One-Loop Six Gluon Amplitude

A Brief Comment on Multi-Gluon Amplitudes and Double Parton Interactions

A simplest case, where a contribution to a color ordered multi-gluon amplitude, describing a three body partonic interaction, can split into two weakly correlated two-body gluon scattering amplitudes and may thus contribute to a Double Parton Interaction, is briefly discussed. We find that the color ordered amplitude is not enhanced in the typical configuration generated by a DPI, where the transverse momenta of final state gluons are compensated pairwise, while a dominant contribution to the multi-gluon amplitude is due to terms proportional to the fusion amplitude of two initial state gluons…

Multi-gluon one-loop amplitudes numerically

A c++ implementation of the D_s-dimensional unitarity cut algorithm for the numerical evaluation of the virtual contribution to NLO QCD amplitudes is presented. The current version includes an arbitrary number of external gluons with gluonic propagators in the loop. The building blocks are tree level color-ordered amplitudes with gluons and with gluons and two scalars in five dimensions. Numerical stability issues are addressed and agreement has been reached with the results published in the literature.

Loops in the gluon emission amplitude: reggeization from the Glauber scattering

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.

LHC multijet events as a probe for anomalous dimension-six gluon interactions

Higher-dimensional multi-gluon interactions affect essentially all effective Lagrangian analyses at the LHC. We show that, contrary to common lore, such operators are best constrained in multi-jet production. Our limit on the corresponding new physics scale in the multi-TeV range exceeds the typical reach of global dimension-6 Higgs and top analyses. This implies that the pure Yang-Mills operator can safely be neglected in almost all specific higher-dimensional analyses at Run II.

Multi gluon collinear limits from MHV amplitudes

We consider the multi-collinear limit of multi-gluon QCD amplitudes at tree level. We use the MHV rules for constructing colour ordered tree amplitudes and the general collinear factorisation formula to derive timelike splitting functions that are valid for specific numbers of negative helicity gluons and an arbitrary number of positive helicity gluons (or vice versa).

One-Loop Corrections to Five-Gluon Amplitudes

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.

The phase between the three gluon and one photon amplitudes in quarkonium decays

The phase between three-gluon and one-photon amplitudes in psi(2S) and psi(3770) decays is analyzed.

Numerical evaluation of multi-gluon amplitudes for High Energy Factorization

We present a program to evaluate tree-level multi-gluon amplitudes with up to two of them off-shell. Furthermore, it evaluates squared amplitudes summed over colors and helicities for up to six external gluons. It employs both analytic expressions, obtained via BCFW recursion, and numerical BCFW recursion. It has been validated numerically with the help of an independent program employing numerical Dyson-Schwinger recursion.

The Five Gluon Amplitude and One-Loop Integrals

We review the conventional field theory description of the string motivated technique. This technique is applied to the one-loop five-gluon amplitude. To evaluate the amplitude a general method for computing dimensionally regulated one-loop integrals is outlined including results for one-loop integrals required for the pentagon diagram and beyond. Finally, two five-gluon helicity amplitudes are given.

Two-Loop Helicity Amplitudes for Quark-Gluon Scattering in QCD and Gluino-Gluon Scattering in Supersymmetric Yang-Mills Theory

We present the two-loop QCD helicity amplitudes for quark-gluon scattering, and for quark-antiquark annihilation into two gluons. These amplitudes are relevant for next-to-next-to-leading order corrections to (polarized) jet production at hadron colliders. We give the results in the `t Hooft-Veltman and four-dimensional helicity (FDH) variants of dimensional regularization. The transition rules for converting the amplitudes between the different variants are much more intricate than for the previously discussed case of gluon-gluon scattering. Summing our two-loop expressions over helicities an…

On Higgs boson plus gluon amplitudes at one loop

We present analytic results for one-loop Higgs boson + $n$-gluon amplitudes for $n\le 5$ in the full theory including all dependence on the (top) quark mass. In this paper we consider only the case where the gluons all have the same helicity. The amplitudes are expressed in simple formula and display similar structure. Their limiting behaviour in small Higgs momentum and large top mass is studied.

