Abstract
We consider the Banfi-Marchesini-Smye (BMS) equation which resums ‘non-global’ energy logarithms in the QCD evolution of the energy lost by a pair of jets via soft radiation at large angles. We identify a new physical regime where, besides the energy logarithms, one has to also resum (anti)collinear logarithms. Such a regime occurs when the jets are highly collimated (boosted) and the relative angles between successive soft gluon emissions are strongly increasing. These anti-collinear emissions can violate the correct time-ordering for time-like cascades and result in large radiative corrections enhanced by double collinear logs, making the BMS evolution unstable beyond leading order. We isolate the first such a correction in a recent calculation of the BMS equation to next-to-leading order by Caron-Huot. To overcome this difficulty, we construct a ‘collinearly-improved’ version of the leading-order BMS equation which resums the double collinear logarithms to all orders. Our construction is inspired by a recent treatment of the Balitsky-Kovchegov (BK) equation for the high-energy evolution of a space-like wavefunction, where similar time-ordering issues occur. We show that the conformal mapping relating the leading-order BMS and BK equations correctly predicts the physical time-ordering, but it fails to predict the detailed structure of the collinear improvement.
Highlights
Axis, within the ‘allowed’ region between the jet and Cout
We shall argue that, when θ0 1, radiative corrections enhanced by the double logarithm ln(E/E0) ln(1/θ02) are generated by successive gluon emissions which accumulate towards Cout: the angles made by these gluons with the central axis of Cout are strongly decreasing from one emission to the one
For the problem at hand, they bring corrections to the kernel of the BMS equation in the form of a series in powers of αs ln2(1/θ02), with αs ≡ αsNc/π. The first such a correction is present in the next-to-leading order (NLO) version of the BMS equation [9], albeit this is perhaps not manifest in the original expressions in ref. [9]. (We shall isolate this contribution from the full NLO kernel in appendix A.) From the experience with the respective space-like evolution — the BFKL equation [13,14,15] and its non-linear generalisations, the Balitsky-Kovchegov (BK) equation [16, 17] and the Balitsky-JIMWLK hierarchy [16, 18,19,20,21,22,23], — where a similar problem arises, we expect such double collinear logs to lead to instabilities in the NLO evolution and, in any case, to jeopardise the convergence of the perturbative expansion
Summary
We shall introduce the leading-order BMS equation and demonstrate that under special circumstances — namely, for the configurations illustrated in figure 2 — this equation resums (anti)-collinear logarithms, on top of the energy logarithms that it was originally meant for. We shall argue that, when these (anti)-collinear logs are sufficiently large, the BMS equation is not boost-invariant anymore: it can still be used as it stands in the di-jet COM frame, but not in a boosted frame where the energy phase-space available to the evolution is much larger
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