TECHNION-PH-2008-39 EFI 08-33 arXiv:0812.4796 December 2008 Doubly CKM-suppressed corrections to CP asymmetries in B0 J/ψK0 → Michael Gronau 9 Physics Department, Technion – Israel Institute of Technology 0 0 32000 Haifa, Israel 2 Jonathan L. Rosner n a Enrico Fermi Institute and Department of Physics J University of Chicago, Chicago, Illinois 60637 3 1 ] A doubly CKM-suppressed amplitude in B0 J/ψK leads to correc- h S → p tions in CP asymmetries S = sin2β,C = 0, which may be enhanced by - p long-distance rescattering. It has been suggested that this enhancement e may lead to several percent corrections. We calculate an upper bound of h [ order 10−3 on rescattering corrections using measured branching ratios for charmless ∆S = 1 B0 decays. 4 | | v 6 PACS codes: 12.15.Hh, 12.15.Ji, 13.25.Hw, 14.40.Nd 9 7 4 . I Introduction 2 1 8 The success of the Kobayashi-Maskawa (KM) model of CP violation [1] in predicting 0 : correctly CP asymmetries in B meson decays has been recently recognized by the v i Nobel committee [2]. The large asymmetry measured in B0 J/ψKS(L) [3, 4] given X → by sin2β, where β φ = arg( V∗V /V∗V ), proved unambiguously that the KM r ≡ 1 − cb cd tb td a phase is the dominant source of CP violation in B decays. This test involves an interference between B0-B¯0 mixing and a B0 decay ampli- ¯ tude [5], consisting of a dominant color-suppressed b c¯cs¯ tree amplitude and a ¯ → small contribution from a b s¯uu¯ penguin amplitude [6, 7, 8]. (We use the unitarity → of the Cabibbo-Kobayashi-Maskawa [1, 9] (CKM) matrix, V∗V = V∗V V∗V , tb ts − cb cs− ub us and will not discuss order 10−3 effects due to CP violation in B0 B0 mixing [10] and − K0 K0 mixing [11].) The magnitude ξ of the ratio of these two amplitudes, which − determines the theoretical precision of this test, involves three suppression factors: A ratio of CKM matrix elements, V∗V / V∗V 0.02 [12], small Wilson coeffi- | ub us| | cb cs| ≃ cients of penguin operators in the effective Hamiltonian, c 0.04 (i = 3,4,5,6) (or i ∼ a QCD loop factor), and a suppression by the Okubo-Zweig-Iizuka (OZI) rule [13]. The parameter ξ is expected to be somewhat larger than the product of these three factors due to color-suppression of the dominant tree amplitude which normalizes ξ. Thus, with ξ 10−3, it has been commonly accepted that the measurement of sin2β ∼ in B0 J/ψK may involve only a very small uncertainty at a level of 10−3, S(L) → 1 or at most a fraction of a percent [6]. This estimate was supported by calculations of ξ using QCD factorization [14] and perturbative QCD [15]. These perturbative calculations are based on the absorptive part of the u-quark loop, assuming a suffi- ciently large momentum transfer in the loop, and applying rather crude methods for evaluating the four quark operator matrix element J/ψK0 (c¯Tac) (¯bTas) B0 . V V−A h | | i The absorptive part of the quark loop was proposed thirty years ago as a mechanism producing a strong phase leading to CP violation in charged B decays [16]. The introduction of a small penguin amplitude, carrying a weak phase γ and a strong phase δ relative to the dominant tree amplitude, affects the time-dependent CP asymmetry in B0 J/ψK , S → Γ(B0 J/ψK ) Γ(B0 J/ψK ) S S A (t) → − → = Ccos(∆mt)+Ssin(∆mt) , (1) CP ≡ Γ(B0 J/ψK )+Γ(B0 J/ψK ) − S S → → where, dropping terms quadratic in ξ [6], C = 2ξsinδsinγ , ∆S S sin2β = 2ξcos2βcosδsinγ . (2) − ≡ − In the limit ξ = 0 one has C = 0,∆ = 0. Current measurements of the two asymme- tries [3, 4, 17] (based on all charmonium decays), C(J/ψK0) = 