Jet axes and universal transverse-momentum-dependent fragmentation

We study the transverse momentum spectrum of hadrons in jets. By measuring the transverse momentum with respect to a judiciously chosen axis, we find that this observable is insensitive to (the recoil of) soft radiation. Furthermore, for small transverse momenta we show that 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 does not involve rapidity divergences and is universal in the sense that it is independent of the type of process and number of jets. These results directly apply to sub-jets instead of hadrons. We discuss potential applications, which include studying nuclear modification effects in heavy-ion collisions and identifying boosted heavy resonances.


Introduction
In the analysis of events from hadron colliders it is common to use jets to organize the final states of hard interactions, making it natural to ask how the QCD confinement of hadrons is realized in this 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 shortdistance scale of the hard-scattering, we have a quark or gluon of a much lower "off-shellness" exiting the hard interaction in a more or less definite direction.The subsequent branching does not change this direction much, but does gives rise to a host of additional partons loosely grouped into a jet.These are the perturbative remains of the slightly off-shell parton.Lastly, these additional partons undergo a "hadronization" process at length scales of 1/Λ QCD , confining themselves into the observed hadrons.Ultimately, to understand the dynamics of confinement within jets, we would like to have a means of comparing the partonically generated momentum distribution inside the jet to the observed hadronic momentum distribution.In addition to momentum, one would also like to understand how quantum numbers, like spin, flavor, or charge, are transported from the hard scattering into the hadronic final state.
The fragmentation function d i→h (z h , µ) describes the distribution of the longitudinal-momentum fraction z h of hadrons of a species h = π + , π − , . . .produced by a parton i = g, u, ū, d, . . .[1][2][3].This allows one to express their production cross section as (see e.g.ref. [4]) where Q is the scale of the hard scattering.A crucial feature of fragmentation is that it is universal, 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 the fragmentation dynamics, and can be factorized from the hard scattering.Thus fragmentation measurements at hadron-hadron, hadron-electron, and electron-positron colliders can all be compared.However, when combining hadron analysis with modern jet algorithms one begins to worry that the definition of the jet itself could potentially spoil this universality, since any given jet 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 accounting 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 exclusive processes (i.e. with a specific number of jets) this was studied using event shapes (hemisphere jets) in refs.[5][6][7][8][9][10] and with a jet algorithm in refs.[11][12][13][14].Inclusive jet production with a jet algorithm was investigated in refs.[15][16][17][18].The applications that were considered range from comparisons to LHC measurements of charged hadron spectra [12] to unravelling quarkonium production channels [13].Multi-hadron fragmentation in jets has also been considered [19][20][21][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 momenta inside the jet, not merely the energy fraction.However, one must be careful, since asking questions about the other components of the hadron's momentum can easily expose one to sensitivity to associated soft processes.While studying these soft processes is an interesting and worthwhile endeavor in and of itself, it can severely complicate any potential claim to universality of these distributions.In the standard terminology the TMDs measure the correlation of transverse momentum 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 instead the measurement of the transverse momentum of a hadron with respect to a jet axis.In the standard TMD correlations one cannot avoid the appearance of rapidity divergences and the consequent regularization and renormalization [23][24][25][26][27][28], which signals a sensitivity to soft physics.
Here we show that one can define a transverse momentum observable which is insensitive to such problems, by a judicious choice of jet axis.The final TMDs will necessarily be different from the 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][30][31].Put loosely, if one uses an axis whose direction is conserved under splittings (e.g. the total momentum of the jet or the thrust axis), this introduces a soft-sensitivity in the axis definition, since a soft emission can displace a collinear one, see fig. 1.Alternatively, one can adopt an axis that itself recoils coherently 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 factorization theorem has a simple multiplicative structure (see e.g.ref. [32]) but the soft radiation suffers from non-global logarithms (NGLs) [33], which arise because of the very different restrictions on the radiation inside an outside the jet.
• Inclusive production with a jet algorithm and a recoil-sensitive axis: This was recently studied in ref. [41].The TMD fragmentation function involves rapidity divergences.One can define a factor consisting of (collinear-)soft modes which cancels these rapidity divergences.However, this (collinear-)soft radiation will displace the jet axis and contaminate the transverse momentum distribution, again introducing a sensitivity to NGLs.1 .
• Inclusive production with a jet algorithm and a recoil-free axis: the observable is purely collinear, making it universal and free of NGLs.This is the case we focus on.
We now briefly outline our framework: For definiteness we focus on hadronic collisions with energetic 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][44][45].Consider the measurement of the longitudinal momentum fraction z h and transverse momentum z h k (with respect to the jet axis) of an energetic hadron h inside a jet. 2 To leading approximation, the soft radiation outside the jet cannot affect the production of a hadron inside a jet.However, as illustrated in fig. 1, the measurement of the transverse momentum of a hadron with respect to the standard jet axis is sensitive to the soft radiation inside the jet.This is not the case when using a recoil-insensitive axis, which is determined by the configuration of the energetic (collinear) radiation.Under the assumption that the jet radius R 1 (the case R ∼ 1 will also be discussed), collinear factorization leads to 2) The partonic cross section σ encodes the hard scattering producing the parton i with transverse momentum p T /x and rapidity η, with respect to the beam axis. (1.3) Since D k→h is a purely collinear object, it is automatically universal, i.e. insensitive to the type of process or number of jets.It also does not involve rapidity divergences, unlike the classical TMD fragmentation functions.The paper is organized as follows: We start by outlining the differences between the classical TMDs and the JTMDs that are considered in this work in sec.2. We also define all ingredients that enter in our fractorization theorems and discuss their renormalization.A discussion of recoilfree jet definitions in the context of a simple example in given in app.A, including a one-loop calculation.The winner-take-all recombination scheme [29,31] that we use to obtain a recoil-free jet 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 not small.We have calculated the one-loop matching coefficients and present these in sec.4. In sec.5 some 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.

