Predictions for all processes e+e− → fermions + γ

NU~LEAI:; P H Y.-.~ I C ~

ELSEVIER

13

Nuclear Physics B 560 (1999) 33-65 www.elsevier.nl/locate/npe

P r e d i c t i o n s for all p r o c e s s e s e + e -

> f e r m i o n s + 3/

A . D e n n e r a, S. D i t t m a i e r b, M . R o t h a,c, D . W a c k e r o t h a a Paul-Seherrer-lnstitut, Wiirenlingen und Villigen, CH-5232 Vitlligen PSI, Switzerland b Theoretische Physik, Universitiit Bielefeld, D-33615 Bielefeld, Germany c Institutfiir Theoretische Physik, ETH-HOnggerberg, CH-8093 Ziirich, Switzerland Received 3 May 1999; accepted 15 July 1999

Abstract The complete matrix elements for e+e - --~ 4 f and e+e - ~ 4 f y are calculated in the Electroweak Standard Model for polarized massless fermions. The matrix elements for all final states are reduced to a few compact generic functions. Monte Carlo generators for e+e - ~ 4 f and e+e - ~ 4 f 7 are constructed. We compare different treatments of the finite widths of the electroweak gauge bosons; in particular, we include a scheme with a complex gauge-boson mass that obeys all Ward identities. The detailed discussion of numerical results comprises integrated cross sections as well as photon-energy distributions for all different final states. @ 1999 Elsevier Science B.V. All rights reserved. PACS: 12.15.Ji; 13.10.+q; 14.70.-e; 14.70.Fro; 13.40.Ks Keywords: Gauge-bosonproduction; Four-fermionproduction; Helicity amplitudes; Monte Carlo integration; Electromagnetic corrections

1. I n t r o d u c t i o n W h e n exceeding the W-pair production threshold, the LEP collider started a new era in the verification of the Electroweak Standard Model ( S M ) : the study of the properties of the W boson and of its interactions. While the most important process at LEP2 in this respect is certainly e+e - ---+ W + W - ~ 4 f , many other reactions have now become accessible. Besides the 4-fermion-production processes, including single W-boson production, single Z-boson production, or Z-boson-pair production, LEP2 and especially a future linear collider allow us to investigate another class of processes, namely e+e - ~ 4 f y . The physical interest in the processes e+e - --+ 4 f y is twofold. First of all, they are an important building block for the radiative corrections to e+e - ~ 4 f , and their effect must be taken into account in order to get precise predictions for the observables 0550-3213/99/$ - see frontmatter (~) 1999 Elsevier Science B.V. All rights reserved. PII S 0 5 5 0 - 3 2 1 3 ( 9 9 ) 0 0 4 3 7 - X

34

A. Denner et al./Nuclear PhysicsB 560 (1999)33-65

that are used for the measurement of the W-boson mass and the triple-gauge-boson couplings. On the other hand, those processes themselves involve interesting physics. They include, in particular, triple-gauge-boson-production processes such as W + W - y , ZZy, or Z y y production and can therefore be used to obtain information on the quartic gauge-boson couplings yyWW, yZWW, and yyZZ. While only a few events of this kind are expected at LEP2, these studies can be performed in more detail at future linear e+e - colliders [ 1]. Important contributions to e+e - ~ 4 f and e+e - --~ 4 f y arise from subprocesses with two resonant gauge bosons. These subprocesses do not only contain interesting physics, such as the quartic gauge-boson self-interactions, but also dominate the cross sections in those regions of phase space where the invariant masses of certain combinations of final-state particles are close to the mass of the corresponding nearly resonant gauge bosons. Therefore, it is sometimes a reasonable approach to consider only the resonant contributions. In the most naive approximation, the produced gauge bosons are treated as stable particles. Most of the existing calculations for triple-gauge-boson production have been performed in this way, such as many calculations for e+e - ~ W + W - T [2]. In an improved approach, the so-called pole expansion [3,4], the resonances are treated exactly, i.e. the decay of the produced gauge bosons is taken into account, but the matrix elements are expanded about the poles of the resonances. Once the non-resonant diagrams have been left out, the expansion is in fact mandatory in order to retain gauge invariance. If only the leading terms in the expansion are kept, this approach is known as the pole approximation, or double-pole approximation in the presence of two resonances. The accuracy of the (double-)pole approximation is, at best, of the order of I'v/Mv, where Mv and / ' v are the mass and the width of the relevant gauge bosons, and thus typically at the level of several per cent. Consequently, this approach is only reasonable if the experimental accuracy is correspondingly low. The error estimate of I'v/Mv is too optimistic in situations in which scales of order Fv, or smaller, are involved. This is, in particular, the case in threshold regions or if photons with energies of the order of /~v are emitted from the resonant gauge bosons. On the other hand, the quality of the pole approximation can be improved by applying appropriate invariant-mass cuts. Note that the double-pole approximation is particularly well-suited for the calculation of the radiative corrections to gauge-boson pair production, since the error resulting from the double-pole approximation is suppressed by an additional factor a/~r in this case. In this approximation the corrections can be classified into factorizable and nonfactorizable corrections [4,5]. The virtual factorizable corrections can be composed of the known corrections to on-shell W-pair production [6] and on-shell W decay [7], and the non-factorizable corrections have recently been calculated [ 8]. In contrast to the virtual factorizable corrections, the real factorizable corrections cannot be simply taken over from the on-shell processes. In the case of real photon emission the definition of the double-pole approximation is non-trivial if the energy of the final-state photon is of the order o f / ' v , since the resonances before and after photon emission are not well separated in phase space. A possible double-pole approximation of the O(a) corrections to four-lepton production has been discussed in Ref. [9].

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

35

In our calculation of e+e - ~ 4 f y we do not use the double-pole approximation since the full calculation is feasible with reasonable effort. Unlike the double-pole approximation, the full calculation is valid for arbitrary processes in the set e+e - ~ 4 f y . Moreover, when using the exact results for the real corrections in an O(a) calculation of 4 fermion-production processes, the reliability of possible approximations can be tested. Some results for e+e - --+ 4 f y with an observable photon already exist in the literature. In Refs. [ 10,11 ] the contributions to the matrix elements involving two resonant W bosons have been calculated and implemented into a Monte Carlo generator. This generator has been extended to include collinear bremsstrahlung [ 12] and used to discuss the effect of hard photons at LEP2 [ 13]. The complete cross section for the process e+e - --+ u a e - ~ e y has been discussed in Ref. [ 14]. In Ref. [15], the complete matrix elements for the processes e+e - ~ 4fy have been calculated using an iterative numerical algorithm without referring to Feynman diagrams. We are, however, interested in explicit analytical results on the amplitudes for various reasons. In particular, we want to have full control over the implementation of the finite width of the virtual vector bosons and to select single diagrams, such as the doubly resonant ones. So far no results for e+e - --+ 4 f y with e+e - pairs in the final state have been published. In order to perform the calculation as efficient as possible we have reduced all processes to a small number of generic contributions. For e+e - ~ 4 f , the calculation is similar to the one in Ref. [ 16], and the generic contributions correspond to individual Feynman diagrams. In the case of e+e - ~ 4 f y we have combined groups of diagrams in such a way that the resulting generic contributions can be classified in the same way as those for e+e - ---+4 f . As a consequence, the generic contributions are individually gauge-invariant with respect to the external photon. The number and the complexity of diagrams in the generic contributions for e+e - --~ 4fy has been reduced by using a nonlinear gauge-fixing condition for the W-boson field [ 17]. In this way, many cancellations between diagrams are avoided, without any further algebraic manipulations. Finally, for the helicity amplitudes corresponding to the generic contributions concise results have been obtained by using the Weyl-van der Waerden formalism (see Ref. [18] and references therein). After the matrix elements have been calculated, the finite widths of the resonant particles have to be introduced. We have done this in different ways and compared the different treatments for e+e - ~ 4 f and e+e - ---+4 f y . In particular, we have discussed a "complex-mass scheme", which preserves all Ward identities and is still rather simple to apply. The matrix elements to e+e - ~ 4 f and e+e - ~ 4 f y exhibit a complex peaking behaviour owing to propagators of massless particles and Breit-Wigner resonances, so that the integration over the 8- and ll-dimensional phase spaces, respectively, is not straightforward. In order to obtain numerically stable results, we adopt the multichannel integration method [ 16,19] and reduce the Monte Carlo error by the adaptive weight optimization procedure described in Ref. [20]. In the multi-channel approach, we define a suitable mapping of random numbers into phase space variables for each arising propagator structure. These variables are generated according to distributions

36

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

that approximate this specific peaking behaviour of the integrand. For e+e - ---, 4 f and e+e - ~ 4 f y we identify up to 128 and 928 channels, respectively, which necessitates an efficient and generic procedure for the phase-space generation. We wrote two independent Monte Carlo programs following the general strategy outlined above. They differ in the realization of a generic procedure for the construction of the phase-space generators. The paper is organized as follows: in Section 2 we describe the calculation of the helicity amplitudes for e+e - ~ 4 f and e+e - ~ 4 f y and list the complete results. The Monte Carlo programs are described in Section 3. In Section 4 the numerical results are discussed, and Section 5 contains a summary and an outlook.

2. Analytical results 2.1. N o t a t i o n and conventions

We consider reactions of the types

e+(p+, 0-+) + e - ( p _ , 0-_) ~ f l (kl, o'1) + f2(k2, 0"2) + f3(k3, 0"3) ÷ f4(k4, 0"4), (2.1) e+ (p+, 0"+) + e - ( p _ , 0"_) -~ fl (kl, 0"1) + fz(k2, 0-2) ÷ f3(k3,0-3)

÷f4(k4, 0-4) -}- '~(ks,/~).

(2.2)

The arguments label the momenta p + , ki and helicities o'i = ± 1 / 2 , a = ±1 of the corresponding particles. We often use only the signs to denote the helicities. The fermion masses are neglected everywhere. For the Feynman rules we use the conventions of Ref. [21]. In particular, all fields and momenta are incoming. It is convenient to use a non-linear gauge-fixing term [ 17] of the form

£~x = -

3 g W + + ie

1 2

A ~ - cw Z g ~ W+ _ iMw~b+ 2 Sw /

( a ~ Z ~ - M z X ) 2 - I(O~A/.)2 ~

(2.3)

where ~b~: and X are the would-be Goldstone bosons of the W ~: and Z fields, respectively. With this choice, the ~b+W~:A vertices vanish, and the bosonic couplings that are relevant for e+e - ---+4 f y read

A. Denner et aL/Nuclear Physics B 560 (1999) 33-65

37

= -iegvww Igup(k- - k+)~ - 2gu~kv, p + 2g~pkv,~] , W~,k_

= -2iegvww g~gp,r,

(2.4)

w# with V = A, Z, and the coupling factors

gA WW = 1,

Cw

gZWW -

Sw .

(2.5)

Note that the gauge-boson propagators have the same simple form as in the 't HooftFeynman gauge, i.e. they are proportional to the metric tensor gu~. This gauge choice eliminates some diagrams and simplifies others owing to the simpler structure of the photon-gauge-boson couplings. The vector-boson-fermion-fermion couplings have the usual form

~

3

=

ieyu Z

gv)ifj w~'

(2.6)

O"

fi where ~o± = ( 1 ~ Ys)/2. The corresponding coupling factors read

gafif~ = - Q i , o"

gzf, fi O"

~

- s w Qi ÷ I3v'i ~ ' - ' Cw

CwSw

l gwfif; - V/~Sw ~'~-' O"

--

-

-

(2.7)

where Qi and 13w,i= ± 1/2 denote the relative charge and the weak isospin of the fermion fi, respectively, and f[ is the weak-isospin partner of fi. The colour factor of a fermion fi is denoted by N),, i.e. N~epton = 1 and Nquark = 3.

