"Exclusive kernels for NLO QCD non-singlet evolution"

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Exclusive Kernels for NLO QCD Non-singlet

Evolution

Aleksander Kusina

PhD thesis written under the supervision of Prof. Stanisław Jadach.

Kraków, 2011

The Henryk Niewodnicza ´nski Institute of Nuclear Physics Polish Academy of Sciences

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1 Introduction

The quest for understanding the most basic laws of nature is the hallmark of our civilization. It really started with the development of the empirical method sometime around the 17-th century. Physics was always the branch of science leading in the use of the empirical method in the research. The Newtonian idea, that basic laws of nature can be described by means of strict mathematics is still successful in a contemporary physics research.

It is the particle physics, which gives us the best insight into the basic laws of nature chang-ing our understandchang-ing of the Universe. The story has started with the concept of chemical basic elements arranged in the periodic table by Mendeleev, which gave a certain view on the basic building blocks of our world. However, this view has evolved a few times with: Dalton’s modern atomic theory, Thomson’s discovery of electron as one of the atom’s components and finally with the discovery of partons (quarks) inside the proton in the SLAC electron–proton scaterring experiment in 1960’s.

The exploration of smaller and smaller components of matter is just one direction where scientists look for the basic laws on nature. Another way is going the opposite direction, by looking at the bigger and bigger scale of the whole Universe. This direction is represented by the Newton gravitation theory, Einstein General Relativity and finally by the Big Bang model of the origin of the Universe.

Today we know that these two directions, which for many years used to be separated, are indeed closely related. We still do not have a consistent theory describing both scales at once (quantum theory of gravity), but both astrophysics and particle physics are getting closer and closer. A good example is the problem of dark matter, which is implicitly seen by the grav-itational effects in cosmology. The most promissing solution to it is proposed by the particle physics in a form of supersymmetric particles.

As we can see, the searches for the basic laws of nature are done using methods of particle physics as well as cosmology, but it is the particle physics that has the empirical tools of collider experiments allowing to look into the most basic phenomena in a systematic way and, at the same time, having the possibility of measuring them very precisely.

In November 2009 the biggest hadron collider ever, the LHC (Large Hadron Collider) started operating in the CERN laboratory in Geneva. It is operating now at the 7 TeV (3.5 TeV per beam) energy. The nominal energy for the LHC 14 TeV in the center of mass will be attained later on. The energies achieved in this machine are much bigger then ever before. For comparison, the second biggest machine is the Tevatron collider (Fermilab US), where the center of mass energy is 1.96 TeV. The LHC is already operating in a stable way and the expectations for physically interesting data, to be collected until the end of the present run next year, are very high.

The huge increase in the available energy gives us hope to see some “new physics” in the LHC. First of all, and most awaited is the Higgs boson, which is the particle responsible for creating masses in the Standard Model (SM) and is still undiscovered. We also hope to be able to observe new supersymmetric particles, which are the potential candidates for the dark matter.

The LHC collider is armed with very sophisticated detectors, which will provide us with experimental data of much better precision than in the previous hadron colliders. The LHC detectors will record events with many particles (jets) measured very precisely, which will require also a much higher precision on the side of theoretical predictions for the experiment. To be able to isolate the signal, originating from “new physics”, from the data, the background

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(due to copious production of many gluons) of the Standard Model needs to be known very precisely, otherwise some new effects can be missed, hidden under this huge background.

A theoretical description of particle physics is given by the well tested theory called Stan-dard Model. StanStan-dard Model consists of two sectors describing two types of interactions be-tween particles. The electroweak sector describing the electromagnetic interactions together with weak interactions given by the Glashow, Salam, Weinberg model [1] (for which they were awarded the Nobel Prize in Physics in 1979). The strong interaction sector is described by Quantum Chromodynamics (QCD) [2, ?] (’t Hooft, Veltman and Gross, Wilczek, Politzer were also rewarded with Nobel Prize in 1999 and 2004). At high energy hadron colliders like LHC, from the practical point of view of the experimental data analysis, the most important effects are from the QCD sector.

The mathematical basis for QCD, as well as the whole SM, is the Quantum Field Theory (QFT). Calculations within the QFT are rather complicated and the most powerful calculation method is the perturbation theory. Perturbative QCD does not describe directly hadrons, which are seen in detectors but rather their constituents, quarks and gluons. The phenomenon of con-finement causes quarks and gluons to be bounded into hadrons, which at low energies cannot be reliably described by means of the perturbative QCD (pQCD). However, pQCD, due to the asymptotic freedom, works perfectly well for high energies. This property of QCD makes the description of high energy (short distance) hadron scattering calculable perturbatively order by order. This is thanks to the methods based on the factorization theorems. QCD factorization is based on separating the effects of two energy scales. The high energy perturbative effects, in form of the hard process (interactions between partons), and the low energy non-perturbative effects, which are described by phenomenological models adjusted to data. The detailed de-scription of factorization will be given in Sec.1.3.

Let us describe briefly what is seen in the detector in the real scattering experiment at hadron collider. There are loads of particles (tracks) going out of the collision center, particles are decaying and forming new particles (more tracks) and in the end everything is recorded in the electronics of the detectors. So what we really get are the reconstructed momenta of the thousands of produced particles, hadrons and leptons.

To compare any theory with the experiment at hadron colliders like the LHC, it is best to have the theoretical calculations encapsulated in the form resembling what is seen in real detectors. This is done by means of the Monte Carlo (MC) generators. The most popular MCs are the general purpose Parton Shower Monte Carlos (PSMC) like PYTHIA [5], HERWIG [6] or SHERPA[7]. PSMCs generate every single collision from the very beginning to the very end, so that each MC event resembles very closely a single experimental collision event. MC generators are rather complex, they consist of many separate modules responsible for various stages of the scattering process.

Let us have a closer look at the stages of the scattering process at a hadron collider. First, there are two incoming hadrons (protons) of the beams. The collision of the most energetic partons, originating from the hadrons, represents potentially new interesting phenomena. This parton collision is referred to as a hard process. Before the hard process occurs, partons inside hadron beams undergo the initial state radiations (ISR) through a cascade emission of partons (quarks, gluons). Then, after the hard process occurs, the newly produced partons are also “cascading” through the final state radiations (FSR), and in the very end, after reaching the transverse momentum scale of about 300 MeV the confinement mechanism forces them to form hadrons, which are seen in the detectors. The process of forming hadrons is referred to as the

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fragmentation/hadronization. The initial and final state radiation cascades are referred to as ISR and FSR. There are still other parts of the scattering process at low transverse momenta, like the underlying event – the interaction of the proton remnants, but they are less important for the general picture. All these building blocks are simulated in a full scale MC for the LHC, the final product is a set of momenta of all particles seen in the detector, which is exactly what is seen in the real life experiments. Such a single set of momenta and charges is called a MC event and represents a single collision of two hadrons in the experiment.

The role, history and importance of the different building blocks of the MC generators are different. The oldest part is the hadronization, which simulates the formation of hadrons. Since hadronization is connected with the phenomena of confinement, the methods of pQCD cannot be applied to describe it, it is done by means of phenomenological models such as the Lund string model [8] or the cluster model [9,10].

As it was already mentioned, for the searches and exploration of any new physics the most important role is played by the most energetic part of the collision, which is the hard process. New heavy particles will be produced in the hard interaction. The high energy collision op-erates in the perturbative region of QCD; the hard process together with the initial and final state radiations lives in the space ranging from several TeVs down to ∼1 GeV in transverse momentum. It is there where it can be described using the powerful methods of pQCD. These two processes, the hard interaction and the radiations, today are the most important ingredient of the MC simulations.

Hard process is calculated within pQCD using Feynman diagram techniques. It must be calculated individually for many different production channels. On the other hand, the ISR and FSR cascades (ladders) are universal1, but they need to be taken from the experiment at one energy scale, then by the methods of pQCD they can be evolved to a scale required by a specific hard process.

The precise description of ISR and FSR in the MC is much more challenging than the de-scription of the hard process. Moreover, in order to have a consistent dede-scription we cannot improve the accuracy only for one part, we need to do it for both parts, the hard process and the ladders. Many hard processes are presently known up to the next-to-leading-order (NLO) or even to the next-to-next-to-leading-order (NNLO) of perturbation theory, whereas the radi-ation part (ladder) in MC is presently treated only in the leading order (LO). This is the point that makes the improvement of the ISR cascades (pQCD ladders) the most important and chal-lenging task for the improvement of PSMCs and this is why it was chosen as the subject of this work.

