Transverse Momentum Dependent Fragmenting Jet Functions with Applications to Quarkonium Production

We introduce the transverse momentum dependent fragmenting jet function (TMDFJF), which appears in factorization theorems for cross sections for jets with an identified hadron. These are functions of $z$, the hadron's longitudinal momentum fraction, and transverse momentum, $\boldsymbol{\mathrm{p}}_{\perp}$, relative to the jet axis. In the framework of Soft-Collinear Effective Theory (SCET) we derive the TMDFJF from both a factorized SCET cross section and the TMD fragmentation function defined in the literature. The TMDFJFs are factorized into distinct collinear and soft-collinear modes by matching onto SCET$_+$. As TMD calculations contain rapidity divergences, both the renormalization group (RG) and rapidity renormalization group (RRG) must be used to provide resummed calculations with next-to-leading-logarithm prime (NLL') accuracy. We apply our formalism to the production of $J/\psi$ within jets initiated by gluons. In this case the TMDFJF can be calculated in terms of NRQCD (Non-relativistic quantum chromodynamics) fragmentation functions. We find that when the $J/\psi$ carries a significant fraction of the jet energy, the $p_T$ and $z$ distributions differ for different NRQCD production mechanisms. Another observable with discriminating power is the average angle that the $J/\psi$ makes with the jet axis.


Introduction
In recent years, jet physics has played a prominent role at high energy colliders, particularly the Large Hadron Collider (LHC). Jets provide an opportunity to test our understanding of Quantum Chromodynamics (QCD) and appear in both Standard Model and beyond the Standard Model cross sections, making them important for searches of new physics as well. Due to the enormous energies available at the LHC, top quarks, W ± , Z 0 , and Higgs bosons are frequently produced with transverse momenta much greater than their mass, and studies of jet substructure have proved essential in identifying these highly boosted particles when they decay hadronically [1,2]. For all these reasons, precision jet calculations have become increasingly important in particle physics. At the heart of analytic calculations of jets are factorization theorems which separate jet cross sections into perturbative and non-perturbative pieces. Non-perturbative functions such as parton distribution functions (PDFs), fragmentation functions (FFs), and fragmenting jet functions (FJFs) offer ways to analytically probe the structure of the proton as well as the nature of hadronization.
TMDPDFs have been used in SCET for studies of Higgs production in the small transverse momentum limit at the LHC [31][32][33][34][35][36]. TMD fragmenting jet functions (TMDFJF) depend on three kinematic variables: the jet energy, ω/2, the fraction of this energy carried by the identified hadron, z, and the hadron transverse momentum with respect to the axis of direction of the original parton, p h ⊥ . The modes that give important contributions to the transverse momentum are collinear-soft: p µ cs ∼ ω(λr, λ/r, λ), λ = p ⊥ /ω collinear: p µ n ∼ ω(λ 2 , 1, λ), (1.3) where collinear-soft modes are soft modes collinear to the direction of the jet axis first introduced in Ref. [37] and r ≡ tan (R/2) for jet cone size R. Similar modes are also studied in [38]. To incorporate contributions from soft-collinear modes, we make use of the SCET + formalism. SCET + and other similar extensions of SCET have been used to study processes with multiple well-separated scales and distinct phase space regions (e.g. [29,[37][38][39]). Recent work [11,13] shows that jet substructure observables may be able to shed light on outstanding puzzles in the production of quarkonia such as J/ψ and Υ. Our modern understanding of quarkonium production comes from non-relativistic QCD (NRQCD) [40], an effective field theory that writes cross sections and decay rates for bound states of heavy quarks as expansions in the strong coupling α s (2m c ) and v, the relative velocity of the quark-antiquark pair. NRQCD provides factorization theorems [41][42][43][44] for cross sections in terms of perturbatively calculable short-distance pieces multiplied by non-perturbative long distance matrix elements (LDMEs). The perturbative piece describes the creation of a heavy quark-antiquark pair in a given color and angular momentum state while the nonperturbative LDMEs describe the hadronization of the heavy quark-antiquark pair into the physical quarkonium state. The different intermediate color and angular momentum states of the pair define different NRQCD production mechanisms for quarkonia.
