The (extended) Standard Model

1. The (extended) Standard Model W-Y. Pauchy Hwang Y.T. Lee Outstanding-Scholar Chair Professor Institute of Astrophysics National Taiwan University

2. My early struggles with the Standard Model • Why do we need Higgs mechanism in SU_L(2) x U(1) electroweak theory but not in SU_c(3) [QCD]? • The toilets in a house – Glashow’s remark at Indiana (early 1980). • A useless paper and rejected (Phys. Lett. B) by H. Georgi:W-Y. P. Hwang, Phys. Rev. D32, 824 (1985). • The idea of “colored Higgs mechanism”.

3. What is the final Standard Model? • Why do we have three generations ? It cannot be “final” without understanding it. • Neutrino oscillations – from one family into another. “Not final” if without understanding it. • Very excited when I conceived (2008 summer night, at U. Penn) why I can’t use the colored Higgs mechanism as the family one.

4. I believe that the God wouldn’t create a particle that is so boring in just knowing only weak interactions. • Neutrinos now have tiny masses these days. So, they, naturally, have 4-components in the Dirac space. Note that in the minimal Standard Model their masses were suspected to vanish. • There is 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 minimum Standard Model. Why?

5. As time goes by, the strange role of neutrinos has become clearer and clearer. • Note that right-handed neutrinos never appear in the construction of the Minimal Standard Model, as though they are unwanted. • The mystery may lie with the neutrinos, which oscillate between different generations and play dual roles between the ordinary-matter world and the dark-matter world.

6. Eleven Science Questions for the New Century: The First Four QuestionsU.S. Science Academy Report, 4/17/2002 • Q1: What is the dark matter? Our Universe has 25% in Dark Matter while only 5% in ordinary matter. The dark-matter particles also tend to cluster – another obstacle to analyze the problem. • Q2: What is the nature of the dark energy? (The overwhelming 70% question, but maybe with uniformly “distributed”.) • Q3: How did the universe begin? • Q4: Did Einstein have the last word on gravity?

7. Cosmology: Eleven Science Questions for the New CenturyU.S. Academy Report, 4/17/2002 • Q1: What is the dark matter? • Q2: What is the nature of the dark energy? • Remark 1: To push our understandings of the Cosmos, I think that question No.1 is far more nontrivial than question No.2. • Remark 2: We don’t even know the dark-matter world – what particles are they?

8. Neutrinos do oscillate !! • Neutrinos oscillate. nu_e <=> nu_mu; nu_mu <=> nu_tau; nu_tau <=> nu_e. • This is occurring at the level of point-like Dirac particles. It differs from the K^0 - \bar K^0 system, a composite system. • Note that neutrino oscillations “move around” different generations, or, changing flavors.

9. In view of the minimal Standard Model, we could write down two working “rules”: • Dirac similarity principle – our struggle of eighty years to describe the point-like Dirac particle such as the electron. • The “minimum Higgs hypothesis” is the other mysterious conjecture – after forty years we finally get glimpse over the SM Higgs particle, and nothing more. • By “induction”, we may write down these two working rules for the much “larger” dark matter world.

10. 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, the size certainly less than 10**(-20) cm in length, these days. • It turns out that, for over eighty years, we recognize only a few point-like particles, those building blocks of the Standard Model. • These “point-like” Dirac particles are described by “quantized” Dirac fields.

11. Finally some Higgs after searching for 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. • Maybe we should work with “the minimum Higgs hypothesis” or “conjecture”.

12. The point is: Without the two working rules, we have too many choices on the extended SM. • Maybe our space-time only allows for “point-like” Dirac particles, those can be created and can communicate among them. • Besides, we use the scalar fields to “modulate” quite a number of things, SSB (the Higgs mechanisms), etc. Their existences appear to be “minimal”.

13. Outline • Language: (Renormalizable) Quantum Fields. (a much powerful than you thought) • No. 1 Question: What is the Dark Matter? • Dirac Similarity Principle • “Minimum Higgs Hypothesis” • My struggles with the Standard Model. • Conclusion

14. The Language: The Axiom Box 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

15. Fields as the generalized coordinates (quantum mechanics of the continuum = quantum fields): • 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.

