Chiral freedomChiral freedomand the and the scale ofscale of
weak interactionsweak interactions
proposal for solution of proposal for solution of gauge hierarchy problemgauge hierarchy problem
model without fundamental scalarmodel without fundamental scalar new antisymmetric tensor fieldsnew antisymmetric tensor fields mass term forbidden by symmetrymass term forbidden by symmetry chiral couplings to quarks and leptonschiral couplings to quarks and leptons chiral couplings are asymptotically freechiral couplings are asymptotically free weak scale by dimensional transmutationweak scale by dimensional transmutation
antisymmetric tensor antisymmetric tensor fieldsfields
two irreducible representations of two irreducible representations of Lorentz – symmetry : (3,1) + (1,3)Lorentz – symmetry : (3,1) + (1,3)
complex representations : (3,1)complex representations : (3,1)** = = (1,3)(1,3)
similar to left/right handed spinorssimilar to left/right handed spinors
chiral couplings to chiral couplings to quarks and leptonsquarks and leptons
most general interaction consistent with most general interaction consistent with Lorentz and gauge symmetry : ß are weak Lorentz and gauge symmetry : ß are weak doublets with hyperchargedoublets with hypercharge
consistent with chiral parity : consistent with chiral parity :
d d RR , e , e RR , ß , ß -- have odd chiral parity have odd chiral parity
no mass term allowed no mass term allowed for chiral tensorsfor chiral tensors
Lorentz symmetry forbids (ßLorentz symmetry forbids (ß++)* ß)* ß++
Gauge symmetry forbids ßGauge symmetry forbids ß++ ß ß++
Chiral parity forbids (ßChiral parity forbids (ß--)* ß)* ß++
kinetic termkinetic term
does not mix ß does not mix ß ++ and ß and ß – –
consistentconsistent with all symmetries, with all symmetries, including chiral parityincluding chiral parity
quartic couplingsquartic couplings
add gauge interactions and gauge invariant kinetic term for fermions …
classical dilatation classical dilatation symmetrysymmetry
action has no parameter with action has no parameter with dimension massdimension mass
all couplings are dimensionlessall couplings are dimensionless
B - basisB - basis
B –fields are unconstrainedB –fields are unconstrained six complex doubletssix complex doublets vectors under space – rotationsvectors under space – rotations irreducible under Lorentz -irreducible under Lorentz -
transformationstransformations
free propagatorfree propagator
inverse propagator has unusual form :
propagator is invertible ! except for pole at q 2 = 0
energy densityenergy density
positive for longitudinal modevanishes for transversal modes ( borderline to stability )unstable secular classical solutions in free theory
asymptotic freedomasymptotic freedom
evolution equations for chiral evolution equations for chiral couplingscouplings
evolution equations for top evolution equations for top couplingcoupling
fermion anomalous dimension
tensor anomalous dimension
no vertex correction
asymptotic freedom !Avdeev,Chizhov ;93
dimensional dimensional transmutationtransmutation
Chiral coupling for top grows large Chiral coupling for top grows large
at chiral scale at chiral scale ΛΛchch
This sets physical scale : dimensional This sets physical scale : dimensional transmutation -transmutation -
similar to similar to ΛΛQCD QCD in strong QCD- gauge interactionin strong QCD- gauge interaction
spontaneous electroweakspontaneous electroweaksymmetry breakingsymmetry breaking
top – anti-top condensatetop – anti-top condensate
large chiral coupling for top leads to large chiral coupling for top leads to large effective attractive interaction for large effective attractive interaction for top quarktop quark
this triggers condensation of top – anti-this triggers condensation of top – anti-top pairstop pairs
electroweak symmetry breaking : electroweak symmetry breaking : effective Higgs mechanism provides mass effective Higgs mechanism provides mass for weak bosonsfor weak bosons
effective Yukawa couplings of Higgs give effective Yukawa couplings of Higgs give mass to quarks and leptonsmass to quarks and leptons
cf : Miranski ; Bardeen, Hill, Lindner
Schwinger - Dyson equationSchwinger - Dyson equationfor top quark massfor top quark mass
solve gap equation for top quark propagatorsolve gap equation for top quark propagator
SDE for B-B-propagatorSDE for B-B-propagator
gap equation for top gap equation for top quark massquark mass
has reasonable solutions for mhas reasonable solutions for mtt
tLtR
two loop SDE for top-two loop SDE for top-quark massquark mass
contract B- exchange to pointlike four fermion interaction
effective interactionseffective interactions
introduce composite field for top- introduce composite field for top- antitop bound stateantitop bound state
plays role of Higgs fieldplays role of Higgs field new effective interactions involving new effective interactions involving
the composite scalar the composite scalar φφ
effective scalar tensor effective scalar tensor interactionsinteractions
chiral tensor – gauge boson chiral tensor – gauge boson - mixing- mixing
and more …
massive chiral tensor massive chiral tensor fieldsfields
effective cubic tensor effective cubic tensor interactionsinteractions
generated by electroweak symmetry breaking
propagator corrections propagator corrections from cubic couplingsfrom cubic couplings
non – local !
effective propagator effective propagator for chiral tensorsfor chiral tensors
massive effective inverse propagator :pole for massive field
mass term :
massive spin one massive spin one particlesparticles
new basis of vector fields:new basis of vector fields:
standard action for massive vector standard action for massive vector fieldsfields
classical stability !classical stability !
classical stabilityclassical stability
massive spin one fields : stablemassive spin one fields : stable free theory : borderline stability/instability,free theory : borderline stability/instability,
actually unstable ( secular solutions , no actually unstable ( secular solutions , no ghosts)ghosts)
mass term moves theory to stable regionmass term moves theory to stable region positive energy densitypositive energy density
phenomenologyphenomenology
new resonances at LHC ?new resonances at LHC ?
production of massive chiral tensors at production of massive chiral tensors at LHC ?LHC ?
signal : massive spin one resonancessignal : massive spin one resonances rather broad : decay into top quarksrather broad : decay into top quarks relatively small production cross section : relatively small production cross section :
small chiral couplings to lowest small chiral couplings to lowest generation quarks , no direct coupling to generation quarks , no direct coupling to gluonsgluons
mixing of charged spin mixing of charged spin one fieldsone fields
modification of W-boson massmodification of W-boson mass similar for Z – bosonsimilar for Z – boson watch LEP – precision tests !watch LEP – precision tests !
mixing between chiral mixing between chiral tensortensor
and photonand photon
Pauli term contributes to Pauli term contributes to g-2g-2
suppressed bysuppressed by inverse mass of chiral tensorinverse mass of chiral tensor small chiral coupling of muon and electronsmall chiral coupling of muon and electron small mixing between chiral tensor and small mixing between chiral tensor and
photonphoton
effective interactions fromeffective interactions fromchiral tensor exchangechiral tensor exchange
solve for Ssolve for Sμμ in presence of other fields in presence of other fields reinsert solutionreinsert solution
general solution
propagator for charged chiral tensorspropagator for charged chiral tensors
momentum dependent momentum dependent Weinberg angleWeinberg angle
new four fermion new four fermion interactionsinteractions
typically rather small effect for lower generationsmore substantial for bottom , top !
mixing of chiral tensors mixing of chiral tensors with with ρρ - meson - meson
could contribute to anomaly in radiative pion decays
conclusionsconclusions chiral tensor model has good chances to chiral tensor model has good chances to
be consistentbe consistent mass generation needs to be understood mass generation needs to be understood
quantitativelyquantitatively interesting solution of gauge hierarchy interesting solution of gauge hierarchy
problemproblem 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 less couplings than in standard model
Predictivity ! Predictivity !
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