mit8_851s13_introtoscet

8/13/2019 MIT8_851S13_IntroToSCET

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2 INTRODUCTIONTOSCET2 Introduction to SCET2.1 What is SCET?TheSoft-CollinearEffectiveTheory isaneffectivetheorydescribingthe interactionsofsoftandcollineardegreesoffreedominthepresenceofahardinteraction. WewillrefertothemomentumscaleofthehardinteractionasQ. ForQCDanotherimportantscaleisQCD,thescaleofhadronizationandnonperturbativephysics,andwewillalwaystakeQQCD.

Softdegrees of freedomwill have momentapsoft, whereQpsoft. Theyhaveno preferred direction,soeachcomponentofp for= 0,1,2,3hasan identicalscaling. SometimeswewillhavepsoftQCDsoftsothatthesoftmodesarenonperturbative(as inHQETforB orDmesonboundstates)andsometimeswewillhavepsoftQCD sothatthesoftmodeshavecomponentsthatwecancalculateperturbatively.

Collineardegreesoffreedomdescribeenergeticparticlesmovingpreferrentiallyinsomedirection(heremotion collinear to a direction means motion near to but not exactly along that direction). In varioussituationsthecollineardegreesoffreedommaybetheconstituentsforoneormoreof

energetichadronswithEHcQQCDmH,

2 energeticjetswithEJcQmJ = p QCD.JBoth the soft and collinear particles live in the infrared, and hence are modes that are described by

fields in SCET. Here we characterize infrared physics in the standard way, by looking at the allowedvaluesofinvariantmassp2 andnotingthatalloffshellfluctuationsdescribedbySCETdegreesoffreedom

2havep Q2. Thus SCET is an EFT which describes QCD in the infrared, but allows for both softhomogeneousandcollinear inhomogeneousmomenta fortheparticles,whichcanhavedifferentdominantinteractions. ThemainpowerofSCETcomesfromthesimplelanguageitgivesfordescribinginteractionsbetweenhardsoftcollinearparticles.

Phenomenologically SCET is useful because our main probe of short distance physics at Q is hardcollisions: e+e stuff, ep stuff, orpp stuff. To probe physics at Q we must disentangle thephysics of QCD that occurs at other scales like QCD, as well as at the intermediatescales like mJ thatareassociatedwithjetproduction. Thisprocessismadesimplerbyaseparationofscales,andthenaturallanguageforthispurposeiseffectivefieldtheory. GenericallyinQCDaseparationofscalesisimportantfordeterminingwhatpartsofaprocessareperturbativewiths1,andwhatpartsarenonperturbativewiths1. Forsomeexamplesthis is fairlystraightforward,thereareonlytworelevantmomentumregions,onewhichisperturbativeandtheothernonperturbative,andwecanseparatethemwithafairlystandardoperator expansion. But many of the most interesting hard scattering processes are not so simple, theyinvolveeithermultiple perturbativemomentumregions, or multiplenonperturbativemomentumregions,orboth. InmostcaseswhereweapplySCETwewillbe interested intwoormoremodes intheeffectivetheory,suchassoftandcollinear,andoftenevenmoremodes,suchassoftmodestogetherwithtwodistincttypesofcollinearmodes.

Part of the power of SCET is the plethora of processes that it can be used to describe. Indeed, it isnotreallyfeasibletogenerateacompletelist. Newprocessesarecontinuouslybeinganalyzedonaregularbasis. SomeexampleprocesseswhereSCETsimplifiesthephysics include

inclusive hard scattering processes: e p eX (DIS),pp Xl+l (Drell-Yan),pp HX, . . .(eitherforthefull inclusiveprocessorforthresholdresummationinthesameprocess)

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2.2 Light-ConeCoordinates 2 INTRODUCTIONTOSCET exclusivejet processes: dijet event shapes in e+e jets,pp H +0-jets,pp W +1-jet,

e pe+1-jet,ppdijets, ... exclusivehardscatteringprocesses: 0,p()p (DeeplyVirtualCompton), . . . inclusiveB-decays: BXs,BXu,BXs+ exclusiveB-decays: BD,B,BK,B,BKK,BJ/K,. . .

