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Adrien Besse

To cite this version:

Adrien Besse. Hard exclusive processes beyond the leading twist. Other [cond-mat.other]. Université

Paris Sud - Paris XI; National centre for nuclear research (Otwock-Świerk), 2013. English. �NNT :

2013PA112106�. �tel-00855281�

Présentée pour obtenir

LE GRADE DE DOCTEUR EN SCIENCES

DE L'UNIVERSITÉ PARIS-SUD XI

en otutelle ave le

NATIONAL CENTRE FOR NUCLEAR RESEARCH

Spéialité: Physique théorique

par

Adrien BESSE

soutenue publiquement leMardi 02juillet2013

Réations dures exlusives au twist sous-dominant

Direteurs de thèse: LehSZYMANOWSKI

Samuel WALLON

Composition du jury

Président du jury: Dr. DamirBECIREVIC

Rapporteurs: Pr. Krzysztof GOLEC-BIERNAT

Dr. Stéphane MUNIER

Examinateurs: Pr. Krzysztof KUREK

Dr. Laurent SCHOEFFEL

Laboratoirede Physique Théorique

(UMR 8627),

Bât. 210, Université Paris-Sud 11,

91405Orsay Cedex

Cette thèse porte sur le alul des amplitudes d'héliités de la leptoprodution dirative

exlusive du méson

ρ

^{dans la limite de Regge} perturbative au-delà du twist dominant. La ompréhension de e proessus et autres proessus exlusifs en terme d'intérations entre

les onstituents fondamentaux de la QCD, onstitue un enjeu majeur pour omprendre la

struture des hadrons. L'approhe suivie par le modèle présentéii est basée d'une part sur

la

k T −

fatorisation à petits

x

^, 'est-à-dire dans la limite des hautes énergies dans le entre de masse

W ∼ √

s

^{et d'autre part sur la} fatorisation olinéairedu méson

ρ

^{dans la limite}

des hautesvirtualités

Q

^{du photonvirtuel} intéragissantave lenuléon.

Dansl'approhedela

k T −

fatorisation,l'amplitudeestsindéeendeuxpièesprinipales, le fateur d'impat orrespondant à la transition du photon virtuel au méson

ρ

⁽

γ ^∗ (λ _γ ) → ρ(λ _ρ )

^{) et le fateur d'impat du nuleon ible. Ces deux fateurs d'impats} intéragissent par l'éhange d'un poméron dans la voie

t

^{qui ontient toute la dépendene en énergie du}

proessus. Lepoméronestdéritàl'ordredominantparl'éhangede deuxgluonsetàl'ordre

dominant en

ln(1/x)

^ave

x ∼ Q ² /W ²

^{par l'éhange d'une éhelle de gluons dans levoie}

t

^.

Lahautevirtualité du photonjustie l'appliationde laQCDperturbativepour aluler

le fateur d'impat

γ ^∗ (λ γ ) → ρ(λ ρ )

^{en utilisant la} fatorisation olinéaire pour séparer les ontributions dominantes au twist 2 et au twist 3. Cette approhe a été employée par

Ginzburg, Panl etSerboen 1985pour alulerlestermes de twist 2des fateursd'impats

des transitions oùle photonvirtuel est polarisé soitlongitudinalementsoittransversalement

etoùleméson

ρ

^estpolarisélongitudinalement. Cestransitionssontdénotées respetivement

"

γ _L ^∗ → ρ L

^{" et "}

γ _T ^∗ → ρ L

^{". L'approhe a ensuite été poussée autwist 3 en 2010 par Anikin,}

Ivanov,Pire,SzymanowskietWallon,pourobtenirletermedetwist3dufateurd'impatde

la transition "

γ _T ^∗ → ρ T

^{" où lephoton virtuel et le méson}

ρ

^{sont polarisés} transversalement.

Ces résultats sont invariants de jauge et font apparaître les ditributions d'amplitudes du

méson

ρ

paramétrisant la prodution du méson à partir des états de Fok intermédiaires quark-antiquarket quark-antiquark-gluon.

Dansette thèse nous présentons un premier modèle sebasant sur es résultatspour les

fateurs d'impats, pour dérire lesrapports d'amplitudes d'héliités assoiés àe proessus

enutilisantunmodèlephénoménologiquepourlefateurd'impatdunuléonible. Onutilise

aussi un modèle pour les distributions d'amplitudes du méson

ρ

^{basé sur le} développement onforme de elles-i. Les résultats de e modèle sont ensuite omparés aux données de

HERA et nous disutons les résultatsobtenus.

Une seonde approhe est présentée où les fateurs d'impats aux twist 2 et 3 des tran-

sitions

γ _L ^∗ → ρ L

^et

γ _T ^∗ → ρ T

^{sont redérivés dans la} représentation des paramètres d'impats.

On montre que es résultats sont équivalents à eux obtenus dans l'approhe dans l'espae

des impulsions et permettent d'avoir une image en terme des ongurations de diples de

ouleurs ontenues dans l'état partonique intermédaire de la transition

γ ^∗ → ρ

^{. Les ampli-}

tudesd'héliités ainsiobtenues sedéomposenten uneonvolutionentrelereouvrementdes

fontions d'onde du photon virtuel et du méson

ρ

^{alulé dans} l'approximation olinéaire,

diusion profondémentinélastique. Nousobtenons ainsi une expression pour les amplitudes

d'héliités où nous pouvons ombiner des modèles d'amplitude de diusion diple-nuléon

ave le reouvrement des fontions d'onde issus des aluls de fatorisation olinéaire aux

twists 2 et 3. Nous présentons les préditions, omparées aux données de HERA, pour les

setions eaes polarisées de la prodution dirative exlusive du méson

ρ

^{obtenues à}

partir des amplitudes d'héliités. Les préditions sont en aord ave les données pour des

virtualitéssupérieuresà5-7GeV

2

. Nousprésentons uneanalysede es résultats, notamment

nous disutons le rle des orretions de twists supérieurs et nous omparons nos résultats

ave des reouvrements de fontionsd'onde obtenus par d'autres modèlesexistants.

Mots-lefs: Proessus exlusifs, Chromodynamique Quantique perturbative, Ampli-

tudes d'héliités, Fatorisation olinéaire,

k T −

fatorisation, Diples de ouleurs.

