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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.

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” [

Let us briefly describe the New Minimal Standard Model [

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.

Fits to spiral galaxy rotation curves [^{+}e^{−} annihilation, is measured to be T h / T = 0.456 − 0.054 + 0.039 ( T h / T = 0.383 − 0.050 + 0.033 ) for scalar (vector) dark matter. All uncertainties have 68% confidence. These numbers are obtained from

m h = 51.2 ( 0.76 km / s v h rms ( 1 ) ) 3 / 4 ( 1 N b ) 1 / 4 eV , (1)

T h T = 0.511 ( v h rms ( 1 ) 0.76 km / s ) 1 / 4 ( 1 N b ) 1 / 4 . (2)

In the case of negligible dark matter elastic scattering, the non-relativistic dark matter retains its ultra-relativistic thermal equilibrium (URTE), i.e. the ultra-relativistic Bose-Einstein momentum distribution, and the measurements are v h rms ( 1 ) = 0.67 ± 0.24 km / s , a ′ h NR = ( 2.23 ± 0.80 ) × 10 − 6 , k fs = 0.37 − 0.08 + 0.17 Mpc − 1 , M fs = 10 13.0 ± 0.4 M ⊙ , and T h / T = 0.367 − 0.038 + 0.029 ( T h / T = 0.309 − 0.033 + 0.024 ) and m h = 124 − 25 + 50 eV ( m h = 104 − 21 + 42 eV ) for scalar (vector) dark matter. The relations between v h rms ( 1 ) and m h and T h / T for zero chemical potential are [

m h = 113 ( 0.76 km / s v h rms ( 1 ) ) 3 / 4 ( 1 N b ) 1 / 4 eV , (3)

T h T = 0.379 ( v h rms ( 1 ) 0.76 km / s ) 1 / 4 ( 1 N b ) 1 / 4 . (4)

For an overview of these measurements see [

The current limit on dark matter self interaction cross-section is σ DM-DM / m DM < 0.47 cm 2 / g with 95% confidence [

The current limits on dark matter particle mass are m h > 70 eV for fermions, and m h > 10 − 22 eV for bosons [

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?

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 m h so tiny compared to the Higgs boson mass M H ?

Notes: For a discussion of tensions between measurements of, and limits on, thermal relic dark matter mass see [

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 [

Therefore, we consider a gauge singlet real Klein-Gordon scalar dark matter field S , with Z 2 symmetry S ↔ − S , and portal coupling to the Higgs boson [

L S = 1 2 ∂ μ S ⋅ ∂ μ S − 1 2 m ¯ S 2 S 2 − λ S 4 ! S 4 + ⋅ ⋅ ⋅ , (5)

and a contact coupling to the Higgs field ϕ :

L S ϕ = − 1 2 λ h S ( ϕ † ϕ ) S 2 . (6)

(We are omitting the metric factor − g .) After electroweak symmetry breaking (EWSB) the Higgs doublet, in the unitary gauge, has the form

ϕ = ( ϕ + ϕ 0 ) = 1 2 ( 0 v h + h ( x ) ) (7)

with real h ( x ) , the interaction Lagrangian becomes

L S ϕ = − 1 4 λ h S ( v h 2 + 2 v h h + h 2 ) S 2 , (8)

and dark matter particles acquire a mass squared

M S 2 = 1 2 λ h S v h 2 + m ¯ S 2 (9)

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 [

σ ( h h ↔ S S ) = λ h S 2 16 π s | p f | | p i | , (10)

σ ( W − W + ↔ h * ↔ S S ) = λ h S 2 M W 4 4 π s | p f | | p i | 1 ( s − M H 2 ) 2 + M H 2 Γ H 2 , (11)

where s ≡ ( p 1 + p 2 ) 2 is the Mandelstam variable. The reaction rates are exponentially suppressed at T ≲ M H or T ≲ M W . These interactions bring dark matter into thermal and diffusive equilibrium with the Standard Model sector at T ≳ M H if | λ h S | ≳ 10 − 6 and 10^{−}^{6}, respectively. The Higgs boson invisible decay rate for M H > 2 M S is

Γ ( h → S S ) = λ h S 2 v h 2 8 π M H . (12)

Requiring this decay rate to be less than the limit on the invisible width of the Higgs boson (≈0.013 GeV [

As an example, take M S = 73 eV and λ h S = ± 10 − 5 , so 1 2 λ h S v h 2 = ± 0.3 GeV 2 . Then there is fine tuning in (9): m ¯ S 2 = 5 × 10 − 15 ∓ 0.3 GeV 2 . Note that to achieve M S as low as 73 eV starting from v h = 246 GeV requires fine tuning between two unrelated input parameters with dimensions of mass.

