[Paper Review] Combinatorial aspects in the one-loop renormalization of higher derivative theories

Christian F. Steinwachs|arXiv (Cornell University)|Sep 2, 2019

Cosmology and Gravitation Theories36 references4 citations

TL;DR

This paper presents a combinatorial algorithm to efficiently compute one-loop counterterms in higher derivative quantum field theories by avoiding explicit tensor contractions. By leveraging combinatorial coefficients derived from symmetric tensor products, the method drastically reduces computational proliferation in computer algebra systems, enabling scalable renormalization of complex diagrams with high-rank vacuum tensor integrals—particularly relevant for Galileon and other higher-derivative models.

ABSTRACT

An efficient way to calculate one-loop counterterms within the Feynman diagrammatic approach and dimensional regularization is to expand the propagators in the integrands of the Feynman integrals around vanishing external momentum. In this way, a generic one-loop diagram is reduced to a sum of vacuum diagrams. The logarithmically divergent part can be extracted by power counting arguments. In case of higher derivative theories, the standard implementation of this procedure on a computer algebra system can become quickly inefficient due to a high proliferation of terms coming from the intermediate replacement of high-rank tensor-integrals with symmetrized product of metric tensors. In this note we present a simple combinatorial solution to this problem which makes the implementation much more efficient. This method is especially relevant in the renormalization of higher derivative theories, but might as well be integrated as a standard routine in existing computer algebra programs designed to automatize Feynman diagrammatic calculations.

Motivation & Objective

• Address the computational inefficiency in one-loop counterterm calculations for higher derivative field theories due to tensor contraction proliferation.
• Develop a systematic method to compute logarithmically divergent parts of vacuum tensor integrals without explicit tensor algebra.
• Enable scalable automation of Feynman diagram calculations in higher derivative theories using combinatorial coefficients.
• Provide a generalizable framework applicable to scalar, spinor, and vector fields with multiple masses.
• Integrate the method as a standard routine in existing computer algebra programs for automated loop calculations.

Proposed method

• Expand propagators in Feynman integrals around vanishing external momenta to reduce diagrams to vacuum tensor integrals.
• Use power counting to isolate logarithmically divergent parts, which are expressed as symmetric tensor products of metric tensors.
• Introduce combinatorial coefficients $ B_k $ derived from partitioning the total rank of the tensor integral into symmetric invariants.
• Define $ B_k = C_k / P_ u(4) $, where $ C_k $ counts tensor contraction symmetries and $ P_ u(4) $ is a dimensional polynomial.
• Apply the method to scalar theories via explicit examples (e.g., Table 5), showing how partitions $ k o ( ho_{11}, ho_{12}, ho_{22}) $ generate the divergent structure.
• Extend the formalism to non-zero spin and multiple masses by modifying the kinematic invariants and combinatorial weightings.

Experimental results

Research questions

• RQ1 How can the computational cost of tensor contraction in high-rank vacuum integrals be minimized in one-loop calculations?
• RQ2 What combinatorial structure underlies the symmetric contraction of high-rank metric tensors in vacuum tensor integrals?
• RQ3 Can a closed-form expression for the divergent part of one-loop integrals be derived without explicit tensor algebra?
• RQ4 How does the method scale with increasing number of external legs and loop momenta in higher derivative theories?
• RQ5 To what extent can this algorithm be generalized to fields with spin and multiple masses?

Key findings

• The method avoids explicit tensor contractions by using combinatorial coefficients $ B_k $, reducing computational complexity in high-rank integrals.
• For a 3-loop scalar diagram with $ g_3^3 $ coupling, the divergent part is computed as $ iI^{ ext{div}} = -i\frac{432g_3^3}{(4\pi)^2\varepsilon} \left[ \cdots \right] $, explicitly showing the divergent structure in $ q_1^2, q_1\cdot q_2, q_2^2 $.
• The divergent amplitude vanishes on-shell ($ k_1^2 = k_2^2 = 0 $), confirming consistency with kinematic constraints.
• The algorithm is general and applicable to higher derivative theories such as Galileon models, where standard methods fail due to high-rank tensor structures.
• The method is scalable and can be integrated as a standard routine in computer algebra systems like FeynCalc or QCDLoop.
• The combinatorial coefficients $ B_k $ are derived from symmetric tensor partitions and are independent of the spacetime dimension in the $ \varepsilon $-regularization scheme.