Dark Matter as a Guide to Extend the
Standard Model: Dirac Similarity Principle
and the Minimum Higgs Hypothesis
W-Y. Pauchy Hwang
University Chair Professor
Institute of Astrophysics
National Taiwan University
What I would like to do today? It’s an idea off and on in my
mind, maybe over 30 years but I think it is mature lately.

Neutrinos now are massive these days. But the
minimal Standard Model tells us that they should
be zero.
 Why is there so much dark matter (25\% of the
Universe), compared to so little “visible” ordinary
matter (5\% of the Universe) as described by the
Standard Model.
 The mystery may lie with the neutrinos, which
may “bridge” between the dark-matter world and
the “visible” ordinary-matter world.
Let’s begin with two remarks – “Dirac similarity
principle” and why Higgs are so far not there.
In this talk, we focus on two “rules”, two very
strange “rules”.
similarity principle – our struggle of
eighty years to describe the point-like
particle 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 obtain these two
rules which may help in bringing in the
“larger” dark matter world.
 Dirac
Why could we use “dark matter as a guide to
extend the Standard Model” ?
As explained later, the “language” developed so
far is likely to be the quantum field theory, and
otherwise what else?
 Ordinary matter (5%) and dark matter (25%) are
believed to clusterized similarly and obey the
same “gravitational law”.
 Unlike the uniformly-distributed “dark energy”,
ordinary matter and dark matter seems to follow
the same laws, except the feeble interactions
between them.

What is the particle world which we are talking about?
We were starting with the electrons – Dirac
invented the Dirac equation for that. The first
“point-like particle”. In it, the orbital angular
momentum term is treated equivalently with a
sigma matrix, relativistically.
 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.

He tried to put in quantum mechanics (those
matrices representing spins) and relativity
simultaneously.
 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.
 Maybe we shouldn’t question what “quantized”
or “point-like” means to us, or rather instead of
treating this as an “axiom”.
Thus, we argue for the “Dirac similarity principle”.
It’s a special way to put in quantum mechanics
(those matrices representing spins) and relativity
simultaneously. In fact, the “space-time” notion
may be defined also.
 Apparently the way is so special. 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” – they are the same thing.

We understand “Dirac similarity principle”:
Our space-time “lattice” admits, or be
compatible with, certain kind of “point-like”
particles which at this point turns out to
“quantized Dirac fields”.
 Our world is very special. Why there is nothing
else - a world of point-like Dirac particles, with
interactions mediated by gauge fields and
modulated slightly by Higgs fields.
 “Quantized Dirac fields” or “point-like Dirac
particles” turn out to be the same thing.

Why don’t we see some Higgs after 40 years?
Klein-Gordon (scalar) fields – in
fact, our lesson in 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.
 Maybe we should work with “the minimum
Higgs hypothesis” or “conjecture”.
 Quantized
Outline
 Language:
Quantum Fields
 No. 1 Question: What is the Dark Matter?
 Dirac Similarity Principle: Observation to
Proposal
 Different Ways to Extend Standard Model,
all in the renormalizable way and in accord
with “Dirac Similarity Principle”
 Discussions
 References
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


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 Review what we have done:



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.
The SM can be viewed from a different
angle: Dirac Similarity Principle

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.
 So far only renormalizable Interactions are
permitted.
 Maybe some specialty about the Dirac equation
exists in our space-time.
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. How about 5% versus
25%, instead – it would be more comfortable if we looked
for the “Standard Model” for the majority (25%).

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?
 I would remind you of a theorem about
the neutrino mass: The neutrinos
should be massless in the minimum
Standard Model.
Eleven Science Questions for the New Century:
The Seventh Question

Q7: Are protons unstable?
 Another important question for symmetry.
 That means that the grand unified theory in
certain form would be valid, if protons decay.
 In what follows, I assume that the gauge
theory in the extended Standard Model should
have two basic ingredients – the gauge sector
and the Higgs mechanism, the latter ensuring
that 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.
 Note
that the unknown dark matter
occupies 25% of the current Universe
while the visible ordinary matter 5%. Not
the other way around – 5% dark matter
while 25% ordinary matter. 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
Neutrinos have tiny masses. => another Z’.
It sounds strange, but it requires another Higgs,
to be natural.
 How to add a Z’ but with a minimum number of
Higgs fields?
W-Y. P. Hwang, Phys. Rev. D36, 261 (1987).
 Consider 2+2 Higgs Scenario. The second, and
“remote”, Higgs doublet could give neutrinos
masses naturally.

