An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach

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1 An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach Satoshi Mishima (Nagoya Univ.) S. M. and A. I. Sanda, accepted in PRD. [hep-ph/31168] S. M. and A. I. Sanda, Prog. Theor. Phys. 11 (23) 549 [hep-ph/3573] Super B Factory Workshop in Hawaii, Jan. 21, 24 Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 1

2 Outline 1. Introduction 2. Perturbative QCD Approach in Exclusive B Decays Review of PQCD 3. MSSM Contribution in B Decays Mass Insertion Approximation Constraint from Br(B X s γ) 4. MSSM Effects on B φk Decays Branching Ratio and CP Asymmetry for B d φk S and B ± φk ± 5. Summary Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 2

3 1. Introduction The time-dependent CP asymmetry a fcp (t) Γ(B (t) f CP ) Γ(B (t) f CP ) Γ(B (t) f CP ) + Γ(B (t) f CP ) A fcp cos( M B t) + S fcp sin( M B t) A fcp λ f CP 2 1 λ fcp 2 + 1, S f CP 2Imλ f CP λ fcp 2 + 1, λ f CP = e 2i φ 1 A(B f CP ) A(B f CP ) B J/ψK S : Tree dominant, A J/ψKS = V cb V csa 1 = S J/ψKS = sin(2φ 1 ), A J/ψKS = B φk S : Pure penguin, A φks = V tb V tsa 2 = S φks = sin(2φ 1 ), A φks = S J/ψKS S φks O(λ 2 ) = New Physics in φk S λ = sinθ c.22 Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 3

4 Experimental Results Charmonium Modes φk S η, K S KKK S OPAL ±.5 ALEPH ±.16 CDF BABAR ±.67 ±.34 Belle ±.57 ±.28 Average (charmonium).736 ±.49 BABAR 3.45 ±.43 ±.7 Belle 3.96 ± BABAR 3.2 ±.34 ±.3 Belle 3.43 ±.27 ±.5 Belle 3.51 ± Average (s penguin).24 ±.15 Average (All).695 ± sin(2β (eff) ) H F A G Summer 23 β = φ 1 In the SM, S J/ψKS = S φks +O(λ 2 ). Belle : S φks indicates a 3.5σ deviation from the SM prediction. There is a 2.1σ discrepancy between BABAR and Belle. = We consider New Physics contributions in B φk. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 4

5 Effective Hamiltonian for B Decays Buchalla, Buras, Lautenbacher, Rev.Mod.Phys.68,1125,1996 H eff = G F 2 i V i CKMC i (µ)o i (µ) µ : Factorization scale New Physics = C 3 6, C 7γ, C 8g Standard Model Tree Penguin Magnetic Penguin Effective Theory O (q) 1 = ( s i q j ) V A ( q j b i ) V A, O (q) 2 = ( s i q i ) V A ( q j b j ) V A, X X O 3,5 = ( s i b i ) V A ( q j q j ) V A, O 4,6 = ( s i b j ) V A ( q j q i ) V A, O 7γ = e 8π 2 m b( sσ µν (1 + γ 5 )b)f µν, q q O 8g = g s 8π 2 m b( s i σ µν (1 + γ 5 )T a ij b j)g a µν Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 5

6 Strong Phases and CP Asymmetries a φks (t) = A φks cos( Mt) + S φks sin( Mt) If there is any new physics in B φk S, A(φK S ) = A SM e iδ SM + A NP e iθ NP e iδ NP A(φK S ) = A SM e iδ SM + A NP e iθ NP e iδ NP δ SM(NP) : Strong phase, (Final-state interaction phase) θ NP : CP violating phase S φks = A φks = sin2φ ANP 2 cos A δ SM sin(θnp + 2φ 1 ) + ANP A SM sin(2θnp + 2φ 1 ) ANP 2 A cos δ cos SM θnp + ANP A SM 2 sin ANP A δ sin SM θnp δ δ ANP 2 SM δ NP A cos SM θnp cos δ + ANP A SM = δ and A NP /A SM are needed. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 6

7 How to Calculate Matrix Elements φk O i (µ) B =? Generalized factorization, QCD factorization, Perturbative QCD, Light-cone QCD sum rules, Lattice QCD, Soft-collinear effective theory, etc. Generalized Factorization Approach QCD Factorization (QCDF or BBNS) (See e.g. Beneke, Buchalla, Neubert, Sachrajda, NPB66,245,21) = These approaches were used for estimating new physics contributions in B φk S decay. (Khalil, Kou, PRD67,559,23, Kane et al. PRL94,14183,23, ) In this study, we use another approach : Perturbative QCD (PQCD) (See e.g. Keum, Li, Sanda, PRD63,548,21) A dominant source of strong phase in PQCD is different from in QCDF. Although it is difficult to discover NP by PQCD, it can be applied as a guide for NP search. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 7

