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Je m'appelle
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Homework Statement
Suppose a proton and an antiproton collide producing a pair of top-antitop quarks.
What would be the minimum required momenta of both proton and antiproton in order for this pair creation to occur?
[tex]p + \bar{p} \rightarrow t + \bar{t} [/tex]
(Answer: [itex]173 \frac{GeV}{c^2}, 59.9 \frac{TeV}{c^2} [/itex])
Homework Equations
I. Energy-momentum relation
[tex]E^2 = (pc)^2 + (m_0 c^2)^2 [/tex]
II. Relativistic kinetic energy equation
[tex]E_{ki} = m_i c^2 \left(\frac{1}{\sqrt{1-\frac{v_i^2}{c^2}}} -1 \right) [/tex]
III. Relativistic momentum equation
[tex]p_i = \frac{m_i v_i}{\sqrt{1-\frac{v_i^2}{c^2}}} [/tex]
IV. Rest masses of proton and top quark
[tex] m_p = 938 \frac{MeV}{c^2}, m_t = 173 \frac{GeV}{c^2}[/tex]
The Attempt at a Solution
1. Conservation of energy:
[tex]E_i = E_f \Rightarrow E_{p} + E_{\bar{p}} = E_{t} + E_{\bar{t}} [/tex]
[tex]\sqrt{ (p_{p}c)^2 + (m_p c^2)^2 } + \sqrt{ (p_{\bar{p}}c)^2 + (m_{\bar{p}} c^2)^2 } = \sqrt{ (p_{t}c)^2 + (m_t c^2)^2 } + \sqrt{ (p_{\bar{t}}c)^2 + (m_{\bar{t}} c^2)^2 } [/tex]
2. Conservation of momentum:
[tex]p_i = p_f \Rightarrow p_{p} + p_{\bar{p}} = p_{t} + p_{\bar{t}} [/tex]
I'm really stuck here because I feel like there isn't enough information and I'm missing something, one thing that occurred to me is that since I'm trying to find the minimum momenta of the proton-antiproton, I could argue that the momenta of the quarks are zero, i.e. they're created at rest, but then on a second thought, I'm not certain about this because it would provide an equal momentum in magnitude for the proton-antiproton, which isn't the case.
Can anyone point me in the right direction here, please?