Jet axes and universal transverse-momentum-dependent fragmentation
PPrepared for submission to JHEP
NIKHEF 2016-051
Jet axes and universal transverse-momentum-dependentfragmentation
Duff Neill, a Ignazio Scimemi, b Wouter J. Waalewijn c,d a Theoretical Division, MS B283, Los Alamos National Laboratory, Los Alamos, NM 87545, USA b Departamento de F´ısica Te´orica II, Universidad Complutense de Madrid, Ciudad Universitaria, 28040Madrid, Spain c Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics, University ofAmsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands d Nikhef, Theory Group, Science Park 105, 1098 XG, Amsterdam, The Netherlands
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We study the transverse momentum spectrum of hadrons in jets. By measuring thetransverse momentum with respect to a judiciously chosen axis, we find that this observable isinsensitive to (the recoil of) soft radiation. Furthermore, for small transverse momenta we showthat the effects of the jet boundary factorize, leading to a new transverse-momentum-dependent(TMD) fragmentation function. In contrast to the usual TMD fragmentation functions, it doesnot involve rapidity divergences and is universal in the sense that it is independent of the typeof process and number of jets. These results directly apply to sub-jets instead of hadrons. Wediscuss potential applications, which include studying nuclear modification effects in heavy-ioncollisions and identifying boosted heavy resonances. a r X i v : . [ h e p - ph ] F e b ontents A.1 One-loop example 18
B Clustering algorithms 20C Results on anomalous dimensions 21
C.1 All-orders anomalous dimension of JTMDFF 21C.2 One-loop anomalous dimensions in moment space 22
References 23
In the analysis of events from hadron colliders it is common to use jets to organize the final statesof hard interactions, making it natural to ask how the QCD confinement of hadrons is realized inthis context. The picture that arises from QCD factorization is that we have the hard scattering,whose calculation is given in terms of partonic degrees of freedom, initiating the jet. At the short-distance scale of the hard-scattering, we have a quark or gluon of a much lower “off-shellness”– 1 –xiting the hard interaction in a more or less definite direction. The subsequent branching does notchange this direction much, but does gives rise to a host of additional partons loosely grouped intoa jet. These are the perturbative remains of the slightly off-shell parton. Lastly, these additionalpartons undergo a “hadronization” process at length scales of 1 / Λ QCD , confining themselves intothe observed hadrons. Ultimately, to understand the dynamics of confinement within jets, wewould like to have a means of comparing the partonically generated momentum distribution insidethe jet to the observed hadronic momentum distribution. In addition to momentum, one wouldalso like to understand how quantum numbers, like spin, flavor, or charge, are transported fromthe hard scattering into the hadronic final state.The fragmentation function d i → h ( z h , µ ) describes the distribution of the longitudinal-momentumfraction z h of hadrons of a species h = π + , π − , . . . produced by a parton i = g, u, ¯ u, d, . . . [1–3].This allows one to express their production cross section as (see e.g. ref. [4])d σ h d z h = (cid:88) i (cid:90) d zz ˆ σ i ( z, Q, µ ) d i → h (cid:16) z h z , µ (cid:17)(cid:104) O (cid:16) Λ QCD Q (cid:17)(cid:105) , (1.1)where Q is the scale of the hard scattering. A crucial feature of fragmentation is that it isuniversal, i.e. insensitive to the underlying hard scattering or the soft background radiation.In field-theoretic terms this means that the same QCD matrix element for d i → h captures thefragmentation dynamics, and can be factorized from the hard scattering. Thus fragmentationmeasurements at hadron-hadron, hadron-electron, and electron-positron colliders can all be com-pared. However, when combining hadron analysis with modern jet algorithms one begins toworry that the definition of the jet itself could potentially spoil this universality, since any givenjet definition will have more or less sensitivity to the underlying event or hard scattering process.As we will see in this paper this can take a rather subtle form.Fragmentation of hadrons inside jets has also been studied extensively, but without account-ing for the transverse momentum dependence of the hadrons. When the jet is sufficiently narrow,its dynamics can be factorized from the hard scattering process. For fragmentation in exclusiveprocesses (i.e. with a specific number of jets) this was studied using event shapes (hemispherejets) in refs. [5–10] and with a jet algorithm in refs. [11–14]. Inclusive jet production with a jetalgorithm was investigated in refs. [15–18]. The applications that were considered range fromcomparisons to LHC measurements of charged hadron spectra [12] to unravelling quarkoniumproduction channels [13]. Multi-hadron fragmentation in jets has also been considered [19–22],to e.g. describe jet charge [19].The observables that we want to construct here are transverse momentum distributions(TMDs). In general, one would like to know the full three-dimensional distribution of momentainside the jet, not merely the energy fraction. However, one must be careful, since asking ques-tions about the other components of the hadron’s momentum can easily expose one to sensitivityto associated soft processes. While studying these soft processes is an interesting and worth-while endeavor in and of itself, it can severely complicate any potential claim to universality ofthese distributions. In the standard terminology the TMDs measure the correlation of transversemomentum of two partons in processes like Drell-Yan, semi-inclusive deep-inelastic scattering(SIDIS), or the production of two hadrons in e + e − collisions. In this work we consider insteadthe measurement of the transverse momentum of a hadron with respect to a jet axis. In the– 2 – tandard jet axisrecoil-free axis ~p coll ? + ~p soft ? = 0 ~p coll ? + ~p soft ? = 0 ~p coll ? + ~p soft ? = 0 Figure 1 . The standard jet axis is sensitive to soft radiation (orange) through recoil effects, whereas arecoil-free axis follows the direction of collinear radiation (blue). standard TMD correlations one cannot avoid the appearance of rapidity divergences and theconsequent regularization and renormalization [23–28], which signals a sensitivity to soft physics.Here we show that one can define a transverse momentum observable which is insensitive tosuch problems, by a judicious choice of jet axis. The final TMDs will necessarily be different fromthe standard ones and thus we coin the name jet TMDs (JTMDs) for this class of observables.The key insight is to adopt an axis definition that is recoil-insensitive [29–31]. Put loosely, if oneuses an axis whose direction is conserved under splittings (e.g. the total momentum of the jetor the thrust axis), this introduces a soft-sensitivity in the axis definition, since a soft emissioncan displace a collinear one, see fig. 