Triple and quartic interactions of Higgs bosons in the two-Higgs-doublet model with CP violation

29 Pages·0.39 MB·English

by  M. N. Dubinin

LAPTH-936/02
Triple and Quartic Interactions of Higgs
3
0 Bosons in the Two-Higgs-Doublet Model
0
2 with CP violation

n
a
J

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2 M.N.Dubinin1, A.V.Semenov2

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v 1Institute of Nuclear Physics, Moscow State University
5
119992 Moscow, Russia
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2 2Laboratoire de Physique Theorique LAPTH
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Chemin de Bellevue, B.P. 110, F-74941, Annecy-le-Vieux, Cedex, France
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2
0
/
h
p
-
p Abstract
e
h
: We consider the two-Higgs-doublet model with explicit CP vio-
v
i lation, where the effective Higgs potential is not CP invariant
X
at the tree-level. Three neutral Higgs bosons of the model are
r
a the mixtures of CP-even and CP-odd bosons which exist in the
CP conserving limit of the theory. The mass spectrum and tree-
level couplings of the neutral Higgs bosons to gauge bosons and
fermions are significantly dependent on the parameters of the
Higgs boson mixing matrix. We calculate the Higgs-gauge bo-
son, Higgs-fermion, triple and quartic Higgs self-interactions in
the MSSM with explicit CP violation in the Higgs sector and
CP violating Yukawa interactions of the third generation scalar
quarks. In some regions of the MSSM parameter space substan-
tial changes of the self-interaction vertices take place, leading to
significant suppression or enhancement of the multiple Higgs bo-
son production cross sections.
1
1 Introduction
General interest to the models with two (and more) Higgs doublets is main-
tained by the absence of a convincing argument in favor of only one genera-
tion of Higgs bosons when there are three known generations of fundamental
fermions. Models with extended Higgs sector provide richer physical possi-
bilities than the standard scheme with one doublet. One of them is the pos-
sibility to introduce CP violation beyond the Cabibbo-Kobayashi-Maskawa
(CKM) mechanism, by means of the Higgs boson exchange amplitudes with
complex Higgs boson-fermion vertices. Complex couplings can be generated
either spontaneously [1], when the vacuum expectation values of the Higgs
fields are complex and couplings of the CP invariant tree-level Higgs poten-
tial are real, or explicitly inserted [2] on the level of SU(2) U(1)-invariant
×
potential terms, when the complex vacuum expectation values of scalar fields
correspond to the minimum of hermitian potential with complex couplings,
which is not CP-invariant (CP-invariance softly broken by the mass terms).
Various representations of the SU(2) U(1)-invariant two-doublet Higgs
×
potentials have been considered in the literature. The two-doublet models
with spontaneous CP violation [1, 3] make use of the potential of general
structure µ2ϕ2 + λϕ4 without the dimension two µ2 -terms. Models with
− 12
explicit CP violation use either the potential with trivial minimization ([2],
see also [4]) or the potential with complex coupling µ2 of the dimension
12
¯ ¯ ¯
two terms and complex couplings λ , λ and λ in front of the dimension
5 6 7
four potential terms [5, 6], similar to effective potential of the minimal su-
pesymmetry (MSSM). Standard transformation (diagonalization) procedure
from the level of primary fields which are the components of scalar doublets
in SU(2) U(1)-invariant potential terms (SU(2) U(1) eigenstates), to
× ×
the physical fields (mass eigenstates) of Higgs bosons should be consistently
performed to respect the SU(2) U(1) invariance and the minimization of
×
the potential. We consider the diagonalization for the two different two-
Higgs-doublet potential forms in much details. A special case of the general
two-Higgs-doublet model is represented by the Higgs sector of MSSM. Sub-
stantial radiative corrections to the Higgs boson masses and couplings are
induced at the m scale mainly by the third generation quarks t, b and the
Z
third generation scalar quarks [7]. In the special case of MSSM the multi-
parameter space of general two-Higgs-doublet model is significantly reduced,
providing possibilities of much less ambiguous phenomenological predictions.
