Differential Top Quark Pair Production

• Differential top quark pair production cross sections are probability densities that quantify t-tbar production as functions of kinematic variables, serving as precision tests of perturbative QCD.
• They are computed using advanced techniques such as NNLL resummation, SCET-inspired factorization, and approximate NNLO corrections to incorporate soft and collinear effects.
• Experimental measurements employ multi-differential analyses and unfolding techniques from collider data to constrain PDFs, tune Monte Carlo generators, and search for new physics.

Differential top quark pair production cross sections quantify the probability density for producing a top–antitop ($t\bar{t}$) pair at hadron colliders as a function of kinematic variables such as the top-pair invariant mass, partonic angle, top quark $p_T$, rapidity, or the properties of the $t\bar{t}$ system and its decay products. The calculation and measurement of these cross sections serve as precision tests of perturbative QCD, constrain parton distribution functions (PDFs), inform Monte Carlo event generator tuning, and are essential for searches involving top quark final states.

1. Theoretical Formalism and Factorization Structure

The differential $t\bar{t}$ production cross section at hadron colliders is described at the parton level by

$$\frac{d^2\sigma}{dM\, d\cos\theta} = \frac{8\pi\,\beta_t}{3 s M} \sum_{i, j} \int_\tau^{1} \frac{dz}{z}\; f_{ij}(\tau/z, \mu_f)\; C_{ij}(z, M, m_t, \cos\theta, \mu_f)\,,$$

where $M$ is the $t\bar{t}$ invariant mass, $\theta$ the partonic scattering angle, $\beta_t=\sqrt{1-4m_t^2/M^2}$, $s$ is the hadronic center-of-mass energy squared, $f_{ij}$ are the partonic luminosities, and $C_{ij}$ are hard-scattering kernels encoding virtual and real corrections (Ahrens et al., 2010).

In the limit of “partonic threshold” ($\hat{s}\to M^2$, or $z \to 1$), the available phase space for extra radiation vanishes. In this regime, the kernels are factorized via soft-collinear effective theory (SCET) into

$$C_{ij}(z, M) = \operatorname{Tr} \left[ H_{ij}(M)\; S_{ij}(\sqrt{\hat{s}(1-z)}) \right] + \mathcal{O}(1-z),$$

where $H_{ij}$ encodes hard (virtual) corrections and $S_{ij}$ describes soft gluon emissions. Both are matrices in color space. The plus-distributions

$$\left[ \frac{\ln^m(1-z)}{1-z} \right]_+, \quad m=0,\ldots,2n-1$$

capture the enhanced threshold logarithms at each perturbative order.

2. Resummation and Higher-Order Corrections

Threshold logarithms arise at every order in $\alpha_s$ and must be resummed to achieve reliable predictions near $z\to1$. The renormalization group equations for $H_{ij}$ and $S_{ij}$ allow their large logarithms to be resummed up to next-to-next-to-leading logarithmic (NNLL) accuracy (Ahrens et al., 2010, Kidonakis, 2013). The resummation introduces independent scales:

• Hard scale $\mu_h$ (virtual corrections),
• Soft scale $\mu_s$ (soft radiation),
• Factorization scale $\mu_f$ (PDF evolution).

To optimize predictions, $\mu_h$, $\mu_s$, and $\mu_f$ are chosen independently and varied to estimate scale uncertainties. NNLL resummed formulas are consistently matched to exact NLO for NNLL+NLO predictions. Incorporating two-loop anomalous dimensions yields approximate NNLO predictions that capture dominant threshold effects (Ahrens et al., 2010, Kidonakis, 2012, Guzzi et al., 2014).

The structure at NNLO is paradigmatic: $$\omega_{ij} = \omega_{ij}^{(0)} + \left(\frac{\alpha_s}{\pi}\right)\omega_{ij}^{(1)} + \left(\frac{\alpha_s}{\pi}\right)^2 \omega_{ij}^{(2)}\,,$$ where the second-order term includes soft, collinear, and virtual two-loop corrections with complex plus-distribution structure (Guzzi et al., 2014).

3. Experimental Measurements: Reconstruction and Unfolding

Collider experiments (ATLAS, CMS, D0, CDF) measure differential $t\bar{t}$ cross sections by reconstructing $t\bar{t}$ events in lepton+jets, dilepton, and all-jets final states (Collaboration, 2012, Collaboration, 2014, Collaboration, 2016, Collaboration, 2017, Collaboration, 13 Feb 2024). Event selection requires isolated leptons/b-jets, missing transverse energy, and jet multiplicities.

Kinematic observables include:

• $p_T$ and $y$ of individual top quarks,
• $p_T$, $y$, and invariant mass $m_{t\bar{t}}$ of the $t\bar{t}$ system,
• Properties of leptons, b-jets, and additional jets,
• Correlations (e.g., $\Delta\phi$ between decay products).

The observed spectrum is unfolded to parton or particle level via regularized inversion of response matrices, often using singular value decomposition or iterative Bayesian methods. Detector effects, resolution, inefficiency, and background contamination are systematically accounted for (Collaboration, 2012, Collaboration, 13 Feb 2024, Kawade et al., 2016). Chi-square statistics quantify goodness-of-fit: $$\chi^2 = \mathbf{R}^T\, \mathbf{Cov}^{-1}\, \mathbf{R}$$ with $\mathbf{R}$ residuals and $\mathbf{Cov}$ the full covariance matrix (Collaboration, 13 Feb 2024).