1. Alternatively, one can adopt an axis that itself recoilscoherently with the soft radiation, and thus follows the direction of the collinear radiation.In describing the transverse momentum of a hadron in a jet, there are a number of choices: • Exclusive production with a jet algorithm with a recoil-sensitive axis: the factorizationtheorem has a simple multiplicative structure (see e.g. ref. [32]) but the soft radiationsuffers from non-global logarithms (NGLs) [33], which arise because of the very differentrestrictions on the radiation inside an outside the jet. • Exclusive production with a global event shape: NGLs are absent for an observable like N -jettiness [34], but potential exchanges between the initial states can spoil the factorizationfor hadronic collisions [35–40]. • Inclusive production with a jet algorithm and a recoil-sensitive axis: This was recentlystudied in ref. [41]. The TMD fragmentation function involves rapidity divergences. Onecan define a factor consisting of (collinear-)soft modes which cancels these rapidity diver-gences. However, this (collinear-)soft radiation will displace the jet axis and contaminatethe transverse momentum distribution, again introducing a sensitivity to NGLs. . • Inclusive production with a jet algorithm and a recoil-free axis: the observable is purelycollinear, making it universal and free of NGLs. This is the case we focus on. From the direct two-loop calculations for related jet shapes in the soft approximation [42], one sees that NGLsare present for all jet radii. Unlike in ref. [42], the out-of-jet radiation is not restricted here, making it effectivelyequal to the partonic center-of-mass energy. – 3 –e now briefly outline our framework: For definiteness we focus on hadronic collisions withenergetic jets that are not particularly close to each other or the beam axis (i.e. central jets).Our approach is easily extendable using e.g. refs. [43–45]. Consider the measurement of thelongitudinal momentum fraction z h and transverse momentum z h k (with respect to the jet axis)of an energetic hadron h inside a jet. To leading approximation, the soft radiation outside thejet cannot affect the production of a hadron inside a jet. However, as illustrated in fig. 1, themeasurement of the transverse momentum of a hadron with respect to the standard jet axis issensitive to the soft radiation inside the jet. This is not the case when using a recoil-insensitiveaxis, which is determined by the configuration of the energetic (collinear) radiation. Underthe assumption that the jet radius R (cid:28) R ∼ σ h d p T d η d k d z h = (cid:88) i (cid:90) d xx ˆ σ i (cid:16) p T x , η, µ (cid:17) G i → h ( x, p T R, k , z h , µ ) (cid:2) O ( R ) (cid:3) . (1.2)The partonic cross section ˆ σ encodes the hard scattering producing the parton i with transversemomentum p T /x and rapidity η , with respect to the beam axis. The fragmenting jet function G describes the fraction x of the parton energy that goes into the jet, as well as the fragmentationof the hadron inside the jet with momentum fraction z h and transverse momentum z h k . Thefunction G obeys a collinear renormalization group equation. A further factorization of this crosssection can be achieved when p T R (cid:29) | k | and/or | k | (cid:29) Λ QCD , which is discussed in detail insec. 3. In particular, for p T R (cid:29) | k | we can separate the effect of the jet boundary B from thefragmentation, leading to a new JTMD fragmentation function D k → h , G i → h ( x, p T R, k , z h , µ ) = (cid:88) k (cid:90) d yy B ik ( x, p T R, y, µ ) D k → h (cid:16) k , z h y , µ (cid:17)(cid:20) O (cid:16) k p T R (cid:17)(cid:21) . (1.3)Since D k → h is a purely collinear object, it is automatically universal, i.e. insensitive to the typeof process or number of jets. It also does not involve rapidity divergences, unlike the classicalTMD fragmentation functions.The paper is organized as follows: We start by outlining the differences between the classicalTMDs and the JTMDs that are considered in this work in sec. 2. We also define all ingredientsthat enter in our fractorization theorems and discuss their renormalization. A discussion of recoil-free jet definitions in the context of a simple example in given in app. A, including a one-loopcalculation. The winner-take-all recombination scheme [29, 31] that we use to obtain a recoil-freejet axis is summarized in app. B. In sec. 3, we show how eq. (1.2) can be further factorized,depending on the hierarchy between p T R , | k | and Λ QCD . We also treat the case when R is notsmall. We have calculated the one-loop matching coefficients and present these in sec. 4. In sec. 5some first numerical results based on a moment analysis are presented. We conclude in sec. 6,discussing the wide range of potential applications of our framework. This definition of k ensures that it is a partonic observable and thus perturbatively calculable. – 4 – Framework
We use light-like vectors n and ¯ n with n · ¯ n = 2 to introduce the light-cone coordinates used here v µ = v − n µ v + ¯ n µ v µ ⊥ , v − = ¯ n · v , v + = n · v . (2.1)The time-like and space-like component of a vector are indicated by ( v , (cid:126)v ), so that v = v − (cid:126)v = v + v − + v ⊥ = v + v − − v . (2.2)In the language of soft-collinear effective theory (SCET) [46–49], if we assign a power countingfor the collinear momenta as p n = (¯ n · p n , n · p n , p n ) ∼ Q (1 , λ , λ ) , , (2.3)where the power counting parameter λ (cid:28) p s ∼ Q ( λ β , λ β , λ β ) . (2.4)For β = 1 this is referred to as soft radiation, and for β = 2 it is called ultra-soft. Before arriving at the formulation of the JTMDs, it is instructive to recall some properties of theclassical unpolarized TMD fragmentation functions [26, 50],∆ q → h ( z h , b T ) = 14 z h N c (cid:88) X (cid:90) d ξ + π e − ip − h ξ + / (2 z h ) (cid:104) | T (cid:104) ˜ W † T n q j (cid:105) a (cid:16) ξ (cid:17) | X, h (cid:105) γ − ij (cid:104) X, h | ¯ T (cid:104) ¯ q i ˜ W T n (cid:105) a (cid:16) − ξ (cid:17) | (cid:105) , ∆ g → h ( z h , b T ) = − − (cid:15) ) p − h ( N c − (cid:88) X (cid:90) d ξ + π e − ip − h ξ + / (2 z h ) ×(cid:104) | T (cid:104) ˜ W † T n F − µ (cid:105) a (cid:16) ξ (cid:17) | X, h (cid:105) g µν (cid:104) X, h | ¯ T (cid:104) F − ν ˜ W T n (cid:105) a (cid:16) − ξ (cid:17) | (cid:105) , (2.5)where ξ = ( ξ + , − , b T ). The variable conjugate to the impact parameter b T is k , which is thetransverse momentum of the hadron divided by its momentum fraction. The sum runs over allintermediate states X , and X does not include the hadron h . The Wilson lines ˜ W T n ( x ) depend onthe coordinate x and continue to the light-cone infinity along the vector n , where it is connectedby a transverse link to the transverse infinity (as indicated by the subscript T ) [51, 52]. Therepresentations of the SU(3) generators inside the Wilson lines correspond to that of the parton(fundamental for quark, adjoint for gluon), and repeated color indices are summed over.It is important to emphasize that implicit in these definitions of the TMDFFs a specificaxis choice has been made, namely that the n direction is along the hadron h . By performing achange of coordinates (or reparametrization [53]), it follows that this corresponds to measuringthe transverse momentum of the hadron with respect to the axis lying along the total momentumof all particles in the intermediate state. For fragmentation in e + e − → hadrons, this axis isequivalent to the thrust axis. – 5 –hese TMDs appear in processes like SIDIS or e + e − → hadrons and involve both ultraviolet(UV) and rapidity divergences that require renormalization through a soft factor. Consequentlythe renormalization group equations obeyed by these TMDs involve a resummation of both theUV and rapidity factorization scales. The factorization of these processes is often described inimpact parameter space and the hadrons in the final state must in principle be detected on thewhole phase space. In the limit of large transverse momentum k , or equivalently b T →
