Transcript The Family Problem: Extension of Standard Model with a

Gauge Symmetry and More ….
Another Solution to Dark Matter?
W-Y. Pauchy Hwang
Institute of Astrophysics
National Taiwan University
Outline
 Introduction
 A Proposal:
Extension of Standard Model
by a Gauge Theory – the SU_f(3) Family
Gauge Theory
 Discussions
 References
Introduction


More than twenty years ago I was curious by the
absence of the Higgs mechanism in the strong
interactions but not in the weak interaction sector[1] – a
question still remains unanswered till today. A
renormalizable gauge theory that does not have to be
massless is already reputed by ‘t Hooft and others, for
the standard model. Maybe our question should be
whether the electromagnetism would be massless.
In fact, this is a deep question – how to write down a
renormalizable theory. During old days, a massive gauge
theory is used to be believed as a nonrenormalizable
theory.
Another clue comes from neutrinos – they are
neutral, massive and mixing/oscillating. These
particles are barely “visible” in the Particle Table.
Maybe these are avenues that connect to those
unknowns, particularly the dark matter in the
Universe.
 In fact, the neutrino sector, with the current
knowledge of masses and mixings[2], presents a
serious basic problem[3] – that is, a theorem that
neutrinos are massless in the minimal standard
model. Any model with at least one massive
neutrino have to be some sort of extended
standard model (i.e. not minimal).
 In the sector of ordinary matter, the only neutral
fermions are neutrinos.

Key References
1.
2.
3.
4.
5.
6.
.
W-Y. P. Hwang, Phys. Rev. D32 (1985) 824; on the “colored Higgs
mechanism”.
Particle Data Group.
Stuart Raby and Richard Slansky, Los Alamos Science, No. 25
(1997) 64.
Notations.
A. Zee, Phys. Lett. B93 (1980) 389; Phys. Lett. B161 (1985) 141;
Nucl. Phys. B264 (1986) 99; on the Zee model.
Ling-Fong Li.



In this talk, I propose that we may add an SU(3) family
gauge theory - the SU_c(3) × SU(2) × U(1) × SU_f(3)
standard model. In addition to QCD and electroweak
(EW) phase transitions there is other SU_f(3) family
phase transition occurring near the familon masses,
maybe above the EW scale (that is, above 1 TeV).
One motivation is that in our Universe there is 25% dark
matter vs 5% ordinary matter – dark matter even though
we cannot see but it may be insufficient to view it as
being “peripheral” only. In the early universe, the
temperature could be as high as that for the familons
such that the Universe could be populated with these
(self-interacting) particles - just like that for QCD. The
Universe would be full of these particles as the dark
matter.
Maybe (nu_e, nu_mu, nu_tau) could serve as the only
“bridge” for ordinary matter.
Why we have three generations? – we provide a “partial”
solution to the “family” problem.

Note that I take a rather conservative position:
If we don’t see the grand unification or
supersymmetry, we try to think about “why not”,
rather than proceeding to speculate.
 So, in our Universe there is 25% dark matter vs
5% ordinary matter – this is taken as a fact to
understand.
 If there is some connection between the
“ordinary matter” sector and the dark matter one,
the neutrinos (nu_e, nu_mu, nu_tau) could be as
the only candidate.
 Note that I have provided only a “partial” solution
to the “family” problem – depending how you
define it.
 For
this SU_f(3) gauge theory, the triplet
(\nu_\tau, \nu_\mu, \nu_e) serves as a
basic connection. Note the up-side-down
position used here because of the way I
wrote the positions of the nonzero vacuum
expectation values.
 For the SU_f(3) gauge theory, should it
exist and we suppose that it is coupled to
neutral fermions, i.e. neutrinos. The
concept of “family” could not mingle with
the other quantum numbers, I.e.
commuting with the others.

Under SU_c(3) × SU(2) × U(1) × SU_f(3), the
neutrino family triplet acts like (1, 1, 0, 3) and all
the others, including quarks, charged leptons,
and ordinary gauge bosons, are assumed to be
family singlets (i.e. no coupling to the family
sector).

I assume that the whole Dirac neutrinos could be
used in the neutrino triplet.

If only the right-handed neutrinos are used in
this context, the dark sector would be completely
dark, making the story a little boring.

