Flavor Structure and Supersymmetric CP-Violation

15 Pages·1.5 MB·English

by  A. Masiero

SISSA/153/99/EP
Flavor Structure and Supersymmetric
CP-Violation
0
0
0 A. Masiero and O. Vives
2

n SISSA – ISAS, via Beirut 2-4, 34013, Trieste, Italy and
a INFN, Sezione di Trieste, Trieste, Italy
J

8
2

1 Abstract
v
8 Inthistalk,weaddressthepossibilityoffindingsupersymmetrythrough
9 indirect searches in theK and B systems. Weprovethat,in theabsence
2 oftheCabibbo–Kobayashi–Maskawaphase,ageneralMinimalSupersym-
1 metricStandardModelwithallpossiblephasesinthesoft–breakingterms,
0 butnonewflavorstructurebeyondtheusualYukawamatrices,cannever
0 give a sizeable contribution to εK, ε′/ε or hadronic B0 CP asymmetries.
0
However, Minimal Supersymmetric models with additional flavor struc-
/
h tures in the soft–supersymmetry breaking terms can produce large devi-
p ations from the Standard Model predictions. Hence, observation of su-
- persymmetriccontributionstoCPasymmetriesinBdecayswouldbethe
p
first sign of the existence of new flavor structures in the soft–terms and
e
h would hint at a non–flavor blind mechanism of supersymmetry breaking.
:
v
i 1 Introduction
X
r
a Beginning with its experimental discovery in K–meson decays, about three
decades ago, the origin of CP violation has been one of the most intriguing
questions in particle phenomenology. Notably, the subsequent experiments in
the searchfor electric dipole moments (EDM) of the neutron and electronhave
observed no sign of new CP–violating effects despite their considerably high
precision. Hence, the neutral K–system remains, so far, the only experimental
information on the presence of CP–violation in nature.
In the near future, this situation will change. Not only the new B factories
will start measuring CP violation effects in B0 CP asymmetries, but also the
experimental sensitivity to the electric dipole moment of the neutron and the
electronwillbesubstantiallyimproved. Thesenewexperimentswillenlargeour
knowledge of CP violation phenomena and, hopefully, will show the existence
of new sources of CP violation from models beyond the Standard Model (SM).
The Standard Model of electroweak interactions is known to be able to ac-
commodate the experimentally observedCP–violationthrough a unique phase,
δ , in the Cabibbo–Kobayashi–Maskawa mixing matrix (CKM). However,
CKM
mostoftheextensionsoftheSMincludenewobservablephasesthatmaysignif-
icantlymodifythepatternofCPviolation. Supersymmetryis,withoutadoubt,
one of the most popular extensions of the SM. Indeed, in the minimal super-
symmetric extensionofthe SM (MSSM), there areadditionalphases whichcan
cause deviations from the predictions of the SM. After all possible rephasings
of the parameters and fields, there remain at least two new physical phases in
the MSSM Lagrangian. These phases can be chosen to be the phases of the
Higgsino Dirac mass parameter (ϕ = Arg[µ]) and the trilinear sfermion cou-
µ
pling to the Higgs, (ϕ = Arg[A ]) [1]. In fact, in the so–called Constrained
A0 0
MinimalSupersymmetricStandardModel(CMSSM),withstrictuniversalityat
the Grand Unification scale, these are the only new phases present.
It was soon realized, that for most of the MSSM parameter space, the ex-
perimental bounds on the electric dipole moments of the electron and neutron
constrained ϕ to be at most (10−2). Consequently these new supersym-
A0,µ O
metricphaseshavebeentakentovanishexactlyinmoststudiesintheframework
of the MSSM.
However, in the last few years, the possibility of having non–zero SUSY
phases has again attracted a great deal of attention. Several new mechanisms
have been proposed to suppress supersymmetric contributions to EDMs below
the experimental bounds while allowing SUSY phases (1). Methods of sup-
O
pressingtheEDMsconsistofcancellationofvariousSUSYcontributionsamong
themselves [2], non universality of the soft breaking parameters at the unifica-
tion scale [3] and approximately degenerate heavy sfermions for the first two
generations [4]. In the presence of one of these mechanisms, large supersym-
metric phases are naturally expected and EDMs should be generally close to
the experimental bounds.
