Factorization analysis for the fragmentation functions of hadrons containing a heavy quark
aa r X i v : . [ h e p - ph ] J un MZ-TH/07-08June 14, 2007
Factorization analysis for the fragmentation functions ofhadrons containing a heavy quark
Matthias Neubert Institut f¨ur Physik (THEP), Johannes Gutenberg-Universit¨atD–55099 Mainz, Germany
Abstract
Using methods of effective field theory, a systematic analysis of the fragmentation func-tions D a/H ( x, m Q , µ ) of a hadron H containing a heavy quark Q is performed (with a = Q, ¯ Q, q, ¯ q, g ). By integrating out pair production of virtual and real heavy quarks,the fragmentation functions are matched onto a single nonperturbative function describ-ing the fragmentation of the heavy quark Q into the hadron H in “partially quenched”QCD. All calculable, short-distance dependence on x is extracted in this step. For x →
1, the remaining fragmentation function can be matched further onto a universalfunction defined in heavy-quark effective theory in order to factor off its residual de-pendence on the heavy-quark mass. By solving the evolution equation in the effectivetheory analytically, large logarithms of the ratio µ/m Q are resummed to all orders inperturbation theory. Connections with existing approaches to heavy-quark fragmenta-tion are discussed. In particular, it is shown that previous attempts to extract ln n (1 − x )terms from the fragmentation function D Q/H ( x, m Q , µ ) are incompatible with a properseparation of short- and long-distance effects. On leave from Laboratory for Elementary-Particle Physics, Cornell University, Ithaca, NY 14853, U.S.A.
Introduction
The production of heavy quarks (bottom or charm) in high-energy collisions and their sub-sequent fragmentation into heavy hadrons is an important process in particle physics. Forinstance, a light Standard Model Higgs boson would preferably decay into a pair of b ¯ b quarks,which then fragment into B mesons or heavier beauty states. The direct production of heavyquarks would be an important background process, so a precise calculation of the heavy-quarkproduction and fragmentation processes becomes mandatory. Since the heavy-quark massprovides a natural infrared cutoff, the calculation of the cross section can, to a large extent,be performed using perturbative QCD.Since the center-of-mass energy √ s of modern particle colliders is much larger than thebottom- or charm-quark masses, the total inclusive heavy-quark production cross section iswell approximated by neglecting the masses m Q of the heavy quarks. However, less inclusiveprocesses such as the differential energy distribution of heavy hadrons produced in e + e − or pp collisions are sensitive to logarithms of the ratio s/m Q . The presence of these logarithmsthreatens the convergence of perturbative expansions. Fortunately, it is possible to showthat up to power corrections in m Q /s the cross section factorizes into mass-independent hardpartonic cross sections convoluted with fragmentation functions, which describe the probabilitythat a parton fragments into a particular hadron containing the heavy quark.Consider, for concreteness, the production of a heavy meson H containing a heavy quark Q via the decay of a vector boson V = γ, Z produced in e + e − annihilation, e + e − → V ( q ) → H ( p H ) + X , (1)where s = q . Working in the center-of-mass frame, the energy of the heavy hadron can bemeasured using the scaling variable x = 2 p H · qq = 2 E H √ s . (2)We will be interested in a situation where s ≫ m Q . At leading power in m Q /s , but to allorders in perturbation theory, the differential production cross section can be factorized as [1](see also [2]) dσ H dx = X a d ˆ σ a dx ( x, √ s, µ ) ⊗ D a/H ( x, m Q , µ ) , (3)where D a/H ( x, m Q , µ ) gives the probability for a parton a to fragment into a heavy meson H carrying a fraction x of the parton’s momentum, and d ˆ σ a /dx is the cross section for producinga massless parton a with energy fraction x defined in analogy with (2). The symbol ⊗ denotesthe convolution A ( x ) ⊗ B ( x ) = Z dz Z dz A ( z ) B ( z ) δ ( z − z z ) = Z x dzz A ( z ) B (cid:16) xz (cid:17) . (4)The factorization formula (3) separates the dependence on the heavy-quark mass m Q fromthe dependence on the center-of-mass energy, which is contained in the partonic cross sec-tions. These hard cross sections can be calculated in the massless approximation. At lead-ing power all mass dependence resides in the fragmentation functions, which are universal1i.e., process-independent) nonperturbative quantities. Like the parton distribution functions,fragmentation functions must be extracted from one set of processes and can then be used tocalculate the rates for other processes. Large logarithms of the ratio s/m Q can be resummedto all orders in perturbation theory by evolving the fragmentation functions from a low scale µ ∼ m Q up to a high scale µ ∼ √ s by solving the DGLAP evolution equations dd ln µ D a/H ( x, m Q , µ ) = X b P a → b ( x, µ ) ⊗ D b/H ( x, m Q , µ ) , (5)where P a → b are the time-like Altarelli-Parisi splitting functions [3, 4, 5, 6]. Alternatively, thepartonic cross section can be evolved from the high scale down to a scale of order m Q . Atleading order in perturbation theory (but not beyond), the functions P a → b are related to thespace-like splitting functions P b ← a governing the evolution of the parton distribution functionsby the Gribov-Lipatov reciprocity relation [7].In contrast to the case of parton distribution functions, the moments of the fragmentationfunctions cannot be related to local operator matrix elements. Even the normalization ofthe leading fragmentation function D Q/H is unknown, since a heavy quark Q can fragmentinto different heavy hadron states H . In the literature on heavy-quark fragmentation onefrequently encounters the notion of “perturbative fragmentation functions” D a/Q [1], whichdescribe the probabilities for partons a to fragment into an on-shell heavy quark Q . Thesefunctions are finite in perturbative QCD, since the heavy-quark mass, or more precisely thescale m Q (1 − x ), provides an infrared cutoff. They are relevant for the discussion of inclusiveheavy-quark production, where one sums over all possible hadron states H containing theheavy quark Q . Indeed, quark-hadron duality suggests that X H D a/H ( x, m Q , µ ) = D a/Q ( x, m Q , µ ) + O (cid:18) Λ QCD m Q (1 − x ) (cid:19) , (6)where the sum is over all hadron states containing the heavy quark Q . However, such a relationcan be expected to hold only if x is not too close to 1, such that the scale m Q (1 − x ) ≫ Λ QCD isin the short-distance regime. The one-loop results for the perturbative fragmentation functionswere obtained a long time ago [1]. They read D Q/Q ( x, m Q , µ ) = δ (1 − x ) + C F α s π " x − x ln µ m Q (1 − x ) − ! + + O ( α s ) ,D g/Q ( x, m Q , µ ) = T F α s π h x + (1 − x ) i ln µ m Q + O ( α s ) , (7)while D a/Q ( x, m Q , µ ) = O ( α s ) for a = Q, g . Throughout our paper (as is the case in most ofthe literature on heavy-quark fragmentation) m Q denotes the pole mass of the heavy quark.If desired, this parameter can be expressed in terms of a short-distance mass parameter usingperturbation theory. The two-loop expressions for D a/Q ( x, m Q , µ ) have been calculated in[8, 9].The strong dynamics binding the heavy quark inside a heavy meson implies that thereexist important nonperturbative effects leading to a prominent peak of the fragmentation2unction D Q/H near the endpoint, 1 − x peak = O (Λ QCD /m Q ). For example, the popular modelof Peterson et al. [10] D Q/H ( x ) = N x (1 − x ) [(1 − x ) + ǫ Q x ] (8)exhibits a peak at x peak ≈ − √ ǫ Q , where ǫ Q ∼ (Λ QCD /m Q ) . It is therefore interestingto study the heavy-quark fragmentation process (1) in the kinematic region x →
