Transcript The Family Problem: Extension of Standard Model with a

Are there Dark-Matter Galaxies ?
W-Y. Pauchy Hwang
Y.T. Lee Outstanding Chair Professor
University Chair Professor
Institute of Astrophysics
National Taiwan University
Now, dark matter are everywhere (25 % of the entire
Universe) and only 5 % in ordinary matter !!

Only about 0.5 % are in (visible) galaxies.

Why is there so much dark matter (25 % of the
Universe), compared to so little “visible” ordinary
matter (5 % of the Universe), the latter as
described by the minimal Standard Model.

My Question: Maybe there should be “invisible”
galaxies (made out of dark matter) in the time
span of 10**9 years, the age of the young
Universe. These dark-matter galaxies could
even host those spiral galaxies – a new story for
galactic formation studies.
Neutrinos: Mysterious Particles in SM !!

Neutrinos now are massive these days. But the
minimal Standard Model tells us that they are
massless. The tiny masses of neutrinos do give
us very serious conceptual problem.

An indirect consequence: Neutrinos are pointlike Dirac particles “naturally”. Because four
components are there.

The neutrinos are rather mysterious -- they
might couple to both the dark-matter world and
the “visible” ordinary-matter world. Its
interactions with the visible world are rather
feeble.
Are dark-matter galaxies there?

If dark-matter galaxies are there playing the
hosts, we could understand easily the spiral
ordinary-matter galaxies such as our Milky Way.

The story of galactic formation is awfully
complicated. First, we have to try to distinguish
the no-seed clustering and the seeded clustering.

The seeded clustering – Starting from atoms,
molecules, complex molecules, and the chunks
of matter, generated from, in the ordinary-matter
world, the strong and electromagnetic forces. I
suspect that it is the way to go.
Possible story for dark-matter galaxies

If dark-matter galaxies were already there in
10**9 years (the time span for a young Universe),
then they could host the formation of ordinarymatter galaxies.

The seeded clustering arising from extra-heavy
dark-matter (“TeV”) particles, different from the
seeded clustering from the chains of baryons,
atoms, molecules, and complex molecules,
could be slightly faster.

The option of the family gauge theory provides
the seeded dark-matter clustering.
Outline
 Dirac
Similarity Principle and minimum
Higgs hypothesis
 Language: Quantum Fields
 No. 1 Question: What is the Dark Matter?
 Different Ways to Extend Standard Model,
all in accord with “Dirac Similarity Principle”
and “minimum Higgs hypothesis”, all are
renormalizable.
 The seeded clusterings
 References
We summarize the minimal Standard Model by
two working “rules”.
similarity principle – our struggle of
eighty years to describe the point-like
particles such as the electron.
 The “minimum Higgs hypothesis” is the
other mysterious conjecture – because we
are looking for Higgs particles for forty
years, but so far none has been found.
 So, by “induction”, we try to write down
these two rules which may help to explore
the “larger” dark matter world.
 Dirac
What is the particle world which we are talking about?
We were starting with the electrons – Dirac
invented the Dirac equation for that. It turned out
to be the first “point-like particle”. In it, the orbital
angular momentum term is treated equivalently
with a 4x4 sigma matrix:
J = r x p + sigma hbar / 2
 Now let’s look at the Standard Model. It’s a world
of point-like Dirac particles, with interactions
mediated by gauge fields and further modulated
by Higgs fields.
 So, to begin with, I would assume, naturally, that
neutrinos are also Dirac particles.

Dirac may be the first “physicist” to formulate
some equation for “point-like” particles.

Dirac didn’t know that the electrons are point-like
particles.

It turns out that, for over eighty years, we
recognize only a few point-like particles, those
building blocks of the Standard Model.

Maybe we should start with “quantized” Dirac
fields or, equivalently, “point-like” Dirac particles.
In other words, a “point-like” particle in the
quantum sense is defined through the quantized
Dirac fields. Less than 10**(-18) cm.
The case for the “Dirac similarity principle”:

The notion of “space-time” may also be defined
accordingly, in some way.

Why there is nothing else - a world of point-like
Dirac particles, with interactions mediated by
gauge fields and modulated slightly by Higgs
fields.

