PQCD Formulations with Heavy Quark Masses and Global Analysis

0.25 MB·English

by  R. S. Thorne

PQCD Formulations with Heavy Quark Masses and Global Analysis∗
RobertSThorne1 andW.K.Tung2.
1 UniversityCollegeLondon, 2 MichiganStateUniversityandUniversityofWashington
Abstract
WecriticallyreviewheavyquarkmasseffectsinDISandtheirimpactonglobal
9
analyses. Welay out all elements ofa properly defined general mass variable
0
flavornumberscheme(GMVFNS)thataresharedbyallmodernformulations
0
2 of the problem. We then explain the freedom in choosing specific implemen-

n tations and spell out, in particular, the current formulations of the CTEQ and
a MSTW groups. We clarify the approximations in the still widely-used zero
J
mass variable flavor scheme (ZM VFNS), mention the inherent flaws in its
1
2 conventional implementation,andconsiderthepossibilityofmendingsomeof
these flaws. Wediscuss practical issues concerning the useofparton distribu-

]
h tions in various physical applications, in view of the different schemes. And
p wecommentonthepossiblepresence ofintrinsicheavyflavors.
-
p
e
h Contents
[

2 1 Introduction 2
v
4
2 GeneralConsiderationsonPQCDwithHeavyFlavorQuarks 2
1
7 2.1 TheFactorization Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0
. 2.2 PartonsandSchemesforGeneralMassPQCD . . . . . . . . . . . . . . . . . . . . . . . 3
9
0 2.3 TreatmentofFinal-state Flavors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
8
2.4 Phase-space Constraints andRescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
0
v: 2.5 Difference between{Ftot}and{FH}StructureFunctions . . . . . . . . . . . . . . . . 6
λ λ
i
X 2.6 Conventions for“LO”,“NLO”,... Calculations . . . . . . . . . . . . . . . . . . . . . . 7
r
a
3 ImplementationsofVFNS:CommonFeaturesandDifferences 7
3.1 Alternative FormulationsoftheZMVFNS . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 PartonDistribution FunctionsinVFNS(ZMandGM) . . . . . . . . . . . . . . . . . . . 8
3.3 TheStructureofaGMVFNS,MinimalPrescription andAdditional Freedom . . . . . . 8
3.4 CTEQImplementation oftheGMVFNS . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.5 MRST/MSTWImplementation oftheGMVFNS . . . . . . . . . . . . . . . . . . . . . 10
3.5.1 Prescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.5.2 Schemevariations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.6 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 UseofPartonDistributionFunctions 15
5 IntrinsicHeavyFlavour 17
∗ContributiontotheHERA-LHCWorkshopProceedings
1 Introduction
The proper treatment of heavy flavours in global QCD analysis of parton distribution functions (PDFs)
isessentialforprecisionmeasurements athadroncolliders. Recentstudies[1–4]showthatthestandard-
candle cross sections for W/Z production at the LHC are sensitive to detailed features of PDFs that
depend onheavy quark masseffects; andcertain standard model aswellasbeyond standard modelpro-
cessesdependcruciallyonbetterknowledgeofthec-quarkpartondensity, inadditiontothelightparton
flavors. These studies also make it clear that the consistent treatment of heavy flavours in perturba-
tiveQCD(PQCD)requiretheoretical considerations thatgobeyondthefamiliartextbookpartonpicture
basedonmasslessquarks andgluons. Therearevarious choices, explicit andimplicit, whichneed tobe
madeinvariousstagesofapropercalculation ingeneralised PQCDincluding heavyquarkmasseffects.
In the global analysis of PDFs, these choices can affect the resulting parton distributions. Consistent
choices are imperative; mistakes may result in differences that are similar to, or even greater than, the
quoted uncertainties due to other sources (such asthe propagation of input experimental errors). Inthis
report, wewillprovide abrief, but full, review of issues related to the treatment ofheavy quark masses
inPQCD,embodied inthegeneralmassvariableflavorscheme(GMVFNS).
In Sec.2, we describe the basic features of the modern PQCD formalism incorporating heavy
quarkmasses. InSec.3,wefirstdelineatethecommonfeaturesofGMVFNS,thenidentifythedifferent
(but self-consistent) choices that have been made in recent global analysis work, and compare their
results. For readers interested in practical issues relating to the use (or choice) of PDFs in various
physics applications, we present a series of comments in Sec. 4 intended as guidelines. In Sec. 5, we
discussthepossibility ofintrinsic heavyflavors.
Wenotethat,thisreviewonGMVFNSandglobalanalysis isnotintended toaddress thespecific
issues pertinent to heavy flavor production (especially the final state distributions). For this particular
process, somewhat different considerations may favor the adoption of appropriate fixed flavor number
schemes (FFNS).Weshall not go into details of these considerations; but willmention theFFNSalong
theway,sincetheGMVFNSisbuiltonaseries ofFFNS’s. Wewillcommentonthisintimate relation-
shipwheneverappropriate.
