Q27 (c)

The simplified model below shows the reactants and products of a proton–antiproton reaction which produces three particles called pions, each having a different charge.

$$p+\bar{p}\rightarrow\pi^+ + \pi^0 + \pi^-$$

There are no other products in this process, which involves only the rearrangement of quarks. No electromagnetic radiation is produced. Assume that the initial kinetic energy of the proton and antiproton is negligible.

Protons consist of two up quarks ($u$) and a down quark ($d$). Antiprotons consist of two up antiquarks ($\bar{u}$) and a down antiquark ($\bar{d}$). Each of the pions consists of two quarks.

The following tables provide information about hadrons and quarks.

Table 1: Hadron information

Particle	Rest mass (MeV/$c^2$)	Charge
proton ($p$)	940	$+1$
antiproton ($\bar{p}$)	940	$-1$
neutral pion ($\pi^0$)	140	zero
positive pion ($\pi^+$)	140	$+1$
negative pion ($\pi^-$)	140	$-1$

Table 2: Quark charges

Particle	Charge
down quark ($d$)	$-\frac{1}{3}$
up quark ($u$)	$+\frac{2}{3}$
down antiquark ($\bar{d}$)	$+\frac{1}{3}$
up antiquark ($\bar{u}$)	$-\frac{2}{3}$

(c) Calculation of the pions’ velocities using classical physics predicts that each pion has a velocity, relative to the point at which the proton–antiproton reaction occurred, which exceeds $3 \times 10^8\ \mathrm{m\,s^{-1}}$.

Explain the problem with this prediction and how it can be resolved.