Are there Dark-Matter Galaxies ?Are there Dark-Matter Galaxies ?
W-Y. Pauchy HwangW-Y. Pauchy HwangY.T. Lee Outstanding Chair ProfessorY.T. Lee Outstanding Chair Professor
University Chair ProfessorUniversity Chair ProfessorInstitute of AstrophysicsInstitute of Astrophysics
National Taiwan UniversityNational Taiwan University
Now, dark matter are everywhere (25 % of the entire Now, dark matter are everywhere (25 % of the entire
Universe) and only 5 % in ordinary matter !! Universe) and only 5 % in ordinary matter !! Only about 0.5 % are in (visible) galaxies. Only about 0.5 % are in (visible) galaxies.
Why is there so much dark matter (25 % of the Why is there so much dark matter (25 % of the Universe), compared to so little “visible” ordinary Universe), compared to so little “visible” ordinary matter (5 % of the Universe), the latter as matter (5 % of the Universe), the latter as described by the minimal Standard Model. described by the minimal Standard Model.
My Question: Maybe there should be “invisible” My Question: Maybe there should be “invisible” galaxies (made out of dark matter) in the time galaxies (made out of dark matter) in the time span of 10**9 years, the age of the young span of 10**9 years, the age of the young Universe. These dark-matter galaxies could even Universe. These dark-matter galaxies could even host those spiral galaxies – a new story for galactic host those spiral galaxies – a new story for galactic formation studies.formation studies.
Neutrinos: Mysterious Particles in SM !! Neutrinos: Mysterious Particles in SM !!
Neutrinos now are massive these days. But the Neutrinos now are massive these days. But the minimal Standard Model tells us that they are minimal Standard Model tells us that they are massless. The tiny masses of neutrinos do give us massless. The tiny masses of neutrinos do give us very serious conceptual problem.very serious conceptual problem.
An indirect consequence: Neutrinos are point-like An indirect consequence: Neutrinos are point-like Dirac particles “naturally”. Because four components Dirac particles “naturally”. Because four components are there. are there.
The neutrinos are rather mysterious -- they might The neutrinos are rather mysterious -- they might couple to both the dark-matter world and the “visible” couple to both the dark-matter world and the “visible” ordinary-matter world. Its interactions with the visible ordinary-matter world. Its interactions with the visible world are rather feeble.world are rather feeble.
Are dark-matter galaxies there?Are dark-matter galaxies there?
If dark-matter galaxies are there playing the hosts, If dark-matter galaxies are there playing the hosts, we could understand easily the spiral ordinary-we could understand easily the spiral ordinary-matter galaxies such as our Milky Way.matter galaxies such as our Milky Way.
The story of galactic formation is awfully The story of galactic formation is awfully complicated. First, we have to try to distinguish the complicated. First, we have to try to distinguish the no-seed clustering and the seeded clustering.no-seed clustering and the seeded clustering.
The seeded clustering – Starting from atoms, The seeded clustering – Starting from atoms, molecules, complex molecules, and the chunks of molecules, complex molecules, and the chunks of matter, generated from, in the ordinary-matter world, matter, generated from, in the ordinary-matter world, the strong and electromagnetic forces. I suspect that the strong and electromagnetic forces. I suspect that it is the way to go.it is the way to go.
Possible story for dark-matter galaxiesPossible story for dark-matter galaxies
If dark-matter galaxies were already there in 10**9 If dark-matter galaxies were already there in 10**9 years (the time span for a young Universe), then years (the time span for a young Universe), then they could host the formation of ordinary-matter they could host the formation of ordinary-matter galaxies.galaxies.
The seeded clustering arising from extra-heavy The seeded clustering arising from extra-heavy dark-matter (“TeV”) particles, different from the dark-matter (“TeV”) particles, different from the seeded clustering from the chains of baryons, seeded clustering from the chains of baryons, atoms, molecules, and complex molecules, could atoms, molecules, and complex molecules, could be slightly faster.be slightly faster.
The option of the family gauge theory provides the The option of the family gauge theory provides the seeded dark-matter clustering.seeded dark-matter clustering.
OutlineOutline
Dirac Similarity Principle and minimum Higgs Dirac Similarity Principle and minimum Higgs hypothesishypothesis
Language: Quantum FieldsLanguage: Quantum Fields No. 1 Question: What is the Dark Matter?No. 1 Question: What is the Dark Matter? Different Ways to Extend Standard Model, all Different Ways to Extend Standard Model, all
in accord with “Dirac Similarity Principle” and in accord with “Dirac Similarity Principle” and “minimum Higgs hypothesis”, all are “minimum Higgs hypothesis”, all are renormalizable. renormalizable.
