Probing Transverse-Momentum Dependent Evolution With Groomed Jets
PPrepared for submission to JHEP
LA-UR-17-31338
Probing Transverse-Momentum Dependent EvolutionWith Groomed Jets
Yiannis Makris, Duff Neill, and Varun Vaidya,
Theoretical Division, MS-248, Los Alamos National Laboratory, Los Alamos, NM 87545
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We propose an observable which involves measuring the properties (transversemomentum p h ⊥ and energy fraction z h ) of an identified hadron inside a groomed jet. The jetis identified with an anti-kT/CA algorithm and is groomed by implementing the modifiedmass drop procedure with an energy cut-off parameter, z cut . The transverse momentumof the hadron inside the jet is measured with respect to the groomed jet axis. We obtaina factorization theorem in the framework of Soft Collinear Effective Theory (SCET), todefine a Transverse Momentum Dependent Fragmenting Jet Function (TMDFJF). TheTMDFJF is factorized into collinear and collinear soft modes by matching onto SCET + .We resum large logarithms in E J /p h ⊥ , where E J is the ungroomed jet energy, to NLLaccuracy and apply this formalism for computing the shape of the p h ⊥ distribution of apion produced in an e + + e − collision. We observe that the introduction of groomingmakes this observable insensitive to non-global logarithms and particularly sensitive tonon-perturbative physics of the transverse momentum dependent evolution at low valuesof p h ⊥ , which can be probed in the variation of the cut-off parameter, z cut , of the groomer.We discuss how this observable can be used to distinguish between non-perturbative modelsthat describe universal TMD evolution and provide a window into the three dimensionalstructure of hadrons. a r X i v : . [ h e p - ph ] A ug ontents A.1 Matching onto the fragmentation functions 17A.2 collinear-soft function, S ⊥ i B An Alternative Path to Factorization of Standard FF 18
The transverse momentum spectrum with respect to a fiducial axis of an energetic or mas-sive color-singlet state has been recognized as an observable of fundamental interest inprobing quantum chromodynamics (QCD) and the factorization of infrared effects fromultraviolet hard processes. In hadron-hadron or hadron-lepton collisions (that is, Drell-Yan like spectra and semi-inclusive production of a hadron in deep inelastic scattering),when the fiducial axis is taken to be the momentum of the hadronic beam(s), these observ-ables give a three-dimensional picture of the single parton dynamics within the hadronicobject. The infrared dynamics can be factored into distribution functions sensitive to theproduced transverse momentum and the energy deposited into the creation of the hardcolor singlet state[1–3] with appropriate soft factors and subtractions [4]. This extendsthe traditional factorization of hadronic structure in terms of collinear Parton Distribu-tion Functions (PDFs)[5–8]. When the color-singlet state is within a final state jet notaligned with the beam (for example, observing the transverse momentum spectrum of afinal state hadron within an e + e − -collision using hemispherical jets [9]), we can likewiseextend the notion of collinear Fragmentation Functions (FFs) to include the relative mo-tion of the hadron with respect to all of the other jet constituents. A critical feature ofall of these transverse momentum spectra are their sensitivity to soft processes, where one– 1 –an show that the equations governing the resummation of large logarithmic correctionsassumes a universal form despite the very different scattering processes (Drell-Yan pro-duction, semi-inclusive deep inelastic scattering, and e + e − to hadrons). This resummationof the generalized transverse-momentum dependent parton distribution (TMDPDFs) ortransverse-momentum dependent fragmentation functions (TMDFFs) is often termed thetransverse-momentum dependent evolution (TMD-evolution) [10]. The goal is to predicthow the spectrum changes with the energy scale of the underlying hard process that cre-ates the massive or energetic color-singlet state. Within perturbation theory, the equationsgoverning this evolution has reached a very precise determination [11–13], allowing one toconfidently investigate the non-perturbative dynamics of the recoiling radiation.However, though formally the TMD-evolution of a TMDFF function is identical tothat of a TMDPDF, at a hadron-hadron collider, such fragmentation processes are oftenstudied within a jet . Since jets are not an intrinsic object to QCD, but rather a pattern ofradiation that is most likely to occur, one must both theoretically and experimentally use aprecise, though ultimately arbitrary, jet definition. While all reasonable jet definitions canbe shown to group together the same energetic radiation into the jet region, one necessarilyselects for different configurations of soft radiation that will be included in the jet, andthis can spoil the equivalence of the TMD-evolution between final state (TMD-FF’s) andinitial state processes (TMD-PDF’s). Generically, one worries about soft correlations thatspan the whole event, entangling the pattern of soft radiation within the jet to either theunderlying event with multiple parton interactions within the colliding hadrons [30–35],or non-global color correlations arising from out-of-jet radiation radiating back into themeasured jet, all of which are color connected back to the hard process [36, 37]. Indeed,both effects could potentially spoil the factorization predictions for TMD-spectra foundin [27, 29]. Thus naively, one suspects that only the TMD-evolution of fragmentationprocesses within hemisphere jets at an e + e − machine could be tied to the TMD-evolutionof the Z-boson spectrum.Developments in jet substructure have shown that the modified mass drop taggingalgorithm (mMDT) or soft-drop grooming procedure robustly removes contamination fromboth underlying event and non-global color-correlations, see Refs. [39–41], and have beenapplied to study a wide variety of QCD phenomenology within jets [42–50]. Exploit-ing this fact, we will give a concrete proposal as to how one can observe the universalTMD-evolution within these groomed jets, where we specify that we study the transversemomentum spectrum of a hadron within the jet with respect to the total momentum ofthe groomed jet, that is, all particles that pass the mMDT or soft-drop procedure.The outline of the paper is as follows: we briefly review the mMDT/soft-drop proce-dure, then we follow the factorization arguments of Ref. [46, 47] and give the factorizationtheorem for the groomed TMD-spectrum of a fragmented hadron, as well as related jetshape observables. We present the structure of the anomalous dimensions for the variousobjects found in the factorization theorem, working in the framework derived in Ref. [3], See Refs. [14–29] for recent work on fragmentation processes both generating and within jets. See Ref. [38] for a comprehensive review. – 2 –hich governs the TMD-evolution. We then show how the normalized and groomed TMD-spectrum gives direct access to the rapidity renormalization group/Collins-Soper evolution,as a function of the grooming parameter z cut . The modified mass-drop procedure [40, 41] or its generalization known as soft-drop [39]removes contaminating soft radiation from the jet by constructing an angular ordered treeof the jet through the Cambridge/Aachen (C/A) clustering algorithm [52–56], and removingthe branches at the widest angles which fail an energy requirement. As soon as a branch isfound that passes, this branch is declared the groomed jet, and all constituents of the branchare the groomed constituents. What is remarkable about the procedure, is that it gives ajet with essentially zero angular area, since at large angles, all collinear energetic radiationis to be found at the center of the jet, and no cone is actually imposed to enclose thiscore. One simply finds the branch whose daughters are sufficiently energetic. Formally thedaughters could have any opening angle, though their most likely configuration is collinear.The strict definition of the algorithm is as follows. Given a ungroomed jet, first webuild the clustering history: we start with a list of particles in the jet. At each stage wemerge the two particles within the list that are closest in angle . This gives a pseudo-particle, and we remove the two daughters from the current list of particles, replacingthem with the merged pseudo-particle. This is repeated until all particles are merged intoa single parent. Then we open the tree back up. At each stage of the declustering, we havetwo branches available, label them i and j . We require:min { E i , E j } E i + E j > z cut , (2.1)where z cut is the modified mass drop parameter, and E i is the energy of the branch i . Ifthe two branches fail this requirement, the softer branch is removed from the jet, and wedecluster the harder branch, once again testing Eq. (2.1) within the hard branch. Thepruning continues until we have a branch that when declustered passes the condition (2.1).All particles contained within this branch, whose daughters are sufficiently energetic, con-stitute the groomed jet. Intuitively we have identified the first genuine collinear splitting.For a hadron-hadron collision, one uses the transverse momentum with respect to thebeam for the condition of Eq. (2.1):min { p T i , p
T j } p T i + p T j > z cut . (2.2)We formally adopt the power counting z cut (cid:28)
1, though typically one chooses z cut ∼ .
