Heavy Quark Fragmenting Jet Functions

Heavy Quark Fragmenting Jet Functions

Christian W. Bauer,    Emanuele Mereghetti cwbauer@lbl.gov, EMereghetti@lbl.gov
Abstract

Heavy quark fragmenting jet functions describe the fragmentation of a parton into a jet containing a heavy quark, carrying a fraction of the jet momentum. They are two-scale objects, sensitive to the heavy quark mass, , and to a jet resolution variable, . We discuss how cross sections for heavy flavor production at high transverse momentum can be expressed in terms of heavy quark fragmenting jet functions, and how the properties of these functions can be used to achieve a simultaneous resummation of logarithms of the jet resolution variable, and logarithms of the quark mass. We calculate the heavy quark fragmenting jet function at , and the gluon and light quark fragmenting jet functions into a heavy quark, and , at . We verify that, in the limit in which the jet invariant mass is much larger than , the logarithmic dependence of the fragmenting jet functions on the quark mass is reproduced by the heavy quark fragmentation functions. The fragmenting jet functions can thus be written as convolutions of the fragmentation functions with the matching coefficients , which depend only on dynamics at the jet scale. We reproduce the known matching coefficients at , and we obtain the expressions of the coefficients and at . Our calculation provides all the perturbative ingredients for the simultaneous resummation of logarithms of and .

Keywords:
QCD, Heavy Quark Physics, Jets
institutetext: Ernest Orlando Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720, U.S.A.

1 Introduction

The production of heavy flavors and heavy flavored jets, where by heavy flavor we here mean charm or bottom, plays an important role in collider experiments. These processes are interesting in themselves, as a probe of QCD dynamics, since for heavy quarks one expects the closest correspondence between calculations at the parton level and experimentally measured hadrons. Furthermore, jets are found in interesting electroweak processes; for instance, one of the most important channel to probe fermionic couplings of the Higgs boson is . Finally, a quick look at the ATLAS and CMS public results pages shows the almost-omnipresence of jets in searches of Beyond Standard Model physics. It is therefore important to have a good theoretical understanding of heavy flavor production (where a heavy flavored hadron is directly observed) and heavy flavored jets (where a jet is tagged by demanding that it contains at least one heavy flavored hadron) in collider experiments.

The current state of the art of fixed order calculations for heavy flavor hadroproduction is next-to-leading order (NLO) accuracy, and such NLO calculations have a long history Nason:1987xz (); Nason:1989zy (); Beenakker:1988bq (); Beenakker:1990maa (). By now, several processes, including heavy quark pair production, associated production of weak bosons and heavy quarks, and Higgs production with decay into , are implemented in the program MCFM Campbell:2000bg () at NLO accuracy, and any distribution can be obtained for these processes.

In such fixed order calculations, the dependence on the heavy quark mass typically enters through the ratio

(1)

were is the transverse momentum of the heavy quark. Many of the available NLO calculations include the full dependence of the heavy quark mass, which is important at small to moderate values of . For large values of , the ratio becomes negligible, and one might want to perform calculations with massless heavy quarks, for which NLO calculations are significantly simpler. However, besides a dependence on powers of , there is also a logarithmic dependence on that ratio, which arises from infrared divergences in the massless calculation which are regulated by the heavy quark mass. Thus, at higher orders in perturbation theory more powers of these logarithms appear, requiring resummation. This is accomplished by introducing a heavy quark fragmentation function Mele:1990cw (). As was discussed for hadroproduction in Ref. Cacciari:1993mq (), the heavy quark fragmentation function can be calculated perturbatively at scales without encountering any large logarithms. Running the fragmentation function from this low scale to using the familiar DGLAP evolution then resums all logarithms of .

A combination of both approaches is needed to describe heavy flavor production for both large and small values of . Such a combination, named “Fixed Order plus Next-to-Leading-Log” (FONLL), has been proposed in Ref. Cacciari:1998it (), and applied to single inclusive production of heavy flavored hadrons. The general idea of FONLL is to add the massive fixed order calculation to the resummed calculation, and then subtract the overlap of the two. The overlap can be calculated either as the massless limit of the fixed order calculation (keeping the logarithms), or as the expansion of the resummed calculation to the appropriate order. The FONLL approach has been successfully compared to Tevatron and LHC data, for a recent discussion see Ref. Cacciari:2012ny ().

