The Higgs Boson Masses and Mixings of the Complex MSSM in the Feynman-Diagrammatic Approach

1.2 MB·English

by  M. Frank

DCPT/06/160
IPPP/06/80 MPP–2006–158
PSI–PR–06–14 hep-ph/0611326
The Higgs Boson Masses and Mixings of the
Complex MSSM in the Feynman-Diagrammatic Approach
7
M. Frank1 , T. Hahn2 , S. Heinemeyer3 , W. Hollik2 ,
0 ∗ † ‡ §
0 H. Rzehak4 and G. Weiglein5
2 ¶ k

n
a 1Institut fu¨r Theoretische Physik, Universit¨at Karlsruhe,
J
D–76128 Karlsruhe, Germany
9 ∗∗
2 2Max-Planck-Institut fu¨r Physik (Werner-Heisenberg-Institut), F¨ohringer Ring 6,

2 D–80805 Mu¨nchen, Germany
v
3Instituto de Fisica de Cantabria (CSIC-UC), Santander, Spain
6
2 4Paul Scherrer Institut, Wu¨renlingen und Villigen, CH–5232 Villigen PSI, Switzerland
3
1 5IPPP, University of Durham, Durham DH1 3LE, UK
1
6
0 Abstract
/
h New results for the complete one-loop contributions to the masses and mixing effects
p
in the Higgs sector are obtained for the MSSM with complex parameters using the
-
p Feynman-diagrammatic approach. The full dependence on all relevant complex phases
e
is taken into account, and all the imaginary parts appearing in the calculation are
h
: treated in a consistent way. The renormalization is discussed in detail, and a hybrid
v
i on-shell/DRschemeisadopted. Wealsoderivethewavefunctionnormalization factors
X
needed in processes with external Higgs bosonsand discusseffective couplings incorpo-
r
a rating leading higher-order effects. The complete one-loop corrections, supplemented
by the available two-loop corrections in the Feynman-diagrammatic approach for the
MSSM with real parameters and a resummation of the leading (s)bottom corrections
for complex parameters, are implemented into the public Fortran code FeynHiggs2.5.
In our numerical analysis the full results for the Higgs-boson masses and couplings are
compared with various approximations, and -violating effects in the mixing of the
CP
heavy Higgs bosons are analyzed in detail. We find sizable deviations in comparison
with the approximations often made in the literature.
∗email: [email protected]
†email: [email protected]
‡email: [email protected]
§email: [email protected]
¶email: [email protected]
kemail: [email protected]
∗∗former address
1 Introduction
A striking prediction of models of supersymmetry (SUSY) [1] is a Higgs sector with at least
one relatively light Higgs boson. In the Minimal Supersymmetric extension of the Standard
Model (MSSM) two Higgs doublets are required, resulting in five physical Higgs bosons: the
light and heavy -even h and H, the -odd A, and the charged Higgs bosons H . The
±
CP CP
Higgs sector of the MSSM can be expressed at lowest order in terms of M (or M ), M
Z W A
(or MH±) and tanβ v2/v1, the ratio of the two vacuum expectation values. All other
≡
masses and mixing angles can therefore be predicted. Higher-order contributions give large
corrections to the tree-level relations.
The limits obtained from the Higgs search at LEP (the final LEP results can be found in
Refs. [2,3]), place important restrictions on the parameter space of the MSSM. The results
obtained so far at Run II of the Tevatron [4–6] yield interesting constraints in particular in
the region of small M and large tanβ (the dependence on the other MSSM parameters has
A
recently been analyzed in Ref. [7]). The Large Hadron Collider (LHC) has good prospects
for the discovery of at least one Higgs boson over all the MSSM parameter space [8–10] (see
e.g. Refs. [11,12] for recent reviews). At the International Linear Collider (ILC) eventually
high-precision physics in the Higgs sector may become possible [13–15]. The interplay of the
LHC and the ILC in the MSSM Higgs sector is discussed in Refs. [16,17].
