Published for SISSA by
Springer
Received: July 11, 2011 Accepted: August 12, 2011 Published: September 6, 2011
Hai-Yang Cheng and Sechul Oh Institute of Physics, Academia Sinica, Taipei 115, Taiwan
E-mail:
[email protected],
[email protected] Abstract: Using flavor SU(3) symmetry, we perform a model-independent analysis of ¯u,d (B ¯s ) → P P, P V decays. All the relevant topological diagrams, including the charmless B presumably subleading diagrams, such as the QCD- and EW-penguin exchange diagrams and flavor-singlet weak annihilation ones, are introduced. Indeed, the QCD-penguin exchange diagram turns out to be important in understanding the data for penguin-dominated decay modes. In this work we make efforts to bridge the (model-independent but less quantitative) topological diagram or flavor SU(3) approach and the (quantitative but somewhat model-dependent) QCD factorization (QCDF) approach in these decays, by explicitly showing how to translate each flavor SU(3) amplitude into the corresponding terms in the QCDF framework. After estimating each flavor SU(3) amplitude numerically using QCDF, we discuss various physical consequences, including SU(3) breaking effects and some useful SU(3) ¯s → P V and B ¯d → P V . relations among decay amplitudes of B Keywords: B-Physics, Rare Decays, Heavy Quark Physics, CP violation
c SISSA 2011
doi:10.1007/JHEP09(2011)024
JHEP09(2011)024
Flavor SU(3) symmetry and QCD factorization in B → P P and P V decays
Contents 1 Introduction
1
2 Flavor SU(3) analysis and QCD factorization 2.1 SU(3)F decomposition of decay amplitudes 2.2 The SU(3)F amplitudes and QCD factorization
2 7 9
4 Conclusion
1
21
Introduction
A large number of hadronic Bu,d decay events have been collected at the B factories which enable us to make accurate measurements of branching fractions (BFs) and direct CP asymmetries for many modes. With the advent of the LHCb experiment, a tremendous amount of new experimental data on B decays is expected to be obtained. In particular, various decay processes of heavier Bs and Bc mesons as well as very rare B decay modes are expected to be observed. In earlier works on hadronic decays of B mesons, the factorization hypothesis, based on the color transparency argument, was usually assumed to estimate the hadronic matrix elements which are inevitably involved in theoretical calculations of the decay amplitudes for these processes. Under the factorization assumption, the matrix element of a four-quark operator is expressed as a product of a decay constant and a form factor. Naive factorization is simple but fails to describe color-suppressed modes. This is ascribed to the fact that color-suppressed decays receive sizable nonfactorizable contributions that have been neglected in naive factorization. Another issue is that the decay amplitude under naive factorization is not truly physical because the renormalization scale and scheme dependence of the Wilson coefficients ci (µ) are not compensated by that of the matrix element hM1 M2 |Oi |Bi(µ). In the improved “generalized factorization” approach [1, 2], nonfactorizable effects are absorbed into the parameter Nceff , the effective number of colors. This parameter can be empirically determined from experiment. With the advent of heavy quark effective theory, nonleptonic B decays can be analyzed systematically within the QCD framework. There are three popular approaches available in this regard: QCD factorization (QCDF) [3, 4], perturbative QCD (pQCD) [5] and softcollinear effective theory (SCET) [6]. In QCDF and SCET, power corrections of order ΛQCD /mb are often plagued by the end-point divergence that in turn breaks the factorization theorem. As a consequence, the estimate of power corrections is generally model
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3 Numerical analysis of flavor SU(3) amplitudes in QCDF 11 3.1 Magnitudes and strong phases of the SU(3)F amplitudes 14 3.2 Estimates of decay amplitudes, SU(3)F breaking effects and SU(3)F relations 19
2
Flavor SU(3) analysis and QCD factorization
It has been established sometime ago that a least model-dependent analysis of heavy meson decays can be carried out in the so-called quark-diagram approach.1 In this diagrammatic scenario, all two-body nonleptonic weak decays of heavy mesons can be expressed in terms 1
It is also referred to as the flavor-flow diagram or topological-diagram approach in the literature.
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dependent and can only be studied in a phenomenological way. In the pQCD approach, the endpoint singularity is cured by including the parton’s transverse momentum. Because a reliable evaluation of hadronic matrix elements is very difficult in general, an alternative approach which is essentially model independent is based on the diagrammatic approach [7–12]. In this approach, the topological diagrams are classified according to the topologies of weak interactions with all strong interaction effects included. Based on flavor SU(3) symmetry, this model-independent analysis enables us to extract the topological amplitudes and sense the relative importance of different underlying decay mechanisms. When enough measurements are made with sufficient accuracy, we can extract the diagrammatic amplitudes from experiment and compare to theoretical estimates, especially checking whether there are any significant final-state interactions or whether the weak annihilation diagrams can be ignored as often asserted in the literature. The diagrammatic approach was applied to hadronic B decays first in [13]. Various topological amplitudes have been extracted from the data in [14–17] after making some reasonable approximations. Based on SU(3) flavor symmetry, the decay amplitudes also can be decomposed into linear combinations of the SU(3)F amplitudes which are SU(3) reduced matrix elements [18– 39]. This approach is equivalent to the diagrammatic approach when SU(3) flavor symmetry is imposed to the latter. In this work we make efforts to bridge these two different approaches, using QCDF and ¯u,d (B ¯s ) → P P, P V decays. For this aim, we first introduce all flavor SU(3) symmetry, in B the relevant topological diagrams, including the presumably subleading diagrams, such as the QCD- and EW-penguin exchange ones and flavor-singlet weak annihilation ones, some of which turn out to be important especially in penguin-dominant decay processes. Then all these decay modes are analyzed by using the intuitive topological diagrams and expressed in terms of the SU(3)F amplitudes. Each SU(3)F amplitude is translated into the corresponding terms in the framework of QCDF. Applying these relations, one can easily find the rather sophisticated results of the relevant decay amplitudes calculated in the QCDF framework. The magnitude and the strong phase of each SU(3)F amplitude are numerically estimated in QCDF. We further discuss some examples of the applications, including the effects of SU(3)F breaking and useful SU(3)F relations among decay amplitudes. This paper is organized as follows. In section II, we introduce topological quark dia¯u,d (B ¯s ) → P P, P V decays and the framework of QCDF. The explicit grams relevant to B SU(3)F decomposition of the decay amplitudes and the relations between the SU(3)F amplitudes and the QCDF terms are presented. In section III, we make a numerical estimation of the SU(3)F amplitudes and discuss its consequences and some applications. Our conclusions are given in section IV.
(i) Tree and penguin amplitudes T : color-favored tree amplitude (equivalently, external W -emission), C: color-suppressed tree amplitude (equivalently, internal W -emission), P : QCD-penguin amplitude, S: singlet QCD-penguin amplitude involving SU(3)F -singlet mesons (e.g., η (′) , ω, φ), PEW : color-favored EW-penguin amplitude, C : color-suppressed EW-penguin amplitude, PEW (ii) Weak annihilation amplitudes E: W -exchange amplitude, A: W -annihilation amplitude, (E and A are often jointly called “weak annihilation”.) P E: QCD-penguin exchange amplitude, P A: QCD-penguin annihilation amplitude, P EEW : EW-penguin exchange amplitude, P AEW : EW-penguin annihilation amplitude, (P E and P A are also jointly called “weak penguin annihilation”.) (iii) Flavor-singlet weak annihilation amplitudes: all involving SU(3)F -singlet mesons3 SE: singlet W -exchange amplitude, 2
Historically, the quark-graph amplitudes T, C, E, A, P named in [22–26] were originally denoted by A, B, C, D, E, respectively, in [10–12]. 3 The singlet amplitudes SE and SA were first discussed in [40–43] as the disconnected hairpin amplitudes and denoted by Eh and Ah , respectively, in [42, 43].
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of six distinct quark diagrams [7, 9–12]:2 T , the color-allowed external W -emission tree diagram; C, the color-suppressed internal W -emission diagram; E, the W -exchange diagram; A, the W -annihilation diagram; P , the horizontal W -loop diagram; and V , the vertical W -loop diagram. (The one-gluon exchange approximation of the P graph is the so-called “penguin diagram”.) For the analysis of charmless B decays, one adds the variants of the penguin diagram such as the electroweak penguin and the penguin annihilation and singlet penguins, as will be discussed below. It should be stressed that these diagrams are classified according to the topologies of weak interactions with all strong interaction effects encoded, and hence they are not Feynman graphs. All quark graphs used in this approach are topological and meant to have all the strong interactions included, i.e., gluon lines are included implicitly in all possible ways. Therefore, analyses of topological graphs can provide information on final-state interactions (FSIs). ¯u,d (B ¯s ) → M1 M2 (with In SU(3)F decomposition of the decay amplitudes for B M1 M2 = P1 P2 , P V, V P ) modes [22–26], we represent the decay amplitudes in terms of topological quark diagram contributions. The topological amplitudes which will be referred to as SU(3)F amplitudes hereafter, corresponding to different topological quark diagrams, as shown in figures 1–3, can be classified into three distinct groups as follows:
M1 W
M1
W B M2
B
M2
(a) T
(b) C
W B
g
B
M2
M2
C (c) P, PEW
g
(d) S, PEW
Figure 1. Topology of possible diagrams: (a) Color-allowed tree [T ], (b) Color-suppressed tree [C], (c) QCD-penguin [P ], (d) Singlet QCD-penguin [S] diagrams with 2 (3) gluon lines for M2 C being a pseudoscalar meson P (a vector meson V ). The color-suppressed EW-penguin [PEW ] and color-favored EW-penguin [PEW ] diagrams are obtained by replacing the gluon line from (c) and all the gluon lines from (d), respectively, by a single Z-boson or photon line.
SA: singlet W -annihilation amplitude, SP E: singlet QCD-penguin exchange amplitude, SP A: singlet QCD-penguin annihilation amplitude, SP EEW : singlet EW-penguin exchange amplitude, SP AEW : singlet EW-penguin annihilation amplitude. Within the framework of QCD factorization [3, 4], the effective Hamiltonian matrix ¯ → M1 M2 (M1 M2 = P1 P2 , P V ) are written in the form elements for B X ¯ , ¯ =G √F λrp hM1 M2 |TA p + TB p |Bi (2.1) hM1 M2 |Heff |Bi 2 p=u,c
∗ with r = s, d. T p where the Cabibbo-Kobayashi-Maskawa (CKM) factor λrp ≡ Vpb Vpr A describes contributions from naive factorization, vertex corrections, penguin contractions and spectator scattering expressed in terms of the flavor operators api , while TB p contains annihilation topology amplitudes characterized by the annihilation operators bpj . The flavor operators api are basically the Wilson coefficients in conjunction with short-distance nonfactorizable corrections such as vertex corrections and hard spectator interactions. In general, they have the expressions [3, 4, 44] i 4π 2 ci±1 ci±1 CF αs h p Vi (M2 )+ ai (M1 M2 ) = ci + Hi (M1 M2 ) +Pip (M2 ) , (2.2) Ni (M2 )+ Nc Nc 4π Nc
where i = 1, . . . , 10, the upper (lower) signs apply when i is odd (even), ci are the Wilson coefficients, CF = (Nc2 − 1)/(2Nc ) with Nc = 3, M2 is the emitted meson and M1 shares
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g
M1
W
M1
g B
M1
M1
W
g
B W
M2
(e) E
M2
(f ) A
W g
B
W
g g
M2
(g) P E, P EEW
M2
(h) P A, P AEW
Figure 2. (e) W -exchange [E], (f) W -annihilation [A], (g) QCD-penguin exchange [P E], (h) QCD-penguin annihilation [P A] diagrams. The EW-penguin exchange [P EEW ] and EW-penguin annihilation [P AEW ] diagrams are obtained from (g) and (h), respectively, by replacing the left gluon line by a single Z-boson or photon line. The gluon line of (e) and (f) and the right gluon line of (g) and (h) can be attached to the fermion lines in all possible ways.
the same spectator quark with the B meson. The quantities Ni (M2 ) = 0 or 1 for i = 6, 8 and M2 = V or else, respectively. The quantities Vi (M2 ) account for vertex corrections, Hi (M1 M2 ) for hard spectator interactions with a hard gluon exchange between the emitted meson and the spectator quark of the B meson and Pi (M2 ) for penguin contractions. ¯ → M1 M2 (M1 M2 = The weak annihilation contributions to the decay B P1 P2 , P V, V P ) can be described in terms of the building blocks bpj and bpj,EW : X X GF X r ¯ = iG √ √F (dj bpj + d′j bpj,EW ). λp hM1 M2 |TB p |Bi λrp fB fM1 fM2 2 p=u,c 2 p=u,c j
(2.3)
The building blocks have the expressions [3, 4] i CF CF h p f f i i i c A , b = c A + c (A + A ) + N c A 1 3 5 c 6 1 1 3 3 3 3 , Nc2 Nc2 i CF CF h b2 = 2 c2 Ai1 , bp4 = 2 c4 Ai1 + c6 Af2 , Nc Nc i h CF bp3,EW = 2 c9 Ai1 + c7 (Ai3 + Af3 ) + Nc c8 Ai3 , Nc CF bp4,EW = 2 c10 Ai1 + c8 Ai2 . Nc b1 =
(2.4)
The subscripts 1,2,3 of Ai,f n denote the annihilation amplitudes induced from (V − A)(V − A), (V − A)(V + A) and (S − P )(S + P ) operators, respectively, and the superscripts i and f refer to gluon emission from the initial and final-state quarks, respectively. We choose
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B
M1
M1
g
M1
M1
g g B
M2
W
M2
B W
(i) SE
(j) SA
M1
g g
B
g
M2
B
W
M2 g
(k) SP E, SP EEW
(l) SP A, SP AEW
Figure 3. (i) Flavor-singlet W -exchange [SE], (j) Flavor-singlet W -annihilation [SA], (k) Flavorsinglet QCD-penguin exchange [SP E], (l) Flavor-singlet QCD-penguin annihilation [SP A] diagrams. The Flavor-singlet EW-penguin exchange [SP EEW ] and flavor-singlet EW-penguin annihilation [SP AEW ] diagrams are obtained from (k) and (l), respectively, by replacing the leftest gluon line by a single Z-boson or photon line. The double gluon lines of (i), (j), (k) and (l) are shown for the case of M1 = P . They are replaced by three gluon lines when M1 = V . Each of the gluon lines of (i), (j), (k) and (l) can be separately attached to the fermion lines in all possible ways.