This thesis, entitled "Hard exlusive proesses beyond the leading twist", deals with the

omputationof the heliityamplitudes ofthe exlusive dirative

ρ −

^mesonleptoprodution intheperturbativeReggelimitbeyondtheleadingtwist. Theunderstandingofsuhexlusive

proessesintermsoftheelementaryonstituentsofQCDisaserioushallengetounderstand

the hadroni struture. The approahwe follow here, rst relies onthe

k T −

fatorization in the small

− x

^{regime, i.e. when there is a high energy}

W ∼ √

s

^{in the enter of mass of}

the photon-proton system. It seondly relies on the ollinear fatorization sheme for large

virtualities

Q

^{of the photon,tofatorize the}

ρ −

^{meson soft part of the proess.}

Within the

k T −

fatorization approah, the amplitude splits in two main piees, the

γ ^∗ (λ _γ ) → ρ(λ _ρ )

^{impat fator, with}

λ _γ

^and

λ _ρ

^the polarizations of the virtual photon and the

ρ −

^{meson, and the nuleon impat fator. The impat fators are interating with the}

exhangeofapomeroninthe

t −

^{hannelwhihorrespondstotheexhangeoftwo}

t −

^hannel

gluons at leadingorder and a ladderof gluons atleading log(1/x) order,with

x ∼ Q ² /W ²

^.

At highvirtualitiesof thephoton,the perturbative QCDtehniquesare justied toom-

pute the

γ ^∗ (λ γ ) → ρ(λ ρ )

^{impat fator using the ollinear} fatorization sheme to get the twist 2 and twist 3 terms. This approah was rst used in 1985 by Ginzburg, Panl and

Serbo to ompute the twist 2

γ ^∗ _L → ρ L

^and

γ _T ^∗ → ρ L

^{impat fators. In 2010 the twist 3}

term of the

γ ^∗ _T → ρ T

^{impat fator was derived by Anikin, Ivanov, Pire,} Szymanowski and Wallon. The results obtained are gauge invariant and they involve the twist 2 and twist 3

distribution amplitudes of the

ρ −

^{meson that} parameterize the meson prodution from the quark antiquarkand the quark antiquarkgluon intermediate Fokstates.

In this thesis we present amodelbased onthese impatfator results toget preditions

for the ratios of heliity amplitudes assoiated to the

ρ −

^{meson dirative} leptoprodution using a phenomenologial model for the proton impat fator. We also use a model for

the distribution amplitudes based on the onformal expansion. The preditions are then

ompared to HERA data and we disuss the results of this approah.

A seondapproah is presented where the twist 2 and twist 3impat fators are derived

in the impat parameter representation. We show that the results are equivalent to the

ones obtained in the momentum spae representation. The results in impat parameter

representation give information about the dipole onguration ontent of the intermediate

state involved in the

γ ^∗ → ρ

^{impat fators. As a result of this approah, the heliity}

amplitudesfatorizeastheonvolutionoftwoparts,the rstone isthe overlapofthe virtual

photon and the

ρ

^{-meson wave funtions omputed in the ollinear} approximation and the seond oneis thedipole-target satteringamplitude. The dipole-targetsattering amplitude

iswelldeterminedonotherproessessuhasdeepinelastisatteringproesses. Combininga

modelforthe dipoleross-setionwith the resultsobtainedwithinthe ollinearfatorization

sheme for theoverlap of the wavefuntions, weget a modelfor heliity amplitudesand the

longitudinal and transverse polarized ross-setions. We ompare our preditions to HERA

data and get a good agreement for virtualities of the photon larger than

Q ² ∼ 5 − 7

^GeV

²

^.

olor dipolepiture.

Keywords: Exlusive proesses, Perturbative quantum hromodynamis, Heliity am-

plitudes, Collinearfatorization,

k T −

fatorization, Color dipoles.

Je voudrais remerier tout d'abord mes direteurs de thèse Samuel Wallon et Leh Szy-

manowski ainsi que notre prohe ollaborateur Bernard Pire pour es trois années de thèse

oùilsm'ontfaitpartagerleur enthousiasmepour lareherhe. J'aiénormémentapprisgrâe

à leurs onseils et à nos disussions et je leur suis profondément reonnaissant autant pour

leur investissement dans mon apprentissage que pour tout le savoir qu'ils ont réussi à me

ommuniquer. Je les remerie d'avoir toujours été disponibles et à l'éoute lorsque j'ai eu

besoindeleur aidequis'esttoujoursrévélée trèspréieusedans l'avanementdemes travaux

de thèse. Cela aura été un vrai plaisir de travailler ave eux ainsi qu'une expériene très

enrihissante.

Je remerie le direteur du laboratoire Henk Hilhorst, pour m'avoir aueilli au LPT et

m'avoir permis d'aller à un grand nombre de onférenes et d'éoles qui m'ont beauoup

apportées. J'aimeraisaussi remeriertoute l'équipeadministrativedu LPT, MireilleCalvet,

Philippe Molle, Joelyne Pueh et Odile Hekenauer, ainsi que l'équipe "informatique" du

laboratoire,PhilippeBouaudetOlivierBrand-Foissa,pourleurdisponibilitéetl'aidequ'ils

m'ont tous apporté durant es trois années.

Je tiens à remerier nos ollaborateurs polonais Krzysztof Gole-Biernat, Leszek Mo-

tyka et Mariusz Sadzikowski pour leur aueil et pour nos disussions lors de mes séjours

à Craovie. Je voudrais remerier Stéphane Munier et Cyrille Marquet pour leurs onseils

onernant mes travaux de thèse, ainsi que Christophe Royon pour ses nombreuses invita-

tions àdes onférenesinternationales. UngrandmeriàHervéMoutardeetFrankSabatié

pour medonnerla hane de ontinuer à travaillersur d'intéressantsproblèmes de physique

hadronique et pour m'avoiraepté ausein de leur nouveau projet.

Meri àtous lesmembres de mon jury pour avoir aepté de prendre de leur tempspour

examiner mathèse. Je lesremerie pour lesremarques très pertinentes qui ontété soulevées

durant la soutenane et qui permettent d'envisager de nouvelles perspetives au travail qui

a été fait dans ette thèse.

Un grand meri aussi aux dotorants du laboratoirepour les bons momentspassés dans

la afèt du LPT. Je remerie partiulièrement Cédri Weiland ave qui j'ai partagé mon

bureau durant es trois années pour son agréable ompagnie et pour toutes les disussions

intéressantes que nous avons eu ensemble.

MeriàmonamiAxelavequinousavonssuivinosétudesdepuisleslassespréparatoires

jusqu'à la thèse et à qui je dois énormément. Je le remerie pour son soutient tout au

long de nos études où nous avons partagé notre passion pour la physique. Je remerie mes

parentsetmonfrèrepourleursinessantsenouragementsduranttoutesmesannéesd'études.

Meri aussi àtousmes amis pourlesbonsmomentspassés ensembledurant estrois années,

Antoine, Gabriel, Olga, Roberto, Béa, Charles, ainsi que tous eux que je ne peux iter ii

mais queje n'oubliepas. Enn je ne peux que remerier mabien-aimée Katyusha pour son

immense soutient durant es deux dernières années de thèse.

Meri à vous tous!

First I would liketo thank my supervisors SamuelWallon and LehSzymanowskias wellas

our lose ollaboratorBernard Pire, forhavingshared withmetheir enthusiasmfor researh

during these three years of thesis. I have learned a lot thanks to their advies and our

disussions and I am deeply grateful for their investment in my formation as well as for all

the knowledge they gave me. I thank them for being always available and attentive when

I needed their assistane whih has always been very helpful in my work. It was a great

pleasure working with them and itwasfor mea veryrewarding experiene.

