Π Decays Within Unparticle Physics ∗ Chuan-Hung Chen A, C.S

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Physics Letters B 671 (2009) 250–255

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Physics Letters B

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Investigation of Bu,d → (π, K )π decays within unparticle physics ∗ Chuan-Hung Chen a, C.S. Kim b, , Yeo Woong Yoon b

a Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan b Department of Physics, Yonsei University, Seoul 120-479, Republic of Korea

article info abstract

Article history: We investigate the implication of unparticle physics on the Bu,d → (π, K )π decays under the constraints Received 12 June 2008 ¯ of the Bd,s–Bd,s mixing. We found that not only the unparticle parameters that belong to the ﬂavor Received in revised form 25 November 2008 changing neutral current (FCNC) processes but also scaling dimension dU could be constrained by the Accepted 3 December 2008 B –B¯ mixing phenomenology. Employing the minimum χ 2 analysis to the B → (π, K )π decays Availableonline7December2008 d,s d,s u,d with the constraints of B mixing, we ﬁnd that the puzzle of large branching ratio for B → 0 0 Editor: M. Cveticˇ d,s d π π and the discrepancy between the standard model estimation and data for the direct CP asymmetry of + + 0 + − B → K π and Bd → π π can be resolved well. However, the mixing induced CP asymmetry of 0 Bd → K S π could not be well accommodated by the unparticle contributions. © 2008 Elsevier B.V. Open access under CC BY license.

1. Introduction SM, e.g. on supersymmetric model [7], extra-dimension model [8], left-right symmetric model [9] and flavor-changing Z model [10]. Although new physics effects will be introduced, however, in phe- Recently some incomprehensible phenomena at B factories nomenological sense we just bring more particles and their related have been explored, especially Bu,d → (π, K )π decays. Firstly, the 0 0 interactions to our system. Recently, Georgi proposed completely observations on the large branching ratio (BR) for Bd → π π de- 0 0 −6 different stuff and suggested that an invisible sector, dictated by cay with the world average B(Bd → π π ) = (1.31 ± 0.21) × 10 + − the scale invariance and coupled weakly to the particles of the SM, and the direct CP asymmetry for Bd → π π with ACP(Bd → + − may exist in our universe [11,12]. Unlike the concept of particles π π ) = 0.38 ± 0.07 [1] are inconsistent with the theoretical es- − in the SM or its normal extensions where the particles own the timations of around 0.5 × 10 6 and 10–20%, respectively. Secondly, + − definite mass, the scale invariant stuff cannot have a definite mass a disagreement in the CP asymmetries (CPAs) for Bd → K π + + unless it is zero. Therefore, if the peculiar stuff exists, it should be and B → K π 0 has been observed to be −0.097 ± 0.012 and made of unparticles [11]. Furthermore, in terms of the two-point 0.050 ± 0.025 [1], respectively, while the naive estimation is ΔCP ≡ + + 0 + − function with the scale invariance, it is found that the unparticle ACP(B → K π ) − ACP(Bd → K π ) ∼ 0. Although many theo- with the scaling dimension dU behaves like a non-integral num- retical calculations based on QCDF [2],PQCD[3] and SCET [4] have ber dU of invisible particles [11]. Based on Georgi’s proposal, the been tried to produce the consistencies with data in the frame- phenomenology of unparticle physics has been extensively studied work of standard model (SM), however, the results have not been in Refs. [11–16]. For illustration, some examples such as t → u + U conclusive yet [5]. For instance, the recent PQCD result for the ΔCP + − + − and e e → μ μ have been introduced to display the unparti- is 0.08 ± 0.09, which is actually consistent with the data. However, + + 0 +0.03 cle properties. In addition, it is also suggested that the unparticle the PQCD prediction ACP(B → K π )PQCD =−0.01− still has 0.05 production in high energy colliders might be detected by search- 1.4σ difference from the current experimental data [6]. In addi- ing for the missing energy and momentum distributions [11–13]. tion, the difference between (sin 2β) 0 and (sin 2β) J/Ψ K in the K S π S Nevertheless, we have to point out that flavor factories with high mixing-induced CPA from the PQCD prediction is 0.065 ± 0.04, luminosities, such as SuperKEKB [17],SuperB[18] and LHCb [19], which shows about 2σ off the data −0.30 ± 0.19. Hence, the in- etc., should also provide good environments to search for the un- consistencies between data and theoretical predictions provide a particle effects in indirect way. strong indication to investigate the new physics beyond SM. Besides the weird property of non-integral number of unparti- There introduced many extensions of the SM, and enormous cles, the most astonished effect is that an unparticle could carry a studies have been done on searching some specific models beyond peculiar CP conserving phase associated with its propagator in the time-like region [12,13]. It has been pointed out that the unparticle phase plays a role like a strong phase and has an important impact Corresponding author. * on direct CP violation (CPV) [14]. In this Letter, we will make de- E-mail addresses: [email protected] (C.-H. Chen), [email protected] → (C.S. Kim), [email protected] (Y.W. Yoon). tailed analysis to examine whether the puzzles in Bu,d (π, K )π

