in order to

[观察]

1. He got up very early in order to / so as to catch the first bus.

2. In order to catch the first bus, he got up very early.

3. He works very hard in order to / so as to support his family.

4. Turn the volume down in order not to / so as not to wake the child.

[归纳] in order to 意为 “为了……”，表示目的；在用法和意义上相当于so as to结构，但是in order to结构可以用于句首、句中，而so as to多用于句中。其否定式分别为：in order not to 和so as not to。

[拓展] in order to和so as to在句中表示目的时，常可以转化成in order that或so that引导的目的状语从句。如:

We should work hard in order to / so as to pass the exam.

→ We should work hard in order that / so that we can pass the exam.

为了能通过考试，我们应该努力学习。

[小试] 翻译下面的句子。

1. He went there early so as to / in order to get a good seat.

He went there early so that / in order that he could get a good seat.

2. In order not to wake the baby we went in quietly.

Key:

1. 他去得早，以便找到个好座位。

2. 为了不惊醒小孩，我们轻轻地走了进去。

第二篇：Threshold resummation to any order in (1-x)

PreprinttypesetinJHEPstyle-PAPERVERSION

arXiv:0710.5693v3 [hep-ph] 13 Nov 2007GeorgesGrunberg?CentredePhysiqueTh?eorique,EcolePolytechnique,CNRS,91128PalaiseauCedex,FranceE-mail:georges.grunberg@pascal.cpht.polytechnique.frAbstract:Asimpleansatzissuggestedforthestructureofthresholdresummationofthemomentumspacephysicalevolutionkernels(‘physicalanomalousdimensions’)atallordersin(1?x),takingasexamplesDeepInelasticScatteringandtheDrell-Yanprocess.Eachtermintheexpansionisassociatedtoadistinctrenormalizationgroupandschemeinvariantperturbativeobject(‘physicalSudakovanomalousdimension’)dependingonasinglemomentumscalevariable.Bothlogarithmicallyenhancedtermsandconstanttermsarecapturedbytheansatzatanyorderintheexpansion.Theansatzismotivatedbyalarge–β0dispersivecalculation.Adispersiverepresentationat?niteβ0ofthephysicalSudakovanomalousdimensionsisalsoobtained,associatedtoasetof‘Sudakove?ectivecharges’whichencapsulatethenon-Abeliannatureoftheinteraction.Itisfoundthatthedispersiverepresentationrequiresanon-trivial,andprocess-dependent,choiceofvariablesinthe(x,Q2)plane.SomeinterestingpropertiesofthephysicalSudakovanomalousdimen-sionsarepointedout.Theensuing1/Nexpansioninmomentspaceisstraightforwardly

derivedfromthemomentumspaceexpansion.Keywords:resummation,renormalons.

1.Introduction

Thresholdresummation,namelytheresummationtoallordersofperturbationtheoryofthelargelogarithmiccorrectionswhicharisefromtheincompletecancellationofsoftandcollineargluonsattheedgeofphasespace,isbynowawelldevelopedtopic[1,2]inperturbativeQCD.Abouttenyearsago,thesubjectwasextended[3–5]tocoveralsotheresummationoflogarithmicallyenhancedtermswhicharesuppressedbysomepowerof(1?x)forx→1inmomentumspace(orbysomepowerof1/N,N→∞inmomentspace),concentratingonthecaseofthelongitudinalstructurefunctionFL(x,Q2)inDeepInelasticScattering(DIS)wherethesecorrectionsareactuallytheleadingterms.AsfarasIamaware,littleworkhasbeenperformedonthissubjectsincethen.Inthispaper,buildingupontherecentwork[6],Iprovideaverysimpleansatzforthestructureofthresholdresummationatallordersin(1?x),workingatthelevelofthemomentumspacephysicalevolutionkernels(or‘physicalanomalousdimensions’,seee.g.[7–10]),whichareinfraredandcollinearsafequantitiesdescribingthephysicalscalingviolation,where

–1–

thestructureoftheansatzappearstobeparticularlytransparent.Ishalldealexplicitlywiththeexamplesofthe(non-singlet)DeepInelasticScattering(DIS)structurefunctionF2(x,Q2),aswellaswiththeDrell-Yanprocess.Theansatzismotivatedbyalarge–β0dispersivecalculation,and,following[6,11–15],easilygeneralizesitselftoageneral?nite–β0dispersiverepresentationofthephysicalevolutionkernelatanyorderinthex→1expansion.

