The black book of quantum chromodynamics : a primer for the LHC era First Edition. Edition Campbell
Visit to download the full and correct content document: https://textbookfull.com/product/the-black-book-of-quantum-chromodynamics-a-prime r-for-the-lhc-era-first-edition-edition-campbell/
More products digital (pdf, epub, mobi) instant download maybe you interests ...
Particle physics in the LHC era 1st Edition Barr
https://textbookfull.com/product/particle-physics-in-the-lhcera-1st-edition-barr/
Delicious Daddy (A DILF For Father's Day Book 1) 1st Edition Barbra Campbell [Campbell
https://textbookfull.com/product/delicious-daddy-a-dilf-forfathers-day-book-1-1st-edition-barbra-campbell-campbell/
A First Book of Quantum Field Theory (Second Revised Edition) Amitabha Lahiri
https://textbookfull.com/product/a-first-book-of-quantum-fieldtheory-second-revised-edition-amitabha-lahiri/
Adrian's Firecracker (The Holiday Firecrackers Book 7) 1st Edition Barbra Campbell [Campbell
https://textbookfull.com/product/adrians-firecracker-the-holidayfirecrackers-book-7-1st-edition-barbra-campbell-campbell/
Quantum Continuous Variables: A Primer of Theoretical Methods 1st Edition Alessio Serafini
https://textbookfull.com/product/quantum-continuous-variables-aprimer-of-theoretical-methods-1st-edition-alessio-serafini/
Ethics for a digital era First Edition Elliott
https://textbookfull.com/product/ethics-for-a-digital-era-firstedition-elliott/
The Little SAS Book A Primer Sixth Edition Lora D. Delwiche
https://textbookfull.com/product/the-little-sas-book-a-primersixth-edition-lora-d-delwiche/
Impossible Duet: A super steamy, opposites attract novella (Taking the Leap Book 8) 1st Edition Barbra Campbell [Campbell
https://textbookfull.com/product/impossible-duet-a-super-steamyopposites-attract-novella-taking-the-leap-book-8-1st-editionbarbra-campbell-campbell/
Secrets & Second Chances (The Law Trilogy: Beyond the Law Book 2) 1st Edition Barbra Campbell [Campbell
https://textbookfull.com/product/secrets-second-chances-the-lawtrilogy-beyond-the-law-book-2-1st-edition-barbra-campbellcampbell/
THEBLACKBOOKOFQUANTUMCHROMODYNAMICS TheBlackBookofQuantum Chromodynamics APrimerfortheLHCEra JohnCampbell
TheoreticalPhysicsDepartment,Fermilab,Batavia,Illinois,USA
JoeyHuston DepartmentofPhysicsandAstronomy,MichiganStateUniversity,EastLansing, Michigan,USA
FrankKrauss InstituteforParticlePhysicsPhenomenology,DurhamUniversity,Durham,UK
GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom
OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©JohnCampbell,JoeyHuston,andFrankKrauss2018
Themoralrightsoftheauthorshavebeenasserted FirstEditionpublishedin2018 Impression:1
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove
Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable
LibraryofCongressControlNumber:2017943893
ISBN978–0–19–965274–7 DOI10.1093/oso/9780199652747.001
LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY
Toourfamilies,withlove.
Tocolleaguesandfriends, whoshapedourunderstandingof particlephysics,withgratitude.
Acknowledgements Wearegreatlyindebtedtoalargenumberofpeople,whoinspiredustodoparticle physics,whodidtheirbesttoteachussomething,whocollaboratedwithus,andwho, byfarandlarge,shapedourviewofQCDattheLHC andothercolliderexperiments: wetrulyarestandingontheshouldersofgiants.
WealsooweadebtofgratitudetoourfriendsandcolleaguesfromCTEQand MCnetwhoputupwithouranticswhilefurtherhelpingourunderstandingofscience incountlessnight-timediscussionsduringgraduateschoolsandothermeetings.We wouldliketothankStefanoCatani,DanieldeFlorian,KeithHamilton,StefanH¨oche, SimonPl¨atzer,StefanPrestel,andMarekSch¨onherrforpatientlyansweringourmany questionsconcerningsomeofthemorespecializedaspectsinthisbook.Youhavebeen ahugehelp!Anyinaccuracyorerrorisnotareflectionofyourexplanationsbutof ourlimitedunderstandingofthem.
WeareextremelygratefultoJoshIsaacson,PavelNadolsky,PetarPetrov,and AlessandroTricoliforcarefullyreadingpartsofthebookwhileitwasbeingwritten, pointingoutconceptualshortcomingsandmisunderstandingsfromourside,thetypical errorswithfactorsoftwo,unfortunateformulations,andmuchmore.Howevergood, forallmistakesleftinthebook,thebuckstopswithus.Alistofupdates,clarifications, andcorrectionsismaintainedatthefollowingwebsite: http://www.ippp.dur.ac.uk/BlackBook
Wewouldliketothankthefollowingforusefulconversationsandforproviding figuresforthebook:SimonBadger,JohannesBellm,KeithEllis,SteveEllis,RickField, JunGao,NigelGlover,StefanH¨oche,SilvanKuttimalai,GionataLuisoni,Matthew Mondragon,IvanPogrebnyak,StefanPrestel,MarekSch¨onherr,CiaranWilliams,Jan Winter,andUn-kiYang.
Thegenesisofthisbookwasareviewarticleco-writtenbytwoofus[320]and wethankbothourco-author,JamesStirling,andtheIOPfortheircollaborationand support.Somepartsofthisbookbenefittedtremendouslyfromusbeingallowedto testthemonunsuspectingstudentsinlecturesduringregularcourses,atgraduate schools,orsummerinstitutes,andonourcolleaguesduringtalksatconferencesand workshops.Thankyou,foryourpatiencewithusandyourfeedback!
Manyoftheplotsinthisbookhavebeencreatedusingthewonderfultoolsapfelweb[331],JaxoDraw[249],RIVET [291],MATPLOTLIB [638],xmGRACE,andXFIG. OnthatoccasionwewouldalsoliketothankAndyBuckley,HolgerSchulzandDavid Grellscheidfortheircontinuoussupportandingenioushelpwithsomeofthefiner issueswithRIVET andLATEX.
Wearealsogratefulforthegreatsupportbyourpublisher,andinparticularbythe teamtakingcareofthismanuscript:SonkeAdlung,HarrietKonishiandHemalatha Thirunavukkarasu.
Finally,ofcoursewewouldliketothankourfamiliesforputtingupwithuswhile weassembledthismanuscript.Surely,wewerenotalwaysthemostprettytowatch ortheeasiestpeopletohavearound.
Introduction 1.1ThephysicsoftheLHC era 1.1.1ParticlephysicsintheLHC era Theturn-onoftheLHC in2008culminatedanalmost20-yeardesignandconstruction effort,resultinginthelargestparticleaccelerator(actuallythelargestmachine)ever built.AtitsinceptionacompetitionstillexistedwiththeTEVATRON which,although operatingatamuchlowerenergy,hadadatasamplewithalargeintegratedluminosityandwell-understooddetectorsandphysics-analysissoftware.TheTEVATRON haddiscoveredthetopquarkandwascontinuingitssearchfortheHiggsboson.Asis wellknown,theLHC sufferedconsiderabledamagefromacryogenicquenchsoonafter turn-onthatresultedinashut-downforabout1.5years.Its(re)turn-onin2010was atamuchlowerenergy(7TeVratherthan14TeV)andatmuchlowerintensities.The smalldatasampleatthelowerenergycanbeconsideredinretrospectasablessingin disguise.TherewasnotenoughdatatoevenconsiderasearchfortheHiggsboson(or evenformuchinthewayofnewphysics),buttherewasenoughdatatoproduce W and Z bosons,topquarks,photons,leptonsandjets—inotherwords,alloftheparticlesoftheStandardModelexceptfortheHiggsboson.Theresultwasthe re-discovery oftheStandardModel (acoinageforwhichoneoftheauthorstakescredit)andthe developmentoftheanalysistoolsandthedetailedunderstandingofthedetectorsthat allowedforthediscoveryoftheHiggsbosononJuly4,2012,withdatafrom7TeV in2011and8TeVin2012.TheLHC turnedoffagaininearly2013forrepairsand upgrades(toavoidthetypeofcatastrophicquenchthatoccurredin2008).TheLHC detectorsalsousedthistwo-yearperiodforrepairsandupgrades.TheLHC ranagain in2015,atanenergymuchclosertodesign(13TeV).Theincreasedenergyallowedfor moredetailedstudiesoftheHiggsboson,butmoreimportantlyofferedamuchgreater reachforthediscoveryofpossiblenewphysics.Atthetimeofcompletionofthisbook, agreatdealofphysicshasbeenmeasuredattheoperatingenergyof13TeV.Given thenewresultscontinuallypouringoutatthisnewenergy,thedecisionwasmadeto concentrateinthisbookonresultsfrom7and8TeVrunning.Thisissufficientfor thedatacomparisonsneededtoillustratethetheoreticalmachinerydevelopedhere.