Let us now check whether non-relativistic dark matter acquires the non-relativistic Bose-Einstein momentum distribution due to elastic scattering. The cross-section at T ≪ M H (neglecting interference with (14)),

σ ( S S → h * → S S ) = 9 λ h S 4 v h 4 16 π s M H 4 , (13)

implies that the mean time between collisions of dark matter particles at T ≲ M S is less than the age of the universe even for λ h S = 10 − 6 , so, in this model, non-relativistic dark matter has non-relativistic thermal equilibrium. The cross-section (neglecting interference with (13)),

σ ( S S → S S ) = λ S 2 16 π s , (14)

also corresponds to collisional dark matter if λ S > 10 − 11 .

Dark matter decouples from the Standard Model sector at T ≈ M H when the Higgs bosons become non-relativistic. As the universe expands and cools, particles and antiparticles that become non-relativistic annihilate heating the Standard Model sector without heating dark matter, or neutrinos if they have already decoupled. For decoupling at M H we expect the temperature of ultra-relativistic dark matter, relative to the photon temperature, after e^{+}e^{−} annihilation, to be T h / T = [ 8 × 43 / ( 385 × 22 ) ] 1 / 3 = 0.344 [

The cross-section limit σ DM-DM / m DM < 0.47 cm 2 / g [

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 m h in two steps.

To the Standard Model Lagrangian we add a complex scalar field S that is invariant with respect to the local U ( 1 ) S transformation S → exp [ i Q S α ( x ) ] S . The corresponding vector gauge boson V μ acquires mass due to the breaking of the U ( 1 ) S symmetry of the ground state. In the present model, V is the dark matter candidate, and S decays to V V . The dark matter sector is known in the literature as the “Abelian Higgs model”.

The relevant part of the Lagrangian is

L ϕ S V = ( D μ ϕ ) † ( D μ ϕ ) + ( D ′ μ S ) † ( D ′ μ S ) − V ( ϕ , S ) , (15)

V ( ϕ , S ) = − μ h 2 ( ϕ † ϕ ) + λ h ( ϕ † ϕ ) 2 − μ s 2 ( S † S ) + λ s ( S † S ) 2 + λ h s ( ϕ † ϕ ) ( S † S ) , (16)

i D μ = i ∂ μ − g τ 2 ⋅ W μ − g ′ 1 2 B μ , (17)

i D ′ μ = i ∂ μ + g V Q S V μ , (18)

V μ → V μ + 1 g V ∂ μ α . (19)

S and the Standard Model sector have no charges in common.

For μ h 2 > 0 , λ h > 0 , μ s 2 > 0 , and λ s > 0 , there is symmetry breaking, and the fields ϕ = ( h + , ( h + i A + v h ) / 2 ) T and S = ( s + i ρ + v s ) / 2 acquire vacuum expectation values [

v h 2 = 2 μ s 2 λ h s − 4 μ h 2 λ s λ h s 2 − 4 λ s λ h and v s 2 = 2 μ h 2 λ h s − 4 μ s 2 λ h λ h s 2 − 4 λ s λ h (20)

if v h 2 > 0 and v s 2 > 0 . In unitary gauge, the real amplitudes A and ρ become the longitudinal components of Z μ and V μ , respectively, and the complex amplitude h ± becomes the longitudinal components of W + and W − . The mass eigenstates are

M V = g V Q S v s , (21)

M ϕ , S 2 = ( λ h v h 2 + λ s v s 2 ) ± ( λ h v h 2 − λ s v s 2 ) 2 + ( λ h s v h v s ) 2 , (22)

and the mixing angle is

tan ( 2 θ ) = λ h s v h v s λ s v s 2 − λ h v h 2 . (23)

To bring S into thermal and diffusive equilibrium with the Standard Model sector without exceeding the limit on the invisible width of the Higgs boson h requires 10 − 6 ≲ λ h s ≲ 0.03 as in Section 3. Some reactions of interest are

Γ ( s → V V ) = g V 4 Q S 4 v s 2 2 π M S , (24)

σ ( s s → h * → W + W − ) = λ h s 2 M W 4 4 π s | p f | | p i | 1 ( s − M H 2 ) 2 + M H 2 Γ H 2 , (25)

σ ( s s → V V ) = g V 4 Q S 4 4 π s | p f | | p i | . (26)

The couplings of V , s , and h (up to order 4) are proportional to h 3 , h 2 s , h s 2 , s 3 , h 4 , h 2 s 2 , s 4 , V 2 s , and V 2 s 2 . The kinematics allow V to decay only to γ’s or ν’s. However, we note that there is no coupling with a single V , so V is absolutely stable.

