Quantum Chromodynamics

Quantum Chromodynamics (CERN-2014-001)

Abstract

I review the basics of perturbative QCD, including infrared divergences and safety, collinear and factorization theorems, and various evolution equations and resummation techniques for single- and double-logarithmic corrections. I then elaborate its applications to studies of jet substructures and hadronic two-body heavy-quark decays.

1 Introduction

One of the important missions of the Large Hadron Collider (LHC) is to search for new physics beyond the standard model. The identification of new physics signals usually requires precise understanding of standard-model background, whose contributions mainly arise from quantum chromodynamics (QCD). Many theoretical approaches have been developed based on QCD, which are appropriate for studies of processes in different kinematic regions and involving different hadronic systems. The theoretical framework for high-energy hadron collisions is known as the perturbative QCD (pQCD). I will focus on pQCD below, introducing its fundamental ingredients and applications to LHC physics. Supplementary material can be found in [1].

The simple QCD Lagrangian reveals rich dynamics. It exhibits the confinement at low energy, which accounts for the existence of various hadronic bound states, such as pions, protons, mesons, and etc.. This nonperturbative dynamics is manifested by infrared divergences in perturbative calculations of bound-state properties like parton distribution functions and fragmentation functions. On the other hand, the asymptotic freedom at high energy leads to a small coupling constant, that allows formulation of pQCD. Therefore, it is possible to test QCD in high-energy scattering, which is, however, nontrivial due to bound-state properties of involved hadrons. That is, high-energy QCD processes still involve both perturbative and nonperturbative dynamics. A sophisticated theoretical framework needs to be established in order to realize the goal of pQCD: it is the factorization theorem [2], in which infrared divergences are factorized out of a process, and the remaining piece goes to a hard kernel. The point is to prove the universality of the infrared divergences, namely, the independence of processes the same hadron participates in. Then the infrared divergences are absorbed into a parton distribution function (PDF) for the hadron, which just needs to be determined once, either from experimental data or by nonperturbative methods. The universality of a PDF guarantees the infrared finiteness of hard kernels for all processes involving the same hadron. Convoluting these hard kernels with the determined PDF, one can make predictions. In other words, the universality of a PDF warrants the predictive power of the factorization theorem.

Though infrared divergences are factorized into a PDF, the associated logarithmic terms may appear in a process, that is not fully inclusive. To improve perturbative expansion, these logarithmic corrections should be organized by evolution equations or resummation techniques. For the summation of different single logarithms, the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation [3] and the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [4] have been proposed. For different double logarithms, the threshold resummation [5, 6, 7] and the resummation [8, 9] have been developed. Besides, an attempt has been made to combine the DGLAP and BFKL equations, leading to the Ciafaloni-Catani-Fiorani-Marchesini (CCFM) equation [10]. Similarly, the threshold and resummations has been unified under the joint resummation [11, 12], which is applicable to processes in a wider kinematic range. A simple framework for understanding all the above evolution equations and resummation techniques will be provided.

After being equipped with the pQCD formalism, we are ready to learn its applications to various processes, for which I will introduce jet substructures and hadronic two-body heavy-quark decays. It will be demonstrated that jet substructures, information which is crucial for particle identification at the LHC and usually acquired from event generators [13], are actually calculable using the resummation technique. Among jet substructures investigated in the literature, the distribution in jet invariant mass and the energy profile within a jet cone will be elaborated. For the latter, it will be shown that the factorization theorem goes beyond the conventional naive factorization assumption [14], and provides valuable predictions for branching ratios and CP asymmetries of hadronic two-body heavy-quark decays, that can be confronted by LHCb data. Specifically, I will concentrate on three major approaches, the QCD-improved factorization [15], the perturbative QCD [16, 17, 18, 19], and the soft-collinear-effective theory [20, 21, 22, 23]. Some long-standing puzzles in meson decays and their plausible resolutions are reviewed. For more details on this subject, refer to [24].

2 Factorization Theorem

The QCD lagrangian is written as

(1)

with the quark field , the quark mass , and the covariant derivative and the gauge field tensor

(2)

respectively. The color matrices and the structure constants obey

(3)

where denotes the fundamental (adjoint) representation. Adding the gauge-fixing term in the path-integral quantization to remove spurious degrees of freedom, Eq. (1) becomes

(4)

with the gauge parameter , and the ghost field . The last term in the above expression comes from the Jacobian for the variable change, as fixing the gauge.

The Feynman rules for QCD can be derived from Eq. (4) following the standard procedures [25]. The quark and gluon propagators with the momentum are given by and in the Feynman gauge, respectively. The quark-gluon-quark vertex and the ghost-gluon-ghost vertex are written as and , respectively, where the subscripts and are associated with the gluon, is the momentum of the outgoing ghost, and () is associated with the outgoing (incoming) ghost. The three-gluon vertex and the four-gluon vertex are given by

(5)

respectively, where the subscripts , , and , , are assigned to gluons counterclockwise. The particle momenta flow into the vertices in all the above Feynman rules.

2.1 Infrared Divergences and Safety

The first step to establish the factorization theorem is to identify infrared divergences in Feynman diagrams for a QCD process at quark-gluon level. We start with the vertex correction to the amplitude , in which a virtual photon of momentum splits into a quark of momentum and an anti-quark of momentum . Given the Feynman rules, one has the loop integral

(6)

where is the loop momentum carried by the gluon, and the inclusion of the corresponding counterterm for the regularization of a ultraviolet divergence is understood. The appearance of infrared divergences becomes more transparent, as performing the contour integration in the light-cone frame, in which the coordinates are defined by

(7)

When an on-shell particle moves along the light cone, only one component of its momentum is large in this frame. For example, the above quark momenta can be chosen as and .

In terms of the light-cone coordinates, Eq. (6) is reexpressed as

(8)

where only the denominators are shown, since infrared divergences are mainly determined by pole structures. The poles of are located, for , at

(9)

As , the contour of is pinched at by the first and third poles, defining the collinear region. As , the contour of is pinched at , defining the soft region. That is, the collinear (soft) region corresponds to the configuration of (), where and denote a large scale and a small scale, respectively. Another leading configuration arises from the hard region characterized by . A simple power counting implies that all the above three regions give logarithmic divergences. Picking up the first pole in Eq. (9), Eq. (8) becomes

(10)

which produces the double logarithm from the overlap of the collinear (the integration over ) and soft (the integration over ) enhancements.

The existence of infrared divergences is a general feature of QCD corrections. An amplitude is not a physical quantity, but a cross section is. To examine whether the infrared divergences really call for attention, we extend the calculation to the cross section of the process , the annihilation into hadrons. A cross section is computed as the square of an amplitude, whose Feynman diagrams are composed of those for the amplitude connected by their complex conjugate with a final-state cut between them. The cross section at the Born level is written as

(11)

where is the number of colors, is the electromagnetic coupling constant, is the invariant mass, and is the quark charge in units of the electron charge. The virtual one-loop corrections, including those to the gluon vertex in Eq. (6) and to the quark self-energy, give in the dimensional regularization [25]

(12)

with the color factor , the strong coupling constant , the renormalization scale , and the Gamma function . The double pole is a consequence of the overlap of the collinear and soft divergences. The one-loop corrections from real gluons lead to [25]

(13)
Figure 1: Final-state cut on self-energy corrections to a virtual photon propagator.

It is a crucial observation that the infrared divergences cancel in the summation over the virtual and real corrections in Eqs. (12) and (13), respectively: the double and single poles have a minus sign in the former, but a plus sign in the latter. It is easy to understand the infrared cancellation by means of self-energy corrections to the propagator of a virtual photon. Since a virtual photon does not involve a low characteristic scale, the loop corrections must be infrared finite. As taking the final-state cut shown in Fig. 1, the imaginary piece of a particle propagator is picked up, , which corresponds to the Feynman rule for an on-shell particle. Because the self-energy corrections are infrared finite, their imaginary part, i.e., the cross section, is certainly infrared finite. The above observation has been formulated into the Kinoshita-Lee-Nauenberg (KLN) theorem [26], which states that a cross section is infrared safe, as integrating over all phase spaces of final states. Combining Eqs. (11), (12), and (13), one derives the cross section up to next-to-leading order (NLO)

(14)

that has been used to determine the strong coupling constant at the scale .

2.2 DIS and Collinear Factorization

Though a naive perturbation theory applies to the annihilation, it fails for more complicated ones, such as the deeply inelastic scattering (DIS) of a nucleon by a lepton, . Even as the momentum transfer squared is large, the quark-level cross section for the DIS suffers infrared divergences at high orders, which reflect the nonperturbative dynamics in the nucleon. A special treatment of the infrared divergences is then required. It will be demonstrated that they can be factorized out of the scattering process, and absorbed into a nucleon PDF.

Consider the two structure functions involved in the DIS, where the Bjorken variable is defined as , and take as an example. We shall not repeat loop integrations, but quote the NLO corrections to the quark-level diagrams [25]:

(15)

where the superscript denotes the initial-state quark, is the Euler constant, and the first term comes from the leading-order (LO) contribution. The subscript represents the plus function, which is understood as a distribution function via

(16)

The integration over generates an infrared divergence, that is regularized in the dimensional regularization with ,

(17)

Hence, the infrared divergence does exist in the perturbative evaluation of the DIS structure function, even after summing over the virtual and real corrections. This divergence arises from the collinear region with the loop momentum being parallel to the nucleon momentum, since it can also be regularized by introducing a mass to the initial-state quark. It is related to the confinement mechanism, and corresponds to a long-distance phenomenon associated with a group of collimated on-shell particles. The other terms in Eq. (15) represent the hard NLO contribution to the structure function. Comparing the results for the DIS and for the annihilation, the former involves the integration over final-state kinematics, but not over initial-state kinematics. This is the reason why the KLN theorem does not apply to the infrared divergences associated with the initial-state nucleon, and the above collinear divergence exists. Note that the soft divergences cancel between virtual and real diagrams due to the fact that a nucleon is color-singlet: a soft gluon with a huge space-time distribution cannot resolve the color structure of a nucleon, so it does not interact with it.

Besides, the collinear gluon emissions modify a quark momentum, such that the initial-state quark can carry various momenta, as it participates in hard scattering. It is then natural to absorb the collinear divergences into a PDF for the nucleon, , which describes the probability for quark to carry certain amount of the nucleon momentum. In other words, the quark-level collinear divergences are subtracted by those in the PDF in perturbation theory, and the remaining infrared finite piece contributes to the hard kernel . We write the quark-level structure function as the following expansion in the strong coupling constant,

(18)

where () is the hard kernel (PDF) of the -th order. The symbol represents a convolution in the parton momentum fraction :

(19)

We are ready to assign each term in Eq. (15) into either or . The first term goes to with the definitions

(20)

which confirm . The second term in Eq. (15) is assigned to and the third term to with

(21)

and the quark splitting function

(22)

The definition of the PDF in terms of a hadronic matrix element is given by

(23)

where denotes the bound state of the nucleon with momentum and spin , is the minus component of the coordinate of the quark field after the final-state cut, the first factor is attributed to the average over the nucleon spin, and the matrix is the spin projector for the nucleon. Here is called the factorization scale, which is similar to a renormalization scale, but introduced in perturbative computations for an effective theory. The Wilson lines are defined by with

(24)

where represents a path-ordered exponential. The Wilson line behaves like a scalar particle carrying a color source. The two quark fields in Eq. (23) are separated by a distance, so the above Wilson links are demanded by the gauge invariance of the nonlocal matrix element. Since Eq. (23) depends only on the property of the nucleon, but not on the hard processes it participates in, a PDF is universal (process-independent). This is the most important observation, that warrants the predictive power of the factorization theorem.

The Wilson line appears as a consequence of the eikonalization of the final-state quark, to which the collinear gluons attach. The eikonalization is illustrated below by considering the loop correction to the virtual photon vertex. Assuming the initial-state quark momentum and the final-state quark momentum , we have the partial integrand

(25)

as the loop momentum is collinear to , where comes from the Feynman rule for the final-state quark, is the photon vertex, and the subleading contribution from the transverse components of has been neglected. Applying the identity and leads the above expression to

(26)

where the dimensionless vector is parallel to , and the subleading contribution from has been restored. The factor and are called the eikonal vertex and the eikonal propagator, respectively.

It is then shown that the Feynman rule for the eikonalized final-state quark is derived from the Wilson line in Eq. (24). Consider the expansion of the path-order exponential in up to order of , and Fourier transform the gauge field into the momentum space,

(27)

where the term has been introduced to suppress the contribution from . The field is contracted with the gauge field from the initial-state quark with interaction to form the gluon propagator . The expansion of the second piece gives the Feynman rules for the eikonal propagator appearing after the final-state cut. In this case the additional exponential factor is combined with , implying that the valence quark after the final-state cut carries the momentum . In summary, the first (second) piece of Wilson lines corresponds to the configuration without (with) the loop momentum flowing through the hard kernel. The above discussion verifies the Wilson lines in the PDF definition.

After detaching the collinear gluons from the final-state quark, the fermion flow still connects the PDF and the hard kernel. To achieve the factorization in the fermion flow, we insert the Fierz identity,

(28)

with being the identity matrix and . At leading power, only the term contributes, in which the structure goes to the definition of the PDF in Eq. (23), and goes into the evaluation of the hard kernel. The other terms in Eq. (28) contribute at higher powers. Similarly, we have to factorize the color flow between the PDF and the hard kernel by inserting the identity

(29)

where denotes the identity matrix, and is a color matrix. The first term in the above expression contributes to the present configuration, in which the valence quarks before and after the final-state cut are in the color-singlet state. The structure goes into the definition of the PDF, and goes into the evaluation of the hard kernel. The second term in Eq. (29) contributes to the color-octet state of the valence quarks, together with which an additional gluonic parton comes out of the nucleon and participates in the hard scattering.

The factorization formula for the nucleon DIS structure function is written as

(30)

with the subscript labeling the parton flavor, such as a valence quark, a gluon, or a sea quark. The hard kernel is obtained following the subtraction procedure for the collinear divergences, and its LO and NLO expressions have been presented in Eqs. (20) and (21), respectively. The universal PDF , describing the probability for parton to carry the momentum fraction in the nucleon, takes a smooth model function. It must be derived by nonperturbative methods, or extracted from data.

2.3 Predictive Power

The factorization theorem derived above is consistent with the well-known parton model. The nucleon travels a long space-time, before it is hit by the virtual photon. As , the hard scattering occurs at point space-time. Relatively speaking, the quark in the nucleon behaves like a free particle before the hard scattering, and decouples from the rest of the nucleon. Therefore, the cross section for the nucleon DIS reduces to an incoherent sum over parton flavors under the collinear factorization. That is, the approximation

(31)

holds, where represents the scattering amplitude for partonic state of the nucleon (it could be a multi-parton state), and represents the infrared finite scattering amplitude for parton .

Comparing the factorization theorem with the operator product expansion (OPE), the latter involves an expansion in short distance . A typical example is the infrared safe , whose cross section can be expressed as a series . The Wilson coefficients and the local effective operators appear in a product in the OPE. A factorization formula involves an expansion on the light cone with small , instead of . A typical example is the DIS structure function, in which the existence of the collinear divergences implies that a parton travels a finite longitudinal distance . It is also the reason why the hard kernel and the PDF appear in a convolution in the momentum fraction.

Figure 2: CT10 NNLO (solid color) and NLO (dashed) parton distribution functions.

The factorization procedure introduces the factorization scale into the hard kernel and the PDF , as indicated in Eq. (30). Higher-order corrections produce the logarithms in and in , which come from the splitting of in the structure function , being a low scale characterizing . One usually sets to eliminate the logarithm in , such that the input for arbitrary is needed. The factorization scale does not exist in QCD diagrams, but is introduced when a physical quantity like the structure function is factorized. The independence of the factorization scale, , leads to a set of renormalization-group (RG) equations

(32)

where denotes the anomalous dimension of the PDF. A solution of the RG equations describes the evolution of the PDF in

(33)

as a result of the all-order summation of . Hence, one just extracts the initial condition defined at the initial scale from data. The PDF at other higher scales is known through the evolution. That is, the inclusion of the RG evolution increases the predictive power of the factorization theorem.

Fitting the factorization formulas for those processes, whose dynamics is believed to be clear, such as Eq. (30) for DIS, one has determined the PDFs for various partons in the proton. The CTEQ-TEA CT10 models at the accuracy of NLO and next-to-next-to-leading order (NNLO) for hard kernels are displayed in Figs. 2 [27, 28]. The increase of the gluon and sea-quark PDFs with the decrease of the momentum fraction is a consequence of more radiations in that region in order to reach a lower . The comparison of the PDFs at GeV and GeV indicates that the valence -quark and -quark PDFs become broader with , while the gluon and sea-quark PDFs increase with .

Note that a choice of an infrared regulator is, like an ultraviolet regulator, arbitrary; namely, we can associate an arbitrary finite piece with the infrared pole in . Shifts of different finite pieces between and correspond to different factorization schemes. Hence, the extraction of a PDF depends not only on powers and orders, at which QCD diagrams are computed, but on factorization schemes. Since perturbative calculations are performed up to finite powers and orders, a factorization scheme dependence is unavoidable. Nevertheless, the scheme dependence of pQCD predictions would be minimized, if one sticks to the same factorization scheme. Before adopting models for PDFs, it should be checked at which power and order, at which initial scale, and in what scheme they are determined.

Figure 3: Comparison of ATLAS data for inclusive jet distribution with a theoretical prediction using CT10 NNLO.

At last, I explain how to apply the factorization theorem to make predictions for QCD processes. A nucleon PDF is infrared divergent, if evaluated in perturbation theory due to the confinement mechanism. The QCD diagram for a DIS structure function involving quarks and gluons as the external particles are also infrared divergent. It has been demonstrated that the infrared divergences cancel between the QCD diagrams and the effective diagrams for , as taking their difference, which defines the hard kernel . One then derives the factorization formula for other processes, such as the Drell-Yan (DY) process , and computes the corresponding hard kernel . The point is to verify that the infrared divergences in the QCD diagrams for DY and in the effective diagrams for the nucleon PDF cancel, and is infrared finite. If it is the case, the universality of the nucleon PDF holds, and the factorization theorem is applicable. If not, the factorization theorem fails. After verifying the factorization theorem, one makes predictions for the DY cross section using the formula . As an example, the predictions for the inclusive jet distribution derived from the factorization theorem [28] are presented in Fig. 3. The consistency between the predictions and the ATLAS data is obvious.

2.4 Factorization

The collinear factorization theorem introduced above has been intensively investigated and widely applied to many QCD processes up to higher powers and orders. The evolution of PDFs from low to high factorization scales is governed by the DGLAP equation. The databases for PDFs have been constructed, such as the CTEQ models. Other nonperturbative inputs like soft functions, jet functions, and fragmentation functions have been all explored to some extent. However, another more complicated framework, the factorization theorem [29, 30, 31], may be more appropriate in some kinematic regions or in semi-inclusive processes. The collinear factorization applies, when the DIS is measured at a finite Bjorken variable . The cross section is written as the convolution of a hard kernel with a PDF in a parton momentum fraction . As , can reach a small value, at which the parton transverse momentum is of the same order of magnitude as the longitudinal momentum , and not negligible. Once is kept in a hard kernel, a transverse-momentum-dependent (TMD) function is needed to describe the parton distribution not only in the momentum fraction , but also in the transverse momentum . The DIS cross section is then written, in the factorization theorem, as the convolution

(34)

The factorization theorem is also applicable to the analysis of low spectra of final states, like direct photon and jet productions, for which is not negligible.

A collinear gluon emission, modifying a parton longitudinal momentum, generates a parton transverse momentum at the same time. The factorization of a TMD from the DIS is similar to that of a PDF, which relies on the eikonal approximation in the collinear region. This procedure results in the eikonal propagator , represented by the Wilson lines similar to that defined in Eq. (24). A naive TMD definition as an extension of the PDF in Eq. (23) is given by

(35)

with the Wilson links . Because the valence quark fields before and after the final-state cut are separated by a transverse distance in this case, the vertical links located at are demanded by the gauge invariance of a TMD [32]. More investigations on the vertical Wilson links can be found in [33].

Though we do need the factorization theorem, many of its aspects have not yet been completely understood. For example, the naive definition in Eq. (35) is actually ill-defined, due to the existence of the light-cone singularity, that arises from a loop momentum parallel to the Wilson line direction . A plausible modification is to rotate the Wilson line away from the light cone, namely, to replace by a vector with . This rotation is allowed, since the collinear divergences are insensitive to the direction as illustrated in Eq. (26) [34]: even when is rotated to , only the minus component is relevant for the evaluation of the collinear divergences. A detailed discussion on this subtle issue can be found in [35]. Besides, a parton is off-shell by , once is retained. Then whether a hard kernel obtained in the factorization theorem is gauge invariant becomes a concern [36]. Dropping the dependence of the hard kernel in Eq. (34), the integration of the TMD over , , can be worked out. How this integral is related to the PDF in Eq. (23) is worth of a thorough study.

3 Evolution and resummation

As stated in the previous section, radiative corrections in pQCD produce large logarithms at each order of the coupling constant. Double logarithms appear in processes involving two scales, such as with being the large longitudinal momentum of a parton and being the small inverse impact parameter, where is conjugate to the parton transverse momentum . In the region with large Bjorken variable , there exists from the Mellin transformation of , for which the two scales are the large and the small infrared cutoff for gluon emissions from a parton. Single logarithms are generated in processes involving one scale, such as and , for which the relevant scales are the large and the small , respectively. Various methods have been developed to organize these logarithmic corrections to a PDF or a TMD: the resummation for [8, 9], the threshold resummation for [5, 6, 7], the joint resummation [11, 12] that unifies the above two formalisms, the DGLAP equation for [3], the BFKL equation for [4], and the CCFM equation [10] that combines the above two evolution equations. I will explain the basic ideas of all the single- and double-logarithmic summations in the Collins-Soper-Sterman (CSS) resummation formalism [8, 9].

3.1 Resummation Formalism

Collinear and soft divergences may overlap to form double logarithms in extreme kinematic regions, such as low and large . The former includes low jet, photon, and boson productions, which all require real gluon emissions with small . The latter includes top pair production, DIS, DY production, and heavy meson decays and [16, 37, 38] at the end points, for which parton momenta remain large, and radiations are constrained in the soft region. Because of the limited phase space for real gluon corrections, the infrared cancellation is not complete. The double logarithms, appearing in products with the coupling constant , such as with the beam energy and , deteriorate perturbative expansion. Double logarithms also occur in exclusive processes, such as Landshoff scattering [39], hadron form factors [40], Compton scattering [41] and heavy-to-light transitions [42] and [43] at maximal recoil. In order to have a reliable pQCD analysis of these processes, the important logarithms must be summed to all orders.

(a)                      (b)                      (c)

Figure 4: (a) Jet subprocess defined in Eq. (36). (b) and (c) LO diagrams of (a).

The resummation of large logarithms will be demonstrated in the covariant gauge [38], in which the role of the Wilson line direction and the key technique can be explained straightforwardly. Take as an example a jet subprocess defined by the matrix element

(36)

where is a light quark field with momentum , and is a spinor. The abelian case of this subprocess has been discussed in [44]. The path-ordered exponential in Eq. (36) is the consequence of the factorization of collinear gluons with momenta parallel to from a full process, as explained in the previous section. For convenience, it is assumed that has a large light-cone component , and all its other components vanish. A general diagram of the jet function is shown in Fig. 4(a), where the path-ordered exponential is represented by a double line along the vector . As explained before, varying the direction does not change the collinear divergences collected by the Wilson line.

It is easy to see that contains double logarithms from the overlap of collinear and soft divergences by calculating the LO diagrams in Fig. 4(b), the self-energy correction, and in Fig. 4(c), the vertex correction. In the covariant gauge both Figs. 4(b) and 4(c) produce double logarithms. In the axial gauge the path-ordered exponential reduces to an identity, and Fig. 4(c) does not exist. The essential step in the resummation technique is to derive a differential equation [38, 16, 42], where the coefficient function contains only single logarithms, and can be treated by RG methods. Since the path-ordered exponential is scale-invariant in , must depend on and through the ratio . The differential operator can then be replaced by using a chain rule

(37)

with the vector being defined via .

Figure 5: Derivative in the covariant gauge.

Equation (37) simplifies the analysis tremendously, because appears only in the Feynman rules for the Wilson line, while may flow through the whole diagram in Fig. 4(a). The differentiation of each eikonal vertex and of the associated eikonal propagator with respect to ,

(38)

leads to the special vertex . The derivative is thus expressed as a summation over different attachments of , labeled by the symbol in Fig. 5. If the loop momentum is parallel to , the factor vanishes, and is proportional to . When this is contracted with a vertex in , in which all momenta are mainly parallel to , the contribution to is suppressed. Therefore, the leading regions of are soft and hard.

Figure 6: (a) examples for the differentiated . (b) Factorization of at . (c) Factorization of at . (d) Factorization of at .

According to this observation, we investigate some two-loop examples exhibited in Fig. 6(a). If the loop momentum flowing through the special vertex is soft but another is not, only the first diagram is important, giving a large single logarithm. In this soft region the subdiagram containing the special vertex can be factorized using the eikonal approximation as shown in Fig. 6(b), where the symbol represents a convoluting relation. The subdiagram is absorbed into a soft kernel , and the remainder is identified as the original jet function , both being contributions. If both the loop momenta are soft, the four diagrams in Fig. 6(a) are equally important. The subdiagrams, factorized according to Fig. 6(c), contribute to at , and the remainder is the LO diagram of . If the loop momentum flowing through the special vertex is hard and another is not, the second diagram in Fig. 6(a) dominates. In this region the subdiagram containing the special vertex is factorized as shown in Fig. 6(d). The right-hand side of the dashed line is absorbed into a hard kernel as an contribution, and the left-hand side is identified as the diagram of . If both the loop momenta are hard, all the diagrams in Fig. 6(a) are absorbed into , giving the contributions.

Extending the above reasoning to all orders, one derives the differential equation

(39)

where the coefficient function has been written as the sum of the soft kernel and the hard kernel . In the above expression is a factorization scale, the gauge factor in is defined as , and a gluon mass has been introduced to regularize the infrared divergence in . It has been made explicit that </