I would like to thank the diretor of the laboratory, Henk Hilhorst, for weloming me

at the LPT and for all the onferenes and the shools that he allowed me to partiipate

and where I have learned a lot. I would like also to thank the administrative team of the

LPT, MireilleCalvet, PhilippeMolle, Joelyne Pueh and Odile Hekenauer, as well as the

"omputer" team, OlivierBrand-Foissa and Philippe Bouaud fortheir availabilityand for

all the assistane they provided meduring these three years.

Iwanttothank ourolleaguesfromPoland,KrzysztofGole-Biernat,Leszek Motykaand

MariuszSadzikowskifortheirhospitalityand forourdisussionsduringmyvisitsinKrakow.

I would like to thank Stéphane Munier and Cyrille Marquet for their advies on my thesis

work and ChristopheRoyonfor hisinvitations tointernationalonferenes. Many thanks to

Hervé Moutarde and Frank Sabatié for giving me the opportunity to ontinue to work on

interesting hadroni physis problems and for aepting meintheir new projet.

Many thanks also to all the members of the jury for taking from their time to examine

mythesis. Ithank themfor allthe relevantremarksthat have beentold duringthe denfense

whih allowtoonsider new perspetivesto the work of my thesis.

A big thank you to all the PhD students of the LPT for the good time we spent in the

afèt of the lab. I ampartiularly thankful to CédriWeiland with whomI was sharingmy

oe during these three years forhis very pleasantompany and the interesting disussions

we have had.

Manythanks tomy friendAxel withwhomwe sharedourtaste forphysissine ourrst

years at the university, and to whom I owe a lot. I thank my parents and my brother for

their onstantsupportallalong mystudies. Manythankstoallmyfriends forthegoodtime

we spent together during these three years, Antoine, Gabriel, Olga, Roberto, Béa, Charles,

and all othersthat I don't name here but who are not forgotten. I nallythank my beloved

Katyusha for her huge support duringthese two lastyears of PhD.

Thank you verymuh!

Introdution 1

1 High energy QCD 5

1.1 Introdution . . . 5

1.1.1 Postulates and onsequenes . . . 5

1.1.2 Regge trajetoriesand the pomeroninterept . . . 8

1.1.3 Cutkosky rules . . . 10

1.2 Sattering amplitudes inthe Regge limit . . . 11

1.2.1 The olor otet exhange . . . 11

1.2.2 The singlet olor exhange in

t −

^{hannel . . . . . . . . . . . . . . . . 14}

1.2.3 Impat fator representation of the quark-quark sattering amplitude 16 1.2.4 The

k T

fatorization sheme . . . 20

1.3 Deep inelastisattering amplitude inthe perturbative Regge kinematis . . 24

1.3.1 Introdutionto DIS observables . . . 24

1.3.2 Impat fators

γ ^∗ _L,T → γ ^∗ _L,T

^{. . . . . . . . . . . . . . . . . . . . . . . . 30}

1.3.3 Color dipolepiture. . . 33

1.3.4 Models for the dipoletarget interations . . . 40

2 Light-Cone Collinear Fatorization applied to the

ρ −

^{meson prodution 51} 2.1 Introdution . . . 51

2.1.1 Dirativeexlusive vetor eletroprodution. . . 51

2.1.2 The underlying ideas of our approah . . . 55

2.2 Light-one ollinearfatorization up totwist 3 auray. . . 56

2.2.1 Soft parts and hard parts . . . 56

2.2.2 Fatorization of the spinor indies . . . 59

2.2.3 Fatorization of the olor indies . . . 62

2.2.4 Fatorization inthe momentum spae aroundthe light one diretion

p

⁶³ 2.3 Parameterizing the vauum torho-meson matrix elements . . . 68

2.3.1 Light-one wave funtionsand distributionamplitudes . . . 69

2.3.2 Lorentzdeomposition and parity analysis . . . 70

2.4 Redution to aminimalset of DAs . . . 77

2.4.1 DA relationsfrom the equations of motionof QCD . . . 77

2.4.3 Wandzura-Wilzekand genuine solutions . . . 82

2.4.4 The ditionary . . . 86

2.5 Conformalexpansion and sale dependene of DAs . . . 88

2.5.1 Goalof the onformal expansion . . . 88

2.5.2 Conformal expansionof the DAs . . . 90

2.5.3 Sale dependene of the DAs . . . 91

2.6 QCD sum rules . . . 98

2.7 Impat fators

γ ^∗ (λ γ ) → ρ(λ ρ )

^{. . . . . . . . . . . . . . . . . . . . . . . . . . 100}

2.7.1 Kinematis . . . 101

2.7.2 The

γ ^∗ _L → ρ L

^{transition. . . . . . . . . . . . . . . . . . . . . . . . . . 102}

2.7.3 The

γ ^∗ _T → ρ L

^{impat fator . . . . . . . . . . . . . . . . . . . . . . . 103}

2.7.4 The

γ ^∗ _T → ρ T

^{impatfator . . . . . . . . . . . . . . . . . . . . . . . 104}

2.8 Heliity amplitudes . . . 109

2.8.1 Measurement ofheliity amplitudes and spin matrix elements . . . . 110

2.8.2 A proton impatfator model . . . 112

2.8.3 Heliity amplitudes

T 11

^and

T 00

^at

t = t min

^{- Comparison of obtained} preditions with H1 data . . . 112

2.8.4 Heliity amplitudes

T 00

^and

T 01

^for

t 6 = t min

^{. . . . . . . . . . . . . . 117}

2.8.5 Disussion of the results . . . 123

3 LCCF in the impat parameter representation 125 3.1 Introdution . . . 125

3.2 The

q q ¯

intermediate state ontributions . . . 126

3.2.1 Equivalent LCCFproedure inimpat parameter representation . . . 126

3.2.2 Impat fator alulation for the

q q ¯

ontribution . . . 128

3.2.3 Interpretationof the resultobtained inthe WW approximation . . . 133

3.2.4 Equivalene of momentum and impat parameter alulations . . . . 135

3.2.5 The impat parameter representation of the

γ _L ^∗ → ρ L

^{impat fator . 135} 3.3 The

q qg ¯

intermediate state ontribution tothe

γ _T ^∗ → ρ T

^{impat fator . . . . 136}

3.3.1 LCCF inimpat parameter representation for the

q qg ¯

^{amplitude . . . 137}

3.3.2 The olor dipoleongurations of the hard part . . . 138

3.3.3 Fourier transformsof the 3-partondiagramsin the ollinearlimit . . 143

3.3.4 Spin non-ip and spin ip

q qg ¯

^{impatfator . . . . . . . . . . . . . . 147}

3.4 The twist 3

γ _T ^∗ → ρ _T

^{impatfator inthe dipole piture. . . . . . . . . . . . 151}

3.4.1 The dipole piturearising fromthe equations of motionof QCD . . . 151

3.4.2 Equivalenewiththe resultsobtainedinmomentumspaeinthelight- one ollinearfatorization sheme . . . 153

3.4.3 Complete twist 3result of the

γ _T ^∗ → ρ T

^{impat fator . . . . . . . . . 155}

3.5 Heliity amplitudes and polarizedross-setions . . . 156

3.6 Comparison with the HERA data . . . 159

3.7.2 Comparison of overlaps . . . 170

3.8 Disussion . . . 173

Conlusions 175

Appendix 177

REFERENCES 182

Inlusive proesses, suh as the deep inelasti sattering (DIS) proesses have provided a

