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√

Sdτ󰀇1

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)=󰀆

ij󰀇1

τ0dτ󰀇1

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,m2󰀁ln2s4Γ(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