Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://theory.sinp.msu.ru/~qfthep04/2004/Proceedings2004/Ginzburg_QFTHEP04_186_203.ps Äàòà èçìåíåíèÿ: Thu Sep 17 21:39:40 2009 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 08:29:05 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 13

Symmetries in Two Higgs Doublet Model, CP violation and heavy Higgs e#ects
Ilya F. Ginzburg a and Maria Krawczyk b
a Sobolev Institute of Mathematics, SB RAS, 630090 Novosibirsk, Russia and
b Institute of Theoretical Physics, Warsaw University, Poland
We apply the invariance of physical picture under a change of Lagrangian -- the reparametrization
invariance in the space of Lagrangians and its particular case -- the rephrasing invariance -- for
analysis of the Two Higgs Doublet Model. We found that some parameters of theory like tan #
are reparametrization dependent and therefore cannot be fundamental. We use the Z2­symmetry
of the Lagrangian, which prevents a #1 # #2 transitions, and the di#erent levels of its violation to
describe a physical content of the model. In general a theory with broken Z2­symmetry describes
a CP violation in the physical Higgs sector. We argue that the 2HDM with a soft breaking of
Z2­symmetry is a natural model in the description of EWSB.
We determine the range of parameters for which CP violation and Flavor Changing Neutral
Current e#ects are naturally small, what corresponds to a small dimensionless mass parameter
# = Rem 2
12 /(2v1v2 ) (in the real vacuum form of Lagrangian -- without phase di#erence among
vacuum expectation values of #1 and #2 ). We discuss how for small # some Higgs bosons can be
heavy, with mass up to about 0.6 TeV, without violation of the unitarity constraints. All physical
Higgs bosons, except one, can be arbitrary heavy if # is large. We analyse main features of this
case, which for # ## corresponds to a decoupling of heavy Higgs bosons.
In the Model II for Yukawa interactions we obtain the set of relations among the couplings to
gauge bosons and to fermions which allows to analyse di#erent physical situations (including CP
violation) in terms of these couplings, instead of parameters of Lagrangian.
I. INTRODUCTION
A spontaneous electroweak symmetry breaking of
SU(2) â U(1) (EWSB) via the Higgs mechanism is
described by the Lagrangian
L = L SM
gf + LH + L Y . (1.1)
Here, L SM
gf describes the SU(2) â U(1) Standard
Model interaction of gauge bosons and fermions, LH
is the Higgs scalar Lagrangian, and L Y describes the
Yukawa interactions of fermions with Higgs scalars.
In the minimal Standard Model (SM) the sin­
gle scalar isodoublet with hypercharge Y = 1 is
implemented. Here LH = (D µ #) + D µ # - V , with
the Higgs potential V = ## 4 /2 -m 2 # 2 /2, its mini­
mum describes the vacuum expectation value v as
### = v/ # 2 = p m 2 /2#. In the SM, the cou­
plings of the physical Higgs boson to gauge bosons
can be expressed via masses as g SM
W = # 2MW /v,
g SM
Z = # 2MZ /v, and Yukawa term has a form:
L Y = X g SM
f Q L #qR + h.c. with g SM
f = # 2m f /v.
Below we consider the simplest extension of the
SM, the Two Higgs Doublet Model (2HDM) ([1],
see e.g. [2] for definitions). We treat a CP violation
in the Higgs sector as a natural feature of theory.
. Sec. II A: The 2HDM contains two doublet
fields, # 1 and # 2 , with identical quantum numbers.
Therefore, its most general form should allow for
global transformations which mix these fields and
change the relative phases. Each this transformation
generates new Lagrangian, with parameters, given
by those of incident Lagrangian and parameters of
transformation (the reparametrization transforma­
tion). Therefore, the physical reality described by
some Lagrangian L (physical model) is also described
by many other Lagrangians. We call this property
by the reparametrization invariance in a space of La­
grangians (with coordinates given by its parameters).
In other words, the Lagrangian describes unambigu­
ously some physical reality as in usual case, however,
given physical reality can be described by a di#er­
ent (reparametrization equivalent) Lagrangians. We
discuss also a particular case of the reparametriza­
tion invariance -- a rephasing invariance.
. Sec. II B: One of the earliest reason for introduc­
ing the 2HDM was to describe the phenomenon of
CP violation [3], an e#ect which can be potentially
large. Glashow and Weinberg [4] realized that the
CP violation and the flavour changing neutral cur­
rents (FCNC) can be naturally suppressed by im­
posing in Lagrangian a Z 2 symmetry, that is the
invariance on the Lagrangian under the interchange
# 1 # # 1 , # 2 # -# 2 or # 1 # -# 1 , # 2 # # 2 . (1.2)
This symmetry forbids the # 1 # # 2 transitions.
The most general Yukawa interaction L Y vio­
lates this Z 2 symmetry leading to the potentially
large flavor--changing neutral­current (FCNC) ef­
fects. The Yukawa interaction can lead (via loop
corrections) to the CP­violation even if CP violation
is absent in the basic Higgs Lagrangian. Imposing
specific constraints on L Y allows to eliminate this
source of CP violation.
Since in Nature both the CP violation and FCNC
e#ects are small, we discuss separately cases of the
exact Z 2 symmetry (then CP is conserved) and of
di#erent levels of its violation, soft and hard. We

consider also a general renormalizability of widely
discussed forms of 2HDM Lagrangians.
. Sec. II C: The EWSB is described by nonzero
vacuum expectation values of fields # 1,2 with gener­
ally di#erent phases. This phase di#erence can be
eliminated by a suitable rephasing transformation,
resulting a real vacuum form of Lagrangian, we use
this form of Lagrangian in the main part of paper.
Here real and imaginary parts of coe#cient at the
mixed quadratic term, describing a soft violation of
Z 2 symmetry, have di#erent properties. The real
part can be treated as a free parameter of theory (in
addition to v.e.v.'s and quartic coe#cients of La­
grangian), while the imaginary part (describing CP
violation) is given by these v.e.v.'s and quartic coef­
ficients of Lagrangian.
. Sec. III: Next we come from # i to the observable
(physical) Higgs particles. The Goldstone modes
and charged H ± are separated easily. In the neu­
tral sector two isotopic doublets give after EWSB
one Goldstone mode, two pure scalars (CP­even)
# 1 , # 2 and one CP­odd pseudoscalar A. These three
states do generally mix leading to the physical states
h i , (i = 1, 2, 3) without definite CP parity. The in­
teraction of matter with these states gives observable
e#ects of CP violation. The analyses of CP violation
e#ects become very transparent if a two--step pro­
cedure of diagonalization is used, with the first step
corresponding to the diagonalization of # 1 , # 2 states.
This leads to the states h and H , discussed usually
in context of the CP--conserving case. It allows us to
consider the general CP nonconserving case (second
step) in terms of states h, H and A, customary in
the case of CP conservation.
. Sect. IV: A most general form of Yukawa in­
teraction violates CP symmetry, gives FCNC, and
breaks Z 2 symmetry in a hard way by loop cor­
rections. A specific form of Yukawa interaction in
which each right--handed fermion isosinglet is cou­
pled to only one scalar field, # 1 or # 2 , guarantees
an absence of the hard violation of Z 2 symmetry if
this violation is absent in proper Higgs Lagrangian
LH . With such Yukawa sector the CP violation is
obliged only by a structure of the Higgs Lagrangian,
and FCNC e#ects can be naturally small. Here we
consider the well known Model II [2] in the explicit
form, which is defined with accuracy up to rephasing
transformation.
In the subsequent discussion it is useful to define
ratios of the couplings of each neutral Higgs boson
h i , i = 1, 2, 3 to the gauge bosons W or Z and to
the quarks or leptons (j = W,Z, u, d, #...), to the
corresponding SM couplings -- relative couplings:
# (i)
j = g (i)
j /g SM
j , (1.3)
As their squared values are in principle measurable,
we treat # (i)
j themselves as measurable quantities.
We present formulae for the relative couplings de­
scribing interactions of observable Higgs bosons with
fermions and gauge bosons, and than derive the set
of relations among these couplings, including ob­
tained by us pattern and linear relations and well
known sum rules. These relations are very useful in
the analyses of di#erent physical situations.
. Sec. V: Parameters of Lagrangian are con­
strained by positivity (vacuum stability) and min­
imum constraints. In most cases the physical phe­
nomena related to the Higgs sector are described
with a good accuracy by the lowest nontrivial order
of the perturbation theory. This should be reliable
for not too large values of parameters of the La­
grangian; we study the relevant unitarity and pertur­
bativity constraints. Most of above constraints were
obtained in literature in the case of soft violation of
Z 2 symmetry. We discuss main new aspects in the
case of hard violation of Z 2 symmetry in sec. V C.
. Sec. VI: In 2HDM there is an attractive pos­
sibility that one of neutral Higgs bosons h 1 is sim­
ilar to that in the SM while others are much heav­
ier. The studies of 2HDM are based often on an as­
sumption of decoupling of these heavy Higgs bosons
from known particles, i.e. e#ects of these additional
Higgs bosons disappear if their masses tend to infin­
ity. This specific property is an natural one in con­
structing of a new theory at extremely large energies
(small distances), since it provides a self­consistent
description of phenomena in our world. However, it
is not necessary for the description of phenomena in
the presence of heavy but not extremely heavy new
particles. In the used real vacuum form of the La­
grangian the mentioned phenomenon is governed by
a singe dimensionless parameter # # Rem 2
12 . For
large # the decoupling limit is realized, i.e. the men­
tioned above additional Higgs bosons can be very
heavy (and almost degenerate in masses) and more­
over such additional Higgs bosons practically decou­
ple from the lighter particles. We analyse properties
of all Higgs bosons and their interactions in this de­
coupling limit. At small #, the masses of such addi­
tional Higgs bosons are bounded from above by the
unitarity constraints. These additional Higgs bosons
can be heavy enough to avoid observation even at
next generation of colliders. Nevertheless, some non­
decoupling e#ects can appear for the lightest Higgs
boson. We present some sets of parameters which
realize this physical picture, respecting the unitar­
ity constraints. We argue that this non--decoupling
option of 2HDM is more natural for weak CP vio­
lation and FCNC (in spirit of t'Hooft's concept of
naturalness [5]).
II. HIGGS LAGRANGIAN
To keep the value # = M 2
W /(M 2
Z cos 2 # W ) equal
to 1 at the tree level, one assumes in 2HDM that
both scalar fields (# 1 and # 2 ) are weak isodoublets
(T = 1/2) with hypercharges Y = ±1 [6]. We use

