Toponium hadroproduction at next-to-next-to-leading order in NRQCD

Technical Summary: NNLO Predictions for Inclusive Color-Singlet Toponium Hadroproduction

Problem and Motivation
Top-quark pair production near the kinematic threshold probes QCD binding dynamics in a regime accessible to perturbation theory. In the attractive color-singlet channel, repeated Coulomb exchange generates would-be bound-state poles, which are smeared by the finite top-quark width into a broad enhancement of the production rate. While these dynamics have been extensively studied in $e^+e^-$ annihilation (reaching N$^3$LO accuracy) and underpin proposed FCC-ee threshold scans, precise Standard Model predictions for hadron colliders have been limited.

At hadron colliders, the leading bound-state enhancement arises from the color-singlet, spin-singlet S-wave state $\eta_t$. Recent observations by CMS and ATLAS of excess events near the $t\bar{t}$ threshold in dilepton and single-lepton channels are consistent with a color-singlet pseudoscalar contribution. However, a complete theoretical description requires control over both hard production and nonrelativistic dynamics. While next-to-leading-order (NLO) QCD corrections to the short-distance coefficient (SDC) for pseudoscalar hadroproduction have long been known, the corresponding complete next-to-next-to-leading-order (NNLO) QCD correction has remained missing. Determining this correction is essential to establish the perturbative stability of the production rate and to disentangle uncertainties in hard production from those associated with binding dynamics.

Methodology
The authors compute the first NNLO predictions for inclusive color-singlet toponium hadroproduction within Non-Relativistic QCD (NRQCD), working in the zero-width limit for the top quark. The calculation adopts the power counting $\alpha_s \sim v \sim \sqrt{\alpha} \ll 1$, where $v$ is the nonrelativistic top-quark velocity.

The cross section is factorized into short-distance coefficients (SDCs) and long-distance matrix elements (LDMEs). Through NNLO, the total cross section includes contributions from:

1. S-wave states: The spin-singlet $\eta_t$ and spin-triplet $\Theta_t$.
2. P-wave states: The spin-triplet $\chi_{tJ}$ ($J=0, 2$).
3. Relativistic corrections: $O(v^2)$ corrections to $\eta_t$ production.

The central new result is the NNLO QCD correction ($\hat{\sigma}^{(2)}_{\eta_t}$) to the SDC for $pp \to \eta_t + X$. This calculation utilizes nested soft-collinear subtraction and two-loop amplitudes from previous literature, employing the history method for integration. To improve perturbative stability, the soft-improved matching (SIM) scheme is used for NRQCD matching.

The methodology also incorporates:

• Electroweak (EW) and Relativistic Corrections: Including NLO EW corrections to $gg \to \eta_t$ and tree-level associated production ($gg \to \eta_t H, \eta_t Z$, etc.).
• Spin-Triplet Contributions: The leading QCD contribution to $\Theta_t$ arises at NLO ($gg \to \Theta_t g$), with NNLO corrections included. Tree-level EW processes ($q\bar{q} \to \gamma^*/Z^* \to \Theta_t$) are also included.
• Matrix Elements: S-wave LDMEs are evaluated including NNLO QCD and NLO QED corrections, with RG evolution applied. The leading top-Yukawa contribution is retained due to the large coupling $y_t \approx 1$.
• Parameters: The calculation uses NNPDF40 NNLO PDFs, $m_t = 172.6$ GeV, and the $G_\mu$ EW scheme. Scale uncertainties are estimated via standard seven-point variations ($\mu_R, \mu_F \in \{m_t/2, m_t, 2m_t\}$), and Coulomb-scale uncertainties are assessed via $\mu_C \in \{\mu_B/2, \mu_B, 2\mu_B\}$.

Key Results
At the LHC with $\sqrt{s} = 13$ TeV, the authors obtain a total inclusive cross section of:
$$\sigma_{\text{NNLO}} = 4.87^{+0.63}_{-0.65} \text{ pb}$$
This value combines scale and parametric uncertainties.

• Perturbative Stability: The NNLO QCD corrections reduce the renormalization- and factorization-scale uncertainties significantly relative to NLO. Specifically, the width of the relative scale variation decreases by more than a factor of three at NLO and by a further factor of approximately four at NNLO, bringing the residual uncertainties to the few-percent level. At 13 TeV, the residual NNLO scale uncertainty is $+0.5\% / -3.0\%$.
• Correction Magnitudes:
  • The NLO QCD correction increases the LO cross section by approximately 28%.
  • The NNLO QCD correction reduces the NLO result by approximately 5% at 13 TeV.
  • The NLO EW correction reduces the LO cross section by approximately 7%, driven primarily by Higgs-exchange contributions proportional to $y_t^2$.
  • Relativistic ($v^2$) and P-wave contributions, as well as associated production channels, remain below 1% in magnitude.
• Uncertainty Breakdown:
  • Dominant Uncertainty: The Coulomb-scale dependence of the bound-state matrix elements accounts for approximately $\pm 12\%$ of the total uncertainty. This uncertainty is largely independent of the perturbative order and collision energy.
  • Parametric Uncertainties: Variations in $\alpha_s(m_Z)$ contribute $\pm 5\%$, and variations in $m_t$ contribute $\pm 0.8\%$.

Significance and Claims
The paper claims to present the first NNLO predictions for inclusive color-singlet toponium hadroproduction within NRQCD. The central new result is the NNLO QCD correction to the short-distance coefficient for pseudoscalar toponium production.

The authors emphasize that these new short-distance coefficients provide a "first necessary step" towards an NNLO prediction for the physical top-quark pair threshold enhancement. The work establishes the perturbative stability of the production rate, demonstrating that higher-order corrections are under control once scale variations are properly assessed. The authors note that while the current calculation is in the zero-width limit, the new coefficients are an essential ingredient for extending finite-width threshold calculations to NNLO within the Green's-function approach, which will connect these improved production coefficients to the physical threshold spectrum measured at the LHC. The dominant remaining uncertainty highlights the priority of evaluating bound-state matrix elements at higher orders (N$^3$LO) for further theoretical improvement.