We consider one-loop five-point QCD amplitudes in next-to-multi-Regge kinematics, and evaluate the one-loop impact factor for the emission of two gluons. This is the last ingredient which is necessary to evaluate the gluon-jet impact factor at NNLO accuracy in $\as$. It is also the first instance in which loop-level QCD amplitudes are evaluated in next-to-multi-Regge kinematics, which requires to apply a different reggeisation ansatz to each colour-ordered amplitude.
We present a much simpler analytic expression for the UV un-renormalized one-loop soft-gluon current which controls the infra-red behavior of one-loop QCD amplitudes with massive fermions when an external gluon becomes soft. The result is given entirely in terms of standard polylogarithms, is of uniform transcendentality and is well-suited for computer implementation.
There is growing evidence that on-shell gluon scattering amplitudes in planar N=4 SYM theory are equivalent to Wilson loops evaluated over contours consisting of straight, light-like segments defined by the momenta of the external gluons. This equivalence was first suggested at strong coupling using the AdS/CFT correspondence and has since been verified at weak coupling to one loop in perturbation theory. Here we perform an explicit two-loop calculation of the Wilson loop dual to the four-gluon scattering amplitude and demonstrate that the relation holds beyond one loop. We also propose an ano…
We briefly review the string technology needed to calculate Yang-Mills amplitudes at two loops, and we apply it to the evaluation of two-loop vacuum diagrams.
We analytically compute all four-loop QCD corrections to the photon-quark and Higgs-gluon form factors involving a closed massless fermion loop. Our calculation of non-planar vertex integrals confirms a previous conjecture for the analytical form of the non-fermionic contributions to the collinear anomalous dimensions of quarks and gluons.
We calculate the two-mass contribution to the 3-loop vacuum polarization of the gluon in Quantum Chromodynamics at virtuality $p^2 = 0$ for general masses and also present the analogous result for the photon in Quantum Electrodynamics.
The generating functional for hard thermal loops in QCD is rewritten in terms of a gauged WZNW action by introducing an auxiliary field. This shows in a simple way that the contribution of hard thermal loops to the energy of the quark-gluon plasma is positive.
We compute the next-to-MHV one-loop n-gluon amplitudes in N=4 super-Yang-Mills theory. These amplitudes contain three negative-helicity gluons and an arbitrary number of positive-helicity gluons, and are the first infinite series of amplitudes beyond the simplest, MHV, amplitudes. We also discuss some aspects of their twistor-space structure.
We present the two-loop pure gauge contribution to the gluon-gluon scattering amplitude with maximal helicity violation. Our construction of the amplitude does not rely directly on Feynman diagrams, but instead uses its analytic properties in 4 - 2 epsilon dimensions. We evaluate the loop integrals appearing in the amplitude through O(epsilon^0) in terms of polylogarithms.
We compute the non-MHV one-loop seven-gluon amplitudes in N=4 super-Yang-Mills theory, which contain three negative-helicity gluons and four positive-helicity gluons. There are four independent color-ordered amplitudes, (- - - + + + +), (- - + - + + +), (- - + + -+ +) and (- + - + - + +). The MHV amplitudes containing two negative-helicity and five positive-helicity gluons were computed previously, so all independent one-loop seven-gluon helicity amplitudes are now known for this theory. We present partial information about an infinite sequence of next-to-MHV one-loop helicity amplitudes, with…
We calculate the four-loop massless QCD corrections with three closed quark lines to quark and gluon form factors. We apply a novel integration by parts algorithm based on modular arithmetic and compute all relevant master integrals for arbitrary values of the space-time dimension. This is the first calculation of a gluon form factor at this perturbative order in QCD.
