Evaluation of CKM matrix elements from exclusive P_{\ell 4} decays
EEvaluation of CKM matrix elements from exclusive P (cid:96) decays C. S. Kim ∗ Department of Physics and IPAP, Yonsei University, Seoul 120-749, Korea
G. L´opez Castro † and S. L. Tostado ‡ Departamento de F´ısica, Centro de Investigaci´on y de Estudios Avanzados,Apartado Postal 14-740, 07000 M´exico Distrito Federal, M´exico
We consider the exclusive P (cid:96) decays, P → ( P P ) V (cid:96)ν (cid:96) , where the subindex V means thatthe invariant mass of the pseudoscalar pair is taken within a small window around the mass ofthe vector meson V . Pole contributions beyond the dominant P → V ( → P P ) (cid:96)ν (cid:96) amplitudeof P (cid:96) decays are identified, which, in turn, affects the determination of the CKM matrixelements | V qq (cid:48) | . We evaluate the effects of those contributions in the extraction of bottom andcharm quark mixings. An application to B → ( ππ ) ρ (cid:96)ν (cid:96) data from Belle collaboration, showsan increase in the extracted value of | V ub | in better agreement with determinations basedon B → π(cid:96)ν (cid:96) decays. The effect of the ρ and D ∗ pole contributions in the determination of | V cd | from the decay D → ππ(cid:96) − ¯ ν (cid:96) , has been also investigated. I. INTRODUCTION
Precise measurements of the Cabibbo-Kobayashi-Maskawa (CKM) [1, 2] matrix elements canshed light on new physics and are among the main targets of flavor factories. Deviations fromthe unitarity property of the CKM matrix would indicate the existence of additional degrees offreedom. While the determination of | V ud | and | V us | has been done with an impressive accuracy of0.02% and 0.3%, respectively, the values of | V ub | and | V cb | are known at the 5% and 2% only [3].Better determinations of | V qb | , among other standard model parameters, are the most importantfor searches of new sources of CP violation beyond the one encoded in the CKM paradigm.Currently, the most precise determinations [3] of the matrix elements, | V cb | and | V ub | , indicate atension between values extracted from exclusive and inclusive decay channels of b -flavored hadrons.Clearly, more theoretical works and refined measurements are required to solve this discrepancyand achieve a better accuracy. One can gain some precision by combining values of | V qb | extractedfrom different decay channels of bottom hadrons, provided their measurements furnish a consistentset of data. While measurements of exclusive channels are better suited from an experimental pointof view, the calculation of their form factors in the whole kinematical regime is still challenging.Among the preferred exclusive channels, the B → ( P, V ) (cid:96)ν (cid:96) decays, with P ( V ) a pseudoscalar(vector) meson, are the simplest ones to describe theoretically and the dominant final states ofcharmfull and charmless semileptonic decays of B mesons. While Lattice QCD provides reliable ∗ Electronic address: [email protected] † Electronic address: glopez@fis.cinvestav.mx ‡ Electronic address: stostado@fis.cinvestav.mx a r X i v : . [ h e p - ph ] A p r results at low recoil of final state mesons [4–7], other methods (like Light Cone sum rules, see forexample [8–10]) are better suited at larger recoil values. Finally, experimental data can be used asa guide to extrapolate between these two domains.In this paper we are concerned with the extraction of | V qq (cid:48) | matrix elements from B, D → V (cid:96)ν (cid:96) and its related observable
B, D → P P (cid:96)ν (cid:96) decay channel. As it was mentioned above, these exclu-sive decays provide complementary information on CKM matrix elements and a consistency test ofvalues extracted from other exclusive and inclusive channels. Furthermore, a good understandingof the dominant exclusive channels is essential to describe how inclusive decays are built out fromexclusive components.While pseudoscalar mesons are quasistable states, some of them directly detectable by experi-ments, vector mesons are highly unstable resonances, which are reconstructed from their detectabledecay products. From the theoretical point of view, using vector mesons as asymptotic states ofthe S -matrix is an approximation which, in principle, is not justified owing to their very short life-times. A theoretical definition, that is consistent with the experimental one, can be used instead.In this paper we will consider P → P P (cid:96)ν (cid:96) ( P (cid:96) ) transitions, where the P P pseudoscalar pairis produced dominantly from a decay of a single vector meson V → P P . The extraction of thedecay observables associated to P → V (cid:96)ν (cid:96) decays is affected by the contributions of subdominant s and d wave configurations of the P P system [11–17] even if one chooses a narrow window in theirinvariant mass distribution around the V resonance mass. For an example, the extraction of the | V ub | matrix element from B → ππ(cid:96)ν (cid:96) considering the resonances, backgrounds and rescatteringeffects in the ππ system, were studied in Refs. [12–14, 18]. Those authors found that these effectscan bring