Thepaperisorganizedasfollows:section2isdevotedtoDIS.Insection2.1,themomentumspaceansatzforthresholdresummationforthenon-singletstructurefunctionF2(x,Q2)isdisplayed,whichintroducesateachorderofthex→1expansionanew‘jetphysicalanomalousdimension’.Theansatzisjusti?edinsection2.2byalarge–β0calculation,whichalsoprovidesadispersiverepresentationforeachofthepreviousphysicalanomalousdimensions.Thesedispersiverepresentationsarethenextendedto?nite–β0insection2.3,whereasetof‘jetSudakove?ectivecharges’whichencapsulatethenon-Abeliannatureoftheinteractionisintroduced.TheDrell-Yancaseisaddressedinsection3inquiteasimilarway.Theansatzfortheanalogousτ→1expansionisgiveninsection3.1,andjusti?edinsection3.2byalarge–β0calculation,whichyieldsthedispersiverepresentationofthecorresponding‘softphysicalanomalousdimensions’speci?ctotheDrell-Yanprocess.Theserepresentationsarethenextendedto?nite–β0insection3.3,whereasetof‘softSudakove?ectivecharges’isintroduced.BothintheDISandintheDrell-Yancase,anon-trivial(process-dependent)choiceofexpansionparameterinthe(x,Q2)(resp.(τ,Q2))planehastobemadeinordertoderivethedispersiverepresentations.TheconclusionsaregiveninSection4.The1/NexpansioninmomentspaceisderivedinastraighforwardwayfromthecorrespondingmomentumspaceexpansionansatzinAppendixA(DIS)andB(Drell-Yan).Thesetwolastappendicesalsoclarifytheconnectionbetweenthede?nitionsofthe‘jet’(DIS)(orthe‘soft’(DY))scalesinmomentumandmomentspacesrespectively.

2.DeepInelasticScatteringcase

2.1Asystematicexpansionforx→1

Thescale–dependenceofthe(?avournon-singlet)deepinelasticstructurefunctionF2canbeexpressedintermsofF2itself,yieldingthefollowingevolutionequation(seee.g.Refs.[7,9,10]):

dF2(x,Q2)K(x/z,Q2)F2(z,Q2).(2.1)z

K(x,Q2)isthemomentumspacephysicalevolutionkernel,orphysicalanomalousdimen-sion;itisrenormalization–groupinvariant.In[6],usingknownresultsofSudakovresum-mationinmomentspace,theresultfortheleadingcontributiontothisquantityinthex→1limitwasderived.Forcompletness,Ireproducethisshortderivationhere.

De?ningmomentsby

?2(N,Q)=F2?1

0dxxN?1F2(x,Q2),(2.2)

–2–

Eq.(2.1)impliesthatthemoment–spacephysicalevolutionkernelis:

?1?2(N,Q2)dlnFN?122?dxxK(x,Q)=K(N,Q)≡

dlnQ2=?1

dx

0xN?1?1

dln?2(2.5)

whereA(αs)isthe‘cusp’anomalousdimension,andB(αs)thestandard‘jet’Sudakovanomalousdimension1.Moreover,theconstanttermcanbewritten[14]intermsofthequarkelectromagneticformfactorF(Q2)[18–21]:

??2)2dlnF(Q2J(?)H(αs(Q2))=(2.6)?2????=G1,αs(Q2),ε=0+Bαs(Q2),

whereeachofthetwotermsinthe?rstlineisseparatelyinfrareddivergent,butthedivergencecancels[14]inthesum;inthesecondlinetheresultisexpressedintermsof?22???222B(αs)andGQ/?,αs(?),ε,whichisthe?nitepartofdlnF(Q)/dlnQ2asde?nedinRef.[21]usingdimensionalregularization.

Comparing(2.3)and(2.4)onetherefore?ndsthefollowingrelation[6]inmomentumspace:

??2??J(1?x)Q0δ(1?x)+O(1?x)K(x,Q2)=dlnQ2

=???2J(1?x)Q

dlnQ2+

???Q2d?20

1?x

1?=+?10??dxF(x)?F(1)???J(1?x)Q2AandBareseparatelyschemedependentquantities.

–3–

whereF(x)isasmoothtestfunction.Thisprescriptionaccountsforthedivergentvir-tualcorrections,whichcancelagainstthesingularitygeneratedwhenintegratingthereal–??emissioncontributionsnearx=1.Onethus?ndsthatJ(1?x)Q2/(1?x)istheleadingtermintheexpansionofthephysicalmomentumspaceevolutionkernelK(x,Q2)inthex→1limitwith(1?x)Q2?xed.Thetermproportionaltoδ(1?x)iscomprisedofpurelyvirtualcorrectionsassociatedwiththequarkformfactor.Thistermisinfrareddivergent,butasindicatedinthesecondlinein(2.7),thesingularitycancelsexactlyuponintegrating??overxwiththedivergenceoftheintegralofJ(1?x)Q2/(1?x)nearx→1.