+ Charmoniumproduction: e eJ/X, . . . Jets inaMedium inheavy-ioncollisions

SomeoftheseexamplescombineSCETwithothereffectivetheories, suchasHQET fortheB-meson, orNRQCDfortheJ/.

Before we dig in, it is useful to stop and askWhatmakesSCETdifferent from otherEFTs?Putanotherway,whataresomeofthethingsthatmakeitmorecomplicatedthanmoretraditionalEFTs?Oranotherway,forthefieldtheoryafficionato,whataresomeoftheinterestingnewtechniquesIcanlearnbystudyingthisEFT?Abrieflist includes:

We will integrate off-shell modes, but not entire degrees of freedom. (This is analogous to HQETwhere lowenergyfluctuationsoftheheavyquarkremainintheEFT.)

Havingmultiplefieldsthataredefinedforthesameparticlen =collinearquarkfield, qs =softquarkfield

whicharerequiredbypowercountingandtocleanlyseparatemomentumscales. IntraditionalEFTwesumoveroperatorswiththesamepowercountingandquantumnumbers. Inp t

SCETsomeofthesesumsarereplacedbyconvolutions, iCiOi dC()O(). ,thepowercountingparameterofSCET, isnotrelatedtothemassdimensionsoffields

t Various Wilson Lines, which are path-ordered line integrals ofgauge fields, Pexp[ig dsnA(ns)],

play an important role in SCET. Some appear from integrating out offshell modes, others fromdynamicsintheEFT,andallarerelatedtotheinterestinggaugesymmetrystructureoftheeffectivetheory.

There are 1/2 divergences at 1-loop which require UV counterterms. This leads to explicit ln()dependence in anomalous dimensions related to the so-called cusp anomalous dimensions, and torenormalizationgroupequationswhosesolutionssumupinfiniteseriesofSudakovdoublelogarithms,p

kak[sln2

(p/Q)]k

.2.2 Light-Cone CoordinatesBefore we get into concepts, which should decide on convenient coordinates. To motivate our choice,consider the decay process B D in the rest frame of the B meson. This decay occurs through theexchangeofaW bosonmediatingbcud,alongwithavalencespectatorquarkthatstartsintheB andendsup intheD meson. Weareconcernedherewiththekinematics. Aligningthe withthez axis itiseasytoworkoutthepionsfourmomentumforthistwo-bodydecay,

p =(2.310GeV,0,0,2.306GeV)cQn, (2.1)6


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2.2 Light-ConeCoordinates 2 INTRODUCTIONTOSCET 2where n = (1,0,0,1) in a 0,1,2,3 basis for the four vector. Here n = 0 is a light-like vector and

QQCD. Thispionhas largeenergyandhasa four-momentumthat isclosetothe light-cone. Withaslightabuseoflanguagewewilloftensaythatthepionismovinginthedirectionn(eventhoughwereallymean

the

direction

specified

by

the

1,

2,

3components

of

n

).

Thenatural

coordinates

for

particles

whose

energyismuch largerthantheirmassarelight-conecoordinates.

We would like to be able to decompose any four vectorp using n as a basis vector. But unlikecartesian coordinates the component along n will not be np, since n2 = 0. If we want to describe thecomponents (wedo) then wewill needanother auxillary light-likevector n. The vector n has a physicalinterpretation, we want to describe particles moving in the n direction, whereas n is simply a devise weintroducetohaveasimplenotationforcomponents.

Thuswestartwith light-conebasisvectorsnandnwhichsatisfythepropertiesn2 = 0, n2 = 0, nn = 2, (2.2)

wherethelastequationisournormalizationconvention. Astandardchoice,andtheonewewillmostoftenuse, istosimplytaken intheoppositedirectionton. Soforexamplewemighthave

n =(1,0,0,1), n =(1,0,0,1) (2.3)Other choices for the auxillary vector workjust as well, e.g. n = (1,0,0,1) with n = (3,2,2,1), andlateronthis freedom indefiningnwillbecodified inareparameterization invariancesymmetry. Fornowwestickwiththechoice inEq.(2.3).

