A Details for jet functions in the single-inclusive case

Tom Kaufmann,   Asmita Mukherjee,   Werner Vogelsang

[5mm] Institute for Theoretical Physics, Tübingen University, Auf der Morgenstelle 14,

72076 Tübingen, Germany

[2mm] Department of Physics, Indian Institute of Technology Bombay,

Powai, Mumbai 400076, India

Abstract

We present an analytical next-to-leading order QCD calculation of the partonic cross sections for the process , for which a specific hadron is observed inside a fully reconstructed jet. In order to obtain the analytical results, we assume the jet to be relatively narrow. We show that the results can be cast into a simple and systematic form based on suitable universal jet functions for the process. We confirm the validity of our calculation by comparing to previous results in the literature for which the next-to-leading order cross section was treated entirely numerically by Monte-Carlo integration techniques. We present phenomenological results for experiments at the LHC and at RHIC. These suggest that should enable very sensitive probes of fragmentation functions, especially of the one for gluons.

## 1 Introduction

Final states produced at high transverse momentum (), such as jets, single hadrons, or prompt photons, have long been regarded as sensitive and well-understood probes of short-distance QCD phenomena. Recently, a new “hybrid” type of high- jet/hadron observable has been proposed and explored theoretically [[1], [2], [3], [4], [5]]. It is defined by an identified specific hadron found inside a fully reconstructed jet, giving rise to a same-side hadron-jet momentum correlation. This correlation may for example be described in terms of the variable , where and are the transverse momenta of the hadron and the jet, respectively. The production of identified hadrons in jets was first considered for the case of annihilation [[1], [2], [3]] and subsequently also for scattering [[4]]. Experimental studies have been pioneered in at the Tevatron [[6]]. At the LHC, the ATLAS [[7], [8]] and CMS [[9]] experiments have studied , and measurements are being carried out by ALICE [[10]]. Measurements of the cross section (and, perhaps, spin asymmetries) should also be possible at RHIC.

There are several reasons why it is interesting to study the production of hadrons inside jets. Perhaps most importantly, the observable provides an alternative window on fragmentation functions [[4]]. The latter, denoted here by , describe the formation of a hadron from a parent parton . The variable is the fraction of the parton’s momentum transferred to the hadron, and denotes the factorization scale at which the fragmentation function is probed. Usually, fragmentation functions for a hadron are determined from the processes or . The power of these processes lies in the fact that they essentially allow direct scans of the fragmentation functions as functions of . The reason for this is that to lowest order (LO) in QCD, it turns out that is identical to a kinematic (scaling) variable of the process. For instance, in one has to LO, where is the momentum of the observed hadron and the momentum of the virtual photon that is produced by the annihilation. NLO corrections dilute this direct “local” sensitivity only little. A drawback of or is on the other hand that the gluon fragmentation function can be probed only indirectly by evolution or higher order corrections.

Being universal objects, the same fragmentation functions are also relevant for describing hadron production in -scattering. So far, one has been using the process as a further source of information on the  [[11], [12], [13]]. Although this process does probe gluon fragmentation, its sensitivity to fragmentation functions is much less clear-cut than in the case of or . This is because for the single-inclusive process the fragmentation functions arise in a more complex convolution with the partonic hard-scattering functions, which involves an integration over a typically rather wide range of already at LO. As a result, information on the is smeared out and not readily available at a given fixed value of .

The process allows to overcome this shortcoming. As it turns out, if one writes its cross section differential in the variable introduced above, then to LO the hadron’s fragmentation function is to be evaluated at . This means that by selecting one can “dial” the value at which the are probed, similarly to what is available in or . Thanks to the fact that in scattering different weights are given to the various fragmentation functions than compared to and , it is clear that has the potential to provide complementary new information on the , especially on gluon fragmentation. Data for should thus become valuable input to global QCD analyses of fragmentation functions. At the very least, they should enable novel tests of the universality of fragmentation functions. We note that similar opportunities are expected to arise when the hadron is produced on the “away-side” of the jet, that is, basically back-to-back with the jet [[14]], although the kinematics is somewhat more elaborate in this case.

