Electroweak baryogenesis in the MSSM and nmssm

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1 Electroweak baryogenesis in the MSSM and nmssm Thomas Konstandin, IFAE Barcelona in collaboration with S. Huber, T. Prokopec, M.G. Schmidt and M. Seco

2 Outline 1 Introduction The Basic Picture 2 Semiclassical Transport Quantum-Transport from Kadanoff-Baym Equations 3 EWBG in the MSSM The BAU EDMs Conclusions 4 EWBG in the nmssm The BAU EDMs Conclusions 5 Conclusions

3 The Basic Picture Sakharov conditions Baryogenesis is one of the cornerstones of the Cosmological Standard Model and tries to explain the observed baryon asymmetry of the Universe (BAU) η = n B n B = s The celebrated Sakharov conditions state the necessary ingredients for baryogenesis: Sakharov conditions Baryon number violation Charge (C) and charge parity conjugation (CP) are no symmetries non-equilibrium

4 The Basic Picture Baryogenesis in the SM at low energies Baryogenesis would be most compelling if it would operate at low energies in the Standard Model. Sakharov conditions condition CP violation non-equilibrium B violation? CKM matrix expansion / PT? No

5 The Basic Picture Baryogenesis in the SM at low energies Baryogenesis would be most compelling if it would operate at low energies in the Standard Model. Sakharov conditions condition CP violation non-equilibrium B violation? CKM matrix expansion / PT? No however No No Sphaleron Electroweak baryogenesis nevertheless aims at producing the BAU at electroweak energy scales by minimally extending the SM (SUSY?).

6 The Basic Picture C- and B-violation: The sphaleron process In the hot universe B- and C-violation is present due to sphaleron processes ( temperature induced instanton ). The effective sphaleron vertex c L s L s L t L b L B = 3, L = 3, N CS = 1 B L conserving B + L violating d L Sphaleron b L Exponentially suppressed after the phase transition (m W ) d L u L ν e ν µ ν τ Topological effect of the SU(2) gauge sector EWPT is last chance of baryogenesis (φ < T).

7 The Basic Picture The electroweak phase transition The Higgs sector is an essential part of the SM that breaks the gauge group and makes the SM particles massive V h at T 100 GeV h V h at T 100 GeV V h at T 100 GeV h V h at T 0 GeV h h

8 The Basic Picture The electroweak phase transition At high temperatures, the free energy is symmetric and the symmetries are unbroken (Weinberg ( 74)) V h at T 100 GeV h V h at T 100 GeV V h at T 100 GeV h V h at T 0 GeV h h

9 The Basic Picture The electroweak phase transition The free energy of the plasma determines the characteristics of the electroweak phase transition V h at T 100 GeV h V h at T 100 GeV h cross-over e.g. SM V h at T 0 GeV V h at T 100 GeV h h

10 The Basic Picture The electroweak phase transition The free energy of the plasma determines the characteristics of the electroweak phase transition V h at T 100 GeV h V h at T 100 GeV h first-order phase transition e.g. MSSM with light stop V h at T 0 GeV V h at T 100 GeV h h

11 The Basic Picture Some comments on the electroweak phase transition Comments on bubble nucleation Order parameter of the phase transition is the Higgs vev h During the phase transition the particles change their masses This is a violent process (v c) For EWBG, the PT has to be strong, φ T In the SM, there is only a cross-over (m Higgs > 60 GeV) Kajantie, Laine, Rummukainen, Shaposhnikov ( 96) A first-order electroweak phase transition requires BSM physics.

12 The Basic Picture CP Violation in Mass Matrices CP violation: Chargino masses in the MSSM and bmssm In the MSSM case the mass matrix of the charged higgsinos/winos is: ( M m = 2 g h 2 (x µ ) ) g h 1 (x µ ) µ c with M 2 and µ c containing a CP-odd complex phase. The Higgs fields are during the phase transition space-time dependent.

