We calculate the two-loop and one-loop/one-emission contributions required for soft gluon evolution at the next-to-leading order. The colour structures are expressed in the colour flow basis, and the kinematic dependence and loop integrals are expressed in terms of multiple cuts and phase-space-like integrals. This directly allows to use them in the resummation of non-global observables and improved parton shower algorithms beyond the leading order and beyond the leading colour limit. Within the colour flow basis it becomes apparent that correlations beyond a dipole picture emerge even in colour-diagonal elements of the virtual corrections.
We calculate the one-loop contributions to the MSSM effective potential when the scalar top fields have non-zero vacuum expectation values and study their impact on charge and colour breaking bounds.
In this letter, we extend the known results for the QCD potential between a static quark and its antiquark by computing the two-loop corrections to the colour-octet state.
We consider the interaction between two static sources in the colour octet configuration and compute the potential to three loops. Special emphasis is put on the treatment of pinch contributions and two methods are applied to reduce their evaluation to diagrams without pinches.
We consider the factorization and resummation of soft and Coulomb gluons for pair-production processes of heavy coloured particles at hadron colliders, and discuss: the construction of a colour basis that diagonalizes the leading soft function to all orders in perturbation theory; the determination of the two-loop soft anomalous dimension needed for NNLL resummations; and a general formula that provides all logarithmically enhanced O(alpha_s^2) correction requiring as process-independent input only the one-loop hard matching coefficients.
We describe the decomposition of one-loop QCD amplitudes in terms of colour-ordered building blocks. We give new expressions for the coefficients of QCD colour structures in terms of ordered objects called primitive amplitudes, for processes with up to seven partons. These results are needed in computations of high-multiplicity scattering cross sections in next-to-leading-order (NLO) QCD. We explain the origin of new relations between multi-quark primitive amplitudes which can be used to optimise efficiency of NLO computations. As a first application we compute the full-colour virtual contribu…
At small inter-parton distances, double parton distributions receive their dominant contribution from the splitting of a single parton. We compute this mechanism at next-to-leading order in perturbation theory for all colour configurations of the observed parton pair. Rapidity divergences are handled either by using spacelike Wilson lines or by applying the delta regulator. We investigate the behaviour of the two-loop contributions in different kinematic limits, and we illustrate their impact in different channels.
We present a factorization formula for the production of pairs of heavy coloured particles in hadronic collisions at the production threshold, which forms the basis for the resummation of soft gluons and Coulomb gluons. We construct a basis in colour space that diagonalizes the soft function appearing in the factorization formula to all orders in perturbation theory. This extends recent results on the structure of soft anomalous dimensions and allows us to determine an analytic expression for the two-loop soft anomalous dimension at threshold for all production processes of interest.
We consider soft gluon evolution in the colour flow basis. We give explicit expressions for the colour structure of the (one-loop) soft anomalous dimension matrix for an arbitrary number of partons, and show how the successive exponentiation of classes of large-N contributions can be achieved to provide a systematic expansion of the evolution in terms of colour supressed contributions.
We analytically compute the four-loop QCD corrections for the colour structure $(d_F^{abcd})^2$ to the massless non-singlet quark form factor. The computation involves non-trivial non-planar integral families which have master integrals in the top sector. We compute the master integrals by introducing a second mass scale and solving differential equations with respect to the ratio of the two scales. We present details of our calculational procedure. Analytical results for the cusp and collinear anomalous dimensions, and the finite part of the form factor are presented. We also provide analytic…
We discuss how colour flows can be used to simplify the computation of matrix elements, and in the context of parton shower Monte Carlos with accuracy beyond leading-colour. We show that, by systematically employing them, the results for tree-level matrix elements and their soft limits can be given in a closed form that does not require any colour algebra. The colour flows that we define are a natural generalization of those exploited by existing Monte Carlos; we construct their representations in terms of different but conceptually equivalent quantities, namely colour loops and dipole graphs,…
We calculate the complete set of two-loop leading-colour QCD helicity amplitudes for $γγj$-production at hadron colliders. Our results are presented in a compact, fully-analytical form.
