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Self­couplings of Higgs bosons in the CP­violating scenarios
E.N. Akhmetzyanova, M.V. Dolgopolov, M.N. Dubinin
Dept. of General and Theoretical Physics, Samara State University,
443011 Samara, Akad. Pavlov str. 1., Russia;
D.V. Skobeltsyn Institute of Nuclear Physics, Moscow State University,
119992 Moscow, Vorobyevi Gory, Russia.
Abstract
The trilinear and the quartic couplings of physical Higgs bosons in the MSSM with explicit
CP violation in the two­doublet Higgs sector are calculated within the framework of the CPX scenario.
The magnitude of an e#ective coupling changes several times with the variation of the CP violating
phase arg(µA), being very sensitive to the radiative corrections taken at the one­loop/two­loop level.
At the phase values around #/2 zeroes of selfcouplings take place.
1 Introduction
General two­Higgs­doublet potential with real parameters # 1 ­# 4 and complex­valued parameters # 5 , # 6 ,
# 7 is not a CP invariant object, thus providing an interesting possibility to introduce to a theory a source
of CP violation di#erent from the standard CKM mixing matrix. This possibility has been extensively
analysed in the framework of the minimal supersymmetric model (MSSM) [1]. The e#ective two­doublet
Higgs potential of the MSSM at the energy scale m top has the form of a general two­Higgs­doublet
potential, where the parameters # 1 ­# 7 can be calculated explicitly [2, 3, 4] and expressed through the
parameters of the MSSM in the sector of scalar quarks--Higgs bosons interaction. In this sense the MSSM
Higgs sector as an e#ective field theory at the scale m top can be embedded in a general two­Higgs­doublet
model (THDM) [5], providing a possibility to interpret general THDM features in the language of the
MSSM parameter space.
In the following we are using the formalism described in [3, 5] for the evaluation of observables with the
explicit CP violation in the MSSM two­doublet Higgs sector. First the mass eigenstates of CP conserving
limit Im# 5,6,7 =0 which are h, H (CP­even scalars), A (CP­odd scalar) and H ± (charged scalar) are
defined in a standard way (using the two mixing angles # and #). If the nonzero imaginary parts of # 5,6,7
are generated in the e#ective potential by the scalar fields interaction with the third generation sfermions,
the mass eigenstates of the CP conserving limit (h, H,A) are rotated by the 3â3 mixing matrix a ij to
define the physical Higgs fields (h 1 , h 2 , h 3 ) [5]. In comparison with other approaches [6, 7] we introduce
the #­rotation of the h and H states respecting the CP conserving scenario, if the parameters are taken
real­valued (e.g. for real parameters h and H are the mass eigenstates of the CP conserving case). In
this sense the CP conserving limit is explicitly constructed. Nevertheless the mass splitting of physical
states (h 1 , h 2 , h 3 ) without a definite CP parity does not depend on the diagonalization procedure [3].
The evaluation of # 1-7 parameters [3] is based on the e#ective field theory approach [4]. The two­
Higgs­doublet e#ective potential parameters are calculated using the MSSM potential of the Higgs bosons ­
scalar quarks interaction and include the contributions coming from the F­terms, leading and nonleading
D­terms, the wave­function renormalization terms, and leading two­loop Yukawa QCD­corrections.
In this proceeding we calculate the trilinear and the quartic couplings of physical Higgs bosons in
the CPX scenario [11] of the MSSM. Continious interest to the self­interactions of Higgs bosons both
in the case of CP conservation [8] and the case of CP violation [6, 9] is motivated by the experimental
accessibility of the two and three Higgs bosons production signals [10] providing possibilities to reconstruct
experimentally the e#ective Higgs potential.
