Ordering Multiple Soft Gluon Emissions

Abstract

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]

Soft gluon evolution and non-global logarithms

“…This intriguing physics may not be so clearly visible in an effective field theory treatment, where the evolution is in an arbitrary renormalization scale. Figure 1 illustrates the key finding of [27]. The upper graph illustrates that the virtual gluon loop integral has its transverse momentum limited by the transverse momenta of the two nearest real emissions.…”

“…momentum ordering should be used [32]. Interestingly, and working in the eikonal approximation, but only for gluons that couple to the original hard partons, explicit calculation of all relevant Feynman diagrams (at one loop) reveals that the corrections associated with the exact triple and four-gluon vertices can be largely subsumed into a Lorentz invariant ordering variable, in a potentially simple extension of the algorithm that we described in [27,32]. This intriguing physics may not be so clearly visible in an effective field theory treatment, where the evolution is in an arbitrary renormalization scale.…”

“…According to the results in [27,32], which were performed to one-loop accuracy, the corresponding differential cross section has a similar structure to eq. (2.1) but with…”

“…We then proceed to reformulate the algorithm in such a way as to make the cancellation of infrared divergences explicit. We also highlight the role of the specific ordering variable first identified in [27].…”

“…(2.2). In [27] we presented the form of this operator in 4 − 2 dimensions after integration over the solid angle, i.e. 34) where c Γ = 1 − γ E , and γ E is the Euler-Mascheroni constant.…”