For the MSSM with real parameters (rMSSM) the status of higher-order corrections to
the masses and mixing angles in the Higgs sector is quite advanced. The complete one-loop
result within the rMSSM is known [18–21]. The by far dominant one-loop contribution is the
(α ) term due to top and stop loops (α h2/(4π), h being the top-quark Yukawa cou-
O t t ≡ t t
pling). The computation of the two-loop corrections has meanwhile reached a stage where
all the presumably dominant contributions are available [22–36]. In particular, the (α α ),
t s
O
(α2), (α α ), (α α ) and (α2) contributions to the self-energies are known for vanish-
O t O b s O t b O b
ing external momenta. For the (s)bottom corrections, which are mainly relevant for large
values of tanβ, an all-order resummation of the tanβ-enhanced term of (α (α tanβ)n)
b s
O
is performed [37–40]. The remaining theoretical uncertainty on the lightest -even Higgs
CP
boson mass has been estimated to be below 3 GeV [41–43]. The above calculations have
∼
been implemented into public codes. The program FeynHiggs [23,44–46] is based on the
results obtained in the Feynman-diagrammatic (FD) approach [22,23,34,41]. It includes all
the above corrections. The code CPsuperH [47] is based on the renormalization group (RG)
improved effective potential approach [26,35,36]. Most recently a full two-loop effective
potential calculation1 (including even the momentum dependence for the leading pieces) has
been published [49]. However, no computer code is publicly available. Besides the masses
in the Higgs sector, also for the couplings of the rMSSM Higgs bosons to SM bosons and
fermions detailed higher-order corrections are available [37–39,50,51].
In the case of the MSSM with complex parameters (cMSSM) the higher order corrections
have yet been restricted, after the first more general investigations [52], to evaluations in the
1InRef. [48]thesymmetryrelationsaffectinghigher-ordercorrectionsintheMSSMHiggssectorhavebeen
analyzedindetail. Ithasbeen shownforthosetwo-loopcorrectionsthat areimplemented inFeynHiggs2.5
thatthe countertermsarisingfrommultiplicative renormalizationpreserveSUSY,sothatthe existingresult
is valid without the introduction of additional symmetry-restoringcounterterms. It is not yet clear whether
the same is true also for the subleading two-loop corrections obtained in Ref. [49].
1
effective potential (EP) approach [53,54] (at one-loop, neglecting the momentum-dependent
effects) and to the RG improved one-loop EP method [55,56]. The latter ones have been
restricted to the corrections arising from the (s)fermion sector and some leading logarith-
mic corrections from the gaugino sector2. Within the FD approach the one-loop leading m4
t
correctionshavebeenevaluatedinRef. [57]. Effectsofimaginarypartsoftheone-loopcontri-
butions to Higgs boson masses and couplings have been considered in Refs. [58–60]. Further
discussions on the effect of complex phases on Higgs boson masses and decays can be found
in Refs. [61–64]. A detailed comparison between the two available computer codes for the
cMSSM Higgs-boson sector, FeynHiggs and CPsuperH, will be performed in a forthcoming
publication.
In the present paper we present the complete one-loop evaluation of the Higgs-boson
masses and mixings in the cMSSM (see Ref. [65] for preliminary results). The full phase
dependence, the full momentum dependence and the effects of imaginary parts of the Higgs-
boson self-energies are taken consistently into account. Our results are based on the FD
approach using a hybrid renormalization scheme where the masses are renormalized on-shell,
while the DR scheme is applied for tanβ and the field renormalizations. The higher-order
self-energy corrections are utilized to obtain wave function normalization factors for external
Higgsbosonsandtodiscuss effectivecouplingsincorporatingleadinghigher-ordereffects. We
provide numerical examples for the lightest cMSSM Higgs boson, the mass difference of the
heavier neutral Higgses and for the mixing between the three neutral Higgs bosons. We
compare our results with various approximations often made in the literature. All results
are incorporated into the public Fortran code FeynHiggs2.5 [23,44–46].