1. See [44] for a study on the magnitude of the power corrections with respect to z cut for For a discussion of the connection between the rapidity renormalization group and the Collins-Soperequation, see [51] This merging is usually taken to be summing the momenta of the particles, though one could usewinner-take-all schemes [57–59]. – 3 –et mass distributions.
The overall impact of the mMDT grooming is that we force ourselves into a regime that isdominated by purely collinear physics. Thus the properties of the jet can be consideredin isolation from the rest of the event in which the jet occurs, and depend only upon theflavor of the initiating parton, a fact that typically is only true for the energy spectrumof hadrons or jets. This is to say, that while the jet is of course color-connected to therest of the event, the color charge and flavor of the initiating hard parton dominate thespectrum of groomed observable. In terms of factorization, the rest of the event appearsas a single wilson pointed in the anti-collinear direction of the jet. Genuine soft colorcorrelations from multiple jets at wide angles are a power suppressed contribution to thegroomed spectrum, due to the grooming procedure putting one in a collinear factorizationregime, see Ref. [47]. This is exactly analogous to the fragmentation spectrum at moderateenergy fractions of the fragmented hadron, which is also set by the color charge and flavorof the parton initiating the fragmentation (as encoded by the fragmentation function), andthe complicated multi-jet soft correlations are power-suppressed. This allows us to writethe following factorization for a mMDT groomed jet: dσd (cid:126)p J d M (cid:16) z cut , R, C (cid:17) = F g (cid:16) Q, R, z cut , (cid:126)p J , C (cid:17) J g (cid:16) M , z cut , R, E J (cid:17) + (cid:88) q F q (cid:16) Q, R, z cut , (cid:126)p J , C (cid:17) J q (cid:16) M , z cut , R, E J (cid:17) + ... . (3.1)The functions F g and F q represent the gluon and quark fractions of the scattering process initiatingthe jet to be studied. These fractions are dependent upon the center of mass energy, Q , ofthe collision, the jet momentum and energy, (cid:126)p J , and E J , the jet radius R , and the groomingparameter, z cut . The parameter C represents any other cuts or constraints one makes on thescattering process outside the groomed jet. The underlying hard scattering process can be eitheran exclusive or inclusive jet cross-section, with various and complicated additional vetoes or observeddecay channels imposed or not.The parameter(s) M represents on the other hand all of the substructure measurements to beperformed upon the groomed jet. Since this is the interesting part of the cross section, henceforth,we will omit writing the differential d (cid:126)p J explicitly. The jet functions J q and J g will be givenan operator definition below, and at this stage in the factorization may contain large logarithms,which would be resummed using an additional factorization within the jet function itself. For ourpurposes we wish to understand the factorization and resummation structure of spectrum of hadronproduction within the groomed jet. Specifically, we will consider the energy fraction spectrum, z h ,of the hadron and its transverse momentum with respect to the direction of the total momentumof groomed jet . Formally, we mean the jet function appearing in Eq. (3.1). The groomed spectra are set by the jet functions which by collinear factorization depend only the colorcharge of the initiating parton, however, the number of initiating quarks and gluons are still sensitive to thesoft correlations in the scattering process, conditioned on the cuts defining the jet. These flavor fractionstherefore cannot in general be considered in isolation of the rest of the event. In hadron-hadron collisions, this is equivalent to the rapidity, azimuth, and transverse momentum withrespect to the beam of a centrally located jet. This is a distinct observable from the transverse momentum of the hadron with respect to a soft- – 4 – et us assume that the parton initiating the measured groomed jet is a quark (the argumentthat follows will apply equally well for any other parton flavor). We measure the transverse mo-mentum of an identified hadron ( p h ⊥ ) inside the jet carrying a fraction z h of the ungroomed jetenergy. We can then write down the jet function in the factorization of Eq. (3.1): G q/h (cid:16) z h , (cid:126)k ⊥ , z cut , R, E J (cid:17) = z h (cid:88) X ∈ Jet( R ) N c δ (2 E J − p − X − p − h )tr (cid:20) / ¯ n (cid:104) | δ (2) ( (cid:126)k ⊥ + (cid:126) P SD ⊥ ) χ n (0) | Xh (cid:105)(cid:104) Xh | ¯ χ n (0) | (cid:105) (cid:21) . (3.2)where (cid:126)k ⊥ is the jet transverse momentum with respect to the direction of motion of the identifiedhadron, h . Although eventually we are interested in the hadron’s transverse momentum with respectto the jet axis, (cid:126)p h ⊥ , we choose to work with (cid:126)k ⊥ since this significantly simplifies our analysis. In thisapproach one needs to project the jet traverse momentum on the hadronic axis without need to worryabout the recoiling hadrons from the fragmentation process. In the collinear limit ( | (cid:126)p h ⊥ | /E J (cid:28) (cid:126)k ⊥ and (cid:126)p h ⊥ using a simple geometric argument. Using | (cid:126)p h | (cid:39) z h E J we then have (cid:126)k ⊥ = (cid:126)p h ⊥ /z h (see Fgure 1). Here X contains all the particles in the jet of radius R (assuming anappropriate jet definition, denoted Jet( R ), like the anti- k t algorithm[61]). The components of (cid:126)k ⊥ are set by the label momentum operator (cid:126) P SD ⊥ which projects the traverse momentum of the subsetof those particles which pass the mMDT/soft-drop grooming procedure of the state | X, h (cid:105) . Thesum over X includes integrals over the phase space of X . Note that the transverse momentum isdefined as the recoil against all the particles which pass