0.005 0.019 , S(J/ψK0) = 0.671 0.024 , (3) ± ± involve experimental errors at a level of 0.02. This error is considerably larger than ± the theoretical uncertainty introduced by the above estimate of the parameter ξ. The estimate ξ 10−3 has been questioned and challenged in Refs. [18] and [19], ∼ arguing that the u-quark penguin amplitude in B0 J/ψK0 may be enhanced by → long distance rescattering effects from intermediate S = 1 charmless states to J/ψK0. A sizable enhancement of a penguin amplitude beyond a perturbative calculation, ar- gued to be due to a large “charming penguin” contribution [20], has been observed in B Kπ. It was therefore argued [18, 19] that similar nonperturbative rescat- → tering effects of intermediate charmless states may enhance ξ, leading to a hadronic uncertainty in ξ at a level of several percent. An uncertainty in ξ at this level implies theoretical uncertainties in the asym- metries C(J/ψK0) and S(J/ψK0) which are comparable to or even larger than the currentexperimentalerrorsinthesemeasurements. ItwaspointedoutinRefs.[18,19] that CP asymmetries C and ∆S inB0 J/ψπ0, proportionaltoξ inthe flavor SU(3) → limit, are enhanced by a factor (1 λ2)/λ2 = 18.6 (λ = 0.2257 [12]) relative to C and − ∆S in B0 J/ψK . Thus, it was suggested to study ξ in B0 J/ψπ0. Unfortu- S → → nately, the decay rate for this process is suppressed by 2λ2/(1 λ2) relative to that − of B0 J/ψK . Consequently one expects errors in the B0 J/ψπ0 asymmetries S → → to be correspondingly larger than in the B0 J/ψK asymmetries. Indeed, current S → measurements [17, 21, 22], C(J/ψπ0) = 0.10 0.13,S(J/ψπ0) = 0.93 0.15, are − ± − ± not sufficiently accurate for providing useful information about ξ. Values of ξ as large as a few percent cannot be ruled out by the two asymmetries. The purpose of this Letter is to calculate upper bounds on ξ in B0 J/ψK0 from → long distance rescattering effects mediated by charmless intermediate states. This 2 provides a re-evaluation of the contribution of the absorptive part associated with cutting the u-quark penguin loop [14, 15, 16] by computing explicitly contributions of charmless intermediate states. Using the rich amount of data for numerous charmless B meson decays obtained in experiments at e+e− B factories, we will show that the estimate ξ 10−3 ismuch more reasonable thanvalues ofξ at a level of a fewpercent. ∼ II Upper bounds on rescattering in B0 J/ψK0 → We write the S matrix in terms of S , which includes strong and electromagnetic 0 interactions, and T, taken to be Hermitian, which corresponds to the effective weak Hamiltonian at a low energy scale, S = S +iT . (4) 0 Unitarity of the S matrix S†S = 1 implies to first order in T, T = S T S . (5) 0 0 Taking matrix elements of the two sides between a B meson state and a final decay state f , and inserting a complete set of intermediate states f, one has 0 f T B = Σ f S f f T B , (6) 0 f 0 0 h | | i h | | ih | | i where we used the fact that B is an eigenstate of S . The matrix elements f T B 0 0 h | | i and f T B are weak decay amplitudes, often denoted A(B f ) and A(B f), 0 h | | i → → while f S f represents a rescattering amplitude from f to f . 