General Mass Scheme for Jet Production in QCD (revised version)

A thesis submitted forthe degreeof Do tor of Philosophy in Physi s

General Mass S heme

for Jet Produ tion in QCD

by

Piotr Kotko

writtenunderthesupervisionof

1 Hadrons, partons and jetsin QCD 13

1.1 Fa torizationtheorems . . . 13

1.2 JetsinQCD. . . 18

1.3 Quarkmassesin QCD . . . 22

2 In lusiveDIS with heavy quarks 25 2.1 Introdu tion. . . 25

2.2 Zero-massvariableavournumbers heme . . . 26

2.3 Fixedavournumbers heme . . . 28

2.4 ACOTs heme . . . 30

3 Massive dipole subtra tion method 35 3.1 Introdu tion. . . 35

3.2 Singularbehaviouroftree-levelmatrixelements. . . 36

3.2.1 Softlimit . . . 36

3.2.2 Quasi- ollinearlimit . . . 40

3.2.2.1 Initialstateemitter ase . . . 40

3.2.2.2 Finalstateemitter . . . 43

3.3 Constru tionofdipoles. . . 44

3.4 Dipolekinemati s. . . 46

3.4.1 FinalStateEmitter-Initial StateSpe tator. . . 46

3.4.1.1 Standarddipolekinemati s . . . 48

3.4.1.2 Kinemati swithadditionalinvariant(gamma-kinemati s) 48 3.4.1.3 Softand ollinearlimitsofthedipole variables . . . 50

3.4.1.4 RelationbetweendipoleandtheSudakovvariables. . . . 51

3.4.1.5 Dipolevariable asafreeparameter . . . 52

3.4.2 InitialStateEmitter-FinalStateSpe tator. . . 53

3.4.2.1 Standardkinemati s . . . 54

3.4.2.2 Gamma-kinemati s . . . 54

3.4.2.3 Dipolevariable asafreeparameter . . . 55

3.4.2.4 Softand ollinearlimits . . . 56

3.4.2.5 RelationbetweendipoleandtheSudakovvariables. . . . 57

3.5 Dipolesplittingfun tions . . . 57

3.5.1 Finalstateemitter-Initial statespe tator . . . 57

3.5.1.1

Q

→ Qg

and

Q

→ Qg

splittings . . . 57

3.5.1.2

g

→ QQ

splitting . . . 59

3.5.2 Initial StateEmitter-FinalStateSpe tator. . . 61

3.5.2.1

Q

→ Qg

and

Q

→ Qg

splittings . . . 61

3.5.2.2

g

→ QQ

splitting . . . 62

3.5.2.3

Q

→ gQ

and

Q

→ gQ

splittings . . . 63

3.6 Phasespa efa torization . . . 64

3.6.1 Preliminaries . . . 64

3.6.1.1 Two-parti lephasespa e . . . 65

3.6.1.2 Two-parti lephasespa eindipolevariables. . . 66

3.6.2 Finalstateemitter-initialstatespe tator. . . 67

3.6.3 Initial stateemitter -nalstatespe tator . . . 68

3.6.4 Fa torizationofthree-parti lephasespa e . . . 68

3.6.4.1 Expli itexamples . . . 71

3.7 Integrationofthedipoles . . . 71

3.7.1 Thenotation . . . 73

3.7.2 Finalstateemitter-Initialstatespe tator . . . 74

3.7.2.1

Q

→ Qg

and

Q

→ Qg

splittings . . . 74

3.7.2.2

g

→ QQ

splitting . . . 78

3.7.2.3

g

→ gg

splitting . . . 81

3.7.3 Initial StateEmitter-FinalStateSpe tator. . . 84

3.7.3.1

Q

→ Qg

and

Q

→ Qg

splittings . . . 84

3.7.3.2

g

→ QQ

splitting . . . 88

3.7.3.3

Q

→ gQ

and

Q

→ gQ

splittings . . . 90

3.7.4 Complete expressionsforintegrateddipoles . . . 92

4 General Mass S heme for Jets 97 4.1 Introdu tion. . . 97

4.2 PartondensitiesinCWZ renormalizations heme . . . 97

4.2.1 Unrenormalizedpartondensities . . . 98

4.2.2 Renormalizationofpartondensities . . . 102

4.3 Quasi- ollinearsubtra tiontermsformassivedipoles . . . 103

4.3.1 Masslesstreatmentoffa torization . . . 103

4.3.2 Fullymassive ase . . . 106

4.4 Masslesslimitand onsisten y he k . . . 110

4.5 Pra ti alappli ationand

MassJet

proje t . . . 113

5 Summaryand outlook 117 A Kinemati s 123 A.1 Thermodynami softheinvariants . . . 123

A.2 Expli itexpressionsforres aledvariables . . . 125

A.2.1 FinalStateEmitter-InitialStateSpe tator. . . 125

A.2.1.1

Q

→ Qg

and

Q

→ Qg

splittings . . . 125

A.2.1.2

g

→ QQ

and

g

→ gg

splittings . . . 126

A.2.2 Initial StateEmitter-FinalStateSpe tator. . . 126

A.2.2.1

Q

→ Qg

and

Q

→ Qg

splittings . . . 126

A.2.2.2

g

→ QQ

splitting . . . 127

A.2.2.3

Q

→ gQ

and

Q

→ gQ

splittings . . . 127

B Mathemati al supplement 131

B.1 Theintegrals . . . 131

B.2 Expansionsin

ε

. . . 136

B.3 Theplusdistribution . . . 138

Quantum hromodynami s(QCD) is aswebelievethe orre ttheory ofthestrong

intera tions,withquarksandgluonsbeingitsfundamentaldegreesoffreedom. Although

there are many puzzles remaining unsolved, it is verysu essful in des ribing various

aspe tsofthemodernhighenergydata. Theoreti alpredi tionsarebasedontwomajor

issues. Themostimportantoneistheasymptoti freedom,whi hassertsthatthevalue

of the strong oupling onstant is relatively low at high energy s ales. It enables us

to use perturbation theory in al ulations on ernings attering amplitudes. However,

there are no freequarksand gluons in the nature they are allbounded in olourless

hadrons,thus theperturbative al ulationsare notthewhole story,asthehadronsare

learly of non-perturbative nature. Therefore, the se ond basi issue are fa torization

theorems, whi h allow fora separationof apro ess to a non-perturbativebound state

physi s and al ulable in QCD hard s attering amplitudes. Te hni ally the former is

des ribed in termsof variousdistribution fun tions and distribution amplitudes,whi h

sofararemostreliablytakenfromexperiments.

Phenomenologi allythemostimportantnon-perturbativeinput omprisesparton

dis-tribution fun tions (PDFs). Histori ally, they appeared in a des ription of in lusive

lepton-hadrondeepinelasti s attering(DIS)asanelementofthepartonmodel.

Nowa-daystheyareusedalsointheotherhighenergyexperimentslikeproton-proton ollisions

atLHC forinstan e. Inorder toobtainPDFsonehasto tthetheoreti ally al ulated

ross se tionwith suitable analyti alparametrizationsof PDFs to thereal data. Most

often the ele tron-proton HERA data are used in this pro edure. There are several

groupsmakinganeortin extra tingPDFs,e.g. CTEQgroup[5℄orMRSTgroup[41℄

tomentiononlythemostknown.

Onemayaskthequestion: whatisthedieren ebetweenvarioussetsofPDFs? There

areatleast afewodds. Therstaredierentfun tionalparametrizationsanddierent

statisti almethodsusedinttingthedata. Howeverthemaindieren eis onne tedto

as hemeinwhi hthea tual rossse tionis al ulated. Thisissueisinseparablyrelated

to heavyquarks and the problem of s alesin QCD. Namely, there is a di ulty with

nite-orderperturbative al ulationsifthereareafewexternalparameters(s ales)that

areverydierent. Thoses ales anbexede.g. byanexternalenergyorbythemasses

ofthequarks. A tuallyitisthe aseinreality,aswehavesixquarkavourswiththreeof

thembeingmarginallyheavierthantheothers. Moreoverthereisalsosubstantialmass

splittingsbetweentheheavyquarks. Insomesituations,thisdi ulty anbesolvedby

means of the renormalizationgroup methods, however there does notexist a uniform

perturbativeexpansionsuitableforallthes ales. Thereforeonehasto hooseaspe i

s heme. Intuitivelyit orresponds toa situation,where our measurementresolution is

toosmall to distinguishsometinydetails and to largeto see the whole thing. We an

s hemesinperturbativeQCD.