Phenomenological consequences of the CP violating Higgs-third genera-
tion squark Yukawa interactions in the Higgs-fermion and the Higgs-gauge
boson sectors have been considered in [5]. We focus mainly on the self-
interactions of Higgs bosons. Experimental observation of the scalar boson
2
signals should be followed by the verification of Higgs mechanism as the
essence of the gauge boson and fermion mass generation. Self-interactions of
the Higgs fields lead to untrivial structure of the vacuum state with nonzero
(andpossibly complex) field tensions, initializing thespontaneous breakdown
of SU(2) U(1) symmetry. Reconstruction of the Higgs self-interaction
×
potential from the data on multiple (mainly double and triple) Higgs bo-
son production cross sections [8] requires the experimental measurements of
triple and quartic Higgs boson self-interaction vertices, which is nontrivial
but valuable task for a future high luminosity colliders, such as LHC and
TESLA.
Insection2wediscussthediagonalizationofCPinvariantHiggspotential,
represented in two different forms, in the general two-Higgs-doublet model
(THDM) and consider the special case of the Higgs sector in the minimal
supersymmetry (MSSM). In section 3 we introduce complex parameters of
the SU(2) U(1) invariant potential terms and discuss the diagonalization of
×
the THDM potential which acquires explicit CP violation. In sections 4 and
5 we calculate the Higgs-gaugeboson, Higgs-fermionand Higgsself-couplings
in the MSSM with CP violation.
2 Diagonalization of the mass matrix in the
general two-Higgs-doublet model
Two representations have beenused forthetwo-doublet Higgspotential. The
first representation [2, 4]
V(ϕ ,ϕ ) = λ (ϕ+ϕ v12)2 +λ (ϕ+ϕ v22)2 (1)
1 2 1 1 1 − 2 2 2 2 − 2
+λ [(ϕ+ϕ v12)+(ϕ+ϕ v22)]2
3 1 1 − 2 2 2 − 2
+λ [(ϕ+ϕ )(ϕ+ϕ ) (ϕ+ϕ )(ϕ+ϕ )]
4 1 1 2 2 − 1 2 2 1
+λ [Re(ϕ+ϕ ) v1v2Re(eiξ)]2 +λ [Im(ϕ+ϕ ) v1v2Im(eiξ)]2
5 1 2 − 2 6 1 2 − 2
where λ arereal constants and the SU(2) doublets ϕ have the components
i 1,2
1 1
ϕ = iw+, (v +h +iz ) , ϕ = iw+, (v +h +iz ) . (2)
1 {− 1 √2 1 1 1 } 2 {− 2 √2 2 2 2 }
w are complex fields and z , h are real scalar fields. At positive λ ,...λ
1,2 1,2 1,2 1 6
each term of the potential V(ϕ ,ϕ ) is obviously positive and its zero min-
1 2
imum is achieved if the vacuum expectation values of < ϕ >, < ϕ > are
1 2
taken in the form
1 1
< ϕ >= 0,v , < ϕ >= 0,v eiξ (3)
1 √2{ 1} 2 √2{ 2 }
3
In the case of λ = λ (corresponding to the CP conserving MSSM-like
5 6
potential, see below) the last two terms in (1) form the modulo squared,
and the phase ξ can be removed from the potential by the U(1) rotation
of ϕ , that does not affect minimization. In this section we will consider
2
the case λ = λ , ξ =0. Substitution of (2) to (1) gives a bilinear form of
5 6
6
the mass term with mixed components w ,h ,z , which can be diagonalized
i i i
by an orthogonal transformation of the fields in order to define the tree
level masses of Higgs bosons. In the CP conserving case the potential terms
involving z ,z fields from the imaginary parts of ϕ , ϕ doublets and h ,h
1 2 1 2 1 2
fields from their real parts do not mix, so the mass terms are diagonalized by
