Soft CP violation in K-meson systems

0.24 MB·

by  J. C. Montero

Soft CP violation in K-meson systems
J. C. Montero,∗ C. C. Nishi,† and V. Pleitez‡
Instituto de F´ısica Te´orica,
Universidade Estadual Paulista
Rua Pamplona, 145
01405-900– S˜ao Paulo
Brazil
O. Ravinez§
6 Universidad Nacional de Ingenieria UNI,
0
Facultad de Ciencias
0
2 Avenida Tupac Amaru S/N apartado 31139

n Lima, Peru
a
J
M. C. Rodriguez¶
7
Fundac¸˜ao Universidade Federal do Rio Grande/FURG,
1
Departamento de F´ısica

2
Av. It´alia, km 8, Campus Carreiros
v
0 96201-900, Rio Grande, RS
0
Brazil
1
1 (Dated: February 7, 2008)
1
Abstract
5
0 We consider a model with soft CP violation which accommodate the CP violation in the neutral
/
h kaons even if we assume that the Cabibbo-Kobayashi-Maskawa mixing matrix is real and the
p sources of CP violation are three complex vacuum expectation values and a trilinear coupling in
-
p the scalar potential. We show that for some reasonable values of the masses and other parameters
e the model allows to explain all the observed CP violation processes in the K0-K¯0 system.
h
:
v
PACS numbers: PACS numbers: 11.30.Er;12.60.-i; 13.20.Eb
i
X
r
a
∗Electronic address: [email protected]
†Electronic address: [email protected]
‡Electronic address: [email protected]
§Electronic address: [email protected]
¶Electronic address: mcrodriguez@fisica.furg.br
1
I. INTRODUCTION
Until some time ago, the only physical system in which the violation of the CP symmetry
was observed was the neutral kaon system [1]. Besides, only the indirect CP violation
described by the ǫ parameter was measured in that system. Only recently, clear evidence
for direct CP violation parametrized by the ǫ′ parameter was observed in laboratory [2].
Moreover, the CP violation in the B-mesons system has been, finally, observed as well [3]. It
is in fact very impressive that all of these observations are accommodated by the electroweak
standard model with a complex Cabibbo-Kobayashi-Maskawa mixing matrix [4, 5] when
QCD effects are also included. In the context of that model, the only way to introduce CP
violation is throughout its hard violation due to complex Yukawa couplings, which imply a
surviving phase in the charged current coupled to the vector boson W± in the quark sector.
In the neutral kaon system, despite the CKM phase being O(1), the breakdown of that
symmetry is naturally small because its effect involves the three quark families at the one
loop level [6]. This is not the case of the B mesons where the three families are involved
even at the tree level and the CP violating asymmetries are O(1) [7].
Notwithstanding, if new physics does exist at the TeV scale it may imply new sources of
CP violation. In this context the question if the CKM matrix is complex becomes nontrivial
since at least part of the CP violation may come from the new physics sector [8]. For
instance, even in the context of a model with SU(2) U(1) gauge symmetry, we may
L Y
⊗
have spontaneous CP violation through the complex vacuum expectation values (VEVs),
this is the case of the two Higgs doublets extension of the standard model if we do not
impose the suppression of flavor changing neutral currents (FCNCs), as in Ref. [9]. The CP
violation may also arise throughout the exchange of charged scalars if there are at least three
doublets and no FCNCs [10]. Truly soft CP violation may also arise throughout a complex
dimensional coupling constant in the scalar potential and with no CKM phase [11]. In fact,
all these mechanisms can be at work in multi-Higgs extensions of the standard model [12].
Hence, in the absence of a general principle, all possible sources of CP violation must be
considered in a given model. However, it is always interesting to see the potentialities of
a given source to explain by itself all the present experimental data. This is not a trivial
issue since, for instance, CP violation mediated by Higgs scalars in models without flavor
changing neutral currents have been almost ruled out even by old data [13, 14, 15, 16, 17].
