The cross section for e+e−→e+e−e+e−

Nuclear Physics B241 (1984) 61-74 O North-Holland Publishing Company

THE

CROSS

SECTION

FOR

e+e---+e+e-e+e -

Ronald KLEISS

Instituut-Lorentz, Leiden, the Netherlands Received 7 November 1983 A calculation of the cross section for the process e+e- ~ e+e-e+e- is presented. The complete set of 36 Feynman diagrams is taken into account and is computed at the amplitude level, neglecting the electron mass. Emphasis is placed on the requirement for any such calculation that it can be done numerically in a fast and simple way. It is indicated how to use the derived helicity amplitudes to describe other scattering processes as well. Results are presented for several four-lepton final states that can be produced in electron-positron collisions.

1. I n t r o d u c t i o n

In this p a p e r I d e s c r i b e a c a l c u l a t i o n of t h e cross section for t h e p r o c e s s e + ( p + ) e - ( p _ ) ~ e+(ql)e+(q2)e-(q3)e-(q4).

(1)

T h e m o t i v a t i o n s for this s t u d y a r e twofold. T h e first o n e is physical: not o n l y is t h e r e a c t i o n (1) i n t e r e s t i n g in its own right as a t w o - p h o t o n s c a t t e r i n g process a n d an e x a m p l e of h i g h - o r d e r Q E D i n t e r a c t i o n s at the t r e e level, but also m a n y o t h e r p r o c e s s e s can be d e s c r i b e d by t a k i n g into a c c o u n t subsets of the collection of F e y n m a n d i a g r a m s c o n t r i b u t i n g to (1). M o s t n o t a b l y , t h e cross section for e + ( p + ) e - ( p _ ) ~ e+(qa)/z+(qz)e (q3)/z-(q4)

(2)

can easily b e o b t a i n e d o n c e the e e e e p r o d u c t i o n is known. By i n t r o d u c i n g t h e a p p r o p r i a t e c o l o r factors, several i n t e r e s t i n g Q C D results can be d e r i v e d as well. M o r e o v e r , p r o c e s s e s like (1) a n d (2) m a y act to p r o d u c e b a c k g r o u n d s to o t h e r physics, such as t h e p r o d u c t i o n of s u p e r s y m m e t r i c d e g r e e s of f r e e d o m . T h e s e c o n d r e a s o n is of a technical n a t u r e : t h e c a l c u l a t i o n of cross sections such as the a b o v e ones is a highly n o n - t r i v i a l a n d c h a l l e n g i n g p r o b l e m b e c a u s e of the large n u m b e r of d i a g r a m s involved, a n d the c o m p l i c a t e d s t r u c t u r e of each d i a g r a m in t e r m s of D i r a c m a t r i c e s . T o m y k n o w l e d g e , a c o m p l e t e c a l c u l a t i o n which t a k e s into a c c o u n t all t h e d i a g r a m s has n e v e r b e e n p e r f o r m e d even for t h e s i m p l e r case (2). A s a side result, t h e t e c h n i q u e s o u t l i n e d in this p a p e r can b e e x p e c t e d to p r o v e useful for o t h e r c a l c u l a t i o n s of this t y p e , such as t h o s e c o n c e r n i n g m u l t i p l e b r e m s s t r a h l u n g [1]. T h e m a i n f e a t u r e of t h e c a l c u l a t i o n p r e s e n t e d h e r e is t h e fact t h a t all f e r m i o n masses a r e c o m p l e t e l y n e g l e c t e d . This is of c o u r s e n o t a g o o d a p p r o x i m a t i o n 61