13 The Basic Picture Picture of Electroweak Baryogenesis Shaposhnikov ( 87) φ symmetric phase broken phase Higgs vev l w v w r

14 The Basic Picture Picture of Electroweak Baryogenesis Shaposhnikov ( 87) symmetric phase broken phase CP-violating effects Higgs vev CP-violating particle density (charginos, quarks, leptons)

15 The Basic Picture Picture of Electroweak Baryogenesis Shaposhnikov ( 87) symmetric phase sphalerons active broken phase sphalerons not active Higgs vev CP-violating particle density

16 The Basic Picture Picture of Electroweak Baryogenesis Cohen, Kaplan, Nelson ( 90) symmetric phase sphalerons active broken phase sphalerons not active Higgs vev diffusion/transport CP-violating particle density

17 The Basic Picture Picture of Electroweak Baryogenesis Cohen, Kaplan, Nelson ( 90) symmetric phase broken phase CP violation & transport Higgs vev CP-violating particle density

18 Outline 1 Introduction The Basic Picture 2 Semiclassical Transport Quantum-Transport from Kadanoff-Baym Equations 3 EWBG in the MSSM The BAU EDMs Conclusions 4 EWBG in the nmssm The BAU EDMs Conclusions 5 Conclusions

19 Quantum-Transport from Kadanoff-Baym Equations Challenge in EWBG Connection between macroscopic and microscopic scales CP violation is a microscopic/quantum effect produced by the interaction of single quanta in the plasma. Transport is a macroscopic/classical effect based on statistical physics and particle densities.

20 Quantum-Transport from Kadanoff-Baym Equations Classical Boltzmann Equations The system is described by the particle distribution function n(x,v,t) on classical phase space. The Boltzmann equations are based on a particle picture. If on the particles in the plasma acts an external force F, one obtains n(x + vδt,v + Fδt,t + δt) n(x,v,t) = collisions/interactions, and thus ( t + v + F v )n(x,v,t) = collisions/interactions. How can CP violation be incorporated in this classical picture? In which semi-classical limit can one obtain a phase-space from a quantum theory?

21 Quantum-Transport from Kadanoff-Baym Equations Summary of Approaches to Transport Semi-classical force / WKB approach Joyce, Prokopec, Turok, hep-ph/ , hep-ph/ Joyce, Cline, Kainulainen, hep-ph/ Huber, Schmidt, hep-ph/ Perturbative mixing / mass insertion approach Carena, Moreno, Quiros, Seco, Wagner, hep-ph/ Carena, Quiros, Seco, Wagner, hep-ph/ Cirigliano, Profumo, Ramsey-Musolf, hep-ph/ Kadanoff-Baym approach Kainulainen, Prokopec, Schmidt, Weinstock, hep-ph/ Konstandin, Prokopec, Schmidt, hep-ph/ Konstandin, Prokopec, Schmidt, Seco, hep-ph/ Huber, Konstandin, Prokopec, Schmidt, hep-ph/

22 Quantum-Transport from Kadanoff-Baym Equations Summary of former Approaches Approach Cline/Joyce Carena/Moreno/Quiros Kainulainen Seco/Wagner CP-violation dispersion relation local source term WKB perturbation theory basis mass eigenbasis flavour eigenbasis quasi-particles charginos higgsinos/winos transport classical classical Boltzmann type diffusion mixing not included in the source not in the diffusion order second order first order comment momentum? basis?, finite? Both approaches fill the gap between classical transport and quantum physics by hand and contain certain ambiguities.

23 Quantum-Transport from Kadanoff-Baym Equations Transport Theory and EWBG Starting point of our formalism are the Kadanoff-Baym equations that are the statistical analogue to the Schwinger-Dyson equations: d 4 z ( S 1 0 (x µ,z µ ) Σ(x µ,z µ ) ) S(z µ,y µ ) = ½δ(x µ y µ ) where the self-energy Σ and the Green function S contain an additional 2 2 structure from the in-in-formalism ( Σ t Σ > ) ( Σ Σ = = ++ Σ + ) ( S t S > ), S =. Σ < Σ t Σ + Σ Of these four entries only two are independent and S < encodes the particle distribution function. S < S t

24 Quantum-Transport from Kadanoff-Baym Equations Transport Theory and EWBG In Wigner space we have X µ = (x µ + z µ )/2, k µ = FT(x µ z µ ). Then the Kadanoff-Baym equations read with e i {S 1 0 Σ,S} = 0. 2 {A,B} := X µa kµ B kµ A X µb and all quantities are functions of X µ and k µ. The super-/subscripts <, >, R, A denote the additional 2 2 structure of the in-in formalism. Without interaction e i {S 1 0 (z),s< } = 0, e i {S 1 0 (z),s A} = 0

25 Quantum-Transport from Kadanoff-Baym Equations Particle densities Using the KMS condition and the correct normalization one obtains in equilibrium is < = 2π sign(k 0 )δ(k 2 m 2 ) n(k 0 ), n(k 0 ) = 1 exp(βk 0 ) 1 The particle density can (also away from equilibrium) be read off from S < dk 0 2π 2ik 0 S < = n(k,t,x). k 0 >0 Thus the Wigner space yields in the semi-classical limit the usual phase space (however, not necessarily positive).

