Inclusive heavy flavor jet production with semi-inclusive jet functions: from proton to heavy-ion collisions

The past several years have witnessed important developments in the QCD theory of jet production and jet substructure in hadronic collisions. In the framework of soft-collinear effective theory, semi-inclusive jet functions and semi-inclusive fragmenting jet functions have allowed us to combine higher order calculations with resummation of potentially large logarithms of the jet radius, $\ln R$. Very recently, the semi-inclusive jet functions for partons fragmenting into heavy flavor jets were computed by Dai, Kim and Leibovich. In this paper we show how the formalism can be extended to c-jet and b-jet production in heavy ion collisions. The semi-inclusive jet functions for heavy flavor jets in a QCD medium are evaluated up to the next-to-leading order in $\alpha_s$ and first order in opacity. For phenomenological applications, we also consider the inclusion of the cold nuclear matter effects and the jet energy dissipation due to collisional interactions in matter. We present the numerical predictions for the cross sections and the corresponding nuclear modification factors in proton-nucleus and nucleus-nucleus collisions and compare our results to data from the Large Hadron Collider.


Introduction
Jet production is one of the cornerstone perturbative Quantum Chromodynamics (QCD) processes in hadronic collisions [1]. It is characterized by large cross sections and has, thus, been measured with unprecedented precision in comparison to other high energy processes. Observables related to jets have served as precision tests of QCD and as tools to search for new physics. In this work, we restrict our discussion on the inclusive heavy flavor-tagged jet production in proton and heavy-ion collisions, which has been measured at the Large Hadron Collider (LHC) [2][3][4][5] and will be measured at the Relativistic Heavy Ion Collider (RHIC) [6] in the near future.
Inclusive jet production is a multiscale problem in both the proton-proton (p+p) and heavy ion (p+A, A+A) collisions. The differential jet cross section versus transverse momentum p T and rapidity η in hadronic collisions is factorized as the convolution of the parton distribution functions (PDFs), the hard kernels, and the semi-inclusive jet functions (SiJFs), or the fragmentation functions to jet [7]: factorization scale µ. The renormalization group (RG) equations are the usual time-like DGLAP evolution equations. The corresponding expressions for the hard part can be found in Refs. [8,9]. The SiJFs have been calculated up to next-to-leading order (NLO), using soft-collinear effective theory (SCET) [10][11][12][13][14] techniques, for light jet [15,16] and very recently for heavy flavor jets [17]. This factorization formula has been used to predict the inclusive and heavy flavor-tagged jet cross sections [7,[15][16][17][18][19]. The first application to heavy-ion collisions for light jet was given by Ref. [20]. In this paper we will extend the formalism to the case of inclusive c-jet and b-jet production in reactions with nuclei at ultra-relativistic energies.
In heavy-ion collisions, one expects the formation of a new deconfined state of matter, known as the quark-gluon plasma (QGP). Energetic parton propagation and shower formation in this strongly interacting matter alter light particle, heavy flavor, and jet observables in heavy-ion relative to proton collisions -a phenomenon known as the jet quenching effect. Studies of jet quenching, especially for the heavy flavor-tagged jets, can reveal the fundamental thermodynamic and transport properties of the QGP. For an experimental perspective on this issue see [21] and references therein. It is well-understood that the jet quenching effects depend on the flavor of the fragmenting parton which, together with heavy quark mass effects, can be studied by light jet and heavy flavor tagged jet (c-jet and b-jet) observables in heavy ion collisions. Mass effects are expected to play a significant role in the small and intermediate transverse momentum regions and are, of course, most pronounced for b-jets [22][23][24]. They can also change the relative importance of radiative and collisional processes in the QGP for the modification of jet cross sections and jet substructure.
