Robust approach to thermal resummation: Standard Model meets a singlet

Perturbation theory alone fails to describe thermodynamics of the electroweak phase transition. We review a technique combining perturbative and non-perturbative methods to overcome this challenge. Accordingly, the principal theme is a tutorial of high­temperature dimensional reduction. We present an explicit derivation with a real singlet scalar and compute the thermal effective potential at two-loop order. In particular, we detail the dimensional reduction for a real-singlet extended Standard Model. The resulting effective theory will impact future non-perturbative studies based on lattice simulations as well as purely perturbative investigations.

Some Mathematics for the Method of Pooled PCR Test

At the time of the worldwide COVID-19 disaster, the author learned about the pooled (RT-) PCR test from the news. From the common sense of individual tests, the idea of mixing multiple samples seems taboo, however in fact many samples can be tested with a smaller number of tests by the method. As a retired researcher of mathematical engineering, the author was deeply interested in the idea and absorbed in the mathematical formulation and intensive analysis of the method. Later, he found that the original basic equation was already proposed in the old (1943) treatise [1] and so many related research works have been done and available as materials on the web [2], although many of those seem to be based on qualitative or intuitive analysis. In that sense, some of the analysis here seems to be already known in the field, but some results might be novel, such as boundary conditions, derivation of limit values, estimation of infection rate and adaptive optimization scheme of pool test, strict extension to multi-stage pool test, and explicit derivation of asymptotic approximate solutions of optimal pooling number and achieved efficiency measure, etc. In any case, he decided to put it together here as a material rather than a formal treatise, hoping that the results here would be useful for deeper mathematical insights into and better understanding of the pool inspection, and also in its actual practice.

Spin and entanglement in general relativity

In a previous paper, we have shown how the classical and quantum relativistic dynamics of the Stueckelberg–Horwitz–Piron [SHP] theory can be embedded in general relativity (GR). We briefly review the SHP theory here and, in particular, the formulation of the theory of spin in the framework of relativistic quantum theory. We show here how the quantum theory of relativistic spin can be embedded, using a theorem of Abraham, Marsden and Ratiu and also explicit derivation, into the framework of GR by constructing a local induced representation. The relation to the work of Fock and Ivanenko is also discussed. We show that in a gravitational field there is a highly complex structure for the spin distribution in the support of the wave function. We then discuss entanglement for the spins in a two body system.

Surface operators in superspace

We generalize the geometrical formulation of Wilson loops recently introduced in [1] to the description of Wilson Surfaces. For N = (2, 0) theory in six dimensions, we provide an explicit derivation of BPS Wilson Surfaces with non-trivial coupling to scalars, together with their manifestly supersymmetric version. We derive explicit conditions which allow to classify these operators in terms of the number of preserved supercharges. We also discuss kappa-symmetry and prove that BPS conditions in six dimensions arise from kappa-symmetry invariance in eleven dimensions. Finally, we discuss super-Wilson Surfaces — and higher dimensional operators — as objects charged under global p-form (super)symmetries generated by tensorial supercurrents. To this end, the construction of conserved supercurrents in supermanifolds and of the corresponding conserved charges is developed in details.

Sparse recovery from extreme eigenvalues deviation inequalities

This article provides a new toolbox to derive sparse recovery guarantees – that is referred to as “stable and robust sparse regression” (SRSR) – from deviations on extreme singular values or extreme eigenvalues obtained in Random Matrix Theory. This work is based on Restricted Isometry Constants (RICs) which are a pivotal notion in Compressed Sensing and High-Dimensional Statistics as these constants finely assess how a linear operator is conditioned on the set of sparse vectors and hence how it performs in SRSR. While it is an open problem to construct deterministic matrices with apposite RICs, one can prove that such matrices exist using random matrices models. In this paper, we show upper bounds on RICs for Gaussian and Rademacher matrices using state-of-the-art deviation estimates on their extreme eigenvalues. This allows us to derive a lower bound on the probability of getting SRSR. One benefit of this paper is a direct and explicit derivation of upper bounds on RICs and lower bounds on SRSR from deviations on the extreme eigenvalues given by Random Matrix theory.

Survival of the frequent at finite population size and mutation rate: bridging the gap between quasispecies and monomorphic regimes with a simple model

In recent years, there has been increased attention on the non-trivial role that genotype-phenotype maps play in the course of evolution, where natural selection acts on phenotypes, but variation arises at the level of mutations. Understanding such mappings is arguably the next missing piece in a fully predictive theory of evolution. Although there are theoretical descriptions of such mappings for the monomorphic (Nμ ≪ 1) and deterministic or very strong mutation (Nμ ⋙ 1) limit, given by developments of Iwasa’s free fitness and quasispecies theories, respectively, there is no general description for the intermediate regime where Nμ ~ 1. In this paper, we address this by transforming Wright’s well-known stationary distribution of genotypes under selection and mutation to give the probability distribution of phenotypes, assuming a general genotype-phenotype map. The resultant distribution shows that the degeneracies of each phenotype appear by weighting the mutation term; this gives rise to a bias towards phenotypes of larger degeneracy analogous to quasispecies theory, but at finite population size. On the other hand we show that as population size is decreased, again phenotypes of higher degeneracy are favoured, which is a finite mutation description of the effect of sequence entropy in the monomorphic limit. We also for the first time (to the author’s knowledge) provide an explicit derivation of Wright’s stationary distribution of the frequencies of multiple alleles.