Bootstrapping an NMHV amplitude through three loops

We extend the hexagon function bootstrap to the next-to-maximally-helicity-violating (NMHV) configuration for six-point scattering in planar ${\cal N}=4$ super-Yang-Mills theory at three loops. Constraints from the $\bar{Q}$ differential equation, from the operator product expansion (OPE) for Wilson loops with operator insertions, and from multi-Regge factorization, lead to a unique answer for the three-loop ratio function. The three-loop result also predicts additional terms in the OPE expansion, as well as the behavior of NMHV amplitudes in the multi-Regge limit at one higher logarithmic accuracy (NNLL) than was used as input. Both predictions are in agreement with recent results from the flux-tube approach. We also study the multi-particle factorization of multi-loop amplitudes for the first time. We find that the function controlling this factorization is purely logarithmic through three loops. We show that a function $U$, which is closely related to the parity-even part of the ratio function $V$, is remarkably simple; only five of the nine possible final entries in its symbol are non-vanishing. We study the analytic and numerical behavior of both the parity-even and parity-odd parts of the ratio function on simple lines traversing the space of cross ratios $(u,v,w)$, as well as on a few two-dimensional planes. Finally, we present an empirical formula for $V$ in terms of elements of the coproduct of the six-gluon MHV remainder function $R_6$ at one higher loop, which works through three loops for $V$ (four loops for $R_6$).Comment: 69 pages, 12 figures, 1 table, 3 ancillary files; v2, minor typo's correcte

Long‐Term Weather Trends and Implications for Grazing Operations in the Mid‐South

Weather plays a large role in determining our agricultural production and management-related decisions throughout the growing season. In a couple examples, it controls when we can cut hay or when we are able/not able to graze (drought 2019). The short‐term shift in the weather pattern is always on the back of any farmers mind and how it’ll impact their own operation, but what about long‐term changes? These can be just as important. This can be tracked by looking at our climate, which is the average weather conditions over a specific period of time (usually 30 years). Unlike weather, which tells us what we will get, climate shows us what we can expect at any point throughout the year. Climate tells us what we can and cannot grow in the Bluegrass State, when we can plant, and long‐term sustainability

Bootstrapping a Five-Loop Amplitude Using Steinmann Relations

The analytic structure of scattering amplitudes is restricted by Steinmann relations, which enforce the vanishing of certain discontinuities of discontinuities. We show that these relations dramatically simplify the function space for the hexagon function bootstrap in planar maximally supersymmetric Yang-Mills theory. Armed with this simplification, along with the constraints of dual conformal symmetry and Regge exponentiation, we obtain the complete five-loop six-particle amplitude.Comment: 5 pages, 2 figures, 1 impressive table, and 2 ancillary files. v2: a few clarifications and references added; version to appear in PR

Bootstrapping six-gluon scattering in planar N=4{\cal N}=4 super-Yang-Mills theory

We describe the hexagon function bootstrap for solving for six-gluon scattering amplitudes in the large $N_c$ limit of ${\cal N}=4$ super-Yang-Mills theory. In this method, an ansatz for the finite part of these amplitudes is constrained at the level of amplitudes, not integrands, using boundary information. In the near-collinear limit, the dual picture of the amplitudes as Wilson loops leads to an operator product expansion which has been solved using integrability by Basso, Sever and Vieira. Factorization of the amplitudes in the multi-Regge limit provides additional boundary data. This bootstrap has been applied successfully through four loops for the maximally helicity violating (MHV) configuration of gluon helicities, and through three loops for the non-MHV case.Comment: 15 pages, 3 figures, 2 tables; contribution to the proceedings of Loops and Legs in Quantum Field Theory, 27 April - 2 May 2014, Weimar, Germany; v2, reference adde

The Double Pentaladder Integral to All Orders

We compute dual-conformally invariant ladder integrals that are capped off by pentagons at each end of the ladder. Such integrals appear in six-point amplitudes in planar N=4 super-Yang-Mills theory. We provide exact, finite-coupling formulas for the basic double pentaladder integrals as a single Mellin integral over hypergeometric functions. For particular choices of the dual conformal cross ratios, we can evaluate the integral at weak coupling to high loop orders in terms of multiple polylogarithms. We argue that the integrals are exponentially suppressed at strong coupling. We describe the space of functions that contains all such double pentaladder integrals and their derivatives, or coproducts. This space, a prototype for the space of Steinmann hexagon functions, has a simple algebraic structure, which we elucidate by considering a particular discontinuity of the functions that localizes the Mellin integral and collapses the relevant symbol alphabet. This function space is endowed with a coaction, both perturbatively and at finite coupling, which mixes the independent solutions of the hypergeometric differential equation and constructively realizes a coaction principle of the type believed to hold in the full Steinmann hexagon function space.Comment: 70 pages, 3 figures, 4 tables; v2, minor typo corrections and clarification

Coil On Plug Ignition: The Wired Differences

The dominant method of providing spark on late model gasoline engines is coil on plug (C.O.P.) ignition systems. Looking underhood on different vehicle makes will reveal that some coils have 2 wires, some have 3 and others have 4. The presentation examines how each of these coils is triggered and what tools and test methods are best. Several scope images and component pictures are included. Also consider viewing the July 2011 article on this topic I wrote for Underhood Service

Variable Camshaft Timing/ Variable Valve Timing

Variable camshaft timing (VCT) and variable valve timing (VVT) have become common strategies used by many automotive manufacturers to increase engine performance, reduce emissions, and improve economy and driveability. This presentation was conducted at the fall 2012 ICAIA conference in Joliet, IL on October 5th, 2012. The two most popular methods include the use of a phaser to alter camshaft position and the use of multiple camshaft lobes with rocker arm coupling and de-coupling (both phaser and lobe switching methods can also be combined). VCT/VVT use can be employed on pushrod engines, DOHC, SOHC engines and can be utilized on just intake valves, just exhaust valves or both intake and exhaust valves. A review of mechanical components, hydraulic controls, and electrical controls is provided along with observations regarding operation. In addition, basic diagnostic trouble codes (DTC\u27s) and troubleshooting points are discussed

Driveability: What is normal?

Like experts in other professions, automotive technicians must sift through a plethora of data to determine course of action. The data itself is meaningless without understanding of what the numbers mean. This fall 2009 ICAIA presentation examines 5 different vehicles across 5 different operating conditions measured with 5 different scan tools in an investigation of normal parameters