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Preprint typeset in JHEP style - HYPER VERSION UCB-PTH-10/01 arXiv:1001.0014 Jet Shapes and Jet Algorithms in SCET Stephen D. Ellis, Christopher K. Vermilion, and Jonathan R. Walsh University of Washington, Seattle, WA 98195-1560, USA E-mail: sdellis@u.washington.edu ,verm@uw.edu ,jrwalsh@u.washington.edu Andrew Hornig and Christopher Lee Theoretical Physics Group, Lawrence Berkeley National Laboratory, and Center for Theoretical Physics, University of California, Berkeley, CA 94720, USA E-mail: ahornig@uw.edu ,clee137@mit.edu Abstract: Jet shapes are weighted sums over the four-momenta of the constituents of a jet and reveal details of its internal structure, potentially allowing discrimination of its par- tonic origin. In this work we make predictions for quark and gluon jet shape distributions inN-jet nal states in e+e collisions, dened with a cone or recombination algorithm, where we measure some jet shape observable on a subset of these jets. Using the framework of Soft-Collinear Eective Theory, we prove a factorization theorem for jet shape distri- butions and demonstrate the consistent renormalization-group 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 Rof the jets, as long as Ris much smaller than the angular separation between jets. We calculate the jet and soft functions for angu- larity jet shapes ato one-loop order ( O(s)) and resum a subset of the large logarithms of aneeded for next-to-leading logarithmic (NLL) accuracy for both cone and k T-type jets. We compare our predictions for the resummed adistribution of a quark or a gluon jet produced in a 3-jet nal state in e+e annihilation to the output of a Monte Carlo event generator and nd that the dependence on aandRis very similar. Keywords: Jets, Factorization, Resummation, Eective Field Theory .arXiv:1001.0014v3 [hep-ph] 15 Nov 2010Contents 1. Introduction 2 1.1 Motivation and Objectives 2 1.2 Soft-Collinear Eective Theory and Factorization 4 1.3 Power Corrections to Factorized Jet Shape Distributions 6 1.4 Resummation and Logarithmic Accuracy 7 1.5 Detailed Outline of This Work 11 2. Jet Shapes and Jet Algorithms 14 2.1 Jet Shapes 14 2.2 Jet Algorithms 14 2.3 Do Jet Algorithms Respect Factorization? 16 3. Factorization of Jet Shape Distributions in e+e toNJets 17 3.1 Overview of SCET 17 3.2 Jet Shape Distribution in e+e !3 Jets 20 3.3 Jet Shapes in e+e !Njets 26 3.4 Do Jet Algorithms Induce Large Power Corrections to Factorization? 27 4. Jet Functions at O(s)for Jet Shapes 30 4.1 Phase Space Cuts 30 4.2 Quark Jet Function 32 4.2.1 Measured Quark Jet 33 4.2.2 Gluon Outside Measured Quark Jet 34 4.2.3 Unmeasured Quark Jet 35 4.3 Gluon Jet Function 35 4.3.1 Measured Gluon Jet 36 4.3.2 Unmeasured Gluon Jet 37 5. Soft Functions at O(s)for Jet Shapes 37 5.1 Phase Space Cuts 37 5.2 Calculation of contributions to the N-Jet Soft Function 38 5.2.1 Inclusive Contribution: Sincl ij 40 5.2.2 Soft gluon inside jet kwithEg>:Sk ij 40 5.2.3 Soft gluon inside measured jet k:Smeas ij(k a) 41 5.3 TotalN-Jet Soft Function in the large- tLimit 42 { 1 {6. Resummation and Consistency Relations at NLL 43 6.1 General Form of Renormalization Group Equations and Solutions 43 6.2 RG Evolution of Hard, Jet, and Soft Functions 46 6.2.1 Hard Function 46 6.2.2 Jet Functions 47 6.2.3 Soft Function 48 6.3 Consistency Relation among Anomalous Dimensions 49 6.4 Refactorization of the Soft Function 50 6.5 Total Resummed Distribution 52 7. Plots of Distributions and Comparisons to Monte Carlo 54 8. Conclusions 61 A. Jet Function Calculations 62 A.1 Finite Pieces of the Quark Jet Function 62 A.2 Finite Pieces of the Gluon Jet Function 65 B. Soft function calculations 67 B.1Sincl ij 67 B.2Si ijandSmeas ij(i a) 69 B.2.1 Common Integrals 69 B.2.2Smeas ij(i a) 70 B.2.3Si ij 70 B.3Smeas ij(k a) andSk ijfork6=i;j 70 B.3.1 Common Integrals 71 B.3.2Sk ij 72 B.3.3Smeas ij(k a) 73 B.3.4Sk ij+Smeas ij(k a) 73 C. Convolutions and Finite Terms in the Resummed Distribution 74 D. Color Algebra for n= 2;3Jets 77 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 { 2 {the other hand, jets contain signatures of exotic physics when produced by the decays of heavy, strongly-interacting particles such as top quarks or particles beyond the Standard Model. Recently, several groups have explored strategies to probe jet substructure to distin- guish 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 iden- tify their origin, such as \ltering" 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 modications of event shapes [ 13] such as thrust. Jet shapes are continuous variables constructed by taking a weighted sum over the four-momenta of all particles constituting a jet. Dierent choices of weighting functions produce dierent jet shapes, and can be designed to probe regions closer to or further from the jet axis with greater sensitivity.1While 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 e+e collision at center-of-mass energy QproducingNjets with an angular size Raccording 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 N-jet nal state looking forward to extension of our results to a hadron collider environment, where such a nal state denition is more typical. For the jet shape observable we choose the angularity aof a jet, dened by (cf. [12,17]), a1 2EJX i2Jpi Te i(1 a); (1.1) whereais a parameter taking values 1<a< 2 (for IR safety, although factorizability will require a < 1), the sum is over all particles in the jet, EJis the jet energy, pT is the transverse momentum relative to the jet direction, and = ln tan(=2) is the (pseudo)rapidity measured from the jet direction. The jet is dened 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 afor jets dened 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 identiable, requiring recalculation only of the observable-dependent 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 R, the measured value of the jet shape such as a, and any other cut or selection parameters introduced by the jet algorithm. 1The original \jet shape," to which the name properly belongs, is the quantity ( r=R), the fraction of the total energy of a jet of radius Rthat is contained in a subjet of radius r[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. { 3 {Precisely this separation of scales, however, allows us to take advantage of the powerful tools of factorization and eective eld 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 s, with the logarithmic accuracy determined by the order to which the anomalous dimensions in the running are calculated [ 20]. Eective eld 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 eective theory facilitate proofs of factorization and achievement of logarithmic resummation at leading order in the power counting and make straightforward the improvement of results order-by-order in power counting and logarithmic accuracy of resummation. 1.2 Soft-Collinear Eective Theory and Factorization Soft-Collinear Eective 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 nand 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 two-jet 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 two-jet 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 demon- strate a factorization theorem for jet shape distributions in e+e !Njet events, d(P1;:::;PN) d1dM=(0)(P1;:::;PN)H(n1;!1;nN;!N;) (1.2) h Jn1;!1(1;)JnM;!M(M;)i Sn1nN(1;:::;M;R;;) JnM+1;!M+1(R;)JnN;!N(R;); where theNjets have three-momenta Pi, andMNof the jets' shapes 1;:::;Mare measured.(0)is the Born cross-section, His a hard function dependent on the directions niand energies !iof theNjets,Jn;!() is the jet function for a jet whose shape is measured to be,Jn;!(R) is the jet function for a jet with size Rwhose shape is not measured, and Sis the soft function connecting all Njets, dependent on all jets' shapes i, sizesR, and total energy that is left outside of all jets. The symbol \ " stands for a set of convolution integrals in the variables ibetween 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 pTof the constituent { 4 {of a jet (relative to the jet direction) divided by the jet energy EJ. This is set either by the measured value of the jet shape afor a measured jet or the algorithm measure R for an unmeasured jet. Thus we encounter in this work the new twist that the size of may be dierent for dierent jets. We will comment on further implications of this in subsequent sections. Still, in each separate collinear sector, the momentum pnof collinear modes in the light-cone direction nin SCET is separated into a large \label" momentum ~ pn containingO(EJ) andO(EJ) components and a \residual" component of O(2EJ), the same size as soft momenta. Eective theory elds 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 dene the jets, we will encounter the need for several additional criteria to ensure the validity of the N-jet factorization theorem. First, to ensure that the algorithm does not group nal-state particles into fewer than Njets, the jets must be \well separated." This allows us to use as the eective theory Lagrangian a sum of Ncopies of the collinear part of the SCET Lagrangian for a single direction nand a soft part, and to construct a basis of N-jet operators built from elds from each of these sectors to produce the nal 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 nd more than Njets, 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 eld producing the jet. Corrections to this identication are subleading in the SCET power counting. Third (and related to the above two), we will assume that the N-jet restriction on the nal state can itself be factorized into a product of N1-jet restrictions, one in each collinear sector, and a 0-jet restriction in the soft sector. We represent the energetic particles in the ith jet by collinear elds in the SCET Lagrangian in the nicollinear sector and soft particles everywhere with elds in the soft part of the Lagrangian. We then stipulate that the jet algorithm acting on states in the nicollinear sector nd exactly one jet in that sector, and when acting on the soft nal state nd no additional jet in that sector. Fourth, the way in which a jet algorithm combines particles in the process of nding a jet must respect the order of steps envisioned by factorization. In particular, fac- torization 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 rst, with soft particles within a radius Rof the jet { 5 {axis being recombined into the jet only afterwards. Jet algorithms in use at experi- ments do not have this precise behavior, but we will discuss in Sec. 3.4the 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 nd that for suciently largeR, infrared-safe cone algorithms and k T-type recombination algorithms satisfy the requirements of factorizability, with anti-k Tallowing smaller values of Rthan k T. After enforcing the above requirements, a key test of the consistency of Eq. ( 1.2) 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, 0 = [ H() + JM+1(R;) ++ JN(R;)](1)(M) + J1(1;)(2)(N) ++(1)(M 1) JM(M;) + S(1;:::;M;R;):(1.3) It seems highly nontrivial that this condition would be satised for any number, size, and avors of jets (and that the soft anomalous dimension be independent of ), but we will demonstrate that it does hold at O(s), up to corrections of O(1=t2) which violate Eq. ( 1.3), wheretis a measure of the separation between jets. In particular, for a pair of jets, i;j, with 3-vector directions separated by a polar angle ij, the separation tijis given by tk;l=tan( k;l=2) tan(R=2): (1.4) Now dene t(no indices) as the minimum of tijover all jet pairs. This quanties the qualitative condition of jets being well-separated, t1, that is required to justify the factorization theorem Eq. ( 1.2). The factorization theorem is valid up to corrections of O() in the SCET power expansion parameter and corrections of O(1=t2) in the separa- tion parameter. As an example of the magnitude of t, for three jets in a Mercedes-Benz conguration ( = 2=3 for all pairs of jets), 1 =t2= 0:04 forR= 0:7 and 1=t2= 0:1 for R= 1, so these corrections are indeed small. More generally, for non-overlapping jets, >2R, we have 1 =t2<1=4. Notice that for back-to-back jets ( =),t!1 . Thus, for all cases previously con- sidered in the literature, the jets are innitely 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 N-jet cross-section described by Eq. ( 1.2), the factorization theorem receives corrections not only in the usual SCET power counting parameter , but also corrections due to jet separation beginning at O(1=t2). 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 ni, which labels that jet in the SCET Lagrangian. This direction niis the direction of the parent parton initiating the jet. The jet observable must be such that the dierence between the parent parton direction and the jet axis { 6 {identied 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 a<1, and we nd the same condition for jet shapes. In the presence of algorithms, however, there are additional power corrections due to the dierence 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 eect of this dierence on the measurement of jet shapes, and nd that for suciently large Rthe power corrections due to the action of the algorithm on soft particles remain small enough not to spoil the factorization for infrared-safe cone and k T-type recombination algorithms. Algorithm-related power corrections to jet momenta were studied more quantitatively in [36], and their estimated Rdependence 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 1=(aQ), enhanced in the endpoint region but suppressed by large energy. The endpoints of our jet shape distribution near a!0, 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 rst 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 back- to-back 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 1 =R. 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 d=d (i.e. Eq. ( 1.2) with two jets and integrated against ( 1 2)), the accuracy of logarithmic resummation [ 43] is typically characterized by counting logs in the exponent ln R() of the \radiator," R() =1 0Z 0d0d d0; (1.5) where they appear in the form n slnmwithmn+ 1. At leading-logarithmic (LL) accuracy all the terms with m=n+ 1 are summed; next-to-leading-logarithmic (NLL) accuracy sums also the m=nterms, and so on. In more traditional methods in QCD, event { 7 {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], theC-parameter in [ 50], and angularities in [17]. Resummation of an event shape distribution using the modern SCET method was rst 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 N3LL resummed thrust distribution in SCET was compared to LEP data to extract a value for the strong coupling sto high precision in [53]. Angularities were resummed to NLL accuracy in SCET in [ 54] directly in a-space instead of in moment space as in [ 17]. Summation of logarithms in eective eld theory is achieved by RG evolution. In the factorized radiator of the thrust distribution Eq. ( 1.5), one nds that the hard function contains logarithms of =Q, the jet functions contain logarithms of =(Qp), and the soft function contains logarithms of =(Q). Thus, evaluating these functions respectively at the hard scale H=Q, jet scaleJ=Qpand soft scale S=Qeliminates large logarithms in each function. They can then be RG-evolved to the common factorization scaleafter calculating their anomalous dimensions. The solutions of the RG evolution equations are of the form that logarithms of are resummed to all orders in sto a logarithmic accuracy determined by the order in sto 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 !=Q] along with a table that lists the order in sto which various quantities must be known in order to achieve a given NkLL accuracy in the exponent of the radiator Eq. ( 1.5). The power of the EFT framework is to organize of the logs of arising in Eq. ( 1.5) 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. ( 1.2), the strategy to sum logs is the same, but is complicated by the presence of additional scales. This also makes trickier the clas- sication of logarithmic accuracy into the standard NkLL scheme. Our aim will be to sum as many logs of the jet shapes aas possible, while not worrying about any others. For instance, phase space cuts induce logs of Rand =!