[Paper Review] Two-loop vacuum diagram through the Symmetries of Feynman Integrals method

Barak Kol|arXiv (Cornell University)|Jul 19, 2018

Algebraic and Geometric Analysis5 references3 citations

TL;DR

This paper applies the Symmetries of Feynman Integrals (SFI) method to the two-loop vacuum diagram (the 'diameter diagram'), demonstrating that the SFI group $GL(2,\mathbb{R})$ acts on the mass and dimension parameters, enabling a complete solution via differential equations and line integrals over simpler diagrams. The method reproduces the known integral value in full generality, with explicit derivation of divergences and a geometric interpretation, validating the approach for complex diagrams.

ABSTRACT

The Symmetries of Feynman Integrals method (SFI) associates a natural Lie group with any diagram, depending only on its topology. The group acts on parameter space and the method determines the integral's dependence within group orbits. This paper analyzes the two-loop vacuum diagram. It is shown how the solution of the SFI equations practically reproduces the most general value of the integral. On the way certain novel derivations are found, a geometrical interpretation is described, and divergences in general dimension are analyzed. These would hopefully be useful for engaging with more involved diagrams.

Motivation & Objective

• To apply the Symmetries of Feynman Integrals (SFI) method to the two-loop vacuum diagram, known as the 'diameter diagram', to test its effectiveness in a non-trivial case.
• To demonstrate that the SFI group $GL(2,\mathbb{R})$ acts on the parameter space of masses and spacetime dimension, with orbits of co-dimension zero, enabling full reconstruction of the integral from a base point.
• To derive the integral’s value in full generality, including divergences in general dimension, and compare it with known results from FJ, Davydov, and Tausk.
• To provide a geometric interpretation of the integral and analyze UV and IR divergences, offering tools for more complex diagrams.

Proposed method

• The SFI method is applied by associating a Lie group $G = GL(2,\mathbb{R})$ to the diagram’s topology, which acts on the parameter space $(x_1, x_2, x_3; d)$ of masses and spacetime dimension.
• The method uses a system of differential equations derived from the SFI formalism: $x_i \partial_{x_i} I - (d-3)I = 0$ and $L_i I + (j_2 - j_1)j_3' = 0$, where $L_i$ are differential operators and $j_i$ are tadpole-derived functions.
• Solutions are found in three ways up to mass-independent terms, with the latter determined by evaluating the integral at specific base points.
• The method reduces the two-loop diagram to a line integral over simpler diagrams (with one edge contracted), leveraging the inductive structure of SFI.
• A geometric interpretation is developed based on the work of Davydov and Delbourgo, relating the integral to a contour in parameter space.
• UV and IR divergences are analyzed by examining the behavior of the integral in general dimension $d$, particularly near poles of the Gamma function.

Experimental results

Research questions

• RQ1 Can the SFI method fully reconstruct the two-loop vacuum diagram’s value in full generality, including divergences in general dimension?
• RQ2 How do the $GL(2,\mathbb{R})$ symmetries of the diagram act on the parameter space, and what is the significance of the orbit co-dimension being zero?
• RQ3 What is the geometric meaning of the integral’s value, and how does it relate to the contour integration used in previous works?
• RQ4 How can the SFI method be used to determine the mass-independent constant term in the solution, and how is it verified?
• RQ5 What are the UV and IR divergence structures of the two-loop vacuum diagram, and how do they match known results?

Key findings

• The SFI method successfully reproduces the full analytical expression for the two-loop vacuum diagram in general dimension $d$, matching known results from FJ, Davydov, and Tausk.
• The solution is derived via three different approaches, with the mass-independent constant determined by evaluating the integral at a base point, confirming consistency.
• The method confirms that the $G$-orbits have co-dimension zero, meaning the integral is completely determined by its value on a single orbit, making the method maximally effective.
• UV and IR divergences are analyzed by examining the poles of the Gamma function $\Gamma(2-d)$ and $\Gamma(3-d)$, with the integral diverging as $d \to 4$ and $d \to 2$, respectively.
• The geometric interpretation reveals that the integral’s value corresponds to a contour integral in parameter space, consistent with the contour $C$ used in FJ1992, thereby validating the SFI approach.
• The final expression for the integral is $I = \pi^d \mu^{d-3} \Gamma(2-d) \frac{2\pi}{\sin(\pi d/2)}$ in the massless limit, confirming positivity and correct analytic structure.