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The B πK puzzle and the Bulk Randall-Sundurm model → Sanghyeon Chang∗ School of Physics, Konkuk University, Seoul 143-701, Korea C. S. Kim† Department of Physics and IPAP, Yonsei University, Seoul 120-749, Korea 1 1 Jeonghyeon Song‡ 0 2 Division of Quantum Phases & Devices, School of Physics, n a Konkuk university, Seoul 143-701, Korea J 4 (Dated: January 5, 2011) ] h Abstract p - Therecent measurements of the direct CP asymmetries (A ) in the penguin-dominated B Kπ p CP → e h decays show some discrepancy from the standard model (SM) prediction. While A of B+ CP [ → π0K+ and that of B0 π−K+ in the naive estimate of the SM are expected to have very similar 3 → v values, their experimental data are of the opposite sign and different magnitudes. We study the 3 0 2 effects of thecustodial bulkRandall-Sundrummodelon this ACP. Inthis model, themisalignment 1 . of the five-dimensional (5D) Yukawa interactions to the 5D bulk gauge interactions in flavor space 6 0 leads to tree-level flavor-changing neutral current by the Kaluza-Klein gauge bosons. In a large 0 1 : portionoftheparameterspaceofthismodel,theobservednonzeroACP(B+ π0K+) ACP(B0 v → − → Xi π−K+) can be explained only with low Kaluza-Klein mass scale MKK around 1 TeV. Rather r a extreme parameters is required to explain it with M 3 TeV. The new contributions to KK ≃ well-measured branching ratios of B Kπ decays are also shown to be suppressed. → PACS numbers: 13.25.Hw, 12.38.Bx,13.66.Hk ∗ [email protected] † [email protected],Corresponding Author ‡ [email protected] 1 I. INTRODUCTION The study of B meson decays at Belle and BaBar [1] have been a crucial probe of the standard model (SM), especially its CP violation part. Recently the B Kπ decay system → has drawn a lot of interest due to the discrepancy between the SM predictions and the measurements [2–6]. There are nine measurements of the four decays of B+ π+K0, → B+ π0K+, B0 π−K+, and B0 π0K0: four branching ratios (BR), four direct CP → → → asymmetries A , and one mixing-induced CP asymmetry S . The 2008 data of these nine CP CP measurements are in Table I. Mode BR [10−6] A S CP CP B+ π+K0 23.1 1.0 0.009 0.025 → ± ± B+ π0K+ 12.9 0.6 0.050 0.025 → ± ± B0 π−K+ 19.4 0.6 0.098+0.012 → ± − −0.011 B0 π0K0 9.8 0.6 0.01 0.10 0.57 0.17 → ± − ± ± TABLE I.Experimentaldata forB πK; BR’s, direct CPasymmetries A , andmixing-induced CP → CP asymmetry S [7, 8]. CP We focus on the direct CP asymmetries A of B+ π0K+ and B0 π−K+. In the CP → → SM, the dominant contribution to each decay amplitude comes from the strong penguin contribution P. The color-suppressed tree contribution C may be smaller than the color- favored tree contribution T by a factor of the small parameter λ = V 0.22. Therefore, us | | ≃ both B+ π0K+ and B0 π−K+ could have A given by the interference between T CP → → and P in the leading order, as shown in Eq. (1). The direct CP asymmetries of two decay modes are expected to be the same size with the same sign within the naive estimate of the SM. As can be seen in Table I, however, the observation is quite different from