Formanyyears,a ompletelymasslesss hemehavebeenthemoststandardintreating

DIS s attering. A tually, thes heme wasmasslessin the sense of negle tingthe mass

parametersin al ulations,but hangingthenumberof avoursin thesametime. We

shallseethedetailslater,howeverevenintuitivelyweseethatsu hanapproa hisvery

limited in a ura y. It an be satisfa tory only in ertain, relatively narrowranges of

kinemati spa e. Thereforealsotheothers hemeswereused,treatingtheheavyquarks

in amoreordered way. Inparti ular, the s hemesfor harm, and bottomquarkswere

used. Those al ulationsareverya uratein asuitablekinemati range, butagain

thoserangesarelimited. Furtherdevelopmentmusthaveledtoa omposites heme,that

isto hangethenumberofavoursononehand (likeinthemasslesss hemementioned

in the beginning of the paragraph) and to keep the masses nite on the other. The

most ommonnamefortheapproa hofthistypeisageneral-masss heme orvariable

avournumbers heme. Weshallseesu hasolutionindetailsbelowinthiswork.Sofar

thegeneral-masss hemeswereusedin in lusivepro esses,bothin extra tingthePDFs

andpredi tingexperimental out ome.

There ishoweveranotherveryimportant lassofhigh energypro esses,namelythe

produ tion of jets. Sin e one measures also the spatial distribution of the outgoing

parti lesit angivemu hmoreinformationaboutunderlyingpartondynami s. Loosely

speaking,ajeta ollimatedbun hofhadronsisaremnantofapartoneje tedfromthe

enterof ollision. Thusbyanalysingthemomentumandenergyofthejet,wegetalmost

dire ta ess tothepartonlevelsubpro ess. Itallowsformorepre ise measurementsof

somequantities, forinstan e strong oupling onstant(e.g. theanalysis performed by

ZEUS ollaboration using dijet produ tion in DIS [1℄ and by H1 ollaboration using

in lusive-jet,dijetsandtrijetanalysis[32℄). Thejetsprodu tionpro essesarealsoused

toobtainthepartondistributionfun tions(togetherwiththein lusivedata). Theoreti al

al ulationsneededtothispro edure areagains hemedependent. In aseofjets,there

ishowevermu hlesstheoreti aldevelopment on erningheavyquarks. Thereareseveral

Monte-Carlo(MC)programsusingmasslessquarks,e.g. NLOJET++[42℄,DISENT[9℄,

bothforhadron-hadronandlepton-hadron ollisions(thelastforneutral urrentwithout

Z

0

-ex hange). Forheavy quarks, there are some al ulationsfor in lusive-jetand two

jets produ tion at NLO [26℄ in a s heme with xed number of avours. It should be

remarked,thatwemeanherestri tQCD al ulations,notamodel-basedones. Forjets,

theformer aremu h moreinvolved andrequirespe ial treatments ofsingularities that

appearat NLO(andhigher)orders.

Inthiswork,weproposeasolutionintendedto llthegapin existing heavyavour

treatments. It is a general-masss heme for jets produ tion pro esses, based on some

solutionsavailableonthemarket. We on entratehereinonDISpro esseswithneutral

urrentintera tions. Furtherextensionsarepossible,asweshortlydis ussinChapter5. The developments we are presenting are essentially theoreti al. However, in order to

support the validity of our al ulationswe givesome samplenumeri al resultsusing a

dedi atedMC program. Itisapartofalargerproje tthat is urrentlyunder

develop-ment.

Thematerialisorganisedasfollows. First,inChapter1were allthebasi formalism weshallusethroughout,in ludingfa torizationtheoremsandjetstreatment. Chapter2

is devoted to existing approa hes to heavy quarks in in lusive DIS pro esses and its

problems. Notably,itintrodu esthegeneral-masssolution,whi hwelaterapplytojets.

re-ofmassivepartons,in ludingpossibleinitialstateheavyquarks. Finallywegatherallthe

pie es and onstru tthe general-masss hemeforjetsin Chapter 4. Weintrodu e tons ofsymbolsthroughoutthis work. Someofthem maylook messy,howeverthisa ounts

forthepre isetheoreti alformulationofthematerial. Inorder tofa ilitatethereading

weput someofthemin aNomen lature. Thete hni aldetailsthat arenotessentialin

themaintextarelistedin theappendi es.

A knowledgements

Inthe rst pla e, I would liketo thank Prof. Woj ie h Sªomi«ski for his patien e,

substantialhelpandmanyhoursofjointdis ussions. Thewordsofgratitudebelongalso

to Prof. Mi haª Praszaªowi z, the head of Parti le Theory Department I had a great

pleasureto beamemberasPHDstudent.

I thankmywife Iwonafor onstantsupport andfaithin me, espe iallywhen

every-thing wasgoing wrong. The sameis trueabouther family: Brunonand Basia, Marek

andElatogetherwiththeir hildren.

I own spe ial thanks to Mirek Tro iuk, my high s hool tea her and Ola Šubni ka,

who reatedthes ienti atmospherethatbroughtmetothis point.

Finally, I am grateful to my parents Maria and Jerzy, my brothers Damian and

Hadrons, partons and jets in

QCD

1.1 Fa torization theorems

Although QCD has in redible amount of su esses, the theory is still not solved. For

instan e, there isa olour onnementhypothesis, stating that all observableparti les

are olour singlets. This onje ture has verystrong experimental eviden e; so far free

quark or gluon has not been found. However, su h a property has not been derived

yetfrom QCD,althoughthere areseveraltheoreti al lues,bothperturbativeand

non-perturbative. Moreover,theredoesnotexista ompletedes riptionof omposedobje ts

like hadrons in terms of the fundamental QCD degrees of freedom (i.e. quarks and

gluons). Forexample,it is known that manyfeatures of aproton anbe explainedby

assuming that it is build of three quarks

u, u, d

. Theirmasses (i.e. the parameters in

QCDlagrangian;massispoorlydenedquantityforanunobservableparti le)areabout

afew

MeV

. Ontheotherhand,theprotonmassiswelldenedand anbemeasuredit

turnsouttobearound

1 GeV

, learlynotaboutthreetimesthemassesof onstituents.

Thisisaneviden eofveryimportantnon-perturbativephenomenon,namelyspontaneous

hiral symmetrybreaking. It generatesso alled onstituentquarkmass,whi h should

beaboutonethirdoftheprotonmass. Su havalue annotbedes ribedbyperturbation

theory,thetoolwhi h atpresentisbest understoodandunder ontrol. Therearemu h

moreproblemsin des ribinghadronswithin perturbativeQCD.

All these features draw hadrons as a very ompli ated, non-perturbative obje ts.

Nevertheless, there are possibilitiesto get ertain insightinto the stru ture ofhadrons

using perturbation theory. As it wasalreadymentioned, QCDhas apropertyof being

asymptoti allyfree, i.e. at veryshort distan es theQCD oupling isveryweak, giving

some han es to useperturbation theory. This is a keyobservationleading to modern

high-energy experiments; ollisionsof parti les with higher energies anprobe smaller

spa e-time volumes. However,sin e weprobeonlyasmall partofa olourlesshadron,

we anhopetohaveperturbativedes riptiononlypartiallytherestmustbesomehow

parametrized,orobtainedbyothermethods. Thisis in fa t averyloosedes riptionof

famous fa torizationtheorems, whi h we nowshall re allin somedetails. Inour

intro-du tionweshalltrytogivemostlyne essaryresults,butwere allalsosome ompletely

We are mainly on entrated on lepton-hadrondeep inelasti pro esses throughout.

Moreover, in this se tion we limit ourselves to in lusive pro esses only. There are

es-sentiallytwopossibleapproa hestofa torization,whi hper olateatsomestages. Both

haveitsown ons andpros.

Firstone,reliesontheoperatorprodu texpansion(OPE)[54℄andishistori allythe

rstapproa hto fa torization[7℄, of ourseex eptFeynman'sparton formalism

onsid-eredbefore QCDhad beenborn. Although OPEallows forverysystemati treatment

ofalltermsthat anappear, itsappli abilityisratherlimitedtothein lusivepro esses

only.

Se ondapproa hisbasedongeneralpower ountingtheorems [40,39℄and methods

developed in [21℄. Letus re all thebasi ideas, astheyshall be importantlater, when

wedis ussmore ompli atedtopi s. Forareviewsee e.g. [17,15℄.

