2x 3 ln( =s)+ln( =(t)) +16 X 6 X

)

3 2 2 2 2 2 2

4y X + 3ln( =(t)) + 8 X + 3 ln( =s)+ln( =(t)) 2 +16 X 3 ;

(2:50 )

4 y 10 50

(2) 2 2 2 2 2 2 2

F = ln ( =s)+ln ( =(t)) + ln( =s)+ln( =(t)) + ; (2:51 )

9 x 3 9

and x, y , X , Y are de ned in eq. (2.40).

(i)

In the t-channel, the functions F are given by the action of the symmetry U of eq. (2.31) on

the s-channel results,

h i

(i) (i)

~ ~

F = U F : (2:52 )

s 15

(i)

In the u-channel, the functions F are given by

h i

(i) (i) (i)

~ ~ ~

F = F + U F ; (2:53 )

u u

where

(0) 2

F V =

4 Li (v )+W Li (v ) +6V Li (v )+Li (w ) +2Li (v ) +

4 3 3 3 3

5 3

2 2

+ Li (v ) V 5VW + W V + W +11

2 2 2

1 13 23 21 31 93 1

3 2 2 2

V V W VW V + VW 5V + W 12 + +

8 3 8 12 4 2 4

2 2

45 47 93 511

+ 20V + V + W 20 109 +

4 3

6 2 2 8 12 16

y (1 x)

+ 5Li (v=w)+6V Li (v )+Li (w )

4 3 3

(3V + W 2)(V W )+5 Li (v )

2 3

1 1 13 9 5

3 2 2

+ V + V W + V VW + V 12 + (V +7) V

4 2 12 4 2 12

+16 Li (v )+W Li (v ) 12V Li (v )+Li (w ) +8Li (v )

4 3 3 3 3

5 9 5 1

3 2 2 2

+ (3V + W 4)(V W )+5 Li (v )+ V +2V W VW + V + VW

3 12 4 6 2

253

+18V 16W + (4V +9W 7) V +8 ;

6 4

(2:54 )

(

y 1370 1

2 (1)

36 Li (v ) V Li (v ) +87ln( =(s)) + 47 + F =

3 2 3

9 x 9

2 2 2

x 6 (V W ) + +2(V W ) ln( =(s))

2 2

+ (13V 31W +10 + 65)(V W )+16

3 2 2 2 2

2y 2V +18V W +12VW ln( =(s)) + 9V +20VW +18(V + W )ln( =(s))

)

+33V + 4V 12 ln( =(s)) 29 ;

(2:55 )

y 10 25 4

2 2 2 (2)

ln ( =(s)) + ln( =(s)) + : (2:56 ) F =

9 x 3 9

Here x, y are de ned in eq. (2.40), whereas v , w , V , W are de ned in eq. (2.27). 16

2.6 Checks on the Result

We p erformed several checks on our calculation. The calculation was p erformed with the computer

algebra programs Maple, Mathematica, and FORM. Tocheck the co de, large parts of the calculation

were p erformed indep endently with alternative programs written in di erent languages. Various

checks were applied to the integral reduction pro cedures describ ed in refs. [12 , 13 , 14 , 15 , 18 , 19 ] and

our implementation of them. For example, we repro duced the double b ox ultraviolet divergences

in d = 8 and d = 10 rep orted in ref. [9], and several other previously calculated double box

integrals [10 ]. An additional check on the non-planar tensor integrals is that unphysical 1=(t u)

p oles o ccur in the representation of these integrals in terms of the master integral basis we used [15 ];

however, in the series expansion in such p oles drop out after delicate cancellations between the

various terms.

Wechecked the gauge invariance of the scattering amplitude by explicitly calculating the Feyn-

man diagrams in a general gauge and observing that the gauge dep endence drops out in the

nal result. This provides a non-trivial check of the diagrams and parts of the integral reduction

pro cedure.

A strong check on the nal result is provided by the matching of the IR divergence structure

of the two-lo op scattering amplitude with Catani's formula (2.9), as discussed in sect. 2.1. A

given integral will contribute to b oth infrared divergences and to nite terms. Thus a checkofthe

divergent terms provides an indirect check that the nite terms have b een correctly assembled.

