# A “Vector-like chiral” fourth family to explain muon anomalies

###### Abstract

The Standard Model (SM) is amended by one generation of quarks and leptons which are vector-like (VL) under the SM gauge group but chiral with respect to a new gauge symmetry. We show that this model can simultaneously explain the deviation of the muon as well as the observed anomalies in transitions without conflicting with the data on Higgs decays, lepton flavor violation, or mixing. The model is string theory motivated and GUT compatible, i.e. UV complete, and fits the data predicting VL quarks, leptons and a massive at the scale, as well as and within reach of future experiments. The Higgs couplings to SM generations are automatically aligned in flavor space.

## 1 Introduction

The Standard Model (SM) is a highly successful theory in predicting and fitting many experimental measurements, with few exceptions. One of the discrepancies between the SM prediction and experimental measurement that has been known for a long time, is the muon anomalous magnetic moment. The discrepancy between the measured value and the SM prediction is [1, 2]

(1) |

More recently, there appeared deviations from the SM predictions in transitions related to tests of lepton flavor universality in the observables and [3, 4], semi-leptonic branching ratios [5, 6, 7], and angular distributions [8, 9, 10, 11, 12, 13, 14]. Most interestingly, all of the more recent anomalies can simultaneously be explained [15, 16, 17, 18, 19, 20, 21, 22, 23] by specific deviations from the SM in one or more of the Wilson coefficients , , and/or of the effective Hamiltonian [24, 25] (see [26] for a possible role of additional tensor operators)

(2) |

where

(3) | ||||||

(4) |

A simple extension of the SM that can explain the discrepancy of the muon are VL leptons that couple exclusively to muons [27, 28, 29]. On the other hand, the anomalies in transitions can be explained by a new massive vector boson of a spontaneously broken gauge symmetry and the introduction of VL quarks [30, 31] (see [32] for a generalization of the new gauge symmetry). Indeed, it has been shown that combining an additional , VL leptons, and VL quarks one can successfully address both the muon and the anomalous physics observables simultaneously [33, 34, 35]. Typically these models predict significant deviations of the SM in [28, 29], [31, 34] and have an upper bound on the mass by keeping oscillations close to their SM value [30].

In the present paper we suggest a holistic way of solving the discrepancies in and . We amend the SM by one complete family of fermions, i.e. a full spinor representation of , which is VL with respect to the SM but chiral with respect to a new spontaneously broken “” gauge symmetry. Under the new gauge group the third SM family and the left-handed part of the new “VL” family have charges and , respectively, while all other fermions are neutral. Our model is motivated by heterotic string orbifold constructions [36, 37, 38, 39, 40, 41, 42, 43], which, in addition to the full MSSM spectrum, typically contain myriads of states which are VL with respect to the SM gauge group, but chiral under new gauge symmetries. In addition, there are many SM singlet scalars that break the additional gauge symmetries, thus giving mass to the vector-like states and the extra gauge bosons. While in earlier constructions these extra states were lifted to the string scale, our model is a prototype of what can happen if at least one extra generation is kept light, i.e. at the scale. Our analysis is not supersymmetric, but it could easily be extended to a supersymmetric model in which case gauge coupling unification is maintained.

We find that the model can simultaneously fit the observed quark and lepton masses, as well as the and anomalies without violating bounds from electroweak precision observables, lepton flavor violating (LFV) decays or mixing. Interestingly, the electroweak singlet Higgs boson couplings in our model are automatically aligned with the SM values to a very high degree. Contrary to [30, 31, 34] there is no upper bound on the mass from the mixing constraint, simply because the “VL” fermions and the simultaneously obtain mass of the order of the breaking scale.

To substantiate our arguments we present two data points that can fit all measured observables while predicting others. The masses of new quarks and leptons, as well as of the new are all at the scale. The in our example has very suppressed couplings to the first family, meaning that production at the LHC is suppressed. mixing is predicted to deviate from the SM at the level of a few percent. There are significant enhancements in and , while is suppressed. Furthermore, our best fit points predict and in reach of upcoming experiments.

## 2 Model

family | |||||||
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[0.3cm] 0 1 0

The model under investigation is the SM with three right-handed neutrinos extended by a complete extra generation of left-chiral fields and a complete extra generation of right-chiral fields. Furthermore, we introduce a new “” gauge symmetry under which the third SM generation as well as the left-chiral part of the fourth generation of particles is charged. The gauge symmetry is spontaneously broken by the vacuum expectation value (VEV) of the new scalar . All fields and their corresponding quantum numbers are summarized in Table 1. The relevant part of the Lagrangian for this study is given by

(5) |

with^{3}^{3}3In our notation is a two component Dirac spinor which can be written
in terms of the two component Weyl spinor, , as .
Then .