The production of specific hadrons inside jets may also provide new insights into of the structure of jets and the hadronization mechanism. Varying and/or the hadron species, one can map out the abundances of specific hadrons in jets. Particle identification in jets becomes particularly interesting in a nuclear environment in scattering, where distributions of hadrons may shed further light on the phenomenon of “jet quenching”. Knowledge of fragmentation functions in jet production and a good theoretical understanding of the process are also crucial for studies of the Collins effect [[15], [16], [17]], an important probe of spin phenomena in hadronic scattering [[18]].

In the present paper, we perform a new next-to-leading order (NLO) calculation of . In contrast to the previous calculation [[4]] which was entirely based on a numerical Monte-Carlo integration approach, we will derive analytical results for the relevant partonic cross sections. Apart from providing independent NLO predictions in a numerically very efficient way, this offers several advantages. In the context of the analytical calculation, one can first of all explicitly check that the final-state collinear singularities have the structure required by the universality of fragmentation functions, meaning that the same fragmentation functions occur for as for usual single-inclusive processes such as . We note that to our knowledge this has not yet been formally proven beyond NLO. Also, as we shall see, the NLO expressions show logarithmic enhancements at high , which recur with increasing power at every order in perturbation theory, eventually requiring resummation to all orders. Having explicit analytical results is a prerequisite for such a resummation. In Ref. [[3]], considering the simpler case of -annihilation, such resummation calculations for large were presented.

Technically, we will derive our results by assuming the jet to be relatively narrow, an approximation known as “Narrow Jet Approximation (NJA)”. This technique was used previously for NLO calculations of single-inclusive jet production in hadronic scattering,  [[19], [20], [21], [22], [23]]. The main idea is to start from NLO “inclusive-parton” cross sections for the processes , which are relevant for the cross section for . They are a priori not suitable for computing a jet cross section, which is evident from the fact that the require collinear subtraction of final-state collinear singularities, whereas a jet cross section is infrared-safe as far as the final state is concerned. Instead, it depends on the algorithm adopted to define the jet and thereby on a generic jet (size) parameter . As was shown in Refs. [[19], [20], [21], [22], [23]], at NLO one may nonetheless go rather straightforwardly from the single-inclusive parton cross sections to the jet ones, for any infrared-safe jet algorithm. The key is to properly account for the fact that at NLO two partons can fall into the same jet, so that the jet needs to be constructed from both. In fact, within the NJA, one can derive the translation between the and the partonic cross sections for jet production analytically. We note that the NJA formally corresponds to the limit , but turns out to be accurate even at values relevant for experiment. In the NJA, the structure of the NLO jet cross section is of the form ; corrections to this are of and are neglected. In this paper, we apply the NJA to the case of , using it to derive the relevant NLO partonic cross sections. In the course of the explicit NLO calculation, we find that the partonic cross sections for and may be very compactly formulated in terms of the single-inclusive parton ones , convoluted with appropriate perturbative “jet functions”. These functions are universal in the sense that they only depend on the type of the outgoing partons that fragment and/or produce the jet, but not on the underlying partonic hard-scattering function. On the basis of the jet functions, the NLO partonic cross sections for and take a very simple and systematic form. In fact, it turns out that for the jet functions have a “two-tier” form, with a first jet function describing the formation of the jet and a second one the fragmentation of a parton inside the jet. We note that the concept of jet functions for formulating jet cross sections is not new but was introduced in the context of soft-collinear effective theories (SCET) [[1], [2], [3], [24], [25], [26]], although applications to have to our knowledge not been given. Jet functions in a more general context of SCET or QCD resummation have been considered in Refs. [[27]] and [[28]], for example. We also note that in Ref. [[14]] the NLO corrections for the case of away-side jet-hadron correlations were presented in the context of a Monte-Carlo integration code.

Our paper is organized as follows. In Sec. 2 we present our NLO calculation. In particular, Sec. 2.3 contains our main new result, the formulation of in terms of suitable jet functions. Section 3 presents phenomenological results for for LHC and RHIC. We finally conclude our work in Sec. 4. The Appendices collect some technical details of our calculations.

