The Mystery of Flavor

After outlining some of the issues surrounding the flavor problem, I present three speculative ideas on the origin of families. In turn, families are conjectured to arise from an underlying preon dynamics; from random dynamics at very short distances; or as a result of compactification in higher dimensional theories. Examples and limitations of each of these speculative scenarios are discussed. The twin roles that family symmetries and GUTs can have on the spectrum of quarks and leptons is emphasized, along with the dominant role that the top mass is likely to play in the dynamics of mass generation. I THE QUESTION OF FLAVOR Flavor is an old problem. I. I. Rabi’s famous question about the muon: “who ordered that?” has now been replaced by an equally difficult question to answer: “why do we have three families of quarks and leptons?” Although qualitatively we understand the issues connected to flavor a lot better now, quantitatively we are as puzzled as when the muon was discovered. When thinking of flavor, it is useful to consider the standard model Lagrangian in a sequence of steps. At the roughest level, neglecting both gauge and Yukawa interactions, the Standard Model Lagrangian Lo = LSM(gi = 0; Γij = 0) has a U(48) global symmetry corresponding to the freedom of being able to interchange any of the 16 fermions of the 3 families of quarks and leptons with one another. If we turn on the gauge interactions, the Lagrangian L1 = LSM(gi 6= 0; Γij = 0) has a much more restricted symmetry [U(3))] corresponding to interchanging fermions of a given type (e.g. the (u, d)L doublet) from one family to the other. When also the Yukawa interactions are turned on, L2 = LSM(gi 6= 0; Γij 6= 0), then the only remaining symmetry of the Lagrangian is U(1)B × U(1)L. In fact, because of the chiral anomaly [1], at the quantum level the symmetry of L2 is just U(1)B−L. The above classification scheme serves to emphasize that there are really three distinct flavor problems. There is a matter problem , a family problem and a mass problem. The first of these problems is simply that of understanding the origin of the different species of quarks and leptons (i.e. why does one have a ν L and a uL state?). The second problem is related to the triplication of the quarks and leptons. What physics forces such a triplication? Finally, the last problem is related to understanding the origin of the observed peculiar mass pattern of the known fermions. The usual approach when thinking about flavor is to try to decouple the above three problems from one another. Thus, for example, one assumes the existence of the quarks and leptons in the Standard Model and asks for the physics behind the replication of families. Although it is difficult to argue cogently on this point, it is certainly true in the examples which we will discuss that the matter problem seems to be unrelated to the question of family replication. Indeed, quite often one also assumes the reverse, namely, that the family replication question is independent of the types of quarks and leptons one has. In fact, it is possible that there is other matter besides the known quarks and leptons and that this matter is also replicated. Certainly, even in the minimal Standard Model there is other matter besides the quarks and leptons, connected to the symmetry breaking sector. This raises a host of questions including that of possible family replication of the ordinary Higgs doublet. One knows, empirically, that this cannot happen if one is to avoid flavor changing neutral currents (FCNC) [2]. However, some replication is needed if there is supersymmetry, but the two different Higgs doublets needed in supersymmetry are connected with different quark charges and need not replicate as families. The above remarks suggests that there are some perils associated with trying to seek the origin for family replication independently from that of the quarks and leptons themselves. Nevertheless, that is the approach usually taken and the one I will follow here. Similarly, one also usually tries to disconnect the problem of mass from that of matter and family. That is, one generally assumes the existence of the three observed families of quarks and leptons, and then tries to postulate (approximate) symmetries of the mass matrices for quarks and leptons which will give interrelations among the masses and mixing parameters for some of these states. This approach usually involves some kind of family symmetry and is sensible provided that: i) There is some misalignment between the mass matrix basis and the gauge interaction basis for the quarks and leptons. Only through such a misalignment will there result a nontrivial mixing matrix: VCKM 6= 1. ii) The family symmetries of the mass matrices are broken (otherwise VCKM ≡ 1) either explicitly or spontaneously. Furthermore, if the breaking is spontaneous, it must occur at a sufficiently high scale to have escaped detection so far. Although the origin of flavor remains a mystery, I want to discuss here three speculative ideas for the origin of families. These ideas are realized up to now only in incomplete ways, in what amount essentially to toy models. Thus, for instance, the issue of family