Over the past decade we have learned how to combine NLO calculations with parton shower algorithms. This provides final states which are fully showered and hadronized, but which still provide NLO accuracy for predicted observables. Since such calculations can be compared much more directly to experimental data, this is used a great deal in analyses. The most popular available methods are MC@NLO Frixione:2002ik () and POWHEG Nason:2004rx (); Frixione:2007vw (), with several other approaches being pursued. Both MC@NLO and POWHEG include heavy flavor production in their list of available processes Frixione:2003ei (); Frixione:2007nw (). Since parton showers resum leading logarithms in their evolution variable , any calculation that is interfaced with such a shower needs to provide at least the same amount of resummation. In fact, as was discussed in detail in Ref. Alioli:2012fc (), any combination of a perturbative calculation with a parton shower algorithm requires at least LL resummation of the dependence on an infrared safe jet resolution variable, however this jet resolution variable does not necessarily have to be equal to the evolution variable of the parton shower. Thus, one can choose a resolution variable for which one has good theoretical control, such as -jettiness. We will denote a general dimensionful -jet resolution variable by and define the dimensionless ratio

(2)

with denoting the transverse momentum of the hadron, or of the jet in which the hadron is found, which we assume to be not too different.

It follows from the above discussion that combining heavy quark production at large with parton shower algorithms requires the simultaneous resummation of logarithms of and .111It should be noted that the leading log resummation in the shower resums a subset of logarithms of the heavy quark mass, for example all terms originating from emissions from the heavy quark or antiquark in the final state. However, not all are included at leading logarithmic accuracy. In particular, gluon splitting in , with almost collinear quark and antiquark, are included only at fixed order. For a full discussion, see Frixione:2003ei (); Frixione:2007nw (). Logarithmic dependence on a second ratio can also arise from explicit experimental cuts restricting the size of the jet resolution variable . For example, observables that explicitly restrict extra jet activity through jet vetoes will have logarithmic dependence on the jet veto scale .

Extending the discussion to heavy-flavor tagged jets, one might expect jet observables to be less sensitive to the heavy quark fragmentation function, and to logarithms of  Frixione:1996nh (). This is because heavy-flavor tagged jets are essentially agnostic to the flavor of the heavy hadron and its energy fraction. Indeed heavy quark jets initiated by heavy quarks produced directly in the hard interaction do not have a logarithmic dependence on . In this case it is not necessary to introduce a fragmentation function, and to resum . The resummation of can be achieved with methods similar to those used for light quark jets.

However, heavy-flavor tagging algorithms also tag jets initiated by gluons or light quarks, where heavy quarks are produced through . In this case, -tagging introduces an infrared dependence on logarithms of , and large uncertainties Banfi:2007gu (). A possible way to deal with final state logarithms is not to label jets with gluon or light quark splittings into as heavy-flavor jets. Banfi, Salam and Zanderighi in Ref. Banfi:2007gu () explored the interesting possibility of using an IR safe jet flavor algorithm Banfi:2006hf (), which would label jets with no net heavy flavor as gluon or light quark jets. Alternatively, one can improve the theoretical description of -tagged jet cross sections by resumming , and, in the presence of another small ratio , by simultaneously resumming .

In this paper we develop a formalism that allows to simultaneously resum the logarithmic dependence on the heavy quark mass as well as on the additional small ratio . This opens the door to deal with vetoed heavy flavor production in a systematic way, and perhaps more importantly to interface FONLL-type calculations with a parton shower. Our formalism is based on the idea of fragmenting jet functions (FJFs), introduced in Refs. Procura:2009vm (); Jain:2011xz (), and first applied to heavy quarks in Ref. Liu:2010ng (). The FJFs describe the fragmentation of a parton inside a jet initiated by the parton , and contain information both on the jet dynamics, and on the parton fragmentation function. Therefore, FJFs encode the dependence on both , as well as . An important property of the heavy flavor FJFs is that the renormalization group evolution is independent of the heavy quark mass, with the anomalous dimension being identical to that of an inclusive jet function. Thus, the dependence on the jet resolution variable can be resummed in the same way as for processes with only light jets.

To perform a simultaneous resummation of and requires to separate these scales in the factorization theorem, and therefore factorize the FJFs themselves. For this is accomplished by integrating out the degrees of freedom responsible for the scale, with the remaining long-distance physics (and therefore the entire dependence) determined by the heavy-quark fragmentation function.