2 GeneralConsiderationsonPQCDwithHeavyFlavorQuarks
Thequark-parton picture isbased onthefactorization theorem ofPQCD.Theconventional proof ofthe
factorization theorem proceeds from the zero-mass limit for all the partons—a good approximation at
energy scales (generically designated by Q) far above all quark mass thresholds (designated by m ).
i
This clearly does not hold when Q/m is of order 1.1 It has been recognised since the mid-1980’s that
i
a consistent treatment of heavy quarks in PQCD over the full energy range from Q . m to Q ≫ m
i i
canbeformulated[5]. In1998,Collinsgaveageneralproofofthefactorizationtheorem(order-by-order
to all orders of perturbation theory) that is valid for non-zero quark masses [6]. The resulting general
theoretical framework is conceptually simple: it represents a straightforward generalisation of the con-
ventionalzero-mass(ZM)modifiedminimalsubtraction(MS)formalismanditcontainstheconventional
approaches asspecial cases intheirrespective regions ofapplicability; thus, itprovides agood basis for
ourdiscussions.
The implementation of any PQCD calculation on physical cross sections requires attention to a
numberofdetails, bothkinematical anddynamical, thatcanaffectboththereliability ofthepredictions.
Physical considerations are important to ensure that the right choices are made between perturbatively
equivalent alternatives that may produce noticeable differences in practical applications. It is important
tomaketheseconsiderations explicit,inordertomakesenseofthecomparisonbetweendifferentcalcu-
lations in the literature. Thisis what weshall do in this section. In subsequent sections, weshall point
outthedifferentchoices thathavebeenmadeinrecentglobalanalysis efforts.
1Heavyquarks,bydefinition,havemi≫ΛQCD. HencewealwaysassumeQ,mi ≫ΛQCD.Inpractice,i=c,b,t.
Heavy quark physics at HERAinvolve mostly charm (c) and bottom (b) production; atLHC,top
(t)production, inaddition, isofinterest. Forsimplicity, weoften focus the discussion ofthetheoretical
issues on the production of a single heavy quark flavor, which we shall denote generically as H, with
mass m . The considerations apply to all three cases, H = c, b, & t. For global analysis, the most
H
important process that requires precision calculation is DIS; hence, for physical predictions, we will
explicitly discuss the total inclusive and semi-inclusive structure functions, generically referred to as
Fλ(x,Q), where λ represents either the conventional label (1,2,3) or the alternative (T,L,3) where
T/Lstandsfortransverse/longitudinal respectively.
2.1 TheFactorization Formula
ThePQCDfactorization theoremfortheDISstructure functions hasthegeneralform
1 dξ χ Q m
F (x,Q2)= f ⊗Cλ = f (ξ,µ)Cλ , , i,α (µ) . (1)
λ k k ξ k k ξ µ µ s
k k Zχ (cid:18) (cid:19)
X X
Here, the summation is over the active parton flavor label k, fk(x,µ) are the parton distributions at the
factorizationscaleµ,CλaretheWilsoncoefficients(orhard-scatteringamplitudes)thatcanbecalculated
k
order-by-order in perturbation theory. The lower limit of the convolution integral χ is determined by
final-statephase-spaceconstraints: intheconventionalZMpartonformalismitissimplyx= Q2/2q·p—
the Bjorken x—but this is no longer true when heavy flavor particles are produced in the final state,
cf. Sec.2.4 below. The renormalization and factorization scales are jointly represented by µ: in most
applications, it is convenient to choose µ = Q; but there are circumstances in which a different choice
becomesuseful.
2.2 PartonsandSchemesforGeneralMassPQCD
InPQCD,the summation over“parton flavor” label k inthefactorization formula, Eq.(1),isdeter-
k
minedbythefactorization schemechosentodefinethePartonDistributions f (x,µ).
k
P
IfmasseffectsofaheavyquarkH aretobetakenintoaccount,thesimplestschemetoadoptisthe
fixedflavornumberscheme(FFNS)inwhichallquarkflavorsbelowH aretreatedaszero-massandone
sumsoverk = g,u,u¯,d,d¯,...upton flavors oflight (massless) quarks. ThemassofH,m ,appears
f H
explicitly in the Wilson coefficients {Cλ}, as indicated in Eq.1. For H = {c,b,t}, n = {3,4,5}
k f
respectively. Historically, higher-order (O(α2))calculations ofthe heavy quark production [7] wereall
S
donefirstintheFFNS.Thesecalculations provide muchimprovedresults whenµ(Q)isoftheorder of
m (bothaboveandbelow),overthoseoftheconventionalZMones(correspondingtosettingm = 0).
