Semi-Inclusive Jet Functions and Jet Substructure in $J_{E_T}^{(I)}$ and $J_{E_T}^{(II)}$ Algorithms

Within the framework of Soft Collinear Effective Theory, we present calculations of semi-inclusive jet functions and fragmenting jet functions at next-to-leading order (NLO) for both quark- and gluon-initiated jets, for jet algorithms of $J_{E_T}^{(I)}$ and $J_{E_T}^{(II)}$ where one maximizes a suitable jet function. We demonstrate the consistency of the obtained results with the standard perturbative QCD calculations for $J_{E_T}^{(I)}$ algorithm, while the results for fragmenting jet functions with the $J_{E_T}^{(II)}$ algorithm are new. The renormalization group (RG) equation for both semi-inclusive jet functions and fragmenting jet functions are derived and shown to follow the time-like DGLAP evolution equations, independent of specific jet algorithms. The RG equation can be used to resum single logarithms of the jet size parameter $\beta$ for highly collimated jets in these algorithms where $\beta \gg 1$.

Within the framework of Soft Collinear Effective Theory, we present calculations of semi-inclusive jet functions and fragmenting jet functions at next-to-leading order (NLO) for both quark-and gluon-initiated jets, for jet algorithms of J

I. INTRODUCTION
In high energy proton-proton and nucleus-nucleus collisions, tremendous amount of collimated jets of hadrons are produced and measured at the Large Hadron Collider (LHC). The studies of the production rate of these jets and their substructures emerged as essential tools to probe the fundamental properties of Quantum Chromodynamics (QCD) [1][2][3][4][5][6] and nucleon structure [7][8][9][10]. In addition, they are also involved in searching for signals of new physics beyond the standard model [11,12], as well as in identifying the properties of the hot dense medium, quark-gluon plasma, created in heavy ion collisions [13][14][15][16][17]. For some recent review, see Refs. [18][19][20] and references therein.
Because of the crucial roles of jets, significant theoretical and experimental efforts have been devoted into the study of jets in both particle and nuclear physics communities. For example, within the framework of Soft Collinear Effective Theory (SCET) [21][22][23][24], the cross section for inclusive jet production in pp collisions can be factorized into a convolution product of initial-state parton distributions functions (PDFs) f , hard-part coefficient H and final state semi-inclusive jet functions (siJFs) J [25][26][27] dσ pp→jet+X ∼ i,j,k The semi-inclusive jet functions J k characterize the probability density of a parton k that is transformed to a jet. Similarly, one can also study the internal structure of the jet by measuring, e.g. the distribution of identified hadrons inside the jet, which is described by the so-called semi-inclusive fragmenting jet functions (siFJFs) G h k [28]. The relevant factorization formula is very similar to that in Eq. (1), but replacing J k → G h k [26,28]. Based on this factorization formula, significant extensions and improvements are established including the study of jet quenching physics for light flavors [29], next-to-leading order (NLO) calculations for heavy flavor jet in vacuum [30,31] and in medium [32], as well as the application to study heavy quarkonium production mechanism [33,34].
In all these works, typical algorithms, for example, cone and/or anti-k T algorithm, are used to match to the experimental analysis. The final results at NLO show single logarithmic structure α n s ln n R with R represents the size of the identified jets. These logarithms spoil the convergence of perturbative expansion, and thus need to be resummed to all orders. This can be realized through the renormalization group (RG) equations of the relevant semiinclusive jet functions. It has been shown that such RG equations are the same as the time-like DGLAP evolution equations [25,26,35].
A while ago, another type of jet algorithm is proposed initially by Georgi [36], where the idea is to cluster jets by maximizing a fixed/suitable function of the total four-momentum of jets. This algorithm was subsequently improved by Bai et al. [37] by using the total transverse energy instead of the energy in the fixed function, which is more appropriate for hadronic collisions such as those at the LHC, because transverse energies are boost invariant. This type of algorithm has been implemented into the standard pQCD calculations at NLO for single-inclusive jet production [38] and jet fragmentation functions [27] at hadron colliders, in which infrared safety of the algorithms is established and comparisons to cone and anti-k T algorithms are presented. Since these algorithms maximize a suitable jet functions, we will refer to them as "maximized jet algorithms" for simplicity.
In this work, within the framework of SCET, we perform explicit calculations at NLO to study siJFs and siFJFs in maximized jet algorithms of J (I) [37] as defined in the next section. We demonstrate the consistency of our results with the standard perturbative QCD calculations for J (I) E T algorithm [27,38]. We also perform the calculations in both light-cone gauge and covariant gauge for cross-checking our results, while the previous calculations are typically performed in covariant gauge, see e.g. [25,39]. For both siJFs and siFJFs, we find exactly the same divergent behavior, which leads to exactly the same RG equations, i.e. time-like DGLAP evolution equations, independent of specific jet algorithms, while the remaining finite parts exhibit the algorithm dependence. These RG equations can be used to perform the resummation of single logarithms of the jet size parameter β for maximized jet algorithms, where β 1 corresponds to highly collimated jets. In this sense, the situation is very similar to the case in the anti-k T algorithm, where the RG equations are used to resum single logarithms of the jet radius R for the narrow jets with R 1 [25]. Note that we focus on fully analytical calculations of the siJFs and siFJFs at NLO in the current work, and we leave phenomenological implementations of these results in pp and AA collisions for future publications.
The remainder of this paper is organized as follows. In section II, we recall the operator definition of siJFs and give introduction to the maximized jet algorithms. In section III, we present explicit calculations of siJFs for quark and gluon jets at NLO by considering both J (I) E T maximized jet algorithms and compare them with the standard pQCD results in the literature. In section IV, we extend the calculation to semi-inclusive fragmenting jet functions at NLO. We conclude our paper in section V.