(6) | ||||

(7) | ||||

(8) | ||||

(9) | ||||

(10) | ||||

(11) | ||||

(12) |

We take all couplings to be real and – in some GUT spirit – set many of them alike. Couplings to the up quark sector electroweak singlets (, , , and ) are not displayed because they will not be constrained by our analysis. It is summed over the repeated indices of the flavor structure of the SM families which can, for example, originate from a flavor symmetry [37, 44, 40, 45, 42, 43]. The first and second families are distinguished by the direction of the breaking VEV . We assume this alignment to happen at a high-scale (one should imagine or ), and the corresponding effective operator coefficient, thus, should be imagined as where is a SM and neutral scalar that gets a VEV around the weak scale. This justifies our assumption here that the first family does not directly mix with the VL states. We will focus on the flavor structure of the second and third generations in this study, remarking that the first family can always be fit in. A more detailed analysis should include all three families and their flavor physics, but that is beyond the scope of the present paper.

### Charged Lepton and Down Quark Masses

The charged lepton mass terms are given by

(13) |

where and the scalar VEVs and couplings are all assumed to be real. Analogously, the down quark mass terms are given by

(14) |

with

(15) | ||||

(16) |

The matrix has exactly the same structure as with the replacements , , , and . Let and be unitary matrices that diagonalize the respective mass matrix,

(17) | ||||

(18) |

The physical fields in the mass basis are then given by

(19) |

### Neutrino Masses

Defining the vectors

(20) |

the neutrino masses can be written as

(21) |

where and . The Dirac mass terms have the same structure as with the replacements , , and . The Majorana mass terms have non-zero elements , and with all other elements being zero. Assuming the hierarchy the neutrino mass matrix can be analytically diagonalized and we give details about that in App. A. The physical states are

(22) |

with corresponding masses

(23) |

up to corrections of the order . There are four sterile neutrinos with mass at the high scale. Furthermore, there are two light active neutrinos with mass of order and one (mostly) Dirac neutrino with a scale mass (cf. App. A)

(24) |

Adding the first generation back in gives one additional high scale sterile neutrino and one additional light active neutrino.

### -Lepton Couplings

The -lepton couplings in the mass basis are

(25) |

with coupling matrices

(26) |

The un-hatted coupling matrices are in the gauge basis and given by

(27) |

where , , and we use the abbreviations , . Since these matrices are not proportional to the identity matrix, the -lepton couplings are not diagonal in the mass basis. Hence, this model has LFV boson decays, which are, however, only effective amongst the heavy VL quarks and leptons.

### -Lepton Couplings

The -lepton couplings in the mass basis are

(28) |

where

and | (29) |

with the coupling matrices of the gauge basis

(30) |

### Couplings

The couplings to charged leptons in the mass basis are

(31) |

where

(32) |

with the charge matrices

(33) |

The couplings here are *not* left-right symmetric, recall the charge assignment Tab. 1,
and our skewed definition of the right-handed states in (13) and (16).
The -down quark couplings in the mass basis are completely analogously given by

(34) |

with

(35) |

The mediated flavor changing neutral currents (FCNC) between the SM generations are naturally suppressed because they only arise from the mixing with the heavy VL states.

The couplings to neutrinos in the mass basis can be written as

(36) |

with the coupling

(37) |

and the gauge basis charge matrix

(38) |

### Higgs-Lepton Couplings

The couplings between the physical Higgs boson, , and the charged leptons in the mass basis are

(39) |

where

(40) |

with the gauge basis couplings

(41) |

A very interesting feature of this model is that the masses of the SM families are to a very high accuracy linear in the Higgs VEV. Thus, the Higgs couplings in the mass basis, , are to a high precision diagonal in the lower block. Hence, the Higgs couplings to the SM states are very much SM-like and there are no significant flavor violating Higgs couplings among the SM states. We give an analytic proof of this feature in App. B. Flavor off-diagonal couplings of the VL states (also to the SM states) can be sizable.

