Genuine complex-region Landau singularities for general Feynman graphs

Landau's 1959 analysis of vertex parts reduced the singularity structure of Feynman amplitudes to the algebraic Landau equations — on-shell conditions for internal lines together with loop-wise momentum equilibrium weighted by Feynman parameters — and classified the nearest (Euclidean-region) singularities geometrically, while explicitly leaving the complex (non-Euclidean) region unclassified for multi-vertex graphs, leaving the fourth-degree relation governing four-point tetrahedral schemes unsolved, and leaving open whether the located candidate singularities are actual singularities of the amplitude. The subsequent literature shows that the catalogue of candidate singularity hypersurfaces (the Landau variety) is now largely computable by algorithmic algebraic geometry for multi-loop, multi-leg graphs, including massless and divergent cases, while the Landau condition has been proven necessary and sufficient only for pinches in the physical region. The surviving open core of Landau's program is a necessary-and-sufficient criterion deciding which complex-region solutions of the Landau equations — leading branches and subleading branches of contracted graphs — are genuine singularities of the Feynman integral, completing, in invariant-based form, the four-point-and-higher classification the source paper left unfinished.

Background

In perturbative quantum field theory, scattering amplitudes and correlation functions are computed as sums of Feynman integrals: for each Feynman graph (a diagram whose internal lines represent virtual particles and whose closed cycles are loop momenta $k, l, \ldots$ integrated over), one integrates a product of propagators $1/(m_i^2 - q_i^2 - i0)$, where $q_i$ and $m_i$ are the four-momentum and mass of the $i$-th internal line and the infinitesimal imaginary part $-i0$ (equivalently $i\varepsilon$) fixes the side from which each propagator is evaluated as a boundary value. A vertex part is the three-point Green function (three external legs). Since the 1950s the driving question has been where these quantities are analytic as functions of the external momenta, because dispersion relations — integral representations connecting real and imaginary parts of amplitudes, which underpin much of scattering theory — are valid only in regions free of singularities.

Landau's 1959 paper, anticipated in part by Nakanishi, made the singularity question algebraic. Using Feynman parameters $\alpha_i \ge 0$ (auxiliary variables introduced so that a product of propagators can be written as a single integral over a combination $\sum_i \alpha_i A_i$, with $A_i = m_i^2 - q_i^2$), and completing the square in the loop momenta, the denominator becomes $\varphi + K$, where $\varphi$ depends only on the $\alpha_i$ and the external momenta and $K$ is a quadratic form in shifted loop variables. Landau showed that a singularity can occur only at an extremum of $\varphi$, which yields the Landau equations

$$\alpha_i (q_i^2 - m_i^2) = 0 \quad \text{for each internal line}, \qquad \sum_{i \in \text{loop } a} \alpha_i q_i = 0 \quad \text{for each independent loop}.$$

Each line either is on shell ($q_i^2 = m_i^2$ with $\alpha_i \ne 0$) or drops out of the graph ($\alpha_i = 0$; the graph contracts to a smaller one, giving subleading singularities); configurations in which loop momenta instead diverge produce singularities of the second type. Landau also translated the conditions geometrically (represent each graph vertex by a polygon whose sides have lengths equal to particle masses, and look for equilibria with positive weights), recovered the anomalous threshold of the triangle vertex graph found by Karplus, Sommerfield and Wichmann (a branch point produced by a triangle of particles going on shell that does not coincide with any normal two-particle threshold), derived the leading singular behavior $\varphi_0^{2v-(n+1)/2}$ (with a logarithm when the exponent is a nonnegative integer) for a graph with $n$ internal lines and $v$ independent loops, and gave the two-point threshold $p^2 = (\sum m_i)^2$. The nearest singularities — those first met when approaching from the physical (Euclidean, real-momenta) region — correspond to solutions with all $\alpha_i > 0$ realized in real Euclidean space.