Framework
We use light-like vectors n and n with n • n = 2 to introduce the light-cone coordinates used here The time-like and space-like component of a vector are indicated by (v 0 , v), so that In the language of soft-collinear effective theory (SCET) [46][47][48][49], if we assign a power counting for the collinear momenta as where the power counting parameter λ 1 is set by the specific measurement, then we can define a power counting for the soft radiation as (2.4) For β = 1 this is referred to as soft radiation, and for β = 2 it is called ultra-soft.

Standard transverse momentum dependent fragmentation functions
Before arriving at the formulation of the JTMDs, it is instructive to recall some properties of the classical unpolarized TMD fragmentation functions [26,50], where ξ = (ξ + , 0 − , b T ).The variable conjugate to the impact parameter b T is k, which is the transverse momentum of the hadron divided by its momentum fraction.The sum runs over all intermediate states X, and X does not include the hadron h.The Wilson lines WT n (x) depend on the coordinate x and continue to the light-cone infinity along the vector n, where it is connected by a transverse link to the transverse infinity (as indicated by the subscript T ) [51,52].The representations 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 specific axis choice has been made, namely that the n direction is along the hadron h.By performing a change of coordinates (or reparametrization [53]), it follows that this corresponds to measuring the transverse momentum of the hadron with respect to the axis lying along the total momentum of all particles in the intermediate state.For fragmentation in e + e − → hadrons, this axis is equivalent to the thrust axis.These 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.Consequently the renormalization group equations obeyed by these TMDs involve a resummation of both the UV and rapidity factorization scales.The factorization of these processes is often described in impact parameter space and the hadrons in the final state must in principle be detected on the whole phase space.In the limit of large transverse momentum k, or equivalently b T → 0, the TMDFFs can be matched onto the standard (integrated) fragmentation functions.These are defined as [3]