It isnowsimpletorepresentstandard4-vectors inthe light-conebasis n n

p = n p+ np+p (2.4)2 2

1 2wherethecomponentsareorthogonaltobothnandn. WiththechoiceinEq.(2.3),p =(0, p ,0)., pItiscustomarytorepresentamomentum inthesecoordinatesby

p + = (p , p, p) (2.5)2 2wherethe lastentry istwo-dimensional,andtheminkowskip isthenegativeoftheeuclideanp (ie. in

2 2=pournotationp ). Herewehavealsodefined

p+ =p+np , =pn p. (2.6)pAs indicatedtheupperor lower indicesmeanthesamething.

Usingthe

standard

(+

)

metric,

the

four-momentum

squared

is

p2 + 2 2+ (2.7)+pp =p p =p p .

Wecanalsodecomposethemetric inthisbasis n n n n = + +g

. (2.8)g

2 2 =nn/2.Finallywecandefineanantisymmetrictensor inthespaceby

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2.3 MomentumRegions: SCETIandSCETII 2 INTRODUCTIONTOSCET2.3 Momentum Regions: SCET I and SCET IILets continuewithourexplorationof the BD decay with the goalof identifyingthe relevantquarkand gluon degrees of freedom (d.o.f.) for designing an EFT to describe this process. Well then do thesameforaprocesswithjets.

Therearedifferentwaysoffindingtherelevant infrareddegreesoffreedom. Wecouldcharacterizeallpossible regions giving rise to infrared singularities at any order in perturbation theory using techniquesliketheLandauequations,andthendeterminethecorrespondingmomentumregions. WecouldcarryoutQCDloopcalculationsusingatechniqueknownasthemethodofregions,wherethefullresultisobtainedbyasumoftermsthatenterfromdifferentmomentumregions. ThenbyexaminingtheseregionswecouldhypothesizethatthereshouldbecorrespondingEFTdegreesof freedom forthoseregionsthatappeartocorrespondto infraredmodesthatshouldbe intheEFT.(Eitheroftheseapproachesmaybeuseful,butnote that when using them we must be careful that the degrees of freedom are appropriate to our truephysical situation, and do not contain artifacts related to our choice of perturbative infrared regulatorsthatarenotpresentinthetruenonperturbativeQCDsituation.) Instead,ourapproachinthissectionwillbebasedsolelyonphysicalinsightofwhattherelevantd.o.f. are,fromthinkingthroughwhatishappeningin the hardscatteringprocess we want tostudy. More mathematical checks thatonehastherightd.o.f.arealsodesirable, andwewilltalkaboutsomeexamplesofhowtodothis lateron. This fallsundertherubericofnotfullytrustingaphysicsargumentwithoutthemaththatbacks itup,andvisaversa.

ForBD intherest frameoftheB,theconstituentsoftheB mesonarethenearlystaticheavybquark,andthesoftquarksandgluonswithmomentaQCD, ie.justthestandarddegreesoffreedomof HQET. Since |pD| = 2.31GeV mD = 1.87GeV the constituents of the D meson are also soft anddescribedbyHQET.Thepionontheotherhandishighlyboosted. Wecanderivethemomentumscalingofthepionconstituentsbystartingwiththe(+,,)scalingof

p(QCD,QCD,QCD) forconstituents inthepionrestframe,and then by boosting alongz by an amount = Q/QCD. The boost is very simple with light conecoordinates,takingpp andp+p+/. Thus

2

QCD,Q,QCD (2.9)cp Q

fortheenergeticpionsconstituentsintheBrestframe. Thisscalingdescribesthetypicalmomentaofthe =(0,Q,0)+O(m2quarksandgluonsthatbind intothepionmovingwith largemomentump /Q),as in

n

The importantfactaboutEq.(2.9)isthat + (2.10)pc pc pc .