TheBlackBookofQuantumChromodynamics:APrimerfortheLHCEra. JohnCampbell,JoeyHuston,andFrankKrauss. ©JohnCampbell,JoeyHuston,andFrankKrauss2018.Publishedin2018byOxfordUniversityPress. DOI10.1093/oso/9780199652747.001.0001
1.1.2ThequestfortheHiggsboson—andbeyond
1.1.2.1FindingtheHiggsboson
TheLHC wasdesignedasadiscoverymachine,withadesigncentre-of-massenergya factorofsevenlargerthanthatoftheTEVATRON.Thishighercollisionenergyopened upawidephasespaceforsearchesfornewphysics,buttherewasonediscoverythat theLHC was guaranteed tomake;thatoftheHiggsboson,oranequivalentmechanism forpreventing WW scatteringfromviolatingunitarityathighmasses.
TheHiggsbosoncouplesdirectlytoquarks,leptonsandto W and Z bosons,and indirectly(throughloops)tophotonsandgluons.ThustheHiggsbosonfinalstates arejustthebuildingblocksoftheSMwithwhichwehavemuchexperience,bothat theTEVATRON andtheLHC.TheATLAS andCMS detectorsweredesignedtofindthe Higgsbosonandtomeasureitspropertiesindetail.
Thecross-sectionforproductionofaHiggsbosonisnotsmall.However,thefinal statesforwhichtheHiggsbosonbranchingratioislarge(suchas bb)havebackgrounds whicharemuchlargerfromothermorecommonprocesses.Thefinalstateswithlow backgrounds(suchas ZZ∗ → + + )sufferfrompoorstatistics,primarilydue tothe Z branchingratiotoleptons.TheHiggs→ γγ finalstatesuffersfromasmall branchingratioandalargeSMbackground.Thusonemightnotexpectthisfinalstate tobepromisingforaHiggsbosonsearch.However,duetotheintrinsicnarrowwidth oftheHiggsboson,adiphotonsignalcanbeobservableiftheexperimentalresolution ofthedetectorisgoodenoughthatthesignalstandsoutoverthebackground.
ThemeasurablefinalstatesoftheHiggsbosondecayswerefurthersubdividedinto differenttopologiessothatoptimizedcutscouldbeusedtoimproveonthesignalto-backgroundratioforeachtopology(forexample,inATLAS thediphotonchannel wasdividedinto12topologies).Theextractedsignalwasfurtherweightedbythe expectationsoftheSMHiggsbosoninthosetopologies.Inthissense,theHiggsboson thatwasdiscoveredin2012wasindeedtheStandardModelHiggsboson.However,as willbediscussedinChapter9,detailedstudieshavedeterminedthepropertiesofthe newparticletobeconsistentwiththisassumption.
1.1.2.2ThetriumphoftheGaugePrinciple ThediscoveryoftheHiggsbosonbytheATLAS andCMS collaboration,reportedin July2012andpublishedin[15,368],isundoubtedlythecrowningachievementofthe LHC endeavoursofar.Itishardtooverestimatetheimportanceofthisdiscoveryfor thefieldofparticlephysicsandbeyond.
TheHiggsbosonistheonlyfundamentalscalarparticleeverfound,whichinitself makesitunique;allotherscalarsuptonowwereboundstates,andthefundamental particlesfoundsofarhavebeenalleitherspin-1/2fermionsorspin-1vectorbosons. Thisdiscoveryisevenmoresignificantasitmarksatriumphofthehumanmind:the HiggsbosonisthepredictedvisiblemanifestationoftheBrout–Englert–Higgs(BEH) mechanism[516,601,619–621,675],whichallowsthegenerationofparticlemassesin agauge-invariantway[580,835,888].Ultimately,thisdiscoveryprovestheparadigm ofgaugeinvarianceasthegoverningprincipleofthesub-nuclearworldatthesmallest
distancesandlargestenergiestestedinalaboratorysofar.Withthisdiscoverya 50-year-oldpredictionconcerningthecharacterofnaturehasbeenproven
ThequestionnowisnotwhethertheHiggsbosonexistsbutinsteadwhatare itsproperties?IstheHiggsbosonperhapsaportaltosomenewphenomena,new particles,orevennewdynamics?Therearesomehintsfromtheoryandcosmology thatthediscoveryoftheHiggsbosonisnotthefinallegofthejourney.
1.1.2.3BeyondtheStandardModel Byfindingthelastmissingparticleandtherebycompletingthemostaccurateand precisetheoryofnatureatthesub-nucleareverconstructed,theparadigmsbywhich ithasbeenconstructedhaveprovedoverwhelminglysuccessful.Despitethisthereare stillfundamentalquestionsleftunanswered.Thesequestionsgobeyondtherealmof theSM,buttheyremainofutmostimportanceforanevendeeperunderstandingof theworldaroundus.
Observationsofmatter—Earth,otherplanetsintheSolarSystemorbeyond, otherstars,orgalaxies—suggestthatthesymmetrybetweenmatterandanti-matter isbroken.Thisisauniversefilledbymatterandpracticallydevoidofanti-matter. Whilenaivelythereisnoobviousreasonwhyoneshouldbepreferredovertheother, atsomepointinthehistoryoftheUniverse—andpresumablyveryearly—this asymmetryhadtoemergefromwhatisbelievedtohavebeenasymmetricinitialstate. Inorderforthistohappen,asetofconditions,thefamous Sakharovconditions [710, 834]hadtobemet.Oneoftheseintricateconditionsistheviolationof CP,which demandsthatthesymmetryunderthecombinedparityandcharge–conjugation(CP) transformationmustbebroken.Experimentally,theexistenceof CP violationhas beenconfirmedandistightlyrelatedtotheexistenceofatleastthreegenerations ofmatterfieldsintheSM.DuetotheBEHmechanism,particlesacquiremasses, andtheirmassandelectroweakinteractioneigenstatesarenolongeralignedafter EWSB.TheexistenceofacomplexphaseintheCKMmatrix,whichparametrizesthe interrelationbetweenthesetwosetofeigenstates,ultimatelytriggers CP violationin thequarksector.However,theamountof CP violationestablishedissubstantially smallerthannecessarytoexplainhowtheuniverseevolvedfromaninitialsymmetric configurationtothematter-dominatedconfigurationseentoday[358].
Likewise,theexistenceofdarkmatter(DM)isnowwellestablished,firstevidenced bytherotationalcurvesofgalaxies[831].DMdenotesmatterwhichinteractsonlyvery weaklywithnormalmatter(describedbytheSM)andthereforecertainlydoesnot interactthroughelectromagnetismorthestrongnuclearforce.Despitenumerousattemptsithasnotbeendirectlydetected.DMinteractsthroughgravityandthereby hasinfluencedtheformationoflarge-scalestructuresintheUniverse.Cosmological precisionmeasurementsbytheWMAP andPLANCK collaborations[125,623,862]concludethatdarkmatterprovidesabout80%ofthetotalmattercontentoftheUniverse. Thisinturncontributesabout25%oftheoverallenergybalance,withtherestofthe energycontentoftheUniverseprovidedbywhatisknownasdarkenergy(DE),which isevenmoremysteriousthanDM.TheonlythingknownisthattheinterplayofDM andDEhasbeencrucialinshapingtheUniverseasobservedtodayandwillcontinue todetermineitsfuture.OnepossibleavenueinsearchesforDMparticlesatcolliderex-
perimentsisthattheyhavenocouplingtoordinarymatterthroughgaugeinteractions butinsteadcouplethroughtheHiggsboson.
TheseexamplesindicatethattheSM,asbeautifulasitis,willdefinitelynotprovide theultimateanswertothequestionsconcerningthefundamentalbuildingblocksofthe worldaroundusandhowtheyinteractattheshortestdistances.TheSMwillhavetobe extendedbyatheoryencompassingatleastenhancedCPviolation,darkmatter,and darkenergy.Anysuchextensionisalreadyseverelyconstrainedbytheoverwhelming successofthegaugeprinciple:thegaugesectoroftheSMhasbeenscrutinizedto incrediblyhighprecision,passingeverytestuptonowwithflyingcolours.Seefor example[179]forarecentreview,combiningdatafrom e e+ andhadroncollider experiments.TheHiggsbosonhasbeenfoundonlyrecently,anditisevidentthatthis discoveryanditsimplicationswillcontinuetoshapeourunderstandingofthemicro–worldaroundus.Thediscoveryitself,andevenmoresothemassofthenewparticle andourfirst,imprecisemeasurementsofitsproperties,alreadyruleoutorplacesevere constraintsonmanynewphysicsmodelsgoingbeyondthewell-establishedSM[515].
Rightnow,wearemerelyatthebeginningofanextensiveprogrammeofprecision testsintheHiggssectoroftheSMorthetheorythatmayrevealitselfbeyondit.It canbeanticipatedthatattheendoftheLHC era,eithertheSMwillhaveprevailed completely,withnewphysicseffectsandtheirmanifestationasnewparticlespossibly beyonddirecthumanreach,oralternatively,wewillhaveforgedanew,evenmore beautifulmodelofparticlephysics.