Definitions for TMD fragmentation inside a jet
We now turn to the operator definitions of the JTMDFFs.The key observation for defining a recoil-free observable, which mitigates its soft sensitivity, is that the recoil of soft radiation translates the whole of the collinear sector coherently in the transverse momentum plane.Therefore, if we define a jet axis that also recoils coherently with the soft radiation, any collinear measurement relative to that axis will be insensitive to these recoil effects.The simplest definition of a recoilfree axis is given via recombination jet algorithms, as summarized in app.B. The basic logic is that given a list of particles, we have a measure to decide what members of the list should be grouped together as if they came from a single hard progenitor.At each stage of the recombination two particles are merged, and we must decide what the direction is of the "particle" formed by the merged particles.In the winner-take-all (WTA) scheme, this is chosen to be the direction of the more energetic of the two daughters [29,31].This scheme is inherently recoil free, since the 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 merger step.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 the jet and those deep inside, depends on the specific measure used to decide which particles will be merged.We will argue in sec.3 that this is the case for the Cambridge/Aachen [54][55][56] and antik T measures [57], provided the transverse momentum is sufficiently small such that the hadron is 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 collinear sector and its gauge-invariant operators.The price paid for a recoil-free axis is that the axis is sensitive to the precise final state configuration of the collinear emissions relative to each other.This is not the case for a thrust axis, which is essentially a conserved quantity under the collinear splittings, and thus independent of the dynamics3 .We can demand that the jet has zero transverse momentum with respect to the fiducial light-cone directions, and if we gave it a non-zero transverse momentum with respect to these directions, we would find that it could be translated away in the course of the calculation.A one-loop example of this phenomena is given in app. A. This captures the notion that the definition of the collinear sector is arbitrary up to translations satisfying a particular power counting, known in the effective theory literature as reparametrization invariance [53,58].Ultimately, it is the measurements imposed on the collinear sector that determine the power counting of the allowed reparametrization: for recoilsensitive measurements, the reparametrizations are restricted to those satisfying an ultra-soft power counting [59].However, for recoil-insensitive measurements, reparametrizations with a soft scaling (see eq. (2.4)) are allowed.
We now present the QCD matrix elements for our fragmenting jet functions and JTMD fragmentation functions.The momentum fraction is defined as where p − h and p − J are the large momentum component of the hadron and jet, respectively.Then we write: Here, the sum runs over the jets J in the final state, with momentum p J .The hadron h is part of J, but its phase-space integral is not included in the sum on J, as indicated by J/h.The unit vector A J along the jet axis is obtained in the WTA scheme, as discussed above and in app.B. In eq.(2.8) the integration over k A = A J • p h , the component of the momentum k along the axis A J , ensures that k picks up the components transverse to this axis.These fragmenting jet functions are a more differential version of the (semi-inclusive) fragmenting jet function [5,11,17,18], see also sec.2.4.When p T R |k| we can perturbatively match the functions G i→h (x, p T R, k, z h ) onto the JTMDFFs D j→h (k, z h ), which are defined as In this expression the boundary of the jet has been expanded to infinity, so X runs over an unrestricted set of states that is independent of the jet definition, and h is not part of X.The only dependence on the jet algorithm is through the definition of the jet axis.Note that the only difference with eq.(2.5) is the axis with respect to which the transverse momentum is measured.

Renormalization
The renormalized fragmentation functions are defined through [3] leading to the following renormalization group equation (RGE) (2.11) The fragmenting jet function G has the same renormalization and thus RGE as the fragmentation function, but in the x variable [17,18,60] (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 of the standard fragmentation function, however it involves a different renormalization factor, Z = Z.The RGE thus has the same structure as eq.(2.11) but the anomalous dimension is modified to γ .
The all-orders anomalous dimensions are given by where P denote the DGLAP splitting functions [61][62][63].At one-loop order this follows directly from our calculation.In app.C, we argue this relationship is true to all orders, and the corresponding expressions in moment space are given to one loop.