Considerageneri unpolarized boson-hadron utamplitude,asshown inFig.1.1A. Wedenoteprotonmomentumas

P

andbosonas

q

,with

q

2

₌

−Q

2

. Moreover,weassume

thatthebosonvirtuality

Q

2

ismu hlargerthanallthequarkmasses(in ludingpossible

heavyquarks)andthattheBjorkenvariable

x

B

= Q

2

_/P

· q

isxed. Thesituationwhere

Q

2

is of the same order as the mass of a given heavy quark will be dis ussed in the

next hapter. Itturnsoutthatalltheleading ontributionstothe utamplitude anbe

hara terizedbythe utamplitudesthat havetheformshowedinFig.1.1B.Theupper blob hasallthe internal momentao-shell by order

Q

2

andthus is alled ahardpart.

Note,thatalthoughsomeoftheinternallinesare utandhen eon-shell,theyee tively

an be treated as o-shell lines by virtue of the opti al theorem. The lower part in

Fig.1.1B, the soft part, in orporates hadroni states and two partoni lines joining it withthehardpart.Thoselinesareeitherquarkorgluonlineswithvirtualitymu hlower

than

Q

2

andmomenta ollineartothehadroni momentum. Itshouldbementionedthat

theinternalblobofthesoftpart, anstillhaveUVsingularities,seebelow.

The ontributions that have stru ture des ribed above are alled twist-2, as they

orrespondinOPElanguagetoaseriesofmatrixelementsoflo aloperatorswithtwist

1

equalto2. Contributionswhi hhavemorethantwolinesjoininghardandsoftpartshave

highertwist. Re allthatsu hhighertwist ontributionsaresuppressedby

m

2

_/Q

2

,where

m

2

isthemassoftheheaviestquarktakenintoa ount. Thereareseveral ompli ations

(seee.g. [15℄),howeverthegeneralpi tureisasjustdes ribed.

Note,that thetwolinesjoining bothparts annot orrespondtoaheavyquarkwith

massoftheorderof

Q

2

,duetotheassertionthat theyhavelowvirtuality omparingto

Q

2

. Thisfa t shallbeimportantlateron.

Now,we ometo morepre ise denitions ofthe softpartand its onne tionto the

restofthe pro ess. As is ommonlyknown, thesoftpart anbeparametrizedinterms

ofpartondistributionfun tions (PDF)(weshallinter hangeably allitparton density)

insideahadron. Inordertopro eedweintrodu elight- one oordinates;anyfour-ve tor

v

anbede omposedas

v

µ

_{= v}

+

_n

_˜

µ

_{+ v}

−

_n

µ

_{+ v}

µ

T

,

(1.1)

where

v

+

= v

_{· n, v}

−

= v

_{· ˜n}

(1.2)

withtwolight-likeve tors

n

,

n

˜

denedas

n =

√

1

2

(1, 0, 0,

−1) , , ˜n =

1

√

2

(1, 0, 0, 1) .

(1.3) 1

A) B) PSfragrepla ements

q

P

PSfragrepla ements

q

P

k

Figure1.1: A)CutFeynmanamplitudeforunpolarizedboson-hadronpro ess. B)

Lead-ingregions ofthe utamplitudeforlargevirtuality oftheboson. Thelines onne ting

upperandlowerpartshavelowvirtualityand anbelightquarksoragluon.

Letusnowassumethatthemomentum

k

joiningthehardandsoftpartsisparametrized

using the light- onevariables and that suitable frame is hosen, su h that

P

+

islarge

∼

p

Q

2

. Then,sin e

k

havesmallvirtuality omparingto

Q

2

its

k

−

and

k

T

omponents

anbenegle tedinthehardpart.Then,the onne tionofthetwoparts anberealisedas

anintegraloverthe

k

+

omponent. Therestofthemomentumintegration(i.etransverse

andminus omponents)areembodiedin thedenition ofPDF, wherethey annotbe

negle ted. Its external lines ( onne ting it with the hard part) ee tively lie on the

light- one.

All these remarks leadto the followingdenitions of the parton distributions. For

thequarkdensitywehave

f

q

(B)

(x) =

1

4π

ˆ

dy

−

e

−ixP

+

y

−

P



ψ

q

y

−

n

γ

+



y

−

n, 0



ψ

q

(0)



 P



(1.4) andforgluon

f

(B)

g

(x) =

1

2πxP

+

ˆ

dy

−

_e

−ixP

+

_y

−

D

P





_F

_A

+µ

y

−

_n



_y

−

_{n, 0}



AB

F

+

B µ

(0)





 P

E

.

(1.5) Letusnowexplaintheabovenotation. First,therearequarkeldoperators

ψ

q

andthe

gluoneldstrengthoperator

F

µν

C

= ∂

µ

A

ν

C

− ∂

ν

A

µ

C

+ g f

CDE

A

µ

D

A

ν

E

. All theseeldsare unrenormalized,thusthePDFsdened in su hawayarethebareones asindi atedby

thesupers ript

2

. Parameter

x

orrespondsto afra tion ofplus omponentofhadron

momentum that is transferred to the hard part, that is we assume

k

+

_{= xP}

+

and is

xed. Next,

y

is a spa e-time point we integrate over, with however xed

y

+

_{= 0}

;

theintegrationover

y

−

indisentangled whiletheone over

y

T

isperformed(orhidden).

Finally,there is agaugelink in orderto makethedenitions gaugeinvariant. It reads

in thepresent ase



y

−

n, 0



= P exp

(

ig

ˆ

y

−

0

dz A

+

_C

(z) t

C

)

,

(1.6)

where thepath joining bothpoints is hosen to be astraightline. Inparti ular, when

weuse light- one gaugedened as

A

· n = 0

the gauge link is a unity operator (it is

2

Sin ewefollowheremainly[15℄andotherpapersofthisauthor,weusethetermbareinthesense

useful in somegeneral onsiderations). The last remark on erning (1.4), (1.5) is that only onne teddiagramsshould betakenintoa ount.

Asalreadymentioned,thepartondistributionfun tionsdenedabove ontainUV

di-vergen es. Requiredrenormalization on ernsnotonlytheelementaryelds,butalsothe

bilo alquarkorgluonoperatorsitself. Asiswellknown,therenormalizationintrodu es

additionaldependen e onaprioriunspe iedmasss ale

µ

r

.

It anbeproved that the relationbetween thebare densities andthe renormalized

oneshastheform[16,15℄

f

a

(R)

x, µ

2

r

=

X

b

ˆ

_dz

z

K

ab

 z

x

, α

s

µ

2

r

; ε



f

_b

(B)

(z) ,

(1.7)

wheretherenormalizationkernel

K

ab

isaperturbatively al ulablequantity. Note that

wehaveintrodu eddimensionalUV regulator

ε

denedas

D = 4

− 2ε,

(1.8)

where

D

is the spa etime dimension. The summation in (1.7) goes over all possible kindsofthelinesjoiningthesoftandhardparts (exa tsetsshallbedenedin thenext

hapter). The kernel

K

ab

an be al ulatedby onsidering thesameobje tsas

f

a

but

withthehadroni statesrepla edbythepartoni ones. Thuswedenethequantity

F

ab

,

whi hwerefertoasadensityofparton

b

insideaparton

a

. Thedenitionisexa tlythe

sameasfor

f

b

withthe hadroni staterepla edby theon-shellstate

a

. Thequantities

F

ab

an be al ulated perturbatively in QCD with the help of spe ial Feynman rules [16, 17℄ we shall use them for massive quarks in Chapter 4.2. A tually, we have to again distinguish between the bare

F

(B)

ab

and the renormalized one

F

(R)

ab

, howeverthe relation between the two remains the same as (1.7). This allows to obtain

K

ab

on e spe i renormalizations hemeis hosen(seealsobelow).

Sin e the bare densities

f

(B)

a

aredened bymeans of the bare elds only, they are ompletelyindependentontherenormalizations ale. Thereforeitisrelatively

straight-forwardtoderiveanevolutionequationforthedensities. Itreads

d

d log µ

r

f

_a

(R)

x, µ

2

_r

=

X

b

ˆ

_dz

z

P

ab

 z

x

, α

s



f

_b

(R)

z, µ

2

_r

,

(1.9)

wheretheevolutionkernel

P

ab

isrelatedto therenormalizationkernelbytheformula

P

ab

 z

x

, α

s



= 2α

s

∂ K

ab,

1

z

_x

, α

s

∂α

s

,

(1.10)

with

K

ab, n

denedbytheLaurentexpansion

K

ab

(z, α

s

; ε) = δ (z

− 1) δ

ab

+

∞

X

n=1



1

ε



n

K

ab, n

(z, α

s

) .

