Sequential Discontinuities of Feynman Integrals

• Sequential discontinuities are higher-order analytic structures defined by the iterative application of discontinuity operators to Feynman integrals, revealing complex branch-cut behavior.
• They employ coaction, intersection theory, and geometric deformations to connect analytic continuations with combinatorial and algebraic constraints in multi-loop computations.
• These methods enforce strict constraints—such as Steinmann and genealogical relations—that streamline amplitude bootstraps and clarify underlying kinematic singularities.

Sequential discontinuities of Feynman integrals are higher-order analytic structures that encode how a Feynman integral responds to multiple, ordered analytic continuations in different kinematic invariants. They generalize the standard notion of a discontinuity (the difference in the value of an integral as a branch cut is crossed—equivalently, the change induced by a single unitarity cut) to discontinuities of discontinuities, probing the deepest structure of multi-valuedness in these functions. The paper of sequential discontinuities is central for the analytic bootstrap of amplitudes, for constraining permissible symbol alphabets, for devising reduction strategies, and for exposing the intricate web of algebraic and geometric constraints at play in quantum field theory.

1. Foundational Principles and Definitions

Sequential discontinuities involve applying multiple discontinuity operators in succession, each associated with a branch cut in a specific Mandelstam invariant or kinematic channel. Given a Feynman integral $F$, the basic single discontinuity in variable $s$ is

$\mathrm{Disc}_s\, F = F(s + i0) - F(s - i0).$

The sequential discontinuity in $s_1, s_2$ is expressed as

$\mathrm{Disc}_{s_2}(\mathrm{Disc}_{s_1} F)$

and analogously for higher orders.

The relationship between discontinuities and cut diagrams is established through generalizations of the Cutkosky rules, where each additional discontinuity corresponds to imposing an additional cut (delta function) on the relevant propagators. (Bourjaily et al., 2020) formalizes this rigorously, providing a master formula connecting sequential discontinuities with multi-cut diagrams, weighted by combinatorial coefficients: $$(\mathrm{Disc}_{s_1})^{m_1} \cdots (\mathrm{Disc}_{s_n})^{m_n} F = \sum_{k_i\ge m_i} \cdots B(k_1,m_1)\cdots B(k_n,m_n)(-1)^{\sum m_i -\sum k_i}[k_i{\text{-cuts}}],$$ where $B(k, m)$ are Stirling numbers of the second kind and $[k_i{\text{-cuts}}]$ denotes the integral where $k_i$ propagators are put on shell in channel $s_i$.

Sequential discontinuities can also be interpreted as monodromy operations in the space of external invariants, implemented geometrically via the deformation of Feynman parameter contours (Bourjaily et al., 2020, Britto et al., 15 Oct 2025). The nilpotence of iterated discontinuities in overlapping channels is a reflection of fundamental causality and group-theoretical properties of the underlying Feynman integral, as captured by the Steinmann relations and their generalizations.

2. Algebraic and Geometric Formalism

The algebraic structure of sequential discontinuities is illuminated by the coaction and coproduct formalisms (Abreu et al., 2017, Abreu et al., 2017, Abreu et al., 2018). The coaction operation maps an integral to a sum over pairs of "master integrals" and "contours," splitting analytic information relevant for discontinuities and derivatives: $$\Delta\left(\int_\gamma \omega\right) = \sum_{i} \left[\int_\gamma \omega_i\right] \otimes \left[\int_{\gamma_i} \omega\right]$$ Discontinuity operators act only on the first entry: $$\Delta(\mathrm{Disc}\, F) = (\mathrm{Disc} \otimes \mathrm{id})\,\Delta(F)$$ This formalizes the combinatorics of sequential (iterated) cuts: the first entry in the tensor product is always a "daughter" integral corresponding to a graph with some lines pinched (contracted), while the second is the corresponding cut integral. This mechanism is realized diagrammatically as a sum over graphs with pinched and cut edges, encoding the propagation of discontinuity data through loop hierarchies.

