A solution to the li problem by the long lived stau
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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.

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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

  • Lifetime of stau

  • Formation time of

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


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