Resolution to the B -> pi K puzzle

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(1)PHYSICAL REVIEW D 72, 114005 (2005). Resolution to the B ! K puzzle Hsiang-nan Li,1,* Satoshi Mishima,2,† and A. I. Sanda3,‡ 1. Institute of Physics, Academia Sinica, Taipei, Taiwan 115, Republic of China and Department of Physics, National Cheng-Kung University, Tainan, Taiwan 701, Republic of China 2 Department of Physics, Tohoku University, Sendai 980-8578, Japan 3 Department of Physics, Nagoya University, Nagoya 464-8602, Japan (Received 11 August 2005; published 9 December 2005) We calculate the important next-to-leading-order contributions to the B ! K, decays from the vertex corrections, the quark loops, and the magnetic penguins in the perturbative QCD approach. It is found that the latter two reduce the leading-order penguin amplitudes by about 10% and modify only the B ! K branching ratios. The main effect of the vertex corrections is to increase the small colorsuppressed tree amplitude by a factor of 3, which then resolves the large difference between the direct CP asymmetries of the B0 ! K and B ! 0 K modes. The puzzle from the large B0 ! 0 0 branching ratio still remains. DOI: 10.1103/PhysRevD.72.114005. PACS numbers: 13.25.Hw, 11.10.Hi, 12.38.Bx. I. INTRODUCTION The B factories have accumulated enough events, which allow precision measurements of exclusive B meson decays. These measurements sharpen the discrepancies between experimental data and theoretical predictions within the standard model, where some puzzles have appeared. The recently observed direct CP asymmetries and branching ratios of the B ! K, decays [1],. ACP B0 ! K 11:5 1:8%; ACP B ! 0 K 4 4%;. (1). P AB0 ! T 1 ei2 ; T p C P 2AB ! 0 T 1 ew ei2 ; T T p P P C ew ei2 ; 2AB0 ! 0 0 T T T T. where T, C, P, and Pew stand for the color-allowed tree, color-suppressed tree, penguin, and electroweak penguin amplitudes, respectively, and 2 is the weak phase defined later. The counting rules in terms of powers of the Wolfenstein parameter 0:22 are then assigned to various decay amplitudes [5–7]. The amplitudes in Eq. (2) obey the counting rules in the standard model,. BB0 ! 5:0 0:4 106 ;. P ; T. BB0 ! 0 0 1:45 0:29 106 ;. are prominent examples. The expected relations ACP B0 ! K ACP B ! 0 K and BB0 ! BB0 ! 0 0 obviously contradict the above data. In this work we shall investigate the indication of Eq. (1) and study whether they can be accommodated in the perturbative QCD (PQCD) approach [2,3]. To explain these puzzles, it is useful to adopt the topological-amplitude parametrization [4] for these decays. The most general parametrization of the B ! decay amplitudes is written as *Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]. 1550-7998= 2005=72(11)=114005(21)$23.00. (2). C ; T. Pew 2 : T. (3). Therefore, the B0 ! 0 0 branching ratio is expected to be of O2 of the B0 ! one. However, Eq. (1) shows that the former is about of O of the latter. The B ! K decay amplitudes are written, up to O2 , as AB ! K 0 P0 ; 0 p P0ew T C0 i3 0 0 ; 2AB ! K P 1 0 0 0 e P P P T0 AB0 ! K P0 1 0 ei3 ; P 0 p P C0 i3 e ; (4) 2AB0 ! 0 K 0 P0 1 ew P0 P0 where the notations T 0 C0 , P0 , and P0ew bear the same meanings as for the B ! decays but with primes, and the weak phase 3 is defined via the Cabibbo-. 114005-1. © 2005 The American Physical Society.

(2) HSIANG-NAN LI, SATOSHI MISHIMA, AND A. I. SANDA. Kobayashi-Maskawa (CKM) matrix element Vub jVub j expi3 [8]. These amplitudes obey the counting rules, T0 ; P0. P0ew ; P0. C0 2 : P0. (5). The data ACP B0 ! K 11% imply a sizable relatively strong phase between T 0 and P0 , which verifies our prediction made years ago using the PQCD approach [2]. Since both P0ew and C0 are subdominant, the approximate equality for the direct CP asymmetries ACP B ! 0 K ACP B0 ! K is expected, which is, however, in conflict with the data in Eq. (1) dramatically. It is then natural to conjecture a large P0ew [7,9–12], which signals a new-physics effect, a large C0 [13–16], or both [17,18] in the B ! K decays. The large C0 proposal seems to be favored by a recent analysis of the B ! K, data based on the amplitude parametrization [13]. Note that the current PQCD predictions for the two-body nonleptonic B decays were derived from the leading-order (LO) formalism. While LO PQCD implies a negligible C0 , it is possible that this supposedly tiny amplitude may receive a significant subleading correction. Hence, before claiming a new-physics signal, one should at least examine whether the next-to-leading-order (NLO) effects could enhance C0 sufficiently. In this paper we shall calculate the important NLO contributions to the B ! K, decays from the vertex corrections, the quark loops, and the magnetic penguins. The higher-power corrections have not yet been under good control, and will not be considered here. We find that the corrections from the quark loops and from the magnetic penguins, being about 10% of the LO penguin amplitude, decrease only the B ! K branching ratios. The vertex corrections tend to increase C0 by a factor of 3. This larger C0 leads to nearly vanishing ACP B ! 0 K without changing the branching ratios, which are governed by P0 . The B ! K puzzle is then resolved. The other NLO corrections, mainly to the B meson transition form factors, can be eliminated by choosing an q b, being a appropriate renormalization scale m hadronic scale and mb the b quark mass. This observation follows the well-known Brodsky-Lepage-Mackenzie (BLM) procedure [19], in which the scale is determined in the way that the vacuum polarization effects are absorbed into the coupling constant s . It has been demonstrated with this procedure that NLO corrections to many exclusive processes are minimized to some extent [19]. Taking the simple pion form factor as an example, the BLM scale has been found to be of the order of the invariant mass of the hard exchanged gluon. The choice of proposed in the PQCD approach [20] is basically in agreement with this procedure: the argument of the coupling constant is set to the invariant masses of internal. PHYSICAL REVIEW D 72, 114005 (2005). q b for the B meson transiparticles, which are of O m tion form factors [21–23]. A general feature of the BLM scale is that it is always much lower than the external kinematic variable, implying that the smallness of the coupling constant is not the only condition for the applicability of perturbation theory. As mentioned before, the observed branching ratio BB0 ! 0 0 1:5 106 is much larger than the LO PQCD prediction 107 [3,24]. The prediction from QCD-improved factorization (QCDF) has the same order of magnitude [25]. Since this mode involves a subdominant color-suppressed tree amplitude as shown in Eq. (2), a larger C certainly helps to resolve the B ! puzzle. We also compute the NLO corrections to these decays and find the similar reduction from the quark loops and the magnetic penguins, which are about 10% of the LO penguin amplitude P. Since P is subdominant, the B0 ! and B ! 0 branching ratios almost remain the same. The enhancement of C from the vertex corrections, leading to BB0 ! 0 0 0:3 106 , is still not sufficient to account for the data. We point out that any new mechanism, introduced to resolve this puzzle, must survive the constraint from the tiny observed branching ratios [1], 6 BB0 ! K 0 K 0 0:960:25 0:24 10 ;. BB0 ! 0 0 < 1:1 106 :. (6). The leading PQCD predictions for BB0 ! K 0 K 0 [26] and for BB0 ! 0 0 [27,28] have been consistent with the experimental data. The proposals of the final-state interaction [29] and of the charming penguin in softcollinear effective theory (SCET) [30] have not yet been applied to the B0 ! 0 0 decay. We review the LO PQCD predictions for the B ! K, decays, including those for the mixing-induced CP asymmetries in Sec. II. The vertex corrections, the quark loops, and the magnetic-penguin amplitudes are computed in Sec. III. We perform the numerical study in Sec. IV, where the theoretical uncertainty is also analyzed. Section V is the conclusion. The explicit factorization formulas for the various topologies of decay amplitudes are collected in the Appendix. II. LEADING-ORDER PREDICTIONS The effective Hamiltonian for the b ! s transition is given by [31] G X

(3) C Oq C Oq Heff pF Vqb Vqs 1 2 1 2 2 qu;c 10 X Ci Oi ; (7) i3. with the Fermi constant GF 1:166 39 105 GeV2 , and the CKM matrix elements V. The four-fermion opera-. 114005-2.