(1.11)

Forexample,inthe

MS

s hemewith

N

f

avoursweobtain

P

ab

(z, α

s

) = δ (z

− 1) δ

ab

+

α

s

2π

P

(1)

ab

(z) +

O α

2

s

,

(1.12) where

P

(1)

ab

arefamouslowestordersplittingfun tions. Theyread

P

qq

(1)

(z) = C

F



1 + z

2

1

_{− z}



+

,

(1.13)

P

gg

(1)

(z) = 2C

A

"

1

1

− z



+

+

1

− z

z

− 1 + z (1 − z)

#

+ δ (1

_{− z)}



11

6

C

A

−

2

3

N

f

T

R



,

(1.14)

P

gq

(1)

(z) = T

R

[1

− 2z (1 − z)] ,

(1.15)

P

qg

(1)

(z) = C

F

1 + (1

_{− z)}

2

z

.

(1.16)

Theplus distributionisdened inastandardwayas

h

+

(z) = h (z)

− δ (1 − z)

ˆ

1

0

dy h (y) .

(1.17)

Note, that the support is

[0, 1]

 we pay attention to this detail, sin e we shall often

use distributions with dierent supports (see also Appendix B.3). Sin e the splitting fun tions

P

(1)

ab

areoftenusedinthis thesiswedropthesupers riptinwhat follows

P

_ab

(1)

(z)

≡ P

ab

(z) .

(1.18)

We turn also attention to our onvention of ordering the subs ripts. The notation

ab

orresponds to a splitting pro ess

a

→ b

, where parton

b

takes the fra tion

z

of the

originalmomentum. Thephysi alinterpretation ofthefun tions

P

ab

isthensu h, that

itgivesaprobabilitydensityforsu h asplitting.

Let us now ome ba k to the fa torization. On e the renormalizationof the PDFs

and of the hard part is done, we an nally write the fa torization formula. In what

followswedroptherenormalizationindi ationinthehadroni PDFs

f

_a

(R)

z, µ

2

_r

_{≡ f}

a

z, µ

2

r

.

(1.19)

Thefa torizationtheoremtakesthefollowingform

dσ P, q; x

B

, Q

2

=

X

a

ˆ

1

x

B

dz

z

f

a

z, µ

2

f

, µ

2

r

dˆ

σ

a

zP, q; Q

2

, µ

2

f

, µ

2

r

+

_O



m

2

Q

2



.

(1.20)

Here

dσ

orrespondstoadierentialDISin lusive rossse tion,while

dˆ

σ

a

isapartoni

ross se tion whi h is infra-red (IR) nite. Besides UV singularities, there are also

divergen eswhi horiginateinzeromassofthegluonsandtherearetwosortsofthem: the

softsingularitiesandthe ollinearones. Theyremainevenafterrenormalization,however

thesoftandmixedsoft- ollineardivergen esare an elledbetweendierent ontributions

(we shalltake upthis issue in thenext se tion). What remains are the ollinear ones.

Thefa torizationpro edureasserts,thatthey anbein ludedinPDFsasitisessentially

anonperturbativeobje tandweshallnever al ulateitusingperturbationtheory. Su h

apro edure isattheexpenseofintrodu ingadditionalfa torizations ale

µ

f

. Aprioriit

isarbitrarys aleandoneoftensetsitequaltotherenormalizations ale. Moreover,there

is ertainfreedomin hoosinga tuallysubtra tedterms. Su hapres riptiondenesthe

thekernel

K

ab

). Thus,wehave

dσ

_a

(R)

p, q; x, Q

2

, µ

2

_r

=

X

b

ˆ

1

x

dz

z

h

S µ

2

_r

, µ

2

_f

_F

_ab

(R)

z, µ

2

_r

+ S

ab

z, µ

2

r

, µ

2

f

i

dˆ

σ

b

zp, q; Q

2

, µ

2

f

, µ

2

r

+

O



m

2

Q

2



,

(1.21)

where the fun tions

F

(R)

ab

are the renormalized densities of parton inside aparton dis- ussedbefore (weindi ated alsothat unsubtra ted rossse tion is renormalized). The

quantities

S

and

S

ab

deneourfa torizations heme,seebelow. Theaboveequation an

besolvedorderbyorder, al ulating

dσ

(R)

a

and

F

(R)

ab

toadesiredorder.

Asanillustration,letus onsider ompletelymassless ase. Choosing

MS

s hemeto

denePDFswegetatthelowestnontrivialorder

F

ab

mMS

x, µ

2

r

= δ (1

_{− x) δ}

ab

+

α

s

µ

2

r

2π



−

1

_ε



P

ab

(x) .

(1.22)

The supers ript

mMS

expli itly indi ates that weuse

MS

renormalizations heme and

ompletelymassless al ulation. Sin e

MS

anbealsoused inamassive ase,wefeela

ne essitytodistinguishbothsituationsasweshallen ountertheminonepla elateron.

Weseethat there isa ollinearpole

1/ε

intheresult, whi h an elsthesimilarpolein

dσ

a

. Next,ifwe hoosethefa torizations hemetobe

MS

,wehave

S

ab

z, µ

2

r

, µ

2

f

= 0,

(1.23)

S µ

2

r

, µ

2

f

=

1

Γ (1

− ε)

4πµ

2

r

µ

2

f

!

ε

.

(1.24)

Let us on lude this se tion by giving some summarizing remarks. First is that

hadroni PDFsareessentiallynonperturbative,andhavetobeobtainedfromexperiment,

latti e al ulationsorlow energyee tivemodels. Most reliablearethose obtainedby

global tsto data (e.g. [38℄). Moreover,PDFs ares hemedependent, andassu h are

unphysi al. Thereforeonehavetobe arefulwhenmixingPDFsobtainedbyonemethod

with al ulationsin someother s heme,asthereminder(

O (. . .)

terms) infa torization

theorem anbe omelarge.

1.2 Jets in QCD

Inthepreviousse tionwehave onsideredthefa torizationtheoremessentiallyfor

in lu-siveDISs attering. Oneoftheelementsofthea tualproofofthefa torizationproperty

isthe an ellation ofthesoftsingularities. Inthisse tion, wetakea loserlookat this

problem. Inparti ular,wedes ribeamethodallowingforthis an ellationin asewhen

thepro essisnotfullyin lusivebut onsistin jets. Thisshallbeaverygeneral

presen-tationofthetopi anditwillevolvethroughoutthewhole dissertation. Wefollow[9℄in

thisintrodu tion.

Beforewestart,letusintrodu esomenotation. The

n

-parti leinvariantphasespa e

(PS)shallbedenoted as

A) B) C) PSfragrepla ements

n

PSfragrepla ements

n + 1

PSfragrepla ements

n

Figure 1.2: Illustrative presentation of the amplitudes for

n

-jet produ tion. A) LO

amplitude,B)realemission orre tions,C)virtual orre tions.

where

p

and

q

arein omingmomenta. Ontherighthandsidewehaveused

mathemati- ians'notationforsets, asitwilloftenallowtomakeformulaeshorter. Thephasespa e

anbeexpressedas

dΦ

n

(p, q;

{p

i

}

n

i=1

) = (2π)

D

δ

(D)

p + q

₋

n

X

i=1

p

i

!

_n

Y

i=1

dΓ

i

(1.26)

in termsoftheinvariantmeasuresforaparti le

i

dΓ

i

≡ dΓ (p

i

) =

d

D

_p

i

δ

+

p

2

_i

− m

2

_i

(2π)

D−1

.

(1.27)

All thedenitions arewrittenin

D

spa e-timedimensions.

A tree-levelamplitude with in omingmomenta

p

,

q

and

n

outgoingstates shall be

denoted as

M

n

(p, q;

{p

i

}

n

i=1

)

. Atthis stage allpossible olour orspinindi es are sup-pressed. Ifrelevant,wewilladorntheamplitudebyvarioussymbolsand/orindi es,for

example we will put ahat if we onsider external fermions to underline that wework

withamatrix. Veryoftenwewillrefertoapartoftheamplitude,forexamplewhentwo

externallegsarerepla edbyone. Thenthereminderisreferredtoasredu edamplitude.

