SISSA 99/2007/EP

December 2007

Probing Majorana Phases and Neutrino Mass Spectrum

in the Higgs Triplet Model at the LHC

A.G. Akeroyd^{*}^{*}*,
Mayumi Aoki^{†}^{†}†
and Hiroaki Sugiyama^{‡}^{‡}‡

1: Department of Physics,

National Cheng Kung University, Tainan, 701 Taiwan

2: National Center for Theoretical Sciences,

Taiwan

3: ICRR, University of Tokyo, Kashiwa 277-8582, Japan

4: SISSA, via Beirut 2-4, I-34014 Trieste, Italy

Doubly charged Higgs bosons () are a distinctive signature of the Higgs Triplet Model of neutrino mass generation. If is relatively light ( GeV) it will be produced copiously at the LHC, which could enable precise measurements of the branching ratios of the decay channels . Such branching ratios are determined solely by the neutrino mass matrix which allows the model to be tested at the LHC. We quantify the dependence of the leptonic branching ratios on the absolute neutrino mass and Majorana phases, and present the permitted values for the channels and . It is shown that precise measurements of these three branching ratios are sufficient to extract information on the neutrino mass spectrum and probe the presence of CP violation from Majorana phases.

PACS index :12.60.Fr,14.80.Cp,14.60.Pq

Keywords : Higgs boson, Neutrino mass and mixing

## 1 Introduction

The firm evidence from a variety of experiments that neutrinos oscillate and possess a small mass below the eV scale necessitates physics beyond the Standard Model (SM). Consequently models of neutrino mass generation which can be probed at present and forthcoming experiments are of great phenomenological interest. In particular, those models which can provide a distinctive experimental signature, such as a New Physics particle with a mass of the TeV scale or less, are especially appealing in light of the approaching commencement of the CERN Large Hadron Collider (LHC).

Doubly charged Higgs bosons () arise in a variety of models of neutrino mass generation as members of , scalar triplets [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] and , scalar singlets [12]. Such particles can be relatively light (i.e., with masses of the electroweak scale) and have impressive discovery potential at hadron colliders due to their low background signature and sizeable cross-sections. The ongoing searches at the Fermilab Tevatron [13, 14] anticipate sensitivity to GeV for the decay channel () with the expected final integrated luminosities of up to 8 fb. LHC simulations [15, 16] show that discovery for TeV is possible with fb, and as little as 1 is needed to probe GeV.

Discovery of with GeV would enable precise measurements of the branching ratios (BRs) of with the anticipated final integrated luminosity at the LHC. Models which predict in terms of experimentally constrained and/or measured parameters are of particular phenomenological interest. In the Higgs Triplet Model (HTM) [3],[4] neutrinos acquire a Majorana mass given by the product of a triplet Yukawa coupling () and a triplet vacuum expectation value . Consequently in the HTM there is a direct connection between and the neutrino mass matrix which gives rise to phenomenological predictions for processes which depend on . Since the coupling determines , this mechanism of neutrino mass generation can be tested if precise measurements of are available [6, 7]. A detailed quantitative study of the dependence of on all the neutrino oscillation parameters has not yet been performed (for previous analyses see [7, 17]).

Of particular interest is the dependence of on the absolute neutrino mass and Majorana phases which is the focus of the present work. Those parameters, which cannot be probed in neutrino oscillation experiments, would significantly affect . We perform a study of the capability of the LHC to probe the neutrino mass spectrum and Majorana phases assuming that neutrino mass is generated solely by the combination in the HTM. In particular, we investigate the possibility to establish and/or CP-violation from Majorana phases at the LHC by means of a analysis with three channels of decays. It is extremely difficult for neutrinoless double beta decay experiments [18] to measure CP-violation from Majorana phases because they affect this process in combination with unmeasured parameters [19], while the absolute neutrino mass can only be measured directly by the future Tritium beta decay experiment [20] if .

Our work is organized as follows: in section 2 we briefly review the HTM; section 3 describes the phenomenology of at hadron colliders; the numerical analysis is contained in section 4 with details of a analysis presented in the appendix; conclusions are given in section 5.