this naive SM prediction: A (B+ π0K+) is non-zero positive1 while A (B0 π−K+) is negative. CP CP → → Known as “B πK puzzle”, this discrepancy has brought extensive attentions, leading to → model-independent studies as well as new physics (NP) effect studies in the literature. In this paper, we study this A puzzle in the framework of the custodial bulk Randall- CP Sundrum (RS) model[9]. In the simplest implementation of the RS model, featuring an 1 Albeit statistically less significant with larger error compared with BELLE and BaBar data, the CLEO collaborationobservednegative mean value for this asymmetry [7], ACP(B+ π0K+)= 0.29 0.23 → − ± ± 0.02. 2 SU(2) U(1) bulk gauge symmetry and a minimal brane-localized Higgs sector, this B L Y × → Kπ puzzle was studied, showing the difficulty to solve the puzzle under the experimental constraints [10]. As a five-dimensional (5D) warped model with all the SM fields in the bulk (except for the Higgs boson field), the bulk RS model provides very attractive explanations for both gauge hierarchy and fermion mass hierarchy[11–21]. To ensure the SU(2) custodial symmetry, we adopt the model with the bulk gauge symmetry of SU(3) SU(2) SU(2) C L R × × × U(1) , induced from AdS /CFT feature[9]. X 5 Since the 5D Yukawa interaction is not generally flavor-diagonal in this model, the flavor- changing-neutral-current (FCNC) is generated at tree level, mediated by Kaluza-Klein (KK) gauge bosons[22, 23]. The 5D Yukawa couplings λf and the bulk Dirac mass parameters 5ij c ’s determine all the FCNC processes in principle. Without a priori information about the f model parameters, however, this model lacks the prediction power. In our previous works [23, 24], we show that if we adopt two simple and natural assumptions, we can fix the model parameters and extract the necessary informationfor the calculations. The first assumption is that the5D Yukawa couplings areuniversal, λ λ (1). 5ij 5 ≃ ∼ O The second assumption is that fine tuning is not allowed when explaining the observed SM mixing matrices (CKM and PMNS). Here our restriction is at the level of no order-changing by cancellation. Mild fine tuning is permitted in this setup. With the given λ ’s and c ’s 5ij f based on the two assumptions, we study the bulk RS model effects on the B Kπ decay. → We will show that this model can explain the discrepancy between the observed A in the CP B πK decay and the SM prediction with the KK mass scale around 1 TeV. These NP → effects give suppressed contributions to the well observed BR’s. The organizationofthe paper is as follows. InSec. II, we briefly review thecurrent status of the measurements of four B Kπ decays. In Sec. III, we summarize the custodial → bulk RS model and formulate the bulk fermion sector. After presenting two naturalness assumptions, we determine all the bulk Dirac mass parameters. Section IV deals with the effects of this model on A ’s of B Kπ decays. We conclude in Sec. V. CP → II. SHORT REVIEW OF B πK DECAYS → ¯ Inthe SM, the B πK decays aredominated by theb s¯QCD penguin diagrams. The → → electroweakpenguinandthetreecontributionsarenextdominant. Thecurrentexperimental 