Adding Dark Matter to the Standard Model

Detailed and redundant measurements of dark matter properties have recently become available. To describe the observations we consider scalar, vector and sterile neutrino dark matter models. A model with vector dark matter is consistent with all current observations.


Introduction
The Standard Model of quarks and leptons is enormously successful, it has passed many precision tests, and is here to stay. However, if the Standard Model were complete, the universe would have no matter: no dark matter, little baryonic matter, and no neutrino masses. "The New Minimal Standard Model" [1] is an extension that aims to "include the minimal number of new degrees of freedom to accommodate convincing (e.g., >5σ) evidence for physics beyond the Minimal Standard Model". But this aim has a moving target: as new data becomes available, the model may need to be amended accordingly. The inclusion of a "minimal number of new degrees of freedom" is in accordance with the absence of new particles at the LHC. The purpose of the present study is to see if the New Minimal Standard Model is consistent with the new data on dark matter that has recently become available, and, if necessary, update the model accordingly.
Let us briefly describe the New Minimal Standard Model [1]. First, the Standard Model Lagrangian is extended to include classical gravity. Next, a gauge singlet real scalar Klein-Gordon field with 2 Z parity is added for dark matter.
Dark energy is described by the cosmological constant Λ . Two gauge singlet Majorana neutrinos are added to account for neutrino masses and mixing (leaving one neutrino massless until data requires otherwise), and also to obtain ba-ryogenesis via leptogenesis. Finally, a real gauge singlet scalar field is included to implement inflation. The outline of this article is as follows. Measurements of dark matter properties are presented in Section 2. Scalar, vector and sterile neutrino dark matter models are studied in Sections 3 to 5. We close with conclusions.

Measured Properties of Dark Matter
Fits to spiral galaxy rotation curves [2] [3] [4], and studies of galaxy stellar mass distributions [5] [6] [7], independently obtain the following dark matter scenario. Dark matter is in thermal and diffusive equilibrium with the Standard Model sector in the early universe, i.e. no freeze-in, and decouples (from the Standard Model sector and from self-annihilation) while still ultra-relativistic, i.e. no freeze-out. The decoupling occurs at a temperature 0.2 GeV C T T > ≈ to not upset Big Bang Nucleosynthesis. Dark matter has zero chemical potential. The root-mean-square velocity of non-relativistic dark matter particles, at expansion parameter a , is In the case of negligible dark matter elastic scattering, the non-relativistic dark matter retains its ultra-relativistic thermal equilibrium (URTE), i.e. the ul- For an overview of these measurements see [8]. To make this article self-contained, Figure 1 presents forty-six independent measurements of NR h a′ from fits to spiral galaxy rotation curves [4]. From we calculate the warm dark matter free-streaming cut-off wavenumber fs k [7]. This cut-off wavenumber is also obtained from galaxy stellar mass distributions as shown in Figure 2 [7].
These independent measurements are consistent!  for bosons [14]. In the present study we will assume this specific dark matter scenario, and ask the following questions. What dark matter interactions lead to this scenario? How is dark matter created? How does dark matter and the Standard Model sector come into thermal and diffusive equilibrium? How do they decouple?  which dark matter particles become non-relativistic (uncorrected for dark matter halo rotation). Each measurement was obtained by fitting the rotation curves of a spiral galaxy in the Spitzer Photometry and Accurate Rotation Curves (SPARC) sample [9] with the indicated total luminosity at 3.6 μm. Full details of each fit are presented in [4]. International Journal of Astronomy and Astrophysics . Figure from [7].
How does dark matter acquire mass? Why is dark matter stable (relative to the age of the universe)? And, why is the measured dark matter particle mass h m so tiny compared to the Higgs boson mass H M ?
Notes: For a discussion of tensions between measurements of, and limits on, thermal relic dark matter mass see [7] [8]. We should mention that the observed galaxy mass distribution presented in Figure 2 is in tension with Lyman-α forest studies [17]. The 3.5σ confidence in favor of boson dark matter mentioned above, based on spiral galaxy rotation curves and galaxy stellar mass distributions, does not include the Tremaine-Gunn limit on fermion dark matter mass [18] [19]. Including this limit would strengthen the confidence. However, the Tremaine-Gunn limit needs to be revised in view of resent observations on dwarf spheroidal "satellites" of the Milky Way [20]

Scalar Dark Matter
The measured dark matter properties allow scalar or vector dark matter, with fermion dark matter disfavored but not ruled out. We begin with the real scalar field S of [1]. To attain thermal and diffusive equilibrium between dark matter and the Standard Model sector we need to add a coupling between the two. The cause the Higgs bosons annihilate, and only the tail of the S particle momentum distribution is above threshold. With to be avoided because it leads to a ratio of number densities S n n φ that depends on T. For this reason, and to obtain a stable S , and to avoid extra parameters in the potential ( ) Therefore, we consider a gauge singlet real Klein-Gordon scalar dark matter field S , with 2 Z symmetry S S ↔ − , and portal coupling to the Higgs boson [1]. Here we present a brief review of the model to see if it can describe the observed properties of dark matter. To the Standard Model Lagrangian we add and a contact coupling to the Higgs field φ : (We are omitting the metric factor g − .) After electroweak symmetry breaking (EWSB) the Higgs doublet, in the unitary gauge, has the form and dark matter particles acquire a mass squared assumed to be >0. We note that S is absolutely stable since there is no interaction term with a single S . The running of coupling parameters to 1-loop or 2-loop order can be found in [24] [25] [26] [27]. Some center of mass cross-sections are Requiring this decay rate to be less than the limit on the invisible width of the Higgs boson (≈0.013 GeV [14]) implies 0.03 hS λ  . In summary, we require Then there is fine tuning in (9) Let us now check whether non-relativistic dark matter acquires the non-relativistic Bose-Einstein momentum distribution due to elastic scattering. The crosssection at (14)), implies that the mean time between collisions of dark matter particles at S T M  is less than the age of the universe even for 6 10 hS λ − = , so, in this model, non-relativistic dark matter has non-relativistic thermal equilibrium. The cross-section (neglecting interference with (13)