Y = +1 for both of doublets (the other choice, Y 1 =
1, Y 2 = -1, is used in the MSSM; this case is also
described by equations below with a trivial change
of variables).
The most general renormalizable Higgs La­
grangian for the fields # 1,2 can be written as
LH = T - V , (2.1a)
where T is the kinetic term with D µ being the co­
variant derivative containing the EW gauge fields,
V is the Higgs potential:
T = (D µ # 1 ) + (D µ # 1 ) + (D µ # 2 ) + (D µ # 2 )
+#(D µ # 1 ) + (D µ # 2 ) + # # (D µ # 2 ) + (D µ # 1 ) ,
(2.1b)
V = # 1
2 (# + 1 # 1 ) 2 + # 2
2 (# + 2 # 2 ) 2
+# 3 (# + 1 # 1 )(# + 2 # 2 ) + # 4 (# + 1 # 2 )(# + 2 # 1 )
+ 1
2 h # 5 (# + 1 # 2 ) 2 + h.c. i
+ nh # 6 (# + 1 # 1 ) + # 7 (# + 2 # 2 ) i (# + 1 # 2 ) + h.c. o
(2.1c)
-
1
2 n m 2
11 (# + 1 # 1 ) +m 2
22 (# + 2 # 2 )
+ h m 2
12 (# + 1 # 2 ) + h.c. io .
(2.1d)
The eq. (2.1d) represents a mass term. Note that
# 1-4 , m 2
11 and m 2
22 are real (by hermiticity of the
potential), while the # 5-7 , m 2
12 and # are in gen­
eral complex parameters. Therefore, this potential
contains 14 independent parameters and the entire
Higgs Lagrangian -- 16. We will see that CP viola­
tion in Higgs sector, which is a very natural feature
of 2HDM, can appear only if some of these coe#­
cients are not real.
A. Reparametrization and rephasing
invariance
1. Reparametrization invariance
Our model contains two fields with identical quan­
tum numbers. Therefore, it can be described both
in terms of fields # i , used in Lagrangian (2.1), and
in terms of fields # # i obtained from # i by a global
unitary transformation “
F of the form:
 # # 1
# # 2
 = “
F  # 1
# 2
 , (2.2)
“
F = e -i#0  cos # e i#/2 sin # e i(#-#/2)
- sin # e -i(#-#/2) cos # e -i#/2  .
. In the # = 0 case the transformation (2.2)
does not change the form of kinetic term but induces
the changes of coe#cients of Lagrangian # i # # # i
,
m 2
ij # (m # ) 2
ij which we call reparametrization
transformation with
# # 1 = c 2 # 1 + s 2 # 2 - cs# - 2cs Re ( ”
# 6 + ” # 7 ),
# # 2 = s 2 # 1 + c 2 # 2 - cs# + 2cs Re ( ”
# 6 + ” # 7 ),
# # 3 = # 3 + cs#, # # 4 = # 4 + cs#,
e 2i# # # 5 = # 5 +
e i#
h cs# + 2is 2 Im ” # 5 - 2ics Im ( ” # 6 - ”
# 7 ) i ,
e i# # # 6 = c 2 # 6 - s 2 # 7 + e i#
2 cs(# 1 - # 2 + #),
e i# # # 7 = c 2 # 7 - s 2 # 6 + e i#
2 cs(# 1 - # 2 - #),
(m # ) 2
11 = c 2 m 2
11 + s 2 m 2
22 - 2csµ 2
12 ,
(m # ) 2
22
= s 2 m 2
11 + c 2 m 2
22 + 2csµ 2
12 ,
e i# (m # ) 2
12 = m 2
12 +
e i#
 cs(m 2
11 -m 2
22 ) - 2s 2 µ 2
12  .
(2.3)
where c = cos #, s = sin #, µ 2
12 = Re (m 2
12 e -i# ),
”
# 5 = # 5 e -2i# , ” # 6,7 = # 6,7 e -i# and
# 0 = # 1 + # 2 - 2(# 3 + # 4 + Re ”
# 5 ),
# = cs# 0 + 2(c 2
- s 2 ) Re ( ”
# 6 - ” # 7 ),
# = (c 2
- s 2 )# 0 - 8cs Re ( ” # 6 - ”
# 7 ) + 2i Im ” # 5 .
By construction, the Lagrangians of the form (2.1)
with coe#cients # i , m 2
ij and that with coe#cients
# # i , (m # ) 2
ij
(2.3) describe the same physical reality.
We call this property a reparametrization invari­
ance.
The set of reparametrization transformations (2.3)
represents the 3--parametrical reparametrization
transformation group in the 16­dimensional
space of Lagrangians with coordinates given by
# 1-4 , Re# 5-7 , Im# 5-7 , m 2
ii , Re (m 2
12 ), Im (m 2
12 ),
Re#, Im# with reparametrization parameters #, #,
# ( similar to the gauge parameter of gauge theo­
ries). The transformations “
F (2.2) represent this
very group in the space of fields # i (the scalar ba­
sis). It contains in addition free parameter # 0 , which
describes an overall phase freedom.
A set of physically equivalent Higgs Lagrangians,
obtained from each other by transformations (2.3)
forms the reparametrization equivalent space, which
is 3­dimensional subspace of the entire space of
Lagrangians. The parameters of Lagrangian can
be determined in principle only with accuracy up
to the reparametrization freedom, the di#erent La­
grangians within the reparametrization equivalent
space are physically equivalent.
Among the reparametrization equivalent La­
grangians one can distinguish such, which possess
some defined property in an explicit form. They oc­
cupy some domain in the reparameterization equiv­

alent space. We call such set of Lagrangians ''a fam­
ily'', adding in front words describing property of
model which are explicit for all Lagrangians of this
family (fig. 1) and use for particular Lagrangian of
this family word ''a form''.
xxxxxx
xxxxxx
xxxxxx
xxxxxx
xxxxxx
xxxxxx
xxxxxx
xxxxxx
xxxxxx
xxxxxx
xxxxxx
xxxxxx
xxxxxx
xxxxxx
xxxxxx
xxxxxx
Higgs basis family of Lagrangians (v 1 =v, v 2 =0)
real vacuum family of Lagrangians (v 1 , v 2 real)
xxxxx
xxxxx
xxxxx
Soft Z 2 violation + Model II family of Lagrangians
FIG. 1: Schematic presentation of reparametrization
equivalent space of Lagrangians. Di#erent strips repre­
sent families with di#erent explicit properties. A partic­
ular case where the soft Z2 violating and Model II La­
grangians families coincide is shown.
. In the # #= 0 case the reparametrization
transformation (2.2) induces in addition transforma­
tion of the kinetic term (2.1b)
T = z -1 1
(Dµ# # 1 ) + (D µ # # 1 ) + z -1 2
(Dµ# # 2 ) + (D µ # # 2 )
+# # (Dµ# # 1 ) + (D µ # # 2 ) + # ## (D µ # # 2 ) + (D µ # # 1 ) (2.4)
with
z -1 1 = 1 - 2cs Re (e -i# #), z -1
2 = 1 + 2cs Re (e -i# #),
# # = e -i# (c 2 # - s 2 e 2i# # # ).
So, in order to restore a canonical form of kinetic
term a field renormalization is needed in addition
to the transformations (2.3). This case will be dis­
cussed in more detail elsewhere [7].
Remark on physical parameters.
Some parameters of theory which are treated often
as physical (and in principle measurable) are in fact
reparametrization dependent. The most important
example provides a ratio of vacuum expectation val­
ues of scalar fields, tan # (2.11). For example, under
the transformation (2.2) with # = # (see eq. (2.11b))
and # = 0, angle # changes to # + #.
2. Rephasing invariance
It is useful also to consider the particular case of
the transformations (2.2) with # = 0. It can be also
treated as a global transformation of fields with the
independent phase rotations (rephasing transforma­
tion of the fields):
# i # e -i# i # i , (i = 1, 2),
# 1 = # 0 -
#
2 , # 2 = # 0 + #
2 , # = # 2 - # 1 .
(2.5)
This transformation leads to a change of phase of
some coe#cients of Lagrangian (rephasing transfor­
mation of the parameters):
# 1-4 # # 1-4 , m 2
11 # m 2
11 , m 2
22 # m 2
22 ,
# 5 # # 5 e -2i# , # 6,7 # # 6,7 e -i# ,
m 2
12 # m 2
12 e -i# , # # # e -i# .
(2.6)
By construction, the Lagrangians of form (2.1) with
coe#cients # i , m 2
ij and with coe#cients given by
eq. (2.6) describe the same physical reality. We call
this property the rephasing invariance, which is
similar to the definition given in [12].
The transformations (2.6) represent the 1--
parametrical rephasing transformation group
with parameter #, called by us the rephasing param­
eter. By construction, this group is a subgroup of
the reparametrization group. The transformations
(2.5) represent the rephasing transformation group
in the space of fields # i .
The 1­dimensional rephasing equivalent space,
being a subspace of the entire (3­dimensional)
reparametrization equivalent space of Lagrangians,
is given by the sets of parameters of Lagrangians
at di#erent #. One can say that the entire
reparametrization equivalent space is sliced to the
rephasing equivalent subspaces (strips in fig. 1).
Two remarks.
. The concept of rephasing invariance is easily ex­
tended to the description of a whole system of scalars
and fermions (4.2). The reparametrization trans­
formations for scalar fields (2.2) evidently induce
changes into the set of Yukawa couplings. This may
hide some properties of Yukawa Lagrangian, which
are explicit in a definite scalar basis (e.g. Model I
or Model II, see sec. IV). The Kobyashi -- Maskawa
matrix represent the reparameterization transforma­
tion from the QCD to electroweak basis for fermions.
. We will see later that CP symmetry is conserved
in the Higgs sector if all parameters of Lagrangian
(2.1) are real. Obviously, the CP--violation does not
appear if the Lagrangian with complex parameters
can be transformed to a form with all real parame­
ters by means of some reparametrization (2.2).

