# Bosonic Seesaw in the Unparticle Physics

###### Abstract

Recently, conceptually new physics beyond the Standard Model has been proposed by Georgi, where a new physics sector becomes conformal and provides “unparticle” which couples to the Standard Model sector through higher dimensional operators in low energy effective theory. Among several possibilities, we focus on operators involving the (scalar) unparticle, Higgs and the gauge bosons. Once the Higgs develops the vacuum expectation value (VEV), the conformal symmetry is broken and as a result, the mixing between the unparticle and the Higgs boson emerges. In this paper, we consider a natural realization of bosonic seesaw in the context of unparticle physics. In this framework, the negative mass squared or the electroweak symmetry breaking vacuum is achieved as a result of mass matrix diagonalization. In the diagonalization process, it is important to have zero value in the -element of the mass matrix. In fact, the conformal invariance in the hidden sector can actually assure the zero of that element. So, the bosonic seesaw mechanism for the electroweak symmetry breaking can naturally be understood in the framework of unparticle physics.

## I Introduction

In spite of the success of the Standard Model (SM) in describing all the existing experimental data, the Higgs boson, which is responsible for the electroweak symmetry breaking, has not yet been directly observed, and is one of the main targets at the CERN Large Hadron Collider (LHC). At the LHC, the main production process of Higgs boson is through gluon fusion, and if Higgs boson is light, say GeV, the primary discovery mode is through its decay into two photons. In the SM, these processes occur only at the loop level and Higgs boson couples with gluons and photons very weakly.

A certain class of new physics models includes a scalar field which is singlet under the SM gauge group. In general, such a scalar field can mix with the Higgs boson and also can directly couple with gluons and photons through higher dimensional operators with a cutoff in effective low energy theory. Even if the cutoff scale is very high, say, 100-1000 TeV, the couplings with gluons and photons can be comparable to or even larger than those of the Higgs boson induced only at the loop level in the SM. This fact implies that if such a new physics exists, it potentially has an impact on Higgs boson phenomenology at the LHC. In other words, such a new physics may be observed together with the discovery of Higgs boson.

As one of such models, in this letter, we investigate a new physics recently proposed by Georgi Georgi:2007ek (), which is described in terms of ”unparticle” provided by a hidden conformal sector in low energy effective theory. A concrete example of unparticle staff was proposed by Banks-Zaks Banks:1981nn () many years ago, where providing a suitable number of massless fermions, theory reaches a non-trivial infrared fixed points and a conformal theory can be realized at a low energy. Various phenomenological considerations on the unparticle physics have been developed in the literature U-propagator (); U-pheno (). It has been found that inclusion of the mass term for the unparticle plays an important role especially in studying about the Higgs-unparticle systems U-Higgs (), indeed we have studied the unparticle physics focusing on the Higgs phenomenology including the effects of the conformal symmetry breaking Kikuchi (), and there are some other studies on the Higgs phenomenology in the literature of the unparticle physics U-Higgs2 (). Inclusion of such effects of the conformal symmetry breaking or the infrared (IR) cutoff is also considered in the literature of hadron collider physics Rizzo (), and in the model of colored unparticles U-color (). There has also been studied on the astrophysical and cosmological applications of the unparticle physics U-astro (), especially, we have proposed a possibility for the unparticle dark matter scenario UDM (). And there are some studies on the more formal aspects of the unparticle physics U-formal () and its effects to the Hawking radiation Dai:2008qn ().

Now we begin with a review of the basic structure of the unparticle physics. First, we introduce a coupling between the new physics operator () with dimension and the Standard Model one () with dimension ,

(1) |

where is a dimension-less constant, and is the energy scale characterizing the new physics. This new physics sector is assumed to become conformal at a energy , and the operator flows to the unparticle operator with dimension . In low energy effective theory, we have the operator of the form,

(2) |

where the dimension of the unparticle have been matched by which is induced the dimensional transmutation, and is the (effective) cutoff scale of low energy effective theory. In this paper, we consider only the scalar unparticle.

