Influence of Kaonic Resonances on the CP Violation in $B\to K^*γ$Like Processes

54 Pages·0.35 MB·English

by  D. Atwood

SLAC-PUB-6425
January 1994
T
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n Influence of Kaonic Resonances on the CP Violation in
a B K γ Like Processes*
J ∗
→

7
2
D. Atwooda and A. Sonib

1
v
7
4 a) Department of Physics, SLAC, Stanford University, Stanford, CA
3
94309, USA b) Department of Physics, Brookhaven National Laboratory,
1
0 Upton, NY 11973, US
4
9
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p
-
p
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:
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a
Submitted to Zeitschrift fu¨r Physik C
*WorksupportedbytheDepartmentofEnergyContractsDE-AC03-76SF00515
(SLAC) and DE-AC02-76CH0016 (BNL) and an SSC Fellowship (for David
Atwood).
Abstract
We consider CP violating effects in decays of the type B k γ Kπγ, K πγ
i ∗
→ →
and Kργ, where k represents a strange meson resonance. We include in our cal-
i
culations five of the low-lying resonances with quantum numbers (JP) 1 , 1+ and
−
2+. At the quark level these decays are driven by the penguin graph as well as
tree graphs. CP violation arises in the Standard Model due to the difference in
the CKM phase between these graphs. We model the final state interaction of the
hadronic system using the low lying k resonances. Bound state effects are incor-
i
porated by using ideas based on the model of Grinstein et al and also in another
bound state model (somewhat similar to the model of Wirbel et al) which we con-
struct designed to take into account relativistic effects better. In these models we
findthatradiativedecays ofBmesonsgiverisetofourofthefivekaonicresonances
at about 1 to 7% of the inclusive b sγ rate. Furthermore, in both bound state
→
models we find that the probability of formation of the other three higher reso-
nances is roughly the same as that of K (892), which was recently seen at CLEO.
∗
In addition to the partial rate asymmetry which arises dueto interference between
resonances of the same quantum numbers, we show how interference between res-
onances of the same parity, and also between resonances of the opposite parity,
resultintwo differenttypes ofenergyasymmetries. CPdifferential asymmetries at
thelevel of afew percentseem possible. We thus obtain CP violating distributions
which may be observed in a sample of about 109B mesons. For concreteness in
±
this paperwe deal with only charged B’s, neutral B’s will be dealt with separately.
1
1 Introduction
Therecent observation[1]ofthelong awaitedprocessB K γ gave another
∗
→
reminder of the possible richness of the physics which could be observed at
B factories.
In this paper we will consider the generalizations of the above decay, i.e.
we consider the decays B k γ where the k denote excited K meson states
i i
→
or resonances. In particular, we wish to investigate if widths of resonances
can be used to enhance CP violation effects in B-decays; their importance in
the context of the top quark has been emphasized in the past few years[2, 3].
The key difference is that in the case of B-decays the resonances are strongly
interacting so that width to mass ratio is much larger than was the case for
the top decays wherein the resonances are electroweak in character.
Inthecaseathand,namelyB k γ,k musthaveJ 1;so,inparticular
i i
→ ≥
we will focus on the five lowest lying such states. We will denote these states
k , k , k , k and k . The first such state which we write k is K (892);
0 1 2 3 4 0 ∗
it has quantum numbers 1 . In a constituent quark model it is a u¯s (or
−
d¯s) quarkonium state 3S . The next two states that we are interested in we
1
will denote as k and k . They both have quantum numbers 1+ and in the
1 2
notation of [4] are written as K (1270) and K (1400). In the quark model
1 1
they should correspond to mixtures of the states 1P and 3P . For these pure
1 1
quark model states we will use the notation kˆ = 1P and kˆ = 3P and so,
1 1 2 1
following [5] the physical states are related to these by a mixing angle θ :