The main part of this paper is an explicit calculation of the heavy flavor FJFs in fixed order perturbation theory. We calculate the heavy quark initiated FJF at , and the gluon and light-quark initiated FJFs at . Besides being an important ingredient to obtain the resummed expressions, it also allows us to check explicitly various properties of the heavy flavor FJFs. In particular, we verify that the heavy quark fragmentation functions reproduce the logarithmic dependence on the heavy quark mass and that the anomalous dimension of the heavy-quark FJFs are independent of . Our calculations are performed using Soft Collinear Effective Theory (SCET) Bauer:2000ew (); Bauer:2000yr (); Bauer:2001yt (); Bauer:2002nz (); Leibovich:2003jd ().

The paper is organized as follows. In Section 2 we recall the main SCET ingredients needed in the rest of the paper. We define and state important properties of heavy quark fragmentation functions in Section 2.1, and of inclusive jet functions in Section 2.2. In Section 2.3 we introduce the fragmenting jet functions, extending the definition of Refs. Procura:2009vm (); Jain:2011xz () to heavy quarks. After reviewing the resummation of in single inclusive observables, and of in jet observables in Section 3, we describe how to achieve the simultaneous resummation of logarithms of the quark mass and the jet resolution variable in Section 4. In Section 5 we calculate the FJFs and at in the massless limit, we give the expressions with full mass dependence in Appendix B. In Sections 6.1 and 6.2 we carry out the calculation of and at . We draw our conclusions in Section 7. In Appendix A we discuss some additional details on how to take the massless limit . In Appendix C we give the analytic expression of the function , defined in Section 6.1.

2 Soft Collinear Effective Theory

In this paper we use the formalism of Soft Collinear Effective Theory (SCET) Bauer:2000ew (); Bauer:2000yr (); Bauer:2001yt (); Bauer:2002nz (), generalized to massive quarks Leibovich:2003jd (). SCET is an effective theory for fast moving, almost light-like, quarks and gluons, and their interactions with soft degrees of freedom. It has been successfully applied to a variety of processes, from physics to quarkonium, and there is a growing body of application to the study of jet physics and collider observables.

We are interested in processes sensitive to three scales, , and . is the hard scattering scale, represented by the of the hardest jet in the event. defines the jet scale, so the typical size of is the jet invariant mass, while is the heavy quark mass. We are interested in the situation . In this case, degrees of freedom with virtuality of order can be integrated out by matching QCD onto SCET. The degrees of freedom of SCET are collinear quarks and gluons, with virtuality , and ultrasoft (usoft) quarks and gluons, with even smaller virtuality . is the SCET expansion parameter, , with the next relevant scale in the problem, e.g. . In SCET different collinear sectors can only interact by exchanging usoft degrees of freedom. An important property of SCET is that usoft-collinear interactions can be moved from the SCET Lagrangian to matrix elements of external operators Bauer:2001yt (), greatly simplifying the proof of factorization theorems. Since the dynamics of different collinear sectors and of usoft degrees of freedom factorize, we can focus in this paper on jets in a single collinear sector.

If there is a large hierarchy between the remaining two scales, , we can further lower the virtuality of the degrees of freedom in the effective theory by integrating out particles with virtuality at the jet scale. This second version of SCET has collinear fields with . The additional matching step allows to factorize the dynamics of the two scales and , and to resum large logarithms of their ratio .

We now summarize some SCET ingredients needed in the rest of the paper. For more details, we refer to the original papers Bauer:2000ew (); Bauer:2000yr (); Bauer:2001yt (); Bauer:2002nz (); Leibovich:2003jd (). We introduce two lightcone vectors and , satisfying , and . The momentum of a particle can be decomposed in lightcone coordinates according to

(3)

Particles collinear to the jet axis have , where is the SCET expansion parameter. Usoft quarks and gluons have all components of the momentum roughly of the same size .

The SCET Lagrangian can be written as

(4)

Each collinear sector is described by a copy of the collinear Lagrangian . For massless quarks, is

(5)

and are collinear quark and gluon fields, labeled by the lightcone direction and by the large components of their momentum . We leave the momentum label mostly implicit, unless explicitly needed. The label momentum operator acting on collinear fields returns the value of the label, for example

(6)

The collinear covariant derivative is defined as

(7)

are Wilson lines, constructed with collinear gluon fields,

(8)

is an usoft gluon field. At leading order in , usoft gluons couple to collinear quarks only through . This coupling can be eliminated from the Lagrangian via the BPS field redefinition Bauer:2001yt ():

(9)
(10)

is a usoft Wilson line in the direction

(11)

with P denoting path ordering. The effect of the field redefinition is to eliminate the usoft gluon in Eq. (5), and to replace the collinear quark and gluon fields and with their non-interacting counterparts. The same field redefinition also decouples usoft from collinear gluons Bauer:2001yt (). From here on we always use decoupled collinear fields, and drop the superscript .