the grooming requirement. This implicitlydefines the groomed jet axis to be the axis such that the sum of all transverse momenta with respectto that axis is zero. ~p J ~p h ~p hT ~p JT ~p h ? ~k ? groomed-jet axishadron axis beam axis Figure 1 . The geometric configuration of the jet and hadron axis relative to beam. Here the jetaxis is defined as the momentum of all the particles clustered by the jet algorithm. The vectors (cid:126)k ⊥ and (cid:126)p h ⊥ are two dimensional arrays with components as measured from the corresponding axis.This jet function contains large logarithmic corrections associated with the scales | (cid:126)p h ⊥ | , and insensitive axis like the winner-take-all axis [60]. Both observables enjoy a form of collinear factorizationand hence universality, though the spectra and resummation structure are distinct. The energy difference between the total energy of the groomed jet constituents and the energy of theungrooomed jet is a collinear unsafe observable [48], however, the additional constraint of the measuredtransverse momentum of the hadron provides a physical collinear cutoff. – 5 – J when we are in the hierarchy: | (cid:126)p h ⊥ | (cid:28) E J . (3.3)Moreover, we must assume that the fragmented hadron is within the set of particles which pass thegrooming, which further requires: λ = (cid:16) | (cid:126)p h ⊥ | E J (cid:17) (cid:28) z cut , R ∼ O (1) . (3.4)We introduce light-cone jet direction n = (1 , ˆ n J ), where ˆ n J is the direction of the total momentumof the particles which pass mMDT/soft-drop. We also have ¯ n = (1 , − ˆ n J ), which is the conjugatedirection, and finally, the transverse momentum plane to these directions. Any momentum p canbe decomposed in terms of these directions as p = (¯ n · p, n · p, p ⊥ ) (3.5)The region of phase space which contributes to our measurement of p h ⊥ , naturally gives two distinctpower counting regions contributing to the observable,collinear : p c ∼ E J (1 , λ , λ )collinear-soft : p cs ∼ E J z cut (cid:16) , λ z , λz cut (cid:17) (3.6)Within this region of phase-space, by following the logic the collinear/collinear-soft factorizationarguments of so-called SCET + [62–64], we may further factorize the jet function in Eq.(3.2) as(leaving the flavor generic): G i/h (cid:16) z h , (cid:126)k ⊥ , z cut , R, E J (cid:17) = z h (cid:90) d (cid:126)k c ⊥ (cid:90) d (cid:126)k s ⊥ δ (cid:16) (cid:126)k ⊥ + (cid:126)k c ⊥ + (cid:126)k s ⊥ (cid:17) ˜ D ⊥ i/h (cid:16) z h , E J , (cid:126)k c ⊥ (cid:17) × S ⊥ i (cid:16) (cid:126)k s ⊥ , z cut (cid:17) + O (cid:16) z cut R (cid:17) (3.7)The operator definitions of these functions are: D ⊥ q/h ( z h , (cid:126)k c ⊥ , E J ) = (cid:88) X z h N c δ (2 E J − p − Xh )tr (cid:104) / ¯ n (cid:104) | δ (2) ( (cid:126)k c ⊥ − (cid:126) P ⊥ ) χ n (0) | Xh (cid:105)(cid:104) Xh | ¯ χ n (0) | (cid:105) (cid:105) (cid:126)p h ⊥ =0 , (3.8) D ⊥ g/h ( z h , (cid:126)k c ⊥ , E J ) = (cid:88) X z h N c − δ (2 E J − p − Xh )tr (cid:104) (cid:104) | δ (2) ( (cid:126)k c ⊥ − (cid:126) P ⊥ ) B µn ⊥ (0) | Xh (cid:105)(cid:104) Xh | B n ⊥ µ (0) | (cid:105) (cid:105) (cid:126)p h ⊥ =0 , (3.9) S ⊥ i ( (cid:126)k s ⊥ , E J , z cut ) = 1 N i tr (cid:104) (cid:104) | T { S in S i ¯ n } (0) δ (2) (cid:16) (cid:126)k s ⊥ − (cid:126) P SD ⊥ (cid:17) ¯ T { S in S i ¯ n } (0) | (cid:105) (cid:105) . (3.10)where the subscript (cid:126)p h ⊥ = 0 indicates that the calculation of these functions is to be done in theframe in which (cid:126)p h ⊥ = 0. Now in the collinear functions, the sum over states is unrestricted by themMDT/soft-drop grooming or jet definition, but still excludes the observed hadron. All particleswithin the collinear function are automatically guaranteed to be within the groomed jet. Thecollinear-soft function S ⊥ i , however, contains particles which may or may not pass the groomingprocedure. The operator (cid:126) P SD gives the total momentum of all particles which are included in the – 6 – roomed jet, so that for a state | X (cid:105) : (cid:126) P SD | X (cid:105) = (cid:88) i ∈ X SD (cid:126)p i | X (cid:105) (3.11)Finally, we use the soft Wilson-line definition: S ik ( x ) = P exp (cid:16) ig (cid:90) ∞ ds k · A a ( x + sk ) T ai (cid:17) . (3.12)Here i denotes the representation of the Wilson-line. The definition of all gauge invariant collinearoperators may be found in Ref. [35]. The substructure parameters M that we wish to measurein this particular case are the transverse momentum of the identified hadron, (cid:126)p h ⊥ , and its energyfraction, z h , with respect to the ungroomed jet energy, E J . So that d M = d (cid:126)p h ⊥ dz h . The p ⊥ Figure 2 . The geometric configuration involved in matching the TMDFF to the standard FF. θ h isthe angle that both the hadron (blue line) with momentum fraction z h of the jet and its initiatingparton (red dotted line) make to the jet axis, and is set by perturbative splittings up to powercorrections. The initiating parton has momentum fraction x of the jet, so that the fragmentedhadron has momentum fraction z h x of the initiating parton.dependent collinear function (TMDFJF), D i/h , also implicitly depends on the hadronization scaleΛ QCD . For perturbative values of p ⊥ , we can separate out the long distance non-perturbativephysics at Λ QCD by matching the TMDFJF onto an ordinary fragmentation function. We illustratethe geometry of this matching in Figure 2: D ⊥ i/h (cid:16) z h , (cid:126)k c ⊥ , E J (cid:17) = (cid:90) z h dxx J ⊥ ij (cid:16) x, (cid:126)k c ⊥ , E J (cid:17) D j/h (cid:16) z h x (cid:17) . (3.13)The operator definition of the fragmentation function is : D q/i (cid:16) z h , E q (cid:17) = z i (cid:88) X N c δ (2 E q − p − X − p − i )tr (cid:20) / ¯ n (cid:104) | χ n (0) | Xi (cid:105)(cid:104) Xi | ¯ χ n (0) | (cid:105) (cid:21) (cid:126)p i ⊥ =0 Keeping in mind that the light-like direction n in the fragmentation function is not the same as thedirection n in the TMDFF, as illustrated