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**Unformatted text preview: **r portion has a clear 1 þ e cos dependence, while the
axial portion has a 1 À ðe =3Þ cos behavior, at least to ﬁrst
order in the nucleon mass. The overall angular effect is
weakly backward scattered for antineutrino-proton interactions, showing that the vector and axial-vector terms both
contribute at equivalent amplitudes. This is less so for cases
where the interaction is almost purely Gamow-Teller in
nature, such as d reactions. In such reactions, the backwards
direction is more prominent. Such angular distributions have
been posited as an experimental tag for supernova detection
(Beacom, Farr, and Vogel, 2002).
The ﬁnal aspect of the cross section that is worthy to note is
that it has a near one-to-one correspondence with the beta
decay of the neutron. We explore this property in greater
detail in the next section.
B. Beta decay and its role in cross section calibration In the above equation, fV , fA , and fP are nuclear vector, axial
vector, and Pauli (weak magnetism) form factors evaluated at
zero-momentum transfer (for greater detail on the form factor
behavior, see Sec. IV.D). To ﬁrst order, the differential cross
section can therefore be written as 4 T -odd triple product
Theoretical phase space factor
Neutron lifetime The weak interaction governs both the processes of decay
and scattering amplitudes. It goes to show that, especially for
simple systems, the two are intimately intertwined, often
allowing one process to provide robust predictions for the
other. The most obvious nuclear target where this takes place
is in the beta decay of the neutron. In much the same way as
muon decay provided a calibration of the Fermi coupling
constant for purely leptonic interactions (Sec. II.D), neutron
beta decay allows one to make a prediction of the inverse beta
decay cross section from experimental considerations alone.
For the case of neutron beta decay, the double differential
decay width at tree level is given by (Nico and Snow, 2005)
d3 À
~
¼ G2 jVud j2 ð1 þ 32 Þjpe jðTe þ me Þ
F
dEe de d
~~
m
p Áp
Â ðE0 À Te Þ2 1 þ a e þ b e
Te E
Te
~
~
~
~
pe
p
pe Â p
:
þ n Á A þ B
~
þD
Te
E
Te E Joseph A. Formaggio and G. P. Zeller: From eV to EeV: Neutrino cross sections . . . ~
~
Here pe and p are the electron and neutrino momenta, Te is
the electron’s kinetic energy, E is the outgoing antineutrino
energy, E0 is the end-point energy for beta decay, and n is
the neutron spin. The deﬁnitions of the other various constants are listed in Table III.
Integrating over the allowed phase space provides a direct
measure of the energy-independent portion of the inverse beta
decay cross section, including internal radiative corrections.
That is, Eq. (35) can also be written as
dðe p ! eþ nÞ
"
22
¼5
Ee pe ð1 þ e cosÞ
d cos
2me fð1 þ R Þn
(36)
þ 32 1 À e cos :
3
The term fð1 þ R Þ is a phase space factor that includes
several inner radiative corrections. Additional radiative...

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