The masses of the neutrino triplet come from the
coupling to some Higgs field - a pair of complex
scalar triplets, as worked out in the previous
publication[1]. Hereafter I ignore the “radiative”
corrections due to gauge bosons. In this case,
the eight components of the Higgs triplets are
absorbed by the eight gauge fields through the
“family” Higgs mechanism via spontaneous
symmetry breaking, while the remaining four
become massive Higgs particles. (In the
previous application, it was referred to as
“colored Higgs Mechanism”[1].)

The neutrino masses do not come from the
minimal Standard Model, but from the Higgs in
the dark sector.
A Proposal


If we think of the role of gauge theories in quantum field
theory, we still have to recognize its unique and
important role. If the standard model is missing
something, a gauge theory sector would be one at the
first guess. On the other hand, in the standard model
there are three generations of quarks and leptons. But
why? It seems to be a first loose point for the standard
model. So, let’s assume that there is an SU_f(3) gauge
theory associated with the story.
I believe that something missing may be a gauge sector,
owing to the successes of SU_c(3) × SU(2) × U(1)
standard model.
 In
fact, an octet of gauge bosons plus a
pair of complex scalar triplets turns out to
be the simplest choice as long as all
gauge bosons become massive while the
remaining Higgs are also massive.
 This is the basic framework. The standard
model is the gauge theory based on the
group SU_c(3) × SU(2) × U(1). Now the
simple extension is that based on SU_c(3)
× SU(2) × U(1) × SU_f(3).
 So, the following story is rather simple.

We may write our “new” basic elements as follows.
Denote the eight family gauge fields (familons) as
F_\mu^a(x). Define F_{\mu\nu}^a ≡ \partial_\mu F_\nu^a
－ \partial_\nu F_\mu^a + \kappa f_{abc} F_\mu^b
F_\nu^c. Then we have[4]
1
L   Fa Fa .
4
(1)
One way to describe the nonabelian nature of the gauge
theory is to add the Fadde’ev-Popov ghost fields
Leff  L  a  x  D a  x  ,
(2)
with D_\mu \phi^a ≡ \partial_\mu \phi^a + \kappa
f_{abc}F_\mu^b \phi^c.

The neutrino triplet \Psi(x) is
Lf    D ,
(3)
with D_\mu ≡ \partial_\mu － i {\kappa\over 2}
\lambda^a F_\mu^a(x). Just like a (triplet) Dirac
field.
 The family Higgs mechanism is accomplished by
a pair of complex scalar triplets. Under SU_f(3),
they transform into the specific forms in the Ugauge:
 

 '  exp i a  a0 u   ,    , 0 ,
 2 
 a 0 
'
   exp i  a u   ,    , 0 .
 2 
(4)
 We
could work out the kinetic terms:
Lscalar    D    D     D    D   V ,
†
†
(5)
such that, by means of choosing,
u      cos ,
   u   sin  ,
(6)
we find, for the familons,
k
M 1  M 2  M 3  k ,
M8 
,
3
k
M 4,5,6,7 
.
2
(7)

That is, the eight gauge bosons all become
massive. On the other hand, by choosing
V 

2
 
2
†


  †   
      
4

†

2

†


2

 2   †     †    ,
(8)
we find that the remaining four (Higgs) particles
are massive (with \mu^2 < 0, we have v^2 = －
\mu^2 / \lambda > 0).
 Because the neutrino-neutrino-Z vertex is now in
our theory augmented by the neutrino-neutrino“dark boson” vertices;
 these dark species should be very massive.

In the model, the couplings to ordinary matter is
only through the neutrinos, the only charged/
neutral fermions that are interacting weakly. This
would make some loop diagrams, involving
neutrinos and familons, very interesting and,
albeit likely to be small, should eventually be
investigated[6]. For example, in the elastic quark
(or charged lepton) - neutrino scattering, the
loop corrections would involve the Z^0 and in
addition the familon loops and if the masses of
the familons were less than that of Z^0 then the
loop corrections due to familons would be bigger.
Thus, we may assume that the familon masses
would be greater than the Z^0 mass, say ≧ 1
TeV.
 The
above argument also implies that we
cannot have the massless familon(s) or
massless family Higgs particle(s).
Otherwise, the loop corrections in some
cases would be dominated by those with
familons.

