M.E. Irizarry-Gelpí

M.E. Irizarry-Gelpí

Physics impostor. Mathematics interloper. Husband. Father.

Threshold and Pseudothreshold Values

In this post I will collect some results related to threshold and pseudothreshold values. I will assume that all masses are real and positive, and that two masses with different labels are always distinct.

Introduction

When working with massive quanta, you find that particular linear combinations of the masses appear quite often. Given \(n + 1\) distinct masses, you can construct a threshold value and \(2^{n} - 1\) pseudothreshold values. The threshold value is always given by the square of the sum of the masses, while the pseudothrehold values are found by taking the square of a combination of sums and differences of masses. Just like for Mandelstam invariants, I will refer to these values as the \((n + 1)\)-threshold and \((n + 1)\)-pseudothreshold values. As I will show below, taking the average over the \((n + 1)\)-threshold and \((n + 1)\)-pseudothreshold values gives the sum of squares of the \(n + 1\) masses.

Two Masses

Given two distinct masses, the 2-threshold value is

$$(m_{I} + m_{J})^{2}$$

and the 2-pseudothreshold value is

$$(m_{I} - m_{J})^{2}.$$

Note that

$$\frac{(m_{I} + m_{J})^{2} + (m_{I} - m_{J})^{2}}{2} = m_{I}^{2} + m_{J}^{2}.$$

Three Masses

Given three distinct masses, the 3-threshold value is

$$(m_{I} + m_{J} + m_{K})^{2}$$

and the three 3-pseudothreshold values are

$$(m_{I} - m_{J} - m_{K})^{2} \qquad (m_{I} - m_{J} + m_{K})^{2} \qquad (m_{I} + m_{J} - m_{K})^{2}.$$

Note that

$$\frac{(m_{I} + m_{J} + m_{K})^{2} + (m_{I} - m_{J} - m_{K})^{2} + (m_{I} - m_{J} + m_{K})^{2} + (m_{I} + m_{J} - m_{K})^{2}}{4} = m_{I}^{2} + m_{J}^{2} + m_{K}^{2}.$$

Four Masses

Given four distinct masses, the 4-threshold value is

$$(m_{I} + m_{J} + m_{K} + m_{L})^{2}$$

and the seven 4-pseudothreshold values are

$$(m_{I} - m_{J} - m_{K} - m_{L})^{2} \qquad (m_{I} - m_{J} + m_{K} + m_{L})^{2} \qquad (m_{I} + m_{J} - m_{K} + m_{L})^{2} \qquad (m_{I} + m_{J} + m_{K} - m_{L})^{2}$$

and

$$(m_{I} + m_{J} - m_{K} - m_{L})^{2} \qquad (m_{I} - m_{J} + m_{K} - m_{L})^{2} \qquad (m_{I} - m_{J} - m_{K} + m_{L})^{2}$$

Note that

$$\frac{(m_{I} + m_{J} + m_{K} + m_{L})^{2} + (m_{I} - m_{J} - m_{K} - m_{L})^{2} + \ldots + (m_{I} + m_{J} - m_{K} - m_{L})^{2} + \ldots}{8} = m_{I}^{2} + m_{J}^{2} + m_{K}^{2} + m_{L}^{2}.$$