references.qmd

---
title: "References and useful material"
---
\
This page collects publications detailing our effort towards the ultimate goal of efficiently computing perturbative cross-sections at arbitrary orders. Needless to say that our work builds upon that of many others whose references can be found in our citations.  

### Loop-Tree Duality
\
Our journey started with a first publication summarizes the muli-loop generalisation of Loop-Tree Duality (LTD)

[Loop-Tree Duality for Multiloop Numerical Integration](https://arxiv.org/pdf/1906.06138.pdf)

It can be used for the numerical evaluation of finite loop integrals, also for physical kinematics when using a contour deformation discussed in this following publication

[Numerical Loop-Tree Duality: contour deformation and subtraction](https://arxiv.org/pdf/1912.09291.pdf)

The original expression of Loop-Tree Duality suffers from spurious singularities that can be algorithmically removed so as to make it *manifestly causal* (cLTD)

[Manifestly Causal Loop-Tree Duality](https://arxiv.org/pdf/2009.05509.pdf)

An alternative approach to regularising theshold singularities at one-loop was also worked out by Dario Kermanschah 

[Numerical integration of loop integrals through local cancellation of threshold singularities](https://arxiv.org/pdf/2110.06869.pdf)

Finally, Zeno Capatti discovered an even more elegant cross-Free Family (cFF) formulation of LTD that is more compact and only retains physical thresholds

[Exposing the threshold structure of loop integrals](https://arxiv.org/pdf/2211.09653.pdf)

All these equivalent formulations LTD, cLTD and cFF can be readily generated from a Mathematica package available at

[Mathematica package for generating LTD expressions](https://github.com/apelloni/cLTD)

### Local Unitarity
\
Equipped with the LTD expressions discussed above, we worked out the original construction of Local Unitarity (LU) is described in detailed at

[Local Unitarity: a representation of differential cross-sections that is locally free of infrared singularities at any order](https://arxiv.org/pdf/2010.01068.pdf)

It is then further refined to detail the handling of self-energy corrections and renormalisation at arbitrary perturbative orders in

[Local unitarity: cutting raised propagators and localising renormalisation](https://arxiv.org/pdf/2203.11038.pdf)

### Miscellaneous
\
A brilliant semester student, Max Hofer, computed the forward-backward asymmetry at NLO for the process $e^+ e^- \rightarrow Q \overline{Q}$ using Local Unitarity.
His report offers a good first introduction to our framework:

[Computing electroweak quantum corrections using Local Unitarity](./resources/MaxHofer_AFB.pdf)

A standalone Mathematica notebook playground allows you to explore an explicit implementation of Local Unitarity for the computation of the inclusive NLO correction to the process $e^+ e^- \rightarrow \gamma \rightarrow d\bar{d}$ yielding the infamous $\alpha_s / \pi$ K-factor

[A Mathematica playground for computing a simple inclusive NLO correction](./resources/LU_epem_a_ddx_inclusive.nb)