Expressions for Sudakov form factors for heavy quarks are presented. They are
used to construct resummed jet rates in electron-positron annihilation.
Predictions are given for production of bottom quarks at LEP and top quarks at
the Linear Collider.Comment: Talk given by German Rodrigo at EPS2003 - Aachen, Germany, July 2003;
  3 pages, 2 figure

Krauss, Frank

Rodrigo, German

English

arXiv

Germán Rodrigo

Frank Krauss

Crossref

Resummed jet rates for heavy quark production in e+e– annihilation

Expressions for Sudakov form factors for heavy quarks are presented. They are used to construct resummed jet rates in e(+)e(-) annihilation. Predictions are given for production of bottom quarks at LEP and top quarks at the Linear Collider

Rodrigo García, Germán Vicente

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:hep-ph/0309325v1  29 Sep 2003EPJ manuscript No.(will be inserted by the editor)CERN-TH/2003-220Resummed jet rates for heavy quark productionin e+e− annihilation ⋆Germa´n Rodrigo a and Frank Krauss bTheory Division, CERN, CH-1211 Geneva 23, Switzerland.Received: September 30, 2003Abstract. Expressions for Sudakov form factors for heavy quarks are presented. They are used to constructresummed jet rates in e+e− annihilation. Predictions are given for production of bottom quarks at LEPand top quarks at the Linear Collider.PACS. 12.38.Bx – 12.38.Cy – 13.66.Bc – 14.65.Fy1 IntroductionThe formation of jets is the most prominent feature ofperturbative QCD in e+e− annihilation into hadrons. Jetscan be visualized as large portions of hadronic energy or,equivalently, as a set of hadrons confined to an angularregion in the detector. In the past, this qualitative defi-nition was replaced by quantitatively precise schemes todefine and measure jets, such as the cone algorithms of theWeinberg–Sterman [1] type or clustering algorithms, e.g.the Jade [2,3] or the Durham scheme (k⊥ scheme) [4]. Arefinement of the latter one is provided by the Cambridgealgorithm [5]. Equipped with a precise jet definition thedetermination of jet production cross sections and theirintrinsic properties is one of the traditional tools to in-vestigate the structure of the strong interaction and todeduce its fundamental parameters. In the past decade,precision measurements, especially in e+e− annihilation,have established both the gauge group structure underly-ing QCD and the running of its coupling constant αs overa wide range of scales. In a similar way, also the quarkmasses should vary with the scale.A typical strategy to determine the mass of, say, thebottom-quark at the centre-of-mass (c.m.) energy of thecollider is to compare the ratio of three-jet productioncross sections for heavy and light quarks [6,7,8,9,10]. Atjet resolution scales below the mass of the quark, i.e. forgluons emitted by the quark with a relative transverse⋆ Work supported by the EC 5th Framework Programmeunder contract numbers HPMF-CT-2000-00989 and HPMF-CT-2002-01663.a e-mail: German.Rodrigo@cern.chb e-mail: Frank.Krauss@cern.ch. Present address: Institutfu¨r Theoretische Physik, TU Dresden, D-01062 Dresden, Ger-many.momentum k⊥ smaller than the mass, the collinear diver-gences are regularized by the quark mass. In this regionmass effects are enhanced by large logarithms ln(mb/k⊥),increasing the significance of the measurement. Indeed,this leads to a multiscale problem since in this kinematicalregion also large logarithms ln(√s/k⊥) appear such thatboth logarithms need to be resummed simultaneously. Asolution to a somewhat similar two-scale problem, namelyfor the average sub-jet multiplicities in two- and three-jetevents in e+e− annihilation was given in [11]. We reporthere on the resummation of such logarithms in the k⊥-like jet algorithms [12] and provide some predictions forheavy quark production. A preliminary comparison withnext-to-leading order calculations of the three-jet rate [6,13,14,15] is presented.2 Jet rates for heavy quarksA clustering according to the relative transverse momentahas a number of properties that minimize the effect ofhadronization corrections and allow an exponentiation ofleading (LL) and next-to-leading logarithms (NLL) [4]stemming from soft and collinear emission of secondarypartons. Jet rates in k⊥ algorithms can be expressed, upto NLL accuracy, via integrated