1.1 The aims

This PHD thesis is a part of a bigger effort, with the final aim of constructing the complete next-to-leading-order (NLO) Parton Shower Monte Carlo (MC) for the initial state radiation (ISR) in Quantum Chromodynamics, see ref. [11,12,13]. The requirements put on this project are strict and ambitious. The aim is that the Monte Carlo is based rigorously on the collinear factorization in terms of Feynman diagrams, so that it relies on the solid theoretical founda-tions. It should implement exactly the NLO DGLAP [14] (Dokshitzer, Gribov, Lipatov, Altarelli, Parisi) evolution in the pQCD ladder parts. The NLO evolution is required to be performed by

1_{It is universal for the commonly used MS or MS factorization scheme but it can be different for example in DIS}

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the Monte Carlo itself, using the unintegrated NLO distributions originating directly from the Feynman diagrams.

Nowadays, all existing Monte Carlo Parton Showers (ladder parts) implement only the leading-order (LO) evolution with incomplete NLO improvements, following the solutions of the early works in the 1980’s [15,10]. Presently there is no single MC program, that implements the complete NLO, in both hard process and ladder parts. The only partial solution of the recent years is the combination of the NLO hard process with the LO evolution, see [16] and [17]2. There are other theoretical concepts for improving the LO Parton Shower. One of them is proposed in ref. [19]. It incorporates quantum interferences, in the case they have leading soft or collinear singularities, and the color and spin information carried by partons, but it is still restricted to the LO. Another idea, based on the Soft Collinear Effective Theory (SCET) [20] is proposed in ref. [21].

This may be a surprising situation, as the NLO evolution kernels are known from the early 1980’s, see refs. [22,23,24] (today they are known even to the NNLO level, see refs. [25,26]), which means that for nearly 30 years the Monte Carlo modeling of the ladder has been stagnant. There are many reasons for this, one of them is that it is a complicated task, as the factorization theorems [27, 28,23,29] are not well suited for the MC implementation, another is that for a long time the quality of the hadron collider experimental data was not good enough to require such a development.

The entire MC project outlined above has to be divided into smaller parts. The hadroniza-tion, underlying event, etc. will be taken from external sources. The factorization theorems dictate the division of the pQCD part into the hard process (coefficient functions) and ladder parts, giving rise to the parton distribution functions (PDFs). The principal aims of the com-plete MC project are: (i) constructing the ISR ladder (PDF) by means of evolution done by the MC itself, (ii) constructing the hard process, and finally (iii) finding the way to do the matching of these two elements.

The construction of the first element (ISR NLO ladder) requires definition and calculation of the new exclusive evolution kernels, which will be the building blocks in its construction. This task is the subject of this PHD thesis. The work presented here is already partly published, see [30,31,32]. Let us now outline the main goals of this work:

• Define the exclusive evolution kernels within the MC factorization scheme.

• Integrate these exclusive distributions (kernels) to obtain the standard inclusive kernels within the MC factorization scheme.

• Calculate the differences between the inclusive kernels in the M S and MC factorization schemes.

• Analyze the dependence of the evolution kernels on the upper phase space limit (factor-ization scale in MC) – this is directly related to the previous point.

• Analyze the infra-red (IR) singularity structure of the exclusive/inclusive evolution ker-nels in the MC scheme.

The above material is presented in the following way. In Sec.1.3we present the general idea of factorization and give details about the Curci, Furmanski, Petronzio (CFP) and Ellis, Georgi,

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Machacek, Politzer, Ross (EGMPR) factorization schemes of refs. [23,27], which are used as a reference in this work. Sec.1.5discusses how the CFP/EGMPR scheme is modified/extended for the purposes of the Monte Carlo, see also ref. [12]. In Sec.2we present and classify Feyn-man diagrams contributing to the NLO DGLAP kernel. Sec.3elaborates on the bremsstrahlung type diagrams: Br of Fig.3(a), and Bx of Fig.3(e). It covers: (i) extraction of exclusive kernel con-tributions (differential discon-tributions), (ii) integration of these discon-tributions (calculating inclusive kernel contributions), (iii) analyzing the dependence on the upper phase space limit (factoriza-tion scale in the MC), (iv) analyzing soft singularity structure on both exclusive and inclusive level. Sec.4presents the results of a similar analysis for the rest of the Feynman diagrams con-tributing to the NLO non-singlet evolution kernel. More emphasis is given to the gluon pair production diagram Vg of Fig.3(b)as there are new singularities to be discussed. Finally, in Sec.5we conclude the results and give some perspectives for the future.

1.2 Glossary/terminology

In the following we summarize the terminology used in this work. This may be helpful as we often extend the existing terminology in order to describe precisely new concepts and objects.

1. 2PI kernel K0:

A set of 2-particle irreducible (2PI) cut-diagrams encountered in the expansion of a ladder, defined as in the EGMPR paper [28].

2. Evolution kernel (inclusive):

A matrix depending on the lightcone variable and two flavor indices which governs pQCD evolution of PDFs. In this work, flavor indices are skipped as we consider only the qqkernels. The inclusive evolution kernel can be either in the M S or the MC factorization scheme.

3. Exclusive evolution kernel:

The exclusive (fully differential) distribution being an integrand of the inclusive evolu-tion kernel, defined explicitly as a distribuevolu-tion taken from the Feynman cut-diagrams contributing to the 2PI kernel K0 with additional terms due to the reorganization of the

ladder into the time ordered exponential. In this work, it is defined within the MC factor-ization scheme.

4. Ladder: Defined as in the EGMPR paper [28]. 5. M S factorization scheme:

Factorization scheme defined in ref. [23] (CFP), using dimensional regularization and axial gauge.

6. Monte Carlo (MC) factorization scheme:

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1.3 Collinear factorization

In this section we describe in more detail the classic collinear factorization schemes of EGMPR (Ellis, Georgi, Machacek, Politzer, Ross [27]) and CFP (Curci, Furmanski, Petronzio [23]), which we use as a reference and a starting point for the variant of the collinear factorization scheme to be used in the Monte Carlo, described in the following in Sec.1.5.

The general result of the factorization theorems is that the cross section for a given process factorizes into a short-distance (high energy 1GeV) contribution and a long-distance (low energy ∼ 0.5GeV) one. The short-distance contribution can be made finite and can be calculated perturbatively, contrary to the long-distance one, which cannot be calculated only from the principles of perturbative QCD. The procedure of factorization in pQCD operates in the energy scales & 1GeV. The factorization procedure is not unique, which means that the division into the hard process (coefficient function) and the pQCD ladder (PDF) part is arbitrary and changes when we change the factorization scheme. An important ingredient is the factorization scale, which is a dimensional constant (dimension of energy), that provides an upper limit for the phase space (in the transverse momentum space). Both the hard process and the ladder parts depend on this scale but this dependence, as well as the dependence of the whole scheme, cancels out for the physical observables order by order.

The calculations presented in this work are based on the collinear factorization formulated in the physical gauge in ref. [27] and later on customized to the dimensional regularization and the M S scheme by CFP [23]. Later on factorization theorems were refined in refs. [29,33].

Factorization theorems in pQCD generally prove, using series of Feynman diagrams to infi-nite order, that the process dependent matrix element (ME) of the hard process can be isolated and made free of collinear singularities, whereas the ladder parts (PDFs), encapsulating all collinear singularities gets factorized off. The ladder parts can be made universal3 and ex-tracted from the experimental data. It means, for example, that the PDFs exex-tracted from the lepton hadron scattering experiments (DIS) like HERA can be used for the calculations of the hadron hadron scattering, for example, for the Large Hadron Collider.

In the following we are going to present in more detail the factorization procedure in the CFP [23] and EGMPR [27] formulation. A general framework is discussed using the DIS process as an example, but it is universal and applies to any other process.

Let us consider a matrix element squared. We concentrate on the DIS process example, but our considerations will hold for the other processes, provided that we calculate the appropri-ate hard process ME (short distance cross-section). Following works of refs. [27,23] it is known that, in the axial gauge, the perturbative expansion of a matrix element squared can be reorga-nized in the form of the generalized ladder expansion (GLE) in terms of 2 particle irreducible (2PI) kernels C0and K0. The GLE separates the part which is free from mass singularities and

the ladder part containing all of them, see the DIS case with the single ladder in Fig.1. In the axial gauge (used by EGMPR, CFP) all the collinear singularities originate from the integration over the vertical lines connecting kernels. In the CFP scheme K0 contains, by definition, the

propagators of those lines. EGMPR state that C0is finite, whereas the ladder part, consisting of

K0, contains singularities. In the generalized ladder expansion the “raw” factorization reads:

M = C0(1 + K0+ K02+ ...) = C0

1 1 − K0

≡ C0Γ0. (1.1)

3_{There are factorization schemes which do not have this property, one of them in DIS factorization scheme, but}

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Figure 1: Graphical presentation of the “raw” factorization appearing directly from GLE ex-pansion in terms of 2PI kernels C0and K0.