H H
Unfortunately, atanyfiniteorderinperturbativecalculation, then -FFNSresultsbecomeincreas-
f
ingly unreliable as Q becomes large compared to m : the Wilson coefficients contain logarithm terms
H
oftheformαnlnm(Q/m ),wherem = 1...n,atordernoftheperturbativeexpansion, implyingthey
s H
are not infrared safe—higher order terms do not diminish in size compared to lower order ones—the
perturbative expansion eventually breaks down. Thus, even if all n -flavor FFNS are mathematically
f
equivalent, in practice, the 3-flavor scheme yields the most reliable results in the region Q . m , the
c
4-flavor scheme in m . Q . m , the 5-flavor scheme in m . Q . m , and, if needed, the 6-flavor
c b b t
schemeinm .Q. (Cf.relateddiscussions laterinthissection.)
t
Thisleadsnaturally tothedefinition ofthemoregeneral variable flavornumberscheme(VFNS):
it is a composite scheme consists of the sequence of n -flavor FFNS, each in its region of validity,
f
for n = 3,4,.. as described above; and the various n -flavor schemes are related to each other by
f f
perturbatively calculable transformation (finite-renormalization) matrices amongthe(running) coupling
α ,therunningmasses{m },thepartondistribution functions{f },andtheWilsoncoefficients{Cλ}.
s H k k
Theserelations ensure thatthereareonlyonesetofindependent renormalization constants, hence make
the definition ofthe composite scheme precise forall energy scale µ(Q); and they ensure that physical
predictionsarewell-definedandcontinuousastheenergyscaletraverseseachoftheoverlappingregions
Q ∼ m where both the n -flavor and the (n + 1)-flavor schemes are applicable. The theoretical
H f f
foundation forthisintuitively obviousschemecanbefoundin[5,6],anditwasfirstappliedindetailfor
structure functions in [8]. Most recent work on heavy quark physics adopt this general picture, in one
formoranother. Weshallmentionsomecommonfeaturesofthisgeneral-mass (GM)VFNSinthenext
few paragraphs; and defer the specifics on the implementation of this scheme, as well as the variations
intheimplementation allowedbythegeneral frameworkuntilSec.3.
Asmentionedabove,then -flavorandthe(n +1)-flavorschemeswithintheGMVFNSshould
f f
bematched atsomematchpoint µ thatisoftheorderofm . Inpractice, thematching iscommonly
M H
chosen to be exactly µ = m , since it has been known that, in the calculational scheme appropriate
M H
for GM VFNS2, the transformation matrices vanish at this particular scale at NLO in the perturbative
expansion [5]; thus discontinuities of the renormalized quantities are always of higher order, making
practical calculations simpleringeneral.
Strictly speaking, once the component n -flavor schemes are unambiguously matched, one can
f
still choose an independent transition scale, µ , at which to switch from the n -flavor scheme to the
T f
(n +1)-flavor scheme in the calculation of physical quantities in defining the GM VFNS. This scale
f
mustagainbewithintheoverlappingregion,butcanbedifferentfromµ [1,6]. Infact,itiscommonly
M
known that, from the physics point of view, in the region above the m threshold, up to ηm with
H H
a reasonable-sized constant factor η, the most natural parton picture is that of n -flavor, rather than
f
(n + 1)-flavor one.3 For instance, the 3-flavor scheme calculation has been favored by most HERA
f
work on charm and bottom quark production, even if the HERADIS kinematic region mostly involves
Q > m ;anditisalsousedinthedynamically generated partonapproach toglobalanalysis[14].
c
Inpractice, almost allimplementations ofthe GMVFNSsimplychoose µ = µ = m (often
T M H
not explicitly mentioning the conceptual distinction between µ and µ = m ). Theself-consistency
T M H
oftheGMVFNSguaranteesthatphysicalpredictionsareratherinsensitivetothechoiceofthetransition
point as long as itis within the overlapping region of validity of the n -and (n +1)-flavor ones. The
f f
simplechoiceofµ = m corresponds tooptingforthelowerendofthisregionfortheconvenience in
T H
implementation. Inthefollowing,weshallusethetermsmatchingpointandtransitionpointinterchange-
ably. As with all definition ambiguities in perturbative theory, the sensitivity to the choice of matching
andtransition pointsdiminishes athigherorders.
2.3 TreatmentofFinal-stateFlavors
Fortotalinclusivestructure functions, thefactorization formula,Eq.(1),contains animplicitsummation
overallpossible quarkflavorsinthefinalstate. Onecanwrite,
C = Cj (2)
k k
j
X
where “j” denotes final state flavors, and {Cj} represent the Wilson coefficients (hard cross sections)
k
for an incoming parton “k” to produce a final state containing flavor “j” calculable perturbatively from
the relevant Feynman diagrams. It is important to emphasize that “j” labels quark flavors that can be
producedphysicallyinthefinalstate;itisnotapartonlabelinthesenseofinitial-statepartonflavorsde-
scribedintheprevioussubsection. Thelatter(labeledk)isatheoreticalconstructandscheme-dependent
2Technically, thismeansemploying theCWZsubtraction scheme [9]incalculating thehigher-order Feynman diagrams.
