arXiv:hep-ph/0006351v1 30 Jun 2000
NCKU-HEP-00-04 hep-ph/0006xxx
Final state interaction and B
→ KK decays in perturbative QCD
Chuan-Hung Chen and Hsiang-nan Li
Department of Physics, National Cheng-Kung University, Tainan, Taiwan 701, Republic of China
PACS numbers: 13.25.Hw, 11.10.Hi, 12.38.Bx, 13.25.Ft
abstract
We predict branching ratios and CP asymmetries of the B → KK decays using perturbative QCD factorization theorem, in which tree, penguin, and annihilation contributions, including both factorizable and nonfactorizable ones, are expressed as convolutions of hard six-quark amplitudes with universal meson wave functions. The unitarity angle φ3= 90oand the B and K meson wave functions extracted from experimental data of the B → Kπ and ππ decays are
employed. Since the B → KK decays are sensitive to final-state-interaction effects, the comparision of our predictions with future data can test the neglect of these effects in the above formalism. The CP asymmetry in the B±→ K±K0
modes and the Bd0→ K±K∓ branching ratios depend on annihilation and nonfactorizable amplitudes. The B → KK
data can also verify the evaluation of these contributions.
I. INTRODUCTION
The conventional approach to exclusive nonleptonic B meson decays relies on the factorization assumption (FA) [1], under which nonfactorizable and annihilation contributions are neglected and final-state-interaction (FSI) effects are assumed to be absent. Factorizable contributions are expressed as products of Wilson coefficients, meson decay con-stants, and hadronic transition form factors. Though analyses are simpler under this assumption, estimations of many important ingredients, such as tree and penguin (including electroweak penguin) contributions, and strong phases are not reliable. Moreover, the above naive FA suffers the problems of scale, infrared-cutoff and gauge dependences [2]. It is also difficult to explain the observed branching ratios of the B → J/ψK(∗) decays in the FA approach, to which
nonfactorizable and factorizable contributions are of the same order [3].
Perturbative QCD (PQCD) factorization theorem for exclusive heavy-meson decays has been developed some time ago [4–6], which goes beyond FA. PQCD is a method to separate hard components from a QCD process, which are treated by perturbation theory. Nonperturbative components are organized in the form of hadron wave functions, which can be extracted from experimental data. This prescription removes the infrared-cutoff dependence in PQCD. Since nonperturbative dynamics has been absorbed into wave functions, external quarks involved in hard amplitudes are on-shell, and gauge invariance of PQCD predictions is guaranteed. Contributions to hard parts from various topologies, such as tree, penguin and annihilation, including both factorizable and nonfactorizable contributions, can all be calculated. Without assuming FA, it is easy to achieve the scale independence in the PQCD approach.
Despite of the above merrits of PQCD, an important subject, final state interaction (FSI), remains unsettled, which is nonperturbative but not universal. FSI effects in two-body decays have been assumed to be small. Though arguments and indications for this assumption have been supplied in [7], experimental justification is necessary. In this paper we shall propose to explore FSI effects by studying B → KK decays. It will be explained in Sec. II that these decays are more sensitive to FSI effects compared to B → Kπ and ππ decays. Employing the meson wave
functions and the unitarity angle φ3= 90odetermined in [7], we predict the branching ratios and the CP asymmetries
of the B± → K±K0, B0
d → K±K∓ and B0d → K0K¯0 modes. The comparision of our predictions with future data
can be used to estimate the importance of FSI effects. In particular, large observed B0
d → K±K∓ branching ratios
and CP asymmetry in the B0
d→ K0K¯0 modes will imply strong FSI effects.
An essential difference between the FA and PQCD approaches is that annihilation and nonfactorizable amplitudes are neglected in the former, but calculable in the latter. It has been shown that annihilation contributions from the operator O5,6 with the (V − A)(V + A) structure, bypassing helicity suppression, are not negligible [7]. These
contributions, being mainly imaginary, result in CP asymmetries in the B → ππ decays, which are much larger than those predicted in FA [8,9]. Hence, measurements of CP asymmetries will distinguish the two approaches [9]. The B± → K±K0 modes contain both annihilation amplitudes from O
5,6 and nonfactorizable annihilation amplitudes
from O1,2, such that they exhibit substantial CP asymmetry. The branching ratios of the B0d → K±K∓ modes,
involving only nonfactorizable annihilation amplitudes, can not be estimated, or are vanishingly small in FA. The data of these two decays can verify PQCD evaluation of annihilation and nonfactorizable contributions.
FSI effects in the B → ππ, Kπ and KK decays are compared in Sec. II. The PQCD formalism for annihilation and nonfactorizable contributions is reviewed in Sec. III. We present the factorization formulas of all the B → KK modes in Sec. IV, and perform a numerical analysis in Sec. V. Section VI is the conclusion.
II. FINAL STATE INTERACTION
FSI is a subtle and complicated subject. Most estimates of FSI effects in the literature [10] suffer ambiguities or difficulties. Kamal has pointed out that the enhancement of CP asymmetry in the B± → K0π± modes from order
0.5 % up to order (10-20)% [11] is due to an overestimation of FSI effects by a factor of 20 [12]. The smallness of FSI effects has been put forward by Bjorken [13] based on the color-transparency argument [14]. The renormalization-group (RG) analysis of soft gluon exchanges among initial- and final-state mesons [15] has also indicated that FSI effects are not important in two-body B meson decays. These discussions have led us to ignore FSI effects in the PQCD formalism. For example, the charge exchange in the rescattering B+→ K+π0→ K0π+, regarded as occuring
through short-distance quark-pair annihilation, is of higher order [7].
As stated in the Introduction, the neglect of FSI effects requires experimental justification. For this purpose, we propose to investigate the B → KK decays, which are more sensitive to FSI effects compared with the B → Kπ and ππ decays. Similar proposals have been presented in the literature [16,17] within the framework of SU (3) symmetry. We make our argument explicit by means of the general expression for the B → ππ, Kπ and KK decay amplitudes,
A = VuT + VuPu+ VcPc+ VtPt. (1)
The factors Vq = VqdVqb∗, q = u, c, and t, are the products of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements,
T denotes the tree amplitude, and Pq denote the penguin amplitudes arising from internal q-quark contributions. FSI
effects have been included in the amplitudes T and Pu,c,t.