0 0 0 h | | i Let us first consider the matrix element between B = B0 and f = J/ψK0 for the 0 effective ∆S = 1,∆C = 0 operator Tu involving a CKM factor V∗V , | | ub us J/ψK0 Tu B0 = Σ J/ψK0 S f f Tu B0 . (7) f 0 h | | i h | | ih | | i Because B is a spinless particle the J/ψ and K0 are in a P-wave. Consequently the states f are all J = 0,P = 1 S = 1 states. Since we are replacing the absorptive − part associated with cutting the u-quark penguin loop by contributions of physical intermediate states, we consider only charmless states. This includes a long list of states,suchasf = K∗+π−,ρ−K+,K∗0π0,ρ0K0,ωK0,K∗0η,K∗0η′,andK∗+(1430)π−, 0 but excludes K+π−,K0π0 in an S-wave state and K∗+ρ−,K∗0ρ0 in S and D waves which have P = +1. Parity and time-reversal symmetry of S imply a reciprocal detailed-balance rela- 0 tion (we are assuming a single polarization state because J = 0), J/ψK0 S f = f S J/ψK0 . (8) 0 0 |h | | i| |h | | i| Upper bounds on matrix elements f S J/ψK0 for each of the above states, f = 0 |h | | i| K∗+π−,...,K∗+(1430)π−, may be obtained using the following considerations. 0 3 We apply Eq. (6) to matrix elements between B0 and the above charmless final states f for the effective ∆S = 1,∆C = 0 operator Tc involving a CKM factor | | V∗V , cb cs f Tc B0 = Σ f S k k Tc B0 . (9) k 0 h | | i h | | ih | | i The left-hand-side is dominated by a penguin amplitude which obtains a sizable “charming penguin” contribution [20]. Assuming that a single intermediate state k = D∗−D+ can at most saturate the sum on the right-hand-side, one has s f S D∗−D+ D∗−D+ Tc B0 f Tc B0 . (10) |h | 0| s i||h s | | i| ≤ |h | | i| On the left-hand-side of (10) we can replace the D∗−D+ state by J/ψK0. One ex- s pects f S J/ψK0 < f S D∗−D+ becausethefirstamplitudeisOZI-suppressed. |h | 0| i| |h | 0| s i| To calculate the ratio of B0 decay amplitudes into D∗−D+ and J/ψK0 we use the s expression for decay rates, p∗ Γ = f f T B0 2 , (11) 8πM2|h | | i| B where p∗ is the momentum of one of the two outgoing particles in the B0 rest frame. f The measured branching ratios and the corresponding momenta are [12] (B0 D∗−D+) = (8.3 1.1) 10−3 , p∗ = 1735 MeV/c , B → s ± × Ds+ (B0 J/ψK0) = (8.71 0.32) 10−4 , p∗ = 1683 MeV/c . (12) B → ± × K0 This leads to D∗−D+ Tc B0 / J/ψK0 Tc B0 = 3.04 0.21. Using the central |h s | | i| |h | | i| ± value, Eq. (10) may be replaced by 1 f Tc B0 f S J/ψK0 < |h | | i| . (13) |h | 0| i| 3 J/ψK0 Tc B0 |h | | i| We denote f Tu B0 r |h | | i| , (14) f ≡ f Tc B0 |h | | i| and note that f Tc B0 is approximately the total B0 decay amplitude into f, h | | i f T B0 ; similarly J/ψK0 Tc B0 J/ψK0 T B0 . Combining Eqs. (8) and (13), h | | i h | | i ≈ h | | i one then obtains the following upper bound on each of the terms contributing to the sum in (7), normalized by the B0 decay amplitude into J/ψK0: J/ψK0 S f f Tu B0 1 f T B0 2 0 ξ |h | | ih | | i| < r |h | | i| . (15) f f ≡ J/ψK0 T B0 3 J/ψK0 T B0 ! |h | | i| |h | | i| This upper bound is a central result in our analysis. It should be considered a strong inequality (in which the factor 1/3 may be replaced by 1/10) because it is based on a conservative inequality (10) and on presumably strong OZI-suppression of f S J/ψK0 relative to f S D∗−D+ . |h | 0| i| |h | 0| s i| 4 Table I: Expressions for matrix elements f Tu B0 and f Tc B0 in B0 VP in h | | i h | | i → terms of graphical amplitudes. Final state f Tu B0 f Tc B0 h | | i h | | i K∗+π− T′ P′ − P − P ρ−K+ T′ P′ − V − P K∗0π0 C′ /√2 (P′ P′ )/√2 − V P − EW,V ρ0K0 C′ /√2 (P′ P′ )/√2 − P V − EW,P ωK0 C′ /√2 (P′ +2S′ + 1P′ )/√2 P V P 3 EW,P K∗0η C′ /√3 (P′ P′ S′ 2P′ )/√3 − V V − P − V − 3 EW,V K∗0η′ C′ /√6 (2P′ +P′ +4S′ 1P′ )/√6 V V P V − 3 EW,V III Numerical upper bounds on rescattering We now study numerical bounds on rescattering parameters ξ for numerous interme- f diate states f in B0 f J/ψK0. We start by discussing S = 1 charmless states → → f = VP consisting of pairs of vector and pseudoscalar mesons. We use f T B0 2 (B0 f) p∗ |h | | i| = B → K0 . (16) |hJ/ψK0|T|B0i|! B(B0 → J/ψK0) p∗f ! Values for the parameter r , the ratio of two amplitudes in B VP involving f → CKM factors V∗V and V∗V , are extracted from a study applying broken flavor ub us cb cs SU(3) to these decays and decays into corresponding S = 0 charmless states [23]. In thelanguageofRef.