Conveners: W. Beenakker and F.A. Berends

Working group: E.N. Argyres, D. Bardin, A. Denner, S. Dittmaier, J. Hoogland, S. Jadach, R. Kleiss, Y. Kurihara, D. Lehner, G. Montagna, T. Munehisa, O. Nicrosini, T. Ohl, G.J. van Oldenborgh, C.G. Papadopoulos, G. Passarino, F. Piccinini, B. Pietrzyk, T. Riemann, Y. Shimizu, M. Skrzypek

1 Introduction

1.1 Goals of W-pair production

The measurements performed at LEP1 have provided us with an extremely accurate knowledge of the parameters of the  gauge boson: its mass, partial widths, and total width. Except perhaps for the and decay widths all data are in perfect agreement with the Standard-Model [1] (SM) predictions [2, 3]. There even is first evidence that the contributions of gauge-boson loops to the gauge-boson self-energies are indeed required [4]. Thus an indirect confirmation of the existence of the triple gauge-boson couplings (TGC’s) has been obtained.

The future measurements of -pair production at LEP2 will add two important pieces of information to our knowledge of the SM. One is a determination of the  mass, which at present is only directly measured at hadron colliders. The envisaged precision of 40–50  gives a significant improvement on the present Tevatron measurements [5]. Since the  mass is one of the key parameters of the electroweak theory, such an improved accuracy makes the tests on the SM more stringent.

The second piece of information that -pair production can provide is the structure of the triple gauge-boson couplings. These couplings now play a role in the tree-level cross-section, in contrast to LEP1 physics where they only enter through loop corrections. So the Yang-Mills character of the TGC’s can be established by studying -pair differential cross-sections at LEP2 energies. Since these couplings are at the heart of the non-abelian gauge theories, this information is essential for a direct confirmation of the SM.

1.1.1 Measurement of the mass of the W boson

With the precise measurement of the mass of the  boson at LEP2 the situation for the electroweak input parameters changes with respect to LEP1. The common practice at LEP1 [6] is to use for  a value derived from the Fermi constant , which is accurately known from muon decay. The relation to obtain  follows from the SM prediction for muon decay


where at tree level and where is - and -dependent when loop corrections are included. Thus,  in LEP1 calculations is - and -dependent through the above procedure. At LEP2, where one wants to measure  and, hence, wants to treat  as a fit parameter, the above relation now primarily acts as a test of the SM. The above relation predicts for any chosen  and measured  a value for  that can be used as input for LEP2 loop calculations and can also be compared with the directly observed top-quark mass from the Tevatron [7].

As to the actual procedure to measure  from -pair production, two methods are advocated [8]. A third one, involving the measurement of the endpoint energy of a charged lepton originating from a decaying  boson, was suggested a few years ago but turns out to be less promising than other methods.

One procedure requires a measurement of the total -pair cross-section close to threshold, where the size of is most sensitive to the  mass. The energy proposed is 161 . For this method the theory should obviously predict with a sufficient accuracy, to wit  .aaaThroughout this report the required theoretical accuracy is taken to be half the expected statistical error. Therefore the radiative corrections (RC’s) to the total cross-section should be under control.

The other method looks at the decay products of the  bosons, in particular at the four- and two-jet production. From the measured momenta of the decay products one tries to reconstruct the  mass. In this reconstruction a good knowledge of the -pair centre-of-mass energy is essential. Since there will be an energy loss due to initial-state radiation (ISR) of primarily photons, the -pair energy will be different from the laboratory energy of the incoming electrons and positrons. So, for this method the ISR should be well under control, i.e. the theoretical error on the average energy loss, , should be less than 30 , which translates into a theoretical error of less than 15  on the reconstructed  mass. This again is an aspect of radiative corrections.

1.1.2 Test of non-abelian couplings

Within the SM the triple gauge-boson couplings have the Yang-Mills (YM) form. Amongst others, this specific form for the TGC’s leads to a proper high-energy behaviour of the -pair cross-section and is a requirement for having a renormalizable theory. Couplings different from the YM form, called anomalous or non-standard couplings, will in general lead to a high-energy behaviour of cross-sections  increasing with energy and thereby violating unitarity. At LEP2 the energy is too low to see such effects and in order to establish the presence of anomalous couplings one therefore has to study in detail the angular distributions of the  bosons and their decay products. In particular, the angular distribution in the -production angle is sensitive to non-standard couplings. Again, the knowledge of RC’s to the tree level SM predictions is required, since they affect the Born-level angular distributions [9].