12
ˆ ˆ
k = cosθ k sinθ k
1 12 1 12 2
−
ˆ ˆ
k = sinθ k +cosθ k . (1)
2 12 1 12 2
ˆ ˆ
Though strictly speaking the states k and k are not eigenstates of charge
1 2
conjugation, in the literature they are sometimes referred to as 1+ and 1++
−
states since they are related by SU(3) to the b (1235) and a (1260) mesons
1 1
with those quantum numbers. In fact the mixing here has been observed to
be close to maximal; in [5] θ is experimentally determined to be 56 3 .
12 ◦
±
The 23S state K (1410) will be denoted as k . In principle k , which is a
1 ∗ 3 3
radially excited 23S state, could also mix with k however since the masses
1 0
are so far apart it is unlikely that this mixing is large. We will therefore
ignore this mixing in our calculation. We also consider the K (1430) state
2
with quantum numbers 2+ which will be designated k . D wave states have
4
2
also been observed around 1700 MeV but we will not include these in our
analysis. We have summarized some of the known properties [4] of these
states in Table 1.
Consider the two decays B k γ; k γ followed by decays of k and k de-
i j i j
→
cay to a common hadronic final state XY. If the two channels have different
CP phases then CP violation could manifest in the momentum distribution
of the mesons making up XY. Thus, for example, the energy distribution of
one of the particles in XY may be different for B decay compared to that in
B¯ decay. If the quantum numbers of the two states are the same, as is the
case for k versus k and k versus k then there also exists the possibility of
1 2 0 3
a partial rate asymmetry (PRA) between B decay and B¯ decay. Of course
in the case of neutral B decay one must also consider these phenomena in
¯
the context of B B oscillations [6].
−
We find that the largest effect occurs in the case of the final state K π.
∗
In this case we estimate that a difference between the distribution of the K
∗
in the decays of B versus B+ may well be observable with about 5 108
−
×
B decays. The final state Kπ seems to require about 5 109 B decays. In
± ±
×
passing, wemention that,whereas wearefocussing oninfluence ofresonances
to CP violation in radiative decays of B mesons to exclusive channels, CP
violation in inclusive, radiative decays of the b quark has been examined by
Soares [7]. The two approaches are therefore somewhat complementary.
The rest of the paper will proceed as follows: In section 2 we will explain
how it could happen that the different k decays could acquire different CP
i
phases, in particular in the Standard Model (SM); we will also estimate the
magnitude these phases could have. In making these estimates, for incorpo-
rating bound states effects, we will use ideas based on references [8, 9]. In
section 3 we will explain how these CP phases will give rise to CP violating
kinematic distributions as well as partial rate asymmetries. In that section
we also estimate the magnitude of various asymmetries to be expected.
3
2 Basic Processes
Any CP violating phase which enters into the process B k γ must have its
i
→
origin either in the electroweak physics which drives the process or in physics
beyond the standard model. In the standard model, three classes of quark
graphs may contribute as shown in Figure 1. Figure 1a shows the penguin
graph for the quark level process b sγ which has been studied extensively
→
[10, 11]. Figure 1(b) shows an annihilation process which is operative only
in the case of B . This process will give a γk state if the two quarks
± i
coalesce into the appropriate k state. Figure 1(c) shows a spectator process
i
which could give rise to γk if the four quarks shown should coalesce into two
i
mesonic states and thence to a k state.
i
Of course to have observable CP violating effects one need not only have
a CP violating phase but there must be the interference of processes with
different CPphases. InthestandardmodelsuchaphaseisgivenbytheCKM
matrix (V). Introducing the standard Wolfenstein [15] parameterization of
the CKM matrix [16]:
1 λ2 λ Aλ3σe iδ
− 2 −
V = λ 1 λ2 Aλ2 (2)
 
