A model of flavorsA model of flavors

Jiří HošekJiří HošekDepartment of Theoretical PhysicsDepartment of Theoretical PhysicsNPI Řež (Prague)NPI Řež (Prague)(Tomáš Brauner and JH, hep-ph(Tomáš Brauner and JH, hep-ph/0407339)/0407339)

Plan of presentationPlan of presentation

Introductory remarksIntroductory remarks Strategy: Role of Strategy: Role of

scalarsscalars Fermion mass Fermion mass

generationgeneration Intermediate-boson Intermediate-boson

mass generationmass generation Concluding remarksConcluding remarks

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1.1. Introductory remarks Introductory remarks

Standard model (SM) is the best what in Standard model (SM) is the best what in theoretical particle physics we have:theoretical particle physics we have:

In operationally well defined framework it In operationally well defined framework it

parameterizes and successfully correlates parameterizes and successfully correlates virtually all electroweak phenomena.virtually all electroweak phenomena.

Objections:Objections: 1. QCD is better1. QCD is better 2. Neutrino masses are different from zero2. Neutrino masses are different from zero

Spontaneous mass generation is a theoretical Spontaneous mass generation is a theoretical necessitynecessity

Hard intermediate boson mass terms ruin Hard intermediate boson mass terms ruin directly renormalizability directly renormalizability

Hard fermion mass terms ruin indirectly Hard fermion mass terms ruin indirectly renormalizabilityrenormalizability

Higgs mechanism is Higgs mechanism is unnatural: unnatural: quadratic mass quadratic mass renormalizationrenormalization

Too many theoretically arbitrary, Too many theoretically arbitrary, phenomenologically phenomenologically vastly differentvastly different parameters parameters

Attempts to solve Attempts to solve disadvantages of SMdisadvantages of SM

SUSYSUSY Weakly coupled theoryWeakly coupled theory The same Higgs mechanismThe same Higgs mechanism no quadratic mass renormalizationno quadratic mass renormalization gauge and fermion masses not relatedgauge and fermion masses not related whole new parallel world of heavy whole new parallel world of heavy

particles particles

TECHNICOLOR-LIKE SCENARIOSTECHNICOLOR-LIKE SCENARIOS Strongly coupled theoryStrongly coupled theory

No quadratic mass renormalizationNo quadratic mass renormalization

Gauge and fermion masses not relatedGauge and fermion masses not related

Plenty of heavy techni-hadronsPlenty of heavy techni-hadrons

LITTLE HIGGSLITTLE HIGGS

Weakly coupled theoryWeakly coupled theory

No quadratic mass renormalization No quadratic mass renormalization at “low energy”; at high energy it at “low energy”; at high energy it reappearsreappears

Gauge and fermion masses not Gauge and fermion masses not relatedrelated

DON’T FORGET UNKNOWNDON’T FORGET UNKNOWN

2. Strategy: Role of 2. Strategy: Role of scalarsscalars

1. Introduce two distinct complex scalar 1. Introduce two distinct complex scalar doublets:doublets:

S = (SS = (S(+)(+) , S , S(0)(0)) with Y(S) = +1 and ) with Y(S) = +1 and ordinary mass squaredordinary mass squared term in the term in the LagrangianLagrangian

N = (NN = (N(0)(0), N, N(-)(-)) with Y(N) = -1 and ) with Y(N) = -1 and ordinary ordinary mass squaredmass squared term in the Lagrangian term in the Lagrangian

NO SPONTANEOUS BREAKDOWN OF NO SPONTANEOUS BREAKDOWN OF SYMMETRY AT TREE LEVELSYMMETRY AT TREE LEVEL

2. For completeness introduce n2. For completeness introduce nff neutrino right-handed SU(2) singletsneutrino right-handed SU(2) singlets

with zero weak hypercharge: hard with zero weak hypercharge: hard Majorana mass term allowed by Majorana mass term allowed by symmetrysymmetry

