B decays and models for CP violation

20 Pages·0.17 MB·

by  Xiao-Gang He

UM-P-95/117, OITS-596
B Decays And Models For CP Violation
Xiao-Gang He
Research Center for High Energy Physics
School of Physics, University of Melbourne
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9
Parkville, Vic 3052, Australia
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and
n
a
J Institute of Theoretical Science

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University of Oregon

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v
Eugene, OR 97403-5203, USA
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2 (December 1995)
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0
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9
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h Abstract
p
-
p
e ThedecaymodesB toππ,ψK ,K−D,πK andηK arepromisingchannelsto
S
h
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v study the unitarity triangle of the CP violating CKM matrix. In this paper I
i
X
studytheconsequences ofthesemeasurementsintheWeinbergmodel. Ishow
r
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that using the same set of measurements, the following different mechanisms
for CP violation can be distinguished: 1) CP is violated in the CKM sector
only; 2) CP is violated spontaneously in the Higgs sector only; And 3) CP is
violated in both the CKM and Higgs sectors.
Typeset using REVTEX
1
I. INTRODUCTION
CP violation is one of the unresolved mysteries in particle physics. The explanation in
the Standard Model (SM) based on Cabibbo-Kobayashi-Maskawa (CKM) matrix [1] is still
not established, although there is no conflict between the observation of CP violation in the
neutral K-system [2] and theory [3], intriguing hints of other plausible explanations emerge
from consideration of baryon asymmetry of the universe [4]. Models based on additional
Higgs bosons [5,6] can equally well explain the existing laboratory data [7] and provide
large CP violation required from baryon asymmetry [4]. It is important to carry out more
experiments tofindout theoriginofCPviolation. Itisforthisreasonthat explorationofCP
violationinB systemissocrucial. TheB system offersseveralfinalstatesthatprovidearich
source for the study of this phenomena [8]. Several methods using B decay modes have been
∗ ∗ ∗ ∗
proposed to measure the phase angles, α = Arg( V V /V V ), β = Arg( V V /V V )
− td tb ub ud − cd cb tb td
∗ ∗
and γ = Arg( V V /V V ) in the unitarity triangle of the CKM matrix [9–14]. It has
− ud ub cb cd
been shown that B¯0(B−) π+π−,π0π0(π−π0) [11], B¯0 ψK [12] and B− K−D [13]
S
→ → →
decays can be used to determine α, β and γ, respectively. Recently it has been shown that
B− π−K¯0, π0K−, ηK− and B− π−π0 can also be used to determine γ [14]. If the
→ →
sum of these three angles is 1800, the SM is a good model for CP violation. Otherwise
new mechanism for CP violation is needed. In this paper I study the consequences of these
measurements in the Weinberg model.
In the Weinberg model CP can be violated in the CKM sector and Higgs sector. If
CP is violated spontaneously, it occurs in the Higgs sector only. I will call the model with
CP violation in both the CKM and Higgs sectors as WM-I, and the model with CP viola-
tion only in the Higgs sector as WM-II. There are many ways to distinguish the SM and
Weinberg model for CP violation. For example the neutron electric dipole moment in the
Weinberg model is several orders of magnitude larger than the SM prediction [15]. However,
the neutron electric dipole moment measurement alone can not distinguish the WM-I from
the WM-II. I show that measurements of CP violation in B decays not only can be used to
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distinguish the SM from the Weinberg model, but can also be used to determine whether
CP is violated in the Higgs sector only or in both the CKM and Higgs sectors.
B Decay Amplitudes In The SM
CP violation in the SM is due to the phase in the CKM mixing matrix in the charged
current interaction,
g
L = U¯γµ(1 γ )V DW+ +H.C. , (1)
−2√2 − 5 KM µ
where U = (u, c, t), and D = (d, s, b). V is the CKM matrix. For three generations, it
KM
is a 3 3 unitary matrix. It has three rotation angles and one non-removable phase which
×
is the source of CP violation in the SM. I will use the Maiani, Wolfenstein and Chau-Keung
∗ ∗
[16] convention for the CKM matrix, in which V has the phase γ, and V has the phase β
ub td
and other CKM elements have no or very small phases.
