Groomed jet mass at high precision

We present predictions of the distribution of groomed heavy jet mass in electron-positron collisions at the next-to-next-to-leading order accuracy matched with the resummation of large logarithms to next-to-next-to-next-to-leading logarithmic accuracy. Resummation at this accuracy is possible through extraction of necessary two-loop constants and three-loop anomalous dimensions from fixed-order codes.

High-energy electron-positron collisions are considered as ideal tools for precision studies of particle interactions. The initial state of the hard scattering event is colorless and known precisely, which eliminates significant sources of uncertainties that are ubiquitous at hadron colliders such as the LHC. For instance, the study of hadronic final states at the Large Electron-Positron collider (LEP) was used extensively to study the dynamics of strong interactions [1][2][3][4][5][6][7][8][9][10][11] and especially to determine the strong coupling α s . Yet, the current state of the art does not support these expectations. Hence, it is somewhat disappointing that presently the second largest spread and uncertainty of determination of α s among seven sub-fields is found in the group of results based on jets and event shapes of hadronic final states in electron-positron annihilation [12]. This failure of fulfilling expectations calls for an investigation of the possible sources.
The comparison of event shape distributions obtained from data collected by the LEP experiments and from theoretical predictions obtained in QCD perturbation theory reveal the possible causes of such a failure [13,14]: (i) the QCD radiative corrections are large, (ii) the hadronization corrections are not well understood from first principles, (iii) the two-types of corrections are strongly anti-correlated for analytic models of hadronization. As a result the systematic theoretical uncertainties are large. In order to decrease these corrections, one has to select the observables used for α s -extraction carefully. For instance, jet rates are expected to be less sensitive to hadronization corrections than event shapes [15], which is supported by a recent Monte Carlo evaluation, resulting in a competitive value for α s [16].
The latter study is based on the highest perturbative order available for two-jet rates: nextto-next-to-next-to-leading order (N 3 LO) matched with the resummation of the first three largest logarithms at all orders (N 2 LL) in perturbation theory.
For precision extraction of the strong coupling the logarithmic accuracy should extend to next-to-next-to-next-to-leading logarithmic order (N 3 LL) that allows for simple additive matching to fixed-order at N 2 LO. Such matched predictions are available for thrust [17] and C-parameter [18], and were used for the extraction of α s from LEP data [19,20].
However, even so high perturbative accuracy does not guarantee small uncertainty for the determination of α s due to lack of good control over the hadronization. One way out is to reduce the latter effect. The analysis techniques broadly referred to as jet grooming have been introduced to mitigate contamination radiation in jets from outside of the jet.
Jet groomers identify such emissions in the jet and remove them from consideration. The modified mass-drop tagger (mMDT) [21,22] and soft drop [23] algorithms are the best understood groomers, due to their unique feature of elimination of non-global logarithms (NGLs) [24] that are the leading correlations between in-jet and out-of-jet scales. Soft drop was indeed found to reduce the hadronization corrections for event shapes in electronpositron annihilation [25].
In this Letter, we present theoretical predictions for the mMDT groomed jet mass in e + e − collisions at N 2 LO matched with N 3 LL accuracy in perturbation theory. Resummation at this accuracy is made possible by the factorization theorem for jet grooming from Ref. [26] and recent extraction of necessary constants and anomalous dimensions at twoand three-loop order [27][28][29]. A demonstration of reduction of scale uncertainties and good convergence of the perturbation series will be presented here, but we leave a detailed study of scale variations and inclusion of non-perturbative corrections to groomed jets established in Ref. [30] for future work.
The modified mass-drop tagger groomer (mMDT) [21], or soft drop with angular exponent β = 0 [23], proceeds as follows: 1. Divide the final state of an e + e − → hadrons event into two hemispheres in any infrared and collinear safe way.
2. Define a clustering metric d ij between particles i and j in the same hemisphere. The metric appropriate for e + e − collisions is with θ ij being the angle between the trajectory of the particles.
3. In each hemisphere, apply the Cambridge/Aachen jet algorithm [31,32] to produce an angular-ordered pairwise clustering history of particles.
4. Starting with one of the hemispheres (say left) and at widest angle, step through the Cambridge/Aachen particle branching tree. At each branching in the tree, test if is satisfied, where i and j are the daughter particles at that branching and z cut is some fixed numerical value where 0 ≤ z cut < 0.5. If the condition (2) is true, then stop and return all particles that remain in the left hemisphere. If it is false, remove the lower energy branch, and continue to the next branching at smaller angle. Repeat the procedure for the other hemisphere.