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

Our modelOur model

SU(2)SU(2)LLx U(1)x U(1)Y Y gauge symmetry is manifest gauge symmetry is manifest No fermion mass terms except of MNo fermion mass terms except of MMM

No gauge-boson mass terms No gauge-boson mass terms Mass scale of the world fixed by MMass scale of the world fixed by MSS and M and MNN

This does not imply that the particles This does not imply that the particles corresponding to their massless fields have corresponding to their massless fields have to stay masslessto stay massless

Breaking SU(2)xU(1) dynamically andBreaking 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 First ASSUME that fermion proper self-energy ΣΣ is generated.is generated.Second, FIND IT SELF-CONSISTENTLY.Second, FIND IT SELF-CONSISTENTLY.

Chirality-changing part of Chirality-changing part of ΣΣ must come out necessarily must come out necessarilyultraviolet-finiteultraviolet-finite – fermion mass counter terms strictly – fermion mass counter terms strictly forbidden by chiral symmetry forbidden by chiral symmetry

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

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

Crucial contribution to the scalar-Crucial contribution to the scalar-field propagator isfield propagator is

Physically observable are then two real Physically observable are then two real spin-0 particles corresponding to real spin-0 particles corresponding to real scalar fields Sscalar fields S11 and S and S22 defined as defined as

The masses and the mixing angle The masses and the mixing angle areare

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

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

ααSNSN is the phase of is the phase of μμSNSN and the mixing angle and the mixing angle θθ is is

Splittings Splittings μμSS22 , , μμNN

22 and and μμSNSN2 2 of the scalar- of the scalar-

particle masses due to particle masses due to yet assumed dynamical yet assumed dynamical fermion massfermion mass generation are both natural and generation are both natural and important:important:

1. They come out 1. They come out UV finiteUV finite due to the large due to the large momentum behavior of momentum behavior of ΣΣ(p(p22 ) (see further). ) (see further).

2. They manifest 2. They manifest spontaneous breakdownspontaneous breakdown

of SU(2)of SU(2)L L xx U(1)U(1)YY symmetry down to U(1) symmetry down to U(1)emem in the in the scalar sector.scalar sector.

3. They will be responsible for the 3. They will be responsible for the UV finiteness of UV finiteness of both the fermion and the intermediate vector both the fermion and the intermediate vector boson masses.boson masses.

3. Fermion mass 3. Fermion mass generationgeneration

Chirality-changing fermion proper self-Chirality-changing fermion proper self-energy energy ΣΣ(p(p22) is bona fide given by the UV ) is bona fide given by the UV finite solution of the Schwinger-Dyson finite solution of the Schwinger-Dyson equation graphically defined for charged equation graphically defined for charged leptons leptons

Explicit form of the equation is not very Explicit form of the equation is not very illuminating. It is, however, easily seen that IF illuminating. It is, however, easily seen that IF

a solution exists it is UV finite:a solution exists it is UV finite:

In order to proceed we are at the In order to proceed we are at the moment forced to resort to moment forced to resort to simplificationssimplifications. . The form of the The form of the nonlinearity is kept unchanged:nonlinearity is kept unchanged:

Neglect fermion mixing (sin 2Neglect fermion mixing (sin 2θθ = = 0).0).

This, unfortunately, implied This, unfortunately, implied neglecting utmost interesting neglecting utmost interesting relation between masses of relation between masses of upper and down fermions in upper and down fermions in doubletsdoublets..

Perform Wick rotation.Perform Wick rotation.

Do angular integrations.Do angular integrations.

Make Taylor expansion in MMake Taylor expansion in M221S 1S – –

MM222S2S (M (M22 – mean value). – mean value).

For a generic (say e) fermion self-For a generic (say e) fermion self-energy in dimensionless energy in dimensionless variables variables ττ = p = p22/M/M22 get get

Numerical analysis done so far by Petr BeneNumerical analysis done so far by Petr Benešš reveals reveals the existence of a solution for the existence of a solution for large values of large values of ββ..