2 Self­interaction of physical Higgs bosons
The e#ective trilinear and quartic couplings of physical Higgs bosons h 1 , h 2 and h 3 (i.e. their mass term
m ij h i h j in the two­doublet potential is diagonal m ij = m h i # ij in the local minimum) can be written in
178

a compact form:
L 3H = v
3
#
i#j#k=1
g h i h j h k
1
N ijk
S
h i h j h k + v
3
# i=1
g h i H+H- h i H + H - , (1)
L 4H =
3
#
i#j#k#l=1
g h i h j h k h l
1
N ijkl
S
h i h j h k h l +
3
#
i#j=1
g h i h j H+H-
1
N ij
S
h i h j H + H - (2)
+ 1
4 g H+H-H+H-
(H + H - ) 2 , (3)
where NS are the combinatorial factors and
g h i h j h k
=
3
#
#####=1
{a #i a #j a #k } g ### , g h i H+H-
=
3
#
#=1
a #i g #H+H- , (4)
g h i h j h k h l
=
3
#
#######=1
{a #i a #j a #k a #l } g #### , g h i h j H+H-
=
3
#
###=1
{a #i a #j } g ##H + H - . (5)
Couplings g### , g #H + H - , g#### and g ##H + H - are the intermediate expressions defined in the unphysical
basis. Expressions in curly brackets in (4)--(5) should be symmetrised in the indexes i, j, k, l, and also the
corresponding combinatorial factors, when two or more indexes coincide, should be taken into account.
For example, {a #i a #j a #k } has the representation
{a #i a #j a #k } #
1
NS # a #i a #j a #k + a #i a #k a #j + a #j a #i a #k + a #j a #k a #i + a #k a #i a #j + a #k a #j a #i # ,(6)
where NS = 6 at i = j = k, NS = 1 at (i, j, k) = (3, 2, 1), and NS = 2 in all other cases.
3 Restrictions on the parameter space from the squark sector
It is well­known that the restrictions on the MSSM parameter space appear from the requirement of
scalar quark masses to be positively defined. In the framework of the CPX scenario [11] when the
MSSM parameters MSUSY , µ and A t,b are large enough and respect the constraints |µ| = 2MSUSY ,
|A t | = |A b | = 4MSUSY , the stop and sbottom quark masses were calculated, see Fig. 1, as a two­
dimensional function of the CP violating phase # = arg(µA t,b ) and the supersymmetry scale MSUSY .
The stop mass ” t 1 is positively defined in large enough regions of the parameter space (#, MSUSY ) at large
tg# (left plot in Fig. 1, while the physical sbottom quark favours small values of tg# (see the right plot).
Two plots in Fig. 2 demonstrate the excluded regions of MSUSY , for large tg# a low supersymmetry scale
MSUSY < 500 GeV is not accepted (right plot) while for a small tg# the region of moderate MSUSY
is excluded at large values of the phase #. For the scenario with MSUSY # 300 GeV positively defined
squark masses are possible only in a strongly restricted regions of the (µ, A) parameter space, see Fig. 3
where we calculate the scalar quark masses at # = #.
4 Higgs boson self­couplings in the CPX scenario
The e#ective trilinear and quartic Higgs boson self­couplings can be written down in the two representa­
tions. First one uses # i parameters and second representation expresses the e#ective couplings by means
of scalar masses in the CP conserving limit # =0. For example, the e#ective charged Higgs triple coupling
in the lambda­basis can be written as
g 1H +H- = Re## 5 s # c # c #+# - Re## 6 c # s 2
# c # + Re## 6 s # s 3
# - Re## 6 s # s 2 # c #
+ Re## 7 c # (s # s # c # - c # # c 2
# - 2 s 2
# ##
-2 s # s 2
# c # # 1 + 2 c # s # c 2
# # 2 - c 3
# s # # 3 + c # s 3
# # 3 - c # s # # 4 c #-#
179

0
1
2
3 300
400
500
600
100
200
300
400
0
1
2
3
0
1
2
3 300
400
500
600
300
400
500
0
1
2
3
Figure 1: Stop mass m t 1 (the left figure for tg# =40) and sbottom mass m b1 (the right figure for tg# =5)
in the CPX scenario, |µ| = 2MSUSY , |A t | = |A b | = 4MSUSY vs the phase # = arg(µA t ) = arg(µA b ),
changing from 0 to #, and MSUSY , changing from 300 GeV to 600 GeV.