the convention that M1 contains an antiquark from the weak vertex and M2 contains a quark from the weak vertex, as in ref. [44]. For the explicit expressions of vertex, hard spectator corrections and annihilation contributions, we refer to [3, 4, 44, 45] for details. In practice, it is more convenient to express the decay amplitudes in terms of the flavor operators αpi and the annihilation operators βjp which are related to the coefficients api and bpj by α1 (M1 M2 ) = a1 (M1 M2 ) , α2 (M1 M2 ) = a2 (M1 M2 ) , ( p p a3 (M1 M2 ) − a5 (M1 M2 ) for M1 M2 = P P, V P , αp3 (M1 M2 ) = ap3 (M1 M2 ) + ap5 (M1 M2 ) for M1 M2 = P V , ( ap4 (M1 M2 ) + rχM2 ap6 (M1 M2 ) for M1 M2 = P P, P V , p α4 (M1 M2 ) = ap4 (M1 M2 ) − rχM2 ap6 (M1 M2 ) for M1 M2 = V P , ( ap9 (M1 M2 ) − ap7 (M1 M2 ) for M1 M2 = P P, V P , p α3,EW (M1 M2 ) = ap9 (M1 M2 ) + ap7 (M1 M2 ) for M1 M2 = P V , ( ap10 (M1 M2 ) + rχM2 ap8 (M1 M2 ) for M1 M2 = P P, P V , p α4,EW (M1 M2 ) = ap10 (M1 M2 ) − rχM2 ap8 (M1 M2 ) for M1 M2 = V P ,
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(2.5)
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M1
W
and βjp (M1 M2 ) =
ifB fM1 fM2 p ¯ 1 ,M2 ) bj (M1 M2 ) , X (BM
(2.6)
where fM is the decay constant of a meson M and the chiral factors rχM2 are given by rχP (µ) =
2m2P , mb (µ)(m2 + m1 )(µ)
rχV (µ) =
2mV fV⊥ (µ) , mb (µ) fV
(2.7)
¯
X (BP1 ,P2 ) ≡ hP2 |J µ |0ihP1 |Jµ′ |Bi = ifP2 (m2B − m2P1 ) F0BP1 (m2P2 ) , ¯
X (BP,V ) ≡ hV |J µ |0ihP |Jµ′ |Bi = 2fV mB pc F1BP (m2V ) , ¯
2 X (BV,P ) ≡ hP |J µ |0ihV |Jµ′ |Bi = 2fP mB pc ABV 0 (mP ) ,
(2.8)
with pc being the c.m. momentum. Here we have followed the conventional Bauer-StechBP and ABV [46, 47]. Wirbel definition for form factors F0,1 0 2.1
SU(3)F decomposition of decay amplitudes
¯u,d (B ¯s ) → M1 M2 modes can be analyzed by the relevant quark The decay amplitudes of B diagrams and written in terms of the SU(3)F amplitudes. The decomposition of the decay amplitudes of these modes is displayed in tables 1–24. In these tables, the subscript M1 (ζ) (or M2 ) of the amplitudes TM1 [M2 ] , · · · , etc., represents the case that the meson M1 (or M2 ) contains the spectator quark in the final state. The superscript ζ of the amplitudes is only applied to the case involving an η (′) or an ω/φ, or both η (′) and ω/φ in the final state and denotes the quark content (ζ = q, s, c) of η (′) and ω/φ with q = u or d. The ¯ 0 (B ¯s ) → η (′) η (′) , two values value of ζ is shown in the parenthesis as (q), (s) or (c). For B of ζ are shown in one parenthesis: e.g., (q, s), where q and s denote the quark content of the first and second η (′) , respectively. A similar rule is also applied to the case of ¯ 0 (B ¯s ) → η (′) ω/φ. On the other hand, to distinguish the decays with |∆S| = 1 from B those with ∆S = 0, we will put the prime to all the SU(3)F amplitudes for the former, (′) ′(ζ) for example TM1 [M2 ] . The SU(3)F -singlet amplitudes SM1 [M2 ] are involved only when the SU(3)F -singlet meson(s) (η, η′ , ω, φ) appear(s) in the final state. ¯0 We will give some examples for illustration. The decay amplitude of B − → π − K ¯u → P P mode with |∆S| = 1 can be written, from tables 4–5, as which is a B 2 1 C′ ′ + A′π + P Eπ′ + P EEW,π . AB − →π− K¯ 0 = Pπ′ − PEW,π 3 3
(2.9)
¯ 0 → η (′) K ¯ ∗0 which is a B ¯d → P V mode with |∆S| = 1 can be The decay amplitude of B
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with fV⊥ (µ) being the scale-dependent transverse decay constant of the vector meson V . The relevant factorizable matrix elements are
K∗
(2.10) (′)
where the superscripts (q), (s) and (c) represent the quark contents of η (′) , such as ηq , (′) (′) ¯ 0 → η (′) ω/φ ηs and ηc , respectively. Likewise, from tables 7–9, the decay amplitude of B ¯d → P V mode with ∆S = 0 is given by which is a B 1 (q, q) 1 C, (q, q) (q, q) (q, q) (q, q) 2 AB¯ 0 →η(′) ω/φ = Cη(′) + 2Sη(′) + Pη(′) + PEW, η(′) − PEW, η(′) 3 3 1 1 (q, q) (q, q) (q, q) (q, q) (q, q) +Eη(′) + P Eη(′) + 2P Aη(′) − P EEW, η(′) + P AEW, η(′) 3 3 2 (q, q) (q, q) (q, q) (q, q) (q, q) 2 +2SEη(′) +2SP Eη(′) +4SP Aη(′) − SP EEW, η(′) + SP AEW, η(′) 3 3 √ (q, s) 1 (q, s) (q, s) (q, s) (q, s) + 2 Sη(′) − PEW, η(′) + SEη(′) + SP Eη(′) + 2SP Aη(′) 3 1 1 (q, s) (q, s) − SP EEW, η(′) + SP AEW, η(′) 3 3 √ 2 (s, q) (s, q) + 2 2SP Aη(′) − SP AEW, η(′) 3 1 1 (s, s) (s, s) (s, s) (s, s) + 2 P Aη(′) − P AEW, η(′) + SP Aη(′) − SP AEW, η(′) 3 3 1 (q, q) 1 C, (q, q) (q, q) (q, q) (q, q) + Cω/φ + 2Sω/φ + Pω/φ + PEW, ω/φ − PEW, ω/φ 3 3 1 1 (q, q) (q, q) (q, q) (q, q) (q, q) +Eω/φ + P Eω/φ + 2P Aω/φ − P EEW, ω/φ + P AEW, ω/φ 3 3 2 (q, q) 2 (q, q) (q, q) (q, q) (q, q) +2SEω/φ +2SP Eω/φ +4SP Aω/φ − SP EEW, ω/φ + SP AEW, ω/φ 3 3 √ (q, s) 1 (q, s) (q, s) (q, s) (q, s) + 2 Sω/φ − PEW, ω/φ + SEω/φ + SP Eω/φ + 2SP Aω/φ 3 1 1 (q, s) (q, s) − SP EEW, ω/φ + SP AEW, ω/φ 3 3 √ (q, c) (q, c) + 2 Cω/φ + Sω/φ √ 2 (s, q) (s, q) + 2 2SP Aω/φ − SP AEW, ω/φ 3 1 1 (s, s) (s, s) (s, s) (s, s) + 2 P Aω/φ − P AEW, ω/φ + SP Aω/φ − SP AEW, ω/φ , (2.11) 3 3
where the superscripts (q ′ , q ′′ ) with q ′ , q ′′ = q, s denote the quark contents of (η (′) , ω/φ), (′) (′) such as (ηq , ωq /φq ), (ηq , ωs /φs ), etc. When ideal mixing for ω and φ is assumed, ωs and
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recast, from tables 10–12, to √ 2 AB¯ 0 →η(′) K¯ ∗0 2 1 ′(q) ′(q) ′(q) ′(q) ′(q) = CK ∗ + 2SK ∗ + PEW, K ∗ + 2SP EK ∗ − SP EEW, K ∗ 3 3 √ ′(s) 1 ′(s) 1 1 C′, (s) ′(s) ′(s) ′(s) + 2 SK ∗ + PK ∗ − PEW, K ∗ − PEW, K ∗ + P EK ∗ − P EEW, 3 3 3 1 ′(s) ′(s) +SP EK ∗ − SP EEW, K ∗ 3 √ ′(c) ′(c) + 2 CK ∗ + SK ∗ 1 C′, (q) 1 ′(q) ′(q) ′(q) + Pη(′) − PEW, η(′) + P Eη(′) − P EEW, η(′) , 3 3
φq terms vanish: i.e., the amplitudes with the superscripts (q, s) or (s, s) for B → η (′) ω and the superscripts (q, q) or (s, q) for B → η (′) φ are set to be zero. 2.2
The SU(3)F amplitudes and QCD factorization
¯u,d (B ¯s ) → M1 M2 (with M1 M2 = P1 P2 , P V, V P ) decays can The SU(3)F amplitudes for B be expressed in terms of the quantities calculated in the framework of QCD factorization as follows:4
(2.12)
where the superscript ζ = q, s, c, which is only applied to the case when M1 (or M2 ) = η (′) or ω/φ, or M1 M2 = η (′) η (′) or η (′) ω/φ. As mentioned before, for |∆S| = 1 decays, we will put the prime to all the SU(3)F amplitudes. The unprimed and primed amplitudes have ∗ with r = d and r = s, respectively. The weak annihilation the CKM factor λrp ≡ Vpb Vpr amplitudes are given by GF (ζ) EM1 [M2 ] = √ λru (ifB fM1 fM2 ) [b1 ]M1 M2 [M2 M1 ] , 2 GF r (ζ) AM1 [M2 ] = √ λu (ifB fM1 fM2 ) [b2 ]M1 M2 [M2 M1 ] , 2 X G (ζ) F λrp (ifB fM1 fM2 ) [bp3 ]M1 M2 [M2 M1 ] , P EM1 [M2 ] = √ 2 p=u,c GF X r (ζ) P AM1 [M2 ] = √ λp (ifB fM1 fM2 ) [bp4 ]M1 M2 [M2 M1 ] , 2 p=u,c 3 p GF X r (ζ) λp (ifB fM1 fM2 ) b , P EEW,M1 [M2 ] = √ 2 3,EW M1 M2 [M2 M1 ] 2 p=u,c 3 p GF X r (ζ) λp (ifB fM1 fM2 ) P AEW,M1 [M2 ] = √ b , 2 4,EW M1 M2 [M2 M1 ] 2 p=u,c 4
¯
The factorizable amplitude X (BM1 ,
M2 )
is denoted by AM1M2 in [44].
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(2.13)
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GF ¯ ¯ (ζ) TM1 [M2 ] = √ λru α1 (M1 M2 ) X (BM1 , M2 ) [X (BM2 , M1 ) ] , 2 GF ¯ ¯ (ζ) CM1 [M2 ] = √ λru α2 (M1 M2 ) X (BM1 , M2 ) [X (BM2 , M1 ) ] , 2 GF X r p ¯ ¯ (ζ) λp α3 (M1 M2 ) X (BM1 , M2 ) [X (BM2 , M1 ) ] , SM1 [M2 ] = √ 2 p=u,c GF X r p ¯ ¯ (ζ) PM1 [M2 ] = √ λp α4 (M1 M2 ) X (BM1 , M2 ) [X (BM2 , M1 ) ] , 2 p=u,c GF X r 3 p ¯ ¯ (ζ) λp α3,EW (M1 M2 ) X (BM1 , M2 ) [X (BM2 , M1 ) ] , PEW,M1 [M2 ] = √ 2 2 p=u,c GF X r 3 p ¯ ¯ C, (ζ) λp α4,EW (M1 M2 ) X (BM1 , M2 ) [X (BM2 , M1 ) ] , PEW,M1 [M2 ] = √ 2 2 p=u,c
and the singlet weak annihilation amplitudes by
(2.14)
where we have used the notation [bpj ]M1 M2 ≡ bpj (M1 , M2 ). Note that the weak annihilation contributions in the QCDF approach given in eq. (2.3) include all the above SU(3)F (′) (′) (′) (′) amplitudes given in eqs. (2.13) and (2.14), such as EMi , AMi , . . . , SEMi , SAMi , · · · , etc. Using the above relations, one can easily translate the decay amplitude expressed in terms of the SU(3)F amplitudes as shown in tables 1–24 into that expressed in terms of the quantities calculated in the framework of QCDF. For example, the decay amplitude ¯ 0 given in eq. (2.9) can be rewritten in terms of the quantities calculated in of B − → π − K QCDF: i 1 GF X s h ¯ p λp δpu β2 + αp4 − αp4,EW + β3p + β3,EW X (Bπ, AB − →π− K¯ 0 = √ 2 2 p=u,c
¯ K)
. (2.15)
¯ 0 → η (′) K ¯ ∗0 in eq. (2.10) now reads Likewise, the decay amplitude of B √ 2 AB¯ 0 →η(′) K¯ ∗0
( i GF X s h 1 ¯ ∗ (′) p = √ λp δpu α2 + 2αp3 + αp3,EW + β3p + β3,EW X (BK , ηq ) 2 2 p=u,c h i √ 1 1 1 p 1 p ¯ ∗ (′) p + 2 αp3 + αp4 − αp3,EW − αp4,EW + β3p − β3,EW X (BK , ηs ) + βS3 − βS3,EW 2i 2 2 2 √ h ¯ ∗ , ηc(′) ) p (BK + 2 δpc α2 + α3 X ) i h (′) ∗ 1 p 1 p ¯ p p (2.16) + α4 − α4,EW + β3 − β3,EW X (Bηq , K ) . 2 2
– 10 –
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GF (ζ) SEM1 [M2 ] = √ λru (ifB fM1 fM2 ) [bS1 ]M1 M2 [M2 M1 ] , 2 GF (ζ) SAM1 [M2 ] = √ λru (ifB fM1 fM2 ) [bS2 ]M1 M2 [M2 M1 ] , 2 GF X r (ζ) SP EM1 [M2 ] = √ λp (ifB fM1 fM2 ) bpS3 M1 M2 [M2 M1 ] , 2 p=u,c GF X r (ζ) λp (ifB fM1 fM2 ) bpS4 M1 M2 [M2 M1 ] , SP AM1 [M2 ] = √ 2 p=u,c GF X r 3 p (ζ) SP EEW,M1 [M2 ] = √ λp (ifB fM1 fM2 ) b , 2 S3,EW M1 M2 [M2 M1 ] 2 p=u,c 3 p GF X r (ζ) λp (ifB fM1 fM2 ) b , SP AEW,M1 [M2 ] = √ 2 S4,EW M1 M2 [M2 M1 ] 2 p=u,c
¯ 0 → η (′) ω/φ in eq. (2.11) is recast to Finally, the decay amplitude of B 2 AB¯ 0 →η(′) GF = √ 2
X
ω/φ
p=u,c
λdp
( h
1 1 δpu (α2 + β1 + 2βS1 ) + 2αp3 + αp4 + αp3,EW − αp4,EW 2 2
(′)
ηs
+2 (−ifB fη(′) fω/φ )
"
bp4
ωq /φq
1 1 − bp4,EW + bpS4 − bpS4,EW 2 2
#
(′)
ηs
ωs /φs
1 1 + δpu (α2 + β1 + 2βS1 ) + 2αp3 + αp4 + αp3,EW − αp4,EW 2 2 i (′) 1 p ¯ p p 1 p p p p p +β3 +2β4 − β3,EW + β4,EW +2βS3 +4βS4 −βS3,EW X (B ωq /φq , ηq ) +βS4,EW 2 2 i √ h (′) 1 p 1 p 1 p ¯ p p X (B ωq /φq , ηs ) − βS3,EW + βS4,EW + 2 δpu βS1 + α3 − α3,EW + βS3 + 2βS4 2 2 i 2 √ h ¯ ωq /φq , ηc(′) ) p (B + 2 δpc α2 + α3 X # " √ p p + 2 (−ifB fη(′) fω/φ ) 2bS4 − bS4,EW h
(′)
ωs /φs , ηq
+2 (−ifB fη(′) fω/φ )
"
bp4
1 1 − bp4,EW + bpS4 − bpS4,EW 2 2
#
ωs /φs ,
(′) ηs
)
.
(2.17)
The above results in eqs. (2.16) and (2.17) obtained by using the relations in eqs. (2.12)– (2.14) are the same as those calculated in QCDF [44]. This fact confirms the validity of our way to bridge the topological diagram or flavor SU(3) approach and the QCDF approach through the explicit relations between them given in eqs. (2.12)–(2.14). ¯ → P P, V P in QCD factorization shown in appendix A All the decay amplitudes of B of ref. [44] can be obtained from the SU(3)F amplitudes displayed in tables 1–24.