It was found in Ref. Georgi:2007ek () that, by exploiting scale invariance of the unparticle, the phase space for an unparticle operator with the scale dimension and momentum is the same as the phase space for invisible massless particles,

(3) |

where

(4) |

Also, based on the argument on the scale invariance, the (scalar) propagator for the unparticle was suggested to be U-propagator ()

(5) |

Because of its unusual mass dimension, unparticle wave function behaves as (in the case of scalar unparticle).

## Ii Unparticle and the Higgs sector

First, we begin with a brief review of our previous work on the Higgs phenomenology in the unparticle physics Kikuchi (). Among several possibilities, we will focus on the operators which include the unparticle and the Higgs sector,

(6) |

where is the Standard Model Higgs doublet and is the Standard Model operator as a function of the gauge invariant bi-linear of the Higgs doublet. Once the Higgs doublet develops the VEV, the tadpole term for the unparticle operator is induced,

(7) |

and the conformal symmetry in the new physics sector is broken U-Higgs (). Here, is the conformal symmetry breaking scale. At the same time, we have the interaction terms between the unparticle and the physical Standard Model Higgs boson () such as (up to coefficients)

(8) | |||||

where GeV is the Higgs VEV. In order not to cause a drastic change or instability in the Higgs potential, the scale of the conformal symmetry breaking should naturally be smaller than the Higgs VEV, . When we define the ‘mass’ of the unparticle as a coefficient of the second derivative of the Lagrangian with respect to the unparticle, , then the mass of the unparticle can be obtained in the following form, .

As operators between the unparticle and the Standard Model sector, we consider

(9) |

where we took into account of the two possible relative signs of the coefficients, and . We will see that these operators are the most important ones relevant to the Higgs phenomenology.

Now let us focus on effective couplings between the Higgs boson and the gauge bosons (gluons and photons) of the form,

As is well-known, in the Standard Model, these operators are induced through loop corrections in which fermions and W-boson are running HHG (). For the coupling between the Higgs boson and gluons, the contribution from top quark loop dominates and is described as

(11) |

where is the QCD coupling, and with the top quark mass and the Higgs boson mass . For the coupling between the Higgs boson and photons, there are two dominant contributions from loop corrections though top quark and W-boson,

(12) |

where with the W-boson mass . In these expressions, the structure functions are defined as

(13) |

with

Note that even though the effective couplings are loop suppressed in the Standard Model, they are the most important ones for the Higgs boson search at the LHC and ILC. In the wide range of the Higgs boson mass TeV, the dominant Higgs boson production process at the LHC is the gluon fusion channel though the first term in Eq. (II). If the Higgs boson is light, , the primary discovery mode of the Higgs boson is on its decay into two photons, in spite of this branching ratio is at most. Therefore, a new physics will have a great impact on the Higgs phenomenology at LHC and ILC. if it can provide sizable contributions to the effective couplings in Eq. (II). Furthermore, the fact that the Standard Model contributions are loop-suppressed implies that it is relatively easier to obtain sizable (or sometimes big) effects from new physics.

Now we consider new contributions to the Higgs effective couplings induced through the mixing between the unparticles and the Higgs boson (the first term in Eq. (8)) and Eq. (9), in other words, through the process or . We can easily evaluate them by using the vertex among the unparticle, the Higgs boson and gauge bosons and the unparticle propagator as

where we replaced the momentum in the unparticle propagator into the Higgs mass, . The unparticle contributions become smaller as and () become larger (smaller) for a fixed . Note that in the limit , the unparticle behaves as a real scalar field and the above formula reduces into the one obtained through the mass-squared mixing between the real scalar and the Higgs boson.

Let us first show the partial decay width of the Higgs boson into two gluons and two photons. Here we consider the ratio of the sum of the Standard Model and unparticle contributions to the Standard Model one,

and we define the event number ratio (),

(16) | |||||

Using Eqs. (11), (12) and (II) we evaluate the ratio of the partial decay widths as a function of . The numerical results in Ref. Kikuchi () show that, even for (1000 TeV), we can see a sizable deviation of from the Standard Model one with . Here, it is shown that the relative sign play an important role in the interference between the unparticle and the Standard Model contributions.

