Jet Shapes and Jet Algorithms in SCET
Abstract:
Jet shapes are weighted sums over the fourmomenta of the constituents of a jet and reveal details of its internal structure, potentially allowing discrimination of its partonic origin. In this work we make predictions for quark and gluon jet shape distributions in jet final states in collisions, defined with a cone or recombination algorithm, where we measure some jet shape observable on a subset of these jets. Using the framework of SoftCollinear Effective Theory, we prove a factorization theorem for jet shape distributions and demonstrate the consistent renormalizationgroup running of the functions in the factorization theorem for any number of measured and unmeasured jets, any number of quark and gluon jets, and any angular size of the jets, as long as is much smaller than the angular separation between jets. We calculate the jet and soft functions for angularity jet shapes to oneloop order () and resum a subset of the large logarithms of needed for nexttoleading logarithmic (NLL) accuracy for both cone and type jets. We compare our predictions for the resummed distribution of a quark or a gluon jet produced in a 3jet final state in annihilation to the output of a Monte Carlo event generator and find that the dependence on and is very similar.
arXiv:1001.0014
1 Introduction
1.1 Motivation and Objectives
Jets provide troves of information about physics within and beyond the Standard Model of particle physics. On the one hand, jets display the behavior of Quantum Chromodynamics (QCD) over a wide range of energy scales, from the energy of the hard scattering, through intermediate scales of branching and showering, to the lowest scale of hadronization. On the other hand, jets contain signatures of exotic physics when produced by the decays of heavy, stronglyinteracting particles such as top quarks or particles beyond the Standard Model.
Recently, several groups have explored strategies to probe jet substructure to distinguish jets produced by light partons in QCD from those produced by heavier particles [1, 2, 3, 4, 5, 6, 7, 8], and methods to “clean” jets of soft radiation to more easily identify their origin, such as “filtering” or “pruning” for jets from heavy particles [5, 9, 10] or “trimming” for jets from light partons [11]. Another type of strategy, explored in [12], to probe jet substructure is the use of jet shapes, which are modifications of event shapes [13] such as thrust. Jet shapes are continuous variables constructed by taking a weighted sum over the fourmomenta of all particles constituting a jet. Different choices of weighting functions produce different jet shapes, and can be designed to probe regions closer to or further from the jet axis with greater sensitivity.^{1}^{1}1The original “jet shape,” to which the name properly belongs, is the quantity , the fraction of the total energy of a jet of radius that is contained in a subjet of radius [14, 15, 16] . This observable falls into the larger class of jet shapes we have described here and for which we have hijacked the name. While such jet shapes may integrate over some of the detailed substructure for which some other methods search, they are better suited to analytical calculation and understanding from the underlying theory of QCD.
In this paper, we consider measuring the shape of one or more jets in an collision at centerofmass energy producing jets with an angular size according to a cone or recombination jet algorithm, with an energy cut on the radiation allowed outside of jets. We use this exclusive characterization of an jet final state looking forward to extension of our results to a hadron collider environment, where such a final state definition is more typical. For the jet shape observable we choose the angularity of a jet, defined by (cf. [12, 17]),
(1) 
where is a parameter taking values (for IR safety, although factorizability will require ), the sum is over all particles in the jet, is the jet energy, is the transverse momentum relative to the jet direction, and is the (pseudo)rapidity measured from the jet direction. The jet is defined by a jet algorithm, such as a cone algorithm, the details of which we will discuss below. We complete the calculation for the jet shape for jets defined by cone or recombination algorithms, but our logic and methods could be applied to a wider spectrum of jet shapes and jet algorithms. We have organized our results in such a way that the pieces independent of the choice of jet shape and dependent only on the jet algorithm are easily identifiable, requiring recalculation only of the observabledependent pieces to extend our results to other choices of jet shapes.
Reliable theoretical prediction of jet observables in the presence of jet algorithms is made challenging by the presence of many scales. Logarithms of ratios of these scales can become large and spoil the behavior of perturbative expansions predicting these quantities. These scales are determined by the jet energy , the cut on the angular size of a jet , the measured value of the jet shape such as , and any other cut or selection parameters introduced by the jet algorithm.
Precisely this separation of scales, however, allows us to take advantage of the powerful tools of factorization and effective field theory. Factorization separates the calculation of a hard scattering cross section into hard, jet and soft functions each depending only on physics at a single scale [18, 19]. Renormalization group (RG) evolution of these functions between scales resums logarithms of these scales to all orders in , with the logarithmic accuracy determined by the order to which the anomalous dimensions in the running are calculated [20]. Effective field theory organizes these concepts and tools into a conceptually simple framework unifying many ingredients going into traditional methods, such as power counting, gauge invariance, and resummation through RG evolution. The rules of effective theory facilitate proofs of factorization and achievement of logarithmic resummation at leading order in the power counting and make straightforward the improvement of results orderbyorder in power counting and logarithmic accuracy of resummation.
1.2 SoftCollinear Effective Theory and Factorization
SoftCollinear Effective Theory (SCET) [21, 22, 23, 24] has been successfully applied to the analysis of many hard scattering cross sections [25] including the production of jets. SCET is constructed by integrating out of QCD all degrees of freedom except those collinear to a lightlike direction and those which are soft, that is, have much lower energy than the energy of the hard scattering or of the jets. Using this formalism, the factorization and calculation of twojet cross sections and event shape distributions in SCET were developed in [26, 27, 28, 29]. Later, these techniques were extended to the factorization of jet cross sections and observables using jet algorithms in [30]. Calculations in SCET of twojet rates using jet algorithms have been performed in [27, 31], and more recently in [32]. Calculations of cross sections with more than two jet directions have been given in [33, 34, 35].
