Supersymmetric Higgs Thermal Inflation and the -term

IntroductionA Minimal Renormalisable Inflationary Supersymmetric Standard Model

Scalar potential and its extremaF-term inflation and (p)reheating

Higgs thermal inflation and gravitinosSummary

Supersymmetric Higgs Thermal Inflation andthe µ-term

Mark Hindmarsh1,2 Tim Jones3

1Department of Physics & AstronomyUniversity of Sussex

2Helsinki Institute of PhysicsHelsinki University

3Dept. of Mathematical SciencesUniversity of Liverpool

COSMO 2012Basbøll, MH, Jones (2011); MH, Jones (2012); MH, Jones (in prep.)

Mark Hindmarsh SUSY Higgs Thermal Inflation & µ




Outline

Introduction

A Minimal Renormalisable Inflationary Supersymmetric Standard Model

Scalar potential and its extrema

F-term inflation and (p)reheating

Higgs thermal inflation and gravitinos





Inflation and the Standard Model

The Standard Model (+ massive ν) now looks complete

Higgs mH =

126.0 ± 0.4 (stat) ± 0.4 (sys) ATLAS125.3 ± 0.4 (stat) ± 0.5 (sys) CMS

Neutrinos∆m2

21 = (7.50 ± 0.20)× 10−5 eV2 sin2(2θ12) = 0.857 ± 0.024∆m2

32 = (2.32+0.20−0.08)× 10−3 eV2 sin2(2θ23) > 0.95

sin2(2θ13) = 0.098 ± 0.013

The very early universe: inflation (WMAP7)Ps(k0) = (2.43± 0.11) × 10−9, ns = 0.963± 0.012 (68%CL),

Reheating: energy Inflaton(s)→ Standard Model (“Big Bang”) How is Inflation sector coupled to SM?

SM ← ? → Inflation





Supersymmetry

xMSSM

Reduces tuning forMew ≪ mP

DM candidates χ0, G

CP, baryo-, leptogenesis.

BUT

Still has tuning No experimental evidence

Supersymmetric inflation(F-terma, D-termb)

Renormalisable field theory Naturally flat potential Corrections under control

BUT

ns = 1− 1Ne

=⇒ ns & 0.98 High Trh → gravitino

problemc

Massive cosmic stringsd

aCopeland et al (1994)bBinetruy, Dvali (1996)cWeinberg (1982)dBattye, Garbrecht, Moss (2006)





A Minimal Renormalisable Inflationary Supersymmetric Standard Model

Incorporates MSSM (with neutrinos, but no more) Incorporates F-term supersymmetric inflation MSSM + 3 fields + U(1)′ symmetry Dynamical explanation of µ-term and RH neutrino masses Second period of Higgs-driven “thermal” inflation Trh ∼ 109 GeV Reduced amount of F-term inflation: ns ≃ 0.975 Neutralino DM (from gravitino decays or freeze-out) Leptogenesis from RH neutrino decays (if MN1 . 109 GeV) Baryogenesis (if electroweak phase transition is 1st order) Cosmic strings, Gµcs ≃ 10−7, consistent with CMB





Fields and symmetries

(ν)MSSM sector: two-parameter family of anomaly free U(1)′

Q U D L E N H1 H2

Y ′−

13 qL −qE −

23 qL qE +

43 qL qL qE −2qL − qE −qE − qL qE + qL

R 1 1 1 1 1 1 0 0

Y : (qL, qE) = (−1, 2). B − L: (qL, qE ) = (−1, 1)

Inflation sector:

Φ Φ S

Y ′ 4qL + 2qE −4qL − 2qE 0

R 0 0 2





Coupling the MSSM to F-term inflation

Superpotential: W = WA + WX + WI

Consisting of: Pure MSSM part: WA = H2QYUU + H1QYDD + H1LYEE + H2LYNN Coupling part: WX = 1

2λ2NNΦ− λ3SH1H2. Pure F-term inflation part: WI = λ1ΦΦS − M2S

µ-term from 〈s〉 RH neutrino masses from 〈φ〉 All other renormalisable terms forbidden by symmetries All B-violating operators forbidden by U(1)′ and R





Related models

Y ′ = B − L 1. Neutrino masses from 〈φ〉, but µ-term notspecified. No Higgs thermal inflation.