[24],matrixelements f Tu B0 involve combinationsofgraphical h | | i amplitudes representing color-favoredand color-suppressed tree amplitudes T′ and V(P) C′ , while f Tc B0 involve penguin amplitudes P′ , singlet penguin amplitudes V(P) h | | i V(P) S′ (corresponding to SU(3) singlet mesons in the final state), and electroweak V(P) penguin amplitudes P′ . The subscript V or P denotes the final-state meson EW,V(P) (vector or pseudoscalar) incorporating the spectator quark. We are neglecting color- suppressed electroweak penguin contributions. SU(3) breaking is included in T′ and V T′ in terms of ratios of pseudoscalar and vector meson decay constants, f /f and P K π fK∗/fρ, respectively. Expressions for the above matrix elements are given in Table I for B0 decays into seven VP states. Note that that while f Tu B0 involves color- h | | i allowed tree amplitudes T′ and T′ for f = K∗+π− and f = ρ−K+, it is governed by P V color-suppressed amplitudes C′ and C′ for all other final states. Consequently the P V values of r in the first two processes are expected to be considerably larger than in f the others. We calculate numerical values for r using entries in the third column of Table V f in Ref. [23], updating some values by fitting to more recent measurements of (B0 B → K∗0π0), (B0 K∗0η)and (B0 K∗0η′). BranchingratiosforthesevenB0 VP B → B → → decays [17], corresponding center-of-mass momenta p∗ [12], and values of r are used f to calculate from Eqs. (15) and (16) upper bounds on ξ for the these intermediate f 5 TableII:Branchingratios[17], center-of-massmomenta[12], parametersr andupper f bounds on ξ for seven charmless intermediate VP states. f Mode p∗ r Upper bound on ξ f f B f (10−6) (MeV) (10−4) K∗+π− 10.3 1.1 2563 0.31 0.03 7.9 1.1 ± ± ± ρ−K+ 8.6 1.0 2559 0.26 0.03 5.6 1.0 ± ± ± K∗0π0 2.4 0.7 2562 0.09 0.04 0.6 0.3 ± ± ± ρ0K0 5.4 1.0 2558 0.04 0.03 0.5 0.4 ± ± ± ωK0 5.0 0.6 2557 0.04 0.03 0.5 0.4 ± ± ± K∗0η 15.9 1.0 2534 0.04 0.02 1.6 0.7 ± ± ± K∗0η′ 3.8 1.2 2471 0.08 0.04 0.8 0.4 ± ± ± VP states. Input values and resulting upper bounds on ξ are summarized in Table f II. The largest upper bounds, ξ < (7.9 1.1) 10−4 and ξ < (5.6 1.0) 10−4, f f ± × ± × are obtained for f = K∗+π− and f = ρ−K+, respectively. Much smaller values, at a level of 10−4, are calculated for all other VP states. Because rescattering effects of the form B0 f J/πK0 increase with (B0 → → B → f), we search for charmless intermediate states f for which this branching ratio is particularly large. We note that the three-body decay mode B0 K0π+π−, with → = (44.8 2.6) 10−6 isdominatedbythequasi-two-bodydecayB0 K∗(1430)+π−, B ± × → 0 involving a scalar and a pseudoscalar meson in a P = 1 S-wave [12]: − [B0 K∗(1430)+π−] = (50+8) 10−6 . (17) B → 0 −9 × The fact that this branching ratio seems to exceed that for the three-body final state indicates strong destructive interference with other amplitudes including B0 → K∗+π−, ρ0K0, f (980)K0, and a non-resonant amplitude [25]. 