Heavy flavor studies are, therefore, central to high-energy nuclear physics since they provide new avenues to explore QCD in the strongly-interacting matter and new diagnostics of its transport properties [25][26][27]. These efforts have been directed toward open heavy flavor, namely D-meson and B-meson production, and quarkonia. Investigation of heavyflavor tagged jet production in heavy ion collisions has been somewhat limited thus far. Inclusive b-jets were first studied using energy the traditional energy loss approach [22] and later with the means of a partonic transport model [28]. Strategies to suppress the contribution from gluon splitting and enhance the fraction of prompt b-jets via energetic photon or B-meson coincident measurements opposite to the b-jet were performed in Ref. [29]. This was generalized to back-to-back b-jets [24,30], moreover heavy flavor dijet mass distributions in heavy ion collisions were also computed [24]. The derivation of full in-medium splitting functions for heavy quarks [31,32] allows us to bridge the gap between high energy and heavy ion theory of jet and heavy flavor production in hadronic and nuclear collisions. Their first application to b-tagged jet substructure [23] has already produced novel results, namely a unique inversion of the mass hierarchy of jet quenching effects as manifested in the stronger modification of the b-jet momentum sharing distributions in comparison to the ones for light jets. These effects are driven by the heavy quark mass and are measurable at the LHC and the future sPHENIX experiment at RHIC [6]. It is, thus, important to pursue improved description of heavy flavor-tagged jet production in heavy-ion collisions using in-medium higher order calculations and resummation. On the experimental side, data in inclusive b-jet production [3] and back-to-back momentum imbalance distributions [33] exist. Measurements of c-jets in lead-lead (Pb+Pb) collisions at the LHC are expected in the near future [34].
With this in mind, in this paper we will present a calculation of the inclusive charm jet and bottom jet cross sections in heavy-ion collisions using the factorization Eq. (1.1) based on semi-inclusive fragmenting jet functions for heavy flavor. We will demonstrate that the final-state medium-induced corrections can be incorporated by modifying the SiJFs, while the short-distance hard part remains the same as p+p collisions. These corrections arise from the emergence of in-medium parton showers as the jet evolving in the QCD medium. We will study these radiative processes with the help of the medium induced splitting functions, which were calculated up to the first order in opacity in the framework of SCET with Glauber gluons (SCET G ) [35,36] and finite mass effects (SCET M,G ) [31]. The in-medium splitting functions capture the full collinear dynamics of energetic parton evolution in a QCD medium. Recent developments based on the formalism of lightcone wavefunctions [32] have allowed us to compute collinear parton branching in QCD media to any order in opacity. For the purpose of this paper, however, we will restrict ourselves to the first order in opacity results where numerical evaluation of the splitting kernels exists. Unlike the vacuum case, in the environment of strongly interacting matter jets can dissipate their energy due to collisional interactions with the medium quasiparticles. The collisional energy loss rate can be obtained, for example, from the divergence of the energy-momentum tensor of the medium induced by the color current generated by the jet [37]. This work is the first study to include this effect in the SiJF formalism. Last but not least, we also consider the cold nuclear matter (CNM) and isospin effects [38][39][40].
The rest of our paper is organized as follows: in Section 2 we discuss the definition of the heavy flavor jet functions in the vacuum. We derive the nuclear modifications of the jet functions in heavy-ion collisions and discuss other relevant nuclear effects. In Section 3 we present our phenomenological results for c-jet and b-jet cross sections in proton and heavy-ion collisions. Finally, we conclude in Section 4.

Semi-inclusive jet functions for heavy flavor
In this section, we discuss in detail the definition of the heavy quark-tagged SiJFs. For completeness, we briefly recall some analytical results from Ref. [17] for the vacuum SiJFs. We then extend these SiJFs to the case of heavy-ion collisions in the perturbative theory.

Jet Functions in vacuum
Soft-collinear effective theory can be generalized to include finite quark masses [41,42] and is often labeled SCET M . Within SCET M , the heavy flavor SiJFs are obtained by calculating the real and virtual contributions inside the jet cone, and the real contributions outside the jet cone which are dependent on the jet algorithm. The NLO SiJFs for any parton to produce a heavy flavor-tagged jet, including the heavy quark mass effects, can be found in Ref. [17]. After renormalization they are given by the following expressions where the flavor singlet J s = J Q + JQ. The function J J Q /Q has both LO and NLO contributions in QCD, while J Js/g starts at NLO. The functions f and g have integral representation and are defined in Ref. [17]. It was argued that, since heavy quark mass does not affect the ultraviolet (UV) behavior of diagrams, the SiJFs evolve according to DGLAP-like equations similar to the ones for light SiJFs. The renormalization group equations read Here, P ij is usual Altarelli-Parisi splitting functions. The evolution equation is solved to leading logarithmic (LL) accuracy using the Mellin moment space approach developed in Ref. [43].
