Heavy quark mass effects in e^+e^- into three jets

5 Pages·0.13 MB·English

by  Mikhail Bilenky

FTUV/98-77
IFIC/98-78
Heavy quark mass effects in e+e− into three jets ∗
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9 Mikhail Bilenkya† Germ´an Rodrigob‡, Arcadi Santamariac and
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n
a Institute of Physics, AS CR, 18040 Prague and
a
J Nuclear Physics Institute, AS CR, 25068 Rˇeˇz(Prague),Czech Republic
4 b INFN-Sezione di Firenze, Largo E. Fermi 2, 50125 Firenze, Italy
2 c Departament de F´ısica Teo`rica, IFIC, CSIC-Universitat de Val`encia, 46100 Burjassot, Val`encia, Spain

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Abstract
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4 Next-to-leading order calculation for three jet heavy quark production in e+e− collisions,
1 including complete quark mass effects, is reviewed. Its applications at LEP/SLC are also
1 discussed.
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∗tobepublishedintheProceedingsoftheHighEnergyPhysicsInternationalEuroconferenceonQuantumChromodinamics
(QCD ’98),Montpellier,France,3-9Jul1998. Ed. S.Narison,NuclPhys. B(Proc. Suppl.).
†OnleavefromJINR,141980Dubna,RussianFederation
‡OnleavefromDepartamentdeF´ısicaTeo`rica,IFIC,CSIC-UniversitatdeVal`encia,46100Burjassot,Val`encia,Spain
Heavy quark mass effects in e+e− into three jets
Mikhail Bilenky a∗, Germ´an Rodrigo b† and Arcadi Santamaria c
a Institute of Physics, AS CR, 18040 Prague and
Nuclear Physics Institute, AS CR, 25068 Rˇeˇz(Prague), Czech Republic
b INFN-Sezione di Firenze, Largo E. Fermi 2, 50125 Firenze, Italy
c Departament de F´ısica Teo`rica, IFIC, CSIC-Universitat de Val`encia, 46100 Burjassot, Val`encia, Spain
Next-to-leading order calculation for three-jet heavy quark production in e+e−-collisions, including complete
quarkmass effects, is reviewed. Its applications at LEP/SLC are also discussed.
The importance of the corrections due to the pects ofthe next-to-leadingorder(NLO)calcula-
mass of the heavy quark in the jet-production in tionofthedecayZ →3jetswithmassivequarks,
e+e−-collisionshasbeenalreadyseenintheearly necessary for the measurements of the bottom-
tests of the flavour independence of the strong quark mass at the Z-peak. Recently such cal-
coupling constant [1,2]. The final high precision culations were performed independently by three
of the LEP/SLC experiments required accurate groups [8–10].
account for the bottom-quark mass in the the- The first question we would like to answer
oretical predictions. If quark mass effects are whether it is not at all surprising that LEP/SLC
neglected, the ratio αb/αuds measured from the observables are sensitive to m as the main scale
s s b
analysisofdifferentthree-jet event-shapeobserv- involved is the mass of the Z-boson, M ≫ m .
Z b
ablesis shifted awayfromunity up to 8%[3](see Indeed, the quark-mass effects for an inclusive
also [4]). observable such as the total width Z → b¯b are
Sensitivity of the three-jet observables to the negligible. Due to Kinoshita-Lee-Nauenberg the-
value of the heavy quark mass allowed to con- orem such observable does not contain mass sin-
sider the possibility [5,6] of the determination of gularities and a quark-mass appears in the ratio
the b-quark mass from LEP data, assuming the m2(M )/M2 ≈ 10−3, where using MS running
q Z Z
universality of α . In a recent analysis of three- massatthe M -scaletakesintoaccountthe bulk
s Z
jet events, DELPHI measured the mass of the of the NLO QCD corrections [11,6].
