Calculation of Power Corrections to Hadronic Event Shapes of Tagged b Events

Abstract

We compute power corrections to mean values of hadronic event shapes - the thrust and the C parameter - of tagged b quark events in electron positron annihilation, using the dispersive approach. We find that the leading power corrections are of the same type of 1/Q corrections as for event shapes in the massless case, with the same non-perturbative coefficient times a perturbatively calculable mass-dependent coefficient. The effect of the mass correction in the power correction is to reduce the latter by 10-30% for tagged b events, for centre-of-mass energies ranging from the Z peak down to 20 GeV. Comment: 14 pages, 4 figures, 1 table, JHEP style a typo in Reference section is corrected

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arXiv:hep-ph/9911353v2 16 Nov 1999
Preprint typeset in JHEP style. - PAPER VERSION
CERN–TH/99-340
hep-ph/9911353
Calculation of Power Corrections to Hadronic
Event Shapes of Tagged b Events
Zolt´ an Tr´ ocs´ anyi∗
Theory Division, CERN, CH-1211 Geneva 23, Switzerland
Abstract: We compute power corrections to mean values of hadronic event shapes
— the thrust and the C parameter — of tagged b quark events in electron positron
annihilation, using the dispersive approach. We find that the leading power correc-
tions are of the same type of 1/Q corrections as for event shapes in the massless
case, with the same non-perturbative coefficient times a perturbatively calculable
mass-dependent coefficient. The effect of the mass correction in the power correction
is to reduce the latter by 10–30% for tagged b events, for centre-of-mass energies
ranging from the Z0peak down to 20GeV.
Keywords: QCD, NLO computations, jets, LEP physics.
∗Sz´ echenyi Fellow of the Hungarian Ministry of Education. On leave from Department of Exper-
imental Physics, Kossuth University and Institute of Nuclear Research of the Hungarian Academy
of Sciences, Debrecen, Hungary

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Contents
1. Introduction1
2. The dispersive approach2
3. Calculations4
4. Merging perturbative and non-perturbative contributions8
5. Conclusion12
1. Introduction
In recent years the idea of an infrared-regular effective strong coupling at low scales
(the strong coupling ‘freezing’ [1] or, more rigorously, the ‘dispersive approach’ of
Dokshitzer, Marchesini and Webber [2]) has been employed for estimating hadroniza-
tion corrections to various hadronic event shapes, using perturbative calculations
[3–7]. These corrections arise in the form of power corrections of order 1/Qp, where
Q is the centre-of-mass energy. Although the magnitude of the correction cannot
be predicted by perturbative means, the power p and the relative coefficients among
different observables can be calculated using the assumption of infrared freezing.
This universality hypothesis has proved to give a phenomenologically fairly consis-
tent picture of power corrections [8–13], and the same results could be derived using
a different method, the so-called renormalon approach [14,15].
In this letter we explore a further check of the universality picture. Assuming
the flavour independence of gluon radiation off quarks, the analytic structure of the
strong coupling in the infrared, defined in an appropriate way, should not depend on
the masses of the quarks, which radiate the gluon. Therefore, in the calculation of
the power corrections to hadronic event shapes of tagged b events, the mass of the
heavy quark enters in a completely perturbative manner. One may expect that the
leading mass correction could be mb/Q times the leading power correction (mbbeing
the heavy-quark mass) with some calculable coefficient. If this coefficient is several
times unity, then the mass corrections in the power corrections are expected to be
significant enough to be a measurable effect.
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A similar problem has already been considered by Nason and Webber in calcu-
lating non-perturbative corrections to heavy-quark fragmentation in e+e−annihila-
tion [16]. Although the calculations are very similar, there is an essential difference
in the results: for the fragmentation function there is a 1/mbpower correction, while
for tagged events shapes there are mass corrections to the magnitude of the leading
1/Qppower correction, which, however, vanish smoothly in the massless limit.
Estimating the mass corrections in the power correction may be interesting from
another point of view, too. Recently, next-to-leading order calculations of three-
jet quantities in electron positron annihilation have been performed in which quark
mass effects have been taken into account explicitly [17–19]. Using this theoretical
input and the value of the b-quark mass, the flavour independence of the strong
interactions was demonstrated by determining the ratio of strong couplings, αb
[20,21]. One can turn around the argument and, assuming flavour independence,
the b-quark mass can be measured from the sensitivity of three-jet event shapes
to mass effects [22–24]. Such a measurement requires an estimate of hadronization
corrections. The traditional way of obtaining such estimates is by Monte Carlo event
generators [25,26]. Those programs use the heavy-quark mass as input. As a result,
the estimate of hadronization corrections brings a significant systematic error into
the measurement of mb. For taking into account the hadronization correction in a
different way, the simple formulae of power corrections presented in this paper can
easily be incorporated in a fit to the b-quark mass, which would hopefully result in
a more stringent mass measurement.
As mentioned above, the presence of the quark mass appears only in the per-
turbative calculation. Therefore, the calculations that lead to the appearance of the
Milan factor in the power corrections [5] can also be performed in the presence of
the quark mass, resulting in a mass-dependent Milan factor. In this paper we make
only the first step and calculate that part of power corrections to the mean value of
two e+e−event shapes, the thrust and the C parameter, which was termed ‘naive
contribution’ in Ref. [5].
s/αuds
s
2. The dispersive approach
In employing the dispersive approach of Ref. [2] one starts with assuming the validity
of the following dispersion relation for the strong coupling:
αs(k2) = −
?∞
0
dµ2
µ2+ k2ρs(µ2) ,ρs(µ2) = −1
2πiDisc
?
αs(−µ2)
?
.(2.1)
Further, it is also assumed that similarly to QED, the dominant effect of the running
of αson some QCD observable F may be represented in terms of the spectral function
2