(where!is a typical hard jet energy), and the presence of multiple jets induces logs of jet separations ninjor ratios of jet energies !i=!j. We will not aim to sum these types logs systematically in this paper, only those of a(though we sum subsets of the others incidentally). In particular, resum- ming the phase space logs of Ror =!is complicated by how the phase space cuts act order-by-order in perturbation theory2, and the fact that a simple angular cut Ris 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 Rof 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 2The JADE algorithm is one well-known example in which resummability even of leading logarithms of the jet mass cut yis spoiled by the dierences in the jet phase space at dierent orders in perturbation theory [ 55]. Another example that will not work is using a k T-type algorithm with Rrandomly chosen for each recombination. This is clearly such that resummation of logarithms of Rcannot be achieved. { 8 {hard scale “unmeasured” jet scale soft scalesµµH=ω µmeas J=ωτ1 2−aaµunmeas J =ωtanR 2 “measured” jet scale µSγmeas Jγunmeas J γS EFT countingmatching/ matrix element LL tree 1-loop tree 1-loop NLL tree 2-loop 1-loop 2-loop NNLL 1-loop 3-loop 2-loop 3-loopΓcusp γH,J,S β[αs] µmeas S =ωτa/tan1−a(R/2)µΛR= 2Λtan(R/2)µΛ=2ΛFigure 1: An illustration of generic scales along with a table of log-accuracy versus perturbative order. A cross section with jets of energy !, radiusR, and energy outside the jets, with some jets' shapes abeing measured and others' shapes left unmeasured, induces measured and unmeasured jet scales at meas J andunmeas J . Dynamics at these scales are described by separate collinear modes in SCET. Soft dynamics occur at several soft scales, andRinduced by the soft out-of-jet energy cut and jet radius R, andmeas S induced by the measured jet shape a. RG evolution in SCET resums logs of ratios of jet scales to the hard scale Hindividually, and logs of the ratio of a \common" soft scale Sto the hard scale. Remaining logs of ratios of soft scales to one another are not resummed in current formulations of SCET. The accuracy of logarithmic resummation of these ratios of scales is determined by the order to which anomalous dimensions and matching coecients or matrix elements (i.e. hard/jet/soft functions) are calculated in perturbation theory. In this paper we perform the RG evolution indicated by the arrows to NLL accuracy. soft radiation from such congurations 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 a. (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. ( 1.2) 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 dierence is the presence now of multiple jet and soft scales. Logarithms in jet functions can still be minimized by choices of individual jet scales, meas J!1=(2 a) a for a jet whose shape a is measured, and unmeas J!tan(R=2) for a jet whose shape is not measured but has a radiusR. Thus logs arising from ratios of these scales to the hard scale can be summed { 9 {completely to a desired NkLL 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 tan1 a(R=2)=(!a) from radiation into measured jets, and logs of =(2) and=(2 tanR 2) from radiation from unmeasured jets cut o by the energy . In addition, though not illustrated in Fig. 1, there can be logs of multiple jet shapes to one another, i a=j a. No single choice of a soft scaleSwill minimize all of these logs. In the present work, we will start with the simple strategy of choosing a single soft scaleS!a=tan1 a(R=2) for a jet whose shape awe 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. ( 1.2) 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 a). In the case that all jets' shapes are measured and are similar to one another, i aj a, our resummation of large logs of i awould 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 O(s) soft functions we calculate. However, one cannot really address mixed logarithms such as log( i a=j a) that arise for multiple jets until O(2 s), the rst order at which two soft gluons can probe two dierent physical regions. This lies beyond the scope of the present work. (We note, however, that our implementation of refactorization using the one-loop 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 \non-global" logarithms described by [ 46,56, 57,58] that spoil the simple characterization of NLL accuracy. In [ 59] these were identied as next-to-leading logs of R2=(!ia) and =Q(whenR1) that appear at O(2 s) in jet shape distributions. These authors organized the radiator for a single jet shape distribution into a \global" and \non-global" part [ 58,59], R(i a;R;;!i;Q) =Rgl i a;R; Q RngR2 !iia; Q : (1.6) In this language, the calculations we undertake in this paper resum logs in the global part to NLL accuracy but not in the non-global part. The rst argument in Rngis 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, R1, and !ii a, the non-global logs vanish. While summing all global and non-global logs to at least NLL accuracy will be impor- tant 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 dened with al- gorithms. We also believe the eective theory approach and the idea of refactorizing the soft function will help us understand and resum many types of non-global logarithms. { 10 {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 O(s) in SCET, and use the renormalization group evolution of these functions to sum global logs of ato NLL accuracy. We consider Njets (dened with a cone or k Talgorithm) produced in an e+e collision, with Mof the jets' shapes (angularities) being measured. The key formal result is our demonstration of Eq. ( 1.3), the consistency of the anomalous dimensions of hard, jet and soft functions to O(s) 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 Rin cone or k T- type algorithms (as long as t1). These results lead to accurate predictions for the shape of the adistribution near the peak, but not necessarily the endpoints for very smalla(where hadronization corrections dominate) and very large a(where xed-order NLO QCD corrections take over, which are not yet calculated and not captured by NLL resummation).3 In Sec. 2we 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 3-jet production where we measure the angularity jet shape of one of the jets, and then perform the straightforward extension to N-jet production with MN 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 N-jet nal states and jet algorithms that must be satised for factorization to hold. In Sec. 3.4we 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 suciently large Rin infrared-safe cone and recombination algorithms, these corrections are suciently small. Next we calculate to O(s) the jet and soft functions corresponding to Ncone or kT-type jets, with Mjets' shapes measured. In Sec. 4we calculate the jet functions for measured quark jets, Jq !(a), unmeasured quark jets,Jq !, measured gluon jets, Jg !(a), and unmeasured gluon jets, Jg !, where!= 2EJ is the label momentum of the collinear jet eld in each jet function. We nd that in collinear sectors for measured jets, the collinear scale (and thus the SCET power counting parameter in that sector i) is given by !i1=(2 a) a , and in unmeasured jet sectors, itan(R=2). In studying power corrections, however, as mentioned above, we nd that Rmust be parametrically larger than a. So, in collinear sectors for measured jets, iis set by the 3Jet shapes were also studied in the QCD factorization approach in [ 60]. In that work QCD jet functions for quark and gluon jets dened with an algorithm and whose jet masses m2 Jare measured were calculated toO(s). The jet mass2corresponds to afora= 0,0=m2=!2(01). A xed-order QCD jet function as dened in [ 60] is given by the convolution of our xed-order SCET jet function and soft function for a measured jet away from a= 0. { 11 {shapeawithR0 i, while in unmeasured jet sectors, itan(R=2). Thus one should understand tan( R=2) to be signicantly less than 1 but much larger than any jet shape a. In Sec. 5we calculate the soft function. To do this, we split the soft function into several contributions from dierent parts of phase space in order to facilitate the calculation and elucidate its intuitive structure. We nd it most convenient to split the soft function into an observable-independent part that arises from soft emission out of the jets, Sunmeas, and a part that depends on our choice of angularities as the observable that arises from soft emission into measured jet i,Smeas(i a).Sunmeasis hence sensitive to the scale while Smeas(i a) is sensitive to the scale !ii a. In Sec. 6, having calculated all the jet and soft function contributions to O(s), we extract the anomalous dimensions and perform renormalization-group (RG) evolution. We nd the hard anomalous dimension from existing results in the literature. The hard, jet, and soft anomalous dimensions have to satisfy the consistency condition Eq. ( 1.3) 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 tEq. ( 1.4) between all pairs of jets is much larger than 1, the condition is satised. Even after requiring t1, the satisfaction of the consistency condition is non-trivial. The hard function knows only about the direction of each jet and the jet function knows only the jet size R; the soft function knows about both. Furthermore, it is not sucient 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 cuto 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. 6by proposing in Sec. 6.4a 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 \non-global" logs of ratios of multiple soft scales that begin at NLL and O(2 s). To help the reader nd the results of the calculations in Sec. 4through Sec. 6just outlined, Table 1provides a summary with equation numbers. In Sec. 7we 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 aect jet shapes. We will illustrate our results in the case of e+e !3 jets, with the jets constrained to be in a conguration where each has equal energy and are maximally separated. In both the eective theory and Monte Carlo, we can take the jets to have been produced by an underlying hard process e+e !qqg. 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 aandR. We nd that the dependencies on aandR of the shapes of the distribution and the peak value of aagree well between the theory and { 12 {Category Contribution Symbol Location measured quark jet function Jq !(a) Eq. ( 4.11) unmeas. quark jet function Jq ! Eq. ( 4.17) measured gluon jet function Jg !(a) Eq. ( 4.25) unmeas. gluon jet function Jg ! Eq. ( 4.26) NLO contributions summary of divergent| Table 2before resummation: parts of soft func. (any t) total universalSunmeasEq. ( 5.20)soft func. (large t) total measuredSmeas(i a) Eq. ( 5.22)soft func. (large t) anomalous dimensions: | | Table 3 measured jet function fi J(i a;i J)Eq. ( 6.42a ) NLO contributions measured soft function fS(i a;i J)Eq. ( 6.42b ) after resummation: unmeas. jet function Ji !(J) Eq. ( 6.43a ) universal soft function Sunmeas( S)Eq. ( 6.43b ) Total NLL Distribution: | | Eq. ( 6.40) Table 1: Directory of main results: the xed-order NLO quark and gluon jet functions for jets whose shapes aare measured or not; the xed-order NLO contributions to the soft functions from parts of phase space where a soft gluon enters a measured jet, Smeas(a), or not,Sunmeas; their anomalous dimensions; the contributions the jet and soft functions make to the nite part of the NLL resummed distributions; and the full NLL resummed jet shape distribution itself. Monte Carlo, with small but noticeable corrections due to hadronization. We can estimate these corrections by comparing output with hadronization turned on or o in Monte Carlo. In Sec. 8, we give our conclusions and outlook. We also collect a number of technical details and results for O(s) nite pieces of jet and soft functions in the Appendices. Our work is, to our knowledge, the rst achieving factorization and resummation of a jet observable distribution in an exclusive N-jet nal state dened by a non-hemisphere jet algorithm.4Having 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 k T- 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 e+e and hadron-hadron collisions. The case of ppcollisions will require a number of modications, including turning two of our outgoing jet functions into incoming \beam functions" introduced in [ 62]. We leave this generalization for future work. The reader wishing to follow the general structure of our ideas and logic and understand the basis of the nal results of the paper without working through all the technical details may read Secs. 1and2, and then skip to Sec. 7. Some short less technical discussion also appears in Sec. 3.4. 4Dijet cross sections for cone jets were factorized and resummed in [ 61]. { 13 {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 nal state by assigning diering weights to events with diering energy distributions. Events that are two-jet like, with two very collimated back-to-back 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 dierent structure in terms of number of jets and the energy distribution within those jets. Jet shapes, which are event shape-like observables applied to single jets, are an eective tool to measure the structure of individual jets. These observables can be used to not only quantify QCD-like events, but study more complex, non-QCD topologies, as illustrated for light quark vs. top quark and Zjets in [ 12,60]. Broad jets, with wide-angle energy depositions, and very collimated jets, with a narrow energy prole, take on distinct values for jet shape observables. In this work, we consider the example of the class of jet shapes called angularities, dened in Eq. ( 1.1) and denoted a. Every value of acorresponds to a dierent jet shape. As adecreases, the angularity weights particles at the periphery of the jet more, and is therefore more sensitive to wide-angle radiation. Simultaneous measurements of the angularity of a jet for dierent values of acan 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 nal 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 pertur- bative calculation, and to combinations of these objects within a jet.) Since the underlying jet is not intrinsically well dened, there is no unique jet algorithm and a wide variety of jet algorithms have been proposed and implemented in experiments. The details of each algo- rithm are motivated by particular properties desired of jets, and dierent algorithms have dierent strengths and weaknesses. In this work we will calculate angularity distributions for jets coming from a variety of algorithms. Because we calculate (only) at next-to-leading 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 sep- aration 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 xed geometric shape, the cone, and nding \stable" cones. A cone contains all of the particles within an angle Rof the cone axis, and the angular parameter Rsets the size of the jet. In found jets (stable { 14 {cones), the direction of the total three-momentum of particles in the cone equals the cone (jet) axis. Dierent cone algorithms employ dierent methods to nd stable cones and deal with dierently the \split/merge" problem of overlapping stable cones. The SISCone algorithm [ 63] is a modern implementation of the cone algorithm that nds all stable cones and is free of infrared unsafety issues. In the next-to-leading order calculation we perform, there are at most two particles in a jet, and we only consider congurations where all jets are well-separated. Therefore, it is straightforward to nd all stable cones, there are no issues with overlapping stable cones, and the phase space cuts of all cone algorithms are equivalent. This simplies all standard cone algorithms to a generic cone-type algorithm, in which each particle is constrained to be within an angle Rof the jet axis. For a two-particle jet, if we label the particles 1 and 2 and the jet axis n, then the cone-like constraints for the two particles to be in a jet are cone jet:1n<R and2n<R: (2.1) This denes our cone-type algorithm. Recombination algorithms build jets by recursively merging pairs of particles. Two distance metrics, dened by the algorithm, determine when particles are merged and when jets are formed. A pairwise metric pair(the recombination metric) denes a distance between pairs of particles, and a single particle metric jet(the beam, or promotion, metric) denes a distance for each single particle. Using these metrics, a recombination algorithm builds jets with the following procedure:5 0. Begin with a list Lof particles. 1. Find the smallest distance for all pairs of particles (using pair) and all single particles (usingjet). 2a. If the smallest distance is from a pair, merge those particles together by adding their four momenta. Replace the pair in Lwith the new particle. 2b. If the smallest distance is from a single particle, promote that particle to a jet and remove it from L. 3. Loop back to step 1 until all particles have been merged into jets. The k T, Cambridge-Aachen, and anti-k Talgorithms are common recombination algo- rithms, and their distance metrics are part of a general class of recombination algorithms. Fore+e colliders, a class of recombination algorithms can be dened by the parameter : pair(i;j) = min E i;E jij R jet(i) =E i; (2.2) 5This denes an inclusive recombination algorithm more typically applied to hadron-hadron colliders. We are applying it here to the simpler case of e+e collisions in order to facilitate the eventual transition to LHC studies. Exclusive recombination algorithms, more typical of e+e collisions, are described along with other jet algorithms in [ 64] and their description in SCET is given in [ 32]. { 15 {where= 1 for k T, 0 for Cambridge-Aachen, and 1 for the anti-k Talgorithm. The parameterRsets the maximum angle between two particles for a single recombination.6 In the multijet congurations we consider the jets are separated by an angle larger than R, so that only the pairwise metric is relevant for describing the phase space constraints for particles in each jet. For a two-particle (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 R: kTjet:12<R: (2.3) This denes a generic recombination algorithm suitable for our calculation. We will denote this as the k T-type algorithm. The congurations with two widely separated energetic particles best distinguish cone- type jets from k T-type jets at NLO. For instance, the case where the two energetic particles are at opposite edges of a cone jet (at an angle 2 Rapart) is not a single k Tjet. However, it is important to note that these congurations will not be accurately described in this SCET calculation for R, as such congurations are power suppressed in our description of jets. Our concern is in accurately describing the congurations with narrow jets (small a), and not the wide angle congurations 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 N-jet event, therefore, is one where Njets have energy larger than the cuto , with any number of jets having energy less than the cuto. 