NLO calculations of the three-jet heavy quark production in e^+e^--annihilation: status and applications

6 Pages·0.17 MB·English

by  Mikhail Bilenky

FTUV/98-79
IFIC/98-80
NLO CALCULATIONS OF THE THREE-JET HEAVY QUARK
PRODUCTION IN e+e -ANNIHILATION:
−
STATUS AND APPLICATIONS
∗
Mikhail Bilenkya†, Germ´an Rodrigob‡ and Arcadi Santamariac
9
9 a Institute of Physics, AS CR, 18040 Prague and
9 Nuclear Physics Institute, AS CR, 25068 Rˇeˇz(Prague),Czech Republic
1
b INFN-Sezione di Firenze, Largo E. Fermi 2, 50125 Firenze, Italy

n c Departament de F´ısica Teo`rica, IFIC, CSIC-Universitat de Val`encia, 46100 Burjassot, Val`encia, Spain
a
J
Abstract

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Next-to-leading order calculations for heavy quark three-jet production in e+e- annihilation are reviewed.

Their applications for the measurement of the b-quark mass at LEP/SLC and for the test of flavour inde-
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v pendence of the strong coupling constant are discussed.
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∗to be published in the Proceedings of the XXIX International Conference on High Energy Physics, Vancouver, B.C. Canada, July 23-29,
1998.
†OnleavefromJINR,141980Dubna,RussianFederation
‡OnleavefromDepartamentdeF´ısicaTeo`rica,IFIC,CSIC-UniversitatdeVal`encia,46100Burjassot,Val`encia,Spain
NLO CALCULATIONS OF THE THREE-JET HEAVY QUARK PRODUCTION IN
e+e−-ANNIHILATION: STATUS AND APPLICATIONS
M. BILENKY
Institute of Physics, AS CR, 18040 Prague and Nuclear Physics Institute, AS CR, 25068 Rˇeˇz(Prague), Czech Republic
G. RODRIGO
INFN-Sezione di Firenze, Largo E. Fermi 2, 50125 Firenze, Italy
A. SANTAMARIA
Departament de F´ısica Teo`rica, IFIC, CSIC-Universitat de Val`encia, 46100 Burjassot, Val`encia, Spain
Next-to-leading order calculations for heavy quark three-jet production in e+e- annihilation are reviewed. Their applications for
the measurement of the b-quark mass at LEP/SLC and for the test of flavour independence of the strong coupling constant are
discussed.
1 Motivation of the strong interactions2,3,4,5 at the Z0-poleb.
Inthistalkwemakeashortreviewofnext-to-leading
Effectsofthebottom-quarkmass,m ,havebeenalready order predictions for e+e− 3jets including effects of
b
→
noticed in the early tests1 of the flavour independence the quark mass and its applications at the Z0-peakc.
ofthestrongcouplingconstant,α ,ine+e−-annihilation
s
at the Z0-peak. They became very significant in the
2 Why are the b-quark mass effects are signifi-
final analysis, which included millions of hadronic Z0-
cant in Z0 3jets?
decays2,3,4,5. For example, if the b-quark mass is ne- →
glected, the ratio αb/αlight, where αb measured from
s s s It might seem surprising that at the Z0-pole, where the
hadronic events with b-quarks in the final state, and
relevant scale is the Z0-boson mass, effects of the quark
αlight fromeventswithlightquarks(uds),isshiftedfrom
s masses can not be neglected. Indeed, if one considers
one3 by 8%. Thus, high LEP/SLC precision requires an inclusive width of the decay Z0 ¯bb, according to the
accurate account for the heavy quark massa in the theo- →
Kinoshita-Lee-Nauenbergtheoremtherearenomasssin-
retical predictions for the e+e−-annihilation into jets at
gularities,andthe only waym enters calculationsd is in
the Z0-pole. b
the ratio m2(M )/M2. Therefore, quark mass effects in
The quark mass effects in the Z0 decays were dis- b Z Z
the best-measuredobservable for b-quarksare negligibly
cussed in the literature6. The leading order (LO) com- small 10−3.
plete Monte-Carlo calculation for e+e− 3jets,4jets ∼
→ But the situation is different in more exclusive pro-
with massive quarks was first done in 7. Later, moti- cesses. Let’s consider the decay Z0 ¯bbg, which con-
vated by the remarkable sensitivity of the three-jet ob- →
tributes to the three-jet final state at the LO. This pro-
servablestothevalueofthequarkmass,thepossibilityof
cess has aninfraredsingularityin the limit when the en-
thedeterminationofm atLEP,assuminguniversalityof
b ergyofradiatedgluonenergyaproacheszero. Tomakea
stronginteractions,wasconsidered8,9. Thisquestionwas
physicalpredictiononehastointroducekinematicalcuts.