2
Among the interesting extensions of the standard model there are the models based on
the SU(3) SU(3) U(1) gauge symmetry called 3-3-1 models for short [18, 19, 20].
C L X
⊗ ⊗
These models have shown to be very predictive not only because of the relation with the
generation problem, some representation content of these models allow three and only three
families when the cancellation of anomalies and asymptotic freedom are used; they also give
some insight about the observed value of the weak mixing angle [21]. The 3-3-1 models are
also interesting context in which new theoretical ideas as extra dimensions [22] and the little
Higgs mechanism can be implemented [23].
In the minimal 3-3-1 model [18] both mechanisms of CP violation, hard [24] and spon-
taneous [25] has been already considered. In this paper we analyze soft CP violation in
the framework of the 3-3-1 model of Ref. [19] in which only three triplets are needed for
breaking the gauge symmetry appropriately and give mass to all fermions. Although it has
been shown that in this model pure spontaneous CP violation is not possible [25], we can
still implement soft CP violation if, besides the three scalar VEVs, a trilinear parameter in
the scalar potential is allowed to be complex. In this case a physical phase survives violating
the CP symmetry. This mechanism was developed in Ref. [26] but there a detailed analysis
of the CP observables in both kaons and B-mesons were not given. Here we will show that
all the CP violating parameters in the neutral kaon system can be explained through this
mechanism leaving the case of the B-mesons for a forthcoming paper.
The outline of this paper is as follows. In Sec. II we briefly review the model of Ref. [26]
in which we will study a mechanism for soft CP violation. In Sec. III we review the usual
parameterization of the CP violating parameters of the neutral kaon system, ǫ and ǫ′, es-
tablishing what is in fact that is being calculated in the context of the present model. In
Sec. IV we calculate ǫ, and in Sec. V we do the same for ǫ′. The possible values for those
parameters in the context of our model are considered in Sec. VI, while our conclusions are
in the last section. In the Appendix A we write some integrals appearing in box and penguin
diagrams.
II. THE MODEL
Here we are mainly concerned with the doubly charged scalar and its Yukawa interac-
tions with quarks since this is the only sector in which the soft CP violation arises in this
3
model [26]. The interaction with the doubly charged vector boson will be considered when
needed (Sec. V). As expected, there is only a doubly charged would be Goldstone boson,
G++, and a physical doubly charged scalar, Y++, defined by
ρ++ 1 v v e−iθχ G++
ρ χ
= | | −| | , (1)
χ++  N  v eiθχ v Y++ 
χ ρ
| | | |
    
where N = ( v 2 + v 2)1/2; the mass square of the Y++ field is given by
ρ χ
| | | |
A 1 1 a
m2 = + 8 v 2 + v 2 , (2)
Y++ √2 v 2 v 2 − 2 | χ| | ρ|
(cid:18)| χ| | ρ| (cid:19)
(cid:0) (cid:1)
where we have defined A Re(fv v v ) with f a complex parameter in the trilinear term
η ρ χ
≡
ηρχ of the scalar potential and a is the coupling of the quartic term (χ†ρ)(ρ†χ) in the scalar
8
potential. For details and notation see Ref. [26]. Notice that since v v , it is ρ++
χ ρ
| | ≫ | |
which is almost Y++.
In Ref. [26] it was shown that all CP violation effects arise from the singly and/or doubly
charged scalar-exotic quark interactions. Notwithstanding, the CP violation in the singly
charged scalar is avoided by assuming the total leptonic number L (or B+L, see below)
conservation and, in this case, only two phases survive after the re-definition of the phases
of all fermion fields in the model: a phase of the trilinear coupling constant f and the phase
of a vacuum expectation value, say v . Among these phases, actually only one survives
χ
because of the constraint equation
Im(fv v v ) = 0, (3)
χ ρ η
which implies θ = θ .
