Lepton flavour universality

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Flavour in the Standard Model

The Standard Model, based on the formulation from Glashow, Weinberg and Salam, can be expressed in terms of a Lagrangian split into two terms


${\cal L} = {\cal L}_{\rm gauge}(A_a,\psi_i) + {\cal L}_{\rm Higgs}(\phi,A_a,\psi_i)~.$

where $\psi_i$ represents a fermion field, $A_a$ a field for the strong or electroweak interactions and $\phi$ is the Higgs field. The first term, the gauge term, deals with the free-fields and their interactions via the strong and electroweak forces. The second term, the Higgs term, gives rise, after spontaneous symmetry breaking, to the mass of the bosons for the weak charged- and neutral-current interactions. This term also gives rise to the mass of the fermions (the quarks and charged leptons) through so-called Yukawa interactions.

In the Standard Model of particle physics (SM), the fermion fields are arranged as left-handed doublets and right-handed singlets. The quark doublets comprise an up- and a down-type quark. The lepton doublets comprise a charged lepton (e.g. an electron) and its corresponding neutrino. The neutrinos are massless in the Standard Model and only the charged leptons appear as right-handed singlets. There are three generations, different left-handed doublets and right-handed singlets, of both quarks and leptons. Consequently there are: three different flavours of charged lepton; three different flavours of neutrino; six different flavours (three up-type and three down-type) of quark. The lightest quarks are the up- and down-type quarks that form the constituents of the proton and neutron. The heavier strange, charm and bottom quarks form short-lived mesons and baryons that can be produced in high-energy particle collisions. The top quark is sufficiently heavy that it decays before it can form a meson or baryon. Without the Higgs (and the resulting Yukawa interaction), all of the particles would be massless and the three generations would be identical copies of each other; there would be an accidental symmetry of the Standard Model. The interaction of the Higgs with different charged fermions (the bottom and top quarks and the muon and $\tau$) has been tested through measurements of Higgs production and decay at the LHC. The ATLAS and CMS experiments both see couplings that are consistent with being proportional to the mass of the fermions.

Table 1: Fermion properties in the Standard Model: mass, charge and weak hypercharge for left- and right-handed fermions. Neutrinos are massless in the the Standard Model and there are no right-handed neutrinos. The quark masses are given in the so-called $\overline{\rm MS}$ scheme (see (Particle Data Group) review for details).
Particle Mass Charge $\mathbf{Y_{\rm L}}$ $\mathbf{Y_{\rm R}}$
Leptons $e^-$ 0.511 ${\rm MeV}/c^{2}$ $-1$ $-1$ $-2$
$\mu^-$ 105.7 ${\rm MeV}/c^{2}$ $-1$ $-1$ $-2$
$\tau^-$ 1.777 ${\rm GeV}/c^{2}$ $-1$ $-1$ $-2$
$\nu_e$ $-1$ $-1$
$\nu_\mu$ $-1$ $-1$
$\nu_\tau$ $-1$ $-1$
Quarks $u$ 2.2 ${\rm MeV}/c^{2}$ $+\tfrac{2}{3}$ $+\tfrac{1}{3}$ $+\tfrac{4}{3}$
$d$ 4.7 ${\rm MeV}/c^{2}$ $-\tfrac{1}{3}$ $+\tfrac{1}{3}$ $-\tfrac{2}{3}$
$c$ 1.3 ${\rm GeV}/c^{2}$ $+\tfrac{2}{3}$ $+\tfrac{1}{3}$ $+\tfrac{4}{3}$
$s$ 96 ${\rm MeV}/c^{2}$ $-\tfrac{1}{3}$ $+\tfrac{1}{3}$ $-\tfrac{2}{3}$
$t$ 173.1 ${\rm GeV}/c^{2}$ $+\tfrac{2}{3}$ $+\tfrac{1}{3}$ $+\tfrac{4}{3}$
$b$ 4.2 ${\rm GeV}/c^{2}$ $-\tfrac{1}{3}$ $+\tfrac{1}{3}$ $-\tfrac{2}{3}$