Our challenge is to choose parameters so that S attains statistical equilibrium with the Standard Model sector at T ≳ M H , but V does not; and we need the decay S → V V to occur after S has decoupled from the Standard Model sector, and while S is still ultra-relativistic, i.e. within the temperature range M S < T < M H ; and that m h = M V = 112 eV .

Case M S < M H : 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 | θ | ≪ 1 , so M H 2 ≈ 2 λ h v h 2 ≈ 2 μ h 2 as in the Standard Model. A bench-mark scenario with M V = 112 eV is M S = 9 × 10 − 4 GeV , λ h s = 10 − 5 , λ s = 0.1 , g V Q S = 6 × 10 − 5 , v s = 2 × 10 − 3 GeV , and μ s = 0.551 GeV . To meet all requirements, there is fine tuning of μ s 2 to lower v s : the relative difference of the two terms in the numerator of (20) is ≈10^{−}^{6}.

The reaction rate of s s ↔ h * ↔ W + W − , relative to the expansion rate of the universe H , is 1 / ( Δ t ⋅ H ) = 500 at T ≈ M H , so this coupling is strong. For s s ↔ h h , 1 / ( Δ t ⋅ H ) = 700 at T ≈ M H , so this coupling is also strong. For s s ↔ V V , 1 / ( Δ t ⋅ H ) = 3 × 10 − 4 (200) at T ≈ M H ( M S ), so V does not attain statistical equilibrium with S , or with the Standard Model sector, at T ≳ M H . The decay rate of s → V V , relative to the expansion rate of the universe H , is Γ ( s → V V ) / H = 3 × 10 − 7 (3 × 10^{4}) at T ≈ M H ( M S ), so indeed we have arranged that the decay occurs after S has decoupled, and while S is still ultra-relativistic, i.e. in the temperature range M S < T < M H .

The cross-section for V V → s * → V V at T ≪ M S is

σ ( V V → s * → V V ) = 9 g V 8 Q S 8 v s 4 π s M S 4 . (27)

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 → V V have correlated polarizations, so the average number of boson degrees of freedom, needed to calculate the dark matter density (see (21) of [

For zero chemical potential, the number of s per unit volume, given by the ultra-relativistic Bose-Einstein distribution, is

n s = N b s ( 2 π ℏ ) 3 ∫ 0 ∞ 4 π p 2 d p exp [ p c k T s ] − 1 , (28)

where the number of boson degrees of freedom of s is N b s = 1 . After the decay

2 n s = n V = N b V ( 2 π ℏ ) 3 ∫ 0 ∞ 4 π p 2 d p exp [ p c k T V ] − 1 . (29)

Each s in 8 orbitals of momentum 2p decays to two V ’s corresponding to one orbital with momentum p, so

2 n s = n V = 2 ⋅ 8 N b s ( 2 π ℏ ) 3 ∫ 0 ∞ 4 π p 2 d p exp [ 2 p c k T s ] − 1 . (30)

Integrating, we obtain T V = ( 4 / 3 ) 1 / 3 T s . So, the predicted ratio is T h / T = ( 4 / 3 ) 1 / 3 ⋅ 0.344 = 0.379 , to be compared with the measured value T h / T = 0.332 − 0.029 + 0.026 .

The cross-section limit σ DM-DM / m DM < 0.47 cm 2 / g [

In summary, the vector model with M S < M H is consistent with all currently measured properties of dark matter. There is fine tuning to obtain the small required symmetry breaking of the ground state of S .

Case M S > M H : Let us now assign the high mass eigenstate to S , and the low mass eigenstate to ϕ . Again, as an example, we consider the case | θ | ≪ 1 , so M H 2 ≈ 2 λ h v h 2 ≈ 2 μ h 2 as in the Standard Model, and M S 2 ≈ 2 λ s v s 2 ≈ 2 μ s 2 . A benchmark solution with M V = 112 eV is M S = 135 GeV , λ h s = 3 × 10 − 5 , λ s = 0.1 , g V Q S = 4 × 10 − 10 , v s = 300 GeV , and μ s = 96 GeV . When particles S become non-relativistic at T ≈ M S > M H , they decay mostly to the Standard Model sector: reactions s s → h * → W + W − are much faster than the universe expansion rate, while s s → V V and s → V V are much slower, so the universe is left with no dark matter.

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.

