S. Albino II. Institut für Theoretische Physik, Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
July 13, 2019
###### Abstract

We review the description of inclusive single unpolarized light hadron production using fragmentation functions in the framework of the factorization theorem. We summarize the factorization of quantities into perturbatively calculable quantities and these universal fragmentation functions, and then discuss some improvements beyond the standard fixed order approach. We discuss the extraction of fragmentation functions for light charged (,  and ) and neutral ( and ) hadrons using these theoretical tools through global fits to experimental data from reactions at various colliders, in particular from accurate reactions at LEP, and the subsequent successful predictions of other experimental data, such as data gathered at HERA, the Tevatron and RHIC from these fitted fragmentation functions as allowed by factorization universality. These global fits also impose competitive constraints on . Emphasis is placed on the need for accurate data from and reactions in which the hadron species is identified in order to constrain the separate fragmentation functions of each quark flavour and hadron species.

###### pacs:
12.38.Cy,12.39.St,13.66.Bc,13.87.Fh

## I Introduction

Fragmentation functions (FFs) constitute one of the most important free inputs required for a comprehensive description of most collider processes to which perturbative QCD is applicable, being a necessary ingredient in any sufficiently complete calculation of processes involving detected hadrons in the final state. They quantify the hadronization of quarks and gluons which must eventually occur in every process in which hadrons are produced. While parton distribution functions (PDFs), another of the important free inputs, are relevant for collisions involving at least one hadron, most importantly at present for collisions at the LHC, FFs are relevant in principle for all collisions, even those without any initial state hadrons such as electron-positron collisions at the future ILC. Furthermore, while knowledge and application of PDFs is limited to the types of hadrons that can be practically used in the initial state, being almost exclusively nucleons, FFs can be constrained by, and/or used for the predictions of, measurements of the inclusive productions of neutral and charged hadron species ranging from the almost massless to the very heavy. Such a large range of processes provides a large range of information on hadronization, and hence provides an important contribution to our understanding of non perturbative physics in general, and allows for a particularly incontrovertible phenomenological determination of the applicability and limitations of the various approximations used in the context of QCD factorization. Such data are also sufficiently accurate to allow for competitive extractions of , the strong coupling constant of QCD evolved for convenience to the -pole mass and the remaining of the most important free inputs, which improves the accuracy of perturbative QCD calculations in general and imposes constraints on new physics. We note here that while other inputs such as higher twist, multi-hadron FFs, fracture functions, and so on become important in certain kinematic limits, FFs are always necessary for a complete description of inclusive hadron production.

Our discussion will be limited to inclusive single unpolarized hadron production, being better understood than its polarized counterpart. Final state partons will on average hadronize to hadrons of all species that are not kinematically forbidden by the particular reaction, but, of all the charged hadrons, partons will mostly hadronize to the 3 lightest ones, being ,  and , so their FFs are the most phenomenologically well constrained. (For a concise summary of the properties of some known mesons and baryons, see Ref. wiki ().) The species of neutral meson and neutral baryon most likely to be produced in the hadronization process are , and respectively, but their FFs are accurately approximated by the FFs for  and due to SU(2) (nuclear) isospin symmetry between the and quark flavours. The next lightest species of meson and baryon is  and  respectively, whose FFs have therefore also been greatly studied. ( is not usually observed in experiment because it takes a long time to decay into charged particles.) Together with , , , , , , , and mesons, whose FFs are unfortunately either unknown or rather poorly known at present, the particles mentioned in this paragraph complete the list of known hadrons which have a mass less than or equal to the  mass and which are the most copiously produced in hadron production, and thus measurements of their production lead to a rather comprehensive picture of the hadronic final state. In this review we shall focus on the productions of these particles only, with the exceptions of the productions of those mesons just listed.

Much of the techniques for global fitting of PDFs can be carried over to global fitting of FFs, for example the treatment of systematic errors on the experimental data and the propagation of experimental errors to the FFs, as well as the techniques used for the minimization of chi-squared. On the theoretical side, much of the formalism of both the fixed order (FO) calculations and its improvement via e.g. large resummation and incorporation of heavy quarks and their masses, is also very similar.