Sum rule
The jet definition restricts the maximum transverse momentum |k| of the hadron.The transverse momentum k may therefore safely be integrated over to yield the (semi-inclusive) fragmenting jet function [17,18].The same is not true for the TMD fragmentation function, which has a different renormalization than the fragmentation function.

Jet factorization and TMD fragmentation
Our starting point is the cross section for producing a jet with transverse momentum p T and rapidity η, containing a hadron with momentum fraction z h and transverse momentum z h k, This observable is insensitive to soft radiation, since the transverse momentum k is measured relative to a recoil-insensitive axis.The above equation thus follows from collinear factorization for R 1.The partonic cross section 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 fragmenting jet function G was defined in eq.(2.8) and describes the longitudinal momentum fraction x of the parton that goes into the jet, as well as the fragmentation of the hadron inside the jet.Depending on the relative hierarchy between p T R, |k| and Λ QCD , eq. (3.1) admits a further factorization.

Factorization of fragmentation from perturbative radiation
Λ QCD , the perturbative dynamics that resolves the jet boundary and generates the transverse momentum factorizes from the nonperturbative fragmentation [5,7], The matching coefficient J ij describes the formation of a jet with momentum fraction x of the initial 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 hadron moving in the same direction with a momentum fraction z h /z × z = z h , see eq. (2.6).

Factorization of TMD fragmentation from jet definition
For p T R |k| Λ QCD , a judicious choice of jet axis enables one to separate the effect of the jet boundary and the generation of the perturbative transverse momentum of the hadron, due to a second collinear factorization at angular scales r = |k|/p T R.This requires the factorization 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 along the direction of such a parton.The hadron will fragment from this parton in order to be enhanced in the small |k| limit.Of course there can be additional partons in the vicinity of the axis.If they are produced as splittings from an initial parton, their effect is captured by C in eq.(3.3).The case 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 R by only considering collinear emissions at angular scales of order R, whereas a more precise determination of the axis position only requires knowledge of radiation within an angular distance r.A more concrete way of thinking about this is illustrated in fig.2: we "pixelate" the measurement into regions of angular size r, and the total energy of each pixel is sufficient to determine the pixel containing the axis.The position of the axis within the pixel only relies on the energy distribution within an angular size r.Collinear splittings inside the pixel only shift the axis an amount of order r and are thus power suppressed by order r/R for radiation at the jet boundary.This guarantees the simple convolution structure in eq.(3.3),where the collinear radiation at angular scales 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 establish two properties: radiation within the pixel that eventually contains the jet axis will be preferentially clustered together first and the configuration of the radiation outside of this pixel cluster winner-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.The Cambridge/Aachen clustering algorithm [54,55] with the winner-take-all recombination scheme naturally has these properties, since it is solely based on angular distances.By definition, most of the radiation within the pixel is at a closer angular distance to each other than to radiation outside 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.Radiation outside 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.Specifically, two particles outside of a pixel can never be recombined to give a "shadow" particle within the 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 is clustered.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 be clustered with that pixel.Nevertheless, the collinear splittings inside the pixel containing the winner-take-all axis still factorize from the splittings at the jet boundary, i.e. changes in the angle of the jet axis due to collinear splittings inside the pixel will be of order r, and the effect at the jet boundary is thus power suppressed by r/R.
Thus we have shown that with the winner-take-all axis, the Cambridge/Aachen and the antik T algorithms satisfy the factorization in eq.(3.3).Note the importance of establishing these all-orders properties, since the one-loop calculation of the matching coefficients in this paper only involve final states with at most two partons, in which case the winner-take-all axis is simply along the most energetic parton.
For p T R |k| ∼ Λ QCD , we can separate the effect of the jet boundary from the fragmentation, but cannot calculate the nonperturbative transverse momentum, (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 of eq.(3.4) for |k| Λ QCD , consistency implies that the same boundary function B enters in these equations and (3.5)

Factorization for large radius jets
So far we have always assumed that the jet radius R is small, allowing for the factorization in eq.(3.1).However, when R is large the jet at scale p T R cannot be factorized from the hard scattering at scale p T .In this case we can still factorize the JTMD fragmentation functions when The partonic cross section σ now describes the hard scattering σi and the jet boundary effects B. Indeed, in the limit R 1,

NLO maching coefficients
In this section we summarize the one-loop matching coefficients that appear in sec.3.