B. Lagrangian and Z2 symmetry
The violation of the Z 2 symmetry (1.2) in the La­
grangian allows for the # 1 # # 2 transitions. The
general Lagrangian LH (2.1) violates Z 2 symmetry
by terms of the operator dimension 2 (with m 2
12 ),
what is called a soft violation of the Z 2 symmetry,
and of the operator dimension 4 (with # 6,7 and #),
called a hard violation of the Z 2 symmetry.
a. An exact Z 2 symmetry. This case is de­
scribed by the Lagrangian LH (2.1) with # 6 = # 7 =
# = m 2
12 = 0 and only one parameter # 5 can be
complex. The rephasing transformation (2.6) with
a suitable phase # allows to get another (rephasing
equivalent) form of Lagrangian with a real # 5 .
b. A soft violation of Z 2 symmetry. In the case
of soft violation of Z 2 symmetry one adds to the Z 2
symmetric Lagrangian the term m 2
12 (# + 1 # 2 ) + h.c.,
with a generally complex m 2
12 (and # 5 ) parame­
ter. This type of violation respects the Z 2 sym­
metry at small distances (much smaller than 1/M)
in all orders of perturbative series, i.e. the ampli­
tudes for # 1 # # 2 transitions disappear at virtuality
k 2
# M 2
# #. That is the reason for the name --
a ''soft'' violation. The di#erent rephasing transfor­
mations (2.6) applied to the Lagrangian with a softly
broken Z 2 symmetry can not change its character;
they generate a whole soft Z 2 violating Lagrangian
family (the crossed ''vertical'' strip in fig. 1).
c. A hard violation of Z 2 symmetry. In the
general case the terms of the operator dimension
4, with generally complex parameters # 6 , # 7 and
#, are added to the Lagrangian with a softly
broken Z 2 symmetry. These new terms break Z 2
symmetry at both large and small distances -- this
is called a hard violation of Z 2 symmetry. We
discuss here the case of the true hard violation
of Z 2 symmetry, which cannot be transformed
to the case of Z 2 conservation, nor its soft viola­
tion, by any reparametrization transformation (2.3).
C. The case of hidden soft Z2 violation
Let us assume that there exists some # 1 , # 2 ba­
sis, in which our physical model is described by the
Lagrangian with exact or softly violated Z 2 symme­
try, L s . The general reparametrzation transforma­
tion (2.3) converts this Lagrangian to a form L hs
with # 6 , # 7 #= 0 and # = 0, which looks like the
Lagrangian with hard Z 2 violation. We call L hs ­ a
Lagrangian with a hidden soft Z 2 violation.
To simplify discussion of such a case we first apply
to L s the rephasing transformation (2.6), eliminat­
ing the phase of # 5 and obtain the Lagrangian L R
s
with real # 5 (in this case m 2
12 can be complex leav­
ing open an opportunity for CP violation). Then
we apply to L R
s a general reparameterization trans­
formation (2.3) and obtain Lagrangian in the form
(2.1) with generally complex # 5 and nonzero # 6,7
(but still # = 0). In this case the equations (2.3)
simplify, we get new quartic couplings in a form
# # 1 = c 2 # 1 + s 2 # 2 - cs#, # # 3 = # 3 + cs#,
# # 2 = s 2 # 1 + c 2 # 2 - cs#, # # 4 = # 4 + cs#,
# # 5 = e -2i# # 5 + e 2i# [cs# + 2is 2 # 5 sin 2# ],
# # 6 = e i(#-#) [cs(# 1 - # 2 ) +A] /2,
# # 7 = e i(#-#) [cs(# 1 - # 2 ) -A] /2,
where A = (c 2
- s 2 )# + 2ics# 5 sin 2#,
# = cs[# 1 + # 2 - 2(# 3 + # 4 + # 5 cos 2# )].
(2.7)
The eq­s (2.7) allow to find parameters of the men­
tioned Lagrangian L R
s . To this goal it is useful to
consider following consequence of combinations of
eq­s (2.7):
# # 6 + # # 7
# # #
6 + # # #
7
= e 2i(#-#) ,
# # 6 + # # 7
# # 1 - # # 2
= e i(#-#) tan 2#
2 , (2.8)
e -i# # 5 = # # 5 - e 2i(#-#) [cs# + +2is 2 sin 2## 5 ].
e -i(#-#) (# # 6 -# # 7 )=(c 2
- s 2 )#+2ics# 5 sin 2#.
(1) The value of # - # is determined from the first
equation. (2) After that one can determine angle
# via second equation. (3) Next one can determine
quantity # and 2cs# 5 sin 2# as the real and imagi­
nary parts of the form in the fourth line. (4) After
that one can determine the angle # and the parame­
ter # 5 via the phase and the module of the quantity
in third line. (5) Now all remaining quantities # 1-4
can be determined easily from the first four equa­
tions (2.7).
Equations in the third and fourth lines repre­
sent two di#erent ways of obtaining the parame­
ter # 5 . Besides, quantity # can be obtained both
via these equations and from basic definition # =
# 1 + # 2 - 2[# 3 + # 4 + # 5 cos 2# ]. The existence of
these two ways can be considered as two constraints
for the Lagrangian. It shows explicitly that in this
case the quartic sector is described by only 8 inde­
pendent parameters (# 1-5 and #, #, #) instead of 10
independent parameters of the general Lagrangian
(2.1) (# 1-4 , Re# 5-7 , Im# 5-7 ).
D. Some features of true hard Z2 violation
Let us discuss briefly what should be done in
the case of hard violation of Z 2 symmetry with the
mixed kinetic terms, i.e. # #= 0 in Eq. (2.1b). These
kinetic terms can be removed by the nonunitary

transformation, e.g.
(# #
1 , # #
2 )# # # # # 1 + # ## 2
2 p |#|(1+|#|) ±
# # # # 1 - # ## 2
2 p |#|(1-|#|)
! .
(2.9)
However, in presence of the # 6 and # 7 terms,
the renormalization of quadratically divergent, non­
diagonal two­point functions leads anyway to the
mixed kinetic terms (e.g. from # # 6 # 1,3-5 and # # 7 # 2-5
loops). It means that # becomes nonzero at the
higher orders of perturbation theory, and vice versa
a mixed kinetic term generates counter­terms with
# 6,7 . Therefore all of these terms should be in­
cluded in Lagrangian (2.1a) on the same footing, i.e.
the treatment of the hard violation of Z 2 symme­
try without # terms is inconsistent (see also [8, 9]).
(The phenomenon is analogous to a need of a quar­
tic coupling of the form ## 4 in the renormalization
of the •
## 5 ## theory [10].) Note that the parameter
# is generally running like parameters #'s. There­
fore, the Lagrangian remains o#--diagonal in fields
# i even at very small distances, above the EWSB
transition. Such theory seems to be unnatural.
To find the signature of this case in the arbitrary
form of Lagrangian, it is useful to consider polar­
ization operator matrix P =  # 11 # 12
# 21 # 22
 . In the
general case the ratio # 12 /(# 11 - # 22 ) is running
quantity at large Higgs boson virtuality k 2 in con­
trast with the case of hidden Z 2 symmetry where
this ratio is not running.
Indeed, let us consider the Lagrangian with soft
violation of Z 2 symmetry, L s , like in sect II C. The
one--loop polarization operator for two fields has a
form P =  # s
1 0
0 # s
2
 k 2 + finite terms, for k 2
# #.
The terms # s
1 , # s
2 describe renormalization of fields
# 1 and # 2 , respectively. There is no mixed kinetic
term, and the # 1 # # 2 transitions at small distances
are absent.
Under reparametrization transformation (2.2),
the L s is converted to the Lagrangian L hs , with
nonzero # 6 and # 7 terms (2.7), still with # = 0.
This Lagrangian gives the nonzero mixed polariza­
tion operator:
P # =  # 11 # 12
# # 12 # 22
 k 2 + finite terms,
# 11 = # s
1 cos 2 # +# s
2 sin 2 #,
# 12 = (# s
1 -# s
2 )e -i# sin # cos #,
# 22 = # s
2 cos 2 # +# s
1 sin 2 #.
Naively, this form of polarization operator suggests
that one should introduce in Lagrangian the mixed
kinetic term # (D µ # 1 ) + D µ # 2 + h.c. However, in­
verse reparametrization transformation restores the
incident form of LH with soft Z 2 symmetry viola­
tion, i.e. without kinetic terms, i.e. without # 1 #
# 2 transitions at small distances. Therefore, the
mentioned relations among parameters of new quar­
tic terms prevent an appearance of the mixed kinetic
term in Higgs Lagrangian in any reparametrization
equivalent form of Lagrangian. The renormaliza­
tion group analysis ensures that for the Lagrangian
L hs the ratio # 12 /(# 11 - # 22 ) at large k 2 is renor­
malization invariant quantity, i.e. it is independent
on normalization scale #. This is in contrast to
the general case with the true hard violation of Z 2
symmetry, where # 1 # # 2 transitions at di#erent
large k 2 cannot be ruled out simultaneously by any
reparameterization transformation (2.3) and the ra­
tio # 12 /(# 11 - # 22 ) at large k 2 is renormalization
invariant the ratio # 12 /(# 11 - # 22 ) at large k 2 is
renormalization dependent quantity.
Besides, it is instructive to consider in more de­
tail the case of soft violation of Z 2 symmetry. The
EWSB procedure (sec. II E) transforms the La­
grangian expressed in terms of fields # i to that writ­
ten in terms of Higgs fields h 1-3 and H ± . In this
form 18 quartic couplings appear but there are some
relations among them, since all of them are obtained
from the initial Lagrangian L s with 6 parameters
(# 1-4 , Re# 5 , Im# 5 ) and the orthogonal transfor­
mation from the (# 1 , # 2 ) basis to (H ± , h 1 , h 2 , h 3 )
basis with the additional 3 parameters. In this La­
grangian a mixed polarization operator may also ap­
pear, however no mixed kinetic term. This is due the
mentioned relations among parameters of new quar­
tic terms which prevent appearance of the mixed
kinetic term in Higgs Lagrangian [11]. The detailed
discussion of these problems will be done elsewhere.
Other aspects of the hard violation of Z 2 symme­
try are related to the description of Yukawa sector.
This will be discussed in sec. IV.
The diagonalization (2.9) is rather special and
it may change even the definitions of #'s, what
would destroy relatively simple relations between the
masses of the Higgs bosons (discussed below).
Although in this paper we present relations for a
case of hard violation of Z 2 symmetry at # = 0 one
should keep in mind that loop corrections can change
results significantly. Such treatment of the case with
hard violation of Z 2 symmetry is as incomplete as
in most of the papers considering this ''most general
2HDM potential''. A full treatment of this problem
goes beyond the scope of the present paper.
E. Vacuum. Used form of the potential
The minimum of the potential defines the vacuum
expectation values (v.e.v) of the fields # i :
#V
## 1
    # 1 =## 1 #,
# 2 =## 2 #
= 0, #V
## 2
    # 1 =## 1 #,
# 2 =## 2 #
= 0. (2.10)
In order to describe the U(1) symmetry of electro­
magnetism and using the overall phase freedom of

the Lagrangian to choose one vacuum expectation
value real [12, 13] we take:
## 1 #=
1
# 2  0
v 1
 and ## 2 #=
1
# 2  0
v 2 e i#
 , (2.11a)
with a relative phase #. These v i (and therefore pa­
rameters of whole Lagrangian) obey SM constraint:
v 2
1 + v 2
2 = v 2 , with v = ( # 2GF ) -1/2 = 246 GeV.
The another parameterization of these v.e.v.'s (via
parameters v and #) is also used:
v 1 =v cos #, v 2 =v sin #, # # (0, #/2) . (2.11b)
The rephasing transformation (2.5) shifts phase
di#erence # as follows
# # # - # . (2.12)
Therefore, the phase di#erence # between the v.e.v.'s
has no physical sense (see e.g. [12]).
Let us take some Lagrangian describing our model
and calculate v.e.v.'s (2.11). Than, by making
rephasing transformation (2.6) with # = #, we get
the real vacuum form of Lagrangian in which the rel­
ative phase of v.e.v.'s equals to zero. In accordance
with eq. (2.6) we get
# 1-4,rv = # 1-4 , # 5,rv = # 5 e -2i# ,
# 6,rv = # 6 e -i# , # 7,rv = # 7 e -i# ,
# rv = #e -i# , m 2
12,rv = m 2
12 e -i# .
(2.13)
where we denote the particular values of parame­
ters of such Lagrangian (potential) by subscript rv.
The following combinations of parameters and new
quantities are useful:
# 345 = # 3,rv + # 4,rv + Re# 5,rv ,
# 67 = 1
2  v 1
v 2
# 6,rv + v 2
v 1
# 7,rv  ,
” # 67 = 1
2  v 1
v 2
# 6,rv - v 2
v 1
# 7,rv  ,
m 2
12,rv = 2v 1 v 2 (# + i#).
(2.14)
The minimum condition (2.10) does not constrain
real part of m 2
12,rv , while this condition expresses
the imaginary part of m 12,rv via Im (# 5-7,rv ):
# = 1
2 Im {# 5,rv
| {z }
soft
+ 2# 67
|{z} hard
} . (2.15)
Here (and in the subsequent equation) the first
and second underbraced term are added to each
other in the case of soft and hard violation of Z 2
symmetry, respectively (the latter term also appear
in the case of hidden soft Z 2 violation). In par­
ticular, in the Z 2 symmetric case m 2
12,rv = 0 and
consequently Im# 5,rv = 0.
Beginning from here all expressions are presented
for the real vacuum form of potential, without ex­
plicit subscript rv. We will specially comment when
other forms of Lagrangian will be discussed.
It is useful for subsequent calculations to change
description of potential, containing three indepen­
dent parameters m 2
ij , to the description with three
other parameters related to the minimum: v 1 , v 2
and #. The eq­s (2.10), (2.10) allow to obtain
m 2
11 =# 1 v 2
1 +# 345 v 2
2
| {z }
Z2 sym
-2#v 2
2
| {z }
soft
+ v 2
v 1
Re 3v 2
1 # 6 +v 2
2 # 7