We call it “raw” factorization, because the 2PI kernels are still connected by spinor indices as well as the dn_k_{integration. For example the product of two kernels is given by:}

C = C_ββαα00(k₂, k₁) = A · B = X γγ0 Z _dn_k (2π)nA αα0 γγ0(k₂, l)Bγγ 0 ββ0(l, k1) (1.2)

In the next step in the factorization procedure the special projection operatorP is introduced. The action of this operator is listed below:

(i) it decouples C0and Γ0in spinor indices,

(ii) partly decouples C0and Γ0 in momentum space (except light-cone variable),

(iii) extracts the singular part of dnkintegrals.

TheP operator of the CFP work [23] is defined in the following.

We start from Eq. (1.1) and proceed recursively. The first step is to factorize the singular part of the last kernel K0:

M = C0 ∞ X i=0 K₀i = C0{1 + ∞ X i=1 K₀i−1(P + (1 − P))K0}, (1.3)

then we do the same with the second K0:

M (1 −PK0) = C0{1 + (1 −P)K0+ ∞

X

i=1

K₀i−1(P + (1 − P))(K0(1 −P)K0)}, (1.4)

and we repeat the same with the third one:

M (1 −PK0−P(K0(1 −P)K0)) = C0{1 + (1 −P)K0+ (1 −P)(K0(1 −P)K0) + ∞ X i=1 K₀i(1 −P)(K0(1 −P)K0)}. (1.5)

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From the above three terms the general pattern is obvious: M 1 −P K0 1 1 − (1 −P)K0 = C0 1 1 − (1 −P)K0 . (1.6)

Now inverting Eq. (1.6) we obtain the following final factorization formula of CFP: M = C0· 1 1 − (1 −P) · K0 ⊗ 1 1 − K ⊗ = C α,Q 2 µ2 ⊗ Γ α,1 , (1.7) where 1 1 − K ⊗ = 1 + K + K ⊗ K + K ⊗ K ⊗ K + ..., K =P K0· 1 1 − (1 −P) · K0 . (1.8)

In the above factorization formula, thanks to subtractions using the projection operatorP, the coefficient function C is finite and coupled with the ladder part Γ only through one-dimensional integral over the light-cone variable. The convolution ⊗ is defined as

f ⊗ g (x) = Z

dudvf (u)g(v)δ(x − uv), (1.9)

where x, u, v are in the (0, 1) range. The new factorization formula is depicted in Fig.2.

Figure 2: Graphical interpretation of the factorization formula.

The projection operator P of CFP consists of three parts: (i) Pspin decoupling kernels in

spinor indices, (ii) Pkin decoupling kernels in 4-momentum space by putting the 4-momenta

towards the hard process (to the left) on-shell and (iii) PP - the pole part operator extracting singular part (in terms of1 poles) from the expression on the right

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The action ofP operator of the CFP factorization scheme is defined below4: C0PK0/p = C0/kPkinPP / n 4knK0/p = Z _dx x C0/k k2₌₀xPP Z _dm_k (2π)mδ(x − pn/kn) / n 4knK0/p , (1.11)

where the contraction of spinor indices and the direction it works is indicated by K0/p.

The above gives both the general idea of the factorization, as well as the set of the explicit formulas, which will be used later on. The above scheme has to be modified for the purposes of the use of the collinear factorization at the NLO level in the Monte Carlo simulations. The collinear factorization scheme suited for the MC is outlined in refs.[12,34] and partially in the following Sec.1.5.

1.4 The framework for the calculation of the CFP inclusive evolution kernels The main intrests of this thesis are the NLO DGLAP evolution kernels, especially their new exclusive (unintegrated) version. Traditionally, the evolution kernels are defined at the inclusive level (depending on the longitudinal momentum fraction x and the energy scale Q), but for the purpose of the construction of the NLO PSMC one needs to use the new exclusive version of the splitting functions, this new version of evolution kernels will be defined and calculated. In this thesis we use scheme of CFP [23] presented in section1.3as a reference and a guide.

As all factorization procedures, see also [28] and [35], CFP factorization works at the inclu-sive level, that is with internal phase space of the K0 kernel integrated over, which prevents

their direct use in the Monte Carlo. To be more specific, in the inclusive kernel calculations the integration over the transverse momentum is performed and even worse, the introduction of projection operator violates the 4-momentum conservation (its transverse component is inte-grated out in CFP or EGMPR). Because of it, for the purpose of the MC, which is exclusive, one needs to go back and define the exclusive kernel in such a way that the singularities are separated before the phase space integration, within the full phase space. The exclusive level is already present at the level of the “raw” factorization of EGMPR, before any projections and integrations were done.

Another important differences between the MC approach and CFP factorization is the fact that MC always operates in 4-dimensional phase space, as the real collider experiment, whereas in the CFP the entire phase space of the real parton emissions is treated in n = 4 + 2 dimen-sions5.

It should be stressed that in the Monte Carlo implementation of the pQCD ladder up to NLO level the MC program will integrate over the internal phase space of the exclusive NLO kernel on its own and in principle we do not need to do this integration analytically. Neverthe-less, we are going to perform this analytical integration because (i) it will be useful for testing MC and (ii) we want to keep a complete control of the differences between pQCD evolution in the MC and in the CFP scheme – this can only be done after such an analytical integration.

First, in Sec.1.4.1we present the explicit formula for the evolution kernel within the CFP factorization scheme, then Sec.1.4.2introduces notation, phase space parametrization, etc. In

4_{The above definition is for the case, when the external lines are the quark lines. It differs if we consider diagrams}

with external gluon lines, but these diagrams contribute to the singlet kernels, not considered in this work.

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Sec.1.5we give the definition of inclusive and exclusive evolution kernels in the Monte Carlo factorization scheme.

1.4.1 Extracting DGLAP evolution kernels from CFP factorization formula

The inclusive evolution kernels are extracted in the CFP scheme directly from the factorization formula of Eq. (1.7) describing the PDF part Γ, which is given by:

Γ = 1 1 − K = 1 1 −P K0·_{1−(1−P)·K}1 ₀ . (1.12)

Since we are intrested in the next-to-leading order kernels it is enough to use Γ expanded only up to NLO terms:

Γ = 1 +PK0+P(K0(1 −P)K0) + (PK0)(PK0) + . . . (1.13)

The general structure of Γ in terms of collinear singularities is given by a series in 1 poles, namely: Γ = Γ0+1Γ1+12Γ2+ . . .. The NLO DGLAP kernel is given by the coefficient in front

of1 pole (the residue of Γ)6. After expansion one can see that we are no longer dealing with the two particle irreducible (2PI) diagrams, for example the termPK2

0 does not have this property.

The termPK02 in Eq. (1.13) is accompanied by a countertermP(K0PK0), which subtracts the

square of the LO contribution. The bare PDF Γ of the M S scheme in a more explicit form, with integrals visible explicitly, takes the following form:

Γ = xPP Z dnk (2π)nδ x −kn pn / n 4knK0/p . (1.14)

The above formula sums up contributions of all relevant Feynman diagrams in K0, see Fig.3for

the two-real contributions to the NLO non-singlet kernel. The contribution of a single Feynman diagram depicted in Fig.3is given by:

Γ = xPP 1 µ4 Z dΨ δ x −qn pn Cg4W (k1, k2, ) Θ s(k1, k2) ≤ Q , (1.15)

where dΨ is a two particle phase space (defined in the next section), C is a color factor, g is the strong coupling, W (k1, k2, )originates from γ-trace of different Feynman graphs contributing

to Γ and µ4_{originates from dimensional regularization (in n = 4+2 dimensions), µ is a formal}

energy scale of M S – the factorization scale. The theta function Θ(s(k1, k2) ≤ Q)encloses the

phase space of emitted particles from above using a dedicated function of kinematical variables s(k1, k2).

The CFP evolution kernel itself is given by twice the residue in = 0 of the bare PDF: P (x) = 2Res0 x 1 µ4 Z dΨ δ x −qn pn Cg4W (k1, k2, ) Θ s(k1, k2) ≤ Q . (1.16)

Let us remind that in general kernels P and bare PDFs Γ have flavor indices. These indices are not written here explicitly, as when we restrict our considerations to the non-singlet case,

6_{Not all terms in Eq. (1.13) contribute to the NLO M S kernel, terms of a form (PK}

0)(PK0)are proportional to

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nearly all considered Feynman diagrams contribute to the qq kernel7. In the following we skip flavor indices and it will be understood implicitly, that we are talking about qq kernel. If we consider other kernels we will note it explicitly.

