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UPRF-2002-12 PreprinttypesetinJHEPstyle. -PAPERVERSION MPI-PhT 2002-34 Soft-Gluon Resummation for Bottom Fragmentation in Top Quark Decay 3 0 0 Matteo Cacciari 2 n Dipartimento di Fisica, Universita` di Parma, Italy, a and INFN, Sezione di Milano, Gruppo Collegato di Parma. J 1 E-mail: [email protected] 2 Gennaro Corcella 2 v Max-Planck-Institut fu¨r Physik, Werner-Heisenberg-Institut, 4 Fo¨hringer Ring 6, D-80805 Mu¨nchen, Germany. 0 2 E-mail: [email protected] 9 0 Alexander D. Mitov∗ 2 0 Department of Physics and Astronomy, University of Rochester, / h Rochester, NY 14627, U.S.A. p E-mail: [email protected] - p e h : v Abstract: We study soft-gluon radiation in top quark decay within the framework i X of perturbative fragmentation functions. We present results for the b-quark energy r distribution, accounting for soft-gluon resummation to next-to-leading logarithmic a accuracy in both the MS coefficient function and in the initial condition of the perturbative fragmentation function. The results show a remarkable improvement and the b-quark energy spectrum in top quark decay exhibits very little dependence on factorization and renormalization scales. We present some hadron-level results in both x and moment space by including non-perturbative information determined B from e+e− data. Keywords: Heavy Quarks Physics, QCD, NLO Computations. ∗ The work of A.D.M. was supported in part by the U.S. Department of Energy, under grant DE-FG02-91ER40685. Contents 1. Introduction 1 2. Perturbative fragmentation and top quark decay 2 3. Soft-gluon resummation 4 4. Energy spectrum of the b quark 9 5. Energy spectrum of b-flavoured hadrons in top decay 13 6. Conclusions 16 1. Introduction Heavy-flavour and in particular top quark physics is presently one of the main fields of investigation in theoretical and experimental particle physics. The current ex- periments at the Tevatron accelerator and, ultimately, at LHC [1] and e+e− Linear Colliders [2] will produce large amounts of top quark pairs, which will allow one to perform improved measurements of top properties, such as its mass. For this purpose, precise calculations for top production and decay processes are mandatory. While fixed-order calculations reliably predict total cross sections or widths, differential distributions typically contain large logarithms associated with soft or collinearpartonradiation. Forheavy-quarkproductionprocesses, althoughthequark mass m (much larger than the QCD scale Λ) acts as a regulator for the collinear sin- gularity, event shapes still contain large αnlnp(Q2/m2) (with p n) terms, Q being S ≤ a typical scale of the process, which make fixed-order predictions unreliable when Q m. Such logarithms can be resummed by using the perturbative fragmentation ≫ approach[3],whichfactorizestherateofheavy-quarkproductionintotheconvolution of a coefficient function, describing the emission of a massless parton, and a perturbative fragmentation function D(µ ,m), where µ is the factorization scale. The F F perturbative fragmentation function expresses the transition of the massless parton into the massive quark, and its value at any scale µ can be obtained by solving the F Dokshitzer–Gribov–Lipatov–Altarelli–Parisi(DGLAP)evolutionequations[4,5]once an initial condition at a scale µ is given. In ref. [3] the large ln(Q2/m2) collinear 0F logarithms were resummed tonext-to-leading logarithmic(NLL) accuracy in thecase 1 of heavy quark production in e+e− collisions and an explicit next-to-leading order (NLO) expression for D(µ ,m), which was argued to be process independent, was F given. More recently, ref. [6] has established the process independence in a more general way. The