Neutrino Oscillations Or how we know most of what we know

June 2005Steve Elliott, NPSS Outline Two-flavor vacuum oscillations Two-flavor matter oscillations Three-flavor oscillations –The general formalism –The “rotation” matrices

June 2005Steve Elliott, NPSS Consider Two Mass States  1 corresponding to m 1  2 corresponding to m 2 Think of  as a Vector

June 2005Steve Elliott, NPSS  is a solution of H

June 2005Steve Elliott, NPSS The Neutrinos Consider the weak eigenstates e, . These are not the mass eigenstates, 1, . The mass eigenstates are propagated via H. The Mixing Matrix: U

June 2005Steve Elliott, NPSS Mixing Weak eigenstates are a linear superposition of mass eigenstates.

June 2005Steve Elliott, NPSS In Vacuum, no potential in H Denote c = cos  s = sin 

June 2005Steve Elliott, NPSS U H U -1

June 2005Steve Elliott, NPSS The energy difference (and Trig.)

June 2005Steve Elliott, NPSS U H U -1 becomes The algebra is going to get involved, so lets define A, B, and D such that:

June 2005Steve Elliott, NPSS The Diff Eq A solution to this equation should have the form:

June 2005Steve Elliott, NPSS Insert proposed solution

June 2005Steve Elliott, NPSS Two Equations

June 2005Steve Elliott, NPSS r + solution r - solution

June 2005Steve Elliott, NPSS  is a superposition of these 2 solutions (D+2A) is a constant so we sweep it into a redefinition of the C’s.

June 2005Steve Elliott, NPSS The solutions To determine the C’s, use =1 and assume that at t=0, we have all e.

June 2005Steve Elliott, NPSS The time dependent solution What is the probability of finding all   at time t?

June 2005Steve Elliott, NPSS Transition probability

June 2005Steve Elliott, NPSS The Answer Complete mixing: large sin2  and long R/L would result in an “average”: that is P=1/2.

June 2005Steve Elliott, NPSS What about MSW? The Sun is mostly electrons (not muons). e can forward scatter from electrons via the charged or neutral current.  can only forward scatter via the neutral current. The e picks up an effective mass term, which acts on the weak eigenstates.

June 2005Steve Elliott, NPSS The MSW H term. This extra term results in an oscillation probability that can have a resonance. Thus even a small mixing angle, , can have a large oscillation probability.

June 2005Steve Elliott, NPSS Similar algebra as before

June 2005Steve Elliott, NPSS Constant Density Solutions Note similar form to vacuum Oscillations. Note that sin 2 2  m can be 1 even when sin 2 2  is small. That is when: L/L 0 = cos2 

June 2005Steve Elliott, NPSS Variable Density Integrate over the changing density (such as in a star).

June 2005Steve Elliott, NPSS Three Formulism

June 2005Steve Elliott, NPSS Transition Probability

June 2005Steve Elliott, NPSS Transition Probability Real U’s

June 2005Steve Elliott, NPSS Complex U’s If U is complex, then we have the possibility

June 2005Steve Elliott, NPSS Oscillation Experiments Appearance: look for  when none are expected Disappearance: look for decrease in flux of 

June 2005Steve Elliott, NPSS Neutrino Sources and Oscillations Solar neutrinos –Few MeV, L~10 11 m –Electron neutrinos –Most are disappearance expts. (Except SNO NC and SK’s slight NC sensitivity) Reactor –Few MeV, L~10m km –Electron neutrino disappearance

June 2005Steve Elliott, NPSS Neutrino Sources Accelerator –30-50 MeV (  decay) –DIF sources can be several GeV –Various appearance and disappearance modes, various baselines Atmospheric –  and  decay –Various energies –Baseline from 20 to 10,000 km

June 2005Steve Elliott, NPSS SolarReactorAtmospheric Maki, Nakagawa, Sakata, Pontecorvo

June 2005Steve Elliott, NPSS PDB parameterization

June 2005Steve Elliott, NPSS CP violation

June 2005Steve Elliott, NPSS The Jarlskog Invariant Note the product of the sin of all the angles. If any angle is 0, CP violation is not observable. Note that I have seen different values of the leading constant. (taken to be 1 here)

June 2005Steve Elliott, NPSS CP violation hep-ph/

June 2005Steve Elliott, NPSS There are only 2 independent  m 2 for 3 This will be important when we discuss LSND.

June 2005Steve Elliott, NPSS Resources Steve Elliott - UW Phys 558 class notes Bahcall Book Many phenomenology papers