DIVERGENCIES AND SYMMETRIES IN HIGGS-GAUGE UNIFICATION THEORIES

1. Outline: • Introduction: motivations for Higgs-gauge unification theories • Gauge theories on orbifolds • Symmetries @ fixed points and localized terms • The residual Of symmetry • Conclusions and outlook Carla Biggio Institut de Física d'Altes Energies Universitat Autonoma de Barcelona DIVERGENCIES AND SYMMETRIES IN HIGGS-GAUGEUNIFICATION THEORIES based on CB & Quirós, Nucl.Phys.B703 (2004) 199 [hep-ph/0407348] (see also hep-ph/0410226) XL Rencontres de Moriond ELECTROWEAK INTERACTIONS & UNIFIED THEORIES La Thuile (IT), 5-12/03/04

2. Standard Model (SM): effective theory with cutoff ΛSM New physics ↔ non-renormalizable (dimension six) operators O A possible motivation: Little Hierarchy Problem (LHP) Barbieri & Strumia 00 Giudice 03 No fine-tuning → ΛSM ≤ 1 TeV Precision tests → ΛLH ≥ 5-10 TeV One order of magnitude of discrepancy: LHP

3. SUSY → no quadratic divergences → (grand) HP solved: SUSY SM (MSSM) can be extended up to MPl ΛSM~ MSUSY If R-parity is conserved → SUSY virtual loops are suppressed → ΛLH ~ 4πΛSM → LHP solved However: • SUSY not yet been observed → fine-tuning • SUSY breaking sector not well defined • … Worthwhile looking for alternative solutions A possible solution: Supersymmetry (SUSY)

4. 4D theory D-dimensional theory UV completion E ΛSM ~ 1/R ~ TeV ΛD ≥ 10 TeV → LHP solved An alternative solution: Higgs-gauge unification Consider a gauge theory in a D-dimensional space-time 4D Lorentz scalars → Higgs fields ! Randjbar-Daemi, Salam & Strathdee 83 4D Lorentz vector they can acquire mass through the Hosotani mechanism Hosotani et al. 83-04 Higgs mass in the bulk is protected by higher-dimensional gauge invariance finite corrections ~ (1/R)2 allowed

5. Torus: Orbifold: Fixed points: invariants under Gorb Gauge theory in D dimensions Invariant under gauge groupG (SO(1,D-1)) Spacetime: MD coord.: xM = (x,yi) Compactification on the orbifold M4xTd/Gorb

6. 4D fields with mass S1/Z2: • zero mode: only for • @ yf Action of Gorb on the fields acts on Lorentz indices acts on gauge and flavour indices unconstrained fixed by requiring invariance of lagrangian  scalars: vectors: it can be used to break symmetries S1:

7. Why looking @ yf? → lagrangian terms localized @ the fixed points can be radiatively generated (if compatibles with symmetries) Georgi, Grant & Hailu 00 Gorb G @ yf Hf Non-zero fields @ yf: (with zero modes) (for some i & â) some derivatives of non-invariant fields (without zero modes)  Residual global symmetry K Gersdorff, Irges & Quirós 02 Gauge symmetry breaking @ yf

8. Forbidden terms: → localized kinetic term for If is Gorb-invariant  a “shift” symmetry forbids a direct mass term: 1 Allowed terms: → localized anomaly → If and are orbifold invariant (D≥6) → localized quartic coupling for → localized kinetic term for Effective 4D lagrangian Lf→ most general 4D lagrangian compatible with symmetries @ yf The symmetries @ yf are: GorbSO(3,1)HfK All these are dimension FOUR operators → renormalize logarithmically

9. If Hf = U(1)ax … and • tadpole for • mass term for can be radiatively generated @ yf and are orbifold invariant • global cancellation of tadpoles SSSW03 … another (worse) allowed term… is invariant underGorb SO(3,1) HfK → This is a dimension TWO operator → quadratic divergencies D≥6 it seems it always exists • D=6 (QFT) Gersdorff, Irges & Quirós 02 Csaki, Grojean & Murayama 02 • Scrucca, Serone, Silvestrini & Wulzer 03 (SSSW03) • D=10 (strings) Groot-Nibbelink et al. 03  How can we avoid this?

10. d-dimensional smooth manifold: at each point can be defined a TANGENT SPACE → SO(d) Gorb when orbifolding: as G Hf such that Gorb so SO(d) Of such that But… another symmetry must be considered CB & Quirós, Nucl.Phys.B703 (2004) 199 The symmetries @ yf are: GorbSO(3,1)HfK Of  Can this Of forbid the tadpole?

11. The tadpole Fij and the symmetry Of If Of=SO(2) x … then the Levi-Civita tensor eij exists → is Of invariant → TADPOLES ARE ALLOWED If Of=SO(p1) x SO(p2) x … (pi>2) then the Levi-Civita tensor is → only invariants constructed with pi-forms are allowed → NO TADPOLES Sufficient condition for the absence of localized tadpoles BQ’04 Of is orbifold-dependent: we studied the Td/ZN case

12. Of depends on RNf: on Td/ZN→ RNf~ diag(r1…ri…rd/2) with ri rotation in the i-plane If Nf>2  which acts on (y2i-1,y2i)  in every subspace (y2i-1,y2i)eIJ exists → If Nf=2 → RNf= -1  [-1,SO(d)] = 0  Of=SO(d) the Levi-Civita tensor is → only invariants constructed with d-forms are allowed valid also for odd D → TADPOLES ONLY FOR d=2 (D=6)  Td/Z2 → explicitely checked @ 1- and 2-loop for any D BQ’04 Orbifolds Td/ZN(d even)

13. Conclusions In Higgs-gauge unification theories (Higgs = Ai) • bulk gauge symmetry G prevents the Higgs from adquiring • a quadratically divergent mass in the bulk • “shift” symmetry K forbids a direct mass @ yf If Hf = U(1)ax … can be radiatively generated @ yf giving rise to a quadratically divergent mass for the Higgs Fija can be generated ↔ it is Of-invariant such that If Of=SO(p1) x SO(p2) x … (pi>2) → NO TADPOLES If Of=SO(2) x … → TADPOLES Td/ZN (d even, N>2): if Nf>2 → Of=SO(2) x … x SO(2) → TADPOLES Td/Z2 (any d): Of=SO(d) → TADPOLES ONLY FOR d=2 (D=6)

14. Outlook • The absence of tadpoles is a necessary but not sufficient condition • for a realistic theory of EWSB without SUSY • Other issues: • REALISTIC HIGGS MASS • D>6 (D=5 no quartic coupling, D=6 tadpoles) • Td/Z2→ d Higgs fields  non-minimal models • → we have to obtain only one SM Higgs • even if this is achieved • → Higgs mass must be in agreement with LEP bounds • FLAVOUR PROBLEM • - matter fermions in the bulk coupled to a background • which localizes them at different locations • Burdman & Nomura 02 • - matter fermions localized and mixed with extra heavy • bulk fermions • Csaki, Grojean & Murayama 02