We present a method for the computation of hepta-cuts of two loop scattering amplitudes. Four dimensional unitarity cuts are used to factorise the integrand onto the product of six tree-level amplitudes evaluated at complex momentum values. Using Gram matrix constraints we derive a general parameterisation of the integrand which can be computed using polynomial fitting techniques. The resulting expression is further reduced to master integrals using conventional integration by parts methods. We consider both planar and non-planar topologies for 2 to 2 scattering processes and apply the method to compute hepta-cut contributions to gluon-gluon scattering in Yang-Mills theory with adjoint fermions and scalars.
We explore maximal unitarity for nonplanar two-loop integrals with up to four massive external legs. In this framework, the amplitude is reduced to a basis of master integrals whose coefficients are extracted from maximal cuts. The hepta-cut of the nonplanar double box defines a nodal algebraic curve associated with a multiply pinched genus-3 Riemann surface. All possible configurations of external masses are covered by two distinct topological pictures in which the curve decomposes into either six or eight Riemann spheres. The procedure relies on consistency equations based on vanishing of in…
We present an alternative reduction to master integrals for one-loop amplitudes using a unitarity cut method in arbitrary dimensions. We carry out the reduction in two steps. The first step is a pure four-dimensional cut-integration of tree amplitudes with a mass parameter, and the second step is applying dimensional shift identities to master integrals. This reduction is performed at the integrand level, so that coefficients can be read out algebraically.
We develop an iterative method for constructing four-dimensional generalized unitarity cuts in $\mathcal{N} = 2$ supersymmetric Yang-Mills (SYM) theory coupled to fundamental matter hypermultiplets ($\mathcal{N} = 2$ SQCD). For iterated two-particle cuts,specifically those involving only four-point amplitudes, this implies simple diagrammatic rules for assembling the cuts to any loop order, reminiscent of the rung rule in $\mathcal{N} = 4$ SYM. By identifying physical poles, the construction simplifies the task of extracting complete integrands. In combination with the duality between color an…
We show how generalised unitarity cuts in D = 4 - 2 epsilon dimensions can be used to calculate efficiently complete one-loop scattering amplitudes in non-supersymmetric Yang-Mills theory. This approach naturally generates the rational terms in the amplitudes, as well as the cut-constructible parts. We test the validity of our method by re-deriving the one-loop ++++, -+++, --++, -+-+ and +++++ gluon scattering amplitudes using generalised quadruple cuts and triple cuts in D dimensions.
Reggeon cuts in QCD amplitudes with negative signature are discussed. These cuts appear in the next-to-next-to-leading logarithmic approximation (NNLLA) and greatly complicate the derivation of the BFKL equation. Feynman diagrams responsible for appearance of these cuts are indicated and the cut contributions are calculated in the two- and three-loop approximation.
We show that one-loop amplitudes in massless gauge theories can be determined from single cuts. By cutting a single propagator and putting it on-shell, the integrand of an n-point one-loop integral is transformed into an (n+2)-particle tree level amplitude. The single-cut approach described here is complementary to the double or multiple unitarity cut approaches commonly used in the literature. In common with these approaches, if the cut is taken in four dimensions, one finds only the cut-constructible parts of the amplitude, while if the cut is in D=4-2 epsilon dimensions, both rational and c…
We report on a technique for evaluating finite unitarity cut for one-loop amplitudes in gauge theories, and discuss its application to the cut-constructible part of six-gluon amplitude in QCD.
The use of complex analysis for computing one-loop scattering amplitudes is naturally induced by generalised unitarity-cut conditions, fulfilled by complex values of the loop variable. We report on two techniques: the cut-integration with spinor-variables as contour integrals of rational functions; and the use of the Discrete Fourier Transform to optimize the reduction of tensor-integrals to master scalar integrals.
Elaborating on the observation that two-particle unitarity-cuts of scattering amplitudes can be computed by applying Stokes' Theorem, we relate the Optical Theorem to the Berry Phase, showing how the imaginary part of arbitrary one-loop Feynman amplitudes can be interpreted as the flux of a complex 2-form.
