Field-Theoretic Analysis of Hadronization Using Soft Drop Jet Mass

One of the greatest challenges in quantum chromodynamics is understanding the hadronization mechanism, which is also crucial for carrying out precision physics with jet substructure. In this Letter, we combine recent advancements in our understanding of the field theory-based nonperturbative structure of the soft drop jet mass with precise perturbative calculations of its multi-differential variants at next-to-next-to-leading logarithmic (NNLL) accuracy. This enables a systematic study of its hadronization power corrections in a completely model-independent way. We calibrate and test hadronization models and their interplay with parton showers by comparing our universality predictions with various event generators for quark and gluon initiated jets in both lepton-lepton and hadron-hadron collisions. We find that hadronization models perform better for quark jets relative to gluon jets. Our results provide the necessary toolkit for precision studies with the soft drop jet mass motivating future analyses using real world collider data. The nontrivial constraints derived in our framework are useful for improving the modeling of hadronization and its interface with parton showers in next generation event generators.

Theoretically, the SD jet mass is the most precisely studied groomed jet observable, with predictions available at next-to-next-to-next-to-leading logarithmic accuracy (N 3 LL) matched to next-to-next-to-leading order (NNLO) predictions for dijets at e + e − collisions [77], and next-to-next-to-leading-logarithmic (NNLL) accuracy [76] for jets at the LHC.At this level of precision, the hadronization power corrections become comparable in size to perturbative accuracy, and cannot be accounted for using hadronization models [78][79][80] that are tuned to lower precision parton showers.In the recent years, there has been significant progress in understanding these hadronization effects in the SD jet mass [81][82][83] using a field theory-based formalism [84][85][86][87][88][89][90], which allows for a model-independent description of nonperturbative (NP) power corrections for precision phenomenology.Furthermore, this formalism imposes powerful constraints on jet-FIG.1: An example of fit for nonperturbative parameters in Pythia 8.306 simulation of groomed jet mass.The insets show distribution of low energy particles as heat maps around the soft drop stopping subjets in the transverse plane.
flavor, kinematics and grooming parameter dependence of these NP corrections.Hence, by comparing these predictions with event generators, we now have a unique opportunity to carry out a nontrivial characterization of these hadronization models and their interplay with parton showers, which is often difficult to interpret and test.In this Letter , using the state-of-the-art theoretical advancements in our understanding of the SD jet mass, we achieve a systematic and complete field theory-based study of hadronization effects and test our predictions with multiple event generators.
Hadronization corrections to groomed jet mass.-Compared to the ungroomed jet mass, the SD jet mass exhibits a much larger region of applicability for pertur-bation theory.This region is referred to as the soft drop operator expansion (SDOE) region, which is defined below and shown in Fig. 1 between the vertical lines.Here, hadronization effects can be studied using factorization in a systematic expansion.
Using soft collinear effective theory (SCET) [91][92][93][94], in Ref. [81] the leading hadronization corrections in the SDOE region were shown to depend on three O(Λ QCD ) NP universal constants {Ω • • 1κ , Υ 1,0κ , Υ 1,1κ }, which solely depend on the parton κ = q, g initiating the jet, and are completely independent of the jet kinematics, such as the jet p T (or E J ), rapidity η J , radius R, and the SD parameters [7], the energy cut z cut and the angular modulation parameter β, such that Here dσ κ and dσ κ , respectively, refer to hadron and parton level groomed jet mass cross sections for flavor κ and Q characterizing the the hard scale of the jet.The weights dσ • •, κ are perturbatively calculable.We note that, in contrast with analytical hadronization models employed in previous work [6,60,75,89], Eq. ( 1) is a model-independent statement and includes hadron mass effects.
