Are there Dark-Matter Galaxies ?

1. Are there Dark-Matter Galaxies ? W-Y. Pauchy Hwang Y.T. Lee Outstanding Chair Professor University Chair Professor Institute of Astrophysics National Taiwan University

2. Now, dark matter are everywhere (25 % of the entire Universe) and only 5 % in ordinary matter !! • Only about 0.5 % are in (visible) galaxies. • Why is there so much dark matter (25 % of the Universe), compared to so little “visible” ordinary matter (5 % of the Universe), the latter as described by the minimal Standard Model. • My Question: Maybe there should be “invisible” galaxies (made out of dark matter) in the time span of 10**9 years, the age of the young Universe. These dark-matter galaxies could even host those spiral galaxies – a new story for galactic formation studies.

3. Neutrinos: Mysterious Particles in SM !! • Neutrinos now are massive these days. But the minimal Standard Model tells us that they are massless. The tiny masses of neutrinos do give us very serious conceptual problem. • An indirect consequence: Neutrinos are point-like Dirac particles “naturally”. Because four components are there. • The neutrinos are rather mysterious -- they might couple to both the dark-matter world and the “visible” ordinary-matter world. Its interactions with the visible world are rather feeble.

4. Are dark-matter galaxies there? • If dark-matter galaxies are there playing the hosts, we could understand easily the spiral ordinary-matter galaxies such as our Milky Way. • The story of galactic formation is awfully complicated. First, we have to try to distinguish the no-seed clustering and the seeded clustering. • The seeded clustering – Starting from atoms, molecules, complex molecules, and the chunks of matter, generated from, in the ordinary-matter world, the strong and electromagnetic forces. I suspect that it is the way to go.

5. Possible story for dark-matter galaxies • If dark-matter galaxies were already there in 10**9 years (the time span for a young Universe), then they could host the formation of ordinary-matter galaxies. • The seeded clustering arising from extra-heavy dark-matter (“TeV”) particles, different from the seeded clustering from the chains of baryons, atoms, molecules, and complex molecules, could be slightly faster. • The option of the family gauge theory provides the seeded dark-matter clustering.

6. Outline • Dirac Similarity Principle and minimum Higgs hypothesis • Language: Quantum Fields • No. 1 Question: What is the Dark Matter? • Different Ways to Extend Standard Model, all in accord with “Dirac Similarity Principle” and “minimum Higgs hypothesis”, all are renormalizable. • The seeded clusterings • References

7. We summarize the minimal Standard Model by two working “rules”. • Dirac similarity principle – our struggle of eighty years to describe the point-like particles 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 try to write down these two rules which may help to explore the “larger” dark matter world.

8. What is the particle world which we are talking about? • We were starting with the electrons – Dirac invented the Dirac equation for that. It turned out to be the first “point-like particle”. In it, the orbital angular momentum term is treated equivalently with a 4x4 sigma matrix: J = r x p + sigma hbar / 2 • Now let’s look at the Standard Model. It’s a world of point-like Dirac particles, with interactions mediated by gauge fields and further modulated by Higgs fields. • So, to begin with, I would assume, naturally, that neutrinos are also Dirac particles.

9. Dirac may be the first “physicist” to formulate some equation for “point-like” particles. • Dirac didn’t know that the electrons are point-like particles. • 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. In other words, a “point-like” particle in the quantum sense is defined through the quantized Dirac fields. Less than 10**(-18) cm.

10. The case for the “Dirac similarity principle”: • The notion of “space-time” may also be defined accordingly, in some way. • 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” – it turns out that they are the same thing.

11. Why don’t we see some Higgs after 40 years? • Quantized Klein-Gordon (scalar) fields – in fact, our first lesson for QFT. • We use the scalar fields to “modulate” quite a number of things, SSB (the Higgs mechanisms), etc. But we still look for them, after 40 years. • But why aren’t they there? Strange !! • In any event, we could work with “the minimum Higgs hypothesis”.

12. 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

13. Simplified Axioms for the various basic concepts: • Classical Mechanical System:“For a given system, we can find a function (lagrangian) of the coordinates and velocities such that the integral (action) between two instants is an extremum for the real motion.” • Quantum Mechanical System: “For the coordinates we can find the conjugate momenta such that the basic (elementary) commutation relations hold.” – Now, they are operators.

14. Classical Field:“For a given system, we can find a function (lagrangian) of the coordinates and velocities such that the integral (action) between two instants is an extremum for the real motion.” – except that quantities take continuum meaning. • Quantum Field: “For the coordinates we can find the conjugate momenta such that the basic (elementary) commutation relations hold.” – except that quantities take continuum meaning and we also generalize the notion to include fermions (I.e. anti-commutation relations).