For fast moving massive particles there are additional mass terms Leibovich:2003jd (),

(12)

We work with one massive quark with mass , massless quarks and assume that quarks heavier than have been integrated out. In this paper we use to denote both heavy and light quarks, when it is not necessary to specify the quark mass. () is used exclusively for heavy quarks (antiquarks), while () denotes the light quarks (antiquarks).

Using the Wilson line it is possible to construct gauge invariant combinations of collinear fields

(13)

Collinear gauge invariant operators are expressed in terms of matrix elements of these building blocks Bauer:2002nz (). In the next subsections, we discuss three such operators, heavy quark fragmentation functions, inclusive quark and gluon jet functions, and heavy quark fragmenting jet functions.

2.1 Heavy Quark Fragmentation Functions

Fragmentation functions describe the fragmentation of a parton into a hadron , which carries a fraction of the parton momentum. In SCET, the operator definitions of the fragmentation function of a quark or a gluon into a hadron are given by Procura:2009vm ()

(14)
(15)

The trace in Eq. (14) is over Dirac and color indices. The sum over denotes the integration over the phase space of all possible collinear final states. In Eqs. (14) and (15), denotes the large component of the momentum of the fragmenting parton, and the frame in which the fragmenting parton has zero has been chosen. The hadron has momentum . The perpendicular component is integrated over, while , or, equivalently, the momentum fraction is measured. is the number of colors, . The definitions in Eqs. (14) and (15) are equivalent to the classical definition of fragmentation function in QCD, in Ref. Collins:1981uw ().

The evolution of the fragmentation functions is governed by the DGLAP equation DGLAP ()

(16)

where the are the time-like splitting functions. The splitting functions are computed in perturbation theory

(17)

with, at one loop, DGLAP ()

(18)
(19)
(20)
(21)

The color factors in Eqs. (18)–(21) are , , , while is the leading order coefficient of the beta function,

(22)

The space-like and time-like splitting functions at are given in Refs. Furmanski:1980cm (); Curci:1980uw (), and nicely summarized in Ref. Ellis:1991qj (). Space-like splitting functions are known to Moch:2004pa (); Vogt:2004mw (). The non-singlet component of the time-like splitting functions is also known to three-loops Mitov:2006ic (), while the singlet is, at the moment, unknown.

The fragmentation functions of light hadrons are non-perturbative matrix elements, which need to be extracted from data. In the case of heavy flavored hadrons, the heavy quark mass is large compared to the hadronization scale . Neglecting corrections of order , one can identify the heavy hadron with a heavy quark or antiquark and the fragmentation function can be computed in perturbation theory at the scale Mele:1990cw (). Expanding in ,

(23)

the fragmentation function for a heavy quark into a quark or a gluon, and for a gluon into a heavy quark at are

(24)
(25)
(26)
(27)

The fragmentation functions of a heavy quark, heavy antiquark or light quark into a heavy quark were computed at in Ref. Melnikov:2004bm (), while the fragmentation of a gluon into a heavy quark in Ref. Mitov:2004du ().

The fixed order expressions for the heavy quark fragmentation functions are reliable at scales , where logarithms are small. The fragmentation functions at an arbitrary scale are obtained by taking the fixed order expressions as initial condition for the DGLAP evolution. The evolution of the one-loop initial condition (24)–(27) with splitting functions resums all leading and next-to-leading logarithms (NLL), that is all terms of the form and . The knowledge of the initial condition at , and of the non-singlet splitting functions at , allows to achieve NNLL accuracy for non-singlet combinations of the quark fragmentation function, for example . The evolution of the gluon distribution and of the singlet distribution require the time-like singlet splitting function to , which, at the moment, is not known.

The picture obtained with the partonic initial conditions (24)–(27) and the DGLAP evolution is a valid description of the fragmentation functions of heavy hadrons, except in the endpoint region, , corresponding to the peak of the quark distribution. In this region soft gluon resummation and non-perturbative effects become important and a model describing hadronization must be included, and fitted to data Cacciari:2001cw (); Cacciari:2005uk (); Neubert:2007je ().