a separate two-dimensional rotations of the z ,z and the h ,h fields. The
1 2 1 2
resulting spectrum of scalars consists of two charged H , three neutral h, H,
±
A0 scalar fields, and three Goldstone bosons G. This procedure is described
in many papers (for instance, [4, 9]). The w sector is diagonalized by the
1,2
rotation of w ,w H,G
1 2
→
w = H s +G c , w = H c +G s (4)
1± − ± β ± β 2± ± β ± β
defined by the angle
v
tgβ = 2 (5)
v
1
and leading to the massless G field and the field of massive charged Higgs
boson H , m2 = λ (v2 + v2)/2. The z sector is diagonalized by the
± H 4 1 2 1,2
rotation z ,z ±A0,G′ defined by the angle β and giving again one massless
1 2
field G′ and th→e field of CP-odd Higgs boson A0 with the mass m2 = λ (v2+
A 6 1
v2)/2. Finally, the h ,h sector is diagonalized by the rotation h ,h h,H
2 1 2 1 2 →
defined by the angle α
2m m m
12 11 22
sin2α = , cos2α = − (6)
(m m )2 +4m2 (m m )2 +4m2
q 11 − 22 12 q 11 − 22 12
where
m = 1[4v2(λ +λ )+v2λ ]
11 4 1 1 3 2 5
m = 1[4v2(λ +λ )+v2λ ]
22 4 2 2 3 1 5
m = 1(4λ +λ )v v
12 4 3 5 1 2
giving two massive fields of CP-even Higgs bosons H,h with masses
m2 = m +m (m m )2 +4m2 (7)
H,h 11 22 ±q 11 − 22 12
4
The diagonal mass term of scalar fields and their triple and quartic self-
interaction vertices can be explicitly obtained by the substitution of the fol-
lowing expressions for λ to the potential V(ϕ ,ϕ ) (1):
i 1 2
λ = 1 1 [sαc m2 cαs m2 ]+ c2βλ (8)
1 2v2 c2β sβ α−β h − sβ α−β H 4c2β 5
λ = 1 1 [cαc m2 + sαs m2 ] c2βλ
2 2v2 s2β cβ α−β h cβ α−β H − 4s2β 5
λ = 1 [ s2αm2 + s2αm2 ] 1λ
3 2v2 −s2β h s2β H − 4 5
λ = 2 m2
4 v2 H±
λ = 2 m2
6 v2 A0
where we used the notation v2 = v2 + v2, s = sinα, c = cosα. Diag-
1 2 α α
onalization of the mass term takes place for arbitrary λ , which is a free
5
parameter of the model.
The second representation of the Higgs potential
U(ϕ ,ϕ ) = µ2(ϕ+ϕ ) µ2(ϕ+ϕ ) µ2 (ϕ+ϕ +ϕ+ϕ ) (9)
1 2 − 1 1 1 − 2 2 2 − 12 1 2 2 1
+λ¯ (ϕ+ϕ )2 +λ¯ (ϕ+ϕ )2 +λ¯ (ϕ+ϕ )(ϕ+ϕ )
1 1 1 2 2 2 3 1 1 2 2
+λ¯ (ϕ+ϕ )(ϕ+ϕ )+ λ¯5[(ϕ+ϕ )(ϕ+ϕ )+(ϕ+ϕ )(ϕ+ϕ )]
4 1 2 2 1 2 1 2 1 2 2 1 2 1
originates from the general SUSY action after the integration over Grassman
variables and introduction of the soft SUSY-breaking terms (see [4]). It is
¯
easy to check that the potentials (1) and (9) are equivalent if parameters λ ,
i
µ2, µ2, µ2 and λ are related by the formulae
1 2 12 i
¯ ¯ ¯
λ = λ +λ , λ = λ +λ , λ = 2λ +λ , (10)
1 1 3 2 2 3 3 3 4
λ λ λ λ v v
λ¯ = λ + 5 + 6, λ¯ = 5 6, µ2 = λ 1 2
4 − 4 2 2 5 2 − 2 12 5 2
and
µ2 = λ v2 +λ v2 +λ v2, µ2 = λ v2 +λ v2 +λ v2 (11)
1 1 1 3 1 3 2 2 2 2 3 1 3 2
Unlike the potential (1) where the minimization is obvious, the symbolic
structure of (9) does not demonstrate evidently its minimum. The substi-
tution of (2) to (9) gives linear terms in the component fields z , h (or
1,2 1,2
physical fields h,H,A) and unless some additional conditions to remove the
linear terms are imposed, we are not in the minimum of the potential. So
the equations (11) which set to zero the terms which are linear in compo-
nent fields are the minimization conditions. The diagonalization of U(ϕ ,ϕ )
1 2
takes place for arbitrary µ2 parameter.
