Transcript Family Gauge Theory

The (extended) Standard Model
W-Y. Pauchy Hwang
Y.T. Lee Outstanding-Scholar Chair Professor
Institute of Astrophysics
National Taiwan University
My early struggles with the
Standard Model

Why do we need Higgs mechanism in SU_L(2) x
U(1) electroweak theory but not in SU_c(3)
[QCD]?
 The toilets in a house – Glashow’s remark at
Indiana (early 1980).
 A useless paper and rejected (Phys. Lett. B) by
H. Georgi:
W-Y. P. Hwang, Phys. Rev. D32, 824 (1985).

The idea of “colored Higgs mechanism”.
What is the final Standard
Model?

Why do we have three generations ? It cannot
be “final” without understanding it.

Neutrino oscillations – from one family into
another. “Not final” if without understanding it.

Very excited when I conceived (2008 summer
night, at U. Penn) why I can’t use the colored
Higgs mechanism as the family one.
I believe that the God wouldn’t create a
particle that is so boring in just knowing
only weak interactions.

Neutrinos now have tiny masses these days. So,
they, naturally, have 4-components in the Dirac
space. Note that in the minimal Standard Model
their masses were suspected to vanish.

There is 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 minimum Standard Model. Why?
As time goes by, the strange role of
neutrinos has become clearer and clearer.

Note that right-handed neutrinos never appear in
the construction of the Minimal Standard Model,
as though they are unwanted.

The mystery may lie with the neutrinos, which
oscillate between different generations and play
dual roles between the ordinary-matter world
and the dark-matter world.
Eleven Science Questions for the New Century:
The First Four Questions
U.S. Science Academy Report, 4/17/2002

Q1: What is the dark matter?
Our Universe has 25% in Dark Matter while
only 5% in ordinary matter. The dark-matter particles also
tend to cluster – another obstacle to analyze the problem.

Q2: What is the nature of the dark
energy?
(The overwhelming 70% question, but
maybe with uniformly “distributed”.)


Q3: How did the universe begin?
Q4: Did Einstein have the last word
on gravity?
Cosmology: Eleven Science Questions for the New
Century
U.S. Academy Report, 4/17/2002


Q1: What is the dark matter?
Q2: What is the nature of the dark
energy?

Remark 1:
To push our understandings of the Cosmos, I
think that question No.1 is far more nontrivial
than question No.2.

Remark 2:
We don’t even know the dark-matter world –
what particles are they?
Neutrinos do oscillate !!

Neutrinos oscillate.
nu_e <=> nu_mu; nu_mu <=> nu_tau;
nu_tau <=> nu_e.

This is occurring at the level of point-like Dirac
particles. It differs from the K^0 - \bar K^0
system, a composite system.

Note that neutrino oscillations “move around”
different generations, or, changing flavors.
In view of the minimal Standard Model, we could
write down two working “rules”:
similarity principle – our struggle of
eighty years to describe the point-like
Dirac particle such as the electron.
 The “minimum Higgs hypothesis” is the
other mysterious conjecture – after forty
years we finally get glimpse over the SM
Higgs particle, and nothing more.
 By “induction”, we may write down these
two working rules for the much “larger”
dark matter world.
 Dirac
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, the size certainly less than 10**(-20)
cm in length, these days.

It turns out that, for over eighty years, we
recognize only a few point-like particles, those
building blocks of the Standard Model.