5.
Once the groomer has terminated, any observable can be measured on the particles that remain in the two hemispheres.
In Ref. [26] a factorization theorem was derived for the cross section differential in the groomed hemisphere masses for mass m i and energy E i of hemisphere i. For τ i z cut 1, the cross section factorizes at all orders in perturbation theory as follows: where σ 0 is the leading-order cross section for e + e − → qq, H(Q 2 ) is the hard function for quark-antiquark production in e + e − collisions, S(z cut ) is the global soft function for mMDT grooming, J(τ i ) is the quark jet function for hemisphere mass τ i , and S c (τ i , z cut ) is the collinear-soft function for hemisphere mass τ i with mMDT grooming. The symbol ⊗ denotes convolution over the hemisphere mass τ i . In the functions we suppressed the dependence on the renormalization scale µ.
Transforming into Laplace space, the cross section assumes a genuine factorized form, In this product form, each function in the factorization theorem satisfies a simple renormalization group equation (RGE), where d F is a constant, µ F is the canonical scale, and γ F is the non-cusp anomalous dimension, all depending on the functionF . Γ cusp is the cusp anomalous dimension for back-to-back light-like Wilson lines in the fundamental representation of color SU(3). Large logarithms of hemisphere masses can be resummed to all orders in α s using this renormalization group equation, whose exact solution is presented explicitly including O(α 3 s ) terms in Ref. [19]. The order to which logarithms can be resummed using the RGE (6) depends on the accuracy to which its components are calculated. For the canonical definition of logarithmic accuracy [33], Tab. I shows the order in α s to which the components of the RGE are needed. The two-loop soft function constants were calculated by the SoftServe collaboration [27,28].
In Ref. [29] we computed the last missing pieces needed for N 3 LL resummation of the distribution of jet masses with mMDT, namely the two-loop constants c mMDT These results enable resummation to N 3 LL accuracy for jet substructure observables that we present here for the first time.
We present predictions in perturbation theory for the single-differential cross section of the groomed heavy hemisphere mass ρ σ 0 dσg dρ , defined as where the subscript g on the cross section indicates that it is groomed. This definition of the heavy hemisphere mass differs from the standard definition of the ungroomed case when the heavy hemisphere mass is defined as: ρ = max(m 2 L ,m 2 R ) Q 2 , with Q being the center-of-mass energy.
When hemispheres are groomed, the grooming eliminates their dominant correlations, and so it is more natural to define the groomed mass with respect to the hemisphere energy, and not the center-of-mass energy.
The CoLoRFulNNLO subtraction method was developed to compute QCD jet cross sections at the N 2 LO accuracy. Currently it is completed for processes without colored particles in the initial states, and it is implemented in the MCCSM code (Monte Carlo for the CoLoRFulNNLO Subtraction Method) [34][35][36][37][38]. This program can be used to compute the differential cross section of the mMDT groomed heavy hemisphere mass at fixed order in perturbation theory. MCCSM calculates directly the ρ-dependent coefficients A, B, and C (times their respective coupling factors) in the differential distribution where α s = α s (µ) is the strong coupling evaluated at the renormalization scale µ = ξQ, β 0 and β 1 are the first two coefficients in the perturbative expansion of the QCD β-function and Q is the center-of-mass collision energy. We present the predictions of MCCSM for the normalized cross section ρ σ 0 dσg dρ at the first three orders in perturbation theory (LO, NLO and N 2 LO) in the top panel of Fig. 1. The lower panels exhibit the K-factors defined as and the ratio K NNLO/NLO . We see that the O(α 3 s ) corrections stabilize the dependence on the renormalization scale for large values of ρ (ρ > 0.1) as expected, while the predictions are clearly not reliable for ρ 0.1. To stabilize the latter we need to resum the large logarithmic contributions.
All functions that appear in the factorization formula Eq. 5 can also be found explicitly in Ref. [26], including their matrix-element definitions. Due to the factorized form of the cross section, each function in the factorization theorem has its own natural scale at which it is defined, and they can be varied independently to provide some estimate of residual scale uncertainties. We leave a detailed scale variation study to future work, and here we just vary can be found in Ref. [29] including the O(α 3 s ) coefficient. For ρ > 0 the δ-functions can be ignored and +-distributions reduce to simple functions of ρ. We compare the D C,g (ρ) function to the C g (ρ) coefficient in the fixed-order expansion in the top panel of Fig. 2 where we show the logarithmic expansion with two assumed values of the three-loop non-cusp anomalous dimension γ (2) S : 0 and our extracted value with uncertainties from Eq. 8. As the value of z cut is decreased, improved agreement between the MCCSM results and the singular distribution is observed at small ρ, down to about ρ ∼ 10 −4 where numerical instabilities in MCCSM become significant.
Subtracting this singular distribution from the sum of the N 2 LO and N 3 LL, we obtain a prediction in perturbation theory with highest available accuracy: which we present in the bottom panel of Fig. 2. Good convergence of the matched predictions is observed for all values of ρ, with the results at N 2 LO+N 3 LL lying within the scale variation bands of the NLO+N 2 LL prediction. We have truncated this perturbative prediction at a value of ρ that lies above the region in which non-perturbative physics dominates the distribution. We have demonstrated the highest precision perturbative predictions for groomed jets in e + e − collisions. These results are sufficiently accurate to enable extraction of α s , when com-bined with leading corrections due to non-perturbative physics. While there is no currentlyrunning e + e − collider, analyses of archived LEP data have been completed [39], and the results presented here motivate further measurements on these archived data. Due to the elimination of soft radiation with mMDT grooming, the collinear-soft and jet functions in the factorization theorem are identical to that for corresponding measurements at hadron colliders. Thus, we anticipate these results can be used to further improve the theory-data comparisons of groomed jet masses measured at ATLAS and CMS [40][41][42], and, along with continual advances in fixed-order predictions, enable precision extractions of fundamental constants at the LHC.
We thank Guido Bell, Jim Talbert