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Analytic and Numerical Approaches for Evaluating 3-Loop Integrals Using Sector Decomposition


Core Concepts
The authors present analytic and numerical methods for evaluating the coefficients of 3-loop two-point function integrals using a simplified sector decomposition approach.
Abstract

The paper focuses on the evaluation of 3-loop two-point function integrals using a simplified sector decomposition (SD) method. The key points are:

  1. The authors derive analytical expressions for the coefficients of the ultraviolet (UV) divergent terms in the Laurent series expansion of the integrals. These coefficients are obtained using the SD method.

  2. For the finite parts of the integrals, the authors employ numerical integration techniques, including adaptive integration, the double-exponential formula, and Quasi-Monte Carlo methods. They use extrapolation methods to handle the singular behavior of the integrands.

  3. Four specific 3-loop two-point function diagrams, referred to as Loop (I), (II), (III), and (IV), are studied in detail. The authors provide the analytical expressions for the UV divergent coefficients and present the numerical results for all the coefficients up to the constant term.

  4. The numerical results are shown to agree well with previous analytical and numerical studies, demonstrating the effectiveness of the combined analytic and numerical approach.

  5. The authors discuss the handling of pseudo-thresholds that appear in the numerical integration and note that further study is required to fully understand their impact on the high-precision computation.

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Stats
The paper provides the following key numerical results: For Loop (I), the coefficients C-2, C-1, C0, and C1 are presented as a function of the squared external momentum s. For Loop (II), (III), and (IV), the coefficients C-2, C-1, C0, and C1 are presented as a function of s. The numerical results are shown to agree well with previous analytical and numerical studies, with the number of digits in agreement ranging from 3 to 12, depending on the specific loop and coefficient.
Quotes
"The sector decomposition (SD) method [1, 2, 3, 4, 5, 6] is well-established for calculating loop integrals." "We have succeeded to manage the singularity from both the UV-divergence and in the physical kinematics below and beyond the threshold in the squared momentum." "The numerical method developed by the authors is able to calculate the coefficients of the Laurent series in the kinematical region both below and above the threshold."

Deeper Inquiries

How can the presented approach be extended to handle more complex multi-loop integrals with non-"complete" U functions or different mass configurations?

The approach presented in the paper can be extended to handle more complex multi-loop integrals by adapting the sector decomposition (SD) method to accommodate non-"complete" U functions and varying mass configurations. For non-"complete" U functions, additional transformation steps may be required to ensure that all necessary monomials are included in the integrals. This can involve identifying the missing monomials and applying further variable transformations to achieve a complete representation of the U function across all integration regions. In cases where the internal masses are not uniform, the integration must be performed region by region, taking into account the specific mass configurations for each loop. This necessitates a more intricate setup of the integration parameters and may require the development of new algorithms to efficiently compute the integrals. The use of advanced numerical techniques, such as adaptive integration methods and extrapolation techniques, can also enhance the accuracy of the results in these more complex scenarios. Moreover, the framework established in the paper can be generalized to include additional loop diagrams and higher-order corrections, allowing for a broader application in theoretical calculations. By systematically addressing the challenges posed by non-"complete" U functions and diverse mass configurations, researchers can leverage the SD method to tackle a wider array of multi-loop integrals in particle physics.

What are the potential applications of the accurate evaluation of these 3-loop two-point function integrals in high-precision calculations for particle physics processes?

The accurate evaluation of 3-loop two-point function integrals has significant implications for high-precision calculations in particle physics. These integrals are crucial for determining radiative corrections in various processes, such as scattering amplitudes and decay rates of particles. As experimental data becomes increasingly precise, the theoretical predictions must match this level of accuracy to validate the Standard Model and search for new physics beyond it. One of the primary applications is in the context of precision electroweak measurements, where 3-loop corrections can influence observables like the W and Z boson masses, as well as the effective weak mixing angle. Additionally, these integrals play a vital role in the calculation of cross-sections for processes at colliders, such as the Large Hadron Collider (LHC), where understanding the contributions from virtual particles is essential for interpreting experimental results. Furthermore, the insights gained from these calculations can aid in the identification of new particles and interactions, as discrepancies between theoretical predictions and experimental outcomes may indicate the presence of new physics. The ability to compute these integrals accurately also enhances the reliability of simulations used in event generation for collider experiments, thereby improving the overall understanding of particle interactions.

What insights can be gained from the study of the pseudo-thresholds observed in the numerical integration, and how can they be better understood and managed?

The study of pseudo-thresholds in numerical integration provides valuable insights into the behavior of loop integrals, particularly in relation to the kinematic regions where singularities may arise. Pseudo-thresholds are indicative of points in the parameter space where the integrand exhibits significant changes in behavior, often leading to complications in numerical evaluations. Understanding these thresholds is crucial for ensuring the accuracy and stability of the numerical integration process. One key insight is that while pseudo-thresholds can introduce challenges, they also highlight the importance of careful treatment of the integration limits and the behavior of the integrand near these critical points. By analyzing the contributions from different regions of integration, researchers can identify patterns in how these thresholds affect the overall integral. This understanding can lead to the development of more sophisticated numerical techniques that specifically address the singular behavior associated with pseudo-thresholds. To better manage these challenges, it is essential to implement robust numerical strategies, such as adaptive integration methods that can dynamically adjust to the behavior of the integrand. Additionally, employing techniques like double extrapolation can help mitigate the impact of pseudo-thresholds on the final results. By systematically studying the effects of these thresholds and refining the numerical methods used, researchers can enhance the precision of their calculations and gain deeper insights into the underlying physics of multi-loop integrals.
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