0370-2693 © 2008 Elsevier B.V. Open access under CC BY license. doi:10.1016/j.physletb.2008.12.007 C.-H. Chen et al. / Physics Letters B 671 (2009) 250–255 251 ¯ = q decays with the Bd,s–Bd,s mixing constraints could be resolved Mq 2 M12 , (5) when the invisible unparticle stuff is introduced to the SM. In order to study the flavor physics associated with scale in- q and the mixing phase φ ≡ arg M can be obtained from the mix- variant stuff, we follow the scheme proposed in Ref. [11].Forthe q 12 ing induced CP asymmetry of b → ccs¯ processes. We summarize system with the scale invariance, there exist so-called Banks–Zaks current experimental data in Table 1. (BZ) fields that have a nontrivial infrared fixed point at a very The SM mixing amplitude reads high energy scale [20]. Subsequently, with the dimensional trans- mutation at the ΛU scale, the BZ operators composed of BZ fields will match onto unparticle operators. We consider only vec- G2 m2 tor unparticle operator in the following analysis. Then, the effective q,SM = F W 2 ˆ ∗ 2 ˆ B M12 mBq f B B Bq Vtq Vtb η S0(xt ), (6) interactions for unparticle stuff and the particles of the SM are 12π 2 q adopted to be B where ηˆ = 0.552 is short distance QCD correction term [25], and q q q q C C = ± L ¯ μ R ¯ μ S0(xt ) 2.35 0.06 is an Inami–Lim function for the t-quark ex- q γμ(1 − γ5)qOU + q γμ(1 + γ5)qOU , (1) dU −1 dU −1 change in the loop diagram [26]. The quantities of f and B ΛU ΛU Bq Bq are non-perturbative parameters which can be obtained from the q q Oμ lattice calculations. We follow the procedure given in Ref. [27] where C L,R are effective coefficient functions and U denotes the spin-1 unparticle operator with scaling dimension dU andisas- for dealing with these non-perturbative parameters. The procedure μ mainly employs the result of lattice calculations in two different sumed to be hermitian and transverse ∂μOU = 0. Since so far the theory for BZ fields and their interactions with SM particles is ways. The one is to use the result of JLQCD Collaboration [28], and q q the other is to combine the results of JLQCD and HPQCD [29] Col- uncertain, here C are regarded as free parameters. With scale L,R laborations. We note that one can obtain the SM mixing phases invariance, the propagator of vector unparticle can be obtained by from Eq. (6) as follows. [12,13] · μ d4xeip x0|T O (x)O ν (0) |0 U U SM = SM =− 2 φd 2β, φs 2λ η, (7) μ ν 2 μν p p −iφU = iΔU p −g + e , (2) p2 where β is an angle of CKM unitarity triangle, λ and η are from = − the Wolfenstein parametrization [30].WeusetheresultofUT- with φU (dU 2)π and ¯ fit [31] from the tree level processes for the β, Rt and η, where 2 2 AdU 1 R ≡ (1 − ρ¯) + η¯ and (ρ¯, η¯) is the apex of the CKM unitary ΔU p2 = , t 2 2−dU 2sin(dU π) (p + i ) triangle. For the |V cb|, we adopt the result of global fit to mo- → 5/2 + ment of inclusive distributions in B Xclνl, which is performed = 16π (dU 1/2) AdU , (3) in the framework of heavy quark expansions with kinetic scheme 2dU dU − 1 2dU (2π) ( )( ) [32]. After putting all the SM input parameters into Eq. (6),theSM where φU could be regarded as a CP conserving phase [12–14]. mixing amplitudes are obtained. And using the experimental data ¯ We note that when unparticle stuff is realized in the framework shownintheTable 1, the NP mixing amplitudes for the Bd–Bd of conformal field theories, the propagators for vector and tensor mixing could be gained through the Eq. (4). The numerical values unparticles should be modified [21]. Although conformal invari- are summarized in Table 2. ance typically implies scale invariance, however, in principle it is As for the unparticle contribution to the mixing amplitude, we not necessary. Unparticle stuff with scalar invariance in 2D space- begin with the effective Hamiltonian for B = 2 processes in un- time has been investigated in Ref. [22]. Hence, in this work, we particle sector such as still concentrate on the stuff built out of only scale invariance. For simplicity, we set the unknown scale factor ΛU to be 1 TeV Table 1 ¯ throughout the analysis. Experimental values for the Bd,s–Bd,s mixings. Even though DØ Collaboration recently have measured φs [23], the data is not used in our analysis because of huge ¯ error of it. 2. Constraints of Bd,s–Bd,s mixing Observables Values Note ± −1 Since the tree level FCNC processes are allowed in the unparti- Md (0.507 0.004) ps HFAG [1] ± −1 cle physics, it is expected that the B –B¯ mixings offer strong Ms (17.77 0.12) ps CDF [24] d,s d,s ◦ ± ◦ db db sb sb φd 43 2 HFAG [1] constraints on the unparticle parameters of C L , C R and C L , C R . Moreover, it will be shown that the scaling dimension dU also can ¯ be constrained by Bd,s–Bd,s mixings. Table 2 − ¯ First of all, we separate the matrix element of B = 2 transition Numerical values for the SM Bd,s Bd,s mixing amplitudes. The values of the NP − ¯ 0 − ¯ 0 = q Bd Bd mixing amplitude are obtained from the experimental data and the Eq. (4). for the Bq Bq mixing (q d, s), which is denoted by M ,into q,NP q,U 12 Thecase(a)denotesJLQCD,whilethecase(b)denotes(HP+JL)QCD. M = M the SM and unparticle contribution as follows. 12 12 should be considered (q = s, d). q q,SM q,NP q,SM iφSM q,NP iφNP M = M + M = M e q + M e q , (4) SM parameters Values NP parameters Values 12 12 12 12 12 + − − 0.75 0.20 ps 1 (a) 0.25 ± 0.26 ps 1 (a) 2|Md,SM| −0.26 2|Md,NP| SM NP 12 ± −1 12 ± −1 where φq and φq represent the phases of mixing amplitudes. 0.97 0.29 ps (b) 0.46 0.29 ps (b) ◦ ◦ ◦ ◦ −130 ± 180 (a) For the second term, we use the superscript ‘NP’ in order to rep- SM 45 2 ± 5 7 NP φd . . φd − ◦ ± ◦ resent general new physics (NP) contribution. Later on, we regard 132 12 (b) −1 q,NP s,SM 16.4 ± 2.8ps (a) s,NP this M as the unparticle mixing amplitude. 2|M | − 2|M | – 12 12 23.8 ± 5.9ps 1 (b) 12 As is well known, the magnitude of total mixing amplitude ◦ ◦ φSM −2.3 ± 0.2 φNP – | q | 0 ¯ 0 s s M12 is given by the Bq –Bq oscillating frequency as follows: 252 C.-H. Chen et al. / Physics Letters B 671 (2009) 250–255