In the projective-geometric approach for one-loop integrals (Gong et al., 2022, Gong et al., 26 Sep 2025, Arkani-Hamed et al., 2017), the analytic structure is tracked by associating 'touching configurations' in Feynman-parameter projective space—where a face of the integration simplex is incident on the singular hypersurface defined by the vanishing of a Symanzik polynomial. Sequential discontinuities are then produced by a hierarchy of projections (point and bi-projections) onto lower-dimensional spaces, and their residues (elementary discontinuities) recursively reconstruct the symbol and thus the full set of discontinuity data.

Methods based on the topological theory of vanishing cycles and Picard-Lefschetz theory (Mühlbauer, 2022, Hannesdottir et al., 2022) further connect the manifestation of sequential discontinuities to the intersection structure and homotopy of integration cycles in compactified parameter space. The Landau equations, which identify critical points at which singularities emerge (pinches), play a central role in determining when and how multi-fold discontinuities can occur.

3. Constraints and Theorems: Steinmann, Extended Steinmann, and Genealogical Constraints

The permitted structure of sequential discontinuities is subject to several powerful constraints.

Steinmann Relations

The original Steinmann relations state that double discontinuities in partially overlapping kinematic channels vanish: $\mathrm{Disc}_{s_1}\mathrm{Disc}_{s_2} F = 0$ when $s_1$ and $s_2$ are not compatible—typically when their cut surfaces cannot be simultaneously satisfied by the Landau equations (Bourjaily et al., 2020, Hannesdottir et al., 2022). This reflects causality and the time-ordering inherent in field theory, also manifest diagram-by-diagram in time-ordered perturbation theory (TOPT).

Extended Steinmann and Cluster-Adjacency

The extended Steinmann property, emerging from amplitude bootstrap studies (He et al., 2021), is a marked empirical observation: for many planar theory amplitudes and their constituent integrals, any two sequential discontinuities in overlapping channels vanish, regardless of whether they are adjacent. Further restrictions, such as cluster-adjacency conditions, associate permissible sequences of symbol entries to algebraic or cluster-theoretic data.

Genealogical and Hierarchical Constraints

Recent developments provide even deeper restrictions. The genealogical constraints (Hannesdottir et al., 10 Jun 2024) introduce an hierarchical structure on allowed sequences of discontinuities: after a minimal cut (the "primogenitor"), only certain "descendant" cuts (sequential discontinuities) are allowed, otherwise the result vanishes even if the channels are not adjacent in the symbol. Formally, for minimal cut $s_1$, the set of permissible next cuts is dictated by the allowed solutions to the Landau equations for subsets including $s_1$, and this family-tree structure can severely reduce the allowed symbol words for high-loop integrals.

This "genealogy" of branch points complements but is not subsumed by extended Steinmann. For example, genealogical constraints disallow certain ordered pairs of discontinuities even if separated by arbitrary other discontinuities. These constraints are robust under dimensional regularization and hold for arbitrary loop order.

4. Methodologies for Computing and Interpreting Sequential Discontinuities

A variety of concrete formalisms are used for the computation and interpretation of sequential discontinuities.

Parametric Integral Deformation and Homology

The analytic continuation of external invariants leads to a deformation of the integration contour in Feynman-parameter space (Britto et al., 15 Oct 2025, Gong et al., 2022). The monodromy (action of a discontinuity operator) replaces the integration domain (usually a positive orthant) with a new domain incorporating additional boundaries where e.g. the Symanzik polynomial $F=0$. Iterated or sequential discontinuities further deform the contour, and if a required boundary is absent after a previous discontinuity, further discontinuity operations must vanish.

Baikov-Lee Representation and Sequential Residues

For multi-loop and multi-scale integrals, the Baikov-Lee representation (Harley et al., 2017) expresses the integrals in terms of scalar-product variables and Gram determinants. Sequential discontinuities correspond to sequential application of Cauchy's residue theorem localizing integration over on-shell conditions, and the sum of all such sequential cuts reproduces the total sequential discontinuity.

The relation between discontinuities and cuts is formalized by

$$\mathrm{Disc}_{s}(I) = -\sum_j I_{s\text{-cut}}^{(j)}$$

where the sum runs over all physically distinct cuts in channel $s$.