The rest of the paper is organized as follows. In Sect. 2 we review all relevant sectors
of the cMSSM. Besides the tree-level structure of the Higgs sector, the renormalization
necessary for the one-loop calculations is explained in detail. In Sect. 3 the evaluation
of the one-loop self-energies is presented. The determination of the Higgs-boson masses
from the propagators and of wave function normalization factors and effective couplings is
described. Our numerical analysis is given in Sect. 4. Information about the Fortran code
FeynHiggs2.5 is provided in Sect. 5, more details about installation and use are given in
the Appendix. We conclude with Sect. 6.
2 Calculational basis
2.1 The scalar quark sector in the cMSSM
The mass matrix of two squarks of the same flavor, q˜ and q˜ , is given by
L R
M2 +m2 +M2 cos2β(Iq Q s2) m X
M = L q Z 3 − q w q q∗ , (1)
q˜ m X M2 +m2 +M2 cos2βQ s2
(cid:18) q q q˜R q Z q w(cid:19)
with
X = A µ cotβ,tanβ , (2)
q q ∗
− { }
2The two-loopresults ofRef. [49]caninprinciple alsobe takenoverto the cMSSM. However,no explicit
evaluation or computer code based on these results exists.
2
where cotβ,tanβ applies for up- and down-type squarks, respectively, the star denotes a
{ }
complex conjugation, and tanβ v /v . In Eq. (2) M2, M2 are real soft SUSY-breaking
≡ 2 1 L q˜R
parameters, while the soft SUSY-breaking trilinear coupling A and the higgsino mass pa-
q
rameter µ can be complex. As a consequence, in the scalar quark sector of the cMSSM
N +1 phases are present, one for each A and one for µ, i.e. N +1 new parameters appear.
q q q
As an abbreviation we will use
ϕ arg(X ), ϕ arg(A ) . (3)
Xq ≡ q Aq ≡ q
One can trade ϕ for ϕ as independent parameter.
Aq Xq
The squark mass eigenstates are obtained by the unitary transformation
q˜ q˜
1 = U L (4)
q˜ q˜ q˜
2 R
(cid:18) (cid:19) (cid:18) (cid:19)
with
c s
Uq˜= sq˜ cq˜ , Uq˜U†q˜= 1l , (5)
(cid:18) − ∗q˜ q˜ (cid:19)
The elements of the mixing matrix U can be calculated as
M2 +m2 +M2 cos2β(Iq Q s2) m2
L q Z 3 − q w − q˜2
c = , (6)
q˜ q
m2 m2
q˜1 − q˜2
q
m X
q q∗
s = . (7)
q˜
M2 +M2 cos2β(Iq Q s2)+m2 m2 m2 m2
L Z 3 − q w q − q˜2 q˜1 − q˜2
q q
Here cq˜ cosθq˜ is real, whereas sq˜ e−iϕXq sinθq˜ can be complex with the phase
≡ ≡
ϕ = ϕ = arg X . (8)
sq˜ − Xq q∗
The mass eigenvalues are given by (cid:0) (cid:1)
1
m2 = m2 + M2 +M2 +IqM2 cos2β (9)
q˜1,2 q 2 L q˜R 3 Z
h
[M2 M2 +M2 cos2β(Iq 2Q s2)]2 +4m2 X 2 , (10)
∓ L − q˜R Z 3 − q w q| q|
q i
and are independent of the phase of X .
q
2.2 The chargino / neutralino sector of the cMSSM
The mass eigenstates of the charginos can be determined from the matrix
M √2sinβM
X = 2 W . (11)
√2cosβM µ
W
(cid:18) (cid:19)
In addition to the higgsino mass parameter µ it contains the soft breaking term M , which
2
can also be complex in the cMSSM. The rotation to the chargino mass eigenstates is done by
3
transforming the original wino and higgsino fields with the help of two unitary 2 2 matrices
×
U and V,
χ˜+ W˜ + χ˜ W˜
1 = V , −1 = U − . (12)
(cid:18)χ˜+2(cid:19) (cid:18)H˜2+(cid:19) (cid:18)χ˜−2(cid:19) (cid:18)H˜1−(cid:19)
These rotations lead to the diagonal mass matrix
m ± 0
χ˜1 = U∗XV†. (13)
0 mχ˜±!