1, where(1 − x ) is treated as a small parameter. In this region the partonic cross sections d ˆ σ a /dx contain large logarithms of (1 − x ), which need to be resummed to all orders in perturbationtheory. The traditional approach to resumming these logarithms proceeds via Mellin momentspace [11, 12]. It has first been applied to the case of the fragmentation process in [13]. More transparently, it is possible to formulate a factorization theorem for the partonic crosssections using methods of effective field theory and to perform the resummation of largelogarithms directly in x space, applying a novel approach developed recently in [14, 15]. Thiswill be discussed in a forthcoming paper [16]. In the x → m Q , m Q (1 − x ),and Λ QCD . The dominant contributions to the cross section come from the region where m Q (1 − x ) ∼ Λ QCD , so that it is natural to assign the power counting (1 − x ) ∼ Λ QCD /m Q .The goal of the present work is to systematically factorize short- and long-distance effects forthe fragmentation functions D a/H ( x, m Q , µ ). The leading singular terms for x → D Q/H ( x, m Q , µ ) describing the conversion of a heavy quark Q intoa heavy hadron H containing that quark.The problem of how to systematically incorporate and describe nonperturbative bound-state effects into the heavy-quark fragmentation function has been addressed previously byseveral authors. The most popular approach is to factorize the fragmentation function intoperturbative and nonperturbative components: D Q/H = D pert Q/H ⊗ D npQ/H [17, 18, 19] (for arecent analysis using this scheme, see e.g. [20]). The first component is identified with theperturbative fragmentation function, D pert Q/H ( x, m Q , µ ) = D Q/Q ( x, m Q , µ ), while for the nonper-turbative component a model such as (8) is adopted. An important observation we make inthe present paper is that such an ansatz is incompatible with a proper factorization of short-and long-distance contributions. For x → m Q (1 − x ) ∼ Λ QCD , which isevident from the form of the logarithmic term in the first relation in (7). Such logarithms arenot controllable in perturbation theory and hence must be factorized into the nonperturbativefunction D npQ/H . Previous attempts to resum the ln n (1 − x ) terms in the fragmentation func-tions have thus led to incorrect formulae for the perturbative Sudakov factors in the x → B -meson decays into light hadrons in the “shape function region”.There as well the systematic factorization of short- and long-distance contributions using ef-fective field-theory methods [21, 22] has helped to identify some long-distance contributions,which had previously been included in the perturbative decay rate [23]. The proper subtraction See also [1] for a discussion of the complications arising in the x →
3f these terms had a significant impact on phenomenology [24].Alternative schemes attempt to model the hadronization process by introducing nonper-turbative effects into the (resummed) perturbative fragmentation functions themselves, forinstance by using a renormalon analysis [25, 26] or by introducing an effective, infrared-safelow-energy QCD coupling constant [27, 28]. A different approach was introduced in [29],where the long-distance effects in the fragmentation process were described using heavy-quarkeffective theory (HQET). A subsequent work followed up on this idea and aimed at model-ing the bound-state effects in the heavy hadron in terms of hadronic form factors [30]. Thecompatibility of the HQET-inspired approaches and those based on perturbative QCD wasstudied in [31]. Only much later the first systematic matching between the two approacheswas performed, and the evolution equation for the HQET fragmentation function was derived[32]. In the same paper, a simple relation with the evolution equation for the B -meson shapefunction [33, 34] was discovered. In fact, there is a close analogy between the factorizationanalyses for the heavy-quark fragmentation process and that of inclusive B -meson decays intolight particles, such as ¯ B → X s γ and ¯ B → X u l ¯ ν [21, 22, 24]. Using the methods developed inthese papers, it would be possible to describe power corrections to the fragmentation functions D a/H in the x → B -meson shape functions studied in [35, 36, 37]. Such an analysisis, however, beyond the scope of the present work, and we argue that it would be of littlephenomenological relevance.The remainder of this paper is organized as follows: In the following section we perform afirst factorization step at the hard scale m Q by integrating out real and virtual heavy-quarkpairs. This results in a “partially quenched” version of QCD, in which the heavy quark thatfragments into the heavy hadron is still present, but no pair production of heavy quarks isallowed. In this first matching step the different fragmentation functions D a/H are matchedonto a single, nonperturbative function describing the fragmentation of the valence heavyquark Q into the heavy hadron H . This matching step can be performed irrespective of thevalue of x . From a practical point of view, it is the most important result of our analysis,since at this stage all calculable dependence on x has been extracted from the fragmentationfunctions. In Section 3 we then consider the limit x → m Q , in which “partially quenched” QCD is matched onto HQET. Thisis appropriate whenever m Q (1 − x ) ≪ m Q , and it is a natural step if m Q (1 − x ) ∼ Λ QCD , as isthe case for the region near the peak of the heavy-quark fragmentation function. We rederivethe relation between the perturbative fragmentation function and the perturbative B -mesonshape function noted in [32] and use it to extract the two-loop matching coefficient relatingthe fragmentation functions defined in HQET and “partially quenched” QCD. We also presentthe exact analytic solution of the evolution equation obeyed by the fragmentation function,which is analogous to the solution of the evolution equation for the shape function obtained in[38]. In Section 4 we illustrate the phenomenological implications of our results with the helpof a simple model for the fragmentation function in HQET. We study the scale dependenceof the fragmentation function D Q/H and derive a heavy-quark symmetry relation between thefragmentation functions of B and D mesons. Section 5 contains a summary and conclusions.Some technical details of our analysis are relegated to three appendices.4igure 1: Examples of fragmentation processes converting a parton a into a heavy hadron H .Heavy quarks are denoted by thick lines, while the heavy hadron is drawn as a double line.Only the first process is allowed in “partially quenched” QCD. The fragmentation functions D a/H ( x, m Q , µ ) receive contributions associated with differentmass scales, such as m Q , m Q (1 − x ), and Λ QCD . Our goal is to factorize short-distance effectsrelated to the hard scale m Q from contributions related to lower scales. It turns out to beuseful to integrate out the heavy-quark scale m Q in two steps. In the first step, QCD with n f = n l + 1 active flavors, in which the factorization formula (3) has been derived, is matchedonto a theory with n l active flavors, which no longer contains Q ¯ Q pair production or heavy-quark loops. As always, light flavors are treated as massless. In the MS scheme, the matchingrelation between the running coupling constants α s ( µ ) in the two theories reads α ( n l +1) s ( µ ) = α ( n l ) s ( µ ) " T F α ( n l ) s ( µ )2 π ln µ m Q + . . . . (9)We would like to write an analogous matching relation for the fragmentation functions D a/H .In the theory where the heavy quark Q is among the active flavors, the fragmentation processcan be initiated by any parton a = Q, ¯ Q, q, ¯ q, g , as illustrated in Figure 1. However, in all casesbut a = Q this will involve the production of a Q ¯ Q pair, which is forbidden in the n l -flavortheory. The most general matching relation for the fragmentation functions defined in the twotheories therefore reads D ( n l +1) a/H ( x, m Q , µ ) = C a/Q ( x, m Q , µ ) ⊗ D ( n l ) Q/H ( x, m Q , µ ) . (10)The meaning of this relation is obvious: the functions C a/Q describe the fragmentation of amassless parton a into a heavy quark Q , while D ( n l ) Q/H describes the fragmentation of this heavyquark into the heavy hadron H . Only the latter process is sensitive to long-distance dynamics.The functions C a/Q can be extracted from the perturbative expressions for the fragmenta-tion functions derived in [8, 9]. We obtain C Q/Q ( x, m Q , µ ) = δ (1 − x ) + c Q ( x ) α ( n l ) s ( µ )2 π ! + O ( α s ) ,C g/Q ( x, m Q , µ ) = α ( n l ) s ( µ )2 π T F h x + (1 − x ) i ln µ m Q + c g ( x ) α ( n l ) s ( µ )2 π ! + O ( α s ) ,C a/Q ( x, m Q , µ ) = c a ( x ) α ( n l ) s ( µ )2 π ! + O ( α s ) ; a = ¯ Q, q, ¯ q. (11)5o the extent they are known, the expressions for the two-loop coefficients c a ( x ) are collected inAppendix A. These coefficients receive contributions from two sources: terms in the fragmen-tation functions of the ( n l + 1)-flavor theory that involve a heavy-quark loop or heavy-quarkpair production, and terms resulting from the replacement of the effective coupling constantin the one-loop fragmentation functions according to (9). The leading terms in the x → c Q and are given by c Q ( x ) = C F T F ( δ (1 − x ) "