62

R . H . P . Kleiss / e+ e

--> e+ e - e + e -

whenever a propagator denominator in one or more Feynman diagrams becomes extremely small. In such a case, however, those particular diagrams will completely dominate the cross section, and an evaluation of all graphs is quite pointless in these regions of phase space. An integration of the cross section over a peaking region should therefore consist of two steps: first, to obtain the correct peaking behaviour using only a small subset of the Feynman graphs in which the necessary mass dependence is kept; and second, to modify this cross section by multiplying it with the ratio of the result for the complete set of diagrams over that of the restricted set. This ratio, which approaches unity as one draws closer to the peak, can be computed with zero mass. It is clear that this approach is extremely well suited for implementation into existing numerical calculations of the Monte Carlo type. If, on the other hand, one is interested in phase spaces that do not contain peaks, the mass of light particles such as electrons and muons can be safely neglected. The application given in sect. 5 is of this type. A second point of importance is the following. The calculation is performed at the amplitude level, i.e. the transition amplitude is calculated numerically before being squared and summed over spins. Scattering processes like (1) and (2) are practically unassailable without the use of such a method. The most naive and straightforward way to do this is to take an explicit spinor representation, and multiply out the appropriate gamma matrices componentwise. Although feasible [2], this procedure is rather slow if many spinors or many diagrams are involved. A more elegant approach is to try to rewrite the amplitude such that it can be evaluated by the standard trace techniques available for amplitude squares. A number of authors have proposed methods for doing so [3]. Many of these papers describe a general formalism rather than an actual calculation of a complicated cross section; the examples given are often processes that could be treated equally well with standard techniques and algebraic manipulation programs such as R E D U C E or S C H O O N S C H I P . In the present paper, the emphasis will be on the calculation of the amplitude, while the more generally applicable tricks will be left to explain themselves during the discussion. The fact that we are looking for a method that can be applied as a computer program has two consequences. In the first place, a major problem in all amplitude calculations is the determination of the complex phases between different possible fermion lines. These are important if one spinor can be a part of more than one fermion line, as in Bhabha scattering. Choosing explicit representations for spinors and gamma matrices obscures this problem even more. The treatment presented here is independent of such a choice of representation, and moreover makes the values of the complex phases irrelevant. Secondly, from the point of view of numerical calculations, quantities wih free Lorentz indices such as vector currents (a(q)y"u(q')) or contractions with the Levi-Civita tensor e~,~o~ are very unattractive. To keep the computer program as fast as possible, the occurrence of such quantities in the final result should be avoided if possible. As will be shown in sect. 3, it is indeed possible to express all amplitudes

R.H.P. Kleiss / e+e-o

63

e+ e e+ e -

for the process (1) in terms of complex scalars only, with no implied summation over free Lorentz indices left in the expressions. The outline of this paper is as follows. In sect. 2, I describe the Feynman diagrams, and the various amplitudes, that contribute to the process (1). In sect. 3 it is shown how these contributions can be reduced at the amplitude level to simple expressions involving only a number of scalars, each of which depends only on the momenta of two particles. In sect. 4 the algorithm for systematic evaluation of all terms in the amplitude for (1) is given. This is subsequently applied, in sect. 5, to calculate cross sections for several four-lepton production processes. Finally, an appendix contains a list of reductions of the amplitude for (1) to other scattering processes.

2. Diagrams and amplitudes The process (1) is described by 36 Feynman diagrams. To deal with this relatively large number we first introduce a systematic ordering of the graphs into six groups of six diagrams. Each group is characterized by the way in which the three spinors (with momenta p_~, q~" and q~) are connected with the three conjugate spinors (with momenta p+~, q~ and q~) to form the three fermion lines in every diagram. Let us denote the connection of the spinor u ( q ~ ) with a(q3) with the bracket [qaql]. Each group can then be described by three of these brackets. Since we are dealing with fermions, three groups will have a minus sign relative to the three other groups. In table 1 we list the six groups, together with their relative sign. We now proceed to order the diagrams within each group. Every diagram contains two fermion lines with only one vertex each. They are connected (by the exchange of a virtual photon) with the remaining fermion line, which has two vertices. We adopt the following convention: the spinor part of the diagram [ k l k 2 ] [ k a k 4 ] [ k s k 6 ] , where kl • • • k6 are a permutation of the six momenta, is defined to be [klkz][kak4][ksk6]-)

(t(kl)y~u(k2)(~(ka)y~'k7yt~u(k4)(t(ks)y~u(k6)

,

(3)

k~ being the momentum flowing in the internal electron line. This convention fixes the virtual photon propagators unambiguously as well. In table 2 the different permutations within a given group are enumerated.

