Tracking discontinuities in parameter space

TL;DR

This work develops a geometric framework in Feynman-parameter space to constrain the analytic structure of Feynman integrals by tracking how integration contours deform under analytic continuation of external kinematics. By interpreting discontinuities and monodromies through modified integration contours and Picard-Lefschetz vanishing cycles, it introduces sequential-discontinuity constraints that complement genealogical/landau-based relations in a bootstrap program. The authors apply the method to a range of one- and two-loop triangles and ice-cream-cone topologies, obtaining concrete symbol-level constraints and explicit integral expressions in several cases, including weight-two and weight-three results in both uniform and non-uniform weight settings, and in integer and non-integer dimensions. The approach offers a principled way to deduce or constrain the analytic structure of amplitudes from first principles, with potential extensions to higher loops, longer discontinuity sequences, and a more complete homological treatment of Feynman integrals. Overall, the paper demonstrates that sequential discontinuities, when read from the contour dynamics in parametric space, are powerful inputs for the perturbative bootstrap of Feynman integrals, enabling precise symbol-level reconstructions and guiding future explorations of the Landau/Steinmann-type constraints in broader quantum field theories.

Abstract

We develop a geometric framework in Feynman-parameter space to determine constraints on the sequential discontinuities of Feynman integrals. Our method is based on tracking the deformation of the integration contour as external kinematics are analytically continued. This procedure imposes powerful constraints on the analytic structure of Feynman integrals, providing crucial inputs for their bootstrap. We demonstrate the usefulness of this framework by applying it to integrals in dimensional regularization, with higher propagator powers, and to examples with non-uniform transcendental weight. The method is illustrated with several one- and two-loop calculations.

Figures (16)

• Figure 1: The analytic structure of $\mathcal{I}$ in the complex $s$ plane.
• Figure 2: Top left. The original integration contour for $\mathcal{I}$ avoids the singularity of the integrand by a deformation in the upper half-plane. Top right. During analytic continuation of $s$ in the complex plane in a counterclockwise circle around $s=0$, the root at $x=-s$ traverses around $x=0$, dragging the integration contour with it. Bottom left. After the analytic continuation in $s$, the integration contour has been suitably modified. Bottom right. After subtracting the original contour, we see that the contour representing the monodromy of $\mathcal{I}$ around $s=0$ is given as a contour around the branch cut.
• Figure 3: The vanishing cycles $\Gamma$ are shaded, for the respective discontinuities in $p^2$, $m_1^2$, and $m_2^2$, each shown in their preferred kinematic region, as seen by the values listed in the plots.
• Figure 4: The contour $\Gamma_{m_1^2}$ near the singularity of $m_1^2 \otimes m_1^2$, as the sign of $m_1^2$ is taken from negative to positive. The region shrinks to a point at $m_1^2=0$ and reappears outside the integration region of the Feynman integral.
• Figure 5: The contour $\Gamma_{m_2^2}$ near the singularity of $m_2^2 \otimes m_1^2$. The region experiences no singularity, and the sequence is absent from the symbol.