12 ln µ m Q −
16 + 2 π ! ln µ m Q + 3139648 − π ζ + 1(1 − x ) +
23 ln µ m Q −
209 ln µ m Q + 5627 ! ) + subleading terms. (12)The remaining two-loop matching corrections in (11) are regular for x →
1. It follows from(5) that the matching coefficients C a/Q and the remaining fragmentation function D ( n l ) Q/H obeythe evolution equations dC a/Q d ln µ = X b P ( n l +1) a → b ⊗ C b/Q − C a/Q ⊗ P ( n l ) Q → Q , dD ( n l ) Q/H d ln µ = P ( n l ) Q → Q ⊗ D ( n l ) Q/H . (13)The first relation can be checked using the explicit results for the Wilson coefficients C a/Q given above along with the known expressions for the Altarelli-Parisi splitting functions.The short-distance coefficients C a/Q can be absorbed into the definition of a partonicheavy-quark production cross section defined in the n l -flavor theory: d ˆ σ ( n l ) Q dx ( x, √ s, m Q , µ ) ≡ X a d ˆ σ ( n l +1) a dx ( x, √ s, µ ) ⊗ C a/Q ( x, m Q , µ ) . (14)The differential heavy-hadron production cross section can then be written in the simple form dσ H dx = d ˆ σ ( n l ) Q dx ( x, √ s, m Q , µ ) ⊗ D ( n l ) Q/H ( x, m Q , µ ) . (15)The advantage is that now only a single fragmentation function remains, which contains allnonperturbative information. The cross section d ˆ σ ( n l ) Q /dx contains large logarithms, which canbe resummed by evolving the partonic cross sections d ˆ σ ( n l +1) a /dx from a high scale down to ascale µ ∼ m Q using the DGLAP evolution equations.From a practical point of view, eqs. (10), (14), and (15) are the most important re-sults of our work. While in the following section we will factorize the fragmentation func-tion D ( n l ) Q/H ( x, m Q , µ ) further by factorizing off its remaining dependence on the heavy-quarkmass, all calculable short-distance dependence on x has already been extracted in the stageabove. For phenomenological applications it would be appropriate to treat the function D ( n l ) Q/H ( x, m Q , µ ) as a hadronic quantity that is extracted from a fit to experimental data. Thediscussion of the next section becomes relevant only when one tries to relate fragmentationfunctions describing different processes (see Section 4).6t this point a comment is in order about a technical detail in the matching calculationwhich, at first sight, appears problematic. When integrating out heavy-quark pairs we com-pute short-distance effects associated with the hard scale m Q . However, the resulting Wilsoncoefficient c Q in (12) contains a term proportional to 1 / (1 − x ) + , which for x → − x ) ∼ Λ QCD /m Q appears to generate an infrared-sensitive logarithm, Z x dx D ( n l +1) Q/H ( x, m Q , µ ) ∋ C F T F α ( n l ) s ( µ )2 π !
23 ln µ m Q −
209 ln µ m Q + 5627 ! × D ( n l ) Q/H ( x , m Q , µ ) ln(1 − x ) , (16)which is not accounted for by the fragmentation function in the low-energy effective theory.But this would be in conflict with the general fact that a low-energy effective theory mustbe identical to the full theory in the infrared domain. The resolution of the puzzle is thatthe ln(1 − x ) term in the relation above should not be interpreted as an infrared-sensitivelogarithm arising from the ratio of the two scales m Q (1 − x ) ∼ Λ QCD and m Q , but as anultraviolet logarithm arising from the presence of the two large scales s and s (1 − x ) in thefragmentation process (1), where s (1 − x ) is the invariant mass squared of the final state X ,which we assume to be much larger than m Q . In fact, these are the only large logarithms thatcan be extracted from the fragmentation functions in a meaningful way. In a forthcoming paperwe will show that the µ -dependent terms in (12) combine with other terms in the resummedexpression for the partonic cross section d ˆ σ Q /dx in such a way that the running coupling α s ( µ ) is converted from the ( n l + 1)-flavor to the n l -flavor theory [16]. The constant piecesgive rise to matching corrections, which are analogous to the matching correction creating adiscontinuity in the running coupling constant at three-loop order.From now on, we focus on the fragmentation function D ( n l ) Q/H defined in the theory with n l active flavors. In the discussion below, quantities without superscript will always be definedin the theory with n l massless flavors, i.e., we use α s ≡ α ( n l ) s and D Q/H ≡ D ( n l ) Q/H . x → The step of integrating out virtual and real heavy-quark pairs discussed in the previous sectionleads to the factorization formula (10), in which the different fragmentation functions of fullQCD are related to a single nonperturbative function D Q/H defined in the “partially quenched” n l -flavor theory. This first matching step can be performed irrespective of the value of x . Inparticular, relation (15) for the fragmentation cross section is exact.We now return to the specific case of the endpoint region, where (1 − x ) ∼ Λ QCD /m Q istreated as a small parameter. We wish to factorize effects associated with the short-distancescale m Q , which are still present in the n l -flavor theory, from long-distance effects associatedwith the scales m Q (1 − x ) and Λ QCD . HQET is the appropriate effective theory for describingthese nonperturbative effects [39]. Matching “partially quenched” n l -flavor QCD onto HQET7e obtain a matching relation of the form D Q/H ( x, m Q , µ ) → C D ( m Q , µ ) ( S Q/H (ˆ ω, µ ) + 1 m Q X i C (1) i ( m Q , µ ) S (1) i (ˆ ω, µ ) + . . . ) , (17)where ˆ ω = O (Λ QCD ) is a momentum variable in the effective theory yet to be defined. TheWilson coefficients C D and C ( n ) i depend on the heavy-quark mass logarithmically through therunning coupling α s ( m Q ). The functions S Q/H and S ( n ) i are effective, leading and subleadingfragmentation functions defined in HQET (see below). The contributions of the subleadingfunctions are suppressed with respect to the leading term by powers of Λ QCD /m Q or (1 − x ).For most of this section we will be concerned with the leading term.We employ the conventional definition of the (bare) fragmentation function (defined in d = 4 − ǫ space-time dimensions) of a heavy quark Q transforming into a heavy hadron H with light-cone momentum fraction x , given by [5] D Q/H ( x, m Q , µ ) = x d − π Z dt e ip · nt X χ / n αβ h | ( ψ Q U ∗ n ) iβ ( tn ) | H ( p H ) χ ih H ( p H ) χ | ( U Tn ¯ ψ Q ) iα (0) | i . (18)Here n µ is a light-like vector, α, β are Dirac indices, and i is a color index. Longitudinal boostinvariance implies that the result should be unchanged under rescalings of the light-like vector n . It follows that the right-hand side of (18) can only depend on the ratio x = n · p H /n · p ∈ [0 , U n ( y ) = P exp (cid:18) ig Z −∞ ds n · A ( y + sn ) (cid:19) , U † n ( y ) = P exp (cid:18) − ig Z −∞ ds n · A ( y + sn ) (cid:19) , (19)where the symbol P means path-ordering, such that gluon fields are ordered from left to rightin the order of decreasing s values, and P means the opposite ordering. In (18) we need therelated objects U ∗ n ( y ) and U Tn ( y ), which are obtained from U † n ( y ) and U n ( y ) by transposition.This reverses the ordering and replaces the gluon fields by the transposed fields A T = A a t Ta ,where t a are the hermitian generators of color SU ( N c ). The transposed Wilson lines appearbecause in (18) the Wilson lines are located on the “wrong” sides of the quark fields.The fragmentation function receives contributions from both hard and soft interactions.In order to separate short- and long-distance effects, we match D Q/H onto a correspondingfunction defined in HQET. Following [29], we define S Q/H (ˆ ω, µ ) = n · v π Z dt e ik · nt X χ s h | ( h v S ∗ n ) iα ( tn ) | H ∞ ( v ) χ s ih H ∞ ( v ) χ s | ( S Tn ¯ h v ) iα (0) | i , (20)where v is the four-velocity of the heavy hadron, h v ( y ) is the two-component heavy-quarkfield in HQET [39], and we have used that between these fields / n can be replaced with n · v . The hadron states H ∞ in HQET are taken to have a mass-independent normaliza-tion h H ∞ | ¯ h v h v | H ∞ i = 1 instead of the conventional normalization to 2 M H employed in (18).In the effective theory we sum over soft states χ s (i.e., states having momenta much less than m Q ), and the QCD Wilson lines U n , U † n are replaced by soft Wilson lines S n , S † n defined inanalogy with (19), but with QCD gluon fields replaced with soft gluon fields. Note that k p = m Q v + k . Boostinvariance implies that the right-hand side must be a function of ω = n · k/n · v ∈ [ ¯Λ , ∞ [,where ¯Λ = M H − m Q is the residual mass of the heavy hadron state in HQET, and the condi-tion ω ≥ ¯Λ follows since the heavy quark must have sufficient energy to produce the heavierhadron state H . The above definitions imply the relation ω = M H /x − m Q , and indeed oneof the main points of [29] was to argue that the fragmentation function should be consideredas a function of this particular combination of x , m Q and M H (see also [31]). We find it moreconvenient to write the fragmentation function instead as a function of the variable ˆ ω = ω − ¯Λ,which takes values between 0 and ∞ and obeys the simpler relationsˆ ω = M H − xx , x = M H M H + ˆ ω . (21)After the decoupling of heavy-quark pairs discussed in the previous section, the matchingrelation between the fragmentation functions in QCD and HQET is local in x and ˆ ω , as shownin (17). Using the usual definition of the HQET spinor h v [39] one readily obtains the tree-level relation D Q/H ( x, m Q , µ ) = M H x S Q/H (ˆ ω, µ ) + . . . , where the factor M H results from thedifferent normalization of the hadron states in (18) and (20), and the factor x is due to the x d − prefactor in (18). Beyond tree level, we write the matching relation in the differentialform D Q/H ( x, m Q , µ ) dxx = g ( x ) C D ( m Q , µ ) S Q/H (ˆ ω, µ ) d ˆ ω + power corrections. (22)Any deviation of the function g ( x ) from 1 is formally a power correction, so instead of the form g ( x ) = x suggested by the tree-level matching relation given above it would be consistent toset g ( x ) = 1 or use any other smooth function obeying the constraint g (1) = 1. We will see inSection 3.4 that a particularly convenient choice is g ( x ) = (1 + x ) / ω →
0, which implies that the QCD fragmentation functionvanishes for x →
1. Note, however, that in general there is no reason why the fragmentationfunction should vanish at x = 0, even though this is built into most phenomenological param-eterizations. Indeed, the perturbative results in (7) and the corresponding two-loop resultsin [8, 9] suggest that the fragmentation functions tend to constants modulo logarithms forsmall x . The hard matching coefficient C D can be calculated in perturbation theory by computing thefragmentation functions in the two theories and requiring that the two sides in (22) be identicalup to power-suppressed terms. Since the effective theory (HQET) exactly matches the fulltheory (“partially quenched” QCD with n l light flavors) in the infrared, the matching can beperformed using on-shell quark and gluon states. Examples of one-loop diagrams contributingto the perturbative fragmentation functions in the two theories are shown in Figure 2.Replacing the heavy hadron by an on-shell heavy quark, we find that at one-loop order in n l -flavor QCD the renormalized fragmentation function D Q/Q in the MS scheme is given by thefirst formula in (7). At the parton level we do not distinguish between the heavy-quark mass9igure 2: Examples of one-loop diagrams contributing to the perturbative fragmentationfunctions. The grey dots represent the Wilson lines in expressions (18) and (20).and the mass of the hadron containing the heavy quark (i.e., we set M H = m Q ). Performingthe corresponding on-shell calculation in HQET, we obtainˆ S Q/Q (ˆ ω, µ ) = δ (ˆ ω ) − C F α s π π ! − C F α s π " ω ωµ ! [ µ ] ∗ , (23)where the star distributions are defined as [22, 23] Z Ω0 d ˆ ω f (ˆ ω ) ln n (ˆ ω/µ )ˆ ω ! [ µ ] ∗ = Z Ω0 d ˆ ω f (ˆ ω ) − f (0)ˆ ω ln n ˆ ωµ + f (0) n + 1 ln n +1 Ω µ . (24)This generalization of the usual plus distributions is required, because in the effective theorythere is no reasonable way of defining a dimensionless variable x ∈ [0 , C D , we integrate these perturbative fragmen-tation functions with an arbitrary test function over an interval x ∈ [ m Q / ( m Q + ω ) ,
1] andˆ ω ∈ [0 , ω ], respectively, where ω ≪ m Q . We then require that the results agree to leadingpower in ω /m Q . This leads to C D ( m Q , µ ) = 1 + C F α s π
12 ln µ m Q + 12 ln µ m Q + 2 + π ! + O ( α s ) . (25)This result agrees with a corresponding expression derived in [32]. We stress again that thematching coefficient is free of ln(1 − x ) terms. All those terms in the perturbative fragmentationfunction in (7) are of long-distance origin and are still contained in the HQET fragmentationfunction S Q/H (ˆ ω, µ ). It is an intriguing observation that the expression (23) for the perturbative HQET fragmen-tation function can be related to the corresponding perturbative expression for the B -mesonshape function. The shape function is the HQET analog of the heavy-quark parton distribu-tion function of the hadron H [33, 34, 40]. In analogy with (20), it is defined as S H (ˆ ω, µ ) = n · v π Z dt e ik · nt h H ∞ ( v ) | (¯ h v S n )(0)( S † n h v )( tn ) | H ∞ ( v ) i . (26)10n this case ω = n · k/n · v is restricted to take values between −∞ and ¯Λ, since the residuallight-cone energy of the heavy quark can at most equal the residual mass ¯Λ of the heavyhadron. We thus define ˆ ω = ¯Λ − ω ∈ [0 , ∞ [. When this is done, our one-loop expressionin (23) coincides with the one-loop expression for the perturbative shape function given ineq. (33) of [22].The agreement between the perturbative fragmentation and shape functions in HQET,when expressed in terms of the proper variables, is not restricted to one-loop order. Rather,it is an all-order identity between the two functions, which follows from their relation tovacuum expectation values of Wilson loops. This has also been observed in [32]. In orderto derive this relation we make use of field redefinitions, which decouple the interactions ofsoft gluons from the effective heavy-quark fields. For the case of the shape function, we use h v ( y ) = S v ( y ) h (0) v ( y ) (see, e.g., [41]), where the soft Wilson line S v is defined in analogy with S n , but with the light-like vector n replaced with the time-like vector v . For the case of thefragmentation function we need the Wilson lines on the “wrong” side of the heavy-quark field,so we use instead h v ( y ) = h (0) v ( y ) S T − v ( y ). The new fields h (0) v are “sterile” and do not coupleto any other fields in the theory. The corresponding field operators simply create or annihilatethe heavy quarks in the external hadron states, giving rise to free two-component Dirac spinors u v satisfying / vu v = u v and ¯ u v u v = 1. We define h (0) iv | H ∞ ( v ) i = u v √ N c | ¯ q iH ( ¯Λ v ) i , (27)where the state | ¯ q iH ( ¯Λ v ) i represents the light degrees of freedom inside the hadron H with colorindex i . This state carries momentum ¯Λ v , which is what remains from the hadron momentum M H v when the heavy quark is removed. We may therefore write S H (ˆ ω, µ ) = 12 πN c Z dt e i (¯Λ − ˆ ω ) v · nt h ¯ q iH ( ¯Λ v ) | h ( S † v S n )(0)( S † n S v )( tn ) i ii | ¯ q iH ( ¯Λ v ) i . (28)After the decoupling transformation, the corresponding expression for the fragmentation func-tion reads S Q/H (ˆ ω, µ ) = 12 πN c Z dt e i (¯Λ+ˆ ω ) v · nt X χ s h | ( S T − v S ∗ n ) ij ( tn ) | ¯ q iH ( ¯Λ v ) χ s ih ¯ q iH ( ¯Λ v ) χ s | ( S Tn S ∗− v ) ji (0) | i . (29)In a partonic picture the spectator quark does not participate in the short-distance process,and to obtain the perturbative shape and fragmentation functions it can simply be droppedfrom the above matrix elements. At the same time the parameter ¯Λ is set to zero, therebyremoving the contribution of the spectator quark to the overall momentum balance. Thisyields for the perturbative shape function S Q (ˆ ω, µ ) = 12 π Z dt e − i ˆ ωv · nt N c h | Tr ( S † v S n )(0)( S † n S v )( tn ) | i . (30) Note that in the original variable ω = n · k/n · v the range of support of the shape function is the complementof the range of support for the HQET fragmentation function.
