What is the $i\varepsilon$ for the S-matrix?

TL;DR

<3-5 sentences>This work develops a modern analytic framework for the perturbative S-matrix that remains causal when complexifying Mandelstam invariants, even in the presence of massless fields, UV/IR issues, and unstable particles. It replaces the traditional Feynman iε prescription with a branch-cut deformation strategy in the Schwinger-parameter (worldline) representation, enabling a causal sheet to be accessed without altering the underlying analytic structure. Central tools include holomorphic unitarity cuts, generalized dispersion relations, and a Lefschetz-thimble viewpoint that clarifies the local behavior of Feynman integrals near thresholds. The results provide practical prescriptions for computing imaginary parts, discontinuities, and unitarity cuts in realistic theories, with explicit one-loop examples and a clear path toward handling higher-multiplicity processes in the Standard Model.

Abstract

Can the S-matrix be complexified in a way consistent with causality? Since the 1960's, the affirmative answer to this question has been well-understood for $2 \to 2$ scattering of the lightest particle in theories with a mass gap at low momentum transfer, where the S-matrix is analytic everywhere except at normal-threshold branch cuts. We ask whether an analogous picture extends to realistic theories, such as the Standard Model, that include massless fields, UV/IR divergences, and unstable particles. Especially in the presence of light states running in the loops, the traditional $i\varepsilon$ prescription for approaching physical regions might break down, because causality requirements for the individual Feynman diagrams can be mutually incompatible. We demonstrate that such analyticity problems are not in contradiction with unitarity. Instead, they should be thought of as finite-width effects that disappear in the idealized $2\to 2$ scattering amplitudes with no unstable particles, but might persist at higher multiplicity. To fix these issues, we propose an $i\varepsilon$-like prescription for deforming branch cuts in the space of Mandelstam invariants without modifying the analytic properties. This procedure results in a complex strip around the real part of the kinematic space, where the S-matrix remains causal. In addition to giving a pedagogical introduction to the analytic properties of the perturbative S-matrix from a modern point of view, we illustrate all the points on explicit examples, both symbolically and numerically. To help with the investigation of related questions, we introduce a number of tools, including holomorphic cutting rules, new approaches to dispersion relations, as well as formulae for local behavior of Feynman integrals near branch points.

Figures (38)

• Figure 1.1: Analytic structure of the matrix element $\mathbf{T}_\mathbb{C}(s,t_\ast)$ for $2\to 2$ scattering of the lightest state of mass $M$ in theories with a mass gap in the complex $s$-plane at sufficiently small fixed $t = t_\ast < 0$. There are two sets of branch cuts (thick lines) corresponding to normal thresholds in the $s$-channel ($s > 4M^2$) and $u$-channel ($u > 4M^2$ or $s < - t_\ast$). The amplitude is real in the Euclidean region between them, which can also feature single-particle poles. The causal way of approaching the physical channels is indicated with arrows. The purpose of this work is to investigate how this picture generalizes to more realistic theories.
• Figure 1.2: Schematic illustration of the first few terms of \ref{['eq:ImT2']}. The orange dashed lines represent holomorphic cuts. The first two terms on the right-hand side are normal thresholds, while the latter two are examples of anomalous thresholds. Note that the constituent $\mathbf{T}$ elements can have disconnected components, as long as they are not $\mathbbold{1}$. In Sec. \ref{['sec:cutting']} we show that these cutting rules carry over to individual Feynman integrals.
• Figure 1.3: Three distinct ways of implementing an $i\varepsilon$ prescription, illustrated on the imaginary part of the bubble diagram with masses $m_1=1$ and $m_2=2$. The physical value, say at $s=12$, is indicated in red. Left: Feynman $i\varepsilon$ displaces the original branch point at $s = (m_1+m_2)^2 = 9$ to $s \approx 9.003 - 0.450i$. Middle: The $s+i\varepsilon$ prescription for approaching the branch cut from the upper-half plane. Right: Branch cut deformation revealing the causal amplitude without modifying analytic properties. We set $\varepsilon = \tfrac{1}{10}$ to make the effects more visible. See Sec. \ref{['sec:primer']} for more details.
• Figure 1.4: Examples of branch cut deformations is the complex $s$-plane at fixed $t=t_\ast < 0$. Causal directions for approaching the physical regions are indicated with red arrows. Left: The simplest case in which cuts are deformed so that analytic continuation between the upper- and lower-half planes is possible, see Sec. \ref{['sec:ExampleI']}. Right: The opposite situation in which the two half-planes are disconnected by branch cuts, see Sec. \ref{['sec:ExampleII']}. Note that, in general, $i\varepsilon$ procedures introduce multiple branch cuts sprouting from every branch point.
• Figure 1.5: Illustration of the analytic structure in the $s$-plane after summing over multiple diagrams involving external unstable particles. The physical sheet is confined to a region near the real axis between the branch cuts, see Sec. \ref{['sec:ExampleIII']}. Once decay widths are included, the branch points move into the complex plane, but the qualitative picture remains the same.