Invariant mass

10.2 Invariant mass

One of the key numbers we can extract from mass and momentum is the invariant mass, a number independent of the Lorentz frame we are in

W^{2} c^{4} = {(\sum_{i} E_{i})}^{2} - {(\sum_{i} p_{i})}^{2} c^{2} .

(10.5)

This quantity takes it most transparent form in the centre-of-mass, where $\sum_{i} p_{i} = 0$ . In that case

W = E_{C M} ∕ c^{2},

(10.6)

and is thus, apart from the factor $1 ∕ c^{2}$ , nothing but the energy in the CM frame. For a single particle $W = m_{0}$ , the rest mass.

Most considerations about processes in high energy physics are greatly simpliﬁed by concentrating on the invariant mass. This removes the Lorentz-frame dependence of writing four momenta. I

As an example we look at the collision of a proton and an antiproton at rest, where we produce two quanta of electromagnetic radiation ( $γ$ ’s), see ﬁg. 10.1, where the anti proton has three-momentum $(p, 0, 0)$ , and the proton is at rest.

kinppbar

Figure 10.1: A sketch of a collision between a proton with velocity

v

and an antiproton at rest producing two

g a m m a

quanta.

The four-momenta are

\begin{array}{rcl} p_{p} & = & (p_{l a b}, 0, 0, \sqrt{m_{p}^{2} c^{4} + p_{l a b}^{2} c^{2}}) \\ p_{\bar{p}} & = & (0, 0, 0, m_{p} c^{2}) . & (10.7) \end{array}

From this we ﬁnd the invariant mass

W = \sqrt{2 m_{p}^{2} + 2 m_{p} \sqrt{m_{p}^{2} + p_{l a b}^{2} ∕ c^{2}}}

(10.8)

If the initial momentum is much larger than $m_{p}$ , more accurately

p_{l a b} ≫ m_{p} c,

(10.9)

we ﬁnd that

W \approx \sqrt{2 m_{p} p_{l a b} ∕ c},

(10.10)

which energy needs to be shared between the two photons, in equal parts. We could also have chosen to work in the CM frame, where the calculations get a lot easier.

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