2.2. Classification of final states for e+e - --~ 4 f The final states for e+e - ~ 4 f have already been classified in Refs. [ 16,22,23]. We introduce a classification that is very close to the one of Refs. [22,23]. It is based on the production mechanism, i.e. whether the reactions proceed via charged-current (CC),

A. Denneret al./NuclearPhysicsB 560 (1999)33-65

38

b) non-abelian graph

a) abelian graph

L (po, ~o) L(p~, O'c)

p

f.(p~, ~d)

fy(p:,cry)

~

fd

fd(Pa,a,~) L(p,,a,) I/(P:,a:)

fb(Pb,~rb) Fig. 1. Generic diagrams f6r e+e - --+ 4f. or neutral-current (NC) interactions, or via both interaction types. The classification can be performed by considering the quantum numbers of the final-state fermion pairs. In the following, f and F denote different fermions ( f 4: F ) that are neither electrons nor electron neutrinos ( f , F ~ e - , ~'e), and their weak-isospin partners are denoted by f~ and U , respectively. We find the following 11 classes of processes (in parenthesis the corresponding classification of Ref. [23] is given): (i) CC reactions: ( CC11 family), (a) e+e - ---* ff'FF", (b) e+e - ~ ~'ee+ff t, ( CC20 family), (CC20 family), (c) e+e - ~ ffle-~e, (ii) NC reactions: (NC32 family), (a) e+e - ~ f f F F , (NC4.16 family), (b) e+e - ---+ f f f f , (NC48 family), (c) e+e - ~ e - e + f f , (d) e+e - ~ e - e + e - e +, (NC4.36 family), (iii) Mixed C C / N C reactions: (mix43 family), (a) e+e - ~ f f f ' f ' , (NC21 family), (b) e+e - ~ UeOeff, (NC4.9 family), (c) e+e - - - ~ pel-'ePePe, ( mix56 family). (d) e+e - ~ ~'e~ee-e +, The radiation of an additional photon does not change this classification.

2.3. Generic diagrams and amplitudes for e+e- --* 4 f In order to explain and to illustrate our generic approach we first list the results for e+e - ---, 4 f . All these processes can be composed from only two generic diagrams, the abelian and non-abelian diagrams shown in Fig. 1. All external fermions f,,,....f are assumed to be incoming, and the momenta and helicities are denoted by Pa,...,f and o'a....,f, respectively. The helicity amplitudes of these diagrams are calculated within the Weyl-van der Waerden ( W v d W ) formalism following the conventions of Ref. [ 18] (see also references therein).

A. Denner et aL/Nuclear PhysicsB 560 (1999) 33-65

39

2.3.1. Leptonic and semi-leptonic final states We first treat purely leptonic and semi-leptonic gauge bosons in the generic graphs of Fig. 1 can trivially leads to a global factor N~ N}3, which the squared amplitude over the colour degrees of amplitudes are ~o-a

final states. In this case, none of the be a gluon, and the colour structure is equal to 1 or 3, after summing freedom. The results for the generic

,(-rb.o-c ,O-d ,O'e ,0" f /

tPa, Pb,Pc, Pd, Pe, PT )

v~ v2

= --4e4¢$o-,~, _o-.,, t$o-c , -tr-~o~ or-S ~o-b_ t,o-b_ t~o-d_ ~,O-: a e,-~V~ .°~ . oV~ o V lfaf~,

2f~fb

fcfd

2fefl

X PV~(Pc + Pd)Pv2 (Pc -}-Pf) O-~,o..... (Pb +Pc + p f ) 2 A2 (Pa,PO,Pc,Pd,Pc,Pu), ~z.~O'a ,O-b,o-c ,O'd, O'e ,O'f

vww

(2.8)

z

tPa, Pb, Pc, Pd, Pe, Pf)

= --4e460-, -o-~6o-¢ +80-a 80-e+~o-s (Qc - Qd)gvwwgvf~fbgwf,.LgwLfi xPv(p<, + Ph)Pw(Pc + Pd)PW(Pe + Pf) A3O-°(Pa,Pb, Pc,Pd,Pe,pf),

(2.9)

where the vector-boson propagators are abbreviated by 1

Pv(p)

p2 __M 2'

V=A,Z,W,g,

MA=Mg=O.

(2.10)

(The case of the gluon is included for later convenience.) The auxiliary functions A~0-~ are expressed in terms of W v d W spinor products, - ...... ~<' and A 3

A~ ++(Pa, Pb, Pc, Pd, Pe, Pf) = (PaPc)(PbPT)* ((PbPd)*(PbPe) + (PdPu)*(PepU)), A++- (Pa, Pb, Pc, Pd, Pe, P f) = A +++(Pa, Pb, Pc, Pd, PU, Pc), A+-+ (Pa, Pb, Pc, Pd, Pe, PT) = A+++(Pa, Pb, Pd, Pc, Pe, PU), a + - - ( p a , p b , P c , p d , p e , p f ) = A~ ++(P,,, Pb, Pd, Pc, PT, Pe), I'x-v,--Ck'c,--(,r¢l I A2 'O-"'O-a(Pa, Pb, Pc',Pd, Pe, Pf) = [,~2 tPa,Pb, Pc,Pd,Pe,pf))*,

(2.1 1)

a+(pa,Pb,Pc,Pd,Pe,pf) = (PbPd)*(PbPf)*(paPb)(PcPe) + (PbPd)* (PdPu)* (PaP~)(PcPd) + (PbPT)* (PdPu)* (PaPc)(PePU),

(2.12)

A3 (Pa, Pb, Pc, Pd, Pe, P f) = A + (Pb, Pa, Pc, Pd, Pc, Pf ) . The spinor products are defined by

(pq) = eASPAqB = 2 pv/fi'0-~ e - i 6 ' COS

sin ~- -- e -kbq cos

sin

,

(2.13)

where PA, qA are the associated momentum spinors for the momenta P~ = Po( 1, sin Op cos ~bp, sin Op sin ~bp, cos Op), q~ = Po ( 1, sin Oqcos ~bq, sin Oqsin ~bq, cos Oq).

(2.14)

40

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

Incoming fermions are turned into outgoing ones by crossing, which is performed by inverting the corresponding fermion momenta and helicities. If the generic functions are called with negative momenta - p , - q , it is understood that only the complex conjugate spinor products get the corresponding sign change. We illustrate this by simple examples: a ( p , q ) = (pq) = a ( p , - q )

= a(-p,q)

= a(-p,-q),

(2.15)

B ( p , q) = (pq) * = - B ( p , - q ) = - B( - p , q) = B ( - p , - q ) .

We have checked the results for the generic diagrams against those of Ref. [ 16] and found agreement. Using the results for the generic diagrams of Fig. 1, the helicity amplitudes for all possible processes involving six external fermions can be built up. It is convenient to construct first the amplitudes for the process types CC(a) and NC(a) (see Section 2.2) in terms of the generic functions (2.8) and (2.9), because these amplitudes are the basic subamplitudes of the other channels. The full amplitude for each process type can be built up from those subamplitudes by appropriate substitutions and linear combinations. We first list the helicity amplitudes for the CC processes: O" F, Or

Adcc a

,O*1,0"2 ,or3 ,O'4

( p +, p - , kl , k2, k3, k4)

0-+ Or-- , - - 0 - 1 , - - 0 - 2 - - 0 - ~ - - 0 - 4

= Mww + Z

(p+, p - , - k l , -k2, -k3, -k4)

["/Mv~v~v-'-Or~ - o - 2 - o r ~ ' - o - 4 ( P + ' P - ' - k l ' - k 2 ' - k 3 ' - k 4 )

V=y,Z --0-1 ,--or2~0-{ , 0 - - - - - 0 " 3 , - - 0 " 4

+.A4VW

'

"

- - o r 3 , - - 0 - 4 0-+ Or-- - - 0 -

--°-2

+Mvw

- - 0 - 1 , - - 0-2 , - - Or3 , - - 0-4 ,or+ ,or --

+2Mwv --03

--0-4,--orl

- - 0 - 2 0-+ Or

-}-Jk4WV . . . .

( - k l , -k2, p+, p - , -k3, -k4) ( -k3, -k4, p+, p - , - k l , -k2) ( - k l , -k2, -k3, -k4, p+, p - ) (-k3, -k4, - k l , - k 2 , p + , p - )

]

,

(2.16)

J 0-4 ,or

A//ccb

,O'l ,or2,or3,0-4

(p+, p - , kl, k2, k3, k4)

0-+ ,or -- ,O-I ,0-2 ,O'3 ,O-4

= .A//cca

-

( p + , p - , kl, k2, k3, k4)

0-4 , - - 0-2 ,orl , - - 0 - - - ,or3 ,O'4

-224cc a

O-f ,or_ ,orl ,or2,0-3,or4

A.4cc~

(p+, -k2, kl, - p - , k3, k4),

( p +, p - , kl , k2, k3, k4)

0-t- , 0 - - - ,or1,0-2,0-3,O-4

= Jk4CCa

"

(p+, p - , kl, k2, k3, k4)

- - 0-3,0- -- ,orl , a ' 2 , - - Or+ ,o-4

--.A-4CCa

(-k3, p - , kl, k2, - p + , k4).

The ones for the NC processes are given by O ' + , 0 - _ ,0-1 O'2,0-3,0-4

"/~NCa =

( p +, p - , kl , k2, k3, k4) ( p + , p - , - k l , -k2, -k3, -k4)

.A4v, v2

V~,v2=7,z I[_M0-t

,0---,--or3,--or4,--O'l

v,v~

(2.17)

"

,--0- 2 /

t P + , P - , - k 3 , - k 4 , - - k l , -k2)

(2.18)

41

A. Denner et al./Nuclear Physics B 560 (1999) 33-65 2[-d~--¢rl

,--0-2,0-~,0---,--O-3,--O'4 (

v~112

t - k l , -k2, p+, p - , -k3, -k4)

--0-3, -- 0-4,0- ~ , 0 -

+Adv~ v2 ~_/~--¢rl,

v, v2

-q'- .A/~ - 0 - 3 , - 0 - 4 , - 0 - 1

(rt ,0-

,0-1,cr2,0-3,0-4

-

--'~NCa "/~NCc

,0-1,0"2,0-3,0-4

= J~NCa

(p+, p - , k3, k2, kl, k4),

---A/[NCa 0-+,0--

---- ,MNC a

(2.20)

( - k l , p - , - p + , k2, k3, k4),

(2.21)

( p +, p - , kl , k2, k3, k4)

0-1 °-2 0"~ 0-4

( p + , p - , kl , k2, k3, k4)

0-+ .0--- ,0-~,0-2 0-1 0-4

--J~NCa

(2.19)

(p+,p-, kl, k2, k3, k4)

--0-1,0--- ,--0-¢ O'2 0"3,0-4

A-4NCa

]

( p + , p - , kl, k2, k3, k4)

0-~,0--- 0-1,0-2,0-q 0-4

0 - t ,¢/--- ,0-1 , ° ' 2 0-'~ 0 - 4

)

(p+, p - , kl, k2, ks, k4)

0- ~ ~0-- ,0-3,0-2 ,O-1,0"4

¢r~ ~0-

(-k3, - k 4 , - k t , - k 2 , p + , p -

( p +, p - , k l , k2, k3, k4)

0-+ ,0--- ,0-1,0-2,0-3 ,O-4

= -A/[NCa

( - k l , -k2, -k3, - k 4 , p + , p - )

,-0-2,0-4,0---

v, v2

.Aqycb

,-0-,,-,,-2 ( - k s , - k 4 , p+, p - , - k l , - k 2 )

- - 0 - 2 , - - 0-'~, - - 0 - 4 , 0 - F , 0 - - -

" ,

( p + , p - , k3, k2, kl, k4)

k --dvtNc a L - - I , P - , - P + , k2, k3, k4) , ~-,,~,0-- ,0-,,0-2.-0-, ,0-4{ k --dvlNC;~ L- 3, P - , kl, k2, - p + , k4) A X - 6 r ] 0 - - ,--O"t 0-2,0"3,0e4 [

--0-1,0"_ ,0-3,0-2, - - 0 " t ,0"4

~-d~NC a

( - k l , p - , k3, k2, - p + , k4)

- - 0 3 , 0 - - , - - 0-+ ,0-2,0-1 71_ J ~ N C a ,0-4(-k3,

p - , - p + , k2, kl, k4).