− − 2
Aλ3(1 σeiδ) Aλ2 1
 − − 
 
where λ = sinθ = 0.22. The amplitude for the penguin graphs [10, 11]
cabibbo
will be proportional to the quantity:
V V F(m ) (3)
i∗s ib i
i=u,c,t
X
where the form of F for the graph in Figure 1a is given in [10] and the QCD
corrected formis given in[11]. Notethat theabovesum willbedominated by
the i = c and i = t terms, hence from(2) we see that in this parameterization
there is little phase in the penguin amplitude. Both the spectator and the
annihilation graphs will, on the other hand, be proportional to the product
V V = Aλ4σe iδ (4)
u∗s ub −
which has a phase of precisely δ in this notation. Thus interference be-
−
tween the penguin graph and either the spectator or the annihilation graphs
potentially can produce observable CP violating effects.
4
To see how this comes about consider, for instance, the two decay chan-
nels B k γ πK γ and B k γ πK γ. The final states are the
2 ∗ 4 ∗
→ → → →
same, hence there could be interference effects between them. If there is a
difference in CP phase as well as a difference in the strong interaction phase
(i.e. CP conserving phase) , there could be a difference in the energy or
other distributions between B and B¯ decay which would clearly violate CP.
We will assume that near these resonances the strong interaction phase is
dominated by Breit-Wigner forms for the k propagators.
i
In order to calculate the CP phase for the decay B k γ we will thus
i
→
need to estimate the relative contribution of each of the three classes of
graphs to the five k resonances that we are considering.
i
In the case of the annihilation and spectator graphs it will be useful to
compare these rates with Γ0 , the tree level inclusive process b uu¯s given
bus →
by:
G2m5 m2 V V 2
Γ0 = 3 F b V V 2 = 3f 1( c) ub u∗s Γ(b eνc) (5)
bus 192π3| ub u∗s| − m2 V →
b (cid:12) cb (cid:12)
(cid:12) (cid:12)
where (cid:12) (cid:12)
(cid:12) (cid:12)
f(x) = 1 8x+8x3 x4 12x2logx (6)
− − −
is the phase space factor defined in [14]; in this case f(m2/m2) = 0.46, where
c b
we are using m = 1.5GeV and m = 4.6GeV.
c b
For a given B decay, via the spectator graph, to the final state X we
define
Γ(X)
r(X) = . (7)
Γ0
bus
2.1 Penguin Graph
The dominant graph as we shall see is likely to be the penguin process de-
picted in Figure 1(a). In this case the k system is formed by the merging
i
of the spectator and the s quark. In order to calculate the probability of
the formation of each of the resonances, one needs a model for the bound
state effects involved. For this purpose we will consider potential models
[8, 9, 17, 18].
In general such a model will relate meson level amplitude to the
m
M
quark level amplitude according to the formula:
q
M
= ΦB(P)Φk(Pˆ)dP (8)
Mm Mq i
Z
5
where the meson wave functions ΦB and Φk are functions of quark momenta
i
and spins with the correct quantum numbers to form the indicated mesons.
Here P represents the momentum of the b quark in the rest frame of the B
meson and Pˆ represents the momentum of the s quark in the frame of the
k . The exact details of how the integral is constructed will depend on the
i
specific model. We will construct two such models which we will designate
A and B in order to get a feel for the accuracy of the predictions.
Model A is similar to the one used in [17] and is based on the quark
model of Grinstein et al. [8] which has been quite successful in semileptonic
charm and bottom decays. Model B is constructed to take relativistic effects
better and is based roughly on the ideas of Wirbel et al. [9]. For model A,
non-relativistic kinematics is used for the quarks and P is taken to be the 3
ˆ
momentum of the b-quark in the B-meson rest frame and P is given by
Pˆ = P~ x K~ (9)
u
−
where x = m /(m +m ) and K~ is the momentum of the k meson in the
u u u b i
B rest frame.
The wave functions for the various meson states are approximated by
harmonic oscillator functions. Thus the momentum dependent part of the
wave functions are:
ΦB = π−34βB−32e−2PβB22
Φ1S = π−43βS−32e−2Pβ2S2
Φ1P(0) = π−34βP−32√β2Pze−2Pβ2P2
P
Φ1P(±1) = π−43βP−32(Pxβ∓PiPy)e−2Pβ2P2
Φ2S = s32π−34βS−32(23Pβ22 −1)e−2Pβ2S2 (10)
S
Here Φ represents the nth excited S-wave state, Φ represents the n th
nS nP(m)
excited P-wave state with angular momentum projection m and φ is the
B
wave function for the B meson. The constants β are determined from the
i
potentialmodelusingavariationalmethod. Thevaluesobtainedinreferences
[17, 8] are
6
β = 0.41 GeV β = 0.34 GeV β = 0.30 GeV (11)
B S P
where they use the constituent masses
m = m = 0.33 GeV m = 0.55 GeV m = 5.12 GeV. (12)
u d s b
With the use of the above momentum distributions above we can specify
the spin-dependent part of the wave function for the B-meson and for the k
i
with J = 1:
z
−
1
B > = Φ (b( )u¯( )+b( )u¯( ))
B
| √2 ↑ ↓ ↓ ↑
k > = Φ s( )u¯( )
0 1S
| ↓ ↓
1
ˆ
k > = Φ (s( )u¯( )+s( )u¯( ))
| 1 √2 1P(−1) ↑ ↓ ↓ ↑
1 1
ˆ
k > = Φ s( )u¯( ) Φ (s( )u¯( ) s( )u¯( ))
| 2 √2 1P(0) ↓ ↓ − 2 1P(−1) ↓ ↑ − ↑ ↓
k > = Φ s( )u¯( )
3 2S
| ↓ ↓
1 1
k > = Φ s( )u¯( )+ Φ (s( )u¯( ) s( )u¯( )) (13)
| 4 √2 1P(0) ↓ ↓ 2 1P(−1) ↓ ↑ − ↑ ↓
Note that for the 1+ states we use the quark model basis kˆ ,kˆ .
1 2
{ }
At the quark level the bsγ coupling is
m
a s¯ σµνq (P + sP ) b (14)
µ R L
m
b
where P = 1(1 + γ ) and P = 1(1 γ ) and a is given in [11]. We will
R 2 5 L 2 − 5
concentrate on the proportional to P which is dominant. Note that this
R
gives rise to left handed photons in the final state.
Let us take the z-axis in the direction of the hadronic momentum so the
4-momentum of the photon is
+1
m2 m2 0
p = B − i   (15)
γ 2 0
 
