How Diffusion Works for Electroweak
Baryogenesis
– and what we Learn for Leptogenesis
Björn Garbrecht
Department of Physics
University of Wisconsin–Madison
Low Energy Precision Electroweak Physics in the LHC Era
INT, University of Washington, Seattle, October 7th 2008
Based on
Daniel J. H. Chung, BG, Michael J. Ramsey-Musolf, Sean Tulin, arXiv:0808.1144 [hep-ph]
Daniel J.H. Chung, BG, Sean Tulin, arXiv:0807.2283 [hep-ph]
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
1/44
Outline
1
Introduction and Overview
2
Diffusion and Chemical Equilibration in
Electroweak Baryogenesis
3
Seattle, 10/07/08
The Effect of the Sparticle Mass Spectrum
on the Conversion of
B–L to B
Björn Garbrecht – UW-Madison
2/44
1
Introduction & Overview
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
3/44
Electroweak Baryogenesis
strong first
order phase
transition
not with
Standard
Model
weak sphaleron
active
B
C
CP
symmetric
phase
Seattle, 10/07/08
[Kuzmin, Rubakov, Shaposhnikov (1985);
McLerran, Shaposhnikov, Turok, Voloshin
(1990), Cohen, Kaplan, Nelson (1990)]
broken
phase
Eq
weak sphaleron
inactive
Björn Garbrecht – UW-Madison
4/44
What Happens at the Wall
Generation of Higgsino
density from CPviolating local (?)
source
Diffusion & chemical
equilibration ahead of
the wall
Strong damping due to
the non-zero Higgs
VEV in the broken
phase
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
5/44
Diffusion Equation for EWBG
inelastic
scatterings
relaxation in
broken phase
[Cohen, Kaplan, Nelson (1994),
Huet, Nelson (1995)]
local (?) CPviolating source
The densities
denote charge densities (particle density minus
antiparticle density).
We denote the particular charge densities by the symbol representing
the particles, e.g.
These equations are (should be) derivable from first principles in the
Schwinger-Keldysh closed time path formalism, which is
appropriate for non-equilibrium problems.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
6/44
Quantum Transport: Closed Time-Path
Vacuum in-out amplitudes:
[Schwinger (1961);
Keldysh (1965)]
General states, in-in expactation values:
introduce closed time path
+
–
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
7/44
Kadanoff-Baym Equations

Schwinger-Dyson equations:
=
+
Diffusion equations
follow in the
coincidence limit.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
8/44
Quantum Source
& Relaxation
[Joyce, Prokopec, Turok (1995); Riotto (1998);
Carena, Moreno, Quiros, Seco, Wagner (2001-03);
Konstandin, Prokopec, Schmidt Weinstock(2001-06);
Cirigliano, Lee, Ramsey-Musolf (2004)]
quantum reflection/
transmission problem
MSSM:
higgsino/chargino
mixing mass-matrix:
source for EWB
different reflection/transmission
coefficients for
and
Eq
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
CP
9/44
Diffusion Operator
Ficks's Law:
Induced by elastic scatterings
Diffusion constants:
[Joyce, Prokopec, Turok (1996)]
In wall-frame coordinates:
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
10/44
Chemical Equilibration: Three Body Interactions
Yukawa and triscalar
Supergauge
[Chung, BG,
Ramsey-Musolf, Tulin (2008)]
[Cirigliano, Lee,
Ramsey-Musolf, Tulin (2006)]
Use thermally averaged three-body rates.
Four-body rates (additional gauge boson radiated) are important, when three
body interaction is kinematically forbidden.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
11/44
2
Diffusion and Chemical Equilibration
in Electroweak Baryogenesis
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
12/44
DIFFUSION
Yukawa
Supergauge
Source
Triscalar
Strong Sphaleron
Weak Sphaleron
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
13/44
How to Solve Diffusion Equations
Analytical:
[Huet, Nelson (1995)]
Assume certain interactions are in equilibrium.
● Use local B conservation.
● Take linear combinations, reduce to one diffusion equation that
can be solved using Green function.
