Heavy quark hadroproduction in perturbative QCD

PHYSICAL REVIEW D, VOLUME 59, 014506

Heavy quark hadroproduction in perturbative QCD

F. I. OlnessDepartment of Physics, Southern Methodist University, Dallas, Texas 75275

and Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, Illinois 60510

R. J. ScaliseDepartment of Physics, Southern Methodist University, Dallas, Texas 75275

Wu-Ki TungDepartment of Physics/Astronomy, Michigan State University, East Lansing, Michigan 48824

~Received 23 December 1997; published 23 November 1998!

Existing calculations of heavy quark hadroproduction in perturbative QCD are either based on the conven-tional zero-mass perturbative QCD theory, valid for energy scales much higher than the quark mass, or on anext-to-leading order~NLO! fixed-flavor-number~FFN! scheme which is appropriate for energy scales com-parable to the quark mass. We formulate this problem in the general mass variable-flavor-number schemewhich incorporates initial~final! state heavy quark parton distribution~fragmentation! functions as well asexact mass dependence in the hard cross section. This formalism has the built-in feature of reducing to the FFNscheme near threshold and to the conventional zero-mass parton picture in the very high energy limit. Makinguse of existing calculations in the NLO FFN scheme, we obtain more complete results on bottom quarkproduction in the general scheme to orderas

3 both for current accelerator energies and for CERN LHC. Modestimprovement over the FFN results is observed in the reduced scale dependence of the cross section and in theincreased magnitude of the cross section, in the direction of experimental measurement. It is shown that thegeneral scheme represents a more efficient way of organizing the perturbation series, since the bulk of the largeNLO (as

3) FFN contribution to the single heavy-quark inclusive cross section is already contained in the~resummed! order as

2 ‘‘heavy flavor excitation’’ term in this scheme. Remaining limitations of the presentcalculation and possible improvements are discussed.@S0556-2821~98!01923-7#

PACS number~s!: 12.38.Bx, 13.85.2t, 14.65.2q

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I. INTRODUCTION

The production of heavy quarks in high energy proceshas become an increasingly important subject of study btheoretically and experimentally@1#. The theory of heavyquark production in perturbative quantum chromodynam~PQCD! is more challenging than that of light parton~jet!production because of the new physics issues brought aby the additional heavy quark mass scale. The correct themust properly take into account the changing role ofheavy quark over the full kinematic range of the relevaprocess from the threshold region~where the quark behavelike a typical ‘‘heavy particle’’! to the asymptotic region~where the same quark behaves effectively like aparton,similar to the well-known light quarks$u,d,s%). Stimulatedby significant recent experimental results on heavy quproduction from the DESYep collider HERA and the Fer-milab Tevatron, a number of theoretical methods have badvanced to improve existing QCD calculations of heaquark production@2–5#, incorporating a dynamic role for thheavy quark parton. The purpose of this paper is to exphow the method of Aivazis, Collins, Olness, and Tu~ACOT! @2# is applied to heavy-quark production in hadrohadron collisions, to compare results of this approach toisting ‘‘next leading order’’~NLO! calculations, and to demonstrate that it satisfies some important consisteconditions.

Let us consider the production of a generic heavy quadenoted byH, with non-zero massmH , in hadron-hadron

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collisions. We will define a quark as ‘‘heavy’’ if its mass isufficiently larger thanLQCD that perturbative QCD is applicable at the scalemH , i.e., thatas(mH) is small. Thus thec,b, and t quarks are regarded as heavy, as usual. LetQ be atypical large kinematic variable in the hard scattering pcess, in this case thepT of the heavy quark~or the associatedheavy flavor hadron!. The details of the physics of the heavquark production process will then depend sensitively onrelative size of the two scalesmH andQ. For simplicity, wewill assume that there is only one heavy quark—H—that weneed to treat. Extending our treatment to the real world cof $c,b,t% quarks with successively higher massesstraightforward.

Conventional PQCD calculations involving heavy quarconsist of two contrasting approaches: the usual QCD paformalism uses thezero-mass approximation(mH50) oncethe hard scale of the problem~say,Q) is greater thanmH ,and treatsH just like the other light partons@6–8#; on theother hand, most recent ‘‘NLO’’ heavy quark production caculations considermH as a large parameter irrespectivethe energy scale of the physical process, and treatH alwaysas aheavy particle, never as a parton@9–11#. We shall referto the former as thezero-mass variable-flavor-number~ZM-VFN! scheme—since the active flavor number varies,pending on the energy scalem;Q—and the latter as thefixed-flavor-number~FFN! scheme—since the parton flavonumber is kept fixed, independent ofQ.

Each of these approaches can only be accurate in a limenergy range appropriate for the approximation involv

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F. I. OLNESS, R. J. SCALISE, AND WU-KI TUNG PHYSICAL REVIEW D59 014506

Q@mH for the ZM-VFN scheme andQ;mH for the FFNscheme. Nonetheless, both approaches have beenwidely beyond their respective regions of natural applicaity: on the one hand, NLO FFN calculations ofc andb pro-duction are invoked from fixed-target to the highest collidenergies@1#, and on the other hand, ZM-VFN calculationdominate practically all other standard model and new phics calculations, including most global PQCD analys~which give rise to commonly used parton distributions@6–8#! and all popular Monte Carlo event generators. The bredown of the approximations beyond the original regionsapplicability of these approaches can lead to unreliablesults, and introduce large theoretical uncertainties. Onesible sign of the latter is excessive dependence of theorepredictions on the unphysical renormalization and factorition scalem—which is well known to be present in the NLOFFN calculation of the hadro-production cross section ocandb @12,1#.

With steadily improving experimental data on a varietyprocesses sensitive to the contribution of heavy quarks~in-cluding the direct measurement of heavy flavor productio!,it is imperative that the two diametrically opposite treatmeof heavy quarks be reconciled in a unified framework whcan also provide reliable theoretical predictions in the intmediate energy region, which in reality may well comprimost of the current experimentally relevant range for chaand bottom physics. This can be achieved in a general-mvariable-flavor-number~GM-VFN! scheme which retains thmH dependence at all energy scales, and which naturreduces to the two conventional approaches in their restive regions of validity@13–15,2#. The method is a development of the one devised by Collins, Wilczek, and Zee@16#.The key point is that one can resum~and factor out! the masssingularities associated with the heavy quark massmH into Hparton distribution and fragmentation functionswithout si-multaneously taking the mH→0 limit in the remaining infra-red safe hard cross section in the overall physical crosstion formula~as is routinely done in the ZM-VFN scheme1!.The resulting general formalism represents the natural exsion of the familiar PQCD framework to includeheavy quarkpartonsboth in the initial and final states of the hard scattings which contribute to high energy processes in the sdard model ~SM! and beyond. For the inclusive singheavy-quark production cross-section calculation, therealso large logarithms of the kind log(s/mH

2 ) which representsanother example of the ‘‘small-x problem’’ @17#. We shallnot be concerned with this problem in this paper.

The principles and the practical application of the GMVFN scheme were described in some detail for~the heavyquark parton contribution to! Higgs boson production in Ref@14# and for lepto-production of heavy quarks in Re

1The resummation of mass singularities into parton distributiand taking the zero-mass limit on the hard cross section are usperformed simultaneously not as a matter of principle, but forincidental reason that the mass singularities are most conveniidentified by using dimensional regularization in the zero-mtheory.