2
Fromthisrelation,itbecomesclearthatthecharginomassesm ± andm ± canbedetermined
χ˜ χ˜
1 2
as the (real and positive) singular values of X. The singular value decomposition of X also
yields results for U and V.
A similar procedure is used for the determination of the neutralino masses and mixing
matrix, which can both be calculated from the mass matrix
M 0 M s cosβ M s sinβ
1 Z w Z w
−
0 M M c cosβ M c sinβ
Y =  2 Z w Z w . (14)
M s cosβ M c cosβ 0 µ
Z w Z w
− −
 M s sinβ M c sinβ µ 0 
 Z w Z w − 
 
This symmetric matrix contains the additional complex soft-breaking parameter M . The
1
diagonalization of the matrix is achieved by a transformation starting from the original
bino/wino/higgsino basis,
χ˜0 B˜0 m 0 0 0
1 χ˜01
χ˜02 = NW˜ 0,  0 mχ˜02 0 0  = N YN . (15)
∗ †
χ˜03 H˜10  0 0 mχ˜03 0 
   
 
χ˜04 H˜20  0 0 0 mχ˜04
   
 
The unitary 4 4 matrix N and the physical neutralino masses again result from a numerical
×
singular value decomposition of Y. The symmetry of Y permits the non-trivial condition
of using only one matrix N for its diagonalization, in contrast to the chargino case shown
above.
2.3 The cMSSM Higgs potential
TheHiggspotentialV containstherealsoftbreakingtermsm˜2 andm˜2 (withm2 m˜2+ µ 2,
H 1 2 1 ≡ 1 | |
m2 m˜2 + µ 2), the potentially complex soft breaking parameter m2 , and the U(1) and
2 ≡ 2 | | 12
SU(2) coupling constants g and g :
1 2
VH = m21H1∗iH1i +m22H2∗iH2i −ǫij(m212H1iH2j +m212∗H1∗iH2∗j)
+ 1(g2 +g2)(H H H H )2 + 1g2 H H 2. (16)
8 1 2 1∗i 1i − 2∗i 2i 2 2| 1∗i 2i|
4
The indices i,j = 1,2 refer to the respective Higgs doublet component (summation over
{ } { }
i and j is understood), and ǫ12 = 1. The Higgs doublets are decomposed in the following
way,
H v + 1 (φ iχ )
= 11 = 1 √2 1 − 1 ,
H1 H φ
(cid:18) 12(cid:19) (cid:18) − −1 (cid:19)
H φ+
= 21 = eiξ 2 . (17)
H2 (cid:18)H22(cid:19) (cid:18)v2 + √12(φ2 +iχ2)(cid:19)
Besides the vacuum expectation values v and v , eq. (17) introduces a possible new phase ξ
1 2
between the two Higgs doublets. Using this decomposition, V can be rearranged in powers
H
of the fields,
V = T φ T φ T χ T χ +
H ···− φ1 1 − φ2 2 − χ1 1 − χ2 2
φ
1
φ φ+
+ 21 φ1,φ2,χ1,χ2 Mφφχχχ2+ φ−1,φ−2 Mφ±φ± φ1+ +··· , (18)
1 (cid:18) 2(cid:19)
(cid:0) (cid:1) χ  (cid:0) (cid:1)
2
 
 
where the coefficients of the linear terms are called tadpoles and those of the bilinear terms
are the mass matrices Mφφχχ and Mφ±φ±. The tadpole coefficients read
T = √2(m2v cosξ m2 v + 1(g2+g2)(v2 v2)v ), (19a)
φ1 − 1 1 − ′| 12| 2 4 1 2 1 − 2 1
T = √2(m2v cosξ m2 v 1(g2 +g2)(v2 v2)v ), (19b)
φ2 − 2 2 − ′| 12| 1 − 4 1 2 1 − 2 2
v
T = √2sinξ m2 v = T 2, (19c)
χ1 ′| 12| 2 − χ2v
1
with ξ ξ +arg(m2 ).