As elsewhere in this volume a detailed report on non-standard TGC’s is given [10], only a few comments will be made here.

When one considers the most general coupling between three gauge bosons allowed for by Lorentz invariance, assuming the gauge bosons to be coupled to conserved currents, one ends up with seven possible couplings for the  and  interaction. Of these seven there are three which are CP violating and one which is CP conserving but C and P violating [11].

In practice it will be impossible to set limits on all these couplings. Therefore usually some assumptions are made to reduce the number of 14 couplings [11, 12]. For instance, one may restrict oneself to CP-conserving non-standard couplings so that 8 couplings are left. Of these the electromagnetic ones can be reduced further by omitting the C and P violating one and requiring the strength of the electromagnetic coupling to be determined by the charge. Two possible anomalous electromagnetic couplings remain and four  anomalous couplings. Even with this reduced number it will be impossible to set experimental limits on all of them simultaneously.

However, there are theoretical arguments that such a purely phenomenological approach is also not required. First of all one might use symmetry arguments, motivated by specific models for the non-standard physics, to find relations between the anomalous couplings [12]. Alternatively, when one considers the electroweak theory as an effective theory originating from a field theory that manifests itself at higher energies, then also some small anomalous couplings may be present at lower energies. In such a gauge-invariant framework the non-standard physics, situated at an energy scale , decouples at low energies and the anomalous TGC’s are suppressed by factors , according to the dimension () of the corresponding operators. This naturally introduces a hierarchy amongst the anomalous TGC’s based on the dimension of the corresponding operators [13].

From the perspective of this report the origin of the non-standard couplings is not so important, but the fact that they often modify angular distributions is relevant. Also SM effects – like RC’s, the finite decay width of the  bosons, and background contributions – provide deviations from the tree-level distributions [9, 10].

1.2 How to obtain accurate predictions?

In order to extract the wanted information from -pair production it is clear from the remarks above that RC’s are needed for total and differential cross-sections and, moreover, for the determination of the energy loss. Anticipating   -pair events, the theoretical accuracy that should be targeted for the SM predictions is  , although specific final states, distributions, or observables in fact often do not require such a precision (like, e.g., at 161  or the energy loss).

Ideally one would like to have the full RC’s to the final state of four fermions, which originate from the two decaying vector bosons. In practice such a very involved calculation does not exist and is hopefully not required in its complete form for the present accuracy. For the discussion of the LEP2 situation and strategy it is useful to distinguish three levels of sophistication in the description of -pair production.

The first level is to consider on-shell -pair production, , which at tree level is described by three diagrams: neutrino exchange in the channel, and and  exchange in the channel. Here the complete RC’s are known, comprising the virtual one-loop corrections and the real-photon bremsstrahlung [14, 15]. When one wants to divide the corrections into different parts the situation differs from LEP1 [9].

As to the bremsstrahlung, a gauge-invariant separation into initial-state radiation, final-state radiation (FSR), and its interference is not possible like, e.g., in -pair production at LEP1. The reason is that the photon should couple to all charged particles in a line of the Feynman diagram. The -channel diagram then makes a separation into ISR and FSR impossible. However, the leading logarithmic (LL) part of ISR is in itself gauge-invariant. This can be combined with the LL QED virtual corrections so that a LL description with structure functions for ISR can be given [16].

A separation of the virtual corrections into a photonic and weak part is also not possible since charged vector bosons are already present at Born level, necessitating an interplay between - and -exchange diagrams in order to preserve gauge invariance.

Once one has a description of on-shell -pair production one can attach to it the on-shell  decay. Again, the RC’s to this decay are known [17][20].

The next level of sophistication is to consider off-shell production of  pairs, which then decay into four fermions [21]. The  propagators with energy-dependent widths can be taken into account. Although this description of four-fermion production through virtual  bosons is a natural extension of the on-shell evaluation, it is not a gauge-invariant treatment. In fact there are more diagrams needed to calculate such a four-fermion process. As to RC’s, the ISR and FSR can be implemented in LL approximation, but the full set of virtual corrections have not yet been calculated.