Building on many of the ideas in these previous studies, in this paper, we will demonstrate a factorization theorem for jet shape distributions in jet events,
(2)  
where the jets have threemomenta , and of the jets’ shapes are measured. is the Born crosssection, is a hard function dependent on the directions and energies of the jets, is the jet function for a jet whose shape is measured to be , is the jet function for a jet with size whose shape is not measured, and is the soft function connecting all jets, dependent on all jets’ shapes , sizes , and total energy that is left outside of all jets. The symbol “” stands for a set of convolution integrals in the variables between the measured jet functions and the soft function. All terms in the factorization theorem depend on the factorization scale .
SCET is typically constructed as a power expansion in a small parameter formed by the ratio of soft to collinear or collinear to hard scales, determined by the kinematics of the process under study. is roughly the typical transverse momentum of the constituent of a jet (relative to the jet direction) divided by the jet energy . This is set either by the measured value of the jet shape for a measured jet or the algorithm measure for an unmeasured jet. Thus we encounter in this work the new twist that the size of may be different for different jets. We will comment on further implications of this in subsequent sections. Still, in each separate collinear sector, the momentum of collinear modes in the lightcone direction in SCET is separated into a large “label” momentum containing and components and a “residual” component of , the same size as soft momenta. Effective theory fields have dynamical momenta only of this soft or residual scale. This fact, along with the fact that soft quarks and soft gluons can be shown to decouple from collinear modes at the level of the Lagrangian [24], makes possible the factorization of a jet shape distribution into hard, jet, and soft functions depending only on the dynamics at those respective scales.
In using SCET for jets in multiple directions and using jet algorithms to define the jets, we will encounter the need for several additional criteria to ensure the validity of the jet factorization theorem.

First, to ensure that the algorithm does not group finalstate particles into fewer than jets, the jets must be “well separated.” This allows us to use as the effective theory Lagrangian a sum of copies of the collinear part of the SCET Lagrangian for a single direction and a soft part, and to construct a basis of jet operators built from fields from each of these sectors to produce the final state. Our calculations will reveal the precise quantitative condition that jets must satisfy to be “well separated”.

Second, to ensure that the jet algorithm does not find more than jets, we place an energy cut on the total energy outside of the observed jets. We will take this energy to scale as a soft momentum so that we will be able to identify the total energy of each jet with the “label” momentum on the SCET collinear jet field producing the jet. Corrections to this identification are subleading in the SCET power counting.

Third (and related to the above two), we will assume that the jet restriction on the final state can itself be factorized into a product of 1jet restrictions, one in each collinear sector, and a jet restriction in the soft sector. We represent the energetic particles in the th jet by collinear fields in the SCET Lagrangian in the collinear sector and soft particles everywhere with fields in the soft part of the Lagrangian. We then stipulate that the jet algorithm acting on states in the collinear sector find exactly one jet in that sector, and when acting on the soft final state find no additional jet in that sector.

Fourth, the way in which a jet algorithm combines particles in the process of finding a jet must respect the order of steps envisioned by factorization. In particular, factorization requires that the jet directions and energies be determined by the collinear particles alone, so that the soft function knows only about the directions and colors of the jets, not the details of any collinear recombinations. Ideally, all energetic collinear particles should be recombined first, with soft particles within a radius of the jet axis being recombined into the jet only afterwards. Jet algorithms in use at experiments do not have this precise behavior, but we will discuss in Sec. 3.4 the extent to which common algorithms meet this requirement and estimate the size of the power corrections due to their failure to do so. In general, we will find that for sufficiently large , infraredsafe cone algorithms and type recombination algorithms satisfy the requirements of factorizability, with anti allowing smaller values of than .
After enforcing the above requirements, a key test of the consistency of Eq. (2) will be the independence of the physical cross section on the factorization scale . This requires the anomalous dimensions of the hard, jet, and soft functions to sum to zero,
(3) 
It seems highly nontrivial that this condition would be satisfied for any number, size, and flavors of jets (and that the soft anomalous dimension be independent of ), but we will demonstrate that it does hold at , up to corrections of which violate Eq. (3), where is a measure of the separation between jets. In particular, for a pair of jets, , with 3vector directions separated by a polar angle , the separation is given by
(4) 
Now define (no indices) as the minimum of over all jet pairs. This quantifies the qualitative condition of jets being wellseparated, , that is required to justify the factorization theorem Eq. (2). The factorization theorem is valid up to corrections of in the SCET power expansion parameter and corrections of in the separation parameter. As an example of the magnitude of , for three jets in a MercedesBenz configuration ( for all pairs of jets), for and for , so these corrections are indeed small. More generally, for nonoverlapping jets, , we have .
Notice that for backtoback jets (), . Thus, for all cases previously considered in the literature, the jets are infinitely separated according to this measure, and no additional criterion regarding jet separation is required for consistency of the factorization and running. A key insight of our work is that for an jet crosssection described by Eq. (2), the factorization theorem receives corrections not only in the usual SCET power counting parameter , but also corrections due to jet separation beginning at .
1.3 Power Corrections to Factorized Jet Shape Distributions
As always, there are power corrections to the factorization theorem which we must ensure are small. One class of power corrections arises from approximating the jet axis of the measured jet with the collinear direction , which labels that jet in the SCET Lagrangian. This direction is the direction of the parent parton initiating the jet. The jet observable must be such that the difference between the parent parton direction and the jet axis identified by the algorithm makes a subleading correction to the calculated value of the jet observable. In the context of angularity event shapes, such corrections were estimated in [17, 29] and found to be negligible for , and we find the same condition for jet shapes.
