Completing the hadronic Higgs boson decay at order αs4\alpha_s^4

We compute four-loop corrections to the hadronic decay of the Standard Model Higgs boson which are induced by effective couplings to bottom quarks and gluons, mediated by the top quark. Our numerical results are comparable in size to the purely massless contributions which have been known for a few years. The results presented in this paper complete the order $\alpha_s^4$ corrections to the hadronic Higgs boson decay.Comment: 15 pages, 2 figure

Hadronic Higgs boson decay at order αs4\alpha_s^4 and αs5\alpha_s^5

We compute corrections to the decay of the Standard Model Higgs boson to hadrons, to the fourth order in the strong coupling constant $\alpha_s$. We use an effective theory in which the Higgs boson couples directly to bottom quarks and to gluons, via top quark--mediated effective couplings. Numerically, our results are of a comparable size to the previously-known "massless" contributions and complete the order $\alpha_s^4$ corrections to the hadronic decay of the Higgs boson. In these proceedings we also provide an independent cross check of the gluonic Higgs boson decay at order $\alpha_s^5$.Comment: 6 pages, 0 figures. Contribution to the proceedings of the XXV International Workshop on Deep-Inelastic Scattering and Related Topics (DIS2017), 3-7 April 2017, University of Birmingham, UK. V2: Reference adde

Real-virtual corrections to Higgs boson pair production at NNLO: three closed top quark loops

We compute the real-radiation corrections to Higgs boson pair production at next-to-next-to-leading order in QCD, in an expansion for large top quark mass. We concentrate on the radiative corrections to the interference contribution from the next-to-leading order one-particle reducible and the leading order amplitudes. This is a well defined and gauge invariant subset of the full real-virtual corrections to the inclusive cross section. We obtain analytic results for all phase-space master integrals both as an expansion around the threshold and in an exact manner in terms of Goncharov polylogarithms.Comment: 25 page

Distribution of permutation statistics across pattern avoidance classes, and the search for a Denert-associated condition equivalent to pattern avoidance

We begin with a discussion of the symmetricity of \maj over \des in pattern avoidance classes, and its relationship to \maj-Wilf equivalence. From this, we explore the distribution of permutation statistics across pattern avoidance for patterns of length 3 and 4. We then begin discussion of Han\u27s bijection, a bijection on permutations which sends the major index to Denert\u27s statistic and the descent number to the (strong) excedance number. We show the existence of several infinite families of fixed points for Han\u27s bijection. Finally, we discuss the image of pattern avoidance classes under Han\u27s bijection, for the purpose of finding a condition which has the same distribution of \den over \exc as pattern avoidance does of \maj over \des

Robust Factorizations and Colorings of Tensor Graphs

Since the seminal result of Karger, Motwani, and Sudan, algorithms for approximate 3-coloring have primarily centered around SDP-based rounding. However, it is likely that important combinatorial or algebraic insights are needed in order to break the $n^{o(1)}$ threshold. One way to develop new understanding in graph coloring is to study special subclasses of graphs. For instance, Blum studied the 3-coloring of random graphs, and Arora and Ge studied the 3-coloring of graphs with low threshold-rank. In this work, we study graphs which arise from a tensor product, which appear to be novel instances of the 3-coloring problem. We consider graphs of the form $H = (V,E)$ with $V =V( K_3 \times G)$ and $E = E(K_3 \times G) \setminus E'$, where $E' \subseteq E(K_3 \times G)$ is any edge set such that no vertex has more than an $\epsilon$ fraction of its edges in $E'$. We show that one can construct $\widetilde{H} = K_3 \times \widetilde{G}$ with $V(\widetilde{H}) = V(H)$ that is close to $H$. For arbitrary $G$, $\widetilde{H}$ satisfies $|E(H) \Delta E(\widetilde{H})| \leq O(\epsilon|E(H)|)$. Additionally when $G$ is a mild expander, we provide a 3-coloring for $H$ in polynomial time. These results partially generalize an exact tensor factorization algorithm of Imrich. On the other hand, without any assumptions on $G$, we show that it is NP-hard to 3-color $H$.Comment: 34 pages, 3 figure

Higgs boson decay into photons at four loops

Future precision measurements of Higgs boson decays will determine the branching fraction for the decay into two photons with a precision at the one percent level. To fully exploit such measurements, equally precise theoretical predictions need to be available. To this end we compute four-loop QCD corrections in the large top quark mass expansion to the Higgs boson–photon form factor, which enter the two-photon decay width at next-to-next-to-next-to-leading order. Furthermore we obtain corrections to the two-photon decay width stemming from the emission of additional gluons, which contribute for the first time at next-to-next-to-leading order. Finally, we combine our results with other available perturbative corrections and estimate the residual uncertainty due to missing higher-order contributions

Housing and Climate Change in the Nigerian Built Environment

The effect of housing and climate change on the Nigerian Environment was assessed employing secondary sources of data.  It revealed among other things that climate change is due to natural force and anthropogenic activities of man especially in the course of building his house.  This alters the Natural equilibrium of the eco-system.  The energy mostly used today in houses is produced from burning of fossil fuels which emits greenhouse gases (GHGs) into the atmosphere.  These cause global warming and by extension climate change.  The paper recommended adaptation and mitigation strategies through sustainable architecture as panacea. It concluded that climate will continue to change; however, man must only tap the natural resources for his survival in a responsible and sustainable manner. Keywords: housing, climate change, built environment, adaptation, mitigation