(2.22)

Finally, the helicity amplitudes for reactions of mixed C C / N C type read 0 - ¢ ,(/"

,0-1 ,~'T2 , O ' 3 , 0 - 4

MCC/NCa

(p+, p--, kl, k2, k3, k4)

6/'+,0-_ ,0-1 ,O-2,0-3,0-4

= A-4NCa

0-+ ,d'T_

-A4cc a 0-t ,0-

,O-1,O'~,0-3,O'4

J~CC/NCb

O" I , 0 - _ . o -

---- .A/[NC a

(p+,p-,

,0-1,0-4,0-3,0"2

kl, k2, k3, k4)

(p+,p-, kl, k4, k3, k2),

(p+, p - , kl, k2, k3, k4)

I 0"2,0" 3 0"4

(p+, p - , kl, k2, k3, k4)

.... -0--.-0-~ ,0"z(-k3, -k4, kl - p - , - p + , k2), - A 4 c -0-~,-0-4 c ~ 0- t , 0 - - - ,0-1,0-2,0-3,0-4

.A/[cC/NCc

(p+,p-,

0" 4 ,(1"_ ,0-3,0"2,0-1 ,(-re

-AdNC a

(2.24)

( p +, p - , kl , k2, k3, k4)

0-1 ,0--- ,0-1,0-2,0-3,0"4

= -A/INCa

(2.23)

kl, k2, k3, k4)

(p+, p - , k3, k2, kl, k4)

..... -0-2,- . . . . 4,0-3,-0-- ( _ k l - k 2 , k3, - p - , - p + , k4) --.]k/[CC a - - 0 - 1 , - - 0-4 , - - 0- ~ , O - 2 , 0 - ~ , - 0 -

+.A'4CCa

--O-3 ~-- 0- 2 , -- 0- ¢ ,0-4,0-I , - - 0 - -

+J~cca

--0-~,--0"4,--0-F,0"2,0-1,--0-_

-.Adcc a

( - k l , -k4, k3, - p - , - p + , k2) (-k3, -k2, kl, - p - , - p + , k4) (-k3, -k4, kl, - p - , - p + , k2),

(2.25)

42

A. Denner et al./Nuclear Physics B 560 (1999) 33-65 O'~ ,O-_ ,O-] ,(/'2,O'3,O'4

'fi'~CC/NCd

"

( p + , p - , kl, k2, k3, k4)

O"~ ,0-_ ,0-1,0"2.0"3,0"4

= "~NCa

"

( p + , p - , kl, k2, k3, k4)

--0-3,0--- .0"1 ,O-2,--O-t ,0"4

-A-4yc~

0"+ ,O-_ ,O"1,O'4 ,O'3 ,O'2

--J~CCa

0-d ,--O-4,0-1 , - - ( r

+.M¢c a

( - k 3 , p - , kl, k2, - p + , k4)

(p+, p - , kl, k4, k3, k2)

,0-3 0"2

"

--0-3,O--- ,O-1,0"4.--0" ÷ ,O'2

+J~CCa

(p+, -k4, kl, - p - , k3, k2)

(-k3, p - , kl, k4, - p + , k2)

--O'3, --O'4,0-1 ,--0--- ,--0- ~ ,O'2

--.h4CCa

(-k3, -k4, kl, - p - , - p + , k2).

(2.26)

The relative signs between contributions of the basic subamplitudes .MCCa and -/~NCa to the full matrix elements account for the sign changes resulting from interchanging external fermion lines. For the CC reactions, the amplitudes .h4CCa are the smallest gauge-invariant subset of diagrams [24]. In the case of NC reactions, the amplitudes ./~NCa a r e composed of three separately gauge-invariant subamplitudes consisting of the first two lines, the two lines in the middle, and the last two lines of (2.19). 2.3.2. Hadronic final states

Next we inspect purely hadronic final states, i.e. the cases where all final-state fermions f i are quarks. This concerns only the channels CC(a), NC(a), NC(b), and CC/NC(a)

given in Section 2.2. The colour structure of the quarks leads to two kinds of modifications. Firstly, the summation of the squared amplitudes over the colour degrees of freedom can become non-trivial, and secondly, the possibility of virtual-gluon exchange between the quarks has to be taken into account. More precisely, there are diagrams of type (a) in Fig. l in which one of the gauge bosons VI,2 is a gluon. The other gauge boson of Vi,2 can only be a photon or Z boson, since this boson has to couple to the incoming e+e - pair. Consequently, there is an impact of gluon-exchange diagrams only for the channels NC(a), NC(b), and CC/NC(a), but not for CC(a). This can be easily seen by inspecting the generic diagrams in Fig. 1: the presence of a gluon exchange requires two quark-antiquark pairs qO in the final state. We first inspect the colour structure of the purely electroweak diagrams. Since the colour structure of each diagram contributing to the basic channels CC(a) and NC(a) is the same, the corresponding amplitudes factorize into a simple colour part and the "colour-singlet amplitudes" .MCCa and -AANca, given in (2.16) and (2.19), respectively. The amplitudes for NC(b) and CC/NC(a) are composed from the ones of CC(a) and NC(a) in a way that is analogous to the singlet case, but now the colour indices ci of the quarks fi have to be taken into account. Indicating the electroweak amplitudes for fully hadronic final states by "had, ew", and writing colour indices explicitly, we get o- ~ ,O- -- ,O- 1,0-2 ,o-3 ,O-4

/

ff~,[ CCa,had,ew,cl,c2,c3,o ~,P+' P - ' kl , k2, k3, k4) =

O- I d'T-- ,0-1 ,O-2,0-3 ,O-4 " (p+,

./k/Ice a

O- ~ ,O"

,O-1,0-2 ,O-3 ,O'4

p-,

kl,

k2, k3, k4)

"A//NCa,had,ew,c~,c2,~'3,c4( P +' P - ' kl , k2, k3, k4)

(~clc2(~c3c.,

(2.27)

A. Denner et aL/Nuclear Physics B 560 (1999) 33-65 (T+ ,(3"

= .A4NCa

0-1,0-2,0"~ ,0-4

= AdNCa

(7 ~ ,0-

--A-4NCa ./~

i'f ~ ,O-

kl, k2, k3, k4)

( p + , p - , kl, k2, k3, k4) C~clc2C~¢3c4

,O'~ ,O"2 ,O- 1 ,CT4

"

JT1 ,¢.r 2,0-3 ,(f4

(2,28)

( p + , p - , kl, k2, k3, k4) C~c~c2Sc~c4,

jVfo-' ,~-,~,o-2,~,3,0-, NCb,had,ew.Q ,cz,c3,Ca( p + , p - , O-F,O'-- O"1,O'2 0-%O'4

43

(p+, p - , k3, k2, kt, k4)

/

(2.29)

(~C3C2(~C1C4 ,

.

CC/NCa,had,ew,c~,c2 c~ c4 ~P+' P - ' kl, k2, k3 k4) (r+ ,0-_ ,0" I ,(T 2 (7%O-4

= .A,'INCa

¢T~,0-

-A//cc a

(p+, p - , kl, k2, k3, k4) C~c,c2(~c~c.

,O-I 0-4dTq,¢T2

(p+, p - , kl, k4, k3, k2)

(2.30)

(~C1C4~C3C2 .

In the calculation of the gluon-exchange diagrams we can also make use of the "colour-singlet" result (2.8) for the generic diagram (a) of Fig. 1, after splitting off the colour structure appropriately. Since each of these diagrams involves exactly one internal gluon, exchanged by the two quark lines, the corresponding matrix elements can be deduced in a simple way from the diagrams in which the gluon is replaced by a photon. The gluon-exchange contributions to the channels NC(b) and CC/NC(a) can again be composed from the ones for NC(a). Making use of the auxiliary function j ~ g . . . . . . . 1,,r: ,o-;,~r4(p+, P - , kl, k2, k3, k4) _

g2 ~ [./~),rl-tr2.o-~,o" . . . . . . -,r4(__kl,__k2,p+,p_ QIQ3e2 v=r,z --O-~ --O-4 0-+ O"

--0-1 --(1"2

+.A/Ivy

_k3 _ k 4 )

( - k 3 , - k 4 , p+, p - , - k l , - k 2 )

, ,o-- ( _ k l , - k 2 , - k 3 , - k 4 , p +, p - ) + 3 4 ~-0-1,-0-2,-~r3,-~4,cr v --0-~ --0-4 --O-i ,--¢)'2,0"+ i T _

+,A-'ITv "

]

( - k 3 , - k 4 , - k l , - k 2 , p+, p - ) ,

(2.31 )

where g~ = ~ is the strong gauge coupling, the matrix elements involving gluon exchange explicitly read ff~/~ 0- F ,(T -- ,0- 1,0"2 ,O-3 ,O'4 / ,C2,C3,C 4

NCa,had,gluon,Q

~P+' P-- ' kl , k2, k3, k4)

(T~ , ( T _ ,O-1,0-2,0-%0-,~

= .A//g

a

(p+, p - , kl, k2, k3, k4) 1 Aac,,,~ a~3c,,

(3" I ,O--- ,0-1 ,O'2 .O-3 ,(f4 ,¢2,C3,C4 ( P - b '

"/~/~NCb,had,gluon,cl

(2.32)

P--' kl, k2, k3, k4)

(Yt ,0--- ,O-1,0-2,O-3, IT,I

(2

= A-48 ( p + , p - , kl, k2, k3, k4) 1 "~c,c2/~a C3C4 ,~.40-~,c-r . . . . . 2,o-1,o"4/ t. t. I. t.'~l/~a /~a --./Vtg " ~//9+, p _ , it3, ~2, ttl, t¢4/ ~ C 3 C 2 6.112.4,

(2.33)