Using the unitarity relation Vc = −Vu− Vt, Eq. (1) is rewritten as
A = Vu(T + Pu− Pc) + Vt(Pt− Pc) , = Vu(T + Pu− Pc) 1 + Vt Vu Pt− Pc T + Pu− Pc , ≡ Vu(T + Pu− Pc) 1 + Vt VuRππ(KK)e iδππ(KK) , (2) or A = Vt(Pt− Pc) 1 +Vu Vt T + Pu− Pc Pt− Pc , ≡ Vt(Pt− Pc) 1 +Vu VtRKπ eiδKπ , (3)
where R are the ratios of different amplitudes and δ the CP-conserving strong phases.
Without FSI, the various amplitudes T and Pu,c,t, namely, the ratios R and the strong phases δ are calculable in
PQCD. If FSI effects are important, they may change branching ratios or induce CP asymmetries of two-body B meson decays by varying R and δ. For the B → ππ decays, the ratio Vt/Vuis of order unity, but Rππ is small because
mode, whose tree amplitude is proportional to the small Wilson coefficient a2 = C1+ C2/Nc. The ratio RKπ may
be large, but its coefficient Vu/Vt ∼ Rbλ2, Rb and λ being the Wolfenstein parameters defined in Sec. IV, is small.
Therefore, FSI effects in the B → ππ and Kπ decays are suppressed by 1/a1 and Vu/Vt, respectively. On the other
hand, the B → ππ and Kπ decays have branching ratios of order 10−5, which are larger than those of the B → KK
decays (of order 10−6 as calculated in Sec. V). It has been also predicted in PQCD that the CP asymmetries in the
B → ππ and Kπ decays are large: 30 − 40% in the former [8,9] and 10 − 15% in the latter [7]. These large values render FSI effects relatively mild.
For the B → KK decays, T arises only from small nonfactorizable annihilation diagrams for the B± → K±K0
and B0
d → K±K∓ modes, and vanishes for the B0d → K0K¯0 modes. Furthermore, Vt/Vu is of order unity. Hence,
there is no suppression from the Wilson coefficients and from the CKM matrix elements, and FSI effects will be more significant. In the PQCD approach RKK is close to unity, corresponding to the branching ratios of order 10−6 for
B± → K±K0 and B0
d → K0K¯0, and 10−8 for Bd0 → K±K∓. CP asymmetry vanishes in the Bd0 → K0K¯0 modes,
because only the penguin operators contribute at leading order. However, if FSI contributes, the above results will be changed dramatically. For example, FSI effects could induce large T and Pu,c via the rescattering of intermediate
states DD and ππ produced from the tree operators, and Ptvia the rescattering of intermediate states KK produced
from the penguin operators. When RKK deviates from unity through rescattering processes, the branching ratios
and CP asymmetries of the B → KK decays could be enhanced.
We show how FSI effects modify amplitudes of various topologies in the B → KK decays in Table I. For more allowed intermediate states, refer to [17]. It is obvious that the rescattering processes DD(ππ) → KK may be important due to the large B → DD(ππ) branching ratios. For example, B(B0
d→ π±π±) is of order 10−5. It is then
possible that FSI effects could be significant enough to increase B(B0
d → K±K∓) from order 10−8 to above 10−7.
For a similar reason, the rescatering processes could induce large Pu,c with the CKM matrix elements Vu,c, which, as
interfered with the penguin contributions, result in sizable CP asymmetry in the B0
d → K0K¯0 modes. Hence, large
CP asymmetry observed in the B0
d → K0K¯0 modes and large deviation of the observed Bd0 → K±K∓ branching
ratios from the PQCD predictions will indicate strong FSI effects.
III. NONFACTORIZABLE AND ANNIHILATION CONTRIBUTIONS
PQCD factorization theorem for exclusive nonleptonic B meson decays has been briefly reviewed in [7]. In this section we simply sketch the idea of PQCD factorization theorem, concentrating on its application to nonfactorizable and annihilation amplitudes in the B → KK decays.
In perturbation theory nonperturbative dynamics is reflected by infrared divergences in radiative corrections. These infrared divergences can be separated and absorbed into a B meson wave function or a kaon wave function order by order [4]. A formal definition of the meson wave functions as matrix elements of nonlocal operators can be constructed, which, if evaluated perturbatively, reproduces the infrared divergences. Certainly, one can not derive a wave function in perturbation theory, but parametrizes it as a parton model, which describes how a parton (valence quark, if a leading-twist wave function is referred) shares meson momentum. The meson wave functions, characterized by the QCD scale ΛQCD, must be determined by nonperturbative means, such as lattice gauge theory and QCD sum rules,
or extracted from experimental data. In the application below small parton transverse momenta kT are included, and
the characteristic scale is replaced by 1/b with b being a variable conjugate to kT.
After absorbing infrared divergences into the meson wave functions, the remaining part of radiative corrections is infrared finite. This part can be evaluated perturbatively as a hard amplitude with six on-shell external quarks, four of which correspond to the four-fermion operators and two of which are the spectator quarks of the B or K mesons. Note that the b quark carries various momenta, whose distribution is described by the B meson wave function introduced above. The six-quark amplitude contains all possible Feynman diagrams, which include both factorizable and nonfactorizable tree, penguin and annihilation contributions. A factorizable diagram involves hard gluon exchanges among valence quarks of the B meson or of a kaon. A nonfactorizable diagram involves hard gluon exchanges bwteeen the valence quarks of different mesons. That is, the PQCD formalism does not rely on FA.
The hard amplitude is characterized by the virtuality t of involved internal particles, which is of order MB, and by
the W boson mass MW. The hard scale t reflects the specific dynamics of a decay mode, while MW serves the scale,
at which the matching conditions of the effective weak Hamiltonian to the full Hamiltonian are defined. The study of the pion form factor has indicated that the choice of t as the maximum of internal particle virtualities minimizes next-to-leading-order corrections to hard amplitudes [18]. Large logarithmic corrections are organized by RG methods. The results consist of the evolution from MW down to t described by the Wilson coefficients, the evolution from t to
1/b, and a Sudakov factor. The two evolutions are governed by different anomalous dimensions, since loop corrections associated with spectator quarks contribute, when the energy scale runs to below t. The Sudakov factor suppresses
the long-distance contributions from the large b region, and vanishes as b = 1/ΛQCD. This suppression guarantees
the applicability of PQCD to exclusive decays around the energy scale of the B meson mass [4].
