We generalize the unifying relations for tree amplitudes to the $1$-loop Feynman integrands. By employing the $1$-loop CHY formula, we construct differential operators which transmute the $1$-loop gravitational Feynman integrand to Feynman integrands for a wide range of theories, include Einstein-Yang-Mills theory, Einstein-Maxwell theory, pure Yang-Mills theory, Yang-Mills-scalar theory, Born-Infeld theory, Dirac-Born-Infeld theory, bi-adjoint scalar theory, non-linear sigma model, as well as special Galileon theory. The unified web at $1$-loop level is established. Under the well known unitarity cut, the $1$-loop level operators will factorize into two tree level operators. Such factorization is also discussed.
Spherical contours introduced in \cite{SphericalContours} translate the concept of "discontinuity across a branch cut" to Feynman parameter space. In this paper, we further explore spherical contours and connect them to the computation of leading IR divergences of 1 loop graphs directly in Feynman parameter space. These spherical contours can be used to develop a Feynman parameter space analog of "Leading Singularities" of loop integrands which allows us to develop a method of determining Feynman parameter integrands with no reference to the momentum space loop integrand. Finally, we explore s…
In this paper, the connections among $1$-loop Feynman integrands of a large variety of theories with massless external states are further investigated. The work includes two parts. First, we construct a new class of differential operators which transmute the $1$-loop gravitational Feynman integrands to $1$-loop Yang-Mills Feynman integrands. The new operators are commutable with the integration of loop momentum, thus the corresponding transmutational relations hold at not only the integrands level, but also the $1$-loop amplitudes level. Secondly, by using $1$-loop level transmutational relati…
An elegant unified web for amplitudes of various theories was given by Cachazo, He and Yuan in the CHY framework a few years ago. Recently, similar web has also been constructed by Cheung, Shen and Wen, which relies on a set of differential operators. In this note, by acting these differential operators on CHY-integrands systematically, we have established the relation between these two approaches. Thus, amplitudes for all theories which have CHY representations, include gravity theory, Einstein-Yang-Mills theory, Einstein-Maxwell theory, pure Yang-Mills theory, Yang-Mills-scalar theory, Born-…
When using dimensional regularization/reduction the epsilon-dimensional numerator of the 1-loop Feynman diagrams gives rise to rational contributions. I list the set of fundamental rules that allow the extraction of such terms at the integrand level in any theory containing scalars, vectors and fermions, such as the electroweak standard model, QCD and SUSY.
Recently the Cachazo-He-Yuan (CHY) approach has been extended to loop level, but the resulting loop integrand has propagators that are linear in the loop momentum unlike Feynman's. In this note we present a new technique that directly produces quadratic propagators identical to Feynman's from the CHY approach. This paper focuses on $Φ^3$ theory but extensions to others theories are briefly discussed. In addition, our proposal has an interesting geometric meaning, we can interpret this new formula as a unitary cut on a higher genus Riemann surface.
We propose a strategy to study the analytic structure of Feynman parameter integrals where singularities of the integrand consist of rational irreducible components. At the core of this strategy is the identification of a selected stratum of discontinuities induced from the integral, together with a geometric method for computing their singularities on the principal sheet. For integrals that yield multiple polylogarithms we expect the data collected in this strategy to be sufficient for the construction of their symbols. We motivate this analysis by the Aomoto polylogarithms, and further check…
The numerical evaluation of multi-loop scattering amplitudes in the Feynman representation usually requires to deal with both physical (causal) and unphysical (non-causal) singularities. The loop-tree duality (LTD) offers a powerful framework to easily characterise and distinguish these two types of singularities, and then simplify analytically the underling expressions. In this paper, we work explicitly on the dual representation of multi-loop Feynman integrals generated from three parent topologies, which we refer to as Maximal, Next-to-Maximal and Next-to-Next-to-Maximal loop topologies. In…
We present the integrand reduction via multivariate polynomial division as a natural technique to encode the unitarity conditions of Feynman amplitudes. We derive a recursive formula for the integrand reduction, valid for arbitrary dimensionally regulated loop integrals with any number of loops and external legs, which can be used to obtain the decomposition of any integrand analytically with a finite number of algebraic operations. The general results are illustrated by applications to two-loop Feynman diagrams in QED and QCD, showing that the proposed reduction algorithm can also be seamless…
The loop-tree duality (LTD) has become a novelty alternative to bootstrap the numerical evaluation of multi-loop scattering amplitudes. It has indeed been found that Feynman integrands, after the application of LTD, display a representation containing only physical information, the so-called causal representation. In this talk, we discuss the all-loop causal representation of multi-loop Feynman integrands, recently found in terms of features that describe a loop topology, vertices and edges. Likewise, in order to elucidate the numerical stability in LTD integrands, we present applications that…
Recently, the Cachazo-He-Yuan (CHY) approach for calculating scattering amplitudes has been extended beyond tree level. In this paper, we introduce a way of constructing CHY integrands for $Φ^3$ theory up to two loops from holomorphic forms on Riemann surfaces. We give simple rules for translating Feynman diagrams into the corresponding CHY integrands. As a complementary result, we extend the $Λ$-algorithm, originally introduced in arXiv:1604.05373, to two loops. Using this approach, we are able to analytically verify our prescription for the CHY integrands up to seven external particles at tw…
In this paper, by treating massive loop momenta to massless momenta in higher dimension, we are able to treat all-loop scattering equations as tree ones. As an application of the new aspect, we consider the CHY-construction of bi-adjoint phi_3 theory. We present the explicit formula for two-loop planar integrands. We discuss carefully how to subtract various forward singularities in the construction. We count the number of terms obtained by our formula and by direct Feynman diagram calculation and find the perfect match, thus provide a strong support for our results.