CWZsubtractionisanelegantextensionof theMSsubtractionschemethatensuresthedecoupling ofheavyquarksathigh
energyscalesorder-by-order. Thisisessentialforfactorizationtobevalidateachorderofperturbationtheory. (Intheoriginal
MSsubtractionscheme,decouplingissatisfiedonlyforthefullperturbationseries—toinfiniteorders.)
3Specifically,thenf-flavorschemeshouldfailwhenαs(µ)ln(µ/mH) = αs(µ)ln(η)ceasestobeasmallparameterfor
theeffectiveperturbationexpansion. However, notheorycantelluspreciselyhow smallisacceptably “small”—hencehow
largeηispermitted. ArdentFFNSadvocatesbelieveeventherangeofthe3-flavorschemeextendstoallcurrentlyavailable
energies,includingHERA[14].ForGMVFNS,seethenextparagraph.
(e.g.it is fixed at three for the 3-flavor scheme); whereas the final-state sum (over j) is over all flavors
that can be physically produced. Furthermore, the initial state parton “k” does not have to be on the
mass-shell, and is commonly treated as massless; whereas the final state particles “j” should certainly
be on-mass-shell in order to satisfy the correct kinematic constraints for the final state phase space and
yield physically meaningful results.4 Thus, in implementing the summation over final states, the most
relevantphysicalscaleisW—theCMenergyofthevirtualComptonprocess—incontrasttothescaleQ
thatcontrols theinitialstatesummationoverpartonflavors.
Thedistinction between thetwosummations isabsent inthesimplest implementation ofthecon-
ventional (i.e., textbook) zero-mass parton formalism: if all quark masses are set to zero to begin with,
then all flavors can be produced in the final state. This distinction becomes blurred in the commonly
usedzero-mass (ZM)VFNS,wheretheheavyquark masses{m }implicitly enterbecause thenumber
H
of effective parton flavors is incremented as the scale parameter µ crosses each heavy quark threshold.
This creates apparent paradoxes in the implementation of the ZM VFNS, such as: for µ = Q < m , b
b
is not counted as a parton, the partonic process γ +g → b¯b would not be included in DIS calculations,
yet physically this can be significant if W ≫ 2m (small x); whereas for µ = Q > m , b is counted
b b
as amassless parton, the contribution of γ +g → b¯b toDIS would be the same asthat ofγ +g → dd¯,
but physically this is wrong for moderate values ofW, and furthermore, itshould be zero if W < 2m
b
(corresponding tolargex). (WeshallreturntothistopicinSec.3.1.)
Theseproblemswerecertainlyoverlookedinconventional globalanalysesfromitsinceptionuntil
thetimewhenissuesonmass-effects inPQCDwerebrought totheforeafterthemid1990’s [8,10–13].
Since then, despite its shortcomings the standard ZM VFNScontinues to be used widely because of its
simplicity and because NLO Wilson coefficients for most physical processes are still only available in
the ZM VFNS.Most groups produce the standard ZM VFNSas either their default set or as one of the
options, and they form the most common basis for comparison between groups, e.g. the “benchmark
study”in[15].
Itisobviousthat,inaproperimplementation ofPQCDwithmass(inanyscheme),thedistinction
betweentheinitial-stateandfinal-statesummationmustbeunambiguously, andcorrectly,observed. For
instance, even in the 3-flavor regime (when c and b quarks are not counted as partons), the charm and
bottomflavorsstillneedtobecounted inthefinalstate—attree-levelviaW++d/s → c,andat1-loop
level via the gluon-fusion processes such as W+ +g → s¯+c or γ + g → cc¯(b¯b), provided there is
enoughCMenergytoproduce theseparticles.
2.4 Phase-spaceConstraintsandRescaling
Theabovediscussionpointstotheimportanceofthepropertreatmentoffinalstatephasespaceinheavy
quark calculations. Once mass effects are taken into account, kinematic constraints have a significant
impact on the numerical results of the calculation; in fact, they represent the dominant factor in the
threshold regions ofthe phase space. InDIS,withheavy flavor produced inthefinalstate, thesimplest
kinematicconstraint thatcomestomindis
W −M > M (3)
N f
f
X
where W is the CM energy of the vector-boson–nucleon scattering process, M is the nucleon mass,
N
andthe right-hand sideisthe sumofall massesinthe finalstate. W isrelated tothefamiliar kinematic
variables (x,Q) by W2 − M2 = Q2(1 − x)/x, and this constraint should ideally be imposed on the
N
right-handsideofEq.(1). Anyapproachachievingthisrepresentsanimprovementovertheconventional
ZMschemecalculations, thatignoresthekinematicconstraint Eq.(3)(resulting inagrossover-estimate
4Strict kinematics would require putting the produced heavy flavor mesons or baryons on the mass shell. In the PQCD
formalism,weadopttheapproximationofusingon-shellfinalstateheavyquarksintheunderlyingpartonicprocess.
ofthecorresponding crosssections). Theimplementation oftheconstraint inthemostusualcaseofNC
processes, say γ/Z +c → c (or any other heavy quark) is not automatic (and is absent in some earlier
definitions of a GM VFNS) because in this partonic process one must account for the existence of a
hidden heavy particle—the c¯—in the target fragment. The key observation is, heavy objects buried in
thetargetfragmentarestillapartofthefinalstate,andshouldbeincludedinthephasespaceconstraint,
Eq.(3).