In the presence of algorithms, however, there are additional power corrections due to the difference in the soft particles that are included or excluded in a jet by the actual algorithm and in its approximated form in the factorization theorem. We study the effect of this difference on the measurement of jet shapes, and find that for sufficiently large the power corrections due to the action of the algorithm on soft particles remain small enough not to spoil the factorization for infraredsafe cone and type recombination algorithms. Algorithmrelated power corrections to jet momenta were studied more quantitatively in [36], and their estimated dependence is consistent with our observations.
We do not address in this work the issue of power corrections to jet shapes due to hadronization. Event shape distributions are known to receive power corrections of the order , enhanced in the endpoint region but suppressed by large energy. The endpoints of our jet shape distribution near , therefore, will have to be corrected by a nonperturbative shape function. Such functions have been constructed for event shapes in [37, 38]. The shift in the first moment of event shape distributions induced by these shape functions was postulated to take a universal form in [39, 40] based on the behavior of single soft gluon emission, and the universality was proven to all orders in soft gluon emission at leading order in the SCET power counting in [41, 42]. This universality relied on the boost invariance of the soft function describing soft gluon radiation from two backtoback collinear jets. The extent to which such universality may survive for jet shapes with multiple jets in arbitrary directions is an open question that must be addressed in order to construct appropriate soft shape function models to deal adequately with the power corrections to jet shapes from hadronization. Nonperturbative power corrections to jet observables from hadronization and the underlying event in hadron collisions were also studied in [36], and hadronization corrections were found to scale like . In this work, we focus only on the perturbative calculation and resummation of large logarithms of jet shapes, and leave inclusion of nonperturbative power corrections for future work.
1.4 Resummation and Logarithmic Accuracy
Knowing the anomalous dimensions of the hard, jet, and soft functions in the factorization theorem allows us to resum logarithms of ratios of the hard, jet, and soft scales. We take this opportunity to explain the order of accuracy to which we are able to resum these logarithms. For an event shape distribution (i.e. Eq. (2) with two jets and integrated against ), the accuracy of logarithmic resummation [43] is typically characterized by counting logs in the exponent of the “radiator,”
(5) 
where they appear in the form with . At leadinglogarithmic (LL) accuracy all the terms with are summed; nexttoleadinglogarithmic (NLL) accuracy sums also the terms, and so on. In more traditional methods in QCD, event shapes that have been resummed include NLL resummation of thrust in [43, 44], jet masses in [43, 45, 46, 47], jet broadening in [48, 49], the parameter in [50], and angularities in [17]. Resummation of an event shape distribution using the modern SCET method was first illustrated with the thrust distribution to LL accuracy in [51]. Heavy quark jet mass distributions were resummed in SCET to NLL accuracy as part of a proposed method to extract the top quark mass in [52]. The NLL resummed thrust distribution in SCET was compared to LEP data to extract a value for the strong coupling to high precision in [53]. Angularities were resummed to NLL accuracy in SCET in [54] directly in space instead of in moment space as in [17].
Summation of logarithms in effective field theory is achieved by RG evolution. In the factorized radiator of the thrust distribution Eq. (5), one finds that the hard function contains logarithms of , the jet functions contain logarithms of , and the soft function contains logarithms of . Thus, evaluating these functions respectively at the hard scale , jet scale and soft scale eliminates large logarithms in each function. They can then be RGevolved to the common factorization scale after calculating their anomalous dimensions. The solutions of the RG evolution equations are of the form that logarithms of are resummed to all orders in to a logarithmic accuracy determined by the order in to which the anomalous dimensions and hard/jet/soft functions are known. This underlying hierarchy of scales is illustrated Fig. 1 [in this case, with only one (measured) jet scale and soft scale and ] along with a table that lists the order in to which various quantities must be known in order to achieve a given NLL accuracy in the exponent of the radiator Eq. (5). The power of the EFT framework is to organize of the logs of arising in Eq. (5) into those that arise from ratios of the jet to the hard scale and those that arise from ratios of the soft to the hard scale, which then allows RG evolution to resum them.
For the multijet shape distribution in Eq. (2), the strategy to sum logs is the same, but is complicated by the presence of additional scales. This also makes trickier the classification of logarithmic accuracy into the standard NLL scheme. Our aim will be to sum as many logs of the jet shapes as possible, while not worrying about any others. For instance, phase space cuts induce logs of and (where is a typical hard jet energy), and the presence of multiple jets induces logs of jet separations or ratios of jet energies . We will not aim to sum these types logs systematically in this paper, only those of (though we sum subsets of the others incidentally). In particular, resumming the phase space logs of or is complicated by how the phase space cuts act orderbyorder in perturbation theory^{2}^{2}2The JADE algorithm is one wellknown example in which resummability even of leading logarithms of the jet mass cut is spoiled by the differences in the jet phase space at different orders in perturbation theory [55]. Another example that will not work is using a type algorithm with randomly chosen for each recombination. This is clearly such that resummation of logarithms of cannot be achieved., and the fact that a simple angular cut is less restrictive than a small jet mass or angularity on how collimated a jet must be. That is, an angular cut allows particles in a jet to be anywhere within an angle of the jet axis regardless of their energy, while a small jet mass or angularity forces harder particles to be closer to the jet axis. The former allows hard particles to lie along the edges of a jet, and soft radiation from such configurations that escapes the jets can lead to logs of that are not captured in our treatment. These are not issues we solve in this paper, in which we focus on resumming logs of jet shapes . (Some exploration of phase space logarithms in SCET was carried out in [31, 32].)