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@15,2#. It has been applied to the analysis of charm quproduction in neutrino scattering@18# and, recently, to a newglobal QCD analysis of parton distributions@19#. In thepresent paper, we apply this general formalism to hequark production in hadron-hadron colliders. For a concsummary of this formalism, and related recent developmesee Ref.@3#; for a systematic proof of the factorization therem which provides the theoretical foundation of this formism, see Ref.@20#. In recent literature on lepto-production oheavy quarks, there have been two other formulations ofVFN scheme with non-zero heavy quark mass: an orderas

scheme by Ref.@4# and an orderas2 scheme by Ref.@5#. For

definiteness, we shall refer to our implementation of the geral principles as the ACOT scheme. Although not uniquerepresents in many ways the simplest and the most naone in relation to the familiar modified minimal subtractio(MS) ZM-VFN scheme@3#. See Ref.@21# for comments onsome aspects of the scheme dependence~in particular, mass-dependent or mass-independent evolution of the partontributions!, as well as on a comparison to the scheme pposed by Ref.@4#.

The following section presents details of our calculatischeme as applied to hadro-production and describesphysical origin of the various terms which appear in the fmalism. Section III contains specific information on how tvarious ingredients of the formalism are calculated. Thisfollowed by numerical comparisons of the new calculatiwith existing FFN scheme results on the inclusive differetial pT cross section for bottom quark production at curreaccelerator and CERN Large Hadron Collider~LHC! ener-gies. In the concluding section, we discuss what remainbe done for a full understanding of heavy quark productin PQCD.

II. HADRO-PRODUCTION OF H IN THE GENERALMASS FORMALISM

Consider the hadro-production of a generic heavy quH:

A1B→H1X ~1!

whereA andB are hadrons, andX includesH along with allother summed-over final state particles. In the ACOT formism for heavy quark production developed in Refs.@13,2,20#,the inclusive cross section for observingH with a given mo-mentump at high energies is given by a factorization fomula of the same form as in the familiar zero-mass QCparton formalism:

sABHX5 (

a,b,cf AB

ab^ sab

c,r^ dc

H , ~2!

where$A,B% denote the initial state hadrons,$a,b% the initialstate partons,f AB

ab the associated parton distribution functio

in the combinationf ABab [ f A

a f Bb , sab

c,r the perturbatively calcu-lable infrared safehard-scattering cross section for$a,b%→$c,r % which is free of large logarithms ofmH over the fullenergy range, anddc

H the fragmentation functions for finding

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HEAVY QUARK HADROPRODUCTION IN PERTURBATIVE QCD PHYSICAL REVIEW D59 014506

FIG. 1. Graphical representation of the leading terms in the factorization formula which correspond to the various productionnisms. The initial state hadron line for the parton distributions are uniformly suppressed.

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H in c. All active partonsare included in the summation ove$a,b% and$c%, includingH provided the renormalization anfactorization scalem is larger thanmH .2

Since we shall compare our results to those of the Fscheme, it is necessary to draw the distinction betweenheavy quarkH and the associatedlight partons; we shalldenote the latter collectively byl for simplicity of notation.By definition, l 5$g,q% whereg is the gluon andq denotesthe light quarks in the sense of the FFN scheme:q5$u,d,s%/$u,d,s,c% for charm and bottom quark productiorespectively. The number of light quark partons will be dnoted bynl .

As mentioned in the Introduction, an important distiguishing feature of this general formalism from the familZM-VFN parton approach is that, after subtraction of masingularities, the fullmH dependence in the perturbativecalculated hard scattering coefficientssab

c,r is retained. Thisallows the theory to maintain accuracy and reproduceFFN scheme results in the threshold region, as requiredphysical considerations. On the other hand, our parton dsities ~and fragmentation functions! are defined in theMSscheme; hence they satisfy mass-independent evoluequations—the same as in the usual zero-mass formaConsiderable simplification then results in the implemention of this scheme since the well-established NLO evolutkernels and evolution programs can be directly usedshould be noted, however, that the parton densities do han implicit dependence on quark masses: to the leadpower inL/mH , this dependence is generated by the mating conditions atm5mH between the parton densities beloand above the threshold3—they are defined in each of th

2For simplicity, we shall use the symbolm to represent collec-tively the renormalization scale as well as the factorization scfor parton distributions and for fragmentation functions.

3Note that equivalent matching conditions can be derived form which is of ordermH . Also, possible non-perturbative heavquarks@22#, as opposed to ‘‘radiatively generated’’ ones~assumedin subsequent discussions!, can be incorporated in the generscheme by allowing for a nonzero heavy quark density inbelow-threshold part of the scheme.

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two regions by the respective renormalization scheadopted for that region@13,20#.

The first few terms in the perturbative expansion of tproduction cross section, Eq.~2!, are schematically

sABHX5 f AB

ll 8 ^2s l l 8

HH^ dH

H : HC0

1 f ABlH

^2s lH

lH^ dH

H1 f ABHl

^2sHl

lH^ dH

H : HE1

1 f ABll 8 ^

3sl l 8

HHl 1^ dH

H : HC1

1 f ABll 8 ^

2sl l 8

l 1l 2^ dl 1

H1 f ABHl

^2sHl

Hl^ dl

H : GF1,2

1 f ABHH

^2sHH

HH^ dH

H : HH2 ~3!

where repeated indicesl, l 8, l 1 , and l 2 are summed over allight flavors (q andg) and the pre-superscriptn on ns de-notes the formal order in the expansion ofas .

The physical interpretation of the various terms, labeby the abbreviations in the last column of this equation, cbe made apparent by the generic diagrams shown in FigThey are the following:

~i! HC0: the orderas2 heavy flavor creationprocessl l 8

→HH followed by heavy-quark fragmentation.~ii ! HE1: the orderas

2 heavy flavor excitationprocesslH→ lH followed by heavy-quark fragmentation.

~iii ! HC1: orderas3 virtual and real corrections to HC0

l l 8→HHl 1 , with fragmentation.~iv! GF1,2: orderas

2 ~light- and heavy-! parton-gluonscattering lg→ lg and Hg→Hg, followed by gluonfragmentation4 into the heavy quarkH.

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4For generality, we have written the summation~over l ) to includeall light-parton fragmentation intoH. In fact, we expect gluon frag-mentation to dominate. Light-quark fragmentation is suppressedcause of the quark structure of the evolution equation. In the fiorder calculations which we will do to compare schemes, fragm

tation of light quarksq and heavy antiquarksH into H are both atleast of orderas

2 .

6-3


FIG. 2. PDFs vsx for m510 and 100 GeV. The gluon PDF is scaled by 1/5.

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~v! HH2: the order as2 HH scattering HH(H)

→HH(H), with fragmentation.For the diagrams in Fig. 1 we use a dashed line for

light partons l ~representing collectively gluons and lighquarks! to distinguish them from the heavy quarkH, repre-sented by a solid line. The square blocks in these diagr

represent the hard cross sectionsns, obtained from all Feyn-man diagrams of the same order with the external lines incated. The round blobs represent the parton distribution fution ~PDF! and fragmentation function. The initial stahadron lines for the parton distribution factors are suppresin all the diagrams. For the differential cross sectids/dpT

H , the four terms involving 2→2 hard scattering processes are tree-level processes. Beyond the tree levehave included only the most important orderas

3 term—HC1—which contains 2→3 real and 2→2 virtual correc-tions to the leading HC0 cross section. This term corsponds to the NLO contribution in the FFN scheme; itknown to be large and it is the only one that has been calated to this order so far~see below!. We will discuss therelative sizes of the various terms in the next paragraph,comment on the possible significance of other terms notcluded here in later parts of the paper.