16. 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 (= Quantum Mechanics for the Continuum): “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).

17. Remarks on the above axioms for QFT: (1) no second quantization (2) no infinite electron sea • This is so far a consistent language framework. Complete ? Too early to tell.

18. 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. If not, what else? • In what follows, we try to sell the SU_c(3) x SU(2) x U(1) x SU_f(3) Standard Model via a renormalizable way.

19. My Struggles with the Standard Model: Part No.1 • How to add a Z’ but with a minimum number of Higgs fields?W-Y. P. Hwang, Phys. Rev. D36, 261 (1987). • When we go beyond the Standard Model, the Higgs sector depends on the sector of gauge bosons – extra Z, then extra Higgs.

20. Thus, …. • Neutrinos have tiny masses. => another Z’. It may sound strange, but it requires another Higgs. • How to add a Z’ but with a minimum number of Higgs fields?W-Y. P. Hwang, Phys. Rev. D36, 261 (1987). • Consider 2+2 Higgs Scenario. The second, and “remote”, Higgs doublet could give tiny neutrinos masses naturally.

21. “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 redundant Higgs multiplet.. • It is a useful “empirical” rule.

22. Another Thought • SU_c(3) × SU_L(2) ×SU_R(2) x U(1) : The missing right-handed sector !! R.N. Mohapatra and J.C. Pati, Phys. Rev. D11, 2558 (1975). • Here we also have an extra Z’ but with another right-handed doublet almost eaten up via SSB. • Mohapatra, Pati, and Salam in fact have many models (by choice of Higgs multiplets) but the “minimum Higgs hypothesis” selects the unique one – unfortunately, it may violate the criterion of choosing “the basic units”.

23. The thought of Dec 2012 • Why the weak interactions break the left-right symmetry is one of the deepest questions. • Reason: When we write the Dirac theory in terms of the left-right basic units, each unit appears once and only once – to ensure that there is one kinetic-energy term. • W-Y. Pauchy Hwang and Tung-Mow Yan, arXiv:1212.4944 [hep-ph] 20 Dec 2012.

24. My Struggles with the Standard Model: Part No. 2 • Why should we have the standard case, i.e., Higgs in the electroweak sector but not in the strong QCD case.W-Y. P. Hwang, Phys. Rev. D32, 824 (1985). • The idea of “colored Higgs mechanism”.

25. Higgs mechanism in QCD? • Almost thirty years ago I was curious by the absence of the Higgs mechanism in the strong interactions but not in the weak interaction sector – 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. • It is a deep question – how to write down a renormalizable theory. Remember that, during old days, a massive gauge theory is used to be believed as a nonrenormalizable theory.

26. No. 3: I started thinking that my miserable life has some meaning. • In 2008 (four years after I was struck by cerebral haemorrhage), I went to U. Penn (my Alma Mater) to attend Lepton-Photon Conference. I woke up one night to ask why the idea of colored Higgs mechanism be “copied” as the family SU(3) gauge theory. • That sets off a series of talks and papers – maybe nobody appreciate the idea.

27. My Struggles with the Standard Model: Part No. 3 • W-Y. Pauchy Hwang, Nucl. Phys. A844, 40c (2012); W-Y. Pauchy Hwang, International J. Mod. Phys. A24, 3366 (2009); W-Y. Pauchy Hwang, Intern. J. Mod. Phys. Conf. Series 1, 5 (2011); W-Y. Pauchy Hwang, AIP 978-0-7354-0687-2/09, pp. 25-30 (2009).

28. In SU(3), 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. • 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) multiplied another independent SU_f(3).

29. 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.

30. 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 too big. Thus, we may assume that the familon masses would be much greater than the Z^0 mass, say ≧ a few TeV.

31. My Struggles with the Standard Model: Part No. 4 • Last June (2012) I went to Groningen to attend SSP2012 and suddenly realized the role of neutrino oscillations in all these.W-Y. Pauchy Hwang, arXiv:1207.6837v1 [hep-ph] 30 Jul 2012; • W-Y. Pauchy Hwang, arXiv:1207.6443v1 [hep-ph] 27 Jul 2012; • W-Y. Pauchy Hwang, arXiv:1209.5488v1 [hep-ph] 25 Sep 2012.