Whenever thecomponents ofpc obey this hierarchy we say ithas acollinear scaling. Its convenient todescribethiscollinearscalingwithadimensionlessparameterbywriting

pc Q(2,1, ) (2.11)

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2.3 MomentumRegions: SCETIandSCETII 2 INTRODUCTIONTOSCET

hard

cn

s

2Q Q

0Q

Q

Q 0

p+

-

p2= 2Q

p2= 2

QCD

Figure1: SCETII example. RelevantdegreesoffreedomforBDwithanenergeticpionintheB restframe.

1where1isasmallparameter. Thisresultisgeneric. ForourBDexamplewehave= QCD/Q.This will be the power counting parameter of SCET. With this notation we can also say how the softmomentaofconstituentsintheB andDmesonscale,

pQ(,,). (2.12)sThusweseethatweneedbothsoftandcollineardegreesoffreedomfortheBD decay.

Itisconvenienttorepresentthedegreesoffreedomwithapicture,asinFig.1. Thispicturehassomeinteresting features. Unlike simpler effective theories SCET requires at least two variables to describe

+ 2 + the d.o.f. The choice ofp andp as the axis here suffices since the-momentum satisfiesp p pand hence does not provide additional information. The hyperbolas in the figures are lines of constantp2 =p+p. The labelled spots indicate the relevant momentum regions. We have included a hyperbolaand a spot for the hard region wherep2Q2, but these are the modes that are actually integrated outwhenconstructingSCET.(ForBD theyarefluctuationsofordertheheavyquarkmasses.) Onthep2QCD2 hyperbola in Fig. 1 we have two types of nonperturbative modes, collinear modes cn for thepion constituents, and soft modes s for the B and D meson constituents. Since these modes live at thesametypicalinvariantmassp2 weneedanothervariable,namelyp/p+,todistinguishthem. Thisvariable

2Y +isrelatedtotherapidity,Y,sincee =p/p+. Putanotherway,weneedbothofthevariablesp andp todefinethemodesfortheEFT.

The example in Fig. 1 is what is knownasan SCETII type theory. Itsdefiningcharacteristic isthatthesoftandcollinearmodes inthetheoryhavethesamescaling forp2,they liveonthesamehyperbola.Thistypeoftheoryturnsouttobeappropriateforawidevarietyofdifferentprocessesandhencewegiveit the generic name SCETII. Essentially this version of SCET is the appropriate one for hard processeswhichproduceenergetic identifiedhadrons,whatweearliercalledexclusivehardscatteringandexclusiveB-decays.

1Pleasedonotbe confused into thinking that youneed to assign aprecisedefinition to. It is onlyused as a scalingparametertodecidewhatoperatorswekeepandwhat termswedrop intheeffectivefield theory, soanydefinitionwhich isequivalentby scaling is equallygood. In the endanypredictionswemake forobservablesdonotdependon thenumericalvalueof. Theonly timeweneedanumber for iswhenmakinganumerical estimate for the sizeof the terms thatarehigherorder inthepowerexpansionwhichwevedropped.

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2.3 MomentumRegions: SCETIandSCETII 2 INTRODUCTIONTOSCETWhenlookingatFig.1weshouldinterpretthecollineardegreesoffreedomaslivingmostlyinaregion

aboutthecn spotandthesoftdegreesoffreedomaslivingmostlyinaregionaboutthesspot. Anobviousquestion is what determines the boundary between these degrees of freedom. In a Wilsonian EFT theanswer

would

be

easy,

there

would

be

hard

cutoffs

that

carve

out

the

regions

defined

by

these

modes.

But

hard cutoffs break symmetries. For SCET the cutoffs must be softer regulators so as to not to breaksymmetrieslikeLorentzinvarianceandgaugeinvariance. Dimensionalregularizationisoneregulatorthatcanbeusedforthispurpose. Ifwewereonlytryingtodistinguishmodeswiththe invariantmassp2 thenthedim.reg. scaleparameterwouldsufficeforthecutoffbetweenUVandIRmodes,andwewouldbesettogo. ButinSCETwealsoneedtodistinguishmodesinanotherdimension,doesnotsufficetoseparateor distinguish the s and cn modes of Fig. 1. We will see how to do this later on without spoiling anysymmetries. Ingeneralitwillrequireacombinationofsubtractionsthatlocalizethemodesintheregionsshown inthefigure,aswellasadditionalcutoffparameters. Thebottom line isthatthephysicalpictureinFig.1forwherethemodesliveisthecorrectonetothinkaboutforthepurposeofpowercounting. Butwhen integratingover loopmomenta inavirtualdiagram involvingoneofthesemodeswe integrateoverallvalueswithasoftregulatortoavoidbreakingsymmetries.

