Two-loop Amplitudes Research Articles

Hidden analytic relations for two-loop Higgs amplitudes in QCD

We compute the Higgs plus two-quark and one-gluon amplitudes () and Higgs plus three-gluon amplitudes () in the Higgs effective theory with a general class of operators. By changing the quadratic Casimir C F to C A , the maximally transcendental parts of the amplitudes turn out to be equivalent to that of the amplitudes, which also coincide with the counterparts in SYM. This generalizes the so-called maximal transcendentality principle to the Higgs amplitudes with external quark states, thus the full QCD theory. We further verify that the correspondence applies also to two-loop form factors of more general operators, in both QCD and scalar-YM theory. Another interesting relation is also observed between the planar amplitudes and the minimal density form factors in SYM.

Soft gluon emission at two loops in full color

The soft emission factor is a central ingredient in the factorization of generic n-particle gauge theory amplitudes with one soft gluon in the external state. We present the complete two-loop soft factor, capturing the leading power behavior in the soft-gluon momentum. At two loops, the color structure and the kinematic dependence of the soft factor become nontrivial as the soft gluon can couple to three hard partons for the first time (tripole terms). The nontrivial kinematic dependence of the tripole terms is of uniform, maximal transcendental weight, and can be expressed (in a “Euclidean” region) in terms of single-valued harmonic polylogarithms. Our results are consistent with the behavior of the recently computed symbol of the two-loop five-particle amplitude in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super-Yang-Mills theory. In the limit where the outgoing soft gluon is also collinear with an incoming hard parton, potentially dangerous factorization-violating terms can arise.

Full color two-loop six-gluon all-plus helicity amplitude

We present the full color two-loop six-point all-plus Yang-Mills amplitude in compact analytic form. The computation uses four dimensional unitarity and augmented recursion.

Direct reduction of multiloop multiscale scattering amplitudes

We propose an alternative approach based on series representation to directly reduce multi-loop multi-scale scattering amplitude into set of freely chosen master integrals. And this approach avoid complicated calculations of inverse matrix and dimension shift for tensor reduction calculation. During this procedure we further utilize the Feynman parameterization to calculate the coefficients of series representation and obtain the form factors. Conventional methodologies are used only for scalar vacuum bubble integrals to finalize the result in series representation form. Finally, we elaborate our approach by presenting the reduction of a typical two-loop amplitude for W boson production.

$1/8$-BPS couplings and exceptional automorphic functions

Unlike the \mathcal{R}^4ℛ4 and \nabla^4\mathcal{R}^4∇4ℛ4 couplings, whose coefficients are Langlands–Eisenstein series of the U-duality group, the coefficient \mathcal{E}^{(d)}_{(0,1)}ℰ(0,1)(d) of the \nabla^6\mathcal{R}^4∇6ℛ4 interaction in the low-energy effective action of type II strings compactified on a torus T^dTd belongs to a more general class of automorphic functions, which satisfy Poisson rather than Laplace-type equations. In earlier work [1], it was proposed that the exact coefficient is given by a two-loop integral in exceptional field theory, with the full spectrum of mutually 1/2-BPS states running in the loops, up to the addition of a particular Langlands–Eisenstein series. Here we compute the weak coupling and large radius expansions of these automorphic functions for any dd. We find perfect agreement with perturbative string theory up to genus three, along with non-perturbative corrections which have the expected form for 1/8-BPS instantons and bound states of 1/2-BPS instantons and anti-instantons. The additional Langlands–Eisenstein series arises from a subtle cancellation between the two-loop amplitude with 1/4-BPS states running in the loops, and the three-loop amplitude with mutually 1/2-BPS states in the loops. For d=4d=4, the result is shown to coincide with an alternative proposal [2] in terms of a covariantised genus-two string amplitude, due to interesting identities between the Kawazumi–Zhang invariant of genus-two curves and its tropical limit, and between double lattice sums for the particle and string multiplets, which may be of independent mathematical interest.

N -point QCD two-loop amplitude

We present an explicit expression for a particular $n$-gluon two loop scattering partial amplitude. Specifically we present an analytic form for the single trace $N_c$ independent colour partial amplitude for the case where all external gluons have positive helicity.