in Figure 2. – 7 – ith z i = E i /E q is the energy fraction of the final state i with respect to the fragmenting quark. D g/i (cid:16) z i , E g (cid:17) = z i (cid:88) X N c − δ (2 E g − p − X − p − i )tr [ (cid:104) | B µn ⊥ (0) | Xi (cid:105)(cid:104) Xi | B n ⊥ µ (0) | (cid:105) ] (cid:126)p i ⊥ =0 . (3.14)The final form of the factorized cross- section can now be written as: z h dσdz h d p h ⊥ = (cid:88) i = g,q F i (cid:16) Q, R, z cut , (cid:126)p J , C, µ (cid:17) (cid:90) d (cid:126) ˜ k c ⊥ (cid:90) d (cid:126)k s ⊥ δ (cid:18) (cid:126)p h ⊥ z h + (cid:126) ˜ k c ⊥ + (cid:126)k s ⊥ (cid:19) × S ⊥ i (cid:16) (cid:126)k s ⊥ , E J , z cut (cid:17) (cid:90) z h dxx J ⊥ ij (cid:16) x, (cid:126)k c ⊥ , E J (cid:17) h.f D j/h (cid:16) z h x (cid:17) + ... . (3.15)We can also arrive at this result by starting off with a more conventional definition of the jet functionas described in Appendix B. Given that the modes in Eq. (3.5) of the factorization have the same invariant mass, the renormal-ization is to be performed within the context of SCET II . The bare functions enjoy both ultra-violetand rapidity divergences that must be renormalized, along the lines of Ref. [3] (see also [1] and [65]).Taking the Fourier transform of all functions, thus going to the so-called b -space distributions, wewrite: D ⊥ i/h (cid:16) z h ,(cid:126)b ⊥ , E J (cid:17) = Z ci (cid:16) E J ν , µ(cid:126)b ⊥ , α s ( µ ) (cid:17) D ⊥ i/h (cid:16) z h , E J ν , µ(cid:126)b ⊥ , α s ( µ ) (cid:17) , (4.1) S ⊥ i (cid:16) (cid:126)b ⊥ , E J , z cut (cid:17) = Z si (cid:16) z cut E J ν , µ(cid:126)b ⊥ , α s ( µ ) (cid:17) S ⊥ i (cid:16) z cut E J ν , µ(cid:126)b ⊥ , α s ( µ ) (cid:17) . (4.2)Likewise, the quark and gluon fractions in Eq. (3.1) are renormalized: F i (cid:16) Q, R, z cut , (cid:126)p J , C (cid:17) = Z Fi (cid:16) z cut E J µ , α s ( µ ) (cid:17) F i (cid:16) Q, R, z cut E J µ , (cid:126)p J , C, α s ( µ ) (cid:17) . (4.3)The full factorization structure of these fractions is in general complicated and unknown (poten-tially suffering from both factorization violating contributions and non-global logarithms), but theydo not essentially effect the predicted shape of the distribution for a quark or a gluon. As thephysical cross-section itself is renormalization group invariant, we have the general constraint onthe renormalization factors: Z ci (cid:16) E J ν , µ(cid:126)b ⊥ , α s ( µ ) (cid:17) Z si (cid:16) z cut E J ν , µ(cid:126)b ⊥ , α s ( µ ) (cid:17) = (cid:16) Z Fi (cid:16) z cut E J µ , α s ( µ ) (cid:17)(cid:17) − , (4.4)where Z Fi is the renormalization factor for the flavor fraction F i . Thus we may write: z h dσdz h d p h ⊥ = (cid:88) i = g,q F i (cid:16) Q, R, z cut , (cid:126)p J , C, µ (cid:17) (cid:90) dbbJ ( b(cid:126)p h ⊥ /z h ) D ⊥ i/h (cid:16) z h , E J ν , µ(cid:126)b ⊥ , α s ( µ ) (cid:17) × S ⊥ i (cid:16) z cut E J ν , µ(cid:126)b ⊥ , α s ( µ ) (cid:17) + ... , (4.5) – 8 – here b = | (cid:126)b ⊥ | and J is the zeroth order Bessel function of the first kind. In general these functionsobey a set of renormalization group (RG) equations of the form: µ ddµ G ( µ, ν ) = γ µ G ( µ, ν ) , (4.6) ν ddν G ( µ, ν ) = γ ν G ( µ, ν ) , (4.7)where G can be either S ⊥ i or D ⊥ i/h . The first is the standard RG equation for the ultraviolet( µ ) anomalous dimension, whereas the second yields the rapidity ( ν ) anomalous dimension. Onlythe functions describing the fragmentation process have rapidity renormalization group running,whereas the parton fractions do not. The renormalized functions possess µ anomalous dimensionsof the following form: γ Fµ,i = − Γ i cusp [ α s ( µ )] ln z + γ Fi [ α s ( µ )] , (4.8) γ D µ,i = Γ i cusp [ α s ( µ )] ln ν (2 E J ) + γ D i [ α s ( µ )] , (4.9) γ Sµ,i = − Γ i cusp [ α s ( µ )] ln ν z (2 E J ) + γ Si [ α s ( µ )] . (4.10)We can also write down the all orders form of the ν anomalous dimensions. The hard factors F i are independent of ν . The ν anomalous dimension for the collinear function ( D ⊥ i/h ) is equal inmagnitude but opposite in sign to that of the collinear soft function ( S ⊥ i ) and obeys the followingconsistency condition: dd ln µ γ Sν,i = dd ln ν γ Sµ,i . (4.11)This allows us to write down γ Sν,i ( µ ) = − (cid:90) µ /b d ln µ (cid:48) Γ i cusp [ α s ( µ (cid:48) )] + γ r (1 /b ) . (4.12)where b is a boundary condition in b space which is usually set to b = be γ E / To resum Large logarithms in p h ⊥ /E J , we first run the collinear soft function in ν from its naturalscale ν s ∼ E J z cut to the scale ν c ∼ E J . Then we run both the collinear and collinear soft functionin µ from an appropriately chosen low scale µ L to the high scale µ H ∼ E J . Ulsing Eq.(4.12) theevolution in ν then yields, S ⊥ i ( µ, ν = 2 E J ) = S ⊥ i ( µ, ν = 2 E J z cut )Exp (cid:104) ln( z cut ) (cid:16) (cid:90) µ /b d ln µ (cid:48) Γ i cusp [ α s ( µ (cid:48) )] + γ r (1 /b ) (cid:17)(cid:105) , (4.13) – 9 – ext we want to evolve D ⊥ i/h and S ⊥ i in µ from the scale µ L to µ H . The µ anomalous dimensionsfor these functions to all orders are, γ D µ,i = +2Γ i cusp [ α s ( µ )] ln (cid:16) ν E J (cid:17) + γ D i [ α s ( µ )] ,γ Sµ,i = − i cusp [ α s ( µ )] ln (cid:16) ν E J z cut (cid:17) + γ Si [ α s ( µ )] . (4.14)The combined µ anomalous dimension for these function is given as γ D µ,i + γ Sµ,i = 2Γ i cusp [ α s ( µ )] ln (cid:16) z cut (cid:17) + γ D i [ α s ( µ )] + γ Si [ α s ( µ )]= 2Γ i cusp [ α s ( µ )] ln (cid:16) z cut (cid:17) − γ Fi [ α s ( µ )] (4.15)The evolution kernel is given as D ⊥ i/h ( µ H , ν = 2 E J ) S ⊥ i ( µ H , ν = 2 E J ) = U i ( µ L , µ H ) × (cid:104) D ⊥ i/h ( µ L , ν = 2 E J ) S ⊥ i ( µ L , ν = 2 E J z cut ) (cid:105) , (4.16)where the resummation exponent U i is: U i ( µ L , µ H ) ≡ Exp (cid:104) − (cid:90) µ H µ L d ln µγ Fi [ α s ( µ )]+2 ln( z cut ) (cid:16) (cid:90) µ H /b d ln µ Γ i cusp [ α s ( µ )]+ γ r (1 /b ) (cid:17)(cid:105) . (4.17)The point to be noted is that the term multiplying ln z cut in the exponent is the all orders rapidityanomalous dimension. The resummation exponent we have to all orders is U i = Exp (cid:104) − (cid:90) µ H µ L d ln µ (cid:16) γ Fi [ α s ( µ )] (cid:17) + 2 ln( z cut ) (cid:90) µ H /b d ln µ (cid:16) Γ i cusp [ α s ( µ )] + γ r (1 /b ) (cid:17)(cid:105) = Exp (cid:104) (cid:90) µ H µ L d ln µ (cid:16) − γ Fi [ α s ( µ )] + 2 ln( z cut )(Γ i cusp [ α s ( µ )] + γ r (1 /b )) (cid:17)(cid:105) × Exp (cid:104) z cut ) (cid:16) (cid:90) µ L /b d ln µ Γ i cusp [ α s ( µ )] + γ r (1 /b ) (cid:17)(cid:105) . (4.18)At NLL, we have, γ Si [ α s ( µ )] = 0 , γ Fi [ α s ( µ )] = − α s ( µ ) C i π ¯ γ i . (4.19)with ¯ γ i given in Eq.