1.4.2 Notation, variables and phase space parametrizations

In the calculations in the light-like axial gauge, the gauge vector n (n2= 0) is used (in both MC approach and in CFP calculations). The gluon propagator is then equal to _k12dαβ(k), where:

dαβ(k) = −gαβ+

kαnβ+ kβnα

kn . (1.17)

The three gluon vertex for all incoming momenta is defined as (without color factors): V (pµ1 1 , p µ2 2 , p µ3 3 ) = (p2− p1) µ3_gµ1µ2 _{+ (p} 3− p2)µ1gµ2µ3 + (p1− p3)µ2gµ3µ1. (1.18)

The notation of this section will be used in both MC approach defined in 4-dimensions and calculations in the CFP scheme, which is using dimensional regularization with n = 4 + 2 dimensions. The differences between 4 and n-dimensions will be the object of a special interest. The incoming particle momentum is denoted as p. We focus on the two-particle real emis-sion diagrams. The four-vectors of two emitted partons are denoted by k1 and k2, with their

transverse components being ki⊥ = (0, ki⊥, 0), additionally we use symbol k for their sum

k = k1 + k2. Since we are in the massless limit of QCD, all external particles are massless,

namely p2 = k2₁ = k2₂ = 0. The Sudakov variables are used for the four-momenta parametriza-tion:

kµ_i = αipµ+ βinµ+ kµ_i⊥, i = 1, 2. (1.19)

Since the emitted gluons are on mass shell and we are in the massless theory, βi are fixed and

equal to βi= k

2 i⊥

2αi(pn). We also use q symbol for the off-shell momentum q = p − k = p − k1− k2.

Instead of the transverse momentum we often will use angular variable: ai =

ki⊥

αi

. (1.20)

The modulus of the above vector variable will be referred to as the angular scale variable. There is a simple relation of the angular scale to the rapidity via ηi = ln |ai| + const.

The typical denominators occurring during the analytical calculations, the virtuality of the emitter parton after two emissions q2 and the effective mass of the produced pair k2, in terms of angular variables are given by:

−q2 = α1α2q˜2(a1, a2), q˜2(a1, a2) = 1 − α2 α2 a2₁+1 − α1 α1 a2₂+ 2a1· a2, k2 = α1α2a2(a1, a2), a2(a1, a2) = a21+ a22− 2a1· a2, (1.21) or in k⊥parametrization: −q2 ₌ 1 − α2 α1 k2_1⊥+1 − α1 α2 k2_2⊥+ 2k1⊥· k2⊥, k2 = α1 α2 k2_2⊥+α2 α1 k2_1⊥− 2k_1⊥· k_2⊥. (1.22)

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The two real-particle phase space of Eq. (1.15), in n-dimensions, is given by: dΨ = d n_k 1 (2π)n2πδ +_(k2 1) dnk2 (2π)n2πδ +_(k2 2). (1.23)

The phase space parametrization in terms of Sudakov variables, using explicitly n = 4 + 2, is given by: dΨk⊥ = 1 4 Ω1+2 (2π)6+4 dα1 α1 dα2 α2 dΩ1+2dk1⊥dk2⊥k1+21⊥ k 1+2 2⊥ . (1.24)

In the above we used the fact, that the angular dependence enters only through the relative angle between two particles φ = φ1−φ2, which allows to integrate out the trivial angles, giving:

Ω1+2 = 2π

1+

Γ(1+). In n dimensions the non-trivial angular dependence is given by: dΩ1+2 =

Ω2(sin φ)2, where φ ∈ (0, π) and Ω2 = 2π

1/2+

Γ(1/2+) (in 4-dimensions we have simply dφ where

φ ∈ (0, 2π)).

The same phase space in angular parametrization takes the following form: dΨa= 1 4 Ω1+2 (2π)6+4 dα1 α1 dα2 α2 α2+2₁ α2+2₂ dΩ1+2da1da2a1+2₁ a1+2₂ . (1.25)

The same formulas for the phase space hold for = 0, but in the Monte Carlo, instead of the Lorentz invariant phase space, we shall use more convenient dimensionless eikonal phase space dΦ defined by:

dΨ = dΦ k 2 1⊥k22⊥ 4(2π)4 = dΦ α2₁α2₂a2₁a2₂ 4(2π)4 , (1.26) where: dΦ = 1 2π dα1 α1 dα2 α2 dk1⊥ k1⊥ dk2⊥ k2⊥ dφ = 1 2π dα1 α1 dα2 α2 da1 a1 da2 a2 dφ. (1.27)

The eikonal phase space is not only dimensionless but it also does not change when we switch between k⊥ and a variables, these two properties will be used extensively. For completeness

we also show n-dimensional eikonal phase space, which is also used in few cases: dΦ= Ω1+2 (2π)2+4 dα1 α1 dα2 α2 dk1⊥ k1⊥ dk2⊥ k2⊥ (k1⊥k2⊥)2dΩ1+2, (1.28) or in angular variables: dΦa = Ω1+2 (2π)2+4 dα1 α1 dα2 α2 (α1α2)2 da1 a1 da2 a2 (a1a2)2dΩ1+2. (1.29)

Note that in n-dimensions the eikonal phase space is no longer dimensionless.

Above we defined the variables and introduced the phase space parametrization. There is still one more thing to be added. In the kernel definition in Eq. (1.15) the s(k1, k2)function,

en-closing the phase space from the above, must be specified. The choice of the phase space enclos-ing is a very important point because it not only determines the parametrization for analytical calculation, but also defines the future evolution time variable in the construction of the MC im-plementation of the ladder. There are many possible choices for the s(k1, k2)function. Our first

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The second choice, which we will consider in this work is s(k1, k2) = max{|k1⊥|, |k2⊥|}, which

corresponds to the transverse momentum evolution time. We will refer to the second choice during the analysis of the kernel dependence on the choice of this enclosing, see also [32]. These two basic choices of the phase space enclosing will be respectively referred to as the phase space with k⊥-ordering and angular-ordering (a-ordering). Other popular choices of

s-function include the total virtualityp−q2_{and the maximum k-minus, max(k}− 1, k

− 2).

1.5 Definition of the exclusive evolution kernels for the Monte Carlo

Construction of the Monte Carlo implementing the NLO ladder requires construction of the exclusive multiparton distributions instead of calculating the inclusive quantities. The inclu-sive evolution kernels in the MC approach can be defined and calculated, but they are rather by-products than basic objects. In the complete MC project the calculation of the inclusive and exclusive kernels is based on the MC factorization scheme [12]. Of course, the detailed knowl-edge of the CFP factorization scheme is required and very helpful. This modified factorization scheme, referred to as the MC factorization scheme, is working at the integrand level, before the phase space integration. In particular the projection operatorP must be redefined. The new projection operatorP0, defined below, see also refs. [12,13], is the key ingredient of the MC scheme:

(i) it works on the integrand level,

(ii) its spin projection part is the same as inP,

(iii) it sets incoming momentum on-shell (in the part of the ladder diagram towards the hard process),

(iv) it encloses the phase space for the real partons (cut lines) with the help of a dedicated kinematical variable Q > s(k1, k2), Q is the factorization scale,

(v) it acts always only on a single-logarithmically divergent object, (vi) it does not contain the pole part operator PP of CFP.

Another important difference between both approaches is the method of regularizing the collinear singularities. The CFP uses dimensional regularization, which cannot be applied di-rectly in the MC since MC for real emissions is always defined in 4-dimensions. A natural reg-ularization method for the MC is the cutoff or mass regreg-ularization. In the calculations within the MC approach we will implicitly assume cutoff of the type s(k1, k2) > q08. We will see later

on that this difference between regularization methods will induce differences in the evolution kernels. Fortunately, these differences are well defined and rather simple. The above definition of the MC factorization scheme is not complete, as it is still being developed. Presently, there is no general definition, all necessary definitions are provided case by case. More information on this subject can be found in refs. [12,13]9_.

8_{For the diagrams with internal singularities another additional cutoffs are required.} 9_{The definition of}_P0

operator is not all that enters the definition of MC factorization scheme. We need to add a kinematical mapping which together with the above definition ofP0enables the 4-momentum conservation. This mapping depends on the singularity structure of diagrams (“case by case”), for more information and explicit examples of such mapping we refer reader to ref. [13].

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Let us now define the evolution kernel in the MC scheme, in both inclusive and exclusive version, giving explicit formulas.