We compute the integrand of the full-colour, two-loop, five-gluon scattering amplitude in pure Yang-Mills theory with all helicities positive, using generalized unitarity cuts. Tree-level BCJ relations, satisfied by amplitudes appearing in the cuts, allow us to deduce all the necessary non-planar information for the full-colour amplitude from known planar data. We present our result in terms of irreducible numerators, with colour factors derived from the multi-peripheral colour decomposition. Finally, the leading soft divergences are checked to reproduce the expected infrared behaviour.
Two-particle unitarity-cuts of scattering amplitudes can be efficiently computed by applying Stokes' Theorem, in the fashion of the Generalised Cauchy Theorem. Consequently, the Optical Theorem can be related to the Berry Phase, showing how the imaginary part of arbitrary one-loop Feynman amplitudes can be interpreted as the flux of a complex 2-form.
We present an extension of the spinor integration formalism of one loop amplitudes from the double-cut to the single-cut case. This technique can be applied for the computation of the tadpole coefficients. Moreover we describe an off-shell continuation of one loop amplitudes that allows a finite evaluation of the unitarity cuts in the channel of a single massive external fermion.
We explain how one-loop amplitudes with massive fermions can be computed using only on-shell information. We first use the spinor-helicity formalism in six dimensions to perform generalised unitarity cuts in $d$ dimensions. We then show that divergent wavefunction cuts can be avoided, and the remaining ambiguities in the renormalised amplitudes can be fixed, by matching to universal infrared poles in $4-2ε$ dimensions and ultraviolet poles in $6-2ε$ dimensions. In the latter case we construct an effective Lagrangian in six dimensions and reduce the additional constraint to an on-shell tree-lev…
We compute the next-to-maximally-helicity-violating one-loop n-gluon amplitudes in N=1 super-Yang-Mills theory. These amplitudes contain three negative-helicity gluons and an arbitrary number of positive-helicity gluons, and constitute the first infinite series of amplitudes beyond the simplest, MHV, amplitudes. We assemble ingredients from the N=4 NMHV tree super-amplitude into previously unwritten double cuts and use the spinor integration technique to calculate all bubble coefficients. We also derive the missing box coefficients from quadruple cuts. Together with the known formula for three…
We present an analytic technique for evaluating single cuts for one-loop integrands, where exactly one propagator is taken to be on shell. Our method extends the double-cut integration formalism of one-loop amplitudes to the single-cut case. We argue that single cuts give meaningful information about amplitudes when taken at the integrand level. We discuss applications to the computation of tadpole coefficients.
Gravitational amplitudes in the Regge limit: waveforms, shock waves and unitarity cuts
INFN, Laboratori Nazionali di Frascati（意大利国家核物理研究所，弗拉斯cati国家实验室）；Queen Mary University of London（伦敦大学玛丽女王学院）；The University of Edinburgh（爱丁堡大学）；Sapienza Università di Roma（罗马第一大学）
By exploiting relations between gravity and gauge theories, we present two infinite sequences of one-loop n-graviton scattering amplitudes: the `maximally helicity-violating' amplitudes in N=8 supergravity, and the `all-plus' helicity amplitudes in gravity with any minimally coupled massless matter content. The all-plus amplitudes correspond to self-dual field configurations and vanish in supersymmetric theories. We make use of the tree-level Kawai-Lewellen-Tye (KLT) relations between open and closed string theory amplitudes, which in the low-energy limit imply relations between gravity and ga…
We obtain the two-loop QCD corrections to the Higgs plus three-parton amplitudes with dimension-seven operators in Higgs effective field theory. This provides the two-loop S-matrix elements for Higgs plus one-jet production at the LHC with top-mass correction. We apply efficient unitarity plus IBP methods which are described in detail. We also study the color decomposition of the fermion cuts and find a connection between fundamental and adjoint representations which can be used to reduce non-planar to planar unitarity cuts in the Higgs to three-gluon amplitudes. We obtain final results in sim…
We present an algorithm for the integrand-level reduction of multi-loop amplitudes of renormalizable field theories, based on computational algebraic geometry. This algorithm uses (1) the Gröbner basis method to determine the basis for integrand-level reduction, (2) the primary decomposition of an ideal to classify all inequivalent solutions of unitarity cuts. The resulting basis and cut solutions can be used to reconstruct the integrand from unitarity cuts, via polynomial fitting techniques. The basis determination part of the algorithm has been implemented in the Mathematica package, BasisDe…
We present analytical results for all six-photon helicity amplitudes. For the computation of this loop induced process two recently developed methods, based on form factor decomposition and on multiple cuts, have been used. We obtain compact results, demonstrating the applicability of both methods to one-loop amplitudes relevant to precision collider phenomenology.