In the jet function J Js/g (z, m, p T R, µ) the contribution proportional to δ(1 − z) comes from the g → QQ splitting inside a jet. Starting only at NLO in QCD, it can be written as the integration of the jet fragmentation function As expected, for heavy flavor a new logarithmic term ln p T R/m arises. When m p T R p T , in addition to ln R the logarithmic term ln p T R/m needs to be resummed. This can be achieved through the jet fragmentation function factorization formula found in Refs. [16,17,44]. In our case, up to NLO, M in−jet g→QQ can be written as whereK l/g is the integrated perturbative kernel at the jet typical scale p T R, whileD Q/l is the integrated parton fragmentation function from parton l to parton Q at the scale of the quark mass. The logarithmic term ln p T R/m can be resummed to all order at the leadinglogarithmic accuracy by running D Q/l (m, µ) from m to p T R analytically. For further details, we direct the reader to Ref. [17].

Medium corrections
In perturbative QCD, both of the LO and NLO vacuum SiJFs receive medium modifications, as illustrated in Fig. 1. At the LO in α s only in-medium jet energy dissipation due collisional interactions is allowed. At NLO in α s there is vacuum radiation along with corrections from the medium-induced parton shower. In the soft gluon approximation, these would correspond to the traditional radiative energy loss in the QCD medium. Here, however, we will calculate the medium-modified jet function up to the NLO in QCD and the first order in opacity with the help of the full in-medium splitting functions [31,32]. We can express it as In this paper we consider for the first time the effects of jet energy dissipation through collisional interactions in the QCD medium in the fragmenting jet function formalism. We will limit our results to the LO contribution J med,(0) J Q /i and defer the interplay between collisional and radiative corrections to future work. Collisional energy loss effects can be included as a transverse momentum shift of the final state jet which, to LO in perturbative QCD, is identified with the final-state hard parton. The corresponding medium modification to the cross section for the Q jet production can be written as where δ iQ is the Kronecker delta symbol and δp i T is the average energy loss for energetic parton i moving through the hot QCD medium. As can be seen from Fig. 1, to include only the medium correction here and avoid double counting one should subtract the vacuum contribution.
We calculate collisional and radiative in-medium effects consistently in a QGP background simulated by 2+1-dimensional viscous event-by-event hydrodynamics [45]. The δp i T in Eq. (2.6) is obtained using the collisional energy loss rate derived from an operator definition [37]. We note that in heavy ion collisions we work with small radius jets. On the other hand, the dissipated jet energy is carried away by hydrodynamic medium excitations at angles O(1) relative to the direction of jet propagation [46]. Similar results were obtained in Monte-Carlo simulations of energy flow distributions inside and outside of jets in Pb+Pb collisions at the LHC [47]. Thus, we consider this energy fully lost from the point of view of jet production and the jet cross section modification can be equivalently expressed in terms of the LO in-medium jet function as where only the diagonal part (i = Q) is effected by the QGP medium.
Following the strategy outlined in Ref. [20], the modification of the NLO SiJFs is defined as the sum of the virtual and real corrections, which can be calculated using the in-medium splitting function derived from SCET M,G . The in-medium splitting functions are expressed as integrals over the medium size and the transverse momenta transferred by the Glauber gluon [31,32]. Those integrations can only be performed numerically and we use the same QGP background [45] to obtain grids for all in-medium splitting functions to first order in opacity. The medium modification of the SiJFs can be evaluated from those grids, where a UV cut-off µ rather than the dimensional regularization scheme is used to regularize the UV poles when the transverse momentum of the vacuum radiation becomes infinity. A similar approach was employed in Ref. [20] to construct the SiJFs for massless quarks and gluons. The NLO medium correction to the Q → J Q SiJF is defined as (2.8) In the above equation the first line corresponds to the contribution with a radiation outside of the jet cone, while the second line represents the combination of the real radiation inside the jet cone and the virtual loop corrections. The singularity that arises when z → 1 is regularized properly by the plus distribution function after combining all the corrections.