g 2H +H- = Re## 5 s # c # s #+# + 2 Re## 6 c # s # c 2
# - Re## 6 c # s 3
# - Re## 6 s # s 2
# c #
- Re## 7 c # (c # s # c # + s # # c 2
# - 2 s 2
# ##
+2 c # s 2
# c # # 1 + 2 s # s # c 2
# # 2 + c # c 3
# # 3 + s # s 3
# # 3 - c # s # # 4 s #+#
g 3H +H- = c 2
# Im## 7 - s # c # Im## 5 + s 2
# Im## 6
The representation in the mass basis can be found in [3]. The magnitude of the coupling g H + H - h1 is
shown in Fig. 4.
Two representations of the e#ective one­ or two­loop Higgs boson self­couplings are equivalent and
use either e#ective # i parameters or radiatively corrected masses of the CP conserving limit, calculated
at the one­ or two­ loop level of precision. The mass representation seems to be more convenient for the
analysis of limiting cases (e.g. the 'decoupling limit', when the h is light and has standard­like couplings
to fermions, and H ,A,H ± are heavy and nearly degenerate in mass). Note that at # # #/2 the coupling
g H + H - h1 is zero, and may be negative and positive at di#erent values of the phase around # #/2. The
# parameter participating in Cardano formula for Higgs mass eigenvalue (see e.g. [3]) has a maximum
at this point of zero self­interaction, when m h1 = m h2 in the intense coupling regime. Some numerical
values of the physical couplings in the CPX scenario are listed in Table 1--3 and displayed in Fig. 5. In the
column denoted by 'SUSY' we give the values of the couplings at the scale MSUSY which is the boundary
condition for the e#ective field theory of the Higgs sector at the scale m top under consideration.
The decay width h 3 # h 1 h 2 has the form
# h3#h1h2 = # (m 2
h3 -m 2
h1 -m 2
h2 ) 2
- 4m 2
h1 m 2
h2
(v g h3h1h2 ) 2
16 # m 3
h3
(7)
The decay width h j # h i h i , j > i, can be written as
# h j #h i h i
= # 1 - (4 m 2
h i
/m 2
h j
) (v g h i h i h j
) 2
32 # m h j
(8)
see Fig. 6. At the charged Higgs boson mass mH ± # 200 GeV the decay at the one­loop and at the
two­loop is not possible kinematically since m h2 < 2m h1 . Decay widths of the two heavier neutral Higgs
bosons h 2 , h 3 are very sensitive to the phase #. Qualitatively, CP violation shows up in the mixing
between the CP--odd Higgs state and, mainly, the heavy CP--even Higgs state with decay patterns of the
180

0
1
2
3 300
400
500
600
0
200
400
0
1
2
3
0
1
2
3 300
400
500
600
0
100
200
0
1
2
3
Figure 2: Stop mass m t 1 (the left figure for tg# =5) and sbottom mass m b1 (the right figure for tg# =40)
in the CPX scenario [11], |µ| = 2MSUSY , |A t | = |A b | = 4MSUSY , vs the phase # = arg(µA t ) =
arg(µA b ) (radians) and MSUSY (GeV), changing from 300 GeV to 600 GeV.
-2000
-1000
0
1000
2000 -1000
-500
0
500
1000
0
100
200
300
-2000
-1000
0
1000
2000
-2000
-1000
0
1000
2000 -1000
-500
0
500
1000
100
150
200
250
300
-2000
-1000
0
1000
2000
Figure 3: Stop mass m t 1 (the left figure for tg# =5) and sbottom mass m b1 (the right figure for tg# =40)
vs µ changing from ­2 TeV to +2 TeV and |A t | = |A b | changing in the interval [­1 TeV, 1 TeV]; MSUSY =
300 GeV, the phase is taken at # = arg(µA t ) = arg(µA b ) = #.
two mass eigenstates determined by the degree of the mixing and di#erent behaviour of each state decay
modes with the phase. For example, the decay h 2 # h 1 h 1 (which is the CP­odd Higgs state decay in
the CP conserving limit) is suppressed for vanishing phase and becomes dominant for some nontrivial
values of the phase # in the CP noninvariant MSSM. Furthermore, new decay modes of h 2 and h 3 may
be opened with the scalar­pseudoscalar HA mixing in comparison with the CP conserving limit, when
they are not allowed due to strict conservation of CP parity.