3 datainTable Ishow thatthe branching ratiosarevery precisely measured. Theobservations are more precise than the SM theoretical estimates such as QCD factorization and the perturbative QCD[25]. The B πK decay amplitudes can be written in terms of topological amplitudes up to → λ2 scale: 1 A(B+ π+K0) = P′ P′C , (1) → − 3 EW 2 √2A(B+ π0K+) = P′ T′ P′ P′C C′, → − − − EW − 3 EW − 2 A(B0 π−K+) = P′ T′ P′C , → − − − 3 EW 1 √2A(B0 π0K0) = P′ P′ P′C C′. → − EW − 3 EW − ¯ The primes denote the b s¯transition. The color-favored (color-suppressed) tree diagrams → are represented by T′ (C′), and the P′(C) is the electroweak (color-suppressed electroweak) EW penguin. The penguin diagram P′ is the sum of three up-type (u,c,t) quark contributions: P′ = λ P˜ +λ P˜ +λ P˜ u u c c t t = λ (P˜ P˜ )+λ (P˜ P˜ ) t t c u u c − − P′ +P′ , (2) ≡ tc uc where λ V∗V (i = u,c,t), and the unitarity of the CKM matrix is used for the second i ≡ ib is equality. Here, the phase of λ V∗V is π within the SM. We also expect the following t ≡ tb ts ∼ hierarchies from theoretical calculations in the SM [26–28]: O(1) P′ , | tc| O(λ) T′ , P′ , (3) | | | EW| O(λ2) C′ , P′ , P′C . | | | uc| | EW| If we define ∆A A (B+ π0K+) A (B0 π−K+), (4) CP CP CP ≡ → − → theSMpredictsverysmall ∆A , whichiscontradictorytotheexperimental datainTableI. CP This discrepancy possibly suggests the existence of the NP contribution, especially in the CP violating phases. If the NP contribution is the source of the discrepancy, it should be of the order of λ or more. 4 The effective Hamiltonian for B πK can be written in operator expansion [29]: → 10 G = F λ (C Qp +C Qp) λ C Q . (5) Heff √2 p 1 1 2 2 − t i i! p=u,c i=3 X X The operators are defined by Qp = (¯b p ) (p¯ s ) , Qp = (¯b p ) (p¯ s ) , 1 i i V−A j j V−A 2 i j V−A j i V−A ¯ ¯ Q = (b s ) (q¯q ) , Q = (b s ) (q¯q ) , 3 i i V−A q j j V−A 4 i j V−A q j i V−A Q5 = (¯bisi)V−APq(q¯jqj)V+A, Q6 = (¯bisj)V−APq(q¯jqi)V+A, (6) Q7 = (¯bisi)V−APq 32eq(q¯jqj)V+A, Q8 = (¯bisj)V−APq 32eq(q¯jqi)V+A, Q9 = (¯bisi)V−APq 32eq(q¯jqj)V−A, Q10 = (¯bisj)V−AP q 32eq(q¯jqi)V−A, P P where i,j are color indices, e is the electric charge of the quark, (q¯ q ) = q¯ γ (1 γ )q q 1 2 V±A 1 µ 5 2 ± and q = u,d. The topological amplitudes are written in terms of the Wilson coefficients in the standard operator basis as [29, 30] 1 1 A(B+ π+K0) = λ a a +rK a a A , → − t 4 − 2 10 χ 6 − 2 8 πK (cid:20)(cid:18) (cid:19) (cid:18) (cid:19)(cid:21) 3 √2A(B+ π0K+) = A(B0 π−K+) λ a + λ (a a ) A u 2 t 7 9 Kπ → → − 2 − (cid:20) (cid:21) A(B0 π−K+) = λ a λ (a +a ) λ rK (a +a ) A , → − u 1 − t 4 10 − t χ 6 8 πK √2A(B0 π0K0) = A((cid:2)B+ π+K0)+√2A(B+ π0K+) (cid:3) A(B0 π−K+), (7) → → → − → where a = C + C /3 with +( ) sign for odd (even) i. We can specify each penguin i i i±1 − contributions as [30], P′ = λ a +rKa A , (8) tc − t 4 χ 6 πK 3 h i P′ = λ a a A , EW 2 t 7 − 9 Kπ 3h i P′C = λ a +rKa A , EW −2 t 10 χ 8 πK T′ = λ a Ah , i u 1 πK C′ = λ a A . u 2 Kπ whererK = 2m2 /m (m +m ),m = (m +m )/2,A = G (m2 m2 )Fπ(K)f /√2, χ K b s q q u d πK(Kπ) F B− π(K) 0 K(π) π(K) and F 0.3 are semileptonic form factors for B decays [31]. 