Fragmenting jet function
The matching coefficients that enter in eq.(3.2) are given by The restriction |k| ≤ p T R encodes the interplay between the jet boundary and |k| measurement at this order.This gets "expanded away" in eq.(3.3) when |k| p T R.

Boundary function
The matching coefficients in eqs.(3.3) and (3.4) describe the effect of the jet boundary.They are not independent, as they can be determined from the matching coefficients J ij and C ij by using eq.(3.3).At tree level and at one-loop order, (4.12) This leads for example to The jet axis is along the most energetic of the two partons at this order.This is reflected in the expressions for the boundary functions, since they vanish for y < 1/2.We have also verified that the k-dependence cancels between J ij and C ij , since these boundary functions are independent of k.

Results for moments
A full-fledged phenomenological analysis will be presented in a forthcoming publication.Here we present some first results, focussing on the transverse momentum dependence and taking moments of z h .To avoid complications from distributions we integrate over the transverse momentum |k| ≤ k c .We will assume p T k c Λ QCD but not make assumptions about the jet radius.Thus, starting from eqs. (3.5) and (3.6), This implies that the transverse momentum dependence is completely governed by the matching coefficients Cij (k c , N, µ), which in fixed-order perturbation theory are constant at leading order and give rise to a ln(k c /µ) at order α s , see eq. (4.6).Note that for N = 1 the expression in eq.(5.1) is purely perturbative, since the dependence on the fragmentation functions drops out due to the momentum sum rule The dependence on k c gets modified by the anomalous dimension of C, which follows from eqs. (2.11) and (2.13) and is multiplicative in moment space (5. 3) The anomalous dimensions in moment space are given at one-loop order in app.C.2.For large values of N , the difference between the anomalous dimensions γ and γ decreases as 2 −N and will cancel in eq. ( 5.3).Thus in that limit the transverse momentum dependence is fully captured by the fixed-order result for C.
To diagonalize the anomalous dimension matrix in eq. ( 5.3) it is convenient to perform the usual singlet/nonsinglet decomposition.Nonsinglet combinations such as C qq − C qq and C qq − C qQ , where Q = q denotes a different quark flavor, do not mix.For such terms the RGE for C in eq. ( 5.3) has as solution At leading order C is independent of k c , so differentiating with respect to k c to determine the k dependence yields |k| −2−∆ .For the singlet contribution eq. ( 5.3) takes the following form The contributions C q q, C qQ and C q Q only enter at two-loop order, but are generated by the RG evolution.There are now four different modifications ∆ of the exponent of k c , that can arise in a linear combination i=1,...,4 These ∆ i are given by the differences of the eigenvalues of the anomalous dimension matrices γ and γ in eq.(5.6).The reason there are not two but four values is because their eigenvectors are not aligned.Denoting the eigenvectors and eigenvalues of γ by v a and λ a and for γ by v b and At leading order C is the identity matrix.Inserting this initial condition in the RGE implies (5.9) For large moments N the eigenvectors start to align, suggesting that two of the four weights would vanish in this limit.However, in the differential spectrum eq.(5.7) leads to i=1,...,4 and these terms have a significantly larger ∆ i that compensates for their small w i .The weights of course also depend on the hard scattering and fragmentation functions, and so their expressions are merely indicative.The exponents ∆ i are shown in fig. 4 at one loop, taking α s = 0.1.For the nonsinglet distributions this is probably too small to observe, but the effect should be noticeable for the singlet distributions.

