Top-Antitop Production and Decay at Threshold at the LHC in QCD Perturbation Theory

Technical Summary: Top-Antitop Production and Decay at Threshold at the LHC in QCD Perturbation Theory

Problem Statement
Recent observations by the ATLAS and CMS collaborations indicate an excess in the $t\bar{t}$ cross-section near the kinematic threshold ($m_{t\bar{t}} \approx 2m_t$), specifically in the pseudoscalar spin channel. This excess has been loosely interpreted as evidence for a $t\bar{t}$ bound state (toponium, $\eta_t$). However, several theoretical ambiguities surround this interpretation:

1. Nature of the Excess: While the $s$-wave state near threshold is predominantly a pseudoscalar singlet due to the Landau-Yang theorem, this enhancement is a feature of perturbative QCD (pQCD) near threshold and does not necessarily imply the existence of a distinct, isolated resonance.
2. Resonance Existence: The top quark's large width ($\Gamma_t \approx 1.5$ GeV) prevents the formation of a narrow toponium resonance. The lifetime of the top quark is comparable to the orbital period of a would-be bound state, meaning any "bump" in the cross-section is smeared by the finite width and phase-space opening.
3. Perturbative Consistency: Claiming an excess "above perturbative QCD" that is "consistent with QCD-predicted toponium" is contradictory. Toponium contributions are intrinsic to the pQCD cross-section, which is known up to NNLO and includes threshold-enhanced terms.
4. Methodological Issues: Experimental analyses often fit data by adding a "toponium" sample (generated via models like the Green's function reweighting method, GFRW) to a standard baseline. This raises concerns about double-counting higher-order corrections already present in the baseline generators and the handling of finite-width effects.

Methodology
The authors present three Next-to-Leading-Order (NLO) event generators capable of interfacing with parton showers (NLO+PS) to compute $t\bar{t}$ production at the LHC, explicitly including threshold-enhanced corrections to all orders in perturbation theory. These generators address the limitations of previous approaches by varying the treatment of finite-width effects and off-shell kinematics:

1. thr1: An extension of the POWHEG-hvq generator. It incorporates threshold corrections up to order $(\alpha_s/v)^6$ (and estimates for higher odd orders) in the narrow-width limit ($\Gamma_t \to 0$). Finite-width effects are introduced via a post-generation mapping of on-shell events to off-shell decays, a method inherited from the original POWHEG-hvq implementation.
2. thr2: A modified POWHEG-hvq where the phase-space generation is exact for off-shell quarks. The matrix elements are computed by projecting the off-shell phase space onto the "nearest" on-shell configuration. This generator includes a rigorous treatment of finite-width effects, including running-width corrections (where the width depends on the virtuality of the decaying particle) and an on-shell mapping scheme that preserves momentum conservation and spin correlations.
3. bb4l: Based on the POWHEG-BOX-RES framework using the full $b\bar{b}4l$ code. This generator accounts for full off-shell effects, including resonant and non-resonant contributions and complete spin correlations, without separating single and double resonant contributions (which is not gauge invariant).

The authors derive a formalism (detailed in Appendix A) for the resummation of threshold effects in the presence of finite widths, valid for fixed invariant mass and fixed virtualities of the decay products. While a full implementation of this exact formula is deferred, the generators use an approximate recipe that reproduces the fully inclusive result of Refs. [3, 4] after integrating over virtualities.

Key Contributions

• High-Order Resummation: The work extends previous calculations (NRR) by including threshold corrections up to $(\alpha_s/v)^6$ and estimating missing odd-order terms, moving beyond the $(\alpha_s/v)^3$ limit of earlier studies.
• Finite-Width Implementation: The paper provides a detailed analysis of how finite top-width effects impact threshold resummation in a Monte Carlo context, distinguishing between narrow-width approximations and exact off-shell treatments.
• Generator Development: The release of three distinct generators (thr1, thr2, bb4l) allows for a systematic comparison of how different treatments of off-shellness and running widths affect the predicted cross-section in the threshold region.
• Spin Decomposition: The generators allow for the separation of spin-singlet and spin-triplet contributions, confirming the dominance of the singlet channel near threshold.

Results

• Convergence of Perturbative Series: The perturbative expansion of the threshold-enhanced corrections converges rapidly. For the ATLAS analysis bin ($m_{t\bar{t}} < 350$ GeV, $p^* < 50$ GeV), the first three orders ($\alpha_s, \alpha_s^2, \alpha_s^3$) capture the vast majority of the enhancement. Higher-order terms contribute less than 1 pb to the total cross-section.
• Finite-Width Effects:
  • In fully inclusive cross-sections, finite-width effects are small (differences between thr1 and thr2 are generally $< 1$ pb).
  • However, when applying fiducial cuts (e.g., restricting top virtuality to a window around the nominal mass), the thr2 and bb4l generators predict a reduction in the cross-section near the nominal threshold compared to thr1. This is attributed to phase-space suppression when one top is highly off-shell; such states are too short-lived to be affected by threshold enhancements. This effect reduces the bound-state contribution by approximately 1 pb in the ATLAS bin.
  • Running Width Effects: Including running-width corrections (where the width scales with virtuality) significantly suppresses single-top contributions in the fully inclusive sample but has a negligible effect on the double-top sample.
• Scale Dependence: The choice of the renormalization scale for the threshold coupling is critical. Using a scale fixed to the invariant mass cut (rather than event-by-event) reduces the predicted threshold enhancement, particularly for larger mass bins.
• Comparison with Experiments: The calculated cross-section enhancement in the ATLAS bin is approximately 4 pb over the standard NLO baseline, leading to a total cross-section of roughly 10 pb. This is consistent with the excess reported by ATLAS and CMS but is significantly lower than the ~6.43 pb "toponium" contribution claimed in some phenomenological fits. The authors note that using an NNLO baseline would further reduce the apparent excess.

Significance and Claims
The paper substantiates the claims made in the NRR work, arguing that:

1. No New Physics Required: The observed excess is consistent with standard perturbative QCD calculations that include threshold resummation and finite-width effects. There is no need to invoke a distinct toponium resonance beyond what is already contained in the pQCD cross-section.
2. Perturbative Adequacy: The perturbative expansion is adequate for describing the cross-section in the ATLAS bin; the expansion does not diverge, and resummation beyond the N3LO level yields negligible corrections.
3. Role of Finite Width: While finite-width effects are small for inclusive observables, they become relevant when specific kinematic cuts are applied, particularly reducing the cross-section for events where the top quarks are far off-shell.
4. Methodological Recommendation: The authors recommend the use of thr2 (with running width effects) or bb4l for phenomenological studies, as thr1 (narrow-width approximation) fails to capture the phase-space suppression effects near the threshold when off-shell cuts are applied.

The work clarifies that the "observation of toponium" is effectively an observation of the well-understood threshold enhancement in pQCD, smeared by the top quark's finite width, and that the magnitude of this effect is smaller than some previous phenomenological estimates suggested.