At the inclusive level, Monte Carlo has its own evolution equation with its own inclusive evolution kernel. Similar to the standard DGLAP kernel, this kernel is defined by taking the logarithmic derivative of the MC parton distribution over the factorization scale Q. The MC distribution G corresponds to Γ, the bare PDF of CFP. For the two-real (2R) kernel contributions, considered in this work, the MC evolution kernel is given by10:

P(x) = ∂ ∂ ln Q(Gb(Q, x) + Ga(Q, x)), (1.30) where Gb(Q, x) = x Z dΨ δ x − qn pn P0_K[2] 0 Θ (Q > s(k1, k2) > q0) (1.31)

represents contribution of 2PI diagramsP0K₀[2], and Ga(Q, x) = x Z dΨ δ x − qn pn ×nP0(K₀[1]K₀[1]) Θ (Q > s(k1, k2) > q0) −P0(K0[1])P 0 (K₀[1]) Θ (Q > s(k1) > s(k2) > q0) o , (1.32) represents contribution of diagrams, which are not 2PI andP0(K₀[1]K₀[1])requires subtraction of the factorization countertermsP0(K₀[1])P0(K₀[1]), see Eq. (1.13) for the full formula. EGMPR kernel contributions K₀[1]and K₀[2]are the diagrams contributing to K0in the first and second order in

the strong coupling αS.

Now, with the help of the newP0 operator, once we defined the factorization counterterms at the integrand level, we can finally define the MC distributions and evolution kernels at the exclusive level.

The contribution of a single 2R Feynman diagram to the inclusive evolution kernel in the MC scheme is given by:

G = Z dΨ4δ x −qn pn xCg4 W (k1, k2, = 0) Θ(Q > s(k1, k2) > q0), (1.33)

where W (k1, k2, = 0) ≡ W (k1, k2)represents either a contribution of a 2PI Feynman diagram

as in Eq. (1.31) or a contribution of a diagram with the subtraction as in Eq. (1.32). G is the analog of Γ, the CFP’s bare PDF of Eq. (1.15), where γ-trace of Feynman diagram W (k1, k2, )

is in both cases the same, up to the -terms, absent in 4-dimensions. As in the Monte Carlo we use the eikonal phase space dΦ of Eq. (1.27), the exclusive evolution kernels are preferred to be defined within this phase space:

G = xCg4 Z dΦ δ x − qn pn k2_1⊥k_2⊥2 4(2π)4 W (k1, k2) Θ(Q > s(k1, k2) > q0). (1.34)

10_{Definition of MC kernel as a derivative over ln Q instead of ln Q}2_{leads to a difference in normalization of NLO}

kernels in the M S and MC schemes, namely: P(x) = 2P (x). The LO kernel normalization is the same for both schemes.

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A new symbol is introduced for the distribution within the eikonal phase space: ˜ W (k1, k2, ) = Cg4x k2_1⊥k_2⊥2 4(2π)4 W (k1, k2, ) = Cx α_S 2π 2 α₁2α2₂a2₁a2₂W (k1, k2, ), (1.35)

which is defined in the same way in n and in 4-dimensions11. The exclusive evolution kernel for the MC scheme is defined using the distribution of Eq. (1.35) for = 0:

˜ W (k1, k2) = Cx α_S 2π 2 α2₁α2₂a2₁a2₂W (k1, k2, = 0), (1.36)

the MC distribution (equivalent of the bare PDF) takes form: G =

Z

dΦ δ(1 − x − α1− α2) ˜W (k1, k2) Θ(Q > s(k1, k2) > q0). (1.37)

The exclusive evolution kernel ˜W (k1, k2), should also contain the theta functions giving the limits

for the phase space. However, they are written separately for the sake of clarity of the following discussions.

2 NLO non-singlet evolution kernels

The aim of this section is to analyze the double real emission diagrams contributing to the NLO non-singlet DGLAP evolution kernel12. This analysis will be done in such a way that the CFP results are reproduced, the building blocks for the MC scheme are defined and the discussion of the differences between the CFP and MC schemes is done.

Fig. 3 shows Feynman cut-diagrams with two emitted on-shell partons (two cut lines), which enter the calculations. We will refer to them as the 2-real (2R) contributions. Other dia-grams contributing to the NLO kernel are diadia-grams with 1-real and 1-virtual emission (1V1R) and purely virtual graphs with 2-virtual emissions (2V). The contribution from the 2V diagrams can be found out using the baryon number conservation, see ref. [23], and will not be consid-ered in the following. The 1V1R diagrams are also beyond the scope of this work but they will be referred to on a few occasions, as they are important for the purpose of combining with the 2R soft contributions.

In Fig.3and in the following the squares and/or interferences of the amplitudes from Feyn-man diagrams are represented as follows:

2 1

+

1 2 2 = 2 1 + 2 1 + 2 2 1 .

Each Feynman (cut-)diagram in Fig.3has additionally a “bullet” vertex at the top (towards the hard process) indicating the presence of the projection operator (either the CFP’sP or P0 introduced for the purpose of the MC, see Sec.1.3and1.5).

11_{The usual notation α}

S= g2/(4π)is used.

12_{Definition of singlet/non-singlet kernels beyond LO can be found in ref. [?]. Possible solutions for}

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(a) (b) (c)

(d) (e) (f) (g)

Figure 3: 2-real diagrams contributing to the NLO non-singlet DGLAP evolution kernel. From the point of view of the analytical calculations of the exclusive/inclusive evolution kernel all the diagrams of Fig.3are in the same class (2R) but from the view point of the MC modeling of the non-singlet evolution the situation is different, see [30]. Especially if our aim is to include into the MC only the non-singlet class of diagrams. This is because not all of the interferences of Fig.3 are accompanied by an appropriate square diagrams and they do not form the square of full amplitude. Consequently, it is not granted that their contributions are positive. Negative distributions generally lead to the negative MC weights, which we want to avoid. For the moment, the solution we propose to addopt in the MC is to include all “prob-lematic” 2R contributions in the integrated form, together with the 1V1R diagrams, as what we call, the unresolved contributions, see [30] and [13]. Having in mind that it is only a temporary solution and that in the future these contributions will be needed, the exclusive distributions for all of the 2R diagrams will be calculated. Of course, as soon as the singlet diagrams are included into the MC, the above problem will disappear.

Let us analyze explicitly which diagrams of Fig.3form a full amplitude square and which need to be temporarily treated as unresolved. The combinatorial coefficients, originating either from taking the amplitude square or from the Bose-Einstein (BE) symmetrization13, accompa-nying each of the contributing cut-diagrams are indicated in the figures. Double gluon emis-sion diagram (Br)3(a), gluonstrahlung interference diagram (Bx) 3(e), gluon pair production diagram (Vg)3(b)and gluon interference diagram (Yg)3(d), all originate from the amplitude

13_{The BE symmetrization prevents us from the double counting of the same physical state. Since the quantum}

particles are indistinguishable we cannot count twice the state with identical particles interchanged. We can either permit only to occupy part of the phase space by each particle (half in case of two particles) or do the symmetrization (add 1/2! coefficient for case of two particles)

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squared: 1 2! + + 2 = = 1 2!   1 2 + 1 2 + 2 + + 2 2 1 + 2 1 2   = 1 2   1 2 + 1 2  + + 1 2 + 2 1 + ₁2 .

Since they form the full amplitude squared, the positiveness of the MC weight is ensured and each of them can be treated as a “MC-complete” 2R contribution. The fermion pair production diagram (Vf)3(c)and the fermion interference diagram (Yf)3(f)originate from the amplitude squared: + 2 = + 2 + .

Since the last diagram (square) on the right in the above figure (q ¯qemission) is beyond the non-singlet kernel, there is no guarantee that the sum of Vf and Yf distributions alone is positive. Of course, Vf, being the amplitude square, is positive on its own, but Yf must stay temporarily in the unresolved form (together with the 1V1R ones) in the MC, while NLO corrections are restricted to the non-singlet. The interference diagram Xf3(g)must also stay in the unresolved part as it is the interference of two diagrams, where both squares

1 2! + 2 = 1 2   1 2 + 2 1  +

contribute to the singlet evolution kernel and are absent in the non-singlet class.

In the figures above, showing the original Feynman amplitudes, the combinatorial coeffi-cients for each diagram entering the NLO non-singlet evolution kernel are also indicated. They are important for the purpose of the analysis of the infra-red (IR) singularity structure in Sec.3.3

and4.7.

2.1 2R kernel contributions

In the following section we analyze in more detail the structure of the MC distributions (bare PDFs of CFP), which allow us to extract kernel contributions of the 2R diagrams in the form suitable for the MC, and show some general steps encountered during the integration. First, we concentrate on the MC scheme but for the sake of systematic comparison between the MC and M S schemes we refer also to the M S scheme.

