aa r X i v : . [ h e p - ph ] S e p UNIVERSIT `A DEGLI STUDI DI MILANOFacolt`a di Scienze Matematiche Fisiche e NaturaliDipartimento di FisicaCorso di Dottorato di Ricerca in Fisica, Astrofisica e Fisica ApplicataCiclo XIXTESI DI DOTTORATO DI RICERCA:
Sudakov resummation in QCD
Settore scientifico: FIS/02
Tutor: Prof. Stefano FORTEReferee: Prof. Giovanni RIDOLFICoordinatore: Prof. Gianpaolo BELLINITesi di dottorato di:Paolo BOLZONIMatr. R05430Anno Accademico 2005-2006
Sudakov resummation in QCD
Paolo Bolzoni
Abstract
In this PhD thesis, we analyze and generalize the renormalization group approachto the resummation of large logarithms in the perturbative expansion due to softand collinear multiparton emissions. In particular, we present a generalization of thisapproach to prompt photon production. It is interesting to see that also with themore intricate two-scale kinematics that characterizes prompt photon production inthe soft limit, it remains true that resummation simply follows from general kine-matic properties of the phase space. Also, this approach does not require a separatetreatment of individual colour structures when more than one colour structure con-tributes to fixed order results. However, the resummation formulae obtained hereturn out to be less predictive than previous results: this depends on the fact thathere neither specific factorization properties of the cross section in the soft limit isassumed, nor that soft emission satisfies eikonal-like relations. We also derive resuma-tion formulae to all logarithmic accuracy and valid for all values of rapidity for theprompt photon production and the Drell-Yan rapidity distributions. We show thatfor the fixed-target experiment E866/NuSea, the NLL resummation corrections arecomparable to NLO fixed-order corrections and are crucial to obtain agreement withthe data. Finally we outline also possible future applications of the renormalizationgroup approach. udakov resummation in QCD
Paolo Bolzoni ontents
Introduction v1 General aspects of perturbative QCD 1 v CONTENTS q ⊥ singularities of soft gluon contributions . . . . . . . . . . . . . 946.4 The resummed exponent in renormalization group approach . . . . . 986.5 Logs of q ⊥ vs. logs of b to all logarithmic orders . . . . . . . . . . . . 1026.6 Resummation in q ⊥ -space . . . . . . . . . . . . . . . . . . . . . . . . 104 N and logarithms of (1 − z ) D.1 Proof of Eq.(42) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135D.2 Proof of Eq.(43) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
E Proof of combinatoric properties of chapter 7 139Acknowledgements 147 ntroduction
The current theory that describes the strong subnuclear interactions is the non-abeliangauge theory of the SU(3) local symmetry group. This theory, called QuantumChromo-Dynamics (QCD), has been tested to great accuracy in many experiments.Its perturbative regime will be crucial at the imminent high-energy proton-protoncollider LHC. The main target of LHC, is to find signals of the Standard Model Higgsboson and of new physics, i.e. supersymmetry, new interactions predicted by greatunified theories, extra-dimensions, new gauge bosons and mini-black holes. In orderto accomplish all this, an excellent understanding of QCD is necessary, both becauseLHC is a proton collider and because QCD backgrounds must be described accurately.In perturbative QCD (pQCD), it is well known that, when one approaches tothe boundary of the phase space, the cross section receives logarithmical-enhancedcontributions of soft and collinear origin at all orders. These large tems at order O ( α ns ) of the strong coupling constant in pQCD have, in general, the form α ns (cid:20) log m (1 − z )1 − z (cid:21) + , m ≤ n − , where z is a parameter that becomes close to 1 near to the phase space boundaryand “+” denotes the plus distribution. These terms become important in the limit z → z is given by the ratio of the searched Higgs mass over thecenter-of-mass energy of the partons in the colliding hadrons and the large logarithmsarise from soft-gluon emissions.These large terms have been resummed a long time ago for the classes of inclusivehadronic processes of the type of deep-inelastic scattering (DIS) and Drell-Yan (DY)[1, 2, 3]. Threshold resummation of inclusive processes can affect significantly crosssections and the extraction of parton densities [4, 5]. For the case of small transversemomentum distributions in Drell-Yan processes, it has been shown that resummationis necessary to reproduce the correct behavior of the cross section [6].These results has been obtained using the eikonal approaximation of Ref.[2] whichgeneralizes to QCD the Sudakov exponentiation of soft photons emissions in electro-dynamics [7] or assuming suitable factorization properties of the QCD cross section[1]. More recently two other approaches to resummation have been proposed. Thefirst is the renormalization group approach of Ref.[8] and the other is the effectivev i Introduction field theoretic (EFT) approach of Refs.[9, 10]. There the resummation of the largelogarithms for full inclusive deep-inelastic and Drell-Yan processes is obtained.In this thesis we will concentrate mostly on the study of the renormalization groupapproach of Ref.[8] and on its applications and generalizations. This approach has theadvantage of being valid to all logarithmic orders, and self-contained, in that it doesnot require any factorization beyond the standard factorization of collinear singular-ities. It relies on an essentially kinematical analysis of the phase space for the givenprocess in the soft limit, which is used to establish the result that the dependence onthe resummation variable only appears through a given fixed dimensionful combina-tion. This provides a second dimensionful variable, along with the hard scale of theprocess, which can be resummed using standard renormalization group techniques.Beyond the leading log level, the resummed result found within this approach turnsout to be somewhat less predictive than the result obtained with the other methods.In the other approaches references resummed results at a certain logarithmic accu-racy is fully determined by a ceratin fixed order computation, whereas a higher fixedorder computation is needed to determine all coefficients in the resummed formulaof Ref.[8]. The more predictive result is recovered within this approach if the depen-dence of the perturbative coefficients on the two dimensionful variables factorizes, i.e.if the two-scale factorization mentioned above holds.We shall show the generalization of the renormalization group approach to theresummation of the inclusive transverse momentum spectrum of prompt photonsproduced in hadronic collisions in the region where the transverse momentum is closeto its maximal value. Prompt photon production is a less inclusive process thanDrell-Yan or deep-inelastic scattering, and it is especially interesting from the pointof view of the renormalization group approach, because the large logs which must beresummed turn out to depend on two independent dimensionful variables, on top ofthe hard scale of the process: hence, prompt photon production is characterized bythree scales. The possibility that the general factorization Ref.[11] might extend toprompt photon production was discussed in Ref.[12], based on previous generaliza-tions [13] of factorization, and used to derive the corresponding resummed results.Resummation formulae for prompt photon production in the approach of Ref.[2] werealso proposed in Ref.[14], and some arguments which might support such resummationwere presented in Ref.[15]. Our treatment will provide a full proof of resummation toall logarithmic orders. Our resummation formula does not require the factorizationproposed in Refs.[12, 14], and it is accordingly less predictive. Because of the presenceof two scales, it is also weaker than the result of Ref.[8] for DIS and Drell-Yan pro-duction. Increasingly more predictive results are recovered if increasingly restrictiveforms of factorization hold.Moreover, we shall prove an all-logarithmic orders resummation formula for differ-ential rapidity of Drell-Yan and prompt photon production processes. The differentialrapidity Drell-Yan cross section is used for the extraction of the ratio ¯ d/ ¯ u of partondensities. The accurate knowledge of these functions is needed to study Higgs bosonproduction and the asymmetry W ± . The resummation of Drell-Yan rapidity distribu-tions was first considered in 1992 [16]. At that time, it was suggested a resummationformula for the case of zero rapidity. Very recently, thanks to the analysis of the full ii NLO calculation of the Drell-Yan rapidity distribution, it has been shown [17], thatthe result given in [16] is valid at next-to-leading logarithmic accuracy (NLL) for allrapidities. In this thesis, we shall give a simple proof of an all-order resummationformula valid for all values of rapidity. To do this, we will use the technique of thedouble Fourier-Mellin moments developed in [18]. In particular, we will show thatthe resummation can be reduced to that of the rapidity-integrated process, which isgiven in terms of a dimensionless universal function for both DY and W ± and Z production. Then, we implement numerically the resummation formula and give pre-dictions of the full rapidity-dependent NLL Drell-Yan cross section for the case of thefixed-target E866/NuSea experiment. We find that resummation at the NLL level isnecessary and that its agreement with the experimental data is better than the NNLOcalculation of Ref.[19]. In this case, we find also that the NLL resummation reducedthe cross section instead of enhancing it for the parameter choices of this experiment.Threshold corrections to Higgs, Z and W ± production rapidity distributions at highenergy hadron colliders have also been studied in [20, 21].Finally, we shall discuss the application of the renormalization group approachto the resummation another class of large logarithms that arise in Drell-Yan pro-cesses for small tranverse momentum distribution of the produces lepton pairs. Theselogarithmic-enhanced terms at order O ( α ns ) have, in general, the form α ns (cid:20) log m ( q ⊥ ) q ⊥ (cid:21) + , m ≤ n − , where q ⊥ is the transverse momentum of the produced Drell-Yan pair. Also in thiscase the resummed results using the renormalization group approach are less pre-dictive than results obtained with the approach of Ref.[1], as it is shown in Ref.[6].Furthermore the conditions that reduce our results to those of Ref.[6] in terms offactorization properties is still an interesting open question.This thesis is organized as follows. In Chapter 1, we review the basics conceptsof pQCD. In particular the construction of the QCD Lagrangian, the asymptoticfreedom of strong interactions, the structure of the cross section when initial statehadrons are present and the evolution equation of the parton densities. In Chapter2, we discuss the importance of resummation at hadron colliders. Then, we describehow in the various approaches the large logarithms are exponentiated and resummed.They are the renormalization group approach, the eikonal approximation approach,the approach of non standard factorization properties and the effective field theoreticapproach. In Chapter 3, we show in the detail the renormalization group approachto the resummation of all inclusive deep-inelastic and Drell-Yan processes and itsgeneralization to the prompt photon process at large transverse photon momentum isshown in Chapter 4. In Chapter 5, we prove the all-logarithmic orders resummationformula for the Drell-Yan and prompt photon production processes. We show alsothe impact of the NLL resummation for the DY process at the E866/NuSea experi-ment and discuss the numerical results. Then in Chapter 6, we turn to discuss thegeneralization of the renormalization group approach to resummation in the case ofthe small transverse momentum differential cross section for the Drell-Yan process.Finally, in the last Chapter, we summarize and determine the predictive power of the iii Introduction resummation formulae obtained with the different approaches, namely, the fixed-ordercomputation needed to determine completely a resummation formula for an arbitrarylogarithmic accuracy. hapter 1General aspects of perturbativeQCD
The quarks (the constituents of hadrons) are Dirac fermions. In the Standard Model(SM), as far as the electroweak interactions are concerned, the properties of quarksand leptons are similar. Indeed, as for the leptons, the six quark flavors are groupedinto three SU L (2) left-handed doublets (cid:18) ud (cid:19) L , (cid:18) cs (cid:19) L , (cid:18) tb (cid:19) L (1)and six SU (2) L singlets, which are the right-handed parts of each flavor. Both, quarksand leptons, interact in a similar way with the electroweak gauge bosons ( γ , W ± and Z ) of the group SU L (2) × U Y (1). The main difference is that each quark flavoreigenstate is a unitary mixing of the quark mass eigenstate, while, according to theSM, this is not the case for charged leptons and massless neutrinos. However, in thisdecade it has been proven that neutrinos have mass and that there is also mixing inthe neutrino sector thanks to observations of their oscillations.The peculiarity of quarks is that they have a specific property, the color charge,which is absent for leptons. Indeed, a quark of a given flavor has three different colorstates with equal masses and electroweak charges. The interaction of the quarksis mediated by the eight gauge bosons (gluons g ) of the color group SU C (3). So,the quarks belong to the fundamental representation of SU C (3) and the gluons tothe adjoint one. The gauge theory of this non-abelian group is called QuantumChromodynamics (QCD) and is the current theory of strong interactions.Specifically, a gauged SU C (3) transformation of a quark ( q a ( x ) with a = 1 , ,
3) isgiven by q a ( x ) → q ′ a ( x ) = U ab ( x ) q b ( x ) (2)¯ q a ( x ) → ¯ q ′ a ( x ) = ¯ q b U † ba ( x ) , (3)where the 3 × U ik ( x ) is the fundamental representation of the SU C (3) group1 General aspects of perturbative QCD that acts on an internal space defined at each space-time coordinate x . It satisfies U U † = U † U = 1 , det( U ) = 1 . (4)In this section the sum over all the repeated indices is implicit and the sum overspinor indices is omitted for brevity. The usual exponential representation of thegauge transformation matrix in terms of the basis of matrices (the generators of SU C (3)) of the corresponding algebra su C (3) is: U ( x ) = e − i ~α ( x ) · ~λ = e − ~α ( x ) · ~t , (5)where ~α ( x ) = ( α ( x ) , . . . , α ( x )) are the eight arbitrary parameters of the gauge trans-formation, ~λ are the eight elements of the basis of the algebra su C (2) (or equivalentlythe eight generators of the group) and ~t are the eight color operators (in analogy tothe spin operators of the group SU (2)). It is clear that, in order to respect Eq.(4),the generators of the group must be hermitian and traceless. The normalization ofthe color operators depends on the representation r tr( t Ar t Br ) = T r δ AB . (6)The form chosen by Gell-Mann for the su C (3) basis in the fundamental representation: λ = , λ = − i i , λ = − ,λ = , λ = − i i , λ = ,λ = − i i , λ = 1 √ − . (7)With these definitions, the color matrices satisfy the following relations[ t A , t B ] = if ABC t C (8)tr( t A t B ) = T F δ AB , T F = 12 (9)where T F is the normalization of the color matrices in the fundamental representationand f ABC are called the structure constants of the algebra su C (3) which are totallyantisymmetric in { A, B, C } . The independent non-vanishing structure constants aregiven by: f = 1 , f = 12 , f = − , f = 12 , f = 12 (10) f = 12 , f = − , f = √ , f = √ . (11) .1 Quarks, Gluons and QCD Furthermore, the structure constants provide the adjoint representation of the su C (3)algebra (the one which has the same dimension of the algebra). Indeed, if we definethe adjoint representation as T ABC = − if ABC , we can verify explicitly that this is theadjoint representatio because[ T A , T B ] = if ABC T C (12)tr( T A T B ) = T A δ AB , T A = 3 (13)The Casimir operator C r (the one which commutes with all elements of the algebra),for a certain representation r , is constructed as t Ar t Ar = C r d r × d r . (14)Now, since the contraction of this last equation is equal to the contraction of Eq.(6),we get d r C r = 8 T r . (15)In particular, we find the Casimir operators are given by C F = 43 , C A = 3 (16)for the fundamental and the adjoint representation respectively.We shall now show that the symmetry with respect to the gauge transformationsEqs.(2,5) (together with the Lorentz invariance), can be used as guiding principle toconstruct the QCD Lagrangian. We start from the usual Dirac free Lagrangian foreach quark mass and color eigenstate L D ( x ) = ¯ ψ fa ( x ) ( i / ∂ − m f ) ψ fa ( x ) , (17)where f is the flavor index and a is the color quark index. This term is not gaugeinvariant. In fact under the gauge transformations Eqs.(2,5), the Dirac free lagrangianEq.(17) transform as L D ( x ) → L D ( x ) + ¯ ψ fb h iU † ba ( x ) ∂ µ U ac ( x ) i γ µ ψ fc ( x ) . (18)To restore gauge invariance, we introduce a gauge field matrix A µ ab ( x ) made up ofeight gauge fields A Aµ ( x ) in this way: A µ ab ( x ) = t Aab A Aµ . (19)We assign to this matrix field the following interaction Lagrangian L I ( x ) = g s ¯ ψ fa A µ ab γ µ ψ fb . (20)Here g s is the gauge dimensionless coupling analogous to the electric charge in QED.In order to cancel the symmetry breaking term of Eq.(18), the gauge transformationof the field matrix A µ ab ( x ) has to be A µ ab ( x ) → U ac ( x ) A µ cd ( x ) U † db ( x ) − ig s ∂ µ U ac ( x ) U † cb ( x ) . (21) General aspects of perturbative QCD
Hence, with the introduction of the field matrix A µ ab , the sum L D + L I is now gaugeinvariant. To complete the Lagrangian one has to add the pure gauge invariant term,which is L G ( x ) = −
12 tr ( G µν ( x ) G µν ( x )) , (22)where the gluon field-strength tensor is given by G µν ab ( x ) = ∂ µ A ν ab − ∂ ν A µ ab − ig s [ A µ , A ν ] ab . (23)Note that the third term of the gluon field-strength gives rise to the self interactionsof gluons and that its origin stands in the fact that the gauge group SU C (3) is non-abelian. The final form of the QCD classical Lagrangian is obtained by adding thethree pieces introduced above and using Eq.(9): L cl QCD ( x ) = − G Aµν ( x ) G A µν ( x ) + ¯ ψ fa ( x )( iD µ ab γ µ − m f δ ab ) ψ fb ( x ) , (24)where G Aµν = ∂ µ A Aν − ∂ ν A Aµ + g s f ABC A Bµ A Cν (25)and D µab = δ ab ∂ µ − ig s A µ ab (26)is the covariant derivative in the sense that D µ ab ψ fb ( x ) transforms as ψ fa ( x ) underthe gauge group.The quantization of the classical gauge invariant theory of QCD Eq.(24), can bedone with the Fadeev-Popov procedure. This procedure takes care of the fact that theequation of motion of the gluon field A Aµ can not be inverted and this prohibits to findthe propagator. However, this is a consequence of gauge invariance which implies thatthe physical massless gluon has only two polarizations/spin states whereas the field A Aµ has four components. To make things work, an additional constraint on the gluonfield is introduced, the so called gauge fixing condition which uses gauge invariance todefine properly the gluon propagator. In QCD, however, this constraint is not linearand one should add specially designed fictitious particles (the so called Fadeev-Popovghosts) which are Lorentz scalar anti-commuting fields and appear only in the loops.After this procedure (usually performed in the functional formalism), we have thatthe QCD Lagrangian from which we can calculate directly the Feynman rules is givenby (see e.g. [22] page 514): L F PQCD ( x ) = L clQCD ( x ) − λ ( ∂ µ ∂ ν A Aµ ( x ) A Aν ( x )) − ¯ c A ∂ µ D ACµ c C , (27)where λ is the gauge fixing parameter, c A is the complex colored scalar ghost fieldand D ABµ is the covariant derivative in the adjoint representation: D ABµ = δ AB ∂ µ − g s f ABC A Cµ . (28)The quark, gluon and ghost propagators and vertices for QCD are collected in Figure1.1. .1 Quarks, Gluons and QCD Figure 1.1:
Feynman rules for QCD in a covariant gauge for gluons (curly lines), quarks (solidlines) and ghosts (dotted lines). Here
A, B, C, D are the color indexes in the adjoint representation, a, b, c in the fundamental one, α, β, γ, δ are the gluon polarization indexes and i, j are the spinorialindexes.
General aspects of perturbative QCD
Feynman diagrams in QCD are obtained employing the vertices and propagators asbuilding blocks. However, the use of diagrams makes sense only if the perturbativeexpansion in g s is meaningful. To respect this condition, the coupling α s = g s π , (29)the QCD analog of the electromagnetic coupling α = e / π , has to be sufficientlysmall. We shall now show that the method of perturbation theory in QCD are usefulat high energy. Indeed, the coupling constant is large at low energy and becomessmaller at high energy (asymptotic freedom).The simplest way to introduce the running coupling, is to consider a dimensionlessphysical observable R which depends on a single energy scale p Q . This is the case,for example of the ratio of the annihilation cross section of electron-positron intohadron with the annihilation into muons where Q = S the center-of-mass energy.We assume that this scale is much bigger than the quark masses that can be thereforeneglected. Now, dimensional analysis should implies that a dimensionless observableis independent of Q . However, higher order corrections produce divergences andso the perturbation series requires renormalization to remove ultraviolet divergencesthat in d = 4 − ǫ dimensions are regularized as 1 /ǫ poles. This poles can be re-moved defining a renormalized coupling constant at a certain renormalization scale µ r . Consequently, in the finite ǫ → R depends in general on the ratio Q /µ r and the renormalized coupling α s depends on µ r ; we call the renormalized couplingat the scale µ r , α µ r . Since the renormalization scale is an arbitrary parameter intro-duced only to define the theory at the quantum level, we conclude that R has to be µ r -independent. Formally this independence is expressed as follows: µ r ddµ r R (cid:18) Q µ r , α µ r (cid:19) = (cid:20) µ r ∂∂µ r + β ( α µ r ) ∂∂α µ r (cid:21) R = 0 , (30)where β ( α µ r ) = µ r ∂α µ r ∂µ r (31)Eq.(30) is a first order differential equation with the initial condition (at Q = µ r ) R (1 , α µ r ). This means that if we find a solution of Eq.(30), it is the only possiblesolution. This solution is easily found defining a function α s ( Q ) such that α s ( µ r ) = α µ r (32)and that ln (cid:18) Q µ r (cid:19) = Z α s ( Q ) α µ r dxβ ( x ) . (33)In fact differentiating this equation, we find that Q ∂α s ( Q ) ∂Q = β ( α s ( Q )) (34) ∂α s ( Q ) ∂α µ r = β ( α s ( Q )) β ( α µ r ) (35) .2 Asymptotic freedom and perturbative QCD and that R (1 , α s ( Q )) is the desired solution of Eq.(30), because ∂∂α µ r = ∂α s ( Q ) ∂α µ r ∂∂α s ( Q ) , µ r ∂∂µ r = − Q ∂∂Q = − Q ∂α s ( Q ) ∂Q ∂∂α s ( Q ) . (36)This shows that all of the scale dependence in R enters through the running of thecoupling constant α s ( Q ). To find explicitly this function, we need to know the β -function so that we can solve Eq.(34). The β -function can be calculate perturbativelyfrom the counterterms of the renormalization precedure and a knowledge to order α n +1 s requires a n -loop computation. The perturbative expansion of the β -function is givenby: β ( α s ) = − α s ∞ X n =0 β n (cid:16) α s π (cid:17) n +1 . (37)At the moment, the QCD β -function is known to order α s [23]: where in the M S scheme, β = 11 − N f , β = 102 − N f , (38) β = 28572 − N f + 32554 N f (39) β = (cid:18) ξ (cid:19) − (cid:18) ξ (cid:19) N f + (cid:18) ξ (cid:19) N f + 1093729 N f , (40)with N f the numbers of flavors and ξ is the Riemann zeta-function ( ξ = 1 . . . . ).The two loop solution of Eq.(34) is given by: α s ( Q ) = α s ( µ r )1 + ( β / π ) α s ( µ r ) log Q µ r (cid:20) − β πβ α s ( µ r ) log(1 + ( β / π ) α s ( µ r ) log Q µ r )1 + ( β / π ) α s ( µ r ) log Q µ r (cid:21) + O ( α k +3 s log k Q µ r ) . (41)For simplicity, in many cases, we will use another parametrization of the coeffi-cients β n , which is obtained with the substitution: β n = b n (4 π ) n +1 . (42)With this parametrization, the perturbative expansion of the β -function Eq.(37) be-comes β ( α s ) = − ∞ X n =0 b n α n +2 s . (43)From Eq.(41), we see that α s ( Q ) is a monotonically decreasing function of Q , be-cause the coefficients β and β are positive (with N f ≤ α s ( Q )has been measured with great accuracy (see Figure 1.2). The fact that at high energy, General aspects of perturbative QCD
Figure 1.2:
The running of the strong coupling constant. The asymptotic freedom is confirmedby the experiments the running coupling becomes small is a peculiarity of non-abelian gauge theories andis called asymptotic freedom.Hence, perturbative QCD can be applied when the relevant scale of a certain pro-cess is high enough such that the running coupling becomes small. A typical exampleis given by the annihilation of two high-energy electrons into hadrons. PerturbativeQCD can also be applied to processes in which hadrons are present also in the initialstate thanks to the factorization theorem. According to this theorem [11], the crosssection for the production of some final state with high invariant mass Q (the scaleat which the running coupling constant is small) with two incoming hadrons is givenby: σ ( P , P , Q ) = X a,b Z dx dx F H a ( x , µ ) F H b ( x , µ )ˆ σ ab ( x P , x P , α s ( µ ) , Q , µ ) . (44)For processes with a single incoming hadron the factorization theorem is simpler. Forexample for the deep inelastic scattering (DIS) of a lepton that exchanges a highsquare momentum Q with the hadron, the cross section takes the form: σ ( P, Q ) = X a Z dxF Ha ( x, µ )ˆ σ a ( xP, α s ( µ ) , Q , µ ) . (45)In Eqs.(44,45), P i is the momentum of the incoming hadron H i . A beam of hadronsof type H i is equivalent to a beam of constituents (or partons) which are quarks .3 NLO DY and DIS cross sections or gluons. These constituents carry a longitudinal momentum x i P i characterized bythe parton densities F H i a ( x i , µ ). That is, given a hadron H i with momentum P i ,the probability density to find in H i the parton a with momentum x i P i is given by F H i a ( x i , µ ). Furthermore, these functions are universal in the sense that they areprocess independent. The parton densities depends also on the so called factorizationscale µ . This scale is introduced to separate off the non-perturbative part of the crosssection (the parton densities) from the perturbative one ˆ σ a ( b ) . This is exactly the crosssection where the incoming particles are the partons a (and b ) and can be calculated asa perturbative expansion in α s ( µ r ). The parton densities have a mild dependence onthe scale µ determined by the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP)equations (see section 1.4). Here, we have chosen the renormalization scale µ r equal tothe factorization scale µ for simplicity. Anyway, in order to reintroduce the scale µ r ,we have only to rewrite α s ( µ ) in terms of µ r (see Eq.(41)) and expand it consistentlywith the order of the calculation. The µ dependence in the parton densities iscompensated by the µ dependence in the partonic cross section ˆ σ . However, with afixed-order computaction of the partonic coefficient function at order α ks the hadroniccross section will still depend on µ with a dependence which should be of order α k +1 s .Hence, this dependence can be used to estimate the theoretical error of a fixed-ordercomputation. A simple discussion about the dependence of the hadronic cross sectionon the factorization and renormalization scale is given in Ref.[24]. We consider for simplicity the classical Drell-Yan (DY) hadronic process for the pro-duction of a dimuon pair through a virtual photon γ ∗ (see Figure 1.3): H ( P ) + H ( P ) = γ ∗ ( Q ) + X ( K ) , (46)where H and H are the colliding hadrons with momementum P and P respectively, Q is the momentum of the virtual photon and X is any number of additional hadronswith total momentum K . For the process of Eq.(46), we define x ≡ Q S , (47)where S = ( P + P ) is the usual Mandelstam invariant, which can be viewed asthe hadronic center-of-mass energy. It is clear that Eq.(47) represents the fraction ofenergy that the hadrons transfer to the photon and, hence, 0 ≤ x ≤ Q differential cross section is given by: dσdQ ( x, Q ) = X i Z dx dx (cid:2) q i ( x , µ )¯ q i ( x , µ ) + ¯ q i ( x ) q i ( x , µ )) (cid:3) d ˆ σ i dQ , (48) d ˆ σ i dQ = σ DY ( Q , x ) Q q i δ ( x x − x ) , σ DY ( Q , x ) = 4 πα Q x (49) General aspects of perturbative QCD
Figure 1.3:
Drell-Yan pair production. Here Q = M . where the functions q i ( x j , µ )(¯ q i ( x j , µ ))are the parton densities of a quark (or ananti-quark) of flavor i in the hadron j = 1 , µ , α is the fine-structureconstant and Q q i is the fraction of electronic charge of the quark q i . Now, if we definethe dimensionaless cross section σ ( x, Q ) as, σ DY ( x, Q ) ≡ σ DY dσdQ ( x, Q ) , (50)and use the identity, δ ( x x − x ) = Z dzδ (1 − z ) δ ( x x z − x ) , (51)then Eqs.(48,49) become: σ DY ( x, Q ) == X i Z dx dx dz [ q i ( x )¯ q i ( x ) + ¯ q i ( x ) q i ( x )] Q q i C qq ( z ) δ ( x x z − x ) (52)= X i Z x dx x Z x/x dx x [ q i ( x )¯ q i ( x ) + ¯ q i ( x ) q i ( x ))] Q q i C qq (cid:18) xx x (cid:19) , (53) C qq ( z ) = δ (1 − z ) , (54)where C qq ( z ) is the LO Drell-Yan coefficient function. From Eq.(52), we see that thenew variable z that we have introduced is in general given by z = xx x . (55)This means that at the partonic level, z can be viewed as the fraction of energy thatthe colliding partons transfer to the virtual photon. At LO it is clear that z = 1as can be explicitely seen from Eq.(54), because there is no emission but the virtualphoton. .3 NLO DY and DIS cross sections Beyond the LO the extra radiated partons in the final state can carry away someenergy (so z <
1) and the gluon channel contributes. The NLO coefficient func-tions C ab ( z ) ( a, b = q, g ) recieves contributions that have infrared and ultraviolet.Infrared singularities cancel out (see e.g. [25]). The ultraviolet ones are reabsobedby renormalization of the bare parameters of the QCD Lagrangian, thus defining arenormalized strong coupling constant α µ r at an arbitraty renormalization scale µ r (see section 1.2). Collinear divergences are cut off by infrared physics. They can beabsorbed multiplicatively in redefinition of the parton densities [26], thus reabsorbingall dependence on soft physics in the parton distributions. The parton densities at acertain scale are determined by a reference process and their scale dependence is de-termined by the DGLAP equations (see section 1.4). However, there is an ambiguityon how to define the reference process, related to the fact that collinear divergencescan always be factorized together with finite terms. The choice of these finite termsdefines a factorization scheme. The most common factorization scheme is the M S scheme in which the collinear divergence (which is in d = 4 − ǫ dimensions a singlepole 1 /ǫ ) is factorized together the finite terms − γ E + log 4 π , where γ E = 0 . ... is the Euler gamma. Now, in order to avoid the perturbative expansion to receivelarge contributions, the factorization and the renormalization scales are expected tobe chosen of the same order of the scale of the process Q . Here, for simplicity, wechoose the factorization scale µ equal to the renormalization scale µ r . We report theNLO Drell-Yan cross section (see e.g. [27, 28]): σ DY ( x, Q ) = X i Q q i Z x dx x Z x/x dx x × (cid:26) (cid:2) q i ( x , µ )¯ q i ( x , µ ) + (1 ↔ (cid:3) C qq (cid:18) z, Q µ , α s ( µ ) (cid:19) + (cid:2) g ( x , µ ) (cid:0) q i ( x , µ ) + ¯ q i ( x , µ ) (cid:1) + (1 ↔ (cid:3) C qg (cid:18) z, Q µ , α s ( µ ) (cid:19) (cid:27) , (56)where, in the M S scheme, C qq (cid:18) z, Q µ , α s ( µ ) (cid:19) = δ (1 − z ) + α s ( µ )2 π (cid:26) (cid:20) (cid:18) π − (cid:19) δ (1 − z )+4(1 + z ) (cid:20) log(1 − z )1 − z (cid:21) + − z − z log z (cid:21) + 83 (cid:20) z [1 − z ] + + 32 δ (1 − z ) (cid:21) log (cid:18) Q µ (cid:19) (cid:27) , (57)and C qg (cid:18) z, Q µ , α s ( µ ) (cid:19) = α s ( µ )2 π (cid:26) (cid:20) ( z + (1 − z ) ) log (1 − z ) z + 12 + 3 z − z (cid:21) + 12 [ z + (1 − z ) ] log (cid:18) Q µ (cid:19) (cid:27) , (58) General aspects of perturbative QCD
Figure 1.4:
Deep inelastic electron-proton scattering where the “+” distribution is defined as follows: Z dzf ( z )[ g ( z )] + ≡ Z dz [ f ( z ) − f (1)] g ( z ) . (59)Also for the case of the deep-inelastic scattering (DIS), we consider the simplestprocess in which a high energy electron scatters from a hadron exchanging with it avirtual photon γ ∗ (see Figure 1.4): H ( P ) + e ( k ) → e ( k ′ ) + X ( K ) , (60)where H is typically a proton with momentum P , e is the scattered electron and X is any collection of hadrons. The standard parametrization of DIS is done in termsof three relevant parameters: Q ≡ − q ≡ − ( k − k ′ ) (61) y ≡ P · qP · k ; 0 ≤ y ≤ x ≡ x Bj = Q P · q = Q ( P + q ) + Q ; 0 ≤ x ≤ , (63)where in the last line we have neglected the proton mass. Q is the virtuality of thephoton exchanged between the electron and the proton and y is the fraction of energythat the incoming electron transfer to the proton. The Bjorken scaling variable x hasa simple physical interpretation: it is the fraction of longitudinal momentum of theLO incoming quark of the partonic subprocess.Indeed, the most general parametrization of the Q differential cross section isgiven by: dσdQ ( x, Q , y ) = 4 πα Q (cid:20) [1 + (1 − y ) ] F ( x, Q ) + (1 − y ) x ( F ( x, Q ) − xF ( x, Q )) (cid:21) , (64) .3 NLO DY and DIS cross sections The functions F are called structure functions and contains the information aboutthe structure of the proton. In fact they are determined by the photon-proton sub-process in this way: F ( x, Q ) = Q π αx [ σ Σ ( γ ∗ P ) + σ L ( γ ∗ P )] (65) F ( x, Q ) = 2 xF ( x, Q ) + F L ( x, Q ) (66) F L ( x, Q ) = Q π α σ L ( γ ∗ P ) , (67)where σ Σ ( γ ∗ P ) and σ Σ ( γ ∗ P ) are the cross sections of the photon-proton process deter-mined summing over all the virtual photon polarization and over only the longitudinalone respectively. At LO F ( x, Q ) = 12 X i Q q i [ q i ( x, Q ) + ¯ q i ( x, Q )] (68) F ( x, Q ) = 2 xF ( x, µ ) , (69)where q i and ¯ q i are the parton densities. In general the structure functions shoulddepend on both x and Q , because these are the relevant kinematic variable of thephoton-proton sub-process. Now we want, as we have done for the Drell-Yan case,rewrite the structure functions in terms of parton densities and of a coefficient functionthat can be computed in perturbative QCD. If we use the identity δ ( y − x ) = Z dz δ (1 − z ) δ ( yz − x ) , (70)we have for the LO structure functions F and F L : F ( x, Q ) = x X i Z dydz (cid:2) q i ( y, µ ) + ¯ q i ( y, µ ) (cid:3) Q q i C q ( z ) δ ( yz − x ) (71)= x X i Z x dyy (cid:2) q i ( y, µ ) + ¯ q i ( y, µ ) (cid:3) Q q i C q (cid:18) xy (cid:19) (72) C q ( z ) = δ (1 − z ) (73) F L ( x, Q ) = 0 , (74)where C q ( z ) is the LO DIS coefficient function for F . From Eq.(71), we see that thevariable z is in general given by z = xy . (75)This means that at the partonic level, z can be viewed as the longitudinal momentumof the incoming parton before it scatters with the virtual photon. At LO it is clearthat z = 1 as can be explicitely seen from Eq.(73), because there is no extra emissions.Beyond the LO the extra radiated partons in the final state can carry some energy(so z <