As discussed before, once the Higgs doublet develops the VEV, the conformal symmetry is broken in the new physics sector, providing the tadpole term in Eq. (7). Once such a tadpole term is induced, the unparticle will subsequently develop the VEV U-Higgs (); U-Higgs2 () whose order is naturally the same as the scale of the conformal symmetry breaking,

(17) |

Here we have introduced a numerical factor , which can be , depending on the naturalness criteria. Through this conformal symmetry breaking, parameters in the model are severely constrained by the current precision measurements. We follow the discussion in Ref. U-Higgs (). From Eq. (9), the VEV of the unparticle leads to the modification of the photon kinetic term,

(18) |

which can be interpreted as a threshold correction in the gauge coupling evolution across the scale . The evolution of the fine structure constant from zero energy to the Z-pole is consistent with the Standard Model prediction, and the largest uncertainty arises from the fine structure constant measured at the Z-pole Yao:2006px (),

This uncertainty (in the scheme) can be converted to the constraint,

(19) |

This provides a lower bound on the effective cutoff scale. For and we find

This is a very severe constraint on the scale of new physics, for example, TeV for .

A similar bound can be obtained by the results on Higgs boson search through two photon decay mode at the Tevatron. With the integrated luminosity 1 fb and the Higgs boson mass around GeV for example, the ratio is constrained to be Wells (). For , this leads to the bound, TeV, which is, as far as we know, the strongest constraint on the cutoff scale by the current collider experiments.

## Iii Bosonic seesaw in the unparticle physics

Now we turn to the discussion of a realization of bosonic seesaw Calmet:2002rf (); Kim:2005qb () in the context of unparticle physics. The basic idea of bosonic seesaw mechanism proposed in Calmet:2002rf (); Kim:2005qb () is to consider a bosonic analogy of the original seesaw mechanism seesaw () in the neutrino physics. In the case of neutrinos, the light neutrino mass eigenvalues are obtained after diagonalizing the matrix for one generation.

(20) |

where is the Dirac neutrino mass matrix and represents the mass of the heavy right handed neutrino.

Assuming the heavy right handed Majorana mass scale , the lightest mass eigenvalue of this mass matrix is given by

(21) |

There is no physical meaning of the minus sign In the case of neutrinos since we can always absorb such a irrelevant phase factor by redefining the fields (rephasing).

However, in the case of scalar fields, such a phase factor possesses a physical meaning, and we cannot erase it by redefinition of the fields. This is a point for the bosonic seesaw works to make a mass squared of scalar fields negative after the diagonalization.

The mass squared matrix of the unparticle-Higgs system is written by

(22) |

where stands for the Higgs self coupling,

(23) |

with .

Assuming the conformal symmetry breaking scale in the hidden sector is smaller than the weak scale, , then the eigenvalue of the mass matrix (22) after diagonalization is given by

(24) |

Thereby, the negative mass squared or the electroweak symmetry breaking vacuum is achieved as a result of the mass matrix diagonalization. In this diagonalization process, it is important to have zero value in the -element of the mass matrix. In fact, the conformal invariance in the hidden sector can actually assure the zero of that element. So, the bosonic seesaw mechanism Calmet:2002rf (); Kim:2005qb () for the electroweak symmetry breaking can naturally be understood in the framework of unparticle physics.

## Iv Summary

In conclusion, we have considered the unparticle physics focusing on the Higgs phenomenology. Once the electroweak symmetry breaking occurs, the conformal symmetry is also broken and this breaking leads to the mixing between the unparticle and the Higgs boson. Providing the operators among the unparticle and the gauge bosons (gluons and photons), the unparticle brings the sizable deviation into effective couplings between the Higgs boson and the gauge bosons, that can be measured at the LHC through the discovery of the Higgs boson.

In this paper, we specifically considered a realization of bosonic seesaw Calmet:2002rf (); Kim:2005qb () in the context of unparticle physics. In this framework, the negative mass squared or the electroweak symmetry breaking vacuum is achieved as a result of mass matrix diagonalization. In the diagonalization process, it is important to have zero value in the -element of the mass matrix. In fact, the conformal invariance in the hidden sector can actually assure the zero of that element. So, the bosonic seesaw mechanism for the electroweak symmetry breaking can naturally be understood in the framework of unparticle physics.

Acknowledgments

We would like to thank N. Okada for his stimulating discussions. The work of T.K. was supported by the Research Fellowship of the Japan Society for the Promotion of Science (#1911329).

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