| {z }
hard
,
m 2
22 =# 2 v 2
2 +# 345 v 2
1
| {z }
Z2 sym
-2#v 2
1
| {z }
soft
+ v 1
v 2
Re v 2
1 # 6 +3v 2
2 # 7

| {z }
hard
.
(2.16)
Using these relations we obtain other form of real
vacuum potential, used in this paper:
V = # 1
2  (# + 1 # 1 ) - v 2
1
2  2
+ # 2
2  (# + 2 # 2 ) - v 2
2
2  2
+# 3 (# + 1 # 1 )(# + 2 # 2 ) + # 4 (# + 1 # 2 )(# + 2 # 1 )
+ 1
2 h # 5 (# + 1 # 2 ) 2 + h.c. i
+ nh # 6 (# + 1 # 1 ) + # 7 (# + 2 # 2 ) i (# + 1 # 2 ) + h.c. o
-
1
2 # 345 + 2 Re# 67
[v 2
2 (# + 1 # 1 ) + v 2
1 (# + 2 # 2 )]
- Re [# 6 (# + 1 # 1 ) + # 7 (# + 2 # 2 )]v 1 v 2
+#(v 2 # 1 - v 1 # 2 ) + (v 2 # 1 - v 1 # 2 )
+2# Im (# + 1 # 2 )v 1 v 2 .
(2.17)
In this form the quartic terms are as those in
the initial potential (2.1) but with particular val­
ues # i equal to # i,rv . The mass term is determined
via v.e.v.'s v 1 , v 2 and the couplings # i plus a sin­
gle free dimensionless parameter #. The quantity
# # Im m 2
12 is constrained in this form by eq. (2.15).
In the above equation the soft Z 2 violating contri­
bution is given by a sum of last two terms of (2.17),
so that the variation of each of them don't influence
v.e.v.'s. This decomposition is less transparent in
Lagrangians with # #= 0.
Remarks.
. The Higgs potential has also a minimum, violat­
ing a charge conservation (''charged vacuum''). At
# > 0 the vacuum energy of this charged vacuum
is higher than that for the neutral vacuum (2.10),
see [13], [14]. That is why we discuss only minimum
(2.10).
. The set of real vacuum Lagrangians (real vac­
uum Lagrangian family) forms a subspace in the en­
tire reparametrization equivalent space, which is pic­
tured in fig. 1 by black horizontal line. In di#erent
points of this subspace the tan # values are di#erent.

III. PHYSICAL HIGGS SECTOR
The fields # 1,2 change under the reparametriza­
tion transformation. Now we introduce the,
in principle, observable Higgs fields and their
couplings. These fields and couplings are ev­
idently reparametrization independent. (The
reparametrization dependent are parameters de­
scribing the transformation to this physical basis,
see below.)
A standard decomposition of the fields # i in terms
of physical fields is made via
# 1 = 0 @
# +
1
v 1 + # 1 + i# 1
# 2
1 A , # 2 = 0 @
# +
2
v 2 + # 2 + i# 2
# 2
1 A .
(3.1)
At # = 0 such decomposition leads to a diago­
nal form of kinetic terms for new fields # +
i , # i , # i ,
while the corresponding mass matrix is o#­diagonal.
The mass squared matrix can be transformed to the
block diagonal form by a separation of the mass­
less Goldstone boson fields, G 0 = cos # # 1 +sin # # 2 ,
G ± = cos # # ± 1 + sin # # ± 2 , and the charged Higgs
boson fields H ± ,
H ± = - sin # # ± 1 + cos # # ± 2 , (3.2)
with the mass squared equal to
M 2
H ± =  # -
1
2 (# 4 + Re# 5 + 2 Re# 67 )  v 2 . (3.3)
. The reparametrization transformation (2.2)
with # = 0, # = -# gives v 1 = v, v 2 = 0. The
set of the obtained Lagrangians forms a Higgs basis
Lagrangian family, it is pictured as the grey vertical
strip in fig. 1 (see [12, 15, 16]).
A. Neutral Higgs sector. General introduction
By definition # i are standard C-- and P -- even
(scalar) fields. The field
A = - sin # # 1 + cos # # 2 , (3.4)
is C--odd (in the interactions with fermions it be­
haves as P -- odd particle, i.e. a pseudoscalar). In
other words, # i and A are fields with opposite CP
parities (see e.g. [2] for details).
The decomposition (3.1) results in the (symmet­
ric) mass--squared matrix M in the # 1 , # 2 , A basis
M = 0 @
M 11 M 12 M 13
M 12 M 22 M 23
M 13 M 23 M 33
1 A , (3.5a)
with
M 11 = h c 2
# # 1 + s 2
# # + s 2
# Re (# 67 + 2 ”
# 67 ) i v 2 ,
M 22 = h s 2
# # 2 + c 2
# # + c 2
# Re (# 67 - 2 ” # 67 ) i v 2 ,
M 33 = [# - Re (# 5 - # 67 )] v 2 , (3.5b)
M 12 = - [# - # 345 - 3 Re# 67 ] c # s # v 2 ,
M 13 = - h # + Im ” # 67 i s # v 2 ,
M 23 = - h # - Im ”
# 67 i c # v 2 ,
where we use abbreviations c # = cos #, s # = sin #.
As we discuss below M 33 is equal to CP--odd Higgs
boson mass in the CP conserving case, namely
M 2
A = M 33 = [# - Re (# 5 - # 67 )] v 2 . (3.5c)
The masses squared M 2
i
of the physical neutral
states h i are eigenvalues of the matrix M. These
states are obtained from fields # 1 , # 2 , A by a unitary
transformation R which diagonalizes the matrix M:
0 @
h 1
h 2
h 3
1 A = R
0 @
# 1
# 2
A
1 A ,
with RMR T = diag(M 2
1 , M 2
2 , M 2
3 ) .
(3.6)
The diagonalizing matrix R can be written as a
product of three rotation matrices described by three
Euler angles # i # (0, #) (we define c i = cos # i ,
s i = sin # i ):
R = R 3 R 2 R 1 , R 1 = 0 @
c 1 s 1 0
-s 1 c 1 0
0 0 1
1 A ,
R 2 = 0 @
c 2 0 s 2
0 1 0
-s 2 0 c 2
1 A , R 3 = 0 @
1 0 0
0 c 3 s 3
0 -s 3 c 3
1 A ,
(3.7a)
R = 0 @
R 11 R 12 R 13
R 21 R 22 R 23
R 31 R 32 R 33
1 A #
0 @
c 1 c 2 c 2 s 1 s 2
-c 1 s 2 s 3 -c 3 s 1 c 1 c 3 -s 1 s 2 s 3 c 2 s 3
-c 1 c 3 s 2 +s 1 s 3 -c 1 s 3 -c 3 s 1 s 2 c 2 c 3
1 A .
(3.7b)
We adopt the convention for masses that M 2 # M 1 ,
but shall not require any other ordering.
In general, the obtained Higgs eigenstates h i (3.6)
have no definite CP parity since they are mixtures
of fields # i and A having opposite CP parities. This
provides a CP nonconservation within the Higgs sec­
tor. The interaction of these bosons with matter
explicitly violates the CP--symmetry. Such mixing
(and violation of CP) is absent if M 13 = M 23 = 0.
B. Diagonalization of the scalar CP­even
sector
It is useful to start with the diagonalization of
scalar #12# sector of matrix M which is given by

rotation matrix R 1 . It results in the neutral, CP­
even Higgs fields which we denote as h and 1 (-H),
while the CP--odd field A remains unmixed, so that
0 @
h
-H A
1 A = R 1
0 @
# 1
# 2
A
1 A with
R 1 MR T
1 = M 1 #
0 @
M 2
h
0 M #
13
0 M 2
H M #
23
M #
13 M #
23 M 2
A
1 A .
(3.8)
Let us stress that in the general CP nonconserving
case the states h, H and A have no direct physi­
cal sense, they are only subsidiary concepts useful in
calculations and discussion. In the case of CP con­
servation (which is realized for M 13 = M 23 = 0) the
fields h, H and A represent physical Higgs bosons,
h 1 = h, h 2 = -H , h 3 = A. This is why we use
instead of # 1 the mixing angle # # (-#/2, #/2),
# = # 1 - #/2 , (3.9)
which is customary for the CP--conserving case.
With this notation we have
H = cos # # 1 + sin # # 2 ,
h = - sin # # 1 + cos # # 2 .
(3.10a)
The diagonalization of the respective #12# corner of
mass­squared matrix M (3.5) results in
M 2
h,H = 1
2 (M 11 +M 22 #N ) ,
N = p (M 11 -M 22 ) 2 + 4M 2
12 .
(3.10b)
C. Complete diagonalization
The above diagonalization keeps in general two
o#­diagonal elements in matrix M 1 (3.8):
M #
13 = c 1 M 13 + s 1 M 23 =
-[# cos(# + #) - Im ” # 67 cos(# - #)]v 2 ,
M #
23 = -s 1 M 13 + c 1 M 23 =
[# sin(# + #) - Im ”
# 67 sin(# - #)]v 2 .
(3.11)
If at least one of these o# diagonal terms di#ers from
zero, the additional diagonalization is necessary, and
the mass eigenstates, being admixtures of CP even
and CP odd states, violate CP symmetry. Therefore
the complexity of some parameters of potential in its
real vacuum form is necessary and su#cient condi­
tion for CP violation in the Higgs sector. For the
1 Sign minus is needed in order to match a standard conven­
tion used for CP­conserving case, see e.g. [2].
other form of Lagrangian (i.e. not for the real vac­
uum form) this necessary and su#cient condition for
CP violation in the Higgs sector can be expressed as
a requirement of complexity of combinations (which
are invariant under rephasing transformation, see
(2.6))
# # 5 (m 2
12 ) 2 , # # 6 m 2
12 , # # 7 m 2
12 . (3.12)
Now we express physical Higgs boson states h i via
h, H , A:
0 @
h 1
h 2
h 3
1 A = R 3 R 2
0 @
h
-H A
1 A with (3.13)
RMR T = R 3 R 2 M 1 R T
2 R T
3 = 0 @
M 2
1 0 0
0 M 2
2 0
0 0 M 2
3
1 A .
This relation allows to discuss a general CP violat­
ing case in terms customary for the CP conserving
case, i.e. with parameters MH , M h , MA and #. The
angles # 2 and # 3 describe mixing between the CP--
even states h, H and the CP--odd state A.
The squared masses M 2
i in Eq.(3.13) are the eigen­
values of the matrix M (3.5). Note, that the trace
of mass­squared matrix does not changed under the
unitary transformations. Therefore, we have mass
sum rule
M 2
1 +M 2
2 +M 2
3 = M 2
h +M 2
H +M 2
A
= M 11 +M 22 +M 33 .
(3.14)
D. Some cases of CP violation
Here we present some important cases.
. If # = 0 and Im ” # 67 = 0, CP symmetry is
not violated: h, H and A are physical Higgs bosons,
with masses given by eqs. (3.10b) and (3.5c), and
# 2 = # 3 = 0.
. If # 13 = |M #
13 /(M 2
A - M 2
h
)| # 1 the Higgs
boson h 1 practically coincides with h (# 2 # 0). The
interaction of h 1 with other particles respects CP--
symmetry (with an accuracy # # 13 ). The diagonal­
ization of residual #23# corner of mass matrix (3.8)
with the aid of rotation matrix R 3 (3.7a) gives states
h 2 and h 3 --- superpositions of H and A states with
mixing angle # 3 , given by
tan 2# 3 # -2M #
23
M 2
A -M 2
H
, # 2 # 0
.
(3.15a)
If MA # MH , the CP violating mixing can be strong
even at small M #
23 /v 2 , the states h 2 and h 3 may
have no definite CP parity and the mass di#erence
M 2
2 -M 2
3 larger than |M 2
H -M 2
A |. For example, at