The aim of this work is to study the holographic dual to the gauge theory with a nonzero gluon condensate. We check for consistency the holographic way of describing the condensate and calculate the expectation value of a small Wilson loop in the presence of the gluon condensate, thus obtaining the relevant coefficient in the operator product expansion of the small loop in different holographic models. We also study the effect of the condensate on the Gross-Ooguri phase transition in the correlator of two circular Wilson loops in parallel and concentric configurations. In the numerical study o…
The two-loop three-gluon vertex is calculated in an arbitrary covariant gauge, in the limit when one of the external momenta vanishes. The differential Ward-Slavnov-Taylor (WST) identity related to this limit is discussed, and the relevant results for the ghost-gluon vertex and two-point functions are obtained. Together with the differential WST identity, they provide another independent way for calculating the three-gluon vertex. The renormalization of the results obtained is also presented.
We calculate the HQET quark-gluon vertex at one loop, for arbitrary external momenta, in an arbitrary covariant gauge and space-time dimension. Relevant results and algorithms for the three-point HQET integrals are presented. We also show how one can obtain the HQET limit directly from QCD results for the quark-gluon vertex.
We compute the photon-quark and Higgs-gluon form factors to four-loop order within massless perturbative Quantum Chromodynamics. Our results constitute ready-to-use building blocks for N${}^4$LO cross sections for Drell-Yan processes and gluon-fusion Higgs boson production at the LHC. We present complete analytic expressions for both form factors and show several of the most complicated master integrals.
Langevin equation describing soft modes in the quark-gluon plasma is reformulated on the loop space. The Cauchy problem for the resulting loop equation is solved for the case when the nonvanishing components of the gauge potential correspond to the Cartan generators of the SU(N)-group and are proportional to a constant unit vector in the Cartan subalgebra. The regularized form of the loop equation with an arbitrary gauge potential is found, and perturbation theory in powers of the 't Hooft coupling is discussed.
We present a conjecture for the $n$-gluon one-loop amplitudes with maximal helicity violation. The conjecture emerges from the powerful requirement that the amplitudes have the correct behavior in the collinear limits of external momenta. One implication is that the corresponding amplitudes where three or more gluon legs are replaced by photons vanish for $n>4$.
We present the one-loop QCD amplitudes for two external massless quarks and three external gluons ($\bar{q}qggg$). This completes the set of one-loop amplitudes needed for the next-to-leading-order corrections to three-jet production at hadron colliders. We also discuss how to use group theory and supersymmetry to minimize the amount of calculation required for the more general case of one-loop two-quark $n$-gluon amplitudes. We use collinear limits to provide a stringent check on the amplitudes.
Using cutting techniques we obtain the two-loop N=4 super-Yang-Mills helicity amplitudes for four-gluon scattering in terms of scalar integral functions. The N=4 amplitudes are considerably simpler than corresponding QCD amplitudes and therefore provide a testing ground for exploring two-loop amplitudes. The amplitudes are constructed directly in terms of gauge invariant quantities and therefore remain relatively compact throughout the calculation. We also present a conjecture for the leading color four-gluon amplitudes to all orders in the perturbative expansion.
Planar gluon amplitudes in N=4 SYM are remarkably similar to expectation values of Wilson loops made of light-like segments. We argue that the latter can be determined by making use of the conformal symmetry of the gauge theory, broken by cusp anomalies. We derive the corresponding anomalous conformal Ward identities valid to all loops and show that they uniquely fix the form of the finite part of a Wilson loop with n cusps (up to an additive constant) for n=4 and n=5 and reduce the freedom in it to a function of conformal invariants for n>=6. We also present an explicit two-loop calculation f…
We present the derivation of the two-loop gluon Regge trajectory using Lipatov's high energy effective action and a direct evaluation of Feynman diagrams. Using a gauge invariant regularization of high energy divergences by deforming the light-cone vectors of the effective action, we determine the two-loop self-energy of the reggeized gluon, after computing the master integrals involved using the Mellin-Barnes representations technique. The self-energy is further matched to QCD through a recently proposed subtraction prescription. The Regge trajectory of the gluon is then defined through renor…
Gluon, ghost, and quark propagators are computed in the Schwinger-Dyson formalism. The full set of one-loop and two-loop contributions to the gap equations are evaluated for the first time. A new and efficient method for dealing with quadratic divergences is presented and a consistent renormalization method is proposed. We find that the two-loop contribution to the propagators is subleading, good fits to lattice data are possible, and reliable vertex models are important to obtaining desired ultraviolet, infrared, and gauge-symmetric behavior.