3
Numerical analysis of flavor SU(3) amplitudes in QCDF
As emphasized before, our major efforts have been made in the previous section, by first introducing a complete set of the topological diagrams, including the QCD- and EWpenguin exchange ones and flavor-singlet weak annihilation ones, followed by the SU(3) ¯u,d (B ¯s ) → P P, P V , and then by presenting decomposition of the decay amplitudes in B the explicit relations to bridge the two different approaches, using the topological diagrams and the QCDF in these decays. In this section we estimate the magnitude of each SU(3)F amplitude in the framework of QCDF by using the relations given in eqs. (2.12), (2.13) and (2.14). From
– 11 –
JHEP09(2011)024
i 1 p 1 p ¯ (′) p p p p X (Bηq , ωq /φq ) +β3p +2β4p − β3,EW + β4,EW +2βS3 +4βS4 −βS3,EW +βS4,EW 2 2 i √ h 1 p 1 p 1 p ¯ (′) p p X (Bηq , ωs /φs ) − βS3,EW + βS4,EW + 2 δpu βS1 + α3 − α3,EW + βS3 + 2βS4 2 2 # "2 √ + 2 (−ifB fη(′) fω/φ ) 2bpS4 − bpS4,EW
F0Bπ (0) = 0.25 ,
F0Bs ηs (0) ∗ ABK (0) 0 Bs φ A0 (0)
F0Bs K (0) = 0.24 , ABρ 0 (0) Bs K ∗ A0 (0)
Bηq
F0BK (0) = 0.35 ,
= 0.303 , = 0.30 ,
F0
(0) = 0.296 ,
= 0.28 , ABω 0 (0) = 0.281 ,
= 0.374 , = 0.32 .
(3.1)
√ ¯ 2, s¯ s and c¯ c labeled by the Here for η (′) we have used the flavor states q q¯ ≡ (u¯ u + dd)/ (′) ηq , ηs and ηc , respectively, and the form factors for B → η are given by ′
F Bη = F Bηq sin θ ,
F Bη = F Bηq cos θ ,
′
F Bs η = −F Bs ηs sin θ ,
F Bs η = F Bs ηs cos θ ,
(3.2)
where the small mixing with ηc is neglected and the ηq -ηs mixing angle θ defined by |ηi = cos θ|ηq i − sin θ|ηs i,
|η ′ i = sin θ|ηq i + cos θ|ηs i,
(3.3)
is (39.3 ± 1.0)◦ in the Feldmann-Kroll-Stech mixing scheme [53, 54]. Note that in this work we shall not consider the admixture of the η and η ′ states with a pseudoscalar glueball. For the decay constants, we use the values (in units of MeV) [53–55] fπ = 132 ,
fK = 160 ,
fηq′
fηq = 89 ,
= 107 ,
fρ = 216 ,
fK ∗ = 220 ,
fBu,d = 210 ,
fBs = 230 ,
fηs′ = −112 ,
fηs = 137 ,
fω = 187 ,
fφ = 215 .
(3.4)
It is known that although physics behind nonleptonic B decays is extremely complicated, it is greatly simplified in the heavy quark limit mb → ∞ as the decay amplitude becomes factorizable and can be expressed in terms of decay constants and form factors. However, this simple approach encounters three major difficulties: (i) the predicted branch¯ → P P, V P, V V decays are systematically below the ing fractions for penguin-dominated B ¯ 0 → K −π+ , B ¯ 0 → K ∗− π + , measurements [44] (ii) direct CP -violating asymmetries for B
– 12 –
JHEP09(2011)024
¯u,d (B ¯s ) → M1 M2 depend eqs. (2.12)–(2.14), it is obvious that the SU(3)F amplitudes for B on the specific final states M1 and M2 , indicating that the SU(3)F breaking effects are automatically involved in these amplitudes through the relevant decay constants, form factors, · · · , etc. i.e., the magnitudes of these SU(3)F amplitudes are inevitably process-dependent. Therefore, for the purpose of illustration to show typical magnitudes of these SU(3)F amplitudes, we shall choose typical decay processes as explained below, and use only the central values of the input parameters. For the numerical analysis, we shall use the same input values for the relevant parameters as those in ref. [48, 49]. Specifically we use the values of the form factors for Bu,d → P and Bu,d → V transitions obtained in the light-cone QCD sum rules [50, 51] and those for Bs → V transitions obtained in the covariant light-front quark model [52] with some modifications:
where Λh is a typical scale of order 500 MeV, and ρA and φA are the unknown real parameters. By adjusting the magnitude ρA and the phase φA in this scenario, all the abovementioned difficulties can be resolved. However, a scrutiny of the QCDF predictions reveals more puzzles with respect to direct CP violation, as pointed out in [48, 49, 56]. While the signs of CP asymmetries in K − π + , K − ρ0 modes are flipped to the right ones in the presence of power corrections from penguin annihilation, the signs of ACP in ¯ 0 → π0π0 , K ¯ ∗0 η will also get reversed in such a way that B − → K − π 0 , K − η, π − η and B they disagree with experiment. This indicates that it is necessary to consider subleading power corrections other than penguin annihilation. It turns out that an additional subleading 1/mb power correction to color-suppressed tree amplitudes is crucial for resolving the aforementioned CP puzzles and explaining the decay rates for the color-suppressed treedominated π 0 π 0 , ρ0 π 0 modes [48, 49, 56]. A solution to the B → Kπ CP -puzzle related to ¯ 0 → K − π + requires a large comthe difference of CP asymmetries of B − → K − π 0 and B plex color-suppressed tree amplitude and/or a large complex electroweak penguin. These two possibilities can be discriminated in tree-dominated B decays. The CP puzzles with π − η, π 0 π 0 and the rate deficit problems with π 0 π 0 , ρ0 π 0 can only be resolved by having a large complex color-suppressed tree topology C. While the New Physics solution to the B → Kπ CP puzzle is interesting, it is most likely irrelevant for tree-dominated decays. We shall use the fitted values of the parameters ρA and φA given in [48, 49]: ¯u,d (B ¯s ) → P P , For B ¯u,d (B ¯s ) → V P , For B
¯u,d (B ¯s ) → P V , For B
ρA = 1.10 (1.00) , ρA = 1.07 (0.90) , ρA = 0.87 (0.85) ,
φA = −50◦ (−55◦ ) ,
φA = −70◦ (−65◦ ) ,
φA = −30◦ (−30◦ ) .
(3.6)
Following [56], power corrections to the color-suppressed topology are parametrized as a2 → a2 (1 + ρC eiφC ),
(3.7)
with the unknown parameters ρC and φC to be inferred from experiment. We shall use [56] φC ≈ −70◦ , −80◦ , 0,
ρC ≈ 1.3 , 0.8 , 0, ¯ → P P, V P, V V decays, respectively. for B
– 13 –
(3.8)
JHEP09(2011)024
¯ 0 → π + π − and B ¯s0 → K + π − disagree with experiment in signs [56], and B − → K − ρ0 , B ¯ → V V decays (iii) the transverse polarization fraction in penguin-dominated charmless B is predicted to be very small [57], while experimentally it is comparable to the longitudinal polarization one. All these indicate the necessity of going beyond zeroth 1/mb power expansion. In the QCDF approach one considers the power correction to penguin amplitudes due to weak penguin annihilation characterized by the parameter β3p or bp3 . However, QCD-penguin exchange amplitudes involve troublesome endpoint divergences and hence they can be studied only in a phenomenological way. We shall follow [3, 4] to model the R1 endpoint divergence X ≡ 0 dx/¯ x in the annihilation diagrams as mB (3.5) XA = ln (1 + ρA eiφA ), Λh
3.1
Magnitudes and strong phases of the SU(3)F amplitudes
From now on, we shall use a notation for the relevant strong and weak phases as follows: (′) e.g., the color-favored tree amplitude for ∆S = 0 (|∆S| = 1) decays is denoted as TP ≡ T (′) T (′) T (′) (′) ∗ . with the strong phase δP and the weak phase θ T (′) = arg Vub Vud(s) |TP | ei δP +θ
V
P
used. The strong phases of the SU(3)F amplitudes are generated from the flavor operators αpi and bpj . As shown in eqs. (2.12)–(2.14), except the amplitudes T (′) , C (′) , E (′) , A(′) , SE (′) and SA(′) , all the other amplitudes including penguin ones are the sum of two terms each of which is proportional to λrp αpi or λrp bpj with p = u, c. Because the CKM factors λru and λrc involve different weak phases from each other, to exhibit the strong phase of each amplitude in the tables, we shall use the approximations, αu3,4(EW) ≃ αc3,4(EW) and bu3,4(EW) ≃ bc3,4(EW) , where the former (latter) relation holds roughly (very well) in QCDF.5 Thus, for instance, the QCD-penguin amplitude can be understood as P (′) ≡ −|P (′) |eiδ (′) (′) ∗ ). strong phase δP and the weak phase θ P = arg(Vtb Vtd(s)
P (′)
eiθ
P (′)
with the
From tables 25–28, hierarchies among the SU(3)F amplitudes are numerically found ¯u,d (B ¯s ) → P P decays, the hierarchical relation is as follows. For ∆S = 0 B (s)
(q)
|TP | > |CP | > |PP | & |P EP | > |EP | > |SP | ∼ |SP | ∼ |PEW, P | ∼ |AP | ∼ |P AP | (c)
C > |PEW, P | > |P EEW, P | ∼ |P AEW, P | ∼ |SP | ,
(3.9)
¯u,d (B ¯s ) → P V , and for B |TP,V | > |CP,V | > |P EP,V | & |PP,V | ∼ |EP,V | > |PEW, C & |PEW,
P,V |
> |P AP,V | & |P EEW,
P,V |
5
P,V |
& |P AEW,
(s)
(q)
& |AP,V | ∼ |SP,V | ∼ |SP,V |
P,V |
(c)
∼ |SV | .
(3.10)
In QCDF the value of αci (i = 3, 4, (3, EW), (4, EW)) differ from that of αui by about (25 − 30)%, while the value of bci is the same as that of bui . It should be emphasized that using eqs. (2.12)–(2.14), one can compute both the magnitude and strong phase of each SU(3)F amplitude without invoking these approximations on αu,c and bu,c i i . In our numerical analysis, we use these approximations only for expressing the strong phases as shown in tables 25–28. All the magnitudes of the amplitudes are obtained without using these approximations.
– 14 –
JHEP09(2011)024
The numerical estimates of the SU(3)F amplitudes are displayed in tables 25–28. In ¯u,d (B ¯s ) → P1 P2 decays with ∆S = 0, the modes B ¯u,d (B ¯s ) → ππ (πK) are used the case of B to numerically compute the relevant SU(3)F amplitudes such as TP , CP , PP and so on, (q,s,c) ¯u,d (B ¯s ) → πη (Kη) are used for. Similarly, for |∆S| = 1 decays except for SP which B ¯ ¯ the processes Bu,d (Bs ) → πK (KK) are used to estimate TP′ , CP′ , PP′ and so on, except ′(q,s,c) ¯u,d (B ¯s ) → Kη (ηη) are used for. In the case of ∆S = 0 B ¯u,d (B ¯s ) → P V which B for SP ¯u,d (B ¯s ) → πρ (πK ∗ , Kρ) are used for the numerical calculation of the decays, the modes B (q,s,c) (q,s,c) ¯u,d (B ¯s ) → for which B and SV amplitudes TP,V , CP,V , PP,V and so on, except for SP πω/φ (Kω/φ) and ηρ (ηK ∗ ), respectively, are used. For |∆S| = 1 decays the processes ¯u,d (B ¯s ) → πK ∗ , Kρ (KK ∗ ) are used to estimate T ′ , C ′ , P ′ and so on, except for B P,V P,V P,V ′(q,s,c) ′(q,s,c) ¯ ¯ for which Bu,d (Bs ) → Kω/φ (ηω/φ) and ηK ∗ (ηφ), respectively, are and S S
¯u,d (B ¯s ) → P P decays, the hierarchical relation is found to be Likewise, for ∆S = 1 B ′(q)
′(s)
′ ′ ′ |PP′ | & |P EP′ | > |TP′ | & |PEW, P | & |CP | & |SP | ∼ |SP | ∼ |P AP | ′(c)
C′ ′ ′ ′ ′ > |PEW, P | > |EP | > |AP | ∼ |P EEW, P | ∼ |P AEW, P | & |SP | ,
(3.11)
¯u,d (B ¯s ) → P V , and for B ′ ′ ′ ′ |P EP,V | & |PP,V | > |TP,V | & |PEW,
′(s)
′(q)
C′ ′ > |CP,V | ∼ |SP,V | ∼ |SP,V | & |PEW,
P,V |
′(c) ′ ′ ′ ′ ′ V | & |P AP,V | ∼ |EP,V | > |AP,V | > |P EEW, P | ∼ |P AEW, P,V | ∼ |SV | .
(3.12)
Several remarks are in order: 1. It is well known that the penguin contributions are dominant in |∆S| = 1 decays due ∗ | and the large top quark mass. to the CKM enhancement |Vcs Vcb∗ | ≈ |Vts Vtb∗ | ≫ |Vus Vub Especially, it is interesting to note that in addition to the QCD-penguin contributions ′ , the QCD-penguin exchange ones P E ′ ¯ ¯ PP,V P,V are large for Bu,d (Bs ) → P P and P V ′ decays. Since the strong phase of P EP (V ) is comparable to that of PP′ (V ) in magnitude E′ ∼ δ P ′ ), the effects from P E ′ ′ with the same sign (i.e., δPP (V ) P (V ) P (V ) and PP (V ) are strongly constructive to each other. It has been shown [48, 49] that in order to accommodate the data including the branching fractions and CP asymmetries of ′ those decays, the QCD-penguin exchange contributions (P EP,V ∝ bp3 ) are important ¯u,d → P P decays, the and play a crucial role. For example, for penguin-dominated B effects of the QCD-penguin exchange dictated by the values of ρA and φA paly a key role in resolving the problems of the smallness of predicted decay rates and of the wrong sign of the predicted direct CP asymmetry ACP (π + K − ). Also, for Bu,d → Kρ and πK ∗ decays, the QCD-penguin exchange contributions will enhance the rates by (15 ∼ 100)% for Kρ modes and by a factor of 2 ∼ 3 for πK ∗ ones. (′)
2. The SU(3)F -singlet contributions SP,V are involved in the decay modes including ¯ → πη (′) , Kη (′) , πω/φ, Kω/φ, · · · , etc. They η (′) , ω, φ in the final state, such as B are expected to be small because of the Okubo-Zweig-Iizuka (OZI) suppression rule (′) (′) which favors connected quark diagrams. Indeed they are found to be: |SP /PP | ≈ ¯u,d(B ¯s ) → ¯u,d (B ¯s ) → P P and |S (′) /P (′) | ≈ (11 ∼ 27)% for B (10 ∼ 24)% for B P,V P,V P V .6 In contrast, in the framework of generalized factorization, the SU(3)F -singlet contribution depend strongly on the parameter ξ ≡ 1/Nc (Nc being the effective ¯u,d → V V decays [27]: e.g., up number of color) and could be large, particularly for B to 77% of the dominant QCD-penguin contribution. In the flavor SU(3) analyses with a global fit of the SU(3)F amplitudes to the data, a large effect from SP′ is also needed for explaining the large BFs of the B → η ′ K modes [58–61]: e.g., |SP′ /PP′ | ≈ 38% [16]. Among the two-body B decays, B → Kη ′ has the largest branching fraction, of order 70 × 10−6 , while B(B → ηK) is only (1 ∼ 3) × 10−6 [62]. This can be qualitatively
6
(′)
(′)
When the effects from P EP,V which are comparable to PP,V are taken into account, the ratio (′) (′) (′) ¯u,d (B ¯s ) → P P and P V . |SP,V /(PP,V + P EP,V )| becomes . 10% for B
– 15 –
JHEP09(2011)024
′ > |P EEW,
P,V |
understood as the interference between the B → Kηq amplitude induced by the b → sq q¯ penguin and the B → Kηs amplitude induced by b → ss¯ s, which is ′ constructive for B → Kη and destructive for B → ηK [63, 64]. This explains the large rate of the former and the suppression of the latter. As stressed in [44, 65], the observed large B → Kη ′ rates are naturally explained in QCDF without invoking large flavor-singlet contributions.