MH # 300 GeV, |MH - MA | # 5 GeV and M #
23 #
0.02v 2 we have |M 2
2 -M 3
3 | # 25 GeV, sin 2# 3 # 0.8.
. If # 23 = |M #
23 /(M 2
A -M 2
H
)| # 1, the Higgs
boson h 2 practically coincides with -H (# 3 # 0).
The interaction of matter with h 2 does not violate
CP--symmetry. Similarly to the previous case, the
diagonalization of #13# minor of mass matrix (3.8)
with the aid of rotation matrix R 2 (3.7a) gives states
h 1 and h 3 --- superpositions of h and A states with
mixing angle # 2 , given by
tan 2# 2 # -2M #
13
M 2
A -M 2
h
, # 3 # 0. (3.15b)
Similar to previous case, if MA # M h , the CP vio­
lating mixing can be strong even at small M #
13 /v 2 .
. Case of weak CP violation combines both
cases (3.15). If both |M #
13 | # |M 2
A - M 2
h | and
|M #
23 | # |M 2
A - M 2
H |, CP--even states h, H are
weakly mixed with the CP--odd state A and param­
eters # 2 and # 3 are simultaneously small:
tan # 2 # s 2 # -M #
13
M 2
A -M 2
h
# # 2 ,
tan # 3 # s 3 # -M #
23
M 2
A -M 2
H
# # 3 .
(3.16a)
To the second order in s 2 and s 3 the corresponding
masses are equal to
M 2
1 = M 2
h - s 2
2 (M 2
A -M 2
h ),
M 2
2 = M 2
H - s 2
3 (M 2
A -M 2
H ), (3.16b)
with M 3 given by the sum rule (3.14) .
. The case of intense coupling regime with
MA # M h # MH [19] may also give strong CP vio­
lating mixing even with small # and Im ” # 67 .
Note that in MSSM, etc. CP symmetry can be
violated by interaction of Higgs fields with di#erent
scalar squarks, etc. In this case the mixed polariza­
tion operators Im#HA and Im#hA appear leading
to the CP violation in Higgs sector even for the CP
conserving Higgs potential. This violation can be
visible if H and A or (and) h and A are almost de­
generate (see e.g. [28] and references therein).
E. Couplings to gauge bosons
The gauge bosons W and Z couple only to the
CP--even fields # 1 , # 2 . For the physical Higgs bosons
h i (3.6) one obtains simple expressions for their cou­
plings, which in terms of relative couplings (1.3) read
# (i)
V = cos # R i1 + sin # R i2 ,
i = 1, 2, 3, and V = W or Z.
(3.17a)
Note that due to the unitarity of transformation
matrix R, the following sum rule takes place [20]
3
X i=1
(# (i)
V
) 2 = 1 . (3.17b)
IV. YUKAWA INTERACTIONS
A. General discussion
In the general case the Yukawa Lagrangian reads
-LY = •
QL [(# 1 # 1 + # 2 # 2 )d R
+(# 1
”
# 1 +# 2
”
# 2 )u R ] + h.c.,
(4.1)
plus similar terms for the leptons. Here, QL refers
to the 3­family vector of the left­handed quark dou­
blets, whereas dR and uR refer to the 3­family
vectors of the the right­handed field singlets (with
q L = (1 - # 5 )q/2), ”
# a = i# 2 # +T a . The matrices # and
# are 3--dimensional matrices in the family space
with generally complex coe#cients.
Obviously the reparametrization transformation
(2.2) induces changes in elements of matrices # i and
# i . In particular, the rephasing invariance is ex­
tended to the full Higgs + Yukawa Lagrangian space
if one supplements the transformations (2.5) of fields
# i by the following transformations of fermion fields
QLk # QLk e i# qk , dRk # dRk e i(# qk +#dk ) ,
uRk # uRk e i(# qk +#uk ) .
(4.2a)
The corresponding transformation of the param­
eters of Yukawa Lagrangian supplemented transfor­
mation (2.6) have form
# i # # i
0 @
e i# d1
e i# d2
e i# d3
1 A e #i# i with - for i = 1,
+ for i = 2 (4.2b)
and similar for # i .
An existence in the Yukawa couplings of the o#­
diagonal (in family index) terms results in the flavor­
changing neutral­currents (FCNC). The rephasing
invariance (2.6), (4.2) allows to make real the diag­
onal elements of only one matrix # and one matrix
#. Complex values of the other elements of matri­
ces # 1,2 and # 1,2 can result in the complex values of
one--loop corrections to some #'s and in consequence
to the CP violation in the Higgs sector discussed
above (even for real bare m 2
12 and #'s).
Note, that in the case when simultaneously # 1 #= 0
and # 2 #= 0 or # 1 #= 0 and # 2 #= 0 (i.e. right­handed
fermion of the type dR or uR interacts with both
fields # 1 and # 2 ), the counter terms corresponding
to the one­loop corrections to the Higgs Lagrangian

contain operators of dimension 4, which violate Z 2
symmetry (1.2) in a hard way. They contribute to
the renormalization of parameters #, # 6 and # 7 [8],
[21]. Therefore ([4, 22]), to have only the soft viola­
tion of Z 2 symmetry (to prevent # 1 # # 2 transitions
at small distances), we demand that
each right­handed fermion couple
to only one scalar field, either # 1 or # 2 .
(4.3)
For example, the case # 2 = # 1 = 0 with diagonal
# 1 , # 2 corresponds to Model II, while # 2 = # 1 = 0 --
to Model I (see e.g. [2]). The rephasing transforma­
tions (2.6), (4.2) with di#erent phases generate from
one Lagrangian of Model II (or I) the Model II (or
I) Lagrangians family in which the property (4.3)
is explicitly seen (general reparametrization trans­
formation (2.2) makes this property latent). If the
Model II (or I) family coincides with the soft Z 2
violated Lagrangian family (crossed vertical strip in
Fig. 1), these properties (soft Z 2 violation and (4.3))
are stable under radiative corrections.
If mentioned families of Lagrangians exist but
don't coincide, the loop corrections transform our
Lagrangian to that of true hard violated Z 2 symme­
try. The appearance of counter--terms, leading to
hard violating Z 2 symmetry, requires adding of the
corresponding terms in the initial Lagrangian, simi­
lar to that it was discussed in sec. II B in context of
the mixed kinetic term.
B. Model II
We limit ourselves to the discussion of Model II,
where the fundamental scalar field # 1 couples to d­
type quarks and charged leptons #, while # 2 couples
to u­type quarks (we take neutrinos to be mass­
less). The rephasing transformation (2.5) induces
only changes of phases of Yukawa couplings, keep­
ing mentioned basic property of Model II immutable.
The reparameterization transformation (2.2) gener­
ally makes above property hidden just as in the case
of soft Z 2 violation. We use below also real vacuum
form of Lagrangian, This form corresponds to the
intersection of crossed and black strips in fig. 1.
Using matrices # 1 = diag(g d1 , g d2 , g d3 ) and
# 2 = diag(g u1 , g u2 , g u3 ) (like in MSSM) with real
g i , as mentioned above, we get
-L II
Y
= P
k=1,2,3
•
QLk h g dk # 1 dRk +g uk
”
# 2 uRk i
+
P
k=1,2,3
g #k
•
# Lk # 1 # Rk + h.c. (4.4)
For the interaction of the charged Higgs bosons
e.g. with t­quark, the Lagrangian (4.4) gives
LH - tb = M t
v # 2
cot # • b(1 + # 5 )H - t
+ M b
v # 2
tan # • b(1 - # 5 )H - t + h.c.
(4.5)
Since v.e.v.'s of scalar fields are responsible for
the fermion mass similarly as in the SM, the relative
Yukawa couplings of physical neutral Higgs bosons
h i (1.3) are identical for all u--type and for all d--
type quarks (and charged leptons). They can be ex­
pressed via elements of the rotation matrix R (3.6):
# (i)
u = 1
sin #
[R i2 - i cos # R i3 ],
# (i)
d = 1
cos #
[R i1 - i sin # R i3 ].
(4.6)
(Note that e.g. the interaction •
QL (g 1 +ig 2 )d R + h.c.
reads as •
d(g 1 - i# 5 g 2 )d for the Dirac fermions.)
In the particular case of weak CP violation (with
small s 2 , s 3 (3.16)) these relative couplings together
with the corresponding ones to gauge bosons are pre­
sented in Table I.
TABLE I: Basic relative couplings in the weak CP­conserving case. The upper lines correspond the case with no CP
violation and second lines (in parenthesis) contain the correction terms # s2 , s3 .
#V #u #d
h1
sin(# - #)
(+0)
cos #
sin #
(-is2 cot #)
-
sin #
cos #
(-is2 tan #)
h2 - cos(# - #)
(+0)
-
sin #
sin #
(-is3 cot #)
- cos #
cos #
(+is3 tan #)
h3 0
(-s2 sin (# - #) + s3 cos (# - #))
-i cot #
(+s3 cos #
sin # - s2 sin #
sin #
)
-i tan #
(+s2 sin #
cos #
+ s3 cos #
cos #
)
. In the case of no CP violation one can express the triple Higgs couplings via relative couplings of