We present the first numerical computation of two-loop amplitudes based on the unitarity method. As a proof of principle, we compute the four-gluon process. We discuss the new method, analyze its numerical properties and apply it to reconstruct the analytic form of the amplitudes. The numerical method is universal, and can be automated to provide multi-scale two-loop computations for phenomenologically relevant signatures at hadron colliders.
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 compute the planar part of the two-loop five gluon amplitude with all helicities positive. To perform the calculation we develop a D-dimensional generalized unitarity procedure allowing us to reconstruct the amplitude by cutting into products of six-dimensional trees. We find a compact form for the integrand which only requires topologies with six or more propagators. We perform cross checks of the universal infra-red structure using numerical integration techniques.
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…
The generating functional for hard thermal loops in QCD is important in setting up a resummed perturbation theory. I review how this functional is related to the eikonal for a Chern-Simons theory, and using an auxiliary field, to the gauged WZNW-action. The induced current due to hard thermal loops, properly incorporating damping effects, is also briefly discussed. (Invited talk at the Third Worshop on Thermal Field Theories, Banff, Canada, August, 1993.)
A scalar cubic action that classically reproduces the self-dual Yang--Mills equations is shown to generate one-loop QCD amplitudes for external gluon all with the same helicity. This result is related to the symmetries of the self-dual Yang--Mills equations.
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.
Virtual two-loop corrections to scattering amplitudes are a key ingredient to precision physics at collider experiments. We compute the full set of planar master integrals relevant to five-point functions in massless QCD, and use these to derive an analytical expression for the two-loop five-gluon all-plus-helicity amplitude. After subtracting terms that are related to the universal infrared and ultraviolet pole structure, we obtain a remarkably simple and compact finite remainder function, consisting only of dilogarithms.
We present the analytic form of the two-loop five-gluon scattering amplitudes in QCD for a complete set of independent helicity configurations of external gluons. These include the first analytic results for five-point two-loop amplitudes relevant for the computation of next-to-next-to-leading-order QCD corrections at hadron colliders. The results were obtained by reconstructing analytic expressions from numerical evaluations. The complexity of the computation is reduced by exploiting physical and analytical properties of the amplitudes, employing a minimal basis of so-called pentagon function…
The rational parts of 6-gluon one-loop amplitudes with scalars circulating in the loop are computed by using the newly developed method for computing the rational parts directly from Feynman integrals. We present the analytic results for the two MHV helicity configurations: $(1^-2^+3^+4^-5^+6^+)$ and $(1^-2^+3^-4^+5^+6^+)$, and the two NMHV helicity configurations: $(1^-2^-3^+4^-5^+6^+)$ and $(1^-2^+3^-4^+5^-6^+)$. Combined with the previously computed results for the cut-constructible part, our results are the last missing pieces for the complete partial helicity amplitudes of the 6-gluon one…
We calculate the four-loop massless QCD corrections with two closed quark lines to quark and gluon form factors. The results for the gluon form factor and the singlet part of the quark form factor are given for the first time. From our analytic expressions for the form factors, we determine the corresponding cusp anomalous dimensions. The relevant Feynman integrals are obtained with novel integral reduction techniques and direct integration methods.
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 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.
We compute the three-loop four-gluon scattering amplitude in maximally supersymmetric Yang-Mills theory, including its full color dependence. Our result is the first complete computation of a non-planar four-particle scattering amplitude to three loops in four-dimensional gauge theory and consequently provides highly non-trivial data for the study of non-planar scattering amplitudes. We present the amplitude as a Laurent expansion in the dimensional regulator to finite order, with coefficients composed of harmonic poly-logarithms of uniform transcendental weight, and simple rational prefactors…
We compute the three-loop helicity amplitudes for $q\bar{q} \to gg$ and its crossed partonic channels, in massless QCD. Our analytical results provide a non-trivial check of the color quadrupole contribution to the infrared poles for external states in different color representations. At high energies, the $qg \to qg$ amplitude shows the predicted factorized form from Regge theory and confirms previous results for the gluon Regge trajectory extracted from $qq' \to qq'$ and $gg \to gg$ scattering.