The seeded clusteringsThe seeded clusterings ReferencesReferences
We summarize the minimal Standard Model by We summarize the minimal Standard Model by two working “rules”.two working “rules”.
Dirac similarity principle – our struggle of eighty Dirac similarity principle – our struggle of eighty years to describe the point-like particles such years to describe the point-like particles such as the electron.as the electron.
The “minimum Higgs hypothesis” is the other The “minimum Higgs hypothesis” is the other mysterious conjecture – because we are mysterious conjecture – because we are looking for Higgs particles for forty years, but looking for Higgs particles for forty years, but so far none has been found.so far none has been found.
So, by “induction”, we try to write down these So, by “induction”, we try to write down these two rules which may help to explore the “larger” two rules which may help to explore the “larger” dark matter world.dark matter world.
What is the particle world which we are talking about?What is the particle world which we are talking about?
We were starting with the electrons – Dirac invented We were starting with the electrons – Dirac invented the Dirac equation for that. It turned out to be the the Dirac equation for that. It turned out to be the first “point-like particle”. In it, the orbital angular first “point-like particle”. In it, the orbital angular momentum term is treated equivalently with a 4x4 momentum term is treated equivalently with a 4x4 sigma matrix:sigma matrix:
J = r x p + sigma hbar / 2J = r x p + sigma hbar / 2 Now let’s look at the Standard Model. It’s a world of Now let’s look at the Standard Model. It’s a world of
point-like Dirac particles, with interactions mediated point-like Dirac particles, with interactions mediated by gauge fields and further modulated by Higgs by gauge fields and further modulated by Higgs fields. fields.
So, to begin with, I would assume, naturally, that So, to begin with, I would assume, naturally, that neutrinos are also Dirac particles.neutrinos are also Dirac particles.
Dirac may be the first “physicist” to formulate Dirac may be the first “physicist” to formulate
some equation for “point-like” particles. some equation for “point-like” particles. Dirac didn’t know that the electrons are point-like Dirac didn’t know that the electrons are point-like
particles.particles.
It turns out that, for over eighty years, we It turns out that, for over eighty years, we recognize only a few point-like particles, those recognize only a few point-like particles, those building blocks of the Standard Model. building blocks of the Standard Model.
Maybe we should start with “quantized” Dirac Maybe we should start with “quantized” Dirac fields or, equivalently, “point-like” Dirac particles. fields or, equivalently, “point-like” Dirac particles. In other words, a “point-like” particle in the In other words, a “point-like” particle in the quantum sense is defined through the quantized quantum sense is defined through the quantized Dirac fields. Less than 10**(-18) cm.Dirac fields. Less than 10**(-18) cm.
The case for the “Dirac similarity principle”:The case for the “Dirac similarity principle”:
The notion of “space-time” may also be defined The notion of “space-time” may also be defined accordingly, in some way. accordingly, in some way.
Why there is nothing else - a world of point-like Why there is nothing else - a world of point-like Dirac particles, with interactions mediated by Dirac particles, with interactions mediated by gauge fields and modulated slightly by Higgs gauge fields and modulated slightly by Higgs fields.fields.
The axiom for “quantized Dirac fields” or “point-The axiom for “quantized Dirac fields” or “point-
like Dirac particles” – it turns out that they are like Dirac particles” – it turns out that they are the same thing.the same thing.
Why don’t we see some Higgs after 40 years?Why don’t we see some Higgs after 40 years?
Quantized Klein-Gordon (scalar) fields – in Quantized Klein-Gordon (scalar) fields – in fact, our first lesson for QFT. fact, our first lesson for QFT.
We use the scalar fields to “modulate” quite We use the scalar fields to “modulate” quite a number of things, SSB (the Higgs a number of things, SSB (the Higgs mechanisms), etc. But we still look for them, mechanisms), etc. But we still look for them, after 40 years.after 40 years.
But why aren’t they there? Strange !! But why aren’t they there? Strange !! In any event, we could work with “the In any event, we could work with “the
minimum Higgs hypothesis”. minimum Higgs hypothesis”.
