Gluon Radiation and Energy Losses in Top Quark Production

Abstract

The emission of energetic gluons in $\tt$ production in $\ee$ annihilation can have important experimental consequences, in particular on top quark mass measurements. We present compact, analytical expressions for the gluon energy distribution and its average value at first order in QCD perturbation theory. Our results are valid for arbitrary masses, collision energies and production currents. We pay particular attention to top quark production near threshold, and show that in certain cases the soft gluon approximation is insufficient to describe the radiation spectrum. Comment: 17 pages, 3 figures (not included)

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arXiv:hep-ph/9405243v1 6 May 1994
DTP/94/14
LU-TP–94–3
March 1994
Gluon Radiation and Energy Losses
in Top Quark Production
Yu.L. Dokshitzer1
Department of Theoretical Physics, University of Lund
S¨ olvegatan 14A, S-22362 Lund, Sweden
V.A. Khoze1
Department of Physics, University of Durham
Durham DH1 3LE, England
and
W.J. Stirling
Departments of Physics and Mathematical Sciences, University of Durham
Durham DH1 3LE, England
Abstract
The emission of energetic gluons in t¯t production in e+e−annihilation can
have important experimental consequences, in particular on top quark mass
measurements. We present compact, analytical expressions for the gluon energy
distribution and its average value at first order in QCD perturbation theory. Our
results are valid for arbitrary masses, collision energies and production currents.
We pay particular attention to top quark production near threshold, and show
that in certain cases the soft gluon approximation is insufficient to describe the
radiation spectrum.
1permanent address: Petersburg Nuclear Physics Institute, Gatchina, St.Petersburg 188350,
Russia