4. Comparison to Theoretical Predictions

Measured differential spectra are compared to fixed-order, resummed, and matched Monte Carlo predictions:

• MC generators: POWHEG, MC@NLO, MG5_aMC@NLO, ALPGEN, interfaced with PYTHIA or HERWIG showers,
• Fixed-order QCD with NNLO and N$^{3}$LO (where available),
• Resummed predictions (NLO+NNLL, NNLO+NNLL, via codes such as DiffTop),
• NNLO+PS (e.g., MiNNLOPS), merging multi-leg NLO matrix elements with parton showers.

Notably, approximate NNLO and NLO+NNLL predictions describe the measured shapes (particularly top $p_T$ and $m_{t\bar{t}}$) more accurately than NLO alone (Collaboration, 2012, Collaboration, 2014, Guzzi et al., 2014, Collaboration, 2017). The uncertainty bands of resummed predictions are narrower, reflecting reduced scale dependence, especially in kinematic regions near threshold.

5. Key Observations and Systematic Deviations

Across experiments and energies, a recurring observation is that the measured top quark $p_T$ spectra are systematically “softer” (steeper) than most Monte Carlo predictions, particularly those based solely on NLO+PS modalities (Collaboration, 2016, Hindrichs, 2018, Martinelli, 2022, Collaboration, 13 Feb 2024). Normalized differential cross sections in high-$p_T$ and high-mass regions typically lie below the generator predictions; this effect persists in boosted topologies ($p_T > 400\,\text{GeV}$) (Collaboration, 2020, Pardos, 2014). NNLO QCD and NLO electroweak corrections bring improved agreement (Hindrichs, 2018).

Discrepancies are more pronounced in multi-differential distributions (e.g., $p_T$ vs. jet multiplicity or invariant mass), where the interplay of additional QCD radiation and kinematics is susceptible to higher-order and shower modeling effects (Collaboration, 13 Feb 2024). Additional jets further modulate the $t\bar{t}$ kinematics, highlighting the incomplete modeling of gluon radiation by state-of-the-art event generators (Collaboration, 2018, Collaboration, 13 Feb 2024).

6. Impact on QCD, PDFs, and Phenomenology

High-precision differential measurements inform and constrain theoretical modeling in several domains:

• Top-pair production as a QCD benchmark: Agreement across many observables and energies validates perturbative QCD and the applicability of threshold resummation, with remaining deviations highlighting where further theoretical work is required (Ahrens et al., 2010, Guzzi et al., 2014).
• PDFs and $\alpha_s$ extraction: $t\bar{t}$ differential data are sensitive to the gluon PDF at moderate and high $x$ and can reduce gluon uncertainties in global fits when incorporated within frameworks like fastNLO/HERAFitter (Guzzi et al., 2014, Martinelli, 2022).
• Top mass and new physics: The high-mass $m_{t\bar{t}}$ and $p_T$ tails can probe heavy resonance production and are sensitive to the value of $m_t$, enabling differentiated new physics searches and increased precision in top mass determination (Collaboration, 2014, Kulesza et al., 2020).
• Monte Carlo tuning and event generator development: Multi-differential spectra and boosted topologies carry information on soft/collinear radiation and promote refinement of parton-shower and matching algorithms (Collaboration, 2018, Collaboration, 2018).

7. Evolving Methodologies and Future Directions

Recent analyses exploit advanced techniques for event reconstruction (deep neural network regressions for missing $p_T$ (Rodríguez, 19 Dec 2024)), analyze kinematic properties of the dineutrino system to probe invisible sectors (Rodríguez, 19 Dec 2024), and introduce pseudo-top observables to anchor measurements to detector-level quantities (Pardos, 2014). Increasingly, measurements are performed in both the particle and parton levels in visible and fiducial phase space, reducing model dependence.

Multi-differential and high-precision absolute and normalized differential cross-section data at 13 TeV and above, spanning high-$p_T$ and high-mass regions, provide a stringent testbed for the Standard Model and for ongoing development of higher-order techniques (e.g., full NNLO+PS, N$^{3}$LO threshold resummation). The systematic deviations observed—particularly the persistent softness of the $p_T$ spectra—indicate the need for improved QCD modeling, optimal scale choices, and further resummation at high multiplicities (Collaboration, 13 Feb 2024).

Table: Theoretical and Experimental Ingredients in Differential $t\bar{t}$ Cross Sections

Aspect	Key Techniques / Observables	Reference Example
Calculational	NNLL resummation, SCET, SCET factorization, NNLO, NNLO+PS	(Ahrens et al., 2010, Guzzi et al., 2014)
Measurement	Unfolding (Bayesian, SVD), kinematic fits, boosted regime	(Collaboration, 2012, Collaboration, 13 Feb 2024)
MC/Comparison	POWHEG, MC@NLO, ALPGEN, MiNNLOPS, DiffTop	(Collaboration, 2016, Guzzi et al., 2014)

Conclusion

The state-of-the-art in differential top quark pair production cross sections combines higher-order perturbative QCD, sophisticated resummation (NNLL, NNLL+NLO, NNLO), advanced experimental methodologies (multi-dimensional unfolding, boosted top reconstruction, deep learning methods for $p_T^{miss}$ inference), and global QCD fits. Experimental results have reached percent-level precision and highlight subtle mismodeling (e.g., persistently soft $p_T$ spectra, especially in multi-differential observables and boosted regimes) that prompt ongoing theory improvements. These measurements remain central for testing QCD, constraining proton structure, and advancing new physics searches at the LHC and beyond (Ahrens et al., 2010, Collaboration, 13 Feb 2024, Rodríguez, 19 Dec 2024).