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\title{Weak Corrections to Three-Jet Production in Electron-Positron 
Annihilations\footnote{Work supported in part by the U.K.\ Particle Physics and
Astronomy Research Council (PPARC),
by the European Union (EU) under contract HPRN-CT-2000-00149 and by the 
Italian Ministero dell'Istruzione, dell'Universit\`a e della Ricerca
(MIUR) under
713\_006.}}
\author{E. Maina\\
Dipartimento di Fisica Teorica -- Universit\`a di Torino\\
Via Pietro Giuria 1, 10125 Torino, Italy\\
and \\
Istituto Nazionale di Fisica Nucleare -- Sezione di Torino\\
Via Pietro Giuria 1, 10125 Torino, Italy\\
E-mail: {\tt maina@to.infn.it}}
\author{S. Moretti\\
CERN -- Theory Division\\
CH-1211 Geneva 23, Switzerland\\
and\\
Institute for Particle Physics Phenomenology -- University of Durham\\
South Road, Durham DH1 3LE, UK\\
E-mails: {\tt stefano.moretti@cern.ch, stefano.moretti@durham.ac.uk}}
\author{D.A. Ross\\
Department of Physics and Astronomy -- University of Southampton\\
Highfield, Southampton SO17 1BJ, UK\\
Email: {\tt dar@hep.phys.soton.ac.uk}} 
\abstract {We report on the calculation of the
 factorisable one-loop weak-interaction corrections
to  the initial and final states for  
 three-jet observables in electron-positron annihilations.
We show that whereas such corrections are negligibly small at LEP energies 
they are significantly enhanced at a future Linear Collider, where we
argue that they may be similar in size to the two-loop QCD effects.
The calculation has been performed using helicity amplitudes so that
it can be applied to the case of polarised beams. The
effects of beam polarisation can be extracted from an analysis
of the angular distributions of the jets relative to the incident  
electron/positron directions.}

\keywords{QCD processes, 
Electroweak effects, Loop calculations, Lepton colliders}
\preprint{{DFTT 27/02}\\ 
{CERN-TH/2002-191}\\
{IPPP/02/48}\\
{DCPT/02/96}\\
{SHEP-02/24}\\ 
{September 2002}}
\begin{document}
\section{Electroweak corrections at high energies}
\label{Sec:EWCorr}

Strong (QCD) and Electroweak (EW) interactions
are two fundamental forces of 
Nature, the latter in turn unifying
weak and electromagnetic (EM) interactions. 
Together they constitute the Standard Model (SM) of particle physics.
A clear hierarchy exists between the strengths of the two interactions
at the energy scales probed by past and present high energy particle 
accelerators (e.g., LEP, SLC, HERA, RHIC and Tevatron): 
QCD forces are stronger than EW ones. This is quantitatively
 manifest if one recalls that the value of the QCD coupling `constant', 
$\alpha_{\mathrm{S}}$, measured at these machines is much larger 
than the EW one, $\alpha_{\mathrm{EW}}$, typically, by an order of
magnitude.

The above-mentioned particle accelerators have been operating at the 
so-called `EW scale', ${\cal O}(M_{\mathrm{EW}})$, around the mass 
of the EW gauge bosons, 
$M_{\mathrm{EW}}\sim M_W\sim M_Z$. (Recall that the energy
at which partons are scattered in hadronic collisions is only
a fraction the actual beam energy.) However, the
future generation of machines, such as the approved proton-proton Large
Hadron Collider (LHC) and the proposed electron-positron Linear
Collider (LC) \cite{LCs}, will enter the multi-TeV energy regime. 
The centre-of-mass (CM) energy of the LHC will in fact be 14 TeV whereas
current designs of LCs consider values up to 3 TeV or so.

At such high energy scales, this hierarchy  between QCD and EW
interactions may no longer be respected. The reason is twofold. On the one
hand, while both coupling constants 
become smaller with increasing energy, the evolution (or `running') with 
energy of $\alpha_{\mathrm{S}}$ is faster than for $\alpha_{\rm{EW}}$.
On the other 
hand, weak corrections to production cross-sections  
are enhanced by double logarithmic factors
which, unlike in QCD, do not cancel for `infrared-safe' observables. 
For example, at one-loop level,
in the case of the inclusive cross-section of $e^+e^-$
into hadrons, the QCD corrections are of  ${\cal O}
(\frac{\alpha_{\mathrm{S}}}{\pi})$, whereas
the EW ones are of ${\cal O}(\frac{\alpha_{\mathrm{EW}}}{4\pi}\log^2
\frac{s}{M^2_{{W}}})$, where $s$ is the collider CM energy
squared \cite{Kuroda:1991wn} --\cite{Ciafaloni:1999xg}.
At $\sqrt s=1.5$ TeV, the former are identical to the latter,
of order 9\% or so. 
The two aspects may combine, giving rise to some
manifestations (both at the LHC and a LC) of the two forces which are of
similar size.

The origin of these `double logs' is understood.
It is due to a lack of cancellation of infrared (both soft
and collinear) virtual and real emission in
higher order contributions. This is in turn a consequence of the 
violation of the Bloch-Nordsieck theorem in non-Abelian theories
\cite{Ciafaloni:2000df}\footnote{Recently, it has been
found that Bloch-Nordsieck violation can also occur in spontaneously
broken Abelian gauge theories, if the incoming particles are mass
eigenstates that do not coincide with gauge eigenstates
\cite{Ciafaloni:2001vt}. In the SM this is particularly 
relevant for incoming longitudinal gauge bosons or Higgs scalars
\cite{Ciafaloni:2001vu}.}.
The problem is in principle present also in QCD. In practice, however, 
it has no observable consequences, because of the final averaging of the 
colour degrees of freedom of partons, forced by their confinement
into colourless hadrons. This does not occur in the EW case,
where, e.g., the initial state has a non-Abelian charge,
dictated by the given collider beam configuration, such as in $e^+e^-$
collisions. Needless to say, the same argument  also holds
for an initial quark doublet in proton-proton scatterings. These
logarithmic corrections (unless the EW process is mass-suppressed)
are universal (i.e., process independent) and are finite (unlike in
QCD), as the masses of the EW gauge bosons provide a physical
cut-off for $W$- and $Z$-emission. Hence, for typical experimental
resolutions, softly and collinearly emitted weak bosons need not be included
in the production cross-section and one can restrict oneself to the calculation
of weak effects originating from virtual corrections and
affecting a purely hadronic final state. Besides, these contributions can  be
isolated in a gauge-invariant manner from purely EM effects
\cite{Ciafaloni:1999xg}, \cite{Beccaria:2000fk} --\cite{Beccaria:2001yf}, 
which may or may not
be included in the calculation, depending on the observable being studied. 

In view of this,  it becomes of crucial importance to assess
the quantitative relevance of such weak corrections
affecting, in particular, key QCD processes studied at present at future 
colliders \cite{Denner:2001mn}. (See 
Refs.~\cite{Beccaria:2000fk} --\cite{Layssac:2001ur}
for a collection of papers dealing with resummed, one- and two-loop 
EW corrections to various high energy processes.)
It is the aim of our paper to report on the computation of the 
one-loop weak effects entering three-jet production in electron-positron
annihilation via the subprocess $e^+e^-\to\gamma^*,Z^{(*)}\to \bar 
qqg$\footnote{See Ref.~\cite{2jet} for the corresponding weak corrections
to the Born process $e^+e^-\to\bar qq$ and Ref.~\cite{4jet} for the
$\sim n_{\rm f}$ component of those to $e^+e^-\to \bar qqgg$ (where 
$n_{\rm f}$ represents the number of light flavours).
For two-loop results on the former, see \cite{Beenakker:2000kb}.},
when the higher order effects arise only from initial or final state
interactions. 
These represent the so-called `factorisable' corrections, i.e.,
those involving loops 
not connecting the initial leptons to the final quarks.
The remainder, `non-factorisable' corrections,
while being negligible at $\sqrt s=M_{Z}$ (where the width 
of the $Z$ resonance provides a natural cut-off for off-shellness
effects), 
are expected to play a quantitatively relevant role as $\sqrt s$ grows
larger. The latter will however be the subject of a future publication. 
Here, we will focus our attention on the former at typical
LEP1/SLC, LEP2 and LC energies, with and without the option of
beam polarisation. While one-loop weak effects are expected to appear
at the percent level at LEP1/SLC, sizable logarithmic enhancement or 
suppression should already be visible at LEP2, while becoming comparable
to the one-loop QCD corrections at LCs. 