approach of perturbative fragmentation has been extensively used for e+e− annihilation [6–10], hadron collisions [11,12], photoproduction [8,13] and, more recently, for bottom quark production in top quark decay t bW [14]. → The initial condition of the perturbative fragmentation function and the coefficient function, though free of the collinear large ln(Q2/m2), contain large logarithms which are due to soft-gluon radiation. The ones contained in the initial condition are process independent [6], and were already included in the top-to-bottom decay process in [14], where the complete (α ) calculation of the t bW(g) process was S O → also performed. The largelogarithms contained inthe coefficient function areinstead process dependent, and have to be evaluated for every specific case. It is precisely the purpose of this paper to extend the analysis of ref. [14] and to present results for soft-gluon resummation in the coefficient function. This will allow, together with the results of [6], to complete the evaluation of soft-gluon effects to NLL accuracy for the top-to-bottom decay process, and to investigate the impact on the b-quark energy distribution. Afterthefragmentationofheavyquarksine+e− collisionsconsideredin[6],thisis the first process whose large logarithms (both collinear and soft) are fully resummed to NLL accuracy within the perturbative fragmentation function formalism. The consistency of this perturbative description with the one used in the e+e− process makes it possible to fit non-perturbative information from e+e− data and use it to make predictions. We shall therefore be able to predict the spectrum for b-flavoured hadron energy distributions in top decay using e+e− experimental data from LEP. The outline of the paper is as follows. In section 2 we review bottom quark production in top quark decay within the framework of perturbative fragmentation. In section 3 we present analytic results for the NLL soft-gluon resummation of the coefficient function in top decay. At the end of the section we also comment on the relation between our results and previous work on soft-gluon resummation in heavy-flavour decay [15–18]. In section 4 we show the b-quark energy distribution in top decay and investigate the impact of soft-gluon resummation. In section 5 we discuss inclusion of non-perturbative effects and present results for b-flavoured hadron spectra in top decay. In section 6 we summarize our main results. 2. Perturbative fragmentation and top quark decay We consider top decay into a bottom quark and a real W boson plus, to order α , a S gluon: t(p ) b(p )W(p )(g(p )) (2.1) t b W g → 2 and define the bottom and gluon normalized energy fractions x and x : b g 1 2p p 1 2p p b t g t x = · , x = · , (2.2) b 1 w m2 g 1 w m2 t t − − where w = m2 /m2. Neglecting the b mass, we have 0 x 1. W t ≤ b,g ≤ Order α corrections to the decay process (2.1) were considered in [14]. It was S observed there that, since m m , one can readily neglect m /m power suppressed b t b t ≪ terms, but on the other hand it is important to resum to all orders terms enhanced, at order α , by the presence of ln(m2/m2). Such a resummation was performed S t b in [14] by employing the perturbative fragmentation formalism [3,6]: The differential width for the production of a massive b quark in top decay is written in terms of the the convolution 1 dΓ 1 dz 1 dΓˆ MS x i b (x ,m ,m ,m ) = (z,m ,m ,µ ) D ,µ ,m b t W b t W F i F b ΓB dxb i Zxb z "ΓB dz # (cid:18) z (cid:19) X + ((m /m )p) , (2.3) b t O where Γ is the width of the Born process t bW, dΓˆ /dz is the differential B i → width for the production of a massless parton i in top decay with energy fraction z, D (x,µ ,m ) is the perturbative fragmentation