The Language: Elementary Particles as The Language: Elementary Particles as Quantum FieldsQuantum Fields
Classical Mechanical Systems Classical Fields
Quantum Mechanical Systems
Quantum Fields
dc
dc
d → c: discreteness to continuum
Dirac CP: Dirac Correspondence Principle
Dirac CP Dirac CP
Simplified Axioms for the various basic concepts: Simplified Axioms for the various basic concepts:
Classical Mechanical System:Classical Mechanical System:“For a given system, we can find a function (lagrangian) of “For a given system, we can find a function (lagrangian) of the coordinates and velocities such that the integral the coordinates and velocities such that the integral (action) between two instants is an extremum for the real (action) between two instants is an extremum for the real motion.” motion.”
Quantum Mechanical System:Quantum Mechanical System: “ “For the coordinates we can find the conjugate momenta For the coordinates we can find the conjugate momenta
such that the basic (elementary) commutation relations such that the basic (elementary) commutation relations hold.” – Now, they are operators.hold.” – Now, they are operators.
Classical Field:Classical Field:“For a given system, we can find a function (lagrangian) of “For a given system, we can find a function (lagrangian) of the coordinates and velocities such that the integral the coordinates and velocities such that the integral (action) between two instants is an extremum for the real (action) between two instants is an extremum for the real motion.” – except that quantities take continuum meaning. motion.” – except that quantities take continuum meaning.
Quantum Field:Quantum Field: “ “For the coordinates we can find the conjugate momenta For the coordinates we can find the conjugate momenta
such that the basic (elementary) commutation relations such that the basic (elementary) commutation relations hold.” – except that quantities take continuum meaning and hold.” – except that quantities take continuum meaning and we also generalize the notion to include fermions (I.e. anti-we also generalize the notion to include fermions (I.e. anti-commutation relations).commutation relations).
Let’s remind ourselves what we have Let’s remind ourselves what we have done for the minimal Standard Model:done for the minimal Standard Model:
All the quarks and leptons are written in terms of Dirac All the quarks and leptons are written in terms of Dirac equations on certain forms. And all the interactions are in equations on certain forms. And all the interactions are in the gauge fields. In reality, nothing more. Even so far no the gauge fields. In reality, nothing more. Even so far no scalar (Higgs) fields. So it’s a world of “pointlike” Dirac scalar (Higgs) fields. So it’s a world of “pointlike” Dirac particles (a Dirac world) with interactions. Maybe this is an particles (a Dirac world) with interactions. Maybe this is an important guideline to follow. (“Dirac Similarity Principle”.)important guideline to follow. (“Dirac Similarity Principle”.)
So far only renormalizable Interactions are permitted. So far only renormalizable Interactions are permitted. (“Renormalizability” means “calculability”.)(“Renormalizability” means “calculability”.)
In other words, we have so many ways to write things In other words, we have so many ways to write things
relativistically, but not all are equally “applicable” for some relativistically, but not all are equally “applicable” for some reasons. reasons.
Ordinary matter as described by the Ordinary matter as described by the minimal Standard Modelminimal Standard Model
Dirac tried to describe the electron by proposing Dirac tried to describe the electron by proposing Dirac equation. Then the quarks and leptons are Dirac equation. Then the quarks and leptons are written in terms of Dirac equations on certain written in terms of Dirac equations on certain forms. And all the interactions are in the gauge forms. And all the interactions are in the gauge fields. In reality, nothing more.fields. In reality, nothing more.
Only renormalizable Interactions are permitted. Only renormalizable Interactions are permitted. Satisfy the Dirac Similarity Principle and Satisfy the Dirac Similarity Principle and
minimum Higgs hypothesis.minimum Higgs hypothesis.
Connecting Connecting Quarks with the Quarks with the
Cosmos:Cosmos:
Eleven Science Eleven Science Questions for the Questions for the New CenturyNew Century
• The report The report released initially released initially on 4/17/2002 by on 4/17/2002 by National Academy National Academy of Sciences, U.S.A.of Sciences, U.S.A.Cosmology as an
Experimental Science for the New Century
Q1:Q1: What is the dark matter? What is the dark matter? Our Universe has 25% in Dark Matter whileOur Universe has 25% in Dark Matter while
only 5% in ordinary matter. 5% - the minimal Standard only 5% in ordinary matter. 5% - the minimal Standard Model. Model.
Q2:Q2: What is the nature of the dark What is the nature of the dark energy? energy? (The overwhelming 70% question !!)(The overwhelming 70% question !!)
Q3:Q3: How did the universe begin? How did the universe begin? Q4:Q4: Did Einstein have the last word Did Einstein have the last word
on gravity? on gravity? (Is geometry everything?)(Is geometry everything?)
