DESY 95–167
hepph/9509abc
September 1995
Off shell pair production in annihilation:
The CC11 process
D. Bardin and T. Riemann
DESY – Zeuthen  
Platanenallee 6, D15738 Zeuthen, Germany 
Bogoliubov Laboratory for Theoretical Physics, JINR  
ul. JoliotCurie 6, RU141980 Dubna, Moscow Region, Russia 
ABSTRACT
The various fourfermion production channels in annihilation are discussed and the CC11 process , , is studied in detail. The cross section , with being the invariant masses squared of the two fermion pairs, may be expressed by six generic functions. All but one may be found in the literature. The cross section, including initial state radiation and the Coulomb correction, is discussed and compared with other calculations from low energies up to = 2 TeV.
Email addresses:  , 
1 Introduction
A measurement of pair production around and above the production threshold will be one of the main tasks of LEP 2 [1] as well as of a future high energy linear collider [2].
Immediately after their creation, the bosons decay and fourfermion () production is observed:
(1.1) 
This double resonating process proceeds via the diagrams shown in figure 1. In addition, there are a lot of Feynman diagrams with the same final state, but different intermediate states, which are single resonant or nonresonant and often called background diagrams. Their number and complexity vary in dependence on the composition of the final state. The total numbers of Feynman diagrams for the different channels are shown in table 1.
43  11  20  10  10  
20  20  56  18  18  
10  10  18  19  9 
One may distinguish three different event classes, all of them containing the CC3 process^{1}^{1}1 In [3], a slightly different classification has been introduced; the relation of both schemes is discussed in [4]. :

The CC11 process.
The two fermion pairs are different, the final state does not contain identical particles nor electrons or electron neutrinos (numbers in table 1 in boldface). The corresponding eleven diagrams are shown in figures 1 and 2. There are less diagrams if neutrinos are produced (CC9, CC10 processes). 
The CC20 process.
The final state contains one together with its neutrino (Roman numbers in table 1); compared to case (i), the additional diagrams have a channel gauge boson exchange. For a purely leptonic final state, a CC18 process results. 
The mix43 and mix56 processes.
Two mutually charge conjugated fermion pairs are produced (italic numbers in table 1). Differing from cases (i) and (ii), the diagrams may contain neutral boson exchanges^{2}^{2}2 We exclude the Higgs boson exchange diagrams from the discussion. In fact, the methods developed here are applicable also for the study of associated Higgs production [6]. (see also table 2 in [5] where ‘neutral current’ type final states are classified). There are less diagrams in the mix43 process if neutrinos are produced (mix19 process).
In this article, we will investigate the simplest topology, case (i), which proceeds via the diagrams of figures 1 and 2:
(1.2) 
As far as observable final states are concerned, the following reactions are covered by this study:

() + ();

() + (), ;

() + ().
We parameterize the sevendimensional four particle phase space as a sequence of two particle phase spaces:
(1.3)  
with the usual definition of the function,
(1.4)  
(1.5) 
In (1.3), the rotation angle around the beam axis has been integrated over already. Variables and are the fourmomenta of electron and positron and are those of the final state particles with ^{3}^{3}3 Fermion masses are retained in the phase space definitions and in the basic kinematical relations. In all subsequent steps of the calculation, they will be neglected compared to invariants , and .. The invariant masses squared of the fermion pairs are:
(1.6)  
(1.7)  
(1.8) 
The production angle in the center of mass system is spanned by the vectors () and . The spherical angle of () in the rest frame of the fermion pair [] ([]) is : . In this frame, the () points into the opposite direction with respect to () . The kinematical ranges of the integration variables are:
(1.9)  
(1.10)  
(1.11)  
(1.12) 
We are interested in analytical formulae for distributions in invariant masses of fermion pairs. Thus, we have to integrate analytically over the five angular variables in (1.3). The squared matrix elements have been derived with CompHEP [7] and the angular integrations were performed with the aid of FORM [8].
The cross section contributions may be grouped into several classes of interferences:
(1.13) 
In (1.13), the CC3 process is accompanied by the interferences of deers with the crayfish diagrams, the interferences of deers with the crab, the interferences of deers of one doublet, and the interferences of deers of different doublets.
Each of these contributions may be described by the product of a coefficient function , which is composed out of coupling constants and gauge boson propagators, and a kinematical function . The latter depends only on the three invariant masses and contains the dynamics of the corresponding hard scattering process. The functions, which are needed for a description of the CC11 process are shown in table 2. The double resonating CC3 process is known for long to be described by three kinematical functions [9]: . The double resonating neutral current NC2 process, , proceeds via two diagrams of the crab type and is characterized by only one kinematical function ^{4}^{4}4 Since not only pairs, but also and intermediate states exist one is faced in practice with an NC8 process with the same topology like NC2. [10]. We also have to mention that the simplest final state configuration of a complete neutral current process has 24 Feynman diagrams and requires, besides , only one additional function [5]. It will be shown below that a complete description of the CC11 process after the angular integrations is possible with adding only one further function .
process  

