The AMPT model

The AMPT model with the string melting mechanism, which is widely used in relativistic heavy-ion collision studies, comprises four main stages [60, 61]: initial conditions, parton cascade, hadronization, and hadronic rescatterings.

(1) Initial conditions. The heavy ion jet interaction generator (HIJING) model [62, 63] provides the initial conditions for pp, pA, and AA collisions. Within the string melting mechanism, the primary interactions comprise two components: a soft contribution modeled by Lund string fragmentation [64-66] and a hard contribution from minijet production. The differential cross section for minijet production is computed using the perturbative QCD factorization as follows:(1)where p_T is the transverse momentum of the produced minijet parton, y₁ and y₂ denote the rapidities of the final produced partons c and d, the factor K accounts for higher-order corrections beyond leading order (LO), and x₁, x₂ are the momentum fractions of the incoming partons a and b. The parton distribution functions (PDFs) fa(x₁, Q²) and fb(x₂, Q²) follow the Duke-Owens parametrization [67] with the factorization scale Q². The parton cross section is determined by the LO matrix element squared , expressed in terms of the Mandelstam variables , , and . A jet-triggering technique is employed in HIJING to generate dijet events, incorporating hard scattering processes such as , , , , , , , and [64, 68].

(2) Parton cascade. The Zhang’s parton cascade (ZPC) model describes partonic interactions via two-body elastic scatterings [69]. The parton cross section for gluon-gluon scattering is evaluated at LO pQCD as(2)where α_s is the strong coupling constant and μ is the Debye screening mass. Varying μ allows for the adjustment of the parton cross section in simulations.

(3) Hadronization. Hadronization is modeled using the quark coalescence mechanism [60], which recombines the nearest two or three partons into mesons and baryons, without considering their relative momenta. Three momentum conservations are satisfied in the coalescence process, and the hadron species are determined by the flavor and invariant mass of the coalescing partons.

(4) Hadronic rescatterings. The relativistic transport (ART) model [70] simulates resonance decays and hadronic reactions, including both elastic and inelastic scatterings for baryon-baryon, baryon-meson, and meson-meson interactions.

In this study, the AMPT model with the string melting mechanism was used to simulate pp and PbPb collisions at TeV. The parton cross section was set to 3 mb, in line with previous studies that have successfully described the collective flow observed in relativistic heavy-ion collisions [71-75]. The AMPT model incorporating jet-medium interactions has been widely utilized in the investigation of various jet observables, including dijet asymmetry [37], γ-jet imbalance [76], jet fragmentation functions [77-80], jet shape [81], jet anisotropies [82], jet transport coefficients [83], the redistribution of lost energy [84-87], and jet mass [59]. For comparison, a parton cross section of 0 mb was also employed to provide a baseline scenario without jet-medium interactions.

Jet reconstruction

To measure the jet substructure observables, the jets were first reconstructed using the anti-k_t algorithm [88] with a radius parameter R = 0.4, as implemented in the FastJet package [89]. In this analysis, jets are required to have pseudorapidity |η_jet| < 1.3. To facilitate a direct comparison with the CMS measurements [43, 48], the masses of the final-state particles were adjusted such that neutral hadrons were treated as massless, whereas charged hadrons were assigned the charged pion mass, in accordance with the experimental setup used by CMS. This treatment has a non-negligible effect on the groomed jet mass distribution and is essential for a consistent comparison between the simulation and data.

In PbPb collisions, the constituent subtraction method [90] is employed to estimate the background, which is characterized by the background density ρ and background mass density ρ_m [89, 91]. The estimation is based on clustering particles using the k_t algorithm [92] with R = 0.4. Massless ghost particles with an area of 0.005 were added to the y-ϕ plane for the event. To reduce contamination from hard jet fragments, the two kt clusters with the highest transverse momentum are excluded. Constituent subtraction is performed on a particle-by-particle basis, enabling the correction of both the jet four-momentum and its substructure. This method was applied to PbPb events generated with the AMPT model using a jet-triggering technique. To reduce the possible artifacts introduced by background subtraction in PbPb events, a smearing procedure similar to that used in the CMS experiment was adopted. Specifically, the background from PbPb collisions (without a jet trigger) is embedded into pp collisions with a jet trigger to produce the so-called “smeared” pp events. The constituent subtraction procedure in PbPb collisions was applied to smeared pp events to maintain consistency.