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 nal state. This imposes phase space cuts: for a gluon radiated outside of all jets in the event, that gluon is required to have energyEg< 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 specic requirements on the structure of jet algorithms used to describe events. As in Eq. ( 1.2), the factorization theorem divides the cross section into separately calculable hard, jet, and soft functions. The hard function depends only on the conguration 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 conguration of jets in an event depends on dynamics across all energy scales. This naively breaks factorization in SCET, since the 6We useRfor both cone and k Talgorithms for ease of notation. For k T, this parameter is sometimes referred to as D. We emphasize that having the same size Rfor dierent algorithms does not in general guarantee the same sized jets. { 16 {conguration 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.3.4we will parameterize the power corrections that arise from various algorithms. The primary constraint on the jet algorithm in order to satisfy the factorization the- orem 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 in- dependent 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 specic way: all collinear particles must be recombined before soft particles. As discussed in Sec. 3.4, commonly used algo- rithms obey this constraint up to power corrections in the observable for measured jets. Of particular note is the anti-kT 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 e+e toNJets In this Section we formulate a factorization theorem for jet shape distributions in e+e annihilation to Njets. All the formal aspects we need to describe an N-jet cross section appear already in the 3-jet cross section, so we will give explicit details only for that case. We will use the framework of Soft-Collinear Eective 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 eective theory. 3.1 Overview of SCET SCET is the eective eld theory for QCD with all degrees of freedom integrated out, other than those traveling with large energy but small virtuality along a light-like trajectory n, and those with small, or soft, momenta in all components. A particularly useful set of coordinates is light-cone coordinates, which uses light-like directions nand n, with n2= n2= 0 andnn= 2. In Minkowski coordinates, we take n= (1;0;0;1) and n= (1;0;0; 1), corresponding to collinear particles moving in the + zdirection. A generic four-vector pcan be decomposed into components p= npn 2+npn 2+p ?: (3.1) In terms of these components, p= (np;np;p?), collinear and soft momenta scale with some small parameter as pn=E(1;2;); psE(2;2;2); (3.2) { 17 {whereEis a large energy scale, for example, the center-of-mass energy in an e+e collision. is then the ratio of the typical transverse momentum of the constituents of the jet to the total jet energy. Quark and gluon elds in QCD are divided into collinear and soft eective theory elds with these respective momentum scalings: q(x) =qn(x) +qs(x); A(x) =A n(x) +A s(x): (3.3) We factor out a phase containing the largest components of the collinear momentum from the eldsqn;An. Dening the \label" momentum ~ p n= n~pnn 2+ ~p ?, where n~pncontains theO(1) part of the large light-cone component of the collinear momentum pn, and ~p?the O() transverse component, we can partition the collinear elds qn;Aninto their labeled components, qn(x) =X ~p6=0e i~pxqn;p(x); A n(x) =X ~p6=0e i~pxA n;p(x): (3.4) The sums are over a discrete set of O(1;) label momenta into which momentum space is partitioned. The bin ~ p= 0 is omitted to avoid double-counting the soft mode in Eq. ( 3.3) [65]. The labeled elds qn;p;An;pnow have spacetime uctuations in xwhich are conjugate to \residual" momenta kof the order E2, describing remaining uctuations within each labeled momentum partition [ 23,65]. It will be convenient to dene label operators P= nPn=2 +P ?which pick out just the label components of momentum of a collinear eld: Pn;p(x) = ~pn;p(x): (3.5) Ordinary derivatives @acting on eective theory elds n;p(x) are of order E2. The nal step to construct the eective theory elds is to isolate the two large compo- nents of the Dirac spinor qn;pfor a fermion with lightlike momentum along n. The large components n;pand the small n;pcan be separated by the projections n;p=n =n = 4qn;p;n;p=n =n = 4qn;p; (3.6) and we have qn;p=n;p+ n;p. One can show, substituting these denitions into the QCD Lagrangian, that the elds n;phave an eective mass of order Eand can be integrated out of the theory. The eective theory Lagrangian at leading order in is [22,23,24] LSCET =L+LAn+Ls; (3.7) where the collinear quark Lagrangian Lis L=n(x) inD+iD =c ?Wn(x)1 inPWy n(x)iD =c ?n = 2n(x); (3.8) whereWnis the Wilson line of collinear gluons, Wn(x) =X permsexp g1 nPnAn(x) ; (3.9) { 18 {the collinear gluon Lagrangian LAnis LAn=1 2g2Trh iD+gA n;iD+gA ni2 + 2 Tr cnh iD;h iD+gA n;cnii +1 Trh iD;A ni ;(3.10) wherecnis the collinear ghost eld and the gauge-xing parameter; and the soft La- grangianLsis Ls= qsiD =sqs(x) 1 2TrG sGs(x); (3.11) which is identical to the form of the full QCD Lagrangian (the usual gauge-xing terms are implicit). In the collinear Lagrangians, we have dened several covariant derivative operators, D=@ igA n igA s; iD c=P+gA n; iD=P+inDn 2: (3.12) In addition, there is an implicit sum over the label momenta of each collinear eld 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 . Mean- while, the coupling of the soft gluon eld to a collinear eld is in the component nAsonly, according to Eqs. ( 3.8) and ( 3.10), which makes possible the decoupling of such interac- tions through a eld redenition of the soft gluon eld given in [ 24]. We will utilize this soft-collinear decoupling to simplify the proof of factorization below. The SCET Lagrangian Eq. ( 3.7) 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. ( 3.8) and ( 3.10) together. The collinear elds in each direction nicon- stitute their own independent set of quark and gluon elds, and are governed in principle by dierent expansion parameters associated with the transverse momentum of each jet, set either by the angular cut Rin the jet algorithm or by the measured value of the jet shapea. Each collinear sector may be paired with its own associated soft eld Aswith momentum of order E2with 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 eld Ascoupling 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 eective theory containing Ncollinear sectors and the soft sector is appropriate to describe QCD processes with strongly-interacting particles collimated in Nwell-separated 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 nd in calculating an N-jet cross section the need for another parameter that guarantees that the dierent directions niare well separated. This latter condition requires t1, wheretis dened in Eq. ( 1.4).7 7This condition is a consequence of our insistence on using operators with exactly Ndirections to create { 19 {3.2 Jet Shape Distribution in e+e !3 Jets Consider a 3-jet cross section dierential in the jet 3-momenta P1;2;3, where we measure the shape1 aof one of the jets, which we will call jet 1. The full theory cross section for e+e ! !3 jets at center-of-mass energy Qis d d1ad3P1;2;3=1 2Q2X XjhXjj(0)j0iLj2(2)44(Q pX)N(J(X));3 1 a a(jet 1)3Y j=13 Pj P(jetj) ;(3.13) where theJ(X) is the jet algorithm acting on nal state X, andN(J(X)) is the number of jets identied by the algorithm [ 30].P(jetj) is the 3-momentum of jet j, and is also a function of the output of the jet algorithm J(X).Lis the leptonic part of the amplitude fore+e ! !qqg. The current jis j=X a;fqa qa; (3.14) summing over colors aand avors f. When the three jet directions are well separated, we can match the QCD current j(x) onto a basis of three-jet operators in SCET [ 34,66]. We build these operators from quark jet eldsn, related to collinear quark elds nbyn=Wy nn, whereWnis given by Eq. ( 3.9), and a gluon jet eld B? nrelated to gluons Anby B? n=1 gWy n(P?+A? n)Wn: (3.15) The matching relation is j(x) =X n1n2n3X ~p1~p2~p3ei(~p1 ~p2+~p3)xC (n1;~p1;n2;~p2;n3;~p3) n1;p1(gB? n3;p3) n2;p2(x); (3.16) with sums over Dirac spinor indices ;and Lorentz index , and over label directions n1;2;3 and label momenta ~ p1;2;3. Sums over colors and avors are implied. We have chosen to produce a quark in direction n1, antiquark in n2, and gluon in n3. The matching coecients C are found by equating QCD matrix elements of jto SCET matrix elements of the right-hand side of Eq. ( 3.16). These coecients have been found at tree level in [ 66]. The number of independent Dirac and Lorentz structures that can actually appear with nonzero coecients is considerably smaller than suggested by Eq. ( 3.16) due to symmetries. We will keep the form of these coecients general to make extension to Njets transparent, which would require the introduction of a basis of Njet elds in Eq. ( 3.16), with specied the nal state. We could move away from the large- tlimit 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 Njet operator and ( N1)-jet operators. This is outside the scope of the present work, where we limit ourselves to kinematics well described by an N-jet operator, and thus, limit ourselves to the large- tlimit. { 20 {numbers of quark, antiquark, and gluon elds. We will not write the details for an N-jet cross section here, but the procedures are obvious extensions of all the steps in factorizing the 3-jet cross section below. As a nal step before factorization, we redene the collinear elds to decouple collinear- soft interactions in the Lagrangian [ 24]: n(x) =Yy n(x)(0) n(x) (3.17a) n(x) = (0) n(x)Yn(x) (3.17b) An(x) =Yn(x)A(0) n(x); (3.17c) whereYnis a Wilson line of soft gluons along the light-cone direction n, Yn(x) =Pexp igZ1 0dsnAs(ns+x) ; (3.18) withAsin the fundamental representation.8Ynis similar but in the adjoint representation. The new elds (0) n;A(0) ndo not have interactions with soft elds in the SCET Lagrangian at leading order in . Henceforth, we use only these redened elds, but for simplicity drop the (0) superscripts. The cross section in SCET can now be written, d d1ad3P1;2;3=NFL2 6Q2X XN(J(X));3(1 a a(jet 1))3Y j=13(Pj P(jetj)) X n1;2;3X ~p1;2;3Z d4xei(Q ~p1+~p2 ~p3)xC (n1;2;3; ~p1;2;3)C (n1;2;3; ~p1;2;3) h0jTn a n2;p2Yab n2YAB n3(gB?B n3;p3)TA bcYycd n1d n1;p1(x)o jXi hXjTn e n1;p1Yef n1YCD n3(gB?D n3;p3)TC fgYygh n2h n2;p2(0)o j0i: (3.19) To proceed to factorize this cross section, it is convenient to rewrite the remaining delta functions that depend on the nal state Xin terms of operators acting on X. Those quantities depending on the jet algorithm Jcan be rewritten in terms of an operator containing the momentum ow operator, E(n) = lim R!1Z1 0dtniTi(t;Rn); (3.20) whereTis the energy-momentum tensor, evaluated at time tand position Rn. The operatorE(n) measures the ow of four-momentum Pin the direction n(cf. [ 29,68,69]), and the jet algorithm Jcan be written as an operator ^Jacting on the momentum ow in all directions [ 30]. Correspondingly we can dene an operator for the 3-momentum of the 8The path choice (0 to 1) in Eq. ( 3.18) is convenient for outgoing particles. The physical cross section is independent of whether the path goes to 1 if the transformation of the external states Xis also taken into account [ 67]. { 21 {jet,^P(Jj(^J)). In addition, the event shape a(jet 1) can also be expressed as an operator ^a(J1(^J)), built from the momentum ow operator, acting on the state jXi(cf. [ 29]): ^a(J1(^J)) =Z de (1 a)ET()( min(J1(^J))): (3.21) The operator is constructed to count only particles actually entering the jet in direction n1 determined by the action of the jet algorithm (for simplicity we will suppress the argument J1(^J) of ^ain the following, but add a superscript for the jet number). Using these operators, we can eliminate the Xdependence in the delta functions in Eq. ( 3.19) and perform the sum over states X, obtaining d d1ad3P1;2;3=L2 6Q2X n1;2;3X ~p1;2;3Z d4xei(Q ~p1+~p2 ~p3)xC (n1;2;3; ~p1;2;3)C (n1;2;3; ~p1;2;3) h0jTn a n2;p2Yab n2YAB n3(gB?B n3;p3)TA bcYycd n1d n1;p1(x)o N(^J);3(1 a ^1 a)3Y j=13(Pj ^P(Jj(^J))) Tn e n1;p1Yef n1YCD n3(gB?D n3;p3)TC fgYygh n2h n2;p2(0)o j0i: (3.22) The matrix element can be calculated as the sum over cuts of time-ordered 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 nal state created by the cut. The operators ^ aand ^Jdepend linearly on the energy-momentum tensor, which itself splits linearly in SCET into separate collinear and soft pieces, T=X iTni+Ts ; (3.23) which will aid us to factorize the full matrix element in Eq. ( 3.22) into separate collinear and soft matrix elements. To achieve this factorization, however, we must make some more approximations: 1. The contribution of soft particles and residual collinear momenta to the momentum P(jetj) of each jet can be neglected, and the jet momentum is just given by the label momentum ~ pjof the collinear state jXji. 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 2compared to its energy. So in this approximation we take the jet energy to be equal to the magnitude of its 3- momentum. On the other hand, we keep the leading non-zero contribution to the angularity even though it is also of order 2. These approximations also require that we treat the energy of any particles outside all of the jets, and thus the cuto , as a soft or residual energy. { 22 {2. 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 ni 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 dierent jets do not overlap. Factoring the restriction on the number of jets in this way is one reason that the parameter tijin Eq. ( 1.4) is required to be large. We describe to what extent the algorithms we consider actually satisfy these two approxi- mations in Sec. 3.4. For now we assume these approximations and facts hold, which allows us to factor the cross section Eq. ( 3.19), d d1adE1;2;3d2 1;2;3=L2 6Q2X n1;2;3X !1;2;3C (n1;2;3;!1;2;3)C (n1;2;3;!1;2;3) Z d4xei(Q !1n1=2+!2n2=2 !3n3=2)xZ dJdS(1 a J S) h0jf n1;!1(x)N(^J);1(J ^n1a)e n1;!1(0)j0i E1 !1 2 2( 1 n1) h0ja n2; !2(x)N(^J);1h n2; !2(0)j0i E2 !2 2 2( 2 n2) h0j(gB?A n3;!3)(x)N(^J);1(gB?B n3;!3(0)j0i E3 !3 2 2( 3 n3) h0jYyab n2Yybc n3TA cdYyde n3Yyef n1(x)N(^J);0(S ^s a)Ygh n1Yhi n3TB ijYjk n3Ykl n2(0)j0i(3.24) We have rewritten the cross section to be dierential in Ei(the magnitude of Pi) and i (the direction of Pi). In the sum over label directions, nican be chosen to align with Pi such that ~p? i= 0. In Eq. ( 3.24) we have written the label momentum as !ini~pi. In Eq. ( 1.1) we approximate the jet axis by this niand the jet energy by ni~pi=2, so that they do not depend on soft momenta at all. The operators ^ n1aand ^s aare dened to count only particles inside the measured jet identied by the algorithm. In the soft matrix element in Eq. ( 3.24), we have introduced the soft Wilson line Ynin the antifundamental representation to remove the time- and anti-time-ordering operators T;Tin Eq. ( 3.19) [27], and related Wilson lines Ynin the adjoint representation to those in the fundamental representation by [ 24] YAB nTB=Yy nTAYn: (3.25) Dening the jet functions by the relations Zd4k1 (2)4e ik1xJn1;!1(J;n1k1)n =1 2 ef=h0jf n1;!1(x)N(^J);1(J ^n1a)e n1;!1(0)j0i (3.26a)Zd4k2 (2)4e ik2xJn2;!2(n2k2)n =2 2 ah=h0ja n2;!2(x)N(^J);1h n2;!2(0)j0i (3.26b) Zd4k3 (2)4e ik3xJn3;!3(n3k3)g ?AB= !3h0j(gB?A n3;!3)N(^J);1(gB?B n3;!3)j0i;(3.26c) { 23 {and the soft function by Zd4r (2)4e irxS(s;r) =1 NCCFTrh0jYy n2Yy n3TAYy n3Yy n1(x)N(^J);0(S ^s a) Yn1Yn3TBYn3Yn2(0)j0i(3.27) we can express the cross section Eq. ( 3.24) as d d1adE1;2;3d2 1;2;3=L2NFNCCF 6Q2X n1;2;3X !1;2;3Z d4xei[Q (!1n1 !2n2+!3n3)=2]x(3.28) C (n1;2;3;!1;2;3)C (n1;2;3;!1;2;3)n =1 2 n =2 2 g ? Z dJdS(1 a J S)3Y i=1 Ei !i 2 2( i ni) Zd4k1 (2)4e ik1xZd4k2 (2)4e ik2xZd4k3 (2)4e ik3xZd4r (2)4e irx Jn1;!1(J;n1k1)Jn2; !2(n2k2)Jn3;!3(n3k3)S(S;r); where now Pi=Ei(1;ni). The quark and antiquark jet functions are now for a single avor, and we have summed over NF avors to obtain the factor in front. The jet functions depend only on the smallest component of momentum nikiin each collinear direction. The residual and soft momenta appearing in the exponentials can be reabsorbed into the label momenta by a series of reparameterizations of the label momenta and directions, under which the SCET Lagrangian is invariant [ 70]. The three-jet operator Eq. ( 3.16) will receive corrections of order 2(which we can drop) under the reparameterizations we perform below, or can be constructed from the outset to be explicitly reparameterization invariant (RPI) [ 66]. First, collect the residual and soft momenta together by dening K=k1+k2+k3+r. We can decompose xinn1light-cone coordinates, so e iKx=e i(n1Kn1x=2+n1Kn1x=2+K?1x?1): (3.29) Performing a type-A transformation (in the language of [ 70]) on the label momentum ~p1=!1n1=2, !1!!1+ n1K; (3.30) and a type-IB transformation on the vector n1itself, n1!n1+ ?;?= 2 !1K?; (3.31) shifts the label momentum on the jet function 1 by !1n1=2!