HW #6 (129A), due Dec 6 , 4pm
1. Calculate the branching fraction of Zboson into ν¯ν,e+e−,µ+µ−,τ+τ−,
and all hadrons (u¯u,d¯
d,s¯s,c¯c,b¯
b) together with the correction factor due to
the additional gluon emission, and compare them to the values in the PDG
booklet.
2. Work out vfrom GF= 1/√2v2,g=e/ sin θW,gZ=e/sin θWcos θW,
and show that the formulae mW=1
2gv,mZ=1
2gZvdo not quite work. On
the other hand, using α(mZ) = e2(mZ)/4π= 1/129 instead of 1/137, show
that they work much better.
3. Show that the relation m2
W=m2
Zcos2θWdoes not quite work. Instead,
one has to take the effect of virtual top quark into consideration, and the
correct relation is m2
W=m2
Zρcos2θW, with ρ= 1 + 3GFm2
t/(8√2π2). Here,
the factor of three comes from the number of colors. Show that this relation
works much better.
4. Plot the survival probability in the neutrino oscillation
Psurv = 1 −sin22θsin2∆m2
4EL, (1)
as a function of the cosine zenith angle for atmospheric neutrinos. Assume
that the neutrinos are produced at the altitude of 20 km. Take the parameters
sin22θ= 1, ∆m2= 3 ×10−3eV2, and E= 1 GeV.