On the subject of higher order QCD
effects, it should be mentioned that a great deal of effort has    
recently been devoted to evaluate two-loop contributions
to the three-jet process\footnote{The one-loop QCD results
have been known for some time \cite{ERT}.} 
(albeit, only at the amplitude level so far,
as there are no numerical results available yet). However, we will argue in our
study that one-loop weak effects are equally important,  particularly as one increases the CM energy of the collider.

\section{One-loop weak effects in three-jets events at leptonic colliders}
\label{Sec:ee}
 
In the case of $e^+e^-$ annihilations, the most important QCD quantity to be 
extracted from multi-jet events is precisely $\alpha_{\mathrm{S}}$.
The confrontation of the measured value of the strong coupling
constant with that predicted by the theory through the 
renormalisation group evolution is an important test of the
SM or else an indication of new physics, whose typical mass scale is larger
than the collider energy, 
but which can manifest itself through virtual effects. 
Jet-shape observables, which offer a handle on non-perturbative
QCD effects via large power corrections, would be affected as well.

A further aspect that should be recalled is that weak corrections naturally
introduce parity-violating effects in jet observables, detectable through
asymmetries in the cross-section, which are often regarded as an indication
of physics beyond the SM. These effects are further enhanced if polarisation
of the incoming beams is exploited.  Although at past and present polarised
colliders (such as SLC and RHIC) parity-violating weak effects are threshold
suppressed because the machines operate close to the EW scale, this will no 
longer be true at future accelerators. While there exists a realistic 
possibility of exploiting beam polarisation at the LHC, this option is
one of the strengths of the LC projects. Comparison of theoretical predictions 
involving parity-violation with future experimental data 
is regarded as another powerful tool for confirming or 
disproving  the existence of some beyond the SM scenarios, such as those 
involving right-handed weak currents and/or new massive gauge bosons.

The plan of the rest of the paper is as follows. In the next Section,
we describe the calculation. Then, in Sect.~\ref{Sec:Results},
we present some numerical results. We conclude in Sect.~\ref{Sec:Conclusions}.

\section{Calculation}
\label{Sec:Calculation}

Since we are considering weak corrections that can be
identified via their induced parity-violating effects and since we wish to
apply our results to the case of polarised electron and/or positron  
beams, it is convenient to work in terms of helicity matrix elements
(MEs). Thus, we define the helicity amplitudes 
${\cal A}^{(G)}_{\lambda_1, \lambda_2, \sigma}$ 
for an off-shell gauge boson of type $G$ (hereafter,
a photon $\gamma$ or a $Z$-boson) of helicity $\lambda_1$ decaying
into a gluon with helicity $\lambda_2$, a massless quark
with helicity $\sigma$ and a massless
 antiquark with opposite helicity\footnote{Note 
that all interactions considered here preserve the helicity along
the fermion line, including
those in which Goldstone bosons appear inside the loop, since these either
 occur in pairs or involve a mass insertion on the fermion line.}.
Since the gauge boson is off-shell, the helicity $\lambda_1$ can take 
the values
 $+1, \, 0, \, -1$, where $0$ indicates the longitudinal polarisation,
whereas $\lambda_2$ and $ \sigma$ can only be equal to $\pm 1$.
 
%At tree level the helicity amplitudes are only functions of the 
%energy fractions $x_1$ and $x_2$ of the antiquark and quark in the final 
%state, respectively,
% the EW coupling $g_j^{(G)}$ of the (anti)quark of type $j$
%(carrying information on both helicity {and} flavour
%of the latter) to the relevant
%gauge boson and the QCD coupling $g_{\rm S}\equiv
%\sqrt{4\pi\alpha_{\mathrm S}}$. Specifically, in case of massless
%(anti)quarks (i.e., $m_q=0$), we have (here, $\tau^a$ represents a colour
%matrix):

The general form of these amplitudes may be written as
\be {\cal A}^{(G)}_{\lambda_1, \lambda_2, \sigma}
 \ = \ \bar{u}(p_2) \Gamma \frac{\left(1+\sigma \gamma^5\right)}{2} v(p_1),
\ee
where $p_1$ and $p_2$ are the momenta of the outgoing antiquark and quark 
respectively and $\Gamma$ stands for a sum of strings
 of Dirac $\gamma-$matrices
with coefficients, which, beyond tree level, 
involve integrals over loop momenta.
Since the helicity $\sigma$ of the fermions is conserved the strings 
must contain an odd number of $\gamma-$matrices. Repeated use of the
 Chrystoffel 
identities\footnote{These identities are only valid in four dimensions.
In our case, where we do not have infrared (i.e., 
soft and collinear) divergences, 
it is a simple matter to isolate the ultraviolet divergent contributions,
which are proportional to the tree-level MEs,
 and handle them separately. However, in $d$ dimensions
one needs to account for the fact that there are $2^{d/4}$ 
helicity states for the fermions and $(d-2)$ for the gauge bosons.
The method described here will {not} correctly trap terms
proportional to $(d-4)$ in coefficients of divergent integrals. 
It is probably for this reason that the formalism of Ref.~\cite{twol3p}
is considerably more cumbersome than that presented here.}
 means that $\Gamma$ can always be expressed in the
form
\be \Gamma \ = \ C_1 \, \gamma \cdot p_1 \ + \ 
C_2 \, \gamma \cdot p_2 \ + \ C_3 \, \gamma \cdot p_3 \ + \ 
C_4 \, \sqrt{Q^2} \, \gamma \cdot n , \label{helicityme1} \ee
where $p_3$ is the momentum of the outgoing gluon, $Q^2=(p_1+p_2+p_3)^2$ is
 the square momentum of the gauge boson,
and  $n$ is a unit space-like vector normal to the jet plane.
The coefficient functions $C_i$ depend on the helicities  
$\lambda_1, \ \lambda_2, \ \sigma$ as well as the 
energy fractions $x_1$ and $x_2$ of the antiquark and quark in the final 
state and on all the couplings and masses of particles that enter into
the relevant perturbative contribution to the amplitude. 

For massless fermions the MEs of the first two terms
of eq.~(\ref{helicityme1})
vanish, and we are left with
\begin{eqnarray} \ {\cal A}^{(G)}_{\lambda_1, \lambda_2, \sigma} &=& 
C_3 \, \bar{u}(p_2) \gamma \cdot p_3 \frac{\left(1+\sigma \gamma^5\right)}{2}
 v(p_1) \, + \, C_4 \sqrt{Q^2}
\, \bar{u}(p_2) \gamma \cdot n \frac{\left(1+\sigma \gamma^5\right)}{2}
 v(p_1), \nonumber \\ &=& 
C_3 \, Q^2 \sqrt{(1-x_1)(1-x_2)} \ - \ i \, \sigma\, C_4 \, Q^2 
  \sqrt{x_1+x_2-1}. \end{eqnarray}


The relevant coefficient functions $C_3$ and $C_4$ 
are scalar quantities and can be projected on a graph-by-graph basis
using the projections
\be C_3 \ = \ {\rm Tr} \left( \Gamma \gamma \cdot v
\frac{\left(1+\sigma \gamma^5\right)}{2}
 \right),\ee  
where $v$ is the vector
$$ v \ = \ \frac{(1-x_2) p_1 + (1-x_1) p_2 - (x_1+x_2-1) p_3}
{2 Q^2 (1-x_1)(1-x_2)},
$$
and
\be C_4 \ = \ -\frac{1}{2\sqrt{Q^2}}{\rm Tr} \left( \Gamma \gamma \cdot n
\frac{\left(1+\sigma \gamma^5\right)}{2}
 \right)  .\ee 