The effective Hamiltonian responsible for ∆C = 0 hadronic B ππ, πK, ηK ,ψK
S
→
decaysatthequarkleveltotheonelooplevelinelectroweakinteractioncanbeparameterized
as,
G
H = F[V V∗(c Oq +c Oq )+V V∗(c Oq +c Oq )
eff √2 ub uq 1 1u 2 2u cb cq 1 1c 2 2c
12
[V V∗cu +V V∗cc +V V∗ct]Oq , (2)
− ub uq i cb cq i tb tq i i
i=3
X
where cf (f = u, c, t) are Wilson Coefficients (WC) of the corresponding quark and gluon
i
operators Oq. The superscript f indicates the internal quarks. q can be d or s quark
i
depending on if the decay is a ∆S = 0 or ∆S = 1 process. The operators Oq are defined
− i
as
Oq = q¯ γ Lf f¯ γµLb , Oq = q¯γ Lff¯γµLb,
1f α µ β β α 2f µ
O = q¯γ LbΣq¯′γµL(R)q′, O = q¯ γ Lb Σq¯′γµL(R)q′ ,
3(5) µ 4(6) α µ β β α
3 3
O7(9) = 2q¯γµLbΣeq′q¯′γµR(L)q′, O8(10) = 2q¯αγµLbβΣeq′q¯β′γµR(L)qα′ ,
g e
O = s m q¯σ RT bGµν , Q = m q¯σ RbFµν , (3)
11 32π2 b µν a a 12 32π2 b µν
3
′
where L(R) = (1 γ ), and q is summed over u, d, s, and c quarks. The subscripts
5
∓
α and β are the color indices. Ta is the color SU(3) generator with the normalization
Tr(TaTb) = δab/2. Gµν and F are the gluon and photon field strengths, respectively.
a µν
O1, O2 are the tree level and QCD corrected operators. O3−6 are the gluon induced strong
penguin operators. O7−10 are the electroweak penguin operators due to γ and Z exchange,
and “box” diagrams at loop level. The WC’s c1−10 have been evaluated at the next-to-
leading-log QCD corrections [17]. The operators O are the dipole penguin operators.
11,12
Their WC’s have been evaluated at the leading order in QCD correction [18], and their
phenomenological implications in B decays have also been studied [19].
One can generically parameterize the decay amplitude of B as
A¯ =< final state Hq B >= V V∗T(q)f +V V∗P(q) , (4)
SM | eff| fb fq tb tq
where T(q) contains the tree and penguin due to internal u and c quark contributions, while
P(q) contains penguin contributions from internal t and c or u quarks. I use A¯ for the decay
¯
amplitude of B meson containing a b quark, and A for a B meson containing a b quark. The
WC’s involved in T are much larger than the ones in P. One expects the hadronic matrix
elements arising from quark operators to be the same order of magnitudes. The relative
strength of the amplitudes T and P are predominantly determined by their corresponding
WC’s in the effective Hamiltonian. In general P , if not zero, is about or less than 10% of
| |
T .
| |
For B¯0 ψK , the decay amplitude can be written as
S
→
¯ ∗ ∗
A (ψK) = V V T +V V P
SM cb cs ψK tb ts ψK
= V V (T P )+ V V∗ e−iγP . (5)
| cb cs| ψK − ψK | ub us| ψK
The second term is about 103 times smaller than the first term and can be safely neglected.
To this level, the decay amplitude for B¯0 ψK does not contain weak CP violating phase.
S
→
This decay mode provides a clean way to measure the phase angle β in the SM [12].
One can parameterize the decay amplitudes for B ππ , Kπ , ηK in a similar way.
→
Further if flavor SU(3) symmetry is a good symmetry there are certain relations among
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the decay amplitudes [20]. I will assume the validity of the SU(3) symmetry in my later
analysis. The operators Qu1,2, O1c,2, O3−6,11,12, and O7−10 transform under SU(3) symmetry
as ¯3 +¯3 +6+15, ¯3, ¯3, and ¯3 +¯3 +6+15, respectively. Flavor SU(3) symmetry predicts
a b a b
√2A¯(π0π0)+√2A¯(π−π0) = A¯(π+π−) , (6)
√2A¯(π0K−) 2A¯(π−K¯0) = √6A¯(η K−) . (7)
8
−
Isospin symmetry also imply eq.(6). These relations form two triangles in the complex plan
which provide important information for obtaining phase angles α and γ [11,14].