− 2 dU 1 d,U −8 s,U −7 U 1 p 1 AU − a = (1.1 ± 1.3) × 10 , a = (2.6 ± 2.3) × 10 , (11) Hq, = 2 e iφU mix mix 4 Λ2 p2 2sindU π U and Eq. (10) leads to qb qb × −q¯ C (1 − ) + C (1 + ) γμ L γ5 R γ5 qb qb q,U C − C = 2 a . (12) L R mix qb qb × bq¯γ μ C (1 − γ ) + C (1 + γ ) b L 5 R 5 qb qb = Therefore, the parameters C L and C R (q d, s) are strongly corre- 1 qb qb ¯ + q¯p/ C (1 − γ ) + C (1 + γ ) b lated under the Bd s–Bd s mixing. In the next section, these strong 2 L 5 R 5 , , p constraints on the unparticle parameters will be used for ﬁtting → × ¯ qb − + qb + the parameters in Bu,d (π, K )π decays. qp/ C L (1 γ5) C R (1 γ5) b . (8) 3. B → ( , K ) decays and the unparticle contributions The factor 2 is from the fact that there are s- and t-channel which u,d π π give same result, and the factor 1/4 is due to the Wick contraction factor [33]. From this effective Hamiltonian, the transition matrix According to the Eqs. (1) and (2), the effective Hamiltonian for → ¯ elements can be shown as b qq q decays is obtained by 2 qb qb U ΔU (p ) − U H =− 2 ¯ + ¯ q, =− iφU 2 ˆ q, U CU q C L (qb)V −A C R (qb)V +A M − e mBq f B B Bq a , (9) 12 2 dU 1 q mix ΛU × q q ¯ + q q ¯ C L (q q )V −A C R (q q )V +A , (13) q,U where the a is deﬁned by ¯ ¯ mix where q = (d, s), q = (u, d, s, c), ( f f )V ±A = f γμ(1 ± γ5) f and