At tree level the helicity amplitudes are only functions of $x_1, \ x_2$,
 the EW couplings $g_j^{(G)}$ of the (anti)quark of type $j$
(proportional to $g_{\rm{W}}\equiv\sqrt {4\pi \alpha_{\rm{EW}}}$,
with $\alpha_{\rm{EW}}=\alpha_{\rm{EM}}/\sin^2\theta_W$, and
carrying information on both helicity {and} flavour
of the latter) to the relevant
gauge boson and the QCD coupling $g_{\rm S}\equiv
\sqrt{4\pi\alpha_{\mathrm S}}$. Specifically, in case of massless
(anti)quarks (i.e., $m_q=0$), we have (here, $\tau^a$ represents a colour
matrix):
\br\label{LO}
{\cal A}^{(G)}_{1,1,1} \ = \ {\cal A}^{(G)}_{-1,-1,-1}
 & = & -2 i g_j^{(G)} g_{\mathrm S} \tau^a \frac{x_1}{\sqrt{(1-x_1)(1-x_2)}},
\nonumber \\
{\cal A}^{(G)}_{1,1,-1} \ = \ {\cal A}^{(G)}_{-1,-1,1}
 &=& -2 i g_j^{(G)} g_{\mathrm S} \tau^a \frac{\sqrt{(1-x_1)(1-x_2)}}{x_1},
\nonumber \\
{\cal A}^{(G)}_{0,-1,1} \ = \ {\cal A}^{(G)}_{0,1,-1}
 &=& - 2 \sqrt{2} i g_j^{(G)} g_{\mathrm S} \tau^a {\sqrt\frac{(1-x_1-x_2)}{x_1}},
\er
with all others being zero. These zero values do {not}, in general,
remain such in the presence of weak corrections 
and this can lead to a relative enhancement of the latter, in comparison
to QCD effects at the same order.

At one-loop level such helicity amplitudes   
acquire higher order corrections from the self-energy insertions on the
fermions and gauge bosons shown in Fig.~\ref{se_graphs},
from the vertex corrections shown in Fig.~\ref{vertex_graphs}
and from the box diagrams shown in Fig.~\ref{box_graphs}. As we have neglected 
here the masses of the final-state quarks, such higher order corrections
depend on the ratio $Q^2/M_{{W}}^2$, where $Q^2$ is the square
momentum of the gauge boson, as well as the EM coupling
constant $\alpha_{\rm EM}$ and the weak mixing angle $s_W\equiv\sin\theta_W$
(with $\alpha_{\rm EW}=\alpha_{\rm EM}/s_W^2$).
Furthermore, in the case where the final state fermions are $b$-quarks, 
the loops involving the exchange of a $W$-boson lead to effects of
virtual $t$-quarks, so that the corrections also depend on 
the ratio $m_t^2/M_{{W}}^2$. (It is only in this case that the graphs 
involving
the exchange of the Goldstone bosons associated with the $W$-boson graphs
are relevant.)

The self-energy and vertex correction graphs contain ultraviolet divergences.
These have been subtracted using the `modified' Minimal Subtraction
($\MSbar$) scheme at
the scale $\mu=M_Z$. Thus the couplings are taken to be
those relevant for such a subtraction: e.g., the EM coupling,
$\alpha_{\mathrm{EM}}$, has been taken to be $1/128$ at the above subtraction
point. The one exception to this
renormalisation scheme has been the case of the self-energy insertions
on external fermion lines, which have been subtracted on mass-shell,
so that the external fermion fields create or destroy particle states
with the correct normalisation.

All these graphs are infrared and collinear convergent so that they
 may be expressed in terms of Passarino-Veltman \cite{VP} functions
which are then evaluated numerically. The expressions for
 each of these diagrams 
have been calculated using FORM \cite{FORM} and checked by an
independent program based on FeynCalc \cite{FeynCalc}. For the numerical
evaluation of the scalar integrals we have relied on FF \cite{FF1.9}. 
A further check on our results has been carried out
by setting the polarisation vector of the photon proportional to its momentum
and verifying that in that case the sum of all one-loop diagrams
vanishes, as required by gauge invariance.
The full expressions for the contributions from these graphs are too
lengthy to be reproduced here.


In terms of the helicity MEs we define the following 
``spin-matrix'' tensors, only depending on the 
polarisation state of the off-shell gauge boson,
\be {\cal T}^{(GG^\prime)}_{\lambda\lambda^\prime}  \ = \ 
 \sum_{\lambda_2,\sigma} 
{\cal A}^{(G)}_{\lambda, \lambda_2, \sigma}
\left({\cal A}^{(G^\prime)}_{\lambda^\prime, \lambda_2, \sigma}\right)^\dagger,
\ee
where the (anti)quark and gluon helicities have been summed over.
These tensor elements are real at tree level, but in general acquire an
imaginary part at one loop arising from the cuts of the
loop integrations above the threshold for the production
of the internal particles. 

Finally, we define the customary nine form-factors, $F_1, ... F_9$,
describing the differential structure of a three-jet final state
in terms of the above spin-matrix tensors, as follows:
\br\label{FFs}
 F_i \ & = & \ \frac{\alpha_{\mathrm{EM}}}{512\pi^3} \left[
\frac{\left(\eta_A^{L(R)}\right)^2}{Q^2} f_i^{AA} +
 \frac{(1-\lambda_e-4 s_W^2)}{4s_W \, c_W}
\frac{\left (\eta_A^{L(R)} \eta_Z^{L(R)} \right)}
 {\Re e\{Q^2-M_Z^2+i\Gamma_Z M_Z\}} \left(f_i^{AZ}+f_i^{ZA}\right)
\right. \nonumber \\ & & \left.  \hspace*{1.25cm} +
 \left(\frac{1-\lambda_e-4s_W^2)}{4s_W \, c_W}\right)^2
\frac{Q^2\left(\eta_Z^{L(R)}\right)^2}
{\left(\Re e\{Q^2-M_Z^2+i\Gamma_Z M_Z\}\right)^2} f_i^{ZZ} \right]
\ \  (i=1, ... 9), \label{eq33}
\er
where
\br f_1^{GG^\prime} &=& \left( {\cal T}^{GG^\prime}_{1,1}
 +{\cal T}^{GG^\prime}_{0,0} +{\cal T}^{GG^\prime}_{-1,-1} \right), \nonumber \\
 f_2^{GG^\prime} &=& {\cal T}^{GG^\prime}_{0,0}, \nonumber \\
 f_3^{GG^\prime} &=& -2 \, \Re e  \left( {\cal T}^{GG^\prime}_{1,1}
 -{\cal T}^{GG^\prime}_{-1,-1} \right), \nonumber \\
 f_4^{GG^\prime} &=& -\sqrt{2} \, \Re e \left( {\cal T}^{GG^\prime}_{1,0}
 +{\cal T}^{GG^\prime}_{-1,0} \right), \nonumber \\
 f_5^{GG^\prime} &=& -2 \, \Re e  {\cal T}^{GG^\prime}_{1,-1}, \nonumber \\
 f_6^{GG^\prime} &=& 2 \sqrt{2}\, \Re e \left( {\cal T}^{GG^\prime}_{1,0}
 - {\cal T}^{GG^\prime}_{-1,0} \right), \nonumber \\
 f_7^{GG^\prime} &=& \sqrt{2} \, \Im m\left( {\cal T}^{GG^\prime}_{0,1}
  -{\cal T}^{GG^\prime}_{0,-1} \right), \nonumber \\
 f_8^{GG^\prime} &=& 2 \, \Im m {\cal T}^{GG^\prime}_{1,-1}, \nonumber \\
 f_9^{GG^\prime} &=& -2 \sqrt{2} \, \Im m \left( {\cal T}^{GG^\prime}_{0,1}
 +{\cal T}^{GG^\prime}_{0,-1} \right), \label{eq34}
\er
with $\eta_G^{L(R)}$ the weak correction factor to the coupling of the
left(right)-handed electron to the gauge boson $G$ and $\lambda_e$
the helicity of the incoming electron beam (assumed always to be
of opposite helicity to the incoming positron beam).

Up to an overall constant, these form-factors are the same as those 
introduced, e.g., in Refs. \cite{BDS,KS}.
The last three ($F_7, ... F_9$) can arise for the first time at the one-loop 
level, since they are proportional to the imaginary parts of the spin-matrix.
Besides, $F_3, \ F_6$ and $F_7$ vanish in the parity-conserving limit and 
can therefore be used
as probes of weak interaction contributions to three-jet production.
(Moreover, $F_3$ and $F_6$ would be exactly 
 zero at tree level if the leading order 
process were only mediated by virtual photons.) 


