10 October 2017

A closure relation relating particles’ masses and Higgs field VEV 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 = LSM + LS

where LSM is the standard model density Lagrangian and LS is the scalar singlet dark matter density Lagrangian

(2) LS = 1/2 μS² S² + 1/2 λHSS²H*H + 1/4 λSS⁴ + 1/2 ∂μSμS.

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 mS is obtained from

(3) mS² = μS² + 1/2 λHS v²,

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

If λHS = 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) mH² + mW² + mZ² + (mS² − µS²) + mt² + mb² + mc² + mτ² + … = 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 mi/v [3], we get from Eqs. (3) and (4)

(5) 2λ + g²/4 + (g² + g’²)/4 + λHS/2 + (yt² + yb² + yc² + yτ² + …)/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 λHS = 0 (mS² − µS² = 0) has been considered in Ref. [3].

With

(6) mH = 125.0 – 125.5 GeV (CMS 2017),

(7) mH = 124.7 – 125.3 GeV (ATLAS 2017),

we guess

(8) mH = 125.0 – 125.3 GeV.

Then Eqs. (3) and (4) imply

(9) mt = 173.6 – 173.9 GeV

to be compared with

mt = 172.0 – 172.9 GeV (CMS 2016) (10)

(11) mt = 172.1 – 173.5 GeV (ATLAS 2017)

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

(12) mt = 172.1 – 172.9 GeV

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

(13) mS² − µS² = 1/2 λHS v² = (15 – 25 GeV)²

or

(14) λHS = 0.008 – 0.021

In Refs. [2], [4] and [5], it has been noted that dark matter phenomenology is driven predominantly by mS and λHS, with viable solutions known to exist in a number of regions, in particular where mS is around mH/2 and where coupling λHS is very small (λHS < 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 λHS is dependent on more precise values of mt and mH.

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.