We establish an efficient polynomial-complexity algorithm for one-loop calculations, based on generalized $D$-dimensional unitarity. It allows automated computations of both cut-constructible {\it and} rational parts of one-loop scattering amplitudes from on-shell tree amplitudes. We illustrate the method by (re)-computing all four-, five- and six-gluon scattering amplitudes in QCD at one-loop.
Characterizing multiloop topologies is an important step towards developing novel methods at high perturbative orders in quantum field theory. In this article, we exploit the Loop-Tree Duality (LTD) formalism to analyse multiloop topologies that appear for the first time at five loops. Explicitly, we open the loops into connected trees and group them according to their topological properties. Then, we identify a kernel generator, the so-called N$^7$MLT universal topology, that allow us to describe any scattering amplitude of up to five loops. Furthermore, we provide factorization and recursion…
We propose a color decomposition for general tree amplitudes in a SU(2) gauge theory which is spontaneously broken via the Higgs mechanism. Working in the unitary gauge, we construct color-ordered amplitudes by explicitly presenting a set of color-ordered Feynman rules. Those primitive amplitudes are gauge-invariant, and they preserve perturbative unitarity in the high-energy limit. Serving as building blocks of color-dressed tree amplitudes, they allow for efficient evaluation of tree-level scattering amplitudes involving gauge bosons and the Higgs boson via the Berends-Giele recursion relati…
We discuss briefly the first numerical implementation of the Loop-Tree Duality (LTD) method. We apply the LTD method in order to calculate ultraviolet and infrared finite multi-leg one-loop Feynman integrals. We attack scalar and tensor integrals with up to six legs (hexagons). The LTD method shows an excellent performance independently of the number of external legs.
The nucleus-nucleus Glauber amplitude in the tree approximation is studied for heavy participant nuclei. It is shown that, contrary to previous published results, it is not unitary for realistic values of nucleon-nucleon cross-sections.
In this talk, we review the most recent developments of the four-dimensional unsubstraction (FDU) and loop-tree duality (LTD) methods. In particular, we make emphasis on the advantages of the LTD formalism regarding asymptotic expansions of loop integrands.
The Loop-Tree Duality (LTD) is a novel perturbative method in QFT that establishes a relation between loop-level and tree-level amplitudes, which gives rise to the idea of treating them simultaneously in a common Monte Carlo. Initially introduced for one-loop scalar integrals, the applicability of the LTD has been expanded to higher order loops and Feynman graphs beyond simple poles. For the first time, a numerical implementation relying on the LTD was realized in the form of a computer program that calculates one-loop scattering amplitudes. We present details on the employed contour deformati…
Loop Tree Duality (LTD) offers a promising avenue to numerically integrate multi-loop integrals directly in momentum space. It is well-established at one loop, but there have been only sparse numerical results at two loops. We provide a formal derivation for a novel multi-loop LTD expression and study its threshold singularity structure. We apply our findings numerically to a diverse set of up to four-loop finite topologies with kinematics for which no contour deformation is needed. We also lay down the ground work for constructing such a deformation. Our results serve as an important stepping…
We construct a modified on-shell BCFW recursion relation to derive compact analytic representations of tree-level amplitudes in QED. As an application, we study the amplitudes of a fermion pair coupling to an arbitrary number of photons and give compact formulae for the NMHV and NNMHV case. We demonstrate that the new recursion relation reduces the growth in complexity with additional photons to be exponential rather than factorial.
We compute in conventional dimensional regularisation the tree-level splitting amplitudes for a gluon parent which splits into four collinear partons. This is part of the universal infrared behaviour of the QCD scattering amplitudes at next- to-next-to-next-to-leading order (N$^3$LO) in the strong coupling constant. Combined with our earlier results for a quark parent, this completes the set of tree-level splitting amplitudes required at this order. We also study iterated collinear limits where a subset of the four collinear partons become themselves collinear.
We report on a new method for the numerical evaluation of loop integrals, based on the Feynman Tree Theorem. The loop integrals are replaced by phase-space integration over fictitious extra on-shell particles. This integration can be performed alongside with the Monte-Carlo integration of ordinary phase-space, directly leading to NLO event generation.
We investigate the construction of tree-level MHV gluon amplitudes in multiplet bases using BCFW recursion. The multiplet basis decomposition can either be obtained by decomposing results derived in (for example) the DDM basis or by formulating the recursion directly in the multiplet basis. We focus on the latter approach and show how to efficiently deal with the color structure appearing in the recursion. For illustration, we also explicitly calculate the four-, five- and six-gluon amplitudes.