These form-factors further generate the double differential cross-section for 
three-jet production in terms of some event shape variable, $S$, 
which is in turn related to  $x_1, \, x_2$ by some function, $s$, i.e.,
$S = s(x_1,x_2) $,
and of the polar and azimuthal angles, $\alpha, \, \beta$, 
between, e.g., the incoming electron beam and the antiquark jet, 
by\footnote{A qualitative difference between the expressions
of the form-factors, $F_i$ ($i=1,...9$),
 used here and those of Refs. \cite{BDS,KS} is that we do {not}
include the sign of the axial vector coupling of the electron
to the exchanged gauge boson in our definitions.
 In this way the difference between the differential cross-sections
for left- and right-handed polarised electron beams is manifest
 in eq.~(\ref{eq35}).}
\br\label{angles}
 \frac{d^3\sigma}{dS \, d\cos\alpha \, d\beta} & =  &\int dx_1 dx_2
 ~\delta \left(S-s(x_1,x_2) \right)    
\left[ (2-\sin^2\alpha) \, F_1  +(1-3\cos^2\alpha) 
\, F_2
\right. \nonumber \\ 
 & &  \hspace*{1cm}  + \, \lambda_e \,
 \cos\alpha \, F_3  + \sin 2\alpha \cos\beta \, F_4 
+ \sin^2\alpha \cos 2\beta \, F_5 + \lambda_e \,
 \sin\alpha \cos\beta \, F_6 \nonumber \\
& & \hspace*{1 cm} + \, \sin 2 \alpha \sin \beta \, F_7 + \sin^2 \alpha \sin 2 \beta \,
 F_8 
\left. + \, \lambda_e \, \sin\alpha \sin\beta \, F_9 \right] . \label{eq35}
 \er  
Note that upon integrating over the antiquark jet angle 
relative to the electron
beam, only the form-factor $F_1$ survives.

In general, it is not possible to distinguish between quark, antiquark
and gluon jets, although the above expression can easily be adapted such that 
the angles $\alpha, \beta$ refer to the leading jet. However, (anti)quark
jets {can} be recognised when they originate from primary 
$b$-(anti)quarks, thanks to rather efficient flavour tagging techniques
(such as  $\mu$-vertex devices). We will therefore consider 
the numerical results for such a case separately.

\section{Numerical results}
\label{Sec:Results}

The processes considered here are the following: 
\begin{equation}\label{procj}
e^+e^-\to \gamma^*,Z^{(*)}\to \bar qqg\quad{\mathrm{(all~flavours)}},
\end{equation}
when no assumption is made on the flavour content of the final state,
so that a summation will be performed over $q=d,u,c,s,b$-quarks, and 
\begin{equation}\label{procb}
e^+e^-\to \gamma^*,Z^{(*)}\to \bar bbg,
\end{equation}
limited to the case of bottom quarks only in the final state. 
As already intimated, all quarks
in the final state of (\ref{procj})--(\ref{procb}) are taken as 
massless\footnote{Mass effects in $e^+e^-\to \gamma^*,Z^{(*)}\to \bar bbg$
have been studied in \cite{BMM} and \cite{bbgNLO}.}.
In contrast, the top quark entering the loops in both reactions has
been assumed to have the mass $m_t=175$ GeV. The $Z$ mass used was
$M_Z=91.19$ GeV and was related to the $W$-mass, $M_W$, via the
SM formula $M_W=M_Z\cos\theta_W$, where $\sin^2\theta_W=0.232$.
(Corresponding widths were $\Gamma_Z=2.5$ GeV and $\Gamma_W=2.08$ GeV.)
For $\alpha_{\rm S}$ we have used the two-loop expression for
$\lms=200$ MeV throughout
(yielding, $\alpha_{\rm S}(M_Z)=0.11$).  

We will be studying processes (\ref{procj})--(\ref{procb}) at typical
LEP1/SLC ($\sqrt s=M_Z$), LEP2 ($\sqrt s=130-210$ GeV) and LC ($\sqrt s=
350$ GeV  $-$ 3 TeV) CM energies. We systematically neglect higher
order effects from EM radiation, including those due to Initial State 
Radiation (ISR) or beamstrahlung. In fact, although these are known 
to be non-negligible
(especially at LC energies), we expect them to have a similar effect on 
both the tree-level and one-loop descriptions, 
hence being irrelevant for our purpose. In this context, we should like
to elaborate further on the purely EM corrections to the final state of
processes (\ref{procj})--(\ref{procb}). 
Those to the form-factor  $F_1$ have already been calculated,
since they can be extracted from the Abelian part of the NLO-QCD corrections
(see \cite{ERT}) by replacing $C_F$ by unity and $\alpha_{\rm S}$ by 
$\alpha_{\rm EM}$.
As was pointed out in Ref.~\cite{DAR}, these corrections are dominated
by a term $\sim\alpha_{\mathrm{EM}} \pi/2$ 
multiplying the tree-level cross-section. This contribution is $\sim 1 \%$
and is independent of the jet event shape. A further correction,
associated with the Sudakov form-factor, acts in the negative direction
and is subdominant  away from the two-jet region (i.e., up to 
values of $\sim0.95$ for the Thrust, see below for its definition).
There is no reason to believe that 
these EM corrections would be enhanced for other form-factors.
Furthermore, unlike the weak contributions, final state EM 
effects are energy independent. Therefore,
while we expect the EM and weak corrections to be comparable at LEP1 energies,
where the mentioned $\log^2(s/M_{{W}}^2)$ enhancement is
negligible and where the $1/\sin\theta_W$ factor entering
the weak couplings may well compensate for the $M_W$- and $M_Z$-driven
mass suppression in the propagators of the weak loops, weak effects should
gradually take over the purely EM ones as the CM energy grows much larger
than $M_{{W}}$.

It is common in the specialised literature to define the $n$-jet fraction
$R_n(y)$ as
%
\beq
\label{fn}
R_n(y)=\frac{\sigma_n(y)}{\sigma_0},
\eeq
%
where $y$  is a suitable
variable quantifying the space-time separation among hadronic objects
and with $\sigma_{0}$ identifying the (energy-dependent) 
Born cross-section for $e^+e^-\to \bar qq$.

For the choice $\mu=\sqrt s$ of the renormalisation scale, 
one can conveniently write the three-jet fraction in the following form:
%
\beq
\label{f3}
R_3(y) =     \left( \frac{\as}{2\pi} \right)    A(y)
           + \left( \frac{\as}{2\pi} \right)^2  B(y) + ... ,
\eeq
where the coupling constant $\as$ and the functions $A(y)$ and $B(y)$ 
are defined in  the $\overline{\mbox{MS}}$ scheme. An experimental fit
of the $R_n(y)$ jet fractions to the corresponding 
theoretical prediction is a powerful
way of determining $\as$ from multi-jet rates.

Through order ${\cal O}(\alpha_{\rm{S}})$
processes (\ref{procj})--(\ref{procb}) are the leading order (LO) perturbative 
contributions to the corresponding three-jet cross-section\footnote{Hereafter,
perturbative contributions are refereed to relatively to the 
${\cal O}(\alpha_{\rm{EM}}^2)$ two-jet rate.}, as defined via
eqs.~(\ref{fn})--(\ref{f3}). 
The LO terms, however, receive  higher order corrections from both QCD
and EW interactions and we are concerned here with the
next-to-leading order (NLO) ones only. 
Whereas at LO all the contributions to the three-jet cross-section
come from the tree-level parton process $e^+e^-\rightarrow \bar qq g$
(which contributes to the $A(y)$ function above),
at NLO the QCD contributions to the three-jet rate (hereafter, denoted
by NLO-QCD) are due to two sources. First, the real emission diagrams for the 
processes $e^+e^-\rightarrow q\bar q gg$ and 
$e^+e^-\rightarrow q\bar q Q\bar Q$, in which
one of the partons is `unresolved'. This can happen when one has either
two collinear partons within one jet or one soft parton outside the
jet. Both these contributions are (in general, positively) divergent. 
Thanks to the Bloch-Nordsieck \cite{BN} and Kinoshita-Lee-Nauenberg
\cite{KLN} theorems, these collinear
and soft singularities are cancelled at the same order in $\as$  by the 
divergent contributions (generally negative) provided by the second source,
namely, the virtual loop graphs. Therefore, after renormalising
the coupling constant $\as$, a finite three-jet cross-section is obtained and
the function $B(y)$ accounts for the above-mentioned three- and four-parton 
QCD contributions\footnote{In order to calculate these,
we make use here of a program based on Ref.~\cite{EERAD}.}. 
While the EM component of the EW corrections may be treated
on the same footing as the QCD one (with the additional photon playing
the role of a second gluon), the weak corrections of interest (hereafter,
labelled as NLO-W) only contribute to three-parton final states. Hence,
in order to account for the latter, it will suffice to make the replacement
\beq
\label{f3EW}
A(y)\to A(y)+A_{\mathrm{W}}(y)
\eeq
in eq.~(\ref{f3}).