NextIobservethateq.(2.7)stronglysuggeststhefollowinggeneralizationtoasys-tematicexpansionforx→1inpowersof1?x,or,moreconveniently(forreasonstobeclari?edinsection2.2),inpowersof

r≡1?x

rJW??2+??2dlnF(Q2)

IalsonotethatnointegrationovertheLandaupoleexplicitlyappearsinthismomen-tumspaceversionofSudakovresummation.Weshallseehoweverthattheperturbative??expansionsofthesubleadingphysicalSudakovanomalousdimensionsJiW2docontain??infraredrenormalons(atthedi?erenceofJW2!).

Onecanalsogivethegeneralizationofeq.(2.7),witharegularizedvirtualcontribution:K(x,Q)=21

dlnQ2

?1δ(1?x)+J0W?Q2

0?2??2??2?+rJ1W+Or=dlnQ2+

??d?2x,W2=1?x

r

whereF(x)isasmoothtestfunction.Theproofthat(2.12)isindeedequivalentto(2.7)forthetwoleadingtermsisgiveninAppendixA,byconsideringthemomentsofthetwoexpressions.

2.2DispersiverepresentationofthephysicalSudakovanomalousdimensions

(largeβ0)

ThecorrectnessoftheansatzEq.(2.11)canbecheckedatlarge–β0,upontakingthex→1expansionofthelarge–β0dispersiverepresentationofthephysicalevolutionkernel.Thisprocedurewillactuallyyieldamorepowerfulresult,namelythe(large–β0)dispersiverep-resentationsofthephysicalSudakovanomalousdimensionsJandJi.

Letusconsiderthelarge–β0butarbitraryx(0<x<1)dispersiverepresentation[8]ofthephysicalevolutionkernel.Atthelevelofthepartoniccalculation,theconvolutiononther.h.sof(2.1)becomestrivialinthelarge–β0limitsincetheO(αs)correctionstoF2(z,Q2)generatetermsthataresubleadingbypowersof1/β0.Therefore,inthislimitK(x,Q2)isdirectlyproportionaltothepartonicdF2(x,Q2)/dlnQ2,so

?∞2?d?K(x,Q2)?largeβ0=CF0??JW2?=+?10?1dxF(x)1?x???2,(2.13)J(1?x)Q

),andaMink(?2)istheV

integratedtime-likediscontinuityoftheone-loopV-schemecoupling(whichcorrespondstothesingledressedgluonpropagator,using‘naivenon-abelization’):d?2

ρV(?)=2daMink(?2)V

m2ρV(m2),(2.15)

–5–

where

ρV(?2)≡

1

π

=

1

lnwith

?

k2/Λ2V

?.Λ2V=Λ2e

5/3

,whereΛ2isde?nedinthe

12CA

?

1

(r)r

F(r)(ξ)+F0(ξ)+rF(r)

1(ξ)+O(r2),

where

ξ≡

?

W2,

with

F(r)(ξ)=?lnξ?

3

1

2

ξ+

2ξlnξ?4ξ+

1

2+2

lnξ?6?9ξlnξ+

1

2

ξ2.

Thus(.≡??2

d

r

F˙(r)(ξ)+F˙(r)0(ξ)+rF˙(r)1

(ξ)+O(r2),with

F˙(r)(ξ)=1?12

ξ2

F˙(r)0

(ξ)=?1?2ξlnξ+2ξ?ξ2

F˙(r)1

(ξ)=32

ξ+2ξ2lnξ?10ξ2,(2.17)(2.18)

(2.20)

(2.21)

(2.22)

(2.23)

(2.24)

and,takingasecondderivative:

¨(r)(?,x)=1xF

¨(r)(ξ)F02=2ξlnξ+2ξ2

¨(r)(ξ)=?9ξlnξ?35F1ξ+ξ2

r

with?FJ(ξ)+FJ0(ξ)+rFJ1(ξ)+O(r)+Vs(?)δ(1?x),2(2.28)

FJ(ξ)=F(r)(ξ)θ(ξ<1)

FJi(ξ)=

?1˙xF(?,x)=(r)Fi(ξ)θ(ξ<1),(2.29)

r

with?¨J(ξ)+F¨J(ξ)+rF¨J(ξ)+O(r)+V¨s(?)δ(1?x),F012(2.32)

