Dark Matter as a Guide to Extend the Dark Matter as a Guide to Extend the Standard Model: Dirac Similarity Principle Standard Model: Dirac Similarity Principle

and the Minimum Higgs Hypothesisand the Minimum Higgs Hypothesis

W-Y. Pauchy HwangW-Y. Pauchy HwangUniversity Chair ProfessorUniversity Chair ProfessorInstitute of AstrophysicsInstitute of Astrophysics

National Taiwan UniversityNational Taiwan University

What I would like to do today? It’s an idea off and on in my 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.mind, maybe over 30 years but I think it is mature lately. Neutrinos now are massive these days. But the Neutrinos now are massive these days. But the

minimal Standard Model tells us that they should minimal Standard Model tells us that they should be zero. be zero.

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) as described by the matter (5\% of the Universe) as described by the Standard Model. Standard Model.

The mystery may lie with the neutrinos, which The mystery may lie with the neutrinos, which may “bridge” between the dark-matter world and may “bridge” between the dark-matter world and the “visible” ordinary-matter world.the “visible” ordinary-matter world.

Let’s begin with two remarks – “Dirac similarity Let’s begin with two remarks – “Dirac similarity principle” and why Higgs are so far not there. principle” and why Higgs are so far not there.

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

Dirac similarity principle – our struggle of eighty Dirac similarity principle – our struggle of eighty years to describe the point-like particle such as years to describe the point-like particle such as the electron.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 obtain these two rules So, by “induction”, we obtain these two rules which may help in bringing in the “larger” dark which may help in bringing in the “larger” dark matter world.matter world.

Why could we use “dark matter as a guide to Why could we use “dark matter as a guide to extend the Standard Model” ? extend the Standard Model” ?

As explained later, the “language” developed so As explained later, the “language” developed so far is likely to be the quantum field theory, and far is likely to be the quantum field theory, and otherwise what else? otherwise what else?

Ordinary matter (5%) and dark matter (25%) are Ordinary matter (5%) and dark matter (25%) are believed to clusterized similarly and obey the believed to clusterized similarly and obey the same “gravitational law”. same “gravitational law”.

Unlike the uniformly-distributed “dark energy”, Unlike the uniformly-distributed “dark energy”, ordinary matter and dark matter seems to follow ordinary matter and dark matter seems to follow the same laws, except the feeble interactions the same laws, except the feeble interactions between them.between them.

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 We were starting with the electrons – Dirac invented the Dirac equation for that. The first invented the Dirac equation for that. The first “point-like particle”. In it, the orbital angular “point-like particle”. In it, the orbital angular momentum term is treated equivalently with a momentum term is treated equivalently with a sigma matrix, relativistically. sigma matrix, relativistically.

Now let’s look at the Standard Model. It’s a world Now let’s look at the Standard Model. It’s a world of (point-like) Dirac particles, with interactions of (point-like) Dirac particles, with interactions mediated by gauge fields and further modulated mediated by gauge fields and further modulated by Higgs fields. by Higgs 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. He tried to put in quantum mechanics (those He tried to put in quantum mechanics (those

matrices representing spins) and relativity matrices representing spins) and relativity simultaneously. simultaneously.

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.

Maybe we shouldn’t question what “quantized” Maybe we shouldn’t question what “quantized” or “point-like” means to us, or rather instead of or “point-like” means to us, or rather instead of treating this as an “axiom”. treating this as an “axiom”.

Thus, we argue for the “Dirac similarity principle”.Thus, we argue for the “Dirac similarity principle”.

It’s a special way to put in quantum mechanics It’s a special way to put in quantum mechanics (those matrices representing spins) and relativity (those matrices representing spins) and relativity simultaneously. In fact, the “space-time” notion simultaneously. In fact, the “space-time” notion may be defined also. may be defined also.

Apparently the way is so special. Why there is Apparently the way is so special. Why there is nothing else - a world of point-like Dirac nothing else - a world of point-like Dirac particles, with interactions mediated by gauge particles, with interactions mediated by gauge fields and modulated slightly by Higgs fields. fields and modulated slightly by Higgs fields.