The medium correction to the channel g → J s = J Q + JQ is where the first and second lines are the contributions with the real radiation outside and inside the jet cone, respectively.
In the application of in-medium splitting functions we use extensively sum rules. Let us take for example the momentum sum rule for the gluon initiated splitting (2.10) In this formulation i runs over all quark and antiquark flavors, therefor we don't have an explicit 2n f factor. Since the sum rules are satisfied by the vacuum part separately, they are also satisfied by the medium-induced splitting kernels. Up to the NLO, in our previous works [15,39] we have implemented the following normalization for the splitting functions x dxP med Qg (x, q ⊥ ) = 0 (2.12) Using the symmetry P Qg (x, q ⊥ ) = P Qg (1 − x, q ⊥ ) we have The equation above implies that there is no additional production of open heavy flavor per nucleon-nucleon collision in heavy ion reactions. This is consistent with experimental measurements [48]. Taking this constraint into account, the function J med,(1) 14) The full in-medium SiJFs are defined as where the vacuum contributions are calculated at the LL accuracy, while only the fixed-order medium corrections are included consistently. In this case we choose the cut-off scale µ as the jet's transverse momentum. In principle the full in-medium SiJFs obey a DGLAP-like evolution in Eq. (2.2) and one can even introduce the medium-induced splitting functions in the DGLAP kernel to fully consider the jet evolution in the QCD medium. We will leave this for a future study. Last but not least, let us mention the cold nuclear matter effects. We evaluate these effects from the multiple elastic, inelastic, and coherent scattering processes in a large nucleus. At the high transverse momenta that we consider, only CNM energy loss effects might play a role [38]. These are a generalization of the Bertsch-Gunion bremsstrahlung [49] to a large nucleus and are computed as in Ref. [50]. They are implemented as shifts in the lightcone momentum fraction of the incident parton in the PDFs [38], which are dependent on its flavor For the case of nuclei we include with relevant weights Z and A − Z the proton PDFs and neutron PDFs, where the latter are constructed using isospin symmetry. Further details relevant to phenomenology can be found in Refs. [39,40].

Numerical results
In this section we present our numerical predictions for inclusive c-jet and b-jet production in hadronic collisions. For our work we choose the CT14NLO PDF sets [51]. The hard part in the factorized cross section is calculated at NLO with massless b-and c-quarks, while the mass effect is included in the c-and b-jet SiJFs. The UV cut-off and the scale of α s in the medium corrections to the jet functions are set to be the factorization scale µ. The default factorization and jet scales are chosen to be µ = p T and µ J = p T R, respectively. The uncertainties are evaluated by varying µ and µ J by a factor of 2 independently. The coupling between the jet and the medium appears in the modification to the SiJFs is set as g = 2. This choice is consistent with many other applications of the medium induced splitting functions. Before we move on to jet production in heavy-ion collisions, we must address inclusive b-jet and c-jet production in p+p collisions. These cross sections set the baseline relative to which cross section modifications with nuclei can be detected. Figure 2 presents the comparison between our theoretical results for the inclusive b-jet cross section (left) and the fraction of b-jets to inclusive light jets (right) for different rapidity intervals as a function of jet transverse momentum p T . The colliding system is p+p with √ s NN =7 TeV and jet reconstruction parameter R=0.5. The ln R resummed cross sections for b-jet changes the NLO predictions by O(10%), but we find larger scale uncertainties. The NLO and NLO+LL p T distributions presented here are consistent with the predictions from Ref. [17] and the experimental measurements [2]. For the b-jet fraction, the difference between NLO+LL and NLO visible in the right panel of Figure 2 can be traced also to the differences in the inclusive jet cross section. The ln R resummation reduces the inclusive jet cross more significantly. The NLO+LL predictions agree very well with the data. We now move on to c-jets and their cross sections on p+p collisions are shown in Fig. 3 for center-of-mass energies 5.02 TeV (left) and 2.76 TeV (right), respectively. Similar to b-jet cross section, the ln R resummation changes the c-jet cross section by about 10% in a p T -dependent fashion, but with larger theoretical uncertainties. We also note that in comparison to b-jets the spectrum appears stiffer, and our calculation agrees better with the experimental measurements at lower p T . This is also reflected in the right panel of Fig. 3 Again, the NLO+LL c-jet fractions agree better with CMS measurements [5] when compared to NLO ones for both collision energies.