8 2. PQCD Approach in B Meson Decays Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 8

9 An Intuitive Picture of the PQCD Approach The spectator quark (d quark) exchanges a hard gluon and it lines itself with the fast ū quark so that it forms π meson. We consider the form factor is dominated by hard gluon exchanges. Are contributions from many soft gluon exchanges actually small? = We calculate other contributions, which are calculable perturbatively, and compare the predictions with data. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 9

10 Review of PQCD Factorization Theorem : (Brodsky, Lepage, PRD22,2157,198) M 2 (P 2 )M 3 (P 3 ) H eff B(P 1 ) = [dx i ]Φ M2 (x 2, P 2 )Φ M3 (x 3, P 3 )C(t)H(x 1, x 2, x 3, t)φ B (x 1, P 1 ) End-Point Singularity in H Light-Cone Coordinate P = (P +, P,P T ) P ± = 1 2 (P ± P 3 ) t O(M B ), x i are the momentum fractions of partons. = 1 (k 1 k 2 ) 1 2 (P 1 k 2 ) 2 1 x 1 x 2 MB 2 k 1T k 2T 2 1 x 1 x 2 2 M2 B P 1 = M B 2 (1, 1, T ) 1 x 2 M 2 B k 2T 2 Φ K (x 2 ) x 2 (1 x 2 ) P 2 = M B 2 (, 1, T ) k 1 = (x 1 P + 1,,k 1T) k 2 = (, x 2 P 2,k 2T) P T = (P 1, P 2 ) x 1,2 1 Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 1

11 Sudakov Factor in QCD Botts, Sterman, NPB325,62,1989 When we retain k T, the large double logarithms are generated from the overlap of collinear and soft divergence in radiative corrections to meson wave functions. = Sudakov factor = Large b (small k T ) is suppressed. Z Ψ(P,k T ) = d 2 b e i k T b Ψ(P,b) 1/Λ QCD Sudakov factor ensures a perturbative calculation of hard part. 1 q 2 = 1 x 1 x 2 MB 2 k 1T k 2T 2 No End-Point Singularity!! α s /π = We have checked self-consistency in our calculations. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p

12 Physical Meaning of Sudakov Factor in b space If a single quark interacts with a hard gluon, it must emit many collinear gluons. = Not exclusive processes!! If a q q pair with small separation b interacts with a hard gluon, it emits no gluons since it is a color singlet. The Sudakov factor gives a probability for no emitted gluons. 1/Λ QCD Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 12

13 Review of PQCD k T Factorization Theorem : Amp. = (Li, Sterman, NPB381,129,1992) [dx i ][db it ]Φ K (x 2, b 2T )Φ φ (x 3, b 3T )C (t) H(x i, b it, t)φ B (x 1, b 1T )e S Non-perturbative part = Meson wave functions : K, φ = Light-cone QCD sum rules B = Model function Φ B ( P + M B )γ 5 x 2 (1 x) 2 exp Theoretical Errors (Ball,JHEP1,1,1999; Ball,Braun,Koike,Tanaka,NPB529,323,1998) " 1 2 «# 2 xmb ω2 B b2 Large theoretical uncertainty comes from Φ B. = These errors are reduced in CP asymmetries. We expect that higher-order corrections are about 3%. ω B 2, ω B = GeV = We suppose these corrections are reduced in CP asymmetries. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 13

14 Review of PQCD The leading order diagrams in PQCD: Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 14

15 Review of PQCD The leading order diagrams in PQCD: Non-factorizable diagrams are calculable. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 14

16 Review of PQCD The leading order diagrams in PQCD: Non-factorizable diagrams are calculable. Annihilation diagrams are also calculable. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 14

17 Review of PQCD The leading order diagrams in PQCD: Non-factorizable diagrams are calculable. Annihilation diagrams are also calculable. Factorizable diagrams are dominant. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 14

18 Review of PQCD The leading order diagrams in PQCD: Non-factorizable diagrams are calculable. Annihilation diagrams are also calculable. Factorizable diagrams are dominant. Factorizable annihilation diagrams generate a large strong phase. 1 (1 x 2 )x 3 M 2 B k 2T k 3T 2 = (Pri. val.) i π δ `(1 x 2 )x 3 M 2 B k 2T k 3T 2 Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 14

