Gram Momentum Invariants | M.E. Irizarry-Gelpí

Thu 28 August 2014
Physics
#kinematics

In this post I will mention some facts about what I call Gram momentum invariants. I will assume that every energy-momentum vector $p_{I}$ satisfies an on-shell constraint of the form

$$p_{I}^{2} = - m_{I}^{2}$$

with the mass $m_{I}$ being real and positive.

Introduction

In a previous post I introduced the Mandelstam momentum invariants. You can construct $n$-Mandelstam invariants with $n$ distinct energy-momentum vectors. With a similar collection of $n$ distinct energy-momentum vectors, you can also construct an $n$-Gram invariant:

$$G_{I_{1} I_{2} \cdots I_{n}} = -\det{ \begin{pmatrix} p_{I_{1}} \cdot p_{I_{1}} & \cdots & p_{I_{1}} \cdot p_{I_{n}} \\ \vdots & \ddots & \vdots \\ p_{I_{1}} \cdot p_{I_{n}} & \cdots & p_{I_{n}} \cdot p_{I_{n}} \end{pmatrix} }.$$

That is, an $n$-Gram invariant correspond to the Gram determinant associated with a collection of $n$ energy-momentum vectors. Hence, by construction, the Gram momentum invariants are sensitive to collinearity of the external energy-momentum vectors (i.e. they vanish whenever a pair of energy-momentum vectors are collinear).

Two Quanta

Given two linearly-independent energy-momentum vectors, the 2-Gram invariant is

$$G_{IJ} \equiv -\det{ \begin{pmatrix} p_{I}^{2} & p_{I} \cdot p_{J} \\ p_{I} \cdot p_{J} & p_{J}^{2} \end{pmatrix} } = -p_{I}^{2} p_{J}^{2} + (p_{I} \cdot p_{J})^{2}.$$

Using

$$p_{I}^{2} = - m_{I}^{2} \qquad (p_{I} + p_{J})^{2} = -s_{IJ} \qquad p_{J}^{2} = - m_{J}^{2}$$

we can write

$$p_{I} \cdot p_{J} = \frac{m_{I}^{2} + m_{J}^{2} - s_{IJ}}{2}$$

This allows you to write the 2-Gram invariant in terms of masses and 2-Mandelstam invariants:

$$G_{IJ} = \frac{1}{4} [s_{IJ} - (m_{I} - m_{J})^{2}] [s_{IJ} - (m_{I} + m_{J})^{2}].$$

In this form, the sign properties of a 2-Gram invariant are obvious. You have $G_{IJ}$ being positive whenever $s_{IJ} < (m_{I} - m_{J})^{2}$ or $s_{IJ} > (m_{I} + m_{J})^{2}$. Part of this region lies above the 2-threshold; the another part lies below the 2-pseudothreshold. On the other hand, you have $G_{IJ}$ being negative whenever $(m_{I} - m_{J})^{2} < s_{IJ} < (m_{I} + m_{J})^{2}$. All of this region lies below the 2-threshold and above the 2-pseudothreshold.

Expanding the expression for $G_{IJ}$ you find a surprisingly symmetric expression:

$$G_{IJ} = \frac{1}{4} \left( s^{2} + m_{I}^{4} + m_{J}^{4} - 2 m_{I}^{2} s - 2 m_{J}^{2} s - 2 m_{I}^{2} m_{J}^{2} \right). $$

This can be rewritten in terms of a symmetric multivariate polynomial:

$$G_{IJ} = \frac{1}{4}\Upsilon_{2}(s_{IJ}, m_{I}^{2}, m_{J}^{2}), \qquad \Upsilon_{2}(x, y, z) = x^{2} + y^{2} + z^{2} - 2 xy - 2 xz - 2 yz .$$

The function $\Upsilon_{2}$ is a Källén polynomial.

Three Quanta

Given three linearly-independent energy-momentum vectors, the 3-Gram invariant is

$$G_{IJK} \equiv -\det{ \begin{pmatrix} p_{I}^{2} & p_{I} \cdot p_{J} & p_{I} \cdot p_{K} \\ p_{I} \cdot p_{J} & p_{J}^{2} & p_{J} \cdot p_{K} \\ p_{I} \cdot p_{K} & p_{J} \cdot p_{K} & p_{K}^{2} \end{pmatrix} } = -p_{I}^{2} p_{J}^{2} p_{K}^{2} + p_{I}^{2} (p_{J} \cdot p_{K})^{2} + p_{J}^{2} (p_{I} \cdot p_{K})^{2} + p_{K}^{2} (p_{I} \cdot p_{J})^{2} - 2 (p_{I} \cdot p_{J})(p_{I} \cdot p_{K})(p_{J} \cdot p_{K}) .$$

In general, this cannot be simplified further in terms of masses and Mandelstam invariants.