(4) RESOLUTION TO THE B ! K PUZZLE. PHYSICAL REVIEW D 72, 114005 (2005). tors are written as Oq 1. At . and. Au f Fe Me fB Fa Ma ;. si qj VA q j bi VA ;. si qi VA q j bj VA ; Oq 2 X O3 si bi VA q 0j q0j VA ;. Ac 0;. X. q 0j q0i VA ;. q0. O5 si bi VA. X. q 0j q0j VA ;. q0. O6 si bj VA. X. (11). t A f FeP MPe fB FaP MPa ;. q0. O4 si bj VA. are decomposed at LO into. (8). q 0j q0i VA ;. q0. where we do not distinguish the color-allowed and colorsuppressed contributions. The factorization formulas for the various contributions to each B ! K, mode are collected in Tables I and II, and in the Appendix, whose dependence on the Wilson coefficients has been made explicit. We define the standard combinations,. X 3 O7 si bi VA eq0 q 0j q0j VA ; 2 q0. a1 C2 . C1 ; Nc. X 3 O8 si bj VA eq0 q 0j q0i VA ; 2 q0. a2 C1 . C2 ; Nc. X 3 O9 si bi VA eq0 q 0j q0j VA ; 2 q0 O10. ai Ci . with the color indices i; j, and the notations q 0 q0 VA q 0 1 5 q0 . The index q0 in the summation of the above operators runs through u, d, s, c, and b. The effective Hamiltonian for the b ! d transition is obtained by changing s into d in Eqs. (7) and (8). According to Eq. (7), the amplitude for a B meson decay transition has the into the final state f through the b ! sd general expression, c

(5)

(6) AB ! f Vub Vusd Au f Vcb Vcsd Af t

(7) V Vtb tsd Af :. For f K, the amplitudes decomposed at LO into. (9) Ac K ,. and. At K. a01 C1 ;. fK Fe Me f FeK MeK fB Fa Ma ;. Ac K. 0;. 0. a0q C4 32eq0 C10 ; 3. 0. a40q C3 32eq0 C9 ;. a4q a4 32eq0 a10 ; 0 a5q 0 a6q . 0. 0. 3 0 2eq a7 ;. 0 a0q 5. C6 . a6 32eq0 a8 ;. 0 a0q 6. C5 32eq0 C7 :. a5 . (13). 3 0 2eq C8 ;. With the amplitude AB ! f being computed using Eq. (9), we derive the two-body nonleptonic B meson decay rates and CP asymmetries. The former are given by B ! f . G2F m3B jAB ! fj2 ; 128. (14). where mB is the B meson mass. The time-dependent CP asymmetry of the B0 ! 0 KS mode is defined as ACP B0 t ! 0 KS BB 0 t ! 0 KS BB0 t ! 0 KS . BB 0 t ! 0 KS BB0 t ! 0 KS . P P P P P At K fK Fe Me f FeK MeK fB Fa. (10). where fB (fK , f ) is the B meson (kaon, pion) decay constant, Fe (Me ) the color-allowed factorizable (nonfactorizable) tree emission contribution, FeK (MeK ) the color-suppressed factorizable (nonfactorizable) tree emission contribution, Fa (Ma ) the factorizable (nonfactorizable) tree annihilation contribution, and those with the additional superscripts P the contributions from the penc guin operators. For f , the amplitudes Au , A ,. a02 C2 ;. a3q a3 32eq0 a9 ;. are. u AK. MPa ;. i 3–10;. where the upper (lower) sign applies, when i is odd (even). The coefficients a and a0 in Tables I and II, besides a1 and a2 given above, are then written as. X 3 si bj VA eq0 q 0j q0i VA ; 2 q0. Au K ,. Ci1 ; Nc. (12). A0 KS cosMd t S0 KS sinMd t;. (15). with the mass difference Md of the two B meson mass eigenstates, and the direct asymmetry and the mixinginduced asymmetry, A0 KS . j0 KS j2 1 ; 1 j0 KS j2. S 0 KS . 2 Im0 KS ; (16) 1 j0 KS j2. respectively. The B0 ! 0 KS decay has a CP-odd final. 114005-3.

(8) HSIANG-NAN LI, SATOSHI MISHIMA, AND A. I. SANDA. PHYSICAL REVIEW D 72, 114005 (2005). TABLE I. B ! K decay amplitudes, whose factorization formulas are presented in the Appendix. p u u 2A0 K A K0 Fe4 a1 Me4 a01 FeK4 a2 MeK4 a02 Fa4 a1 Ma4 a01 p t 2A0 K. 0 0 0 0 Fa4 a1 Ma4 a01 . Fe Me FeK MeK Fa Ma. At K 0 Fe4 a4d Fe6 ad 6 0d Me4 a4 Me6 a0d 6 0 0 Fa4 a4u Fa6 a6u 0u Ma4 a0u 4 Ma6 a6 . FeP MPe P FeK MPeK FaP MPa. Au K Fe4 a1 Me4 a01 0 0 0 0 At K . 0 0 FeK4 a2 MeK4 a02 0 0 p t 2A0 K0. Fe4 a4u Fe6 au 6 0u Me4 a4 Me6 a0u 6 0 0 Fa4 a4d Fa6 a6d 0d Ma4 a0d 4 Ma6 a6 . d Fe4 ad 4 Fe6 a6 0d Me4 a0d 4 Me6 a6 u d u FeK4 a3 a3 a5 a5d 0d 0u 0d MeK4 a0u 3 a3 a5 a 5 d d Fa4 a4 Fa6 a6 0d Ma4 a0d 4 Ma6 a6 . Fe Me FeK MeK Fa Ma. FeP MPe P FeK MPeK FaP MPa. state, and the corresponding factor, 0 KS . e2i1. u Fe4 au 4 Fe6 a6 0u Me4 a0u 4 Me6 a6 u d u FeK4 a3 a3 a5 a5d 0d 0u 0d MeK4 a0u 3 a3 a5 a 5 u u Fa4 a4 Fa6 a6 0u Ma4 a0u 4 Ma6 a6 p u 2A0 K0. P0 P0ew C0 ei3 ; P0 P0ew C0 ei3. A (17). S . 2 Im ; 1 j j2. (19). respectively, and the factor,. where the weak phase 1 is defined via Vtd jVtd j expi1 . The time-dependent CP asymmetry of the B0 ! mode is defined by. e2i2. T Pei2 ; T Pei2. (20). where the weak phase 2 comes from the identity 2 180 1 3 . In addition, the direct CP asymmetry for is defined by a charged B meson decay B ! fB ! f. ACP B0 t ! BB 0 t ! BB0 t ! . BB 0 t ! BB0 t ! A cosMd t S sinMd t;. j j2 1 ; 1 j j2. ACP . (18). with the direct asymmetry and the mixing-induced asymmetry,. BB ! f BB ! f BB ! f : BB ! f. (21). The PQCD predictions for the branching ratios and the CP asymmetries of the B ! K, decays in the naive. 114005-4.

(9) RESOLUTION TO THE B ! K PUZZLE TABLE II. Appendix.. PHYSICAL REVIEW D 72, 114005 (2005). B ! decay amplitudes, whose factorization formulas are presented in the u A . Fe4 a1 Me4 a01 0 Ma4 a02 . Fe Me Fa Ma. At FeP MPe FaP MPa. u Fe4 au 4 Fe6 a6 0u Me4 a4 Fa6 ad 6 0u 0d 0u 0d Ma4 a3 a3 a0d 4 a5 a5 p u 2A 0. Fe Me Fa Ma. Fe4 a1 a2 Me4 a01 a02 0 0 p t 2A 0. FeP MPe FaP MPa. u d u d u d Fe4 a3u ad 3 a4 a4 a5 a5 Fe6 a6 a6 0d 0u 0d 0u 0d Me4 a0u 3 a3 a4 a4 a5 a5 0 0 p u 2A0 0. Fe Me Fa Ma. Fe4 a2 Me4 a02 0 Ma4 a02 p t 2A0 0. FeP MPe FaP MPa. d u d d Fe4 a3u ad 3 a4 a5 a5 Fe6 a6 0d 0d 0u 0d Me4 a0u 3 a 3 a4 a5 a5 d Fa6 a6 0u 0d 0u 0d Ma4 a3 a3 a0d 4 a5 a5 . dimensional regularization (NDR) scheme are listed in Tables III, IV, V, and VI. Using the LO and NLO Wilson coefficients, we obtain the values in the columns labeled by LO and LONLOWC , respectively. It is noticed that some of the NLO Wilson coefficients, like C5 , diverge at a low scale. To derive the above tables, we have frozen the values Ci at Ci 0 0:5 GeV, whenever runs to below the scale 0 , since the renormalization-group (RG) evolution is not reliable for < 0 . Note that 0 is also the scale, which sets the starting point of the RG evolution of the Gegenbauer coefficients in the meson distribution ampli-. tudes [32]. We have kept the corrections in higher orders of the electroweak coupling to the Wilson evolution, which were neglected in [33]. Because the effect of the NLO Wilson coefficients is to enhance the penguin amplitude, the branching ratios of the penguin-dominated B ! K modes increase, and the magnitudes of the direct CP asymmetries decrease a bit accordingly. As shown in Eq. (2), the enhanced penguin amplitude P, being destructive to the color-allowed tree amplitude T, allows the B ! branching ratios to vary toward the direction favored by the data. The larger subdominant penguin amplitude. 114005-5.