Eleven Science Questions for the New Century: Eleven Science Questions for the New Century: The First Four QuestionsThe First Four Questions
CPU/BPA/NRC Report, 4/17/2002CPU/BPA/NRC Report, 4/17/2002
Q5:Q5: What are the masses of the What are the masses of the neutrinos, and how have they neutrinos, and how have they shaped the evolution of the shaped the evolution of the universe? universe?
It is likely that neutrinos are also point-It is likely that neutrinos are also point-like Dirac particles, since they are like Dirac particles, since they are massive (and have tiny masses). The massive (and have tiny masses). The reason for us to propose “Dirac reason for us to propose “Dirac similarity principle”.similarity principle”.
Eleven Science Questions for the New Century: Eleven Science Questions for the New Century:
The Fifth QuestionThe Fifth Question
Q7:Q7: Are protons unstable? Are protons unstable? Another important question for symmetry.Another important question for symmetry. Q7 means that the grand unified theory in Q7 means that the grand unified theory in
certain form would be valid, if protons decay. certain form would be valid, if protons decay.
We assume that, through proper Higgs We assume that, through proper Higgs mechanism, all particles in the dark sector are mechanism, all particles in the dark sector are massive. massive.
Eleven Science Questions for the New Century: Eleven Science Questions for the New Century: The Seventh Question The Seventh Question
Now, “What is the dark matter?” Could we Now, “What is the dark matter?” Could we describe it or them? If yes, what would be the describe it or them? If yes, what would be the language? The first guess would be to use the language? The first guess would be to use the language which we set up for the Standard language which we set up for the Standard Model – a gauge theory with/without Higgs Model – a gauge theory with/without Higgs Mechanism. Mechanism.
Generalizing the SU_c(3) x SU(2) x U(1) Generalizing the SU_c(3) x SU(2) x U(1) standard model via a renormalizable way by standard model via a renormalizable way by adding particles which we have not seen – it adding particles which we have not seen – it turns out that there are many ways.turns out that there are many ways.
““Minimum Higgs hypothesis” implies that Minimum Higgs hypothesis” implies that extensions in Higgs sector is less favored than extensions in Higgs sector is less favored than those in gauge sector.those in gauge sector.
The ordinary-matter world and the dark-matter The ordinary-matter world and the dark-matter world jointly defines the extended Standard world jointly defines the extended Standard Model.Model.
Candidates for the dark-matter seeds: Long life Candidates for the dark-matter seeds: Long life
time (> 1 Gyr at least), heavy; it doesn’t have time (> 1 Gyr at least), heavy; it doesn’t have ordinary strong and electromagnetic ordinary strong and electromagnetic interactions.interactions.
Dark matter particles: They don’t participate Dark matter particles: They don’t participate (directly) in the ordinary strong and (directly) in the ordinary strong and electromagnetic forces.electromagnetic forces.
Note that the unknown dark matter Note that the unknown dark matter occupies 25% of the current Universe occupies 25% of the current Universe while the visible ordinary matter 5%. We while the visible ordinary matter 5%. We can describe the 5% but 25% unknowns.can describe the 5% but 25% unknowns.
Fortunately if we view the world from the Fortunately if we view the world from the symmetry point of view, it probably does symmetry point of view, it probably does not matter in this 25%-5% upside-down; not matter in this 25%-5% upside-down; but the symmetry of certain kind has to be but the symmetry of certain kind has to be there. there.
First thoughtFirst thought
We could adopt “Dirac similarity principle” and We could adopt “Dirac similarity principle” and the “minimum Higgs hypothesis” as our working the “minimum Higgs hypothesis” as our working rules, when we use the extended Standard rules, when we use the extended Standard Model to describe the dark-matter and ordinary-Model to describe the dark-matter and ordinary-matter particles.matter particles.
If the gauge group, SU_c(3) x SU_L(2) x U(1) x If the gauge group, SU_c(3) x SU_L(2) x U(1) x G, is fixed, the two working rules guarantee the G, is fixed, the two working rules guarantee the uniqueness of the model.uniqueness of the model.
We haven’t seen Higgs after 40 years !!We haven’t seen Higgs after 40 years !!
Years ago (in 1987), I tried to add Z’ and Years ago (in 1987), I tried to add Z’ and realized immediately that we have to add realized immediately that we have to add additional Higgs multiplet(s), too.additional Higgs multiplet(s), too.