All One-loop Maximally Helicity Violating Gluonic Amplitudes in QCD

We use on-shell recursion relations to compute analytically the one-loop corrections to maximally-helicity-violating n-gluon amplitudes in QCD. The cut-containing parts have been computed previously; our work supplies the remaining rational parts for these amplitudes, which contain two gluons of negative helicity and the rest positive, in an arbitrary color ordering. We also present formulae specific to the six-gluon cases, with helicities (- + - + + +) and (- + + - + +), as well as numerical results for six, seven, and eight gluons. Our construction of the n-gluon amplitudes illustrates the r…

Two-Loop Helicity Amplitudes for Gluon-Gluon Scattering in QCD and Supersymmetric Yang-Mills Theory

We present the two-loop helicity amplitudes for the scattering of two gluons into two gluons in QCD, which are relevant for next-to-next-to-leading order corrections to jet production at hadron colliders. We give the results in the `t Hooft-Veltman and four-dimensional helicity variants of dimensional regularization. Summing our expressions over helicities and colors, and converting to conventional dimensional regularization, gives results in complete agreement with those of Glover, Oleari and Tejeda-Yeomans. We also present the amplitudes for 2 to 2 scattering in pure N=1 supersymmetric Yang-…

Gluon Wavefunctions and Amplitudes on the Light-Front

We investigate the tree level multi-gluon components of the gluon light cone wavefunctions in the light cone gauge keeping the exact kinematics of the gluon emissions. We focus on the components with all helicities identical to the helicity of the incoming gluon. The recurrence relations for the gluon wavefunctions are derived. In the case when the virtuality of the incoming gluon is neglected the exact form of the multi-gluon wavefunction as well as the fragmentation function is obtained. Furthermore we analyze the 2 to N tree-level gluon scattering in the framework of light-front perturbatio…

Differences in binarity between gluon and graviton scattering amplitudes

Both 3- and 4-point scattering amplitudes for spin-1 massless particles (gluons) and spin-2 massless particles (gravitons) are reviewed through self-contained step-by-step derivation. Gluon and graviton interactions are computed from on-shell diagrams by starting from complex momenta and introducing spinor-helicity formalism. By Fourier transforming spinor variables in momentum space, binarity is revealed in pure Yang-Mills scattering amplitudes while absent to a certain extent for gravity indicating fundamental differences in the probability spaces spanned by the two interactions.

Massive Quark-Gluon Scattering Amplitudes at Tree Level

Results for four-, five-, and six-parton tree amplitudes for massive quark-antiquark scattering with gluons are calculated using the recursion relations of Britto, Cachazo, Feng, and Witten. The required diagrams are generated using shifts of the momenta of a pair of massless legs to complex values. Checks verifying the calculations are described, and a simple formula for the shifted spinors of an internal gluon is presented.

Recursion in multiplet bases for tree-level MHV gluon amplitudes

We investigate the construction of tree-level MHV gluon amplitudes in multiplet bases using BCFW recursion. The multiplet basis decomposition can either be obtained by decomposing results derived in (for example) the DDM basis or by formulating the recursion directly in the multiplet basis. We focus on the latter approach and show how to efficiently deal with the color structure appearing in the recursion. For illustration, we also explicitly calculate the four-, five- and six-gluon amplitudes.

Contribution of third generation quarks to two-loop helicity amplitudes for W boson pair production in gluon fusion

We compute the contribution of third generation quarks ($t,\ b$) to the two-loop amplitude for on-shell $W$ boson pair production in gluon fusion $gg \to WW$. We present plots for the amplitude across partonic phase space as well as reference values for two kinematic points. The master integrals are efficiently evaluated by numerically solving a system of ordinary differential equations.

One-loop helicity amplitudes for H -> gluons: the all-minus configuration

We use twistor inspired rules to compute the one-loop amplitude for a Higgs boson coupling to any number of negative helicity gluons in the large top mass limit.