(10) HSIANG-NAN LI, SATOSHI MISHIMA, AND A. I. SANDA. PHYSICAL REVIEW D 72, 114005 (2005). 106 ).. TABLE III. Branching ratios in the NDR scheme ( The label LONLOWC means the LO results with the NLO Wilson coefficients, and VC, QL, MP, and NLO mean the inclusions of the vertex corrections, the quark loops, the magnetic penguin, and all the above NLO corrections, respectively. The errors in the parentheses were defined in the context. Data [1]. LO. LONLOWC. VC. QL. MP. NLO. 0. 24:1 1:3. 17:0. 32:3. 31:0. 34:2. 24:1. 24:513:612:9 8:17:8. B ! 0 K . 12:1 0:8. 10:2. 18:4. 17:4. 19:4. 14:0. 13:910:07:0 5:64:2. B0 ! K . 18:9 0:7. 14:2. 27:7. 26:7. 29:4. 20:5. 20:915:611:0 8:36:5. B0 ! 0 K 0. 11:5 1:0. 5:7. 12:1. 11:8. 12:8. 8:7. 9:15:65:1 3:32:9. B0 ! . 5:0 0:4. 7:0. 6:8. 6:6. 6:9. 6:7. 6:56:72:7 3:81:8. B ! 0. 5:5 0:6. 3:5. 4:1. 4:0. 4:1. 4:1. 4:03:41:7 1:91:2. B0 ! 0 0. 1:45 0:29. 0:12. 0:27. 0:37. 0:29. 0:21. 0:290:500:13 0:200:08. Mode . . B ! K. TABLE IV. Data [1]. LO. LONLOWC. VC. QL. MP. NLO. 0. 0:02 0:04. 0:01. 0:01. 0:01. 0:00. 0:01. 0:00 0:000:00. 0 K . 0:04 0:04. 0:08. 0:06. 0:01. 0:05. 0:08. 0:010:030:03 0:050:05. 0:12. 0:08. 0:09. 0:06. 0:10. 0:090:060:04 0:080:06. Mode . . B ! K B. !. Direct CP asymmetries in the NDR scheme.. B0 ! K . 0:115 0:018. B0 ! 0 K 0. 0:02 0:13. 0:02. 0:00. 0:07. 0:00. 0:00. 0:070:030:01 0:030:01. B0 ! . 0:37 0:10. 0:14. 0:19. 0:21. 0:16. 0:20. 0:180:200:07 0:120:06. B ! 0. 0:01 0:06. 0:00. 0:00. 0:00. 0:00. 0:00. 0:00 0:000:00. B0. 0:280:40 0:39. 0:04. 0:34. 0:65. 0:41. 0:43. !. 0 0. 0:630:350:09 0:340:15. TABLE V. Topological amplitudes in units of 105 GeV for the B ! K, decays in the NDR scheme. Topology. LO. VC. LONLOWC. QL. MP. NLO. 0. P T0 C0 P0ew. 36:6e 6:9ei0:0 0:5ei2:5 5:8ei3:1. 50:6e 6:6ei0:0 0:6ei0:4 5:8ei3:1. 49:6e 6:6ei0:1 1:9ei1:3 5:4ei3:0. 52:1e 6:6ei0:0 0:6ei0:2 5:8ei3:1. 43:7e 6:6ei0:0 0:6ei0:4 5:8ei3:1. 44:1ei2:9 6:6ei0:1 1:7ei1:3 5:4ei3:0. T P C Pew. 24:3ei0:0 4:7ei0:4 0:8ei2:6 0:7ei0:0. 23:5ei0:0 6:5ei0:4 2:2ei0:2 0:7ei0:0. 23:1ei0:0 6:3ei0:3 4:8ei1:1 0:7ei0:1. 23:6ei0:1 6:7ei0:3 2:3ei0:4 0:7ei0:0. 23:5ei0:0 5:7ei0:4 2:2ei0:2 0:7ei0:0. 23:2ei0:0 5:6ei0:4 4:3ei1:1 0:7ei0:1. i2:9. i2:9. TABLE VI. Data S0 KS S. LO. i2:9. i2:9. i2:8. Mixing-induced CP asymmetries in the NDR scheme. LONLOWC. VC. QL. MP. NLO. 0:31 0:26. 0:70. 0:73. 0:74. 0:73. 0:73. 0:740:020:01 0:030:01. 0:50 0:12. 0:34. 0:49. 0:47. 0:51. 0:41. 0:421:000:05 0:560:05. also increases the magnitudes of the direct CP asymmetries in the B ! decays due to the stronger interference with the dominant tree amplitudes. As stated before, the LO PQCD predictions for the B ! K branching ratios are consistent with the data, viewing the range spanned by the columns LO and LONLOWC in. Table III. However, the prediction for the B0 ! 0 0 branching ratio is too small compared to the measured value. Those for the direct CP asymmetries of the B ! K, decays, except ACP B ! 0 K , are all in good agreement with the data as shown in Table IV. The LO direct CP asymmetry of the B0 ! 0 0 mode differs in. 114005-6.

(11) RESOLUTION TO THE B ! K PUZZLE. PHYSICAL REVIEW D 72, 114005 (2005). sign from the result obtained in [3], because we have employed the different pion distribution amplitudes (see Sec. IV). It simply implies that the theoretical uncertainty for the modes with tiny branching ratios is huge. Note that the predictions from QCDF [25] for the direct CP asymmetries usually have signs opposite to those from PQCD. It has been realized that the set ‘‘S4’’ with nonuniversal parameters, such as the different annihilation phases for the B ! PP, PV, and VP decays, must be adopted in order for QCDF to accommodate the data [33–36]. The above two discrepancies associated with the B0 ! 0 0 branching ratio and with the B ! 0 K direct CP asymmetry lead to the puzzles mentioned in the Introduction. We prepare Table V for the various topological amplitudes, whose definitions are referred to [6]. The values in the columns LO and LONLOWC follow the power counting rules in Eqs. (3) and (5) exactly, explaining why the B ! K, puzzles appear. After obtaining the values of the various topological amplitudes, we compute the mixing-induced CP asymmetries through Eqs. (17) and (20). Since C0 is of O2 compared to P0 , it is expected that the LO PQCD results of S0 KS are close to that extracted from the b ! ccs decays, Sccs sin2 . 0:685, as shown in Table VI. 1 On the contrary, P is of O of T in the B0 ! decays, such that a larger deviation of S from Sccs is found. The LO PQCD results of S are consistent with the data, but those of S0 KS are not. Moreover, PQCD predicts S0 KS S0 KS Sccs > 0, opposite to the observed values. This result is in agreement with those obtained in the literature [15,37,38]. Hence, the measurement of the mixing-induced CP asymmetries in the penguindominated modes provides an opportunity of discovering new physics. Currently, the data of S0 KS still suffer significant errors. On the other hand, the NLO corrections and the theoretical uncertainty, which concern the allowed range of the PQCD predictions, need to be analyzed. A more clear picture will be attained, after we complete these analyses. III. NEXT-TO-LEADING-ORDER CORRECTIONS We explain the consistent power countings in s and in large logarithms, before computing the NLO corrections. A PQCD formula of leading power in 1=mb is written symbolically as exp0 s LWC exp0 s L2S 1 2s L2S 0 exp0 q s LRG H s 0 ;. (22). where the first, second, and third exponentials represent the Wilson coefficient, the Sudakov factor, and the RG factor [39], with the notations LWC lnmW =t, LS lnxPb, and LRG lntb, xP being a fractional parton momenq b a characteristic hard scale, and b the tum, t m. conjugate variable to the parton transverse momentum kT , and , , and q the corresponding anomalous dimensions. The RG factor governs the evolution from t down to 1=b. The evolution from 1=b down to the cutoff 0 , which characterizes the meson distribution amplitude , has been neglected. This formula is complete at LO, since the hard kernel H is evaluated to Os , and at next-to-leading logarithm (NLL), since the Wilson coefficient, the Sudakov factor, and the RG factor have been resummed up to the next-to-leading logarithms s LWC , s LS , and s LRG , respectively, (s L2S is the leading logarithm). In all our previous works we used the one-loop running coupling constant s , which is, strictly speaking, a NLO effect. This effect takes into account the potential large NLO corrections to the B meson transition form factors through the BLM procedure (see the Introduction). Next, we add subleading corrections to Eq. (22), which include (1) H 0 s ! H 0 s H 1 2s .—This is what we are going to do in this section, where the NLO hard kernel H 1 contains the vertex corrections, the quark loops, and the magnetic penguin. (2) exp0 s LWC ! exp0 s LWC 1 2s LWC .—The LO Wilson coefficient is replaced by the NLO one, for which the corresponding anomalous dimension is calculated to two loops: 0 s ! 0 s 1 2s . According to our counting rules, the NLO anomalous dimension leads to the summation of the next-to-next-to-leading logarithm (NNLL) 2s LWC . (3) exp0 s L2S 1 2s L2S ! exp0 s L2S 1 2s L2S 2 3s L2S .—This means the accuracy of the summation up to NNLL (3s L2S ). Unfortunately, it requires a three-loop evaluation of the corresponding anomalous dimension for the Sudakov factor, which is not yet available in the literature. 0 1 2 (4) exp0 q s LRG ! expq s LRG q s LRG .—Since LRG and LS are of the same order of magnitude, and the NNLL Sudakov resummation is not available, this NNLL piece of subleading corrections (2s LRG ) cannot be included consistently. The power countings in s and in various large logarithms are independent in principle. Based on the above classification, we shall extend Eq. (22) by considering the subleading corrections from the first and second pieces. With the one-loop running s , the NLO corrections to the hard kernel are complete (assuming that the corrections to the form factors have been minimized by our choice of the hard scale). It is not necessary to adopt the two-loop s as in [33], whose effect is next-to-next-to-leading order (NNLO). Because of LWC LS ; LRG , the NNLL term 2s LWC is much more essential than those from the third and fourth pieces. The LO PQCD results for the B ! K, decays from using the NLO Wilson coefficients have. 114005-7.