¨J(ξ)=F¨(r)(ξ)θ(ξ<1),Fii¨J(ξ)=F¨(r)(ξ)θ(ξ<1)F(2.33)

(r)˙(r)(ξ),F˙(r)(ξ)in(2.20)andwhereIusedthefactthatallthetermsF(r)(ξ),Fi(ξ)andFi

(2.23)vanishatξ=1(whichallowstotreattheθfunctione?ectivelyasamultiplicativeconstantwhentakingthederivatives).Thesefeaturesactuallyfollowfromthestronger

–7–

˙(r)(?,x),bothpropertythattheexactfunctionF(r)(?,x),aswellasits?rstderivativeFvanishatξ=1,i.e.for?=1?x

4r??(1+r)(3+r)(1?ξ)2+O(1?ξ)3.(2.34)

Ifurthernotethatthetermswhichvanishforξ→0inFJi(butnotinFJ!)arelog-arithmicallyenhanced,hencenon-analytic,whichimplies(seebelow)thatthesubleadingphysicalSudakovanomalousdimensionsJidohaverenormalons(atthedi?erenceoftheleadingphysicalanomalousdimensionJ!).

ReportingtheresultEq.(2.32)into(2.14),onethusobtainsthesmallrexpansion(r=(1?x)/x)ofthephysicalevolutionkernel:

?1?2?K(x,Q)largeβ0=CF?∞

?Mink2¨2a(?)FJ1(ξ)+O(r)?2V

?∞2d?+δ(1?x)CF0Mink2¨a(?)FJ(ξ)?2V+d?20

?Ji(W)?largeβ0=CF2??2∞¨J(ξ)aMink(?2)FVd?2

dlnQ2?????=CF

largeβ0?∞0d?2

dlnQ2+?Q2

0d?2aMink(?2)V2???2222¨˙Vs(?/Q)+FJ(?/Q),

(2.38)

wheretheintegralonther.h.s.isindeedbothinfraredandultravioletconvergent.

–8–

2.3DispersiverepresentationofthephysicalSudakovanomalousdimensions

(?niteβ0)

Following[6,13],itisstraightforwardtogivethegeneralizationofEqs.(2.36)and(2.37)at?niteβ0:

2J(W)=CF?∞0d?2(2.39)

?2¨J(ξ),aMink(?2)FJii

where,infullanalogywith(2.15),

ρJ,Ji(?)=22daMinkJ,Ji(?)m2ρJ,Ji(m2),(2.40)

and,similarlyto(2.16),ρJ,Ji(?2)correspondtothetimelikediscontinuitiesofsome“Eu-

clidean”e?ectivecharges,originallyde?nedforspacelikemomenta:

?∞2?∞d?d?2Eucl2aJ,Ji(k)==?(1+?2/k2)200

?∞

0dlnQ2

?=CFd?2

Q2

dlnQ2+d?2

wheretermshavebeenarrangedproperlytohavebothinfraredandultravioletconver-gence:InotethattheintegralsoveraMink(?2)andaMink(?2)in(2.43),althoughinfraredFJconvergent,areseparatelyultravioletdivergent.

Fromthesedispersiverepresentations,theperturbativeexpansionsofthe‘jet’Sudakove?ectivechargesaMink(?2)andaMink(?2)canbederivedinastraightforwardway[6]JJi

orderbyorderinthefullnon-Abeliantheory,giventheexpansionsofJ(W2)andJi(W2)(andsimilarlyforaMink(?2)using(2.43)and(2.6)).F??22¨Vs(?/Q)+1?2???Mink222˙+aJ(?)FJ(?/Q)?1,aMink(?2)F?(2.43)

3.Drell-Yancase

3.1Theτ→1expansion

ThecrosssectionoftheDrell–Yanprocess,ha+hb→e+e?+X,is:

??1dxidσfi/ha(xi,?F)fj/hb(xj,?F)gij(τ,Q2,?2F)29Qsxj0i,j(3.1)

–9–

wheresisthehadroniccenter–of–massenergy,s?=xixjsisthepartonicone,Q2isthesquaredmassoftheleptonpairandτ=Q2/s?.Thepartonicthresholdτ→1ischaracter-izedbySudakovlogarithms.