(′)
(′)
(′)
4. The W -exchange EP,V , W -annihilation AP,V and QCD-penguin annihilation P AP,V contributions are small, as expected because of a helicity suppression factor of fB /mB ≈ 5% arising from the smallness of the B meson wave function at the origin [22–26]; they are at most only a few % (or up to 12% in the case of P A′A ) of ′ . the dominant contributions TP,V or PP,V Finally let us compare the numerical values of the SU(3)F amplitudes computed in ¯u,d → P P and B ¯u,d → P V QCDF with those obtained from global fits to charmless B decays. The ratios of the SU(3)F amplitudes extracted from global fits to charmless ¯u,d → P P modes [16] are7 B C (′) P (′) = 0.67 (0.67), T P S (′) /λd(s) P t (′) d(s) = 0.065 (0.053), T /λu P
P (′) /λd(s) P t (′) d(s) = 0.17 (0.14), T /λu P (′) d(s) PEW,P /λt (′) d(s) = 0.020 (0.016), TP /λu
(3.13)
∗ (q = u, t and r = d, s), and the relative strong phases are with λrq ≡ Vqb Vqr C(′)
δP
S(′)
δP
T (′)
− δP
T (′)
− δP
P (′)
= −68.3◦ ,
δP P
= −42.9◦ ,
δP EW
(′)
T (′)
− δP
T (′)
− δP
= −15.9◦ ,
= −57.6◦ .
(3.14)
¯u,d → P P and of “Scheme B2” in [17] for B ¯u,d → P V We only show the cases of “Scheme 4” in [16] for B below, since these cases take into account the largest set of SU(3) breaking effects among the four schemes presented in [16] and [17]. For comparison to our results, only the central values are shown. 7
– 16 –
JHEP09(2011)024
3. In ∆S = 0 decays, as expected, the tree contributions TP,V dominate and the color-suppressed tree amplitudes CP,V are larger than the penguin ones. Among the penguin contributions, the QCD-penguin ones PP,V and the QCD-penguin exchange ones P EP,V are comparable. Large strong phases in the decay amplitudes are needed ¯ decay processes. For tree-dominated to generate sizable direct CP violation in B ◦ ◦ ¯ ¯ Bu,d (Bs ) → P P decays we have CP /TP ≈ 0.63 e−i56 (0.83 e−i53 ) which is larger than the naive expectation of CP /TP ∼ 1/3 in both magnitude and phase. Recall that a large complex color-suppressed tree topology C is needed to solve the rate deficit problems with π 0 π 0 and π 0 ρ0 and give the correct sign for direct CP violation ¯ ∗0 η, π 0 π 0 and π − η [48, 49]. in the decays K − π 0 , K − η, K
¯u,d → P V modes, the ratios of the SU(3)F amplitudes extracted Likewise, for charmless B from global fits [17] are
and the relative strong phases are C(′)
δP
P (′)
δP
S(′)
δP P
δP EW
(′)
T (′)
T (′)
= 149.0◦ ,
δV
T (′)
= −2.6◦ ,
δV
− δP − δP
T (′)
− δP
T (′)
− δP
= −139.8◦ ,
= 59.0◦ ,
P (′) S(′)
δV P
δV EW
(′)
T (′)
= 0.6◦ ,
T (′)
= 172.5◦ ,
T (′)
= −47.7◦ ,
− δP − δP − δP
T (′)
− δP
C(′)
δV
T (′)
− δP
(3.15)
= −75.9◦ ,
= −111.0◦ .
(3.16)
In the above the numerical values outside (inside) parentheses correspond to ∆S = 0 ¯u,d → P P with |∆S| = 1, the primed amplitudes were (|∆S| = 1) decays. In the case of B obtained by including the SU(3) breaking factor fK /fπ for both |TP′ | and |CP′ | and a univer′ ¯u,d → P V , sal SU(3) breaking factor ξ = 1.04 for all the amplitudes except PEW,P . But, in B the primed amplitudes were extracted by imposing partial SU(3) breaking factors on T and C only: i.e., including fK ∗ /fρ for |TP′ | and |CP′ |, and fK /fπ for |TV′ | and |CV′ |. Also, for ¯u,d → P P and P V , the top penguin dominance was assumed, which is equivalent both B to the assumption that αu3,4(EW) ≃ αc3,4(EW) in QCDF. For the strong phases, exact flavor ¯u,d → P V , SU(3) symmetry was assumed in the fits so that δPT = δPT ′ , δPC = δPC′ , etc. In B all the relative strong phases were found relative to the strong phase of TP (i.e., δPT ). ¯u,d → P P On the other hand, from table 25, the ratios of the SU(3)F amplitudes for B are given by P (′) /λd(s) P t = 0.091 (0.097), (′) d(s) T /λu P (′) d(s) PEW,P /λt (′) d(s) = 0.013 (0.016), TP /λu
C (′) P (′) = 0.63 (0.64), T P S (′)(q) /λd(s) P t (′) d(s) = 0.014 (0.012), T /λu P P E (′) /λd(s) t P (′) d(s) = 0.061 (0.060), T /λu ) P
– 17 –
(3.17)
JHEP09(2011)024
C (′) V (′) = 0.76 (0.76), T V P (′) /λd(s) V t (′) d(s) = 0.056 (0.046), T /λu V S (′) /λd(s) V (′) td(s) = 0.041 (0.034), T /λu V (′) d(s) PEW,V /λt (′) d(s) = 0.074 (0.061), TV /λu
C (′) P (′) = 0.15 (0.15), T P P (′) /λd(s) P t (′) d(s) = 0.11 (0.11), T /λu P S (′) /λd(s) P t (′) d(s) = 0.018 (0.018), T /λu P (′) d(s) PEW,P /λt (′) d(s) = 0.039 (0.039), TP /λu
and the relative strong phases are C(′)
δP δPS
(′)(q)
T (′)
− δP
T (′)
P E(′)
δP
− δP
T (′)
− δP
P (′)
= −55.7◦ (−57.5◦ ) ,
δP P
= 158.2◦ (149.4◦ ) ,
δP EW
= −147.1◦ (−147.1◦ ) .
P (′)
δP
S(′)
δP P
δP EW
(′)
P E(′)
δP
T (′) T (′)
− δP
T (′)
− δP
T (′)
− δP
T (′)
− δP
(3.18)
(3.19)
V
and the relative strong phases are − δP
= −179.8◦ (−179.8◦ ) ,
C (′) V (′) = 0.39 (0.33), T V P (′) /λd(s) V t (′) d(s) = 0.041 (0.039), T /λu V S (′) /λd(s) V t (′) d(s) = 0.010 (0.007), T /λu V (′) d(s) PEW,V /λt (′) d(s) = 0.013 (0.014), TV /λu P E (′) /λd(s) t (′)V d(s) = 0.051 (0.049), T /λu
P
C(′)
T (′)
− δP
= −158.6◦ (−158.3◦ ) ,
C(′)
= −16.8◦ (−19.8◦ ) ,
δV
P (′)
= −145.5◦ (−145.2◦ ) ,
δV
S(′)
= −5.5◦ (−6.4◦ ) ,
δV P
= 179.8◦ (179.8◦ ) ,
δV EW
= −124.4◦ (−125.0◦ ) ,
(′)
P E(′)
δV
T (′)
− δP
T (′)
− δP
= −52.5◦ (−56.2◦ ) , = 7.6◦ (7.1◦ ) ,
T (′)
= 150.7◦ (133.4◦ ) ,
T (′)
= −179.7◦ (−179.7◦ ) ,
− δP
− δP
T (′)
− δP
= 4.5◦ (4.2◦ ) ,
(3.20)
where the numerical values outside (inside) parentheses correspond to ∆S = 0 (|∆S| = 1) decays. In our case, δPT = δVT = δPT ′ = δVT ′ . In comparison of eqs. (3.13)–(3.16) [“fitting case”] with eqs. (3.17)–(3.20) [“QCDF (′) (′) ¯u,d → P P in the fitting case are very case”], it is found that the values of |CP /TP | for B (′) similar to those of our QCDF case: both results show the large magnitudes of CP together ¯u,d → P V , the with large strong phases, as discussed in the above “remark 3”. But, for B (′) (′) (′) (′) values of |CP,V /TP,V | from both cases are different: in the fitting case, the ratios |CV /TV | (′)
(′)
(′)
are significantly larger than |CP /TP |, though the values of |CP | include large errors (′) (′) (′) (′) in [17], while in the QCDF case |CV /TV | ∼ |CP /TP |. For the penguin amplitudes, the d(s) (′) d(s) (′) results from both cases are also different. The values of |(PP /λt )/(TP /λu )| for both ¯u,d → P P and P V in the fitting case are larger than those in the QCDF case. In the latter B (′) (′) case, the effects from P EP are comparable to and contribute constructively to those of PP , as discussed in the above “remark 1”. Interestingly it is found that the combined effects
– 18 –
JHEP09(2011)024
¯u,d → P V , we obtain Likewise, for B C (′) P (′) = 0.30 (0.35), T P P (′) /λd(s) P t = 0.030 (0.036), (′) d(s) T /λu P S (′) /λd(s) P t (′) d(s) = 0.005 (0.006), T /λu P (′) d(s) PEW,P /λt (′) d(s) = 0.014 (0.019), TP /λu P E (′) /λd(s) t (′)P d(s) = 0.037 (0.040), T /λu δP
(′)
T (′)
− δP
(′)
(′)
(′)
from PP and P EP obtained in the QCDF case are comparable to that of PP determined (′) (′) in the fitting case. In contrast, the combined effects from PV and P EV found in the QCDF (′) case are (roughly two times) larger than that of PP obtained in the fitting case. Also, d(s) (′) d(s) (′) ¯u,d → P V decays, the ratio |(P (′) /λd(s) )/(T (′) /λd(s) )/(TV /λu )| for B u )| ∼ |(PV /λt t P P d(s) (′) d(s) d(s) (′) (′) d(s) (′) in the QCDF case, while |(PP /λt )/(TP /λu )| ∼ 2|(PV /λt )/(TV /λu )| in the fitting case. For the SU(3)F -singlet contributions, as discussed in the above “remark (′) 2”, SP,V obtained in the fitting case are much larger than those found in the QCDF ¯u,d → P V , the ratio |S (′) /P (′) | ≈ 73% in the fitting case, in contrast to case: e.g., for B 3.2
+
(′) P EV )|
. 10% [or
V (′) (′) |SV /PV |
V
≈ (18 − 24)%] in the QCDF case.
Estimates of decay amplitudes, SU(3)F breaking effects and SU(3)F relations
¯u,d (B ¯s ) → P P, P V Using tables 25–28, one can easily estimate the decay amplitudes of B − − 0 numerically. For example, the decay amplitude of B → π π is obtained as √ C 2AB − →π− π0 = Tπ + Cπ + PEW,π + PEW,π = (1.52 − i 24.94) × 10−9 GeV ,
(3.21)
¯ 0 given in eq. (2.9) is estimated as and the decay amplitude of B − → π − K AB − →π− K¯ 0 = (−49.71 − i 24.77) × 10−9 GeV .
(3.22)
¯s → K 0 π 0 is found to be Likewise, the decay amplitude of B 1 C 1 AB¯s →K 0π0 = CK − PK + PEW,K + PEW,K − P EK + P EEW,K 3 3 = (−16.32 − i 17.91) × 10−9 GeV ,
(3.23)
¯s → K + K ∗− is given by and the decay amplitude of B 2 C′ ′ ′ ′ ′ ′ ′ AB¯s →K +K ∗− = TK + PK + PEW,K + EK ∗ + P EK + P AK + P AK ∗ 3 1 1 2 ′ ′ − P EEW,K − P A′EW,K + P EEW,K ∗ 3 3 3 = (−30.20 − i 4.97) × 10−9 GeV .
(3.24)
In the above, the color-suppressed and color-favored tree amplitudes are, for example, C C T ′ T ′ ′ ≡ |T ′ | ei δK +θ CK ≡ |CK | ei δK +θ and TK , respectively, with the strong phases K C and δ T ′ and the weak phases θ C = arg V V ∗ T ′ = arg V V ∗ . δK The ub ud and θ ub us K QCD- and EW-penguin and weak annihilation amplitudes have the similar form, such (′)
(′)
P (′)
P (′)
E′
E′
′ | ei δK ∗ +θ , etc, where the strong and EK ∗ ≡ |EK as PK ≡ |PK | ei δK +θ ∗ P 6= δ P ′ 6= δ E′ in general and the weak phases θ P = arg − V V ∗ phases δK tb td and K K∗ θ P ′ = θ E′ = arg − Vtb Vts∗ . By using eqs. (3.21)–(3.24), and noting that each SU(3)F amplitude and its CP -conjugate one are the same except for having the weak phase with C
C
opposite sign to each other (e.g., the CP -conjugate amplitude to CK is |CK | ei δK −θ ), the estimation of direct CP asymmetries as well as the decay rates can be easily obtained.
– 19 –
JHEP09(2011)024
(′) (′) |SV /(PV
¯u,d → P V decays, Likewise, for B Vud T ′ P (V ) = 1.02 (1.23), Vus TP (V ) Vtd P ′ P (V ) = 1.07 (1.12), Vts PP (V )
Vud C ′ P (V ) = 1.19 (1.02), Vus CP (V ) Vtd P E ′ P (V ) = 1.10 (1.18), Vts P EP (V )
¯s → P V decays, and for B Vud T ′ P (V ) = 1.02 (1.22), Vus TP (V ) Vtd P ′ P (V ) = 1.07 (1.15), Vts PP (V )
(3.26)
Vud C ′ P (V ) = 1.00 (1.28), Vus CP (V ) Vtd P E ′ P (V ) = 1.11 (1.18). Vts P EP (V )
(3.27)
In the above, we have factored out the relevant CKM matrix element from each SU(3)F amplitude. The results show that the SU(3)F breaking is up to 28% for the tree and color-suppressed tree amplitudes, and 19% for the QCD-penguin and QCD-penguin exchange amplitudes. In refs. [22–27], a number of SU(3)F linear relations among various decay amplitudes were presented. These relations were suggested to be used in testing various assumptions made in the SU(3)F analysis and extracting CP phases and strong final-state phases and so on. In the previous studies, certain diagrams, such as the QCD-penguin exchange P E ′ ′ and the EW-penguin exchange P EEW , were ignored. As we have discussed in the previous subsection, the contribution from the P E ′ diagram turns out to be important in |∆S| = 1 decay processes. Because of its topology, the QCD-penguin exchange amplitude P E ′ always appears in the decay amplitude together with the QCD-penguin one P ′ . Thus, all the SU(3)F linear relations obtained in [22–27] still hold after replacing P ′ by (P ′ +P E ′ ). However, due to this replacement, the relevant strong phase of P ′ should be changed as follows: P′
P ′ = |P ′ |eiδ eiθ
P′
P′
→ P ′ + P E ′ = |P ′ |eiδ eiθ ′
′
P′
˜′
+ |P E ′ |eiδ
P E′
eiθ
P E′
˜′
˜′
P P ≡ |P˜ ′ |eiδ eiθ , (3.28)
where the weak phases θ P = θ P E = θ P under the assumption that the top quark dominates the penguin amplitudes in the relevant processes. Apparently, the strong phase
– 20 –
JHEP09(2011)024
The SU(3)F breaking effects in the amplitudes arise from the decay constants, masses of the mesons and the form factors in addition to the CKM matrix elements. For example, ¯u,d → P P , the ratio of T ′ and taking into account the effects of SU(3)F breaking in B P TP is estimated by |TP′ /TP | ≈ [|Vus | fK (m2B − m2K )F0BK ]/[|Vud | fπ (m2B − m2π )F0Bπ ]. From tables 25–28, the numerical estimates of the SU(3)F breaking effects in the amplitudes can ¯u,d (B ¯s ) → P P decays with ∆S = 0 and |∆S| = 1, be obtained. For both B Vud TP′ Vud CP′ Vus TP = 1.22 (1.21), Vus CP = 1.25 (1.26), Vtd PP′ Vtd P EP′ = 1.18 (1.18), (3.25) Vts PP Vts P EP = 1.19 (1.19).