A salient feature of PQCD factorization theorem is the universality of nonperturbative wave functions. Because of universality, meson wave functions extracted from some decay modes can be employed to make predictions for other modes. We have determined the B and K meson wave functions from the experimental data of the B → Kπ and ππ decays [7], and the unitarity angle φ3= 90o from the CLEO data of the ratio [19],
R = B(B
0
d→ K±π∓)
B(B±→ K0π±) = 0.95 ± 0.30 , (4)
where B(B0
d → K±π∓) represents the CP average of the branching ratios B(Bd0 → K+π−) and B( ¯B0d → K−π+).
It has been emphasized that the B → ππ data can be explained using the same angle φ3= 90o in PQCD, contrary
to the conclusion in [20,21], where an angle larger than 100o must be adopted. In this work we shall predict the
branching ratios and CP asymmetries of the B → KK decays in the PQCD formalism employing the above meson wave functions and the unitarity angle.
Factorizable annihilation contributions correspond to the time-like kaon form factor. It is known that annihilation contributions from the O1−4 operators with the (V − A)(V − A) structure vanish because of helicity suppression.
However, those from the O5,6 operators with the (V − A)(V + A) structure bypass helicity suppression, and turn
out to be comparible with penguin contributions [7]. Without FSI in PQCD, strong phases arise from non-pinched singularities of quark and gluon propagators in annihilation and nonfactorizable diagrams. Especially, annihilation amplitudes are the main source of strong phases [7]. In the FA and Beneke-Buchalla-Neubert-Sachrajda (BBNS) [22,23] approaches, where annihilation diagrams are not taken into account, strong phases come from the Bander-Silverman-Soni (BSS) mechanism [24] and from the extraction of the scale dependence from hadronic matrix elements [25]. As shown in [9], these sources are in fact next-to-leading-order. As a consequence, CP asymmetries predicted in FA and BBNS are smaller than those predicted in PQCD. Nonfactorizable amplitudes have been also considered in the BBNS approach, which are, however, treated in a different way. For example, they are real because of some approximation in [22], but complex in PQCD [26]. Generally speaking, nonfactorizable contributions are less important compared to factorizable ones except in the cases where factorizable contributions are proportional to the small Wilson coefficients a2 or vanish.
As stated before, the B± → K±K0 decays involve both annihilation amplitudes from O
5,6 and nonfactorizable
annihilation amplitudes from O1,2. Their interference then leads to substantial CP asymmetry in PQCD. The B0d→
K±K∓decays involve only nonfactorizable annihilation amplitudes from tree and penguin operators, such that their
branching ratios can not be estimated, or are vanishingly small in the FA and BBNS approaches. These quantities mark the essential differences among FA, BBNS and PQCD. The comparision of our predictions for the CP asymmetry in the B±→ K±K0decays and for the B0
d→ K±K∓ branching ratios with future data will justify our evaluation of
annihilation and nonfactorizable contributions, and distinguish the FA, BBNS and PQCD approaches.
IV. FACTORIZATION FORMULAS
We present the factorization formulas of the B → KK decays in this section. The effective Hamiltonian for the flavor-changing b → d transition is given by [27]
Heff = GF √ 2 X q=u,c Vq " C1(µ)O1(q)(µ) + C2(µ)O2(q)(µ) + 10 X i=3 Ci(µ)Oi(µ) # , (5)
with the CKM matrix elements Vq = Vqd∗Vqb and the operators
O(q)1 = ( ¯diqj)V −A(¯qjbi)V −A, O(q)2 = ( ¯diqi)V −A(¯qjbj)V −A,
O3= ( ¯dibi)V −A X q (¯qjqj)V −A, O4= ( ¯dibj)V −A X q (¯qjqi)V −A, O5= ( ¯dibi)V −A X q (¯qjqj)V +A, O6= ( ¯dibj)V −A X q (¯qjqi)V +A, O7= 3 2( ¯dibi)V −A X q eq(¯qjqj)V +A, O8= 3 2( ¯dibj)V −A X q eq(¯qjqi)V +A, O9=3 2( ¯dibi)V −A X q eq(¯qjqj)V −A, O10=3 2( ¯dibj)V −A X q eq(¯qjqi)V −A, (6)
i and j being the color indices. Using the unitarity condition, the CKM matrix elements for the penguin operators O3-O10can also be expressed as Vu+ Vc= −Vt. The unitarity angle φ3 is defined via
Vub = |Vub| exp(−iφ3) . (7)
Adopting the Wolfenstein parametrization for the CKM matrix upto O(λ3),
Vud Vus Vub Vcd Vcs Vcb Vtd Vts Vtb ! = 1 − λ2 2 λ Aλ 3(ρ − iη) −λ 1 −λ22 Aλ2
Aλ3(1 − ρ − iη) −Aλ2 1
, (8)
we have the parameters [28]
λ = 0.2196 ± 0.0023 , A = 0.819 ± 0.035 , Rb≡
p
ρ2+ η2= 0.41 ± 0.07 . (9)
For the B±→ K±K0decays, the operators O(u)
1,2 contribute via the annihilation topology, in which the fermion flow
forms two loops as shown in Fig. 1. O(c)1,2 do not contribute at leading order of αs. O3−10 contribute via the penguin
topology with the light quark q = s and via the annihilation topology with q = u, in which the fermion flow forms one loop. As evaluating hard amplitudes, an additional minus sign should be associated with the O(u)1,2 contributions, that
contain two fermion loops. O1,3,5,7,9give both factorizable and nonfactorizable (color-suppressed) contributions, while
O2,4,6,8,10 give only factorizable ones because of the color flow. The electroweak penguin contributions from O7−10
have been included in the same way as those from the QCD penguins O3−6. Obviously, the electroweak penguins are
less important because of the small electromagnetic coupling. The diagrams for the B0
d → K±K∓decays are displayed in Fig. 2. The operators O (u)
1,2 contribute via the annihilation
topology, which contain one fermion loop. O(c)1,2 do not contribute at leading order of αs. O3−10 contribute via the
annihilation topology with the light quark q = s or u, in which the fermion flow forms two loops. In these modes O2,4,6,8,10 give both factorizable and nonfactorizable contributions, while O1,3,5,7,9give only factorizable ones because
of the color flow. Only the operators O3−10 contribute to the Bd0 → K0K¯0 modes via the penguin topology with
the light quark q = s and via the annihilation topology with the light quark q = s or d. The penguin contributions contain one fermion loop. The q = s annihilation amplitudes involve two fermion loops, while the q = d annihilation amplitudes contain both cases of one fermion loop and of two fermion loops as shown in Fig. 3.