For electrically charged fermions For electrically charged fermions m = m = ΣΣ(p(p22 = m = m22).).

SO FAR ONLY A GENERIC MODELSO FAR ONLY A GENERIC MODEL. It can pretend to . It can pretend to phenomenological relevance only after demonstrating phenomenological relevance only after demonstrating strong strong amplificationamplification of fermion masses to small changes of Yukawa of fermion masses to small changes of Yukawa couplings. couplings.

Generation of neutrino masses is more Generation of neutrino masses is more subtle and requires more work:subtle and requires more work:

Without Without ννRR neutrinos would be massless in our model. neutrinos would be massless in our model.

With With ννRR the mechanism just described generates UV-finite Dirac the mechanism just described generates UV-finite Dirac ΣΣνν..

There is a hard mass termThere is a hard mass term

Due to MDue to MM M there is a UV-finite left-handed Majorana mass matrix.there is a UV-finite left-handed Majorana mass matrix.

As a result the model describes 2nAs a result the model describes 2nff massive Majorana massive Majorana neutrinos with generic sea-saw spectrumneutrinos with generic sea-saw spectrum

4. Intermediate-boson mass 4. Intermediate-boson mass generationgeneration

Dynamically generated fermion proper self-Dynamically generated fermion proper self-energies energies ΣΣ(p(p22) break spontaneously SU(2)) break spontaneously SU(2)LL x x U(1)U(1)YY down to U(1) down to U(1)emem. .

Consequently, there are just Consequently, there are just three COMPOSITE three COMPOSITE Nambu-Goldstone bosonsNambu-Goldstone bosons in the spectrum if the in the spectrum if the gauge interactions are switched off. gauge interactions are switched off.

When switched on, the W and Z boson should When switched on, the W and Z boson should acquire masses. acquire masses.

To determine their values it is necessary to To determine their values it is necessary to calculate calculate residues at single massless poles of residues at single massless poles of their polarization tensorstheir polarization tensors

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

From the pole terms in From the pole terms in ΓΓ we extract the we extract the effective two-effective two-leg vertices between the gauge and three multi-leg vertices between the gauge and three multi-component ‘would-be’ NG bosonscomponent ‘would-be’ NG bosons. They are given in . They are given in

terms of the UV-finite loopterms of the UV-finite loop

As a result the gauge-boson masses are expressed in As a result the gauge-boson masses are expressed in terms of sum rulesterms of sum rules

If If ΣΣU U and and ΣΣDD were degenerate the relation were degenerate the relation

mmWW22/m/mZZ

22 coscos22 ΘΘWW= 1 would be fulfilled. = 1 would be fulfilled. Illustrative analysis with a particular Illustrative analysis with a particular model for model for ΣΣ shows that the departure shows that the departure from this relation is very small.from this relation is very small. Knowledge of detailed form of Knowledge of detailed form of ΣΣ(p(p22) is ) is indispensable.indispensable.

5. Concluding remarks5. Concluding remarks

Genuinely quantum and non-perturbative mechanism of Genuinely quantum and non-perturbative mechanism of mass generation is rather rigid.mass generation is rather rigid.

Not yet quantitative; yet strong-coupling.Not yet quantitative; yet strong-coupling. Quadratic scalar mass renormalization can be avoided.Quadratic scalar mass renormalization can be avoided. Relates fermion masses with each other.Relates fermion masses with each other. Relates fermion masses to the intermediate boson masses.Relates fermion masses to the intermediate boson masses. There is no generic weak-interaction mass scale v = 246 There is no generic weak-interaction mass scale v = 246

GeV.GeV. Mass scale of the world is fixed by MMass scale of the world is fixed by MNN, M, MSS and M and MMM..

There should exist four electrically neutral real There should exist four electrically neutral real scalar bosons, and two charged ones. They scalar bosons, and two charged ones. They should be heavy, but not too much (O(10should be heavy, but not too much (O(1066 GeV)). GeV)).