Gauge term of the Standard Model lagrangian

The gauge term of the Standard Model lagrangian is based on symmetries and the so-called gauge principle; the interaction of the free-fields with the different forces can be specified by requiring that the lagrangian is unchanged under a continuous symmetry. The gauge term has the structure:


$ \begin{split} {\cal L}_{\rm gauge} = & i \bar{\psi}_L \gamma^{\mu} \left( \partial_\mu + i g_s \lambda_a G_{\mu}^{a} + i g \sigma_i W^{i}_{\mu} + i Y_{L} g' B_\mu \right) \psi_L + \\ & i \bar{\psi}_{R}\gamma^{\mu} (\partial_\mu + i g_s \lambda_a G_{\mu}^{a} + i Y_{R} g' B_\mu) \psi_R - \dfrac{1}{4} F_{a}^{\mu\nu}F_{a\,\mu\nu}~, \end{split} $

where $\psi_L$ and $\psi_R$ represent left- and right-handed fermion fields, $G^a_{\mu}$ are the gluon fields following from a SU(3) local gauge symmetry, $W^{i}_\mu$ are the fields associated with a SU(2) symmetry, $B_\mu$ is the field associated with a U(1) symmetry. The indices $\mu$ and $\nu$ are space-time indices and the $\gamma^{\mu}$ are the Dirac $\gamma$-matrices. The tensors $F_{a}^{\mu\nu}$ are the field-strength tensor associated with the gauge bosons. After electroweak symmetry breaking, $W^{3}$ and $B$ become the $Z^0$ and photon fields and $W^{1,2}$ give rise to the $W^\pm$ for the charged-current interaction.

The interactions between the particle fields and the gauge bosons are governed by three coupling strengths: $g_s$ (for the strong interaction), $g$ and $g'$. The parameter $Y$ is the so-called weak hypercharge and takes the values $Y_L = +1$ $(-1)$ for charged leptons (neutrinos) and $Y_L = +\tfrac{1}{3}$ for left-handed quark doublets, and $Y_R = -2$ for charged-leptons and $Y_R = +\tfrac{4}{3}$ ($-\tfrac{2}{3}$) for right-handed up- (down-) type singlets. The charged-current interaction involves only left-handed doublets. The couplings, $g_s$, $g$ and $g'$ are universal, i.e. they are identical independent of the fermion field that is considered. The gauge term therefore has identical couplings to the different flavours of lepton. Lepton flavour universality is only broken in the Standard Model by the Yukawa interaction, which gives rise to radically different masses for the different leptons (the $\tau$ lepton mass is $1776\,\text{MeV}/c^{2}$, the muon mass is $106\,\text{MeV}/c^{2}$ and the electron mass $0.5\,\text{MeV}/c^{2}$). The fermion masses, electric charge and weak hypercharge are summarised in Table 1.

Fermion mass in the Standard Model

Before considering the Yukawa terms for the charged leptons it's illustrative to consider the case for the quark fields. The Yukawa interaction between a quark field and the Higgs can be written as:


$ \begin{align} {\cal L}_{\rm Yukawa} = -\bar{Q}_L^{i} Y_{U}^{ij}\phi^{\dagger} u^j_{R} - \bar{Q}_L^{i}Y_D^{ij} \phi d_{R}^{j} + {\rm h.c}~, \end{align} $

where $Q_L$ is a left-handed quark doublet, $u_R$ is a right-handed up-type singlet, $d_R$ is a right-handed down-type singlet field and $\phi$ is the Higgs doublet. The most general interaction can involve different flavours of quark, labelled with index $i$ or $j$. Replacing the Higgs by its vacuum expectation value, $\nu_{\rm Higgs}$, yields mass-like terms in the Lagrangian:


$ {\cal L}_{\rm Yukawa} = -\bar{u}_L^{i} m_{U}^{ij} u^j_{R} - \bar{d}_L^{i}m_D^{ij} d_{R}^{j} $

where $m_U^{ij} = \nu_{\rm Higgs} Y_{U}^{ij}$ and $m_D^{ij} = \nu_{\rm Higgs} Y_{D}^{ij}$. This expression now contains the left-handed fields for the up- and down-type quarks indicated by $u_L$ and $d_L$, respectively. The mass matrices, $m_U$ and $m_D$ are $3\times 3$ complex-matrices whose coefficients are not predictions of the Standard Model. We can chose to diagonalise the matrices by redefining the fields in terms of mass eigenstates through $3\times 3$ unitary transformations:


$ \begin{split} u^{i}_{L} = \hat{U}^{ij}_{L} u_{L}^{\prime j} \quad, & \quad d^{i}_{L} = \hat{D}^{ij}_{L} d_{L}^{\prime j}\quad, \\ u^{i}_{R} = \hat{U}^{ij}_{R} u_{R}^{\prime j} \quad, & \quad d^{i}_{R} = \hat{D}^{ij}_{R} d_{R}^{\prime j}. \end{split} $

This results in $m_{U}^{\rm diag} = \hat{U}^{\dagger}_{L} m_{U} \hat{U}_{\rm R}$ and $m_{D}^{\rm diag} = \hat{D}^{\dagger}_{\rm L}m_{D}\hat{D}_{R}$.

The impact of the misalignment between the mass and weak eigenstates depends on the type of interaction considered. The electromagnetic interaction is associated with a current, for an up-type quark, that has the form $\bar{u}^i \gamma^{\mu} u^i$. Changing from $u$ to $u'$ leaves the current unchanged. On the other hand, the charged-current interaction is associated with a current


$ \bar{u}^{i}_{L} \gamma^{\mu} d^{i}_{L} = \bar{u}'_{L} (U^{\dagger}_{L} D_{L})^{ij} d'^{j}_{L} = \bar{u}'_{L} V^{ij} d'^{j}_{L} $

The matrix, $V$, is the Cabibo-Kobayashi-Maskawa quark-mixing matrix. Like the mass matrices, the complex matrix $V$ is not a prediction of the Standard Model and its coefficients must be determined by experiment. A complex phase associated with $V$ gives rise to CP violation in the Standard Model of particle physics. In the quark sector, the strong, electromagnetic and neutral current interactions are: flavour universal, i.e. they have the same strength for different flavours of quark; and flavour conserving, i.e. they only involve quarks of the same flavour. Conversely, charged-current interactions are flavour-changing and depend on the structure of $V$.

The same process could be followed to obtain mass terms for the leptons. However, because neutrinos are treated as massless in the Standard Model of particle physics, the charged leptons and neutrinos can be diagonalised by the same matrix. Consequently, there is no equivalent to $V$ and the charged-current interaction is flavour conserving. The interaction is also flavour universal because the couplings $g$ and $g'$ do not distinguish between the different types of charged lepton.

While neutrinos are treated as massless in the Standard Model of particle physics, there is evidence (through neutrino oscillation phenomena) that the neutrinos have a small mass. The mass mechanism for neutrinos is unknown. The Yukawa mechanism could be used to generate a non-zero mass by introducing right-handed neutrino singlets. These singlets must only participate in the Yukawa interaction but not in any of the gauge interactions. Because the neutrino is electrically neutral, and there are no obvious quantum numbers that distinguish neutrinos from antineutrinos, it is also possible to generate its mass through the so-called Majorana mass mechanism. The introduction of neutrino mass terms does not change the expectation that charged leptons should be treated universally by gauge interactions.

The flavour structure of the Standard Model is puzzling for a number of reasons:

  1. why do we have three generations of quarks and leptons?

  2. why do particle masses span many orders of magnitude, with a pattern that appears very hierarchical?

  3. why are neutrinos light compared to the charged leptons?

The Standard Model is unable to answer these questions. It is hoped that through either precise measurements of particle decays, or through searches for new heavy particles, that we might find evidence for a breakdown of the Standard Model. This could lead us to a new theory, which addresses some of these fundamental questions. Tests probing the universality of lepton couplings could help to understand the flavour structure of the Standard Model.