Earlyattemptstoaddress thisissuewereeitherapproximate orrathercumbersome, andcouldnot
benaturally extendedtohighorders.5 Amuchbetterphysicallymotivatedapproachisbasedontheidea
of rescaling. The simplest example is given by charm production in the LO CC process W +s → c.
Itis well-known that, when the final state charm quark is put onthe mass shell, kinematics requires the
momentumfractionvariablefortheincomingstrangeparton,χinEq.(1)tobeχ = x(1+m2/Q2)[16],
c
rather than the Bjorken x. This is commonly called the rescaling variable. The generalization of this
ideatothemoreprevalent caseofNCprocesses tookalongtimetoemerge[17,18] whichextended the
simple rescaling tothe more general case of γ/Z +c → c+X, where X contains only light particles,
it was proposed that the convolution integral in Eq.(1) should be over the momentum fraction range
χ < ξ < 1,where
c
4m2
χ = x 1+ c . (4)
c Q2
(cid:18) (cid:19)
Inthemostgeneralcasewherethereareanynumberofheavyparticlesinthefinalstate,thecorrespond-
ingvariableis(cf.Eq.(3))
(Σ M )2
f f
χ = x 1+ . (5)
Q2
!
Thisrescaling prescription hasbeenreferredtoasACOTχintherecentliterature [17–19].
Rescaling shifts the momentum variable in the parton distribution function fk(ξ,µ) in Eq.(1) to
a higher value than in the zero-mass case. For instance, at LO, the structure functions F (x,Q) are
λ
given by some linear combination of fk(x,Q) in the ZM formalism; but, with ACOTχ rescaling, this
becomes fk(χ ,Q). In the region where (Σ M )2/Q2 is not too small, especially when f(ξ,µ) is a
c f f
steep function of ξ, this rescaling can substantially change the numerical result of the calculation. It
is straightforward to show that, when one approaches a given threshold (M + Σ M ) from above,
N f f
the corresponding rescaling variable χ → 1. Since generally fk(ξ,µ) −→ 0 as ξ → 1, rescaling
ensures a smoothly vanishing threshold behavior for the contribution of the heavy quark production
term to all structure functions. This results in a universal6, and intuitively physical, realization of the
thresholdkinematicconstraint forallheavyflavorproduction processesthatisapplicable toallordersof
perturbation theory. Forthisreason, mostrecentglobalanalysis effortschoosethismethod.
2.5 Differencebetween{Ftot}and{FH}StructureFunctions
λ λ
In PQCD, the most reliable calculations are those involving infra-red safe quantities—these are free
from logarithmic factors that can become large (thereby spoiling the perturbative expansion). The total
inclusive structure functions {Ftot} defined in the GM VFNS are infrared safe, as suggested by the
λ
discussion ofSec.2.2andproveninRef. [6].
Experimentally, the semi-inclusive DIS structure functions for producing a heavy flavor particle
inthefinalstateisalsoofinterest. Theoretically,itisusefultonotethatthestructurefunctions{FH}for
λ
5In[8],thethresholdviolationwasminimizedbyanartificialchoiceofthefactorizationscaleµ(mH,Q). In[12,13]the
kinematic limit was enforced exactly by requiring continuity of the slope of structure functions across the matching point,
resultinginarathercomplicatedexpressionforthecoefficientfunctionsinEq.(1).
6Sinceitisimposedonthe(universal)partondistributionfunctionpartofthefactorizationformula.
producing heavyflavorH arenotaswelldefinedasFtot.7 Toseethis,considertherelationbetweenthe
λ
two,
Ftot = Flight+FH , (6)
λ λ λ
where Flight denotes the sum of terms withonly light quarks inthe finalstate, and FH consist of terms
λ λ
withatleastoneheavyquarkH inthefinalstate. Unfortunately,FH(x,Q,m )is,strictlyspeaking,not
λ H
infraredsafebeyondorderα (1-loop): theycontainresiduallnn(Q/m )termsathigherorders(2-loop
s H
and up). The same terms occur in Flight due to contributions from virtual H loops, with the opposite
λ
sign. Only the sum of the two, i.e.the total inclusive quantities Ftot are infra-red safe. This problem
λ
could be addressed properly by adopting a physically motivated, infrared-safe cut-off on the invariant
massoftheheavyquarkpair,corresponding tosomeexperimentalthreshold[20]inthedefinitionofFH
λ
(drawing onsimilar practises injetphysics). Inpractice, uptoorder α2,theresult isnumerically rather
s
insensitive to this, and different groups adopt a variety of less sophisticated procedures, e.g. including
contributions with virtual H loops within the definition of FH. Nonetheless, it is prudent to be aware
λ
that the theoretical predictions on FH are intrinsically less robust than those for Ftot when comparing
λ λ
experimental resultswiththeorycalculations.