Higgs bosons to gauge bosons and fermions. Denot­
ing by # either h or H , we find
g### = 3
2v h # #
u +# #
d -# #
V # #
u # #
d  (M 2
# -#v 2 )
+# #
V #v 2
i ,
## 1 #= # 2 # g #1#2#2 =
-
1
2v # #1
V  # #2
u # #2
d (2M 2
#2 +M 2
#1 -3#v 2 )-#v 2
 ,
g#AA = 1
v h (2M 2
A -M 2
# )# #
V
+(M 2
# - #v 2 )(# #
u + # #
d ) i ,
g A#1#2 = g AAA = g AH + H - = 0 ,
g #H + H - = 1
v h 2M 2
H ± -M 2
#  # #
V
+(M 2
# - #v 2 )(# #
u + # #
d ) i .
(4.7)
(These results are valid also in the case of weak CP
violation).
C. Useful relations in Model II
The unitarity of the mixing matrix R allows to ob­
tain a number of useful relations [20, 23, 24] between
the relative couplings of neutral Higgs particles to
gauge bosons (3.17a) and fermions (4.6) (basic rela­
tive couplings), which can be treated as measurable
quantities.
1. The pattern relation among basic relative cou­
plings holds of each neutral Higgs particle h i [23, 24]:
(# (i)
u + # (i)
d )# (i)
V = 1 + # (i)
u # (i)
d . (4.8)
2. The relations (4.6) allow also to write for each
neutral Higgs boson h i a horizontal sum rule [25]:
|# (i)
u | 2 sin 2 # + |# (i)
d | 2 cos 2 # = 1 . (4.9)
These sum rules guarantee that the cross section to
produce each neutral Higgs boson h i (or h, H,A) of
the 2HDM, in the processes involving Yukawa inter­
action, cannot be lower than that for the SM Higgs
boson with the same mass [25].
3. A vertical sum rule for each relative coupling
# j for all three neutral Higgs bosons h i is given by:
3
X i=1
(# (i)
j ) 2 = 1 (j = V, d, u) . (4.10)
4. Besides, the useful linear relation follows di­
rectly from Eqs. (3.17a), (4.6):
# (i)
V = cos 2 # # (i)#
d + sin 2 # # (i)
u =
= cos 2 # # (i)
d + sin 2 # # (i)#
u #
8 < :
# (i)
V
= Re  cos 2 ## (i)
d
+ sin 2 ## (i)
u  ,
Im  cos 2 ## (i)
d - sin 2 ## (i)
u  = 0.
(4.11)
By excluding from (4.11) the parameter # (with the
aid of the (4.9)), one obtains a new relation:
(1-|# (i)
d | 2 ) Im# (i)
u +(1-|# (i)
u | 2 ) Im# (i)
d
= 0 . (4.12)
The relations (4.8), (4.10), (4.12) are reparametriza­
tion independent while (4.9) and (4.11) take place
only for the Model II form of Lagrangian.
5. One can also express tan #, which is a basic
parameter of the 2HDM defined for the Model II
Lagrangian family, via the basic relative couplings:
tan 2 #=
(# (i)
V -# (i)
d
) #
# (i)
u -# (i)
V
= Im# (i)
d
Im# (i)
u
= 1-|# (i)
d | 2
|# (i)
u | 2
-1
. (4.13)
d. Some applications. Let us remind that the
relative couplings to quarks are generally complex
while coupling to gauge bosons are real.
. From (4.9) we get that,
if |# (i)
u | # 1 # tan # # 1 ;
if |# (i)
d | # 1 # tan # # 1 .
(4.14)
It is instructive to consider now consequences of the
relations (4.8)--(4.10) for the case when some basic
relative couplings of a Higgs boson are close to ±1.
. In virtue of (4.9) for moderate tan # we have
if |# (i)
u | # 1 # |# (i)
d | # 1. (4.15)
Note, that if tan # is extremely large or extremely
small, horizontal sum rule allows |# (i)
d | to di#er
strong from 1 or |# (i)
u | to di#er strong from 1, re­
spectively (in agreement with (4.14)).
. From (4.10),
if # (2)
u # ±1 # # (1)
u # ±i# (3)
u ,
if # (2)
d # ±1 # # (1)
d # ±i# (3)
d .
(4.16)
. For # (2)
V # ±1 (analogously for h 1 and h 3 )
if # (2)
V # ±1
#
8 > > > > > > < > > > > > > :
(a) # (2)
u # # (2)
V or # (2)
d # # (2)
V ,
(b) # (2)
u # # (2)
d # # (2)
V ,
(c) # (1)
V # # (3)
V # 0 ,
(d) # (1)
u # (1)
d # # (3)
u # (3)
d # -1 .
(4.17)
The property (a) obtained from (4.8b), means
that the coupling of h 2 to at least one fermion type
is close to the # (2)
V
. The property (b) follows from
property (a) and (4.11), at moderate tan #. The fact
that the couplings of Higgs bosons to gauge bosons
are real leads, with the aid of (4.10), in the prop­
erty (c). Taking into account property (c) and the
pattern relation (4.8a) we obtain property (d): the

product of Yukawa couplings for other Higgs bosons
(not h 2 ) is close to the corresponding product for
pseudoscalar A in the CP conserving case. (Cer­
tainly, properties similar to (4.16), (4.17) take place
as well in the cases when # (1)
i # ±1 or # (3)
i # ±1.)
V. CONSTRAINTS FOR HIGGS
LAGRANGIAN
A. Positivity (vacuum stability) and minimum
constraints
. To have a stable vacuum, the potential must
be positive at large quasi--classical values of fields
|# i | (positivity constraints) for an arbitrary direc­
tion in the (# 1 , # 2 ) plane. Let (# + 1 # 1 ) = x 1 # 0,
(# + 2 # 2 ) = x 2 # 0. Than (# + 1 # 2 ) = # x 1 x 2 ce i# with
|c| # 1 (due to Schwartz theorem). In the case of soft
Z 2 violation the potential is the quadratic form in
x 1 , x 2 . It should be positive at large x i at di#erent c
and #. This condition gives (see e.g. [26, 27, 29, 30])
# 1 > 0 , # 2 > 0, # 3 + # # 1 # 2 > 0,
# 3 + # 4 - |# 5 | + # # 1 # 2 > 0.
(5.1)
(At x 2 = 0 or x 1 we obtain two first conditions. At
c = 0 the third inequality is derived. At c = ±1
with variation of # in respect to the phase of # 5 we
obtain the latter constraint.)
. The condition for vacuum (2.10) describes the
extremum of potential but not obligatory the mini­
mum. The minimum constraints are the conditions
ensuring that above extremum is a local minimum
for all directions in (# 1 , # 2 ) space, except the Gold­
stone modes (the physical fields provide the basis
in the coset). This condition is realized if the mass
matrix squared for the physical fields is positively
defined, which mean that its eigenvalues, the physi­
cal mass squared, are positive: M 2
h i
, M 2
H ± > 0.
B. Unitarity constraints
The quartic terms of Higgs potential are trans­
formed to the quartic self--couplings of the physical
Higgs bosons. They lead, in the tree approximation,
to s--wave Higgs­Higgs and WLWL and WLH , etc.
scattering amplitudes for di#erent elastic channels.
These amplitudes should not overcome unitary limit
for this partial wave -- that is the tree­level unitarity
constraint.
The unitarity constraint was obtained first in the
minimal SM, with one Higgs doublet and Higgs po­
tential V = (#/2)(# + # - v 2 /2) 2 [4]. In this model
Higgs­boson mass MH = v # #; its width #H grow
with MH as M 3
H . Unitarity limit corresponds si­
multaneously the case when #H # MH , so that
the physical Higgs boson disappears, besides the
strong interaction in the Higgs sector is realized for
# s > v # # # v # 8# # 1.2 TeV, as a strong interac­
tion of WL and ZL . Therefore, the unitarity limit
is a boundary (in the # space) between two physical
regimes. Below the unitarity limit we have more or
less narrow Higgs boson with well known properties.
Above the unitarity limit the Higgs boson disappears
as a particle and strong interaction in the Higgs sec­
tor becomes essential.
Akeroyd et al. [31] have derived the unitarity con­
straints for the 2HDM without a hard violation of
Z 2 symmetry for the CP conserving case, i.e. for real
# 1-5 . In the general CP nonconserving case the pa­
rameter # 5 is complex. The application of rephasing
transformation (2.5) allows to eliminate phase of # 5 ,
coming to the rephasing equivalent Lagrangian with
real # s
5 # |# 5 | (m 2
12 remains complex). Use of one of
this Lagrangian allows to extend the above results
to the CP nonconserving case [27]. It is useful to
present unitarity constraints as the bounds for the
eigenvalues # Z2parity
Y # of the high energy Higgs--Higgs
scattering matrix for the di#erent quantum numbers
of an initial state: total hypercharge Y , weak isospin
# and Z 2 parity. These bounds given separately for
the Z 2 ­even (# 1 # 1 and # 2 # 2 ) and Z 2 ­odd (# 1 # 2 ) ini­
tial states are [27]:
|# Z2
Y #± | < 8# with
# even
21± = 1
2  # 1 + # 2 ± p (# 1 - # 2 ) 2 + 4|# 5 | 2
 ,
# odd
21 = # 3 + # 4 , # odd
20 = # 3 - # 4 ,
# even
01± = 1
2  # 1 + # 2 ± p (# 1 - # 2 ) 2 + 4# 2
4  ,
# odd
01± = # 3 ± |# 5 | , (5.2)
# even
00± = 3(# 1 + # 2 ) ± p 9(# 1 - # 2 ) 2 + 4(2# 3 + # 4 ) 2
2 ,
# odd
00± = # 3 + 2# 4 ± 3|# 5 | .
At small # these constraints result in moderately
large upper bound of 600 Â 700 GeV for MH , MA ,
MH ± (see examples in Table II of sec. VI B), see also
e.g. [31] for CP conserving case. At large #, all MH ,
MA , MH ± can be large without violation of unitary
constraints (5.2).
The correspondence between a violation of the
tree­level unitarity limit and a lack of realization of
the Higgs field as a particle, like in the minimal SM,
takes place in the 2HDM only in the case when all
constraints (5.2) are violated simultaneously. If only
some of these constraints are violated the physical
picture become more complex.
C. The case of hard Z2 violation
The analysis of the case with hard Z 2 violation
(i.e. the potential with # 6,7 terms) is more com­

plicated. One can say definitely that the positivity
constraints (5.1) are valid for some particular direc­
tions of growth of quasi­classic fields #. Similarly,
unitarity constraints (5.2) hold for such transition
amplitudes which don't violate Z 2 symmetry.
With hard violation of Z 2 symmetry one should
consider new directions in the # i phase space, which
violate the Z 2 symmetry. Moreover, an existence
of the # term makes some directions in the #'s pa­
rameter space equivalent, because di#erent sets of
these parameters can be obtained from each other
by transformations like (2.9). This subject is be­
yond a scope of this paper.
VI. HEAVY HIGGS BOSONS IN 2HDM
Many analyses of 2HDM assume a SM--like physi­
cal picture: the lightest Higgs boson h 1 is similar to
the Higgs boson of the SM and other Higgs bosons
escape observation since they are too heavy. Besides,
many authors assume in addition that masses of
other Higgs bosons M are close to the scale of new
physics, and the theory should possess an explicit
decoupling property [32], i.e. the correct descrip­
tion of the observable phenomena must be valid for
the unphysical limit M # # [30, 33--36]. However,
the 2HDM allows also for another realization of the
mentioned SM--like physical picture.
A. Decoupling of heavy Higgs bosons
In 2HDM the decoupling case corresponds to
# # |# i | . (6.1)
In this case equations for masses and mixing angles
# i simplify. With accuracy up to #/# terms we ob­
tain first the masses of the subsidiary Higgs states
obtained at the first stage of diagonalization (3.8­
3.11). From eqs. (3.10b) we derive
M 2
h
v 2
= c 4
# # 1 + s 4
# # 2 + 2s 2
# c 2
# # 345
| {z }
soft
+ 8s 2
# c 2
# Re# 67
| {z }
hard
,
M 2
H
v 2
= # + s 2
# c 2
# (# 1 + # 2 - 2# 345 )
| {z }
soft
(6.2)
- [2(c 2
# - s 2
# ) Re ”
# 67 + (8s 2
# c 2
# - 1) Re# 67 ]
| {z }
hard
.
In the proper decoupling limit # # # we have
# -# # #/2. It is useful to characterize a deviation
from this limiting value by introducing parameter
# ## = #/2 - (# - #). Using s 2# = sin 2# , c 2# =
cos 2#, we get from the second line of eq. (3.10):
# ## = -
L a s 2#
2# , (6.3)
L a =s 2
# # 2 -c 2
# # 1 +c 2# # 345
| {z }
soft
+2Re (2c 2# # 67 - ” # 67 )
| {z }
hard
.
The lightest Higgs boson h 1 . The condition
(6.1) together with constraint (2.15) shows that the
element M #
13 of the matrix (3.8), responsible for the
mixing of scalar h with A, is small as compared to
the mass di#erence M 2
A -M 2
h # #v 2 . Therefore, the
state h 1 is very close to h. The mixing angle # 2 ,
describing the CP--odd admixture in this state, is
given by s 2 # |#|/# (3.16a).
Since for # # # the # ## # 0 the scalar h 1 cou­
ples to the gauge bosons and to the quarks and lep­
tons in Model II as in the SM (with accuracy |#|/#).
Besides, h 1 practically decouples from H ± , since the
quantity # (1)
H ± = g h
H ± v/M 2
H ± # O(|#|/#) (4.7).
Higgs bosons h 2 , h 3 and H ± . The Eqs. (3.2),
(3.5c), (6.2) show that M 2
H ± # M 2
A # M 2
H =
v 2 #  1 +O  |#|
#
 , i.e. H ± , H and A (and there­
fore h 2 and h 3 ) are very heavy and almost degener­
ate in masses,
M 2
H ± # M 2
2 # M 2
3 # v 2 #  1 +O  |#|
#
 . (6.4)
That is one of the reasons to consider the condition
of the decoupling regime (6.1)) in the form, used e.g.
in ref. [30], M 2
A # |#|v 2 .
In the considered case the CP violating mixing
between H and A can be strong, i.e. mixing angle # 3
given by Eq. (3.15a) can be large as it was discussed
in sec. III D.
Since # (h)
V # 1, coupling of H to gauge bosons is
very small, while A does not couple to gauge bosons
(Table I). With mixing between H, A states given
by angle # 3 , we have from (6.2)
# (H)
V = cos(# - #) # # ## , # (A)
V = 0 #
# (2)
V # - cos # 3 # ## , # (3)
V # sin # 3 # ## .
(6.5a)
Besides, couplings of H and A to u­ and d­type
quarks coincide in their modules (see Table I), so
that also the corresponding couplings for h 2,3 have
equal modules, while their phases, related to the CP
violation in the (•uh 2,3 u) and ( •
dh 2,3 d) vertices, are
given by the mixing angle # 3 . Using eqs. (4.6) and
(3.7) we obtain
# (2)
u = i# (3)
u = cot # e -i#3 ,
# (2)
d = -i# (3)
d = - tan # e i#3 .
(6.5b)
The corresponding Higgs decay widths are given
mainly by fermions,
#A # #H # # 2 # # 3
= 3
16# cot 2 #  1 + m 2
b
m 2
t
tan 4 #  MH
with #A - #H
#H # m t
MH
.
(6.5c)

