HW6: Physics 124 (Fall 2023)
(due 11:59 PM Pacific Time Tuesday, December 5, 2023, and submit via Gradescope please)
1. A neutrino experiment detects ν ̄e neutrinos at a distance of 200 m from a nuclear reactor and finds that the flux is (90 ± 10)% of that expected if there were no oscillations. Assuming a two-component model with maximal mixing (θ = 45◦) and a mean neutrino energy of 3 MeV, use this result to estimate the squared mass difference of the ν ̄e and its oscillating partner. What is the possible range for such a squared mass difference?
2. Assuming we have a proton beam moving with four-momentum p1 = (E1/c,p⃗1) and p⃗1 along +z direction, and an anti-proton beam with four momentum p2 = (E2/c,p⃗2) and p⃗2 along −z direction. If this proton beam is colliding with the anti-proton beam, one usually defines so-called center-of-mass energy as √s, with s = (p1 + p2)2c2.
(a) Now assuming |p⃗1| = |p⃗2| = 270GeV/c, please calculate its center-of-mass (CM) energy √s. This is so-called collider mode.
(b) Assuming |p⃗1| = 270GeV/c and |p⃗2| = 0, please calculate its CM energy √s. This is so-called fixed-target mode.
(c) In the fixed-target mode, how large |p⃗1| has to be in order to achieve the same √s in the collider mode?
3. Particle A with energy E hits particle B at rest, producing particles C1, C2, · · · , Cn:
A + B → C1 + C2 + · · · + Cn, (1)
whereparticlemasscanbedenotedasmA,mB,m1,m2, ···,mn,respectively.Pleasecalculate the threshold (i.e. minimum E) for this reaction to happen, in terms of the various particle masses.
4. A particle has mass m, and in its own rest frame, it has a mean life τ. Now assuming its momentum is p = |p⃗|, what would be the mean distance l of this particle before it decays? Write your answer in terms of τ, p, m. Now for π mesons, they have life time τπ = 2.6 × 10−9 seconds, its mass mπ = 140 MeV/c2 and momentum |p⃗π| = 140 MeV/c. Please calculate the mean distance lπ before it decays. Write your results in terms of centimeter (cm).
5. In the inelastic electron-proton scattering at high energies,
e(k) + p(P) → e(k′) + X(PX), (2)
where the four-momenta k = (E/c,⃗k), k′ = (E′/c,⃗k′), and P = (Ep/c,P⃗). A variable ν can be defined as follows:
2Mν = Wc2 + Q2 − M2c2, (3)
where M is the proton mass, W is the invariant mass of the final-state hadrons, i.e., W c2 = PX2 . Please use energy-momentum conservation to show that the variable ν can be written as ν = E − E′ in the rest frame of the proton. With such a result at hand, please demonstrate that the Bjorken xB = Q2/(2Mν) lies in the range 0 ≤ xB ≤ 1 if the mass of the electron and proton is neglected.