m2 q,U qb 2 qb 2 2 5 Bq U 2 a ≡ C + C − 2 Δ (q ) −iφU mix L R 2 CU q = e . (14) 3 12 p 2 dU −1 (ΛU ) m2 qb qb 5 7 Bq + C C − + , (10) Based on this effective Hamiltonian, we study the unparticle con- L R 3 6 p2 tributions to the decay amplitudes for Bu,d → (π, K )π . It is known 2 = 2 2 that the most uncertain theoretical calculations for two-body ex- with p mB . ΔU (p ) is given in Eq. (3).Inordertogetthe q clusive decays are the QCD hadronic transition matrix elements. To constraints on the unparticle parameters from the NP parameter q,U deal with the hadronic matrix elements, we adopt recent pertur- regions shown in the Table 2, we ﬁrst consider the phase of M . 12 bative QCD (PQCD) calculations for the SM amplitudes and naive −iφU It depends on the scaling dimension dU through the e term factorization (NF) approach for the amplitudes of unparticle con- d,U and the sign of sin dU in the U p2 . The plot of arg M ( π) Δ ( ) ( 12 ) tributions. The SM amplitudes for Bu,d → (π, K )π decays can be versus dU is shown in Fig. 1. parameterized in the context of quark diagram approach (QDA) d,U It is interesting that arg(M12 ) takes only positive value as dis- [34] as follows: played in Fig. 1. Therefore, as we can see from Table 2.(HP+JL)QCD √ d,U SM + + 0 iγ iγ −iβ case cannot give a solution of dU while all value of arg(M ) 2A B → π π =−Te − Ce − PEWe , (15) 12 is possible in JLQCD case. So, we take only the JLQCD case. Al- SM + − iγ −iβ A Bd → π π =−Te − Pe , (16) though we cannot determine dU in the JLQCD case presently, if √ ¯ SM 0 0 iγ −iβ −iβ the SM prediction for the Bd–Bd mixing amplitude becomes more 2A Bd → π π =−Ce + Pe − PEWe , (17) precise in future, it can give strong constraint on dU from the SM + → 0 + = d,U = A B K π P , (18) plot of arg(M12 ) versus dU . Here, we assume dU 1.5forthe SM + − iγ remaining analysis. Then, the magnitude of unparticle transition A Bd → K π =−P − T e , (19) s,U √ matrix element M can be obtained through Eq. (4) and the + + 12 2ASM B → K π 0 =−P − T eiγ − C eiγ − P , (20) values in Table 2 for the JLQCD case with the experimental data √ EW | s,U |= ± −1 SM → 0 0 = − iγ − as 2 M12 7.6 6.6ps . Using this value and the value for 2A Bd K π P C e PEW, (21) d,U q,U 2|M | in Table 2, we can see that the mixing parameter a 12 mix where T ( ) and C( ) denote the tree color-allowed and color- should be strongly suppressed as follows: suppressed amplitudes for Bu,d → π(K )π , respectively, while () () P (PEW) is gluonic (electroweak) penguin amplitude. All CP- conserving phases are included in these parameters. The phase γ (β) is the CP violating phase in the SM and from V ub(Vtd). Table 3 shows the recent PQCD result for the values of each topological parameters [6,35]. For deriving the unparticle contributions, the deﬁnitions for rel- evant decay constants and form factors are given by

Table 3 Recent PQCD predictions for the topological parameters of Bu,d → (π, K )π in unit − of 10 5 GeV. The phases are indicating strong phases of the parameters in radian unit. The predictions include NLO calculation.