In this note, we introduce and study a new class of "half integrands" in Cachazo-He-Yuan (CHY) formula, which naturally generalize the so-called Parke-Taylor factors; these are dubbed Cayley functions as each of them corresponds to a labelled tree graph. The CHY formula with a Cayley function squared gives a sum of Feynman diagrams, and we represent it by a combinatoric polytope whose vertices correspond to Feynman diagrams. We provide a simple graphic rule to derive the polytope from a labelled tree graph, and classify such polytopes ranging from the associahedron to the permutohedron. Furthe…
In this note, we study the $\mathcal{Q}$-cut representation by combining it with BCFW deformation. As a consequence, the one-loop integrand is expressed in terms of a recursion relation, i.e., $n$-point one-loop integrand is constructed using tree-level amplitudes and $m$-point one-loop integrands with $m\leq n-1$. By giving explicit examples, we show that the integrand from the recursion relation is equivalent to that from Feynman diagrams or the original $\mathcal{Q}$-cut construction, up to scale free terms.
In this work we explore the properties of four-dimensional gravity integrands at large loop momenta. This analysis can not be done directly for the full off-shell integrand but only becomes well-defined on cuts that allow us to unambiguously specify labels for the loop variables. The ultraviolet region of scattering amplitudes originates from poles at infinity of the loop integrands and we show that in gravity these integcrands conceal a number of surprising features. In particular, certain poles at infinity are absent which requires a conspiracy between individual Feynman integrals contributi…
The notion of a unique integrand does not a priori makes sense in field theory: different Feynman diagrams have different loop momenta and there should be no reason to compare them. In string theory, however, a global integrand is natural and allows, for instance, to make explicit the separation between left and right-moving degrees of freedom. However, the significance of this integrand had not really been investigated so far. It is even more important in view of the recently discovered loop monodromies that are related to the duality between color and kinematics in gauge and gravity loop amp…
The method of canonical differential equations is an important tool in the calculation of Feynman integrals in quantum field theories. It has been realized that the canonical bases are closely related to $d$-dimensional $d\log$-form integrands. In this work, we introduce the concept of generalized loop-by-loop Baikov representation, and clarify its relation and difference with Feynman integrals using the language of intersection theory. We then utilize the generalized Baikov representation to construct $d$-dimensional $d\log$-form integrands, and discuss how to convert them to Feynman integral…
Institute for Particle and Nuclear Physics, Charles University（查尔斯大学粒子与核物理研究所）；Center for Quantum Mathematics and Physics (QMAP), University of California, Davis（加州大学戴维斯分校量子数学与物理中心）
We recently proposed the Halohedron to be the 1-loop Amplituhedron for planar $ϕ^3$ theory. Here we prove this claim by showing how it is possible to extract the integrand for the partial amplitude $m^1_n(1,\dots,n|1,\dots,n)$ from the canonical form of an Halohedron which lives in an abstract space. This space is just a step away from ordinary kinematical space at 1-loop, because it is composed by abstract variables associated to propagators of 1-loop Feynman diagrams. Such variables, however, are unbound from momentum conservation relations that would give problems such as double poles. As a…
Feynman integrals in quantum field theory evaluate to special functions and numbers that are usefully described by the notion of transcendental weight. In this paper, we propose a way of projecting a given dimensionally-regularised Feynman integral, for example contributing to a scattering amplitudes, onto its maximal weight part. The method uses insights into the singularity structure of space-time loop integrands, and is complementary to usual generalised unitarity approaches. We describe the method and give a proof-of-principle application to the two-loop scattering amplitudes $gg \to H$ in…
A Systematic Approach to Finite Multiloop Feynman Integrals
Universität Zürich（苏黎世大学）；Universidad Autónoma de Sinaloa（西马洛阿自治大学）；Universitat de València – Consejo Superior de Investigaciones Científicas（瓦伦西亚大学–西班牙国家研究委员会）；Universidad de Salamanca（萨拉曼卡大学）
Integrand Analysis, Leading Singularities and Canonical Bases beyond Polylogarithms
Technical University of Munich（慕尼黑工业大学）；University of Oxford（牛津大学）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（阿尔伯特·爱因斯坦研究所））