19 Applications of the PQCD approach PQCD has been applied to various charmless B decays: Kπ ππ KK Kη ( ) ρπ, ωπ Keum, Li, Sanda, Phys.Lett.B54,6,21:Phys.Rev.D63,548,21 Lu, Ukai, Yang, Phys.Rev.D63,749,21 Chen, Li, Phys.Rev.D63,143,21 Kou, Sanda, Phys.Lett.B525,24,22 Lu, Yang, Eur.Phys.J.C23,275,22 ρk, ωk Chen, Phys.Lett.B525,56,22 φk φπ K π Keum, hep-ph/21127 φk K γ Mishima, Phys.Lett.B521, 252,21: Chen, Keum, Li, Phys.Rev.D64,1122,21 Melic, Phys.Rev.D59,745,1999 Chen, Keum, Li, Phys.Rev.D66,5413,22 M. Matsumori will talk on this workshop. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 15

20 Branching ratios (1 6 ) and Direct CP Asymmetries Y. Y. Keum and A. I. Sanda, econf C3452, WG42 (23). BABAR Belle PQCD B π + π 4.7 ±.6 ± ±.6 ± B ± π ± π ±.6 5. ± 1.2 ± B π π 2.1 ±.6 ± ±.6 ± B K ± π 17.9 ±.9 ± ± 1. ± B ± K ± π ± ± B ± K π ± 22.3 ± 1.7 ± ± 1.9 ± B K π 11.4 ± 1.7 ± ± A CP (π ± π ).19 ±.19 ±.5.77 ±.27 ± A CP (π ± π ) ±.2.14 ± A CP (K ± π ).17 ±.41 ± ±.35 ± A CP (K ± π ).9 ±.9 ±.1.23 ± A CP (K π ± ).5 ±.8 ± Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 16

21 PQCD Prediction for A K π ± Direct CP asymmetry for B K π ± Exp. : BABAR Belle CLEO Average.17 ±.41 ± ±.35 ±.18.4 ±.16 ±.2.95 ±.28 PQCD :.4.2 A K +- π Belle BaBar φ 3 PQCD prediction is negative and in agreement with exp. data. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 17

22 B φk Decays in the PQCD Approach Mishima, Phys.Lett.B521,252,21 Chen, Keum, Li, PRD64,1122,21 (a) and (b) are dominant. Branching ratios are larger than those in factorization approaches. Mode Exp.(BABAR) Exp.(Belle) Result B φk ( ±.5) 1 6 ( ±.7) 1 6 ( ) 1 6 B ± φk ± ( ±.5) 1 6 (9.4 ± 1.1 ±.7) 1 6 ( ) 1 6 = PQCD predictions are in agreement with exp. data. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 18

23 3. MSSM Contribution in B Decays Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 19

24 SUSY Contributions in B Decays We consider supersymmetric contribution (MSSM) in B decays. There are new sources of CP Violation and Flavor Changing Neutral Current. Gluino contributions Chargino contributions Neutralino contributions Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 2

25 SUSY Contributions in B Decays We consider supersymmetric contribution (MSSM) in B decays. There are new sources of CP Violation and Flavor Changing Neutral Current. Gluino contributions Chargino contributions Neutralino contributions Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 2

26 SUSY Contributions in B Decays We consider supersymmetric contribution (MSSM) in B decays. There are new sources of CP Violation and Flavor Changing Neutral Current. Gluino contributions Chargino contributions Neutralino contributions We use Mass Insertion Approximation. (Hall, Kostelecky, Raby, NPB267,415,1986) Off-diagonal elements in the squark mass matrix produce FCNC. Squark mass matrix : M 2 d = ( ) m 2 d,ll m 2 d,lr m 2 d,rl m 2 d,rr = δd LL,ij (V d m2 d,ll V d ) ij m 2 q 1 (i j) LL LR, RL, RR Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 2