We present a calculation of the helicity amplitudes for the process $gg\toγγ$ in three-loop massless QCD. We employ a recently proposed method to calculate scattering amplitudes in the 't Hooft-Veltman scheme that reduces the amount of spurious non-physical information needed at intermediate stages of the computation. Our analytic results for the three-loop helicity amplitudes are remarkably compact, and can be efficiently evaluated numerically. This calculation provides the last missing building block for the computation of NNLO QCD corrections to diphoton production in gluon fusion.
Exploiting the gauge-invariant properties of the PT-BFM truncation scheme for the gluon Schwinger-Dyson equation, we estimate the individual non-perturbative contribution of the "one-loop dressed" ghost loop to the gluon propagator. Using the available quenched lattice data for the gluon and the ghost propagators in d=4 and d=3, we determine how the overall shape of the gluon propagator is affected by the removal of the ghost loop, and what are the consequences on the phenomenon of gluon mass generation.
We compute the two loop corrections to the gluon propagator for low momentum in a gluon mass model. This model has recently been proposed as an alternative to the Gribov construction in the way it handles Gribov copies in the gauge fixing. The corrections provide improvements for estimating the point where the gluon propagator freezes in relation to lattice data.
The rational parts of 5-gluon one-loop amplitudes are computed by using the newly developed method for computing the rational parts directly from Feynman integrals. We found complete agreement with the previously well-known results of Bern, Dixon and Kosower obtained by using the string theory method. Intermediate results for some combinations of Feynman diagrams are presented in order to show the efficiency of the method and the local cancellation between different contributions.
We obtain an exact result for the soft gluon production and its p_T distribution due to a gluon loop in a constant chromo-electric background field E^a with arbitrary color. Unlike Schwinger's result for e^+e^- pair production in QED which depends only on one gauge invariant quantity, the Electric field E, we find that the p_T distribution of the gluons depend on two gauge invariant quantities, E^aE^a and [d_{abc}E^aE^bE^c]^2.
In this paper analytical results are presented for higher order corrections to coefficient functions of the operator product expansion (OPE) for the correlator of two pseudoscalar gluonium operators \tilde{O}_1=G^{μν}\tilde{G}_{μν}. The Wilson coefficient in front of the scalar gluon condensate operator O_1=-1/4 G^{μν}G_{μν} is given at three-loop accuracy. The leading coefficient C_0 in front of the unity operator O_0=1 has been calculated up to three-loop order some time ago but has been checked independently in this work. It is interesting to see that the coefficient C_1 in the pseudoscalar…
We compute the three-loop leading-colour corrections to the Yang-Mills Regge trajectory and gluon impact factor. Conjecturing that, in analogy with $\mathcal{N}=4$ super Yang-Mills (SYM), in a suitable scheme $N_c$-subleading terms are absent from the three-loop Regge trajectory, we understand our result as the first computation of the pure gauge, or $n_f = 0$, part of the QCD three-loop Regge trajectory. The results are presented both for the bare and renormalised amplitudes and are consistent with predictions from infrared factorisation along with reproducing known results in planar $\mathca…
We study the influence of light quark loops on the Wigner phase by solving coupled Dyson-Schwinger equations for quark propagator and gluon propagator. We take the gluon propagator in the Nambu phase from $N_f$ = 2 unquenched lattice QCD and choose various phenomenological models for the quark-gluon vertex. The gluon propagator in Winger phase is assumed to be different from that in the Nambu phase only due to the vacuum polarization of the quark loop. We obtain the Wigner solution of the coupled equations, compared with that from solving only the equation of the quark propagator. We discussed…
We present further evidence for a dual conformal symmetry in the four-gluon planar scattering amplitude in N=4 SYM. We show that all the momentum integrals appearing in the perturbative on-shell calculations up to five loops are dual to true conformal integrals, well defined off shell. Assuming that the complete off-shell amplitude has this dual conformal symmetry and using the basic properties of factorization of infrared divergences, we derive the special form of the finite remainder previously found at weak coupling and recently reproduced at strong coupling by AdS/CFT. We show that the sam…
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.