We conclude this section by mentioning two important sum rules obeyed by the fragmentation functions. The first is the momentum conservation sum rule,

(28)

The sum is extended over a complete set of states. Eq. (28) is the statement that the total energy carried off by all the fragmentation products sums to that of the original parton. At the perturbative level, . Eq. (28) can be readily verified using the one loop results for the quark distributions in Eqs. (24)–(26). Using the one loop expression for the splitting functions, and the DGLAP equation (16), one can also check that the momentum conservation sum rule is not spoiled by renormalization. This is true at all orders Collins:1981ta ().

In addition, there are flavor conservation sum rules. For heavy quarks,

(29)

Eq. (29) is a consequence of the fact that QCD interactions do not change the flavor of the fragmenting quark, and therefore the number of quark minus antiquark in the fragmentation products is always equal to one. At vanishes and Eqs. (24) and (25) explicitly satisfy the flavor sum rule. The fragmentation functions at also satisfy Eq. (29) Melnikov:2004bm (). DGLAP evolution does not modify the flavor sum rule.

2.2 Inclusive Jet Functions

The gauge invariant quark and gluon fields, Eq. (13), are a natural ingredient for the description of jets in SCET. Quark and gluon inclusive jet functions are defined as matrix elements of and Bauer:2002ie (); Bauer:2003pi (); Fleming:2003gt (),

(30)
(31)

We work in a frame where the momentum is aligned with the jet direction, . is the large component of the momentum, of the size of the jet , and the jet invariant mass is . To simplify the notation, in the rest of the paper we drop the superscript on the plus component of the jet momentum. The quark and gluon inclusive jet functions are infrared finite quantities, insensitive to the scale , and can be computed in perturbation theory. In the case the quark field is massive, the quark jet function (30) depends on the quark mass, but the dependence is not singular Fleming:2007qr (); Fleming:2007xt ().

Beyond leading order, the quark and gluon jet functions are UV divergent and require renormalization. The dependence on the renormalization scale is governed by the renormalization group equation (RGE)

(32)

The anomalous dimension is

(33)

where the plus distribution of the dimensionful variable is defined as

(34)

and it is independent of the arbitrary cut-off .

and are the quark and gluon cusp anomalous dimensions Korchemsky:1987wg (); Korchemskaya:1992je (), which are known to three loops Moch:2004pa (). Up to this order, they are related by . is the non-cusp component of the anomalous dimension, known to Becher:2006qw (); Becher:2009th ().

The form of the anomalous dimension (33), in particular its dependence on , allows to resum Sudakov double logarithms. The RGE (32) can be solved analytically, and, given an initial condition at the scale , the jet function at the scale is

(35)

where is an evolution function, given, for example in Ref. Fleming:2007xt ().

In hadronic collisions, in addition to collinear radiation from final state particles, one has to account for initial state radiation from the incoming beams. Initial state radiation is described by beam functions Stewart:2009yx (); Stewart:2010qs (). The beam functions depend on the invariant mass and also on the momentum fraction of the incoming parton. They satisfy the same RGE as final state jets, Eq. (32). Large Sudakov logarithms induced by collinear radiation from the incoming beams are resummed in the same way as logarithms in the jet functions,

(36)

Notice that the beam function evolution does not change the distribution in the momentum fraction . The beam functions are perturbatively related to the parton distributions Stewart:2009yx (); Stewart:2010qs (), according to

(37)

In this case, the initial condition for the evolution (36) cannot be computed purely in perturbation theory, but it is obtained convoluting the perturbative matching coefficients with the parton distributions evaluated at the scale .

The last ingredient in factorization theorems for jet cross sections is a soft function, describing soft interactions between jets, and between jets and the beams. The precise definition of the soft function depends on the observable in consideration, but in general its RGE is of similar form as Eq. (32), and resums Sudakov double logarithms.

2.3 Heavy Quark Fragmenting Jet Functions

Fragmenting jet functions were introduced in Refs. Procura:2009vm (); Jain:2011xz () to describe the fragmentation of a hadron inside a quark or gluon jet. A first application to heavy quarks was discussed in Ref. Liu:2010ng (). FJFs combine the fragmentation function, given in Eqs. (14) and (15), with the inclusive jet function, given in Eqs. (30) and (31). This can be explicitly seen in their definition Procura:2009vm (); Jain:2011xz ():

(38)
(39)

and are the gauge-invariant fields defined in Eq. (13). Antiquark FJFs are defined in a similar way, by exchanging the fields and in Eq. (38). As in Eqs. (30) and (31), the large component of the jet momentum is , and is the jet invariant mass. However, differently from inclusive jets, in the definition of FJF a heavy hadron in the final state is singled out, and its momentum is measured. The FJFs thus depend on the jet invariant mass , on the momentum fraction and on the heavy quark mass .