I parameterize the decay amplitudes in the SM as
A¯SM(π−π0) = |VubVu∗d|e−iγTπ−π0 +|VtbVt∗d|eiβPπ−π0 ,
A¯SM(π+π−) = |VubVu∗d|e−iγTπ+π− +|VtbVt∗d|eiβPπ+π− ,
A¯SM(K−π0) = |VubVu∗s|e−iγTK−π0 +|VtbVt∗s|PK−π0 ,
A¯ (K¯0π−) = V V∗ e−iγT + V V∗ P , (8)
SM | ub us| K¯0π− | tb ts| K¯0π−
The decay amplitudes A¯ (π0π0) and A¯ (K−η) are obtained by the SU(3) relations in
SM SM
eqs.(6) and (7).
I would like to point out that A¯ (π−π0) and √2A¯(K−π0) A¯(K¯0π−) only receive
SM
−
contributions from the effective operators which transform as 15 [14,22],
A¯ (π−π0) = V V∗CT +V V∗CP ,
SM ub ud 15 tb td 15
A¯ (K−π0) A¯ (K¯0π−)/√2 = V V∗CT +V V∗CP , (9)
SM − SM ub us 15 tb ts 15
whereC istheinvariantamplitudeduetooperatorstransformas15underSU(3)symmetry.
15
Thisisanimportantpropertyusefulformylaterdiscussions. ThesecondterminA¯ (π−π0)
SM
is less than 3% of the first term [21]. For all practical purposes it can be neglected. However,
the second term on the right hand side of the second equation in eq.(9) can not be neglected
∗ ∗
because there is an enhancement factor V V / V V which is about 50 [22].
| tb ts| | ub us|
The effective Hamiltonian responsible for B DK decay is given by
→
5
G
H = F[V V∗(c u¯αγ Lb s¯βγµLc +c u¯γ Lbs¯γµLc)
eff √2 ub cs 1 µ β α 2 µ
+ V V∗(c c¯αγ Lb s¯βγµLu +c c¯γ Lbs¯γµLu)] . (10)
cb us 1 µ β α 2 µ
The decay amplitudes for B− K−D0 and B− K−D¯0, respectively, are given by
→ →
A¯ (K−D0) = V V∗ a e−iγ ,
SM | ub cs| KD
A¯ (K−D¯0) = V V∗ b . (11)
SM | cb us| KD
− −
From the above, one easily obtains the decay amplitude for B K D with D =
CP CP
→
(D0 D¯0)/√2 being the CP even eigenstate,
−
1
A¯ (K−D ) = [A¯ (K−D0) A¯ (K−D¯0)] . (12)
SM CP SM SM
√2 −
This relation form a triangle in the complex plan which is useful in determining the phase
angle γ in the SM [13].