Topology Abs Arg Topology Abs Arg +10.8 +0.1 +8.0 ± P 43.6−8.0 2.9−0.2 T 23.2−6.1 0.0 0.0 +2.4 ± +1.2 − +0.2 T 6.5−1.8 0.1 0.0 P 5.6−0.8 0.4−0.1 +1.4 − ± +2.1 − ± PEW 5.4−1.0 1.3 0.1 C 4.3−1.5 1.1 0.0 + + U 0.9 − ± 0.1 − ± d, − ◦ ◦ C 1.7−0.6 3.0 0.0 PEW 0.7−0.1 0.1 0.0 Fig. 1. Plot of arg(M12 ) versus dU within the range of ( 180 , 180 ). C.-H. Chen et al. / Physics Letters B 671 (2009) 250–255 253

Table 4 The definition of coefficients for the unparticle amplitudes in Bu,d → (π, K )π decays within NF approach. U Decay mode a dec + − π π − 1 Cdb C uu − Cdb C uu + 2rπ Cdb C uu − Cdb C uu Nc L L R R 1 L R R L + 0 − √ 1 db uu − dd − db uu − dd π π C L C L C L C R C R C R 2Nc + π db uu − dd − db uu − dd 2r2 C L C R C R C R C L C L C (q2) − √ U 2 db + db uu − dd − uu + dd Fig. 2. Typical flavor diagrams for B → (π, K )π decays mediated by unparticle with 2 C L C R C L C L C R C R 2CU (q1 ) q= u and d,whereq1,2 are the momenta of unparticle. 0 0 √ 1 db dd − db dd + π db dd − db dd π π C L C L C R C R 2r2 C L C R C R C L 2Nc C (q2) ¯ | = − √ U 2 db + db uu − dd − uu + dd P(p) qγμγ5u 0 ifP pμ, 2 C L C R C L C L C R C R 2CU (q1 ) ¯ | = 0 0 − 1 sb dd sb dd K sb dd sb dd P(p) qγ5u 0 ifP mP , K π C C − C C + 2r C C − C C Nc L L R R 1 L R R L 2 m + − − 1 sb uu − sb uu + K sb uu − sb uu ¯ ¯ B BP 2 K π N C L C L C R C R 2r1 C L C R C R C L P(p) qγμb B(pB ) = (pB + p)μ − qμ F q c q2 1 + 0 − √ 1 sb uu − sb uu + K sb uu − sb uu K π C L C L C R C R 2r1 C L C R C R C L 2 2Nc m C (q2 ) F BK (m2 ) + B BP 2 − √ U 2 fπ 0 π sb + sb uu − dd − uu + dd q F q , (22) Bπ 2 C C C C C C 2 μ 0 2CU (q2 ) f K F (m ) L R L L R R q 1 0 K 0 0 √ 1 sb dd − sb dd + K sb dd − sb dd = = − 0 = 2 + K π C L C L C R C R 2r2 C L C R C R C L with P (π, K ), q pB p and mP mP /(mq mu). Here, due 2Nc CU (q2 ) F BK (m2 ) 2 → − √ 2 fπ 0 π C sb +C sb C uu −Cdd −C uu +Cdd to mP mB ,wehaveneglectedthem effects in B P tran- 2 f K Bπ 2 L R L L R R P 2CU (q1 ) F0 (mK ) sition matrix element. Subsequently, by considering various flavor diagrams in which the typical diagrams mediated by unparticle are db db sb sb C L , C R , C L , C R , illustrated in Fig. 2, the unparticle amplitudes for Bu,d → (π, K )π uu uu dd dd decays within NF approach are obtained to be C L , C R , C L , C R , (28) U → i j = 2 2 Bπ 2 U,π i π j which denote the couplings of unparticle to SM particles. First A B π π CU q1 fπ mB F0 mπ adec , (23) ¯ four parameters are strongly correlated by Bd,s–Bd,s mixing phe- U i j 2 2 Bπ 2 U,K i π j A B → K π = CU q f K m F m a , (24) nomena as shown in Eq. (12). If we regard all these parameters 1 B 0 K dec = = U to be real number and set ΛU 1 TeV and dU 1.5aswe where the coefficients a saredefinedinN = 3 is the number ¯ dec c do in the Bd,s–Bd,s analysis, 8 free parameters are involved for 2 = 2 π K → of colors, q2 mπ , and the chiral enhanced factor r(1,2) and r(1,2) Bu,d (π, K )π in the unparticle physics. are defined by In order to fit to the data for the Bu,d → (π, K )π decays with these8parameters,weperformtheminimumχ 2 analysis. Accord- m2 m2 rπ = π , rπ = π , ing to current experimental observations, the amount of available 1 m (m + m ) 2 m (m + m ) b u d b d d data for Bu,d → (π, K )π decays is 17 as their world averages are m2 m2 K = K K = K displayed in Table 5. Besides the BRs, the important quantities to r1 , r2 . (25) mb(mu + ms) mb(md + ms) display the new physics effects are the direct and mixing induced

