Explaining the bizarre pattern of fermion types and masses has led theorists to explore more complex symmetry groups than the Standard Model’s, with recent work suggesting that the “flavour scale” could be at a much lower energy than previously thought.

The origin of the three families of quarks and leptons and their extreme range of masses is a central mystery of particle physics. According to the Standard Model (SM), quarks and leptons come in complete families that interact identically with the gauge forces, leading to a remarkably successful quantitative theory describing practically all data at the quantum level. The various quark and lepton masses are described by having different interaction strengths with the Higgs doublet (figure 1, left), also leading to quark mixing and charge-parity (CP) violating transitions involving strange, bottom and charm quarks. However, the SM provides no understanding of the bizarre pattern of quark and lepton masses, quark mixing or CP violation.

In 1998 the SM suffered its strongest challenge to date with the decisive discovery of neutrino oscillations resolving the atmospheric neutrino anomaly and the long-standing problem of the low flux of electron neutrinos from the Sun. The observed neutrino oscillations require at least two non-zero but extremely small neutrino masses, around one ten millionth of the electron mass or so, and three sizeable mixing angles. However, since the minimal SM assumes massless neutrinos, the origin and nature of neutrino masses (i.e. whether they are Dirac or Majorana particles, the latter requiring the neutrino and antineutrino to be related by CP conjugation) and mixing is unclear, and many possible SM extensions have been proposed.

The discovery of neutrino mass and mixing makes the flavour puzzle hard to ignore, with the fermion mass hierarchy now spanning at least 12 orders of magnitude, from the neutrino to the top quark. However, it is not only the fermion mass hierarchy that is unsettling. There are now 28 free parameters in a Majorana-extended SM, including a whopping 22 associated with flavour, surely too many for a fundamental theory of nature. To restate Isidor Isaac Rabi’s famous question following the discovery of the muon in 1936: who ordered *all* of that?

**A theory of flavour **

There have been many attempts to formulate a theory beyond the SM that can address the flavour puzzles. Most attempt to enlarge the group structure of the SM describing the strong, weak and electromagnetic gauge forces: SU(3)_{C} × SU(2)_{L} × U(1)_{Y} (see “A taste of flavour in elementary particle physics” panel). The basic premise is that, unlike in the SM, the three families are distinguished by some new quantum numbers associated with a new family or flavour symmetry group, G_{fl}, which is tacked onto the SM gauge group, enlarging the structure to G_{fl} × SU(3)_{C} × SU(2)_{L} × U(1)_{Y}. The earliest ideas dating back to the 1970s include radiative fermion-mass generation, first proposed by Weinberg in 1972, who supposed that some Yukawa couplings might be forbidden at tree level by a flavour symmetry but generated effectively via loop diagrams. Alternatively, the Froggatt–Nielsen (FN) mechanism in 1979 assumed an additional U(1)_{fl} symmetry under which the quarks and leptons carry various charges.

To account for family replication and to address the question of large lepton mixing, theorists have explored a larger non-Abelian family symmetry, SU(3)_{fl}, where the three families are analogous to the three quark colours in quantum chromodynamics (QCD). Many other examples have been proposed based on subgroups of SU(3)_{fl}, including discrete symmetries (figure 2, right). More recently, theorists have considered extra-dimensional models in which the Higgs field is located at a 4D brane, while the fermions are free to roam over the extra dimension, overlapping with the Higgs field in such a way as to result in hierarchical Yukawa couplings. Still other ideas include partial compositeness in which fermions may get hierarchical masses from the mixing between an elementary sector and a composite one. The possibilities are seemingly endless. However, all such theories share one common question: what is the scale, M_{fl}, (or scales) of new physics associated with flavour?

Since experiments at CERN and elsewhere have thoroughly probed the electroweak scale, all we can say for sure is that, unless the new physics is extremely weakly coupled, M_{fl} can be anywhere from the Planck scale (10^{19}^{ }GeV), where gravity becomes important, to the electroweak scale at the mass of the W boson (80 GeV). Thus the flavour scale is very unconstrained.

## A taste of flavour in elementary particle physics

The origin of flavour can be traced back to the discovery of the electron – the first elementary fermion – in 1897. Following the discovery of relativity and quantum mechanics, the electron and the photon became the subject of the most successful theory of all time: quantum electrodynamics (QED). However, the smallness of the electron mass (m_{e} = 0.511 MeV) compared to the mass of an atom has always intrigued physicists.