In theories with unstable particles, unitarity is satisfied by the inclusion of only stable states in unitarity sums. Hence unitarity cuts are not to be taken through unstable particles. This raises a challenge to the generalized unitarity method, whose aim is to reconstruct amplitudes by analyzing sets of unitarity cuts. Nevertheless, under some general physical conditions, and perhaps some methodological modifications, we prove that the method is still reliable for one-loop amplitudes containing resonances. We discuss some simple examples which illustrate these features.
In this letter, we exploit generalised unitarity in order to calculate the cut-constructible part of one-loop amplitudes in non-supersymmetric Yang-Mills theory. In particular, we rederive the n-gluon MHV amplitudes for both the adjacent and non-adjacent gluon helicity configurations from 3- and 4-particle cuts alone.
This article reviews on-shell methods for analytic computation of loop amplitudes, emphasizing techniques based on unitarity cuts. Unitarity techniques are formulated generally but have been especially useful for calculating one-loop amplitudes in massless theories such as Yang-Mills theory, QCD, and QED.
We review some recent additions to the tool-chest of techniques for finding compact integrand representations of multiloop gauge-theory amplitudes - including non-planar contributions - applicable for N=4 super-Yang-Mills in four and higher dimensions, as well as for theories with less supersymmetry. We discuss a general organization of amplitudes in terms of purely cubic graphs, review the method of maximal cuts, as well as some special D-dimensional recursive cuts, and conclude by describing the efficient organization of amplitudes resulting from the conjectured duality between color and kin…
We present a set of algebraic functions for evaluating the coefficients of the scalar integral basis of a general one-loop amplitude. The functions are derived from unitarity cuts, but the complete cut-integral procedure has been carried out in generality so that it never needs to be repeated. Where the master integrals are known explicitly, the results here can be used as a black box with tree-level amplitudes as input and one-loop amplitudes as output.
We review recent progress in calculations of one-loop QCD amplitudes. By imposing the consistency requirements of unitarity and correct behavior as the momenta of two legs become collinear, we construct ansatze for one-loop amplitudes with an arbitrary number of external legs. For supersymmetric amplitudes, which can be thought of as components of QCD amplitudes, the cuts uniquely specify the amplitude.
We elaborate on a non-perturbative formulation of scattering amplitudes/null polygonal Wilson loops in planar N=4 Super-Yang-Mills theory. The construction is based on a decomposition of the Wilson loop into elementary building blocks named pentagon transitions. Our discussion expands on a previous letter of the authors where these transitions were introduced and analyzed for the so-called gluonic excitations. In this paper we revisit these transitions and extend the analysis to the sector of scalar excitations. We restrict ourselves to the single particle transitions and bootstrap their finit…
We present a compact analytic expression for the leading colour two-loop five-gluon amplitude in Yang-Mills theory with a single negative helicity and four positive helicities. The analytic result is reconstructed from numerical evaluations over finite fields. The numerical method combines integrand reduction, integration-by-parts identities and Laurent expansion into a basis of pentagon functions to compute the coefficients directly from six-dimensional generalised unitarity cuts.
We express the planar five- and six-gluon two-loop Yang-Mills amplitudes with all positive helicities in compact analytic form using D-dimensional local integrands that are free of spurious singularities. The integrand is fixed from on-shell tree amplitudes in six dimensions using D-dimensional generalised unitarity cuts. The resulting expressions are shown to have manifest infrared behaviour at the integrand level. We also find simple representations of the rational terms obtained after integration in 4-2epsilon dimensions.
