Effects of higher twists and kT -factorization in the Drell-Yan process

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Institute of Physics

Faculty of Physics, Astronomy and Applied Computer Science Jagiellonian University

Doctoral Dissertation

Effects of higher twists and

k

T

-factorization in the Drell-Yan

process

Tomasz Stebel

Supervisor: dr hab. Leszek Motyka

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Wydziaª Fizyki, Astronomii i Informatyki Stosowanej Uniwersytet Jagiello«ski

O±wiadczenie

Ja ni»ej podpisany Tomasz Stebel (nr indeksu: 1028430) doktorant Wydzi-aªu Fizyki, Astronomii i Informatyki Stosowanej Uniwersytetu Jagiello«skiego o±wiadczam, »e przedªo»ona przeze mnie rozprawa doktorska pt. "Eects of higher twists and kT-factorization in the Drell-Yan process" jest oryginalna

i przedstawia wyniki bada« wykonanych przeze mnie osobi±cie, pod kierunk-iem dr hab. Leszka Motyki. Prac¦ napisaªem samodzielnie.

O±wiadczam, »e moja rozprawa doktorska zostaªa opracowana zgodnie z Ustaw¡ o prawie autorskim i prawach pokrewnych z dnia 4 lutego 1994 r. (Dziennik Ustaw 1994 nr 24 poz. 83 wraz z pó¹niejszymi zmianami).

Jestem ±wiadom, »e niezgodno±¢ niniejszego o±wiadczenia z prawd¡ ujawniona w dowolnym czasie, niezale»nie od skutków prawnych wynikaj¡cych z ww. ustawy, mo»e spowodowa¢ uniewa»nienie stopnia nabytego na podstawie tej rozprawy.

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Acknowledgements

I would like to express my highest gratitude to my supervisor dr. Leszek Motyka for enormous help and patience during writing this thesis and thor-ough supervision over the past 3 years. I also thank for a very friendly at-mosphere, numerous conversations (not always about physics) and of course playing a ping-pong!

I would like to thank Prof. Michaª Praszalowicz for supervising me during my M.Sc. studies and rst year of my PHD which introduced me to the eld of high energy physics.

I am very grateful to my collaborators Prof. Mariusz Sadzikowski and Dawid Brzemi«ski for scientic contribution in the subjects discussed in this thesis.

I also thank to my other collaborators Prof. Anna Kulesza and dr Vincent Theeuwes for inviting me to their project and fruitful collaboration.

I want to thank my parents and my sister for the love and great support they provide to me.

The last, but not least, I want to thank my girlfriend Asia for here love, patience and many wonderful moments.

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Abstract

This dissertation is devoted to an analysis of DrellYan (DY) lepton pro-duction at high energies. This process, which occurs in hadronhadron scat-tering is a very important probe of Quantum Chromodynamics (QCD) ef-fects. The dierential DY cross section may be described in terms of DY structure functions and they are the main observables investigated in this thesis. The standard description of the DY process within QCD is based on the leading twist collinear approximation. In this thesis QCD eects beyond this approximation are addressed. An analysis of possible higher twist eects and eects of parton transverse momentum is carried out. In the higher twist analysis of the forward DY process models are employed that describe mul-tiple scattering of hadrons, the Golec-BiernatWüstho (GBW) saturation model and a dipole model interpolating a leadinglogarithmic Balitsky-Fadin-Kuraev-Lipatov (BFKL) evolution. These two models are found to have essentially dierent higher twist content. The GBW model leads to sizable higher twist eects at small x and moderate DY masses, whereas the BFKL based model yields negligible higher twist eects. We give explicit predictions for the Large Hadron Collider (LHC) experiments and conclude that eects of higher twists predicted in the GBW model may be observed assuming a good control of a leptonic background.

In the analysis of the forward DY process another dierence between the GBW and BFKL models is found that occurs already at the leading twist. The source of it is the parton transverse momentum which is sizable in the BFKL approach and negligible in the GBW model. In the last part of the thesis central production of the Z0 boson at the LHC is investigated within

the kT-factorization approach. We focus on recent ATLAS data for the Lam

Tung combination of the structure functions which are not well described within the collinear QCD approach up to the next-to-next-to leading order (NNLO). It is found that the ATLAS data may be explained by the parton transverse momentum eects if the parton transverse momentum distribution is suciently hard.

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Streszczenie

Poni»sza praca po±wi¦cona jest analizie wysokoenergetycznego rozprasza-nia DrellaYana (DY) w zderzerozprasza-niach dwóch hadronów. Gªównymi obserw-ablami badanymi w rozprawie s¡ funkcje struktury opisuj¡ce ró»niczkowy przekrój czynny na rozpraszanie DY. Standardowy opis tego procesu w QCD oparty jest na przybli»eniu kolinearnym w wiod¡cym twi±cie. W rozprawie tej analizujemy efekty b¦d¡ce skutkiem wyj±cia poza te przybli»enia. W szczególno±ci rozwa»amy efekty wy»szych twistów i niezerowego p¦du poprze cznego partonów. W analizie wy»szych twistów dla procesu DY do przodu u»yto dwóch modeli dipolowych, mianowicie modelu Golca-BiernataWüst-hoa (GBW) i modelu opartego na ewolucji Balitsky'ego-Fadina-Kuraeva-Lipatova (BFKL) w przybli»eniu wiod¡cych logarytmów. Oba modele okazuj¡ si¦ mie¢ caªkowicie inny skªad twistowy. Model GBW posiada znacz¡ce poprawki od wy»szych twistów dla maªego x-a i ±rednich mass, natomiast w modelu BFKL te poprawki okazuj¡ si¦ by¢ zaniedbywalnie maªe. W poni»szej pracy podajemy przewidywania dla eksperymentów Wielkiego Zderzacza Had-ronów (LHC). Okazuje si¦, »e model GBW przewiduje mierzalne w LHC efekty wy»szych twistów pod warunkiem dobrej kontroli eksperymentalnego tªa dla leptonów.

W analizie procesu DY do przodu ró»nica pomi¦dzy modelami GBW i BFKL mo»e by¢ spowodowana innymi efektami ni» wy»sze twisty. S¡ to efekty zwi¡zane s¡ z niezerowym p¦dem poprzecznym partonów, który, jak si¦ okazuje, jest znacz¡cy w modelu BFKL i zaniedbywalnie maªy w modelu GBW. W ostatniej cz¦±ci rozprawy rozwa»ana jest centralna produkcja bo-zonu Z0 w formalizmie k

T faktoryzacji. Szczególny nacisk poªo»ony jest na

niedawno opublikowane dane dotycz¡ce kombinacji funkcji struktury, nazy-wanej kombinacj¡ LamTunga. Dane eksperymentalne dotycz¡ce tej kom-binacji nie s¡ wystarczaj¡co dobrze opisywane przez kolinearne podej±cie QCD nawet w drugim rz¦dzie rachunku zaburze« (NNLO). Znacznie lep-szy opis danych do±wiadczalnych uzyskuje si¦ uwzgl¦dniaj¡c efekty p¦du poprzecznego partonów. Jednak»e warunkiem konicznym jest u»ycie odpowied-nio szerokiego w p¦dzie poprzecznym rozkªadu gluonów.

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Chapter 1 Introduction

DrellYan (DY) lepton production is historically one of the most impor-tant processes in high energy physics. It takes place when two hadrons collide producing a neutral gauge boson which in turn decays into a pair of leptons. It was rst proposed by S. Drell and T.-M. Yan in 1970 [1] and then conrmed experimentally by J.H. Christenson et al. in protonuranium collisions [2] in the same year.

At the leading order of the perturbative QCD expansion in powers of αs

this process can be described as an annihilation of a quarkantiquark pair coming from the colliding hadrons. The annihilation results in the emission of virtual photon γ∗_{, which in turn decays into the leptonantilepton pair.}

At higher orders of the perturbative expansion the simple annihilation of the quarkantiquark pair is supplemented by other partonic channels like gluon quark or gluongluon scattering. Moreover, if the energy is suciently large one should take into account that the lepton-pair production can be mediated by the neutral electroweak boson Z0_.

From the theoretical point of view the DY process has multiple advan-tages. The presence of hard electromagnetic (or electroweak) probe allows an eective application of the QCD perturbative expansion. Moreover, since the emitted boson and the produced leptons do not interact strongly one may separate the boson decay into leptons from the hadronic interactions. The boson decay can be precisely described using perturbative theory of elec-troweak (or even electromagnetic) interactions since the coupling constant is

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very small. In this way the DY process may serve as an eective probe of the hadron structure and QCD dynamics.