The mystery of the electron mass was compounded by the discovery in 1936 of the muon with a mass of 207 m_{e} but otherwise seemingly identical properties to the electron. This led Isidor Isaac Rabi to quip “who ordered that?”. Four decades later, an even heavier version of the electron was discovered, the tau lepton, with mass m_{τ} = 17 m_{μ}. Yet the seemingly arbitrary values of the masses of the charged leptons are only part of the story. It soon became clear that hadrons were made from quarks that come in three colour charges mediated by gluons under a SU(3)_{C} gauge theory, quantum chromodynamics (QCD). The up and down quarks of the first family have intrinsic masses m_{u} = 4 m_{e} and m_{d} = 10 m_{e}, accompanied by the charm and strange quarks (m_{c} = 12 m_{μ} and m_{s} = 0.9 m_{μ}) of a second family and the heavyweight top and bottom quarks (m_{t} = 97 m_{τ} and m_{b} = 2.4 m_{τ}) of a third family.

It was also realised that the different quark “flavours”, a term invented by Gell-Mann and Fritzsch, could undergo mixing transitions. For example, at the quark level the radioactive decay of a nucleus is explained by the transformation of a down quark into an up quark plus an electron and an electron antineutrino. Shortly after Pauli hypothesized the neutrino in 1930, Fermi proposed a theory of weak interactions based on a contact interaction between the four fermions, with a coupling strength given by a dimensionful constant G_{F}, whose scale was later identified with the mass of the W boson: G_{F} ∝ 1/m_{W}^{2}.

After decades of painstaking observation, including the discovery of parity violation, whereby only left-handed particles experience the weak interaction, Fermi’s theory of weak interactions and QED were merged into an electroweak theory based on SU(2)_{L} × U(1)_{Y} gauge theory. The left-handed (L) electron and neutrino form a doublet under SU(2)_{L}, while the right-handed electron is a singlet, with the doublet and singlet carrying hypercharge U(1)_{Y }and the pattern repeating for the second and third lepton families. Similarly, the left-handed up and down quarks form doublets, and so on. The electroweak SU(2)_{L} × U(1)_{Y} symmetry is spontaneously broken to U(1)_{QED} by the vacuum expectation value of the neutral component of a new doublet of complex scalar boson fields called the Higgs doublet. After spontaneous symmetry breaking, this results in massive charged W and neutral Z gauge bosons, and a massive neutral scalar Higgs boson – a picture triumphantly confirmed by experiments at CERN.

To truly shed light on the Standard Model’s flavour puzzle, theorists have explored higher and more complex symmetry groups than the Standard Model. The most promising approaches all involve a spontaneously broken family or flavour symmetry. But the flavour-breaking scale may lie anywhere from the Planck scale to the electroweak scale, with grand unified theories suggesting a high flavour scale, while recent hints of anomalies from LHCb and other experiments suggest a low flavour scale.

To illustrate the unknown magnitude of the flavour scale, consider for example the FN mechanism, where M_{fl} is associated with the breaking of the U(1)_{fl} symmetry. In the SM the top-quark mass of 173 GeV is given by a Yukawa coupling times the Higgs vacuum expectation value of 246 GeV divided by the square root of two. This implies a top-quark Yukawa coupling close to unity. The exact value is not important, what matters is that the top Yukawa coupling is of order unity. From this point of view, the top quark mass is not at all puzzling – it is the other fermion masses associated with much smaller Yukawa couplings that require explanation. According to FN, the fermions are assigned various U(1)_{fl} charges and small Yukawa couplings are forbidden due to a U(1)_{fl} symmetry. The symmetry is broken by the vacuum expectation value of a new “flavon” field <φ>, where φ is a neutral scalar under the SM but carries one unit of U(1)_{fl} charge. Small Yukawa couplings then originate from an operator (figure 1, right) suppressed by powers of the small ratio <φ>/M_{fl} (where M_{fl} acts as a cut-off scale of the contact interaction).