χ f
−
Let us briefly recall the representation content of the model [26] with a little modification
in the notation. In the quark sector we have Q = (d , u ,j )T (3,3∗, 1/3), i = 1,2
iL i i i L ∼ −
Q = (u ,d ,J)T (3,3,2/3); U (3,1,2/3), D (3,1, 1/3), α = 1,2,3, j
3L 3 3 L ∼ αR ∼ αR ∼ − iR ∼
(3,1, 4/3) and J (3,1,5/3), and the Yukawa interactions are written as:
R
− ∼
= Q (F ρ∗U +F˜ D η∗)+Q (F U η +F˜ D ρ)
iL iα αR iα αR 3L 3α αR 3α αR
−L
iα
X
+ λ Q j χ∗ +λ Q J χ+H.c., (4)
im iL mR 3 3L R
im
X
where all couplings in the matrices F,F˜ and λ’s are in principle complex. Although the
fields in Eq. (4) are symmetry eigenstates we have omitted a particular notation. Here we
4
will assume that all the Yukawa couplings in Eq. (4) are real in such a way that we may be
able to test to what extension only the phase θ can describe the CP violation parameters
χ
in the neutral kaon system, ǫ and ǫ′.
In order to diagonalize the mass matrices coming from Eq. (4), we introduce real and
orthogonal left- and right-handed mixing matrices defined as
U′ = u U , D′ = d D , (5)
L(R) OL(R) L(R) L(R) OL(R) L(R)
withU = (u,c,t)T etc; theprimedfieldsdenotesymmetryeigenstatesandtheunprimedones
masseigenstates, being theCabibbo-Kobayashi-Maskawa matrixdefined asV = uT d.
CKM OL OL
In terms of the mass eigenstates the Lagrangian interaction involving exotic quarks, the
known quarks, and doubly charged scalars is given by [26]:
v Md v m
= √2J¯ e−iθχ| χ| α R e+iθχ| ρ| J L ( d) d Y++ +H.c. , (6)
−LY − N v − N v OL 3α α
(cid:20) | ρ| | χ| (cid:21)
where N is the same parameter appearing in Eq (1), i.e., N = ( v2 + v2 )1/2 and now,
| ρ| | χ|
unlike Eq. (4), all fields are mass eigenstates, L = (1 γ )/2, R = (1 + γ )/2, with m =
5 5 J
−
λ v /√2. In writing the first term of Eq.(6) we have used F˜ = √2( dMd dT) / v ,
3| χ| 3α OL OR 3α | ρ|
where Md is the diagonal mass matrix in the d-quark sector and we have omitted the
summation symbol in α so that d = d,s,b. The Eq. (6) contains all CP violation in the
α
quark sector once we have assumed that all the Yukawa couplings are real. Unlike in multi-
Higgs extensions of the standard model [9, 10, 11, 12, 13, 14, 15, 16, 17] there is no Cabibbo
suppression since in this model only one quark, J, contributes in the internal line, i.e., we
have the replacement u,c,t, J.
→
Notice that in Eq. (6) the suppression of the mixing angle in the sector of the doubly
charged scalars [see Eq. (1)] has been written explicitly. We will use as illustrative values
>
v 246 GeV and v 1 TeV. In this situation the CP violation in the neutral kaon
ρ χ
| | ≤ | | ∼
system will impose constraints only upon the masses m , m , and, in principle, on m
J Y U
the mass of the doubly charged vector boson. Although j has free parameters since the
OL
masses m are not known, the exotic quarks j do not play any role in the CP violation
j1,2 1,2,
phenomena of K mesons.
We should mention that it was implicit in the model of Ref. [26] the conservation of the
quantum number B+L defined inRefs.[19, 20]. Only inthis circumstance (orby introducing
appropriately a Z symmetry) we can avoid terms like ǫ(Ψ )cΨ η and (l )cE , where
2 aL bL aL bR
5
Ψ , l andE denote the left-handed lepton triplet, and the usual right-handed components
L R R
for usual and exotic leptons. These interactions imply mixing among the left- and right-
handed components of the usual charged leptons with the exotic ones [27]. The quartic
term χ†ηρ†η in the scalar potential which would imply CP violation throughout the single
charged scalar exchange is also avoided by imposing the B+L conservation. In fact, this
model has the interesting feature that when a Z symmetry is imposed, the Peccei-Quinn
2
U(1), the total lepton number, and the barion number are all automatic symmetries of the
classic Lagrangian [28].