lot of informationabout the nature of strong interations and the nuleon struture. These

proesses rstdesribed by thenaivepartonmodelproposed by Feynmanand Bjorken [1,2℄

to explain the approximate Bjorken salingobserved at SLAC inlate 60's, allowed to disen-

tangle the hadroni struture as made of elementary asymptotiallyfree onstituents alled

"partons". The mysterious fats that in a strongly bound hadroni state the partons are

ating likefreeand the fatthat quarks withouttheir olor degrees of freedom are violating

the Pauliexlusion priniplewere solved with the apparitionof the quantum hromodynam-

is (QCD) to desribe the strong interations. Indeed, QCD whih is a non-abelian gauge

quantum eld theory basedon the SU(3) olorgroup, isan asymptotiallyfree theory given

the number of avors we know, as demonstrated in 1973 by Wilzek, Politzer and Gross

[3, 4,5℄. This isdue tothe non-abelianharater of QCDand the runningof

α _s

^{is very well}

reproduedby the data.

Anotherimportantfeatureof QCDisthe onnementofquarks andgluonsintoolorless

hadronistateswhihmakesthediretobservationofpartonsasexternalpartilesimpossible.

The experimentalevidenefor gluonsatPETRA in 1979omes from3-jetevents,due toan

energeti gluon radiation

q q ¯ → q qg ¯

^{in the hard sub-proess}

e ⁻ e ⁺ → q q ¯

^{. The onnement of}

the emittedquarkantiquarkandgluonleads totheobservationof3-jetevents. Theseevents

are alsoused todetermined the ouplingonstant of the strong interation

α _s

^.

Many tehniques exist to study the QCD properties. The perturbative QCD (pQCD)

approahis oneof themand itrelies onthe fatorizationofa proessintoa hardpart where

large energy sales are involved and a soft part involvingthe long distane dynamis of the

partonsinsidethehadrons. Thepreseneofahardsale

Q

^{intheollisionisneededtojustify}

the perturbativeexpansionin

α s (Q)

^{of thehardpartandthe} fatorizationintohard andsoft piees. Under kinematiassumptions, one an derivepQCD evolutionequationssuh asthe

Dokshitzer-Gribov-Lipatov-Altarelli-Parisi(DGLAP),Efremov-Radyushkin-Brodsky-Lepage

(ERBL) or Balitsky-Fadin-Kuraev-Lipatov (BFKL) equations, for the soft parts but pQCD

annot provide information of non-pertubative aspets of soft parts. Other tehniques an

supplyinformationonnon-perturbativequantitiessuhaslattieQCD,eetiveeldtheories

or QCDsum rules tehniques.

Inlusiveproesseshavealsoprovidedadeepunderstandingofthestrutureofthehadrons

and thepartonidistributionfuntions(PDFs),whihareknown onawidekinematirange.

They have been the testing ground of theoretial innovations suh as the operator produt

expansion(OPE) formalismrst introdued inpartilephysis by Wilson inthe 70's[6℄and

then applied to DIS [7, 4℄. However inlusive proess observables give only information on

the forward kinematis where there is no momentum transfered in

t −

^{hannel. With the}

inreasing improvementof the experiments,the measurementsonexlusiveproesses, where

one is interesting to aspei nal state, have begun tobring additionalinformationonthe

hadronistruture. Forexample,thegeneralizedpartondistributions(GPDs)parameterizing

of the partoni distributions but also the skewness dependene. The exlusive proesses

suh as the dirative produtionof vetor mesons, orthe deep virtual Compton sattering

(DVCS) have been studiedfor more than 25 years and are still the subjet of many studies

and experiments. For our purpose, we should name more partiularly the HERA ollider

ollaborations H1 and ZEUS as they have provided data for very small values of

x

^and

moderate

Q ²

^{, whih is the kinemati region of interest in this thesis. The low}

− x

^{physis is}

an interesting limit of QCD. Alternative approahes from the usual ollinear fatorization

shemearebasedon

k _T −

fatorization,suhasthedipolemodelsbyNikolaev, Zakharov[8,9℄

and Mueller[10, 11℄orthe CGCformalism[12,13,14,15,16,17℄. Suhapproahes areused

to understand the transitionfrom a diluted to a dense partoni system due to the emission

of gluons by Bremsstrahlung whih takes plae in the small

− x

^{limit. This transition poses}

the interesting question of saturation eets inside the hadrons.

In this thesis we developed a model for the dirative

ρ −

^{meson prodution in the per-}

turbativeRegge limit,i.e. atsmall

x

^{and at high enough}

Q ²

^{to use pQCD tehniques. This}

approahwillbepresentedinhapter2and hapter3,whiletherst hapterwillbedevoted

to introdue the main toolsof this treatment ona DIS proess.

Inthehapter1,wewillintroduebasisofdierenttehniquesthatareusedinthisthesis.

Wewillpresentthe

k _T −

fatorizationonthesimplestexamplestoexplainhowtheamplitudes an be fatorized in the high energy limit, in sub-proesses alled "impat fators". Next,

after a brief general introdution toDIS, we will fous on a DIS proess to show how these

impatfatorsanbeinterpretedinthelanguageofdipolemodels. Thispermitsustodisuss

theimportaneanddierentwaysofinorporationintothedipolemodelofsaturationeets.

In the hapter 2,we willpresent the Light-Cone CollinearFatorization(LCCF) sheme

beyondtheleadingtwistanditsappliationtotheomputationoftheimpatfator

Φ ^{γ ∗ (λ γ ) → ρ(λ ρ )}

of the transition of the virtual photon of heliity

λ γ

^{into a}

ρ −

^{meson of heliity}

λ ρ

^{. In this}

approah,the softpartassoiatedtothe produtionofthe

ρ −

^mesonisparameterized bythe distribution amplitudes(DAs)of the

ρ −

^{meson. Wewilldisuss theenergy saledependene}

of the DAsand the QCDsum rule tehniquetoget non-perturbativeparameters thatenters

the DAs. Finally we will present a phenomenologial model to get preditions on heliity

amplitudes of the dirative

ρ −

^{meson produtionat HERA. This model willnaturallylead}

us tothe next hapter topi.