1.1.3LHC:Acceleratoranddetectors 1.1.3.1LHC,themachine
TheLHC notonlyistheworld’slargestparticleacceleratorbutitisalsotheworld’s largestmachine,at27kmincircumference.TheLHC isaproton-protoncollider(althoughitalsooperateswithcollisionsofprotonsonnuclei,andnucleionnuclei), locatedapproximately100mundergroundandstraddlingtheborderbetweenFrance andSwitzerland.TheLHC occupiesthetunnelformerlyusedfortheLEP accelerator inwhichelectronsandpositronscollidedatcentre-of-massenergiesupto209GeV. TheLHC contains9593magnets,including1232superconductingdipolemagnets,capableofproducingmagneticfieldsoftheorderof8.3T,andamaximumprotonbeam energyof7TeV(trillionelectron-volts),leadingtoamaximumcollisionenergyof14 TeV.Thusfar,theLHC hasrunatcollisionenergiesof7TeV(2010,2011),8TeV (2012)and13TeV(2015,2016),greatlyexceedingthepreviousrecordoftheFermilabTEVATRON of1.96TeV.1 ThelargeradiusoftheLHC isnecessitatedbecauseof thedesiretoreachashighabeamenergyaspossible(7TeV)usingdipoleswiththe largestmagneticfieldspossible(inanaccelerator).Runningatfullenergy,thepower consumption(includingtheexperiments)is750GWhperyear.Atfullpower,theLHC willcollide2808protonbunches,eachapproximately30cmlongand16micronsin diameterandcontaining1 15 × 1011 protons,leadingtoaluminosityof1034cm 2/s andabillionproton-protoncollisionspersecond.Thespacingbetweenthebunchesis 25nsleadingtocollisionsoccurringevery25ns;thus,atfull luminosity therewill
1UnliketheLHC,theTEVATRON wasaproton-antiprotoncollider.
Fig.1.1 A3DlayoutoftheLHC,showingthelocationofthefourmajor experiments.ReprintedwithpermissionfromCERN. beonaverage25interactionseverybeamcrossing,mostofwhichwillberelatively uninteresting.Thehighluminosityforthemachineisneededtoproduceeventsfrom processeswithsmallcross-sections,forexampleinvolvingphysicsattheTeVscale.
TherearesevenexperimentsrunningattheLHC (ATLAS,CMS,LHCB,ALICE, TOTEM,LHCfandMoEDAL),withATLAS andCMS beingthetwogeneral-purpose detectors.AschematicdrawingoftheLHC,indicatingthepositionofthefourlarger experimentsisshowninFig.1.1.
1.1.3.2Thedetectors Itseemsparadoxicalthatthelargestdevicesareneededtoprobethesmallestdistance scales.TheATLAS detector,forexample,is46mlong,25mindiameterandweighs 7000tonnes.TheCMS detector,althoughsmallerthanATLAS at15mindiameter and21.5minlength,istwiceasmassive,at14, 000tonnes.Thiscanbecompared totheCDFdetectorattheTEVATRON whichwas only 12m×12m×12m(and5000 tonnes).ThekeytothesizeandcomplexityoftheLHC detectorsistheneedto measurethefour-vectorsofthelargenumberofparticlespresentinLHC events,whose momentacanextendtotheTeVrange.Thelargeparticlemultiplicityrequiresvery finesegmentation;theATLAS detector,forexample,has160millionchannelstoread out,halfofwhichareinthepixeldetector.Thelargeenergies/momentarequire,in additiontofinesegmentation,largemagneticfieldsandtrackingvolumesandthick calorimetry.
BothATLAS andCMS arewhatareknownasgeneral-purpose4π detectors,meaning thattheyattempttocoverasmuchofthesolidanglearoundthecollisionpointas
possible,inordertoreconstructasmuchinformationabouteacheventaspossible.2 Thereisa universal cylindricallysymmetricconfigurationfora4π detector,embodied, forexample,intheATLAS detector,asshowninFig.1.2.Collisionstakeplaceinthe centreofthedetector.Particlesproducedineachcollisionfirstencounterthepixel detector(6)andthesilicontrackingdetector(5).Thefirstlayerofthepixeldetector isactuallymountedonthebeam-pipeinordertobeasclosetotheinteractionpoint aspossible.Thebeam-pipeitself,intheinteractionregion,iscomposedofberylliumin ordertopresentaslittlematerialaspossibletotheparticlesproducedinthecollision. Theproximityofthepixelandsilicondetectorstothecollisionpointandtheveryfine segmentation(50×400 µm forthepixeldetectorand70 µm forthesilicondetector) allowforthereconstructionofsecondaryverticesfrombottomandcharmparticles, whichcantraveldistancesofafewmmfromtheinteractionpointbeforedecaying. Thenexttrackingdevice(4),thetransitionradiationdetector,isastraw-tubedetector thatprovidesinformationnotonlyonthetrajectoryofthechargedparticlebutalso onthelikelihoodoftheparticlebeinganelectron.Allthreetrackingdevicessitinside thecentralmagneticfieldof2Tproducedbythesolenoid(3).
Theenergiesoftheparticlesproducedinthecollision(bothneutralandcharged) aremeasuredbytheATLAS calorimeters,thelead-liquidargonelectromagneticcalorimeter(7)andtheiron-scintillatorhadroniccalorimeter(Tilecal)(8).BoththeATLAS and CMS electromagneticcalorimeterdesignsemphasizedgoodresolutionforthemeasurementoftheenergiesofphotonsandelectrons,primarilytobeabletodistinguishthe Higgsbosonto γγ signalfromthemuchlargerdiphotonbackground.Thewidthofa lightHiggsbosonismuchlessthantheexperimentalresolution,soanyimprovement intheresolutionwillleadtoabetterdiscriminationoverthebackground.
Energeticmuonscanpassthroughthecalorimetry,whileotherparticlesareabsorbed.Thetoroidalmagnets(2),inboththecentralandforwardregions,producean additionalmagneticfield(4T)inwhichasecondmeasurementofthemuonmomentumcanbecarriedoutusingthemuontrackingchambers(1),usingseveraldifferent technologies.OneoftheuniquecharacteristicsoftheATLAS detector(andpartofits acronym)isthepresenceoftheair-coretoroidalmuonsystem.Therelativelysmall amountofmaterialinthetrackingvolumeleadstolessmultiplescatteringandthus amoreprecisemeasurementofthemuon’smomentum.Themuonmomentumcanbe measuredtoaprecisionof10%atatransversemomentumvalueof1TeV.
1.1.3.3Challenges Touseapopularanalogy,samplingthephysicsattheLHC issimilartotryingtodrink fromafirehose.Over1billionproton-protoncollisionsoccureachsecond,butthe limitofpracticaldatastorageisontheorderofhundredsofeventspersecondonly. Thus,theexperimentaltriggershavetoprovideareductioncapabilityofafactorofthe orderof107,whilestillrecording bread-and-butter signaturessuchas W and Z boson production.Thisrequiresahighlevelofsophisticationfortheon-detectorhardware triggersandaccesstolargecomputingresourcesforthehigher-leveltriggering.Timing
2Themainlimitationforthesolid-anglecoverageisintheforward/backwarddirections,wherethe instrumentationiscutoffbythepresenceofthebeampipe.
Fig.1.2 AlayoutoftheATLAS detector,showingthemajordetector components,from en.wikipedia.org/wiki/ATLAS experiment.Original imagefromCERN.ReprintedwithpermissionfromCERN.
isalsoanimportantissue.TheATLAS detectoris25mindiameter.Withabunchcrossingtimeof25ns,thismeansthatasnewinteractionsareoccurringinonebunch crossing,theparticlesfromthepreviousbunchcrossingarestillpassingthroughthe detector.Eachcrossingproduces25interactions.Experimentalanalysesthusfaceboth in-timepileupandout-of-timepileup.Thelattercanbelargelycontrolledthroughthe readoutelectronics(modulosubstantialvariationsinthepopulationoftheindividual bunches),whiletheformerrequiressophisticatedtreatmentinthephysicsanalyses.
ThedynamicrangesattheLHC arelargerthanattheTEVATRON.Leptonsfrom W bosondecaysontheorderoftensofGeVarestillimportant,butsoaremulti-TeV leptons.Precisecalibrationandthemaintenanceoflinearityarebothcrucial.Tosome extent,theTEVATRON hasservedasabootcamp,providingalearningexperiencefor physicsattheLHC,albeitatlowerenergiesandintensities.Cominglater,theLHC has benefitedfromadvancesinelectronics,incomputing,andperhapsmostimportantly, inphysicsanalysistools.Thelattercomprisebothtoolsfortheoreticalpredictionsat higherordersinperturbativeQCDandtoolsforthesimulationofLHC finalstates.
