Electroweak Symmetry Breaking with Holomorphic Supersymmetric - PowerPoint PPT Presentation

Electroweak Symmetry Breaking with Holomorphic Supersymmetric Nambu–Jona-Lasinio Model — Talk at PHENO 2010 OTTO C. W. KONG — Nat’l Central U, Taiwan

1 Nambu–Jona-Lasinio Model :- Otto Kong (NCU) — 09ntnu-1 • dynamical symmetry breaking • four-fermion interaction ψ − σ µ ∂ µ ψ − + g 2 ¯ L ψ = i ¯ ψ + σ µ ∂ µ ψ + + i ¯ ψ + ¯ ψ − ψ + ψ − → L ψ − ( µφ † + gψ + ψ − )( µφ + g ¯ ψ + ¯ − ψ − ) ψ − σ µ ∂ µ ψ − − µ 2 φ † φ − µg ( φ † ¯ = i ¯ ψ + σ µ ∂ µ ψ + + i ¯ ψ + ¯ ψ − + φψ + ψ − ) • auxiliary scalar field φ (no kinetic term) • EL-eq for φ † gives φ as composite φ = − g/µ ¯ ψ + ¯ ψ − • � φ � � = 0 = ⇒ symmetry breaking and fermion mass

2 → low energy effective field theory :- Otto Kong (NCU) — 09ntnu-2a • 1-loop effective potential for φ gives gap equation � φ � � = 0 solution for g 2 Λ 2 8 π 2 > 1 • Dirac fermion mass m = µg � φ � for ψ + – ψ − g 2 � ¯ g 2 ¯ ψ + ¯ ψ + ¯ � ψ − ψ + ψ − = ⇒ ψ − ψ + ψ − • kinetic term for φ through wave-function renormalization fermion-loop propagator with Yukawa vertices ( m ≪ Λ) � � c µ 2 g 2 ln Λ 2 Z = N M 2 + O (1) 16 π 2 • Higgs with mass 2 m and a Goldstone boson

3 √ φ − → φ/ Z :- Otto Kong (NCU) — 09ntnu-2b L ψ = i ¯ ψ + σ µ ∂ µ ψ + + i ¯ ψ − σ µ ∂ µ ψ − + ∂ µ φ † ∂ µ φ • ˜ 2 ( φ † φ ) 2 − ˜ µ 2 φ ∗ φ − λ − ˜ yφψ + ψ − + h.c. µg 4 π 1 √ — y = ˜ Z = √ √ N ln (Λ 2 /M 2 ) c � � µ 2 = c g 2 − (Λ 2 − M 2 ) 8 π 2 2 — ˜ ln (Λ 2 /M 2 ) N 32 π 2 ˜ — λ = N c ln (Λ 2 /M 2 ) • condition for � φ � � = 0 gives gap equation result

Supersymmetrizing the NJL Model (Naively):- • i ¯ d 4 θ ¯ � ψ + σ µ ∂ µ ψ + Φ + Φ + − → • g 2 ¯ d 4 θ g 2 ¯ ψ + ¯ Φ + ¯ � ψ − ψ + ψ − Φ − Φ + Φ − − → d 2 θ µg ΦΦ + Φ − � • − µg φψ + ψ − − → • − µ 2 φ ∗ φ � d 2 θ µ 2 ΦΦ − → BUT :- • φ = − g/µ ¯ ψ + ¯ ψ − implies µ 2 φ ∗ φ = − µg φψ + ψ − = g 2 ¯ ψ + ¯ ψ − ψ + ψ − (no SUSY !) • no nontrivial vacuum without SUSY breaking

The Supersymmetric NJL Model :- • i ¯ d 4 θ ¯ Φ + Φ + (1 − m 2 θ 2 ¯ � θ 2 ) ψ + σ µ ∂ µ ψ + − → • g 2 ¯ d 4 θ g 2 ¯ ψ + ¯ Φ + ¯ � ψ − ψ + ψ − − → Φ − Φ + Φ − d 2 θ µg Φ 2 Φ + Φ − � • − µg φψ + ψ − − → d 4 θ ¯ • − µ 2 φ ∗ φ d 2 θ µ Φ 1 Φ 2 + � � − → Φ 1 Φ 1 BUT :- • EL-eq for Φ 2 gives Φ 1 = − g Φ + Φ − implies d 4 θ g 2 ¯ d 4 θ ¯ Φ + ¯ � � Φ 1 Φ 1 = Φ − Φ + Φ − • Φ 2 not the composite Φ 1 plays the Higgs superfield � Φ 1 � = 0