(12) HSIANG-NAN LI, SATOSHI MISHIMA, AND A. I. SANDA. been listed in Tables III, IV, V, and VI. When investigating the NLO corrections from the vertex corrections, the quark loops, and the magnetic penguin to the hard kernel below, we shall always use the NLO Wilson coefficients. After obtaining the decay amplitudes AB ! f up to NLO, we employ Eq. (14) to evaluate the corresponding decay rates. A. Vertex corrections It has been known that the vertex corrections, reducing the dependence of the Wilson coefficients on the renormalization scale , play an important role in a NLO analysis. Since the nonfactorizable contributions are negligible [40], we concentrate only on the vertex corrections to the factorizable amplitudes. For charmless B meson decays, these corrections do not involve the end-point singularities from vanishing momentum fractions in collinear factorization theorem (QCDF [25]). Therefore, there is no need to employ the kT factorization theorem (PQCD [2,3,39,41,42]) here. This claim can be justified by recalculating one of the nonfactorizable amplitudes, Me4 , for the B ! K decays in the collinear factorization theorem, which is also free of the end-point singularity. It is found that the results for Me4 from the two calculations (with and without the parton transverse momentum kT in the kaon) differ only by 10%. For more detail, refer to the Appendix. After justifying the neglect of the parton trans-. PHYSICAL REVIEW D 72, 114005 (2005). verse degrees of freedom, we simply quote the QCDF expressions for the vertex corrections. An important remark is that the light quark from the b quark transition is assumed to carry the full momentum of the associated meson in QCDF [25]. Strictly speaking, this light quark carries the fractional momentum, whose dependence should appear in the PQCD formalism for the vertex corrections. Because it is indeed an energetic quark, the assumption is reasonable. The vertex corrections modify the Wilson coefficients in Eq. (12) into [25] s C CF 1 V1 M; 4 Nc C V2 M; CF 2 a2 ! a2 s 4 Nc C C i1 Vi M; ai ! ai s 4 F Nc a1 ! a1 . i 3–10; (23). with M being the meson emitted from the weak vertex. For the B ! K decays, M denotes the kaon for the vertex functions V1;4;6;8;10 and the pion for V2;3;5;7;9 . In the NDR scheme Vi M are given by [25]. p 8 2 2Nc R1 mb > 18 dxAM xgx; 12 ln > fp > M 0 < R Vi M 12 lnmb 6 2 2Nc 10 dxAM xg1 x; > f M > p > : 2 2N R 6 fM c 10 dxPM xhx;. for i 1–4; 9; 10; for i 5; 7;. (24). for i 6; 8;. where fM is the decay constant of the meson M, and AM x [PM x] the twist-2 (twist-3) meson distribution amplitude given in Sec. IV, x being the parton momentum fraction. The hard kernels are 1 2x 2 lnx 2 lnx i 2Li2 x ln x 3 2i lnx x $ 1 x ; (25) gx 3 1x 1x hx 2Li2 x ln2 x 1 2i lnx x $ 1 x: The expressions of Vi M in the ’t Hooft-Veltman scheme can be found in [43]. The factorization formulas for the various B ! K, decay amplitudes are still the same as in Tables I and II. The dependence of the Wilson coefficients ai on the renormalization scale modified by the vertex corrections is exhibited in Fig. 1 for both the real and the imaginary parts. It is found that the dependence of most of ai is moderated by the vertex corrections (with the generation of the imaginary parts). The dependence of a6;8 is, however, not altered. It has been known that their dependence will be moderated after being combined with the dependence of the chiral scale m0K associated with the kaon [25]. The most dramatic changes arise from a2;3;10 .. (26). Because of the smallness of a3 (a10 ) compared to the Wilson coefficient a4;6 (a9 ) for the QCD (electroweak) penguins, the only significant effect appears in the colorsuppressed tree amplitude C0 , which is governed by a2 . For other ai , the vertex qcorrections amount only up to 70% at b 1:5 GeV. The above observation is the scale m manifest in Table V: most of the topological amplitudes for the B ! K, decays change a little, while C0 and C are enhanced by factors of 3 and 2 (viewing the values in the columns LONLOWC and VC), respectively. It is then understood that the B ! K branching ratios, dominated by the penguin contributions from a4;6 , vary only slightly under the vertex corrections, as indicated in. 114005-8.

(13) RESOLUTION TO THE B ! K PUZZLE 1.5. PHYSICAL REVIEW D 72, 114005 (2005) 0.2. 0.02. 0.1. 0.015. 0. 0.01. -0.1. 0.005. 1. 0.5. 0. -0.2 1. 2. 3. 4. 5. 0 1. 2. µ (GeV). 3. 4. 5. 1. 2. µ (GeV). (a) a1. (b) a2. 0. 0. -0.025. -0.01. -0.05. -0.02. -0.075. -0.03. 3. 4. 5. µ (GeV). (c) a3 0. -0.05. -0.1. -0.1. -0.04 1. 2. 3 µ (GeV). 4. 5. -0.15 1. 2. (d) a4. 3 µ (GeV). 4. 5. 1. 2. (e) a5. 0.0006. 3 µ (GeV). 4. 5. 4. 5. (f) a6. 0.002. 0. 0.0015. -0.003. 0.0004 0.001. -0.006. 0.0005. 0.0002. -0.009 0 0. -0.012 1. 2. 3 µ (GeV). 4. 5. (g) a7. 1. 2. 3 µ (GeV). (h) a8. 4. 5. 1. 2. 3 µ (GeV). (i) a9. 0.002. 0.001. 0. -0.001. -0.002 1. 2. 3 µ (GeV). 4. 5. (j) a10. FIG. 1. Real parts of ai for the B ! K decays without the vertex corrections (dotted lines) and with the vertex corrections (solid lines), and imaginary parts with the vertex corrections (dot-dashed lines) in the NDR scheme.. Table III. However, the direct CP asymmetries of the B ! 0 K and B0 ! 0 K 0 modes, related to C0 , are modified significantly, as shown in Table IV: ACP B ! 0 K has increased from 0:06 to 0:01, and ACP B0 ! 0 K 0 A0 KS has decreased from 0.00 to 0:07. ACP B0 ! K , determined solely by the colorallowed tree amplitude T 0 , does not change much. The. effect from the vertex corrections on the LO PQCD predictions for the B ! decays can also be understood by means of the enhanced color-suppressed tree amplitude C: the B0 ! 0 0 branching ratio increases by 30%, and the direct CP asymmetry changes from 0:34 to 0:65. The sign flip of the direct CP asymmetry is attributed to a huge change of the strong phase of C caused by the vertex. 114005-9.

(14) HSIANG-NAN LI, SATOSHI MISHIMA, AND A. I. SANDA 0. . . . . PHYSICAL REVIEW D 72, 114005 (2005) 0. corrections. The predicted B ! and B ! branching ratios, to which C remains subdominant, decrease only a bit. The NLO effect, though increasing jCj by a factor of 2, is not enough to resolve the B ! puzzle. Perhaps, the penguin amplitude is also larger than expected [30,44]. Nevertheless, the vertex corrections do improve the consistency between the theoretical predictions and the experimental data of the B ! decays. Though the vertex corrections have been included in QCDF [25], they do not help resolve the B ! K puzzle. We neglect the electroweak penguin P0ew for convenience in the following explanation. Table V shows that the penguin amplitude P0 is in the second quadrant, and the colorallowed tree amplitude T 0 is roughly aligned with the positive real axis. The color-suppressed tree amplitude C0 is enhanced by the vertex corrections and becomes almost imaginary. It then orients the sum T 0 C0 into the fourth quadrant, such that T 0 C0 and P0 more or less line up (and point to the opposite directions). This is the reason the magnitude of ACP B ! 0 K , proportional to the sine of the angle between T 0 C0 and P0 , becomes smaller in PQCD. The situation in QCDF is different, where P0 is preferred to be in the third quadrant [40]. That is, the predicted ACP B0 ! K has a wrong sign. Then the modified C0 , still orienting T 0 C0 into the fourth quadrant, cannot reduce the magnitude of ACP B ! 0 K . The three types of NLO corrections considered here have been extended up to O2s 0 in QCDF recently [45], which, however, make the QCDF predictions for ACP B ! 0 K deviate more from the data. Another O2s piece from the b ! sg

(15) g

(16) transition was included into QCDF [46], which enhances the B ! K branching ratios, but leaves their direct CP asymmetries intact. The B ! K puzzle cannot be resolved in SCET either [47]: the leading SCET formalism requires the ratio C0 =T 0 to be real, so that C0 , being parallel to T 0 , cannot orient the sum T 0 C0 into the fourth quadrant, and that the magnitude of ACP B ! 0 K remains large. We have found that the color-suppressed tree amplitude C0 could be enhanced a few times by the vertex corrections in the standard model. It is then worthwhile to investigate whether the mixing-induced CP asymmetry S0 KS in the B ! 0 KS decays deviates from Sccs substantially under a large C0 according to Eq. (17). A similar investigation of the large C effect applies to S in the B0 ! decays according to Eq. (20). The results are collected in Table VI, which indicates that the deviation is still small and positive. It is known that the leading deviation caused by C0 is proportional to cos C0 P0 , if neglecting P0ew . Because the vertex corrections also rotate the orientation of C0 , it becomes more orthogonal to P0 as shown in Table V, and S0 KS is not increased much. The mixinginduced CP asymmetry S , depending only on T and P, is insensitive to the vertex corrections, which mainly affect C.. B. Quark loops For the B ! K and B ! decays, the dominant penguin amplitude P0 jVtb Vts

(17) jC4 and tree amplitude

(18) T jVub Vud jC2 are both of O4 [13]. Hence, the charm-quark loop amplitude, proportional to

(19) s jVcb Vcs jC2 s 2 in the former and to

(20) s jVcb Vcd jC2 s 3 in the latter, could be an important source of NLO corrections. Its effect is expected to be larger in the B ! K decays. On the other hand, the up

(21) quark loop amplitude, proportional to s jVub Vus jC2 5 s [13] for B ! K, seems to be negligible. For B ! , the up-quark loop amplitude, proportional to

(22) s jVub Vud jC2 s 4 [13], might be comparable to the charm-quark one. Therefore, we shall include both quark loops in the following analysis. For completeness, we shall also include the quark-loop amplitudes from the QCDpenguin operators, whose contributions are proportional to s jVtb Vts

(23) jCi s 4 , i 3; 4; 6. They have the order of magnitude the same as or larger than the up-quark one, and can influence the direct CP asymmetries of the B ! K modes. The quark loops from the electroweak penguins will be neglected due to their smallness. Note that the CKM factors of these corrections differ among the loop amplitudes and between the b ! sd and b ! sd transitions. For the b ! s transition, the contributions from the various quark loops are given by. Heff . X X GF

(24) s Cq ; l2 p Vqb Vqs 2 2 qu;c;t q0. s 1 5 T a bq 0 T a q0 ;. (27). l2 being the invariant mass of the gluon, which attaches the quark loops in Fig. 2. For the b ! d transition, the quarkloop corrections are obtained by substituting d for s in Eq. (27). The functions Cq ; l2 are written as Cq ; l2 Gq ; l2 23C2 ;. (28). for q u, c, and K. K s. s l b. b. π. B. 114005-10. l. (a). FIG. 2.. π. B (b). Quark-loop amplitudes..

