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 m & 1TeV � Λ QCD in the early universe V ∼ α s Coulomb potential r E B ∼ α 2 Binding energy s m & 10GeV a − 1 ∼ α s m & 100GeV inverse Bohr radius 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. χ 1

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. χ 1 Consider produced thermally. χ 1 dn 1 1 � n eq 2 dt + 3 Hn 1 = �h σ v i 11 ( n 2 ) 1

Standard DM relic abundance calculation equilibrium n/s= constant density comoving 
 DM 
 number density freeze out [Kolb, Turner ’90] time larger annihilation cross section -> smaller relic abundance

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. χ 1 Consider produced thermally. χ 1 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. χ 1 Consider produced thermally. χ 1 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. χ 1 Consider produced thermally. χ 1 Specifically, consider LSP coannihilating with an almost mass-degenerate R-odd SUSY particle (not necessarily the second χ 2 lightest neutralino). Coannihilation becomes vital.

[Griest, Seckel ’91] How coannihilation works? conditions: χ 2 has large annihilation cross section with itself or χ 1 χ 2 χ 2 ↔ SMSM χ 2 χ 1 ↔ SMSM

[Griest, Seckel ’91] How coannihilation works? conditions: χ 2 has large annihilation cross section with itself or χ 1 χ 2 χ 2 ↔ SMSM χ 2 χ 1 ↔ SMSM can convert to efficiently. χ 2 χ 1 χ 2 SM ↔ χ 1 SM

Boltzmann equations For simplicity, consider dn 1 dt + 2 Hn 1 = �h σ v i 11 ( n 2 1 � n 2 1 eq ) dn 2 2 � n eq 2 dt + 3 Hn 2 = �h σ v i 22 ( n 2 ) 2 fast conversion means that 1 = g 2 m 3 / 2 n 2 /n 1 = n eq 2 /n eq 2 exp( − ( m 2 − m 1 ) /T ) g 1 m 3 / 2 1 = g i ( m i T/ 2 π ) 3 / 2 e − m i /T n eq note that i

Boltzmann equations χ 2 SM ↔ χ 1 SM assuming fast conversion defining n ≡ n 1 + n 2 n eq i n eq 2 dn n 2 � n 2 j X � � dt + 3 Hn = � h σ v i ij → SM eq n 2 eq i,j =1 h σ v i e ff call this dn χ ⇣ 2 ⌘ χ � n eq n 2 dt + 3 Hn χ = �h σ v i χχ → SM compare with χ without coannihilation

n eq i n eq 2 dn n 2 � n 2 j X � � dt + 3 Hn = � h σ v i ij → SM eq n 2 eq i,j =1 h σ v i e ff call this Two limits m 2 � m 1 : h σ v i e ff ' h σ v i 11 → SM m 2 = m 1 : h σ v i e ff = g 2 1 h σ v i 11 → SM + g 2 2 h σ v i 22 → SM + 2 g 1 g 2 h σ v i 12 → SM ( g 1 + g 2 ) 2 = g i ( m i T/ 2 π ) 3 / 2 e − m i /T n eq note that i

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

If is colored (squark or gluino in MSSM) � χ 2 QCD Sommerfeld effect is important g χ 2 tree-level annihilation χ 2 g χ 2 g non-perturbative (Sommerfeld) 
 effect that modifies 
 the initial-state wave function χ 2 g see e.g. [De Simone et al. ‘14]

If is colored (squark or gluino in MSSM) � χ 2 formation of QCD bound state of � χ 2 could be important as well [Ellis et al. ‘15] g ↔ ˜ Rg, ˜ for gluino g ˜ ˜ R ↔ gg ˜ t ˜ t ↔ ˜ η g, ˜ for stop η ↔ gg e − p ↔ H γ Compare recombination process

If is colored (squark or gluino in MSSM) � χ 2 formation of QCD bound state of � χ 2 could be important as well [Ellis et al. ‘15] g ↔ ˜ Rg, ˜ for gluino g ˜ ˜ R ↔ gg ˜ t ˜ t ↔ ˜ η g, ˜ for stop η ↔ gg e − p ↔ H γ Compare recombination process note: bound state formation is important only when Γ ann & Γ ˜ t/ ˜ g bound state decay rate annihilation rate

