# Collinear limit The collinear limit is an example of [[0295 Infrared divergences in scattering amplitude|infrared divergences in scattering amplitude]]. Two external particles are collinear if their 4-momenta are proportional to each other, i.e., they are aligned. When this happens, the amplitude becomes divergent. Interestingly, for many theories, this divergence is universal, and this universality has played an important role in [[0010 Celestial holography|celestial holography]], where the collinear limit is dual to [[0114 Celestial OPE|celestial OPE]]. ## Refs - for gluons, reviewed in chapter 8 of [Taylor notes](https://arxiv.org/pdf/1703.05670.pdf) - for gravitons - first derived in [[1998#Bern, Dixon, Perelstein, Rozowsky (Nov)]] - further developments in [[White2011]] and [[AkhourySaotomeSterman2011]] ## What happens - in the collinear limit, an $(n+1)$ point function factorises into a $n$-point function times a **universal splitting function** - [[0079 Mellin transform|Mellin transforming]] this splitting function, one obtains the leading term in the [[0114 Celestial OPE|celestial OPE]] ## External v.s. internal - when a particle (e.g. photon) connects to an external leg, a collinear singularity appears - when a particle connects to an internal leg, can check that no singularity appears ## Subleading collinear - [[NandanPlefkaWormsbecher2016]] ## Loop corrections - graviton - no loop correction [[1998#Bern, Dixon, Perelstein, Rozowsky (Nov)]] - gauge theory - correction at every loop order [[Kosower1999]] and [[FeigeSchwartz2014]] - more refs - [[1993#Bern, Chalmers, Dixon, Kosower]] - [[BernDixonDunbarKosower1994]] - [[BernDixonKosower1994]][](https://arxiv.org/abs/hep-ph/9409393) - [[BernChalmers1995]] ## See also - [[0077 Multi-collinear limit]]