We discuss jet vetoes as a means of probing colour flow in hard-scattering processes in hadronic collisions. As an example, we describe a calculation of the dijet cross-section with a jet veto, which resums the leading logarithms of the veto scale and is matched to a fixed-order computation. We compare this prediction to the measurement performed by the ATLAS collaboration. Finally, we outline future developments in this research area.
Top-Yukawa contributions to $pp\to b\bar{b}H$: two-loop leading-colour amplitudes
Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Pohang University of Science and Technology（浦项科技大学）；The Henryk Niewodniczański Institute of Nuclear Physics（亨利克·涅沃德尼茨基核物理研究所）
We discuss the non-perturbative computation and interpretation of a colour adjoint static potential based on Wilson loops with generator insertions. Numerical lattice results for SU(2) gauge theory are presented and compared to corresponding perturbative results.
In this talk, we review recent developments towards the calculation of multi-loop scattering amplitudes. In particular, we discuss how the colour-kinematics duality can provide new integral relations at one-loop level via the Loop-Tree duality formalism. On the other hand, in order to compute scattering amplitudes at one- and two-loop level, numerically and analytically, we describe the preliminary automation of the adaptive integrand decomposition algorithm. We show preliminary results on the analytic reduction of the $μe$-elastic scattering at one- and two-loop level.
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 present the two-loop leading colour QCD helicity amplitudes for the process $pp\to W(\to lν)γ+j$. We implement a complete reduction of the amplitudes, including the leptonic decay of the $W$-boson, using finite field arithmetic, and extract the analytic finite remainders using a recently identified basis of special functions. Simplified analytic expressions are obtained after considering permutations of a rational kinematic parametrisation and multivariate partial fractioning. We demonstrate efficient numerical evaluation of the two-loop colour and helicity summed finite remainders for phys…
Using an iterative solution of rapidity evolution equations, we compute partonic $2\to 2$ gauge theory amplitudes at four loops in full colour up to the Next-to-Next-to-Leading Logarithms (NNLL) in the Regge limit. By contrasting the resulting amplitude with the exponentiation properties of soft singularities we determine the four-loop correction to the soft anomalous dimension at this logarithmic accuracy, which universally holds in any gauge theory. We find that the latter features quartic Casimir contributions beyond those appearing in the cusp anomalous dimension. Finally, in the case of $…
We study the soft limit of one-loop QCD amplitudes and we derive the process-independent factorization formula that controls the singular behaviour in this limit. This is obtained from the customary eikonal factorization formula valid at tree (classical) level by introducing a generalized soft-gluon current that embodies the quantum corrections. We compute the explicit expression of the soft-gluon current at one-loop order. It contains purely non-abelian correlations between the colour charges of each pair of hard-momentum partons in the matrix element. This leads to colour correlations betwee…
We present the master integrals relevant for computing the complete set of leading-colour analytic helicity amplitudes for top-quark pair production via gluon fusion at two-loops in QCD. We include corrections due to massive fermion loops which give rise to integrals involving elliptic curves. We also elaborate on the structure of singularities that play an important role in the numerical evaluation of the iterated integrals and their analytic continuation.
We present the leading colour and light fermionic planar two-loop corrections for the production of two photons and a jet in the quark-antiquark and quark-gluon channels. In particular, we compute the interference of the two-loop amplitudes with the corresponding tree level ones, summed over colours and polarisations. Our calculation uses the latest advancements in the algorithms for integration-by-parts reduction and multivariate partial fraction decomposition to produce compact and easy-to-use results. We have implemented our results in an efficient C++ numerical code. We also provide their …
We compute the two-loop QCD helicity amplitudes for the production of a Higgs boson in association with a bottom-quark pair at a hadron collider. We take the approximations of leading colour and work in the five flavour scheme, where the bottom quarks are massless while the Yukawa coupling is non-zero. We extract analytic expressions from multiple numerical evaluations over finite fields and present the results in terms of an independent set of special functions that can be reliably evaluated over the full phase space.