(A.4). The cusp anomalous dimension can be written to all orders in perturba-tion theory as follows, Γ i cusp = C i (cid:88) n =1 (cid:16) α s π (cid:17) n Γ n . (4.20) – 10 – elevant for the NLL result are Γ and Γ given by,Γ = 4 , Γ = 4 C A (cid:18) − π (cid:19) − T R n f . (4.21)In order to proceed we do one more approximation. Assuming that ln( µ L b ) is small, we cando an expansion in the second exponent in this log keeping only the leading order term at NLLand ignoring non-perturbative contributions to the anomalous dimension. Then at NLL we are leftwith: U i ( µ L , µ H ) = Exp (cid:104) (cid:90) µ H µ L d ln µ (cid:16) z cut )Γ i cusp ( α s ( µ )) − γ Fi [ α s ( µ )] (cid:17)(cid:105) Exp (cid:104) Γ i cusp [ α s ( µ L )] ln( z ) ln( µ L b ) (cid:105) = e K i ( µ L ,µ H ) ( µ L b ) ω iJ ( µ L ,z ) , (4.22)with ω iJ = ln( z )Γ i cusp [ α s ( µ L )]. The function K i is evaluated including the running of α s to twoloops and is defined as K i ( µ , µ ) = − C i ln( z ) Γ β (cid:110) ln r + α s ( µ )4 π (cid:18) Γ Γ − β β (cid:19) (cid:111) − γ β ln r , (4.23)where we have defined r = α s ( µ ) / α s ( µ ) and γ D i = α s ( µ ) / (4 π ) γ . We can now go to momentumspace by doing the inverse Fourier transform: (cid:90) d be − i (cid:126)q T /z h · b = 2 π (cid:90) bdbJ ( bq T /z h ) , (4.24)˜ U ( q T ) = − πe K i ( µ L ,µ H ) z h ω iJ q T (cid:18) z h µ L e γ E q T (cid:19) ω iJ Γ (cid:104) ω iJ (cid:105) Γ (cid:104) − ω iJ (cid:105) . (4.25)In the perturbative regime, we set the scale µ L = q T / ( z h e γ E ) and µ H ∼ E J so that we are leftwith: ˜ U i ( q T ) = − πe K i ( µ L ,E J ) z h ω iJ q T Γ (cid:104) ω iJ (cid:105) Γ (cid:104) − ω iJ (cid:105) . (4.26)There are no fixed order terms to be included at NLL. The cross section now looks like: z h dσdz h d p h ⊥ = (cid:88) i = g,q F i (cid:16) Q, R, z cut , (cid:126)p J , C, µ (cid:17) ˜ U i ( p h ⊥ ) D i/h ( z h , µ L ) . (4.27) In this section we discuss how non-perturbative contributions can be incorporated in our formalismin Fourier space. We illustrate that for k T (cid:38) – 11 – omentum space as was done in the previous section at NLL accuracy. The cross section looks like z h dσdz h d p h ⊥ = (cid:88) i = g,q F i (cid:16) Q, R, z cut , (cid:126)p J , C, µ (cid:17) (cid:90) d (cid:126)k c ⊥ (cid:90) d (cid:126)k s ⊥ δ (cid:18) (cid:126)p h ⊥ z h + (cid:126)k c ⊥ + (cid:126)k s ⊥ (cid:19) × S ⊥ i (cid:16) (cid:126)k s ⊥ , z cut (cid:17) D ⊥ i/h (cid:16) z h , (cid:126)k c ⊥ (cid:17) + ... . (5.1)For convenience of notation we use (cid:126)k ⊥ = (cid:126)p h ⊥ /z h so that the cross section becomes dσdz h d (cid:126)k ⊥ = (cid:88) i = g,q F i (cid:16) Q, R, z cut , (cid:126)p J , C, µ (cid:17) G i/h ( z h , (cid:126)k ⊥ , E J , z cut ; µ ) , (5.2)where: G i/h ( z h , (cid:126)k ⊥ , E J , z cut ; µ L ) = (cid:90) d (cid:126)k c ⊥ (cid:90) d (cid:126)k s ⊥ δ (cid:16) (cid:126)k ⊥ + (cid:126)k c ⊥ + (cid:126)k s ⊥ (cid:17) S ⊥ i (cid:16) (cid:126)k s ⊥ , z cut (cid:17) D ⊥ i/h (cid:16) z h , (cid:126)k c ⊥ (cid:17) . (5.3)Taking the Fourier transform of G ( z h , (cid:126)k ⊥ , E J , z cut ) with respect to (cid:126)k ⊥ we get, G i/h ( z h , b, E J , z cut ; µ L ) ≡ (cid:90) d(cid:126)k ⊥ (2 π ) e − i(cid:126)k ⊥ · (cid:126)b ⊥ G i/h ( z h , (cid:126)k ⊥ , E J , z cut ; µ L )= D ⊥ i/h ( z h , b, E J ; µ L , ν = 2 E J ) S ⊥ i ( b, E J , z cut ; µ L , ν = 2 E J ) , (5.4)where µ L is taken to be a perturbative scale and the soft function S ⊥ i is evolved in rapidity spacefrom 2 E J z cut to 2 E J . This evolution is described as before using the RRG anomalous dimension, γ Sν,i ( b, µ ), S ⊥ i ( b, E J , z cut ; µ L , ν = 2 E J ) = S ⊥ i ( b, E J , z cut ; µ L , ν = 2 E J z cut )Exp (cid:16) − γ Sν,i ( b, µ L ) ln z cut (cid:17) , (5.5)where γ Sν,i ( b, µ ) ≡ (cid:90) d (cid:126)p ⊥ (2 π ) Exp (cid:104) i(cid:126)p ⊥ · (cid:126)b ⊥ (cid:105) γ Sν,i ( (cid:126)p ⊥ , µ ) , (5.6)The all orders expression for γ Sν,i is (Eq. 4.12) γ Sν,i ( µ ) = − (cid:90) µ /b d ln µ (cid:48) Γ i cusp [ α s ( µ (cid:48) )] + γ r (1 /b ) . (5.7)The choice of b that minimizes all logarithms is b = be γ E /
2. However, this choice enters the non-perturbative regime for b (cid:38) Λ QCD . In the large b region we need a model function for the unknownnon-perturbative physics which ultimately needs to be extracted from experiment. To that end, weseparate out the perturbative contribution to the anomalous dimension from the non-perturbativeone by defining a new b dependent scale µ = µ b ≡ γ E ) b ∗ , (5.8) We have essentially recombined the factorized functions to reform, up to power corrections, the initialjet function in Eq. (3.2). In order to not introduce more functions, we keep the same symbol, simplydropping functional dependencies that are power suppressed. – 12 – here b ∗ = b/ (cid:112) b/b max ) , and b max is chosen such that µ b is a perturbative scale for all valuesof b . At low values of b, b ∗ is just b while at large values this approaches the fixed scale b max .The replacement b → b ∗ is compensated with a non-perturbative model function, g K ( b ), to bedetermined from experimental data. That is, γ Sν,i ( µ ) = − (cid:90) µµ b d ln µ (cid:48) Γ i cusp [ α s ( µ (cid:48) )] + γ if ( µ ) − g K ( b ; b max ) . (5.9)where γ if ( µ ) is the perturbative non-cusp rapidity anomalous dimension of the collinear-soft func-tion. This term only starts at two loops and hence we will set it to 0 for our analysis at NLLaccuracy. What we have done is to put in all the non-perturbative parts of the anomalous di-mension into the function g K . Notice that this function depends on the precise choice of b max which decides the boundary between perturbative and non-perturbative physics. Also, in order toreproduce the perturbative result for small b , we need to impose g K ( b →