0, theTMDFFs can be matched onto the standard (integrated) fragmentation functions. These aredefined as [3] d q → h ( z h ) = 14 z h N c (cid:88) X (cid:90) d ξ + π e − ip − h ξ + / (2 z h ) (cid:104) | T (cid:104) ˜ W † T n q j (cid:105) a (cid:16) ξ + (cid:17) | X, h (cid:105) γ − ij (cid:104) X, h | ¯ T (cid:104) ¯ q i ˜ W T n (cid:105) a (cid:16) − ξ + (cid:17) | (cid:105) ,d g → h ( z h ) = − − (cid:15) ) p − h ( N c − (cid:88) X (cid:90) d ξ + π e − ip − h ξ + / (2 z h ) × (cid:104) | T (cid:104) ˜ W † T n F − µ (cid:105) a (cid:16) ξ + (cid:17) (cid:88) X | X, h (cid:105) g µν (cid:104) X, h | ¯ T (cid:104) F − ν ˜ W T n (cid:105) a (cid:16) − ξ + (cid:17) | (cid:105) . (2.6) We now turn to the operator definitions of the JTMDFFs. The key observation for defining arecoil-free observable, which mitigates its soft sensitivity, is that the recoil of soft radiation trans-lates the whole of the collinear sector coherently in the transverse momentum plane. Therefore, ifwe define a jet axis that also recoils coherently with the soft radiation, any collinear measurementrelative to that axis will be insensitive to these recoil effects. The simplest definition of a recoil-free axis is given via recombination jet algorithms, as summarized in app. B. The basic logic isthat given a list of particles, we have a measure to decide what members of the list should begrouped together as if they came from a single hard progenitor. At each stage of the recombina-tion two particles are merged, and we must decide what the direction is of the “particle” formedby the merged particles. In the winner-take-all (WTA) scheme, this is chosen to be the directionof the more energetic of the two daughters [29, 31]. This scheme is inherently recoil free, sincethe winners of the axis direction are always the most energetic clusters of particles in the jet.Having a recoil-free axis in a recombination algorithm is then simply a matter of the mergerstep. Thus any specific recombination algorithm can be made recoil free, and satisfies eq. (1.2).However, whether one can further factorize collinear splittings landing near the boundary of thejet and those deep inside, depends on the specific measure used to decide which particles will bemerged. We will argue in sec. 3 that this is the case for the Cambridge/Aachen [54–56] and anti- k T measures [57], provided the transverse momentum is sufficiently small such that the hadronis not at the edge of the jet.In what follows, we call the light-cone directions n, ¯ n introduced in eq. (2.1) the fiducial light-cone directions. These are not dynamical, and are simply necessary to define the collinearsector and its gauge-invariant operators. The price paid for a recoil-free axis is that the axisis sensitive to the precise final state configuration of the collinear emissions relative to eachother. This is not the case for a thrust axis, which is essentially a conserved quantity under the– 6 –ollinear splittings, and thus independent of the dynamics . We can demand that the jet haszero transverse momentum with respect to the fiducial light-cone directions, and if we gave ita non-zero transverse momentum with respect to these directions, we would find that it couldbe translated away in the course of the calculation. A one-loop example of this phenomena isgiven in app. A. This captures the notion that the definition of the collinear sector is arbitraryup to translations satisfying a particular power counting, known in the effective theory literatureas reparametrization invariance [53, 58]. Ultimately, it is the measurements imposed on thecollinear sector that determine the power counting of the allowed reparametrization: for recoil-sensitive measurements, the reparametrizations are restricted to those satisfying an ultra-softpower counting [59]. However, for recoil-insensitive measurements, reparametrizations with asoft scaling (see eq. (2.4)) are allowed.We now present the QCD matrix elements for our fragmenting jet functions and JTMDfragmentation functions. The momentum fraction is defined as z h = p − h p − J , (2.7)where p − h and p − J are the large momentum component of the hadron and jet, respectively. Thenwe write: G q → h ( x, p T R, k , z h ) = 14 xN c (cid:88) X (cid:88) J/h (cid:90) d ξ + π e − ip − J ξ + / (2 x ) δ (cid:16) z h − p − h p − J (cid:17) (cid:90) d k A δ (3) (cid:16) (cid:126)k − (cid:126)p h z h (cid:17) (2.8) × (cid:104) | T (cid:104) ˜ W † T n q j (cid:105) a (cid:16) ξ + (cid:17) | X, h ∈ J (cid:105) γ − ij (cid:104) X, h ∈ J | ¯ T (cid:104) ¯ q i ˜ W T n (cid:105) a (cid:16) − ξ + (cid:17) | (cid:105) , G g → h ( x, p T R, k , z h ) = − − (cid:15) ) p − J ( N c − (cid:88) X (cid:88) J/h (cid:90) d ξ + π e − ip − J ξ + / (2 x ) δ (cid:16) z h − p − h p − J (cid:17) (cid:90) d k A δ (3) (cid:16) (cid:126)k − (cid:126)p h z h (cid:17) × (cid:104) | T (cid:104) ˜ W † T n F − µ (cid:105) a (cid:16) ξ + (cid:17) | X, h ∈ J (cid:105) g µν (cid:104) X, h ∈ J | ¯ T (cid:104) F − ν ˜ W T n (cid:105) a (cid:16) − ξ + (cid:17) | (cid:105) , Here, the sum runs over the jets J in the final state, with momentum p J . The hadron h is partof J , but its phase-space integral is not included in the sum on J , as indicated by J/h . The unitvector (cid:126)A J along the jet axis is obtained in the WTA scheme, as discussed above and in app. B. Ineq. (2.8) the integration over k A = A J · p h , the component of the momentum (cid:126)k along the axis (cid:126)A J ,ensures that k picks up the components transverse to this axis. These fragmenting jet functionsare a more differential version of the (semi-inclusive) fragmenting jet function [5, 11, 17, 18], seealso sec. 2.4.When p T R (cid:29) | k | we can perturbatively match the functions G i → h ( x, p T R, k , z h ) onto the Indeed, from a factorization point of view, this is what makes the thrust axis natural. The light-cone directionsused to define the collinear sector should not depend on the specific configuration of collinear particles, since thefactorization itself is unphysical (e.g. it depends on a specific renormalization point). However, the only physical jet axis that is independent of the collinear final state is the