Ref. [13] studied the dependence of the cross section for the production of J/ψ within jets initiated by gluons on z, the fraction of the jet's energy, E J , carried by the J/ψ. The authors showed that the z dependence is sensitive to the underlying quarkonium production mechanism. Thus, simultaneously measuring the z and E J dependence of the cross section for J/ψ production within jets provides a new and independent way of extracting the values of the LDMEs. Ref. [11] extended these results to J/ψ production in e + e − collisions where the angularity of the jet was probed. Ref. [11] also found that NLL' resummed analytic calculations of the z distributions were quite different from those predicted by PYTHIA simulations. The authors attributed this large discrepancy to an unrealistic modeling of the shower radiation from color-octet quarkonium production mechanisms.
Intuitively, one might expect color-octet quark-antiquark intermediate states to radiate more gluons relative to color-singlet pairs. This would result in J/ψ produced with higher p ⊥ relative to the jet axis. Also, since different color-octet production mechanisms have different FFs in NRQCD, FJFs should be able to distinguish between the different coloroctet production mechanisms. In addition to generalizing FJFs to TMD distributions, this paper also shows that these TMDFJFs do in fact provide discriminating power between the different mechanisms.
In Section 2, we give a definition of the TMDFJF and show how it emerges from definitions of TMDFFs in the literature. We then perform a matching calculation at next-to-leading order (NLO) onto SCET + and derive a result that is completely factorized into hard, collinear, collinear-soft, and ultra-soft modes. We present a calculation of the matching coefficients J ij between the TMDFJF and the more traditionally studied FFs. Additionally, we present a perturbative calculation of the corresponding collinear-soft function at NLO. In Section 3, we use renormalization group (RG) and rapidity renormalization group (RRG) techniques to resum logarithms to next-to-leading-log-prime (NLL') accuracy. The TMDFJF formalism is applied to the production of J/ψ in gluon jets where the FFs are calculated to LO in NRQCD. We find that distributions in p ⊥ and z as well as the average angle of J/ψ relative to the axis of the jet can discriminate between the various NRQCD production mechanisms. Conclusions are given in section 4. Appendix A gives calculational details of the matching of the TMDFJF onto the FF, Appendix B has an alternative derivation of the TMDFJF from an SCET factorization theorem for a jet cross section, and Appendix C has details about the RG and RRG evolution.

Transverse momentum dependent fragmenting jet function
In this section we will present the definition of the TMDFJF, connecting it with definitions of TMDFFs from the literature. We first show the matching calculation of the TMDFJF onto SCET + and its factorization into pure collinear, soft-collinear, and hard pieces. We then present perturbative calculations of the matching coefficients, J i/j , from matching the pure collinear function onto the FF as well as the one-loop expression for the soft-collinear function.
As we show in Appendix B, the expression for the TMDFJF given in Eq.(2.4) is closely related to the FJF introduced in Ref. [3].
As discussed in the introduction, the TMDFJF receives contributions from two different modes, collinear and colllinear-soft or csoft . In order to make the contribution of the csoft modes explicit, we now match our expression onto SCET + , where are Wilson lines of csoft fields (the csoft analogue of W n ) and C + (µ) are SCET + matching coefficients. In order to decouple the collinear fields A n from the csoft gluons, we now perform field redefintions similar to those of the BPS procdure [37] This allows us to factorize the TMDFJF into three pieces where H + is proportional to the square of the matching coefficient from G q/h in SCET I to SCET + , and D q/h and S C are the contributions collinear and the collinear-soft modes of SCET + to the TMDFJF, respectively. These are defined by where the Tr is over Dirac and color indices in D q/h and color indices in S C . From now on, we drop the (0) and (0, 0) superscripts since the different collinear, soft-collinear, and ultrasoft modes are now factorized. We also employ the following shorthand for the convolution in the ⊥ components Analogously for gluon fragmentation we have where the collinear gluon field is and iD n⊥ = P µ n⊥ + gA µ n⊥ is the standard ⊥-collinear covariant derivative in SCET. At this point, only the purely collinear term D i/h contains information about the hadron h. The collinear-soft function (S C ) and the hard function (H + ) are universal functions dependent on the fragmenting parton i but not on the hadron h. Additionally, in the limit that |p ⊥ | Λ QCD , we may use the operator product expansion to factorize D i/h into short distance coefficients and the more commonly studied FFs, D j/h , via, where J i/j are the short distance coefficients that do not depend on the final hadron and can be calculated order by order in perturbation theory.