′ ≡ 12
With the help of a Peccei-Quinn transformation [66] µ and m2 can be redefined [67]
12
such that the complex phase of m2 vanishes. In the following we will therefore treat m2 as
12 12
a real parameter, which yields
m2 = m2 , ξ = ξ. (20)
| 12| 12 ′
The real, symmetric 4 4-matrix Mφφχχ and the hermitian 2 2-matrix Mφ±φ± contain
× ×
the following elements,
M M
φ φχ
M = , (21a)
φφχχ
M†φχ Mχ !
m2 + 1(g2 +g2)(3v2 v2) cosξm2 1(g2 +g2)v v
M = 1 4 1 2 1 − 2 − 12 − 2 1 2 1 2 , (21b)
φ
−cosξm212 − 12(g12 +g22)v1v2 m22 + 41(g12 +g22)(3v22 −v12) !
0 sinξm2
12
M = , (21c)
φχ
sinξm2 0 !
− 12
5
m2 + 1(g2 +g2)(v2 v2) cosξm2
M = 1 4 1 2 1 − 2 − 12 , (21d)
χ
cosξm2 m2 + 1(g2 +g2)(v2 v2)!
− 12 2 4 1 2 2 − 1
m2 + 1g2(v2 v2)+ 1g2(v2 +v2) eiξm2 1g2v v
Mφ±φ± = 1 4 1 1 − 2 4 2 1 2 − 12 − 2 2 1 2 . (21e)
−e−iξm212 − 21g22v1v2 m22 + 14g12(v22 −v12)+ 41g22(v12 +v22)!
The non-vanishing elements of M lead to -violating mixing terms in the Higgs potential
φχ
CP
between the -even fields φ and φ and the -odd fields χ and χ if ξ = 0. The mass
1 2 1 2
CP CP 6
eigenstates in lowest order follow from unitary transformations on the original fields,
h φ
1
H φ H φ
  = U  2, ± = U ±1 . (22)
A n(0) · χ1 (cid:18)G±(cid:19) c(0) ·(cid:18)φ±2(cid:19)
G χ 
   2
   
The matrices U and U transform the neutral and charged Higgs fields, respectively,
n(0) c(0)
such that the resulting mass matrices
MdhiHagAG = Un(0)MφφχχU†n(0) and MdHia±gG± = Uc(0)Mφ±φ±U†c(0) (23)
are diagonal in the basis of the transformed fields. The new fields correspond to the three
neutral Higgs bosons h, H and A, the charged pair H and the Goldstone bosons G and
±
G .
±
The lowest-order mixing matrices can be determined from the eigenvectors of M and
φφχχ
Mφ±φ±, calculated under the additional condition that the tadpole coefficients (19) must
vanish in order that v and v are indeed stationary points of the Higgs potential. This
1 2
automatically requires ξ = 0, which in turn leads to a vanishing matrix M and a real,
φχ
symmetric matrix Mφ±φ±. Therefore, no -violation occurs in the Higgs potential at the
CP
lowest order, and the corresponding mixing matrices can be parametrized by real mixing
angles as
sinα cosα 0 0
−
cosα sinα 0 0 sinβ cosβ
U =  , U = − c c . (24)
n(0) 0 0 sinβ cosβ c(0) cosβ sinβ
n n c c
− (cid:18) (cid:19)
 0 0 cosβ sinβ 
 n n
 
The mixing angles α, β and β can be determined from the requirement that this transfor-
n c
mation results in diagonal mass matrices for the physical fields. It is necessary, however, to
determine theelements of themassmatrices without inserting theexplicit formofthe mixing
angles and keeping the dependence on the complex phase ξ, since these expressions will be
needed for the renormalization of the Higgs potential and the calculation of the tadpole and
mass counterterms at one-loop order.