Experimental tests of lepton flavour universality

While the couplings in the Standard Model are universal, the different flavours of charged lepton look very different in experiments. The most striking difference is that the muon and $\tau$ are not stable particles and are able to decay. Muons decay almost exclusively to electrons, a muon neutrino and an antielectron neutrino, with a long (on particle physics timescales) lifetime of 2.2 microseconds. The $\tau$ lepton is sufficiently heavy that it can decay to muons, electrons and to hadronic final-states, with a much shorter lifetime of 0.3 picoseconds. Because of their short lifetime, $\tau$ leptons are not usually seen directly by experiments. Their presence must instead be inferred from their decay in the detector.

The different masses of the charged-leptons also has important consequences for how they interact as they pass through detector material. Due to their small mass, electrons are usually highly relativistic in experiments and interact more readily than muons or taus. This leads to larger coulomb scattering effects and to larger Bremsstrahlung emission as the charged leptons pass close to nuclei in material. These effects need to be accounted for by experiments when measuring the momentum and the energy of electrons.

W and Z boson decays

Figure 1: Summary of tests of lepton flavour universality in electroweak boson decays.

An obvious check of the gauge sector of the Standard Model is to compare the direct decays of the electroweak bosons between experiment and theory. The expectation is that the $Z^0$ and the $W^\pm$ should have almost identical probabilities to decay to the different types of charged lepton (have equal branching fractions) due to the universal nature of $g$ and $g'$ and the similar available phase space (number of momentum states available) in the decays. The most precise measurements of the ratios of different branching fractions are shown in Figure 1.

Measurements of the $Z^{0}$ are dominated by results from the four LEP experiments that ran at CERN and the SLD experiment at the Stanford Linear Collider, using data that was collected in the 1990's (LEP, 2005). Comparable precision for the $W^{\pm}$ decays has been achieved by the LEP and LHC experiments. The data are consistent with flavour universality, i.e. with ratios of decay rates being very close to one.

There has historically been a small tension in the measurements by the LEP experiments of the ratios of $W^{\pm}$ boson decays to $\tau$ leptons versus the two lighter flavours (LEP, 2013), where an excess of $W^{\pm}$ decays to $\tau$ leptons was seen. This discrepancy is, however, not seen in a more recent measurements by the ATLAS and CMS collaborations. Any remaining difference must be small.


Semileptonic decays of strange, charm and beauty hadrons

Figure 2: Feynman diagram for $b \to c \ell \bar{\nu}_{\ell}$ transition.
Figure 3: Summary of tests of lepton flavour universality using charm and strange meson decays. The SM expectation, where available, is indicated by the band.

Lepton flavour universality has been studied extensively in semileptonic strange-, charm- and beauty-hadron decays at a range of experiments over the last couple of decades. In these processes, particles decay via the charged-current interaction to a lighter hadron, a lepton and its corresponding neutrino (as illustrated in Figure 2). The most precise measurements of strange meson decays come from data collected by the CLOE experiment at the Frascati laboratory and the NA48 experiment at the super-proton-synchrotron (SPS) at CERN. The most precise measurements of charm hadron decays come from the $B$-factory experiments (the BaBar experiment at the Stanford linear accelerator and the Belle experiment at the KEK accelerator) and the BES III experiment at the Beijing electron-positron collider. The most precise measurements of $b$-quark hadron decays come from the $B$-factory experiments and from the LHCb experiment at the LHC.

Figure 4: Summary of tests of lepton flavour universality using semileptonic $b$-hadron decays. The $R$ ratios are computed as ratios between final states containing $\tau$ leptons and those containing light leptons, e.g $R(D)$ is the ration between the rates of $B^+ \to \bar{D}^{0} \tau^+\nu_\tau$ and $B^+ \to \bar{D}^{0} \mu^+\nu_\mu$ decays. The SM expectation, where available, is indicated by the band.

A summary of lepton flavour universality tests in semileptonic $s$- to $u$-quark and $c$- to $s$-quark processes is shown in Figure 4 The tests using decays of $K^+$ and the neutral $K_{\rm L}$ mesons, yield ratios close to 66% due to the different masses of the electron and muon and the available phase space in the decays. The large masses of the $D^+$ and $D^0$ mesons, compared to those of the $K^+$ and $K_{\rm L}$, leads to ratios between decays to muons and electrons that are much closer to one. All of the measurements of semileptonic charm and strange hadron decays are found to be consistent with lepton flavour universality.