One-loop radiative corrections to the helicity amplitudes of QCD processes involving four quarks and one gluon

We present analytic results for the one-loop corrections of the helicity amplitudes of the QCD five-parton subprocesses involving four quarks and one gluon obtained with a standard Feynman diagram calculation using dimensional reduction.

Parke-Taylor amplitudes in the multi-Regge kinematics

The Parke-Taylor multigluon amplitudes are examined in the multi-Regge kinematics, which assumes strong rapidity ordering of the produced gluons, and are used to compute the $n$-gluon production cross section and the gluon-gluon total cross section.

Top quark contribution to two-loop helicity amplitudes for $Z$ boson pair production in gluon fusion

We compute the top quark contribution to the two-loop amplitude for on-shell $Z$ boson pair production in gluon fusion, $gg \to ZZ$. Exact dependence on the top quark mass is retained. For each phase space point the integral reduction is performed numerically and the master integrals are evaluated using the auxiliary mass flow method, allowing fast computation of the amplitude with very high precision.

On the Numerical Evaluation of One-Loop Amplitudes: the Gluonic Case

We develop an algorithm of polynomial complexity for evaluating one-loop amplitudes with an arbitrary number of external particles. The algorithm is implemented in the Rocket program. Starting from particle vertices given by Feynman rules, tree amplitudes are constructed using recursive relations. The tree amplitudes are then used to build one-loop amplitudes using an integer dimension on-shell cut method. As a first application we considered only three and four gluon vertices calculating the pure gluonic one-loop amplitudes for arbitrary external helicity or polarization states. We compare ou…

Gluon cascades and amplitudes in light-front perturbation theory

We construct the gluon wave functions, fragmentation functions and scattering amplitudes within the light-front perturbation theory. Recursion relations on the light-front are constructed for the wave functions and fragmentation functions, which in the latter case are the light-front analogs of the Berends-Giele recursion relations. Using general relations between wave functions and scattering amplitudes it is demonstrated how to obtain the maximally-helicity violating amplitudes, and explicit verification of the results is based on simple examples.

Calculating gluon one-loop amplitudes numerically

This note reports on an independent implementation of calculating one-loop amplitudes semi-numerically using generalized unitarity techniques. The algorithm implemented in form of a C++ code closely follows the method by Ellis, Giele, Kunszt and Melnikov. For the case of gluons, the algorithm is briefly reviewed. Double-precision results are presented documenting the accuracy and efficiency of this computation.

A technique for loop calculations in non-Abelian gauge theories - with application to five gluon amplitude

A powerful tool for calculations in non-Abelian gauge theories is obtained by combining the background field gauge, the helicity basis and the color decomposition methods. It has reproduced the one-loop calculation of the five-gluon amplitudes in QCD, is applicable to electroweak processes and extendable to two-loop calculations.

One-loop amplitudes with an off-shell gluon

I report on recent advancements regarding calculations that require initial-state partons with non-vanishing transverse momentum, causing it to be space-like and thus off-shell.

On hard electroproduction of mesons: helicity flip amplitudes with tensor gluon contributions

We consider hard electroproduction of mesons at twist-3 level and focus our attention on helicity flip amplitudes corresponding to the production of transversely polarized mesons by transverse photons. We demonstrate that helicity-flip amplitude for the transversely polarized particles are closely related to the contributions of the matrix elements of the tensor gluon operator which can serve as pure gluonic probe of the hadrons. Using standard factorisation approach and Wanzura-Wilczek approximation for the twist-3 matrix elements we find that such amplitudes do not have divergencies from the…

Gluon poles and photon distribution amplitudes in Drell-Yan-like processes

We calculate the hadron tensor related to the photon-induced Drell-Yan process, with incoming nucleon being transversally polarised. We predict the new single spin asymmetry which probes gluon poles and which is expressed in terms of photon (both chiral-odd and chiral-even) distribution amplitudes and chiral-odd nucleon function stemming from the nucleon transverse polarization.