We introduce the computer code Recola for the recursive generation of tree-level and one-loop amplitudes in the Standard Model. Tree-level amplitudes are constructed using off-shell currents instead of Feynman diagrams as basic building blocks. One-loop amplitudes are represented as linear combinations of tensor integrals whose coefficients are calculated similarly to the tree-level amplitudes by recursive construction of loop off-shell currents. We introduce a novel algorithm for the treatment of colour, assigning a colour structure to each off-shell current which enables us to recursively co…
We present a complete set of analytic helicity amplitudes for top quark pair production via gluon fusion at two-loops in QCD. For the first time, we include corrections due to massive fermion loops which give rise to integrals over elliptic curves. We present the results of the missing master integrals needed to compute the amplitude and obtain an analytic form for the finite remainders in terms of iterated integrals using rationalised kinematics and finite field sampling. We also study the numerical evaluation of the iterated integrals.
$N$-subjettiness ratios are in wide use for tagging heavy boosted objects, in particular the ratio of 2-subjettiness to 1-subjettiness for tagging boosted electroweak bosons. In this article we introduce a new, \emph{dichroic} ratio, which uses different regions of a jet to determine the two subjettiness measures, emphasising the hard substructure for the 1-subjettiness and the full colour radiation pattern for the 2-subjettiness. Relative to existing $N$-subjettiness ratios, the dichroic extension, combined with SoftDrop (pre-)grooming, makes it possible to increase the ultimate signal signif…
The one-loop polarisation tensor and the propagation of ``in-medium'' photons of colour superconductors in the 2SC and CFL phase is discussed. For a study of thermal corrections to the low energy effective theory in the 2SC phase, a classical transport theory for fermionic quasiparticles is invoked.
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.
By direct Monte-Carlo simulations it is shown that the anisotropic flows can be successfully described in the colour string picture with fusion and percolation provided anisotropy of particle emission from the fused string is taken into account. Quenching of produced particles in the strong colour field of the string is the basic mechanism for this anisotropy. The concrete realization of this mechanism is borrowed from the QED. Due to dependence of this mechanism on the external field strength the found flows grow with energy, with values for $v_2$ at LHC energies greater by ~15% than at RHIC …
In this talk I discuss the application and generalization of the antenna subtraction method to processes involving incoherent interferences of partial amplitudes, which are generically present for the sub-leading colour contributions to processes involving more than five partons. The approach makes use of the known infrared (IR) singularity structure of one- and two-loop matrix elements to guide the construction of the subtraction terms. A set of integrated dipoles are defined which can be used to express the poles of one- and two-loop matrix elements in terms of integrated antennae. The unint…
We present the computer code RECOLA for the computation of EW and QCD amplitudes in the Standard Model at next-to-leading order. One-loop amplitudes are represented as linear combinations of tensor integrals whose coefficients are calculated by means of recursive relations similar to Dyson-Schwinger equations. A novel treatment of colour enables us to recursively construct the colour structure of the amplitude efficiently. RECOLA is linked with the library COLLIER for the computation of the tensor integrals.
We present first calculations of QCD matrix-elements in perturbation theory at finite Nc beyond 2 loops in the eikonal approximation for e+ e- annihilation processes. For the emission of n soft energy-ordered gluons we solve both the colour and kinematic structures at a given order in perturbation theory by means of a Mathematica program that relies solely on a recently developed Mathematica code, ColorMath, that evaluates the trace of products of colour matrices. At large Nc, our squared amplitudes reduce to those already known in the literature.
We compute the soft contribution to the static energy of two heavy colour sources interacting via a N=4 supersymmetric Yang-Mills theory. Both singlet and octet colour configurations are considered. Our calculations complete recent considerations of the ultrasoft contributions.
