Sphaleron-induced baryogenesis
] 21
amplitude for transmission is given by the WKB formula in which the amplitude
is suppressed by an exponential factor
(4.161)
T = ex p ( -1:' P dx )
where p(x) = ,J2m[V(x) - E] and the limits of integration Xo and Xl are the
two points at which the kinetic energy of the classical particle is zero, so that
V(xo) = E = V(Xl). It is convenient to choose the zero of energy such that
E = O. Suppose that Xo is a minimum of V(x), so that we are considering the
probability of a particle in classical equilibrium tunnelling through the potential
barrier. In several dimensions, (minus) the exponent in (4. ]61) generalizes [38]
to
b ==
l
X1 ds J2mV(x)
(4.]62)
Xo
(where ds 2 = dx . dx) and the integral is to be evaluated along the path for which
b is a minimum. The required path r(r), therefore, satisfies
d 2 r
m-=VV
(4.163)
dr 2
with
1 dx dr _ V(x) = O.
2
m-·
dr
(4.164)
dr
Equation (4.163) is just the Euler -Lagrange equation for the imaginary-( or
Euclidean-)time version of Hamilton's principle, in which the formal substitution
r = ; t is made. In other words, it minimizes the Euclidean action
(4.]65)
SE = f dr LE
where
1 dr dx
LE == -m-· - + V(x).
(4.166)
2 dr dr
Equivalently, it describes the motion of a particle in time r moving in the inverted
potential - V (x). It is clear then that the classical equilibrium point xo can only
be reached asymptotically, as r -+ -00
lim r = ro.
(4.167)
~ ..... -oo
We can choose the time at which the particle reaches r" where dr/dr = 0 next,
to be r = O. Then the exponent b may be written as
l x,
1°
b = ds J2mV(x) = dr LE.
(4.168)
Xo
-00