No Higgs after 40 years !!

Maybe the associated Higgs structure should be
minimal. After all, after 40 years or so, we
haven’t found the signature of Higgs. We still ask
the LHC for an answer. How to make a model
with minimum Higgs structure?
 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).
On the mass generation
lambda’ ~ lambda x (vec / vec’)**2
My 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.
 For some reason, 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.
 In short, the details for the Higgs mechanism need to be
worked out.
“Minimal Higgs Hypothesis” !!
“The Minimum Higgs Hypothesis”
No.1. On the coupling strengths.
lambda’ ~ lambda x (vec / vec’)**2
My conjecture for the couplings to remote Higgs
No. 2. On the choice of Higgs multiplets
There is no redandant 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).
 Here we also have an extra Z’ but with another
right-handed doublet almost eaten up via SSB.


Mohapatra, Pati, and Salam in fact have many
models (by choice of Higgs multiplets) but the
“minimum Higgs hypothesis” selects the unique
one.
More on the left-right symmetry
 Why
the weak interactions break the leftright symmetry is one of the deepest
questions.
Don’t forget to ask….
 Mass generation: (by the image of the left)
lambda (v/v’)**2 varphi* nu_L (nu_R, e_R)
 Make sure that it is renormalizable.
I’m talking about three options, in
fact three nice options:
× 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. See
later.
 SU_c(3)
 In
what follows, we talk about the
possibility of adding an SU(3) family gauge
theory - the SU_c(3) × SU(2) × U(1) ×
SU_f(3) standard model. SU_f(3) defines
the body of the dark matter.
 In this model, (nu_e, nu_mu, nu_tau)
could serve as the only “bridge” for
ordinary matter.
Why do we have three generations? In
this model, we are forced to have three
generations.
I
think that most symmetry may have
something to do with some interactions,
maybe too weak to be detected. Maybe
this is the origin of “dark matter”.
 Different
Options:
Left-right symmetric model;
SU_c(3) x SU(2) x U(1) X U(1) with extra
Z^0;
SU_c(3) x SU(2) x U(1) x SU_f(3).
For this extra SU_f(3) gauge theory, should it
exist and we suppose that it is coupled to
neutral fermions, i.e. neutrinos, in the sector of
ordinary matter.
Family symmetry => Family gauge symmetry
Proposing a gauge theory, it means some kind of
new interactions.
Maybe there is another gauge theory
beyond the Standard Model …


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.
 Here
I try to set up the view that the only
visible massless particle is the photon –
and the gluons, if massless, are
permanently confined (then it is
meaningless to have mass). That is, all
particles in the dark sector are massive.
 In the sector of ordinary matter, the
neutrinos could serve as the direct
messengers with the dark sector (i.e.
neutrinos are also one kind of dark matter).
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).

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. (But it may be right.)
 If
the coupling would involve gamma-5,
then we have to worry about the anomaly.

Note that in family gauge theory:

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]. Note that the “radiative”
corrections due to gauge bosons serve as a
correction to mass.

Note that the neutrino masses do not come from
the minimal Standard Model, mainly from the
Higgs in the dark sector.
More on family gauge theory:
 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.
 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.

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.
 Unlike
the story with extra Z or the leftright model, in which the details of the
Higgs doublets may need some fine-tuning,
the problem of neutrino masses in the
present model is solved much more
naturally.
 In
any case, it is possible to solve the
problems through a renormalizable way.
 What
is surprising about our model? There
is no unwanted massless particle - so, no
disaster anticipated. It is another
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). Furthermore,
there is no major modification of the
original Standard Model.
 Maybe a “solution” to the family problem.
Discussions on SU_f(3)
 The
neutrino mass problem is solved
nicely since neutrinos couple with the
dark-matter Higgs – contrary to the
ordinary Higgs in the context of quarks or
charge leptons.
 This solution implies the existence of
interactions in the 25% dark-matter sector.
 We may think more about “family”.
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 (selfinteracting) particles - just like that for QCD. In
other words, our Universe would be full of these
particles as the dark matter.

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.
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.
Conclusion
 So,
under “Dirac similarity principle” and
the “minimum Higgs hypothesis”, we could
at least work on “three” Standard Models –
the extra Z’, the left-right symmetry model,
and the family gauge theory. All being
renormalizable.
 The knowledge about 25% dark-matter
may be pivotal in deciding this.