Multi-gluon one-loop amplitudes using tensor integrals

An efficient numerical algorithm to evaluate one-loop amplitudes using tensor integrals is presented. In particular, it is shown by explicit calculations that for ordered QCD amplitudes with a number of external legs up to 10, its performance is competitive with other methods.

A compact representation of the 2 photon 3 gluon amplitude

A compact representation of the loop amplitude gamma gamma ggg -> 0 is presented. The result has been obtained by using helicity methods and sorting with respect to an irreducible function basis. We show how to convert spinor representations into a field strength representation of the amplitude. The amplitude defines a background contribution for Higgs boson searches at the LHC in the channel H -> gamma gamma + jet which was earlier extracted indirectly from the one-loop representation of the 5-gluon amplitude.

$ZH$ production in gluon fusion: two-loop amplitudes with full top quark mass dependence

We present results for the two-loop helicity amplitudes entering the NLO QCD corrections to the production of a Higgs boson in association with a $Z$-boson in gluon fusion. The two-loop integrals, involving massive top quarks, are calculated numerically. Results for the interference of the finite part of the two-loop amplitudes with the Born amplitude are shown as a function of the two kinematic invariants on which the amplitudes depend.

The singular behavior of one-loop massive QCD amplitudes with one external soft gluon

We calculate the one-loop correction to the soft-gluon current with massive fermions. This current is process independent and controls the singular behavior of one-loop massive QCD amplitudes in the limit when one external gluon becomes soft. The result derived in this work is the last missing process-independent ingredient needed for numerical evaluation of observables with massive fermions at hadron colliders at the next-to-next-to-leading order.

LHC dijet angular distributions as a probe for the dimension-six triple gluon vertex

In the absence of a concrete discovery of new physics at the LHC, global analyses of the standard model effective field theory (SMEFT) are important to find and describe the impact of new physics beyond the energy reach of the LHC. Among the SMEFT operators that can be constrained via various measurements, the dimension six triple gluon operator involves neither the Higgs boson nor the top quark, yet its variation can have measurable effects on top and Higgs production. Without independent constraints on its impact, the sensitivity of measurements in the top and Higgs sectors to new physics is…

Soft gluon contributions to the B --> K eta' amplitude in a low energy bosonization model

Intriguing B --> K eta' decays provide a unique opportunity to study a joining of two-gluon configurations arising from the penguin b --> sG and b --> sGG transitions, with those inherent to the eta' particle. We employ the heavy-light chiral quark model, applied previously to a somewhat related B --> D eta' decay, as a calculational tool accounting for the nonperturbative soft gluon contributions to the amplitude at hand. Thereby we arrive at a novel contribution to the singlet penguin amplitude, which within our model accounts for ~ 10 % of the measured B --> K eta' amplitude.

Multi-gluon helicity amplitudes with one off-shell leg within high energy factorization

Basing on the Slavnov-Taylor identities, we derive a new prescription to obtain gauge invariant tree-level scattering amplitudes for the process g*g->Ng within high energy factorization. Using the helicity method, we check the formalism up to several final state gluons, and we present analytical formulas for the the helicity amplitudes for N=2. We also compare the method with Lipatov's effective action approach.

eta（eta'）-gamma跃迁形状因子与胶子-胶子分布振幅

The eta (eta') gamma transition form factor and the gluon-gluon distribution amplitude

该研究在领头扭度、NLO微扰QCD框架下结合eta-eta'混合与实验数据，提取夸克-反夸克及胶子-胶子分布振幅的Gegenbauer系数，并预测g*g*eta'顶角函数。

One-loop Higgs plus four gluon amplitudes: Full analytic results

We consider one-loop amplitudes of a Higgs boson coupled to gluons in the limit of a large top quark mass. We treat the Higgs as the real part of a complex field phi that couples to the self-dual field strengths and compute the one-loop corrections to the phi-NMHV amplitude, which contains one gluon of positive helicity whilst the remaining three have negative helicity. We use four-dimensional unitarity to construct the cut-containing contributions and a hybrid of Feynman diagram and recursive based techniques to determine the rational piece. Knowledge of the phi-NMHV contribution completes th…

Two-Loop Amplitudes for Gluon Fusion into Two Photons

We present the two-loop matrix elements for the scattering of two gluons into two photons in QCD. These matrix elements will enter into improved estimates of the QCD background to Higgs production at the LHC when the Higgs decays into two photons. The photon mode is especially important if M_H < 140 GeV. We also give the amplitudes for the crossed process, glue gamma to glue gamma.