In measurements of beauty hadron decays, the mass difference between the beauty hadron and the charm hadron is large enough that all three flavours of lepton can be produced. However, the comparatively large mass of the $\tau$ lepton limits the available phasespace for the decay and leads to an expectation that muons and electrons are produced about three times as often as $\tau$ leptons in these decay processes. A summary of tests of lepton flavour universality in beauty hadron decays by the BaBar, Belle and LHCb experiments is shown in Figure 4, along with predictions for the ratios based on the Standard Model of particle physics (HFLAV, 2019) (Bernlocher et. al, 2018). The data indicate that $\tau$ leptons are produced more frequently than expected, with a significance of around three standard deviations (HFLAV, 2019). The semileptonic decay rates with electrons and muons in the final state are found to be very similar and are consistent with lepton flavour universality. The first evidence of the discrepancy in the decay rate to $\tau$ leptons was reported by the BaBar collaboration in 2012 (BaBar, 2012). Since then, the BaBar measurement has been confirmed by both the Belle and LHCb collaborations. Although, the significance of the discrepancy has not increased significantly as new measurements have been added.

Leptonic decays of strange, charm and beauty hadrons

It is also possible for $K^{\pm}$, $D^{\pm}$ and $B^\pm$ mesons to decay via the charged-current interaction to only a lepton and its antineutrino. In this case, since the decaying particle is a pseudoscalar, the lepton and neutrino must have the same helicity. Because the charged-current interaction couples to left-handed states, the decay must therefore only involve the right-handed helicity component of the left-handed chiral state of the charged lepton. This component is proportional to the charged leptons mass. Neglecting radiative effects, the ratio of decay rates to different flavours of leptons just depends on the lepton masses,


$ R(\ell_1/\ell_2) = \left(\frac{m_{\ell_1}}{m_{\ell_2}}\right)^2 \sqrt{\frac{m^2 - 4 m_{\ell_1}^2}{m^2 - 4 m_{\ell_2}^2}} $

where $m$ is the mass of the decaying meson, and $m_{\ell_1}$ and $m_{\ell_2}$ are the masses of the two leptons. Only the $B^+ \to \tau^+ \nu_\tau$ decay has been seen experimentally. The $\tau$ and muon decays have been seen for the lighter $D_s^-$ meson by the Belle experiment, with a tau-to-muon ratio of $(10.73 \pm 0.69 {}^{+0.56}_{-0.53})$ (Belle, 2013). The muon and electron decays have been measured precisely for the $K^-$ decay by the NA62 experiment at the SPS at CERN, with a electron-to-muon ratio of $(2.488 \pm 0.010)\times 10^{-5}$ (NA62, 2012). These measurements are consistent with the expectation based on the mass differences between the particles.

Rare decays of beauty hadrons

Figure 5: Feynman diagram for $B^+ \to K^+ e^+ e^-$ and $B^+ \to K^+ \mu^+ \mu^-$ decays.

Lepton flavour universality has also been tested using rare $b$-quark hadron decay processes. The term rare $b$-quark hadron decays is typically used to refer to processes involving $b$- to $s$-quark (or $b$- to $d$-quark) transitions. In the Standard Model of particle physics, these processes are suppressed and proceed via loop-order Feynman diagrams (such as the one shown in Figure 5). Here, we are particularly interested in a subset of processes in which the $b$-quark hadron decays to a lighter hadron and a pair of leptons of the same flavour (for example the $B^+ \to K^+ \mu^+\mu^-$ decay). Processes of this type were observed for the first time by the Belle collaboration in 2001 (Belle, 2001), but have since been studied extensively by the $B$-factory experiments and by the LHCb experiment at the LHC. The large $\tau$-lepton mass and the need to reconstruct the decay products of two taus means that decays involving $b\to s\tau^+\tau^-$ transitions have not yet been seen.

Figure 6: Summary of tests of lepton flavour universality in rare $b$-hadron decays. The $R$ ratios correspond to the ratio between the rates of decays to final states with muons compared with final states with electrons. The measurements are performed in regions of dilepton mass squared ($q^2$) where the Standard Model expectation is close to 1.0.

