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Neutrino Oscillations Or how we know most of what we know

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June 2005Steve Elliott, NPSS 20052 Outline Two-flavor vacuum oscillations Two-flavor matter oscillations Three-flavor oscillations –The general formalism –The “rotation” matrices

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June 2005Steve Elliott, NPSS 20053 Consider Two Mass States 1 corresponding to m 1 2 corresponding to m 2 Think of as a Vector

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June 2005Steve Elliott, NPSS 20054 is a solution of H

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June 2005Steve Elliott, NPSS 20055 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

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June 2005Steve Elliott, NPSS 20056 Mixing Weak eigenstates are a linear superposition of mass eigenstates.

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June 2005Steve Elliott, NPSS 20057 In Vacuum, no potential in H Denote c = cos s = sin

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June 2005Steve Elliott, NPSS 20058 U H U -1

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June 2005Steve Elliott, NPSS 20059 The energy difference (and Trig.)

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June 2005Steve Elliott, NPSS 200510 U H U -1 becomes The algebra is going to get involved, so lets define A, B, and D such that:

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June 2005Steve Elliott, NPSS 200511 The Diff Eq A solution to this equation should have the form:

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June 2005Steve Elliott, NPSS 200512 Insert proposed solution

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June 2005Steve Elliott, NPSS 200513 Two Equations

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June 2005Steve Elliott, NPSS 200514 r + solution r - solution

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June 2005Steve Elliott, NPSS 200515 is a superposition of these 2 solutions (D+2A) is a constant so we sweep it into a redefinition of the C’s.

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June 2005Steve Elliott, NPSS 200516 The solutions To determine the C’s, use =1 and assume that at t=0, we have all e.

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June 2005Steve Elliott, NPSS 200517 The time dependent solution What is the probability of finding all at time t?

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June 2005Steve Elliott, NPSS 200518 Transition probability

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June 2005Steve Elliott, NPSS 200519 The Answer Complete mixing: large sin2 and long R/L would result in an “average”: that is P=1/2.

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June 2005Steve Elliott, NPSS 200520 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.

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June 2005Steve Elliott, NPSS 200521 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.

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June 2005Steve Elliott, NPSS 200522 Similar algebra as before

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June 2005Steve Elliott, NPSS 200523 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

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June 2005Steve Elliott, NPSS 200524 Variable Density Integrate over the changing density (such as in a star).

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June 2005Steve Elliott, NPSS 200525 Three Formulism

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June 2005Steve Elliott, NPSS 200526 Transition Probability

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June 2005Steve Elliott, NPSS 200527 Transition Probability Real U’s

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June 2005Steve Elliott, NPSS 200528 Complex U’s If U is complex, then we have the possibility

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June 2005Steve Elliott, NPSS 200529 Oscillation Experiments Appearance: look for when none are expected Disappearance: look for decrease in flux of

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June 2005Steve Elliott, NPSS 200530 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 - 300 km –Electron neutrino disappearance

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June 2005Steve Elliott, NPSS 200531 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

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June 2005Steve Elliott, NPSS 200532 SolarReactorAtmospheric Maki, Nakagawa, Sakata, Pontecorvo

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June 2005Steve Elliott, NPSS 200533 PDB parameterization

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June 2005Steve Elliott, NPSS 200534 CP violation

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June 2005Steve Elliott, NPSS 200535 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)

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June 2005Steve Elliott, NPSS 200536 CP violation hep-ph/0306221

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June 2005Steve Elliott, NPSS 200537 There are only 2 independent m 2 for 3 This will be important when we discuss LSND.

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June 2005Steve Elliott, NPSS 200538 Resources Steve Elliott - UW Phys 558 class notes Bahcall Book Many phenomenology papers

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