Ryuichiro Kitano (SLAC), "Moduli Decays and Gravitinos" - cosmo 06

cosmo06.ucdavis.edu

Ryuichiro Kitano (SLAC), "Moduli Decays and Gravitinos" - cosmo 06

hep-ph/0604140Moduli Decays and GravitinosRyuichiro Kitano (SLAC)collaboration with M.Dine, A.Morisse and Y.ShirmanSeptember 25, 2006, Cosmo06


Moduli ProblemModuli: light and long-lived particle in String theoryCosmological history:ρρ R ∝ a −4 ρ φ ∝ a −3decayradiation dominateφ − dominateaIf m φ ∼ m 3/2 ∼ 100 GeVτ φ ∼⎡⎢⎣14πm 3 φM 2 Pl⎤⎥⎦−1∼ 10 7 sec--> Excluded by BBNWe need radiation dominated universe at τ ∼ 10 −2 sec


m φ ≫ 100 TeV τ φ ∼ 10 −2 sec --> BBN OKBut... φ → ψ 3/2 ψ 3/2decaydangerous for BBN!!decayLSPLSPoverclosure[Endo, Hamaguchi,Takahashi 06][Nakamura, Yamaguchi 06]This is studied by Moroi and Randall[Moroi, Randall 99]Moroi - RandallΓ 3/2 ∼ 1 m 2 3/2m φ4π MPl2B 3/2 ∼ O⎛⎜⎝m 2 3/2m 2 φ⎞⎟⎠Recent calculationΓ 3/2 ∼ 14πm 3 φM 2 PlB 3/2 ∼ O (1)Are we doomed?We found that the branching ratio depends on details ofSUSY breaking sector and microscopic physics of moduli.


Interaction of gravitinos(Unitary gauge calculation)[Endo, Hamaguchi,Takahashi 06][Nakamura, Yamaguchi 06]L int = −e G/2 [ψ µ σ µν ψ ν + h.c.](G = K + ln |W | 2 )Taylor expansion− eG/22 G φφ [ψ µ σ µν ψ ν + h.c.]G φ ∼F φm 3/2∼ m 3/2m φSupersymmetric mass termψ µ = k µm 3/2u(k): Longitudinal modee G/2 = m 3/2⇒ M 3/2 ∼ m φ⇒ Γ 3/2 ∼ m3 φM 2 Plquite generally⇒ B 3/2 ∼ O(1)No suppression by m 3/2


Simple modelK = φ † φ + a(φ + φ † ) + z † zW = m φ2 φ2 + µ 2 z + cSupersymmetric mass termV = e G (|G φ | 2 + |G z | 2 − 3)(G = K + ln |W | 2 )SUSY breaking sectorF z = µ 2 ∼ m 3/2V φ = V z = V = 0⇒ G φ = − 3am 3/2m φG z = √ 3Naive estimation looks fine......


Caution!Calculation in unitary gauge is extremely dangerous for order estimateAnalogy: U(1) gauge theoryV = m 2 H|H| 2 + λ 4(|h| 2 − v 2) 2+ ǫ 2 (H † h + h.c.)(m H ≫ v)Let’s consider H → W L W L decayNaive guess:W LHg 2W LHW L〈H〉W LM ∼ g 2 v⎛⎜⎝ǫ 2m 2 H⎞⎟⎠⎛⎜⎝k µgv⎞ ⎛⎟ ⎜⎠ ⎝gv⎞k ′µ⎟⎠ ∼ m2 Hv⎛⎜⎝ǫ 2m 2 H⎞⎟⎠But this is incorrect. Equivalence theorem saidsW LHW L= HM ∼ λv⎛⎜⎝⎞ǫ 2m 2 ⎟⎠ ∼ m2 hH v⎛⎜⎝ǫ 2m 2 H⎞⎟⎠GGH〈h〉hλGG


Acturally in unitary gaugeHW LHW L= + HhW LW L〈H〉W L〈h〉W Lcancellation!O(m 2 H) − O(m 2 H) = O(m 2 h)Lesson: Goldstone picture is safer.


Goldstino LagrangianL χχ = − 1 ⎛2 eG/2 ⎜⎝G zz + 1 ⎞3 G zG z − Γ z zzG z − Γ φ ⎟zzG φ ⎠ψ z ψ z⇒ L χχφ = − 1 ⎛2 eG/2 ⎜⎝G zzφ + 2 3 G zφG z=O(1)−Γ z zz,φG z − Γ z zzG zφ − Γ φ zz,φG φ − Γ φ zzG φφ)φψz ψ z=0⇒ M 3/2 ∼ m 3/2not m φThere is a suppression.⇒ B 3/2 ∼⎛⎜⎝⎞2m 3/2⎟⎠m φIt seems that unitary gauge calculation overestimated the branching ratio!!


What’s wrong in unitary gauge calculation?=> the same reason as that in U(1) exampleφψ 3/2G φ+ψ 3/2zφψz3/2G zψ 3/2mass eigenstate of heavy scalar:cancellation! O(1) − O(1) = O(m 3/2 /m φ )Φ = φ + G¯φ¯z z (up to O(m 2 3/2/m 2 φ) )G¯φ¯φFrom V φ = 0⇒ G φ = − G¯φ¯z G zG¯φ¯φ⇒ G Φ = G φ + G¯φ¯z G z = 0G¯φ¯φΦψ 3/2G Φψ 3/2= 0


Is decay rate always suppressed by⎛⎜⎝⎞2m 3/2⎟⎠m φ?There are two possibilities for non-suppressed decay:1.δK = − (z† z) 2Λ 2L ∋ − 1 2 eG/2 Γ z zz,zG z z † ψ z ψ z==> large SUSY breaking mass term for z⎛ ⎞M⎜ Pl ⎟m z ∼ m 3/2⎝ ⎠ ⇒ Φ = φ + m 3/2m φzΛm 2 z→ − 1 2 m 1 m 3/2 m φ3/2 Φ † ψΛ 2 z ψ zm 2 z(For m z ≫ m φ )⇒M ∼ m φ∼ m φIf z has large SUSY breaking mass term as is happening indynamical SUSY breaking models, there is no suppression.


2.δK = κφ † zzL ∋ − 1 2 eG/2 Γ φ zzG φφ φψ z ψ z= κ= m φ /m 3/2⇒M ∼ κm φIf there is direct coupling between moduli and SUSY breakingfield of this type, there is no suppression.


SummaryModuli --> gravitino decay branching ratio can be of order⎛⎜⎝⎞2m 3/2⎟⎠m φ--> Gravitino problem from moduli decay is model dependentBut if it is suppressed,There is always lighter moduli z. This may cause another cosmological problem.But this again depends on SUSY breaking models.Branching ratio is O(1) when1. m z ≫ m φ2.orδK = κφ † zzwithκ ∼ O(1)These give constraints on SUSY breaking models.

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