Erratum to: Vacuum stability of a general scalar potential of a few fields
Eur. Phys. J. C
Erratum to: Vacuum stability of a general scalar potential of a few fields
Kristjan Kannike 0
0 NICPB , Rävala pst. 10, 10143 Tallinn , Estonia
−(λ3 + λ4 − λ5)[λ5(λ3 + λ4 − λ5)

In the original, it was erroneously assumed in the derivation
of the vacuum stability conditions for the two Higgs doublet
model (2HDM) with real couplings that in the case of ρ = 1,
it is sufficient to consider only cos φ = ±1. In fact, the
solution with ρ = 1, cos φ = ±1 may exist, yielding an
extra condition.
The minimisation equations for φ, h1, h2 and λ in the case
of ρ = 1 are
0 = h1h2 2λ5h1h2 cos φ + λ6h12 + λ7h22 sin φ,
λh1 = 4λ1h13 + 2(λ3 + λ4 + λ5 cos 2φ) h1h22
+ 6λ6 cos φ h12h2 + 2λ7 cos φ h23,
λh2 = 4λ2h23 + 2(λ3 + λ4 + λ5 cos 2φ)h12h2
+ 2λ6 cos φ h13 + 6λ7 cos φ h1h22,
2 2
1 = h1 + h2.
Their solutions with cos φ = ±1 are given by
cos φρ=1 = −
λ6h12 + λ7h22
2λ5h1h2
(1)
(2)
(3)
(4)
(5)
(6)
(7)
−2λ6λ7]]/[2λ5(λ1 + λ2 − λ3 − λ4 + λ5)
−(λ6 − λ7)2].
Altogether, the conditions for the 2HDM potential with
real couplings to be bounded from below are
Vρ=0 > 0 ∧ Dcos φ=±1, ρ=1
∧ (Qcos φ=±1, ρ=1 > 0 ∨ Rcos φ=±1, ρ=1 > 0)
2 2
∧ (0 < h1,ρ=1 < 1 ∧ 0 < h2,ρ=1 < 1
∧ 0 < cos2 φρ=1 < 1
⇒ Vmin,ρ=1 > 0)
∧ (0 < h21 < 1 ∧ 0 < h22 < 1
∧ 0 < ρ2 < 1
⇒ Vmin > 0),
which includes the new condition involving cos φρ=1, h12,ρ=1,
2
h2,ρ=1 and Vmin,ρ=1.
Figures 3 and 4 that present examples of the allowed
parameter space for the 2HDM remain unaffected by the
change.
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