Collider physics at the precision frontier

Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman integrals. With this application, the practical and efficient computation of intersection numbers of twisted cocycles becomes a topic of interest. An existing algorithm for the computation of intersection numbers of twisted cocycles requires in intermediate steps the introduction of algebraic extensions (for example square roots), although the final result may be expressed without algebraic extensions. In this article I present an improvement of this algorithm, which avoids algebraic extensions.

On the computation of intersection numbers for twisted cocycles

We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers, and provide three different approaches for a direct decomposition of Feynman integrals, which we dub the straight decomposition, the bottom-up decomposition, and the top-down decomposition. These algorithms exploit the unitarity structure of Feynman integrals by computing intersection numbers supported on cuts, in various orders, thus showing the synthesis of the intersection-theory concepts with unitarity-based methods and integrand decomposition. We perform explicit computations to exemplify all of these approaches applied to Feynman integrals, paving a way towards potential applications to generic multi-loop integrals.

/pdf/decomposition-of-feynman-integrals-by-multivariate-5c1pqo1wl8.pdf

Decomposition of Feynman integrals by multivariate intersection numbers

/pdf/the-three-loop-form-factor-in-n-4-super-yang-mills-1totgx1kiw.pdf

The three-loop form factor in N=4 super Yang-Mills

: We review recent progress and future prospects for harnessing powerful tools from theoretical high-energy physics, such as scattering amplitudes and eﬀective ﬁeld theory, to develop a precise and systematically improvable framework for calculating gravitational-wave signals from binary systems composed of black holes and/or neutron stars. This eﬀort aims to provide state-of-the-art predictions that will enable high-precision measurements at future gravitational-wave detectors. In turn, applying the tools of quantum ﬁeld theory in this new arena will uncover theoretical structures that can transform our understanding of basic phenomena and lead to new tools that will further the cycle of innovation. While still in a nascent stage, this research direction has already derived new analytic results in general relativity, and promises to advance the development of highly accurate waveform models for ever more sensitive detectors.

Snowmass White Paper: Gravitational Waves and Scattering Amplitudes

Constructing canonical Feynman integrals with intersection theory

https://link.springer.com/content/pdf/10.1007/JHEP07(2022)066.pdf

Baikov representations, intersection theory, and canonical Feynman integrals

In this paper, we explore the recursive structure of Baikov representations for Feynman integrals. We demonstrate that the various Baikov representations for all sectors of an integral family can be organized in a tree-like structure. Using this structure, we show that the symbol letters of one-loop Feynman integrals can be written in terms of minors of a matrix associated with the top sector. Non-trivial relations among these symbol letters can then be easily discovered using results from linear algebra.

The recursive structure of Baikov representations I: generics and application to symbology

The brief presents an efficient resonant pole inverter for the sake of the efficiency improvement of the inverter. Its design process is elaborated briefly as follows: the same auxiliary units are added in parallel with three bridge arms and auxiliary units on each bridge arm only consist of one auxiliary switch, two auxiliary diodes, two resonant capacitors and two coupled resonant inductors; when the auxiliary units work, the voltage across main switches can decrease to zero periodically before main switches are switched on. Thus, main switches can acquire zero-voltage switching, making for the reduction of switching loss. In particular, the notable advantage of the presented inverter is that only one switch is included in the auxiliary units of each phase, which is beneficial to the simplification of the control strategy and the restriction on the hardware cost. Moreover, the single auxiliary switch of each phase can also acquire zero-current switching. The brief illustrates each working state in a switching period. The relative experimental waveform indicates that switching devices attain soft-switching. Furthermore, the efficiency of the presented inverter achieves 98% at rated output power, increasing by 0.3%, compared with that in the latest literature. Consequently, the presented inverter is more advantageous in efficiency than other similar soft-switching inverters. 

An Efficient Resonant Pole Inverter With One Single Auxiliary Switch of Each Phase for AC Motor Drive

An efficient resonant pole inverter is presented in this brief for the efficiency optimization. Main switches and auxiliary switches can acquire zero-voltage switching and zero-current switching, respectively, achieving the remarkable decline of switching loss. In especial, no constant circulating current flows through resonant inductors after the voltage across resonant capacitors parallel with main switches stops changing in a switching cycle, eliminating constant circulating current loss of resonant inductors in the novel inverter. Each working state in a switching period is illustrated in this brief. The relative experimental waveform implies that switching devices attain soft-switching. Furthermore, the efficiency of the novel inverter achieves 99.0% at rated output power. Thus, the novel inverter can have efficiency superiority over other resonant pole inverters.

Xuhang Jiang

Papers

Constructing canonical Feynman integrals with intersection theory

Baikov representations, intersection theory, and canonical Feynman integrals

The recursive structure of Baikov representations I: generics and application to symbology

An Efficient Resonant Pole Inverter With One Single Auxiliary Switch of Each Phase for AC Motor Drive

An Efficient Resonant Pole Inverter Without Constant Circulating Current Loss in the Auxiliary Circuit