The decision as to whether two hadronic 
objects are unresolved or otherwise is usually taken
through the application onto the hadronic final state
of a so-called `jet clustering algorithm',
wherein the number of {\sl clusters}\footnote{Here
  and in the following, the word `cluster' refers to hadrons or
  calorimeter cells in the real experimental case, to partons in the
  theoretical perturbative calculations and also to intermediate jets
  during the clustering procedure.}  
is reduced one at a time by combining the two most (in some sense)
nearby ones. The joining procedure is stopped by testing against some
criterion and the final clusters are called jets. 

As jet clustering schemes\footnote{We 
acknowledge here the well admitted abuse of referring to the various
jet `finders'
both as algorithms and as schemes, since the last term was originally
intended to identify the composition law of four-momenta when pairing two
clusters: in our case, $p^\mu_{ij}=p^\mu_{i}+p^\mu_{j}$.}, 
we have used a selection of
the `binary' ones, in which only two objects are clustered together at any 
step. Given two clusters
labelled as $i$ and $j$, the measure of their `distance' is normally
denoted by $y_{ij}$ and the minimal separation allowed by $y_{\rm{cut}}$.
The algorithms are the following:
the JADE (J) one \cite{jade}, which uses as a measure of separation 
the quantity
\begin{equation}\label{J}
y_{ij}^{\rm J} = {{2 E_i E_j (1-\cos\theta_{ij})}\over{s}};
\end{equation}
the Durham (D) \cite{durham} and the Cambridge (C)
\cite{cambridge}  ones, both using\footnote{The Cambridge algorithm in 
fact only modifies the clustering procedure of the Durham jet finder and the
two implementations coincide for $n\le 3$ parton final states.}
\begin{equation}\label{D_C}
y_{ij}^{\rm D} \equiv y_{ij}^{\rm C} 
= {{2\min (E^2_i, E^2_j)(1-\cos\theta_{ij})}\over{s}};
\end{equation}
the Geneva (G) one \cite{BKSS},  for which one has
\begin{equation}\label{G}
y_{ij}^{\rm G} = \frac{8}{9} \frac{E_i E_j (1 - \cos\theta_{ij})}%
{(E_i + E_j)^2}.
\end{equation}

In eqs.~(\ref{J})--(\ref{G}), $E_i$ and $E_j$ are the energies
 and $\theta_{ij}$ the angular separation
of any pair $ij$ of clusters in the final state. 
The choice of these particular schemes has a simple motivation.
The D and C ones are different versions of `transverse-momentum'
based algorithms, whereas the J and G ones use an `invariant-mass' measure
(see \cite{schemes} for a review).
In fact, these two categories are those that have so far been employed 
most in phenomenological studies of jet physics
in electron-positron collisions, with the former
gradually overshadowing the latter, thanks to their reduced
scale dependence in higher order QCD (e.g., in the case of the
${\cal{O}}(\as^2)$ three- \cite{schemes} --\cite{mb}
and ${\cal{O}}(\as^3)$ four-jet rates 
\cite{as4}) and to smaller hadronisation effects in the same contexts
(see Refs.~\cite{schemes,BKSS}).

Fig.~\ref{fig:y_LEP1} presents the $A(y)$, $-A_{\rm{W}}(y)$ 
and $B(y)$ coefficients 
entering eqs.~(\ref{f3})--(\ref{f3EW}), as a function of 
$y(\equiv y_{\rm{cut}})$ for the four above
jet algorithms at LEP1\footnote{Notice that $A(y)$ and $A_{\rm{W}}(y)$ for the
C scheme are identical to those for the D one (recall the previous footnote)}.
%Notice that
%the sign of $A_{\mathrm{W}}$ has been reversed for better presentation.
Upon multiplying $B(y)$ by $\left( \frac{\as}{2\pi} \right)$, a comparison
to $A(y)$ and $A_{\rm{W}}(y)$ reveals that the NLO-W corrections are
negative and remain
indeed at the percent level, i.e., of order $\frac{\aem}{2\pi s_W^2}$
without any logarithmic enhancement (since $\sqrt s\approx M_{{W}}$).
They give rise to corrections to $\sigma_3(y)$ of around 2\%,
and
thus are generally much smaller than the NLO-QCD ones. In this context, no
systematic difference is seen with respect to the choice of jet clustering
algorithm, over the typical range of application of the latter at LEP1 (say, 
$\ycut\gsim0.005$ for D, C and $\ycut\gsim0.01$ for G, J).

Table.~\ref{tab:y_LEP2} reports the total three-jet cross-section, 
i.e., the numerator of the right-hand side of
eq.~(\ref{fn}) for $n=3$, at LO and the two corrections to it, NLO-QCD and
NLO-W, at the representative LEP2 energy of $\sqrt s=200$ GeV. 
Again, given the relatively
small collider energy compared to the EW scale $M_{{W}}$, the
pattern already seen at LEP1 for the relative size of the NLO effects
is repeated here.

\begin{table}[h]
\begin{center}
\begin{tabular}{|c||c|c|c|}
\hline
\multicolumn{4}{|c|}{$\sigma_3(e^+e^-\to \bar qqg)$ (pb)}\\
\hline
$\sqrt s$ (GeV) & LO & NLO-QCD & NLO-W \\
\hline
200 & 13. & $-3.26$ & $+0.24$ \\
\hline
\hline
\multicolumn{4}{|c|}{D scheme \quad\quad\quad\quad\quad\quad\quad $\ycut=0.001$}\\
\hline
\end{tabular}
\end{center}
\caption{Total cross-sections for the tree-level process
(\ref{procj}) alongside the NLO-QCD and NLO-W corrections, at 
$\sqrt s=200$ GeV, for the D scheme with $\ycut=0.001$.}
\label{tab:y_LEP2}
\end{table}


Figs.~\ref{fig:scan_3jet} and \ref{fig:scan_bbg} show the energy dependence of
the total cross-section $\sigma_3(y)$ at LO and the corresponding NLO-W
correction alongside the relative size of the latter for processes
(\ref{procj}) and (\ref{procb}), respectively, over the energy range
200 GeV $\le \sqrt s\le$ 3 TeV, that is, from LEP2 up to typical CLIC
energies \cite{LCs}. 
One can appreciate the doubly logarithmic growth with energy
of the NLO-W term, whose size increases up to 15--20\% of the LO 
rate at the upper end of the energy spectrum. 
Indeed, well above 500 GeV, the correction term in 
Figs.~\ref{fig:scan_3jet} and \ref{fig:scan_bbg} is substantially
reproduced by the expression $\frac{\alpha_{\rm{EW}}}{4\pi} \log^2
\frac{s}{M_W^2}$.
Furthermore, one may clearly notice at 
$\sqrt s\approx 2m_t$ the threshold effect
of the one-loop weak diagrams in which a pair of virtual top quarks
is produced in triangle- and box-type graphs. The effect is, as expected,
more pronounced for process (\ref{procb}) than for process (\ref{procj}). 
Rates are plotted here only for the D scheme with $\ycut=0.001$,
for reference, as the pattern seen for other choices of algorithm and/or
resolution parameter is very similar. 

