Effects of QCD bound states on relic abundance Seng Pei Liew (U. - - PowerPoint PPT Presentation

Effects of QCD bound states on relic abundance

Seng Pei Liew (U. of Tokyo)

(based on...)

work in progress with F. Luo (IPMU) Fermilab 2016 9/27

Consider colored particle with mass V ∼ αs r m & 1TeV ΛQCD Coulomb potential EB ∼ α2

sm & 10GeV

a−1 ∼ αsm & 100GeV Binding energy inverse Bohr radius in the early universe

we consider (perturbatively) QCD bound state way before QCD phase transition occurs, and its interaction with dark matter.

As an example, consider R-parity conserving Minimal Supersymmetric Standard Model (MSSM) Consider the R-odd lightest SUSY particle (LSP) as the lightest neutralino and is the dark matter. χ1

As an example, consider R-parity conserving Minimal Supersymmetric Standard Model (MSSM) Consider the R-odd lightest SUSY particle (LSP) as the lightest neutralino and is the dark matter. Consider produced thermally. χ1 χ1 dn1 dt + 3Hn1 = hσvi11(n2

1 neq2 1

)

time comoving
 DM 
 number density

equilibrium density n/s= constant

larger annihilation cross section -> smaller relic abundance freeze out Standard DM relic abundance calculation [Kolb, Turner ’90]

As an example, consider R-parity conserving Minimal Supersymmetric Standard Model (MSSM) Consider the R-odd lightest SUSY particle (LSP) as the lightest neutralino and is the dark matter. Consider produced thermally. χ1 χ1 Wino-like neutralino: ~3 TeV Higgsino-like neutralino: ~1 TeV

As an example, consider R-parity conserving Minimal Supersymmetric Standard Model (MSSM) Consider the R-odd lightest SUSY particle (LSP) as the lightest neutralino and is the dark matter. Consider produced thermally. χ1 χ1 Bino? depends on the masses of squarks & sleptons usually bino is overproduced if sfermions are heavy

As an example, consider R-parity conserving Minimal Supersymmetric Standard Model (MSSM) Consider the R-odd lightest SUSY particle (LSP) as the lightest neutralino and is the dark matter. Consider produced thermally. Specifically, consider LSP coannihilating with an almost mass-degenerate R-odd SUSY particle (not necessarily the second lightest neutralino). Coannihilation becomes vital. χ1 χ1 χ2

How coannihilation works? [Griest, Seckel ’91] χ2 has large annihilation cross section with itself or conditions: χ1 χ2χ2 ↔ SMSM χ2χ1 ↔ SMSM

How coannihilation works? χ1 χ2 has large annihilation cross section with itself or conditions:

can convert to efficiently.

χ2 χ1 χ2χ2 ↔ SMSM χ2χ1 ↔ SMSM χ2SM ↔ χ1SM [Griest, Seckel ’91]

Boltzmann equations dn1 dt + 2Hn1 = hσvi11(n2

1 n2 1eq)

dn2 dt + 3Hn2 = hσvi22(n2

2 neq2 2

) For simplicity, consider fast conversion means that neq

i

= gi (miT/2π)3/2 e−mi/T note that n2/n1 = neq

2 /neq 1 = g2m3/2 2

g1m3/2

1

exp(−(m2 − m1)/T)

Boltzmann equations dn dt + 3Hn =

2

X

i,j=1

hσviij→SM neq

i neq j

n2

eq

• n2 n2

eq

• assuming fast conversion

χ2SM ↔ χ1SM dnχ dt + 3Hnχ = hσviχχ→SM ⇣ n2

χ neq χ 2⌘

compare with without coannihilation

hσvieff

call this

n ≡ n1 + n2 defining

Two limits dn dt + 3Hn =

2

X

i,j=1

hσviij→SM neq

i neq j

n2

eq

• n2 n2

eq

• hσvieff

call this

m2 m1 : hσvieff ' hσvi11→SM

m2 = m1 : hσvieff = g2

1hσvi11→SM + g2 2hσvi22→SM + 2g1g2hσvi12→SM

(g1 + g2)2

neq

i

= gi (miT/2π)3/2 e−mi/T note that

We consider dark matter accompanied by an almost mass-degenerate colored particle.