In conventional applications of PQCD with light partonit is common to distinguish the order of terms in a crosection by treating the~non-perturbative! parton distributionsin f AB

ab as all being of order 1, and then counting powersas in the hard scattering. With the inclusion of heavy quapartons, one may expect heavy quark distribution and frmentation functions to be of orderas in the energy regionnot too far above threshold, assuming they are purely ratively generated.5 We know that at largex the valenceu andd quarks dominate all the other partons, while at smallx, it isthe gluon density that dominates. For instance, in the laregion, we compare in Fig. 2 the relative sizes of the varisea partons at two energy scales to the gluon~scaled downby a factor of 5!. In all cases it appears safe to regard tfully evolved heavy quark densityf A

H to be of effective orderas with respect to the dominant parton density~gluon orvalence quark!, i.e.,

5As mentioned in footnote II, our formalism can accommodnon-perturbative, i.e. non-radiative or ‘‘intrinsic’’ heavy quarkHowever, we shall not get into that possibility in this paper.

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Note, however, that the evolved charm quark density comwithin a factor of 2 of the other sea quark densities at smx.

The same argument leads to the following expectatifor the fragmentation functions~note that we confine ourselves to the production of the heavy flavor partonH, ratherthan the associated hadron!:

dHH;d~12z!1O~as!

dgH;O~as! ~5!

dq,HH

;O~as2!.

These order-of-magnitude estimates are, of course, expeto become inapplicable at very large scales. The gluon–heavy-quark fragmentation function, in particular, can bcome substantial at large scales.

Nevertheless, let us use these estimates as the first gto orders of magnitudes of the terms in Eq.~3!. The first term~HC0! is of orderas

2 ; the HE1, and GF1, and HC1 terms aof effective orderas

3 ; and the GF2 and the HH2 terms areeffective orderas

4 . ~This naive counting of effective ordeexplains the choice of numerical suffixes in the labelsthese terms.! The actual relative numerical importance of thvarious terms will also depend on other considerations sas large color factors or dynamical effects~e.g. spin-1t-channel exchange contribution at high energies!. This is wellknown for the perturbative expansion of various processincluding heavy quark production calculated in the FFscheme, where the orderas

3 NLO term ~corresponding toHC1 without heavy quark mass subtractions! is actuallylarger than the orderas

2 LO term ~HC0!. We will examine indetail the numerical significance of the various terms in SIV. The results are revealing, since they show simple ftures which are not accessible from the conventional Fscheme perspective.

To specify fully the calculation in our scheme, we needspecify the perturbative hard cross sectionssab

c,r . FollowingRef. @2#, we obtain these by applying the factorization fomula, Eq. ~2!, to cross sections involving partonic beamthat is to sab

c,r ~without the caret!, calculated from the rel-evant Feynman diagrams. Then we solve for the hard c

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sectionssabc,r order by order inas . Specifically, in a given

orderasn , the factorization theorem implies that the parton

cross section has the form

nsabHX5(

n1f abab

^n2sab

c,r^

n3dcH , ~6!

where the sum is over the values ofn1 , n2 , andn3 that givethe correct order, i.e.,n5n11n21n3 . On the right-handside, this equation differs from Eq.~2! in two respects. Firstthe parton distributions are relative to an on-shell partonget, instead of a hadron target. Second, we have expathe parton-level distribution and fragmentation functionspowers ofas , with, for example,n1f a

a denoting the term ofas

n1 in the distribution function. These coefficients are calclated order by order inas , but are not infrared safe; they arused only in the intermediate steps toward the derivationsab

c,r .The most general method to obtain the hard scatte

coefficients is directly from Eq.~6!. All of the full partoniccross sections, on the left-hand side, and the parton densand fragmentation functions at the partonic level, onright-hand side, can be computed from definite Feynmrules. From them, Eq.~6! gives the hard scattering coefficients. However, at the orderas

3 level to which we are work-ing, it is convenient to carry out this calculation in two stepthe relation to the coefficient functions of the other schemin the literature is then readily obtained.

First, the unsubtracted cross sections are calculated fall relevant Feynman diagrams with the use of dimensioregularization~with the familiar parameterse and m), withall light quark masses set to zero and with the heavy qumass non-zero. After ultraviolet renormalization~with MScounterterms! and cancellation of infrared divergences~aftercombining real and virtual diagrams!, the cross section formula will depend on the renormalized massmH and the un-physical parameters (e andm), in addition to the kinematicvariables. The collinear singularities associated with theepoles can be factorized into a kind ofdistribution function of

light partons in a light parton faa(e,m) using theMS con-

vention. Since the cross section is inclusive with respecthe light partons, we need no light parton fragmentatfunctions. Thus,6

nsabHX~mH ,e,m!5 (

n11n25n

n1 f abab ~e,m! ^

n2sabHX~mH ,m!,

~7!

where f abab represents the product of two singular light part

distributions@cf. Eq. ~2!#, and all non-essential~kinematicaland convolution! variables have been suppressed. This ccludes the first step of the construction, where the se

6For simplicity, we have used the symbolm to collectively repre-sent both the factorization scale and the renormalization scale.

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intermediate finite cross sectionsnsabHX(mH ,m) is obtained

by systematically subtracting the collinear singularities.At first sight, this appears to imply that the sub-set

nsabHX(mH ,m) with $a,b%5$ l ,l 8% ~light partons! is the same

as the conventional FFN scheme cross sections. This ifact true at the lowest non-trivial order, which is all that wwill consider in this paper. However, at sufficiently high oder extra heavy-quark loops come in, and these are renorized differently in the FFN scheme and in theMS schemethat we use whenm.mH . Hence the singularities to be subtracted differ by some kind of renormalization-group tranformation. In this paper, we do not treat these higher orgraphs.

The second stage of our calculation starts from the obvation that although they are finite, the cross sectionsab

HX(mH ,m) contain logarithmic mass-singularities, i.powers of ln(mH /m), in themH /m→0 limit. The second stepof the derivation consists of factoring these singularitiesto arrive at thefully infrared safe hard cross sectionsnsab

c,r

of Eq. ~6!. Explicitly,

nsabHX~mH ,m!5( n1 f ab

ab„ln~mH /m!…^

n2sabc,r~mH ,m!

^n3dc

H„ln~mH /m!…. ~8!

Here the logarithmically singular terms in themH→0 limitare factored intof and d. These are partonic level partodensities and fragmentation functions with subtractions mto remove the singularities associated with light partoThese subtractions exactly correspond to the subtractused to obtains from s. The remaining infrared safemH

dependence is kept ins ~in contrast to the conventional approach, wheremH is set to zero!.

As with all calculations of hard scattering coefficientthere is a freedom to choose exactly how to define the padensities and fragmentation functions. The choice definesfactorization scheme~cf. @21#!, and thus determines howmuch of the finitemH dependence is included inf and d.

We use the ACOT scheme@13,2,20#. In this scheme, theparton densities and fragmentation functions in the regm.mH are determined by the requirement that all ultraviodivergences be renormalized by theMS scheme—both theultraviolet divergences in the renormalization of the interations of QCD and the ultraviolet divergences that make finthe parton densities~in hadrons!. The hard scattering coefficientss in Eq. ~6! are then well defined. In the first stageour calculation, the factorization of light parton singularitiewe define the intermediate coefficientss in Eq. ~7! by sub-traction of collinear singularities in theMS scheme.

