1. DIVERGENCIES AND SYMMETRIES IN HIGGS-GAUGE UNIFICATION THEORIES Carla Biggio Institut de Física d'Altes Energies Universitat Autonoma de Barcelona XL Rencontres de Moriond ELECTROWEAK INTERACTIONS & UNIFIED THEORIES La Thuile (IT), 5-12/03/04 Outline:1.Introduction: motivations for Higgs-gauge unification theories 2.Gauge theories on orbifolds 3.Symmetries @ fixed points and localized terms 4.The residual O f symmetry 5.Conclusions and outlook based on CB & Quirós, Nucl.Phys.B703 (2004) 199 [hep-ph/0407348] (see also hep-ph/0410226)

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

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

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

5. 5 Invariant under gauge group G (SO(1,D-1)) Gauge theory in D dimensions Compactification on the orbifold M 4 xT d / G orb Fixed points: invariants under G orb Spacetime: M D coord.: x M = (x ,y i ) Torus: Orbifold:

6. 6 Action of G orb on the fields acts on gauge and flavour indices acts on Lorentz indices scalars: vectors: fixed by requiring invariance of lagrangian unconstrained  it can be used to break symmetries 4D fields with mass S1:S1: zero mode: only for @ y f S 1 /Z 2 :

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

8. 8 Effective 4D lagrangian L f → most general 4D lagrangian compatible with symmetries @ y f The symmetries @ y f are: G orb SO(3,1) H f K Ifis G orb -invariant  a “shift” symmetry forbids a direct mass term: 1 Forbidden terms: → If and are orbifold invariant → localized kinetic term for → localized anomaly → localized quartic coupling for (D≥6) → localized kinetic term for Allowed terms: All these are dimension FOUR operators → renormalize logarithmically

9. 9 If H f = U(1)  x … and and are orbifold invariant … another (worse) allowed term… is invariant under G orb SO(3,1) H f K → tadpole for mass term for can be radiatively generated @ y f This is a dimension TWO operator → quadratic divergencies  How can we avoid this? 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 D≥6 it seems it always exists 1.global cancellation of tadpoles SSSW03 2.

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

11. 11 The tadpole F ij and the symmetry O f Sufficient condition for the absence of localized tadpoles If O f =SO(2) x … then the Levi-Civita tensor  ij exists is O f invariant → TADPOLES ARE ALLOWED If O f =SO(p 1 ) x SO(p 2 ) x … (p i >2) then the Levi-Civita tensor is → NO TADPOLES → → only invariants constructed with p i -forms are allowed O f is orbifold-dependent: we studied the T d /Z N case BQ’04

12. 12 Orbifolds T d /Z N (d even) If N f >2  which acts on (y 2i-1,y 2i )  in every subspace (y 2i-1,y 2i )  IJ exists →  T d /Z 2 → explicitely checked @ 1- and 2-loop for any D BQ’04 If N f =2 → R Nf = -1  [-1,SO(d)] = 0 → only invariants constructed with d-forms are allowed → TADPOLES ONLY FOR d=2 (D=6) valid also for odd D the Levi-Civita tensor isO f =SO(d)  O f depends on R Nf : on T d /Z N → R Nf ~ diag(r 1 …r i …r d/2 ) with r i rotation in the i-plane

13. 13 If H f = U(1)  x … can be radiatively generated @ y f giving rise to a quadratically divergent mass for the Higgs Conclusions In Higgs-gauge unification theories (Higgs = A i ) bulk gauge symmetry G prevents the Higgs from adquiring a quadratically divergent mass in the bulk “shift” symmetry K forbids a direct mass @ y f such that If O f =SO(2) x … → TADPOLES If O f =SO(p 1 ) x SO(p 2 ) x … (p i >2) → NO TADPOLES T d /Z N (d even, N>2): if N f >2 → O f =SO(2) x … x SO(2) → TADPOLES T d /Z 2 (any d): O f =SO(d) → TADPOLES ONLY FOR d=2 (D=6) F ij  can be generated ↔ it is O f -invariant

14. 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) T d /Z 2 → 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