Download Document

A model of flavors
Jiří Hošek
Department of Theoretical Physics
NPI Řež (Prague)
(Tomáš Brauner and JH, hep-ph/0407339)
Plan of presentation
 Introductory
remarks
 Strategy: Role of
scalars
 Fermion mass
generation
 Intermediate-boson
mass generation
 Concluding remarks
1. 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
3
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
The same Higgs mechanism
no quadratic mass renormalization
gauge and fermion masses not related
whole new parallel world of heavy
particles
 TECHNICOLOR-LIKE SCENARIOS
Strongly coupled theory
No quadratic mass renormalization
Gauge and fermion masses not related
Plenty of heavy techni-hadrons
 LITTLE HIGGS
Weakly coupled theory
No quadratic mass renormalization at
“low energy”; at high energy it
reappears
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
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 nf
neutrino right-handed SU(2) singlets
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
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




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 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
Crucial contribution to the scalar-field
propagator is
Physically observable are then two real spin-0
particles corresponding to real scalar fields S1
and S2 defined as
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
αSN is the phase of μSN
and the mixing angle θ is
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 Σ(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.
3. Fermion mass generation
 Chirality-changing fermion proper selfenergy Σ(p2) 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.
Perform Wick rotation.
Do angular integrations.
Make Taylor expansion in M21S – M22S
(M2 – mean value).
For a generic (say e) fermion selfenergy in dimensionless variables τ
= p2/M2 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(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.
Generation of neutrino masses is more subtle
and requires more work:

Without νR neutrinos would be massless in our model.

With νR the mechanism just described generates UV-finite Dirac Σν.

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
4. Intermediate-boson mass
generation
 Dynamically generated fermion proper self-energies
Σ(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
‘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 ΣU and ΣD were degenerate the relation
mW2/mZ2 cos2 ΘW= 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(p2) 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 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)).