function for a parton i to fragment i F b into a massive b quark, µ is the factorization scale. The term ((m /m )p) on the F b t O right-hand side stands for contributions that are suppressed by some power p (p 1) ≥ of m in the m m regime. Of course, non-perturbative corrections of the type b b t ≪ Λ/m and Λ/m are understood on the right-hand side of eq. (2.3). We shall use t b everywhere a branching fraction B(t bW) = 1, and only include i = b in the above → summation. The massless differential distribution (1/Γ ) dΓˆ /dz (which is what we shall also B i refer to as “coefficient function”) is defined in the MS factorization scheme after subtraction of the collinear singularities. It has been calculated at order α in [14]. S In the following we shall often use its Mellin moments, defined by 1 1 dΓˆ Γˆ = dz zN−1 (z). (2.4) N Γ dz Z0 B In moment space the convolution (2.3) can then be rewritten as Γ (m ,m ,m ) = Γˆ (m ,m ,µ )D (µ ,m ). (2.5) N t W b N t W F b,N F b The perturbative fragmentation function D (x,µ ,m ) at any scale µ can be b F b F obtained by solving the DGLAP equations. As shown in [6], as long as one can neglect contributions proportional to powers of (m /m )p, the initial condition for b t the perturbative fragmentation function, which we evaluate at a scale µ , is process 0F independent (but scheme dependent). In the MS scheme it reads [3]: α (µ2)C 1+x2 µ2 Dini(x,µ ,m ) = δ(1 x)+ S 0 F ln 0F 2ln(1 x) 1 . (2.6) b 0F b − 2π " 1 x m2 − − − !# − b + 3 The solution of the DGLAP equations in the non-singlet sector, for the evolution from the scale µ to µ , is given in moment space by: 0F F P(0) α (µ2 ) D (µ ,m ) = Dini (µ ,m )exp N ln S 0F b,N F b b,N 0F b 2πb α (µ2)  0 S F α (µ2 ) α (µ2) 2πb + S 0F − S F P(1) 1P(0) , (2.7) 4π2b0 " N − b0 N #) (0) (1) In eq. (2.7)P and P arethe Mellin transforms of the leading and next-to-leading N N order Altarelli-Parisi splitting vertices, and their explicit expression can be found, e.g., in [3]. b and b are the first two coefficients of the QCD β-function 0 1 33 2n 153 19n f f b = − , b = − , (2.8) 0 12π 1 24π2 which enter the following expression for the strong coupling constant at a scale Q2: 1 b ln[ln(Q2/Λ2)] α (Q2) = 1 1 . (2.9) S b0ln(Q2/Λ2) ( − b20ln(Q2/Λ2) ) Equation (2.7) resums to all order terms containing large ln(µ2/µ2 ). In particu- F 0F lar, leading (αnlnn(µ2/µ2 )) and next-to-leading (αnlnn−1(µ2/µ2 )) logarithms are S F 0F S F 0F resummed. Setting, as done in [14], µ m and µ m , one resums the large F t 0F b ≃ ≃ ln(m2/m2) terms with NLL accuracy. These are indeed the large collinear logarithms t b exhibited by the fixed-order calculation with a massive b quark [14]. 3. Soft-gluon resummation In this section we address the problem of soft-gluon resummation in top quark decay. The MS coefficient function computed in [14] and the initial condition of the perturbative fragmentation function (2.6) contain terms behaving like 1/(1 x) or + − [ln(1 x)/(1 x)] , which become arbitrarily large when x approaches one. This is + − − equivalent to contributions proportional to lnN and ln2N in moment space, as can be seen by writing the MS coefficient function [14] in the large-N limit1: α C m2 Γˆ (m ,m ,µ ) = 1+ S F 2ln2N + 4γ +2 4ln(1 w) 2ln t lnN N t W F 2π ( " E − − − µ2F # 1 + K(m ,m ,µ )+ (3.1) t W F O(cid:18)N(cid:19)) 1Following [18], we note that, by defining n = Nexp(γ ), we could rewrite this expression in E terms of ln(n) rather than lnN, with no γ terms explicitly appearing. E 4 where γ = 0.577... is the Euler constant and w = m2 /m2, as defined in section 1. E W t In eq. (3.1) we have introduced the function K(m ,m ,µ ), which contains terms t W F which are constant with respect to N. It reads: 3 m2 K(m ,m ,µ ) = 2γ ln t +2γ2 +2γ [1 2ln(1 w)] t W F 2 − E µ2 E E − − (cid:18) (cid:19) F 1 w 2w + 2lnwln(1 w) 2 − ln(1 w) lnw − − 1+2w − − 1 w − π2 + 4Li (1 w) 6 . (3.2) 2 − − − 3 The x 1 (N ) limit corresponds to soft-gluon radiation in top decay. → → ∞ These soft logarithms need to be resummed to all orders in α [19,20] to improve S our prediction. Softlogarithmsintheinitialconditionoftheperturbativefragmentationfunction are process independent. We can hence resum them with NLL accuracy using the result presented in [6], which we do not report here for the sake of brevity. We present instead the results for the NLL resummation of process-dependent soft-gluon contributions in the MS coefficient function. In order to resum the large terms in eq. (3.1), we follow standard techniques [19], evaluate the amplitude of the process in eq. (2.1) at (α ) in the eikonal S O approximation and exponentiate the result. The eikonal current reads: m2 (p p ) J(p ,p ,p ) 2 = t 2 t · b . (3.3) t b g | | (cid:12)(pt pg)2 − (pt pg)(pb pg)(cid:12) (cid:12) · · · (cid:12) (cid:12) (cid:12) For the sake of comparison with [1(cid:12)9], we express the (α ) w(cid:12)idth in the soft ap- (cid:12) S (cid:12) O proximation as an integral over the variables2 q2 = (p + p )2x and z = 1 x , b g g g − with 0 q2 m2(1 w)2(1 z)2 and 0 z 1. The limits z 1 and q2 0 ≤ ≤ t − − ≤ ≤ → → correspond to soft and collinear emission respectively. In soft approximation, z x , b ≃ the b-quark energy fraction. We obtain: Γˆ (m ,m ,µ ) = CF 1dzzN−1 −1 m2t(1−w)2(1−z)2 dq2α N t W F S π Z0 1−z "Zµ2F q2 1 m2t(1−w)2(1−z)2 dq2α . (3.4) − m2t(1−w)2(1−z)2 Z0 S# Ineq.(3.4)wehaveregularizedthecollinearsingularitysettingthecutoffq2 µ2. At ≥ F NLLaccuracylevel, thisisequivalenttoMSsubtractionindimensionalregularization [6,21]. 2We point out that our definition of the integration variable q2 is analogous to the quantity (1 z)k2 to which the authors of ref. [19] set the scale for α for soft-gluon resummation in S − Drell–Yan and Deep-Inelastic-Scattering processes. For small-angle radiation, q2 q2, the gluon ≃ T transversemomentumwithrespecttotheb-quarkline. Thevariablezisanalogoustoz =1 E /E g q − of ref. [19]. 5 In order to perform soft-gluon resummation to NLL accuracy a number of opera- tions have to be performed on this expression. We set the argument of α equal to q2 S and, as far as the collinear-divergent term is concerned, we perform the replacement C α (q2) A[α (q2)] F S S , (3.5) π q2 → q2 where the function A(α ) was introduced in [19] and is detailed below. Moreover, S the integral over q2 of the non-collinear divergent term can be written, up to terms beyond NLL accuracy, as 1 m2t(1−w)2(1−z)2 dq2α (q2) = α m2(1 w)2(1 z)2 . (3.6) m2(1 w)2(1 z)2 S S t − − t − − Z0 (cid:16) (cid:17) This term describes soft radiation at large-angle, i.e. not collinear enhanced, and it is characteristic of processes where a heavy quark (the top quark in our case, the bottom quark in [15–18]) is present. It can be generalized to all orders by replacing eq. (3.6) according to: C Fα m2(1 w)2(1 z)2 S α m2(1 w)2(1 z)2 . (3.7) − π S t − − → S t − − (cid:16) (cid:17) h (cid:16) (cid:17)i This function is called Γ(α ) in [15], B(α ) in [16], S(α ) in [17], D(α ) in [18]. We S S S S now insert Eqs. (3.5-3.7) into eq. (3.4) and exponentiate the result. We obtain: 1 zN−1 1 m2t(1−w)2(1−z)2 dq2 ln∆ = dz − A α (q2) N S Z0 1−z (Zµ2F q2 h i + S α m2(1 w)2(1 z)2 . (3.8) S t − − ) h (cid:16) (cid:17)i Wewouldliketoevaluateeq.(3.8)toNLLlevel. ThefunctionA(α )canbeexpanded S as follows: ∞ α n A(α ) = S A(n). (3.9) S π n=1(cid:18) (cid:19) X The first two coefficients are needed at NLL level and are given by [19,22]: A(1) = C , (3.10) F 1 67 π2 5 A(2) = C C n , (3.11) F A f 2 " 18 − 6 !