(!1+ n1K)n1=2 +K?1. The summation variables n1;!1can then be shifted to eliminate n1KandK?1from the exponentials entirely. We drop all corrections suppressed by 2due to these shifts. It remains to absorb the n1Kcomponent of residual and soft momentum, appearing in the exponential factor e in1Kn1x=2. This cannot be achieved by RPI transformations { 24 {in then1sector since this momentum is purely residual|there is no label momentum in this direction. However, in a multijet cross section, n1can be written as a linear combination of n2;n3, and, say, n?2(a unit vector transverse to n2;n2), so that RPI transformations on !2;!3andn2similar to those above can absorb n1Kinto the label momenta also. Then, the x-dependent residual and soft exponentials all disappear, and we can combine the nine super uous nikiandk?iintegrals with the nine discrete sums over label directions and momenta to give continuous integrals over total momenta. Performing these with the remaining energy and direction delta functions in Eq. ( 3.28) and thex integral with the remaining exponential gives the momentum conservation delta function 4(Q E1n1 E2n2 E3n3). The resulting cross section Eq. ( 3.28) takes the form d d1adE1;2;3d2n1;2;3=d(0) dE1;2;3d2n1;2;3H(n1;2;3;E1;2;3)Z dJdS(1 a J S) Zdn1k1 2Zdn2k2 2Zdn3k3 2Zd4r (2)4 Jn1;2E1(J;n1k1)Jn2;2E2(n2k2)Jn3;2E3(n3k3)S(S;r);(3.32) where we used that the matching coecients C (ni; ~pi) are such that, by construction, the right-hand side at tree-level is simply the Born cross section (denoted by (0)) for e+e !qqgtimes(1 a). The hard function H= 1 +O(s) is determined by calculating the matching coecients Corder-by-order in perturbation theory. The above may be easily modied to describe the antiquark or gluon jet angularities, by moving the appropriate delta function (i a a(jeti)) into the J2orJ3jet functions. In addition, we may choose from among various jet algorithms. The choice determines what -function restrictions must be inserted into the nal state phase space integrations created by cutting the jet and soft diagrams to determine which particles make it into the jet. As long as the algorithm is such that the approximations enumerated above hold, it will not violate factorization of the jet shape cross section. We will discuss factorization and its potential breakdown in the context of particular jet algorithms in more detail in Sec.3.4. Another check of the validity of the factorization theorem is that the factorized jet and soft functions be separately IR safe, which is a stronger condition than the full cross section being IR safe. If the observable [ 54,71] or algorithm [ 32] too strongly weights nal states with narrow jets whose invariant masses are the same as the virtuality of soft particles, then the jet and soft functions for the observable in standard SCET with dimensional regularization become IR divergent. When this occurs it does not necessarily mean that factorization is not possible; but at least not in the standard form derived from the version of SCET we utilized above. It could be, for example, that a scheme to further separate modes by dening the theory with an explicit cuto [ 32] or by factorizing modes by rapidity instead of virtuality [ 65] can restore a version of the factorization theorem. We leave an explicit study of which algorithms and observables give IR safe jet and soft functions in SCET in dimensional regularization for a separate publication. However, we note here that { 25 {the algorithms and observables ( afora <1) that we consider in this paper, at least at NLO, do give rise to IR safe jet and soft functions. 3.3 Jet Shapes in e+e !Njets To generalize the result Eq. ( 3.32) to an arbitary number Nof jets, we simply add more quark and gluon jet elds to the operator matching in Eq. ( 3.16), resulting in the corre- sponding number of additional quark and gluon jet functions in Eq. ( 3.32), along with a hard coecient and a soft function for Njet directions. Consider an event with 2 Nqquark and antiquark jets and Nggluon jets, where 2 Nq+Ng=N. Furthermore, we can choose to measure the angularity shape for any number of these jets. Achieving a factorization theorem that remains consistent for any of these combinations is a nontrivial task and thus a powerful test of the validity of the eective theory. For anN= 2Nq+Ngjet event, we generalize the matching of the QCD current in Eq. ( 3.16) to: j(x) =X n1nNX ~p1~pNei(~p1++~pN)xCa1aNqb1bNqA1ANg 1Nq1Nq1Ng(n1;~p1;;nN;~pN) NqY i=1iaini;pi(x)NqY j=1jbj nj; ~pj(x)NgY k=1(gB?kAk nk; ~pk)(x);(3.33) with implicit sums over Dirac spinor indices i;j, Lorentz indices k, (anti-)fundamental color indices ai;bj, and adjoint color indices Ak. TheNjet cross section dierential in M jet shapes, with M <N , factors in SCET into the form d(E1;n1;;EN;nN) d1a1dMaM=(0)(E1;n1;;EN;nN)HaibjAk(n1;E1;nN;EN) MY l=1Z dl Jdl S(l a l J l S)Zdn1k1 2ZdnNkN 2 Jf1 n1;2E1(1 J;n1k1)JfM nM;2EM(M J;nMkM) JfM+1 nM+1;2EM+1(nM+1kM+1)JfN nN;2EN(nNkN) Zd4r (2)4SaibjAk(1 S;:::;M S;r); (3.34) where(0)is the Born cross section for e+e toNqquarks,Nqantiquarks, and Nggluons; the color indices on the hard and soft functions HandSallow for color mixing; and fiis the avor of each jet function (quark, antiquark, or gluon). Hitself is determined by calculating the matching coecients Cin Eq. ( 3.33). The jet functions have the same denitions given in Eq. ( 3.26), and the soft function is given by the appropriate generalization of Eq. ( 3.27) with Wilson lines in the directions and color representations corresponding to the choice of elds in Eq. ( 3.33). We rearrange the order of avor and color indices in the hard and soft functions to agree with the choices of avor indices on the jet functions. { 26 {3.4 Do Jet Algorithms Induce Large Power Corrections to Factorization? In this section we explore when power corrections to the factorization theorem above be- come large, in particular those that are due to the action of jet algorithms. We will argue that power corrections to jet angularities induced by the commonly-used cone and recom- bination algorithms remain suppressed as long as Ris suciently large. In particular, we need in general that RsatisesR&and, for the case of the k Talgorithm, we require R.9These power corrections are associated with assumptions we made about the action of the jet algorithm on nal states in deriving Eq. ( 3.34). In general, the size of these power corrections depends both on the algorithm and the observable. Power corrections to the pTof a jet arising from perturbative emissions (as well as from hadronization and the underlying event in ppcollisions) for various jet algorithms were explored in [ 36]. These power corrections arise for similar reasons as those we discuss below, namely, perturbative emissions changing which partons get combined into the jet. Ref. [ 36] nds that such power corrections scale like ln Rfor smallR. This result is consistent with our qualitative discussion below, where we argue that power corrections to jet angularities arising from the jet algorithms we use are minimized when Ris suciently large. For us, Rshould be at least O() (and for the case of the k Talgorithm,R). One set of power corrections that is independent of the choice of algorithm arises from the approximation of setting the jet axis equal to the label direction n. Since this neglects the eects of soft particles, it is valid up to O(2) corrections. It was argued in Refs. [ 17,29,41] for the case of hemisphere jet algorithms that these corrections in turn induce corrections to the angularity aof order2(2 a), which, for a2, are subleading fora < 1. Essentially the same arguments can be applied to all of the algorithms we consider. Jet algorithm-dependent power corrections arise from the dierence in soft particles included in a jet by a given algorithm and those included in the soft function in the factorization theorem. The algorithms also dier amongst themselves in which soft particles they include in a jet. For observables that scale as O(1), such as the jet energies and 3- momenta, the contribution of soft momenta can be neglected since they scale as O(2). Clearly then, these observables are not dependent on our choice of jet algorithm and so the assumptions we made about factorization of the algorithm in deriving Eq. ( 3.34) are trivially satised. However, for observables that scale as O(2) such as angularities, soft contributions become important and so the details of the algorithms we consider become relevant. We now estimate how closely the phase space region included in the soft function in the SCET factorization theorem approximates the region of soft particles actually included by the jet algorithm. We will argue that unless R&O() for the anti-k Tand infrared-safe cone algorithms, and Rfor the k Talgorithm, the mismatch in areas is suciently large 9We noted earlier that there may be dierent 's for the SCET modes describing dierent jets. For measured jets, 2a, while for unmeasured jets, tan(R=2), and for soft gluons outside jets, 2=Q. In this section, we mean by the expansion parameter associated with measured jets, and ensuring Ris much larger than this . But ifRistoolarge, the separation parameter t/1=tan(R=2) becomes too small. We will consider R0:4 to 1 to be safe. { 27 {(B)Rθij}θij anti-k Talg .Parent Location (before collinear splitting)Additional Soft Region of AlgorithmSoft Region Common to Both Algorithm and SCET Factorization Thm. Daughter Location (after collinear splitting) (A)θij} RR kTalg .Figure 2: Dierence between regions of soft radiation included in the SCET factorization theorem and the actual (A) k Tand (B) anti-k Talgorithms. We illustrate how the regions of soft radiation included by the algorithms change when a single, energetic parent particle splits into two collinear daughters. Both the algorithm and the soft function merge soft particles contained in the large white circle. The algorithm also merges the hatched area and hence contains a region of phase space which is dierent than that included in the SCET soft function (since, as explained in the text, the shape and size of the region used in SCET cannot depend on the details of collinear splittings). so as to cause a leading-order power correction to the measured jet angularities. For R larger than these bounds, the corrections are negligible. This miscounting arises due to the fact that factorization requires that collinear particles be combined rst, and that the soft function only knows about the parent collinear direction. When algorithms do not obey this ordering, factorization may be violated. To determine the size of the soft particle phase space region for each jet algorithm that is not included by the factorization theorem, we consider the situation depicted in Fig. 2. A parent collinear particle splits into two daughter collinear particles. In the factorization theorem, since collinear particles are combined rst, the region of phase space where soft particles are combined into the jet is a circle of radius Rabout the parent particle direction. However, in a jet algorithm, soft particles in an additional region outside of this may also be combined into the jet (the hatched regions in Fig. 2). If the area of this region is of the same order as the area included by the factorization theorem, then the power corrections to jet angularities induced by the jet algorithm will be leading-order. Because soft particles have momenta that are parametrically smaller than collinear particle momenta, we determine the omitted region of soft particle phase space by con- sidering the dominant action of the jet algorithm. The k Talgorithm serves as a useful example. The k Tmetric between a pair of soft particles is O(2)ss, the metric between a soft particle and collinear particle is O(2)cs, and the metric between a pair of collinear particles isO(0)cc. Therefore, collinear-collinear recombinations only occur if the angle ccbetween the collinear particles is smaller than the separation between any soft parti- cle and its nearest neighbor by a factor of O(2). Given that the typical angle between collinear particles is O(), the dominant action of the jet algorithm is to rst merge all soft particles with their nearest neighbors, while collinear-collinear recombinations occur late in the operation of the algorithm. This description will suce to accurately determine the area of the omitted soft phase space for the k Talgorithm. Since collinear particles are combined last, on average soft particles within circles of radius Rabout the collinear { 28 {daughter particles are included in the k Talgorithm jet,10as shown in Fig. 2A. The area that the k Talgorithm includes that the soft function does not is represented by the hatched region, which is an area of O(ijR). This area must be parametrically smaller than the area included by the soft function (of O(R2)) for the associated power corrections to be small. We thus require that Rijin the SCET power counting. The anti-k Talgorithm combines particles in a manner that is closer to respecting factorization. It nds the hardest particle rst and merges particles at successively larger distances from this particle. For the example of two collinear daughters, it will merge all soft particles with the hardest daughter that are closer than the distance to the softer daughter before merging the two daughters and then merging all soft particles a distance Rfrom the merged daughters (i.e., the parent particle), as shown in Fig. 2B. As the Figure illustrates, the hatched area of the anti-k Tjet tends to be smaller than that of the k T jet. In fact, for R > 3ij=2, this region vanishes completely (and this case of having only two collinear daughters is a worst-case scenario). This leads us to expect that, for any number of collinear splittings, for R&(i.e., not necessarily parametrically larger), power corrections due the action of the anti-k Talgorithm vanish. Cone algorithms such as the SISCone algorithm can also include regions that dier from the lowest-order region at higher orders in perturbation theory. We now argue that an arithmetic bound R&is sucient to minimize the power correction from these dierences, as for the anti-k Talgorithm. This situation arises due to the fact that stable solutions may exist with overlapping cones when collinear splittings are larger than the cone radius, i.e., R<ij. In these cases, the amount of radiation that falls into the overlapping region is used to decide whether the cones are split or merged. In either case, the boundary of the resulting jet(s) has roughly the appearance of Fig. 2A and the dierence between the region of soft radiation assumed in SCET and that by the actual algorithm is O(1). However, for R > ijfor the case of a single collinear splitting, all of the collinear radiation lies within a region of size Rand there will always be a stable cone that includes this radiation and thus the algorithm and the SCET soft function will be sensitive to soft particles in the same region of phase space. In summary, we have argued that for all the algorithms we consider (k T, anti-k T, infrared-safe cone), power corrections are negligible for suciently large R. While anti-k T and cone allow simply R&instead ofRas for k T, we will in fact always consider R(for thein a measured jet sector) in the remainder of this paper, guaranteeing small power corrections for all these algorithms. ( Rstill determines the scale in an unmeasured jet sector.) Our focus will remain on resumming logs of jet shapes such as angularities ain the presence of jet algorithms, without worrying about resumming logs ofRthemselves.11 10Soft particles in this region can also be removed from this region by merging with other soft particles outside of the region and vice-versa, but this average area suces for our discussion. 11Because small R(.0.3) jets cannot be well resolved in current experiments, resummation of logarithms ofRis not of primary practical importance in the near future. { 29 {4. Jet Functions at O(s)for Jet Shapes In this section, we calculate the quark and gluon jet functions for jet shapes at next-to- leading order in perturbation theory. The jet functions can be divided into two categories: those for measured jets, which are xed to have a specic angularity a, and those for unmeasured jets, which are not. We will denote the quark jet function by Jq !, the gluon jet function by Jg !, where!is the label momentum, and a jet function Jq;g(a) with an argument of adenotes a measured jet. We will calculate the jet functions for the two classes of jet algorithms, k T-type and cone-type algorithms. In the course of these calculations, we will demonstrate the crucial role of zero-bin subtractions [ 65] from collinear jet functions in obtaining the consistent anomalous dimen- sions and the correct nite parts. In this case zero-bin subtractions are not merely scaleless integrals converting IR to UV divergences, but in fact contribute part (sometimes the most important part) of the correct nonzero result, as was already pointed out by [ 32,72]. The relation of zero-bin subtractions in SCET to eikonal jet subtractions from soft functions in traditional methods of QCD factorization was explored in [ 41,73,74]. In addition, we nd that the zero-bin subtraction removes the contribution of collinear emissions that escape a jet, leaving only power-suppressed pieces in =!i. 4.1 Phase Space Cuts To calculate the jet functions for a particular algorithm, Figure 3: A representa- tive diagram for the NLO quark and gluon jet func- tions. The incoming mo- mentum isl=n 2!