We review the recent developments of the loop-tree duality method, focussing our discussion on analysing the singular behaviour of the loop integrand of the dual representation of one-loop integrals and scattering amplitudes. We show that within the loop-tree duality method there is a partial cancellation of singularities at the integrand level among the different components of the corresponding dual representation. The remaining threshold and infrared singularities are restricted to a finite region of the loop momentum space, which is of the size of the external momenta and can be mapped to t…
We review the recent progress on the numerical implementation of the Loop-Tree Duality Method (LTDM) for the calculation of scattering amplitudes. A central point is the analysis of the singularities of the integrand. In the framework of the LTDM some of these singularities cancel out. The ones left over are dealt with by contour deformation. We present details on how to achieve this as well as first results.
The Loop-Tree Duality (LTD) is a novel perturbative method in QFT that establishes a relation between loop-level and tree-level scattering amplitudes. This is achieved by directly applying the Residue Theorem to the loop-energy-integration. The result is a sum over all possible single cuts of the Feynman diagram in consideration integrated over a modified phase space. These single-cut integrals, called Dual contributions, are in fact tree-level objects and thus give rise to the opportunity of bringing loop- and tree-contributions together, treating them simultaneously in a common Monte Carlo e…
We present a first numerical implementation of the Loop-Tree Duality (LTD) method for the direct numerical computation of multi-leg one-loop Feynman integrals. We discuss in detail the singular structure of the dual integrands and define a suitable contour deformation in the loop three-momentum space to carry out the numerical integration. Then, we apply the LTD method to the computation of ultraviolet and infrared finite integrals, and present explicit results for scalar integrals with up to five external legs (pentagons) and tensor integrals with up to six legs (hexagons). The LTD method fea…
The possibility of treating colour in one-loop amplitude calculations alike the other quantum numbers is briefly discussed for semi-numerical algorithms based on generalized unitarity and parametric integration techniques. Numerical results are presented for the calculation of virtual corrections in multi-gluon scattering.
This article discusses the occurences of one-loop amplitudes within a next-to-next-to-leading order calculation. In an NNLO calculation the one-loop amplitude enters squared and one would therefore naively expect that the O(epsilon)- and O(epsilon^2)-terms of the one-loop amplitudes are required. I show that the calculation of these terms can be avoided if a method is known, which computes the O(epsilon^0)-terms of the finite remainder function of the two-loop amplitude.
The infrared structure of (multi-loop) scattering amplitudes is determined entirely by the identities of the external particles participating in the scattering. The two-loop infrared structure of pure \qcd\ amplitudes has been known for some time. By computing the two-loop amplitudes for $\bar{f}\,f\longrightarrow X$ and $\bar{f}\,f\longrightarrow V_1\,V_2$ scattering in an $SU(N)\times SU(M)\times U(1)$ gauge theory, I determine the anomalous dimensions which govern the infrared structure for any massless two-loop amplitude.
I report on progress in the calculation of one-loop amplitudes with a space-like gluon. Amplitudes with space-like partons are relevant for factorization procedure s that require initial-state partons with non-vanishing transverse momentum components.
We discuss the duality theorem, which provides a relation between loop integrals and phase space integrals. We rederive the duality relation for the one-loop case and extend it to two and higher-order loops. We explicitly show its application to two- and three-loop scalar master integrals and discuss the structure of the occurring cuts.
We relate a $l$-loop Feynman integral to a sum of phase space integrals, where the integrands are determined by the spanning trees of the original $l$-loop graph. Causality requires that the propagators of the trees have a modified $iδ$-prescription and we present a simple formula for the correct $iδ$-prescription.
We present a new formulation of the loop-tree duality theorem for higher loop diagrams valid both for massless and massive cases. $l$-loop integrals are expressed as weighted sum of trees obtained from cutting $l$ internal propagators of the loop graph. In addition, the uncut propagators gain a modified $i δ$-prescription, named dual-propagators. In this new framework one can go beyond graphs and calculate the integrand of loop amplitudes as a weighted sum of tree graphs, which form a tree-like object. These objects can be computed efficiently via recurrence relations.
Using loop-tree duality, we relate a renormalised $n$-point $l$-loop amplitude in a quantum field theory to a phase-space integral of a regularised $l$-fold forward limit of a UV-subtracted $(n+2l)$-point tree-amplitude-like object. We show that up to three loops the latter object is easily computable from recurrence relations. This defines an integrand of the loop amplitude with a global definition of the loop momenta. Field and mass renormalisation are performed in the on-shell scheme.