15. Let’s remind ourselves what we have done for the minimal Standard Model: • All the quarks and leptons are written in terms of Dirac equations on certain forms. And all the interactions are in the gauge fields. In reality, nothing more. Even so far no scalar (Higgs) fields. So it’s a world of “pointlike” Dirac particles (a Dirac world) with interactions. Maybe this is an important guideline to follow. (“Dirac Similarity Principle”.) • So far only renormalizable Interactions are permitted. (“Renormalizability” means “calculability”.) • In other words, we have so many ways to write things relativistically, but not all are equally “applicable” for some reasons.

16. Ordinary matter as described by the minimal Standard Model • 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. • Only renormalizable Interactions are permitted. • Satisfy the Dirac Similarity Principle and minimum Higgs hypothesis.

17. 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

18. Eleven Science Questions for the New Century: The First Four QuestionsCPU/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. 5% - the minimal Standard Model. • 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?)

19. 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? • It is likely that neutrinos are also point-like Dirac particles, since they are massive (and have tiny masses). The reason for us to propose “Dirac similarity principle”.

20. Eleven Science Questions for the New Century: The Seventh Question • Q7: Are protons unstable? • Another important question for symmetry. • Q7 means that the grand unified theory in certain form would be valid, if protons decay. • We assume that, through proper Higgs mechanism, all particles in the dark sector are massive.

21. 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. • “Minimum Higgs hypothesis” implies that extensions in Higgs sector is less favored than those in gauge sector.

22. The ordinary-matter world and the dark-matter world jointly defines the extended Standard Model. • Candidates for the dark-matter seeds: Long life time (> 1 Gyr at least), heavy; it doesn’t have ordinary strong and electromagnetic interactions. • Dark matter particles: They don’t participate (directly) in the ordinary strong and electromagnetic forces.

23. Note that the unknown dark matter occupies 25% of the current Universe while the visible ordinary matter 5%. 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.

24. First thought • We could adopt “Dirac similarity principle” and the “minimum Higgs hypothesis” as our working rules, when we use the extended Standard Model to describe the dark-matter and ordinary-matter particles. • If the gauge group, SU_c(3) x SU_L(2) x U(1) x G, is fixed, the two working rules guarantee the uniqueness of the model.

25. We haven’t seen Higgs after 40 years !! • Years ago (in 1987), I tried to add Z’ and realized immediately that we have to add additional Higgs multiplet(s), too. • How to add a Z’ but with a minimum number of Higgs fields?References: W-Y. P. Hwang, Phys. Rev. D36, 261 (1987). • Crucial to have the mass-generation mechanism spelled-out.

26. On the mass generation lambda’ ~ lambda x (vev / vev’)**2 The 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. • 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. !

27. “The Minimum Higgs Hypothesis” No.1. On the coupling strengths. lambda’ ~ lambda x (vev / vev’)**2 My conjecture for the couplings to remote Higgs No. 2. On the choice of Higgs multiplets There is no redundent Higgs multiplet.. • It is a useful “empirical” rule.

28. 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). • Mohapatra, Pati, and Salam in fact have many models (by choice of Higgs multiplets) but the “minimum Higgs hypothesis” selects the unique one. • The missing left-right symmetry should be understood some day.

29. More on the left-right symmetry • Why the weak interactions break the left-right symmetry is one of the deepest questions. Are these stuffs in the 25% dark matter? • Mass generation: (by the image of the left) lambda (v/v’)**2 varphi* nu_L (nu_R, e_R) • Again, it is renormalizable.

30. In fact, we could talk about three unique 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. W-Y. P. Hwang, Intern. J. Mod. Phys. A24, 3366 (2009).

31. Three or two unique options: SU_c(3) × SU(2) × U(1) × G • As the group G is fixed, the extended standard model is fixed. Thanks to “Dirac similarity principle” and “minimum Higgs hypothesis”!! • We should pay more attention to the Mohapatra-Pati-Salam left-right symmetric model and the family gauge theory – because of the symmetry reasons.

32. Maybe we have to “rethink” what we are doing – trying to set our tone. • We have so much of dark matter (25 % of the current Universe) --- the “final” theory, if the SU_c(3) × SU_L(2) × U(1) × SU_f(3) x SU_R(2) extended Standard Model would describe our ordinary-matter and dark-matter world, would still be “minimal”. • We have already seen partially the SU_f(3) part but we still don’t have the clue about the missing SU_R(2) part.

33. The extended Standard Model is naively renormalizable. • The extended Standard Model, based on the group SU_c(3) × SU(2) × U(1) × SU_f(3) x SU_R(2), only requires the presence of point-like Dirac particles (“Dirac similarity principle”) and the minimum presence of Higgs particles (“minimum Higgs hypothesis”). • We may “say” that the left-handed neutrinos belongs to the ordinary-matter world while (nu_tau, nu_mu, nu_e)_(right-handed) belongs to the dark-matter world.