26 Quantum-Transport from Kadanoff-Baym Equations A Simple Example Free bosonic theory with one flavour and a constant mass: e i {S 1 0,S< } = 0, S 1 0 = k 2 m 2. The hermitian/anti-hermitian parts are the so called constraint/kinetic equations: (k 2 m 2 )S < = 0, k µ Xµ S < = 0 which at low energies k µ = (m,mv) is of Boltzmann type m( t + v )S < = 0.

27 Quantum-Transport from Kadanoff-Baym Equations Gradient Expansion Consider e i {S 1 0 (z),s< } = 0, S 1 0 = k 2 m 2 (z) Since the background in the MSSM is weakly varying (l w 20/T c ) the Moyal star product can be simplified by the semi-classical approximation k X 1 T c l w e i 1 i The simplest example for a transport equation in a varying background is for one bosonic flavour with real mass (up to first order in ) (k 2 m 2 )S < = 0 (k µ µ 1 2 ( zm 2 ) kz )S < = 0.

28 Quantum-Transport from Kadanoff-Baym Equations CP violation Prokopec, Schmidt, Weinstock ( 01) Consider a fermion with complex mass term and S 1 0 = k/ P L m(z) P R m (z) m(z) = m(z) e iθ(z). Then the kinetic equation up to second order in reads [ k µ µ 1 2 ( zm 2 ) kz s 2k 0 z (m 2 z θ) kz ] S < s (k µ,z) = 0 which leads to CP-violating particle densities. This agrees with the findings of Cline et al. in the WKB framework.

29 Quantum-Transport from Kadanoff-Baym Equations Fermionic Systems After spin projection the fermionic system of equations reads 2i k 0 k 0 t + k S s 0 k (2iskz + s z) Ss 3 2im he i 2 z kz s S 1 2imae i 2 z kz s S 2 = 0 0 2i k 0 k 0 t + k S1 s k (2skz is z) Ss 2 2im he i 2 z kz s S0 + 2m i ae 2 z kz s S3 = 0 0 2i k 0 k 0 t + k S s 2 k + (2skz is z) Ss 1 2m he i 2 z kz s S 3 2imae i 2 z kz s S 0 = 0 0 2i k 0 k 0 t + k S3 s k (2iskz + s z) Ss 0 + 2m he i 2 z kz s S2 2m i ae 2 z kz s S1 = 0, 0 where S 0...S 3 are 2 2 matrices in flavour space and s denotes the spin.

30 Quantum-Transport from Kadanoff-Baym Equations Transport in the Chargino sector Konstandin, Prokopec, Schmidt ( 04) The chargino transport equations for the left/right handed densities are of the form Comments k µ µ S < + i 2[ m 2,S <] + sources/forces = collisions S < is a 2 2 flavor matrix The term i 2[ m 2,S <] will lead to an oscillatory behaviour of the off-diagonal particle densities, similar to neutrino oscillations with frequency (m 2 1 m2 2 )/k z. The source contains first order contributions that correspond to the sources in the approach of Carena et al. The source contains second order contributions that correspond to the sources in the approach of Cline et al.

31 Quantum-Transport from Kadanoff-Baym Equations Sources for EWBG This approach resembles two mechanisms of EWBG from former approaches Joyce, Prokopec, Turok ( 96) Cline, Joyce, Kainulainen ( 97, 00) Fromme, Huber ( 06) The dispersion shift source from the WKB approach: { S (2) m m m m, kz S <}. Carena, Moreno, Quiros, Seco, Wagner ( 00) Carena, Quiros, Seco, Wagner ( 02) Cirigliano, Profumo, Ramsey-Musolf ( 06) Cirigliano, Ramsey-Musolf, Tulin, Lee ( 06) Sources from flavor mixing effects, e.g. [ S (1) m m m m, kz S <]. CP violation from mixing appears only on the off-diagonal in the mass eigenbasis.