A way to understand how we sum logs and which ones we capture is presented in Fig. 1. The factorization theorem Eq. (2) organizes logs in the multijet cross section into those in the hard function, those in measured jet functions, those in unmeasured jet functions, and those in the soft function, much like for the simple thrust distribution. The difference is the presence now of multiple jet and soft scales. Logarithms in jet functions can still be minimized by choices of individual jet scales, for a jet whose shape is measured, and for a jet whose shape is not measured but has a radius . Thus logs arising from ratios of these scales to the hard scale can be summed completely to a desired NLL order. The complication is in the soft function. The soft function is sensitive to soft radiation into measured and unmeasured jets and outside of all jets. As we will see by explicit calculation, this induces logs of from radiation into measured jets, and logs of and from radiation from unmeasured jets cut off by the energy . In addition, though not illustrated in Fig. 1, there can be logs of multiple jet shapes to one another, . No single choice of a soft scale will minimize all of these logs.
In the present work, we will start with the simple strategy of choosing a single soft scale for a jet whose shape we are measuring and logs of which we aim to resum. We will calculate hard/jet/soft functions and anomalous dimensions corresponding to “NLL” accuracy listed in Fig. 1. By this we do not mean all potentially large logs in Eq. (2) are resummed to NLL, but only those logs of ratios of a jet scale to the hard scale or of the (common) soft scale to the hard scale. We do not attempt to sum logs of ratios of soft scales to one another completely to NLL accuracy (which can contain ). In the case that all jets’ shapes are measured and are similar to one another, , our resummation of large logs of would be complete to NLL accuracy.
We will nevertheless venture to propose a framework to “refactorize” the soft function into further pieces dependent on only a single soft scale at a time and perform additional RG running between these scales to resum the additional logarithms, and will implement it at the level of the soft functions we calculate. However, one cannot really address mixed logarithms such as that arise for multiple jets until , the first order at which two soft gluons can probe two different physical regions. This lies beyond the scope of the present work. (We note, however, that our implementation of refactorization using the oneloop soft function does already seem to tame logarithmic dependence on in our numerical studies of jet shape distributions.)
These issues are related to some types of “nonglobal” logarithms described by [46, 56, 57, 58] that spoil the simple characterization of NLL accuracy. In [59] these were identified as nexttoleading logs of and (when ) that appear at in jet shape distributions. These authors organized the radiator for a single jet shape distribution into a “global” and “nonglobal” part [58, 59],
(6) 
In this language, the calculations we undertake in this paper resum logs in the global part to NLL accuracy but not in the nonglobal part. The first argument in is related to ratios of soft scales illustrated in Fig. 1, and the second argument arises when there are unmeasured jets. In the case that all jets are measured, , and , the nonglobal logs vanish.
While summing all global and nonglobal logs to at least NLL accuracy will be important for precision jet phenomenology, what we contribute in this paper are key developments and calculations necessary to resum even global logs of jet shapes for jets defined with algorithms. We also believe the effective theory approach and the idea of refactorizing the soft function will help us understand and resum many types of nonglobal logarithms.
1.5 Detailed Outline of This Work
In this paper, we will formulate and prove a factorization theorem for distributions in the jet shape variables we introduced above, calculate the jet and soft functions appearing in the factorization theorem to in SCET, and use the renormalization group evolution of these functions to sum global logs of to NLL accuracy. We consider jets (defined with a cone or algorithm) produced in an collision, with of the jets’ shapes (angularities) being measured. The key formal result is our demonstration of Eq. (3), the consistency of the anomalous dimensions of hard, jet and soft functions to for any number of total jets, any numbers of quark and gluon jets, any number of these jets whose shapes are measured, and any value of the distance measure in cone or type algorithms (as long as ). These results lead to accurate predictions for the shape of the distribution near the peak, but not necessarily the endpoints for very small (where hadronization corrections dominate) and very large (where fixedorder NLO QCD corrections take over, which are not yet calculated and not captured by NLL resummation).^{3}^{3}3Jet shapes were also studied in the QCD factorization approach in [60]. In that work QCD jet functions for quark and gluon jets defined with an algorithm and whose jet masses are measured were calculated to . The jet mass corresponds to for , (). A fixedorder QCD jet function as defined in [60] is given by the convolution of our fixedorder SCET jet function and soft function for a measured jet away from .
In Sec. 2 we describe in detail the jet shapes and jet algorithms that we use. We describe features of an “ideal” jet algorithm that would respect exactly the order of operations envisioned in the factorization theorem derived in SCET, and the extent to which cone and recombination algorithms actually in use resemble this idealization.
In Sec. 3, using the tools of SCET, we will derive in detail a factorization theorem for exclusive jet production where we measure the angularity jet shape of one of the jets, and then perform the straightforward extension to jet production with measured jets. We will give a review of the necessary technical details of SCET in Sec. 3.1. In the process of justifying the factorization theorem, we identify the new requirements listed above on jet final states and jet algorithms that must be satisfied for factorization to hold. In Sec. 3.4 we explore in some detail the power corrections to the factorization theorem due to soft radiation and the action of jet algorithms that cause tension with these requirements, and argue that for sufficiently large in infraredsafe cone and recombination algorithms, these corrections are sufficiently small.
Next we calculate to the jet and soft functions corresponding to cone or type jets, with jets’ shapes measured.
In Sec. 4 we calculate the jet functions for measured quark jets, , unmeasured quark jets, , measured gluon jets, , and unmeasured gluon jets, , where is the label momentum of the collinear jet field in each jet function. We find that in collinear sectors for measured jets, the collinear scale (and thus the SCET power counting parameter in that sector ) is given by , and in unmeasured jet sectors, . In studying power corrections, however, as mentioned above, we find that must be parametrically larger than . So, in collinear sectors for measured jets, is set by the shape with , while in unmeasured jet sectors, . Thus one should understand to be significantly less than 1 but much larger than any jet shape .