●
What's new? [Chung, Garbrecht, Ramsey-Musolf and Tulin (2008)]
Numerical:
Full simulation of diffusion at the classical level in the symmetric
phase.
● We did not address issues arising when approaching the broken
phase: quantum source & relaxation, weak isospin breaking.
●
Analytical
Use the exact three body rates to determine systematically which
reactions are fast.
● Particle – sparticle equilibrium good almost everywhere.
● b-Yukawa equilibrium for
, -Yukawa for
● Sparticle mass-dependence of the baryon asymmetry.
●
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
14/44
Diffusion Equation: Asymptotic Behavior
Assume that all particle densities are proportional to the Higgs-density
(we will see to what extent this is true).
Then, there is a linear combination of diffusion equations, which takes the
form:
Solutions in absence of the source
outside the bubble wall.
Relaxation term
Inside the wall,
ahead of the wall (vanishing Higgs VEV).
Ahead of wall:
Inside wall:
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
15/44
Diffusion Time Scale
Within a time
:
charges diffuse
wall expands
At what time-scale does chemical equilibration happen?
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
16/44
k-Factors
Linear relation between charge
for
Seattle, 10/07/08
and chemical potential
Björn Garbrecht – UW-Madison
17/44
Three Body Rates
For example:
Can be derived from Kadanoff-Baym equations, but to this end classical (up to
quantum statistical factors).
Can analytically be reduced to one-dimensional integral without approximations.
[Cirigliano, Lee, Ramsey-Musolf, Tulin (2006)]
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
18/44
When is a Process fast?
“fast”
forces this
term to be zero.
compare
with
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
19/44
Three Body Chemical Equilibration Rates
Yukawa
Triscalar
In particular, Yukawa rates of order
ahead of the wall.
Seattle, 10/07/08
may still ensure chemical equilibrium
Björn Garbrecht – UW-Madison
20/44
Yukawa Rates
top-quark:
bottom-quark:
-lepton:
always in equilibrium
may be in equilibrium if
(depending on the details)
may be in equilibrium if
For example,
mediates
Equilibrium implies
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
21/44
Particle – Sparticle Equilibrium
Gaugino interactions equilibrate particle and sparticle
chemical potentials.
What happens if e.g. gluinos are very heavy or the
conversion
is suppressed because of
?
See, for example, Yukawa and triscalar interactions
of the top (s)quark:
Particle – sparticle equilibrium is generically maintained if
Yukawa- and triscalar interactions are fast.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
22/44
Rate Dependence on
and
are suppressed for heavy
.
At tree-level,
For large
, back to the usual top-mediated
scenario (but this also suppresses the source).
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
23/44
Strong Sphaleron
strong sphaleron
(thermal QCD instanton)
baryon number conservation
(before weak sphaleron)
if
for all (s)quarks
“Shaposhnikov suppression”
does not follow if we break
squark mass degeneracy.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
24/44
Summary of our Equilibrium
Assumptions
Particles and sparticles are in equilibrium, independent of
sparticle masses.
● Strong sphaleron is in equilibrium.
● Gauge interactions are in equilibrium, in particular weak
interactions. Therefore, we use common Chemical
potentials for isodoublets and do not need chemical
potentials for
. Questionable as we
proceed into the broken phase.
● Top quark Yukawa coupling is in equilibrium.
● Depending on
, bottom and tau Yukawa couplings
are in equilibrium.
●
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
25/44
Imposing Particle – Sparticle
Equilibrium
Introduce common charge densities and k-factors, e.g.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
26/44
Analytic Approximation: Reduced set of Diffusion Equations
t- and b- Yukawa equilibrium: No production of 1st and 2nd
generation quarks through strong sphalerons (in contrast to earlier
studies).
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
27/44
Analytic Solution to Diffusion Equations
For the set of equations on the preceding slide, assume
●
are fast
● local baryon number conservation (same
for all (s)quarks)
● local lepton number conservation (requires
)
Then, can express all densities as proportional to Higgs(ino) density.