II. DEFINITIONS AND MAXIMIZED JET ALGORITHMS
In this section we start by giving the definition of the semi-inclusive quark and gluon jet functions in SCET, which can be constructed from the corresponding gauge invariant quark and gluon fields. The siJFs are interpreted as the probability density of the parton to transform into a jet. In light-cone coordinates, they are given by the following operator definitions [25] for the quark and gluon jets, respectively where E J = zE is the jet energy, and E is the energy of the parton initiating the jet, P is the label momentum operator, and the state |JX represents the final-state observed jet J and unobserved particles X. χ n and B n⊥µ are gauge invariant n-collinear quark and gluon fields, respectively. Note that the light-cone vector n µ is defined along the jet axis, and its conjugate vector isn µ . In the frame where the jet has no transverse momentum, we can write n µ = (1, 0, 0, 1) andn µ = (1, 0, 0, −1), which satisfies n 2 =n 2 = 0 and n ·n = 2. In such a frame, we define ω =n · P and ω J =n · P J as the large light-cone components of the momenta for the parton initiating the jet (P ) and the jet itself (P J ), respectively. For a collimated jet, we have ω ≈ 2E and ω J ≈ 2E J . In this work, we neglect the nonperturbative hadronization effect and consider only the perturbative aspect of the jet functions, thus one can calculate both quark and gluon initiated jets perturbatively. At leading order (LO) in which one parton forms the jet, the jet functions are independent of specific jet algorithms and are simply delta functions where the superscript (0) denotes the LO result. At next-to-leading order (NLO), one has to consider the phase space constraints for the radiated parton according to specific jet clustering algorithms. It is this constraint that leads to the algorithm dependence of the jet functions. There are so far two classes of broadly defined algorithms: cone algorithms [40] and successive recombination algorithms [41]. Cone algorithms include the Snowmass and SIS cone algorithms [42], while recombination algorithms include the Cambridge-Aachen, k T and anti-k T algorithms [43,44]. In Refs. [25,45], the semi-inclusive jet functions in cone and anti-k T algorithms have been calculated up to NLO. In the present paper, we will extend the calculation to two other jet finding methods J Here, E set and P µ set is the total energy and four-momentum of a given subset of the final-state particles, and m set is its invariant mass, i.e. P 2 set = m 2 set . Ref. [37] improves the above definition by using the transverse energy defined as (E ⊥ set ) 2 ≡ P 2 ⊥ + m 2 set instead of the energy E set , where P ⊥ is the magnitude of the transverse momentum. Thus we have the J (I) Apparently the definition in Eq. (5) is more suitable for jet production in e + e − collisions where the energy of the jet is relevant. On the other hand, the definition in Eq. (6) uses transverse energies which are boost-invariant and hence more suitable for the application to hadronic scattering, such as pp collisions at the LHC. Note that for collimated jets, so-called narrow jet approximation (NJA) applies, and one could replace the transverse energies E ⊥ by the transverse momenta P ⊥ . In our calculations below, we choose a frame in which the jet has zero transverse momentum, and thus we will follow Eq. (5) in most of our calculations. However, once our calculations is done, translating from Eqs. (5) to (6) will correspond to simply replacing the jet energy E J in our final expressions by the jet transverse momentum P J⊥ for studying jet production in pp collisions at the LHC. For this new jet algorithm, the parameter β ≥ 1 specifies the algorithm and is introduced to determine the geometric size of the jet. By maximizing the J In other words, when the jet is formed, the corresponding total four-momentum of the subset of the final-state particles P set gives the jet momentum P J , and E ⊥ J and m J are the transverse energy and invariant mass of the jet. The algorithm is iterative, i.e., once a jet has been found, the algorithm proceeds by removing the subset from the list of particles in the event, and apply iteratively to the remaining ones.
One may further vary the function J (I) E T by changing the weighted functions [38]. For instance, one can define the J (n) E T algorithm as follows where n = 1 and n = 2 correspond to the J (I) E T algorithms 1 , respectively. In the present paper, we will focus on J (I) E T algorithms to explicitly show the algorithm dependence of siJFs. We will derive the NLO siJFs by considering highly collimated jets, i.e., β 1. In this case, one can take the so-called narrow jet approximation and obtain fully analytical expressions.