We consider the maximal cut of a three-loop four point function with massless kinematics. By applying Groebner bases and primary decomposition we develop a method which extracts all ten propagator master integral coefficients for an arbitrary triple-box configuration via generalized unitarity cuts. As an example we present analytic results for the three loop triple-box contribution to gluon-gluon scattering in Yang-Mills with adjoint fermions and scalars in terms of three master integrals.
In this paper we propose a dispersive method to describe two-body scattering with unitarity imposed. This approach is applied to elastic $ππ$ scattering. The amplitudes keep single-channel unitarity and describe the experimental data well, and the low-energy amplitudes are consistent with that of chiral perturbation theory. The pole locations of the $σ$, $f_0(980)$, $ρ(770)$ and $f_2(1270)$ and their couplings to $ππ$ are obtained. A virtual state appearing in the isospin-two S-wave is confirmed. The correlations between the left (and right) hand cut and the poles are discussed. Our results sh…
In four-dimensional theories with massless particles, the one-loop amplitudes can be expressed in terms of a basis of scalar integrals and rational terms. Since the scalar bubble integrals are the only UV divergent integrals, the sum of the bubble coefficients captures the 1-loop UV behavior. In particular, in a renormalizable theory the sum of the bubble coefficients equals the tree-level amplitude times a proportionality constant that is related to the one-loop beta function coefficient beta_0. In this paper, we study how this proportionality is achieved from the viewpoint of on-shell amplit…
We present formulas for the coefficients of 2-, 3-, 4- and 5-point master integrals for one-loop massive amplitudes. The coefficients are derived from unitarity cuts in D dimensions. The input parameters can be read off from any unitarity-cut integrand, as assembled from tree-level expressions, after simple algebraic manipulations. The formulas presented here are suitable for analytical as well as numerical evaluation. Their validity is confirmed in two known cases of helicity amplitudes contributing to gg -> gg and gg -> gH, where the masses of the Higgs and the fermion circulating in the loo…
Scattering amplitudes at loop level can be reduced to a basis of linearly independent Feynman integrals. The integral coefficients are extracted from generalized unitarity cuts which define algebraic varieties. The topology of an algebraic variety characterizes the difficulty of applying maximal cuts. In this work, we analyze a novel class of integrals whose maximal cuts give rise to an algebraic variety with irrational irreducible components. As a phenomenologically relevant example we examine the two-loop planar double-box contribution with internal massive lines. We derive unique projectors…
The pion - nucleon scattering phase shifts in $s$ and $p$ waves are analyzed using PKU unitarization approach that can separate the phase shifts into different contributions from poles and branch cuts. It is found that in $S_{11}$ and $P_{11}$ channels, there exist large and positive missing contributions when one compares the phase shift from known resonances plus branch cuts with the experimental data, which indicates that those two channels may contain sizable effects from $N^*(1535)$ and $N^*(1440)$ shadow poles. Those results are obtained using tree level results of the $πN$ amplitude.
Unitarity cuts are widely used in analytic computation of loop amplitudes in gauge theories such as QCD. We expand upon the technique introduced in hep-ph/0503132 to carry out any finite unitarity cut integral. This technique naturally separates the contributions of bubble, triangle and box integrals in one-loop amplitudes and is not constrained to any particular helicity configurations. Loop momentum integration is reduced to a sequence of algebraic operations. We discuss the extraction of the residues at higher-order poles. Additionally, we offer concise algebraic formulas for expressing coe…
In this letter, we construct the recursion relations for one-loop planar integrands in the SU(N) non-linear sigma model. This generalizes the soft recursions for tree-level amplitudes in a variety of quantum field theories with special soft limits. The main ingredient is the definition of the one-loop planar integrand, which is fixed by cuts in the sense of generalized unitarity and by requiring the Adler zero on all external legs. We show that this does not uniquely fix the integrand, so additional constraints on the soft behavior of the loop momentum have to be imposed. Our work is the first…
One-loop amplitudes of gluons in supersymmetric Yang-Mills are four-dimensional cut-constructible. This means that they can be determined from their unitarity cuts. We present a new systematic procedure to explicitly carry out any finite unitarity cut integral. The procedure naturally separates the contributions from bubble, triangle and box scalar integrals. This technique allows the systematic calculation of N=1 amplitudes of gluons. As an application we compute all next-to-MHV six-gluon amplitudes in N=1 super-Yang-Mills.