Perturbative results
The O(α s ) diagrams contributing to the gluon and quark TMDFJFs are shown in Figs. 1 and 2, respectively. At NLO, the matching coefficients J i/j are directly related to the  matching coefficients I i/j between TMDPDFs and the more commonly studied PDFs calculated in Refs. [29,31] by the substitution I i/j → J j/i . See Appendix A for additional details of the matching calculation. Following Ref. [31], a rapidity regulator is used to regulate rapidity divergences in the perturbative calculation. This is implemented by first modifying the form of the collinear and collinear-soft Wilson lines with similar modifications to U n . This introduces a regulator η, a bookkeeping parameter w, and a new dimensionful parameter ν. The dependence of our results on ν should of course cancel amongst the terms in our factorization theorem. The renormalized results for the J i/j in the MS scheme can be written, and For convenience we use the following shorthand notation for the vector plus-distributions, Performing the convolutions in the energy ratio parameter z we get, Real gloun emission diagrams that contribute to the collinear-soft function The gluons passing through the shaded oval indicate they are contained within the phase-space of the jet.
At NLO, the collinear-soft function, defined by Eq. (2.17), receives contributions from the two diagrams shown in Fig. 3. The real gluon is contained within a jet defined by a cone or k T -type jet algorithm with cone size parameter R. A global soft funciton of similar form has been calculated at NLO in Ref. [31] and at NNLO in Ref. [33] in studies of Higgs p T spectrum. The two diagrams in Fig. 3 yield identical contributions and thier sum is given by, where Θ alg defines the jet algorithm, r ≡ tan(R/2), and C q = C F , C g = C A . After an expansion in η followed by an expansion in and summing both diagrams we get, The renormalized result (where we have now set w → 1)in the MS scheme is thus (2.32) While in general this expression receives contributions from virtual gluon emission diagrams at NLO, these diagrams yield scaleless integrals when using this particular set of regulators. Thus virtual diagrams are neglected and all singularities from these real emission diagrams are interpreted as UV divergences. We also verified, using a set of regulators where such virtual gluons give non-zero contributions, that the result is identical. 1 Note if pure dimensional regularization is used for ultraviolet and infrared divergences then H + = (2π) 2 N c as discussed in Ref. [29].

Renormalization Group (RG) and Rapidity Renormalization Group (RRG)
Individual diagrams for the collinear-soft function S C and the matching coefficients J i/j suffer from infra-red (IR), ultra-violet (UV) and rapidity divergences (RD). We use dimensional regularization and a rapidity regulator (as introduced and developed in Ref. [31,47]) to regulate these divergences. IR divergences in the collinear-soft function cancel when summing over all diagrams. In the matching coefficients J i/j , IR divergences cancel in the matching of the collinear functions D i/h onto traditional FFs, D j/h . The remaining poles (UV and rapidity), are removed by renormalization. In addition to the scale µ introduced by dimensional regularization our use of a rapidity regulator requires the introduction of an additional scale, ν. With this scale are associated rapidity renormalization group (RRG) equations which can be used to resum large logarithms by evolving each function from its canonical scale to a common scale. Bare and renormalized quantities are related through the following convolution with the renormalization factor Z, where F can be either D i/h or S i C and satisfies the following RG and RRG equations, Here γ F µ and γ F ν are the anomalous dimensions associated to RG and RRG respectively and are defined by, For the renormalization factors we find, The µ anomalous dimensions are found using Eq. (3.3), For the ν anomalous dimensions, our bookkeeping parameter w plays an analogous role to the coupling g for the case of the µ anomalous dimension, although w itself is not a coupling, such that, The anomalous dimensions satisfy where γ J is the anomalous dimension of the unmeasured quark jet function [48] and In order to resum our results to NLL' accuracy we evolve the purely collinear function and the collinear-soft function from their characteristic scales where logarithms are minimized to common scales in µ and ν using the RG and RRG respectively. To perform the evolution, we first solve the Fourier transforms of both the RRG and RG equations. We then perform the evolution using the RG and RRG before finally performing the inverse Fourier transform. The simplest resummation procedure is, in this case, to first evolve our collinear-soft function in RRG space and choose the common scale to be ν = ν D . We then evolve both functions in RG space to the common scale µ = ωr. Notice that S C and D have the same characteristic renormalization scale µ S C = µ D ≡ µ C . The equivalence of the virtualities of the soft and collinear modes is a defining feature of SCET II .
