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

Work with others at your table on these activities. Argue about the answers but work efficiently!

Goal: Understand neutrino interactions at the microscopic level.

**Electroweak interactions**- 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.)

*Draw the appropriate diagram for the beta decay of a proton.*

*Draw an alternative for the second diagram with a different intermediate state.*

*On the second slide, why does one redefine the down quark but not the up quark?*

*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?)

*Which factors should appear at the quark-W vertices in the diagrams on pages 1 and 2 of the "Neutrinos 2" slides?*

- Look at the diagrams on page 1 of the "Neutrinos 2" slides.
**Neutrino oscillations**- 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?*

- 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!)

*Mathematically, what does it mean to be a "mass eigenstate," and what determines its time evolution?*

- 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 |ν_{1}> and |ν_{2}>?

- The relativistic energy-momentum relation is
*E*^{2}=*p*^{2}+*m*^{2}.*What are the non-relativistic and ultra-relativistic limits, respectively, and which one would you use here?*

*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.

- Finally:
*Why does the observation of neutrino oscillations imply that they have a nonzero mass? What can be inferred about the masses from oscillations?*

- Page 4 of the "Neutrinos 2" slides shows the CKM matrix that
describes the mixing of quark eigenstates.

koenig.389@osu.edu furnstahl.1@osu.edu