## 2 The Higgs Triplet Model

In the Higgs Triplet Model (HTM) [3],[4] a complex isospin triplet of scalar fields is added to the SM Lagrangian. Such a model can provide a Majorana mass for the observed neutrinos without the introduction of a right-handed neutrino via the gauge invariant Yukawa interaction:

(1) |

Here is a complex and symmetric coupling, is the Dirac charge conjugation operator, is a Pauli matrix, is a left-handed lepton doublet, and is a representation of the complex triplet fields:

(2) |

A non-zero triplet vacuum expectation value gives rise to the following mass matrix for neutrinos:

(3) |

The necessary non-zero arises from the minimization of the most general invariant Higgs potential, which is written as follows [6, 7] (with ):

(4) | |||||

Here in order to ensure which spontaneously breaks to , and is the mass term for the triplet scalars. In the model of Gelmini-Roncadelli [21] the term is absent, which leads to spontaneous violation of lepton number for . The resulting Higgs spectrum contains a massless triplet scalar (majoron, ) and another light scalar (). Pair production via would give a large contribution to the invisible width of the and this model was excluded at LEP. The inclusion of the term ) explicitly breaks lepton number when is assigned , and eliminates the majoron [3, 4]. Thus the scalar potential in eq. (4) together with the triplet Yukawa interaction of eq. (1) lead to a phenomenologically viable model of neutrino mass generation. The expression for resulting from the minimization of is:

(5) |

In the scenario of light triplet scalars () within the discovery reach of the LHC, eq. (5) leads to . In extensions of the HTM the term ) may arise in various ways: i) the vev of a Higgs singlet field [22, 23]; ii) be generated at higher orders in perturbation theory [7]; iii) be generated in the effective Lagrangian [10]; iv) originate in the context of extra dimensions [6, 8].

An upper limit on can be obtained from considering its effect on the parameter . In the SM at tree-level, while in the HTM one has (where ):

(6) |

The measurement leads to the bound , or GeV. At the 1-loop level must be renormalized and explicit analyses lead to bounds on its magnitude similar to those derived from the tree-level analysis [24].

The HTM has seven Higgs bosons . The doubly charged is entirely composed of the triplet scalar field , while the remaining eigenstates are in general mixtures of the doublet and triplet fields. Such mixing is proportional to the triplet vev, and hence small even if assumes its largest value of a few GeV. Therefore are predominantly composed of the triplet fields, while is predominantly composed of the doublet field and plays the role of the SM Higgs boson. The masses of are of order with splittings of order . For TeV of interest for direct searches for the Higgs bosons at the LHC, the couplings are constrained to be or less by a variety of processes such as etc. which are reviewed in [25, 26].

## 3 Production and Decay of at Hadron Colliders

The most distinct and experimentally accessible decay mode of is to two same-sign charged leptons [27]. Without loss of generality one can work in the basis in which the charged lepton mass matrix is diagonal i.e., are the mass eigenstates. Then the decay rate for () is given by:

(7) |

where for (). Clearly depends crucially on the absolute values of the , where is related to the neutrino mass matrix via eq. (3). However, if no other decay modes for are open kinematically the leptonic BRs are determined solely by the relative values of . Other decay modes for can be important in regions of parameter space e.g., i) , which is potentially sizeable for [17, 28], and ii) , which is proportional to the triplet vev, . In this work, we assume (which precludes ) and , which suppresses sufficiently in the HTM (e.g., see [29]). Then, can be regarded as the sole decay mode for and one has:

(8) |

### 3.1 Searches for at the Tevatron

In the year 2003 the Fermilab Tevatron performed the
first search for
at a hadron collider.^{1 }^{1 }1 Direct searches for
have also been performed at
LEP [30] and HERA [31].
The D0 collaboration [13]
searched for
while the CDF collaboration [14]
searched for 3 final states: . The assumed production mechanism for
is , which
proceeds via gauge strength couplings and
depends on only one unknown parameter, [25, 32].