The axiom for “quantized Dirac fields” or “point-The axiom for “quantized Dirac fields” or “point-like Dirac particles” – they are the same thing.like Dirac particles” – they are the same thing.

We understand “Dirac similarity principle”:We understand “Dirac similarity principle”:

Our space-time “lattice” admits, or be Our space-time “lattice” admits, or be compatible with, certain kind of “point-like” compatible with, certain kind of “point-like” particles which at this point turns out to particles which at this point turns out to “quantized Dirac fields”. “quantized Dirac fields”.

Our world is very special. Why there is nothing Our world is very special. Why there is nothing else - a world of point-like Dirac particles, with else - a world of point-like Dirac particles, with interactions mediated by gauge fields and interactions mediated by gauge fields and modulated slightly by Higgs fields. modulated slightly by Higgs fields.

““Quantized Dirac fields” or “point-like Dirac Quantized Dirac fields” or “point-like Dirac particles” turn out to be the same thing.particles” turn out to be 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 lesson in QFT. fact, our lesson in QFT.

We use the scalar fields to “modulate” We use the scalar fields to “modulate” quite a number of things, SSB (the Higgs quite a number of things, SSB (the Higgs mechanisms), etc. But we still look for mechanisms), etc. But we still look for them, after 40 years.them, after 40 years.

Maybe we should work with “the minimum Maybe we should work with “the minimum Higgs hypothesis” or “conjecture”. Higgs hypothesis” or “conjecture”.

OutlineOutline

Language: Quantum FieldsLanguage: Quantum Fields No. 1 Question: What is the Dark Matter?No. 1 Question: What is the Dark Matter? Dirac Similarity Principle: Observation to Dirac Similarity Principle: Observation to

Proposal Proposal Different Ways to Extend Standard Model, Different Ways to Extend Standard Model,

all in the renormalizable way and in accord all in the renormalizable way and in accord with “Dirac Similarity Principle” with “Dirac Similarity Principle”

DiscussionsDiscussions ReferencesReferences

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

Classical Mechanical System:Classical Mechanical System:“For a given system, we can find a function “For a given system, we can find a function (lagrangian) of the coordinates and velocities such (lagrangian) of the coordinates and velocities such that the integral (action) between two instants is an that the integral (action) between two instants is an extremum for the real motion.” extremum for the real motion.”

Quantum Mechanical System:Quantum Mechanical System: “ “For the coordinates we can find the conjugate For the coordinates we can find the conjugate

momenta such that the basic (elementary) momenta such that the basic (elementary) commutation relations hold.” – Now, they are commutation relations hold.” – Now, they are operators.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 Review what we have done:Let’s Review what we have done:

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 particles (a Dirac world) with interactions. Maybe this is an important guideline to follow. (“Dirac Similarity an important guideline to follow. (“Dirac Similarity Principle”.)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 relativistically, but not all are equally “applicable” for some reasons. some reasons.

The SM can be viewed from a different The SM can be viewed from a different angle: Dirac Similarity Principleangle: Dirac Similarity Principle

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.

So far only renormalizable Interactions are So far only renormalizable Interactions are permitted. permitted.

Maybe some specialty about the Dirac equation Maybe some specialty about the Dirac equation exists in our space-time.exists in our space-time.

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. How about 5% versus only 5% in ordinary matter. How about 5% versus 25%, instead – it would be more comfortable if we looked 25%, instead – it would be more comfortable if we looked for the “Standard Model” for the majority (25%). for the “Standard Model” for the majority (25%).

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?

I would remind you of a theorem about I would remind you of a theorem about the neutrino mass: The neutrinos the neutrino mass: The neutrinos should be massless in the minimum should be massless in the minimum Standard Model. Standard Model.