The most precise measurements of the properties of rare $b$-hadron decays currently come from the LHCb experiment in $\mu^+\mu^-$ final-states. These measurements show an interesting pattern of deviations from Standard Model predictions. Notably, the rates of the process appear to be systematically lower than expected in the Standard Model. This is a common feature seen across all of the different measurements. It is unclear if this points to evidence for a breakdown of the Standard Model, or a systematic problem with the predictions or the experimental measurements.

A summary of the measurements of different lepton flavour universality testing ratios is presented in figure Figure 6. Because of the rarity of the process, the measurements have much larger experimental uncertainties than those of the semileptonic decays. Decays involving $b$- to $d$-quark transitions are even rarer due to the structure of the Cabibbo-Kobayashi-Maskawa matrix and so far neither the $B$-factory experiments or LHCb have been able to provide a measurement of a final state with an electron-positron pair. If the mass of the lepton pair is sufficiently large, then decays to $e^+e^-$ and $\mu^+\mu^-$ final states are expected to happen at roughly the same rate in the Standard Model. The latest measurements are consistent with this expectation. Earlier measurements by LHCb collaboration had shown discrepancies with Standard Model predictions at the level of around three standard deviations. Those measurements had indicated a deficit in the rate of decays to $\mu^+\mu^-$ final-states compared to those with $e^+e^-$ final-states. A reexamination of sources of experimental background in the $e^+e^-$ data set in the 2022 analysis of $B^+ \to K^+ \ell^+\ell^-$ and $B^0\to K^{*0}\ell^+\ell^-$ decays lead to ratios consistent with one (LHCb, 2022). In the latest measurements the rates of decays to both $\mu^+\mu^-$ and $e^+e^-$ final-states are lower than expected in the Standard Model.

Lepton flavour universality in extensions of the Standard Model

As discussed above, only the interaction with the Higgs has a non-universal coupling to the different flavours of charged-leptons in the Standard Model. A natural starting place to explain the excess of semileptonic $\tau$ decays would be an extension of the Standard Model Higgs sector. Introducing a second Higgs doublet could lead to the presence of a charged Higgs boson that preferentially couples to the heavy $\tau$ lepton. This type of extension can naturally arise in supersymmetric extensions of the Standard Model. In these models the largest effects are typically seen amongst the heaviest (third generation) particles but smaller effects could be seen in the lighter generations. These models could also provide solutions to some of the other open questions in particle physics, e.g. providing a dark matter candidate in the case of supersymmetric extensions of the Standard Model.

The pattern of discrepancies that existed before the LHCb's 2022 measurement $B^+ \to K^+ \ell^+\ell^-$ and $B^0\to K^{*0}\ell^+\ell^-$ decays led to the study of different classes of model that may have a deeper connection with the flavour structure of the Standard Model. This included models with hypothesised leptoquarks (Buttazzo et. al., 2017), which carry both quark and lepton quantum numbers. In many cases, models predicting non-universal couplings will also generate processes that are lepton flavour violating. There is no experimental evidence of charged lepton flavour violation. This would be an unambiguous sign for physics beyond the Standard Model.

Outlook

Lepton flavour universality has been tested in a wide variety of strange-, charm- and beauty-hadron decays. The majority of the measurements are found to be consistent with the picture expected in the Standard Model. However, a long-standing discrepancy exists in measurements of semileptonic beauty-hadron decays that appear to have a larger probability to decay to $\tau$ leptons than predicted by the Standard Model. In order to understand the pattern of measurements more data will be needed. Fortunately, there are two running experiments that will provide much larger data sets in the next 5-10 years. A significant upgrade has been made to the Belle detector at KEK. Belle 2 started taking data in 2018 and aims to collect a data set that is roughly 50 times larger than the original Belle experiment. The LHCb experiment at the LHC has also recently gone through a major upgrade programme and has recently restarted taking data. These new data sets will allow more precise tests of the decay rates and new measurements comparing the angular distributions of decays to the different charged-leptons.


References

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