analyzed in detail in9, where the necessity of the next- In the e+e−-annihilation this is usually done by apply-
to-leadingorder(NLO)calculationforthemeasurements ing one of the so-called jet clustering algorithms14. The
of the mb was also emphasized. phase-space for Z0 ¯bbg is split into two parts, two-jet
TheNLOcalculationsfortheprocesse+e− 3jets, and three-jet one, a→nd this separation is defined by the
→
with complete effects of the quark mass, were performed jet-resolution parameter, y . Therefore, instead of two
c
independently bythreegroups10,11,12. Thesepredictions
are in agreement with each other and were successfully bDue to the large correlation between αs and mb, either mb or
used in the measurements of the b-quarkmass far above αs wastreatedasafreeparameter. Thevalueofanother onewas
takenfromothermeasurements.
threshold2,5 andinthe precisiontests ofthe universality cWe wouldliketo note that elements of these calculations can be
alsoappliedforthee+e−→t¯t+···.
aIt is mainly related to the bottom-quark. Effects of the charm- dTaking the MS running quark mass one includes the principal
quarkmassaresmaller,roughlybythefactorm2/m2. partoftheNLOQCDcorrections tothetotalwidth13,9.
c b
scales in the inclusive process, M and m , one has here can usee, for example, the pole mass M 4.6GeV or
Z b b
≈
in addition the new scale, √ycMZ. The jet-resolution the MS-running massmb(µ)atanyscalerelevantto the
parametercanbe rather small,in the range10−2 10−3 problem, m µ M , with m (m ) 4.13GeV and
b Z b b
− ≤ ≤ ≈
and effects of the quark mass appears as m2b/(√ycMZ)2 mb(MZ)≈2.83GeV.
and can reach several percents.
1
bd
3 The three-jet observable 0.99 R3 LO-mb(mZ)
The convenientobservable for studies of the mass effects 0.98 NLO
in the three-jet final state is defined as follows
0.97
Γb (y )/Γb 0.96
R3bd = Γd33jj(ycc)/Γd (1) 0.95 LO-Mb
α Jade
=1+r b (y ,r )+ sb (y ,r ) + (α2) 0.94
b(cid:16) 0 c b π 1 c b (cid:17) O s
where Γq3j and Γq are the three-jet and the total de- 0.93 yc
0.92
cay widths of the Z0-boson into a quark pair of flavour 0 0.02 0.04 0.06 0.08 0.1
q, r = m2/M2. Both the LO function, b , and the
NLObfunctibon,bZ,dependonthejet-clustering0algorithm. Figure1: TheratioRb3d (Jade)asafunctionofyc.
1
Note that although the leading r -dependence is factor-
b
1
ized for convenience, the above expression is not an ex-
bd
pansion in rb. 0.99 R3 LO-mb(mZ)
This observable has both experimental and theoret-
NLO
ical advantages. It is a relative quantity, therefore many 0.98
experimentaluncertaintiesduetothenormalizationdrop
out. In addition, in the ratios Γq /Γq the bulk of elec- 0.97
3j
troweak corrections is cancelled. The double ratio (1) 0.96 LO-Mb
was measuredby DELPHI2 and usedfor the determina-
tion of the m . In practice one uses normalization with 0.95
b
respect to all light flavours, uds. Such quantity then Durham
differs from Rbd, mainly due to the contribution of the 0.94 y
3
triangle diagrams 15. The difference is, however, very c
0.93
small numerically. Similar ratio normalized on udsc was 0 0.02 0.04 0.06 0.08 0.1
considered in11.
Figure2: TheratioRb3d (Durham)asafunctionofyc.
Another observable is the differential two-jet rate,
D defined as
2 The results for the ratio Rbd, Eq.(1), for Jade17 and
3
[Γb (y +∆y /2) Γb (y ∆y /2)]/Γb Durham18 schemes are presented by solid curves in the
D2bd = [Γd2j(yc+∆yc/2)−Γd2j(yc−∆yc/2)]/Γd . (2) Figs.1,2. Theuppercurvecorrespondstomb =2.83GeV,
2j c c − 2j c− c the lower one to m = 4.6GeV. The difference between
b
with ∆y taken to be sufficiently small. The two-jet thetwocurvesgivesthesizeoftheuncertaintyoftheLO
c
widthisobtainedfromtherelation: Γq =Γq Γq Γq , prediction. ToimprovethesituationtheNLOcalculation
where Γq is the four-jet width. 2j − 3j− 4j is necessary.