12
Inverse transformation (10) has the form
¯ ¯ ¯ ¯ ¯ ¯
λ λ λ λ λ λ λ λ
¯ 3 4 5 5 ¯ 3 4 5 5
λ = λ + , λ = λ + (12)
1 1 2 2
− 2 − 2 − 2 2 − 2 − 2 − 2 2
5
¯ ¯ ¯
λ λ λ λ
3 4 5 5 ¯ ¯ ¯
λ = + + , λ = λ λ +λ , λ = 2λ +λ
3 4 4 5 5 6 5 5
2 2 2 − 2 − − −
so masses of the CP-even scalars and their mixing angle α (6),(7) in the case
of potential U(ϕ ,ϕ ) can be easily obtained using
1 2
µ2
m +m = v2λ¯ +v2λ¯ + 12, m m = v2λ¯ v2λ¯ ctg2βµ2 , (13)
11 22 1 1 2 2 s 11 − 22 1 1 − 2 2 − 12
2β
2m = v v (λ¯ +λ¯ +λ¯ ) µ2 .
12 1 2 3 4 5 − 12
ThediagonalformofU(ϕ ,ϕ )andthephysicalscalarbosonself-interaction
1 2
¯
vertices are obtained by substitution of the following expressions for λ and
i
µ to (9):
i
λ¯ = 1 [(sα)2m2 +(cα)2m2 sβµ2 ] (14)
1 2v2 cβ h cβ H − c3β 12
λ¯ = 1 [(cα)2m2 +(sα)2m2 cβµ2 ] (15)
2 2v2 sβ h sβ H − s3β 12
λ¯ = 1 [2m2 µ212 + s2α(m2 m2)] (16)
3 v2 H± − sβcβ s2β H − h
λ¯ = 1 ( µ212 +m2 2m2 ) (17)
4 v2 sβcβ A − H±
λ¯ = 1 ( µ212 m2) (18)
5 v2 sβcβ − A
µ2 = λ¯ v2 +(λ¯ +λ¯ +λ¯ )v22 µ2 tgβ (19)
1 1 1 3 4 5 2 − 12
µ2 = λ¯ v2 +(λ¯ +λ¯ +λ¯ )v12 µ2 ctgβ (20)
2 2 2 3 4 5 2 − 12
Conditions (14)-(18) ensure the diagonal form of the mass term expressed in
physical fields h, H, A, H and (19)-(20) are the minimization conditions.
±
Two parametrizations for the Higgs boson self-interaction vertices can be
used in THDM. In the first parametrisation [10] µ2 is a free parameter and
12
λ¯ is defined by (18). In the second one λ¯ is a free parameter and µ2 is
5 5 12
equal to s c (v2λ¯ +m2). Complete sets of Feynman rules (unitary gauge)
β β 5 A
forthetriple(µ2 andλ parametrisations) andquartic(µ2 parametrisation)
12 5 12
Higgs boson interactions in the general two-Higgs-doublet model with CP-
conservation are shown in Tables 1-2 1. In the case of MSSM potential at
the scale M (see (29)) λ¯ =0 and it follows from (8),(10),(11) that µ2
SUSY 5 12
is fixed and equal to m2s c .
A β β
Two additional terms of the dimension 4 can be constructed using the
complete set of SU(2) U(1) invariants ϕ+ϕ , ϕ+ϕ , Reϕ+ϕ and Imϕ+ϕ
× 1 1 2 2 1 2 1 2
1These sets were obtained by means of the LanHEP package [11], see
http://theory.sinp.msu.ru/~semenov/lanhep.html Six misprints in sign of the sec-
ondterminthefactor(c3c s3s )thathavetakenplacein[10],arecorrectedinTable3
intheexpressionsforvertαicβes−hhαhhβ,AAAA,hhAA,H+H−H+H−,H+H−AA,H+H−hh.
6
(a detailed discussion of all possible potental forms can be found in [12]).
