Definable henselian valuation rings

We give model theoretic criteria for $\exists \forall$ and $\forall \exists$- formulas in the ring language to define uniformly the valuation rings $\mathcal{O}$ of models $(K, \mathcal{O})$ of an elementary theory $\Sigma$ of henselian valued fields. As one of the applications we obtain the existence of an $\exists \forall$-formula defining uniformly the valuation rings $\mathcal{O}$ of valued henselian fields $(K, \mathcal{O})$ whose residue class field $k$ is finite, pseudo-finite, or hilbertian. We also obtain $\forall \exists$-formulas $\varphi_2$ and $\varphi_4$ such that $\varphi_2$ defines uniformly $k[[t]]$ in $k((t))$ whenever $k$ is finite or the function field of a real or complex curve, and $\varphi_4$ does the job if $k$ is any number field

Combining states without scale hierarchies with ordered parton showers

We present a parameter-free scheme to combine fixed-order multi-jet results with parton-shower evolution. The scheme produces jet cross sections with leading-order accuracy in the complete phase space of multiple emissions, resumming large logarithms when appropriate, while not arbitrarily enforcing ordering on momentum configurations beyond the reach of the parton-shower evolution equation. This requires the development of a matrix-element correction scheme for complex phase-spaces including ordering conditions as well as a systematic scale-setting procedure for unordered phase-space points. The resulting algorithm does not require a merging-scale parameter. We implement the new method in the Vincia framework and compare to LHC data.Comment: updated to version published in EPJ

Merging Multi-leg NLO Matrix Elements with Parton Showers

We discuss extensions the CKKW-L and UMEPS tree-level matrix element and parton shower merging approaches to next-to-leading order accuracy. The generalisation of CKKW-L is based on the NL3 scheme previously developed for e+e- -annihilation, which is extended to also handle hadronic collisions by a careful treatment of parton densities. NL3 is further augmented to allow for more readily accessible NLO input. To allow for a more careful handling of merging scale dependencies we introduce an extension of the UMEPS method. This approach, dubbed UNLOPS, does not inherit problematic features of CKKW-L, and thus allows for a theoretically more appealing definition of NLO merging. We have implemented both schemes in Pythia8, and present results for the merging of W- and Higgs-production events, where the zero- and one-jet contribution are corrected to next-to-leading order simultaneously, and higher jet multiplicities are described by tree-level matrix elements. The implementation of the procedure is completely general and can be used for higher jet multiplicities and other processes, subject to the availability of programs able to correctly generate the corresponding partonic states to leading and next-to-leading order accuracy.Comment: updated references, submitted to JHE

The midpoint between dipole and parton showers

We present a new parton-shower algorithm. Borrowing from the basic ideas of dipole cascades, the evolution variable is judiciously chosen as the transverse momentum in the soft limit. This leads to a very simple analytic structure of the evolution. A weighting algorithm is implemented, that allows to consistently treat potentially negative values of the splitting functions and the parton distributions. We provide two independent, publicly available implementations for the two event generators Pythia and Sherpa.Comment: 23 pages, 9 figure

Uniform definability of henselian valuation rings in the Macintyre language

We discuss definability of henselian valuation rings in the Macintyre language $\mathcal{L}_{\rm Mac}$, the language of rings expanded by n-th power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian residue field are uniformly $\exists$-$\emptyset$-definable in $\mathcal{L}_{\rm Mac}$, and henselian valuation rings with value group $\mathbb{Z}$ are uniformly $\exists\forall$-$\emptyset$-definable in the ring language, but not uniformly $\exists$-$\emptyset$-definable in $\mathcal{L}_{\rm Mac}$. We apply these results to local fields $\mathbb{Q}_p$ and $\mathbb{F}_p((t))$, as well as to higher dimensional local fields

Unitarising Matrix Element + Parton Shower merging

We revisit the CKKW-L method for merging tree-level matrix elements with parton showers, and amend it with an add/subtract scheme to minimise dependencies on the merging scale. The scheme is constructed to, as far as possible, recover the unitary nature of the underlying parton shower, so that the inclusive cross section is retained for each jet multiplicity separately.Comment: 30 pages, 13 figures, updated references, submitted to JHE

Combining parton showers and NNLO matrix elements

In this talk, we discuss recent developments in combining parton showers and fixed-order calculations. We focus on the UNNLOPS method for matching next-to-next-to-leading order computations to the parton shower, and we present results from Sherpa for Drell-Yan lepton-pair and Higgs-boson production at the LHC.Comment: 4 pages, 2 figures, proceedings for Moriond QCD 201

Higgs-boson production through gluon fusion at NNLO QCD with parton showers

We discuss how the UN2LOPS scheme for matching NNLO calculations to parton showers can be applied to processes with large higher-order perturbative QCD corrections. We focus on Higgs-boson production through gluon fusion as an example. We also present an NNLO fixed-order event generator for this reaction.Comment: 9 pages, 3 figures, 1 tabl