III. CP VIOLATION IN THE NEUTRAL KAONS
First of all let us say that in the present model there are tree level contributions to
the mass difference ∆M = 2ReM (where M = K0 K¯0 /2m ). This is because
K 12 12 eff K
h |H | i
the existence of the flavor changing neutral currents in the model in both the scalar sector
and in the couplings with the Z′0. The H0’s contributions to ∆M have been considered
K
in Ref. [25]. For m 150 GeV the constraint coming from the experimental value of
H
∼
∆M implies ( d) ( d) . 0.01. There are also tree level contributions to ∆M coming
K OL dd OL ds K
from the Z′ exchange which were considered in Ref. [18, 29]. However, since there are 520
diagrams contributing to ∆M , we will use in this work the experimental value for this
K
parameter. In this vain a priori there is no constraints on the matrix elements of d.
OL
The definition for the relevant parameters in the neutral kaon system is the usual one [30,
31, 32, 33]:
ǫ′ = ei(δ2−δ0+π2)ReA2 ImA2 ImA0 , ǫ = eiπ4 ImA0 + ImM12 , (7)
√2 ReA ReA − ReA √2 ReA ∆M
0 (cid:20) 2 0(cid:21) (cid:20) 0 K (cid:21)
We shall use the ∆I = 1/2 rule for the nonleptonic decays which implies that ReA /ReA
0 2
≃
22.2 and that the phase δ δ π is determined by hadronic parameters following
2 − 0 ≃ −4
Ref. [34] and it is, therefore, model independent.
The ǫ parameter has been extensively measured and its value is reported to be [33]
ǫ = (2.284 0.014) 10−3 . (8)
exp
| | ± ×
More recently, the experimental status for the ǫ′/ǫ ratio has stressed the clear evidence for
a non-zero value and, therefore, the existence of direct CP violation. The present world
6
average (wa) is [33]
ǫ′/ǫ = (1.67 0.26) 10−3 , (9)
wa
| | ± ×
where the relative phase between ǫ and ǫ′ is negligible [35]. These values of ǫ and ǫ′/ǫ
| | | |
imply
ǫ′ = 3.8 10−6 . (10)
| exp| ×
On the other hand, we can approximate
1 ImM 1 ImA
ǫ 12 (a), ǫ′ 0 (b). (11)
| | ≈ √2 ∆M | | ≈ 22.2√2 ReA
(cid:12) K (cid:12) (cid:12) 0(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
In the prediction of ǫ′/ǫ, (cid:12)ReA an(cid:12)d ∆M are taken fro(cid:12)m exp(cid:12)eriments, whereas ImA
(cid:12) 0 (cid:12) K (cid:12) (cid:12) 0
and ImM are computed quantities [36]. The experimental values used in this work are
12
ReA = 3.3 10−7 GeV and ∆M = 3.5 10−15 GeV.
0 K
× ×
Let us finally consider the condition with which we will calculate the parameters ǫ and ǫ′.
The main ∆S = 1 contribution for the ǫ′ parameter comes from the gluonicpenguin diagram
in Fig.1 that exchanges a doubly charged scalar. The electroweak penguin is suppressed as
in the SM and will not be considered. On the other hand the ∆S = 2 and CP violating
parameter ǫ has only contributions coming from box diagrams involving two doubly charged
scalars Y++ (see Fig.2a) and box diagrams involving one doubly charged scalar and one
vector boson U++ (see Fig.2b). The relevant vertices for the calculations are given in
Eq. (6) and we will use the unitary gauge in our calculations. In other renormalizable R
ξ
gauges we must to take into account the would be Goldstone contributions and notice that,
according to Eq. (1), the component of χ++ O(1)G++.