The B meson momentum in light-cone coordinates is chosen as P1= (MB/
√
2)(1, 1, 0T). Momenta of the two kaons
are chosen as P2= (MB/
√
2)(1, 0, 0T) and P3= P1− P2. We shall drop the contributions of order (MK/MB)2∼ 5%,
MK being the kaon mass. The B meson is at rest with the above parametrization of momenta. We define the
momenta of light valence quark in the B meson as k1, where k1 has a plus component k1+, giving the momentum
fraction x1= k+1/P1+, and small transverse components k1T. The two light valence quarks in the kaon involved in the
B → K transition form factor carry the longitudinal momenta x2P2 and (1 − x2)P2, and small transverse momenta
k2T and −k2T, respectively. The two light valence quarks in the other kaon carry the longitudinal momenta x3P3
and (1 − x3)P3, and small transverse momenta k3T and −k3T, respectively.
The Sudakov resummations of large logarithmic corrections to the B and K meson wave functions lead to the exponentials exp(−SB), exp(−SK2) and exp(−SK3), respectively, with the exponents
SB(t) = s(x1P1+, b1) + 2 Z t 1/b1 d¯µ ¯ µ γ(αs(¯µ)) , SK2(t) = s(x2P + 2 , b2) + s((1 − x2)P2+, b2) + 2 Z t 1/b2 d¯µ ¯ µ γ(αs(¯µ)) , SK3(t) = s(x3P − 3 , b3) + s((1 − x3)P3−, b3) + 2 Z t 1/b3 d¯µ ¯ µ γ(αs(¯µ)) . (10)
The variable b1, b2, and b3 conjugate to the parton transverse momentum k1T, k2T, and k3T represents the transverse
extent of the B and K mesons, respectively. The exponent s is written as [29–31] s(Q, b) = Z Q 1/b dµ µ ln Q µ A(αs(µ)) + B(αs(µ)) , (11)
where the anomalous dimensions A to two loops and B to one loop are A = CFαs π + 67 9 − π2 3 − 10 27f + 2 3β0ln eγE 2 αs π 2 , B = 2 3 αs π ln e2γE−1 2 , (12)
with CF = 4/3 a color factor, f = 4 the active flavor number, and γE the Euler constant. The one-loop expression of
the running coupling constant,
αs(µ) =
4π β0ln(µ2/Λ2QCD)
, (13)
is substituted into Eq. (11) with the coefficient β0 = (33 − 2f)/3. The anomalous dimension γ = −αs/π describes
the RG evolution from t to 1/b.
The decay rates of B±→ K±K0 have the expressions
Γ = G
2 FMB3
128π |A|
2. (14)
The decay amplitudes A+and A− corresponding to B+→ K+K0and B− → K−K0, respectively, are written as
A+ = fKVt∗F P (s) 46 + Vt∗M P (s) 46 + fBVt∗F P (u) a6 + Vt∗M P (u) a46 − Vu∗Ma1 , (15)
A− = fKVtF46P (s)+ VtMP (s)46 + fBVtFa6P (u)+ VtMP (u)a46 − VuMa1, (16)
with the kaon decay constant fK. The notation F (M) represents factorizable (nonfactorizable) contributions, where
the indices a and P (q) denote the annihilation and penguin topologies, respectively, with the q quark pair emitted from the electroweak penguins, and the subscripts 1, 4 and 6 label the Wilson coefficients appearing in the factorization formulas. The nonfactorizable amplitude Ma1 are from the operators O(u)1,2.
The decay rates of B0
d → K±K∓ have the similar expressions with the amplitudes
A = V∗ t (M P (u) a35 + M P (s) a35 ) − Vu∗Ma2, (17) ¯ A = Vt(MP (u)a35 + M P (s) a35 ) − VuMa2, (18) for B0
d → K+K− and ¯B0d → K−K+, respectively. The notations are similar to those in Eqs. (15) and (16). The
decay amplitudes for B0
d → K0K¯0 and ¯Bd0→ K0K¯0 are written as
A′= fKVt∗F P (s) 46 + Vt∗M P (s) 46 + fBVt∗F P (d) a6 + Vt∗(M P (d) a46 + M P (d) a35 + M P (s) a35 ) , (19) ¯ A′= f KVtF46P (s)+ VtMP (s)46 + fBVtFa6P (d)+ Vt(MP (d)a46 + M P (d) a35 + M P (s) a35 ) , (20) respectively.
The factorizable contributions are written as F46P (s) = F4P (s)+ F6P (s), F4P (s) = 16πCFMB2 Z 1 0 dx1dx3 Z ∞ 0 b1db1b3db3φB(x1, b1) ×{[(1 + x3)φK(x3) + rK(1 − 2x3)φ′K(x3)] Ee4(s)(t(1)e )he(x1, x3, b1, b3) +2rKφ′K(x3)Ee4(s)(te(2))he(x3, x1, b3, b1)} , (21) F6P (s) = 32πCFMB2 Z 1 0 dx1dx3 Z ∞ 0 b1db1b3db3φB(x1, b1) ×rK{[φK(x3) + rK(2 + x3)φ′K(x3)] Ee6(s)(t(1)e )he(x1, x3, b1, b3) + [x1φK(x3) + 2rK(1 − x1)φ′K(x3)] Ee6(s)(t(2)e )he(x3, x1, b3, b1)} , (22) Fa6P (q) = 32πCFMB2 Z 1 0 dx2dx3 Z ∞ 0 b2db2b3db3 ×rK{[x3φK(1 − x2)φ′K(1 − x3) + 2φ′K(1 − x2)φK(1 − x3)] Ea6(q)(t (1) a )ha(x2, x3, b2, b3) + [2φK(1 − x2)φ′K(1 − x3) + x2φ′K(1 − x2)φK(1 − x3)] Ea6(q)(t(2)a )ha(x3, x2, b3, b2)} , (23)
for the light quarks q = u and d. The evolution factors are given by
E(s)ei (t) = αs(t)a(s)i (t) exp[−SB(t) − SK3(t)] , (24)
Eai(q)(t) = αs(t)a(q)i (t) exp[−SK2(t) − SK3(t)] . (25)
Notice the arguments 1 − x2 and 1 − x3 of the kaon wave functions φK and φ′K in Eqs. (26) and (23). The explicit
expressions of the kaon wave functions will be given in Sec. V, where x represents the momentum fraction of the light u or d quark. However, to render the annihilation contributions for q = s and for q = u or d have the same hard parts, we have labelled the s quark momentum by x in the latter case, and changed the arguments of the kaon wave functions to 1 − x.