The final level for the study of -pair production would be a full -corrected evaluation of all possible four-fermion final states. At tree level there now exist evaluations where, besides the three off-shell -pair diagrams, all other diagrams for a specific four-fermion final state have been included [22]. On top of that ISR can be taken into account. Again, one has to be aware of possible gauge-invariance problems. In particular, the introduction of energy-dependent widths in the -propagators will destroy electromagnetic gauge invariance and may introduce dramatically wrong cross-sections in certain regions of phase space.

As long as a full -corrected evaluation of four-fermion production is not available, certain approximative schemes, like for instance the ‘pole scheme’ [23][25], may be useful. This goes beyond the treatments where only ISR through structure functions is taken into account. Actual numerical results from a complete ‘pole-scheme’ evaluation are not yet available.

2 On-Shell W-Pair Production and W Decay

Although the actual process that will be probed at LEP2 is , we first focus on the production and subsequent decay of on-shell  bosons, being basic building blocks in some of the schemes for handling off-shell  bosons. In contrast to off-shell -pair production the on-shell processes are not plagued by the problem of gauge invariance for unstable particles and a complete set of  radiative corrections is available. Consequently, they are well suited for studying the typical sizes of various radiative corrections. Moreover, many of the important features of the production and decay of off-shell  bosons are already contained in the on-shell limit. We indicate explicitly where the width of the  bosons radically changes the on-shell predictions.

2.1 Notation and conventions

We use the Bjørken–Drell metric and fix the totally antisymmetric tensor by . The matrix is defined as and the helicity projectors , which are used to project on right- and left-handed massless fermions, as


First we set the conventions for the process


where the arguments indicate the momenta and helicities of the incoming fermions and outgoing bosons (, ). Note that we sometimes use the shorthand version in certain sub- and superscripts. In the centre-of-mass (CM) system of the pair, which we will refer to as the laboratory (LAB) system in the following, the momenta read


with denoting the beam energy, the scattering angle between the and the , and the velocity of the  bosons. The Mandelstam variables used in the following are given by


In order to define helicity amplitudes we need to introduce the corresponding polarization vectors for the  and  bosonbbbNote that the helicity of the massive  bosons is not Lorentz-invariant. We define it in the LAB system.


Because we are neglecting the electron mass, the helicity of the positron is opposite to the helicity of the electron


Henceforth we refer to the helicity amplitudes for -pair production as . CP invariance implies the relation:


This holds in the SM if we neglect the CP-violating phase in the quark-mixing matrix. Even in the presence of this CP-violating phase, the CP breaking occurs at  in SM -pair production and is additionally suppressed by the smallness of the mixing angles between the quarks. Consequently in the SM one effectively has only 12 independent helicity-matrix elements instead of 36. In the presence of substantial (non-standard) CP violation this number should be increased to 18. This allows for a decomposition of the matrix elements in terms of an explicit set of 12 (18) independent basic matrix elements multiplied by purely kinematical invariant functions (coefficients) [9, 14, 15].

From the helicity amplitudes the differential cross-sections for explicit -boson polarization and various degrees of initial-state polarization can be constructed. For example the differential cross-section for unpolarized electrons, positrons, and  bosons is given by


For the numerical evaluations we have to fix the input parameters. We use the following default set [2, 26]:

The masses of the light quarks are adjusted in such a way that the experimentally measured hadronic vacuum polarization [27] is reproduced. In the actual calculations either these light quark masses are used or the dispersion-integral result for the hadronic vacuum polarization of [27]. The strong coupling constant is calculated according to the parametrization of [28], using as input. The -pair threshold region is very sensitive to  and consequently, if  were to be calculated from the other parameters using the muon decay width including radiative corrections (like at LEP1), to the masses of the top quark and Higgs boson. This is of course not very natural, since we want to use  as a model-independent fit parameter. In view of this we use , ,  as inputcccWe do not eliminate  or , as  is needed in the RC’s and  reduces the size of the RC’s to the production and decay of the  bosons., treat  as free fit parameter, and calculate  from muon decaydddIt should be noted that the RC’s associated with the muon decay make that this parameter set is not overcomplete.. This calculated value of  can then be confronted with the direct bounds from Fermilab (weighted average [7]) and the indirect ones from the precision measurements at LEP1/SLC (

At present the error on this value as a result of the 160  error on  is roughly 27 . The precise  measurement at LEP2 will lead to a reduction of this error by a factor of four. Of course the so-obtained top-quark mass will become - and -dependent through the RC’s associated with the muon decay. We come back to that point in section 2.3.3 and only mention here that the  dependence of  amounts to roughly 33  in the range , which clearly exceeds the above-mentioned expected error from  (see table 1).

