Advances in High Energy Physics

Volume 2018, Article ID 8670954, 9 pages

https://doi.org/10.1155/2018/8670954

## Dark Matter in the Standard Model Extension with Singlet Quark

Research Institute of Physics, Southern Federal University, 344090 Rostov-on-Don, Pr. Stachky 194, Russia

Correspondence should be addressed to Vladimir Kuksa; ur.liam@74askukv

Received 29 September 2018; Accepted 3 December 2018; Published 16 December 2018

Academic Editor: Jouni Suhonen

Copyright © 2018 Vitaly Beylin and Vladimir Kuksa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We analyze the possibility of hadron Dark Matter carriers consisting of singlet quark and the light standard one. It is shown that stable singlet quarks generate effects of new physics which do not contradict restrictions from precision electroweak data. The neutral and charged pseudoscalar low-lying states are interpreted as the Dark Matter particle and its mass-degenerated partner. We evaluate their masses and lifetime of the charged component and describe the potential asymptotes of low-energy interactions of these particles with nucleons and with each other. Some peculiarities of Sommerfeld enhancement effect in the annihilation process are also discussed.

#### 1. Introduction

The problem of Dark Matter (DM) explanation has been in the center of fundamental physics attention for a long time. The existence of the DM is followed from astrophysical data and remains the essential phenomenological evidence of new physics’ manifestations beyond the Standard Model (SM) [1, 2]. Appropriate candidates as DM carriers should be stable particles which weakly interact with ordinary matter (so called, WIMPs). Such particles usually are considered in the framework of supersymmetric, hypercolor, or other extensions of the SM (see, for instance, review [3]). The last experimental rigid restrictions on cross section of spin-independent WIMP-nucleon interaction [4] exclude many variants of WIMPs as the DM carriers. So, another scenarios are discussed in literature, such as quarks from fourth generation, hypercolor quarks, dark atoms, and axions [3]. In spite of some theoretical peculiarities, the possibility of hadronic DM is not excluded and considered, for example, in [5–11]. The possibility of new hadrons existence, which can be interpreted as carriers of the DM, was analyzed in detail within the framework of the SM chiral-symmetric extension [11].

Principal feature of the hadronic DM structure is that the strong interaction of new stable quarks with standard ones leads to the formation of neutral stable meson or baryon heavy states. Such scenario can be realized in the extensions of the SM with extra generation [5–9], in mirror and chiral-symmetric models [11, 12] or in extensions with singlet quark [13–17]. The second variant was considered in detail in [11], where the quark structure and low-energy phenomenology of new heavy hadrons were described. It was shown that the scenario does not contradict cosmochemical data, cosmological tests, and known restrictions for new physics effects. However, the explicit realization of the chiral-symmetric scenario faces some theoretical troubles, which can be eliminated with the help of artificial assumptions. The extensions of SM with fourth generation and their phenomenology were considered during last decades in spite of strong experimental restrictions which, for instance, follow from invisible Z-decay channel, unitary condition for CM-matrix, FCNC, etc. The main problem of 4th generation is the contribution of new heavy quarks to the Higgs boson decays [18]. The contribution of new heavy quarks to vector boson coupling may be compensated by the contribution of 50 GeV neutrino [19–21]; however, such assumption seems artificial. In this paper, we analyze the hypothesis of hadronic Dark Matter which follows from the SM extension with singlet quark.

The paper is organised as follows. In the second section we describe the extension of the SM with singlet quark and consider the restrictions on its phenomenology, following from precision electroweak data. Quark composition and interaction of new hadrons with the standard ones at low energies are analyzed in the third section. The masses of new hadrons, decay properties of charged partner of the DM carrier, and annihilation cross section are analyzed in the fourth section.

#### 2. Standard Model Extension with Stable Singlet Quark

There is a wide class of high-energy extensions of the SM with singlet quarks which are discussed during many decades. Here, we consider the simplest extension of the SM with singlet quarks as the framework for description of the DM carrier. Singlet (or vector-like) quark is defined as fermion with standard and gauge interactions but it is singlet under transformations. The low-energy phenomenology of both down- and up-type quarks (D and U) was considered in detail in large number of works (see, for instance, [10, 22–24] and references therein). As a rule, singlet quark is supposed to be unstable due to the mixing with the ordinary ones. This mixing leads to the FCNC appearing at the tree level. As a consequence, we get additional contributions into rare processes, such as rare lepton and semileptonic decays, and mixing in the systems of neutral mesons ( oscillations). The current experimental data on new physics phenomena give rigid restrictions for the angles of ordinary-singlet quark mixing. In this work, we consider alternative aspect of the extensions with singlet quark , namely, the scenario with the absence of such mixing. As a result, we get stable singlet quark which has no decay channels due to absence of nondiagonal -quark currents. More exactly, due to confinement, the singlet quark forms bound states with the ordinary ones, for instance , and the lightest state is stable. In this work, we consider some properties of such particles and analyze the possibility of interpreting the stable neutral meson as the DM carrier.