## 2 Associated jet-plus-hadron production in the NJA

Our formalism is best developed by first considering the process , where a hadron is observed at large transverse momentum , but no requirement of a reconstructed hadronic jet is made. This is of course a standard reaction, for which the NLO corrections have been known for a long time [[29], [30]]. The factorized cross section at given hadron and rapidity reads

 dσH1H2→hXdpTdη = 2pTS∑abc∫1xminadxaxafH1a(xa,μF)∫1xminbdxbxbfH2b(xb,μF) (1) × ∫1zmincdzcz2cd^σcab(^s,^pT,^η,μF,μ′F,μR)vdvdwDhc(zc,μ′F),

with the usual parton distribution functions , the fragmentation functions , and the hard-scattering cross sections for the partonic processes , denoting an unobserved partonic final state. Defining

 V≡1−pT√Se−η,W≡p2TSV(1−V), (2)

with the hadronic c.m.s. energy, we have

 xmina=W,xminb=1−V1−VW/xa,zminc=1−Vxb+VWxa. (3)

The are functions of the partonic c.m.s. energy , the partonic transverse momentum and the partonic rapidity . Since only depends on , the last integral in Eq. (1) takes the form of a convolution. The variables and in (1) are the partonic counterparts of and :

 v≡1−^pTe−^η√^s,w≡^p2T^sv(1−v). (4)

One customarily expresses and by and :

 ^p2T=^svw(1−v),^η=12log(vw1−v). (5)

Finally, the various functions in Eq. (1) are tied together by their dependence on the initial- and final-state factorization scales, and , respectively, and the renormalization scale .

The partonic hard-scattering cross sections may be evaluated in QCD perturbation theory. We write the perturbative expansion to NLO as

 d^σcabdvdw=d^σc,(0)abdvδ(1−w)+αs(μR)2πd^σc,(1)abdvdw+O(α2s(μR)), (6)

where we have used that for leading-order (LO) kinematics (since the unobserved partonic final state consists of a single parton), equivalent to . The NLO terms have been presented in Refs. [[29], [30]].

### 2.2 Translation to single-inclusive jet cross section via jet functions

As shown in Refs. [[19], [20], [21], [22]], one can transform the cross section for single-inclusive hadron production to a single-inclusive jet one. References [[19], [20], [21], [22]] explicitly constructed this translation at NLO. We may write the jet cross section as

where and are the jet’s transverse momentum and rapidity, and where denotes a parameter specifying the jet algorithm. For the jet cross section we still have

 xmina=W,xminb=1−V1−VW/xa, (8)

as in (3), but with and now defined by

 V≡1−pjetT√Se−ηjet,W≡(pjetT)2SV(1−V). (9)

Likewise, and are as in (4) but with . Furthermore, in analogy with the inclusive-hadron case, . We note that the partonic cross sections relevant for jet production depend on the algorithm used to define the jet. They do not carry any dependence on a final-state factorization scale.

In order to go from the inclusive-parton cross sections to the jet ones , the idea is to apply proper correction terms to the former. The have been integrated over the full phase space of all final-state partons other than . Therefore, they contain contributions where a second parton in the final state is so close to parton that the two should jointly form the jet for a given jet definition. One can correct for this by subtracting such contributions from and adding a piece where they actually do form the jet together. At NLO, where there can be three partons in the final state, one has after suitable summation over all possible configurations:

 d^σjetab = [d^σcab−d^σc(d)ab−d^σc(e)ab]+[d^σdab−d^σd(c)ab−d^σd(e)ab]+[d^σeab−d^σe(c)ab−d^σe(d)ab] (10) + d^σcdab+d^σceab+d^σdeab.

Here is the cross section where parton produces the jet, but parton is so close that it should be part of the jet, and is the cross section when both partons and jointly form the jet. The decomposition (10) is completely general to NLO. It may be applied for any jet algorithm, as long as the algorithm is infrared-safe. As mentioned before, a property of the is that all dependence on the final-state factorization scale , which was initially present in the , must cancel. This cancellation comes about in (10) because the possess final-state collinear singularities that require factorization. This introduces dependence on in exactly the right way as to compensate the -dependence of the .