TABLE 1 Group number

Fermion brackets

Relative sign

1 2 3 4 5 6

[P÷P ][q3ql][qaq2] [ P+P-][q3qz][q4ql ] [P+q,][qaq2][q4P-] [P÷q2][q3ql][q4P-] [P+qz][q3P ][q4ql] [ P+ql ][q3P-][q4q2]

+ + -

+ -

+

64

R . H . P . Kleiss / e e

-._>

+

-

+

e e e e

-

TABLE 2 Number in group

Permutation

1 2

[k 1 k2][k3k4][ksk6] [ksk6][k3k4][k 1 k2]

3 4

[k3 k4][k5 k6][kl k2] [ k 1k2] [ k 5k6] [ k 3k4]

5

[ksk6][k 1 k2][k3k4]

6

[ k 3k4] [ k 1k2] [ k 5k6]

Notice that no relative signs are connected with these permutations. Moreover, the sum of diagram numbers 1 + 2 is gauge invariant, and similarly the sums 3 + 4 and 5+6. The above systematic approach enables us to denote each diagram by the symbol (i, j), where i stands for the group number, and j for the number within the group. In fig. 1 all the diagrams are given explicitly. We still have to define the helicity amplitudes for the process (1). As a first step, we introduce helicities for the various spinors. Since we have neglected the fermion mass, the difference between electrons and positrons is quite irrelevant for our calculation, and we will denote every spinor by the symbol u (as we have already done in eq. (3)). We define positive and negative helicity states up to a complex

4

4

4

4

2

2

1

1

(I,I)

(!,3)

(1,5)

(1,2)

(2,1)

(2,2)

3

3

3

3

1

1

2

2

(1,4)

(2,3)

(1,6)

(2,5)

(2,4)

(2,6)

Fig. 1. The 36 Feynman diagrams contributing to the process e + e - ~ e+e÷e-e -. The symbols + , - , 1, 2, 3, 4 denote the particle momenta p÷, p_, ql, q2, q3 and q4, respectively.

R.H.P. Kleiss/e

65

. e . ~. e . e. e. e 2

-

-

(3,2)

(3,1) +

h

(4,1)

+

I

I

(4,2J

+

2

÷

/

3

\2

(3,4)

(3,31

3

(4,3)

I /<:

4

/m~ +

I

(3,6) 2

2

i

~

(5,2)

l

?

-

4

(4,6)

+

_

(5,1)

,

(4,5)

+

3

f J:

+

4

(3,5)

(4,4)

3

j~ 1

+

4

+

~

4

(6,1)

I

h

4

(6,2)

4

4 2

-

3

(5,3)

~J: (5,5)

(5,4)

(6,3)

/
f

(5,6)

(6,4)

// 4

f°

(6,5)

(6,6)

Fig. 1 (continued)

phase by specifying the corresponding spin-projection operator:

u±(k)a.(k)=~o~k,

o3± =1(1 ± 75),

(4)

where again k" stands for either a p" or a q". We are now able to enumerate the different helicity configurations which give non-zero amplitudes. In total, 20 configurations can occur; but each diagram can contribute to only 8 helicity amplitudes. In table 3 we list 10 non-vanishing helicity amplitudes together with the groups (as defined in table 1) that contribute to them. We have fixed the helicity of the incoming positron to be positive. The 10 remaining amplitudes can be obtained from these by performing the parity transformation 7 s ~ - 7 s.

66

R.H.P. Kleiss / e+ e--> e+ e - e + e -

TABLE 3

P+

Helicity of fermion Pql q2 q3

q4

1

+

+

+

+

+

+

X

+

+

-t-

-

+

-

x

+

+

-

+

+

-

+

-

+

+

+

-

+

+

+

-

-

+

+

+

-

+

-

+

+

-

+

+

-

+

+

+

.

+

-

+

-

-

-

+

--

--

+

--

--

.

.

Contributing group no. 2 3 4 5 x

x

X

x

x x

x

X

X

x

x x

X

x x

x

.

