The flavour puzzle and the B-decay anomalies

Riccardo Barbieri

1 Introduction and the flavour puzzle

At the end of 1973 the Standard Model (SM) of elementary particles was fully formulated, with many different contributions put together in its Lagrangian and its particle content. An effective, although partial, way to see the progress that has been made since then to compare the SM with observations is illustrated in fig. 1, which shows all the particles of the SM with the dates of their experimental discoveries. Only less than half of these particles had been seen by the end of 1973, whereas the table was filled up in 2012 with the discovery of the Higgs boson. What the figure does not show is the amount of experimental results that have so far confirmed all the interactions contained in the SM Lagrangian, both in the extension of their manifestations as in the precision of the corresponding predictions.

The true paradox is that, in spite of these clear successes, there are reasons to think that the SM is not a complete theory. To justify this statement one normally quotes two different orders of reasons: i) phenomena non accounted for, like Dark Matter or the generation of the matter-antimatter asymmetry or neutrino masses; ii) phenomena included in a conceptually unsatisfactory way. In my own view the second reason is as important as the first one, with the spectrum and the mixings of the spin-1/2 constituents of matter as one of the main issues.

In the SM the differences between the quarks and leptons with the same gauge quantum numbers, colour, weak and electric charges (the vertical columns in the J = 1/2 particles of fig. 1 with three elements each) are exclusively attributed to their Yukawa interactions with the Higgs boson. The main consequences of this picture can be summarized as follows:

– The interactions of quarks and leptons with the gauge bosons (the J = 1 particles) are ”universal”, i.e. described by flavour-independent gauge couplings.

– The interaction with the W-boson is the only one that produces transitions between particles of different generations, the quarks of charge 2/3, ui, and −1/3, dj , with an amplitude proportional to the elements Vij of the Cabibbo-Kobayashi-Maskawa (CKM) unitary matrix.

– The masses of all the charged fermions, i.e. every fermion except the neutrinos, are linearly related to their respective couplings to the Higgs boson.

There is mounting direct evidence, again extended and precise, for the first two of these features. Even the third one is receiving support from measurements performed at LHC, although limited, for the time being, to the heaviest particles and to 20–30% precision. The problem comes with the numerical values both of the masses and of the ”mixing angles” Vij: they are all pure parameters. The values of the masses are graphically represented in fig. 2. The main feature of the CKM matrix is the smallness of the entries connecting the third heaviest generation to the two lighter ones at or below 5 · 10–2 level. The inability of the theory to predict any of these fundamental quantities is in sharp contrast with the fact that almost all of them are measured with significant precision, up to 10–7 in the case of the electron mass. This contrast, in fact, may introduce skepticism on the possibility of throwing light on this "flavour puzzle" without experimental inputs of a new kind that would show deviations from the flavour pattern of the SM.

2 A $U ( 2 )^{5}$ flavour symmetry

Most if not all the attempts to try to attack this problem are based on symmetries, which I do not try to describe here. Rather I focus on one such symmetry that does not determine per se any of the flavour parameters but is suggested by their generic pattern and, more importantly, can hide deviations from the SM that are there, waiting to be discovered.

At the origin of the phenomenological features of flavour physics outlined above there is the group of unitary transformations, $G = U ( 3 )^{5}$, that leave unchanged the SM Lagrangian in the limit in which the Yukawa couplings between the Higgs boson and the fermions are formally switched off. This symmetry arises because, as we said, the three fermions of given gauge quantum numbers, altogether making five different irreducible representations of the SM gauge group, are precisely only distinguished by the Yukawa couplings themselves. To account for the Yukawa couplings, i.e. for the masses and the mixing angles, is equivalent to control the precise breaking of this symmetry.

While we do not know what gives rise to this breaking, a significant feature emerges from the data. In the limit in which one neglects the masses of the first two generations and their mixings with the third one, both numerically small, a $U ( 2 )^{5} $ subgroup of the full $G = U ( 3 )^{5} $ remains unbroken, which acts on the lighter two generations as doublets and on the third generation fermions as singlets: $U ( 2 )^{5} $ is an approximate symmetry of the observed flavour parameters. Once again to test quantitatively the relevance of $U ( 2 )^{5} $ would require knowing the origin of its small breaking. A different indirect possibility emerges, however, if there are other interactions than the SM ones with suitable $U ( 2 )^{5} $ properties. Assuming that this be the case, the deviations they would produce in flavour physics relative to the SM could have been kept below observability so far by the small amount of $U ( 2 )^{5} $ breaking, but nevertheless be there and possibly discovered by further more precise observations. New observables could also give more insight into the flavour puzzle itself.