Lets consider a second example involving QCDjets. Jets are collimated sprays of hadrons producedbytheshoweringprocessofanenergeticquarkorgluonas itundergoesmultiplesplittings. Thesplittingis enhanced in the forward direction by the presence of collinear singularities. The simplest process is

+ +e edijets,whichat lowestorder istheprocesse eqq witheachofthe lightquarksq andq formingajet. Letq bethemomentumofthe, then inthecenter-of-momentum frame(CM frame)q =(Q,0,0,0) and sets the hard scale. If there are only twojets in the final state then by momentumconservationtheywillbeback-to-backalongthehorizontal z axis:

ultrasoft articles

n-collinear

jetn-collinear

jet

nn

21

p

a b

Thexyplanedefinestwohemispheresaandb,andweconsideraprocesswithonejetineachofthem.TheenergyineachhemisphereisQ/2andispredominantlycarriedbythecollimatedparticlesinthejets.Todescribethedegreesoffreedomweneedtwocollineardirections. Wealignn1 withthedirectionofthefirstjet and n2 withthe second. (Thesedirections can be defined by using ajet algorithm to determinetheparticles insideajet,or indirectlyfromtheprocessofcalculatingajeteventshape likethrust.)

Letsfirstconsidertheenergeticconstituentsofthen1-jet. Sincetheseconstituentsarecollimatedtheyhavea-momentumthatisparametricallysmallerthantheirlargeminusmomentum,ppQ.InorderthatwehaveajetofhadronsandnotasinglehadronorsmallnumberofhadronswemusthaveQCD. Thusthejetsconstituentshave(+,,)momentawithrespecttotheaxesn1 =(1,z)andn1 =(1, z)thathaveacollinearscaling

2 ,Q, =Q(2,1, ). (2.13)pn1 Q

As usual the scaling of the +-momentum is determined by noting that we are considering fluctuations2 + 2aboutp =0, sop p /p. Here the power counting parameter is = /Q1. Note that thejet

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2.3 MomentumRegions: SCETIandSCETII 2 INTRODUCTIONTOSCETconstituentshavethesamescalingastheconstituentsofacollinearpion,butcarry largeroffshellnessp2.IfwemakesolargethatQthenwenolongerhaveadijetconfiguration,andifwemakesosmallthatQCD thentheconstituentswillbindintoone(ormore)individualhadronsratherthanthelargecollection

of

hadrons

that

make

up

the

jet.

Another

way

to

characterize

the

presence

of

the

jet

is

through

2 2thejet-massm , sinceajetwillhaveQ2m 2 Forourexampleherewecan makeuseoftheJ J QCD.

a-hemispherejet-mass, 2

2 + m p p p 2Q2. (2.14)Ja i n1 n1ia

Fortheconstituentsofthen2-jetwesimplyrepeatthediscussionabove,butwithparticlescollimatedaboutthedirection,n2 =n1 =(1, z). Achoicethatmakesthissimpleisn2 =n1 =(1,z),sincethenwecansimplytakethen1-jetanalysisresultswith+ . Usingthesame(+,,)componentsas forthen1-jetwethenhave

2

p Q, , =Q(1, 2, ). (2.15)n2 QAgainameasurementoftheb-hemispherejet-masscanbeusedtoensurethatthereisonlyonejetinthatregionjet-mass,