¯ ¯
These terms are usually added to the U(ϕ ,ϕ ) with the couplings λ and λ
1 2 6 7
U¯(ϕ ,ϕ ) = U(ϕ ,ϕ ) (21)
1 2 1 2
+λ¯ (ϕ+ϕ )[(ϕ+ϕ )+(ϕ+ϕ )]+λ¯ (ϕ+ϕ )[(ϕ+ϕ )+(ϕ+ϕ )]
6 1 1 1 2 2 1 7 2 2 1 2 2 1
¯
The diagonal formofU(ϕ ,ϕ ) at the localminimum takes place at arbitrary
1 2
µ2 , λ¯ , λ¯ and can be achieved by means of the substitution (14)-(20) with
12 6 7
¯ ¯
additional λ , λ terms in the right-hand side:
6 7
1 s c s 1
λ¯ = [( α)2m2 +( α)2m2 βµ2 ]+ (λ¯ tg3β 3λ¯ tgβ) (22)
1 2v2 c h c H − c3 12 4 7 − 6
β β β
1 c s c 1
λ¯ = [( α)2m2 +( α)2m2 βµ2 ]+ (λ¯ ctg3β 3λ¯ ctgβ) (23)
2 2v2 s h s H − s3 12 4 6 − 7
β β β
1 µ2 s λ¯ λ¯
λ¯ = [2m2 12 + 2α(m2 m2)] 6ctgβ 7tgβ (24)
3 v2 H± − s c s H − h − 2 − 2
β β 2β
1 µ2 λ¯ λ¯
λ¯ = ( 12 +m2 2m2 ) 6ctgβ 7tgβ (25)
4 v2 s c A − H± − 2 − 2
β β
1 µ2 λ¯ λ¯
λ¯ = ( 12 m2) 6ctgβ 7tgβ (26)
5 v2 s c − A − 2 − 2
β β
v2 v2s2
µ2 = λ¯ v2 +(λ¯ +λ¯ +λ¯ ) 2 µ2 tgβ + β(3λ¯ ctgβ +λ¯ tgβ)(27)
1 1 1 3 4 5 2 − 12 2 6 7
v2 v2c2
µ2 = λ¯ v2 +(λ¯ +λ¯ +λ¯ ) 1 µ2 ctgβ + β(λ¯ ctgβ +3λ¯ tgβ)(28)
2 2 2 3 4 5 2 − 12 2 6 7
¯ ¯
Our expressions for the redefined λ and λ are the same as given in [13].
4 5
The potentials (1) and (9) can be reduced to the MSSM potential in some
regions of the parameter space which we are going to discuss. The potential
V(ϕ ,ϕ ) (1) has eight parameters: two vev’s v , v and six couplings λ
1 2 1 2 i
(i=1,...6). Eight parameters of the potential U(ϕ ,ϕ ) (9) µ , µ , µ and
1 2 1 2 12
¯
λ (i=1,...5) can be found using (10),(11). From the other side, in the Higgs
i
sector we have eight physical parameters: the mixing angle β and W-boson
mass m , mixing angle α, the parameter µ2 and four masses of scalars m ,
W 12 h
mH, mA, mH±. The mW is fixed experimentally maintaining the constraint
on the v , v , v2 = v2+v2 = 4m2 /e2 sin2θ which follows from the Higgs
1 2 1 2 W · W
kinetic term D ϕDµϕ (g = e/sinθ , θ is the Weinberg angle). So the
µ W W
Higgs sector of THDM with the potentials (1) or (9) is described by a seven-
dimensional parameter space. In the case of superpotential five additional
¯
constraints are imposed, relating all Higgs boson self-couplings λ , (i=1,...5)
i
7
to the gauge coupling constants at the energy scale M [14]:
SUSY
g2 +g2 g2 g2 g2
λ¯SUSY = λ¯SUSY = 1, λ¯SUSY = − 1, λ¯SUSY = , λ¯SUSY = 0.
1 2 8 3 4 4 − 2 5
(29)
It follows that the four Higgs boson masses and the two mixing angles are
defined by two independent parameters. One can choose, for example, the
r ,r parametrization [15] (r = m2 /m2) or the well-known m , tgβ
1 2 1,2 h,H Z A
parametrization. In order to reduce the general two-Higgs-doublet model
vertices to MSSM at the scale M it is convenient to use the α, β
SUSY
parametrization:
s c s
m2 = m2c α+β, m2 = m2c α+β, m2 = m2 2(α+β), µ2 = m2s c .
h Z 2β s H Z 2β c A Zs 12 A β β
α β α β 2(α β)
− − −
(30)
Substitution of these expressions to the vertex factors in Tables 1,2 after
trivial trigonometric transformations reduces them [10] to a simpler MSSM
factors (see [4]). However, (30) are no longer valid at the energy scale m
W
where λ¯SUSY couplings and masses of Higgs bosons are significantly changed
i
by radiative corrections and the effective two-Higgs-doublet potential should
be described in the complete seven dimensional parameter space. Practi-
cal calculations of the radiatively corrected masses and/or couplings can
be conveniently carried out using results of the two approaches, renormal-
ization group (the HMSUSY package [16] or the analytical representation
[17]) and diagrammatic (the FeynHiggsFast package, see [18]). Two different
parametrizations can be used for these approaches.
