Table Of ContentAUE-02-94
EHU-01-94
US-94-01
January 1994
Neutron Electric Dipole Moment
in Two Higgs Doublet Model
4
9 Takemi HAYASHI(a), Yoshio KOIDE(b) 1
9
1 and
n Masahisa MATSUDA(c), 2 Morimitsu TANIMOTO(d)
a
J
6 (a) Kogakkan University, Ise, Mie 516, JAPAN
2
(b) Department of Physics, University of Shizuoka, 52-1 Yada, Shizuoka 422, JAPAN
1
v
1 (c) Department of Physics and Astronomy, Aichi University of Education
3
3 Kariya, Aichi 448, JAPAN
1
0
4 (d) Science Education Laboratory, Ehime University, Matsuyama 790, JAPAN
9
/
h
ABSTRACT
p
-
p
e We study the effect of the ”chromo-electric” dipole moment on the electric dipole
h
:
v moment(EDM) of the neutron in the two Higgs doublet model. We systematically
i
X
˜ ˜
investigate the Weinberg’s operator O = GGG and the operator O = qσGq, in
r 3g qg
a
the cases of tanβ 1, tanβ 1 and tanβ 1. It is shown that O gives the
sg
≫ ≪ ≃
main contribution to the neutron EDM compared to the other operators, and also
that the contributions of O and O cancel out each other. It is pointed out that
ug 3g
the inclusion of second lightest neutral Higgs scalar adding to the lightest one is of
essential importance to estimate the neutron EDM. The neutron EDM is considerably
reduced due to the destructive contribution with each other if the mass difference of
the two Higgs scalars is of the order O(50GeV).
1E-mail:[email protected]
2E-mail:[email protected]
1. Introduction
The physics of CP violation has attracted one’s attention in the circumstance that
the B-factory will go on in the near future. In the experiments of the B decay asymme-
try, the central subject is the test of the standard Kobayashi-Maskawa model(SM)[1] as
an origin of CP violation. On the other hand, the electric dipole moment(EDM) of the
neutron is of central importance to probe a new origin of CP violation, because it is
very small in SM. Begining with the papers of Weinberg[2], there has been considerable
renewed interest in the neutron EDM induced by CP nonconservation of the neutral
Higgs sector. Some studies[3,4,5] revealed numerically the importance of the ”chromo-
electric” dipole moment, which arises from the three-gluon operator GGG˜ by found
Weinberg[2] and the light quark operator qσG˜q introduced by Gunion and Wyler[3],
in the neutral Higgs sector. Thus, it is important to study the effect of these operators
systematically in the model beyond SM. In this paper, we study the contribution of
above two operators to the neutron EDM in the two Higgs doublets model(THDM)[6].
The 3 3 mass matrix of the neutral Higgs scalars is carefully investigated in the
×
typical three cases of tanβ 1, tanβ 1 and tanβ 1, where tanβ v /v with
2 1
≫ ≃ ≪ ≡ | |
v φ0 and φ and φ couple with down- and up-quark sectors, respectively. In
i ≡ h iivac 1 2
these restricted regions of tanβ, the Higgs mass matrix becomes very simple, and then
we can easily estimate the CP violation parameters of the neutral Higgs sector, which
lead to the neutron EDM in THDM. We found that the neutron EDM follows mainly
from the two light neutral Higgs scalar exchanges. Due to the opposite signs of the
two contributions, the neutron EDM is considerably reduced if the mass difference of
the two Higgs scalars is in the order of O(50GeV).