Despitethedifficulties,theLHC hashadgreatsuccessduringitsinitialrunning, culminatinginthediscoveryoftheHiggsboson,but,alas,notinthediscoveryofnew physics.Theresultsobtainedsofarcompriseasmallfractionofthetotaldatataking plannedfortheLHC.Newphysicsmaybefoundwiththismuchlargerdatasample, butdiscoveringitmayrequirepreciseknowledgeofSMphysics,includingQCD.
1.2Aboutthisbook ThereaderisassumedtobealreadyfamiliarwithtextbookmethodsforthecalculationofsimpleFeynmandiagramsattreelevel,theevaluationofcross-sectionsthrough phase-spaceintegrationwithanalyticterms,andtheideasunderlyingtheregularizationandrenormalizationofultravioletdivergenttheories;however,forashortreview,
readersarereferredtoAppendixB.1,andforamorepedagogicalintroductiontothese issuestoawealthofoutstandingtextbooksonvariouslevels,includingthebooksby PeskinandSchr¨oder[803],HalzenandMartin[606],Ramond[822],Field[525]and others.ForareviewofQCDatcolliderexperiments,thereaderisreferredtothe excellentbooksbyEllis,Stirling,andWebber[504]andbyDissertori,Knowles,and Schmelling[467].Ofcourse,forarealunderstandingofvariousaspectsitishardto beattheoriginalliterature,andreadersareencouragedtousethereferencesinthis bookasastartingpointfortheirjourneythroughparticlephysics.
Thisbookaimstoprovideanintuitiveapproachastohowtoapplytheframework ofperturbativetheoryinthecontextofthestronginteractiontowardspredictionsat theLHC andultimatelytowardsanunderstandingofthesignalsandbackgroundsat theLHC.Thus,evenwithoutthebackgrounddiscussedatthebeginningofthissection, thisbookshouldbeusefulforanyonewishingforabetterunderstandingofQCDat theLHC.
Theideasforthisbookhavebeendevelopedovervariouslectureseriesgivenat graduatelevellecturesoratadvancedschoolsonhigh-energyphysicsbytheauthors. Theauthorshopethatthisbookturnsouttobeusefulinsupportingtheself-study ofyoungresearchersinparticlephysicsatthebeginningoftheircareeraswellas moreadvancedresearchersasaresourcefortheiractualresearchandasmaterialfor agraduatecourseonhigh-energyphysics.
1.2.1Contents Chapter2providesafirstoverviewofthecontentofthisbookandaimsatputting varioustechniquesandideasintosomecoherentperspective.Firstofall,aphysical pictureunderlyinghadronicinteractions,andespeciallyscatteringreactionsathadron colliders,isdeveloped.Toarriveatthispicture,theideasunderlyingtheall-important factorizationformalismareintroducedwhich,intheend,allowstheuseofperturbative conceptsinthediscussionofthestronginteractionathighenergiesandthecalculation ofcross-sectionsandotherrelatedobservables.Theseconceptsarethenusedina specificexample,namelytheinclusiveproductionof W bosonsathadroncolliders. There,theirproductioncross-sectioniscalculatedatleadingandatnext-to-leading orderinthestrongcouplingconstant,therebyremindingthereaderoftheingredients ofsuchcalculationsandfixingthenotationandconventionsusedinthisbook.This partalsoincludesafirstdiscussionofobservablesrelevantforthephenomenologyof stronginteractionsathadroncolliders.Inaddition,somegenericfeaturesandissues relatedtosuchfixed-ordercalculationsaresketched.Inasecondpart,theperturbative conceptsalreadyemployedinthefixed-ordercalculationsareextendedtoalsoinclude dominanttermstoallordersthroughtheresummationformalism.Genericfeaturesof analyticalresummationareintroducedthereandsomefirstpracticalapplicationsfor W productionathadroncollidersarebrieflydiscussed.Asasomewhatalternativeuse ofresummationtechniques,jetproductioninelectron–positronannihilationsandin hadroniccollisionsisalsodiscussedand,especiallyinthelatter,somecharacteristic patternsaredeveloped.
Thenextchapter,Chapter3,isfairlytechnical,asitcomprisesapresentationof mostofthesometimesfairlysophisticatedtechnologythatisbeingusedinorderto
evaluatecross-sectionatleadingandnext-toleadingorderintheperturbativeexpansionofQCD.Italsoincludesabriefdiscussionofemergingtechniquesforevenhigher ordercorrectionsinQCD.Inaddition,theinterplaybetweenQCDandelectroweak correctionsistoucheduponinthischapter.Startingwithadiscussionofgenericfeatures,suchasameaningfuldefinitionofperturbativeordersforvariouscalculations, thecorrespondingtechnologyisintroduced,representingthecurrentstateoftheart. Assimpleillustrativeexamplesforthemethodsemployedinsuchcalculations,again inclusive W bosonproductionanditsproductioninassociationwithajetareemployed.Thecalculationsareworkedoutinsomedetailatbothleadingandnext-to leadingorderintheperturbativeexpansioninthestrongcoupling.
Theoverallpictureandphenomenaencounteredinhadron–hadroncollisions,developedinChapter2,isdiscussedinthecontextofspecificprocessesinChapter4. Theprocessesdiscussedhererangefromthecommonplace(e.g.jetproduction)to someofthemostrare(e.g.productionofHiggsbosons).Ineachcasetheunderlying theoreticaldescriptionoftheprocessisdescribed,typicallyatnext-toleadingorder precision.Specialemphasisisplacedonhighlightingphenomenologicallyrelevantobservablesandissuesthatariseinthetheoreticalcalculations.Thechaptercloseswith asummaryofwhatisachievablewithcurrenttechnologyandanoutlookofwhatmay becomeimportantandrelevantinthefuturelifetimeoftheLHC experiments.
FollowingthelogicoutlinedinChapter2,inChapter5thediscussionoffixed-order technologyisextendedtotheresummationofdominantterms,connectedtolargelogarithms,toallorders.Afterreviewinginmoredetailstandardanalyticresummation techniques,anddiscussingtheirsystematicimprovementtogreaterprecisionbythe inclusionofhigher-orderterms,theconnectiontootherschemesishighlighted.Inthe secondpartofthischapter,numericalresummationasencodedinpartonshowersisdiscussedinsomedetail.Thephysicalpictureunderlyingtheirconstructionisintroduced, somestraightforwardimprovementsbyintroducingsomegenerichigh–ordertermsare presentedanddifferentimplementationsarediscussed.Sincethepartonshowersare attheheartofmoderneventsimulation,bridgingthegapbetweenfixed-orderperturbationtheoryathighscalesandphenomenologicalmodelsforhadronizationand thelikeatlowscales,theirimprovementhasbeeninthefocusofactualresearchin thepastdecade.Therefore,somespaceisdevotedtothediscussionofhowthesimple partonshowerpictureissystematicallyaugmentedwithfixed-orderprecisionfromthe correspondingmatrixelementsinseveralschemes.
InChapter6,animportantingredientforthesuccessofthefactorizationformalismunderlyingtheperturbativeresultsintheprevioustwochaptersisdiscussedin moredetail,namelythepartondistributionfunctions.Havingbrieflyintroducedthem, mostlyatleadingorder,inChapter2,andpresentedsomesimpleproperties,inthis chapterthefocusshiftsontheirscalingbehaviouratvariousordersandhowthiscan beemployedtoextractthemfromexperimentaldata.Variouscollaborationsperform suchfitswithslightlydifferentmethodologiesandslightlydifferentbiasesinhowdata areselectedandtreated,leadingtoavarietyofdifferentresultingpartondistributions.Theyarecomparedforsomestandardcandlesinthischapteraswell,witha specialemphasisonhowtheintrinsicuncertaintiesinexperimentaldataandthemore theoreticalfittingproceduretranslatesintosystematicerrors.
Thetourofingredientsforacompletepictureofhadronicinteractionsterminatesin Chapter7,wheredifferentnon-perturbativeaspectsarediscussed.Mostoftheideasto addressthemarefairlyqualitativeandcanbeembeddedinphenomenologicalmodels only.Therefore,ratherthanpresentingindetailalldevelopmentsinthisfield,the bookfocusesmoreongenericfeaturesandbasicstrategiesunderlyingtheirtreatment indifferentcontexts.Issuesdiscussedthereincludehadronization,thetransitionfrom thepartonsofperturbationtheoryofthestronginteraction,quarksandgluons,tothe experimentallyobservablehadrons,andtheirdecaysintostableones,theunderlying event,whichisduetosofterfurtherinteractionsbetweenthehadronicstructuresof theincidentparticles,anditsconnectiontoveryinclusiveobservablessuchastotal andelasticcross-sections.