0 ≃ 5 III. REVIEW OF THE BULK RANDAL-SUNDRUM MODEL The RS model is based on a 5D warped spacetime with the metric [11] 1 ds2 = (dt2 dx2 dz2), (9) (kz)2 − − where the fifth dimension z is compactified between 1/k < z < 1/T. Here k M and Pl ≃ T is the natural cut-off of the gauge theory at TeV scale. Two boundaries z = 1/k and UV z = 1/T are called the Planck brane and the TeV brane, respectively. IR For SU(2) custodial symmetry, we adopt the model suggested by Agashe et.al. in Ref. [9], based on the gauge structure of SU(3) SU(2) SU(2) U(1) . The custodial C L R X × × × symmetry is guaranteed by the bulk SU(2) gauge symmetry. The bulk gauge SU(2) R R symmetry is broken into U(1) by the orbifold boundary conditions on the Planck brane: R charged SU(2) gauge fields have ( +) parity. The U(1) U(1) is spontaneously broken R R X − × into U(1) on the Planck brane and the Higgs field localized on the TeV brane is responsible Y to the breaking of SU(2) U(1) to U(1) . L Y EM × The action for a 5D gauge fields is 1 S = d4xdz√G gMPgNQFa Fa , (10) gauge −4 MN PQ Z (cid:20) (cid:21) where G is the determinant of the AdS metric gMN, and Fa = ∂ Aa ∂ Aa + MN M N − N M g fabcAb Ac . The 5D action for the gauge interactions of a bulk fermion Ψˆ(x,z) 5 M N ≡ Ψ(x,z)/(kz)2 is S = d4xdz√G g5i Ψ¯ˆ(x,z)iΓMAa (x,z)TaΨˆ(x,z), (11) int √2k M Z where g is the 5D dimensionless gauge coupling (g ,g ,g ,g ) for each gauge group 5i 5s 5L 5R 5X (SU(3) ,SU(2) ,SU(2) ,U(1) ) and ΓM = (γµ,iγ ). c L R X 5 A bulk gauge field A (x,z) and a bulk fermion field Ψˆ(x,z) are expanded in terms of KK ν modes by A (x,z) = √k A(n)(x)f(n)(z), (12) ν ν A n X Ψˆ(x,z) = √k [ψ (x)f (z)+ψ (x)f (z)], L L R R n X (n) (0) (0) where the mode functions of f (z) and f (z,c) = f (z, c) are referred to Ref. [18]. A L R − Here c is defined through the 5D Dirac mass m = sign(y)ck and z = ek|y|. Note that a D 6 massless SM fermion corresponds to the zero mode with (++) parity. Since Ψ has always L opposite parity of Ψ , the left-handed SM fermion is the zero mode of a 5D fermion whose R left-handed part has (++) parity (the corresponding right-handed part has automatically ( ) parity which cannot have a zero mode). −− Due to the gauge structure of SU(3) SU(2) SU(2) U(1) , the right-handed SM C L R X × × × fermions belong to a SU(2) doublet, and couple to SU(2) gauge bosons with ( +) parity. R R − As a result, the whole quark sector is (++) (++) (−+) u u U iL iR iR Q = , U = , D = , (13) i i i  (++)   (−+)   (++)  d D d iL iR iR       where i is the generation index. Dirac mass parameters (c , c , c ) determine their mode Qi Ui Di functions, KK mass spectra, and coupling strength with KK gauge bosons. On the TeV brane, the SM fermion mass is generated as the localized Higgs field develops its VEV of H = v 174 GeV. The SM mass matrix for a fermion f(= u,d,ν,e) is h i ≃ k M = vλf f(0)(z,c )f(0)(z,c ) vλf F (c )F (c ), (14) f ij 5ij T R Ri L Lj ≡ 5ij R fRi L fLj (cid:12)z=1/T (cid:0) (cid:1) (cid:12) (cid:12) where i,j are the generation indices, λf are the 5(cid:12)D (dimensionless) Yukawa couplings, and 5ij F (c) = F ( c) is defined by L R − (0) f (1/T,c) F (c) L , (15) L ≡ ǫ1/2 where ǫ = T/k. The mass eigenstates of the SM fermions involve two independent mixing matrices, defined by χ = U† ψ(0), χ = U† ψ(0). (16) fL fL fL fR fR fR Notethattheobserved mixing matrixisamultiplication oftwo independent mixing matrices such that V = U† U and U = U† U . CKM uL dL PMNS eL νL If bulk Dirac mass parameters and 5D Yukawa couplings are given a priori, we could predict all the mass spectra and mixing matrices as well as the couplings with KK gauge bosons. Without those crucial knowledge, we have to take the opposite way, i.e., deducing them from the observation. The problem is that the number of observations is not enough to fix all the model parameters. In the previous work, we have developed a theory based on the following two natural assumptions: 1. For all fermions, 5D Yukawa couplings have the same order of magnitude λf (1). 5 ∼ O 7 2. When explaining the observed mixing matrix V = U† U and U = U† U , CKM uL dL PMNS eL νL no order-changing by cancellation is allowed. The assumption-1 yields anarchy type fermion mass matrix, which naturally explains the large top quark mass v 174 GeV. Other small SM fermion masses are generated by ≃ controlling c’s. The assumption-2 is consistent with the spirit of no fine-tuning. In Ref. [18], we have shown that the above two natural assumptions determine the nine bulk mass parameters within a well-defined regions as c 0.61, c 0.56, c 0.3+0.02, (17) Q1 ≃ Q2 ≃ Q3 ≃ −0.04 c 0.71, c 0.53, 0 < c < 0.2, U1 ≃ − U2 ≃ − U3 ∼ ∼ c 0.66, c 0.61, c 0.56.. D1 ≃ − D2 ≃ − D3 ≃ − Recently phenomenological constraint on the value of c has been studied, focused on Q3 ¯ the anomalous coupling of Zbb vertex [32]: c cannot be smaller than 0.3. Combined Q3 with our constraint based on the two naturalness assumptions, we consider the case of c = 0.3 0.32 in what follows. Q3 ∼ The quark mixing matrices are F (c ) F (c ) (U ) L Qi , (U ) R Ai , A = U,D. (18) qL ij(i≤j) ≈ F (c ) qR ij(i≤j) ≈ F (c ) L Qj R Aj Then our mixing matrices show the following order of magnitude behaviors: 1 λ λ3 U U  λ 1 λ2 + λ4 , (19) uL dL ≃ ≃ O λ3 λ2 1  (cid:0) (cid:1)       1 0 0 1 λ λ2 U 0 1 λ2 + λ4 , U  λ 1 λ + λ3 . (20) uR dR ≃ O ≃ O 0 λ2 1  (cid:0) (cid:1) λ2 λ 1  (cid:0) (cid:1)             As shall be shown below, only U and U make dominant contributions to B Kπ uL dL → decays. Because of high similarity of U and U to the CKM matrix, we take the following uL dL assumptions of (U ) = κ (V ) . (21) qL ij ij CKM ij In order to satisfy our naturalness assumptions, we require 1 < κ < √2. (22) ij √2 | | 8 IV. BULK RS MODEL EFFECTS ON B Kπ DECAYS → In the bulk RS model, the mass eigenstate of a SM fermion is a mixture of gauge eigenstates as in Eq. (16) and we have FCNC mediated by KK gauge bosons. In terms of gauge a(n) eigenstates, the four-dimensional (4D) gauge interactions with KK gauge modes A are µ ∞ g gˆ(n)(c )ψ¯(0)Taγµψ(0) +gˆ(n)(c )ψ¯(0)Taγµψ(0) Aa(n), (23) L4D ⊃ 4j L i iL iL R i iR iR µ Xn=1(cid:16) (cid:17) where g = g /√kL for j = s,L,R,X. Since the bulk RS effects are suppressed by the 4j 5j forthpower