In the RG approach it seems convenient to use the two-Higgs-doublet
model parameter space m , tgβ, λ¯ ,...λ¯ . In the following we shall take into
A 1 5
¯ ¯
account the λ and λ terms defined by (21), so the parameter space will be
6 7
¯
nine-dimensional. RG evolution of the coupling constants λ from the energy
i
¯ ¯
scale M to the electroweak scale m defines the λ ,...λ in (22)-(26) and
SUSY W 1 5
the parameters λ¯ , λ¯ . At a given m , tgβ, λ¯ , λ¯ the parameter µ2 and
6 7 A 6 7 12
mH± are fixed by the conditions (25) and (26), the parameters µ21 and µ22
are fixed by (27) and (28), α, m and m can be found using the equations
h H
(22)-(24). If we denote the deviation from the coupling λ¯SUSY at the MSSM
i
¯
scale by ∆λ
i
2(λ¯SUSY λ¯ ) = ∆λ¯ , λ¯SUSY λ¯ = ∆λ¯ , λ¯ = ∆λ¯
1,2 − 1,2 1,2 3,4 − 3,4 3,4 − 5,6,7 5,6,7
we find the mixing angle (introducing the notation g2 + g2 = g2m2/m2 ,
1 Z W
g2 g2 = g2(2 m2/m2 ) while using (22)-(26))
− 1 − Z W
s (m2 +m2)+v2((∆λ¯ +∆λ¯ )s +2c2∆λ¯ +2s2∆λ¯ )
2β A Z 3 4 2β β 6 β 7
tg2α= (31)
c (m2 m2)+v2(∆λ¯ c2 ∆λ¯ s2 ∆λ¯ c +(∆λ¯ ∆λ¯ )s )
2β A − Z 1 β − 2 β − 5 2β 6 − 7 2β
8
CP-even Higgs boson masses and the µ2 parameter
12
m2 = c2 m2 +s2 m2 (32)
H α+β Z α β A
−
v2(∆λ¯ c2c2 +∆λ¯ s2s2 +2(∆λ¯ +∆λ¯ )c c s s +∆λ¯ (c2s2 +s2c2))
− 1 α β 2 α β 3 4 α β α β 5 α β α β
¯ ¯
+2s (∆λ c c +∆λ s s )
α+β 6 α β 7 α β
m2 = s2 m2 +c2 m2
h α+β Z α β A
−
v2(∆λ¯ s2c2 +∆λ¯ c2s2 2(∆λ¯ +∆λ¯ )c c s s +∆λ¯ (s2s2 +c2c2))
− 1 α β 2 α β − 3 4 α β α β 5 α β α β
¯ ¯
2c (∆λ s c ∆λ c s )
α+β 6 α β 7 α β
− −
v2
m2 = m2 +m2 (∆λ¯ ∆λ¯ ) (33)
H± W A − 2 5 − 4
v2
µ2 = s c [m2 (2∆λ¯ +∆λ¯ ctgβ +∆λ¯ tgβ)]
12 β β A − 2 5 6 7
with the minimization conditions
1
µ2 = m2c µ2 tgβ (34)
1 2 Z 2β − 12
v2 s3
[∆λ¯ c2 +(∆λ¯ +∆λ¯ +∆λ¯ )s2 +3∆λ¯ s c +∆λ¯ β]
− 2 1 β 3 4 5 β 6 β β 7c
β
1
µ2 = m2c µ2 ctgβ
2 −2 Z 2β − 12
v2 c3
[∆λ¯ s2 +(∆λ¯ +∆λ¯ +∆λ¯ )c2 +∆λ¯ β +3∆λ¯ s c ]
− 2 2 β 3 4 5 β 6s 7 β β
β
These expressions can be straighforwardly used to calculate the radiatively
corrected masses of Higgs bosons and the mixing angle α in the MSSM with
¯ ¯
the help of a solution of the RG equations for λ ,...λ . Apparently, in the
1 7
¯
RG approach Feynman rules in terms of λ couplings are more convenient
i
than rules in terms of Higgs particle masses.