In this work we will study the effects of these phases in CP violation ob-
servables as ε , ε′/ε and B0 CP asymmetries. We will show that the presence
K
of large susy phases is not enough to produce sizeable supersymmetric con-
tributions to these observables. In fact, in the absence of the CKM phase, a
general MSSM with all possible phases in the soft–breaking terms, but no new
flavor structure beyond the usual Yukawa matrices, can never give a sizeable
contribution to ε , ε′/ε or hadronic B0 CP asymmetries. However,as recently
K
emphasized[5,3],assoonasoneintroducessomenewflavorstructureinthesoft
Susy–breaking sector, even if the CP violating phases are flavor independent,
it is indeed possible to get sizeable CP contribution for large Susy phases and
δ = 0. Then, we can rephrase our sentence above in a different way: A
CKM
new result in hadronic B0 CP asymmetries in the framework of supersymmetry
would be a direct prove of the existence of a completely new flavor structure in
thesoft–breakingterms. ThismeansthatB–factorieswillprobetheflavorstruc-
ture ofthe supersymmetrysoft–breakingtermsevenbeforethe directdiscovery
of the supersymmetric partners [6].
2
2 Soft–breaking flavor structure
As announced in the introduction, the presence of new flavor structure in the
soft–breaking terms is necessary to obtain sizeable contributions to flavor–
changing CP observables (i.e. ε , ε′/ε and hadronic B0 CP asymmetries).
K
To prove this we will consider any MSSM, i.e. with the minimal supersym-
metric particle content, with generalcomplex soft–breakingterms, but with a
flavorstructurestrictlygivenbythetwofamiliarYukawamatricesoranymatrix
strictly proportional to them. In these conditions, the most general structure
of the soft–breaking terms at the large scale, that we call M , is,
GUT
(m2) =m2 δ (m2) =m2 δ
Q ij Q ij U ij U ij
(m2 ) =m2 δ (m2) =m2 δ
D ij D ij L ij L ij
(m2) =m2 δ m2 m2
E ij E ij H1 H2
m eiϕ3 m eiϕ2 m eiϕ1
g˜ W˜ B˜
(AU)ij =AU eiϕAU (YU)ij
(AD)ij =AD eiϕAD (YD)ij
(AE)ij =AE eiϕAE (YE)ij. (1)
where all the allowed phases are explicitly written and one of them can be
removed by an R–rotation. All other numbers or matrices in this equation are
alwaysreal. Noticethatthisstructurecovers,notonlytheCMSSM[7],butalso
mostofTypeIstringmotivatedmodelsconsideredsofar[8,9],gaugemediated
models [10], minimal effective supersymmetry models [11, 12], etc.
Experiments of CP violation in the K or B systems only involve supersym-
metric particles as virtual particles in the loops. This means that the phases in
thesoft–breakingtermscanonlyappearintheseexperimentsthroughthe mass
matrices of the susy particles. Then, the key point in our discussionwill be the
role played by the susy phases and the soft–breaking terms flavor structure in
the low–energy sparticle mass matrices.
It is important to notice that, even in a model with flavor–universal soft–
breakingtermsatsomehighenergyscale,asthisisthecase,someoff–diagonality
in the squark mass matrices appears at the electroweak scale. Working on the
basis where the squarks are rotated parallel to the quarks, the so–called Super
CKM basis (SCKM), the squark mass matrix is not flavor diagonal at M .
W
This is due to the fact that at M there are always two non-trivial flavor
GUT
structures, namely the two Yukawa matrices for the up and down quarks, not
simultaneously diagonalizable. This implies that through RGE evolution some
flavor mixing leaks into the sfermion mass matrices. In a general Supersym-
metric model, the presence of new flavor structures in the soft breaking terms
would generate large flavor mixing in the sfermion mass matrices. However,
in the CMSSM, the two Yukawa matrices are the only source of flavor change.
Always in the SCKM basis, any off-diagonal entry in the sfermion mass matri-
ces at M will be necessarily proportional to a product of Yukawa couplings.
