Loop-Tree Duality in Quantum Field Theory

• Loop-Tree Duality is a formalism that transforms L-loop Feynman integrals into sums of tree-level on-shell configurations with a Lorentz-covariant i0 prescription.
• It employs sequential Cauchy-residue integrations and dual propagators to systematically expose causal threshold singularities while canceling noncausal divergences.
• The method unifies virtual and real corrections at the integrand level, enabling efficient high-loop numerical computations and automation in perturbative quantum field theory.

Loop-Tree Duality (LTD) is a formalism in perturbative quantum field theory that recasts L-loop Feynman integrals into sums of phase-space–like integrals over tree-level (cut) configurations by systematically integrating out the energy components of the loop momenta. Through a sequence of Cauchy-residue operations on the loop energies, each loop integral is “opened” into tree-like terms involving single on-shell cuts per loop, with the uncut propagators carrying a modified—Lorentz-covariant—$i0$ prescription. The result is a representation of multiloop amplitudes that exhibits manifest causal threshold singularities exclusively and enables direct integrand-level cancellation of noncausal singularities, leading to improved numerical stability and unification of virtual and real contributions in four-dimensional integration domains (Aguilera-Verdugo et al., 2020, Aguilera-Verdugo et al., 2021, Runkel et al., 2019).

1. Fundamental Principles and Lorentz-Covariant Prescription

In the standard Feynman representation, an $L$-loop amplitude is given by: $$A_n^{(L)} = \int_{\ell_1, ..., \ell_L} N(\ell_i, p_j) \prod_{i=1}^n [G_F(q_i)]^{a_i}$$ where $G_F(q_i) = 1/(q_i^2-m_i^2 + i0)$, $a_i \ge 1$, and the $q_i$ are loop plus external momentum combinations.

The loop-tree duality is implemented by integrating out one energy component per loop via the residue theorem, utilising a future-like vector $\eta$ $(\eta_0>0)$. This “cuts” one propagator per loop, replacing $G_F(q)$ by $2\pi i\delta_+(q^2-m^2)$, and the remaining propagators become “dual propagators”: $$G_D(q_i; q_j) = \frac{1}{q_j^2-m_j^2 - i0\,\eta\cdot(q_j-q_i)}$$ This prescription is Lorentz-covariant, and the sign of the $i0$ follows from the flow of the energy integration contour.

By iteratively applying this procedure for all loops, the $L$-loop integrand is converted into a sum over tree-like configurations where each internal loop line is on-shell, and all remaining denominators carry an explicit momentum-shifted $i0$ that enforces causal ordering (Aguilera-Verdugo et al., 2020, Bierenbaum et al., 2010).

2. Topological Classification and Factorization

Rather than handling each diagram individually, LTD classifies multiloop topologies according to the coupling between sets of propagators, which are grouped into “loop-lines.” Each set contains propagators sharing the same linear combination of loop momenta.

Three canonical families are identified:

• Maximal Loop Topology (MLT): $n=L+1$ sets, each depending on a unique loop momentum and one depending on $-\sum_{s=1}^L \ell_s$ (“sunrise” topology).
• Next-to-Maximal (NMLT): $L+2$ sets, introducing one mixed set depending on sums of loop momenta.
• Next-to-Next-Maximal (N${}^2$MLT): $L+3$ sets, with two mixed sets coupling adjacent loop momenta.

LTD expressions depend only on the labelling of these sets—not on detailed kinematics or routing. Factorization is manifest: NMLT topologies decompose into convolutions of MLT subtopologies, and this recursive structure persists to arbitrary loops (Aguilera-Verdugo et al., 2020, Ramírez-Uribe et al., 2022, Bobadilla, 2021).

3. Manifestly Causal Integrands and Cancellation of Noncausal Singularities

In naive multiloop representations, partial fractioning can introduce “noncausal” (hyperboloid/H-surface) pinch singularities. LTD regulates these configurations via the dual $i0$ prescription. Crucially, when all dual terms are summed, residues of noncausal poles cancel algebraically, leaving only causal threshold and infrared (ellipsoid/E-surface) singularities (Aguilera-Verdugo et al., 2020, Capatti et al., 2020).

The remaining singularities correspond to physically allowed multi-particle thresholds, appearing on ellipsoidal surfaces in loop three-momentum space. For massless theories, these collapse to line segments encoding soft/collinear IR limits. Spurious singularities, associated with unphysical energy assignments, are locally cancelled at the integrand level.

Manifestly causal forms can be systematically derived via partial fractioning and combinatorial identities. These forms coincide at the integrand level with those obtained from time-ordered perturbation theory (TOPT), where denominators involve only sums of positive-energy solutions. The “cLTD” (manifestly causal LTD) representation preserves four-dimensional UV scaling for every term, ensuring numerical stability throughout momentum space (Capatti et al., 2020).

4. Algorithmic Construction and Independence from Momentum Routing

LTD integrand constructions are independent of the initial assignment of loop-momentum flows. The only trace of momentum flow reversal appears in the notation for off-shell sets. Any change in momentum routing relabels terms but does not alter the sum.

