# Effects of Family Nonuniversal Boson on Leptonic Decays

of Higgs and Weak Bosons

###### Abstract

Though not completely a surprise according to the standard model and existing indirect constraints, the Higgs-like particle, , of mass around 125 GeV recently observed at the LHC may offer an additional window to physics beyond the standard model. In particular, its decay pattern can be modified by the existence of new particles. One of the popular scenarios involves a boson associated with an extra Abelian gauge group. In this study, we explore the potential effects of such a boson with family-nonuniversal couplings on the leptonic decays of , both flavor-conserving and flavor-changing. For current constraints, we take into account leptonic decays at the pole, LEP II scattering data, limits on various flavor-changing lepton transitions, and lepton magnetic dipole moments. Adopting a model-independent approach and assuming that the has negligible mixing with the boson, we find that present data allow the effects to reach a few percent or higher on decays into a pair of leptons. Future measurements on at the LHC or a linear collider can therefore detect the contributions or impose further constraints on its couplings. We also consider -mediated four-lepton decays of the and bosons.

###### pacs:

12.60.-i, 14.70.Pw, 14.80.Bn, 12.60.Cn## I Introduction

The recent discovery at the LHC lhc () of a new particle having mass about 125 GeV and other properties compatible with those of the standard model (SM) Higgs boson undoubtedly has far-reaching implications for efforts to search for new physics beyond the SM. Particularly, all new-physics models would have to include such a particle, hereafter denoted by , as one of their ingredients. In general, different models would have different production rates and decay patterns for because of contributions from and/or mixing with other new particles. It is therefore important to have a detailed study of the characteristic quantum numbers of and its interactions with known SM particles. It is hoped that through such an analysis, the newly discovered particle will provide us with hints of new physics.

One of the possible scenarios for new physics is the existence of an extra U(1) gauge group involving a massive gauge particle, the boson. Such a gauge symmetry may have its origin from various grand unified theories, string-inspired models, dynamical symmetry breaking models, and little Higgs models Nardi:1993ag (); Langacker:2008yv (), just to name a few. The boson in different representative models has been directly searched for at colliders as well as indirectly probed via a variety of precision data brooijmans-pdg (); delAguila:2011yd (), putting limits on its gauge coupling and/or mass. Generally speaking, the couplings to SM fermions can be family universal or nonuniversal. The latter case has especially attracted a lot of interest in recent years due to its many interesting phenomenological implications Yue:2002ja (); Chiang:2011cv ().

In this work, we focus on family-nonuniversal interactions of the with the neutrinos and charged leptons and explore constraints on its relevant couplings from a number of experiments on transitions involving leptons in the initial and final states, plus possibly a photon. These processes suffer less from QCD corrections and hadronic uncertainties than processes involving hadrons. Moreover, many of such experiments have been performed at a high precision, imposing relatively stringent constraints on any possible new interactions.

More specifically, we assume that the boson arises from a new U(1) gauge symmetry, interacts with leptons in a family-nonuniversal way, and has negligible mixing with the boson for simplicity, but otherwise adopt a model-independent approach to make the analysis as general as possible. Due to the family nonuniversality, such a boson would feature flavor-changing leptonic couplings at tree level, leading to the distinctive signature of lepton-flavor violation. However, lepton-flavor violating processes have been searched for at colliders with null results. We therefore examine a number of flavor-conserving and flavor-changing processes to evaluate constraints on the leptonic couplings. The results are then used to estimate contributions to both flavor-conserving and flavor-changing decays of the Higgs boson into a pair of leptons, , at the one-loop level. As will be probed with increasing precision at the LHC in coming years, and even more so if a Higgs factory is built in the future, the acquired data could reveal the signals of the boson which we consider, assuming that is a SM-like Higgs boson. Last but not least, we will compute the effects on several four-lepton decays of the and bosons and make comparison with the data if available. The LHC may also be sensitive to such indirect indications of the presence. The information on the gained from the , , and measurements would be complementary to that from the direct searches.