W
Then, a typical estimate for the element (i,j) in the L–L down squark mass
3
matrix at the electroweak scale would necessarily be (see [7] for details),
(m2(D)) c m2 YuYu∗, (2)
LL ij ≈ Q ik jk
withcaproportionalityfactorbetween0.1and1. Thisroughestimateprovides
the orderofmagnitude ofthe differententriesinthe sfermionmassmatrices. It
isimportanttonoticethatifthephasesoftheseelementswere (1),duetosome
O
of the phases in equation (1), we would be able to give sizeable contributions,
or even saturate, the different CP observables [13]. Then, it is clear that the
relevantquestionforCPviolationexperimentsisthepresenceofimaginaryparts
in these off–diagonal entries.
As explained in [7, 14], once we have solved the Yukawa RGEs, the RGE
equations of all soft–breaking terms are a set of linear differential equations.
Then, they can be solved as a linear function of the initial conditions,
m2(M )= η(φi)m2 + η(gi) m2
Q W Pi Q φi Pi Q gi
+Pi6=j(cid:16)ηQ(gij)eiϕij +ηQ(gij)Te−iϕij(cid:17)mgimgj
+Pij(cid:16)ηQ(gAij)eiϕiAj +ηQ(gAij)Te−iϕiAj(cid:17)mgiAj
+Pi6=j(cid:16)ηQ(Aij)eiϕAiAj +ηQ(Aij)Te−iϕAiAj(cid:17)AiAj
+ η(Ai)A2, (3)
Pi Q i
where φ refers to any scalar, g to the different gauginos, A to any tri–linear
i i i
coupling and the phases ϕ = (ϕ ϕ ). In this equation, the different η
ab a b
−
matrices are 3 3 matrices, strictly real and all the allowedphases have been
×
explicitly written. Regardingthe imaginaryparts,due to the hermiticity of the
sfermion mass matrices, any imaginary part will always be associated to the
non–symmetric part of the η(gigj), η(AiAj) or η(giAj) matrices. To estimate the
Q Q Q
size of these anti–symmetric parts, we can go to the RGE equations for the
scalar mass matrices, where we use the same conventions and notation as in
[7, 14]. Taking advantage of the linearity of these equations we can directly
write the evolution of the anti–symmetric parts, mˆ2 =m2 (m2)T as,
Q Q− Q
dmˆ2
Q = 1 (Y˜ Y˜† + Y˜ Y˜†) mˆ2
dt −(cid:16)2 U U D D Q
+ 1 mˆ2 (Y˜ Y˜† + Y˜ Y˜†)
2 Q U U D D
+ Y˜ mˆ2 Y˜† + Y˜ mˆ2 Y˜†
U U U D D D
+ 2 i A˜ A˜† + A˜ A˜† , (4)
ℑ{ U U D D}(cid:17)
where, due to the reality of Yukawa matrices, we have used YT = Y†, and
following [14] a tilde over the couplings (Y˜, A˜, ...) denotes a re–scaling by a
factor 1/(4π). In the evolution of the R–R squark mass matrices, m2 and m2 ,
U D
onlyoneofthe twoYukawamatrices,the onewithequalisospintothe squarks,
4
is directly involved. Then, it is easy to understand that these matrices are in a
very good approximation diagonal in the SCKM basis once you start with the
initial conditions given in equation (1). Hence, for the sake of clarity, we can
safely neglect the last two terms in equation (4) and forgetabout mˆ2 and mˆ2 .
U D
However, if needed, we could always apply to estimate their anti–symmetric
parts an analogous reasoning as the one we show below to mˆ2.
Q
From equation (1), the initial conditions for mˆ2 at M are identically
Q GUT
zero. This means that the only source for mˆ2 in equation (4) is necessarily
Q
A A† +A A† .
ℑ{ U U D D}
The next step is then to analyze the RGE for the tri–linear couplings,
dA˜U = 1 16 α˜ +3 α˜ + 1 α˜ A˜
dt 2 (cid:16) 3 3 2 9 1(cid:17) U
16 α˜ M +3 α˜ M + 1 α˜ M Y˜
− (cid:16) 3 3 3 2 2 9 1 1(cid:17) U
2 A˜ Y˜†Y˜ +3 Tr(A˜ Y˜†)Y˜
− (cid:16) U U U U U U
+ 5 Y˜ Y˜†A˜ + 3 Tr(Y˜ Y˜†)A˜
2 U U U 2 U U U
+ A˜ Y˜†Y˜ + 1 Y˜ Y˜†A˜ (5)
D D U 2 D D U(cid:17)
with an equivalent equation for A . With the general initial conditions in
D
equation (1), A is complex at any scale. However, we are interested in the
U
imaginary parts of A A† . At M this combination is exactly real, but, due
U U GUT
to different renormalization of different elements of the matrix, this is not true
any more at a different scale.