direction of total momentum flow, since it is conserved. – 7 –TMDFFs D j → h ( k , z h ), which are defined as D q → h ( k , z h ) = 14 z h N c (cid:88) X (cid:90) d ξ + π e − ip − h ξ + / (2 z h ) (cid:90) d k A δ (3) (cid:16) (cid:126)k − (cid:126)p h z h (cid:17) × (cid:104) | T (cid:104) ˜ W † T n q j (cid:105) a (cid:16) ξ + (cid:17) | X, h (cid:105) γ − ij (cid:104) X, h | ¯ T (cid:104) ¯ q i ˜ W T n (cid:105) a (cid:16) − ξ + (cid:17) | (cid:105) ,D g → h ( k , z h ) = − − (cid:15) ) p − h ( N c − (cid:88) X (cid:90) d ξ + π e − ip − h ξ/ (2 z h ) (cid:90) d k A δ (3) (cid:16) (cid:126)k − (cid:126)p h z h (cid:17) × (cid:104) | T (cid:104) ˜ W † T n F − µ (cid:105) a (cid:16) ξ + (cid:17) | X, h (cid:105) g µν (cid:104) X, h | ¯ T (cid:104) F − ν ˜ W T n (cid:105) a (cid:16) − ξ + (cid:17) | (cid:105) , (2.9)In this expression the boundary of the jet has been expanded to infinity, so X runs over anunrestricted set of states that is independent of the jet definition, and h is not part of X . Theonly dependence on the jet algorithm is through the definition of the jet axis. Note that the onlydifference with eq. (2.5) is the axis with respect to which the transverse momentum is measured. The renormalized fragmentation functions are defined through [3] d bare i → h ( z h ) = (cid:88) j (cid:90) d zz Z ij (cid:16) z h z , µ (cid:17) d j → h ( z, µ ) , (2.10)leading to the following renormalization group equation (RGE) µ dd µ d i → h ( z h , µ ) = (cid:88) j (cid:90) d zz γ ij (cid:16) z h z , µ (cid:17) d j → h ( z, µ ) ,γ ij ( z h , µ ) = − (cid:90) d zz Z − ik (cid:16) z h z , µ (cid:17) µ dd µ Z kj ( z, µ ) . (2.11)The fragmenting jet function G has the same renormalization and thus RGE as the fragmen-tation function, but in the x variable [17, 18, 60] G bare i → h ( x, p T R, k , z h , µ ) = (cid:88) j (cid:90) d x (cid:48) x (cid:48) Z ij (cid:16) xx (cid:48) , µ (cid:17) G j → h ( x (cid:48) , p T R, k , z h , µ ) . (2.12)The RGE of the matching coefficients J in eq. (3.2) follows from inserting eqs. (2.11) and (2.12)in eq. (3.2), and thus involves a DGLAP evolution in both x and z .The renormalization of the JTMD fragmentation function has the same structure as that ofthe standard fragmentation function, D bare i → h ( k , z ) = (cid:88) j (cid:90) d z (cid:48) z (cid:48) Z (cid:48) ij (cid:16) zz (cid:48) , µ (cid:17) D j → h ( k , z (cid:48) , µ ) , (2.13)however it involves a different renormalization factor, Z (cid:48) (cid:54) = Z . The RGE thus has the samestructure as eq. (2.11) but the anomalous dimension is modified to γ (cid:48) .– 8 –he all-orders anomalous dimensions are given by γ ij ( z, µ ) = P ji ( z, µ ) ,γ (cid:48) ij ( z, µ ) = θ (cid:16) z ≥ (cid:17) P ji ( z, µ ) , (2.14)where P denote the DGLAP splitting functions [61–63]. At one-loop order this follows directlyfrom our calculation. In app. C, we argue this relationship is true to all orders, and the corre-sponding expressions in moment space are given to one loop. The jet definition restricts the maximum transverse momentum | k | of the hadron. The transversemomentum k may therefore safely be integrated over (cid:90) d k G i → h ( x, p T R, k , z h , µ ) = G i → h ( x, p T R, z h , µ ) , (2.15)to yield the (semi-inclusive) fragmenting jet function [17, 18]. The same is not true for the TMDfragmentation function, which has a different renormalization than the fragmentation function. Our starting point is the cross section for producing a jet with transverse momentum p T andrapidity η , containing a hadron with momentum fraction z h and transverse momentum z h k ,d σ h d p T d η d k d z h = (cid:88) i (cid:90) d xx ˆ σ i (cid:16) p T x , η, µ (cid:17) G i → h ( x, p T R, k , z h ) (cid:2) O ( R ) (cid:3) . (3.1)This observable is insensitive to soft radiation, since the transverse momentum k is measuredrelative to a recoil-insensitive axis. The above equation thus follows from collinear factorizationfor R (cid:28)
1. The partonic cross section ˆ σ encodes the hard scattering that produces the parton i with transverse momentum p T /x and rapidity η , with respect to the beam axis. The fragmentingjet function G was defined in eq. (2.8) and describes the longitudinal momentum fraction x of theparton that goes into the jet, as well as the fragmentation of the hadron inside the jet. Dependingon the relative hierarchy between p T R , | k | and Λ QCD , eq. (3.1) admits a further factorization. If p T R ∼ | k | (cid:29) Λ QCD , the perturbative dynamics that resolves the jet boundary and generatesthe transverse momentum factorizes from the nonperturbative fragmentation [5, 7], G i → h ( x, p T R, k , z h , µ ) = (cid:88) j (cid:90) d zz J ij (cid:16) x, p T R, k , z h z , µ (cid:17) d j → h ( z, µ ) (cid:20) O (cid:16) Λ k (cid:17)(cid:21) . (3.2)The matching coefficient J ij describes the formation of a jet with momentum fraction x of theinitial parton i , containing a parton j with momentum fraction z h /z and transverse momentum k . The (standard) fragmentation function d j → h describes how this parton j produces a hadronmoving in the same direction with a momentum fraction z h /z × z = z h , see eq. (2.6).– 9 – et boundary winner-take-all axis Figure 2 . Factorization of the axis finding between the angular scale r = | k | /p T and R , with r (cid:28) R . For p T R (cid:29) | k | (cid:29) Λ QCD , a judicious choice of jet axis enables one to separate the effect of thejet boundary and the generation of the perturbative transverse momentum of the hadron, J ij ( x, p T R, k , z, µ ) = (cid:88) k (cid:90) d yy B ik ( x, p T R, y, µ ) C kj (cid:16) k , zy , µ (cid:17)(cid:20) O (cid:16) k p T R (cid:17)(cid:21) , (3.3)due to a second collinear factorization at angular scales r = | k | /p T (cid:28) R . This requires thefactorization of the amplitude and the measurement, which we discuss in turn.For the amplitude to factorize, there must be an energetic parton within an angular distance r of the axis. This is ensured for the winner-take-all axis, which by construction is always alongthe direction of such a parton. The hadron will fragment from this parton in order to be enhancedin the small | k | limit. Of course there can be additional partons in the vicinity of the axis. If theyare produced as splittings from an initial parton, their effect is captured by C in eq. (3.3). Thecase where independent emissions at angular scales R randomly happen to be within a distance r is power suppressed by r/R .For the measurement to factorize as in eq. (3.3), the axis finding must be “recursively local”.What we mean is that the jet axis can be determined within a angular distance of r (cid:28) R by onlyconsidering collinear emissions at angular scales of order R , whereas a more precise determinationof the axis position only requires knowledge of radiation within an angular distance r . A moreconcrete way of thinking about this is illustrated in fig. 2: we “pixelate” the measurement intoregions of angular size r , and the total energy of each pixel is sufficient to determine the pixelcontaining the axis. The position of the axis within the pixel only