Observations of spiral galaxy rotation curves and of galaxy stellar mass distributions favor boson over fermion dark matter with a significance of 3.5σ [

We extend the Standard Model with a gauge singlet neutrino ν R with a Majorana mass M = 107 − 20 + 36 eV . This is the measured mass for the case of fermion dark matter retaining ultra-relativistic thermal equilibrium (URTE), see

To include Weyl spinors into the Standard Model, it is convenient to use 4-component Dirac spinor notation. Our metric is d i a g ( η μ ν ) = ( 1, − 1, − 1, − 1 ) . The matrices A and C are defined, in any basis, as A γ μ = γ μ † A , and γ μ C = − C γ μ T [

In a Weyl basis [

ψ = ( ν L ν R ) , ψ ˜ = ( ν R † , ν L † ) , (31)

γ 0 = ( 0 σ 0 σ 0 0 ) , γ k = ( 0 σ k − σ k 0 ) , C = ( − i σ 2 0 0 i σ 2 ) , (32)

γ 5 = ( − σ 0 0 0 σ 0 ) , ψ L = ( ν L i σ 2 ν L * ) , ψ R = ( − i σ 2 ν R * ν R ) . (33)

Note that ψ L c = ψ L , and ψ R c = ψ R , so these are Majorana fields. With this notation the Majorana fields ψ L and ψ R c can mix. Note however that ψ L and ψ R c are distinct: ψ L has weak interactions while ψ R c does not. ψ ˜ R ψ R , ψ ˜ L c ψ R c , ψ ˜ R ψ L , ψ ˜ R c ψ R , and ψ ˜ R c ψ R c are scalars with respect to the proper Lorentz group. The neutrino mass term after electroweak symmetry breaking has the form [

L ν m a s s = − m 2 ψ ˜ L c ψ R c − m 2 ψ ˜ R ψ L − M 2 ψ ˜ R ψ R c + H . c ., (34)

where m = Y v h / 2 is a Dirac mass ( Y is a Yukawa coupling), and M is a Majorana mass. We consider the case | m / M | ≪ 1 . The mass eigenstates are [

ψ a = i cos θ ψ L − i sin θ ψ R c , with mass m a = m 2 M , ψ s = sin θ ψ L + cos θ ψ R c , with mass m s = M , (35)

where tan ( θ ) = m / M , ν L a ( t ) = ν L a ( 0 ) exp [ ∓ i E t ± i E 2 − m a 2 x ] , and ν R s ( t ) = ν R s ( 0 ) exp [ ± i E t ∓ i E 2 − M 2 x ] .

Let us now consider dark matter production. We are interested in the reactions ν e e + → W + * → ν s e + , or u u ¯ → Z * → ν s ν e . First, we verify that the produced ψ L is a coherent superposition of ψ a and ψ s . The coherence factor is [

ε coh = exp [ − Δ M 2 / ( 8 σ E 2 ) ] ⋅ exp [ − Δ t 2 / t coh 2 ] , (36)

with Δ M 2 ≡ M 2 − m a 2 . Since we are interested in energy E of ν L of order M W / 2 , we take its uncertainty to be σ E ≈ Γ W ≫ M , so the first factor is 1. Δ t is the mean time between ν L interactions. The propagation time of ν a and ν s over which their wave packets cease to overlap is the decoherence time [

t coh = 2 2 2 E 2 | Δ M 2 | σ t . (37)

Taking the wave packet duration σ t ≈ 1 / Γ W , we estimate Δ t ≪ t coh for the small value of M being considered. In conclusion, ε coh ≈ 1 , and ν a and ν s do not become decoherent between ν L interactions, so we must take into account their oscillations.

Consider initial conditions for ψ L production to be ψ L ( 0 ) ∝ 1 and ψ R c ( 0 ) ∝ 0 . Then, from (35), we obtain the probabilities P L ( Δ t ) = | ψ L ( Δ t ) | 2 to observe a Weyl_L neutrino, and P s ( Δ t ) = | ψ R c ( Δ t ) | 2 to create a sterile neutrino,

P s ( Δ t ) = 1 − P L ( Δ t ) = 4 sin 2 θ cos 2 θ sin 2 ( Δ M 2 4 E x ) , (38)

with x ≈ Δ t , and E = s / 2 . Note that P s ( Δ t ) = 2 m a / M for Δ M 2 Δ t / ( 4 E ) ≫ π , but this probability is suppressed by a factor 2 sin 2 [ Δ M 2 Δ t / ( 4 E ) ] for small Δ t . Equation (38) describes the oscillation between the active and sterile neutrinos. Similar phenomenology has been confirmed in neutrino flavor oscillation experiments.

The cross-section σ ( ν e e + → W + * → ν e e + ) is given by Eq. (50.25) of [

The production channel W + W − → h * → ψ L ψ R is negligible.

Accurate, detailed and redundant measurements of dark matter properties have recently become available [

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 neutrinos!) 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 τ 2 ( k / k fs ) [

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

Hoeneisen, B. (2021) Adding Dark Matter to the Standard Model. International Journal of Astronomy and Astrophysics, 11, 59-72. https://doi.org/10.4236/ijaa.2021.111004