In the NJA, the correction terms and may be computed analytically. At NLO, they both receive contributions from real-emission diagrams only. For the NJA one assumes that the observed jet is rather collimated. This in essence allows to treat the two outgoing partons and as collinear. The relevant calculations for the standard cone1 and (anti-) [[32], [33], [34]] algorithms were carried out in Refs. [[21], [22]], while Ref. [[23]] addressed the case of the “” algorithm proposed in [[35], [36]]. We note that we always define the four-momentum of the jet as the sum of four-momenta of the partons that form the jet. This so-called “ recombination scheme” [[37]] is the most popular choice nowadays.

By close inspection of (10), we have found that in the NJA the jet cross section may be cast into a form that makes use of the single-inclusive parton production cross sections :

 dσH1H2→jetXdpjetTdηjet = 2pjetTS∑abc∫1xminadxaxafH1a(xa,μF)∫1xminbdxbxbfH2b(xb,μF) (11) × ∫1zmincdzcz2cd^σcab(^s,^pT,^η,μF,μ′F,μR)vdvdwJc(zc,RpjetTμ′F,μR),

with inclusive jet functions and . We have , and and  are now as in (3) and (4), respectively. Equation (11) thus states that one can go directly from the cross section for single-hadron production to that for jet production by replacing the fragmentation functions in (1) by the jet functions . The latter are such that any dependence on disappears from the cross section. They depend on the jet algorithm and hence on a jet parameter . For the cone and (anti-) algorithms is just given by the usual jet size parameter introduced for these algorithms, while for the jet algorithm of [[35], [36]] we have with the “maximization” parameter defined for this algorithm. In the NJA we generally assume and neglect contributions. The jet functions then read explicitly

 Jq(z,λ≡RpjetTμ′F,μR) = δ(1−z)−αs(μR)2π[2CF(1+z2)(log(1−z)1−z)++Pqq(z)log(λ2) +δ(1−z)Ialgoq+CF(1−z)] −αs(μR)2π[Pgq(z)log(λ2(1−z)2)+CFz], Jg(z,λ≡RpjetTμ′F,μR) = δ(1−z)−αs(μR)2π[4CA(1−z+z2)2z(log(1−z)1−z)+ (12) +Pgg(z)log(λ2)+δ(1−z)Ialgog] −αs(μR)2π2nf[Pqg(z)log(λ2(1−z)2)+z(1−z)],

where , and is the number of active flavors, and where the LO splitting functions as well as the “plus”-distribution are defined in Appendix A. The dependence on the jet algorithm is reflected in the terms and , which are just numbers that we also collect in Appendix A.

Equation (11) evidently exhibits a factorized structure in the final state for the jet cross section in the NJA. Its physical interpretation is essentially that the hard scattering produces a parton that “fragments” into the observed jet via the jet function , the jet carrying the fraction of the produced parton’s momentum. At NLO, the factorization is in fact rather trivial. To get a clear sense of it, it is instructive to see how one recovers (7),(10) from (11). To this end, we combine (6) and (2.2) and expand to first order in the strong coupling. The products of the with the LO terms in just reproduce the single-inclusive parton cross sections at , i.e. the terms in (10). The only other terms surviving in the expansion to are the products of the LO terms of (6) with the terms in the jet functions. These precisely give the remaining contributions (plus the other combinations) in (10). Because of the convolution in in (11), the -function in the Born cross section actually fixes to the value . Based on our NLO calculation, we evidently cannot prove the factorization shown in (11) to beyond this order. We note, however, that similar factorization formulas have been derived using Soft Collinear Effective Theory (SCET) techniques [[25], [24], [26]], for the case of jet observables in annihilation. In particular, functions closely related to our inclusive jet functions may be found in Ref. [[25]], where they are termed “unmeasured” quark (or gluon) jet functions. We shall return to comparisons with SCET results below.

