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

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.

Dark matter 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 … Beyond the standard model Yamanaka’s talk by WMAP

Which model is the answer? 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. e.g. Moriyama-san’s talk One criterion : Constraint for model parameter In this work, we calculate wino relic abundance precisely.

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

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

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

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

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

Enhancement Temperature dependence with fixed m Perturbative Non-perturbative [cm 3 /s] m = 2.8 TeV Thermally averaged cross section m / T Freeze-out Decoupling The cross section is increased by 20-30% even at the freeze- out temperature For m/T = , the cross section is increased. At m/T = 10 5, charged wino is decoupled.

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

Relic abundance of Wino Allowed region : 2.7 TeV < m < 3.0 TeV Perturbative Non-perturbative WMAP m (TeV)

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 coannihilationat most 1%  KK dark matter in UED modelwithin 4%

Other dark matter candidates 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 O(10)% at most 1% within 4%

Enhancement factor Mass dependence of the cross section normalized by the perturbative one m/T = 2000 m (TeV) m/T = 200 m/T = 20 The resonance appears at m=2.4TeV due 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. The enhancement is more significant for smaller temperature

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

Schwinger-Dyson eq. For Wino-like DM pair MSSM action Forward Scattering amplitude. Annihilation cross section Im. part Optical theorem Derivation of the Schwinger-Dyson eq. Integrate all field except  0 and   fields. Derive the Schwinger-Dyson eq. for the 2-body states. Schwinger-Dyson equation Schrödinger equation Expanding  0 and   by their velocities (NR-Lagrangian is produced) Introducing auxiliary fields for the pairs composed of  0 and  , and derive the 2-body states effective action by integrating out  0 and  –. Strategy to calculation

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

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.