Resummation of non-global logarithms

Most of what we know about the laws of physics at the sub-nuclear length scale is derived from analyzing the outcomes of high-energy collisions of particles. Size and sophistication of such experiments have steadily increased, producing more accurate and precise measurements. To interpret these and to understand the underlying physics we need to develop and improve state-of-the-art theoretical tools to predict collider observables in a more refined way. Such a tool is Soft-Collinear Effective Theory, which is a theoretical framework for calculations that involve particles at widely separated energy scales. In a collider experiment, such hierarchies are very common. In the analysis of detector patterns after a collision event, one observes regions where a lot of energetic particles were measured close together, and a small amount of energy that is uniformly distributed in the entire detector. This name-giving collinear and soft radiation is captured by this specific effective theory. When calculating the cross sections of such multi-scale problems, corrections that are normally suppressed by orders of the coupling constant are multiplied by a logarithm of the ratio of the scales. In the case of the interjet energy flow, for example, we have particles of high energy Ein ~ Q produced at the hard interaction inside the jets and soft particles of energy Eout ~ βQ outside the jets. If these scales are far separated, the logarithms ln(β) become big and can ruin the perturbative expansion in the coupling constants. One needs to resum these logarithmically enhanced terms to all orders. The so-called non-global logarithms that arise in the presence of hard phase-space cuts are especially difficult to resum. A simple exponentiation of the logarithmically enhanced terms is in this case not possible. Starting from a factorization theorem in Soft-Collinear Effective Theory, we derive a parton shower equation for the resummation of non-global logarithms. We implement the shower in a dedicated, easy-to-use computer code NGL_RESUM to resum such logarithms at leading logarithmic accuracy in the large-Nc limit. We use tree-level event files in the common Les Houches format as input and explicitly calculate observables such as the interjet energy flow and photon isolation cone cross sections at leading-logarithmic accuracy. Since our parton shower is derived from first principles and based on renormalization group evolution, it is clear what ingredients we have to include to perform the resummation at subleading logarithmic accuracy. After adding the observable-dependent next-to-leading order corrections for di-jet processes, we are able to take a first step towards the higher-logarithmic resummation for the interjet energy flow and the jet mass. To reach full next-to-leading logarithmic accuracy, one would also have to include the two-loop anomalous dimension. This is indeed very challenging and has not been reached thus far. In a final step, we extend the framework to get away from the high-energy limit and include massive quarks. At this point, NGL_RESUM becomes a general-purpose parton shower to resum non-global logarithms at leading logarithmic accuracy for a plethora of observables. We apply it to compute the interjet energy flow in tt̅ -production

Non-global logarithms in jet and isolation cone cross sections

Starting from a factorization theorem in effective field theory, we derive a parton-shower equation for the resummation of non-global logarithms. We have implemented this shower and interfaced it with a tree-level event generator to obtain an automated framework to resum the leading logarithm of non-global observables in the large-$N_c$ limit. Using this setup, we compute gap fractions for dijet processes and isolation cone cross sections relevant for photon production. We compare our results with fixed-order computations and LHC measurements. We find that naive exponentiation is often not adequate, especially when the vetoed region is small, since non-global contributions are enhanced due to their dependence on the veto-region size. Since our parton shower is derived from first principles and based on renormalization-group evolution, it is clear what ingredients will have to be included to perform resummations at subleading logarithmic accuracy in the future.Comment: 39 pages, 13 figures. v2: journal version with new result (4.18) for narrow isolation cone

Resummation of non-global logarithms in cross sections with massive particles

A factorization formalism for jet processes involving massive colored particles such as the top quark is developed, extending earlier results for the massless case. The factorization of soft emissions from the underlying hard process is implemented in an effective field theory framework, which forms the basis for the resummation of large logarithms. The renormalization group evolution giving rise to non-global logarithms is implemented into a parton shower code in the large-$N_c$ limit. After a comparison of the massive and massless radiations patterns, the cross section for $t\bar{t}$ production with a veto on additional central jet activity is computed, taking into account radiation both from the production and the decay of the top quarks. The resummation of the leading logarithms leads to an improved description of ATLAS measurements at $\sqrt{s}=7\,{\rm TeV}$.Comment: 40 pages, 10 figure

NLL′{'} resummation of jet mass

Starting from a factorization theorem in effective field theory, we present resummed results for two non-global observables: the invariant-mass distribution of jets and the energy distribution outside jets. Our results include the full next-to-leading-order corrections to the hard, jet and soft functions and are implemented in a parton-shower framework which generates the renormalization-group running in the effective theory. The inclusion of these matching corrections leads to an improved description of the data and reduced theoretical uncertainties. They will have to be combined with two-loop running in the future, but our results are an important first step towards the higher-logarithmic resummation of non-global observables.Comment: 32 pages, 12 figures. v2: journal versio

Resummation of non-global logarithms in cross sections with massive particles

A factorization formalism for jet processes involving massive colored particles such as the top quark is developed, extending earlier results for the massless case. The factorization of soft emissions from the underlying hard process is implemented in an effective field theory framework, which forms the basis for the resummation of large logarithms. The renormalization group evolution giving rise to non-global logarithms is implemented into a parton shower code in the large-Nc limit. After a comparison of the massive and massless radiation patterns, the cross section fo

Solutions to Problems at Les Houches Summer School on EFT

International audienceThis final chapter provides details of worked solutions to the various problems set by the lecturers during the course of the school; some of these problems appear within the chapters of this book. This chapter also contains further exercises that were added after the school are not solved here; these are left as a challenge for the enterprising reader. Problems run the range of topics covered. These problems and solutions are associated with topics that include the introduction to EFT, renormalization theory, nuclear and atomic physics, Nambu–Goldstone modes, inflation, and large-scale structure, and how each topic relates to EFTs