The axiom for “quantized Dirac fields” or “pointlike Dirac particles” – it turns out that they are
the same thing.
Why don’t we see some Higgs after 40 years?
Klein-Gordon (scalar) fields – in
fact, our first lesson for QFT.
 We use the scalar fields to “modulate”
quite a number of things, SSB (the Higgs
mechanisms), etc. But we still look for
them, after 40 years.
 But why aren’t they there? Strange !!
 In any event, we could work with “the
minimum Higgs hypothesis”.
 Quantized
The Language: Elementary Particles as
Quantum Fields
Classical Mechanical
Systems
dc
Classical Fields
Dirac CP
Dirac CP
dc
Quantum
Mechanical Systems
Quantum Fields
d → c: discreteness to continuum
Dirac CP: Dirac Correspondence Principle

Simplified Axioms for the various basic concepts:

Classical Mechanical System:
“For a given system, we can find a function
(lagrangian) of the coordinates and velocities
such that the integral (action) between two
instants is an extremum for the real motion.”

Quantum Mechanical System:
“For the coordinates we can find the conjugate
momenta such that the basic (elementary)
commutation relations hold.” – Now, they are
operators.


Classical Field:
“For a given system, we can find a function
(lagrangian) of the coordinates and velocities
such that the integral (action) between two
instants is an extremum for the real motion.” –
except that quantities take continuum meaning.
Quantum Field:
“For the coordinates we can find the conjugate
momenta such that the basic (elementary)
commutation relations hold.” – except that
quantities take continuum meaning and we also
generalize the notion to include fermions (I.e.
anti-commutation relations).
Let’s remind ourselves what we have
done for the minimal Standard Model:

All the quarks and leptons are written in terms of Dirac
equations on certain forms. And all the interactions are in
the gauge fields. In reality, nothing more. Even so far no
scalar (Higgs) fields. So it’s a world of “pointlike” Dirac
particles (a Dirac world) with interactions. Maybe this is
an important guideline to follow. (“Dirac Similarity
Principle”.)

So far only renormalizable Interactions are permitted.
(“Renormalizability” means “calculability”.)

In other words, we have so many ways to write things
relativistically, but not all are equally “applicable” for
some reasons.
Ordinary matter as described by the
minimal Standard Model


Dirac tried to describe the electron by proposing
Dirac equation. Then the quarks and leptons are
written in terms of Dirac equations on certain
forms. And all the interactions are in the gauge
fields. In reality, nothing more.
Only renormalizable Interactions are permitted.
 Satisfy the Dirac Similarity Principle and
minimum Higgs hypothesis.
Connecting Quarks
with the Cosmos:
Eleven Science
Questions for the
New Century
• The report released
initially on 4/17/2002
by National Academy
of Sciences, U.S.A.
Cosmology as an
Experimental Science
for the New Century
Eleven Science Questions for the New Century:
The First Four Questions
CPU/BPA/NRC Report, 4/17/2002

Q1: What is the dark matter?
Our Universe has 25% in Dark Matter while
only 5% in ordinary matter. 5% - the minimal Standard
Model.

Q2: What is the nature of the dark
energy?
(The overwhelming 70% question !!)
 Q3: How did the universe begin?
 Q4: Did Einstein have the last word
on gravity?
(Is geometry everything?)
Eleven Science Questions for the New Century:
The Fifth Question
 Q5:
What are the masses of the
neutrinos, and how have they
shaped the evolution of the
universe?
 It is likely that neutrinos are also pointlike Dirac particles, since they are
massive (and have tiny masses). The
reason for us to propose “Dirac
similarity principle”.
Eleven Science Questions for the New Century:
The Seventh Question

Q7: Are protons unstable?
 Another important question for symmetry.
 Q7 means that the grand unified theory in
certain form would be valid, if protons decay.