In order to give reliable predictions, one needs the improvement on the accuracy
2
of the description of the strong-interaction hadronic effects. Weinberg employed the
”naive dimensional analyse”(NDA) as developed by Georgi and Manohar[7] in comput-
˜
ing the effect of the GGG operator on the neutron. However, this method admittedly
provides at best the order-of-magnitude estimation. Moreover, when gluon fields are
present, there occurs an indeterminable factor of 4π, which depends on whether one as-
sociates a factorg or 4πg with each gluonfield factor in the interaction Lagrangian[3].
s s
Recently, Chemtob[8] proposed a systematic approach which gives the hadronic matrix
elements of the higher-dimension operators involving the gluon fields by using the large
N current-algebra. In his model, the hadronic matrix elements of the operators are
c
approximated by the intermediate states with the single nucleon pole and the nucleon
plus one pion. So, this approach may be a realistic one. We employ his model to get
the hadronic matrix elements of the relevant operators in this work. The comparision
between results by this approach and NDA will be discussed briefly in the last section.
In section 2, the neutral Higgs mass matrix is analyzed and then the magnitudes
of the CP violation factors ImZ are estimated. In section 3, the formulation of the
i
neutron EDM with the hadronic matrix elements of the CP violating operators are
discussed. Section 4 is devoted to the numerical results of the neutron EDM and some
remarks and conclusion are given in section 5.
2. CP violation parameter in THDM
The simple extension of SM is the one with the two Higgs doublets[6]. This model
has the possibility of the soft CP violation in the neutral Higgs sector, which does not
contribute to the flavor changing neutral current in the B, D and K meson decays.
Weinberg[9] has given the unitarity bounds for the dimensionless parameters of the CP
3
nonconservation inTHDM. However, values of these parameters arenot always close to
theWeinberg’s bounds[9]. Actually, theCP violationparameter ImZ (thisdefinition is
1
given later) is suppressed by 1/tanβ compared with the Weinberg’s bound at the large
tanβ as pointed out by Barr[5]. Chemtob[10] has predicted CP violation parameters
by the use of the renormalization approach under the assumption that the coupling
constants of the Yukawa couplings and self-coupling scalar mesons interactions reach
infraredfixedpointsattheelectroweak scale. Thisinfraredfixedpointsapproacheither
leads to the large top quark mass m 230GeV, which is unfavourable to the recent
t
∼
electroweak precision test, or leads to the existence of the unobserved fourth generation
quarks. Thus, it is difficult to estimate the reliable magnitudes of the CP violation
