PHY357S:
Answers to Problem Set 5
Fermion decay rates depend on the fermion mass to the fifth power
(see F&H equation 11.53 and Lectures 20 & 21). F&H
Table 5.7 gives the beauty to charm mass ratio to be about 4000/1300,
so we would naively expect the b quark lifetime to be about 0.0036
as long as the charm lifetime. This naive ratio is further increased
because the W+ boson in charm decays can only decay into nee+,
nmm+, and 3 flavours of up+antidown quarks (i.e. a total
5 fermion-antifermion possibilities); the W- boson in beauty decays
can also decay in tau+neutrino and strange+anticharm (i.e. a total
9 fermion-antifermion possibilities). This further reduces the
naive expectation for the relative charm to beauty decay rate
by a factor of 5/9, giving a total ratio of 0.002. Beauty quarks
cannot, however, decay into their preferred partner, the top quark,
because top is heavier than beauty. Beauty can only decay into
charm and up quarks, with rates proportional to (sinq13)2
=(0.003)2 and (sinq23
cosq13)2=(0.04)2
(1-(0.003)2). (How come nobody pointed out the typo
to me in the problem set! It should have read " sinq13=0.003".)
The total decay rate for beauty is hence reduced by a factor of
(0.003)2+(0.04)2 (1-(0.003)2)=0.0016
due to the values of the Cabibbo-Kobayashi-Maskawa (CKM) matrix
elements. Charm has no CKM suppression since it can decay into
its preferred partner, the strange quark. The CKM suppression
factor (0.0016) cancels out the mass factor (0.002), so charm
and beauty have similar lifetimes.
(a) The neutron is heavier than the proton because the d quark is heavier than the up quark by about 4 MeV (see F&H Table 5.7), but the extra mass of the proton due to the fact it is electrically charged is only about 1 MeV (see F&H equations 16.7, 8.37, 16.11).
(b) The p+ meson
is heavier than the p0 meson
because the p+ has extra
mass because it is electrically
(a) The Coulomb potential and the gravitational potential have the same form, so a gravitational binding energy term is just like the Coulomb term (F&H Equation 16.7), with Newton's constant (GN) and the nuclear mass (M = u A) replacing the fine structure constant and the nuclear charge, i.e.
Note that since gravity is attractive and the coulomb term is repulsive (since all protons have like charges), the gravitational term has the opposite sign to the Coulomb term.This term should be added to F&H 16.9.
(b) The binding energy per particle (with gravity added
to F&H equation 16.10, using 16.11) is
So neutron stars could be significantly lighter than the sun, if one could figure out how to make them. Normally one needs supernovae to create them, and these usually produce neutron stars slightly heavier than the sun, since only massive stars go supernova.