Lecture

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Lecture 7, April 17,2006 Neutrino
oscillations in matter
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Effect of matter on neutrino propagation

• That is a preposterous idea the mean free path
  is too long
• 1/Ns, N number density N0r 1024 cm-3,
• cross section 10-43 cm2 for low energy
  neutrinos ,
• thus, l 1019 cm 10 light years.
• The s is GF2 therefore is so small, But
  interaction energy
• with matter is GF and might affect the relative
  phase
• of states that are not energy eigenstates.
• Recall that
• oscillation
• effects arise
• because of
• phase
• differences

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In matter the phase e-iEt, where E p
m2/2p should be replaced by E ltHeffgt, where
ltHeffgt represents the expectation value of the
weak interaction between the neutrinos and the
constituents of matter. Thus in matter
schematically Eeff E0 m2/2E0 21/2GFNe
This term is present only for ne and has a minus
sign for ne
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All neutrinos interact equally through Z0
exchange (NC) with electrons and quarks
Electron neutrinos interact with electrons by Z0
and W- exchange
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In practical units VC 21/2GFNe 7.6x10-14 r(g
cm-3) Ye (eV) (see HW4), where electron
fraction Ye Ne/(Np Nn)
Thus ne propagating in matter of density Ne
acquires a phase ne(L) ne(0)exp(-iVcL) where L
is the distance of propagation. Other neutrino
flavors do not have this phase, so a mixed state
will experience interferences, leading to matter
oscillations.
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The matter oscillation length is therefore L0
2p/21/2GFNe 1.7x107 (meters)/Yer(g cm-3) L0 is
independent of energy, For typical densities on
Earth L0 Earth diameter so matter effects

are small However, in Sun or other astrophysical
objects they are decisive.
So, we can have two kinds of neutrino
oscillations, vacuum and matter. To see which of
them dominates, compare the two phases
or oscillation lengths Losc/L0 23/2GFNe
En/Dm2 0.22En(MeV)rYe(100g
cm-3)7x10-5/Dm2(eV2) If this ratio is gtgt 1
matter oscillations dominate, if it is lt1,
vacuum oscillations dominate.
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Remembering that Losc 2p 2E/Dm2, and
subtracting from the diagonal cos2q/2Losc we can
now obtain the equation of motion in the flavor
basis including the matter effects (2p/L0 VC
21/2GFNe)
The effect of matter is only visible in the
lt1H1gt diagonal matrix element here.
Or, equivalently (subtracting 221/2EGFNe/4E)
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The equations of motion can be also rewritten in
the vacuum mass eigenstate basis,
Now matter effects are present in all entries.
In either case the matter eigenstates depend on
L0, Losc, and on the vacuum mixing angle qv
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• For this constant density case there are three
  distinct regimes
• Low density, L0 gtgt Losc matter has little
  effect on oscillations
• High density, L0 ltlt Losc ne -gt nH and
  oscillations are suppressed
• (since the amplitude sin22qm)
• Resonance, X(Ne) -gt 0,
• i.e. 221/2EGFNe -gt Dm2cos2qv
• oscillations are enhanced since qm -gt p/4
  independently of qv.
• Note that the resonance condition
  depends on the sign of
• Dm2, and whether neutrinos or
  antineutrinos are involved.

The most interesting case is the case of
neutrinos propagating through an object of
varying density (e.g. the Sun) from the high
density regime to the low density regime.
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How do we treat the case of varying density? We
can integrate the equations of motion
numerically, which is relatively easy but not
very enlightening. We can also try to solve them
analytically.
Nec Dm2/2E cos2qv/21/2GF (critical or resonance
density)
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What are the conditions from matter effects to
matter? Clearly, when neutrinos are produced in
the high density regime, and propagate into the
low density regime, the matter effects have a
profound influence. From above it follows that
the corresponding condition is Losc/L0
23/2GFNe En/Dm2
0.22En(MeV)rYe(100g cm-3)7x10-5/Dm2(eV2) gt
1 If En/Dm2 is such that this condition is not
fulfilled, matter effects become unimportant.
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Edges of matter effects
Edges of nonadiabacity