III. THE SEMI-INCLUSIVE JET FUNCTIONS IN MAXIMIZED ALGORITHM
In this section, we present the detailed calculations at NLO for semi-inclusive jet functions for both quark and gluon jets in the maximized jet algorithm. As we have already mentioned, we perform the calculations in both light-cone gauge and covariant gauge for cross-checking our results, while the previous calculations are typically performed in covariant gauge, see e.g. [25,39]. We denote the incoming parton with momentum = ( − = ω, + , 0 ⊥ ) splits into two partons with one of them carry momentum q = (q − , q + , q ⊥ ).

A. The semi-inclusive quark jet function
For quark jet function, the total contributions from the relevant diagrams, as shown in Fig. 1, have been written down explicitly in d = 4−2 dimensions in Ref. [25]. We rewrite it here for the completeness of showing the calculation In this case, q denotes the momentum for the radiated gluon, Θ alg is determined by the jet algorithm and by the kinematics of the radiated parton whether it is inside the jet. As shown in Fig. 1, there are three situations that we need to take into consideration. The diagram (A) is for the case that both quark and gluon are inside the jet, diagrams (B) and (C) are for only quark is inside the jet, and only gluon is inside the jet, respectively. Each case has different Θ alg to be specified below, the combination of these three cases gives the final result. In the situation that both quark and gluon are inside the jet, the incoming quark energy E is all converted to the jet energy E J , which leads to z = E J /E = 1. In this case, we can perform the integration over quantities + and q + in Eq. (9), thus the quark jet function becomes, where the subscript "q → qg" stands for the situation that both the radiated quark and gluon are inside the jet, x = ( − − q − )/ − is the momentum fraction of the initial quark carried by the final-state quark, and m 2 = 2 is the invariant mass for the final-state parton pair, i.e. q + g. Of course, since q and g together forms the jet, m 2 = m 2 J is just the jet invariant mass, and it is related to the radiated gluon transverse momentum q ⊥ by m 2 = In the channel q → qg, to make sure that the final q and g actually forms one jet, one has to require that the value of the J (n) E T function constructed from the two partons together is larger than the J (n) E T value constructed for each parton individually. When both q and g forms the jet, we have where we remind that E is the energy of the incoming quark and is the same as the jet energy E J = E when q and g together forms the jet. On the other hand, for the case in which either q or g forms the jet, we have where we have used the fact that the invariant mass of the final-state quark q or g vanishes. Thus the requirements of the maximized jet algorithm J (n) E T (g) lead to the following constraint, Realizing E q = xE and E g = (1 − x)E, we obtain the following constraint for m 2 , This leads to the following algorithm constraint where Θ is the step function. Let us first consider the J E T algorithm will follow the same procedure. Apply the above constraint to the jet function with n = 1, we arrive at where Expand I (1) qq in , we get the explicit expression Substituting the above expression to Eq. (20), we obtain the quark jet function when both quark and gluon are inside the jet for the J (I) withL defined asL For highly collimated jet, i.e., β 1, the above logarithmic term becomes very large and needs to be resummed. This is similar to that in k t -type algorithms [25], where large logarithmic terms of R needs to be resummed for small jet radius R. Now let's consider the situation that only the final state quark forms the jet. The corresponding diagram is presented in Fig. 1(B). In this case, the final-state quark forms the jet with jet energy E J = zE, namely only a fraction z of the initial parton energy E falls inside the jet. Follow the same calculation as before, we obtain where the subscript "q → q(g)" represents the situation that only the quark q is inside the jet while the gluon g is radiated outside the jet. Here m is again the invariant mass for the final-state parton pair, i.e. q + g. Since now only q is inside the jet while the gluon g is outside, m is different from the jet invariant mass.