∼
The hadronic matrix elements will be taken fromliterature and whenever possible we also
take, for the reasons we expose at the beginning of this section, from the experimental data
or as free parameters. One of the features of this model is that there is no GIM mechanism
since the only CP violation source comes from the vertices involving a d-type quark, an
exotic quark, and a single doubly charged scalar.
IV. DIRECT CP VIOLATION
The dominant contributions to the ǫ′ parameter come from the penguin diagram showed
in the Fig. 1 [32, 37]. The part of the Lagrangian that takes into account this amplitude is
7
obtained from Eq. (6) and the corresponding imaginary effective interaction is given by
g λ m 1
ImLǫ′ = 16π2sN2 Cdsms s¯σµν 2a L− mdR d Gaµν 2 [h(x)−xh′(x)]sin2θχ, (12)
(cid:20) (cid:18) s (cid:19) (cid:21)
where we have defined C = ( d) ( d) , and Ga in the context of the effective inter-
ds OL 3d OL 3s µν
actions is just Ga = ∂ Ga ∂ Ga, x = m2 /m2 and the function h(x) is given in the
µν µ ν − ν µ Y J
Appendix, and the prime denotes first derivative.
Neglecting the γ,Z contributions, i. e., the amplitudes with I = 2, and using the values
for the other parameters given above, Eq. (11b) leads to
1 1 ImA ImA
ǫ′ | 0| 9.6 104 | 0|, (13)
| | ≈ √222.2 ReA ≈ × 1GeV
0
| |
where we have used ReA = 3.3 10−7GeV−1, with,
0
×
g m 1 m
ImA = √3 s s C (h(x) xh′(x)) P dP sin2θ . (14)
| 0| 16π2N2 ds 2 − L − m R χ
(cid:12) (cid:12)(cid:12) s (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
We can write ǫ′ as follows: (cid:12) (cid:12)(cid:12) (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
| | ǫ′
| | = C A(x)sin2θ , (15)
ǫ′ ds χ
| exp|
g m m 1 9.6 104
A(x) = √3 s s P sP (h(x) xh′(x)) × , (16)
(4π)2 ǫ′ N2 L − m R 2 − 1GeV
| exp| (cid:12) d (cid:12)(cid:12) (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
where we have defined the matrix ele(cid:12)ments (cid:12)(cid:12) (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
λa λa
P = ππ(I = 0) (s¯σµνL d) Ga K0 , P = ππ(I = 0) (s¯σµνR d) Ga K0 . (17)
L h | 2 µν | i R h | 2 µν | i
Using the bag model (BM) it has been obtained that P = 0.5GeV2 [15]. The other
L
−
term in Eq. (14) with the matrix element P is negligible [even if P O( P )] since it
R R L
| | ≈ | |
has a m factor. We will also use the following values: m = 498 MeV, m /m = 1/20,
d K d s
m = 120 MeV, and α = 0.2. The function h(x) xh′(x) has its maximum equal to one at
s s
| − |
x = 0. Both P and P matrix elements can be considered as free parameters, for instance
L R
in Fig. 3 we use P = (1/2)P (BM). Of course, there is also a solution if we use the bag
L L
model value of P .
L
V. INDIRECT CP VIOLATION
The contributing diagrams for the ǫ parameter are of two types, one with the exchange
of two Y++ and the other with one U++ and one Y++. They are shown in the Figs. 2a and
8
Y++
d s¯
J J
g
FIG. 1: Dominant CP violating penguin diagram contributing to the decay K0 ππ.