Here we take into account that v 2 /m 2
t # 2. The
gauge boson contributions to these widths are negli­
gibly small (# L 2
a /#). In this case the equations for
# 3 , M 2,3 and # 2,3 include shift of A, H poles due
to their proper widths (6.5c). The obtained mass
squared matrix is non­hermitian, therefore, the mix­
ing angle # 3 become complex, and the states h 2 , h 3
non­orthogonal (see [11, 39] for more details). It is
seen from the corrected eq. (3.15):
tan 2# 3 #
2M #
23
M 2
A -M 2
H - i# , # = MA#A-MH#H .
(6.6)
Note that with this strong overlapping of states
the experimental distinguishing of states h 2 , h 3
seems to be di#cult. The visible e#ects of CP viola­
tion in the fermion interaction will be very similar in
both quite di#erent cases: of a true CP violation and
of a strong overlapping of H and A states without
CP violating mixing.
B. Heavy Higgs bosons without decoupling
The option, where except of one neutral Higgs bo­
son h 1 (or h), all other Higgs bosons are reasonable
heavy, can also be realized in 2HDM for relatively
small #, i.e. beyond decoupling limit. In this case
possible masses of heavy Higgses are bounded from
above by unitarity constraints for #'s. In the case of
CP violation the corresponding bounds can be gen­
erally enhanced since the constraints (5.2) put limit
on parameter |# 5 | while equations for masses contain
Re# 5 . For a non­decoupling case we present in Ta­
ble II some particular examples of sets of parameters
of potential, satisfying constraints (5.2), for light h
(mass 120 GeV) and heavy H , H ± .
The first three lines presents set of #, # values for
the case without CP violation with reasonably heavy
H , H ± , A. One sees that these masses can be
TABLE II: Sets of parameters of potential for light h (mass 120 GeV) and heavy H, H ± satisfying constraints (5.2)
in the nondecoupling case.
tan # #1 #2 #3 #4 #5 # Mh MH MA M H ± s2 s3
50 1 6 5.5 ­6 ­6 0.24 120 600 600 600 ­ ­
0.02 6 1 5.5 ­6 ­6 0.24 120 600 600 600 ­ ­
1 6.25 6.25 6.25 ­6 ­6 0 120 600 600 600 ­ ­
10 4 8 4.4 ­9 -0.5 + 0.3i 0.24 120 700 206 556 0.09 0.02
obtained for very large or small tan # and reasonably
small # # (M h /v) 2 , as well as for tan # # 1, # # 0.
The fourth line of the Table II presents an ex­
ample of the natural set of parameters (see below),
with heavy H and H ± in the weak CP violation
case. Since here mixing angles # 2 , # 3 are small, the
physical states h 1 , h 2 , h 3 are close to the states h,
-H and A, existing in the CP conserved case.
In the considered non­decoupling case couplings
of the lightest Higgs boson to gauge bosons, quarks
and leptons can be either close to their SM values
or di#er from these values. The case when all basic
couplings of the lightest Higgs boson are close to
those of SM Higgs boson is discussed in detail in
paper [41], see also [23, 24]. Note that also in such
case some non­decoupling e#ects due to heavy Higgs
bosons may appear.
C. Natural set of parameters of 2HDM
First of all, we remind of our discussion of the
hard violation of Z 2 symmetry. In addition to dif­
ficulties with the mixed kinetic term, in this case
Yukawa sector cannot be described by simple mod­
els of type I or II, i.e. models like Model III should be
realized. However in such models the FCNC e#ects
(and CP violation in the Higgs sector) are naturally
large. That is an additional reason why the natu­
ral set of parameters of 2HDM is considered as such
which corresponds to the case of softly broken Z 2
symmetry.
The eqs. (3.5), (3.8), (3.11) show explicitly that
the CP violation in the Higgs sector can be weak
for the arbitrary mixing angles # and # only if
Im (m 2
12 ) # #v 2 is small enough, i.e.
Imm 2
12 #
(a) |M 2
A -M 2
h |,
(b) |M 2
A -M 2
h |, |M 2
A -M 2
H |.
(6.7)
If the case (b) is realized, CP is weakly violated for
all h i , if the case (a) is realized, CP is weakly violated
only for h 1 .
This simple condition is valid only for the real
vacuum form of potential, sec. II E. For all other
rephasing equivalent Lagrangians the condition (6.7)
becomes more complex including both Imm 2
12 and

Rem 2
12 . The generalization of the condition (6.7)
should be formulated independently on the rephas­
ing parameter. Therefore, for the natural set of pa­
rameters of 2HDM we require that both Imm 2
12
and Rem 2
12 are small for all rephasing equivalent
Lagrangians. In virtue of (2.14), (2.15), (3.11) in
the case of soft violation of Z 2 symmetry the same
requirements should be transmitted to Re# 5 and
Im# 5 . It means that we define a natural set of pa­
rameters as follows
|#|, |# 5 | # |# 1-4 | . (6.8)
Contrary, in the decoupling case, where h 1 can di#er
from h by a small admixture of A (while H and A
can still be mixed strong) the term m 2
12 has the un­
natural property, namely Rem 2
12 # Imm 2
12 . From
this point of view the decoupling case of 2HDM (6.1)
is unnatural.
For the natural set of parameters of 2HDM the
breaking of the Z 2 symmetry is governed by a small
parameter (#). Due to the existence of a limit when
Z 2 symmetry holds, a small soft Z 2 violation in the
Higgs Lagrangian and the Yukawa interaction re­
mains small also at the loop level. In this respect
we use term natural in the same sense as in ref. [5].
In accordance with Eq. (3.5), for the natural set
of parameters also MA cannot be too large (see Ta­
ble II). This opportunity is not ruled out by data.
VII. SUMMARY
. In our approach we introduce the 16­
dimensional space of Lagrangians with coordinates
given by the Lagrangian parameters. Within this
space there is the 3­dimensional subspace -- the
reparametrization equivalent space, containing La­
grangians which can be obtained from a chosen
one by the reparametrization transformations (2.3).
All the Lagrangians from this subspace describe
the same physical reality. Di#erent properties of
physical model can either be explicit or hidden for
di#erent families of Lagrangians in the mentioned
reparametrization equivalent space. Di#erent forms
of these Lagrangians are suitable for the study of dif­
ferent properties of the model. Obviously, all mea­
surable quantities characterizing a system (like the
coupling constants and masses) are reparametriza­
tion invariant while many other parameters of the­
ory (like tan #) are reparametrization dependent.
The reparametrization equivalent space is natu­
rally sliced to the rephasing equivalent subspaces,
which are described by transformations (2.2) with
# = 0 (the rephasing transformations) represented
by vertical strips in fig. 1. One can characterize these
subspaces e.g. by the value of ratio of v.e.v.'s tan#.
The CP violation in Higgs sector leads to an ex­
istence of physical neutral Higgs bosons without a
definite CP parity. The necessary condition for such
CP violation is that some of coe#cients of the Higgs
Lagrangian are complex. However, complex param­
eters can appear also in the CP conserving case
if an unappropriate reparametrization form of La­
grangian is chosen. We found a specific, real vacuum
form of Lagrangian in which complexity of the pa­
rameters of Higgs Lagrangian becomes a necessary
condition for the CP violation in Higgs sector. This
condition is also su#cient for the CP violation in
Higgs sector, except for very specific relations among
mentioned imaginary parts.
. The 2HDM provides mechanism of the EWSB
which allows for potentially large CP violation and
FCNC e#ects. These phenomena are controlled to a
large extent by the Z 2 symmetry under transforma­
tion (1.2) and various levels of its violation. If the
Z 2 invariance holds, then the considered doublets
of scalar fields # 1,2 are the true fundamental basic
fields before EWSB. The soft violation of Z 2 symme­
try is given by the mixed mass term m 2
12 in the Higgs
potential. In this case doublets # 1 and # 2 mix near
EWSB scale but they don't mix at smaller distances.
The reparametrization transformation converts such
Higgs Lagrangian L s , to the form which looks like
a general Lagrangian with terms as for a hard vio­
lation of the Z 2 symmetry (hidden soft Z 2 violation
form of Lagrangians). However, in this case the pa­
rameters of Higgs potential are related as it is given
by eq. (2.7) (see also eqs. (2.8)). It prevents an ap­
pearance of a running coe#cient at the mixed kinetic
term.
In the case of true hard violation of Z 2 symmetry
even the discussion of Higgs potential alone is in­
complete, since it is necessary to consider more gen­
eral Higgs Lagrangian with a mixed kinetic term.
The coe#cient of this mixed term of Lagrangian, #
(2.1b), generally runs due to the loop corrections.
At some fixed distance (renormalization scale) the
kinetic part of the Lagrangian can be removed by
diagonalization like (2.9) but this term is restored
at other distances (renormalization scales) due to
loop corrections from hard terms of a Higgs poten­
tial. We did not find a fully consistent formulation
of 2HDM in the case when mixed kinetic term is
present. We argue, that due to the mentioned rela­
tion to the phenomena at small distances, the case
with soft violation of Z 2 symmetry looks much more
attractive and natural.
In our calculation we keep separately contribu­
tions of soft and hard violation of Z 2 symmetry.
Nevertheless, our discussion of a hard violation of
Z 2 symmetry is incomplete just as all other analy­
ses known to us, since e#ects related to the running
coe#cient of the mixed kinetic term should be anal­
ysed in addition.
. The EWSB appears at the minimization of a
Higgs potential giving the vacuum expectation val­
ues for two scalar fields # 1 and # 2 . Generally, phases
of these v.e.v.'s di#er from each other. However,