b-quark, m , for the first time far above the pro- However,the situationwithmoreexclusiveob-
b
duction threshold [4]. This result is in a good servables is different. Let’s consider the simplest
agreement with low energy determinations of m process, Z → bbg, which contributes to three-jet
b
using QCD sum rules and lattice QCD from Υ final state at the leading order (LO). When the
and B-mesons spectra (for recent results see e.g. energyoftheradiatedgluonapproacheszero,the
[7]) The agreement between high and low energy process has an infrared (IR) divergence and in
determinations of the quark mass is rather im- order to have an IR-finite prediction, some kine-
pressive as non-perturbative parts are very dif- matical restriction should be introduced in the
ferent in the two cases. phase-space integration to cut out the trouble-
In this contribution we will discuss some as- someregion. Ine+e−-annihilationthatisusually
donebyapplyingtheso-calledjet-clusteringalgo-
∗OnleavefromJINR,141980Dubna,RussianFederation. rithmwithajet-resolutionparameter,yc (see[12]
†On leave from Departament de F´ısica Teo`rica, IFIC, for recent discussion of jet-algorithms in e+e−).
CSIC-UniversitatdeVal`encia, 46100 Burjassot,Val`encia, Then the transition probability in the three-jet
Spain.
partofthe phase-spacewillhavecontributionsas lationsinthemassivecase[6]. Themainobserva-
large as 1/y ·(m2/M2), where y can be rather tion from fig. 1 is that for y >0.05, b is almost
c b Z c c 0
small,intherange10−2−10−3. Thenonecanex- independent of the value of m for all schemes.
b
pectasignificantenhancementofthequark-mass Although, this remains true alsofor smaller y in
c
effects, which can reach several percents. DURHAM and E schemes, there is a noticeable
The convenient observable for studies of the mass dependence in JADE and EM schemes.
mass effects in the three-jet final state, proposed Note that b is positive for E-scheme. That
0
some time ago [1,6], is defined as follows contradicts the intuitive expectations that a
heavy quark should radiate less than a light one.
Γb (y )/Γb
Rbd = 3j c (1) This unusual behavior is due to the definition
3 Γd3j(yc)/Γd of the resolution parameter in E-scheme, yij =
α (p +p )2/s, whichhas significantly different val-
s i j
=1+r b (y ,r )+ b (y ,r )
b 0 c b π 1 c b ues for partons with the same momenta in the
(cid:16) (cid:17)
massiveandmasslesscases,anditcanbe usedas
where Γq and Γq are three-jet and total decay
3j a consistency check of the data.
widths of the Z-boson into quark pair of flavour
In what follows we restrict ourselves to
q, r = m2/M2. Note that above expression is
b b Z DURHAM scheme, the one used in the experi-
not an expansion in r .
b mental analysis [4], and b can be interpolated
TheLOfunction,b ,isplottedinfig.1forfour 0
0 as: b = b(0) +b(1)lny +b(2)ln2y . In the LO
different jet algorithms. 0 0 0 c 0 c
calculations we can not specify what value of the
b-quarkmassshouldbetakeninthecalculations:
LO functions b
0 allquark-massdefinitions areequivalent(the dif-
-10
-8 -12 ference is due to the higher orders in αs). One
-10 --1164 canuse,forexample,thepolemassMb ≈4.6GeV
-18 or the MS-running mass m (µ) at any scale rel-
-12 b
-20
-14 EM --2242 JADE emva(nmt t)o≈t4h.e13pGreoVbleamnd, mmb(M≤ )µ≈≤2.8M3GZe,Vw. iAths
-16 -26 b b b Z
-28 a result, the spread in LO predictions for differ-
-18 -30
ent values of b-quark mass is significant, the LO
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1
y y prediction is not accurate enough and the NLO
40 c -8 c
calculation should be done.
-10
35
-12 At the NLO there are two different contribu-
30 E -14
tions: from one-loop corrections to the three-
-16
25
-18 parton decay, Z →bbg and tree-levelfour-parton
20 -20 decay, Z → bbgg and Z → bbqq, q = u,d,s,c,b
-22
15
-24 DURHAM integrated over the three-jet region of the four-
10 -26
parton phase-space. In the NLO calculation one
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 has to deal with divergences, both ultraviolet 3
y y
c c
and infrared which appear at the intermediate
Figure 1. LO contribution to the ratio Rbd as a stages. The sum of the one-loop and tree-level
3 contributions is, however, IR finite. We would
function of y (see eq.(1) for the definition) for
c
like to stress that the structure of the NLO cor-
m = 3GeV (dashed curve) and m = 5GeV
b b
rectionsinthemassivecaseiscompletelydifferent
(solid curve).