(25) RESOLUTION TO THE B ! K PUZZLE. PHYSICAL REVIEW D 72, 114005 (2005). 2 Ct ; l2 Gs ; l2 C3 3 X 00 q G ; l2 C4 C6 : q00 u;d;s;c. (29) The constant term 2=3 in the above expressions arises from the Fierz transformation of the four-fermion operators in D dimensions with the anticommuting Dirac matrix 5 in the NDR scheme. The contribution proportional to the Wilson coefficient C5 , being purely ultraviolet, should be combined with that from the magnetic penguin to form the effective Wilson coefficient C8g C5 [31]. Since our. q b 1:5 GeV, the characteristic hard scale is of order m b quark is not an active one, and does not contribute to Eq. (29). Except for this difference, our expressions are basically the same as in [25]. The function Gc ; l2 for the loop of the massive charm quark is given by Gc ; l2 4. Z1. dxx1 x ln. 0. m2c x1 xl2 ; 2 (30). mc being the charm-quark mass, whose real and imaginary parts are. 8 q2 > 4m q2 > 1 2c 1 > > 4mc l > q ln 1 > > l2 > 4m2c > 1 2 1 > > l > q q > > 2 2 2 2 4m 4m2c < 1 c 2 5 4m m 2 2m 1 cot 1 2 c c c 2 l l2 Re Gc ; l2 2 ln 2 1 2 4m2c 3 3 3 l l > > 21 l2 > > > q > > > q2 1 14m2c > > 4m l2 > > 1 l2 c ln q > 2 > 4m : 1 1 c. 1 < l2 < 0; 0 l2 < 4m2c ;. (31). l2 4m2c ; 4m2c < l2 < 1;. l2. and s 2 2 2m 4m2 4m2 1 2 c 1 2 c 1 2 c ; Im Gc ; l2 3 l l l (32) respectively. For the loops of the light quarks u, d, and s, we have the expressions similar to Eq. (30) but with mc being replaced by mu , md , and ms , respectively. Because their contributions are insensitive to the light quark masses, we simply adopt the same mass m for the three quark loops. Varying m from mu 4:5 MeV to ms 100 MeV, the branching ratios change by less than 1%. To picture the quark-loop effect, we display in Fig. 3 the

(26)

(27) dependence of Vqb Vqsd =Vtb Vtsd Cq , q u; c; t, on the renormalization scale for a given l2 m2B =4 in the NDR scheme. The real part of the up-quark loop is indeed negligible compared to that of the charm-quark loop in the b ! s transition as indicated in Fig. 3(a). However, in the other transitions described by Figs. 3(b)–3(d), the up- and charm-loop corrections are comparable as argued above. The quark loops from the QCD-penguin operators are in fact essential. Figures 3(a) and 3(c) [and also Figs. 3(b) and 3(d)] imply that the weak phases cause different dependences between the b ! s and b ! d transitions in the cases of the up and charm loops, but not in the case of the QCD-penguin loops. The quark-loop amplitudes depend on the gluon invariant mass l2 , which is assumed to be an arbitrary constant. hl2 i in the naive factorization assumption (FA). Since the topology displayed in Fig. 2 is the same as the penguin one, its contribution can be absorbed into the Wilson coefficients a4;6 ,. 0.02. 0.01. 0.01. 0. 0. -0.01. -0.01. -0.02. -0.02. -0.03 1. 2. (a) Re. 3. 4. 5. µ (GeV) ∗ αs (µ) Vqb Vqs ∗ 9π Vtb Vts. C (q) (µ, l2 ). 1. 2. . (b) Im. 0.02. 0.01. 0.01. 0. 0. -0.01. -0.01. -0.02. -0.02. 3. 4. 5. µ (GeV) ∗ αs (µ) Vqb Vqs ∗ 9π Vtb Vts. C (q) (µ, l2 ). . -0.03 1. 2. (c) Re. 3. 4. µ (GeV) ∗ αs (µ) Vqb Vqd 9π Vtb V ∗ td. C (q) (µ, l2 ). 5. . 1. 2. (d) Im. 3. 4. µ (GeV) ∗ αs (µ) Vqb Vqd 9π Vtb V ∗ td. C (q) (µ, l2 ). 5. . FIG. 3. Quark-loop contributions to the b ! s [(a),(b)] and b ! d [(c),(d)] transitions for l2 m2B =4 with the solid, dotted, and dot-dashed lines corresponding to the up-quark, charmquark, and QCD-penguin loops, respectively.. 114005-11.

(28) HSIANG-NAN LI, SATOSHI MISHIMA, AND A. I. SANDA. PHYSICAL REVIEW D 72, 114005 (2005). TABLE VII. a4;6 including the quark loops and the magnetic penguin for l2 m2B =4 in the NDR scheme. LONLOWC. QL (b ! s). QL (b ! d). MP. a4 (1.5 GeV) a6 (1.5 GeV). 0:0601 0:0952. 0:0659 i0:0152 0:1010 i0:0152. 0:0500 i0:0131 0:0850 i0:0131. 0:0492 0:0843. a4 (4.4 GeV) a6 (4.4 GeV). 0:0336 0:0428. 0:0454 i0:0036 0:0546 i0:0036 QL (b ! d). 0:0279 0:0371. LONLOWC. 0:0545 i0:0048 0:0637 i0:0048 QL (b ! s). a4 (1.5 GeV) a6 (1.5 GeV). 0:0601 0:0952. 0:0646 i0:0150 0:0997 i0:0150. 0:0804 i0:0180 0:1155 i0:0180. 0:0492 0:0843. a4 (4.4 GeV) a6 (4.4 GeV). 0:0336 0:0428. 0:0537 i0:0047 0:0630 i0:0047. 0:0628 i0:0065 0:0720 i0:0065. 0:0279 0:0371. a4;6 ! a4;6 .

(29) s X Vqb Vqsd q C ; hl2 i;

(30) 9 qu;c;t Vtb Vtsd. (33) with the other ai unmodified. The resultant values of a4;6 at 1:5 and 4.4 GeV are listed in Table VII. As 1:5 GeV, the quark-loop corrections do not change a4;6 much for b ! s and b ! s, while they are destructive As (constructive) to a4;6 for b ! d (b ! d). 4:4 GeV, these corrections are always constructive for the different b quark transitions. Besides, the quark-loop ! Au;c Mu;c Au;c K ; K 0 K 0 1 At ! At p Mt K ; 0 K 0 K 2 1 u;c u;c u;c A 0 K 0 ! A0 K 0 p MK ; 2 ! At Mt At ; 1 ! Au;c p Mu;c Au;c ; 0 0 0 0 2. MP. corrections are mode dependent. For example, they cancel 0 between p the uu and dd 0 components of uu dd= 2 in the B ! decays, but not in others. The assumption of a constant gluon invariant mass in FA introduces a large theoretical uncertainty as making predictions. In the more sophisticated PQCD approach, the gluon mass is related to the parton momenta unambiguously (see the Appendix). Because of the absence of the end-point singularities associated with l2 , l02 ! 0 in Figs. 2(a) and 2(b), respectively, we have dropped the parton transverse momenta kT in l2 , l02 for simplicity. The amplitudes in Eq. (9) become. t t At ! A K 0 MK ; K 0. 1 u;c u;c u;c A 0 K ! A0 K p MK ; 2. Au;c ! Au;c Mu;c K ; K K 1 t t t A 0 K 0 ! A0 K 0 p MK ; 2. At ! At Mt K ; K K Au;c ! Au;c Mu;c ; . (34). Au;c;t ! Au;c;t ; 0 0 1 t t t A 0 0 ! A0 0 p M ; 2. c t where Mu f , Mf , and Mf denote the up-, charm-, and QCD-penguin-loop corrections, respectively, and the minus sign for the final state 0 K 0 comes from the dd component in 0 . The factorization formulas for Mu;c;t K and Mu;c;t are presented in the Appendix. As indicated in Eq. (33), the quark-loop corrections affect the penguin contributions, but have a minor impact on other topological amplitudes. This observation is clear in Table V: jP0 j (jPj) has increased from 50.6 to 52.1 (6.5 to 6.7) in the NDR scheme. Since the B ! K decays are penguin dominated, their branching ratios receive an enhancement (see Table III). The increase of the branching ratios then reduces the magnitude of the direct CP asymmetries in the B ! K modes slightly as shown in. Table IV. It is also easy to understand the insensitivity of the mixing-induced CP asymmetry S0 KS to the quark-loop corrections (see Table VI), viewing the small change in the dominant amplitude P0 in Eq. (17). On the contrary, the penguin contribution is subdominant in the B ! decays, so the branching ratios do not vary much. However, the direct CP asymmetries ACP B0 ! and ACP B0 ! 0 0 , and the mixing-induced CP asymmetry S , directly related to the penguin amplitude, change sizably. C. Magnetic penguins We then discuss the NLO corrections from the magnetic penguin, whose weak effective Hamiltonian contains the. 114005-12.