6

X

x x

x x

X

3. Evaluation of the diagrams In sect. 2 it was indicated how every diagram of fig. 1 could be denoted unambiguously by specifying a permutation of the six m o m e n t a p+, p_, q~, q2, q3, and q4. We will now concentrate on a single graph and derive for it an expression that is suitable for numerical evaluation. Let ki ( i = 1 . . . . . 6) be a permutation of the particle momenta, and let ai = + be the corresponding helicity. To this permutation we let correspond the following diagram:

M ( k l , al, k2; k3, a3, k4; ks, as, k6) = _ie4

~ ( k l , a l , / 2 ; k3, a3, k4; ks, as, k6)

( bl kl + b2kz)2( bl kl + b2k2 + b3k3)2( bsk5 + b6k6) 2' bi -= b(k~) ~

{+i' _ ,

if ki is one of the outgoing m o m e n t a if ki is one of the incoming m o m e n t a .

(5)

The spinorial part J is that which was already given in eq. (3): at(k1, al, k2; k3, a3, k4; ks, as, k6) = ~a,(kl)~lo~Ua,(k2)aa3(k3)~la(blkl

-~- b 2 k 2

-t-

b3](3)'YCua3(k4)uas(ks)Ycuas(k6),

(6)

where we have made use of m o m e n t u m conservation, and conservation of helicity for massless fermions. It is the evaluation of eq. (6) which forms the core of the problem of calculating the cross section. We will attack this problem in two steps, the first being the elimination of the repeated indices a and/~ in eq. (6), and the second the evaluation of the resulting expression, avoiding taking traces as much as possible. A well-known method by which explicit indices in different fermion lines can be made to disappear is repeated application of the so-called Chisholm identities [4]. However, the resulting expressions will consist of four terms, each about as long as eq. (6) itself.

R.H.P. Kleiss / e+e-~ e+e-e+e-

67

It has been noted by various authors (and actually done in ref [5]) that one can improve somewhat on this using Kahane algorithms, or some other trick. The method by which we will get rid of the repeated indices was first introduced in ref. [1]. Using a little Diracology, we write for instance a+(kl) y'~u+(k2) t/+(k3) y,, -

u+ ( k 1) y~u+(k2) a+ (k2)J( ] u+(k3) u+(k3) Y~ a+(k2)Xlu+(k3)

= U+( kl ) T'~J(2J(,J(3 T~/ a+( kz)J( 1u+(k3) = - 4 ( k l " k3)~+( kl)J(z/ Ct+(k2).l(, u+( k3) . The various other possibilities are treated in the same way. We find

a+(k,)y~u+(k2)gt+(k3)3", = - 4 ( k l " k3)a+(kl)J(2/ct+(k2)J(,u+(k3),

(8.1)

g~ (k,)3"~'u (k2)a+(k3)%, = 4 ( k 2 " k 3 ) o _ ( k O / o _ ( k z ) u + ( k 3 ) ,

(8.2)

3"~u+(k4)a+(ks)3"~u+(k6) = - 4 ( k 4 " k6)gsu+(k6)/a+(k4)X6u+(ks),

(8.3)

3"~u+( kn)O-( ks)3"t3u_( k6) = 4 ( k 4 " k5)u_( k6)/ a+( k4)u ( k5) .

(8.4)

As before, the remaining cases can be obtained from eqs. (8.1)-(8.4) by replacing 3,5 by _ y s . The occurrence of spinor sandwiches in the denominator, which is usually avoided, does not bother us here, as will become clear later on. The second step, i.e. the reduction of ~ to simpler quantities, is made by expressing all spinors in terms of one single spinor only. To this end we introduce two lightlike vectors p~ and p~, that must not be proportional to any of the k~ or to each other, but may otherwise be arbitrary. We define an auxiliary spinor Uo by

UoCto= PoW+ .

(9.1)

The positive and negative helicity spinors for any massless momentum k~' can be defined as follows:

u+(k~) = ~Uo/[2(po" k,)] ]/2 ,

(9.2)

u_(k~) = ~ p l U o / [ 4 ( p o " Pl)(P," k~)]1/2.