27 Wilson Coefficients in MSSM Gabbiani, Gabrielli, Masiero, Silvestrini, NPB477,321,1996 Gluino contributions in Penguin and Magnetic Penguin : C 3 (M S ) 2α 2 s 4G F V tb V C 4 (M S ) 2α 2 s 4G F V tb V C 5 (M S ) 2α 2 s 4G F V tb V C 6 (M S ) 2α 2 s 4G F V tb V 2αs π C 7γ (M S ) 6G F V tb Vts m2 q 2αs π C 8g (M S ) 2G F V tb Vts m2 q ts m2 q ts m2 q ts m2 q ts m2 q (δ d LL ) 23 (δ d LL ) 23 (δ d LL ) 23 (δ d LL ) 23» (δll d ) 8 23» (δ d LL ) 23» 1 9 B 1(x) 5 9 B 2(x) 1 18 P 1(x) 1 2 P 2(x)» 7 3 B 1(x) B 2(x) P 1(x) P 2(x)» 1 9 B 1(x) B 2(x) 1 18 P 1(x) 1 2 P 2(x)» 2 3 B 1(x) B 2(x) P 1(x) P 2(x) 3 M 1(x) 3 M 3(x)+(δLR d ) m g 8 23 m b «1 3 M 3(x)+3M 4 (x) +(δlr d ) 23 x m 2 g /m2 q New Operators: C i O i L R C i Õ i m g m b «1 3 M 1(x)+3M 2 (x) Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 21

28 b s FCNC Ciuchini, Franco, Masiero, Silvestrini, PRD67,7516,23 We consider single mass insertions: LR insertion : (δlr d ) 23 RL insertion : (δrl d ) 23 LL insertion : (δll d ) 23 RR insertion : (δrr d ) 23 and constrain them from b s FCNC processes: Br(B X s γ) A CP (B X s γ) B s B s Mixing: M s = The constraint from Br(B X s γ) is strong. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 22

29 Constraint from B X s γ Kagan, Neubert, Eur.Phys.J.C7,5,1999 Br(B X s γ) = (3.3 ±.4) 1 4 (PDG23) Br(B X s γ) C 7γ (m b ) 2 + C 7γ (m b ) 2 C 7γ (m b ) = C7γ SM (m b ) + C7γ NP (m b ) C 7γ (m b ) = C 7γ NP (m b ) C NP 7γ (M S ) 2α s π 6G F V tb Vts m2 q [ ] (δll d ) M 3(x)+(δLR d ) 23 m g 8 m b 3 M 1(x) m g = m q = 5 GeV We take (δ d ) 23 in 2σ region. LR RL LL RR Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 23

30 4. MSSM Effects on B φk Decays Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 24

31 Magnetic Penguin in PQCD Mishima, Sanda, Prog.Theor.Phys.11(23)549 [hep-ph/3573] Magnetic Penguin is important to a search for New Physics. O 8g = g s 8π 2 m b( s i σ µν (1 + γ 5 )T a ijb j )G a µν + Above diagrams are dominant. Strong phase comes from their absorptive part. A relative strong phase between Penguin and MP is large. = If there is a new CP-violating phase in MP, then A φk might change significantly from the SM prediction. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 25

32 Branching Ratios 7e-5 2e-5 2e-5 5e-5 Belle (δ RL ) 23 =.86 Belle (δ LL,RR ) 23 =1. 3e-5 1e-5 1e-5 1e-5 -π Belle BaBar -π/2 π/2 π θ r (rad) -π (δ RL ) 23 =.1 BaBar -π/2 π/2 π θ (rad) -π (δ LL,RR ) 23 =.5 BaBar -π/2 π/2 π θ (rad) 7e-5 2e-5 2e-5 5e-5 3e-5 1e-5 -π BaBar Belle -π/2 π/2 π θ r (rad) 1e-5 -π BaBar (δ RL ) 23 =.1 (δ RL ) 23 =.86 Belle -π/2 π/2 π θ (rad) 1e-5 -π BaBar (δ LL,RR ) 23 =.5 (δ LL,RR ) 23 =1. Belle -π/2 π/2 π θ (rad).2.1 LR : (δ d LR ) 23 =.15 + r e iθ r Im(δ d LR ) θr Im(δ d RL ) 23.5 RL, LL, RR : (δ d ) 23 = (δ d ) 23 e iθ θ Re(δ d LR ) 23 Re(δ d RL ) 23 Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 26

33 Direct CP Asymmetries A φk 1 1 (δ RL ) 23 =.86 1 A φks BaBar Belle A φks BaBar Belle (δ RL ) 23 =.1 A φks BaBar (δ LL,RR ) 23 =.5 (δ LL,RR ) 23 =1. Belle -π -π/2 π/2 π θ r (rad) -π -π/2 π/2 π θ (rad) -π -π/2 π/2 π θ (rad) 1 1 (δ RL ) 23 =.86 1 A φk BaBar Belle A φk BaBar Belle (δ RL ) 23 =.1 A φk BaBar (δ LL,RR ) 23 =.5 (δ LL,RR ) 23 =1. Belle -π -π/2 π/2 π θ r (rad) -π -π/2 π/2 π θ (rad) -π -π/2 π/2 π θ (rad).85 A φk.85 A φks A φk ± A φk are generated from the interference between penguin and magnetic-penguin. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 27