(31) RESOLUTION TO THE B ! K PUZZLE. PHYSICAL REVIEW D 72, 114005 (2005). b ! sg transition, G Heff pF Vtb Vts

(32) C8g O8g ; 2 with the magnetic-penguin operator, g O8g 2 mb si

(33) 1 5 Tija Ga

(34) bj ; 8. (35). (36). i, j being the color indices. The Hamiltonian for the b ! d transition is obtained by changing s into d in Eq. (35). The topology of the magnetic-penguin operator is similar to that of the quark loops. If regarding the invariant mass l2 of the gluon emitted from the operator O8g as a constant hl2 i, the magnetic-penguin contribution to the B ! K, decays can also be included into the Wilson coefficients, similar to Eq. (33), a4;6 ! a4;6 . s 2mB eff p C8g ; 9 hl2 i. (37). with the effective Wilson coefficient Ceff 8g C8g C5 [31]. The resultant Wilson coefficients a4;6 for 1:5 and 4.4 GeV have been presented in Table VII. The cancellation between the real parts of the quark-loop corrections and of the magnetic penguin is obvious, except in the case of the b ! d transition for 1:5 GeV. In the PQCD approach the gluon invariant mass l2 is related to the parton momenta, such that the corresponding factorization formulas involve the convolutions of all three meson distribution amplitudes. Because the nonfactorizable contributions are negligible, we calculate only the magnetic-penguin corrections to the factorizable amplitudes, which modify only At f in Eq. (9): At K 0. !. At K 0. . except those of the tree-dominated B0 ! and B ! 0 modes, and intends to increase the magnitude of most of the direct CP asymmetries. The mixing-induced CP asymmetry S0 KS is stable under the magnetic-penguin correction for the same reason. The magnitude of S decreases due to the smaller penguin pollution. Because the quark-loop corrections are smaller than the magnetic penguin, the pattern of their combined effect is similar to that of the latter. In summary, the above two pieces of NLO corrections reduce the LO penguin amplitudes by about 10% in the B ! K, decays, and the B ! K and B0 ! 0 0 branching ratios by about 20%. The direct CP asymmetries are not altered very much, which are mainly affected by the vertex corrections, as shown by the similarities between the columns VC and NLO in Table IV.. Mg K ;. IV. THEORETICAL UNCERTAINTY In this section we explain in detail how to derive the results in Tables III, IV, V, and VI, and discuss their theoretical uncertainty. The PQCD predictions depend on the inputs for the nonperturbative parameters, such as decay constants, distribution amplitudes, and chiral scales for pseudoscalar mesons. For the B meson, the model wave function has been proposed in [21]: 1 xmB 2 !2B b2 B x; b NB x2 1 x2 exp ; 2 !B 2 (39) where the Gaussian form was motivated by the oscillator model in [48], and the normalization constant NB is related to the decay constant fB through. 1 g ! At p MK ; At 0 K 0 K 2. Z1. g t t A K ! A K MK ;. 1 ! At p Mg At K ; 0 K 0 0 K 0 2. 0. (38). g t t A ! A M ;. ! At ; At 0 0 1 ! At p Mg At : 0 0 0 0 2 The explicit expressions for the magnetic-penguin amplig tudes Mg K and M are referred to the Appendix. Since an end-point singularity arises, as the invariant mass l2 approaches zero, we have employed the kT factorization theorem, i.e., the PQCD approach in this case. The effect of the magnetic penguin is just opposite that of the quark-loop corrections as indicated in Tables III, IV, and V: it decreases all the B ! K, branching ratios,. fB : dxB x; b 0 p 2 2Nc. (40). There are certainly other models of the B meson wave function available in the literature (see [49,50]). It has been confirmed that the model in Eq. (39) and the model derived in [51] with a different functional form lead to similar numerical results for the B ! transition form factor [52]. The twist-2 pion (kaon) distribution amplitude AK , and the twist-3 ones PK and TK have been parametrized as fK AK x p 6x1 x1 a1K C3=2 1 2x 1 2 2Nc. 114005-13. K 3=2 a2K C3=2 C4 2x 1; 2 2x 1 a4. (41).

(35) HSIANG-NAN LI, SATOSHI MISHIMA, AND A. I. SANDA. fK 5 PK x p 1 30 3 2K C1=2 2 2x 1 2 2 2Nc 9 3 3 !3 2K 1 6a2K 20 2x 1 ; (42) C1=2 4. TK x. fK 1 p 1 2x 1 6 5 3 3 !3 2 2 2Nc 7 2 3 2 K 2 K K a2 1 10x 10x ; 20 5 (43). with a the mass ratio K mu 1 0, mds =mK mK =m0K and the Gegenbauer polynomials C

(36) n t, 1 2 C1=2 2 t 3t 1; 2. 1 2 4 C1=2 4 t 3 30t 35t ; 8 3 2 C3=2 C3=2 1 t 3t; 2 t 5t 1; 2 15 1 14t2 21t4 : (44) C3=2 4 t 8. In the above kaon distribution amplitudes the momentum fraction x is carried by the s quark. For both the pion and kaon, we choose 3 0:015 and !3 3 [53]. Because we did not employ the equations of motions for the twist-3 meson distribution amplitudes [25], we are allowed to include the higher Gegenbauer terms, which are in fact important. However, we drop the derivative term with respect to the transverse parton momentum kT in TK . It has been observed that the contribution from this derivative term to the B ! form factor is negligible [54]. In our previous works we adopted the models of the pion and kaon distribution amplitudes derived from QCD sum rules in [53]. Fixing the B meson decay constant fB 190 MeV from lattice QCD (see [55]), the shape parameter of the B meson wave function was determined to be !B B 0 0:3 in 0:4 GeV [21] from the B ! form factor F light-cone sum rules [56,57]. The chiral scales were chosen as m0 1:3 GeV for the pion and m0K 1:7 GeV for the kaon [2]. The renormalization scale was set to the off shellness of the internal particles, consistent with the BLM procedure. The resultant PQCD predictions [2] have been confirmed by the observed B ! K branching ratios and B0 ! K direct CP asymmetry. The consistency indicates not only that the above inputs are reasonable, but that the short-distance QCD dynamics has been described correctly in PQCD. In this paper we employ the updated models of the pion and kaon distribution amplitudes in [58]. Since the updated Gegenbauer coefficient a 2 0:115 is smaller than the previous one 0.44 for the twist-2 pion distribution ampli-. PHYSICAL REVIEW D 72, 114005 (2005) B F 0. tude [53], reduces compared to that obtained in [24]. To compensate this reduction, we increase the B meson decay constant up to fB 210 MeV, which is consistent with the recent lattice result [59], in order to maintain the B ! K, branching ratios. For the same reason, the penguin annihilation amplitudes, which involve the -K or - timelike form factor, decrease. The magnitude of the resultant direct CP asymmetries of the B ! K, decays, which is not compensated by the overall decay constant fB , then becomes smaller than in [24] as shown in the column LO of Table IV. The smaller B0 ! K direct CP asymmetry is in better agreement with the data, implying that the data could be covered by the theoretical uncertainty at LO of PQCD. All the above nonperturbative inputs suffer uncertainties, and it is necessary to investigate how these uncertainties propagate into the predictions for nonleptonic B meson decays. Here we shall constrain the shape parameter !B and the Gegenbauer coefficients of the twist-2 pion distribution amplitude A using the experimental error of the semileptonic decay B ! l

(37) . The sufficient uncertainties will be assigned to the Gegenbauer coefficients of the twist-2 kaon distribution amplitude AK . The other inputs, such as the B meson decay constant, the twist-3 distribution amplitudes, and the chiral scale associated with the pion and the kaon will be fixed. On one hand, the considered sources of theoretical uncertainties have been representative enough. On the other hand, it is impossible to constrain all the inputs with the currently available data. The spectrum of the semileptonic decay B ! l

(38) in the lepton invariant mass q2 has been measured [60]: R8. 0 d=dq. total. 2. dq2. 0:43 0:11 104 ;. (45). with the total decay rate total 4:29 0:04 1013 GeV [61]. Assuming that the above error is uniform in the region 0 < q2 < 8 GeV2 , we derive the uncertainty of d=dq2 jq2 0 by solving the equation 8 0:11 104 total , where we take only the central value of total for simplicity. With the allowed range of jVub j 3:67 0:47 103 [61], is translated into the uncertainty of the B ! form factor, B 0 0:24 0:05; F. (46). whose central value comes from our choice of the inputs. Equation (46) is consistent with 0:23 0:04 extracted in [62] from a global fit to the above CLEO data, lattice QCD B 2 results of F q , etc. A numerical analysis indicates that B F 0 is more sensitive to !B than to the Gegenbauer coefficients of A . Therefore, we propose the following: (1) The shape parameters for the distribution amplitudes,. 114005-14.

(39) RESOLUTION TO THE B ! K PUZZLE. PHYSICAL REVIEW D 72, 114005 (2005). The resultant B ! ; K transition form factors,. !B 0:40 0:04 GeV; a 2 0:115 0:115; aK 1. 0:17 0:17; aK 4. a 4 0:015; aK 2. B F 0 0:240:05 0:04 ;. 0:115 0:115;. 0:015;. (47). that is, the Gegenbauer coefficients can vary by 100%. We do not consider the uncertainty from K the coefficients a 4 and a4 , to which our predictions are insensitive. Note that the first Gegenbauer coefK ficients aK 1 0:10 0:12 and a1 0:05 0:02 have been found to be smaller in [63,64], respectively. A hint on the effect from the evolution of the meson distribution amplitudes from 1=b down to the cutoff 0 (see Sec. III) can also be obtained through the above variation of the Gegenbauer coefficients. (2) The CKM matrix elements, Vud 0:9734;. Vus 0:2200;. jVub j 3:67 0:47 103 ; Vcs 0:996;. Vcd 0:224;. Vcb 0:0413;. (48). where we consider only the representative source of theoretical uncertainties from jVub j [61]. This source is essential for estimating the uncertainty of the predicted direct CP asymmetries. Vcb 41:3 1:5 103 [61] has a smaller uncertainty, and the other matrix elements have been known

(40) more precisely. The unitarity condition Vtb Vtsd

(41)