22 + m p p p 2Q2. (2.16)Jb i n2 n2

ibFinally injet processes there are also soft homogeneous modes that account for soft hadrons that

appearbetweenthecollimatedjetradiation(aswellaswithinit). Theprecisemomentumofthesedegreesof freedomdependsontheobservablebeingstudied,andtherestrictions it imposesonthisradiation. In

+ 2 2oure edijetsexamplewecanconsidermeasuringthatm andm areboth2. InthiscasetheJa Jbhomogeneousmodesareultrasoftwithmomentumscalingas2 2 2p , , =Q(2, 2, 2). (2.17)us Q Q Q

2To derive this we consider the restrictions that mJa 2 imposes on the observed particles, noting inparticularthatwithacollinearandultrasoftparticle inthea-hemispherewehave

2 2(pn1 +pus)2 =p + 2pn1 pus+p 2. (2.18)n1 us + +The term 2pn1 pus =pn1pus plus higher order terms, sopus 2/pn1 2/Q, which is the ultrasoft

+

momentumscale

given

in

Eq.

(2.17).

Any

larger

momentum

for

p is

forbidden

by

the

hemisphere

mass

usmeasurement. Thescalingoftheotherultrasoftmomentumcomponentsthenfollowsfromhomogeneity.

If we draw the degrees of freedom, then for the double hemisphere mass distribution measurement+ + of e e dijets in thep -p plane we find Fig. 2. Again we have labelled hard modes with momenta

p2Q2 that are integrated out inconstructing theEFT(here they correspond tovirtual corrections atthejetproductionscale). Inthe lowenergyeffectivetheorywehavetwotypesofcollinearmodescn andcn,one foreachjet,which liveonthep22 hyperbola. Finallytheultrasoftmodes liveonadifferent

2 2hyperbola withp 4/Q2. The collinear and ultrasoft modes all havep Q22 and are degrees offreedom inSCET,whilemodeswithp2Q22 are integratedout. Whenweare inasituation likethisone,wherethecollinearandhomogeneousmodes liveonhyperbolaswithparametricallydifferentscalingforp2,thentheresultingSCET isknownasanSCETI typetheory. Notethatthecn andusmodeshave

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hardcn

us

2Q Q

0Q

Q

2Q

Q 0

p+

p-

p2=4

p2= 2Q

p2=

Q

cn

QCD

QCD

2

/ 2

+

Figure2:

SCETI

example.

Relevant

degrees

of

freedom

for

dijet

production

e e

dijets

with

measured

hemisphere invariantmassesm2 andm2 .Ja Jb

p+ momentaofthesamesize,whereasthecnandusmodeshavep momentaofthesamesize. Thenamescollinear and ultrasoft denote the fact that these modes live on different hyperbolas.2 Once again thesedegreesoffreedomcaptureregionsofmomentumspace,whicharecenteredaroundthespotsindicatedandeachofthemextend intothe infrared.

It is important to note in this dijet example that 4/Q2 2 , so in general the nonperturbativeQCDultrasoftmodescanliveonanevensmallerhyperbolap2QCD2 thantheperturbativecontributionsfromultrasoftmodesthathavep24/Q2. Anadditionalp22QCD hyperbola isshown ingreen inFig.2.If 4/Q22 then the yellow and green hyperbolas are not distinguishable by power counting, andQCDhence are equivalent. If on the other hand we are in a situation where 4/Q2 2 then when weQCDsetuptheSCETI theoryboththeperturbativeultrasoftmodeswithp24/Q2 andthenonperturbativeultrasoftmodeswithp22 willbepartofoursingleultrasoftdegreeoffreedom. This isconvenientQCDbecausewecanfirstformulatethe/Q1expansionwiththecn,cnandusd.o.f.,andonlylaterworryabout making another expansion in QQCD/2 1 to separate the two types of ultrasoft modes thatwould liveontheyellowandgreenhyperbolas.

IfwecompareFig.1andFig.2weseethatitistherelativebehaviourofthecollinearandsoft/ultrasoftmodes that determine whether we are in an SCETI or SCETII type situation. (There are also SCETIIexampleswhich involvejetswithmeasurementsratherthanjetmasses,andwewillmeettheselateronin

Section

11.3

and

11.4.)