From the experimental side, the signal from the large invariant mass dilep-ton state can be precisely measured. In particular the angular distributions of the produced leptons provide valuable information about the structure of hadron. A measurement of the hadronic state associated to the lepton production is more complicated and also its theoretical description requires more advanced methods. We will focus in this thesis only on the inclusive DY process where the nal hadronic state is ignored and only the dilepton state is measured.

The most eective description of QCD eects in the inclusive DY produc-tion is obtained using socalled structure funcproduc-tions. They describe the boson interaction with hadrons so contain only information about the hadron struc-ture. There are at least two denitions of the structure functions which are used in the DY process description [3, 4]: the invariant1 _{and helicity structure}

functions. Both of them may be related to angular coecients of the lepton distributions in some frame. The helicity structure functions are coecients of lepton distributions in the lepton center-of-momentum (COM) frame. On the other hand in the hadron COM frame angular lepton distributions are parametrized in terms of invariant structure functions. So measuring the lepton angular distributions and their momenta one can directly determine the structure functions dependence on kinematic variables. The number of the structure functions depends on the boson which mediates the lepton pro-duction. For a general γ∗_/Z0mediated process there are nine independent

structure functions [5] (invariant or helicity) for a γ∗_{-mediated production}

this number reduces to four. Even though the DY structure functions are in-dependent, it was already proven by Lam and Tung that in the parton model they are related by the certain relations. One of the relations, named the LamTung relation after the authors is particularly useful since its breaking is strictly connected to non-collinear eects in QCD.

The DY structure functions may be expanded into a socalled twist-series using the Operator Product Expansion [6]. The leading twist contribution

1_{Name "invariant" is related to the property of Lorentz invariance. On the other hand}

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can be computed using standard techniques of QCD. The higher twist oper-ators which are suppressed by negative powers of the hard scale are poorly known theoretically [7]. On the other hand the higher twist eects are visible in high-precision experimental data in the Deep Inelastic Scattering (DIS) from the HERA collider [8, 9]. Therefore, the DY process is a natural can-didate for an extension of higher twists searches. One needs to choose a kinematic region where the power suppression of higher twists is compen-sated by dierent eects. It was shown [10, 11] that such an enhancement of higher twist eects can be found in the small Bjorken x region of high energy scattering. The low-x and a moderate hard scale regime is a very interesting region in high energy physics, many interesting phenomena may be found there, like the small-x resummation leading to the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [12, 13, 14], parton saturation or multiple scattering eects [15, 16, 17].

In order to circumvent the theoretical problems connected to the higher twists we follow the standard approach in the context of the higher twist analysis we relay on models. A resumation of multiple scattering ef-fects assuming independent pomeron exchanges leads to a well known model of saturation by Golec-Biernat and Wüstho (GBW) [16, 17]. This model formulated in the color dipole language [18] was proven to provide a very ef-fective description of data for many processes at low x [16, 17, 19]. The twist expansion within this model was used for the DIS [10, 11] and the dirac-tive DIS [8]. Moreover the model was used to perform the twist expansion of the total cross section in the forward DY process [20]. The forward DY scattering is a natural choice when one searches for small-x eects since the strong asymmetry of the x values of partons guarantees probing this region. We shall follow these ideas and extend the results of [20] to all the four DY structure functions. We shall do it in two setups of kinematic variables: de-pendent on the boson transverse momentum or integrated over this variable. Since in the analysis of the higher twist content of the DY structure functions we have no rigorous theoretical approach it is important to exploit dierent possible models of higher twists. The model which has encoded an essentially dierent dynamics of multiple hard scattering than the GBW model is based on the mechanism of the BFKL pomeron exchange. This

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model was already applied to the twist expansion for the DIS structure func-tions [21] and a very dierent picture of the higher twist contribufunc-tions was found than in the GBW model. In this thesis we compare the predictions of the DY structure functions in both of the models.

The framework which allows us to apply the models of higher twists is the kTfactorization approach [12, 13, 14, 15, 22, 23]. The application of this

framework to the forward DY production was proposed in [24] where the color dipole representation [18] was also used. Since then many applications were made of the kTfactorization approach to the forward DY process, also

including the LHC data description (see e.g. [25, 26, 27, 28, 29]).

The kTfactorization framework was used not only to describe the forward

DY process. For the central DY production this formalism was also applied to the analysis of q∗_q_¯∗ _{[30, 31, 32, 33, 34] and g}∗_g∗ _{[35, 36] partonic channels}

with oshell partons q∗_{, ¯q}∗ _{and g}∗_{. In this thesis we will use the g}∗_g∗_channel

to predict the structure functions in a general DY process.

When the Large Hadron Collider (LHC) started to operate a new era of hadron physics has come. Since the energy of the colliding particles is much higher than for any of the previous colliders, a new region of the kinematic space has opened. The high energy DY production is one of the most precise probes of QCD eects in the LHC physics. In particular the smallx region which was measured extensively at HERA may be extended to even smaller x values, down to x ' 10−6. Particularly sensitive for exploring the low-x domain is the LHCb experiment which allows to measure the particles at high rapidities, up to y ≈ 5. The DY process in this experiment is expected to be probed to a very small invariant mass M of the dilepton system, down to M ≈ 2.5GeV [37]. One should note however that this region of DY masses is very dicult to measure experimentally since a large background is present. On the other hand, because of very small x and moderate M values it is particularly interesting from the theoretical point of view. First attempts of DY measurements in the small mass region were performed at the LHCb [38] and ATLAS experiments [39].

At central rapidities and larger masses the DY process is already measured with a good precision for several setups in the ATLAS and CMS detectors. In particular the mass distributions [39, 40, 41], transverse momentum and

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rapidity distributions for the Z0 _{mass [42, 43] and the coecients of the}

lepton angular distributions at the Z0 _{peak [44, 45] were measured. The}

last observables are particularly interesting since, as we will show, eects of parton transverse momentum may be tested using these data.

The thesis is organized as follows. In Chapter 2 we describe the DY high energy scattering and dene the structure functions. Later we discuss the main concepts of the Operator Product Expansion and factorization schemes in high energy physics. Eects of the small x resummation resulting in the BFKL equation and parton saturation phenomenon are reviewed. The new results are given in Chapters 3 and 4. The twist decomposition for the forward DY structure functions within two proposed models and predictions for the experiments are given in Chapter 3. In Chapter 4 we study the eects of kT factorization in the Z0 production and show comparisons of the

obtained theoretical predictions with selected experimental data. Finally, the summary are given in Chapter 5.

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Chapter 2 Basics

2.1 Drell-Yan process

2.1.1 Cross-section

The Drell-Yan production is a process:

N + N → l++ l−+ X (2.1)

where N represents a nucleon (or a nucleus), l− _(l+_{) is a negative (positive)}

lepton and X is a hadronic state (not measured). We will consider electron or muon pairs. In this thesis we will concentrate on the situation where both the initial hadrons are protons. This choice is motivated by the fact that nowadays the most of experimental data in high energy physics comes form the Large Hadron Collider (LHC) where proton-proton scattering is (mostly) measured.

In Fig. 2.1 we show schematically the DY process: we denote the pro-tons momenta P1 and P2, then a square of energy in the protons'

center-of-momentum frame is S = (P1+P2)2. Since we work in the high energy regime,

namely √S is much larger than other quantities, we treat the proton four-momenta as light-like: P1 = (

√

S/2, 0, 0, −√S/2), P2 = (

√

S/2, 0, 0,√S/2). The intermediate neutral boson may be either a virtual photon γ∗ _{or Z}0

(virtual or real)1_{. The boson four-momentum we denote by q and its square}

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Figure 2.1: A general Drell-Yan process considered in this thesis: two protons with four-momenta P1 and P2 collide, a neutral boson with four-momentum

q is produced and it decays into a lepton-antilepton pair with the invariant mass M; X is a hadronic state not measured in the experiment.

q2 _{= M}2_{. Since q = l}+_{+ l}− _{we have an interpretation of M as an invariant}

mass of the lepton pair. It is useful to dene also a vector κ = l+ _{− l}−_,

κ2 _{= −M}2 _{if we neglect the lepton mass .}

In the high energy regime it is common to use Sudakov decomposition of the four vectors:

v = αvP1+ βvP2+ v⊥, (2.2)

and the light-cone coordinates: v = (v+_{, v}−_{; v}

⊥) where v± = v0 ± vz and

v⊥ = (vx, vy). In the light-cone notation P1 = (0,

√

S; 0), P2 = (

√

S, 0; 0) so v⊥ = (0, 0; v⊥). The transverse momentum of the boson is dened as

q2

T = −q⊥2 ≥ 0. Feynman x of the virtual photon (or the DY pair) is

xF = q+/P2+.