An Alternative Supersymmetrization ? • i ¯ Φ + Φ + (1 − m 2 θ 2 ¯ d 4 θ ¯ ψ + σ µ ∂ µ ψ + � θ 2 ) − → � d 2 θ µg Φ 0 Φ + Φ − • − µg φψ + ψ − − → µ • − µ 2 φ ∗ φ � d 2 θ 2 Φ 0 Φ 0 − → (¯ Φ + Φ + + ¯ Φ − Φ − )(1 − m 2 θ 2 ¯ � d 4 θ � θ 2 ) � = L = ⇒ 0 + √ µG Φ 0 Φ + Φ − � µ � d 2 θ 2 Φ 2 � + + h.c. W = G • consider superpotential 2 Φ + Φ − Φ + Φ − √ √ 2 ( √ µ Φ 0 + G Φ + Φ − )( √ µ Φ 0 + → W − 1 G Φ + Φ − ) −

7 With Holomorphic Four-Chiral Superfield Otto Kong (NCU) — 09ntnu-6 Interaction :- 2 Φ + Φ − Φ + Φ − contains no g 2 ¯ ψ + ¯ • W = G ψ − ψ + ψ − � • EL-eq for auxiliary superfield Φ 0 gives Φ 0 = − G/µ Φ + Φ − √ µG µ 2 Φ 2 Φ 0 Φ + Φ − = G implies 0 = − 2 Φ + Φ − Φ + Φ − 2 G • � Φ 0 � = ⇒ 2 � Φ + Φ − � Φ + Φ − Dirac mass for Φ + –Φ − • kinetic term for Φ 0 from wave-function renormalization through Φ + –Φ − loop with Yukawa vertices

8 → low energy effective field theory :- Otto Kong (NCU) — 09ntnu-7 1 + (2 m 2 + A 2 ) θ 2 ¯ d 4 θ Z 0 ¯ Φ 0 e 2 V Φ Φ 0 � θ 2 � � • (gauged-)kinetic term � � ln Λ 2 where Z 0 = N c µG M 2 + O (1) 16 π 2 0 = − (2 m 2 + A 2 ), tachyonic soft mass ( cf. radiative EWSB) m 2 — ˜ √ µG 4 π 1 √ — y = ˜ √ Z 0 = √ N ln (Λ 2 /M 2 ) c Z 0 = 16 π 2 µ 1 — µ = ˜ ln (Λ 2 /M 2 ) N c G 2 Φ 2 term = • µ ⇒ Φ in real representation of symmetry

9 Condensate/Mass Generation — A Comparison :- Otto Kong (NCU) — 09ntnu-8 • NJL : g 2 � ¯ ψ + ¯ � ψ − ψ + ψ − − → − µg � φ � ψ + ψ − — symmetry breaking with bi-fermion condensate � φ � d 4 θ g 2 � ¯ Φ + ¯ � � • SNJL : Φ − Φ + Φ − � � � � F † F † − → − g [ A + F − + A − F + − ψ + ψ − ] 1 = − µA 2 1 — � F 1 � = − g � A + F − + A − F + − ψ + ψ − � , sbi-fermion condensate d 2 θ − G � Φ + Φ − � Φ + Φ − � • HSNJL : √ µG � A 0 � [ A + F − + A − F + − ψ + ψ − ] + √ µG � F 0 � A + A − − → � — � A 0 � = − G/µ � A + A − � , a bi-scalar condensate

10 Otto Kong (NCU) — 09ntnu-9 T owards EW Symmetry Breaking

11 NJL Model → SM :- Otto Kong (NCU) — 09ntnu-10 g 2 ¯ Q ¯ t c Qt c • four-fermion interaction • Higgs doublet as top-composite φ = − g/µ ( ¯ Q ¯ t c ) • top condensate breaks EW symmetry → fermion masses — gives top quark mass at to infared quasi-fixed point (Λ ∼ 10 19 GeV ) • high m t ∼ 218 GeV Bardeen, Hill, Lindner 90 (Λ ∼ 10 15 − 10 19 GeV ) m t ∼ 214 − 202 GeV Marciano 89,90 m t ∼ 253 GeV Miransky, Tanabashi, Yamawaki 89; King & Mannan 90,91 • extensions, e.g. two-Higgs-doublet model

