Extracting a Short Distance Top Mass with Light Grooming

We propose a kinematic method for precisely measuring the top quark mass $m_t$ in $pp$ collisions using boosted top jets with light soft drop grooming. Light grooming retains a universal description of the top mass scheme and decay effects and removes soft contamination. One can obtain $m_t$ in a short distance scheme by fitting our hadron level jet mass distributions to data or by Monte-Carlo calibration. Lepton+jets and all-jet samples can be used. The peaked distributions for $pp$ and $e^+e^-$ collisions are very similar, up to sensitivity to underlying event which is reduced by soft drop.

The top quark mass m t is one of the most important Standard Model (SM) parameters. It significantly affects studies of the SM vacuum stability [1] and the electroweak precision observables [2]. The most precise top mass measurements are based on kinematic reconstruction, yielding results such as m MC t = 172.44(49) GeV (CMS) [3], m MC t = 172.84(70) (ATLAS) [4] and m MC t = 174.34(64) GeV (Tevatron) [5]. These measurements are based on Monte Carlo (MC) simulations and determine the mass parameter m MC t of the MC generator, which depends on the shower dynamics and its interface with hadronization. Identifying these values with a Lagrangian top-mass scheme m t induces an additional ambiguity at the 0.5-1.0 GeV level [6,7]. We propose a factorization approach to remove this uncertainty in pp → tt by constructing an observable that has high kinematic sensitivity to m t and at the same time allows for hadron level predictions from QCD employing a short distance top-mass. It can be used to extract m t from experimental data, or to calibrate the parameter m MC t as was done for 2-Jettiness in e + e − collisions [8].
We consider boosted tops whose decay products are collimated in a single jet region, enabling a simultaneous theoretical description of both the top production and decay [9]. This requires Q m t Γ t where Γ t 1.4 GeV is the top width and Q is twice the large momentum along the boost direction. For pp collisions Q = 2p T cosh(η) with p T and η being the jet's transverse momentum and pseudo-rapidity, respectively. Recently an experimental analysis along these lines was carried out by CMS [10]. For e + e − → tt a hadron level factorization theorem for a distribution with high kinematic sensitivity to a short distance m t was derived in [9,11]. So far an analogous approach has been missing for pp → tt, due to theory complications in controlling external radiation, parameters like the jet radius R, and soft contamination from initial state radiation and underlying event (UE), which is often modeled in MC simulations by multiple particle interactions (MPI).
Our method relies on the measurement of the jet mass M J on a jet of radius R ∼ 1 with light soft drop grooming in a boosted top sample. The soft drop algorithm [12,13] removes peripheral soft radiation by comparing subsequent jet constituents i, j in an angular ordered cluster tree until is satisfied. Here R ij is the angular distance in the rapidity-φ plane, and z cut and β are fixed soft drop parameters. When Eq. (1) is satisfied all subsequent constituents in the tree are kept, thus setting a new jet radius R g < R for the groomed jet. This retains strong kinematic sensitivity to m t as in the template method, grooms away contamination from other parts of the collision, and allows for a factorization based description [14]. It also reduces tuning dependence in MC simulations [15]. We make use of the Soft-Collinear Effective Theory [16] to derive peak region factorization formulae for the crosssection, with the modes pictured in Fig. 1a. Our calculation requires light grooming which satisfies The first constraint enables a simple treatment of the top-decay products and ensures Γ t /m t (Q/2m t ) β z cut so that boosted ultra-collinear (UC) massless radiation associated with the top quark is not modified. These effects are then described by the same inclusive jet function J B (ŝ, δm, Γ t , µ) as in [9,11], providing control over the scheme for m t through δm = m pole t − m t . This constraint is significantly stronger than that needed to retain the decay products, (Q/2m t ) β z cut . The second constraint ensures that wide angle soft radiation (y-axis of Fig. 1a above the green dot) is groomed away, isolating the jet, and removing the majority of soft contamination. The remaining perturbative collinear-soft (CS) radiation is then captured by the same function S C defined earlier for soft-dropped massless quark jets in [14,17] where z = 2E/Q for energy E, and θ is the polar angle relative to the top jet-axis. (b) Allowed values of zcut which are strong enough to isolate the jet from contaminating radiation (above red band), but not so strong as to invalidate the factorization formulae we derive (below blue band). Fig.1a). The final components are the factorization theorem's description of non-perturbative hadronization corrections (F C ), and our modeling of underlying event, to be discussed below. We assume that pileup corrections can be handled experimentally without significant modifications. We take β = 2 as our default. The allowed z cut region satisfying Eq. (2) is shown as a function of p T in Fig. 1b (red line replaces "a b" by "a > 3b"). For p T 750 GeV this is 0.02 > ∼ z cut and z 1/4 cut .073 which is satisfied by z cut 0.01. This is an order of magnitude smaller than typically used for jets at the LHC, but as we will see, is still very effective for m t measurements. For smaller β the allowed region is more constrained, so for experimentally accessible p T s the expansions used to derive the factorization formulae are less convergent.