Soft Drop

In addition, jet grooming is performed using the Soft Drop (SD) algorithm [93], which removes the soft components and isolates the hard components of the jet, thereby enhancing the sensitivity to medium-induced modifications. The procedure reclusters jet constituents using the Cambridge-Aachen (C/A) algorithm [94] to form a pairwise clustering tree, followed by recursive declustering. At each declustering step, the jet is separated into two subjets by reversing the last C/A clustering stage. The subjets are required to satisfy the Soft Drop condition:(3)where p_T,1 and p_T,2 are the transverse momenta of the subjets, is the distance between the subjets in the y-ϕ plane, and R = 0.4 is the jet radius. The parameters z_cut and β control the grooming procedure. If the Soft Drop condition is satisfied, the corresponding pair of subjets is retained as the final groomed jet. Otherwise, the subjet with a higher p_T is further declustered, and the procedure is repeated until the condition is satisfied. The final groomed subjets were used to measure the jet substructure properties.

In this study, groomed jet observables, including the jet splitting momentum fraction z_g [43] and the ratio of the groomed jet mass to the ungroomed jet transverse momentum M_g / p_T,jet [48], were calculated using the AMPT model with a jet-triggering technique. The jet splitting momentum fraction z_g is defined as(4)where p_T,1 and p_T,2 are the transverse momenta of the two subjets. This definition is identical to that on the left side of Eq. (3) when the subjets satisfy the Soft Drop condition. The groomed jet mass M_g is defined as(5)where Ei and are the energy and three-momentum vectors of the subjets, respectively. To reconstruct the jet substructure, two sets of Soft Drop grooming parameters are employed, each probing distinct regions of the subjet phase space in the Lund plane [95, 96]. The first setup, with (z_cut = 0.1 and β = 0.0), imposes a grooming condition based solely on the energy fraction between subjets to capture both the jet core and peripheral modification. It is utilized for analyzing both z_g and M_g / p_T,jet observables. The second setup, which employs a stronger grooming condition (z_cut = 0.5, β = 1.5), reduces the contribution from large-angle subjets, thereby enhancing the sensitivity to the jet core region. This configuration is employed exclusively in the M_g / p_T,jet analysis. An additional angular requirement of ΔR₁₂ > 0.1 is imposed across all measurements to suppress unphysical contributions, following the CMS experimental procedure [43, 48]. Furthermore, the z_g and M_g / p_T,jet distributions were normalized by the number of groomed jets (N_g,jet). We have verified that treating jet constituents as either massive or massless results in very similar z_g and M_g / p_T,jet distributions. To facilitate a more direct comparison between the two grooming configurations, Fig. 1 presents the joint distributions of z_g and ΔR₁₂ for jets that satisfy the Soft Drop condition defined in Eq. (3). The distributions include an angular cut of ΔR₁₂ > 0.1 and are normalized by . The comparison is performed within the jet transverse momentum range of 140 < p_T,jet < 160 GeV in pp collisions at TeV using the AMPT model. Relative to the first setup (z_cut = 0.1, β = 0.0), the stronger grooming configuration (z_cut = 0.5, β = 1.5) exhibits a clearly depleted region in the upper-left part of the distribution, corresponding to a reduction in subjets with large angular separation. This indicates an increased focus on the jet core region.

