Mixing Patterns from the Groups Sigma (n phi)

We survey the mixing patterns which can be derived from the discrete groups Sigma (36 x 3), Sigma (72 x 3), Sigma (216 x 3) and Sigma (360 x 3), if these are broken to abelian subgroups Ge and Gnu in the charged lepton and neutrino sector, respectively. Since only Sigma (360 x 3) possesses Klein subgroups, only this group allows neutrinos to be Majorana particles. We find a few patterns that can agree well with the experimental data on lepton mixing in scenarios with small corrections and that predict the reactor mixing angle theta_{13} to be 0.1 <= theta_{13} <= 0.2. All these patterns lead to a trivial Dirac phase. Patterns which instead reveal CP violation tend to accommodate the data not well. We also comment on the outer automorphisms of the discussed groups, since they can be useful for relating inequivalent representations of these groups.Comment: 1+28 pages, 6 tables, no figures; v2: matches version published in J. Phys. A: Math. Theo

NLO Renormalization in the Hamiltonian Truncation

Hamiltonian Truncation (a.k.a. Truncated Spectrum Approach) is a numerical technique for solving strongly coupled QFTs, in which the full Hilbert space is truncated to a finite-dimensional low-energy subspace. The accuracy of the method is limited only by the available computational resources. The renormalization program improves the accuracy by carefully integrating out the high-energy states, instead of truncating them away. In this paper we develop the most accurate ever variant of Hamiltonian Truncation, which implements renormalization at the cubic order in the interaction strength. The novel idea is to interpret the renormalization procedure as a result of integrating out exactly a certain class of high-energy "tail states". We demonstrate the power of the method with high-accuracy computations in the strongly coupled two-dimensional quartic scalar theory, and benchmark it against other existing approaches. Our work will also be useful for the future goal of extending Hamiltonian Truncation to higher spacetime dimensions.Comment: 28pp + appendices, detailed version of arXiv:1706.0612

High-Precision Calculations in Strongly Coupled Quantum Field Theory with Next-to-Leading-Order Renormalized Hamiltonian Truncation

Hamiltonian Truncation (a.k.a. Truncated Spectrum Approach) is an efficient numerical technique to solve strongly coupled QFTs in d=2 spacetime dimensions. Further theoretical developments are needed to increase its accuracy and the range of applicability. With this goal in mind, here we present a new variant of Hamiltonian Truncation which exhibits smaller dependence on the UV cutoff than other existing implementations, and yields more accurate spectra. The key idea for achieving this consists in integrating out exactly a certain class of high energy states, which corresponds to performing renormalization at the cubic order in the interaction strength. We test the new method on the strongly coupled two-dimensional quartic scalar theory. Our work will also be useful for the future goal of extending Hamiltonian Truncation to higher dimensions d >= 3.Comment: 8 pages, 4 figures; v2: published versio

The local Callan-Symanzik equation: structure and applications

The local Callan-Symanzik equation describes the response of a quantum field theory to local scale transformations in the presence of background sources. The consistency conditions associated with this anomalous equation imply non-trivial relations among the $\beta$-function, the anomalous dimensions of composite operators and the short distance singularities of correlators. In this paper we discuss various aspects of the local Callan-Symanzik equation and present new results regarding the structure of its anomaly. We then use the equation to systematically write the n-point correlators involving the trace of the energy-momentum tensor. We use the latter result to give a fully detailed proof that the UV and IR asymptotics in a neighbourhood of a 4D CFT must also correspond to CFTs. We also clarify the relation between the matrix entering the gradient flow formula for the $\beta$-function and a manifestly positive metric in coupling space associated with matrix elements of the trace of the energy momentum tensor.Comment: v2: Modified discussion of the amplitude; v3: typos fixe

Conformal Truncation of Chern-Simons Theory at Large NfN_f

We set up and analyze the lightcone Hamiltonian for an abelian Chern-Simons field coupled to $N_f$ fermions in the limit of large $N_f$ using conformal truncation, i.e. with a truncated space of states corresponding to primary operators with dimension below a maximum cutoff $\Delta_{\rm max}$. In both the Chern-Simons theory, and in the $O(N)$ model at infinite $N$, we compute the current spectral functions analytically as a function of $\Delta_{\rm max}$ and reproduce previous results in the limit that the truncation $\Delta_{\rm max}$ is taken to $\infty$. Along the way, we determine how to preserve gauge invariance and how to choose an optimal discrete basis for the momenta of states in the truncation space.Comment: 32+25 pages, 8 figures. v2: updated ref

A naturally light dilaton

Goldstone's theorem does not apply straightforwardly to the case of spontaneously broken scale invariance. We elucidate under what conditions a light scalar degree of freedom, identifiable with the dilaton, can naturally arise. Our construction can be considered an explicit dynamical solution to the cosmological constant problem in the scalar version of gravity.Comment: v2: published versio

Deterrence in Competition Law

This paper provides a comprehensive discussion of the deterrence properties of a competition policy regime. On the basis of the economic theory of law enforcement we identify several factors that are likely to affect its degree of deterrence: 1) sanctions and damages; 2) financial and human resources; 3) powers during the investigation; 4) quality of the law; 5) independence; and 6) separation of power. We then discuss how to measure deterrence. We review the literature that use surveys to solicit direct information on changes in the behavior of firms due to the threats posed by the enforcement of antitrust rules, and the literature based on the analysis of hard data. We finally argue that the most challenging task, both theoretically and empirically, is how to distinguish between “good” deterrence and “bad” deterrence

Measuring the deterrence properties of competition policy: the Competition Policy Indexes

The aim of this paper is to describe in detail a set of newly developed indicators of the quality of competition policy, Competition Policy Indexes, or CPIs. The CPIs measure the deterrence properties of a competition policy in a jurisdiction, where for competition policy we mean the antitrust legislation, including the merger control provisions, and its enforcement. The CPIs incorporate data on how the key features of a competition policy regime score against a benchmark of generally-agreed best practices and summarise them so as to allow cross-country and cross-time comparisons. The CPIs have been calculated for a sample of 13 OECD jurisdictions over the period 1995-2005