What the 1959 paper explicitly did not settle: (a) beyond the nearest singularity, in the complex (non-Euclidean) region, the extrema of $\varphi$ were classified only for the basic triangle graph — the algebraic surface in the cosines $\mu_a, \mu_b, \mu_c$ of the three on-shell directions consists of one bounded Euclidean sheet and three unbounded non-Euclidean sheets meeting at three points — while for multi-vertex graphs, and especially four-point graphs, no such classification was given: for tetrahedral (four-external-leg) schemes the equilibrium condition becomes an equation of the fourth degree in the cosines between four unit vectors, which Landau stated but did not solve, adding that the complex-region situation is more complicated and would not be discussed; (b) whether every solution of the algebraic conditions is an actual singularity of the amplitude (sufficiency) was argued only heuristically and only for the nearest case; (c) for the physical amplitude viewed as a function of a single variable $x$, the Feynman prescription produces two boundary functions $a(x+i0)$ and $b(x-i0)$, and Landau asserted that singularities met when continuing $a$ into the lower or $b$ into the upper (foreign) half-plane — virtual states and resonances — cannot in principle be predicted from general considerations but require a dynamical theory.

The subsequent development: Cutkosky (1960) derived the rules for the discontinuity of an amplitude across each located singularity; Fairlie, Landshoff, Nuttall and Polkinghorne (1962) analyzed the second-type singularities; Eden, Landshoff, Polkinghorne and Taylor (1961) studied the real geometry of the Landau curves of four-point graphs, in particular their acnodes (isolated real points of a real algebraic curve) and cusps (points at which a curve turns back on itself); Coleman and Norton (1965) proved that a physical-region Landau singularity admits a spacetime interpretation as a classical process of on-shell particles. In the modern era the catalogue half of the problem — determining the candidate singularity hypersurfaces (the Landau variety, the algebraic set in the space of external invariants on which the Landau equations admit a solution) — has been largely algorithmized: by elimination and homotopy continuation (Landau discriminants), through principal A-determinants of the parametric representation, by Whitney stratifications (a decomposition of a map separating singularity types), by recursive use of unitarity, and by Euler-characteristic computations over finite fields, including massless graphs and graphs whose integrals diverge at large (ultraviolet) or small (infrared) loop momenta. What has not been settled is the other half: which candidate loci are genuine singularities of the integral once one leaves the physical region, and on which Riemann sheet (branch of the multivalued integral) each singularity appears. Collins (2020) gave the first complete proof that the Landau condition is necessary and sufficient for pinches of the integration contour in the physical region (a pinch arises when the integration contour is trapped between two singular hypersurfaces of the integrand that approach each other); the complex-region statement remains open, and the sheet structure of multivalued Feynman integrals is an active topic (coamoeba methods, i.e., images of the complex phases of the parameters; the breakdown of the standard $i\varepsilon$ prescription in theories with massless or unstable particles; the discovery that on the second sheet infinitely many ladder-type Landau singularities can accumulate into a natural boundary — a curve beyond which analytic continuation is impossible).

Problem Statement

The open problem is the surviving core of the program that Landau's 1959 paper left unfinished: determining the actual singularities of Feynman-graph amplitudes — not only their candidate locations — beyond the nearest (Euclidean/physical-region) singularities.

For a general Feynman graph with $n$ internal lines of masses $m_i$ and $v$ independent loops, contributing to an amplitude viewed as a function of the external kinematic invariants, solve jointly:

1. Actuality in the complex region. Give a necessary and sufficient criterion — valid for kinematics outside the Euclidean/physical region and covering the solution branches of the Landau equations within the source's framework of graphs whose internal lines are stable massive particles: leading branches with all $\alpha_i \ne 0$ and subleading branches of contracted graphs ($\alpha_i = 0$) — deciding which components of the Landau variety are genuine singularities of the Feynman integral and which are spurious solutions of the algebraic conditions. Extensions to massless or ultraviolet/infrared-divergent graphs, to second-type branches with divergent loop momenta, and to mixed or degenerate scalings of the Feynman parameters (the "generalized" Landau equations of the modern literature) go beyond the source's framework and are not required.

2. Four-point and higher graphs. Apply the criterion of part 1 to the four-point-and-higher case the source left unfinished: exhibit the classification explicitly in invariant-based form for the tetrahedral four-point scheme — the source's fourth-degree relation among cosines, with the complex region explicitly set aside, resolved into the Landau variety together with its actuality data for leading and subleading branches — and state explicitly how the criterion extends this classification to higher-point graphs.

The criterion is sought for a definite class of integrals with specified regularization and continuation conventions, so that genuine singularity is well defined. The dynamical prediction of resonance and virtual-state positions on unphysical sheets — which the source paper itself asserted to be impossible from general graph-level considerations — is not part of this graph-level task; nor is the assignment of genuine singularities to Riemann sheets of the multivalued integral, an active direction of the modern literature recorded below that the source did not pose as a task.

References

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