Sum of the left handed quark and lepton densities:
For comparison, not taking account of
:
The Higgs(ino) density itself can be calculated from integrating the
diffusion Green function over the source.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
28/44
Lepton-Mediated Electroweak
Baryogenesis
after inclusion of
thermal masses
In the region of Shaposhnikov suppression, there may still be a
sizable left-handed charge in form of third generation (s)leptons.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
29/44
Light Stop & Sbottom Baryogenesis
In the region of Shaposhnikov suppression, there may still be a
sizable left-handed charge in form of third generation (s)leptons.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
30/44
Light Stop Baryogenesis
[e.g. Carena, Nardini, Quiros, Wagner (2008)]
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
31/44
Sign Change
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
32/44
Summary
General numerical solution to diffusion equations.
● Analytic solution taking account of sparticle masses. In
devisong solution, found that
● supergauge equilibrium holds independent of gaugino
masses
●
Yukawa couplings cannot be neglected in general
● (S)bottom and (s)tau three-point interactions work against the
(s)top interactions when it comes to the transfer
of
.
●
Sign of the baryon asymmetry can depend on the
sparticle masses.
● Room four improvement:
, isospin breaking,
source & relaxation.
●
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
33/44
3
The Effect of the Sparticle Mass
Spectrum on the Conversion of
B–L to B
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
34/44
B–L to B Conversion
Global
symmetry of the (MS)SM is anomalous.
It is violated by the sphaleron (weak thermal instanton),
Up to the EWPT, this process is in equilibrium.
is non-anomalous.
Can be produced through all manner of mechanisms:
(Leptogenesis, Affleck & Dine, GUT-baryogenesis...)
NB: for EWBG
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
35/44
Situation before EWSB
Chemical equilibration occurs also far from the bubble
wall.
● Timescale
all interactions of the MSSM are in equilibrium (includes
quark flavor violation), see our discussion on diffusion
for particle – sparticle equilibrium, equilibrium of heavy
particles.
● Depending on the model, lepton flavor violating
interactions may or may not be in equilibrium.
●
•
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
36/44
The Standard Approach
[Harvey, Turner (1990)]
Assume first order phase transition, sphaleron transitions
end abruptly, can consider equilibration in the broken
phase.
● 2nd order phase transition and crossover have been
discussed by [Khlebnikov, Shaposhnikov (1996); Laine, Shaposhnikov (1999)].
● Harvey and Turner consider a non-supersymmetric
model with N Higgs doublets.
● In most papers on supersymmetric models, the nonSUSY formula for
, regardles of the sparticle
spectra. [See e.g. Davidson, Ibarra (2002); Davidson, Nardi, Nir (2008)]
●
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
37/44
Derivation of Conversion for MSSM in
Case of Lepton-Flavor Equilibrium
(following [Harvey, Turner (1990)])
Charge neutral Universe
-neutrality automatically implemented (no distinction between
isopartners)
Hypercharge:
We define the k-factors such that they do not include color- or
isospin multiplicities.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
38/44
Derivation Continued
Weak sphaleron in equilibrium:
Eliminate Q.
Fast three point interactions:
Eliminate H.
Only L is left.
(B–L) sets the absolute value.
Formula applicable whenever lepton number is violated or when or
when all generation have an identical particle mass spectrum.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
39/44
Examples
Standard
Model:
Non-SUSY
2 Higgs doublet:
[often used by leptogenesists for SUSY]
mSUGRA
motivated:
moderately light sleptons, e.g.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
40/44
Parameter Space for Lepton-Flavor
Violating Case
Trend: larger asymmetry for lighter left-handed squarks.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
41/44
Lepton Flavor Conserving Case
In general, lepton asymmetry is not evenly distributed between weak
lepton flavors.
There may be different asymmetries in different flavours, perhaps of
different sign.
are conserved seperately.
Have also derived analytic expression for this situation.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
42/44
Parameter Space for Lepton Flavor
Conserving Case
Have set
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
43/44
Conclusions
Sparticle masses (or, more generally the spectrum
beyond the Standard Model) close to the Electroweak
Scale have an order one impact on the baryon
asymmetry.
● Knowledge of this spectrum may or may not be
necessary to rule out a model.
● Will be necessary for checking consistency of a model or
answering the question whether a certain CP-violating
observable determines why the world is made of matter
rather than antimatter.