3 Putative anomalies in B-decays and lepton flavour universality

The hypothesis put forward in the last paragraph of the previous section is a daring one. It motivates, however, to give consideration to a number of experimental results that have progressively emerged in the last five years and seem to indicate altogether an anomaly in B-meson decays. The quantities of main interest are the two ratios

(1) $R_{D}^{ ( * ) } = \frac{BR ( B \rightarrow D^{ ( * )}\tau \nu ) }{BR ( B \rightarrow D^{ ( * ) } l \nu ) }$, $l = e, \mu ,$

(2) $R_{K}^{ ( * ) } = \frac{BR ( B \rightarrow K^{ ( * ) }\mu^{+}\mu^{-} ) }{BR ( B \rightarrow K^{ ( * ) } e^{+} e^{-} ) }$,

between branching ratios of B-decays only differing in the lepton appearing in the final states: τ versus e or μ in the first case; μ versus e in the second case. One focuses on these ratios because the uncertainties due to the treatment of B, D or K quark-antiquark bound states drop out when predicting them in the SM. The experimental results relevant to the two cases are shown in tables 1 and 2, respectively and compared with the SM expectations. As one can see, while no single experimental result is precise enough to allow the claim of a discovered anomaly –a reason to be cautious– their global consideration is suggestive. If confirmed, they would bring evidence against Lepton Flavour Universality (LFU), expected in the SM and always respected in experiments so far.

In fact, if one wants to consider the anomalies in $R_{D}^{ ( * ) }$ and $R_{K}^{ ( * ) }$ together, it is important to note the different nature of the two decays at quark level: the "charged current" process $b \rightarrow c\tau\nu$ in the case of the B to D transition (table 1), versus the "neutral current" process $b \rightarrow s\mu\mu$ in the B to K case (table 2). This introduces an important difference between the two processes: while the flavour transition in the first case is due to a tree level exchange of the W-boson (fig. 3A), the flavour change in the neutral current process can only occur at second order in the electroweak interactions (fig. 3B). In turn this manifests itself in the values of the two branching fractions: $B\rightarrow D\tau\nu$ is more than four orders of magnitude larger than $B\rightarrow K\mu\mu$. In passing we recall that other measurements involving the $b\rightarrow s$ transition indicate possible deviations from the SM of size consistent with the anomalies shown in table 2. Not involving deviations from LFU, however, their prediction in the SM is at the moment more subject to theoretical uncertainties.

If confirmed by future measurements, are there qualitative reasons to think that these anomalies might be related to $U ( 2 )^{5}$? On general grounds $U ( 2 )^{5}$ introduces a natural distinction between semi-leptonic K and π decays on one side, where successful tests of LFU are ubiquitous, and B semi-leptonic decays on the other side, where LFU violations seem to emerge. More specifically, suppose that the deviations from the SM in both the charged and neutral current modes be due to the exchange of a massive bosonic mediator around the TeV scale. If this mediator is coupled only to third-generation fermions in the limit of unbroken $U ( 2 )^{5}$, this can explain the comparable relative deviations from the tree-level versus loop-level SM amplitudes, which are apparently being observed: $b\rightarrow c\tau\nu$ only involves a single second-generation particle, i.e. a single $U ( 2 )^{5}$ breaking, whereas $b\rightarrow s\mu\mu$ has three secondgeneration fermions in it or three $U ( 2 )^{5}$-breaking suppressions.

4 A lepto-quark mediator of the anomalies

To a good approximation the four-fermion effective operators that give rise to $b\rightarrow c\tau\nu $ and $b\rightarrow s\mu\mu $ in the SM are all made of left-handed fields. This is because only the W-exchange may give flavour transitions and because of the left-handed nature of the weak interactions. Respecting Lorentz and SM gauge invariance, a number of mediators can give rise, through their exchange, to effective amplitudes capable to interfere with these operators without having to pay for small mass suppressions: colourless vectors or colour-triplet scalars or vectors. There is however a single mediator that has this property and is not conflicting with the non-observation, so far, of $B\rightarrow K^{ ( * )} \nu \nu$: the vector $U_{\mu}^{a}$ transforming under the $SU ( 3 ) \times SU ( 2 ) \times U ( 1 ) $ SM gauge group as (3, 1, 2/3), i.e. a charge 2/3 lepto-quark. The relevant gauge invariant interaction is

(3) $L_{U} = g_{U} ( \bar{q}^{a}_{L} \gamma^{\mu} I_{L} ) U_{\mu}^{a}$

where $q$ and $l$ are left-handed $SU ( 2 ) $-doublets with implicit indices contracted and a is a colour index. The flavour indices in $q$ and $l$ are not specified. However, with unbroken $U ( 2 )^{5}$, the allowed coupling would only be the one to the third generation, as anticipated and desired, since the first two generations of $q$ and $l$ transform as doublets under different $U ( 2 ) $ factors of the full flavour group. After $U ( 2 )^{5}$ breaking, the relevant amplitudes are shown in fig. 3C and 3D. They can account for the observed anomalies with a ratio between the coupling and the mass of the vector lepto-quark of about $ g_{U} / m_{U} \sim 2 $/TeV and with $U ( 2 )^{5}$-breaking parameters of similar size to the ones occurring in the spectrum and mixings of the SM fermions at about 5 · 10–2 level. Note that, at the elementary level, $B\rightarrow K^{ ( * ) }\nu\nu$ is $b\rightarrow s \nu\nu$, which is not produced by U-exchanges.