12 SNJL Models → MSSM (why SUSY ?):- Otto Kong (NCU) — 09ntnu-11 • SM → MSSM — hierarchy/fine-tuning problem scalar field is somewhat sick • SM fermion spectrum sort of fixed (anomaly cancelation) e.g. OK 96 scalar content — only part arbitrary ( cf. gauge symmetry) • SUSY — technically natural hierarchy scalar as (part of) chiral superfield (constrained as fermions) Vs Georgi’s survival hypothesis • BUT µ -problem — vectorlike pair of Higgs superfields • SNJL models solve our problems — and avoid fine-tuning of four-quark coupling(s)

13 Towards the MSSM :- Q α ˆ U c ˆ Q ′ β ˆ D c (1 + Bθ 2 ) Otto Kong (NCU) — 09ntnu-12 αβ ˆ • consider W = G ε Q ′ ˆ → W − µ ( ˆ H d − λ u ˆ Q ˆ U c )( ˆ H u − λ d ˆ D c )(1 + Bθ 2 ) W − U c + y d ˆ Q ′ ˆ = ( − µ ˆ H d ˆ H u + y u ˆ Q ˆ H u ˆ H d ˆ D c )(1 + Bθ 2 ) Q ′ ˆ H u = y d ˆ µ ˆ H d = y u ˆ Q ˆ ˆ D c U c • two composites — and µ • low energy effective theory looks like MSSM ( A = B ) � � • symmetric role for ˆ H u and ˆ also : µλ u λ d = y u y d H d = G µ — numerical lifted through non-universal soft masses — expect � h u � > ∼ � h d � (Vs UBB in D -flat)

14 Holomorphic Vs Old Model (for MSSM) :- Otto Kong (NCU) — 09ntnu-13 • bottom together with (vs only) top mass at quasi-fixed point ⋆ both (vs one) Higgs superfields as composites • large (vs small) tan β • A t ≃ A b ≃ B (vs A t ≃ 0) • m 2 d ≃ − ( m 2 Q + m 2 b + | A b | 2 ) H plus (vs only) m 2 u ≃ − ( m 2 Q + m 2 t + | A t | 2 ) H ⋆ full W [= G ijkh Q i U c j Q k D c h (1 + Aθ 2 ) + G e 3 U c 3 L i E c j (1 + Aθ 2 )] ij Q — non-holomorphic case needs similar holomorphic terms for Yukawa couplings of down-type quarks and charged leptons • sbottom and stop condensates for u i and d i + ℓ i masses (vs top condensate and stop condensates for u i and d i + ℓ i masses)

15 Numerical (RG analysis) Results :- Otto Kong (NCU) — 09ntnu-14 • earlier MSSM t − b − τ quasi-fixed point analysis Froggatt et.al 93 (without background model) m t = 184 . 3 ± 6 . 8 GeV, m h = 121 . 8 ± 4 . 3 GeV • ? m t = 171 . 2 ± 2 . 1 GeV • old SNJL: MSSM t quasi-fixed point analysis Carena et.al 92 — high Λ and large tan β lower m t • infared quasi-fixed point NOT necessary SNJL – y t blows up at Λ HSNJL – y t and y b blows up at ∼ Λ

16 Our Solution :- Otto Kong (NCU) — 09ntnu-14p 75 M s = 10 TeV M s = 1 TeV 70 M s = 200 GeV 65 60 tan β 55 50 45 40 35 10 4 10 6 10 8 10 10 10 12 10 14 10 16 Λ [GeV]

17 Illustrative y t and y b :- Otto Kong (NCU) — 09ntnu-14r M s = 1 TeV 5 y b 4.5 tan β = 57.8 tan β = 42.8 y t Λ b = 10 4 GeV Λ b = 10 10 GeV 4 3.5 3 y t , y b 2.5 2 1.5 1 0.5 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 µ [GeV]

18 Mass of the lightest Higgs boson :- Otto Kong (NCU) — 09ntnu-14h 130 m A ≥ 100 GeV m A = 140 GeV 125 m A = 130 GeV 120 m A = 120 GeV m A = 110 GeV 115 m A = 100 GeV m h [GeV] 110 105 100 95 90 10 3 10 4 200 M s [GeV]

19 Final Remarks :- Otto Kong (NCU) — 09ntnu-15 • SNJL model with holomorphic term works • may provide more interesting version of MSSM — SUSY : scalar → chiral superfield — problematic MSSM superfield spectrum — vectorlike Higgs superfields, turn up as composites — four-superfield ( G ) term from integrated out heavy Higgs superfields ? — more natural B (and A ) term, and all Yukawa coupling • chiral symmetry explicitly broken

20 Otto Kong (NCU) — end T HANK Y OU ! well done Otto !