Phys. Lett. B646 (2007) 34, (hep-ph/0610249)

1. Non-perturbative effect on thermal relic abundance of dark matter Masato Senami (University of Tokyo, ICRR) Collaborated with Junji Hisano (ICRR) Shigeki Matsumoto (KEK) Minoru Nagai (ICRR) Osamu Saito (ICRR, KEK) Phys. Lett. B646 (2007) 34, (hep-ph/0610249)

2. Previously, wino dark matter mass is believed as , if wino is thermal relic dark matter. But, this is not true If we include nonperturbative effects (Sommerfeld enhancement) for thermal relic wino dark matter.

3. Dark matter by WMAP Beyond the standard model Yamanaka’s talk • Non-baryonic cold dark matter • No candidate in the standard model • Supersymmetric (SUSY) model • Lightest SUSY particle (LSP) : Bino, Wino, Gravitino … • Universal extra dimension (UED) model • Lightest Kaluza-Klein(KK) particle (LKP) : KK photon …

4. Which model is the answer? e.g. Moriyama-san’s talk One criterion : Constraint for model parameter In this work, we calculate wino relic abundance precisely. • Direct and indirect detection • Collider signature (LHC, ILC) • Prediction from thermal relic scenario • Precise data by WMAP (within 10%) • Precise calculation of relic density is required.

5. Wino dark matter SU(2) triplet Mass spectrum (Mixing with other neutralino is suppressed by heavy wino mass ) Mass Other superparticles Non-thermal production • Superpartner of W boson • Pure wino LSP • Anomaly mediation • Thermal relic scenario • SU(2)L gauge interaction • Degeneracy : neutral and charged wino

6. Thermal relic scenario Thermal averaged Freeze out Increasing n/s = constant (Net dark matter density) Comoving number density • Large cross section reduces relic abundance. • Degeneracy between and • Coannihilation should be considered. Equilibrium density equilibrium 1 1000 10 100 m/T (time ) Cross section : average by weighted with degree of freedom

7. Annihilation cross section Sommerfeld enhancement : SU(2) : SU(2), U(1)em SU(2) interaction is important if wino is much heavier than the weak gauge bosons. is important. (Thermally averaged effective annihilation cross section) If dark matter (or coannihilating particle) particle has a gauge charge, non-perturbative effects are important.

8. Sommerfeld enhancement : U(1) annihilation + Enhancement factor - annihilation Photons Wave functions are affected by attractive force and modified from plane wave. This enhances the annihilation cross section. Coulomb correction

9. Sommerfeld enhancement : SU(2) Diagrams have an additional factor a2m/mw for each W boson exchange DM ● ● ● + + + W W W DM Non-perturbative effects are important. m : wino mass. • W-boson exchange • For heavy wino, W-boson mass is negligible. • W-boson exchanges modify wave functions.

10. Enhancement Thermally averaged cross section The cross section is increased by 20-30% even at the freeze-out temperature. 10-26 m = 2.8 TeV [cm3/s] Decoupling Non-perturbative 10-25 For m/T = 102 - 105, the cross section is increased. Perturbative 10-24 1 102 104 106 108 At m/T = 105, charged wino is decoupled. m / T Freeze-out Temperature dependence with fixed m

11. n/nTree Delayed freeze-out 1 At the freeze-out,the enhancement of thecross section is about 20%. m = 2.8 TeV 0.8 n/nTree The abundance is reduced by about 20%. 0.6 Late time annihilation 1 108 106 102 104 For m/T = 102 - 105, the cross section is increased. m / T The abundance is reduced by more than 40% compared to perturbative results. The abundance is reduced by about 20%. Since the cross section depends on the temperature in a non-trivial way, we should solve the Boltzmann equation numerically.

12. Relic abundance of Wino Perturbative 0.2 WMAP 0.1 Non-perturbative 0 3 2 1 m (TeV) Allowed region : 2.7 TeV < m < 3.0 TeV

13. Summary and Discussion • Wino dark matter in thermal relic scenario • Nonperturbative reduces the relic abundance • 2.7 TeV < m < 3.0 TeV(c.f. perturbative result 1.9 TeV < m < 2.3 TeV) • Other dark matter candidates • Higgsino about 10% • Bino-stau coannihilation at most 1% • KK dark matter in UED model within 4%

14. Other dark matter candidates O(10)% at most 1% within 4% • Higgsino LSP (SU(2) and U(1) charge) • Higgsino is doublet in SU(2)x1/4 compared with wino • Bino-Stau coannihilation (U(1) charge for only stau) • almost cancel each other • KK dark matter in UED model (U(1) charge for E(1)) • Gluino NLSP • Strong nonperturbative effects by QCD • Involved by QCD phase transition

15. Enhancement factor The resonance appears at m=2.4TeVdue to the bound state, which are composed by and pairs. For m=2.4TeV, the binding energy of the bound state is almost zero. So, resonances appear at these masses. 10 m/T = 2000 5 m/T = 200 m/T = 20 The enhancement is more significant for smaller temperature. 1 1 2 3 m (TeV) Mass dependence of the cross section normalized by the perturbative one

16. Annihilation of Winos Only the s-wave annihilation is relevant to the DM phen. S = 0 S = 1 S = 0 S = 1 Only S = 0 Only S = 0 Annihilation processes we have to calculate.

17. Strategy to calculation Schwinger-Dyson eq. For Wino-like DM pair Forward Scattering amplitude. Annihilation cross section Im. part Optical theorem Derivation of the Schwinger-Dyson eq. MSSM action Integrate all field except c0 and c- fields. Expanding c0 and c- by their velocities (NR-Lagrangian is produced) Introducing auxiliary fields for the pairs composed of c0 and c- , and derive the 2-body states effective action by integrating out c0 and c–. Schrödinger equation Derive the Schwinger-Dyson eq. for the 2-body states. Schwinger-Dyson equation

18. Schrödinger equations for Wino DM Schrödinger equation (S = 0) (S = 0, 1) (S = 1) & (S = 0) det V < 0

19. Sommerfeld factor Cross section formula Sommerfeld factor If we neglect the non-perturbative effect (V = 0), the factors become 1 and annihilation cross sections coincide with perturbative results.