InChapters8and9theoreticalresultsfromanalyticcalculationsandsimulation toolsarecomparedwithahostofexperimentaldata.Chapter8focusesondataespeciallyfromtheTEVATRON,3 wherethefoundationsofourcurrentunderstandingof theSMandinparticularthedynamicsofthestronginteractionhavebeenshaped. InChapter9themostsophisticatedcalculationsandsimulationsarecomparedwith themostrecent,mostpreciseandmostchallengingdatasofar,takenattheLHC duringRun I.Thiscomparisonrangesfrominclusiveparticleproductionoverevent shapeobservablestodatatestingthedynamicsoftheSM—andpotentiallybeyond —overscalesrangingovertwoorderofmagnitudeinthesameprocess.Thisisthe mostchallengingtestofourunderstandingofnatureatitsmostfundamentallevel everperformed.Itisfairtostatethatwhileourmostup-to-datetools,analyticalcalculationsandsimulationsfareamazinglywellinthiscomparison,somefirstcracksare showingthatwillmotivatethecommunitytopushevenfurtherintheyearstocome.
1.2.2Auser’sguide ThisbookismeanttoprovidePhDstudentsinexperimentalparticlephysicsworking attheLHC whohaveakeeninterestintheoreticalissues,aswellasPhDstudents workinginparticletheorywithanemphasisonphenomenologyatcolliders,astarting pointfortheirresearch.Itismeanttointroduceandexposethereadertoallrelevant conceptsincurrentcolliderphenomenology,introduceandexplainthetechnologythat bynowisroutinelyusedintheperturbativetreatmentofthestronginteraction,and provideanintegratedperspectiveontheresultsofsuchcalculationsandsimulations andthecorrespondingdata.
Thebookconsistsofthreeparts.Thefirstpartisanoverviewoftherelevant terminologyandtechnology,workedoutthroughonestandardexampleandproviding acoherentperspectiveonhadronicinteractionsathighenergies.Readersandteachers, usingthisbookforlectures,areinvitedtostudyChapter2firstbeforeembarkingon amorein-depthdiscussionofvarioustheoreticalorexperimentalaspects.Theother twopartsconsistofamoredetaileddiscussionofvariousaspectsoftheperturbative treatmentofthestronginteractioninhadronicreactionsinthesecondpartofthebook, inChapters3–7.Whilethesechaptersfrequentlyreferbacktotheoverviewchapter, Chapter2,theyarefairlyindependentfromeachotherandcouldinprinciplebeusedin
3ExperiencesfromLEP andHERA havealsobeenimportantbutarenotincludedduetospace limitations.
anysequencethereaderorteacherfindsmostbeneficial.Thethirdpart,Chapters8and 9,wherecoreexperimentalfindingsareconfrontedwiththeoreticalpredictions,again isindependentofthesecondpart,althoughforabetterunderstandingoftheoretical subtletiesitmaybeadvantageoustobeacquaintedwithcertainaspectsthere.
Finally,alistofupdates,clarificationsandcorrectionstothisbookismaintained atthefollowingwebsite:
http://www.ippp.dur.ac.uk/BlackBook
HardScatteringFormalism Beforeembarking,inthischapter,onafirstdiscussionofthefactorizationformula andsomeofitsimmediateconsequencesintermsofactualphenomenaandcalculations,inSection2.1anintuitivepictureofhigh-energyreactionsinvolvinghadronsin theinitialstatewillbedeveloped.Thispictureinfactformsthephysicalbackground ofthefactorizationformalism,whichinturnprovidesthetheoreticalfoundationsof thisbook.
Inthenextsection,Section2.2,theideasformulatedintheprevioussectionwill befurtherformalizedandcondensedintoadiscussionoftheperturbativetreatmentof high-energyreactionsathadroncollidersatfixedorder.Toillustratethefactorization formalisminaction,thecaseofinclusive W -bosonproductionwillbeanalysedat leadingandatnext-to-leadingorder.
InSection2.3theperturbativeformalismdevelopedsofarwillbefurtherexpanded toincludethemostimportanteffectstoallperturbativeorders.Thisisachieved throughtheidentificationandsubsequentresummationofthecorrespondingleading terms.Again,thecaseof W bosonproductionservesasthemainillustrativeexample.
Finally,inSection2.4,somegeneralthoughtsandissuesrelatedtothedescription ofhardprocessesthroughperturbationtheorywillbedeepened.
2.1Physicalpictureofhadronicinteractions 2.1.1Electromagneticanalogy
Toillustratethephysicalpictureunderlyingfactorizationinprocessesinitiatedbycollidingstronglyinteractingparticlessuchasprotonsathighenergies,thesimplercase ofcollisionswithleptonsintheinitialstateisconsideredfirst.Inthis,anintuitive understandingofthetheoryofstronginteractions,QCD,quantumchromodynamics, willbedevelopedincloseanalogywiththesimplertheoryofQED,quantumelectrodynamics.Inbothcases,althoughultimatelyverydifferent,thecollidingparticleswill emitsecondaryquantainanintricateradiationpattern,whichmustbetakeninto accounttogainfulltheoreticalcontroloverthecollisiondynamics.
Experimentally,forexampleincollisionsofelectron–positronpairs,itisofcourse typicallynotdifficulttorequirethattheenergyintheactualcollisionisclosetothe
TheBlackBookofQuantumChromodynamics:APrimerfortheLHCEra. JohnCampbell,JoeyHuston,andFrankKrauss. ©JohnCampbell,JoeyHuston,andFrankKrauss2018.Publishedin2018byOxfordUniversityPress. DOI10.1093/oso/9780199652747.001.0001
centre-of-massenergyofthecollidingbeams,thuseffectivelyreducingtheamount ofenergycarriedawayfromtheleptonsthroughelectromagneticradiation.However, mostofthetime,especiallywhentheircombinedinitialinvariantmassisabovethe massofaresonance,suchasthe Z boson,theleptonswillreactwithactualenergies thatarereducedwithrespecttotheirfullavailableenergy.Thiseffectissometimes called“radiativereturn”Thecorrespondingenergylossisduetotheemissionof photonsfromtheincidentleptons,aprocessdenotedasQEDinitial-stateradiation (ISR)..
WhileinQEDthetreatmentofISRpotentiallyisatediousbutessentiallystraightforwardexercise,tractablewithperturbationtheory,inQCDtheproblemismuchmore involvedandfundamentallydifferent.ThisisbecauseinQCD,thecollidingparticles cannotbeinterpretedasthefundamentalquantaofthetheorybutratherasbound states,hadronssuchasprotons,whichcannotbequantitativelyunderstoodanddescribedthroughthelanguageofperturbationtheory.Thisconceptualgapnecessitates theconstructionofaframeworktoprovidedirectandsystematicallyimprovablecontactbetweentheprovenlanguageofperturbativecalculationsofcorrespondingcrosssectionsandthenon-perturbativestructureofthehadronicboundstates.Thissection isdevotedtodevelopinganintuitivepictureofhowthis factorization framework actuallyworks,bydwellingonthelimitedanalogywiththeemissionsofsecondary quantainQEDandthedifferencesbetweenelectromagneticandstronginteractions whenconsideringsuchcollisionsingreaterdetail.
2.1.1.1Equivalentquanta Classically,thephenomenonofinitial-stateradiationcanbeunderstoodwiththe equivalentphotonpicture [523,884,892].Initsownrestframe,theleptonactsas thepoint-likesourceofapurelyradialelectricfield,withnomagneticfieldpresent. ThissituationisdepictedintheleftpanelofFig.2.1.Boostingtheleptonintoany framewhereitisnotatrestand,inparticular,intothelaboratoryframetransforms thestaticsourceofanelectricfieldintoanelectromagneticcurrent,whichinturn producesalsoamagneticfield.Withincreasingleptonvelocity, v → c,thetwofields becomeincreasinglyconfinedtoaplaneperpendiculartotheaxisofmotionandthey alsobecomemoreandmoreperpendiculartoeachother,withtheelectricalfieldradialandthemagneticfieldcirculararoundtheaxisofmotion, cf. themiddlepanelof Fig.2.1.Thisorthogonalityallowstheidentification,inastraightforwardway,ofthis classicalconfigurationwithquantaoftheelectromagneticinteraction,thephotons, shownintherightpanelofFig.2.1.Theaccumulatedenergyflux–thatis,thetotal energycarriedbythefieldsaccompanyingthelepton–isobtainedbyintegratingthe Poyntingvectorovertheplaneorthogonaltothelepton’saxisofmotion.Itisthus paralleltothelepton’smotion.Interpretingthecontinuousenergyfluxwithaflux ofequivalentquanta,thephotons,thenumberdensityofaccompanyingphotonsper energyintervalanddistancefromtheleptoncanbededucedas
Fig.2.1 Electricalandmagneticfieldsinblueandgreenofaleptonat rest(v =0,leftpanel)andwithavelocity v ≈ c (middlepanel).The equivalentphotonsaredepictedontherightpanel.
Here ω istheenergyoftheequivalentphoton,andtheconstantofproportionality isobtainedbyintegratingoverthetransverseplane,parameterizedbytheimpact factor b⊥,andbytheelectromagneticcouplingconstant.Thelattergetsmodifiedby therelativechargeoftheelectron, ee,whichofcourseequals-1.Thereisamaximal energyavailablefortheseequivalentphotons;naivelyitisgivenby ωmax = E,the energyofthelepton.