Subtracting the terms which containf and d with non-trivial mass singularities fromnsab

HX(mH ,m), we obtain the

fully infrared safe hard cross sectionsnsabc,r(mH ,m) which

we need in Eq.~2!. The relevant non-vanishing perturbativparton distributionsnf up to orderas are

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0 f ab~x!5da

bd~12x!

1 f gH~x,m!5

as~m!

2plnS m2

mH2 D Pg→q~x!

~9!

1 f HH~x,m!5

as~m!

2plnS m2

mH2 D Pq→q~x!

1 f gg~x,m!5

as~m!

2plnS m2

mH2 D d~12x!

wherea,b5$g,q,q,H%, andPg→q(x) and Pq→q(x) are theusual first order splitting functions. The only nonzero terat orderas in Eq. ~9! come from contributions with a heavquark loop. All one-loop corrections that involve light patons are zero, because of the well-known cancellation ofand UV singularities. For the fragmentation functions,have@23#

0dHH~z!5d~12z!

1dHH~z!5

as~m!CF

2p F11x2

12x S lnm2

MH2

22ln~12x!21D G1

~10!

1dgH~z,m!5

as~m!

2plnS m2

mH2 D Pg→q~z!.

We can now substitute Eqs.~9! and ~10! into Eq. ~8! tosolve for nsab

c,r . To orderas2 , only 0 f a

b and 0dab are needed

and we obtain the trivial identities

2s l l 8HH

52s l l 8HH

2s lHlH52s lH

lH

~11!

2sHHHH52sHH

HH

2sHHHH

52sHHHH

which reflect the fact that the orderas2 cross sections are

given by tree graphs: i.e. the2s ’s are infrared safe, in themH→0 limit. Hence they do not need any subtraction. Flowing the same procedure to orderas

3 , we obtainschematically:7

7It should be observed that the only contribution to1 f gg , in Eq.~9!

corresponds to a gluon self-energy graph on the external gluon

Therefore when obtainings from s in Eq. ~12!, the effect of the1 f g

g terms is simply to cancel external line corrections.

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HX5SX

3s l l 8HX

1@d lg1 f g

g^

2sgl8HH

1d l 8g1 f g

g^

2s lgHH#

12s l l 8HH

^1dH

H

1@1 f l 8H

^2s lH

lH11 f lH

^2sHl 8

Hl 8#

12sl l 8

gl1^

1dgH . ~12!

For convenience, we have used a single equation to covevarious possible initial states. Not all terms on the right-haside are applicable to all cases: for gluon-gluon scatter

$ l l 8%5$gg%, all terms are present; for$ l l 8%5$qq%, only the1st, 3rd and 5th terms contribute; and for$ l l 8%5$gq(q)%,only the 1st, 4th and 5th terms contribute.

This equation can easily be inverted to obtain the orderas3

hard cross section in terms of the calculated finite intermdiate cross sections3s ~with non-zero massmH) and varioussubtraction terms which remove the mass singularities aciated with the heavy quark degree of freedom:

SX3s l l 8

HX5SX

3s l l 8~real1virtual!HX : HC1-FFN

21 f gg

^2sgg8

HH21 f g8

g8^

2sgg8HH : Fln-Sub

22s l l 8HH

^1dH

H : HF1-Sub

21 f l 8H

^2s lH

lH21 f lH

^2sHl 8

Hl 8 : HE1-Sub

22sl l 8

gl1^

1dgH : GF1-Sub. ~13!

Here, we have replaced all tree-level2s in Eq. ~11! by thecorresponding2s because they are the same; cf. Eq.~11!.The content of the terms on the right-hand side of this eqtion, labeled by the abbreviations in the last column, canseen more easily from the diagrams in Fig. 3 where,clarity, the four possible initial and final state channels aseparately shown.

The terms in Eq.~13! and Fig. 3 are the following:~i! HC1-FFN: the usual orderas

3 FFN scheme result, dueto contributions of the NLO-HC~virtual and real! diagrams,with infrared and collinear singularities associated with ligpartons cancelled or subtracted in the conventional way~inthe MS scheme!.

~ii ! Fln-SUB: the correction to the orderas2 gluon-gluon

HC process due to the difference in the definition of tgluon distribution between the below-threshold (nf5nl)scheme and the above-threshold (nf5nl11) scheme.

~iii ! HE-SUB: the subtraction of large logarithms of thheavy mass contained in1 f g

H @cf. Eq. ~9!#, due to a flavor-excitation configuration.

~iv! HF-SUB ~GF-SUB!: the subtraction of large logarithms of the heavy mass in the final state heavy-quark~glu-on! fragmentation residing in1dH,g

H @cf. Eq. ~10!#.

e.

6-6

t

inuc

rh

qson

-eteii

e

.b

1-

ehealio

E/e

inge

thec-ne

hs,

is

ingti-

lar

neetio

ion,ic

tion

are


After these subtractions,3s l l 8HX is free from all large loga-

rithms associated with potential mass singularities; i.e., iinfrared safe with respect tomH /Q→0. When this result isused in Eq.~3!, the hadronic cross section is well behavedthe high energy limit, in contrast to the FFN scheme reswhich would diverge because of the large logarithms. In fain this limit Eq. ~3! reduces to the usual zero-massMS as

3

parton formula with theH quark counted just like the othelight partons. This is, of course, the correct limit at higenergies.

Some insight can be gained by explicitly substituting E~11!, ~13! in Eq. ~3! and obtaining the hadronic cross sectiin terms of the intermediate partonic cross sections$2s,3s%~which are all finite for non-zeromH) along with the necessary subtraction terms which represent the overlap betwthe two sets of cross sections and which remove the potially dangerous mass singularities. The full set of termsmost clearly displayed in diagrammatic form, as shownFig. 4. The first column lists contributions from all th2→2 tree-level diagrams summarized in Eq.~3! and Fig. 1plus 1-loop virtual corrections from Eq.~13!, the last columnlists contributions from all the 2→3 terms contained in Eq~13!, and the middle column contains all the relevant sutraction terms. These terms can most easily be obtainedsubstituting the terms shown in Fig. 3 in those in Fig.summingl over$g,q%. With the exception of uniformly suppressing the initial state parton distribution factors~repre-sented by dark blobs in Fig. 1!, these diagrams contain all thingredients needed to write down the full formula for tcross section. In between the last two columns, we havedrawn lines to indicate the origins of the various subtractterms in the 2→3 diagrams, according to Fig. 3.

FIG. 3. Graphical representation of the terms in the right-haside of Eq.~13!. Collinear singularities due to light partons havalready been subtracted. The vertices represented by a dot arheavy quark parts of the perturbative distribution and fragmentafunctions, as in Eqs.~9! and ~10!.

01450

is

ltt,

.

enn-sn

-by,

son

In this way of organizing the results, the orderas subtrac-tion terms in Fig. 4 are shown next to the relevant HGF/HF contributions in the same row, making explicit thphysical origin of the subtractions: these terms containlarge logarithms~present in the un-regulated FFN schemcalculations! also represent the low-order components ofQCD evolved parton distribution and fragmentation funtions. As an example, consider the first two terms in row oof HE1 of Fig. 4:

~2→2s !2~mH subtraction!uHE1gH→gH

5~ f AH2 f A

g^

1 f gH! f B

g^

2sHgHg

^ dHH . ~14!

