Dynamical Mass GenerationDynamical Mass Generation by Strong Yukawa Interactions by Strong Yukawa Interactions

TomTomáš Brauner and Jiří Hošek, Phys.Rev.D72: áš Brauner and Jiří Hošek, Phys.Rev.D72: 045007(2005)045007(2005)

Petr Beneš, Tomáš Brauner and Jiří HošekPetr Beneš, Tomáš Brauner and Jiří Hošek::hep-ph/0605147 hep-ph/0605147

CERN Yellow Book : hep-ph/0608079 CERN Yellow Book : hep-ph/0608079

Lagrangian and its propertiesLagrangian and its propertiesFermion mass generation and scalar boson mass Fermion mass generation and scalar boson mass

splittingsplittingWhere is the NambuWhere is the Nambu-Goldstone boson ?-Goldstone boson ?

Gauge boson mass generationGauge boson mass generationSymmetry-breaking loop-generated verticesSymmetry-breaking loop-generated vertices

SU(2)xU(1) generalizationSU(2)xU(1) generalization

1. Lagrangian and its 1. Lagrangian and its propertiesproperties

remember arrows of the fieldsremember arrows of the fields

Symmetry:Symmetry:

• Why two fermion species: no axial Why two fermion species: no axial anomalyanomaly

• With MWith M22>O no symmetry breakdown >O no symmetry breakdown in scalar sector itselfin scalar sector itself

• Comparison with Higgs mechanism Comparison with Higgs mechanism by heart (see the Lagrangian) by heart (see the Lagrangian)

II. Fermion mass generation and scalar II. Fermion mass generation and scalar boson mass splittingboson mass splitting

• ASSUME that Yukawa ASSUME that Yukawa interactions generate interactions generate the chiral-symmetry the chiral-symmetry breaking fermion breaking fermion proper self-energies proper self-energies ΣΣ::

THEN Yukawa interactions generate the THEN Yukawa interactions generate the symmetry-breaking scalar proper self-symmetry-breaking scalar proper self-energy energy ΠΠ

Yukawa interactions GENERATE Yukawa interactions GENERATE ΣΣProvided solutions for anomalous proper Provided solutions for anomalous proper

self-energies self-energies ΠΠ exist exist

Explicit form of the coupled Schwinger-Explicit form of the coupled Schwinger-Dyson equationsDyson equations (not very (not very illuminating)illuminating)

NICELY CONVERGENT KERNEL NICELY CONVERGENT KERNEL (corresponding counter terms prohibited by (corresponding counter terms prohibited by

symmetry)symmetry)

• By dimensional By dimensional argument:argument:

• Compare with Higgs:Compare with Higgs:

Boson mass splittingBoson mass splitting

• In particular case of In particular case of real real ΠΠ found found numericallynumerically

the real and imaginary the real and imaginary parts of parts of ΦΦ are the are the mass eigenstatesmass eigenstates with masseswith masses

Solutions found for LARGE YUKAWA Solutions found for LARGE YUKAWA COUPLINGS COUPLINGS

numerically numerically (exploratory stage (!!!))(exploratory stage (!!!))1. Line of critical couplings1. Line of critical couplings

2.Typical shape of UV-finite solutions 2.Typical shape of UV-finite solutions ΣΣ and and ΠΠ . . They are found numerically upon Wick They are found numerically upon Wick rotation in SD equations.rotation in SD equations.

3. Large amplification of the fermion 3. Large amplification of the fermion mass ratiosmass ratios

Large amplification of fermion mass ratios as a Large amplification of fermion mass ratios as a response to small changes in Yukawa coupling response to small changes in Yukawa coupling ratiosratios

Explicit knowledge of non-analytic dependences of Explicit knowledge of non-analytic dependences of masses upon couplings – masses upon couplings –

ULTIMATE DREAMULTIMATE DREAM

Coupling constant Coupling constant λλ ignored as unimportant for ignored as unimportant for

non-perturbative mass generationnon-perturbative mass generation

III. Where is the Nambu-Goldstone III. Where is the Nambu-Goldstone boson ?boson ?

• Axial-vector currentAxial-vector current

• Axial-vector Ward Axial-vector Ward identities for proper identities for proper verticesvertices

For For ΣΣ, , ΠΠ non-zero the identities imply the non-zero the identities imply the massless pole in proper vertices massless pole in proper vertices ΓΓ

Basic quantities to be calculated are the UV Basic quantities to be calculated are the UV finite vectorial tadpoles I:finite vectorial tadpoles I:

Effective NG couplings are related to Effective NG couplings are related to ΣΣ and and ΠΠ::

The overall normalization is given by The overall normalization is given by tadpoles I:tadpoles I:

IV. Gauge boson mass generationIV. Gauge boson mass generation

• Gauge boson mass Gauge boson mass squared is the residue at squared is the residue at the massless pole of the the massless pole of the gauge field polarization gauge field polarization tensortensor

(Schwinger):(Schwinger):

• In Higgs (complete In Higgs (complete polarization polarization tensor):tensor):

In strongly coupled models no control on the In strongly coupled models no control on the bound-state spectrum except NG. bound-state spectrum except NG.

Consequently, only the longitudinal part can Consequently, only the longitudinal part can be computed. By transversalitybe computed. By transversality

V. Symmetry-breaking loop-generated UV-V. Symmetry-breaking loop-generated UV-finite vertices: genuine (albeit gedanken) finite vertices: genuine (albeit gedanken)

predictionspredictions

An illustration: Three-gluon vertex An illustration: Three-gluon vertex (G is computed)(G is computed)

VI. Outlook (bona fide)VI. Outlook (bona fide)

- Common source of fermion and gauge-boson mass - Common source of fermion and gauge-boson mass generationgeneration

- Hope for natural description of wide, sparse and Hope for natural description of wide, sparse and irregular fermion mass spectrumirregular fermion mass spectrum

- Basic SU(2)xU(1) generalization exists:Basic SU(2)xU(1) generalization exists: T. Brauner, J.H., A model of flavors,T. Brauner, J.H., A model of flavors, hep-ph/0407339 : introduce hep-ph/0407339 : introduce TWO DISTINCT TWO DISTINCT

massive scalar SU(2)xU(1) doublets. massive scalar SU(2)xU(1) doublets. I LIKE I LIKE SCALARS-THEY FEEL FLAVORS.SCALARS-THEY FEEL FLAVORS.

- Phenomenological viability is to be investigated- Phenomenological viability is to be investigated