from the ones in the massless case [13]. That is
Together with well known JADE, E and
DURHAMschemesweconsiderthe so-calledEM
3The ultraviolet divergences in the loop-contribution are
algorithm [6] with a resolution parameter y =
ij cancelled after the renormalization of the parameters of
2pipj/s and which was used for analytical calcu- theQCDLagrangian.
due to the fact that in the massive case, part of aboutm . Thenwecanuseone-looprenormaliza-
b
the collinear divergences, those associated with tion group improved equation in order to define
the gluonradiationfromthe quarks,aresoftened the quarkmassatthe higherscales. Substituting
into lnr and only collinear divergences associ- eq.(2) into definition eq.(1) we have
b
ated with gluon-gluon splitting remain.
Rbd(y ,m (µ),µ)=
In the NLO calculations one should specify 3 c b
the quark mass definition. It turned out tech- 1+r (µ) b + αs(µ) b −2b lnMZ2 (3)
nically simpler to use a mixed renormalization b (cid:20) 0 π (cid:18) 1 0 µ2 (cid:19)(cid:21)
scheme which uses on-shell definition for the
with b =b +b (8/3−2lnr ) andr =m2/M2.
quarkmassandMS definitionforthestrongcou- 1 1 0 b b b Z
In fig. 2 we show the NLO function b (y ,r )
pling. Therefore, physical quantities are origi- 1 c b
calculated for three different values of the quark
nally expressed in terms of the pole mass. It can
mass : 3GeV (open circles), 4GeV (squares) and
beperfectlyusedinperturbationtheory,however,
5GeV (triangles).
in contrast to the pole mass in QED, the quark
pole mass is not a physical parameter. The non- 1
bd
perturbative corrections to the quark self-energy R
0.99 3
bring an ambiguity of order ≈ 300MeV (hadron
size) to the physical position of the pole of the 0.98
quark propagator. Above the quark production
0.97
threshold, it is natural to use the running mass
definition (we use MS). The advantage of this 0.96
definition is that m (µ) can be used for µ≫m .
b b 0.95
-100
– Durham
b 0.94
-120 1 y
c
-140 0.93
0 0.02 0.04 0.06 0.08 0.1
-160
-180
Figure 3. The ratio Rbd (eq. (1)). Solid curves
3
-200 - LO predictions, dashed curves give the NLO re-
-220 sults (see text for details).
Durham
-240 y In contrast to the b0, one sees a significant
-260 c residual mass dependence in b , which can not
1
be neglected. The solid lines in fig. 2 represent a
0 0.02 0.04 0.06 0.08 0.1
(0) (1) (2)
fitbythefunction: b =b +b lny +b lnr
1 1 1 c 1 b
performedinthe range0.01≤y ≤0.1Thequal-
c
Figure 2. NLO function b for different m (see
1 b ityofthisinterpolationisverygoodandthemain
eqs.(1),(3) for the definition and text for details).
residualm dependenceinb istakenintoaccount
b 1
The errors are due to numerical integrations.
by lnr term. Inclusion of higher powers of lnr
b b
The pole, Mb, and the running masses of the does not improve the fit.
quark are perturbatively related Fig. 3 presents theoretical predictions in the
DURHAM scheme for the Rbd observable mea-
α 4 m2 3
M =m (µ) 1+ s −ln b . (2) sured by DELPHI[4]. The solid lines are
b b (cid:20) π (cid:18)3 µ2 (cid:19)(cid:21)
LO predictions for the b-quark mass, m =
b
We use this one-loop relation to pass from the m (M ) = 2.83GeV (upper curve) and m =
b Z b
pole massto the running one,which is consistent M = 4.6GeV (lower curve). The dashed curves
b
withourNLOcalculations. Tomatchneededpre- give NLO results for different values of scale
cisionwehavetousethisequationforvaluesofµ µ : 10, 30, 91GeV. One sees that NLO curve
for large scale is naturally closer to LO curve for the warmhospitality during his stay at Montpel-
m (M ) and for smaller scale is closer to the LO lier.
b Z
one with m =M .
b b
Fig. 4 illustrates the scale dependence of Rbd
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