For example, suppose that the ratio <φ>/M_{fl} is identified with the Wolfenstein parameter λ = sinθ_{C }= 0.225 (where θ_{C} is the Cabibbo angle appearing in the CKM quark-mixing matrix). Then the fermion mass hierarchies can be explained by powers of this ratio, controlled by the assigned U(1)_{fl} charges: m_{e}/m_{τ} ∼ λ^{5}, m_{μ}/m_{τ} ∼ λ^{2}, m_{d}/m_{b} ∼ λ^{4}, m_{s}/m_{b} ∼ λ^{2}, m_{u}/m_{t} ∼ λ^{8} and m_{c}/m_{t} ∼ λ^{4}. This shows how fermion masses spanning many orders of magnitude may be interpreted as arising from integer U(1)_{fl} charge assignments of less than 10. However, in this approach, M_{fl} may be anywhere from the Planck scale to the electroweak scale by adjusting <φ> such that the ratio λ = <φ>/M_{fl} is held fixed.

One possibility for M_{fl}, reviewed by Kaladi Babu at Oklahoma State University in 2009, is that it is not too far from the scale of grand unified theories (GUTs), of order 10^{16} GeV, which is the scale at which the gauge couplings associated with the SM gauge group unify into a single gauge group. The simplest unifying group, SU(5)_{GUT}, was proposed by Georgi and Glashow in 1974, following the work of Pati and Salam based on SU(4)_{C} × SU(2)_{L} × SU(2)_{R}. Both these gauge groups can result from SO(10)_{GUT}, which was discovered by Fritzsch and Minkowski (and independently by Georgi), while many other GUT groups and subgroups have also been studied (figure 2, left). However, GUT groups by themselves only unify quarks and leptons within a given family, and while they may provide an explanation for why m_{b}_{ }= 2.4 m_{τ}, as discussed by Babu, they do not account for the fermion mass hierarchies.

**Broken symmetries**

A way around this, first suggested by Ramond in 1979, is to combine GUTs with family symmetry based on the product group G_{GUT} × G_{fl}, with symmetries acting in the specific directions shown in the figure “Family affair”. In order not to spoil the unification of the gauge couplings, the flavour-symmetry breaking scale is often assumed to be close to the GUT breaking scale. This also enables the dynamics of whatever breaks the GUT symmetry, be it Higgs fields or some mechanism associated with compactification of extra dimensions, to be applied to the flavour breaking. Thus, in such theories, the GUT and flavour/family symmetry are both broken at or around M_{fl} ∼ M_{GUT }∼ 10^{16} GeV, as widely discussed by many authors. In this case, it would be impossible given known technology to directly experimentally access the underlying theory responsible for unification and flavour. Instead, we would need to rely on indirect probes such as proton decay (a generic prediction of GUTs and hence of these enlarged SM structures proposed to explain flavour) and/or charged-lepton flavour-violating processes such as μ → eγ (see *CERN Courier* May/June 2019 p45).

New ideas for addressing the flavour problem continue to be developed. For example, motivated by string theory, Ferruccio Feruglio of the University of Padova suggested in 2017 that neutrino masses might be complex analytic functions called modular forms. The starting point of this novel idea is that non-Abelian discrete family symmetries may arise from superstring theory in compactified extra dimensions, as a finite subgroup of the modular symmetry of such theories (i.e. the symmetry associated with the non-unique choice of basis vectors spanning a given extra-dimensional lattice). It follows that the 4D effective Lagrangian must respect modular symmetry. This, Feruglio observed, implies that Yukawa couplings may be modular forms. So if the leptons transform as triplets under some finite subgroup of the modular symmetry, then the Yukawa couplings themselves must transform also as triplets, but with a well defined structure depending on only one free parameter: the complex modulus field. At a stroke, this removes the need for flavon fields and ad hoc vacuum alignments to break the family symmetry, and potentially greatly simplifies the particle content of the theory.

**Compactification **

Although this approach is currently actively being considered, it is still unclear to what extent it may shed light on the entire flavour problem including all quark and lepton mass hierarchies. Alternative string-theory motivated ideas for addressing the flavour problem are also being developed, including the idea that flavons can arise from the components of extra-dimensional gauge fields and that their vacuum alignment may be achieved as a consequence of the compactification mechanism.

The discovery of neutrino mass and mixing makes the flavour puzzle hard to ignore

Recently, there have been some experimental observations concerning charged lepton flavour universality violation which hint that the flavour scale might not be associated with the GUT scale, but might instead be just around the corner at the TeV scale (*CERN Courier* May/June 2019 p33). Recall that in the SM the charged leptons e, μ and τ interact identically with the gauge forces, and differ only in their masses, which result from having different Yukawa couplings to the Higgs doublet. This charged lepton flavour universality has been the subject of intense experimental scrutiny over the years and has passed all the tests – until now. In recent years, anomalies have appeared associated with violations of charged lepton flavour universality in the final states associated with the quark transitions b → c and b → s.