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

call the colored particle X and bound state η one needs to solve the coupled Boltzmann eq. � η including the bound state dn η n η X − n eq 2 dt + 3 Hn η = − Γ η ( n η − n eq η ) + Γ bsf ( n 2 ) n eq X η bound state bound state bound state annihilation rate formation rate dissociation rate

Solving the coupled Boltzmann equations dn 1 dt + 2 Hn 1 = �h σ v i 11 ( n 2 1 � n 2 1 eq ) dn X n η X � n eq 2 X � n eq 2 + 3 Hn X = �h σ v i XX ( n 2 X ) � Γ bsf ( n 2 ) n eq X dt η dn η n η X − n eq 2 dt + 3 Hn η = − Γ η ( n η − n eq η ) + Γ bsf ( n 2 ) n eq X η

Solving the coupled Boltzmann equations dn 1 dt + 2 Hn 1 = �h σ v i 11 ( n 2 1 � n 2 1 eq ) dn X n η X � n eq 2 X � n eq 2 + 3 Hn X = �h σ v i XX ( n 2 X ) � Γ bsf ( n 2 ) n eq X dt η dn η n η X − n eq 2 dt + 3 Hn η = − Γ η ( n η − n eq η ) + Γ bsf ( n 2 ) n eq X η 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)

Solving the coupled Boltzmann equations dn 1 dt + 2 Hn 1 = �h σ v i 11 ( n 2 1 � n 2 1 eq ) dn X n η X � n eq 2 X � n eq 2 + 3 Hn X = �h σ v i XX ( n 2 X ) � Γ bsf ( n 2 ) n eq X dt η dn η n η X − n eq 2 dt + 3 Hn η = − Γ η ( n η − n eq η ) + Γ bsf ( n 2 ) n eq X η

Then, the Boltzmann equation is modified by adding � the following terms: n eq i n eq 2 dn n 2 � n 2 j X � � dt + 3 Hn ' � h σ v i ij → SM eq n 2 eq i,j =1 h Γ i η → gg X � n eq 2 ( n 2 �h σ v i XX → η g X ) h Γ i η → gg + h Γ i η g → XX bound state bound state formation rate bound state dissociation rate annihilation rate

Then, the Boltzmann equation is modified by adding � the following terms: n eq i n eq 2 dn n 2 � n 2 j X � � dt + 3 Hn ' � h σ v i ij → SM eq n 2 eq i,j =1 h Γ i η → gg X � n eq 2 ( n 2 �h σ v i XX → η g X ) h Γ i η → gg + h Γ i η g → XX η g → XX η → gg becomes unimportant at low temperature compared to because gluon is not energetic enough to dissociate the bound state

Then, the Boltzmann equation is modified by adding � the following terms: n eq i n eq 2 dn n 2 � n 2 j X � � dt + 3 Hn ' � h σ v i ij → SM eq n 2 eq i,j =1 h Γ i η → gg X � n eq 2 ( n 2 �h σ v i XX → η g X ) h Γ i η → gg + h Γ i η g → XX (at temperature T < binding energy) η g → XX η → gg becomes unimportant at low temperature compared to because gluon is not energetic enough to dissociate the bound state

Then, the Boltzmann equation is modified by adding � the following terms: n eq i n eq 2 dn n 2 � n 2 j X � � dt + 3 Hn ' � h σ v i ij → SM eq n 2 eq i,j =1 h Γ i η → gg X � n eq 2 ( n 2 �h σ v i XX → η g X ) h Γ i η → gg + h Γ i η g → XX (at temperature T < binding energy) η g → XX η → gg becomes unimportant at low temperature compared to because gluon is not energetic enough to dissociate the bound state late-time “annihilation” is important! One needs to solve 
 the Boltzmann eqs. numerically

Calculation of bound state formation/dissociation rate

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

Use Coulomb approximation with

Consider photoelectric effect as an analogy H γ → e − p photoelectric effect:

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

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

Consider photoelectric effect as an analogy H γ → e − p photoelectric effect: 1 p + e ~ A ) 2 H = 2 m ( ~ Electromagnetic Hamiltonian H ≈ p 2 2 m + e ~ A · ~ p m h � f | e ~ A · ~ p | � i i calculate the matrix m free particle 
 bound state 
 wave function wave function