Fig.~\ref{fig:y_LC} is the counterpart of Fig.~\ref{fig:y_LEP1} for
a LC with $\sqrt s=1$ TeV. Since in this case one has that $\sqrt s\gg 
M_{{W}}$,
the NLO-W corrections are rather large and comparable to the NLO-QCD ones
over the interesting $y$ range, hence well above typical experimental
uncertainties entering the determination of three-jet observables
at LC energies. Besides, contrary to the lower energy case,
NLO-W effects are here always positive.
At these energies, the NLO-W corrections are of order 10\%. Such 
corrections are likely to be similar in size to the 
next-to-next-to-leading order (NNLO) corrections from QCD.
 Indeed, if the speculation in Ref.~\cite{DAR}
that the dominant contribution to the QCD correction exponentiates, then at
 $\sqrt s=1$ TeV, the NNLO-QCD corrections should be in the
same range.

In view of this, it is then worthwhile considering the effects
of NLO-W corrections to some other `infrared-safe' jet observables typically
used in the determination
of $\as$, the so-called `shape variables' \cite{KunsztNason}. 
A representative quantity in this respect is the Thrust (T)
distribution \cite{thrust}. This is defined as the sum of 
the longitudinal momenta relative to the (Thrust) axis $n_{\rm T}$ chosen
to maximise this sum, i.e.:
\begin{equation}\label{thrust}
\thrust = {\rm max} \frac{\sum_i |\vec{p_i}\cdot\vec{n_{\mathrm{T}}}|}
                         {\sum_i |\vec{p_i}|} ,
\end{equation} 
where $i$ runs over all final state clusters.
This quantity is identically one at Born level, getting
the first non-trivial
contribution through ${\cal O}(\as)$ from events of the
type (\ref{procj})--(\ref{procb}). Also notice that any other higher 
order contribution will affect this observable. Through ${\cal O}(\as^2)$,
for the choice $\mu =\sqrt s$ of the renormalisation scale, 
the T distribution can be parametrised in the following form:
\begin{equation}\label{T}
(1-{\rm{T}})\frac{d\sigma}{d\thrust}\frac{1}{\sigma_0} = 
\left(\frac{\as}{2\pi}\right)   A^{\Thrust}(\thrust)+
\left(\frac{\as}{2\pi}\right)^2 B^{\Thrust}(\thrust).
\end{equation} 
Again, the replacement 
\begin{equation}\label{TW}
A^{\Thrust}(\thrust)\to A^{\Thrust}(\thrust)+A^{\Thrust}_{\rm{W}}(\thrust) 
\end{equation} 
accounts for the inclusion of the NLO-W contributions.

We plot the terms $\left(\frac{\as}{2\pi}\right)A^{\Thrust}(\thrust)$,
$\left(\frac{\as}{2\pi}\right)A^{\Thrust}_{\rm{W}}(\thrust)$ and 
$\left(\frac{\as}{2\pi}\right)^2B^{\Thrust}(\thrust)$ in Fig.~\ref{fig:thrust},
always at  $\sqrt s=1$ TeV,
alongside the relative rates of the NLO-QCD and NLO-W terms 
with respect to the LO contribution. Again, it can be seen that
the NLO-W effects can reach the level of $+10\%$ or so. We then combine 
in Fig.~\ref{fig:thrustNNLO} the
above expressions for the Thrust coefficient functions into the
complete event shape distribution, according to eq.~(\ref{T}).
Here, alongside the LO, NLO-QCD and NLO-W results, we add an estimate of the 
NNLO-QCD effects based on the considerations of Ref.~\cite{DAR}, all plotted
over a restricted T range, where experimental fits are usually performed.   
Here, it can be appreciated that the NLO-W effects are 
indeed of the same size as  those expected from NNLO-QCD.
Exact calculations of the latter are
currently being performed  by several groups \cite{EERAD}, 
\cite{ERT} --\cite{twol3p}.  

The ability to polarise electron (and possibly, positron) beams
renders future LCs a privileged environment in which to test the
differential structure of hadronic samples. In fact, 
our earlier remark should be recalled: 
that the latter may well carry the distinctive hallmark of some new
and heavy strongly interactive particles (such as squarks
and gluinos in Supersymmetry), whose rest mass is too
large for these to be produced in pairs
as real states but that may enter as virtual 
objects into multi-jet events. 
Similar effects may however also be induced by the NLO-W corrections tackled 
here. Therefore, as a benchmark for future studies along the above lines, we
reproduce in Figs.~\ref{fig:FFs-L}--\ref{fig:FFs-R} the differential
structure of the nine form-factors given in eq.~(\ref{FFs}) for
processes (\ref{procj})--(\ref{procb}),
as a function of the energy fractions $x_1$ and $x_2$, at $\sqrt
s=1$ TeV (at this energy, the shape is basically the 
same for both final states in (\ref{procj})--(\ref{procb})), 
for the case of a left- and right-handed 
incoming electrons, respectively. As already intimated
in the previous Section, $F_7$ to $F_9$ are identically zero at 
LO\footnote{This is strictly true only for massless quarks, as, for $m_q\ne 0$,
Ref.~\cite{BDS} has shown that $F_9$ becomes non-zero.}, even prior to 
any integration in $\alpha,\beta$
and/or averaging over the $e^+e^-$ helicities. Besides, $F_7$ would remain
zero unless corrections involve parity-violating interactions. As for 
$F_1,... F_6$, we should mention that the NLO-W corrections to the
corresponding tree-level distributions were found of the order
10--15\% on average, with some maxima/minima of order 50\% in
specific regions of the $x_1,x_2$ plane: e.g.,
for $F_5$ when $x_1,x_2\to 1$ and for $F_6$ when $x_1\to0$ and $x_2\to 1$,
for both the left- and right-handed electron 
polarisations\footnote{For reason of 
space, we refrain
from presenting here the LO dependence of $F_1,...F_6$ in term
of $x_1$ and $x_2$. This can easily be reproduced starting from
the formulae in (\ref{LO}).}.

Notice that all of the nine 
distributions in Figs.~\ref{fig:FFs-L}--\ref{fig:FFs-R}, 
in the presence of a precise determination
of $\alpha$ and $\beta$ (or, for that matter,
any other combination of angles used to parametrise the event orientation),
are directly observable in the case the
$\bar bbg$ final state if also the charge (other than the flavour)
of the quark is known, e.g., via high-$p_T$ lepton tagging
from $B$-meson decays or via global jet charge determination.
If not, all distributions in Figs.~\ref{fig:FFs-L}--\ref{fig:FFs-R}
have to be symmetrised around the $x_1=x_2$ direction. In the case
of the full hadronic sample $\bar qqg$, when no flavour tagging
is available, one normally identifies the two most energetic jets
in the three-jet sample with those originating from the
quark-antiquark pair. Although not always true, this approach
is known to be a good approximation for most studies (see, e.g.,
Ref.~\cite{ordering}). Hence, even in this case one may be able
to verify to a good approximation
the shape of the form-factors $F_1,...F_9$ in terms
of the energy fractions.

Finally, Fig.~\ref{fig:allthrust} presents the shape 
and normalisation of what we call the `unintegrated' (or `oriented') Thrust 
distributions associated to each of the form-factors in
eq.~(\ref{angles}) (wherein $S={\rm T}$),  at any fixed values of $\alpha$ and $\beta$,
alongside the absolute value of the 
relative size of the NLO-W corrections with respect
to the LO case,
for the form-factors  $F_1,...F_6$, which are non-zero at the Born level.
 Here, the form-factors have been averaged over the electron
 beam polarisations. For these form-factors 
NLO-W corrections can be either positive
or negative, depending on the form-factor being considered, and 
can be as large as 10--12\% or so.

\section{Conclusions}
\label{Sec:Conclusions}

On the basis of our numerical findings in the previous Section, we should like
to conclude as follows.

\begin{itemize}
\item At LEP1 and LEP2 energies the size of the NLO-W corrections is
rather small, of order percent or so, hence confirming that determinations
of $\as$ at such colliders are stable in this respect and that the
SM background to parity-violating effects possibly induced by new physics
is well under control.