1) and also the gluon channel contributes. At NLO we have ultraviolet,infrared and collinear singularities. They must be regularized and treated as in the General aspects of perturbative QCD
Drell-Yan case. We report the NLO structure functions (see e.g. [27, 28]) with therenormalization scale equal to the factorization scale: F ( x, Q ) = x X i Q q i Z x dyy (cid:26) (cid:2) q i ( y, µ ) + ¯ q i ( y, µ ) (cid:3) C q (cid:18) z, Q µ , α s ( µ ) (cid:19) (cid:27) + x X i Q q i Z x dyy g ( y, µ ) C g (cid:18) z, Q µ , α s ( µ ) (cid:19) (cid:27) (76)where, in the M S scheme, C q (cid:18) z, Q µ , α s ( µ ) (cid:19) = δ (1 − z ) + α s ( µ )2 π (cid:26) (cid:20) (cid:20) ln(1 − z )1 − z (cid:21) + − (cid:20) − z (cid:21) + − (1 + z ) ln(1 − z ) − z − z ln z − (cid:18) π (cid:19) δ (1 − z )+3 + 2 z (cid:21) + 43 (cid:20) z [1 − z ] + + 32 δ (1 − z ) (cid:21) log (cid:18) Q µ (cid:19) (cid:27) , (77) C g (cid:18) z, Q µ , α s ( µ (cid:19) = α s ( µ )2 π (cid:26) (cid:20) (cid:0) (1 − z ) + z (cid:1) ln (cid:18) − zz (cid:19) − z +8 z − z + (1 − z ) ] log (cid:18) Q µ (cid:19) (cid:21)(cid:27) (78)and F L ( x, Q ) = x X i Q q i Z x dyy (cid:2) q i ( y, µ ) + ¯ q i ( y, µ ) (cid:3) α s ( µ )2 π (cid:26) z + 43 (cid:20) z [1 − z ] + + 32 δ (1 − z ) (cid:21) log (cid:18) Q µ (cid:19) (cid:27) + x X i Q q i Z x dyy g ( y, µ ) α s ( µ )2 π (cid:26) z (1 − z )+ 12 [ z + (1 − z ) ] log (cid:18) Q µ (cid:19) (cid:27) , (79)is factorization scheme independent at the lowest non-vanishing order. The coefficient function and the parton densities depend on the factorization scalein such a way that the resulting hadronic cross section is µ -independent. Theequations that fix the µ -dependence of parton densities (the so called Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations) can be found imposing the µ -independence of the DY cross section or of the DIS structure functions. For example, .4 The DGLAP equations imposing this condition to the explicit expression for the NLO F (see Eqs.(76,77,78),we find the LO DGLAP evolution equations for the quark parton densities: µ ∂q i ( x, µ ) ∂µ = α s ( µ )4 π (cid:26) X j Z x dyy (cid:20) P (0) q i q j (cid:18) xy (cid:19) q j ( y, µ ) + P (0) q i ¯ q j (cid:18) xy (cid:19) ¯ q j ( y, µ ) (cid:21) + Z x dyy P (0) q i g (cid:18) xy (cid:19) g ( y, µ ) (cid:27) + O ( α s ) , (80)where P (0) q i q j ( z ) = δ ij P (0) qq ( z ) , (81) P (0) q i ¯ q j ( z ) = 0 (82) P (0) qq ( z ) = 83 (cid:20) z [1 − z ] + + 32 δ (1 − z ) (cid:21) , (83)and P (0) q i g ( z ) = 1 N f P (0) qg ( z ) = z + (1 − z ) , (84)with N f the number of active flavors. The functions P (0) pp ′ ( z ) are called LO splittingfunctions. They can be viewed as the probability per unit of ln( µ /Q ) to find aparton p in a parton p ′ . The LO evolution equation for the gluon can be calculatedfrom the LO splitting diagrams for a quark into another quark and a gluon andfor a gluon into two gluons. Furthermore, we simplify the notation introducing theconvolution product ⊗ , defined in this way:( f ⊗ f ⊗ · · · ⊗ f n )( x ) = Z dx dx · · · dx n f ( x ) f ( x ) · · · f n ( x n ) δ ( x x · · · x n − x ) . (85)We report here the full result for the DGLAP evolution equations: µ ∂∂µ (cid:18) q i ( z, µ ) g ( z, µ ) (cid:19) = X q j , ¯ q j (cid:18) P q i q j ( z, µ ) P q i g ( z, µ ) P gq j ( z, µ ) P gg ( z, µ ) (cid:19) ⊗ (cid:18) q j ( z, µ ) g ( z, µ ) (cid:19) , (86)where q i can be also a quark or anti-quark and where the splitting functions P pp ′ havethe following perturbative expansion: P q i q j ( z, µ ) = P ¯ q i ¯ q j ( z, µ ) = α s ( µ )4 π δ ij P V (0) qq ( z )+ ∞ X k =1 (cid:18) α s ( µ )4 π (cid:19) k +1 (cid:0) δ ij P V ( k ) qq ( z ) + P S ( k ) qq ( z ) (cid:1) , (87) P q i ¯ q j ( z, µ ) = P ¯ q i q j ( z, µ ) = (cid:18) α s ( µ )4 π (cid:19) (cid:16) δ ij P V (1) q ¯ q + P S (1) qq (cid:17) + ∞ X k =1 (cid:18) α s ( µ )4 π (cid:19) k +2 (cid:16) δ ij P V ( k +1) q ¯ q ( z ) + P S ( k +1) q ¯ q ( z ) (cid:17) , (88) General aspects of perturbative QCD P q i g ( z, µ ) = P ¯ q i g ( z, µ ) = 1 N f ∞ X k =1 (cid:18) α s ( µ )4 π (cid:19) k P ( k − qg , (89) P gq i ( z, µ ) = P g ¯ q i = ∞ X k =1 (cid:18) α s ( µ )4 π (cid:19) k P ( k − gq , (90) P gg ( z, µ ) = ∞ X k =1 (cid:18) α s ( µ )4 π (cid:19) k P ( k − gg , (91)where N f is the number of active flavors. Eq.(86) represents a system of 2 N f + 1integro-differential equations equations. The solution to this system however can becalculated analytically for a certain fixed-order. In fact it can be translated into asystem of ordinary differential equation performing a Mellin transform: F p ( N, µ ) = Z dz z N − F p ( z, µ ) , (92) µ ∂F p ( N, µ ) ∂µ = X p ′ γ APpp ′ ( N, µ ) F p ′ ( N, µ ) , (93)where p, p ′ = q i , ¯ q j , g and γ APpp ′ ( N, µ ) = Z dzP pp ′ ( z, µ ) . (94)After that, these equations can be decoupled searching linear combinations of par-ton densities that depends on the independent splitting functions of Eqs.(68-77) andthat diagonalize the system. For example, at LO, there are 4 independent splittingfunctions which are P V (0) qq = P (0) qq , P (0) qg (given in Eqs.(83,84) respectively) and P (0) gq ( z ) = 83 (cid:20) − z ) z (cid:21) (95) P (0) gg ( z ) = 12 (cid:20) z [1 − z ] + + 1 − zz + z (1 − z ) (cid:21) + (cid:18) − N f (cid:19) δ (1 − z ) . (96)At NLO there are 6 independent splitting functions which are for example P Vqq ( z ), P Sqq ( z ), P Vq ¯ q ( z ), P qg ( z ), P gq ( z ) and P gg ( z ). They are given in Ref.[27] pages 111 and112. The LO and the NLO solution to the DGLAP equations (93) in Mellin space iscomputed in the next Section. In this Section, we want to solve the NLO DGLAP equations (Eq.(93) of section 1.4): µ ∂F p ( N, µ ) ∂µ = X p ′ γ pp ′ ( N, µ ) F p ′ ( N, µ ) , (97) .5 NLO solution of the DGLAP evolution equations where all the splitting functions defined here (and in the following) have the sameperturbative expansion: γ pp ′ ( N, µ ) = α s ( µ )4 π γ (0) pp ′ ( N ) + (cid:18) α s ( µ )4 π (cid:19) γ (1) pp ′ ( N ) + O ( α s ) . (98)This is a system of 2 N f + 1 coupled equations with N f the number of active flavors.At NLO, there are 6 independent splitting functions defined through the followingequations: γ gq i = γ g ¯ q i ≡ γ gq (99) γ q i g = γ ¯ q i g ≡ /N f γ qg (100) γ q i q k = γ ¯ q i ¯ q k ≡ δ ik γ Vqq + γ Sqq (101) γ q i ¯ q k = γ ¯ q i q k ≡ δ ik γ Vq ¯ q + γ Sqq (102) γ gg ≡ γ gg , (103)where i, k are a flavor index. We omit the dependence on N and µ for brevityof notation. Note that beyond the NLO, there is one more independent splittingfunction. In fact in Eq.(102) we should substitute γ Sqq with γ Sq ¯ q which are differentbeyond the NLO [29, 30]. We note also that at LO γ Sqq = γ Sq ¯ q = γ Vq ¯ q = 0 and hence atLO there are only 4 independent splitting functions.Now, we define the 2 N f − q ± ( NS ) k = k X i =1 ( q i ± ¯ q i ) − k ( q k ± ¯ q k ); k = 2 , . . . , N f (104) q V ( NS ) = N f X i =1 ( q i − ¯ q i ) (105)and the 2 so called singlet (S) combinations: g and q ( S ) = N f X i =1 ( q i + ¯ q i ) . (106)With this definitions, from Eq.(97) and Eqs.(99-103), we find that for the non-singletcombinations µ ∂q ± ( NS ) k ∂µ = γ ± q ± ( NS ) k (107) µ ∂q V ( NS ) ∂µ = γ − q V ( NS ) , (108)where γ ± = γ Vqq ± γ Vq ¯ q . (109) General aspects of perturbative QCD
For the 2 remaining singlet combinations, we find in the same way that µ ∂∂µ (cid:18) q ( S ) g (cid:19) = (cid:18) γ qq γ qg γ gq γ gg (cid:19) (cid:18) q ( S ) g (cid:19) , (110)where γ qq = γ + + γ P S , γ
P S ≡ N f γ Sq ¯ q . (111)The NLO Mellin splitting functions can be found in Ref.[31] written in termsof harmonic sums. In many cases, however, their analytic continuation to all thecomplex plane is useful (see e.g. [32, 33]) For the NNLO solution of the DGLAPequations and the NNLO splitting functions we refer to [29, 30]. The techniques forthe analytic continuations of the NNLO splitting functions can be found in Ref.[34].For the NS combinations, the solution is easy to obtain. Indeed, making thechange of variable dµ µ = dα s ( µ ) β ( α s ( µ )) , (112)where β ( α s ) is the β function defined in section 1.2, we get: q ± ( NS ) k ( µ ) q ± ( NS ) k ( µ ) = (cid:18) α s ( µ ) α s ( µ ) (cid:19) − γ (0) ± /β (cid:20) (cid:18) γ (1) ± β − β γ (0) ± β (cid:19) (cid:18) α s ( µ )4 π − α s ( µ )4 π (cid:19)(cid:21) (113)and q V ( NS ) ( µ ) q V ( NS ) ( µ ) = (cid:18) α s ( µ ) α s ( µ ) (cid:19) − γ (0) − /β (cid:20) (cid:18) γ (1) − β − β γ (0) − β (cid:19) (cid:18) α s ( µ )4 π − α s ( µ )4 π (cid:19)(cid:21) , (114)where we have omitted the N -dependence of the splitting functions for brevity. Forthe S combinations, some linear algebra is needed. We, first, define the singlet vectorand the splitting matrix: ~q S ≡ (cid:18) q ( S ) g (cid:19) , ˜ γ S ≡ (cid:18) γ qq γ qg γ gq γ gg (cid:19) . (115)Using the NLO splitting matrix and the change of variable Eq.(112), we find im-meditely the formal solution, which is ~q S ( µ ) = exp (cid:26) − R ln α s ( µ ) α s ( µ ) + R (cid:18) α s ( µ )4 π − α s ( µ )4 π (cid:19)(cid:27) ~q S ( µ ) , (116)where R = ˜ γ (0) S β , R = ˜ γ (1) S β − β ˜ γ (0) S β . (117)The two matrices R and R in Eq.(116) cannot be diagonalized simultaneously, asthey do not commute. Hence, in order to extract the NLO solution from Eq.(116),we use the following Ansatz: ~q S ( µ ) = U ( α s ( µ )) (cid:18) α s ( µ ) α s ( µ ) (cid:19) − R U − ( α s ( µ )) ~q S ( µ ) , (118) .5 NLO solution of the DGLAP evolution equations where the matrix U has the perturbative expansion: U ( α s ( µ )) = 1 + α s ( µ )4 π U + O ( α s ) . (119)The condition that the matrix U should satisfy can be easily obtained imposing thatthe derivative with respect to α s ( µ ) of Eq.(116) and of Eq.(118) are equal at NLO.Thus, we get [ U , R ] = U + R . (120)We write R in terms of its 2 eigenvalues R ± = 12 β (cid:20) ( γ (0) qq + γ (0) gg ) ± q ( γ (0) qq − γ (0) gg ) + 4 γ (0) qg γ (0) gq (cid:21) (121)and of the 2 corresponding eigenspaces projectors P ± : R = R + P + + R − P − . (122)The explicit expression for the projectors can be obtained using the completenessrelation P + + P − = 1. We find P ± = 1 R ± − R ∓ [ R − R ∓ ] . (123)Now, writing the matrices U and R in terms of these projectors U = P − U P − + P − U P + + P + U P − + P + U P + (124) R = P − R P − + P − R P + + P + R P − + P + R P + , (125)substituting them and Eq.(122) into Eq.(120) and comparing each matrix element,we find U = − ( P − R P − + P + R P + ) + P + R P − R − − R + − P − R P + R + − R − − . (126)Thanks to this result, we can now write the NLO solution of the singlet doublet ina form which is useful for practical calculations. Indeed, if we substitute Eq.(126) inEq.(118), we get (at NLO) ~q S ( µ ) = (cid:26) (cid:18) α s ( µ ) α s ( µ ) (cid:19) − R − (cid:20) P − + (cid:18) α s ( µ )4 π − α s ( µ )4 π (cid:19) P − R P − − α s ( µ )4 π − α s ( µ )4 π (cid:18) α s ( µ ) α s ( µ ) (cid:19) R − − R + ! P − R P + R + − R − − (cid:21) +(+ ↔ − ) (cid:27) ~q S ( µ ) . (127) General aspects of perturbative QCD
After the evolution of the NS and S combinations has been performed from a certainscale µ to the scale µ , we need to return to the parton distributions for all the quarksbut the gluon. These are obtained straightforwardly with the following relations q k + ¯ q k = 1 N f q S − k q +( NS ) k + N f X i = k +1 i ( i − q +( NS ) i , k = 1 , . . . , N f (128) q k − ¯ q k = 1 N f q V ( NS ) − k q − ( NS ) k + N f X i = k +1 i ( i − q − ( NS ) i , k = 1 , . . . , N f . (129)However, Eqs.(113,114,127) represent the NLO solution of the DGLAP equationsEq.(97) in the case when the number of active flavors N f has been kept fixed. Thisis the so called fixed flavor scheme solution. If we want to take into account thethresholds of the heavy quark flavors, we can evolve up the NS a S combinations fromthe scale µ (with a certain number N f of active flavors) to the scale of production ofa new flavor. Then, we can take the result of this evolution as the starting point of asecond evolution (with N f + 1 active flavors this time) above the production scale ofthe new flavor, assuming that the new flavor vanishes at threshold. This is the mostsimple way to generate dynamically a new flavor.Finally, we note that the procedure outlined in this appendix can be easily gen-eralized beyond the NLO order [35]. Furthermore, in many cases, it is interestingto study the dependence on the renormalization scale, in order to estimate the the-oretical error of the evolution. Here the renormalization scale µ r has been chosenequal to the factorization one µ for simplicity. To restore the implicit µ r -dependencein parton densities, we need only to rewrite the running coupling constant α s ( µ )in terms of µ r (see Eq.(41) in section 1.2) in the splitting functions. Making thissubstitution, we have that the perturbative expansion of a generic splitting functionEq.(130) becomes γ pp ′ ( N, µ , k ′ ) = α s ( k ′ µ )4 π γ (0) pp ′ ( N ) + (cid:18) α s ( k ′ µ )4 π (cid:19) ( γ (1) pp ′ ( N ) + β γ (0) pp ′ ln k ′ ) + O ( α s ) , (130)where k ′ = µ r /µ . Hence, in Eqs.(113,114,127), we should perform the followingsubstitutions: γ (1) → γ (1) + β γ (0) ln k ′ , α s ( µ ) → α s ( k ′ µ ) , α s ( µ ) → α s ( k ′ µ ) (131)and use k ′ µ as reference scale for new flavors production. hapter 2High order QCD and resummation In section 1.3, we have discussed briefly the analytic NLO calculation of the fullinclusive DIS and DY cross sections. However, in many cases, the NLO pQCD com-putation turns out not to be enough. This is, for example, often the case at LHCwhere the Higgs boson production has to be distinguished from the background. Acomputation beyond the NLO is needed also when the NLO corrections are largeand higher-order calculation permit us to test the convergence of the perturbativeexpansion. In figure 2.1 the total cross section of the production of the Higgs bosonat LHC [36] is plotted and we note convergence in going from LO to NLO and toNNLO.This can also happen when a new parton level subprocess first appear at NLO.This is the case for example for the rapidity DY distributions at Tevatron (shown inFigure 2.1:
Total cross section for the Higgs boson production at LHC at (from bottom to top)at LO, NLO, NNLO in the gluon fusion channel [36]. High order QCD and resummation
Figure 2.2:
DY rapidity distribution for proton anti-proton collisions at Tevatron at (from bottomto top) LO, NLO, NNLO, together with the CDF data [37].
Figure 2.3:
DY rapidity distribution for proton proton collisions at fixed-target experimentE866/NuSea at (from bottom to top) LO, NLO, NNLO, together with the data [19, 38]. figure 5.4) and at the fixed-target experiment E866/NuSea (shown in figure 2.3). Theagreement with the data of figure 5.4 has represented an important test of the NNLOsplitting functions [29, 30]. We note also that going from the LO to the NNLO thefactorization scale dependence is significantly reduced.Calculations beyond the NLO can be important also in processes which involvelarge logaritms when different significant scales appear. In these cases, these largelogarithms should be resummed and this is the topic of this thesis. A first example ofthese large logarithms has appeared in section 1.3. In fact, from Eqs.(57,77)of section1.3, we see that there are contributions that become large when z → .1 When is NLO not enough? function F in the DIS case. These are the terms proportional to α s (cid:20) log(1 − z )1 − z (cid:21) + , α s (cid:20) − z (cid:21) + . (1)The terms of the type of Eq.(1) arise from the infrared cancellation between virtualand real emissions. It can be shown that enhanced contributions of the same typearise at all orders. In fact, at order O ( α ns ) there are contributions proportional to[1, 2, 3]: α ns (cid:20) log m (1 − z )1 − z (cid:21) + , m ≤ n − . (2)These terms become important in the limit z → z → n radiated extra partons with momenta k , . . . , k n , the squaring of four-momentumconservation ( p + p = Q + k + · · · + k n ) implies x x S (1 − z ) = n X i,j =1 k i · k j + 2 n X i =1 Q · k i (3)= n X i,j =1 k i k j (1 − cos θ ij ) + 2 n X i =1 k i ( q Q + | ~Q | − | ~Q | cos θ i ) , (4)where θ ij is the angle between ~k i and ~k j and θ i is the angle between ~k i and ~Q .Since all the terms in the first sum of Eq.(4) are positive semi-definite and the terms( q Q + | ~Q | − | ~Q | cos θ i ) in the second sum are positive for all possible values of θ i ,we have that the limit z = 1 is achieve only for k i = 0 for all i . This means thatwhen z approaches 1 all the emitted partons in the Drell-Yan processes are soft andthat we have reached the threshold for the production of a virtual photon or a realvector boson.In the DIS case, at the partonic level, we have p + q = k + · · · + k n + k n +1 , (5)where k n +1 is the LO outgoing parton. If we square this last equation, we get Q (1 − z ) z = n +1 X i,j =1 k i k j (1 − cos θ ij ) , (6)where θ ij is the angle between ~k i and ~k j . Eq.(6) tells us that in the z → High order QCD and resummation
Figure 2.4:
Total cross section for the Higgs boson production at LHC at (from bottom to top)at NNLO and NNLO improved with NNLL resummation in the gluon fusion channel [39].
Figure 2.5:
Total cross section for the transverse momentum Higgs boson production at LHC atLO and LO improved with NLL resummation in the gluon fusion channel [40]. of partons collinear to each other. However, in Section 3.2 we will show with a moredetailed analysis of the DIS kinematics and phase space that the collinear partonsare also soft in the z → limit for the deep-inelastic process.An example of the impact of resummation can be seen in figure 2.4. There, thetotal cross section for the Higgs production at LHC is plotted at NNLO with itsNNLL resummation improvement improvement [39]. The scale uncertainty reducedto about 15% at NNLO is further reduced to 10% by the NNLL resummation. Theresummed large logaritms in this case are of the class of the DY-like soft emissions.Resummation of another class of large logarithms plays a crucial role in transversemomentum distributions. Indeed, in figure 2.5, we observe that resummation changessubstantially the behavior of the cross section for the production of the Higgs bosonat small transverse momentum. In these case the large logarithms of q ⊥ /M H with M H the Higgs mass are resummed.The state of art of QCD predictions for Higgs boson production at LHC is reportedin figure 2.6 as it was summarized by Laura Reina at the CTEQ summer shool 2006 .1 When is NLO not enough? Figure 2.6:
State of art of QCD predictions for Higgs boson production at hadron colliders. on QCD analysis and phenomenology, where also the Monte Carlo event generatorsare indicated.Furthermore, at LHC, multi-particles/jet production will be the inescapable back-ground to Higgs searches and searches for new physics. This means that we shouldhave a precise knowledge of the QCD background. As seen previously, we know manyQCD processes up to the NNLO. However, we have at the moment limited NLOknowledge of some important final states that will constitute background. They are → W/Z + jets (2 j ) → W W/ZZ/W Z + jets (0 j ) → W W W/ZZZ/W ZZ + jets (0 j ) → Q ¯ Q + jets (0 j ) → γ + jets (1 j ) → γγ + jets → Zγγ + jets , where in parenthesis is indicated the NLO knowledge.Finally, we also note that in higher order contributions to the splitting functions( P gg , P gq for example), it can be shown that there can appear also terms proportionalto α ns ln m z ; m ≤ n. (7)These contributions spoil the convergence when z → m = n defines a LL z resummation, the inclusion of also the terms with m = n − N LL z resum-mation. This resummation is realized by the Balitsky-Fadin-Kuraev-Lipatov (BFKL)equation. Anyway, in this thesis, we will not concentrate on this resummation. We High order QCD and resummation will give a briefly description of the various techniques to resum the large soft logsgiving attention to the renormalization group approach and studying in detail itsapplications.
The aim of resummation is to include all the logarithmic enhanced terms of the formof Eq.(2) of Section 2.1 with a certain hierarchy of logarithms that we shall define inthe current section.From Eqs.(56,76) of Section 1.3 we see that the QCD cross section (up to dimen-sional overall factors) can in general be written as a convolution of the parto densities F H i a ( x i , µ ) and of the dimensionless partonic cross section, i.e. the coefficient function C ( z, Q /µ , α s ( µ )): σ DY ( x, Q ) = X a,b (cid:2) F H a ( µ ) ⊗ F H b ( µ ) ⊗ C ab ( Q /µ , α s ( µ )) (cid:3) ( x ) , (8)for the DY case; and σ DIS ( x, Q ) = X a (cid:2) F Ha ( µ ) ⊗ C a ( Q /µ , α s ( µ )) (cid:3) ( x ) , (9)for the DIS case. The convolution product ⊗ has been defined in Eq.(85) of Section1.4. Performing the Mellin transformation σ ( N, Q ) = Z dx x N − σ ( x, Q ) (10)we turn the convolution products of Eqs.(8,9) into ordinary products: σ DY ( N, Q ) ≡ X a,b σ ab ( N, Q )= X a,b F H a ( N, µ ) F H b ( N, µ ) C ab ( N, Q /µ , α s ( µ )) , (11) σ DIS ( N, Q ) ≡ X a σ a ( N, Q ) = X a F Ha ( N, µ ) C a ( N, Q /µ , α s ( µ )) , (12)where C a ( b ) (cid:18) N, Q µ , α s ( µ ) (cid:19) = Z dzz N − C a ( b ) (cid:18) z, Q µ , α s ( µ ) (cid:19) , (13) F H j a ( N, µ ) = Z dxx N − F H j a ( x, µ ) , (14)and where the second index in brackets ( b ) is involved only when there are two hadronsin the initial state as the the DY case. .2 The renormalization group approach to resummation The large logs of 1 − z of Eq.(2) in Section 2.1 are mapped to the large logs of N by the Mellin transform. This fact and the relations between the large logs of 1 − z and the large logs of N are shown in detail in Appendix A.Whereas the cross section σ ( N, Q ) is clearly µ -independent, this is not the casefor each contribution σ a ( b ) ( N, Q ). However, the µ dependence of each contributionto the sum over a, ( b ) in Eqs.(11,12) is proportional to the off-diagonal anomalousdimensions γ qg and γ gq . In the large N limit, these are suppressed by a power of N in comparison to γ gg and γ qq , or, equivalently, the corresponding splitting functionsare suppressed by a factor of 1 − z in the large z limit (see for example Eqs.(83,84)in Section 1.4). Hence, in the large N limit each parton subprocess can be treatedindependently, specifically, each C a ( b ) is separately renormalization-group invariant.Because we are interested in the behaviour of C a ( b ) ( N, Q /µ , α s ( µ )) in the limit N → ∞ we can treat each subprocess independently.Because resummation takes the form of an exponentiation, we define a so-calledphysical anomalous dimension defined implicitly through the equation Q ∂σ a ( b ) ( N, Q ) ∂Q = γ a ( b ) ( N, α s ( Q )) σ a ( b ) ( N, Q ) . (15)The physical anomalous dimensions γ a ( b ) Eq.(15) is independent of factorization scale,and it is related to the diagonal standard anomalous dimension γ APcc , defined by µ ∂F c ( N, µ ) ∂µ = γ AP cc ( N, α s ( µ )) F c ( N, µ ) , (16)according to γ a ( b ) ( N, α s ( Q )) = ∂ ln C a ( b ) ( N, Q /µ , α s ( µ )) ∂ ln Q = γ AP aa ( N, α s ( Q )) (17)+ γ AP( bb ) ( N, α s ( Q )) + ∂ ln C a ( b ) ( N, , α s ( Q )) ∂ ln Q . (18)We recall that both the standard anomalous dimensions (Altarelli-Parisi splittingfunctions) and the coefficient function are computable in perturbation theory. Hence,the physical anomalous dimensions differs from the standard anomalous dimensionsonly beyond the LO in α s as can be seen directly from Eq.(18). In terms of thephysical anomalous dimensions, the cross section can be written as σ ( N, Q ) = X a, ( b ) K a ( b ) ( N ; Q , Q ) σ a ( b ) ( N, Q ) (19)= X a, ( b ) exp (cid:2) E a ( b ) ( N ; Q , Q ) (cid:3) σ a ( b ) ( N, Q ) , (20)where E a ( b ) ( N ; Q , Q ) = Z Q Q dk k γ a ( b ) ( N, α s ( k )) (21)= Z Q Q dk k [ γ AP aa ( N, α s ( k )) + γ AP( bb ) ( N, α s ( k ))]+ ln C a ( b ) ( N, , α s ( Q )) − ln C a ( b ) ( N, , α s ( Q )) . (22) High order QCD and resummation
We now concentrate on the single subprocess with incoming partons a, ( b ). Re-summation of the large logs of N in the cross section is obtained performing theirresummation in the physical anomalous dimension: σ res ( N, Q ) = exp (Z Q Q dk k γ res ( N, α s ( k )) ) σ res ( N, Q ) . (23)This shows how in general the large logs of N can be exponentiated. For the DY caseonly the quark-anti-quark channel should be resummed and in the DIS case only thequark one. This is a consequence of the fact that the off-diagonal splitting functionsare suppressed in the large N limit as discussed before.The accuracy of resummation here depends on the accuracy at which the physicalanomalous dimension γ is computed. We say that the physical anomalous dimensionis resummed at the next k − -to-leading-logarithmic accuracy ( N k − LL ) when all thecontributions of the form α n + ms ( Q ) ln m N ; n = 0 , . . . , k − d = 4 − ǫ dimensionsthus finding where the large logs are originated in the coefficient function and in thephysical anomalous dimension. The second consists in resumming them imposing therenormalization group invariance of the physical anomalous dimension.So, let’s consider a generic phase space measure dφ n for the emission of n masslesspartons with momenta p , . . . , p n . In Appendix B, we show that this phase spacecan be decomposed in terms of two-body phase space. Roughly speaking, the phasespaces measure for the emission of n partons can be viewed as the emission of twopartons (one with momentum p n and the other with momentum P n = p + · · · + n − and invariant mass P n times the phase space measure where the momentum P n isincoming and the momenta p , . . . , p n are outgoing. The price to pay for this is theintroduction of an integration over the invariant mass P n . Then using recursivelythis procedure, we obtain that the n -body phase space measure is decomposed in n − n -body phase space measure to the study of the soft emissionof two-body phase spaces. .2 The renormalization group approach to resummation The two-body phase space with an incoming momentum P and two outgoingmomenta Q and p in d = 4 − ǫ dimensions (derived explicitly in Eq.(16) of SectionB) is given by dφ ( P ; Q, p ) = N ( ǫ )( P ) − ǫ (cid:18) − Q P (cid:19) − ǫ d Ω d − , N ( ǫ ) = 12(4 π ) − ǫ , (25)where d Ω d − is the solid angle in d − p . Thus, we haveDIS-like emission: P ∝ (1 − z DIS ); Q = 0 (26)DY-like emission: (cid:18) − Q P (cid:19) ∝ (1 − z DY ); P = s DY , (27)where z DIS , z DY are close to one for a soft emission. Hence, we have that the two-bodyphase space measure for a single soft emission contributes with a factor (1 − z ) − aǫ with a = 1 for a DIS-like emission and a = 2 for DY-like emission. In the case ofthe prompt-photon process, we will see in Chapter 4 that there are both types ofemission. The large logs of 1 − z are originated by the interference with the infraredpoles in ǫ = 0 in the square modulus amplitude in the ǫ → ǫ (1 − z ) − aǫ = 1 ǫ − ln(1 − z ) a + O ( ǫ ) , (28)for the case of interference with a pole of order 1.Now, since each factor of (1 − z ) − aǫ that comes from the phase space measureis associated to a single real emission then it will appear in the coefficient functiontogether with a power of the bare strong coupling constant α . In d -dimensions, thecoupling constant is dimensionful, and thus on dimensional grounds each emission isaccompanied by a factor α (cid:2) Q (1 − z ) a (cid:3) − ǫ , (29)where Q is now the typical perturbative scale of a certain process. Upon Mellintransformation, this becomes α (cid:20) Q N a (cid:21) − ǫ . (30)Furthermore an analysis of the structure of diagrams shows that in the soft (large N ) limit, all dependence on N appears through the variable Q /N a also in the am-plitude. Finally, a renormalization group argument shows that all this dependencecan be reabsorbed in the running of the strong coupling. Indeed, the first orderrenormalization of the bare coupling constant at the renormalization scale µα = µ ǫ α s ( µ ) + O ( α s ) (31)and the renormalization group invariance of the physical anomalous dimension implythat α (cid:20) Q N a (cid:21) − ǫ = α s ( µ ) (cid:20) Q µ N a (cid:21) − ǫ + O ( α s )= α s ( Q /N a ) + O ( α s ) , (32) High order QCD and resummation where α is the bare coupling, α s ( µ ) the renormalized coupling and the higher orderterms contain divergences which cancel those in the cross section. Following thisline of argument one may show that the finite expression of the renormalized crosssection in terms of the renormalized coupling is a function of α s ( Q ) and α s ( Q /N a )with numerical coefficients, up to O (1 /N ) corrections. We shall see this in detail inChapter 3. The exponentiation of the large soft logs and their resummation has been demon-strated in QCD with the eikonal approximation [2] or thanks to strong non-standardfactorization properties of the cross section in the soft limit [1]. Recently, also theeffective field theoretic (EFT) approach has been applied to QCD resummation inRefs.[9, 10] for DIS and DY and in Ref.[41] for the B meson decay B → X s γ . In thisSection, we shall only give a brief description of these alternative approaches to theresummation of the large perturbative logarithms. We first consider the simpler case of QED. In QED the exponentiation of the large softlogs has been proved thanks to the eikonal approximation in Ref.[7]. We report thebasic steps of the proof for the QED case and a brief description of the generalizationto the QCD case.Consider a final fermion line with momentum p ′ of a generic QED Feynman dia-gram. We attach n soft photons to this fermion line with momenta k , . . . , k n . Forthe moment we do not care whether these are external photons, virtual photons con-nected to each other, or virtual photons connected to vertices on other fermion lines.The amplitude for such a diagram has the following structure in the soft limit:¯ u ( p ′ )( − ieγ µ ) i / p ′ p ′ · k ( − ieγ µ ) i / p ′ p ′ · ( k + k ) · · · ( − ieγµ n ) i / p ′ p ′ · ( k + · · · + k n ) i M h , (33)where e = −| e | is the electron charge and i M h is the amplitude of the hard part ofthe process without the final fermion line we are considering. We note that here wehave neglected the electron mass. Then we can push the factors of / p ′ to the left anduse the Dirac equation ¯ u ( p ′ )/ p ′ = 0:¯ u ( p ′ ) γ µ / p ′ γ µ / p ′ · · · γ µ n / p ′ = ¯ u ( p ′ )2 p ′ µ γ µ / p ′ · · · γ µ n / p ′ = ¯ u ( p ′ )2 p ′ µ p ′ µ · · · p ′ µ n . (34)Thus Eq.(33) becomes e n ¯ u ( p ′ ) (cid:18) p ′ µ p ′ · k (cid:19) (cid:18) p ′ µ p ′ · ( k + k ) (cid:19) · · · (cid:18) p ′ µ n p ′ · ( k + · · · + k n ) (cid:19) i M h . (35)Still working with only a final fermion line, we must now sum over all possible order-ings of momenta k , . . . , k n . There are n ! different diagrams to sum, corresponding to .3 Alternative approaches the n ! permutations of the n photon momenta. Let P denote one such permutation,so that P ( i ) is the number between 1 and n that i is taken to. Now, using the identity X P p · k P (1) p · ( k P (1) + k P (2) ) · · · p · ( k P (1) ) + · · · + k P ( n ) ) = 1 p · k · · · p · k n , (36)the sum over all the permutations of the photons of Eq.(35) is: e n ¯ u ( p ′ ) (cid:18) p ′ µ p ′ · k (cid:19) (cid:18) p ′ µ p ′ · k (cid:19) · · · (cid:18) p ′ µ n p ′ · k n (cid:19) i M h . (37)At this point, we consider an initial fermion line with momentum p . In this casethe photon momenta in the denominators of the fermion propagators have an oppositesign. Therefore, if we sum over all the diagrams containing a total of n soft photons,connected in any possible order to an arbitrary number of initial and final fermionlines, Eq.(37) becomes: e n i M n Y r =1 X i η i p µ i p i · k r , (38)where i M is the full amplitude of the hard part of the process and where the index r runs over the radiated photons and the index j runs over the initial and final fermionlines with η i = (cid:26) − Y = Z d k (2 π ) k e X i η i p i p i · k ! (40)in the final cross section. If n real photons are emitted, we get n such Y factorsEq.(40), and also a symmetry factor 1 /n ! since there are n identical bosons in thefinal state. The cross section resummed for the emission of any number of soft photonsis therefore σ res ( i → f ) = σ ( i → f ) ∞ X n =0 Y n n ! = σ ( i → f ) e Y , (41)where σ ( i → f ) is the cross section for the hard process without extra soft emissions.This result shows that all the possible soft real emissions exponentiate and that onlythe single emission contributes to the exponent. However, this is not the end ofthe story, because the exponent Y Eq.(40) is infrared divergent. Indeed, to obtain areliable finite result, we must include also loop corrections to all orders. For a detailedanalysis about the inclusion of loops see for example Ref.[42]. Here, we just give thefinal result which reads: σ res ( i → f ) = σ ( i → f ) e σ (1) , (42) High order QCD and resummation where σ (1) is the cross section relative to the single soft emission from the hard process.Clearly, the accuracy of this resummation formula for soft photon emission Eq.(42)depends on the accuracy at which the exponent for the single emission is computed.In Ref.[2] the exponentiation of the soft emissions, here outlined for QED, is gen-eralized to the QCD case. Differently from QED, QCD is a non-abelian gauge theoryand this implies that this generalization is highly non-trivial. Indeed, the gluonscan interact with each other. This fact makes the exponentiation mechanism muchmore difficult since the three gluon vertex color factor is different from that of thequark-gluon vertex. In order to exponentiate the single emission cross section (as ithappens in QED), one should prove that these gluon correlations cancels out order byorder in perturbation theory. This is shown for example in Ref.[43]. According to thisresult, it has been shown in Ref.[2] how the the exponentiation of soft emission worksin QCD resummation. We report here the result for the NLL resummed coefficientfunction in Mellin space for inclusive DIS and DY processes in the M S scheme in acompact form: C NLL ( N, Q /µ , α s ( µ )) = exp (cid:26) a Z dx x N − − − x (cid:20) Z Q (1 − x ) a µ dk k A ( α s ( k ))+ B ( a ) ( α s ( Q (1 − x ) a )) (cid:21)(cid:27) , (43)where A ( α s ) = A α s + A α s + . . . (44) B ( a ) ( α s ) = B ( a )1 α s + . . . (45)with A = C F π , A = C F π (cid:20) C A (cid:18) − π (cid:19) − N f (cid:21) , B ( a )1 = − (2 − a )3 C F π . (46)Here a = 1 for the DIS structure functio F and a = 2 for the DY case. How thisresult is strictly connected to the resummed results that can be obtained with therenormalization group approach will be discussed in Section 3.4. This is the approach of Ref.[1]. Also in this approach the results given in Eq.(43)of Section 2.3.1 are recovered. Here we give only a rough description of this methodbased on strong factorization properties of the QCD cross section.It is essentially assumed that at the boundary of the phase space, the cross sectionis factorized in a hard and in a soft part and eventually in an other factor associatedto final collinear jets as in the DIS case where there is an outgoing emitting quark.The final result is then obtained exponentiating the soft and collinear factors. Thisis done solving their evolution equations.In Ref.[1] it is shown that the semi-inclusive cross section can be factorized in threefactors relative to the three different regions in the momentum space of the process: .3 Alternative approaches the off-shell partons that participate to the partonic hard process, the collinear andsoft on-shell radiated partons. The cross section is given by σ ( w ) = H (cid:18) p µ , p µ , ζ i (cid:19) Z dw w dw w dw w J (cid:18) p · ζ µ , w (cid:18) Qµ (cid:19) a (cid:19) J (cid:18) p · ζ µ , w (cid:18) Qµ (cid:19) a (cid:19) S (cid:18) w s Qµ , ζ i (cid:19) δ ( w − w − w − w ) , (47)where a is the number of hadrons in the initial state, µ is the factorization scale and ζ i are gauge-fixing parameters; the integration variables w , w and w are referredto the two collinear jets and to soft radiation respectively. Each factor of Eq.(47) isevaluated at the typical scale of the momentum space region which is associated to.The delta function imposes that w = w + w + w = (cid:26) − x Bj , for the DIS case1 − Q /S for the DY case (48)The convolution of Eq.(47) is turned into an ordinary product performing the Mellintransform: σ ( N ) = Z ∞ dw e − Nw σ ( w ) = H (cid:18) p µ , p µ , ζ i (cid:19) S (cid:18) QµN , ζ i (cid:19) × J (cid:18) p · ζ µ , QµN /a (cid:19) J (cid:18) p · ζ µ , QµN /a (cid:19) . (49)Each factor H , J i , S satisfy the following evolution equations µ ∂∂µ ln H = − γ H ( α s ( µ )) , (50) µ ∂∂µ ln S = − γ S ( α s ( µ )) , (51) µ ∂∂µ ln J i = − γ J i ( α s ( µ )) , (52)where the physical anomalous dimensions γ H ( α s ), γ J i ( α s ) and γ S ( α s ) are calcula-ble in perturbation theory and must satisfy, according to the renormalization groupinvariance of the cross section the relation γ H ( α s ) + γ S ( α S ) + X i =1 γ J i ( α s ) = 0 . (53)Solving Eqs.(50-52) and imposing renormalization group invariance Eq.(53), the re-summed section can be written in the form σ ( N ) = exp (cid:26) D ( α s ( Q )) + D (cid:18) α s (cid:18) Q N (cid:19)(cid:19) − a − Z QN /aQN dξξ ln (cid:18) ξNQ (cid:19) A ( α s ξ ) − Z Q QN /a dξξ (cid:20) ln (cid:18) Qξ (cid:19) A ( α s ( ξ )) − B ( α s ( ξ )) (cid:21) , (54)where the functions A ( α s ), B ( α s ), D i ( α s ) are determined in terms of the anomalousdimensions and the beta function. Finally, it can be shown that this result can becasted in the form of Eq.(43) in Section 2.3.1 for the resummed coefficient function. High order QCD and resummation
