1. Chiral freedom and the scale of weak interactions

2. proposal for solution of gauge hierarchy problem model without fundamental scalar model without fundamental scalar new anti-symmetric tensor fields new anti-symmetric tensor fields mass term forbidden by symmetry mass term forbidden by symmetry chiral couplings to quarks and leptons chiral couplings to quarks and leptons chiral couplings are asymptotically free chiral couplings are asymptotically free weak scale by dimensional transmutation weak scale by dimensional transmutation

3. antisymmetric tensor fields two irreducible representations of Lorentz – symmetry : (3,1) + (1,3) two irreducible representations of Lorentz – symmetry : (3,1) + (1,3) complex representations : (3,1) * = (1,3) complex representations : (3,1) * = (1,3) similar to left/right handed spinors similar to left/right handed spinors

4. chiral couplings to quarks and leptons most general interaction consistent with Lorentz and gauge symmetry : ß are weak doublets with hypercharge most general interaction consistent with Lorentz and gauge symmetry : ß are weak doublets with hypercharge consistent with chiral parity : consistent with chiral parity : d R, e R, ß - have odd chiral parity d R, e R, ß - have odd chiral parity

5. no local mass term allowed for chiral tensors Lorentz symmetry forbids (ß + )* ß + Lorentz symmetry forbids (ß + )* ß + Gauge symmetry forbids ß + ß + Gauge symmetry forbids ß + ß + Chiral parity forbids (ß - )* ß + Chiral parity forbids (ß - )* ß +

6. kinetic term does not mix ß + and ß – does not mix ß + and ß – consistent with all symmetries, including chiral parity consistent with all symmetries, including chiral parity

7. quartic couplings add gauge interactions and gauge invariant kinetic term for fermions …

8. classical dilatation symmetry action has no parameter with dimension mass action has no parameter with dimension mass all couplings are dimensionless all couplings are dimensionless

9. consistency of chiral tensors ?

10. B - basis B –fields are unconstrained B –fields are unconstrained six complex doublets six complex doublets vectors under space – rotations vectors under space – rotations irreducible under Lorentz -transformations irreducible under Lorentz -transformations

11. free propagator free propagator inverse propagator has unusual form : propagator is invertible ! except for pole at q 2 = 0

12. energy density positive for longitudinal mode b 3 vanishes for transversal modes b 1,2 ( borderline to stability ) unstable secular classical solutions in free theory quantum theory : free Hamiltonian is not bounded

13. no consistent free theory !

14. interacting chiral tensors can be consistently quantized Bounded Hamiltonian permits canonical quantization Bounded Hamiltonian permits canonical quantization Interactions will decide on which side of the borderline between stability and instability the model lies. Interactions will decide on which side of the borderline between stability and instability the model lies. Vacuum not perturbative Vacuum not perturbative Non – perturbative generation of mass: Non – perturbative generation of mass: stable massive spin one particles ! stable massive spin one particles ! Chirons Chirons

15. asymptotic freedom

16. evolution equations for chiral couplings

17. evolution equations for top coupling fermion anomalous dimension tensor anomalous dimension no vertex correction asymptotic freedom ! Similar observation in abelian model :Avdeev,Chizhov ‘93

18. dimensional transmutation Chiral coupling for top grows large at chiral scale Λ ch This sets physical scale : dimensional transmutation - similar to Λ QCD in strong QCD- gauge interaction

19. spontaneous electroweak symmetry breaking

20. top – anti-top condensate large chiral coupling for top leads to large effective attractive interaction for top quark large chiral coupling for top leads to large effective attractive interaction for top quark this triggers condensation of top – anti-top pairs this triggers condensation of top – anti-top pairs electroweak symmetry breaking : effective Higgs mechanism provides mass for weak bosons electroweak symmetry breaking : effective Higgs mechanism provides mass for weak bosons effective Yukawa couplings of Higgs give mass to quarks and leptons effective Yukawa couplings of Higgs give mass to quarks and leptons cf : Miranski ; Bardeen, Hill, Lindner