How to add a Z’ but with a minimum number of How to add a Z’ but with a minimum number of Higgs fields?Higgs fields?References: W-Y. P. Hwang, Phys. Rev. D36, References: W-Y. P. Hwang, Phys. Rev. D36, 261 (1987).261 (1987).
Crucial to have the mass-generation mechanism Crucial to have the mass-generation mechanism spelled-out.spelled-out.
On the mass generationOn the mass generation
lambda’ ~ lambda x (vev / vev’)**2lambda’ ~ lambda x (vev / vev’)**2 The conjecture for the couplings to “remote” The conjecture for the couplings to “remote”
HiggsHiggs
On the mass generation by the first Higgs doublet, the On the mass generation by the first Higgs doublet, the size are of the same order and of O(v), with v the size are of the same order and of O(v), with v the vacuum expected value. vacuum expected value.
The mass generation for the second Higgs doublet is The mass generation for the second Higgs doublet is down by order O((v/v’)^alpha), with alpha greater than down by order O((v/v’)^alpha), with alpha greater than unity.unity.
In what follows, we take alpha = 2. !In what follows, we take alpha = 2. !
““The Minimum Higgs Hypothesis”The Minimum Higgs Hypothesis”
No.1. On the coupling strengths.No.1. On the coupling strengths.
lambda’ ~ lambda x (vev / vev’)**2lambda’ ~ lambda x (vev / vev’)**2 My conjecture for the couplings to remote Higgs My conjecture for the couplings to remote Higgs
No. 2. On the choice of Higgs multipletsNo. 2. On the choice of Higgs multiplets
There is no redundent Higgs multiplet..There is no redundent Higgs multiplet..
It is a useful “empirical” rule.It is a useful “empirical” rule.
Another ThoughtAnother Thought SU_c(3) × SU_L(2) ×SU_R(2) x U(1) : The SU_c(3) × SU_L(2) ×SU_R(2) x U(1) : The
missing right-handed sector !!missing right-handed sector !! R.N. Mohapatra and J.C. Pati, Phys. Rev. D11, R.N. Mohapatra and J.C. Pati, Phys. Rev. D11,
2558 (1975).2558 (1975).
Mohapatra, Pati, and Salam in fact have many Mohapatra, Pati, and Salam in fact have many models (by choice of Higgs multiplets) but the models (by choice of Higgs multiplets) but the “minimum Higgs hypothesis” selects the unique “minimum Higgs hypothesis” selects the unique one.one.
The missing left-right symmetry should be The missing left-right symmetry should be understood some day.understood some day.
More on the left-right symmetryMore on the left-right symmetry
Why the weak interactions break the left-Why the weak interactions break the left-right symmetry is one of the deepest right symmetry is one of the deepest questions.questions.
Are these stuffs in the 25% dark matter?Are these stuffs in the 25% dark matter? Mass generation: (by the image of the left) Mass generation: (by the image of the left)
lambda (v/v’)**2 varphi* nu_L (nu_R, e_R)lambda (v/v’)**2 varphi* nu_L (nu_R, e_R) Again, it is renormalizable. Again, it is renormalizable.
In fact, we could talk about three In fact, we could talk about three unique options:unique options:
SU_c(3) × SU(2) × U(1) × G SU_c(3) × SU(2) × U(1) × G How to add a Z’ but with a minimum How to add a Z’ but with a minimum
number of Higgs fields?number of Higgs fields?References: W-Y. P. Hwang, Phys. Rev. References: W-Y. P. Hwang, Phys. Rev. D36, 261 (1987).D36, 261 (1987).
To make Mohapatra-Pati-Salam left-right To make Mohapatra-Pati-Salam left-right model minimal in the Higgs sector.model minimal in the Higgs sector.
G = SU_family(3) is also possible. W-Y. P. G = SU_family(3) is also possible. W-Y. P. Hwang, Intern. J. Mod. Phys. A24, 3366 Hwang, Intern. J. Mod. Phys. A24, 3366 (2009).(2009).
Three or two unique options:Three or two unique options:
SU_c(3) × SU(2) × U(1) × G SU_c(3) × SU(2) × U(1) × G As the group G is fixed, the extended As the group G is fixed, the extended
standard model is fixed.standard model is fixed. Thanks to “Dirac similarity principle” and Thanks to “Dirac similarity principle” and
“minimum Higgs hypothesis”!!“minimum Higgs hypothesis”!!
We should pay more attention to the We should pay more attention to the Mohapatra-Pati-Salam left-right symmetric Mohapatra-Pati-Salam left-right symmetric model and the family gauge theory – model and the family gauge theory – because of the symmetry reasons.because of the symmetry reasons.