From now on we shall focus on the γ∗_{-mediated process which is simpler}

due to lack of the parity violation. Then, as we shall see below, the cross section may be described by four independent structure functions. For the DY process with the parity violation (mediated by Z0_{or Z}0∗_{boson) one needs}

additional ve structure functions. However in this thesis we will focus only on the structure functions representing the parity-conserving contribution, then Z0∗_{-mediated DY might be calculated in the same way as for γ}∗_.

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2.1 Drell-Yan process 21

following form (see for example [46]): dσ = 1 4 X pol X X (2π)4δ4(P1+ P2− pX − l+− l−) × 1 2S|hl + l−X|T |P1P2i|2 d3_l+ (2π)3 _2E l+ d3_l− (2π)3 _2E l− , (2.3)

where the sums P_pol and PX are over polarizations of the incoming protons

and the nal hadronic states, respectively; El± is the energy of (anti-)lepton.

The amplitude (shown on Fig. 2.1) is given by:

hl+_l− X|T |P1P2i = ¯u(l−)γµv(l+) e2 q2hX|j µ_(0)|P 1P2i, (2.4)

where ¯u and v are Dirac spinors for the leptons, e is the elementary electric charge and jµ is an electromagnetic current. Puting (2.4) into (2.3) and

separating the leptonic and the hadronic part one gets [3]:

dσ = α 2 em SM2L µν_W µν d3_l−_d3_l+ (2π)4_E l+El− , (2.5)

where we introduced the leptonic tensor: Lµν = −gµν+

κµ_κν

κ2 +

qµ_qν

q2 , (2.6)

and the hadronic tensor: Wµν =

Z

d4xe−iqxhP1P2|jµ(x)jν(0)|P1P2i. (2.7)

Note that the current conservation ∂µjµ= 0 implies the relation:

qµWµν = Wµνqν = 0, (2.8)

so one can omit the last term in (2.6).

2.1.2 Structure functions

It is natural to follow the approach from the Deep Inelastic Scattering (DIS) and to introduce structure functions which contain all information about the hadronic tensor Wµν, hence also about the interaction between the

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protons and the emission of photon. Here we follow a classical paper by Lam and Tung [3] where two types of structure functions were considered.

Invariant structure functions Tiare dened as a decomposition coecients

of Wµν into independent tensors:

Wµν = −T1 g˜µν + T2 P˜₍₊₎µ P˜(+)ν − T3 1 2 ˜P µ (+)P˜ ν (−)+ ˜P µ (−)P˜ ν (+) + T4 P˜ µ (−)P˜ ν (−), (2.9) where: ˜ P_(±)µ = P_(±)µ − (q · P(±)/M2)qµ, P_(±)µ = P1µ± P µ 2, (2.10) ˜ gµν = gµν− qµ_qν_/q2_. _(2.11)

Note that the vectors ˜P_(±)µ and the tensor ˜g are dened to be orthogonal to the photon four-momentum q, in this way property (2.8) is explicitly satis-ed. As we have stated earlier we have four independent structure functions (comparing to two in the DIS). The structure functions Ti can be measured

using angular distributions of the leptons in the protons' COM frame: dσ dl1dl2dθ1dθ2dφ = α 2 em 32π3_S2 cos2(Θ/2)

sin4(Θ/2)sin θ1sin θ2 (2.12) × T2+ 2 tan2 Θ 2T1 + cos θ1+ cos θ2 1 + cos Θ T3+ 1 + cos Θ0 1 + cos ΘT4 ,

where we have chosen the following variables: lengths of the leptons three-momenta l1 and l2, polar angles of the leptons w.r.t. the beam axis θ1

and θ2 and their relative azimuthal angle φ (see Fig. 2.2a). The angle

be-tween leptons' momenta Θ can be calculated from: cos Θ = cos θ1cos θ2 +

sin θ1sin θ2cos φ, the conjugate angle Θ0 is given by cos Θ0 = cos θ1cos θ2 −

sin θ1sin θ2cos φ.

2.1.3 Helicity structure functions

Even though Ti are Lorentz invariants, hence convenient from theory

point of view, more popular in the literature are so-called helicity structure functions (HSFs), which we denote Wi, i = L, T, T T, L. They describe

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Figure 2.2: a) Dilepton production in the protons COM frame, angles used in (2.12) are denoted. b) Denition of the CollinsSoper frame in terms of protons' momenta.

angles one needs to choose axes X, Y and Z in the leptons COM frame, their orientation w.r.t. physical (protons) momenta is arbitrary we shall call a particular choice as the helicity frame. Vectors X, Y and Z satisfy the following conditions: X · q = Y · q = Z · q = 0 (they span the hyperspace transverse to q), X2 _{= Y}2 _{= Z}2 _{= −1} _{(they are normalized and spacelike),}

X · Y = Y · Z = Z · X = 0 (they are orthogonal to each other).

In the literature several helicity frames are used, it is a common conven-tion to choose the Y vector as perpendicular to the reacconven-tion plane (spanned by the momenta of the protons and the boson). In this thesis we will use two frames:

• GottfriedJackson frame [47] (called also the t-channel helicity frame): the Z axis is anti-parallel to the momentum of one of the protons Z = −−→p1

|−→p1| and X xed by the requirement of orthogonality to Y and Z.

In terms of ˜P_(±)µ vectors (2.10) this may be written as:

Xµ = M 2_a (+)+ qT2a(−) qT √ S(M2_{+ q}2 T) ˜ P₍₊₎µ − M 2_a (−)+ qT2a(+) qT √ S(M2_{+ q}2 T) ˜ P₍₋₎µ , Zµ = −M √ S a(+)+ a(−) ˜_Pµ (+)+ ˜P µ (−) . (2.13)

• CollinsSoper frame [48]: the Z axis is an external bisector of the angle between the momenta of the two protons (see Fig. 2.2 b) and X is xed

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by the orthogonality conditions: Xµ = − M SqTMT a(+)P˜₍₊₎µ + a(−)P˜₍₋₎µ , Zµ = 1 SMT a(−)P˜ µ (+)+ a(+)P˜ µ (−) , (2.14)

where we have introduced a(±) = ±q · P(±) and qT = |qT|. Note that these

helicity frames dier only by a rotation in the hadron reaction plane (a rotation w.r.t. the Y axis).

With the denition of axes in the leptons COM frame we may dene the photon polarization vectors :

µ₍₀₎ = Zµ, µ_(±)= √1 2(±X

µ_{− iY}µ_). _(2.15)

The HSFs are dened as contractions of the hadronic tensor with the photon polarization vectors: WL = µ₍₀₎ ? Wµνν(0), (2.16) WT = µ₍₊₎?Wµνν(+), (2.17) WT T = µ₍₊₎ ? Wµνν(−), (2.18) WLT = 1 √ 2 µ₍₊₎?Wµνν(0)+ µ (0) ? Wµνν(+) , (2.19)

where "?_{" means the complex conjugation (note that}

(0)? = (0) and (±)? =

−(∓)).

One may rewrite the Wµν tensor explictly in terms of the axis vectors:

Wµν = −˜gµν(WT + WT T) − XµXνWT T + ZµZν(WL− WT − WT T)

− (Xµ_Zν _{+ Z}µ_{Xν) W}

LT, (2.20)

where ˜gµν is dened in (2.11). It is easy to prove this relation by contracting

both sides with the photon polarization vectors according to denitions (2.16-2.19).

