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Unformatted text preview: e symmetries (Weinberg, 1979),
L5 ¼ cij iT
ðL ÞCðT Lj Þ þ H:c:;
Ã (1) where Li is the lepton doublet ﬁeld of the ith generation and
is the Higgsdoublet ﬁeld [the 2 Â 2 matrix and the 4 Â 4
chargeconjugation matrix C are present to ensure invariance
under SUð2ÞL and Lorentz transformations, respectively].
When the Higgs doublet acquires a vacuumexpectation value
pﬃﬃﬃ
hi ¼ ð0; v= 2Þ (v ¼ 246 GeV), this term gives rise to a
(Majorana) mass for neutrinos,
ij
L5 ¼ À1M iT Cj þ H:c:
2 (2) ij
where M ¼ cij v2 =Ã is the neutrino mass matrix. Because
of the small inferred masses of neutrinos, the scale Ã lies
around 1015 GeV, assuming cij is not much less than order
unity.
There are other indications that the SM is not a complete
description of nature, most of them related to gravitation and
cosmology. Even with massive neutrinos included, the SM
particles only constitute 4.6% of the present Universe, with
the remainder in mysterious dark matter (23%) and dark
energy (72%). Neither dark matter nor dark energy are
accommodated in the SM. There is no adequate mechanism
for baryogenesis (the observed excess of baryons over Rev. Mod. Phys., Vol. 84, No. 4, October–December 2012 antibaryons) or inﬂation, which is the simplest explanation
of the observed temperature ﬂuctuations in the cosmic microwave background. The SM also provides no explanation of
the strong CP problem: the lack of observed CP violation in
the strong interaction, which is allowed by the SM.
If physics beyond the SM lies at an energy scale less than
1 TeV, then we should be able to observe it directly at highenergy colliders. If it lies at a scale greater than 1 TeV, then
we can parametrize its effects via higherdimension operators, suppressed by inverse powers of the scale of new physics
Ã exactly as in the case of neutrino masses described above.
Other than the dimensionﬁve operator responsible for
neutrino masses, the lowestdimension operators are of
dimension six, and are therefore suppressed by two inverse
powers of Ã. If Ã is of order 1015 GeV, as suggested by
neutrino masses, then these operators are so suppressed that
they are unobservable, with the possible exception of baryonnumber violating operators that mediate nucleon decay.
However, there could be more than one scale of new physics,
and if this scale is not much greater than 1 TeV, its effects
could be observable via dimensionsix operators. Operators
of a dimension greater than six are suppressed by even more
inverse powers of Ã and can be neglected.
There are many dimensionsix operators allowed by the
SM gauge symmetry (Buchmuller and Wyler, 1986). There
are three ways to detect the presence of these operators. The
ﬁrst is to observe phenomena that are absolutely forbidden (or
extremely suppressed) in the SM, such as nucleon decay. The
second is to make measurements with such great precision
that the small effects of the dimensionsix operators manifest
themselves. The t...
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This document was uploaded on 09/28/2013.
 Fall '13
 Energy

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