260 300 1000
80.10 119.1 137.3 154.2
80.18 133.9 151.6 168.0
80.26 148.1 165.3 181.2
80.34 161.7 178.3 193.9
80.42 174.4 190.7 206.1
Table 1: Calculated  for and different Higgs- and -boson masses, using the state-of-the-art calculation described in section 2.3.3. The theoretical error in  is roughly 1–2 .

Finally, the sine and cosine of the weak mixing angle are defined by


2.2 Lowest order

In the SM there are three lowest-order diagrams (figure 1), if we omit a Higgs-exchange diagram that is suppressed by a factor and thus completely negligible.

Figure 1: Lowest-order diagrams for .

The -channel diagram involving the exchange only contributes for left-handed electrons. The -channel diagrams, containing the non-abelian triple gauge-boson couplings, contribute for both helicities of the electron. The corresponding matrix element reads


with for left-handed electrons and for right-handed electrons, and


After inserting the explicit form of the -boson–fermion couplings


we can organize the lowest-order amplitude into two gauge-invariant subsets:




The gauge invariance of the two contributions and can be simply inferred from the fact that they are accompanied by different coupling constants, one of which involving the electromagnetic coupling constant , the other the charged-current coupling constant . Whereas corresponds to the pure contribution, the parity-conserving contribution is a result of the symmetry-breaking mechanism.

The lowest-order cross-section determines the essential features of -pair production. The threshold behaviour is important for the determination of the  mass from the measurement of the cross-section in a single energy point very close to threshold [8], i.e. at . For small the matrix elements behave as


for fixed scattering angles. Consequently, the -channel matrix elements vanish at thresholdeeeThis holds for arbitrary CP-conserving -channel contributions in the limit of vanishing electron mass [11]. and the -channel graph dominates in the threshold region. For the differential cross-section for unpolarized beams and  bosons is given by [9]


where the leading term originates from the -channel diagram only. Note that the leading term is angular-independent. In the total cross-section


all terms drop out and the -channel and the -interference contributions are proportional to . This is the consequence of CP conservation, fermion-helicity conservation in the initial state, and the orthogonality of different partial waves [9]. Hence in the threshold region the channel is dominant and the cross-section for  is not very sensitive to the triple gauge-boson couplings.

In table 2 we give the integrated cross-section for different centre-of-mass energies and different polarizations of the electrons (+,). Positrons are assumed to be unpolarized.

Table 2: Integrated lowest-order cross-section in pb for different polarizations of the electrons and different centre-of-mass energies.

Using right-handed electrons one could study a pure triple-gauge-coupling process, but this would require longitudinally polarized electron beams, the prospect of which looks rather unfavourable for LEP2. Furthermore, for all energies the cross-section for right-handed electrons is suppressed by two orders of magnitude compared with the dominant left-handed mode, mainly because there is no -channel contribution. Therefore essentially only the latter can be investigated at LEP2. With transverse beam polarization, however, one could obtain information on the right-handed matrix element via its interference with the left-handed one.

As stated in section 2.1, for on-shell -pair production and unpolarized electrons and positrons there are nine independent helicity-matrix elements or six if CP is conserved. These yield nine or six independent observables. Taking into account the decay of the  bosons there are many more observables: 81 or, if CP is conserved, 36 products of the various helicity-matrix elements [11]. Because the structure of the  decays is well established, these observables can be extracted in a model-independent way from the data for the five-fold differential cross-section 

2.3 Radiative corrections

As has been argued in section 1, the SM theoretical predictions for -pair production should have an uncertainty of about 0.5% (2% at ) in order to obtain reasonable limits on the structure of the gauge-boson self-couplings, and to determine the -boson mass with the envisaged precision of roughly 40–50 . In this context radiative corrections are indispensable.

Much effort has been made in recent years to obtain such precise theoretical predictions for -pair production. In the following we discuss the existing results for the virtual and real electroweak corrections in the on-shell case. In addition we discuss the quality of an improved Born approximation (IBA) that contains all familiar, LEP1-like leading corrections for the -pair production cross-section at LEP2 energies. Such a discussion is in particular relevant as the present off-shell LEP2 Monte Carlos often make use of such an approximation.

