6805: 1094 Activities 14

Write your name and answers on this sheet and hand it in at the end.

Neutrino masses and oscillations

Goal: Understand neutrino interactions at the microscopic level.

1. Electroweak interactions
   1. Look at the diagrams on page 1 of the "Neutrinos 2" slides. Which processes are shown in the first two diagrams? (Hint: Draw appropriate time axes.)
      
      
      
      
      
      
      
      
      
      
   2. Draw the appropriate diagram for the beta decay of a proton.
      
      
      
      
      
      
      
      
      
      
   3. Draw an alternative for the second diagram with a different intermediate state.
      
      
      
      
      
      
      
      
      
      
   4. On the second slide, why does one redefine the down quark but not the up quark?
      
      
      
      
      
      
      
      
      
      
   5. Why is the relation between mass and weak quark eigenstates for two generations expressed in terms of a single angle? (Hint: write the transformation in matrix form. What property does the matrix have to have, yielding how many free parameters? Of these, how many are physically significant?)
      
      
      
      
      
      
      
      
      
      
      
   6. Which factors should appear at the quark-W vertices in the diagrams on pages 1 and 2 of the "Neutrinos 2" slides?
      
      
      
      
      
      
2. Neutrino oscillations
   1. Page 4 of the "Neutrinos 2" slides shows the CKM matrix that describes the mixing of quark eigenstates. What would an analogous matrix for neutrino oscillations look like?
      
      
      
      
      
      
      
   2. Let us now take a more quantitative look at oscillations. For simplicity, we work with only two flavors |ν_α,β> (and thus two mass eigenstates, call them 1 and 2). What does the mixing matrix look like in this case? (Hint: recall what we had for two quark generations!)
      
      
      
      
      
      
   3. Mathematically, what does it mean to be a "mass eigenstate," and what determines its time evolution?
      
      
      
      
      
      
   4. Suppose you start with an initially pure |ν_α> beam. What is its composition after a time t? In other words, what is |ν_α(t)>, expressed in terms of |ν₁> and |ν₂>?
      
      
      
      
      
      
   5. The relativistic energy-momentum relation is E² = p² + m². What are the non-relativistic and ultra-relativistic limits, respectively, and which one would you use here?
      
      
      
      
      
      
      
   6. Calculate the probability to measure a |ν_β> after travelling a distance L, i.e., derive the formula given in p. 5 of the "Neutrinos 2" slides.
      
      
      
      
      
      
      
      
      
      
      
      
      
   7. Finally: Why does the observation of neutrino oscillations imply that they have a nonzero mass? What can be inferred about the masses from oscillations?