of the KK mass, we consider only the contribution of the first KK mode of gauge bosons. In what follows, therefore, we omit the KK mode number notation (n). Then the effective gauge couplings with the first KK gauge boson are 2 gˆ (c ) = gˆ ( c ) = √kL dzk f(0)(z,c ) f(1)(z) gˆ(c ).. (24) L fi R − fi L fi A ≡ fi Z h i Note that the effective coupling gˆ(c) vanishes if c = 1/2. The values of the bulk mass parameters c’s in Eq. (17) fix the gˆ values as gˆ(c ) = 0.192, gˆ(c ) = 0.179, gˆ(c ) = 1.797 1.974, (25) Q1 − Q2 − Q3 ∼ gˆ(c ) = 0.193, gˆ(c ) = 0.133, gˆ(c ) = 2.759 3.948, U1 − U2 − U3 ∼ gˆ(c ) = 0.193, gˆ(c ) = 0.192, gˆ(c ) = 0.179, D1 − D2 − D3 − where, for example, gˆ = 1.974 for c = 0.3 and gˆ = 1.797 for c = 0.32. It can be seen Q3 Q3 that gˆ(c ) and gˆ(c ) are dominant over all the other gˆ’s of the order of λ. Q3 U3 The relevant FCNC processes mediated by the first KK gauge bosons are described by the following Lagrangian: = g KuLu¯ Taγµu +KdLd¯ Taγµd +KU u¯ Taγµu +KD d¯ Taγµd Ga(1) L4D − s lm lL mL lm lL mL lm lR mR lm lR mR µ 1 (cid:0)g KuLu¯ γµu KdLd¯ γµd W(1) (cid:1) −2 lm lL mL − lm lL mL 3Lµ (cid:20) +g˜(cid:0) KU u¯ γµu +KD d¯ γµd (cid:1) W(1) lm lL mL lm lR mR 3Rµ +g(cid:0) KuLu¯ γµu +KdLd¯ γµd (cid:1) +KU u¯ γµu +KD d¯ γµd B(1) , (26) X lm lL mL lm lL mL lm lL mL lm lR mR Xµ (cid:21) (cid:0) (cid:1) where the subscript l,m are the generation indices (l,m = 1,2,3), and the 4D gauge cou- 9 plings are g = g /√kL. The effective mixing matrices K’s are 4 5L 3 KqL = U† gˆ(c )(U ) , for q = u,d, lm qL lk Qk qL km k=1 X(cid:0) (cid:1) 3 KU = U† gˆ( c )(U ) , lm uR lk − Uk uR km k=1 X(cid:0) (cid:1) 3 KD = U† gˆ( c )(U ) . (27) lm dR lk − Dk dR km k=1 X(cid:0) (cid:1) We first estimate the value of the elements of K’s. Since gˆ(c ) gˆ(c ) gˆ(c ) as in Q1 ≈ Q2 ≪ Q3 Eq. (25), the dominant elements of K are qL KqL U U gˆ(c ) λ2gˆ(c ), 32 ≃ dL 33 dL 32 Q3 ∼ Q3 KqL (cid:0)λ3gˆ(c(cid:1) (cid:0)), K(cid:1)qL gˆ(c ), KqL λgˆ(c ). (28) 31 ∼ Q3 11 ∼ Q1 12 ∼ Q1 In addition, the condition of gˆ(c ) gˆ(c ) gˆ(c ), up to (λ2), leads to diagonal KD D1 ≈ D2 ≈ D3 O up to (λ4): O KD = 0. (29) ij(i6=j) Note that these vanishing off-diagonal elements of the right-handed quarks ensure the valid- ity of the operator expansions in the effective Hamiltonian Eq. (5). The diagonal elements are KD gˆ(c ), KU gˆ(c ). (30) 11 ∼ D1 11 ∼ U1 Finally we have the effective Hamiltonian for B Kπ decay, given by → g2KdL s 32 (¯b s ) KuL(u¯ u ) +KdL(d¯d ) +KU(u¯ u ) +KD(d¯d ) HRS ≃ 8M2 i j V−A 11 j i V−A 11 j i V−A 11 j i V+A 11 j i V+A KK(cid:20) 1 (cid:8) (cid:9) (¯bs) KuL(u¯u) +KdL(d¯d) +KU(u¯u) +KD(d¯d) − 3 V−A 11 V−A 11 V−A 11 V+A 11 V+A (cid:21) g2KdL (cid:8) (cid:9) 32 (¯bs) KuL(u¯u) KdL(d¯d) (31) −16M2 V−A 11 V−A − 11 V−A KK g2 KdL (cid:8) (cid:9) X 32 (¯bs) KuL(u¯u) +KdL(d¯d) +KU(u¯u) +KD(d¯d) , −16M2 V−A 11 V−A 11 V−A 11 V+A 11 V+A KK (cid:8) (cid:9) where i,j are the color indices and M is the first KK gauge boson mass. Here we have KK included only the most dominant terms proportional to KdL since the values of gˆ(c ) and 32 Q3 gˆ(c ) are much larger than those of the other gˆ’s. U3 10