B Decay Amplitudes In The Weinberg Model
In the Weinberg model, besides the CP violating phase in the CKM matrix, CP violation
for hadronic B decays can also arise from the exchange of charged Higgs at tree and loop
levels, and also neutral Higgs at loop levels. In this model, there are two physical charged
HiggsparticlesandthreeneutralHiggsparticles. TheneutralHiggscouplingstofermionsare
flavor conserving and proportional to the fermion masses. Flavor changing decay amplitude
can only be generated at loop level. For the cases in consideration, all involve light fermions,
the CP violating amplitude generated by neutral Higgs exchange is very small and can
be neglected. The exchange of charged Higgs may generate sizable CP violating decay
amplitudes, however. The charged Higgs couplings to fermions are given by [25]
L = 27/4G1/2U¯[V M (α H+ +α H+)R+M V (β H+ +β H+)L]D +H.C. , (13)
F KM D 1 1 2 U KM 1 1 2 2
where M are the diagonal up and down quark mass matrices. The parameters α and β
U,D i i
are obtained from diagonalizing charged Higgs masses and can be written as,
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α = s c /c , α = s s /c ,
1 1 3 1 2 1 3 1
β = (c c c +s s eiδH)/s c , β = (c c s s c eiδH)/s c , (14)
1 1 2 3 2 3 1 2 2 1 2 3 2 3 1 2
−
where s = sinθ and c = cosθ with θ being the rotation angles, and δ is a CP violating
i i i i i H
phase. The decay amplitudes due to exchange of charged Higgs at tree level will be propor-
tional to VfbVf∗′q(mbmf′/m2Hi)αiβi∗. Therefore if a decay involves light quark, the amplitude
will be suppressed. However, at one loop level if the internal quark masses are large, sizable
CP violating decay amplitude may be generated. The leading term is from the strong dipole
penguin interaction with top quark in the loop [26],
˜
L = fO ,
DP 11
G 2 m2 m4 m2 m2 1
f˜= F α∗β V V∗ t ( Hi ln Hi Hi ) . (15)
16√2 i i tb tqm2 m2 (m2 m2)2 m2 − m2 m2 − 2
Xi Hi − t Hi − t t Hi − t
This is not suppressed compared with the penguin contributions in the SM. There is also a
similar contribution from the operator O . However the WC of this operator is suppressed
12
by a factor of α /α and its contribution to B decays can be neglected. I write the O
em s 11
contribution to B decays as
A¯ = V V∗a eiαH , (16)
final tb tq final
˜ ˜
where α is the phase in f which is decay mode independent, and a = f <
H final
| |
final state O B > which is decay mode dependent. Note that L transforms as ¯3
11 DP
| |
under SU(3) symmetry. It does not contribute to A¯(π−π0) and √2A¯(K−π0) A¯(K¯0π−).
−
The decay amplitudes in the Weinberg model can be written as
A¯ (π+π−) = A¯ (π+π−)+V V∗eiαHa ,
W SM tb td ππ
A¯ (π−π0) = A¯ (π−π0) ,
W SM
1
A¯ (K−π0) = A¯ (K−π0)+ V V∗eiαHa ,
W SM √2 tb ts Kπ
A¯ (K¯0π−) = A¯ (K¯0π−)+V V∗eiαHa ,
W SM tb ts Kπ
A¯ (ψK ) = A¯ (ψK )+V V∗eiαHa . (17)
W S SM S tb ts ψK
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In the SU(3) limit a = a .
ππ Kπ
Thedecay amplitudesforB KD onlyhavecontributionsfromtreeoperators. Because
→
the CP violating amplitude from tree level charged Higgs exchange is negligibly small, to a
good approximation,
A¯ (KD) = A¯ (KD) . (18)
W SM
The decay amplitudes for both the WM-I and WM-II have the same form given in
eqs. (17) and (18). In the WM-I CP is violated in both the CKM and Higgs sectors with
αβγα = 0. In the WM-II CP is violated only in the Higgs sector with α = β = γ = 0, but
H
6
α = 0. I will drop the asterisk of the CKM matrix elements in the WM-II.
H
6
II. CP VIOLATION IN B DECAYS
B ππ Decays
→
In the time evolution of the rate asymmetry for B¯0 π+π− and B0 π−π+, there are
→ →
two terms varying with time, one varies as a cosine function and the other as a sine function.
The coefficients of these two terms can be measured experimentally. The coefficient of the
sine term is given by [12]
qA¯(π+π−)
Imλ = Im , (19)
pA(π−π+)!
where p and q are the mixing parameters defined by
B >= p B¯0 > +q B0 > , B >= q B¯0 > p B0 > , (20)
H L
| | | | | − |
where B > are the heavy and light mass eigenstates, respectively.
H,L
|
In the SM, the mixing is dominated by the top quark loop in the box diagram, and
∗
q V V
= tb td = e−2iβ . (21)
∗
p V V
tb td
One obtains
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eiγA¯ (π+π−)
Imλ = Im e−2i(β+γ) SM
e−iγASM(π−π+)!