2.6 Conventionsfor“LO”,“NLO”,... Calculations
It is also useful to point out that, in PQCD,the use of familiar terms such as LO,NLO,... is often am-
biguous, depending onwhichtypeofphysical quantities areunder consideration, andontheconvention
usedbytheauthors. Thiscanbeasourceofconsiderable confusion whenonecomparesthecalculations
ofFtot andFH bydifferentgroups (cf.nextsection).
λ λ
One common convention is to refer LO results as those derived from tree diagrams; NLO those
from 1-loop calculations, ... and so on. This convention is widely used; and it is also the one used in
the CTEQ papers. Another possible convention is to refer to LO results as the first non-zero term in
the perturbative expansion; NLOasone order higher inα , ... and so on. This convention originated in
s
FFNS calculations of heavy quark production; and it is also used by the MRST/MSTWauthors. It is a
process-dependent convention, and it depends apriori on the knowledge of results of the calculation to
thefirstcoupleofordersinα .
s
Whereas the twoconventions coincide for quantities such as Ftot; they lead todifferent designa-
2
tions forthelongitudinal structure function Ftot andthen -flavor FH,nf,sincethetree-level results are
L f 2
zeroforthesequantities. Thesedesignations, bythemselves,areonlyamatterofterminology. However,
mixing the two distinct terminologies in comparing results of different groups can be truly confusing.
Thiswillbecomeobviouslater.
3 ImplementationsofVFNS:CommonFeaturesandDifferences
In this section, we provide some details of the PQCD basis for the GM VFNS, and comment on the
differentchoicesthathavebeenmadeinthevariousversionsofthisgeneralframework,implementedby
twoofthemajorgroupsperforming globalQCDanalysis.
3.1 AlternativeFormulationsoftheZMVFNS
Aspointed out in Sec.2.3, the ZMVFNS,ascommonly implemented, represents anunreliable approx-
imation to the correct PQCDin some kinematic regions because of inappropriate handling of the final-
state counting and phase-space treatment, in addition to the neglect of heavy-quark mass terms in the
7Inthefollowingdiscussion, weshall overlooklogarithmicfactorsnormallyassociated withfragmentationfunctionsfor
simplicity. Thesearesimilartothoseassociatedwithpartondistributions,butarelessunderstoodfromthetheoreticalpointof
view—e.g.thegeneralproofoffactorizationtheorem(withmass)[6]hasnotyetbeenextendedtocoverfragmentation.
Wilson coefficients. Whereas the latter is unavoidable to some extent, because the massive Wilson co-
efficientshavenotyetbeencalculated evenat1-looplevelformostphysicalprocesses (exceptforDIS),
theformer(whichcanbemoresignificantnumericallyincertainpartsofphasespace)canpotentially be
remediedbyproperlycountingthefinalstatesandusingtherescalingvariables,asdiscussedinSecs.2.3
and 2.4under general considerations. Thus, alternative formulations oftheZMVFNSarepossible that
only involve the zero-mass approximation in the Wilson coefficient. This possibility has not yet been
explicitly explored.
3.2 PartonDistributionFunctionsinVFNS(ZMandGM)
InPQCD,thefactorization schemeisdeterminedbythechoicesmadeindefiningthepartondistribution
functions (as renormalized Green functions). In a GM VFNS based on the generalized MS subtraction
(cf.footnote 2) the evolution kernel of the DGLAP equation is mass-independent; thus the PDFs, so
defined,applytoGMVFNScalculations astheydofortheZMVFNS.
In the VFNS, the PDFs switch from the n -flavor FFNS ones to the (n +1)-flavor FFNS ones
f f
at the matching point µ = m (cf.Sec.2.2); the PDFs above/below the matching point are related,
H
order-by-order inα ,by:
s
fVF(µ → m+)≡ f(nf+1)FF = A ⊗fnfFF ≡ A ⊗fVF(µ → m−), (7)
j H j jk k jk j H
+/−
wherem indicatethattheµ → m limitistakenfromabove/below,andwehaveusedtheshorthand
H H
VF/FFforVFNS/FFNSinthesuperscripts. Thetransition matrix elements A (µ/m ), representing a
jk H
finite-renormalization between the two overlapping FFNSschemes, can be calculated order by order in
α ; they are known to NNLO,i.e. O(α2) [10,11]. (Note that A is not a square matrix.) It turns out,
s S jk
atNLO,A (µ = m )= 0[6];thusfVF arecontinuous withthischoiceofmatchingpoint. Thereisa
jk H k
rathersignificantdiscontinuity inheavyquarkdistributions andthegluondistribution atNNLO.