We note that since q1 in Fig. 2(a) involves different mesons, the CPAs, where they could be briefly defined through estimation of q2 should have ambiguity. To understand the typi- 1 |λ |2 − 1 2Im(λ ) 2 A ≡ f ≡ f cal value of q ,wewritetheq1 = pB − k2 − k3 with k2,3 being f , S f , (29) 1 1 +|λ |2 1 +|λ |2 themomentaofvalencequarksinsidethelightmesons.Interms f f of momentum fraction of valence quark and light-cone coordi- = iφd ¯ with λ f e A ¯f /A f , respectively. For direct CPA, f could be any nates and by neglecting√ the transverse√ momentum, one can get possible final states; however, for mixing induced CPA, f could = = k2 (0,mB x2/ 2, 0⊥) and k3 (mB x3/ 2, 0, 0⊥).Asaresult,we only be CP eigenstates. Since the PQCD prediction for the SM de- 2 = 2 − − have q1 mB (1 x2)(1 x3). According to the behavior of lead- cay amplitudes has sizable error as shown in Table 3,itshouldbe tw−2 ing twist wave function of light meson, Φ ∝ x(1 − x) which considered for the minimum χ 2 analysis. Therefore, we define the is calculated by QCD sum rules [37], it is known that the maxima χ 2 to be of x2,3 occur at x2 = x3 ∼ 1/2. Therefore, for numerical estima- n 2 tions, the value for momentum transfer could be roughly taken as (ti − ei) χ 2 = . (30) 2 2 exp 2 q ≈ m /4withmB = 5.28 GeV. Since the contributions of Fig. 2(a) + thr 2 1 B i=1 σi σi are color-suppressed, we emphasize that the corresponding results 2 ei and ti denote the experimental data for ith observable and its are insensitive to the adopted value of q1. We discard irrelevant factor i in the NF and match the sign with the QDA parametriza- theoretical prediction within unparticle contribution, respectively. exp thr tion; then, the total amplitude is σi is the experimental error of ith observable, while σi is the- U oretical error propagated from the errors of topological parameters A B → f = ASM B → f + A B → f (26) ( ) κ f ( ) ( ). obtained from PQCD. Since there are 8 free parameters involved

Here, the κ f is the ratio of phase space factor coming from the in our analysis, the degree of freedom (d.o.f) for the fitting is − = = +15 ◦ difference of notation of decay amplitude between NF and PQCD (17 8) 9. As for the angle γ ,weusethevaluesofγ (63−12 ) group and is defined by from the PDG 2006 [36]. Consequently, by imposing the mixing db sb constraints of C L(R) and C L(R) displayed in Eq. (12),wefindthe G2 m3 F b p f −4 optimized values of unparticle parameters as follows: κ f ≡ = 1.15 × 10 . (27) 128π 8πm2 B − − Cdb = 3.3 × 10 4, Cdb = 4.6 × 10 4, From Table 4, we see that besides dU and ΛU , the introduced new L R sb = × −4 sb = × −4 free parameters are C L 7.6 10 , C R 11.2 10 , 254 C.-H. Chen et al. / Physics Letters B 671 (2009) 250–255