Towards an efficient method to compute two-loop scalar amplitudes

We report on an ongoing work initiated by Prof. Shimizu, proposing a method to numerically compute two-loop scalar integrals as sums of two-dimensional integrals of generalised one-loop N-point functions analytically computed and integrated over some simple weight functions. The analytic computation of the generalised one-loop N-point functions in a systematic way motivates a novel approach sketched in this talk.

Two-loop non-leptonic penguin amplitude in QCD factorization

We complete the calculation of the QCD penguin amplitude at next-to-next- to-leading order in the QCD factorization approach to non-leptonic B-meson decays. This provides the last missing piece in the computation of the QCD correction to direct CP asymmetries at leading power in the heavy-quark expansion.

All-Multiplicity Nonplanar Amplitude Integrands in Maximally Supersymmetric Yang-Mills Theory at Two Loops.

We give a prescriptive representation of all-multiplicity two-loop maximally-helicity-violating (MHV) amplitude integrands in fully-color-dressed (nonplanar) maximally supersymmetric Yang-Mills theory.

Two-loop octagons, algebraic letters and Q¯ equations

We compute the symbol of the first two-loop amplitudes in planar $\mathcal{N}=4$ SYM with algebraic letters, the eight-point Next to Maximally Helicity Violating (NMHV) amplitude (or the dual octagon Wilson loops). We show how to apply $\overline{Q}$ equations of [S. Caron-Huot and S. He, J. High Energy Phys. 07 (2012) 174] for computing the differential of two-loop $n$-point NMHV amplitudes and present the result for $n=8$ explicitly. The symbol alphabet for octagon consists of 180 independent rational letters and 18 algebraic ones involving Gram-determinant square roots. We comment on all-loop predictions for final entries and aspects of the result valid for all multiplicities.

Two-loop n-point anomalous amplitudes in supergravity.

We compute the anomalous two-loop four-point amplitudes in pure supergravity, using unitarity and the double-copy construction. We also present all terms determined by four-dimensional cuts in two all-multiplicity two-loop anomalous superamplitudes. This result provides the first two-loop n-point gravity amplitude, up to a class of undetermined terms, which we determine completely at four points. We show that a recently proposed finite counterterm cancels these amplitudes to this order. We argue that the counterterm does not spoil the three-loop finiteness of anomalous amplitudes in the theory.

NNLO QCD + NLO EW with Matrix+OpenLoops: precise predictions for vector-boson pair production

We present the first combination of NNLO QCD and NLO EW corrections for vector-boson pair production at the LHC. We consider all final states with two, three and four charged leptons, including resonant and non-resonant diagrams, spin correlations and off-shell effects. Detailed predictions are discussed for three representative channels corresponding to W+W−, W±Z and Z Z production. Both QCD and EW corrections are very significant, and the details of their combination can play a crucial role to achieve the level of precision demanded by experimental analyses. In this context we point out nontrivial issues that arise at large transverse momenta, where the EW corrections are strongly enhanced by Sudakov logarithms and the QCD corrections can feature so-called giant K -factors. Our calculations have been carried out in the Matrix+OpenLoops framework and can be extended to the production of an arbitrary colour singlet in hadronic collisions, provided that the required two-loop QCD amplitudes are available. Combined NNLO QCD and NLO EW predictions for the full set of massive diboson processes will be made publicly available in the next release of Matrix and will be instrumental in advancing precision diboson studies and new-physics searches at the LHC and future hadron colliders.

Color dressed unitarity and recursion for Yang-Mills two-loop all-plus amplitudes

We present a direct computation of the full color two-loop five-point all-plus Yang-Mills amplitude using four dimensional unitarity and recursion. We present the $SU(N_c)$ amplitudes in compact analytic forms. Our results match the explicit expressions previously computed but do not require full two-loop integral methods.

Infrared and transcendental structure of two-loop supersymmetric QCD amplitudes

Using a careful choice of infrared (IR) subtraction scheme, we demonstrate cancellation of all terms with transcendental weights 0, 1, 2 from the finite part of the full-color two-loop four-gluon mathcal{N} = 2 supersymmetric QCD amplitude, with Nf massless supersymmetric quarks. This generalizes the previously observed cancellation of weight-2 terms in the superconformal theory, where Nf = 2Nc for gauge group SU(Nc). The subtraction scheme follows naturally both from general IR factorization principles and from an integrand-level analysis of divergences in this amplitude. The divergences are written in terms of scalar triangle integrals whose expressions are known to all orders in the dimensional regulator ϵ = (4 − D)/2. We also present integrated expressions for the full-color two-loop four-point amplitudes with both matter and vectors on external legs in which lower-weight terms also cancel using an appropriate IR scheme. This provides us with values for the two-loop cusp, gluonic, and quark anomalous dimensions in mathcal{N} = 2 supersymmetric QCD, which are cross-checked between the three different amplitudes.