In this case, it is the single quark that forms the jet, rather than quark and gluon together forms a jet. Following the algorithm constraint, we should have This can be understood as follows. If J (n) E T (g) , and satisfies Eq. (26), then we would have a single gluon form the jet. Since the jet algorithm is iterative, the gluon jet will be removed from the event. As a result, we are left with only a single quark, which will automatically form another jet. On the contrary, if J (n) E T (g) , the single quark first forms the jet. In other words, for a two-parton configuration, as long as the larger one among single-parton J E T functions is greater than the J E T function for the two-parton set, we will have the quark form the jet. With Eq. (26) at hand and from Eqs. (15) and (16), we thus have the following constraint where we have used E = E J /z for this configuration. Implementing this constraint in Eq. (25), and considering the case n = 1, we obtain Performing the -expansion, we find that the bare jet function is given as follows where L =L + ln z = ln We realize that Eq. (29) is the same as that in the anti-k T algorithm, except for the slightly different definition of L. The universality of this term is caused by the NJA, and the algorithm dependence is hidden in power suppressed terms.
Likewise, for the case that only the gluon falls inside the jet, as illustrated in Fig. 1(C), the calculation is very similar to the case of q → q(g). For n = 1, it can be expressed as Summing Eqs. (23,29,31) together, one obtains the full expression for the semi-inclusive quark jet function in the J (I) E T algorithm at NLO, where P ij (z) are the standard Altarelli-Parisi splitting functions, with β 0 = 11 3 C A − 4 3 T F n f , and the "plus"-distributions are defined as usual Note that the double pole term (∝ 1/ 2 ) as shown in q → qg channel cancel with that in q → q(g), and at the same time the q → (q)g channel is free of double-pole terms, thus we are left with only single-pole term (accordingly single logarithm L) in the final result. The single-pole term is universal and independent of jet algorithm, and it is this single-pole term that leads to the DGLAP evolution of the semi-inclusive jet functions [25]. Likewise, the jet function for the J (II) E T algorithm with n = 2 can be derived by following exactly the same procedure as that for n = 1. The final result is where Li 2 (1/4) = 0.267653 . . . is the dilogarithm function. We find that all the terms in the last two rows are algorithm dependent, while the rest of the terms are independent of the jet algorithm.

B. The semi-inclusive gluon jet function
The calculation of the semi-inclusive gluon jet function J g (z, E J ) is very similar to that for the quark jet case. As illustrated in Fig. 2, the semi-inclusive gluon jet function receives four contributions, including both g → gg and g → qq splittings. For the case that both final-state partons are inside the jet, the semi-inclusive gluon jet function can be written as where n f represents for the number of flavors of the final-state quark, andP qg (x, ) andP gg (x, ) are given by Eqs. (13) and (14), respectively. In the situation when both the final state partons are inside the jet, the constraint is shown in Eq. (19). Apply this constraint to Eq. (39), we obtain the semi-inclusive gluon jet function where Expanding to O( ), we obtain Insert the above expressions into Eq. (40) and perform the -expansion, we find We also need to consider the situation that only one of the partons forms the jet and the corresponding Feynman diagrams are shown in Fig. 2 (C) and (D). Summing these two diagrams together, we obtain J g→g(g)+q(q) (z, E J ) = α s 2π With the constraint from the J Finally, adding all contributions together, we obtain the following expression for the semi-inclusive gluon jet function where P gg (z) and P qg (z) are the standard Altarelli-Parisi splitting functions defined in Eqs. (35) and (36), respectively. Similar to the case for the quark jet, the double pole 1/ 2 and the double logarithms L 2 cancel, and we are left with only a single pole 1/ and a single logarithm L.
Using the same procedure, we can also derive the semi-inclusive gluon jet function for J (II) E T algorithm. The final expression is given by where the last two rows in the above equation are algorithm dependent, while all the rest terms are algorithm independent.