splitting functions andSudakov form factors [4]. For a better description of thejet properties, however, the matching with fixed order cal-culations is mandatory. Such a matching procedure wasfirst defined for event shapes in [16]. Later applicationsinclude the matching of fixed-order and resummed expres-sions for the four-jet rate in e+e− annihilation into mass-less quarks [17,18]. A similar scheme for the matching oftree-level matrix elements with resummed expressions inthe framework of Monte Carlo event generators for e+e−2 Germa´n Rodrigo, Frank Krauss: Resummed jet rates for heavy quark production in e+e− annihilationprocesses was suggested in [19] and extended to generalcollision types in [20].We shall recall here the results obtained in [12] forheavy quark production in e+e− annihilation. In the quasi-collinear limit [21,22], the squared amplitude at tree-levelfulfils a factorization formula, where the splitting func-tions Pab for the branching processes a → b + c, with atleast one of the partons being a heavy quark, are given inD = 4− 2ǫ dimensions byPQQ(z, q) = CF[1 + z21− z − ǫ (1− z)−2z(1− z)m2q2 + (1− z)2m2],PgQ(z, q) = TR[1− 2z(1− z)1− ǫ +2z(1− z)m2(1− ǫ)(q2 +m2)], (1)where z is the usual energy fraction of the branching, andq2 is the space-like transverse momentum. As expected,these splitting functions match the massless splitting func-tions in the limit m → 0 for q2 fixed. Furthermore, forq ≪ (1− z) m the splitting function PQQ is not any moreenhanced at z → 1, which is the known “dead cone” [23]effect. The splitting functionPgg(z) = CA[z1− z +1− zz+ z(1− z)]. (2)obviously does not get mass corrections at the lowest or-der.Branching probabilities are defined through [12]ΓQ(Q, q,m) =1−q/Q∫q/Qdzq2q2 + (1 − z)2m2 PQQ(z, q)= ΓQ(Q, q,m = 0) + CF[12− qmarctan(mq)−2m2 − q22m2ln(m2 + q2q2)],Γf (Q, q,m) =1−q/Q∫q/Qdzq2q2 +m2PgQ(z, q)= TRq2q2 +m2[1− 13q2q2 +m2],Γg(Q, q) =1−q/Q∫q/Qdz Pgg(z) = 2CA(lnQq− 1112), (3)and the Sudakov form factors, which yield the probabilityfor a parton experiencing no emission of a secondary par-ton between transverse momentum scales Q down to Q0,read∆Q(Q,Q0) = exp−Q∫Q0dqqαs(q)πΓQ(Q, q) ,∆g(Q,Q0) = exp−Q∫Q0dqqαs(q)π(Γg(Q, q) + Γf (Q, q)) ,∆f (Q,Q0) = [∆Q(Q,Q0)]2/∆g(Q,Q0) , (4)where Γf (Q, q) accounts for the number n(l,h)f of activelight or heavy quarks. Jet rates in the k⊥ schemes canbe expressed by the former branching probabilities andSudakov form factors. For the two- and the three-jet ratesR2 = [∆Q(Q,Q0)]2 , (5)R3 = 2 [∆Q(Q,Q0)]2Q∫Q0dqqαs(q)πΓQ(Q, q)∆g(q,Q0) ,where Q is the c.m. energy of the colliding e+e−, andQ20 = ycutQ2 plays the role of the jet resolution scale.Single-flavour jet rates in Eq. (5) are defined from theflavour of the primary vertex.In order to catch which kind of logarithmic correctionsare resummed with these expressions it is illustrative tostudy the above formulae in the kinematical regime suchthat Q ≫ m ≫ Q0. Expanding in powers of αs, jet ratescan formally be expressed asRn = δn2 +∞∑k=n−2(αs(Q)π)k 2k∑l=0c(n)kl , (6)where the coefficients c(n)kl are polynomials of order l inLy = ln(1/ycut) and Lm = ln(m2/Q20). The coefficientsfor the first order in αs are given byc(2)12 = −c(3)12 = −12CF (L2y − L2m) ,c(2)11 = −c(3)11 =32CFLy +12CFLm . (7)The coefficients for the second order in αs can be foundin [12], as well as the corresponding result for the four-jetrate.The impact of mass effects can be highlighted by twoexamples, namely by the effect of the bottom quark massin e+e− annihilation at the Z-pole (Fig. 1, up), and bythe effect of the top quark mass at a potential Linear Col-lider operating in the TeV region (Fig. 1, down). In Fig. 1,leading order (LO) and next-to-leading order (NLO) pre-dictions for three-jet rates are compared with the NLLresult as obtained by numerical integration from Eq. (5).For a comparison with the dead cone approximation seealso [12]. While in the case of bottom quarks at LEP1 en-ergies the overall effect of the quark mass is at the few-per-cent level, this effect becomes tremendous for top quarksat the Linear Collider. Fixed order predictions for b-quarkproduction clearly fail at very low values of ycut, by givingunphysical values for the jet rate, while the NLL predic-tions keep physical and have the correct shape. The latteris an indication of the necessity for performing such kindof resummations. Fixed order predictions work well forGerma´n Rodrigo, Frank Krauss: Resummed jet rates for heavy quark production in e+e− annihilation 300.10.20.30.40.50.60.70.80.9110-410-3ycutthree jet rateLO masslessLO mb=5GeVNLO masslessNLO mb=5GeVNLL masslessNLL mb=5GeVLEPI 91 GeV0.050.10.150.20.250.30.3510-410-3ycutthree jet rate1 TeV, mt=175 GeVLONLONLLFig. 