Inamorequantumfield-theoreticalwayofthinkingaboutthis,theimpactparameterisreplacedwiththetransversemomentum, b⊥ ←→ k⊥,throughaFourier transform,andtheequivalentphotonsareconsideredtobepartofthelepton’swave function.Theirspectrumthenreads
Insuchapicture,thephysicalleptonisgivenbyasuperpositionofstateswithvarying photonmultiplicity,
wherethephotonshavedifferentenergiesandtransversemomenta.Duetomomentum conservation,theyaretypicallyofftheirmassshell.Thislimitsthelifetimeofthe quantum-fluctuationslike |eγ or |eγγ
Toseehowthisworksinsomewhatmoredetail,considerthecaseofthe |eγ Fock state.Assumingmasslesson-shellelectronsintheinitialandfinalstate,butallowing thephotonstogooff-shell,thekinematicsofasplitting e(P ) → e(p)+ γ(k)canbe writtenas
Parameterizingthesplittingoftheenergy, E of P suchthat ω ≡ Eγ = xE and therefore Ee =(1 x)E fortheenergyoftheoutgoingelectron,leadsto
inthecasewhere x issmall, i.e.,forthebulkofthephotons.Thisalsoimpliesthat inthislimitthemomentumcomponentofthephotonparalleltotheelectronisabout k ≈ ω,theenergyofthephoton.Thefactthattheemittedphotons(ortheelectrons orboth)areoff-shellisequallytrueformassiveelectrons,anditlimitsthelifetimeof the(eγ)-componentofthewavefunctionthroughtheuncertaintyprinciple.Withthe photonmomentumgivenby kµ ≈ (ω, k⊥,ω),theenergyshiftnecessarytomoveiton itsmassshellisgivenby δω ≈ k2 ⊥/2ω yielding
asthe lifetimeofthefluctuation.Thisimpliesthatsuchfluctuationslivelonger, astheenergyofthephotonincreasesandtherelativetransversemomentumofthe photonandtheelectrondecrease.Notethatinthisbooknaturalunitsarebeingused, soeffectively = c =1.
Inaddition,ofcourse,thephotonscansplitintoafermion–anti-fermionpair,asyet anotherquantumfluctuation,whichinturnmayemitfurtherphotons.Thiscomplicatesthewave-functionpictureevenmore.However,altogether,thedistancefromthe originalleptonandthelifetimeofallthesevirtualparticles,thequantumfluctuations whichtheymanifest,aregivenbytheamounttheyareofftheirmassshellandbythe amountofenergytheycarry.
2.1.1.2Dokshitser–Gribov–Lipatov–Altarelli–Parisiequations
Thewave-functionideacanberelated,atleadingorder,byprobabilitiesoffinding theoriginallepton,thephotons,orsecondaryleptonsatgivenenergiesandtransverse momenta.Duetothenatureoftheelectromagneticinteraction,theseprobabilities (or,theleptonwavefunction),canbecalculatedfromfirstprinciples.Inthefollowing theprobabilityoffindingaleptonorphotonatenergyfraction x (withrespectto theoriginallepton)andattransversemomentum k⊥ willbedenotedas (x,k⊥)and γ(x,k⊥),respectively.At leading orderin α,thatiswithoutanyemissionofsecondary quanta,theyareofcoursegivenby
(x,k2 ⊥ =0)= δ(1 x)and γ(x,k2 ⊥ =0)=0 (2.7)
Theradiationthatactuallygivesrisetotheaforementionedsecondaryparticlescan nowbedescribed,toallorders,byequationsthatareknownas evolutionequations, sincetheyrelatetheprobabilitytofindquantawithcertainkinematics x and k2 ⊥ to similarprobabilitiesatotherscales,throughemissionofsecondaries.Ingeneral,they canbeobtainedindifferentapproximationsrelatedtodifferentkinematicalsituations, whichcanbeintimatelyrelatedtodifferent factorizationschemes.Inthe collinear factorizationscheme,whichwillbeemployedinmostoftheremainderofthis
book,theenergyandespeciallytransversemomentumofthesecondaryparticlesare consideredtobesmallwithrespecttotheenergyoftheoriginallepton,andtherefore thekinematicaleffectmerelyamountstoasuccessivereductionoftheoriginallepton energy.Inthisscheme,theprobabilitydensitiesevolvewiththelogarithmofthe transversemomentum, cf. Eq.(2.2),as
(2.8)
wheretheeffectofphotonssplittingintovirtuallepton–anti-leptonpairshasbeen omitted.Thefirstlineoftheseequationsexhibithowtheprobabilitydensityofleptons withsmallerenergyfraction x isdrivenbyleptonswithalargerenergyfraction x/ξ> x,whichcanbeinterpretedastheleptonsontherighthandsideoftheequationlosing anenergyfraction1 ξ intheemissionofaphoton.Inthekinematicalapproximation employedhere,termslike P (x/ξ,α),thesplittingkernels,canbeunderstoodas reducedmatrixelementsfortheemissionofonephotonoffthelepton.Ingeneral, Pba denotessuchkernelsforatransitionofaparticleoftype a toaparticleoftype b,while emittingaparticleoftype c,whichisnotbeenmadeexplicit.Thesekernelshavebeen takenatleadingorder,asmanifestbytheexplicitorderin α infront,butofcourse theycanalsobeevaluatedtohigherordersinaperturbativeexpansioninpowersof thecouplingwithcorrespondingcoefficientfunctions.
Closerinspectionrevealsthatthissetofequationsisnothingbutthecelebrated Dokshitser–Gribov–Lipatov–Altarelli–Parisi(DGLAP)equation,specifiedfor QED.Inthisequation,the splittingkernels atleadingorderareindependentofthe couplingconstantandread
Here,thenotationof“+”–functionshasbeenemployed,whichwillcropupagain atvariousplaces,especiallyinconjunctionwithsplittingkernelssuchastheones discussedhere.Theyaredefinedthroughtheirintegraltogetherwithatestfunction g(z)suchthat
Forfurtherdetails,thereaderisreferedtoAppendixA.1.2.Togainamoreintuitive understanding,considerthisprescriptiontoworkinsuchawaythatthepolefor z → 1in P isexcludedinthe+-functionandreinsertedthroughthesecondpartof
thesplittingkernel,proportionaltothe δ-function.Itwillbeseenlater,howsuchterms becomenecessarytoensurethecorrectphysicalbehaviourofthesplittingfunctions, suchassatisfyingmomentumconservation.
However,theseequationswillberevisitedanddiscussedinmoredetailinlater chaptersofthebook.Tofollowthereasoninginthismoreintroductorychapterit shouldsufficetomentionthatthe1/ω polepresentinthenaiveclassicalpicturehas itscounterpartinthe1/(1 z)or1/z termsappearinginthesplittingfunction.
2.1.1.3Initial-stateradiation Turningawayfromthedetailsofhowtheleptonwavefunctionisevaluatedandback tothephysicalconsequencesoftheexistenceofthefluctuations,theobviousquestion is:whathappensinacollision?Cantheemergingpicturebeputtostringenttests andcantheexistenceofsuchfluctuationsbequantifiedbysuitableprobes?
Toanswerthis,atleastqualitatively,andtogainsomeinsightintothephysical processestakingplaceinacollision,considerfirstthecaseofnocollisionatall.There, thefluctuationshaveafinitelifetimeanddistancetotheoriginallepton,relatedto thekinematicsofthevirtualparticles.Four-momentumconservationthusforcesthese fluctuationstoeventuallycollapsebackintotheoriginallepton,guaranteeingthequantumcoherenceofthelepton–itmustremainintact.If,ontheotherhand,acollision takesplace,oneormoreofthequantumexcitations,theparticlesformingthelepton’s Fockstate,may,throughtheexchangeoffour-momentumwiththeincidentprojectile, goontheirmassshell.Inthiswaytheyacquireaninfinitelifetime.Insuchasituation, thecorrespondingparticleswillnotfallbackontotheoriginalleptonandthecoherenceofthefluctuationsisthereforebroken.Inthisway,oneormoreofthehitherto virtualparticlesmaybecomerealandhencephysicallyobservableinthefinalstateof thecollision.
Toillustratethis,considerthecaseof e+e collisions,whichhavebeeninvestigated ingreatdetailforexampleatLEP,andassumethatitistheelectronandpositronthat takepartinthecollision.Thenthecentre-of-massenergy ˆ Ec m ofthisactualreaction maybereducedwithrespecttotheenergyobtainedfromthenominalbeamenergies, ˆ Ec m <Ec m =2Ebeam. 1 Thedifferenceisstoredintheenergiesofthephotons accompanyingtheelectron–positronpair;theirdistributionintransversemomenta andenergiesisgivenapproximatelybyEq.(2.2).Inmoremodernlanguagethese photonswouldbeattributedto initial-stateradiation offtheincident“original” pair,ratherthantothequantummanifestationofthebreakdownofcoherenceof theFockstatesdescribingtheleptonsandtheiraccompanyingelectromagneticfields. Nevertheless,thisradiationwould,ofcourse,bedescribedbyequationsverysimilar totheapproximateone.