The $S$- and $P$- wave phase shifts of low-energy pion-nucleon scatterings are analysed using Peking University representation, in which they are decomposed into various terms contributing either from poles or branch cuts. We estimate the left-hand cut contributions with the help of tree-level perturbative amplitudes derived in relativistic baryon chiral perturbation theory up to $\mathcal{O}(p^2)$. It is found that in $S_{11}$ and $P_{11}$ channels, contributions from known resonances and cuts are far from enough to saturate experimental phase shift data -- strongly indicating contributions f…
In this paper we perform an amplitude analysis of $η'\toπ^+π^-γ$ and confront it with the latest BESIII data. Based on the final-state interaction theorem, we represent the amplitude in terms of an Omnés function multiplied by a form factor that corresponds to the contributions from left-hand cuts and right-hand cuts in the inelastic channels. We also take into account the isospin violation effect induced by $ρ-ω$ mixing. Our results show that the anomaly contribution is mandatory in order to explain the data. Its contribution to the decay width of $Γ(η'\toππγ)$ is larger than that induced by …
We present analytic results for all planar two-loop Feynman integrals contributing to five-particle scattering amplitudes with one external massive leg. We express the integrals in terms of a basis of algebraically-independent transcendental functions, which we call one-mass pentagon functions. We construct them by using the properties of iterated integrals with logarithmic kernels. The pentagon functions are manifestly free of unphysical branch cuts, do not require analytic continuation, and can be readily evaluated over the whole physical phase space of the massive particle production channe…
In this work we explore general leading singularities of one-loop amplitudes in higher-derivative Yang-Mills and quadratic gravity. These theories are known to possess propagators which contain quadratic and quartic momentum dependence, which leads to the presence of an unstable ghostlike resonance. However, unitarity cuts are not to be taken through unstable particles and therefore unitarity is still satisfied. On the other hand, this could engender issues when calculating leading singularities which are generalizations of unitarity cuts. Nevertheless, we will show with explicit examples how …
Among the unitarity cuts of 4-loop massless propagators two kinds are currently fully known: the 2-particle cuts with 3 loops corresponding to form-factors, and the 5-particle phase-space integrals. In this paper we calculate master integrals for the remaining ones: 3-particle cuts with 2 loops, and 4-particle cuts with 1 loop. The 4-particle cuts are calculated by solving dimensional recurrence relations. The 3-particle cuts are integrated directly using 1->3 amplitudes with 2 loops, which we also re-derive here up to transcendentality weight 7. The results are verified both numerically, and …
We propose a first implementation of the integrand-reduction method for two-loop scattering amplitudes. We show that the residues of the amplitudes on multi-particle cuts are polynomials in the irreducible scalar products involving the loop momenta, and that the reduction of the amplitudes in terms of master integrals can be realized through polynomial fitting of the integrand, without any apriori knowledge of the integral basis. We discuss how the polynomial shapes of the residues determine the basis of master integrals appearing in the final result. We present a four-dimensional constructive…
We show how to compute the coefficients of the double box basis integrals in a massless four-point amplitude in terms of tree amplitudes. We show how to choose suitable multidimensional contours for performing the required cuts, and derive consistency equations from the requirement that integrals of total derivatives vanish. Our formulae for the coefficients can be used either analytically or numerically.
The most general structure of an elastic partial wave amplitude when the unphysical cuts are neglected is deduced in terms of the N/D method. This result is then matched to lowest order, ${\mathcal{O}}(p^2)$, Chiral Perturbation Theory($χ$PT) and to the exchange (consistent with chiral symmetry) of resonances in the s-channel. The extension of the method to coupled channels is also given. Making use of the former formalism, the $ππ$ and $Kπ$(I=1/2) P-wave scattering amplitudes are described without free parameters when taking into account relations coming from the 1/$N_c$ expansion and unitari…
We present a general method to account for full colour dependence Yang-Mills amplitudes at loop level. The method fits most naturally into the framework of multi-loop integrand reduction and in a nutshell amounts to consistently retaining the colour structures of the unitarity cuts from which the integrand is gradually constructed. This technique has already been used in the recent calculation of the two-loop five-gluon amplitude in pure Yang-Mills theory with all positive helicities, arXiv:1507.08797. In this note, we give a careful exposition of the method and discuss its connection to loop-…
We review techniques for more efficient computation of perturbative scattering amplitudes in gauge theory, in particular tree and one-loop multi-parton amplitudes in QCD. We emphasize the advantages of (1) using color and helicity information to decompose amplitudes into smaller gauge-invariant pieces, and (2) exploiting the analytic properties of these pieces, namely their cuts and poles. Other useful tools include recursion relations, special gauges and supersymmetric rearrangements.