U(1)′ → SU(2)R2. Higgs thermal inflation not noticed.

FD inflation3. 〈s〉 gives both µ and (TeV-scale) N masses. Alchemical inflation4. Singlet couples to Higgs and inflaton.

1Buchmüller, Domcke, Schmitz, Vertongen (2010-2012)2Dvali, Lazarides, Shafi (1997)3Garbrecht, Pallis, Pilftsis (2006)4Nakayama, Takahashi (2012)





The scalar potential.

V (s, φ, φ, h1, h2) = VF + VD + Vsoft + ~∆V1

VF =|λ1φφ − λ3h1h2 −M2|2 +

[

λ21(|φ|2 + |φ|2) + λ2

3(|h1|2 + |h2|2)]

|s|2

VD = 12g′2

(

qΦ(φ∗φ− φ

∗φ) + qH(h

†1h1 − h†

2h2))2

+

18 g2

2

∑

a(h†1σ

ah1 + h†2σ

ah2)2 + 1

8 g21(h

†1h1 − h†

2h2)2

No FI-term - unproblematic embedding in supergravity5

5Komargodski & Siegel (2009)Mark Hindmarsh SUSY Higgs Thermal Inflation & µ




The minimum and its geometry.

VD = 0 when |φ| = |φ|, |h1| = |h2|, h†1h2 = 0.

VF = 0 when s = 0, λ1φφ− λ3h1h2 = M2

Minimum parameters: angles χ ∈ [0, π/2], ϕ ∈ [0, 2π], SU(2):

φ =M√λ1

sinχeiϕ, hT1 =

(

M√λ3

cosχeiϕ, 0)

Angle ϕ is associated with U(1)′′ gauge symmetry generated byY ′′ = Y ′ − (qL + qE )Y

Unbroken symmetries :χ = 0 U(1)em×U(1)′′ h-vacuum

0 < χ < π/2 U(1)em

χ = π/2 SU(2)×U(1)Y φ-vacuum





Inflationary flat direction.

Charged fields vanish: |φ| = |φ| = |h1| = |h2| = 0. Flat direction s with VF = M4

0

0.5

1

1.5

0

0.5

1

1.50

0.5

1

1.5

2

φ/φ0

s/sc

V/M

4





SUSY-breaking

Vsoft = m2φ|φ|2 + m2

φ|φ|2 + m2

h1|h1|2 + m2

h2|h2|2 + m2

s |s|2 +ρM2m 3

2(s + s∗) + (hλ1φφs + hλ3 h1 · h2s + c.c.).

Soft terms are all proportional to m 32, the gravitino mass.

SUSY-breaking lifts vacuum degeneracy, gives µ-term (〈s〉H1H2)

〈s〉 ≃ − hλ1λ2

1−

m 32ρ

λ1(in φ-vacuum).





F-term inflation

0

0.5

1

1.5

0

0.5

1

1.50

0.5

1

1.5

2

φ/φ0

s/sc

V/M

4

Recall s flat direction (all other fields vanishing): SUSY is broken, radiative corrections contribute

V (s) ≃ M4[

1 + α ln 2s2

s2c

]

, when sc ≪ s

α = λ2

16π2 , λ =√

λ21 + 2λ2

3, s2c = M2/λ

OK to neglect linear soft term if ρ≪ λ3

16π2sc

m 32

OK to neglect c|s|4/m2P in Kähler if c ≪ 105α





End of inflation and (p)reheating

M2φ,φ

= λ1(|s|2 − λ1M2), M2h1,h2

= λ3(|s|2 − λ3M2)

λ1 < λ3: inflation ends at s2

c1 = M2/λ1 Transition to φ-vacuum (〈|φ|2〉 = M2/λ1) U(1)” symmetry broken: cosmic strings formed

λ3 < λ1: inflation ends at s2

c3 = M2/λ3 Transition to h-vacuum (〈|h1,2|

2〉 = M2/λ3) U(1)′′ symmetry preserved: no strings formed (yet)

Reheating time: M−1. Much smaller than expansion time H−1. Thermal potential keeps universe in false h-vacuum





Predictions from F-term inflation

Predictions: Ps(k) ≃ 4Nk3

(

scmp

)4, ns ≃

(

1− 1Nk

)

. 6

WMAP7 (neglecting cosmic string contribution):Ps(k0) = (2.43± 0.11) × 10−9, ns = 0.963± 0.012 (68% CL),

Normalisation: scmp≃ 2.9× 10−3

(

27Nk0

)14, Nk0 = 27+13

−7 .