### 2.3 Hadrons produced inside jets

We are now ready to tackle the case that we are really interested in, where the hadron is observed inside a reconstructed jet and is part of the jet. Our strategy for performing an analytical NLO calculation will be to use the NJA and the same considerations as those that gave rise to Eq. (10). Subsequently, we will again phrase our results in a simple and rather general way in terms of suitable jet functions.

The cross section we are interested in is specified by the jet’s transverse momentum and rapidity , and by the variable

 zh≡pTpjetT, (13)

where as in section 2.1 refers to the transverse momentum of the produced hadron. As we are working in the NJA, we consider collinear fragmentation of the hadron inside the jet. Thus, the observed hadron and the jet have the same rapidities, , since differences in rapidity are effects and hence suppressed in the NJA.

The factorized jet-plus-hadron cross section is written as

 dσH1H2→(jeth)XdpjetTdηjetdzh = 2pjetTS∑a,b,c∫1xminadxaxafH1a(xa,μF)∫1xminbdxbxbfH2b(xb,μF) (14) × ∫1zhdzpzpd^σ(jetc)ab(^s,pjetT,^η,μF,μ′F,μR,R,zp)vdvdwdzpDhc(zhzp,μ′F),

where , , and are as for the single-inclusive jet cross section, and where is the partonic analog of . In other words, the are the partonic cross sections for producing a final-state jet (subject to a specified jet algorithm), inside of which there is a parton with transverse momentum that fragments into the observed hadron. The argument of the corresponding fragmentation functions is fixed by and hence, using (13), is given by . Thus the new partonic cross sections are in convolution with the fragmentation functions. Note that all other variables and have the same definitions as in the single-inclusive jet case; see Eq. (9).

At lowest order, there is only one parton forming the jet, and this parton also is the one that fragments into the observed hadron, implying . The partonic cross sections hence have the perturbative expansions

 d^σ(jetc)abdvdwdzp=d^σc,(0)abdvδ(1−w)δ(1−zp)+αs(μR)πd^σ(jetc),(1)abdvdwdzp+O(α2s(μR)), (15)

with the same Born terms as in (6).

In order to derive the NLO partonic cross sections , we revisit Eq. (10). Since we now “observe” a parton in the final state (the fragmenting one), we must not sum over all possible final states, but rather consider only the contributions that contain parton :

 d^σcab−d^σc(d)ab−d^σc(e)ab+d^σcdab+d^σceab. (16)

However, for each term we now need to derive its proper dependence on before combining all terms. For the terms and this is trivial since for all of these terms parton alone produces the jet and also is the parton that fragments. As a result, all these terms simply acquire a factor . This becomes different for the pieces . Following [[21], [22], [23]], in the NJA we may write the NLO contribution to any as

 d^σcd,(1)abdvdw=αsπNab→K(v,w,ε)δ(1−w)∫10dzpz−εp(1−zp)−ε~P

where we have used dimensional regularization with space-time dimensions. Equation (17) is derived from the fact that the leading contributions in the NJA come from a parton splitting into partons and “almost” collinearly in the final state. We therefore have an underlying Born process (with some unobserved recoil final state ), whose -dimensional cross section is contained in the “normalization factor” , along with some trivial factors. The integrand then contains the -dimensional LO splitting functions , where the superscript “” indicates that the splitting function is strictly at , that is, without its contribution that is present when . The functions are defined in Eq. (A.3) in Appendix A. The argument of the splitting function is the fraction of the intermediate particle’s momentum (equal to the jet momentum) transferred in the splitting. In the NJA it therefore coincides with our partonic variable . In the second integral in (17) is the invariant mass of the jet. The explicit factor in the denominator represents the propagator of the splitting parton . The integral over the jet mass runs between zero and an upper limit , which in the NJA is formally taken to be relatively small. As indicated, depends on the algorithm chosen to define the jet. We have [[21], [22], [23]]

 Misplaced & (18)