This is the approach of Refs.[9, 10]. This EFT methodology to resum thresholdlogarithms is made concrete due to the recently developed “soft collinear effectivetheory” (SCET) [44, 45, 46, 47]. The SCET describes interactions between soft andcollinear partons.The starting point (considering the DY case as an example) is the collinearlyfactorized inclusive cross section in Mellin space: σ ( N, Q ) = σ C ( N, Q /µ , α s ( µ )) F ( N, µ ) F ( N, µ ) , (55)where σ is the born level cross section and F i ( N, µ ) are the parton densities atthe factorization scale µ . Here, the basic idea is to write the coefficient function C ( N, Q /µ , α s ( µ )) in terms of an intermediate scale µ I : C ( N, Q /µ , α s ( µ )) = C ( N, Q /µ I , α s ( µ I )) exp " − Z µ µ I dk k γ AP qq ( N, α s ( k )) (56)and then to compute C ( N, Q /µ I , α s ( µ I )) with the “soft collinear effective theory”with the intermediate scale µ I equal to the typical scale of the soft-collinear emission,i.e. µ I = Q /N in the DY case and µ I = Q /N in the DIS case. In SCET, C ( N, Q /µ I , α s ( µ I )) has the following general structure: C (cid:18) N, Q µ I , α s ( µ I ) (cid:19) = (cid:12)(cid:12)(cid:12)(cid:12) ˜ C (cid:18) Q µ I , α s ( µ I ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) M ( N, α s ( µ I )) , (57)Here ˜ C ( Q /µ I , α s ( µ I )) is the effective coupling that matches the full QCD theorycurrents J QCD = ˜ C ( Q /µ I , α s ( µ I )) J eff ( µ I ) . (58)We note that the effective coupling ˜ C contains the perturbative contribution betweenthe scale Q and µ I and J eff ( µ I ) contains the soft and collinear contributions belowthe scale µ I . Then M ( N, α s ( µ I ) is the matching coefficient that guarantees that theEFT used generates the full QCD results in the appropriate kinematical limit. InSCET the matching coefficient M can be computed perturbativly and is free of anylogarithms.The effective coupling ˜ C satisfy to a certain evolution equation µ ∂∂µ ln ˜ C ( Q /µ , α s ( µ )) = − γ ( α s ( µ )) (59)where the physical anomalous dimension γ is computable perturbativly in SCET.Finally, solving the evolution equation Eq.(59) one finds that Eq.(57) becomes C (cid:18) N, Q µ I , α s ( µ I ) (cid:19) = (cid:12)(cid:12)(cid:12) ˜ C (1 , α s ( Q )) (cid:12)(cid:12)(cid:12) e E ( Q /µ I ,alpha s ( Q )) ××M ( N, α s ( Q )) e E ( Q /µ I ,α s ( Q )) , (60) .3 Alternative approaches where E (cid:18) Q µ I , α s ( Q ) (cid:19) = − Z Q µ I dk k γ ( α s ( k )) , (61) E (cid:18) Q µ I , α s ( Q ) (cid:19) = Z Q µ I dk k β ( α s ( k )) d ln M ( N, α s ( k )) d ln α s ( k ) , (62)with β ( α s ) the beta function of Eq.(43) of Section 1.2. ˜ C (1 , α s ( Q )) contains the non-logarithmic contribution of the purely virtual diagrams and the exponent E containsall the logarithms originating from the same type of diagrams. E encodes all thecontributions due to the running of the coupling constant between the scale µ I and Q . All the logarithms appear only in the exponents (see Eqs.(56,60)) and the term | ˜ C (1 , α s ( Q )) | M ( N, α s ( Q )) is free of any large logarithms.The various approaches can be related one to the other according to factorizationproperties of the QCD cross section in the soft limit. In this way, it is possible to showthat all the approaches are equivalent except for the renormalization group approachthat produces correct but less restrictive results. hapter 3Renormalization groupresummation of DIS and DY In this chapter, we analyze in detail how the renormalization group approach toresummation works in the case of the all-inclusive Drell-Yan (DY) and deeep inelasticscattering (DIS). We recall that this is done only for the quark-anti-quark channel inthe DY case and only for the quark channel in the DIS case as discussed in Section2.2. First, we determine the N dependence of the regularized coefficient functionin the large- N limit. Then we show that, given this form of the N -dependenceof the regularized cross section, renormalization group invariance fixes the all-orderdependence of the physical anomalous dimension in such a way that the infiniteclass of leading, next-to-leading etc. resummations can be found in terms of fixedorder computations. This approach will lead us to resummation formulae valid to alllogrithmic orders. In the case of deep-inelastic scattering, the relevant parton subprocesses are: γ ∗ ( q ) + Q ( p ) → Q ( p ′ ) + X ( K ) (1) γ ∗ ( q ) + G ( p ) → Q ( p ′ ) + X ( K ) , (2)where Q is a quark or an anti-quark, G is a gluon and X is any collection of quarksand gluons. We are interested in the most singular parts in the limit z → m = 1 recursively, we can express the phasespace for a generic process in terms of two-body phase space integrals. For the DISprocesses Eqs.(1,2) with n extra emissions ( K = k + · · · + k n ) we have dφ n +1 ( p + q ; k , . . . , k n , p ′ )= Z s dM n π dφ ( p + q ; k n , P n ) dφ n ( P n ; k , . . . , k n − , p ′ )37 Renormalization group resummation of DIS and DY = Z s dM n π dφ ( p + q ; k n , P n ) × Z M n dM n − π dφ ( P n ; k n − , P n − ) dφ n − ( P n − ; k , . . . , k n − , p ′ )= Z s dM n π dφ ( P n +1 ; k n , P n ) Z M n dM n − π dφ ( P n ; k n − , P n − ) × · · · × Z M dM π dφ ( P ; k , P ) dφ ( P ; k , P ) , (3)where we have defined M n ≡ P n , P ≡ p ′ , P n +1 ≡ p + q and s ≡ M n +1 = ( p + q ) = Q (1 − z ) z ; z = Q p · q . (4)Now, according to Eq.(16) in Appendix B we have for each two-body phase space dφ ( P i +1 ; k i , P i ) = N ( ǫ )( M i +1 ) − ǫ (cid:18) − M i M i +1 (cid:19) − ǫ d Ω i ; i = 1 , . . . n, (5)where N ( ǫ ) = 12(4 π ) − ǫ (6)and Ω i is the solid angle in the center-of-mass frame of P i +1 . We perform the changeof variables z i = M i M i +1 ; M i = sz n . . . z i ; i = 2 , . . . , n. (7)From the fact that M i +1 ≥ M i (we have one more real particle in P i +1 than in P i ),it follows that 0 ≤ z i ≤ . (8)From Eq.(7) we get dM n dM n − · · · dM = det (cid:18) ∂M i ∂z j (cid:19) dz n dz n − · · · dz , (9)where det (cid:18) ∂M i ∂z j (cid:19) = ∂M n ∂z n · · · ∂M ∂z = s n − z n − n z n − n − · · · z . (10)Furthermore, in these new variables the two-body phase space becomes dφ ( P i +1 ; k i , P i ) = N ( ǫ ) s − ǫ ( z n z n − · · · z i +1 ) − ǫ (1 − z i ) − ǫ d Ω i . (11)Substituting Eqs.(9,10,11) into the generic phase space Eq.(3), we finally get dφ n +1 ( p + q ; k , . . . , k n , p ′ ) = 2 π (cid:20) N ( ǫ )2 π (cid:21) n s n − − nǫ d Ω n · · · d Ω × Z dz n z ( n − − ( n − ǫn (1 − z n ) − ǫ · · · Z dz z − ǫ (1 − z ) − ǫ . (12) .1 Kinematics of inclusive DIS in the soft limit The dependence of the phase space on 1 − z comes entirely from the prefactor of s n − − nǫ according to Eq.(4). Indeed the dependence on z and Q has been entirelyremoved from the integration range thanks to the change of variables of Eq.(7).Now, the amplitude whose square modulus is integrated with the phase spaceEq.(12) is in general a function: A n +1 = A n +1 ( Q , s, z , . . . , z n , Ω , . . . , Ω n ) . (13)The number of independent variables for a process with 2 incoming particle (one on-shell and the other virtual) and n + 1 outgoing real particles is given by the numberof parameters minus the on-shell conditions and the ten parameters of the Poincare’group: 4( n + 3) − ( n + 2) −
10 = 3 n. (14)These 3 n variable correspond in this case to Q , s, z , . . . , z n , Ω , . . . , Ω n , (15)where an azimutal angle is arbitrary. In the z → s → s vanishes. Because ofcancellation of infrared singularities [48, 49], | A n +1 | ∼ s − n + O ( ǫ ) when s →
0. Indeed,a stronger singularity would lead to powerlike infrared divergences and a weakersingularity would lead to suppressed terms in the z → s − n + O ( ǫ ) contribute inthe z → d dimensions, these pick up an s − − nǫ + O ( ǫ ) prefactor from thephase space Eq.(12). Let’s consider the simplest case, that is the tree level case wherewe have only purely real soft emission. In this case O ( ǫ ) = 0 and thus we get thatthe contribution to the coefficient function from the tree level diagrams with n extraradiated partons behaves as: | A n +1 | dφ n +1 ∼ s − − nǫ Z dz n · · · dz z ( n − − ( n − ǫn (1 − z n ) − ǫ · · · z − ǫ (1 − z ) − ǫ × Z d Ω · · · d Ω n . (16)We note that each z integration can produce at most a 1 /ǫ pole from the soft regionand that each angular integration can produce at most an additional 1 /ǫ pole fromthe collinear region (see Eq.(23) in Appendix B). Therefore, from the contribution oftree level diagrams with n extra radiated partons there come at most1 ǫ n − ǫ n = 1 ǫ n − (17)poles in ǫ = 0. All this means that we can write the O ( α ns ) to the bare coefficientfunction in d dimensions in the following form: C ′ (0) n ( z, Q , ǫ ) = ( Q ) − nǫ C (0) nn ( ǫ )Γ( − nǫ ) (1 − z ) − − nǫ , (18) Renormalization group resummation of DIS and DY where the factor ( Q ) − nǫ is due to elementary dimensional analysis, C (0) nn ( ǫ ) are coef-ficients with poles in ǫ = 0 of order at most of 2 n and the Γ function factor Γ − ( − nǫ )has been introduced for future convenience. For the LO tree level case (see Eq.(73)of Chapter 1) we have that C ′ (0)0 ( z, Q , ǫ ) = δ (1 − z ) . (19)Hence, the tree level coefficient function C (0)tree in the z → C (0)tree ( z, Q , α , ǫ ) = ∞ X n =0 α n C ′ (0) n ( z, Q , ǫ ) (20)= δ (1 − z ) + ∞ X n =1 α n ( Q ) − nǫ C ′ (0) nn ( ǫ )Γ( − nǫ ) (1 − z ) − − nǫ + O ((1 − z ) ) , (21)where α is the bare i.e. the unrenormalized strong coupling constant.We will now study how the result of Eq.(21) is modified by the inclusion of loops.To this purpose, we notice that a generic amplitude with loops can be viewed as atree-level amplitude formed with proper vertices. Contributions to the dimensionlesscoefficient function with powers of s ǫ can only arise from loop integrations in theproper vertices. We thus consider only purely scalar loop integrals, since numeratorsof fermion or vector propagators and vertex factors cannot induce any dependenceon s ǫ . Let us therefore consider an arbitrary proper diagram G in a massless scalartheory with E external lines, I internal lines and V vertices. It can be shown (seee.g. section 6.2.3 of [50] and references therein) that, denoting with P the set of E external momenta and P E the set of independent invariants, the correspondingamplitude ˜ A G ( P E ) has the form˜ A G ( P E ) = K (2 π ) d δ d ( P ) A G ( P E ) ,A G ( P E ) = i I − L ( d − (4 π ) dL/ Γ( I − dL/ × I Y l =1 (cid:20)Z dβ l (cid:21) δ (cid:16) − P Il =1 β l (cid:17) [ P G ( β )] d ( L +1) / − I [ D G ( β, P E )] I − dL/ . (22)Here, β l are the usual Feynman parameters, P G ( β ) is a homogeneous polynomial ofdegree L in the β l , D G ( β, P E ) is a homogeneous polynomial of degree L + 1 in the β l with coefficients which are linear functions of the scalar products of the set P E , i.e.with dimensions of (mass) , and K collects all overall factors, such as couplings andsymmetry factors.The amplitude ˜ A G ( P E ) Eq.(22) depends on s only through D G ( β, P E ), which,in turn is linear in s . We can determine in general the dependence of A G ( P E ) byconsidering two possible cases. The first possibility is that D G ( β, P E ) is independentof all invariants except s , i.e. D G ( β, P E ) = sd G ( β ). In such case, A G ( P E ) depends on s as A G ( P E ) = (cid:18) s (cid:19) I − dL/ a G = (cid:18) s (cid:19) I − L + Lǫ a G , (23) .2 Kinematics of inclusive DY in the soft limit where a G is a numerical constant, obtained performing the Feynman parameters in-tegrals. The second possibility is that D G ( β, P E ) depends on some of the otherinvariants. In such case, A G ( P E ) is manifestly an analytic function of s at s = 0, andthus it can be expanded in Taylor series around s = 0, with coefficients which dependon the other invariants. In the former case, Eq.(23) implies that the s dependenceinduced by loops integration in the square amplitude is given by integer powers of s − ǫ . In the latter case, the s dependence induced by loops integrations in the squareamplitude is given by integer positive powers of s .Therefore, we conclude that each loop integration can carry at most a factor of s − ǫ and that Eq.(21), after the inclusion of loops, becomes: C (0) ( z, Q , α , ǫ ) = ∞ X n =0 α n C (0) n ( z, Q , ǫ ) , (24) C (0) n ( z, Q , ǫ ) = ( Q ) − nǫ " C (0) n ( ǫ ) δ (1 − z ) + n X k =1 C (0) nk ( ǫ )Γ( − kǫ ) (1 − z ) − − kǫ + O ((1 − z ) ) , (25)where again for future convenience we have defined the coefficients of (1 − z ) − − kǫ interms of Γ − ( − kǫ ) and where O ((1 − z ) ) denotes terms which are not divergent as z → ǫ → Z dz z N − (1 − z ) − − kǫ = Γ( N )Γ( − kǫ )Γ( N − kǫ ) (26)and the Stirling expansion Eq.(5) of Appendix A we get that the Mellin transform ofEqs.(55,25) in the large- N limit is given by C (0) ( N, Q , α , ǫ ) = ∞ X n =0 α n C (0) n ( N, Q , ǫ ) , (27) C (0) n ( N, Q , ǫ ) = n X k =0 C (0) nk ( ǫ )( Q ) − ( n − k ) ǫ (cid:18) Q N (cid:19) − kǫ + O (cid:18) N (cid:19) . (28)The content of this result is that, in the large- N limit, the dependence of the regular-ized cross section on N only goes through integer powers of the dimensionful variable( Q /N ) − ǫ . In the Drell-Yan case the argument follows in an analogous way with minor modi-fication which account for the different kinematics. In this case the relevant partonsubprocesses are: Q ( p ) + Q ( p ′ ) → γ ∗ ( Q ) + X (29) Q ( p ) + G ( p ′ ) → γ ∗ ( Q ) + X (30) G ( p ) + G ( p ′ ) → γ ∗ ( Q ) + X . (31) Renormalization group resummation of DIS and DY
The recursive application of Eq.(12) in Appendix B with m = 1, in this case gives: dφ n +1 ( p + p ′ ; Q, k , . . . , k n )= Z s dM n π dφ ( p + p ′ ; k n , P n ) dφ n ( P n ; Q, k , . . . , k n − )= Z s dM n π dφ ( p + p ′ ; k n , P n ) × Z M n dM n − π dφ ( P n ; k n − , P n − ) dφ n − ( P n − ; Q, k , . . . , k n − )= Z s dM n π dφ ( P n +1 ; k n , P n ) Z M n dM n − π dφ ( P n ; k n − , P n − ) × · · · × Z M dM π dφ ( P ; k , P ) dφ ( P ; k , P ) , (32)where now we have defined P n +1 ≡ p + p ′ , so M n +1 = s and P ≡ Q . The change ofvariables which separates off the dependence on (1 − z ), where now z = Q /s , is z i = M i − Q M i +1 − Q ; i = 2 , . . . , n (33) M i − Q = ( s − Q ) z n · · · z i . (34)Also here all z i range between 0 and 1, because M i ≤ M i +1 ≤ Q . From Eq.(33), weget: dM n dM n − · · · dM = det (cid:18) ∂ ( M i − Q ) ∂z j (cid:19) dz n dz n − · · · dz , (35)wheredet (cid:18) ∂ ( M i − Q ) ∂z j (cid:19) = ∂ ( M n − Q ) ∂z n · · · ∂ ( M − Q ) ∂z = ( s − Q ) n − z n − n z n − n − · · · z . (36)In this case in the new variables Eq.(33) the two-body phase space becomes dφ ( P i +1 ; k i , P i )= N ( ǫ )( M i +1 ) − ǫ (cid:2) ( M i +1 − Q ) − ( M i − Q ) (cid:3) − ǫ d Ω i = N ( ǫ )( Q ) − ǫ ( s − Q ) − ǫ ( z n · · · z i +1 ) − ǫ (1 − z i ) − ǫ d Ω i , (37)where in the last step we have replaced ( M i +1 ) − ǫ by ( Q ) − ǫ in the z → dφ n +1 ( p + p ′ ; Q, k , . . . , k n ) = 2 π (cid:20) N ( ǫ )2 π (cid:21) n ( Q ) − n (1 − ǫ ) ( s − Q ) n − − nǫ d Ω n · · · d Ω × Z dz n z ( n − − ( n − ǫn (1 − z n ) − ǫ · · · Z dz z − ǫ (1 − z ) − ǫ . (38) .2 Kinematics of inclusive DY in the soft limit The dependence on 1 − z is now entirely contained in the phase space prefactor( Q ) − n (1 − ǫ ) ( s − Q ) n − − nǫ = z − n +2 nǫ Q (1 − z ) (cid:2) Q (1 − z ) (cid:3) n − nǫ . (39)As before, this proves that the coefficient function for real emission at tree level isgiven by C (0)tree ( z, Q , α , ǫ ) = ∞ X n =0 α n C ′ (0) n ( z, Q , ǫ ) (40)= δ (1 − z ) + ∞ X n =1 α n ( Q ) − nǫ C ′ (0) nn ( ǫ )Γ( − nǫ ) (1 − z ) − − nǫ + O ((1 − z ) ) , (41)In this case the introduction of loops requires more care than for the deep-inelastic-scattering case. We shall now show the main difference between the DIS case and theDY case as far as the introduction of loops is concerned. The two-body kinematics(see Eq.(14) in Appendix B together with Eqs.(3,7) states that the radiated partonsin the DIS case are all soft: k i = M i +1 (cid:18) − M i M i +1 (cid:19) = √ s z n · · · z i +1 ) / (1 − z i ); 1 ≤ i ≤ n − k n = √ s (cid:18) − M n s (cid:19) = √ s − z n ); s = Q − zz . (43)This confirms the validity of the argument of Section 3.1 for the introduction of loops,because all the invariants that can appear in the function D G ( β, P E ) in Eq.(22) canbe expresses in terms of the following ones q = − Q (44) p = p ′ = k i = 0 (45) p · p ′ ∼ p · k i ∼ Q (46) k i · k j ∼ Q (1 − z ) , (47)which are either constant or proportional to 1 − z , i.e. to s. Here we have usedEqs.(42,43) of this Section, Eq.(6) in Chapter 2 and the fact that the definition of z in the DIS case Eq.(4) implies that( p ) = Q z (1 − z ) . (48)In the DY case things are quite different, because in this case two-body kinematics(see Eq.(14) in Appendix B together with Eqs.(32,33) gives k i = √ s − z ) z n · · · z i +1 (1 − z i ) + O ((1 − z ) ); 1 ≤ i ≤ n − k n = √ s − z )(1 − z n ); s = Q z (50) Renormalization group resummation of DIS and DY and this implies that all the invariants that can appear in the function D G ( β, P E ) inEq.(22) can be expresses in terms of the following ones q = Q (51) p · p ′ = s/ p · k i ∼ p ′ · k i ∼ s (1 − z ) (53) k i · k j ∼ s (1 − z ) . (54)Hence, we see that, in general, both odd and even powers of (1 − z ) − ǫ may ariseadding the loops contribution to the tree level coefficient function Eq.(41). Here, inthis thesis, we will assume that odd powers of (1 − z ) − ǫ do not arise, because it canbe shown by explicit computations that it is the case up to order O ( α s ). However,there are possible indications that at higher orders this assumption could not be true.Anyway the investigations of these aspects is beyond the aim of this thesis.Thus, after the inclusions of loops and with our assumptions, Eq.(41) becomes C (0) ( z, Q , α , ǫ ) = ∞ X n =0 α n C (0) n ( z, Q , ǫ ) , (55) C (0) n ( z, Q , ǫ ) = ( Q ) − nǫ " C (0) n ( ǫ ) δ (1 − z ) + n X k =1 C (0) nk ( ǫ )Γ( − kǫ ) (1 − z ) − − kǫ + O ((1 − z ) ) . (56)Its Mellin transform can be written in a compact way together with that of the DIScase Eq.(28): C (0) ( N, Q , α , ǫ ) = ∞ X n =0 n X k =0 C (0) nk ( ǫ )[ α ( Q ) − ǫ ] ( n − k ) " α (cid:18) Q N a (cid:19) − ǫ k + O (cid:18) N (cid:19) , (57)where a = 1 for the DIS case and a = 2 for the DY case and where the coefficientsare those that could be obtained from the parton-level cross sections for the partonicsubprocesses that contribute to the given process. In this section, we want to impose the restrictions that renormalization group in-variance imposes on the cross section. Our only assumption is that the coefficientfunction can be multiplicatively renormalized. This means that all divergences canbe removed from the bare coefficient function C (0) ( N, Q , α , ǫ ) Eq.(57) by defining arenormalized running coupling α s ( µ ) according to the implicit equation α ( µ , α s ( µ ) , ǫ ) = µ ǫ α s ( µ ) Z ( α s ) ( α s ( µ ) , ǫ ) (58) .3 Resummation from renormalization group improvement and a renormalized coefficient function C (cid:18) N, Q µ , α s ( µ , ǫ ) (cid:19) = Z ( C ) ( N, α s ( µ ) , ǫ ) C (0) ( N, Q , α , ǫ ) , (59)where µ is the renormalization scale (here chosen equal to the factorization one) and Z ( α s ) ( α s ( µ ) , ǫ ) and Z C ( N, α s ( µ ) , ǫ ) are computable in perturbation theory and havemultiple poles in ǫ = 0. The renormalized coefficient function C ( N, Q /µ , α s ( µ , ǫ ))is finite in ǫ = 0 and it can only depend on Q through Q /µ , because α s ( µ ) isdimensionless.The physical anomalous dimension is given by γ ( N, α s ( Q ) , ǫ ) = Q ∂∂Q ln C (cid:18) N, Q µ , α s ( µ ) , ǫ (cid:19) = Q ∂∂Q ln C (0) (cid:0) N, Q , α , ǫ (cid:1) = − ǫ ( α Q − ǫ ) ∂∂ ( α Q − ǫ ) ln C (0) (cid:0) N, Q , α , ǫ (cid:1) , (60)where we have exploited the fact that C (0) Eq.(57) depends on Q through the com-bination α Q − ǫ . This implies that the physical anomalous dimension γ has thefollowing perturbative expression: γ ( N, α s ( Q ) , ǫ ) = ∞ X i =0 n X j =0 γ ij ( ǫ )[ α ( Q ) − ǫ ] ( i − j ) " α (cid:18) Q N a (cid:19) − ǫ j + O (cid:18) N (cid:19) . (61)The renormalized expression of the physical anomalous dimension is found ex-pressing in this equation the bare coupling constant in terms of the renormalized oneby means of Eq.(58). Now, the functions( Q ) − ǫ α = (cid:18) Q µ (cid:19) − ǫ α s ( µ ) Z ( α s ) ( α s ( µ ) , ǫ ) (62)( Q /N a ) − ǫ α = (cid:18) Q /N a µ (cid:19) − ǫ α s ( µ ) Z ( α s ) ( α s ( µ ) , ǫ ) (63)are manifestly renormalization group invariant, i.e. µ -independent. Thus, it followsthat ( Q ) − ǫ α = α s ( Q ) Z ( α s ) ( α s ( Q ) , ǫ ) (64)( Q /N a ) − ǫ α = α s ( Q /N a ) Z ( α s ) ( α s ( Q /N a ) , ǫ ) . (65)The renormalized physical anomalous dimension is then found by substituting Eqs.(64,65)into Eq.(61) and re-expanding Z ( α s ) in powers of the renormalized coupling. We ob-tain: γ ( N, α s ( Q ) , ǫ ) = ∞ X m =1 m X n =0 γ R mn ( ǫ ) α m − ns ( Q ) α ns ( Q /N a ) + O (cid:18) N (cid:19) . (66)At this point, we cannot yet conclude that the four-dimensional physical anoma-lous dimension admits an expression of the form of Eq.(66), because the coefficients Renormalization group resummation of DIS and DY γ R mn ( ǫ ) are not necessarily finite as ǫ →
0. In order to understand this, it is convenientto separate off the N -independent terms in the renormalized physical anomalous di-mension, i.e. the terms with n = 0 in the internal sum in Eq.(66). Namely, wewrite γ ( N, α s ( Q ) , ǫ ) = ˆ γ ( l ) ( α s ( Q ) , α s ( Q /N a ) , ǫ ) + ˆ γ ( c ) ( α s ( Q ) , ǫ ) + O (cid:18) N (cid:19) , (67)where we have definedˆ γ ( l ) ( α s ( Q ) , α s ( Q /N a ) , ǫ ) = ∞ X m =0 ∞ X n =1 γ R m + nn ( ǫ ) α ms ( Q ) α ns ( Q /N a ) (68)ˆ γ ( c ) ( α s ( Q ) , ǫ ) = ∞ X m =1 γ R m ( ǫ ) α ms ( Q ) . (69)Whereas γ ( N, α s ( Q ) , ǫ ) is finite in the limit ǫ →
0, where it coincides with the four-dimensional physical anomalous dimension, ˆ γ ( l ) and ˆ γ ( c ) are not necessarily finite as ǫ →
0. However, Eq.(67) implies that ˆ γ ( l ) and ˆ γ ( c ) can be made finite by adding andsubtracting the counterterm Z ( γ ) ( α s ( Q ) , ǫ ) = ˆ γ ( l ) ( α s ( Q ) , α s ( Q ) , ǫ ) . (70)In this way the physical anomalous dimension Eq.(67) becomes γ ( N, α s ( Q ) , ǫ ) = γ ( l ) ( α s ( Q ) , α s ( Q /N a ) , ǫ ) + γ ( c ) ( α s ( Q ) , ǫ ) + O (cid:18) N (cid:19) , (71)where γ ( l ) ( α s ( Q ) , α s ( Q /N a ) , ǫ ) = ˆ γ ( l ) ( α s ( Q ) , α s ( Q /N a ) , ǫ ) + − ˆ γ ( l ) ( α s ( Q ) , α s ( Q ) , ǫ ) , (72) γ ( c ) ( α s ( Q ) , ǫ ) = ˆ γ ( c ) ( α s ( Q ) , ǫ ) + ˆ γ ( l ) ( α s ( Q ) , α s ( Q ) , ǫ ) . (73)Now, γ ( c ) is clearly finite in ǫ = 0, because at N = 1 γ ( l ) vanishes and it is N -independent. This also implies that γ ( l ) is finite for all N , because γ shoul be finitefor all N . Therefore, γ ( l ) provides an expression of the resummed physical anomalousdimension in the large N limit, up to non-logarithmic terms: γ ( N, α s ( Q ) , ǫ ) = γ ( l ) ( α s ( Q ) , α s ( Q /N a ) , ǫ ) + O ( N ) . (74)It is apparent from Eq.(73) that γ ( c ) is a power series in α s ( Q ) with finite coefficientsin the ǫ → γ ( l ) as well,define implicitly the function g ( α s ( Q ) , α s ( Q /n ) , ǫ ) as γ ( l ) ( α s ( Q ) , α s ( Q /N a ) , ǫ ) = Z N a dnn g ( α s ( Q ) , α s ( Q /n ) , ǫ ) , (75) .3 Resummation from renormalization group improvement where g ( α s ( Q ) , α s ( µ ) , ǫ ) ≡ − µ ∂∂µ ˆ γ ( l ) ( α s ( Q ) , α s ( µ ) , ǫ ) (76)= − β ( d ) ( α s ( µ ) , ǫ ) ∂∂α s ( µ ) ˆ γ ( l ) ( α s ( Q ) , α s ( µ ) , ǫ ) , (77)with β ( d ) ( α s ) is the d -dimensional beta function β ( d ) ( α s ( µ ) , ǫ ) − ǫα s ( µ ) + β ( α s ( µ )) (78)and where we have performed the change of variable n = Q /µ . It immediatelyfollows from Eqs.(68-77) that g is a power series in α s ( Q ) and α s ( µ ) with finitecoefficients in the limit ǫ → ǫ → g ( α s ( Q ) , α s ( µ ) , ǫ ) ≡ g ( α s ( Q ) , α s ( µ ))= ∞ X m =0 ∞ X n =1 g mn α ms ( Q ) α ns ( µ ) . (79)Hence, our final result for the four-dimensional all-rder resummed physical anomalousdimension is given by γ res ( N, α s ( Q )) = Z N a dnn g ( α s ( Q ) , α s ( Q /n )) + O ( N ) . (80)This result can be compared to the all-order resummation formula derived inRef.[51]. This resummation has the form of Eq.(80), but with g a function of α s ( µ )only, i.e. with all g mn = 0 when m > g mn . The predictive powerof the resummation formulae is analyzed in detail in Chapter 7. According to themore restrictive result of Ref.[51], Eq.(80) becomes γ res ( N, α s ( Q )) = Z N a dnn g ( α s ( Q /n )) + O ( N ) , (81)where g ( α s ( µ )) = ∞ X n =1 g n α ns ( µ ) . (82)The conditions under which the more restrictive result of Ref.[51] holds can beunderstood by comparing to our approach the derivation of that result. The approachof Ref.[51] is based on assuming the validity of the factorization formula Eq.(47) ofSec.2.3.2 which is more restrictive than the standard collinear factorization. Thisfactorization was proven for a wide class of processes in Ref.[11], and implies that theperturbative coefficient function Eq.(57) in the large N limit can be factored as: C (0) ( N, Q , α , ǫ ) = C (0 ,l ) ( Q /N a , α , ǫ ) C (0 ,c ) ( Q , α , ǫ ) . (83) Renormalization group resummation of DIS and DY
We notice that this can happen if and only if the coefficients C (0) nk ( ǫ ) in Eq.(57) canbe written in the form C (0) nk ( ǫ ) = F k ( ǫ ) G n − k ( ǫ ) . (84)The validity of factorization Eq.(47) of Sec.2.3.2 to all orders and for various processesis based on assumptions whose reliability will not be discussed here. Anyway, Eq.(83)implies that the physical anomalous dimension Eq.(60) becomes γ ( N, α s ( Q ) , ǫ ) = γ ( l ) ( α s ( Q /N a ) , ǫ ) + γ ( c ) ( α s ( Q ) , ǫ ) . (85)Then, proceeding as before, one then ends up with the resummation formula Eq.(81). In this Section, we shall give explicit expressions of the reummation formulae at NLLfor the deep-inelastic structure function F and for the Drell-Yan cross section. Theseexplicit expressions are useful for practical computations and we shall use them inChapter 5.The expression of the resummed physical anomalous dimension in Eq.(80) can beused to compute the resummed evolution factor K res ( N ; Q , Q ) Eq.(19) in Section2.2. At NLL, we get K NLL ( N ; Q , Q ) = exp (cid:2) E NLL ( N ; Q , Q ) (cid:3) = exp Z Q Q dk k γ resNLL ( N, α s ( k )) , (86)where γ resNLL ( N, α s ( k )) = Z N a dnn (cid:2) g α s ( k /n ) + g α s ( k /n ) (cid:3) ; g = 0 . (87)The fact that the resummation coefficient g = 0 for both the deep-inelastic and theDrell-Yan case can be shown by explicit computations of the fixed-order anomalousdimension (see Ref.[8] in Section 4.3). This shows that at NLL level Eq.(81) holds.However this does not mean that this is the case at all logarithmic orders.Many times in literature, the resummed results are given in terms of the Mellintransform of a resummed physical anomalous dimension in z space. To rewrite Eq.(87)as the Mellin transform of a function of x , we can use the all-orders relations betweenthe logs of N and the logs of 1 − z that are given in Appendix A. In particular, wecan use Eqs.(23,24) in Appendix A to rewrite Eq.(87) at NLL in the following form: γ resNLL ( N, α s ( k )) = a Z dx x N − − − x (cid:2) ˆ g α s ( k (1 − x ) a ) + ˆ g α s ( k (1 − x ) a ) (cid:3) , (88)where ˆ g = − g , ˆ g = − ( g + aγ E b g ) . (89) .4 NLL resummation and where we have used the definition of the beta function Eq.(43) in Section 1.2.As a consequence, the NLL resummed exponent in Eq.(86) can be rewritten in thefollowing form E NLL ( N ; Q , Q ) = a Z dx x N − − − x Z Q (1 − x ) a Q (1 − x ) a dk k ˆ g ( α s ( k )) , (90)where ˆ g ( α s ) = ˆ g α s ( k ) + ˆ g α s ( k ) (91)Beyond leading order the standard anomalous dimension differs from the physicalone, so ˆ g receives a contribution both from the standard anomalous dimension andfrom the coefficient function. It is thus natural to rewrite the resummation formulaEq.(90) separating off the contribution the contribution which originates from theanomalous dimension γ AP Eq.(18) of Section 2.2. This is done defining two functionsof α s , A ( α s ) and B a ( α s ) in such a way thatˆ g ( α s ) = A ( α s ) + ∂B ( a ) ( α s ( k )) ∂ ln k , A ( α s ) = A α s + A α s , B ( a ) ( α s ) = B ( a )1 α s . (92)It is clear that the constant A i are obtained directly form the coefficients of the1 / [1 − x ] + terms of the i -loop quark-quark splitting functions and that the coefficients B ai depends on the particular process ( a = 1 for DIS and a = 2 for DY). To the NLLorder, we find that ˆ g = A , ˆ g = A − b B ( a )1 (93)and that E NLL ( N ; Q , Q ) = a Z dx x N − − − x (cid:20) Z Q (1 − x ) a Q (1 − x ) a dk k A ( α s ( k ))+ B ( a ) ( α s ( Q (1 − x ) a )) − B ( a ) ( α s ( Q (1 − x ) a )) (cid:21) . (94)We can then rewrite the resummed cross section σ NLL ( N, Q ) = exp (cid:2) E NLL ( N ; Q , Q ) (cid:3) σ NLL ( N, Q ) (95)in a factorized form according to Eqs.(11,12) in Section 2.2 by collecting all Q -dependent contributions to the resummation Eq.(94) into a resummed perturbativecoefficient function C NLL : σ NLL ( N, Q ) = C NLL ( N, Q /µ , α s ( µ )) F ( N, µ ) , (96)where C NLL ( N, Q /µ , α s ( µ )) = exp (cid:26) a Z dx x N − − − x (cid:20) Z Q (1 − x ) a µ dk k A ( α s ( k ))+ B ( a ) ( α s ( Q (1 − x ) a )) (cid:21)(cid:27) , (97) Renormalization group resummation of DIS and DY which ha the same form of the resummed results discussed in Section 2.3.1. Theprecise definition of the parton distribution F and the factorization scale µ willdepend on the choice of factorization scheme: according to the choice of scheme, theresummed terms will be either part of the hard coefficient function C NLL , or of theevolution of the parton distribution F . In the M S scheme the NLL coefficients A , A and B ( a )1 are given in Eq.(46) of Section 2.3.1 and the NNLL ones are given forexample in Ref.[52]. These coefficients can also be obtained comparing a fixed ordercomputation of the physical anomalous dimension with a fixed order expansion ofEq.(80) as is shown explicitly in Section 4.3 of Ref.[8].To compute explicitly Eq.(97), we first exploit Eq.(20) in Appendix A at NLLlevel, thus finding C NLL ( N, Q /µ , α s ( µ )) = exp ( − Z N a dnn "Z Q nµ dk k A ( α s ( k /n )) + ˜ B ( a ) ( α s ( Q /n )) , (98)where ˜ B ( a ) ( α s ) = B ( a ) ( α s ) − aγ E A α s . The explicit expression of Eq.(97) is thenobtained performing the changes of variables, dk k = dα s ( k /n ) β ( α s ( k /n )) , dnn = − dα s ( Q /n ) β ( α s ( Q /n )) , (99)to evaluate the integrals in Eq.(98) and using the two loop solution of the renormalitazion-group equation for the running of α s given in Eq.(41) of Section 1.2, Now, after somealgebra we find for the integral in Eq.(98): − Z N a dnn "Z Q nµ dk k (cid:18) A α s ( k n ) + A α s ( k n ) (cid:19) + ˜ B ( a )1 α s ( Q n ) = log N g ( λ, a ) + g , ( λ, a ) (100)where λ = b α s ( µ r ) log N and g ( λ, a ) = A b λ [ aλ + (1 − aλ ) log(1 − aλ )] (101) g ( λ, a ) = − A aγ E − B ( a )1 b log(1 − aλ ) + A b b [ aλ + log(1 − aλ ) + 12 log (1 − aλ )] − A b [ aλ + log(1 − aλ )] + log (cid:18) Q µ r (cid:19) A b log(1 − aλ )+ log (cid:18) µ µ r (cid:19) A b aλ, (102)where a = 1 for the DIS case and a = 2 for the DY case. Evidently, in Eqs.(101,102)there is a dependence on the renormalization scale. To obtain the desired result, wesimply have to keep the renormalization scale equal to the factorization scale µ r = µ .Thus, for the explicit analytic expression of Eq.(98), we get C NLL (cid:18)
N, Q µ , α s ( µ ) (cid:19) = [exp { log N g ( λ, a ) + g ( λ, a ) } ] µ r = µ , (103) .4 NLL resummation with the resummation coefficients given in Eq.(46) in Section 2.3.1 for the M S factor-ization scheme choice. However, a general analysis of the factorization scheme choicesand changes for the resummation formulae is given for example in Section 6 of Ref.[8]. hapter 4Renormalization groupresummation of prompt photonproduction
In this chapter, we prove the all-order exponentiation of soft logarithmic corrections toprompt photon production in hadronic collisions, by generalizing the renormalizationgroup approach of chapter 3. Here, we will show that all large logs in the softlimit can be expressed in terms of two dimensionful variables. Then, we use therenormalization group to resum them. The resummation formulae that we obtainare more general though less predictive than those that can be obtained with otherapproaches discussed in chapter 2.