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. That means that the grand unified theory in That means that the grand unified theory in

certain form would be valid, if protons decay. certain form would be valid, if protons decay. In what follows, I assume that the gauge theory In what follows, I assume that the gauge theory

in the extended Standard Model should have in the extended Standard Model should have two basic ingredients – the gauge sector and two basic ingredients – the gauge sector and the Higgs mechanism, the latter ensuring that the Higgs mechanism, the latter ensuring that all particles in the dark sector are massive. all particles in the dark sector are 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 describe it or them? If yes, what would be the language? The first guess would be to the language? The first guess would be to use the language which we set up for the use the language which we set up for the Standard Model – a gauge theory Standard Model – a gauge theory with/without Higgs Mechanism. with/without Higgs 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 standard model via a renormalizable way by adding particles which we have not by adding particles which we have not seen – it turns out that there are many seen – it turns out that there are many ways.ways.

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%. Not while the visible ordinary matter 5%. Not the other way around – 5% dark matter the other way around – 5% dark matter while 25% ordinary matter. We can while 25% ordinary matter. We can describe the 5% but 25% unknowns.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

Neutrinos have tiny masses. => another Z’. Neutrinos have tiny masses. => another Z’. It sounds strange, but it requires another Higgs, It sounds strange, but it requires another Higgs,

to be natural.to be natural. 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?W-Y. P. Hwang, Phys. Rev. D36, 261 (1987).W-Y. P. Hwang, Phys. Rev. D36, 261 (1987).

Consider 2+2 Higgs Scenario. The second, and Consider 2+2 Higgs Scenario. The second, and “remote”, Higgs doublet could give neutrinos “remote”, Higgs doublet could give neutrinos masses naturally. masses naturally.

No Higgs after 40 years !!No Higgs after 40 years !!

Maybe the associated Higgs structure should be Maybe the associated Higgs structure should be minimal. After all, after 40 years or so, we minimal. After all, after 40 years or so, we haven’t found the signature of Higgs. We still haven’t found the signature of Higgs. We still ask the LHC for an answer. How to make a ask the LHC for an answer. How to make a model with minimum Higgs structure?model with minimum Higgs structure?

Important Question: How to add a Z’ but with a Important Question: How to add a Z’ but with a minimum number of Higgs fields?minimum number of Higgs fields?References: W-Y. P. Hwang, Phys. Rev. D36, References: W-Y. P. Hwang, Phys. Rev. D36, 261 (1987).261 (1987).

On the mass generationOn the mass generation

lambda’ ~ lambda x (vec / vec’)**2lambda’ ~ lambda x (vec / vec’)**2 My conjecture for the couplings to remote HiggsMy conjecture for the couplings to remote Higgs

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.

For some reason, the mass generation for the second For some reason, the mass generation for the second Higgs doublet is down by order O((v/v’)^alpha), with Higgs doublet is down by order O((v/v’)^alpha), with alpha greater than unity.alpha greater than unity.

In what follows, we take alpha = 2. In what follows, we take alpha = 2. In short, the details for the Higgs mechanism need to be In short, the details for the Higgs mechanism need to be

worked out. worked out. ““Minimal Higgs Hypothesis” !!Minimal Higgs Hypothesis” !!

““The Minimum Higgs Hypothesis”The Minimum Higgs Hypothesis”

No.1. On the coupling strengths.No.1. On the coupling strengths.

lambda’ ~ lambda x (vec / vec’)**2lambda’ ~ lambda x (vec / vec’)**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 redandant Higgs multiplet..There is no redandant 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). Here we also have an extra Z’ but with another Here we also have an extra Z’ but with another

right-handed doublet almost eaten up via SSB. right-handed doublet almost eaten up via SSB.

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.

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.

Don’t forget to ask….Don’t forget to ask…. 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) Make sure that it is renormalizable. Make sure that it is renormalizable.

I’m talking about three options, in I’m talking about three options, in fact three nice options:fact three nice 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. See G = SU_family(3) is also possible. See later.later.