To make the cross section differential in we now just need to drop the integration over in (17). We next expand the resulting expression in . The integration produces a collinear singularity in . It also contributes a factor at large which may be combined with the explicit factor in (17). In the presence of a diagonal splitting function in the integrand we hence arrive at a term , which may be expanded in to give a further pole in and “plus”-distributions in . The double poles arising in this way cancel against double poles in . The remaining single poles are removed by collinear factorization into the fragmentation function for parton . For non-diagonal splitting functions there are only single poles which are directly subtracted by factorization. We note that the original is in fact needed both for the cross section with parton fragmenting and also for the one where fragments. This is reflected in the fact that the -integral in (17) runs from to , while for the limit is never reached as long as . For parton fragmenting, however, we need to use which differs from only by a change of the splitting function. In case of a quark splitting into a quark and a gluon, this change is from for an observed quark to for an observed gluon. Because of one precisely recovers the old expression for the inclusive-jet cross sections when all final states are summed over. Likewise, if a gluon splits into a or , the relevant splitting functions , are by themselves symmetric under .

From this discussion, and combining with Eqs. (15),(16), we obtain to NLO in the NJA:

 d^σ(jetc)abdvdwdzp=⎡⎣d^σcabdvdw−d^σc(d)abdvdw−d^σc(e)abdvdw⎤⎦δ(1−zp)+d^σcdabdvdwdzp+d^σceabdvdwdzp. (19)

Computing and inserting all ingredients of this expression, we find that the cross section may be cast into a form that again makes use of the single-inclusive parton production cross sections , similar to the case of inclusive-jet production in Eq. (11):

where and are as given in (3), with and defined in terms of jet transverse momentum and rapidity. Furthermore, as in (11) we have . The (jet algorithm dependent) functions are new “semi-inclusive” jet functions that describe the production of a fragmenting parton inside a jet that results from a parton produced in the hard scattering. For the “transition” we find

 Kq→q(z,zp,λ=RpjetTμ′F,κ=RpjetTμ′′F,μR) = δ(1−z)δ(1−zp)+αs(μR)2π[ −δ(1−zp){2CF(1+z2)(log(1−z)1−z)++Pqq(z)log(λ2)+CF(1−z)} +δ(1−z){2CF(1+z2p)(log(1−zp)1−zp)++Pqq(zp)log(κ2)+CF(1−zp)+Ialgoqq(zp)}],

where is a function that depends on the jet algorithm. Since we will write our new jet functions in a more compact form below, we do not present the other functions here but collect them in Appendix B, along with the .

As indicated, carries dependence on two (final-state) factorization scales, and . The former is the same as we encountered in the case of single-inclusive jets in Eqs. (11),(2.2). It was originally introduced in the collinear factorization for the single-inclusive parton cross sections, but now has to cancel exactly between the and the . As in the case of single-inclusive jets, the cancelation of dependence on is just a result of the fact that we foremost define our observable by requiring a jet in the final state. In this sense, is simply an artifact of the way we organize the calculation and is not actually present in the final answer. The scale , on the other hand, arises because we now also require a hadron in the final state. Technically it arises when we subtract collinear singularities from the . The logarithms in are thus just the standard scale logarithms that compensate the evolution of the fragmentation functions at this order. We also note that there are two sum rules that connect the inclusive and the semi-inclusive jet functions [[1], [2]]:

 ∫10dzpzp[Kq→q(z,zp;λ,κ,μR)+Kq→g(z,zp;λ,κ,μR)] =Jq(z,λ,μR), ∫10dzpzp[Kg→g(z,zp;λ,κ,μR)+Kg→q(z,zp;λ,κ,μR)] =Jg(z,λ,μR). (22)

Both are fulfilled by our expressions. Furthermore, reproduces the quark splitting contributions to , i.e. the first two lines in Eq. (2.2).

We may actually go one step further and decompose the functions into products of jet functions that separate the dependence on and . We define two sets of functions:

 jq→q(z,λ,μR) ≡δ(1−z)−αs(μR)2π[2CF(1+z2)(log(1−z)1−z)++Pqq(z)log(λ2) +δ(1−z)Ialgoq+CF(1−z)], jq→g(z,λ,μR)</