Recently, we proposed a new approach using a punctured Elliptic curve in the CHY framework in order to compute one-loop scattering amplitudes. In this note, we further develop this approach by introducing a set of connectors, which become the main ingredient to build integrands on $\mathfrak{M}_{1,n}$, the moduli space of n-punctured Elliptic curves. As a particular application, we study the $Φ^3$ bi-adjoint scalar theory. We propose a set of rules to construct integrands on $\mathfrak{M}_{1,n}$ from $Φ^ 3$ integrands on $\mathfrak{M}_{0,n}$, the moduli space of n-punctured spheres. We illustr…
I analyze the algebraic patterns underlying the structure of scattering amplitudes in quantum field theory. I focus on the decomposition of amplitudes in terms of independent functions and the systems of differential equations the latter obey. In particular, I discuss the key role played by unitarity for the decomposition in terms of master integrals, by means of generalized cuts and integrand reduction, as well as for solving the corresponding differential equations, by means of Magnus exponential series.
We present recent developments on the topic of the integrand reduction of scattering amplitudes. Integrand-level methods allow to express an amplitude as a linear combination of Master Integrals, by performing operations on the corresponding integrands. This approach has already been successfully applied and automated at one loop, and recently extended to higher loops. We describe a coherent framework based on simple concepts of algebraic geometry, such as multivariate polynomial division, which can be used in order to obtain the integrand decomposition of any amplitude at any loop order. In t…
Calculation of amplitudes in perturbative quantum field theory involve large loop integrals. The complexity of those integrals, in combination with the large number of Feynman diagrams, make the calculations very difficult. Reduction methods proved to be very helpful, lowering the number of integrals that need to be actually calculated. Especially, the reduction at the integrand level technique, improves the speed and set-up of these calculations. In this article we demonstrate, by counting the numbers of tensor structures and independent coefficients, how to write such relations at the integr…
In this article a first step is made towards the extension of Britto-Cachazo-Feng-Witten (BCFW) tree level on-shell recursion relations to integrands and integrals of scattering amplitudes to arbitrary loop order. Surprisingly, it is shown that the large BCFW shift limit of the integrands has the same structure as the corresponding tree level amplitude in any minimally coupled Yang-Mills theory in four or more dimensions. This implies that these integrands can be reconstructed from a subset of their `single cuts'. The main tool is powercounting Feynman graphs in a special lightcone gauge choic…
Recently, loop integrands for certain Yang-Mills scattering amplitudes and correlation functions have been shown to be systematically expressible in dlog form, raising the possibility that these loop integrals can be performed directly without Feynman parameters. We do so here to give a new description of the planar 1-loop MHV amplitude in N = 4 super Yang-Mills theory. We explicitly incorporate the standard Feynman i epsilon prescription into the integrands. We find that the generic MHV diagram contributing to the 1-loop MHV amplitude, known as Kermit, is dual conformal invariant up to the ch…
We present a construction of the integrand of the correlation function of four stress-tensor multiplets in N=4 SYM at weak coupling. It does not rely on Feynman diagrams and makes use of the recently discovered symmetry of the integrand under permutations of external and integration points. This symmetry holds for any gauge group, so it can be used to predict the integrand both in the planar and non-planar sectors. We demonstrate the great efficiency of graph-theoretical tools in the systematic study of the possible permutation symmetric integrands. We formulate a general ansatz for the correl…
In this review, we discuss recent developments concerning efficient calculations of multi-loop multi-leg scattering amplitudes. Inspired by the remarkable properties of the Loop-Tree Duality (LTD), we explain how to reconstruct an integrand level representation of scattering amplitudes which only contains physical singularities. These so-called causal representations can be derived from connected binary partitions of Feynman diagrams, properly entangled according to specific rules. We will focus on the detection of flux orientations which are compatible with causality, describing the implement…
The low-energy expansion of one-loop amplitudes in type II string theory generates a series of world-sheet integrals whose integrands can be represented by world-sheet Feynman diagrams. These integrands are modular invariant and understanding the structure of the action of the modular Laplacian on them is important for determining their contribution to string scattering amplitudes. In this paper we study a particular infinite family of such integrands associated with three-loop scalar vacuum diagrams of tetrahedral topology and find closed forms for the action of the Laplacian. We analyse the …