0 In order to evaluate an upper bound for ξ based on the u-quark amplitude’s f contribution to the K∗(1430)+π− intermediate state, we must obtain an estimate 0 of the value of r for this state. This quantity (the subscript P denotes the final f pseudoscalar meson incorporating the spectator quark), T′ (B0 K∗+π−) r | P → 0 | , (18) f ≡ P′ (B0 K∗+π−) | P → 0 | is the ratio of the u-quark tree amplitude and the c-quark penguin amplitude in B0 K∗(1430)+π−. While the latter amplitude dominates this process, the former → 0 maybeestimatedtoagoodapproximationassumingfactorization. Asimilarsituation occurs in B0 K+π−, where the ratio of tree and penguin amplitudes has been → determined within a global flavor SU(3) fit to all B Kπ and B ππ decays [26], → → T′(B0 K+π−) 0.281(16.1 2.0) | → | = ± = 0.094 0.012 . (19) P′(B0 K+π−) 48.2 1.0 ± | → | ± 6 The ratio of the two penguin amplitudes dominating B0 K+π− and B0 → → K∗(1430)+π− is obtained from the corresponding partial rates, 0 P′(B0 K+π−) Γ(B0 K+π−) p∗(K∗+) | → | → 0 = 0.60 0.05 . (20) P′(B0 K∗+π−) ≈ vΓ(B0 K∗+π−) p∗(K+) ± | P → 0 | uu → 0 t Herewehaveused(17)with[12] (B0 K+π−) = (19.4 0.6) 10−6, p∗(K+,K∗+) = B → ± × 0 (2615,2445) MeV/c. In the factorization approximation the tree amplitudes T′(B0 K+π−) and → TP′ (B0 → K∗+π−) involve respectively the K and K0∗ decay constants, fK and fK0∗, and the B to π form factors at q2 = m2 and m2 which are assumed to be approxi- K K∗ 0 mately equal. Thus, |TP′(B0 → K∗+π−)| fK0∗ . (21) T′(B0 K+π−) ≈ f K | → | Note that the scalar K∗ couples to the weak vector current through a coupling pro- 0 portional to the K0∗ decay constant fK0∗ which vanishes by G-parity in the SU(3) symmetry limit [27, 28, 29]. SU(3) breaking leads to a nonzero value, expected to be of order (m m )/Λ relative to the K meson decay constant. Theoretical s d QCD − calculations of fK∗ lead to values in the range [30, 31, 32, 33] 0 fK∗ = 40 6 MeV , (22) 0 ± to be compared with f = 155.5 0.8 MeV [34]. K ± Taking a product of the three factors in (19), (20), and (21), we find r = 0.015 0.003 , f = K∗(1430)+π− . (23) f ± 0 An upper bound on ξ for f = K∗(1430)+π− is then obtained using Eqs. (15) and f 0 (16) while taking into account correlated errors, 1 [B0 K∗(1430)+π−] p∗(K0) ξ < r B → 0 f 3 f (B0 J/ψK0) p∗(K∗+) B → 0 = (1.9 0.4) 10−4 , f = K∗(1430)+π− . (24) ± × 0 Thus, in spite of the large branching ratio measured for this decay mode, this upper bound is about four times smaller than the largest value obtained for the correspond- ing VP state K∗+π− in Table II. There are good prospects for replacing the theoretical estimate (22) with an ex- perimentally determined value. The partial width for the decay τ− M−ν, where → M is a strange scalar or pseudoscalar meson, is f2 (m2 m2 )2 Γ(τ− M−ν) = G2 V 2 M τ − M . (25) → F| us| 16π m τ With G = 1.16637(1) 10−5 GeV−2, m = 1.77684(17) GeV/c2, m = 0.493677(16) F τ K × GeV/c2, V = 0.2255(19), and f = 0.1555(8) GeV [12] (for the last two, see Ref. us K | | 7 [34]), this yields a prediction Γ(τ− K−ν) = (1.59 0.03) 10−14 GeV. The → ± × lifetime of the τ is (290.6 1.0) 10−15 s [12], implying the prediction (τ− ± × B → K−ν) = (7.02 0.14) 10−3, in satisfactory agreement with the experimental value ± × [12] (6.95 0.23) 10−3. ± × The prediction (25) can be applied to the K∗(1430) in the narrow-width approxi- 0 mation, permitting onetoobtainthe ratioofscalar andpseudoscalar decay constants: fK0∗ = m2τ −m2K B(τ → K0∗ν) . (26) f m2 m2 v (τ Kν) K τ − K0∗uu B → t With the experimental upper limit [35] [τ K∗(1430)ν] < 5 10−4 (95% c.l.) and B → 0 × using the central value of m[K∗(1430)] = 1425 50 MeV/c2 [12], this ratio is less 0 ± than 0.69, entailing fK∗ < 