11n the case of the perturbative fragmentation function we can now perform the sum over thesoft states χ s , since after the removal of the spectator quark this is a complete sum over statesin the effective theory. The result is S Q/Q (ˆ ω, µ ) = 12 π Z dt e i ˆ ωv · nt N c h | Tr ( S †− v S n ) ∗ ( tn )( S † n S − v ) ∗ (0) | i = 12 π Z dt e i ˆ ωv · nt N c h | Tr ( S †− v S n )( tn )( S † n S − v )(0) | i ∗ = 12 π Z dt e i ˆ ωv · nt N c h | Tr ( S †− v S n )(0)( S † n S − v )( tn ) | i = 12 π Z dt e i ˆ ωv · nt N c h | Tr ( S † v S n )(0)( S † n S v )( − tn ) | i = S Q (ˆ ω, µ ) . (31)The last step follows since the position-space Wilson loop h | Tr ( S †− v S n )(0)( S † n S − v )( tn ) | i isa function of the variable itn · v , so replacing the sign of v is equivalent to changing the signof t . It follows that the perturbative fragmentation function coincides with the perturbativeshape function.Armed with this result, the short-distance coefficient C D in (22) can be computed totwo-loop order using existing calculations. The two-loop expression for the fragmentationfunction in the “partially quenched” n l -flavor theory can be extracted from [8] by retainingthe leading terms in the x → D Q/Q ( x, m Q , µ ) and accounting for the first matching step (10) using relations (11)and (12). The integral over the two-loop perturbative fragmentation function in HQET isobtained from the corresponding integral over the two-loop perturbative shape function givenin eq. (38) of [42]. Matching the two expressions, we find C D ( m Q , µ ) = 1 + C F α s ( µ )2 π L L π ! + C F α s ( µ )2 π ! h C F H F + C A H A + T F n f H f i + O ( α s ) , (32)where L = ln( µ /m Q ), and the expansion coefficients at two-loop order are H F = L L
98 + π ! L + − π
24 + 6 ζ ! L + 24132 + (cid:18) − (cid:19) π − π − ζ ,H A = 11 L
36 + − π ! L + π − ζ ! L + 128772592 + (cid:18) (cid:19) π − π
720 + 8936 ζ ,H f = − L − L − π ! L − − π − ζ . (33)12f desired, the two matching steps discussed above and in Section 2 can be combined intoa single matching relation between the original fragmentation functions D a/H and the HQETfragmentation function S Q/H . In the resulting relation, which is of convolution type, theWilson coefficients C a/Q in (10) are simply multiplied by the coefficient C D in (32). While the perturbative fragmentation function in HQET coincides with the perturbative shapefunction to all orders in perturbation theory, such a simple relation will no longer be truenonperturbatively. This can be seen from the different ways in which the spectator quarkappears in expressions (28) and (29). It is also apparent from the fact that the residualmomentum k of the heavy quark obeys different kinematical restrictions in the two cases. Inthe case of the shape function, n · k varies in the range ] −∞ , ¯Λ] (setting n · v = 1 for simplicity),and the equation of motion iv · D h v = 0 of HQET enforces the condition h n · k i = 0 [33, 34, 40].As a result, the first moment of the shape function is given by h ˆ ω i S H = ¯Λ = M H − m Q , andindeed this relation can be used to define a short-distance, running heavy-quark mass calledthe “shape-function mass” to all orders in perturbation theory [22, 43].For the fragmentation function, on the other hand, the variable n · k is always positiveand varies in the range [ ¯Λ , ∞ [, so that the average value h n · k i > ¯Λ is a nontrivial hadronicparameter. It follows that the first moment of the fragmentation function, h ˆ ω i S Q/H ≡ ǫ H >
0, is not related to any known parameter and must be regarded as an unknown hadronicquantity. This, along with the fact that even the normalization of the fragmentation functionsis unknown, implies that there is much less theoretical handle on the fragmentation functionthan there is on the shape function.Nevertheless, the relation between the perturbative shape and fragmentation functionsmight be employed to motivate some simple models for the fragmentation function. In thecase of the shape function, a renormalon-inspired model based on an analysis of the divergencestructure of large-order perturbation theory appears to provide a rather good description ofdifferent decay spectra in the inclusive processes ¯ B → X s γ and ¯ B → X u ℓ ¯ ν [44, 45]. This modelstarts from the perturbative shape function and introduces nonperturbative effects through asingle model parameter set by the first moment. This suggests that it might be reasonableto use the shape function extracted from the ¯ B → X s γ photon spectrum as a model for theHQET fragmentation function. However, we must account for the fact that the normalizationsand the first moments of the two functions are different. As a simple model, we thus proposethe form S model Q/H (ˆ ω, µ ) = N H S model H (ˆ ω, µ ) (cid:12)(cid:12)(cid:12) ¯Λ → ǫ H . (34)A rather flexible, two-parameter model for the shape function, which is known to provide goodfits to the experimental data, consists of an exponential times a power of ˆ ω [46]. This leadsto the form (with b > F (ˆ ω ) ≡ S model H (ˆ ω, µ ) (cid:12)(cid:12)(cid:12) ¯Λ → ǫ H = b b Γ( b ) ˆ ω b − ǫ bH e − b ˆ ω/ǫ H . (35)13his function obeys the moment relations Z ∞ d ˆ ω F (ˆ ω ) = 1 , Z ∞ d ˆ ω ˆ ω F (ˆ ω ) = ǫ H , Z ∞ d ˆ ω (cid:16) ˆ ω − h ˆ ω i (cid:17) F (ˆ ω ) = ǫ H b . (36)The parameters ǫ H and b can be tuned to fit the data for the first and second moment of thefragmentation function. So far, we have discussed the leading term in the matching relation (22) for the fragmentationfunction, while in general there is an infinite series of power-suppressed contributions as shownin (17). Like the leading term, these power corrections can be