"AdCCiNC'a,~ga~,2g"~o'n~;..c2,c3,c4(P +' P - ' k, , k2, k3, k4 ) (T t

= Adg

(f_

~TI,O- 2 0"~,0" 4

( p + , p - , kl, k2, k3, k4) 1 ,taCl C2 ,~aC3C4 " -

(2.34)

The colour structure is easily evaluated by making use of the completeness relation a a = --g(~i.j(~kl 2 Ai.ihkl -]- 2~il~i k for the Gell-Mann matrices a~.. The complete matrix elements for the fully hadronic channels result from the sum of the purely electroweak and the gluon-exchange contributions,

44

A. Denner et a l . / N u c l e a r Physics B 560 (1999) 33-65 MO'+,fZ

,O'1,O'2,O'3,O"4 -- ~.~O'+,O'_,O'1,O'2,O'3,O- 4 O-~ ,O'_,O'1,O'2,O'3~O" 4 -...,had,ew,cl ,c2,c3,c4 ÷ J~...,had,gluon,c,,c2,c3,c4"

.... had,cq,c=,c3,c4

(2.35)

The gluon-exchange contributions are separately gauge-invariant. For clarity, we explicitly write down the colour-summed squared matrix elements for the fully hadronic channels. Abbreviating A4¢? . . . . ~.,,~b.... ~d (p +, p _ , ka, kb, kc, kd ) by A4... (a, b, c, d) we obtain I.AAcca,ha~( 1,2, 3,4)12

"=-- 91Mcca(

1,2, 3,4)[2,

(2.36)

colour

Z

I'MlNCa'had(l'2' 3' 4) ]Z = 9I'A4NCa( 1' 2' 3' 4) t2 + 2IA4g(I' 2' 3' 4) [2'

(2.37)

colour

IMNcb,had( 1,2, 3,4)12 = 9I.A4NCa(1,2, 3, 4)12 + 9IAdNCa(3, 2, 1,4)12 colour

- 6 Re {.L4NCa( 1,2, 3,4).A4~ca (3, 2, 1,4) } + 2).L4~( l, 2, 3, 4)12 +21~4g(3, 2, 1,4)12 + 4~ Re (~.4g ( 1,2, 3, 4)2~4~(3, 2, 1,4)} - 8 Re {.A.4NCa(1,2, 3,4).L4~ (3, 2, 1,4)} --8 Re {.L4NCa(3, 2, 1,4).L4~ ( 1,2, 3, 4)},

(2.38)

]J~CC/NCa,had ( 1,2, 3,4)]2 = 9[JL4NCa ( 1,2, 3,4)12 + 9[.A'lCCa( l, 4, 3, 2)12 colour

- 6 Re {A/INCa( 1,2, 3, 4).M~Ca( 1,4, 3, 2) } +2lAAg ( 1,2, 3, 4)12 - 8 Re {3Acca ( 1,4, 3, 2)Adg( 1,2, 3,4)}.

(2.39)

Owing to the colour structure of the diagrams, a non-zero interference between purely electroweak and gluon-exchange contributions is only possible if the four final-state fermions can be combined into one single closed fermion line in the squared diagram. This implies that fermion pairs must couple to different resonances in the electroweak and the gluon-exchange diagrams, leading to a global suppression of such interference effects in the phase-space integration (see Section 4.5). 2.4. Generic functions and amplitudes f o r e+e - --+ 4 f y

The generic functions for e+e - -~ 4 f 7 can be constructed in a similar way. The idea is to combine the contributions of all those graphs to one generic function that reduce to the same graph after removing the radiated photon. These combined contributions to e+e - ---. 4 f y are classified in the same way as the diagrams for the corresponding process e+e - ~ 4f, i.e. the graphs of Fig. 1 also represent the generic functions for e+e - ~ 4f7. Finally, all amplitudes for e+e - --+ 4 f y can again be constructed from only two generic functions. Note that the number of individual Feynman diagrams ranges between 14 and 1008 for the various processes. We note that the generic functions can in fact be used to construct the amplitudes for all processes involving exactly six external fermions and one external photon, such as e e --~ 4 f y and ey ~ 5f.

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

45

As a virtue of this approach, the so-defined generic functions fulfill the QED Ward identity for the external photon, i.e. replacing the photon polarization vector by the photon momentum yields zero for each generic function. This is simply a consequence of electromagnetic charge conservation. Consequently, in the actual calculation in the WvdW formalism the gauge spinor of the photon drops out in each contribution separately. Assuming the external fermions as incoming and the photon as outgoing, the generic functions read .A/~(%a,(fh,O'c,O-d,(l"e,O'f

v~v2

,~ t'~

i~1

t ~a, ~b, Qc, Qd, Qe, Q u , pc,, Pb, Pc., Pd, Pe, P f , k)

= -4v~e58,~ _,~,(3,~ _~.~6,~ _o.I g~Lfg;'fj,,g;lfcfdg;ffe,] a,

~

0-

Cr

x ,a2 . . . .

(r

~

e,

J

~

'

-

'

A z ~

ttda, ~db, Q~, Qa, Q,, Q f, Pa, PO, Pc, Pa, Pe, P f, k),

'a,(%b~fTc,(fd.O'e~()'/./~ d ~

vww

c,

(2.40)

f~

t~a, ~o, Qc, Qd, Qe, Qf, Pa,ph,pc,Pd,Pe,pf, k ) O"b

--

--

x AO-,,,a 3 (Qa, QI, Qc, Qd, Qe,Qf,p~,pb,pc,Pd,pe,pf, k),

(2.41)

with the auxiliary functions

A++++ (Qa, Qb, Qc, Qd, Qc, QT,Pa,Pt,,Pc,Pd,p¢,Pu, k) -{PaPc)

IPvI (Pc + Pd)Pv2 (Pe 4- pf)

[ (PbPf )* ( Qc - Qd (PaP~) x [ (pak) (Pb +Pc 4-pf)2 (pck) ((PbPa)*(PbP~) 4- (PaPf)*(PePf)) o f - Qe

(PaPa)((PaPd)*(PaPc) 4- (PcPa)*(PcPe)))

(pa "~-pc q--pd) 2 (pek)

Ob ( (PaPa)" (PaPc) 4- (PcPa)* (PcPe) ) ((PbPf)*(PaPb) -- (pfk)* (pak) ) (Pa 4- Pc" @ Pd) 2 (pak) (pbk)

( Qa + Qc -- Qa ) (PbPT) * (PcPa )* (PaPc) ( (pok )* (PbPe) -- (p f k ) * (PePT) ) ] ~ a -+ ~c "+ ~ P - ~ ~P f )-~a~ QJ - (Qc -- Qd)2(pd" k)Pvt (Pc + Pa) (Pb + Pe + pf)2 PVl (Pc. + Pa -- k)Pv2 (Pc + PT) (PbPT)* × [ (PcPd)( (P~,Pa)* (PbPe) + (PaPT)* (PePU)) + {pck) ( (pbk)* (PbPc) -- (pfk)* (PcPT)) ]/{ (pck)(pak) } 4 Qf - (Qe -- Q f ) 2 ( p f . k)Pv2 (Pe + Pf) Pv, (Pc + Pd)Pv: (Pe + PU -- k) (Pa 4-Pc 4- Pd) 2

X

( (PhPf)* (P,~Pf) + (pbk)* (pek)) ( (PaPd)* (PaPc) + (PcPd)* (PcPe)) ] (pek)(pfk) f'

A ++-+ (Qa, Qb, Qc, Qa, Qe, Qu, Pa, Pb, Pc, Pd, Pe, P f , k) -- Af+++ (Qa, Q~',,Qc, Qd, - Q f , -Qe,Pa,Pb,Pc,Pa,PT,Pe, k ),

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

46

A-~-++(Qa, Qb, Qc, Qd, Qe, Qf,Pa,Pb,Pc,Pd,Pe,Pf, k) = A ++++(Qa, Qb, -Qcl, -Qc, Qe, Q f, Po, Pb, Pd, Pc, Pe, Pf, k), A+--+ (Qa, Qb, Qc, Qa, Qe, Qf , Pa,Pb,Pc,PcI,Pe,PI, k) = A++++(Qa, Qb, -Qa, -Qc, - Q f , -Qe,Pa,Pb, Pd,Pc,Pf,Pe, k), A2 +++(Qa, Qb, Qc, Qd, Qe, QT, Pa, Pb, Pc, Pa, Pe, PT, k) a + + + + t z'~

= 'a2

t~b, Q a , - Q e , - Q f , - Q c , - Q a , p b , p a , P e , p f , p c , p a , k ) ,

A~ +- +( Qa, Q~, Qc, Q~, Qe, Q f , Pa, Pb, Pc, Pd, Pe, PT, k) = A ++++(Qb, Qa, QT, Qe, -Qc, -Qa, Pb, Pa, PT, Pe, Pc, Pa, k), A ~ - ++(Q., Qb, Qc, Q,t, Qe, Qu, pa, pb, pe, p,t, Pc, Pf, k) = A ++++( Qb, Qa, -Qe, - Q f, Od, Qc, Pb, Pa, Pe, Pf, Pd, Pc, k), A 2 - - + ( Qa, Qb, Qc, Qd, Qe, Q f, Pa, Pb, Pc, Pd, Pe, P f, k) = A ++++( Qb, Qa, Qf, Qe, Qa, Qc, Pb, Pa, Pf, Pe, Pa, Pc, k), a2,,,o',,,o-,,,- ( Qa, Qb, Qc, Qd, Qe, Q f, Pa, Pb, Pc, Pal,Pe, P f, k) = ( a 2 m,,-,~.-o-,,,+ ( Qa, Qb, Qc, Qd, Qe,

Pd, Pe, Pf, k)

). Pvl.2 (P) ~P~,2 (p)'

Ol, Pc,, Pb, Pc, (2.42)

and

A~ + (Q~, Qb, Qc, QJ, Q~, Q f, p~, p~, pc, pa, Pe, P f, k) = Pv(Pa + Pb)PW(Pc + Pa)Pw(Pe + Pf)

( Qc - Qa )(PcPe) (pck)(pek}

X ((PbPd}*(PbPf)* (PaPb)(PcPe) + (PbPd)* (PdPf)*(PaPe} (PcPd) +(PbPT}* (PdPT)* (PaPc) (PePT}) Qb +Pv(Pa + Pb -- k )Pw(Pc + Pd)PW(Pe + Pf ) (pak)(pbk) x { (PdPf}* [(PaPe)(PcPd}( (PbPd}* (PaPb) -- (p~k}* (pak)) + (PaPc)(PePT)((PbPf}* (PaPb} -- (pfk)*(pak))] .~_(PcPe)((PbPd) (PaPb) -- (pdk)*(pak))((pbpf)*(paPb) *

I

(pfk}*(pak)) f

+Pv(pa + Pb)PW(Pc + P~t -- k )Pw(Pe + pf ) Qa - (Qc - Qd)2(pd " k )Pw(Pc +Pal) X (pck)(pak) x { (PbPf)* [ {PaPc)(PePy) ( (PaPf)* {PcPa} - (pfk}* (pck)) + (PoPe)(PaPb) ( (PbPa}* (PcP~*) + IPhk}* (pck}) ] + (P,,P~) ( (PdPU)* (PcPd) - (pfk)* (pck)) ( (PbP~t)*(PcPa) + (pbk}* (pck))