Intersection Theory and Maximal Cuts

In the intersection-theoretic approach (Frellesvig et al., 2019), sequential discontinuities connect directly to the structure of intersection numbers between twisted cycles (integration regions with specified singularities) and cocycles (differential forms). Decomposition-by-intersection enables the reduction of maximal-cut integrals and direct access to the algebraic part of symbols, even for integrals involving elliptic (non-polylogarithmic) structure.

5. Analytic Implications, Bootstrap, and Applications

Sequential discontinuities encode crucial information about the analytic and algebraic structure of Feynman integrals.

• Symbol Constraints. The allowed pattern of symbol entries for an integral is largely determined by which sequential discontinuities do not vanish. This is central in modern amplitude bootstraps, drastically reducing the space of possible functions that can occur (He et al., 2021, Hannesdottir et al., 10 Jun 2024, Britto et al., 15 Oct 2025).
• Differential Equations. The algebraic structure revealed by coactions enables the direct derivation of the Picard-Fuchs differential equations for integrals, as only a few "maximal" or "next-to-maximal" sequential cuts need to be computed explicitly (Abreu et al., 2017, Abreu et al., 2018).
• Numerical and Analytical Evaluation. Decomposing an integral into a basis of (quasi-)finite master integrals (Manteuffel et al., 2014) relegates all divergence to an overall factor, making both the extraction and numerical evaluation of sequential discontinuities tractable and directly linked to well-behaved parametric integrals.
• Generalizations to Event Shapes and Jet Observables. Methods for reducing integrals with irregular integration regions (Chen, 25 Aug 2025, Chen, 2020) facilitate analytic studies and allow sequential discontinuities to be tracked systematically even in complex observables like energy correlators. The pattern of w-functions and their derivatives systematically encodes the sequence of cuts.

6. Examples and Case Studies

• Vacuum and Two-Loop Diagrams. The algebraic locus approach (1604.07827) relates threshold sequential discontinuities to the vanishing of Baikov polynomials, leading to algebraic relations among integrals expressed directly at the branch point.
• One-Loop Hexagon and Box Integrals. Explicit geometric analysis in Feynman-parameter space (Gong et al., 26 Sep 2025, Arkani-Hamed et al., 2017) shows how both point and bi-projections yield all elementary (symbol-level) discontinuities, with the full symbol constructed recursively from this data.
• Ladder Integrals and Multi-Loop Boxes. The application of genealogical constraints in high-loop settings (Hannesdottir et al., 10 Jun 2024) systematically prohibits non-genealogical sequences and matches direct computation, dramatically reducing the space of permissible weight $n$ symbol letters.

Table: Overview of Sequential Discontinuity Constraints

Constraint Type	Condition on Channels	Method of Enforcement
Steinmann relations	Overlapping (adjacent)	Landau equation compatibility, monodromy
Extended Steinmann	Any overlap	Observed in planar $\mathcal{N}=4$ SYM, etc.
Genealogical	Non-descendant chains	Minimal cut and family-tree structure

These constraints hold to all orders in $\epsilon$ (the dimensional regulator) and for arbitrary loop order in the correspondingly formulated integral.

7. Outlook and Future Directions

The geometric, algebraic, and combinatorial structure exposed by sequential discontinuities is active terrain for ongoing research. Current directions include:

• Pushing geometric and intersection-theoretic methods to elliptic and more general non-polylogarithmic integrals (Harley et al., 2017, Frellesvig et al., 2019).
• Exploiting genealogical constraints across a broader class of amplitudes and integrals with highly intricate Landau singularity structure (Hannesdottir et al., 10 Jun 2024).
• Automating analysis and reconstruction of symbols by recursive computation of elementary discontinuities in projective Feynman parameter space (Gong et al., 26 Sep 2025, Gong et al., 2022).
• Systematic integration of sequential-discontinuity constraints with cluster adjacency and related combinatorial restrictions for emerging bootstraps in higher-loop and higher-multiplicity amplitudes (He et al., 2021).

These threads point toward a unified analytic-geometric theory that will further organize Feynman integral computation and classification, with sequential discontinuities as a central organizing principle.