2.4 Higgs mass terms and tadpoles
In order to specify our notation and the conventions used in this paper we write out the
Higgs mass terms and tadpole terms in detail. The terms in V , expressed in the mass
H
6
eigenstate basis, which are linear or quadratic in the fields are denoted as follows,
V = const. T h T H T A T G
H h H A G
− · − · − · − ·
m2 m2 m2 m2 h
h hH hA hG
m2 m2 m2 m2 H
+ 1 h,H,A,G  hH H HA HG  + (25)
2 · m2 m2 m2 m2 · A
hA HA A AG
(cid:0) (cid:1) m2 m2 m2 m2  G
 hG HG AG G   
   
m2 m2 H+
+ H ,G H± H−G+ + .
− − · m2 m2 · G+ ···
(cid:18) G−H+ G± (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
Our notation for the Higgs masses in this paper is such that lowest-order mass parameters
are written in lower case, e.g. m2, while loop-corrected masses are written in upper case, e.g.
h
M2.
h
For the gauge-fixing, affecting terms involving Goldstone fields in Eq. (25), we have cho-
sen the ’t Hooft–Feynman gauge. In the renormalization we follow the usual approach where
the gauge-fixing term receives no net contribution from the renormalization transformations.
Accordingly, the counterterms derived below arise only from the Higgs potential and the ki-
netic terms of the Higgs fields but not from the gauge-fixing term.
In order to perform the renormalization procedure in a transparent way, we express the
parameters in V in terms of physical parameters. In total, V contains eight independent
H H
real parameters: v , v , g2, g2, m2, m2, m2 and ξ, which can be replaced by the parameters
1 2 1 2 1 2 12
MZ, MW, e, mH± (or mA), tanβ, Th, TH and TA. Thereby, the coupling constants g1 and
g are replaced by the electromagnetic coupling constant e and the weak mixing angle θ in
2 w
terms of c cosθ = M /M ,s = 1 c2,
w ≡ w W Z w − w
p
e = g c = g s , (26)
1 w 2 w
while the Z boson mass M and tanβ substitute for v and v :
Z 1 2
v
M2 = 1(g2 +g2)(v2 +v2), tanβ = 2. (27)
Z 2 1 2 1 2 v
1
The W boson mass is then given by
1
M2 = g2(v2 +v2). (28)
W 2 2 1 2
The tadpole coefficients in the mass-eigenstate basis follow from the original ones (19) by
transforming the fields according to Eq. (22),
T = √2( m2v cosα m2v sinα +cosξm2 (v sinα +v cosα) (29a)
H − 1 1 − 2 2 12 1 2
1(g2 +g2)(v2 v2)(v cosα v sinα)),
− 4 1 2 1 − 2 1 − 2
T = √2(+m2v sinα m2v cosα +cosξm2 (v cosα v sinα) (29b)
h 1 1 − 2 2 12 1 − 2
+ 1(g2+g2)(v2 v2)(v sinα +v cosα)),
4 1 2 1 − 2 1 2
T = √2sinξm2 (v cosβ +v sinβ ), (29c)
A − 12 1 n 2 n
7
T = tan(β β )T . (29d)
G n A
− −
Using Eqs. (26) – (29) the original parameters can be expressed in terms of e, tanβ, M ,
Z
M , T , T , T and either the mass of the neutral A boson, m , or the mass of the charged
W h H A A
Higgs boson, mH± (it should be noted that Eqs. (29a)–(29d) yield only three independent
