134
Baryogenesis
passes through. The most direct method of baryogenesis WOUld, of course, be to
use the left-chiral quarks themselves, since any CP-violating contributions to their
distributions would directly bias the sphaleron interactions and thereby generate
a baryon asymmetry. However, in the MSSM (but not in all two-Higgs doublet
models) the Higgs potential is real at tree level and CP-violating contributions
to quark masses arise only at one-loop order. Moreover, such contributions are
potentially suppressed by the GIM mechanism, as in the standard model. We are,
therefore, forced to consider CP-violating perturbations in other particle densities.
Specifically, we consider squarks, which couple to quarks strongly via the strong
supergauge interactions, and charginos, which couple strongly to third-generation
quarks via Yukawa interactions. Neutralinos also contribute but their coupling to
fennions is weaker than that of the charginos, so the transport of any asymmetry to
the quark sector is much less efficient and we neglect them. The coupled diffusion
equations that control the densities with CP-violating sources have the general
form
,
,
D;~i + vW~i + r;(~i + ~j + ... ) = S;
(4.232)
where ~j == I-£;/T with 1-£; the chemical potential of the ith species, Dj is a
diffusion constant, Vw is the velocity of the bubble wall, r; is the inelastic rate
converting species i into other species j, ... , and S; is the CP-violating source
term created at the bubble wall; the prime denotes differentiation with respect
to the z-direction in which the bubble wall propagates. Different approaches
have been proposed for calculating the source terms [31,63,66], and it is unclear
whether they agree. We shall use the classical force method [63,64,67,68], in
which the particles move in the plasma under the influence of a classical force
exerted on them by the spatially varying Higgs field. Because of CP-violation,
particles and antiparticles experience (slightly) different forces. The source terms
in the diffusion equations are proportional to the thermal average of this CPviolating component of the force.
For illustrative purposes, we shall follow the treatment of Cline et al [69].
The set of coupled diffusion equations can be simplified considerably by taking
account of the hierarchy of inelastic reaction rates rj that change the particle
species i into other species j, .... The electroweak sphaleron rate rb, of order
a~T (see (4.20S», is the slowest and can be ignored until we are ready to compute
the actual baryon asymmetry. In contrast, the various gauge interaction rates, of
order aa T, are fast and can be taken to be in equilibrium on the time scale for
particles to diffuse in front of the bubble wall:
Dj
aa T » ~ .
(4.233)
w
Then, in particular, the chemical potentials of the weak bosons are zero, so the
chemical potentials of quarks in the same doublet are equal:
~'L = ~bL == ~q3·
(4.234)
Baryogenesis
passes through. The most direct method of baryogenesis WOUld, of course, be to
use the left-chiral quarks themselves, since any CP-violating contributions to their
distributions would directly bias the sphaleron interactions and thereby generate
a baryon asymmetry. However, in the MSSM (but not in all two-Higgs doublet
models) the Higgs potential is real at tree level and CP-violating contributions
to quark masses arise only at one-loop order. Moreover, such contributions are
potentially suppressed by the GIM mechanism, as in the standard model. We are,
therefore, forced to consider CP-violating perturbations in other particle densities.
Specifically, we consider squarks, which couple to quarks strongly via the strong
supergauge interactions, and charginos, which couple strongly to third-generation
quarks via Yukawa interactions. Neutralinos also contribute but their coupling to
fennions is weaker than that of the charginos, so the transport of any asymmetry to
the quark sector is much less efficient and we neglect them. The coupled diffusion
equations that control the densities with CP-violating sources have the general
form
,
,
D;~i + vW~i + r;(~i + ~j + ... ) = S;
(4.232)
where ~j == I-£;/T with 1-£; the chemical potential of the ith species, Dj is a
diffusion constant, Vw is the velocity of the bubble wall, r; is the inelastic rate
converting species i into other species j, ... , and S; is the CP-violating source
term created at the bubble wall; the prime denotes differentiation with respect
to the z-direction in which the bubble wall propagates. Different approaches
have been proposed for calculating the source terms [31,63,66], and it is unclear
whether they agree. We shall use the classical force method [63,64,67,68], in
which the particles move in the plasma under the influence of a classical force
exerted on them by the spatially varying Higgs field. Because of CP-violation,
particles and antiparticles experience (slightly) different forces. The source terms
in the diffusion equations are proportional to the thermal average of this CPviolating component of the force.
For illustrative purposes, we shall follow the treatment of Cline et al [69].
The set of coupled diffusion equations can be simplified considerably by taking
account of the hierarchy of inelastic reaction rates rj that change the particle
species i into other species j, .... The electroweak sphaleron rate rb, of order
a~T (see (4.20S», is the slowest and can be ignored until we are ready to compute
the actual baryon asymmetry. In contrast, the various gauge interaction rates, of
order aa T, are fast and can be taken to be in equilibrium on the time scale for
particles to diffuse in front of the bubble wall:
Dj
aa T » ~ .
(4.233)
w
Then, in particular, the chemical potentials of the weak bosons are zero, so the
chemical potentials of quarks in the same doublet are equal:
~'L = ~bL == ~q3·
(4.234)