We have so much of dark matter (25 % of the We have so much of dark matter (25 % of the current Universe) --- the “final” theory, if the current Universe) --- the “final” theory, if the SU_c(3) × SU_L(2) × U(1) × SU_f(3) x SU_R(2) SU_c(3) × SU_L(2) × U(1) × SU_f(3) x SU_R(2) extended Standard Model would describe our extended Standard Model would describe our ordinary-matter and dark-matter world, would ordinary-matter and dark-matter world, would still be “minimal”. still be “minimal”.
We have already seen partially the SU_f(3) part We have already seen partially the SU_f(3) part but we still don’t have the clue about the missing but we still don’t have the clue about the missing SU_R(2) part.SU_R(2) part.
Maybe we have to “rethink” what we are doing – trying to set our tone.
The extended Standard Model, based on the The extended Standard Model, based on the group SU_c(3) × SU(2) × U(1) × SU_f(3) x group SU_c(3) × SU(2) × U(1) × SU_f(3) x SU_R(2), only requires the presence of point-like SU_R(2), only requires the presence of point-like Dirac particles (“Dirac similarity principle”) and Dirac particles (“Dirac similarity principle”) and the minimum presence of Higgs particles the minimum presence of Higgs particles (“minimum Higgs hypothesis”).(“minimum Higgs hypothesis”).
We may “say” that the left-handed neutrinos We may “say” that the left-handed neutrinos belongs to the ordinary-matter world while belongs to the ordinary-matter world while (nu_tau, nu_mu, nu_e)_(right-handed) belongs (nu_tau, nu_mu, nu_e)_(right-handed) belongs to the dark-matter world.to the dark-matter world.
The extended Standard Model is naively renormalizable.
We also introduce 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) – i.e. the SU_c(3) × SU(2) × U(1) × SU_f(3) standard model. SU_f(3) defines the body of the standard model. SU_f(3) defines the body of the dark matter.dark matter.
The only SU_f(3) triplet from the ordinary-matter The only SU_f(3) triplet from the ordinary-matter world, (nu_tau, nu_mu, nu_e)_(right-handed), or world, (nu_tau, nu_mu, nu_e)_(right-handed), or just (nu_tau, nu_mu, nu_e). just (nu_tau, nu_mu, nu_e).
Question: Why do we have three generations?Question: Why do we have three generations?
In what follows, we explain briefly the SU(3) family gauge theory.
An octet of gauge bosons plus a pair of An octet of gauge bosons plus a pair of complex scalar triplets turns out to be the complex scalar triplets turns out to be the simplest choice as long as all gauge simplest choice as long as all gauge bosons become massive while the bosons become massive while the remaining Higgs are also massive.remaining Higgs are also massive.
Now the simple extension is that based on Now the simple extension is that based on SU_c(3) × SU(2) × U(1) × SU_f(3).SU_c(3) × SU(2) × U(1) × SU_f(3).
The rest is straightforward.The rest is straightforward.
There are 8 gauge bosons: There are 8 gauge bosons: Denote the eight family gauge fields (familons) as F_\Denote the eight family gauge fields (familons) as F_\
mu^a(x). Define F_{\mu\nu}^a ≡ \partial_\mu F_\nu^a 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_\\partial_\nu F_\mu^a + \kappa f_{abc} F_\mu^b F_\nu^c. Then we have[4]nu^c. Then we have[4]
One way to describe the nonabelian nature of the gauge One way to describe the nonabelian nature of the gauge theory is to add the Fadde’ev-Popov ghost fieldstheory is to add the Fadde’ev-Popov ghost fields
with D_\mu \phi^a ≡ \partial_\mu \phi^a + \kappa with D_\mu \phi^a ≡ \partial_\mu \phi^a + \kappa f_{abc}F_\mu^b \phi^c.f_{abc}F_\mu^b \phi^c.
1. (1)
4a aL F F
, (2)a aeffL L x D x
The neutrino triplet \Psi(x) isThe neutrino triplet \Psi(x) is
with D_\mu ≡ \partial_\mu with D_\mu ≡ \partial_\mu － － i {\kappa\over 2} \i {\kappa\over 2} \lambda^a F_\mu^a(x). Just like a (triplet) Dirac lambda^a F_\mu^a(x). Just like a (triplet) Dirac field.field.