Vector Dark Matter
To reduce the dark matter-dark matter elastic scattering cross-section, and to relieve the fine tuning in the model of Section 3, we attempt reaching the small h m in two steps.
To the Standard Model Lagrangian we add a complex scalar field S that is invariant with respect to the local ( )  . The corresponding vector gauge boson V µ acquires mass due to the breaking of the ( ) 1 S U symmetry of the ground state. In the present model, V is the dark matter candidate, and S decays to VV . The dark matter sector is known International Journal of Astronomy and Astrophysics in the literature as the "Abelian Higgs model".
The relevant part of the Lagrangian is S and the Standard Model sector have no charges in common. : Let us assign the high mass eigenstate to φ , and the low mass eigenstate to S (the opposite case will be considered below). A particular solution of interest has 1 θ  , so  (20) This cross-section implies that the mean dark matter particle interaction rate is much less than the expansion rate of the universe H at all temperatures, so, in this model, non-relativistic dark matter retains the ultra-relativistic Bose-Einstein momentum distribution. The two V 's in the decay S VV → have correlated polarizations, so the average number of boson degrees of freedom, needed to calculate the dark matter density (see (21)  Assigning charges Q S to Standard Model particles, to enhance or replace the contact interaction between S and φ , does not lead to compelling alternative models.

Sterile Neutrino Dark Matter
Observations of spiral galaxy rotation curves and of galaxy stellar mass distributions favor boson over fermion dark matter with a significance of 3.5σ [7], so we should not yet rule out fermion dark matter. Sterile neutrinos have been studied extensively as dark matter candidates [29] [30] [31]. In this section we briefly review sterile neutrinos and see if they are consistent with the measured properties of dark matter presented in Section 2.
We extend the Standard Model with a gauge singlet neutrino R ν with a Majorana mass 36 20 107 eV M + − = . This is the measured mass for the case of fermion dark matter retaining ultra-relativistic thermal equilibrium (URTE), see Table 4 of [7]. We will refer to the two irreducible representations of the proper Lorentz group of dimension 2 as "Weyl_L" and "Weyl_R". For simplicity we focus on one generation. L ν and R ν are two-component Weyl_L and Weyl_R fields, respectively. In a Weyl basis [30], , , Let us now consider dark matter production. We are interested in the reactions

Conclusions
Accurate, detailed and redundant measurements of dark matter properties have recently become available [7]. We have studied scalar, vector and sterile neutrino dark matter models in the light of these measurements. The vector dark matter model presented in Section 4 is (arguably) the renormalizable model with the least number of new degrees of freedom that is consistent with all current observations, and replaces the scalar dark matter model of Section 3 [1] that is ruled out. The sterile neutrino dark matter production mechanism studied in Section 5 did not meet experimental constraints. New insights pose new questions. If nature has chosen the vector dark matter of Section 4, why do the two terms in the numerator of (20) cancel to 1 part in 10 6 ? Similar questions can be made regarding the cosmological constant Λ , or the strong CP phase θ . Do the scalars φ and/or S participate/cause inflation? Baryogenesis via leptogenesis (arguably) requires sterile Majorana neutrinos. How are they produced? What is the origin, if any, of their masses?
How can we move forward? A signal in direct dark matter searches would rule out the vector model. Indirect searches may find an excess of photons (or neu-International Journal of Astronomy and Astrophysics trinos!) with energy ≈36 eV, ≈53 eV, or ≈62 eV, if dark matter is unstable and decays. Such a signal would also rule out the vector dark matter model. Collider experiments may discover an invisible Higgs decay width. Further progress will come from the cosmos: more studies of disk galaxy rotation curves, and galaxy stellar mass distributions (these studies can enhance the boson/fermion discrimination, and perhaps can observe the predicted tail of the boson warm dark matter power spectrum cut-off factor ( ) 2 fs k k τ [7]), galaxy formation simulations, the "small scale crisis" (missing satellites, too big to fail, galaxy core vs. cusp, large voids), super massive black holes at galaxy centers (Einstein condensation may occur at the galaxy center), revised constraints on fermion dark matter mass from the Tremaine-Gunn limit, and tighter constraints on dark matter self-interactions. It is necessary to understand the tensions between the Lyman-α forest studies and the observed galaxy stellar mass distributions, see Figure 2.
Studies of dark matter halo rotation in disk galaxies are also needed.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.