related to nonperturbative,subleading fragmentation functions S ( n ) i in HQET. Once a basis of such functions has beenconstructed, their Wilson coefficients C ( n ) i can be determined perturbatively by matchingperturbative expressions for the HQET fragmentation functions with the subleading terms inthe perturbative fragmentation function D Q/Q (as well as more complicated functions such as D Q/gQ ) in n l -flavor “partially quenched” QCD, in analogy with our treatment in Section 3.1.Using the relations between plus and star distributions collected in Appendix B, we findthat at one-loop order in perturbation theory the power-suppressed terms not accounted forby the leading contribution in the matching relation (22) are given by D Q/Q ( x, m Q , µ ) − g ( x ) C D ( m Q , µ ) S Q/Q (ˆ ω, µ ) x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˆ ωdx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = C F α s π ( − x x − g ( x ) ! ln µ m Q (1 − x ) − ! − g ( x ) ln x − x ) . (37)The argument of the logarithm once again indicates that this difference is associated withphysics at the low scale m Q (1 − x ) ∼ Λ QCD . A particularly simple expression is obtained if weuse the special form g ( x ) = (1 + x ) / S Q/H (ˆ ω, µ ) is not constrained by any useful conditions. Unlike the case of the shapefunction, the moments of the fragmentation function are not related to local HQET operators.For practical purposes we may therefore define our model for the function S Q/H (ˆ ω, µ ) to includeall power-suppressed terms in (17) – in any case this model must be tuned to fit experimentaldata. This definition introduces some subleading dependence on the heavy-quark mass m Q into the fragmentation function, while the original S Q/H (ˆ ω, µ ) was strictly a universal, m Q -independent function. 14 .5 Renormalization-group evolution As discussed earlier, the fragmentation function D Q/H in the n l -flavor QCD theory obeys thesecond evolution equation in (13). The relevant splitting function takes the form [15] P Q → Q ( x, µ ) = Γ cusp ( µ )(1 − x ) + + γ φ ( µ ) δ (1 − x ) + subleading terms, (38)where the regular terms are subleading for x →
1. Here Γ cusp is the cusp anomalous dimensionof Wilson loops with light-light segments [47], which is known to three-loop order [48]. Thequantity γ φ can be identified with a combination of anomalous dimensions of operators definedin soft-collinear effective theory and is known to the same order [14, 15, 48].It follows from the discussion of Section 3.2 that the fragmentation function in HQETobeys the same integro-differential renormalization-group equation as the shape function. Itreads dd ln µ S Q/H (ˆ ω, µ ) = − Z ˆ ω d ˆ ω ′ Γ(ˆ ω, ˆ ω ′ , µ ) S Q/H (ˆ ω ′ , µ ) , (39)where to all orders in perturbation theory the kernel has the form [22, 49]Γ(ˆ ω, ˆ ω ′ , µ ) = − Γ cusp ( α s ) (cid:18) ω − ˆ ω ′ (cid:19) [ µ ] ∗ + γ S ( α s ) δ (ˆ ω − ˆ ω ′ ) . (40)The anomalous dimension γ S is known to two-loop order [32, 42]. Finally, from (38) and (40)it follows that the hard matching coefficient obeys the evolution equation dd ln µ C D ( m Q , µ ) = " Γ cusp ( α s ) ln µm Q + γ φ ( α s ) + γ S ( α s ) C D ( m Q , µ ) . (41)The renormalization-group equations (39) and (41) can be solved analytically and in closedform [38]. The solution for the Wilson coefficient reads C D ( m Q , µ ) = exp h − S ( µ h , µ ) − a γ φ ( µ h , µ ) − a γ S ( µ h , µ ) i m Q µ h ! a Γ ( µ h ,µ ) C D ( m Q , µ h ) , (42)where µ h ∼ m Q is a hard matching scale, at which the initial condition C D ( m Q , µ h ) is free oflarge logarithms. The evolution functions are given by S ( ν, µ ) = − α s ( µ ) Z α s ( ν ) dα Γ cusp ( α ) β ( α ) α Z α s ( ν ) dα ′ β ( α ′ ) , a Γ ( ν, µ ) = − α s ( µ ) Z α s ( ν ) dα Γ cusp ( α ) β ( α ) , (43)and similarly for the functions a γ φ and a γ S . Here β = dα s /d ln µ is the QCD β -function.The exponential in (42) accomplishes the resummation of large logarithms to all orders inperturbation theory. The perturbative expansions of the anomalous dimensions are collectedin Appendix C. The Sudakov exponent S should not be confused with the fragmentation function. S Q/H (ˆ ω, µ ) = exp h S ( µ , µ ) + 2 a γ S ( µ , µ ) i e − γ E η Γ( η ) Z ˆ ω d ˆ ω ′ S Q/H (ˆ ω ′ , µ ) µ η (ˆ ω − ˆ ω ′ ) − η , (44)where η = 2 a Γ ( µ, µ ). Here µ serves as a low reference scale still in the perturbative domain,at which a model for the HQET fragmentation function is provided. With the help of theabove relation, the fragmentation function can then be evolved to a higher scale µ . It should bestressed that the solutions (42) and (44) are formally independent of the two matching scales µ h and µ . In practice, a scale dependence remains when one truncates the perturbativeexpansions of the evolution functions.The renormalization-group improved expressions for the Wilson coefficient and the HQETfragmentation function can be combined and simplified to obtain the final result for the frag-mentation function D Q/H in “partially quenched” QCD. We find D Q/H ( x, m Q , µ ) = M H g ( x ) x exp h S ( µ , µ h ) + 2 a γ S ( µ , µ h ) + 2 a γ φ ( µ, µ h ) i µ m Q ! a Γ ( µ,µ h ) × C D ( m Q , µ h ) e − γ E η Γ( η ) Z ˆ ω d ˆ ω ′ S Q/H (ˆ ω ′ , µ ) µ η (ˆ ω − ˆ ω ′ ) − η ; η = 2 a Γ ( µ, µ ) , (45)where ˆ ω = M H (1 − x ) /x on the right-hand side. This relation allows us to derive the frag-mentation function D Q/H from a primordial function S Q/H defined at a low renormalizationscale.
We now illustrate our results for the case of the B -meson fragmentation function D b/B ( x, m b , µ )with the help of a phenomenological model. We use the function (35) with the parameterchoices ǫ B = 0 .