this phase di#erence can be eliminated by a suit­
able rephasing transformation giving the mentioned
above the real vacuum Lagrangian. We use in our
analysis this very form of Lagrangian. We prefer
to express the mass terms of Higgs potential via
v.e.v.'s, v 1,2 and the free dimensionless parameter
# # Rem 2
12 , (2.14).
The particular form of Lagrangian written for
fields # i is reparametrization dependent. On the
contrary the description of Lagrangian in terms of
the observable Higgs states h i is reparametrization
invariant. For the neutral Higgs sector, the transi­
tion to the basis of observable Higgs bosons is rather
complicated. We have performed this in two steps.
First, we diagonalize the CP­even part of the mass­
squared matrix. For our real vacuum form of La­
grangian this step is identical to the one used in the
CP conserving case. It allows to describe the general
CP violating case in terms of well known states h, H
and A treated here as the subsidiary states (i.e. hav­
ing no direct physical meaning). Using these states
it becomes evident that the existence of complex co­
e#cients in the Higgs potential in the real vacuum
form of Lagrangian is necessary and su#cient condi­
tion for the CP violation in Higgs sector. Our proce­
dure allows to analyze easily various important cases
when one of neutral Higgs boson is almost the CP­
even one, while two other neutral Higgs bosons can
strongly mix, leading to a strong CP violation in the
processes with exchange of these Higgs bosons.
. Considering the Yukawa interactions we note
that for a case of hard violation of Z 2 symmetry the
most general form of this interaction (e.g. Model
III) should be implemented. We limit ourselves to
models in which each fermion isosinglet couples to
only one Higgs field and discuss the flavor structure
of such couplings. Next we consider in detail the
Model II, for which the well known form appears
only in the Model II Lagrangian family defined in
sec. IV. Here we assume that this family coincides
with mentioned above family of Lagrangians with
explicit softly violated Z 2 symmetry. In addition we
use the real vacuum form of Lagrangian which cor­
responds in fig. 1 to the intersection of the crossed
vertical strip and black horizontal strip.
. In this paper which is based on [1] we extend
our approach introduced earlier for the CP conserv­
ing case in [23, 24] to the analysis of the CP noncon­
serving case. This approach relies on using the mea­
surable (in principle) Higgs boson masses and basic
relative couplings (1.3) plus parameter # (2.14) in­
stead of variety of parameters #'s and mixing angles
# i , #. This way phenomenological analyses simplify
considerably.
We present a series of relations between di#er­
ent relative couplings of each Higgs boson, (4.8)--
(4.12), both the reparametrization invariant and
reparametrization dependent ones. Among these re­
lations there are well known sum rules, the pattern
relation (obtained by us in [23] for the CP conserv­
ing case and in [24] for the CP violation), and new
linear relations. Eq. (4.13) represents the formulae
which allow to determine the quantity tan # for the
Model II family of Lagrangians.
Using these relations we obtained various useful
relations among couplings of Higgs bosons to quarks
and gauge bosons in the case when some of these
couplings (or their absolute values) are close to the
corresponding couplings in the SM (4.14)--(4.17).
. As the obtained relations between relative cou­
plings are of great phenomenological importance it
was crucial to check how the radiative corrections in­
fluence them; it was found that radiative corrections
change only weakly the considered relations.
. Next we combine and discuss di#erent types of
constraints on the parameters of the Higgs poten­
tial, like the positivity condition or the vacuum sta­
bility condition at large quasiclassical values of # i ,
the existence of a minimum, the tree­level unitarity
constraint from the Higgs-- Higgs scattering matrix,
both in the CP conserving and CP violating cases.
Some of them were known till now only in the CP
conserving case. All known results in this field were
obtained for the case of soft violation of Z 2 symme­
try only. We ascertained that these results are valid
also in the case of hard violation of Z 2 symmetry, as
a part of more general system of constraints.
. We perform the detailed discussion about an op­
portunity that in the 2HDM there is one light Higgs
boson, while others are much heavier, so that they
can escape observation. As it was already claimed
in [24],[30] such situation can be realized in the dif­
ferent regions of #. At # # |# i | we have decoupling
case in which the lightest Higgs boson h 1 is very sim­
ilar to the SM Higgs boson, while other Higgs bosons
except h 1 are very heavy and almost degenerate in
masses. We found simple expressions for their cou­
plings which hold for a possible strong CP violating
mixing among them (6.5).
At small # the reasonably heavy Higgs bosons,
lighter however than 600 GeV, may appear without
violation of unitarity constraints. This small # op­
tion looks more natural in context of the rephas­
ing invariance. Here one can expect some non­
decoupling e#ects due to the heavy Higgs bosons
[23, 24, 37]. The detailed analysis of various SM­
like realizations and some non­decoupling e#ects is
presented in [41].
. Note that the CP violation can be implemented
in a model in di#erent ways. In this paper we con­
sider mainly the CP violation given by complex val­
ues of some parameters of Higgs Lagrangian. How­
ever, there are other ways of implementation of CP
violation. The first one, mentioned in sec. IV A rely
on complex values of terms of Yukawa coupling ma­
trices. The second way, used in fact in many analy­
ses of MSSM is related to the CP nonconservation in
the coupling of Higgs bosons to some other particles

like superpartners. The general reasons of renor­
malizability demand to add in these cases the CP
violated terms also in Higgs Lagrangian.
Acknowledgments
Some of results presented here were obtained
together with Per Osland [37], [23, 24] and we
are very grateful to him for a fruitful collabora­
tion. We express our gratitude to A. Djouadi, J.
Gunion, H. Haber, M. Spira, P. Chankowski, B.
Grz›adkowski, W. Hollik, I. Ivanov, M. Vychugin,
M. Dubinin, M. Dolgopolov, J. Kalinowski, R. Nev­
zorov, A. Pilaftsis for various valuable discussions.
This research has been supported by grants RFBR,
NSh­2339.2003.2 in Russia, INTAS, European Com­
mission ­5TH Framework contract HPRN­CT­2000­
00149 (Physics in Collision), by Polish Committee
for Scientific Research Grant No. 1 P03B 040 26 and
115/E­343/SPB/DESY/P­03/DWM517/2003­2005.
[1] I. F. Ginzburg and M. Krawczyk, hep­ph/0408011.
[2] J.F. Gunion, H.E. Haber, G. Kane, S. Dawson, The
Higgs Hunter's Guide (Addison­Wesley, Reading,
1990).
[3] T. D. Lee, Phys. Rev. D 8, 1226 (1973).
[4] S. L. Glashow and S. Weinberg, Phys. Rev. D 15,
1958 (1977).
[5] G. 't Hooft, in ``Recent Developments In Gauge
Theories'', Proc. Nato Adv. Study Inst., Cargese,
France, August 26 ­ September 8, 1979, edited by
G. 't Hooft et al. NY, USA: Plenum (1980) (Nato
Adv. Study Inst. Ser.: Series B, Physics, 59);
S. Dimopoulos, S. Raby and L. Susskind, Nucl.
Phys. B 173 (1980) 208.
[6] A. Mendez and A. Pomarol, Phys. Lett. B 272, 313
(1991).
[7] I.F. Ginzburg, I.P. Ivanov. In preparation
[8] I.F. Ginzburg, Sov. J. Nucl. Phys. 25 (1977) 227.
[9] S. Weinberg, Phys.Rev. D 42 (1990) 860.
[10] N. N. Bogoliubov and D. V. Shirkov, Introduction to
the Theory of Quantized Fields, 3­rd Edition, John
Wiley & Sons, (1980), p. 378.
[11] A. Pilaftsis, Nucl. Phys. B 504, 61 (1997)[hep­
ph/9702393].
[12] G. C. Branco, L. Lavoura and J. P. Silva,``CP vio­
lation,'' (Oxford Univ. Press, 1999).
[13] J. L. Diaz­Cruz and A. Mendez, Nucl. Phys. B380
(1992) 39,
J. L. Diaz­Cruz and G. Lopez Castro, Phys. Lett.
B301 (1993) 405.
[14] P. M. Ferreira, R. Santos and A. Barroso, Phys.
Lett. B 603 (2004) 219, hep­ph/0406231].
[15] H. Georgi and D. V. Nanopoulos, Phys. Lett. B 82,
95 (1979).
[16] L. Lavoura and J. P. Silva, Phys. Rev. D 50, 4619
(1994), hep­ph/9404276.
[17] M. N. Dubinin and A. V. Semenov, hep­ph/0206205,
Eur. Phys. J. C (2003)
[18] A.G. Akeroyd, S. Kanemura, Y. Okada, E. Sehana,
hep­ph/0409318
[19] E. Boos, A. Djouadi and A. Nikitenko, Phys. Lett.
B 578, 384 (2004), hep­ph/0307079.
[20] J. F. Gunion, H. E. Haber and J. Wudka, Phys. Rev.
D 43, 904 (1991).
[21] C. D. Froggatt, R. G. Moorhouse and I. G. Knowles,
Nucl. Phys. B 386 (1992) 63.
[22] E. A. Paschos, Phys. Rev. D 15, 1966 (1977).
[23] I. F. Ginzburg, M. Krawczyk and P. Osland,
Linear Collider Note LC­TH­2001­026, In 2nd
ECFA/DESY Study 1998--2001, 1705, hep­
ph/0101208; Nucl. Instrum. Meth. A472 (2001)
149, hep­ph/0101229, hep­ph/0101331.
[24] I. F. Ginzburg, M. Krawczyk and P. Osland, hep­
ph/0211371.
[25] B. Grzadkowski, J. F. Gunion and J. Kali­
nowski, Phys. Rev. D 60, 075011 (1999), hep­
ph/9902308; Phys. Lett. B 480, 287 (2000)
[arXiv:hep­ph/0001093].
[26] N. G. Deshpande and E. Ma, Phys. Rev. D 18, 2574
(1978).
[27] I. F. Ginzburg and I. P. Ivanov, hep­ph/0312374.
[28] J. Ellis, J, S. Lee, A. Pilaftsis. hep­ph/0404167 and
0411379
[29] B. M. Kastening, hep­ph/9307224.
[30] J. F. Gunion and H. E. Haber, hep­ph/0207010.
[31] A. G. Akeroyd, A. Arhrib and E. M. Naimi, Phys.
Lett. B 490, 119 (2000)
[32] T. Appelquist and J. Carazzone, Phys. Rev. D 11,
2856 (1975).
[33] S. Kanemura, T. Kubota and H. A. Tohyama, Nucl.
Phys. B 483, 111 (1997) [Erratum­ibid. B 506,
548 (1997)], hep­ph/9604381; S. Kanemura and
H. A. Tohyama, Phys. Rev. D 57, 2949 (1998), hep­
ph/9707454]
[34] P. Ciafaloni and D. Espriu, Phys. Rev. D 56 (1997)
1752, hep­ph/9612383.
[35] M. Malinsky, hep­ph/0207066.
[36] S. Kanemura, S. Kiyoura, Y. Okada, E. Senaha and
C. P. Yuan, hep­ph/0209326; S. Kanemura, S. Kiy­
oura, Y. Okada, E. Senaha and C. P. Yuan, Phys.
Lett. B 558 (2003) 157 hep­ph/0211308.
[37] I. F. Ginzburg, M. Krawczyk and P. Osland,
Proc. 4th Int. Workshop on Linear Colliders, April
28­May 5, 1999; Sitges (Spain) p. 524, hep­
ph/9909455; I. F. Ginzburg, Nucl. Phys. Proc.
Suppl. 82, 367 (2000), hep­ph/9907549.
[38] I. F. Ginzburg and M. V. Vychugin, Phys. Atom.
Nucl. 67, 281 (2004) [Yad. Fiz. 67, 298 (2004)].
[39] S. Y. Choi, J. Kalinowski, Y. Liao and P. M. Zerwas,
hep­ph/0407347.
[40] F. J. Botella, M. Nebot and O. Vives, hep­
ph/0407349.
[41] I.F. Ginzburg, M. Krawczyk, in preparation.