1. 2012
Nov
24

Sudakov parameters

Sudakov parameters are a common mathematical tool in the realm of high-energy physics — so I was surprised not to find a web page describing them at the top of a list of Google search results.

$$k_1^\mu = \frac{1}{\sqrt{2}}(K, K, 0, 0)$$
$$k_2^\mu = \frac{1}{\sqrt{2}}\biggl(\frac{1}{K}, -\frac{1}{K}, 0, 0\biggr)$$

Since these are both null vectors, $$k_1^2 = k_2^2 = 0$$, and they also satisfy $$k_1\cdot k_2 = 1$$. They form two components of a complete basis. Any arbitrary four-vector can thus be parametrized as

$$A^\mu = \alpha k_1^\mu + \beta k_2^\mu + A_\perp^\mu$$

where $$A_\perp^\mu$$ is a four-vector that contains only two nonzero components, the ones orthogonal to $$k_1^\mu$$ and $$k_2^\mu$$. This is the Sudakov parametrization of the vector.

The Sudakov parametrization is useful because for an object moving at or near the speed of light, if the directions of the coordinate axes are appropriately chosen, only one of $$\alpha$$ or $$\beta$$ will be large, and the other will be nearly zero, as will both components of $$A_\perp^\mu$$.

Source: appendix B of Transverse spin physics by Vincenzo Barone and Philip …