In what follows, we talk about the In what follows, we talk about the possibility of adding an SU(3) family gauge possibility of adding an SU(3) family gauge theory - the SU_c(3) × SU(2) × U(1) × theory - the SU_c(3) × SU(2) × U(1) × SU_f(3) standard model. SU_f(3) defines SU_f(3) standard model. SU_f(3) defines the body of the dark matter. the body of the dark matter.

In this model, (nu_e, nu_mu, nu_tau) In this model, (nu_e, nu_mu, nu_tau) could serve as the only “bridge” for could serve as the only “bridge” for ordinary matter.ordinary matter.

Why do we have three generations? In Why do we have three generations? In this model, we are forced to have three this model, we are forced to have three generations.generations.

I think that most symmetry may have I think that most symmetry may have something to do with some interactions, something to do with some interactions, maybe too weak to be detected. Maybe maybe too weak to be detected. Maybe this is the origin of “dark matter”.this is the origin of “dark matter”.

Different Options:Different Options: Left-right symmetric model; Left-right symmetric model; SU_c(3) x SU(2) x U(1) X U(1) with extra SU_c(3) x SU(2) x U(1) X U(1) with extra

Z^0;Z^0; SU_c(3) x SU(2) x U(1) x SU_f(3).SU_c(3) x SU(2) x U(1) x SU_f(3).

For this extra SU_f(3) gauge theory, should it For this extra SU_f(3) gauge theory, should it exist and we suppose that it is coupled to exist and we suppose that it is coupled to neutral fermions, i.e. neutrinos, in the sector of neutral fermions, i.e. neutrinos, in the sector of ordinary matter.ordinary matter.

Family symmetry => Family gauge symmetryFamily symmetry => Family gauge symmetry

Proposing a gauge theory, it means some kind of Proposing a gauge theory, it means some kind of new interactions.new interactions.

Maybe there is another gauge theory Maybe there is another gauge theory beyond the Standard Model …beyond the Standard Model …

More than twenty years ago I was curious by the More than twenty years ago I was curious by the absence of the Higgs mechanism in the strong absence of the Higgs mechanism in the strong interactions but not in the weak interaction sector[1] – a interactions but not in the weak interaction sector[1] – a question still remains unanswered till today. A question still remains unanswered till today. A renormalizable gauge theory that does not have to be renormalizable gauge theory that does not have to be massless is already reputed by ‘t Hooft and others, for massless is already reputed by ‘t Hooft and others, for the standard model. Maybe our question should be the standard model. Maybe our question should be whether the electromagnetism would be massless.whether the electromagnetism would be massless.

In fact, this is a deep question – how to write down a In fact, this is a deep question – how to write down a renormalizable theory. During old days, a massive gauge renormalizable theory. During old days, a massive gauge theory is used to be believed as a nonrenormalizable theory is used to be believed as a nonrenormalizable theory.theory.

Here I try to set up the view that the only Here I try to set up the view that the only visible massless particle is the photon – visible massless particle is the photon – and the gluons, if massless, are and the gluons, if massless, are permanently confined (then it is permanently confined (then it is meaningless to have mass). That is, all meaningless to have mass). That is, all particles in the dark sector are massive. particles in the dark sector are massive.

In the sector of ordinary matter, the In the sector of ordinary matter, the neutrinos could serve as the direct neutrinos could serve as the direct messengers with the dark sector (i.e. messengers with the dark sector (i.e. neutrinos are also one kind of dark neutrinos are also one kind of dark matter). matter).

Another clue comes from neutrinos – they are Another clue comes from neutrinos – they are neutral, massive and mixing/oscillating. These neutral, massive and mixing/oscillating. These particles are barely “visible” in the Particle Table. particles are barely “visible” in the Particle Table. Maybe these are avenues that connect to those Maybe these are avenues that connect to those unknowns, particularly the dark matter in the unknowns, particularly the dark matter in the Universe.Universe.