The decay width h i # H + H -
# h i #H + H - = # 1 - (4 m 2
H ± /m 2
h i
) (v g H + H - h i
) 2
16 # m h i
(9)
181

0 1 2 3 4 5 6
-50
0
50
100
Figure 4: Triple Higgs boson interaction vertex v · g H + H - h1 (GeV) vs the phase arg(µA) in the e#ective
one­loop approximation at the parameter values MSUSY = 500 GeV, tg# =5, A t,b =1000 GeV, µ =
2000 GeV. Long dashed line -- mH ± = 300 GeV, solid line -- mH ± = 200 GeV, short dashed line --
mH ± = 190 GeV.
SUSY # = 0 #/6 #/3 #/2 2#/3 5#/6 #
g h1h1h1 --0.34 --0.56 --0.59 --0.65 --0.70 --0.66 --0.55 --0.49
g h1h1h2 --0.17 --0.36 0.17 0.12 --0.07 --0.35 --0.61 --0.71
g h1h1h3 0 0 0.30 0.29 0.28 0.24 0.15 0
g h1h2h2 0.06 --0.22 --0.01 0.20 0.46 0.64 0.70 0.70
g h1h2h3 0 0 --0.04 --0.03 --0.02 0.01 0.02 0
g h1h3h3 0.12 --0.07 --0.18 --0.11 0.01 0.14 0.23 0.27
g h2h2h2 0.15 0.70 --0.68 --0.96 --1.04 --0.92 --0.75 --0.68
g h2h2h3 0 0 --0.16 --0.11 --0.09 --0.07 --0.04 0
g h2h3h3 0.05 0.24 --0.22 --0.31 --0.35 --0.34 --0.31 --0.30
g h3h3h3 0 0 --0.45 --0.28 --0.18 --0.12 --0.06 0
Table 1: The e#ective trilinear couplings g h i h j h k
vs the phase # = arg (µA t,b ) at the one­loop and
the one­loop with leading two­loop corrections to parameters # i included. GF = 1.174 · 10 -5 GeV -2 ,
#EM (mZ ) = 0.007812, #S (mZ ) = 0.1172, tg# = 5, MSUSY = 500 GeV, |A t | = |A b | = A = 1000 GeV,
|µ| = 2000 GeV, mH ± = 300 GeV.
5 Summary
The structure of Higgs boson self­interactions in the theories with explicitly CP violating potentials is
extremely strongly sensitive to radiative corrections and the phase of e#ective parameters. Some detailed
numerical evaluations illustrating this sensitivity were performed in the framework of the CPX scenario.
The calculation of the trilinear Higgs boson self­couplings g h1H + H - and g h1h2h2 with the h 2 # h 1 h 1 decay
width at the one­ and two­loop level demonstrate changes of the couplings in sign at the phase # #/2
and in the absolute value by a factor of 3--5 in comparison with the CP conserving limit, the magnitude
of this factor depending essentially on the value taken for the phase of # 5,6,7 parameters. For example,
at moderate mH ± the decay h 2 # h 1 h 1 is sensitively suppressed at the two­loop level of precision by the
kinematical restriction m h2 < 2m h1 .