Letusde?netheMellintransformofthequark–antiquarkpartoniccrosssectionin

222(3.1),e.g.inelectromagneticannihilationgqq?(τ,Q,?F)=eq[δ(1?τ)+O(αs)],by

?12222(3.2)dττN?1gqqGqq?(τ,Q,?F).?(N,Q,?F)≡

IntheN→∞limitonecanderive(seee.g.[14])theanalogueofEq.(2.4):

22dlnGqq?(N,Q,?F)

1?τ

whereKDY(τ,Q2)isthemomentumspacephysicalDrell–Yanevolutionkernelde?nedforarbitraryτ,andtheN-dependenttermsarecontrolledbythe‘soft’SudakovphysicalanomalousdimensionS:

S(?2)=A(αs(?2))+1

dln?2,(3.4)?2SQ2(1?τ)2+HDY(αs(Q2))+O(1/N),?(3.3)

whereDisthestandard(scheme-dependent)‘soft’SudakovanomalousdimensionrelevanttotheDrell-Yanprocess.Moreover,theconstanttermcanbeexpressedintermsoftheanalytically–continuedelectromagneticquarkformfactor[14]:

????22?dlnF(?Q)?2HDY(αs(Q))=S(?2),(3.5)2?

wheretheinfraredsingularitiescancel[14]inthesum,asin(2.6):

????22?dlnF(?Q)?2S(?2)HDY(αs(Q))=2????dR=G1,αs(Q2),ε=0+β(αs)

withR(αs(Q2))?2?=ln?F(?Q)

????2dln?F(?Q2)?

?(3.7)2?Dαs(Q2),?(3.6)1?τ1?τ

?????dln?F(?Q2)?2

+????+?+infrared?nite???02?δ(1?τ)+O(1?τ)S(?)DY??2?

–10–

WeseethatthephysicalSudakovanomalousdimensionScontrolstheleadingtermintheexpansionofthemomentumspacephysicalDrell–Yankernel(3.3)nearthreshold.Theτ→1limitistakensuchthatEDY=Q(1?τ),correspondingtothetotalenergycarriedbysoftgluonstothe?nalstate,iskept?xed.Thevirtualcontribution,proportionaltoδ(1?τ),isdeterminedbythequarkformfactor,analytically–continuedtothetime–likeaxis.Thistermisinfraredsingular,butuponperforminganintegraloverτthissingularitycancels

?2?2withtheonegeneratedbyintegratingthereal-emissionterm2SQ(1?τ)/(1?τ)near

τ→1.

SimilarlytotheDIScase,Eq.(3.7)suggestsageneralizationtoanexpansionforτ→1inpowersof1?τat?xedEDY.AsintheDIScase,however,itturnsout(seesection3.2)thatinordertoderiveadispersiverepresentationsuchanexpansionisnottheappropriateone.Instead,oneshouldconsideranexpansioninpowersofthealternativevariable:

rDY≡2

1?τ

τ

(3.8)

2,with:(properlynormalizedsothatrDY?1?τforτ→1)at?xedWDY

22

WDY≡rDYQ2,

(3.9)

namely:KDY(τ,Q2)=

2

dlnQ2

?2??2??2?

δ(1?τ)+S0WDY+rDYS1WDY+OrDY,

(3.10)

wherethecoe?cientsSandSiarethephysical‘soft’anomalousdimensionsappropriatetotheDrell-Yanprocess.Afterregularizingthevirtualcontribution,Eq.(3.10)canbewrittenas:KDY(τ,Q)=

?

2

2

dlnQ2

2

dlnQ2

δ(1?τ)+S0WDY+rDYS1WDY

+??

?

Q2

?

2

??

2

??2?+OrDY

=

d?2

τ+

√

rDY

2

SWDY

?

??

=

+

?

1

?

1

dτF(τ)

1?τ

S(1?τ)2Q

?

(3.12)

??2

,

whereF(τ)isasmoothtestfunction.

–11–

3.2DispersiverepresentationofthephysicalSudakovanomalousdimensions

(largeβ0)

InquiteasimilarwaytotheDIScase,westartfromthelarge–β0butarbitraryτ(0<τ<1)dispersiverepresentationofthephysicalDrell–Yanmomentumspaceevolutionkernel,

?2?KDY(τ,Q)largeβ0=CF?∞d?20

where?=?2/Q2,andexpandingtherealcontributionusingtheexplicitexpression3for(r)FDY(?,τ)in[8](Eq.(4.87)there),oneobtains:

FDY(?,τ)=(r)??)2+Vt(?)δ(1?τ),(3.14)41?4?22EDY???11?4?21?4?2

τ+√

2rDY=?2

rDY2FDY(ξDY)+F0,DY(ξDY)+rDYF1,DY(ξDY)+O(rDY),(r)(r)(r)(3.17)

with

FDY(ξDY)=2tanh(r)?1

1?4ξDY?(1?4ξDY)tanh?2??19+84ξDY?32ξDY2?1+(18+16ξDY+32ξDY)tanh(81?4ξDY?(?1?(

rDY

Takingonederivative(.=??2

rDY

with

(r)˙DYF(ξDY)=?d1+11+11?4ξDY+....(3.19)(r)˙(r)(ξDY)+rDYF˙DY˙(r)(ξDY)+O(r2),(ξDY)+FFDY0,DY1,DY(3.20)

˙(r)(ξDY)=F0,DY11?4ξDY12ξDY?1?4ξDY

8√1?4ξDY?4ξDYtanh?1?((3.21)1?4ξDY),(r)˙DY˙(r)(ξDY)arenowsingular,butintegrable,atthephase-whereallthetermsF(ξDY)andFi,DY

spaceboundary4ξDY=1.Thisresultfollowsimmediatelyfrom(3.19)whichyields:

2(r)˙DY(?,τ)=F42rDY

Usingtheseresultsin(3.14),andnotingthat:

?√θ(1?τ(1+

?FS(ξDY)+FS0(ξDY)+rDYFS1(ξDY)+O(rDY)+Vt(?)δ(1?τ),(3.24)2??2rDY1+....1?4ξDY(3.22)

rDY

with

FS(ξDY)=FDY(ξDY)θ(4ξDY<1)

FSi(ξDY)=(r)Fi,DY(ξDY)θ(4ξDY(r)<1),(3.25)

–13–

and

˙DY(?,τ)=F?2

1?4ξDY)??1

τ+√

rDY

??2

∞

0+rDYd?2??˙˙ρV(?)FS0(ξDY)?FS0(0)2?∞d?20(3.29)

?2¨t(?),aMink(?2)VV

whereIperformedintegrationbypartsinthevirtualcontribution,inordertoobtainat

˙t(0)beingin?nite(theintegralitselfisinfrareddivergentsinceleasta?niteintegrand,V2):¨t(0)=?1).Comparingwith(3.10)Ideduce(ξDY=?2/WDYV

?2?S(WDY)largeβ0=CF

andalso(?=?2/Q2):

????22?dlnF(?Q)?¨t(?),aMink(?2)VV(3.31)?∞0d?2??2˙S(ξDY)?F˙S(0),ρV(?)Fii(3.30)?2?2

–14–

whichchecksEq.(3.10)inthelarge–β0limit,andinadditiongivesthe(large–β0)dispersiverepresentationsofthephysicalSudakovanomalousdimensionsSandSi,aswellasthe(large–β0)dispersiverepresentationofthetime-likequarkformfactor[14].Finally,thedispersiverepresentationoftheregularizedvirtualcontributionin(3.11)isgivenby:

????2dln?F(?Q2)?S(?)=CF2

?2?∞d?20

Si(WDY)=CF2?∞

0?2d?2??˙˙ρS(?)FS(ξDY)?FS(0)2

dln?2;aMinkS,Si(?)≡?2?∞?2dm2

?2aMinkS,Si(?)2?2/k2?2+k2ρS,Si(?2).(3.35)

Fromthesedispersiverepresentations,theperturbativeexpansionsofthe‘soft’Sudakove?ectivechargesaMink(?2)andaMink(?2)canbederivedinastraightforwardway[6]orderSSi2)andS(W2).byorderinthefullnon-Abeliantheory,giventheexpansionsofS(WDYiDY

Furthermore,thegeneralizationof(3.31)is

????2dln?F(?Q2)?

whereasthatof(3.32)is:

????2dln?F(?Q2)?S(?)=CF2?2¨t(?),aMink(?2)VF(3.36)

?2?∞0d?2

4.Conclusions

IpresentedasimplemomentumspaceansatzforthestructureofasystematicexpansionofthephysicalevolutionskernelsaroundtheSudakovlimit,focussingonDISandtheDrell-Yanprocess.Eachorderintheexpansionisgivenbyapeculiar‘jet’or‘soft’(dependingontheprocess)physicalSudakovanomalousdimension.

Theansatzhasbeenderivedfromalarge–β0dispersiverepresentation,whichcanbereadilyextendedtothefullnon-Abeliantheoryat?niteβ0.Clearlymorejusti?cationsandchecksarestillneeded.Inthisrespect,thegeneralOPEmethodintroducedin[3,4]maygivetheappropriatetoolforasystematicderivation.Anotherpossibility(whichshallbethesubjectofafutureinvestigation)istoperformacheckoftheansatzbymatchingitwithexisting[24,25]?xedordercalculations;asaby-productonewouldget(iftheansatzturnsouttobecorrect)theperturbativeexpansionoftherelevant‘jet’and‘soft’physicalanomalousdimensionswhichoccurascoe?cientsintheseexpansions.