 1 
 − 
 
7
and we take the 3-momentum of the b-quark to be P~. Expanding the ampli-
tude for small P ,P we obtain:
x y
= a√8(m m )P
++ b s
M − − −
= 0
+
M −
= a√8(m2 m2)
M−+ − b − s
= +a√2m 1(m2 m2)P (16)
M−− −b b − s −
where is the amplitude for b quark with spin projection S = j/2 going
ij z
M
to s-quark with spin projection S = i/2. The quantity P = P iP .
z x y
± ±
Using the above expansion we can obtain analytic expressions for the
meson amplitudes:
3
βˆ2 2
(k ) = 2a (m2 m2)e ∆S
M 0 − βBβS! b − s −
3
1 βˆ2 βˆ2 2
(kˆ ) = a (m m )2e ∆P
1 b s −
M −s2 mbβP βBβP! −
5
βˆ2 2 m2 m2
(kˆ ) = a β 1x B − i (m2 m2)
M 2 − " βBβP! B− u 2mB ! b − s
3
1 βˆ2 2 βˆ2
+ (m m )(3m +m ) e ∆P
b s b s −
2 βBβP! βPmb − #
3
2 1 βˆ2 2 3(β4 β4)+2β2(x K)2
(k ) = a 2 B − S S u (m2 m2)e ∆S
M 3 − (cid:18)3(cid:19) βBβS! (βB2 +βS2)2 b − s −
5
βˆ2 2 m2 m2
(k ) = a β 1x B − i (m2 m2)
M 4 − " βBβP! B− u 2mB ! b − s
3
1 βˆ2 2 βˆ2
(m m )(3m +m ) e ∆P (17)
b s b s −
−2 βBβP! βPmb − #
where
x2(m2 m2)2 2β2β2
∆ = u B − i βˆ = B i (18)
2m2 (β2 +β2) i vβ2 +β2
B B i u B i
u
t
8
We can now calculate the ratio
Γ(B k γ)
i
R = → . (19)
i
Γ(b sγ)
→
The results are shown in Table 2. Note that our results are slightly different
than those obtained in [17] because here we expanded the matrix element
only to first order in P~.
One thing which is worrisome about the model constructed in this way
is that the transformation from P to Pˆ is not relativistic. The velocity of
the mesons however is, since v ranges from .85 in the case of k to .95 in
c 2
the case of K . This motivates us to consider a modification which respects
∗
relativity.
In model B, we consider wave functions Ψ which are functions of 4-
i
momentum and are related to Φ by
i
E(m E) (E−E0)2
Ψi(P) = NiΦi(P~) i − e− 2βi2 . (20)
s m
i
In this equation P (E,P~) is the 4-momentum of the quark in the meson i
≡
and
m2 +m2 m2
E = i q − q¯ (21)
0
2m
i
m and m being the constituent masses. We define the wave function to be
q q¯
0 if E is outside of the physical range 0 E m and the normalization
i
≤ ≤
constant N is determined by the condition
i
Ψ(P) 2d4P = 1. (22)
Z0≤E≤mi|| ||
For the form of the E dependent part of the wave function we are motivated
by the similar form in [9].
Thus, in the reaction B k γ the relation between Pˆ and P becomes,
i
→
Pˆ = L(P P ) where
γ
−
γ 0 0 βγ
−
0 1 0 0
L =   (23)
0 0 1 0
 
 βγ 0 0 γ 
 − 
 
9