= Im e2iα |VubVu∗d|Tπ+π− +|VtbVt∗d|Pπ+π−ei(β+γ)
|VubVu∗d|Tπ+π− +|VtbVt∗d|Pπ+π−e−i(β+γ)!
A¯ (π+π−)
SM
= |A (π−π+)|sin(2α+θ+−) . (22)
SM
| |
The ratio A¯ / A can be determined from time integrated rate asymmetry at symmet-
SM SM
| | | |
ric [24] and asymmetric colliders [8]. If θ+− can be determined, the phase angle α can be
determined. To determine θ+−, Gronau and London [11] proposed to use the isospin relation
in eq.(6),
√2A¯(π0π0)+√2A¯(π−π0) = A¯(π+π−) , (23)
and normalize the amplitudes A¯ = √2eiγA (π−π0) and A = √2e−iγA (π+π0) on the
2 SM 2 SM
real axis. The triangle is shown in Figure 1. It is easy to see from eq.(22) that the angle θ+−
is given by phase angle difference between A¯ = eiγA¯ (π+π−) and A = e−iγA (π−π+).
1 SM 1 SM
It can be easily read off from Figure 1.
In the Weinberg model, similar measurement will obtain different result. In the WM-I,
in addition to the phase β, there is also a phase β in q/p due to charged Higgs exchange
H
in the box diagram. One obtains q/p = e−2i(β+βH), and
A¯ (π+π−)+V V∗ei(αH+γ)a
Imλ = Im e−2i(β+γ+βH) SM tb td ππ
ASM(π+π−)+VtbVt∗de−i(αH+γ)aππ!
A¯ (π+π−)
= | W |sin(2α 2β +θH ) . (24)
A (π−π+) − H +−
W
| |
This equation has the same form as eq.(22) for the SM. The determination of α β
H
−
is exactly the same as α in the SM except that in this case A¯ = eiγA¯ (π+π−) and
1 W
A = e−iγA (π−π+). The phase β can be neglected because it is suppressed by a fac-
1 W H
tor of m m /m2 . The measurement proposed here still measures α even though there is an
b t H
additional CP violating phase in B¯0 π+π−(π0π0) decay amplitudes.
→
If CP is violated spontaneously, the result will be dramatically different. Here the CP
violating weak phases in A¯ are all zero. The amplitude A¯ = √2A¯ (π−π0) is equal to
SM 2 W
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A = √2A (π+π0), and can be normalized to be real. Now using the isospin triangle in
2 W
Figure 1, one easily obtains the phase in A¯ (π+π−)/A (π−π+), and therefore determine
W W
thephase angle β . Onewould obtaina very smallvalue. This will bea test forspontaneous
H
CP violation model WM-II.
Using the isospin triangle in Figure 1, the CP violating amplitude a 2sin2α in the
ππ H
| |
WM-II can also be determined. It is given by
L2
a 2sin2α = (25)
| H| H 4 V V 2
tb td
| |
This measurement will also serve as a test for the WM-II. I will come back to this later.
B ψK Decay
S
→
IntheSM,thecleanestwaytomeasureβ istomeasuretheparameterImλforB¯0 ψK
S
→
decay [12]. In this case,
qA¯ (ψK )
SM S
Imλ = Im . (26)
ψK
pASM(ψKS)!
∗
Neglecting the small term proportional to V V , one obtains
cb cs
∗
qV V
Imλ = Im cb cs = sin(2β) . (27)
ψK ∗
pVcbVcs! −
This is a very clean way to measure the phase angle β in the SM.
In the Weinberg model, the same measurement will give different result. In the WM-I,
one has
A¯ (ψK )+V V∗eiαHa
Imλ = Im e−2i(β+βH) SM S tb ts ψK . (28)
ψK ASM(ψKS)+VtbVt∗se−iαHaψK!
The amplitude from the new contribution proportional to a is expected to be about 10%
ψK
of the SM contribution. Even though β is small, Imλ in the WM-I will be different
H ψK
from sin(2β). This measurement alone will not be able to distinguish the SM and WM-I.
−
However combining the result from this measurement and knowledge about α determined
from the previous section and γ to be determined in the next section, one can distinguish
10