The searches performed in [13, 14] seek at least one pair of same-sign leptons with high invariant mass i.e., the search is sensitive to single production of . The SM background can be reduced to negligible proportions with suitable cuts. Single production mechanisms which involve a dependence on potentially small parameters such as the Yukawa coupling or triplet vev are subdominant at Tevatron energies (e.g., see [33]). In [17] it was suggested that this search strategy is also sensitive to the mechanism [34], which has a cross-section comparable in magnitude to that of . The following inclusive single cross-section was introduced, which would extend the search sensitivity to larger values of and strengthen the mass limits on derived in [13, 14]:

(9) |

Here the factor of 2 accounts for the CP conjugate process .

In 2006 the CDF collaboration searched for decays involving [35]. The strategy of searching for one pair of same-sign leptons () is not effective due to the larger SM backgrounds, and instead three () and four () lepton searches were performed. The production mechanism was assumed. The process never contributes to the signature, but can contribute to the signature if decays leptonically, .

In Table 1 the mass limits for from the Tevatron searches are summarized. A blank entry signifies that no search has yet been performed. The displayed mass limits assume production via for a belonging to a triplet with . Moreover, in a given channel is assumed. The ultimate sensitivity at the Tevatron is expected to be GeV in the and channels.

2l | GeV | GeV | GeV | |||

3l | GeV | GeV | ||||

4l | GeV | GeV |

### 3.2 Simulations of production at the LHC

Several simulations have been performed for () at the LHC [15, 16, 29, 36, 37]. The production mechanism is assumed to be followed by . The LHC sensitivity to considerably extends that at the Tevatron due to the increased cross-sections and larger luminosities e.g., the analysis of [16] shows that a can be discovered for GeV assuming and fb. Importantly, all the above simulations suggest that as little as 1 fb is needed for discovery of GeV if one of BR() is large, and hence such a light would be found very quickly at the LHC.

The sensitivity of the LHC to single production of for has only been performed in [37], and importantly the SM background was shown to be negligible in the signal region of high invariant mass. It was concluded that such a search strategy allows more events than the 4 lepton search since the event number is linear (and not quadratic) in . Therefore the search is more effective at probing small . Importantly, the addition of the channel (eq. (9)) would further enhance the event number for a given .

In Table 2 we show approximate expected numbers of and events arising from pair and singly produced at the LHC. We only consider the decay channels which offer the greatest discovery potential. A detection efficiency of 0.5 is assumed, which is slightly less than the values given in [16] for . The theoretical cross-section is multiplied by this detection efficiency and the SM background is taken to be negligible. The number of events for a specific is denoted by , assuming integrated luminosities of fb and fb. The displayed numbers are for in a given channel. In the HTM, is necessarily (eq. (3) and eq. (7)) and hence must be multiplied by . The final column shows the number of events () obtained by adding the contribution from the mechanism as defined in eq. (9). We take which increases the number of singly produced events by a factor of around 2.8 for [17]. For , the numbers presented in Table 2 are scaled as shown in Appendix A.

It is clear from Table 2 that early discovery of at the LHC with GeV would allow large event numbers for with the expected integrated luminosities of fb. This would enable precise measurements of BR() for the dominant channels. Sensitivity to BR or less would also be possible in the channel.

(GeV) | (30 fb) | (300 fb) | (300 fb) |
---|---|---|---|

200 | 1500 | 15000 | 42000 |

300 | 300 | 3000 | 8400 |

400 | 90 | 900 | 2500 |

## 4 Numerical Analysis

The mass matrix for three Dirac neutrinos is diagonalized by the MNS (Maki-Nakagawa-Sakata) matrix [38] for which the standard parametrization is:

(10) |

where and , and is the Dirac phase. For Majorana neutrinos, two additional phases appear and then the mixing matrix becomes

(11) |

where and are referred to as the Majorana phases [3, 39]. Since we are working in the basis in which the charged lepton mass matrix is diagonal, the neutrino mass matrix is then diagonalized by . Using eq. (3) one can write the couplings as follows [6, 7]:

(12) |

Then, eq. (8) becomes

(13) |

Note that the branching ratios are independent of and , and given by neutrino parameters only.