(42) Vub Vusd Vcb Vcsd is then employed as evaluating the penguin contributions. (3) The weak phases, 1 21:6 ;. 3 70 30 ;. (49). where the range of the well-measured 1 with sin21 0:685 0:032 [65] has been neglected, and the range of 3 is hinted by the determinations [65,66], 3 6814 15 13 11 Belle; Dalitz; 70 311214 1011 BaBar; Dalitz; 6315 13 CKM fitter; 64 18UT fit:. (50). We fix the other parameters, such as the meson decay constants fB 210 MeV, fK 160 MeV, f 130 MeV, the meson masses mB 5:28 GeV, mK 0:49 GeV, m 0:14 GeV, the charm-quark mass mc 1:5 GeV, and the B meson lifetimes B0 1:528 1012 s, B 1:643 1012 s [1]. We also fix the chiral scales m0 1:3 GeV and m0K 1:7 GeV, where the value of m0 (m0K ) is close to that (larger than 1:25 0:15 GeV) obtained in the recent sum-rule analysis [67].. BK F 0 0:360:09 0:07 ;. (51). respect Eq. (46) from the measurement, and are consistent with the estimation from light-cone sum rules [64]. If further including the variation of m0K as a source of theoretical uncertainties, we just enlarge the range of the B ! K branching ratios, but not of the other quantities. We have tested the dependence of our predictions on the cutoff 0 , which is found to be weak. The above inputs lead to Tables III, IV, V, and VI, where the theoretical uncertainties are displayed only in the columns NLO. The errors (not) in the parentheses represent those from (all sources) the first source of uncertainties. It indicates that the nonperturbative inputs, i.e., the first source, contribute to the theoretical uncertainties more dominantly in the B ! K decays than in the B ! decays, because the former depend on the less controllable parameters associated with the kaon. We also observe that ACP B0 ! K and ACP B ! 0 K always increase or decrease simultaneously, when varying the nonperturbative inputs. Hence, the B ! K puzzle cannot be resolved by tuning these parameters. After including the uncertainties, the predicted B0 ! 0 0 branching ratio and mixing-induced CP asymmetry S0 KS are still far from the data. A more transparent comparison between the predictions and the data is made by considering the ratios of the branching ratios. The following three ratios of the B ! K branching ratios have been widely studied in the literature, R. BB0 ! K B 0:85 0:06; BB ! K 0 B0. BB ! 0 K 1:00 0:08; BB ! K 0 1 BB0 ! K 0:82 0:08; Rn 2 BB0 ! 0 K 0 Rc 2. (52). whose values are quoted from [1]. We have confirmed that these ratios depend on the nonperturbative inputs weakly. Therefore, their deviation from the standard-model predictions could reveal a signal of new physics, such as a large electroweak penguin amplitude. Table III shows that for 3 70 , the ratio R increases slightly from 0.90 to 0.92, when the NLO Wilson coefficients are adopted, beyond which the various NLO corrections do not change R much. The ratio Rc (Rn ) decreases from 1.20 (1.25) to 1.14 (1.14), when the NLO Wilson coefficients are adopted, and settles down at this value as indicated by the column NLO. The different types of NLO corrections cause only small fluctuations. Comparing the columns LO and NLO, the consistency between the PQCD predictions and the data has been improved.. 114005-15.

(43) HSIANG-NAN LI, SATOSHI MISHIMA, AND A. I. SANDA. PHYSICAL REVIEW D 72, 114005 (2005). FIG. 4. Dependence of R, Rc , and Rn on 3 from NLO PQCD with the bands representing the theoretical uncertainty. The two dashed lines represent 1 bounds from the data.. Varying the weak phase 3 and the inputs, we find that the PQCD predictions for R and Rn are in good agreement with the data in Eq. (52), which is obvious from Fig. 4. However, the predictions for Rn exhibit a tendency of overshooting the data, which is attributed to the smaller PQCD results for the B0 ! 0 K 0 branching ratio. A A smaller Gegenbauer coefficient a 2 of enhances Rn . That is, when using the updated pion distribution amplitudes from [58], the consistency of the predictions for Rn with the data deteriorates. A smaller !B enhances Rn . This is the reason we do not lower !B in order to compensate the reduction from the smaller a 2 . Note that m0K has an effect on the electroweak penguin amplitude, i.e., on the B0 ! 0 K 0 branching ratio. Hence, we have also studied the dependence of Rn on the chiral scale m0K . A smaller mK 0 indeed reduces Rn , but does not help much: choosing m0K 1:3 GeV causes only a few percent reduction of Rn . It has been known that the B0 ! 0 K 0 branching ratio can be significantly increased by rotating the electroweak penguin amplitude P0ew away from the penguin amplitude P0 (their values in Table V are roughly parallel to each other). Therefore, we cannot rule out the possibility that P0ew acquires an additional phase from new-physics effects [9,68,69]. However, our theoretical uncertainty is representative, and the actual uncertainty could be larger, such that the discrepancy is not serious at this moment. We do not discuss the ratios relevant to the B ! decays, because the PQCD predictions for the B0 ! 0 0 branching ratio are currently far below the measured values. V. CONCLUSION The LO PQCD has correctly predicted the direct CP asymmetry ACP B0 ! K , but failed to explain another one ACP B ! 0 K [2]. Phenomenologically, the substantial difference between ACP B ! 0 K and ACP B0 ! K has led to the conjecture of new physics [7,9]. However, the difference can also be attributed to a large color-suppressed tree amplitude C0 as pointed out in [13]. Theoretically, an examination of NLO effects is always demanded for a systematic approach like PQCD. Since C0 itself is a subdominant contribution, it is easily. affected by subleading corrections. Hence, before claiming a new-physics signal in the B ! K data, one should at least check whether the NLO effects could enhance C0 sufficiently. This is one of our motivations to perform the NLO calculation in PQCD for the B ! K, decays here. Another motivation comes from the mixing-induced CP asymmetries in the penguin-dominated modes, some of which also depend on the color-suppressed tree amplitudes. To estimate the deviation of S0 KS from Sccs within 0 the standard model, one must compute C reliably. In this paper we have calculated the NLO corrections to the B ! K, decays from the vertex corrections, the quark loops, and the magnetic penguin in the PQCD approach. The results for the branching ratios and CP asymmetries in the NDR scheme have been presented in Tables III, IV, V, and VI, and discussed in Sec. III. It has been shown that the corrections from the quark loops and from the magnetic penguin come with opposite signs and sum to about 10% of the LO penguin amplitudes. Their effect is to reduce the B ! K branching ratios, to which the penguin contribution is dominant, by about 20%. They have a minor influence on the B ! branching ratios, and CP asymmetries. The vertex corrections play an important role in modifying direct CP asymmetries, especially those of the B ! 0 K , B0 ! 0 K 0 , and B0 ! 0 0 modes, by increasing the color-suppressed tree amplitudes a few times. The larger color-suppressed tree amplitude leads to nearly vanishing ACP B ! 0 K , resolving the B ! K puzzle within the standard model. Our analysis has also confirmed that the NLO corrections are under control in PQCD. The NLO corrections, though increasing the colorsuppressed tree amplitudes significantly, are not enough to enhance the B0 ! 0 0 branching ratio to the measured value. A much larger amplitude ratio jC=Tj 0:8 must be obtained in order to resolve this puzzle [13]. Nevertheless, the NLO corrections do improve the consistency of our predictions with the data: the predicted B0 ! (B0 ! 0 0 ) branching ratio decreases (increases). Viewing the consistency of the PQCD predictions with the tiny measured B0 ! K 0 K 0 and B0 ! 0 0 branching. 114005-16.

(44) RESOLUTION TO THE B ! K PUZZLE. PHYSICAL REVIEW D 72, 114005 (2005). ratios, we think that our NLO results for the B ! decays are reasonable. In SCET [30], the large jC=Tj comes from a fit to the data, instead of from an explicit evaluation of the amplitudes. The amplitude C was indeed found to be increased in SCET by the NLO jet function (the short-distance coefficient from matching SCETI to SCETII ) [70], if the parameter set ’’S4’’ in QCDF [25] was adopted. The large measured B0 ! 0 0 branching ratio was then explained. However, we emphasize again that the same analysis should be applied to the B ! decays for a check. Hence, the B ! puzzle still requires more investigation. The tendency of overshooting the observed ratio Rn has implied a possible new-physics phase associated with the electroweak penguin amplitude P0ew . This additional phase can render P0ew orthogonal to the penguin amplitude, and enhance the B0 ! 0 K 0 branching ratio. We have also computed the deviation S0 KS of the mixing-induced CP asymmetry, and found that the NLO effects push it toward the even larger positive value. Therefore, it is difficult to understand the observed negative deviation without physics beyond the standard model. ACKNOWLEDGMENTS We thank T. Browder, H. Y. Cheng, C. K. Chua, W. S. Hou, Y. Y. Keum, M. Nagashima, A. Soddu, and A. Soni for useful discussions. This work was supported by the National Science Council of the R.O.C. under Grant No. NSC-94-2112-M-001-001, and by the Grants-in-aid. Fe4 a 16CF m2B. Z1 0. dx1 dx2. Z1 0. from the Ministry of Education, Culture, Sports, Science and Technology, Japan under Grant No. 14046201. H, N. L. acknowledges the hospitality of the Department of Physics, Hawaii University, where this work was initiated. APPENDIX: FACTORIZATION FORMULAS We first define the kinematics for the B ! M2 M3 decay, where M2 (M3 ) denotes the light pseudoscalar meson involved in the B meson transition (emitted from the weak vertex). In the rest frame of the B meson, the B (M2 , M3 ) meson momentum P1 (P2 , P3 ), and the corresponding spectator quark momentum k1 (k2 , k3 ) are taken, in the light-cone coordinates, as m P1 pB 1; 1; 0T ; 2 mB P2 p 1; 0; 0T ; 2 mB P3 p 0; 1; 0T ; 2. k1 0; x1 P 1 ; k1T ; k2 x2 P 2 ; 0; k2T ; k3 0; x3 P 3 ; k3T ;. where the light meson masses have been neglected. We also define the ratio r2 m02 =mB (r3 m03 =mB ) associated with the meson M2 (M3 ), m02 (m03 ) being the chiral scale. The factorization formulas for the B ! M2 M3 decay amplitudes appearing in Tables I and II are collected below:. b1 db1 b2 db2 B x1 ; b1 f1 x2 A2 x2 r2 1 2x2 P2 x2 T2 x2 . Ee the A; B; b1 ; b2 ; x2 2r2 P2 x2 Ee t0 he A0 ; B0 ; b2 ; b1 ; x1 g;. Fe6 a 32CF m2B. Z1 0. dx1 dx2. Z1 0. Z1 0. dx2 dx3. Z1 0. (A2). b1 db1 b2 db2 B x1 ; b1 fr3 A2 x2 r2 x2 P2 x2 T2 x2 2r2 P2 x2 . Ee the A; B; b1 ; b2 ; x2 2r2 r3 P2 x2 Ee t0 he A0 ; B0 ; b2 ; b1 ; x1 g;. Fa4 a 16CF m2B. (A1). (A3). b2 db2 b3 db3 fx3 A2 x2 A3 x3 2r2 r3 P2 x2 fP3 x3 T3 x3 x3 P3 x3 . T3 x3 gEa the A; B; b2 ; b3 ; x3 1 x2 A2 x2 A3 x3 2r2 r3 f2P2 x2 x2 P2 x2 T2 x2 g P3 x3 Ea t0 he A0 ; B0 ; b3 ; b2 ; x2 g;. Fa6 a 32CF m2B. Z1 0. dx2 dx3. (A4) Z1 0. b2 db2 b3 db3 f2r2 P2 x2 A3 x3 r3 x3 A2 x2 P3 x3 . T3 x3 Ea the A; B; b2 ; b3 ; x3 r2 1 x2 P2 x2 T2 x2 A3 x3 2r3 A2 x2 P3 x3 Ea t0 he A0 ; B0 ; b3 ; b2 ; x2 g;. 114005-17. (A5).