For implementation, the procedure consists of:

• Identifying all sets of loop-lines.
• Applying the residue theorem for each loop energy.
• Constructing dual amplitudes for each cut configuration.
• Summing terms to obtain a compact causal representation that is independent of routing conventions.

These principles have been codified and automated in the Lotty package (Bobadilla, 2021), which generates dual and causal integrands for arbitrary multi-loop topologies up to high loop orders.

5. Numerical Implementation, Contour Deformation, and Performance

Numerical integration over dual representations requires addressing threshold (ellipsoid) singularities that reside on real spatial-momentum domains. Stable evaluation is achieved by contour deformation in three-momentum space: $\vec\ell \to \vec\ell + i\,\vec\kappa(\vec\ell)$ where $\vec\kappa$ is constructed as a sum over all ellipsoid singularities, with functional forms enforcing correct imaginary directions per the dual $i0$ prescription and vanishing at infinity.

The LTD C++ implementations employ adaptive multidimensional integration routines, such as those in the CUBA library (Cuhre, VEGAS), and include analytic Jacobian factors. Benchmarks show competitive performance and stability, with precision stable across thresholds for scalar and tensor integrals up to six legs. CPU time scales mildly with the number of legs, and agreement with traditional codes is at the $10^{-5}$–$10^{-8}$ level (Buchta et al., 2015, Buchta, 2015, Buchta, 2015).

6. Integrand-Level Combination of Virtual and Real Contributions

A defining feature of LTD is its unification of virtual and real corrections. The singular regions of dual virtual integrands correspond, via explicit momentum mapping, to those of real emission phase space. Through local mapping, measured in terms of on-shell loop momenta, infrared divergences are cancelled pointwise at the integrand level. This "Four-Dimensional Unsubtraction" (FDU) strategy enables NLO and higher-order computations to be performed numerically in strictly four dimensions, sidestepping the need for analytic continuation in $d=4-2\epsilon$ (Rentería-Estrada, 2024, Aguilera-Verdugo et al., 2021, Sborlini, 2016).

Differential cross sections and decay rates can be assembled from a single $\mathbb{R}^{3L}$ integral over dual kernels, with IR and UV counterterms also implemented locally.

7. Generalization to Higher Loops, Multiple Poles, and Parametric Representations

The LTD framework extends to arbitrary loop orders, with explicit recurrence relations for integrand construction based on nested residues and combinatorial weights. For integrals beyond simple poles, integration-by-parts (IBP) reduction is employed to express higher-order pole integrals in terms of single-pole primitives, and the duality formula is applied to the reduced basis (Bierenbaum et al., 2012, Buchta, 2015). Weighted sums over trees correspond to all possible cut configurations forming a connected tree after pinching $L$ lines in the graph.

Moreover, variants of LTD have been formulated in Schwinger parametric representation, where the integration domain is partitioned into cells indexed by spanning trees, each yielding a fiber-bundle structure and a loop-tree dual form after fiber integration (Berghoff, 2022).

Table: Key Properties of Loop-Tree Duality (LTD)

Aspect	Description	Core References
Dual prescription	$$G_D(q_i; q_j) = [q_j^2-m_j^2 - i0\,\eta\cdot(q_j-q_i)]^{-1}$$	(Aguilera-Verdugo et al., 2020, Bierenbaum et al., 2010)
Singularities	Only causal (E-surface/ellipsoid) thresholds survive; H-surface (hyperboloid) singularities cancel locally	(Capatti et al., 2020, Aguilera-Verdugo et al., 2021)
Topological factorization	MLT, NMLT, N${}^2$MLT, etc.—recursive convolution structure	(Ramírez-Uribe et al., 2022, Bobadilla, 2021)
Independence from routing	Integrand-level expressions invariant under momentum flow assignments	(Aguilera-Verdugo et al., 2020, Bobadilla, 2021)
Contour deformation	Sum over radial deformations aligned with threshold surfaces	(Buchta et al., 2015, Buchta, 2015)
Integration domain	Euclidean $\mathbb{R}^{3L}$ after energy integration; suited for numerical methods	(Aguilera-Verdugo et al., 2021, Capatti et al., 2019)
Combination of real/virtual	IR cancellation at integrand level; four-dimensional unsubtracted scheme	(Rentería-Estrada, 2024, Sborlini, 2016, Aguilera-Verdugo et al., 2021)
Extension to parametric representation	Partition into cells indexed by spanning trees; fiber bundle structure	(Berghoff, 2022)

8. Applications and Outlook

Recent work demonstrates efficient calculation of multi-leg and multi-loop amplitudes, including up to five-loop topologies via the universal N${}^7$MLT kernel (Ramírez-Uribe et al., 2022), and full automation of causal representations up to high loop orders (Bobadilla, 2021). The approach enables direct asymptotic expansions at integrand level (Plenter et al., 2020) and links the graph-theoretic structure of amplitudes to the combinatorics of spanning trees and causal partitions.

LTD offers a foundational framework for all-orders integrand-level classification, giving a profound topological and combinatorial understanding of Feynman amplitudes, causal structure, and the local cancellation of spurious singularities—providing new routes for the automation and efficient numerical evaluation of quantum field theory observables.