The family Higgs mechanism is accomplished by The family Higgs mechanism is accomplished by a pair of complex scalar triplets. Under SU_f(3), a pair of complex scalar triplets. Under SU_f(3), they transform into the specific forms in the U-they transform into the specific forms in the U-gauge:gauge:
, (3)fL D
' 0
' 0
exp , ,0 ,2
exp , ,0 . (4)2
aa
aa
i u
i u
We could work out the kinetic terms:We could work out the kinetic terms:
such that, by means of choosing,such that, by means of choosing,
we find, for the familons,we find, for the familons,
† †, (5)scalarL D D D D V
cos , sin , (6)u u
1 2 3 8
4,5,6,7
, ,3
. (7)2
kM M M k M
kM
That is, the eight gauge bosons all become That is, the eight gauge bosons all become massive. On the other hand, by choosingmassive. On the other hand, by choosing
we find that the remaining four (Higgs) particles we find that the remaining four (Higgs) particles are massive (with \mu^2 < 0, we have v^2 = are massive (with \mu^2 < 0, we have v^2 = －－ \\mu^2 / \lambda > 0).mu^2 / \lambda > 0).
Because the neutrino-neutrino-Z vertex is now in Because the neutrino-neutrino-Z vertex is now in our theory augmented by the neutrino-our theory augmented by the neutrino-neutrino-“dark boson” vertices;neutrino-“dark boson” vertices; these dark species should be very massive. these dark species should be very massive.
2† †
2 2† † † †
2
2 , (8)4
V
In the SU_f(3) model, the couplings to ordinary In the SU_f(3) model, the couplings to ordinary matter is only through the neutrinos. matter is only through the neutrinos.
This would make some loop diagrams, involving This would make some loop diagrams, involving neutrinos and familons, very interesting and, neutrinos and familons, very interesting and, albeit likely to be small, should eventually be albeit likely to be small, should eventually be investigated[6]. For example, in the elastic quark investigated[6]. For example, in the elastic quark (or charged lepton) - neutrino scattering, the (or charged lepton) - neutrino scattering, the loop corrections would involve the Z^0 and in loop corrections would involve the Z^0 and in addition the familon loops and if the masses of addition the familon loops and if the masses of the familons were less than that of Z^0 then the the familons were less than that of Z^0 then the loop corrections due to familons would be loop corrections due to familons would be bigger. Thus, we may assume that the familon bigger. Thus, we may assume that the familon masses would be greater than the Z^0 mass, masses would be greater than the Z^0 mass, say 1 TeV.≧say 1 TeV.≧
In other words, there are loop corrections In other words, there are loop corrections involving familons and other dark-matter involving familons and other dark-matter particles, which should be suppressed to particles, which should be suppressed to protect the validity of the minimal Standard protect the validity of the minimal Standard Model. So, the masses of familons and Model. So, the masses of familons and family Higgs have to be greater that a family Higgs have to be greater that a certain value, such as 1 TeV.certain value, such as 1 TeV.
Neutrino mass generation is through the coupling Neutrino mass generation is through the coupling between the neutrino triplet and the family Higgs triplets:between the neutrino triplet and the family Higgs triplets:
resulting a mass matrix which is off diagonal (but is resulting a mass matrix which is off diagonal (but is perfectly acceptable). in a form similar to the Zee perfectly acceptable). in a form similar to the Zee matrix[5], it can easily be fitted to the observed data[2]. matrix[5], it can easily be fitted to the observed data[2].
In other words, the “origin” of the tiny neutrino masses In other words, the “origin” of the tiny neutrino masses comes from the family Higgs and from the loop of family comes from the family Higgs and from the loop of family gauge bosons. It is different from those for quarks and gauge bosons. It is different from those for quarks and charged leptons, a nice way to escape the theorem charged leptons, a nice way to escape the theorem mentioned earlier[3].mentioned earlier[3].
The tiny neutrino masses are generated in the dark The tiny neutrino masses are generated in the dark sector, and in a renormalizable way. This is a very sector, and in a renormalizable way. This is a very interesting solution.interesting solution.
, (9)
SU_f(3) so similar to SU_c(3): SU_f(3) so similar to SU_c(3): One important consequence of the SU_c(3) × One important consequence of the SU_c(3) × SU(2) × U(1) × SU_f(3) standard model is that in SU(2) × U(1) × SU_f(3) standard model is that in addition to QCD and electroweak (EW) phase addition to QCD and electroweak (EW) phase transitions there is other SU_f(3) family phase transitions there is other SU_f(3) family phase transition occurring near the familon masses, transition occurring near the familon masses, maybe above the EW scale (that is, above 1 maybe above the EW scale (that is, above 1 TeV). TeV).