Table 5 constraints on unparticle parameters as well as scaling dimen- → The experimental data for Bu,d (π, K )π decays and comparison between the sion dU .The(HP+JL)QCD case could not give a solution for dU , experimental data and the theoretical predictions with and without unparticle con- −6 while all value of dU is possible in JLQCD case. However, we see tribution. The BRs are order of 10 . The data is updated by September 2007. ‘w/o’ 0 ¯ 0 means ‘without unparticle contribution’. χ 2 contributions of each observables are that more accurate SM prediction for the Bq –Bq mixing ampli- shown. tude in future would give strong constraint on dU .Inorderto understand whether the unparticle effects could satisfy all mea- Observables Data Theory (w/o) Theory χ 2 (w/o) χ 2 surements in exclusive B → ( , K ) decays, we utilize the min- 0 + u,d π π B(K π ) 23.1 ± 1.023.5 ± 12 23.1 ± 11 0.001 0.0 2 + imum χ analysis to search for the solutions of free parameters. B(K π 0) 12.9 ± 0.613.0 ± 6.212.7 ± 6.00.001 0.001 + − = B(K π ) 19.4 ± 0.619.7 ± 10 20.3 ± 10 0.001 0.007 Interestingly, after we fix dU 1.5 for the specific unparticle scal- B(K 0π 0) 9.9 ± 0.68.8 ± 4.99.5 ± 5.10.046 0.006 ing dimension, we find that the unsolved problem of large BR for → 0 0 0 + Bd π π could be explained excellently in the framework of ACP(K π ) 0.009 ± 0.025 0.0 ± 0.00.0 ± 0.00.13 0.13 + 0 unparticle physics. Moreover, the discrepancy between the stan- ACP(K π ) 0.050 ± 0.025 −0.017 ± 0.068 0.074 ± 0.068 0.84 0.11 + − + → + 0 ACP(K π ) −0.097 ± 0.012 −0.099 ± 0.073 −0.11 ± 0.075 0.001 0.03 dard model estimation and data for the direct CPA of B K π A 0 0 − ± − ± − ± + − CP(K π ) 0.14 0.11 0.065 0.040 0.058 0.036 0.41 0.50 and Bd → π π could be reconciled very well. However, the puz- → 0 ± ± ± zle of the mixing induced CPA of Bd K S π could not be resolved S K π 0 0.38 0.19 0.74 0.08 0.74 0.07 3.13.2 S well in unparticle physics. B + 0 +0.41 ± ± (π π ) 5.59−0.40 4.03 2.53 4.05 2.53 0.37 0.36 + − B(π π ) 5.16 ± 0.22 6.80 ± 4.43 7.11 ± 4.43 0.14 0.19 Acknowledgements B(π 0π 0) 1.31 ± 0.21 0.23 ± 0.13 1.33 ± 0.30 19 0.002

+ 0 ACP(π π ) 0.06 ± 0.05 0.00 ± 0.01 0.055 ± 0.018 1.60.01 + − This work of C.H.C. is supported by the National Science Coun- ACP(π π ) 0.38 ± 0.07 0.17 ± 0.10 0.38 ± 0.14 2.80.0 0 0 +0.32 cil of R.O.C. under Grant No. NSC-95-2112-M-006-013-MY2. C.H.C. ACP(π π ) 0.48− 0.64 ± 0.23 0.53 ± 0.13 0.17 0.024 0.31 would like to thank Dr. Shao-Long Chen for useful discussions. The + − − ± − ± − ± Sπ π 0.61 0.08 0.55 0.44 0.55 0.42 0.021 0.023 work of C.S.K. was supported in part by CHEP-SRC and in part by the KRF Grant funded by the Korean Government (MOEHRD) Table 6 No. KRF-2005-070-C00030. The work of Y.W.Y. was supported by Numerical values of unparticle amplitudes for each decay mode. ‘Abs’ represents the − the KRF Grant funded by the Korean Government (MOEHRD) No. magnitude of the amplitude in unit of 10 5 GeV. ‘Arg’ represents the CP conserving phase of unparticle in radian unit. KRF-2005-070-C00030. Y.W.Y thank Jon Parry for his comment on mixing constraints. 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