69 GeV, b = 2 . ǫ B = 0 .
82 GeV, b = 3 . S Q/H (ˆ ω, µ ) at the scale µ = 1 . D b/B ( x, m b , µ ) in the n l -flavor theory at a scale µ ≥ µ .The evolution from the low scale to a higher scale is performed at next-to-next-to-leadingorder in renormalization-group improved perturbation theory, using the results collected inAppendix C. In order to test the stability of the perturbative approximation, we vary thehard matching scale µ h between m b / √ √ m b . Our results are shown in Figure 3. Weobserve that the evolution to higher scales softens the fragmentation function and introducesa significant radiation tail. The dependence on the choice of µ h is very weak. Depending onthe set of input parameters the functional form of the fragmentation function can be varied.Parameter set 1 provides a good description of the B -meson shape function [22]. Parameterset 2 is chosen such that the resulting fragmentation function at the scale µ = m b resemblesthat obtained from a recent phenomenological fit to e + e − data [52].For practical purposes it does not matter whether a model for the fragmentation function isspecified at a low or at a high scale. As long as one is interested in, say, B -meson fragmentation16 .6 0.7 0.8 0.9 10246810 PSfrag replacements x D b / B ( x , m b , µ ) / N B Set 1
PSfrag replacements x D b / B ( x , m b , µ ) / N B Set 2
Figure 3: Scale dependence of the heavy-quark fragmentation function D b/B ( x, m b , µ ) obtainedfrom (45) using the model functions specified in the text. The overall normalization N B isleft free. In each figure, the three sets of curves refer to µ = m b = 4 . µ = 3 GeV(red), and µ = µ = 1 . µ h . The dotted line shows the phenomenological fit at µ = m b obtained in [52].in e + e − annihilation, one might as well stick with formula (15) for the cross section, evolvethe partonic cross section down to a scale µ = m b (for details see [16]), and model thefragmentation function D b/B ( x, µ = m b ) directly. At this stage, all perturbative logarithmsof (1 − x ) have already been extracted and absorbed into the partonic cross section. Thefactorization of the fragmentation function itself and its scale evolution become importantonly if one attempts to connect the fragmentation functions extracted in different processeswith each other. In particular, treating the charm quark as a heavy quark, it is possibleto relate the b -quark fragmentation function of B mesons to the charm-quark fragmentationfunction of D mesons. In comparing the two functions we work for simplicity at next-to-leading order in α s , so that we do not need to worry about the decoupling of real and virtualcharm-quark pairs.The resulting relation is simplest if we choose to evaluate both fragmentation functions ata common scale µ and ignore resummation effects. In this case, we obtain at leading power inΛ QCD /m c,b D b/B ( x, µ ) = C D ( m b , µ ) C D ( m c , µ ) M B M D D c/D ( y, µ ) ; y = M D M D + M B − xx . (46)The relation becomes more complicated when the two functions are evaluated at differentscales and resummation effects are taken into account. We then obtain D b/B ( x, µ = m b ) = U ( m c , m b ) M B M D e − γ E η Γ( η ) (cid:18) M D m c (cid:19) η Z y dy ′ y ′ D c/D ( y ′ , µ = m c ) (cid:16) − yy − − y ′ y ′ (cid:17) − η , (47)where renormalization-group effects are included in the quantities η = 2 a Γ ( m b , m c ) and U ( m c , m b ) = C D ( m b , m b ) C D ( m c , m c ) exp h S ( m c , m b ) + 2 a γ S ( m c , m b ) i , (48)17 .7 0.75 0.8 0.85 0.9 0.95 100.511.522.5 PSfrag replacements x D c / D ( x , µ = m c ) D D + Figure 4: Heavy-quark symmetry prediction for the charm-quark fragmentation function D c/D ( x, µ = m c ) (solid blue line) obtained from the inverse of relation (47) and the phe-nomenological fit result for the b -quark fragmentation function D b/B ( x, µ = m b ) determinedin [52] and overlaid as a dotted line. For comparison we show as red dashed lines the fit resultsfor the charm-quark fragmentation functions of D + and D mesons obtained in [54].which we evaluate in 4-flavor QCD. Using m b = 4 . m c = 1 . η ≈ .
322 and U ( m c , m b ) ≈ .
965 at next-to-leading order.At present the extraction of the charm-quark fragmentation functions into D mesons ap-pears to be affected by large uncertainties, at least as far as the overall normalization isconcerned. For instance, the authors of [20] and [54] find a factor of almost 3 difference in thenormalization of the fragmentation functions of D + and D meson. A reason is, perhaps, thata significant fraction of D mesons is produced indirectly via decays of D ∗ mesons, and thesemust be subtracted in order to obtain the sample of primary D mesons. From a theoreticalperspective, a large difference between the fragmentation functions of D + and D mesonswould be very difficult to explain in QCD, as the two functions must coincide up to smallelectromagnetic and isospin-breaking corrections. We will therefore use the inverse of relation(47) to derive the charm-quark fragmentation function from the fragmentation function of b quarks. The corresponding relation is obtained by replacing m c ↔ m b and M D ↔ M B everywhere. Since the exponent η = 2 a Γ ( m c , m b ) is negative in this case, it is necessary toregularize the integral using a star distribution (see eq. (47) in [15] for details). The evolutionfunctions now evaluate to η ≈ − .
322 and U ( m c , m b ) ≈ .
715 at next-to-leading order.Relations (46) and (47) are model-independent consequences of heavy-quark symmetry [39],which are analogous to the relations between semileptonic form factors of D and B mesonsobtained by Isgur and Wise [53]. Identical relations hold for other pairs of fragmentationfunctions, such as those for D ∗ and B ∗ or D s and B s mesons. In practice, we expect theserelations to receive sizable power corrections, since the charm quark is not very heavy on theQCD scale.To illustrate relation (47) and its inverse we use the result for D b/B ( x, µ = m b ) obtained in[52] from a fit to e + e − data and derive from it a prediction for the charm-quark fragmentationfunction D c/D ( x, µ = m c ). In Figure 4, we compare this prediction with the phenomenologicalfit results for the charm-quark fragmentation functions of D + and D mesons obtained in1854]. In the region of large x values our prediction comes close to the phenomenological fitfor D mesons. Note that for values of x below 0.79 the parameter ˆ ω exceeds 0.5 GeV for thecase of charm, so that we would expect significant corrections from subleading fragmentationfunctions neglected in our analysis. We have performed a systematic factorization analysis for the fragmentation functions of ahadron H containing a heavy quark Q . Short- and long-distance contributions associated withdifferent momentum scales have been separated from each other using effective field-theorymethods. In several aspects our approach goes beyond previous analyses of heavy-hadronfragmentation functions and introduces some novel features. In particular, we have shown thatthe popular decomposition of the heavy-quark fragmentation function into perturbative andnonperturbative components, in which the first component is identified with the perturbativefragmentation function D Q/Q in (7), does not provide a proper factorization of short- andlong-distance effects.We found it useful to integrate out heavy-quark pair production (virtual and real) bymatching the set of fragmentation functions D a/H ( x, m Q , µ ) defined in QCD with n l + 1 activeflavors onto a single nonperturbative fragmentation function D Q/H ( x, m Q , µ ) defined in “par-tially quenched” QCD with n l active flavors, in which the heavy quark is still present but pairproduction of heavy quarks is forbidden. Besides the reduction to a single hadronic entity,which offers practical advantages for the modeling of fragmentation functions, in this step allcalculable, short-distance dependence on x is extracted from the fragmentation function. Wehave stressed that most logarithms of (1 − x ) in the expression for the perturbative fragmen-tation function D Q/Q ( x, m Q , µ ) are of long-distance origin and thus cannot be extracted andresummed in a meaningful way. We have completed this first matching step at two-loop orderwith the exception of a single coefficient function, c Q ( x ) in (12), for which the terms that areregular for x → − x )is a small parameter, typically of order Λ QCD /m Q . In this case the residual dependenceof the fragmentation function on the heavy-quark mass m Q , which remains after the firststep, can be extracted by matching “partially quenched” QCD onto heavy-quark effectivetheory. We have derived evolution equations for the relevant quantities in the low-energyeffective theory and solved them analytically. The resulting relation (45) allows us to controlthe scale dependence of the fragmentation function in the large- x region analytically, andto derive symmetry relations between the fragmentation functions for charm and bottomhadrons. Examples of such relations, which are model-independent consequences of QCD inthe heavy-quark limit, have been discussed in Section 4.The results obtained in this paper provide the first step in a systematic analysis of frag-mentation of heavy hadrons at large x in e + e − or pp collisions. In a forthcoming paper, wewill perform the threshold resummation of large logarithms in the fragmentation cross sectionnear x → D b/H ( x, m b , µ ) and D c/H ( x, m c , µ ) from fits to experimental data. Once thesefunctions have been determined at a scale µ ∼ m Q , the entire set of fragmentation functions D a/H can be obtained using relation (10). These functions can then be evolved up to higherscales by solving the DGLAP evolution equations (5). Acknowledgments
I am grateful to Alexander Mitov and Hubert Spiesberger for useful discussions, and to IgnazioScimemi for collaboration during the early stages of this work.
