Charm and Bottom Quark Production Cross Sections Near Threshold
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arXiv:hep-ph/9609388v1 18 Sep 1996ITP-SB-95-60
LBL-38282
CharmandBottomQuarkProductionCross
SectionsNearThreshold
J.Smith
InstituteforTheoreticalPhysics,
StateUniversityofNewYorkatStonyBrook,
StonyBrook,NY11794-3840
R.Vogt1
NuclearScienceDivision,
LawrenceBerkeleyNationalLaboratory,
Berkeley,California94720
and
PhysicsDepartment,
UniversityofCaliforniaatDavis,
Davis,California95616USA
April1996
Abstract
Thecrosssectionsforcharmandbottomquarkproductioninthethresholdregionarediscussed.Weconsidertheeffectsofanallorderresummationofinitialstatesoft-plus-virtualgluonradiationonthetotalcrosssectionscomparedtotheorderα3sresults.Heavyquarkproductionhaslongbeenatopicofinterest,bothexperi-
mentallyandtheoretically.Earlymeasurementsofthetotalc
S≤63GeVsuggestedthatthecalculatedBorn(LO)cross
sectionunderpredictedthedatabyafactoroftwotothree[1,2],knownas
theKfactorafterasimilarsituationinDrell-Yanproduction.Ingeneral,
Kexp=σdata(AB→Q
σtheory(AB→Q
Q)
Q),(2)
whereσ(0)istheLOcrosssectionandσNLOisthesumoftheLOandthe
exactO(αs)correction,σNLO=σ(0)+σ(1)|exact.
TheheavyquarkproductioncrosssectioniscalculatedinQCDbyas-
sumingthevalidityofthefactorizationtheorem[6]andexpandingthecon-
tributionstotheamplitudeinpowersofthecouplingconstantαs(µ2).The
hadronicproductioncrosssectionathadroniccenterofmassenergy√
Sdτ1
τdx
x,µ2)σ(k)ij(τS,m2,µ2),(3)
wherefhi(x,µ2)arethescale-dependentpartondensitiesofhadronheval-
uatedatthescaleµ2andσ(k)ijisthekthorderpartoniccrosssectionfor
ij→Qsection.ThesensitivitytoevenhighertermsintheQCDexpansionisoften
demonstratedbyvaryingµbetweenm/2and2m.Thismaynotbevery
meaningful,especiallyforcharm,sinceavariationofanorderofmagnitude
ormoreisobserved.Itisthereforenotclearthatthenext-to-next-to-leading
order(NNLO)correctionsarenotatleastaslargeastheNLOcorrections,
particularlywhenm≪√
cproductioncrosssection,σtotc
Qandhigh
energybproduction.Neitherapproachisfullysatisfactory:eitheranun-
comfortablysmallcharmquarkmassisneededortheparametersusedto
describelowenergyproductionareincompatiblewiththoseusedatcollider
energies.
Althoughacompletecalculationofstillhigherordertermsisnotpossible
forallvaluesofSandm,improvementsmaybemadeinspecifickinematical
regions.Investigationshaveshownthatnearthresholdtherecanbelargelogarithmsintheperturbativeexpansionwhichmustberesummedtomake
morereliabletheoreticalpredictions.Theselargelogarithmsarisefroman
imperfectcancellationofthesoft-plus-virtual(S+V)terms.In[17]anap-proximationwasgivenfortheS+Vgluoncontributionsandtheanalogywith
theDrell-Yanprocess,studiedin[18,19],wasexploitedtoresumtheseto
allordersofperturbationtheory.Thesameresummationprocedurewasalso
appliedtoσtottpcollisionsatthe
FermilabTevatron[17,20,21].Fortopproductioninpqan-
nihilationisthedominantprocess,fortunatesincetheexponentiationofthe
S+Vterms[17]isbetterunderstoodbecausethesimplecolorstructurehas
aclosecorrespondencetotheDrell-Yanstudies[18,19].
TheresummationoftheleadingS+Vterms[17]modifieseq.(3)sothat
σres(S,m2)=
ij1
τ0dτ1
τdx
x,µ2)σresij(τS,m2),(4)
where
σresij(τS,m2)=−s−2ms1/2
s0ds4f(s4µ2)d
ds4.(5)
Theintegrationvariables4,aninvariantwhichmeasuresthefour-momentum
carriedawaybythefinal-stategluonintheprocessi(k1)+j(k2)→Q(p1)+
σ(0)ij(s,s4,m2)is
theangle-averagedBorncrosssectionand
m2,m2
πm2,m2ln2s4Γ(1+η)exp(−ηγE),(6)
whereγEistheEulerconstant,η=(8Cij/β0)ln(1+(β0αs(µ2)/4π)ln(m2/µ2)),
β0=11−2nf/3forSU(3),andAand
MSscheme,A=2and
S,(7)
4wheres0=m2(µ20/µ2)3/2inthe
bproduction
inppinteractions[23]atHERA-B(√qchannelincbproduction.Howevercharmproductionmustbe
treatedwithsomecare.Thedataisavailablefor√
S/m),isnotsmall.However,ifthemodel
isreliableforcharmproduction,abetterunderstandingofc
MSscheme,weareforcedtousethisscheme
forboththeq
qchannelislikelytobelargerthanfound
previously[17,20,21,23].Ourresultsfortheexact,approximate,andre-
summedhadroniccrosssectionswillbecalculatedusingthesedistributions
alongwiththetwo-loopuncorrectedrunningcouplingconstant4.Thisset
hastheadditionaladvantageofarathersmallinitialscalesothatwecan
useµ=mcinourcalculations.Wethereforetakemc=µ=1.5GeV/c2and
mb=µ=4.75GeV/c2,alongwiththeGRVHOpartondensitiesandthe
3Quark-gluonscattering,producingafinalstatequarkorantiquark,cannotbere-summedbythismethodsincethereisnoequivalentBornterm.4ThedifferencebetweentheuncorrectedrunningcouplingconstantandthecorrectedvaluegivenbyPDFLIB[12]issmall,≈3-4%.
5