However, a careful analysis of equation (5) is enough to convince ourselves
that these imaginary parts are extremely small. Let us, for a moment, neglect
the terms involving A˜ Y˜† or Y˜ Y˜† from the above equation. Then, the only
D D D D
flavor structure appearing in equation (5) at M is Y . We can always go
GUT U
to the basis where Y is diagonal and then we will have A exactly diagonal
U U
at any scale. In particular this means that A A† would always exactly
ℑ{ U U}
vanish. A completely parallel reasoning can be applied to A and A A† .
D ℑ{ D D}
Hence, simply taking into account the flavor structure, our conclusion is that,
necessarily, any non–vanishing element of [A A† +A A† ] and hence of mˆ2
ℑ U U D D Q
must be proportional to (Y˜ Y˜†Y˜ Y˜† H.C.). So, we can expect them to be,
D D U U −
(mˆ2) K Y Y†Y Y† H.C.
Q i<j ≈ (cid:16) D D U U − (cid:17)i<j
(mˆ2) Kcos−2β (h h λ5)
Q 12 ≈ s t
(mˆ2) Kcos−2β (h h λ3)
Q 13 ≈ b t
(mˆ2) Kcos−2β (h h λ2), (6)
Q 23 ≈ b t
where h = m2/v2, with v = v2+v2 the vacuum expectation value of the
i i p 1 2
Higgs, λ = sinθ and K is a proportionality constant that includes the effects
c
5
of the running from M to M . To estimate this constant we have to
GUT W
keep in mind that the imaginary parts of A A† are generated through the
U U
RGE running and then these imaginary parts generate mˆ2 as a second order
Q
effect. This means that roughly K (10−2) times a combination of initial
≃ O
conditions as in equation (3). So, we estimate these matrix elements to be
(cos−2β 10−12,6 10−8,3 10−7 ) times initial conditions. This was exactly
{ × × }
the result we found for the A–g terms in [7] in the framework of the CMSSM.
In fact, now it is clear that this is the same for all the terms in equation (3),
g –A , g –g and A –A , irrespectively of the presence of an arbitrary number
i j i j i j
of new phases.
As we have already said before, the situation in the R–R matrices is still
worse because the RGE of these matrices involves only the corresponding Yu-
kawa matrix and hence, in the SCKM, they are always diagonal and real in
extremely good approximation.
Hence, so far, we have shown that the L–L or R–R squark mass matrices
are still essentially real. The only complex matrices, then, will still be the L–R
matrices that include, from the very beginning, the phases ϕ and ϕ . Once
Ai µ
more,the sizeofthese entriesisdeterminedby the Yukawaelements withthese
two phases providing the complex structure. However, this situation is not
new for these more general MSSM models and it was already present even in
the CMSSM. We can conclude, then, that the structure of the sfermion mass
matrices at M is not modified from the familiar structure already present in
W
the CMSSM, irrespective of the presence of an arbitrary number of new susy
phases.
In the next sectionwe analyzethe different indirect anddirect CP violation
observables in this general MSSM without new flavor structure.
3 CP observables
3.1 Indirect CP violation
In first place, we will consider indirect CP violation both in the K and B
systems. In the SM neutral meson mixing arises at one loop through the well–
knownW–box. However,inthe MSSM,there arenewcontributionsto ∆F =2
processescoming fromboxes mediatedby supersymmetricparticles. These are:
chargedHiggsboxes(H±),charginoboxes(χ±)andgluino-neutralinoboxes(g˜,
χ0). – ¯ mixing is correctlydescribedby the ∆F =2 effective Hamiltonian,
M M
∆F=2, which can be decomposed as,
Heff
G2M2
∆F=2 = F W (V∗V )2 (C (µ) Q (µ)
Heff − (2π)2 td tq 1 1
+ C (µ) Q (µ) + C (µ) Q (µ)). (7)
2 2 3 3
With the relevant four–fermion operators given by
Q = d¯αγµqα d¯βγ qβ,
1 L L· L µ L
6
Q = d¯αqα d¯βqβ,
2 L R· L R
Q = d¯αqβ d¯βqα, (8)
3 L R· L R
whereq =s,bfortheK andB–systemsrespectivelyandα,βascolorindices. In
theCMSSM,thesearetheonlythreeoperatorspresentinthelimitofvanishing
m . The Wilson coefficients, C (µ), C (µ) and C (µ), receive contributions
d 1 2 3
from the different supersymmetric boxes,
C (M ) = CW(M )+CH(M ) (9)
1 W 1 W 1 W
+ Cg˜,χ0(M )+Cχ(M )
1 W 1 W
C (M ) = CH(M )+Cg˜(M )
2 W 2 W 2 W
C (M ) = Cg˜,χ0(M )+Cχ(M )
3 W 3 W 3 W
Both, the usual SM W–box and the charged Higgs box contribute to these
operators. However, with δ = 0, these contributions do not contain any
CKM
complex phase and hence cannot generate an imaginary part for these Wilson
coefficients.