After a brief review of the BFKL approach to Regge processes in QCD and in supersymmetric (SUSY) gauge theories we propose a strategy for calculating the next-to-next-to-leading order corrections to the BFKL kernel. They can be obtained in terms of various cross-sections for Reggeized gluon interactions. The corresponding amplitudes can be calculated in the framework of the effective action for high energy scattering. In the case of N=4 SUSY it is also possible to use the Bern-Dixon-Smirnov (BDS) ansatz. For this purpose the analytic properties of the BDS amplitudes at high energies are invest…
We develop a unitarity method to compute one-loop amplitudes with massless propagators in d=4-2*epsilon dimensions. We compute double cuts of the loop amplitudes via a decomposition into a four-dimensional and a -2*epsilon-dimensional integration. The four-dimensional integration is performed using spinor integration or other efficient techniques. The remaining integral in -2*epsilon dimensions is cast in terms of bubble, triangle, box, and pentagon master integrals using dimensional shift identities. The method yields results valid for arbitrary values of epsilon.
We review recent progress in D-dimensional integrand reduction algorithms for two loop amplitudes and give examples of their application to non-planar maximal cuts of the five-point all-plus helicity amplitude in QCD.
One-loop amplitudes are to a large extent determined by their unitarity cuts in four dimensions. We show that the remaining rational terms can be obtained from the ultraviolet behaviour of the amplitude, and determine universal form factors for these rational parts by applying reduction techniques to the Feynman diagrammatic representation of the amplitude. The method is valid for massless and massive internal particles. We illustrate this method by evaluating the rational terms of the one-loop amplitudes for gg --> H, the 4-photon and 4-gluon amplitude, the amplitude for 2 photons plus 3 gluo…
Analytic decomposition of two-body electroweak processes with left-hand cuts
Ruhr-Universität Bochum（波鸿鲁尔大学）；University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；CSIC‐Universitat de Valéncia（西班牙国家研究委员会-瓦伦西亚大学）
This thesis is focused on the development of new mathematical methods for computing multi-loop scattering amplitudes in gauge theories. In this work we combine, for the first time, the unitarity-based construction for integrands, and the recently introduced integrand-reduction through multivariate polynomial division. After discussing the generic features of this novel reduction algorithm, we will apply it to the one- and two-loop five-point amplitudes in ${\cal N}=4$ sYM. The integrands of the multiple-cuts are generated from products of tree-level amplitudes within the super-amplitudes forma…
A production representation of partial-wave $S$ matrix is utilized to construct low-energy elastic pion-nucleon scattering amplitudes from cuts and poles on complex Riemann sheets. Among them, the contribution of left-hand cuts is estimated using the $\mathcal{O}(p^3)$ results obtained in covariant baryon chiral perturbation theory within the extended-on-nass-shell scheme. By fitting to data on partial-wave phase shifts, it is indicated that the existences of hidden poles in $S_{11}$ and $P_{11}$ channels, as conjectured in our previous paper~\citep{Wang:2017agd}, are firmly established. Speci…
The success of the experimental program at the Tevatron re-inforced the idea that precision physics at hadron colliders is desirable and, indeed, possible. The Tevatron data strongly suggests that one-loop computations in QCD describe hard scattering well. Extrapolating this observation to the LHC, we conclude that knowledge of many short-distance processes at next-to-leading order may be required to describe the physics of hard scattering. While the field of one-loop computations is quite mature, parton multiplicities in hard LHC events are so high that traditional computational techniques be…