Ihaveshownthattheconnectionoftheansatzwiththedispersiverepresentationrequiresanon-trivial,andprocess-dependent,choiceofexpansionparameterintheSudakovlimit.WhiletheadequateparameterrintheDIScaseisstandard,andreferstothe?nalstatejetmassscale,themeaningofthecorrespondingparameterrDYintheDrell-Yancaseremainstobeunderstood:itdoesnotcorrespond(ascouldbenaivelyexpected)tothetotalenergyEDYradiatedinthe?nalstate.

2))momentumspaceItwasfoundthat,althoughtheleadingjet(J(W2))andsoft(S(WDY

physicalanomalousdimensionsdonot[6]haverenormalons(atleastatlarge–β0),renor-

2))physicalmalonsdostarttoappearinthesubleadingjet(J0(W2))andsoft(S0(WDY

anomalousdimensions.

GiventhemultiplicityofemergingphysicalSudakovanomalousdimensions,itwouldbeinterestingto?ndoutwhetherthereexistsanyrelationshipbetweenthem.Inparticular,onemightwonderwhetheranyexact,orapproximate,universalitypropertyholdsamong

Minkthevarious‘Sudakove?ectivecharges’aMinkJ,JiandaS,Si.Actually,fromtheobservationthat

thesumJ(W2)+J0(W2)isatotalderivativeatlarge–β0(whichfollowsfromthe?nitnessofFJ(ξ)+FJ0(ξ)atξ=0,see(2.22)),andmakingtheassumptionthatthisproperty

2stillholdsat?niteβ0,onecanalreadyreadilydeduceintheDIScasethataMinkJ(?)and

2aMinkJ0(?)haveidenticalexpansionsuptoNLO,andtherefore[6]coincidewiththecusp

anomalousdimensionuptothisorder.

Thetransitiontoan1/Nexpansioninmomentspacehasbeenshowntobestraightforward,andtheconnectionbetweenthemomentumandmomentspacejet(orsoft)scaleshasbeenclari?ed.Fromtheexpansionsinmomentspace,onecaneasilyobtainthemomentspacedispersiverepresentations,followingthemethodin[6],bysubstitutingthedispersiverepresentationsofthephysicaljetandsoftphysicalanomalousdimensionsinthemomentspaceexpansion.

ThelongitudinalstructurefunctionFL(x,Q2),whichhasnotbeenaddressedinthispaper,needsaspecialtreatment(tobereportedelsewhere).Indeed,itappearsthedispersiveap-proachdoesnotworkinastraightforwardwayinthiscase,essentiallybecauseatlarge–β0thelongitudinalphysicalevolutionkernelKL(x,Q2)doesnotcoincideanymorewiththe

–16–

derivativedFL(x,Q2)/dQ2.Thisisduetothefactthatthelongitudinalcoe?cientfunctionisO(αs),ratherthenO(α0s)asinthecaseofF2(whichcouldbereplacedatlarge–β0byitsleadingtermδ(1?z)ontherighthandsideofEq.(2.1)).Inparticular,KL(x,Q2)isnotsuppressedforx→1,contrarytoFL(x,Q2)itself.Preliminaryinvestigationnever-thelessseemstoindicatethatanansatzoftheformofEq.(2.11)mightstillbevalidforKL(x,Q2)(withadi?erentcoe?cientoftheδ(1?x)term),althoughtheconnectionwiththedispersiveapproachislost(excepteventuallyatlarge–β0,wheretheleadingSudakov??physicalanomalousdimensionJLW2couldbeatotalderivative).

Finally,althoughIhavestudiedherethesimplestexampleswhereonlyonescale(‘jet’or‘soft’)ispresent,theansatzcanbereadilyextendedtoothercases[6]wherebothofthesescalesdooccursimultaneously.

Acknowledgements

IwishtothankEinanGardiforanumberofpastdiscussionsonthedispersiveapproach.