We have computed the contributions of the quartic Casimir invariants to the four-loop anomalous dimensions of twist-2 spin-N operators at N =< 16. The results provide new information on the structure of the next-to-next-to-next-to-leading order (N^3LO) splitting functions P_{ik}^(3)(x) for the evolution of parton distributions, and facilitate approximate expressions which include the quartic-Casimir contributions to the (light-like) gluon cusp anomalous dimension. These quantities turn out to be closely related, by a generalization of the lower-order `Casimir scaling', to the corresponding qua…
We study the Polyakov loop and the correlator of two Polyakov loops at finite temperature in the weak-coupling regime. We calculate the Polyakov loop at order g^4. The calculation of the correlator of two Polyakov loops is performed at distances shorter than the inverse of the temperature and for electric screening masses larger than the Coulomb potential. In this regime, it is accurate up to order g^6. We also evaluate the Polyakov-loop correlator in an effective field theory framework that takes advantage of the hierarchy of energy scales in the problem and makes explicit the bound-state dyn…
We present compact analytic formulae for the one-loop amplitudes for Higgs + 4 parton scattering, $0 \to g g g g h$, $0 \to \bar{q} q gg h$ and $0\to \bar{q} q \bar{q}^\prime q^\prime h$, mediated by a loop of massive coloured quarks. We exploit the correspondence with a theory in which a massive coloured scalar circulates in the loop to avoid a proliferation in the number of terms in the result. In addition, we use momentum twistors and high precision numerical evaluations to simplify the expressions. The analytic results in this paper, in terms of spinor products, allow construction of an ef…
We present the contributions to electric dipole moments (EDMs) induced by the Yukawa couplings of an additional electroweak doublet of colour-octet scalars. The full set of one-loop diagrams and the enhanced higher-order effects from Barr-Zee diagrams are computed for the quark (chromo-)EDM, along with the two-loop contributions to the Weinberg operator. Using the stringent experimental upper limits on the neutron EDM, constraints on the parameter space of this model are derived.
The soft function in non-abelian gauge theories exponentiate, and their logarithms can be organised in terms of the collections of Feynman diagrams called Cwebs. The colour factors that appear in the logarithm are controlled by the web mixing matrices. Direct construction of the diagonal blocks of Cwebs using the new concepts of Normal ordering, basis Cweb and Fused-Web was recently carried out in~\cite{Agarwal:2022wyk}. In this article we establish correspondence between the boomerang webs introduced in ~\cite{Gardi:2021gzz} and non-boomerang Cwebs. We use this correspondence together with Un…
We investigate the next-to-leading-colour (NLC) contributions to the colour matrix in the fundamental and the colour-flow decompositions for tree-level processes with all gluons, one quark pair and two quark pairs. By analytical examination of the colour factors, we find the non-zero elements in the colour matrix at NLC. At this colour order, together with the symmetry of the phase-space, it is reduced from factorial to polynomial the scaling of the contributing dual amplitudes as the number of partons participating in the scattering process is increased. This opens a path to an accurate tree-…
I formulate in a colour-friendly way the FKS method for the computation of QCD cross sections at the next-to-leading order accuracy. This is achieved through the definition of subtraction terms for squared matrix elements, constructed with single colour-dressed or pairs of colour-ordered amplitudes. The latter approach relies on the use of colour flows, is exact to all orders in $N$, and is thus particularly suited to being organized as a systematic expansion in 1/N.