47

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

+Pv(Pa + Pb)Pw(Pc + Pa)Pw(Pe + Pf - k) Qf + (Of - Oe)2(pf. k)ew(Pe + p f ) x

(pek)(pfk)

× { (PbP~)* ](P~Pe)(PaPb)( (PbPu)* (P~Pu) + (pbk)* (p~k}) + (P~,P~)(PcPd)( (PdPu)* (P~Pu) + (pdk)*(p~k)) ] + (PaPc)( (PbPT)* (PePU} + (pbk)* (pek)) ((PdPT)*(P~PT) + (pdk)* (p~k)) }, a3+ ( Qa, Qb, Qc, Qd, Qe, Qf , Pa, Pb, Pe, Pd, Pe, PU, k) = Af+(-Qb, -Qa, Qc, Qd, Qe, QU,Pb,Pa,Pc,Pd, Pe, PT, k), " a

3

,

-

-

(Qa, Qb, Qc, QJ,Qe, Qf,pa,pt,,pc,pa,pe,pf, k)

(2.43)

= t/A \ 3...... + (Oa, Q ~ , - Q a , - Q c , - O f , - Q e , p a , p b , p a , p c , p f , p e , k ) , * , /

IP~w(p)---+P~w(p)

The replacements Pv ~ PC after the complex conjugation in the last lines of (2.42) and (2.44) ensure that the vector-boson propagators remain unaffected. Note that the vector-boson masses do not enter explicitly in the above results, but only via Pv. In gauges such as the 't Hooft-Feynman or the unitary gauge this feature is obtained only after combining different Feynman graphs for e - e - ~ 4 f y ; in the non-linear gauge (2.3) this is the case diagram by diagram. The helicity amplitudes for e+e - --~ 4 f y follow from the generic functions .AAv.v2 and .A4vww of (2.41) in exactly the same way as described in Section 2.3 for e+e - --+ 4 f . This holds also for the gluon-exchange matrix elements and for the colour factors. Moreover, the classification of gauge-invariant sets of diagrams for e+e - -~ 4 f immediately yields such sets for e+e - --, 4 f y , if the additional photon is attached to all graphs of a set in all possible ways. We have checked analytically that the electromagnetic Ward identity for the external photon is fulfilled for each generic contribution separately. In addition, we have numerically compared the amplitudes for all processes with amplitudes generated by MADGRAPH [25] for zero width of the vector bosons and found complete agreement. We could not compare our results with Madgraph for finite width, because Madgraph uses the unitary gauge for massive vector-boson propagators and the 't Hooft-Feynman gauge for the photon propagators, while we are using the non-linear gauge (2.3). Therefore, the matrix elements differ after introduction of finite vector-boson widths. While the calculation with Madgraph is fully automated, in our calculation we have full control over the matrix element and can, in particular, investigate various implementations of the finite width. A comparison of our results with those of Refs. [10,11], which include only the matrix elements that involve two resonant W bosons, immediately reveals the virtues of our generic approach.

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

48

2.5. Implementation of finite gauge-boson widths We have implemented the finite widths of the W and Z bosons in different ways: (i) fixed width in all propagators: Pv(P) = [p2 _ M~ + i M v F v ] - l , (ii) running width in time-like propagators:

Pv(p) = [p2 _ M~ + ip2(Fv/Mv)O(p 2) ] -1, (iii) complex-mass scheme: complex gauge-boson masses everywhere, i.e. [ M 2 v iMvFv]-l instead of My in the propagators and in the couplings. This results, in particular, in a constant width in all propagators,

Pv(P) = [p2 _ M 2 + i M v - r v ] - l ,

(2.44)

and in a complex weak mixing angle: 2

1

Cw :

2 -- Sw :

M~v - iMw Fw M~ - iMzFz

(2.45)

The virtues and drawbacks of the first two schemes have been discussed in Ref. [26]. Both violate SU(2) gauge invariance, the running width also U( 1 ) gauge invariance. The complex-mass scheme obeys all Ward identities i and thus gives a consistent description of the finite-width effects in any tree-level calculation. While the complex-mass scheme works in general, it is particularly simple for e+e - ~ 4 f y in the non-linear gauge (2.3). In this case, no couplings involving explicit gauge-boson masses appear, and it is sufficient to introduce the finite gauge-boson widths in the propagators [cf. (2.44)] and to introduce the complex weak mixing angle (2.45) in the couplings. We note that a generalization of this scheme to higher orders requires to introduce complex mass counterterms in order to compensate for the complex masses in the propagators [28]. We did not consider the fermion-loop scheme [26,29], which is also fully consistent for lowestorder predictions, since it requires the calculation of fermionic one-loop corrections to e+e - ~ 4 f y which is beyond the scope of this work.

3. The Monte Carlo programs

The cross section for e+e - ~ 4 f ( y ) is given by

do'=

2s

d4ki6(k2i)O(k°i)

6(4)

ki

P+ + P - i=1

x I.Ad(p+, p_, kl . . . . . k,) [2,

(3.1)

I In the c o n t e x t o f e l e c t r o m a g n e t i c g a u g e invariance, the i n t r o d u c t i o n o f a c o m p l e x m a s s h a s b e e n p r o p o s e d in Ref. [ 2 7 ]

A. Denner et aL/Nuclear Physics B 560 (1999) 33-65

49

where n = 4, 5 is the number of outgoing particles. The helicity amplitudes .A4 for e+e - ~ 4 f ( y ) have been calculated in Sections 2.3 and 2.4, respectively. The phasespace integration is performed with the help of a Monte Carlo technique, since the Monte Carlo method allows us to calculate a variety of observables simultaneously and to easily implement cuts in order to account for the experimental situation. The helicity anaplitudes in (3.1) exhibit a complicated peaking behaviour in different regions of the integration domain. In order to obtain a numerically stable result and to reduce the Monte Carlo integration error we use a multi-channel Monte Carlo method [ 16,19], which is briefly outlined in the following. Before turning to the multi-channel method, we consider the treatment of a single channel. We choose a suitable set ~ of 3n - 4 phase-space variables to describe a point in phase space, and determine the corresponding physical region V and the relation ki(~I~ ) between the phase-space variables q~ and the momenta kl . . . . k,. The phasespace integration of (3.1) reads

l,,=f d~=f O~p(ki(~))f ( k i ( ~ ) ) ,

(3.2)

V

f(ki(~))

-

(27r)4-3" [M ( p + , p - , k 1 ( ~ ) , 2s

,k,,(q~))l 2 . . . .

where p is the phase-space density. For the random generation of the events, we further transform the integration variables ~ to 3n - 4 new variables r = (ri) with a hypercube as integration domain: q~ = h ( r ) with 0 ~< ri ~ 1. We obtain 1

I. =

d~p(ki(~))

f(ki(~))=

V

drg(ki(h(r))), 0

where g is the probability density of events generated in phase space, defined by 1 g (ki(O)) - p (ki(O))

dh(r) Or r = h - ~ .

(3.4)

If f varies strongly, the efficiency of the Monte Carlo method can be considerably enhanced by choosing the mapping of random numbers r into q~ in such a way that the resulting density g mimics the behaviour of Ifl. For this importance sampling, the choice of q~ is guided by the peaking structure of f , which is determined by the propagators in a characteristic Feynman diagram. We choose the variables • in such a way that the invariants corresponding to the propagators are included. Accordingly, we decompose the n-particle final state into 2 2 scattering processes with subsequent 1 --~ 2 decays. The variables q~ consist of Lorentz invariants si, ti, defined as the squares of time- and space-like momenta, respectively, and of polar and azimuthal angles Oi, fbi, defined in appropriate frames. A detailed description of the parameterization of an n-particle phase space in terms of invariants and angles can be found in Ref. [30]. The parameterization of the invariants si, ti in

50

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

= h ( r ) is chosen in such a way that the propagator structure of the function f is compensated by a similar behaviour in the density g. More precisely, if f contains BreitWigner resonances or distributions like sZ ~, which are relevant for massless propagators, appropriate parameterizations of si are given by: (i) Breit-Wigner resonances: si = M 2 + M v F v tan[yl + (Y2 - yl )ri] with Yl,2 = arctan

(3.5)

( Smin,rnax = M2 ~

\

Mvrv

J

(ii) propagators of massless particles:

v 4= 1 : u= 1 :

1 --u 1 --u si = [Srnax ri + SrO, n (1 -- ri) ]1/(1-")

si=exp[ln(smax)ri+ln(srnin)(1

--ri)].

(3.6)

The parameter l, can be tuned to optimize the Monte Carlo integration and should be chosen > 1. The naive expectation v = 2 is not necessarily the best choice, because the propagator poles of the differential cross section are partly cancelled in the collinear limit. The remaining variables in q) = h ( r ) , i.e, those for which f is expected not to exhibit a peaking behaviour, are generated as follows: Si = Smaxri -t-

Stain (1 -- r i ) ,

q~i = 2"rrri,

COS Oi = 2ri -- 1.

(3.7)

The absolute values of the invariants ti are generated in the same way as si. The resulting density g of events in phase space is obtained as the product of the corresponding Jacobians, as given in (3.4). In the appendix, we provide an explicit example for an event generation with a specific choice of mappings ki(q~) and h ( r ) , and for the calculation of the corresponding density g. The differential cross sections of the processes e+e - ---, 4 f and especially e+e - --~ 4 f y possess very complex peaking structures so that the peaks in the integrand f ( ~ ) in (3.3) cannot be described properly by only one single density g(q~). The multichannel approach [ 16,19] suggests a solution to this problem. For each peaking structure we choose a suitable set q~k, and accordingly a mapping of random numbers ri into ~k: ~ k = h k ( r ) with 0 ~< r i ~< l, so that the resulting density g~ describes this particular peaking behaviour of f . All densities gk are combined into one density gtot that is expected to smoothen the integrand over the whole phase-space integration region. The phase-space integral of (3.3) reads

M f (ki(CI~k)) I, = ~_~ ; drI)kp k ( k i ( ~ k ) ) gk (ki(CI)k) ) gtot ( ki( ~l~k) ) k=l ~/ I

=~-" f dr f(l~(h~(r))) ~__1 Jo with

gtot(ki(hk(r)))'

(3.8)

A. Denner et al./Nuclear Physics B 560 (1999) 33-65 M g~ ( ki( ~k ) ) , gtot ( ki( rrPk) ) = ~--~

t:I

l

gl ( ki( ~llk ) )

- Pt ( ki( dP~) ) 3hl(r)

51

r = h ~ I (q~l,.)

