# Phenomenological constraints on minimally coupled exotic lepton triplets

###### Abstract

By introducing a set of new triplet leptons (with nonzero hypercharge) that can Yukawa couple to their Standard Model counterparts, new sources of tree-level flavor changing currents are induced via mixing. In this work, we study some of the consequences of such new contributions on processes such as the leptonic decays of gauge bosons, and which violate lepton flavor, and - conversion in atomic nuclei. Constraints are then placed on the parameters associated with the exotic triplets by invoking the current low-energy experimental data. Moreover, the new physics contribution to the lepton anomalous magnetic moments is calculated.

###### pacs:

12.60.-i, 13.40.Em, 14.60.Hi## I Introduction

The discovery of neutrino oscillations neutrinos_exp has long been suggestive of new physics in the lepton sector. It provides compelling evidence for nonzero neutrino masses, and hints of possible lepton flavor violation (LFV). However, it is well-known that the minimal Standard Model (SM) cannot incorporate these new ingredients, so it must be extended in one way or the other as a result. Clearly, there is a huge variety of approaches for introducing new physics. Nevertheless, from the point of view of phenomenological studies, the most essential part of any models is the effective couplings induced between ordinary SM particles and the exotic ones. Therefore, even without specifying the underlying mechanisms (or UV completions) that give rise to these operators, a lot of useful analyses on the new particles can be studied. This is the approach we shall adopt in this work.

Whilst there are potentially many different new effective operators which can lead to interesting phenomenologies, our main focus here is motivated by the generic minimal couplings of the form

(1) |

where denotes the coupling strength. Such minimal interactions are of interest because it is relatively simple and may lead to well-defined collider signatures DelNobile:2009st which may be seen at the LHC in the near future.
Since we would like to concentrate on the lepton sector alone, we take all particles in (1) to be *uncolored* (in the sense) but allowing the ‘‘exotic particle’’ to be either a scalar boson, a fermion or a vector boson.
With these choices and the requirement of renormalizability, there are five distinct types of interaction with the SM fields (schematically)^{1}^{1}1We have ignored terms like (SM Higgs)(SM Higgs)(new boson) because they do not involve any type of leptons.,

(2) |

where is the left-handed (LH) lepton doublet, is the right-handed (RH) lepton singlet, and denotes the SM Higgs doublet. Suppose these interaction terms must also obey Lorentz and SM gauge symmetries, then there are only 13 types of exotic multiplets (see Table 1) which fit either one of the setups in (I).
Furthermore, it is perhaps obvious that the majority of the new particles implied by these minimal couplings
have already been closely studied due to other motivations. However, to the best of our knowledge, the exotic lepton triplets with *nonzero* hypercharge, , and the doublets, (see Table 1 for their transformation properties) have received very little attention.^{2}^{2}2Recently, the -like triplets were mentioned in the context of neutrino mass generation involving a triply charged Higgs Babu:2009aq .

So, the aim of this work is to fill part of that gap by investigating in some details the implications of introducing to the SM.^{3}^{3}3The analysis of the exotic doublets, , shall be presented elsewhere. We begin by elucidating the formalism used to analyze the system in the next section,
before deriving various experimental constraints on the relevant new physics parameters in subsequent sections (with a summary of all constraints and fits collected in Sec. V). Processes such as and decays (Sec. III), LFV decays (e.g. , ) and - conversion in atomic nuclei (Sec. IV) are considered as a result, while discussion on the new physics contribution to the lepton anomalous magnetic moments will also be included (Sec. VI).

[new] | spin | type | SM fields involved | studied in | ||

0 | 2 | (ii) | multi-Higgs doublet models multi_Higgs_models ; rad_seesaw_eg | |||

0 | 1 | (i) | dilepton/Babu-Zee models rad_seesaw_eg ; dileptons1 ; dileptons2 ; babu_zee | |||

0 | 3 | (i) | dilepton/Type-II seesaw dileptons1 ; doubly_Higg ; type2seesaw ; Abada:2007ux | |||

0 | 1 | 2 | (iii) | dilepton/Babu-Zee models dileptons1 ; dileptons2 ; babu_zee ; babu_zee2 | ||

1/2 | 1 | (iv) | Type-I seesaw Abada:2007ux ; type1seesaw ; Biggio:2008in ; type1_3 ; type1set2 | |||

