ICRR-Report-523-2005-6

KEK-TH-1055

February 21, 2021

Efficient Coannihilation Process through Strong Higgs Self-Coupling in LKP Dark Matter Annihilation

Shigeki Matsumoto^{1}^{1}1
and
Masato Senami^{2}^{2}2

Theory Group, KEK, Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan

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

## 1 Introduction

There are some compelling evidence to require an extension of the standard model (SM), for example, the existence of non-baryonic cold dark matter, neutrino masses and the baryon number asymmetry in the universe. Many models beyond the standard model are proposed to solve one or a few of these problems. In particular, extensions to explain the existence of dark matter are promising, because many direct and indirect detection experiments for dark matter are now on going, and many future experiments are also proposed. Therefore the extensions will be tested in near future.

Many models including a dark matter candidate have been proposed. Among those, models with a weakly interacting massive particle (WIMP) are attractive. This is because the WIMP can naturally provide the correct relic abundance of dark matter in the present universe in addition to the successful explanation of the large scale structure of the universe.

A famous candidate for WIMP is the lightest supersymmetric particle (LSP) in a supersymmetric extension of the SM [1], which is stabilized by the R-parity. Recently, the lightest Kaluza-Klein particle (LKP) in the flat universal extra dimension (UED) scenario [2] has been proposed as an alternative candidate for WIMP. In UED models, all particles in the SM propagate in the compact extra dimensions. The momentum along an extra dimension is interpreted as the Kaluza-Klein (KK) mass in the four dimensional point of view. The KK mass spectrum is quantized in terms of KK number as , where is the size of the extra dimension. The momentum conservation along the extra dimension leads to the KK number conservation. Since UED models must contain the SM as a low energy effective theory, extra dimensions are compactified on an orbifold for deriving chiral fermions. This orbifolding breaks the KK number conservation down to the KK-parity conservation, in which the SM particles and even KK particles carry charge while odd KK particles carry charge. Due to the KK-parity conservation, the LKP is stable and a good candidate for dark matter.

The thermal relic abundance of the LKP dark matter is calculated in several papers [3, 4, 5, 6] and the results indicate the compactification scale consistent with observations such as WMAP [7] is in the range of 500 to 700 GeV in the minimal UED (MUED) model. The MUED model is defined in the five space-time dimensions in which the extra dimension is compactified on an orbifold. The one-loop corrected KK mass spectrum of the model are calculated in Ref. [8].

On the other hand, the scale is constrained by electroweak precision measurements (EWPM) [2, 9, 10, 11]. The constraints in most of the previous papers [2, 9, 10] are satisfied for 300 GeV, however, a severe constraint is recently reported as 700 GeV (for = 120 GeV) at the % confidence level [11]. This result seems to indicate that the MUED model is inconsistent with the observed abundance of dark matter.

However, we found a parameter region reconciling the relic abundance of dark matter with the stringent constraint reported in Ref. [11] at the 3 level in the MUED model. The parameter region is where the Higgs mass is slightly heavy as 200 GeV. In the region, the self-coupling of the SM Higgs field (and its KK particles) becomes large. Therefore the annihilation cross sections of the KK Higgs particles are enhanced. Furthermore, the first KK particles of the Higgs field are degenerated with the LKP (the first KK photon) in mass, which is less than level in the MUED model. As a result, the thermal relic abundance of the LKP is significantly changed through the coannihilation process including the first KK Higgs bosons. To be more precise, the first KK charged and pseudo Higgs bosons are important for the coannihilation process, which are the first KK particles of the SM Goldstone bosons. Since the first KK particle of the neutral scalar Higgs is heavier as the SM Higgs is heavier, it does not contribute significantly to the coannihilation process.

This paper is organized as follows. In the next section, we briefly review the MUED model, especially we focus on the masses of the LKP and the first KK Higgs bosons. Next we discuss the thermal relic abundance of the LKP dark matter in the section 3. We evaluate the compactification scale consistent with observations of the relic abundance depending the value of the SM Higgs boson mass. Section 4 is devoted to summary and discussion.