Appendix A: Two-loop matching coefficients
We collect the two-loop expressions for the matching coefficients C a/Q defined in (10) andgiven in perturbative form in (11). The two-loop coefficients c a can be extracted from [8, 9].Since the analytic expressions obtained in these two papers are lengthy, we will not reproducethem here but rather refer to the equations where they can be found.We begin with the two-loop coefficient c g describing the conversion of a massless gluon intoa heavy quark Q . We obtain c g ( x ) = d (2) g ( x ) + 23 T F h x + (1 − x ) i ln µ m Q − C F T F ln µ m Q h x + (1 − x ) i ⊗ " x − x ln µ (1 − x ) m Q − ! + , (49)where d (2) g ( x ) is given in eq. (19) of [9]. The second term arises from the application of thematching relation (9) for the running coupling constant in the one-loop expression for thegluon fragmentation function, while the convolution term arises from the one-loop cross termin (10).The matching coefficients c a with a = q, ¯ q, ¯ Q for the conversion of a light quark, light anti-quark, or heavy anti-quark to a heavy quark start at two-loop order and are directly relatedto the corresponding perturbative fragmentation functions. We have c q ( x ) = c ¯ q ( x ) = d (2) q ( x ) , c ¯ Q ( x ) = d (2)¯ Q ( x ) , (50)where d (2) q ( x ) and d (2)¯ Q ( x ) are given in eqs. (54) and (56) of [8], respectively.We finally focus on the coefficient c Q , for which we obtain c Q ( x ) = C F T F F ( C F T F ) Q ( x ) + 23 ln µ m Q " x − x ln µ (1 − x ) m Q − ! + + C F (cid:18) C F − C A (cid:19) ∆ F Q ( x ) , (51)20 a) (b) (c) Figure 5: (a,b) Diagrams involving heavy-quark pair production, which yield the contribution∆ F Q ( x ) in eq. (51). (c) A diagram with identical color structure, which does not involve pairproduction and hence does not contribute to c Q .where F ( C F T F ) Q ( x ) is given in eq. (62) of [8]. The term ∆ F Q ( x ) contains those parts of thecoefficients F ( C F ) Q and F ( C A C F ) Q in eq. (59) of [8] that are related to the Q → QQ ¯ Q fragmentationchannel. They are given by the cuts of the first two diagrams shown in Figure 5. Due to thepresence of other graphs with identical color structure, these contributions cannot be extractedwithout a dedicated calculation. Appendix B: Relations between distributions
Using the usual definition of plus distributions together with the definition of the star distri-butions in (24), it is possible to derive general relations between the two types of distributions,which are valid for test functions depending on variables x or ˆ ω related by ˆ ω = m Q (1 − x ) /x .We obtain (cid:18) ω (cid:19) [ µ ] ∗ d ˆ ω = ((cid:20) − x (cid:21) + − ln µm Q δ (1 − x ) ) dxx , (52) ln(ˆ ω/µ )ˆ ω ! [ µ ] ∗ d ˆ ω = (" ln(1 − x )1 − x + − ln µm Q (cid:20) − x (cid:21) + + 12 ln µm Q δ (1 − x ) − ln x − x ) dxx . Expressions involving plus distributions of complicated functions of x can be simplifiedusing the identity " f ( x )1 − x ln n (1 − x ) + = f ( x ) (cid:20) − x ln n (1 − x ) (cid:21) + + δ (1 − x ) Z dx f (1) − f ( x )1 − x ln n (1 − x ) , (53)which is valid for integer n ≥
0. Applying this result, the one-loop expression for the per-turbative heavy-quark fragmentation function in the first equation in (7) can be rewrittenas D Q/Q ( x, m Q , µ ) = δ (1 − x ) " C F α s π
32 ln µ m Q + 2 ! + C F α s π (1 + x ) " − x ln µ m Q (1 − x ) − ! + . (54)21sing our results (23) and (25), and expressing the star distributions in terms of plus distri-butions with the help of (52), we find g ( x ) C D ( m Q , µ ) S Q/Q (ˆ ω, µ ) x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˆ ωdx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = δ (1 − x ) " C F α s π
32 ln µ m Q + 2 ! (55)+ C F α s π g ( x ) " − x ln µ m Q (1 − x ) − ! + + 4 ln x − x . Appendix C: Renormalization-group functions
The exact solutions (42) and (44) to the evolution equations can be evaluated by expandingthe anomalous dimensions and the β -function as perturbative series in the strong coupling. Wework consistently at next-to-next-to-leading order (NNLO) in renormalization-group improvedperturbation theory, keeping terms through order α s in the final expressions for the Sudakovexponent S and the functions a Γ , a γ φ , and a γ S . We define the expansion coefficients asΓ cusp ( α s ) = Γ α s π + Γ (cid:18) α s π (cid:19) + Γ (cid:18) α s π (cid:19) + . . . ,β ( α s ) = − α s " β α s π + β (cid:18) α s π (cid:19) + β (cid:18) α s π (cid:19) + . . . , (56)and similarly for the other anomalous dimensions. In terms of these quantities, the NNLOexpression for the renormalization-group function a Γ in (43) is given by a Γ ( ν, µ ) = Γ β ( ln α s ( µ ) α s ( ν ) + Γ Γ − β β ! α s ( µ ) − α s ( ν )4 π + " Γ Γ − β β − β β Γ Γ − β β ! α s ( µ ) − α s ( ν )32 π + . . . ) . (57)The NNLO expression for the Sudakov exponent S in (43) has been given in eq. (94) of [15].We do not reproduce this lengthy formula here.In the following we list the relevant expansion coefficients for the anomalous dimensions andthe QCD β -function, which are required at NNLO. All results refer to the MS renormalizationscheme. For the convenience of the reader we also quote numerical values corresponding to n f = 4 massless quark flavors, as relevant for studies of b -quark fragmentation.The expansion of the cusp anomalous dimension Γ cusp to two-loop order was obtained along time ago [47], while the three-loop coefficient has been calculated in [48]. The results areΓ = 4 C F = 163 , Γ = 4 C F " − π ! C A − T F n f ≈ . , = 4 C F " C A − π
27 + 11 π
45 + 223 ζ ! + C A T F n f − π − ζ ! + C F T F n f (cid:18) −
553 + 16 ζ (cid:19) − T F n f ≈ . . (58)One also needs the four-loop cusp anomalous dimension, for which the Pad´e approximationΓ ≈ n f = 4) was obtained in [55].The anomalous dimension γ φ is know to three-loop order [48]. The expansion coefficientsare γ φ = 3 C F = 4 ,γ φ = C F (cid:18) − π + 24 ζ (cid:19) + C F C A
176 + 22 π − ζ ! − C F T F n f
23 + 8 π ! ≈ . ,γ φ = C F
292 + 3 π + 8 π ζ − π ζ − ζ ! + C F C A − π − π
135 + 8443 ζ + 8 π ζ + 120 ζ ! + C F T F n f −
46 + 20 π π − ζ ! + C F C A − π − π − ζ + 40 ζ ! + C F C A T F n f − π
81 + 2 π
45 + 4009 ζ ! + C F T F n f −
689 + 160 π − ζ ! ≈ . . (59)The soft anomalous dimension γ S is known to two-loop order [32, 42]. Its expansioncoefficients are γ S = − C F = − ,γ S = C F " π − ζ ! C A +
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