The factorizable annihilation contribution associated with the Wilson coefficient a(q)4 from Fig. 1(c) is identical to
zero because of helicity suppression as indicated by Fa4P (q) = 16πCFMB2 Z 1 0 dx2dx3 Z ∞ 0 b2db2b3db3 ×{−x3φK(1 − x2)φK(1 − x3) − 2rK2(1 + x3)φ′K(1 − x2)φ′K(1 − x3) ×Ea4(q)(t(1)a )ha(x2, x3, b2, b3) +x2φK(1 − x2)φK(1 − x3) + 2r2K(1 + x2)φ′K(1 − x2)φ′K(1 − x3) ×Ea4(q)(t(2)a )ha(x3, x2, b3, b2)} . (26)
The helicity suppression does not apply to the annihilation contributions associated with a(q)6 , and the two terms in Eq. (23) are constructive. It is easy to confirm these observations by interchaning the integration variables x2 and
x3 in the second terms of Eqs. (23) and (26). The factorization formulas for Fa1 from Fig. 1(a) and for Fa2 from
Fig. 2(a), associated with the Wilson coefficient a1(ta) and a2(ta), respectively, are the same as Fa4P (q), i.e., vanish. The
expressions of Fa35P (q) from Figs. 2(b), 2(c), 3(c), and 3(d), associated with the Wilson coefficients a (q)
3 (ta) + a(q)5 (ta),
are also the same as Fa4P (q) and vanish.
The hard functions h’s in Eqs (21)-(23), are given by he(x1, x3, b1, b3) = K0(√x1x3MBb1) × [θ(b1− b3)K0(√x3MBb1) I0(√x3MBb3) +θ(b3− b1)K0(√x3MBb3) I0(√x3MBb1)] , (27) ha(x2, x3, b2, b3) = iπ 2 2 H0(1)(√x2x3MBb2) ×hθ(b2− b3)H0(1)( √x 3MBb2) J0(√x3MBb3) +θ(b3− b2)H0(1)( √x 3MBb3) J0(√x3MBb2) i . (28)
The derivation of h, from the Fourier transformation of the lowest-order H, is similar to that for the B → Dπ decays [3,26]. The hard scales t are chosen as the maxima of the virtualities of internal particles involved in b quark decay amplitudes, including 1/bi: t(1) e = max(√x3MB, 1/b1, 1/b3) , t(2)e = max(√x1MB, 1/b1, 1/b3) , t(1)a = max(√x3MB, 1/b2, 1/b3) , t(2) a = max(√x2MB, 1/b2, 1/b3) , (29)
which decrease higher-order corrections. The Sudakov factor in Eq. (10) suppresses long-distance contributions from the large b (i.e., large αs(t)) region, and improves the applicability of PQCD to B meson decays.
For the nonfactorizable amplitudes, the factorization formulas involve the kinematic variables of all the three mesons, and the Sudakov exponent is given by S = SB + SK2+ SK3. The integration over b3 can be performed trivially,
MP (s)46 = M P (s) 4 + M P (s) 6 , MP (s)4 = −32πCFp2NcMB2 Z 1 0 [dx] Z ∞ 0 b1db1b2db2φB(x1, b1)φK(x2) ×{[(x1− x2)φK(x3) + rKx3φ′K(x3)] E(s)′e4 (t (1) d )h (1) d (x1, x2, x3, b1, b2, b1) + [(1 − x1− x2+ x3)φK(x3) − rKx3φ′K(x3)] Ee4(s)′(t (2) d )h (2) d (x1, x2, x3, b1, b2, b1)} , (30) MP (s)6 = −32πCF p 2NcMB2 Z 1 0 [dx] Z ∞ 0 b1db1b2db2φB(x1, b1)φ′K(x2) ×rK{[(x1− x2)φK(x3) + rK(x1− x2− x3)φ′K(x3)] ×Ee6(s)′(t (1) d )h (1) d (x1, x2, x3, b1, b2, b1) + [(1 − x1− x2)φK(x3) + rK(1 − x1− x2+ x3)φ′K(x3)] ×Ee6(s)′(t (2) d )h (2) d (x1, x2, x3, b1, b2, b1)} , (31) MP (q)a46 = M P (q) a4 + M P (q) a6 , MP (q)a4 = 32πCFp2NcMB2 Z 1 0 [dx] Z ∞ 0 b1db1b2db2φB(x1, b1) ×{x3φK(1 − x2)φK(1 − x3) − rK2(x1− x2− x3)φ′K(1 − x2)φ′K(1 − x3) ×Ea4(q)′(t (1) f )h (1) f (x1, x2, x3, b1, b2, b2) −(x1+ x2)φK(1 − x2)φK(1 − x3) + r2K(2 + x1+ x2+ x3)φ′K(1 − x2)φ′K(1 − x3) ×Ea4(q)′(t (2) f )h (2) f (x1, x2, x3, b1, b2, b2)} , (32) MP (q)a6 = 32πCFp2NcMB2 Z 1 0 [dx] Z ∞ 0 b1db1b2db2φB(x1, b1) ×{[−rKx3φK(1 − x2)φ′K(1 − x3) − rK(x1− x2)φ′K(1 − x2)φK(1 − x3)] ×Ea6(q)′(t (1) f )h (1) f (x1, x2, x3, b1, b2, b2) − [rK(2 − x3)φK(1 − x2)φ′K(1 − x3) − rK(2 − x1− x2)φ′K(1 − x2)φK(1 − x3)] ×Ea6(q)′(t (2) f )h (2) f (x1, x2, x3, b1, b2, b2)} , (33)
with the definition [dx] ≡ dx1dx2dx3 and q = u and d.