5 A Pati-Salam $SU ( 4 ) $ model

Can one make sense of a massive vector lepto-quark at more than an effective field theory level? To this end the observation comes obviously to mind that the quantum numbers of the lepto-quark under the SM gauge group are the same as for the lepto-quark living inside the adjoint representation of the Pati-Salam $SU ( 4 ) $ group. This group, first proposed in 1974, unifies colour with the (B-L) charge by treating lepton number as a fourth colour. As a consequence, the vectors (generators) of $SU ( 4 ) $ are grouped in representations of $SU ( 3 ) \times U ( 1 )_{B-L}$ as

(4) $G_{\mu} \rightarrow G_{\mu} ( 8,0 ) \oplus U_{\mu} ( 3, 2/3 ) \oplus U_{\mu}^{+} ( \bar{3}, - 2/3 ) \oplus B_{\mu} ( 1, 0 )$,

graphically represented in fig. 4.

The inclusion of fermions in this picture faces, however, an immediate difficulty. Since the quarks and the leptons of a given helicity are normally unified in 4-plets of $SU ( 4 )$, the ones of the first two generations cannot transform anymore as doublets of two different $U ( 2 )$ factors of the flavour group, as advocated above. Consequently, the unsuppressed exchange of the lepto-quark between the first two families, with $g_{U} / m_{U} \sim 1.5$/TeV as for the third one, would generate a $K_{L}\rightarrow \mu e$ transition at a rate exceeding the experimental bouand by several orders of magnitude. This difficulty can be avoided by enlarging the $SU ( 4 ) $ group and/or by introducing extra heavy vector-like fermions to which the SM fermions are suitably mixed. Different implementations of this features lead to a number of observable consequences, both in flavour physics and in collider physics. Here we observe that the constraint $g_{U}/ m_{U} \sim 1.5$/TeV and the lower limits on the mass $m_{U}$ by direct collider searches, at about 1.5 TeV, require a relatively large coupling gU of the lepto-quark, $g_{U > 2.5}$, to the third generation of fermions. In turn this suggests to hypothesize a composite picture of the lepto-quark, as resulting from a new strong interaction possessing an $SU ( 4 ) $ global symmetry and capable to produce at the same time a composite pseudo-Goldstone Higgs boson.

6 Summary

The inability to predict the masses of any of its particles, with the exception of the massless gluon and photon, is a drawback of the SM. This problem is particularly acute in the fermion sector, with the greatest number of particles and the equal limitation to predict the mixing angles. I have referred to this altogether as the flavour puzzle.

Given the current precise experimental knowledge of most of the flavour parameters, one is inclined to think that progress on this issue might only come from new measurements showing deviations from the SM in the flavour sector itself. One can legitimately ask, however, how likely is this to happen in view of the increasing confirmations that the SM keeps receiving, lastly especially at LHC. It has been argued that the observed approximate $U ( 2 ) ^{5}$ flavour symmetry may have played a role in this, acting to keep hidden so far possible deviations from the SM even in the presence of new interactions at the TeV scale. In turn, since $U ( 2 ) ^{5}$ introduces a symmetry distinction between the third and the first two generations of quarks and leptons, the putative anomalies emerging in B-meson decays represent an obvious phenomenon to watch.

The reality of these anomalies needs confirmation. Luckily the experimental opportunities are there both to improve the precision in the key observables and to extend the searches in related directions. These related directions, in other flavour observables and in direct searches of possible mediators as well, only partially depend on the modelling of the anomalies themselves. This is particularly the case for the models briefly outlined in sects. 4 and 5. The eventual confirmation of the anomalies will tell if any of these models is relevant or it will indicate the different direction to explain them. All this is without forgetting the pending resolution of the flavour puzzle.


I am deeply indebted for their collaboration/discussions on these matters to Dario Buttazzo, Roberto Contino, Gino Isidori, Paolo Lodone, Chirstopher Murphy, Andrea Pattori, Maurizio Pierini, Filippo Sala, Fabrizio Senia, David Straub, Andrea Tesi, Riccardo Torre.