A model of flavors Jiří Hošek Department of Theoretical Physics NPI Řež (Prague) (Tomáš Brauner and JH, hep-ph/ )

Plan of presentation  Introductory remarks  Strategy: Role of scalars  Fermion mass generation  Intermediate-boson mass generation  Concluding remarks

3 1.Introductory remarks Standard model (SM) is the best what in theoretical particle physics we have: Standard model (SM) is the best what in theoretical particle physics we have: In operationally well defined framework it parameterizes and successfully correlates virtually all electroweak phenomena. In operationally well defined framework it parameterizes and successfully correlates virtually all electroweak phenomena. Objections: Objections: 1. QCD is better 1. QCD is better 2. Neutrino masses are different from zero 2. Neutrino masses are different from zero

Spontaneous mass generation is a theoretical necessity  Hard intermediate boson mass terms ruin directly renormalizability  Hard fermion mass terms ruin indirectly renormalizability  Higgs mechanism is unnatural: quadratic mass renormalization  Too many theoretically arbitrary, phenomenologically vastly different parameters

Attempts to solve disadvantages of SM  SUSY Weakly coupled theory Weakly coupled theory The same Higgs mechanism The same Higgs mechanism no quadratic mass renormalization no quadratic mass renormalization gauge and fermion masses not related gauge and fermion masses not related whole new parallel world of heavy particles whole new parallel world of heavy particles

 TECHNICOLOR-LIKE SCENARIOS Strongly coupled theory Strongly coupled theory No quadratic mass renormalization No quadratic mass renormalization Gauge and fermion masses not related Gauge and fermion masses not related Plenty of heavy techni-hadrons Plenty of heavy techni-hadrons

 LITTLE HIGGS Weakly coupled theory Weakly coupled theory No quadratic mass renormalization at “low energy”; at high energy it reappears No quadratic mass renormalization at “low energy”; at high energy it reappears Gauge and fermion masses not related Gauge and fermion masses not related

 DON’T FORGET UNKNOWN

2. Strategy: Role of scalars  1. Introduce two distinct complex scalar doublets: S = (S (+), S (0) ) with Y(S) = +1 and ordinary mass squared term in the Lagrangian S = (S (+), S (0) ) with Y(S) = +1 and ordinary mass squared term in the Lagrangian N = (N (0), N (-) ) with Y(N) = -1 and ordinary mass squared term in the Lagrangian N = (N (0), N (-) ) with Y(N) = -1 and ordinary mass squared term in the Lagrangian  NO SPONTANEOUS BREAKDOWN OF SYMMETRY AT TREE LEVEL

 2. For completeness introduce n f neutrino right-handed SU(2) singlets with zero weak hypercharge: hard Majorana mass term allowed by symmetry with zero weak hypercharge: hard Majorana mass term allowed by symmetry

 Yukawa couplings of scalars distinguish between otherwise identical fermion families and break down explicitly all unwanted and dangerous inter-family symmetries:

Our model Our model  SU(2) L x U(1) Y gauge symmetry is manifest  No fermion mass terms except of M M  No gauge-boson mass terms  Mass scale of the world fixed by M S and M N  This does not imply that the particles corresponding to their massless fields have to stay massless

Breaking SU(2)xU(1) dynamically and Breaking SU(2)xU(1) dynamically and non-perturbatively. non-perturbatively. In perturbation theory the symmetry is preserved In perturbation theory the symmetry is preserved order by order. order by order. First ASSUME that fermion proper self-energy Σ is generated. Second, FIND IT SELF-CONSISTENTLY. Chirality-changing part of Σ must come out necessarily ultraviolet-finite – fermion mass counter terms strictly forbidden by chiral symmetry

Assumed fermion mass insertions give rise to generically new contributions of the scalar field propagators

Problem reduces to finding the spectrum of the bilinear Lagrangian Problem reduces to finding the spectrum of the bilinear Lagrangian

Crucial contribution to the scalar-field propagator is

Physically observable are then two real spin-0 particles corresponding to real scalar fields S 1 and S 2 defined as

The masses and the mixing angle are The masses and the mixing angle are

The case of the charged scalars is similar: Only particles with the same charge can mix, and they really do:

The masses and the field transformations are The masses and the field transformations are

α SN is the phase of μ SN and the mixing angle θ is α SN is the phase of μ SN and the mixing angle θ is

Splittings μ S 2, μ N 2 and μ SN 2 of the scalar-particle masses due to yet assumed dynamical fermion mass generation are both natural and important: 1. They come out UV finite due to the large momentum behavior of Σ(p 2 ) (see further). 2. They manifest spontaneous breakdown of SU(2) L x U(1) Y symmetry down to U(1) em in the scalar sector. of SU(2) L x U(1) Y symmetry down to U(1) em in the scalar sector. 3. They will be responsible for the UV finiteness of both the fermion and the intermediate vector boson masses.

3. Fermion mass generation  Chirality-changing fermion proper self- energy Σ(p 2 ) is bona fide given by the UV finite solution of the Schwinger-Dyson equation graphically defined for charged leptons

Explicit form of the equation is not very illuminating. It is, however, easily seen that IF a solution exists it is UV finite:

In order to proceed we are at the moment forced to resort to simplifications. The form of the nonlinearity is kept unchanged:  Neglect fermion mixing (sin 2θ = 0). This, unfortunately, implied neglecting utmost interesting relation between masses of upper and down fermions in doublets. This, unfortunately, implied neglecting utmost interesting relation between masses of upper and down fermions in doublets. Perform Wick rotation. Perform Wick rotation. Do angular integrations. Do angular integrations. Make Taylor expansion in M 2 1S – M 2 2S (M 2 – mean value). Make Taylor expansion in M 2 1S – M 2 2S (M 2 – mean value). For a generic (say e) fermion self- energy in dimensionless variables τ = p 2 /M 2 get For a generic (say e) fermion self- energy in dimensionless variables τ = p 2 /M 2 get

Numerical analysis done so far by Petr Beneš reveals the existence of a solution for large values of β. For electrically charged fermions m = Σ(p 2 = m 2 ).  SO FAR ONLY A GENERIC MODEL. It can pretend to phenomenological relevance only after demonstrating strong amplification of fermion masses to small changes of Yukawa couplings.

Generation of neutrino masses is more subtle and requires more work:  Without ν R neutrinos would be massless in our model.  With ν R the mechanism just described generates UV-finite Dirac Σ ν.  There is a hard mass term  Due to M M there is a UV-finite left-handed Majorana mass matrix.  As a result the model describes 2n f massive Majorana neutrinos with generic sea-saw spectrum

4. Intermediate-boson mass generation  Dynamically generated fermion proper self-energies Σ(p 2 ) break spontaneously SU(2) L x U(1) Y down to U(1) em.  Consequently, there are just three COMPOSITE Nambu-Goldstone bosons in the spectrum if the gauge interactions are switched off.  When switched on, the W and Z boson should acquire masses.  To determine their values it is necessary to calculate residues at single massless poles of their polarization tensors

‘Would-be’ NG bosons are visualized as massless poles in proper vertex functions of W and Z bosons as necessary consequences of Ward-Takahashi identities

From the pole terms in Γ we extract the effective two-leg vertices between the gauge and three multi-component ‘would-be’ NG bosons. They are given in terms of the UV-finite loop  As a result the gauge-boson masses are expressed in terms of sum rules

If Σ U and Σ D were degenerate the relation m W 2 /m Z 2 cos 2 Θ W = 1 would be fulfilled. Illustrative analysis with a particular model for Σ shows that the departure from this relation is very small. Knowledge of detailed form of Σ(p 2 ) is indispensable.

5. Concluding remarks  Genuinely quantum and non-perturbative mechanism of mass generation is rather rigid.  Not yet quantitative; yet strong-coupling.  Quadratic scalar mass renormalization can be avoided.  Relates fermion masses with each other.  Relates fermion masses to the intermediate boson masses.  There is no generic weak-interaction mass scale v = 246 GeV.  Mass scale of the world is fixed by M N, M S and M M.  There should exist four electrically neutral real scalar bosons, and two charged ones. They should be heavy, but not too much (O(10 6 GeV)).