Since the helicity frames dier by a rotation, the transformation of HSFs between frames is linear. As an example we express the CollinsSoper HSFs

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using HSFs from the GottfriedJackson frame:

W_L(CS) = M2_W(GJ ) L + 2M qTW (GJ ) LT + W_T(GJ )− W_{T T}(GJ )q2 T M2 T , W_T(CS) = M2_W(GJ ) T − M qTW (GJ ) LT + W_L(GJ )+ W_T(GJ )+ W_{T T}(GJ )q2 T/2 M2 T , W_{T T}(CS) = M2W_{T T}(GJ )+ M qTW (GJ ) LT + −W_L(GJ )+ W_T(GJ )+ W_{T T}(GJ )q_T2/2 M2 T , W_LT(CS) = (q2 T − M2)W (GJ ) LT + M qT W_L(GJ )− W_T(GJ )+ W_{T T}(GJ ) M2 T . (2.21)

Let us sketch the proof of these relations. We start with substituting the GottfriedJackson expressions (2.13) into (2.20), in this way we express Wµν

in terms of tensors Pµ

(±)P

ν

(±) and ˜gµν and the coecients are linear

combina-tions of HSFs. We know that (2.9) is a similar expansion but the coecients are the invariant structure functions Ti. Comparing both the expansions

we get Ti as the linear combinations of W (GJ ) i : T (GJ ) i = P jR (GJ ) ij W (GJ ) j ,

which can be written using matrix notation as T(GJ ) _{= R}(GJ )_W(GJ ). The

same procedure we apply to the CollinsSoper expression(2.14) obtaining T(CS) _{= R}(CS)_W(CS). From the construction T

i are invariants w.r.t. the

rotation (they are Lorentz invariants) so TGJ _{= T}CS which leads us to the

transformation law W(CS) _{= (R}(CS)₎−1_R(GJ )_W(GJ ).

Decomposition (2.20) may be used to obtain the angular distribution of the negative lepton in the photon rest frame. Contracting this with the leptonic tensor (2.6) we get [49]:

dσ dxFdM2dΩd2qT = α 2 em 2(2π)4_M4 (1 − cos 2_θ)W L+ (1 + cos2θ)WT

+ (sin2θ cos 2φ)WT T + (sin 2θ cos φ)WLT , (2.22)

where Ω = (θ, φ) are spherical angles of the three-momentum −→l− w.r.t. the XY Z axes in the photon rest frame and dΩ is the corresponding angular integration measure. The scalar products of κ = l+ _{− l}− _{with the basis}

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vectors have the following form: κ · X = 2 − → l− sin θ cos φ, (2.23) κ · Y = 2 − → l− sin θ sin φ, (2.24) κ · Z = 2 − → l− cos θ. (2.25)

We have used obvious relations in the leptons COM frame: −→l+ _{= −}−→_l−_{, −}→_{κ =}

−2−→l− leading to: κ · X = −−→κ−→X = 2−→l−−→X = 2

− → l−

sin θ cos φ and similarly for Y and Z.

Integrating (2.22) over dΩ one gets: dσ dxFdM2d2qT = α 2 em 2(2π)4_M4 16π 3 (WT + WL/2) . (2.26) It is fruitful to transform (2.22) to a slightly dierent form. First we rewrite the term in front of WL: 1 − cos2θ = 1₂(1 + cos2θ) + 1₂(1 − 3 cos2θ).

Then we shall pull out (WT + WL/2) from the bracket:

dσ dxFdM2dΩd2qT = α 2 em 2(2π)4_M4 (WT + WL/2)(1 + cos 2_θ) + 1 2(1 − 3 cos 2 θ) WL WT + WL/2 (2.27) + (sin 2θ cos φ) WLT WT + WL/2 +1 2(sin 2_{θ cos 2φ)} 2WT T WT + WL/2 . This may be written using (2.26) as:

dσ dxFdM2dΩd2qT = dσ dxFdM2d2qT 3 16π (1 + cos2θ) + 1 2A0(1 − 3 cos 2_θ) +A1sin 2θ cos φ + 1 2A2sin 2_{θ cos 2φ} , (2.28)

where we dened coecients [48]:

A0 = WL WT + WL/2 , A1 = WLT WT + WL/2 , A2 = 2WT T WT + WL/2 . (2.29) Note that Ai's measure a relative size of the structure functions to the

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those coecients are not sensitive to the overall normalization of the structure functions.

In general the structure functions are independent, however in the parton model it was shown rst by Lam and Tung [3, 4] that the following relation is satised:

WL− 2WT T = 0. (2.30)

It turns out that the relation holds also at the next-to-leading order in collinear QCD. In this thesis the Lam-Tung relation will play an important role. We shall show that breaking of this relation gives a good insight into eects beyond collinear QCD.

At the end let us rewrite (2.30) in terms of A coecients:

A0− A2 = 0. (2.31)

The relations (2.30) and (2.31) hold both in the CS and GJ frames.

2.1.4 Integrated helicity structure functions

Up to now we have considered the DY cross section σ dependent on xF,

M2_{, Ω and q}

T, so fully dierential in the leptons' momenta. It may be

useful to study also partially integrated cross sections. For instance we shall consider the cross section which is integrated over the photon transverse momentum qT: dσ dxFdM2dΩ = α 2 em 2(2π)3_M2 h ˜WL(1 − cos 2_{θ) + ˜}_W T(1 + cos2θ)

+ ˜WT T(sin2θ cos 2φ) + ˜WLT(sin 2θ cos φ)

i

. (2.32) Comparing with (2.22) we obtain the relation between helicityintegrated and unintegrated structure functions:

˜ Wi = 1 2πM2 Z Wi d2qT. (2.33)

We dene ˜Ai coecients analogously to the unintegrated case:

˜ A0 = ˜ WL ˜ WT + ˜WL/2 , A˜1 = ˜ WLT ˜ WT + ˜WL/2 , A˜2 = 2 ˜WT T ˜ WT + ˜WL/2 . (2.34) The LamTung relation has obviously the form:

˜

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2.2 Operator product expansion and

factoriza-tion

In this Section we will focus on the hadronic tensor: Wµν =

Z

d4xe−iq·xhP1P2|jµ(x)jν(0)|P1P2i. (2.36)

The presentation will follow Ref. [46]. Similary to the DIS case we are interested in the "hard" Drell-Yan process, namely the virtuality of photon is large compared to the proton mass M2 _M2

p. It turns out that for M2 → ∞

the dominant part of integral (2.36) comes from the light-cone region x2 _{∼ 0}

[46].

Looking at (2.36) we see that for x → 0 operator jµ(x)jν(0) becomes a

product of two operators in the same point. This means that we are dealing with a composite operator. It is known from quantum eld theory that such operators are indenite.

The Operator Product Expansion (OPE) which was rst proposed by Wilson in 1969 [6] is a way of dealing with composite operators. The general idea is to expand a composite operator ˆA(x) ˆB(0) into a series:

ˆ A(x) ˆB(0) = ∞ X n=1 Cn(x) ˆOn(x), (2.37)

where ˆOn are local operators with nite matrix elements and the

singular-ity of the composite operator at x = 0 is fully contained in the coecient functions Cn(x).

In our case we need an expansion of jµ(x)jν(0)near the light cone x2 ∼ 0.

The general expression is given by [6]: jµ(x)jν(0) = ∞ X n=1 X k C_n(k)(x2) xµ1_{. . . x}µn_Oˆ(k) µ1...µn;µν(0), (2.38)

where the index n labels the spin of the operator and k distinguishes the dierent types of the operators with the same spin.

For a free eld the asymptotic behavior of C(k)

n (x2) for x → 0 may be

obtained by the dimensional counting: C_n(k)(x2) ∼ 1 x2

dk−(dO−n)/2

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2.2 Operator product expansion and factorization 29

where dj = 3and dO are dimensions of the jµ current and ˆO(k) operators in

the mass units. The singularity of coecient function is determined by the twist of the operator:

τO = dO− n. (2.40)

The lowest twist (which we call the leading twist) represents the most sin-gular coecient functions, hence the dominant contribution in (2.38). From (2.36) we see that the hadron tensor can be expanded in the power series of 1/M2 (recall that q2 = M2) and the twist value is labeling the terms of the expansion.

For interacting elds scaling (2.39) is modied by additional logarithmic corrections. To include those eects one should consider the renormalization group equations (RGE) for the coecient functions.

The matrix elements of operators ˆO(k)_{are divergent so renormalization is}

necessary. The renormalization group method might be applied to the OPE in a similar way as for standard quantum eld theory. Renormalization of the operators ˆO(k) _{is performed by multiplying by a renormalization matrix:}

ˆ O(k0) µ1...µn;µν = X k ˆ O(k)unren. µ1...µn;µν(Z −1 n )kk0, (2.41)

where we allow that renormalization mixes operators of dierent types (dis-tinguished by k).

We introduce an anomalous dimension matrix: γn= µ

∂Zn

∂µ Z

−1

n , (2.42)

and the Fourier transform of the coecient functions: qµ1_{. . . q}µn_C˜(k) n (q 2_{) =} Z d4x eiq·xxµ1_{. . . x}µn_C(k) n (x 2_). _(2.43)

Then the RGE for ˜Cn(k) have the following form:

µ ∂ ∂µ + β(g) ∂ ∂g δk0_k− γ_n(g)_k0_k ˜ C_n(k)(q2, g) = 0, (2.44) where β is the Gell-MannLow function and g is the QCD coupling constant.