These “point-like” Dirac particles are described
by “quantized” Dirac fields.
Finally some Higgs after searching for 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.
 Maybe we should work with “the minimum
Higgs hypothesis” or “conjecture”.
 Quantized
The point is: Without the two working rules, we
have too many choices on the extended SM.
 Maybe
our space-time only allows for
“point-like” Dirac particles, those can be
created and can communicate among
them.
 Besides, we use the scalar fields to
“modulate” quite a number of things, SSB
(the Higgs mechanisms), etc. Their
existences appear to be “minimal”.
Outline
 Language:
(Renormalizable) Quantum
Fields. (a much powerful than you thought)
 No. 1 Question: What is the Dark Matter?
 Dirac Similarity Principle
 “Minimum Higgs Hypothesis”
 My
struggles with the Standard Model.
 Conclusion
The Language: The Axiom Box
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
Fields as the generalized coordinates (quantum
mechanics of the continuum = quantum fields):


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 (= Quantum Mechanics for the
Continuum):
“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).
 Remarks
on the above axioms for QFT:
(1) no second quantization
(2) no infinite electron sea
 This
is so far a consistent language
framework. Complete ? Too early to tell.
 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. If not, what
else?
 In
what follows, we try to sell the SU_c(3)
x SU(2) x U(1) x SU_f(3) Standard Model
via a renormalizable way.
My Struggles with the Standard
Model: Part No.1

How to add a Z’ but with a minimum number of
Higgs fields?
W-Y. P. Hwang, Phys. Rev. D36, 261 (1987).

When we go beyond the Standard Model, the
Higgs sector depends on the sector of gauge
bosons – extra Z, then extra Higgs.
Thus, ….
Neutrinos have tiny masses. => another Z’.
It may sound strange, but it requires another
Higgs.
 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 tiny neutrinos
masses naturally.

“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 redundant 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 – unfortunately, it may violate the criterion of
choosing “the basic units”.
The thought of Dec 2012
 Why
the weak interactions break the leftright symmetry is one of the deepest
questions.
 Reason: When we write the Dirac theory in
terms of the left-right basic units, each unit
appears once and only once – to ensure
that there is one kinetic-energy term.
 W-Y. Pauchy Hwang and Tung-Mow Yan,
arXiv:1212.4944 [hep-ph] 20 Dec 2012.
My Struggles with the Standard
Model: Part No. 2

Why should we have the standard case, i.e.,
Higgs in the electroweak sector but not in the
strong QCD case.
W-Y. P. Hwang, Phys. Rev. D32, 824 (1985).

The idea of “colored Higgs mechanism”.
Higgs mechanism in QCD?

Almost thirty years ago I was curious by the absence of
the Higgs mechanism in the strong interactions but not in
the weak interaction sector – 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.

It is a deep question – how to write down a
renormalizable theory. Remember that, during old days,
a massive gauge theory is used to be believed as a
nonrenormalizable theory.
No. 3: I started thinking that my
miserable life has some meaning.

In 2008 (four years after I was struck by cerebral
haemorrhage), I went to U. Penn (my Alma
Mater) to attend Lepton-Photon Conference. I
woke up one night to ask why the idea of colored
Higgs mechanism be “copied” as the family
SU(3) gauge theory.

That sets off a series of talks and papers –
maybe nobody appreciate the idea.
My Struggles with the Standard
Model: Part No. 3

W-Y. Pauchy Hwang, Nucl. Phys. A844, 40c
(2012); W-Y. Pauchy Hwang, International J.
Mod. Phys. A24, 3366 (2009); W-Y. Pauchy
Hwang, Intern. J. Mod. Phys. Conf. Series 1, 5
(2011); W-Y. Pauchy Hwang, AIP 978-0-73540687-2/09, pp. 25-30 (2009).
 In
SU(3), 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.
 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)
multiplied another independent SU_f(3).

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 too big. Thus, we may
assume that the familon masses would be much
greater than the Z^0 mass, say ≧ a few TeV.
My Struggles with the Standard
Model: Part No. 4

Last June (2012) I went to Groningen to attend
SSP2012 and suddenly realized the role of
neutrino oscillations in all these.
W-Y. Pauchy Hwang, arXiv:1207.6837v1 [hepph] 30 Jul 2012;
 W-Y. Pauchy Hwang, arXiv:1207.6443v1 [hepph] 27 Jul 2012;
 W-Y. Pauchy Hwang, arXiv:1209.5488v1 [hepph] 25 Sep 2012.