− 9 # where C = 4/3, C = 3 and n is the number of quark flavours, which we shall F A f take equal to five for bottom production. The function S(α ) can be expanded according to: S ∞ α n S(α ) = S S(n). (3.12) S π n=1(cid:18) (cid:19) X 6 At NLL level, we are just interested in the first term of the above expansion, which is given by: S(1) = C . (3.13) F − The integral in eq. (3.8) can be performed, up to NLL accuracy, by making the following replacement [19]: e−γE zN−1 1 Θ 1 z , (3.14) − → − − N − ! Θ being the Heaviside step function. This leads to writing the following result for the function ∆ : N ∆ (m ,m ,α (µ2),µ,µ ) = exp lnNg(1)(λ)+g(2)(λ,µ,µ ) , (3.15) N t W S F F h i with λ = b α (µ2)lnN , (3.16) 0 S and the functions g(1) and g(2) given by A(1) g(1)(λ) = [2λ+(1 2λ)ln(1 2λ)] , (3.17) 2πb λ − − 0 A(1) m2(1 w)2 g(2)(λ,µ,µ ) = ln t − 2γ ln(1 2λ) F 2πb0 " µ2F − E# − A(1)b + 1 4λ+2ln(1 2λ)+ln2(1 2λ) 4πb3 − − 0 h i 1 A(2) µ2 [2λ+ln(1 2λ)] +A(1)ln − 2πb0 − πb0 µ2F! S(1) + ln(1 2λ). (3.18) 2πb − 0 In eq. (3.15) the term lnNg(1)(λ) accounts for the resummation of leading logarithms αnlnn+1N in the Sudakov exponent, while the function g(2)(λ,µ,µ ) resums NLL S F terms αnlnnN. S Furthermore, we follow ref. [6] and in our final Sudakov-resummed coefficient function we also include the constant terms of eq. (3.2): α (µ2) C ΓˆS(m ,m ,α (µ2),µ,µ ) = 1+ S FK(m ,m ,µ ) N t W S F " 2π t W F # exp lnNg(1)(λ)+g(2)(λ,µ,µ ) . (3.19) F × h i One can check that the (α ) expansion of eq.(3.19) yields eq. (3.1). S O We now match the resummed coefficient function to the exact first-order result, so that also 1/N suppressed terms, which are important in the region x < 1, are b 7 taken into account. We adopt the same matching prescription as in [6]: we add the resummed result to the exact coefficient function and, in order to avoid double counting, we subtract what they have in common, i.e. the up-to- (α ) terms in the S O expansion of eq. (3.19). Our final result for the resummed coefficient function reads3: Γˆres(m ,m ,α (µ2),µ,µ ) = ΓˆS(m ,m ,α (µ2),µ,µ ) N t W S F N t W S F ΓˆS(m ,m ,α (µ2),µ,µ ) − N t W S F αS h i + Γˆ (m ,m ,α (µ2),µ,µ ) , (3.20) N t W S F αS h i where [ΓˆS] and[Γˆ ] arerespectively the expansion of eq. (3.19)up to (α ) and N αS N αS O S the full fixed-order top-decay coefficient function at (α ), evaluated in Appendix S O B of ref. [14]. Before closing this section we would like to add more comments on the comparison of our resummed expression with other similar results obtained in heavy quark decay processes [15–18]. Besides the obvious replacement ofa bottomquark with a topinthe initial state, our work presents other essential differences. We have resummed large collinear logarithms α ln(m2/m2), while Refs. [15–18] just address the decay of heavy quarks S t b into massless quarks. Moreover, thiswork stilldiffers ina criticalissue. Those papers are concerned with observing the lepton produced by the W decay or the photon in the b X γ process, while we wish instead to observe the outgoing b quark. This s → is immediately clear from the choice of the x variable whose x 1 endpoint leads → to the Sudakov logarithms. In our case it is the normalized energy fraction of the outgoing bottom quark; in [15–18] it is instead related to the energy of either the lepton or the radiated photon. The most evident effect of this different perspective is that an additional scale, namely the invariant mass of therecoiling hadronic jet, enters theresults [15–18], but itisinsteadabsent inourcase. Anadditionalfunction(calledγ(α )in[15,16],C(α ) S S in [17], B(α ) in [18]) appears in those papers. The argument of α in this function S S is related to the invariant mass of the unobserved final state jet constituted by the