Acknowledgments
182

SUSY # = 0 #/6 #/3 #/2 2#/3 5#/6 #
g h1h1h1h1 --0.33 --0.53 --0.56 --0.63 --0.69 --0.62 --0.47 --0.38
g h1h1h1h2 --0.17 --0.34 0.17 0.12 --0.08 --0.41 --0.71 --0.83
g h1h1h1h3 0 0 0.29 0.28 0.28 0.25 0.15 0
g h1h1h2h2 0.05 --0.26 --0.04 0.17 0.47 0.72 0.81 0.82
g h1h1h2h3 0 0 --0.06 --0.06 --0.04 --0.01 0.01 0
g h1h1h3h3 0.11 --0.09 --0.21 --0.13 0.01 0.16 0.28 0.32
g h1h2h2h2 0.17 0.46 --0.58 --0.90 --1.06 --0.99 --0.82 --0.74
g h1h2h2h3 0 0 --0.10 --0.06 --0.06 --0.05 --0.03 0
g h1h2h3h3 0.06 0.17 --0.18 --0.29 --0.35 --0.37 --0.34 --0.33
g h1h3h3h3 0 0 --0.26 --0.13 --0.08 --0.06 --0.03 0
g h2h2h2h2 --0.33 2.69 2.46 2.14 1.56 0.94 0.52 0.38
g h2h2h2h3 0 0 0.06 0.07 0.06 0.04 0.02 0
g h2h2h3h3 --0.11 0.86 0.82 0.71 0.54 0.36 0.24 0.20
g h2h3h3h3 0 0 0.06 0.07 0.06 0.05 0.03 0
g h3h3h3h3 --0.35 2.48 2.47 2.09 1.65 1.25 0.98 0.89
Table 2: The e#ective couplings g h i h j h k h l
vs the phase # = arg (µA t,b ), at the one­loop and the one­loop
with leading two­loop corrections to parameters # i included. GF = 1.174 · 10 -5 GeV -2 , #EM (mZ ) =
0.007812, #S (mZ ) = 0.1172, tg# = 5, MSUSY = 500 GeV, |A t | = |A b | = A = 1000 GeV, |µ| = 2000 GeV,
mH ± = 300 GeV.
SUSY # = 0 #/6 #/3 #/2 2#/3 5#/6 #
g h1H + H - --0.10 --0.21 --0.19 --0.12 --0.03 0.06 0.12 0.14
g h2H + H - 0.05 0.23 --0.22 --0.31 --0.35 --0.35 --0.33 --0.32
g h3H + H - 0 0 --0.15 --0.09 --0.06 --0.03 --0.02 0
g h1h1H + H - --0.10 --0.22 --0.20 --0.13 --0.02 0.09 0.16 0.19
g h1h2H + H - 0.05 0.15 --0.19 --0.29 --0.36 --0.38 --0.36 --0.35
g h1h3H + H - 0 0 --0.09 --0.04 --0.02 --0.01 --0.01 0
g h2h2H + H - --0.11 0.86 0.80 0.69 0.53 0.36 0.24 0.20
g h2h3H + H - 0 0 0.02 0.02 0.02 0.02 0.01 0
g h3h3H + H - --0.12 0.83 0.80 0.69 0.55 0.42 0.33 0.30
g H + H - H + H - --0.24 1.65 1.57 1.36 1.08 0.83 0.65 0.59
Table 3: The e#ective coulings g h i H+H-
, g h i h j H+H- g H+H-H+H-
vs the phase # = arg (µA t,b ), at
the one­loop and the one­loop with leading two­loop corrections to parameters # i included. GF =
1.174 · 10 -5 GeV -2 , #EM (mZ ) = 0.007812, #S (mZ ) = 0.1172, tg# = 5, MSUSY = 500 GeV, |A t | =
|A b | = A = 1000GeV, |µ| = 2000GeV, mH ± = 300GeV.
E. Akhmetzyanova thanks the ''Dynasty'' foundation and ICPPM for partial financial support. The work
of M. Dolgopolov and M. Dubinin was partially supported by RFBR grant 04­02­17448. The work of
M. Dubinin was partially supported by INTAS 03­51­4007, UR 02.03.028 and NS 1685.2003.2.
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1 2 3 4 5 6
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