32 Quantum-Transport from Kadanoff-Baym Equations Determination of the BAU Huet, Nelson ( 95) The missing parts to determine the baryon asymmetry of the universe are: q s L s L t L ~ h c L b L Y and Sphaleron b L d L ~ q d L u L ν e ν µ ν τ

33 Quantum-Transport from Kadanoff-Baym Equations Diffusion and the Sphaleron The originally used system of diffusion equations is of the form (for a recent treatment see Chung, Garbrecht, Tulin ( 08)) [ v w n Q = D q n Q Γ nq Y n T n ] [ H + n h nq Γ m n ] T k Q k T k H k Q k T [ 6Γ ss 2 n Q n T + 9 n ] Q + n T + Sources k Q k T k B where n Q,n T,n H,n h denote particle densities, Γ ss,γ m,γ Y interaction rates, k Q,k T,k H statistical factors and D q a diffusion constant. However, these equations are classical and back-reactions on the charginos cannot be taken into account in our approach.

34 Quantum-Transport from Kadanoff-Baym Equations Advantages and Disadvantages No ambiguities No divergences WKB and mixing effects Flavor oscillations

35 Quantum-Transport from Kadanoff-Baym Equations Advantages and Disadvantages No ambiguities No divergences WKB and mixing effects Flavor oscillations No quantum transport in quark sector No back-reactions on the charginos

36 Quantum-Transport from Kadanoff-Baym Equations Advantages and Disadvantages No ambiguities No divergences WKB and mixing effects Flavor oscillations No quantum transport in quark sector No back-reactions on the charginos Almost no diffusion in for the mixing contributions in the chargino sector

37 Outline 1 Introduction The Basic Picture 2 Semiclassical Transport Quantum-Transport from Kadanoff-Baym Equations 3 EWBG in the MSSM The BAU EDMs Conclusions 4 EWBG in the nmssm The BAU EDMs Conclusions 5 Conclusions

38 The BAU EWBG in the MSSM CP violation: Chargino masses in the MSSM In the MSSM case the mass matrix of the charged higgsinos/winos is: ( M m = 2 g h 2 (x µ ) ) g h 1 (x µ ) µ c with M 2 and µ c containing a CP-odd complex phase. The Higgs fields are during the phase transition space-time dependent. Phase transition in the MSSM In the MSSM the strength of the phase transition depends mostly on the loop effects of the bosons. A strong phase transition fulfilling the current mass bounds on the Higgs is possible if the stops are relatively light, m top m stop.

39 The BAU Numerical Results Konstandin, Prokopec, Schmidt, Seco ( 05) Parameters chosen: v w = 0.05, l w = 20/T c, CP-phase maximal. m A µ c M2 Due to the flavor oscillations, EWBG requires in the MSSM quasi-degenerate chargino masses.

40 EDMs eedm: One Loop Contributions One-loop contributions to the EDMs are of the following form in the MSSM γ d L d R dl dr d g d d g d These contributions have to be suppressed by several orders Small CP phases? Large sfermion masses? (Split SUSY)

41 EDMs eedm: Two Loop Contributions Chang, Chang, Keung ( 02), Pilaftsis ( 02) Y. Li, S. Profumo and M. Ramsey-Musolf ( 08) Potentially dangerous diagrams are of the following form with a chargino in the loop γ(k, µ) l + p l + q ϕ (p) l γ(q, ν) f L f R (q) f R These contribution depend on the chargino masses and tan(β) and EDM Arg(M 2 µ) BAU.

42 Conclusions Conclusions in the MSSM MSSM electroweak baryogenesis is a constrained scenario A light stop to acquire a strong first-order phase transition The condition µ c M GeV of the a priori unrelated parameters M 2 and µ c A large CP-violating phase that is testable by next generation EDM experiments

43 Outline 1 Introduction The Basic Picture 2 Semiclassical Transport Quantum-Transport from Kadanoff-Baym Equations 3 EWBG in the MSSM The BAU EDMs Conclusions 4 EWBG in the nmssm The BAU EDMs Conclusions 5 Conclusions

44 The BAU Why is the nmssm interesting? Panagiotakopoulos, Pilaftsis ( 02) The nearly Minimal Supersymmetric Standard Model has the following effective superpotential W nmssm = λŝĥ1 Ĥ2 m2 12 λ Ŝ + W MSSM, and has the virtues to solve the µ-problem of the MSSM by introducing a dynamical µ-term µ = λ S. In this model singlet self-couplings are forbidden by a R -symmetry. The resulting model has neither problems with the stability of the hierarchy nor with domain walls (but λ might develop a Landau pole).

