N-jettiness Subtractions for NNLO QCD Calculations

Abstract: We present a subtraction method utilizing the N-jettiness observable, Tau_N, to perform QCD calculations for arbitrary processes at next-to-next-to-leading order (NNLO). Our method employs soft-collinear effective theory (SCET) to determine the IR singular contributions of N-jet cross sections for Tau_N -> 0, and uses these to construct suitable Tau_N-subtractions. The construction is systematic and economic, due to being based on a physical observable. The resulting NNLO calculation is fully differential and in a form directly suitable for combining with resummation and parton showers. We explain in detail the application to processes with an arbitrary number of massless partons at lepton and hadron colliders together with the required external inputs in the form of QCD amplitudes and lower-order calculations. We provide explicit expressions for the Tau_N-subtractions at NLO and NNLO. The required ingredients are fully known at NLO, and at NNLO for processes with two external QCD partons. The remaining NNLO ingredient for three or more external partons can be obtained numerically with existing NNLO techniques. As an example, we employ our method to obtain the NNLO rapidity spectrum for Drell-Yan and gluon-fusion Higgs production. We discuss aspects of numerical accuracy and convergence and the practical implementation. We also discuss and comment on possible extensions, such as more-differential subtractions, necessary steps for going to N3LO, and the treatment of massive quarks.

• The paper introduces an NNLO subtraction method using the N-jettiness observable to isolate and control infrared singularities in QCD calculations.
• It leverages Soft-Collinear Effective Theory to factorize cross sections into beam, jet, and soft functions, ensuring proper matching of divergent and finite contributions.
• Numerical tests on Drell-Yan and Higgs production validate the robustness of the approach, promising enhanced precision for collider phenomenology.

Overview of "N-jettiness Subtractions for NNLO QCD Calculations"

The paper, N-jettiness Subtractions for NNLO QCD Calculations, presents a method utilizing the $N$-jettiness observable, denoted as $T_N$, to compute next-to-next-to-leading order (NNLO) QCD corrections for various processes. The authors, Gaunt, Stahlhofen, Tackmann, and Walsh, leverage Soft-Collinear Effective Theory (SCET) to systematically address infrared (IR) singularities that plague higher-order QCD calculations. Their approach is grounded on $N$-jettiness, a physically motivated observable that partitions the event phase space into distinct jet and beam regions, allowing for a natural subtraction of the IR singularities.

Technical Approach

The core idea behind this subtraction method is to use $N$-jettiness to control the IR behavior of the cross sections. Utilizing SCET, the authors derive a factorization theorem that effectively decomposes the cross sections into convolutions over beam, jet, and soft functions. These functions capture the radiative behavior of QCD interactions in their respective collinear and soft limits. An essential component of this approach is the matching of the full QCD amplitudes onto SCET operators, enabling a clear separation of divergent and finite contributions.

The NNLO subtractions are constructed by first determining the singular part of the $N$-jettiness distribution. This involves identifying and isolating the terms that become singular as $T_N \rightarrow 0$. The singular contributions are then used to define subtraction terms that can be integrated with the virtual loop corrections. By obtaining explicit expressions for these subtraction terms at both leading and next-to-leading orders, the authors provide a solid foundation for achieving accurate NNLO calculations.

Numerical Implementation and Results

In terms of numerical implementation, the integration over phase space is divided into two regions based on the value of $T_N$: those contributing to $T_N > T_N^{cut}$ are computed by lower-order calculations, coupled with the subtractions calculated for $T_N \leq T_N^{cut}$. The resulting subtracted cross sections are fully differential, making them particularly suitable for interfacing with modern parton shower and resummation techniques.

To demonstrate the effectiveness of their subtraction scheme, the authors apply it to calculate the NNLO rapidity spectrum for two critical processes: Drell-Yan production and gluon-fusion Higgs production. These examples substantiate the robustness and accuracy of the $N$-jettiness subtraction method across different processes involving multiple external partons.

Implications and Future Directions

The implications of this work are profound for both theoretical and practical aspects of QCD computations. The ability to perform NNLO QCD calculations efficiently and with reduced computational complexity holds promise for better precision in collider experiment phenomenology. This precision is crucial for deciphering potential signals of new physics beyond the Standard Model at colliders such as the LHC.

Furthermore, the authors hint at the feasibility of extending their subtraction method to next-to-next-to-next-to-leading order (N$^3$LO), which would address power corrections and other subtle effects that emerge at higher perturbative orders. There is also potential for incorporating massive quarks and more complex final-states, which would broaden the applicability of their method to a wider array of processes of interest in high-energy physics.

Overall, the paper opens several avenues for future research, including refining the subtraction scheme through factorization refinements and exploring the integration of these techniques into Monte Carlo event generators for comprehensive and direct phenomenological applications.