2σ discrepancy with Hot Big Bang: Nk0 ≃ 55 + ln(Trh/1015 GeV) Solution: Nθ ≃ 17 e-foldings of thermal inflation Reduces high scale inflation Nk0 ≃ 40: hence ns ≃ 0.975

6Nk is number of e-foldings of inflation after aH = kMark Hindmarsh SUSY Higgs Thermal Inflation & µ




Thermal inflation from the MSSM Higgs

Write difference in vacuum energies asVh − Vφ ≡ ∆V ≃ v2

+m2sb

v+ is U(1)′ symmetry-breaking scale O(1016) GeV msb is supersymmetry-breaking scale O(1) TeV

Inflation re-starts when ρ(T ) ≃ ∆V or Ti ≃ 109( msb

1 TeV

)

12 GeV.

Inflation ends in thermal phase transition at Te ∼ msb7

U(1)′′ symmetry broken, cosmic strings form

7Lyth & Stewart (1996)Mark Hindmarsh SUSY Higgs Thermal Inflation & µ




(P)reheating after thermal inflation

Inflaton is modulus field χ

Near χ = 0 mostly φ, φ Near χ = π/2 mostly h1, h2

Efficient reheat (Higgs) Reheat temperature

Trh2 =(

30geffπ

2 ∆V)

14 ≃ Ti

Hence

Trh2 ≃ 109( msb

1 TeV

)

12 GeV.

OK for leptogenesis (just)a

aBuchmüller, Di Bari, Plümacher(2004)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

χ/(π/2)

∆ V

/∆ V

max





(Re)generation of gravitinos

e-foldings of thermal inflation: Nθ ≃ 12 ln

(

v+msb

)

≃ 17− ln( msb

1 TeV

)

Gravitinos from high-scale inflation diluted

Gravitinos regenerated by reheating to Trh2 ≃ 109( msb

1 TeV

)

12 GeV

Gravitinos are either LSP, or decay into LSP

ΩLSPh2 ≃ 6× 10−3ωGmLSP

100 GeV

( msb1 TeV

)

12 8

ωG: O(1) theoretical uncertainty factor9

8see e.g. Kawasaki et al (2008)9Bolz, Brandenberger, Buchmüller (2001); Pradler, Steffen (2006); Rychkov,

Strumia (2007)Mark Hindmarsh SUSY Higgs Thermal Inflation & µ




Gravitino constraints (AMSB)

ΩLSPh2 ≃ 6× 10−3ωGmLSP

100 GeV

( msb1 TeV

)

12 ≃ 0.11

Hindmarsh, Jones (2012) AMSB empirical formula: mLSP ≃ 3.3× 10−3m 3

2

m2sb ≃ m2

32

(

g′2

16π2

)2Q[

q2φ − q2

H

(

λ1λ3

)]

Correct µ-parameter (EW vacuum ): q2φg′2 ≃ λ3

λ1

Hence m 32≃ 300

(

1ω2

G

qφ√Q

λ3λ1

)13

TeV

Allowed range: 130ω− 2

3

GTeV . m 3

2. 210ω

− 23

GTeV

Higgs mass 125 GeV with m 32= 140 TeV





Gravitino constraints (GMSB & mSUGRA)

GMSB: Requires λ3/λ1 ≪ 1 for TeV-scale superparticles e.g. for m 3

2∼ 1 TeV, Mgaugino ∼ 10 TeV, λ3/λ1 ∼ 10−3

GUT-scale messenger mass - disfavoured by NLSP decays10

mSUGRA: Non-thermal gravitino decays produce negligible LSP density Trh ≃ 109 GeV means m 3