˜′
′
(′)
(′)
(′)
Also, from tables 26 and 28, we see that |Eπ,ρ,K (∗) |, |P Aπ,ρ,K (∗) |, |P AEW,π,ρ,K (∗) | ≪ (′)
′ ′ |Tπ,ρ,K (∗) | and the dominant contributions |TK ∗ (K) | ≃ |Tρ(π) |, |PK ∗ (K) | ≃ |Pρ(π) | and ′ ′ |P EK ∗ (K) | ≃ |P Eρ(π) |. Thus, it is expected from eqs. (3.24) and (3.29) that
AB¯s →K +ρ− ≃ AB¯d →π+ ρ− ,
AB¯s →π− K ∗+ ≃ AB¯d →π− ρ+ ,
AB¯s →K +K ∗− ≃ AB¯d →π+ K ∗− .
AB¯s →K −K ∗+ ≃ AB¯d →K −ρ+ ,
(3.30)
Consequently, we obtain the relations for the BFs and the direct CP asymmetries: ¯s → π − K ∗+ ) ≃ B(B ¯ d → π − ρ+ ) , B(B ¯s → K − K ∗+ ) ≃ B(B ¯ d → K − ρ+ ) , B(B ¯s → π − K ∗+ ) ≃ ACP (B ¯ d → π − ρ+ ) , ACP (B
¯s → K + ρ− ) ≃ B(B ¯d → π + ρ− ) , B(B ¯s → K + K ∗− ) ≃ B(B ¯d → π + K ∗− ) , B(B ¯s → K + ρ− ) ≃ ACP (B ¯d → π + ρ− ) , ACP (B
¯s → K − K ∗+ ) ≃ ACP (B ¯d → K − ρ+ ) , ACP (B ¯s → K + K ∗− ) ≃ ACP (B ¯d → π + K ∗− ) . ACP (B (3.31) Numerically the above SU(3)F relations are generally respected [48, 49].
4
Conclusion
Based on flavor SU(3) symmetry, we have presented a model-independent analysis of ¯u,d (B ¯s ) → P P, P V decays. Based on the topological diagrams, all the decay ampliB tudes of interest have been expressed in terms of the the SU(3)F amplitudes. In order
– 21 –
JHEP09(2011)024
δP is generally not the same as δP , although they differ not much because roughly ′ ′ ˜′ |P ′ | ∼ |P E ′ | and δP ∼ δP E , as shown in eqs. (3.17)–(3.20). In fact, δP arises from the u,c different flavor operators αu,c 4 and b3 in QCDF. For completeness, we present some useful SU(3)F relations among the decay amplitudes ¯d (B ¯s ) → P V which are not given in [22–26]. From tables 7–12 and 19–24, we find of B 2 C 1 AB¯s →π− K ∗+ = TK ∗ + PK ∗ + PEW,K P EEW,K ∗ , ∗ + P EK ∗ − 3 3 2 C 1 AB¯d →π− ρ+ = Tρ + Pρ + PEW,ρ + Eπ + P Eρ + P Aπ + P Aρ − P EEW,ρ 3 3 2 1 + P AEW,π − P AEW,ρ , 3 3 2 C 1 AB¯s →K +ρ− = TK + PK + PEW,K + P EK − P EEW,K , 3 3 2 C 1 AB¯d →π+ ρ− = Tπ + Pπ + PEW,π + Eρ + P Eπ + P Aρ + P Aπ − P EEW,π 3 3 1 2 + P AEW,ρ − P AEW,π , 3 3 2 ′ ′ ′ ′ ′ ′ AB¯s →K − K ∗+ = TK P C′ ∗ + EK + P EK ∗ + PK ∗ + ∗ + P AK ∗ + P AK 3 EW,K 1 2 1 ′ P A′EW,K ∗ + P A′EW,K , − P EEW,K ∗ − 3 3 3 2 1 C′ ′ AB¯d →K − ρ+ = Tρ′ + Pρ′ + PEW,ρ + P Eρ′ − P EEW,ρ , (3.29) 3 3 2 C′ 1 ′ AB¯d →π+ K ∗− = Tπ′ + Pπ′ + PEW,π + P Eπ′ − P EEW,π . 3 3
– 22 –
JHEP09(2011)024
to bridge the topological-diagram approach (or the flavor SU(3) analysis) and the QCDF approach, we have explicitly shown how to translate each SU(3)F amplitude involved in these decay modes into the corresponding terms in the framework of QCDF. This is practically a way to easily find the rather sophisticated results of the relevant decay amplitudes calculated in QCDF by taking into account the simpler and more intuitive topological diagrams of relevance. For further quantitative discussions, we have numerically computed each SU(3)F amplitude in QCDF and shown its magnitude and strong phase. In our analysis, we have used the complete set of the topological diagrams, including the presumably subleading diagrams, such as the QCD- and EW-penguin exchange ones (P E and P EEW ) and flavor-singlet weak annihilation ones (SE, SA, SP E, SP A, SP EEW , SP AEW ). Among them, the contribution from the QCD-penguin exchange diagram plays a crucial role in understanding the branching fractions and direct CP asymmetries for ¯u,d → π + K − , Kρ, πK ∗ decays. Nupenguin-dominant decays with |∆S| = 1, such as B (′) ¯u,d (B ¯s ) → πη (′) , Kη (′) , π ω/φ, merically the SU(3)F -singlet amplitudes SP,V involved in B K ω/φ, etc, are found to be small as expected from the OZI suppression rule. On the other hand, the color-suppressed tree amplitude C is found to be large and complex: e.g., ¯u,d (B ¯s ) → P P decays, CP /TP ≈ 0.63 e−i56◦ (0.83 e−i53◦ ) which is for tree-dominated B larger than the naive expectation of CP /TP ∼ 1/3 in phase and magnitude. This large complex C is needed to understand the experimental data for the branching fractions of ¯d → π 0 π 0 , π 0 ρ0 and the direct CP asymmetries in B ¯u,d → K − π 0 , K − η, K ¯ ∗0 η, π 0 π 0 , B π − η modes. We have also compared our results with those obtained from global fits to ¯u,d → P P , are con¯u,d → P P, P V decays. Certain results, such as the effects of C (′) for B B P (′) (′) sistent with each other, but some other results, such as the contributions of PP,V and SP,V ¯u,d → P P, P V , are different from each other. These differences stem mainly from the for B ¯u,d → P P , P V in these two approaches, different ways of explaining the current data of B depending on which SU(3)F amplitudes become more important in a particular mode. As an example of the applications, we have discussed the SU(3)F breaking effects. Our results show that the SU(3)F breaking is up to 28% for the tree and color-suppressed tree amplitudes and 19% for the QCD-penguin and QCD-penguin exchange ones. Using the SU(3)F amplitudes, we have also derived some useful relations among the decay amplitudes ¯s → P V and B ¯d → P V . These SU(3)F relations are expected to be tested in future of B experiments such as the upcoming LHCb one.
B − → π− π0 ¯0 B ¯0 B B−
factor √1 2
→
π− π+
1
→
π0π0
− 12
→
K −K 0
1
– 23 –
¯ 0 → K −K + B
1
¯0 → K ¯ 0K 0 B
1
B−
→
π − η (′)
√1 2
(ζ)
TP1
(ζ)
[P2 ]
¯ 0 → η (′) η (′) B
− 12 1 2
(ζ)
SP 1
[P2 ]
(ζ)
PP1
[P2 ]
(ζ)
[P2 ]
PEW,
P1 [P2 ]
C, (ζ) P1 [P2 ]
PEW,
0
1
0
−1
1
[1]
[0]
[0]
[1]
[0]
1
0
0
1
0
1 3 [ 32 ] 2 3
[0]
[0]
[0]
[0]
[0]
[0]
0
1
0
1
[0]
[1]
[0]
−1
[−1]
[1]
0
0
0
1
0
[0]
[0]
[0]
[0]
[0]
1 3 [ 31 ] − 31
0
0
0
0
0
0
[0]
[0]
[0]
[0]
[0]
[0]
0
0
0
1
0
[0]
[0]
− 31
0 [1(q)]
¯ 0 → π 0 η (′) B
CP1
0 [0] 0 [0]
1(q) +
√
[0] 2(c)
2(q) +
√
2(q) +
√
[0] 1(q) +
√
[0]
2(s) +
√
2(c)
[0] 2(c)
[−1(q)] 1(q, q)
√ + 2(q, c) [1(q, q) √ + 2(q, c)]
[1(q)]
2(s) +
√
2(c)
[0] 2(q, q) +
√
1(q)
[0] 1 (q) 3
1(q) [1(q)]
2(q, s)
1(q, q)
√ [2(q, q) + 2(q, s) √ + 2(q, c)]
[1(q, q)]
√ + 2(q, c)
−
√ 2 (s) 3
[0] 1 (q) 3
−
√ 2 (s) 3
[−1(q)]
[0]
[0]
− 13 (q) [ 23 (q)]
− 13 (q)
[− 13 (q)]
(q, q)
− 13 (q, q)
[ 13 (q, q)
[− 31 (q, q)]
1
3 √ − 32 (q, s) √ − 32 (q, s)]
¯ → P1 P2 ( ∆S = 0 ). Table 1. Coefficients of SU(3)F amplitudes in B
JHEP09(2011)024
¯ → P1 P2 B
B − → π− π0 ¯0 B
→
π− π+
factor √1 2
1
– 24 –
¯ 0 → π0 π0 B
− 12
B− → K −K 0
1
¯0 B
1
→
K −K +
¯0 → K ¯ 0K 0 B
1
B − → π − η (′)
√1 2
¯0 B ¯0 B
→
π 0 η (′)
− 12
→
η (′) η (′)
1 2
(ζ)
EP1
(ζ)
[P2 ]
AP1
(ζ)
[P2 ]
P EP1
(ζ)
[P2 ]
P AP1
0
−1
−1
0
[0]
[1]
[1]
[0]
0
0
1
1
[1]
[0]
[0]
[1]
−1
0
−1
−2
(ζ)
[P2 ]
P EEW,
P1 [P2 ] − 23 [ 23 ] − 13
(ζ)
P AEW,
P1 [P2 ]
0 [0] − 31
[ 23 ]
[0]
− 31
[−1]
[0]
[−1]
[−2]
0
1
1
0
1 3 1 [3] 2 3
[0]
[0]
[0]
[0]
[0]
[0]
1
0
0
1
0
2 3
[0]
[0]
[0]
[1]
[0]
[− 13 ]
0
0
1
1
[0]
[0]
[0]
[1]
− 13
− 31
0
1(q)
1(q)
0
[0]
[1(q)]
[1(q)]
[0]
−1(q)
0
1(q)
0
[0]
[1(q)]
[0]
[−1(q)] 1(q, q)
0
1(q, q)
2(q, q) + 2(s, s)
[1(q, q)]
[0]
[1(q, q)]
[2(q, q) + 2(s, s)]
[− 13 ] 0
[0]
[− 13 ]
2 3 (q) [ 32 (q)] − 31 (q) [− 31 (q)] − 13 (q, q) [− 13 (q, q)]
0 [0] −1(q)
[−1(q)] 1 (q, q) 3
− 23 (s, s)
[ 13 (q, q) − 23 (s, s)]
Table 2. (Continued from table 1) Weak annihilation contributions.
JHEP09(2011)024
¯ → P1 P2 B
¯ → P1 P2 B
factor
B − → π − η (′)
√1 2
¯ 0 → π 0 η (′) B
− 21
¯ 0 → η (′) η (′) B
1 2
(ζ)
SEP1
(ζ)
(ζ)
(ζ)
(ζ)
(ζ)
SAP1 [P2 ] SP EP1 [P2 ] SP AP1 [P2 ] SP EEW, P1 [P2 ] SP AEW, P1 [P2 ] √ √ √ 2 2 4 0 2(q) + 2(s) 2(q) + 2(s) 0 0 3 (q) + 3 (s) [0] [0] [0] [0] [0] √ [0] √ √ √ 2 2 −2(q) − 2(s) 0 2(q) + 2(s) 0 − 3 (q) − 3 (s) −2(q) − 2(s) [0] [0] [0] [0] [0] [0]√ √ 2 2 q) + 32 (q, s) 2(q, q) 0 2(q, q) 4(q, q) + 2 2(q, s) −√3 (q, q) 3 (q, √ √ √ √ 2 (s, s) + 2(q, s) + 2(q, s) +2 2(s, q) + 2(s, s) − 32 (q, s) − 2 3 2 (s, q) − √ 3 √ 2 2 2 [2(q, q) [0] [2(q, q) [4(q, q) + 2 2(q, s) [−√3 (q, q) [ 3√(q, q) + 3 (q, s) √ √ √ 2 + 2(q, s)] + 2(q, s)] +2 2(s, q) + 2(s, s)] − 3 (q, s)] − 2 3 2 (s, q) − 23 (s, s)] [P2 ]
Table 3. (Continued from table 2) Singlet weak annihilation contributions.
factor 1
B − → π0 K −
√1 2
¯ 0 → π+ K − B
1
¯ 0 → π0 K ¯0 B
√1 2
B−
→
K − η (′)
¯0 → K ¯ 0 η (′) B
√1 2 √1 2
′(ζ) [P2 ]
TP1
0 [0] 1 [0] 1 [0] 0 [0] 0 [1(q)] 0 [0]
′(ζ) [P2 ]
′(ζ) [P2 ]
CP1
0 [0] 0 [1] 0 [0] 0 [1] 1(q) +
√
1(q) +
[0]
PP1
0 [0] 0 [0] 0 [0] 0 [0] 2(c)
2(q) +
√
2(q) +
√
[0] √
′(ζ) [P2 ]
SP 1
2(s) +
√
2(c)
[0] 2(c)
2(s) +
[0]
√
2(c)
1 [0] 1 [0] 1 [0] −1 [0] √ 2(s) [1(q)] √ 2(s) [1(q)]
′(ζ)
PEW,
P1 [P2 ]
0 [0] 0 [1] 0 [0] 0 [1] 1 (q) 3
− −
− 31 [0] [0] 2 3
[0] 1 3
√
2 (s) 3
[0]
1 (q) 3
C′, (ζ) P1 [P2 ]
PEW,
2 3
√
2 (s) 3
[0]
¯ → P1 P2 ( |∆S| = 1 ). Table 4. Coefficients of SU(3)F amplitudes in B
JHEP09(2011)024
– 25 –
¯ → P1 P2 B ¯0 B − → π− K
[0] − 32 (s) [ 23√(q)] − 32 (s) [− 13 (q)] √
¯ → P1 P2 B ¯0 B − → π− K B− ¯0 B ¯0 B B−
factor
′(ζ) [P2 ]
E P1
1 √1 2
→
π0 K −
→
π+ K −
1
→
¯0 π0 K
√1 2
→
K − η (′)
√1 2 √1 2
[P2 ]
′(ζ) [P2 ]
′(ζ) [P2 ]
′(ζ)
P E P1
P A P1
P EEW,
′(ζ)
P AEW,
P1 [P2 ]
P1 [P2 ]
0
1
1
0
2 3
0
[0] 0
[0] 1
[0] 1
[0] 0
[0] 2 3
[0] 0
[0]
[0]
[0]
[0]
[0]
[0]
0 [0]
0 [0]
1 [0]
0 [0]
− 31 [0]
0 [0]
0
0
0
1 3
0
[0]
[0] √ 2(s)
[0]
[0]
[0]
0
[0] √ 2(s)
−1
[0]
[1(q)]
[0]
0
0
[1(q)] √ 2(s)
[0]
[0]
[1(q)]
[0]
√ 2 2 3 (s) 2 [ 3 (q)] √ − 32 (s) [− 31 (q)]
0 0
0 [0] 0 [0]
Table 5. (Continued from table 4) Weak annihilation contributions.