0) = 0.Following Eq. 4.16, we can write the full resummed result as G i/h ( z h , b, E J , z cut ; µ ) = G FO i/h ( z h , b, E J ; µ L )Exp (cid:104) γ Sν,i ( b, µ ) ln( z cut ) − (cid:90) µµ L d ln( µ (cid:48) ) γ Fi [ α s ( µ (cid:48) )] (cid:105) , (5.10)To proceed, we need to make a choice for the scale µ L . Naturally we would like to make thecanonical choice which minimizes the large logarithms in the perturbative expansion of D ⊥ i/h and S i , but unfortunately in Fourier space this scale is 2 exp( − γ E ) /b and the perturbative expansionfails for large b . For this reason we choose µ L = µ b and compensate (as we did for γ S i ν ) with anon-perturbative input function g i/h ( z h , b ) . G i/h ( z h , b, E J , z cut ; µ ) = G FO i/h ( z h , b, E J ; µ L = µ b )Exp (cid:104) γ Sν,i ( b, µ ) ln( z cut ) − g i/h ( z h , b ) − (cid:90) µµ b dµ (cid:48) µ (cid:48) γ Fi [ α s ( µ (cid:48) )] (cid:105) , (5.11)This is our final result for the maximum perturbative input. In contrast with the function g i/h ,which is specific to the particular fragmentation process, g K is universal for all TMD distributionsand controls the non-perturbative evolution in rapidity. In this work we are primarily interestedin g K and for what follows we take g i/h = 0. As an explicit example, we consider the process e + + e − → dijets. We groom one of the jets and identify a pion in that jet. In this case then,we only have quark initiated jets and henceforth we will assume the parton initiating the jet tobe a quark. The normalization factor F q ( Q, R, z cut ) in this case can be factorized for exclusivehemisphere jets, and has been evaluated to two-loop accuracy in Ref. [46]. Since gluon jets will notappear at leading power, the particular value of the normalization will not matter for the shape ofthe TMD-distribution. Then we can make predictions in the low p h ⊥ regime for extracting out thenon-perturbative physics.In Figure 3 we compare the resummed distribution evaluated in momentum space directlyagainst our final result in Eq.(5.11). The inverse Fourier transform was performed numericallyafter integrating analytically over the azimuthal angle. For the non-perturbative model function,we use the CSS model, g K ( b ; b max ) = 12 g ( b max ) b , (5.12)where g ( b max ) is a free parameter to be determined by fitting to data. The values for theparameters we use are extracted from experimental data in Ref. [66] and are as follows: g = 0 . Note that the implicit dependence of g K on the parameter b max is absorbed into the value of g , that isto say the choice of b max influences the fit of g . An alternative approach is to fit g and b max simultaneously.This approach was implemented by the Pavia 2016 fits. – 13 – nd b max = 1 .
123 GeV. We find that for k ⊥ (cid:38) Figure 3 . The up-quark to charged pions TMDFJF at NLL for z h = 0 . z h = 0 . D u/π + ( x ; µ ) are taken from Ref. [67]We note that the only explicit dependence on z cut is in the exponent through a logarithm witha coefficient which is the rapidity anomalous dimension γ S i ν ( b, µ ). This suggests that the TMDFJF,and thus the corresponding cross section, is sensitive to the rapidity anomalous dimension throughvariations of z cut . . We exploit this property through the normalized logarithmic derivative, d/d ln z cut , to discriminate between various non-perturbative models suggested previously in theliterature. We evaluate the logarithmic derivative of the cross section as a function of the transverseenergy for a fixed value of the energy fraction, z h , for four different parametrizations of the function g K ( b ; b max ). We consider three fits of the CSS model (see Eq.(5.12)): 1) BNLY from Ref. [72], 2)KN from Ref. [73], and 3) Pavia from Ref. [66]. The values for the parameters of these fits are givenin Table 5. We also consider the following functional form, g K ( b ; b max ) = g ( b max ) b (cid:16) b b (cid:17) , (5.13)which was suggested in Ref. [74] and we refer to as the AFGR model. For the latter, there are nofits to the free parameters ( g and b NP ) from data. Hence, we use the approximate values suggestedby the authors in the corresponding publication. Our results are illustrated in Figure 4. We notethat for large values of the transverse momentum, the four models merge to the same distributionas expected since in that regime the perturbative anomalous dimension dominates the evolutionof the cross section. In contrast, for small values of the transverse momentum the four modelsgive clearly distinguishable distributions. These results suggest that the normalized logarithmicderivative can be used for accurate and precise extraction of the model function g K describing thenon-perturbative part of the rapidity anomalous dimension of TMD observables. For recent work on the theoretical considerations of the non-perturbative corrections to the rapidityresummation exponent, also called the collinear anomaly, see Refs. [68, 69]. One would need to also considerthe effects that grooming would have on the effective rapidity range of the non-perturbative corrections.For the non-perturbative contributions including renormalon effects, see Ref. [70], and other field theoreticconsiderations see Ref. [71]. – 14 –
Figure 4 . The logarithmic derivative of TMDFJF for the three models. All the models agree inthe perturbative regime but show significant differences in the non-perturbative region.