2& (few) TeV to avoid BBN constraints

10Gherghetta, Giudice, Riotto (1998)Mark Hindmarsh SUSY Higgs Thermal Inflation & µ




Cosmic string constraints

Cosmic strings are formed at the end of thermal inflation Mass per unit length µcs ∝ v2

+, v+ ≃ 1016 GeV CMB limit11 Gµcs < 0.42× 10−6 (95%CL) Ordinary Abelian Higgs strings:

Gµcs ≃ 3B(β)× 10−6, with β = λ1/q2φg′2

B(β) . 0.1 means λ1/q2φg′2 . 10−10

Small λ1 (very flat potential) means ns → 1, tuned Kählerpotential12. Also important linear soft term.

Here, cosmic strings have a Higgs condensate: β ∝ m2

sb/v2+ ∼ 10−2613 gives B ≃ 0.04

Gµcs ≃ 10−7, no constraint on inflaton couplings λ1, λ3 CMB limit satisfied

11Urrestilla et al (2011), Abelian Higgs strings. See also Dunkley et al (ACT, 2010)12Battye, Garbrecht, Moss (2006)13Barreiro et al (1996)





Summary

A minimal inflationary supersymmetric standard model Mechanisms for Higgs µ-term and RH neutrino masses Two phases of inflation: (a) F-term at 1015 GeV and (b) Higgs

thermal at 109 GeV DM from thermally produced gravitinos (AMSB) Cosmic string Gµcs ≃ 10−7 satisfies CMB constraints Reduced e-foldings of F-term inflation: ns ≃ 0.975 Temperature high enough for leptogenesis

Outlook Leptogenesis in detail? Other string constraints (gravitational waves, cosmic rays)?


Appendix

Higg thermal inflation potential

V (χ) ≃ −M2

2

(

hλ1 sin2 χ+ hλ3 cos2 χ)2

λ1 sin2 χ+ λ3 cos2 χ+M2

(

m2φ

λ1sin2 χ+

m2h

λ3cos2 χ

)

,

(1)where hλ1 =

hλ1λ1

, hλ3 =hλ3λ3

, m2φ = m2

φ + m2φ

and m2h = m2

h1+ m2

h2.

v2+ = 4M2/λ1

m2sb ≃

m232

(

g′2

16π2

)2Q[

q2φ − q2

H

(

λ1λ3

)]

, (AMSB),

Λ2sλ1λ3

(

g′2

16π2

)2 [

q2H

(

λ1λ3

)

− q2φ

]

(GMSB),

12 (m

20 − A2/4)

[

λ1λ3− 1]

(mSUGRA).


Appendix

GMSB gravitino constraint

SU(2) gaugino mass M2, messenger index Nmi

(

m 32

1 TeV

)2

≃ 50√

Nmi

√

λ3λ1

1ω2

G

(

M210 TeV

)−1


Appendix

Vacuum degeneracy and the µ-term

Define: vφ = 〈|φ|〉/√

2, v1 = 〈|h1|〉/√

2, vs = 〈|s|〉/√

2

µ-term: vs ≃ − hλ1√2λ2

1

−m 3

2ρ

√2λ1

(Neglect ρ term from now)

φ-vacuum energy: Vφ = M2

λ1

(

m2φ + m2

φ− h2

λ1

2λ21

)

h-vacuum energy: Vh = M2

λ3

(

m2h1+ m2

h2− h2

λ3

2λ23

)

Must have Vh > Vφ for minimum at electroweak vacuum Constraint on SUSY-breaking scenarios


Appendix

Constraints in 3 common SUSY-breaking scenarios

Anomaly-mediated Supersymmetry-breaking (AMSB) Assume Q = Tr(Y ′2) ≫ 1

λ1

(

qHqΦ

)2. λ3.

Gauge-mediated Supersymmetry-breaking (GMSB) Assume Q = Tr(Y ′2) ≫ 1

λ1

(

qHqΦ

)2& λ3.

Minimal Supergravity (mSUGRA) Universal scalar soft mass m0, universal trilinear coupling A [2m2

0 − A2/2 > 0 and λ1 > λ3 ] or [2m20 − A2/2 < 0 and λ1 < λ3]


Documents

Supersymmetric Higgs Thermal Inflation and the -term