4j
Different event shapes observables were consid- It is worth mentioning that the residual mass de-
ered12,11,3 in the literature as well. pendence in the function b (y ,r ) is small in the region
0 c b
0.005<y <0.1,especiallyfortheDurhamscheme. The
c
4 The leading order calculation simpleinterpolation: b0 =b(00)+b(01)lnyc+b(02)ln2yc can
be used in practice9,19.
The LO contribution to the decay Z0 3jets is given
by the process Z0 ¯bbg. At the LO w→e can not spec-
→
ify what value of m should be taken in the calcula-
b
tions: all definitions of the quark mass are equivalent eThe values of the b-quark masses are taken from the recent sum
(the difference is due to the higher orders in αs). One rulesandlatticeQCDanalyses16 oftheΥandB mesonsspectra.
5 The next-to-leading order corrections In the NLO calculations one can, and have to, spec-
ify the quark mass definition. It turned out that techni-
At the NLO there are two different contributions. One callyitis simplerto useamixedrenormalizationscheme
comes from one-loop corrections to the three-parton de- with on-shell definition for the quark mass and MS def-
cay, Z0 bbg. Another one comes from tree-level four- inition for the strong coupling. In this case observables
→
partondecaysZ0 bbgg and Z0 bbqq, q =u,d,s,c,b are originally expressed in terms of the pole mass. This
→ →
integrated over the three-jet region of the four-parton definitionofthequarkmasscanbeperfectlyusedinper-
phase-space. The main difficulty of the NLO calcula- turbation theory. However, in contrast to the pole mass
tionisthe presenceofultraviolet(UV)andinfrared(IR) inQED,thequarkpolemassisnotaphysicalparameter.
divergences at the intermediate stages. The UV diver- Thenon-perturbativecorrectionstothequarkself-energy
gences in the one-looppart are removedby renormaliza- bring an unavoidable ambiguity of order 300MeV
≈
tion of the QCD parameters. The IR singularities in the (hadron size) to the physical position of the pole of the
virtual part are due to massless gluons in the loops. In quarkpropagator24. Abovethequarkproductionthresh-
thefour-partonprocesstheyappearwhenoneoftheglu- old, it is natural to use the running mass definition (we
ons is soft or two gluons are collinear. The sum of the use MS). The advantageof this definition is that m (µ)
b
virtualandtree-levelcontributionsis,however,IRfinite. canbeusedforanyscale,µ m . Thepole,M ,andthe
b b
≫
In addition,the finite quarkmass makesthe NLO calcu- running masses of the quark are perturbatively related
lation technically much more involved,comparing to the
massless one, known since many years20,21. M =m (µ) 1+ αs 4 lnm2b . (4)
The one-loop part is calculated analytically and di- b b (cid:20) π (cid:18)3 − µ2 (cid:19)(cid:21)
mensional regularization is used to regularize both UV
Althoughthisrelationisknowntohigherordersinα ,we
s
and IR divergences. The IR singularities appear as sim-
useitsone-loopversiontopassfromthepolemasstothe
ple and double poles in ǫ, D =4 2ǫ, where D is space-
− running one, which is consistent with NLO calculations.
time dimension. The singularpartis proportionalto the
Substituting Eq.(4) into Eq.(1) we have
tree-leveltransitionprobabilityanditiscancelledbythe
IR-singular part of the four-parton contribution. Rbd(y ,m (µ),µ)=
3 c b
There are several methods of analytical cancelation α (µ) M2
ofIRsingularities. In10,11 theso-calledslicingmethod22 1+rb(µ)(cid:20)b0+ sπ (cid:18)b1−2b0ln µ2Z(cid:19)(cid:21) (5)
was used. In this case the analytical integration over
a thin slice at the border of phase-space is performed in with b =b +b (8/3 2lnr ) and r =m2/M2.
D-dimensions. Theintegrationovertherestofthephase- To1relat1equ0arkm−assesatbdifferenbtscalebs,wZeusethe
space, defined by a particular jet-algorithm, is done nu- one-loop renormalization group equation, e.g. m (µ) =
b
merically in D = 4. In the third NLO calculation12, a 2γ0
de.igff.e2r0e,n23t)awpparsouascehd,.theso-calledsubtractionmethod(see wmibt(hMnZ)=hα5αs.(sM(µZ))iβ0 , where γ0 =2 and β0 =11− 32nf
f
We would like to stress that the structure of the
NLO result for the Z0 3jets in the massive case is 6 Numerical results and discussion.
→
completely different from the massless one20,21. In the
massivecasethecollineardivergencesassociatedwiththe In Figs.1,2 we show NLO results, obtained by using
gluon radiation from the quarks, are softened into lnr Eq.(5)parameterizedintermsofm (M )=2.83GeV for
b b Z
and only collinear divergences due to gluon-gluon split- different values of the renormalization scale µ = 10GeV
tingremain. Therefore,thetestperformedin10byrecov- - dashed line, µ = 30GeV - dotted line and µ = M
Z
ering the massless limit from the result with finite mass, - dashed-dotted line. As expected, the NLO curve for
is a rather non-trivial one. largescaleis closerto LO curvefor m (M ), while NLO
b Z
In contrast to the LO function, b , the NLO func- result for smaller scale is closer to the LO one for M .