Let us now swit h to the a tual matter of this se tion. We start with the very

s hemati des riptionof NLO al ulation for

n

-jets. Suppose for simpli ity that there

arenoinitial statehadrons,e.g. ele tron-positronannihilation. A detailed formulaefor

DISshallbegivenin Se tion4.3.

ToNLOa ura y,thetotal rossse tion anbewrittenas

σ

n

= σ

LO

n

+ σ

NLO

n

.

(1.28)

Theleadingorder ontributionreads (Fig.1.2A)

σ

_n

LO

=

ˆ

dΦ

n

|M

n

|

2

F

n

,

(1.29)

where

M

n

and

dΦ

n

are explainedabove(we suppressallmomenta dependen e), while

F

n

is ertain(generalized) fun tionthat givesusanobservableweareinterestedin(i.e it may in lude step-fun tions for kinemati uts, delta fun tions for dierential ross

se tion, jet algorithms et .). We shall refer to

F

n

as a jet fun tion. We des ribe its

propertiesin detaillater.

Thenext-to-leadingordertermhasin turnthefollowingform

σ

NLO

where

σ

R

represents the real orre tions, i.e the ones onne ted to the emissions of

additionalon-shellparti lesin thenalstate(Fig. 1.2B).Next,

σ

V

orrespondstoloop

orre tionsto

M

n

(Fig.1.2C).Thelast anbewrittenas

σ

n

V

=

ˆ

dΦ

n

M

(loop) 2

n

F

n

.

(1.31)

Thenotationissymboli here,

M

(loop) 2

n

isa tuallyaninterferen ebetweenthetreelevel amplitudeandtheone ontainingloop orre tions. Forthereal orre tionswewrite

σ

n

R

=

ˆ

dΦ

n+1

|M

n+1

|

2

F

n+1

.

(1.32)

As alreadystated in thepreviousse tion, higherorder al ulationsin QCDlead to

divergen es.First,thereareUVsingularities,whi hareremovedbyrenormalizationand

we donot onsider them here any more. Se ond, there are mentionedIR singularities

oming from vanishing propagators due to almost zero energy of masslessparti les or

ollinear emissions. We shall dene them pre isely in Se tion 3.2. Both kinds of sin-gularities appear in

σ

R

and

σ

V

and are regularized e.g. dimensionally. However the

physi al rossse tion,whi hdoesnotdistinguishbetweenthesoftor ollinearemissions,

hasto benite. Therefore IR singularities haveto an elbetween both termsin ross

se tions (ex ept possible pure ollinear singularities onne ted with initial state

emis-sionswhi h are removed byfa torization). It is pre isely statedby meansof the KLN

theorem (Kinoshita-Lee-Nauenberg)and itsextensions, see e.g. [48, 49℄and referen es

thereintotheoriginalpapers. Inwhatfollowsweassumethatthejet rossse tionunder

onsiderationisinfra-redsafe,that isitfulls alltheassumptionsoftheKLNtheorem.

This howeverrequires to impose somerestri tions on the jet fun tions. Namely, if

oneof thenal stategluonsin

(n + 1)

-parti lephasespa e is soft(itsfour-momentum

vanishes)we musthave

F

n+1

= F

n

. Similarly, iftwoofthe nalstate partonsbe ome

ollinear,their

F

n+1

fun tionmustalso oin idewith

F

n

. Ontheotherhand,ifweenter

asingularregionin

n

-parti lephasespa e

F

n

mustvanish. Thoserules anbeextended

toinitialstatepartonsandmassivepartonsaswell.

Now,sin eweknowthat IRsingularities an el,thereremainstheproblem of

te h-ni al nature, whi h however is of great importan e. Namely, both orre tions

σ

R

and

σ

V

areintegratedoverdierentphasespa eswithdierentjetfun tions. Analyti al

al- ulations arehereextremelydi ult andimpra ti al, thus oneoftenusesMonte Carlo

methods. The problem is now to an elthe singularity that appears during numeri al

integrationin

σ

R

withanalyti alsingularitiesin

σ

V

,e.g.

1/ε

poles.

Histori ally the rst method was so alled phase spa e sli ing method. It an be

illustratedbysimplemathemati al example(e.g. [36℄). Suppose wehavethe following

niteexpression

I = lim

κ→0

ˆ

1

0

dx

h (x)

x

1−κ

−

1

κ

h (0)



,

(1.33)

where the dependen e on

x

in

h

is very ompli ated but su h that the integralexists.

Thersttermin urlybra ket orrespondstoareal ontributionregularized

dimension-ally, while the se ond termis the orresponding softpole in virtual orre tion. Both

singularities an elasa tuallytherealvalueoftheintegralis

I =

ˆ

1

0

dx

h (x)

− h (0)

Supposehowever,that wewantto an elthemnumeri ally. Tothisend, wedividethe integration domain

´

1

0

. . . =

´

δ

0

. . . +

´

1

δ

. . .

, with

δ

≪ 1

. Sin e

h (x)

is regular enough, we anapproximate

h (x)

≈ h (0)

for

x

∈ [0, δ]

. Then,after simplestepsweget

I

_{≈ h (0) log δ +}

ˆ

1

δ

dx

h (x)

x

.

(1.35)

Note,thatthesingularities an elledandtheintegral anbenowperformednumeri ally

withremovedregularization, i.e. weset

κ = 0

. Thereis howeveradisadvantageasthe

resultisapproximate.

Another method, advo ated in thiswork,is thesubtra tion method [36℄. One

on-stru tsanauxiliary rossse tion

σ

sub

=

ˆ

dΦ

n+1



M

sub

n+1





2

F

n

(1.36)

whi hmimi sallthesingularitiesof

σ

R

,i.e

σ

sub

_{= σ}

R

inthesingularregionsofPS(note,

there is

F

n

for

n

partons). Besidethose pointsof phasespa e it an be anything that

havethepropertiesof a rossse tion. Onthe otherhand, itmust be hosenin su h a

way,thattheanalyti alintegrationoverone-parti lesubspa eispossible. Thatis, ifwe

writePS s hemati allyas

dΦ

n+1

= dΦ

n

⊗ dφ,

(1.37)

we must be able to perform

´ dφ



M

sub

n+1





2

analyti ally. It leads then in dimensional

regularizationto poles ofthe form

1/ε

whi h an el those in virtual orre tionsdue to

the KLN theorem. Thepro edure of al ulating NLO ontribution using this method

anbesummarizedasfollows

σ

NLO

_{= σ}

R

− σ

sub

+ σ

V

_{+ σ}

sub

=

ˆ

dΦ

n+1



|M

n+1

|

2

F

n+1

−



M

sub

n+1





2

F

n



+

ˆ

dΦ

n



M

(loop) 2

n

+

ˆ

dφ



M

sub

n+1





2



F

n

.

(1.38)

Inthese ondline,duetoIRpropertiesofthejetfun tions,we anperformtheintegration

infourdimensionsanditisnite. Inthethirdlinea an ellationofthepolestakespla e

andafterthat we anset

D = 4

.

This methodhasanobviousadvantage,namelyit isexa t. Se ond,alltheintegrals

overone-parti lesubspa ehavetobemadeonlyon e andtheyareuniversal. This an

bealsogeneralizedtohigherorders,wehoweverneedmu hmoresubtra tionterms.

A parti ular hoi efor

σ

sub

isrealizedin [9, 25℄ formasslesspartons,andin [10℄for

massivequarksinthe nalstate(withsomerestri tionsdis ussed in3.1). Thisspe i hoi e is alled dipole subtra tion term. A tually, asolid part of this work is devoted

to generalizingthisapproa hto ompletely massive ase, su h that one anpra ti ally

applymassivefa torizationpro eduredes ribedin theChapter2.

Thedipole methodhas,however,alsosomedrawba ks. First,itisrelatively

ompli- ated,asweshallsee. Moreover,itisunlikelytobegeneralizedeasily tohigherorders.

Thereasonisthatitoperatesontheamplitudessquaredandthenumberofsubtra tion

whi ha tually on ernsthesubtra tionpro edureingeneral,isthatofnumeri alnature.

Namely, depending on implementation, there may be someproblems when performing

theintegrationinthese ondlineof(1.38). Thusee tively,onemaybefor edto usea supportin aformofasli ing-likemethod.