We compute the leading colour contributions to five-gluon scattering at two loops in massless QCD. The integrands of all independent helicity amplitudes are evaluated using d-dimensional generalised unitarity cuts and finite field reconstruction techniques. Numerical evaluation of the integral basis is performed with sector decomposition methods to obtain the first benchmark results for all helicity configurations of a 2 to 3 scattering process in QCD.
At small momentum transfer, the quark-gluon scattering cross section dσ/dt has a power-law divergence in the backward scattering region where the outgoing quark is nearly collinear to the incoming gluon. In this Regge limit |t|<< s, the leading behavior of the 2 2 amplitude can be described by the exchange of Glauber quarks. In Soft-Collinear Effective Theory (SCET) at leading power, Glauber quark exchange is given by five non-local Glauber quark operators, of which only one is generated at tree-level. We show that at leading power the QCD amplitude for quark-gluon backscattering at one-loop c…
In this article we study the unquenched three-gluon and ghost-gluon vertex in all momentum range going from the ultraviolet to the infrared regime using the Curci-Ferrari model at one-loop in Landau gauge as an extension of the results presented in \cite{Pelaez:2013cpa}. Results are compared with recent lattice data for $SU(3)$ in the unquenched case. This calculation is a pure prediction of the model because it does not require fixing any parameter once two-point functions are fitted. An analysis of the influence of dynamical quarks in the position of the zero crossing of the three-gluon vert…
We test gauge/string duality by evaluating expectation values of small Wilson loops in pure Yang-Mills theories. On the gauge-theory side, there exists rich phenomenology. The dual formulation provides with a universal language to evaluate the gluon condensate and quadratic correction in terms of the metric in the fifth coordinate. Quantitatively, the estimated value of the gluon condensate is approximately 0.010 GeV^4.
We apply the solution to the recursion relation for the double-off-shell quark current to the problem of computing one loop amplitudes with an arbitrary number of gluons. We are able to compute amplitudes for photon-gluon scattering, electron-positron annihilation to gluons, and gluon-gluon scattering via a quark loop in the case of like-helicity gluons. In addition, we present the result for the one-loop gluon-gluon scattering amplitude when one of the gluons has opposite helicity from the others.
The two-loop three-gluon vertex is calculated in an arbitrary covariant gauge, in the limit when two of the gluons are on the mass shell. The corresponding two-loop results for the ghost-gluon vertex are also obtained. It is shown that the results are consistent with the Ward-Slavnov-Taylor identities.
It is shown that in the Glauber scattering of a fast quark in the external field loop corrections to the gluon emission amplitude due to virtual softer gluon after renormalization coincide with a correction due to reggeization of the exchanged gluon in the BFKL picture.
We present the one-loop helicity amplitudes with five external gluons. The computation employs string-based methods, new techniques for performing tensor integrals, and improvements in the spinor helicity method.
We compute the pole mass of the gluon in QCD from the local composite operator formalism at two loops in the MSbar renormalization scheme. For Yang-Mills theory an estimate of the mass at two loops is 2.13 Lambda_MSbar.
We present results for the six-gluon scattering amplitude at one-loop. Since our method is semi-numerical, it yields the result for arbitrary momenta and helicities of the external gluons. We evaluate the colour-ordered sub-amplitudes with gluons, fermions and scalars running in the internal loop. This is more than sufficient to give a complete description of six-gluon scattering at one-loop in QCD. Combination of these results into amplitudes with ${\cal{N}}=4$ and ${\cal{N}}=1$ multiplets of supersymmetric Yang-Mills or with a complex scalar in the internal loops allows comparison with analy…
