Number Of Gluons Research Articles

On gauge amplitudes first appearing at two loops

We study scattering amplitudes in massless non-abelian gauge theory where all outgoing gluons have positive helicity. It has been argued recently by Costello that for a particular fermion representation (8 fundamentals plus one antisymmetric-tensor representation in SU(N)) the one-loop amplitudes vanish identically. We show that this vanishing leads to previously-observed identities among one-loop color-ordered partial amplitudes. We then turn to two loops, where Costello has computed the all-plus amplitudes for this theory, as rational functions of the kinematics for any number of gluons using the celestial chiral algebra (CCA) bootstrap. We show that in dimensional regularization, these two-loop amplitudes are not rational, and they are not even finite as ϵ → 0. However, the finite remainder for four gluons agrees with the formula by Costello. In addition, we provide a mass regulator for the infrared-divergent loop integrals; with this regulator, the CCA bootstrap formula is recovered exactly. Finally, we use the CCA bootstrap to compute the double-trace terms in the theory at two loops for an arbitrary number of gluons.

Kinematic Hopf algebra for amplitudes from higher-derivative operators

Recently it has been shown that Bern-Carrasco-Johansson (BCJ) numerators of colour-kinematic duality for tree-level scattering amplitudes in Yang-Mills theory (coupled with scalars) can be determined using a quasi-shuffle Hopf algebra. In this paper we consider the same theory, but with higher-derivative corrections of the forms α′F3 and α′2F4, where F is the field strength. In the heavy mass limit of the scalars, we show that the BCJ numerators of these higher-derivative theories are governed by the same Hopf algebra. In particular, the kinematic algebraic structure is unaltered and the derivative corrections only arise when mapping the abstract algebraic generators to physical BCJ numerators. The underlying kinematic Hopf algebra enables us to obtain a compact expression for the BCJ numerators of any number of gluons and two heavy scalars for amplitudes with higher-derivative operators. The pure gluon BCJ numerators can also be obtained from our results by a simple factorisation limit where the massive particles decouple.

Investigating the partonic entanglement entropy at low Bjorken‐x within Gluon saturation approach

AbstractIn this work, the entanglement entropy is examined within the context of deep inelastic scattering in collisions. The calculation is based on a formalism where the partonic state at small‐ is maximally entangled, consisting of a large number of micro‐states occurring with equal probabilities. Analytical expressions for the number of gluons, , are considered, derived from gluon saturation models for dipole‐target amplitudes within the framework of the Quantum Chromodynamics (QCD) color dipole picture. A comparison of the entanglement entropy with thermodynamic entropy measured in and collisions at high energies is done.

Research on electron and positron spectrum in the high-energy region based on the gluon condensation model

Electron (positron), proton and nuclei can be accelerated to very high energy by local supernova remnants (SNR). The famous excesses of electron and proton (nuclei) potentially come from such kind of local sources. Recently, the DAMPE experiment measured the electron spectrum (including both electrons and positrons) of cosmic rays with high-accuracy. It provides an opportunity to further explore the excess of electrons. According to the gluon condensation (GC) theory, once GC occurs, huge number of gluons condense at a critical momentum, and the production spectra of electron and proton showing typical GC characteristics. There are exact correlations between the electron and proton spectrum from a same GC process. It is possible to interpret the power-law break of cosmic rays in view of GC phenomenon, and predict one from another based on the relations between electron and proton spectrum. In this work, we point out the potential existence of a second excess in the electron spectrum, the characteristics of this excess is derived from experimental data of proton. We hope that the future DAMPE experiments will confirm the existence of this second excess and support the result of GC model.

Kinematic Hopf algebra for amplitudes and form factors

We propose a kinematic algebra for the Bern-Carrasco-Johansson (BCJ) numerators of tree-level amplitudes and form factors in Yang-Mills theory coupled with bi-adjoint scalars. The algebraic generators of the algebra contain two parts: the first part is simply the flavour factor of the bi-adjoint scalars, and the second part that maps to non-trivial kinematic structures of the BCJ numerators obeys extended quasi-shuffle fusion products. The underlying kinematic algebra allows us to present closed forms for the BCJ numerators with any number of gluons and two or more scalars for both on-shell amplitudes and form factors that involve an off-shell operator. The BCJ numerators constructed in this way are manifestly gauge invariant and obey many novel relations that are inherited from the kinematic algebra.

Chiral Approach to Massive Higher Spins.

We propose a new, chiral description for massive higher-spin particles in four spacetime dimensions, which facilitates the introduction of consistent interactions. As proof of concept, we formulate three theories, in which higher-spin matter is coupled to electrodynamics, non-Abelian gauge theory, or gravity. The theories are chiral and have simple Lagrangians, resulting in Feynman rules analogous to those of massive scalars. Starting from these Feynman rules, we derive tree-level scattering amplitudes with two higher-spin matter particles and any number of positive-helicity photons, gluons, or gravitons. The amplitudes reproduce the arbitrary-multiplicity results that were obtained via on-shell recursion in a parity-conserving setting, and which chiral and nonchiral theories thus have in common. The presented theories are currently the only examples of consistent interacting field theories with massive higher-spin fields.