1. Effect of a b-mass of 5 GeV on the single-flavourthree-jet rate at LEP1 energies as a function of the jet res-olution parameter in the Cambridge algorithm (upper plot).Effect of a t-mass of 175 GeV on the single-flavour three-jetrate as a function of the jet resolution parameter, at a poten-tial Linear Collider operating at c.m. energies of 1 TeV (lowerplot). Predictions are provided at LO, NLO and NLL accuracy.top production at the Linear Collider, a consequence ofthe strong cancellation of leading logarithmic corrections,and are fully compatible with our NLL result.3 ConclusionsSudakov form factors involving heavy quarks have beenemployed to estimate the size of mass effects in jet ratesin e+e− annihilation into hadrons. These effects are size-able and therefore observable in the experimentally rel-evant region. A preliminary comparison with fixed orderresults have been presented, and showed good agreement.Matching between fixed-order calculations and resummedresults is in progress [24].AcknowledgementsWe would like to thank Z. Trocsanyi for his kind invita-tion to present these results at the HEP2003 EurophysicsConference in Aachen. G.R. acknowledges partial supportfrom Generalitat Valenciana under grant CTIDIB/ 2002/24and MCyT under grants FPA-2001-3031 and BFM 2002-00568.References1. G. Sterman and S. Weinberg, Phys. Rev. Lett. 39 (1977)1436.2. W. Bartel et al. [JADE Collaboration], Z. Phys. C 33(1986) 23.3. S. Bethke et al. [JADE Collaboration], Phys. Lett. B 213(1988) 235.4. S. Catani, Y. L. Dokshitzer, M. Olsson, G. Turnock andB. R. Webber, Phys. Lett. B 269 (1991) 432.5. Y. L. Dokshitzer, G. D. Leder, S. Moretti and B. R. Web-ber, JHEP 9708 (1997) 001 [hep-ph/9707323].6. G. Rodrigo, A. Santamaria and M. S. Bilenky, Phys. Rev.Lett. 79 (1997) 193 [hep-ph/9703358].7. P. Abreu et al. [DELPHI Collaboration], Phys. Lett. B 418(1998) 430.8. A. Brandenburg, P. N. Burrows, D. Muller, N. Oishi andP. Uwer, Phys. Lett. B 468 (1999) 168 [hep-ph/9905495].9. R. Barate et al. [ALEPH Collaboration], Eur. Phys. J. C18 (2000) 1 [hep-ex/0008013].10. G. Abbiendi et al. [OPAL Collaboration], Eur. Phys. J. C21 (2001) 411 [hep-ex/0105046].11. S. Catani, B. R. Webber, Y. L. Dokshitzer and F. Fiorani,Nucl. Phys. B 383 (1992) 419.12. F. Krauss and G. Rodrigo, hep-ph/0303038;hep-ph/0309326.13. G. Rodrigo, Nucl. Phys. Proc. Suppl. 54A (1997) 60[hep-ph/9609213].14. M. S. Bilenky, S. Cabrera, J. Fuster, S. Marti, G. Ro-drigo and A. Santamaria, Phys. Rev. D 60 (1999) 114006[hep-ph/9807489].15. G. Rodrigo, M. S. Bilenky and A. Santamaria, Nucl. Phys.B 554 (1999) 257 [hep-ph/9905276]; Nucl. Phys. Proc.Suppl. 64 (1998) 380 [hep-ph/9709313].16. S. Catani, L. Trentadue, G. Turnock and B. R. Webber,Nucl. Phys. B 407 (1993) 3.17. L. J. Dixon and A. Signer, Phys. Rev. D 56 (1997) 4031[hep-ph/9706285].18. Z. Nagy and Z. Trocsanyi, Phys. Rev. D 59 (1999) 014020[Erratum-ibid. D 62 (2000) 099902] [hep-ph/9806317].19. S. Catani, F. Krauss, R. Kuhn and B. R. Webber, JHEP0111 (2001) 063 [hep-ph/0109231].20. F. Krauss, JHEP 0208 (2002) 015 [hep-ph/0205283].21. S. Catani, S. Dittmaier and Z. Trocsanyi, Phys. Lett. B500 (2001) 149 [hep-ph/0011222].4 Germa´n Rodrigo, Frank Krauss: Resummed jet rates for heavy quark production in e+e− annihilation22. S. Catani, S. Dittmaier, M. H. Seymour and Z. Trocsanyi,Nucl. Phys. B 627 (2002) 189 [hep-ph/0201036].23. Y. L. Dokshitzer, V. A. Khoze and S. I. Troian, J. Phys.G 17 (1991) 1602.24. F. Krauss, G. Rodrigo, A. Scha¨licke and P. Tortosa, inpreparation.

Resummed jet rates for heavy quark production in e+e- annihilation

Expressions for Sudakov form factors for heavy quarks are presented. They are used to construct resummed jet rates in e
+
e
– annihilation. Predictions are given for production of bottom quarks at LEP and top quarks at the Linear Collider.
        
PACS: 12.38.Bx – 12.38.Cy – 13.66.Bc – 14.65.F

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