In Sec. 5 we calculate the soft function. To do this, we split the soft function into several contributions from different parts of phase space in order to facilitate the calculation and elucidate its intuitive structure. We find it most convenient to split the soft function into an observableindependent part that arises from soft emission out of the jets, , and a part that depends on our choice of angularities as the observable that arises from soft emission into measured jet , . is hence sensitive to the scale while is sensitive to the scale .
In Sec. 6, having calculated all the jet and soft function contributions to , we extract the anomalous dimensions and perform renormalizationgroup (RG) evolution. We find the hard anomalous dimension from existing results in the literature. The hard, jet, and soft anomalous dimensions have to satisfy the consistency condition Eq. (3) in order for the physical cross section to be independent of the arbitrary factorization scale . Our calculations reveal that, as long as the jet separation parameter Eq. (4) between all pairs of jets is much larger than 1, the condition is satisfied.
Even after requiring , the satisfaction of the consistency condition is nontrivial. The hard function knows only about the direction of each jet and the jet function knows only the jet size ; the soft function knows about both. Furthermore, it is not sufficient only to include regions of phase space where radiation enters the measured jets. We learn from our results in this Section that it is crucial to include soft radiation outside of all jets with an upper energy cutoff of . Only after including all of these contributions from the various parts of phase space do the jet, hard, and soft anomalous dimensions cancel and we arrive at a consistent factorization theorem.
We conclude Sec. 6 by proposing in Sec. 6.4 a strategy to sum logs due to a hierarchy of scales in the soft function, by “refactorizing” it into multiple pieces, each sensitive to a single scale, as suggested by the discussion surrounding Fig. 1. Our current implementation of this procedure does tame the logarithmic dependence of jet shape distributions on the ratio in our numerical studies, but we leave for further development the resummation of all “nonglobal” logs of ratios of multiple soft scales that begin at NLL and .
To help the reader find the results of the calculations in Sec. 4 through Sec. 6 just outlined, Table 1 provides a summary with equation numbers.
Category  Contribution  Symbol  Location 
measured quark jet function  Eq. (4.2.1)  
unmeas. quark jet function  Eq. (60)  
measured gluon jet function  Eq. (4.3.1)  
unmeas. gluon jet function  Eq. (69)  
NLO contributions  summary of divergent  —  Table 2 
before resummation:  parts of soft func. (any )  
total universal  Eq. (89)  
soft func. (large )  
total measured  Eq. (5.3)  
soft func. (large )  
anomalous dimensions:  —  —  Table 3 
measured jet function  Eq. (134)  
NLO contributions  measured soft function  Eq. (134)  
after resummation:  unmeas. jet function  Eq. (135a)  
universal soft function  Eq. (135)  
Total NLL Distribution:  —  —  Eq. (132) 
In Sec. 7 we compare our resummed perturbative predictions for jet shape distributions to the output of a Monte Carlo event generator. We test both the accuracy of these predictions and assess the extent to which hadronization corrections affect jet shapes. We will illustrate our results in the case of , with the jets constrained to be in a configuration where each has equal energy and are maximally separated. In both the effective theory and Monte Carlo, we can take the jets to have been produced by an underlying hard process . After placing cuts on jets to ensure each parton corresponds to a nearby jet, we measure the angularity jet shape of one of the jets. We compare our resummed theoretical predictions with the Monte Carlo output for quark and gluon jet shapes with various values of and . We find that the dependencies on and of the shapes of the distribution and the peak value of agree well between the theory and Monte Carlo, with small but noticeable corrections due to hadronization. We can estimate these corrections by comparing output with hadronization turned on or off in Monte Carlo.
In Sec. 8, we give our conclusions and outlook. We also collect a number of technical details and results for finite pieces of jet and soft functions in the Appendices.
Our work is, to our knowledge, the first achieving factorization and resummation of a jet observable distribution in an exclusive jet final state defined by a nonhemisphere jet algorithm.^{4}^{4}4Dijet cross sections for cone jets were factorized and resummed in [61]. Having demonstrated the consistency of this factorization for any number of quark and gluon jets, measured and unmeasured jets, and phase space cuts in cone and type algorithms, and having constructed a framework to resum logarithms of jet shapes in the presence of these phase space cuts, we hope to have provided a starting point for future precision calculations of many jet observables both in and hadronhadron collisions. The case of collisions will require a number of modifications, including turning two of our outgoing jet functions into incoming “beam functions” introduced in [62]. We leave this generalization for future work.
2 Jet Shapes and Jet Algorithms
2.1 Jet Shapes
Event shapes, such as thrust, characterize events based on the distribution of energy in the final state by assigning differing weights to events with differing energy distributions. Events that are twojet like, with two very collimated backtoback jets, produce values of the observable at one end of the distribution, while spherical events with a broad energy distribution produce values of the observable at the other end of the distribution. While event shapes can quantify the global geometry of events, they are not sensitive to the detailed structure of jets in the event. Two classes of events may have similar values of an event shape but characteristically different structure in terms of number of jets and the energy distribution within those jets.