34. In what follows, we explain briefly 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) standard model. SU_f(3) defines the body of the dark matter. • The only SU_f(3) triplet from the ordinary-matter world, (nu_tau, nu_mu, nu_e)_(right-handed), or just (nu_tau, nu_mu, nu_e). Question: Why do we have three generations?

35. 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. • Now the simple extension is that based on SU_c(3) × SU(2) × U(1) × SU_f(3). • The rest is straightforward.

36. There are 8 gauge bosons: Denote the eight family gauge fields (familons) as F_\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_\nu^c. Then we have[4]One way to describe the nonabelian nature of the gauge theory is to add the Fadde’ev-Popov ghost fieldswith D_\mu \phi^a ≡ \partial_\mu \phi^a + \kappa f_{abc}F_\mu^b \phi^c.

37. The neutrino triplet \Psi(x) iswith D_\mu ≡ \partial_\mu － i {\kappa\over 2} \lambda^a F_\mu^a(x). Just like a (triplet) Dirac field. • The family Higgs mechanism is accomplished by a pair of complex scalar triplets. Under SU_f(3), they transform into the specific forms in the U-gauge:

38. We could work out the kinetic terms:such that, by means of choosing,we find, for the familons,

39. That is, the eight gauge bosons all become massive. On the other hand, by choosingwe find that the remaining four (Higgs) particles are massive (with \mu^2 < 0, we have v^2 = －\mu^2 / \lambda > 0). • Because the neutrino-neutrino-Z vertex is now in our theory augmented by the neutrino-neutrino-“dark boson” vertices; these dark species should be very massive.

40. In the SU_f(3) model, the couplings to ordinary matter is only through the neutrinos. • This would make some loop diagrams, involving neutrinos and familons, very interesting and, albeit likely to be small, should eventually be investigated[6]. For example, in the elastic quark (or charged lepton) - neutrino scattering, the loop corrections would involve the Z^0 and in addition the familon loops and if the masses of the familons were less than that of Z^0 then the loop corrections due to familons would be bigger. Thus, we may assume that the familon masses would be greater than the Z^0 mass, say ≧ 1 TeV.

41. In other words, there are loop corrections involving familons and other dark-matter particles, which should be suppressed to protect the validity of the minimal Standard Model. So, the masses of familons and family Higgs have to be greater that a certain value, such as 1 TeV.

42. Neutrino mass generation is through the coupling between the neutrino triplet and the family Higgs triplets:resulting a mass matrix which is off diagonal (but is perfectly acceptable). in a form similar to the Zee matrix[5], it can easily be fitted to the observed data[2]. • In other words, the “origin” of the tiny neutrino masses comes from the family Higgs and from the loop of family gauge bosons. It is different from those for quarks and charged leptons, a nice way to escape the theorem mentioned earlier[3]. • The tiny neutrino masses are generated in the dark sector, and in a renormalizable way. This is a very interesting solution.

43. SU_f(3) so similar to SU_c(3): 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). • In the early universe, the temperature could be as high as that for the familons such that the Universe could be populated with these massive dark-matter particles – giving rise to the so-called “seeded clusterings”.

44. Initial 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. • 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, and Pei-Ming Ho for useful conversations, but the errors remain to be mine.

45. An important conclusion: • So, under “Dirac similarity principle” and the “minimum Higgs hypothesis”, we could work on “three” extended Standard Models – the family gauge theory, the left-right symmetric model and the extra Z’ (in our order), all in a unique version. All being renormalizable. • 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 Model.

46. Clusterings • “Unseeded” gravitational clusterings supposed to happen in a long time, in a time much longer than the age of the Universe. • So, in a time span of the young Universe, neutrino masses contribute very little and we need the seeds (for the clustering) to catalyze the processes.

47. The seeded clusterings • From hadrons, atoms, molecules, complex molecules, and chunks of the matter, to begin clustering, in a time span of 1 Gyr, it grows into our present visible Universe – the so-called “seeded clusterings”. • Similar things could happen if we have SU_f(3) family gauge theory with a normal coupling. So, there might be dark-matter galaxies.

48. Conclusions • “Dirac similarity principle” and the “minimum Higgs hypothesis” allows us to work on the SU_c(3) x SU_L(2) x U(1) x SU_f(3) x SU_R(2) extended Standard Model. It is “naively” renormalizable. • The “seeded” clustering might exist in the dark-matter world if SU_f(3) is indeed there. => There are dark-matter galaxies.