C. The RG evolution for semi-inclusive jet functions
The bare quark and gluon jet functions J q,g (z, E J ) are divergent as they contain poles of 1/ , which need to be renormalized. We follow the standard procedure and define the renormalized semi-inclusive jet functions J q,g (z, E J , µ) as follows where Z ij is the renormalization matrix. Taking derivative with respect to µ on both sides of the above equation, we can obtain the renormalization-group equation for the semi-inclusive jet function where γ J ij is anomalous dimension related to the renormalization matrix with the inverse of the renormalization factor (Z) −1 ik defined as At LO, the renormalization matrix Z (0) ij is purly δ-function, which is given by Z from which we obtain the anomalous dimension Thus the renormalization group equations for the renormalized siJFs is just the time-like DGLAP evolution equation for the usual fragmentation functions The divergence of Z (1) ij cancel exactly with the divergence in the bare semi-inclusive jet functions, and eventually leads to finite renormalized semi-inclusive jet functions. For the maximized algorithm with n = 1, i.e. J (I) E T , we have the following expressions for the renormalized siJFs at NLO On the other hand, for the maximized algorithm with n = 2, i.e. J

(II)
E T , the siJFs at NLO can be written as follows It is important to point out that the natural scale µ J for the siJFs can be derived from Eq. (30), Thus by solving the DGLAP evolution equation Eq. (55) and thus evolving the siJFs from its natural scale µ J to the hard scattering scale µ ∼ E J , we can resum large logarithms of the jet size parameter β. This is similar to the small jet radius ln R resummation developed in [25]. It might be worthwhile to point out that the momentum fraction z also plays some role in this evolution, and it would be valuable to study numerically how important such an effect is. Finally, it is interesting to point out that the above renormalized semi-inclusive jet functions satisfy the following momentum sum rule [26,45] This momentum conservation rule provides us an alternative way to check our results. Note that the above semiinclusive jet functions were first derived in Ref. [27] via standard pQCD techniques 2 .

IV. THE SEMI-INCLUSIVE FRAGMENTING JET FUNCTIONS
We now turn to the evaluation of jet substructure with maximized jet algorithms. In this section, we focus on the so-called semi-inclusive fragmenting jet functions (siFJFs) as initially introduced in Ref. [28]. The siFJFs describe the longitudinal momentum distribution of hadrons inside the jet. Comparing to the siJFs, we need one more variable, ω h ≈ 2E h , to represent for the light-cone momentum of the observed hadron. Accordingly, z h = E h /E J denotes the fraction of the jet energy carried by the observed hadron. In this section, we will first introduce the definition of siFJFs in SCET and we then present their NLO calculations for both the J E T algorithm as the procedure is very similar. Similar to the siJFs, one can define the siFJFs for quark and gluon jets in terms of gauge invariant quark and gluon fields in SCET [28] where (d − 2) is the number of polarizations for initial gluons in d-dimension. The state |(Jh)X represents the final-state identified hadron (h) inside the jet J as denoted by (Jh), X stands for unobserved particles. On the other hand, ω, E J and z have exactly the same meaning as those in siJFs. Different from the semi-inclusive jet functions as discussed in the previous section, the siFJFs involve hadronic scale, thus contains non-perturbative information and is in principle not calculable in pQCD. However, the matching coefficients between siFJFs and standard fragmentation functions can be determined perturbatively, thus one can simply replace the hadronic states in Eq. (62) by the corresponding partonic states, which allows us to use the methodology of pQCD. In the following, we will show explicitly the calculation of G j i with i the initial parton and j the fragmenting parton. Similar to siJFs, the siFJFs at LO involve only delta functions The last two delta functions correspond to the case that the total energy of the initiating parton is transfered to the jet, and the fragmenting parton carries the whole energy of the jet.