+n 2l+and particles in the loop carry momentum q(\particle 1") andl q(\particle 2").we must impose phase space restrictions in the matrix ele- ment. From the jet function denitions, Eq. ( 3.26), these cuts take two forms. One kind, imposed by the operator N(^J);1 in Eq. ( 3.26), is common to every jet function. It is the set of phase space restrictions related to the jet algorithm, and requires exactly one jet to arise from each collinear sector of SCET. The other, imposed by the operator (a ^a), is im- plemented only on measured jets and restricts the kinematics of the cut nal states to produce a xed value of the jet shape. In this section we describe these phase space cuts in detail. The typical form of the NLO diagrams in the jet functions is shown in Fig. 3. As shown in the gure, the momentum owing through the graph has label momentum l nl=!and residual momentum l+nl, and the loop momentum is q. We will label \particle 1" as the particle in the loop with momentum qand \particle 2" as the particle in the loop with momentum l q. For the quark jet, we take particle 1 as the emitted gluon and particle 2 as the quark. As usual, the total forward scattering matrix element can be written as a sum over all cuts. Cutting through the loops corresponds to the interference of two real emission diagrams, each with two nal state particles, whereas cutting through a lone propagator that is connected to a current corresponds to the interference between a tree-level diagram and a virtual diagram, each with a single nal state particle. Thus, the phase space restrictions and measurements we impose act dierently depending on where the diagrams { 30 {are cut. In addition, since we will be working in dimensional regularization (with d= 4 2), which sets scaleless integrals to zero, the only diagrams that contribute are the cuts through the loops. This means that we only need to focus on the form of phase-space restrictions and angularities in the case of nal states with two particles. The regions of phase space for two particles created by cutting through a loop in the jet function diagrams can be divided into three contributions: 1. Both particles are inside the jet. 2. Particle 1 exits the jet with energy E1<. 3. Particle 2 exits the jet with energy E2<. In contributions (2) and (3), the jet has only one particle, which is the remaining particle withE > . It is well known that collinear integrations of jet functions can be allowed to extend over all values of loop momenta so long as a \zero-bin subtraction" is taken from the result to avoid double counting the soft region already accounted for in the soft function [ 65]. We will demonstrate that contributions (2) and (3) are power suppressed by O(=!), which scales as2, after the zero-bin subtraction. The phase space cuts that enforce both particles to be in the jet depend on the jet algorithm. There are two classes of jet algorithms that we consider, cone-type algorithms and (inclusive) k T-type algorithms, and all the algorithms in each class yield the same phase space cuts. We label the phase space restrictions as coneand kT, generically alg. For the cone-type algorithms, conecone(q;l+) = tan2R 2>q+ q tan2R 2>l+ q+ ! q : (4.1) These functions demand that both particles are within Rof the label direction. For the kT-type algorithms, the only restriction is that the relative angle of the particles be less thanR: kTkT(q;l+) = 0 @cosR<~ q~l q2 qq l2+q2 2~ q~l1 A = tan2R 2>q+!2 q (! q )2 : (4.2) In the second line we took the collinear scaling of q(q+q ). While this is not strictly needed, it makes the calculations signicantly simpler. For the phase space restrictions of zero-bin subtractions, we take the soft limit of the above restrictions. The zero-bin subtractions are the same for all the algorithms we consider. For the case of particle 1, which has momentum q, the zero-bin phase space cuts are given by (0) alg= (0) cone= (0) kT= tan2R 2>q+ q : (4.3) { 31 {(B) (A) (D) (C) (A) (A)Figure 4: Diagrams contributing to the quark jet function. (A) and (B) Wilson line emission diagrams; (C) and (D) QCD-like diagrams. The momentum assignments are the same as in Fig. 3. The zero bin of particle 2 is given by the replacement q!l q. For all the jet algorithms we consider, the zero-bin subtractions of the unmeasured jet functions are scaleless integrals.12However, for the measured jet functions, the zero-bin subtractions give nonzero contributions that are needed for the consistency of the eective theory. In the case of a measured jet, in addition to the phase space restrictions we also demand that the jet contributes to the angularity by an amount awith the use of the delta function R=(a ^a), which is given in terms of qandlby RR(q;l+) = a 1 !(! q )a=2(l+ q+)1 a=2 1 !(q )a=2(q+)1 a=2 :(4.4) In the zero-bin subtraction of particle 1, the on-shell conditions can be used to write the corresponding zero-bin -function as (0) R= a 1 !(q )a=2(q+)1 a=2 ; (4.5) (and for particle 2 with q!l q). 4.2 Quark Jet Function The diagrams corresponding to the quark jet function are shown in Fig. 4. The fully inclusive quark jet function is dened as Z d4xeilxh0ja n;!(x)b n;!(0)j0iabn = 2 Jq !(l+); (4.6) and has been computed to NLO (see, e.g., [ 75,76]) and to NNLO [ 77]. Below we compute the quark jet function at NLO with phase space cuts for the jet algorithm for both the measured jet, Jq !(a), and the unmeasured jet, Jq !. As discussed above, we will nd that the only nonzero contributions come from cuts through the loop when both cut particles are inside the jet. 12Note that algorithms do exist that give nonzero zero-bin contributions to unmeasured jet functions [ 32]. { 32 {4.2.1 Measured Quark Jet The measured quark jet function includes contributions from naive Wilson line graphs (A) and (B) and QCD-like graphs (C) and (D) in Fig. 4. The sum of these contributions is ~Jq !(a) =g22CFZdl+ 21 (l+)2Zddq (2)d 4l+ q + (d 2)l+ q+ ! q 2(q q+ q2 ?) (q )(q+)2 l+ q+ q2 ? ! q (! q )(l+ q+) algR:(4.7) The contribution proportional to d 2 comes from the QCD-like graphs (C) and (D) in Fig.4. Only the Wilson line graphs have a nonzero zero-bin limit, which comes from taking the scaling limit q2of the naive contribution: Jq(0) !(a) = 4g22CFZdl+ 21 l+Zddq (2)d1 q 2(q q+ q2 ?)(q )(q+) 2 l+ q+ (l+ q+) (0) alg(0) R:(4.8) All jet algorithms that we use yield the same zero-bin contribution, since the phase space cuts are the same. To evaluate these integrals, we can analytically extract the coecient of (a) by integrating over aand using the fact that the remainder is a plus distribution, as dened in Eq. ( A.2). We nd the naive contribution is ~Jq !(a) =sCF 21 (1 ) 42 !2tan2R 2!1 2+3 2 (a) +s 2~Jq alg(a): (4.9) The only dierence between the jet algorithms that we consider resides in the nite distri- bution ~Jq alg(a), which is a complicated function of athat we give in Appendix A. Note that the divergent part of the naive contribution is proportional to (a). This is due to the fact that the jet algorithm regulates the distribution for a>0. The divergent plus distributions come entirely from the zero-bin subtraction, which is given by Jq(0) !(a) =sCF 1 (1 ) 42tan2(1 a)R 2 !2!1 (1 a)1 1+2a: (4.10) Adding the leading-order contribution to all of the NLO graphs and expanding in powers of, adopting the MS scheme, we nd the total quark jet function, Jq !(a) =(a) +~Jq !(a) Jq(0) !(a) =( 1 +sCF " 1 a 2 1 a1 2+1 a 2 1 a1 ln2 !2+3 4#) (a) sCF " 1 1 1 a(a) a# ++s 2Jq alg(a):(4.11) This agrees with the standard jet function J(k+) given in [ 75,76] by setting a= 0 and k+=!a. We have shown the divergent terms explicitly, and collect the nite pieces in Jq alg(a), which we give in Eq. ( A.14). Note that there is no jet algorithm dependence in the divergent parts of the jet function at this order in perturbation theory. { 33 {4.2.2 Gluon Outside Measured Quark Jet In this section we calculate the contribution to the quark jet function from the region of phase space in which the gluon exits the jet carrying an energy Eg<. This cut causes the contribution to be power suppressed by =!, which scales as 2. However, we elect to evaluate this case explicitly as it provides a clear example of the zero-bin subtraction giving the proper scaling to the total contribution. We only evaluate this contribution for the cone algorithm; the details of the k Talgorithm calculation are similar. Note that the contribution when the quark is out of the jet is power suppressed at the level of the Lagrangian given in Sec. 3.1, in which soft quarks do not couple to collinear partons at leading order in . For the cone algorithm, the gluon exits the jet when the angle between the jet axis, n1, and the gluon is greater than R. When the gluon is not in the jet, the cone axis is the quark direction, and so it makes no contribution to the angularity. Therefore, this region of phase space contributes only to the (a) part of the angularity distribution. For the naive contributions, requiring the gluon to be outside the jet and have energy less than , we have the integral ~Jq;out !(a) =g22CFZdl+ 21 (l+)2Zddq (2)d 4l+ q + (d 2)l+ q+ ! q 2(q q+ q2 ?) (q )(q+)2 l+ q+ q2 ? ! q (! q )(l+ q+) q+ q tan2R 2 2 q (a): (4.12) Note that the theta function requiring q <2 is more restrictive than q <!. Evaluating Eq. ( 4.12) yields a contribution that scales with only below the leading term in 1 =: ~Jq;out !(a) = sCF 21 (1 ) 42 (2 tanR 2)2! (a)1 2+1 4 ! 22 !2 +8 ! (4.13) The zero-bin subtraction of Eq. 4.12 is ~Jq;out(0) ! (a) =g22CFZdl+ 21 (l+)2Zddq (2)d 4l+ q + (d 2)l+ q+ ! q 2(q q+ q2 ?) (q )(q+)2 l+ q+ q+ q tan2R 2 2 q (a):(4.14) Evaluating Eq. ( 4.14), we nd the zero bin will exactly remove the leading term in 1 =: ~Jq;out(0) ! (a) = sCF 21 (1 ) 42 (2 tanR 2)2! (a)1 2(4.15) Therefore, the dierence is power suppressed only after the zero bin is included. Because other contributions when one particle is outside of the jet are similarly power suppressed, we will drop them in our remaining discussion of the jet functions. { 34 {4.2.3 Unmeasured Quark Jet When the angularity of a jet is not measured, the jet function has no adependence. The naive and zero-bin contributions are the same as Eqs. ( 4.7) and ( 4.8) except for the factor ofR. The zero-bin contribution is Jq(0) != 2g22CFnnZdl+ 21 l+Zddq (2)d1 q 2(q q+ q2 ?)(q )(q+) 2 l+ q+ (l+ q+) (0) alg:(4.16) This integral is scaleless and therefore equal to 0 in dimensional regularization. This implies that the NLO part of the quark jet function for an unmeasured jet is just the naive result. We nd, making the divergent part explicit, in the MS scheme, Jq != 1 + ~Jq != 1 +sCF 2( 1 2+3 2+1 ln 2 !2tan2R 2!) +s 2Jq alg; (4.17) where the nite part Jq algis given in Eq. ( A.18).13 4.3 Gluon Jet Function The diagrams needed for the gluon jet function at NLO are shown in Fig. 5. The fully inclusive jet function, dened as Z d4xeilxh0jBA ?;!(x)BB ?;!(0)j0i 1 !g ?ABJg !(l+); (4.18) (withJg !(l+) normalized to 2 (l+) at tree-level) has been calculated to NLO in Feynman gauge in [ 34,78,79] and was reported to give the same result in both Rand light-cone gauges in [ 35]. Since our phase space restrictions and the observables act dierently on cuts through loops and on cuts through external propagators, it is useful to reproduce these results by explicitly cutting the diagrams. After some algebra, we nd that the sum of all cuts through the loops of the na ve collinear graphs gives Zdl+ 2~Jg !(l+) =22g2 (2)d 1Zdl+ l+Z ddq(q2)((l q)2) (! q ) TRNf 1 2 1 q+q !l+ CA 2 ! q ! ! q q+q !l+ :(4.19) We also record the zero bin that needs to be subtracted from Eq. ( 4.19). To leading-power it is given by Zdl+ 2Jg(0) !(l+) =2CA2g2 (2)d 1Zdl+ l+Z ddq (q2)(l+ q+)(q )1 q +((l q)2)(q+)(! q )1 ! q : (4.20) 13The unmeasured jet function Eq. ( 4.17) is not simply obtained by integrating the measured jet function Eq. ( 4.11) overa. This is due to the dierent relative scaling of Rwith the SCET expansion parameter iin a measured and unmeasured jet sector, as noted earlier. Namely, R0 iin a measured jet sector (wherepa) whilektan(R=2) in an unmeasured jet sector. { 35 {Figure 5: Diagrams contributing to the gluon jet function. (A) sunset and (B) tadpole gluon loops; (C) ghost loop; (D) sunset and (E) tadpole collinear quark loops; (F) and (G) Wilson line emission loops. Diagrams (F) and (G) each have mirror diagrams (not shown). The momentum assignments are the same as in Fig. 3. Without inserting any additional constraints, this integral is scaleless and zero in dimen- sional regularization. Therefore, in the absence of phase-space restrictions, the na ve inte- gral Eq. ( 4.19) gives the standard (inclusive) gluon jet function Jg !(l+) 2!=s 42(!l+) 1 TRNf4 3+20 9 CA4 +11 3+67 9 2 ;(4.21) in the MS scheme. The measured and unmeasured jet functions are obtained by inserting algRand alg, respectively, into Eqs. ( 4.19) and ( 4.20). 4.3.1 Measured Gluon Jet The naive contribution to the measured gluon jet can be written as ~Jg !(a) =s 21 (1 )42 !21 1 a 21 a1+2 2 aZ1 0dx(xa 1+ (1 x)a 1)2 2 a(4.22) TRNf 1 2 1 x(1 x) CA 2 1 x(1 x) x(1 x) alg(x); wherexq =!. This gives ~Jg !(a) =s 21 (1 ) 42 !2tan2R 2! (a)" CA1 2+11 61 2 3TRNf# +s 2~Jg alg(a); (4.23) where, as for the quark jet function, the nite distributions ~Jg alg(a) dier among the algorithms we consider. They are given in Appendix A. The zero-bin result is Jg(0) !(a) =sCA 1 (1 ) 42tan2(1 a)R 2 !2!1 a1+21 (1 a): (4.24) { 36 {Subtracting the zero-bin from the naive integral and adding the leading-order contribution, we obtain in MS Jg !(a) =(a) +~Jg !(a) Jg(0) !(a) =( 1 +sCA " 1 a=2 1 a1 2+1 ln2 !2 +11 121 # s 3TRNf1 ) (a) sCA 1 1 a1 (a) a ++s 2Jg alg(a): (4.25) The nite distribution Jg alg(a) is given in Eq. ( A.14). 4.3.2 Unmeasured Gluon Jet As for the quark jet function, for unmeasured jets the naive and zero-bin contributions are given by the measured jet contributions without the Rconstraint. Also, as it was for the quark jet function, the zero-bin contribution to the unmeasured jet function is a scaleless integral. Thus, the nal answer is just the naive result, which is given by Jg != 1 +s 2" CA 1 2+11 61 +1 ln2 !2tan2R 2! 2 3TRNf# +s 2Jg alg; (4.26) with the nite part Jg alggiven in Eq. ( A.29) in the Appendix. 5. Soft Functions at O(s)for Jet Shapes In this section we compute the soft function for a generic Njet event. In Sec. 5.1, we describe the phase space cuts that we impose on soft particles emitted into the nal state. We then give an outline of how we disentangle the various contributions to the N-jet soft function in Sec. 5.2and proceed to calculate these contributions in the remaining subsections. 5.1 Phase Space Cuts Soft particles in the nal state must satisfy the phase space cuts required by the operator N(^J);0in Eq. ( 3.27), which constrains the soft particles to not create an extra jet. A soft particle is allowed in the nal state if it is emitted into one of the jets as dened by the jet algorithm, or if it is not in a jet but has energy less than a cuto . At NLO, only a single soft gluon can be emitted. Therefore, for both cone-type and k T-type algorithms, the constraint that the soft gluon be in a jet is simply that the angle of the gluon with respect to the jet axis satises gJ<R. Thus, our requirement on soft gluons is that they obey one of the two following conditions: in jeti:gJi<R out of all jets: Eg< andgJi>R for alli: (5.1) { 37 {Figure 6: Soft function real-emission diagrams. Diagrams (A) and (C) are interference diagrams of Wilson line emission from lines iandjand (B) and (D) are from lines iandk. The shaded region in the center represents the region of phase space corresponding to jet kdened by the jet algorithm, and so the gluons in diagrams (A) and (B) are inside jet kand those in (C) and (D) are not. Each diagram has a corresponding mirror diagram (not shown). These conditions can be written in terms of theta functions on the gluon momentum k. We denote the energy restriction for out-of-jet gluons as (k0<); (5.2) and we denote the requirement that a gluon be in jet iin terms of the light-cone components kabout the direction of jet i,ni, as i Rk+ k <tan2R 2 : (5.3) For the case when the soft gluon is in a measured jet, we demand that it contributes an amount ato the angularity of a jet with label momentum !with the use of the delta function R a 1 !