, ,  ;  ;  ; 
In the next section, notations and the formulae for the CC3 process are introduced. In section 3, the contributions from the background diagrams of figure 2 are presented. The photonic initial state and Coulomb corrections are added in section 4 and section 5 contains numerical results and a discussion. Several appendices are devoted to technical details of the calculation.
2 The Cc3 process
2.1 The total cross section
The double resonating cross section is:
The kinematical functions are known from [9]:
(2.2)  
(2.3)  
(2.4) 
In the superscript, the ‘3’ indicates to the matrix element with a three gauge boson vertex and ‘’ to the diagram with fermion propagator. The function has been slightly simplified compared to [4, 9]. The logarithm in (2.3) and (2.4) arises from the integration over the fermion propagator in the channel (see appendix C):
(2.5) 
Some properties of and are collected for the convenience of the reader in appendix A.
For the coefficient functions, we choose generic definitions. At first glance this seems to lead to artificially complicated constructions. At a later stage, however, the usefulness will become quite evident. We begin with
(2.6) 
The coefficient has been introduced for the neutral current NC2 process in eq. (11) of [5]; with a slight change of notion for the current presentation, it becomes:
(2.7)  
The labels denote the corresponding members of weak isodoublets. For the charged current, all left and righthanded fermion couplings are equal:
(2.8)  
(2.9) 
Below we will also need neutral current couplings where flavors will cause differences for the couplings.
The boson propagator is:
(2.10) 
With these definitions and using the relation , one easily verifies that
(2.11) 
where
(2.12) 
and
(2.13) 
The off shell width is used throughout this paper. It contains a sum over all open fermion decay channels at energy , and BR() is the corresponding branching ratio. Equation (2.11) makes the presence of the two BreitWigner factors explicit. These are normalized such that .
The other two coefficient functions are:
(2.14)  
(2.15)  
(2.16)  
with
(2.17) 
and , .
2.2 The angular distribution
It is quite useful to know not only the invariant mass distributions and the total cross section, but in addition also the distribution in the production angle of one of the bosons:
(2.18)  
with
(2.19)  
(2.20)  
(2.21) 
and
(2.22)  
(2.23) 
The scattering angle is contained in the denominator of the channel propagator:
(2.24) 
With the aid of appendix C, it is trivial to integrate over :
(2.25) 
The distribution (2.18) is used for the description of pair production in PYTHIA [11]. It also may be used for the calculation of moments like
(2.26) 
as a check on the accuracy of Monte Carlo integrations [12]. The angular distribution is more interesting in the context of anomalous couplings of gauge bosons [13].
3 Background contributions
We now come to the contributions from the eight background diagrams of figure 2 and from their interferences with the double resonating diagrams of figure 1. For this purpose, we use a classification, which was introduced for neutral current processes in [5]. A single resonant diagram with a virtual fermion is called an deer. Strictly speaking, the deers are double resonating diagrams as crabs are: besides the one resonance, they contain the channel (or ) propagator. One of the invariant masses, in which the diagram may become resonating, is ‘eaten’ by the fermion line. In the crab it is , while in a deer either or . In this language, the crab is an deer. In appendix C it is made plausible that in fact this observation may be used for a treatment of all the contributions on an equal footing.
3.1 The crayfishdeer interferences
The interferences of the crayfish diagram with the four types of deers are similar to the crayfishcrab interference:
(3.1) 
3.2 The deer interferences
The contribution from the square of two deers, which belong to one doublet occurs twice:
(3.3) 
with the coefficient function
(3.4) 
The is defined in (2.7).
The kinematical functions are different for the pure squares of diagrams, where (2.4) is to be used, and for their interference. The latter may be expressed by known functions. Let us refer for a moment to the neutral current case. There, all couplings are equal and one gets the following relation: . With , and , one gets the requested identity,
(3.5) 
with the neutral current function from [10]; see appendix D.
3.3 The deer interferences
There are four interferences among deer diagrams belonging to different doublets:
(3.6) 
The coefficient function may be traced back to that introduced in [5]:
(3.7) 
The function is defined as follows:
Wherever a neutral gauge boson couples, the corresponding fermion or etc. will replace the doublets , in the arguments of the couplings .
For the kinematical functions, again a relation to a neutral current function may be established. The function , explicitly given in appendix D, describes a sum of interferences analogue to those considered here. Since the couplings are equal in that case, we have: . Furthermore, using the symmetry relations and , we may write . This relation allows to determine once is known:
(3.9) 
By an explicit calculation, we obtained:
(3.10) 
where .
For numerical applications, it is helpful to know the limit :