The high energy jet production in proton-nucleus collisions can place constraints on the cold nuclear matter effects. Early ATLAS and PHENIX measurements suggested that the suppression of inclusive jet cross sections, especially at high p T and in central p+Pb collisions 2 , can be large [52,53]. The ALICE collaboration as further studied the event activity in semi-inclusive hadron-jet distrbutions [54]. In minimum bias collisions, the modification of jet cross sections, if any, is smaller. Within the statistical and systematic uncertainties many measurements are consistent with a range of possibilities -from no  nuclear effects to ±10% cross section modification.  The rapidity cut in the center-of-mass frame is |η CM | < 1.5 for c-jets and |η CM | < 0.5 for b-jets.
We can use proton-lead collisions at the LHC to check the validity of our theoretical model. At the high jet transverse momenta under investigation, we take into account the initial-state cold nuclear matter energy loss [38] In Fig. 4, we compare our calculated nuclear modification factor R pA for heavy flavor tagged-jets to the CMS experimental measurements [4,5] in 5.02 TeV p+Pb collisions. The scale dependence of the cross sections in p+Pb collisions is almost the same as the one in p+p collisions. Therefore, the NLO+LL ratio R pA change very little with scale variation. Furthermore there is not an obvious difference between the predicted c-jet and b-jet R pA , which is between 0.9 and 0.95 for the jet transverse momentum in the range of 50 GeV < p T < 400 GeV. The NLO+LL jet cross section modification R pA for c-jet, displayed in the top panel, describes the CMS data [5] very well. In the bottom panel, the prediction is on the lower edge of the experimental error bar [4] and cannot describe the fluctuation of the data around p T ∼ 150 GeV. Given large uncertainties, however, there is no significant deviation between predictions and the measurements. It is worth noticing that in Pb+Pb collisions cold nuclear matter effects will be amplified when compared to p+Pb collisions because there is one more nucleus in the initial state. To proceed to nucleus-nucleus reactions, we include final-state interactions. In Fig. 5 we show the theoretical model predictions for the in-medium suppression factor R AA for bjets with |η| < 2, reconstructed with the anti-k T algorithm with R=0.3 at √ s NN =5.02 TeV.
The effect of the medium-induced parton shower is represent by the green band. Compared to the light jet results from Ref. [20], the effect of in-medium radiative processes on b-jets is noticeably smaller. The reason for that lies in the strength of the medium-induced parton shower contribution to b-jet production, which is predominantly proportional to the second Casimir in the color representation of the parent parton and is smaller for quark-initiated jets. The difference between the blue and green bands in Fig. 5 represents the jet energy dissipation in the medium due to collisional processes. It is of the same order as the mediuminduced out-of cone radiation and is more important when the jet transverse momentum is small. The CNM effects, represented by the difference between the blue and red bands, are more important in the high energy regime, especially when the attenuation due to final-state effects become smaller. As expected, they are about twice larger than the R pA in Fig. 4. The full nuclear modification factor R AA is about 0.3 for p T ∼ 50 GeV and about 0.6 for p T ∼ 250 GeV. Even though the b-jet R AA is found to be qualitatively consistent with that of inclusive jets from measurements with the same collision energy [55,56], the underlying physics might be different. In Fig. 6 we compare our numerical calculations to the measurements [3] of b-jet production in p+p and Pb+Pb collisions with √ s NN = 2.76 TeV. The left plot shows the inclusive b-jet p T distributions, where the cross sections in heavy-ion collisions with centrality 0-100%, 0-10% and 30-50% are scaled by powers of 10 for visibility. The right plot presents the nuclear modification factor R AA with the same settings. In general R AA decreases (larger suppression) with increasing collision centrality (toward head-on nuclear collisions). From the data, the attenuation factor R AA seems less dependent on the centrality when compared to the well-known light jet modification. The predictions agree very well with the data for both the inclusive cross sections and the nuclear modification factors. TeV. Since the CNM effects and collisional energy loss do not depend on the jet radius for small