Neutrino oscillation experiments involving solar [40], atmospheric [41], accelerator [42], and reactor neutrinos [43] are sensitive to the mass-squared differences and the mixing angles, and give the following preferred values:

(14) | |||

(15) |

The small mixing angle has not been measured yet and hence the value of in completely unknown. Since the sign of is also undetermined at present, distinct neutrino mass hierarchy patterns are possible. The case with is referred to as Normal hierarchy (NH) where and the case with is known as Inverted hierarchy (IH) where . Information on the mass of the lightest neutrino and the Majorana phases cannot be obtained from neutrino oscillation experiments. This is because the oscillation probabilities are independent of these parameters, not only in vacuum but also in matter. If , future H beta decay experiment [20] can measure it. Experiments which seek neutrinoless double beta decay [18] are sensitive to only a combination of neutrino masses and phases. Certainly, extracting information on Majorana phases alone from these experiments seems extremely difficult, if not impossible [19]. Therefore it is worthwhile to consider other possibilities.

Multiplying out eq. (12) one obtains the following explicit expressions for :

One may express in terms of
two neutrino mass-squared differences ()
and the mass of the lightest neutrino .
The are functions of nine parameters:

,,,
three mixing angles
and three complex phases .
Four of the above parameters have been measured well
experimentally (see eq. 15)
and thus the HTM already provides numerical predictions for
as a function
of the unmeasured five parameters.
Four cases
corresponding to no CP violation from Majorana phases
can be defined as follows:
Case I ;
Case II ;
Case III ;
Case IV .
These four cases have been studied
in [7, 17] for values
of or (1) eV. In this work we will quantify the
dependence of on , and .
Those 3 parameters essentially determine , with subdominant
corrections from the neutrino oscillation parameters.
Hence multiple signals for
at the LHC would probe , and ,
in the context of the HTM.

We note here that such collider probes of the Majorana phases are particular to models in which lepton number violation (which leads to the Majorana neutrino mass) is associated with New Physics particles at the TeV scale. Analogous collider probes are not possible in models where the scale of lepton number violation is much higher e.g., supersymmetric models with very heavy right-handed neutrinos with masses of order . However, in such models the Majorana phases (which are also required for successful leptogenesis) can significantly affect the rates for the lepton flavour violating (LFV) decays and [44]. Likewise, in the HTM and would also affect the rates for LFV decays which depend explicitly on e.g., and [7]. Similar studies of the effects of Majorana phases on these LFV decays in other models which contain a TeV scale and an analogous coupling have been performed in [45].

### 4.1 Dependence of on the neutrino mass spectrum

In Fig. 1, we plot as a function of for and (Case I) in NH and IH. The values of the oscillation parameters are fixed in the figures as follows:

(17) | |||

(18) |

For NH with smaller , the
-related modes are suppressed
because and are small
and the contribution from the heaviest mass
is also small due to the tiny .
, , and
are roughly equal ^{2 }^{2 }2
The naïve expectation is that
is twice as large as the other two modes
because of .
However, this difference is accidentally compensated
by the effect of .
,
and they dominate for smaller
because does not appear with .
Note that
and
can be understood as the approximate symmetry
for - exchange by virtue of
and tiny .
For larger ,
dominate
because all coefficients of are positive
for with in Case I
and there is no strong cancellation among these terms
even for .
For , the
results for NH and IH are almost identical
because of almost degenerate masses.
Since BR are given by ratios of ,
they converge for large .
This is an attractive feature
because HTM can predict certain ranges of BR
without restricting .

On the other hand, IH case gives rather simple results. This is because and include the larger scale without any suppression from and consequently the effects of small and are hidden. In this case, becomes the dominant channel even for smaller ; one naïvely expects for with eq. (LABEL:hij_expressions).

In order to quantify the effect of the unmeasured on , we show for , and 4 values of the Dirac phase () for Cases I to IV in Fig. 2. Other parameters are same as Fig. 1. Several lines are coincident in the figure, since and always appear as a combination in . For example, Case I with and Case II with give the same lines. Although the contribution of to is considerably smaller than the effect of varying or the Majorana phases (as expected by quadratic suppression with small ), it is not negligible. Consequently, it is also not negligible for other the BR. Fig. 2 also shows that the HTM predicts small in case III and IV (cases with ) for any value of , , and in both of NH and IH. Thus, Fig. 2 indicates that some information on Majorana phases may be extracted without knowledge of the sign of and the values of , , and .

