A model of flavors

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A model of flavors

• Jirí Hošek
• Department of Theoretical Physics
• NPI Rež (Prague)
• (Tomáš Brauner and JH, hep-ph/0407339)

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Plan of presentation

• Introductory remarks
• Strategy Role of scalars
• Fermion mass generation
• Intermediate-boson mass generation
• Concluding remarks

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Introductory remarks

• 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.
• Objections
• 1. QCD is better
• 2. Neutrino masses are different from zero

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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

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Attempts to solve disadvantages of SM

• SUSY
• Weakly coupled theory
• The same Higgs mechanism
• no quadratic mass renormalization
• gauge and fermion masses not related
• whole new parallel world of heavy particles

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

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

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• DONT FORGET UNKNOWN

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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
• N (N(0), N(-)) with Y(N) -1 and ordinary
  mass squared term in the Lagrangian
• NO SPONTANEOUS BREAKDOWN OF SYMMETRY AT TREE LEVEL

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

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

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Our model

• SU(2)Lx U(1)Y gauge symmetry is manifest
• No fermion mass terms except of MM
• No gauge-boson mass terms
• Mass scale of the world fixed by MS and MN
• This does not imply that the particles
  corresponding to their massless fields have to
  stay massless

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Breaking SU(2)xU(1) dynamically and
non-perturbatively. In perturbation
theory the symmetry is preserved
order by order. First ASSUME that fermion
proper self-energy S is generated. Second, FIND
IT SELF-CONSISTENTLY. Chirality-changing part
of S must come out necessarily ultraviolet-finite
fermion mass counter terms strictly forbidden
by chiral symmetry
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Assumed fermion mass insertions give rise to
generically new contributions of the scalar field
propagators
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Problem reduces to finding the spectrum of the
bilinear Lagrangian
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Crucial contribution to the scalar-field
propagator is
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Physically observable are then two real spin-0
particles corresponding to real scalar fields S1
and S2 defined as
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The masses and the mixing angle are
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The case of the charged scalars is similar Only
particles with the same charge can mix, and they
really do
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The masses and the field
transformations are
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aSN is the phase of µSN and the mixing angle ?
is
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Splittings µS2 , µN2 and µSN2 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 S(p2 ) (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.
• 3. They will be responsible for the UV finiteness
  of both the fermion and the intermediate vector
  boson masses.

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3. Fermion mass generation

• Chirality-changing fermion proper self-energy
  S(p2) is bona fide given by the UV finite
  solution of the Schwinger-Dyson equation
  graphically defined for charged leptons

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Explicit form of the equation is not very
illuminating. It is, however, easily seen that IF
a solution exists it is UV finite
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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 2? 0).
• This, unfortunately, implied neglecting
  utmost interesting relation between masses of
  upper and down fermions in doublets.
• Perform Wick rotation.
• Do angular integrations.
• Make Taylor expansion in M21S M22S (M2
  mean value).
• For a generic (say e) fermion self-energy in
  dimensionless variables t p2/M2 get

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Numerical analysis done so far by Petr Beneš
reveals the existence of a solution for
large values of ß.For electrically charged
fermions m S(p2 m2).

• 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.

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Generation of neutrino masses is more subtle and
requires more work

• Without ?R neutrinos would be massless in our
  model.
• With ?R the mechanism just described generates
  UV-finite Dirac S?.
• There is a hard mass term
• Due to MM there is a UV-finite left-handed
  Majorana mass matrix.
• As a result the model describes 2nf massive
  Majorana neutrinos with generic sea-saw spectrum

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4. Intermediate-boson mass generation

• Dynamically generated fermion proper
  self-energies S(p2) 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

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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
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From the pole terms in G 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

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If SU and SD were degenerate the relation
mW2/mZ2 cos2 TW 1 would be fulfilled.
Illustrative analysis with a particular model for
S shows that the departure from this relation is
very small. Knowledge of detailed form of S(p2)
is indispensable.
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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 MN, MS and
  MM.
• There should exist four electrically neutral real
  scalar bosons, and two charged ones. They should
  be heavy, but not too much (O(106 GeV)).