(9.3)

It is easily verified that with this definition the spinors automatically satisfy eq. (4). Consequently, a change in the choice of p~ or p~' can only lead to a shift in the overall complex phase of the spinors. We now define the basic ingredients for the helicity amplitudes, namely the two scalar quantities S and T:

S(q,, qs) =- a+(q~)u (qj),

(10.1)

T(q,, qj) =- •_(q,)u+(qj) = S(qs, q~)* ,

(10.2)

where i and j run from 1 to 6, and q5 = P+, q6--P-. With the definitions in eqs.

68

R . H . P . Kleiss / e+ e - ~ e+ e e+ e -

(9.1)-(9.3), the quantity S(q~, qj) can readily be evaluated, and we find

S(qi, qj) = a o q ~ p , Uo/[8(qi" Po)(Po"

Pl)(Pl

"

qj)],/2,

ftoqiqjpl Uo = Tr ( qiqfl¢ 1poO)+) = 2{(q~ "qj)(Po' P,)+(Po" q~)(Pl " q j ) - ( P o ' q j ) ( P , " • •

I~

v

p

q~)

o-

- le,~o~.poplq i qj }.

(11)

A particularly simple form for S can be obtained if we can choose Po,1 to have non-zero components in the x-direction only: for instance

= ( pO,

poL p0 = ( 1,1, 0, 0 ) ,

p~=(p,~, p,, p,, p,)3 _ _- (1, - 1 , 0, 0). 1

2

(12)

Inserting eq. (12) into the result (11) we find

S(qi, qj) -

( qO_ q~ )( qO + q) ) _ ( q~ + iq3)( q2 _ iq~) [(qO_q~)(qO+q~)]l/2

(13)

A few remarks are in order here. In the first place, the quantity S seems to be a very fundamental entity for this type of amplitude, and to be even more basic for spinor products than the Minkowski product qTqj~. Indeed, we have

S(qi, qj) T(qj, qi) = 2q,. qj.

(14)

In the second place, the form (13) for S (and its complex conjugate for T) seems at first sight not to be very attractive since it depends on the explicit components of q~ and qJ*. We should keep in mind, however, that quick and easy numerical computation is our aim, and that in computer programs the m o m e n t a q/~ always occur in terms of explicit components anyway. A form like eq. (13), which consists of only a few simple manipulations that are the same for all q~.j, is actually much to be preferred (from a p r o g r a m m e r ' s point of view) over one like eq. (11). O u r computation of J , defined in eq. (6), is essentially finished if we realize that as soon as we have S and T as in eqs. (10.1, 10.2) we can write any sandwich in our problem immediately in terms of these two scalars: a+(kl),]~2,](3,](4... ,k"n l u + ( k n )

=S(k,,k2)T(k2,

k3)S(k3, k4)...S(k,,_2, kn-,)T(k~

1, k,,),

(15)

if n is odd, while for even n the string on the r.h.s, ends with S instead of T (of course, the spinor u(kn) must have the other helicity in this case). It is now a trivial matter to write out J in terms of S and T. For completeness, in table 4 are given the eight non-vanishing helicity configurations with the resultant expression for J . In this table, s12 stands for S(kl, k2), and so on.

69

R.H.P. Kleiss / e + e --* e+e-e + e TABLE 4

al

a3 a5

+

+

+

+

+

-

-

+

+

_

q-

-

+

-

+

+

J(kl, al, k2; k 3 ,

a3,

k4; ks, as, k6)

4 s3t t64s12( bl t21s15 + b2t22s25 + b3t23s35)156/ ( s21165) -4s31154s12(bltels16 + b2t22s26 + b3t23s36)/s21 -4s32/64(bl tl 1 s1 s -1- b2ta 2s25 + b3tl 3s35) t56/t65 4s32tsa( bl tl i s16 + bztl 2s26 + b3t13s36) 4132s54(bls11t16 + b2s12126 + b3s13136) - 4 / 3 2 s 6 4 ( b l sl i tl 5 + b2sl 2t25 + b3sl 3/35)$56/$65 -4t31s54t12( bl s21116 + b2s22t26 + b3s23136)/ t21 4131s64t12(bls21115 -F b2s22t25 + b3s23135)s56/(t21s65)

4. The computational algorithm With the results of the preceding sections, it is now easy to calculate the cross section for the process (1), which is given by dSo.-

(27) 8 . 1 Z [ M 1 2 6 4 ( p + + P - - q l - q z - q 3 - q 4 ) 4(p+ • p ) spins

[d3q,~ [I \2qO].