32. So, let’s come back to neutrino oscillations: • The origin of neutrino masses comes from the coupling between the neutrino triplet and the family Higgs triplets:resulting a mass matrix which is off diagonal (but is perfectly acceptable). Note that alpha = i eta, which is needed to make it hermitian. • This turns out to be that it is not a standard matrix operation, but a unique SU(3) operation – the unique singlet out of three given triplets. And from left- and right-handed as usual. • This is the new way to add a renormalizable term, a curl-dot term.

33. So, in the SU(3) family gauge theory, we write down the renormalizable term by using a curl-dot product, a new term indeed. • 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]. • The neutrino masses do not come from the minimal Standard Model, but mainly from the Higgs in the dark sector.

34. At this point, we had to ask two difficult questions: • Why do we need to make the SU(3) family gauge theory massive? • Whether is (nu_tau, nu_mu, nu_e) the only triplet from the ordinary-matter world (as described by the minimal Standard Model)?

35. In the construction of the minimal Standard Model, the right-handed neutrinos do not appear. So, we could construct an independent SU_f(3) gauge theory with the neutrinos as a triplet, to the least. • We should write the extended SU_c(3) x SU_L(2) x U(1) x SU_f(3) theory altogether – each of the basic units has one kinetic-energy term and only one. W-Y. Pauchy Hwang and Tung-Mow Yan, arXiv:1212.4944 [hep-ph] 20 Dec 2012.

36. My Struggles with the Standard Model: Part No. 5 • I invited Tung-Mow Yan to lecture on the Standard Model last Fall. On one day, it was clear to us that we could put the three lepton doublets, six of them altogether, as an SU_f(3) triplet. • W-Y. Pauchy Hwang and Tung-Mow Yan, arXiv:1212.4944v1 [hep-ph] 20 Dec 2012.

37. My Struggles with the Standard Model: Part No. 6 • Finally, I worked out the “combined” Higgs mechanisms – using the scalar/Higgs fields Phi(3,2), Phi(3,1), and the standard Phi(1,2); it is like a magic. • W-Y. Pauchy Hwang, arXiv:1304.4705v1 [hep-ph] 17 April 2013.

38. After Thoughts • I think that the SU_f(3), a GeV or sub-Fermi family gauge theory, is meant to protect the lepton world from the QED Landau ghost and it is asymptotically free. • So, this Standard Model has SU(3) everywhere. Indeed, it looks like a magic. • W-Y. Pauchy Hwang, arXiv:1301.6464v5 [hep-ph] 29 April 2014.

39. Ultraviolet divergences Fig. 1: The cancelations for all quadratic and logarithmic divergences.

40. Fig. 2

41. Fig. 3

42. Fig. 4: Cancelations of divergences essential to make a theory a true theory.

43. What is the “complete” theory? • I believe that ultraviolet divergences could be there for some diagrams, owing to uncountable infinite DOF’s but altogether they should be canceled out. • We should work on the SU_c(3) x SU_L(2) x U(1) x SU_f(3) Standard Model, as a complete theory.

44. Conclusion • Ifinally arrived at the Standard Model that have SU_f(3) to protect the lepton world and have neutrino oscillations in a natural way. • This is the SU_c(3) x SU_L(2) x U(1) x SU_f(3) extended Standard Model. It is renormalizable.

45. 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, Nucl. Phys. A844, 40c (2010); Intern. J. Mod. Phys. A24, 3366 (2009); Intern. J. Mod. Phys. Conf. Series 1, 5 (2011). • W-Y. P. Hwang and Tung-Mow Yan, The Universe, 1-1, 5 (2013); arXiv:1212.4944 [hep-ph] 20 Dec 2012. • W-Y. Pauchy Hwang, arXiv:1304.4705 [hep-ph] 17 april 2013. • W-Y. Pauchy Hwang, arXiv:1301.6464v5 [hep-ph] 29 April 2014.