Recursion relations for scattering amplitudes with massive particles II: Massive vector bosons

Using the recently introduced recursion relations with covariant massive-massless shift, we study tree-level scattering amplitudes involving a pair of massive vector bosons and an arbitrary number of gluons in the massive spinor-helicity formalism. In particular, we derive compact expressions for cases in which i) all gluons are of the same helicity and ii) one gluon has flipped helicity and is colour adjacent to one of the massive particles. We provide numerous consistency checks of our results including the exact match of high energy limits with well known MHV and NMHV amplitudes in pure Yang-Mills theory. As a corollary, we obtain an alternative novel representation of the NMHV amplitude.

Dijet production at EIC and interplay of Sudakov and saturation effects in Weizsacker-Williams TMD gluon distribution

We study production of dijets at the EIC using the small-x Improved Transverse Momentum Dependent factorization (ITMD), which is a framework based on the Color Glass Condensate theory. Dijet production in DIS is the simplest process directly coupled to the Weizsäcker-Williams TMD gluon distribution, which is the only small-x gluon distribution possessing the gluon number density interpretation. We study various observables sensitive to the interplay of the Sudakov effects and the nonlinear effects, in particular the azimuthal correlations between the jet system and the scattered electron.

Kinematic Hopf Algebra for Bern-Carrasco-Johansson Numerators in Heavy-Mass Effective Field Theory and Yang-Mills Theory.

We present a closed formula for all Bern-Carrasco-Johansson (BCJ) numerators describing D-dimensional tree-level scattering amplitudes in a heavy-mass effective field theory with two massive particles and an arbitrary number of gluons. The corresponding gravitational amplitudes obtained via the double copy directly enter the computation of black-hole scattering and gravitational-wave emission. Our construction is based on finding a kinematic algebra for the numerators, which we relate to a quasishuffle Hopf algebra. The BCJ numerators thus obtained have a compact form and intriguing features: gauge invariance is manifest, locality is respected for massless exchange, and they contain poles corresponding to massive exchange. Counting the number of terms in a BCJ numerator for n-2 gluons gives the Fubini numbers F_{n-3}, reflecting the underlying quasishuffle Hopf algebra structure. Finally, by considering an appropriate factorization limit, the massive particles decouple, and we thus obtain a kinematic algebra and all tree-level BCJ numerators for D-dimensional pure Yang-Mills theory.

All-multiplicity amplitudes with four massive quarks and identical-helicity gluons

We explore the on-shell recursion for tree-level scattering amplitudes with massive spinning particles. Based on the factorization structure encoded in the same way by two different recursion relations, we conjecture an all-multiplicity formula for two gauged massive particles of arbitrary spin and any number of identical-helicity gluons. Specializing to quantum chromodynamics (QCD), we solve the on-shell recursion relations in the presence of two pairs of massive quarks and an arbitrary number of identical-helicity gluons. We find closed-form expressions for the two distinct families of color-ordered four-quark amplitudes, in which all gluons comprise a single color-adjacent set. We compare the efficiency of the numerical evaluation of the two resulting analytic formulae against a numerical implementation of the off-shell Berends-Giele recursion. We find the formulae for both amplitude families to be faster for large multiplicities, while the simpler of the two is actually faster for any number of external legs. Our analytic results are provided in a computer-readable format as two files in the supplementary material.

Probing gluon number density with electron-dijet correlations at EIC

We propose a novel way of studying the gluon number density (the so-called Weizsäcker–Williams gluon distribution) using the planned Electron Ion Collider. Namely, with the help of the azimuthal correlations between the total transverse momentum of the dijet system and the scattered electron, we examine an interplay between the effect of the soft gluon emissions (the Sudakov form factor) and the gluon saturation effects. The kinematic cuts are chosen such that the dijet system is produced in the forward direction in the laboratory frame, which provides an upper bound on the probed longitudinal fractions of the hadron momentum carried by scattered gluons. Further cuts enable us to use the factorization formalism that directly involves the unpolarized Weizsäcker–Williams gluon distribution. We find this observable to be very sensitive to the soft gluon emission and moderately sensitive to the gluon saturation.

All-plus helicity off-shell gauge invariant multigluon amplitudes at one loop

We calculate one loop scattering amplitudes for arbitrary number of positive helicity on-shell gluons and one off-shell gluon treated within the quasi-multi Regge kinematics. The result is fully gauge invariant and possesses the correct on-shell limit. Our method is based on embedding the off-shell process, together with contributions needed to retain gauge invariance, in a bigger fully on-shell process with auxiliary quark or gluon line.