In the hapter 3, we will onnet the impat fator

Φ ^{γ ∗ (γ) → ρ λρ}

^{obtained in the previous}

hapterinthe ollinearapproximation,tothe olordipolepiture. From thisresult, onean

get phenomenologial models by ombining our results for the impat fators with dipole

models that are already known from DIS analysis and that ontains the

x −

^dependene

of the heliity amplitudes. These dipole models inlude the saturation dynamis of the

nuleon target. We ompare the preditions of the polarized ross-setions of the

ρ −

^meson

eletroprodutionwith HERA data and disuss our results.

In the hapters 2 and 3, some parts are based on our own ontributions like the phe-

nomenologial model [18℄ at the end of the hapter 2, and the hapter 3 whih is based on

High energy QCD

In this hapter we present basis of the onepts and tools neessary to takle the phe-

nomenology of hadroni reations inthe small

− x

^physis.

AfteranintrodutionontheReggetheoryandthepomerontrajetoryse.1.1,weexplain

on the quark-quark sattering the

k T −

fatorization proedure, rst in the ase of one gluon exhangedin

t −

^{hannelandthenintheaseofaolorsingletexhange(twogluonexhange)}

inse. 1.2. Weshowhowthe impat fatorsemergefromthis pitureand briey disussthe

resummation atleading log(1/x)of the gluonladder exhange in the

t −

^hannel.

We present some basis of DIS proess in se. 1.3, and show how the amplitude an be

fatorized inthe dipolepiture intothe photonwave funtions and the dipoleross-setion.

Wepresent nallydierentmodels ofdipoleross-setionthatinludethe saturationeets,

aswellastheequationsthatgovernstheenergy dependene ofthedipoleross-setioninthe

diluted and dense regimes.

1.1 Introdution

1.1.1 Postulates and onsequenes

Before QCD was applied to desribe the strong interations, physiists relied on the basi

postulates of the Lorentz invariane, the unitarity and the analytiity of the

S

^{-matrix in}

order to get informationonthe hadroni sattering.

Lorentzinvarianeofthe

S −

^{matriximplies thatthe}

S −

^{matrixelement}orrespondingto the proess

a(p A , λ A ) + b(p B , λ B ) → c(p C , λ C ) + d(p D , λ D ) ,

^(1.1)

an be expressed in terms of Lorentz invariant quantities suh as the Mandelstam variables

and the masses of the partiles. For the partiular ase of the proess (1.1) where two

partiles in the initial state give two partiles in the nal state, the sattering amplitude

an be expressed in terms of the Mandelstam variables

s = (p A + p B ) ²

^,

t = (p A − p C ) ²

^and

u = (p A − p D ) ²

^{whih satisfy}

s + t + u = X

i

m ² _i ,

^(1.2)

where

m i

^{denotes the mass of the partile}

i

^.

The unitarity ondition of the

S −

^matrix

S ^† S = SS ^† = 1 ,

^(1.3)

expresses the fat that the probability for an initialstate to give any nal state is equal to

one. Let usonsider anin-state

| a i

^{and anout-state}

| b i

^{whih are} respetively states of free partiles atthe times

t → −∞

^and

t → ∞

^{. The} orresponding

S

^{-matrix element is}

S ab = h b | a i .

^(1.4)

Letusintroduenowthe

T −

^{matrixelementsuhas}

S = 1+iT

^{,andthesatteringamplitude}

A ab

^{and the} ross-setion

σ ab

^{assoiated tothis proess,}

S ab = δ ab + iT ab = δ ab + i(2π) ⁴ δ ⁴ ( X

a

p a − X

b

p b ) A ab .

^(1.5)

The ross-setion

σ ab

^{of theevent}

a → b

^{isrelatedtothe} probabilityofthis event tohappen, it isthen proportionalto the square of the sattering amplitude,

σ _ab = 1 F

Z

dΠ _b |A ab | ² ,

^(1.6)

with

F

^{the ux fator and}

Π b

^{the phase spae of the}

n −

^{body partiles of the}

b

^{nal state.}

The ux fator in the ase of the proess (1.1) is given by

F = 2 q

λ(s, m ² _A , m ² _B )

^(1.7)

where

λ(s, m ² _A , m ² _B )

^{isthe standard kinemativariable,}

λ(s, m ² _A , m ² _B ) = s − (m A + m B ) ²

s − (m A − m B ) ²

.

^(1.8)

The expression (1.7) for the ux fator remainstrue for the produtionof

n

^{partiles inthe}

nal state from a two-partile initialstate. Note that in the large

s

^{limitwhere the masses}

an benegleted ompared to

s

^{, the ux fator is just}

F = 2s

^.

Theunitarityonditionofthe

S −

^{matrix(1.3)impliesthenthe followingonditiononthe}

T −

^{matrix elements}

X

c

(δ _ac + iT _ac )

δ _cb − iT _cb ^†

= δ _ab i

T _ab ^† − T _ab

= X

c

T _ac T _cb ^† ,

^(1.9)

where

c

^{is any physial state, i.e. the partiles of this state are on the} mass-shell. In terms of the sattering amplitudes,using the fat that

2i I m A ab = A ab − A ^† ab ,

^(1.10)

the relation(1.9) reads

2 I m A ab = (2π) ⁴ δ ⁴ ( X

a

p a − X

b

p b ) X

c

A ac A ^† cb .

^(1.11)

Thisrelationhasveryimportantonsequenesasitleads tothe Cutkosky rules.f. se. 1.1.3

and, in the speial ase where one put idential in- and out- states, it leads to the optial

theorem. The theorem reads

2 I m A ^aa (s, t = 0) = (2π) ⁴ δ ⁴ ( X

a

p a − X

b

p b ) X

c

|A ^ac | ² .

^(1.12)

Asaonsequeneoftheoptialtheorem,thetotalross-setion

σ tot

^{,assoiatedtotheproess}

"

a →

^anyphysialstate",is given up toaoeientby the imaginarypartof the amplitude

A aa (s, t = 0)

^,

2 I m A ^aa (s, t = 0) = F σ tot .

^(1.13)

The third postulate is the analytiity of the

S −

^{matrix elements, meaning that the}

S −

^{matrix is ananalytial funtionof the Lorentzinvariantsseen as omplexvariables. An-}

alytiity has been shown to be a onsequene of the ausality, whih prevents two regions

separated by a spae-like distane to inuene on eah other. Some onsequenes of the

analytiity are:

•

^{the rossing symmetry of the sattering} amplitudes,

•

^{the dispersion relations whih allows to get the real part of the amplitude from the}

imaginary part.