UC and CS radiation contribute to
We present here the key aspects of the factorized crosssections, postponing a detailed discussion to elsewhere. There are two relevant formulas depending on the dominant non-perturbative modes Λ. Shown in Fig.1a is the case (called "decay") when soft drop stops when comparing decay products, so Λ is determined by the intersection of the brown p 2 = Λ 2 QCD line and the dashed line at the angle θ d between the jet-axis and the last decay product to be re-clustered by soft drop. This occurs for and the jet mass as the sum over all constituents in the jet of radius R after soft-drop has been applied, After soft drop the dependence on the jet-algorithm and R are power suppressed. Then the "decay" groomed top-jet mass cross section is Here N contains factors that affect only the normalization like parton distributions, the global soft function, the hard function, as well as the other t ort jet, and may also be factorized and computed explicitly. For our predictions below we compute N using N-Jettiness with XCone or anti-k T jets [18][19][20][21][22] and a loose jet-veto following Ref. [23], though beyond capturing the Born Φ J dependence our analysis is insensitive to this choice. For larger Q than in Eq. (3) the top-decay products are well inside the groomed jet and the dashed line in Fig. 1a moves to the right. In this case (called "high-p T ") the Λ modes are at the intersection of the brown line and orange line for Eq. (1), and the cross section is In Eqs. (4) and (5) only M J associated to either the hadronically decaying t ort is measured, while the other can decay hadronically or semi-leptonically. In fully hadronic decays both jets can be sampled independently. (5) is a non-perturbative function which is identical to that for a soft dropped jet initiated by a massless quark in [17], and can be determined by fitting its first few moments Ω (β) n = dk k n F C (k, β). The induced nonperturbative function in Eq. (4) is also determined by the same F C (k, β) with β = 1, Here Φ d are the 5 independent kinematic variables of the top-decay t → bqq in its rest frame (4 angles, one energy fraction), and d t (Φ d , mt Q ) is the angular dependence of the top-decay with dΦ d d t (Φ d , mt Q ) = 1. The h function is defined by the angle tan(θ d /2) = (m t /Q) h(Φ d , m t /Q). The m t /Q prefactor pulls out the dominant dependence on the boost. It cancels out the Q/m t boost factor in the argument of J B in Eq. (4), which largely eliminates the Q dependence of the peak position observed in the ungroomed case. If we take the n-th moment then Ω n , so the h n causes the effective moments to only have residual m t /Q dependence. We implement Eq. (6) by computing h and h 2 exactly and using the resulting Ω (1)eff 1 and Ω (1)eff 2 to specify the functionF C .
The result in Eq. (5) is a direct generalization of the results in Refs. [9,11,14,17,23], whereas Eq. (4) is more involved. To derive it we first show that cut , β, µ , (7) where this J B (ŝ, δm, µ) is the stable top jet-function and D t (ŝ , Φ d , m t /Q) encodes the angular cross-section of the top-decay products. The calculation of D t requires a geometric sum of decay product bubbles, where one hadronically decaying bubble is cut. In the non-cut bubbles we just keep Γ t yielding the resonant contribution For the calculation of d t (Φ d , m t /Q) we can setŝ = 0, leading to the factorized structure in Eq. (8). Thus we can do the integral overŝ in Eq. (7) which gives back the unstable jet function J B (ŝ, δm, Γ t , µ) [9]. Changing variable to k = h(Φ d )k then turns Eq. (7) into Eq. (4) with Eq. (6) for the functionF C . Eqs. (4) and (5) determine the M J spectrum as a Breit-Wigner distribution smeared by non-perturbative corrections and dressed by perturbative corrections including resummed large Sudakov double logarithms from the hierarchy p T m t Γ t > Λ QCD . As a default we take p T ≥ 750 GeV, |η| < 2.5, z cut = .01, β = 2, jets with radius R = 1, p veto T = 200 GeV, and plot spectra normalized over the displayed range. In Fig. 2 we test the factorization theorem predictions using default Pythia 8.219 including hadronization and MPI effects, and the soft drop plugin in FastJet [13,24]. In Fig. 2a we show the dependence on z cut , and observe a dramatic shift to smaller M J at precisely the small z cut ∼ 0.005 predicted by Eq. (2), see Fig.1b's red line. Increasing z cut further does not groom soft radiation inside the radius determined by the top decay products, leaving the peak position quite stable even beyond the limit in Eq. (2), unlike for massless jets. In Fig. 2b we demonstrate that the light groomed spectrum becomes