To make the interpretation of our plots easier, we study the quantity G i/h (p ⊥ , z, µ) which is related to the TMDFJF by the change of variables from vector transverse momenta (p ⊥ ) to the amplitude (p ⊥ = |p ⊥ |). Performing the evolutions described above we find, where b is the Fourier conjugate variable of p ⊥ , J 0 is a Bessel function of the first kind, , (3.14) and (3.15) are the evolution kernels resulting from solving the RG and RRG equations respectively. The pure collinear term D i/h in Eq.(3.13) involves the convolution of the perturbatively calculated short distance coefficients and the standard fragmentation functions evolved from their canonical scale to the canonical scale of the collinear term in momentum space, µ = p ⊥ . The form of the fragmentation functions is fixed during the Fourier transforms in Eq.(3.13). The scales µ F , ν F and m F for each of the functions are given in Table 1 and more details of the RG and RRG evolution are provided in Appendix C.

Applications to quarkonium production
In this section we apply our TMDFJF formalism to the production of quarkonium in jets. We will focus on J/ψ production within jets initiated by gluons, though our results can be easily generalized to Υ or other quarkonia and jets initiated by quarks. For J/ψ production the leading production mechanism in the NRQCD v expansion is 3 S indicates the color and angular momentum quantum numbers of the cc produced in the short-distance process. This mechanism scales as v 3 , whereas the leading coloroctet mechanisms, 3 S  Table 2 shows this scaling along with numerical values of the corresponding LDME extracted from the fits in Ref. [49,50] (which we use below). The extracted LDME are consistent with the v 4 suppression expected from NRQCD. As was done for the FJF's in Ref. [11] we use the leading order NRQCD [40] FFs for gluon fragmentation to J/ψ for each of the four mechanisms. In the α s expansion the leading order contribution to gluon fragmentation to J/ψ via the 3 S 1 mechanism scales as α s (2m c ) 3 , while for 1 S [8] 0 and 3 P [8] J the leading contribution scales as α s (2m c ) 2 and for the 3 S 1 mechanism the fragmentation function scales as α s (2m c ). Thus for gluon fragmentation the v 4 suppression of color-octet mechanisms is compensated for by fewer powers of α s and all four contributions are roughly the same size. Our goal is to see if the z and p ⊥ dependence of the TMDFJF can discriminate between these competing mechanisms.    The TMDFJF as a function of p ⊥ for fixed z, for z = 0.3, 0.5, 0.7, and 0.9, are shown Figs. 4 and 6, for jet energies of 100 GeV and 500 GeV, respectively. In order to make it easier to view all distributions simultaneously, we have rescaled the 3 S J production mechanisms, with p ⊥ = 10 GeV for E J = 100 and 500 GeV. Theoretical uncertainties are calculated by varying the renormalization scales by factors of 1/2 and 2. µ = ωr. Though we plot our distributions in the range 0 < p ⊥ < 20 GeV, it is important that to keep in mind that our calculations are only reliable for p ⊥ ≥ 2m c = 3 GeV.