Jet shapes, which are event shapelike observables applied to single jets, are an effective tool to measure the structure of individual jets. These observables can be used to not only quantify QCDlike events, but study more complex, nonQCD topologies, as illustrated for light quark vs. top quark and jets in [12, 60]. Broad jets, with wideangle energy depositions, and very collimated jets, with a narrow energy profile, take on distinct values for jet shape observables. In this work, we consider the example of the class of jet shapes called angularities, defined in Eq. (1) and denoted . Every value of corresponds to a different jet shape. As decreases, the angularity weights particles at the periphery of the jet more, and is therefore more sensitive to wideangle radiation. Simultaneous measurements of the angularity of a jet for different values of can be an additional probe of the structure of the jet.
2.2 Jet Algorithms
A key component of the distribution of jet shapes is the jet algorithm, which builds jets from the final state particles in an event. (We are using the term “particle” generically here to refer to actual individual tracks, to cells/towers in a calorimeter, to partons in a perturbative calculation, and to combinations of these objects within a jet.) Since the underlying jet is not intrinsically well defined, there is no unique jet algorithm and a wide variety of jet algorithms have been proposed and implemented in experiments. The details of each algorithm are motivated by particular properties desired of jets, and different algorithms have different strengths and weaknesses. In this work we will calculate angularity distributions for jets coming from a variety of algorithms. Because we calculate (only) at nexttoleading order, there are at most 2 particles in a jet, and jet algorithms that implement the same phase space cuts at NLO simplify to the same algorithm. At this order the two standard classes of algorithms, cone algorithms and recombination algorithms, each simplify to a generic jet algorithm at NLO. At NLO the cone algorithms place a constraint on the separation between each particle and the jet axis, while the recombination algorithms place a constraint on the separation between the two particles.
Cone algorithms build jets by grouping particles within a fixed geometric shape, the cone, and finding “stable” cones. A cone contains all of the particles within an angle of the cone axis, and the angular parameter sets the size of the jet. In found jets (stable cones), the direction of the total threemomentum of particles in the cone equals the cone (jet) axis. Different cone algorithms employ different methods to find stable cones and deal with differently the “split/merge” problem of overlapping stable cones. The SISCone algorithm [63] is a modern implementation of the cone algorithm that finds all stable cones and is free of infrared unsafety issues. In the nexttoleading order calculation we perform, there are at most two particles in a jet, and we only consider configurations where all jets are wellseparated. Therefore, it is straightforward to find all stable cones, there are no issues with overlapping stable cones, and the phase space cuts of all cone algorithms are equivalent. This simplifies all standard cone algorithms to a generic conetype algorithm, in which each particle is constrained to be within an angle of the jet axis. For a twoparticle jet, if we label the particles and and the jet axis , then the conelike constraints for the two particles to be in a jet are
(7) 
This defines our conetype algorithm.
Recombination algorithms build jets by recursively merging pairs of particles. Two distance metrics, defined by the algorithm, determine when particles are merged and when jets are formed. A pairwise metric (the recombination metric) defines a distance between pairs of particles, and a single particle metric (the beam, or promotion, metric) defines a distance for each single particle. Using these metrics, a recombination algorithm builds jets with the following procedure:^{5}^{5}5This defines an inclusive recombination algorithm more typically applied to hadronhadron colliders. We are applying it here to the simpler case of collisions in order to facilitate the eventual transition to LHC studies. Exclusive recombination algorithms, more typical of collisions, are described along with other jet algorithms in [64] and their description in SCET is given in [32].

Begin with a list of particles.

Find the smallest distance for all pairs of particles (using ) and all single particles (using ).

If the smallest distance is from a pair, merge those particles together by adding their four momenta. Replace the pair in with the new particle.

If the smallest distance is from a single particle, promote that particle to a jet and remove it from .

Loop back to step 1 until all particles have been merged into jets.
The , CambridgeAachen, and anti algorithms are common recombination algorithms, and their distance metrics are part of a general class of recombination algorithms. For colliders, a class of recombination algorithms can be defined by the parameter :
(8) 
where for , for CambridgeAachen, and for the anti algorithm. The parameter sets the maximum angle between two particles for a single recombination.^{6}^{6}6We use for both cone and algorithms for ease of notation. For , this parameter is sometimes referred to as . We emphasize that having the same size for different algorithms does not in general guarantee the same sized jets. In the multijet configurations we consider the jets are separated by an angle larger than , so that only the pairwise metric is relevant for describing the phase space constraints for particles in each jet. For a twoparticle (NLO) jet, the only phase space constraint imposed by this class of recombination algorithms is that the two particles be separated by an angle less than :
(9) 
This defines a generic recombination algorithm suitable for our calculation. We will denote this as the type algorithm.
The configurations with two widely separated energetic particles best distinguish conetype jets from type jets at NLO. For instance, the case where the two energetic particles are at opposite edges of a cone jet (at an angle apart) is not a single jet. However, it is important to note that these configurations will not be accurately described in this SCET calculation for , as such configurations are power suppressed in our description of jets. Our concern is in accurately describing the configurations with narrow jets (small ), and not the wide angle configurations above.
Because jets are reliable degrees of freedom and provide a useful description of an event when they have large energy, in the description of an event we impose a cut on the minimum energy of jets. An jet event, therefore, is one where jets have energy larger than the cutoff , with any number of jets having energy less than the cutoff. In our calculation, we impose the same constraint: any jet with energy less than is not considered when we count the number of jets in the final state. This imposes phase space cuts: for a gluon radiated outside of all jets in the event, that gluon is required to have energy to maintain the same number of jets in the event. The proper division of phase space in calculating the jet and soft functions is a key part of the discussion below, and careful treatment of the phase space cuts is needed.