outgoing quark and the gluon(s). It is worth noting that an identical function, called B[α (Q2(1 z))], also appears in the e+e− [6] and DIS [19] massless coefficient S − functions, where it is again associated with the invariant mass of the unobserved jet. We do not have instead any B(α ) contribution in our result. In fact, this function S contains collinear radiation associated with an undetected quark, which we do not have since the b quark is observed. We also observe that to order α the coefficient S(1) coincides with the corre- s sponding H(1) of the function H [α (m2(1 z)2)], which resums soft terms in the S b − 3Alternativewaysofmatching,identicaluptoorderα anddifferinginhigher-order,subleading S terms, are of course possible. 8 initial condition of the perturbative fragmentation function [6]. It will be very interesting to compare the functions S(α ) and H(α ) at higher orders as well. S S One final comment we wish to make is that, as expected, in our final result, eq.(2.5), which accounts for NLL soft resummation in both the coefficient function andtheinitialconditionoftheperturbativefragmentationfunction,αnlnn+1N terms S do not appear, since they are due to soft and collinear radiation. Both the quarks being heavy, only the former leads to a logarithmic enhancement. Double logarithms are generated by a mismatch in the lower and upper q2 integration limit over the A[α (q2)] function in the exponent of the resummation expression. In our case both S of them have the same functional dependence with respect to z, i.e. (1 z)2 (see − eq. (3.8) and eq. (69) of ref. [6]). The cancellation of the αnlnn+1N term can be S explicitly seen atorderα by comparing thelarge-N limit forthecoefficient function, S eq. (3.1), and the initial condition (eq. (45) of ref. [6]): the ln2N terms have identical coefficients and opposite signs. 4. Energy spectrum of the b quark Inthissectionwepresent resultsfortheb-quarkenergydistributionintopdecay. The b-quark spectrum in N-space Γ (m ,m ,µ ) is given by eq. (2.5). In the following N t W F we shall normalize Γ to the full NLO width Γ [23], so that Γ = 1 will always hold. N 1 Results in x -space will be obtained by inverting numerically eq. (2.5) via contour b integration in the complex plane, using the minimal prescription [24] to avoid the Landau pole. In order to estimate the effect of the NLL soft-gluon resummation, we compare our result with ref. [14] and use the same values for the parameters: m = 175 GeV, t m = 5 GeV, m = 80 GeV and Λ(5) = 200 MeV. b W In figure 1 we present the x distribution according to the approach of perturba- b tive fragmentation, with and without NLL soft-gluon resummation. For the scales appearing in Eqs. (2.6), (2.7) and (3.1) we have set µ = µ = m andµ = µ = m . F t 0 0F b We note that the two distributions agree for x <0.8, while for larger x values the b ∼ b resummation of large terms x 1 smoothens out the distribution, which exhibits b → the Sudakov peak. Both distributions become negative for x 0 and x 1. As b b → → discussed in [14], the negative behaviour at small x can be related to the presence b of unresummed α lnx terms in the coefficient function. At large x , we approach S b b instead the non-perturbative region, and resumming leading and next-to-leading logarithms is still not sufficient to correctly describe the spectrum for x close to 1. In b fact, the range of reliability of the perturbative calculation has been estimated to be x <1 Λ/m 0.95 [6]. b ∼ b − ≃ It is interesting to investigate the dependence of phenomenological distributions ontherenormalizationandfactorizationscales which enter thecoefficient function (µ and µ ) and the initial condition of the perturbative fragmentation function (µ and F 0 9