2.3.1 Radiative electroweak  corrections

The radiative corrections can be naturally divided into three classes, the virtual, soft-photonic, and hard-photonic contributions. Since the process involves the charged current in lowest order, the corresponding radiative corrections cannot be separated on the basis of Feynman diagrams into electromagnetic and weak contributions in a gauge-invariant way. Like we have already observed in equation (2.2), gauge invariance requires an interplay between the - and -exchange diagrams.

The complete radiative electroweak  corrections have been calculated independently by two groups [14, 15]gggThe process has also been calculated in [29].. For the unpolarized case they have been checked and found to agree within a couple of per-mil (i.e. within the integration error).

The virtual corrections contain infra-red (IR) divergences, which result from virtual photons exchanged between external charged particles. They are compensated for by adding the cross-section for the process . If the energy of the emitted photon is small compared with the detector resolution (soft photons), this process cannot be distinguished experimentally from the non-radiative -pair production process. In practical experiments the soft-photon approximation is in general not sufficient and one has to include the radiation of hard photons, too. When adding these real-photon effects to the contribution of the virtual corrections, not only the IR singularities but also the large Sudakov double logarithms drop out.

Still there are various sources of potentially large  corrections left at LEP2 energies. First of all there are large QED corrections of the form with , originating from collinear photon radiation off the electron or positron (see appendix A). They form a gauge-invariant subset of QED corrections and amount to roughly 6% at LEP2, not taking into account possible enhancements from the corresponding coefficients (like, e.g., close to thresholds).

From the renormalization two sets of potentially large fermionic (formally weak) corrections arise. The first set is associated with the charge renormalization at zero momentum transfer, where the relevant scale is set by the fermion masses entering the vacuum polarization. In high-energy experiments, however, the running charge should be evaluated at scales much larger than the masses of the light fermions . This leads to large logarithmic (‘mass singular’) contributions of the form with , which can amount to a shift in of 8% at LEP2 energies. Related to the top quark, corrections will occur. They show up as universal corrections via the renormalization of the  and  masses (or equivalently ) if the corresponding renormalization scales are small compared with the mass splitting in the isospin doublet.

Finally the long-range electromagnetic interaction between the slowly moving  bosons leads to the so-called Coulomb singularity [30]. This singularity yields an  correction factor , resulting in an -corrected cross-section  that does not vanish at threshold for left-handed electrons. The right-handed cross-section remains suppressed by at least . This correction factor exhibits the fact that the free-particle approximation is inadequate near the -pair production threshold in the presence of the long-range Coulomb interaction. We want to stress here that the Coulomb singularity is changed substantially by effects that effectively truncate the range of the interaction, like the off-shellness and the decay of the  bosons. This will be treated in section 3.4.2 and will lead to the conclusion that higher-order Coulombic corrections to the total cross-section are not important.

The sensitivity of the total unpolarized cross-section to and the unknown mass of the Higgs boson is illustrated in table 3.

300 300 300 260 300 1000
0.117 0.123 0.129 0.123 0.123 0.123
164.80 165.26 165.73 148.14 165.26 181.20
2161.0 22.473 22.472 22.470 22.514 22.472 22.460
2175.0 14.471 14.465 14.459 14.548 14.465 14.422
2184.0 16.619 16.613 16.607 16.671 16.613 16.568
2190.0 17.263 17.257 17.251 17.301 17.257 17.213
2205.0 17.683 17.677 17.671 17.698 17.677 17.637
Table 3: Integrated unpolarized cross-section including radiative electroweak  corrections for different values for the Higgs-boson mass and , at various centre-of-mass energies. The theoretical error in  is roughly 1–2 .

The dependence on , originating from the calculation of , is completely negligible (). Compared with the lowest-order cross-sections of table 2, a variation of  between 60 and 1000 , however, influences the total cross-section by around 0.5% at the three highest energy points, by about 0.8% at , and by 1.4% at threshold. These numbers are lowered by about 0.5% if the radiative corrections are calculated in the so-called  representation, which absorbs the universal  and  effects present in the  wave-function factors (see section 2.4.2). In this representation a variation of  between 300 and 1000  has a negligible impact on the cross-section. Keeping in mind that we would like to reach a theoretical accuracy of 2% (0.5%) at (), it should be clear that the Higgs-mass dependence constitutes a major uncertainty. A more detailed investigation [31] revealed that this Higgs-boson-mass dependence is the result of the Yukawa interaction between the two slowly moving  bosons (mediated by the Higgs boson). As such the effect is largest close to threshold and for the lightest Higgs masses, which yield the shortest range of the interaction. For instance, upon increasing the lower Higgs-mass bound from 60 to 90 , the resulting uncertainty is reduced by roughly a factor of twohhhAt the start of the 161  run no significant change in the  bound is expected. So, only after the higher-energy LEP2 runs have taken place the improved knowledge on  can be used for an a posteriori reduction of the  dependence of the 161  run..