parameters ImZ (i = 1,2). However, we found that the Higgs mass matrix is simplified
i
in the extreme cases of tanβ 1, tanβ 1 and tanβ 1, in which the CP violation
≪ ≃ ≫
parameters are easily calculated.
TheCP violationwilloccurviathescalar-pseudoscalarinterferencetermsinvolving
the imaginary parts of the scalar meson fields normalization constants, Z , which are
i
column vectors in the neutral Higgs scalar vector space, defined in terms of the tree
level approximation to the two-point function as follows:
1 φ0φ0 , 1 φ0φ0 , 1 φ0φ0 , 1 φ0φ∗0
v12h 1 1iq v22h 2 2iq v1v2h 2 1iq v1∗v2h 2 1 iq
h 3 √2G i
= F [Z(n), Z(n), Z˜(n), Z(n)] , (1)
q2 m2 1 2 0 0
nX=1 − Hn
where v φ0 . The CP violation factors ImZ(n) are deduced to
i ≡ h iivac i
1 tanβ
(k) (k) (k) (k) (k) (k)
ImZ = u u , ImZ = u u , (2)
2 tanβsinβ 2 3 1 −cosβ 1 3
1 1 1 1 1 1
ImZ˜(k) = u(k) u(k) u(k) , ImZ(k) = u(k) + u(k) u(k) ,
0 2 sinβ 1 − cosβ 2 ! 3 0 2 sinβ 1 cosβ 2 ! 3
(k)
where u denotes the i th component of the k th normalized eigenvector of the
i
− −
4
(k)
Higgs mass matrix. Let us estimate u by studying the symmetric Higgs mass matrix
i
2
M whose components are
ξ +Re(hv∗2v2)
M2 = 2g v 2 +g′ v 2 + 1 2 ,
11 1| 1| | 2| v 2
1
| |
ξ +Re(hv∗2v2)
M2 = 2g v 2 +g′ v 2 + 1 2 ,
22 2| 2| | 1| v 2
2
| |
ξ Re(hv∗2v2)
M2 = ( v 2 + v 2) g′ + − 1 2 ,
33 | 1| | 2| " v1v2 2 #
| |
Re(hv∗2v2) ξ
M2 = v v (2g +g′)+ 1 2 − , (3)
12 | 1 2| v v
1 2
| |
v 2 + v 2
M2 = | 1| | 2| Im(hv∗2v2) ,
13 −q v2v 1 2
| 1 2|
v 2 + v 2
M2 = | 1| | 2| Im(hv∗2v2) ,
23 −q v v2 1 2
| 1 2|
which are derived from the Higgs potential
1 1
V = g (φ†φ v 2)2 + g (φ†φ v 2)2
2 1 1 1 −| 1| 2 2 2 2 −| 2|
+ g(φ†φ v 2)(φ†φ v 2)
1 1 −| 1| 2 2 −| 2|
+ g′ φ†φ v∗v 2 +Re[h(φ†φ v∗v )2]
| 1 2 − 1 2| 1 2 − 1 2
†
φ φ φ φ
1 2 1 2
+ ξ . (4)
"v1 − v2# "v1 − v2#
As a phase convension, we take h to be real and
v∗2v2 = v 2 v 2exp(2iφ) . (5)
1 2 | 1| | 2|
2
Now, the Higgs mass matrix M is rotated so as to make the (1,3)(and then (3,1))
component zero by the orthogonal matrix U0 as
cosβ sinβ 0
U0 = sinβ cosβ 0 . (6)
−
0 0 1
5
Then, the transformed matrix M′2 = Ut0M2U0 is given as
M′2 = 2g cos4β +2g sin4β +4(ξ g)sin2βcos2β ,
11 1 2 −
M′2 = 2(g +g +2g 2ξ)sin2βcos2β +g′ +ξ +hcos2φ ,
22 1 2 −
M′2 = g′ +ξ hcos2φ ,
33 −
M′2 = sinβcosβ cos2β(2g 2ξ +g +g )+g g , (7)
12 − 1 2 1 − 2
h i
M′2 = 0 ,
13
M′2 = hsin2φ ,
23 −
inthev2 v 2+ v 2 unitandtheparameterξ isdefinedasξ = ξ/ v v 2. Thismatrix
1 2 1 2
≡ | | | | | |
cannot be diagonalized in the analytic form generally, unless special relations among
the parameters of the mass matrix are satisfied. The parameters are only constrained
by the positivity condition as follows[10,11]:
g > 0 , g > 0 , h < 0 , h+g′ < 0 , g +g′ +h > √g g . (8)
1 2 1 2
−
However, we can simply diagonalize the Higgs mass matrix in the extreme cases of
tanβ 1, tanβ 1 and tanβ 1.
