A model of soft mass A model of soft mass generationgeneration

Jiří HošekJiří HošekDepartment of Theoretical PhysicsDepartment of Theoretical Physics

Nuclear Physics InstituteNuclear Physics InstituteŘež (Prague)Řež (Prague)

hosekhosek@ujf.cas.cz@ujf.cas.cz

1. AN OLD-FASHIONED REASONING1. AN OLD-FASHIONED REASONING

• In (mysterious) SU(2)In (mysterious) SU(2)LLxU(1)xU(1)YY : IF fermion masses are : IF fermion masses are generated dynamically without elementary scalars generated dynamically without elementary scalars by a new interaction the intermediate masses follow by a new interaction the intermediate masses follow (Schwinger). Hence, fermion masses are of primary (Schwinger). Hence, fermion masses are of primary concern. concern. L-R and interfamily connections needed.L-R and interfamily connections needed.

• Composite (bound state) ‘would-be’ NG bosons => Composite (bound state) ‘would-be’ NG bosons => New interaction must be strong.New interaction must be strong.

• Massiveness is a low-momentum phenomenon: Massiveness is a low-momentum phenomenon: Dynamical mass generation should be low-Dynamical mass generation should be low-momentum dominated. ‘Right choice of the degrees momentum dominated. ‘Right choice of the degrees of freedom: of freedom: Massive complex vector field.Massive complex vector field.

2. THE MODEL 2. THE MODEL

• LLSM SM is the SM Lagrangian of three fermion families is the SM Lagrangian of three fermion families supplemented with neutrino right-handed singletssupplemented with neutrino right-handed singlets

• LLFF : Heavy complex vector field C: Heavy complex vector field Cμμ interacting interacting with family-changing current of the left- and right-with family-changing current of the left- and right-handed fermion electroweak multipletshanded fermion electroweak multiplets

SU(2)SU(2)LLxU(1)xU(1)YY invariance is manifest invariance is manifest

NEW GLOBAL U(1)NEW GLOBAL U(1)FF

• RenormalizabilityRenormalizability

• No one-loop vertex correctionsNo one-loop vertex corrections

• Hard right-handed Majorana mass term only for Hard right-handed Majorana mass term only for one familyone family

• M>10M>1077 GeV GeV CHALLENGE !!!CHALLENGE !!!

3. FERMION MASS GENERATION3. FERMION MASS GENERATIONself-consistent variational (nonperturbative)self-consistent variational (nonperturbative)

ASSUME ASSUME ΣΣ are dynamically are dynamically generatedgenerated

ΣΣ implies nondiagonal part of C boson implies nondiagonal part of C boson propagator D propagator D

effective bilinear Lagrangianeffective bilinear Lagrangian

Nondiagonal part of the propagatorNondiagonal part of the propagator

FERMION SCHWINGER-DYSON EQUATIOS FERMION SCHWINGER-DYSON EQUATIOS extra convergent kernels; all fermion extra convergent kernels; all fermion

masses in the world relatedmasses in the world related

A HOPE A HOPE

• Approximations: hard fermion masses; p=0; Approximations: hard fermion masses; p=0; ΠΠ=M=M

• Integral equations turn into UV finite calculable Integral equations turn into UV finite calculable integralsintegrals

System of implicit coupled equations for the fermion System of implicit coupled equations for the fermion masses can be analyzedmasses can be analyzed

abbreviateabbreviate

Once we solve the equation f(y)=0 the Once we solve the equation f(y)=0 the fermion masses are given. Solutions are fermion masses are given. Solutions are

found.found.

4. W AND Z BOSON MASS GENERATION4. W AND Z BOSON MASS GENERATION

• Dynamically generated Dynamically generated ΣΣ break break SU(2)SU(2)LLxU(1)xU(1)Y Y down to U(1)down to U(1)emem

• The ‘would-be’ NGs (composite !) are visualized in WT identities:The ‘would-be’ NGs (composite !) are visualized in WT identities:

From the pole terms we extract the effective vertices between From the pole terms we extract the effective vertices between the gauge and three multicomponenet NG bosons. These the gauge and three multicomponenet NG bosons. These

vertices give rise to the longitudinal parts of W and Z vertices give rise to the longitudinal parts of W and Z polarization tensors with massless poles (Schwinger). Their polarization tensors with massless poles (Schwinger). Their

residues areresidues are

• W and Z boson masses are related to the fermion W and Z boson masses are related to the fermion masses by sum rulesmasses by sum rules

• From a model of From a model of ΣΣ the sum rules are saturated by the sum rules are saturated by the top quark massthe top quark mass

• Weinberg’s relation tolerably violatedWeinberg’s relation tolerably violated

5. SPONTANEOUS BREAKDOWN OF THE 5. SPONTANEOUS BREAKDOWN OF THE GLOBAL U(1)GLOBAL U(1)FF SYMMETRY SYMMETRY

• ΣΣ and and ΠΠ dynamically generated by strong dynamically generated by strong C boson-fermion interaction break C boson-fermion interaction break spontaneously the global U(1)spontaneously the global U(1)FF

• Massless NG pseudoscalar –Massless NG pseudoscalar – phenomenologically dangerous phenomenologically dangerous

• Multicomponent bound state of both fermions and Multicomponent bound state of both fermions and CCii

μμ

• Effective couplings to fermions negligibly small Effective couplings to fermions negligibly small due to C admixturedue to C admixture

• Weinberg-Wilczek axion ? Weinberg-Wilczek axion ?

6. CONCLUDING COMMENTS (work in 6. CONCLUDING COMMENTS (work in progress)progress)

• We believe in stiff dynamics underlying SMWe believe in stiff dynamics underlying SM

• Admiration for WilsonAdmiration for Wilson

• Strongly coupled model of Kroll, Lee, ZuminoStrongly coupled model of Kroll, Lee, Zumino

• Model Model requiresrequires several families several families