Diagrammatic proof of the BCFW recursion relation for gluon amplitudes in QCD

We present a proof of the Britto-Cachazo-Feng-Witten tree-level recursion relation for gluon amplitudes in QCD, based on a direct equivalence between BCFW decompositions and Feynman diagrams. We demonstrate that this equivalence can be made explicit when working in a convenient gauge. We exhibit that gauge invariance and the particular structure of Yang-Mills vertices guarantees the validity of the BCFW construction.

Properties of the quark-antiquark-gluon Fock state in the η_c meson

In this paper the twist-3 distribution amplitude of the quark-antiquark-gluon Fock state in the η_c meson is studied. To calculate the moments of this distribution amplitude QCD sum rules is applied. Using the results of the calculation the model of this distribution amplitude is built. In addition NRQCD matrix elements which determine the properties of the quark-antiquark-gluon Fock state are determined. In particular, the probability amplitude to find the quark-antiquark pair in the color octet state, the mean gluon energy and the fraction of momentum carried by gluon, the relative velocity …

Tree-level splitting amplitudes for a gluon into four collinear partons

We compute in conventional dimensional regularisation the tree-level splitting amplitudes for a gluon parent which splits into four collinear partons. This is part of the universal infrared behaviour of the QCD scattering amplitudes at next- to-next-to-next-to-leading order (N$^3$LO) in the strong coupling constant. Combined with our earlier results for a quark parent, this completes the set of tree-level splitting amplitudes required at this order. We also study iterated collinear limits where a subset of the four collinear partons become themselves collinear.

Leading-twist two gluon distribution amplitude and exclusive processes involving eta and eta' mesons

I briefly review the formalism for treating the leading-twist two-gluon states appearing in processes which involve eta and eta' mesons. The constraints on the size of the lowest order Gegenbauer coefficients of the two-gluon distribution amplitude are obtained from the fit to the eta and eta' transition form factor data. The results are applied to chi -> eta eta (eta' eta') decays and deeply virtual electroproduction of eta and eta' mesons.

The Dimension Six Triple Gluon Operator in Higgs+Jet Observables

Recently a lot of progress has been made towards a full classification of new physics effects in Higgs observables by means of effective dimension six operators. Specifically, Higgs production in association with a high transverse momentum jet has been suggested as a way to discriminate between operators that modify the Higgs-top coupling and operators that induce an effective Higgs-gluon coupling---a distinction that is hard to achieve with signal strength measurements alone. With this article we would like to draw attention to another source of new physics in Higgs+jet observables: the tripl…

Unitarization of Gluon Exchange Amplitudes and Rapidity Gaps at the Tevatron

Rapidity gaps between two hard jets at the Tevatron have been interpreted as being due to the exchange of two gluons which are in an overall color-singlet state. We show that this simple picture involves unitarity violating amplitudes. Unitarizing the gluon exchange amplitude leads to qualitatively different predictions for the fraction of $t$-channel color singlet exchange events in forward $qq$, $qg$ or $gg$ scattering, which better fit Tevatron data.

Helicity Amplitudes for Charmonium Production in Hadron-Hadron and Photon-Hadron Collisions

We present the gluon-gluon and photon-gluon helicity amplitudes for color singlet and octet charmonium production in polarized and unpolarized hadron-hadron and photon-hadron collisions.

