v2: Normalisation of MSbar poles clarified; v3: Version published on JHEP; References added; Further comments on use of master formula (5.61) in Sect. 6; New footnote on scale dependence in Sect. 5; Typos fixed and/or presentation improved in eqs. (4.8), (4.20), (5.2), (6.5), (6.7), (6.8), (A.2), (A.4); Results unchanged
The advent of efficient numerical algorithms for the construction of one-loop amplitudes has played a crucial role in the automation of NLO calculations, and the development of similar algorithms at two loops is a natural strategy for NNLO automation. Within a numerical framework the numerator of loop integrals is usually constructed in four dimensions, and the missing rational terms, which arise from the interplay of the $(D-4)$-dimensional parts of the loop numerator with $1/(D-4)$ poles in $D$ dimensions, are reconstructed separately. At one loop, such rational terms arise only from UV divergences and can be restored through process-independent local counterterms. In this paper we investigate the behaviour of rational terms of UV origin at two loops. The main result is a general formula that combines the subtraction of UV poles with the reconstruction of the associated rational parts at two loops. This formula has the same structure as the R-operation, and all poles and rational parts are described through a finite set of process-independent local counterterms. We also present a general formula for the calculation of all relevant two-loop rational counterterms in any renormalisable theory based on one-scale tadpole integrals. As a first application, we derive the full set of two-loop rational counterterms for QED in the $R_ξ$-gauge.
We present a calculation of the rational terms in two-loop all-plus gluon amplitudes using $D$-dimensional unitarity. We use a conjecture of separability of the two loops, and then a simple generalization of one-loop $D$-dimensional unitarity to perform calculations. We compute the four- and five-point rational terms analytically, and the six- and seven-point ones numerically. We find agreement with previous calculations of Dalgleish, Dunbar, Godwin, Jehu, Perkins, and Strong. For a special subleading-color amplitude, we compute the eight- and nine-point results numerically, and find agreement…
All four-point mixed gluon-graviton amplitudes in pure Einstein-Yang-Mills theory with at most one state of negative helicity are computed at one-loop order and maximal powers of the gauge coupling using D-dimensional generalized unitarity. The resulting purely rational expressions take very compact forms. We comment on the color-kinematics duality picture and a relation to collinear limits of pure gluon amplitudes.
Scattering amplitudes in $D$ dimensions involve particular terms that originate from the interplay of UV poles with the $D-4$ dimensional parts of loop numerators. Such contributions can be controlled through a finite set of process-independent rational counterterms, which make it possible to compute loop amplitudes with numerical tools that construct the loop numerators in four dimensions. Building on a recent study [1] of the general properties of two-loop rational counterterms, in this paper we investigate their dependence on the choice of renormalisation scheme. We identify a nontrivial fo…
Rational counterterms are a key ingredient for the automation of loop calculations through numerical methods. Building on the recently established properties of rational terms of UV origin at two loops, in this paper we present a systematic method for the determination of rational counterterms within spontaneously broken theories. In particular we introduce a generalised vev-expansion approach that makes it possible to obtain the rational counterterms of UV origin for a spontaneously broken theory by means of calculations in the unbroken phase. The drastic simplifications that result from the …
We present an extension of the renormalisation procedure based on the R-operation in $D$ dimensions at two-loop level, in which the numerators of all Feynman diagrams can be constructed in four dimensions, and the rational terms stemming from the interplay of $(D-4)$-dimensional numerator parts and UV poles are fully reconstructed from a finite set of universal local counterterms. This represents an extension of the concept of rational terms of type $R_2$ to two loops. We provide a general method to compute one and two-loop rational counterterms from massive one-scale tadpole integrals. Finall…
We present work on two-loop amplitudes in pure Yang-Mills theory with all gluons of identical helicity. We show how to obtain their rational terms -- the hardest parts to compute -- via well-understood one-loop unitarity techniques.
We study Feynman integrals and scattering amplitudes in ${\cal N}=4$ super-Yang-Mills by exploiting the duality with null polygonal Wilson loops. Certain Feynman integrals, including one-loop and two-loop chiral pentagons, are given by Feynman diagrams of a supersymmetric Wilson loop, where one can perform loop integrations and be left with simple integrals along edges. As the main application, we compute analytically for the first time, the symbol of the generic ($n\geq 12$) double pentagon, which gives two-loop MHV amplitudes and components of NMHV amplitudes to all multiplicities. We repres…
We present advances in the development of the numerical unitarity method for the computation of multi-loop amplitudes in QCD. As an application, we show results for all the leading-color two-loop five-gluon helicity amplitudes. The amplitudes are reduced to a linear combination of master integrals by employing unitarity-compatible integration-by-parts identities, and the corresponding integral coefficients are computed in an exact manner on rational phase-space points through finite fields arithmetics.
We use rational approximants to study missing higher orders in the massless scalar-current quark correlator. We predict the yet unknown six-loop coefficient of its imaginary part, related to $Γ(H\to b \bar b)$, to be $c_5=-6900\pm 1400$. With this result, the perturbative series becomes almost insensitive to renormalization scale variations and the intrinsic QCD truncation uncertainty is tiny. This confirms the expectation that higher-order loop computations for this quantity will not be required in the foreseeable future, as the uncertainty in $Γ(H\to b \bar b)$ will remain largely dominated …
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 describe on-shell methods for computing one- and two-loop anomalous dimensions in the context of effective field theories containing higher-dimension operators. We also summarize methods for computing one-loop amplitudes, which are used as inputs to the computation of two-loop anomalous dimensions, and we explain how the structure of rational terms and judicious renormalization scheme choices can lead to additional vanishing terms in the anomalous dimension matrix at two loops. We describe the two-loop implications for the Standard Model Effective Field Theory (SMEFT). As a by-product of th…
The evaluation of multi-loop Feynman integrals is one of the main challenges in the computation of precise theoretical predictions for the cross sections measured at the LHC. In recent years, the method of differential equations has proven to be a powerful tool for the computation of Feynman integrals. It has been observed that the differential equation of Feynman integrals can in many instances be transformed into a so-called canonical form, which significantly simplifies its integration in terms of iterated integrals. The main result of this thesis is an algorithm to compute rational transfo…
We present the two-loop master integrals relevant to the ${\mathcal O}(αα_s)$-corrections to the decay $H \rightarrow b \bar{b}$ through a $H t \bar{t}$-coupling. We keep the full dependence on the heavy particle masses, but neglect the $b$-quark mass. The occurring square roots can be rationalised and the result is expressed in terms of multiple polylogarithms.