In fact, the neutrino sector, with the current In fact, the neutrino sector, with the current knowledge of masses and mixings[2], presents a knowledge of masses and mixings[2], presents a serious basic problem[3] – that is, a theorem serious basic problem[3] – that is, a theorem that neutrinos are massless in the minimal that neutrinos are massless in the minimal standard model. Any model with at least one standard model. Any model with at least one massive neutrino have to be some sort of massive neutrino have to be some sort of extended standard model (i.e. not minimal).extended standard model (i.e. not minimal).

I assume that the whole Dirac neutrinos I assume that the whole Dirac neutrinos could be used in the neutrino triplet. could be used in the neutrino triplet.

If only the right-handed neutrinos are used If only the right-handed neutrinos are used in this context, the dark sector would be in this context, the dark sector would be completely dark, making the story a little completely dark, making the story a little boring. (But it may be right.)boring. (But it may be right.)

If the coupling would involve gamma-5, If the coupling would involve gamma-5, then we have to worry about the anomaly. then we have to worry about the anomaly.

Note that in family gauge theory: Note that in family gauge theory:

The masses of the neutrino triplet come from the The masses of the neutrino triplet come from the coupling to some Higgs field - a pair of complex coupling to some Higgs field - a pair of complex scalar triplets, as worked out in the previous scalar triplets, as worked out in the previous publication[1]. Note that the “radiative” publication[1]. Note that the “radiative” corrections due to gauge bosons serve as a corrections due to gauge bosons serve as a correction to mass. correction to mass.

Note that the neutrino masses do not come from Note that the neutrino masses do not come from the minimal Standard Model, mainly from the the minimal Standard Model, mainly from the Higgs in the dark sector.Higgs in the dark sector.

More on family gauge theory:More on family gauge theory:

If we think of the role of gauge theories in If we think of the role of gauge theories in quantum field theory, we still have to quantum field theory, we still have to recognize its unique and important role. If recognize its unique and important role. If the standard model is missing something, the standard model is missing something, a gauge theory sector would be one at the a gauge theory sector would be one at the first guess.first guess.

I believe that something missing may be a I believe that something missing may be a gauge sector, owing to the successes of gauge sector, owing to the successes of SU_c(3) × SU(2) × U(1) standard model. SU_c(3) × SU(2) × U(1) standard model.

In fact, an octet of gauge bosons plus a In fact, an octet of gauge bosons plus a pair of complex scalar triplets turns out to pair of complex scalar triplets turns out to be the simplest choice as long as all be the simplest choice as long as all gauge bosons become massive while the gauge bosons become massive while the remaining Higgs are also massive.remaining Higgs are also massive.

This is the basic framework. The standard This is the basic framework. The standard model is the gauge theory based on the model is the gauge theory based on the group SU_c(3) × SU(2) × U(1). Now the group SU_c(3) × SU(2) × U(1). Now the simple extension is that based on SU_c(3) simple extension is that based on SU_c(3) × SU(2) × U(1) × SU_f(3).× SU(2) × U(1) × SU_f(3).

So, the following story is rather simple.So, the following story is rather simple.

In the model, the couplings to ordinary matter is In the model, the couplings to ordinary matter is only through the neutrinos, the only charged/ only through the neutrinos, the only charged/ neutral fermions that are interacting weakly. This neutral fermions that are interacting weakly. This would make some loop diagrams, involving 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.≧

The above argument also implies that we The above argument also implies that we cannot have the massless familon(s) or cannot have the massless familon(s) or massless family Higgs particle(s). massless family Higgs particle(s). Otherwise, the loop corrections in some Otherwise, the loop corrections in some cases would be dominated by those with cases would be dominated by those with familons.familons.