¯ → P1 P2 B − B → K − η (′)
factor
¯0 → K ¯ 0 η (′) B
√1 2
√1 2
′(ζ) [P2 ]
SEP1
′(ζ)
′(ζ)
0
SAP1 [P2 ] √ 2(q) + 2(s)
SP EP1 [P2 ] √ 2(q) + 2(s)
[0] 0
[0] 0
[0]
[0]
[0]
2(q) +
√
[0]
2(s)
′(ζ) [P2 ]
SP AP1
0 [0] 0 [0]
′(ζ)
SP EEW, 4 (q) 3
+
P [P2 ] √1 2 2 (s) 3
[0] − 32 (q)
−
′(ζ)
SP AEW,
√
2 (s) 3
[0]
Table 6. (Continued from table 5) Singlet weak annihilation contributions.
JHEP09(2011)024
– 26 –
¯0 → K ¯ 0 η (′) B
′(ζ)
A P1
0 [0] 0 [0]
P1 [P2 ]
B
¯ → PV B − − 0
→π ρ
factor √1 2
B − → π 0 ρ−
√1 2
¯ 0 → π + ρ− B
1
¯ 0 → π − ρ+ B
1
¯ 0 → π 0 ρ0 B
− 12
B − → K − K ∗0
1
B − → K 0 K ∗−
1
¯ 0 → K − K ∗+ B
1
¯0
+
B →K K
∗−
1
– 27 –
¯0 → K ¯ 0 K ∗0 B
1
¯0 → K0K ¯ ∗0 B
1
B
−
(′) −
→η ρ
√1 2
¯ 0 → η (′) ρ0 B
− 12
B − → π − ω/φ
√1 2
¯ 0 → π 0 ω/φ B
− 12
¯ 0 → η (′) ω/φ B
1 2
(ζ)
(ζ)
(ζ)
(ζ)
(ζ)
TP [V ]
CP [V ]
SP [V ]
PP [V ]
PEW, P [V ]
0 [1] 1 [0] 1 [0] 0 [1] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 1(q)
1 [0] 0 [1] 0 [0] 0 [0] 1 [1] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 √ [1(q) + 2(c)] −1(q) √ [1(q) + 2(c)] 1(q) [0] 1(q) [−1(q)] 1(q, q) √ [1(q, q) + 2(q, c)]]
0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0
−1 [1] 1 [−1] 1 [0] 0 [1] −1 [−1] 1 [0] 0 [1] 0 [0] 0 [0] 1 [0] 0 [1] 1(q)
1 [0] 0 [1] 0 [0] 0 [0] 1 [1] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0
√ 2(s) + 2(c)] 0 √ √ [2(q) + 2(s) + 2(c)] √ 2(q) + 2(s) [0] √ 2(q) + 2(s) [0] √ 2(q, q) + 2(q, s) √ √ [2(q, q) + 2(q, s) + 2(q, c)]]
[1(q)] 1(q) [1(q)] 1(q) [1(q)] 1(q) [1(q)] 1(q, q) [1(q, q)
[ 13 (q) − 32 (s)] −1(q) √ [ 13 (q) − √32 (s)] 1 (q) − 32 (s) 3 [0]√ 1 2 (q) − (s) 3 3 [−1(q)] √ 1 (q, q) − √32 (q, s) 3 1 [ 3 (q, q) − 32 (q, s)]
[0] 0 [0] 0 [1(q)] 0 [0] 0 [0]
[2(q) +
√
√
C, (ζ)
PEW, P [V ] 1 3 [ 32 ] 2 3 [ 31 ] 2 3
[0] 0 [ 32 ]
1 3 [ 31 ] − 13
[0] 0 [− 31 ] 0 [0] 0 [0] − 13 [0] 0 [− 31 ] 2 (q) 3 [− 13 (q)] − 13 (q) [− 13 (q)] − 13 (q) [ 23 (q)] − 13 (q) [− 13 (q)] − 13 (q, q) [− 13 (q, q)]
¯ → P V ( ∆S = 0 ). When ideal mixing for ω and φ is assumed, i) for B − → π − ω (π − φ) Table 7. Coefficients of SU(3)F amplitudes in B ¯ 0 → π 0 ω (π 0 φ), set the coefficients of SU(3)F amplitudes with the subscript π and the superscript ζ = s (q) to zero: i.e., for B ¯ → πω, and B (s) (s) (q) (q) (q) 0 (′) (′) ¯ ¯ Sπ = PEW,π = 0, and for B → πφ, Cπ = Sπ = Pπ = · · · = 0, and ii) for B → η ω [η φ], set the coefficients of SU(3)F amplitudes with (q,s) ¯ 0 → η (′) ω, S (q,s) ¯ 0 → η (′) φ, the subscript η (′) and the superscript ζ = (q, s) or (s, s) [(q, q) or (s, q)] to zero: i.e., for B =P (′) (′) = 0, and for B η
(q,q)
(q,q)
EW,η
(q,q)
Cη(′) = Sη(′) = Pη(′) = · · · = 0.
JHEP09(2011)024
factor
B − → π 0 ρ−
√1 2
¯ 0 → π + ρ− B
1
¯ 0 → π − ρ+ B
1
¯ 0 → π 0 ρ0 B
− 12
B − → K − K ∗0
1
B − → K 0 K ∗−
1
¯ 0 → K − K ∗+ B
1
¯ 0 → K + K ∗− B
1
¯0 → K ¯ 0 K ∗0 B
1
¯0 → K0K ¯ ∗0 B
1
B − → η (′) ρ−
√1 2
¯ 0 → η (′) ρ0 B
− 12
B − → π − ω/φ
√1 2
¯ 0 → π 0 ω/φ B
− 12
¯ 0 → η (′) ω/φ B
1 2
√1 2
(ζ)
EP [V ] 0 [0] 0 [0] 0 [1] 1 [0] −1 [−1] 0 [0] 0 [0] 1 [0] 0 [1] 0 [0] 0 [0] 0 [0] −1(q) [−1(q)] 0 [0] −1(q) [−1(q)] 1(q, q) [1(q, q)]
(ζ)
AP [V ] −1 [1] 1 [−1] 0 [0] 0 [0] 0 [0] 1 [0] 0 [1] 0 [0] 0 [0] 0 [0] 0 [0] 1(q) [1(q)] 0 [0] 1(q) [1(q)] 0 [0] 0 [0]
(ζ)
P EP [V ] −1 [1] 1 [−1] 1 [0] 0 [1] −1 [−1] 1 [0] 0 [1] 0 [0] 0 [0] 1 [0] 0 [1] 1(q) [1(q)] 1(q) [1(q)] 1(q) [1(q)] 1(q) [1(q)] 1(q, q) [1(q, q)]
(ζ)
P AP [V ] 0 [0] 0 [0] 1 [1] 1 [1] −2 [−2] 0 [0] 0 [0] 1 [1] 1 [1] 1 [1] 1 [1] 0 [0] 0 [0] 0 [0] 0 [0] 2(q, q) + 2(s, s) [2(q, q) + 2(s, s)]
(ζ)
P EEW, P − 32 [ 23 ]
(ζ)
[V ]
2 3 [− 32 ] − 31
[0] 0 [− 31 ] 1 3 1 [3] 2 3
[0] 0 [ 23 ] 0 [0] 0 [0] − 31 [0] 0 [− 31 ] 2 (q) 3 [ 32 (q)] − 31 (q) [− 31 (q)] 2 (q) 3 [ 32 (q)] − 31 (q) [− 31 (q)] − 13 (q, q) [− 13 (q, q)]
P AEW, P 0 [0] 0 [0] − 13 [ 23 ]
[V ]
2 3
[− 13 ] − 13 [− 13 ] 0 [0] 0 [0] 2 3 [− 13 ] − 31 [ 23 ] − 13 [− 13 ] − 31 [− 13 ]
0 [0] −1(q) [−1(q)] 0 [0] −1(q) [−1(q)] 1 (q, q) − 32 (s, s) 3 [ 13 (q, q) − 32 (s, s)]
Table 8. (Continued from table 7) Weak annihilation contributions. When ideal mixing for ω and φ is assumed, the same rules as used in table 7 are applied.
JHEP09(2011)024
– 28 –
¯ → PV B − B → π − ρ0
¯ → PV B
B−
→
η (′) ρ−
factor √1 2
(ζ) [V ]
SEP
0
B−
→
η (′) ρ0
→
π0
→
η (′)
ω/φ
√1 2
0 [−2(q) − 0
[0] ¯0 B ¯0 B
ω/φ
− 12
ω/φ
1 2
√
−2(q) −
√
0
0 √ 2(s)] [0] [2(q) + 2(s)] √ √ 2(q) + 2(s) 2(q) + 2(s) [0] 2(s)
[0]
(ζ) [V ]
SP AP 0 [0] [0]
[− 32 (q) −
[0]
0 [0]
[0]
[0] √ 4(q, q) + 2 2(q, s) √ +2 2(s, q) + 2(s, s) √ [4(q, q) + 2 2(q, s) √ +2 2(s, q) + 2(s, s)]
0
2(q, q) √ + 2(q, s)
[2(q, q) √ + 2(q, s)]
[0]
[2(q, q) √ + 2(q, s)]
[ 34 (q) +
0
4 3 (q) +
SP AEW,
P [V ]
0
√ 2 2 3 (s)]
0
0
(ζ)
P [V ]
0
0
[0] √ 2(q) + 2(s)
2(q, q) √ + 2(q, s)
(ζ)
SP EEW,
√
2 (s)] √3 2 2 3 (s)
[0]
√ − 32 (q) − 32 (s)
[0]
− 32 (q, q) √ − 32 (q, s) [− 23 (q, q) √ − 32 (q, s)]
[0] 0 [−2(q) −
√
2(s)]
√
2(s)
0
[0] −2(q) −
[0]
√ 2 q) + 32 (q, s) 3 (q, √ − 2 3 2 (s, q) − 32 (s, s) √ [ 23 (q, q) + 32 (q, s) √ − 2 3 2 (s, q) − 32 (s, s)]
Table 9. (Continued from table 8) Singlet weak annihilation contributions. When ideal mixing for ω and φ is assumed, the same rules as used in table 7 are applied.
JHEP09(2011)024
– 29 –
→
π−
− 12
(ζ) [V ]
SP EP
0 0 √ √ [2(q) + 2(s)] [2(q) + 2(s)]
[0] ¯0 B
(ζ) [V ]
SAP
factor 1
B − → π 0 K ∗−
√1 2
¯ 0 → π + K ∗− B
1
¯ 0 → π0K ¯ ∗0 B
√1 2
¯ 0 ρ− B− → K
1
B − → K − ρ0
√1 2
¯ 0 → K − ρ+ B
1
¯0 → K ¯ 0 ρ0 B
√1 2
B − → η (′) K ∗−
√1 2
¯ 0 → η (′) K ¯ ∗0 B
√1 2
B − → K − ω/φ
√1 2
¯0 B
√1 2
→
¯0 K
ω/φ
′(ζ) [V ]
TP
0 [0] 1 [0] 1 [0] 0 [0] 0 [0] 0 [1] 0 [1] 0 [0] 1(q) [0] 0 [0] 0 [1(q)] 0 [0]
′(ζ) [V ]
′(ζ) [V ]
CP
0 [0] 0 [1] 0 [0] 0 [1] 0 [0] 1 [0] 0 [0] 1 [0] 0
PP
0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0
√ [ 2(s)]
[1(q) + 2(c)] 1(q)
[2(q) + 2(s) + 2(c)] √ 2(q) + 2(s)
√ [ 2(s)] √ 2(s)
[0]
[0]
[1(q)]
2(c)]
[2(q) +
√
0 √
1(q)
[0]
2(s) +
√
0 √
1(q) √
2(q) +
√
[0]
′(ζ)
PEW,
1 [0] 1 [0] 1 [0] −1 [0] 0 [1] 0 [1] 0 [1] 0 [−1] 1(q) 2(c)]
[1(q) +
√
′(ζ) [V ]
SP
2(s)
√
2(s)
[1(q)]
P [V ]
0 [0] 0 [1] 0 [0] 0 [1] 0 [0] 1 [0] 0 [0] 1 [0] 0 [ 31 (q) −
1 (q) 3
2 3
[0] 1 3
[0]
√
2 (s)] 3
√
2 (s)] √3 2 (s) 3
[0]√
−
− 31 [0] [0]
0
[ 31 (q) − 1 (q) − 3
C′, (ζ) P [V ]
PEW,
2 3
2 (s) 3
[0]
[− 31 ] 0 [ 32 ] 0 [ 32 ] 0 [ 31 ] 2 3 (q) [−
√
2 (s)] 3
− 31 (q) √
[− 32 (s)] √ − 32 (s)
[ 23√(q)]
−
2 (s) 3
[− 31 (q)]
¯ → P V ( |∆S| = 1 ). When ideal mixing for ω and φ is assumed, for B − → K − ω (K − φ) Table 10. Coefficients of SU(3)F amplitudes in B 0 0 0 ¯ → K ω (K φ), set the coefficients of SU(3)F amplitudes with the subscript K and the superscript ζ = s (q) to zero: i.e., for B ¯ → Kω, and B ′(s) ′(s) ′(q) ′(q) ′(q) ¯ SK = PK = · · · = 0, and for B → Kφ, CK = SK = PEW,K = 0.