relies on the energy distributionwithin an angular size r . Collinear splittings inside the pixel only shift the axis an amount oforder r and are thus power suppressed by order r/R for radiation at the jet boundary. Thisguarantees the simple convolution structure in eq. (3.3), where the collinear radiation at angularscales R and r only communicate through a single variable: the energy fraction of the “pixel”containing the winner-take-all axis.When we argue for this recursively local picture of the axis determination, we must es-tablish two properties: radiation within the pixel that eventually contains the jet axis will bepreferentially clustered together first and the configuration of the radiation outside of this pixel– 10 – lusterwinner-take-all standard recombination Figure 3 . The standard recombination scheme allows particles outside a pixel to be clustered into a pixel(blue), without being clustered with the pixel. This is not the case for the winner-take-all scheme (red). does not interfere with the constituents of the pixel, except perhaps at the boundary. TheCambridge/Aachen clustering algorithm [54, 55] with the winner-take-all recombination schemenaturally has these properties, since it is solely based on angular distances. By definition, mostof the radiation within the pixel is at a closer angular distance to each other than to radiationoutside the pixel, and this will be recombined first, except for possible splittings at the boundary.Radiation far from the pixel, e.g. at the jet boundary, will not be clustered in too early. Radiationoutside the pixel that is clustered together will not interfere with the clustering history inside,since the winner-take-all axis always lies on a particle at each step in the recombination. Specifi-cally, two particles outside of a pixel can never be recombined to give a “shadow” particle withinthe pixel, as illustrated in fig. 3, regardless of the ordering in which particles get recombined.The key difference between anti- k T and Cambridge/Aachen is the order in which radiation isclustered. As is well established, anti- k T clusters the most energetic radiation first. By definition,the pixel which will contain the winner-take-all axis in anti- k T will be clustered preferentially,since this is the most energetic region and is where the algorithm will start to cluster. However,radiation around this pixel may not first be clustered with each other but could directly beclustered with that pixel. Nevertheless, the collinear splittings inside the pixel containing thewinner-take-all axis still factorize from the splittings at the jet boundary, i.e. changes in the angleof the jet axis due to collinear splittings inside the pixel will be of order r , and the effect at thejet boundary is thus power suppressed by r/R .Thus we have shown that with the winner-take-all axis, the Cambridge/Aachen and the anti- k T algorithms satisfy the factorization in eq. (3.3). Note the importance of establishing theseall-orders properties, since the one-loop calculation of the matching coefficients in this paper onlyinvolve final states with at most two partons, in which case the winner-take-all axis is simplyalong the most energetic parton.For p T R (cid:29) | k | ∼ Λ QCD , we can separate the effect of the jet boundary from the fragmenta-tion, but cannot calculate the nonperturbative transverse momentum, G i → h ( x, p T R, k , z h , µ ) = (cid:88) k (cid:90) d yy B ik ( x, p T R, y, µ ) D k → h (cid:16) k , z h y , µ (cid:17)(cid:20) O (cid:16) k p T R (cid:17)(cid:21) . (3.4)The JTMD fragmentation function D that arises here is the universal object anticipated before,and is defined through eq. (2.9). As we may also obtain eq. (3.3) by a further factorization ofeq. (3.4) for | k | (cid:29) Λ QCD , consistency implies that the same boundary function B enters in these– 11 –quations and D k → h ( k , z h , µ ) = (cid:88) j (cid:90) d zz C kj (cid:16) k , z h z , µ (cid:17) d j → h ( z, µ ) (cid:20) O (cid:16) Λ k (cid:17)(cid:21) . (3.5) So far we have always assumed that the jet radius R is small, allowing for the factorization ineq. (3.1). However, when R is large the jet at scale p T R cannot be factorized from the hardscattering at scale p T . In this case we can still factorize the JTMD fragmentation functions when k (cid:28) p T , d σ h d p T d η d k d z h = (cid:88) k (cid:90) d yy ¯ σ k ( p T , η, R, y, µ ) D k → h (cid:16) k , z h y , µ (cid:17)(cid:20) O (cid:16) k p T (cid:17)(cid:21) . (3.6)The partonic cross section ¯ σ now describes the hard scattering ˆ σ i and the jet boundary effects B . Indeed, in the limit R (cid:28) σ k ( p T , η, R, y, µ ) = (cid:88) i (cid:90) d xx ˆ σ i (cid:16) p T x , η, µ (cid:17) B ik ( x, p T R, y, µ ) (cid:2) O ( R ) (cid:3) . (3.7) In this section we summarize the one-loop matching coefficients that appear in sec. 3.
The matching coefficients that enter in eq. (3.2) are given by J (0) ij ( x, p T R, k , z, µ ) = δ ij δ ( k ) δ (1 − x ) δ (1 − z ) , (4.1) J (1) qq ( x, p T R, k , z, µ )= α s C F π (cid:18) π µ k /µ ) + δ (1 − x ) θ (cid:16) ≥ z (cid:17) θ (cid:0) p T R ≥ | k | (cid:1) z − z − δ ( k ) δ (1 − z ) θ (1 − x ) (cid:26) (1 + x ) (cid:104)(cid:16) − x (cid:17) + ln (cid:16) p T R µ (cid:17) + 2 (cid:16) ln(1 − x )1 − x (cid:17) + (cid:105) + 1 − x (cid:27) + δ ( k ) δ (1 − x ) (cid:26) θ (cid:16) ≥ z ≥ (cid:17)(cid:20) (1 + z ) (cid:104)(cid:16) − z (cid:17) + ln (cid:16) p T R z µ (cid:17) + 2 (cid:16) ln(1 − z )1 − z (cid:17) + (cid:105) + 1 − z (cid:21) + θ (cid:16) ≥ z (cid:17)(cid:104) z − z ln (cid:0) z (1 − z ) (cid:1) + (1 − z ) (cid:105)(cid:27)(cid:19) , (4.2) J (1) qg ( x, p T R, k , z, µ )= α s C F π (cid:18) π µ k /µ ) + δ (1 − x ) θ (cid:16) ≥ z (cid:17) θ (cid:0) p T R ≥ | k | (cid:1) − z ) z − δ ( k ) δ (1 − z ) θ (1 − x ) (cid:26) − x ) x ln (cid:16) p T R (1 − x ) µ (cid:17) + x (cid:27) + δ ( k ) δ (1 − x ) (cid:26) θ (cid:16) ≥ z ≥ (cid:17)(cid:104) − z ) z ln (cid:16) p T R z (1 − z ) µ (cid:17) + z (cid:105) – 12 – θ (cid:16) ≥ z (cid:17)(cid:104) − z ) z ln (cid:0) z (1 − z ) (cid:1) + z (cid:105)(cid:27)(cid:19) , (4.3) J (1) gg ( x, p T R, k , z, µ )= α s C A π (cid:18) π µ k /µ ) + δ (1 − x ) θ (cid:16) ≥ z (cid:17) θ (cid:0) p T R ≥ | k | (cid:1) − z + z ) z (1 − z ) − δ ( k ) δ (1 − z ) θ (1 − x ) 2(1 − x + x ) x (cid:26)(cid:18) − x (cid:19) + ln p T R µ + 2 (cid:18) ln(1 − x )1 − x (cid:19) + (cid:27) + δ ( k ) δ (1 − x ) (cid:26) θ (cid:16) ≥ z ≥ (cid:17) − z + z ) z (cid:20)(cid:18) − z (cid:19) + ln p T R z µ + 2 (cid:18) ln(1 − z )1 − z (cid:19) + (cid:21) + θ (cid:16) ≥ z (cid:17) − z + z ) z (1 − z ) ln (cid:0) z (1 − z ) (cid:1)(cid:27)(cid:19) , (4.4) J (1) gq ( x, p T R, k , z, µ )= α s T F π (cid:18) π µ k /µ ) + δ (1 − x ) θ (cid:16) ≥ z (cid:17) θ (cid:0) p T R ≥ | k | (cid:1) (cid:0) z + (1 − z ) (cid:1) − δ ( k ) δ (1 − z ) θ (1 − x ) (cid:26)(cid:0) x + (1 − x ) (cid:1) ln p T R (1 − x ) µ + 2 x (1 − x ) (cid:27) + δ ( k ) δ (1 − x ) (cid:26) θ (cid:16) ≥ z ≥ (cid:17)(cid:104)(cid:0) z + (1 − z ) (cid:1) ln p T R z (1 − z ) µ + 2 z (1 − z ) (cid:105) + θ (cid:16) ≥ z (cid:17)(cid:104) (cid:0) z + (1 − z ) (cid:1) ln (cid:0) z (1 − z ) (cid:1) + 2 z (1 − z ) (cid:105)(cid:27)(cid:19) . (4.5)The restriction | k | ≤ p T R encodes the interplay between the jet boundary and | k | measurementat this order. This gets “expanded away” in eq. (3.3) when | k | (cid:28) p T R . The matching coefficients for the universal JTMD fragmentation function in eq. (3.5) are C (0) ij ( k , z, µ ) = δ ij δ ( k ) δ (1 − z ) , (4.6) C (1) qq ( k , z, µ ) = α s C F π θ (cid:16) ≥ z (cid:17)(cid:26) π µ k /µ ) + z − z + δ ( k ) (cid:20) z )1 − z ln (cid:0) z (1 − z ) (cid:1) + 1 − z (cid:21)(cid:27) , (4.7) C (1) qg ( k , z, µ ) = α s C F π θ (cid:16) ≥ z (cid:17)(cid:26) π µ k /µ ) + − z ) z + δ ( k ) (cid:104) − z ) ) z ln (cid:0) z (1 − z ) (cid:1) + z (cid:105)(cid:27) , (4.8) C (1) gg ( k , z, µ ) = α s C A π θ (cid:16) ≥ z (cid:17)(cid:26) π µ k /µ ) + − z + z ) z (1 − z )+ δ ( k ) 4(1 − z + z ) z (1 − z ) ln (cid:0) z (1 − z ) (cid:1)(cid:27) , (4.9) C (1) gq ( k , z, µ ) = α s T F π θ (cid:16) ≥ z (cid:17)(cid:26) π µ k /µ ) + (cid:0) z + (1 − z ) (cid:1) – 13 – δ ( k ) (cid:104) (cid:0) z + (1 − z ) (cid:1) ln (cid:0) z (1 − z ) (cid:1) + 2 z (1 − z ) (cid:105)(cid:27) . (4.10) The matching coefficients in eqs. (3.3) and (3.4) describe the effect of the jet boundary. They arenot independent, as they can be determined from the matching coefficients J ij and C ij by usingeq. (3.3). At tree level B (0) ij ( x, p T R, y, µ ) = δ ij δ (1 − x ) δ (1 − y ) , (4.11)and at one-loop order, J (1) ij ( x, p T R, k , z, µ ) = (cid:104) δ ( k ) B (1) ij ( x, p T R, z, µ ) + δ (1 − x ) C (1) ij ( k , z, µ ) (cid:105)(cid:20) O (cid:16) k p T R (cid:17)(cid:21) . (4.12)This leads for example to B (1) qq ( x, p T R, y, µ ) = α s C F π (cid:18) − δ (1 − y ) θ (1 − x ) (cid:26) (1 + x ) (cid:104)(cid:16) − x (cid:17) + ln (cid:16) p T R µ (cid:17) + 2 (cid:16) ln(1 − x )1 − x (cid:17) + (cid:105) + 1 − x (cid:27) + δ (1 − x ) θ (cid:16) ≥ y ≥ (cid:17) × (cid:26) (1 + y ) (cid:104)(cid:16) − y (cid:17) + ln (cid:16) p T R y µ (cid:17) + 2 (cid:16) ln(1 − y )1 − y (cid:17) + (cid:105) + (1 − y ) (cid:27)(cid:19) . (4.13)The jet axis is along the most energetic of the two partons at this order. This is reflected in theexpressions for the boundary functions, since they vanish for y < /
2. We have also verified thatthe k -dependence cancels between J ij and C ij , since these boundary functions are independentof k . A full-fledged phenomenological analysis will be presented in a forthcoming publication. Here wepresent some first results, focussing on the transverse momentum dependence and taking momentsof z h . To avoid complications from distributions we integrate over the transverse momentum | k | ≤ k c . We will assume p T (cid:29) k c (cid:29) Λ QCD but not make assumptions about the jet radius.Thus, starting from eqs. (3.5) and (3.6), (cid:90) | k | 1) ¯ C qQ + ( n f − C q ¯ Q ¯ C qg n f ¯ C gq ¯ C gg (cid:33) , ¯ γ (cid:48) = (cid:32) ¯ γ (cid:48) qq ¯ γ (cid:48) qg n f ¯ γ (cid:48) gq ¯ γ (cid:48) gg (cid:33) , ¯ γ = (cid:32) ¯ γ qq ¯ γ qg n f ¯ γ gq ¯ γ gg (cid:33) . The contributions C q ¯ q , C qQ and C q ¯ Q only enter at two-loop order, but are generated by the RGevolution. There are now four different modifications ∆ of the exponent of k c , that can arise ina linear combination (cid:88) i =1 ,..., w i k − ∆ i c . (5.7)– 15 – - - - - � ��� � � �� � � � Δ - - - - � � � �� � � � Δ Figure 4 . The dependence of the cross section on the transverse momentum is given by | k | − − ∆ where ∆is controlled by anomalous dimensions. The one-loop exponent ∆ is shown for nonsinglet (left) and singlet(right) distributions, with α s = 0 . These ∆ i are given by the differences of the eigenvalues of the anomalous dimension matrices ¯ γ (cid:48) and ¯ γ in eq. (5.6). The reason there are not two but four values is because their eigenvectors arenot aligned. Denoting the eigenvectors and eigenvalues of ¯ γ by (cid:126)v a and λ a and for ¯ γ (cid:48) by (cid:126)v (cid:48) b and λ (cid:48) b with a, b = 1 , 2, ∆ i = λ (cid:48) b − λ a , i = 2( a − 1) + b . (5.8)At leading order ¯ C is the identity matrix. Inserting this initial condition in the RGE implies w i ∝ (cid:126)v (cid:48) b · (cid:126)v a . (5.9)For large moments N the eigenvectors start to align, suggesting that two of the four weightswould vanish in this limit. However, in the differential spectrum eq. (5.7) leads to (cid:88) i =1 ,..., ∆ i w i | k | − − ∆ i , (5.10)and these terms have a significantly larger ∆ i that compensates for their small w i . The weights ofcourse also depend on the hard scattering and fragmentation functions, and so their expressionsare merely indicative. The exponents ∆ i are shown in fig. 4 at one loop, taking α s = 0 . 1. For thenonsinglet distributions this is probably too small to observe, but the effect should be noticeablefor the singlet distributions. In this paper we introduced a new definition of TMD fragmentation in jets, where the transversemomentum k is measured with respect to a jet axis that is insensitive to the recoil of soft radiation.We derived factorization theorems for the regimes:1 (cid:29) R (cid:38) | k | /p T , (cid:29) R (cid:29) | k | /p T , (cid:38) R (cid:29) | k | /p T , (6.1)where p T is the jet transverse momentum and R is the jet radius parameter. Angular scalesthat have a large hierarchy are described by different ingredients in the factorization theorem.– 16 –he factorization in the latter two cases relied on the winner-take-all recombination schemefor Cambridge/Aachen or anti- k T , because having a recoil-free axis was insufficient. We havecalculated all the (process-independent) matching coefficients at one-loop order.The latter two cases in eq. (6.1) involve a new jet TMD fragmentation function (in the firstcase this cannot be separated from the jet boundary). This JTMDFF is independent of the pro-cess or the number of jets and does not involve rapidity (light-cone) divergences, because our axischoice guarantees that our observable is insensitive to soft radiation. When the transverse mo-mentum k is perturbative, the JTMD fragmentation function can be matched onto the standardfragmentation functions.One can also consider the fragmentation of subjets instead of hadrons. One particular contextwhere this could prove fruitful is in the area of jet substructure (see e.g. refs. [64–66] for anoverview of developments in this field). One of the key applications of jet substructure is toidentify hadronic decays of boosted heavy resonances. The boost causes the decay products to becollimated, yielding a fat jet containing subjets. Understanding the distribution of these subjetswithin the fat jet is critical to distinguish the desired signal from the overwhelming backgroundof normal QCD jets. Our approach would provide analytical control over the transverse momenta(i.e. angles) of subjets. To extend our formalism to subjets is trivial when the reclustering scale R sub (cid:28) | k | /p T , but requires additional calculations for other hierarchies.The case studied in this work treated only unpolarized hadrons/partons. The