One-loop phi-MHV amplitudes using the unitarity bootstrap

We consider a Higgs boson coupled to gluons via the five-dimensional effective operator H Tr G_{munu}G^{munu} produced by considering the heavy top quark limit of the one-loop coupling of Higgs and gluons in the standard model. We treat H as the real part of a complex field phi that couples to the selfdual gluon field strengths and compute the one-loop corrections to amplitudes involving phi, two colour adjacent negative helicity gluons and an arbitrary number of positive helicity gluons - the so-called phi-MHV amplitudes. We use four-dimensional unitarity to construct the cut-containing contr…

Analytic Form of the Three-loop Four-gluon Scattering Amplitudes in Yang-Mills Theory

We present analytic forms of three-loop four-gluon planar amplitudes in pure Yang-Mills theory in this letter. Gauge invariant bases and a set of proper master integrals are chosen such that the amplitudes are explicitly invariant under cyclic permutations of external particles. The $D$-dimensional unitarity method and integration-by-parts reductions with cut conditions are performed to determine the coefficients of the master integrals. Helicity amplitudes in the conventional dimension regularization scheme are obtained by setting gauge invariant bases to helicity configurations. After renorm…

One-loop phi-MHV amplitudes using the unitarity bootstrap: the general helicity case

We consider a Higgs boson coupled to gluons via the five-dimensional effective operator H tr G_{μν}G^{μν}. We treat H as the real part of a complex field phi that couples to the selfdual gluon field strengths and compute the one-loop corrections to the phi-MHV amplitudes involving phi, two negative helicity gluons and an arbitrary number of positive helicity gluons. Our results generalise earlier work where the two negative helicity gluons were constrained to be colour adjacent. We use four-dimensional unitarity to construct the cut-containing contributions and the recently developed recursion…

多重态基下带离壳递归的全胶子振幅

All-gluon amplitudes with off-shell recursion in multiplet bases

针对高部分子数下色求和QCD振幅计算效率低的问题，提出基于正交多重态基、结合Wigner 6j系数与缓存优化的离壳递归框架，将n胶子色求和平方振幅计算复杂度降至𝒪(17ⁿ)，展现出高多重数过程的应用潜力。

Using HERA Data to Determine the Infrared Behaviour of the BFKL Amplitude

We determine the infrared behaviour of the BFKL forward amplitude for gluon-gluon scattering. Our approach, based on the discrete pomeron solution, leads to an excellent description of the new combined inclusive HERA data at low values of x (<0.01) and at the same time determines the unintegrated gluon density inside the proton, for squared transverse momenta of the gluon less than 100 GeV^2. The phases of this amplitude are sensitive to the non-perturbative gluonic dynamics and could be sensitive to the presence of Beyond-the-Standard-Model particles at very high energies.

胶子运动学与Sudakov形状因子对偶极振幅的影响

Effects of gluon kinematics and the Sudakov form factor on the dipole amplitude

本文研究精确胶子运动学对两类饱和模型参数的影响，发现偶极截面归一化存在差异，并将重新拟合的模型用于研究电子离子对撞机能标下Sudakov形状因子对双喷注产生过程的影响。

New Color Decompositions for Gauge Amplitudes at Tree and Loop Level

Recently, a color decomposition using structure constants was introduced for purely gluonic tree amplitudes, in a compact form involving only the linearly independent subamplitudes. We give two proofs that this decomposition holds for an arbitrary number of gluons. We also present and prove similar decompositions at one loop, both for pure gluon amplitudes and for amplitudes with an external quark-antiquark pair.

Two-loop helicity amplitudes for the production of two off-shell electroweak bosons in gluon fusion

We compute the part of the two-loop virtual amplitude for the process $gg \to V_1 V_2 \to (l_1 \bar l'_{1}) (l_2 \bar l'_2)$, where $V_{1,2}$ are arbitrary electroweak gauge bosons, that receives contributions from loops of massless quarks. Invariant masses of electroweak bosons are allowed to be different from each other. Our result provides an important ingredient for improving the description of gluon fusion contribution to the production of four-lepton final states at the LHC.