generation is in general disconnected from the question of SU(2) × U(1) breaking and, often, also from trying to explicitly calculate the Yukawa couplings. As a result, in all of these attempts at trying to understand flavor, the question of mass is approached from a much more phenomenological viewpoint. One guesses certain family or GUT symmetries, and their possible patterns of breaking, and then one checks out these guesses by testing their predictions experimentally. In all of these considerations, the top mass, because it is the dominant mass in the spectrum, plays a fundamental role. In my lectures [3], I will begin by describing three speculative ideas for the origin of families. Specifically, I will consider in turn the generation of families dynamically; through short distance chaotic dynamics; and as a result of geometry. After this speculative tour, I will discuss briefly the issue of mass generation. In particular, I will illustrate the twin roles that family symmetries and GUTs can have for the spectrum of quarks and leptons. I will conclude by commenting on the profound role that the top mass is likely to have on the detailed dynamics of mass generation. II GENERATING FAMILIES DYNAMICALLY The underlying idea behind this approach to the flavor problem is that familiies of quarks and leptons result because they are themselves composites of yet more fundamental ingredients–preons. There is a nice isotope analogy [4] which serves to illustrate this point. Think of the three isotopes of Hydrogen as three distinct families. Just like the families of quarks and leptons, all three isotopes have the same interactions–their chemistry being determined by the electromagnetic interactions of the proton. Deuterium and tritium, however, have different masses than the proton because they have, respectively, 1 and 2 neutrons. Of course, the analogy is not perfect since H and H are fermions and H is a boson! Nevertheless, it is tempting to suppose that the 3 families of quarks and leptons, just like the Hydrogen isotopes, result from the presence of different “neutral” constituents. I will illustrate how to generate families dynamically by using as an example some recent work of Kaplan, Lepeintre and Schmaltz [4]. By using essentially the isotope analogy, these authors constructed an interesting toy model of flavor. Their simplest toy model is based on an underlying supersymmetric gauge theory based on the symplectic group Sp(6). The fundamental constituents in this model are 6 preons Qα transforming according to the fundamental representation of Sp(6) and one preon Aαβ transforming according to the 2-rank antisymmetric representation. Such a theory has three families of bound states distinguished by their Aαβ content, plus a pair of (neutral) exotic states. To wit, the bound states of the model are the 15 flavor states F [i,j] 3 ∼ Q i αQ j α; F [i,j] 2 ∼ Q i αAαβQ j β ; F [i,j] 1 ∼ Q i αAαβAβγQ j γ (1) plus the two neutral exotic states T2 ∼ TrA 2 ; T3 ∼ TrA 3 . (2) The six Qα preons act as the protons in the isotope analogy. In principle, one could imagine having the SU(3)× SU(2)×U(1) interactions act on the Qα states, while the Aαβ preons act as the neutrons. Furthermore, there is clearly a family U(1)F in the spectrum which counts the number of Aαβ fields. Finally, one should note that, because of the supersymmetry, each of the states in Eqs. (1) and (2) contain both fermions and bosons. Although the number of bound states per family (15) is encouraging, these states cannot really be the ordinary quarks and leptons (minus the right-handed neutrinos). It turns out that one cannot properly incorporate the SU(3)×SU(2)×U(1) gauge interactions with only 6 Qα preons. To do that, in fact, one has to at least triplicate the underlying gauge theory [4] from Sp(6) to Sp(6)L×Sp(6)R×Sp(6)H. Each of these Sp(6) groups has again six Qα and one Aαβ preon. To obtain the desired quarks and leptons the Q preons are assumed to have the following SU(3)× SU(2)× U(1) assignments: QL : (3, 1)0 ⊕ (1, 2)1/6 ⊕ (1, 1)−1/3 QR : (3̄, 1)0 ⊕ (1, 1)−2/3 ⊕ 2(1, 1)1/3 QH : (1, 2)−1/6 ⊕ (1, 1)1/3 ⊕ (1, 1)2/3 ⊕ 2(1, 1)−1/3 (3) Because of the preon group triplication, instead of having 15 F [i,j] bound states per family, one now has 45 such states. Per family, these states now include 16 states with the quantum numbers of the observed quarks and leptons, plus 29 exotic states which, however, sit in vector-like representations of the Standard Model group. Specifically, the quark doublet (u, d)L is a bound state of Sp(6)L; u c L and d c L are bound states of Sp(6)R; while the lepton states (ν, e)L, ν c L and e c L are bound states of Sp(6)H. Among the exotic states one finds as bound states of Sp(6)H two states with the quantum numbers of the Higgs doublets of a supersymmetric theory: H1 ∼ (h o 1, h − 1 ) and H2 ∼ (h + 2 , h o 2). So, in this model, there is a natural family repetition of the Higgs states. Naively, this could cause problems with FCNC. It turns out, however, that when one calculates the dynamical superpotential of the theory [5] one can show [4] that there is a ground state where only one of the three families of Higgs states are left light. So, in fact, there are no FCNC