In figure 2 we display the influence of the full  corrections on the total unpolarized cross-section near the -pair production threshold.

Figure 2: Unpolarized total cross-section in the threshold region. The dotted curve corresponds to Born, the dashed one to -Born, and the solid one to the results including full radiative electroweak   corrections.

As has been stated before, it should be noted that, unlike at LEP1, the presence of the charged current at lowest order in  complicates the separation between QED and weak corrections. So, in order to have a cleaner look at the QED corrections we present in addition to the Born cross-section also the lowest-order cross-section in the so-called  representation, i.e. the Born cross-section with replaced by , which already contains an important part of the leading weak effects discussed aboveiiiAs the channel is dominant at LEP2 energies, the -Born describes the leading weak corrections reasonably well for the default Higgs mass =300 . (see also section 2.3.3). We will refer to this cross-section as -Born. As a result of the steep drop with decreasing centre-of-mass energy of the -pair cross-section  close to threshold, large and predominantly soft QED effects can be observed. Compared with -Born the  corrections amount to more than in the direct vicinity of the threshold (), apart from the region very close to threshold where the positive Coulomb-singularity contribution takes over (see close-up in figure 2), and still () at (190 ). The size of these effects necessitates the inclusion of higher-order QED corrections in order to end up with an acceptable theoretical uncertainty. A discussion of these higher-order QED corrections can be found in section 2.3.3. The finite  width has a drastic impact on the effects related to the Coulomb singularity (see section 3.4.2), but the large soft QED effects are merely smoothened and stay sizeable.

In figure 3 the effect of the  corrections on the unpolarized differential cross-section  is shown for . Here stands for the polar angle of the  boson with respect to the incoming positronjjjIn the presence of hard-photon radiation in general ..

Figure 3: Unpolarized differential cross-section  at . The same signature as adopted in figure 2.

Apart from the expected normalization effects that are already observed in the total cross-section, also a distortion of the distribution occurs. As this is exactly the type of signature one might expect from anomalous gauge-boson couplings, this underlines again the importance of having a profound knowledge of the SM radiative corrections. The origin of the distortion can be traced back to hard initial-state photonic corrections. Hard-photon emissions boost the centre-of-mass system of the  pair. As a result of that, events that are forward in the centre-of-mass system of the produced  bosons can show up as backward events in the LAB system and vice versa. Since the cross-section in the backward direction is substantially lower than in the forward direction, the net effect of this redistribution (migration) of events will be a distortion of the differential distribution with respect to the lowest-order one. Of course these boost effects are much more pronounced at high energies [9].

2.3.2 Approximations in the LEP2 region

The complete analytic results for the electroweak  corrections are very lengthy and complicated, resulting in huge and rather slow computer programs. Moreover, the formulae are completely untransparent. In view of this, simple approximative expressions are desirable. Apart from providing fast computer programs, which are useful for many applications, simple transparent formulae should reveal the physical content and the origin of the dominant radiative corrections. Furthermore, if these approximative expressions represent the exact corrections adequately, one might consider implementing them in the existing LEP2 Monte Carlos.

Owing to the lack of a calculation of the complete corrections, the present LEP2 Monte Carlos for off-shell -pair production include only the known leading universal corrections. In order to assess the theoretical uncertainty inherent in this approach, the on-shell case can be used as guideline. The size of the non-leading corrections in this case should provide a reasonable estimate for the corresponding left-out non-leading corrections in the off-shell case.