34 Indirect CP Asymmetries S φk LR RL LL or RR 1 1 (δ RL ) 23 =.1 1 S φk S BaBar Belle S φk S (δ RL ) 23 =.86 BaBar Belle S φk S BaBar (δ LL,RR ) 23 =.5 (δ LL,RR ) 23 =1. Belle π -π/2 π/2 π θ r (rad) -π -π/2 π/2 π θ (rad) -π -π/2 π/2 π θ (rad) S φks.28 It is difficult to explain the current Belle data. We attempted to maximize MSSM effects, S φks had been larger than about.4. S φk 1.5 BaBar Belle π -π/2 π/2 π θ r (rad) m g = 2GeV m q = 2 TeV Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 28

35 5. Summary We calculated MSSM contributions with a mass insertion in B φk decays using the PQCD approach..85 A φk.85 A φks A φk ± S φks.28 It is difficult to explain the current Belle data. Experimental Error (only statistical) 23 : 14 fb 1 δs φks.5 25 : 3 fb 1 δs φks : 1 ab 1 δs φks <.1 We need more theoretical study in B φk decays. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 29

36 Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 3

37 Backup Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 31

38 Meson Wave Functions B meson wave function: = δ Z ij 2Nc K meson wave function: b βj ()d αi (z) B(P 1 ) i, j : color indices α, β : spinor indices dx 1 d 2 k 1T e i(x 1P 1 z+ k 1T z T ) [( P 1 + M B )γ 5 φ B (x 1,k 1T )] αβ K(P 2 ) d βj ()s αi (z) m K = M2 K m d +m s = 1.7 GeV, = iδ Z 1 ij 2Nc φ meson wave function: n µ z µ /z, v µ P 2µ /P + 2 h i dx 2 e ix 2P 2 z γ 5 P 2 φ A K (x 2) + m K φ P K (x 2) + m K ( v n 1)φ T K (x 2) αβ = δ ij 2Nc Z 1 φ(p 3 ) s βj ()s αi (z) h i dx 3 e ix 3P 3 z M φ ɛ φ φ φ (x 3 )+ ɛ φ P 3 φ t φ (x 3) + M φ φ s φ (x 3) αβ Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 32

39 Meson Distribution Amplitudes " φ B (x, b) = N B x 2 (1 x) 2 exp 1 «# 2 xmb ω2 B b2, ω B = GeV 2 ω B 2» φ A K (x) = f K 2 2 6x(1 x) 1 + a 1 C 3 2 2N 1 (1 2x) + a 2 C 3 2 (1 2x), ρ K = (m d + m s )/M K c» φ P K (x) = f K η 3 5 «j 2N c 2 ρ2 K C (1 2x) 3 η 3 ω ff 2 ρ2 K (1 + 6a 2) φ T K (x) = f K 2 2N c (1 2x) φ φ (x) =» Cn ν (x) : Gegenbauer polynomial 5η η 3ω ρ2 K 3 5 ρ2 K a 2 a 1 =.18, a 2 =.16, η 3 =.15, ω 3 = 3. = f φ φ t φ (x) = f T φ 2 2N c φ s φ (x) = f T φ 4 2N c 2 6x(1 x) + 3 2N c 2 δ +» 3(1 2x) ζt 3 {3 3(1 2x)2 + 35(1 2x) 4 }» n o (1 2x) 6 + 9δ ζ3 T (1 1x + 1x 2 ) + 3δ + log ζ T 3 =.24, δ + =.46 = Light-cone QCD sum rules «(1 1x + 1x 2 ) C (1 2x) Light-cone QCD sum rules Ball, JHEP.1:1,1999 j 1 (1 2x) log 1 x x x 1 x ff Ball, Braun, Koike, Tanaka, Nucl.Phys.B529:323,1998 Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 33

40 q 2 -dependence of Magnetic Penguin Amplitudes The distribution of q 2 for ReM MP a : = q 2 = 6.3 GeV 2 M 2 B /4 The shape of this graph is not simple. Satoshi Mishima (Nagoya Univ.) An Analysis of Supersymmetric Effects on B φk Decays in the PQCD Approach p. 34