The other important point is the coupling between the neutrino triplet The other important point is the coupling between the neutrino triplet and the family Higgs triplets:and the family Higgs triplets:

resulting a mass matrix which is off diagonal (but is perfectly resulting a mass matrix which is off diagonal (but is perfectly acceptable). In other words, the mass matrix, being proportional to acceptable). In other words, the mass matrix, being proportional to －－ \bar{\nu}_e(v_{+} + \epsilon v_\bar{\nu}_e(v_{+} + \epsilon v_ －－ )\nu_\tau )\nu_\tau

+ \bar{\nu}_e(u_{+} + \epsilon u_+ \bar{\nu}_e(u_{+} + \epsilon u_ －－ )\nu_\mu )\nu_\mu + \bar{\nu}_\tau (v_{+} + \epsilon v_+ \bar{\nu}_\tau (v_{+} + \epsilon v_ －－ )\nu_e)\nu_e －－ \bar{\nu}_\mu(u_{+} + \epsilon u_\bar{\nu}_\mu(u_{+} + \epsilon u_ －－ )\nu_e , is off-diagonal, in )\nu_e , is off-diagonal, in

the form similar to the Zee matrix[5], and can easily be fitted to the 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.)observed data[2]. (And i is needed to make it hermitian.)

In other words, the “source” of the neutrino masses comes from the In other words, the “source” of the neutrino masses comes from the family Higgs and is different from those for quarks and charged family Higgs and is different from those for quarks and charged leptons, a nice way to escape the theorem mentioned earlier[3].leptons, a nice way to escape the theorem mentioned earlier[3].

The neutrino masses are obtained from the dark sector, but in a The neutrino masses are obtained from the dark sector, but in a renormalizable way. This is a very interesting solution.renormalizable way. This is a very interesting solution.

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Unlike the story with extra Z or the left-Unlike the story with extra Z or the left-right model, in which the details of the right model, in which the details of the Higgs doublets may need some fine-Higgs doublets may need some fine-tuning,tuning,

the problem of neutrino masses in the the problem of neutrino masses in the present model is solved much more present model is solved much more naturally.naturally.

In any case, it is possible to solve the In any case, it is possible to solve the problems through a renormalizable way.problems through a renormalizable way.

What is surprising about our model? There What is surprising about our model? There is no unwanted massless particle - so, no is no unwanted massless particle - so, no disaster anticipated. It is another disaster anticipated. It is another renormalizable extension of the standard renormalizable extension of the standard model idea. Coming back to the neutrino model idea. Coming back to the neutrino sector, we now introduce the mass terms sector, we now introduce the mass terms in a renormalizable way (with the help in a renormalizable way (with the help from SU_f(3) gauge theory). Furthermore, from SU_f(3) gauge theory). Furthermore, there is no major modification of the there is no major modification of the original Standard Model.original Standard Model.

Maybe a “solution” to the family problem.Maybe a “solution” to the family problem.

Discussions on SU_f(3)Discussions on SU_f(3)

The neutrino mass problem is solved The neutrino mass problem is solved nicely since neutrinos couple with the nicely since neutrinos couple with the dark-matter Higgs – contrary to the dark-matter Higgs – contrary to the ordinary Higgs in the context of quarks or ordinary Higgs in the context of quarks or charge leptons.charge leptons.

This solution implies the existence of This solution implies the existence of interactions in the 25% dark-matter sector.interactions in the 25% dark-matter sector.

We may think more about “family”.We may think more about “family”.

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). The exact scale is hard to decide, for the TeV). The exact scale is hard to decide, for the moment.moment.

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 (self-Universe could be populated with these (self-interacting) particles - just like that for QCD. In interacting) particles - just like that for QCD. In other words, our Universe would be full of these other words, our Universe would be full of these particles as the dark matter.particles as the dark matter.

ReferencesReferences1.1. 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. Ling-Fong Li, private communications.Ling-Fong Li, private communications.

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.

ConclusionConclusion

So, under “Dirac similarity principle” and So, under “Dirac similarity principle” and the “minimum Higgs hypothesis”, we could the “minimum Higgs hypothesis”, we could at least work on “three” Standard Models – at least work on “three” Standard Models – the extra Z’, the left-right symmetry model, the extra Z’, the left-right symmetry model, and the family gauge theory. All being and the family gauge theory. All being renormalizable.renormalizable.

The knowledge about 25% dark-matter The knowledge about 25% dark-matter may be pivotal in deciding this. may be pivotal in deciding this.