(16)

i=1

The spin sum of IMI 2 can be evaluated as soon as the momenta p+~ and q~" have been completely specified. An algorithm to do this is the following: (a) Calculate all possible scalars S(q~, qj) and T(q~, qj) (in principle there are 72 such quantities. However, we have to calculate only 30, for S(q~, q~)= 0 and T is obtained from S using eq. (10.2)). (b) Pick one of the helicity configurations of table 3 (the 10 possibilities not present in the table are taken into account by replacing in eq. (16) the factor ~ by ½). (c) Pick a group of diagrams contributing to the amplitude (in most cases, only two groups have to be done. Only when all helicities are equal do we have to calculate all six groups). (d) Pick a diagram and calculate it using table 4 (the choice of a diagram consists of nothing more than determining which vector is represented by each of the k~ in eq. (5); this then defines which S and T correspond to the sq and to in table 4. This means that we have to do only some permutations of labels (indices) and very limited arithmetic with the complex numbers for S and T). (e) Repeat steps (c) and (d), summing the results. (f) Square the result of step (e). (g) Repeat steps (b) through (f), summing the results. The algorithm as outlined here has several nice features. In the first place it is extremely simple to program. In the second place, the contribution of each diagram is clearly identified, and the cross section for the various subsets of diagrams, like the cross section for the process (2), is obtained along with that for the process (1). Also the ratio for the contribution from a restricted set of diagrams to the complete set, mentioned in section 1, can easily be found.

R.H.P. Kleiss/ e+e--+ e + e - e

70

+ e -

5. An application: four-lepton production at large angles In this section the methods which were outlined in this paper will be applied in orer to actually calculate some cross sections. Because the emphasis here is on the calculation of IMI 2 itself and not so much on the treatment of peaks and collinear singularities, I have chosen to study four-lepton production under the following experimental conditions:

~.(p±,q,)>~Oo,

(qi+qj)Z>~So,

(17)

i.e. all final-state particles are produced at large scattering angles, and with large invariant masses. Since a number of events of this type has been observed at both P E T R A and P E P [6], the cross section for this kind of signature (which is also relevant for exotic particle searches) is a quantity of m o r e than academic interest. Also, processes like these are an excellent illustration of the approach advocated here, since on the one hand no collinear peaks or effects of a small but finite fermion mass occur, and on the other hand no particular diagram, or set of diagrams, can a priori be expected to dominate the cross section. The integration of the amplitude square proceeds as follows. Let us define two vectors Q~ and Q~ as

Q~=q~+q~,

Q~=q~+q~.

(18)

The Lorentz-invariant phase-space element appearing in eq. (16) can then be written in the following form:

64(p++p--ql-q2-q3-q4)

H {d3qi'~ \2q°J

i=1 ~. 1 / 2 ( S , S1, $ 2 )

-

512s

dSl ds2 d/2(Ol) d/2(qO d/2(q3)

A(x, y, z) = x2+ y2+ z 2 - 2 ( x y + xz + yz) , s = (p++p_)2,

sl = Q 2 ,

s2 = Q22 •

(19)

In this formula, the solid angle ~ ( Q I ) is the direction of Q1 in the laboratory frame (where p++p_ = 0 ) , w h i l e / 2 ( q l ) and J~(q3) are defined as the directions of ql and q3 in the rest frames of Q~ and Q2, respectively. The invariants s~ and s2 must satisfy sL2 i> So,

( , / ~ + , / Z ) 2 ~< s.

(20)

The eight-dimensional integration element in eq. (19) is normalized to

A(s, So) =

I~-~2 ~-~2 I dSl

~'So

ds 2

~$o

-- ~ - ~ ( ~ - 12S~o+ 16~o4~).

4 qr

dg~(O,)

~

7r

d/2(ql)

d~2(q3) 7r

(21)

R.H.P. Kleiss / e + e - ~ e+e e+ e -

71

TABLE 5

Process no.