We start out by investigating the structure of the matrix element for . Whereas it involves only three different tensor structures in lowest order, at twelve independent tensor structures occur, each of which is associated with an independent invariant function, which can be considered as an - and -dependent effective coupling. The dominant radiative corrections, as e.g. those that are related to UV, IR, or mass singularities, in general have factorization properties and are at  restricted to those invariant functions that appear already at lowest order. Therefore the contributions of the other invariant functions should be relatively small. Indeed, a numerical analysis reveals that in a suitably chosen representation for the basic set of independent matrix elements only the three Born-like invariant functions plus one extra right-handed piece, related to , are relevant for a sufficiently good approximation [32, 33]. This results in the following approximation for the matrix element:


with and defined in equation (2.2). The term involving is needed for right-handed electrons because of contributions , originating from the interference of with particular 1-loop matrix elements, which are not present at lowest order. Neglecting the other invariant functions in the basis given in [32] introduces errors well below the per-cent level (see table 4). This is of course only true for observables where is not suppressed or absent. All this demonstrates that improved Born approximations are possible. Note, however, that in contrast to the situation at LEP1 the invariant functions are both energy- and angular-dependent.

In order to construct an improved Born approximation (IBA) one has to specify simple expressions for the invariant functions , which reproduce the corresponding exact expressions with sufficient accuracy. In the LEP2 energy region the following expressions can be used as a reasonable ansatz [32]


The terms containing and incorporate all the leading universal corrections associated with the running of and the corrections associated with the parameter (see also section 2.3.3). As these are linked to the renormalization of the electric charge at zero momentum transfer and of the weak mixing angle, they only contribute to the structures already present at lowest order. The corresponding leading  contributions can be recovered from equation (2.3.2) by substituting




The term describes the effect of the Coulomb singularity, which for yields a simple correction factor to the lowest-order cross-section:


The factor is introduced by hand to switch off the Coulomb singularity fast enough. In this way one avoids corrections that are too large away from threshold, where the Coulomb singularity should not play a role anymore. As has been studied in [32], heavy-mass contributions and that are not covered by equation (2.3.2) have a negligible impact on the approximation. Since the above IBA analysis has been performed for the default Higgs-boson mass , the large light-Higgs-boson corrections close to threshold, shown in table 3, are absent. By adding a simple approximation for these corrections [31] to the IBA, however, the full  dependence of the exact  corrections can be reproduced.

In addition to the contributions described so far, one has to include the leading logarithmic QED corrections. These can be calculated using the structure-function method as described in appendix A. They comprise all contributions . In the following numerical analysis the scale has been used and, in order to extract the effect of the non-leading virtual corrections, hard-photon radiation is left out.

As we want to compare the approximation with an calculation we do not use the square of the matrix element defined in equations (19) and (2.3.2) for the numerical analysis, but only the square of the terms involving and and the interference of these terms with the others. Moreover, in this interference and are replaced by the corresponding lowest-order expressions in terms of and . Nevertheless, the so-obtained approximation still contains higher-order contributions through the squares of and . To allow for a meaningful comparison these have been included in the numbers for the exact one-loop results given in table 4 in the same way (see [32] for more details).

unpolarized right-handed left-handed
total 1.45 0.00 1.56 0.01 1.45 0.00
10 1.63 0.00 4.41 0.00 1.63 0.00
90 1.44 0.00 1.57 0.01 1.44 0.00
170 1.26 0.00 7.52 0.00 1.26 0.00
total 1.27 0.02 2.09 0.01 1.28 0.02
10 1.67 0.00 0.49 0.00 1.67 0.00
90 1.17 0.02 2.09 0.02 1.18 0.02
170 0.75 0.00 4.64 0.00 0.77 0.00
total 1.26 0.03 2.58 0.02 1.28 0.03
10 1.71 0.00 0.18 0.00 1.71 0.00
90 1.03 0.05 2.59 0.03 1.06 0.05
170 0.59 0.00 5.30 0.00 0.69 0.00
total 1.02 0.04 2.80 0.03 1.06 0.04
10 1.57 0.00 2.17 0.01 1.57 0.00
90 0.67 0.08 2.82 0.05 0.72 0.08
170 0.10 0.00 7.72 0.01 0.32 0.00
total 1.24 0.03 2.91 0.04 1.28 0.03
10 1.67 0.00 0.59 0.01 1.67 0.00
90 0.95 0.06 2.92 0.06 1.01 0.06
170 0.58 0.00 6.32 0.00 0.83 0.00
total 1.60 0.00 3.11 0.09 1.65 0.00
10 1.77 0.00 1.68 0.01 1.77 0.00
90 1.55 0.00 3.14 0.12 1.64 0.00
170 1.61 0.00 4.37 0.00 1.94 0.00
Table 4: Quality of the IBA (first column per polarization) and the form-factor approximation (19) (second column per polarization), given in per cent relative to Born. Results are given for the total cross-section (total) and the differential cross-section at 10, 90, and .