The starting point for the integration is the formula of Eq. (1.37) G =

Z

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or in case of the M S scheme the formula of Eq. (1.15) using the eikonal phase space of Eq. (1.29): Γ =PP 1 µ4 Z dΦδ(1 − x − α1− α2) ˜W (k1, k2, ) Θ Q > s(k1, k2) . (2.2) ˜

W is dimensionless in both 4 (MC) and n dimensions (CFP) (depends only on the ratio a1/a2,

the relative angle φ between a1and a2and the longitudinal components α1, α2). This specific

form of ˜W enables immediate factorization of the singularity due to the overall scale (either in form of the logarithm or 1 pole). To do it, it is convenient to introduce additional integration variable ˜Qusing the identity Θ Q > s(k1, k2)f (k1, k2) ≡

RQ

0 d ˜Q δ ˜Q = s(k1, k2)f (k1, k2). The

remaining integral is parametrized using the dimensionless variables yi = a_Q˜i and choosing

s(k1, k2) = max{a1, a2}: Γ =PP ( 1 µ4 Ω1+2 (2π)2+4 Z dα1 α1 dα2 α2 (α1α2)2δ(1 − x − α1− α2) Z Q 0 d ˜Q ˜Q4−1 × Z dΩ1+2 Z 1 0 dy1 y1 dy2 y2 (y1y2)2 W (y˜ 1, y2, φ, α1, α2, ) δ 1 − max{y1, y2} ) . (2.3)

Collinear singularity is now explicitly factorized off in a form of the integralRQ

0 d ˜Q ˜Q 4−1 ₌ Q4

4 . The same integral in 4 dimensions takes form

RQ q0 d ˜Q ˜ Q = ln(Q/q0). The δ 1 − max{y1, y2}

function can be integrated out leading to the division of yiphase space into two parts:

Γ =PP ( 1 4 Q4 µ4 Ω1+2 (2π)2+4 Z _dα 1 α1 dα2 α2 (α1α2)2δ(1 − x − α1− α2) × Z dΩ1+2 Z 1 0 dy1 y1 y2₁ W (y˜ 1, 1, φ, α1, α2, ) + Z 1 0 dy2 y2 y₂2 W (1, y˜ 2, φ, α1, α2, ) ) . (2.4)

The additional 1 poles (or logarithms) may possibly arise from the internal singularities of the integrands of Feynman diagrams. They are always connected with the integrations over the transverse degrees of freedom (yi). The longitudinal components can also lead to the infra-red

(IR) singularities, when αi → 0, but this type of singularities does not lead to additional1poles,

because in the MC approach as well as in the CFP factorization scheme they are regularized in a non-dimensional manner. For regularization of the IR singularities the principal value prescription (P V ) is used:

1

α →

α

α2_{+ δ}2. (2.5)

The following notation of CFP for the divergent integrals will be used: Z 1 0 dα α α2_{+ δ}2 ≡ I0, Z 1 0 dα ln α α α2_{+ δ}2 ≡ I1. (2.6)

Before the α-integration we need to perform -expansion and introduce (implicitly) the P V regularization prescription.

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Only some of the diagrams entering the NLO kernels feature internal singularities and for the ones which do not, both calculations, within the MC approach and in the CFP scheme, are practically the same. In the CFP case Eq. (2.4) simplifies to14_:

Γ = 1 4 1 2π Z dα1 α1 dα2 α2 δ(1 − x − α1− α2) × Z 2π 0 dφ Z 1 0 dy1 y1 ˜ W (y1, 1, = 0) + Z 1 0 dy2 y2 ˜ W (1, y2, = 0) . (2.7)

The same equation in the MC approach takes the following form: G = ln(Q/q0) 1 2π Z _dα 1 α1 dα2 α2 δ(1 − x − α1− α2) × Z 2π 0 dφ Z 1 0 dy1 y1 ˜ W (y1, 1) + Z 1 0 dy2 y2 ˜ W (1, y2) . (2.8)

In the following sections, for the purpose of the calculation of the inclusive evolution ker-nels (integration) we could start the calculation from Eq. (2.4). However, for the exclusive (un-integrated) MC scheme it is mandatory to write down the whole integrand for each Feynman diagram contributing to the kernel, in order to define explicitly the new exclusive evolution kernel, see next sections.

3 Bremsstrahlung type diagrams

In this section we show step by step the analysis of the exclusive NLO DGLAP evolution kernel for the bremsstrahlung diagrams. The choice of the bremsstrahlung type of diagrams for this purpose is not accidental. These diagrams have considerably easy analytical formulas and yet they feature nearly all kinds of problems that one has to deal with while defining the exclusive kernel and integrating it to obtain the standard inclusive version. In particular, they feature internal singularities cancelled by the factorization counterterms.

There are two Feynman diagrams: the double ladder diagram (Br) of Fig.4and the ladder interference diagram (Bx) of Fig.5and one counterterm graph (Ct) of Fig.4 to be analyzed. The counterterm is induced by the factorization, its role is to subtract the LO part of the double bremsstrahlung (Br) diagram. Only the remnant left after the subtraction does contribute to the NLO evolution kernel. The precise form of the factorization counterterm is given by the factorization scheme and the definition of the projection operator, eitherP0of Sec.1.5(MC) or P of Sec.1.3(CFP).

Another reason for choosing the bremsstrahlung set of diagrams is that it allows to profit from the relation to the case of QED where we have photons instead of gluons.

For the sake of the clarity of presented material we divide this section into subsections. First, in Sec.3.1we show the exclusive kernels – differential distributions, which will be used in the MC and their integration, which gives the standard inclusive kernels. We need to remember that the most important part of it are the exclusive kernel contributions. The inclusive kernels are not our primary interest but it is important to show the integration procedure for the sake of further analysis. The integration is done both in the MC scheme and in the reference scheme

14_{We use shorthand notation ˜}_{W (y}

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Figure 4: Br - ladder graph and its counterterm graph - Ct.

Figure 5: Bx - ladder interference graph.

of CFP, which allows for the easy comparison of both results. Sec.3.2 shows the analysis of the dependence of the NLO kernel on the upper phase space limit. In Sec.3.3we analyze the infra-red singularity structure of the kernels obtained in the MC factorization scheme.

3.1 Exclusive kernel for bremsstrahlung and its integration

As a warm-up exercise we analyze the genuine contribution to the K0 kernel of the EGMPR of

Bx diagram, see Fig.5. It is considerably easier from the Br case as it is free from the internal singularities, which makes the phase space integration in both 4 and n-dimensions almost the same. We omit all the coefficients originating from either the Bose-Einstein symmetrization or the fact that we consider interference diagrams. They are restored when we sum up all contributions.

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3.1.1 Ladder interference diagrams - Bx

The original Feynman diagram distribution of ladder interference graph WBxreads:

WBx= 1 4(qn) 1 q4 1 (p − k1)2 1 (p − k2)2 Trh/pγα(/p − /k1)γµ/q /n/qγβ(/p − /k2)γν i dαβ(k1)dµν(k2). (3.1)

Upon calculating the trace15_{introducing the rapidity related variables (a}

i= ki⊥/αi), restricting

only to 4 dimensions (no- terms) and applying the additional factor of Eq. (1.36), we obtain the following exclusive kernel contribution:

˜ WBx = 4C α_S 2π 2 a2 1a22 ˜ q4_(a 1, a2) T0+ T1 a1· a2 a2 1 + T2 a1· a2 a2 2 + T3 (a1· a2)2 a2 1a22 , (3.2) where: T0 = 2x 1 + x2 1 − x 1 α1 + 1 α2 − 2x, T1= 1 + 2x2 α1 − 1 + x − x2, T2 = 1 + 2x2 α2 − 1 + x − x2, T3= 2(1 + x2), (3.3)

and the color factor C = C_F2 − 1₂CACF. The contribution from the Bx diagram to the MC

distribution (analog of the bare PDF) reads: GBx= Z dΦδ (1 − x − α1− α2) ˜WBxΘ (max{a1, a2} < Q) = Z _dα 1 α1 dα2 α2 δ(1 − x − α1− α2) Z ∞ 0 da1 a1 Z ∞ 0 da2 a2 1 2π Z 2π 0 dφ × 4CαS 2π 2 a2 1a22 ˜ q4_(a 1, a2) T0+ T1 a2 a1 cos φ + T2 a1 a2 cos φ + T3cos2φ Θ (Q > max{a1, a2} > q0) . (3.4) The same expression in n-dimensions is completely analogical, instead of cutoff q0, for the

regularization of the overall scale singularity, it features (a1a2)2term.

The calculation proceeds in the manner presented in Sec.2.1. First, the scale singularity is extracted and dimensionless variables yi = ai/ ˜Qare introduced:

GBx = ln (Q/q0) 4C α_S 2π 2Z dα₁ α1 dα2 α2 δ(1 − x − α1− α2) 1 2π Z 2π 0 dφ Z 1 0 dy1 y1 Z 1 0 dy2 y2 × y 2 1y22 ˜ q4_(y 1, y2) T0+ T1 y2 y1 cos φ + T2 y1 y2 cos φ + T3cos2φ δ (1 − max{y1, y2}) . (3.5)

Then, after performing the remaining δ-function we obtain: GBx = ln (Q/q0) 4C α_S 2π 2Z dα₁ α1 dα2 α2 δ(1 − x − α1− α2) 1 2π Z 2π 0 dφ × Z 1 0 dy1 1 ˜ q4_(y 1, 1)

T0y1+ T1cos φ + T2y21cos φ + T3y1cos2φ

+ Z 1 0 dy2 1 ˜ q4_{(1, y} 2)

T0y2+ T1y22cos φ + T2cos φ + T3y2cos2φ

.