2.3 Do Jet Algorithms Respect Factorization?
The factorization theorem places specific requirements on the structure of jet algorithms used to describe events. As in Eq. (2), the factorization theorem divides the cross section into separately calculable hard, jet, and soft functions. The hard function depends only on the configuration of jets, while the jet and soft functions describe the degrees of freedom in each jet in terms of the observable . While the soft function is global, the jet function depends only on the collinear degrees of freedom in a single jet. The limited dependence of the hard and jet functions implies constraints on the jet algorithm.
Because jets are built from the long distance degrees of freedom arising from evolution of energetic partons to lower energies, the configuration of jets in an event depends on dynamics across all energy scales. This naively breaks factorization in SCET, since the configuration of jets in the hard function would depend on dynamics at low energy in the soft function. However, we can describe a jet algorithm that respects factorization, and in Sec. 3.4 we will parameterize the power corrections that arise from various algorithms.
The primary constraint on the jet algorithm in order to satisfy the factorization theorem is that the phase space cuts on the collinear particles in the jet are determined only by the collinear degrees of freedom. This essentially ensures that the jet functions are independent of dynamics in the soft function. Correspondingly the soft function can only know about the jet directions and their color representations. The direction of the jet is naturally set by the collinear particles, as soft particles have energy parametrically lower than the collinear ones and change the jet direction by a power suppressed amount. The further restriction that the phase space cuts on the collinear degrees of freedom are independent of the soft degrees of freedom places a strong constraint on the action of the jet algorithm. Cone algorithms already implement this constraint: the jet boundary (the cone) is determined by the location of the jet axis, which is the direction of the sum of all collinear particles up to a power correction. Recombination algorithms, however, are constrained by the factorization theorem to operate in a specific way: all collinear particles must be recombined before soft particles. As discussed in Sec. 3.4, commonly used algorithms obey this constraint up to power corrections in the observable for measured jets. Of particular note is the antikT algorithm, which exhibits behavior very close to what is required by the factorization theorem (similarly to the way cone algorithms behave).
3 Factorization of Jet Shape Distributions in to Jets
In this Section we formulate a factorization theorem for jet shape distributions in annihilation to jets. All the formal aspects we need to describe an jet cross section appear already in the 3jet cross section, so we will give explicit details only for that case. We will use the framework of SoftCollinear Effective Theory (SCET), developed in [21, 22, 23, 24], to formulate the factorization theorem. We begin with a basic review of the relevant aspects of the effective theory.
3.1 Overview of SCET
SCET is the effective field theory for QCD with all degrees of freedom integrated out, other than those traveling with large energy but small virtuality along a lightlike trajectory , and those with small, or soft, momenta in all components. A particularly useful set of coordinates is lightcone coordinates, which uses lightlike directions and , with and . In Minkowski coordinates, we take and , corresponding to collinear particles moving in the direction. A generic fourvector can be decomposed into components
(10) 
In terms of these components, , collinear and soft momenta scale with some small parameter as
(11) 
where is a large energy scale, for example, the centerofmass energy in an collision. is then the ratio of the typical transverse momentum of the constituents of the jet to the total jet energy. Quark and gluon fields in QCD are divided into collinear and soft effective theory fields with these respective momentum scalings:
(12) 
We factor out a phase containing the largest components of the collinear momentum from the fields . Defining the “label” momentum , where contains the part of the large lightcone component of the collinear momentum , and the transverse component, we can partition the collinear fields into their labeled components,
(13) 
The sums are over a discrete set of label momenta into which momentum space is partitioned. The bin is omitted to avoid doublecounting the soft mode in Eq. (12) [65]. The labeled fields now have spacetime fluctuations in which are conjugate to “residual” momenta of the order , describing remaining fluctuations within each labeled momentum partition [23, 65]. It will be convenient to define label operators which pick out just the label components of momentum of a collinear field:
(14) 
Ordinary derivatives acting on effective theory fields are of order .
The final step to construct the effective theory fields is to isolate the two large components of the Dirac spinor for a fermion with lightlike momentum along . The large components and the small can be separated by the projections
(15) 
and we have . One can show, substituting these definitions into the QCD Lagrangian, that the fields have an effective mass of order and can be integrated out of the theory. The effective theory Lagrangian at leading order in is [22, 23, 24]
(16) 
where the collinear quark Lagrangian is
(17) 
where is the Wilson line of collinear gluons,
(18) 
the collinear gluon Lagrangian is
(19) 
where is the collinear ghost field and the gaugefixing parameter; and the soft Lagrangian is
(20) 
which is identical to the form of the full QCD Lagrangian (the usual gaugefixing terms are implicit). In the collinear Lagrangians, we have defined several covariant derivative operators,
(21) 
In addition, there is an implicit sum over the label momenta of each collinear field and the requirement that the total label momentum of each term in the Lagrangian be zero.
Note the soft quarks do not couple to collinear particles at leading order in . Meanwhile, the coupling of the soft gluon field to a collinear field is in the component only, according to Eqs. (17) and (19), which makes possible the decoupling of such interactions through a field redefinition of the soft gluon field given in [24]. We will utilize this softcollinear decoupling to simplify the proof of factorization below.
The SCET Lagrangian Eq. (16) may be extended to include collinear particles in more than one direction [25]. One adds multiple copies of the collinear quark and gluon Lagrangians Eqs. (17) and (19) together. The collinear fields in each direction constitute their own independent set of quark and gluon fields, and are governed in principle by different expansion parameters associated with the transverse momentum of each jet, set either by the angular cut in the jet algorithm or by the measured value of the jet shape . Each collinear sector may be paired with its own associated soft field with momentum of order with the appropriate . For the purposes of keeping the notation tractable while proving the factorization theorem in this section, we will for simplicity take all ’s to be the same, with a single soft gluon field coupling to collinear modes in all sectors. In Sec. 6.4, we will discuss how to “refactorize” the soft function further into separate soft functions each depending only on one of the various possible soft scales.
