Radiative Lifting of Flat Directions of the MSSM during Inflation - PowerPoint PPT Presentation

Outline Flat Directions Radiative Corrections in SUSY Radiative Corrections in de Sitter Background Corrections for F -term hybrid inflation Summary Radiative Lifting of Flat Directions of the MSSM during Inflation Björn Garbrecht School of Physics & Astronomy The University of Manchester COSMO 07, Brighton, August 21st 2006 BG, Phys. Rev. D 74 (2006) 043507, [arXiv:hep-th/0604166] BG, Nucl. Phys. B. (in press), [arXiv:hep-ph/0612011] BG, C. Pallis and A. Pilaftsis, JHEP 0612 (2006) 038, [arXiv:hep-ph/0605264]

Outline Flat Directions Radiative Corrections in SUSY Radiative Corrections in de Sitter Background Corrections for F -term hybrid inflation Summary Outline Flat directions of the MSSM are lifted during inflation. Usually considered origins of lifting: SUGRA corrections nonrenormalizable superpotential terms both contributions in general unknown or arbitrary In this talk: There are calculable corrections of competitive magnitude to the aforementioned ones. Two types of radiative corrections: a generic in the curved de Sitter background a particular one, arising in F -term hybrid inflation

Outline Flat Directions Radiative Corrections in SUSY Radiative Corrections in de Sitter Background Corrections for F -term hybrid inflation Summary MSSM Flat Directions Combination of Higgs, squark and slepton scalar fields which are gauge invariant ( D -flat). have vanishing potential arising from superpotential ( F -flat). For example u d d may contain       ϕ 0 0 ˜ ˜  ,  ,  . t R = s ∗ ˜ R = d ∗ R = 0 ϕ ∗ 0    0 0 ϕ ∗ These compose a massless scalar field as 1 R + ˜ √ � ˜ � Φ = t R + ˜ s ∗ d ∗ . R 3 φ = | ϕ | is the canonically normalized modulus field and V ( φ ) ≡ 0 .

Outline Flat Directions Radiative Corrections in SUSY Radiative Corrections in de Sitter Background Corrections for F -term hybrid inflation Summary Flat Directions in Cosmology There is a large number of flat directions, giving rise to exhaustively studied scenarios. Affleck-Dine baryogenesis (Affleck & Dine (1985)). Baryonic isocurvature perturbations (Enqvist & McDonald (1999)). Q -balls (Coleman (1985)). Curvaton Scenario (Enqvist & Sloth; Lyth & Wands (2002)). Thermal history of the Universe (Mazumdar, Allahverdi (2005)). During inflation, they can acquire large VEVs. VEV is determined by lifting contributions that break the flatness. Critical mass for overdamped regime: m 2 = 16 H 2 . 9

Outline Flat Directions Radiative Corrections in SUSY Radiative Corrections in de Sitter Background Corrections for F -term hybrid inflation Summary Non-calculable contributions to the lifting SUGRA (Dine, Randall, Thomas (1995)) For F � = 0 , typical mass terms of order H 2 . Depend on the unknown Kähler potential. These corrections are absent or highly suppressed when imposing certain symmetries on the Kähler potential. (Gaillard, Murayama & Olive (1995)) Also absent in D -term inflation. Nonrenormalizable superpotential terms (see e.g. Ghergetta, Kolda, Martin (1996)) Higher dimensional, Planck scale suppressed superpotential terms. Purpose: Stabilizing the potential for VEVs towards the Planck scale.

Outline Flat Directions Radiative Corrections in SUSY Radiative Corrections in de Sitter Background Corrections for F -term hybrid inflation Summary One-Loop Effective Potentials Sum of all mass insertions. VEV φ of the flat direction generates masses via the Yukawa couplings h from the superpotential. via the gauge coupling g (super-Higgs mechanism). NB: These corrections vanish when SUSY exact (nonrenormalization). Gauge Contributions Yukawa Contributions Bosons: Bosons: gauge bosons, Higgs/squark/sfermion mixing state D -term scalars Fermions: Fermions: higgsino/quark/lepton mixing state gaugino/higgsino mixing state tr m 2 B = tr m 2 F ∝ h 2 φ 2 tr m 2 B = tr m 2 F ∝ g 2 φ 2

Outline Flat Directions Radiative Corrections in SUSY Radiative Corrections in de Sitter Background Corrections for F -term hybrid inflation Summary SUSY breaking during inflation Breaking through the curved de Sitter background. BG, Phys. Rev. D 74 (2006) 043507, [arXiv:hep-th/0604166] BG, Nucl. Phys. B. (in press), [arXiv:hep-ph/0612011] The ususal mechanism of spontaneous SUSY breaking: For certain models, the MSSM fields couple via loops to the vacuum energy driving inflation. F -term hybrid inflation. BG, C. Pallis and A. Pilaftsis, JHEP 0612 (2006) 038, [arXiv:hep-ph/0605264]

Outline Flat Directions Radiative Corrections in SUSY Radiative Corrections in de Sitter Background Corrections for F -term hybrid inflation Summary Effective Potentials for Fermions & Scalars in de Sitter Effective potentials in curved spacetime are generalizations of the Coleman Weinberg potential. Additional corrections of order H 2 . Calculable by using position space techniques. UV cutoff length ̺ , de Sitter invariant. Dirac fermion contribution: � � V ψ = − m 2 1 1 − m 4 log ( ̺ 2 m 2 ) − 2 H 2 m 2 log ( ̺ 2 m 2 ) ̺ 2 + 2 π 2 16 π 2 Candelas, Raine (1975); corrected form in BG (2006) and Miao, Woodard (2006) Real scalar contribution: � � m 2 1 1 φ 4 m 4 � ̺ 2 m 2 � − H 2 m 2 � ̺ 2 m 2 � V φ = 8 π 2 ̺ 2 + φ log φ log φ φ 16 π 2 Candelas, Raine (1982)