These plots show that the TMDFJF does in fact provide discriminating power amongst the four mechanisms. For z = 0.3, all four distributions look similar for both E J = 100 GeV and 500 GeV. The distributions peak at roughly the same location and they have same slope for large p ⊥ . For z ≥ 0.5, the color-singlet 3 S [1] 1 mechanism and the color-octet 1 S [8] 0 mechanism peak at lower values of p ⊥ and fall more steeply with p ⊥ than the 3 S J color-octet mechanisms. The 3 P [8] J mechanism has the peculiar feature that in order to obtain a positive FF we need to have a negative LDME, as is found in the fits of Refs. [49,50]. The peaks in the p ⊥ distribution for the 3 S 1 and 1 S [8] 0 mechanisms are at very low p ⊥ where perturbation theory is not reliable. On the other hand, the peaks of the 3 S [8] 1 and 3 P [8] J distributions are at larger values of p ⊥ ∼ 6 − 8 GeV where perturbation theory can be trusted. The 3 P [8] J gives a slightly harder p ⊥ distribution than 3 S [8] 1 mechanism, and both are significantly harder than the other mechanisms.
It is interesting to study the dependence of the TMDFJF as a function of z with p ⊥ fixed to be a perturbative scale. In Fig. 5 we plot the TMDFJF as a function of z for p ⊥ = 10 GeV for jets with energy E J = 100 and 500 GeV. Large logarithms and shape function effects will affect these distributions in both the z → 0 and z → 1 limits, but our calculations should be reliable for intermediate values of z. While for z < 0.5 the distributions have similar shapes, in the range 0.5 < z < 0.9, the shapes of all four mechanisms are different. The z dependence of the TMDFJF for fixed p ⊥ can be used to differentiate between the NRQCD production mechanisms.
The TMDFJF formalism also allows us to calculate the angle at which J/ψ are pro-  Figure 6. The TMDFJF as a function of the p ⊥ of the J/ψ for the 3 S duced relative to the jet axis. The average production angle for the J/ψ is given by Using the small angle approximation the differential cross section can be written as Substituting this into Eq. 3.16 yields θ (z) = 2 dp ⊥ p ⊥ (dσ/dp ⊥ dz) zω dp ⊥ (dσ/dp ⊥ dz) . (3.18) As discussed in Appendix B, the cross section dσ/dθdz can be factorized into hard, soft and collinear terms in SCET. In general the hard and soft contributions will not cancel because there is a sum over partonic channels in both the numerator and denominator of Eq. 3.18. However, they will if gluon fragmentation dominates production, then the expression above can be written as (3.19) where G g/h (p ⊥ , z, µ) is the gluon TMDFJF. Fig. 7 the function f J/ψ ω (z) is plotted at points z = 0.3, 0.5, 0.7, and 0.9 for ω = 2E J = 200 GeV and 1 TeV for J/ψ with p ⊥ ∈ [5, 20] GeV and p ⊥ ∈ [5, 60] GeV, respectively. As was done earlier we have fixed the scale µ = ωr. Note the typical angles are small enough that the small angle approximation is justified. The dashed lines in figure show the results of a fit to the functional form, C 0 exp(−z C 1 ), the values of C 0 and C 1 for each mechanism at each energy are shown in Table 4. Again we see that differences between the various NRQCD mechanisms become more pronounced as z increases. This shows that the average angle does in fact yield some discriminating power between the octet mechanisms. In particular the slope on the semilog plot, which is determined by the parameter C 1 in Table 4, differs by as much as 20% between the various NRQCD mechansims for E J = 100 GeV and and as much as 40% for E J = 500 GeV. Note however that 1 S [8] 0 and 3 S   Figure 7. The function f J/ψ ω (z) (as defined in the text) as a function of z relative to the jet axis for each NRQCD production mechanism where the jet has E J = ω/2 = 100 GeV(left) and 500 GeV (right). The J/ψ is restricted to have p ⊥ ∈ [5,20] GeV in the 100 GeV jet and p ⊥ ∈ [5, 60] GeV in the 500 GeV jet.