We consider the singular behaviour of one-loop QCD matrix elements when several external partons become simultaneously parallel. We present a new factorization formula that describes the singular collinear behaviour directly in colour space. The collinear singularities are embodied in process-independent splitting matrices that depend on the momenta, flavours, spins and colours of the collinear partons. We give the general structure of the infrared and ultraviolet divergences of the one-loop splitting matrices. We also present explicit one-loop results for the triple collinear splitting, $q \t…
Any loop QCD amplitude at full colour is constructed from kinematic and gauge-group building blocks. In a unitarity-based on-shell framework, both objects can be reconstructed from their respective counterparts in tree-level amplitudes. This procedure is at its most powerful when aligned with flexible colour decompositions of tree-level QCD amplitudes. In this note we derive such decompositions for amplitudes with an arbitrary number of quarks and gluons from the same principle that is used to bootstrap kinematics - unitarity factorisation. In the process we formulate new multi-quark bases and…
We determine the complete set of independent dimension six and eight Lorentz scalar operators in Yang-Mills theory for an arbitrary colour group. The anomalous dimension mixing matrix is determined at one loop.
We calculate the one-loop contributions to the effective potential for the minimal supersymmetric model when scalar fields other than the Higgses have vacuum expectation values. The importance of these contributions for studies of charge and colour breaking bounds is discussed.
The high-energy scattering amplitude of two colour-singlet quark-antiquark pairs is governed by the correlation function of two Wilson loops, which follow the classical straight lines for quark (antiquark) trajectories. This quantity is expected to be free of IR divergences, differently from what happens for the parton-parton scattering amplitude, described, in the high-energy limit, by the expectation value of two Wilson lines. We shall explicitly test this IR finitness by a direct non-perturbative computation of the loop-loop scattering amplitudes in the (pedagogic, but surely physically int…
The static potentials for systems of a heavy quark and anti-quark, two gluinos and a quark and a gluino are computed for all possible colour configurations of a $SU(N_c)$ gauge group.
Jet pull is an observable designed to probe colour flow between jets. Thus far, a particular projection of the pull vector, the pull angle, has been employed to distinguish colour flow between jets produced by a colour singlet or an octet decay. This is of particular importance in order to separate the decay of a Higgs boson to a pair of bottom quarks from the QCD background. However, the pull angle is not infra-red and collinear (IRC) safe. In this paper we introduce IRC safe projections of the pull vector that exhibit good sensitivity to colour flow, while maintaining calculability. We calcu…
We present the Fortran95 program Recola for the perturbative computation of next-to-leading-order transition amplitudes in the Standard Model of particle physics. The code provides numerical results in the 't Hooft-Feynman gauge. It uses the complex-mass scheme and allows for a consistent isolation of resonant contributions. Dimensional regularization is employed for ultraviolet and infrared singularities, with the alternative possibility of treating collinear and soft singularities in mass regularization. Recola supports various renormalization schemes for the electromagnetic and a dynamical …
New coloured scalars in the Manohar-Wise model give sizeable contributions to the Electric Dipole Moment (EDM) of the neutron through one-loop and two-loop diagrams, computed in Reference [1]. Contributions from neutral scalars cancel out for degenerate values of the masses, whose difference is related to the oblique parameters $S$ and $T$. The Manohar-Wise model is able to explain the recent value of the $W$-boson mass published by the CDF collaboration. We show, however, that the case of total degeneracy of masses is strongly disfavoured in this model, whose parameter space is substantially …
Guided by the colour-diagonal structure of collinear singularities, we identify a set of kinematic basis functions that are well suited to the simplification of soft gluon emission amplitudes. In particular, these basis functions, which emerge naturally in the colour flow basis, isolate the subleading colour contributions and improve the statistical convergence of the CVolver amplitude-evolution code. They also allow us to extend current angular-ordered parton showers beyond the azimuthally-averaged, two-jet limit.
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.
We calculate the colour averaged and colour singlet free energies of static quark anti-quark sources placed in a thermal gluonic heat bath. We discuss the renormalization of these free energies using the short distance properties of the zero temperature heavy quark potential. This leads to the definition of the renormalized Polyakov loop as an order parameter for the deconfinement phase transition of the SU(3) gauge theory which is well behaved in the continuum limit.