The nonfactorizable amplitudes MP (q)a35 are written as
MP (q)a35 = M P (q) a3 + M P (q) a5 , MP (q)a5 = −32πCFp2NcMB2 Z 1 0 [dx] Z ∞ 0 b1db1b2db2φB(x1, b1) ×{(x1− x2)φK(1 − x2)φK(1 − x3) + r2K(x1− x2− x3)φ′K(1 − x2)φ′K(1 − x3) ×Ea5(q)′(t (1) f )h (1) f (xi, bi) +x3φK(1 − x2)φK(1 − x3) + r2K(2 + x1+ x2+ x3)φ′K(1 − x2)φ′K(1 − x3) ×Ea5(q)′(t (2) f )h (2) f (xi, bi)} , (34)
for q = u and d. The evolution factors are given by
Eei(s)′(t) = αs(t)a(s)′i (t) exp[−S(t)|b3=b1] , (35)
Eai(q)′(t) = αs(t)a(q)′i (t) exp[−S(t)|b3=b2] . (36)
The expressions of Ma1, Ma2 and MP (q)a3 are the same as M P (q)
a4 but with the Wilson coefficients a′1(tf), a′2(tf), and
a′
3(tf), respectively. The expressions of MP (s)a35 are the same as M P (q)
a35 but with the kaon wave functions φ (′)
K(1 − xi)
replaced by φ(′)K(xi), i = 2 and 3, and with the q quark replaced by the s quark. Notice the difference between the hard
parts of MP (q)a6 and M P (q)
fermion flow from the b quark to the d quark in the kaon (the ¯d quark in the B meson), i.e., the case with one fermion loop (two fermion loops). That is, the nonfactorizable contributions associated with the structure (V − A)(V + A) distinguish these two cases, while those associated with the structure (V − A)(V − A) do not.
The functions h(j), j = 1 and 2, appearing in Eqs. (30)-(33), are written as
h(j)d = [θ(b1− b2)K0(DMBb1) I0(DMBb2) +θ(b2− b1)K0(DMBb2) I0(DMBb1)] ×K0(DjMBb2) , for Dj2≥ 0 , ×iπ2H0(1)( q |D2 j|MBb2) , for D2j ≤ 0 , (37) h(j)f = iπ 2 h θ(b1− b2)H0(1)(F MBb1) J0(F MBb2) +θ(b2− b1)H0(1)(F MBb2) J0(F MBb1) i ×K0(FjMBb1) , for Fj2≥ 0 , ×iπ2H0(1)( q |F2 j|MBb1) , for Fj2≤ 0 , (38)
with the variables
D2= x1x3, D2 1= F12= (x1− x2)x3, D22= −(1 − x1− x2)x3, F2= x2x3, F2 2 = x1+ x2+ (1 − x1− x2)x3. (39)
For details of the derivation of h(j), refer to [26]. The hard scales t(j) are chosen as
t(1)d = max(DMB, q |D2 1|MB, 1/b1, 1/b2) , t(2)d = max(DMB, q |D2 2|MB, 1/b1, 1/b2) , t(1)f = max(F MB, q |F2 1|MB, 1/b1, 1/b2) , t(2)f = max(F MB, q |F2 2|MB, 1/b1, 1/b2) . (40)
In the above expressions the Wilson coefficients are defined by a1= C2+ C1 Nc , a′ 1= C1 Nc , a2= C1+C2 Nc , a′2= C2 Nc , a(q)3 = C3+C4 Nc + 3 2eq C9+C10 Nc , a(q)′3 = 1 Nc C4+ 3 2eqC10 , a(q)4 = C4+ C3 Nc +3 2eq C10+ C9 Nc , a(q)′4 = 1 Nc C3+ 3 2eqC9 ,
a(q)5 = C5+ C6 Nc + 3 2eq C7+ C8 Nc , a(q)′5 = 1 Nc C6+ 3 2eqC8 , a(q)6 = C6+ C5 Nc +3 2eq C8+ C7 Nc , a(q)′6 = 1 Nc C5+ 3 2eqC7 . (41)
Both QCD and electroweak penguin contributions have been included as shown in Eq. (41). It is expected that electroweak penguin contributions are small, as concluded in [32].
The pseudovector and pseudoscalar kaon wave functions φK and φ′K are defined by
φK(x) = Z dy+ 2π e −ixP− 3y +1 2h0|¯u(y +)γ−γ 5s(0)|πi , (42) m0K P3− φ′K(x) = Z dy+ 2π e −ixP− 3y +1 2h0|¯u(y +)γ 5s(0)|πi , (43)
respectively, satisfying the normalization Z 1 0 dxφK(x) = Z 1 0 dxφ′ K(x) = fK 2√2Nc . (44) The factor rK, rK =m0K MB , m0K = M 2 K ms+ md , (45)
with ms and md being the masses of the s and d quarks, respectively, is associated with the normalization of the
pseudoscalar wave function φ′
K. Note that we have included the intrinsic b dependence for the heavy meson wave
function φB but not for the kaon wave functions [7]. As the transverse extent b approaches zero, the B meson
wave function φB(x, b) reduces to the standard parton model φB(x), i.e., φB(x) = φB(x, b = 0), which satisfies the
normalization Z 1 0 φB(x)dx = fB 2√2Nc . (46) V. NUMERICAL ANALYSIS
In the factorization formulas derived in Sec. IV, the Wilson coefficients evolve with the hard scale t that depends on the internal kinematic variables xi and bi. Wilson coefficients at a scale µ < MW are related to the corresponding
ones at µ = MW through usual RG equations. Since the typical scale t of a hard amplitude is smaller than the b
quark mass mb = 4.8 GeV, we further evolve the Wilson coefficients from µ = mb down to µ = t. For the scale t
below the c quark mass mc = 1.5 GeV, we still employ the evolution function with f = 4, instead of with f = 3, for
simplicity, since the matching at mc is less essential. Therefore, we set f = 4 in the RG evolution between t and 1/b
governed by the quark anomalous dimension γ. The explicit expressions of Ci(µ) are referred to [7].