### 4.2 Dependence of on Majorana phases

In this section we show the dependence of on Majorana phases with the values of the oscillation parameters given in (17) and (18). For , only the relative phase determines in NH. In Fig. 3 (left) we plot as a function of for NH with . Tiny suppresses the dependence of on , and then the HTM gives a clear prediction of very small for this case. The other change non–negligibly e.g., and vary by or more, which could be larger than the experimental error if sufficiently large numbers of are produced. In IH for , does not depend on because it always appears multiplied by . Fig. 3 (right) shows the dependence of on for this case. One can see that the dependence on is even more pronounced because for in IH () is larger than that in NH (). In particular, and seem to be very useful for extracting information on because they have large and opposite dependence. This dependence on can be understood by the relative sign of the terms of and in eq. (LABEL:hij_expressions), which is for and for , neglecting .

For eV, where neutrino masses are almost degenerate, Fig. 4 shows that the phase has a large effect on all . The figure also shows that the dependence for and is sizeable and overcomes the suppression from . However, the latter suppression factor ensures that has a very small dependence on , and so this channel is crucial for extracting clear information on and/or without contamination from the effect of .

It is clear that the Majorana phases have a large effect on in the HTM, and their inclusion is required in order to quantify the allowed regions of .

### 4.3 Sensitivity to and .

As we have seen, both the neutrino mass spectrum and the Majorana phases have large effects on in the HTM, but the model also gives a clear prediction for . Therefore we can expect to obtain some information on the neutrino mass spectrum and/or the Majorana phases by observing . In our analysis, we use for which the LHC expects greatest sensitivity. Naïvely, three measurements of are sufficient to extract information on the three parameters.

In this section we consider the possibility to determine and/or to exclude . Hereafter, we use (17) and

(19) |

Non-zero values of and affect
as was shown in Fig. 2.
Moreover, deviation of from
especially affects in our analysis
because a rather wide range is allowed.
In Fig. 5 the
allowed regions of in the HTM are shown
by the shaded (light and dark) regions; the dark shaded
area corresponds to the overlap of the allowed
regions for NH and IH. The area above the dotted line is unphysical
because the sum of BR exceeds unity.
It is clear that
the HTM predicts ,
, and .
The areas outside the solid line are unreachable,
and are particular to the HTM
(in which neutrino mass is solely given by eq. (3))
and will differ from the corresponding disallowed regions,
if any,^{3 }^{3 }3 If are arbitrary parameters,
as in the Left-Right symmetric model of [1],
the unreachable region would vanish.
in other models which can contain a light .
Hence in the event of signals for
these allowed regions can be compared
with the experimental measurements of
in order to test the HTM.
If measured BR are outside the shaded region
the HTM is disfavored.
If given signals for
are compatible with the reachable regions in the HTM,
the data can be interpreted in the context of this model and
one can try to extract information on parameters
of neutrino mass matrix.

Furthermore, the area inside the dashed and dash-dotted lines in Fig. 5 is possible for NH and IH, respectively. NH gives the additional bounds and , while IH gives and . These are also particular features of the HTM. If BR in the light shaded region are measured, NH or IH is disfavored in the HTM.

Next, let us consider the possibility to exclude . Fig. 6 shows the attainable regions of BR with . As in Fig. 5, the area inside the thin solid line is reachable in the HTM, and the region above the dotted line is unphysical. The dashed line corresponds to the BR which can be obtained in NH, and the dash-dotted line is BR in IH, both with . Note that the area inside these lines can also be obtained with but outside is impossible for . This behaviour can also be seen in Fig. 3. The bold solid line denotes a analysis for measurements of signals in the channels (, , and ) of decays (see Appendix A for detail). If BR in the shaded region are measured, one can exclude at 90%CL in HTM. NH with gives a very clear prediction of small as can be seen in Fig. 3. The bold line shows that can be excluded at 90%CL if