independent of the jet radius R for R > ∼ 0.9, as expected, in contrast to the strong dependence on R for ungroomed jets. The light groomed spectrum is also independent of an anti-k T jetveto cut p veto T (for jets beyond the two with largest p T ) once p veto T > ∼ 50 GeV, as shown in Fig. 2c. An important prediction of the light soft dropped top factorization theorems is an insensitivity to parts of the event outside the groomed top jet. Thus the same factorization theorems apply for top-jets from e + e − → tt and pp → tt, with only changes to the meaning of Q and the function N . To obtain a reasonable comparison with Pythia8 we take the ee center-of-mass energy Q = 2400 GeV to approximate the spectrum weighted average Q for pp with |η| < 2.5 and p T ≥ 750 GeV. We see in Fig. 2d that the spectra differ without soft drop (dotted green and dot-dashed blue curves), but agree quite well with soft drop (solid green and dashed blue curves). Also shown is the impact of MPI on the pp spectra. Without soft drop adding MPI shifts the peak of the spectrum by 4.5 GeV (dotted red versus dotted green), whereas with light soft drop the shift is only 1.1 GeV (solid red versus green). Formally effects from UE are outside the framework of factorization. However, in Ref. [25] it was shown that MPI in Pythia for the ungroomed jet mass spectrum can be well modeled by simply changing F C . This occurs because the dominant impact of MPI is to populate the jet with uncorrelated soft radiation of somewhat higher energy than that associated to the soft hadronization. We adopt this approach to account for hadronization plus UE, replacing Ω (β) n → Ω (β)MPI n . Estimating this treatment of UE is uncertain at the < ∼ 30% level, this induces a residual uncertainty of ∆m t < ∼ 0.3 GeV for our soft drop top mass extraction, compared to ∆m t < ∼ 1.4 GeV without soft drop. With additional dedicated studies this uncertainty may be further reduced.
In Fig. 3 we show a comparison of Pythia8 results with the "decay" and "high-p T " factorization formulae in Eqs. (4) and (5) with all ingredients taken at treelevel with next-to-leading-logarithmic (NLL) order resummation and α s (m Z ) = 0.118. In the factorization theorems we adopt the MSR short distance top mass scheme m MSR t (R) [26,27] and include it's leading logarithmic evolution from a reference scale R = 1 GeV to the scale µ in J B . As fit parameters we have the MSR mass m limit of Eq. (3), and we find that both factorization theorems reproduce the Pythia8 results accurately in the fit range. There is a noticeable difference between the factorization theorem results and Pythia8 for the tail on the left of the peak, which is affected by events where the decay products are outside the R = 1 jet cone, or are close to the boundary within the cone. Likely both the factorization and Pythia8 predictions could be improved in this region. The m MSR t fit values are within 0.3 GeV of the input m MC t , which is compatible with the e + e − calibration result in [8]. We also observe that the fit values of m MSR t are compatible between the "decay" and "high-p T " results (within 0.2 GeV), and between results with and without MPI effects (within 0.3 GeV). As anticipated, the dominant effect of adding MPI is to significantly increase the scale of the hadronization parameter Ω (β) 1 and modify x (β) 2 . This result is crucial and enables a precision m t to be obtained from this method. In the supplemental material we present the corresponding fit results for the pole mass scheme, obtaining values 0.4-0.7 smaller than m MC t , which is also compatible with [8]. Examining a bin with p T ∈ [550, 750] GeV we find poorer agreement with Pythia8, which likely indicates that higher order terms in the soft drop factorization expansions are becoming important.
Since the proposed approach is systematically improvable, we foresee that the perturbative and hadronization uncertainties on extracting m t will be below a GeV. Experimentally requiring p T above 700 GeV limits the data sample, however this is somewhat mitigated by the light soft drop method not requiring other cuts, like those on p veto T or the angle between decay products used in Ref. [10]. We leave a more detailed analysis to future work, including results at one higher order, precise estimates of all uncertainties, and exploring the smaller p T region. We anticipate that these results can be used for direct fits to LHC data, to calibrate MC, and even to make predictions independent of fitting hadronic parameters by exploiting the fact that universality allows Ω (β)MPI 1 and x (β)MPI 2 to be determined by fits to light or b quark soft dropped jet data.