We assume that, through proper Higgs
mechanism, all particles in the dark sector are
massive.
 Now,
“What is the dark matter?” Could we
describe it or them? If yes, what would be
the language? The first guess would be to
use the language which we set up for the
Standard Model – a gauge theory
with/without Higgs Mechanism.
 Generalizing the SU_c(3) x SU(2) x U(1)
standard model via a renormalizable way
by adding particles which we have not
seen – it turns out that there are many
ways.
 “Minimum Higgs hypothesis” implies that
extensions in Higgs sector is less favored
than those in gauge sector.

The ordinary-matter world and the dark-matter
world jointly defines the extended Standard
Model.

Candidates for the dark-matter seeds: Long life
time (> 1 Gyr at least), heavy; it doesn’t have
ordinary strong and electromagnetic interactions.

Dark matter particles: They don’t participate
(directly) in the ordinary strong and
electromagnetic forces.
 Note
that the unknown dark matter
occupies 25% of the current Universe
while the visible ordinary matter 5%. We
can describe the 5% but 25% unknowns.
 Fortunately
if we view the world from the
symmetry point of view, it probably does
not matter in this 25%-5% upside-down;
but the symmetry of certain kind has to be
there.
First thought

We could adopt “Dirac similarity principle” and
the “minimum Higgs hypothesis” as our working
rules, when we use the extended Standard
Model to describe the dark-matter and ordinarymatter particles.

If the gauge group, SU_c(3) x SU_L(2) x U(1) x
G, is fixed, the two working rules guarantee the
uniqueness of the model.
We haven’t seen Higgs after 40 years !!
Years ago (in 1987), I tried to add Z’ and
realized immediately that we have to add
additional Higgs multiplet(s), too.
 How to add a Z’ but with a minimum number of
Higgs fields?
References: W-Y. P. Hwang, Phys. Rev. D36,
261 (1987).
 Crucial to have the mass-generation mechanism
spelled-out.

On the mass generation
lambda’ ~ lambda x (vev / vev’)**2
The conjecture for the couplings to “remote”
Higgs

On the mass generation by the first Higgs doublet, the
size are of the same order and of O(v), with v the
vacuum expected value.

The mass generation for the second Higgs doublet is
down by order O((v/v’)^alpha), with alpha greater than
unity.
In what follows, we take alpha = 2. !
“The Minimum Higgs Hypothesis”
No.1. On the coupling strengths.
lambda’ ~ lambda x (vev / vev’)**2
My conjecture for the couplings to remote Higgs
No. 2. On the choice of Higgs multiplets
There is no redundent Higgs multiplet..

It is a useful “empirical” rule.
Another Thought

SU_c(3) × SU_L(2) ×SU_R(2) x U(1) : The
missing right-handed sector !!
R.N. Mohapatra and J.C. Pati, Phys. Rev. D11,
2558 (1975).

Mohapatra, Pati, and Salam in fact have many
models (by choice of Higgs multiplets) but the
“minimum Higgs hypothesis” selects the unique
one.

The missing left-right symmetry should be
understood some day.
More on the left-right symmetry
 Why
the weak interactions break the leftright symmetry is one of the deepest
questions.
Are these stuffs in the 25% dark matter?
 Mass generation: (by the image of the left)
lambda (v/v’)**2 varphi* nu_L (nu_R, e_R)
 Again, it is renormalizable.
In fact, we could talk about three
unique options:
SU_c(3) × SU(2) × U(1) × G
 How to add a Z’ but with a minimum
number of Higgs fields?
References: W-Y. P. Hwang, Phys. Rev.
D36, 261 (1987).
 To make Mohapatra-Pati-Salam left-right
model minimal in the Higgs sector.
 G = SU_family(3) is also possible. W-Y. P.
Hwang, Intern. J. Mod. Phys. A24, 3366
(2009).
Three or two unique options:
SU_c(3) × SU(2) × U(1) × G
 As the group G is fixed, the extended
standard model is fixed.
Thanks to “Dirac similarity principle” and
“minimum Higgs hypothesis”!!
 We
should pay more attention to the
Mohapatra-Pati-Salam left-right symmetric
model and the family gauge theory –
because of the symmetry reasons.
Maybe we have to “rethink” what we are
doing – trying to set our tone.