In the SDOE region, the leading hadronization corrections are driven by a two-pronged dipole, which consists of an energetic collinear subjet at the core of the jet and a collinear-soft (c-soft) subjet that is responsible for stopping the grooming algorithm.The corrections represented by the ellipsis '. ..' in Eq. (1) involve higher power corrections of Λ QCD and corrections from configurations that distort the two-pronged catchment area.The latter correction is a next-to-leading-logarithmic effect, and therefore Eq. ( 1) can also be seen as a factorization of NP effects at leading-logarithmic accuracy, where the strong ordering of angles ensures the two-pronged geometry.As the jet mass decreases, we enter the soft drop non-perturbative (SDNP) region, where the c-soft mode becomes nonperturbative and correspondingly the nonperturbative effects are of O (1).The transition between these two regions is clearly visible in Fig. 1, where the insets show the distribution of low-energy NP particles in the transverse plane of the jet [81].
The statement of NP factorization in Eq. (1) presents us with a singular opportunity to probe hadronization in jets in a rich setting.As can be seen from Eq. (1), the consistency of the formalism requires that the three constants be sufficient to describe data measured from high energy colliders over a wide range of energies.The highly constraining structure given by Eq. (1) (constants being of O(Λ QCD ), having a β proportional coefficient Υ 1,1κ , z cut -independence, etc.) makes this far from a trivial feat FIG.2: NNLL results for perturbative weights in Eq. ( 1) of hadronization corrections (shown here for gluon jets).Bands denote perturbative uncertainty and vertical lines the extent of the fit region (see Eq. ( 7)).The factor of 1/Q is included to illustrate the size of hadronization corrections.and hence useful for calibrating hadronization models.In this work, we demonstrate how the universality structure strongly constrains the NP parameters, allowing them to be accurately determined by considering various combinations of soft drop and kinematic parameters.This, for example, improves the prospects for measuring the strong coupling constant α s at the LHC.
Calculation of perturbative weights.-Characterizing the two-pronged configuration of the collinear and the c-soft subjet in the SDOE region requires auxiliary measurements of the groomed jet radius R g and soft subjet energy fraction z g [7,34,35,95], which after marginalizing give [82] 1 σκ 1 σκ As the NP constants in Eq. ( 1) are independent of the jet kinematics and grooming parameters, all these dependencies are encapsulated by dσ • •, κ .The appearance of r g = R g /R in Eq. ( 2) is analogous to how jet radius R appears in hadronization corrections for the ungroomed jet mass in the tail and for the jet p T [89,90]: where Ω 1κ , Υ 1κ ∼ Λ QCD are NP parameters and hatted variables are parton level values.In the case of the SD jet mass, the dynamically determined groomed jet radius R g plays the role of R. The term in Eq. ( 1) with dσ • • κ is analogous to the ungroomed jet mass shift correction in the tail, but is now described by a different constant 1κ .The term in the second line in Eq. ( 1) with dσ κ is called the boundary correction.This effect is similar to the migration of events across p T -bins due to hadronization.Near the "boundary" of the c-soft subjet passing/failing soft drop, i.e. when z g ≈ z cut r β g , the partonic values ẑg and rg are modified due to hadronization as Here, Υ 1,0κ characterizes the shift in the p T of the csoft subjet analogous to jet p T shift in Eq. ( 3), and Υ 1,1κ describes the change in the subjet location relative to the collinear subjet.The combination of the two gives rise to the linear structure Υ 1κ = Υ 1,0κ + βΥ 1,1κ as shown in Eq. ( 1), and constitutes a nontrivial prediction.Finally, it is useful to factor out the parton level groomed jet mass cross section from dσ • •, : This definition is convenient as it will allow us to combine analytical calculation of the coefficients C κ 1,2 (m 2 J ) with parton shower jet mass cross section dσ κ as discussed below.
In Ref. [81], C κ 1,2 (m 2 J ) were computed in the coherent branching framework at LL accuracy.The first big step towards improving the accuracy of these coefficients was achieved in Ref. [82] by recasting them as moments of doubly differential cross section as in Eq. ( 2) and computing them at NLL accuracy in the SDOE region.In this work, we employ a further improved calculation at NNLL accuracy described in the companion paper in Ref. [83], where the matching of the doubly differential cross section in the ungroomed region is included for correct treatment of the soft drop cusp location at NNLL.In Fig. 2, we show calculations of dσ  2), we see that the leading hadronization corrections can be as large as 10% for small jet masses.