In table 4 we show the difference between the exact and approximated virtual and soft  radiative corrections to the total and differential cross-section for right-handed, left-handed, and unpolarized electrons. The positrons are assumed to be unpolarized. In the LEP2 energy region the relative difference between the exact result and the approximation can be as large as 1–2% for left-handed or unpolarized electrons, and reach 3–8% for right-handed electrons. As there is no obvious reason why these remaining non-leading corrections should be smaller in the case of off-shell  bosons or , their inclusion in the LEP2 data analysis seems to be unavoidable.

2.3.3 Higher-order corrections

In section 2.3.1 we have encountered large  corrections. A short description of the way to include the corresponding higher-order corrections is hence in place.

In two distinct ways the higher-order corrections enter the calculation of the distributions for -pair production. First of all there is the calculation of  from  and the input parameters, either to be used in the calculation of the -pair RC’s or to be confronted with the Tevatron data. As  enters the relation between , , , and , resulting from the muon decay width, at the 1-loop level, the highest precision possible is required for this relation. As this relation relies on calculations performed at for the muon decay width and at the subtraction points for the renormalization procedure, we can use the state-of-the-art calculation developed for the LEP1 analysis [6]. This yields


Usually is written in the form


where contains all the one-loop corrections to the muon decay width with the inclusion and the proper arrangement of the higher-order terms. Next we subdivide as introduced in equation (25) into a leading term, , and remainder terms, , as follows:




This contains all the terms known at present: the complete one-loop corrections (two-, three-, four-point functions) and complete two-loop insertions to two-point functions, from which the leading and terms are subtracted. The Born version of equation (24), i.e. , relates  directly to , , and , hence leaving no room for using  as additional independent input parameter. However, having introduced radiative corrections to the Born relation we can solve equation (24) in terms of the top quark mass and, in turn, this particular value for  will be used throughout the rest of the calculation. One should notice that this procedure is not free of ambiguities since is also a function of , but with much smaller sensitivity due to the well known screening. So in the end we should also allow for some variation in . Moreover, is also a function of through higher-order corrections, for instance , and, as a consequence, also some variation in the strong coupling constant should be allowed.

The leading terms and appearing in equation (27) are given by


where is the renormalized  transverse photon self-energy originating from fermion loops, excluding top-quark loops.

The term in equation (28) is a next-to-leading order term, whose proper treatment is rather important:


where denotes the unrenormalized transverse self-energy of the V gauge boson. The ultra-violet (UV) divergences are removed according to the renormalization scheme with .

In contrast to , the leading contribution appearing in equation (26) is normalized by  rather than by , as is required by the resummation proposed in [34]:


where and


The coefficients and describe the first- and second-order QCD corrections for the leading contribution to , calculated in [35, 36]. Correspondingly:


with the total number of flavours (). The function describes the leading two-loop electroweak correction to , calculated first in the approximation in [37] and later in [38] for an arbitrary relation between  and .

It should be noted that the higher-order radiative corrections discussed above are usually calculated in the limit of a heavy top mass, i.e. usually only the leading part of the corrections is under control. This often raises the question of what effect can be estimated from the sub-leading terms, since . As far as QCD and corrections are concerned the sub-leading terms are well under control. In the pure electroweak sector, however, there has been, so far, no calculation of higher-order sub-leading terms at an arbitrary scale. The only available piece of calculation concerns the parameter at [39], therefore relevant only for scattering. If however one is willing to extrapolate the result to a higher scale, by assuming that the ratio of leading to next-to-leading corrections is representative for the corresponding theoretical uncertainty, then the answer is next-to-leading leading.

Even before considering the actual impact of the higher-order terms on , we should mention at this point that the way in which the non-leading terms can be treated and the exact form of the leading–remainder splitting give rise to several possible options in the actual implementation of radiative corrections. This in turn becomes a source of theoretical uncertainty. For instance, for we can introduce the decomposition into leading and remainder. Since we know how to proceed with all objects in the leading approximation, the only ambiguity is due to the treatment of the remainders. Clearly, after the splitting there are in principle several possible ways of handling the remainder: