Affleck-Dine baryogenesis
139
To see how, recall that the superpotential W is expected to have higher-order,
non-renonnalizable tenns. These must be gauge and R-parity invariant. Thus.
focusing on a single flat direction. in general
J..
le
J..
n
Wor = - - x = --,p
(4.253)
nMn-3
nM,,-3
where n = mk; k is even if X has odd R-parity, as in the previous example, and M
is some large mass. This gives F -tenns that are non-zero in the specifed direction.
Note, however, that although the superpotential Wor does not conserve U ( 1 ) B - L.
the scalar potential V derived from it does:
v. = ~ 1.1.1 211 - 2
(4.254)
or
M2n-6 'I"
since it is proportional to (r/J*,p)Iat-I.
We shall see later, in chapter 7, that there is good reason for believing that
in the early universe there was a period of 'inflation', during which the Hubble
constant H was approximately constant. The second, and most important, point
is that global supersymmetry is necessarily broken during inflation. Since the
vacuum energy
(V) = 8~H2m~
(4.255)
is non-zero, there must be non-zero F and/or D components for some matter
fields. and a soft potential develops. This includes a mass tenn for the (erstwhile
massless) moduli field lp, of order H. the (instantaneous) Hubble constant [72].
In fact. H - 10 13 - 15 GeV during inflation, as can be seen from (7.41) with
V(,p) given by (7.115) in order to explain the observed cosmic microwave
background. A mass tenn too conserves B - L. The supersymmetry breaking
also induces (soft) A-tenns in the scalar potential. These have the same fonn as
the superpotential W. so, in our example. they have the fonn
A "
VA = - - 3tP +h.c.
(4.256)
M"Like the mass tenn, the scale of A is of order the current Hubble constant H.
Note that such tenns do not conserve B - L and this is the source of the (B - L)violation necessary to generate a net B - L in the evolution of the flat direction.
The Sakharov criteria. necessary for baryogenesis, also require CP-violation and
we shall see later that a CP-violating phase difference between this A-tenn and
that arising from the hidden-sector supersymmetry breaking is essential to getting
a non-zero baryon asymmetry at the end.
The equation of motion for the field tP is
..
. av
,p+3H,p+- =0
(4.257)
fJr/J*
139
To see how, recall that the superpotential W is expected to have higher-order,
non-renonnalizable tenns. These must be gauge and R-parity invariant. Thus.
focusing on a single flat direction. in general
J..
le
J..
n
Wor = - - x = --,p
(4.253)
nMn-3
nM,,-3
where n = mk; k is even if X has odd R-parity, as in the previous example, and M
is some large mass. This gives F -tenns that are non-zero in the specifed direction.
Note, however, that although the superpotential Wor does not conserve U ( 1 ) B - L.
the scalar potential V derived from it does:
v. = ~ 1.1.1 211 - 2
(4.254)
or
M2n-6 'I"
since it is proportional to (r/J*,p)Iat-I.
We shall see later, in chapter 7, that there is good reason for believing that
in the early universe there was a period of 'inflation', during which the Hubble
constant H was approximately constant. The second, and most important, point
is that global supersymmetry is necessarily broken during inflation. Since the
vacuum energy
(V) = 8~H2m~
(4.255)
is non-zero, there must be non-zero F and/or D components for some matter
fields. and a soft potential develops. This includes a mass tenn for the (erstwhile
massless) moduli field lp, of order H. the (instantaneous) Hubble constant [72].
In fact. H - 10 13 - 15 GeV during inflation, as can be seen from (7.41) with
V(,p) given by (7.115) in order to explain the observed cosmic microwave
background. A mass tenn too conserves B - L. The supersymmetry breaking
also induces (soft) A-tenns in the scalar potential. These have the same fonn as
the superpotential W. so, in our example. they have the fonn
A "
VA = - - 3tP +h.c.
(4.256)
M"Like the mass tenn, the scale of A is of order the current Hubble constant H.
Note that such tenns do not conserve B - L and this is the source of the (B - L)violation necessary to generate a net B - L in the evolution of the flat direction.
The Sakharov criteria. necessary for baryogenesis, also require CP-violation and
we shall see later that a CP-violating phase difference between this A-tenn and
that arising from the hidden-sector supersymmetry breaking is essential to getting
a non-zero baryon asymmetry at the end.
The equation of motion for the field tP is
..
. av
,p+3H,p+- =0
(4.257)
fJr/J*