In the early universe, the temperature could be In the early universe, the temperature could be as high as that for the familons such that the as high as that for the familons such that the Universe could be populated with these massive Universe could be populated with these massive dark-matter particles – giving rise to the so-dark-matter particles – giving rise to the so-called “seeded clusterings”.called “seeded clusterings”.
Initial ReferencesInitial References1.1. W-Y. P. Hwang, Phys. Rev. D32 (1985) 824; on the “colored Higgs W-Y. P. Hwang, Phys. Rev. D32 (1985) 824; on the “colored Higgs
mechanism”.mechanism”.2.2. Particle Data Group, “Review of Particle Physics”, J. Phys. G: Nucl. Part. 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.Phys. 33 (2006) 1; on neutrino mass and mixing, see pp. 156 - 164.3.3. For example, see Stuart Raby and Richard Slansky, Los Alamos Science, For example, see Stuart Raby and Richard Slansky, Los Alamos Science,
No. 25 (1997) 64.No. 25 (1997) 64.4.4. For notations, see T-Y. Wu and W-Y. Pauchy Hwang, Relativistic Quantum For notations, see T-Y. Wu and W-Y. Pauchy Hwang, Relativistic Quantum
Mechanics and Quantum Fields (World Scientific, Singapore, 1991).Mechanics and Quantum Fields (World Scientific, Singapore, 1991).5.5. A. Zee, Phys. Lett. B93 (1980) 389; Phys. Lett. B161 (1985) 141; Nucl. A. Zee, Phys. Lett. B93 (1980) 389; Phys. Lett. B161 (1985) 141; Nucl.
Phys. B264 (1986) 99; on the Zee model.Phys. B264 (1986) 99; on the Zee model.6.6. W-Y. P. Hwang, Intern. J. Mod. Phys. A24, 3366 (2009).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, 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.and Pei-Ming Ho for useful conversations, but the errors remain to be mine.
An important conclusion:An important conclusion:
So, under “Dirac similarity principle” and the So, under “Dirac similarity principle” and the “minimum Higgs hypothesis”, we could work on “minimum Higgs hypothesis”, we could work on “three” extended Standard Models – the family “three” extended Standard Models – the family gauge theory, the left-right symmetric model gauge theory, the left-right symmetric model and the extra Z’ (in our order), all in a unique and the extra Z’ (in our order), all in a unique version. All being renormalizable.version. All being renormalizable.
Or, we may work with the SU_c(3) x SU_L(2) x 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 U(1) x SU_f(3) x SU_R(2) extended Standard Model.Model.
ClusteringsClusterings
““Unseeded” gravitational clusterings Unseeded” gravitational clusterings supposed to happen in a long time, in a supposed to happen in a long time, in a time much longer than the age of the time much longer than the age of the Universe.Universe.
So, in a time span of the young Universe, So, in a time span of the young Universe,
neutrino masses contribute very little and neutrino masses contribute very little and we need the seeds (for the clustering) to we need the seeds (for the clustering) to catalyze the processes.catalyze the processes.
The seeded clusteringsThe seeded clusterings
From hadrons, atoms, molecules, complex From hadrons, atoms, molecules, complex molecules, and chunks of the matter, to begin molecules, and chunks of the matter, to begin clustering, in a time span of 1 Gyr, it grows into clustering, in a time span of 1 Gyr, it grows into our present visible Universe – the so-called our present visible Universe – the so-called “seeded clusterings”.“seeded clusterings”.
Similar things could happen if we have SU_f(3) Similar things could happen if we have SU_f(3) family gauge theory with a normal coupling. So, family gauge theory with a normal coupling. So, there might be dark-matter galaxies. there might be dark-matter galaxies.
ConclusionsConclusions
““Dirac similarity principle” and the Dirac similarity principle” and the “minimum Higgs hypothesis” allows us to “minimum Higgs hypothesis” allows us to work on the SU_c(3) x SU_L(2) x U(1) x work on the SU_c(3) x SU_L(2) x U(1) x SU_f(3) x SU_R(2) extended Standard SU_f(3) x SU_R(2) extended Standard Model. It is “naively” renormalizable.Model. It is “naively” renormalizable.
The “seeded” clustering might exist in the The “seeded” clustering might exist in the dark-matter world if SU_f(3) is indeed dark-matter world if SU_f(3) is indeed there. => There are dark-matter galaxies. there. => There are dark-matter galaxies.