Model:Fits g b max [GeV − ] b NP [GeV − ]CSS:BNLY 2003 [72] 0.68 0.5 n.a.CSS:KN 2006 [73] 0.18 1.5 n.a.CSS:Pavia 2016 [66] 0.12 1.123 n.a.AFGR: n.a. [74] 0.10 0.5 2.0 Table 1 . Parameters for models of the non-perturbative part of the rapidity anomalous dimension.
In this paper, we propose an observable which measures the transverse momentum of an identifiedhadron inside a groomed jet. We use the modified mass-drop/soft-drop grooming algorithm withan energy cut-off parameter z cut . The radiation that recoils against the hadron is factorized intocollinear and collinear- soft modes in the framework of SCET + . The transverse momentum ofthe hadron is measured with respect to the groomed jet axis which is determined by the totalmomentum of the radiation in the jet that passes the soft-drop condition. For perturbative valuesof p h ⊥ , we separate out the non-perturbative physics by matching onto a fragmentation function.Apart from dimensional regularization, we also need to introduce a rapidity regulator to handledivergences that arise from separating two modes( the collinear and collinear-soft) that have thesame invariant mass. Consequently we have RG equation in two scales µ and ν . We use RGevolution in these scales to resum large logarithms in p h ⊥ /E J . The all orders expression for theresummed result reveals that the co-efficient of ln z cut in the exponent is the all orders rapidityanomalous dimension γ ν . This anomalous dimension is universal, in that the same term appears inthe resummation of TMDPDF’s, and more traditional TMD-fragmentation processes.However, conventional TMD observables (those that do not include grooming) are suppressedby double logarithms in ln( µb ) in the exponent, i.e., the exponent in b space has the LL form Exp [ − ln ( E J b )]. Since b is the conjugate parameter to p h ⊥ , the b space cross-section is sensitive tonon-perturbative physics only at large values of b. However, the presence of the double logarithmsuppresses any non-perturbative effects since the cross-section is vanishing due to perturbative – 15 – ontributions alone.The effect of including grooming is two-fold. First, it makes the shape of the p h ⊥ distributioninsensitive to non-global logarithms. Second, it removes from the resummed exponent, one power ofln( E J b ), replacing it with a − ln( z cut ). For the typical values z cut that are used in experiment, thisis much smaller than ln E J b at large (b ≥ − ) values of b. The physical interpretation of thisreplacement is that the grooming algorithm effectively cuts off the soft radiation at a specific energyfraction of the jet that is much larger than the transverse momentum scales that normally set thesoft scaling. In standard TMD-observables this soft part can have, in principle, an arbitrarily smallenergy fraction, up to the kinematics requiring it to be within the jet and the on-shell conditions.This automatically means that the cross-section is much more sensitive to non-perturbative physicsas compared to the corresponding ungroomed observable, and in particular, this sensitivity can beprobed by comparing the groomed jet with different values of z cut .We take advantage of this sensitivity to test the effect of several different non-perturbativemodels that have been proposed in literature to modify the behavior of γ ν at large b. We then useour formalism to give a prediction for the shape of the p ⊥ distribution of a pion. An analysis of theslope (with respect to ln z cut ) at low values of p ⊥ is a measure of the rapidity anomalous dimension.We compare the results using different non-perturbative models as shown in Fig 4. The significantdifferences between the predictions is indicative of the discriminating power of this observable.An important consideration for hadron-hadron colliders (currently the Large Hadron Collideror the Relativistic Heavy Ion Collider, where there is significant interest in transverse momentumdependent observables, see Refs. [75–77]) is the z cut dependence of the quark and gluon fractions.It may be the precision of determining the quark and gluon fractions for a given process that willlimit the ability to probe the non-perturbative contributions to rapidity evolution. Unlike the e + e − case considered in the previous sections, it cannot be simply normalized away. Finding theTMD-distributions at several z cut is a more or less straightforward re-clustering of the same events.Experimental collaborations at both colliders have measured TMD-observables without grooming,and in particular TMD-fragmentation has been measured at ATLAS (Ref. [78]) down to transversemomenta scales of the order of 1 GeV, or related observables such as the jet-shape [79]. Ouranalysis is easily extended to next-to-next leading logarithm accuracy in the resummation of thegroomed fragmenting jet function, using the results of Ref. [47]. The largest unknown, then, is thequark and gluon fraction functions. They could potentially be extracted directly from experiment ata specific renormalization point using groomed jet mass measurements and theoretical calculationsof Refs. [44, 47, 80], and one may also be able to provide a robust theoretical estimation of thesefractions for jets with a moderate R , using single inclusive jet production results while resummingthe jet radius logarithms and perhaps even the z cut dependence. Acknowledgments
We would like to thank Wouter Waalewijn for many enlightening discussions on fragmentation injets. This work was supported by the U.S. Department of Energy through the Office of Science,Office of Nuclear Physics under Contract DE-AC52-06NA25396 and by an Early Career ResearchAward, through the LANL/LDRD Program, and within the framework of the TMD Topical Col-laboration. Though looking at jets with different underlying hard processes, like pp → Z + j + X versus singleinclusive jet production, and examining different jet p T bins can help give sufficiently diverse set of fractionsfavoring quarks or gluons. For our purposes, this is the appropriate moment with respect to z h of our TMD-fragmentation func-tions. – 16 – One loop results
In this section we gather together one loop results for the matching coefficient J ij of the Collinearfunction D ⊥ i/h on to the fragmentation function, as well as the Collinear- Soft function S ⊥ i . A.1 Matching onto the fragmentation functions