problems. This nice result is tempered by other troublesome features of the model which render it unrealistic–but not uninteresting. For example, to break the [UF(1)] 3 family symmetry of the model, it is necessary to introduce by hand some heavy fields (with masses μ > Λ–the dynamical scale of the preon theories) which serve to couple the preon groups together. The simplest possibility is afforded by having 3 such fields: v αHβR, v 2 αRβL , v αLβH with indices spanning 2 of the preon groups, interacting through a superpotential W = avvv + bHv vAH + b 1 Rv vAR + b 2 Rv vAR + b 2 Lv vAL + bLv vAL + b 3 Hv vAH (4) The a-term above ties the preon theories together, while the various b-terms serve to break the family symmetries. Although Eq. (4) is introduced by hand, integrating out the effects of the heavy v fields gives effective Yukawa couplings of different strengths, much in the way originally suggested by Froggatt and Nielsen [6]. This is illustrated schematically in Fig. 1 for the Yukawa coupling of uL with (c, s)L via the Higgs state H (3) 2 of the third family–which is the only one which is assumed to get a VEV. One finds [4] Γ (3) 12 ∼ a b 1 Rb 2 Rb 3 L(Λ/μ) 6 ∼ ǫ (5) Although the various elements in the up-and-down quark mass matrices are hierarchial, unfortunately there is no resulting quark mixing since Mu ∼ Md. This follows because the model has an unbroken global SU(2) symmetry at the preon level corresponding to the interchange of the (1, 1)−2/3 and (1, 1)1/3 assignments in Eq. (3). Furthermore, for the lepton sectors there is a dynamically generated set of Yukawa couplings [5] which are typically unsuppressed. As a result, naively, one expects mτ ≫ mt. Both of these results make the [Sp(6)] 3 model as presented above unrealistic. By further complicating the model, Kaplan, Lepeintre and Schmaltz [4] are able to obtain both a non-trivial CKM matrix and re-establish the top as the heaviest bound state. However, these “improved” models are not particularly attractive and represent, more than anything else, a “proof of principle”. In addition, even after these problems are resolved, the models still lack mechanisms for breaking SU(2)×U(1) and supersymmetry, features which must be understood to make contact with reality. These negative remarks should not obscure the considerable achievement of these dynamical models for understanding the origin of flavor. Families in these models 1) In the model [4] the lightest family has the most Aαβ fields–c.f. Eq. (1). FIGURE 1. Effective Yukawa coupling generated in the [Sp(6)] preon model. arise as a result of hidden degrees of freedom in some underlying confining dynamics. Furthermore, the presence of heavy excitations in this same dynamics can result in hierarchial patterns of Yukawa couplings, once all family symmetries are explicitly broken. Unfortunately, it is difficult to see how one can obtain real evidence for these kinds of schemes, barring the discovery of some of the exotic bound states they predict–in the example discussed, the T2 and T3 states or the vector-like partners of the quarks and leptons. III FAMILIES FROM SHORT-DISTANCE RANDOM DYNAMICS A radically different scheme for the origin of families has been proposed and elaborated by Holgar Nielsen and his collaborators [7]. The basic idea that Nielsen has put forth is that there exist both order and chaos at very short distances. He imagines that at scales much smaller than the inverse of the Planck mass there is actually a lattice structure of scale length a ≪ 1/MPlanck. However, both the dynamics on the lattice as well as the structure of the lattice is random. In particular, the lattice is amorphous with sites at random positions. Furthermore, characteristic of the random dynamics, the interactions on each of the links are governed by different groups, with the groups varying from link to link. Remarkably, even starting from these very general assumptions, one can arrive at some conclusions. Generally, one naively would imagine that no group could survive the random dynamics. That is, that the gauge group will end up by breaking down spontaneously, producing supermassive fields of mass M ∼ a−1 ≫ MPlanck. In fact, as Brene and Nielsen [8] showed, there are special groups Gsurv. on the links which survive the random dynamics–i.e., the associated vector bosons are massless. What Brene and Nielsen [8] showed is that the groups which survive must have a center which is non-trivial and connected. By taking values in the center the links are effectively gauge-invariant. However, the center cannot be simply the unit matrix because the random nature of the dynamics would then end up by averaging out the effects of all links. The connectedness of the center, finally, is necessary to insure that the Bianchi identities are satisfied. Specifically, it turns out that Gsurv. is a product of “prime” groups with a certain discrete group Dprime, generated from the center, removed: Gsurv. = U(1)× SU(2)× SU(3)× SU(5)× . . . SU(prime)/Dprime (6) From the above, it appears that Nielsen’s random dynamics allows the Standard Model group to survive, with a restriction: GSM = SU(3)× SU(2)× U(1)/D3 . (7) Here the discrete group D3 is given by powers of the center element h =