(45) HSIANG-NAN LI, SATOSHI MISHIMA, AND A. I. SANDA. PHYSICAL REVIEW D 72, 114005 (2005). p Z1 2Nc 2 Z 1 M e4 a0 32CF mB dx1 dx2 dx3 b1 db1 b3 db3 B x1 ; b1 A3 x3 f1 x3 A2 x2 r2 x2 PK x2 Nc 0 0 TK x2 E0e thn A; B; b1 ; b3 x2 x3 A2 x2 r2 x2 P2 x2 T2 x2 E0e t0 hn A0 ; B0 ; b1 ; b3 g; (A6) p Z1 2Nc 2 Z 1 Me6 a 32CF mB r3 dx1 dx2 dx3 b1 db1 b3 db3 B x1 ; b1 f1 x3 A2 x2 P3 x3 T3 x3 Nc 0 0 0. r2 1 x3 P2 x2 T2 x2 P3 x3 T3 x3 r2 x2 P2 x2 T2 x2 P3 x3 T3 x3 E0e thn A; B; b1 ; b3 x3 A2 x2 P3 x3 T3 x3 r2 x3 P2 x2 T2 x2 P3 x3 T3 x3 r2 x2 P2 x2 T2 x2 P3 x3 T3 x3 E0e t0 hn A0 ; B0 ; b1 ; b3 g; Ma4 a0 32CF. (A7). p Z1 2Nc 2 Z 1 mB dx1 dx2 dx3 b1 db1 b3 db3 B x1 ; b1 f1 x2 A2 x2 A3 x3 Nc 0 0. r2 r3 f1 x2 P2 x2 T2 x2 P3 x3 T3 x3 x3 P2 x2 T2 x2 P3 x3 T3 x3 g E0a thn A; B; b3 ; b1 x3 A2 x2 A3 x3 r2 r3 f4P2 x2 P3 x3 1 x3 P2 x2 T2 x2 P3 x3 T3 x3 x2 P2 x2 T2 x2 P3 x3 T3 x3 gE0a t0 hn A0 ; B0 ; b3 ; b1 g;. (A8). p Z1 2Nc 2 Z 1 mB dx1 dx2 dx3 b1 db1 b3 db3 B x1 ; b1 fr2 1 x2 P2 x2 T2 x2 A3 x3 Ma6 a 32CF Nc 0 0 0. r3 x3 A2 x2 P3 x3 T3 x3 E0a thn A; B; b3 ; b1 r2 1 x2 P2 x2 T2 x2 A3 x3 r3 2 x3 A2 x2 P3 x3 T3 x3 E0a t0 hn A0 ; B0 ; b3 ; b1 g; 0 The evolution factors E0 e and Ea are given by. where we have adopted the notations x2 1 x2 and x3 1 x3 , and ignored the mass difference between mB and mb . FeKi and MeKi are obtained by choosing M2 (M3 ) to be the kaon (pion) in Fei and Mei , respectively. The invariant masses A, B, A0 , and B0 of the virtual quarks and gluons involved in the above hard kernels are functions of x1 , x2 , and x3 , as in Table VIII. The hard scales are chosen as q q t max jA2 j; jB2 j; 1=bi ; q q t0 max jA02 j; jB02 j; 1=bi ;. (A10). with the index i 1; 2 for Fe4;e6 , i 2; 3 for Fa4;a6 , and i 1; 3 for the nonfactorizable amplitudes. TABLE VIII.. Fe4;e6 Me4;e5;e6 Fa4;a6 Ma4;a5;a6. (A9). Ee t s tat expSB t S2 t; Ea t s tat expS2 t S3 t; E0e t s ta0 t expSB t S2 t S3 tjb2 b1 ; E0a t s ta0 t expSB t S2 t S3 tjb2 b3 ; (A11) where a0 represent the combination of the Wilson coefficients appearing in Tables I and II. The Sudakov exponents associated with the various mesons are written as 5 Z t d SB t exp sx1 P ; 1 ; b1 3 1=b1 q s (A12). The invariant masses A, B, A0 , and B0 in the hard kernels.. A p x2 mB p i x2 x3 x1 mB p i x3 mB p x1 x3 x2 mB. B p x1 x2 mB p x1 x2 mB p i x2 x3 mB p i x2 x3 mB. 114005-18. A0 p x1 mB p x2 x1 x3 mB p i x2 mB p 1 x2 x3 x1 mB. B0 p x1 x2 mB p x1 x2 mB p i x2 x3 mB p i x2 x3 mB.

(46) RESOLUTION TO THE B ! K PUZZLE. PHYSICAL REVIEW D 72, 114005 (2005). S2 t exp sx2 P 2 ; b2 s1 x2 P2 ; b2 Z t d q s ; (A13) 2 1=b2 . with the quark anomalous dimension q s =. The formula for the exponential S3 is the same as S2 but with the kinematic variables of meson 2 being replaced by those of meson 3. The explicit expression of the exponent s can be found in [20,71,72]. The variable b1 , conjugate to the parton transverse momentum k1T , represents the transverse extent of the B meson. The transverse extents b2 and b3 have a similar meaning for mesons 2 and 3, respectively. For the running coupling constant s , we employ the one-loop expression, and the QCD scale 4 QCD 0:250 GeV. The Sudakov exponential decreases fast in the large b region, such that the long-distance contribution to the decay amplitude is suppressed.. The hard functions are written as he A; B; b1 ; b2 ; xi b1 b2 K0 Ab1 I0 Ab2 b2 b1 K0 Ab2 I0 Ab1 K0 Bb1 St xi ;. (A14). hn A; B; b1 ; b3 K0 Ab3 b1 b3 K0 Bb1 I0 Bb3 b3 b1 K0 Bb3 I0 Bb1 ;. (A15). where St resums the threshold logarithm ln2 x appearing in the hard kernels to all orders. It has been parametrized as [73] St x . 212c 3=2 c p x1 xc ; 1 c. (A16). with c 0:3. The factorization formulas for Mq K , q u; c; t, involve the convolutions of all three meson distribution amplitudes:. 2 Z1 Z1 2 CF Mq dx1 dx2 dx3 b1 db1 b2 db2 B x1 ; b1 f1 x2 A x2 AK x3 K 16mB p 2Nc 0 0. r 1 2x2 P x2 T x2 AK x3 2rK A x2 PK x3 2r rK 2 x2 P x2 x2 T x2 PK x3 Eq tq ; l2 he A; B; b1 ; b2 ; x2 2r P x2 AK x3 4r rK P x2 PK x3 Eq t0q ; l02 he A0 ; B0 ; b2 ; b1 ; x1 g; (A17) with the evolution factor, Eq t; l2 s t2 Cq t; l2 expSB t S t:. (A18). The hard scales are chosen as q q p q q q tq max jA2 j; jB2 j; x2 x3 mB ; 1=bi ; t0q max jA02 j; jB02 j; jx3 x1 jmB ; 1=bi ; (A19) p p with the index i 1; 2. The additional scales x2 x3 mB and jx3 x1 jmB , compared to those appearing in Eq. (A10), come from the gluon invariant masses l2 1 x2 x3 m2B and l02 x3 x1 m2B in Figs. 2(a) and 2(b), respectively. The formulas for Mu;c;t are derived from Eq. (A17) by substituting the mass ratio r for rK , and the distribution amplitudes A and P for AK and PK , respectively. The magnetic-penguin amplitude is written, for the B ! K modes, as [74] Z1 C2F Z 1 g MK 16m4B p dx1 dx2 dx3 b1 db1 b2 db2 b3 db3 B x1 ; b1 f1 x2 f2A x2 r 3P x2 2Nc 0 0 T x2 r x2 P x2 T x2 gK x3 rK 1 x2 x3 A x2 3PK x3 TK x3 rK r 1 x2 P x2 T x2 3PK x3 TK x3 rK r x3 1 2x2 P x2 T x2 3PK x3 TK x3 Eg tq hg A; B; C; b1 ; b2 ; b3 ; x2 4r P x2 K x3 2rK r x3 P x2 3PK x3 TK x3 Eg t0q hg A0 ; B0 ; C0 ; b2 ; b1 ; b3 ; x1 g; with the evolution factor Eg t, Eg t s t2 Ceff 8g t expSB t SK t S t: (A21). (A20). Since the terms proportional to rK r develop the end-point singularities as the invariant mass of the gluon from O8g vanishes (x3 ! 0), we have kept the transverse momentum k3T . This is the reason the Sudakov factor associated with. 114005-19.