PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

Appendix
A: The Scalar Potential and Higgs Spectrum
Our model has two SU(2) complex Higgs doublets Φ 1 and Φ 2 , with hypercharge Y = 1.
The scalar potential can then be written as
( ) 2 ( ) 2 ) 2
V = λ 1 Φ † 1Φ 1 −v 2 +λ2 Φ † 2Φ 2 −v ′2 +λ3
((Φ † 1Φ 1 −v 2 )+(Φ † 2Φ 2 −v ′2 )
( ) 2
+λ 4
((Φ † 1Φ 1 )(Φ † 2Φ 2 )−(Φ † 1Φ 2 )(Φ † 2Φ 1 )
)+λ 5 Re(Φ † 1Φ 2 )−vv ′ cosξ
where
+λ 6
(Im(Φ † 1Φ 2 )−vv ′ sinξ) 2,
(A1)
〈Φ 1 〉 =
( ( )
0 0
, 〈Φ
v)
2 〉 =
v ′ e iξ , and tanβ = v′
v .
(A2)
According to our Z 2 charge assignment, Φ 1 carries charge +1, while Φ 2 has −1 charge.
Therefore, the λ 5 term is zero when the symmetry is exact. We will discuss the phenomenological
consequences of this and argue in favor of a mild breaking of this Z 2 symmetry.
With the scalar potential Eq. (A1) it is straightforward to obtain the Higgs mass
matrix and obtain the corresponding mass spectrum. The physical degrees of freedoms
contain the charged Higgs H ± and the neutral Higgs H 0 , h 0 , and A 0 . While H 0 and h 0
are CP even, A 0 is CP odd. If we work in a simplified scenario where ξ is taken as zero,
then it is is quite straightforward to derive the mass of the charged Higgs H ± and the
CP-odd Higgs A 0 . The masses are given as
M 2 H ± = λ 4(v 2 +v ′2 ), and M 2 A 0 = λ 6(v 2 +v ′2 ),
(A3)
respectively. The mass matrix for the neutral CP-even Higgs is
( )
4v
M ′ =
2 (λ 1 +λ 3 )+v ′2 λ 5 (4λ 3 +λ 5 )vv ′
(4λ 3 +λ 5 )vv ′ 4v ′2 (λ 2 +λ 3 )+v 2 . (A4)
λ 5
The mixing angle, obtained from diagonalizing the above matrix is given by
and the corresponding masses are
tan2α = 2M 12
M 11 −M 22
,
M 2 H 0 ,h 0 = 1 2 {M 11 +M 22 ±
(A5)
√
(M 11 −M 22 ) 2 +4M 2 12. (A6)
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