Both from this formula and from the corresponding grapone can see that the subtraction term containingf A

g^

1 f gH

represents that part of the NLO-FFN contribution whichalready included in the~fully evolved! parton distributionf A

H . The latter, of course, represents the result of resummall powers of asln(m2/mH

2 ), and hence contains importanphysics not included in the NLO-FFN calculation, in addtion to being well behaved asm2/mH

2 becomes large—according to the renormalization group equation. Simi

d

then

FIG. 4. Graphical representation of the physical cross sectEq. ~3! and Fig. 1, written in terms of the intermediate parton

cross sectionsns ’s and the attendant heavy quark mass subtracterms which represent the overlap between the 2→2 and 2→3cross sections. Initial state parton distribution function factorsuniformly suppressed.

6-7

-l t

-the

s, in


comments apply to the other 2→2s terms and the corresponding subtraction. The GF1 terms will be proportiona(dg

H21dgH

^ dHH) wheredg,H

H are the fully evolved and1dgH

the perturbative fragmentation functions.The following ~terms in! parton distribution and fragmen

tation functions f AH , 1 f g

H , 1 f gg , and dg

H all vanish at the‘‘threshold,’’ m5mH , by calculation @cf. Eqs.~9!, ~10! andRefs.@13,2##. In addition, in the threshold region,

dHH;d~12z!1O~as! ~15!

ceoell

sto

aine

erE

pa

ls-t

efifis

01450

of A

H2 f Ag

^1 f g

H;O~as2!

dgH21dg

H^ dH

H;O~as2!.

Hence, the differences between the first two columns ofterms represented in Fig. 4—(2→2s)—~heavy quark masssubtraction!—vanish even faster than the individual termapproaching the threshold. Using these results, we obtainthis limit,

sABHX →

threshold

f ABll 8 ^

2s l l 8HH : LO-FFN

1 f ABll 8 ^ ~3s l l 8~virtual!

HH13s l l 8~real!

HHl 9 ! : NLO-FFN

5sABHX : full NLO-FFN scheme ~16!

e of

i-OT

ary

be--

nark

-e-ion

tual

al

nseadtherart

That is, the hadronic cross section in this formalism reduto the full flavor creation~HC! result of the FFN scheme torder as

3 . In this region, there is effectively only one largmomentum scale (pT;mH), and the FFN scheme is wesuited to represent the correct physics.

We see, therefore, that the ACOT formalism providenatural generalization of the familiar light-parton PQCDthe case including quarks with non-zero mass which contthe right physics over the entire energy range. At high engies, Eq.~3! gives the most natural description of the undlying physics. The mass subtraction terms appearing in~13! for the as

3 hard cross section correspond to thee21

poles arising from collinear singularities in themH50 QCDparton formalism which are usually removed byMSregularization.8 On the other hand, at energy scales comrable to the quark massmH , consistency with the physicallysensible NLO-FFN calculation in the threshold region is aguaranteed by keeping the fullmH dependence in the partonic cross sections appearing in Fig. 4, as described inprevious paragraph.

III. CALCULATIONS

Our numerical calculations are carried out using Eq.~3!,with hard cross sections given by Eqs.~11! and ~13!. ~Theequations are presented graphically in Fig. 4.! We briefly

8Consistency of this generalized formalism with the usualMSscheme at high energies is ensured by adopting the precise dtions of as(m) and f A

g,H , dG,HH in the renormalization scheme o

Refs.@13,16#. As mentioned earlier, in this scheme, the parton dtributions and fragmentation functions satisfy the familiarMS evo-lution equations above the respective thresholds.

s

a

sr--q.

-

o

he

describe how the various quantities on the right-hand sidthe equation are obtained.

For the parton distributionsf Aa(x,Q), we use the

CTEQ3M set@8# for definiteness. The CTEQ parton distrbution sets, in general, are evolved according to the ACscheme described in Refs.@2,13,16,20# which is the one weuse in the heavy quark production theory. This is necessfor obtaining consistent results~a point which has not beenentirely obvious to all users of the scheme!. The reason isthat the expected compensation in the threshold regiontween the QCD-evolvedf A

H(x,Q) and the perturbatively gen

erated subtractionf AH(x,Q)SUB[ f A

g(x,Q) ^1 f g

H(x,Q) @cf.Eq. ~14!# will not take place unless the choice of evolutioscheme and the choice of the location of the heavy quthreshold match properly.9 This is illustrated in Fig. 5 whichcomparesf A

H(x,m) and f AH(x,m)SUB for charm and bottom as

a function ofm at x50.1 andx50.01. Each curve individu-ally vanishes at the thresholdm5mH as required in this general scheme. Asm increases, both grow at a rapid rate bcause the evolution is driven by the large gluon distribut~through the mass-independent splitting kernel!; however,the difference between the two, which determines the accorrection to the main contribution in this region~due to theHC process!, grows slowly as one would expect on physic

ni-

-

9As an example, MRS distributions usem52mH as the thresholdfor evolving f H(x,m), from zero in contrast tom5mH which isrequired by our renormalization scheme. Using MRS distributioin this framework represents a mismatch of schemes, and will lto unphysical results as described below. If a matching point othan m5mH is used, then the heavy quark distribution must stfrom a non-zero value.

6-8


FIG. 5. Comparison of the evolved PDFs,f H(x,m) ~labeled PDF!, and perturbative PDFs,1f H(x,m) ~labeled SUB!, as a function of therenormalization scalem in GeV for charm atx50.1 ~a! and x50.01 ~b!, and for bottom atx50.1 ~c! and x50.01 ~d!. This shows thecompensation between fully evolved heavy quark parton distribution and the first order perturbative contribution~which is the only partcontained in the FFN scheme calculation!.

beulr

r-

io

,th

ada

gce

at

istri-nc-

in

s-

calethesenpend

Ex-

u-

imth1

grounds. A failure to ensure the proper compensationtween these terms due to a mismatch of schemes or thcation of the threshold can lead to quite unphysical resbecause then the difference would be of the same ordemagnitude as the individual terms.10

In calculating the partonic cross sections which appeaEqs. ~11! and ~13!, the formula for the LO-FFN cross sections 2s l l 8

HH and 2s lHlH are well known. For the NLO-FFN

cross section, both virtual3s l l 8HH and real3s l l 8

HHl 9 , we usedthe FORTRAN codes from Ref.@11# and Ref.@24#.

To calculate the gluon and heavy-quark fragmentatfunctions appearing in Eq.~3!, we need also the QCDevolved fragmentation functionsdg,H

H . As mentioned earlierfor the purpose of this paper, we restrict ourselves toH-quark production cross section; hencedg,H

H are parton-to-parton fragmentation functions. To obtain heavy flavor hron production cross sections, it is necessary to performadditional convolution with the appropriate hadronic framentation functions. The QCD evolved fragmentation funtions dg,H

H (z,m) are generated by solving numerically th

10f AH(x,m) and f A

H(x,m)SUB are not expected to cancel at largem;the latter, in fact contains the divergent ln(m/mH) factor in that limit.In that region, this divergent subtraction term plays the otherportant role of cancelling the corresponding large logarithm inNLO-FC term to render the latter infrared safe, as shown in Eq.

01450

e-lo-tsof

in

n

e

-n

--

QCD evolution equation in our scheme, using as inputm05mH the perturbative formula; cf. Refs.@23,25#. Thecomments about proper compensation between parton dbutions made above also apply to the fragmentation fu

tionsdg,HH and 1dg,H

H . These features have been examineddetail during our calculation.