The rest of this review is structured as follows. The basic results of the QCD factorization theorem, which forms the starting point of all calculations of inclusive hadron production, are given in section II, in particular the separation from the overall cross section of the process dependent parts, which can be calculated using the FO approach to perturbation theory. (The derivation of these results is outlined in appendix A.) Then in section III we discuss the extraction of FFs from accurate data and from various well motivated non perturbative assumptions. The ability of calculations using these fitted FFs to reproduce measurements is studied in section IV. Then we turn to more recent progress: Improvements to the standard FO approach that have not been incorporated into nearly all global fits are given in section V, namely hadron mass effects and large resummation. The treatment of experimental errors and their propagation to fitted quantities, which has been rather comprehensively implemented in PDF fits but to a somewhat lesser degree in FF, is discussed in section VI. The 3 most recent global fits, in which the implementation of these improvements can be found, are discussed and compared in section VII. Finally, in section VIII we examine the improvement of the standard FO approach at small by resummation of soft gluon logarithms, which has so far been successfully tested at LO, and we consider what is needed for a full treatment of soft gluon logarithms to NLO. In section IX we predict the future experimental and theoretical progress of FF extraction, and give a summary of the current progress in section X. The LO splitting functions are given in appendix B for reference, the relevant Mellin space formulae in appendix C, and a summary of all data that can be reasonably reliably calculated and which can be used to provide constraints on FFs in appendix D. Some of these issues have also been discussed very recently in Ref. Arleo:2008dn (), with an aim towards the modification of fragmentation in QCD media.

## Ii Results of the QCD factorization theorem

Intuitively, the detected hadron of inclusive single hadron production events is produced in the jet formed by a parton produced at a short distance , and carries away a fraction of the momentum of . This “probing scale” should be of order of the energy scale of the process, whose precise choice is somewhat conventional. The probability density in for this to take place is given by the FF . The hadronic cross section is then equal to the cross section with replaced by , hereafter referred to as the partonic cross section, weighted with the FF of and summed and integrated over all degrees of freedom such as and , i.e. eq. (1) below. According to the factorization theorem, which follows from QCD in a model independent way, the leading twist component of any inclusive single hadron production cross section takes this intuitive form, where the FFs are process independent or universal among different initial states. The factorization theorem also asserts that all processes in the partonic cross section that have energy scale below the factorization scale factor out of the partonic cross section and are accounted for by the FFs, and that the resulting factorization scale dependence of these FFs may be calculated perturbatively. In this section, we highlight the results of the factorization theorem. An outline of the formal derivation of the factorization theorem is given in appendix A for the interested reader.

### ii.1 Factorized cross sections

In general, as well as , an inclusive single hadron production process depends on the fraction of the available momentum or energy carried away by the detected hadron . For example, in , whose kinematics are specified in Fig. 1 (left), and . We will assume for now that it is reasonable to neglect the effect of the hadron mass , which is of when , where is the momentum of the detected hadron. When the effect of hadron mass, to be discussed in section V.1, is taken into account in the calculations, the scaling variable of the factorization theorem and the fractions and of the available momentum and energy respectively of the process taken away by the detected hadron must be distinguished from one another. Note that the cross section may depend on other variables in addition to and , but we will not indicate this explicitly unless necessary. The factorization theorem asserts that the cross section takes the form of a convolution

 (1)

up to higher twist terms, which are suppressed relative to the overall cross section by a factor or more. The parton label for the gluon, while for (anti)quarks, where corresponds respectively to the flavours , , , , and . The fact that necessarily will be explained below. is a renormalization scheme-dependent mass associated with parton , which will be taken to be its pole mass, and is the expansion parameter in perturbative series. The are the equivalent partonic cross sections obtained by replacing the detected hadron with a real on-shell parton moving in the same direction but with momentum , and the sum over unobserved hadrons replaced with a sum over unobserved partons. In e.g. , is completely calculable in perturbation theory, while for processes involving initial state hadrons, such as at HERA and at the LHC (RHIC), will be convolutions of perturbatively calculable quantities with PDFs for each initial state hadron, a result which also follows from the factorization theorem. The are otherwise perturbatively calculable if all subprocesses with energy scale below some arbitrary factorization scale are factored out of them and into the FFs (and PDFs if applicable) according to the factorization theorem. While the differ from process to process, the FFs are universal and therefore, through them, measurements of one process impose constraints on others in which the same hadron is produced.