The FJFs have several important properties, which were proven for light partons in Ref. Procura:2009vm (); Jain:2011xz (); Procura:2011aq () and which we now discuss briefly.

The first relationship states that after integrating over and summing over all the possible emitted particles, one should recover the inclusive jet function. This is guaranteed by the momentum Procura:2009vm (); Jain:2011xz () and flavor Procura:2011aq () sum rules obeyed by the FJFs. The momentum conservation sum rule states that

(40)

where the sum is over a complete set of states. The flavor sum rule for a quark is Procura:2011aq ()

(41)

These relations are valid both in the approximation , and in the regime . In the former case, are the inclusive quark and gluon jet functions, computed with massless quarks Bauer:2002ie (); Fleming:2003gt (); Becher:2006qw (); Becher:2010pd (). If , is the massive jet function of Ref. Fleming:2007qr (); Fleming:2007xt (). In both cases, the mass dependence of the jet function on the r.h.s. of Eqs. (40) and (41) is not singular.

The second property arises again due to the similarity of FJFs with inclusive jet functions. In the UV, the FJFs look like inclusive jet functions, initiated by the parton . In particular, the restriction on the final state, requiring the identification of the hadron , does not affect the UV poles of the FJF, so that have the same renormalization group equation as quark or gluon inclusive jet functions Procura:2009vm (); Jain:2011xz ()

(42)

The anomalous dimension is identical to the inclusive case, given in Eq. (33), and in particular is independent of the momentum fraction and the mass of the heavy quark . The resummation of proceeds as in the inclusive case discussed in Section 2.2, albeit with a different initial condition.

The final relation is due to the fact that the IR sensitivity of the FJFs is completely captured by the unpolarized fragmentation functions. Therefore, if the jet scale and the heavy quark mass are well separated, , at leading power in one can factorize the dynamics at the two scales by matching the FJFs onto fragmentation functions

(43)

The coefficients depend on the jet invariant mass, and on the momentum fraction, but are independent of the heavy quark mass, up to power corrections of the size . Eq. (43) is very similar to the relation between the beam functions and the parton distributions in Eq. (37).

We will further discuss the properties (40), (41), (42), and (43), and illustrate them with examples at and in Secs. 5, 6.1 and 6.2.

3 Review of how to resum logarithms of and

Consider single inclusive production of one (light) hadron in collisions, . Beyond leading order, the partonic cross section contains collinear divergences, when additional emissions become collinear to initial or final state partons. The divergences are physically cut off by non-perturbative physics, and they need to be absorbed into non-perturbative matrix elements, parton distribution functions for the partons in the initial state, and fragmentation functions for the final state.

In the case of single inclusive hadroproduction of heavy hadrons, , the final state collinear divergences in the partonic cross section are cut off by the heavy quark mass , a perturbative scale. Identifying the heavy hadron with a heavy quark, the cross section for the production of a heavy hadron, differential in the hadron and rapidity , can be expressed as a convolution of the partonic cross section for the production of a heavy quark and two parton distribution functions for the incoming partons Nason:1989zy ()

(44)

Here the functions denote the standard parton distributions functions, while denotes the partonic cross section for a parton and parton to scatter into a heavy quark with transverse momentum and rapidity . We have omitted the dependence of the short-range cross section on the momentum fractions , and on the heavy quark rapidity.

The final state collinear divergences present in the massless case manifest as logarithms of the heavy quark mass in Eq. (44). As the energy increases, logarithms of in the partonic cross section become large, threatening the validity of the perturbative expansion. In order to resum them, one needs to factorize the partonic cross section into two separate pieces, each of which depends on only one of the two scales and . This is achieved by introducing a fragmentation function Cacciari:1993mq ()

(45)

In Eq. (45), the short-range cross section is the cross section for the production of the parton with transverse momentum and rapidity in the collision of partons and , computed with all partons considered massless. The parton then fragments into a heavy hadron , carrying a transverse momentum , and the same rapidity as the original parton . If the short-range cross section and the fragmentation function are evaluated at their characteristic scale, respectively and , no large logarithms arise in the perturbative expressions. Of course, in the end all functions have to be evaluated at a common scale , and one therefore has to use the RGE to evolve each function to this scale. The RG evolution of the fragmentation function is determined by the DGLAP equation, Eq. (16). By evolving the fragmentation function from to one can sum the logarithms of .