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If we assume no operator mixing the solution of (2.44) has the following form: ˜ C_n(k)(µ2, g) = ˜C_n(k)(µ2₀, g(µ0)) exp " Z µ2 µ2 0 dµ02 µ02 γn(g(µ 0 )) # . (2.45)

Expanding β(g) and γn(g)in the series of g one can calculate the ˜C (k) n

coef-cients order by order using perturbative QCD.

We shall now briey discuss applicability of the OPE in the DIS and in the Drell-Yan processes. In fact the OPE is a very fruitful tool in the DIS. The following momentum sum rules can be proved at the leading twist [50]:

Z 1 0 dx xn−2F (x, Q2) =X k ˜ C_n(k)(Q2)A(k)_n , (2.46)

where F is a DIS structure function (FL or F2) depending on the Bjorken

variable2 _x_{and photon virtuality Q}2_{. A}(k)

n are coecients of matrix elements

(|P i is a proton state): hP | ˆO(k)

µ1...µn|P i = A

(k)

n Pµ1. . . Pµn+terms containg gµiµj. (2.47)

Relation (2.46) provides a straightforward expression of the structure functions through the OPE coecients and the matrix elements. In fact it also factorizes the cross section into the shortdistance (large momentum), perturbative-in-nature coecients functions C and the longdistance (small momentum), nonperturbative coecients A. For practical applications the factorization has a key importance it allows to disentangle the perturba-tive part of the process (which can be calculated) from the nonperturbaperturba-tive one (given as an input from the experiment).

In the DrellYan scattering the situation is dierent. Substituting the light-cone expansion (2.38) into (2.36) we see that the matrix elements of operators are of the form

hP1P2| ˆOµ(k)1...µn;µν|P1P2i, (2.48)

2_{Note that up to now by x we denoted the position four-vector, from now on we shall}

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2.2 Operator product expansion and factorization 31

hence the two-proton states are involved. This complicates the situation signicantly and the counterpart of (2.46) does not exist. The OPE in the standard form (2.38) is not particularly useful in practical applications in the DY process. In particular, its existence does not guarantee the factorization at the leading twist as in the DIS. Fortunately it was proven (either by generalization of OPE or so-called separation of mass singularities) that the hard factorization also holds in the DY scattering [51, 52, 53] :

dσ =X i,j Z dx1dx2 ℘i(x1, µF2)℘j(x2, µ2F) dˆσ ij part, (2.49)

where ℘i(x1, µ2F) is a parton distribution function (pdf) (for the parton of

type i) and dˆσij

part is a partonic cross section. The factorization scale µF is a

stipulated energy scale which separates the high energy, perturbative regime represented by the parton interaction with the cross section dˆσpart from the

low energy, nonperturbative input represented by the pdfs. The pdfs depend only on the fraction xi of the proton's momentum taken by the partons. This

is just a direct implication of the collinearity condition:

pi = xiPi and p2i = 0. (2.50)

We call (2.49) the collinear factorization formula.

kT factorization

Relation (2.50) is of the form of Sudakov decomposition (2.2) where the transverse momentum and the second Sudakov factor were neglected. It is natural to try generalize (2.50) to include the transverse momentum of partons:

pi = xiPi+ piT. (2.51)

In this approach the partons are o-shell, p2

i = −p2i 6= 0.

We can generalize the collinear factorization formula (2.49) to the follow-ing expression [22, 23]: dσ = X i,j Z dx1dx2 Z d2_k 1 πk2₁ d2_k 2 πk2₂ Fi(x1, k 2 1, µ 2 F)Fj(x2, k22, µ 2 F) dˆσ ∗ pij, (2.52)

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where dˆσ_p is called an o-shell partonic cross section since it describes scattering of two o-shell partons.

The Fj(x2, k22, µ2F) are called Transverse Momentum Dependent (TMD)

parton distribution functions. They are playing the role of pdfs in the kT

factorization approach. We dene also the Unintegrated Parton Densities (UPD) which will be useful later3_:

fi(x1, k2T, µ2F) = k 2

TFi(x1, k2T, µ2F). (2.53)

At the leading logarithmic (see the next Section) accuracy one can prove that integrating f one obtains the collinear parton distribution function:

x℘i(x, Q2) = Z Q2 0 d2kT πk2_T fi(x, k 2 T, Q 2 ), (2.54)

which justies the name of UPD.

There is a natural question if this picture is consistent. In particular one should guarantee that the gauge invariance is satised. Indeed, it can be shown that considering high energy limit, namely neglecting all terms non-leading in 1/S powers, one gets gauge invariant expressions. We shall therefore consider only the leading behavior in the powers of energy. For this reason (2.52) formula is sometimes called the high energy factorization.

The kTfactorization written in the form (2.52) could be an useful

ex-pression in analyses of many QCD processes. However there is a question of universality of the TMDs. One can always write formula similar to (2.52) but the predictive power of such expression comes from the fact that we can use as an input the process-independent nonperturbative TMDs extracted from data. In contrary to the collinear case, there is no strict proof that such pro-cedure is possible. In fact a famous example of kT-factorization breaking was

found in predictions for a transverse single spin asymmetry (SSA) in a polar-ized DY process and for a semi-inclusive DIS process (SIDIS). Within the kT

factorization in both processes the Sivers function enters that describes a kT

dependent unpolarized parton density in a transversely polarized proton. It was shown in [54] however, that the Sivers function enters to the SSAs and SIDIS with the opposite sign, so it is not universal.

3_{Note that there are several conventions of naming F and f functions in the literature}

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2.3 BFKL equation 33

On the other hand kTfactorization was applied in the numerous high

energy processes and gives reasonable predictions (see [55] for the review). It was also used in the context of the DY process [30, 31, 32, 33, 34, 35, 36] and we shall also use it in this thesis.

2.3 BFKL equation

We consider a phase space region where S M2_{, q}2

T Mp2, hence the

total energy of collision is much larger than other scales of the process which, in turn, are much larger than the proton mass. One can easily show (see for example equation (3.32)) that it corresponds to the limit x → 0. This regime is well known in the high energy physics as the lowx physics. It has been extensively studied since the HERA collider started to operate.

In the limit x → 0 some additional problems arise. It can be shown that at order n of the perturbation theory large logarithms (log 1

x)

noccur (see [56]

for example). This spoils the perturbative expansion since (αslogx1)

n _{∼ 1}

at all orders of the perturbative series. One should then resum terms of such form to all orders in αs this is called the leading logarithmic (LL)

approximation since terms of the form αs(αslog 1x)

n_{, α}2

s(αslog 1x)

n_{, . . . ,} are

neglected. Resuming the LL corrections we get the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [12, 13, 14]. We shall now briey describe the steps leading to this equation. In this Section we summarize discussion from Ref. [57].

Let us consider a ladder diagram for 2 → 2 + n scattering (see Fig. 2.3a) where in the initial state we have two quarks and n additional gluons are produced in the nal state. We can use Sudakov decomposition for the emitted gluons:

pi = αiP1+ βiP2+ piT, (2.55)

where P1 and P2 are light-like quark's momenta. The rapidity of gluon might

be expressed in terms of the Sudakov coecients: yi = 1₂ log(αi/βi). The

resummation can be performed within socalled multi-regge kinematics where the rapidities of partons are strongly ordered:

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Figure 2.3: a) A ladder diagram contributing to 2 → 2 + n process. b) the 2 → 2 + n ladder diagram including the eective Lipatov vertex (the black dot) and the resummed virtual corrections (the reggeised gluons denoted by the zigzag line). The light dot represents the eikonal quarkgluon vertex (2.57).

Note that there is no ordering in transverse momenta, namely p2

iT ∼ s0 where

s0 is an energy scale much smaller than S.