(3.9) The different mappings hk(r ) are called channels, and M is the number of all channels. In order to reduce the Monte Carlo error further, we adopt the method of weight optimization of Ref. [20] and introduce a priori weights ak, k = 1 . . . . . M ( ak ~ 0 and } - ~ l oek = 1 ). The,' channel k that is used to generate the event is picked randomly with probability crk, i.e. M

I,, = k=l 1

cek / d~k p/~ ( ki( q~k ) ) gk ( ki( ~Pk) ) f ( ki( ~k ) ) gtot ( ki ( ¢I~t() ) V

M

f dro ~ O(ro- B~-,)O(Bk- ro) 0

k=l

× [d~kpk

(ki(~k))gk (ki(~k))

, I

V

I

M

1

= / dro ~ 2 0 ( r o 0

f (ki(~)) g~o, ( k~ (q,k) )

t~k-l)O(~k--ro)

k=l

f dr f ( k i ( h k ( r ) ) ) gtot (ki( hk(r ) ) )'

(3.1o)

0

where/3o = 0,/3 i = }-~-~=lak, j = 1 . . . . . M - 1, /3M = ~ M

c~k = 1, and

M gtot

( ki( q[~k) ) =

~ cetgl ( ki( ~k ) ) ,

(3.11)

/=1

is the total density of the event. For the processes e+e - ---, 4 f we have between 6 and 128 different channels, for e+e - --+ 4 f y between 14 and 928 channels. Each channel smoothens a particular combination of propagators that results from a characteristic Feynman diagram. We have written phase-space generators in a generic way for several classes of channels determined by the chosen set of invariants si, ti. The channels within one class differ in the choice of the mappings (3.5), (3.6), and (3.7) and the order of the external particles. We did not include special channels for interference contributions. The cek-dependence of the quantity N

W.a( ) =

±N Z\--'[w(,-~, r J)12, --'

(3.12)

j=1

where w f/gtot is the weight assigned to the Monte Carlo point (r~, r J) of the jth event, can be exploited to minimize the expected Monte Carlo error =

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

52

6[n = ~/W(a)-N- Fn ,

(3.13)

with the Monte Carlo estimate of In 1

N

L' : -N Z

w ( rio' ri )

(3.14)

j=l

by trying to choose an optimal set of a-priori weights. We perform the search for an optimal set of crk by using an adaptive optimization method, as described in Ref. [20]. After a certain number of generated events a new set of a-priori weights a~,ew is calculated according to

new I 1 £ gk(ki(hk(rJ)))[w(rio, rJ)] 2 ~k 0~k N j=l gtot(ki(hk(rJ)))

M

= 1.

(3.15)

k=l

Based on the above approach, we have written two independent Monte Carlo programs. While the general strategy is similar, the programs differ in the explicit phasespace generation.

4. Numerical results

If not stated otherwise we use the complex-mass scheme and the following parameters: a = 1/128.89, Mw = 80.26 GeV, Mz = 91.1884 GeV,

as = 0.12, Fw = 2.05 GeV, Fz = 2.46 GeV.

(4.1)

In the complex-mass scheme, the weak mixing angle is defined in (2.45), in all other schemes it is fixed by Cw = Mw/Mz, s 2w= 1 - c 2w. The energy in the centre-of-mass (CM) system of the incoming electron and positron is denoted by x/-~. Concerning the phase-space integration, we apply the canonical cuts of the A D L O / T H detector, O(/,beam)> 0(y, beam)> Er > m(q,q') >

10 °, 1°, 0.1 GeV, 5 GeV,

O(l,l') > 5 ° , 0(y,I) > 5° , Et > 1 GeV,

O(1, q) > 5 ° , O(T,q) > 5 °, Eq > 3 GeV,

(4.2)

where O(i,j) specifies the angle between the particles i and j in the CM system, and l, q, y, and "beam" denote charged leptons, quarks, photons, and the beam electrons or positrons, respectively. The invariant mass of a quark pair qq' is denoted by m(q, qt). The cuts coincide with those defined in Ref. [23], except for the additional angular cut between charged leptons. The canonical cuts exclude all collinear and infrared singularities from phase space for all processes.

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

53

Although our helicity amplitudes and Monte Carlo programs allow for a treatment of arbitrary polarization configurations, we consider only unpolarized quantities in this paper. All results are produced with 10 7 events. The calculation of the cross section for e % - --+ e + e - / x + / z - requires about 50 minutes on a DEC A L P H A workstation with 500 MHz, the calculation of the cross section for e+e - ~ e + e - # + # - y takes about 5 hours. The results of our two Monte Carlo programs agree very well. The numbers in parentheses in the following tables correspond to the statistical errors of the results o f the Monte Carlo integrations.

4.1. Comparison with existing results In order to compare our results for e+e - --~ 4 f with Tables 6 - 8 of Ref. [ 31 ], we use the corresponding set o f phase-space cuts and input parameters, i.e. the canonical cuts defined in (4.2), a C M energy of v G = 190 GeV, and the parameters a = ce(2Mw) = 1/128.07, o~s = 0.12, Mw = 80.23 GeV, F w = 2.0337 GeV, Mz = 91.1888 GeV, and F z = 2.4974 GeV. The value of Sw, which enters the couplings, is calculated from ~ ( 2 M w ) / ( 2 s 2) = G~M~v/(Trv~) with Gu = 1.16639 x 10 -5 GeV -2. In Table 1, we list the integrated cross sections for various processes e+e - ~ 4 f with running widths and constant widths, and for the corresponding processes e+e - --+ 4 f y with constant widths. For processes involving gluon-exchange diagrams we give the cross sections resulting fi'om the purely electroweak diagrams and those including the gluonexchange contributions. The latter results include also the interference terms between purely electroweak and gluon-exchange diagrams. In Table 1 we provide a complete list of processes for vanishing fermion masses. All processes e+e - ~ 4 f ( y ) not explicitly listed are equivalent to one of the given processes. For NC processes e+e - --, 4 f with four neutrinos or four quarks in the final state we find small deviations of roughly 0.2% between the results with constant and running widths. A s s u m i n g that a running width has been used in Ref. [31], we find very good agreement. Unfortunately we cannot compare with most of the publications [ 1 2 - 1 5 ] for the bremsstrahlung processes e+e - --~ 4fy. In those papers, either the cuts are not ( c o m pletely) specified, or collinear photon emission is not excluded, and the corresponding fermion-mass effects are taken into account. Note that the contributions of collinear photons dominate the results given there. We have compared our results with the ones given in Refs. [10,11], where the total cross sections for e+e - --~ 4 f y have been calculated for the purely leptonic and the semi-leptonic final states. As done in Refs. [ 10,11] only diagrams involving two resonant W bosons have been taken into account for this comparison. Table 2 contains our results corresponding to Table 2 o f Ref. [ I 0 ] . Based on Refs. [ 10,11], we have chosen ~f~ = 200 GeV and the input parameters oe = 1/137.03599, Mw = 80.9 GeV, F w = 2.14 GeV, Mz = 91.16 GeV, F z = 2.46 GeV, Sw obtained from ce/(2S2w) = G~M2w/(Trx/2) with G~ = 1.16637 x 10 -5 GeV -2, and constant gauge-boson widths.

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

54

Table 1 Integrated cross sections for all representative processes e+e - ---+4 f with running widths and constant widths and for the corresponding processes e+e - ---, 4 f y with constant widths. If two numbers are given, the first results from pure electroweak diagrams and the second involves in addition gluon-exchange contributions (r/ fb

e+e - ---+4 f Running width

e+e - --+ 4 f Constant width

e+e - ---+4 f y Constant width

ve 9ee- e + up #+ e - ~e

256.7 ( 3 ) 227.4 ( l ) 228.7( 1) 218.55 (9) 109.1(3) 116.6(3) 5.478(5) 11.02(1) 14.174(9) 17.78(6) 10.108(8) 4.089( 1) 8.354(2) 4.069( 1) 8.241 (2) 693.5(3) 666.7(3) 86.87(9) 43.02(4) 24.69(2) 23.73( 1) 24,00(2) 20.657(8) 21.080(5) 19.863(5) 2064.1(9), 2140.8(9) 2015.2(8) 25.738(7), 71.28(4) 23.494(6), 51.35(3) 51.61(1), 144.72(9) 49.68(1), 126.52(8) 47.13(1), 104.79(6)

257.1 ( 7 ) 227.5 ( 1) 228.8( I ) 218.57(9) 109.4(3) 116.4(3) 5.478(5) 11.02(1) 14.150(9) 17.73(6) 10.103(8) 4.082( 1) 8.337(2) 4.057( 1) 8.218(2) 693.6(3) 666.7(3) 86.82(9) 42.95(4) 24.69(2) 23.73( 1) 23.95(2) 20.62( 1) 21.050(5) 19.817(5) 2064.3(9), 2141(1) 2015.3(8) 25.721(7), 71.30(4) 23.448(6), 51.32(3) 51.57(1), 144.75(9) 49.62(1), 126.52(8) 47.02(1), 104.74(6)

89.4 ( 2 ) 79.1 ( 1) 81.0(2) 76.7( 1 ) 38.8(4) 43.4(4) 3.37( 1) 6.78(3) 5.36(1) 6.63(2) 4.259(9) 0.7278(7) 1.512(1) 0.7434(7) 1.511 ( 1) 220.8(4) 214.5 (4) 32.3(2) 16.17(8) 12.70(4) 10.43(2) 6.84( 1) 4.319(6) 6.018(9) 4.156(5) 615(1), 672(1) 598( 1) 9.78(2), 42.1(1) 5.527(7), 28.68(4) 19.61(4), 86.1(2) 15.17(2), 75.1(2) 11.10(2), 59.2(1)

UU~UI x - IX+ ut,#+r - Jr

e-e+e-e + e-e+ix-ix + # - # + # - IX+ ix-tx+r-r +

e - e+ u,, bu ue~cix- Ix+ v~TIx- Ix+ vc~cuc ~e ~c~eu~bl, v,,f,~,v#~ t, v~bt, U~br

u d e - ~c u d Ix- 9u c - e + u fi e-e+dd u uix Ix+ d d ix- ix+ uc~cu ~ uc~cd a u fl vuf, u

d d u~,P,z ufida ua se ufiufi ddda u0c6 ufisg ddsg

T h e e n e r g y o f the p h o t o n is r e q u i r e d to b e l a r g e r t h a n ET,min, a n d the a n g l e b e t w e e n the p h o t o n a n d a n y c h a r g e d f e r m i o n m u s t b e l a r g e r t h a n 0r,min. A m a x i m a l p h o t o n e n e r g y is r e q u i r e d , E r < 6 0 G e V , in o r d e r to e x c l u d e c o n t r i b u t i o n s f r o m the Z r e s o n a n c e . O u r results are c o n s i s t e n t w i t h t h o s e o f Refs. [ 10,11 ] w i t h i n t h e statistical e r r o r o f 1% g i v e n there. In s o m e c a s e s w e find d e v i a t i o n s o f 2 % . 2

2 Note that the input specified in Refs. [ 10,11] is not completely clear even if the information of both publications is combined.