1.3 Quark masses in QCD

Inthepreviousse tionswedidnotpayspe ialattentionto thequarkmasses. Here we

re allsomebasi fa ts onne tedwiththeir in lusionin perturbative al ulations. The

followingmaterialisessentialto thewhole work. Insomepartswerelyon[13℄.

Todayweknowsixavoursofquarkswiththefollowingmasses

3

[43℄:

m

u

= 1.7

-

3.1 MeV,

m

d

= 4.1

-

5.7 MeV,

m

s

≈ 100 MeV,

(1.39)

m

c

≈ 1.29 GeV, m

b

≈ 4.19 GeV, m

t

≈ 172.9 GeV.

(1.40) Re allnow,thatthebasi requirementtobeinaperturbativeregime,isthatthetypi al

energys ale,say

Q

, satises

Q

≫ Λ

QCD

. Sin e

Λ

QCD

≈ 200 MeV

we ansafelynegle t

themasses of

u

,

d

,

s

quarksin perturbative al ulations. Ifthes aleishighenough,we

analsomakesu hanapproximationwiththeotherquarks.

On the other hand, aprioriwe do notknownif there exist heavierquarks. Similar

situationusedtobebeforethedis overyofthetopquark. Thus,thequestionwasabout

therelevan eofeldtheoreti al ulation,wheresomeofthequarksarepossiblymissed.

Thesolutiontothisproblemisformulatedbymeansofso alledde ouplingtheorem[4℄.

Itstatesthat for aFeynmanamplitudewith atypi almomentum s ale

Q

we androp

all thediagrams with quark mass

m

≫ Q

, doing error

O (Q/m)

. Let us now assume,

thattheremainingnumberofquarkavoursis

N

f

,thusalltherenormalizedparameters

(masses, ouplings et .) in su h an ee tive theory are al ulated using this number.

Ingeneral,therenormalizedparametersin theee tivetheorywith

N

f

+ 1

avoursare

dierent.

Theproblemhoweverarises,whenthemassesarenotextremelydierent,asa tually

happensfor harmandbottomquarks.Forinstan e,whenthes aleis loseto

m

c

,we an

makeamistakeof theorder

m

c

/m

b

≈ 30%

(foranexamplesee e.g. [13℄). Fortunately,

there is a better method than su h an un ontrolled de oupling. It redu es to thelast

in thelimitof verylarge masses. It is aspe ial renormalizations hemeexisting in the

literatureas CWZ (Collins-Wil zek-Zee)renormalizations heme[19, 45, 14℄. In order

to dene its basi slet us introdu ean a tive number of quarks

N

a

. It is anumber of

quarkslighterthanthexedexternalenergys ale(note,thatwedonothavetosetthose

masses to zero). TheCWZ s heme onsist in the subs hemes hara terizedby

N

a

. In

ea h subs hemetherenormalizationisdonea ordingtothefollowingpoints:

a) thegraphswithinternallines beinga tivearerenormalizedusing

MS

b) thegraphswithat leastoneinternalheavyquarkline(ina tive) arerenormalized

byzero-momentumsubtra tion

) massesofheavyquarksareusually denedasthepolemasses

3

Asthefreequarkstatesareunobservable,thesearejustparametersobtainedin

MS

s hemeats ale about

2 GeV

.

Thiss hemepossessesseveralimportantproperties(seee.g. [13℄). Forustwoofthemare

themostimportant. Firstisthatitsatisesmanifestde oupling. Thatis,iftheexternal

s aleismu hsmallerthanthemassesofina tivequarks,therenormalizedparametersof

asubs hemewith

N

a

a tiveavoursarethesameasinee tivetheory with

N

f

= N

a

.

Hen e we anjust drop all the diagrams with ina tive quarks. The se ond important

propertyisthattheevolutionoftherenormalizedparametersinea hsubs hemeisexa tly

thesameasin

MS

with

N

f

= N

a

,inparti ular theevolutionkernelsaremassless.

This last property isof great importan e in thisthesis. Aswehaveseenin Se tion

1.1,theoperationaldenitionofpartondistributionfun tionsin ludesarenormalization s heme. Sin ewearegoingtotreatfa torizationwiththeheavyquarksitis onvenient

todene PDFsin CWZs heme. Then, duetothese ond property,su h PDFsundergo

thestandardDGLAPevolutionequationinea hsubs heme. Weshalldis ussitindetails

in Se tion2.4,whilein Se tion4.2we al ulatesomeofthem inthiss heme.

Thereisonemore ommentinorder. Thepurposeofintrodu ingsu has heme,isto

beabletoevolveagivenparameterthroughallappli ables aleswithoutloosinga ura y.

Itis realizedbyswit hing thes hemesatgivenswit hingpoints. Therefore,wehaveto

state amat hing onditionsat those points

4

in order to have astartingparameters in

evolution. Su h onditionswereobtainedevenupto threeloopsforthe oupling(using

ee tivetheoryformalism[12℄)anduptotwoloopsforPDFs[8℄.

In theend, letus introdu esomemorenotationwe shallusethroughout. First,we

oftenneedtodistinguishbetweenheavyandlightavours. Thuswedene

N

f

= N

q

+N

Q

,

where

q

is ageneri lightquark, while

Q

orrespondsto heavy quarks. Sometimes we

referto lightpartons number,whi h issimply

N

l

= N

q

+ 1

,asgluonis alwayslight. If

wewantto referto allthe quarkavours,but in ludinggluon, we usethesymbol

N

′

f

. Forallthedenedsymbols,weintrodu ethesets, ontaining orrespondingavoursand

theiranti-avours. Thesetsshallbedenotedbybla kboardfont,forinstan e

N

f

,

N

l

et .

4

Ingeneral,oneshoulddistinguishbetweentheswit hingpointandamat hingpoint. Therstisthe

In lusive DIS with heavy quarks

2.1 Introdu tion

As we haveseenin Se tion 1.3, there are ertainlysome ompli ationswhen there are heavy quarkswith masses that are neither marginally largenor negligibly small. The

problemsareevenmoreevidentin thepro esseswhi h requirefa torization. InSe tion

1.1were alledthefa torizationtheoremassumingthatthemasses anbenegle ted. In ase,when they annot, su h atreatmentis obviouslyveryina urate. Inthis hapter,

weshallanalysethisissueinmoredetailsinthe ontextofin lusiveDISs attering.

First,inthenextse tionwere allthesimplestpossiblewayofin ludingheavyquarks,

a tuallytreatingthemasmasslesspartons. Thiss heme,often alledzero-massvariable

avournumbers heme(ZM-VFNS) ismostoftenused in phenomenologi alanalysis of

DIS pro esses. However, as weshall see, it is ina urate in non-asymptoti regions of

energy s ale. That se tion is also devoted to introdu ing some notationwhi h weuse

in thisandthenext hapters. Further,inSe tion 2.3webriey des ribemorea urate treatment,howeveraimingat ompletelydierentkinemati regimethanthelatter. This

se ond solution is oftenreferred to asxed-avour numbers heme (FFNS), and takes

alltheee ts ofheavyquarksintoa ount. Theproblemishowever,that astheenergy

s alein reases,su hapredi tionbe omeslessa urate,unlesswegotohigherordersof

perturbation theory. Needless to say, su h a massive high-order al ulationsare mu h

moreinvolvedandtime- onsumingthanthemasslessones,nottomentiongeneralizations

toex lusivepro esses.

Therefore,itisdesirabletohaveas hemewhi hisappli ableatintermediateenergy

s alesand ontainsbothaboves hemesasalimiting ases. Su hsolutionswere indeed

developed[2,50, 8℄, howeverwithexpli it treatment ofin lusive pro essesonly. What

isworthemphasizing,theapproa h itedas[2℄wasprovedto allordersofperturbation

theory[15℄. Weshallbrieydes ribethisapproa h,referredtoasACOT

(Aivazis-Collins-Olness-Tung) s heme, in Se tion 2.4. It is based on CWZ renormalizations heme for partondensitiesand an beeasilygeneralizedtoanotherIR safe rossse tions.

Forashortreviewofthementionedtreatmentsofheavyquarkprodu tioninin lusive

DISseee.g. [51,52℄.

We remark,that although this hapter is onsidered to be introdu tory, wedis uss

2.2 Zero-mass variable avour number s heme

Letusstartbydeningourobje tofinterestin this hapter. Weshallbe on entrated

heremainly onthestru turefun tions parametrizingthe rossse tionforin lusiveDIS

pro esses, notably

F

2

x

B

, Q

2

, and its dependen e on the photon virtuality

Q

2

.

