Many-Body Sudakov Decompositions | M.E. Irizarry-Gelpí

In spacetime you can have null energy-momentum vectors, with vanishing Minkowski norm,

$$ \left\Vert p \right\Vert^{2} \equiv -E^{2} + \left\vert \mathbf{p} \right\vert^{2} = 0; $$

and non-null energy-momentum vectors. The non-null momenta can either be slow,

$$ \left\Vert p \right\Vert^{2} \equiv -E^{2} + \left\vert \mathbf{p} \right\vert^{2} = - m^{2}, \qquad m > 0; $$

or fast,

$$ \left\Vert p \right\Vert^{2} \equiv -E^{2} + \left\vert \mathbf{p} \right\vert^{2} = w^{2}, \qquad w > 0. $$

Massive quanta carry slow momenta; tachyons carry fast momenta. It is possible, however, to use null momenta to describe non-null momenta. If $a$ and $b$ are null vectors with $a \cdot b \neq 0$, then a non-null vector $p$ can be written as

$$ p = a + \frac{\left\Vert p \right\Vert^{2}}{2 \left(a \cdot b \right)} b. $$

Here $a$ and $b$ are called Sudakov vectors and this expression for $p$ is called a Sudakov (null) decomposition. As $\left\Vert p \right\Vert^{2}$ approaches zero, the non-null vector $p$ approaches the null vector $a$, so the null limit is well-defined. Note that the "content" of $p$ is carried by both $a$ and $b$.

In many-body scattering processes you typically have $M + N$ momenta with $M$ of these being non-null and $N$ being null. Naively, you can introduce two null vectors for each non-null vector and end with a total of $2M + N$ null vectors. This seems like too much. Alternatively, you can "recycle" some of the null vectors once such that the total number of null vectors is at most $M + N$. The process of taking $M$ non-null vectors and introducing $M$ null vectors to describe the $M$ non-null vectors is called a Sudakov $M$-decomposition.

In this page you will find my notes on many-body Sudakov decompositions.

Two-Body

The simplest example is the Sudakov 2-decomposition:

The Sudakov 2-decomposition can be used in any scattering process with more than two external quanta:

Three-Body

With three non-null momenta you have two options. You can perform a contiguous Sudakov 3-decomposition:

Or you can perform a recursive Sudakov 3-decomposition:

The difference between these two Sudakov 3-decompositions is that the recursive one involves composite momenta. I will mostly study the contiguous case. The Sudakov 3-decomposition can be used in any scattering process with more than two external quanta:

Three-Point Sudakov 3-Decomposition

Four-Body

Things get very complicated with four non-null momenta:

Four-Body Sudakov Decompositions

The Sudakov 4-decomposition can be used in any scattering process with more than three external quanta:

Four-Point Non-Contiguous Sudakov 4-Decomposition