107.5 MeV if the predicted value of (τ− K−ν) is used. 0 B → The large width Γ[K∗(1430)] = 270 80 MeV leads to a small positive correction of 0 ± 1.127 to the predicted partial width for τ K∗(1430)ν, reducing this upper bound → 0 slightly to fK∗ < 101 MeV. This is about a factor of 2.5 larger than the theoretical 0 estimates summarized in Eq. (22). Using those estimates and including the finite- width correction, we predict [τ K∗(1430)ν] = (7.8 2.3) 10−5. As stressed in B → 0 ± × Ref. [28], this should be accessible in present experiments. The intermediate state f = K∗(1430)0π0 is fed by a color-suppressed u-quark tree 0 amplitude. Its penguin-dominated branching ratio is expected to be about half of that measured for B0 K∗(1430)+π−. Consequently the upper bound on ξ for → 0 f f = K∗(1430)0π0 is considerably smaller than (24). 0 IV Conclusion We have calculated upper bounds on contributions to the doubly-CKM-suppressed parameter ξ from rescattering B0 f J/ψK0 through charmless S = 1 inter- → → mediate states f. We have derived in Eq. (15) a general conservative upper bound on ξ which increases with (B0 f) and with the ratio r of tree and penguin f f B → amplitudes in B0 f. The actual upper bound may involve a factor 1/10 instead of → 1/3 because OZI suppression in f J/ψK0 has not been included in (15). → The highest upper bounds on ξ , somewhat below 10−3, were obtained for f = f K∗+π− and ρ−K+, while other intermediate states with neutral vector and pseu- doscalar mesons involve much smaller rescattering contributions. This applies also to the state K∗(1430)+π−, which has the largest quasi-two-body decay branching ratio 0 measured so far in B0 decays, [B0 K∗(1430)+π−] = (50+8) 10−6. We noted de- B → 0 −9 × structive interference between B0 K∗(1430)+π− and other modes contributing to → 0 B0 K0π+π−. Thisindicates potentialdestructive interference between rescattering → contributions to ξ of these intermediate states. One may wonder whether larger contributions to ξ may originate in charmless intermediate states with multiplicity larger than three. Only two S = 1 charmless branching ratios comparable to that of B0 K∗(1430)+π− have been measured [36], → 0 (B0 K∗0π+π−) = (54.5 5.2) 10−6, (B0 K∗0K+K−) = (27.5 2.6) 10−6. B → ± × B → ± × 8 These involve quasi-three-body decays leading to four particles in the final state. P = 1 projections of these states, with smaller branching ratios, may rescatter into − J/ψK0. Although it is difficult to calculate or measure the tree-to-penguin ratio r for these states, we expect it to be no more than 0.1. The isospin relation [37] f Γ(B0 K∗0π+π−) = Γ(B+ K∗+π+π−) which holds within 1.5σ [17] is consistent → → withr = 0intheseprocesses. Wedonotanticipateconstructive interference between f rescatteringcontributionsoftheintermediatestatesK∗0π+π− andK∗0K+K− andthe above calculated contributions of K∗+π− and ρ−K+ which are probably larger. Rescattering from intermediate states with two vector mesons in a P = 1 P- − wave state, including K∗+ρ−,K∗0ρ0 and K∗0φ, involve branching ratios considerably smaller than those of B0 K∗0π+π− and B0 K∗0K+K− [12] and very small → → u-quark tree amplitudes. For instance, the tree amplitude in B0 K∗+ρ− is related → by flavor SU(3) to the amplitude dominating B0 ρ+ρ−. Approximately 100% → longitudinal polarization has been measured in this process [17], corresponding to a combination of S and D waves but no P wave. This implies a negligible u-quark P wave amplitude in B0 K∗+ρ−. 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