Gluino and neutralino contributions are specifically supersymmetric. They
involvethesuperpartnersofquarksandgaugebosons. Here,thesourceofflavor
mixing is not directly the usual CKM matrix. It is the presence of off–diagonal
elements in the sfermion mass matrices, as discussed in section 2. From the
pointofviewofCPviolation,wewillalwaysneeda complexWilsoncoefficient.
In the SCKM basis all gluino vertices are flavor diagonal and real. This means
thatinthe massinsertion(MI) approximation,we needa complexMI in oneof
the sfermion lines. Only L–L mass insertions enter at first order in the Wilson
coefficientC g˜,χ0(M ). Fromequation(6),theimaginarypartsoftheseMIare
1 W
at most (10−6) for the b–s transitions and smaller otherwise [7]. Comparing
O
these values with the phenomenological bounds required to saturate the mea-
sured values of these processes [13] we can easily see that in this model we are
always several orders of magnitude below.
In the case of the Wilson coefficients C g˜(M ) and C g˜(M ), the MI are
2 W 3 W
L–R. However these MI are always suppressed by light masses of right handed
squark, or in the case of b–s transitions directly constrained by the b sγ
→
decay. Hence, gluino boxes, in the absence of new flavor structures, can never
give sizeable contributions to indirect CP violation processes [7].
The chargino contributions to these Wilson coefficients were discussed in
great detail in the CMSSM framework in reference [7]. In this more general
MSSM, as we have explained in section 2, we find very similar results due to
the absence of new flavor structure.
Basically, in the chargino boxes, flavor mixing comes explicitly from the
CKM mixing matrix, although off–diagonality in the sfermion mass matrix in-
troduces a small additional source of flavor mixing.
Cχ(M )= 2 6
1 W Pi,j=1Pk,l=1Pαγα′γ′
7
C)1 2
m(
30 I 1.5
1
1
0.5
0
-0.5
-1
-1.5
-2
-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0
Re(C)
1
Figure 1: Imaginary and Real parts of the Wilson coefficient Cχ in B mixing.
1
V∗ V V∗ V
α′d αq γ′d γq G(α,k)iG(α′,k)j∗
(V∗V )2
td tq
G(γ′,l)i∗G(γ,l)j Y (z ,z ,s ,s ) (10)
1 k l i j
whereV G(α,k)i representthecouplingofcharginoandsquarkktoleft–handed
αq
down quark q, z =M2 /M2 and s =M2 /M2 . The explicit expressions for
k u˜k W i χ˜i W
these couplings and loop functions can be found in reference [7]. G(α,k)i are
in general complex, as both ϕ and ϕ are present in the different mixing
µ Ai
matrices.
The main part of Cχ in equation (10) will be given by pure CKM flavor
1
mixing, neglecting the additional flavor mixing in the squark mass matrix [15,
16]. This means, α = α′ and γ = γ′. In these conditions, using the symmetry
of loopfunction Y (a,b,c,d) under the exchangeof any two indices it is easy to
1
prove that Cχ would be exactly real [11]. This is not exactly true either in the
1
CMSSM or in our more general MSSM, where there is additional flavor change
inthe sfermionmassmatrices. Here,some imaginarypartsappearin the Cχ in
1
equation (10). In figure 1 we show in a scatter plot the size of imaginary and
real parts of Cχ in the B system for a fixed value of tanβ = 40. We see that
1
this Wilsoncoefficientisalwaysrealuptoapartin103. Inanycase,this isout
of reach for the foreseen B–factories.