For the B meson wave function, we adopt the model [7] φB(x, b) = NBx2(1 − x)2exp " −1 2 xMB ωB 2 −ω 2 Bb2 2 # , (47)
with the shape parameter ωB = 0.4 GeV [33]. The normalization constant NB= 91.7835 GeV is related to the decay
φK(x) = 3 √ 2Nc fKx(1 − x)[1 + 0.51(1 − 2x) + 0.3(5(1 − 2x)2− 1)] , (48) φ′ K(x) = 3 √ 2Nc fKx(1 − x) . (49)
φK is derived from QCD sum rules [34], where the second term 1 − 2x, rendering φK a bit asymmetric, corresponds
to SU (3) symmetry breaking effect. The decay constant fK is set to 160 MeV (in the convention fπ = 130 MeV).
The wave funcitons φB and φ′K were determined from the data of the B → Kπ decays [7].
We employ GF = 1.16639 × 10−5 GeV−2, the Wolfenstein parameters λ = 0.2196, A = 0.819, and Rb = 0.38, the
unitarity angle φ3 = 90o, the masses MB = 5.28 GeV, MK = 0.49 GeV, and ms = 100 MeV, which correspond to
m0K = 2.22 GeV [7], and the ¯Bd0 (B−) meson lifetime τB0 = 1.55 ps (τB− = 1.65 ps) [28]. Our predictions for the
branching ratio of each mode are
B(B+→ K+K0) = 1.47 × 10−6, B(B−→ K−K0) = 1.84 × 10−6, B(Bd0→ K+K−) = 3.27 × 10−8, B( ¯Bd0→ K−K+) = 5.90 × 10−8, B(Bd0→ K0K¯0) = 1.75 × 10−6, B( ¯Bd0→ K0K¯0) = 1.75 × 10−6. (50)
The above values are lower than those of the B → ππ decays [8,9]. Since the B0
d → K±K∓ modes involved only
nonfactorizable annihilation amplitudes, their branching ratios are much smaller than those of the B± → K±K0
and B0
d → K0K¯0 modes. As explained in Sec. II, a large deviation of future experimental data from the predicted
B0
d → K±K∓ branching ratios will imply the existence of large FSI effects.
So far, CLEO gives only the upper bound of the B → KK decays [19]: B(B± → K±K0
) < 5.1 × 10−6,
B(Bd0→ K±K∓) < 2.0 × 10−6. (51)
We also quote the upper bound
B(B0d→ K0K0) < 1.7 × 10−5, (52)
from [28]. Obviously, our preictions are consistent with the above data. The CP asymmetries are defined by
ACP =B( ¯B → KK) − B(B → KK)
B( ¯B → KK) + B(B → KK). (53)
Employing the above set of parameters and φ3= 90o, we predict
ACP(B± → K±K0) = 0.11 ,
ACP(Bd0→ K±K∓) = 0.29 ,
ACP(Bd0→ K0K0) = 0 . (54)
Basically, the values are of the same order of those in the B → Kπ decays [7]. The CP asymmetry in the K0K¯0modes
vanishes, because they involve only penguin contributions. Measurements of the CP asymmetry in the Bd±→ K±K0
can justify the PQCD evaluation of annihilation and nonfactorizable contributions to two-body B meson decays, and distinguish the FA, BBNS and PQCD approaches. The significant CP asymmetry observed in the B0
d → K0K0 will
indicate strong FSI effects.
The dependences of the B → KK branching ratios on the angle φ3are displayed in Fig. 4. The branching ratios of
the K±K0 modes increase with φ
3, while those of the K±K∓ modes decrease with φ3. The branching ratios of the
K0K¯0 modes are insensitive to the variation of φ
3. The variation with φ3 is mainly a consequence of the inteference
between the penguin contributions and the nonfactorizable annihilation contributions Ma from the tree operators.
Since Ma1in Eqs. (15) and (16) and Ma2in Eqs. (17) and (18) contain the Wilson coefficients a′1and a′2, respectively,
which are opposite in sign, the behaviors of the branching ratios with φ3 in Figs. 4(a) and 4(b) are different.
The dependences of the CP asymmetries on the angle φ3are displayed in Fig. 5. The CP asymmetry in the K0K¯0
modes remains vanishing. The CP asymmetry in the K±K∓ modes drop suddenly from 70% to zero near the high
end of φ3. Since their branching ratios and the denominator in the definition of ACP are small, the variation with φ
is amplified. Figures 4 and 5 can be employed to determine the range of the angle φ3, when compared with future
VI. CONCLUSION
In this paper we have predicted the branching ratios and the CP asymmetries of all the B → KK modes using PQCD factorization theorem. The unitarity angle φ3 = 90o and the universal B and K meson wave functions
extracted from the data of the B → Kπ and ππ decays have been employed. The dependences of the branching ratios and the CP asymmetries on the angle φ3 have been also presented. These predictions can be confronted with future
experimental data. We believe that these modes can be observed in B factories, which have started their operation recently.
The B → KK decays are very important for understanding dynamics of nonleptonic two-body B meson decays, such as FSI, annihilation and nonfactorizable effects. In the PQCD formalism FSI effects have been assumed to be small. As explained in Sec. II, the B → KK decays are more sensitive to these effects compared to the B → Kπ and ππ decays. Hence, the comparision of our predictions, especially for the CP asymmetry in the B0
d → K0K0 decays
and for the B0
d → K±K∓ branching ratios, with future data provides a justification of the assumption. In PQCD
the CP asymmetry of the B± → K±K0 modes depends on annihilation amplitudes. It has been argued that CP
asymmetries in the B → KK decays are small in the FA and BBNS approaches, where annihilation contributions have been neglected. Therefore, experimental data of CP asymmetries will distinguish the FA, BBNS and PQCD approaches. The B0
d→ K±K∓modes involve only nonfactorizable annihilation amplitudes, such that their branching
ratios can not be estimated in FA and BBNS. Future data of these modes can also verify the PQCD evaluation of the above contributions.