\item As the CM energy of the collider grows, NLO-W effects on the
inclusive three-jet cross-section become more and more visible,
being competitive with the NLO-QCD ones starting from the TeV region (the
one that will be explored by future LCs). Indeed, at such high energies,
they could be  larger than NNLO-QCD effects. Moreover, when
$\sqrt s\approx 2m_t$, the NLO-W corrections show a characteristic
kinematic feature, a threshold effect depleting their contribution to the
cross-section locally by
up to a factor of four (in the case of $b\bar bg$ final states), induced
by virtual $t\bar t$ production. This peculiar SM effect should not be 
mistaken for the onset of some new physics phenomena.

\item Exclusive observables in three-jet events are
 also affected by large 
NLO-W effects  at the TeV scale: e.g., the jet fractions and 
the Thrust distribution. The
experimental error expected at LCs in the determination of
$\as$ from such quantities is of the order of percent,
so that  the
inclusion of NLO-W effects in the corresponding theory predictions 
is then mandatory.

\item Since the exploitation of beam polarisation effects will be a
key feature of experimental analyses of hadronic events at future LCs,
we have presented the full differential structure of three-jet processes
in the presence of polarised electrons and positrons, in terms of the energy 
fractions of the two leading jets and of two angles describing the final
state orientation.
The cross-sections were parametrised by means of nine independent 
form-factors, some of which carrying parity-violating
effects which are absent in ordinary QCD and that should appropriately 
be subtracted from hadronic samples in the search for physics beyond the SM.

\item All our results were presented for the case of the factorisable
NLO-W effects, i.e., for corrections to the initial and final states only.
Whereas these should be sufficient to describe adequately
the phenomenology of three-jet events at LEP1 and LEP2 energies, at LC energy
scales one expects comparable effects due to the non-factorisable corrections,
in which weak gauge bosons connect via one-loop diagrams electrons and positrons
to quarks and antiquarks. Their computation is currently in progress and we 
will report on it in due time.

\end{itemize}

\section*{Acknowledgements}

EM and DAR are grateful to the CERN Theory Division and SM to the KEK
Theory Division for hospitality while
part of this work was been carried out. SM and DAR are grateful to John
Ellis for illuminating discussions during the early stages of this project.
%
This research is supported in part
by a Royal Society Joint Project within
the European Science Exchange Programme (Grant No. IES-14468).


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\newpage

\begin{figure}
\begin{center}
\begin{picture}(365,225)
\Photon(0,175)(50,175){4}{4}
\ArrowLine(150,125)(125,137) \Line(125,137)(100,150)
\ArrowLine(100,150)(75,162) \ArrowLine(75,162)(50,175)
\ArrowLine(50,175)(150,225) \Gluon(115,142)(150,150){4}{3}
\PhotonArc(87,155)(12,337,150){3}{5} \put(87,175){$W,Z$}
\put(10,185){$Z,\gamma$}

\Photon(200,175)(250,175){4}{4}
\ArrowLine(350,125)(325,137) \ArrowLine(325,137)(300,150)
\ArrowLine(300,150)(275,162) \ArrowLine(275,162)(250,175)
\ArrowLine(250,175)(350,225) \Gluon(275,162)(350,175){4}{8}
\PhotonArc(312,143)(12,334,150){3}{5} \put(324,152){$W,Z$}
\put(210,185){$Z,\gamma$}

\Photon(0,50)(50,50){4}{4}
\ArrowLine(150,0)(100,25) \ArrowLine(100,25)(50,50)
\ArrowLine(50,50)(90,72) \ArrowLine(90,72)(110,82)
\ArrowLine(110,82)(150,100) \Gluon(100,25)(150,35){4}{5}
\PhotonArc(100,76)(12,32,206){3}{5} \put(87,95){$W,Z$}
\put(10,60){$Z,\gamma$}

\Photon(200,50)(240,50){4}{4} \Photon(265,50)(300,50){4}{4}
\ArrowLine(350,0)(325,25) \ArrowLine(325,25)(300,50)
\ArrowLine(300,50)(350,100) \Gluon(321,29)(350,45){4}{3}
\GCirc(252,50){12}{.5}
\put(210,60){$Z,\gamma$}

\end{picture}
\vspace{1.0truecm}
\caption{Self-energy insertion graphs. The shaded blob on the incoming
wavy line represents all the contributions to the gauge boson 
self-energy and is dependent on the Higgs mass (hereafter, we will use
$M_H=115$ GeV for the latter). In this and all subsequent
figures the graphs in which the exchanged gauge boson is a $W$-boson
is accompanied by corresponding graphs in which the  $W$-boson is replaced
by its corresponding Goldstone boson. Since the Yukawa couplings are
proportional to the fermion masses, such graphs are only significant
 in the case of $b$-quark jets. There is a similar set of diagrams
in which the direction of the fermion line is reversed.}
\label{se_graphs}
\end{center}
\end{figure}


\newpage

\begin{figure}
\begin{center}
\begin{picture}(365,225)

\Photon(100,175)(150,175){4}{4}
\ArrowLine(250,125)(225,137) \ArrowLine(225,137)(200,150)
\ArrowLine(200,150)(175,162) \ArrowLine(175,162)(150,175)
\ArrowLine(150,175)(250,225) \Gluon(200,150)(250,170){4}{5}
\PhotonArc(200,150)(24,154,330){3}{8} \put(187,115){$W,Z$}
\put(110,185){$Z,\gamma$}



\Photon(0,50)(50,50){4}{4}
\ArrowLine(150,0)(125,12) \ArrowLine(125,12)(87,30)
 \ArrowLine(87,30)(50,50)
\ArrowLine(50,50)(87,70) \ArrowLine(87,70)(150,100)
 \Gluon(115,17)(150,25){4}{3}
\Photon(97,25)(97,75){3}{5} \put(102,50){$W,Z$}
\put(10,60){$Z,\gamma$}


\Photon(200,50)(250,50){4}{4}
\ArrowLine(350,0)(325,12) \ArrowLine(325,12)(297,25)
 \Photon(297,25)(250,50){3}{5}
\Photon(250,50)(297,75){-3}{5} \ArrowLine(297,75)(350,100)
 \Gluon(315,17)(350,25){4}{3}
\ArrowLine(297,25)(297,75) \put(275,73){$W$} \put(275,19){$W$}
\put(210,60){$Z,\gamma$}




\end{picture}
\vspace{1.0truecm}
\caption{Vertex correction  graphs.  Again, same considerations 
as in the previous figure apply for the case of Goldstone bosons and
there is a similar set of graphs
in which the direction of the fermion line is reversed} \label{vertex_graphs}
\end{center}
\end{figure}

\vspace{3.0truecm}

\begin{figure}
\begin{center}
\begin{picture}(365,105)

\Photon(0,50)(50,50){4}{4}
\put(10,60){$Z,\gamma$}
\ArrowLine(150,29)(116,38) \ArrowLine(116,38)(83,44) \ArrowLine(83,44)(50,50)
\ArrowLine(50,50)(116,83) \ArrowLine(116,83)(150,100)
\Gluon(83,44)(150,0){-4}{8} \Photon(116,38)(116,83){4}{5}
\put(122,55){$W,Z$}

\Photon(200,50)(250,50){4}{4}
\ArrowLine(350,0)(307,20) 
 \Photon(307,20)(250,50){3}{5}
\Photon(250,50)(307,80){-3}{5} \ArrowLine(307,80)(350,100)
 \Gluon(307,50)(350,50){4}{5}
\ArrowLine(307,20)(307,50) \ArrowLine(307,50)(307,80)
 \put(275,73){$W$} \put(275,19){$W$}
\put(210,60){$Z,\gamma$}




\end{picture}
\vspace{1.0truecm}
\caption{Box  graphs. Again, same considerations as in the previous 
two figures apply for the case of Goldstone bosons. Here, the first 
graph is accompanied by a similar graph with the direction of the 
fermion line reversed whereas for the second
graph this reversal does {not} lead to a distinct
Feynman diagram.} \label{box_graphs}
\end{center}
\end{figure}

\newpage

\begin{figure}
\begin{center}
\epsfig{file=yCD_LEP1_new.ps,height=160mm, width=100mm, angle=90}
\epsfig{file=yGJ_LEP1_new.ps,height=160mm, width=100mm, angle=90}
\end{center}
\vskip -0.5cm
\caption{The $A(y)$, $-A_{\mathrm{W}}$ and $B(y)$ coefficient functions
of eqs.~(\ref{f3})--(\ref{f3EW}) for the Cambridge, Durham, Geneva and
Jade jet clustering algorithms, at $\sqrt s=M_Z$. (In the plot
the sign of $A_{\mathrm{W}}$ has been reversed for better presentation.)}
\label{fig:y_LEP1}
\end{figure}