IV. RESULTS

We now present typical results for theb quark productioncross section (ds/dpT

2dy)uy50 vs pT and vs the QCD scaleparameterm at collider energies. For simplicity, as is cutomary in the literature, we use a single scale parameterm torepresent the renormalization scale, the factorization sfor the parton distributions, and the factorization scale forfragmentation functions. In principle, these could be choas independent; the hard cross sections would then deon all three scales. As a rule, we shall express the scalem asmultiples of the natural physical scaleMT[ApT

21mH2 , al-

though, again, other choices could also be considered.cept for explicit discussions concerning them-dependence ofthe cross sections, our default choice of scale ism25MT

2/2.In order to be able to discern clearly the various contribtions to the steeply falling function (ds/dpT

2dy)(pT), we

shall in general use the scaled cross sections(pT)[pT

5(ds/dpT2dy)uy50 when examining itspT behavior. We

-e3.

6-9


FIG. 6. Contributions to the scaled cross sectionpT5ds/dpT

2/dyuy50 (nb GeV3) vs pT for b production at 1800 GeV withm5MT /A2organized in the ACOT formalism.~a! The curves correspond to the separate terms of Eq.~3!. ~b! The curves are leading-order (as

2)~LO-VFN!, next-to-leading-order (as

3) ~NLO-VFN!, and the total result~TOT-VFN!.

-

ot

l

c

thom

th

ap

eN

usg

nnh

theflo

C0

men-

be-

etionllyith

g-ein

wi

en-iallyoneand

the

ble.of

dis-ly

avor

tion

concentrate mostly onb-production at the Tevatron for definiteness.

A. Inclusive pT distribution and comparison of heavy partonpicture in the FFN scheme

Figure 6a showss(pT) vs pT for b production at 1800GeV, including the individual terms on the right-hand sideEq. ~3!.11 Over the range 10,pT,100 GeV, the two largescontributing terms are ‘‘leading order’’ (2→2, tree-level!HE1 (s lH→ lH) and HC0 (s l l 8→HH); the other tree-leveterms and the ‘‘next-to-leading’’ term, HC1 (s l l 8→ l 9HH ,mass-subtracted!, constitute about 10–25 % of the cross setion, depending on the value ofpT . The left-hand-side plotshows the contributions from the individual terms, andright-hand-side plot compares the relative sizes of the cbined LO-VFN ~i.e. tree-level! contribution, the NLO-VFNcontribution and the total cross section TOT-VFN whencalculation is organized in this scheme.

Two interesting features are worth noting. First,the LO-VFN contributions (tree processes) give a reasonableproximation to the full cross section; the NLO-VFN correc-tion is relatively small.~This is in sharp contrast to thsituation in the familiar FFN scheme where the NLO-FFterm is bigger than the LO-FFN one. Cf. Fig. 7 and discsions below.! This is, of course, an encouraging result, sugesting that the heavy quark parton picture represents aficient way to organize the perturbative QCD series. Secothe HE1 contribution is comparable to, and even somewlarger than, the HC0 one—in spite of the smaller heavyquark parton distribution in the initial state compared togluon distribution. Closer examination reveals that twofects contribute to this non-apparent result: a larger cofactor for the HE1 process and the presence oft-channel

11In terms of Fig. 4, the mass-subtraction terms are combinedthe associated orderas

3 HC1 terms~from which they originate! to

yield the infrared safe hard cross sectionss.

01450

f

-

e-

e

-

--ef-d,at

e-r

gluon exchange diagrams which is absent in the Hprocess.12

It is useful to compare the above situation with the saresults organized in a way more familiar from the convetional FFN scheme point of view. For this purpose, one

gins with the intermediate cross sectionss for heavy flavor

creation processes only, HC0 (s l l 8→HH) and HC1

(s l l 8→ l 9HH , no mass-subtraction!. Corrections to the FFNscheme calculationsin the full scheme then consist of thremaining terms on the right-hand side of the cross secformula depicted in Fig. 4, which are now most naturaorganized with the mass-subtraction terms combined wthe corresponding 2→2 cross sections in the same row. Fiures 7~a!,7~b! shows(pT) vs pT in the same format as in thprevious plot but with individual contributions organizedthis way.13 The largest term is now the ‘‘NLO’’ HC1(s l l 8→ l 9HH) followed by the ‘‘LO’’ HC0 of the conventionalFFN scheme. The fact thatthe NLO~orderas

3) term sHC1 is

much larger than the LO term~order as2) sHC0—the ‘‘K-

th

12The precise values of the HE1 contribution are somewhat ssitive to the choice of factorization scheme and scale, especclose to the threshold region, as will be shown below. However,of the important features of our formalism is that any schemescale dependence in HE1 will be closely matched by changes inHC1 contribution~through the corresponding subtraction term!, sothat the combined inclusive cross section remains relatively staCf. discussions in the previous section about the matchingevolved and perturbative parton distributions. For the currentcussion, we adoptm5MT /A2 as a central choice, given commonused ranges ofm such as@MT/2,2MT# and @MT/4,MT#.

13For the purposes of this comparison, we use the same 5-flPDF’s for both Figs. 6~a!,6~b! and Figs. 7~a!,7~b! so that the onlydifference is how we combine the terms. Using 4-flavor PDFs~aswould be appropriate for the FFN scheme, without the subtracterms! yields virtually indistinguishable curves, differing by;1%at pT510 GeV, and;3% atpT5100 GeV.

6-10

rmseled. Note


FIG. 7. Contributions to the scaled cross sectionpT5ds/dpT

2/dyuy50 (nb GeV3) vs pT for b production at 1800 GeV withm5MT /A2organized according to the FFN scheme.~a! The curves correspond to the orderas

2 ~LO-FFN! and orderas3 ~NLO-FFN! heavy-flavor

creation~HC! contributions~without heavy mass subtractions!. Also shown are the corrections due to the HE, GF1, GF2, and HH2 tewith associated subtractions as given in Fig. 4. The last three are numerically negqligible and appear at the bottom of the plot unlabthat ~i! the NLO term is 2 timeslarger then the LO one;~ii ! the contributions from are small and unlabeled.~b! LO-FFN and NLO-FFNcontributions along with the total result~TOT-FFN!. Cf. Fig. 6 for a comparison with the ACOT scheme case.

wer

gatKe

tiotall

ire

lynomFN

vy

eli

re

y

y tobe-

r-

he

u-

ark

ross

inN

teen-

n-chGF

e

e

setoth

aton

factor’’ is typically of the order;2.5—is disturbing fromthe perturbation theory point of view, as has been knosince the orderas

3 calculations were first done. On the othhand, we see from Fig. 7~a! that the corrections to the FFNscheme terms, consisting of the other terms in Fig. 4,arepositive but not very large—again of the order of&20%.This means that the effects of resumming the collinear lorithms, represented by these additional terms, are modesthis case—a non-obvious result on account of the largefactor and the significantm-dependence of the FFN schemcalculations~see next subsection!. The net effect of thesecorrection terms is to increase the theoretical cross secThis is encouraging since the NLO FFN scheme resulknown to be systematically smaller than the experimentmeasured cross section at the Tevatron. However, thiscrease appears to fall short of the current observed discancy @1,26#. Cf. Fig. 8.