(k )a=2(k+)1 a=2 : (5.4) 5.2 Calculation of contributions to the N-Jet Soft Function The topology of the various graphs that we need to compute the soft function is shown in Fig. 6. The next-to-leading order part S(1)of the soft function Sis the sum of the interference of soft gluon emissions from Wilson lines in directions iandj,Sij, over all linesiandjwithi6=j(since fori=j, the diagram is proportional to n2 i= 0), S(1)=X i6=jSij: (5.5) It is conceptually straightforward to see that Sijis just the sum of the following three classications of the nal state cut gluon: The gluon is in a measured jet and thus contributes to the jet angularity. The gluon is outside of all the jets and has energy Eg<. The gluon is in an unmeasured jet and has any energy. { 38 {However, the second contribution is technically dicult to compute due to the complicated shape of the space with all jets cut out of it, like Swiss cheese. A division of phase space leading to a simpler calculation is the following: Smeas ij(k a): The gluon is in measured jet kand contributes to the jet's angularity k a. Sk ij: The gluon is in jet kwith energy Eg> (and does not contribute to k a). Sk ij: The gluon is in jet kwith energy Eg< (and does not contribute to k a). Sincl ij: The gluon is anywhere with Eg< (and does not contribute to any angularity). In terms of these pieces, the NLO soft function with Mmeasured jets and N Munmea- sured jets is given by S(1)(1 a;2 a;:::;M a) =X i6=j2 664X k2measSmeas ij(k a)MY l=1 l6=k(l a)3 775 +X i6=j" Sincl ij X k2measSk ij+X k=2measSk ij!MY l=1(l a)3 5:(5.6) From the denitions above, it is easy to see that the term in large parentheses on the second line is equivalent to the sum of the last two contributions on the original list above, i.e., the contributions from a gluon not in any jet with Eg< and from a gluon in an unmeasured jet with any energy. We can simplify this expression by noting that the contribution from a gluon in jet kwith no energy restriction involves a scaleless integral over the energy that vanishes in dimensional regularization and thus Sk ij+Sk ij= 0: (5.7) Using this relation, the soft function simplies to S(1)(1 a;:::;M a) =Sunmeas (1)MY l=1(l a) +X k2measSmeas (1)(k a)MY l=1 l6=k(l a); (5.8) where the rst term in Eq. ( 5.8) is a universal contribution that is needed for every N-jet observable, dened as Sunmeas (1)X i6=j Sincl ij+NX k=1Sk ij : (5.9) The second term, dened as, Smeas (1)(k a)X i6=jSmeas ij(k a); (5.10) { 39 {depends on our choice of angularities as the observable. We now proceed to set up the one-loop expressions for the contributions in Eq. ( 5.8). The integrals involved are highly nontrivial and so in this section we simply quote the result of each integral, referring the reader to Appendix Bfor details. Most of these integrals are most easily written in terms of the variable tij, dened in Eq. ( 1.4) astijtan ij 2=tanR 2, where ijis the angle between jet directions iandj. (That is, ninj= 1 cos ij.) In Table 2, we summarize the divergent parts of the soft function. The Feynman rules for the emission of a soft gluon from fundamental- and adjoint- representation Wilson lines (corresponding to quark and gluon jets, respectively) have the same kinematic structure. The dierence is entirely encoded in the color charge of the Wilson lines which, using the color space formalism of [ 80,81], we denote as Tifor emission from Wilson line i. Thus, we can consider the N-jet soft function without specifying the color representation of the nal-state jets. 5.2.1 Inclusive Contribution: Sincl ij The contribution to the soft function from a gluon going in any direction with energy Eg< is given by the integral Sincl ij= g22TiTjZddk (2)dninj (nik)(njk)2(k2)(k0) : (5.11) We evaulate this integral in Sec. B.1of the Appendix and nd Sincl ij= s 2TiTj (1 )42 421 2 1 lnninj 2 2 6 Li2 1 2 ninj :(5.12) Note that this calculation is related to the inclusive, timelike soft function that has applications elsewhere (see, e.g., [ 82,83,84]), generalized for non back-to-back jets: dSincl ij d= g22TiTjZddk (2)dninj (nik)(njk)2(k2)(k0)(k0 ): (5.13) 5.2.2 Soft gluon inside jet kwithEg>:Sk ij Using Eq. ( 5.7), the contribution Sk ijfrom a gluon emitted into jet kfrom linesiandjis given by the integral Sk ij=g22TiTjZddk (2)dninj (nik)(njk)2(k2)(k0) k R: (5.14) Much like for the Smeas ij contribution, if k=i;j, there is an additional divergence (arising fromnkk!0) relative to the case k6=i;j, and so we evaluate these two cases separately below. { 40 {Case 1:k=iorjThe integrals for this case are performed in Sec. B.2of the Appendix, with the result that Sj ijis Sj ij=Si ij=sTiTj 4" 1 21 (1 )42 42 t2 ij t2 ij 1tan2R 2! + Li 2 1 t2 ij 1 + 2 Li 2 1 cos2 ij 2(t2 ij 1)# : (5.15) Case 2:k6=i;j These contributions are at most 1 =divergent since the matrix element does not have the nkk!0 singularity. We show in Appendix B.3.2 that the result takes the form Sk ij k6=i;j= s 4TiTj1 lnt2 ikt2 jk 2tiktjkcosij+ 1 (t2 ik 1)(t2 jk 1) +F(tik;tjk;ij) ;(5.16) whereijis the angle between the i-kandj-kplanes and the nite function Fis given in Eq. ( B.33) and isO(1=t2). 5.2.3 Soft gluon inside measured jet k:Smeas ij(k a) The contribution of a gluon emitted into jet 1 (the measured jet) from lines iandjis given by the integral Smeas ij(k a) = g22TiTjZddk (2)dninj (nik)(njk)2(k2)(k0) k RR: (5.17) The singularity structure of this integral depends on whether or not k=iorj. Thus, we evaluate the case k=iorjand the case k6=i;jseparately below. Case 1:k=iorjWe consider rst Smeas ij(i a). Using the results of Sec. B.2of the Appendix, we obtain the result in terms of tij, Smeas ij(i a) =Smeas ji(i a) = s 2TiTj1 1 1 a1 ia1+21 (1 )42 !2t2 ij t2 ij 1tan2R 2(1 a) +1 +a 2(i a) Li2 1 t2 ij 1 : (5.18) Case 2:k6=i;j The remaining contributions to the observed jet angularity are Smeas ij fork6=i;j. Using the results from Sec. B.3.3 in the Appendix, this contribution is Smeas ij(k a) i;j6=k= s 2TiTj1 ka1+2 lnt2 ikt2 jk 2tiktjkcosij+ 1 (t2 ik 1)(t2 jk 1) +(k a)G(tik;tjk;ij) ; (5.19) whereGisO(1=t2) and is given in Eq. ( B.36) and, again, ijis the angle between the i-k andj-kplanes. { 41 {contribution divergent terms Sincl ij 1 s 2TiTj 1 lnninj 2+ ln2 42 Si ij1 s 4TiTj 1 lnt2 ijtan2(R=2) t2 ij 1+ ln2 42 Sk ij 1 s 4TiTjlnt2 ikt2 jk 2tiktjkcosij+1 (t2 ik 1)(t2 jk 1) Sunmeas (1)1 s 2hPN i=1T2 iln tan2(R=2) +P i6=jTiTjln(ninj=2)i +O(1=t2) Smeas ij(i a)1 s 4TiTjh 1 1 a 1 + ln2 !2 i + lnt2 ijtan2(R=2) t2 ij 1 (i a) 2 1 a 1 ia +i Smeas ij(k a)1 s 4TiTjlnt2 ikt2 jk 2tiktjkcosij+1 (t2 ik 1)(t2 jk 1)(k a) Smeas (1)(i a) 1 s 2T2 ih 1 1 a 1 + ln2 !2 i + ln tan2(R=2) (i a) 2 1 a 1 ia +i +O(1=t2) Table 2: Summary of the divergent parts of the soft function at NLO. The rst block contains the the observable-independent contributions: soft gluons emitted by jets iandjin any direction with energyEg< in row 1; soft gluons entering jet kwithEg> (withk=iorjin the second row andk6=i;jin the third). In the last row of the rst block, we summed over iandjand took the large-tlimit to get the total Sunmeas (1). Similarly, in the second block we give the contributions to the angularities k a(withk=iorjin the rst row and k6=i;jin the second) and summed over i andjand took the large- tlimit to get Smeas (1)in the third row. 5.3 Total N-Jet Soft Function in the large- tLimit In this section, we add together the necessary ingredients calculated above to obtain the N-jet soft function from Eq. ( 5.8). The results for the divergent pieces are summarized in Table 2. In Sec. 6we use Table 2to show that the consistency of factorization is explicitly violated by terms of order 1 =t2, and so in this section we give the full soft function (including the nite terms) to O(1=t2). Using color-conservation (P iTi= 0), we nd that the observable-independent part, Sunmeas (1), dened in Eq. ( 5.9), is given for large tby Sunmeas (1) =s 2X iT2 i" 1 ln2 42 1 ln 2 42tan2R 2! (5.20) +1 2ln22 42 1 2ln2 2 42tan2R 2! 2 6# +s 2X i6=jTiTj" 1 lnninj 2+ ln2 42 lnninj 2 + Li 2 1 2 ninj# +O(1=t2): { 42 {We see that the nite parts of this contribution are sensitive to two scales, 2 and 2 tanR 2. For simplicity, in this paper, since we take tan( R=2)O(1), we will choose only a single scale Sto attempt to minimize logs in Eq. ( 5.20), where S2 tan1=2R 2; (5.21) chosen as the geometric mean of the two. The remaining part of the soft function that is dependent on angularities as our choice of jet observable is the sum over Smeas (1)(i a) (dened in Eq. ( 5.10)) for each jet i, where Smeas (1)(i a) is given by Smeas (1)(i a) = s 2T2 i1 1 a(1 2+1 ln2 !2 itan2(1 a)R 2 2 12 +1 2ln22 !2 itan2(1 a)R 2 (i a) (5.22) 2" 1 + ln2tan2(1 a)R 2 (!iia)2! (i a) ia# +) +O(1=t2): The nite part of this contribution is sensitive to the scale i S, where i S!ii a tan1 aR 2; (5.23) which, in principle, diers for each jet iand from the scale Sin the unmeasured part of the soft function Eq. ( 5.20). After discussing resummation of large logarithms through RG evolution, we will describe in Sec. 6.4a framework to \refactorize" the soft function into pieces depending on multiple separated soft scales. This framework will enable us to resum logarithms of all of these potentially disparate scales. 6. Resummation and Consistency Relations at NLL The factorized cross section Eq. ( 3.34) is written in terms of hard, jet, and soft functions evaluated at a factorization scale . Since the physical cross section is independent of , the anomalous dimensions of these functions are closely related. We derive and verify this relation in Sec. 6.3. In Sec. 6.1and Sec. 6.2, we work out the form of the renormalization- group equations (RGEs) satised by the hard, jet, and soft functions, and obtain their solutions so that we can express each function at the scale in terms of its value at a scale 0where logarithms in it are minimized. In Sec. 6.4, we present a framework to refactorize the soft function and give the total resummed distribution in Sec. 6.5. 6.1 General Form of Renormalization Group Equations and Solutions We will build solutions for the hard, jet, and soft functions from two forms of RGEs. The rst form is for a function which does not depend on the observable aand is multiplicatively renormalized, Fbare=ZF()F(); (6.1) { 43 {and satises the RGE, d dF() = F()F(); (6.2) where the anomalous dimension Fis found from the Z-factor by the relation F() = 1 ZF()d dZF(); (6.3) and takes the general form, F() = F[] ln2 !2+ F[]: (6.4) We call F[] the \cusp part" of the anomalous dimension and F[] the \non-cusp part". They have the perturbative expansions F[s] =s 4 0 F+s 42 1 F+ (6.5) and F[s] =s 4 0 F+s 42 1 F+: (6.6) The RGE Eq. ( 6.2) has the solution F() =UF(; 0)F(0); (6.7) where the evolution kernel UFis given by UF(; 0) =eKF(;0)0 !!F(;0) ; (6.8) whereKFand!Fwill be dened below in Eq. ( 6.15). The second form of RGE is for a function dependent on the jet shape aand is renor- malized through a convolution, Fbare(a) =Z d0 aZF(a 0 a;)F(0 a;); (6.9) and satisfying the RGE d dF(a;) =Z d0 a F(a 0 a;)F(0 a;); (6.10) with an anomalous dimension calculated from F(a;) = Z d0Z 1 F(a 0 a;)d dZF(0 a;); (6.11) and taking the general form F(a;) = F[s]2 jF(a) a + ln2 !2(a) + F[s](a): (6.12) { 44 {The solution of an RGE of the form Eq. ( 6.10) has the solution [ 82,85,86,87,52] F(a;) =Z d0UF(a 0 a;; 0)F(0 a;0); (6.13) where the evolution kernel UFis given to all orders in sby the expression UF(a;; 0) =eKF+ E!F ( !F)0 !jF!F(a) (a)1+!F +; (6.14) where Eis the Euler constant. In Eqs. ( 6.8) and ( 6.14), the exponents !FandKFare given in terms of the cusp and non-cusp parts of the anomalous dimensions by the expressions !F(; 0)2 jFZs() s(0)d [] F[]; (6.15a) KF(; 0)Zs() s(0)d [] F[] + 2Zs() s(0)d [] F[]Z s(0)d0 [0]: (6.15b) In the case of Eq. ( 6.8) or if F[] happens to be zero, we dene jFto be 1. To achieve NLL accuracy in the evolution kernels UF, we need the cusp part of the anomalous dimension to two loops and the non-cusp part to one loop, in which case the parameters !F;KFin Eq. ( 6.15) are given explicitly by !F(; 0) = 0 F jF0" lnr+ 1 cusp 0cusp 1 0! s(0) 4(r 1)# ; (6.16a) KF(;0) = 0 F 20lnr 2 0 F (0)2r 1 rlnr s() + 1 cusp 0cusp 1 0! 1 r+ lnr 4+1 80ln2r : (6.16b) Herer=s() s(0), and0;1are the one-loop and two-loop coecients of the beta function, [s] =ds d= 2s 0s 4 +1s 42 + ; (6.17) where (with TR= 1=2) 0=11CA 3 2Nf 3and1=34C2 A 3 10CANf 3 2CFNf: (6.18) The two-loop running coupling s() at any scale in terms of its value at another scale Qis given by 1 s()=1 s(Q)+0 2ln Q +1 40ln 1 +0 2s(Q) ln Q : (6.19) { 45 {In Eq. ( 6.16), we have used the fact that, for the hard, jet, and soft functions for which we will solve, the cusp part of the anomalous dimension F[s] is proportional to thecusp anomalous dimension cusp[s], where cusp[s] =s 4 0 cusp+s 42 1 cusp+: (6.20) The ratio of the one-loop and two-loop coecients of cuspis [88] 1 cusp 0cusp=67 9 2 3 CA 10Nf 9: (6.21) 1 cuspand1are needed in the expressions of !FandKFfor complete NLL resummation since we formally take 2 slnaO(s). 6.2 RG Evolution of Hard, Jet, and Soft Functions 6.2.1 Hard Function The hard function is related to the matching coecient CNof theN-jet operator in Eq. ( 3.33). If there are multiple operators with dierent color structures, CNis a vec- tor of coecients. The hard function is then a matrix Hab=Cy bCa. The hard function is contracted in the cross section Eq. ( 3.34) with a matrix of soft functions. The anomalous dimensions of the matching coecients Cahave been calculated in the literature (for example, Table III of Ref. [ 89]). For an operator with Nlegs with color charges T2 i, the anomalous dimension is CN(s) = NX i=1 T2 i (s) ln !i+1 2 i(s) 1 2 (s)X i6=jTiTjln ninj i0+ 2 ; (6.22) where Tiis a matrix of color charges in the space of operators, and iis given for quarks and gluons by q=3sCF 2; g=s 11CA 2Nf 6: (6.23) The coecient ( s) is the cusp anomalous dimension and is given to one-loop order by (s) =s=. The anomalous dimension of the hard function itself is given by H= y CN+ CN, and takes the form of Eq. ( 6.4), with cusp and non-cusp parts H[s] and H[s] given to one loop in Table 3, with the result H(s) = (s)T2ln2 !2 H NX i=1 i(s) (s)X i6=jTiTjlnninj 2; (6.24) where T2=PN i=1T2 iis the sum of all the Casimirs, and the eective hard scale !H appearing as the scale !in the logarithm in Eq. ( 6.4) is given by the color-weighted average of the jet energies, !H=NY i=1!T2 i=T2 i (6.25) { 46 { F[s] F[s] jF! H P iT2 i P i i P i6=jTiTjlnninj 21 !H Ji(i a) T2 i2 a 1 a i 2 a!i meas S(i a) T2 i1 1 a0 1!itan 1+aR 2 Ji T2 i i 1!itanR 2 unmeas S 0 P iT2 iln tan2R 2+ P i6=jTiTjlnninj 21 | O(1=t2) 0 P i6=jTiTjh i=2meas2 lnt2 ij t2 ij 11 | + P k6=i;j k=2measlnt2 ikt2 jk 2tiktjkcosij+1 (t2 ik 1)(t2 jk 1)i Table 3: Anomalous dimensions for the jet and soft functions. We give the cusp and non-cusp parts of the anomalous dimensions, F[s] and F[s]. is the cusp anomalous dimension itself, equal tos=at one loop. iis given for quark and gluon jets in Eq. ( 6.23). The third column gives the value of jFappearing in Eq. ( 6.12) or Eq. ( 6.15). The last column gives the values of !appearing in the logarithm ln 2=!2multiplying the cusp part of the anomalous dimension in Eqs. ( 6.4) and ( 6.12). The scale !His the color-weighted averages of all jet energies dened in Eq. ( 6.25). All rows except for the last are given in the large- tlimit and in the last row we give the remaining terms that are present for arbitrary t. This last row explictly violates consistency at O(1=t2). The rst group of rows are needed for measured jets and the second group for unmeasured jets. In the large- tlimit, for any number of measured and unmeasured jets, the consistency relation Eq. ( 6.33) is satised. 6.2.2 Jet Functions There are two forms of jet functions, those for measured and those for unmeasured jets. Unmeasured jet functions Jq;g !() satisfy multiplicative RGEs, with anomalous dimensions given by the innite parts of Eqs. ( 4.17) and ( 4.26), Ji= (s)T2 iln2 !2 itan2R 2+ i; (6.26) which falls into the general form Eq. ( 6.4), with cusp and non-cusp parts of the anomalous dimension given in Table 3, and the scale !in Eq. ( 6.4) being!itanR 2. The part iis given by Eq. ( 6.23). Measured jet functions satisfy RGEs of the form Eq. ( 6.10), with anomalous dimensions given by the innite parts of Eqs. ( 4.11) and ( 4.25), Ji(i a) = T2 i (s)2 a 1 aln2 !2 i+ i (i a) 2 (s)T2 i1 1 a(i a) ia +; (6.27) { 47 {which takes the form Eq. ( 6.12) with cusp and non-cusp parts of the anomalous dimension split up as in Table 3, and the scale !in Eq. ( 6.12) being!i. 6.2.3 Soft Function The totalN-jet soft function is given by Eq. ( 5.20) for unmeasured jets added to the sum over measured jets of Eq. ( 5.22). This soft function depends on the Mjet shapes 1 a;:::;M a, and satises the RGE d dS(1;:::;M;) =Z d0 1d0 M S(1 0 1;:::;M 0 M;)S(0 1;:::;0 M;):(6.28) From the innite parts of the soft function given in Table 2, we nd the anomalous di- mension S(1;:::;M;) decomposes, as required by the consistency condition Eq. ( 6.33) given below, into a sum of terms, S(1;:::;M;) = unmeas S ()(1)(M) +MX k=1 meas S(k;)Y j6=k(j); (6.29) where the pieces unmeas S () and meas S(k;) are given in terms of their cusp and non-cusp parts in Table 3, with the result unmeas S () =NX i (s)T2 iln tan2R 2+ (s)X i6=jTiTjlnninj 2; (6.30) which takes the form of Eq. ( 6.4) with no cusp part, and meas S(k;) = (s)T2 k1 1 a ln 2tan2(1 a)R 2 !2 k! (k) 2(k) k + ; (6.31) which takes the form of Eq. ( 6.12) with no non-cusp part, and the scale !in Eq. ( 6.12) being!k=tan1 aR 2. The solution of the RGE Eq. ( 6.28) is given by S(1;:::;M;) =Z d0 1d0 MS(0 1;:::;0 M;0)Uunmeas S (; 0)MY k=1Uk S(k 0 k;; 0); (6.32) whereUunmeas S is given by the form of Eq. ( 6.8) andUk S(k a) by the form of Eq. ( 6.14). The solution Eq. ( 6.32) is appropriate if all the scales appearing in the soft function are similar, and thus all large logarithms in the nite part can be minimized at a single scale0. As we noted in Sec. 5.3, however, the potentially disparate scales !ii a, set by the jet shapes of the measured jets, and , set by the cuto on particles outside jets, appear together in the soft function, and logarithms of ratios of these scales may be large. In this case, the soft function should be \refactorized" into pieces depending only on one of these scales at a time. We describe a framework for doing so below in Sec. 6.4. But rst, we verify the consistency of the anomalous dimensions for the hard, jet, and soft functions to the order we have calculated them. { 48 {6.3 Consistency Relation among Anomalous Dimensions We summarize the anomalous dimensions of the hard, jet, and soft functions in Table 3. We separate contributions to the jet and soft anomalous dimensions that arise from measured jets, from unmeasured jets, and those that are universally present. In all rows except the last row, we take the large- tlimit and give the additional terms that arise (from the soft function) for arbitrary