Withthematchingconditions, Eq.7,{fVF(µ)}areuniquely definedforallvaluesofµ. Weshall
j
omitthesuperscriptVFinthefollowing. Moreover,whenthereisaneedtofocusonf (µ)inthevicinity
j
+/−
ofµ = m ,wheretheremaybeadiscontinuity, weusef (µ)todistinguish theabove/below branch
H j
ofthefunction. AsindicatedinEq.7,f−correspondtothen -flavorPDFs,andf+tothe(n +1)-flavor
j f j f
ones.
3.3 TheStructureofaGMVFNS,MinimalPrescription andAdditionalFreedom
Physical quantities should be independent of the choice of scheme; hence, in a GM VFNS, we must
requirethetheoretical expressions forthestructurefunctions tobecontinuous acrossthematchingpoint
µ = Q = m toeachorderofperturbative theory:
H
F(x,Q) = C−(m /Q)⊗f−(Q) = C+(m /Q)⊗f+(Q) (8)
k H k j H j
≡ C+(m /Q)⊗A (m /Q)⊗f−(Q). (9)
j H jk H k
+/−
wherewehavesuppressedthestructurefunctionlabel(λ)onF’sandC’s,andusedthenotationC to
k
denotetheWilsoncoefficientfunctionC (m /Q)above/belowthematchingpointrespectively. Hence,
k H
theGMVFNScoefficient functions arealso,ingeneral, discontinuous, andmustsatisfythetransforma-
tionformula:
C−(m /Q) = C+(m /Q)⊗A (m /Q). (10)
k H j H jk H
order-by-order inα . Forexample,atO(α ),A = α P0 ln(Q/m ),thisconstraint implies,
s S Hg s qg H
C−,1(m /Q) = α C+,0 (m /Q)⊗P0 ln(Q/m )+C+,1(m /Q). (11)
H,g H s H,H H qg H H,g H
where the numeral superscript (0,1) refers to the order of calculation in α (for P , the order is by
s jk
standard convention one higher then indicated), and the suppressed second parton index on the Wilson
coefficients (cf.Eq.2) has been restored to make the content of this equation explicit. Eq.(11) was
implicitly used in defining the original ACOT scheme [8]. The first term on the RHS of Eq.11, when
movedtotheLHS,becomesthesubtraction termofRef. [8]thatservestodefinetheWilsoncoefficient
C+,1(m /Q) (hence the scheme) at order α , as well as to eliminate the potentially infra-red unsafe
H,g H s
logarithm inthegluonfusionterm(C−,1(m /Q))athighenergies.
H,g H
The GM VFNS as described above, consisting of the general framework of [5,6], along with
transformation matrices {A } calculated to order α2 by [10,11], is accepted in principle by all recent
jk s
workonPQCDwithmass. Together, theycanberegardedastheminimalGMVFNS.
The definition in Eq.10 was applied to find the asymptotic limits (Q2/M2 → ∞) of coeffi-
H
cient functions in [10,11], but it is important to observe that it does not completely define all Wil-
son coefficients across the matching point, hence, there are additional flexibilities in defining a specific
scheme [6,12,13,22]. Thisisbecause, as mentioned earlier, thetransition matrix {A }isnot asquare
jk
matrix—it is n × (n + 1). It is possible to swap O(m /Q) terms between Wilson coefficients on
f f H
the right-hand-side of Eq.10 (hence redefining the scheme) without violating the general principles of
a GM VFNS. For instance, one can swap O(m /Q) terms between C+,0(m /Q) and C+,1(m /Q)
H H H g H
whilekeeping intact therelation (11)thatguarantees thecontinuity ofF(x,Q)according toEq.8. This
general feature, applies to(10)toallorders. Itmeans, inparticular, thatthereisnoneedtocalculate the
coefficient function C+,i(m /Q), for anyi–itcanbechosen asapart ofthedefinition ofthescheme.
H H
Also, it is perfectly possible to define coefficient functions which do not individually satisfy the con-
straintinEq.3,sinceEq.10guarantees ultimatecancellation ofanyviolationsbetweenterms. However,
thiswillnotoccurperfectly atanyfiniteordersomoderndefinitionsdoincludetheconstraint explicitly,
asoutlined inSec.2.4.
The additional flexibility discussed above has been exploited to simply the calculation, as well
as to achieve some desirable features of the prediction of the theory by different groups. Of particular
interest andusefulness isthegeneral observation that, givenaGMVFNScalculation of{C+}, onecan
j
alwaysswitchtoasimplerschemewithconstant {C˜+}
j
C˜+(m /Q) = C+(0) (12)
H H H
Thisisbecausetheshift(C+(m /Q)−C+(0))vanishesinthem /Q → 0limit,andcanbeabsorbed
H H H H
into a redefinition of the GM scheme as mentioned above. The detailed proof are given in [6,22].
By choosing the heavy-quark-initiated contributions to coincide with the ZM formulae, the GM VFNS
calculation becomesmuchsimplified: giventhebetterknownZMresults, weonlyneedtoknowthefull
m -dependent contributions from the light-parton-initiated subprocesses; and these are exactly what is
H
provided by the n -flavor FFNS calculations available in the literature. This scheme is known as the
f
SimplifiedACOTscheme,orSACOT[6,22].
