1. A Solution to the Li Problem by the Long Lived Stau7
Masato Yamanaka
Collaborators
Toshifumi Jittoh, Kazunori Kohri, Masafumi Koike, Joe Sato, Takashi Shimomura
Phys. Rev. D78 : , and arXiv : 0904.××××

2. 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

3. Introduction

4. 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) ]

5. Li problem Li problem ( 4.15 )×10 ( 1.26 )×107
Li problem
Theory
( )×10
＋0.49
－10
－0.45
A. Coc, et al., astrophys. J. 600, 544(2004)
Observation
( )×10
＋0.29
－10
－0.24
P. Bonifacio, et al., astro-ph/
Predicted Li abundance ≠ observed Li abundance 7 7 7
Li problem

6. t Requirement for solving the Li problem ~ Stau7
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 t ~
Stau
Able to survive still the BBN era
Superpartner of tau lepton
Exotic processes are introduced

7. The small m scenario and long lived stau

8. Setup
Minimal Supersymmetric Standard Model (MSSM) with R-parity
Lightest Supersymmetric Particle (LSP) c ～
Lightest neutralino c ～ B ～ ～ W ～ H u ～ H d = N 1 + N 2 + N 3 + N 4
Next Lightest Supersymmetric Particle (NLSP) ～ t
Lighter stau ～ t ～ t e -i g t ～ t =
cos q t +
sin q t L R

9. Dark matter and its relic abundance
Dark Matter (DM)
Neutral, stable (meta-stable), weak interaction, massive
Good candidate : LSP neutralino
DM relic abundance W DM h 2
= ±
m n ∝ × DM DM m DM =
(constant) 1 n

10. m Naïve calculation of neutralino relic density m = n
DM reduction process
Neutralino pair annihilation c ～
SM particle
Not enough to reduce the DM number density m DM =
(constant) 1 n
Not consistent ! m c ～
> 46 GeV
[ PDG (J. Phys. G 33, 1 (2006)) ]

11. + Coannihilation mechanism m < ~ 10 % t – DM reduction process
[ K. Griest and D. Seckel PRD43(1991) ]
DM reduction process
Neutralino pair annihilation
+ stau-neutralino coannihilation c ～
SM particle c ～
SM particle t +
Possible to reduce DM number density efficiently !
Requirement for the coannihilation to work m
LSP
NLSP –
< ~
10 %

12. Long lived stau dm ≡ NLSP mass ｰ LSP mass < tau mass
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 !!

13. t BBN era survive until BBN era
Stau lifetime
BBN era ～ t
survive until BBN era
Stau provides additional processes to reduce the primordial Li abundance !! 7

14. Solving the Li problem by the long lived stau7

15. Destruction of nuclei with free stau～ t
A(Z)
A(Z±1) n c
Negligible due to
Cancellation of destruction
Smallness of interaction rate
Interaction time scale (sec)

16. Stau-nucleus bound state
Key ingredient for solving the Li problem 7
Negative-charged stau can form a bound state with nuclei
nuclear radius ~
･･stau ・・nucleus
Formation rate
Solving the Boltzmann Eq.
New processes
Stau catalyzed fusion
Internal conversion in the bound state

17. Weakened coulomb barrier
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

18. Constraint from stau catalyzed fusion
Standard BBN process
Catalyzed BBN process
Catalyzed BBN cause over production of Li
Constraint on stau life time

19. UP UP Internal conversion t Hadronic current
Closeness between stau and nucleus UP
Overlap of the wave function : UP
Interaction rate of hadronic current : t ~ +
does not form a bound state
No cancellation processes

20. G Internal conversion rate | | y | | y = | | y p a a ( ) s v ( ) s v =
The decay rate of the stau-nucleus bound state
: The overlap of the wave functions
: Cross section × relative velocity
| | y 2
( ) s v G IC
| | y 2 ・ =
( ) s v
The bound state is in the S-state of a hydrogen-like atom
| | y 2 = 1 p a
nucl 3 a
nucl
= nuclear radius = ～
1.2 × A
1/3
( ) s v
is evaluated by using
ft-value
( ) s v ∝
( ft )
– 1
ft-value of each processes
7Be → 7Li ・･･ ft = sec (experimental value)
7Li → 7He ･･･ similar to 7Be → 7Li (no experimental value)

21. Interaction rate of Internal conversion
Interaction time scale (s)
Very short time scale
significant process for reducing Li abundance ! 7

22. Scattering with background particles
New interaction chain reducing Li and Be 7
Internal conversion 7 He ～
Internal conversion t ～ c , n t 7 Be 7 Li
Scattering with background particles ～ ～ t c , n t 4
He, 3
He,
D, etc
proton, etc

23. Y n Y n Numerical result for solving the Li problem t t t s t s7
0.01
0.1
Mass difference between stau and neutralino (GeV) 10
-16
-12
-13
-14
-15
-11 Y t ~
m = 300 GeV DM t ~ n Y t ~ s t ~ n
: number density of stau s
: entropy density 1

24. Allowed region Y t Y t d m = (100 – 120) MeV = (7 – 10) 10 (7 – 10) 10
(7 – 10) ×
-13
MeV
The Li problem is solved 7 Y
Strict constraint on m and d t ~ Y
-13 d m =
(100 – 120) MeV t ~ =
(7 – 10) 10 ×

25. Allowed region
( Be) 7 t ~
Internal conversion of occur when back ground protons are still energetic
Produced Li are destructed by energetic proton 7
Back ground particles are not energetic when Li are emitted 7
Produced Li are destructed by internal conversion 7

26. Insufficient reduction
Forbidden region
Overclosure of the universe
Stau decay before forming a bound state
Lifetime of stau　
Formation time of
Bound states are not formed sufficiently
Over production of Li due to stau catalyzed fusion 6
Insufficient reduction

27. Relic density of stau at the BBN era

28. 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

29. The evolution of the number density
Boltzmann Eq. for neutralino and stau
i, j
: stau and neutralino
X, Y
: standard model particle
n (n ) eq
: actual (equilibrium) number density H
: Hubble expansion rate

30. 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

31. 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

32. The evolution of the number density
Boltzmann Eq. for neutralino and stau
Decay and inverse decay processes i
j Y

33. 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

34. ≠ Significant process for stau relic density calculation n , <<
Boltzmann Eq. for neutralino and stau
Annihilation process
Exchange process n i j X Y ,
<<
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

35. m t t Exchange process ~ g c c g T 20 ~ ~ DM freeze out temperature
5 GeV - 50 GeV T DM ~ m 20
( DM mass GeV GeV )
Exchange process for NLSP stau and LSP neutralino t ~ g c t ~ g c
After DM freeze out, number density ratio is still varying

36. n Numerical calculation Y s n s
Boltzmann Eq. for the number density of stau and neutralino Y n s i n i
: number density of particle i s
: entropy density

37. 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

38. Tau decoupling temperature
Condition ( tau is in thermal bath )
[ Production rate ] > [ decay rate, Hubble expansion rate ]
Production processes of tau g t t l t q
Tau decoupling temperature
~ 120 MeV

39. Summary

40. Stau-catalyzed fusion Internal conversion process
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

41. Appendix

42. Effective Lagrangian mixing angle between and CP violating phase
Weak coupling
Weinberg angle

43. Bound ratio

44. 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