Finally, chargino boxes contribute also to the quirality changing Wilson co-
8
efficient Cχ(M ),
3 W
Cχ(M )= 2 6
3 W Pi,j=1Pk,l=1Pαγα′γ′
V∗ V V∗ V m2
α′d αq γ′d γq q
(V∗V )2 2M2 cos2β
td tq W
H(α,k)iG(α′,k)j∗G(γ′,l)i∗H(γ,l)j
Y (z ,z ,s ,s ) (11)
2 k l i j
wherem /(√2M cosβ) V H(α,k)i isthecouplingofcharginoandsquarkto
q W αq
theright–handeddownqu·arkq· [7]. UnliketheCχ Wilsoncoefficient,duetothe
1
differences between H and G couplings, Cχ is complex even in the absence of
3
intergenerationalmixinginthesfermionmassmatrices[11]. Then,thepresence
of flavor violating entries in the up–squark mass matrix hardly modifies the
results obtained in their absence [15, 16, 7]. In fact, in spite the presence
of the Yukawa coupling squared, m2/(2M2 cos2β), this contribution could be
q W
relevant in the large tanβ regime. For instance, in B0–B¯0 mixing we have
m2/(2M2 cos2β)thatfortanβ > 25islargerthan1andso,itisnotsuppressed
b W
at all when compared with the∼Cχ Wilson Coefficient. This means that this
1
contribution can be very important in the large tanβ regime [11] and could
have observable effects in CP violation experiments in the new B–factories.
However,we will show next, that when we include the constraints coming from
b sγ these chargino contributions are also reduced to an unobservable level.
→
Thecharginocontributestotheb sγdecaythroughtheWilsoncoefficients
→
and , corresponding to the photon and gluon dipole penguins respectively
7 8
C C
[14, 17, 7]. In the large tanβ regime, we can approximate these Wilson coeffi-
cients as [7],
χ±(M )= 6 2 VαbVβ∗s
C7 W Pk=1Pi=1Pα,β=u,c,t VtbVt∗s
mb H(α,k)iG∗(β,k)i Mχi F7(z ,s )
√2M cosβ mb R k i
W
χ±(M )= 6 2 VαbVβ∗s
C8 W Pk=1Pi=1Pα,β=u,c,t VtbVt∗s
mb H(α,k)iG∗(β,k)i Mχi F8(z ,s )
√2M cosβ mb R k i
W
(12)
Now, if we compare the chargino contributions to these Wilson coefficients
andtothecoefficientC ,equations(11)and(12),wecanseethattheyaredeeply
3
related. In fact, in the approximation where the two different loop functions
involved are of the same order, we have,
m2
C (M ) ( (M ))2 q (13)
3 W ≈ C7 W M2
W
9
C)71
m(
I0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5
Re(C)
7
Figure 2: Experimental constraints on the Wilson Coefficient
7
C
In figure 2, we show a scatter plot of the allowed values of Re( ) versus
7
C
Im( ) in the CMSSM for a fixed value of tanβ = 40 [7] with the constraints
7
C
fromthe decayB X γ takenfromthe reference[18]. Notice that arelatively
s
→
large value of tanβ, for example, tanβ > 10, is needed to compensate the W
andchargedHiggscontributionsandcove∼rthewholeallowedareawithpositive
and negative values. However, the shape of the plot is clearly independent of
tanβ, only the number of allowed points and its location in the allowed area
depend on the value considered. Then, figure 3 shows the allowed values for a
re–scaled Wilson coefficient C¯ (M )= M2 /m2C (M ) corresponding to the
3 W W q 3 W
same allowed points of the susy parameter space in figure 2. As we anticipated
previously, the allowed values for C¯ are close to the square of the values of
3 7
C
in figure 2 slightly scaled by different values of the loop functions.
We can immediately translate this result to a constraint on the size of the
chargino contributions to ε .
M
ε = G2FMW2 (VtdVtq)2
M 4π2√2 ∆M 24
M
M2
F2 M M
M M m2(µ)+m2(µ)
q d
η (µ) B (µ) Im[C ] (14)
3 3 3
In this expression M , ∆M and F denote the mass, mass difference and
M M M
decay constant of the neutral meson 0. The coefficient η (µ) = 2.93 [19]
3
M
10