The rossing symmetry inthe ase of the two to two partileproess (1.1) reads

A a+¯ c → ¯ b+d (s, t) = A a+b → c+d (t, s)

^(1.14)

A a+ ¯ d → ¯ b+c (s, u) = A a+b → c+d (u, s)

^(1.15)

where

¯ b

^,

¯ c

^and

d ¯

^{are the} antipartilesassoiated to

b

^,

c

^and

d

^{. In the ase where}

I m A (s, t)

fallstozerowhen

z → ∞

^{,thedispersionrelationwhihrelatestheamplitudetoitsimaginary}

part is obtained by deforming the integration ontour whih surround the uts,

A (s, t) = 1 π

Z _∞

s ⁺ _th

ds ^′ I m A (s ^′ , t) s ^′ − s + 1

π Z s ⁻ _th

−∞

ds ^′ I m A (s ^′ , t)

s ^′ − s ,

^(1.16)

where

s ^{th +}

^and

s ^{th −}

^{arethe thresholds ofpartileprodutionalong thereal positiveandreal}

negative axis. If the asymptoti behavior of the integrand when

| s | → ∞

^{is not falling fast}

enoughthenthedispersionrelation(1.16)isnotvalidandshouldbereplaedbyasubtrated

dispersionrelationwheretheintegrandisdividedbyasmanyfators

(s ^′ − s ₀ )

^{asitisneessary}

to ensure the onvergene of the integrand with

s 0

^{an arbitrary point. For the addition of}

one of the fator

s ^′ − s 0

^{,the subtrated dispersion relation reads}

A (s, t) = A (s 0 , t) + (s − s 0 ) π

Z _∞

s ⁺ _th

ds ^′ I m A (s ^′ , t)

(s ^′ − s)(s ^′ − s ₀ )

^(1.17)

+ s − s 0

π

Z s ⁻ _th

−∞

ds ^′ I m A (s ^′ , t) (s ^′ − s)(s ^′ − s 0 ) .

Note thatthese relationsrequire the knowledgeof the asymptotibehavior of the sattering

amplitudes whihis the subjet of the Reggetheory.

Theseso-alled"bootstrap"relations,thatrelate the imaginarypart ofthe amplitudeto

the amplitude itself and to the sum of produt of other amplitudes due to the analytiity

and unitarity postulates, are obtained without for now speifying the underlying quantum

eld theory and are very general onsiderations.

1.1.2 Regge trajetories and the pomeron interept

In thehigh energylimit

s → ∞

^{with xed}

t

^{,alledthe Reggelimit,the asymptotibehavior}

of the amplitudeof the proess

a + b → c + d ,

^(1.18)

isonnetedtotheangularmomentum

l

^{ofthepartileexhangedin}

s −

^{hanneloftherossed}

hannelproess,

a + ¯ c → ¯ b + d .

^(1.19)

The partial wave expansionof the amplitudeof the rossedproess (1.19),

A a+¯ c → ¯ b+d (s, t) = X

l=0

(2l + 1)a l (s)P l (1 + 2 t

s ) ,

^(1.20)

allowstodeouplethe ontributiongiven by elementarypartileofangularmomentum

l

^and

mass

M

^{exhanged in the}

s −

^{hannel. The rossing symmetry implies that for the proess}

a +b → c+d

^{wheretheroleoftheMandelstamvariablesareexhanged,}

s ↔ t

^{,theamplitude}

is essentially given by the resonane and takes the form,

A ^ab → cd (s, t = M ² ) = A a¯ c → ¯ bd (t = M ² , s)

^(1.21)

= A l (t)P l (1 + 2 s

t ) = G a¯ c (t)G ¯ bd (t)

t − M ² (σ t + ( − 1) ^l )P l (1 + 2s/t) ,

where

σ t

^{is the signature whih is}

1

^{for rossing even amplitudes and}

− 1

^{for rossing odd}

amplitudes,

G _{a¯ c} (t)

^{isthevertex ofthe partileexhangedinthe}

t −

^{hannelwiththe external}

partiles. Theproess

t = M ²

^{isnotinthephysialregionofthe}

s −

^{hannelandineq.(1.21)}

an analytialontinuationof the Legendre polynomials inthe physial regionof the proess

(1.18) allows to derivethe asymptotibehavior of the amplitude of the proess,

A a+b → c+d (s, t) = g ac (t)g bd (t)

t − M ² s ^l .

^(1.22)

Note that the fat that the verties

G ij

^{do not depend on}

s

^{at high energy is an universal}

feature thatwe willsee alsowhendesribingthe impatfator approahin

k T −

fatorization sheme. The amplitude depends on

s

^{onlythrough the partilesexhanged in}

t −

^hannel.

This asymptotibehavior violates the unitarity of the theory. Indeed it wasproven long

ago by Froissart[21℄ using unitarity andpartial waveexpansionthat hadroni ross-setions

has toinrease slowerthan

ln ² (s)

^,

σ _tot < A ln ² (s) ,

with

A ∼ 60

^{mb. This isequivalent tobound the asymptoti amplitudes by,}

A(s, t) < s ln ² (s) ,

whih from(1.22) islearly violated for

l > 1

^.

The way to solve this problemis to use the Sommerfeld-Watson integral transformation

to express the partial wave expansion. The pole struture in the omplex variable

l

^{of the}

partial wave amplitude

A l (t) = A(l, t)

^{will then x the omplex angular momentum of the}

resonane. The resonane angularmomentum given by the pole

α _R (t)

^{of maximalreal value}

willdominate the asymptoti power behaviorof the amplitude, this poleis alledthe Regge

pole and the eetive "resonane" assoiated to this pole, of omplex angular momentum

l = α R (t)

^{is alled reggeon. The underlying assumption is that poles are simple poles, but}

in pratie logarithmsappearingin theperturbationtheory an gives branhuts. The pole

α R (t)

^{is a Regge trajetory and}

α R (0)

^{the reggeon interept. The} trajetories

l = α R (t)

^are

universal objets that only depends on the quantum numbers of the partile exhanged in

t −

^hannel.

For

t < 0

^,the

t −

^{dependene ofthe Regge polean be}experimentallyobtained by tting the energydependene ofthe

s −

^hannelamplitudes. Asexplainedabove,thereggeonan be seen as resonanesat

t = M ²

^{of angularmomentum}

l

^{. The idea of so-alledChew Frautshi}

plotswasthentoshowthemassesofknownresonanes

ρ, ω, · · ·

^{,asafuntionoftheirangular}

momentum. It turns out that the data are aligned onstraight-linesand by extrapolating to

the physial region

t < 0

^{, the} straight-lines give a relatively good desriptions of the data obtained from experiments, leadingto linear Reggetrajetories

α R (t = M ² ) = α R (0) + α ^′ _R t .

The Regge theory allows to omplete the bootstrap relation as it allows to obtain the

asymptoti behaviorof the amplitude.

Usingthe optial theorem, the

s −

^{power like dependene of the total} ross-setion is

σ tot ∝ I m A (s, t = 0) ∝ s ^{α R (t=0) − 1} .