Full-Color Two-Loop Four-Gluon Amplitude in N=2 Supersymmetric QCD.

We present the fully integrated form of the two-loop four-gluon amplitude in N=2 supersymmetric quantum chromodynamics with gauge group SU(N_{c}) and with N_{f} massless supersymmetric quarks (hypermultiplets) in the fundamental representation. Our result maintains full dependence on N_{c} and N_{f}, and relies on the existence of a compact integrand representation that exhibits the duality between color and kinematics. Specializing to the N=2 superconformal theory, where N_{f}=2N_{c}, we obtain remarkably simple amplitudes that have an analytic structure close to that of N=4 super-Yang-Mills theory, except that now certain lower-weight terms appear. We comment on the corresponding results for other gauge groups.

A numerical evaluation of planar two-loop helicity amplitudes for a W-boson plus four partons

We present the first numerical results for the two-loop helicity amplitudes for the scattering of four partons and a W-boson in QCD. We use a finite field sampling method to reduce directly from Feynman diagrams to the coefficients of a set of master integrals after applying integration-by-parts identities. Since the basis of master integrals is not yet fully known analytically, we identify a set of master integrals with a simple divergence structure using local numerator insertions. This allows for accurate numerical evaluation of the amplitude using sector decomposition methods.

Analytic Form of the Full Two-Loop Five-Gluon All-Plus Helicity Amplitude.

We compute the full-color two-loop five-gluon amplitude for the all-plus helicity configuration. In order to achieve this, we calculate the required master integrals for all permutations of the external legs, in the physical scattering region. We verify the expected divergence structure of the amplitude and extract the finite hard function. We further validate our result by checking the factorization properties in the collinear limit. Our result is fully analytic and valid in the physical scattering region. We express it in a compact form containing logarithms, dilogarithms, and rational functions.

Hto boverline{b}j at next-to-next-to-leading order accuracy

We present the calculation of the decay Hto boverline{b}j at next-to-next-to-leading order (NNLO) accuracy in QCD. We treat the bottom quarks as massless with a non-zero Higgs Yukawa coupling yb. We consider contributions in which the Higgs boson couples directly to bottom quarks, i.e. our predictions are accurate to order mathcal{O}left({alpha}_s^3{y}_b^2right) . We calculate the various components needed to construct the NNLO contribution, including an independent calculation of the two-loop amplitudes. We compare our results for the two-loop amplitudes to an existing calculation finding agreement. We present additional checks on our two-loop expression using the known infrared factorization properties as the emitted gluon becomes soft or collinear. We use our results to construct a Monte Carlo implementation of Hto boverline{b}j and present jet rates and differential distributions in the Higgs rest frame using the Durham jet algorithm.

Analytic form of the planar two-loop five-parton scattering amplitudes in QCD

We present the analytic form of all leading-color two-loop five-parton helicity amplitudes in QCD. The results are analytically reconstructed from exact numerical evaluations over finite fields. Combining a judicious choice of variables with a new approach to the treatment of particle states in D dimensions for the numerical evaluation of amplitudes, we obtain the analytic expressions with a modest computational effort. Their systematic simplification using multivariate partial-fraction decomposition leads to a particularly compact form. Our results provide all two-loop amplitudes required for the calculation of next-to-next-to-leading order QCD corrections to the production of three jets at hadron colliders in the leading-color approximation.

Two-loop doubly massive four-point amplitude involving a half-BPS and Konishi operator

The two-loop four-point amplitude of two massless SU(N) colored scalars and two color singlet operators with different virtuality described by a half-BPS and Konishi operators is calculated analytically in maximally supersymmetric Yang-Mills theory. We verify the ultraviolet behaviour of the unprotected composite operator and exponentiation of the infrared divergences with correct universal values of the anomalous dimensions in the modified dimensional reduction scheme. The amplitude is found to contain lower transcendental weight terms in addition to the highest ones and the latter has no similarity with similar amplitudes in QCD.