As already noted, the short-range cross section in (45) is calculated in the limit . Thus, while this approach correctly resums the logarithms of , it does not contain any dependence on powers of the same ratio. Since the power dependence is correctly reproduced using (44), one can obtain an expression that correctly reproduces both the logarithmic and the power dependence on by combining the two ways of calculating. This is the approach taken in FONLL Cacciari:1998it ().

Now consider jet cross sections. As in the previous case, the starting point for a resummation of the large logarithms that arise in cross sections that are differential in a jet resolution parameter is the separation of the dimensionful variables whose ratio gives the value of . This is achieved by a factorization of the cross section. There are many jet resolution variables one can choose, and a large body of literature how to obtain the relevant factorization theorems. Since all approaches in the end contain the same physics, and the final factorization theorems look very similar, we simply state the result here for one specific resolution variable, namely -jettiness Stewart:2010tn (). For the purposes of this discussion, the only relevant part of the definition of -jettiness is that has dimension one, as we approach pencil-like jets, and that is linear in the contributions from each jet (both from initial and final state radiation) and soft physics .

The factorization theorem can be written schematically as Stewart:2010tn ()

(46)

and we have only included the dependence on terms that are relevant for our discussion. is the hard function for the production of the partons with flavor , and it depends on the of the signal jets. Collinear radiation in the final state is described by the inclusive jet functions , while two beam functions describe initial state radiation from the incoming beams, initiated by the partons of flavor and . The jet and beam functions are function of the jet invariant mass , where is of the size of the hard scattering scale. In addition, the beam functions depend on the momentum fraction of the incoming partons. The soft function describes soft interactions between jets, and between jets and the beams. It depends on soft momenta .

After evolving the parton distribution functions to the jet scale, as discussed in Section 2, each function in the factorization theorem (46) only depends on a single scale, thus one can again calculate each term at its characteristic scale without encountering any large logarithms, and then evolve them to a common scale using the RGEs discussed in Section 2.2.

The factorization formula in (46) is derived in the limit , such that no power corrections of the ratio can be included. In order to derive an expression that is valid in both the limits of small and large , one needs to combine the resummed result with the known fixed order expression, which includes this power dependence.

4 A combined resummation of and

In this section we give the factorization theorems that are required to combine both types of resummation, such that one can study the production of heavy flavor at high energy in the presence of jet vetoes, or perhaps more importantly, such that one can combine calculations which resum the dependence on the heavy quark mass with parton shower algorithms. The later sections in this paper are then devoted to calculating the new ingredients in the resulting factorization theorems perturbatively.

We will consider two separate cases. The first is the production of identified heavy flavored hadrons in hadronic collisions, with measured momentum of the heavy hadron. Examples are or the associated production of a heavy flavored hadron and a weak boson, . In particular, we consider the case in which the heavy hadron is part of an identified jet, and a jet veto limits the total number of jets in the event. The momentum of the heavy hadron is characterized by its fraction of the total jet momentum it is part of. A second interesting application is the production of jets (identified by a regular jet algorithm), which are tagged as jets, and again extra jet activity is vetoed. Since -tagging algorithms rely on the presence of at least one weakly decaying -flavored hadron, the situation is related to the previous case. The main difference is that the momentum of the hadron is not measured in this case, and the momentum fraction is therefore integrated over.

The factorization theorem is based on the FJFs, defined in Refs. Procura:2009vm (); Jain:2011xz () for light hadrons, and reviewed in Section 2.3. The main new ingredient in this work is to extend the idea of a FJF to the case of heavy quarks, in which case infrared singularities that were present in the light FJFs manifest themselves as a logarithmic dependence on the heavy quark mass.

In addition to cases we discuss, logarithmic dependence on appears in the flavor excitation channel. In this channel, one heavy quark is present in the initial state and enters the hard collision. This is similar to the cases discussed above, with the difference that the production of the heavy flavor happens in the initial rather than the final state. In this case are resummed by introducing a perturbative -quark parton distribution at the scale , and running it with the DGLAP equation up to the hard scattering scale. Initial state radiation at a scale can be studied using the same techniques developed in this paper, by introducing a heavy quark beam function. We leave a detailed discussion for future work, and will not discuss initial state splitting any further.

4.1 The production of an identified heavy hadron

We consider first the case of production of an identified heavy flavored hadron in the presence of a veto on extra jet activity. As already discussed, the momentum of the heavy hadron is measured to have a fraction of the momentum of the jet it is part of. The extra jet activity is vetoed using a jet resolution variable , where is defined such that it goes to zero when there are at most pencil-like jets present. Phenomenologically interesting applications are the two-jettiness cross section in , or the one- and two-jettiness cross sections for .