^(1.23)

It was demonstrated by Pomeranhuk that the ross-setion vanishes asymptotially in the

ase where there is aharge exhange in the

t −

^{hannel. A Reggetrajetorywith}

α R (0) > 1

orresponds thentoareggeonthatarriesthe vauumquantumnumbersandwhihisalled

the "pomeron" (for a pedagogial review on the pomeron in QCD see [22℄). The pomeron

interept is denoted

α P (0)

^{. Donnahie and Landsho [23℄ have proposed a t of the total}

ross-setions for

pp

^and

p p ¯

^ollisionsas

σ tot = Xs ^ǫ + Y s ^{− η} ,

where the rst term an beinterpretedas the exhange of apomeron whilethe seond term

orresponds tothe exhange of a reggeon. The best ts were

σ _tot ^pp = 21.7 s ^0.08 + 56.1 s ^{− 0.45} , σ _tot ^{p¯ p} = 21.7 s ^0.08 + 98.4 s ^{− 0.45} .

These ts highlights the fat that the pomeron ouplings to the antiproton and the proton

are the same whih is due to the fat that the pomeron arries vauum quantum numbers.

The value

η = 0.45

^{,orresponds to the Regge trajetorylose to the one given by the linear}

ts of ChewFrautshiplots based on the spetrum of

{ ρ, ω · · · }

^resonanes.

The pomeron interept

α P (0) = 1.08

^{violates the unitarity bound from the Froissart}

theorem butoneanshowthatwiththis valueofthe pomeroninterept,the violationours

only atthe Plank sale.

The quark and gluon ontent of the pomeron an be studied in dirative dissoiation

proesses wherefor example in

ep

^{ollision,the pomeron is seen likea parton of the proton}

that interatswith the eletron togiveany nal state X. This reationis analogous to deep

inelastisatteringwherethepomeronreplaes theprotonwhihallowstostudy itspartoni

ontent.

1.1.3 Cutkosky rules

In the ase of QED or QCD one an hek that the imaginary part of an amplitude

A (s, t)

ariseswhenavirtualpartilegoeson-shellduetothe

iǫ

^{terminthe propagator}denominators

p ² + iǫ

^{. Branh uts appear for}

s

^{real suh as}

s > s 0

^with

s 0

^{the threshold wherea physial}

state an beprodued. Due toanalytiitywe have the relations

R e A (s + iǫ, t) = R e A (s − iǫ, t) ,

^(1.24)

I m A (s + iǫ, t) = −I m A (s − iǫ, t) ,

^(1.25)

the disontinuity of the amplitude around the branh ut along the real axis reads

D

^is

s A (s, t) = Lim ǫ → 0 ( A (s + iǫ, t) − A (s − iǫ, t)) = 2i I m A (s + iǫ, t) .

^(1.26)

It an beshown that the disontinuity of the amplitude an be obtained by replaing inthe

propagators

1

p ² + iǫ → − 2iπδ(p ² − m ² ) θ(p ⁰ ) .

^(1.27)

The

θ(p ⁰ )

^{ensures that the partile has positive energy, i.e. is a physial partile. For any}

diagram the disontinuity an be diretly obtained by following the so-alled "Cutkosky

rules" [24℄,

1. the diagramsmust beutinallpossibleways suhthatthe utpropagatorsanbeput

on shell simultaneously,

2. the ut propagatorsare replaed following eq. (1.27),

3. the disontinuity is given by the sum of all the ut diagrams.

We will use these rules in the following parts in order to get the imaginary part of the

amplitudes by omputingtheir disontinuities with the Cutkosky rules.

1.2 Sattering amplitudes in the Regge limit

In this setion, we introdue the approximations to get the dominant ontribution of the

amplitudes in the perturbative Regge limit, using the fat that in this limit

s/ | t |

^{is very}

large. We rst onsider the quark-quark sattering amplitude with one gluon exhange in

the

t −

^{hanneltoshowthekinematisofthedominant}ontributioninpowersof

1/s

^{. Thenwe}

ompute thequark-quark amplitude ofaolorsingletexhange in

t −

^{hannelinvolvingatwo}

gluon exhange in

t −

^{hannel. This example is} partiularly relevant for hadroni proesses in the perturbativeRegge limit,as the olor singletexhange dominates the olorless states

sattering. We nally show how the amplitude an be fatorized into the so-alled"impat

fators" and the

t −

^{hannelgluons Greenfuntion. Notethat the approahpresented inthis}

setion, isbasedonFeynman gaugealulationsand the alulationsbeyond theBorn order

approximationwouldbedierentwithinanothergauge. Ofourse,thenalresultsforgauge

invariantquantities are gauge independent.

1.2.1 The olor otet exhange

Atleading orderthe sattering of twoquarks inQCDis given by the tree diagramshown in

g. 1.1, where agluon arryingthe olor harge

a

^{is exhangedbetween the two quarks. We}

willassumethat ahard salejusties the use ofpQCD for example

| t | ≫ Λ ² _QCD

^{and the fat}

that

s ≫ | t |

^.

PSfrag replaements

p _A ∼ p ₁

p B ∼ p 2

a ∆

Figure 1.1: Quark-quark sattering amplitude at the tree level with an otet exhange in

t −

^hannel.

We denoterespetively

p A

^and

p B

^{the momentaof the upper quarkand lowerquarkand}

m _A

^,

m _B

^{theirmasses. TheMandelstamvariable}

S _AB = (p _A + p _B ) ²

^{islargebyassumptionand}

we an negletthe massesof thequarks andthen assumethat their momenta

p A

^and

p B

^are

verylosetotwolight-likevetors

p 1

^and

p 2

^{ofoppositediretions suhas}

S AB ∼ s = 2p 1 · p 2

^,

where

s

^{is the large sale. We an expand}

p A

^and

p B

^{onthis Sudakov basis as,}

p A = p 1 + m ² _A

s p 2 , p B = p 2 + m ² _B s p 1 ,

S AB = (p A + p B ) ² = m ² _B + m ² _A + 2p A · p B ∼ 2p 1 · p 2 = s .

The momentum of the gluon exhanged in

t −

^{hannel an also be deomposed on this basis}

as,

∆ = αp 1 + βp 2 + k _⊥ .

^(1.28)

Itisonventionaltouseatwo-dimensionaleulidean vetor,thatweunderline(

x

^),toreplae

the Minkowskian transverse vetor

x _⊥

^{, suh as}

x ² _⊥ = − x ²

^{. We will use this onvention all}

along the manusript.

Assuming that the partiles are on the mass-shell (we neglet now the masses of the

quarks), one has the two following onditions,

(p _A − ∆) ² = 0

^(1.29)

(p B + ∆) ² = 0

^(1.30)

whih lead to

− (1 − α)β + ∆ ² _⊥

s = 0 ,

^(1.31)

(1 + β)α + ∆ ² _⊥

s = 0 .