To resum the logarithms one should rst remove powerlike corrections in the energy (suppressed by powers of 1/S), this is done by the eikonal approximation. Then standard QCD quarkgluon vertex is replaced by its high energy limit,

−igλa2qµ_i δσσ0, (2.57)

where g is the strong coupling constant and σ, σ0 _{are the polarizations of}

quarks. In a similar way one can simplify the QCD threegluon vertex. In the Fig. 2.4 a) we show the simplest diagram of the ladder type, namely without real gluon emissions. The radiative corrections to this diagram are shown in Fig. 2.4 b) f). Within the LL accuracy those onegluon corrections may be gathered into the eective Lipatov vertex [12, 13]:

Γµνσ = 2P ν 2 P1σ S α1+ 2k2_1T β2S P₁µ+ β2+ 2k2_1T α1S P₂µ− (k1⊥+ k2⊥)2 , (2.58)

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Figure 2.4: a) The leading diagram for the qq → qq scattering. b) f) real corrections to the leading diagram. The light dot represents the eikonal quarkgluon vertex (2.57).

which replaces the threegluon vertex of QCD. One can generalize the ar-gument to larger ladders and prove that each threegluon vertex in Fig 2.3 should be replaced by (2.58).

Finally the virtual (loop) corrections of the exchanged gluons may be in-cluded by a socalled reggeisation of gluon, which simplies to a modication of the propagator by the Regge factor [58]:

1 k2 i → 1 k2 i si,i−1 s0 ω(k2i) , (2.59)

where si,i−1 = (pi + pi−1)2 and ω(k2) = 3αs/(4π)2R d2l [k2/l2(l − k)2] is

a gluon Regge trajectory. So in conclucison, the naive 2 → 2 + n ladder diagram Figure 2.3 a) is replaced by the BFKL ladder diagram Figure 2.3 b), which incorporates all possible emission topologies and the resummed virtual correction at the LL accuracy.

The BFKL equation is obtained when we consider the color singlet am-plitude A(1)_{(s, t)}_{for the diagrams in Fig. 2.3 b). The concept of an exchange}

of the vacuum quantum numbers between two colored particles were stud-ied within the Regge theory and this mechanism is known as a pomeron exchange. We introduce the Mellin transform of the imaginary part of the amplitude: ˜ A(1) qq(ω, t) = Z ∞ 1 d s s0 s s0 −ω−1 Im A(1)(s, t) s , (2.60)

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where s and t are the Mandelstam variables.

This Mellin transform can be rewritten in terms of a ˜F function: ˜ A(1) qq(ω, t) = 4iα 2 sδσ1σ01δσ2σ02 N2 c − 1 4N2 c Z d2_k 1d2k2 k2₂(k1− q)2 ˜ F (ω, k1, k2, q), (2.61) where δσ0

iσi reects the helicity conservation of the eikonal vertex and q is a

transverse momentum transfer between the quarks, q2 _{= −t}.

The BFKL equation for the ˜F function has the following form:

ω ˜F (ω, k1, k2, q) = (2.62) δ2(k1− k2) + α_sNc 2π2 Z d2κ −q2 (κ − q)2_k2 1 ˜ F (ω, κ, k2, q) + 1 (κ − k1)2 ˜ F (ω, κ, k2, q) − k2₁F (ω, k˜ 1, k2, q) κ2_{+ (κ − k} 1)2 ! + 1 (κ − k1)2 (k1− q)2κ2F (ω, κ, k˜ 2, q) k2₁(κ − q)2 − (k1 − q)2F (ω, k˜ 1, k2, q) (κ − q)2_{+ (κ − k} 1)2 !# . In Fig. 2.5 a) we show a schematic representation of this equation.

For the zero momentum transfer q = 0 this equation simplies signi-cantly: ω ˜F (ω, k1, k2, 0) = (2.63) δ2(k1− k2) + α_sNc 2π2 Z _d2_κ (κ − k1)2 ˜ F (ω, κ, k2, 0) − k2₁F (ω, k˜ 1, k2, 0) κ2_{+ (κ − k} 1)2 ! . Up to now we considered the qq → qq scattering. We would like to generalize this idea to the case of two hadrons. Then the pomeron coupling to the hadrons is included by introduction of an impact factors Φ(k, q). Then (2.61) can be generalized as:

˜ A(1)_hh(ω, t) = G (2π)4 Z d2_k 1d2k2 k2₂(k1 − q)2 Φ1(k1, q) ˜F (ω, k1, k2, q) Φ2(k2, q), (2.64)

where G stands for the color factor. In Figure 2.5 b) we show a schematic representation of this formula. Note that the impact factors include the infor-mation about distribution of partons inside the proton and their calculation would require nonperturbative methods.

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Figure 2.5: Schematic representations of: a) the BFKL equation (2.63), b) the pomeron exchange between two hadrons.

For our purposes it is useful to derive the BFKL equation for an gluon UPD. Let us perform the inverse Mellin transform of the ˜F function with q = 0: F (x, k1, k2, 0) = Z C dω 2πi x −ω _˜ F (ω, k1, k2, 0). (2.65)

Then the gluon UPD (2.53) is :

f (x, k2₁) = 1 (2π)3 Z _d2_k 2 k2₂ Φ2(k2, 0)k 2 1F (x, k1, k2, 0). (2.66)

Note that this represents a substitution Φ1(k1, q) → 2πk41δ2(k1 − k2) in

equation (2.64) so f(x, k2

1) can be depicted as the diagram in Fig. 2.5 b)

with one impact factor amputated.

Rewriting BFKL equation (2.63) in terms of f(x, k2

) one obtains: f (x, k2) = f0(x, k2) + α_sNc π2 Z 1 x dx0 x0 Z ∞ 0 d2κ (k − κ)2 f (x0, κ2)k 2 κ2 −f (x0, k2) k 2 κ2_{+ (κ − k)}2 , (2.67) where f0(x, k2) represents the exchange of two perturbative gluons.

This equation is often written as:

f = f0+ K ⊗ f, (2.68)

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Equation (4.16) or (2.68) can be solved in the Mellin space conjugated to k2, we have: K ⊗ (κ2)γ = αsNc π χ(γ)(k 2 )γ, (2.69)

and χ(γ) is the BFKL characteristic function. At the LL accuracy we have: χ(γ) = 2ψ(1) − ψ(−γ) − ψ(1 + γ) (2.70) and ψ is the Euler digamma function.

At the end of this Section let us comment about a possible extension of the BFKL equation. The substantial progress was made in this matter. In particular nexttoleading logarithmic (NLL) corrections to the BFKL evolution were calculated [59, 60, 61, 62, 63]. We are not going to address this issue in this thesis.

2.4 Saturation

This Section is based on the argumentation from Ref. [64].

2.4.1 Unitarity violation

It can be shown that the solution of (2.63) leads to a powerlike behavior of the cross section as a function of s:

σtot ∼ sω0 at s → ∞, (2.71)

where ω0 = 4Ncαsln 2/π is called the Pomeron intercept [65], ω0 ≈ 0.4 for

α_s = 0.15. However, it is a known fact that the unitarity of theory does not hold if the cross section grows asymptotically faster then log2

s (so called FroissartMartin bound). Clearly the LL BFKL equation cannot be a complete description of the hadrons interaction in the limit s → ∞.

The powerlike behavior (2.71) can be traced down to the proliferation of the soft gluon emissions when the energy is growing. From the physical point of view it is clear that this proliferation should be somehow tamed since the gluon density cannot be arbitrarily large. This leads us to the concept of the parton saturation. It is based on an assumption that there exists an energy

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2.4 Saturation 39

scale Qs(x), called the saturation scale, such that for Q < Qs(x) the system

is dense enough that the gluon recombination becomes important. In the (x, Q) plane the Q = Qs(x)curve separates the dilute phase from the dense

one. Since the saturation scale rises with the decreasing x one can still safely use the perturbative approach at the low x since then Qs ΛQCD.

For phenomenological studies of the saturation the dipole representation is very useful. Before introducing this formalism let us reformulate the uni-tarity problem using a general quantum mechanical approach. We consider a scattering of two particles in the impact parameter space b. The optical the-orem states that total cross section is given by the imaginary part of forward scattering amplitude:

σ_tot = 2 Z

d2b Im A(s, b), (2.72)

which is a direct consequence of the Smatrix unitarity. Using S = I+i T we can get A(s, b) = i [1 − S(s, b)] where S(s, b) is the forward matrix element of Smatrix. We can then rewrite (2.72) as:

σ_tot = 2 Z

d2b [1 − Re S(s, b)] . (2.73) This results in the following bound:

d2σ_tot

db2 ≤ 2, (2.74)

which is nothing else than the black disc limit in the quantum mechanics. The BFKL equation leads to a power like growth with s at each impact parameter, which clearly breaks (2.74).