In the CHY-frame for the tree-level amplitudes, the bi-adjoint scalar theory has played a fundamental role because it gives the on-shell Feynman diagrams for all other theories. Recently, an interesting generalization of the bi-adjoint scalar theory has been given in arXiv:1708.08701 by the "Labelled tree graphs", which carries a lot of similarity comparing to the bi-adjoint scalar theory. In this note, we have investigated the Labelled tree graphs from two different angels. In the first part of the note, we have shown that we can organize all cubic Feynman diagrams produces by the Labelled tr…
In the low-energy effective action of string theories, non-abelian gauge interactions and supergravity are augmented by infinite towers of higher-mass-dimension operators. We propose a new method to construct one-loop matrix elements with insertions of operators $D^{2k} F^n$ and $D^{2k} R^n$ in the tree-level effective action of type-I and type-II superstrings. Inspired by ambitwistor string theories, our method is based on forward limits of moduli-space integrals using string tree-level amplitudes with two extra points, expanded in powers of the inverse string tension $α'$. Similar to one-loo…
Intersection matrices associated to geometric-ordered bases of Feynman integrals
Johannes Gutenberg-Universität Mainz（约翰内斯·古腾堡美因茨大学）；Universität Bonn（波恩大学）；University of Zurich（苏黎世大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Xiamen University（厦门大学）
Elaborating on the novel formulation of the loop-tree duality, we introduce the Mathematica package Lotty that automates the latter at multi-loop level. By studying the features of Lotty and recalling former studies, we discuss that the representation of any multi-loop amplitude can be brought in a form, at integrand level, that only displays physical information, which we refer to as the causal representation of multi-loop Feynman integrands. In order to elucidate the role of Lotty in this automation, we recall results obtained for the calculation of the dual representation of integrands up-t…
We construct a diagrammatic coaction acting on one-loop Feynman graphs and their cuts. The graphs are naturally identified with the corresponding (cut) Feynman integrals in dimensional regularization, whose coefficients of the Laurent expansion in the dimensional regulator are multiple polylogarithms (MPLs). Our main result is the conjecture that this diagrammatic coaction reproduces the combinatorics of the coaction on MPLs order by order in the Laurent expansion. We show that our conjecture holds in a broad range of nontrivial one-loop integrals. We then explore its consequences for the stud…
Multi-loop scattering amplitudes constitute a serious bottleneck in current high-energy physics computations. Obtaining new integrand level representations with smooth behaviour is crucial for solving this issue, and surpassing the precision frontier. In this talk, we describe a new technology to rewrite multi-loop Feynman integrands in such a way that non-physical singularities are avoided. The method is inspired by the Loop-Tree Duality (LTD) theorem, and uses geometrical concepts to derive the causal structure of any multi-loop multi-leg scattering amplitude. This representation makes the i…
The R*-operation by Chetyrkin, Tkachov, and Smirnov is a generalisation of the BPHZ R-operation, which subtracts both ultraviolet and infrared divergences of euclidean Feynman graphs with non-exceptional external momenta. It can be used to compute the divergent parts of such Feynman graphs from products of simpler Feynman graphs of lower loops. In this paper we extend the R*-operation to Feynman graphs with arbitrary numerators, including tensors. We also provide a novel way of defining infrared counterterms which closely resembles the definition of its ultraviolet counterpart. We further expr…
The two point integrals contributing to the self energy of a particle in a three dimensional quantum field theory are calculated to two loop order in perturbation theory as well as the vacuum ones contributing to the effective potential to three loop order. For almost every integral an expression in terms of elementary and dilogarithm functions is obtained. For two integrals, the master integral and the Mercedes integral, a one dimensional integral representation is obtained with an integrand consisting only of elementary functions. The results are applied to a scalar $λϕ^4$ theory.
The numerical unitarity approach has been important for obtaining reliable QCD predictions for the LHC. Here I discuss the extension of the approach beyond the leading quantum corrections for computing multi-loop amplitudes. The numerical unitarity approach requires a suitable parametrisation of the loop integrands as a sum of terms that integrate to zero (surface terms) and master integrands. The construction and classification of suitable surface terms which match the propagators structures of Feynman amplitudes is the main technical advance discussed. A number of spin-offs for integral redu…