This paper is organized as follows. We present the interactions of the boson with the leptons in Section II, allowing for in particular flavor-nonuniversal couplings. In Section III, we study constraints on the couplings of the from -pole data, cross sections of scattering into lepton-antilepton pairs measured at LEP II, various experimental limits on low-energy flavor-changing processes involving charged leptons, and measurements of their anomalous magnetic moments. In Section IV, we proceed to make predictions on both flavor-conserving and flavor-changing decays of into a pair of charged leptons. We also explore the contributions to and decays into four leptons. We summarize our findings in Section V.

## Ii Interactions

The Lagrangian describing the interactions of the boson with neutrinos and charged leptons can be expressed as

(1) |

where summation over is implied, the primes of the lepton fields refer to their
interaction eigenstates, , and the parameters
are generally different from one another, reflecting
the family nonuniversality.^{1}^{1}1Note that throughout the paper, we use to denote
the triplet of charged leptons, ,
and to refer to an individual charged lepton in general.
The Hermiticity of requires these coupling constants to be real.
Since each of the left-handed neutrinos and its charged counterpart form a SM weak doublet,
they share the same .
Since we are concerned with processes below the electroweak scale, we do not consider right-handed
neutrinos in the low-energy spectrum.
The may also have couplings to quarks and other nonstandard fermions, but we do not
address them in this analysis.

The interaction states are related to the mass eigenstates and by

(2) |

where for fermion and the 33 matrices are unitary. In terms of the mass eigenstates, one can then write

(3) |

where summation over is implied and

(4) |

It follows that

(5) |

where . Hence family nonuniversality implies that the interactions with the leptons can be flavor violating at tree level. Furthermore, and are generally unequal.

## Iii Constraints

### iii.1 High-energy observables

We begin with the determination of constraints from the existing data on the -boson decays and . For the former, the amplitude takes the form

(6) |

where and contain both SM and contributions and are given by

(7) | |||

(8) |

with as usual the weak coupling constant and
involving the Weinberg angle.
In the absence of - mixing,^{2}^{2}2This is a reasonable approximation based on
the findings of various analyses that the mixing parameter typically has an upper bound
inferred from data of a few times for
GeV or lower for greater
masses Langacker:2008yv (); Chiang:2011cv (); Hook:2010tw ().
Moreover, there are scenarios in which - mass-mixing is absent, because no Higgs bosons
in the theory carry both the electroweak and extra-U(1) quantum numbers, and kinetic mixing
between the hypercharge and extra-U(1) gauge bosons is naturally small Carone:1994aa ().
the effects modify the vertex and
leptonic self-energy diagrams at the one-loop level.
We obtain

(9) | |||||

where Li is the dilogarithm. The expression for the real part of has been derived previously Carone:1994aa (). The relevant observables here are the forward-backward asymmetry at the pole and decay rate

(10) |

where

(11) |

In , the -loop contributions are analogous to those in the charged-lepton case, but without the right-handed couplings. Since the unobserved neutrinos in the final state may belong to different mass eigenstates, we can express the amplitude as

(12) |

Since and are related according to Eq. (5), summing over the final neutrinos then results in the decay rate

(13) |

Accordingly, this channel may offer a complementary probe for . From the formula for in Eq. (III.1), we can then see that these decay modes are potentially sensitive to not only the flavor-conserving couplings, but also the flavor-changing ones.

These -pole observables have been measured with good precision. The experimental and SM values of are pdg (), in MeV,

(14) |

while those of are pdg ()

(15) |

For , we have pdg (), also in MeV,

(16) |

Numerically, for the SM contributions we employ the tree-level formulas in Eqs. (6)-(8) and (12) along with the effective values

(17) |

which allow us to reproduce the SM numbers in Eqs. (III.1) and (III.1) within their errors and obtain MeV in agreement with , indicating that other SM invisible modes are negligible. To extract the upper limit on , we assume it to be the only nonvanishing coupling subject to the 90% confidence-level (CL) ranges

(18) |

the rate numbers being in MeV.^{3}^{3}3We have taken the lower (upper) bound of
to be its SM lower (upper) value because is
above is below
by 2 sigmas.

