Automatic evaluation of UV and R2 terms for beyond the Standard Model Lagrangians: a proof-of-principle

Abstract: The computation of renormalized one-loop amplitudes in quantum field theory requires not only the knowledge of the Lagrangian density and the corresponding Feynman rules, but also that of the ultraviolet counterterms. More in general, and depending also on the methods used in the actual computation of the one-loop amplitudes, additional interactions might be needed. One example is that of the R2 rational terms in the OPP method. In this paper, we argue that the determination of all elements necessary for loop computations in arbitrary models can be automated starting only from information on the Lagrangian at the tree-level. In particular, we show how the R2 rational and ultraviolet counterterms for any renormalizable model can be computed with the help of a new package, which we name NLOCT and builds upon FeynRules and FeynArts. To show the potential of our approach, we calculate all additional rules that are needed to promote a Two Higgs Doublet Model Lagrangian to one-loop computations in QCD and electroweak couplings.

• The paper introduces the NLOCT package to automate the computation of UV and R2 counterterms in renormalizable BSM Lagrangians.
• The paper leverages FeynRules, FeynArts, and FormCalc to renormalize models and validate results against established SM and MSSM corrections.
• The paper demonstrates that automated loop computations streamline theoretical predictions and support experimental analyses at colliders.

Overview of the Automated Evaluation of UV and $R_2$ Terms for BSM Lagrangians

This paper addresses a critical aspect of modern particle physics: the automation of loop computations for beyond the Standard Model (BSM) theories. It specifically focuses on the calculation of ultraviolet (UV) and $R_2$ rational terms essential for one-loop corrections in quantum field theory scenarios. The author endeavors to demonstrate a method that integrates multiple computational tools to achieve this automation, emphasizing the utility of the new package, NLOCT.

Key Contributions and Methodology

The development of the NLOCT package is at the heart of this research. This package facilitates the automatic computation of UV and $R_2$ counterterms for any renormalizable Lagrangian expressed in the Feynman gauge. The computations are achieved by leveraging existing software packages: \texttt{FeynRules} (FR), \texttt{FeynArts} (FA), and \texttt{FormCalc} (FC). Specifically:

• FeynRules is utilized to implement the renormalization of the models, taking advantage of its ability to process Lagrangians and compute interaction vertices.
• NLOCT builds upon the capabilities of FeynRules to perform the renormalization and output Next-to-Leading Order (NLO) vertices.
• \texttt{FeynArts} and \texttt{FormCalc} are used for writing the amplitudes, integral evaluation, and performing the algebraic operations necessary for extracting loop-induced contributions.

The strength of this approach lies in its capacity to handle arbitrary BSM models automatically without the need for manual intervention, thus enabling researchers to focus more deeply on the phenomenological implications of their theories.

Numerical Results and Validation

The paper provides numerical results for the Standard Model (SM) and the Minimal Supersymmetric Standard Model (MSSM) as test beds for the developed framework. For the SM, the calculations of $R_2$ terms due to QCD and electroweak (EW) corrections were cross-verified with existing results in the literature, showing consistency and correctness. For the MSSM, a similar approach was taken, particularly focusing on strong interaction corrections, and confirming their alignment with known theoretical predictions.

Furthermore, the NLOCT package's capability was demonstrated using the Two Higgs Doublet Model (2HDM). Extensive $R_2$ and UV counterterm computations were presented for both the QCD and EW sectors, highlighting the package's applicability to non-trivial BSM scenarios.

Theoretical and Practical Implications

This research offers significant implications for both theoretical investigations and practical applications in high energy physics. The automated computation of loop corrections affects:

• Theoretical Development: By eliminating the bottleneck of manual loop corrections, theorists can explore more complex models with less computational overhead. It enhances the ability to test various BSM theories against experimental data, particularly in the contexts where precise NLO corrections are essential.
• Experimental Relevance: For ongoing and future collider experiments like the LHC, having precise theoretical predictions including loop corrections is crucial for interpretation of data and identification of potential signals of new physics.

Future Directions

Future developments in this domain might involve extending the framework to encompass effective field theories where non-renormalizable operators are of interest at the loop level. Additionally, enhancing the capability of NLOCT to handle different gauges could provide more comprehensive tools for the community. Such advancements would further bridge the gap between theoretical predictions and experimental observations in particle physics.

In conclusion, this paper presents a significant advancement in the automation of NLO computations for BSM theories, providing a robust framework that promises to enhance theoretical exploration and experimental validation in the search for new physics.