1/2 | 3 | (iv) | Type-III seesaw Abada:2007ux ; type1_3 ; type3seesaw ; Abada:2008ea | |||

1/2 | 2 | (v) | 4th generation leptons 4thgen | |||

1/2 | 1 | (iv) | 4th generation leptons 4thgen | |||

1/2 | 3 | (iv) | ( only) | — rarely discussed — |
||

1/2 | 2 | (v) | ( only) | — rarely discussed — |
||

1 | 1 | (i) & (iii) | & | |||

1 | 2 | (ii) | GUT/dilepton boson models dileptons1 ; dilepton_boson | |||

1 | 3 | (i) |

## Ii Model with exotic lepton triplets,

In order to identify and study the new phenomenologies arising from the mixing with the exotic lepton triplets (and to establish the notations), we shall begin by describing the model in detail. Consider adding to the SM two sets of new leptons (RH plus LH) which transform as triplets in and all carrying hypercharge of (where we have defined ). We can conveniently group them together in a matrix representation as follows

(3) |

where and are independent fields and both transform as under the SM gauge group.^{4}^{4}4As a result of the identical transformation properties for RH and LH fields, chiral anomalies cancel automatically. In the following, we shall also introduce three RH neutrino fields, , so that neutrinos can have a Dirac mass. However, we do *not* include a Majorana mass term (e.g. ) in the Lagrangian (and hence no seesaw mechanism^{5}^{5}5A full discussion on the mixing effects due to seesaw models can be found in Abada:2007ux ; Antusch:2006vwa .) for simplicity. Thus, the Lagrangian of interest is given by

(4) |

where , and the covariant derivative,

(5) |

with and being the operators for electric charge and the 3rd component of isospin respectively. In (4), the Yukawa term involving defines the minimal coupling between SM leptons and while sets the energy scale of the new physics. It is worth pointing out that SM symmetries forbid a similar type of minimal couplings for with other SM leptons, and hence enters into this picture only via the mass terms. Writing out all the relevant interactions in (4), we have

(6) |

where

(7) | ||||

(8) | ||||

(9) | ||||

(10) |

In getting (9) and (10), we have written , where is the Higgs vacuum expectation value, and are the would-be Goldstone bosons. Also, we have defined and .

To deduce the mixing between SM leptons and the components of the exotic triplet, it is convenient to package the LH and RH fields in the following way

(11) |

and rewriting (7) to (10) in matrix forms. In particular, for we obtain

(12) |

Without loss of generality, one can choose to work in the basis where and are real and diagonal (which is what we have already assumed in writing out (12) above). All fields are related to their mass eigenbasis via the unitary transformations

(13) |

where the subscript indicates the mass basis. In general, and are matrices with denoting the number of generations for the exotic fields. To , the transformation matrices are given by^{6}^{6}6In the definition of , the neutrino right diagonalisation matrix (from ) has already been absorbed into in (12). In other words, .

(14) | ||||

(15) |

where

(16) |

are and matrices in flavor space respectively, while is the unitary matrix that transforms into its mass eigenbasis. At this order, may be identified as the usual neutrino mixing matrix, .

Hence, and with respect to the mass eigenbasis become

(17) | ||||

(18) | ||||

(19) |

with the new generalized coupling matrices given by (to leading order)

(20) | ||||

(21) | ||||

(22) | ||||

(23) |

Note that each upper-left -block in (20) to (23) corresponds to the modified mixing matrix for the respective interaction involving SM leptons. In particular, we observe that new contributions to tree-level flavor changing currents would be provided by the nonzero off-diagonal entries of matrix . Furthermore, these -submatrices that define the new mixings between ordinary leptons are now in general non-unitary.

Suppose we define the non-unitary mixing matrix which is responsible for charged current mixing as

(24) |

then we note that, at first order in , observable effects mediated by and may be conveniently re-cast as follows

(25) | ||||

(26) |

Expressions (25) and (26) are analogous to those derived and subsequently analyzed in Abada:2007ux for seesaw models. It is worth pointing out that the structure displayed in (26), though looks similar to its counterpart in Abada:2007ux , they are definitely *not* identical in form. The small difference comes from the fact that these exotic triplets carry nonzero hypercharges which resulted in the assignments being different from the seesaw situations.

