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**Unformatted text preview: **llected for
analysis. Stopped pion beams have also been used for these
electroweak tests at the Los Alamos National Laboratory in
the Los Alamos Meson Physics Facility (LAMPF) (Allen
et al., 1993) and Liquid Scintillator Neutrino Detector
(LSND) (Auerbach et al., 2001) experiments. Table II provides a summary of the types of measurements made using
pure neutrino-lepton scattering.
D. Radiative corrections and GF Upon inspection of the cross section formalisms discussed
above, it is clear that, with the exception of ratios, one is
critically dependent on certain fundamental constants, such as
the strength of the weak coupling constant GF . Ideally, one
wants to separate the dependence on the weak mixing angle
from the Fermi constant strength. Fortunately, measurements
of the muon lifetime provides such a possibility, as it is
inversely proportional to the coupling strength GF and the
muon mass m ,
G2 m 5
3 m2
½1 þ ÁðÞ:
(22)
ð ÞÀ1 ¼ F 3 fðÞ 1 þ
2
5 MW
192
2
In the above expression, fðÞ is a phase factor, the m2 =MW
factor encapsulates the W -boson propagator, and ÁðÞ encodes the QED radiative ﬁeld corrections. For completeness,
we list these correction factors: fðÞ ¼ 1 À 8 þ 83 À 4 À 122 ln ’ 0:999 813;
(23) " sin2 MS ¼ 1 À
W MW ðÞ2
:
MZ ðÞ2 (25) Such radiative corrections, although small, often need to be
accounted for in order to properly predict the sin2 W value.
Theoretical compilation of such radiative effects can be found
in a variety of papers [see, for example, Marciano and Parsa
(2003)].
III. THRESHOLDLESS PROCESSES: E $ 0–1 MeV Having established the formalism of basic neutrino interactions, we turn our attention toward describing neutrino
interactions across the various energy scales. The ﬁrst step
in our journey involves thresholdless interactions, which can
be initiated when the neutrino has essentially zero momentum. Such processes include coherent scattering and neutrino
capture.2 25 2
À
À ð9 þ 42 þ 12 lnÞ
8
2
þ 162 3=2 þ Oð2 Þ þ Oð2 Þ þ . . . ; Á¼ precision electroweak tests of the standard model. The best
measurement of the muon lifetime to date has been made by
the MuLan experiment (Webber et al., 2011), yielding a
value for GF of 1:166 378 8ð7Þ Â 10À5 GeVÀ2 , a precision of
0.6 ppm.
At tree level, knowing GF (and and MZ ) it is possible to
exactly predict the value of sin2 W and test this prediction
against the relevant cross section measurements. However,
once one introduces one-loop radiative contributions, dependencies on the top and Higgs masses are also introduced.
The size of these corrections depends partially on the choice
of the normalization scheme. The two commonly used
renormalization schemes include the Sirlin on-shell model
(Sirlin, 1980) and the modiﬁed minimal subtraction scheme
(Marciano and Sirlin, 1981). In the latter method, the
Weinberg angle is deﬁned by MW and MZ at some arbitrary
renormalized mass sca...

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