A. Denner et at/Nuclear Physics B 560 (1999) 33-65

55

Table 2 Comparison with Table 2 of Ref. [10]: Cross sections resulting from diagrams involving two resonant W bosons for purely leptonic and semi-leptonic final states and several photon separation cuts E~, min =

l GeV

E,g,min =

Oy.min leptonic process

semi-leptonic process

5 GeV

E,?. rain =

10 GeV

Ey.min =

15 GeV

o'/fb

1° 5° 0° 5°

53.54(8) 32.65(4) 23.48(3) 18.03(2)

27.57(3) 16.98(3) 12.30(2) 9.5l(2)

16.96(2) 10.48(2) 7.61(2) 5.90(1)

11.22(2) 6.94(1) 5.04(I) 3.90(1)

1° 5° 10° 15 ~

141.9(2) 86.8(1) 62.29(7) 47.42(5)

71.90(8) 44.25(6) 31.92(5) 24.50(4)

43.56(5) 26.78(4) 19.40(4) 14.97(3)

28.26(4) 17.40(3) 12.61(3) 9.77(2)

Table 3 Comparison of different width schemes for several processes and energies

¢r/ fb

x/~ = 189 GeV v'~ = 500 GeV v'~ = 2 TeV v,~ = 10 TeV constant width 703.5(3) running width 703.4(3) complex-mass scheme 703.1(3)

237.4( 1) 238.9( 1) 237.3(I)

13.99(2) 34.39(3) 13.98(2)

0.624(3) 498.8( l ) 0.624(3)

e+e - ~ u d # - ~ y

constant width 224.0(4) running width 224.6(4) complex-mass scheme 223.9(4)

83.4(3) 84.2(3) 83.3(3)

6.98(5) 19.2(1) 6.98(5)

0.457(6) 368(6) 0.460(6)

e+e - --~ ude-Dc

constant width 730.2(3) running width 729.8(3) complex-mass scheme 729.8(3)

395.3(2) 396.9(2) 395.1 (2)

211.0(2) 231.5(2) 210.9(2)

32.38(6) 530.2(6) 32.37(6)

e+e - --~ uae-.~ey

constant width 230.0(4) running width 230.6(4) complex-mass scheme 229.9(4)

136.5(5) 137.3(5) 136.4(5)

84.0(7) 95.7(7) 84.1(6)

16.8(5) 379(6) 16.8(5)

e+e

~ u d/x-.~,~

4.2. Comparison o f finite-width schemes As discussed in Refs. [26,29], particular care has to be taken when i m p l e m e n t i n g the finite g a u g e - b o s o n widths. D i f f e r e n c e s between results obtained with running or constant widths can already be seen in Table 1, where a typical L E P 2 energy is considered. In Table 3 we c o m p a r e predictions for integrated cross sections obtained by using a constant width, a running width, or the c o m p l e x - m a s s s c h e m e for several energies. We c o n s i d e r two s e m i - l e p t o n i c final states for e + e - --~ 4 f ( y ) .

The numbers show that the

constant width and the c o m p l e x - m a s s s c h e m e yield the same results within the statistical accuracy for e + e - ---, 4 f and e+e - ~

4 f y . In contrast, the results with the running

width p r o d u c e totally w r o n g results for high energies. The difference o f the running width with respect to the other i m p l e m e n t a t i o n s o f the finite width is up to 1% already for 500 G e V . Thus, the running width should not be used for linear-collider energies.

56

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

Table 4 Photon energies Erv~v2 corresponding to thresholds •/s/ GeV VLg2 Evlv2/ GeV

189 WW 26.3

ZZ 6.5

500 7Z 72.5

7y

94.5

WW 224

ZZ 217

yZ 242

yy

250

As already stated above, our default treatment of the finite width is the complex-mass scheme in the following. 4.3. Survey o f photon-energy spectra

In Fig. 2 we show the photon-energy spectra of several processes for the typical LEP2 energy of 189 GeV and a possible linear-collider energy of 500 GeV. The upper plots contain CC and C C / N C processes, the plots in the middle and the lower plots contain NC processes. Several spectra show threshold or peaking structures. These structures are caused by diagrams in which the photon is emitted from the initial state. The two important classes of diagrams are shown in Fig. 3. The first class, shown in Fig. 3a, corresponds to triple-gauge-boson-production subprocesses which yield dominant contributions as long as the two virtual gauge bosons V~ and V2 can become simultaneously resonant. If the real photon takes the energy Er, defined in the CM system, only the energy v/~, with s' = s - 2 v / s E t ,

(4.3)

is available for the production of the gauge-boson pair VI V2. If at least one of the gauge bosons is massive, and if the photon becomes too hard, the two gauge bosons cannot be produced on shell anymore, so that the spectrum falls off for E r above the corresponding threshold E~ v~. Using the threshold condition for the on-shell production of the V1V2 pair, > Mv~ + Mv2,

(4.4)

the value of E vlv2 is determined by fir, v2 = s -

(My, + My2) 2

2v/~

(4.5)

The values of the photon energies that cause such thresholds can be found in Table 4. The value E~~ corresponds to the upper endpoint of the photon-energy spectrum, which is given by the beam energy v ~ / 2 . Since ~ is fully determined by s and E~, the contribution of the V1V2-production subprocess to the E~ spectrum qualitatively follows the energy dependence of the total cross section for VIV2 production (cf. Ref. [31], Fig. 1 ) above the corresponding thresholds. The cross sections for y y and yZ production strongly increase with decreasing energy, while the ones for ZZ and WW production are comparably flat. Thus, the y y and yZ-production subprocesses introduce contributions

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

189 GeV 100 ~

i

i

i

i

i

500 GeV

!

I

I

I

I

I

I

e+e -

10 ~1[""~.

e+e -

f[

I.

. . . .

",.,,.

•

o o~

~, i

0.001 100

~

I

I

I

I

I

I i!:. I

I

I

I

I

I

I

I

I

I

I

I

i

I

('+e

10

e+e -

.lli i

I

u f i c c ' / .......... --+ U t] tl u ~1 - - ~ -+

-+ e-e+uQ7

.....

e+e - ~ #-,u+ufi7 .......

:ti 1: :a 4

L'~

1

---

=

I

fb

.........

uae-,~,r

%:. -227--=._.

e+e -

da /

I

-+ u~dd3'

e+e - ~. u d # - 5 ~ ' y . . . . e + e - -+ uet"ee- e + v ....... e+e - -+ UuPu#-It.+ 7

i'

1 -

57

t..: " "%

0.1

0.01

0.001

100

I

I

I

I

I

I

I

I

I i!

I

I

I

I

I

I

I

I

I

I

I

I

I

I !

e+e - -~ pePeu fi'7 ........ e+e -

10 do

/rb

1

"t •:

-'"-L

i '-._ ......;? ........

0.0001

.-',.

t ..-.]

'%. "i' !

-%;ii:::: ............

I

I "]". t t i P: t I 10 20 30 40 50 60 70 80 90 100 0 E.J G e V

0

--_

~kx .... "--- . . . .

0.001 t

~. e~e+e-e+~

e+e - -+ /,'e/~eUeOe'~ . . . . . e + e - -+ P # + # - # + 7 ......

dE-~ / GeV

I

I

I

50

100

150

I ': 200

i 250

E.~/ GeV

Fig. 2. P h o t o n - e n e r g y spectra for several processes and for v G = ! 89 G e V and 500 GeV.

a)

b)

Fig. 3. D i a g r a m s for important subprocesses, where Vl, V2 = W, Z, y, and V3 = y, g.

58

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

in the photon-energy spectra with resonance-like structures, whereas the ones with ZZ or WW pairs yield edges. The second class of important diagrams, shown in Fig. 3b corresponds to the production of a photon and a resonant Z boson that decays into four fermions. These diagrams are important if the gauge boson V3 is also resonant, i.e. a photon or a gluon with small invariant mass. In this case, the kinematics fixes the energy of the real photon to E~=E~ z= s-M 2 2x/~ ,

(4.6)

which corresponds to the yZ threshold in Table 4. This subprocess gives rise to resonance structures at E~z, which are even enhanced by a s i a in the presence of gluon exchange. In the photon-energy spectra of Fig. 2 all these threshold and resonance effects are visible. The effect of the yZ peak can be nicely seen in different photon-energy spectra, in particular in those where gluon-exchange diagrams contribute (cf. also Fig. 5). The effect of the WW threshold is present in the upper two plots of Fig. 2. In the plot for ~ = 189 GeV the threshold for single W production causes the steep drop of the spectrum for the pure CC processes above 70 GeV. Note that the CC cross sections are an order of magnitude larger than the NC cross sections if the WW channel is open. The ZZ threshold is visible in the middle and lower plots for x/~ = 500 GeV. The yZ threshold (resulting from the graphs of Fig. 3a) is superimposed on the yZ peak (resulting from the graphs of Fig. 3b) and therefore best recognizable in those channels where the yZ peak is absent or suppressed, i.e. where a neutrino pair is present in the final state or where at least no gluon-exchange diagrams contribute. Processes with four neutrinos in the final state do not involve photonic diagrams and are therefore small above the ZZ threshold. The effects of the triple-photon-production subprocess appear as a tendency of some photon-energy spectra to increase near the maximal value of Er for two charged fermion-antifermion pairs in the final state. 4.4. Triple-gauge-boson-production subprocesses

In Fig. 4 we compare predictions that are based on the full set of diagrams with those that include only the graphs associated with the triple-gauge-boson-production subprocesses, i.e. the graphs in Fig. 3a. In addition we consider the contributions of the ZZy-production subprocess alone. For CC processes, the photon-energy spectra resulting from the W+W-y-production subprocess are close to those resulting from all diagrams at LEP2 energies, but large differences are found for higher energies and e + in the final state. Note that the spectra are shown on a logarithmic scale. Even at LEP2 energies the differences between the predictions for different final states may be important, as can be seen, for instance, in Table 1 by comparing the cross sections of e+e - --~ u a / x - ~ y and e+e - --~ u a e - ~ e y . In the case of NC processes, already for 189 GeV the contributions from ZZy, ZTT, and yy'y production are not sufficient: in the vicinity of the yZ peak sizeable contributions result from the yZ-production subprocess (Fig. 3b) even for the /z+/x u u y final state. For e+e - ~ e - e + u f i y other

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

da /

2TeV

500 GeV

189 GeV

1oot ,,,

I

I

I

I

e+e - + u~]#-0u7 e+e - --+ u de-Oe7 e+e - ~ WW7

fb

10 1

X

L

0.Ol

da/

?

~,~

0.001

I

t

t ~ I

i

i

i

I

10

I

!

i

I

I

I

I

fb

I

~,l

.......

0.00010"001t 0.00001 0

i

I I

...... i~'=°~"T ....... I I

!

I

e+e - - + # #+ufi? ........... e+e- -+ e - e + u u 7 . . . . e+e --+ ~/~V27 e+e - ~+ ZZ'y .....

'l

0.01

59

',),

-i

'X%

""% ':~:

I 20

i

40

i:; 60

i

80

100 0

I 50

I I I ": 100 150 200 250 0 E~/ GeV

200 400 600 800 1000

Fig. 4. Photon-energy spectra resulting from the triple-gauge-boson-production subprocesses compared to those resulting from all diagrams (V1V2 includes ZZ, yZ, and yy).

diagrams become dominating everywhere. The contribution of ZZy production is always small and could only be enhanced by invariant-mass cuts. Note that the triple-gaugeboson-production diagrams form a gauge-invariant subset for NC processes, while this is not the case for CC processes.