If is colored (squark or gluino in MSSM) QCD Sommerfeld effect is important χ2 χ2 χ2 χ2 χ2 g g g g tree-level annihilation non-perturbative (Sommerfeld)
 effect that modifies 
 the initial-state wave function

see e.g. [De Simone et al. ‘14]

If is colored (squark or gluino in MSSM) formation of QCD bound state of could be important as well χ2 χ2 ˜ g˜ g ↔ ˜ Rg, ˜ R ↔ gg [Ellis et al. ‘15] ˜ t˜ t ↔ ˜ ηg, ˜ η ↔ gg

for gluino for stop

Compare recombination process e−p ↔ Hγ

If is colored (squark or gluino in MSSM) formation of QCD bound state of could be important as well χ2 χ2 ˜ g˜ g ↔ ˜ Rg, ˜ R ↔ gg [Ellis et al. ‘15] ˜ t˜ t ↔ ˜ ηg, ˜ η ↔ gg

for gluino for stop

Compare recombination process e−p ↔ Hγ Γann & Γ˜

t/˜ g

note: bound state formation is important only when

bound state annihilation rate decay rate

˜ g˜ g ↔ ˜ Rg, ˜ R ↔ gg ˜ t˜ t ↔ ˜ ηg, ˜ η ↔ gg note that bound state annihilation removes 2 R-odd particles, thus helps reducing DM density Bound state

gluino bound state

• stop bound state

call the colored particle X and bound state dnη dt + 3Hnη = −Γη(nη − neq

η ) + Γbsf(n2 X − neq2 X

nη neq

η

)

η

• ne needs to solve the coupled Boltzmann eq.

including the bound state

η

bound state annihilation rate bound state formation rate bound state dissociation rate

dnη dt + 3Hnη = −Γη(nη − neq

η ) + Γbsf(n2 X − neq2 X

nη neq

η

) dn1 dt + 2Hn1 = hσvi11(n2

1 n2 1eq)

dnX dt + 3HnX = hσviXX(n2

X neq2 X ) Γbsf(n2 X neq2 X

nη neq

η

) Solving the coupled Boltzmann equations

dnη dt + 3Hnη = −Γη(nη − neq

η ) + Γbsf(n2 X − neq2 X

nη neq

η

) dn1 dt + 2Hn1 = hσvi11(n2

1 n2 1eq)

dnX dt + 3HnX = hσviXX(n2

X neq2 X ) Γbsf(n2 X neq2 X

nη neq

η

) Solving the coupled Boltzmann equations

bound state number density is exponentially suppressed. 
 One can set LHS to zero as an approximation. (the validity of this approx. has been checked numerically)

dnη dt + 3Hnη = −Γη(nη − neq

η ) + Γbsf(n2 X − neq2 X

nη neq

η

) dn1 dt + 2Hn1 = hσvi11(n2

1 n2 1eq)

dnX dt + 3HnX = hσviXX(n2

X neq2 X ) Γbsf(n2 X neq2 X

nη neq

η

) Solving the coupled Boltzmann equations

dn dt + 3Hn '

2

X

i,j=1

hσviij→SM neq

i neq j

n2

eq

• n2 n2

eq

• Then, the Boltzmann equation is modified by adding

the following terms:

hσviXX→ηg hΓiη→gg hΓiη→gg + hΓiηg→XX (n2

X neq2 X )

bound state annihilation rate bound state formation rate bound state dissociation rate

dn dt + 3Hn '

2

X

i,j=1

hσviij→SM neq

i neq j

n2

eq

• n2 n2

eq

• becomes unimportant at low temperature compared to

Then, the Boltzmann equation is modified by adding the following terms:

because gluon is not energetic enough to dissociate the bound state

hσviXX→ηg hΓiη→gg hΓiη→gg + hΓiηg→XX (n2

X neq2 X )

η → gg ηg → XX

dn dt + 3Hn '

2

X

i,j=1

hσviij→SM neq

i neq j

n2

eq

• n2 n2

eq

• becomes unimportant at low temperature compared to

Then, the Boltzmann equation is modified by adding the following terms:

because gluon is not energetic enough to dissociate the bound state

(at temperature T < binding energy)

hσviXX→ηg hΓiη→gg hΓiη→gg + hΓiηg→XX (n2

X neq2 X )

η → gg ηg → XX

dn dt + 3Hn '

2

X

i,j=1

hσviij→SM neq

i neq j

n2

eq

• n2 n2

eq

• becomes unimportant at low temperature compared to

Then, the Boltzmann equation is modified by adding the following terms:

because gluon is not energetic enough to dissociate the bound state late-time “annihilation” is important! One needs to solve 
 the Boltzmann eqs. numerically

hσviXX→ηg hΓiη→gg hΓiη→gg + hΓiηg→XX (n2

X neq2 X )