2.4.2 Dipole picture

Let us consider scattering of a virtual photon on a nucleus. In the nucleus rest frame the photon dissociates into a quarkantiquark pair (a color dipole) which then interacts with the hadron. We denote the transverse size of the dipole (the distance between the quark and antiquark in the transverse position space) as r. Then the relative change of dipole size during the interaction is [64]:

∆r

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where M and R are the mass and the transverse size of the nucleus. So for x 1 the relative size of the dipole does not change after the interaction with the nucleus. We conclude that at high energies the dipoles are the eigenstates of the scattering matrix [66, 67, 68, 69].

The cross section for the scattering of the dipole with the size r on the nucleus may be written using (2.72) as:

ˆ

σ(x, r) = 2 Z

d2b N (x, r, b), (2.76) where we have denoted the imaginary part of the forward scattering am-plitude by N(x, r, b). This amam-plitude can be calculated assuming that the dipole couples to a nucleon by two gluons (there are four elementary dia-grams describing the coupling of two gluons to the quarkantiquark pair). If we assume that the dipole scatters on one nucleon we get:

N1(x, r, b) =

α_sπ2 2Nc

T (b) xg(x, 1/r2) r2, (2.77) where the nuclear prole function T (b) describes nucleon distribution inside the nucleus and xg(x, 1/r2₎ is the gluon density of the nucleon. We can

easily see that this formula violates the black disc limit for r → ∞ (very large dipoles).

It turns out however that if we consider multiple scattering we can cure the problem of unitarity breaking. One can prove that the amplitude for multiple scattering of the dipole on independent nucleons is given by the following GlauberGribovMueller (GGM) formula [68]:

Nms(x, r, b) = 1 − exp −αsπ 2 2Nc T (b) xg(x, 1/r2) r2 . (2.78) We see that Nms(x, r, b) ≤ 1so that black disk bound is conserved. For r → 0

we have the color transparency Nms(x, r, b) → 0. This natural property for

the zero size dipole since then the color structure of quark and antiquark dipole cannot be resolved and the pair is colorless.

A natural improvement of the discussed dipole picture is an inclusion of higher order QCD corrections. These was done by Mueller in the large Nc

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2.4 Saturation 41

dipole model [70, 71, 72]. Resummation of multiple scattering of the Mueller dipole cascade leads to a nonlinear Balitsky Kovchegov (BK) equation [73, 74] which may be treated as an extension of the BFKL equation into the high density regime. We shall not go into the details of this equation.

2.4.3 Golec-Biernat Wüstho model

The dipole cross section could be connected to the gluon TMD:

ˆ σ(r) = 2παs 3 Z d2kT F (xg, k2T) k4_T 1 − exp(−ikT · r) 2 . (2.79) The mechanisms of multiple independent scattering leading to the GGM formula (2.78) was contained in the saturation model proposed by Golec-Biernat and Wüstho (GBW) [16, 17]. The following form of the dipole cross section was assumed:

ˆ σ(x, r) = σ0 1 − exp − r 2 4R2 0(x) , (2.80)

where the saturation radius is given by:

R0(x) = 1 GeV x x0 λ/2 . (2.81)

The GBW model has three parameters σ0 = 23.03 mb, λ = 0.288 and x0 =

3.04 · 10−4 which were tted to the DIS data [16]. The factor 1/GeV gives the dimension to R0.

The connection between the GBW dipole cross section (2.76) and the GGM formula (2.78) can be easily obtained by introducing an eective radius of the proton Rp:

2 Z

d2b N (x, r, b) ≡ 2πR2_pN (x, r), (2.82) where we have neglected the bdependence of the imaginary part of forward scattering amplitude N for |b| < Rp. Substituting GGM (2.78) into (2.82)

and using the optical theorem (2.76) we get (2.80) by equating 2πR2

p ≡ σ0 and αsπ2 2NcT xg(x, 1/r 2_{) ≡ (4R}2 0(x)) −1_.

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1 0 - 1 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 - 7 1 0 - 6 1 0 - 5 1 0 - 4 1 0 - 3 1 0 - 2 1 0 - 1 1 0 0 1 0 a ) F 2 / Q 2 Q 2 [ G e V 2/ c 2] 1 0 - 2 1 0 - 1 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 - 7 1 0 - 6 1 0 - 5 1 0 - 4 1 0 - 3 1 0 - 2 1 0 - 1 1 0 0 1 0 b )  = 0 . 3 2 9 F 2 / Q 2 

Figure 2.6: The geometrical scaling for the combined HERA data. a) The photonproton cross section σγ∗_p = F₂/Q2 as functions of Q2 for xed x.

Dierent points correspond to dierent Bjorken x's. b) The same but plotted as a function of scaling variable τ for λ = 0.329. Points in the right end of the plot correspond to large x (due to a kinematical correlation of the HERA phase space), and therefore show explicitly a violation of the geometrical scaling. Figure was taken from [80].

The GBW model inherits the properties of the color transparency [66, 18, 75, 76, 77] and the saturation: ˆσ(x, r) ∼ r2 _{when r → 0 and ˆσ(x, r) → σ}

0

for r → ∞. The transition between those two regimes is dened by the saturation radius r ≈ R0. It is natural then to connect this radius with

saturation scale: Qs(x) = 1 R0(x) =GeV x x0 −λ/2 . (2.83)

The power behavior of the saturation scale may be justied by looking at the DIS far from the saturation region Q Qs, then the transverse structure

function is proportional to the square of saturation scale:

FT ∼ Q2s(x). (2.84)

For the GBW model we get FT ∼ x−λ which is the BFKL type of behavior

(2.71).

Given the simplicity, the GBW model remarkably well describes the DIS data, both inclusive and diractive (see [16, 17]).

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2.4 Saturation 43

Sta±to, GolecBiernat and Kwieci«ski [78] noticed a simple consequence of the existence of saturation scale Qs the geometrical scaling4 (GS). The

easiest way of justify this property is to note that the DIS γ∗_p _{cross section}

σγ∗_p depends only on x and Q variables and the mass of quarks can be

neglected (since charm production is a small fraction of the DIS cross section). Since we have no additional mass parameters (σ0 is multiplicative constant

giving the dimension to the total crosssection) we conclude that the cross section depends only on the dimensionless ratio τ:

σγ∗_p(x, Q2) = σ₀F (τ ), and τ =

Q2

s(x)

≡ Q2xλ, (2.85) where F (τ) is a dimensionless function. Note that we omit unimportant constants in τ. In Figure 2.6 we show combined HERA [79] data for σγ∗_p =

F2/Q2 (up to some constant factor): on the left as a function of Q2, dierent

colors distinguish x values; on the right as a function of τ (2.85). It turns out that the scaling holds up to x ≈ 0.1 so to much higher values than the GBW applicability region. This shows that the GS may be more general property than the saturation. In particular the GS may be obtained also from the DGLAP evolution equation. Note however, that in contrast to the GBW model the DGLAP evolution is not applicable at the low Q2_.

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Chapter 3 Twist expansion of forward

DrellYan structure functions

In this chapter we are interested in the twist expansion for the DrellYan process. As we have mentioned in Section 2.2 the standard formulation of OPE known from the DIS is not particularly useful in the DY process. We are not going to consider higher twists in the strict theoretical manner but instead we use the dipole approach. Basing on two models of the dipole cross section we will give explicit predictions for the higher twists contributions. This chapter is based on two papers [28, 81].

At present the LHC experiments are the most promising sources of data that may unveil the structure of the higher twists at small x. Thus we focus on the predictions for this experiment. As we know the higher twist contributions are suppressed by powers of 1/M so it is natural to consider the small M region of the phase space. This leads us to the regime of large rapidities of produced particles (socalled forward physics) and the low-x eects. So the ideas discussed in Chapter 2 will be very useful. One should note that the LHCb experiment was designed so that it can probe the region of even smaller x's then in the HERA collider (down to x ≈ 10−6_{) [38].}

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Figure 3.1: Dominant diagrams for the forward Drell-Yan process in the t-channel helicity frame (in which the target is at rest).

3.1 Impact factors

In the forward region at the LHC there is a strong asymmetry in parton x-variables of the colliding partons x2 x1 (we always denote the "fast"

quark as q2 and its momentum fraction x2). Since x2 is a sizable fraction of

the proton momentum (x2 ≈ 0.1), the parton distribution is dominated by

the quarks. On the other hand since x1 is very small the gluon distribution

dominates for the partons coming from proton P1. We therefore conclude

that the dominant channel for this process should be qg∗ _{[24] and the so}

called hybrid factorization should be applied (one parton is collinear and the other has nonzero kT).

