A model of soft mass generation Jiří Hošek Department of Theoretical Physics Nuclear Physics Institute Řež (Prague)

1. AN OLD-FASHIONED REASONING In (mysterious) SU(2) L xU(1) Y : IF fermion masses are generated dynamically without elementary scalars by a new interaction the intermediate masses follow (Schwinger). Hence, fermion masses are of primary concern. L-R and interfamily connections needed. In (mysterious) SU(2) L xU(1) Y : IF fermion masses are generated dynamically without elementary scalars by a new interaction the intermediate masses follow (Schwinger). Hence, fermion masses are of primary concern. L-R and interfamily connections needed. Composite (bound state) ‘would-be’ NG bosons => New interaction must be strong. Composite (bound state) ‘would-be’ NG bosons => New interaction must be strong. Massiveness is a low-momentum phenomenon: Dynamical mass generation should be low-momentum dominated. ‘Right choice of the degrees of freedom: Massive complex vector field. Massiveness is a low-momentum phenomenon: Dynamical mass generation should be low-momentum dominated. ‘Right choice of the degrees of freedom: Massive complex vector field.

2. THE MODEL L SM is the SM Lagrangian of three fermion families supplemented with neutrino right-handed singlets L SM is the SM Lagrangian of three fermion families supplemented with neutrino right-handed singlets L F : Heavy complex vector field C μ interacting with family-changing current of the left- and right-handed fermion electroweak multiplets L F : Heavy complex vector field C μ interacting with family-changing current of the left- and right-handed fermion electroweak multiplets

SU(2) L xU(1) Y invariance is manifest NEW GLOBAL U(1) F

Renormalizability Renormalizability No one-loop vertex corrections No one-loop vertex corrections Hard right-handed Majorana mass term only for one family Hard right-handed Majorana mass term only for one family M>10 7 GeV CHALLENGE !!! M>10 7 GeV CHALLENGE !!!

3. FERMION MASS GENERATION self-consistent variational (nonperturbative) ASSUME Σ are dynamically generated ASSUME Σ are dynamically generated

Σ implies nondiagonal part of C boson propagator D effective bilinear Lagrangian Nondiagonal part of the propagator

FERMION SCHWINGER-DYSON EQUATIOS extra convergent kernels; all fermion masses in the world related

A HOPE Approximations: hard fermion masses; p=0; Π=M Approximations: hard fermion masses; p=0; Π=M Integral equations turn into UV finite calculable integrals Integral equations turn into UV finite calculable integrals

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

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

4. W AND Z BOSON MASS GENERATION Dynamically generated Σ break SU(2) L xU(1) Y down to U(1) em Dynamically generated Σ break SU(2) L xU(1) Y down to U(1) em 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 the gauge and three multicomponenet NG bosons. These vertices give rise to the longitudinal parts of W and Z polarization tensors with massless poles (Schwinger). Their residues are

W and Z boson masses are related to the fermion masses by sum rules W and Z boson masses are related to the fermion masses by sum rules From a model of Σ the sum rules are saturated by the top quark mass From a model of Σ the sum rules are saturated by the top quark mass Weinberg’s relation tolerably violated Weinberg’s relation tolerably violated

5. SPONTANEOUS BREAKDOWN OF THE GLOBAL U(1) F SYMMETRY Σ and Π dynamically generated by strong C boson-fermion interaction break spontaneously the global U(1) F Σ and Π dynamically generated by strong C boson-fermion interaction break spontaneously the global U(1) F Massless NG pseudoscalar – phenomenologically dangerous Massless NG pseudoscalar – phenomenologically dangerous

Multicomponent bound state of both fermions and C i μ Multicomponent bound state of both fermions and C i μ Effective couplings to fermions negligibly small due to C admixture Effective couplings to fermions negligibly small due to C admixture Weinberg-Wilczek axion ? Weinberg-Wilczek axion ?

6. CONCLUDING COMMENTS (work in progress) We believe in stiff dynamics underlying SM We believe in stiff dynamics underlying SM Admiration for Wilson Admiration for Wilson Strongly coupled model of Kroll, Lee, Zumino Strongly coupled model of Kroll, Lee, Zumino Model requires several families Model requires several families