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ρs(µ2) and a characteristic function F(µ2):
F = αs(0)F(0;...) +
?∞
?∞
F(µ2;...) − F(0;...)
0
dµ2
µ2ρs(µ2)F(µ2;...)(2.2)
=
0
dµ2
µ2ρs(µ2)
??
,(2.3)
where we used Eq. (2.1) to express αs(0). The characteristic funtion F(µ2;...) is a
function of dimensionless ratios of ε ≡ µ2/Q2, ρ ≡ 4m2
quark mass and {y}, the collection of any further relevant dimensionless parameters,
e.g. the jet shape S in our present considerations. We obtain F by computing the
one-loop graphs, corresponding to the physical process under consideration, with a
non-zero gluon mass ε ?= 0, and dividing by αs:
?
where dΦ({xi};ρ,ε) denotes the phase space in terms of the independent variables
{xi}, M is the proper squared amplitude divided by αs, and S stands for the event
shape variable. Introducing the effective coupling αeff(µ2), defined in terms of the
spectral function by
ρs(µ2) = µ2dαeff
b/Q2, where mbis the heavy-
F(ε;ρ,S) = dΦ({xi};ρ,ε)M({xi};ρ,ε)S({xi};ρ,ε) ,(2.4)
dµ2,(2.5)
we can integrate Eq. (2.3) by parts to obtain
F =
?∞
0
dµ2
µ2αeff(µ2)˙F(µ2) =
?∞
0
dε
εαeff(Q2ε)˙F(ε) ,(2.6)
where
˙F(ε) = −εdF
dε.
(2.7)
Using the definition Eq. (2.5) and the dispersion relation Eq. (2.1), we can deduce
that in the perturbative domain, αs≪ 1, the standard and effective couplings are
approximately the same [2]:
αeff(µ2) = αs(µ2) + O(α3
s) .(2.8)
Thus we may interpret αeff as an effective coupling that extends the physical per-
turbative coupling into the non-perturbative domain. If the effective coupling has a
non-perturbative component δαeff(µ2), with support limited to low values of µ2, the
corresponding contribution to F,
δF =
?∞
0
dε
εδαeff(Q2ε)˙F(ε) ,(2.9)
will have a Q2dependence determined by the small-ε behaviour of˙F.
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We shall be interested in the leading power behaviour of δF, which will be called
the leading power correction to F,
δF = Fpow+ O
?
1/Qp?
. (2.10)
In order to determine Fpow, we recall that power-suppressed contributions to δF can
arise from only those terms in F that are non-analytic at ε = 0 [2]. Therefore, the
leading power correction Fpowwill be obtained from the leading non-analytic term
at ε = 0 in˙F, which we denote by CFδF/2π.
3. Calculations
In calculating the function δF for the e+e−→Q¯Qg event shapes, we find three sources
of quark-mass dependence: (i) restriction of phase space; (ii) mass corrections in the
matrix element; (iii) mass corrections in the definition of the event shape. The double
differential cross section for the production of a massive quark antiquark pair and a
massive gluon given in terms of the scaled quark, antiquark and gluon energies, x, ¯ x
and xg= 2−x−¯ x was derived in Ref. [16]. This cross section is an analytic function
of the gluon mass and, therefore, in calculating δF, the ε-dependent terms can be
dropped completely. In the case of the vector current contribution the cross section
for zero gluon mass is given by
1
σV
dσV
dxd¯ x=CF
β
αs
2π
?−λ(x2− ρ, ¯ x2− ρ,x2
8(1 − x)2(1 − ¯ x)2
+
1 +1
g)
(3.1)
1
2ρ
?
1
(1 − x)2+
1
(1 − ¯ x)2
??
;
for the axial vector current contribution we have
1
σA
dσA
dxd¯ x=CF
β
αs
2π
?−λ(x2− ρ, ¯ x2− ρ,x2
8(1 − x)2(1 − ¯ x)2
+
1 − ρ
g)
(3.2)
1
?
1
(1 − x)2+
1
(1 − ¯ x)2+ρ
2
x2
g
(1 − x)(1 − ¯ x)
? ?
,
where β =√1 − ρ is the quark velocitey and λ(x,y,z) = x2+y2+z2−2xy−2yz−
2z x. In Eqs. (3.1) and (3.2)
σV = σ0
?
1 +1
2ρ
?
β ,(3.3)
and
σA= σ0(1 − ρ) β(3.4)
are the Born cross sections for heavy-quark production by a vector and an axial
vector current respectively, σ0 being the massless quark Born cross section. The
corresponding phase space was also given in Ref. [16]:
dΦ(x, ¯ x;ε,ρ) = dxd¯ xΘ
?
−1
8λ(x2− ρ, ¯ x2− ρ,x2
g− 4ε)
?
(3.5)
4