Gauge Symmetry and More …. Another Solution to Dark Matter?

1. Gauge Symmetry and More …. Another Solution to Dark Matter? W-Y. Pauchy Hwang Institute of Astrophysics National Taiwan University

2. Outline • Introduction • A Proposal: Extension of Standard Model by a Gauge Theory – the SU_f(3) Family Gauge Theory • Discussions • References

3. Introduction • More than twenty years ago I was curious by the absence of the Higgs mechanism in the strong interactions but not in the weak interaction sector[1] – a question still remains unanswered till today. A renormalizable gauge theory that does not have to be massless is already reputed by ‘t Hooft and others, for the standard model. Maybe our question should be whether the electromagnetism would be massless. • In fact, this is a deep question – how to write down a renormalizable theory. During old days, a massive gauge theory is used to be believed as a nonrenormalizable theory.

4. Another clue comes from neutrinos – they are neutral, massive and mixing/oscillating. These particles are barely “visible” in the Particle Table. Maybe these are avenues that connect to those unknowns, particularly the dark matter in the Universe. • In fact, the neutrino sector, with the current knowledge of masses and mixings[2], presents a serious basic problem[3] – that is, a theorem that neutrinos are massless in the minimal standard model. Any model with at least one massive neutrino have to be some sort of extended standard model (i.e. not minimal). • In the sector of ordinary matter, the only neutral fermions are neutrinos.

5. Key References • W-Y. P. Hwang, Phys. Rev. D32 (1985) 824; on the “colored Higgs mechanism”. • Particle Data Group. • Stuart Raby and Richard Slansky, Los Alamos Science, No. 25 (1997) 64. • Notations. • A. Zee, Phys. Lett. B93 (1980) 389; Phys. Lett. B161 (1985) 141; Nucl. Phys. B264 (1986) 99; on the Zee model. • Ling-Fong Li. .

6. In this talk, I propose that we may add an SU(3) family gauge theory - the SU_c(3) × SU(2) × U(1) × SU_f(3) standard model. 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). • One motivation is that in our Universe there is 25% dark matter vs 5% ordinary matter – dark matter even though we cannot see but it may be insufficient to view it as being “peripheral” only. In the early universe, the temperature could be as high as that for the familons such that the Universe could be populated with these (self-interacting) particles - just like that for QCD. The Universe would be full of these particles as the dark matter. • Maybe (nu_e, nu_mu, nu_tau) could serve as the only “bridge” for ordinary matter. Why we have three generations? – we provide a “partial” solution to the “family” problem.

7. Note that I take a rather conservative position: If we don’t see the grand unification or supersymmetry, we try to think about “why not”, rather than proceeding to speculate. • So, in our Universe there is 25% dark matter vs 5% ordinary matter – this is taken as a fact to understand. • If there is some connection between the “ordinary matter” sector and the dark matter one, the neutrinos (nu_e, nu_mu, nu_tau) could be as the only candidate. • Note that I have provided only a “partial” solution to the “family” problem – depending how you define it.

8. For this SU_f(3) gauge theory, the triplet (\nu_\tau, \nu_\mu, \nu_e) serves as a basic connection. Note the up-side-down position used here because of the way I wrote the positions of the nonzero vacuum expectation values. • For the SU_f(3) gauge theory, should it exist and we suppose that it is coupled to neutral fermions, i.e. neutrinos. The concept of “family” could not mingle with the other quantum numbers, I.e. commuting with the others.

9. Under SU_c(3) × SU(2) × U(1) × SU_f(3), the neutrino family triplet acts like (1, 1, 0, 3) and all the others, including quarks, charged leptons, and ordinary gauge bosons, are assumed to be family singlets (i.e. no coupling to the family sector). • I assume that the whole Dirac neutrinos could be used in the neutrino triplet. • If only the right-handed neutrinos are used in this context, the dark sector would be completely dark, making the story a little boring.

10. The masses of the neutrino triplet come from the coupling to some Higgs field - a pair of complex scalar triplets, as worked out in the previous publication[1]. Hereafter I ignore the “radiative” corrections due to gauge bosons. In this case, the eight components of the Higgs triplets are absorbed by the eight gauge fields through the “family” Higgs mechanism via spontaneous symmetry breaking, while the remaining four become massive Higgs particles. (In the previous application, it was referred to as “colored Higgs Mechanism”[1].) • The neutrino masses do not come from the minimal Standard Model, but from the Higgs in the dark sector.

11. A Proposal • If we think of the role of gauge theories in quantum field theory, we still have to recognize its unique and important role. If the standard model is missing something, a gauge theory sector would be one at the first guess. On the other hand, in the standard model there are three generations of quarks and leptons. But why? It seems to be a first loose point for the standard model. So, let’s assume that there is an SU_f(3) gauge theory associated with the story. • I believe that something missing may be a gauge sector, owing to the successes of SU_c(3) × SU(2) × U(1) standard model.