Itshouldbestressed,however,that,takenonitsown,thisinitial-stateradiationis amanifestationofthebreakdownofquantumcoherenceofcomplicatedFockstates, bindingthephotonstotheoriginalleptons.
1Here,andthroughoutthebook,kinematicalquantities q relatedtothecollidingpartonsare supplementedwithaˆ q,whilethoserelatedtothe(beam-)particlesareleftwithoutitas q
2.1.1.4Final-stateradiation
Asimilarpicturealsoemergeswhenchargedparticlesareproducedinthefinalstate. Asanexample,considerthecaseof,say,muonpairproductioninelectron-positron annihilations, e+e → µ+µ .Classically,theproductionofthemuonscanbeunderstoodastheirinstantaneousaccelerationafteremergingfromafiniteenergydensity relatedtothepreviousannihilationoftheelectronandpositronintoelectromagnetic fields,theintermediatephotoninquantumfieldtheory.Inclassicalelectrodynamics suchanaccelerationofchargedparticlestriggerstheradiationofadditionalphotons offthecharges,knownas Bremsstrahlung 2 Interpretingthemuonpairasacurrentgoingfromavelocity v to v attheoriginofthecoordinatesystem,thedouble differentialclassicalradiationspectrum I inthedirection n(Ω)withpolarization reads
inthedominantregionofsmallenergies,wherethesquaredtermisknownasthe radiationfunction W[645].Formasslessparticlestravellingatthespeedoflight, theradiationfunctioncanberewrittenas
Castinitscovariantformandinterpretedasaphotonspectrumandtacitlyinserting α = e2/(4π),theradiationspectrumbecomes
forthenumberofphotons N emittedintheprocess.Here, k and denotethephoton’s four-momentumandpolarizationfour-vector,while p and p denotethefour-momenta ofthemuons.Theformabove,
isalsoknownasthe eikonal form,anditisidenticalwiththeresultofafull-fledged calculationinquantumfieldtheory,asfollows.
Forthecaseathand,considerthephotonemissionpartoffthemuons,whichare assumedtobemassless.ThemuonsareproducedthroughavertexfactorcalledΓ.The leading-orderFeynmandiagramsaredepictedinFig.2.2.Thecorrespondingmatrix elementfor X → µ (p)µ+(
)γ(k)isgivenby
2Inthequantumpicture,equivalently,thistranslatesintotheradiationofBremsstrahlungphotons.
Fig.2.2 Feynmandiagramsfortheemissionofonephotonbyamuon pairatthelowestperturbativeorder.
wheretheequationsofmotion
formasslessparticlesaswellastheanti-commutatorrelation {γµ,γν } =2gµν have beenused.Notethattheonlyeffectofthemuonspinistheoccurrenceofemissions throughmagneticterms, 1 2 [γµ,k/],whiletheterms2pµ kµ donotbearanymemory ofthemuonshavingspin.Goingtothelimitofsoftphotonemission,terms ∝ k inthe numeratorvanish,resultingin
Inotherwords,inthelimitofsoftphotonemission,theemissiontermcompletely factorizesfromtheproductionofthesystememittingthephotonandisjustgivenby the eikonal term W(p,p ; k, ).
Itisworthnotingthatsoftphotonemissionisindependentfromthespinofthe emittingparticlesandotherinternalpropertiesandbyitsverynatureitisaclassical phenomenon.Thisistheessenceof Low’stheorem [734].Asaconsequence,soft photonscanbethoughtofasessentiallynotcarryinganyquantumnumbers,and, therefore,theemissionofsoftphotonswillnotliftanyveto–forexample,dueto C-parityorangularmomentum–foratransitiontohappen.Suchtransitions,inthe caseathand,forinstance,fromonehelicitytoanother,necessitatetheemissionof a hard photon,describedbytheterms ∝ k.Incontrasttothesoft,classicalterms, theseessentiallyquantum-mechanicalcontributionsdonotexhibitanysoftdivergence dω/ω butratherbehavelikedω ω.
Thesimpleclassicalpictureofsoftphotonemissioninprincipleallowsadescription ofthepatternofphotonradiationfromthemuonpair,byiteratingemissionsthrough
theeikonalterm.Thisimplicitlyassumesthattheindividualemissionsareindependentfromoneother.Thekeypointhereistheintroductionofthenotionofphoton resolution.Thereasonforthisistheobservationthattheeikonalfactordivergesfor k → 0,thesoftdivergence,orfor k parallelto p or p ,thecollineardivergence.This isnothingbutthewell-knowninfrared-catastropheofQED,apatternthatoccursfor everytheorywithmasslessspin-1bosonssuchasQEDorQCD.Essentiallyitcanbe explainedbythefactthatitmakesnophysicalsensetoaskhowmanyphotonsare emittedbyaparticle,withoutspecifyinghowthephotonsaremeasured.Inpractice, photonscanbetoosoftforadetectortorespond;atthesametime,iftheyaretoo parallelwiththeemittingparticletheywillendupinthesamedetectorcell,which musthaveafinitesize,andthuswillnotbedistinguished.Therefore,thephasespace ofthephotonsmustbeconstrainedtodetectablephotonstomakeanysense.This isachievedbyappropriatecuts,forinstancebydemandingaminimalenergyanda minimalanglewithrespecttotheemittingparticle.
Therearenowinprincipletwowaysofhowthefullphotonradiationpatterncan bedescribed:inamoredirectapproach,theintegraloftheeikonalfactoroverthe constrainedphotonphasespacecouldbeusedastherelevantterminaPoissonian distributionofthenumberofphotonsbeingemitted,theindependenceoftheindividualemissionsguaranteeingthePoissoniancharacterofthisdistribution.Foreach photonthen,thephasespacecouldbeindividuallyfixed.Alternatively,theemissions couldbeorderedin,say,theenergyoftheemittedphotons,or,asasomewhatpreferredchoice,therelativetransversemomentumwithrespecttotheemitter.This wouldallowaredefinitionoftheradiationpatternthroughaprobabilisticpicture, drivenbythe Sudakovformfactor.Inordertoseehowthisworksinmoredetail, cf. Section2.3andChapter5.
Inanycase,itisworthnotingthattheeikonalpicturehere,condensedinEq.(2.17) canbedirectlytranslatedtotheequivalentquantapictureaboveandtheslightlymore sophisticatedversionencodedintheDGLAPevolutionequations,Eq.(2.8).
2.1.2Boundstates,stronginteractionsand αs Thissimple,qualitativepicturewillnowbeextendedtothecaseofincidenthadrons -forsimplicitytheywillbeassumedtobe(anti-)protons.Naively,theyconsistof three valencequarks,whichcarrythequantumnumbersoftheproton—electrical charge,spin,andisospin.Inanaivepicture,wherethequarksjuststicktogether withnodiscernibleinteractionsresponsibleforit,itwouldbeafairguesstoassume thattheyareallequallyimportantandthusallcarrythesameenergyfractionofthe proton,namely1/3.
Switchingonnaiveinteractions,actinglikerubberbandsgluingthevalencequarks together,onecouldassumethattheoriginalsharpdistributioniswashedout-but thattheenergyfractiondistributionofthevalencequarksstillhasameanvalueof1/3. Ofcourse,inviewofthepreviousdiscussionoftheQEDcase,andbecausetheQCD couplingconstantismuchlargerthantheQEDone,thissimplisticpicturecannot holdtrue.Infact,takingintoaccountquantumcorrectionstothestrongcoupling constant, αs,whicharemediatedbyloopdiagramsdepictedinFig.2.3,thepictureis abitmorecomplicated.
Fig.2.3 TheleadingquantumcorrectionstotherunningoftheQCD couplingconstant αs:gluonselfenergyatonelooporder(ghostdiagrams areignored).
2.1.2.1Therunningof αs
Includingsuchcorrections,anddenotingallcouplings—QEDandQCD—collectively with α = g2/(4π)theyvarywiththescale µR,alsoknownas (renormalization) scale,as
wherethe β–function β(α)canbeexpandedinaperturbativeseriesandreads
Herethecustomaryrelation
S /(4π),and,similarly,
/(4π)hasbeen assumed.
In SU (N )gaugetheory,thefirstcoefficients βi oftheperturbativeexpansionof the β-functionread
wherethenumberofactivefermionsisgivenby nf ,andwherethe Casimiroperators ofthegaugegroupinits fundamental and adjoint representationsare CF and CA. Itisworthstressingherethatthefirsttwocoefficients, β0 and β1 arerenormalization scheme–independent,whileallfurthercontributions,startingwith β2,dependonthe renormalizationscheme;theresultgivenherefor β2 istheoneinthe MSscheme. Foraverybriefreview, cf. AppendixB.1.
InQCD,theyaregivenby
Another random document with no related content on Scribd:
When Mr. Bushy Tail scrambled out of the fire-place, he found himself in a strange little underground room, from which low passage ways branched out in every direction.