Calibrating hadronization models.-Withstate-of-theart NNLL perturbative results for C κ 1,2 (m 2 J ), we are in position to carry out a precise calibration of hadronization models.Furthermore, by incorporating NNLL perturbative uncertainty, we are able to significantly improve upon the analysis of Ref. [82] with LL predictions lacking uncertainty estimates.We simulate e + e − → gg, e + e − → q q, pp → Z + q jet and pp → Z + g jet processes using Pythia 8.306 [78], Vincia 2.3 [96] and Herwig 7.2.3 [80] parton showers with their default hadronization models.We reconstruct anti-k T [97] jets with R = 0.8 using Fastjet [98], and analyze them using jet analysis software JETlib written by two of the FIG.3: Weighted cross sections for hadronization corrections normalized to parton level jet mass spectrum as defined in Eq. ( 5) for z cut = 0.1 and β = 1.
authors [99].For e + e − collisions, we sample both jets in the dijet configuration, while only using the leading jet in pp collisions.As NP parameters are explictly predicted to be independent of the jet kinematics and grooming parameters, we carry out analysis using a wide range of kinematic and grooming parameter choices.In e + e − collisions, we analyze events at center of mass energies Q = 500, 750, 1000 GeV, while in pp, we use jets with p T ∈ {[400, 600], [600, 800], [800, 1000]} GeV and soft drop parameters z cut ∈ {0.05, 0.1, 0.15, 0.2} and β ∈ {0, 0.5, 1, 1.5, 2}.
We begin by explicitly defining the SDOE region where our analysis is carried out.We first define a dimensionless variable ξ ≡ m 2 J /Q 2 , where In terms of ξ, the SDOE region is then defined as ξ ∈ ξ SDOE , ξ 0 , where Here ξ 0 is the location of the soft drop cusp [76,83]: while ζ is defined by such that ξ 0 in Eq. ( 7) is the soft-wide angle transition point of the NNLL calculation.We set Λ QCD → 1 GeV, the typical scale of transition from parton showers to hadronization.The parameter ρ in Eq. ( 7) determines GeV) χ 2 min /dof.e + e − → q q 0.55  the onset of the SDOE region, and we set ρ = 4.5.In principle, any choice satisfying ρ 1 is acceptable.We explore other choices of ρ in the Supplemental Material.
In Fig. 3 we show a comparison of the NNLL computation of C κ 1,2 with partonic Pythia and Herwig.The parton level results for from Vincia are found to be almost identical to Pythia.We find a good agreement of the NNLL C κ 1 with MC for all four processes.The unusually small errors for C κ 1 result from cancellation between correlated uncertainties in the two factors in Eq. ( 5).For pp, the agreement for the boundary term is poor for jet masses close to the cusp due to the initial-state radiation (ISR) contribution.However, as seen in Fig. 2, the NP corrections in the cusp-region are relatively suppressed, and NP corrections from ISR are also expected to be smaller as they involve subleading r 2 g moment of the boundary cross section [83].Consequently, these effects do not significantly impact the analysis below.
Finally, we perform a least-squares fit for the NP parameters by defining our χ 2 statistic as Here, σ X is a vector of cross section values for n bins = 10 bins in the fit range and all permutations of p T (or E J ), z cut , and β values considered above.We denote the hadron level MC groomed jet mass cross section as σ MC κ,had , and define σ κ,part+NP by including the NP constants in Eq. ( 5) to the parton level MC spectrum dσ MC κ following Eq.( 1).The uncertainty in the denominator is defined as where, guided by the size of perturbative uncertainties in Fig. 2, we have assigned 5% and 25% uncertainty respectively to the weighted cross sections for shift and boundary corrections respectively.The NP constants Ω • • 1κ , Υ 1,0κ , Υ 1,1κ are then varied to minimize this χ 2 statistic.An example of the fit for mass distribution is shown in Fig. 1.