Fig. 1Full-screen View

(Color online) Comparison of two Soft Drop grooming parameter settings: z_cut = 0.1 and β = 0.0 (left panel) and z_cut = 0.5 and β = 1.5 (right panel), with an angular cut of ΔR₁₂ > 0.1, for jets in the transverse momentum range 140 < p_T,jet < 160 GeV in pp collisions at TeV

Jet splitting momentum fraction

We compare the z_g distributions obtained using the first set of Soft Drop grooming parameters (z_cut = 0.1, β = 0.0) from the AMPT model with CMS measurements [43] at TeV, as shown in Fig. 2. The left panel shows the comparison for pp collisions in the jet transverse momentum range of 160 < p_T,jet < 180 GeV. The AMPT results are in good agreement with the CMS data, providing a reliable baseline. The right panel presents the z_g distributions for 0–10% most central smeared pp and PbPb collisions in the same p_T,jet range. Here, AMPT simulations were performed with partonic interactions of 3 mb. The centrality of smeared pp events is determined based on the multiplicity distributions from the PbPb collisions without applying jet triggers. Compared to the CMS data, the AMPT predictions exhibited a slightly flatter z_g distribution.

Fig. 2Full-screen View

Distributions of the z_g in pp collisions (left panel) and in 0–10% smeared pp and PbPb collisions (right panel), within the jet transverse momentum range of 160 < p_T,jet < 180 GeV at TeV. The Soft Drop grooming parameters are set to z_cut = 0.1 and β = 0.0, with an angular cut of ΔR₁₂ > 0.1. Lines represent the AMPT model results, while data points correspond to CMS measurements [43], with statistical uncertainties shown as error bars and systematic uncertainties as shaded bands

Figure 3 displays the ratios of z_g distributions between PbPb and smeared pp collisions at TeV, shown for different centrality bins (left panel) and jet transverse momentum ranges (right panel). Two scenarios are considered: simulations with partonic interactions for 3 mb and without partonic interactions for 0 mb. In the left panel, for jets with 160 < p_T,jet < 180 GeV, no modification is observed across all centrality bins when partonic interactions are turned off. In contrast, simulations with partonic interactions exhibit negligible changes in peripheral PbPb collisions, whereas a slight enhancement in asymmetric splitting is observed in more central PbPb collisions. These results are qualitatively consistent with the CMS measurements, within the uncertainties. The right panel illustrates the z_g modification for different p_T,jet ranges in the 0–10% most central collisions. For partonic interactions of 0 mb, the z_g distributions remain largely unmodified across the p_T,jet range of 140–200 GeV, with only minor deviations observed at higher jet momenta (200 < p_T,jet < 250 GeV). When partonic interactions are included, slight modifications appear throughout the full p_T,jet range, indicating a weak sensitivity of z_g to jet-medium interactions, particularly at low p_T,jet.

Fig. 3Full-screen View

(Color online) Ratios of the z_g distributions between PbPb and smeared pp collisions at TeV, for different centrality bins (left panel) and jet transverse momentum ranges (right panel). The Soft Drop grooming parameters are set to z_cut = 0.1 and β = 0.0, with an angular cut of ΔR₁₂ > 0.1. Solid (3 mb) and dashed (0 mb) lines represent the AMPT model results, while data points correspond to CMS measurements [43], with statistical uncertainties shown as error bars and systematic uncertainties as shaded bands

Furthermore, several theoretical studies have explored the behavior of the z_g observable in the high jet transverse momentum regime (p_T,jet > 140 GeV) in comparison with CMS measurements [43]. Simulations using JEWEL with the four-momentum subtraction method [51], which incorporates medium response effects, predict a mild shift of the z_g distribution toward smaller values. Similarly, calculations based on the SCET incorporating Glauber gluon interactions [52] also revealed a slight enhancement in the probability of asymmetric splittings. At lower jet transverse momentum (60 < p_T,jet < 100 GeV), the ALICE experiment [45] observed no significant modification of the z_g distribution in PbPb collisions relative to pp collisions when using strong grooming settings (z_cut = 0.2, β = 0.0). These results are consistent with predictions from JETSCAPE [53], which combines the MATTER model [97] for high-virtuality parton evolution and the LBT model [42, 98] for low-virtuality in jet-medium interactions. Interestingly, the higher twist formalism [54], which is based on the coherent energy loss assumption for the two split subjets, predicts a non-monotonic dependence of the z_g modification on the jet energy: the strongest modification occurs at intermediate jet energies, while both lower and higher energies exhibit weaker effects.