Cross section and error (in pb) a)

Final state

(1)

e+(ql)e+(q2)e-(q3)e-(q4) e+(ql)e-(q3)~+(q2)l~-(q4) (12 diagrams) e+(ql)e-(q3)~+(q2)tt-(q4) (2 diagrams) tz +(ql)tz +(q2)tz-(q3)l~-(q4) tx+(qa)lz-(q3)~+(q2) e (q4)

(2) (3) (4) (5)

0.030 + 0,002 0.035 ±0,002 0.0023 ± 0,0001 0.0038 ± 0.0002 0.0082+0.0005

a) The error estimate on the integral is purely statistical. It is the standard deviation of the distribution of the values of F(ql) in the sample.

In order to obtain the cross section, integrated under the restrictions of eq. (17), we pick r a n d o m values for O ( 0 1 ) , O ( q 0 , O(q3), Sl and s2 (satisfying eq. (20)), and construct from them the f o u r - m o m e n t a q~'. The quantity D36 , which is defined as the spin sum and average of IMI 2 without the overall factor e 8, is then calculated as indicated in sect. 4. By repeating this a large number of times we find the cross section as the average value of the following function:

F(qi) =

47r ~0,

4s z

(a'

sl, s2)A(s, So),

if the q~ satisfy eq. (17) (22)

otherwise.

The method of integration outlined here is, as a Monte Carlo procedure, quite crude. H o w e v e r , since we avoid all peaks in the cross section, an adaptive integration routine is not really necessary. If the phase-space integral does run over some peak, it becomes of course necessary to use an event generator, and apply the procedure discussed in the introduction. It has already been mentioned that processes different from ee-~ eeee can be studied by restricting the set of Feynman diagrams contributing to the cross section. By doing so (following the rules described in the appendix) I have obtained values for the cross section for various processes, which are listed in table 5. These were obtained by sampling 10 000 non-zero values for F(qi). The parameters were chosen as follows: s = 900 G e V 2, So = 1 G e V 2, 00 = 45 °. Concerning this table, a few remarks are in order. The third entry is the cross section for ee--, ee/x/z, calculated with only the two multiperipheral diagrams of fig. 2. The cross section for the complete set of diagrams (entry 2) comes out quite different, indicating that the multiperipheral approximation is not at all appropriate for this kind of experimental condition (which was, of course, to be expected). In the last entry, the symbol ~ stands for a third lepton species, for instance the tau lepton if we neglect its mass also. The n u m b e r given here holds for a light lepton with unit charge. Finally, it should be noted that the cross sections for the eeee and

72

R . H . P . K l e i s s / e+ e - ~

e+(P+) " " - - - . . . . ~ . . . . . . ~ ~

e+(ql ) e+(p+) ~

> ~ . - - - - - - - - ~ ~'

e-(p_) ~

e+e-e+ e -

~q4~

e+(ql)

~---

~ e-(q 3) e-(p) -/~

~+~q2I

~--e (q3)

Fig. 2. The two Feynman diagrams describing the process e+e---,e-e-~z+tz- in the multiperipheral scattering approximation. iztx/z/x final states have been multiplied by a statistical factor of ¼, due to the occurrence of two pairs of identical particles in the produced system. The hospitality of SLAC, where this work was initiated, is gratefully acknowledged. The author also wants to thank Dr. F. Berends for suggesting this investigation, Drs. D. Berg and S. Yamada for stimulating discussions and Dr. P.H. Daverveldt for his valuable assistance with the numerical work.

Appendix REDUCTION TO OTHER PROCESSES Below are listed several processes that can be examined simultaneously with the four-electron production. This is done by dropping some diagrams and some trivial manipulations: e+( p+)e-(p_) -~ e+(q~)e+(q2)e-(q3)e-(q4),

(A. 1)

the basic scattering process, all groups of diagrams contribute; e+(p+)e-(p_) ~ e+~qx)e-(q3) p~+(q2) ~-(q4),;

(A.2)

groups 1 and 6 contribute, the two multiperipheral diagrams, which are often used to approximate the cross section, are numbers (6, 3) and (6, 4) of fig. 1; e+(p+)e-(p_) -~ i,~+(ql)~+(q2)#-(q3)p~-(q4),

(A.3)

groups 1 and 2 contribute, the effect of the statistics obeyed by the muons can be investigated by changing the relative sign between the two groups from - to + , thereby forcing the muons to have Bose rather than Fermi statistics; e+(p+)e-(p_) group 1 only contributes.

