10 October 2017

 

A closure relation relating particles’ masses and Higgs field VEV
in the scalar singlet dark matter model

 

Jean Pestieau
Centre for Cosmology, Particle Physics and Phenomenology (CP3)
Université catholique de Louvain, Belgium

 

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_S² S² + 1/2 λ_HSS²H*H + 1/4 λ_SS⁴ + 1/2 ∂_μS∂^μS.

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_S² = μ_S² + 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_H² + m_W² + m_Z² + (m_S² − µ_S²) + m_t² + m_b² + m_c² + m_τ² + … = 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_t² + y_b² + y_c² + y_τ² + …)/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_S² − µ_S² = 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_S² − µ_S² = 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.

References

[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.