Dark Matter as a Guide to Extend the Standard Model: Dirac Similarity Principle and the Minimum Higgs Hypothesis

1
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

2
What I would like to do today? Its 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.
• Lets begin with two remarks Dirac similarity
  principle and why Higgs are so far not there.

3
In this talk, we focus on two rules, two very
strange rules.

• Dirac 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.

4
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.

5
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 lets look at the Standard Model. Its 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.

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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 shouldnt question what quantized or
  point-like means to us, or rather instead of
  treating this as an axiom.

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Thus, we argue for the Dirac similarity
principle.

• Its 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
  point-like Dirac particles they are the same
  thing.

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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.

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Why dont we see some Higgs after 40 years?

• Quantized 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.

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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

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The Language Elementary Particles as Quantum
Fields
d?c
Classical Mechanical Systems
Classical Fields
Dirac CP
Dirac CP
d?c
Quantum Mechanical Systems
Quantum Fields
d ? c discreteness to continuum Dirac CP Dirac
Correspondence Principle
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• Classical Mechanical SystemFor 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.

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• Classical FieldFor 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).

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Lets 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 its 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.

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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.

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• 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
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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?)

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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.

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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.

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• 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.

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• 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.

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First thought

• Neutrinos have tiny masses. gt 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 22 Higgs Scenario. The second, and
  remote, Higgs doublet could give neutrinos
  masses naturally.

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No Higgs after 40 years !!

• Maybe the associated Higgs structure should be
  minimal. After all, after 40 years or so, we
  havent 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).

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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 !!

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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.

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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.

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More on the left-right symmetry

• Why the weak interactions break the left-right
  symmetry is one of the deepest questions.
• Dont 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.

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Im talking about three options, in fact three
nice 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. See later.

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• 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.

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• 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 Z0
• SU_c(3) x SU(2) x U(1) x SU_f(3).

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• 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 gt Family gauge symmetry
• Proposing a gauge theory, it means some kind of
  new interactions.

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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
  sector1 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.

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• 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).

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• 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 mixings2, presents a
  serious basic problem3 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).

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• 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.

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• 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
  publication1. 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.

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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.

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• 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.

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• 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
  investigated6. For example, in the elastic
  quark (or charged lepton) - neutrino scattering,
  the loop corrections would involve the Z0 and in
  addition the familon loops and if the masses of
  the familons were less than that of Z0 then the
  loop corrections due to familons would be bigger.
  Thus, we may assume that the familon masses would
  be greater than the Z0 mass, say ? 1 TeV.

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• 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.

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• The other important point is the coupling between
  the neutrino triplet and the family Higgs
  tripletsresulting 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
  matrix5, and can easily be fitted to the
  observed data2. (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 earlier3.
• The neutrino masses are obtained from the dark
  sector, but in a renormalizable way. This is a
  very interesting solution.

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• Unlike the story with extra Z or the left-right
  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.

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• 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.

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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.

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• 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.

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References

• 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.

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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.