A.N→∞expansioninmomentspace(DeepInelasticScattering)

TheN→∞expansionofthemomentspacephysicalevolutionkernelEq.(2.3)canbestraightforwardlyobtainedbytakingthemomentsofthemomentumspaceexpansionEq.(2.11)(r=1?x

??2?2?dlnF(Q2)JW+r

N1+i?∞0dt?1N?N+1tJiti?Q2

1+tNe?tt

N1+i?∞0dte?ti

Oneshouldnotethattheintegralonther.h.s.of(A.4)containsconstantterms,aswellaslogarithmsofN.Moreover,theintegralbeingconvergent,theconstanttermsdetermine??2?tJitW+O(1/N2+i).(A.4)

–17–

thelogarithmicallydivergentterms,giventhebetafunctioncoe?cients:seetheargumentfollowingEq.4in[13]–whichimpliesthattheknowledgeofthelogarithmictermsatagivensubleadingorderofthe1/Nexpansionalso?xestheconstanttermsatthesameorder.Inparticular,theN→∞expansionofthenexttoleadingrealemissionterminEq.(A.1)is:

?1

?2?1N?1

dxxJ0W=

r

JW

and,takingtheN→∞limitwithQ2/N?xedinsidetheintegralonther.h.s.,theanalogue

ofEq.(A.4):

?1??

2N?11?+O(1/N).(A.7)JtWdxx

t0inEq.(A.1)(wheretheIRdivergences

?2

?Q2d?2

Q

cancell,aswehaveseeninsection(2.1)).Since0tJt

dlnQ2

??2

=

?

∞

dt

?

1

N

?N+1

1

N

?

,(A.6)

TheinfrareddivergenceinEq.(A.7)canberegularizedasinEq.(2.12)bysubtractingfrom

?Q22

ther.h.s.of(A.7)theIRdivergentpiece0d?

r

JW

??2

?

=

+

??+

∞

dte?t

1

t

?

1

2

Combining(A.5)with(A.8)onethusgetsthelargeNexpansionofthe?rsttwoleadingrealemissiontermsinEq.(A.1):

???1??N1?1?2?dtJtWdxxN?1

t00

???∞?1?t2?dte+J0tW

N0

?

?2+O(1/N2).(t?2)JtW

?2JtW

?

??

(A.8)

The?rsttermonther.h.sof(A.9)canalsobewritten,uptoexponentiallysmallcorrectionsatlarge

RN

N,as0dt(e?t?1)1

?Q2

5

dlnQ2

d?2

+

Ialsonotethattakingthemomentsoftheleadingterminthe?rstlineofEq.(2.7)onegets(settingt=N(1?x)):

?

1

dxxN?1

1

=?

N

∞0

dte?t

1

?N?1

1

N

?

τ+

√

rDY

????

?2?dln?F(?Q2)?2SWDY+

N1+i

?

∞0

du

?

1

2N

?2N+1

uSiu

i

?

2Q

2

N1+i

?

∞0

due

?u

ThustheN→∞expansionofthenexttoleadingrealemissionterminEq.(B.1)is:

?

1

??

2?2

uSiuWDY+O(1/N2+i).

i

(B.3)

dττ

N?1

S0WDY=

?

2

?

1

rDY

?2?SWDY=2

?

∞0

du

?

1

2N

?2N+1

1

N2

?

,(B.5)

–19–

and,takingtheN→∞limitwithQ2/N2?xedinsidetheintegralonther.h.s.of(B.5),theanalogueofEq.(B.3):

?

1

dττ

N?1

2

u

TheinfrareddivergenceinEq.(B.6)canberegularizedasinEq.(3.11)bysubtractingfrom

?Q22

ther.h.s.of(B.6)theIRdivergentpiece0d?

dlnQ2

?2?SuWDY+O(1/N).?

2

(B.6)

cancell,aswehaveseeninsection(3.1)).Since

?Q2

inEq.(B.1)(wheretheIRdivergences

?

d?22Q2

Suu

???

2?DYSu2W

(B.7)

rDY

?2?

SWDY

?

+

??=2+

∞0

due?u

1

?

u

2

1

2

?2?(u?2)SuWDY+O(1/N2).

Combining(B.4)with(B.7)onethusgetsthelargeNexpansionofthe?rsttwoleading

realemissiontermsinEq.(B.1):?

1

dττ

N?1

??

2

?

∞0

N

due

?u

u?

??12?2

S0uWDY+

???

2?DY?2Su2W

N

du

1

?Q2

dlnQ2

d?2

+

1?τ

??22

S(1?τ)Q=2

?

N0

?udu1?

u

which,comparedto(B.6),checkstheequivalenceoftheleadingtermsin(3.7)and(3.11).

“

2fDYu2W

u

??

2?2

SuWDY+O(1/N),

?2

2QSu

(B.9)

u

S

”

.

–20–

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