In Tab.I, we present the fit results for the NP constants with scale variations of C κ 1,2 for Pythia.As an-0.5 1.0 1.5 2.0 1 (GeV) ticipated, the parameters are 1 GeV.We also find similar parameter values for quark jets within the two quark processes within uncertainties.Even when NP parameters for quark jets are simultaneously fit for in e + e − and pp process, we find an excellent χ 2 value of 0.840/dof.This is expected, as soft drop isolates the jet from surrounding radiation.To further investigate this, we show correlations between Ω • • 1κ and Υ 1,0κ for the four processes in Fig. 4 where each ellipse represents a 1σ deviation.To account for perturbative uncertainties, we repeat the fit by varying C κ 1,2 up and down within the uncertainty band shown in Fig. 3.We observe an excellent agreement within uncertainties between the NP parameters for quark jets in pp and e + e − collisions in Pythia simulations, and a moderate agreement for Vincia and Herwig.In contrast, while Herwig exhibits similar levels of agreement for gluon jets and quark jets at both colliders, Pythia and Vincia show significant disagreement.This shows that contrary to the expectation for groomed jets, hadronization modeling of gluon jets in isolation in e + e − collisions in Pythia and Vincia differs significantly from jets in hadron colliders.Additionally, the differing results between Pythia and Vincia point to the interplay of parton showers with hadronization models.In the Supplemental Material we show correlations in the other two combinations of NP parameters which show similar behavior as well as numerical fit results for Herwig and Vincia.
Next, we test the grooming parameters independence of these NP constants.We follow the same procedure as Ref. [81] and test this behavior by comparing the fit results for individual z cut and β values with the global fit.In Fig. 5, we demonstrate the linear β-dependence of the boundary correction by fitting for a single parameter Υ 1κ (β) for each value of β.Because of degeneracy in the NP parameters, we fix Ω • • 1κ to its global-fit value in this case.The error bars take into account perturbative uncertainty in C κ 1,2 by re-fitting with minimum and maximum variations.We find that all the three simulations perform well in each of the four cases.In Fig. 6, we repeat the same procedure to test z cut -independence of NP parameters.We find here that the three event generators pass the test for both quark and gluon jets in e + e − collisions, but exhibit a linear trend in z cut for both flavors in pp collisions.The larger χ2 values for gluon jets, as seen in Tab.I for Pythia (also true for Herwig and Vincia) suggest that modeling of hadronization in gluon jets is less consistent with our field theory predictions.Finally, our analysis of the e + e − → q q process using NNLL predictions of C κ 1,2 demonstrates significant improvement in the universality behavior of z cut and β, compared to Ref. [81] where LL predictions were used. 1n conclusion, while our universality tests of the NP parameters generally display expected behaviors in all the cases considered, they also reveal some tension with the hadronization models, pointing to interesting avenues for further improvement 2 and motivate the use of real-world collider data for further analyses.
Conclusions.-In this Letter, we have presented a systematic framework for analyzing nonperturbative corrections in soft drop jet mass by bringing together earlier work on nonperturbative factorization and high precision calculations of multi-differential soft drop cross sections.Our analysis with hadronization models successfully demonstrates that nonperturbative parameters exhibit the universal behaviors predicted by field theory.Our analysis is also directly applicable to precision phenomenology involving soft drop jet mass.For example, in Ref. [76] our results are used to assess the impact of the NP corrections on the sensitivity and ultimate precision achievable on α s at the LHC using SD jet mass.Findings in Ref. [76] indicate that the hadronization effects in the β = 1 case, for instance, are 3% (8%) for quark (gluon) jets when nonperturbative parameters in Eq. ( 1) are left unconstrained, which are of the same size as the NNLL perturbative uncertainty.We anticipate that with high precision calculations for the soft drop jet mass and the boundary correction (C κ 2 in Fig. 3), it will be possible to significantly constrain some or all of the NP constants, and hence improve the ultimate precision achievable on α s -determination at the LHC.In summary, our work thus provides crucial understanding of hadronization corrections necessary for precision measurements with soft drop jet mass, a benchmark tool for improving hadronization modeling in MC event generators, and motivation for analyses with real world collider data.