12. In fact, an octet of gauge bosons plus a pair of complex scalar triplets turns out to be the simplest choice as long as all gauge bosons become massive while the remaining Higgs are also massive. • This is the basic framework. The standard model is the gauge theory based on the group SU_c(3) × SU(2) × U(1). Now the simple extension is that based on SU_c(3) × SU(2) × U(1) × SU_f(3). • So, the following story is rather simple.

13. We may write our “new” basic elements as follows. 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.

14. 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:

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

16. 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.

17. In the model, the couplings to ordinary matter is only through the neutrinos, the only charged/ neutral fermions that are interacting weakly. This would make some loop diagrams, involving neutrinos and familons, very interesting and, albeit likely to be small, should eventually be 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.

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

19. The other important point is the coupling between the neutrino triplet and the family Higgs triplets:resulting a mass matrix which is off diagonal (but is perfectly acceptable). In other words, the mass matrix, being proportional to －\bar{\nu}_e(v_{+} + \epsilon v_－)\nu_\tau + \bar{\nu}_e(u_{+} + \epsilon u_－)\nu_\mu + \bar{\nu}_\tau (v_{+} + \epsilon v_－)\nu_e －\bar{\nu}_\mu(u_{+} + \epsilon u_－)\nu_e , is off-diagonal, in the form similar to the Zee matrix[5], and can easily be fitted to the observed data[2]. (And i is needed to make it hermitian.) • In other words, the “source” of the neutrino masses comes from the family Higgs and is different from those for quarks and charged leptons, a nice way to escape the theorem mentioned earlier[3]. • The neutrino masses are obtained from the dark sector, but in a renormalizable way. This is a very interesting solution.

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

21. Discussions • Our life during the next stage seems to be rather difficult. Neutrinos, albeit abundant, are very elusive. We use neutrinos as the basic bridge to construct the SU_f(3) gauge theory for the family in the building blocks of matter. If the only coupling has to go through neutrinos, then the detection (from the visible side of the matter) would be extremely difficult. Of course, we should look for the potential couplings to other sectors such as quarks or charge leptons. • Well, the neutrino mass problem, if coupled with the renormalizability requirement, leads to our solution.

22. One important consequence of the SU_c(3) × SU(2) × U(1) × SU_f(3) standard model is that in addition to QCD and electroweak (EW) phase transitions there is other SU_f(3) family phase transition occurring near the familon masses, maybe above the EW scale (that is, above 1 TeV). The exact scale is hard to decide, for the moment. • In the early universe, the temperature could be as high as that for the familons such that the Universe could be populated with these (self-interacting) particles - just like that for QCD. In other words, our Universe would be full of these particles as the dark matter - at this point, it is believed that our Universe has 25% in dark matter while only 5\% in visible matter. • Well, this may solve the 25% dark matter problem.

23. Another Thought • SU_c(3) × SU(2) × U(1) × GG: U(1) or n U(1) the extra Z^0 (bottom-up or up-down strategy) • Important Question: How to add a Z’ but with a minimum number of Higgs fields?References: W-Y. P. Hwang, Phys. Rev. D36, 261 (1987); W-Y. P. Hwang, D38, 3427 (1988). • The present paper, with G = SU_f(3), brings out another important generalization, solving the neutrino mass problem and perhaps the dark matter issue simultaneously. • I think that renormalizability, even though we really don’t understand it, does provide a useful guide.

24. Another Discussion • It is used to think that, to start with the E_8 group, we have SU(3) x E_6. E_6 breaks down to U(1) x SO(10), and then to U(1) x U(1) x SU(5), and to our standard model. • What is wrong with this line of thinking? • Maybe we have to introduce gauge bosons and Higgs particles to make sure that no unwanted particles exist – an argument against “arbitrary” grand unified theories.

25. References • W-Y. P. Hwang, Phys. Rev. D32 (1985) 824; on the “colored Higgs mechanism”. • Particle Data Group, “Review of Particle Physics”, J. Phys. G: Nucl. Part. Phys. 33 (2006) 1; on neutrino mass and mixing, see pp. 156 - 164. • For example, see Stuart Raby and Richard Slansky, Los Alamos Science, No. 25 (1997) 64. • For notations, see T-Y. Wu and W-Y. Pauchy Hwang, Relativistic Quantum Mechanics and Quantum Fields (World Scientific, Singapore, 1991). • A. Zee, Phys. Lett. B93 (1980) 389; Phys. Lett. B161 (1985) 141; Nucl. Phys. B264 (1986) 99; on the Zee model. • Ling-Fong Li, private communications.