We consider soft-gluon evolution at the amplitude level. Our evolution algorithm applies to generic hard-scattering processes involving any number of coloured partons and we present a reformulation of the algorithm in such a way as to make the cancellation of infrared divergences explicit. We also emphasise the special role played by a Lorentzinvariant evolution variable, which coincides with the transverse momentum of the latest emission in a suitably defined dipole zero-momentum frame. Handling large colour matrices presents the most significant challenge to numerical implementations and we present a means to expand systematically about the leading colour approximation. Specifically, we present a systematic procedure to calculate the resulting colour traces, which is based on the colour flow basis. Identifying the leading contribution leads us to re-derive the BanfiMarchesini-Smye equation. However, our formalism is more general and can systematically perform resummation of contributions enhanced by the t'Hooft coupling α s N ∼ 1, along with successive perturbations that are parametrically suppressed by powers of 1/N . We also discuss how our approach relates to earlier work.
Expressions for SO(4) invariant Euclidean QCD generating functionals are introduced which should produce nonvanishing gluon condensates. We consider first the loop expansion of the corresponding effectíve action searching for a description differing from the usual perturbation theory. At this level, we consider special free propagators which have off-diagonal long range order. The calculation of the polarization tensor leads to a gluon mass term which is proportional to the squared root of the finite value for <G2>. The summation of all the one-loop contributions to the energy having only mass insertions, indicates the spontaneous generation of the condensate from the perturbative ground state in a way resembling the similar effect in the case of chromomagnetic field models.
We present an expression for the QCD amplitude for a general hard scattering process with any number of soft gluon emissions, to one-loop accuracy. The amplitude is written in two different but equivalent ways: as a product of operators ordered in dipole transverse momentum and as a product of loop-expanded currents. We hope that these results will help in the development of an all-orders algorithm for multiple emissions that includes the full color structure and both the real and imaginary contributions to the amplitude. DOI: 10.1103/PhysRevLett.116.212003 Soft gluon factorization (e.g., see [1]) is an important property of perturbative QCD. It is an essential ingredient in the construction of state-of-the-art Monte Carlo event generators [2][3][4] and all-orders logarithmic resummations (e.g., see [5]), and it allows control over infrared poles in computations of cross sections at fixed order in the strong coupling (e.g., see [6]). In this Letter, we consider the one-loop amplitude for any number of soft gluon emissions off a general n-parton hard process. We work in the eikonal approximation for the coupling of the soft gluons to the hard partons and are able to derive an interesting identity in the limit where the gluons are ordered in softness. Specifically, we show that the N emission amplitude may be written in two equivalent ways: either in terms of an ordered evolution of the hard scattering amplitude (in which intermediate infrared divergences cancel) or in terms of a product of loop-expanded emission operators acting on a loop-expanded matrix element (both of which are infrared divergent). It is our hope that Eq. (1), below, will form the basis for a future all-orders amplitude-level parton shower algorithm. This would be a major improvement over cross-section-level algorithms, as implemented in the existing parton shower event generators, not least because it would include full color evolution and Coulomb gluon exchange. Other work towards this goal can be found in Refs. [7][8][9][10]. As we will see, our calculations also indicate how successive real emissions constrain the intermediate (and finite) loop integrals by imposing an ordering fixed by the transverse momenta of adjacent real emissions, defined with respect to the directions of the partons involved in the virtual exchange.In the ordered evolution approach, the one-loop amplitude for a total of N soft gluon emissions, with four-momenta q i , from a hard process with n legs, with four-momenta p i , is [11]
We consider the Lorentz-violating extended QED involving all nonminimal dimension-5 additive CPT-odd terms. For this theory, we investigate the generation of the Carroll–Field–Jackiw (CFJ) term and its higher-derivative counterparts of the first order in any of these nonminimal couplings. The CFJ term is demonstrated to vanish in the dimensional regularization scheme. We also study the question of higher-derivative divergent contributions and demonstrate that they can be eliminated by considering a given proportionality between the coefficients.
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