**Puzzle solving**

In the case of b → c transitions, the final states involving τ leptons appear to violate charged lepton universality. In particular B → D^{(*) }ℓ ν_{ℓ} decays where the charged lepton ℓ is identified with τ have been shown by Babar and LHCb to occur at rates somewhat higher than those predicted by the SM (the ratios of such final states to those involving electrons and muons being denoted by R_{D} and R_{D*}). This is quite puzzling since all three types of charged leptons are predicted to couple to the W boson equally, and the decay is dominated by tree-level W exchange. Any new-physics contribution, such as the exchange of a new charged Higgs boson, a new W′ or a leptoquark, would have to compete with tree-level W exchange. However, the most recent measurements by Belle, reported at the beginning of 2019 (*CERN Courier* May/June 2019 p9), measure R_{D} and R_{D*} to be closer to the SM prediction.

In the case of b → s transitions, the LHCb collaboration and other experiments have reported a number of anomalies in B → K^{(*)} ℓ^{+}ℓ^{–} decays such as the R_{K} and R_{K*} ratios of final states containing μ^{+}μ^{–} versus e^{+}e^{–}, which are measured deviate from the SM by about 2.5 standard deviations. Such anomalies, if they persist, may be accounted for by a new contact operator coupling the four fermions b_{L}s_{L}μ_{L}μ_{L} suppressed by a dimensionful coefficient M^{2}_{new} where M_{new} ~30 TeV, according to a general operator analysis. This hints that there may be new physics arising from the non-universal couplings of leptoquark and/or a new Z′ whose mass is typically a few TeV in order to generate such an operator (where the 30 TeV scale is reduced to just a few TeV after mixing angles are taken into account). However, the introduction of these new particles increases the SM parameter count still further, and only serves to make the flavour problem of the SM worse.

**Link-up**

Motivated by such considerations, it is tempting to speculate that these recent empirical hints of flavour non-universality may be linked to a possible theory of flavour. Several authors have hinted at such a connection, for example Riccardo Barbieri of Scuola Normale Superiore, Pisa, and collaborators have related these observations to a U(2)^{5} flavour symmetry in an effective theory framework. In addition, concrete models have recently been constructed that directly relate the effective Yukawa couplings to the effective leptoquark and/or Z′ couplings. In such models the scale of new physics associated with the mass of the leptoquark and/or a new Z′ may be identified with the flavour scale M_{fl} defined earlier, except that it should be not too far from the TeV scale in order to explain the anomalies. To achieve the desired link, the effective leptoquark and/or Z′ couplings may be generated by the same kinds of operators responsible for the effective Higgs Yukawa couplings (figure 3).

In such a model the couplings of leptoquarks and/or Z′ bosons may be related to the Higgs Yukawa couplings, with all couplings arising effectively from mixing with a vector-like fourth family. The considered model predicts, apart from the TeV scale leptoquark and/or Z′, and a slightly heavier fourth family, extra flavour-changing processes such as τ → μμμ. The model in its current form does not have any family symmetry, and explains the hierarchy of the quark masses in terms of the vector-like fourth family masses, which are free parameters. Crucially, the required TeV scale Z′ mass is given by M_{Z′} ~ <φ> ~ TeV, which would fix the flavour scale M_{fl} ~ few TeV. In other words, if the hints for flavour anomalies hold up as further data are collected by the LHCb, Belle II and other experiments, the origin of flavour may be right around the corner.

### Further reading

K S Babu 2009 arXiv:0910.2948.

R Barbieri *et al.* 2016 *Eur. Phys. J. C* **76** 67.

F Feruglio 2017 arXiv:1706.08749.

F Feruglio 2015 *Eur. Phys. J. C* **75** 373.

C D Froggatt and H B Nielsen 1979 *Nucl. Phys. B* **147** 277.

H Georgi and S L Glashow 1974 *Phys. Rev. Lett*. **32** 438.

S F King 2019 arXiv:1905.02660.

S F King and C Luhn 2013 *Rept. Prog. Phys.* **76** 056201.

S F King and G G Ross 2001 *Phys. Lett B* **520** 243.

J C Pati and A Salam 1974 *Phys. Rev. D* **10** 275.