We find that the constraint on from is weaker (stronger) than that from for . Assuming that only one flavor-conserving coupling is nonzero at a time, we show the results for TeV in Fig. 1(a), where the displayed curves represent the stronger limit in each case. The blue dotted (dotted-dashed) curve refers to , the green short-dashed (long-dashed) curve , and the red solid curve . The curve for coincides with that for because of the constraint. The horizontal solid straight line for above 500 GeV marks the perturbativity limit, , which we have imposed as an extra requirement on the couplings. We present another view on the results in Fig. 1(b), which has the corresponding limits on the couplings divided by the mass.

If instead only one of the flavor-changing couplings is nonvanishing at a time, we can get its upper limit also from Fig. 1, in view of in Eq. (III.1). Accordingly, the limits on , , and are the same as those on , , and , respectively, for or . As we will see later, there may be stronger bounds on some of these individual flavor-changing couplings from other measurements, depending on the mass.

Since -mediated diagrams can contribute at tree level to scattering, its data can provide additional restrictions on the couplings. Here we will use LEP-II measurements at various center-of-mass energies above the pole, from 130 to 207 GeV Alcaraz:2006mx (). In the absence of - mixing, the amplitude if is

where is the electromagnetic fine-structure constant, the lepton masses have been neglected, denote the total widths, the plus sign of the -channel term follows from Fermi statistics, , and .

In the absence of flavor-changing couplings, thus the last line of Eq. (III.1), the resulting cross-section and forward-backward asymmetry , with , are known in the literature (see, e.g., Chiang:2011cv (); Kors:2005uz ()). As mentioned in Ref. Chiang:2011cv () (which also has the more general formulas in the presence of - mixing), the expressions for these observables imply that in a model-independent study their experimental values cannot lead to restrictions on the individual flavor-conserving couplings, assumed to be free parameters, but can nevertheless still restrain their products, .

Since we have left the total width, , unspecified in concentrating on
its couplings to leptons and since it would be needed to compute the cross sections
if , we evaluate the limits on from
the LEP II data only for values starting from 210 GeV up to 3 TeV.
Using as before Eq. (17), along with the effective value
, and the 90%CL ranges of
the experimental numbers Alcaraz:2006mx (),^{4}^{4}4A few of
the and measurements disagree with their
SM predictions by about sigmas or more. In each of those cases, we take the lower (upper)
bound of the required range of the relevant observable to be its SM lower (upper) value if
the measurement is above (below) the SM prediction. we draw
Fig. 2(a) for the upper limits on the products
(green long-dashed curve), (green dotted-dashed curve),
(green short-dashed curve), (red dotted curve),
(red double-dotted-dashed curve), and
(red solid curve).
Since the positive and negative limits on these coupling products are not generally symmetric
with respect to zero, we present Fig. 2(b) for the negative limits.
The plots on the right (c and d) depict the corresponding limits for
.

If one of the flavor-changing couplings in Eq. (III.1) does not vanish, we can constrain it separately, assuming . In that case, the cross-sections have the expressions given in the Appendix, and so is not required in the calculation. Using the LEP II data again, we obtain the upper limits on for TeV in Fig. 3. The second plot displays the corresponding limits on . These results are evidently more stringent than the bounds on inferred from Fig. 1.

### iii.2 Low-energy processes

Turning our attention now to constraints on the couplings from low-energy data, we will consider a number of lepton flavor-violating processes and leptonic anomalous magnetic moments. The relevant formulas are available from the expressions given in Ref. Chiang:2011cv () in the more general case with - mixing.

We begin by mentioning an additional process that can restrict separately, namely the muonium-antimuonium conversion, . In this case, the constraints are the same as those determined in Ref. Chiang:2011cv (),

(20) |

These are stricter than their counterparts in Fig. 3 if goes below 30 GeV.

We next look at various constraints on the products of a pair of different couplings of the , at least one of them being flavor changing, from the experimental limits pdg () for flavor-changing leptonic 3-body and radiative 2-body decays of the and leptons. The leptonic decays arise from tree-level -mediated diagrams, whereas the radiative decays proceed from loop diagrams containing the and an internal lepton. Assuming that only the two couplings in each of the products are present at a time, we collect the results in Table1.