R, consequently, the radius dependence of R AA is reduced relative to the case where only in-medium radiative processes contribute. For example, in the earlier study of inclusive light jet modification [20] collisional energy losses were not included and, consequently, the calculated radius dependence was larger. In p+p collisions, the ratio of the cross section with R = 0.4 to that with R = 0.3 practically does not depend on jet p T . On the other hand, there is small dependence on jet p T in Pb+Pb collisions. It can be seen from Fig. 7 that the smaller radius jet tends to dissipate more energy in the medium. There is no significant difference between the c-jet and b-jet due to the high transverse momentum where heavy quark mass effects are small to negligible. In the small transverse momentum region where the uncertainty of our calculations is relatively large, the contribution from higher orders in opacity [32] might play an important role. which can be included either by calculating higher order splitting functions [57] or by solving the renormalization group equation with the medium-corrected DGLAP kernel [39]. This can be one of the interesting applications of this framework in the future.

Conclusions
In this work, we presented a formalism to study heavy flavor jet production in hadronic and heavy-ion collisions using the heavy flavor semi-inclusive jet functions. This approach relies on hard-collinear factorization, where the cross section is expressed as the convolution of the PDFs, the hard kernel, and jet functions. For light-flavor jets, similar formalism has been applied to the inclusive jet production and jet substructure yielding gains in the accuracy of theoretical predictions. It has also helped place jet calculations in heavy ion collisions on the firmer theoretical ground. With this in mind, we presented the first calculation of heavy flavor jets in heavy ion collisions using the heavy flavor semi-inclusive jet functions technique. For jets produced in hadronic collisions, the important ln R terms were resummed up to leading logarithmic accuracy using a DGLAP-like evolution of the vacuum jet fragmenting functions. In heavy ion collisions the medium corrections are included consistently up to next-to-leading order in QCD and first order in opacity. This limits the applicability of our approach at small jet transverse momenta, where such medium corrections can become large and need to also be numerically resummed. We defer this study to future work.
We compared the theoretical cross section results for inclusive c-jet and b-jet production to the experimental data from p+p and Pb+Pb collisions and found very good agreement between data from the Large Hadron Collider and theory. For the more complex heavy ion collisions, we did include cold nuclear matter effects and, for the first time, collisional energy losses in the jet fragmentation function formalism. These were shown to play an important role in the overall suppression of the heavy flavor jet cross sections. We further presented our calculations of c-jet and b-jet cross sections and their modification R pA in the cold nuclear matter. Comparison to data at √ s NN =5.02 TeV demonstrates that, while experimental error bars are large, the CNM effects employed in this calculation are compatible with measurement. We finally showed the predictions of R AA for the production and attenuation of c-jets and b-jets of different radii in the highest center-of-mass energy Pb+Pb collisions.
In future, we plan to further investigate heavy flavor-tagged jet substructure observables. Energy correlators inside jets [58] are being evaluated in the framework of SCET [59]. It would be interesting to calculate them in heavy ion collisions, and for such observables inclusion of higher orders-in-opacity corrections in the medium [32] might be important. We further expect that resummation if the in-medium branchings will improve the predictions in the small-p T region, allowing access to the kinematic domain where mass effects on the heavy flavor jet production and propagation in a dense QCD are most pronounced and can lead to novel phenomena [23]. With a proper extension, we expect that this formalism will be well-suited to investigate jet shapes in hadronic [60][61][62] and heavy ion collisions [63]. Other observables of interest are collinear [64] and transverse momentum fragmentation functions [65] for a hadron production inside jets, extended to heavy flavor 3 . We finally remark that our approach is also applicable to heavy flavor jet production at a future electron-ion collider.