We have so much of dark matter (25 % of the
current Universe) --- the “final” theory, if the
SU_c(3) × SU_L(2) × U(1) × SU_f(3) x SU_R(2)
extended Standard Model would describe our
ordinary-matter and dark-matter world, would
still be “minimal”.

We have already seen partially the SU_f(3) part
but we still don’t have the clue about the missing
SU_R(2) part.
The extended Standard Model is
naively renormalizable.

The extended Standard Model, based on the
group SU_c(3) × SU(2) × U(1) × SU_f(3) x
SU_R(2), only requires the presence of pointlike Dirac particles (“Dirac similarity principle”)
and the minimum presence of Higgs particles
(“minimum Higgs hypothesis”).

We may “say” that the left-handed neutrinos
belongs to the ordinary-matter world while
(nu_tau, nu_mu, nu_e)_(right-handed) belongs
to the dark-matter world.
In what follows, we explain briefly
the SU(3) family gauge theory.

We also introduce the SU(3) family gauge
theory – i.e. the SU_c(3) × SU(2) × U(1) ×
SU_f(3) standard model. SU_f(3) defines the
body of the dark matter.

The only SU_f(3) triplet from the ordinary-matter
world, (nu_tau, nu_mu, nu_e)_(right-handed), or
just (nu_tau, nu_mu, nu_e).
Question: Why do we have three generations?
 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.
 Now
the simple extension is that based on
SU_c(3) × SU(2) × U(1) × SU_f(3).
 The
rest is straightforward.

There are 8 gauge bosons:
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 SU_f(3) model, the couplings to ordinary
matter is only through the neutrinos.

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.
 In
other words, there are loop corrections
involving familons and other dark-matter
particles, which should be suppressed to
protect the validity of the minimal Standard
Model. So, the masses of familons and
family Higgs have to be greater that a
certain value, such as 1 TeV.

Neutrino mass generation is through 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 a form similar to the Zee
matrix[5], it can easily be fitted to the observed data[2].

In other words, the “origin” of the tiny neutrino masses
comes from the family Higgs and from the loop of family
gauge bosons. It is different from those for quarks and
charged leptons, a nice way to escape the theorem
mentioned earlier[3].

The tiny neutrino masses are generated in the dark
sector, and in a renormalizable way. This is a very
interesting solution.

SU_f(3) so similar to SU_c(3):
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).

In the early universe, the temperature could be
as high as that for the familons such that the
Universe could be populated with these massive
dark-matter particles – giving rise to the socalled “seeded clusterings”.
Initial 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.
W-Y. P. Hwang, Intern. J. Mod. Phys. A24, 3366 (2009).
I would like to thank my colleagues, Tony Zee, Ling-Fong Li, Xiao-Gang He,
and Pei-Ming Ho for useful conversations, but the errors remain to be mine.
An important conclusion:
 So,
under “Dirac similarity principle” and
the “minimum Higgs hypothesis”, we could
work on “three” extended Standard
Models – the family gauge theory, the leftright symmetric model and the extra Z’ (in
our order), all in a unique version. All
being renormalizable.
 Or, we may work with the SU_c(3) x
SU_L(2) x U(1) x SU_f(3) x SU_R(2)
extended Standard Model.
Clusterings
 “Unseeded”
gravitational clusterings
supposed to happen in a long time, in a
time much longer than the age of the
Universe.
 So,
in a time span of the young Universe,
neutrino masses contribute very little and
we need the seeds (for the clustering) to
catalyze the processes.
The seeded clusterings
 From
hadrons, atoms, molecules, complex
molecules, and chunks of the matter, to
begin clustering, in a time span of 1 Gyr, it
grows into our present visible Universe –
the so-called “seeded clusterings”.
 Similar
things could happen if we have
SU_f(3) family gauge theory with a normal
coupling. So, there might be dark-matter
galaxies.
Conclusions
 “Dirac
similarity principle” and the
“minimum Higgs hypothesis” allows us to
work on the SU_c(3) x SU_L(2) x U(1) x
SU_f(3) x SU_R(2) extended Standard
Model. It is “naively” renormalizable.
 The
“seeded” clustering might exist in the
dark-matter world if SU_f(3) is indeed
there. => There are dark-matter galaxies.