The effective theory containing collinear sectors and the soft sector is appropriate to describe QCD processes with stronglyinteracting particles collimated in wellseparated directions. Thus, in addition to the power counting in the small parameter within each sector, guaranteeing that the particles in each direction are well collimated, we will find in calculating an jet cross section the need for another parameter that guarantees that the different directions are well separated. This latter condition requires , where is defined in Eq. (4).^{7}^{7}7This condition is a consequence of our insistence on using operators with exactly directions to create the final state. We could move away from the large limit and account for corrections to it by using a basis of operators with arbitrary numbers of jets and properly accounting for the regions of overlap between an jet operator and jet operators. This is outside the scope of the present work, where we limit ourselves to kinematics well described by an jet operator, and thus, limit ourselves to the large limit.
3.2 Jet Shape Distribution in
Consider a 3jet cross section differential in the jet 3momenta , where we measure the shape of one of the jets, which we will call jet 1. The full theory cross section for at centerofmass energy is
(22) 
where the is the jet algorithm acting on final state , and is the number of jets identified by the algorithm [30]. is the 3momentum of jet , and is also a function of the output of the jet algorithm . is the leptonic part of the amplitude for . The current is
(23) 
summing over colors and flavors .
When the three jet directions are well separated, we can match the QCD current onto a basis of threejet operators in SCET [34, 66]. We build these operators from quark jet fields , related to collinear quark fields by , where is given by Eq. (18), and a gluon jet field related to gluons by
(24) 
The matching relation is
(25) 
with sums over Dirac spinor indices and Lorentz index , and over label directions and label momenta . Sums over colors and flavors are implied. We have chosen to produce a quark in direction , antiquark in , and gluon in . The matching coefficients are found by equating QCD matrix elements of to SCET matrix elements of the righthand side of Eq. (25). These coefficients have been found at tree level in [66]. The number of independent Dirac and Lorentz structures that can actually appear with nonzero coefficients is considerably smaller than suggested by Eq. (25) due to symmetries. We will keep the form of these coefficients general to make extension to jets transparent, which would require the introduction of a basis of jet fields in Eq. (25), with specified numbers of quark, antiquark, and gluon fields. We will not write the details for an jet cross section here, but the procedures are obvious extensions of all the steps in factorizing the 3jet cross section below.
As a final step before factorization, we redefine the collinear fields to decouple collinearsoft interactions in the Lagrangian [24]:
(26a)  
(26b)  
(26c) 
where is a Wilson line of soft gluons along the lightcone direction ,
(27) 
with in the fundamental representation.^{8}^{8}8The path choice (0 to ) in Eq. (27) is convenient for outgoing particles. The physical cross section is independent of whether the path goes to if the transformation of the external states is also taken into account [67]. is similar but in the adjoint representation. The new fields do not have interactions with soft fields in the SCET Lagrangian at leading order in . Henceforth, we use only these redefined fields, but for simplicity drop the superscripts.
The cross section in SCET can now be written,
(28) 
To proceed to factorize this cross section, it is convenient to rewrite the remaining delta functions that depend on the final state in terms of operators acting on . Those quantities depending on the jet algorithm can be rewritten in terms of an operator containing the momentum flow operator,
(29) 
where is the energymomentum tensor, evaluated at time and position . The operator measures the flow of fourmomentum in the direction (cf. [29, 68, 69]), and the jet algorithm can be written as an operator acting on the momentum flow in all directions [30]. Correspondingly we can define an operator for the 3momentum of the jet, . In addition, the event shape can also be expressed as an operator , built from the momentum flow operator, acting on the state (cf. [29]):
(30) 
The operator is constructed to count only particles actually entering the jet in direction determined by the action of the jet algorithm (for simplicity we will suppress the argument of in the following, but add a superscript for the jet number). Using these operators, we can eliminate the dependence in the delta functions in Eq. (3.2) and perform the sum over states , obtaining
(31) 
The matrix element can be calculated as the sum over cuts of timeordered Feynman graphs, with the delta function operators inside the matrix element enforcing phase space restrictions from the jet algorithm and jet shape measurement on the final state created by the cut.
The operators and depend linearly on the energymomentum tensor, which itself splits linearly in SCET into separate collinear and soft pieces,
(32) 
which will aid us to factorize the full matrix element in Eq. (3.2) into separate collinear and soft matrix elements. To achieve this factorization, however, we must make some more approximations:

The contribution of soft particles and residual collinear momenta to the momentum of each jet can be neglected, and the jet momentum is just given by the label momentum of the collinear state . Thus the energy and jet axis of each jet is approximated to be that of the parent collinear parton initiating the jet. In particular, the squared mass of the jet is order compared to its energy. So in this approximation we take the jet energy to be equal to the magnitude of its 3momentum. On the other hand, we keep the leading nonzero contribution to the angularity even though it is also of order . These approximations also require that we treat the energy of any particles outside all of the jets, and thus the cutoff , as a soft or residual energy.

The Kronecker delta restricting the total number of jets to 3 can be factored into three separate Kronecker deltas restricting the number of jets in each collinear direction to 1, and one Kronecker delta restricting the soft particles not to create an additional jet. This approximation requires the separation between jets to be much larger than the size of any individual jet so that different jets do not overlap. Factoring the restriction on the number of jets in this way is one reason that the parameter in Eq. (4) is required to be large.
We describe to what extent the algorithms we consider actually satisfy these two approximations in Sec. 3.4. For now we assume these approximations and facts hold, which allows us to factor the cross section Eq. (3.2),