The other important point is the coupling between the neutrino triplet
and the family Higgs triplets:
      ,


(9)
resulting a mass matrix which is off diagonal (but is perfectly
acceptable). In other words, the mass matrix, being proportional to
－\bar{\nu}_e(v_{+} + \epsilon v_－)\nu_\tau
+ \bar{\nu}_e(u_{+} + \epsilon u_－)\nu_\mu
+ \bar{\nu}_\tau (v_{+} + \epsilon v_－)\nu_e
－\bar{\nu}_\mu(u_{+} + \epsilon u_－)\nu_e , is off-diagonal, in the
form similar to the Zee matrix[5], and can easily be fitted to the
observed data[2]. (And i is needed to make it hermitian.)
In other words, the “source” of the neutrino masses comes from the
family Higgs and is different from those for quarks and charged
leptons, a nice way to escape the theorem mentioned earlier[3].
The neutrino masses are obtained from the dark sector, but in a
renormalizable way. This is a very interesting solution.

What is surprising about our model? There is no
unwanted massless particle - so, no disaster
anticipated. It is the renormalizable extension of
the standard model idea. Coming back to the
neutrino sector, we now introduce the mass
terms in a renormalizable way (with the help
from SU_f(3) gauge theory) - previously a
headache problem in the old-day Standard
model. Furthermore, there is no major
modification of the original Standard Model.
 This is the end of the story
Discussions


Our life during the next stage seems to be rather difficult.
Neutrinos, albeit abundant, are very elusive. We use
neutrinos as the basic bridge to construct the SU_f(3)
gauge theory for the family in the building blocks of
matter. If the only coupling has to go through neutrinos,
then the detection (from the visible side of the matter)
would be extremely difficult. Of course, we should look
for the potential couplings to other sectors such as
quarks or charge leptons.
Well, the neutrino mass problem, if coupled with the
renormalizability requirement, leads to our solution.



One important consequence of the SU_c(3) × SU(2) ×
U(1) × SU_f(3) standard model is that in addition to QCD
and electroweak (EW) phase transitions there is other
SU_f(3) family phase transition occurring near the
familon masses, maybe above the EW scale (that is,
above 1 TeV). The exact scale is hard to decide, for the
moment.
In the early universe, the temperature could be as high
as that for the familons such that the Universe could be
populated with these (self-interacting) particles - just like
that for QCD. In other words, our Universe would be full
of these particles as the dark matter - at this point, it is
believed that our Universe has 25% in dark matter while
only 5\% in visible matter.
Well, this may solve the 25% dark matter problem.
Another Thought
SU_c(3) × SU(2) × U(1) × G
G: U(1) or n U(1) the extra Z^0
(bottom-up or up-down strategy)
 Important Question: How to add a Z’ but with a minimum
number of Higgs fields?
References: W-Y. P. Hwang, Phys. Rev. D36, 261 (1987);
W-Y. P. Hwang, D38, 3427 (1988).
 The present paper, with G = SU_f(3), brings out another
important generalization, solving the neutrino mass
problem and perhaps the dark matter issue
simultaneously.
 I think that renormalizability, even though we really don’t
understand it, does provide a useful guide.

Another Discussion

It is used to think that, to start with the E_8
group, we have SU(3) x E_6. E_6 breaks down
to U(1) x SO(10), and then to U(1) x U(1) x
SU(5), and to our standard model.
 What is wrong with this line of thinking?
 Maybe we have to introduce gauge bosons and
Higgs particles to make sure that no unwanted
particles exist – an argument against “arbitrary”
grand unified theories.
References
1.
2.
3.
4.
5.
6.
W-Y. P. Hwang, Phys. Rev. D32 (1985) 824; on the “colored Higgs
mechanism”.
Particle Data Group, “Review of Particle Physics”, J. Phys. G: Nucl.
Part. Phys. 33 (2006) 1; on neutrino mass and mixing, see pp. 156 164.
For example, see Stuart Raby and Richard Slansky, Los Alamos
Science, No. 25 (1997) 64.
For notations, see T-Y. Wu and W-Y. Pauchy Hwang, Relativistic
Quantum Mechanics and Quantum Fields (World Scientific,
Singapore, 1991).
A. Zee, Phys. Lett. B93 (1980) 389; Phys. Lett. B161 (1985) 141;
Nucl. Phys. B264 (1986) 99; on the Zee model.
Ling-Fong Li, private communications.