We consider the process H ( P ) + H ( P ) → γ ( p γ ) + X, (1)of two colliding hadrons H and H with momentum P and P respectively into a realphoton with momentum p γ and any collection of hadrons X . More specifically, we areinterested in the differential cross section p ⊥ dσdp ⊥ ( x ⊥ , p ⊥ ), where p ⊥ is the transversemomentum of the photon with respect to the direction of the colliding hadrons H and H , and x ⊥ = 4 p ⊥ S ; S = ( P + P ) . (2)The scaling variable x can be viewed as the squared fraction of transverse energythat the hadrons transfer to the outgoing particles (hence 0 ≤ x ⊥ ≤
1) and S isthe hadronic center-of-mass energy. We parametrize the momentum of the photon interms of its partonic center-of-mass pseudorapidity and its transverse momentum ~p ⊥ .The pseudorapidiry of a massless particle is defined in terms its scattering angle θ inthe center-of-mass frame as follows η = − ln(tan( θ/ . (3)53 Renormalization group resummation of prompt photon production
So, in the partonic center-of-mass frame, we can write: p γ = ( p ⊥ cosh ˆ η γ , ~p ⊥ , p ⊥ sinh ˆ η γ ) . (4)In the same frame, the incoming partons’ momenta can be written as p = x P = √ s , ~ ⊥ , , p = x P = √ s , ~ ⊥ , − , (5)where x are the longitudinal fraction of momentum of the parton 1(2) in the hadron H and s = ( p + p ) = x x S is the center-of-mass energy of the partonic process.The relation between the hadronic center-of-mass pseudorapidity and the partonicone is obtained performing a boost along the collision axis:ˆ η γ = η γ −
12 ln x x . (6)The factorized expression for this cross section in perturbative QCD is p ⊥ dσdp ⊥ ( x ⊥ , p ⊥ ) = X a,b Z dx dx dz x F H a ( x , µ ) x F H b ( x , µ ) × C ab (cid:18) z, Q µ , α s ( µ ) (cid:19) δ ( x ⊥ − zx x ) , (7)where F H a ( x , µ ), F H b ( x , µ ) are the distribution functions of partons a, b in thecolliding hadrons. Here we have defined the perturbative scale Q and the partonicscaling variable z as the squared fraction of transverse energy that the parton a, b transfer to the outgoing partons (0 ≤ z ≤ Q = 4 p ⊥ , (8) z = Q s = Q x x S . (9)The coefficient function C ab ( z, Q µ , α s ( µ )) is defined in terms of the partonic crosssection for the process where partons a , b are incoming as C ab (cid:18) z, Q µ , α s ( µ ) (cid:19) = p ⊥ d ˆ σ ab dp ⊥ . (10) The hard-scattering subprocesses that contribute to the amplitude of the prompt-photon prodution at the leading order are: q ( p ) + ¯ q ( p ) → g ( p ′ ) + γ ( p γ ) (11) q ( p ) + g ( p ) → q ( p ′ ) + γ ( p γ ) (12)¯ q ( p ) + g ( p ) → ¯ q ( p ′ ) + γ ( p γ ) (13) .2 Leading order calculation Figure 4.1: Feynman graphs that contribute to the amplitude of the partonic processat the leading orderThe corresponding Feynman graphs are shown in figure 4.1.We now want to obtain the coefficient function for these two elementary sub-processes at the leading order. Using the QCD Feynman rules to evaluate the firstamplitude in figure 4.1, we have: i M q ¯ q → γg = − i ¯ v ( p ) Q q eg (cid:20) γ ρ / p γ γ µ − γ ρ p µ p · p γ + γ µ / p ′ γ ρ − γ µ p ρ p · p ′ (cid:21) ǫ ∗ µ ( p γ ) ǫ ∗ ρ ( p ′ ) t aji u ( p ) , (14)where e = | e | is the electrical charge, g is the strong charge and Q q is the electricalcharge of the quark q in units of e . Taking the square modulus of equation (14) andaveraging over the two quarks polarizations and colours we obtain:14 1 N C X pol,col |M q ¯ q → γg | = 2 C F N C Q q e g (cid:20) p · p γ p · p ′ + p · p ′ p · p γ (cid:21) , (15)where N C is the number of quark colours and C F ≡ ( N C − / N C is the Casimiroperator with respect to the colour matrix t aji .Proceeding in the same way for the Feynman graphs contributing to the secondamplitude in figure 4.1 we could obtain the averaged square modulus for this othersubprocess, but, we note that it can be immediately obtained using crossing simmetry.More precisely, we only have to substitute p ′ with p and take into account the factthat we must now average not over two quarks’ colours but over the colours of a quark Renormalization group resummation of prompt photon production and a gluon. Therefore we arrive at the following expression for the averaged squaremodulus of the second amplitude of figure 4.1:14 12 N C C F N C X pol,col |M q (¯ q ) g → γq (¯ q ) | = 14 12 N C C F N C X pol,col |M q ¯ q → γg | (cid:12)(cid:12)(cid:12)(cid:12) p ′ → p == 1 N C Q q e g (cid:20) p · p γ p · p + p · p p · p γ (cid:21) . (16)We will now rewrite equations (15) and (16) in terms of the kinematic parametersdefined in section 4.1. For this purpose we first note that momentum conservationand the parametrizations defined in Eqs.(4,5) imply (in the centre of mass of theincident partons): p ′ = ( p ⊥ cosh ˆ η γ , − ~p ⊥ , − p ⊥ sinh ˆ η γ ) (17) s = ( p + p ) = ( p γ + p ′ ) = 4 p ⊥ cosh ˆ η γ (18) z = 4 p ⊥ s = 1cosh ˆ η γ . (19)From the last equation we see that in the limit z →
1, ˆ η γ →
0. Physically this isbecause in this limit all the centre of mass energy is transverse and so the photoncannot have a non zero pseudorapidity. Now, using again Eqs.(4,5) and the equationfor p ′ (17) we have: p · p = s p · p ′ = √ s p ⊥ e ˆ η γ (20) p · p γ = √ s p ⊥ e − ˆ η γ A combination of equations (19) and (20) yields our final result for the averagedsquare modulus of the amplitudes:14 1 N C X pol,col |M q ¯ q → γg | = 4 C F N C Q q e g (2 − z ) z (21)14 12 N C C F N C X pol,col |M q (¯ q ) g → γq (¯ q ) | = 12 N C Q q e g (cid:20) ± √ − z + 41 ± √ − z (cid:21) , (22)where the plus sign has to be chosen for positive values of the pseudorapidity ˆ η γ andthe minus sign for negative values. The two-body phase space is: dφ ( p + p ; p γ , p ′ ) = d p γ (2 π ) E p γ d p ′ (2 π ) E p ′ (2 π ) δ (4) ( p ′ + p γ − p − p )= d p γ E p γ E p ′ π ) δ (1) ( E p ′ + E p γ − E p − E p ) . (23) .2 Leading order calculation Imposing the conservation of spatial momentum ( E p γ = E p ′ = | ~p γ | ) the two-bodyphase space becomes: dφ ( p + p ; p γ , p ′ ) = d p γ | ~p γ | (2 π ) δ (1) (2 | ~p γ | − √ ˆ s )= 116 π d cos θd | ~p γ | δ (1) ( | ~p γ | − √ ˆ s/ π δ (1) ( p ⊥ cosh ˆ η γ − √ ˆ s/ d cos θd | ~p γ | , (24)where θ is the scattering angle of the photon with respect to the collision axis. Becauseof the fact that we want to integrate over the pseudorapidity of the photon η γ at fixed p ⊥ , we must perform a change of variables. In particular we must rewrite the two-body phace space, expressed in terms of cos θ and | ~p γ | , in terms of the new variablescosh ˆ η γ and p ⊥ . This change of variables is given by the equations: (cid:26) cos θ = tanh ˆ η γ | ~p γ | = p ⊥ cosh ˆ η γ . The determinant of the Jacobian matrix is easily obtained and is given by: | J | = (cid:13)(cid:13)(cid:13)(cid:13) ∂ (cos θ, | ~p γ | ) ∂ (cosh ˆ η γ , p ⊥ ) (cid:13)(cid:13)(cid:13)(cid:13) = 1cosh ˆ η γ q cosh ˆ η γ − . (25)Thanks to equation (19), we obtain this determinant in term of z : | J | = z √ − z (26)Using this last result, equation (24) becomes: dφ ( p + p ; p γ , p ′ ) = 116 π z √ − z δ (1) (cosh ˆ η γ − √ s/ p ⊥ ) dp ⊥ p ⊥ d cosh ˆ η γ = 116 π z √ − z [ δ (1) (ˆ η γ − ˆ η + ) + δ (1) (ˆ η γ − ˆ η − )] × dp ⊥ p ⊥ d ˆ η γ , (27)where ˆ η + and ˆ η − are the two solutions (one positive and one negative) of the equationimposed by the delta function p ⊥ cosh ˆ η γ = √ s/ η ± = ln (cid:18) √ s p ⊥ ± r s p ⊥ − (cid:19) . (28)The flux factor Φ is immediately obtained from equations (5) and (19):Φ ≡ p · p ) = 2 s = 8 p ⊥ z . (29) Renormalization group resummation of prompt photon production
Remembering the definition of the QED and QCD coupling constants, α ≡ e π ; α s ≡ g π , and putting together expressions (21), (22), (27), (29) and performing the integrationover ˆ η γ , we obtain our final result for the coefficient function at the leading order forthe two subprocesses in figure 4.1: C ( LO ) q ¯ q → γg ( z, α s ) = αα s Q q π C F N C z √ − z (2 − z ) (30) C ( LO ) q (¯ q ) g → γq (¯ q ) ( z, α s ) = αα s Q q π N C z √ − z (cid:16) z (cid:17) . (31) We will study the cross section Eq. (7) in the threshold transverse limit, when thetransverse transverse energy of outgoing particles is close to its maximal value ( x ⊥ → z → σ ( N, Q ) = X a,b σ ab ( N, Q ) (32)= X a,b F H a ( N + 1 , µ ) F H b ( N + 1 , µ ) × C ab (cid:18) N, Q µ , α s ( µ ) (cid:19) . (33)As discussed in Section 2.2, in the large N limit each parton subprocess canbe treated independently, specifically, each C ab is separately renormalization-groupinvariant.At this point, it is interesting to discuss the differences in the large N behaviorof the partonic subprocesses. The cross sections for the partonic channels with twoquarks of different flavors ( ab = q ¯ q ′ , ¯ qq ′ , qq , qq ′ , ¯ q ¯ q , ¯ q ¯ q ′ ) vanish at LO and are hencesuppressed by a factor of α s with respect to the subprocesses with ab = q ¯ q , qg , ¯ qg .Moreover, in the large N limit this relative suppression is further enhanced by afactor of O (1 /N ) because the processes with two different quark flavors involve theoff-diagonal Alatarelli-Parisi splitting functions. Therefore, these partonic channelsdo not contribute in the large N limit. The partonic channel ab = gg has a differentlarge N behavior. It begins to contribute at NLO via the partonic process g + g → γ + q + ¯ q , which again leads to a suppression effect of O (1 /N ) with respect to theLO suprocesses. However, owing to the photon-gluon coupling through a fermionbox, the partonic subprocess g + g → γ + g is also permitted. This subprocessis logarithmically-enhanced by multiple soft-gluon radiation in the final state, but itstarts to contribute only at NNLO in perturbation theory. It follows that the partonic .3 The soft limit channel ab = gg is suppressed by a factor α s with respect to the LO partonic channels ab = q ¯ q , qg , ¯ qg and it will enter the resummed cross secion only at NNLL order. Inconclusion, the partonic channels that should be resummed are ab = q ¯ q , qg , ¯ qg , gg ,where the last channel is that coupled to the gluon via a fermion box and enetersresummation only at NNLL.Furthermore, on top of Eqs. (7, 33) the physical process Eq. (1) receives anotherfactorized contribution, in which the final-state photon is produced by fragmentationof a primary parton produced in the partonic sub-process. However, the cross sectionfor this process is also suppressed by a factor of N in the large N limit. This isdue to the fact that the fragmentation function carries this suppression, for the samereason why the anomalous dimensions γ qg and γ gq are suppressed. Therefore, we willdisregard the fragmentation contribution.According to Eqs.(19-22) of Section 2.2, the cross section can be written in termsof the physical anomalous dimensions: σ ( N, Q ) = X a,b K ab ( N ; Q , Q ) σ ab ( N, Q ) (34)= X a,b exp (cid:2) E ab ( N ; Q , Q ) (cid:3) σ ab ( N, Q ) , (35)where E ab ( N ; Q , Q ) = Z Q Q dk k γ ab ( N, α s ( k )) (36)= Z Q Q dk k [ γ AP aa ( N, α s ( k )) + γ AP bb ( N, α s ( k ))]+ ln C ab ( N, , α s ( Q )) − ln C ab ( N, , α s ( Q )) . (37)In the large- x ⊥ limit, the order- n coefficient of the perturbative expansion ofthe hadronic cross section is dominated by terms proportional to h ln k (1 − x ⊥ )1 − x ⊥ i + , with k ≤ n −
1, that must be resummed to all orders. Upon Mellin transformation, theselead to contributions proportional to powers of ln N . In the sequel, we will considerthe resummation of these contributions to all logarithmic orders, and disregard allcontributions to the cross section which are suppressed by powers of (1 − x ⊥ ), i.e.,upon Mellin transformation, by powers of N .The resummation is performed in two steps as in chapter 3. First, we show that theorigin of the large logs is essentially kinematical: we identify the configurations whichcontribute in the soft limit, we show by explicit computation that large Sudakov logsare produced by the phase-space for real emission with the required kinematics as logsof two dimensionful variables, and we show that this conclusion is unaffected by virtualcorrections. Second, we resum the logs of these variables using the renormalizationgroup.The l -th order correction to the leading O ( α s ) partonic process receives contri-bution from the emission of up to l + 1 massless partons with momenta k , . . . , k l +1 . Renormalization group resummation of prompt photon production
Four-momentum conservation implies: p + p = p γ + k + . . . k l +1 . (38)In the partonic center-of-mass frame, according to Eqs.(4,5), we have( p + p − p γ ) = Q z (1 − √ z cosh ˆ η γ ) = l +1 X i,j =1 k i k j (1 − cos θ ij ) ≥ , (39)where θ ij is the angle between ~k i and ~k j . Hence,1 ≤ cosh ˆ η γ ≤ √ z . (40)Therefore, l +1 X i,j =1 k i k j (1 − cos θ ij ) = Q − z ) + O (cid:2) (1 − z ) (cid:3) . (41)Equation (41) implies that in the soft limit the sum of scalar products of momenta k i of emitted partons Eq. (39) must vanish. However, contrary to the case of deep-inelastic scattering or Drell-Yan, not all momenta k i of the emitted partons can be softas z →
1, because the three-momentum of the photon must be balanced. Assume thusthat momenta k i , i = 1 , . . . , n ; n < l + 1 are soft in the z → k i , i > n are non-soft. For the sake of simplicity, we relabel non-soft momenta as k ′ j = k n + j ; 1 ≤ j ≤ m + 1; m = l − n. (42)The generic kinematic configuration in the z = 1 limit is then k i = 0 1 ≤ i ≤ nθ ij = 0; m +1 X j =1 k ′ j = p ⊥ ≤ i, j ≤ m + 1 . (43)for all n between 1 and l , namely, the configuration where at least one momentum isnot soft, and the remaining momenta are either collinear to it, or soft.With this labelling of the momenta, the phase space can be written, using twicethe phase space decomposition of Eq.(12) in Appendix B, as dφ n + m +2 ( p + p ; p γ , k , . . . , k n , k ′ , . . . , k ′ m +1 )= Z s dq π dφ n +1 ( p + p ; q, k , . . . , k n ) × Z q dk ′ π dφ m +1 ( k ′ ; k ′ , . . . , k ′ m +1 ) dφ ( q ; p γ , k ′ ) . (44)We shall now compute the phase space in the z → d = 4 − ǫ dimensions.Consider first the two-body phase space dφ in Eq. (44). In the rest frame of q we .3 The soft limit have dφ ( q ; p γ , k ′ ) = d d − k ′ (2 π ) d − k ′ d d − p γ (2 π ) d − p γ (2 π ) d δ ( d ) ( q − k ′ − p γ )= (4 π ) ǫ π Γ(1 − ǫ ) P − ǫ p q sin − ǫ θ γ d | ~p γ | d cos θ γ δ ( | ~p γ | − P ) , (45)where P = p q (cid:18) − k ′ q (cid:19) . (46)Because momenta k i , i ≤ n are soft, up to terms suppressed by powers of 1 − z , therest frame of q is the same as the center-of-mass frame of the incoming partons, inwhich | ~p γ | = p ⊥ cosh ˆ η γ (47)cos θ γ = tanh ˆ η γ . (48)Hence, dφ ( q ; p γ , k ′ ) = (4 π ) ǫ π Γ(1 − ǫ ) ( Q / − ǫ p q dp ⊥ d ˆ η γ δ cosh ˆ η γ − P p Q ! . (49)The conditions cosh ˆ η γ = 2 P p Q ≥ k ′ ≥ , (50)together with Eq.(46), restrict the integration range to Q ≤ q ≤ s (51)0 ≤ k ′ ≤ q − p Q q . (52)It is now convenient to define new variables u, vq = Q + u ( s − Q ) = Q [1 + u (1 − z )] + O ((1 − z ) ) (53) k ′ = v ( q − p Q q ) = Q uv (1 − z ) + O ((1 − z ) ) (54)0 ≤ u ≤ ≤ v ≤ , (55)in terms of which P = p Q (cid:20) u (1 − v )(1 − z ) (cid:21) + O (cid:2) (1 − z ) (cid:3) . (56)Thus, the two-body phase space Eq. (49) up to subleading terms is given by dφ ( q ; p γ , k ′ ) = (4 π ) ǫ π Γ(1 − ǫ ) ( Q / − ǫ p Q dp ⊥ d ˆ η γ δ (ˆ η γ − ˆ η + ) + δ (ˆ η γ − ˆ η − ) p u (1 − v )(1 − z ) , (57) Renormalization group resummation of prompt photon production where ˆ η ± = ln P p Q ± s P Q − ! = ± p u (1 − v )(1 − z ) . (58)We now note that the phase-space element dφ n +1 ( p + p ; q, k , . . . , k n ) containsin the final state a system with large invariant mass q ≥ Q , plus a collection of n soft partons: this same configuration is encountered in the case of Drell-Yan pairproduction in the limit z DY = q /s →
1, discussed in Section 3.2 . Likewise, the phasespace for the set of collinear partons dφ m +1 ( k ′ ; k ′ , . . . , k ′ m +1 ) is the same as the phasespace for deep-inelastic scattering (discussed in Section 3.1), where the invariant massof the initial state k ′ vanishes as 1 − z (see Eq. (54)). We may therefore use theresults obtained in chapter 3, where, in the case of deep-inelastic scattering, one ofthe outgoing parton momenta ( k ′ m +1 , say) was identified with the momentum of theleading-order outgoing quark p ′ . Hence Eq. (60) is obtained from the correspondingresult in chapter 3 for deep-inelastic scattering by the replacement p ′ → k ′ m +1 : dφ n +1 ( p + p ; q, k , . . . , k n ) = 2 π (cid:20) N ( ǫ )2 π (cid:21) n ( q ) − n (1 − ǫ ) ( s − q ) n − − nǫ × d Ω ( n ) ( ǫ ) (59) dφ m +1 ( k ′ ; k ′ , . . . , k ′ m +1 ) = 2 π (cid:20) N ( ǫ )2 π (cid:21) m ( k ′ ) m − − mǫ d Ω ′ ( m ) ( ǫ ) , (60)where N ( ǫ ) = 1 / (2(4 π ) − ǫ ) and d Ω ( n ) ( ǫ ) = d Ω . . . d Ω n Z dz n z ( n − n − − ǫ ) n (1 − z n ) − ǫ . . . × Z dz z − ǫ (1 − z ) − ǫ (61) d Ω ′ ( m ) ( ǫ ) = d Ω ′ . . . d Ω ′ m Z dz ′ m z ′ ( m − − ( m − ǫm (1 − z ′ m ) − ǫ . . . × Z dz ′ z ′ − ǫ (1 − z ′ ) − ǫ . (62)Here, the variables z i , z ′ i are defined as in chapter 3 for the Drell-Yan and deep-inelasticscattering respectively.Equations (53,54) imply that the phase space depends on (1 − z ) − ǫ through thetwo variables k ′ ∝ Q (1 − z ) (63)( s − q ) q ∝ Q (1 − z ) , (64)where the coefficients of proportionality are dimensionless and z -independent. Byexplicitly combining the two-body phase space Eq. (57) and the phase spaces for soft .3 The soft limit radiation Eq. (59) and for collinear radiation Eq. (60) we get dφ n + m +2 ( p + p ; p γ , k , . . . , k n , k ′ , . . . , k ′ m +1 )= ( Q ) n + m − ( n + m +1) ǫ dp ⊥ p ⊥ (1 − z ) n + m − (2 n + m ) ǫ √ − z × − m + mǫ (16 π ) − ǫ Γ(1 − ǫ ) (cid:20) N ( ǫ )2 π (cid:21) n + m d ˆ η γ d Ω ( n ) ( ǫ ) d Ω ′ ( m ) ( ǫ ) × Z du u m − mǫ (1 − u ) n − − nǫ √ u Z dv v m − − mǫ √ − v [ δ (ˆ η γ − ˆ η + ) + δ (ˆ η γ − ˆ η − )] . (65)In the limiting cases n = 0 and m = 0 we have dφ ( p + p ; q ) = 2 πδ ( s − q ) = 2 πQ (1 − z ) δ (1 − u ) (66) dφ ( k ′ ; p ′ ) = 2 πδ ( k ′ ) = 4 πQ u (1 − z ) δ ( v ); (67)the corresponding expressions for the phase space are therefore obtained by simplyreplacing (1 − u ) − d Ω ( n ) ( ǫ ) → δ (1 − u ); v − d Ω ′ ( m ) ( ǫ ) → δ ( v ) (68)in Eq. (65) for n = 0, m = 0 respectively.The logarithmic dependence of the four-dimensional cross section on 1 − z is dueto interference between powers of (1 − z ) − ǫ and ǫ poles in the d -dimensional crosssection. Hence, we must classify the dependence of the cross section on powers of(1 − z ) − ǫ . We have established that in the phase space each real emission producesa factor of [ Q (1 − z ) ] − ǫ if the emission is soft and a factor of [ Q (1 − z )] − ǫ if theemission is collinear. The squared amplitude can only depend on (1 − z ) − ǫ becauseof loop integrations. This dependence for a generic proper Feynman diagram G willin general appear, as discussed in chapter 3, through the coefficient (see Eq.(22) ofSection 3.1) [ D G ( P E )] dL/ − I , (69)where L and I are respectively the number of loops and internal lines in G , and D G ( P E ) is a linear combination of all scalar products P E of external momenta. Inthe soft limit all scalar products which vanish as z → Q (1 − z ) or to Q (1 − z ) as shown in Eqs.(47,53,54) of Section 3.2 . Equation (69)then implies that each loop integration can carry at most a factor of [ Q (1 − z ) ] − ǫ or [ Q (1 − z )] − ǫ .This then proves that the perturbative expansion of the bare coefficient function,for each sub-process which involves partons a, b , takes the form C (0) ( z, Q , α , ǫ ) = αα ( Q ) − ǫ ∞ X l =0 α l C (0) l ( z, Q , ǫ ) (70) C (0) l ( z, Q , α , ǫ ) = ( Q ) − lǫ Γ(1 / √ − z l X k =0 l − k X k ′ =0 C (0) lkk ′ ( ǫ )(1 − z ) − kǫ − k ′ ǫ , (71) Renormalization group resummation of prompt photon production where the factor 1 / Γ(1 /
2) was introduced for later convenience and terms C (0) lkk ′ with k + k ′ < l at order α ls are present in general because of loops. The coefficients C (0) lkk ′ have poles in ǫ = 0 up to order 2 l . To understand this, we have to count the indepen-dent variables for the prompt photon process. We have 2 incoming particles and l + 2outgoing partons (a leading-order parton, the photon and l extra emissions). There-fore, imposing the on-shell conditions and the constraints due to Poincar invariance,we get 4( l + 4) − ( l + 4) −
10 = 3 l + 2 , (72)independent variables. Now, we need to understand which are these independentvariables: the general expression of the phase space in the threshold limit Eq.(65) iswritten in terms of 3 l + 3 variables which are s, p ⊥ , u, v, z , . . . , z n , z ′ , . . . , z ′ m , Ω , . . . , Ω n , Ω ′ , . . . , Ω ′ m , ˆ η γ , (73)where n + m = l and each solid angle depends on two parameters. Clearly, one of themmust be a function of some of the others because of Eq. 72. In fact, from Eq.(58),we know that ˆ η γ depends on u, v, p ⊥ , s . Thus, the 3 l + 2 independent variables onwhich depends the square modulus amplitude can be chosen as s, p ⊥ , u, v, z , . . . , z n , z ′ , . . . , z ′ m , Ω , . . . , Ω n , Ω ′ , . . . , Ω ′ m . (74)Now, each of the l integrations over a solid angle can produce a pole 1 /ǫ from thecollinear region. Furthermore, each of the l integrations over a dimensionless variable u, v, z , . . . , z n , z ′ , . . . , z ′ m can produce a pole 1 /ǫ from the soft region. This explainswhy the coefficients C (0) lkk ′ can have poles in ǫ = 0 up to order 2 l . The Mellin transform of Eq. (70) can be performed using Z dzz N − (1 − z ) − / − kǫ − k ′ ǫ = Γ(1 / √ N N kǫ N k ′ ǫ + O (cid:18) N (cid:19) , (75)with the result C (0) ( N, Q , α , ǫ )= αα ( Q ) − ǫ √ N ∞ X l =0 l X k =0 l − k X k ′ =0 C (0) lkk ′ ( ǫ ) "(cid:18) Q N (cid:19) − ǫ α k "(cid:18) Q N (cid:19) − ǫ α k ′ × (cid:2) ( Q ) − ǫ α (cid:3) l − k − k ′ + O (cid:18) N (cid:19) . (76)Equation (76) shows that indeed as N → ∞ , up to N corrections, the coefficientfunction depends on N through the two dimensionful variables Q N and Q N . The argu-ment henceforth follows in the same way as in chapter 3, in this more general situation. .4 Resummation from renormalization group improvement The argument is based on the observation that, because of collinear factorization, thephysical anomalous dimension γ ( N, α s ( Q )) = Q ∂∂Q ln C ( N, Q /µ , α s ( µ )) (77)is renormalization-group invariant and finite when expressed in terms of the renor-malized coupling α s ( µ ), related to α by α ( µ , α s ( µ )) = µ ǫ α s ( µ ) Z ( α s ) ( α s ( µ ) , ǫ ) , (78)where Z ( α s ) ( α s ( µ ) , ǫ ) is a power series in α s ( µ ). Because α is manifestly indepen-dent of µ , Eq. (78) implies that the dimensionless combination ( Q ) − ǫ α ( α s ( µ ) , µ )can depend on Q only through α s ( Q ):( Q ) − ǫ α ( α s ( µ ) , µ ) = α s ( Q ) Z ( α s ) ( α s ( Q ) , ǫ ) . (79)Using Eq. (79) in Eq. (76), the coefficient function and consequently the physicalanomalous dimension are seen to be given by a power series in α s ( Q ), α s ( Q /N ) and α s ( Q /N ): γ ( N, α s ( Q ) , ǫ ) = ∞ X m =0 ∞ X n =0 ∞ X p =0 γ Rmnp ( ǫ ) α ms ( Q ) α ns ( Q /N ) α ps ( Q /N ) . (80)Even though the anomalous dimension is finite as ǫ → N , the individualterms in the expansion Eq. (57) are not separately finite. However, if we separate N -dependent and N -independent terms in Eq. (57): γ ( N, α s ( Q ) , ǫ ) = ˆ γ ( c ) ( α s ( Q ) , ǫ ) + ˆ γ ( l ) ( N, α s ( Q ) , ǫ ) , (81)we note that the two functions γ ( c ) ( α s ( Q ) , ǫ ) ≡ ˆ γ ( c ) ( α s ( Q ) , ǫ ) + ˆ γ ( l ) (1 , α s ( Q ) , ǫ ) (82) γ ( l ) ( N, α s ( Q ) , ǫ ) ≡ ˆ γ ( l ) ( N, α s ( Q ) , ǫ ) − ˆ γ ( l ) (1 , α s ( Q ) , ǫ ) (83)must be separately finite. In fact, γ ( N, α s ( Q ) , ǫ ) = γ ( c ) ( α s ( Q ) , ǫ ) + γ ( l ) ( N, α s ( Q ) , ǫ ) , (84)is finite for all N and γ ( l ) vanishes for N = 1. This implies that γ ( c ) ( α s ( Q ) , ǫ ) isfinite in ǫ = 0 and that γ ( l ) ( N, α s ( Q ) , ǫ ) is also finite in ǫ = 0 for all N because ofthe N -independence of γ ( c ) ( α s ( Q ) , ǫ ).We can rewrite conveniently γ ( l ) ( N, α s ( Q ) , ǫ ) = Z N dnn g ( α s ( Q ) , α s ( Q /n ) , α s ( Q /n ) , ǫ ) , (85)where g ( α s ( Q ) , α s ( Q /n ) , α s ( Q /n ) , ǫ ) = n ∂∂n ˆ γ ( l ) ( n, α s ( Q ) , ǫ ) . (86) Renormalization group resummation of prompt photon production is a Taylor series in its arguments whose coefficients remain finite as ǫ →
0. In fourdimension we have thus γ ( N, α s ( Q )) = γ ( l ) ( N, α s ( Q ) ,
0) + γ ( c ) ( N, α s ( Q ) ,
0) + O (cid:18) N (cid:19) = γ ( l ) ( N, α s ( Q ) ,