\newpage

\begin{figure}
\begin{center}
\epsfig{file=scan_3jet.ps,height=160mm, width=140mm, angle=90}
\end{center}
\vskip -0.5cm
\caption{The LO cross-section and NLO-W correction for process (\ref{procj})
(top) and the relative size of the latter
(bottom) as a function of the collider CM energy.}
\label{fig:scan_3jet}
\end{figure}

\newpage

\begin{figure}
\begin{center}
\epsfig{file=scan_bbg.ps,height=160mm, width=140mm, angle=90}
\end{center}
\vskip -0.5cm
\caption{The LO cross-section and NLO-W correction for process (\ref{procb})
(top) and the relative size of the latter
(bottom) as a function of the collider CM energy.}
\label{fig:scan_bbg}
\end{figure}

\newpage

\begin{figure}
\begin{center}
\epsfig{file=yCD_LC_new.ps,height=160mm, width=100mm, angle=90}
\epsfig{file=yGJ_LC_new.ps,height=160mm, width=100mm, angle=90}
\end{center}
\vskip -0.5cm
\caption{The $A(y)$, $A_{\mathrm{W}}$ and $B(y)$ coefficient functions
of eqs.~(\ref{f3})--(\ref{f3EW}) for the Cambridge, Durham, Geneva and
Jade jet clustering algorithms, at $\sqrt s=1$ TeV.}
\label{fig:y_LC}
\end{figure}

\newpage

\begin{figure}
\begin{center}
\epsfig{file=   thrust_3jet_LC.ps,height=140mm, width=100mm, angle=90}
\end{center}
\vskip -0.5cm
\caption{The LO, NLO-QCD and NLO-W  
contributions to the coefficient functions entering the
integrated Thrust distribution, see eq.~(\ref{T}), for
process (\ref{procj}) (top) and the relative 
size of the two NLO corrections (bottom), at $\sqrt s=1$ TeV.}
\label{fig:thrust}
\end{figure}

\newpage

\begin{figure}
\begin{center}
\epsfig{file=   estimate.ps,height=150mm, width=90mm, angle=90}
\end{center}
\vskip -0.5cm
\caption{The integrated full Thrust distribution, see eq.~(\ref{T})
for process (\ref{procj}), at $\sqrt s=1$ TeV. 
We show separately the LO result and the additional contributions from
the exact NLO-QCD, an estimate of the NNLO-QCD and the exact 
NLO-W corrections, respectively.}
\label{fig:thrustNNLO}
\end{figure}

\newpage

\begin{figure}
\begin{center}
\centerline{}
\begin{minipage}[b]{.33333\linewidth}
\centerline{$+F_1$}
\vspace{-0.75truecm}\centering\epsfig{file=ff1_L_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\begin{minipage}[b]{.33333\linewidth}
\centerline{$+F_2$}
\vspace{-0.75truecm}\centering\epsfig{file=ff2_L_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\begin{minipage}[b]{.33333\linewidth}
\centerline{$-F_3$}
\vspace{-0.75truecm}\centering\epsfig{file=ff3_L_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\centerline{}
\begin{minipage}[b]{.33333\linewidth}
\centerline{$-F_4$}
\vspace{-0.75truecm}\centering\epsfig{file=ff4_L_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\begin{minipage}[b]{.33333\linewidth}
\centerline{$+F_5$}
\vspace{-0.75truecm}\centering\epsfig{file=ff5_L_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\begin{minipage}[b]{.33333\linewidth}
\centerline{$+F_6$}
\vspace{-0.75truecm}\centering\epsfig{file=ff6_L_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\centerline{}
\begin{minipage}[b]{.33333\linewidth}
\centerline{$+F_7$}
\vspace{-0.75truecm}\centering\epsfig{file=ff7_L_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\begin{minipage}[b]{.33333\linewidth}
\centerline{$+F_8$}
\vspace{-0.75truecm}\centering\epsfig{file=ff8_L_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\begin{minipage}[b]{.33333\linewidth}
\centerline{$+F_9$}
\vspace{-0.75truecm}\centering\epsfig{file=ff9_L_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\centerline{}

\noindent
\caption{The nine form-factors defined in eq.~(\ref{FFs}) as a function
of the antiquark ($i=1$) and quark ($i=2$) 
energy fractions $x_i=\frac{2E_i}{\sqrt s}$ at $\sqrt s=1$ TeV
in units of nb, for left-handed incoming electrons.
(Note that in some cases we plot the opposite of the form-factor.)}
\label{fig:FFs-L}
\end{center}
\end{figure}

\newpage

\begin{figure}
\begin{center}
\centerline{}
\begin{minipage}[b]{.33333\linewidth}
\centerline{$+F_1$}
\vspace{-0.75truecm}\centering\epsfig{file=ff1_R_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\begin{minipage}[b]{.33333\linewidth}
\centerline{$+F_2$}
\vspace{-0.75truecm}\centering\epsfig{file=ff2_R_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\begin{minipage}[b]{.33333\linewidth}
\centerline{$-F_3$}
\vspace{-0.75truecm}\centering\epsfig{file=ff3_R_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\centerline{}
\begin{minipage}[b]{.33333\linewidth}
\centerline{$-F_4$}
\vspace{-0.75truecm}\centering\epsfig{file=ff4_R_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\begin{minipage}[b]{.33333\linewidth}
\centerline{$+F_5$}
\vspace{-0.75truecm}\centering\epsfig{file=ff5_R_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\begin{minipage}[b]{.33333\linewidth}
\centerline{$+F_6$}
\vspace{-0.75truecm}\centering\epsfig{file=ff6_R_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\centerline{}
\begin{minipage}[b]{.33333\linewidth}
\centerline{$+F_7$}
\vspace{-0.75truecm}\centering\epsfig{file=ff7_R_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\begin{minipage}[b]{.33333\linewidth}
\centerline{$-F_8$}
\vspace{-0.75truecm}\centering\epsfig{file=ff8_R_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\begin{minipage}[b]{.33333\linewidth}
\centerline{$-F_9$}
\vspace{-0.75truecm}\centering\epsfig{file=ff9_R_bit.ps,angle=0,height=8cm,width=\linewidth}
\vskip-2.0cm
\centerline{
$x_2$
\qquad\qquad\qquad\qquad
$x_1$}
\end{minipage}\hfil
\centerline{}

\noindent
\caption{The nine form-factors defined in eq.~(\ref{FFs}) as a function
of the antiquark ($i=1$) and quark ($i=2$) 
energy fractions $x_i=\frac{2E_i}{\sqrt s}$ at $\sqrt s=1$ TeV
in units of nb, for right-handed incoming electrons.
(Note that in some cases we plot the opposite of the form-factor.)}
\label{fig:FFs-R}
\end{center}
\end{figure}

\newpage

\begin{figure}
\begin{center}
\epsfig{file=allthrust_3jet_LC.ps,height=140mm, width=100mm, angle=90}
%\vskip -0.5cm
\caption{The unintegrated Thrust distributions for the nine
component of the cross-section associated to the form-factors
in eq.~(\ref{FFs}) for the NLO-W process (\ref{procj}) (top and middle) 
and the relative size of the
six components which are non-zero at LO (bottom), at $\sqrt s=1$ TeV.
Labels are as follows: (top) $F_1$ (yellow),
                             $F_2$ (magenta),
                             $F_5$ (blue),
                             $F_6$ (green),
                             $F_8$ (red);
                    (middle) $F_3$ (magenta),
                             $F_4$ (blue),
                             $F_7$ (green),
                             $F_9$ (red);
                    (bottom) $F_1$ (fine-dotted),
                             $F_2$ (dashed),
                             $F_3$ (dot-dashed),
                             $F_4$ (dotted),
                             $F_5$ (fine-dashed),
                             $F_6$ (solid).}
\label{fig:allthrust}
\end{center}
\end{figure}

\end{document}



