--> t z + ( q l ) t z - ( q 3 ) C + ( q 2 ) g - ( q 4 )

,

(A.4)

R.H.P. Kleiss / e+e - ~ e+e-e+e -

73

By taking fractional charges and color effects into account, Q C D processes with h i g h - m o m e n t u m transfer can also be included. As an example: e+(p+)e-(p_) ~ e+ ( q l ) e - ( q3)q( q2)q( q4) ,

(A.5)

like the process (A.2), except for the difference in quark charges. Let the quark have a charge O in units of the electron charge. The diagrams (6, 3) and (6, 4) will then have an additional factor 0 2, and all the other diagrams a factor O. Also, the cross section gets an overall factor 3 from colour. Finally, it is a trivial matter to transform initial-state particles into final-state ones, and vice versa, by changing the values of the b's introduced in eq. (5). Some examples: e+(p+)e+(q3) ~ e+(p_)e+(ql)e+(q2)e-(q4),

(A.6)

is o b t a i n e d from process (A.1) by interchanging the values of b(p_) and b(q3), b(p_) goes from - to + , and b(q3) goes from + to - (of course, also the numerical values of the f o u r - m o m e n t a p~ and q~ are interchanged); e - ( p - ) / x - ( q 2 ) ~ e-(p+)tz-(q4)e+(ql)e-(q3),

(A.7)

comes from process (A.2), by interchanging b(p+) and b(q2), and the values of the corresponding m o m e n t a ; e-(p_)q(q2) ~ e-(p+)q(q4)e+(qa)e-(q2).

(A.8)

This m o r e interesting possibility, which may be studied at H E R A , is obtained from process (A.5) by interchanging q~ and p+", and b(q2) and b(p+). It should be noted that by shifting particles from the initial state to the final state or back, the statistical factor connected with the cross section may change as well. By going from process (A. 1) to process (A.6), the factor changes from ¼to ~-; going from process (A.2) to process (A.7) we have to add a statistical factor of 1, and the same factor occurs in the transition from process (A.5) to process (A.8).

References [1] CALKUL collaboration: F.A. Berends, R. Kleiss, P. de Causmaecker, R.Gastmans and T.T. Wu, Phys. Lett. 103B (1981) 124; P. de Causmaecker, R. Gastmans, W. Troost and T.T. Wu, Phys. Lett. 105B (1981) 215; Nucl. Phys. B206 (1982) 53; F.A. Berends, R. Kleiss, P. de Causmaecker, R. Gastmans, W. Troost and T. T. Wu, Nucl. Phys. B206 (1981) 61; P. de Causmaecker, Ph.D.thesis (Leuven, 1982) [2] J.A.M. Vermaseren, private communication [3] J.D. Bjorken and M.C. Chen, Phys. Rev. 154 (1967) 1335; O. Reading Henry, Phys. Rev. 154 (1967) 1534; H. W. Fearing and R.R. Silbar, Phys. Rev. D6 (1972) 471; K.J.F. Gaemers and D. Gounaris, Z. Phys. C1 (1979) 259; J.O. Eeg, J. Math. Phys. 21 (1980) 170;

74

R . H . P . Kleiss / e÷ e---> e+ e e÷ e -

M. Caffo and E. Remiddi, Helv. Phys. Acta 55 (1982) 339; R. Farrar and F. Neri, Rutgers University preprint RU-83-20; G. Passarino, SLAC preprint SLAC-PUB-3125 [4] E.R. Caianello and S. Fubini, Nuovo Cim. 9 (1952) 1218; J.S.R. Chisholm, Nuovo Cim. 30 (1963) 426 [5] J. Kahane, J. Math. Phys. 9 (1968) 1732; J.S.R. Chisholm, Comp. Phys. Comm. 4 (1972) 205 [6] S. Yamada, private communication; D. Berg, private communication