21. Schwinger - Dyson equation for top quark mass solve gap equation for top quark propagator

22. SDE for B-B-propagator

23. gap equation for top quark mass has reasonable solutions for m t : somewhat above the chiral scale

24. tLtL tRtR two loop SDE for top-quark mass contract B- exchange to pointlike four fermion interaction

25. effective interactions introduce composite field for top- antitop bound state introduce composite field for top- antitop bound state plays role of Higgs field plays role of Higgs field new effective interactions involving the composite scalar φ new effective interactions involving the composite scalar φ

26. effective scalar tensor interactions

27. chiral tensor – gauge boson - mixing and more …

28. massive chiral tensor fields

29. massive spin one particles new basis of vector fields: new basis of vector fields: standard action for massive vector fields standard action for massive vector fields classical stability ! classical stability ! Z(q) = 1 + m 2 / q 2

30. classical stability massive spin one fields : stable massive spin one fields : stable free theory : borderline stability/instability, free theory : borderline stability/instability, actually unstable ( secular solutions, no ghosts) actually unstable ( secular solutions, no ghosts) mass term moves theory to stable region mass term moves theory to stable region positive energy density positive energy density

31. non – perturbative mass term m 2 : local in S - basis, non-local in B – basis m 2 : local in S - basis, non-local in B – basis cannot be generated in perturbation theory in absence of electroweak symmetry breaking cannot be generated in perturbation theory in absence of electroweak symmetry breaking plausible infrared regularization for divergence of inverse quantum propagator as chiral scale is approached plausible infrared regularization for divergence of inverse quantum propagator as chiral scale is approached in presence of electroweak symmetry breaking : generated by loops involving chiral couplings in presence of electroweak symmetry breaking : generated by loops involving chiral couplings

32. effective cubic tensor interactions generated by electroweak symmetry breaking

33. propagator corrections from cubic couplings non – local !

34. effective propagator for chiral tensors effective propagator for chiral tensors massive effective inverse propagator : pole for massive field mass term :

35. phenomenology

36. new resonances at LHC ? production of massive chirons at LHC ? production of massive chirons at LHC ? signal : massive spin one resonances signal : massive spin one resonances rather broad : decay into top quarks rather broad : decay into top quarks relatively small production cross section : small chiral couplings to lowest generation quarks, no direct coupling to gluons relatively small production cross section : small chiral couplings to lowest generation quarks, no direct coupling to gluons

37. mixing of charged spin one fields modification of W-boson mass modification of W-boson mass similar for Z – boson similar for Z – boson watch LEP – precision tests ! watch LEP – precision tests !

38. mixing between chiral tensor and photon

39. Pauli term contributes to g-2 suppressed by inverse mass of chiral tensor inverse mass of chiral tensor small chiral coupling of muon and electron small chiral coupling of muon and electron small mixing between chiral tensor and photon small mixing between chiral tensor and photon for M c ≈ 300 GeV : Δ(g-2) ≈ 5 10 -9 for muon for M c ≈ 300 GeV : Δ(g-2) ≈ 5 10 -9 for muon

40. effective interactions from chiral tensor exchange solve for S μ in presence of other fields solve for S μ in presence of other fields reinsert solution reinsert solution

41. general solution propagator for charged chiral tensors

42. electroweak precision tests compatible with LEP experiments compatible with LEP experiments for M c > 300 GeV for M c > 300 GeV

43. mixing of chiral tensors with ρ - meson could contribute to anomaly in radiative pion decays

44. conclusions chiral tensor model has good chances to be consistent chiral tensor model has good chances to be consistent mass generation needs to be understood quantitatively mass generation needs to be understood quantitatively interesting solution of gauge hierarchy problem interesting solution of gauge hierarchy problem phenomenology needs to be explored ! phenomenology needs to be explored ! if quartic couplings play no major role: if quartic couplings play no major role: less couplings than in standard model predictivity ! less couplings than in standard model predictivity !

45. end

46. new four fermion interactions typically rather small effect for lower generations more substantial for bottom, top !

47. momentum dependent Weinberg angle