(3.6)

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The phase space of y1 and y2 integration can be combined. It is done by means of a simple

change of variables y1 = y, y2= 1/y16:

GBx = ln (Q/q0) 4C α_S 2π 2Z dα₁ α1 dα2 α2 δ(1 − x − α1− α2) 1 2π Z 2π 0 dφ × Z ∞ 0 dy 1 ˜ q4_{(y, 1)}

T0y + T1cos φ + T2y2cos φ + T3y cos2φ

.

(3.7)

Next, the integration over transverse degrees of freedom φ and y is performed: GBx= ln (Q/q0) 4C α_S 2π 2Z dα₁ α1 dα2 α2 δ(1 − x − α1− α2) × T0 α1α2 2x − T1 α2₁α2 2x(1 − α1) − T₂ α1α 2 2 2x(1 − α2) + T3 1 4ln x (1 − α1)(1 − α2) +α1α2 2x . (3.8) Finally, the α-integration is done, remembering about the PV prescription for regularization of the IR singularities and using the notation of Eq. (2.6) for the IR divergences, we get the contribution of the ladder interference diagram Bx to the bare PDF (kernel) in the MC scheme:

GBx(x, Q) = ln (Q/q0)PBx(x), PBx(x) = C_F2 − 1 2CACF α_S 2π 2 1 + x2 1 − x 8I0+ 8 ln(1 − x) − 2 ln 2_{(x) + 4(1 + x) ln(x)} . (3.9) We see that Bx diagram features a single logarithmic IR singularity in form of the I0 term. The

detailed analysis of soft singularity structure using the exclusive distributions before phase space integration will be done in a separate section3.3.

In the M S scheme the bare PDF (kernel) contribution of Bx graph reads: ΓBx= 1 4 C_F2 −1 2CACF α_S 2π 2 1 + x2 1 − x 8I0+ 8 ln(1 − x) − 2 ln 2_{(x) + 4(1 + x) ln(x)} , (3.10) and PBx(x) = C_F2 −1 2CACF α_S 2π 2 1 + x2 1 − x 4I0+ 4 ln(1 − x) − ln 2_{(x) +2(1+x) ln(x)} . (3.11) Both kernel contributions in the MC and M S factorization schemes are the same, up to an overall factor 2. The overall factor 2 is a matter of convention in the definition of the kernel (depending if we define it in the MC scheme as_{∂ ln Q}∂ Gor _{∂ ln Q}∂ 2G) and is not important. What

is important is that the inclusive kernels in both schemes are the same. This holds for all the diagrams contributing to the NLO kernel, which do not have any internal singularities!

16_{The same trick does not work when the considered diagram features internal singularity. It is because, then}

some lower cutoff is present and one will not get nice integration limits (0, ∞), as in the case of single logarithmic divergent diagrams. In case of dimensional regularization, combining the phase space is not possible because of additional y

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3.1.2 Double ladder diagrams - Br and its counterterm

Now we analyze the case of the double bremsstrahlung ladder diagram Br and its countert-erm.17 For this diagram the MC and M S factorization schemes are significantly different. First, we perform the integration in the MC scheme and then do it in the M S scheme. When we stay in 4 dimensions (MC scheme) it is much more convenient to calculate the difference between the double bremsstrahlung diagram and its counterterm since this difference is free from dou-ble logarithmic singularity18. Lack of the double logarithmic singularity is present because in 4 dimensions we use the modified prescription ofP0 operator (according to the Monte Carlo approach, see Sec.1.5), instead of the CFP’s P operator. The P0 operator assures the proper cancellation of the double logs, in fact counterterm in 4 dimensions includes all the double logarithmic parts of the double bremsstrahlung diagram, contrary to the counterterm in M S scheme, which introduces oversubtractions and features both double and single poles.

The γ-traces for double ladder diagram Br and its counterterm are given by: WBr = 1 4(qn) 1 q4 1 (p − k1)4 Trh/pγα(/p − /k1)γµ/q /n/qγν(/p − /k1)γβ i dαβ(k1)dµν(k2) (3.12) and Wct = 1 4(qn) 1 q4Tr h / q1γα/q /n/qγβ i dαβ(k2) q2 1=0 1 4(q1n) 1 q4 1 Tr / pγµq/1n //q1γν dµν(k1). (3.13)

The exclusive kernel contributions of these graphs in the MC scheme read: ˜ WBr= 4C α_S 2π 2 a2 1a22 ˜ q4_(a 1, a2) T0+ T1 a1a2 a2₁ + T2(0) a2₂ a2₁ (3.14) and ˜ Wct = 4C α_S 2π 2 α2 1 (1 − α1)2 T2(0), (3.15) where: T0 = 1 + x2+ (1 − α1)2, T1= 2 1 − α1 α1 (1 + x2+ (1 − α1)2), T2() = T2+ T20, T2= 1 α2 1 1 + (1 − α1)2 x2+ (1 − α1)2 , T₂0 = 1 α2₁α 2 1 x2+ (1 − α1)2 + α22 1 + (1 − α1)2 , (3.16)

and the color factor for both diagrams is equal to C = C2

F. Note that the most singular term T2

is a product of two LO kernels T2() = α2 α1 (1 − α1)Pqq(0)(z1, )Pqq(0)(z2, ), P_qq(0)(z, ) ≡ 1 + z 2_{+ (1 − z)}2 1 − z = P (0) qq (z) + Pqq0(0)(z), (3.17)

17_{Bremsstrahlung diagram Br is specific as it is not a 2PI graph, it is a square of the kernel K}2

0 and because of it,

it is accompanied by a factorization counterterm.

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where z1 = 1−α1and z2= (1−α1−α2)/(1−α1). When we are in 4 dimensions the counterterm

˜

Wctis also proportional to this product of LO kernels:

˜ Wct= 4C α_S 2π2 2 α₁α₂ 1 − α1 P_qq(0)(z1)Pqq(0)(z2). (3.18)

The contribution of the Br diagram and the counterterm to the PDF is following: GBr = Z dΦδ (1 − x − α1− α2) ˜WBr Θ (Q > max{a1, a2} > q0) (3.19) and Gct= Z dΦδ (1 − x − α1− α2) ˜WctΘ (Q > a2 > q0) Θ (a2 > a1) . (3.20)

Let us stress the difference in the integration limits caused by the additionalP0operator of the counterterm19. There is no limit for a1variable from below (Θ(a1 > q0)) because there is

noth-ing to be regularized when we have difference “Br − Ct”. Actually, theP0operator acts only on objects which are single logarithmic divergent:P0(K0(1−P0)K0) 6=P0K02−P0(K0P0K0), which

means that in principle in our 4-dimensional prescription we cannot calculate separately Br di-agram and its counterterm, we need to take it as a whole such that theP0 operator is always acting on single divergent object like K0(1 −P0)K0.

The subtracted bremsstrahlung contribution takes the following form: GBr−ct = Z dα1 α1 dα2 α2 δ(1 − x − α1− α2) Z ∞ 0 da1 a1 Z ∞ 0 da2 a2 1 2π Z 2π 0 dφ × ˜ WBr(a1/a2, α1, α2, φ)Θ (Q > max{a1, a2} > q0) − ˜Wct(α1, α2)Θ (Q > a2 > q0) Θ (a2 > a1) , (3.21) where the arguments in the differential distributions of Eqs. (3.14) (3.15) are written explicitly. The above expression for the subtracted bremsstrahlung diagram features only single loga-rithmic divergence, which is regularized by the cutoff q0. Now, by introducing an additional

integration variable Θ Q > max{a1, a2} ≡ R₀Qd ˜Q δ ˜Q − max{a1, a2} the logarithm

repre-senting this singularity is extracted. First, the a1 integration region is divided into two parts:

a1 < a2 and a1 > a2. Due to the Θ (a2 > a1)function counterterm does not contribute to the

second region and we obtain: GBr−ct = Z dα1 α1 dα2 α2 δ(1 − x − α1− α2) 1 2π Z 2π 0 dφ Z Q 0 d ˜Q × Z Q 0 da2 a2 Z a2 0 da1 a1 δ( ˜Q − a2) ˜ WBr(a1/a2, α1, α2, φ) − ˜Wct(α1, α2) Θ (Q > a2 > q0) + Z Q 0 da1 a1 Z a1 0 da2 a2 δ( ˜Q − a1) ˜WBr(a1/a2, α1, α2, φ) Θ (Q > a1> q0) . (3.22)