^(1.32)

Substituting ineq. (1.32) the expression of

β

^by,

β = ∆ ² _⊥

s(1 − α) ,

^(1.33)

leads to aseond order equation in

α

^,

α ² − α − ∆ ² _⊥

s = 0 .

^(1.34)

The two ouplesof solutionsfor

α

^and

β

^{up torst order in}

^∆ _s ^{2 ⊥}

^are,

α = 1 + ∆ ² _⊥

s , β = − 1 ,

^(1.35)

and

α = − ∆ ² _⊥

s , β = ∆ ² _⊥

s .

^(1.36)

The rst ouple of solutions is not relevant as it would imply that

t = ∆ ² ∼ − s

^{, whih}

violates our rst assumption

s ≫ − t

^{. The seondouple of solutiongives,}

∆ = − ∆ ² _⊥

s p 1 + ∆ ² _⊥

s p 2 + ∆ _⊥ .

^(1.37)

We get then that

t = ∆ ² ∼ ∆ ² _⊥ = − ∆ ²

^{. Note that it justies a posteriori that}

∆ ² _⊥ /s ∼ t/s

an benegleted.

We will now introdue another approximation to simplify the vertex expression, alled

the eikonal approximation. The uppervertex givesthe ontribution,

igu r (p 1 − ∆)γ ^µ t ^a _ij u s (p 1 ) ,

^(1.38)

where we put expliitlythe spinor indies

r

^,

s

^{, of the Dira spinors. The spinor}

u r (p 1 − ∆)

depends onthe vetor

p 1 − ∆

^whihisapproximatelyequalto

p 1

^as

| β | ∼ | α | ∼ | ∆ ² _⊥ | /s ≪ 1

^.

Thus, the uppervertex simplies as,

igu r (p 1 )γ ^µ t ^a _ij u s (p 1 ) = 2igp ^µ ₁ δ r,s t ^a _i,j ,

^(1.39)

where we haveused the Gordonidentity,

u r (p ^′ )γ ^µ u s (p) = 1

2m u ¯ r (p ^′ ) ((p ^{′ µ} + p ^µ ) + iσ ^µν (p ^′ − p) ν ) u s (p) ,

^(1.40)

with

m

^{the mass of the fermion and}

σ ^µν = i

2 [γ ^µ , γ ^ν ] ,

^(1.41)

for

p ^′ = p = p 1

^{, and the} normalizations of the spinors

u r (p)u s (p) = 2mδ r,s

^{. This approx-}

imation is known as the "eikonal approximation" and an be used as long as a soft gauge

partile is exhanged. Finally, using for the lower vertex the same approximation one gets

for the sattering amplitude,

i M = ig ² (2p ^µ ₁ ) g µν

∆ ² (2p ^ν ₂ )δ r 1 ,s 1 δ r 2 ,s 2 t ^a _ij t ^a _kl

= i8πα s

s

t δ r 1 ,s 1 δ r 2 ,s 2 t ^a _ij t ^a _kl .

^(1.42)

Notethat the upperand lowervertiesare respetivelyproportionalto

p ^µ ₁

^and

p ^ν ₂

^{, thusif we}

deompose the metri tensorintothe followingtensor omponents

g µν = 2

s p 2µ p 1ν + 2

s p 1µ p 2ν + g _µν ^⊥ ,

^(1.43)

only the omponent

2

s p 2µ p 1ν

^{gives a} non-vanishing ontribution. As the metri tensor is oming from the sum over the polarizations of the propagator of the gluon, this omponent

an beseen asthe tensor produt of the so-alled"non-sense" polarizations,

ε

^up

_µ = r 2

s p 2µ , ε

^down

_ν = r 2

s p 1ν ,

^(1.44)

suhas

g µν

^{anbereplaeddue tothe eikonal} approximationby

ε

^up

_µ ε

^down

_ν

^{.Nowthe amplitude}

M

^reads

i M = − i

∆ ² (ig) ² u λ ^′ (p 1 )/ε

^up

t ^a _ij u λ (p 1 )

u λ ^′ (p 2 )/ε

^down

t ^a _kl u λ (p 2 )

.

^(1.45)

(a) (b)

PSfrag replaements

k 1 k 2

p ₁

p 2

+

Figure1.2: Diagrams of the singlet exhange at Bornorder.

1.2.2 The singlet olor exhange in

t −

^hannel

Indirativeproessesthequantumnumbersexhangedin

t −

^{hannelarethoseofthevauum}

and onsequently, wehave toonsider asingletolorexhangein

t −

^{hannel. Letusonsider}

the olorsinglet exhange on the quark-quark sattering amplitude.

A olor singlet exhange in

t −

^{hannel involves at least two gluons. At Born order, the}

sattering of two quarksis given by thetwodiagramsshown ing. 1.2. Thesetwodiagrams

are related by rossing symmetry. Let us dene

∆ = k 1 − k 2

^{the momentum exhanged in}

t −

^{hannel. Thediagram(b) anbeobtained fromdiagram(a)results,up totheolorfator}

that are dierent by

A ^(b) (s, t, u) = A ^(a) (u, t, s) ≈ A ^(a) ( − s, t, s) ,

^(1.46)

where we use for the lastequality, the fat that atlarge

s

^{and xed}

t

^,

s ≈ − u .

The olor fator for a singlet exhange of the diagrams(a)and (b) are equaland given by

(t ^a t ^b ) ij

δ ij

N (t ^a t ^b ) kl

δ kl

N = 1 N ²

δ ^ab 2

δ ^ab 2

= N ² − 1

4N ² .

^(1.47)

Let usompute the imaginarypart of the diagram (a)by using the Cutkosky rules,

I m A = 1 2

N ² − 1 4N ²

Z

dΠ

^Cut.

₂ A

^tree

(k 1 ) A

^tree

^† ( − k 2 = ∆ − k 1 ) .

^(1.48)

In g. 1.3the ut of the fermioni lineof diagram(a) isrepresented by the dashed line.

Theolorfatorsare putapartof theamplitude

A

^{tree. Theexpression of}

A

^tree

(k)

^isgiven

by (1.42),

A

^tree

(k) = − 8πα s

s

k ² .

^(1.49)

The integralmeasure

dΠ

^Cut.

₂

^{on the phase spae isgiven by,}

Z

dΠ

^Cut.

₂ =

Z d ⁴ l 1

(2π) ⁴ d ⁴ l 2

(2π) ⁴ (2π)δ(l ² ₁ )(2π)δ(l ² ₂ ) (2π) ⁴ δ ⁽⁴⁾ (p 1 + p 2 − l 1 − l 2 )

=

Z d ⁴ k 1

(2π) ⁴ d ⁴ l 2 (2π)δ((p 1 + k 1 ) ² )(2π)δ(l ₂ ² )δ ⁽⁴⁾ (p 2 − l 2 − k 1 )

=

Z d ⁴ k 1

(2π) ² δ((p 1 + k 1 ) ² )δ((p 2 − k 1 ) ² ) ,

^(1.50)

(1.51)