●
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
44/44
How Diffusion Works for Electroweak
Baryogenesis
– and what we Learn for Leptogenesis
Björn Garbrecht
Department of Physics
University of Wisconsin–Madison
Low Energy Precision Electroweak Physics in the LHC Era
INT, University of Washington, Seattle, October 7th 2008
Based on
Daniel J. H. Chung, BG, Michael J. Ramsey-Musolf, Sean Tulin, arXiv:0808.1144 [hep-ph]
Daniel J.H. Chung, BG, Sean Tulin, arXiv:0807.2283 [hep-ph]
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
1/44
Outline
1
Introduction and Overview
2
Diffusion and Chemical Equilibration in
Electroweak Baryogenesis
3
Seattle, 10/07/08
The Effect of the Sparticle Mass Spectrum
on the Conversion of
B–L to B
Björn Garbrecht – UW-Madison
2/44
1
Introduction & Overview
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
3/44
[Kuzmin, Rubakov, Shaposhnikov (1985);
Shaposhnikov, Turok, Voloshin
Electroweak Baryogenesis McLerran,
(1990), Cohen, Kaplan, Nelson (1990)]
strong first
order phase
transition
not with
Standard
Model
weak sphaleron
active
B
C
CP
symmetric
phase
Seattle, 10/07/08
broken
phase
Eq
weak sphaleron
inactive
Björn Garbrecht – UW-Madison
4/44
What Happens at the Wall
Generation of Higgsino
density from CPviolating local (?)
source
Diffusion & chemical
equilibration ahead of
the wall
Strong damping due to
the non-zero Higgs
VEV in the broken
phase
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
5/44
Kaplan, Nelson (1994),
Diffusion Equation for EWBG [Cohen,
Huet, Nelson (1995)]
inelastic
scatterings
relaxation in
broken phase
local (?) CPviolating source
The densities
denote charge densities (particle density minus
antiparticle density).
We denote the particular charge densities by the symbol representing
the particles, e.g.
These equations are (should be) derivable from first principles in the
Schwinger-Keldysh closed time path formalism, which is
appropriate for non-equilibrium problems.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
6/44
Quantum Transport: Closed Time-Path
Vacuum in-out amplitudes:
[Schwinger (1961);
Keldysh (1965)]
General states, in-in expactation values:
introduce closed time path
+
–
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
7/44
Kadanoff-Baym Equations

Schwinger-Dyson equations:
=
+
Diffusion equations
follow in the
coincidence limit.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
8/44
Quantum Source
& Relaxation
[Joyce, Prokopec, Turok (1995); Riotto (1998);
Carena, Moreno, Quiros, Seco, Wagner (2001-03);
Konstandin, Prokopec, Schmidt Weinstock(2001-06);
Cirigliano, Lee, Ramsey-Musolf (2004)]
quantum reflection/
transmission problem
MSSM:
higgsino/chargino
mixing mass-matrix:
source for EWB
different reflection/transmission
coefficients for
and
Eq
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
CP
9/44
Diffusion Operator
Ficks's Law:
Induced by elastic scatterings
Diffusion constants:
[Joyce, Prokopec, Turok (1996)]
In wall-frame coordinates:
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
10/44
Chemical Equilibration: Three Body Interactions
Supergauge
Yukawa and triscalar
[Chung, BG,
Ramsey-Musolf, Tulin (2008)]
[Cirigliano, Lee,
Ramsey-Musolf, Tulin (2006)]
Use thermally averaged three-body rates.
Four-body rates (additional gauge boson radiated) are important, when three
body interaction is kinematically forbidden.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
11/44
2
Diffusion and Chemical Equilibration
in Electroweak Baryogenesis
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
12/44
DIFFUSION
Yukawa
Supergauge
Source
Triscalar
Strong Sphaleron
Weak Sphaleron
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
13/44
How to Solve Diffusion Equations
Analytical:
[Huet, Nelson (1995)]
Assume certain interactions are in equilibrium.
● Use local B conservation.
● Take linear combinations, reduce to one diffusion equation that
can be solved using Green function.