One should note that in the kT factorization framework the expansion in

terms of powers of αs in the hard matrix element should be performed

care-fully. Consider the qvalg∗ → qγ∗ matrix element, where by qval we understand

a collinear valence quark. It contains the NLO collinear qvalg → qγ∗ matrix

element but also the LO collinear qvalq¯sea → γ∗ contribution1. In addition

some contributions from the exact (non-collinear) kinematics are present that may contribute to the higher orders in the collinear picture. On the other hand in this approach we do not include the virtual and real corrections to qvalq¯sea → γ∗ which enter at the NLO.

In Figure 3.1 we show two diagrams contributing to q(p2)g(k) → q(p02)γ

∗_{(q) →}

q l+_l− _{process at the NLO. We are going to calculate them in the target rest}

frame (where by target we understand proton P1 with small x1 ≡ xg). The

1_{In the k}_T_{factorization picture the q}_val_q_¯_sea _{→ γ}∗ _{scattering is realized by the earlier}

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3.1 Impact factors 47

collinear quark distribution function is denoted by ℘. Since the quark in the initial state is not specied the ℘ contains contribution of all avors (except t). Therefore ℘(x) = P_fe2

f℘f(x)where ef is a fraction of elementary charge

carried by quark and ℘f(x) is pdf of quark.

The gluon carries a sizable transverse momentum so we describe it using the gluon TMD F(xg, k2), with gluon xvariable xg = k−/P1−. By z = q+/p

+ 2

we denote the longitudinal momentum fraction of the fast quark q(p2) taken

by γ∗_.

3.1.1 Impact factors in the momentum space

In this chapter we will use only the GottfriedJackson frame. It is a natural helicity frame when the target has much smaller momentum than the projectile. The polarization vectors have the simple form:

(0) = q + M, − M q+; 0 and (±) = 0, 0; (±)_T , (±)_⊥ = √1 2(±1, −i) . (3.1) We start with a calculation of amplitudes desribing the γ∗ _{emission from}

quarks. The amplitudes for the three photon polarizations are: A(0)_λ 1,λ2(qT, z, kT) = e 4√π3 √ 1 − z z xF(P1· P2) M δλ1,λ2 (3.2) × M2(1 − z) M2_{(1 − z) + q} 2 T − M 2_{(1 − z)} M2_{(1 − z) + (q} T − zkT)2 , A(±)_λ 1,λ2(qT, z, kT) = e 8√π3 √ 1 − z z xF(P1· P2)δλ1,λ2(2 − z ∓ λ1z) × _⊥(±)· _−q T M2_{(1 − z) + q} 2 T − −(qT − zkT) M2_{(1 − z) + (q} T − zkT)2 , where we denoted helicities of the incoming and outgoing quark by λ1 and

λ2. We dene helicity dependent γ∗ impact factors:

˜ Φσσ0(q T, z, kT) = X λ1,λ2=+,− A(σ)_λ 1,λ2(qT, z, kT) † A(σ_λ0) 1,λ2(qT, z, kT). (3.3)

The photon splitting into leptons is described by the leptonic tensor (2.6), which we write in the helicity basis by contracting with the photon polariza-tion vectors:

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We are now ready to write the forward DY cross section given by diagrams from Figure 3.1: dσ dxFdM2dΩd2qT = αem (2π)2_(P 1· P2)2 M2 x2_F(1 − z) Lσσ0(Ω) Z 1 xF dz ℘(xF/z) × Z d2kT 2παs 3 F (xg, k2T) k4_T ˜ Φσσ0(q_T, z, k_T). (3.5)

We are now going to write this formula in the transverse position space and apply the color dipole approach. In the DY scattering we have no as clear picture of the dipole model as in the DIS. Here the eective color dipole emerges through an interference of amplitudes of the virtual photon emission before and after the quark scattering o the target proton. The eective color dipole size corresponds to a displacement of the quark position in the transverse space due to the γ∗ _emission.

3.1.2 Impact factors in the position space

We start with the Fourier transform of amplitudes A: ˜ A(σ)_λ 1,λ2(r, z, kT) = 1 2π Z A(σ)_λ 1,λ2(qT, z, kT) e −i qT·r _d2_q T. (3.6) It is convenient to rewrite it as ˜ A(σ)_λ 1,λ2(r, z, kT) = 1 − e −i kT·r ˜a(σ) λ1,λ2(r, z, kT), (3.7) with: ˜ a(0)_λ 1,λ2 = e 2√π(2π)2 √ 1 − z z xF(P1· P2)δλ1,λ2M (1 − z)K0 √ 1 − zM r , ˜ a(±)_λ 1,λ2 = i e 4√π(2π)2 √ 1 − z z xF(P1· P2)δλ1,λ2 ×(2 − z ∓ λ1z)M √ 1 − zK1 √ 1 − zM r rx√± iry 2 r . (3.8)

We have omitted the arguments of ˜a(i)

λ1,λ2 and introduced the length of vector

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3.1 Impact factors 49

The integrated gluon density may be replaced by the dipole cross section using inverted relation (2.79):

2παs 3 F (xg, k2T) k2_T = 1 2 Z d2r eikT·r_∇2_{σ(r) =}_ˆ 1 2 Z d2r eikT·r _σ(r)(−k_ˆ 2 T ), (3.9) where we omitted the x dependence of ˆσ to simplify the notation.

We can rewrite (3.5) in the position space as: dσ dxFdM2dΩd2qT = αem (2π)2_(P 1· P2)2 M2 x2F(1 − z) Lσσ0(Ω) × Z 1 xF dz ℘(xF/z) Z d2r ˆσ(r)Φσσ0(q_T, r, z),(3.10) where Φσσ0(q T, z, r) = − 1 2 Z d2kT ei kT·rΦ˜σσ0(q T, z, kT) = = 1 2 X λ1,λ2=+,− Z d2r1d2r2 ˜ a(σ)_λ 1,λ2 † ˜ a(σ_λ 0) 1,λ2 e −i qT·(r1−r2) × δ(r − r1) + δ(r − r2) − δ(r − (r1− r2)) . (3.11) In the second equality we substituted (3.6) into (3.5) and integrated over d2_k

T.

We parameterize the DY structure functions Wi in terms of leptonic

im-pact factors Φi, Wi = 2(2π)4_M4 α2 em Z 1 xF dz ℘(xF/z) Z d2r ˆσ(r)Φi(qT, z, r) (3.12)

for i = {L, T, T T, LT }. Comparing (2.22) and (3.10) one gets the following expressions for Φi in terms of the Φσσ0 impact factors :

L00Φ00 ≡ (1 − cos2θ)ΦL, (3.13)

L++Φ+++ L−−Φ−− ≡ (1 + cos2θ)ΦT,

L+−Φ+−+ L−+Φ−+ ≡ (sin2θ cos 2φ)ΦT T,

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3.1.3 Impact factors in the Mellin space

Our nal aim is to rewrite (3.12) in the Mellin space. To this end we dene the Mellin transform of the dipole cross section2_:

˜ σ(s) = Z ∞ 0 dρ2 ρ2 ρ 2s ˆ σ(ρ), (3.14)

and its inverse:

ˆ σ(r) = Z C ds 2πi Λ2 4 r 2 −s ˜ σ(s). (3.15)

Scale Λ is an intrinsic scale of the dipole model, for the GBW model it is a saturation scale (2.83). The variable ρ = rΛ/2 is dimensionless. The integration contour is chosen as C = (−1/2 − i∞, −1/2 + i∞).

The Mellin representation of the leptonic impact factors is dened as:

ˆ Φi(qT, z, s) = 2(2π)4_M4 α2 em Z d2r η 2 z 4z2 r s Φi(qT, z, r), (3.16)

where we have denoted η2

z = M2(1 − z). After its integration over d2r and

d2r1d2r2 we get the following results [28]:

ˆ ΦL(qT, s, z) = 2 z2 2Γ2_{(s + 1)} 1 + q2 T/η2z 2F1 s + 1, s + 1, 1, −q 2 T η2 z − Γ(s + 1)Γ(s + 2) 2F1 s + 1, s + 2, 1, −q 2 T η2 z , ˆ ΦT(qT, z, s) = 1 + (1 − z)2 2z2_{(1 − z)} ( 2q_T2/η2_z 1 + q2 T/ηz2 Γ(s + 1)Γ(s + 2)2F1 s + 1, s + 2, 2, −q 2 T η2 z −Γ(s + 1)2 2F1 s + 1, s + 1, 1, −q 2 T η2 z −(s + 1) 2F1 s + 1, s + 2, 1, −q 2 T η2 z ) ,

2_{Note that we have assumed that the dipole cross section depends only on the length}