Groomed jet mass

We further investigate the ratio of the groomed jet mass to ungroomed jet transverse momentum, M_g / p_T,jet, using two sets of Soft Drop grooming parameters at TeV, and compare our results with the CMS data [48]. Figure 4 presents the M_g / p_T,jet distributions in pp collisions (top panels) and in both smeared pp and PbPb collisions with partonic interactions of 3 mb (bottom panels), for jets in p_T,jet range of 160–180 GeV. In the top panels, our simulations agree well with the CMS results within uncertainties, providing a reliable baseline for further comparison. Additionally, the M_g / p_T,jet distributions with the Soft Drop parameters z_cut = 0.5 and β = 1.5 (top right panel) appear steeper than those with z_cut = 0.1 and β = 0.0 (top left panel), reflecting a stronger grooming constraint focused on the jet core. In the bottom left panel, corresponding to the first grooming condition, we observe a broadening of the M_g / p_T,jet distribution in PbPb collisions compared to smeared pp collisions. The smeared pp result exhibits a minor deviation, while the PbPb result presents an enhancement at high M_g / p_T,jet in comparison with CMS data. For the second grooming condition, shown in the bottom right panel, the smeared pp result remains consistent with the CMS data within uncertainties, while the PbPb events display a deviation in the M_g / p_T,jet distribution compared to the CMS measurement.

Fig. 4Full-screen View

Distributions of the M_g / p_T,jet in pp collisions (top panels) and in 0–10% smeared pp and PbPb collisions (bottom panels) within the jet transverse momentum range of 160 < p_T,jet < 180 GeV at TeV. The Soft Drop grooming parameters are set to z_cut = 0.1, β = 0.0 (left panels) and z_cut = 0.5, β = 1.5 (right panels), with an angular cut of ΔR₁₂ > 0.1. Lines represent the AMPT model results, while data points correspond to CMS measurements [48], with statistical uncertainties shown as error bars and systematic uncertainties as shaded bands

Figure 5 shows the ratios of M_g / p_T,jet distributions between PbPb and smeared pp collisions in the jet transverse momentum range of 160 < p_T,jet < 180 GeV at TeV, for different centrality bins, using two Soft Drop grooming parameters: a condition sensitive only to the energy fraction between subjets (z_cut = 0.1, β = 0.0) in the left panel, and a stronger grooming setup that emphasizes the jet core structure (z_cut = 0.5, β = 1.5) in the right panel. Simulations were performed both with (3 mb) and without (0 mb) partonic interactions, consistent with the approach taken in the z_g analysis. In the left panel, when partonic interactions are turned off, the ratio remains flat across all centrality classes, except for fluctuations in the highest M_g / p_T,jet bin. However, when partonic interactions are included, a hint of enhancement emerges in the high M_g / p_T,jet region, particularly in more central collisions, indicating medium-induced broadening of the groomed jet mass. In contrast, the stronger grooming condition shown in the right panel leads to no significant modification in any centrality bin, regardless of the presence of partonic interactions. This confirms that the observed enhancement is predominantly associated with the medium response located at larger angles from the jet axis.

Fig. 5Full-screen View

(Color online) Ratios of the M_g / p_T,jet distributions between PbPb and smeared pp collisions in the jet transverse momentum range of 160 < p_T,jet < 180 GeV at TeV, for different centrality bins. The Soft Drop grooming parameters are set to z_cut = 0.1, β = 0.0 (left panel) and z_cut = 0.5, β = 1.5 (right panel), with an angular cut of ΔR₁₂ > 0.1. Solid (3 mb) and dashed (0 mb) lines represent the AMPT model results, while data points correspond to CMS measurements [48], with statistical uncertainties shown as error bars and systematic uncertainties as shaded bands