●
What's new? [Chung, Garbrecht, Ramsey-Musolf and Tulin (2008)]
Numerical:
Full simulation of diffusion at the classical level in the symmetric
phase.
● We did not address issues arising when approaching the broken
phase: quantum source & relaxation, weak isospin breaking.
●
Analytical
Use the exact three body rates to determine systematically which
reactions are fast.
● Particle – sparticle equilibrium good almost everywhere.
● b-Yukawa equilibrium for
, -Yukawa for
● Sparticle mass-dependence of the baryon asymmetry.
●
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
14/44
Diffusion Equation: Asymptotic Behavior
Assume that all particle densities are proportional to the Higgs-density
(we will see to what extent this is true).
Then, there is a linear combination of diffusion equations, which takes the
form:
Solutions in absence of the source
outside the bubble wall.
Relaxation term
Inside the wall,
ahead of the wall (vanishing Higgs VEV).
Ahead of wall:
Inside wall:
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
15/44
Diffusion Time Scale
Within a time
:
charges diffuse
wall expands
At what time-scale does chemical equilibration happen?
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
16/44
k-Factors
Linear relation between charge
for
Seattle, 10/07/08
and chemical potential
Björn Garbrecht – UW-Madison
17/44
Three Body Rates
For example:
Can be derived from Kadanoff-Baym equations, but to this end classical (up to
quantum statistical factors).
Can analytically be reduced to one-dimensional integral without approximations.
[Cirigliano, Lee, Ramsey-Musolf, Tulin (2006)]
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
18/44
When is a Process fast?
“fast” forces this
term to be zero.
compare
with
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
19/44
Three Body Chemical Equilibration Rates
Yukawa
Triscalar
In particular, Yukawa rates of order
ahead of the wall.
Seattle, 10/07/08
may still ensure chemical equilibrium
Björn Garbrecht – UW-Madison
20/44
Yukawa Rates
top-quark:
bottom-quark:
-lepton:
always in equilibrium
may be in equilibrium if
(depending on the details)
may be in equilibrium if
For example,
mediates
Equilibrium implies
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
21/44
Particle – Sparticle Equilibrium
Gaugino interactions equilibrate particle and sparticle
chemical potentials.
What happens if e.g. gluinos are very heavy or the
conversion
is suppressed because of
?
See, for example, Yukawa and triscalar interactions
of the top (s)quark:
Particle – sparticle equilibrium is generically maintained if
Yukawa- and triscalar interactions are fast.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
22/44
Rate Dependence on
and
are suppressed for heavy
.
At tree-level,
For large
, back to the usual top-mediated
scenario (but this also suppresses the source).
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
23/44
Strong Sphaleron
strong sphaleron
(thermal QCD instanton)
baryon number conservation
(before weak sphaleron)
if
for all (s)quarks
“Shaposhnikov suppression”
does not follow if we break
squark mass degeneracy.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
24/44
Summary of our Equilibrium
Assumptions
Particles and sparticles are in equilibrium, independent of
sparticle masses.
● Strong sphaleron is in equilibrium.
● Gauge interactions are in equilibrium, in particular weak
interactions. Therefore, we use common Chemical
potentials for isodoublets and do not need chemical
potentials for
. Questionable as we
proceed into the broken phase.
● Top quark Yukawa coupling is in equilibrium.
● Depending on
, bottom and tau Yukawa couplings
are in equilibrium.
●
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
25/44
Imposing Particle – Sparticle
Equilibrium
Introduce common charge densities and k-factors, e.g.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
26/44
Analytic Approximation: Reduced set of Diffusion Equations
t- and b- Yukawa equilibrium: No production of 1st and 2nd
generation quarks through strong sphalerons (in contrast to earlier
studies).
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
27/44
Analytic Solution to Diffusion Equations
For the set of equations on the preceding slide, assume
●
are fast
● local baryon number conservation (same
for all (s)quarks)
● local lepton number conservation (requires
)
Then, can express all densities as proportional to Higgs(ino) density.
