7. A Solution to the Li Problem by the Long Lived Stau. Masato Yamanaka Collaborators Toshifumi Jittoh, Kazunori Kohri, Masafumi Koike, Joe Sato, Takashi Shimomura. Phys. Rev. D78 : 055007, 2008 and arXiv : 0904.××××. Contents. 1, Introduction.
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7
A Solution to the Li Problem by the Long Lived Stau
Masato Yamanaka
Collaborators
Toshifumi Jittoh, Kazunori Kohri, Masafumi Koike, Joe Sato, Takashi Shimomura
Phys. Rev. D78 : 055007, 2008 and arXiv : 0904.××××
Contents
1, Introduction
2, The small m scenario and long lived stau
d
3, Solving the Li problem by the long lived stau
7
4, Relic density of stau at the BBN era
5, Summary
Introduction
Big Bang Nucleosynthesis (BBN)
Yellow box
Observed light element abundances
[CMB] vertical band
Cosmic baryon density obtained from WMAP data
Color lines
Light element abundances as predicted by the standard BBN
Prediction of the light elements abundance
consistent !
Primordial abundance obtained from observations
[ B.D.Fields and S.Sarkar (2006) ]
7
Li problem
Theory
＋0.49
－10
( 4.15 )×10
－0.45
A. Coc, et al., astrophys. J. 600, 544(2004)
Observation
－10
( 1.26 )×10
＋0.29
－0.24
P. Bonifacio, et al., astro-ph/0610245
7
7
Predicted Li abundance ≠ observed Li abundance
7
Li problem
~
t
Requirement for solving the Li problem
7
Able to destroy a nucleus
Requirement for solving the Li problem
To occur at the BBN era
Not a Standard Model (SM) process
Possible to be realized in a framework of minimal supersymmetric standard model !
Coupling with a hadronic current
Stau
Able to survive still the BBN era
Superpartner of tau lepton
Exotic processes are introduced
The small m scenario and long lived stau
d
～
～
c
c
N
1
～
N
W
2
N
3
～
0
～
H
～
u
B
0
H
d
N
4
～
～
～
～
t
t
t
t
q
q
t
t
g
-i
e
t
Setup
Minimal Supersymmetric Standard Model (MSSM) with R-parity
Lightest Supersymmetric Particle (LSP)
Lightest neutralino
=
+
+
+
Next Lightest Supersymmetric Particle (NLSP)
Lighter stau
cos
sin
=
+
L
R
W
h
2
= 0.1099 ± 0.0062
DM
1
m
(constant)
=
n
DM
DM
Dark matter and its relic abundance
Dark Matter (DM)
Neutral, stable (meta-stable), weak interaction, massive
Good candidate : LSP neutralino
DM relic abundance
m n
×
∝
DM
DM
http://map.gsfc.nasa.gov
SM particle
～
～
c
c
SM particle
1
m
m
(constant)
=
> 46 GeV
～
n
DM
c
DM
Naïve calculation of neutralino relic density
DM reduction process
Neutralino pair annihilation
Not enough to reduce the DM number density
Not consistent !
[ PDG 2006 (J. Phys. G 33, 1 (2006)) ]
SM particle
SM particle
～
～
～
c
c
c
SM particle
SM particle
～
t
–
<
~
10 %
m
m
LSP
LSP
m
NLSP
Coannihilation mechanism
[ K. Griest and D. Seckel PRD43(1991) ]
DM reduction process
+ stau-neutralino coannihilation
Neutralino pair annihilation
+
Possible to reduce DM number density efficiently !
Requirement for the coannihilation to work
Long lived stau
Attractive parameter region in coannihilation case
dm ≡ NLSP mass ｰ LSP mass < tau mass
(1.77GeV)
NLSP stau can not two body decay
Stau has a long lifetime due to phase space suppression !!
Stau lifetime
BBN era
～
t
survive until BBN era
Stau provides additional processes to reduce the primordial Li abundance !!
7
Solving the Li problem by the long lived stau
7
Interaction time scale (sec)
A(Z±1)
～
t
±
A(Z)
n
t
～
c
Destruction of nuclei with free stau
Negligible due to
Cancellation of destruction
Smallness of interaction rate
~
nuclear radius
･･stau ・・nucleus
Stau-nucleus bound state
Key ingredient for solving the Li problem
7
Negative-charged stau can form a bound state with nuclei
Formation rate
Solving the Boltzmann Eq.
New processes
Stau catalyzed fusion
Internal conversion in the bound state
Stau catalyzed fusion
[ M. Pospelov, PRL. 98 (2007) ]
･･stau ・・nucleus
Weakened coulomb barrier
Nuclear fusion
( ): bound state
Ineffective for reducing 7Li and 7Be
∵ stau can not weaken the barrieres of Li3+ and Be4+ sufficiently
Constraint from stau catalyzed fusion
Standard BBN process
Catalyzed BBN process
Catalyzed BBN cause over production of Li
Constraint on stau life time
~
t
Internal conversion
Hadronic current
Closeness between stau and nucleus
UP
Overlap of the wave function :
UP
Interaction rate of hadronic current :
+
does not form a bound state
No cancellation processes
: The overlap of the wave functions
: Cross section × relative velocity
G
=
・
IC
2
2
2
| |
| |
| |
y
y
y
1
=
p
a
3
nucl
a
– 1
( ft )
∝
nucl
1/3
1.2 × A
( )
( )
( )
( )
s
s
s
s
v
v
v
v
Internal conversion rate
The decay rate of the stau-nucleus bound state
The bound state is in the S-state of a hydrogen-like atom
～
= nuclear radius =
is evaluated by using
ft-value
ft-value of each processes
7Be → 7Li ・･･ ft = 103.3 sec (experimental value)
7Li → 7He ･･･ similar to 7Be → 7Li (no experimental value)
Interaction rate of Internal conversion
Interaction time scale (s)
Very short time scale
significant process for reducing Li abundance !