0 b
tion b has a significant residual mass dependence10,19 Like in the massless case14, the NLO corrections for the
1
andthephenomenologicalinterpolation,e.g. forDurham Jade scheme are larger than in the Durham and depend
scheme, can be chosen as follows significantly on y for small values, y <0.025.
c c
The µ-dependence is a reflection of the fixed order
b =b(0)+b(1)lny +b(2)lnr . (3) perturbative calculation and the size of the variation
1 1 1 c 1 b
of the observable is usually used to estimate higher or-
TheapproximationEq.(3)workswellintheregion0.01 der corrections. The study of the µ-dependence for the
≤
y 0.1 and extra powers of lnr and/or mixed terms Durham scheme is presented in Fig.3. The Rbd is shown
c ≤ b 3
lnr lny do not improve its quality. as a function of scale µ for the fixed value of y = 0.02.
b c c
Thelower,solidcurveisobtainedbyusingEq.(1),i.e. pa- the totalwidth: the mainradiativecorrectionsaretaken
rameterizedintermsofthepolequarkmass. Inthiscase into account by the running of the QCD parameters to
thescaledependence isduetotherenormalizedcoupling the M -scale. Note also that for small, but still reason-
Z
constant,α . Othercurvesshowµ-dependencewhenRbd able value of y 0.01, the mass effects in D are as
s 3 c ≈ 2
is parameterizedin terms of the running mass,m (M ), large as 10% (although one has to remember that this
b Z
Eq.(5), with different definitions of the quark mass used is a differential rate and statistical errors are larger here
in the logarithms: M , m (M ), m (µ). too).
b b Z b
1 bd
0.974 Rbd 0.98 D LO-mb(mZ)
2
3
0.972 NLO-m(m ) 0.96
b Z
0.94
0.97
0.92
0.968 Durham (y=0.02) 0.9 LO-Mb
c
0.966
0.88
Cambridge
0.964 0.86
NLO-M
b
0.84 NLO-m(m )
0.962 b Z
m 0.82 NLO-M y
0.96 (GeV) b c
0.8
0 20 40 60 80 100 0.005 0.01 0.015 0.02 0.025
Figure3: TheratioRb3d (Durham)asafunctionofµ. Figure5: TheratioD2bd (Cambridge)asafunctionofyc.
Theconservativeestimatefofthetheoreticalerrorfor The scale dependence for the Cambridge scheme is
the Rbd is to take the whole spread given by the curves shown in Fig.6 for fixed yc = 0.01. The result is also
3
in Fig.3. The uncertainty in Rbd induces an error in quite remarkable. The ratio expressed in terms of run-
3
m (M ): ∆Rbd = 0.004 ∆m 0.23GeV, which is, ning mass is very stable with respect to scale variation.
b Z 3 → b ≃ ThisbehaviorintheCambridgeschemeisratherpromis-
however, below current experimental errors, dominated
ingwithrespecttoimprovementsoftheDELPHIresult2
by fragmentation.
for m (M ). We refer to26 for detailed discussionof the
In the Figs.4,5 we show results for the fixed value of b Z
scale,µ=M ,fortheCambridgealgorithm25,arecently NLO predictions in this new scheme and to27 for the
Z
first analysis of the LEP data applying the Cambridge
proposed modification of the Durham scheme.
algorithm.
1 bd
R 0.97 bd
LO-m(m ) R
3 b Z
0.98 0.968 3
0.966 NLO-mb(mZ)
0.96
0.964
LO-M Cambridge
b
0.94 0.962 (y=.01)
c
Cambridge
0.96
0.92 NLO-mb(mZ) 0.958 NLO-Mb
NLO-Mb y m
c 0.956 (GeV)
0.9
0.005 0.01 0.015 0.02 0.025
0 20 40 60 80 100
Figure4: TheratioRb3d (Cambridge)asafunctionofyc. Figure6: TheratioRbd (Cambridge)asafunctionofµ.
3
It is remarkable that in this scheme and for two
observables, Rbd and two-jet differential ratio, D (see
3 2 7 Conclusions
Eq.(2)), the LO result for the running mass, m (M ) is
b Z
very close to the NLO one. In a sense, it is similar to Three-jets observables at the Z0-peak (jet rates, differ-
fTheJadeschemegivessignificantlylargererror,seeFig.1. ential jet-rates, event-shape observables etc.) have sig-
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