Introduction to Particle Physics I Exercise Sheet 2 - Wed 29.1. University of Helsinki - Spring 2014 Lectures: Prof. Risto Orava, Exercises: Mikael Mieskolainen Return the exercises by Tuesday 18.00 to the box in the Physicum 2nd floor. Task 1 [Natural Units (c = ~ = 1)] In Natural Units one can express most quantities in terms of [GeV]n , where n is some integer. 1. Why would anyone use Natural Units? 2. Express time (s), length (m), cross section (m2 ), mass (kg) and momentum (kg·m/s) in terms of energy (GeV), i.e. find out both n and the numerical conversion factor. 3. Express gravitational constant G = 6.67 · 10−11 N·m2 ·kg−2 in units of energy (GeV). 4. Express temperature (K) in units of energy (GeV), when Boltzmann constant is set to kB = 1. 5. Express the classic radius of proton (∼ 1 fm = 1 fermi) in units of energy (GeV). 6. Finally, what does the unit electronvolt (eV) mean? Task 2 [Conservation of energy and momentum - Kinematics] 1. Under what fundamental symmetries (invariance under transformations) one gets the conservation of energy and momentum? 2. Show that the scattering (annihilation) process of e+ e− → γ, where γ is one physical photon, is not allowed, but the final state γγ is allowed. Hint: Go to the center of mass Lorentz frame of the electron-positron pair and use explicitly 4-momentum (energymomentum) conservation. Task 3 [Lorentz scalars and the Mandelstam variables] 1. Write down the definitions of the Mandelstam variables s, t and u for a 2 → 2 scattering process p1 + p2 → p3 + p4 (1) and draw the corresponding Feynman diagrams for s-, t- and u-channels. Express in words the physical meaning of each Mandelstam variable. 1 2. The Mandelstam variables s, t and u are Lorentz scalars, i.e. they have the same value in every Lorentz frame and are thus invariant under Lorentz transformations. Show that in a case of 2 → 2 scattering process they obey s+t+u= 4 X m2i , (2) i=1 where corresponding particle masses are denoted with mi . 3. Initial and final state particles of the Mandelstam variables obey here the on-shell condition p2i = m2i . (3) Derive E 2 = |~ p|2 + m2 from the on-shell condition by calculating the scalar product of µ 4-vectors p pµ = p2 open. Why is m2 known as the invariant mass and how is that a useful quantity? 2