Worked exercise — The Higgs boson at 0.8c

Problem

The Higgs boson has rest mass $m = 125\ \text{GeV}/c^2$ (natural units used in particle physics). Suppose that, after production in an accelerator, it travels at $v = 0{,}8\,c$. Calculate (a) $\gamma$, (b) total energy $E$, (c) kinetic energy $E_k$, (d) momentum $p$, (e) de Broglie wavelength.

Solution

(a) Lorentz factor. $\gamma = \frac{1}{\sqrt{1-0{,}64}} = \frac{1}{\sqrt{0{,}36}} = \frac{5}{3} \approx 1{,}667$

(b) Total energy. In natural units $mc^2 = 125$ GeV, so $E = \gamma\,mc^2 = \frac{5}{3}\cdot 125\ \text{GeV} = \ev{E \approx 208{,}3\ \text{GeV}}$

(c) Kinetic energy. $E_k = (\gamma - 1)\,mc^2 = \left(\frac{5}{3}-1\right)\cdot 125\ \text{GeV} = \frac{2}{3}\cdot 125\ \text{GeV} \approx \ev{E_k \approx 83{,}3\ \text{GeV}}$

(d) Momentum. $p = \gamma\,mv = \frac{5}{3}\cdot 125\ \frac{\text{GeV}}{c^2}\cdot 0{,}8\,c = \frac{5}{3}\cdot 100\ \frac{\text{GeV}}{c} \approx \ev{p \approx 166{,}7\ \text{GeV}/c}$

(e) De Broglie wavelength. We have $\lambda = h/p$. Converting the momentum to SI units, $p \approx 167\ \text{GeV}/c \approx 8{,}9\cdot 10^{-17}$ kg·m/s, so $\lambda = \frac{h}{p} = \frac{6{,}63\cdot 10^{-34}}{8{,}9\cdot 10^{-17}} \approx \ev{\lambda \approx 7{,}5\cdot 10^{-18}\ \text{m}}$

Interpretation. This wavelength is roughly $10\,000$ times smaller than the proton radius: the Higgs, accelerated this way, is an extremely localised object.

Links

Topics: Special relativity Concepts: Mass-energy equivalence · Relativistic momentum Skills: Symbolic setup