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Published byTheodore Rich Modified about 1 year ago

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Chiral freedom and the scale of weak interactions

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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 local mass term forbidden by symmetry local 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

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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

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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

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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 +

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

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

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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

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flavor and CP violation chiral couplings can be made diagonal and real by suitable phases for fermions chiral couplings can be made diagonal and real by suitable phases for fermions Kobayashi – Maskawa Matrix Kobayashi – Maskawa Matrix same flavor violation and CP violation as in standard model same flavor violation and CP violation as in standard model additional CP violation through quartic couplings possible additional CP violation through quartic couplings possible

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asymptotic freedom

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evolution equations for chiral couplings

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

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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

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spontaneous electroweak symmetry breaking

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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

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

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SDE for B-B-propagator

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

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

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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

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effective scalar tensor interactions

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chiral tensor – gauge boson - mixing and more …

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massive chiral tensor fields

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chirons irreducible representation for anti-symmetric tensor fields has three components irreducible representation for anti-symmetric tensor fields has three components in presence of mass : little group SO(3) in presence of mass : little group SO(3) with respect to SO(3) : anti-symmetric tensor equivalent to vector with respect to SO(3) : anti-symmetric tensor equivalent to vector massive chiral tensors = massive spin one particles : chirons massive chiral tensors = massive spin one particles : chirons

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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

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classical stability massive spin one fields : stable massive spin one fields : stable free theory for chiral tensors: borderline stability/instability, free theory for chiral tensors: 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 for solutions of field equations positive energy density for solutions of field equations

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consistency of chiral tensors ?

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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

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

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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 for plane waves :

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secular instability solutions grow linearly with time !

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no consistent free theory !

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mechanical analogue d x / d t 2 = ε x d x / d t 2 = ε x ε > 0 : exponentially growing mode ε > 0 : exponentially growing mode ( tachyon or ghost ) ( tachyon or ghost ) ε < 0 : stable mode ε < 0 : stable mode ε = 0 : borderline ( secular solution growing ε = 0 : borderline ( secular solution growing linearly with time ) linearly with time ) even tiny ε decides on stability ! even tiny ε decides on stability ! interactions will decide on stability ! interactions will decide on stability !

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interacting chiral tensors can be consistently quantized Hamiltonian permits canonical quantization 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

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chiron mass

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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

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effective cubic tensor interactions generated by electroweak symmetry breaking

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propagator corrections from cubic couplings non – local !

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

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phenomenology

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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

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effects at low energy mixing with gauge bosons is important mixing with gauge bosons is important also direct four fermion interactions with tensor structure also direct four fermion interactions with tensor structure

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mixing between chiral tensor and photon photon remains massless but acquires new tensor interaction

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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 and small chiral couplings : for M c ≈ 300 GeV and small chiral couplings : Δ(g-2) ≈ 5 10 -9 for muon Δ(g-2) ≈ 5 10 -9 for muon larger chiral couplings : M c ≈ few TeV larger chiral couplings : M c ≈ few TeV

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anomalous magnetic moment of muon

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electroweak precision tests chiron exchange and mixing: chiron exchange and mixing: compatible with LEP experiments compatible with LEP experiments for M c > 300 GeV for M c > 300 GeV rough estimate : for M c =1 TeV:

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composite scalars two composite Higgs doublets expected two composite Higgs doublets expected mass 400 -500 GeV mass 400 -500 GeV loop effects ? loop effects ?

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

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generation of light fermion masses involves chiral couplings and chiron – gauge boson mixing

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chiron – photon - mixing

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effective tensor vertex of photon contributes to g-2

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determination of chiral couplings restricts g-2

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for characteristic value … and neglecting chiron – mixing large chiron mass above LHC range

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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 !

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end

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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

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general solution propagator for charged chiral tensors

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effective propagator correction

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

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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 !

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momentum dependent Weinberg angle

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