0) + O (cid:0) N (cid:1) = Z N dnn g ( α s ( Q ) , α s ( Q /n ) , α s ( Q /n )) + O (cid:0) N (cid:1) , (87)where g ( α s ( Q ) , α s ( Q /n ) , α s ( Q /n )) ≡ lim ǫ → g ( α s ( Q ) , α s ( Q /n ) , α s ( Q /n ) , ǫ ) (88)is a generic Taylor series of its arguments.Renormalization group invariance thus implies that the physical anomalous dimen-sion γ Eq. (77) depends on its three arguments Q , Q /N and Q /N only through α s . Clearly, any function of Q and N can be expressed as a function of α s ( Q ) and α s ( Q /N ) or α s ( Q /N ). The nontrivial statement, which endows Eq. (87) with pre-dictive power, is that the log derivative of γ , g ( α s ( Q ) , α s ( Q /n ) , α s ( Q /n )) Eq. (86),is analytic in its three arguments. This immediately implies that when γ is computedat (fixed) order α ks , it is a polynomial in ln N of k -th order at most.In order to discuss the factorization properties of our result we write the function g as g ( α s ( Q ) , α s ( Q /n ) , α s ( Q /n )) = g ( α s ( Q ) , α s ( Q /n ))+ g ( α s ( Q ) , α s ( Q /n ))+ g ( α s ( Q ) , α s ( Q /n ) , α s ( Q /n )) , (89)where g ( α s ( Q ) , α s ( Q /n )) = ∞ X m =0 ∞ X p =1 g m p α ms ( Q ) α ps ( Q /n ) (90) g ( α s ( Q ) , α s ( Q /n )) = ∞ X m =0 ∞ X n =1 g mn α ms ( Q ) α ns ( Q /n ) (91) g ( α s ( Q ) , α s ( Q /n ) , α s ( Q /n )) = ∞ X m =0 ∞ X n =1 ∞ X p =1 g mnp α ms ( Q ) × α ns ( Q /n ) α ps ( Q /n ) . (92)The dependence on the resummation variables Q , Q /N and Q /N is fully factorizedif the bare coefficient functions has the factorized structure C (0) ( N, Q , α , ǫ ) = C (0 ,c ) ( Q , α , ǫ ) C (0 ,l )1 ( Q /N, α , ǫ ) C (0 ,l )2 ( Q /N , α , ǫ ) . (93) .4 Resummation from renormalization group improvement This is argued to be the case in the approach of Refs.[14, 12]. If this happens,the resummed anomalous dimension is given by Eq. (87) with all g mnp = 0 except g n , g p : γ ( N, α s ( Q )) = Z N dnn g (0 , α s ( Q /n )) + Z N dnn g (0 , α s ( Q /n )) . (94)We recall that the coefficient function depends on the parton subprocess in which theincoming partons are a, b (compare Eq. (7)). So, the factorization Eq. (93) appliesto the coefficient function corresponding to each subprocess, and the decompositionEq. (94) to the physical anomalous dimension computed from each of these coefficientfunctions.A weaker form of factorization is obtained assuming that in the soft limit the N -dependent and N -independent parts of the coefficient function factorize: C (0) ( N, Q , ǫ ) = C (0 ,c ) ( Q , α , ǫ ) C (0 ,l ) ( Q /N , Q /N, α , ǫ ) . (95)This condition turns out to be satisfied [8] in Drell-Yan and deep-inelastic scatteringto order α s . It holds in QED to all orders [53] as a consequence of the fact that eachemission in the soft limit can be described by universal (eikonal) factors, independentof the underlying diagram. This eikonal structure of Sudakov radiation has beenargued in Refs. [2, 14] to apply also to QCD. If the factorized form Eq. (95) holds,the coefficients g mnp Eqs. (90,91,92) vanish for all m = 0, and the physical anomalousdimension takes the form γ ( N, α s ( Q )) = Z N dnn g (0 , α s ( Q /n )) + Z N dnn g (0 , α s ( Q /n ))+ Z N dnn g (0 , α s ( Q /n ) , α s ( Q /n )) . (96)It is interesting to observe that in the approach of Refs. [14, 12] for processes wheremore than one colour structure contributes to the cross-section, the factorizationEq. (93) of the coefficient function is argued to take place separately for each colourstructure. This means that in such case the exponentiation takes place for each colourstructure independently, i.e. the resummed cross section for each parton subprocessis in turn expressed as a sum of factorized terms of the form of Eq. (93). Thishappens for instance in the case of heavy quark production [14, 54]. In promptphoton production different colour structures appear for the gluon-gluon subprocesswhich starts at next-to-next-to-leading order, hence their separated exponentiationwould be relevant for next-to-next-to-leading log resummed results.When several colour structures contribute to a given parton subprocess, the coef-ficients of the perturbative expansion Eq. (76) for that process take the form C (0) lkk ′ ( ǫ ) = C (0) lkk ′ ( ǫ ) + C (0) lkk ′ ( ǫ ) , (97)(assuming for definiteness that a colour singlet and octet contribution are present)so that the coefficient function can be written as a sum C (0) = C (0) + C (0) . The Renormalization group resummation of prompt photon production argument which leads from Eq. (76) to the resummed result Eq. (87) then impliesthat exponentiation takes place for each colour structure independently if and only if γ ≡ ∂ ln C (0) /∂ ln Q , γ ≡ ∂ ln C (0) /∂ ln Q (98)are separately finite.This, however, is clearly a more restrictive assumption than that under whichwe have derived the result Eq. (87), namely that the full anomalous dimension γ isfinite. It follows that exponentiation of each colour structure must be a special caseof our result. However, this can only be true if the coefficients g ijk of the expansionEq. (90) of the physical anomalous dimension satisfy suitable relations. In particular,at the leading log level, it is easy to see that exponentiation of each colour structureis compatible with exponentiation of their sum only if the leading order coefficientsare the same for the given colour structures: g = g and g = g . This isindeed the case for heavy quark production (where g = 0).Note that, however, if the factorization holds for each colour structure separatelyit will not apply to the sum of colour structures. For instance, the weaker form offactorization Eq. (95) requires that C (0) lkk ′ ( ǫ ) = F k + k ′ ( ǫ ) G l − k − k ′ ( ǫ ) , (99)but F k + k ′ ( ǫ ) G l − k − k ′ ( ǫ ) + F k + k ′ ( ǫ ) G l − k − k ′ ( ǫ ) = F k + k ′ ( ǫ ) G l − k − k ′ ( ǫ ) . (100)Hence, our result Eq. (87) for the sum of colour structures is more general than theseparate exponentiation of individual colour structures, but it leads to results whichhave weaker factorization properties. In this section, we want to make a comparison with the resummation formula forprompt photon production previously released. In order to do this, we need to rewritethe NLL result of Ref.[14] in our formalism. The physical anomalous dimension canbe obtained performing the Q -logarithmic derivative of the NLL resummed exponentin the M S scheme of Ref.[14]. We obtain for a particular partonic sub-process: γ ( N, α s ( Q )) = Z dx x N − − − x (cid:2) ˆ g α s ( Q (1 − x ) ) + ˆ g ′ α s ( Q (1 − x ) )+ˆ g α s ( Q (1 − x )) + ˆ g ′ α s ( Q (1 − x − )) (cid:3) , (101) .5 Comparison with previous results where ˆ g = A (1) d + A (1) b − A (1) d π (102)ˆ g ′ = A (2) a + A b − A (2) d π − β π " A (1) a + A (1) b − A (1) d π ln 2 (103)ˆ g = A (1) d π (104)ˆ g ′ = A (2) d π − β π B (1) d π . (105)Here, A ia is the coefficient of ln(1 /N ) in the Mellin transform of the P aa Altarelli-Parisisplitting function at order α is , β is the α s coefficient of the β function (Eq.(38) insection 1.2) and B (1) d is a constant to be determined from the comparison with thefixed-order calculation. In Eqs.(102-105) a, b are the incoming partons (on which thecoefficient function implicitly depends) and d is the LO outgoing parton uniquely de-termined by the incoming ones. For completeness, we list explicitly these coefficients: A (1) a = q, ¯ q = 43 , A a = g = 3 (106) A (2) a = 12 A (1) a K, K = 676 − π − N f (107) B (1) d = q, ¯ q = − , B (1) d = g = −
112 + 13 N f , (108)where N f is the number of active flavors. Now, performing the change of variable n = 11 − x (109)in the integral Eq.(101), we obtain at NLL γ ( N, α s ( Q )) = Z dnn (cid:2) g α s (cid:18) Q n (cid:19) + g α s (cid:18) Q n (cid:19) + g α s (cid:18) Q n (cid:19) + g α s (cid:18) Q n (cid:19) (cid:3) , (110)where g = − ˆ g , g = − (cid:18) ˆ g ′ − γ E β π ˆ g (cid:19) (111) g = − ˆ g , g = − (cid:18) ˆ g ′ − γ E β π ˆ g (cid:19) , (112)with γ E the Euler constant. Hence, according to Ref.[14], we know exactly the valueof the coefficients g , g , g and g . This enables us to compute predictions ofhigh-order logarithmic contributions to the physical anomalous dimension performinga fixed order expansion of γ . Renormalization group resummation of prompt photon production
We shall now show that the resummation formula of Ref.[14] predicts the coef-ficient of α s ln (1 /N ) of the fixed order expansion of γ , while in our approach it isrequired in order to perform a NLL resummation. We need to expand Eq.(110) toorder α s and this is obtained using the change of variable dnn = − dα s ( Q /n a ) aβα s , a = 1 , α s (see Eq.(43) insection 1.2). We find γ = [ − ( g + g )] α s ( Q ) ln 1 N +[ − ( g + g ) − ( b /b )( g + g )] α s ( Q ) ln 1 N + (cid:20) b g + 2 g ) (cid:21) α s ( Q ) ln N + [ − ( g + g )] α s ( Q ) ln 1 N +[(3 b / g + 2 g ) + b ( g + 2 g )] α s ( Q ) ln N +[ − ( b / g + 4 g )] α s ln N + O ( α s ) . (114)In our approach, in order to determine the NLL resummation coefficients g , g , g and g , we must compare this expansion to a fixed order computation of thephysical anomalous dimension, which in the general has the form γ FO ( N, α s ) = k X i =1 α is i X j =1 γ ij ln j N + O ( α k +1 s ) + O ( N ) , (115)where k is the fixed-order at which it has been computed (see Chapter 7 for a gen-eral discussion about the determination of the resummation coefficients). Hence, wedetermine the 4 NLL resummation coefficients through the following 4 independentconditions: g + g = − γ (116) g + 2 g = 2 b γ (117) g + g = − γ − b b ( g + g ) (118) g + 2 g = 1 b γ − b b ( g + 2 g ) . (119)Thus, according to our formalism, all the coefficients γ , γ , γ and γ should beknown. The first three coefficients are all known thanks to the explicit O ( α s ) calcu-lation of the prompt photon cross section [55, 56, 57]. The last one ( γ ), is not yetknown from explicit O ( α s ) calculation, but, according to the approach of Ref.[14], it .5 Comparison with previous results is predicted using Eqs.(119,111,112): γ = − b " A (2) a + A (2) b ) − A (2) d π − b (2 ln 2 + 4 γ E ) A (1) a + A (1) b π (120)+ b (2 ln 2 + 3 γ E ) A (1) d π − b B (1) d π − b " A (1) a + A (1) b ) − A (1) d π . The correctness of this result could be tested by an order α s calculation. If it were tofail, the more general resummation formula with g determined by Eq.(119) shouldbe used, or one of the resummations which do not assume the factorization Eq.(93). hapter 5Resummation of rapiditydistributions In this chapter, we present a derivation of the threshold resummation formula for theDrell-Yan and the prompt photon production rapidity distributions. Our argumentsare valid for all values of rapidity and to all orders in perturbative QCD. For the caseof the Drell-Yan process, resummation is realized in a universal way, i.e. both for theproduction of a virtual photon γ ∗ and the production of a vector boson W ± , Z . Wewill show that for the fixed-target proton-proton Drell-Yan experiment E866/NuSeaused in current parton fits, the NLL resummation corrections are comparable to NLOfixed-order corrections and are crucial to obtain agreement with the data. This meansthat the NLL resummation of rapidity distributions is necessary and turns out to givebetter results than high-fixed-order calculations. We consider first the resummationof the DY case and its phenomenology and then the resummation of the promptphoton case. We consider the general Drell-Yan process in which the collisions of two hadrons( H and H ) produce a virtual photon γ ∗ (or an on-shell vector boson V ) and anycollection of hadrons (X): H ( P ) + H ( P ) → γ ∗ ( V )( Q ) + X ( K ) . (1)In particular, we are interested in the differential cross section dσdQ dY ( x, Q , Y ), where Q is the invariant mass of the photon or of the vector boson, x is defined as usualas the fraction of invariant mass that the hadrons transfer to the photon (or to thevector boson) and Y is the rapidity of γ ∗ ( V ) in the hadronic centre-of-mass-frame: x ≡ Q S , S = ( P + P ) , Y ≡
12 ln (cid:18) E + Q z E − Q z (cid:19) , (2)73 Resummation of rapidity distributions where E and p z are the energy and the longitudinal momentum of γ ∗ ( V ) respectively.In this frame, the four-vector Q of γ ∗ ( V ) can be written in terms of its rapidity andits tranverse momentum Q = ( Q , ~Q ) = ( q Q + Q ⊥ cosh Y, ~Q ⊥ , q Q + Q ⊥ sinh Y ) , (3)or in terms of the scattering angle θQ = ( Q , ~Q ) = ( q Q + | ~Q | , ~Q ⊥ , | ~Q | cos( θ )) . (4)For completeness, we recall that when Q = 0 (which is not our case), the rapidity Y defined in Eq.(2) reduces to the pseudorapidity η according to Eq.(4): η = − ln(tan( θ/ . (5)At the partonic level, a parton 1(2) in the hadron H ( H ) carries a fraction ofmomentum x ( x ): p = x P = x √ S , ~ ⊥ , , p = x P = x √ S , ~ ⊥ , − . (6)It is clear that the hadronic center-of-mass frame does not coincide with the partonicone, because x is, in general, different from x . Furthermore, from Eq.(2), we seethat the rapidity is not an invariant. Hence, in order to define the rapidity in thepartonic center-of-mass frame ( y ), we have to perform a boost of Y which connectsthe two frames. This provides us a relation between the rapidity in these two frames: y = Y −
12 ln( x x ) . (7)In order to understand the kinematic configurations in terms of rapidity, it isconvenient to define a new variable u , u ≡ Q · p Q · p = e − y = x x e − Y . (8)With no partons radiated as in the case of the LO, the rapidity is obviously zero.Beyond the LO, one or more partons can be radiated. Now, if these partons areradiated collinear with the incoming parton 2, then the partonic rapidity reaches itsmaximum value and u its minimum one. Similarly the minimum value of y and themaximum value of u is achieved when the radiated partons are collinear with theincoming parton 1. To be more precise, suppose that in the first case the radiatedpartons (collinear with the parton 2) carry away a fraction of momentum equal to(1 − z ) p , so that by momentum conservation Q = p + zp . In this case, we obtainimmediately the lower bound for u , which is z . In the second case the collinearradiated partons have momentum (1 − z ) p , hence Q = zp + p and the upper boundof u is 1 /z . So, z can be interpreted as the fraction of invariant mass that incomingpartons transfer to γ ∗ ( V ). In fact: z = Q p · p = Q ( p + p ) = xx x , (9) .1 Threshold resummation of DY rapidity distributions where we have neglected the quark masses. Therefore, we have that the kinematicconstraints of u are: z ≤ u ≤ z . (10)Then, since x < x <
2, the lower and upper bounds of z are: x ≤ z ≤ . (11)Thanks to Eqs.(7,8), the first relation can be translated directly into a relation forthe upper and lower limit of the partonic center-of-mass rapidity:12 ln z ≤ y ≤
12 ln 1 z . (12)Now, we need to obtain the boundaries of the hadronic center-of-mass rapidity. Sub-stituting Eqs.(8, 9) into the two conditions u ≥ z and u ≤ /z , we obtain the lowerkinematical bound for x and x : x ≥ √ xe Y ≡ x , x ≥ √ xe − Y ≡ x (13)and the obvious requirment that x ≤ x ≤ Y ≤
12 ln 1 x . (14)
According to standard factorization of collinear singularities of perturbative QCD,the expression for the hadronic differential cross section in rapidity has the form, dσdQ dY = X i,j Z x dx Z x dx F H i ( x , µ ) F H j ( x , µ ) × d ˆ σ ij dQ dy (cid:18) x , x , Q µ , α s ( µ ) , y (cid:19) , (15)where y depends on Y , x and x according to Eq.(7). The sum runs over all possiblepartonic subprocesses, F (1) i , F (2) j are respectively the parton densities of the hadron H and H , µ is the factorization scale (chosen equal to renormalization scale forsimplicity) and d ˆ σ ij / ( dQ dy ) is the partonic cross section. Even if the cross sectionEq.(15) is µ -independent, this is not the case for each parton subprocess. However,the µ -dependence of each contribution is proportional to the off-diagonal anoma-lous dimensions (or splitting functions), which in the threshold limit, ( z →
1) aresuppressed by factors of 1 − z . Therefore, each partonic subprocess can be treatedindependently and is separately renormalization-group invariant. Furthermore, thesuppression, in the threshold limit, of the off-diagonal splitting functions implies also Resummation of rapidity distributions that only the gluon-quark channels are suppressed. So, in order to study resumma-tion, we will consider only the quark- anti-quark channel, which can be related tothe same dimensionless coefficient function C ( z, Q /µ , α s ( µ ) , y ) for both, the pro-duction of a virtual photon and the production of a on-shell vector boson. In fact, iffor the production of a virtual photon, we define C ( z, Q /µ , α s ( µ ) , y ) through theequation, x x d ˆ σ γ ∗ q ¯ q ′ dQ dy (cid:18) x , x , Q µ , α s ( µ ) , y (cid:19) = 4 πα c q ¯ q ′ Q S C (cid:18) z, Q µ , α s ( µ ) , y (cid:19) , (16)where the prefactor x x has been introduced for future convenience, we find that forthe case of the production of a real vector boson, x x d ˆ σ Vq ¯ q ′ dQ dy (cid:18) x , x , Q µ , α s ( µ ) , y (cid:19) = πG F Q √ c q ¯ q ′ S δ ( Q − M V ) × C (cid:18) z, Q µ , α s ( µ ) , y (cid:19) , (17)where G F is the Fermi constant, M V is the mass of the produced vector boson. Thecoefficients c q ¯ q ′ , for the different Drell-Yan processes, are given by: c q ¯ q ′ = Q q δ q ¯ q for γ ∗ , (18) c q ¯ q ′ = | V qq ′ | for W ± , (19) c q ¯ q ′ = 4[( g qv ) + ( g qa ) ] δ q ¯ q for Z . (20)Here, Q q is the square charge of the quark q , V qq ′ are the CKM mixing factors for thequark flavors q, q ′ and g qv = 12 −
43 sin θ W , g qa = 12 for an up-type quark , (21) g qv = −
12 + 23 sin θ W , g qa = −
12 for a down-type quark , (22)with θ W the Weinberg weak mixing angle. As a consequence of these facts, resumma-tion has to be performed only for the quark-anti-quark channels omitting the over-all dimensional factors of C ( z, Q /µ , α s ( µ ) , y ) in the different Drell-Yan processes.Thus, we are left with the following dimensionless cross section, which has the form: σ ( x, Q , Y ) ≡ Z x dx x Z x dx x F H ( x , µ ) F H ( x , µ ) × C (cid:18) z, Q µ , α s ( µ ) , y (cid:19) , (23)where F and F are quark or anti-quark parton densities in the hadron H and H respectively. This shows the universality of resummation in Drell-Yan processes inthe sense that only the renormalization-group invariant quantity defined in Eq.(23)has to be resummed. .1 Threshold resummation of DY rapidity distributions For the case of the rapidity-integrated cross section, resummation is usually done inMellin space transforming the variable x into its conjugate variable N , because theMellin transformation turns convolution products into ordinary products. Further-more, the Mellin space is the natural space where to define resummation of leading,next-to-leading and so on logarithmic contribution, because in this space momentumconservation is respected as shown in [58]. In the case of the rapidity distribution,the Mellin transformation is not sufficient. In fact, rewriting Eq.(23) in this form σ ( x, Q , Y ) = Z dx dx dzF H ( x , µ ) F H ( x , µ ) × C (cid:18) z, Q µ , α s ( µ ) , y (cid:19) δ ( x − x x z ) , (24)we see that the Mellin transform with respect to x , σ ( N, Q , Y ) ≡ Z dxx N − σ ( x, Q , Y ) , (25)does not diagonalize the triple integral in Eq.(24). This is due to the fact thatthe partonic center-of-mass rapidity y depends on x and x through Eq.(7). Theordinary product in Mellin space can be recovered performing the Mellin transformwith respect to x of the Fourier transform of Eq.(24) with respect to Y . Calling theFourier moments M , using Eq.(7) the relations (12,14) and the identity C (cid:18) z, Q µ , α s ( µ ) , Y −
12 ln x x (cid:19) = Z ln 1 / √ z ln √ z dyC (cid:18) z, Q µ , α s ( µ ) , y (cid:19) × δ (cid:18) y − Y + 12 ln x x (cid:19) , (26)we find that σ ( N, Q , M ) ≡ Z dxx N − Z ln 1 / √ x ln √ x dY e iMY σ ( x, Q , Y ) (27)= F H ( N + iM/ , µ ) F H ( N − iM/ , µ ) × C (cid:18) N, Q µ , α s ( µ ) , M (cid:19) , (28)where F H i i ( N ± iM/ , µ ) = Z dxx N − ± iM/ F H i i ( x, µ ) , (29) C (cid:18) N, Q µ , α s ( µ ) , M (cid:19) = Z dzz N − Z ln 1 / √ z ln √ z dye iMy × C (cid:18) z, Q µ , α s ( µ ) , y (cid:19) . (30) Resummation of rapidity distributions
Eq.(28) shows that performing the Mellin-Fourier moments of the hadronic dimen-sionless cross section Eq.(23), we recover an ordinary product of the Mellin-Fouriertrnsform of the coefficient function and the Mellin moments of the parton densitiestranslated outside the real axis by ± iM/
2. Because the coefficient function is sym-metric in y , we can rewrite Eq.(30) in this way: C (cid:18) N, Q µ , α s ( µ ) , M (cid:19) = 2 Z dzz N − Z ln 1 / √ z dy cos( M y ) × C (cid:18) z, Q µ , α s ( µ ) , y (cid:19) . (31)From this last equation and Eq.(29), we see that the dependence on M , the Fourierconjugate of the rapidity y , originates from the parton densities, that depend on N ∓ iM/
2, and from the factor of cos(
M y ) in the integrand of Eq.(31).
In this section, we show that the resummed expression of Eq.(28) is obtained bysimply replacing the coefficient function C (cid:16) N, Q µ , α s ( µ ) , M (cid:17) with its integral over y , resummed to the desired logarithmic accuracy. This is equivalent to saying thatthe factor of cos( M y ) in Eq.(31) is irrelevant in the large-N limit. Indeed, one canexpand cos(
M y ) in powers of y ,cos( M y ) = 1 − M y O ( M y ) . (32)and observe that the first term of this expansion leads to a convergent integral (therapidity-integrated cross section), while the subsequent terms are suppressed by pow-ers of (1 − z ), since the upper integration bound in Eq.(31) isln 1 √ z = 12 (1 − z ) + O ((1 − z ) ) . (33)Hence, up to terms suppressed by factors 1 /N , Eq.(30) is equal to the Mellin transformof the rapidity-integrated Drell-Yan coefficient function that we call C I ( N, Q /µ , α s ( µ )).This completes our proof. We get σ res ( N, Q , M ) = F H ( N + iM/ , µ ) F H ( N − iM/ , µ ) × C resI (cid:18) N, Q µ , α s ( µ ) (cid:19) . (34)This theorethical result is very important: it shows that, near threshold, theMellin-Fourier transform of the coefficient function does not depend on the Fouriermoments and that this is valid to all orders of QCD perturbation theory. Furthermorethis result remains valid for all values of hadronic center-of-mass rapidity, because we .1 Threshold resummation of DY rapidity distributions have introduced a suitable integral transform over rapidity. The resummed rapidity-integrated Drell-Yan coefficient function to NLL order has been studied in Section3.4. It is given by C resI (cid:18) N, Q µ , α s ( µ ) (cid:19) = [exp { ln N g ( λ,
2) + g ( λ, } ] µ r = µ (35)where λ = b α s ( µ r ) ln N and where the resummation functions g ( λ,
2) and g ( λ, M S scheme given in Eq.(46) of Section 2.3.1.Now, we want to arrive to a NLO and NLL expression of the rapidity-dependentdimensional cross section. This is achieved firstly taking the Mellin and Fourierinverse transforms of σ res ( N, Q , M ) Eq.(34) in order to turn back to the variables x and Y : σ res ( x, Q , Y ) = Z ∞−∞ dM π e − iMY Z C + i ∞ C − i ∞ dN πi x − N σ res ( N, Q , M ) . (36)In principle the contour in the complex N -space of the inverse Mellin transform inEq.(36) has to be chosen in such a way that the intersection of C with the real axis liesto the right of all the singularities of the integrand. In practice, this is not possible,because the resummed coefficient function Eqs.(101,102) of section 3.4 has a branchcut on the real positive axis for N ≥ N L ≡ e ab αs ( Q , (37)which corresponds to the Landau singularity of α s ( Q /N a ) (see Eq.(41) in Appendix1.2). This is due to the fact that if the N -space expression is expanded in powers of α s ,and the Mellin inversion is performed order by order, a divergent series is obtained.The “Minimal Prescription” proposed in [58] gives a well defined formula to obtain theresummed result in x -space to which the divergent series is asymptotic and is simplyobtained choosing C = C MP in such a way that all the poles of the integrand are tothe left, except the Landau pole Eq.(37). Recently, another method has been prposedin Ref.[59]. Here, we will adopt the “Minimal Prescription ” formula, deforming thecontour in order to improve numerical convergence and to avoid the singularities ofthe parton densities of Eq.(34) which are transated out of the real axis by ± iM/ N -integral in Eq.(36) over a curve Γ given by:Γ = Γ + Γ + Γ (38)Γ ( t ) = C MP − i M t (1 + i ) , t ∈ ( −∞ ,
0) (39)Γ ( s ) = C MP + is M , s ∈ ( − ,
1) (40)Γ ( t ) = C MP + i M − t (1 − i ) , t ∈ (0 , + ∞ ) (41)The double inverse tranform of Eq.(36) over the curve Γ then becomes: σ res ( x, Q , Y ) = 1 π Z dmm cos( − Y ln m ) σ res ( x, Q , − ln m ) , (42) Resummation of rapidity distributions where we have done the change of variable M = − ln m . The factor σ res ( x, Q , M ) ofthe integrand in Eq.(42) is given by σ res ( x, Q , M ) = (43)1 π Z dss ℜ (cid:20) x − C MP − ln s + i ( M/ σ res ( C MP + ln s − i ( M/ , Q , M ) × (1 − i ) + sM x − C MP − isM/ σ res ( C MP + isM/ , Q , M ) (cid:21) , where we have done another change of variables ( t = − ln s ). Eqs.(42,43) are theexpressions that we use to evaluate numerically the resummed adimensional crosssection in the variables x and Y Eq.(36). Furthermore, we need to know the analyticcontinuations to all the complex plane of the parton densities at the scale µ inEq.(34). Here, we need to evolve up a partonic fit taken at a certain scale solving theDGLAP evolution equations in Mellin space. The solution of the evolution equationsis given in Section 1.5.Finally, we want to obtain a NLO determination of the cross section improved withNLL resummation. In order to do this, we must keep the resummed dimensionalesspart of the cross section Eq.(42), multiply it by the correct dimensional prefactorsEqs.(57,19,20) and parton densities, add to the resummed part the full NLO crosssection and subtract the double-counted logarithmic enhanced contributions. Thus,we have dσdQ dY = dσ NLO dQ dY + dσ res dQ dY − (cid:20) dσ res dQ dY (cid:21) α s =0 − α s (cid:20) ∂∂α s (cid:18) dσ res dQ dY (cid:19)(cid:21) α s =0 . (44)The first term is the full NLO cross section given in [17, 60, 61, 62]. We report thecomplete expression in Appendix C.The third and the fourth terms in Eq.(44) are obtained in the same way as thesecond one, but with the substitutions C resI (cid:18) N, Q µ , α s ( µ ) (cid:19) → , (45) C resI (cid:18) N, Q µ , α s ( µ ) (cid:19) → α s ( µ )2 A (cid:26) ln N + ln N (cid:20) γ E − ln (cid:18) Q µ (cid:19)(cid:21)(cid:27) , respectively. The terms that appear in the second in the second line of Eqs.(45) areexactly the O ( α s ) logarithmic enhanced contributions in the M S scheme.We note that the resummed cross section Eq.(44) is relevant even when the variable x is not large. In fact, the cross section can get the dominant contributions from theintegral in Eq.(23) for values of z Eq.(9) that are near the threshold even when x isnot close to one, because of the strong suppression of parton densities F i ( x i , µ ) when x i are large. To show the importance of this resummation, we have calculated the Drell-Yan rapid-ity distribution for proton-proton collisions at the Fermilab fixed-target experiment .1 Threshold resummation of DY rapidity distributions Figure 5.1:
Y-dependence of d σ/ ( dQ dY ) in units of pb / GeV . The curves are, from top tobottom, the NLO result (red band), the LO+LL resummation (blue band) and the LO (blackband). The bands are obtained varying the factorization scale between µ = 2 Q and µ = 1 / Q . E866/NuSea [38]. The center-of-mass energy has been fixed at √ S = 38 .
76 GeV andthe invariant mass of the virtual photon γ ∗ has been chosen to be Q = 64 GeV inanalogy with [19]. Clearly the contribution of the virtual Z can be neglected, becauseits mass is much bigger than Q . In this case x = 0 . Y Eq.(14) are given by ± . µ = 1 GeV ) in order to compare toRef.[19], where the NNLO calculation is performed. However, results obtained usingmore modern parton sets should not be very different. The LO parton set is given in[63] with α LOs ( m Z ) = 0 .
130 and the NLO set is given in [64] with α NLOs ( m Z ) = 0 . µ has been performed in the variableflavor number scheme. The quarks has been considered massless and, at the scaleof the transition of the flavor number ( N f → N f + 1), the new flavor is generateddynamically. The resummation formula Eq.(34) together with Eqs.(101-35) has beenused with the number of flavors N f = 4.In figure 5.1, we plot the rapidity-dependence of the cross section at LO, NLO andLO improved with LL resummation. The effect of LL resummation is small comparedto the effect of the full NLO correction. We see that, at leading order, the impact ofthe resummation is negligible in comparison to the NLO fixed-order correction. Thismeans that, at leading order, resummation is not necessary.The LO, the NLO and its NLL improvement cross sections are shown in figure5.2. The effect of the NLL resummation in the central rapidity region is almost aslarge as the NLO correction, but it reduces the cross section instead of enhancing itfor not large values of rapidity. The origin of this suppression will be discussed inthe next Section. Going from the LO result to the NLO with NLL resummation, we Resummation of rapidity distributions
Figure 5.2:
Y-dependence of d σ/ ( dQ dY ) in units of pb / GeV . The curves are, from top tobottom, the NLO result (red band), the NLO+NLL resummation (green band) and the LO (blackband). The bands are obtained as in figure 5.1. note a reduction of the dependence on the factorization scale i.e. a reduction of thetheoretical error.Now, we want to establish if the leading logarithmic terms that are included inthe resummed exptonent represent a good approximation to the exact fixed ordercomputation. Only if this is the case, we can believe that our resummation is reliablein perturbative QCD. In order to do this, we compare the full NLO DY rapiditycross section with the one obtained including only the large- N leading terms of thecoefficient function. For simplicity, we choose the factorization scale µ to the scaleof the process Q . The leading large- N coefficient function is given by: C lead ( N, α s ( Q )) = 1 + α s ( Q )2 A (cid:18) ln N + 2 γ E ln N + γ E − π (cid:19) , (46)where we have added the constant terms at O ( α s ) which are not resummed. For anexplicit derivation of these constant terms see for example Ref. [17] Section 3. Weplot the result in Figure 5.3. We see that the leading terms Eq.(46) represent a goodapproximation to the exact NLO computation, because they account for more than90% of the full NLO computation for all relevant rapidities.In Figure 5.4, we plot only the O ( α s ) correction. Here we see that the O ( α s )contribution of Eq.(46) represent a good approximation to the exact O ( α s ) NLOcontribution, because it accounts for more than 80% of the full O ( α s ) NLO correctionfor all relevant rapidities.In figure 5.5, we plot the experimental data of Ref.[38] converted to the Y variabletogether with our NLO and NLL resummed predictions.The data in Ref.[38] are tabulated in invariant Drell-Yan pair mass p Q andFeynman x F bins. To convert the data to the hadronic rapidity Y , we have used the .1 Threshold resummation of DY rapidity distributions -1.5 -1 -0.5 0.5 1 1.50.050.10.150.20.25 Figure 5.3: d σ/ ( dQ dY ) in units of pb / GeV for the full NLO computation (upper red line)and for the leading terms of Eq.(46) (lower black line). It has been calculated for one value of thefactorization scale µ = Q . -1.5 -1 -0.5 0.5 1 1.50.020.040.060.08 Figure 5.4: d σ/ ( dQ dY ) in units of pb / GeV for O ( α s ) correction of the full NLO computation(upper red line) and for the leading O ( α s ) term of Eq.(46) (lower black line). Resummation of rapidity distributions
Figure 5.5:
Y dependence of d σ/ ( dQ dY ) in units of pb / GeV . The curves are, from top tobottom, the NLO result (red band) and the NLO+NLL resummation (green band) together withthe E866/NuSea data. The bands are obtained as in figure 5.1. definition of the Feynman x F which is x F ≡ Q z √ S = 2 p Q + Q ⊥ sinh Y √ S , (47)where we have used Eq.(3). Solving Eq.(47) in Y we have Y = ln h H + √ H + 1 i , H = x F √ S p Q + Q ⊥ . (48)With this equation and with the aid of the Q ⊥ distribution, which is also givenin Ref.[38], we have converted the data from x F to Y . Furthermore, for each x F bin, we have done the weighted average of three p Q bins (7 . ≤ p Q ≤ . . ≤ p Q ≤ . . ≤ p Q ≤ . In this section, we shall show that the suppression of the cross section of the NLLcorrection with the parameter choices of the experiment E866 is due to the shift in .2 The origin of suppression the complex plane of the dominant contribution of the resummed exponent. We shalldo it using a simplified toy-model.Consider the collision of only two quarks with parton density F ( x ) = (1 − x ) . (49)Its Mellin transform is given by: F ( N ) = Z dx x N − (1 − x ) = Γ( N )Γ(3)Γ( N + 3) = 2 N ( N + 1)( N + 2) . (50)Furthermore, we take the double-log approximation (DLA) which is obtained per-forming the limit λ → σ ( N, M ) = σ F O ( N, M ) + | F ( N + iM/ | ∆ σ DLA ( N ) , (51)where σ F O ( N, M ) are the exact NLO Mellin-Fourier moments and where∆ σ DLA ( N ) = h e α s A ln N − − α s A ln N i . (52)If there is a suppression, this means that the quantity σ ( N, M ) − σ F O ( N, M ) = 4∆ σ DLA ( N ) (cid:2) N + M (cid:3) (cid:2) ( N + 1) + M (cid:3) (cid:2) ( N + 2) + M (cid:3) , (53)should produce a negative contribution in performing the inverse Mellin and Fouriertransform. It is given by the integral Z ∞−∞ dM π e − iMY Z C + i ∞ C − i ∞ dN πi x − N σ DLA ( N ) (cid:2) N + M (cid:3) (cid:2) ( N + 1) + M (cid:3) (cid:2) ( N + 2) + M (cid:3) . (54)The integrand function of this expression has not only a cut on the negative real axis(as it happens in the inclusive case), but also poles that are shifted in the complexplane: − n ± iM n = 0 , , . (55)Because of the factor x − N in the inverse Mellin integral in Eq.(54), its dominantcontribution comes from the poles with n = 0 in Eq.(55). The contribution of thepole at + iM/ Z ∞−∞ dM π e − iM ( Y +ln √ x ) σ DLA ( N = iM/ iM ( iM + 1)( iM + 2) , (56)where ∆ σ DLA ( N = iM/
2) = exp (cid:20) α s A (cid:18) ln | M | − π iπ ln | M | (cid:19)(cid:21) + (57) − − α s A (cid:18) ln | M | − π iπ ln | M | (cid:19) . Resummation of rapidity distributions
The important thing to notice of this contribution is the fact that the imaginarypole has produced an oscillating prefactor in front of the resummed exponent which,together with the oscillating factor of the Fourier inverse integral in Eq.(56) at zerohadronic rapidity Y , is given byexp (cid:20) i (cid:18) α s πA ln | M | − iM ln √ x (cid:19)(cid:21) . (58)We note that, with the inclusion of the contribution of the other pole at − iM/
2, thereal and immaginary part of Eq.(57) contribute to the integral of Eq.(54). The realand immaginary part of Eq.(57) are given by: ℜ [∆ σ DLA ( iM/ e α s A “ ln | M | − π ” cos (cid:18) α s πA ln | M | (cid:19) (59) − − α s A (cid:18) ln | M | − π (cid:19) ℑ [∆ σ DLA ( iM/ e α s A “ ln | M | − π ” sin (cid:18) α s πA ln | M | (cid:19) (60) − α s πA ln | M | M = M where the phase of Eq.(58) is stationary and is given by: | M | = α s πA | ln √ x | . (61)Substituting this in Eqs.(59,60), we find a suppression of about 35% for the parameterchoice of E866 experiment and a suppression of about 10% for the W boson productionat RHIC with a center-of-mass energy of √ S = 500GeV which has more or less thesame value of x . We should now recall that this is a rough estimation and that thereis also the contribution of the cut on the negative real axis which usually produces anenhancement. However, comparing this estimation with the result for the W bosonproduction at RHIC (see e.g. figure 1 and 2 in reference [17]), where resummationproduces an enhancement of about 4% we can believe that the ignored contributionin this section produce an enhancement of about 15%, thus giving a suppressio ofabout 20% at E866 experiment. Here, we consider the rapidity distribution of the prompt photon process discussedin chapter 4, H ( P ) + H ( P ) → γ ( p γ ) + X. (62) .3 Resummation of prompt photon rapidity distribution Specifically, we are interested in the differential cross section p ⊥ dσdp ⊥ dY ( x ⊥ , p ⊥ , η γ ),where as in section 4.1 of chapter 4 p ⊥ is the transverse momentum of the photon, η γ is its hadronic center-of-mass pseudorapidity and x ⊥ = 4 p ⊥ S . (63)The pseudorapidity of the direct real photon in the partonic center-of-mass frame ˆ η γ is related to η γ through Eq.(6) in section 4.1:ˆ η γ = η γ −
12 ln x x . (64)Furthermore, as in chapter 4, we use the following parametrizations of the photonand of the incoming partons’ momenta p γ = ( p ⊥ cosh ˆ η γ , ~p ⊥ , p ⊥ sinh ˆ η γ ) , (65) p = x P = √ s , ~ ⊥ , , (66) p = x P = √ s , ~ ⊥ , − . (67)The transverse energy that the partons can transfer to the outgoing partons must beless than the partonic center-of-mass energy √ s = √ x x S . This means that z cosh ˆ η γ ≤ , (68)where we have defined the parton scaling variable z = Q s = x ⊥ x x , (69)with, as in chapter 4, Q = 4 p ⊥ . Eq.(68) implies that the upper and lower boundariesfor the partonic center-of-mass pseudorapidity are given byˆ η − ≤ ˆ η γ ≤ ˆ η + , (70)where ˆ η ± = ln √ z ± r z − ! = ± ln √ z + r z − ! . (71)Using Eq.(64), we can rewrite the transverse energy condition Eq.(68) as a conditionfor the lower bound of x : x ≥ x √ x ⊥ e − η γ x − √ x ⊥ e η γ ≡ x . (72)Now, the requirement that x ≤ x : x ≥ √ x ⊥ e η γ − √ x ⊥ e − η γ ≡ x . (73) Resummation of rapidity distributions
The upper and lower bounds of the hadronic center-of-mass pseudorapidity can befound with the obvious condition that x ≤
1. In this way, we find, η − ≤ η γ ≤ η + , (74)where η ± = ln (cid:18) √ x ⊥ ± r x ⊥ − (cid:19) = ± ln (cid:18) √ x ⊥ + r x ⊥ − (cid:19) . (75) The expression with the factorization of collinear singularities of this cross section inperturbative QCD is p ⊥ dσdp ⊥ dη γ ( x ⊥ , p ⊥ , η γ ) = X a,b Z x dx Z x dx x F H a ( x , µ ) x F H b ( x , µ ) × C ab (cid:18) z, Q µ , α s ( µ ) , ˆ η γ (cid:19) δ ( x ⊥ − zx x ) , (76)where F H a ( x , µ ), F H b ( x , µ ) are the distribution functions of partons a, b in thecolliding hadrons and µ is the factorization scale equal to the renormalization scale.ˆ η γ is a function of η γ , x and x as defined by Eq.(64). The coefficient function C ab ( z, Q µ , α s ( µ ) , ˆ η γ ) is defined in terms of the partonic cross section for the processwhere partons a , b are incoming as C ab (cid:18) z, Q µ , α s ( µ ) , ˆ η γ (cid:19) = p ⊥ d ˆ σ ab dp ⊥ dη γ . (77)To allow the Mellin transform to deconvolute Eq.(76), we first perform the Fouriertransform with respect to η γ , thus obtaining σ ( N, Q , M ) = Z dx ⊥ x N − ⊥ Z η + η − dη γ p ⊥ dσdp ⊥ dη γ ( x ⊥ , p ⊥ , η γ ) (78)= X a,b F H a ( N + 1 + iM/ , µ ) F H b ( N + 1 − iM/ , µ ) × C ab (cid:18) N, Q µ , α s ( µ ) , M (cid:19) . (79)where F H i c ( N + 1 ± iM/ , µ ) = Z dxx N ± iM/ F H i i ( x, µ ) , (80) C ab (cid:18) N, Q µ , α s ( µ ) , M (cid:19) = 2 Z dzz N − Z ˆ η + d ˆ η γ cos( M ˆ η γ ) × C (cid:18) z, Q µ , α s ( µ ) , ˆ η γ (cid:19) . (81) .3 Resummation of prompt photon rapidity distribution A resummed expression of Eq.(83) in the threshold limit for the transverse energy( z → N → ∞ ) is obtained in the same way as we have done at thebeginning of section 5.1.4, since the upper integration bound of ˆ η γ in Eq.(81) isˆ η + = ln √ z + r z − ! = √ − z − (1 − z ) + O ((1 − z ) / ) . (82)Thus, up to terms suppressed by factors 1 /N , the resummed expression of Eq.(83) is: σ res ( N, Q , M ) = X a,b F H a ( N + 1 + iM/ , µ ) F H b ( N + 1 − iM/ , µ ) × C resI ab (cid:18) N, Q µ , α s ( µ ) (cid:19) , (83)where C resI ab is the resummed pseudorapidity-integrated coefficient function for theprompt photon production for the subprocess which involves the initial partons a, b .These resummed coefficient function has been studied in chapter 4 and in Refs. [14,65]. This result is analogous to that of the Drell-Yan rapidity distributions case,in the sense that the resummed formula is obtained through a translation of theparton densities’ moments by 1 ± iM/ hapter 6Renormalization groupresummation of tranversedistributions We prove the all-order exponentiation of soft logarithmic corrections at small trans-verse momentum to the distribution of Drell-Yan process. We apply the renormaliza-tion group approach developed in the context of integrated cross sections. We showthat all large logs in the soft limit can be expressed in terms of a single dimensionalvariable, and we use the renormalization group to resum them. The resummed resultthat we obtain is, beyond the next-to-leading log accuracy, more general and less pre-dictive than those previously released. The origin of this could be due to factorizationproperties of the cross section. The understanding of this point is a work in progress.
We consider the Drell-Yan process H ( P ) + H ( P ) → γ ∗ ( Q ) + X ( K ) , (1)and, in particular, the differential cross section dσdq ⊥ dY ( Q , q ⊥ , x , x ), where q ⊥ is thetransverse momentum with respect to colliding axis of the hadrons H and H , Q isthe virtuality of photon and x , x are useful dimensionless variables, that, in termsof the hadronic center-of-mass squared energy S = ( P + P ) and the photon center-of-mass rapidity Y , are given by x = r Q + q ⊥ S e Y ; x = r Q + q ⊥ S e − Y . (2)The relation between these two variables and the fraction of energy carried by thevirtual photon is x + x E γ ∗ √ S . (3)91 Renormalization group resummation of tranverse distributions
According to standard factorization of perturbative QCD, the expression for the dif-ferential cross section is dσdq ⊥ dY ( Q , q ⊥ , x , x ) = Z z min dz Z z min dz f ( z , µ ) f ( z , µ ) × d ˆ σdq ⊥ dy ( Q , q ⊥ , s, y, µ , α s ( µ )) , (4)where f ( z , µ ) , f ( z , µ ) are the parton distribution functions of the colliding quarkand anti-quark in the hadrons H and H respectively. The arbitrary scale µ is thefactorization scale, which, for simplicity, is chosen to be equal to the renormalizationscale. The condition that the invariant mass of the emitted particles K cannot benegative, imposes that ( z − x )( z − x ) ≥ q ⊥ S and, taking the small q ⊥ limit, weobtain that z min = x , z min = x . (5)The partonic center-of-mass squared energy s and rapidity y are related to thehadronic ones by a scaling and a boost along the collision axis with respect to thelongitudinal momentum fraction z , z of the incoming partons: s = z z S ; y = Y −
12 ln z z . (6)We define analogous variables to that of Eqs.(2) at the partonic level ξ ≡ x z = r Q + q ⊥ s e y ; ξ ≡ x z = r Q + q ⊥ s e − y , (7)with inverse relations s = Q + q ⊥ ( x /z )( x /z ) ; y = 12 ln x /z x /z . (8)Now, thanks to these equations, we can define a dimensionless differential cross sectionand coefficient function W ( q ⊥ /Q , x , x ) = Q x x dσdq ⊥ dY ( Q , q ⊥ , x , x ) , (9)ˆ W (cid:18) Q µ , q ⊥ Q , x z , x z , α s ( µ ) (cid:19) = Q ( x /z )( x /z ) d ˆ σdq ⊥ dy ( Q , q ⊥ , s, y, µ , α s ( µ )) , (10)in such a way that Eq.(4), together with the conditions Eqs.(5), takes the useful formof a convolution product W ( q ⊥ /Q , x , x ) = Z x dz z Z x dz z f ( z , µ ) f ( z , µ ) × ˆ W ( Q /µ , q ⊥ /Q , x /z , x /z , α s ( µ )) , (11)which is valid only for small q ⊥ . .2 The role of standard factorization It is known that this expression (or equivalently Eq.(4)) is originated by the factor-ization of collinear divergences in the impact parameter ( ~b ) which is conjugate uponFourier transformation to the transverse momentum ( ~q ⊥ ): W ( Q b , x , x ) = Z d q ⊥ e i~q ⊥ ~b W ( q ⊥ /Q , x , x ) . (12)In d = 4 − ǫ dimensions this factorization has the formˆ W ( Q /µ , Q b , x , x , α s ( µ )) == Z x dz z Z x dz z Z ( z , α s ( µ ) , ǫ ) Z ( z , α s ( µ ) , ǫ ) ˆ W (0) ( Q , b , x z , x z , α , ǫ ) . (13)Note that the universal function Z that extracts the collinear divergences from thebare coefficient function doesn’t depend on the Fourier conjugate ( b ) of the transversemomentum Ref.[11]. In Fourier space Eq.(11), becomes W ( Q b , x , x ) = Z x dz z Z x dz z f ( z , µ ) f ( z , µ ) × ˆ W ( Q /µ , Q b , x /z , x /z , α s ( µ )) (14)Furthermore, Eq.(11) tells us that the differential cross section is a convolutionalproduct which is diagonalized by a double Mellin transform. Thus, performing thedouble Mellin and Fourier transform, the coefficient function takes the simple factor-ized form: W ( Q b , N , N ) = f ( N , µ ) f ( N , µ ) ˆ W ( Q /µ , Q b , N , N , α s ( µ )) . (15)Our goal is to resum the large logarithms ln Q b to all logarithmic orders. These logsare present to all orders in the contributions to this differential cross section. Theycome from the kinematical region of soft and collinear emissions. However, we knowfrom Eq.(13) that collinear divergences that arises in the limit q ⊥ → Q b that come from soft contributions.We define the usual physical anomalous dimension: Q ∂∂Q W ( Q b , N , N ) = γ ( W ) ( Q b , N , N , α s ( Q )) W ( Q b , N , N ) . (16)It is clear that γ ( W ) ( Q b , N , N , α s ( Q )) is a renormalization group invariant andwe will show that it is also independent of N and N when choosing the arbitraryscale µ equal to 1 /b and taking into account only soft contributions. Thus, in thesoft limit and with the convenient choice µ = 1 /b , we can write γ SOF T ( W ) ( Q b , N , N , α s ( Q )) = γ (1 , Q b , α s ( Q )) (17) Renormalization group resummation of tranverse distributions
So, the resummed expression for the cross section Eq.(11) in Fourier space, in whichthe collinear contributions to the large ln Q b are separated from the soft ones, hasthe general form: W res ( Q b , x , x ) = Z x dz z Z x dz z f ( z , /b ) f ( z , /b ) K res ( b , Q , Q ) × ˆ W res ( Q b , Q b , x /z , x /z , α s (1 /b )) , (18)where K res ( Q b , Q , Q ) = exp (Z Q Q d ¯ µ ¯ µ Γ res ( Q ¯ µ , ¯ µ b , α s (¯ µ )) ) , (19)The scale Q , must be larger than the lower limit of the perturbative analysis ( Q > Λ QCD ). Hence, in order to absorb the possible large correction of the type ln Q b we will always choose Q = 1 /b . Accordingly, the condition Q > Λ QCD becomes b < / Λ QCD and the resummed exponent of Eq.(19) is related to the resummedphysical anomalous dimension γ res through the logarithmic derivative: γ res (1 , Q b , α s ( Q )) = Q ∂∂Q Z Q /b d ¯ µ ¯ µ Γ res ( Q ¯ µ , ¯ µ b , α s (¯ µ )) . (20)It is now clear that resummation of collinear emissions is realized by the partondistribution evolution thanks to the fact that the factorization scale µ is arbitrary.Resummation of soft gluon emissions can be achieved by the resummation of the ex-ponent that appears in this expression. This is the subject of the next section. q ⊥ singularities of soft gluon contributions We now proceed through the calculation of the resummed exponent using kinematicsanalysis and renormalization group improvement. According to the Appendix B, thephase space measure in d = 4 − ǫ dimensions for n extra emissions of the partonicDrell-Yan subprocess can be written for n = 0 and n ≥ dφ ( p + p ; q ) dq ⊥ dy = 1 Q δ (1 − ξ ) δ (1 − ξ ) δ (ˆ q ⊥ ) (21) dφ n +1 ( p + p ; q, k , . . . , k n ) dq ⊥ dy = N ( ǫ )( q ⊥ ) − ǫ Z ( √ s − √ Q ) dM π dφ n ( k ; k , . . . , k n ) × δ ( M − M ); k = M ; M = Q ξ ξ [(1 − ξ )(1 − ξ ) + ˆ q ⊥ (1 − ξ − ξ )] , (22) .3 The q ⊥ singularities of soft gluon contributions where N ( ǫ ) = 1 / (2(4 π ) − ǫ ), ξ i = x i /z i , ˆ q ⊥ = q ⊥ /Q and d Ω n − ( ǫ ) stands for theintegration of n − z i , i = 1 , . . . , n − ξ and ξ are relatedto the partonic center-of-mass rapidity (y) and energy (s) by the relations: s = Q + q ⊥ ξ ξ ; y = 12 ln ξ ξ . (23)The phase space measure dφ n ( k ; k , . . . , k n ) is the same as the phase space measureof the DIS process with an incoming momentum with a nonzero invariant mass ( k = M ) and n outgoing massless particles. This phase space has been analyzed in Section3.1 and is given by, dφ ( k ; k ) = 2 πδ ( M ) , n = 0 (24) dφ n ( k ; k , . . . , k n ) = 2 π (cid:20) N ( ǫ )2 π (cid:21) n − ( M ) n − − ( n − ǫ d Ω n − ( ǫ ) , n ≥ . (25)According to this, we can rewrite Eqs.(21),(22) in this form: dφ ( p + p ; q ) dq ⊥ dy = 1 Q δ (1 − ξ ) δ (1 − ξ ) δ (ˆ q ⊥ ) (26) dφ ( p + p ; q, k ) dq ⊥ dy = ( q ⊥ ) − ǫ N ( ǫ ) δ ( M ) (27) M = Q ξ ξ [(1 − ξ )(1 − ξ ) + ˆ q ⊥ (1 − ξ − ξ )] . (28)for n = 0 , dφ n +1 ( p + p ; q, k , . . . , k n ) dq ⊥ dy = ( q ⊥ ) − ǫ π (cid:20) N ( ǫ )2 π (cid:21) n ( M ) n − − ( n − ǫ d Ω n − ( ǫ ) (29)for n ≥
2. The dependence of the phase space on ˆ q ⊥ comes entirely from the factors:( q ⊥ ) − ǫ δ ( M ) , n = 1 (30)( M ) n − ( q ⊥ ) − ǫ ( M ) − ( n − ǫ − , n ≥ . (31)The phase space measure must be combined with the square modulus of theamplitude, in order to determine the logarithmic singularities in q ⊥ = 0 which areregularized in d = 4 − ǫ dimensions. Studying the behavior of the invariants thatcan be constructed with the external momenta, we can establish in which kinematicalregion the square modulus of the amplitude can be singular in ˆ q ⊥ →
0. From thestudy of the DIS-like emissions (see Section 3.2) we know that k i = p M z n − · · · z i +1 ) / (1 − z i ) , ≤ i ≤ n − k n − = p M − z n − ) (33) k n = k . (34) Renormalization group resummation of tranverse distributions
This means that all the invariants that can appear in the function D G ( β, P E ) inEq.(22) of Section 3.1 can be expressed in terms of the following ones: q = Q , p = p = k i = 0 , p · p = s k i · k j ∼ M = Q ξ ξ [(1 − ξ )(1 − ξ ) + ˆ q ⊥ (1 − ξ − ξ )] (36) p · k i ∼ p · k i ∼ q sM (37)= Q (cid:20) (1 + ˆ q ⊥ ) ξ ξ [(1 − ξ )(1 − ξ ) + ˆ q ⊥ (1 − ξ − ξ )] (cid:21) / . (38)In the case n = 1, the single emission squared amplitude at tree level has a 1 /q ⊥ singularity for q ⊥ →
0. This can be easily seen by an explicit O ( α s ) computation(see for example Refs.[66, 67, 68]). Hence, in the general case, we expect that in the q ⊥ → | A n +1 | ∼ q ⊥ ( M ) n ( M ) n ǫ g n n ( ξ , ξ ) , (39)where N and k are integer or half-integer numbers (see Eqs.(36,37)). However, asdiscussed in Section 3.2, here we will assume that only the integer powers of M contribute. Then, we know that phase space contributes the factors of Eqs.(30,31)and, hence, Eq.(39) implies that a generic contribution to the coefficient functionˆ W (0) has the folowing structure:(ˆ q ⊥ ) − − ǫ δ ( M )( M ) n ( M ) n ǫ g nn ( ξ , ξ ); n = 1 , (40)(ˆ q ⊥ ) − − ǫ ( M ) − − ( n − n ′ − ǫ ( M ) n − n g nn ′ ( ξ , ξ ); n > , (41)where for the moment we do not care about the overall dimensional factor. Now, weare interested in taking into account only the 1 / ˆ q ⊥ singularity, because more singularterms are forbidden and less singular ones are suppressed. As proven in Appendix D,in the limit q ⊥ → δ ( M ) = ξ ξ Q (cid:20) δ (1 − ξ )(1 − ξ ) + + δ (1 − ξ )(1 − ξ ) + − ln ˆ q ⊥ δ (1 − ξ ) δ (1 − ξ ) (cid:21) + O (ˆ q ⊥ ) , (42)and, for η = − ( n − n ′ − ǫ (with ǫ < n − n ′ − > − ξ )(1 − ξ ) + ˆ q ⊥ (1 − ξ − ξ )] η − = (43)= (1 − ξ ) η − (1 − ξ ) η − − (ˆ q ⊥ ) η η δ (1 − ξ ) δ (1 − ξ ) + O (ˆ q ⊥ ) . Therefore, n = − n + 1, n ′ < n −
1. Furthermore, we note that the terms notproportional to δ (1 − ξ ) δ (1 − ξ ) are divergent (in the q ⊥ → f ( z , µ ) and f ( z , µ ) when they are evaluated .3 The q ⊥ singularities of soft gluon contributions at the scale µ = 1 /b in Fourier space (see Eq.(18)). As a first conclusion, we obtainthat the contributions that must be resummed in the limit ˆ q ⊥ → δ (1 − ξ ) δ (1 − ξ ) and, thus, belong to the kinematical region of onlythe soft extra emissions. Hence, we obtain that the soft part of the coefficient functionthat must be resummed has, after the inclusion of loops, the following general form:ˆ W ( Q , q ⊥ , ξ , ξ , α , ǫ ) = ∞ X n =0 α n ˆ W n ( Q , q ⊥ , α , ǫ ) δ (1 − ξ ) δ (1 − ξ ) , (44)with ˆ W n ( Q , q ⊥ , α , ǫ ) = ( Q ) − nǫ (cid:2) C (0) n ( ǫ ) δ (ˆ q ⊥ ) + n X k =2 C (0) nk ( ǫ )(ˆ q ⊥ ) − − kǫ + n X k =1 C ′ (0) nk ( ǫ )(ˆ q ⊥ ) − − kǫ ln ˆ q ⊥ (cid:3) , (45)where the factor of ( Q ) − nǫ has been introduced for dimensional reasons. We, now,perform the double Mellin transform and the Fourier transform using the fact that: Z d ˆ q ⊥ e i ˆ b · ˆ q ⊥ δ (ˆ q ⊥ ) = π, (46) Z d ˆ q ⊥ e i ˆ b · ˆ q ⊥ (ˆ q ⊥ ) − − kǫ = πF k ( ǫ )(ˆ b ) kǫ , (47) Z d ˆ q ⊥ e i ˆ b · ˆ q ⊥ (ˆ q ⊥ ) − − kǫ ln ˆ q ⊥ = − πF k ( ǫ )(ˆ b ) kǫ ln ˆ b − π F ′ k ( ǫ ) k (ˆ b ) kǫ , (48)ˆ b ≡ Q b ; F k ( ǫ ) = − − kǫ kǫ Γ(1 − kǫ )Γ(1 + kǫ ) . (49)According to this, Eq.(44) has, after Mellin and Fourier transform, this structure:ˆ W (0) ( Q , b , α , ǫ ) = ∞ X n =0 n X k =0 ˜ C (0) nk ( ǫ )[( Q ) − ǫ α ] n − k [(1 /b ) − ǫ α ] k + (50)+ ln Q b ∞ X n =1 n X k =1 ˜ C ′ (0) nk ( ǫ )[( Q ) − ǫ α ] n − k [(1 /b ) − ǫ α ] k Renormalization group resummation of tranverse distributions
At this point, we calculate the resummed exponent that appears in Eq.(19): Z Q /b d ¯ µ ¯ µ Γ (0) ( Q , ¯ µ , b , α , ǫ ) = ln ˆ W (0) ( Q , b , α , ǫ )ˆ W (0) (1 /b , b , α , ǫ ) ! = (51)= " ∞ X n =1 n − X k =0 E (0) nk ( ǫ )[(¯ µ ) − ǫ α ] n − k [(1 /b ) − ǫ α ] k ¯ µ = Q ¯ µ =1 /b + ln (cid:18) Q b ∞ X n =1 n X k =1 ˜ E (0) nk ( ǫ )[( Q ) − ǫ α ] n − k × [(1 /b ) − ǫ α ] k (cid:19) , where the last term has not been expanded because we must take into account thatin the bare coefficient function Eq.(44) there is only one explicit logarithm. From theexplicit calculation to order O ( α ), we find that E (0)10 ( ǫ ) = 2 π π ) ǫ Γ(1 − ǫ ) (cid:18) − ǫ − ǫ + π − (cid:19) , (52)˜ E (0)11 ( ǫ ) = 4 π π ) ǫ Γ(1 − ǫ ) F ( ǫ ) . (53)Now, we want to rewrite Eq.(51) in a renormalized form. To do this, we use, asexpalined in Chapter 3, the fact that ( Q ) − ǫ α and (1 /b ) − ǫ α are renormalizationgroup invariant. Consequently, we may write:( Q ) − ǫ α = α s ( Q ) Z ( α s ) ( α s ( Q ) , ǫ ) , (54)(1 /b ) − ǫ α = α s (1 /b ) Z ( α s ) ( α s (1 /b ) , ǫ ) , (55)where Z ( α s ) ( α s ( µ ) , ǫ ) has multiple poles at ǫ = 0 and µ is the renormalization scalewhich for simplicity has been chosen equal to the factorization scale. Furthermore wenote that the universal functions Z ( ˆ W ) ( N , α s ( µ ) , ǫ ) Z ( ˆ W ) ( N , α s ( µ ) , ǫ ) that extractthe collinear poles from the coefficient function simplify in the first line of Eq.(51).Thus the renormalized expression of Eq.(51) has the form: Z Q /b d ¯ µ ¯ µ Γ( Q ¯ µ , ¯ µ b , α s (¯ µ ) , ǫ ) = " ∞ X m =1 n − X n =0 E Rmn ( ǫ ) α m − ns (¯ µ ) α ns (1 /b ) ¯ µ = Q ¯ µ =1 /b (56)+ ln (cid:18) Q b ∞ X m =1 m X n =1 ˜ E Rmn ( ǫ ) × α s ( Q ) m − n α ns (1 /b ) (cid:19) . .4 The resummed exponent in renormalization group approach The resummed exponent is clearly pole-free and so we can exploit the cancellation ofthe poles that could be present in the coefficients E Rmn ( ǫ ) and ˜ E Rmn ( ǫ ). Furthermorewe want to perform a comparison with previously released resummation formulaegiven in [6, 67]. In order to do these two things we rewrite Eq.(56) in terms of therenormalized physical anomalous dimension Γ( Q ¯ µ , ¯ µ b , α s (¯ µ ) , ǫ ) and therefore, wecalculate, according to Eq.(20), the logarithmic derivative of Eq.(56): γ (1 , ¯ µ b , α s (¯ µ ) , ǫ ) = (57)= β ( d ) ( α s (¯ µ )) ∂∂α s (¯ µ ) ∞ X m =1 n − X n =0 E Rmn ( ǫ ) α m − ns (¯ µ ) α ns (1 /b )+ P ∞ m =1 P mn =1 ˜ E Rmn ( ǫ ) α s (¯ µ ) m − n α ns (1 /b )1 + ln ¯ µ b P ∞ m =1 P mn =1 ˜ E Rmn ( ǫ ) α s (¯ µ ) m − n α ns (1 /b )+ ln ¯ µ b β ( d ) ( α s (¯ µ )) ∂/∂α s P ∞ m =1 P mn =1 ˜ E Rmn ( ǫ ) α s (¯ µ ) m − n α ns (1 /b )1 + ln ¯ µ b P ∞ m =1 P mn =1 ˜ E Rmn ( ǫ ) α s (¯ µ ) m − n α ns (1 /b ) , where β ( d ) ( α s (¯ µ )) = − ǫα s (¯ µ ) + β ( α s (¯ µ )) , (58)and β ( α s (¯ µ )) = − β α s (¯ µ ) + O ( α s ) is the usual four-dimensional β -function. Now,in order to isolate the terms which contain the explicit logs from the rest we add andsubtract the term ∞ X m =1 m X n =1 ˜ E Rmn ( ǫ ) α s (¯ µ ) m − n α ns (1 /b ) . (59)After this, we re-expand the various terms in powers of α s (¯ µ ) and α s (1 /b ), but notin powers of ln ¯ µ b . The result that we find in this way has the following structure: Z Q /b d ¯ µ ¯ µ Γ( Q ¯ µ , ¯ µ b , α s (¯ µ ) , ǫ ) = Z Q /b d ¯ µ ¯ µ ∞ X m =1 m X n =0 Γ Rmn ( ǫ ) α m − ns (¯ µ ) α ns (1 /b ) ! + Z Q /b d ¯ µ ¯ µ ln ¯ µ b P ∞ m =2 P m − n =1 ˜Γ Rmn ( ǫ ) α m − ns (¯ µ ) α ns (1 /b )1 + ln ¯ µ b P ∞ m =1 P mn =1 ˜ E Rmn ( ǫ ) α m − ns (¯ µ ) α ns (1 /b ) ! (60)To show the cancellation of divergences we rewrite the integrand separating off the b -independent terms as in Chapter 3 and Chapter 4: γ (1 , ¯ µ b , α s (¯ µ ) , ǫ ) = ˆΓ ( c ) ( α s (¯ µ ) , ǫ ) + ˆΓ ( l ) ( α s (¯ µ ) , α s (1 /b ) , ǫ )+ˆΓ ′ ( l ) (ln ¯ µ b , α s (¯ µ ) , α s (1 /b ) , ǫ ) , where ˆΓ ( c ) ( α s (¯ µ ) , ǫ ) = ∞ X m =1 Γ Rm ( ǫ ) α ms (¯ µ ) (61)ˆΓ ( l ) ( α s (¯ µ ) , α s (1 /b ) , ǫ ) = ∞ X m =0 ∞ X n =1 Γ Rm + nn ( ǫ ) α ms (¯ µ ) α ns (1 /b ) (62) Renormalization group resummation of tranverse distributions ˆΓ ′ ( l ) (ln ¯ µ b , α s (¯ µ ) , α s (1 /b ) , ǫ ) = ln ¯ µ b P ∞ m =2 P m − n =1 ˜Γ Rmn ( ǫ ) α m − ns (¯ µ ) α ns (1 /b )1 + ln ¯ µ b P ∞ m =1 P mn =1 ˜ E Rmn ( ǫ ) α m − ns (¯ µ ) α ns (1 /b )(63)We know that α s (1 /b ) = f ( α s (¯ µ ) , ln ¯ µ b ) , (64)so, in principle, it is possible to invert this relation in order to obtain:ln ¯ µ b = g ( α s (¯ µ ) , α s (1 /b )) . (65)The function g has the following perturbative expression: g ( α s (¯ µ ) , α s (1 /b )) = 1 β (cid:18) α s (¯ µ ) − α s (1 /b ) (cid:19) ∞ X r =1 l r α rs (¯ µ ) ! . (66)Substituting this expression in Eq.(63) and re-expanding in powers of α s (¯ µ ) and α s (1 /b ), we obtain that:ˆΓ ′ ( l ) (ln ¯ µ b , α s (¯ µ ) , α s (1 /b ) , ǫ ) = ¯Γ ( l ) ( α s (¯ µ ) , α s (1 /b ) , ǫ ) − ¯Γ ( l ) ( α s (¯ µ ) , α s (¯ µ ) , ǫ ) , (67)where ¯Γ ( l ) ( α s (¯ µ ) , α s ( µ ) , ǫ ) = ∞ X m =1 ∞ X n =1 ¯Γ Rm + nn ( ǫ ) α ms (¯ µ ) α ns ( µ ) . (68)We choose as counterterm, Z (Γ) ( α s (¯ µ ) , ǫ ) = ˆΓ ( l ) ( α s (¯ µ ) , α s (¯ µ ) , ǫ ) . (69)With this choice we obtain: γ (1 , ¯ µ b , α s (¯ µ ) , ǫ ) = ˆΓ ( c ) ( α s (¯ µ ) , ǫ ) + ˆΓ ( l ) ( α s (¯ µ ) , α s (¯ µ ) , ǫ ) ++ˆΓ ( l ) ( α s (¯ µ ) , α s (1 /b ) , ǫ ) − ˆΓ ( l ) ( α s (¯ µ ) , α s (¯ µ ) , ǫ ) ++¯Γ ( l ) ( α s (¯ µ ) , α s (1 /b ) , ǫ ) − ¯Γ ( l ) ( α s (¯ µ ) , α s (¯ µ ) , ǫ ) . (70)The first line is a power series with coefficients which are pole-free for each b , becausethe second and the third lines vanish when b = 1 / ¯ µ . Hence, the sum of the secondand the third line must be finite at ǫ = 0, but it is not necessarily analytic in α s ( µ ).To find its perturbative expression in powers of α s ( µ ) we rewrite the last two linesof Eq.(70) as Z ¯ µ /b dµ µ (cid:18) ∂∂ ln µ ˆΓ ( l ) ( α s (¯ µ ) , α s ( µ ) , ǫ ) + ∂∂ ln µ ¯Γ ( l ) ( α s (¯ µ ) , α s (1 /b ) , ǫ ) (cid:19) . (71)There could be other residual cancellations of ǫ = 0 poles between these two terms,but their sum must be finite at ǫ = 0 and analytic in α s ( µ ) and α s (1 /b ). Thus, .4 The resummed exponent in renormalization group approach we get a perturbative expression of the physical anomalous dimension with finitecoefficient: γ (1 , ¯ µ b , α s (¯ µ )) = − Z ¯ µ /b dµ µ A ( α s ( µ )) − B ( α s (¯ µ )) −− Z ¯ µ /b dµ µ C ( α s (¯ µ ) , α s ( µ )) , (72)where A ( α s ( µ )) = ∞ X n =1 A n α ns ( µ ) (73) B ( α s (¯ µ )) = ∞ X m =1 B m α ms (¯ µ ) (74) C ( α s (¯ µ ) , α s ( µ )) = ∞ X m =1 ∞ X n =1 C mn α ms (¯ µ ) α ns ( µ ) (75)(76)After an integration by parts of the first term we obtain an expression for the all-ordersresummed exponent: Z Q /b d ¯ µ ¯ µ Γ res ( Q ¯ µ , ¯ µ b , α s (¯ µ )) = − Z Q /b d ¯ µ ¯ µ (cid:20) ln Q ¯ µ A ( α s (¯ µ )) + B ( α s (¯ µ )) +(77)+ Z ¯ µ /b dµ µ C ( α s (¯ µ ) , α s ( µ )) (cid:21) To obtain a LL resummation we need only a coefficient ( A ) and to obtain aNLL resummation we need four coefficients A , A , B , C . However it has beendemonstrated by explicit calculations [67, 68] that C = 0 . (78)Therefore for the resummation at the NLL level we need only three coefficients A , A , B . Our general formula reduces to that of Ref.[6] when C mn = 0 . (79)This restriction could be a consequence of the factorization of soft emissions from thehard part of the coefficient function, but this remains unproven.The result for the anomalous dimension in Eq.(72) can be rewritten in our for-malism performing the change of variable n ′ = ¯ µ µ . (80) Renormalization group resummation of tranverse distributions
We get γ (1 , ¯ µ b , α s (¯ µ )) = − Z ¯ µ b dn ′ n ′ G ( α s (¯ µ ) , α s (¯ µ /n ′ )) + ˜ G ( α s (¯ µ )) , (81)where G ( α s (¯ µ ) , α s ( µ )) = ∞ X m =0 G mn α ms (¯ µ ) α ns ( µ ) (82)˜ G ( α s (¯ µ )) = ∞ X m =1 ˜ Gα ms (¯ µ ) . (83)The case of the resummation formula of Ref.[6] is obtained when G mn is non-vanishingonly when m = 0. In this case, we have γ (1 , ¯ µ b , α s (¯ µ )) = − Z ¯ µ b dn ′ n ′ G ( α s (¯ µ /n ′ )) + ˜ G ( α s (¯ µ )) . (84) q ⊥ vs. logs of b to all logarithmic or-ders Large logarithms of q ⊥ appear in the perturbative coefficients in the form of plusdistributions. We define (cid:20) ln p (ˆ q ⊥ )ˆ q ⊥ (cid:21) + (85)in such a way that Z d ˆ q ⊥ (cid:20) ln p (ˆ q ⊥ )ˆ q ⊥ (cid:21) + = 0 . (86)Let us consider the Fourier transforms I p ( Q b ) = 1 Q Z d q ⊥ e i~q ⊥ · ~b (cid:20) ln p (ˆ q ⊥ )ˆ q ⊥ (cid:21) + = 2 π Z ∞ ˆ q ⊥ d ˆ q ⊥ J (ˆ q ⊥ ˆ b ) (cid:20) ln p (ˆ q ⊥ )ˆ q ⊥ (cid:21) + , (87)where we have used the definition of the 0-order Bessel function J : J ( z ) = 12 π Z π dθe iz cos θ . (88)We now exploit the definition of the plus distribution: I p ( Q b ) = 2 π Z d ˆ q ⊥ [ J (ˆ q ⊥ ˆ b ) −
1] ln p ˆ q ⊥ ˆ q ⊥ + 2 π Z ∞ d ˆ q ⊥ J (ˆ q ⊥ ˆ b ) ln p ˆ q ⊥ ˆ q ⊥ . (89) .5 Logs of q ⊥ vs. logs of b to all logarithmic orders Writing ln p ˆ q ⊥ as the p th α -derivative of (ˆ q ⊥ ) α at α = 0, we get I p ( Q b ) = 2 π ∂ p ∂α p (cid:26)Z d ˆ q ⊥ [ J (ˆ q ⊥ ˆ b ) − q α − ⊥ + Z ∞ d ˆ q ⊥ J (ˆ q ⊥ ˆ b )ˆ q α − ⊥ (cid:27) = 2 π ∂ p ∂α p (cid:20)Z ∞ d ˆ q ⊥ J (ˆ q ⊥ ˆ b )ˆ q α − ⊥ − Z d ˆ q ⊥ ˆ q α − ⊥ (cid:21) = π ∂ p ∂α p "(cid:18) Q b (cid:19) − α Γ( α )Γ(1 − α ) − α , (90)where the last equality follows from the identity Z ∞ dxx µ J ν ( ax ) = 2 µ a − µ − Γ(1 / ν/ µ/ / ν/ − µ/
2) (91) a > − Re ν − < Re µ < / . (92)From Eq.(90), we read off the generating function G ( α ) of I p G ( α ) = πα "(cid:18) Q b (cid:19) − α Γ(1 + α )Γ(1 − α ) − , (93)in the sense that I p ( Q b ) = (cid:20) d p dα p G ( α ) (cid:21) α =0 . (94)Now, the generating function of logarithms of Q b is ( Q b ) − α in the sense that L p ≡ ln p (1 / ( Q b )) = (cid:20) d p dα p ( Q b ) − α (cid:21) α =0 . (95)Inverting Eq.(93), we find the relation between the generating function of L p and thegenerating function of I p , which is( Q b ) − α = 1 π S ( α )[ αG ( α ) + π ] , (96)where S ( α ) = 14 α Γ(1 − α )Γ(1 + α ) . (97)Performing the Taylor expansion of the r.h.s. of Eq.(96) around α = 0 and usingEq.(94), we obtain:( Q b ) − α = 1 π ∞ X m =0 α m m ! m X i =0 (cid:18) mi (cid:19) iI i − S ( m − i ) (0) , (98) Renormalization group resummation of tranverse distributions where S ( j ) (0) is the j-th derivative of S ( α ) evaluated at α = 0. Now, using Eq.(95),we the relation between L p and I p : L p = 1 π p X i =1 (cid:18) pi (cid:19) iI i − S ( p − (0) = 1 π p X k =1 (cid:18) p − k − (cid:19) pS ( k − I p − k . (99)Thanks to the first equality in Eq.(87) and to the fact thatln p − k ˆ q ⊥ = 1 p ( p − · · · ( p − k + 1) d k d ln k ˆ q ⊥ ln p ˆ q ⊥ , (100)we arrive at a relation to all logarithmic orders between the logs of b and the logs of q ⊥ : L p = 1 π p X k =1 S ( k − (0)( k − Z d q ⊥ Q e i~q ⊥ · ~b (cid:20) q ⊥ d k d ln k ˆ q ⊥ ln p ˆ q ⊥ (cid:21) + . (101)This relation allows us to derive the relation between a generic function of ln(1 /Q b )and a function of ln ˆ q ⊥ . Indeed, given a function h (cid:18) ln 1 Q b (cid:19) = ∞ X p =0 h p ln p Q b , (102)Eq.(101) implies: h (cid:18) ln 1 Q b (cid:19) = 1 π ∞ X k =1 S ( k − (0)( k − Z d q ⊥ Q e i~q ⊥ · ~b (cid:20) q ⊥ d k d ln k ˆ q ⊥ h (ln ˆ q ⊥ ) (cid:21) + . (103)The r.h.s. of Eq.(103) can be viewed as the Fourier transform of a function (moreproperly a distribution) ˆ h (ln ˆ q ⊥ ): h (cid:18) ln 1 Q b (cid:19) = Z d q ⊥ Q e i~q ⊥ · ~b ˆ h (ln ˆ q ⊥ ) , ˆ h (ln ˆ q ⊥ ) = 1 π ∞ X k =1 S ( k − (0)( k − (cid:20) q ⊥ d k d ln k ˆ q ⊥ h (ln ˆ q ⊥ ) (cid:21) + . (104) q ⊥ -space In this section, we investigate the consequences of our general result Eq.(104) forthe resummation at the NLL level of logarithmic accuracy. According to eq(19) andthe discussion below and according to Eq.(77), we have that our resummation factorformula in Fourier space is: K res ( Q b , /b , Q ) = exp (cid:8) E res ( Q b , /b , Q ) (cid:9) , (105) .6 Resummation in q ⊥ -space where E res ( Q b , /b , Q ) = Z Q /b d ¯ µ ¯ µ Γ res ( Q ¯ µ , ¯ µ b , α s (¯ µ )) , (106)and where at NLL levelΓ resNLL ( Q ¯ µ , ¯ µ b , α s (¯ µ )) = − ln Q ¯ µ (cid:2) A α s (¯ µ ) + A α s (¯ µ ) (cid:3) − B α s (¯ µ ) − C β α s (¯ µ ) ln α s (1 /b ) α s (¯ µ ) , (107)where we have used the definition of the β -function: µ ddµ α s ( µ ) = βα s = − β α s − β α s + O ( α s ) (108)and where we have used the change of variable dµ µ = dα s β ( α s ) (109)to compute the integral that appears in the last term of Eq.(77). Now, thanks toEq.(104), we can rewrite the resummed exponent in b -space Eq.(106) in terms of aresummed exponent defined in q ⊥ -space. Thus, up to NNLL terms, we obtain: E resNLL ( Q b , /b , Q ) = Z d q ⊥ e i~q ⊥ · ~b " ˆΓ resNLL (ˆ q ⊥ , q ⊥ , Q ) q ⊥ + , (110)where ˆΓ resNLL (ˆ q ⊥ , q ⊥ , Q ) = = − ln ˆ q ⊥ (cid:2) ˆ A α s ( q ⊥ ) + ˆ A α s ( q ⊥ ) (cid:3) − ˆ B α s ( q ⊥ ) + − ˆ C β α s ( q ⊥ ) ln α s ( Q ) α s ( q ⊥ ) (111)and where the relation of the constant coefficients of this last equation and the ofEq.(107) is ˆ A = A π (112)ˆ A = − (cid:18) A π + A π β ln e γ E (cid:19) (113)ˆ B = B π − A π ln e γ E C = C π . (115)Here γ E is the usual Euler gamma. Now, we want to define a resummation factorin q ⊥ -space. Looking at Eq.(18), we note that large ln Q b of collinear nature are Renormalization group resummation of tranverse distributions resummed by the parton distribution function. So, in order to define a resummation in q ⊥ -space, we must take them into account. For simplicity, we consider the resummedpart of non-singlet cross section, because the non-singlet parton distribution functions,which are defined as f ′ a ′ ( N, µ ) = f a ( N, µ ) − f b ( N, µ ) a, b = g, (116)evolve independently. In particular, in Mellin moments N they satisfy the followingevolution equations: µ ∂∂µ f ′ a ′ ( N, µ ) = γ ′ ( N, α s ( µ )) f ′ a ′ ( N, µ ) . (117)Hence, the non-singlet parton distribution functions evaluated at µ = 1 /b are relatedto the ones evaluated at µ = Q by, f ′ a ′ ( N, /b ) = exp ( − Z Q /b d ¯ µ ¯ µ γ ′ ( N, α s (¯ µ )) ) f ′ a ′ ( N, Q ) . (118)Thus, the resummed part of the cross section with the non-singlet parton distributionfunctions evaluated at µ = Q becomesexp ( − Z Q /b d ¯ µ ¯ µ X j =1 γ ′ ( N, α s (¯ µ )) ) K res ( b , /b , Q ) = (119)= exp (Z Q /b d ¯ µ ¯ µ " Γ res ( Q ¯ µ , ¯ µ b , α s (¯ µ )) − X j =1 γ ′ ( N j , α s (¯ µ )) The general relation between a function of ln Q b and its Fourier anti-transformEq.(104), immediately enables us to define a resummed exponent of the non-singletpart of the cross section in q ⊥ -space, which is: K res (ˆ q ⊥ , q ⊥ , Q ) = 1 π ∞ X k =1 S ( k − (0)( k − × ( q ⊥ d k d ln k ˆ q ⊥ exp "Z Q q ⊥ d ¯ µ ¯ µ Γ res ( Q ¯ µ , ¯ µ /q ⊥ , α s (¯ µ )) − X j =1 γ ′ ( N j , α s (¯ µ )) ! + = 1 π ∞ X k =1 S ( k − (0)( k − × ( dd ˆ q ⊥ d k − d ln k − ˆ q ⊥ exp "Z Q q ⊥ d ¯ µ ¯ µ Γ res ( Q ¯ µ , ¯ µ /q ⊥ , α s (¯ µ )) − X j =1 γ ′ ( N j , α s (¯ µ )) ! + = lim η → + π ∞ X k =0 S ( k ) (0) k ! d k d ln k ˆ q ⊥ d ˆ q ⊥ (cid:26) θ (ˆ q ⊥ − η ) exp (cid:20) Z Q q ⊥ d ¯ µ ¯ µ × Γ res ( Q ¯ µ , ¯ µ /q ⊥ , α s (¯ µ )) − X j =1 γ ′ ( N j , α s (¯ µ )) ! (cid:21)(cid:27) , (120) .6 Resummation in q ⊥ -space where the last equation defines implicity the q ⊥ -space resummation exponent: K res (ˆ q ⊥ , q ⊥ , Q ) = exp (cid:8) E res (ˆ q ⊥ , q ⊥ , Q ) (cid:9) . (121)All the previously released expressions for this exponent given in [69, 70, 71] are par-ticular cases of this general expression. They differ essentially in the criteria accordingto which the subleading terms are kept.We want to calculate the NLL result in q ⊥ -space. Thus, keeping only the termsup to NNLL in Eq.(120) we obtain K resNLL (ˆ q ⊥ , q ⊥ , Q ) = 1 π d ˆ q ⊥ (cid:26) θ (ˆ q ⊥ − η ) exp (cid:20) Z Q q ⊥ d ¯ µ ¯ µ (cid:18) Γ resNLL ( Q ¯ µ , ¯ µ /q ⊥ , α s (¯ µ )) − X j =1 γ ′ ( N j , α s (¯ µ )) (cid:19)(cid:21) ∞ X k =0 S ( k ) (0) k ! [ − ln ˆ q ⊥ A α s ( q ⊥ )] k (cid:27) . (122)In order to compare this result at NLL to that of [70], we define a new variable h : h ≡ q ⊥ A α s ( q ⊥ ) . (123)In terms of this variable and using Eq.(97) the series that appears in Eq.(122) can becomputed: ∞ X k =0 S ( k ) (0) k ! [ − ln ˆ q ⊥ A α s ( q ⊥ )] k = S ( − h/
2) = 2 h Γ(1 + h/ − h/ . (124)In conclusion, we obtain that the NLL resummation factor becomes K resNLL (ˆ q ⊥ , q ⊥ , Q ) = 1 π d ˆ q ⊥ (cid:26) θ (ˆ q ⊥ − η ) exp (cid:20) Z Q q ⊥ d ¯ µ ¯ µ (cid:18) Γ resNLL ( Q ¯ µ , ¯ µ /q ⊥ , α s (¯ µ )) − X j =1 γ ′ ( N j , α s (¯ µ )) (cid:19)(cid:21) h Γ(1 + h/ − h/ (cid:27) , (125)which gives the same result given in Ref.[70] in the case that the coefficient C thatappears in Eq.(107) is equal to zero and that the arbitrary constants c and c alsodefined in [70] are equal to one. It is clear that the last two terms of the exponentialof our result let the non-singlet parton distribution densities, which enter the ˆ q ⊥ derivative, evolve from the scale Q to the scale q ⊥ .We conclude the chapter noting that also in this case the resummed results usingthe renormalization group approach are less predictive than results obtained with theapproach of Ref.[1], as it is shown in Ref.[6]. Furthermore the conditions that reduceour results to those of Ref.[6] in terms of factorization properties is still an interestingopen question. hapter 7Predictive power of theresummation formulae We have shown that the renormalization group resummed expressions are less predic-tive than those obtained with other approaches discussed in Sec.2.3. In this Chapterwe shall compare the various approaches quantitatively. We will show that all the re-summation coefficients can be determined by a fixed order computation. This can beuseful, because the determination of the resummation coefficients from a fixed ordercomputation represents a possible way to check the correctness of the resummationformulae with strong factorization properties.In particular, in this chapter, we will show how the resummation coefficients g mnp (for the prompt photon case), g mn (for the DIS and DY cases) and G mn , ˜ G m (forthe DY transverse momentum distribution) can be determined. For the rapiditydistributions of DY and DIS they are the same of the all-inclusive cases (see Chapter5) The resummation coefficients are determined by comparing the expansion of theresummed anomalous dimension γ in powers of α s ( Q ) with a fixed-order calculation,which in general has the form: γ FO ( N, α s ) = k min X i =1 α is i X j =1 γ ij ln j N + O ( α k min +1 s ) + O ( N ) , (1)where γ FO ( N, α s ) is the physical anomalous dimension for each individual partonicsubprocess for the promt photon case, for the qq channel in the DY case and forthe q channel in the DIS case. For the case of the small transverse momentum DYdistribution it ha the form: γ FO (¯ µ b , α s ) = k min X i =1 α is i X j =0 ˜ γ ij ln j µb + O ( α k min +1 s ) + O ( 1¯ µ b ) . (2)The number k min is the minimum order at which the anomalous dimension must becalculated in order to determine its N k − LL resummation.For prompt photon production, the number of coefficients N k that must be deter-mined at each logarithmic order, and the minimum fixed order which is necessary in109 Predictive power of the resummation formulae
Prompt photonEq.(94) sec.4.4 Eq.(96) sec.4.4 Eq.(87) sec.4.4 N k k k ( k +3)2 k ( k +1)( k +5)6 k min k + 1 2 k k − N k and minimum order of the required perturbativecalculation k min for inclusive prompt photon N k − LL resummation.DIS/DYEq.(81) sec.3.3 Eq.(80) sec.3.3 N k k k ( k +1)2 k min k k − N k and minimum order of the required perturbativecalculation k min for inclusive DIS and DY N k − LL resummation.DY transverse distributionEq.(84) sec.6.4 Eq.(81) sec.6.4 N k k − k +3 k − k min k k − N k and minimum order of the required perturbativecalculation k min for small transverse momentum DY N k − LL resummation.order to determine them are summarized in Table 7.1, according to whether the coeffi-cient function is fully factorized [Eq.(94) sec.4.4 ], or has factorized N -dependent and N -independent terms [Eq.(96) sec.4.4], or not factorized at all [Eq. (87) sec4.4]. In theapproach of Refs.[12, 14] the coefficient function is fully factorized, and furthermoresome resummation coefficients are related to universal coefficients of Altarelli-Parisisplitting functions, so that k min = k . For prompt-photon production, available re-sults do not allow to test factorization, and test relation of resummation coefficientsto Altarelli-Parisi coefficients only to lowest O ( α s ).The results for DIS and Drell-Yan, according to whether the coefficient functionhas factorized N -dependent and N -independent terms as in Refs.[51, 2, 1] [Eq.(81)sec.3.3] or no factorization properties as in [(Eq.80) sec.3.3], are reported in table 7.2.Current fixed-order results support factorization for Drell-Yan and DIS only to thelowest nontrivial order O ( α s ).In table 7.3, we report also the results for the small transverse DY resummation.We list N k and k min for the approach of Ref.[6] [Eq.(84) sec.6.4] and for the renormal-ization group approach [Eq.(81) sec.6.4]. If the two cases are related by factorizationproperties of the cross section is not yet understood even if probable.In the following, we present all the proofs of these results. .1 Prompt photon production in the strongest factorization case This is the case of Eq.(94) in section 4.4. In this case there are N k = 2 k non-vanishingcoefficients g i and g i i = 1 , , . . . , k . The resummed expression of the anomalousdimension at N k − LL is given by: γ ( N, α s ( Q )) = Z N dnn k X i =1 g i α is ( Q /n ) ! + Z N dnn k X i =1 g i α is ( Q /n ) ! . (3)We consider first the second integral in Eq.(3). Noting that: dnn = − dα s ( Q /n ) β ( α s ) , (4)where β ( α s ) = − b α s − b α s + O ( α s ) (5) b ≡ β π b ≡ β (4 π ) , (6)with β and β given in Eq.(38) in section 1.2, we can rewrite it in the form: Z α s ( Q /N ) α s ( Q ) dα s P ki =1 g i α is β α s (cid:16) β β α s + β β α s + · · · (cid:17) . (7)Now, we expand up to order α k − s each term that compares in the integrand of thislast expression and collect all the coefficients that correspond to the same power of α s . Doing this, we have that the integral (7) can be rewritten in the following form:1 b ( Z α s ( Q /N ) α s ( Q ) dα s (cid:20) g α s + ( b g + g ) + ( b g + b g + g ) α s + · · · ++( b k − g + · · · + b k − k − g k − + g k ) α k − s (cid:21)) , (8)where k > b ji are build up with the coefficients of the β function.Now, we perform the integral over α s . We get: Z N dnn k X i =1 g i α is ( Q /n ) ! == 1 b ( g ln (cid:18) α s ( Q /N ) α s ( Q ) (cid:19) + ( b g + g )[ α s ( Q /N ) − α s ( Q )] ++ 12 ( b g + b g + g )[ α s ( Q /N ) − α s ( Q )] + · · · ++ 1 k − b k − g + · · · + b k − k − g k − + g k )[ α k − s ( Q /N ) − α k − s ( Q )] ) (9) Predictive power of the resummation formulae
To perform the first integral of Eq.(3), we it is sufficient to note that in this case dnn = − dα s ( Q /n )2 β ( α s ) . (10)and proceed as before. The result that we obtain is: Z N dnn k X i =1 g i α is ( Q /n ) ! == 12 b ( g ln (cid:18) α s ( Q /N ) α s ( Q ) (cid:19) + ( b g + g )[ α s ( Q /N ) − α s ( Q )] ++ 12 ( b g + b g + g )[ α s ( Q /N ) − α s ( Q )] + · · · ++ 1 k − b k − g + · · · + b k − k − g k − + g k )[ α k − s ( Q /N ) − α k − s ( Q )] ) (11)At this point, we take the first term of Eq.(9) together with the first term of Eq.(11)in order to isolate the first contributions of Eq.(3). We have:12 b (cid:20) g ln (cid:18) α s ( Q /N ) α s ( Q ) (cid:19) + 2 g ln (cid:18) α s ( Q /N ) α s ( Q ) (cid:19)(cid:21) (12)From this contribution, we want to extract the firs two LL terms. Hence, using theone loop running of α s ( Q /N a ) , a = 1 , α s ( Q ) ln 1 N [ − ( g + g )] + α s ( Q ) ln N [ b / g + 2 g )]+ O ( α s ln( N )) + O ( α is ln j ( N )) , (13)where i ≥ , ≤ j ≤ i + 3. Now, we take the second term of Eq.(9) together withthe second term of Eq.(11) in order to keep the second the second contributions ofEq.(3) and we have:12 b (cid:2) ( b g + g )( α s ( Q /N ) − α s ( Q )) + 2( b g + g )( α s ( Q /N ) − α s ( Q )) (cid:3) . (14)From this contribution, we want to extract the first two NLL terms. In order todo this, we use the one loop running of α s ( Q /N a ) , a = 1 , g and g are modified by the last two terms of Eq.(13). We get: α s ( Q ) ln(1 /N )[ − ( c g + d g + g + g )] + α s ( Q ) ln (1 /N )[ b (2˜ c g + ˜ d g + 2 g + g )]+ O ( α s ln( N )) + O ( α is ln j ( N )) , (15)where c , d , ˜ c , ˜ d are coefficients (that are of no concern to us) and i ≥ , ≤ j ≤ i + 3. This procedure can be repeated for all the other contributions. So, we take the .1 Prompt photon production in the strongest factorization case k-th term of Eq.(9) together with the k-th term of Eq.(11) in order to keep the k-thcontributions of Eq.(3). For the general k-th contribution, we get:12 b (cid:20) k − b k − g + · · · + b k − k − g k − + g k )( α k − s ( Q /N ) − α k − s ( Q )) ++ 2 k − b k − g + · · · + b k − k − g k − + g k )( α k − s ( Q /N ) − α k − s ( Q )) (cid:21) . (16)From this contribution, we want to extract the first two N k − LL terms. In order todo this, again, we use the one loop running of α s ( Q /N a ) , a = 1 , g i and g i with i = 1 , , . . . , k − α ks ln 1 N [ − ( c k − g + · · · + c k − k − g k − + d k − g + · · · + d k − k − g k − + g k + g k )] ++ α k +1 s ln N [ b k/ c k − g + · · · + 2˜ c k − k − g k − + ˜ d k − g + · · · + ˜ d k − k − g k − +2 g k + g k )] + O ( α k +1 s ln( N )) + O ( α k +2+ is ln j ( N )) , (17)where i ≥ , ≤ j ≤ i + 3.To summarize, Eqs. (13,15,17) tell us that from the expression of the physicalanomalous dimension Eq.(3), we can extract the following linear combinations of thecoefficients g , . . . , g k , g , . . . , g k : l = − ( g + g ) l = b g + 2 g ) l = − ( c g + d g + g + g ) l = b (2˜ c g + ˜ d g + 2 g + g ) · · · l k − = − ( c k − g + · · · + c k − k − g k − + d k − g + · · · + d k − k − g k − + g k + g k ) l k = kb c k − g + · · · + 2˜ c k − k − g k − + ˜ d k − g + · · · + ˜ d k − k − g k − + 2 g k + g k ) , with k ≥ l i , i = 1 , . . . , k the known terms. These are 2 k independent lin-ear combinations that determine the 2 k coefficients g , . . . , g k , g , . . . , g k of a N k − LL resummation comparing them with the correspondent terms of Eq.(2) up toorder α k +1 s . This is a direct consequence of the fact that the two vectors (1 ,
1) and(1 ,
2) are independent. This shows that, in order to obtain a N k − LL resummationin the case of the strongest factorization (Eq.(94) in section 4.4) , we need to know a N k +1 LO fixed order calculation of the physical anomalous dimension. Hence, in thiscase k min = k + 1. Predictive power of the resummation formulae
This is the case of Eq.(96) in section 4.4. In this case in order to perform a LLresummation, we need two coefficients ( g , g ); to perform a NLL resummationthree more coefficients are added ( g , g , g ); in general to perform a N k − LL resummation k + 1 coefficients are added ( g ij , i + j = k ) to those of the N k − LL resummation. Hence, in order to perform a N k − LL resummation, we need to deter-mine N k = k X p =1 ( p + 1) = k ( k + 3)2 , (18)coefficients. We want to determine k min in Eq.(2) so that all the k ( k + 3) / k + 1 from the N k − LL . The N k − LL expression of thephysical anomalous dimension in this case is given by: γ ( N, α s ( Q )) = Z N dnn k X i =1 g i α is ( Q /n ) ! + Z N dnn k X i =1 g i α is ( Q /n ) ! ++ Z N dnn k X s =2 s − X i =1 g is − i α is ( Q /n ) α s − is ( Q /n ) . (19)The first two LL contributions have been already computed and are given in Eq.(13).Now, we extract the first 3 NLL contributions of Eq.(19). Recalling the deriva-tion of Eq.(14), we obtain that these contributions are contained in the followingexpression:12 b (cid:2) ( b g + g )( α s ( Q /N ) − α s ( Q )) + 2( b g + g )( α s ( Q /N ) − α s ( Q )) (cid:3) + Z N dnn g α s ( Q /n ) α s ( Q /n ) . (20)Since α s ( Q /p a ) = α s ( Q )1 + aβ α s ( Q ) ln p + O ( α is ln j p ) , i ≥ , ≤ j ≤ i
11 + ab α s ( Q ) ln p = ∞ X j =0 ( − ) j a j b j α js ( Q ) ln j p , (21)and keeping in mind that corrections to the NLL come from Eq.(13), we have that .2 Prompt photon production in the weaker factorization case Eq.(20) become:12 b ∞ X j =1 ( − ) j [2 j ( c j g + g ) + 2( d j g + g )] b j α j +1 s ( Q ) ln j N + (22)+ Z N dnn g α s ( Q ) ∞ X j =0 ( − ) j b j α js ( Q ) ln j n ! ∞ X i =0 ( − ) i i b i α is ( Q ) ln i n ! . The Cauchy product of the two series in Eq.(22) is given by( ∞ X j =0 ( − ) j b j α js ( Q ) ln j n )( ∞ X i =0 ( − ) i i b i α is ( Q ) ln i n ) == 12 b ∞ X j =1 ( − ) j − j X i =1 i ! b j α j − s ( Q ) ln j − n (23)and j X i =1 i = 2(2 j − . (24)Now, because dnn = − d ln 1 n , (25)we can perform the integration. We get12 b X j =1 ( − ) j (cid:20) j ( c j g + g ) + 2( d j g + g ) + 2(2 j − j g (cid:21) b j α j +1 s ( Q ) ln j n , (26)where c j , d j are certain coefficients we don not need to worry about. The first threeNLL contributions are given by j = 1 , , k + 1 N k − LL contributions that come fromEq.(19). Recalling how Eq.((16)) was computed, we have that the desired contribu-tions are contained in the following expression:12 b (cid:20) k − b k − g + · · · + b k − k − g k − + g k )( α k − s ( Q /N ) − α k − s ( Q )) ++ 2 k − b k − g + · · · + b k − k − g k − + g k )( α k − s ( Q /N ) − α k − s ( Q )) (cid:21) ++ Z N dnn k − X i =1 g ik − i α is ( Q /n ) α k − is ( Q /n ) (27)We use the following relations α rs ( Q /p a ) = α rs ( Q )(1 + ab α s ( Q ) ln p ) r + O ( α is ln j p ) , i ≥ , ≤ j ≤ i + 1 , ab α s ( Q ) ln p ) r = ∞ X m =0 ( − ) m (cid:18) r + m − m (cid:19) a m b m α ms ( Q ) ln m p , (28) Predictive power of the resummation formulae where (cid:0) rm (cid:1) are the usual binomial coefficients. With this, we can compute the integralin Eq.(27) performing the Cauchy product of the two series expansion of α s ( Q /n )and of α s ( Q /n ) and performing explicitly the integral using the change of variableEq.(3). We get: Z N dnn k − X i =1 g ik − i α is ( Q /n ) α k − is ( Q /n ) = k − X i =1 g ik − i ∞ X m =0 C ( i,k − i ) m b m α k + ms ( Q ) ln m +1 N , (29)where C ( i,j ) m = ( − m +1 m + 1 m X l =0 l (cid:18) l + i − i − (cid:19) (cid:18) m − l + j − j − (cid:19) , (30)and where (cid:18) n − (cid:19) = Γ( n + 1)Γ(0)Γ( n + 2) (31)is equal to 1 for n = − k + 1 N k − LL contributions of Eq.(27), we have k X m =0 (cid:26) ( − ) m +1 k − (cid:18) k + m − m + 1 (cid:19) [2 m ( c k − m g + · · · + c k − k − m g k − + g k ) ++( d k − m g + · · · + d k − k − j g k − + g k ) + k − X t =2 t − X i =1 g it − i f ( k − itm ] ++ k − X i =1 g ik − i C ( i,k − i ) m (cid:27) b m α k + ms ( Q ) ln m +1 N , (32)where c ik − m , d ik − m , f ( k − itm are certain coefficients we do not have to worry about. Atthis point, we can make some simplifications. In fact, since C (0 ,k ) m = ( − m +1 m + 1 (cid:18) m + k − k − (cid:19) = ( − m +1 k − (cid:18) m + k − m + 1 (cid:19) (33)and (see Appendix E) C ( k, m = 2 m C (0 ,k ) m , (34)we can write Eq.(32) in the following form: k X m =0 (cid:26) [ C ( k, m ( c k − m g + · · · + c k − k − m g k − ) ++ C (0 ,k ) m ( d k − m g + · · · + d k − k − j g k − ) + C (0 ,k ) m k − X t =2 t − X i =1 g it − i f ( k − itm ] ++ k X i =0 g ik − i C ( i,k − i ) m (cid:27) b m α k + ms ( Q ) ln m +1 N , (35) .2 Prompt photon production in the weaker factorization case
What this result tells us, is that passing from the N k − LL to the N k − LL resum-mation, k + 1 new resummation coefficients are added. In Eq.(35), we have k + 1conditions for this coefficients (one for each m ) to be set equal to the correspond-ing fixed order contribution of Eq.(2) . We shall now show that this conditions areindependent. This is equivalent to showing that the k + 1 linear combinations k X i =0 g ik − i ˜ C ( i,k − i ) m m = 0 , , . . . , k (36)with ˜ C ( i,j ) m ≡ ( − ) m +1 ( m + 1) C ( i,j ) m = m X l =0 l (cid:18) l + i − i − (cid:19)(cid:18) m − l + j − j − (cid:19) (37)are independent. Moreover, this is equivalent to showing that for each k the columnsof the ( k + 1) × ( k + 1) matrix A kmi ≡ ˜ C ( i,k − m are independent vectors. To show this,we need to use two identities proved in Appendix E:˜ C ( i, m = 2 m ˜ C (0 ,i ) m (38)˜ C ( i,j ) m = 2 ˜ C ( i,j − m − ˜ C ( i − ,j ) m ; i, j ≥ , (39)˜ C (0 ,i ) m = (cid:18) m + i − i − (cid:19) (40)which allows us to to compute the columns of the matrix A ( k ) mj explicitly for all k . Toshow their independence, we use induction on k , i.e. we demonstrate the independenceof the columns for k = 1 and then we assume that the property is valid for k − k . In the case k = 1, we have a 2 × C (0 , m ˜ C (1 , m ) = (1 2 m ) , (41)and now it is clear the two columns are independent, because 1 and 2 m are independentfunctions of m . Now, using the induction hypothesis k − X i =0 α i ˜ C ( i,k − − i ) m = 0 ⇐⇒ α i = 0 , i = 0 , . . . , k − , (42)we want to show that it is sufficient to prove that: k X i =0 β i ˜ C ( i,k − i ) m = 0 ⇐⇒ β i = 0 , i = 0 , . . . , k. (43) Predictive power of the resummation formulae
Using the relations (38,39), we have: k X j =0 β j ˜ C ( j,k − j ) m = ( β + 2 m β k ) ˜ C (0 ,k ) m + k − X j =1 β j ˜ C ( j,k − j ) m = ( β + 2 m β k ) ˜ C (0 ,k ) m + 2 k − X j =1 β j ˜ C ( j,k − − j ) m − k − X j =1 β j ˜ C ( j − ,k − j ) m = ( β + 2 m β k ) ˜ C (0 ,k ) m + 2 k − X j =1 β j ˜ C ( j,k − − j ) m − k − X j ′ =0 β j ′ +1 ˜ C ( j ′ ,k − − j ′ ) m = ( β + 2 m β k ) ˜ C (0 ,k ) m − β ˜ C (0 ,k − m + k − X j =1 (2 β j − β j +1 ) ˜ C ( j,k − − j ) m +2 β k − ˜ C ( k − , m = 0 (44)Now, from Eq.(40), we know that ˜ C (0 ,k ) m is a degree-( k −
1) polynomial in m . Fur-thermore, from Eq.(39), we know that the vectors ˜ C ( j,k − − j ) m with j = 0 , . . . , k − k − m . Consequently, Eq.(44) can be satisfied ifand only if ( β + 2 m β k ) ˜ C (0 ,k ) m = 0 , (45) − β ˜ C (0 ,k − m + k − X j =1 (2 β j − β j +1 ) ˜ C ( j,k − − j ) m + 2 β k − ˜ C ( k − , m = 0 . (46)From the first, it follows that: β = β k = 0 , (47)while from the second, thanks to the induction hypothesis Eq.(42), we have that β = β k − = 0 , (48)and that β = 12 β = 14 β = · · · = 12 k − β k − . (49)In conclusion from Eqs.(47,48,49) it follows that β j = 0 , j = 0 , . . . , k. (50)This completes the proof that the columns of the squared ( k + 1) × ( k + 1) matrices A ( k ) mj ≡ ˜ C ( j,k − j ) m are independent for all k . This shows that, in order to obtain a N k − LL resummation in the case of the weaker factorization (Eq.(96) in section 4.4), we needto know a N k LO fixed order calculation of the physical anomalous dimension. Hence,in this case k min = 2 k . .3 Prompt photon in the general case Let us now consider the most general case, in which the coefficient function doesnot satisfy any factorization property. This is the case of Eq.(87) in section 4.4. Inthis case, in order to perform a LL resummation we need 2 coefficients ( g , g );to perform a NLL resummation 5 coefficients are added ( g , g , g , g , g ) andto perform a N k − LL resummation k ( k + 3) / g mnp with m + n + p = k but without n = p = 0). Thus, in order to perform a N k − LL resummation, we need to determine N k = k X p =1 p ( p + 3)2 = k ( k + 1)( k + 5)6 (51)coefficients. The N k − LL expression of the physical anomalous dimension is given,in this case, by γ ( N, α s ( Q )) = Z N dnn k X s =1 s − X i =0 s − i X j =0 g ijs − i − j α is ( Q ) α js ( Q /n ) α s − i − js ( Q /n ) . (52)Now, we proceed in the same way as we have done in section 7.2 and we find that the k ( k + 3) / N k − LL to the N k − LL contributions appear only in the following combinations: k − X i =0 k − i X j =0 g ijk − i − j ∞ X m =0 C ( j,k − i − j ) m b m α s ( Q ) k + m ln m +1 N . (53)Each term with fixed m in the expansion Eq. (66) provides a new condition on thesecoefficients. However, these conditions are not linearly independent for all choices of m . Indeed, let us define the matrix C ( j,k − i − j ) m ≡ D ( k ) m ( i,j ) , where the lines are labelledby the index m and the columns by the multi-index ( i, j ). This matrix gives the linearcombination of the coefficients g mnp in Eq.(66) to be determined and it turns out tobe of rank rg ( D ( k ) m ( i,j ) ) = 2 k ≤ k ( k + 3)2 . (54)We shall now prove this statement: D ( k ) m ( i,j ) is a M × k ( k +3)2 matrix , whose columns are the M -component vectors D ( k ) m = C ( j,k − i − j ) m ; 0 ≤ i ≤ k −
1; 0 ≤ j ≤ k − i ; 0 ≤ m ≤ M. (55)We use induction on k . For k = 1, D (1) is a 2 × D (1) m = (cid:0) C (0 , m , C (1 , m (cid:1) = ( − m +1 m + 1 (1 , m ) , (56)that are linearly independent; the rank of D (1) is 2. Let us check explicitly also thecase k = 2. In this case D (2) m = (cid:0) C (0 , m , C (1 , m , C (0 , m , C (1 , m , C (2 , m (cid:1) . (57) Predictive power of the resummation formulae
The first two columns are the same as in the case k = 1: they span a 2-dimensionalsubspace. The last three columns are independent as a consequence of Eq.(43) with k = 1. Furthermore, C (0 , m and C (2 , m = 2 m C (0 , m are independent of all other columns,because they are the only ones that are proportional to a degree-1 polynomial in m .Finally, C (1 , m is a linear combination of the first two columns, as a consequence ofEqs.(37,39) with i = j = 1. Thus, the rank of D (2) is 2 + 2 = 4.We now assume that D ( k − has rank 2( k − D ( k ) as D ( k ) m = (cid:0) C ( j,k − − i − j ) m , C ( l,k − l ) m (cid:1) (58)0 ≤ i ≤ k − , ≤ j ≤ k − − i ≤ l ≤ k. (59)By the induction hypothesis, only 2( k −
1) of the columns C ( j,k − − i − j ) m are independent.The columns C ( l,k − l ) m are all independent as a consequence of Eq.( 43); among them,those with 1 ≤ l ≤ k − C ( j,k − − i − j ) m byEq. (39). Only C (0 ,k ) m and C ( k, m are independent of all other columns because they areproportional to a degree-( k −
1) polynomial in m , while all others are at most of degree( k − k − C ( j,k − − i − j ) m , and the rank of D ( k ) is2( k −
1) + 2 = 2 k. (60)It follows that each individual terms in the sum over m in Eq. (66) depends only on2 k independent linear combinations of the coefficients g ijk − i − j , , ≤ i ≤ k − , ≤ j ≤ k − i .This means that the N k − LL order resummed result depends only on 2 k indepen-dent linear combinations of the k ( k +3) / N k − LL resummation to the N k − LL one and that the remaining coefficients arearbitrary. Because a term with fixed m in Eq.(66) is of order α k + ms , this implies that acomputation of the anomalous dimension up to fixed order k min = 3 k − k − LL resummation, because m = 0 , , . . . , k −
1. Note that when goingfrom N k − LL to N k LL, at this higher order, in general some new linear combinationsof the k ( k + 3) / N k − LL to the N k − LL ,will appear through terms depending on β . Hence, some of the combinations of co-efficients that were left undetermined in the N k − LL resummation will now becomedetermined. However, this does not affect the value k min of the fixed-order accuracyneeded to push the resummed accuracy at one extra order. In conclusion, even inthe absence of any factorization, despite the fact that now the number of coefficientswhich must be determined grows cubically according to Eq. (51), the required orderin α s of the computation which determines them grows only linearly. This is the case of Eq.(80) in section 3.3. Here we discuss the case without assumingany factorization property, because the factorized case will be recovered as a particular .5 DY small transverse momentum distribution case. So, in the general case, at LL we need to determine 1 coefficient ( g ); atNLL 2 coefficients are added ( g , g ) and at N k − LL , k coefficients ( g ik − i with i = 0 , , . . . , k −
1) are added to the N k − LL ones . Thus, in order to perform a N k − LL resummation, we need to determine N k = k X p =1 p = k ( k + 1)2 (61)coefficients. The N k − LL expression of the physical anomalous dimension is given by γ ( N, α s ( Q )) = a Z N dnn k X s =1 s − X i =0 g is − i α is ( Q ) α s − is ( Q /n a ) , (62)where a = 1 for DIS and a = 2 for DY. Now, we proceed in the same way as we havedone in the previous sections and we find that the k new coefficients that are addedpassing from the N k − LL to the N k − LL contributions appear only in the followingcombinations: k − X i =0 g ik − i ∞ X m =0 a m +1 C (0 ,k − i ) m b m α s ( Q ) k + m ln m +1 N . (63)Again, each term with fixed m in this expansion provides a new condition on thesecoefficients. These conditions are all linearly independent for any choice of m , becausethe k × k matrix A ( k ) mi ≡ C ,k − im with m, i = 0 , , . . . , k − m . This implies that, in order to determine a N k − LL resummation,a computation of the physical anomalous dimension up to order k min = 2 k − g ,k . This means that going from the N k − LL to the N k − LL we needto take only one combination of the expansion Eq.(63). Hence, in this more restrictivecase, N k = k and k min = k . This is the case of Eq.(81) in section 6.4 obtained with the renormalization groupapproach. The result of the approach of Ref.[6] (reported in Eq.(81) of section 6.4)will be recovered as a particular case. In the most general case, at LL we need todetermine 1 coefficient ( G ); at NLL 3 more coefficients are added ( G , G , ˜ G ) andat N k − LL , k + 1 coefficients ( G ik − i with i = 0 , , . . . , k − G k − ) are added.Therefore, in order to perform a N k − LL resummation, we need to determine N k = k X p =1 ( p + 1) − k + 3 k −
22 (64) Predictive power of the resummation formulae coefficients. The N k − LL expression of the physical anomalous dimension is given by γ (1 , ¯ µ b , α s (¯ µ )) = − Z ¯ µ b dn ′ n ′ k X s =1 s − X i =0 G is − i α is (¯ µ ) α s − is (¯ µ /n ′ )+ k − X i =1 ˜ G i α is (¯ µ ) . (65)Now, we proceed as before and we find that the k new coefficients that are addedpassing from the N k − LL to the N k − LL contributions appear only in the followingcombinations: − k − X i =0 G ik − i ∞ X m =0 C (0 ,k − i ) m b m α s (¯ µ ) k + m ln m +1 (cid:18) µ b (cid:19) + ˜ G k − α k − s (¯ µ ) . (66)As before, each term with fixed m in this expansion provides a new independentcondition on this coefficients. This implies that, in order to determine a N k − LL resummation, a computation of the physical anomalous dimension up to order k min =2 k − G ,k and ˜ G k − . This means that going from the N k − LL to the N k − LL only 2 coefficients are added but the LL where we have only onecoefficient. Hence, in this more restrictive case, N k = 2 k − k min = k . hapter 8Conclusions In this thesis, we have studied the renormalization group approach to resummation forall inclusive deep-inelastic and Drell-Yan processes. The advantage of this approachis that it does not rely on factorization of the physical cross section, and in fact itsimply follows from general kinematic properties of the phase space. Then we haveanalyzed some of its generalizations.In particular, we have presented a generalization to prompt photon productionof the approach to Sudakov resummation which has been described in Chapter 3for deep-inelastic scattering and Drell-Yan production. It is interesting to see thatalso with the more intricate two-scale kinematics that characterizes prompt photonproduction in the soft limit, it remains true that resummation simply follows fromgeneral kinematic properties of the phase space. Also, this approach does not re-quire a separate treatment of individual colour structures when more than one colourstructure contributes to fixed order results.The resummation formulae obtained here turn out to be less predictive than pre-vious results: a higher fixed-order computation is required in order to determine theresummed result. This depends on the fact that here neither specific factorizationproperties of the cross section in the soft limit is assumed, nor that soft emissionsatisfies eikonal-like relations which allow one to determine some of the resummationcoefficients in terms of universal properties of collinear radiation. Currently, fixed-order results are only available up to O ( α s ) for prompt photon production. An order α s computation is required to check nontrivial properties of the structure of resum-mation: for example, factorization, whose effects only appear at the next-to-leadinglog level, can only be tested at O ( α s ). The greater flexibility of the approach pre-sented here would turn out to be necessary if the prediction obtained using the morerestrictive resummation were to fail at order α s .We have also proved a resumation formula for the Drell-Yan rapidity distributionsto all logarithmic accuracy and valid for all values of rapidity. Isolating a universaldimensionless coefficient function, which is exactly that ones of the Drell-Yan rapidity-integrated, we have shown a general procedure to obtain resummed results to NLLfor the rapidity distributions of a virtual photon γ ∗ or of a real vector boson W ± , Z .Furthermore, we have outlined a general method to calculate numerical predictionsand analyzed the impact of resummation for the fixed-target experiment E866/NuSea.123 Conclusions
This shows that NLL resummation has an important effects on predictions of differ-ential rapidity cross sections giving an agreement with data that is better than NNLOfull calculations. We have found a suppression of the cross section for not large valuesof hadronic rapidity instead of enhancing it. This suppression arises due to the shiftin the complex plane of the dominant contribution of resummed exponent. Theseleaves open questions for future studies about possible suppression of the rapidityintegrated cross sections at small x .The study of the renormalization group resummation applied to the case of smalltransverse momentum distribution of Drell-Yan pairs has opened further interestinglyaspects about the relation between factorization properties of the cross section andthe final structure of the resummed results which has been not yet well understood.Furthermore, because of its generality, renormalization group resummation lendsitself naturally to some important future applications. They are the factorization ofresummation of rare meson decay processes (like B → X s γ ) and the resummation ofgeneralized parton densities in deeply virtual proton Compton scattering. ppendix ARelations between logarithms of N and logarithms of (1 − z ) In this Appendix we want to find the general relations between the logarithms of N and the logarithms of (1 − z ). Let’s consider a generic logarithmical enhanced termin z space (cid:20) ln p (1 − z )1 − z (cid:21) + (1)and take its Mellin transform: I p ≡ Z dz z N − − − z ln p (1 − z ) . (2)To find a general relation between the terms of the type of Eq.(1) and the logs of N ,we first notice that all integrals I p Eq.(2) can be obtained from a generating function G ( η ): I p = d p dη p G ( η ) | η =0 , (3) G ( η ) = Z dz ( z N − − e ( η −
1) ln(1 − z ) = Γ( N )Γ( η )Γ( N + η ) − η . (4)Now, using the Stirging expansion of the Γ function at large N Γ( N + 1) = √ πN e N ln N − N + O (cid:18) N (cid:19) , (5)we get G ( η ) = 1 η (cid:20) Γ(1 + η ) N η − (cid:21) + O (cid:18) N (cid:19) . (6)At this point, we notice that N − η is just the generating function of the log of N : L p ≡ ln p N = d p dη p e η ln 1 /N | η =0 . (7)125 Relations between logarithms of N and logarithms of (1 − z )Hence, Eq.(6) can be viewed as a relation between the generating function for I p andfor L p . In particular, Taylor-expanding Γ(1 + η ) in Eq.(6) leads to leading, next-to-leading,. . . ln N relations: G ( η ) = 1 η " N η ∞ X k =0 Γ ( k ) (1) k ! η k − = ∞ X k =0 Γ ( k ) (1) k ! 1 η d k d ln k /N [ e η ln 1 /N − − ∞ X k =0 Γ ( k ) (1) k ! d k d ln k /N Z − N dze ( η −
1) ln(1 − z ) . (8)Now, if we put this last equation in Eq.(3), we obtain: I p = − ∞ X k =0 Γ ( k ) (1) k ! d k d ln k /N Z − N dz ln p (1 − z )1 − z + O (cid:18) N (cid:19) (9)= 1 p + 1 p +1 X k =0 (cid:18) p + 1 k (cid:19) Γ ( k ) (1) (cid:18) ln 1 N (cid:19) p +1 − k + O (cid:18) N (cid:19) , (10)where in the last equality we have used the identity d k d ln k /N Z − N dz ln p (1 − z )1 − z = − p + 1 k ! (cid:18) p + 1 k (cid:19) (cid:18) ln 1 N (cid:19) p +1 − k . (11)This last expression is equal to zero when k > p + 1. This result can be expressed interms of derivatives with respect to ln(1 − z ) with the identity d k d ln k /N Z − N dz ln p (1 − z )1 − z = Z − N dz − z d k ln p (1 − z ) d ln k (1 − z ) (12)thus obtaining for I p Eq.(9) the following all-logarithmic-order relation: I p = − p X k =0 Γ ( k ) (1) k ! Z − N dz − z d k ln p (1 − z ) d ln k (1 − z )+ Γ ( p +1) (1) p + 1 + O (cid:18) N (cid:19) . (13)The inverse result, expressing L p in terms of I p , can be analogously found invertingthe relation between the generating functions Eq.(6): N − η = ηG ( η ) + 1Γ(1 + η ) . (14) Proceeding as before, we getln n N = n X i =1 (cid:18) ni (cid:19) ∆ ( n − i ) (1) iI i − + ∆ ( n ) (1) + O (cid:18) N (cid:19) (15)= n X k =1 ∆ ( k − (1)( k − Z dz z N − − − z d k ln n (1 − z ) d ln k (1 − z )+∆ ( n ) (1) + O (cid:18) N (cid:19) , (16)where ∆ ( k ) ( η ) is the k th derivative of∆( η ) ≡ η ) . (17)Because of the p -independence of the coefficients of the expansion Eq.(13), wecan determine explicitly the Mellin transform of a generic logarithmical enhancedfunction (cid:20) ˆ g (ln(1 − z ) a )1 − z (cid:21) + = ∞ X p =0 ˆ g p (cid:20) ln p (1 − z ) a − z (cid:21) + . (18)Indeed, its Mellin transform up to non-logarithmical terms is given by Z dz z N − − − z ˆ g (ln(1 − z ) a ) = ∞ X p =0 ˆ g p a p I p = ∞ X p =0 ˆ g p a p p + 1 p +1 X k =0 (cid:18) p + 1 k (cid:19) Γ ( k ) (1) (cid:18) ln 1 N (cid:19) p +1 − k (19)= − ∞ X k =0 Γ ( k ) (1) k ! Z − N dz − z d k d ln k (1 − z ) a ˆ g (ln(1 − z ) a )= Z − N dz − z g (ln(1 − z ) a ) = 1 a Z N a dnn g (ln 1 n ) , (20)where in the last equality we have done the change of variable n = (1 − z ) − a andwhere g (ln K ) ≡ − ∞ X k =0 Γ ( k ) (1) a k k ! d k d ln k K ˆ g (ln K ) . (21)The inverse relation can be analogously derive. Namely, we can cast the integral ofany function g (ln(1 − z ) a ) = ∞ X p =0 g p ln p (1 − z ) a (22)as a Mellin transform, up to non-logarithmic terms:1 a Z N a dnn g (ln 1 n ) = Z dz z N − − − z ˆ g (ln(1 − z ) a ) , (23) Relations between logarithms of N and logarithms of (1 − z )where ˆ g (ln K ) ≡ − ∞ X k =0 ∆ ( k ) (1) a k k ! d k d ln k K g (ln K ) . (24)For completeness, we recall that Γ ′ (1) = − γ E = − . n = n ′ a so that we have1 a Z N a dnn g (ln 1 n ) = Z N dn ′ n ′ g (ln 1 n ′ a ) (25) ppendix BPhase space The n -body phase space can be expressed in terms of ( n − m )-body and ( m + 1)-bodyphase spaces. To prove this statement, let us consider the definition of the phasespace in d = 4 − ǫ dimensions for a generic process with incoming momentum P and n on-shell particles in the final state with outgoing momenta p , p , . . . , p n : dφ n ( P ; p , . . . , p n ) = d d − p (2 π ) d − p . . . d d − p n (2 π ) d − p n (2 π ) d δ d ( P − p − · · · − p n ) . (1)The momentum-conservation delta function can be rewritten as δ d ( P − p − · · · − p n ) = Z d d Qδ d ( P − Q − p − · · · − p m ) × δ d ( Q − p m +1 − · · · − p n ) . (2)The integration measure of this equation can be rewritten in this way: d d Q = (2 π ) d d d − Q (2 π ) d − Q d ( Q ) π = (2 π ) d d d − Q (2 π ) d − Q dQ π , (3)where Q = ( Q ) − | ~Q | (4)= ( p m +1 + · · · + p n ) (5)= ( P − p − · · · − p m ) . (6)We shall now find the minimum and the maximum value of the variable Q . If weuse Eq.(5) in the center-of-mass frame of the momentum p m +1 + · · · + p n , we have Q = ( p m +1 + · · · + p n ) . (7)The minimum value of Q is achieved when all the energies p m +1 , . . . , p n are equal totheir invariant masses ( m i = p p i ). Hence Q min = ( q p m +1 + · · · + p p n ) , (8)129 Phase space because Q is a Lorentz ivariant. From Eq.(6), we have that in the center-of-massframe of the momentum P − p − · · · − p m , Q = ( P − p − · · · − p m ) . (9)Now, the minimum value of p + · · · + p m is achieved when all these terms reduces totheir invariant masses ( m i = p p i ). In this case ~p = · · · = ~p m = 0 (10)and this implies that ~P = 0 and that P = √ P . Therefore, we obtain Q max = ( √ P − q p − · · · − p p m ) , (11)again as a consequence of the Lorentz invariance of Q . We obtain immediately thegeneral phase space decomposition formula using Eqs.(1,2,3) together: dφ n ( P ; p , . . . , p n ) = Z Q max Q min dQ π dφ m +1 ( P ; Q, p , . . . , p m ) × dφ n − m ( Q ; p m +1 , . . . , p n ) , (12)where Q min and Q max are given by Eq.(8) and Eq.(11) respectively. Using recursivelyEq.(12) with m = 1, it is possible to rewrite a n -body phase space in terms of n two-body phase spaces. We shall now compute the two-body phase space in thecenter-of-mass frame. We have dφ ( P ; Q, p ) = d d − Q (2 π ) d − Q d d − p (2 π ) d − p (2 π ) d δ d ( P − Q − p )= (2 π ) ǫ − d d − pQ p δ ( P − Q − p ) . (13)In the center-of-mass frame ~P = 0 and thus P = √ P . If we can neglect the invariantmass of p , we have that in this frame | ~Q | = | ~p | = p and that δ ( P − Q − p ) = p ( p ) + Q √ P δ (cid:18) p − P − Q √ P (cid:19) = Q √ P δ (cid:18) p − P − Q √ P (cid:19) . (14)Now, d d − p = | ~p | d − d | ~p | d Ω d − = ( p ) d − dp d Ω d − ., (15)where d Ω d − is the solid angle in d − p , we obtain dφ ( P ; Q, p ) = (2 π ) ǫ − p ) − ǫ √ P d Ω d − = N ( ǫ )( P ) − ǫ (cid:18) − Q P (cid:19) − ǫ d Ω d − , (16) where N ( ǫ ) = 12(4 π ) − ǫ . (17)Finally, we want to calculate d Ω d − . To do this, we use its definition d Ω d − = dθ d − sin d − θ d − d Ω d − , (18)which can be applied recursively i times till d − − i > ≤ θ d − i ≤ π . Thenormalization of the solid angles can be obtained performing the gaussian integral inspherical coordinates thus givingΩ d − = Z d Ω d − = 2 π ( d − / Γ(( d − / , (19)where Γ is the usual gamma functionΓ( α ) = Z ∞ dt e − t t α − (20)In our case ( d = 4 − ǫ ) we can use Eq.(18) two times and we odtain d Ω d − = d Ω − ǫ = sin − ǫ θ dθ sin − ǫ φ dφ d Ω − ǫ . (21)In many cases it is useful to rewrite this equation in terms of other variables variables y = 1 + cos θ , y = 1 + cos φ , ≤ y i ≤ . (22)Doing this change of variables and recalling that the two-body squared amplitudecannot depend on more than two angles, we obtain d Ω − ǫ = 4 − ǫ π / − ǫ Γ(1 / − ǫ ) Z dy [ y (1 − y )] − ǫ Z dy [ y (1 − y )] − / − ǫ . (23)We recall for completeness that the integrals in Eqs.(12,23) are indicated only toremind the integration range. They can be performed without the matching withthe square amplitude of the corrispondig process only for the determination of theirnormalization. We can check explicitly that Eq.(23) is correct performing its integral.This is easily done using the definition of the B function B ( z, w ) ≡ Z dt t z − (1 − t ) w − = Γ( z )Γ( w )Γ( z + w ) , (24)and the Legendre duplication formulaΓ(2 z ) = (2 π ) − / z − / Γ( z )Γ( z + 1 / . (25)We find Ω − ǫ = 2 π / − ǫ Γ(3 / − ǫ ) , (26)which is in agreement with Eq.(19). ppendix CFull NLO expression for DYrapidity distributions We report here the complete expression of the NLO DY distributions given in [17,60, 61, 62] with the factorization scale equal to the renormalization scale: dσ NLO dQ dY = N ( Q ) X q,q ′ c qq ′ Z x dx x Z x dx x × (cid:26) (cid:20) C (0) q ¯ q ( x , x , Y ) + α s ( µ )2 π C (1) q ¯ q (cid:18) x , x , Y, Q µ (cid:19)(cid:21) × (cid:8) q ( x , µ ) ¯ q ′ ( x , µ ) + ¯ q ( x , µ ) q ′ ( x , µ ) (cid:9) + α s ( µ )2 π C (1) gq ( x , x , Y ) g ( x , µ ) (cid:8) q ′ ( x , µ ) + ¯ q ′ ( x , µ ) (cid:9) + α s ( µ )2 π C (1) qg ( x , x , Y ) (cid:8) q ( x , µ ) + ¯ q ( x , µ ) (cid:9) g ( x , µ ) (cid:27) , (1)where N ( Q ) = 4 πα Q S for γ ∗ , (2) N ( Q ) = πG F Q √ S δ ( Q − M V ) for Z and W ± , (3)133 Full NLO expression for DY rapidity distributions and where C (0) q ¯ q ( x , x , Y ) = x x δ ( x − x ) δ ( x − x ) , x = √ xe ± Y , (4) C (1) q ¯ q (cid:18) x , x , Y, Q µ (cid:19) = x x C F (cid:26) δ ( x − x ) δ ( x − x ) (cid:20) π − ( x ) + 2Li ( x ) + ln (1 − x ) + ln (1 − x ) + 2 ln x − x × x − x (cid:21) + (cid:18) δ ( x − x ) (cid:20) x − x x − x + x x ( x − x ) × ln x x + x + x x (cid:18) ln(1 − x /x ) x − x (cid:19) + + x + x x × x − x ) + ln 2 x (1 − x ) x ( x + x ) (cid:21) + (1 ↔ (cid:19) + G A ( x , x , x , x )[( x − x )( x − x )] + + H A ( x , x , x , x ) + ln Q µ × (cid:26) δ ( x − x ) δ ( x − x ) (cid:2) − x x + 2 ln 1 − x x (cid:3) + (cid:18) δ ( x − x ) x + x x x − x ) + + (1 ↔ (cid:19) (cid:27)(cid:27) , (5) C (1) gq ( x , x , Y ) = x x T F (cid:26) δ ( x − x ) x (cid:20) ( x + ( x − x ) ) × ln 2( x − x )(1 − x )( x + x ) x + 2 x ( x − x ) (cid:21) + G C ( x , x , Y )( x − x ) + + H C ( x , x , Y ) + ln Q µ (cid:26) δ ( x − x ) x ( x + ( x − x ) ) (cid:27)(cid:27) , (6) C (1) qg ( x , x , Y ) = C (1) gq ( x , x , − Y ) , (7)with Li ( x ) = − Z x dt ln(1 − t ) t , (8) G A ( x , x , Y ) = 2( x x + x x )( x x + x x ) x x ( x + x )( x + x ) , (9) H A ( x , x , Y ) = − x x ( x x + x x ) x x ( x x + x x ) , (10) G C ( x , x , Y ) = 2 x ( x x + ( x x − x x ) )( x x + x x ) x x ( x x + x x )( x + x ) , (11)and H C ( x , x , Y ) = 2 x x ( x x + x x )( x x x + x x ( x x + 2 x x )) x x ( x x + x x ) . (12) ppendix DProof of some identities of chapter6 D.1 Proof of Eq.(42)
We compute in the limit ˆ q ⊥ → δ ( M ) where M is given by Eq.(28).Firstly we have δ ( M ) = ξ ξ Q δ ((1 − ξ )(1 − ξ ) + ˆ q ⊥ (1 − ξ − ξ )) . (1)Then taking a generic test function f ( ξ , ξ ), Z dξ Z dξ f ( ξ , ξ ) δ ((1 − ξ )(1 − ξ ) + ˆ q ⊥ (1 − ξ − ξ ))= Z dξ Z dξ f ( ξ , ξ )1 − ξ + ˆ q ⊥ δ (cid:18) − ξ − ξ ˆ q ⊥ − ξ + ˆ q ⊥ (cid:19) = Z dξ Z dξ f ( ξ , ξ ) − f ( ξ , − ξ + ˆ q ⊥ δ (cid:18) − ξ − ξ ˆ q ⊥ − ξ + ˆ q ⊥ (cid:19) + Z dξ f ( ξ , Z dξ − ξ + ˆ q ⊥ δ (cid:18) − ξ − ξ ˆ q ⊥ − ξ + ˆ q ⊥ (cid:19) = Z dξ Z dξ f ( ξ , ξ ) − f ( ξ , − ξ + ˆ q ⊥ δ (cid:18) − ξ − ξ ˆ q ⊥ − ξ + ˆ q ⊥ (cid:19) + Z dξ f ( ξ , − ξ + ˆ q ⊥ Z dξ δ (cid:18) − ξ − ξ ˆ q ⊥ − ξ + ˆ q ⊥ (cid:19) = Z dξ Z dξ f ( ξ , ξ ) − f ( ξ , − ξ + ˆ q ⊥ δ (cid:18) − ξ − ξ ˆ q ⊥ − ξ + ˆ q ⊥ (cid:19) + Z dξ f ( ξ , − ξ + ˆ q ⊥ = Z dξ Z dξ f ( ξ , ξ ) − f ( ξ , − ξ + ˆ q ⊥ δ (cid:18) − ξ − ξ ˆ q ⊥ − ξ + ˆ q ⊥ (cid:19) + Z dξ f ( ξ , − f (1 , − ξ + ˆ q ⊥ + f (1 ,
1) ln 1 + ˆ q ⊥ ˆ q ⊥ . (2)135 Proof of some identities of chapter 6
Now, taking the limit ˆ q ⊥ → δ ( M ) = ξ ξ Q (cid:20) δ (1 − ξ )(1 − ξ ) + + δ (1 − ξ )(1 − ξ ) + − ln ˆ q ⊥ δ (1 − ξ ) δ (1 − ξ ) (cid:21) + O (ˆ q ⊥ ) , (3)which is exactly Eq.(42). D.2 Proof of Eq.(43)
Let us compute for a generic test function f ( ξ , ξ ) the distribution T ˆ q ⊥ = Z dξ Z dξ [(1 − ξ )(1 − ξ ) + ˆ q ⊥ (1 − ξ − ξ )] η − f ( ξ , ξ ) (4)in the small-ˆ q ⊥ limit with η = a | ǫ | , a > ǫ = 0. The integration range is fixedby the requirement M = Q ξ ξ [(1 − ξ )(1 − ξ ) + ˆ q ⊥ (1 − ξ − ξ )] ≥ , (5)which gives 0 ≤ ξ ≤ ≤ ξ ≤ (1 − ξ )(1 + ˆ q ⊥ )1 − ξ + ˆ q ⊥ = ¯ ξ (6)or equivalently 0 ≤ ξ ≤ ≤ ξ ≤ (1 − ξ )(1 + ˆ q ⊥ )1 − ξ + ˆ q ⊥ = ¯ ξ . (7)We observe also that(1 − ξ )(1 − ξ ) + ˆ q ⊥ (1 − ξ − ξ ) = (1 − ξ + ˆ q ⊥ ) (cid:18) − ξ − ξ ˆ q ⊥ − ξ + ˆ q ⊥ (cid:19) = (1 − ξ + ˆ q ⊥ ) (cid:18) − ξ − ξ ˆ q ⊥ − ξ + ˆ q ⊥ (cid:19) . (8)Next, we decompose T ˆ q ⊥ = T q ⊥ + T q ⊥ (9)where T q ⊥ = Z dξ (1 − ξ + ˆ q ⊥ ) η − Z ¯ ξ dξ (cid:18) − ξ − ξ ˆ q ⊥ − ξ + ˆ q ⊥ (cid:19) η − [ f ( ξ , ξ ) − f ( ξ , T q ⊥ = Z dξ (1 − ξ + ˆ q ⊥ ) η − Z ¯ ξ dξ (cid:18) − ξ − ξ ˆ q ⊥ − ξ + ˆ q ⊥ (cid:19) η − f ( ξ , . (10) .2 Proof of Eq.(43) The ξ integral in T q ⊥ is immediately performed: Z ¯ ξ dξ (cid:18) − ξ − ξ ˆ q ⊥ − ξ + ˆ q ⊥ (cid:19) η − = 1 η (cid:20) (1 − ξ )(1 + ˆ q ⊥ )1 − ξ + ˆ q ⊥ (cid:21) η . (11)Therefore, T q ⊥ = (1 + ˆ q ⊥ ) η η Z dξ f ( ξ ,
1) (1 − ξ ) η − ξ + ˆ q ⊥ . (12)Proceeding as above, we regularize the ξ integral: T q ⊥ = T q ⊥ + T q ⊥ , (13)where T q ⊥ = (1 + ˆ q ⊥ ) η η Z dξ (1 − ξ ) η − ξ + ˆ q ⊥ [ f ( ξ , − f (1 , T q ⊥ = (1 + ˆ q ⊥ ) η η f (1 , Z dξ (1 − ξ ) η − ξ + ˆ q ⊥ = f (1 ,
1) (1 + ˆ q ⊥ ) η η (1 + ˆ q ⊥ ) η − (ˆ q ⊥ ) η η . (15)The two distributions T q ⊥ and T q ⊥ are now well defined as ˆ q ⊥ →
0. Similarly, T q ⊥ = T q ⊥ + T q ⊥ , (16)where T q ⊥ = Z dξ (1 − ξ + ˆ q ⊥ ) η − Z ¯ ξ dξ (cid:18) − ξ − ξ ˆ q ⊥ − ξ + ˆ q ⊥ (cid:19) η − (17) { [ f ( ξ , ξ ) − f ( ξ , − [ f (1 , ξ ) − f (11)] } T q ⊥ = Z dξ (1 − ξ + ˆ q ⊥ ) η − Z ¯ ξ dξ (cid:18) − ξ − ξ ˆ q ⊥ − ξ + ˆ q ⊥ (cid:19) η − [ f (1 , ξ ) − f (1 , T q ⊥ can be computed by changing the order of integration and exploitingEq.(8) and using eq(11), thus obtaining T q ⊥ = (1 + ˆ q ⊥ ) η η Z dξ (1 − ξ ) η − ξ + ˆ q ⊥ [ f (1 , ξ ) − f (1 , . (19)Collecting Eqs.(14,15,17,19), taking the limit ˆ q ⊥ → − ξ ) η − = (1 − ξ ) η − + 1 η δ (1 − ξ ) , (20)we obtain the following identity in sense of distributions:[(1 − ξ )(1 − ξ ) + ˆ q ⊥ (1 − ξ − ξ )] η − = (21)= (1 − ξ ) η − (1 − ξ ) η − − (ˆ q ⊥ ) η η δ (1 − ξ ) δ (1 − ξ ) + O (ˆ q ⊥ )which is exactly Eq.(43). ppendix EProof of combinatoric properties ofchapter 7 Let us consider the coefficients defined by C ( i,j ) m = ( − m +1 m + 1 m X l =0 l (cid:18) l + i − i − (cid:19)(cid:18) m − l + j − j − (cid:19) . (1)We shall now prove that C ( i, m = 2 m C (0 ,i ) m (2) C ( i,j ) m = 2 C ( i,j − m − C ( i − ,j ) m ; i, j ≥ , (3)or, equivalently, that˜ C ( i, m = 2 m ˜ C (0 ,i ) m (4)˜ C ( i,j ) m = 2 ˜ C ( i,j − m − ˜ C ( i − ,j ) m ; i, j ≥ , , (5)where ˜ C ( i,j ) m ≡ ( − ) m +1 ( m + 1) C ( i,j ) m = m X l =0 l (cid:18) l + i − i − (cid:19)(cid:18) m − l + j − j − (cid:19) . (6)Eq.(4) follows immediately. Indeed:˜ C ( i, m = 2 m (cid:18) m + i − i − (cid:19) (7)˜ C (0 ,i ) m = (cid:18) m + i − i − (cid:19) . (8)We turn now to the proof of Eq.(5). In the case i = j = 1 it follows againimmediately. In fact we have:˜ C (0 , m = 1 , ˜ C (1 , m = 2 m , ˜ C (1 , m = m X l =0 l = 2 m +1 − . (9)139 Proof of combinatoric properties of chapter 7
In the cases i = 1 , j = 2 e i = 2 , j = 1 it is quite easy:˜ C (1 , m = 2 m +1 − , ˜ C (0 , m = m + 1 , ˜ C (2 , m = 2 m ( m + 1) (10)˜ C (1 , m = 2(2 m +1 − − ( m + 1) , ˜ C (2 , m = 2 m +1 ( m + 1) − (2 m +1 − , (11)where we have use the equality: m X l =0 l l = lim α → ddα m X l =0 ( e α ln 2 ) l = 2 m +1 ( m + 1) − m +1 − . (12)In the other cases, i.e. for i, j ≥
2, it follows in a straightforward way:2 ˜ C ( i,j − m − ˜ C ( i − ,j ) m == 2 m X l =0 l (cid:18) l + i − i − (cid:19)(cid:18) m − l + j − j − (cid:19) − m X l =0 l (cid:18) l + i − i − (cid:19)(cid:18) m − l + j − j − (cid:19) = ˜ C ( i,j ) m − m X l =0 l +1 (cid:18) l + i − i − (cid:19)(cid:18) m − l + j − j − (cid:19) + m X l =0 l (cid:18) l + i − i − (cid:19)(cid:18) m − l + j − j − (cid:19) = ˜ C ( i,j ) m − m +1 X l ′ =1 l ′ (cid:18) l ′ + i − i − (cid:19)(cid:18) m − l ′ + j − j − (cid:19) + m X l =0 l (cid:18) l + i − i − (cid:19)(cid:18) m − l + j − j − (cid:19) = ˜ C ( i,j ) m − m +1 (cid:18) m + i − i − (cid:19)(cid:18) j − j − (cid:19) + (cid:18) i − i − (cid:19)(cid:18) m + j − j − (cid:19) = ˜ C ( i,j ) m , (13)where l ′ = l + 1 and where in the second, the third and the last line we have used thefollowing identities: (cid:18) n + r − r − (cid:19) = (cid:18) n + r − r − (cid:19) − (cid:18) n + r − r − (cid:19) , (14) (cid:18) r − r − (cid:19) = Γ( r − r )Γ(0) = 0 . (15)We note that these last two properties are valid only for r ≥ i = j = 1 e i = 1 , j = 2 and i = 2 , j = 1 have been treated separately. ibliography [1] George Sterman. Summation of large corrections to short distance hadroniccross-sections. Nucl. Phys. , B281:310, 1987.[2] S. Catani and L. Trentadue. Resummation of the QCD perturbative series forhard processes.
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