Parton showers are widely used to generate fully exclusive final states needed to compare theoretical models to experimental observations. While, in general, parton showers give a good description of the experimental data, the precise functional form of the probability distribution underlying the event generation is generally not known. The reason is that realistic parton showers are required to conserve four-momentum at each vertex. In this paper we investigate in detail how four-momentum conservation is enforced in a standard parton shower and why this destroys the analytic control of the probability distribution. We show how to modify a parton shower algorithm such that it conserves four-momentum at each vertex, but for which the full analytic form of the probability distribution is known. We then comment how this analytic control can be used to match matrix element calculations with parton showers, and to estimate effects of power corrections and other uncertainties in parton showers.

Using the known resummation of virtual corrections together with knowledge of the

leading-log structure of real radiation in a parton shower, we derive analytic expressions

for the resummed real radiation after they have been integrated over all of phase space.

Performing a numerical analysis for both the 13 TeV LHC and a 100 TeV pp collider, we

show that resummation of the real corrections is at least as important as resummation of

the virtual corrections, and that this resummation has a sizable eect for partonic center of

mass energies exceeding

p

s = O(few TeV). For partonic center of mass energies

p

s & 10

TeV, which can be reached at a 100 TeV collider, resummation becomes an O(1) eect and

needs to be included even for rough estimates of the cross-sections.

We compute the leading-order evolution of parton distribution functions for all Standard

Model fermions and bosons up to energy scales far above the electroweak scale, where electroweak

symmetry is restored. Our results include the 52 PDFs of the unpolarized proton,

evolving according to the SU(3), SU(2), U(1), mixed SU(2)U(1) and Yukawa interactions.

We illustrate the numerical eects on parton distributions at large energies, and show that

this can lead to important corrections to parton luminosities at a future 100 TeV collider.

We present a resummation of those double-logarithmically enhanced electroweak correction

that arise in pp colliders because protons are not SU(2) singlets, by solving DGLAP

equations in the full Standard Model. We then show how to match these results with those

of xed-order electroweak calculations. At a 100 TeV pp collider, contributions beyond order

are 10% at partonic center-of-mass energies of a few TeV. These are mainly due to

initial states with massive vector bosons.

Factorization is the central ingredient in any theoretical prediction for collider experiments. I introduce a factorization formalism that can be applied to any desired observable, like event shapes or jet observables, for any number of jets and a wide range of jet algorithms in leptonic or hadronic collisions. This is achieved by using soft-collinear effective theory to prove the formal factorization of a generic fully-differential cross section in terms of a hard coefficient, and generic jet and soft functions. The factorization formula for any such observable immediately follows from our general result, including the precise definition of the functions appropriate for the observable in question.

As a first application, I present a new prediction of angularity distributions in e+e- annihilation. Angularities tau_a are an infinite class of event shapes which vary in their sensitivity to the substructure of jets in the final state, controlled by a continuous parameter a < 2. I calculate angularity distributions for all a < 1 to first order in the strong coupling alpha_s and resum large logarithms in these distributions to next-to-leading logarithmic (NLL) accuracy.

I then apply SCET to the more exclusive case of jet shapes. In particular, I make predic- tions for quark and gluon jet shape distributions in N-jet final states in e+e- collisions, defined with a cone or recombination algorithm, where I measure some jet shape observable on a subset of these jets. I demonstrate the consistent renormalization-group running of the functions in the factoriza- tion theorem for any number of measured and unmeasured jets, any number of quark and gluon jets, and any angular size R of the jets, as long as R is much smaller than the angular separation between jets. I calculate the jet and soft functions for angularity jet shapes tau_a to next-to-leading order (NLO) in alpha_s and resum large logarithms of tau_a to next-to-leading logarithmic (NLL) accuracy for both cone and kT -type jets.

Finally, I apply SCET to the case of threshold resummmation at hadron colliders. Factor- ization theorems for processes at hadron colliders near the hadronic endpoint have largely focused on simple final states with either no jets (e.g., Drell-Yan) or one inclusive jet (e.g., deep inelastic scattering and prompt photon production). Factorization for the former type of process gives rise to a soft function that depends on timelike momenta whereas the soft function for the latter type depends on null momenta. I derive a factorization theorem that allows for an arbitrary number of jets, where the jets are defined with respect to a jet algorithm, together with any number of non- strongly interacting particles. I find the soft function in general depends on the null components of the soft momenta inside the jets and on the timelike component of the soft momentum outside of the jets. This generalizes and interpolates between the soft functions for the cases of no jets and one inclusive jet.

Matrix-element calculations and parton shower programs are both crucial tools in the analysis of data at modern particle physics experiments at colliders. Finding the most effective ways to combine these complementary, but sometimes conflicting, approaches to simulating physical events has been the subject of much work in the recent decade. This thesis investigates state-of-the-art ways in which the precision of the matrix elements can be extended in combination with the parton shower. We identify three dimensions along which precision can be improved and describe how progress can be made along each one.

First, we present a general method to match fully differential next-to-next-to-leading-order (NNLO) calculations to parton shower (PS) programs, which represents an extension of the successful LO+PS (leading order) and NLO+PS (next-to-leading order) frameworks to NNLO+PS. We discuss in detail the perturbative accuracy criteria a complete NNLO+PS matching has to satisfy, and we give an explicit and general construction of the input "Monte Carlo cross sections" satisfying all required criteria.

Next, we describe how augmenting an NLO calculation with higher-order resummation of large Sudakov logarithms allows one to extend the lowest-order matching of tree-level matrix elements with parton showers to give a complete description at the next higher perturbative accuracy in α_s, at both small and large jet resolutions. As a byproduct, this combination naturally leads to a smooth connection of the NLO calculations for different jet multiplicities. We focus on the general construction of our method and present results of an implementation in the GENEVA Monte Carlo framework. For leptonic collisions, we apply our construction to e⁺e^- → jets and obtain good agreement with LEP data for a variety of 2-jet observables. For hadronic collisions, we look at Drell-Yan production.