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1Introduction
The discovery of the top quark – one of the basic components of the Standard Model
– is one of the most important goals for present and future experiments. Indirect
evidence for the existence of top is very strong, see for example [1], and the chances
that it will be detected at the Fermilab p¯ p collider in the next few years are very high.
Once top is discovered, the next challenge for experiments is to measure its parameters
– in particular its mass M – as precisely as possible.
An unambiguous interpretation of experimental data and the determination of the
top quark parameters relies on a clear quantitative understanding of the details of the
production process, including the effects of gluon bremsstrahlung at the production
stage [2, 3]. To quantify these effects, one can study the average amount of energy lost
by the top quark – the simplest infrared-safe characteristic of gluon bremsstrahlung.
Such a quantity is of potential practical importance in top mass determinations and,
for energies above threshold, can be calculated quite straightforwardly without undue
complications from non-perturbative phenomena or top-width effects [2, 3, 4]. The key
point here is the large top width, Γt[5]. Since we know from experiment that the top
mass is greater than about 120 GeV [6], the decay width is greater than the typical
hadronic scale, Γt> µ ∼ 1 fm−1, and so the top quark decays before non-perturbative
effects like the formation of t-flavoured hadrons become important [5, 7].
The bremsstrahlung of gluons off the top quark can have an important impact
(“radiation damage”) on various characterstics of the t¯t final state. Taking e+e−→ t¯t
as an example, it obviously affects the kinematical constraint Et= Ebeam, over and
above the well-known effects of initial radiation.2It also influences the form of the t¯t
production vertex and measurements of the spectra of secondary leptons from t-decay
[10, 11].
Since one of the primary goals of a future e+e−collider would be to obtain an
accurate measurement of the top mass M, it is worth considering the impact of gluon
radiation on such a measurement. There are two scenarios which have been considered
[12]. If the top is not too heavy, it can have sizeable kinetic energy and a mass
measurement can be obtained from the invariant mass per hemisphere. In practice,
the distributions in the heavy (MH) and light (ML) hemisphere masses can be fitted to
a Monte Carlo simulation with M as a free parameter. However, a gluon radiated from
the t¯t system will distort this measurement. In particular the average difference in the
hemisphere masses squared is proportional (at O(αs)) to the average gluon energy,
?M2
H− M2
L? =√s?Eg?.(1)
The more energy is carried away by the gluon, the more asymmetric the hemisphere
2These initial state radiation effects can be incorporated in a standard way [8, 9] and will not be
addressed further here.
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masses. In practice, as we shall see, ?Eg? can be O(10 GeV), which is much larger
than the estimated statistical errors on M from such analyses. A Monte Carlo which
does not take full account of gluon radiation at the t¯t production stage could yield
misleading results on the precision on M.
Another method which is more suited to a heavier top quark is to reconstruct the
mass directly from the three-jet final state [12]. To avoid combinatorial and other
backgrounds, the energy of the three-jet system is typically constrained to be close to
the beam energy. But once again, the emission before the top decay of an energetic
gluon can distort this constraint, since naively
?E3 jet? − Ebeam= −1
2?Eg?. (2)
Once again we see the importance of taking the effects of gluon radiation into account
in reconstructing the top mass.
In this paper we present results for gluon bremsstrahlung in e+e−→ t¯t which are
of practical interest for physics studies of future linear e+e−colliders. We focus in
particular on the gluon energy spectrum which, as we have seen, can have important
implications for the top mass measurement. It is also straightforward, at least in
principle, to extend our results to the more complicated case of hadronic t¯t production.
Since centre-of-mass energies√s at which multiple gluon radiation should be taken
into account [2, 13] (log(√s/M) ≫ 1) appear unrealistic for the foreseeable future,
we need only consider a first-order perturbative analysis. This is calculationally quite
straightforward, and in fact many parts of the calculation can be taken over from
existing QED analyses [14, 15] (see also [16]).
The remainder of the paper is organised as follows. We first set up the theoretical
framework which allows us to calculate the gluon radiation distribution for arbitrary
production currents. We derive compact, analytical expressions for the gluon energy
spectrum and its average value for vector, axial vector, scalar and pseudoscalar pro-
duciton currents. We then present some numerical calculations which illustrate the
size of the effects for typical experimental parameters. Finally, we summarize our
results and present our conclusions.
2Calculation of QCD bremsstrahlung distributions
In this section we describe the calculation of the cross section for single primary gluon
emission in the process e+e−→ t¯t. Our calculation is valid for any collision enegy
√s > 2M, and we take both photon and Z exchanges into account.
We begin by defining some variables:
energy and M is the mass of the top quark. The momenta of the initial and final
state particles are labelled by e−(k1) + e+(k2) → t(p1) +¯t(p2) + g(k) and we define
√q2≡√s = 2E is the total centre-of-mass
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qµ= pµ
centre-of-mass frame are
1+ pµ
2+ kµ. The energy fractions shared by the t,¯t and gluon in the e+e−
zi =
2q · pi
q2
2q · k
q2
,i = 1,2
z =
,z = 2 − z1− z2.(3)
The quark velocity in the t¯t centre-of-mass frame is
β = β(z) ≡
?
1 − ρ0=
?
1 −
4γ
1 − z≤ v =
?
1 − 4γ ,(4)
where
γ =M2
q2≤1
4,ρ0=
4γ
1 − z.
(5)
The final-state radiation is limited by the maximum kinematically allowed energy
0 ≤ z ≤ zmax= v2.(6)
The virtualities of the top quark and antiquark before gluon emission, κi, can be
written in terms of the energy fractions:
κ1 = 2k · p1= q2(1 − z2) ,
κ2 = 2k · p2= q2(1 − z1) .(7)
It is also convenient to introduce the “angular” variable
ρ =κ1
zq2=1 − z2
z
.(8)
In the calculations which follow, we neglect the electron mass me. Since we are con-
cerned with energy losses caused by gluon emission at the t¯t production stage, we may
treat the t-quarks as stable objects. Only very soft gluons with energy Eg
are affected by the top width, and these make a negligible contribution to the average
energy loss.3
<∼ΓtE/M
2.1 Lowest-order cross section
The total non-radiative cross section for t¯t production in e+e−collisions can be ex-
pressed in terms of the vector and axial contributions (see for example [18, 19]):
σt¯t= σV V(s) + σAA(s) = Rt¯t(s)σpt(s) ,(9)
3A detailed discussion of the potential impact of the top quark instability on gluon emission can
be found in Refs. [2, 3, 4, 17].
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where
σpt(s) =4πα2(s)
3s
(10)
is the point cross section for e+e−→ µ+µ−with the QED coupling evaluated at the
scale s, and
Rt¯t(s) = RV V(s) + RAA(s) = v (ζVτV(s) + ζAτA(s))(11)
with
ζV
ζA = v2= 1 − 4γ .
=
1
2(3 − v2) = 1 + 2γ , (12)
(13)
The electroweak factors τC(C = A,V ) are defined as
τV
=
4
3− 4vevtκ
s(s − M2
Z)2+ (sΓZ/MZ)2
Z)
(s − M2
e)v2
+3(v2
e+ a2
tκ2
s2
(s − M2
Z)2+ (sΓZ/MZ)2,
s2
Z)2+ (sΓZ/MZ)2,
(14)
τA
= 3(v2
e+ a2
e)a2
tκ2
(s − M2
(15)
with
κ =
√2GFM2
4πα(s)
2+ 2sin2θW,
2−4
Z
,
ve= −1
vt=1
ae= −1
at=1
2,
3sin2θW,
2. (16)
The functions τV(s) and τA(s) are shown in Fig. 1. Except for energies close to the Z0
peak, which are not relevant for t¯t production, the axial current piece is numerically
small compared to the vector current piece. At very high energies the ratio of τAto τV
tends to a constant value of approximately 0.26. Furthermore, close to t¯t threshold the
axial current contribution is further suppressed by two powers of v, see Eqs. (12,13).
We should also note that near threshold, the t¯t cross section is strongly modified by
Coulomb enhancement effects, see Refs. [20, 21, 22]. However the extreme threshold
region is not of primary interest in the present study, since the perturbation radiation
is strongly suppressed there.
2.2Dalitz plot distribution for the radiative process
We consider in this section the calculation of the differential cross section for the
emission of a primary gluon of momentum k in t¯t production. After integrating over
4