Re-member,that thestru turefun tions areobtainedbymeansof asuitableproje tionof

hadroni tensor

W

µν

,denedasusualintermsofmatrixelementofele troweak urrents

sandwi hedbetweenhadronstates

W

µν

_{q, P ; x}

B

, Q

2

=

1

4π

X

spin

X

P

x

ˆ

dΦ

1

(q, P ; P

X

)

P



j

† µ

₍₀₎



 _P

X



hP

X

|j

ν

(0)

| P i ,

(2.1)

wherethese ondsumgoesoverallnalstates

P

X

. Theproje tionismadeusingsuitable

basetensorsmadeoftheve tors

P

,

q

andthemetri tensor. Negle tingthehadronmass

wegetfor

F

2

F

2

x

B

, Q

2

=

2 x

B

D

− 2



−W

µ

µ

+ (D

− 1)

2 x

B

P

· q

W

µν

P

µ

_P

ν



.

(2.2)

If we repla e the hadroni state by a parton, su h a tensor is alled the partoni

tensor. We shall denote it as

w

µν

. Both tensors are related by meansof fa torization

theorem  we shall give some examplesbelow. We do not give further details related

to otherstru ture fun tions and relatedissues astheyare allstandard(forthe pre ise

denitions in orporatingquarkand target masses see [3℄). Su h limited onsiderations

are ompletely enoughtoelu idate thebasi problemswith heavyquarkmasses,aswe

shallsee.

Beforewepro eed,letusre all,thatwedenoteageneri heavyquarkbysymbol

Q

.

Thelightquarksaredenoted as

q

,there should beno onfusion sin ethis isonly used

inthismeaningasasubs ript.

Letusstartfurther onsiderationsbynoting,that thesimplestpossibleapproa hto

heavy quarksis when

Q

2

→ ∞

with

x

B

xed, su h that all the existing heavyquark masses an be negle ted. Then, the pre ise predi tions are given by the fa torization

theorem (1.20), whi h is exa t. All the quarks (in luding heavy quarks) are treated asmassless partons having orresponding PDFs. Su h situation is obviously not very

plausible. Inpra ti etheenergys alesdonottendtoinnity,moreovermanyinteresting

phenomenaexistatlowers ales. Se ondly,wehaveseveralheavyquarkswithlargemass

splittings,asdis ussedinSe tion1.3. Ontheotherhand,when

Q

2

ismu hsmallerthan

themassofagivenheavyquark,itmaybedroppedfrom al ulationsduetode oupling

theoremmentionedalsoinSe tion1.3.

These twomarginally dierent situations (

Q

2

_{≫ m}

2

Q

and

m

2

Q

≫ Q

2

)motivate the followingsimplests hemeoftreatingheavy quarks:

a) ompletelyde ouplegivenheavyquark

Q

when

m

2

Q

> Q

2

,i.e. treatitasinnitely

heavy

b) treat

Q

asamasslesspartonwithasso iatedPDF, when

Q

2

_{> m}

2

Q

Wehaveassumedherethat thefa torizationandrenormalizations alesareequalto

Q

.

Ifthereareseveralheavyquarks,wehavethe omposites heme,withsubs hemes

har-a terizedbyana tivenumberofavours

N

a

. Thuswehaveasetofpartondistribution

fun tions

f

(N

a

)

a

and ouplings

α

(N

a

)

s heme mentioned earlier, in whi h PDFs are dened. All the masses are however set

tozero. Sin eCWZsatisesmanifest de oupling,wejustdroptheina tivequarks,and

ea hsubs hemeisee tivelya

MS

s hemewith

N

a

avoursand orrespondingDGLAP

masslessevolutionof PDFs. As already mentioned, the s hemes with dierent

N

a

are

a tually dierent renormalization s hemes, they dier by nite terms and a relation

betweens hemeswith

N

a

and

N

a

+ 1

avours anbestated.

Here theswit hing pointis usually hosentobe

µ

th

= m

Q

, where

Q

is

(N

a

+ 1)

-th

avour. It is onvenient, sin e then the heavy quark density

f

(N

a

+1)

Q

is zero at the threshold

1

. Itfollowsfromtwofa ts. First isjustapre iseformoftherelationbetween

PDFs in two subs hemes [18℄. Se ond is that below

µ

th

it is suppressed by power of

Λ

QCD

/m

Q

due tode ouplingtheorem. Thuswehavethe ontinuity ondition

f

(N

a

)

Q

z, µ

2

th

= f

(N

a

+1)

Q

z, µ

2

th

= 0

(2.3)

Then, abovethe threshold it is evolved using DGLAP equations with

N

a

+ 1

avours

startingfromzerovalue.

Asalreadymentionedintheintrodu tion,su has hemeis alledzero-massvariable

avournumbers heme(ZM-VFNS).Correspondingfa torizationtheoremtakestheform

W

(N

a

)

µν

q, P ; x

B

, Q

2

=

X

a∈N

a

f

(N

a

)

a

µ

2

f

⊗ ˆ

w

(N

a

)

µν

q, p

a

;

Q

2

µ

2

f

!

,

(2.4)

whereweexpli itlydenotedthedependen eonthefa torizations ale(equalheretothe

renormalizations ale). Wealsointrodu edthe onvolutionsymbol,whi hsimpliesthe

notation;itisdenedhereas

f

⊗ w =

ˆ

1

x

B

dξ

ξ

f (ξ) w



x

B

ξ



.

(2.5)

In (2.4)

p

a

= ξP

, nevertheless we leave

p

a

as this notation is more general. As we vary the s ale, thea tivenumberof partons hanges. Su h aformula isa tually valid

upto orre tions oforder

O m

2

N

a

/Q

2

, where

m

N

a

would bethemass ofthe heaviest a tive quark, if we did not set it to zero. Therefore, in reality su h an approa h is

unreliablefor

Q

2

aroundthemasses ofheavyquarks. Moreover,aswerea htheregion

of validity of (2.4) for one heavy quark, say harm, we simultaneously an enter the regionof inappli ability forthebeautyquark. Thus, onlyat really asymptoti regimes

thiss hemeis orre t,asweremarkedearlier.

To illustrate this approa h, onsider now a al ulation of

F

2

stru ture fun tion in

thiss hemeuptoorder

α

s

. Letusassumeweworkin thes hemewith

N

a

= 4

,that is

besidesgluon,

u, d

and

s

quarks,whi harealwaysmassless,wehavealso harm

c

N

_a

₌



_{g, u, u, d, d, s, s, c, c}

_.

(2.6)

Then,tothis order

1

x

B

F

2

x

B

, Q

2

=

X

a∈N

a

f

a

µ

2

f

⊗

"

C

a

(0)

Q

2

µ

2

f

!

+ C

a

(1)

Q

2

µ

2

f

!#

.

(2.7) 1

A) B)

Figure 2.1: A)Feynmandiagrams ontributing to stru turefun tions in ZM-VFNS up

toorder

α

1

s

;B)ThesameforFFNS.Thereisonlyboson-gluonfusionatorder

α

1

s

. Thi k line orrespondstoaheavyquark.

Wearea tuallyinterestedin harm ontributionto

F

2

,whi h anbepi kedupfromthe

aboveequation. Setting

µ

2

f

= Q

2

(andthesameforrenormalizations ale)wehave

F

2

c

x

B

, Q

2

=



f

c

Q

2

+ f

c

Q

2



⊗

h

C

c

(0)

+ C

c

(1)

i

+ f

g

Q

2

⊗ C

(1)

g

,

(2.8)

where theresultfor oe ients

C

i

in massless

MS

s hemeis well known (e.g. [28℄, for

orrespondingdiagramsseeFig.2.1A)andreads

C

c

(0)

(z) = e

2

c

zδ (1

− z) ,

(2.9)

C

_c

(1)

(z) = e

2

_c

α

s

2π

C

F



1 + z

1

_{− z}



log

1

− z

z

−

3

4



+

1

4

(9 + 5z)



+

,

(2.10)

C

g

(1)

(z) = e

2

c

α

s

2π



2P

gq

(z) log

1

_{− z}

z

+ 8z (1

− z) − 1



.

(2.11)

Thesplitting fun tion

P

gq

and plusdistribution weredened inSe tion1.1.