## 2 Mass difference between LKP and first KK Higgs bosons

The simplest version of UED model i.e. MUED model has one extra dimension compactified on an orbifold. Thus the model is described by the SM with extra particle contents which are the KK modes of the SM particles in the four-dimensional point of view. The SM particles and their KK particles have identical charges and couplings relevant for KK particles are completely determined by those of SM. Hence, the MUED model has only two new input parameters, the compactification scale and the cutoff scale . The cutoff scale is usually taken to be [2]. In this paper, we adopt the value, , since the changing this value only slightly modifies our results. In addition to these parameters, we have one undetermined parameter, that is the mass of the SM Higgs boson, . This parameter is very important for our studies because the masses of the KK Higgs bosons and the self-coupling between them depend on this value.

The mass spectra of KK particles are determined by and the mass of the corresponding SM particle at tree level. One of the typical features of the MUED model is that all particles at each KK level are degenerated in mass. Thus, the radiative corrections to the masses play an important role when we consider the mass difference between KK particles [8]. Below, we summarize the masses including the radiative corrections for the first KK photon and first KK Higgs bosons, which are important for our discussion.

The LKP is the first KK photon in most of the parameter region in the MUED model. Its mass is obtained by diagonalizing the mass squared matrix described in the basis,

(1) |

where is the SU(2)(U(1)) gauge coupling constant and GeV is the vacuum expectation value of the SM Higgs field. The radiative corrections to the massive KK gauge bosons are given by

(2) | |||||

(3) |

The difference between diagonal elements, , exceeds the off-diagonal ones when . Hence, the weak mixing angle of the first KK gauge bosons is small and the KK photon is dominantly composed by the first KK particle of the hypercharge gauge boson.

Let us turn to the masses of the first KK Higgs particles. Since the KK modes of Higgs field are not eaten by the SM gauge bosons, all of neutral scalar , pseudoscalar and charged scalar remain as physical states. The latter three states are the KK particles of the Goldstone modes in the SM. The KK Higgs boson masses turn out to be

(4) | |||||

(5) | |||||

(6) |

where and are and boson masses, respectively. The radiative correction is given by

(7) |

where is the Higgs self-coupling defined as
As increasing , becomes large
and the negative contribution in Eq. (7) increases.
Hence, for large ,
the annihilation cross sections of the KK Higgs bosons are significantly enhanced
and the mass differences between the LKP and become small.
However, the mass differences are negative when is too large.
Thus the LKP becomes the charged KK Higgs boson
and this case is not allowed from the point of view of dark matter
^{3}^{3}3We oversighted the charged LKP region in the previous papers
[4] and noticed it recently..

In Fig. 1, we depict the mass degeneracy between the LKP and the charged KK Higgs boson, . It is clear form this figure that the mass difference between the first KK charged Higgs boson and the first KK photon is very small for GeV. On the other hand, is identified with the LKP for GeV, so that this parameter region should be discarded from our discussion.

Here, we should address the KK particle of the graviton. Since a radiative correction to the mass of the KK graviton is extremely small due to the gravitational interaction, the mass is given by with high accuracy. Furthermore, the value, , is positive for GeV, the LKP is the first KK graviton in this region. This fact leads us to a serious problem, that is, the graviton LKP dark matter with mass less than O(10) TeV is excluded due to the late time decay of the NLKP (next LKP) to the KK graviton and it is severely constrained by cosmological observation for cosmic microwave background [12].

Fortunately, the region we are interested in is GeV for GeV, in which the relic abundance of dark matter is consistent with cosmological observations. In the region the KK graviton is not the LKP, and the problem discussed above is replaced with the problem caused by the late time decay of the KK graviton to the LKP. This is avoided if the reheating temperature of the universe is low enough [14].