To further investigate the sensitivity to the jet transverse momentum, Fig. 6 displays the modification ratio of M_g / p_T,jet distribution for the 0–10% most central PbPb collisions relative to smeared pp collisions for different p_T,jet. For the first grooming condition (left panel), simulations without partonic interactions again show no modification, except for minor fluctuations in the highest M_g / p_T,jet bin at intermediate p_T,jet. In contrast, when partonic interactions are included, a noticeable enhancement is observed in the low p_T,jet region, consistent with CMS measurements. This suggests that the elastic interactions between the jets and medium partons contribute to an increase in the groomed jet mass. No significant changes are observed across all p_T,jet intervals under the stronger grooming condition (right panel), further confirming that medium-induced modifications are primarily attributed to the large-angle scattering within the AMPT framework.

Fig. 6Full-screen View

(Color online) Ratios of the M_g / p_T,jet distributions between 0-10% PbPb and smeared pp collisions at TeV, for different jet transverse momentum ranges. The Soft Drop grooming parameters are set to z_cut = 0.1, β = 0.0 (left panel) and z_cut = 0.5, β = 1.5 (right panel), with an angular cut of ΔR₁₂ > 0.1. Solid (3 mb) and dashed (0 mb) lines represent the AMPT model results, while data points correspond to CMS measurements [48], with statistical uncertainties shown as error bars and systematic uncertainties as shaded bands

To elucidate the origin of these modifications, we further analyzed the evolution of the ratio of M_g / p_T,jet distribution at four dynamic stages in the AMPT model: initial state, after parton cascade, after hadronization, and after hadronic rescatterings, as shown in Fig. 7. The analysis focuses on jets with 160 < p_T,jet < 180 GeV in 0–10% most central PbPb and smeared pp events. For the first grooming condition (top left panel), no modification is observed in the initial state, while a pronounced enhancement in the ratio of M_g / p_T,jet distribution emerges immediately after the parton cascade stage (3 mb), confirming that jet-medium interactions are the primary drivers of the modification. The hadronization and hadronic rescattering stages retain the enhancement, but their additional effects are minor, consistent with the ability of the grooming procedure to suppress non-perturbative contributions. In contrast, for simulations without partonic interactions (bottom left panel), no noticeable change is observed across any stage, reaffirming the absence of jet quenching effects. The results using the second grooming condition are shown in the right panels, where only minor deviations are observed, further emphasizing that the groomed jet mass is sensitive to the medium response located at larger angles from the jet axis.

Fig. 7Full-screen View

Ratios of the M_g / p_T,jet distributions between 0–10% PbPb and smeared pp collisions in the jet transverse momentum range of 160 < p_T,jet < 180 GeV within four dynamical evolution stages at TeV. The Soft Drop grooming parameters are set to z_cut = 0.1, β = 0.0 (left panels) and z_cut = 0.5, β = 1.5 (right panels), with an angular cut of ΔR₁₂ > 0.1. Lines represent the AMPT model results with parton cross section of 3 mb (top panels) and 0 mb (bottom panels), while data points correspond to CMS measurements [48], with statistical uncertainties shown as error bars and systematic uncertainties as shaded bands

In summary, our analysis demonstrates that jet-medium interactions lead to a significant enhancement in the groomed jet mass. This effect has been investigated using several theoretical frameworks. JEWEL simulations [55] fail to describe the smeared pp and PbPb results, in comparison with CMS data [48]. Nevertheless, the medium response in JEWEL still leads to a noticeable enhancement at large M_g / p_T,jet values [48]. Similarly, the LBT model [56] predicts a substantial enhancement in the tail of the groomed jet mass distribution owing to the medium response. When comparing with ALICE data [50], Hybrid model [57], based on a strongly coupled AdS/CFT framework, shows that the elastic Molière scattering [58] generates large-angle jet constituents and broadens the groomed jet mass distributions. These results collectively highlight the strong sensitivity of the groomed jet mass to jet-medium interactions, making it a powerful observable for probing QGP properties.