t. Consistency of the eective theory requires that the anomalous dimensions satisfy 0 = H() + unmeas S () +X i=2meas Ji() (i a) +X i2meas Ji(i a;)) + meas S(i a;) : (6.33) From the results tabulated in Table 3, up to corrections of O(1=t2), we see that, remark- ably, this relation is indeed satised! This is highly nontrivial, as jet and soft anomalous dimensions depend on the jet radius Rand the jet shape a, while the hard function does not; in addition, the hard and soft functions know the directions niof allNjets, while the jet functions do not. These dependencies cancel precisely between the R-dependent pieces of unmeasured jet contributions to the jet and soft functions, between a-dependent pieces of the measured jet contributions, and between ninj-dependent pieces of the hard and soft functions. The sum of all jet and soft anomalous dimensions then precisely matches the hard anomalous dimensions, satisfying Eq. ( 6.33). Making the satisfaction of consistency even more nontrivial, individual contributions to the innite part of the soft function, and therefore its anomalous dimension, given by Table 2depend on the energy cut parameter as well. However, these terms cancel in the sum over the contributions Sincl ijandSi ijin the rst two rows of Table 2. The double poles of the form1 ln arise from regions of phase space where a soft gluon can become both collinear to a jet direction (giving a 1 =) and soft (giving a ln ). These regions exist in the integral over all directions giving Sincl ijbut are subtracted back out in the contributions Si ij. In the surviving \Swiss cheese" region the regions giving these double poles are cut out. TheO(1=t2) terms that violate consistency arise whenever there are unmeasured jets. While this limit is not required for the contribution of measured jets to the anomalous dimension to satisfy the consistency condition Eq. ( 6.33), the nite parts of measured jet contributions to the soft function contain large logarithms of != that can not be minimized with a scale choice but are suppressed by O(1=t2) (cf. Eq. ( B.37) of Appendix B). This is the manifestation of the fact that jets need to be well-separated, as explained in Sec. 3. For the remainder of the paper, we work strictly in the large- tlimit. It may seem mysterious that the calculations of the hard, jet, and soft functions them- selves and requiring their consistency lead to the condition of a large separation parameter t. Although we already specied qualitatively in the proof of factorization the requirement of well-separated jets, it may not be immediately apparent where it is implemented in the actual calculations. It enters in the denition of the collinear jet functions. In the large- t limit, theNjets are innitely separated from one another according to the measure given { 49 {by Eq. ( 1.4). And indeed, when N-jet operators are constructed in SCET, each collinear jet eld contains a Wilson line Wn, dened below in Eq. ( 3.9), of collinear gluons in the directionnthat were emitted from the back-to-back direction n, for which the separation measuret!1 . (This is similar to QCD factorization proofs of hard scattering cross sections, e.g. in [ 17], in which this direction nis chosen to be along some arbitrary path that is separated by an O(1) amount from the jet direction n.) Furthermore, the ni- collinear jet function knows only its own color representation, and not those of the other jets. Meanwhile, the hard and soft functions we calculate \know" about all Njets and their precise directions and color representations. Therefore it is no surprise that, when we actually calculate the anomalous dimensions of these functions, we nd that they are consistent with one another only in the limit that the separation parameter t!1 . 6.4 Refactorization of the Soft Function Our results for the soft function in Sec. 5.3make clear that in general there are multiple scales that appear in the soft function: the 1 S;:::;M Sassociated with the Mmeasured jets and the scale Sassociated with the out-of-jet cuto (see Eq. ( 5.21)). When these scales are all comparable, we can RG evolve the soft function from the single scale S. However, when any of them dier considerably from the others, we need to \refactorize" the soft function into multiple contributions, each of which is sensitive to a single energy scale. For illustration, take the scales i Sto be such that 1 S2 SM Sas in Fig.7. We also take l 1 S Sl Sfor our discussion, but the result is independent of whether Slies in the range 1 S< S<M Sor not. We can express the soft function appearing in Eq. ( 3.34) as Figure 7: Soft scales.S(1 a;2 a;:::;M a;) =h0jOy S( ^)MY i=1(i a ^i a)OSj0i; (6.34) where the operator i areturns the contribution to aof nal-state soft particles entering jet i, and ^ returns the energy of nal-state soft particles outside of all Njets. We have kept the dependence on the scales i Sand on implicit on the left-hand side. There areNWilson lines appearing in the operator OS, OS=Y1:::YMYM+1:::YN; (6.35) corresponding to the Mmeasured jets and N Munmeasured jets. The scales associated with soft gluons entering the Mmeasured jets whose shapes are measured to be 1;:::;M are1 S;:::;M S, given by Eq. ( 5.23). The scale associated with soft gluons outside of measured jets is Sgiven by Eq. ( 5.21). We have illustrated the ladder of these scales in Fig. 7. Each of these soft scales can be associated with dierent soft elds Ai swhose momenta scale as 2 i!iwhereiis associated with the typical transverse momentum i!i of the collinear modes for the ith jet. For measured jets, iis determined by i a, while for unmeasured jets itan(R=2). For soft gluons outside jets, however, the soft momentum is set by the cuto scale , which is why Sappears in the ladder of Fig. 7. { 50 {At a scalelarger than all i Sand S, the soft function is calculated as we presented in Sec. 5. In particular, we take the quantities i aand to be nonzero and nite. At a scale below1 S, we integrate out soft gluons of virtuality 1 Sand match onto a theory with soft gluons of virtuality 2 S. The scale 1 Sassociated with 1 ais taken to innity, and phase space integrals for soft gluons entering the measured jet 1 become zero (see, e.g., Eq. ( B.17)). Therefore, the matching coecient from the theory above 1 Sto the theory below is just the measured jet 1 contribution Smeas(1 a) to the soft function given by Eq. ( 5.22). The same occurs when matching from the theory above each scale i Sfor soft gluons entering measured jet ito the scale below i S, giving a matching coecient Smeas(i a). When going through the scale S, in the theory above this scale, the calculation of unmeasured contributions to the soft function gives the result Eq. ( 5.20), by treating as a nonzero, nite cuto. In the theory below S, we take to innity, making all phase space integrals originally cuto by to be scaleless and thus zero. So the matching coecient between the theory above and below Sis justSunmeas. After performing the above matchings all the way down to the lowest soft scale in Fig. 7, we nd that the original soft function S(1 a;:::;M a;) can be expressed to all orders as S(1 a;:::;M a;) =Sunmeas()MY i=1Smeas(i a;)h0jOy SOSj0i; (6.36) where to next-to-leading order SmeasandSunmeasare given by Sunmeas() = 1 +Sunmeas (1) () (6.37) Smeas(i a;) =(i a) +Smeas (1)(i a;); (6.38) whereSunmeas (1)is given by Eq. ( 5.20) andSmeas (1)is given by Eq. ( 5.22). Note that no operators restricting the jet shape or the phase space appear in the nal matrix element of soft elds living at the lowest scale on the ladder in Fig. 7. If all the scales on the ladder are at a perturbative scale, we can now just use hOy SOSi= 1 to eliminate the nal matrix element. If any scale is nonperturbative, we should have stopped the matching procedure before that scale, and dened the surviving soft matrix element still containing additional delta function operators as a nonperturbative shape function. Since the factors Sunmeas() andSmeas(i a;) are now matching coecients between two theories above and below the respective scales Sandi S, we can run each of the individual factors separately from their natural scale, instead of from a single soft scale 0 as in Eq. ( 6.32). The result for the RG-evolved soft function is then Eq. ( 6.36) where each factor at NLO is given by the solution of its RGE, Sunmeas() =Uunmeas S (; S)Sunmeas( S) (6.39a) Smeas(i a;) =Z d0Ui S(i a 0;;i S)Smeas(0;i S): (6.39b) These solutions allow us now to resum logarithms of all of the scales appearing in the ladder in Fig. 7when these scales are widely disparate. However, the result we obtained in Eq. ( 6.28) when we took all scales to be of the same order and had a single soft scale { 51 {has the form Eq. ( 6.39) at NLL accuracy. We will use equation Eq. ( 6.39) in all cases to interpolate between these two extremes. 6.5 Total Resummed Distribution Collecting together the above results for the running of hard, jet, and soft functions in the factorized cross section Eq. ( 3.34), we obtain the RG-improved N-jet cross section dierential in Mjet shapes, 1 (0)dN d1a1dMaM=H(H)H !H!H(;H)NY k=M+1Jk !k(k J) k J !ktanR 2!!k J(;k J) Sunmeas( S) MY i=1( 1 +fi J(i a;i J) +fi S(i a;i S) i Stan1 aR 2 !i!!i S(;i S) i J !i(2 a)!i J(;i J)1 [ !i J(;i J) !i S(;i S)]1 (ia)1+!i J(;i J)+!i S(;i S)) + exph K(;H;1;:::;N J;1;:::;M S; S) + E (;1;:::;M J;1;:::;M S)i ; (6.40) where !His dened by Eq. ( 6.25), the evolution parameters !F(;F) andKF(;F) are dened in Eq. ( 6.15), and we have dened the collective parameters, K(;H;1;:::;N J;1;:::;M S; S) =KH(;H) +NX i=1Ki J(;i J) +MX j=1Kj S(;j S) +Kunmeas S (; S)(6.41a) (;1;:::;M J;1;:::;M S) =MX i=1 i(;i J;i S)MX i=1[!i J(;i J) +!i S(;i S)]:(6.41b) Using results from Appendix C, we obtain the functions fi J;Sgenerated by the nite pieces of the measured jet and soft functions, fi J(i a;i J) =s(i J)T2 i 2(max a i a)4 2a 1 aln2i J !i(ia)1 2 a+1 1 a1 1 a 22 6 (1)( i) + ci+2 1 aH( 1 i)" 2 lni J !i(ia)1 2 a+1 2 aH( 1 i)# + (4 2a) ln2tanR 2 2ciln tanR 2 +s(i J) 2dJ(i a) (6.42a) fi S(i a;i S) = s(i S)T2 i 1 1 a(" lni Stan1 aR 2 !iia+H( 1 i)#2 +2 6 (1)( i)+dS) ; (6.42b) { 52 {whereci= 3=2 for quark jets and 0=(2CA) for gluon jets. max ais the upper limit on i a found in the nite part of the na ve jet function, given in Appendix A.H( 1 i) is the harmonic number function, with igiven by Eq. ( 6.41b ). (1)is the rst derivative of the digamma function, (1)(z) = (d=dz)[ 0(z)= (z)]. The terms dJ;Sare additional contribu- tions from the nite parts of jet and soft functions that do not contain any logarithms, wheredS= 2=24, anddJis given in Eq. ( C.6) in the Appendix. dJ;Sare not needed at NLL accuracy. Similarly, the terms containing large logarithms in the unmeasured jet functions and unmeasured contribution to the soft function are Ji !(J) = 1 + (s(J))T2 iln2J !tanR 2+ k[s(J)] lnJ !tanR 2+di J (6.43a) Sunmeas( S) = 1 + (s( S))X iT2 i ln S 2 tan1=2R 2 ln tan2R 2 2 8 + (s( S))X i6=jTiTj ln S 2lnninj 2+ Li 2 1 2 ninj ; (6.43b) wheredi Jis the part of the unmeasured jet function containing no large logarithms (given in Eqs. ( A.19) and ( A.30) in the Appendix). The nite parts of the measured and unmeasured jet and soft functions given in Eqs. ( 6.42) and ( 6.43) show that to minimize large logarithms in the O(s) nite parts in the resummed distribution Eq. ( 6.40), we should choose initial scales for the running to be H= !H (6.44a) i J=!i(i a)1 2 a; k J=!ktanR 2(6.44b) i S=!ii a tan1 aR 2; S= 2 tan1=2R 2: (6.44c) These choices eliminate all large logarithms in the O(s) hard, jet, and soft functions. They still leave logs of tanR 2andninjin the unmeasured part of the soft function, and logs of tanR 2in the measured jet function, but we already take Rnumerically ofO(1)14 to minimize power corrections from our implementation of the jet algorithm as discussed in Sec. 3.4, andninj1 since the jet separation parameter tijis large compared to 1. All logs ofR, , andi aare eliminated in the unmeasured jet function and measured part of the soft function. 14We still consider tan( R=2) to be of order kin the collinear sectors describing unmeasured jets, as implied by Eq. ( 6.44). This means kis eectively much larger than the parameter iin a measured jet sector. In fact, note that Eq. ( 6.44) tells us that tanR 2must be parametrically larger than ( i a)1 2 a; otherwise, the jet scale falls below the soft scale in the measured jet sectors, invalidating the use of SCET and, thus, the validity of the factorization theorem. { 53 {7. Plots of Distributions and Comparisons to Monte Carlo Having resummed the jet shape distributions in ato NLL accuracy, in this section we plot the distributions for various values of aandR, compare to Monte Carlo simulated events, and perform scale variation on the resummed distribution. We use the process e+e !3 jets to study our predictions of jet shapes, where the jets arise from partons in the \Mercedes-Benz" conguration, with each parton having equal energy. In these congurations, the partons lie in a plane and are equally separated with a pairwise angle of 2=3. This allows us to study event shape distributions of well-separated jets where tis reasonably large. We choose three values of Rto study,R= 1:0, 0.7, and 0.4. With these values of R, the 1 =t2suppression factor for corrections to the large- tlimit are 0.10, 0.044, and 0.014 respectively. We will measure the angularity of only one of the three jets; the other two jets will be unmeasured. In general, the TiTjcolor correlations in the soft and hard functions lead to operator mixing in color space under RG evolution. This implies that the RG kernels USandUH are matrices in color space and must be studied on a process-by-process basis (see, e.g., [89,90,91,92,93,94]). For the case of N= 2;3 jets there is only one color-singlet operator and hence no mixing. This can be seen, for example, by noting that all color correlations reduce to the Casimir invariants ( CFandCA) in this case (cf. Appendix D). We have restricted the example process we use in this work to N= 3 jets, avoiding the additional complication of color-correlations that comes with a larger number of jets. The NLL resummed distribution for one quark or gluon jet shape (jet 1) in a three-jet nal state, written as the derivative of the radiator Eq. ( 1.5), is 1 (0)d3 d1a=H !H!H(;H)1 J !1(2 a)!1 J(;1 J) 2 J !2tanR 2!!2 J(;2 J) 3 J !3tanR 2!!3 J(;3 J) 1 Stan1 aR 2 !1!!1 S(;1 S) exp K(;H;1 J;2 J;3 J;1 S; S) + E (;1 J;1 S) [1 + ^fJ(1 a) +^fS(1 a)]1 [ (;1 J;1 S)]" 1 (1a)1+ (;1 J;1 S)# +; (7.1) where the various evolution parameters !i J;S; ;Kare all dened in Eqs. ( 6.15) and ( 6.41), and ^fJ;Sare given by fJ;Sin Eq. ( 6.42) with thedJ;Sterms set to zero (accurate to NLL). The best scale choices Eq. ( 6.44) for this case are H= !T2 1 1!T2 2 2!T2 3 3 1 2CF+CA(7.2a) 1 J=!1(1 a)1 2 a; 2;3 J=!2;3tanR 2(7.2b) 1 S=!11 a tan1 aR 2; S= 2 tan1=2R 2: (7.2c) In Eq. ( 7.1) we have used tree-level initial conditions for the hard, jet, and soft functions at these scales. Eq. ( 7.1) evolves these functions to the arbitrary scale at NLL accuracy. { 54 {τaq jet R=1g jet R=1 q jet R=0.7g jet R=0.7 q jet R=0.4g jet R=0.41 σ(0)dσ dτa a=0a=-1/2 a=-1/4 a=1/4a=1/2 τa0.000 0.002 0.004 0.006 0.008 0.010050100150200250300 0.000 0.005 0.010 0.015 0.020010203040506070 0.000 0.002 0.004 0.006 0.008 0.0100100200300400500 0.000 0.005 0.010 0.015 0.020020406080100120 0.000 0.002 0.004 0.006 0.008 0.0100200400600800 0.000 0.005 0.010 0.015 0.020050100150200Figure 8: Quark and gluon jet shapes for several values of aandR. The NLL resummed distribu- tion in Eq. ( 7.1) is plotted for a= 1 2; 1 4;0;1 4;1 2for quark and gluon jets with R= 1;0:7;0:4. The plots are for jets produced in e+e annihilation at center-of-mass energy Q= 500 GeV, with three jets produced in a Mercedes-Benz conguration with equal energies EJ= 150 GeV, and minimum threshold = 15 GeV to produce a jet. With these choices, we plot Eq. ( 7.1) in Fig. 8for several values of aandRfor a quark or gluon jet shape in a three-jet nal state in e+e annihilation at center-of-mass energy Q= 500 GeV.15The jets are chosen to be in a Mercedes-Benz conguration, where all jets have equal energies of 150 GeV. We choose the jet energy cuto to be 15 GeV. We choose the factorization scale to be =H. 15The distributions plotted with the ^fJ;Sterms included in Eq. ( 7.1) exhibit a small negative dip near a= 0 (not shown) that can be cured by convolving with a nonperturbative shape function with a renormalon- free gap parameter [ 38,54]. This is beyond the scope of the present work, so we only plot the perturbative distributions where they are positive. { 55 {τa τa1 σ(0)dσ dτa a=1/2, R=0.4 τaa=0, R=0.4 a=-1/2, R=0.4a=1/2, R=0.7 a=0, R=0.7 a=-1/2, R=0.7a=1/2, R=1 a=0, R=1 a=-1/2, R=1Legend Quark jets (blue) Gluon jets (red)Theory NLLMonte Carlo hadronization offMonte Carlo hadronization on 0.000 0.002 0.004 0.006 0.008 0.0100501001502002500.000 0.002 0.004 0.006 0.008 0.010010203040506070 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.01401002003004005000.000 0.002 0.004 0.006 0.008 0.010 0.012 0.0140501001500.000 0.002 0.004 0.006 0.008 0.010 0.012 0.0140102030405060 0.000 0.005 0.010 0.015 0.0200501001502002503000.000 0.005 0.010 0.015 0.0200204060801001201400.000 0.005 0.010 0.015 0.0200102030405060 0.000 0.002 0.004 0.006 0.008 0.0100200400600800Figure 9: Quark vs. gluon jet shapes with comparison to Monte Carlo. Solid, straight curves represent the resummed jet shape distribution in Eq. ( 7.1), and jagged curves are histograms from the Monte Carlo, normalized