## 3 Observables

### Lepton Non-Universality

Generally, our model gives rise to lepton non-universality in the operators by tree-level exchange. The corresponding effective contributions to the Wilson coefficients are

(42) |

with the couplings

(43) | ||||||

(44) |

These couplings are expressible solely through mixing matrix-elements, for example

(45) | ||||

(46) |

While we focus on the coupling to muons in order to explain the observed anomalies, our model also modifies the effective Wilson coefficients and leading to lepton non-universality also in and . Quantitative results for these observables have been obtained using the formulas given in [46] from where we also adopt the SM prediction (cf. also [47, 48]). The NP contributions to the Wilson coefficients affect the SM prediction for and we have followed [49, 50] to quantify these effects in our model.

### Muon Anomalous Magnetic Moment

The , and contributions to the anomalous magnetic moment of the muon are very close to their SM values and we do not detail them here. On the other hand, the contribution can be sizeable, despite its -scale mass. This is due to off-diagonal muon- couplings to VL leptons, allowing them to significantly contribute to the loop. Since the VL leptons and masses are of the same scale, the leading order contribution has one power of less than the naive flavor diagonal contribution (cf. e.g. the discussion in [51, sec. 7.2]). Parametrically, the dominant modification of in our model is of the size

(47) |

where the sum goes over the VL leptons. Naively this points to a scale , but the FC couplings to the VL leptons can easily be . In addition, the contributions of individual VL leptons can partly cancel against one another and we will see this effect to be at work in our numerical analysis below. We give detailed formulas for in App. C.

### Mixing

There is a new tree-level contribution to mixing due to exchange. We adopt the results and numerical factors from [30, 52] and estimate the relative change of the mixing matrix element

(48) |

The most recently updated theoretical uncertainty shows that a deviation from the SM of can currently not be excluded [53]. This gives an important constraint on the down sector flavor changing couplings. In our model the coupling is suppressed because it only arises from the mixing with the heavy VL states such that the can be kept at the scale consistent with the bound derived in [54].

### Lepton Flavor Violating Decays

Tree-level exchange also induces the decay . We follow Ref. [55, 56] to estimate

(49) |

There is also a new contribution to due to and the new VL leptons in the loop, which is enhanced by a power of the VL mass. Using the results of [57] (cf. also [58, 59, 60]) we estimate the leading order contribution to be

(50) |

where the sum is over internal VL leptons. For the numerical analysis we have used a more detailed result which we present in App. D.

### Other Observables

Flavor violating couplings of the or the SM scalar to the SM families are generally suppressed far below their experimental thresholds. In contrast, flavor changing couplings to the heavy VL leptons can be large. Since our is heavy, neutrino trident production [30, 61] does not give any important constraints. Lepton unitarity bounds (cf. e.g. [62, 63]) are easily fulfilled. In addition, there are no constraints coming from the branching ratios for or via loop diagrams, since these contributions are suppressed by factors of (see e.g. [64, 65, 66, 59]).

## 4 Analysis

### Strategy

This model can explain the anomalies in the muon and transitions without obviously conflicting with other experimental data. To demonstrate this, we have constructed a -function including the appropriate errors to simultaneously fit the anomaly in with the value given in (1), and to reproduce two of the best-fit values of [15] (cf. also [16, 17, 18, 19, 20, 21, 22, 23]) for a consistent explanation of anomalies:

(51) | ||||||||||

(52) |

Furthermore we fit the masses , , , and at the weak scale [67] while requiring
to be consistent with the data and theoretical error of mixing.
All other observables are not constrained in the fit, i.e. they arise as predictions of our best fit points.
However, there are currently more parameters than observables and so it is not excluded that there are more points that fit the data well
with different predictions. The first family could always be included in the analysis in a straightforward way without affecting our conclusions.^{4}^{4}4Including the first family in the most straightforward way the Yukawa couplings and hence resulting mass matrices are extended to (and similarly for neutrinos and down quarks)

### Results

We have found two points which give a very good fit for the cases (I.) and (II.), they are listed in Tab. 2 (and in Tab. 4 in App. 4). We cannot find a good fit for .

Best fit point I. | |||||
---|---|---|---|---|---|

= | = | ||||

= | = | ||||

= | = | ||||

= | = | ||||

= | = | ||||

= | = | ||||

= | = | ||||

= |

The predictions of the best fit points are listed in Tab. 3 together with current experimental bounds. Effects on other observables such as , , , neutrino trident production or PMNS unitarity violation have also been considered, but they are robustly suppressed in this model and so we do not discuss them in detail. While is practically absent at tree level, the dominant contribution arises from a one-loop diagram involving and VL leptons in the loop. A rough estimate of this shows that nonetheless comes out well below the current bound of [2].