4.5. Relevance of gluon-exchange contributions In the analytical calculation of the matrix elements for e+e - --* 4 f ( y ) in Section 2 we have seen that NC processes with four quarks in the final state involve, besides purely electroweak, also gluon-exchange diagrams. Table 5 illustrates the impact of these diagrams on the integrated cross sections for a CM energy of 500 GeV. The results for the interference are obtained by subtracting the purely electroweak and the gluon contribution from the total cross section. For pure NC processes the contributions of gluon-exchange diagrams dominate over the purely electroweak graphs. This can be understood from the fact that the gluon-exchange diagrams are enhanced by the strong coupling constant, and, as discussed in Section 4.3, that the diagrams with gluons

60

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

Table 5 Full lowest order cross section (ew and gluon) and contributions of purely electroweak diagrams (ew), of gluon-exchange diagrams (gluon), and their interference for 500 GeV tr/ fb e+e e+e e+e e+e e+e e+e -

~ ~ ~ ~ ~ ~

u0cE u~c~y u Qu fi u~u~y daufi dduQy

Ew and gluon

Purely ew

Gluon

Interference

52.98(4) 29.8(1) 26.25(2) 14.83(7) 901.2(6) 290(1)

21.560(6) 10.38(4) 10.765(3) 5.16(2) 876.4(5) 275(1)

31.38(3) 19.6(1) 15.34( 1) 9.52(5) 24.24(2) 14.82(8)

0.04(5) -0.1(1) 0.14(2) 0.15(9) 0.6(8) 0(1)

replaced by photons yield a sizeable contribution to the cross section. For the mixed C C / N C processes the purely electroweak diagrams dominate the cross section. Here, the contributions from the W + W - y - p r o d u c t i o n subprocess are large compared to all other diagrams, even if the latter are enhanced by the strong coupling. At 500 GeV the gluonexchange diagrams contribute to the cross section at the level of several per cent. The interference contributions are relatively small. As discussed at the end of Section 2.3.2, this is due to the fact that interfering electroweak and gluon-exchange diagrams involve different resonances. Note that the interference vanishes for e+e - --~ u tic ~ y, and the corresponding numbers in Table 5 are only due to the Monte Carlo integration error. In Fig. 5 we show the photon-energy spectra for the processes e+e - ~ u fi d d y and e+e - --+ u fi u ~ y together with the separate contributions from purely electroweak and gluon-exchange diagrams. The pure electroweak contributions are similar to the ones for e+e - ~ u a / z - ~ y and e+e - ---, e - e + u f i y in Fig. 2. For the NC process e+e u fi u fi % the photon-energy spectrum is dominated by the gluon-exchange contribution, which shows a strong peak at 72.5 GeV owing to the yZ-production subprocess. For the C C / N C process e+e - ~ u rid a y, the electroweak diagrams dominate below the W W threshold, whereas the gluon-exchange diagrams dominate at the y Z peak and above. The interference between purely electroweak and gluon-exchange diagrams is generally small.

5. Summary and outlook The class of processes e+e - ~ 4 fermions + y has been discussed in detail. After classifying the different final states according to their production mechanism, the sets o f all Feynman graphs are reduced to two generic subsets that are related to the two basic graphs of the non-radiative processes e+e - ~ 4 fermions. In this way, all helicity matrix elements are expressed in terms of two generic functions. Using the W e y l - v a n der Waerden spinor formalism, we have given compact expressions for these functions. We wrote two independent Monte Carlo programs, both using the multichannel integration technique and an adaptive weight optimization procedure to reduce the Monte Carlo error. The results obtained with the two Monte Carlo programs agree

61

A. D e n n e r et a l . / N u c l e a r P h y s i c s B 5 6 0 (1999) 3 3 - 6 5

100 [ I1\

1

I

I

I

I

e + e - --+ u O d d 7

I

i

I

I

e + e - ~" u f i u f i y

10 / fb dhg.y / GeV

'I !~,

da

l

ii (!

:"',..., ....... ~%':=~,_.:~ .......... ::,:/''ii~ ~ -. ........ >~....

0.1

,~}.~

0.01

"\L

0.001 0.0001 20

iI

I

I

I

40

60

80

E~/CeV

~/

q ~! / 100 0

all ........... electroweak - - gluon-exchange ........

~ ~ ". . . . . , i h"

i

i

i

i

20

40

60

80

]i 100

E~/CeV

Fig. 5. E l e c t r o w e a k a n d g l u o n - e x c h a n g e contributions to the p h o t o n - e n e r g y spectra for e + e - ~ e + e - ~ u Q u f i y at ,,/3,,i = 189 G e V .

u fi d a y and

well within the integration error. The detailed discussion of numerical results comprises a survey of integrated cross sections and photon-energy spectra for all different final states. Moreover, we have numerically compared different ways to introduce finite decay widths of the massive gauge bosons. Similar to the known results for e+e - --~ 4 fermions, we find that the application of running gauge-boson widths leads to totally wrong results for the radiative processes. Using constant widths consistently, leads to meaningful predictions. For the considered observables, the results for constant widths practically coincide with those obtained in a complex-mass scheme that fully respects gauge invariance. In the latter scheme gauge-boson masses are treated as complex parameters everywhere, in particular, leading to complex couplings. Similar to the situation for the well-known non-radiative processes, we find that for precise predictions the diagrams corresponding to triplegauge-boson-production subprocesses are not sufficient and the inclusion of the other graphs is mandatory Finally, we have investigated the relevance of gluon-exchange contributions. In general, both the purely electroweak contributions and the gluon-exchange contributions are relevant and either one can be dominant depending on the process and the considered observable. The interference of both contributions is at most at the per-cent level. In this paper, we have assumed that the radiated photon appears as a detectable particle in the final state, i.e. soft and collinear photons are excluded by cuts. The inclusion of soft and collinear photons is, however, necessary if e+e - ---, 4 fermions+y is considered as a correction to four-fermion production. The analytical results of this paper and the constructed Monte Carlo programs will be used as a building block in the evaluation of four-fermion production in e+e - collisions including O(o~) corrections.

A. Denneret al./NuclearPhysicsB 560 (1999)33-65

62

~+(p+) W(q~)

l¥(ql) e-(p_) .

~

,-(k~) ~(k~) •

~+(k4) uo(k2)

Fig. A.I. An example for a multi-peripheral diagram contributing to the process e+e - ~ UeOe/Z-/z+T.

Appendix A. An example for phase-space generation To illustrate the phase-space generation in one channel, we choose the multi-peripheral diagram, shown in Fig. A.I. We determine the momentum configuration ki from the phase-space variables si, ti, qSi, and Oi with a suitable choice of mappings and calculate the resulting probability density g. First we study the propagator structure of the Feynman diagram of Fig. A.1 and choose a set • of 1 1 phase-space variables that is suitable for importance sampling. Our choice of q' consists of three time-like invariants S35 = k25 = (k3 %. ks) 2, $345 = k3245= (k3 + k4 %. ks)2, 2 =

S1345= k1345

(kl %- k3 %- k4 %- ks) 2,

(A.1)

two space-like invariants tl =q2 = (p_

_

k2)2,

t 2 = q 2 = (p+ -- kl) 2,

(A.2)

and six azimuthal and polar angles ~1,2,3,4 and 03,4, respectively. As worked out in detail in Ref. [30], the 11-dimensional phase-space integral in (3.1) can be written as follows:

R5(s)=f[rId4ki~(k2)O(k°i)]~(4)(

i=1

/]

= J ds35 ds345 ds1345 R 2 ( s ) R2(s1345) R2(s345) R2(s35),

(A.3)

v

where v ~ denotes the CM energy with s = (p_ %.p+)2 The phase space of the 2 --~ 5 scattering process is composed of the phase spaces of two 2 ---, 2 scattering processes with two subsequent 1 ---* 2 decays described by the following functions: R2(s): First, we consider the 2 --+ 2 scattering process p+ + p _ ~ k1345 + k2 in the system p+ + p_ = 0 where the direction of p+ is chosen to be the positive z axis.

A. Denner et al./Nuclear Physics B 560 (1999) 33-65

63

R 2 ( s 1 3 4 5 ) : Then, we boost and rotate to the system k/345 = 0 and describe the 2 -~ 2 scattering process p~ + q~ ---, k~45 -J- ktl , where the direction of p ' + is chosen to be the positive z axis. R2($345): Next, we define the 1 --+ 2 decay of the Z boson k~ 5 --, k~ + k~' in its rest frame k~] 5 = 0. R2 (s35): Finally, we treat the subsequent 1 ---, 2 decay of the virtual muon k ~ -+ k'"3+sk"' in its rest frame k ~ = 0. For massless external fermions, Rs(s) explicitly reads 3

// Rs(s) =

/

ds35 S35,cut

)<

ds345 $35

.... ds1345

$345

- - dtl ~

4.)tl/2(Sl345, t l , 0 )

t 1,min

dt2

dq~l 0

d~b2

t2,min

d cos 03

/~1/2( $345 ' $35,0) )<

d~b3

d COS 04

dq~4

(A.4)

8S345 -1

--1

0

with the kinematical function A(x,y,z)

= x 2 + y2 + z e _ 2 x y -

(A.5)

2yz - 2zx.

The limits on t~,2 are the physical boundaries of the 2 --, 2 scattering processes tl,min = S1345 -- S,

/l,max = 0,

tZ,min = ( t l -- S1345) (S1345 -- $345)/S1345,

t2,max = 0.

In our example, we only have to cope with massive space-like propagators, so that we generate the invariants s35 and s345 according to (3.5) and (3.6), but do not apply importance sampling in the variables s1345, tl,2, q~1,2,3,4, and cos 03,4, which are generated according to (3.7). The resulting density g of (3.9) for this specific channel reads for t : 4 = I, l

g

/~1/2 ( $345 ' $35, O)

4282ss345Al/2(s1345,tl,0) P

x s3s

sl-,

l - v t [($345 _ M 2 ) 2 + M z2F z 2 ] _ $35,cu

1- v

MzFz

× (S -- S345) Ill,mini [t2,rninl 4 ( 2 7 r ) 4

(Y2 - Yl ) (A.6)

with YI,2 = arctan[ (Sl,2 - M ~ ) / ( M z F z ) ] , sj = s35, and s2 = s. Finally, the event characterized by the four-momentum configuration ki, i = 1 . . . . . 5, and the beam momenta are defined in terms of the generated invariants and angles as follows: 3 As usual we set f = 0 outside the imposed cuts. However, to improve the efficiency of the phase-space generation, cuts can be included already in the physical boundaries of phase space. In our example we have introduced a lower cut on s~s,

64

A. Denner et al./Nuclear Physics

B

560 (1999) 33-65

yv'~-(1, O, O, 1), = "/s

o,0,-1),

k~ = AV2( s' S1345,0) 2v G

(1, - cos~bl sin 01, - sin ~bl sin 01, - cos 01)

with c o s 0 j = (tl + 2 p ° _ k ° ) / ( 2 p °

(A.7)

k°),

ktltz : .A1/2( s1345, $345,0) ( 1, cos q~2 sin 02, sin ~b2 sin 02, cos 02) w i t h c o s O 2 = ( t 2 + 2 p + k ,o ,o 1 ) / ( 2 p + ,o k ,o 1 ) a n d p + ,o = (s1345 - t l ) / ( 2 sgE ktt# 4

=

,,~1/2($345, $35 , 0) 2 sx/~-~345

( 1, -- COS ~b3 sin 03, - sin ~b3 sin 03, - cos 03),

(A.8) ),

(A.9)

and

k tlttz - x / ~ 3

2

( 1, cos 1~4 sin 04, sin ~4 sin 04, cos 04),

kttrl, 5 = ( k13tto, - k m 3 ).

(A.10)

The f o u r - m o m e n t a kl, k3, k4, and k5 in the C M system are obtained from the fourm o m e n t a Uj, k~", k~', and k~" by a p p l y i n g the appropriate Lorentz transformations, as for instance described in Ref. [30].

Acknowledgements We thank D. G r a u d e n z and R. Pittau for useful discussions about M o n t e Carlo integration and T. Stelzer for discussions c o n c e r n i n g Madgraph.

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