In more detail, the perturbative expansion of the unfactorized in the limit contains potential mass singularities, being logarithms of the form raised to powers of integers and, because the largest power in a given order in grows with the order, they may spoil the convergence of the series even when . These potential mass singularities, which arise from energy processes much below , may be factored out of the . There is clearly some freedom in choosing whether to place each process of energy scale around in or , and this choice defines the factorization scheme. (This is a physical definition — in practice the scheme is fixed by the choice of subtraction terms in the factorization.) Although the leading twist component of is formally independent of the choices of factorization scale and scheme, its perturbative approximation discussed above will depend on these choices, which must therefore reflect the physics of the overall process. The theoretical error in this approximation can also be estimated by varying these choices. The potential mass singularities from (anti)quarks with mass are dampened by factors of , i.e. they approximately decouple Appelquist:1974tg () and so must not be factorized to avoid introducing large uncanceled counterterms in . Therefore, the active partons, being those partons whose potential mass singularities are factored out of the and into the FFs, and to which the summation over in eq. (1) is restricted so that labels the number of species of active quarks, should be limited to include only those partons whose masses . This will always include the gluon and, since perturbative QCD is only valid in the region , also the light quarks, defined to be those quarks whose masses are of or less, i.e. the 3 lightest quarks , and or . The intrinsic light hadron PDF of a quark is expected to be of or less Witten:1975bh (), but the same property may not necessarily hold for FFs. Therefore, it may also be necessary to always include some of the heavy quarks, defined to be those quark whose mass , i.e. , and or , in the list of active partons, in which case the cross section will not be perturbatively calculable if . For example, it could happen that intrinsic charm quark fragmentation is deemed important in a cross section, but it cannot be incorporated into the cross section’s calculation if the only appropriate scheme is the 3 flavour one. This problem would be avoided if a method was known for correctly incorporating the intrinsic FF of quark in a cross section when this quark is not active. This issue will be discussed further at the end of subsection II.3 in the light of matching conditions between quantities defined with different numbers of active partons.

Note that there is another type of potential mass singularity appearing in the cross section, which behaves like a power of for any heavy quark mass . These can be absorbed into the strong coupling constant by using a renormalization scheme for which quark is not active.

The th scheme is a renormalization and factorization scheme for which the number of active quark flavours is . Results are usually presented in the th Collins-Wilczek-Zee (CWZ) scheme Collins:1978wz (), also known as the decoupling scheme, which reduces to the scheme for only massless quark flavours in the limit that the active quark masses vanish and the remaining quarks’ masses become infinite so that they completely decouple from the theory. The unsubtracted partonic cross sections with heavy quarks and their masses included and their subtraction terms in the CWZ schemes have been calculated for and reactions in Refs. Nason:1993xx (); Kneesch:2007ey () and Kretzer:1998nt () respectively. The inclusion of heavy quarks and their masses in the partonic cross sections for reactions has been calculated in Ref. Nason:1989zy (), and their subtraction terms in Ref. Kniehl:2004fy ().

### ii.2 DGLAP evolution

Roughly speaking, factorization replaces each logarithm in the partonic cross section with . These artefacts of factorization may spoil the accuracy of the perturbation series in the same way that the potential mass singularities did unless we ensure , in which case the dependence of the must also be known. Fortunately, unlike for the dependence, this dependence is perturbatively calculable, provided : Writing the dependence of the FFs in the form of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation Gribov:1972ri (); Lipatov:1974qm (),

 ddlnM2fDhi(z,M2f)=nf∑j=−nf∫1zdz′z′Pij(zz′,as(M2f))Dhj(z′,M2f), (2)

with , then the splitting functions are each perturbatively calculable as a series in . Specifically, denoting the square matrix with components by , their expansion in takes the form

 P(z,as)=∞∑n=1P(n−1)(z)ans, (3)

where the are non-singular even in the limits for which any active parton mass vanishes. In the th CWZ scheme, the are independent of all parton masses, and thus can be obtained by taking the limit discussed at the end of subsection II.1 and performing factorization in the scheme. Equations (2) and (3) serve similar purposes to the evolution of ,

 ddlnμ2as(μ2)=β(as(μ2)), (4)

and the perturbative expansion of the function,

 β(as)=−∞∑n=2βn−2ans, (5)

respectively, which are used to resum powers of the logarithm in perturbatively calculated quantities, where is the renormalization scale. In other words, the ultimate purpose of the DGLAP equation is to resum powers of the logarithm for all in the partonic cross section. The physical interpretation of eq. (2) is most easily made from its solution, which takes the form

 Dhi(z,M2f)=nf∑j=−nf∫1zdz′z′Eij(zz′,as(M2f),as(M20))Dhj(z′,M20). (6)

Each quantity , which also obeys eq. (2) on taking , may be interpreted as the FF of parton at resolution scale into parton at resolution scale carrying a fraction of parton ’s momentum. It is subject to the boundary condition and depends only on the of eq. (3), although its analytic dependence on may not be calculated. We will see in subsection II.5 that an analytic form for the perturbative calculation of may be obtained in Mellin space.

### ii.3 Changing the number of active partons

So far, everything we have discussed applies for a given assignment of partons as active. Using always the same FFs at some initial input value of the factorization scale, we would like to be able to calculate cross sections at any energy (in this review, , and are much greater than unless otherwise stated). However, to ensure that the perturbative calculation of gives a reliable approximation, all partons of mass must be assigned as active, even if they are not active at . Consequently, this assignment must be allowed to vary and the relationship between quantities defined in a scheme in which the lightest quarks are active (and of course the gluon, which must always be active) must be related to similar quantities for which quarks are active. This is trivial for the factorized partonic cross sections , and the necessary matching conditions for are known Marciano:1983pj (). The matching conditions between the FFs of the th scheme, with , and the FFs of the th scheme, with , take the form

 Dh′i(z,M2f)=nf∑j=−nf∫1zdz′z′Aij⎛⎝zz′,m2kM2f,as(M2f)⎞⎠Dhj(z′,M2f) (7)

where the matrix is perturbatively calculable for , where this matching should therefore be done. Otherwise, the precise choice of this matching threshold is arbitrary because it is non physical, and should be distinguished from non arbitrary physical thresholds such as that for the production of a charm quark. In the CWZ scheme, depends on but not the other and is now known to NLO Cacciari:2005ry (). The only non zero components at NLO are , which vanishes if the matching is done at , and . The latter quantity is not needed if the whole FF of the th flavour quark, both intrinsic and extrinsic, is treated as one single function to be fitted to data, which is usually the case. This is in contrast to the spacelike case: because intrinsic heavy quark PDFs for the proton can normally be neglected, the quantity is necessary for obtaining the PDF of the th flavour quark, being almost completely extrinsic.

According to eq. (7), a heavy quark FF with a negligible intrinsic component (i.e. its FF vanishes when it is not active) is fully determined from the other FFs. Therefore, if the intrinsic components of the heavy quarks’ FFs are negligible, their FFs are perturbatively completely determined by only the gluon and the light quarks in the scheme: Using eq. (7) to convert to scheme gives the charm quark FF in terms of the gluon and light quark FFs, i.e. charm quark fragmentation proceeds via perturbative fragmentation to a gluon or light quark, which then fragments non perturbatively to the detected hadron. Similarly the bottom quark FF for active quark flavours is fully determined from the gluon, light and charm quark FFs.

On the other hand, it may not be a good approximation to neglect the intrinsic heavy quark FFs even for light hadrons, and the intrinsic charm FF is certainly not negligible for charmed hadrons such as mesons. In many calculations for light hadron production, the charm quark FF is treated as an unknown function at the matching scale where , just as the gluon and light quark flavour FFs at the initial scale are, and the gluon and all 4 quark FFs are evolved from there according to the DGLAP equation of the scheme. A similar procedure is performed for the bottom quark FF. While this incorporation of intrinsic charm in the scheme is consistent with the factorization scheme, the conventional approach of setting the complete charm quark FF to zero in the cross section calculated in the scheme is not, because intrinsic charm quark fragmentation effects should not depend on the choice of scheme (although the precise definition of the intrinsic charm quark FF itself is scheme dependent). Consequently, this approach contains an inconsistency: When the energy, and therefore the factorization, scale is close to the charm quark mass so that both the 3 and 4 flavour number schemes should be valid, they will lead to very different cross sections in the case that the intrinsic charm quark FF is large. In practice, this inconsistency of the 3 flavour scheme, and 4 flavour scheme when one also considers a possibly important bottom quark FF, does not matter since most cross sections of interest are of a sufficiently high energy scale that the 5 flavour scheme should suffice for all calculations.

### ii.4 Various treatments of quarks with non negligible mass

In the above discussion, we have assumed that all quarks have non negligible but finite masses. In practice, the energy scale is usually not close to any quark mass, so all perturbative results are usually approximated by the zero mass variable flavour number “scheme” (ZM-VFNS), where the masses of all quarks above are set to infinity so that these quarks decouple from the theory, which introduces a relative error of to the cross section, while all quarks with masses below are treated as active and their masses are set to zero, which introduces a relative error of . Such an approach therefore fails when the energy scale is of the order of a quark mass, but is otherwise a reasonable approximation. However, the procedure we have been discussing in the previous subsections is more general and is called the general mass variable flavour number scheme (GM-VFNS), where both quarks and their masses are treated in a similar fashion to their treatment in the Aivazis-Collins-Olness-Tung (ACOT) scheme Aivazis:1993pi (); Collins:1998rz () of spacelike factorization. Such a procedure has for example been explicitly implemented for heavy flavour hadrons in photo- Kramer:2001gd () and hadroproduction Kniehl:2004fy () in reactions, and more recently for reactions Kneesch:2007ey (). The usual simplifications and improvements that have been made to the ACOT scheme since its inception may also be applied to fragmentation, at least in principle. For example, assuming that the intrinsic fragmentation of heavy quarks is negligible, the power suppressed terms in the NLO factorized cross section for the inclusive production of this heavy quark appearing in the overall hadronic cross section are in fact arbitrary, since the only purpose of the former cross section in this case is to complete the cancellation of all potential mass singularities in the overall hadronic cross section such that there are no singularities in the massless limit Collins:1998rz (). In the S-ACOT scheme Kramer:2000hn (), these power suppressed terms are set to zero for simplicity, while in the ACOT() scheme Tung:2001mv (), the accuracy of the perturbative calculation relative to the original ACOT scheme is improved: The heavy quark production cross section is again chosen to be that for the production of a massless quark as in the S-ACOT scheme, but its LO part is also multiplied by a dependent step function which is equal to that multiplying the gluon production cross section’s potential mass singularity, to ensure it and its counterterm only ever appear at the same time.

### ii.5 Analytic Mellin space solution of the DGLAP equation

The integrations over on the right hand sides of eqs. (1), (2) and (6) are known as convolutions, and are a typical feature of results of the factorization theorem. The Mellin transform, discussed in appendix C, which does not destroy any information because this transform is invertible using the inverse Mellin transform of eq. (236), converts these convolutions into simple products, and therefore analytic work is usually performed in Mellin space. For example, eq. (2) becomes

 ddlnM2fD(N,M2f)=P(N,as(M2f))D(N,M2f), (8)

where for further simplification we are now also omitting parton labels , and the more trivial hadron label , it being understood in eq. (8) that the matrix acts on the column vector according to the usual matrix product definition. The elements of for integer are equal to the anomalous dimensions of the twist two, spin gauge invariant operators in , which are needed in the operator product expansion. In the solution to eq. (8),

 D(N,M2f)=E(N,as(M2f),as(M20))D(N,M20), (9)

which is the Mellin transform of eq. (6), the elements of may be expressed analytically in terms of , and up to the same accuracy as Furmanski:1981cw (); Ellis:1993rb (): Define the series

 U(N,as)=1+∞∑n=1U(n)(N)ans (10)

such that

 E(N,as,a0)=U(N,as)ELO(N,as,a0)U−1(N,a0), (11)

where

 ELO(N,as,a0)=exp[−P(0)(N)β0lnasa0] (12)

is the LO result for , being an exact solution to eq. (8) with . The purpose of in eq. (11) is to ensure the boundary condition . To all orders, eq. (8) can be converted via eq. (11) to an evolution equation for :

 dUdas=−Rβ0+1β0as[U,P(0)], (13)

where . The exponentiation in eq. (12) and the commutator in eq. (13) are handled by choosing a specific basis for the FFs in a factorization scheme in which the symmetries of QCD are obeyed. Such a basis will be given in subsection II.6.

As a final remark, the Mellin space formalism makes clear the importance of the DGLAP equation in the application of perturbation theory to cross section calculations: Any set of functions of the variables and obeys the form of the DGLAP equation, eq. (2), by choosing , which follows from its Mellin transform, eq. (8). (Here, can be regarded as a matrix whose columns consist of the for a given hadron species which varies from column to column, i.e. there is some freedom in the definition of .) The importance of the formalism behind the DGLAP equation is that is constrained in QCD to depend on purely partonic graphs, and furthermore to be perturbatively calculable (these results follow from eq. (219)), and therefore the dependence of the FFs is completely and calculably constrained. In particular, when expanded in it takes the form in eq. (3) with non singular coefficients. However, eq. (2) is not the only way to evolve the FFs in . Alternatives to eq. (2) may be preferable for certain physical reasons, such as the double logarithmic approximation (DLA), to be discussed in subsection VIII.3, which is more apt for the small region, or the evolution proposed in Ref. Dokshitzer:2005bf (), whose equivalent splitting functions may exhibit certain physical properties not seen in the usual DGLAP splitting functions beyond LO, such as the Gribov-Lipatov relations and the expected large behaviour beyond leading order due to purely multi-parton quantum fluctuations when the physical strong coupling constant is used as the expansion parameter. Such equations are similar to the DGLAP equation but with both occurrences of in eq. (2) multiplied by (different) powers of to ensure that the inverse of this modified scale truly represents the fluctuation lifetimes of successive virtual parton states pertinent to the kinematic region being studied. However, if required, it is always possible to recast such alternative evolution equations back into the form of eq. (2), and thereby use them to obtain an alternative expansion for in the kinematic region of interest to that in eq. (3). To put this in an alternative, but equivalent, way, eq. (3) is not the only possibility for approximating : In general, in certain limits of and may be better approximated in the form , where and are each in general a suitably chosen function of both and . We will see an example of such an alternative expansion in subsection VIII.2, namely eq. (148) (where ).

### ii.6 Symmetries

Although FFs are not physical, the factorization scheme should be chosen such that they respect the symmetries of QCD in order to keep results as simple as possible, and we will assume that this has been done. Indeed, the commonly used scheme is such a scheme. Consider first the charge conjugation symmetry of QCD. Writing the cross section or FF for the production of (or ) as , where or respectively, and then defining charge-sign unidentified and charge-sign asymmetry quantities as and respectively (or more generally as and which includes also neutral hadrons), this symmetry implies that each of these two combinations of cross sections only depends on FFs that have been combined in the same way (or they vanish). The calculation of such cross sections is therefore simpler than that of the production of hadrons of a given charge-sign, which depends on both charge-sign unidentified and charge-sign asymmetry FFs. Lorentz invariance implies a similar feature for the polarization of the hadron and of the partons, assuming that there are only 2 possibilities for the hadron’s polarization.

We now study the simplifications to the evolution of FFs when the charge conjugation symmetry is taken into account. Letting () denote the FF for the hadron of the (anti)quark of flavour , this symmetry is accounted for by the results that follow from charge conjugation symmetry,

 Dh+(or h−)qI=Dh−(or h+)¯qI. (14)

Equation (14) implies that charge-sign unidentified FFs (the separate brackets “” and “” imply that the changes and respectively may be made, and may be made independently of one another) where, omitting the superscript “” for brevity from now on, the sums

 DqI/¯qI=DqI+D¯qI. (15)

Due to charge conjugation symmetry, these FFs and the gluon FF mix only with each other on evolution, but not with the valence quark FFs defined in eq. (19) below. Furthermore, in a scheme for which is explicitly independent of quark masses, such as the CWZ scheme, the SU symmetry for active quark flavours of equal mass that follows from QCD implies that the gluon FF will mix via eq. (2) with the quark FFs combined into one singlet quark FF

 DΣ=1nfnf∑I=1DqI/¯qI, (16)

In other words, eq. (2) is obeyed with and

 P=(PΣΣPΣgPgΣPgg). (17)

This SU symmetry also implies that every non singlet quark FF, being any linear combination of the that vanishes when they are all equal, will only mix with itself on evolution, i.e. it obeys eq. (2) but with reduced to the single quantity which is the same for all non singlets. The non singlets can be chosen such that they and the singlet form a linearly independent set of FFs, so that after evolution the FFs of quarks of each flavour, or any other alternative basis of FFs, can be extracted by taking appropriate linear sums. A common choice of the set of non singlet FFs is

 DqI,NS=DqI/¯qI−DΣ. (18)

Equation (14) also implies that charge-sign asymmetry FFs where, again omitting the superscript “”, the differences

 DΔcqI/¯qI=DqI−D¯qI, (19)

which we refer to as the valence quark FFs. Although valence quark FFs are the same as charge-sign asymmetry FFs, we distinguish between them depending on the context — when working with the symmetries of quarks as we do in this subsection, we shall refer to them as valence quark FFs. Due to charge conjugation symmetry, valence quark FFs mix with each other on evolution but not with the summed FFs defined in eq. (15) above. As for charge-sign unidentified FFs, this mixing is further simplified by the SU symmetry introduced above: The singlet valence quark FF

 DΔcΣ=1nfnf∑I=1DΔcqI/¯qI (20)

and the non singlet valence quark FFs, which, similarly to the definition of non singlet quark FFs in eq. (18), we take as

 DI,ΔcNS=DΔcqI/¯qI−DΔcΣ, (21)

will obey eq. (2) but with reduced to the single quantity and respectively. As for , is the same quantity for all non singlet valence quark FFs. At NLO, , which leads to the simpler evolution in which each valence quark FF only mixes with itself: obeys eq. (2) with (or ).

In the non singlet and valence quark sector, can be calculated in the form given in eq. (12) and the commutator in eq. (13) vanishes. In the singlet quark and gluon sector, eq. (12) and this commutator are both handled by the “diagonalization” Furmanski:1981cw (); Ellis:1993rb () of the matrix , where are the eigenvalues of and are projection operators, i.e. they obey where , and . Then eq. (12) reduces to the calculable form , and eq. (13) is equivalent to the result

 U(n)=∑ij1λj−λi−β0nMiR(n)Mj, (22)

where the are defined in eq. (10), and the immediately after eq. (13). Note that the right hand side of eq. (22) only depends on the for , so is constructed order by order. In general, the zeroes in the numerator of eq. (22) lead to singularities in for complex values of . In the calculation of the cross section via the inverse Mellin transform defined by eq. (236), it is not necessary that the contour should lie to the right of these singularities. Although this arbitrariness due to the dependence on the choice of is of higher order than the order of the calculation, for numerical purposes it is preferable to eliminate the singularities, by choice of the spurious higher order terms. For example, at NLO, is free of such singularities when eq. (11) is calculated in the form , which we note obeys the desired boundary condition . Alternatively, the singularities can be made to cancel simply by choosing in eq. (11) to be exactly equal to the inverse of Blumlein:1997em (), instead of expanding it in .

### ii.7 Sum rules

Intuitively, FFs as probability densities will be constrained by conservation laws. For example, from momentum conservation, the momentum of a parton must equal the total momentum of all hadrons to which it fragments, giving the momentum sum rule

 ∑h∫10dz zDhi(z,M2f)=1 (23)

for every parton . Similarly, charge conservation implies the charge sum rule

 ∑h∫10dz ehDhi(z,M2f)=ei, (24)

where is the electric charge of hadron (parton ). In fact, whether this probability density interpretation is correct, eqs. (23) and (24) hold in the scheme Collins:1981uw (): They are true for the bare FFs of appendix A, for which the probability density interpretation is valid, which is seen by applying the operation to the matrix element definition of the bare quark FFs in eq. (202) (and the similar definition for the bare gluon FF), and identifying the number operator for hadrons of species in an infinitesimal region of momentum space, . Then eqs. (23) and (24) follow for factorized FFs at all values of in schemes for which no subtraction is made on bare quantities that are free of divergences, such as the scheme. Such intuitive but theoretically solid results may be regarded as “physical”.

In practice, eqs. (23) and (24) are not directly used to impose a precise constraint between FFs. Firstly, the summation here is over all hadron species , so that they impose no constraint in global fits of FFs in the case of a specific hadron. Secondly, FFs at low are poorly constrained because soft gluon logarithms render the FO approximation of the splitting functions inaccurate at small . This is in contrast to the momentum sum rule for PDFs, which imposes a constraint between them for any given hadron because the summation is over parton species instead of over hadron species, and PDFs are better understood at small momentum fraction. Instead, eq. (23) may serve as an upper bound on FFs and therefore as a check on phenomenological extractions of them because, when the sum over is limited to just a few hadrons and the lower bound for the integral over is replaced by a value sufficiently greater than zero, it will be less than one if all FFs are positive. However, this positivity condition only follows from the probabilistic interpretation for FFs, which is not quite correct.

Equations (23) and (24) do impose constraints on partonic cross sections, examples of which will be seen in subsection II.9, which give a useful check on their perturbative calculations.

### ii.8 Properties of splitting functions

The LO coefficients of the splitting functions (the coefficients in eq. (3)) of the quark singlet and gluon sector are given in appendix B. Because of the Gribov-Lipatov relations Gribov:1972ri (), they are equal to the spacelike ones after the interchange is made. The LO coefficients of the splitting functions , , and are equal to one another. The NLO coefficients (the coefficients in eq. (3)) have been calculated in Ref. Curci:1980uw () (see also Refs. Gluck:1992zx (); Ellis:1991qj () for the correction to a misprint therein). Although the simple Gribov-Lipatov relation does not hold beyond LO, the timelike and spacelike splitting functions are related by analytic continuation of the form factor from the spacelike to the timelike case Curci:1980uw (); Stratmann:1996hn (); Mitov:2006ic (). Of the NNLO coefficients , the non singlet and singlet valence, the non singlet and the two diagonal singlet splitting functions have been calculated in Ref. Mitov:2006ic () using this continuation. Only the off-diagonal splitting functions and need to be calculated before a full NNLO timelike evolution of all FFs will be possible. The resulting reduction in the theoretical error on all calculations when upgraded from NLO to NNLO should lead to a reduction on the total errors on FFs obtained in global fits, although this reduction will be generally somewhat smaller than the current experimental errors on FFs.

Using the momentum sum rule of eq. (23) in eq. (8) with and all hadron species summed over gives a constraint on the splitting functions in the singlet and gluon sector:

 ∫10dxx(2P(n)ΣΣ(x)+P(n)Σg(x))=∫10dxx(2P(n)gΣ(x)+P(n)gg(x))=0. (25)

The factors of 2 account for the identical contributions of quarks and antiquarks. Similar momentum sum rule constraints exist for the spacelike splitting functions after the interchange is made. Similarly, the charge sum rule of eq. (24) and eq. (8) with implies that the valence quark splitting functions obey

 ∫10dxP(n)ΔcNS(x)=∫10dxP(n)ΔcΣ(x)=0, (26)

which are similar to the valence sum rule constraints on the spacelike splitting functions.

### ii.9 The simplest case: e+e−→h+X

We are now in a position to highlight the main features of the perturbative calculation of the above process, which serves as a simple illustration of the calculation of factorized cross sections in general. In this subsection we assume for simplicity that there are only flavours of quarks, which are all massless. This process proceeds via . For clarity, we will neglect the boson for now and discuss the modifications due to its effects later. Furthermore, the (anti)quark () at the electroweak vertex, called the primary quark, of specific flavour is often tagged in experiment, and we will assume for generality that this is the case. The process is calculated by factorization of the modulus squared of the diagram in Fig. (1) (left), in which the resulting partonic kinematics are also shown (right).

We work to LO in electroweak theory. Taking and in eq. (1), where is the tagged quark, and then making the replacement to ensure only differentials in and not appear, the cross section can be written

 dσhqJdx(x,s)=∫1xdzz[dσNSqJdz(z,s,M2f)DhqJ/¯qJ(xz,M2f)+1nfnf∑I=1dσPSqJdz(z,s,M2f)DhqI/¯qI(xz,M2f)+dσgqJdz(z,s,M2f)Dhg(xz,M2f)] (27)

up to higher twist terms of or less. Each partonic cross section may be written

 dσXqJdz(z,s,M2f)=σ0(s)NcQqJ(s)CX⎛⎝z,as(s),lnM2fs⎞⎠ for X=NS, PS and g. (28)

The quantity is the leading order (LO) cross section for the process , and the coupling of the tagged quark at the photon vertex is accounted for by , where is the electric charge of the quark and that of the electron/positron. Note that becomes dependent on when the effects of the boson, which will be discussed later, are included. The are the coefficient functions whose NLO Altarelli:1979kv () and NNLO Rijken:1996vr () terms are known. For the choice , the are given to NLO by

 CNS(z,as)=δ(1−z)+asCF[(2π23−92)δ(1−z)−32[11−z]++(1+z2)[ln(1−z)1−z]++1+21+z21−zlnz+32(1−z)],CPS(z,as)=O(a2s)   andCg(z,as)=asCF[21+(1−z)2z(ln(1−z)+2lnz)]. (29)

Note that the pure singlet contribution only enters at NNLO. The coefficient functions in the case where the masses of partonic heavy quark are not neglected and the remaining heavy quarks are not decoupled (i.e. their masses are not set to infinity) are presented in Ref. Nason:1993xx ().

The non singlet partonic cross section contains only and all those contributions in which the “detected” (fragmenting) quark is part of the same quark line as that for the tagged connected to the electroweak vertex. The pure singlet partonic cross section contains all other contributions, i.e. those for which the tagged that goes through the electroweak vertex is not part of the same quark line as the quark which fragments. It is independent of the flavour of the “detected” quark and gives the same result when this quark is replaced by an antiquark. Finally, contains all contributions in which the “detected” parton is a gluon.

In terms of singlets and non singlets,

 dσhqJdx(x,s)=∫1xdzz[dσNSqJdz(z,s,M2f)DhqJ,NS(xz,M2f)+dσSqJdz(z,s,M2f)DhΣ(xz,M2f)+dσgqJdz(z,s,M2f)Dhg(xz,M2f)], (30)

where the singlet and non singlets are defined in eqs. (16) and (18) respectively, and the singlet partonic cross sections

 dσSqJdz=dσNSqJdz+dσPSqJdz. (31)

The full untagged cross section can always be obtained by summing the tagged quarks in eq. (27) over all flavours,

 dσhdx(x,s)=nf∑J=1dσhqJdx(x,s). (32)

Conversely, eq. (27) may be obtained from eq. (32) simply by setting all except . Thus the tagged cross sections are physical at least in the sense that they are factorization scheme and scale independent.

Now including the effects of the boson, i.e. for all processes ,

 dσhqJdx(x,s)=dσhγ∗,Z;qJdx(x,s)+dσhqJ,Fdx(x,s). (33)

where contains the contributions from all processes which couple to the virtual photon and which couple to the boson in the same way as to the virtual photon. It is therefore equal to the with virtual photon effects only but with the electroweak coupling