In the limit of small , the factorization theorem for the cross section differential in and in the and rapidity of the observed hadron can schematically be written as

(47)

This factorization theorem is almost identical to the one given in Eq. (46), and, as in that case, it holds up to power corrections in . The only difference is that a hadron is observed inside the jet initiated by the parton , and its and rapidity are measured. To be able to describe this extra information, the inclusive jet function needs to be replaced by the FJF . In addition to the argument , describing the contribution of the inclusive jet to the jet resolution variable, the fragmenting jet function depends on the mass of the heavy quark as well as the momentum fraction of the heavy hadron .

As discussed in Section 2.3 the RGE of the FJF, Eq. (42), is identical to that of an inclusive quark or gluon jet function, so that the resummation of proceeds as in the inclusive case.

The FJFs are two-scale objects, sensitive to the jet invariant mass and to the heavy quark mass . Differently from the light parton FJFs discussed in Refs. Procura:2009vm (); Jain:2011xz (), the heavy quark FJFs can be computed purely in perturbation theory. If the jet scale in Eq. (47) is close to , the fixed order expression for the FJFs at the scale does not contain large logarithms, and the evolution (35) resums logarithms of . On the other hand, if , there is no choice of initial scale the minimizes the logarithms in the FJFs, and the initial condition for the jet evolution is still plagued by large logarithms. However, the IR sensitivity of the FJFs is completely captured by the unpolarized fragmentation functions. Therefore, if the jet scale and the heavy quark mass are well separated one can factorize the dynamics at the two scales by matching the FJFs onto heavy quark fragmentation functions , as in Eq. (43). Since each term on the right-hand side of Eq. (43) depends only on a single scale, the logarithms of of the FJFs are reproduced through logarithms of and on the right hand side, such that they can be resummed through RG evolution. Evolving the fragmentation function from the mass scale to a scale of order , no large logarithms are left in the initial condition for the FJF evolution. Then, running the hard, beam, soft and jet functions to a common scale, all large logarithms in Eq. (47) are correctly resummed.

By matching the FJFs onto fragmentation functions and the beam functions onto parton distributions, we can recast Eq. (47) in a form that stresses the relation to single inclusive production discussed in Section 3.

This form is very similar to Eq. (45), the only difference being that the partonic short range cross section in Eq. (45) has been further separated into different pieces, each of them dependent on a single scale. If , the hard, jet and soft scales become equal and the resummation of is turned off. The fragmentation function and the parton distributions are evolved up to the hard scale, resumming , at the desired logarithmic accuracy. In this situation, Eq. (4.1) reduces to the jet limit of Eq. (45).

4.2 The production of tagged heavy flavor jets

In this section, we consider the impact of logarithms of the quark mass on observables involving jets containing heavy flavor. The most important application of this is for the description of -tagged jets. Let us start by giving a closer look to the experimental definition of jets. In high energy experiments, like ATLAS or CMS, jets are tagged using a variety of techniques based on the long lifetime of weakly decaying heavy flavored hadrons inside the jets ATLAS:2011qia (); Chatrchyan:2012jua (). These techniques have in common the requirement of the presence of a weakly decaying -flavored hadron, within a certain distance from the jet axis, in rapidity-azimuthal angle space, and with a minimum . Typical choices are , and GeV. tagging algorithms do not differentiate between jets containing or quarks.

Our goal is to define a heavy quark tagged (-tagged) jet function with these features, such that one can use the factorization formula given in Eq. (46) and simply replace the standard jet function by its heavy quark tagged version. A heavy quark tagged jet function is agnostic as to which type of hadron gives rise to the long decay time and therefore the secondary vertex. Furthermore, it is insensitive to the momentum fraction of the heavy hadron, as long as the transverse momentum is above the minimum transverse momentum imposed in the -tagging algorithm. Therefore, the -tagged jet function can be obtained from the heavy flavor FJF by summing over all heavy flavored hadrons as well as integrating over the momentum fraction of the heavy hadron (down to a cutoff related to the minimum ).

As is the case for a fragmentation function, the hard interaction does not necessarily need to involve the production of a heavy quark , since this can be produced from the splitting in the radiation happening within the jet. Thus, heavy quark tagged jets can be initiated by any possible flavor. In the case of heavy quark initiated jets we define