7
7
7
New interaction chain reducing Li and Be
Internal conversion
7
He
～
Internal conversion
～
c
t
n
,
t
7
Be
7
Li
Scattering with background particles
～
～
c
t
n
,
t
4
He,
3
He,
D, etc
proton, etc
-11
10
-12
10
Y
~
t
-13
10
n
n
m = 300 GeV
Y
~
~
~
DM
t
t
t
-14
10
-15
10
-16
10
0.01
0.1
Mass difference between stau and neutralino (GeV)
Numerical result for solving the Li problem
7
s
: number density of stau
s
: entropy density
1
~
~
t
t
-13
(7 – 10) 10
×
Allowed region
100 - 120 MeV
7
The Li problem is solved
Y
Strict constraint on m and
d
-13
Y
(7 – 10) 10
(100 – 120) MeV
=
d
m
=
×
( Be)
7
~
t
Allowed region
Internal conversion of occur when back ground protons are still energetic
7
Produced Li are destructed by energetic proton
Back ground particles are not energetic when Li are emitted
7
7
Produced Li are destructed by internal conversion
Forbidden region
Overclosure of the universe
Stau decay before forming a bound state
Bound states are not formed sufficiently
6
Over production of Lidue to stau catalyzed fusion
Insufficient reduction
Relic density of stau at the BBN era
For the prediction of parameter point
Stau relic density at the BBN era
Not a free parameter !
Value which should be calculated from other parameters
For example
DM mass, mass difference between stau and neutralino, and so on
Calculating the stau relic density at the BBN era
We can predict the parameters more precisely, which provides the solution for the Li problem
7
The evolution of the number density
Boltzmann Eq. for neutralino and stau
i, j
: stau and neutralino
: standard model particle
X, Y
n (n )
eq
: actual (equilibrium) number density
: Hubble expansion rate
H
The evolution of the number density
Boltzmann Eq. for neutralino and stau
Annihilation and inverse annihilation processes
i j
X Y
Sum of the number density of SUSY particles is controlled by the interaction rate of these processes
The evolution of the number density
Boltzmann Eq. for neutralino and stau
Exchange processes by scattering off the cosmic thermal background
i X
j Y
These processes leave the sum of the number density of SUSY particles and thermalize them
The evolution of the number density
Boltzmann Eq. for neutralino and stau
Decay and inverse decay processes
i
j Y
Can we simplify the Boltzmann Eq. with ordinal method ?
In the calculation of LSP relic density
All of Boltzmann Eq. are summed up
( all of SUSY particles decay into LSP )
∴
Single Boltzmann Eq. for LSP DM
Absence of exchange terms ( cancel out )
∴
In the calculation of relic density of long lived stau
We are interesting to relic density of NLSP
Obviously we can not use the method !
We must solve numerically a coupled set of differential Eq. for stau and neutralino
n
i
n
Y
n
n
X
j
,
<<
,
Significant process for stau relic density calculation
Boltzmann Eq. for neutralino and stau
Annihilation process
Exchange process
Due to the Boltzmann factor,
Annihilation process rate << exchange process rate
≠
Freeze out temperature of sum of number density of SUSY particles
Freeze out temperature of number density ratio of stau and neutralino
m
~
T
20
DM
DM
~
~
~
g
c
t
t
t
~
t
c
g
Exchange process
DM freeze out temperature
5 GeV - 50 GeV
( DM mass 100 GeV - 1000 GeV )
Exchange process for NLSP stau and LSP neutralino
After DM freeze out, number density ratio is still varying
n
Y
i
s
i
: number density of particle
i
Numerical calculation
Boltzmann Eq. for the number density of stau and neutralino
n
i
s
: entropy density
Numerical result (prototype)
Tau decoupling temperature (MeV)
60
60
80
80
Mass difference between tau and neutralino (MeV)
50
100
50
100
Ratio of stau relic density and current DM density
15 %
3 %
8 %
1.5 %
Significant temperature for the calculation of stau number density
・
Tau decoupling temperature
・
Freeze out temperature of exchange processes
g
g
t
t
t
t
l
l
t
t
q
q
Tau decoupling temperature
Condition ( tau is in thermal bath )
[ Production rate ] > [ decay rate, Hubble expansion rate ]
Production processes of tau
Tau decoupling temperature
~ 120 MeV
Summary
We investigated solution of the Li problem in the MSSM, in which the LSP is lightest neutralino and the NLSP is lighter stau
Long lived stau can form a bound state with nucleus and provides new processes for reducing Li abundance
Stau-catalyzed fusion
Internal conversion process
We obtained strict constraint on the mass difference between stau and neutralino, and the yield value of stau
Stau relic density at the BBN era strongly depends on the tau decoupling temperature and mass difference between stau and neutralino
Our goal is to predict the parameter point precisely by calculating the stau relic density and nucleosynthesis including new processes
Appendix
Effective Lagrangian
mixing angle between and
CP violating phase
Weak coupling
Weinberg angle
Bound ratio
Red : consistent region with DM abundance and mass difference < tau mass
Yellow : consistent region with DM abundance and mass difference > tau mass
The favored regions of the muon anomalous magnetic moment at 1s, 2s, 2.5s, 3s confidence level are indicated by solid lines