So, let’s come back to neutrino oscillations:

The origin of neutrino masses comes from the coupling
between the neutrino triplet and the family Higgs triplets:
       ,
(9)
resulting a mass matrix which is off diagonal (but is perfectly
acceptable). Note that alpha = i eta, which is needed to make it
hermitian.

This turns out to be that it is not a standard matrix operation,
but a unique SU(3) operation – the unique singlet out of three
given triplets. And from left- and right-handed as usual.

This is the new way to add a renormalizable term, a curl-dot
term.

So, in the SU(3) family gauge theory, we write
down the renormalizable term by using a curldot product, a new term indeed.

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].

The neutrino masses do not come from the
minimal Standard Model, but mainly from the
Higgs in the dark sector.
 At
this point, we had to ask two difficult
questions:
 Why
do we need to make the SU(3) family
gauge theory massive?
 Whether
is (nu_tau, nu_mu, nu_e) the only
triplet from the ordinary-matter world (as
described by the minimal Standard Model)?
 In
the construction of the minimal
Standard Model, the right-handed
neutrinos do not appear. So, we could
construct an independent SU_f(3) gauge
theory with the neutrinos as a triplet, to the
least.
 We should write the extended SU_c(3) x
SU_L(2) x U(1) x SU_f(3) theory
altogether – each of the basic units has
one kinetic-energy term and only one. W-Y.
Pauchy Hwang and Tung-Mow Yan,
arXiv:1212.4944 [hep-ph] 20 Dec 2012.
My Struggles with the Standard
Model: Part No. 5

I invited Tung-Mow Yan to lecture on the
Standard Model last Fall. On one day, it was
clear to us that we could put the three lepton
doublets, six of them altogether, as an SU_f(3)
triplet.

W-Y. Pauchy Hwang and Tung-Mow Yan,
arXiv:1212.4944v1 [hep-ph] 20 Dec 2012.
My Struggles with the Standard
Model: Part No. 6

Finally, I worked out the “combined” Higgs
mechanisms – using the scalar/Higgs fields
Phi(3,2), Phi(3,1), and the standard Phi(1,2); it is
like a magic.

W-Y. Pauchy Hwang, arXiv:1304.4705v1 [hepph] 17 April 2013.
After Thoughts

I think that the SU_f(3), a GeV or sub-Fermi
family gauge theory, is meant to protect the
lepton world from the QED Landau ghost and it
is asymptotically free.
 So, this Standard Model has SU(3) everywhere.
Indeed, it looks like a magic.

W-Y. Pauchy Hwang, arXiv:1301.6464v5 [hepph] 29 April 2014.
Ultraviolet divergences
Fig. 1: The cancelations for all quadratic and logarithmic
divergences.
Fig. 2
Fig. 3
Fig. 4: Cancelations of divergences essential to make a
theory a true theory.
What is the “complete” theory?
I
believe that ultraviolet divergences could
be there for some diagrams, owing to
uncountable infinite DOF’s but altogether
they should be canceled out.
 We should work on the SU_c(3) x SU_L(2)
x U(1) x SU_f(3) Standard Model, as a
complete theory.
Conclusion
I
finally arrived at the Standard Model that
have SU_f(3) to protect the lepton world
and have neutrino oscillations in a natural
way.
 This is the SU_c(3) x SU_L(2) x U(1) x
SU_f(3) extended Standard Model. It is
renormalizable.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
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, Nucl. Phys. A844, 40c (2010); Intern. J. Mod. Phys. A24,
3366 (2009); Intern. J. Mod. Phys. Conf. Series 1, 5 (2011).
W-Y. P. Hwang and Tung-Mow Yan, The Universe, 1-1, 5 (2013);
arXiv:1212.4944 [hep-ph] 20 Dec 2012.
W-Y. Pauchy Hwang, arXiv:1304.4705 [hep-ph] 17 april 2013.
W-Y. Pauchy Hwang, arXiv:1301.6464v5 [hep-ph] 29 April 2014.