angular dis-tribution of hadrons can certainly be affected by the measure of the spin and/or helicity of theproduced final state. We postpone to a future work the study of the sensitivity of the jet axisto the spin/helicity of final states and the relative measure of hadron spin-dependent transversemomentum.Another application of our framework is the study of medium effects in heavy-ion collisions.Here the modification of the momentum fraction distribution of hadrons has already been studiedextensively, see e.g. refs. [67, 68]. Our approach would allow one to study the modification ofthe (relative) transverse momentum of collinear hadrons. The insensitivity of our observable tothe abundant background of soft radiation present in heavy ion collisions is crucial to make thisobservable robust, and to be able to make meaningful comparisons to proton-proton collisions. Acknowledgments We thank A. Papaefstathiou, M. Procura, J. Thaler and L. Zoppi for comments on this manuscript.I.S. is supported by the Spanish MECD grant FPA2014-53375-C2-2-P. W.W. is supported by theERC grant ERC-STG-2015-677323 and the D-ITP consortium, a program of the NetherlandsOrganization for Scientific Research (NWO) that is funded by the Dutch Ministry of Education,Culture and Science (OCW). D.N. acknowledges support from US Department of Energy con-tract DE-AC52-06NA25396 and through the LANL/LDRD Program. We also thank the ErwinSchr¨odinger Institute program “Challenges and Concepts for Field Theory and Applications inthe Era of the LHC Run-2”, where this work was initiated. A perhaps more robust observable is the fragmentation of subjets in the heavy ion context. – 17 – Defining recoil-free jet functions To make the paper as self-contained as possible, we will define the general criteria a jet functionmust satisfy to be recoil free, and explicitly illustrate the insensitivity to soft recoil in a one-loopexample. Many different measurements can be made recoil free, and for an extensive discussionin the context of jet shapes, see ref. [30]. We start with a typical jet function, defined as J n ( Q, q , τ ) = N tr (cid:10) (cid:12)(cid:12) Φ n (0) δ ( Q − ¯ n · P ) δ (2) ⊥ (cid:0) q − (cid:126) P ⊥ (cid:1) δ ( τ − ˆ O )Φ n (0) (cid:12)(cid:12) (cid:11) , (A.1)where Φ n is either a quark or gluon field operator, with appropriate Wilson lines in the ¯ n directionfor gauge invariance. ˆ O is the observable imposed on the final state of the jet function, and τ isits value. The trace is over the appropriate color and spin indicies (including the leading-powerDirac structures in the case of a quark), and N normalizes the function. We have included deltafunctions of the momentum operator P that constrain the final state of the jet function to have atotal large momentum component Q , and a total transverse momentum q . The fiducial light-conedirection n need not be aligned with the axis n τ used to define the measurement ˆ O . All we needis that the axis implicit in ˆ O is within a reparameterization transformation of n [53]. That is, ifthe collinear sector has assigned power counting p n ∼ Q (1 , λ , λ ) , (A.2)then 1 − ˆ n · ˆ n τ = O ( λ ) . (A.3)That is, the angle between ˆ n and the measurement axis is of order λ . Definition: the jet function J n is recoil free, if the measurement of τ satisfies: J n ( Q, q , τ ) = J n ( Q, , τ ) + O (cid:18) | q | Q (cid:19) , (A.4)otherwise we call it recoil sensitive . As an example of a recoil sensitive jet function, take theinclusive jet function found in jet mass or thrust calculations. Then ˆ O = n · P , where n is alignedwith the thrust axis of the event, J n ( Q, q , τ ) = J n (cid:0) Q, , τ − q /Q (cid:1) (A.5)This structure appears at all orders, and we immediately see that it fails condition (A.4). Wecan only expand out the injected transverse momentum if q (cid:28) Qτ [59]. A.1 One-loop example We will now show explicitly to one-loop order that if we disturb the fiducial light-cone directionby an injection of soft recoil q , this has no effect on the measured transverse momentum k . Foran all-orders discussion, see ref. [30].First we derive the form of the transverse momentum with respect to the recoil-free axis ina jet with two particles. To see that the corrections really do scale as indicated in eq. (A.4), we– 18 –alculate the winner-take-all axis as a function of the two particle state momenta exactly, thenexpand in the collinear power counting. Since k , k are the only two momenta in the jet, thewinner-take-all axis b is determined by the particle with larger energy:if k > k : b ( k , k ) = k k , ¯ b ( k , k ) = ( n + ¯ n ) − k k , if k > k : b ( k , k ) = k k , ¯ b ( k , k ) = ( n + ¯ n ) − k k . (A.6)For the conjugate ¯ b the sign of the spatial components is flipped, which is accomplished by theabove expressions since n + ¯ n = (2 ,(cid:126) n · k > ¯ n · k : b ( k , k ) · k = 2 k · k ¯ n · k , ¯ b ( k , k ) · k = ¯ n · k , if ¯ n · k > ¯ n · k : b ( k , k ) · k = 2 k · k ¯ n · k , ¯ b ( k , k ) · k = ¯ n · k , (A.7)where expanding in the collinear power counting explicitly gives corrections that scale as thesmall component of the momenta k i over the large momentum fraction (not the transverse scale).The relative transverse momentum of k with respect to the winner-take-all axis isif ¯ n · k > ¯ n · k : | k | = 0 , if ¯ n · k > ¯ n · k : | k | = 1 z h (cid:112) b · k ¯ b · k = 1 z h (cid:16) n · k ¯ n · k k · k (cid:17) / + ... (A.8)We now carry out the calculation of the one-loop JTMDFF given in eq. (2.9), but only tothe point where we can see the independence of the recoil against the injected soft momentum.Exploiting azimuthal symmetry, we may simply consider the measurement of k . To inject softmomentum, we write the matrix element in eq. (2.9) so that the position of the field operatorsacquire a transverse displacement b T as in the standard TMDFF of eq. (2.5). Then we takethe Fourier transform at a momentum q with respect to b T , and integrate over the fiducialtransverse momentum of the hadron. Taking Q to be the large momentum component, theone-loop JTMDFF has the form D (1) i → h ( k , z h ; q ) = g (cid:16) µ e γ E π (cid:17) (cid:15) (cid:90) d d k (2 π ) d − θ (¯ n · k ) δ ( k ) (cid:90) d d k (2 π ) d − θ (¯ n · k ) δ ( k ) (A.9) × (2 π ) d − δ ( Q − ¯ n · k − ¯ n · k ) δ ( d − ( k + k − q ) 4 QC i P gi ( z h )( k + k ) × δ (cid:16) z h − ¯ n · k Q (cid:17) π (cid:20) θ (cid:16) z h − (cid:17) δ ( k ) + θ (cid:16) − z h (cid:17) δ (cid:18) k − (cid:16) n · k z h ¯ n · k (cid:17) k · k (cid:19)(cid:21) Here we are integrating over the on-shell phase space of the two final-state partons, with momenta k and k . The phase space is simple to interpret: The large components of the two particlessum to Q , while they have a non-trivial total transverse momentum q with respect to the fiducialcollinear direction n . The key point will be that the recoil-free axis is only sensitive to the relativetransverse momentum of the two particles. We assume that k is the momentum of the observed– 19 –ragmented particle, which for conciseness we take to be a gluon with splitting function P gi . Thecolor factor C i is C F for quarks and C A for gluons. From the delta functions in eq. (A.9) we infer:¯ n · k = Qz h , ¯ n · k = Q (1 − z h ) ,n · k = k Qz h , n · k = k Q (1 − z h ) , k = q − k , ( k + k ) = 1 z h (1 − z h ) (cid:0) k − z h k · q + z h q (cid:1) . (A.10)Performing the integrations in eq. (A.9) yields D (1) i → h ( (cid:126)k, z h ; q ) = g (cid:16) µ e γ E π (cid:17) (cid:15) C i (cid:90) d − (cid:15) k (2 π ) − (cid:15) P gi ( z h ) k − z h k · q + z h q (A.11) × π (cid:20) θ (cid:16) z h − (cid:17) δ (cid:0) k (cid:1) + θ (cid:16) − z h (cid:17) δ (cid:18) k − k − z h k · q + z h q z h (1 − z h ) (cid:19)(cid:21) . We can immediately see that this function is recoil free, since the injected transverse momenta q always appears in the same combination with k . Thus we can just perform a variable changeand get rid of it, k → k + z h q , (A.12)making the jet function manifestly independent of q . B Clustering algorithms We give a brief review of jet recombination algorithms. A more extensive discussion can be foundin e.g. ref. [69]. We need a metric d α ( p i , p j ; R ) ≡ d αij ( R ) that measures the distance betweentwo particles with momenta p i , p j in momentum space, where R is the jet radius parameter. Inaddition we need a single particle metric d αjet ( p i ) ≡ d αjet ( i ) that will decide whether a particle canbe considered a jet or not. The class of metrics of interest are: e + e − collision pp collision d αij ( R ) = min (cid:16) ( p i ) α , ( p j ) α (cid:17) θ ij R d αij ( R ) = min (cid:16) p αT i , p αT j (cid:17) R ij Rd αjet ( i ) = ( p i ) α d αjet ( i ) = p αT i (B.1)In the case of e + e − collisions, θ ij is the angle between the two particles’ 3-momenta, and in thecase of pp collisions, R ij = (cid:113) ( η i − η j ) + ( φ i − φ j ) (B.2)is the euclidean distance between them in rapidity and azimuthal space. Note that the subscript T refers to the transverse momentum with respect to the beam axis. The commonly used k T [70, 71], Cambridge/Aachen [54–56], and anti- k T [57] algorithms correspond to α = 1 , , − P = { p , ..., p n } with momenta p i , and an empty list of jets J = {} , the recombination algorithmproceeds as follows:1. If P is empty, stop, output J . If P nonempty, continue.2. Compute d αij ( R ) for all i, j ∈ P , and d αjet ( i ) for all i ∈ P .3. Select the pair or the particle whose distance measure is smallest.4. If the selection with smallest measure is a single particle, i , delete p i from the list P , moveit to the list J .5. If the selection with smallest measure is a pair of particles, i, j , delete both from P , merge (i,j) them into one particle p new , and append P with p new .6. Go back to step 1.The particles inside a jet are simply all the particles that got clustered into the “particle” thatwinds up in the list J . The merge (i,j) procedure is usually one of the following procedures: • E-scheme: p new = p i + p j . • Winner-take-all scheme [29, 31]: writing p i = ( p i , (cid:126)p i ) , p j = ( p j , (cid:126)p j ), then p new = p i + p j , ˆ p new = (cid:126)p i | (cid:126)p i | if p i > p j(cid:126)p j | (cid:126)p j | if p j > p i p new = p new (1 , ˆ p new ) (B.3)The E-scheme results in a jet axis that aligns with the total jet momentum. Thus many propertiesof the thrust axis commonly used in event shape descriptions of jets also hold true for an E-schemeaxis. The WTA-scheme generally has a jet axis displaced from the total jet momenta. In the caseof the JTMDFF, eq. (2.9), we apply the clustering algorithm assuming the final states remain inthe jet. That is, we wish to only find the axis, and the jet algorithm is expanded in the limit thatall particles are collinear enough, that they would always cluster before being promoted to a jet.In that case, we do not apply the single particle jet measure, and merely recombine pairwise allthe particles until only one particle remains in the list P . That remaining particle gives the jetaxis. C Results on anomalous dimensions C.1 All-orders anomalous dimension of JTMDFF The one-loop anomalous dimension of the JTMDFF D ( k , z h , µ ), defined in eq. (2.9), is given by γ (cid:48) (1) ij ( z, µ ) = θ (cid:16) z ≥ (cid:17) P (1) ji ( z ) . (C.1)– 21 –e will now argue this relation holds to all orders in perturbation theory, that is γ (cid:48) ij ( z, µ ) = θ (cid:16) z ≥ (cid:17) ∞ (cid:88) (cid:96) =1 P ( (cid:96) ) ji ( z ) , (C.2)where P ( (cid:96) ) ji is the DGLAP splitting kernel at order α (cid:96)s . First we observe that if the partonmomentum fraction z > / 2, the winner-take-all axis will be along its direction and k = 0. Thusthe transverse momentum measurement does not impose a restriction on the phase space andthe calculation of the JTMDFF is identical to the standard fragmentation function in this case.In particular, the IR and the UV divergences exactly match between the fragmentation functionand the JTMDFF.For z < / k . The transverse momentum of the observed parton now hasan explicit upper bound due to the k measurement, since for large parton transverse momentathe WTA axis will be along one of the other partons. This implies that this parton’s momentumcannot scale into the UV with all the other momenta to produce a UV divergence. The only UVdivergences that can occur are subdivergences corresponding to the strongly-ordered limit, whichare renormalized by appropriate lower-order counter terms.Of course there can be new IR divergences introduced at each order, since the k measurementdoes not prevent the transverse momenta of the partons from scaling into the IR (in a non-strongly ordered limit). Indeed, the IR divergences must exactly match those in the standardfragmentation function, including for z < / 2, due to eq. (3.5). Note that we do not need to beconcerned that virtual corrections will convert a 1 /(cid:15) IR into a 1 /(cid:15) UV , since they are located at z = 1. C.2 One-loop anomalous dimensions in moment space The one-loop anomalous dimensions are in moment space given by¯ γ (1) qq ( N, µ ) = α s ( µ ) C F π (cid:104) − H ( N ) − N + 1 − N + 2 + 32 (cid:105) , ¯ γ (1) qg ( N, µ ) = α s ( µ ) C F π (cid:104) N − N + 1 + 1 N + 2 (cid:105) , ¯ γ (1) gg ( N, µ ) = α s ( µ ) C A π (cid:104) − H ( N + 1) + 2 N − N + 1 + 2 N + 2 − N + 3 (cid:105) + α s ( µ ) β π , ¯ γ (1) gq ( N, µ ) = α s ( µ ) T F π (cid:104) N + 1 − N + 2 + 2 N + 3 (cid:105) , ¯ γ (cid:48) (1) qq ( N, µ ) = ¯ γ (1) qq ( N, µ ) − α s ( µ ) C F π (cid:2) − H / ( N ) − H / ( N + 2) + 2 ln 2 (cid:3) , ¯ γ (cid:48) (1) qg ( N, µ ) = ¯ γ (1) qg ( N, µ ) − α s ( µ ) C F π − N − N + 17 N + 16 N ( N + 1)( N + 2) , ¯ γ (cid:48) (1) gg ( N, µ ) = ¯ γ (1) gg ( N, µ ) − α s ( µ ) C A π (cid:20) − H / ( N + 1) + 2 ln 2 + 2 − N − N + 33 N + 68 N + 48 N ( N + 1)( N + 2)( N + 3) (cid:21) , ¯ γ (cid:48) (1) gq ( N, µ ) = ¯ γ (1) gq ( N, µ ) − α s ( µ ) T F π − N − N + 5 N + 8( N + 1)( N + 2)( N + 3) , (C.3)– 22 –here H ( N ) = N (cid:88) i =1 i , H / ( N ) = N (cid:88) i =1 i i = ln 2 − − N − Φ (cid:16) , , N + 1 (cid:17) , (C.4)and Φ is the Lerch transcendent function. 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