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**Unformatted text preview: **le , which is typically set to the
electroweak scale MZ A. Coherent scattering (24) where ¼ ðme =m Þ2 and is the ﬁne structure constant.
Radiative QED corrections have been calculated to second
order and higher in electroweak theory paving the way to
Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 Coherent scattering involves the neutral-current exchange
where a neutrino interacts coherently with the nucleus,
2 Technically, neutrino elastic scattering off of free electrons also
falls within this deﬁnition, as discussed earlier in this paper. Joseph A. Formaggio and G. P. Zeller: From eV to EeV: Neutrino cross sections . . . þ AZ ! þ AÃZ :
N
N (26) Shortly after the discovery of neutral-current neutrino
reactions, Freedman, Schramm, and Tubbs pointed out that
neutrino-nucleus interactions should also exist (Freedman,
Schramm, and Tubbs, 1977). Furthermore, one could take
advantage of the fact that at low energies the cross section
should be coherent across all of the nucleons present in the
nucleus. As a result, the cross section grows as the square of
the atomic number A2 . Such an enhancement is possible if the
momentum transfer of the reaction is much smaller than the
inverse of the target size. Letting Q represent the momentum
transfer and R the nuclear radius, the coherence condition is
satisﬁed when QR ( 1. Under these conditions, the relevant
phases have little effect, allowing the scaling to grow as A2 .
Given a recoil kinetic energy T and an incoming neutrino
energy E , the differential cross section can be written compactly as the following:
d G2 2
MT
¼ F Q W MA 1 À A 2 F ðQ 2 Þ2 ;
dT
4
2E (27) where MA is the target mass (MA ¼ AMnucleon ), FðQ2 Þ is the
nucleon form factor, and QW is the weak current term
QW ¼ N À Zð1 À 4sin2 W Þ: (28) The cross section essentially scales quadratically with
neutron (N ) and proton (Z) number; the latter highly suppressed due to the 1 À 4sin2 W ’ 0 term. The form factor
FðQ2 Þ encodes the coherence across the nucleus and drops
quickly to zero as QR becomes large.
Despite the strong coherent enhancement enjoyed by this
process, this particular interaction has yet to be detected
experimentally. Part of the obstacle stems from the extremely
small energies of the emitted recoil. The maximum recoil
energy from such an interaction is limited by the kinematics
of the elastic collision,
Tmax ¼ E
;
1 þ M A =2E (29) similar to that of any elastic scatter where the mass of the
incoming particle is negligible. Several experiments have
been proposed to detect this interaction, often taking advantage of advances in recoil detection typically utilized by dark
matter experiments (Scholberg, 2006; Formaggio, FigueroaFeliciano, and Anderson, 2012). The interaction has also been
proposed as a possible mechanism for cosmic relic neutrinos,
due to its nonzero cross section at zero momentum. However,
the G2 suppression makes detection beyond the reach of any
F
realizable experim...

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