Although this viewpoint of linking the new physics to the non-unitary in weak mixing can be useful, for the purpose of our investigation here, we have found it more convenient to use the expressions written in (17) to (19) for doing the calculations. Henceforth, we shall present all our discussions in terms of rather than .

## Iii Constraints from and decays

As hinted earlier, elements of the matrix (see (16) for definition) which encapsulate all the essential information regarding triplets , are the key to any new physics contributions to the electroweak processes considered in this paper. Amongst them, the most basic interactions are the tree-level and decays into SM leptons. As we shall see, these processes can provide constraints on all elements of although the restriction for the off-diagonal entries are not as stringent as those obtained from other LFV interactions (see Sec. IV). Nonetheless, their constraints for the diagonal elements of will be useful in the later analysis of the anomalous magnetic moments (see Sec. VI).

### iii.1 decays

The rate for decaying into a lepton of flavor plus a neutrino may be straightforwardly obtained by invoking the relevant interaction terms in (17). Using the usual approximation of negligible final lepton masses and in the centre of mass frame, one gets

(27) |

where we have only kept the leading order terms in . In (27), is the mass of while is the Fermi constant extracted from muon decay when assuming *only* SM physics.

In order for (27) to be a useful bound on the elements of , one must also study the modification to the value of “” as measured from muon decay experiments ( + missing energy) in the presence of the new physics due to triplets . It is not difficult to see from (20) and (21) that additional tree-level flavor changing currents mediated by and are expected to give rise to a new definition for the Fermi constant. In terms of the SM version of , we have to leading order

(28) |

Using (28) in (27), one can rearrange the expression to obtain a global constraint on and in terms of experimental parameters:

(29) |

Putting in the respective values from Nakamura:2010zzi ,^{7}^{7}7Note that what we have labelled as is simply the experimentally measured Fermi constant GeV. we arrive at the following bounds for :

(30) |

As expected, this quantity (within experimental uncertainties) is very close to 1 — the limit where the new physics is decoupled.

### iii.2 decays

In this subsection, we investigate the bounds for the elements of coming from decaying into charged leptons: . The cases will place restrictions on ’s whereas for , the off-diagonal entries can be constrained.

Applying the usual formalism on the modified couplings in (21), the decay rate, , may be easily written down (in the usual massless limit for the final state leptons) as

(31) |

where we have again included the correction to the Fermi constant. and are the usual Weinberg angle and boson mass respectively. Putting the decay widths obtained from experiments Nakamura:2010zzi into (31) for each lepton flavor , one gets a system of three equations in the ’s. Solving these simultaneously then yields

(32) |

These results should be checked against the values obtained in the decays for consistency. Taking into account the uncertainties in , we have found that the bounds displayed in (32) are compatible with those in (30). Although one may worry about the negative sign in front of , this outcome is not unexpected given that there is also a minus sign in the definition of (24).

Next, we turn our attention to the case where . The decay rate is given by

(33) |

Clearly, in the limit , this rate disappears. This is in accordance with the fact that there is no flavor changing neutral currents (FCNC) at tree-level in the SM. Writing this as a branching ratio and keeping only the leading order terms in the denominator, one has

(34) |

From this, we can derive the following bounds for :^{8}^{8}8Note that the LFV branching ratios quoted in Nakamura:2010zzi is in fact the experimental values for . Therefore, the expression in (34) must be multiplied by a factor of 2 before applying the experimental numbers.

(35) | |||

(36) | |||

(37) |

Notice that since is hermitian (as we are working in the basis where is real and diagonal), necessarily holds.

## Iv Constraints from LFV decays of charged leptons and - conversion in atomic nuclei

Some of the strongest constraints on the new physics come from the studies of lepton flavor violating decays of ordinary charged leptons. Therefore, in the following two subsections, we present our analysis of the contributions induced by the exotic triplets on charged lepton processes like and . Furthermore, in the third subsection, we shall take a look at the bound coming from experiments studying the muon-to-electron conversion in atomic nuclei as it is well-known Bernabeu:1993ta that such process can give rise to a very strong constraint on the -- vertex.