45 The BAU Higgs potential of the nmssm Including soft SUSY breaking terms, the Higgs potential reads V 0 = m 2 1 H 1 H 1 + m 2 2 H 2 H 2 + m 2 s S 2 + λ 2 H 1 H 2 2 +λ 2 S 2 (H 1 H 1 + H 2 H 2) + ḡ2 8 (H 2 H 2 H 1 H 1) 2 + g2 2 H 1 H t s (S + h.c.) + (a λ SH 1 H 2 + h.c.) m12(h 2 1 H 2 + h.c.). Compared to the MSSM, the new singlet terms help to strengthen the EWPT and introduce additional sources of CP violation.

46 The BAU CP violation: Chargino masses in the nmssm Huber, Konstandin, Prokopec, Schmidt ( 06) In the nmssm the µ term contains a z-dependent complex phase µ(z) = λ S = λφ s (z)e iqs(z) In the nmssm second order sources dominate The dynamical parameter µ = λ S leads to novel and dominating sources Charginos are generically non-degenerate (M 2 µ) Thin wall profiles

47 The BAU Phase Transition due to Tree Level Dynamics In the MSSM, a first order phase transition requires a light stop. Menon, Morrissey, Wagner ( 04) In a simplified model without CP violation and the thermal effective potential V T = αt 2 φ 2, the criterion for a first order phase transition is ms 2 < 1 λ λ 2 t s m s ã m s. This criterion seems to be decisive, even if one-loop effects and CP violation is taken into account 5 spectrum+phase transition Eq. (27) 5 spectrum Eq. (27) lhs 3 lhs rhs rhs

48 The BAU Thin Bubble Walls In the MSSM, the bubble wall is rather thick l w T c (Moreno, Quiros, Seco ( 98)). In the nmssm, bubble walls are comparably thin, what enhances the second order sources in the gradient expansion 100 # of models 80 # l w / T c For numerical methods of multi dimensional phase transitions see Konstandin, Huber ( 06).

49 EDMs eedm: One Loop Contributions One-loop contributions to the EDMs are of the following form γ d L d R dl dr d g d As in the MSSM, these contributions have to be suppressed by large sfermion masses in the first two generations. This restriction can be relaxed, since in the nmssm CP violation could be small in the broken phase. BAU EDM Arg(M 2 µ). d Arg(µ) in the nmssm, Arg(M 2 µ) in the MSSM, g d

50 EDMs eedm: Two Loop Contributions Chang, Chang, Keung ( 02), Pilaftsis ( 02) Potentially dangerous diagrams are of the following form with a chargino in the loop γ(k, µ) l + p l + q ϕ (p) l γ(q, ν) f L f R (q) f R However in the nmssm, these contributions are not as severe as in the MSSM, since usually tan(β) O(1).

51 EDMs Numerical Results A numerical analysis of the BAU leads to the following result (sets passed LEP constraints and have a first order phase transition) # of models # of models 200 # of models # of models # # η 10 The left (right) plot shows the generated BAU for M 2 = 1 TeV (M 2 = 200 GeV). 50% (63%) of the models are in accordance with observation. The lower models fulfill the the EDM bounds with 1 TeV sfermion masses, 4.8 % (6.2 %). η 10

52 Conclusions Conclusions in the nmssm EWBG in the nmssm is very promising Strong first order phase transition due to tree-level dynamics η 10 1 for most of the parameter space EDMs eventually small due to small Arg(M 2 µ c ) two loop EDMs relatively small due to tan(β) O(1)

53 Outline 1 Introduction The Basic Picture 2 Semiclassical Transport Quantum-Transport from Kadanoff-Baym Equations 3 EWBG in the MSSM The BAU EDMs Conclusions 4 EWBG in the nmssm The BAU EDMs Conclusions 5 Conclusions

54 Final Remarks on EWBG Compared to other baryogenesis mechanisms, EWBG has the disadvantages that Requirements EWBG requires a non-trivial extension of the Standard Model. In particular a strong first-order phase transition is a quite demanding and not well motivated requirement. Technicalities A quantitative analysis of EWBG is complicated and many details are not yet settled.

55 Final Remarks on EWBG Compared to other baryogenesis mechanisms, EWBG has the virtues that Testability=Predictivity EWBG can be tested in the near future. EDM experiments of the next generation have the potential to rule out (almost) all models of EWBG. Back on the envelope EWBG predicts quite generically the correct BAU: η = Γ sphaleron l w T 10 9 δ CP 4π exp( m χ/t)