Furthermore, there is the mechanism that makes only the KK graviton become heavy without changing other sectors [13]. The mechanism is based on higher dimensional setup than that used in the MUED model, and KK particles in the MUED are assumed to be localized in the five dimensional space-time. With the use of the mechanism, the LKP dark matter is identified with the KK photon in the MUED model even if is less than 800 GeV.

## 3 Relic abundance of the LKP dark matter revisited

We are now in a position to calculate the thermal relic abundance of the LKP dark matter including the coannihilation with the first KK Higgs bosons, especially when the SM Higgs is slightly heavy. Since the LKP is also degenerate with the first KK leptons in mass, the coannihilation processes including these particles should be taken into consideration as well. For the detailed formulae of the mass spectra, refer to Ref. [8].

We use the method developed in Ref. [15] to calculate the relic abundance including the coannihilation effects. Under reasonable assumptions, the relic density of the LKP obeys the following Boltzmann equation,

(8) |

where and . The number density is defined as the sum of the number density of each species as . The entropy density is given by , with being the relativistic degrees of freedom at the decoupling. The Hubble parameter is , where GeV is the Planck mass. The abundance in the equilibrium is written as

(9) |

where is the number of the effective degrees of freedom and defined by

(10) |

The number of the internal degrees of freedom for species is denoted by .

The effective annihilation cross section governs the relic density of the LKP dark matter and is given as the sum of , which denotes the coannihilation cross section between species and ,

(11) |

The annihilation cross section, , in each process has been already calculated. For the explicit expressions, see Refs. [3, 5, 6]. By solving the Boltzmann equation numerically, we obtain the present abundance of dark matter, . It is useful to express the relic density in terms of , which is the ratio of the dark matter density to the critical density . The small letter denotes the scaled Hubble parameter, .

The results of the calculation are shown in Fig. 2. The thermal relic abundance of the LKP dark matter is depicted as a function of with the SM Higgs mass, , and GeV. Two horizontal dashed lines denote the allowed region from the WMAP observation at the 2 level, [7].

It is found that the larger compactification scale is allowed for the larger Higgs mass. The is because the large Higgs self-coupling is derived for the larger Higgs mass. The large Higgs self-coupling induces two effects. First, the mass differences between and become small for the larger Higgs self-coupling. Therefore the Boltzmann suppression factor in the effective annihilation cross section in Eq. (11) become negligible. Second, the annihilation cross sections between the first KK Higgs bosons are significantly enhanced. As a result, the effective annihilation cross section also increases, and the compactification scale as large as TeV can be consistent with the observed relic abundance of dark matter.

## 4 Summary and discussion

In this paper, we have investigated the dependence of the relic abundance of the LKP dark matter on the SM Higgs mass in the MUED model. It is found that the effective annihilation cross section governing the abundance is drastically enhanced when GeV and the compactification scale consistent with the observed abundance increases. The key ingredient is the strong Higgs self-coupling, which allows the LKP dark matter to annihilate very efficiently in the early universe through the coannihilation processes including the first KK Higgs bosons. Due to the enhancement of the coannihilation processes for GeV, the relic abundance consistent with cosmological observations could be produced without conflicting the bound reported by Ref. [11] at the 3 level.

Finally, we address the implication of the parameter region we investigated to the Large Hadron Collider (LHC), which starts at CERN in . In order to satisfy both bounds from the abundance and the EWPM reported by Ref. [11], a slightly heavy Higgs is required. Since large mass of the light Higgs scalar in the minimal supersymmetric model is not favored, it may be a signature of the UED models if we observe a heavy Higgs boson and missing momentums at the LHC.

## Acknowledgements

We are grateful to M. Kakizaki for useful discussions. The work of SM was supported in paart by a Grant-in-Aid of the Ministry of Education, Culture, Sports, Science, and Technology, Government of Japan, No. 16081211.

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