Sum of the left handed quark and lepton densities:
For comparison, not taking account of
:
The Higgs(ino) density itself can be calculated from integrating the
diffusion Green function over the source.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
28/44
Lepton-Mediated Electroweak
Baryogenesis
after inclusion of
thermal masses
In the region of Shaposhnikov suppression, there may still be a
sizable left-handed charge in form of third generation (s)leptons.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
29/44
Light Stop & Sbottom Baryogenesis
In the region of Shaposhnikov suppression, there may still be a
sizable left-handed charge in form of third generation (s)leptons.
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
30/44
Light Stop Baryogenesis
[e.g. Carena, Nardini, Quiros, Wagner (2008)]
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
31/44
Sign Change
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
32/44
Summary
General numerical solution to diffusion equations.
Analytic solution taking account of sparticle masses. In
devisong solution, found that
● supergauge equilibrium holds independent of gaugino
masses
●
Yukawa couplings cannot be neglected in general
● (S)bottom and (s)tau three-point interactions work against the
(s)top interactions when it comes to the transfer
of
.
●
●
Sign of the baryon asymmetry can depend on the
sparticle masses.
● Room four improvement:
, isospin breaking,
source & relaxation.
●
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
33/44
3
The Effect of the Sparticle Mass
Spectrum on the Conversion of
B–L to B
Seattle, 10/07/08
Björn Garbrecht – UW-Madison
34/44
B–L to B Conversion
Global
symmetry of the (MS)SM is anomalous.
It is violated by the sphaleron (weak thermal instanton),
Up to the EWPT, this process is in equilibrium.
is non-anomalous.
Can be produced through all manner of mechanisms:
(Leptogenesis, Affleck & Dine, GUT-baryogenesis...)
NB: for EWBG
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Situation before EWSB
Chemical equilibration occurs also far from the bubble
wall.
● Timescale
all interactions of the MSSM are in equilibrium (includes
quark flavor violation), see our discussion on diffusion
for particle – sparticle equilibrium, equilibrium of heavy
particles.
● Depending on the model, lepton flavor violating
interactions may or may not be in equilibrium.
●
•
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The Standard Approach
[Harvey, Turner (1990)]
Assume first order phase transition, sphaleron transitions
end abruptly, can consider equilibration in the broken
phase.
● nd
2 order phase transition and crossover have been
discussed by [Khlebnikov, Shaposhnikov (1996); Laine, Shaposhnikov (1999)].
● Harvey and Turner consider a non-supersymmetric
model with N Higgs doublets.
● In most papers on supersymmetric models, the nonSUSY formula for
, regardles of the sparticle
spectra. [See e.g. Davidson, Ibarra (2002); Davidson, Nardi, Nir (2008)]
●
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Derivation of Conversion for MSSM in
Case of Lepton-Flavor Equilibrium
(following [Harvey, Turner (1990)])
Charge neutral Universe
-neutrality automatically implemented (no distinction between
isopartners)
Hypercharge:
We define the k-factors such that they do not include color- or
isospin multiplicities.
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Derivation Continued
Weak sphaleron in equilibrium:
Eliminate Q.
Fast three point interactions:
Eliminate H.
Only L is left.
(B–L) sets the absolute value.
Formula applicable whenever lepton number is violated or when or
when all generation have an identical particle mass spectrum.
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Examples
Standard
Model:
Non-SUSY
2 Higgs doublet:
[often used by leptogenesists for SUSY]
mSUGRA
motivated:
moderately light sleptons, e.g.
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Parameter Space for Lepton-Flavor
Violating Case
Trend: larger asymmetry for lighter left-handed squarks.
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Lepton Flavor Conserving Case
In general, lepton asymmetry is not evenly distributed between weak
lepton flavors.
There may be different asymmetries in different flavours, perhaps of
different sign.
are conserved seperately.
Have also derived analytic expression for this situation.
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Parameter Space for Lepton Flavor
Conserving Case
Have set
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Conclusions
Sparticle masses (or, more generally the spectrum
beyond the Standard Model) close to the Electroweak
Scale have an order one impact on the baryon
asymmetry.
● Knowledge of this spectrum may or may not be
necessary to rule out a model.
● Will be necessary for checking consistency of a model or
answering the question whether a certain CP-violating
observable determines why the world is made of matter
rather than antimatter.
●
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