relations because of the linear dependence of TG on TA). The masses mA and mH± are
related to the original parameters by
m2 = m2sin2β +m2cos2β +sin2β cosξm2
A 1 n 2 n n 12
cos2β 1(g2 +g2)(v2 v2), (30a)
− n4 1 2 1 − 2
m2 = m2sin2β +m2cos2β +sin2β cosξm2
H± 1 c 2 c c 12
cos2β 1(g2 +g2)(v2 v2)+ 1g2(v cosβ +v sinβ )2 . (30b)
− c4 1 2 1 − 2 2 2 1 c 2 c
Choosing m as the independent parameter yields the following relations,
A
√2 cosβs c M
w w Z
v = (31)
1
e
√2 sinβs c M
w w Z
v = (32)
2
e
g = e/c (33)
1 w
g = e/s (34)
2 w
1
m2 = M2 cos(2β)+m2 sin2β/ cos2(β β )
1 −2 Z A − n
eThcosβn (cid:0) (cid:1)
+ (cosβcosβ sinα+sinβ(cosαcosβ +2sinαsinβ ))
n n n
2c s M
w w Z
h
eT cosβ
H n (cos(α+β)cosβ +2cosαsinβsinβ ) / cos2(β β ) (35)
n n n
− 2c s M −
w w Z
1 i (cid:0) (cid:1)
m2 = M2 cos(2β)+m2 cos2β/ cos2(β β )
2 2 Z A − n
eTH sinβn (cid:0) (cid:1)
(sinαsinβsinβ +cosβ(2cosβ sinα cosαsinβ ))
n n n
− 2c s M −
w w Z
h
eT sinβ
+ h n (2cosαcosβcosβ +sin(α+β)sinβ ) / cos2(β β ) (36)
n n n
2c s M −
w w Z
i
m2 = (f2 +f2) (cid:0) (cid:1) (37)
12 m s
sinξ →pfs/ fm2 +fs2 (38)
cosξ → fm/p fm2 +fs2, (39)
where p
1 eT
f = m2 sin2β + h (cos(β +α)+cos(β α)cos(2β ))
m 2 A 4c s M − n
w w Z
h
eT
+ H (sin(β +α) sin(β α)cos(2β )) / cos2(β β ) , (40)
n n
4c s M − − −
w w Z
i
eT (cid:0) (cid:1)
A
f = . (41)
s
−2s c M cos(β β )
w w Z n
−
8
We now give the bilinear terms of Eq. (25) in this basis, expressed in terms of m or
A
mH±, depending on which parameter leads to more compact expressions. For the charged
Higgs sector this yields, apart from mH± itself,
m2 = m2 tan(β β ) (42a)
H−G+ − H± − c
e
T sin(α β )/cos(β β )
H c c
− 2M s c − −
Z w w
e
T cos(α β )/cos(β β )
h c c
− 2M s c − −
Z w w
e
iT /cos(β β ),
A n
− 2M s c −
Z w w
m2 = (m2 ) , (42b)
G−H+ H−G+ ∗
m2 = m2 tan2(β β )
G± H± − c
e
T cos(α+β 2β )/cos2(β β ) (42c)
H c c
− 2M s c − −
Z w w
e
+ T sin(α+β 2β )/cos2(β β ). (42d)
h c c
2M s c − −
Z w w
The neutral mass matrix is more easily parametrized by m , as can be seen from the
A
2 2 sub-matrix of the A and G bosons:
×
m2 = m2 tan(β β ) (43a)
AG − A − n
e
T sin(α β )/cos(β β )
H n n
− 2M s c − −
Z w w
e
T cos(α β )/cos(β β ),
h n n
− 2M s c − −
Z w w
m2 = m2 tan2(β β ) (43b)
G A − n
e
T cos(α+β 2β )/cos2(β β )
H n n
− 2M s c − −
Z w w
e
+ T sin(α+β 2β )/cos2(β β ). (43c)
h n n
2M s c − −
Z w w
The -violating mixing terms connecting the h-/H- and the A-/G-sector are
CP
e
m2 = T sin(α β )/cos(β β ), (44a)
hA 2M s c A − n − n
Z w w
e
m2 = T cos(α β )/cos(β β ), (44b)
hG 2M s c A − n − n
Z w w
m2 = m2 , (44c)
HA − hG
e
m2 = T sin(α β )/cos(β β ). (44d)
HG 2M s c A − n − n
Z w w
Finally, the terms involving the -even h and H bosons read:
CP
m2 = M2 sin2(α+β) (45a)
h Z
+m2 cos2(α β)/cos2(β β )
A − − n
e
+ T cos(α β)sin2(α β )/cos2(β β )
H n n
2M s c − − −
Z w w
9