Further uses of the freedom to reshuffle O(m /Q) terms between Wilson coefficients, as well
H
as adding terms of higher order in the matching condition (without upsetting the accuracy at the given
order)havebeenemployed extensively bytheMRST/MSTWgroup, aswillbediscussed inSec.3.5.
3.4 CTEQImplementationoftheGMVFNS
The CTEQ group has always followed the general PQCD framework as formulated in [5,6]. Up to
CTEQ6.1, the default CTEQ PDF sets were obtained using the more familiar ZM Wilson coefficients,
because,thevastmajorityofHEPapplicationscarriedoutbyboththeoristsandexperimentalistsusethis
calculational scheme. Forthoseapplications thatemphasizedheavyquarks,specialGMVFNSPDFsets
werealsoprovided;thesewerenamedasCTEQnHQ,wheren = 4,5,6.
The earlier CTEQ PDFsare now superseded by CTEQ6.5[1] and CTEQ6.6 [3] PDFs; these are
based on a new implementation of the general framework described in previous sections, plus using
the simplifying SACOT choice of heavy quark Wilson coefficients [8,21] specified by Eq.12 above.
TherearenoadditionalmodificationoftheformulaeoftheminimalGMVFNS,asdescribedinprevious
sections. CTEQuses the convention of designating tree-level, 1-loop, 2-loop calculations as LO,NLO,
andNNLO,forallphysical quantities, Ftot,FH,... etc.,cf.Sec.2.6.
λ λ
With these minimal choices, this implementation is extremely simple. Continuity of physical
predictions across matching points in the scale variable µ = Q is guaranteed by Eqs.8 and 10; and
continuity across physicalthresholds inthephysical variable W,forproducing heavyflavorfinalstates,
areguaranteed bytheuseofACOT-χrescaling variables5,asdescribed inSec.2.4.
Forexample,toexaminethecontinuity ofphysicalpredictions toNLOinthisapproach, wehave,
forthebelow/abovematchingpointcalculations:
F2−H(x,Q2) = αsC2−,H,1g ⊗gnf (13)
F2+H(x,Q2) = αsC2+,H,1g ⊗gnf+1+(C2+,H,0H +αsC2+,H,1H)⊗(h+h¯)
where non-essential numerical factors have been absorbed into the convolution ⊗. The continuity of
FH(x,Q2) in the scaling variable µ = Q is satisfied by construction (Eq.9) because the relation be-
2
tweenthePDFsgivenbyEq.7andthatbetweentheWilsoncoefficientsgivenbyEq.8involvethesame
transformationmatrix{A }(calculatedin[10,11,20]). Infact,tothisorder,A =α P0 ln(Q/m ),
jk Hg s qg H
hence
h(h¯) = 0
gnf+1 = gnf
+,1 −,1
C = C ,
2,Hg 2,Hg
atthematching point µ = Q = m . Thus,thetwolines inEq.13givethesameresult, andFH(x,Q2)
H 2
is continuous. The separate issue of continuity of FH(x,Q2) in the physical variable W across the
2
production threshold ofW = 2m issatisfiedautomatically byeachindividual term(using theACOT-
H
χprescription forthequarktermsandstraightforward kinematicsforthegluonterm).
In the CTEQ approach, all processes are treated in a uniform way; there is no need to distin-
guish between neutral current (NC) and charged current (CC) processes in DIS, (among others, as in
MRST/MSTW).AllCTEQglobal analyses so far are carried out up to NLO.This isquite adequate for
currentphenomenology, givenexistingexperimentalandothertheoretical uncertainties. BecauseNNLO
results hasbeen known toshowsigns ofunstable behavior oftheperturbative expansion, particularly at
small-x, they are being studied along with resummation effects that can stabilize the predictions. This
studyisstillunderway.
3.5 MRST/MSTWImplementationoftheGMVFNS
3.5.1 Prescription
IntheTRheavyflavourprescriptions,describedin[12,13]theambiguityinthedefinitionofCVF,0(Q2/m2 )
2,HH H
was exploited by applying the constraint that (dFH/d lnQ2) wascontinuous atthe transition point (in
2
thegluonsector). However,thisbecomestechnically difficultathigherorders. Hence,in[19]thechoice
of heavy-flavour coefficient functions for FH was altered to be the same as the SACOT(χ) scheme
2
described above. This choice of heavy-flavour coefficient functions has been used in the most recent
MRST/MSTWanalysis, inthefirstinstance in[2]. Tobeprecisethechoiceis
CVF,n(Q2/m2 ,z) = CZM,n(z/x ). (14)
2,HH H 2,HH max
This is applied up to NNLO in [19] and in subsequent analyses. For the first time at this order satis-
fying the requirements in Eq.(10) leads to discontinuities in coefficient functions, which up to NNLO