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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