Inthediagrammaticapproachestocalculationoftheradiativelycorrected
masses [18] the corrections to mh, mH, mA and mH± are extracted from the
renormalized Higgs boson self-energies (usually radiative corrections to only
theCP-even Higgsbosonmasses arecalculated). The set of7+2 independent
parameters inherent for the diagrammatic approaches could be m , tgβ, α,
A
µ212, mh, mH, mH±, andλ¯6, λ¯7. At a given mA, tgβ, λ¯6, λ¯7 the µ212 parameter
can be fixed at the value m2s c , and α can be calculated using the renor-
A β β
malized self-energies correction [18] to the relation valid at the M scale
SUSY
m2 +m2 = s /s (m2 m2). Then λ¯ is defined by (25) and λ¯ ,...λ¯ can
A Z − 2α 2β H− h 4 1 3
be found using (22)-(24). In the diagrammatic calculations Feynman rules
in terms of the radiatively corrected Higgs boson masses look more natural.
Substitution of the radiatively corrected Higgs masses to the tree-level Higgs
9
vertex factorsisexpected to give results very close tothose obtained fromthe
loop corrections to Higgs vertex at the SUSY scale (see the discussion in the
last ref. [8]). It has been shown in [19] on the example of hhh and hhhh ver-
tices (and for the case of diagonal third generation squark mass matrix) that
large radiative corrections to the vertex factors calculated diagrammatically
can be absorbed in the radiatively corrected Higgs boson masses.
Other parametrizations in the two-Higgs-doublet modelare of course pos-
sible, but they should be carefully introduced to respect the minimization
and diagonalization conditions (22)-(28). The introduction of scalar particle
masses and mixing angles inconsistent with them violates either diagonal-
izaton of the potential or its SU(2) invariance, even if the minimization
conditions remain valid.
3 CP violation in the two-Higgs-doublet model
CP transformation of the scalar doublet CPϕP+C+ =ζ∗ ϕ+ (the phase
CP
factor ζ =1) changes the sign of imaginary part Im(ϕ+ϕ ) in the λ term
| CP| 1 2 6
of potential (1), so if λ = λ , λ =0 and ξ =0, CP symmetry is broken there
5 6 6
6 6 6
explicitly. In other words, the dimension two terms of (1) appear with the
complex parameter µ2
12
λ λ
5[ϕ+ϕ +ϕ+ϕ v v cosξ]2 + 6[ i(ϕ+ϕ ϕ+ϕ ) v v sinξ]2 (35)
4 1 2 2 1 − 1 2 4 − 1 2 − 2 1 − 1 2
λ λ λ λ
( 5 6)[(ϕ+ϕ )2 +(ϕ+ϕ )2]+( 5 + 6)ϕ+ϕ ϕ+ϕ
⇒ 4 − 4 1 2 2 1 2 2 1 2 2 1
v v v v
1 2(λ cosξ iλ sinξ)ϕ+ϕ 1 2(λ cosξ +iλ sinξ)ϕ+ϕ
− 2 5 − 6 1 2 − 2 5 6 2 1
so we find
v v
µ2 = 1 2(λ cosξ iλ sinξ). (36)
12 2 5 − 6
2
CP is (softly) broken by the µ2 ϕ+ϕ + µ∗ ϕ+ϕ terms. In this special case
12 1 2 12 2 1
when the same phase ξ is involved both in the potential and the vacuum ex-
pectation value of ϕ , diagonalization and minimization of CP transformed
2
potential become less transparent. It is more convenient to analyse the po-
tential form (21) for the general case of complex parameters with arbitrary
phases. In the following we shall consider the hermitian potential which is
the generalization of (21)
U¯(ϕ ,ϕ ) = 1[ µ2(ϕ+ϕ ) µ∗2 (ϕ+ϕ ) µ2(ϕ+ϕ ) µ∗2 (ϕ+ϕ )] (37)
1 2 2 − 1 1 1 − 1 1 1 − 2 2 2 − 2 2 2
10