as described in the text. The solid histogram has no hadronization, while the dashed histogram includes the eects of hadronization. The distributions are plotted fora= 1 2;0;1 2with quark (blue) and gluon (red) jets compared on the same plot, for jets of sizeR= 1:0;0:7;0:4. Gluon jet shape distributions peak at larger values of athan quark jets, indicative of their broader shape. The plots are for jets in e+e annihilation at center-of-mass energy Q= 500 GeV, with three jets produced with equal energies EJ= 150 GeV, angular separation = 2=3 between all pairs of jets, and minimum threshold = 15 GeV to produce a jet. We compare the results of a jet algorithm implemented on Monte Carlo simulated events with our NLL resummed predictions for a variety of aandRvalues in Fig. 9. Because the resummed NLL distribution we choose to study applies to an exclusive process, three- jet events in the Mercedes-Benz conguration, we implement cuts on the simulated events to obtain a sample that matches onto this conguration. We use MadGraph/MadEvent v.4.4.21 [ 95] to generate parton-level e+e !qqgevents at a center-of-mass energy Q= 500 GeV, with cuts imposed to obtain partons in the Mercedes-Benz conguration. We shower and hadronize the events with Pythia v.6.414 [ 96] usingpT-ordered parton showers. The process of hadronization will induce a shift in the angularity distribution, which we do { 56 {not model in our resummed distribution. Therefore, we produce two samples: one sample with only QCD nal-state showering, no initial-state radiation, and no hadronization, and another sample with complete showering, initial-state radiation, and hadronization. The anti-k Tjet algorithm is run on the nal state particles from Pythia, and we use FastJet [ 97] to implement the jet algorithm. The anti-k Talgorithm is particularly well suited for this comparison, as very few particles at an angle >R to the jet axis are included in the jet. With anti-k T, the phase space cut on an individual particle matches well with the phase space cuts in the next-to-leading order calculation. To select a sample of events to compare to our resummed distributions, we make cuts on the nal state jets, requiring each of the three hard, well-separated partons from MadGraph to be associated with a jet. This involves a cut on the jet direction and momentum: ppartonpjet jppartonjjpjetj>0:9 andjjppartonj jpjetjj jppartonj<0:15: (7.3) We analyze events passing these cuts, and tag each associated jet as coming from a quark or a gluon based on which parton it matches onto. The angularity value for each jet is computed from the constituent particles of the jet, using the matching parton direction as the jet axis. The jet direction only diers from the parton direction by a power correction (see Sec. 3.2). In Fig. 9, we isolate some of the quark and gluon jet shapes in Fig. 8and compare to Monte Carlo events. The relative normalization between the distribution of Monte Carlo events and the NLL resummed angularity distribution requires some explanation. Both our calculation and the Monte Carlo simulation are most accurate in the region that includes the peak of the distribution and the larger- side of the peak, but both are inaccurate as !0 and in the tail region. Therefore, each will dier in the relative normalization between the peak region and the tail region. An accurate prediction of the tail region requires matching onto a calculation at xed-order in sin full QCD as in [ 43,53,54]. In Fig. 9, we choose to normalize the area of the Monte Carlo distribution to the total area of the NLL aτpeak a gluon jets quark jetsR=1 R=0.7 R=0.4 R=1 R=0.7 R=0.4 /Minus1.0 /Minus0.5 0.0 0.50.0000.0020.0040.0060.0080.0100.0120.014 Figure 10: Location of peak of jet shape distribution as a function of afor quark and gluon jets. We plot the value of aat the peak of the jet shape distribution for abetween -1.0 and 0.8 for quark vs. gluon jets, using R= 1;0:7;0:4. { 57 {quark jets1 σ(0)dσ dτagluon jets hard scale variation hard scale variation scale variation scale variation unmeasured jet scale variation unmeasured jet scale variation measured jet scale variation measured jet scale variation measured soft scale variation measured soft scale variationµΛ µΛ τa τafactorization scale variation factorization scale variation 0.000 0.002 0.004 0.006 0.008 0.010050100150200 0.000 0.005 0.010 0.015 0.020010203040500.000 0.005 0.010 0.015 0.02001020304050 0.002 0.004 0.006 0.008 0.0100501001502000.000 0.005 0.010 0.015 0.02001020304050 0.002 0.004 0.006 0.008 0.0100501001502000.000 0.005 0.010 0.015 0.020010203040506070 0.002 0.004 0.006 0.008 0.0100501001502002500.000 0.005 0.010 0.015 0.0200102030405060 0.002 0.004 0.006 0.008 0.0100501001502002500.000 0.005 0.010 0.015 0.02001020304050 0.002 0.004 0.006 0.008 0.010050100150Figure 11: Scale variation of quark and gluon jet shapes. For a= 0 andR= 0:7, we display the variation of the NLL resummed jet shape distributions with the hard scale H, the jet cuto scale S, the unmeasured jet scales 2;3 J, the measured jet scale 1 J(a), and the measured soft scale S(a). In each case we vary the scale between 1 =2 and 2 times the natural choices in Eq. ( 6.44), except for the measured soft scale, which we varied between 1 and 2 times the choice in Eq. ( 6.44). We keep the factorization scale xed at the default hard scale given by Eq. ( 7.2),=!i. { 58 {resummed theory distribution. We nd the area under the theory curves for quark and gluon jets to be approximately 0.3 for R= 0:4, 0.5 forR= 0:7, and 0.7 for R= 1. A more accurate prediction of the normalizations may require summing remaining unsummed logs of the phase space cuts in the theory and Monte Carlo predictions. These plots should be interpreted as comparisons of the predictions of the shapes in aand these shapes' scaling as we vary aandR, rather than the overall normalization. The shapes of the theory and Monte Carlo distributions are largely similar, though they display noticeable dierences at the leftmost endpoint near a= 0 and in the \sharpness" of the peak. These may be due to the dierent ways the two approaches deal with the growth of the strong coupling for small a, the dierent orders of log resummation (LL vs. NLL) and the need to match the tails onto xed-order QCD predictions. Since neither the Monte Carlo nor theory partonic predictions without inclusion of hadronization eects is yet a prediction of a physically observable quantity, we use this comparison as an intermediate diagnostic rather than a conclusive test of either method. Nevertheless, comparing the way the shapes of the distributions and locations of the peaks vary over the range of values of aandRthat we sample, the behavior agrees very well between the theory distributions and the Monte Carlo distributions without hadronization for both quark and gluon jets. In Fig. 10we plot the location of the peak of the jet shape distributions as a function ofafor three values of R, displaying the dierent variation of the peak of quark and gluon jet shape distributions. The peak value increases with increasing Randa, as wide angle radiation is included (increasing R) and less suppressed (increasing a). Although the dierence in the peak value between the quark and gluon jet angularity distributions is large, the width of each distribution creates substantial overlap in angularity values between quark and gluon jets. Distinguishing between quark and gluon jets using jet angularities is a complex task which we will explore in future work; for now, we note only that the NLL resummed distributions indicate that discrimination between quark and gluon jets using jet angularities is possible. As a rough estimate of the theoretical uncertainty in our NLL resummed predictions, we show in Fig. 11the change in the a= 0 quark and gluon ajet shape distributions for R= 0:7 when the various scales that appear in the resummed cross section Eq. ( 7.1) are varied. These are the initial scales at which the hard, jet, and soft functions are evaluated to minimize logarithms in the NLO xed-order part, from which the evolution kernels run them to the common factorization scale . In the top row of Fig. 11, we varybetween !H=2 and 2!H. The tiny variation is a sign of the consistency condition satised by the anomalous dimensions in Eq. ( 6.33). In the next four rows, we vary the hard scale H, the soft jet energy cuto scale S, the unmeasured jet scales 2;3 J, and the measured jet scale1 J(1 a) between half and twice the natural values given in Eq. ( 7.2). In the last row, we vary the measured soft scale 1 S(1 a) between one and two times the value in Eq. ( 7.2). This is because too low a value of 1 S(1 a) asa!0 brings it into the nonperturbative region where s(1 S) blows up, so that the perturbative estimate of uncertainty is not so meaningful. We note that, while the uncertainty in the vertical scale of the distributions is considerable in some cases, the location of the peak is much more stable. Finally, in Fig. 12we give a sense for how robust our theoretical predictions are for other { 59 {0.000 0.001 0.002 0.003 0.004 0.0050100200300400 0.000 0.001 0.002 0.003 0.004 0.0050501001502002503001 σ0dσ dτ1 0 τ1 0 τ1 0ψnear=π/2 ψnear=π/3 Measured “near” jets Measured “far” jetsτ1 0 τ1 0ψnearblue q near g green q near q red g near q blue q far red g far0.000 0.001 0.002 0.003 0.004 0.005050100150200250300 0.000 0.001 0.002 0.003 0.004 0.005050100150200250Figure 12: Jet shapes for other kinematic congurations. We compare our theoretical predictions to Monte Carlo simulations for the shape 1 0(a= 0) for a quark or gluon jet found in a three-jet conguration where the two jets with narrowest separation angle nearhave equal energy. We consider the two cases near==2 and=3. In the rst row, we plot shapes of one of the jets in the \near" pair. The blue solid curve is the shape of quark jet found near a gluon jet, the green dotted curve is a quark found near an antiquark, and the red solid curve is a gluon found near a quark. In the second row, we compare shapes of a quark or gluon jet found far from the near pair. kinematic congurations. We consider e+e !qqgevents where the angle nearbetween two partons is either =2 or=3, and these partons have equal energy. We nd jets using the anti-kT algorithm with R= 0:4, and plot jet shapes for a= 0. The selection cuts to choose events from the Monte Carlo are the same as the Mercedes-Benz conguration. In these events there are ve distinct characterizations for a single parton. If the event has the quark (or antiquark) as the \far" (most well separated) parton, then each parton in the event is distinct: there is the far quark, the near quark, and the near gluon. If the event has the gluon as the far parton, then there are only two distinct types of partons: the far gluon and the near quark (antiquark). In Fig. 12, we plot all these congurations for both near==2 and near==3. The agreement between the theory predictions and the Monte Carlo are as good as in the Mercedes-Benz case, a good indication that our calculation applies to a broad range of kinematic congurations of multijet events. Additionally, we observe features consistent with our intuition about the relative dierences between the jet shape distributions between dierent jets in these congurations. As one would expect, the distribution of near jet shapes is weighted more heavily towards larger athan the far jet shapes, due to the enhanced soft radiation in the near jet system. When the near quark is near a gluon, the distribution is weighted more heavily towards larger athan when the near quark is near an antiquark, due to the enhanced radiation coming from a gluon rather than a quark. These distributions serve as further evidence that jet shapes may be an eective discriminant between quark and gluon jets. { 60 {8. Conclusions In this work, we have factorized an N-jet exclusive cross section dierential in MN jet observables and resummed global logarithms of the jet observable ato NLL accuracy, leaving summation of non-global logarithms to future work. We demonstrated that the anomalous dimensions of the hard, jet, and soft functions in the factorization theorem satisfy the nontrivial consistency condition Eq. ( 6.33) toO(s), for any number of quark and gluon jets, any number of jets whose shapes are measured, and any size Rof the jets, as long as the jets are well-separated, meaning t1. This condition ensures the validity of an eective theory with Ncollinear directions that are assumed to be distinct. We identied and estimated important power corrections to the factorized form of the cross section. We also illustrated that zero-bin subtractions give nonzero contributions to the anomalous dimensions crucial for consistency. Armed with consistent factorization and the xed-order jet and soft functions, we resummed large logarithms in the jet shape distribution by running each individual function from the scale where logs in it are minimized to the common factorization scale . We thereby resummed, to NLL accuracy, global logs of the jet shape aand logs of the scale =EJof soft radiation outside of jets, but leaving some non-global logs and logs of the angular cut R(but we took Rto be numerically of order 1). This is the rst such calculation of a resummed jet shape distribution in an exclusive multijet cross section. We constructed a framework to deal with all the scales that appear in the multijet soft function which depends on the values i aof allMjet shapes and the phase space cuts ;R. By refactorizing the full soft function into individual pieces depending on one of these scales at a time, we were able to sum logs of ratios of these scales. We demonstrated the accuracy of our results by comparing our resummed prediction for quark and gluon jet shapes in e+e !3 jets to the output of Monte Carlo event generators, MadGraph/MadEvent and Pythia. We compared our predictions with the Monte Carlo output without hadronization. The changes in shape and location of the peak value as functions of aandRmatch quite well between the theory and Monte Carlo. Our results provide a basis for future studies of other jet observables at both e+e and hadron colliders, requiring recalculation of those parts of our jet and soft functions that depend on the choice of observable. Studying jets at hadron colliders requires constructing observables appropriate for that environment and the switching of two of our outgoing jets to incoming beams, which can be described by beam functions in SCET [ 62]. Precision calculations of jet shapes will allow improved discrimination of jets of dierent origins. We are applying the results of our predictions of light quark and gluon jet shapes to distinguish quark and gluon jets with greater eciency than achieved before. Extensions to the shapes of heavy jets and calculations of other types of jet shapes such as the ( r=R) shape introduced in [ 14,15,16] can also be performed. Note added in nal preparation: As this paper was being completed, Ref. [ 98] appeared reporting the calculation of a quark jet function for a jet dened with a Sterman-Weinberg algorithm and whose invariant mass sis measured. This jet function is related to our { 61 {measured jet function Jq !(a) for a cone jet at a= 0 given in Eq. ( 4.11), sinces=!20. We have checked that the corresponding results between the two papers agree. Acknowledgments We are grateful to C. Bauer for valuable discussions and review of the draft. The authors at the Berkeley CTP and in the Particle Theory Group at the University of Washington thank one another's groups for hospitality during portions of this work. AH was supported in part by an LHC Theory Initiative Graduate Fellowship, NSF grant number PHY-0705682. The work of AH and CL was supported in part by the U.S. Department of Energy under Contract DE-AC02-05CH11231, and in part by the National Science Foundation under grant PHY-0457315. The work of SDE, CKV, and JRW was supported in part by the U.S. Department of Energy under Grants DE-FG02-96ER40956. A. Jet Function Calculations A.1 Finite Pieces of the Quark Jet Function Measured Quark Jet Function The nite pieces the jet functions, which depend on the jet algorithm, share common features. For cone-type algorithms, the nite piece of the naive part of the quark jet function, ~Jq alg(a), is given by ~Jq cone(a) =CF7 2+ 3 ln 2 2 3 (a) +CF 1 a 2 Iq cone(a)(max a a) a +(A.1) where in this Appendix, plus distributions are dened by [ 62] [(x)g(x)]+= lim !0d dx[(x )G(x)]; withG(x) =Zx 1dx0g(x0); (A.2) dened so as to satisfy the boundary conditionR1 0dx[(x)g(x)]+= 0. The quantity Iq cone depends implicitly on aandRand is given by Iq cone=Z1 xcone xconedx2(1 x) +x2 x= 2 log1 xcone xcone 3 2+ 3xcone: (A.3) The parameter xcone=xcone(a) is the lower limit on the x=q =!scaled gluon momentum integral imposed by the cone restriction. It is given by the solution of the equation fcone(xcone) =a tan2 aR 2; (A.4) wherefcone(x) is dened as fcone(x)x2 a[x 1+a+ (1 x) 1+a] (A.5) in the range 0 < x < 1=2, which is plotted in Fig. 13A. The limit max ais given by the maximum value over xof Eq. ( A.5). Similarly, for k T-type algorithms, ~Jq kT(a) is given by ~Jq kT(a) =CF13 2 22 3 (a) +CF 1 a 2 Iq kT(a)(max a a) a +: (A.6) { 62 {x(A) (B)x2a−22a−2 τa tan(2−a)R/2x fcone (x) fkT(x) fkT(x) 0.2 0.4 0.6 0.8 1.00.050.100.150.200.25 x1 1−x1 x2 x1 1−x1 1−x20.2 0.4 0.6 0.8 1.00.010.020.030.040.050.06 0.2 0.4 0.6 0.8 1.00.20.40.60.81.0 xcone 1−xcone (C)x xFigure 13: Regions of integration for the (A) cone and k T-type algorithms for (B) a > |