We analyze the inclusive decay mode in the unparticle model, where an unparticle can also serve as the missing energy. We use the Heavy Quark Effective Theory in the calculation. The analytical result of the decay width in the free quark limit and that of the differential decay rate to the order of are presented. Numerical results of the inclusive mode show interesting differences from those of the exclusive modes. Near the lower endpoint region, the unparticle has very different behavior from the Standard Model particles.

## 1 Introduction

The rare decay is very small in the Standard Model (SM)[1] so that it might be very sensitive to the new physics beyond the SM. Furthermore, in the final state the missing energy carried by the neutrino-anti-neutrino pair might be polluted experimentally by other missing-energy-like states in the presence of new physics.

Recently, the Unparticle Model suggested by Georgi[2] provides a possible candidate for the missing energy in . In the Unparticle Model, below a scale the interaction of the unparticle with the SM sector takes a form like[2]

(1) |

where is a coefficient function and is a large mass scale of the particles mediating the interaction between the SM fields and the unparticle fields. If is large enough, the unparticle stuff does not couple strongly to the ordinary particles. People have introduced many kinds of couplings[2, 3, 4], including the Yukawa and the partial differential couplings. A lot of works have been done to study the possible consequence of the unparticle, most of which focus on the exclusive processes. Here we will discuss the inclusive mode , treating the unparticle as part of the missing energy. We will use Heavy Quark Effective Theory (HQET) in our analysis. The results are constrained by the data [5].

The organization of this paper is as follows. At first, we apply the HQET results to unparticle model and give out the general form of decay rates. Then we present in the analytical forms the decay width in the free quark approximation and the differential decay width versus the missing energy to the order. Numerical results will be given. We will summarize at the end.

## 2 HQET application in Unparticle Model

In the Unparticle Model, the effective Hamiltonian for at quark level is given by

(2) |

for the scalar unparticle, or

(3) |

for the vector unparticle. Here and are the dimensionless coupling constant between the quark current and unparticle fields. The is a non-integral number severing as the dimension of unparticle operators. The can not be small than 1 for the unitary of the theory[2]. If one impose the conformal symmetry on the vector unparticle fields, the primary, gauge invariant vector unparticle operators could only have dimension [6, 7].

We will also introduce the two dimensional coefficients corresponding to scalar and vector unparticles

(4) |

The most general forms of differential decay rates for the scalar and vector unparticle final states are

(5) | |||||

where is defined as[2]

(6) |

The factor in (5) counts for the unparticle phase space[2], over which the integrations can be performed in the rest frame of the meson. The rest part in (5) containing the matrix element squared are conventionally written, in analogy with those in the semileptonic decays in the SM, as the product of the hadron and the unparticle tensors analytical [8, 9, 10, 11, 12],

(7) | |||||

where the unparticle tensors are

(8) |

and the hadronic tensor is defined by

(9) |

with .

The most general form of is[9]

(10) |

The scalar structure functions are functions of and , where is the four velocity of the heavy bottom quark. They are related to ’s by applying , where

(11) |

In general,

(12) |

The HQET provides a systematical tool in investigating the Heavy-light hadrons such like the mesons[8, 9]. ’s can be expanded in using HQET, their forms to can be found in Ref.[11, 12]. There emerge some problems near the endpoint region of the energy spectrum[10], which can be avoided by introducing suitable cuts in this work.

## 3 Differential decay rates and the width

The inclusive differential decay rates are calculated using (13) and (14) by taking . Terms with derivatives of function are evaluated using integrating by parts. Then the differential decay width for the scalar unparticle emission is

(15) | |||||

and for the vector unparticle emission, it is

(16) | |||||

where and are parameters in the heavy quark expansion. Here we introduce the dimensionless variable , which serves as unparticle energy modulated by the heavy quark mass.

Further integration over will not generally give analytical results for an arbitrary because of the factor in Eq.15(or 16). However, when taking the limit we can integrate over . We get

(17) |

and

(18) |

Eqs.(17) and (18) are the decay widths for in the free quark approximation.

We find that, even if , the scalar unparticle in the final state gives a finite contribution in (17), as a result of the fact that the singularity in the factor is compensated by the factor . This is not a common feature as in the exclusive processes such like in Ref.[13]. We cannot give a simple analytical formula like (18) for the vector unparticle in the final state when . But under the assumption of exact conform symmetry must be bigger than 3[6], then Eq.(18) is enough at the first order.

## 4 Numerical results

Numerical results are needed in order to exhibit the unparticle effects. Both the unparticle and neutrino-anti-neutrino pairs serve as the missing energy . We can not distinguish them in the experiments, so the process may contain both and in the final states. The decay width is

(19) |

where comes from the SM and come from either the scalar or the vector unparticle contribution.

Present calculation in SM[1, 14] gives

(20) |

The experimental bound[5],

(21) |

is about one order larger than the SM calculation. This large difference allows new physics to provide candidates as the missing energy. In the unparticle model the candidate is the unparticle.

There are two kinds of singularities brought by the unparticle: one comes from the and the other from . At the quark level, the endpoint of unparticle energy spectrum is singular. But the true endpoint are at and , depending on the masses of the hadrons. Near or when the HQET fails, we introduce some cuts. For the scalar unparticle model, we set the , i. e. we take the MeV that around the HQET fails[10]. But for the vector unparticle theory, the case is different because the term in (16) goes through zero when , so it severs as our cut. We have taken the mass parameters as[5, 15]

(22) |

and the heavy quark expansion parameters as [16]

(23) |

There are very distinctive differences between the unparticle model and the SM in the differential decay widths when . Near the lower endpoint region , the SM differential width approach zero, while the unparticle model gives finite results. This comes from in the phase space of the unparticle, see Eqs.(13) and (14). When goes to zero, the phase space goes to infinity. But the SM final state has no such an enhancement. For the vector unparticle model , the differential width also gives a finite result at when . Here, the enhancement comes from the vector unparticle tensor in Eq.(8). appears in the denominator.

Near the endpoint , the scalar unparticle model results go to infinity, which comes from the HQET. There are also such kinds of singularity in SM[9, 18, 19]. For the vector unparticle model, the results go to negative infinity when goes to 1. And the vector unparticle contributions vanish near . This turning comes the from the form of matrix element in (14) and HQET.

We plot the spectra for the scalar unparticle model in Fig. 1 for and Fig. 2 for , respectively. The spectra for vector unparticle model are given in Fig. 3. for , and in Fig. 4 for , respectively. Note that is allowed if we give up the strict conformal symmetry.

The branching ratios of scalar and vector unparticle emission processes show many resemblances if one neglect the conformal constraint on . They are presented in Fig. 5 for the scalar unparticle, and in Fig. 6 and 7 for the vector unparticle. Figs. 6 is allowed if one gives up the strict conformal symmetry, with the unparticle contribution peaking at . In this case the branching ratio goes down when becomes bigger, as is exhibited in (17) or (18). This is quite different from the previous results gained from the similar exclusive decay processes[13, 17], where the branching ratios go up as is increaing. It is also important to note the different definitions of the couplings from Ref.[17].

## 5 Summary

In this paper we have discussed the process Missing Energy in the unparticle model and given some analytical results of the decay withes in free quark limit and the differential decay rates to the order. If , the unparticle stuffs is most likely to be tested. If one regards the conformal symmetry[6], the vector unparticle has the most distinctive effect around . Near the lower endpoint region in the spectrum, the unparticle model show very distinctive behavior from the SM. This is very possible to be tested in experiments.

This work was supported in part by the National Natural Science Foundation of China (NSFC) under the grant No. 10435040.

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