The anomalous magnetic moments and of the electron and muon have been measured very precisely and therefore can provide extra constraints. In contrast, the experimental information on is still too limited to provide significant bounds pdg (). For the contributions to and , we will retain only the terms induced by the lepton in the loop, as they are enhanced by the mass compared to the other lepton terms Chiang:2011cv (),

(21) |

The SM prediction for is compatible with its most recent measurement, the difference between them being Aoyama:2012wj (). Consequently, we can impose the 90%CL range . In contrast, the SM and experimental values of presently differ by nearly 3 sigmas, Aoyama:2012wk (). This suggests that we may require . It follows that

(22) |

The limits are complementary to the individual bounds on illustrated in Fig. 3.

Decay | Measured | Derived limits on products of |
---|---|---|

mode | limits pdg () | in GeV |

## Iv Predictions

The results above allow us to explore how the effects may modify the leptonic decays of the newly found particle, , assumed to be a SM-like Higgs boson, and also those of the weak bosons ( and ). We will deal with both flavor-conserving and -violating channels involving two and four leptons in the final states.

### iv.1 Two-body decays

We first look at the flavor-violating decay . It proceeds from a -loop diagram having an and two lepton- vertices, at least one of the latter being flavor changing, and one-loop diagrams with flavor-changing couplings. We calculate the amplitude to be

(23) |

where GeV is the Higgs vacuum expectation value,

(24) |

with being the fermion in the loop. The rate for is then

(25) |

To predict the largest rates, we observe from the bounds derived in the last section and Eq. (IV.1) that the most important contribution comes from the internal lepton and that the rates are maximized for final states with one . Thus, we take and from the bounds in Table1. For definiteness, we take GeV, compatible with the average mass of GeV from the LHC measurements lhc (). With only one nonzero product of couplings being present in each case and MeV lhctwiki (), we obtain for TeV

(26) |

the upper bound occurring at TeV, and a somewhat smaller number for . Clearly these -induced flavor-violating Higgs decays will not be observable in the near future.

We next consider the impact on the flavor-conserving decay . The contribution follows from Eq. (23) after setting . Combining the result with the SM tree-level contribution, we then get

(27) |

leading to

(28) |

We expect again that the internal lepton in the loop yields the maximal impact. Accordingly, for and we take, respectively, the and ranges in Eq. (22), assuming that the coupling products are purely real or dominated by the real part. The graphs in Fig. 4 depict the resulting fractional change

(29) |

in the rate due to the presence of exclusively via these flavor-changing couplings. Although the contribution can reduce the rate sizeably [blue-shaded areas in Fig. 4(a)], this decay mode, with a branching ratio of in the SM, may be beyond reach for a long time. Much more interesting is , which has a SM branching ratio of and therefore may be measurable in the not-so-distant future with precision possibly sensitive to the effect, indicated by the green-shaded areas in Fig. 4(b).

As it turns out, potentially more considerable modifications to the rate can be induced by the flavor-conserving couplings , which enter as the product . To estimate their maximal impact, we focus on the cases. In each of them, we set all the other couplings to zero and scan the values of satisfying the requirements in Eq. (III.1) as well as the perturbativity condition . We find that the coupling values allowed by these constraints can translate into substantial that are positive or negative. Especially, in the GeV region, the decrease in the rate could reach a few tens percent, whereas the increase could exceed 100%, even up to 300%, as can be seen in Fig. 5.

The decay has begun to be observable at the LHC. The latest signal strengths for this channel reported by the ATLAS and CMS Collaborations are and , respectively, at GeV h2tt (). Obviously, these early findings already disfavor some parts of the parameter space implied by Fig. 5(b), although it is still too soon to be quantitative about the exclusion zones in view of the sizable uncertainties of these current data. As their precision continues to improve, the upcoming experiments can either uncover a signal or restrain its couplings further. A similar expectation can be stated regarding the couplings from future measurements of the mode.

The flavor-violating decays are not yet observed, but have been searched for, the experimental upper-limits on their branching ratios being pdg ()

(30) |

at 95%CL, where . These modes get -loop contributions analogously to those in and hence may provide further limits on the couplings if the experimental limits are less than the predicted values derived from the upper limits on the couplings. To make the predictions for the , , and final states, we take the biggest values