He ran down one of the passage ways, but finding no one, he came back and tried another. At the end of this one, in a cosy little room, he came unexpectedly upon an old acquaintance, Mrs. Mole, who was taking a comfortable afternoon nap.
She awoke with such a start of surprise at seeing Mr. Bushy Tail, that she nearly fell out of her rocking-chair.
“I did not hear you knock,” said she.
“I did not knock, I dropped,” said he.
Then he told her of his accident, and apologized most politely, for falling so unceremoniously down her chimney.
Mrs. Mole assured him that he was a welcome visitor at any time, and only regretted that her chimney had tripped him up.
She was very sorry that her husband and sons were away on business, but urged him strongly to stay to supper.
With many thanks Mr Bushy Tail was obliged to decline her polite invitation, but he assured her that, considering his hungry family, he must hurry home with his bag of food as soon as possible, and begged her to kindly show him the nearest way out of her maze-like house.
When, after following Mrs. Mole through a number of long, winding, passages, Mr. Bushy Tail came at last to the surface of the ground, it was snowing hard, and the dreaded North wind was blowing half a gale.
He found himself outside the shelter of the woods, on a broad plain, and he felt that his only safety lay in getting back among the trees.
He started off at full speed, and had gone some distance, when suddenly the North wind struck him, and lifted him completely off his four little paws.
There was no use in struggling, so he lay quite still, and was whirled away, faster and faster. Miles and miles was he blown, until finally he fell asleep from sheer exhaustion and fright.
When he awoke it was night, and still the wild wind was carrying him far, far away.
His precious bag was still on his shoulder, for he had clung to it even in his terror, but the string had become untied, and most of the food had been blown away.
He ate a little of the corn, but he was too frightened to be hungry, and very soon, numb and dazed with the cold, he fell asleep again.
At the Top of the South Pole All the next day and night, the fierce North wind kept on blowing a gale, but towards morning of the third day, it seemed to Mr. Bushy Tail that he was going more slowly, and just as the sun rose, he was suddenly dropped.
What he rested on he could not tell, but in a few minutes, as it grew lighter, he looked about him, and saw it was a very high wooden post. Then he knew he was sitting on top of the South Pole, where the North wind has to stop blowing, or else it becomes a South wind.
It was beautifully warm, and poor Bushy Tail stretched himself most comfortably in the sunshine, and thawed out his half-frozen little paws.
He felt very much shaken-up and alone in the world, and, with tears in his eyes, he thought of his little wife and hungry babies, and wondered if he should ever get home to them.
In the first place, he could not see any possible way of getting down from his lofty perch. The sides of the Pole were very smooth and slippery, and the Pole itself was much too high for comfort or safety, if you tried jumping off.
While he was pondering ways and means of descent, he heard a high squeaky voice say:
“Try the middle course.”
Peering over the edge of the Pole, Mr. Bushy Tail espied on the ground, far below, a funny little creature, such as he had never seen
before. It was covered with long blue quills, and moved slowly, and with much dignity.
“What is the middle course?” asked Mr. Bushy Tail, timidly
“Why the middle of the Pole, you goose!” replied the squeaky voice.
Mr. Bushy Tail thought this mode of address not strictly polite, especially to a stranger, but he said nothing, and looked about him on top of the Pole.
Sure enough, right in the middle was a little winding staircase, down which he scrambled into darkness.
A Quilly Acquaintance The little staircase seemed very dark and pokey to Mr. Bushy Tail, and he devoutly hoped that nothing would jump out and bite him.
After a number of windings, however, he saw a faint light below him, and a few moments later, he stepped through a low doorway, and found himself close beside his quilly acquaintance.
“How do you do, and where did you drop from?” asked Squeaky Voice.
“I am sure I don’t know where I dropped from,” replied Mr. Bushy Tail, “but I came a long distance, at a very high rate of speed.”
“Have a few ants for luncheon?” inquired Mr. Quills.
“Have a few what?” asked the astonished Mr. Bushy Tail.
“Ants,” replied the Quilly One. “Red ants! White ants! Black ants! Speckled ants!—just any kind of ants. They are all excellent, both as food and appetizers.”
“No, thank you,” said Mr. Bushy Tail, in rather a disgusted voice. “I do not care for ants. Do you eat nothing else?”
“Nothing!” answered the other proudly. “I am the world famous Quilly Ant-Eater, of whom you have doubtless heard.”
Now Mr. Bushy Tail had never heard of this celebrated personage, but he was too wise to say so. He only inquired where he could find a few nuts, for he was half starved, and also the nearest way to the North.
The squeaky Mr. Quilly Ant-Eater led him to a charming wood, where nuts grew in abundance, (as well as ants) for here at the South Pole it was summer, and the flowers and trees were in full bloom.
Mr. Bushy Tail’s winter coat felt much too warm, and as he could not shed it until Spring came in the North, he concluded that he had better travel home as speedily as possible, or he might melt away entirely.
Mr. Quills did not know the way North, but he directed Mr. Bushy Tail to an intimate friend of his, Mr. Ring-Tailed Snorter, who was a great traveller, and would undoubtedly be able to help him.
Mr. Bushy Tail felt rather timid about meeting a person with such a fearsome name, but he felt that he must hurry home to his perchance starving family.
So, after thanking Mr. Quilly Ant-Eater for his kindness, he took his courage in his paws, and started off to find Mr. Snorter.
The Snortling of the Ring-Tailed Snorter After travelling for several miles, Mr Bushy Tail entered a lovely glade full of flowers and ferns.
He had heard, as he approached, a most peculiar noise, such as he had never heard in his life. As he drew nearer, the sounds grew much louder, and finally he saw the strangest looking object seated on a tree trunk.
It had the body and legs of a kangaroo, and the head of a monkey. Its tail was extremely long, with furry rings around it, and was tasseled at the end; and to keep it out of the dust, it had been tied in a bow-knot around the animal’s neck.
The noise, which Mr. Bushy Tail had heard, proceeded from this person.
“Who are you?” he inquired, as Mr Bushy Tail approached. “I am the celebrated Ring-Tailed Snorter, and you may now have the privilege of hearing me snortle,” which he proceeded to do most vigorously.
Poor Mr. Bushy Tail was scared almost out of his wits, for never had he heard such terrible sounds.
After a few minutes the Snorter stopped snortling, and said:
“Now, what do you think of that?”
“Never have I heard anything to equal it,” replied the tactful Mr. Bushy Tail, and he certainly never had.
“If you’ll stay with me, I’ll do it for you every day,” said Mr. Ring Tail.
“Thank you kindly,” said Mr. Bushy Tail, “but much as I should enjoy it, I must go home as soon as I can to my starving family.”
He then told Mr Snorter what had befallen him, and of his great desire to travel back to the North, in the speediest manner.
“Come on!” said Mr. Ring Tail. “Just jump on my back, and I will take you to a friend of mine, who can whisk you there in no time.”
“Hadn’t I better fill my bag first with these fine nuts?” asked Mr. Bushy Tail.
“No, no, don’t bother about that; you’ll find nuts all the way home,” answered the other. So on jumped Mr. Bushy Tail, and away they went.
“Hang on tight to my tail,” said the Snorter, as he leaped along.
Never had Mr. Bushy Tail travelled at such a peculiar gait. It was like leap frog, only more so, and he felt as though he were on a ship at sea. However, he held on tight, and hoped for the best.
The Gentle South Wind Towards evening, they reached the top of a high hill, where Mr. Ring-Tailed Snorter stopped leaping, and gazed towards the South.
“Here comes my friend,” said he. “Good bye,” and without a “by your leave,” or the chance of a “Thank you again,” the South wind had caught up Mr. Bushy Tail and was whirling him rapidly Northwards.
For nearly a week they travelled on, but much more pleasantly than with the fierce North wind.
At times the wind would stop blowing, and Mr. Bushy Tail would be gently dropped in some pretty wood or meadow, where he could find plenty to eat and to drink.
He filled his bag to overflowing with the most delicious squirrel’s food, and only regretted that the bag was not bigger
At last one evening, the wind softly dropped him, and blew on alone. Mr. Bushy Tail looked about him, and saw that he was in his own woods, only a short distance from home.
How fast he scampered toward his house tree. He scarcely noticed that since he had gone away Spring had come, and the first soft green shoots were covering the trees. The grass was full of flowers, and the birds were singing merrily.
Quickly he ran up into his nest, and there they all were, the dear wee family; Mrs. Bushy Tail, the children, and dear old Grandmother Chipmunk.
How delighted they were to see him. Poor little Mrs. Bushy Tail quite broke down and cried with joy, for she had never expected to see her husband alive again.
Mr. Chipmunk and Mr. Red Squirrel had come over, a few days after Mr. Bushy Tail’s visit, to inquire if he had reached home safely, and to bring great bags of provisions. When they heard that their friend was still missing, they had looked very anxious and sad.
The snow storm, in which Mr. Bushy Tail was blown away, had turned into such an awful blizzard, that every one thought he had been buried in the deep snow and frozen.
