10 October 2017

in the scalar singlet dark matter model

One of the simplest viable models for dark matter is, next to the standard model, an additional scalar singlet S, permitted by ℤ2 symmetry.

The Lagrangian density L is expressed then as [1]

(1)
L = L_{SM} + L_{S}

where L_{SM} is the standard model density Lagrangian and
L_{S} is the scalar singlet dark matter density Lagrangian

(2)
L_{S} = 1/2 μ_{S}² S²
+ 1/2 λ_{HS}S²H*H
+ 1/4 λ_{S}S⁴
+ 1/2 ∂_{μ}S∂^{μ}S.

From left to right, we have the bare S mass, the Higgs-scalar singlet coupling, the S quartic self-coupling, and the S kinetic term.

The singlet mass m_{S} is obtained from

(3)
m_{S}² = μ_{S}² + 1/2 λ_{HS} v²,

where v = 246.22 GeV is the VEV of H, the Higgs field.

If λ_{HS} = 0, dark matter has no interaction with ordinary matter except through
gravitational field.

We propose a sum rule or a closure relation relating particle masses and v :

(4)
m_{H}² + m_{W}² + m_{Z}² + (m_{S}² − µ_{S}²)
+ m_{t}² + m_{b}² + m_{c}² + m_{τ}² + … = v²

Noticing that, in the electroweak standard model, the couplings of all particles to the
H field are proportional to 1/v, namely
of the form m_{i}/v [3], we get from Eqs. (3) and (4)

(5)
2λ + g²/4 + (g² + g’²)/4 + λ_{HS}/2
+ (y_{t}² + y_{b}² + y_{c}² + y_{τ}² + …)/2 = 1

In fact, what we propose is that v² is exclusively built on the particles’ masses. v² is a paved parquet exclusively composed of particles’ masses.

Case with λ_{HS} = 0 (m_{S}² − µ_{S}² = 0) has been considered in Ref. [3].

With

(6)
m_{H} = 125.0 – 125.5 GeV (CMS 2017),

(7)
m_{H} = 124.7 – 125.3 GeV (ATLAS 2017),

we guess

(8)
m_{H} = 125.0 – 125.3 GeV.

Then Eqs. (3) and (4) imply

(9)
m_{t} = 173.6 – 173.9 GeV

to be compared with

m_{t} = 172.0 – 172.9 GeV (CMS 2016)
(10)

(11)
m_{t} = 172.1 – 173.5 GeV (ATLAS 2017)

Now from Eqs. (10) and (11), we guess

(12)
m_{t} = 172.1 – 172.9 GeV

With Eqs (3), (4), (8) and (12), we get

(13)
m_{S}² − µ_{S}² = 1/2 λ_{HS} v² = (15 – 25 GeV)²

or

(14)
λ_{HS} = 0.008 – 0.021

In Refs. [2], [4] and [5], it has been noted that dark matter phenomenology is driven predominantly
by m_{S} and λ_{HS}, with viable solutions known to exist in a number of regions,
in particular where m_{S} is around m_{H}/2 and where coupling λ_{HS}
is very small (λ_{HS} < 0.01). Furthermore the scalar singlet can constitute all
the observed dark matter.

In the framework of the scalar singlet dark matter model — assuming Eq.(4) — a more precise
value of λ_{HS} is dependent on more precise values of m_{t} and m_{H}.

[1] A. Silveira and A. Zee, “Scalar Phantoms”, Phys. Lett. B 161 (1985) 136-170.

[2] For recent updates and references, see : The Gambit Collaboration, “Status of the scalar singlet dark matter model”, Eur. Phys. J. C 77, 568 (2017), https://arxiv.org/abs/1705.07931.

[3] G. López Castro and J. Pestieau, “Relation between masses of particles and the Fermi constant in the electroweak Standard Model” (2013), https://arxiv.org/abs/1305.4208.

[4] J. M. Cline, K. Kainulainen, P. Scott, and C. Weniger, “Update on scalar singlet dark matter”, Phys. Rev. D 88 (2013) 055025, https://arxiv.org/abs/1306.4710.

[5] A. Beniwal, F. Rajec, et. al., “Combined analysis of effective Higgs portal dark matter models”, Phys. Rev. D 93 (2016) 115016, https://arxiv.org/abs/1512.06458.