A comparison of Figs. 6 and 7 shows that, interestingthe HE1 (sHE1) contribution to the heavy quark productiocross section in the heavy quark parton picture is quite cparable to the HC1 contribution in the complementary Fscheme view (sHC1 , no mass-subtraction!—to within about10%. Thus, at least for this energy range,the heavy quarkparton picture overlaps considerably with the FFN heaflavor creation picture as far as the inclusive pT distributionis concerned14—these two pictures are complementary raththan mutually exclusive, as sometimes perceived in theerature. It is, of course, much easier to calculate the tlevel HE1 cross section~a textbook case! than the HC1 one~a tour de force!. Thus, for this physical quantity, the heav

14An equivalent way of saying this is that the subtraction termwhich represent the overlap between the two, are a reasonablproximation to both in this energy range. Thus the ‘‘correction’’either one, represented by the combination of the other withcorresponding subtraction, are relatively small—as demonstrabove.

01450

n

-for-

n.isyn-p-

,

-

rt-e-

quark parton picture represents a much more efficient waarrive at the right answer. This approximate equivalencetween the HE1 and HC1 contributions to theinclusive pTcross section cannot, of course, be taken literally.15 The twocontributions do not have the same (s,pT ,m,y)dependence—in fact, them dependence can be rather diffeent, as we will discuss next.

B. Scale dependence of the cross section

In Fig. 9 we show a representative plot ofs(pT ,m) vs mat pT520 GeV. The tree-level HC0 and HE1 terms give tdominant contributions. In addition to the commonas

2(m)factor, HC0 is predominantly driven by the gluon distribtion, and this is a decreasing function ofm. On the otherhand, the tree-level HE1 term is driven by the heavy-qudistribution, and this is a increasing function ofm. These twocomponents compensate each other. Thus the full LO csection in the VFN ACOT scheme has a moderatem depen-dence.

The situation is different for the FFN scheme, shownFig. 10. Here one finds that both the LO-FFN and NLO-FFresults ~proportional to light parton distributions! are de-creasing functions ofm, resulting in a steepm dependencefor the combined result, TOT-FFN. If we were to compuhigher order corrections in the FFN scheme, we would evtually observe compensating terms to reduce them depen-dence~as we know the ‘‘all-orders’’ result must be indepedent ofm). The corrections to the FFN scheme result whihave been resummed in the ACOT scheme into HE andcontributions~minus subtractions! represent a part of theshigher-order effect.

In Fig. 11 we compare directly the TOT-FFN and thTOT-VFN results. The TOT-VFN result ranges from;5%,ap-

eed

15The HC1 diagram, of course, contains a lot more informationdetailed differential distributions~such as non-back-to-back jets!which is not contained in the HE1 diagram.

6-11

N

.y

ch, b

eudat

esfo

e,

gerf

FFN

elts.en-ableun-vy-

tbe

ionvi-ef.

her

-also

n-

s

n

-en

-


to ;25% above the TOT-FFN result form in the rangeMT/2 to 2MT . In the plot, the lower range ofm has beenextended belowMT/2 (.10 GeV, for pT520 GeV) toverify that the TOT-VFN result reduces to that of TOT-FFfor m5mH55 GeV. The additional resummed terms~la-beledDifference! is seen to improve them dependence in theregionm;MT . ~See also Fig. 13 below for LHC energies!

Figure 12 displays the scaled cross sections vs the phcal variablepT in both the VFN and FFN schemes. In eacase, the scale uncertainty is represented, as customarytheory band bounded by an upper curve correspondingm5MT/2 and a lower one tom52MT . We see the increasin the central value of the cross section as well as the redtion in the scale dependence in the VFN scheme comparethe FFN scheme. The improvement is however not dramin either case.

In the low pT region, one might expect the two schemto coincide. This happens for the upper curves, but not

FIG. 8. Comparison with experimentalb-production data at theTevatronAs51800 GeV,uyu,1 @27,28#. The dashed lines represent them-variation in the FFN scheme, and solid lines represthem-variation in the VFN scheme. To gauge them scale variation,we choosem5MT/2 for upper curves andm52MT for lowercurves.

FIG. 9. Scale (m in GeV! dependence of the leading (as2) order

contributions to the cross sectionds/dpT2/dyuy50 (nb GeV22) for

b production at 1800 GeV withpT520 GeV in the ACOT formal-ism.

01450

si-

y ato

c-toic

r

the lower ones. The reason is that, atpT510 GeV, the scaleparameterm5MT/2 ~upper curves! is A102152/2 GeV.5 GeV.mH , at which the two schemes do agrewhereasm52MT ~lower curves! is .10 GeV.2mH atwhich the difference between the two schemes is no lonnegligible; cf. Fig. 11. A close look at them dependences othe additional terms of the VFN scheme~similar to that de-scribed for thepT520 case in Figs. 9,10! easily explainswhy the scale dependence in reduced compared to theresult in this region.

In the largepT region, one would expect the VFN schemto produce the most improvement over the FFN resuHowever, Fig. 12 indicates that the reduction in scale depdence turns out to be rather marginal. We have not beento pinpoint the reason for this. One possible source ofcompensatedm dependence is the next higher order heaflavor excitation contribution NLO-HE~HE2!, e.g., the sub-processgH→gHg, which is not yet included in the currencalculation, although one may regard this contribution toof one effective order higher inas than HC1~because of theinitial state heavy quark parton distribution!, the large size ofthe HE1 term may be an indication that the HE2 contributis in fact needed to achieve full consistency. Supporting edence for this explanation is provided by a related work, R@25#, discussed below, which includes this contribution~butleaves out mass effects!. The full HE2 contribution, includ-ing heavy quark mass effects, is anO(as

3) process which hasnot yet been computed.

One expects the VFN results to get even better at higenergies. Figure 13 shows the inclusivepT cross-section atthe LHC energy ofAs514 TeV. We see the same qualitative features as seen in Fig. 12. For completeness, wepresent results forb-production at the CERN SppS energy ofAs5630 GeV in Fig. 14.

In the preceding, we have uniformly omitted the last covolution with dH

H for simplicity of calculation; however,there are some subtleties regarding this effect. WeredH

H(z,m)independent ofm, the fragmentation would shift all termuniformly. While dH

H(z,m) has only a logarithmicm-dependence~recall this is roughly a delta function plus a

t

FIG. 10. Scale (m in GeV! dependence of the main contributions to the cross sectionds/dpT

2/dyuy50 (nb GeV22) for b pro-duction at 1800 GeV withpT520 GeV organized in the FFNscheme.

6-12

e

t

th

Thebl

ti-ce

eherea.dityd

edi-onig.he

a-u-

by

on

on

n

n

n

me


order as logarithmic term!, the steep behavior of thpT-distribution can magnify the effect of thism-variation.

We display the finalpT-distribution convolved withdHH in

Fig. 15. We observe that the shift in the curves~as comparedwith Fig. 12! is minimal at lowpT wheredH

H is close to adelta-function. At the highestpT , the effect is more pro-nounced because the softerdH

H(z,m52MT) distributionyields a greater reduction to the highpT cross section thanthe harderdH

H(z,m51/2MT). The net effect is that the lowespT region is largely unchanged, but the spread of them51/2MT andm52MT curves in the highestpT region willincrease by;15%. While the effect ofdH

H(z,m) is in prin-ciple a higher order contribution, this example illustratespotential effects of large logarithms.

C. Rapidity distribution

Although the inclusive HE1 contribution in the ACOformalism was roughly comparable to the HC1 term in tFFN schemes, the dependence on the individual varia

FIG. 11. Comparison of the total cross sectids/dpT

2/dyuy50 (nb GeV22) in the ACOT and FFN formalisms vsm in GeV for b production at 1800 GeV withpT520 GeV. TheDifferencecurve represents the additional resummed contributiincluded in the ACOT result.