Conclusions
In this paper we introduced a new definition of TMD fragmentation in jets, where the transverse momentum 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: where p T is the jet transverse momentum and R is the jet radius parameter.Angular scales that have a large hierarchy are described by different ingredients in the factorization theorem.
The factorization in the latter two cases relied on the winner-take-all recombination scheme for Cambridge/Aachen or anti-k T , because having a recoil-free axis was insufficient.We have calculated 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 first case this cannot be separated from the jet boundary).This JTMDFF is independent of the process or the number of jets and does not involve rapidity (light-cone) divergences, because our axis choice guarantees that our observable is insensitive to soft radiation.When the transverse momentum k is perturbative, the JTMD fragmentation function can be matched onto the standard fragmentation functions.
One can also consider the fragmentation of subjets instead of hadrons.One particular context where this could prove fruitful is in the area of jet substructure (see e.g.refs.[64][65][66] for an overview of developments in this field).One of the key applications of jet substructure is to identify hadronic decays of boosted heavy resonances.The boost causes the decay products to be collimated, yielding a fat jet containing subjets.Understanding the distribution of these subjets within the fat jet is critical to distinguish the desired signal from the overwhelming background of 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 |k|/p T , but requires additional calculations for other hierarchies.The case studied in this work treated only unpolarized hadrons/partons.The angular distribution of hadrons can certainly be affected by the measure of the spin and/or helicity of the produced final state.We postpone to a future work the study of the sensitivity of the jet axis to the spin/helicity of final states and the relative measure of hadron spin-dependent transverse momentum.
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 studied extensively, see e.g.refs.[67,68].Our approach would allow one to study the modification of the (relative) transverse momentum of collinear hadrons. 4The insensitivity of our observable to the abundant background of soft radiation present in heavy ion collisions is crucial to make this observable robust, and to be able to make meaningful comparisons to proton-proton collisions.fragmented particle, which for conciseness we take to be a gluon with splitting function P gi .The color factor C i is C F for quarks and C A for gluons.From the delta functions in eq.(A.9) we infer: Performing the integrations in eq.(A.9) yields .
We can immediately see that this function is recoil free, since the injected transverse momenta q always appears in the same combination with k 1 .Thus we can just perform a variable change and get rid of it, 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 found in e.g.ref. [69].We need a metric d α (p i , p j ; R) ≡ d α ij (R) that measures the distance between two particles with momenta p i , p j in momentum space, where R is the jet radius parameter.In addition we need a single particle metric d α jet (p i ) ≡ d α jet (i) that will decide whether a particle can be considered a jet or not.The class of metrics of interest are: e + e − collision pp collision In the case of e + e − collisions, θ ij is the angle between the two particles' 3-momenta, and in the case of pp collisions, 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][55][56], and anti-k T [57] algorithms correspond to α = 1, 0, −1.

Figure 1 .
Figure1.The standard jet axis is sensitive to soft radiation (orange) through recoil effects, whereas a recoil-free axis follows the direction of collinear radiation (blue).

Figure 2 .
Figure 2. Factorization of the axis finding between the angular scale r = |k|/p T and R, with r R.

Figure 4 .
Figure 4.The dependence of the cross section on the transverse momentum is given by |k| −2−∆ where ∆ is controlled by anomalous dimensions.The one-loop exponent ∆ is shown for nonsinglet (left) and singlet (right) distributions, with α s = 0.1.
The fragmenting jet function G describes the fraction x of the parton energy that goes into the jet, as well as the fragmentation of the hadron inside the jet with momentum fraction z h and transverse momentum z h k.The function G obeys a collinear renormalization group equation.A further factorization of this cross section can be achieved when p T R |k| and/or |k| Λ QCD , which is discussed in detail in sec.3.In particular, for p T R |k| we can separate the effect of the jet boundary B from the fragmentation, leading to a new JTMD fragmentation function D k→h ,