Thebehaviourof thissolutionwill be expli itlydemonstratedin Se tion 2.4, where wepresentsomeplots omparingthisNLO al ulationtoothers hemes.

Although  aswe havejust seen su h a s heme is very simplied, it is still most

ommonlyused in PDFsglobal tsto data (e.g. CTEQ ts [38℄and earlier). Its great

advantageis simpli ity and pra ti ality. It should bealso mentionedthat it wasvery

su essfulindes ribinglargeamountofmodernhighenergydata.

2.3 Fixed avour number s heme

Letus now present anotherapproa h,whi h is appli ablewhen

Q

2

isabout theheavy

quark mass

m

2

Q

. A tually, it is a generalization of the previous s heme, where

Q

is ina tive, but has nite mass. Thus we have

N

a

masslesspartons undergoing massless

evolutionandoneheavyavour,whi h anbeonlyprodu eddynami ally. Forexample,

atLOinDISitis theboson-gluonfusion(BGF)pro essdepi tedinFig.2.1B. Thefa torizationtheoreminthis asetakestheform

W

µν

q, P ; x

B

, Q

2

, m

2

Q

=

X

a∈N

a

f

a

µ

2

f

⊗ ˆ

w

µν

q, p

a

;

Q

2

µ

2

f

,

m

2

Q

µ

2

f

!

+

_O

Λ

2

QCD

m

2

Q

!

.

(2.12)

Here,thehard s aleisgiven bytheheavyquarkmassand onthe ontraryto (2.4)

N

a

doesnot hange. Thereforesu has hemeis alledxedavournumbers heme(FFNS)

and was pioneered in [28, 27, 37, 29℄. We note, that here the onvolution symbol is

dened as in (2.5)but the integrationlimitsdepend onquarkmasses (weshall seethe examplebelow).

In order to dis uss someof itsproperties, letus again onsider the expli it result,

namelythe ontributionto

F

2

omingfromthe harmquark. Asalreadymentioned,the

situationwhere harm onne tsthehardandsoftpartsissuppressedby

Λ

2

QCD

/m

2

c

,thus theperturbative al ulationfor

F

c

2

startsat

α

1

s

withBGFpro ess(Fig. 2.1)

1

x

B

F

c

2

x

B

, Q

2

= f

g

Q

2

⊗ C

(1)

g



m

2

c

Q

2



,

(2.13)

with oe ientgivenby(e.g. [47℄)

C

(1)

g

(z, ρ) = e

2

c

α

s

2π

(



2P

gq

(z) + 4ρz

2

(1

− 3z) − 8ρ

2

z

3



log

1 + v

1

− v

+ (4z (1

_{− z) (2 − ρ) − 1) z v}

)

,

(2.14) where we abbreviated

ρ = m

2

c

/Q

2

and

v =

p

1

_{− 4ρz/ (1 − z)}

is the velo ity of the harm quarkin thephoton-gluonCM frame. Nowthelowerlimitonthe onvolutionis

z

min

= x

B

(1 + 4ρ)

.

Letusdis ussnowthisresult. First,letusnotethatit ontainsthepowersof

m

2

c

/Q

2

, whi harela kinginZM-VFNStreatment(highertwists). Thereforeindeeditisreliable

al ulationwhen

Q

2

isoftheorderof

m

2

c

. Now,thequestioniswhatisthebehaviourof thissolutionwhenthes aleismu hlarger. Inthis ase,wendthat

C

_g

(1)

(z, ρ) = e

2

_c

α

s

2π

P

gq

(z) log ρ +

O (ρ) .

(2.15)

Thus we see, that wehave apotentiallylarge logarithm of the heavy quarkmass and

thehards aleratio. Su hlogarithmsappearineverynextorderofperturbationtheory,

typi ally

C

_a

(m)

=

m

X

k=0

c

(m)

_{a, k}

log

k

ρ,

(2.16)

what makes su h an expansion unreliable. The solution is to resum all the powersof

α

s

in front ofthe givenpowerof logarithm,i.e. to suitably rearrangetheaboveseries. Then, we a tuallyarriveat thezero-masss heme with harm beingamasslessparton.

However, onehas to bear in mind that it happens at apri e of loosing ontrol of the

terms

O m

2

c

/Q

2

(a tually,ifwedonottra khighertwistterms,whi hisnoteasyand

sofarhasnot beensolved). In thenextse tionweshallpresentsomeplots omparing

thiss hemeto ZM-VFNS.

There isonemore ommentin order. One anaskwhenthis xedavourapproa h

fails, sin e logarithm is a veryslowly in reasing fun tion. In [29℄ it was argued, that

the rossse tions al ulatedinthisapproa hatNLOarestableevenforrelativelylarge

s ales,howeveronehastouseaspe ialsets ofPDFs,namelyso alleddynami alPDFs

2.4 ACOT s heme

As wealreadyanti ipated in theintrodu tion,there exist solutionswhi h ontain

ZM-VFNS and FFNS asaspe ial ases. In thefollowingse tion we des ribeone ofthem,

theso alledACOTs heme[2,15℄. Webelieveitisthebest solutionthat anbeeasily

generalized to less in lusive pro esses, in parti ular jets. This is in ontrast to other

approa hes like [50, 8℄. As we havealready remarked,it has been proved forin lusive

DIStoallordersin[15℄.

Basi assumption ofthe s hemeis that thePDFs aredened using CWZ

renormal-izations hemeand thatthemassesrelevantto a tualenergys alearekeptnite. This

resultsinthehighertwisterrorsoftheorder of

Λ

2

QCD

/Q

2

overthewhole kinemati ally allowedregionof

Q

2

. Weshallseehowitworksinpra ti ebelow.

Consideragainthehadroni tensor

W

µν

andsupposeforsimpli itythatthereisonly

oneheavyquark

Q

. Thefa torizationisrealiseda tuallybytwodierenttheorems[15℄.

Therstoneis essentiallythesameas(2.12),i.e. itisappli ablewhen

Q

2

_.

_m

2

Q

. The se ond one, is when

Q

2

_&

_m

2

Q

, that is both theorems have anoverlap region. Let us analysethese ond ase. Thetheoremunder onsiderationhasthefollowingform

W

µν

q, P ; x

B

, Q

2

, m

2

Q

=

X

a∈N

a

f

a

µ

2

f

⊗ ˆ

w

µν

q, p

a

;

Q

2

µ

2

f

,

m

2

Q

µ

2

f

!

+

_O

Λ

2

QCD

Q

2

!

.

(2.17)

Super ially itisalmostthesameas(2.12),howevertherearedieren es. Firstis that in thiss heme (i.e. abovesomeswit hing point

µ

th

∼ m

Q

)theset of a tivequarks

N

a

does in lude the quark

Q

. Se ond dieren e is subtle. It is onne ted with IR nite

partoni tensor. Toseethis letus al ulate itto therstorderin

α

s

. Re all, thatit is

donewithfa torization(2.17),butonthepartoni level(letussetallthes alesequalto

Q

)

w

µν

a

q, p

a

; Q

2

, m

2

Q

=

X

b∈N

a

F

CWZ

ab

Q

2

m

2

Q

!

⊗ ˆ

w

µν

_b

q, p

b

;

m

2

Q

Q

2

!

.

(2.18)

Wedenoted thatthe partondensitiesinside aparton arerenormalizedusing

CWZ

. To

the rst order it be omes (below wedrop all the arguments, ve tor indi es and CWZ

supers riptfortransparen y)

w

(0)

a

+ w

a

(1)

=

X

b∈N

a



F

ab

(0)

+

F

(1)

ab



⊗



w

ˆ

(0)

_b

+ ˆ

w

_b

(1)



+

_{O α}

2

s

.

(2.19)

Thus,thezerothorderpartoni tensorisIR safe

w

a

(0)

= ˆ

w

a

(0)

.

(2.20)

Solvingfurtherthere urren ewegetforalightquark

w

q

(1)

=

F

qq

(1)

⊗ w

(0)

q

+ ˆ

w

q

(1)

,

(2.21) forheavyquark

w

(1)

Q

=

F

(1)

QQ

⊗ w

(0)

Q

+ ˆ

w

(1)

Q

(2.22)

andforagluon

w

(1)

g

=

F

gq

(1)

⊗ w

(0)

q

+

F

(1)

gQ

⊗ w

(0)

Q

+ ˆ

w

(1)

g

.

(2.23)