Acknowledgements.-We would like to thank Mrinal Dasgupta, Michael Seymour for helpful discussions.We are grateful to Simon Plätzer for many discussions and support with analysis with Herwig.We thank Holmfridur Hannesdottir, Johannes Michel and Iain Stewart for numerous discussions and feedback on the manuscript.We provide a numerical implementation of the NNLL calculation in C++ building on core classes of SCETlib [100] which will be made available as a part of the scetlib::sd module [101].We thank Johannes Michel for support with above-mentioned implementation in SCETlib.KL was supported by the LDRD program of LBNL and the U.S. DOE under contract number DE-SC0011090.AP acknowledges support from DESY (Hamburg, Germany), a member of the Helmholtz Association HGF.AP was a member of the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics, which is supported by the UK Science and Technology Facilities Council (STFC) under grant number ST/T001038/1.AF also gratefully acknowledges support from the above-mentioned grant.
GeV) χ 2 min /dof.e + e − → q q 0.62 +0.39  −0.10 −0.69   consistent with the global-fit value within uncertainty, with the exception of β = 0 result of Ω • • 1g for gluon jets in Fig. 1 where we notice a significant deviation.
In order to better visualize the level of agreement for quark and gluon jets in e + e − and pp collisions for the three MC event generators considered we show in Fig. 4 the 1σ contours for correlations between Υ 1,1κ and Ω • • 1κ , and between Υ 1,0κ and Υ 1,1κ .The dominant uncertainty results from variation in C κ 2 .Here we see for all three hadronization models a better agreement of correlations between Ω • • 1κ and Υ 1,0κ for quark jets in the two colliders than Υ 1,0κ and Υ 1,1κ .As noted earlier in the main text, Pythia and Vincia results for gluon jets for the two collider settings significantly disagree, whereas the corresponding results for Herwig display similar trend as quark jets.
other two NP parameters due to the reduction in perturbative uncertainty on C κ 1,2 moments.At the same time, increasing ρ also reduces the available fit range which can impact the quality of the fit.As a result, we see in Fig. 5 that the reduced χ 2 value initially improves as ρ is increased but eventually saturates as the fit range becomes too small.In order to find an optimal balance between perturbative uncertainty and fit range, we used the e + e − → q q process as a benchmark due to the ab-sence of ISR effects.We chose ρ = 4.5 as in all the results presented above, it is at this point the errors and χ 2 values stabilize while still allowing for a reasonable fit range.We also see this trend reflected in the values of Ω • • 1κ for the e + e − → q q case.In the pp cases, however, we observe a linear dependence of Ω • • 1κ on the fit range, which is within the uncertainty for quark jets but not for gluon jets.

TABLE I :
Fit results for NP constants in Pythia 8.306 for quark and gluon jets in e + e − and pp collisions.

TABLE I :
Fit results for NP constants in Vincia 2.3.

TABLE II :
Fit results for NP constants in Herwig 7.2.3.In this Supplemental material, we present additional results for the calibration exercise of the MC hadronization model.Tables I and II present the results of fitting to Vincia 2.3 and Herwig 7.2.3,respectively.Similar to fits to Pythia 8.306 in Tab.I discussed in the main text, the χ 2 values in Tab.I and Tab.II show that the fits for quark jets are much better constrained than gluon jets for both Vincia and Herwig, with those of Herwig Testing z cut -independence of Υ 1,0κ for quark and gluon jets while fixing Υ 1,1κ and Ω • • 1κ paramaters to their global-fit values.Υ 1,0κ , and Υ 1,1κ .The horizontal lines in these figures represent the global-fit values, while the markers with error bars represent fits for individual z cut or β parameters.We observe that for each process the results of fits to individual z cut or β values are FIG.4: 1σ contours showing correlations between Υ 1,1κ and Ω • • 1κ (left column) and between Υ 1,1κ and Υ 1,0κ (right column) for Pythia 8.306 (top row), Vincia 2.3 (middle row) and Herwig 7.2.3 (bottom row).