Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
p
k
νe e −
e −
W ±
Figure 1.3: Charged-current interactions bewteen electrons and neutrinos.
where x is the spatial variable. In order to separate the neutrino and electron
contributions, we apply the Fierz transformation and obtain:
νe
H CC GF
eff (x) = √2 [¯νe(x)γ µ (1 − γ5)νe(x)] [ē(x)γµ(1 − γ5)e(x)]. (1.27)
To obtain the coherent forward scattering contribution to the energy of νe in matter
(i.e. the matter-induced potential for νe) we fix the variables corresponding
to νe and integrate over all the variables that correspond to the electron. For
coherent forward scattering we have p = p ′ = pν and therefore k = k ′ = pe. The
helicities of the electrons also remain unchanged after the scattering because the
interaction must leave the medium unchanged in order to contribute coherently
to the neutrino potential 7 . The average of the effective Hamiltonian over the
electron background in the medium rest frame is given by
HCC GF
eff (x) = √2 ¯νe(x)γ µ
(1 − γ5)νe(x) d 3 pef(Ee, T) (1.28)
× 1
〈e
2
− (pe, he) | e(x)γµ(1 − γ5)e(x) | e − (pe, he)〉.
he=±1
For simplicity, we consider for the electron background a finite normalization
volume V with the one electron states | e − (pe, he)〉 :
k ′
p ′
| e − (pe, he)〉 = 1
2EeV ahe†
e (pe) | 0〉 (1.29)
The function f(Ee, T) is the statistical distribution of the electron energy Ee,
which depends on the temperature T of the electron background and is normalized
7 Indeed the spin of the electrons is not changed therefore the helicities are conserved.
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