Outline Flat Directions Radiative Corrections in SUSY Radiative Corrections in de Sitter Background Corrections for F -term hybrid inflation Summary Effective Potential for Chiral Multiplets Within supersymmetry, one massive Dirac fermion is accompanied by four real scalars of the same mass. These can be constructed from two chiral multiplets. Two-Chiral Multiplet Effective Potential V chiral = 4 V φ + V ψ = − 3 8 π 2 H 2 m 2 log � ̺ 2 m 2 � Flat space contributions cancel, as they should. Non-vanishing contribution ∝ H 2 due to the curvature.

Outline Flat Directions Radiative Corrections in SUSY Radiative Corrections in de Sitter Background Corrections for F -term hybrid inflation Summary Example Consider again t s d . Superpotential W contains W ⊃ h t ttH 0 u . Neglect other Yukawa couplings, h t ≫ h s ≫ h d . Four real scalars from H 0 u and ˜ t L . � t L � One Dirac fermion . ˜ H 0 u All these particles have the mass square | h t φ | 2 . Lifting Potential 8 π 2 H 2 | h t φ | 2 log V chiral = − 3 � ̺ 2 | h t φ | 2 � ̺ needs to be fixed by renormalization condition. Need to check whether also the gauge coupling g mediates lifting.

Outline Flat Directions Radiative Corrections in SUSY Radiative Corrections in de Sitter Background Corrections for F -term hybrid inflation Summary Effective Potential for the Higgs Mechanism Need to add gauge boson A , Goldstone G and ghost η contributions. Gauge-fixing parameter ξ . � M 2 ̺ 2 M 2 �� 1 3 M 4 + 12 H 2 M 2 + ξ 2 M 4 − 4 H 2 ξ M 2 � � � V A = tr 8 π 2 ̺ 2 ( 3 + ξ ) + log , 64 π 2 � M 2 �� 1 � � � G ξ 2 M 4 G − 4 H 2 ξ M 2 ̺ 2 M 2 V G = tr 8 π 2 ̺ 2 ξ + log , G G 64 π 2 � − M 2 ̺ 2 M 2 �� 1 2 ξ 2 M 4 − 8 H 2 ξ M 2 � � � V η = tr 4 π 2 ̺ 2 ξ − log . 64 π 2 Using tr M 2 G = tr M 2 , we find the net result, which is independent of ξ : � 3 M 2 ̺ 2 M 2 �� 1 3 M 4 + 12 H 2 M 2 � � � V gauge = V A + V G + V η = tr 8 π 2 ̺ 2 + log 64 π 2 First derived in Landau gauge, ξ = 0 , by Allen (1982); Ishikawa (1982).

Outline Flat Directions Radiative Corrections in SUSY Radiative Corrections in de Sitter Background Corrections for F -term hybrid inflation Summary Effective Potential for the Super-Higgs Mechanism Within SUSY, have additional fermionic contributions from Higgsinos/Gauginos. − → One set of Dirac fermions with mass matrix M ψ satisfying ψ = tr M 2 . Effective potential contribution V ψ . tr M 2 And one set of real scalars with mass matrix M 2 arising from the D -terms, yielding contribution V D . Effective Potential for the Super-Higgs Mechanism V SH = V gauge + V D + V ψ = 0 (disappointingly, up to possible corrections of order H 4 ) This completes the possible contributions to curvature-induced lifting.

Outline Flat Directions Radiative Corrections in SUSY Radiative Corrections in de Sitter Background Corrections for F -term hybrid inflation Summary Spontaneous SUSY-breaking in F -term inflation Superpotential κ SXX − κ SM 2 + λ SH u H d During inflation � S � � = 0 . For definitenenss, calculate corrections due to H u & H d . In general, X and X break a GUT-symmetry and also couple to the MSSM-fields. Higgs Bosons and Higgsinos, squarks and quarks acquire different masses. To be specific, we again consider the u ¯ d ¯ d -direction. u R corresponds to the right handed stop ˜ Assume that ˜ t R . Can then expand in terms of the top-quark Yukawa coupling h = h t .

Outline Flat Directions Radiative Corrections in SUSY Radiative Corrections in de Sitter Background Corrections for F -term hybrid inflation Summary Effective potential for the stop � � � � u R ) = κ 2 λ 2 M 4 λ 2 | S | 2 h 2 κ 4 M 8 h 4 κ 2 M 4 − 3 1 1 u R | 2 + V ( 1 ) (˜ u R | 4 − λ 2 | S | 6 | ˜ | ˜ ln 8 π 2 Q 2 2 48 π 2 16 π 2 λ 2 | S | 4 � h 2 κ 2 M 4 � h 2 κ 2 M 4 � 2 � 1 u R | 2 u R | 2 O ( h 6 | ˜ u R | 6 ) + λ 2 | S | 4 | ˜ ln λ 4 | S | 6 | ˜ + 16 π 2 The ˜ u R -dependent terms are indpendent of the renormalization scale Q . Unique vaccuum expectation value κ � ˜ u R � = √ M 6 h Unique mass term h 2 κ 4 1 M 2 λ 2 M 2 u R = ˜ 24 π 2