### iv.1 tree-level decays

Given three generations of ordinary leptons, there are only three generic types of final lepton states possible for a charged lepton decaying into three lighter ones: , and , where , with denoting the flavor of the decaying lepton. For all of these cases, the mediating particle can be either the gauge boson or the Higgs boson . However, the amplitude associated with the Higgs is suppressed by a factor of , where and denote the lepton and Higgs masses respectively. Thus, we may ignore their contributions to a good approximation.

Extracting the relevant coupling from (21), and invoking the usual assumption of negligible final state masses, we get the following formulae for the branching ratios:

(38) | |||||

and | |||||

(39) | |||||

(40) |

where we have kept only the leading order terms.

For (38), there are three kinematically allowed processes (), which lead to the constraints

(41) | |||

(42) | |||

(43) |

while (39) has two possibilities ( and ), yielding

(44) | |||

(45) |

Finally, we have

(46) | |||

(47) |

from another two possibilities ( and ) allowed by (40). Note that in deriving (41) to (47), we have used the branching ratios from Nakamura:2010zzi .

### iv.2 radiative decays via one loop

Another type of LFV processes that has received enormous amount of attention is the radiative decays of charged leptons (). There is continually much experimental effort on improving the bounds associated with these rare interactions.^{9}^{9}9For a review, see for example Marciano:2008zz . The current MEG experiment exp_MEG located at PSI is expected to reach a sensitivity of for the branching ratio, which is a significant improvement compare to the current limit of Brooks:1999pu . In addition, the Super KEKB project exp_SuperB will provide the platform for investigating LFV decays at an unprecedented precision. As a result, the bounds on and are also expected to tighten.

Generically (because of gauge invariance), the transition amplitude for is given by the dimension-5 operator of the form

(48) |

where and correspond to the transition magnetic and electric dipole form factors^{10}^{10}10It is understood that and are dimensionful quantities when written in this form. Also, we have absorbed the extra factor of which is usually associated with the definition of the electric dipole moment into . respectively. In writing this down, we have used the on-shell condition, , and , where and denote respectively, the photon 4-momentum and polarization.
In the SM with neutrino masses (), it is well-known that electroweak interactions involving the bosons in a loop (see Fig. 1a) can give rise to a finite value for this amplitude although its size turns out to be vanishingly small because Cheng:1985bj ; mu2eg_SM . However, the situation may change drastically when there are new couplings to the SM leptons, such as those involving the exotic triplets studied here.

From Lagrangians (17) to (19), we can identify all the new interactions and subsequently calculate the corresponding loop amplitudes from the definitions of the modified coupling matrices given in (20) to (23). Working in the unitary gauge where only diagrams associated with the physical degrees of freedom are relevant, there are three types of one-loop graphs which may contribute to the LFV process (see Fig. 1). As a result of the direct involvement of the triplet particles and in these diagrams, stringent constraints on can be derived. These expressions are particularly useful when the expected improvement in experimental bounds are realized in the near future.

In calculating the amplitude for the lowest-order graphs in Fig. 1, we note that any terms in (48) that is proportional to (or ) will not contribute to the final answer as they cannot be transformed into the electromagnetic moment form Cheng:1985bj . We can separate out this unwanted component from (48) using Gordon identity, and get

(49) |

where we have again used when simplifying the expression. In (49), is the momentum of while denotes the mass. Working in the limit where the final state lepton is assumed to be massless (), one finds that amplitudes and become identical to leading order in , and thus in the explicit computation, we simply require to evaluate the coefficient of the terms for all graphs.

Because we wish to work in the unitary gauge where there are less diagrams to consider,^{11}^{11}11One drawback is that some of the intermediate expressions/steps would be considerably more complicated than in other approaches. our strategy is to perform the calculations in the notations of the generalized renormalizable () gauge Fujikawa:1972fe , and at the end of the computation, we take the limit to obtain the desired results.^{12}^{12}12We have adopted the definition of as used in modern textbooks Cheng:1985bj ; Peskin:1995ev , which is equivalent to the parameter as appeared in Fujikawa:1972fe . Moreover, we will work exclusively in the and limits (where represents the mass of the internal -flavor SM lepton) and will only keep the leading order terms.

After the dust has settled, we obtain the following expressions for the amplitudes of the one-loop contributions shown in Fig. 1 (superscripts and subscripts denote the type of internal leptons and bosons involved respectively):

(50) | ||||

(51) | ||||

(52) | ||||

(53) |

(54) | ||||

(55) |

with

(56) | ||||