We thank X.G. He, Y.Y. Keum and A.I. Sanda for useful duscussion. This work was supported in part by the National Center of Theoretical Science, by the National Science Council of R.O.C. under the Grant No. NSC-89-2112-M-006-004, and by the Grant-in Aid for Scientific Exchange from the Ministry of Education, Science and Culture of Japan.
[1] M. Bauer, B. Stech, M. Wirbel, Z. Phys. C 34, 103 (1987); Z. Phys. C 29, 637 (1985). [2] H.Y. Cheng, H-n. Li, and K.C. Yang, Phys. Rev. D 60, 094005 (1999).
[3] T.W. Yeh and H-n. Li, Phys. Rev. D 56, 1615 (1997).
[4] H-n. Li and H.L. Yu, Phys. Rev. Lett. 74, 4388 (1995); Phys. Lett. B 353, 301 (1995); Phys. Rev. D 53, 2480 (1996). [5] H-n. Li, Phys. Rev. D 52, 3958 (1995).
[6] C.H. Chang and H-n. Li, Phys. Rev. D 55, 5577 (1997).
[7] Y.Y. Keum, H-n. Li, and A.I. Sanda, hep-ph/0004004; hep-ph/0004173. [8] C. D. L¨u, K. Ukai, and M. Z. Yang, hep-ph/0004213.
[9] Y.Y. Keum and H-n. Li, hep-ph/0006001.
[10] R. Fleischer, Eur. Phys. J. C 6, 451 (1999), Phys. Lett. B435, 221 (1998); M. Neubert and J. Rosner, Phys. Lett. B441, 403 (1998); M. Neubert, Phys. Lett. B424, 152(1998); H. Jin, hep-ph/9805235; D.S. Du, X.Q. Li, Z.T. Wei and B.S. Zou, Eur. Phys. J. A 4, 91 (1999); A.F. Falk, A.L. Kagan, Y. Nir and A.A. Petrov, Phys. Rev. D57, 4290 (1998); M. Neubert, JHEP 9902, 014 (1999); D. Atwood and A. Soni, Phys. Rev. D 58, 036005 (1998).
[11] G. Kramer, W.F. Palmer and H. Simma, Z. Phys. C 66, 429 (1995). [12] A.N. Kamal, Phys. Rev. D 60, 094018 (1999).
[13] J.D. Bjorken, Nucl. Phys. (Proc. Suppl.) B11, 325 (1989). [14] G.P. Lepage and S.J. Brodsky, Phys. Rev. D 22, 2157 (1980). [15] H-n. Li and B. Tseng, Phys. Rev. D 57, 443 (1998).
[16] X.G. He, Eur. Phys. J. C 9, 443 (1999); N.G. Deshpande, X.G. He, and J.Q. Shi, hep-ph/0002260.
[17] M. Gronau and J.L. Rosner, Phys. Rev. D 58, 113005 (1998); M. Gronau and D. Pirjol, Phys. Rev. D 61, 013005 (2000). [18] B. Melic, B. Nizic and K. Passek, Phys. Rev. D 60, 074004 (1999).
[19] CLEO Coll., D. Cronin-Hennessy et al., hep-ex/0001010.
[20] N.G. Deshpande, X.G. He, W.S. Hou and, S. Pakvasa, Phys. Rev. Lett. 82, 2240 (1999); W.S. Hou, J.G. Smith, and F. W¨urthwein, hep-ex/9910014.
[22] M. Beneke, G. Buchalla, M. Neubert, and C.T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999); hep-ph/0006124. [23] D. Du, D. Yang, and G. Zhu, hep-ph/0005006; T. Muta, A. Sugamoto, M.Z. Yang, and Y.D. Yang, hep-ph/0006022. [24] M Bander, D. Silverman and A. Soni, Phys. Rev. Lett. 43, 242 (1979).
[25] A. Ali and C. Greub, Phys. Rev. D 57, 2996 (1998).
[26] C.Y. Wu, T.W. Yeh, and H-n. Li, Phys. Rev. D 53, 4982 (1996).
[27] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Review of Modern Physics, 68, 1125 (1996). [28] Particle Data Group, C. Caso et al., Eur. Phys. J. C 3, 1 (1998).
[29] J.C. Collins and D.E. Soper, Nucl. Phys. B193, 381 (1981). [30] J. Botts and G. Sterman, Nucl. Phys. B325, 62 (1989). [31] H-n. Li and G. Sterman Nucl. Phys. B381, 129 (1992). [32] D. Atwood and A. Soni, Phys. Lett. B 466, 326 (1999). [33] M. Bauer and M. Wirbel, Z. Phys. C 42, 671 (1989). [34] P. Ball, JHEP 9809, 005 (1998).
[35] CLEO Coll., Y. Kwon et al., hep-ex/9908039.
modes intermediate affected branching ratios for data of
states topologies intermediate states [35,28] branching ratios B+→ K+¯ K0 ¯ D+ D0 Pc <6.7 × 10−3 <5.1 × 10−6 B0 d→K + K− π+π− T, P u 4.7+1.8 −1.5 ±0.6 × 10−6 <2.0 × 10−6 π0π0 T, Pu <9.3 × 10−6 K0¯ K0 Pt <1.7 × 10−5 B0 d→K 0¯ K0 D+ D− P c <5.9 × 10−3 <1.7 × 10−5 π+π− Pu 4.7+1.8 −1.5 ±0.6 × 10−6 π0 π0 Pu <9.3 × 10−6
TABLE I. FSI effects in the B → KK decays.
Figure Captions
1. Fig. 1: Feynman diagrams for the B±→ K±K0 decays.
2. Fig. 2: Feynman diagrams for the B0
d → K±K∓ decays.
3. Fig. 3: Feynman diagrams for the B0
d → K0K¯0 decays.
4. Fig. 4: Dependences of the branching ratios on φ3 for (a) the B± → K±K0 modes, (b) the B0d → K±K∓
modes and (c) the B0
d → K0K¯0modes. The upper (lower) lines correspond to the ¯B (B) meson decays.
5. Fig. 5: Dependence of CP asymmetries on φ3 for (a) the B± → K±K0 modes, (b) the Bd0→ K±K∓ modes