≫ ≃ ≪
At first, we consider the case of tanβ 1. By retaining the order of cosβ and by
≫
setting cos2β = 0, sinβ = 1, the mass matrix becomes
2g 2cosβ(ξ g g ) 0
2 2
− −
2cosβ(ξ g g ) g′ +ξ +hcos2φ hsin2φ . (9)
2
− − −
0 hsin2φ g′ +ξ hcos2φ
− −
In the limit of cosβ = 0, this mass matrix is diagonalized by only rotating φ on the
(2-3) plane. However, due to the non-vanishing tiny M′2 component, this rotation is
12
slightly deviated from the (2-3) plane. The orthogonal matrix U1 to diagonalize the
6
Higgs mass matrix of Eq.(9) is approximately obtained as:
1 0 0 1 ǫcosφ 0 1 0 ǫsinφ
U1 0 cosφ sinφ ǫcosφ 1 0 0 1 0 , (10)
≃ −
0 sinφ cosφ 0 0 1 ǫsinφ 0 1
− −
where, neglecting hcos2φ,
2(ξ g g )
2
ǫ − − cosβ . (11)
≃ ξ +g′ 2g
2
−
2
Then, the eigenvectors of M in Eq.(1) are
u(1) = cosβ ǫsinβ, sinβ, 0 ,
{ − − }
u(2) = sinβcosφ, (cosβ ǫsinβ)cosφ, sinφ , (12)
{ − − }
u(3) = sinβsinφ, (cosβ ǫsinβ)sinφ, cosφ ,
{ − }
with the order of O(cos2β) being neglected. The diagonal masses are given as
M2 = 2g +O(cos2β), M2 = g′+ξ+h+O(cos2β), M2 = g′+ξ h+O(cos2β), (13)
1 2 2 3 −
in the v2 unit. The lightest Higgs scalar to yield CP violation is the second Higgs
scalar with the mass M since h is negative and ξ is positive. The Higgs scalar with
2
(1)
M does not contribute to CP violation because of u = 0. The absolute values of
1 3
g′ is expected to be O(1), but h seems to be small as estimated in some works[11].
Therefore, the masses M and M may be almost degenerated. Then, CP violation
2 3
is reduced by the cancellation between the two different Higgs exchange contributions
(2) (3) (2) (2) (3) (3)
ImZ andImZ sinceu u andu u (i=1,2)havesamemagnitudeswithopposite
i i i 3 i 3
signs. Thus, it is noted that the lightest single Higgs exchange approximation gives
miss-leading of CP violation in the case of tanβ 1.
≫
(2,3)
In order to get the magnitudes of u , we estimate ǫ, which depends on the value
2
of ξ. The parameter ξ is determined by the charged Higgs mass as follows:
M±2 = ξv2 . (14)
7
Wehave alreadystudied thecharged Higgsscalar effect inTHDMthroughtheinclusive
decay B X γ[12], as to which the upper bound of the branching ratio was recently
s
→
given by the CLEO collaboration[13]. We obtained 300GeV for the lower bound of the
charged Higgs scalar mass in the case of m = 150GeV and m = 5GeV. This lower
t b
bound means ξ > 3. In the limit of the large ξ with retaining other parameters to be
O(1), ǫ/cosβ reachs 2 as seen in Eq.(11). Actually, g , g , g and g′ are around 1 in
1 2
| |
some numerical studies[11]. Then, if we take M± = 400(350)GeV, which corresponds
to ξ = 5(4), the value of ǫ/cosβ becomes 3(4). In the following calculations, we fix to
(2)
be ǫ = 3cosβ. By use of these resulting u values, we can calculate the CP violation
i
(2)
factors ImZ = ImZ . We show the numerical results together with the Weinberg
i i
bounds for ImZ and ImZ in Figs.1(a) and 1(b), where φ = π/4 is taken. Although
1 2
the Weinberg bounds give nothing for these signs, our estimates determine the relative
sign between ImZ and ImZ . For ImZ , our result reachs the Weinberg bound, but
1 2 1
for ImZ the our calculated value is suppressed compared with the Weinberg bound in
2
the order of 1/tanβ.