JHEP09(2011)024
– 30 –
¯ → PV B − ¯ ∗0 B → π− K
¯ → PV B − ¯ ∗0 B → π− K B−
→
π 0 K ∗−
factor 1 √1 2
¯ 0 → π + K ∗− B
1
¯ 0 → π0 K ¯ ∗0 B
√1 2
B−
¯ 0 ρ− →K
1 √1 2
¯ 0 → K − ρ+ B
1
¯0 → K ¯ ∗0 ρ0 B
√1 2
η (′) K ∗−
√1 2
¯ 0 → η (′) K ¯ ∗0 B
√1 2
B−
B−
→
→
K−
ω/φ
¯0 → K ¯ 0 ω/φ B
√1 2 √1 2
′(ζ) [V ]
AP
′(ζ) [V ]
P EP
′(ζ) [V ]
P AP
′(ζ)
P EEW,
P [V ]
2 3
′(ζ)
P AEW,
0 [0] 0
1 [0] 1
1 [0] 1
0 [0] 0
[0] 0 [0] 0
[0] 0 [0] 0
[0] 0 [0] 0
[0] − 13 [0] 1 3
[0] 0 [0] 0
[0] 0 [0] 0
[0] 0 [1] 0
[0] 1 [0] −1 [0] 0 [1] 0
[0] 0 [0] 0
[0] 0 [ 32 ] 0
[0] 0 [0] 0
[0] 0 [0] 0
[1] 0 [0] 0
[1] 0 [1] 0
[0] 0 [0] 0
[ 32 ] 0 [− 31 ] 0
[0] 0 [0] 0
[0] 0
[0] √ 2(s)
[0] 0
[1(q)] 0
[1(q)] √ 2(s)
[0] 0
[0]
[0]
[1(q)]
[0]
[ 31 ] 2 3 (q) √ 2 2 [ 3 (s)] − 31 (q) √ [−√32 (s)] 2 2 3 (s) 2 [ 3√(q)] − 32 (s) [− 31 (q)]
[0] 0
[0] 0
[−1] 1(q) √ [ 2(s)] 1(q) √ [ 2(s)] √ 2(s)
[0] 0
[0] 0
[0] 1(q) √ [ 2(s)] 0
[0] 0 [0] 0
[0] 2 3
P [V ]
0 [0] 0
[0] 0 [0] 0 [0] 0 [0]
Table 11. (Continued from table 10) Weak annihilation contributions. When ideal mixing for ω and φ is assumed, the same rules as used in table 10 are applied.
JHEP09(2011)024
– 31 –
B − → K − ρ0
′(ζ) [V ]
EP
¯ → PV B
B − → η (′) K ∗− ¯ 0 → η (′) K ¯ ∗0 B B−
→
K−
ω/φ
¯0 → K ¯ 0 ω/φ B
factor
′(ζ) [V ]
′(ζ) [V ]
′(ζ) [V ]
′(ζ) [V ]
SEP
SAP
SP EP
SP AP
0
0
0
0
√1 2
[0] √1 2
[2(q) +
0 [0] 0
√1 2
2(s)]
[2(q) +
0 2(q) +
√
0
[0]
[0]
[0]
2(s)]
2(s)
[2(q) + 2(q) +
√
2(s)
[0] 2(q) +
[ 34 (q) +
√
4 (q) 3
[0]
P [V ]
[0] 0
√
− 32 (s)] √ + 2 3 2 (s)
[0] 0
[0]
[0]
− 32 (q)
0
[0]
√ 2 2 (s)] 3
0
[0] 2(s)
′(ζ)
SP AEW,
0
[− 32 (q)
[0] 0
2(s)]
P [V ]
0
0
√
[0]
0
√
0
[0]
[0] √1 2
√
′(ζ)
SP EEW,
√
2 (s) 3
−
0
[0]
[0]
¯s → P1 P2 B ¯s → K +π− B
factor
¯s → K 0 π 0 B
√1 2
¯s → K 0 η (′) B
√1 2
1
(ζ)
TP1
(ζ)
[P2 ]
CP1
(ζ)
SP 1
[P2 ]
(ζ)
PP1
[P2 ]
(ζ)
[P2 ]
PEW,
P1 [P2 ]
C, (ζ) P1 [P2 ]
PEW,
2 3
1 [0]
0 [0]
0 [0]
1 [0]
0 [0]
[0]
0
1
0
1
1 3
[0] 0
[0]
−1
[0]
1(q) +
√
[0]
[0] 2(c)
2(q) +
√
2(s) +
[0]
√
2(c)
[0] 1(q) √ [ 2(s)]
[0] 1 (q) 3
−
√
2 (s) 3
[0]
¯s → P1 P2 ( ∆S = 0 ). Table 13. Coefficients of SU(3)F amplitudes in B
JHEP09(2011)024
– 32 –
Table 12. (Continued from table 11) Singlet weak annihilation contributions. When ideal mixing for ω and φ is assumed, the same rules as used in table 10 are applied.
[0] − 31 (q)
[−
√ 2 3 (s)]
¯s → P1 P2 B ¯s → K + π − B
factor 1
¯s → K 0 π0 B
√1 2
¯s → K 0 η (′) B
√1 2
(ζ)
(ζ)
EP1 [P2 ] 0 [0] 0 [0] 0
AP1 [P2 ] 0 [0] 0 [0] 0
[0]
[0]
(ζ)
P EP1 [P2 ] 1 [0] −1 [0] 1(q) √ [ 2(s)]
(ζ)
(ζ)
P AP1 [P2 ] 0 [0] 0 [0] 0
P EEW, P1 − 31 [0]
(ζ)
[P2 ]
1 3
[0] − 13 (q)
[0]
[−
√
P AEW, P1 0 [0] 0 [0] 0
2 3 (s)]
[P2 ]
[0]
Table 14. (Continued from table 13) Weak annihilation contributions.
¯s → P1 P2 B ¯s → K 0 η (′) B
(ζ)
factor
SEP1
√1 2
(ζ)
(ζ)
SAP1
[P2 ]
0 [0]
[P2 ]
0 [0]
(ζ)
SP EP1
√[P2 ] 2(q) + 2(s) [0]
SP AP1
(ζ)
[P2 ]
SP EEW, − 23 (q) −
0 [0]
(ζ)
P1 [P2 ] √ 2 (s) 3
SP AEW,
P1 [P2 ]
0 [0]
[0]
Table 15. (Continued from table 14) Singlet weak annihilation contributions.
factor 1
¯s → π 0 π 0 B
1 2
¯s → K ¯ 0K 0 B
1
¯s → K − K + B
1
¯s → π 0 η (′) B
1 2
¯s → η (′) η (′) B
1 2
′(ζ) [P2 ]
TP1
0 [0] 0 [0] 0 [0] 0 [1] 0 [0] 0
′(ζ) [P2 ]
CP1
0 [0] 0 [0] 0 [0] 0 [0] √0 [ 2(s)] √
2(s, q)
+2(s, c) [0]
√ [ 2(s, q)
+2(s, c)]
′(ζ) [P2 ]
SP 1
0 [0] 0 [0] 0 [0] 0 [0] 0 [0]
√ 2 2(s, q) + 2(s, s)
′(ζ) [P2 ]
PP1
0 [0] 0 [0] 0 [1] 0 [1] 0 [0] 2(s, s)
+2(s, c)
√ [2 2(s, q) + 2(s, s)
+2(s, c)]
[2(s, s)]
′(ζ)
PEW,
P1 [P2 ]
C′, (ζ) P1 [P2 ]
PEW,
0 [0] 0 [0] 0 [0] 0 [0] √0 [√ 2(s)] 2 (s, q) 3
− 32 (s, s)
2 (s, q) 3
[− 23 (s, s)]
−√23 (s, s) [
0 [0] 0 [0] 0 [− 31 ] 0 [ 32 ] 0 [0]
− 23 (s, s)]
¯s → P1 P2 ( |∆S| = 1 ). Table 16. Coefficients of SU(3)F amplitudes in B
JHEP09(2011)024
– 33 –
¯s → P1 P2 B ¯ Bs → π + π −
¯s → P1 P2 B ¯s → π + π − B
factor 1
¯s → π0 π0 B
1 2
¯s → K ¯ 0K 0 B
1
¯s → K −K + B
1
¯s → π 0 η (′) B
1 2
¯s → η (′) η (′) B
1 2
′(ζ) [P2 ]
E P1
0 [1] 1 [1] 0 [0] 1 [0] 1(q) [1(q)] 1(q, q)
′(ζ) [P2 ]
A P1
′(ζ) [P2 ]
P E P1
0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0
′(ζ) [P2 ]
P A P1
– 34 –
0
P1 [P2 ]
0 [0] 0 [0] 0 [1] 0 [1] 0 [0]
1 [1] 2 [2] 1 [1] 1 [1] 0 [0]
0 [0] 0 [0] 0 [− 13 ] 0 [− 13 ] 0 [0]
2(s, s)
2(q, q)
− 32 (s, s)
+2(s, s) [1(q, q)]
′(ζ)
P EEW,
[2(s, s)]
[2(q, q)
′(ζ)
P AEW,
P1 [P2 ]
− 13 [ 23 ]
1 3 [ 13 ] − 13 [− 13 ] 2 3 [− 13 ]
1(q) [1(q)] 1 (q, q) 3
− 23 (s, s)
[− 32 (s, s)]
[ 13 (q, q)
− 32 (s, s)]
+2(s, s)] Table 17. (Continued from table 16) Weak annihilation contributions.
factor
¯s → η (′) η (′) B
1 2
1 2
′(ζ)
SEP1 [P2 ] √ 2(q) + 2(s) [0] 2(q, q) √ + 2(q, s) [2(q, q) √ + 2(q, s)]
′(ζ)
SAP1
0 [0] 0 [0]
[P2 ]
′(ζ) [P2 ]
SP EP1
0 [0] √ 2 2(s, q) +2(s, s) √ [2 2(s, q) +2(s, s)]
′(ζ) [P2 ]
SP AP1
0 [0] √ 4(q, q) + 2 2(q, s) √ +2 2(s, q) + 2(s, s) √ [4(q, q) + 2 2(q, s) √ +2 2(s, q) + 2(s, s)]
′(ζ)
SP EEW,
P1 [P2 ]
0 [0] √ − 2(s, q) − 32 (s, s) √ [− 2(s, q) − 32 (s, s)]
Table 18. (Continued from table 17) Singlet weak annihilation contributions.
JHEP09(2011)024
¯s → P1 P2 B ¯s → π 0 η (′) B
′(ζ)
SP AEW, P1 [P2 ] √ 2(q) + 2(s) [0]√ 2 (q, q) + 32 (q, s) 3√ 2 − 2(s, q) − (s, s) √ 3 2 2 [ 3 (q, q) + 3 (q, s) √ − 2(s, q) − 32 (s, s)]
¯s → P V B
factor
¯s → π − K ∗+ B
1
¯s → π 0 K ∗0 B
√1 2
¯s → η (′) K ∗0 B
√1 2
(ζ) [V ]
TP
– 35 –
1
¯s → K 0 ρ0 B
√1 2
¯s → K 0 ω/φ B
√1 2
(ζ) [V ]
(ζ) [V ]
SP
PP
(ζ)
PEW,
P [V ]
C, (ζ) P [V ]
PEW,
0
0
0
0
0
0
[1]
[0]
[0]
[1]
[0]
[ 32 ]
0
0
0
0
0
0
[0]
[1]
[0]
[1]
0
0
[−1] √ 2(s)
1
0
0
1
0
[1] √3 − 32 (s) [− 13 (q)] 2 3
[0]
[0]
[0]
[0]
[0]
[0]
0
1
0
−1
1
1 3
[0]
[0]
[0]
[0]
[0] ¯s → K + ρ− B
(ζ) [V ]
CP
[1(q) +
√
0
1(q)
[0]
[0]
0 2(c)]
[2(q) +
√
2(s) +
2(q) +
√
[0]
√
2(s)
2(c)]
[1(q)]
1(q) √ [ 2(s)]
0 [ 13 (q) −
√ 2 (s)] 3
[0] 1 (q) 3
−
√ 2 (s) 3
[0]
[0] − 13 (q) [−
√
2 3 (s)]
¯ s → P V ( ∆S = 0 ). When ideal mixing for ω and φ is assumed, for B ¯s → K 0 ω (K 0 φ), set Table 19. Coefficients of SU(3)F amplitudes in B (s) 0 ¯s → K ω, S = P (s) the coefficients of SU(3)F amplitudes with the subscript K and the superscript ζ = s (q) to zero: i.e., for B K EW,K = 0, and for (q) (q) (q) 0 ¯s → K φ, C = S = P = · · · = 0. B K
K
JHEP09(2011)024
K
¯s → P V B
factor
¯s → π − K ∗+ B
1
¯s → π 0 K ∗0 B
√1 2
¯s → B
η (′) K ∗0
√1 2
¯s → B
K + ρ−
(ζ) [V ]
EP
1 √1 2
¯s → K 0 ω/φ B
√1 2
(ζ) [V ]
(ζ) [V ]
P EP
P AP
(ζ)
P EEW,
(ζ)
P [V ]
P AEW,
0 [0] 0
0 [0] 0
0 [1] 0
0 [0] 0
0 [− 13 ] 0
0 [0] 0
[0] 0
[0] 0
[−1] √ 2(s)
[0] 0
[0] 0
[0] 0 [0] 0
[0] 0 [0] 0
[0] 0 [0] 0
[0] 0
[0] 0
[1(q)] 1 [0] −1
[1] √3 − 32 (s)
[0]
[0]
[0] 1(q) √ [ 2(s)]
− 13 [0] 1 3
[0] 0
[0]
[0] 0
− 31 (q)
[0]
[−
P [V ]
[0] 0 [0] 0
[− 31 (q)]
√
2 3 (s)]
[0]
Table 20. (Continued from table 19) Weak annihilation contributions. When ideal mixing for ω and φ is assumed, the same rules as used in table 19 are applied.
¯s → P V B ¯s → η (′) K ∗0 B
factor
¯s → K 0 ω/φ B
√1 2
√1 2
(ζ) [V ]
(ζ) [V ]
SEP
SAP
0
0
[0] 0 [0]
(ζ) [V ]
SP EP
(ζ) [V ]
SP AP
(ζ)
SP EEW,
[0] 0
0 √ [2(q) + 2(s)] √ 2(q) + 2(s)
0 [0] 0
[− 23 (q) −
[0]
[0]
[0]
[0]
(ζ)
P [V ]
SP AEW,
0 − 23 (q)
−
P [V ]
0 √
2 (s)] 3 √ 2 (s) 3
[0] 0 [0]
Table 21. (Continued from table 20) Singlet weak annihilation contributions. When ideal mixing for ω and φ is assumed, the same rules as used in table 19 are applied.
JHEP09(2011)024
– 36 –
¯ s → K 0 ρ0 B
(ζ) [V ]
AP
– 37 –
¯s → P V B ¯ Bs → π + ρ−
factor 1
¯s → π − ρ+ B
1
¯s → π 0 ρ0 B
1 2
¯s → K ¯ 0 K ∗0 B
1
¯s → K 0 K ¯ ∗0 B
1
¯s → K − K ∗+ B
1
¯s → K + K ∗− B
1
¯s → π 0 ω/φ B
1 2
¯s → η (′) ρ0 B
1 2
¯s → η (′) ω/φ B
1 2
′(ζ) [V ]
TP
′(ζ) [V ]
CP
0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [1] 1 [0] 0 [0] 0 [0] 0
0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 √ [ 2(s)] √ 2(s) [0]
[0]
′(ζ) [V ]
SP
′(ζ) [V ]
PP
0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0]
0 [0] 0 [0] 0 [0] 0 [1] 1 [0] 0 [1] 1 [0] 0 [0] 0 [0]
2(s, q)
√ 2 2(s, q) + 2(s, s)
2(s, s)
√ [ 2(q, s)
√ [2 2(q, s) + 2(s, s)
√
+2(c, s)]
[2(s, s)]
′(ζ)
PEW,
P [V ]
C′, (ζ) P [V ]
PEW,
0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 √ [ 2(s)] √ 2(s) [0] √
2 (s, q) 3
− 23 (s, s) [
√
2 (q, s) 3
0 [0] 0 [0] 0 [0] 0 [− 13 ] − 13 [0] 0 [ 32 ] 2 3
[0] 0 [0] 0 [0] − 23 (s, s) [− 32 (s, s)]
− 23 (s, s)]
+2(c, s)]
¯s → P V ( |∆S| = 1 ). When ideal mixing for ω and φ is assumed, i) for B ¯s → π 0 ω (π 0 φ), set Table 22. Coefficients of SU(3)F amplitudes in B ′(s) ¯s → π 0 ω, SEπ = SP A′(s) = 0 [See the coefficients of SU(3)F amplitudes with the subscript π and the superscript ζ = s (q) to zero: i.e., for B EW,π ′(q) ′(q) 0 (′) (′) ¯ ¯ Tabel 24.], and for Bs → π φ, Eπ = P Aπ = · · · = 0 [See Tabel 23.], and ii) for Bs → η ω [η φ], set the coefficients of SU(3)F amplitudes with ′(q,s) ′(s,s) ¯s → η (′) φ, ¯s → η (′) ω, Cω′(q,s) = S ′(s,s) the superscript ζ = (s, s) or (q, s) [(s, q) or (q, q)] to zero: i.e., for B = Sω = Sω = · · · = 0, and for B (′) η
′(s,q)
= Sη(′)
′(s,q)
= PEW,η(′) = 0.