Finally, we observed for small scattering angles a suppression of the leading logarithms, `

ln( =4), e.g. in the limit s ! 0 in the t-channel for pro cess (2.14). In other small-angle limits

(those not enhanced by the photon propagator pole) the leading power-law b ehavior is of course

4 3

less singular, but it is dressed by large logarithms of the typ e ` and N ` . But in the t-channel

s ! 0 limit it cancels down to ` and N `. This b ehavior is in accord with a generalized eikonal

representation for small-angle scattering [2].

3 Conclusions

+ +

In this pap er we presented the two-lo op QED corrections to e e ! and to Bhabha scat-

tering. We presented the results in terms of two-lo op amplitudes interfered with tree amplitudes

and summed over spins in the context of conventional dimensional regularization. In these results

wehave set the small electron and muon masses to vanish. (This is an excellentapproximation for

the highest energy current and future electron-p ositron colliders.)

The two-lo op amplitudes presented in this pap er are infrared divergent. Tomake use of them in a

Monte Carlo program for the NNLO terms in the cross section, they must b e combined with lower-

lo op matrix elements including photon emission, which should be computed using conventional

dimensional regularization, at least in the singular regions of phase space. In particular, the pieces

that need to b e computed (for the Bhabha case) are 17

+ +

the e e e e one-lo op amplitude interfered with itself.

+ +

the e e e e one-lo op amplitude interfered with a ve-p oint tree amplitude, and

+ +

the e e e e tree-level squared matrix element,

The interference of the dimensionally regularized one-lo op four-p oint amplitude with itself do es

not app ear to b e in the literature. Nevertheless, it should b e relatively straightforward to obtain,

given that it involves only one-lo op amplitudes with four-p oint kinematics. The required integrals

are given to suciently high order in in eq. (2.26).

The QED one-lo op ve-p oint amplitude interfered with the ve-p ointtree is a rather involved

ob ject to compute from scratch. However, the closely related one-lo op helicity amplitudes for one

photon and two quark pairs are known [33 , 34 , 35 ], and it is a relatively simple matter to mo dify the

color factors to obtain the corresp onding QED amplitudes. The one-lo op helicity amplitudes are

in the 't Ho oft-Veltman scheme. They can b e converted to conventional dimensional regularization

by altering the tree amplitude app earing in the co ecient of their singular terms [36 ]. Thus the

+ + + +

e e and e e e e one-lo op amplitudes may be extracted from the known literature

through O ( ).

Because of the 1= infrared divergences that are encountered in the phase-space integral, in

regions where the photon is soft or collinear, one might seem to require the one-lo op ve-p oint

amplitude through O ( ). However, this is not necessary [21 ]. Instead, one can replace the ve-

point amplitudes in singular phase-space regions bya combination of four-p oint amplitudes (which

are given in this pap er to the required order in the dimensional regularization parameter) and

splitting amplitudes [37 , 38 ]. The one-lo op splitting amplitudes for QCD are enumerated to the

required order in refs. [21 ]; the case of QED follows as usual by an appropriate conversion of color

factors.

+ + + +

The tree-level helicity amplitudes for e e and e e e e have been known for a

while [39 ]. (They also can be converted from the four-quark two photon amplitudes in ref. [35 ],

for example.) In infrared-divergent regions of phase space one must include higher order in

contributions from the matrix elements. Systematic discussion of these regions, where two particles

can b e soft or three collinear, has b een presented in refs. [22 ] for the case of QCD. Once again the

results for QED can b e obtained bya conversion of the color factors.

Even with all of these matrix element ingredients assembled, it is a nontrivial task to devise a

numerically stable metho d for carrying out the singular phase-space integrations. Nevertheless, this

task is very analogous to that required to obtain QCD jet predictions at next-to-next-to-leading

order, so it is likely that it will b e attacked so on.

Besides the obvious application of the present pap er to re ned theoretical predictions for Bhabha

scattering and for electron-p ositron annihilation into muons, it also serves as a further test of

metho ds that can be applied to analogous QCD pro cesses. We are con dent that many more

multi-particle two-lo op amplitudes will b e calculated b efore long. 18

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