Figs.1(a) 1(f)
∼
CP violation in the case of tanβ 1 is similar to the one of tanβ 1. By
≪ ≫
retaining the order of sinβ and by setting sin2β = 0, cosβ = 1, the mass matrix
becomes
2g 2sinβ(ξ g g ) 0
1 1
− − −
2sinβ(ξ g g ) g′ +ξ +hcos2φ hsin2φ . (15)
1
− − − −
0 hsin2φ g′+ξ hcos2φ
− −
Except for replacing g with g and cosβ with sinβ, the mass matrix is the same one
2 1
−
8
as of Eq.(9) in the case of tanβ 1. The eigenvectors are easily obtained as follows:
≪
u(1) = cosβ, (sinβ +ǫ′cosβ), 0 ,
{ − }
u(2) = (ǫ′ +sinβ)cosφ, cosβcosφ, sinφ , (16)
{ − }
u(3) = (ǫ′ +sinβ)cosφ, cosβsinφ, cosφ ,
{ }
with the order of O(sin2β) being neglected. The ǫ′ parameter is defines as
2(ξ g g )
ǫ′ = − − 1 sinβ , (17)
−ξ +g′ 2g +hcos2φ
1
−
whichistakentobeǫ′ = 3sinβ asdiscussed inEq.(11). Takingu(2) astheeigenvector
i
−
of the lightest Higgs scalar, we show ImZ and ImZ in Figs.1(c) and 1(d). For ImZ ,
1 2 2
ourresult reachs theWeinberg bound, while forImZ thecalculated valueissuppressed
1
from the Weinberg bound in the order of tanβ. The relative sign between ImZ and
1
ImZ is just same as in the case of tanβ 1.
2
≫
The last case to consider is of tanβ 1. Setting cos2β = 0, we get the Higgs mass
≃
matrix as
1g + 1g +ξ g 1(g g ) 0
2 1 2 2 − 2 1 − 2
1(g g ) 1g + 1g +g +g′ +hcos2φ hsin2φ . (18)
2 1 − 2 2 1 2 2 −
0 hsin2φ g′ +ξ hcos2φ
− −
The off diagonal components are very small compared to the diagonal ones because
g g is suggested by some analyses[11] and h is small. Then, we get the approximate
1 2
≃
eigenvectors as follows:
u(1) = cosβ sinβsinθ cosθ , sinβ cosβsinθ cosθ , sinθ sinθ ,
12 23 12 23 12 23
{ − − − }
u(2) = sinβcosθ +cosβsinθ , cosβcosθ sinβsinθ , sinθ ,
23 12 23 12 23
{ − − }
u(3) = (sinβsinθ , cosβsinθ , cosθ , (19)
23 23 23
{ }
9
where
g g
2 1
tan2θ = − ,
12
ξ 2g g′ hcos2φ
− − −
4hsin2φ 2hsin2φ
tan2θ = v2 . (20)
23 g +g +2g 2ξ +4hcos2φ ≃ M2 M2
1 2 − 2 − 3
The Higgs scalar mass M is expected to be the heaviest one and the M to be the
1 2
lightest one because of ξ > 3.0 and g g g g′ O(1). We estimate the effect
1 2
∼ ∼ ∼ | | ≃
of CP violation by considering the Higgs scalar with M being lightest one and then
2
add the effect of the one with M . Since θ is expected to be of O(10−2)[11], we neglect
3 12
terms with sinθ in Eq.(19). We can calculate the CP violation parameters ImZ by
12 i
fixing both values of h and M /M . We show ImZ and ImZ in Figs.1(e) and 1(f)
2 3 1 2
taking M = 200GeV, M = 250GeV and h = 0.1. For both ImZ and ImZ , the
2 3 2 1
−
calculating values are roughly 1/3 of the Weinberg bounds. The relative sign between
ImZ and ImZ is opposite to the one in the cases of tanβ 1 and tanβ 1.
1 2
≫ ≪
3. Formulation of the neutron EDM
The low energy CP-violating interaction is described by an effective Lagrangian
L , which isgenerally decomposed into thelocalcomposite operatorsO of thequarks
CP i
and gluons fields,
L = C (M,µ)O (µ) . (21)
CP i i
i
X
Some authors pointed out[3,8] that the three gluon operator with the dimension six
and the quark-gluon operator with the dimension five dominate EDM of the neu-
tron in THDM. So, we study the effect of these two operators on the neutron EDM.
Various techniques have been developed to estimate the strong-interaction hadronic
effects[7,8,14]. The simplest one is the NDA approach[7], but it provides at best the
order-of-magnitudeestimates. The systematic technique has been given by Chemtob[8]
10