Investigating entanglement entropy at small x in DIS off protons and nuclei

In this work we analyse the entanglement entropy in deep inelastic scattering off protons and nuclei. It is computed based on the formalism where the partonic state at small-x is maximally entangled with proton being constituted by large number of microstates occuring with equal probabilities. We consider analytical expressions for the number of gluons, N_{gluon}, obtained from gluon saturation models for the dipole-target amplitudes within the QCD color dipole picture. In particular, the nuclear entanglement entropy per nucleon is studied. We also study the underlying uncertainties on these calculations and compare the results to similar investigations in literature.

Double inclusive small- x gluon production and their azimuthal correlations in a biased ensemble

We consider double $gg\rightarrow g$ production in the presence of a bias on the unintegrated gluon distribution of the colliding hadron or nuclei. Such bias could be due to the selection of configurations with a greater number of gluons or higher mean transverse momentum squared or, more generally, due to a modified spectral shape of the gluon distribution in the hadrons. Hence, we consider reweighted functional averages over the stochastic ensemble of small-x gluons. We evaluate explicitly the double inclusive gluon transverse momentum spectrum in high-energy collisions, and their azimuthal correlations, for a few simple examples of biases.

Multi-quark colour decompositions from unitarity

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 provide closed-form expressions for the new decompositions. We then elaborate upon their application in colour decompositions of loop multi-quark amplitudes.

String correlators: recursive expansion, integration-by-parts and scattering equations

We further elaborate on the general construction proposed in [1], which connects, via tree-level double copy, massless string amplitudes with color-ordered QFT amplitudes that are given by Cachazo-He-Yuan formulas. The current paper serves as a detailed study of the integration-by-parts procedure for any tree-level massless string correlator outlined in the previous letter. We present two new results in the context of heterotic and (compactified) bosonic string theories. First, we find a new recursive expansion of any multitrace mixed correlator in these theories into a logarithmic part corresponding to the CHY integrand for Yang-Mills-scalar amplitudes, plus correlators with the total number of traces and gluons decreased. By iterating the expansion, we systematically reduce string correlators with any number of subcycles to linear combinations of Parke-Taylor factors and similarly for the case with gluons. Based on this, we then derive a CHY formula for the corresponding (DF)2 + YM + ϕ3 amplitudes. It is the first closed-form result for such multitrace amplitudes and thus greatly extends our result for the single-trace case. As a byproduct, it gives a new CHY formula for all Yang-Mills-scalar amplitudes. We also study consistency checks of the formula such as factorizations on massless poles.

TMD gluon distributions for multiparton processes

We derive gauge invariant operators entering definitions of the Transverse Momentum Dependent (TMD) gluon distributions, for all five and six parton processes. Our calculations utilize color decomposition of amplitudes in the color flow basis. In addition, we find the general result for multi-gluon process (with arbitrary number of gluons) at large N_{c}. On phenomenological ground our results may be used for multi-jet production in the small-x regime, where the TMD gluon distributions can be derived from the Color Glass Condensate effective theory.

Direct photon production at LHCb

At small Bjorken-x, the large gluon number density in the nucleon leads to gluon recombination competing with gluon splitting, which could result in saturation of the gluon PDF. This gluon saturation has yet to be conclusively observed. Direct photon production provides sensitivity to gluon densities in protons and nuclei, and the forward acceptance of LHCb detector allows for measurements of this process at low Bjorken-x, providing an ideal probe of saturation effects. Progress towards the measurement of forward direct photon production using the LHCb detector is presented.

Cyclic Mario worlds \u2014 color-decomposition for one-loop QCD

We present a new color decomposition for QCD amplitudes at one-loop level as a generalization of the Del Duca-Dixon-Maltoni and Johansson-Ochirov decomposition at tree level. Starting from a minimal basis of planar primitive amplitudes we write down a color decomposition that is free of linear dependencies among appearing primitive amplitudes or color factors. The conjectured decomposition applies to any number of quark flavors and is independent of the choice of gauge group and matter representation. The results also hold for higher-dimensional or supersymmetric extensions of QCD. We provide expressions for any number of external quark-antiquark pairs and gluons.

General expressions for extra-dimensional tree amplitudes and all-plus 1-loop integrands in mathcal {Q}-cut representation

In this paper, we give the general expressions for a special series of tree amplitudes of the Yang–Mills theory. This series of amplitudes have two adjacent massless spin-1 particles with extra-dimensional momenta and any number of positive helicity gluons. With special helicity choices, we use the spinor helicity formalism to express these n-point amplitudes in compact forms, and find a clever way to use the BCFW recursion relations to prove the results. Then these amplitudes are used to form the complete 1-loop all-plus integrand with any number of gluons, expressed in the mathcal {Q}-cut representation.