Tree amplitudes of any gauge theory and gravity can be factorized into primitive three-particle amplitudes by the BCFW recursion relations. We show that the amplitudes at any perturbation order are given by tree amplitudes with additional pairs of particles in the forward limit. As an example we compute the four-gluon all-plus helicity amplitude at next-to-leading order.
We investigate relations between loop and tree amplitudes in quantum field theory that involve putting on-shell some loop propagators. This generalizes the so-called Feynman tree theorem which is satisfied at 1-loop. Exploiting retarded boundary conditions, we give a generalization to L-loop expressing the loops as integrals over the on-shell phase space of exactly L particles. We argue that the corresponding integrand for L>2 does not involve the forward limit of any physical tree amplitude, except in planar gauge theories. In that case we explicitly construct the relevant physical amplitude.…
We study scalar one-loop amplitudes in massive $ϕ^3$-theory within causal loop-tree duality. We derive a recurrence relation for the integrand of the amplitude. The integrand is by construction free of spurious singularities on H-surfaces. Taking renormalisation and contour deformation into account, we obtain the (integrated) amplitude by Monte Carlo integration. We have checked up to seven points that our results agree with analytic results.
Recently it has been shown that in gauge theories amplitudes to any perturbation order can be obtained by glueing together simple three-point on-shell amplitudes. These three-point amplitudes in turn are fixed by locality and Lorentz invariance. This factorization into three-point on-shell amplitudes follows from the BCFW recursion relations and the Feynman-tree theorem. In an explicit example, that is, the four-gluon amplitude with all plus helicities, we illustrate the method. In conventional calculation this amplitude corresponds to one-loop box diagrams.
We consider aspects of tree and one-loop behavior in a generic 4d EFT of massless scalars, fermions, and vectors, with a particular eye to the high-energy limit of the Standard Model EFT at operator dimensions 6 and 8. First, we classify the possible Lorentz structures of operators and the subset of these that can arise at tree-level in a weakly coupled UV completion, extending the tree/loop classification through dimension 8 using functional methods. Second, we investigate how operators contribute to tree and one-loop helicity amplitudes, exploring the impact of non-renormalization theorems t…
We present an extension of the duality theorem, previously defined by S. Catani et al. on the one-loop level, to higher loop orders. The duality theorem provides a relation between loop integrals and tree-level phase-space integrals. Here, the one-loop relation is rederived in a way which is more suitable for its extension to higher loop orders. This is shown in detail by considering the two-loop N-leg master diagram and by a short discussion of the four master diagrams at three loops, in this sketching the general structure of the duality theorem at even higher loop orders.
Multiloop scattering amplitudes describing the quantum fluctuations at high-energy scattering processes are the main bottleneck in perturbative quantum field theory. The loop-tree duality is a novel method aimed at overcoming this bottleneck by opening the loop amplitudes into trees and combining them at integrand level with the real-emission matrix elements. In this Letter, we generalize the loop-tree duality to all orders in the perturbative expansion by using the complex Lorentz-covariant prescription of the original one-loop formulation. We introduce a series of mutiloop topologies with ar…
We review the recent developments of the Loop-Tree Duality method, focussing our discussion on the first numerical implementation and its use in the direct numerical computation of multi-leg Feynman integrals. Non-trivial examples are presented.
We study the three-particle and four-particle scattering amplitudes for an arbitrary, finite number of massive scalars, spinors and vectors by employing the on-shell massive spinor formalism. We consider the most general three-particle amplitudes with energy growing behavior at most of $\mathcal{O} (E)$. This is the special case of the requirement of tree unitarity, which states that the $N$-particle scattering amplitudes at tree-level should grow at most as $\mathcal{O} (E^{4-N})$ in the high energy hard scattering limit, i.e. at fixed non-zero angles. Then the factorizable parts of the four-…
The duality relation between one-loop integrals and phase-space integrals, developed in a previous work, is extended to higher-order loops. The duality relation is realized by a modification of the customary +i0 prescription of the Feynman propagators, which compensates for the absence of the multiple-cut contributions that appear in the Feynman tree theorem. We rederive the duality theorem at one-loop order in a form that is more suitable for its iterative extension to higher-loop orders. We explicitly show its application to two- and three-loop scalar master integrals, and we discuss the str…
It is well-known that the standard BCFW construction cannot be used for on-shell amplitudes in effective field theories due to bad behavior for large shifts. We show how to solve this problem in the case of the SU(N) non-linear sigma model, i.e. non-renormalizable model with infinite number of interaction vertices, using scaling properties of the semi-on-shell currents, and we present new on-shell recursion relations for all on-shell tree-level amplitudes in this theory.