19_{In case of the CFP scheme the phase space limits for the counterterm are different, which causes}

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The delta and theta functions are integrated out and the dimensionless variables: yi = ai/ ˜Qare introduced: GBr−ct = Z _dα 1 α1 dα2 α2 δ(1 − x − α1− α2) 1 2π Z 2π 0 dφ Z Q q0 d ˜Q ˜ Q × Z 1 0 dy1 y1 ˜ WBr(y1, α1, α2, φ) − ˜Wct(α1, α2) + Z 1 0 dy2 y2 ˜ WBr(1/y2, α1, α2, φ) . (3.23)

In this way, the single logarithm, representing the overall scale singularity, is extracted in the form of integralR_qQ

0

d ˜Q ˜

Q = ln(Q/q0). The internal singularity of Br diagram, when one of the

gluons is collinear, is cancelled by the counterterm and we obtain the final finite object. The integrals over the transverse degrees of freedom (φ, yi) are performed providing:

GBr−ct= ln (Q/q0) 4C α_S 2π 2Z dα₁ α1 dα2 α2 δ(1 − x − α1− α2) × T2(0) 2 − c1c2 2c2₂(c1c2− 1) + 1 2c2₂ ln c2₂ c1c2− 1 + T0 α1α2 2x − T1 α2₁α2 2x(1 − α1) , (3.24)

where: c1= 1−α_α₂2 and c2= 1−α_α₁1, and the α-integral reads:

GBr−ct = ln (Q/q0) CF2 α_S 2π 2 1 + x2 1 − x −8I0− 8 ln(1 − x) + 4 ln 2_(x) + (1 − x) (6 − 2 ln(x)) + (1 + x) (ln(x) − 2) ln(x) , (3.25)

or taking the derivative with respect to ln Q we obtain the inclusive DGLAP kernel contribu-tion: PBr−ct= CF2 α_S 2π 2 1 + x2 1 − x −8I0− 8 ln(1 − x) + 4 ln 2_(x) + (1 − x) (6 − 2 ln(x)) + (1 + x) (ln(x) − 2) ln(x) . (3.26)

As in the case of Bx diagram one can see that apart from the overall scale singularity encap-sulated in ln(Q/q0)there is also an infra-red (IR) divergent term represented by I0. The detail

analysis of IR structure of the separate diagrams Br, Ct, Bx and their sum, on both exclusive and inclusive level, is given in Sec.3.3.

3.1.3 Double ladder diagrams in n dimensions

For the sake of comparison we are going to reproduce also the CFP calculations in n sional prescription and determine the differences with the MC scheme. First of all, in n dimen-sional CFP scheme the difference Br - Ct features double pole due to the oversubtractions embedded into it. In CFP Br and Ct are integrated separately.

First, we perform the calculation for Br graph. The structure of Br distribution is the same as in 4 dimensions, see Eq. (3.14). The only difference is due to the terms from traces of gamma matrices and from the phase space. The additional terms from the T2(0) → T2() = T2+ T20

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The calculation of the kernel contribution of the Br diagram to the bare PDF in n dimensions starts from the following:

ΓBr =PP ( α_S 2π 24C2 F µ4 Ω1+2 (2π)2+4 Z dα1 α1 dα2 α2 (α1α2)2δ(1 − x − α1− α2) × Z dΩ(12)₁₊₂ Z ∞ 0 da1da2 (a1a2)1+2 ˜ q4_(a 1, a2) T0+ T1 a2 a1 cos θ + T2() a2₂ a2 1 θ (Q − max{a1, a2}) ) . (3.27) The overall pole is extracted in the same manner as in Sec.2.1, see Eq. (2.4):

ΓBr=PP ( α_S 2π 2C2 F Q4 µ4 π (2π)1+4_{Γ(1 + )} Z dα1 α1 dα2 α2 (α1α2)2δ(1 − x − α1− α2) × Z dΩ(12)₁₊₂ Z 1 0 dy1 y₁2 ˜ q4_(y 1, 1) T0y1+ T1cos θ + T2() 1 y1 + Z 1 0 dy2 y₂2 ˜ q4_{(1, y} 2) T0y2+ T1y22cos θ + T2()y23 ) . (3.28)

Next, the integration over angles and yivariables are performed. A special attention is paid to

the part proportional to T2coefficient. This term leads to the double pole and must be evaluated

in n dimensions, so that all the terms are correctly taken into account. ΓBr = α_S 2π 2C2 F Z dα1 α1 dα2 α2 δ(1 − x − α1− α2) × T0 α1α2 2x − T1 α2₁α2 2x(1 − α1) + T2 α3₁α2 2x(1 − α1)2 + α 2 1 2(1 − α1)2 ln α2(1 − α1) 2 α1x − 1 +αS 2π 2C2 F 2 Z _dα 1 α1 dα2 α2 δ(1 − x − α1− α2) × α 2 1 (1 − α1)2 T2 1 + 2 ln Q2 4πµ2 + 2γ + 2 ln(α1α2) + T₂0 . (3.29) Finally, the α-integrals are performed using the P V prescription for IR singularities:

ΓBr= α_S 2π 2 C2 F 2 × 1 2+ ln Q2 4πµ2 + γ 81 + x 2 1 − x (I0+ ln(1 − x)) − 4(1 − x) + 2(1 + x) ln(x) + (1 − x) (4I0− 4 ln(1 − x) − ln(x) + 3) + (1 + x) 2π2 3 − 3 ln(x) + 3 2ln 2_{(x) − 4Li} 2(x) +1 + x 2 1 − x 8I1− 4I0+ 8I0ln(1 − x) − 4 ln(1 − x) + 12 ln2(1 − x) + 2 ln2(x) − 4 3π 2 . (3.30)

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The factorization counterterm depicted by graph of Fig. 4 also features double pole. The counterterm features an additionalP operator, which eliminates partly some of the terms from γ-traces. The additionalP operator also decouples spinor indices of the two parts of the graph and simplifies the kinematics, for instance, the angular integrals become trivial. The integration over the angular dependence leads to:

Γct= CF2g4PP Ω1+2 (2π)3+2 1 µ2 Z _dα 2 α1−2₂ Z Q 0 da2 a1−2₂ 2 − 4α1− 2α2+ 2α1α2+ 2α21+ α22+ α22 1 − α1 × PP Ω1+2 (2π)3+2 1 µ2 Z _dα 1 α1−2₁ Z Q1 0 da1 a1−2₁ α1 1 − α1 2 α1 − 2 + α1+ α1 δ (1 − x − α1− α2) , (3.31) where the limits on ai integrals are set separately, since there are twoP operators. The upper

limit on the internal integration over a1 does not matter as the dependence on Q1is eliminated

by additional pole-part operator. The integrals over aivariables are done and the action of the

internal PP operator provides us with: Γct=PP C2 Fg4 42 Ω1+2 (2π)3+2 Q2 µ2 Z _dα 2 α1−2₂ 1 1 − α1 2 − 4α1− 2α2+ 2α1α2+ 2α21+ α22+ α22 × 1 (2π)2 Z dα1 α1 α1 1 − α1 2 α1 − 2 + α1 δ (1 − x − α1− α2) . (3.32) The last two steps in calculation are the expansion and the α integration:

Γct= α_S 2π 2 C2 F 1 + ln Q2 4πµ2 + γ 41 + x 2 1 − x (I0+ ln(1 − x)) − 2(1 − x) + (1 + x) ln(x) +1 + x 2 1 − x 4I1+ 4I0ln(1 − x) + 6 ln2(1 − x) − 2 3π 2 − 2 ln(x) − (1 + x) 2Li2 x − 1 x − 2 ln(x) ln(1 − x) + (1 − x)2I0− 2 ln(1 − x) − ln(x) . (3.33) The NLO contribution to the bare PDF of CFP is obtained after subtracting the counterterm from the Br diagram:

ΓBr−ct= C_F2 2 α_S 2π 2 − 1 2 81 + x 2 1 − x (I0+ ln(1 − x)) − 4(1 − x) + 2(1 + x) ln(x) +1 + x 2 1 − x − 4I0− 4 ln(1 − x) + 2 ln2(x) + 3(1 − x) (1 + ln(x)) − (1 + x) 1 2ln 2_{(x) + ln(x)} . (3.34) The CFP kernel contribution, calculated by taking twice the residue of Γ, is following:

PBr−ct= α_S 2π 2 C_F2 1 + x 2 1 − x − 4I0− 4 ln(1 − x) + 2 ln2(x) + 3(1 − x) (1 + ln(x)) − (1 + x) 1 2ln 2_{(x) + ln(x)} . (3.35)