JHEP09(2011)024
′(s,q)
Cη(′)
factor 1
¯s → π − ρ+ B
1
¯s → π 0 ρ0 B
1 2
¯s → K ¯ 0 K ∗0 B
1
¯s → K 0 K ¯ ∗0 B
1
¯s → K − K ∗+ B
1
¯s → K + K ∗− B
1
¯s → π 0 ω/φ B
1 2
¯s → η (′) ρ0 B
1 2
¯s → η (′) ω/φ B
1 2
′(ζ) [V ]
EP
0 [1] 1 [0] 1 [1] 0 [0] 0 [0] 1 [0] 0 [1] 1(q) [1(q)] 1(q) [1(q)] 1(q, q)
′(ζ) [V ]
AP
0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0 [0] 0
′(ζ) [V ]
P EP
′(ζ) [V ]
P AP
0
P [V ]
0 [0] 0 [0] 0 [0] 0 [1] 1 [0] 0 [1] 1 [0] 0 [0] 0 [0]
1 [1] 1 [1] 2 [2] 1 [1] 1 [1] 1 [1] 1 [1] 0 [0] 0 [0]
0 [0] 0 [0] 0 [0] 0 [− 13 ] − 13 [0] 0 [− 13 ] − 13 [0] 0 [0] 0 [0]
2(s, s)
2(q, q)
− 23 (s, s)
+2(s, s) [1(q, q)]
′(ζ)
P EEW,
[2(s, s)]
[2(q, q)
+2(s, s)]
[− 23 (s, s)]
′(ζ)
P AEW,
P [V ]
− 31 [ 23 ] 2 3
[− 31 ] 1 3 [ 13 ] − 31 [− 13 ] − 13 [− 31 ] 2 3 [− 13 ] − 13 [ 23 ]
1(q) [1(q)] 1(q) [1(q)] 1 (q, q) 3
− 32 (s, s) [ 31 (q, q)
− 32 (s, s)]
Table 23. (Continued from table 22) Weak annihilation contributions. When ideal mixing for ω and φ is assumed, the same rules as used in table 22 are applied.
JHEP09(2011)024
– 38 –
¯s → P V B ¯s → π + ρ− B
¯s → P V B
¯s → π 0 ω/φ B ¯s → B
η (′)
ω/φ
1 2 1 2 1 2
′(ζ) [V ]
SEP
′(ζ) [V ]
′(ζ) [V ]
0
0
[0]
[0]
[0]
[0]
[0]
[0]
0 √ [2(q) + 2(s)]
0
0
0
0
[0]
[0] √ 4(q, q) + 2 2(q, s) √ +2 2(s, q) + 2(s, s) √ [4(q, q) + 2 2(s, q) √ +2 2(s, q) + 2(s, s)]
0 √ [2(q) + 2(s)]
2(q, q) √ + 2(q, s)
0
[0] √ 2 2(s, q)
[2(q, q) √ + 2(s, q)]
[0]
+2(s, s) √ [2 2(q, s) +2(s, s)]
SP EEW,
′(ζ)
0
2(s)
SP AP
′(ζ)
0
√
SP EP
′(ζ) [V ]
SP AEW, P [V ] √ 2(q) + 2(s)
2(q) +
SAP
P [V ]
[0]
√ − 2 3 2 (s, q) − 32 (s, s) √ [− 2 3 2 (q, s) − 23 (s, s)]
√ 2 2 (q, q) + 3√ 3 (q, s) − 2 3 2 (s, q) − 32 (s, s) √ [ 23 (q, q) + 32 (s, q) √ − 2 3 2 (q, s) − 23 (s, s)]
Table 24. (Continued from table 23) Singlet weak annihilation contributions. When ideal mixing for ω and φ is assumed, the same rules as used in table 22 are applied.
JHEP09(2011)024
– 39 –
¯s → B
η (′) ρ0
factor
∆S = 0
Numerical values 0.9◦ )
TP
(24.52,
CP
(15.47, − 54.8◦ )
(q)
(0.87, 159.1◦ )
SP
(s)
(0.89, 159.1◦ )
SP
(c)
(0.02, 159.1◦ )
PP
(5.59, − 157.7◦ )
SP
P
C PEW,
P
(0.82, − 178.9◦ ) (0.17, 163.9◦ )
EP
(1.96, 52.7◦ )
AP
(0.61, − 127.3◦ ) (3.79, − 146.2◦ )
P EP
(0.61, − 127.3◦ )
P AP P EEW,
P
P AEW,
P
(0.02, − 34.1◦ ) (0.03, 52.7◦ )
TP′ CP′ ′(q) SP ′(s) SP ′(c) SP PP′ ′ PEW, P C′ PEW, P EP′ A′P P EP′ P A′P ′ P EEW, P ′ P AEW, P
Numerical values (6.90, 0.9◦ ) (4.48, − 56.6◦ ) (4.11, 150.3◦ ) (4.21, 150.3◦ ) (0.09, 150.3◦ ) (34.25, − 157.4◦ ) (5.48, − 178.9◦ ) (1.05, 163.1◦ ) (0.54, 52.9◦ ) (0.17, − 127.1◦ )
(21.19 , − 146.2◦ ) (3.46, − 127.1◦ ) (0.13, − 36.1◦ ) (0.15, 52.9◦ )
¯ u,d → P1 P2 decays with ∆S = 0 and |∆S| = 1 calculated in QCD factorization. The Table 25. Numerical values of the SU(3)F amplitudes of B −9 magnitude (in units of 10 GeV) and strong phase (in degrees) of each SU(3)F amplitude are shown in order within the parenthesis: e.g., for the tree amplitude TP ≡ |TP | ei(δP +θP ) with δP and θP being the strong and weak phases, respectively, its magnitude and strong phase are shown as (|TP |, δP ).
JHEP09(2011)024
– 40 –
PEW,
|∆S| = 1
∆S = 0
Numerical values
|∆S| = 1
Numerical values
( TP ; TV )
( 40.82, 0.8◦ ; 30.16, 0.8◦ )
( TP′ ; TV′ )
( 9.63, 0.8◦ ; 8.54, 0.8◦ )
( C P ; CV )
( 12.30, − 16.0◦ ; 11.84, − 51.7◦ )
( CP′ ; CV′ )
( 3.38, − 19.0◦ ; 2.79, − 55.4◦ )
( 0.59, − 4.6◦ ; 0.81, 151.5◦ )
( SP ; SV
( 3.04, − 144.7◦ ; 3.12, 8.4◦ )
( PP′ ; PV′ )
(q)
(q)
(s)
(s)
(c)
(c)
′(q)
′(q)
)
′(s)
′(s)
)
′(c)
′(c)
)
( 0.50, − 4.7◦ ; 0.79, 151.5◦ )
( SP ; SV
( −; 0.02, 151.5◦ )
( SP ; SV
( 1.41, − 179.4◦ ; 1.01, − 178.9◦ )
′ ′ ( PEW, P ; PEW,
V)
( 0.38, 165.0◦ ; 0.34, 164.7◦ )
C′ C′ ( PEW, P ; PEW,
V)
( EP ; EV )
( 2.46, 70.1◦ ; 2.29, 38.9◦ )
( EP′ ; EV′ )
( 0.60, 69.4◦ ; 0.64, 39.2◦ )
( AP ; AV )
( 0.77, − 109.9◦ ; 0.72, − 141.1◦ )
( A′P ; A′V )
( 0.19, − 110.6◦ ; 0.20, − 140.8◦ )
( SP ; SV ) ( SP ; SV ) ( SP ; SV ) ( PP ; PV ) V)
C C ( PEW, P ; PEW,
V)
( P EP ; P EV ) ( P AP ; P AV ) ( P EEW, P ; P EEW,
V)
( P AEW, P ; P AEW,
V)
( 3.22, − 5.5◦ ; 3.11, 134.2◦ ) ( −; 0.07, 134.2◦ )
( 17.99, − 144.4◦ ; 17.02, 7.9◦ )
( 9.39, − 179.4◦ ; 6.00, − 178.9◦ ) ( 1.76, 165.1◦ ; 1.87, 160.6◦ )
( 3.82, − 123.6◦ ; 3.83, 5.3◦ )
( P EP′ ; P EV′ )
( 0.12, 70.1◦ ; 0.11, 38.9◦ )
( P A′P ; P A′V )
( 0.02, − 57.8◦ ; 0.14, − 157.3◦ )
′ ′ ( P EEW, P ; P EEW,
V)
( P A′EW, P ; P A′EW,
V)
( 0.02, 70.1◦ ; 0.02, 38.9◦ )
( 19.83, − 124.2◦ ; 21.20, 5.0◦ ) ( 0.61, 69.4◦ ; 0.66, 39.2◦ )
( 0.10, − 50.7◦ ; 0.77, − 157.0◦ ) ( 0.08, 69.4◦ ; 0.09, 39.2◦ )
¯u,d → P V decays: e.g., for the tree amplitudes (TP ; TV ) where TP,V ≡ |TP,V | ei(δP,V +θP,V ) with δP,V Table 26. Same as table 25 except for B and θP,V being the strong and weak phases, respectively, their magnitudes (in units of 10−9 GeV) and strong phases (in degrees) are shown as (|TP |, δP ; |TV |, δV ).
JHEP09(2011)024
– 41 –
( PEW, P ; PEW,
( 2.75, − 5.6◦ ; 3.04, 134.2◦ )
∆S = 0
Numerical values 0.9◦ )
TP
(23.54,
CP
(19.57, − 51.6◦ )
(q)
(1.31, 166.6◦ )
SP
(s)
(1.35, 166.6◦ )
SP
(c)
(0.03, 166.6◦ )
PP
(5.64, − 158.1◦ )
SP
P
C PEW,
P
(0.78, − 178.9◦ ) (0.25, 170.0◦ )
EP
(2.35, 51.2◦ )
AP
(0.74, − 128.8◦ ) (4.31, − 149.1◦ )
P EP
(0.74, − 128.8◦ )
P AP P EEW,
P
P AEW,
P
(0.03, − 46.2◦ ) (0.03, 51.2◦ )
TP′ CP′ ′(s,q) SP ′(s,s) SP ′(s,c) SP PP′ ′ PEW, P C′ PEW, P EP′ A′P P EP′ P A′P ′ P EEW, P ′ P AEW, P
Numerical values (6.61, 0.8◦ ) (5.72, − 50.5◦ ) (3.53, 162.5◦ ) (3.62, 162.5◦ ) (0.08, 162.5◦ ) (34.69, − 157.9◦ ) (4.46, − 178.8◦ ) (1.58, 171.0◦ ) (0.65, 51.4◦ ) (0.20, − 128.6◦ )
(24.12, − 149.1◦ ) (4.18, − 128.6◦ ) (0.15, − 48.4◦ ) (0.18, 51.4◦ )
¯s → P1 P2 decays. Table 27. Same as table 25 except for B
JHEP09(2011)024
– 42 –
PEW,
|∆S| = 1
∆S = 0
Numerical values 0.9◦ ;
( TP ; TV )
( 39.33,
( CP ; CV )
( 15.49, − 12.5◦ ; 12.90, − 50.6◦ )
(
( 0.76, − 3.6◦ ; 0.94, 155.3◦ )
(
( 2.82, − 142.7◦ ; 3.51, 7.7◦ )
(
(q)
(q)
(s)
(s)
30.59,
0.8◦ )
(
( 0.64, − 3.6◦ ; 0.91, 155.3◦ )
(
( −; 0.02, 155.3◦ )
(
( 1.38, − 179.4◦ ; 1.03, − 178.9◦ )
(
( 0.48, 168.6◦ ; 0.36, 164.9◦ )
(
( EP ; EV )
( 3.24, 70.5◦ ; 2.44, 45.0◦ )
(
( AP ; AV )
( 1.01, − 109.5◦ ; 0.76, − 135.0◦ )
(
( SP ; SV ) (c)
(c)
( SP ; SV ) ( PP ; PV ) – 43 –
( PEW, P ; PEW,
V
)
C C ( PEW, P ; PEW,
V
)
( P EP ; P EV ) ( P AP ; P AV ) ( P EEW, P ; P EEW,
V
)
( P AEW, P ; P AEW,
V
)
( 4.87, − 123.4◦ ; 4.01, 16.3◦ )
(
( 0.16, 70.5◦ ; 0.12, 45.0◦ )
(
( 0.03, − 63.1◦ ; 0.15, − 148.7◦ )
(
( 0.02, 70.5◦ ; 0.02, 45.0◦ )
(
TP′ ; TV′ ) CP′ ; CV′ ) ′(s,q) ′(q,s) SP ; SV ) ′(s,s) ′(s,s) SP ; SV ) ′(s,c) ′(c,s) SP ; SV ) PP′ ; PV′ ) ′ ′ PEW, P ; PEW, V ) C′ C′ PEW, P ; PEW, V ) EP′ ; EV′ ) A′P ; A′V ) P EP′ ; P EV′ ) P A′P ; P A′V ) ′ ′ P EEW, P ; P EEW, V P A′EW, P ; P A′EW, V
Numerical values ( 9.28, 0.9◦ ; 8.60, 0.9◦ ) ( 3.59, − 13.0◦ ; 3.82, − 50.7◦ ) ( 1.90, − 4.3◦ ; 4.56, 155.1◦ ) ( 2.24, − 4.2◦ ; 4.68, 155.1◦ ) ( −; 0.10, 155.1◦ )
( 16.71, − 142.1◦ ; 19.83, 6.7◦ )
( 6.61, − 179.4◦ ; 5.89, − 178.9◦ ) ( 2.30, 168.2◦ ; 2.10, 164.6◦ ) ( 0.79, 69.8◦ ; 0.68, 45.4◦ ) ( 0.25, − 110.2◦ ; 0.21, − 134.6◦ ) ( 25.33, − 124.0◦ ; 22.29, 16.1◦ ) ( 0.81, 69.8◦ ; 0.70, 45.4◦ ) ) )
( 0.13, − 56.1◦ ; 0.82, − 148.4◦ ) ( 0.11, 69.8◦ ; 0.09, 45.4◦ )
¯s → P V decays. Table 28. Same as table 26 except for B
JHEP09(2011)024
( SP ; SV )
|∆S| = 1
Acknowledgments This work was supported in part by the National Science Council of R.O.C. under Grants Numbers: NSC-97-2112-M-001-004-MY3 and NSC-99-2811-M-001-038.
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