→
2b, respectively. The imaginary part for this class of diagrams has been derived in Refs. [16]
and [17]. The Higgs scalar-quark interaction is given in Eq.(6) and the gauge boson-quark
Lagrangian interaction is
g
= J¯( d) γµLd U++ +H.c. (18)
LW −√2 OL 3α α µ
The contributions to the effective Lagrangian of diagrams like that shown in Fig. 2a are
given by
C2 2m2 m2 sin4θ m2
Im YY = ds K s χ[(s¯Ld)2 d(s¯Rd)2] g (x)
Lǫ (4π)2 N2 N2 m2 − m2 0
(cid:26)(cid:20) K s (cid:21)
v ↔ m 3
| ρ| sin2θ s¯γµLi ∂ ds¯ L dR d 5g (x)+ xg′(x)
− 4 v χ µ − m 0 2 0
| χ| (cid:18)(cid:20) (cid:18) s (cid:19) (cid:21)(cid:20) (cid:21)
m ↔ 3
+ s¯ L dR i ∂ ds¯γµLd g (x)+ xg′(x) , (19)
− m µ 0 2 0
(cid:18) s (cid:19) (cid:20) (cid:21)(cid:19)(cid:27)
where g (x) is given in the Appendix.
0
On the other hand, the contributions to the effective Lagrangian of diagrams like that
9
shown in Fig. 2.b are given by
C2 2m2 m2 g2 N2 sin2θ m ↔
Im UY = ds K s χ s¯γµγν L dR i ∂ d (s¯γ Ld) E (x,y)
Lǫ (4π)2 N2 N2 2 4m2 m m2 − m µ ν 1
(cid:18) J(cid:19) s K (cid:26)(cid:20) (cid:18) s (cid:19) (cid:21)
↔ m m ↔
+ s¯γ Li ∂ ds¯γµγν L dR dE (x,y)+ s¯ L dR i ∂ d(s¯γµLd)
ν µ 2 µ
− m − m
(cid:18) s (cid:19) (cid:20) (cid:18) s (cid:19)
↔ m m
+ s¯γµLi ∂ ds¯ L dR d E (x,y) i∂ s¯γµγν L dR d s¯γ Ld
µ 3 µ ν
− m − − m
(cid:18) s (cid:19) (cid:21) (cid:20) (cid:20) (cid:18) s (cid:19) (cid:21)
m m
+ i∂ (s¯γ Ld)s¯γµγν L d d E (x,y) i∂ s¯ L dR d s¯γµLd
µ ν 4 µ
− m − − m
(cid:18) s(cid:19) (cid:21) (cid:18) (cid:20) (cid:18) s (cid:19) (cid:21)
m
+ i∂ (s¯γµLd)s¯ L dR d E (x,y) , (20)
µ 5
− m
(cid:18) s (cid:19) (cid:19) (cid:27)
where y = m2/m2 and the functions E are defined in the Appendix.
U J 1,2,3,4,5
Taking into account both contributions in Eqs.(19) and (20) and using
Im K¯0 (0) K0
ǫ
ImM = h |L | i, (21)
12
2m
K
we obtain
C2 m2 f2 m −2 5 m2
ImM = ds K K 1+ d sin4θ 1 d g (x)
12 −(4π)2 N2 2N2 m 6 χ − m2 0
(cid:18) s(cid:19) (cid:26) (cid:18) s(cid:19)
v 5 3
| ρ| sin2θ 5g (x) g′(x)
− 2 v χ 12 0 − 2 0
| χ| (cid:20) (cid:18) (cid:19)
2
1 1 m +m 3
1+ s d g (x)+ xg′(x)
− 3 4 m 0 2 0
" (cid:18) K (cid:19) #(cid:18) (cid:19)#
g2 N2 2 2 m +m 2
+ sin2θ (E′(x,y)+E (x,y)) s d [E (x,y)+E (x,y)]
2 2m2 χ 3 1 3 − 3 m 1 4
J " (cid:18) K (cid:19)
2
1 m +m
s d
+ 1 (E (x,y)+E (x,y)) . (22)
2 4
12 − m
" (cid:18) K (cid:19) # #)
Thus, we can calculate ǫ from Eq. (11a) using f = 161.8 MeV and ∆M = 3.5 10−15
K K
| | ×
10