The operator definition of the collinear function D ⊥ i/h is given by Eq.3.8. The k c ⊥ dependentcollinear function D ⊥ i/h is matched onto a fragmentation function D i/h , using the relation (Eq.3.13) D ⊥ i/h (cid:16) z h , (cid:126)k c ⊥ , E J (cid:17) = (cid:90) z h dxx J ⊥ ij (cid:16) x, (cid:126)k c ⊥ , E J (cid:17) D j/h (cid:16) z h x (cid:17) . (A.1)The matching is done at the parton level, i.e., by replacing h by an appropriate final state parton.Moreover we are working in a frame in which the final state parton has zero transverse momentum.The one loop results for J ij , are identical to those obtained in Ref. [24]. J ⊥ i/j ( z, (cid:126)p ⊥ , E J ; µ, ν ) = δ ij δ (1 − z ) δ (2) ( (cid:126)p ⊥ ) + α s T ij π (cid:110)(cid:104) δ ij δ (1 − z )2 ln (cid:16) E J ν (cid:17) + ¯ P ji ( z ) (cid:105) L ( p ⊥ , µ )+ c ij ( z ) δ (2) ( (cid:126)p ⊥ ) (cid:111) , (A.2)with T qq = T qg = C F , T gg = C A , T gq = T F . The anomalous dimensions in momentum space are γ D µ,i = α s C i π (cid:104) (cid:16) ν E J (cid:17) + ¯ γ i (cid:105) ,γ D ν,i = − (8 π ) α s C i L ( p ⊥ , µ ) , (A.3) D ⊥ i/h includes the anomalous dimensions of D j/h and J ij . with¯ γ q = 3 / γ g = β / (2 C A ) , (A.4)In fourier(b) space, γ µ remains unchanged while γ ν becomes γ D ν,i = 2 α s C i π ln (cid:16) µbe γ E (cid:17) . (A.5) A.2 collinear-soft function, S ⊥ i The operator definition of the collinear soft function is given by Eq.3.8 S ⊥ i = δ ( (cid:126)p ⊥ ) + 2 α s π C i ln (cid:16) ν E J z cut (cid:17) L ( p ⊥ , µ ) , (A.6)which leads to the µ and ν anomalous dimensions γ Sµ,i = − α s π C i (cid:16) ν E J z cut (cid:17) ,γ Sν,i = (8 π ) α s C i L ( p ⊥ , µ ) . (A.7)In Fourier space, the ν anomalous dimension changes to γ Sν,i = − α s C i π ln (cid:16) µbe γ E (cid:17) . (A.8) – 17 – e then see immediately that the consistency condition for rapidity RG invariance at one loop issatisfied γ Sν,i + γ D ν,i = 0 . (A.9) B An Alternative Path to Factorization of Standard FF
In this section, we provide an alternative way by which we can come to the final form of thefactorized cross-section given in Eq. 3.15. We start off with a more conventional definition of thejet function: G q/h (cid:16) z h , (cid:126)p h ⊥ , z cut , R, E J (cid:17) = z h (cid:88) X ∈ Jet( R ) N c δ (2 E J − p − X − p − h ) δ (2) ( (cid:126)p h ⊥ + (cid:126)p X SD ⊥ )tr (cid:20) / ¯ n (cid:104) | χ n (0) | Xh (cid:105)(cid:104) Xh | ¯ χ n (0) | (cid:105) (cid:21) . (B.1) X here contains all the particles in the jet of radius R excluding the hadron, X SD are the subsetof those particles which pass the mMDT/soft-drop grooming procedure of the state | X, h (cid:105) . Inthis case its clear that the transverse momentum of the hadron is the recoil against the transversemomentum of all the particles in the jet which pass the grooming procedure. As before, we mayfurther factorize the jet function in Eq. (B.1) as (leaving the flavor generic): G i/h (cid:16) z h , (cid:126)p h ⊥ , z cut , R, E J (cid:17) = z h (cid:90) d (cid:126)k c ⊥ (cid:90) d (cid:126)k s ⊥ δ (cid:16) (cid:126)p h ⊥ + (cid:126)k c ⊥ + (cid:126)k s ⊥ (cid:17) ˜ D ⊥ i/h (cid:16) z h , E J , (cid:126)k c ⊥ (cid:17) × S ⊥ i (cid:16) (cid:126)k s ⊥ , z cut (cid:17) . (B.2)The operator definitions of these functions are exactly the same as those in Eq. 3.8. For perturbativevalues of p ⊥ , we can separate out the long distance non-perturbative physics at Λ QCD by matchingthe TMDFJF onto an ordinary fragmentation function:˜ D ⊥ i/h (cid:16) z h , E J , (cid:126)k c ⊥ (cid:17) = (cid:90) z h dxx J ⊥ ij (cid:16) x, (cid:126)k c ⊥ , z h , (cid:126)p h ⊥ (cid:17) D j/h (cid:16) z h x (cid:17) . (B.3)Apart from a dependence on (cid:126)k c ⊥ , the matching co-efficient J ⊥ ij also depends on (cid:126)p h ⊥ and z h via the angle ( (cid:126)θ h = (cid:126)p h ⊥ / ( z h E J )) that the final state hadron makes with the groomed jet axis.While it is possible to do the matching calculation directly(i.e., in the frame where the transversemomentum is measured with respect to the groomed jet axis), it is particularly convenient to do soin a frame in which the hadron has zero transverse momentum (In this frame the matching closelyresembles the matching of the TMDPDF onto a PDF). Let us call this the hadron-frame (h.f). Inthis frame, we are guaranteed by construction that J ⊥ ij only depends on (cid:126)k c ⊥ and x. The details ofthe matching in h.f are given in Section A.1. We then rotate back to the frame of our experimentvia an inverse rotation by (cid:126)θ h .˜ D ⊥ i/h (cid:16) z h , E J , (cid:126)k c ⊥ (cid:17) = (cid:90) z h dxx J ⊥ ij (cid:16) x, (cid:126)k c ⊥ − (cid:126)θ h E J (1 − z h ) (cid:17) h.f D j/h (cid:16) z h x (cid:17) . (B.4)where J ij ( (cid:126)p ⊥ ) h.f are the matching coefficients evaluated in the hadron frame. We have also usedthe fact that the total energy of all the collinear final state particles except the hadron is just – 18 – − z h ) E J up to power corrections. Defining (cid:126) ˜ k c ⊥ = (cid:126)k c ⊥ − (cid:126)θ h E J (1 − z h ), our jet function Eq. 3.7becomes G i/h (cid:16) z h , (cid:126)p h ⊥ , z cut , R, E J (cid:17) = (cid:90) d (cid:126) ˜ k c ⊥ (cid:90) d (cid:126)k s ⊥ δ (2) (cid:18) (cid:126)p h ⊥ z h + (cid:126) ˜ k c ⊥ + (cid:126)k s ⊥ (cid:19) S ⊥ i (cid:16) (cid:126)k s ⊥ , z cut (cid:17) . (B.5)The final form of the factorized cross- section can now be written as z h dσdz h d p h ⊥ = (cid:88) i = g,q F i (cid:16) Q, R, z cut , (cid:126)p J , C, µ (cid:17) (cid:90) d (cid:126) ˜ k c ⊥ (cid:90) d (cid:126)k s ⊥ δ (cid:18) (cid:126)p h ⊥ z h + (cid:126) ˜ k c ⊥ + (cid:126)k s ⊥ (cid:19) × S ⊥ i (cid:16) (cid:126)k s ⊥ , z cut (cid:17) (cid:90) z h dxx J ⊥ ij (cid:16) x, (cid:126) ˜ k c ⊥ (cid:17) h.f D j/h (cid:16) z h x (cid:17) + ... . (B.6)which is exactly the same Eq. 3.15. The key point to be noted is that we can do the matching inany reference frame. Obviously , the matching co-efficient will change depending on which framewe choose. However, at the end, as long as we rotate back to the frame in which the total groomedjet momentum has zero transverse momentum, we will always arrive at the same result. References [1] J. Collins,
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