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Stars as Laboratories for Fundamental Physics - MPP Theory Group

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296 Chapter 8<br />

Explicitly one finds <strong>for</strong> the mixing angle and the oscillation length<br />

in the medium the following expressions,<br />

where<br />

tan 2θ = sin 2θ 0<br />

cos 2θ 0 − ξ ,<br />

sin 2θ =<br />

sin 2θ 0<br />

[sin 2 , (8.30)<br />

2θ 0 + (cos 2θ 0 − ξ) 2 ]<br />

1/2<br />

ξ ≡<br />

√<br />

2 GF n e 2ω<br />

m 2 2 − m 2 1<br />

= 1.53×10 −7 Y e ρ<br />

g cm −3<br />

ω<br />

MeV<br />

eV 2<br />

, (8.31)<br />

m 2 2 − m 2 1<br />

and<br />

l osc =<br />

4π ω<br />

m 2 2 − m 2 1<br />

sin 2θ<br />

sin 2θ 0<br />

. (8.32)<br />

For m 2 > m 1 these functions are shown in Fig. 8.8; they exhibit a<br />

“resonance” <strong>for</strong> cos 2θ 0 = ξ.<br />

The dispersion relation h<strong>as</strong> two branches which in vacuum correspond<br />

to ω 1,2 = (k 2 − m 2 1,2) 1/2 . In the relativistic limit they are<br />

ω 1,2 − k = m2 1 + m 2 2<br />

4k<br />

+ √ 2 G F n B (Y e − 1 2 )<br />

± m2 2 − m 2 1<br />

4k<br />

[<br />

sin 2 2θ 0 + (cos 2θ 0 − ξ) 2] 1/2<br />

, (8.33)<br />

a result schematically shown in Fig. 8.8. The “resonance” of the mixing<br />

angle corresponds to the crossing point of the two branches of the<br />

dispersion relation. Of course, the levels do not truly cross, but rather<br />

show the usual “repulsion.”<br />

Because the medium effect changes sign <strong>for</strong> antineutrinos, a resonance<br />

occurs between ν e and ν µ if m 2 > m 1 , while none occurs between<br />

ν e and ν µ . If the m<strong>as</strong>s hierarchy is the other way round, a resonance<br />

occurs <strong>for</strong> ν e and ν µ , but not <strong>for</strong> ν e and ν µ .<br />

8.3.2 Oscillations in Homogeneous Media<br />

In a homogeneous medium the treatment of neutrino oscillations is<br />

exactly <strong>as</strong> in vacuum except that one must use the effective medium<br />

mixing angle and oscillation length given above. If the medium is sufficiently<br />

dilute it will not affect the oscillations at all. This is the c<strong>as</strong>e

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