Conclusions
In this paper we introduce the transverse momentum dependent fragmenting jet function (TMDFJF) in the framework of SCET and show how it is related to the previously introduced TMDFFs and fragmenting jet functions (FJFs). TMDFJFs describe the transverse as well as longitudinal momentum distribution of an identified hadron within a jet. TMD-FJFs evolve with the renormalization group (RG) scale µ and obey RG equations similar to jet functions. Using SCET + we show that this new distribution can be further factorized into soft and purely collinear terms. The purely collinear factor can be written as a  Table 3. Results of fits of ln (f ω (z)) shown in Fig. 7 to the function C 0 exp(−z C 1 ). convolution of perturbatively calculable short distance coefficients and the standard FFs, where the soft factor is given by a vacuum matrix element of product of Wilson lines. This factorization introduces rapidity divergences that are regulated with the rapidity regulator. We check that at NLO the regulator dependance vanishes in the final product. Associated with rapidity divergences are rapidity renormalization group (RRG) equations. By evolving the collinear and soft terms separately using the RG and RRG equations all orders resummation of large logarithms in the TMDFJF can be performed.
As an example we implement this formalism for the case of quarkonium production. In the case of quarkonia the TMDFJF can be calculated in terms of the NRQCD FFs which are perturbatively calculable at the scale 2m Q . For the gluon TMDFJF for J/ψ, we study the p ⊥ and z dependence predicted by the four production mechanisms: 3 S J . We use the leading order (in α S ) NRQCD FF for each of these mechanisms, and the RG and RRG equations are used to calculate the TMDFJFs to next-to-leadinglogarithmic-prime (NLL') accuracy. We find that the z dependence (for fixed p ⊥ ) is different for all four mechanisms. We also find that the dependence on p ⊥ and the average angle of the J/ψ relative to the jet axis can discriminate between the various NRQCD production mechanisms.

A Matching calculation
In this Appendix we provide details for the evaluation of the matching coefficients, J i/j . From the sum of diagrams in Figs. 2a) and 2b) we get: where we define c qq (z) = (1 − z)/2. The superscripts B and R denote bare and renormalized quantities, respectively, and the superscript (1) indicates that this is the O(α S ) contribution. The NLO matching coefficient is given by The 1/ pole appearing in the FF is interpreted as an infrared divergence. Although for extracting the renormalized matching coefficients J i/j we can ignore scaleless integrals and interpret the finite terms as the renormalized result to that particular order, here we are interested in the origin of the poles since this will allow us to extract the anomalous dimensions. Performing the matching we get: For the coefficient J q/g we simply perform the replacement z → (1 − z) and drop δ(z) and plus-distributions since these functions are always integrated for values of z greater than zero. Thus where c qg (z) = c qq (1 − z) = z/2. For the gluon splitting we get Expanding in η and we have and since the corresponding FF is given by: where the 1/ pole is an infrared divergence, we have where c gq (z) = z(1 − z). Performing the matching and since the corresponding FF is where again the 1/ pole is an infrared divergence, we get

B Factorization Theorems in SCET
Much like the standard FJFs, TMDFJFs appear in factorization theorems for cross-sections that are differential in z, the fraction of a jet initiating parton's energy carried by an identified hadron, and p ⊥ , the transverse momenta of the hadron measured from the parton's momentum. It is shown in Ref. [48] that the cross-section for the production of two jets in electron-positron annihilation can be written as, where dσ (0) is the Born cross section, H 2 (µ) is the hard function resulting from matching a 2-jet operator in full QCD onto the corresponding SCET operators, S Λ (µ) is a soft function that describes soft scale cross-talk between the jets and the soft out-of-jet radiation is constrained via E out < Λ, and J n (ω, µ) is a jet function that describes collinear radiation within a jet in then direction that has energy E J = ω/2 (here ω = E cm ). The jet function can be defined in SCET as To study jets with identified hadrons, we insert the following expression for the identity where h is an identified hadron within the jet. Performing the integration over x, which is the Fourier conjugate of the residual momenta, and the residual k + yields Insetrting this back to Eq.(B.1) we have which directly implies This suggests a rather powerful rule (already known to be true for the standard FJFs) for constructing the factorization theorem in SCET with identified hadron with measured transverse momenta :
(C.8) The solultion to the RGE is thus given by where again U F (µ, µ 0 , m F ) = exp (K F (µ, µ 0 )) µ 0 m F ω F (µ,µ 0 ) (C.14) and the exponents K F and ω F are given in terms of the anomalous dimension, 16) and for up to NLL and NLL' accuracy are given by