Dy-YaRN: Drell–Yan Resummation & Computation

• Dy-YaRN is a computational framework that combines NNLL resummation with NNLO fixed-order methods for precise Drell–Yan lepton-pair production modeling.
• The approach utilizes factorized phase space integration, efficient quadrature methods, and OpenMP parallelization to significantly speed up computations.
• It delivers accurate predictions for key observables, supporting PDF fits, Standard Model parameter extraction, and background modeling in new physics searches.

Dy-YaRN refers to the intersection of Drell–Yan (DY) QCD theory and computational frameworks for rapid, high-fidelity predictions in collider processes—specifically as implemented in the DYTurbo program and related resummation and fixed-order matching methodologies (Camarda et al., 2019). The term highlights both the deep theoretical structure of Drell–Yan lepton-pair production (including its resummed and exclusive aspects) and the numerical strategies that allow precise and scalable evaluation of its observables. This article surveys core principles, technical frameworks, methodologies, and the interpretative context for Dy-YaRN as a research area.

1. Theoretical Foundations of Drell–Yan Production

The Drell–Yan process concerns the production of lepton pairs ($e^+e^-$, $\mu^+\mu^-$, etc.) via electroweak bosons ($\gamma^*/Z/W$) in hadronic collisions. Its theoretical precision and phenomenological importance stem from several factors:

• The hard scattering cross section is perturbatively calculable to high order in QCD ($\mathcal{O}(\alpha_S^2)$ and beyond).
• The process serves as a “standard candle” for parton density determination and for the extraction of Standard Model parameters.
• The inclusive and exclusive observables (e.g., boson rapidity $y$, dilepton invariant mass $m$, transverse momentum $q_T$) are sensitive to both hard and soft QCD effects.

Modern approaches to Drell–Yan incorporate matched resummation and fixed-order predictions, enabling reliable calculation from the high $q_T$ regime (dominated by hard radiation) down to the soft/collinear region where resummation of logarithms is necessary.

2. Fast Computational Approaches: The DYTurbo Program

DYTurbo (Camarda et al., 2019) is the culmination of algorithmic and theoretical improvements for Drell–Yan differential cross sections and distributions. Its key innovations are:

• Combined Resummed and Fixed-order Formalism: The cross section is decomposed as

$$d\sigma^{(V)}_{NNLL+NNLO} = d\sigma^{res}_{NNLL} - d\sigma^{asy}_{NNLO} + d\sigma^{f.o.}_{NNLO}$$

where $d\sigma^{res}$ captures Sudakov-resummed logarithms, $d\sigma^{asy}$ removes double counting in the small $q_T$ limit, and $d\sigma^{f.o.}$ supplies the fully differential fixed-order correction.

• Factorization and Integration Strategies: The phase space is factorized into production variables (e.g., $q_T$, $y$, $m$) and decay angles. Integration over the angular variables is done using Clenshaw–Curtis and double exponential quadrature, improving both speed and numerical stability relative to previous Monte Carlo approaches.
• Optimized Software Design: DYTurbo employs modular C++ code, OpenMP multithreading, and fork-based parallelization (Cuba library), achieving at least fivefold speedup compared to predecessor codes such as DYqT and DYRes.

Table: Key Features of DYTurbo

Aspect	Technical Realization	Impact
Algorithm	Factorized phase space, quadrature	Efficient integration
Resummation Order	NNLL accuracy (impact parameter space)	Small $q_T$ stability
Matching	NNLO fixed-order terms	Full kinematic fidelity
Parallelization	OpenMP, fork/wait (Cuba)	Scalable on multicores

3. Resummation and Observables

The accurate prediction of the Drell–Yan $q_T$ spectrum—in particular, at small transverse momentum—requires next-to-next-to-leading logarithmic (NNLL) resummation. This is performed in impact parameter $b$-space, using a Sudakov exponentiation: $$d\sigma^{res}(q_T) \sim \int_0^\infty b\,db\,J_0(b q_T)\,\exp\{\mathcal{G}(b)\}\,\mathcal{H}^{(V)}(m^2)$$ Here $\mathcal{G}$ defines the resummed Sudakov factor (including soft, collinear, and hard scales), $\mathcal{H}^{(V)}$ is the hard function, and $J_0$ denotes the Bessel function for Fourier inversion. Matching to fixed-order is achieved via the $q_T$-subtraction formalism, guaranteeing that power corrections vanish as $q_T \to 0$.

Angular distributions are factorized and computed via efficient quadrature. Lepton spin correlations and finite-width effects are included via full kinematic treatments, enabling comparison to experimental shape and normalization data.

4. Applications in Standard Model Precision, PDF Fits, and New Physics Searches

Dy-YaRN methodology as realized in DYTurbo addresses several experimental requirements:

• PDF Determination: Rapid evaluation of theoretical cross sections for global fits, enabled by code integration with grid-based frameworks (e.g., APPLGRID) for fast reweighting.
• Standard Model Parameter Extraction: Observables such as the $W$ mass, weak mixing angle, and $Z$ production rate rely on the accuracy of the predicted distributions for extraction from collider measurements.
• Background Modeling: In new physics searches, both the normalization and shape of Drell–Yan backgrounds (in high mass regions or in channels with jet vetoes, for example) are essential; the full $q_T$, $y$, $m$, and angular dependence control systematic uncertainties.

Comparison with ATLAS, CMS, E605, and other datasets demonstrates the high fidelity and numerical precision of the DYTurbo approach—often with per-mille level numerical accuracy and runtime scaling that is practical for large scale analyses.

5. Numerics, Validation, and Extensions

Computational correctness is validated by closure tests: fixed-order expansions of the resummed component are numerically checked to reproduce the singular behavior at low $q_T$. Global unitarity constraints are met with deviation well below $10^{-6}$. Dedicated tests ensure kinematic coverage and precision in all relevant differential distributions.

Performance benchmarks highlight reduction of computational time by factors of five or more over previous codes, with error bars (from quadrature/interpolation) well-controlled. The modular structure and threading ensure scalability with processor count.

Possible extensions include further reduction of bottlenecks in the fixed-order term, inclusion of additional electroweak corrections, and improved modeling of nonperturbative effects (e.g., more sophisticated treatment within the Sudakov exponent).

6. Interpretative Context and Future Directions

Dy-YaRN marks a convergence between advanced QCD theory (NNLL/NNLO matching, resummation, subtraction) and high-performance computational physics. The precise factorization and resummation frameworks ensure that all major components—production, decay, and angular correlations—are evaluated both rapidly and accurately, suitable for precision LHC analyses and rapid phenomenological paper.

Future research is likely to pursue tighter code integration with global fitting tools, incorporation of augmented (beyond NNLO) electroweak and QCD corrections, and continued algorithmic development to exploit emerging hardware architectures. Improvements in nonperturbative modeling may further enhance the reliability of ultra-precise collider predictions.

The Dy-YaRN framework will remain central to contemporary high-energy physics, providing the computational backbone for Standard Model studies and background estimations in discovery-oriented experiments (Camarda et al., 2019).