(at temperature T < binding energy) ηg → XX η → gg

Calculation of bound state formation/dissociation rate

Use Coulomb approximation to describe the bound state with SU(3) quadratic casimir of constituent particle SU(3) quadratic casimir of bound state Calculation of bound state formation/dissociation rate

Use Coulomb approximation with

Hγ → e−p photoelectric effect: Consider photoelectric effect as an analogy

Hγ → e−p photoelectric effect: H = 1 2m(~ p + e ~ A)2 Electromagnetic Hamiltonian Consider photoelectric effect as an analogy

Hγ → e−p photoelectric effect: H = 1 2m(~ p + e ~ A)2 Electromagnetic Hamiltonian H ≈ p2 2m + e m ~ A · ~ p Consider photoelectric effect as an analogy

Hγ → e−p photoelectric effect: H = 1 2m(~ p + e ~ A)2 Electromagnetic Hamiltonian H ≈ p2 2m + e m ~ A · ~ p calculate the matrix hf| e m ~ A · ~ p |ii Consider photoelectric effect as an analogy bound state
 wave function free particle
 wave function

Hγ → e−p photoelectric effect: H = 1 2m(~ p + e ~ A)2 Electromagnetic Hamiltonian H ≈ p2 2m + e m ~ A · ~ p calculate the matrix hf| e m ~ A · ~ p |ii rescale with appropriate color factors Consider photoelectric effect as an analogy

Bound state formation rate is related to the
 dissociation rate via the Milne relation 
 (or principle of detailed balance)

bound state formation rate bound state dissociation rate

scalar triplet bound state (Stoponium) ˜ t˜ t → gη˜

t

we consider


• nly the 


ground state formation rate dissociation rate

500 1000 1500 2000 2500 3000 3500 4000 10 20 30 40 50

m[GeV] mass splitting m [GeV]

Scalar triplet (stop) coannihilation DM mass DM-stop mass splitting Colored bands
 show parameter
 region matching
 the observed
 DM relic abundance

500 1000 1500 2000 2500 3000 3500 4000 10 20 30 40 50

m[GeV] mass splitting m [GeV]

no Sommerfeld no bound-state effect Sommerfeld effect no bound-state effect Sommerfeld plus bound-state effect Sommerfeld plus 2x bound-state effect

Scalar triplet (stop) coannihilation Colored bands
 show parameter
 region matching
 the observed
 DM relic abundance

2000 4000 6000 8000 10000 12000 14000 20 40 60 80 100 120 140

m[GeV] mass splitting m [GeV]

500 1000 1500 2000 2500 10 20 30 40 50 60

m[GeV] mass splitting m [GeV]

2000 4000 6000 8000 10000 50 100 150 200

m[GeV] mass splitting m [GeV]

fermion triplet fermion octet (gluino) scalar octet (fast conversion implicitly assumed) Coannihilation with other types


• f colored particle

gluino coannihilation (with conversion taken into account appropriately) [Ellis et al. ‘15]

[Low, Wang ‘14] bino/stop coan. 5-sigma discovery becomes impossible at 100 TeV collider previous estimate DM mass significance a short comment on 100 TeV collider prospects

[Low, Wang ‘14] bino/stop coan. 5-sigma discovery becomes impossible even at 100 TeV collider

500 1000 1500 2000 2500 3000 3500 4000 10 20 30 40 50

m[GeV] mass splitting m [GeV]

DM mass significance

Other implications of bound-state effects: BBN constraints on long-lived particles

500 1000 5000 104 10-17 10-16 10-15 10-14 10-13 10-12

m˜

t[GeV]

n˜

t/s

τ˜

t ∼ 0.1 − 102s

τ˜

t ∼ 102 − 107s

no Sommerfeld no bound-state effect Sommerfeld effect Sommerfeld plus bound-state effects

see e.g. [Kawasaki et al ‘04]

Summary Bound state of the colored particles can increase the effective annihilation cross section significantly We have considered dark matter accompanied by an almost mass-degenerate colored particle.

Backup

for gluino for stop

σbsfvrel Sann(σannvrel) ∼ 1.4 (vrel → 0)

0.01 0.10 1 10 0.01 0.10 1 10 100 EB/T <bsfv>N

Attractive-repulsive

= = =

How large are bound-state effects? (κ = 8)

hσbsfvreli σannvrel

binding energy/temperature