The calculation of 2-loop self-energy diagrams by the sector decomposition

Technical Summary: Calculation of 2-Loop Self-Energy Diagrams by Sector Decomposition

Problem Statement
The paper addresses the computational challenge of evaluating 2-loop self-energy integrals for scalar particles, a critical component in high-precision radiative corrections within the electro-weak theory. Specifically, the work targets the one-particle irreducible (1PI) self-energy diagrams ($S_1$ through $S_8$) and tadpole diagrams ($T_1$ through $T_3$) relevant to the study of the Higgs particle. The primary difficulty lies in managing two types of singular behaviors inherent in loop integrals: ultraviolet (UV) divergences and threshold singularities. While the Standard Model (SM) features 3- and 4-particle vertices, the presence of massive particles in loop diagrams necessitates robust methods for multi-loop integration. The paper focuses on diagrams $S_1, S_2, S_3,$ and $S_4$, as the remaining diagrams can be reduced to products of 1-loop contributions or evaluated at zero momentum transfer ($s=0$).

Methodology
The author employs the Sector Decomposition (SD) method to systematically isolate UV divergences from finite parts. The approach utilizes dimensional regularization ($n = 4 - 2\epsilon$), where UV divergences manifest as inverse powers of $\epsilon$.

1. Integral Formulation:
   The loop integrals are expressed using the Nakanishi-Kinoshita-Cvitanovi´c formula, converting momentum space integrals into Feynman parameter space. The integral $I$ is decomposed into a prefactor involving Gamma functions and a parametric integral involving the Symanzik polynomials $U$ and $V$ (where $V = M^2 - sW/U$).

2. Variable Transformation and Sector Decomposition:

   • Initial Transformation: Specific variable transformations are applied to the Feynman parameters ($x_j$) for each diagram type ($S_1$ to $S_4$) to simplify the structure of the $U$ and $W$ functions.
   • Sector Partitioning: The integration domain is divided into $3! = 6$ sectors based on permutations of the variables $y_1, y_2, y_3$. Within each sector $(k, l, m)$, new variables $(s_k, t_l, u_m)$ are introduced. This transformation renders the $U$ function "complete" (containing all possible monomials), which is advantageous for SD.
   • Singularity Extraction: The transformation isolates the singular behavior into a single variable $t$. The $U$ function scales as $s_k^2 t_l f$, and $W$ scales as $s_k^3 t_l q$, where $f$ is non-singular. This allows the UV divergence to be explicitly factored out as a power of $t$ in the integrand.

3. Handling Numerators:
   The paper extends the scalar integral formalism to cases with non-trivial numerators dependent on loop momenta ($\ell_1, \ell_2$). Using a transformation described in Appendix A, the numerator is rewritten in terms of the Feynman parameters. The resulting integrand is expanded into terms involving powers of $V/U$. The coefficients of these terms ($F_0, F_2, F_4$) are derived from the specific momentum flow assignments for each diagram.

4. Integration Strategy:
   The final integrals are categorized by the power of $t$ in the denominator ($t^{r-\epsilon}$).

   • $J_0$ type: No singularity at $t=0$; integrable directly.
   • $J_1, J_2$ types: Contain singularities; handled via specific integration procedures (detailed in Appendix C) to extract the $\epsilon$-expansion.
   • The paper notes that $J_3$-type integrals do not appear in these specific 2-loop self-energy calculations.

Key Contributions

• Explicit Formulae: The paper provides detailed, ready-to-use formulae for the 2-loop self-energy integrals of diagrams $S_1, S_2, S_3,$ and $S_4$, covering both scalar cases and those with non-trivial numerators.
• Systematic SD Application: It demonstrates a "simple form" of the SD method applied to 2-loop integrals, successfully separating UV divergences from finite parts without requiring complex adaptive integration strategies for the initial separation.
• Massive Particle Handling: The formalism is explicitly constructed to handle massive propagators, a necessity for electro-weak theory calculations involving the Higgs and gauge bosons.
• Dual Computation Path: The resulting expressions are structured to support both analytic expansion (for leading terms) and numerical computation (for finite results).

Results and Computational Approach
The derived expressions allow the calculation of the loop integral $I$ as a series in $\epsilon$. The required order of the expansion depends on the diagram:

• $S_1$: Requires the $O(1)$ term.
• $S_2$: Requires $O(\epsilon^{-1})$ and $O(1)$ terms.
• $S_3, S_4$: Require $O(\epsilon^{-1})$, $O(1)$, and $O(\epsilon)$ terms.

The paper outlines two numerical strategies for obtaining results:

1. Symbolic Expansion: Expanding the formula in $\epsilon$ symbolically and integrating each term numerically.
2. Extrapolation: Computing the formula numerically for several values of $\epsilon$ and estimating the series coefficients via extrapolation or linear solvers.
   The author states that both methods have been tested in preceding research and yield good results.

Significance
The paper claims that the presented detailed formulae enable the efficient calculation of 2-loop self-energies for scalar particles, specifically aiding the study of the Higgs particle within the electro-weak theory. By effectively managing UV divergences through sector decomposition, the method provides a pathway to renormalize these divergences and obtain finite physical results. The work is positioned as a foundational tool for further studies of radiative corrections at the 2-loop level, particularly in scenarios involving massive particles where standard methods may be cumbersome. The formalism is noted to be extendable to fermions and vector particles with appropriate form factors.