A. Fragmenting jet functions at NLO
At NLO, the initiating parton splits into two partons, which leads to two contributions for siFJFs: one is for the case when both partons are inside the jet and another case is when only one parton is inside the jet. In the situation that both partons are inside the jet, the corresponding Feynman diagrams for quark and gluon initiated jet are shown in Figs. 1(A) and Fig. 2(A), respectively. In this case, all the initial parton energy E stays inside the jet, thus z = E J /E = 1. Different with the semi-inclusive jet functions, we now observe a particular fragmenting parton inside the jet, where the fragmenting parton carries only part of the jet energy as characterized by z h = E h /E J < 1. In MS scheme, the one-loop bare siFJFs can be written as [28] G jk,(1) In comparison with the siJFs in Eq. (10), the above expression is only different by a factor δ(x − z h ), since we now measure the momentum fraction carried by the final-state parton. Similar to that in semi-inclusive jet functions, "jk" indicates the situation that both splitted partons j and k remain inside the jet. The functionsP ji (x, ) are given by Eqs. (11), (12), (13), and (14).
To proceed with the NLO calculation, we implement the jet algorithm constraint, given already in Eq. (19), and at the same time integrate x with the help of the δ(x − z h ) function in Eq. (64), we can derive the final result. It turns out that the result depends on the value of z h . When z h ≤ 1/2, we have While for z h ≥ 1/2, we have the following expression Substituting these two integrals into Eq. (64), we obtain the following result for the case when both partons are inside the jet Now we turn to the case that only one parton is inside the jet. The corresponding diagrams are shown in Fig. 1  (B) and (C) for the quark-initiated jet and Fig. 2 (C) and (D) for the gluon-initiated jet. In this case, the final-state fragmenting parton forms the jet, thus we have z < 1. On the other hand, within the jet, there is only one parton at NLO which is eventually converted to the fragmenting parton. This leads to an overall delta function ensuring z h = 1, i.e., all the jet energy is translated into the fragmenting parton energy. Performing the evaluation of diagrams Fig. 1 (B), (C) and Fig. 2 (C), (D), we obtain the bare siFJFs when only parton j forms the jet where the superscript "j(k)" indicates that parton j is inside the jet while k exits the jet. We can now implement the jet algorithm constraint Eq. (27) in the m 2 integration, and we obtain for the J (I) With the above result at hand, we can write the second contribution of the bare siFJFs as follows Summing the two contributions shown in Eqs. (67) and (70), we obtain the following results for the bare siFJFs at NLO Performing -expansion, we obtain the explicit expression of the siFJFs for the J (I) G q,(I) g,bare (z, z h , E J ) = α s 2π − 1 − L P qg (z h ) δ(1 − z) + α s 2π 1 + L P qg (z)δ (1 − z h ) where the logarithm L is defined in Eq. (30), and the Altarelli-Parisi splitting functions are defined in Eqs. (33), (34), (35) and (36).
The results for the bare siFJFs at NLO for the J (II) E T algorithm can be obtained similarly, and they are given by G q,(II) g,bare (z, z h , E J ) = α s 2π − 1 − L P qg (z h ) δ(1 − z) + α s 2π 1 + L P qg (z)δ (1 − z h ) whereP qq (z) andP gg (z) are the Altarelli-Parisi splitting functions defined in Eqs. (33) and (36), but with the δ(1−z)term removed. For later convenience, we also defineP gq (z) = P gq (z) andP qg (z) = P qg (z). Notice that the pole terms are universal for different jet algorithms. This is why all these semi-inclusive jet functions follow the same time-like DGLAP evolution equations [28].

B. Renormalization and matching
From the final expressions of the bare siFJFs G j i (z, z h , E J ), single pole terms still remain, which include both infrared (IR) poles as identified in association with P ji (z h )δ(1 − z), and the ultraviolet (UV) poles that multiplied by P ji (z)δ(1 − z h ). The standard procedure to subtract the UV poles is realized by the definition of renormalized