Much

of

our

discussion

will

be

devoted

to

studying

these

two

examples

of

SCET,

since they are already quire rich and cover a wide variety of processes. In general however one shouldbeaware thatamorecomplicatedprocess or setofmeasurementsmaywellrequireamoresophisticatedpattern of degrees of freedom. For example, we could have soft or collinear modes on more than onehyperbola, or might require modes with a new type of scaling. Indeed, this is not even uncommon, thecolliderphysicsexampleofppdijetsintheCMframerequiresbothSCETII typecollinearmodesfortheincoming protons, and SCETI type collinear modes for thejets. Nevertheless, after having studied bothSCETI andSCETII wewillseethatoftenthesemorecomplicatedprocessesdonotreallyrequireadditionalformalism,butrathersimplyrequirecarefuluseofthetoolswehavealreadydevelopedinstudyingSCETI

2Incertainsituations inthe literaturetousethenameshard-collinearandsofttodenotethesamething,andwewillfindoccasiontoexplainwhywhendiscussinghowSCETI canbeusedtoconstructSCETII.

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hardcn

us

2Q Q

0Q

Q

2Q

Q 0

p+

-

p2= 2

QCD

p2= 2Q

p2=QQCD

Figure3:

Another

SCETI

example.

Relevant

degrees

of

freedom

for

B

Xs

in

the

endpoint

region.

andSCETII.A comment is also in order about the frame dependence of our degrees of freedom. In both of our

exampleswefounditconvenienttodiscussthedegreesoffreedominaparticularframe(theBrestframe,+ore e CMframe). Typicallythere isanaturalreferenceframetothinkabouttheanalysisofaprocess,

but of course the final result describing the dynamics of a process will actually not be frame dependent.Thusitisnaturaltoaskwhatthed.o.f. andcorrespondingmomentumregionswouldlooklikeinadifferentframe. Asimpleexampletodiscussisaboostoftheentireprocessalongthezaxis. Allthemodesthenslidealongtheirhyperbolas(sincep2 isunchanged). Theimportantpointisthattherelativesizeofmomentaofdifferent

d.o.f.

is

unchanged

by

this

procedure:

the

p

+momenta

of

collinear

and

ultrasoft

modes

in

SCETI+will bethesamesize evenaftertheboost, andthep momentum ofasoftparticle willalwaysbe larger

+than thep momentum of a collinear particle in SCETII. In B D such a boost can take us to thepionrestframe,whereitsconstituentsarenowsoft,andtheconstituentsoftheBandDarenowboosted.Some components of the SCET analysis may look abit different if we use different frames, but the finalEFTresultsfordecayratesandcrosssectionswillobeytheexpectedoverallboostrelations. Ingeneralitis only the relative scaling of the momenta of various degrees of freedom that enter into expansions andthe final physical result. The relative placement of the spots for our d.o.f. in SCETI and SCETII is notaffectedbythe z boost.

Before finishing our discussion of d.o.f. we consider one final example. For the purpose of studyingSCETI itisusefultohaveanexamplewithonejetratherthantwo,sothed.o.f. becomesimplycn andus.This

can

occurs

for

the

process

B

Xs

or

for

B

Xue.

The

underlying

processes

here

are

the

flavor

changing neutral current proess b s or the semileptonic decay b ue. For these inclusive decayswe sum over any collection of hadronic s tates Xs or Xu that can be produced from the s or u quark.

2 2In the B rest frame, the total energy of the or (e) is E = (m m )/(2mB) and ranges from 0 toB X2 2(m m )/(2mB)wheremHmin isthesmallestappropriatehadronmass,eithermHmin =mK ormB Hmin

forXs orXu respectively. An interestingregiontoconsiderfortheapplicationofSCETis2 22 XQ2 =m (2.19)QCDm B

wherethephotonor(e)recoilsagainstajetofhadronswhicharetheconstituentsofX. ForBXsthe picture is (double line being the b-quark, yellow lines are soft particles, and red lines are collinearparticles):

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8.851 Effective Field Theory

Spring 2013

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