FIG. 12. Variation of the total cross sectiopT

5ds/dpT2/dyuy50 (nb GeV3) in the ACOT and FFN formalisms

vs pT for b production at 1800 GeV. To gauge them scale variation,we choosem5MT/2 for upper curves, andm52MT for lowercurves.

01450

e

es

(s,pT ,m,y) can be quite different. Having already invesgated them-dependence, we turn to the rapidity dependenof the underlying processes.

In Fig. 16 we compare the rapidity distribution for thHE1 and the HCO” processes. To more easily compare trelative shape, we have scaled the two curves to equal aWe observe that the HE1 process yields a broader rapidistribution than the HCO” processes. In part, this is expecteas the HE1 process includes at-channel gluon exchangwhich can give an enhanced contribution in the forwardrection. To see the relative effect on the rapidity distributifor the complete next-to-leading order calculations, in F17 we display the ratio of the TOT-VFN compared to tTOT-FFN cross section as a function of the rapidity.

D. Comment on related work

It is worth mentioning that the resummation of large logrithms in the fixed-order calculations into parton distribtions and fragmentation functions has also been studiedCacciari and Greco@25#. In the notation of Eq.~2!, their

s



vs pT for b production at 14 TeV. To gauge them scale variation,we choosem5MT/2 for upper curves, andm52MT for lowercurves.



vs pT for b production at 630 GeV. The curves band the extrerange of the cross sections as the scale is varied fromm5MT/2 tom52MT .

6-13

th

g

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approach amounts to adopting the following ansatz forhadro-production cross section:

sABHX5 (

a,b,cf $AB%

$ab%^ sab

c,r~mH50! ^ dcH ~17!

where (a,b,c) are summed over all parton flavors includinH, and sab

c,r(mH50) is given by the zero-mass~i.e. light-parton! NLO jet cross section calculation. This is a gooapproximation in the asymptotic regionpT@mH , but it doesnot reproduce the right physics whenpT is not significantlylarger than the quark mass. The main result of their calction was that the predictedb-production cross section hanoticeably less scale dependence than the FFN schemsult, but it lies within the uncertainty band of the latteHence their predictions lie substantially below the expemental measurement, like the FFN results, and somewbelow our results. Our theory resums the same large lorithms associated with final state collinear singularities. Tadvantages of our treatment are that~i! we treat initial stateparton distributions and final state fragmentations ofheavy quark symmetrically;~ii ! by keeping the heavy quar

FIG. 15. Same as for the TOT-VFN result in Fig. 12, but wthe final convolution ofdH

H included.

FIG. 16. Comparison of the rapidity~y! distribution for the HCO”and HE1 processes forb production at 1800 GeV withpT

520 GeV andm5MT . The curves are scaled to equal areafacilitate comparison of the shapes.

01450

e

a-

re-

-ata-e

e

mass in the hard cross section according to the generaltorization theorem, our results are, in principle, applicaover the entire energy range. On the other hand, by usingorder as

3 jet cross section in Eq.~17!, Ref. @25# includeshigher order corrections to the corresponding terms incalculation@HE1, GF1 and HF1 terms in Eq.~3!#. Some ofthese higher-order terms are already shown to be small aorder we calculated~GF1 and HF1!, and hence should nomake any difference. However, as mentioned earlier,clear improvement in scale dependence of this calculatiolarge pT over both the FFN and our general mass VFNsults, plus our finding that the HE1 term is numerically largthan expected, is indicative that the orderas

3 heavy-flavorexcitation contribution~HE2 in our notation! can play a non-negligible role overall. To be sure about this point, a mocomplete calculation in the general mass VFN schemecluding the HE2 contribution needs to be carried out~see‘‘note added’’ below!.

V. CONCLUDING REMARKS

In this paper, we have systematically developedtheory for hadro-production of heavy quarks according tonatural PQCD scheme of Refs.@13,15,2,20# which general-izes the conventional~zero-mass! ‘‘improved QCD partonmodel’’ to include quark mass effects. This formalism hthe advantage that it contains the correct physics over a wrange of energies andQ (pT), and it reduces to the conventional results both at the low and the high energy limits.particular, it coincides with the widely used FFN schemeits natural region of applicability whereQ is or of the sameorder of magnitude asmH—the one-large-scale region. Improvement over the FFN scheme results become imporwhenQ@mH . This is manifested both in the reduced theretical uncertainty and in the increased cross section.improvement comes at the price of somewhat more comcated calculations. As seen in Eq.~13! and Fig. 4, in additionto the FFN scheme contributions, one needs to computeother terms involving flavor excitation and fragmentati

FIG. 17. Comparison of the total cross sectionds/dpT2/dy in

the ACOT and FFN formalism vsy for b production at 1800 GeVwith pT520 GeV andm5MT . The curves are scaled by the totFFN cross section to facilitate comparison of the relative magtude.

6-14

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processes and their subtractions. As emphasized in Secthese calculations must be done with care, due to the delicancellations required.

This more complete and consistent theory is, however,a cure-for-all. Within our scheme, the residue scale depdence seen in Sec. IV may suggest that certain nnegligible higher order terms still need to be included. Ocross section predictions, although higher than the exisFFN ones, still fall somewhat short of theb cross sectionmeasured at the Tevatron. Besides, the scale dependenlarge pT is still too large to have full confidence in the improved results. Some physics effects not included in ourmalism could be important: notably, higher-order heavflavor excitation contributions and those related to lalogarithms of the type ln(Q2/s),ln(mH

2 /s);ln x which needs tobe separately resummed. The latter is another example osmall-x problem@17#. In this paper, we also have not adressed questions concerning the hadronization of the hquark which is not yet fully understood.

As an important physical process involving the interplof several large scales, heavy quark production poses anificant challenge for further development of QCD theory

Following the completion of this work, we learned ofrelated calculation by Cacciari, Greco, and Nason@29# which

in,

g,

r-.

g,

n,d

d.

d.

h,-

01450

III,te

otn-n-rg

e at

r--e

he

vy

ig-

merges the fragmentation formalism of Ref.@25# with themassive NLO FFN calculation of Refs.@10,11#. The result isto obtain significantly improved renormalization and factoization scale dependence in the largepT region, and the cal-culation reduces to the NLO FFN calculation in the smallpTregion. We also thank the authors for private communitions detailing this result.

ACKNOWLEDGMENTS

We are indebted to J. C. Collins for his participationthe early stages of this work, and for his insightful formultion of the factorization scheme that we have applied tohadro-production process. We would also like to thank R.Ellis, B. Harris, M. L. Mangano, P. Nason, J. Smith, andE. Soper for useful discussions. We are indebted to B. Haand J. Smith, and to M. L. Mangano, P. Nason, andRidolfi for the use of theirFORTRAN code in calculating theorder as

3 fixed-flavor-number scheme cross sections. F.Ithanks the Fermilab Theory Group for their kind hospitalduring the period in which this research was carried out. Twork is partially supported by the National Science Fountion, the U.S. Department of Energy, and the Lightner-SaFoundation.

D

t..

w-

ett.

@1# S. Frixione, M. L. Mangano, P. Nason, and G. Ridolfi,‘‘Heavy Flavours II,’’ edited by A. J. Buras and M. Lindnerhep-ph/9702287 ~unpublished!; M. L. Mangano,hep-ph/9711337.

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50, 3085~1994!.@16# J. Collins, F. Wilczek, and A. Zee, Phys. Rev. D18, 242

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Heavy quark hadroproduction in perturbative QCD