Now, we examine the minimal variants of the SM extension with singlet quark , where subscript denotes up- or down- type with charge . According to the definition, the field is singlet with respect to group and has standard transformations under abelian and color groups. So, the minimal additional gauge-invariant Lagrangian has the formwhere is charge in the case of singlet , and denotes phenomenological mass of quark. Note, singlet quark (SQ) cannot get mass term from the standard Higgs mechanism because the Higgs doublet is fundamental representation of group. Abelian part of the interaction Lagrangian (1), which will be used in further considerations, includes the interactions with physical photon and boson:where , , , and is Weinberg angle of mixing. Note, the left and right parts of the singlet field have the same transformation properties, interaction (2) has vector-like (chiral-symmetric) form, and singlet quark usually is named vector-like quark [23, 24].

First of all, we should take into account direct and indirect restrictions on new physics (NF) manifestations which follow from the precision experimental data. The additional chiral quarks, for instance from standard fourth generation, are excluded at the level by LHC data on Higgs searches [22]. As the vector-like (nonchiral) singlet fermions do not receive their masses from a Higgs doublet, they are allowed by existing experimental data on Higgs physics. The last limits on new colored fermions follow from the jets data from the LHC [25]. The corresponding limits for effective colored factors are about 200 GeV, 300 GeV, and 400 GeV. Note that these limits are much less than the estimation of quark mass which follows from the DM analysis (see the fourth section). Theoretical and experimental situation for long-lived heavy quarks were considerably discussed in the review [10], where it was noted that vector-like new heavy quarks can elude experimental constraints from LHC.

Indirect limits on new fermions follow from precision electroweak measurements of the effects, such as flavor-changing neutral currents (FCNC) and vector boson polarization, which take place at the loop level in the SM. Because we consider the case of stable singlet quark, there is no mixing with ordinary quarks and, consequently, FCNC effects are absent. The NF manifestations in polarization effects of gauge bosons are usually described by oblique Peskin-Takeuchi parameters [26] (PT parameters). From (2), it follows that the singlet quark gives nonzero contributions into polarization of - and -bosons which are described by the values of . As -boson does not interact with the SQ, corresponding contribution into polarization operator is zero, . These parameters are expressed in terms of vector bosons polarization , where . Here, we use the definition and the expressions for PT oblique parameters from [27]. In the case under consideration, and PT parameters can be represented by the following expressions:In (3) polarization , where , in one-loop approach can be represented in simple form (for the case of SQ with ):In (4) the function contains divergent terms in the one-point, , and two-point, , Veltman functions which are exactly compensated in physical parameters (3). Using standard definitions of the functions and and the equality , by straightforward calculations we get simple expressions for oblique parameters:where . We check that in the limit the values of and go to zero as in accordance with well-known results for the case of vector-like interactions [2, 27]. From (5) it follows that beginning from GeV the parameter and the remaining nonzero parameters have nearly the same values. These values are significantly less than the current experimental limits [28]: ; that is, the scenario with up-type singlet quark satisfies the restrictions on indirect manifestations of heavy new fermions. Note that the parameters describe the contributions of new fermions with masses close to the electroweak scale. In the case of down-type singlet quark, having charge , the contributions into all polarization and, consequently, into PT parameters are four times smaller.

In the quark-gluon phase (QGP) of the Universe evolution, stable SQ interacts with standard quarks through exchanges by gluons , , and according to (1). So, we have large cross section for annihilation into gluons and quarks, and correspondingly, and also small additional contributions in electroweak channels . These cross sections can be simply derived from the known expressions for the processes and (see review in [28]) by time inversion. Two-gluon cross section in the low-energy limit looks likewhere is mass of -quark and is strong coupling at the corresponding scale. Two-quark channel in the massless limit is as follows:So, the two-gluon channel dominates. We should note that the cross section of SQ-annihilation is suppressed by large in comparison with the annihilation of standard quarks.

After the transition from quark-gluon plasma to hadronization stage, the singlet quarks having standard strong interactions (gluon exchange) form coupled states with ordinary quarks. New heavy hadrons can be constructed as coupled states which consist of heavy stable quark and a light quark from the SM quark sector. Here, we consider the simplest two-quark states, neutral and charged mesons. The lightest of them, for instance, neutral meson , is stable and can be considered as the carrier of cold Dark Matter. Possibility of existence of heavy stable hadrons was carefully analyzed in [11], where it was shown that this hypothesis does not contradict cosmochemical data and cosmological test. This conclusion was based on the important property of new hadron, namely, repulsive strong interaction with nucleons at large distances. The effect will be qualitatively analyzed for the case of and interactions in the next section.

#### 3. Quark Composition of New Hadrons and Their Interactions with Nucleons

At the hadronization stage, heavy SQ form the coupled states with the ordinary light quarks. Classification of these new heavy hadrons was considered in [11], where quark composition of two-quark (meson) and three-quark (fermion) states was represented for the case of up- and down-types of quark . Stable and long-lived new hadrons are divided into three families of particles with characteristic values of masses M, 2M and 3M, where M is the mass of -quark. Quantum numbers and quark content of these particles for the case of up-type quark are represented in Table 1.