the determination of | V ub | from four-body semileptonic decays in better agreement withthe value extracted from B → π(cid:96)ν (cid:96) decay. Also, additional kinematical distributions accessible infour-body semileptonic decays, as compared to three-body decays, allows to explore further ob-servables sensitive to new physics [13]. A study of D + → K − π + e + ν e decays that incorporate thestrong interaction dynamics of the Kπ system was recently reported in Ref. [19].Here, we consider the effects of an additional pole contribution P ∗ in the observables associatedto P → P P ∗ → P P (cid:96)ν (cid:96) decays. Although four-body decays of heavy mesons have been consideredbefore including refinements in the treatment of the s -wave and excited resonances in the p -wave offinal state mesons [11–18], the effects of the P ∗ pole has not been considered in the literature. Thispollution can affect the different invariant mass distribution of the P P system and can modify thevalues of CKM matrix elements extracted from P → V (cid:96)ν (cid:96) transitions. Examples of these decays are B → ( Dπ, ππ ) (cid:96)ν or D → ( Kπ, ππ ) (cid:96)ν which are dominated by the ( D ∗ , ρ ) and ( K ∗ , ρ ) resonances,respectively. The presence of additional B ∗ and D ∗ poles can affect the determination of the CKMmatrix elements to a few percent level, which are important for present and future studies. Wepresent the effects of these additional pole contributions in the invariant mass distribution of themeson pair and in the branching fractions, to estimate their effects in the values of the relevantCKM matrix elements. II. FOUR-BODY SEMILEPTONIC DECAYS OF PSEUDOSCALAR MESONS
Let us consider the generic P ( p ) → P ( p ) P ( p ) (cid:96) ( p ) ν (cid:96) ( p ) decay, denoted as P (cid:96) ( P P ), in-duced by the quark level transition q → q (cid:48) (cid:96)ν (cid:96) , with ( p, p i ) the particle four-momenta subject tothe on-shell conditions ( p = M , p i = m i ). At the lowest level, using the local approximation(infinitely heavy W boson), the decay amplitude can be written as M = G F √ V qq (cid:48) H µ (cid:96) µ , (1)where V qq (cid:48) is the quark mixing CKM matrix element, (cid:96) µ is the leptonic V − A charged current, and H µ = (cid:104) P ( p ) P ( p ) | j µ | P ( p ) (cid:105) = V µ − A µ (2)is the hadronic matrix element of the V − A quark current.Following Ref. [20], we can write the most general vector and axial-vector pieces of the hadronicmatrix element as follows: V µ = − HM (cid:15) µνρσ L ν P ρ Q σ , (3) A µ = − iM [ F P µ + GQ µ + RL µ ] . (4)The form factors H, F, G, R depend on the square of the momentum transfer to leptons and on twoadditional independent Lorentz scalars [20, 21]. The hadronic vertices (3,4) depend upon three-independent Lorentz vectors which we chose as P = p + p , Q = p − p , L = p + p = p − p − p .Conservation of energy-momentum implies p = P + L . This choice is useful to fix the set of fiveindependent kinematical variables to describe the four-body decay: ( s = P , s = L , θ P , θ (cid:96) , φ )(see definitions in Refs. [20, 21]). The corresponding limits of integration are given by (for masslessneutrinos m = 0): ( m + m ) ≤ s ≤ ( M − m ) ; m ≤ s ≤ ( M − √ s ) ; 0 ≤ θ P , θ (cid:96) ≤ π and 0 ≤ φ ≤ π [20, 21].One can get the decay rates by integrating over s , the invariant mass distribution of the pairof final state pseudoscalar mesonsΓ( P → P P (cid:96)ν (cid:96) ) = (cid:90) s +12 s − ds d Γ( P → P P (cid:96)ν (cid:96) ) ds . (5)In the case that the invariant mass distribution is fully dominated by a single intermediate resonance R , namely P → R ( → P P ) (cid:96)ν (cid:96) , we can restrict the integration to the region defined by s ± =( m R ± ∆) , where m R is the mass of the resonance, and typically ∆ = Γ R / R , with Γ R itsdecay width. In the case of a very narrow resonance (Γ R → P → P P (cid:96)ν (cid:96) ) = Γ( P → R(cid:96)ν (cid:96) ) × B ( R → P P ) . (6)This result is also a good approximation for wider resonances, provided no other contributions tothe decay amplitude are present. It will be modified, however, by the contribution of additionalpole contributions to the decay amplitude. ( a ) ( b ) ( c ) P P ∗ π P P ∗ π P P P π P FIG. 1: Contributions to the hadronic vertex in P → P π(cid:96) − ¯ ν (cid:96) decays. Double-lines are used for theintermediate vector resonances. The solid dot indicates the hadronic weak vertex. As shown in Figure 1, there are three different contributions to the hadronic vertex of the P → P P (cid:96)ν decay (here we have chosen P = π , for definiteness). We will assume that the twodominant contributions are given by single pole contributions, namely: P → P ∗ π → P π(cid:96)ν (Figure1a) and P → P ∗ (cid:96)ν → P π(cid:96)ν (Figure 1b). Additional resonances in these channels can contributeas well and their contributions can be trivially added to our results; notice that usually the effectsof heavier resonances similar to the one of interest in the P ∗ channel are taken into account andestimated as background in simulations. For simplicity, we will make the reasonable assumptionthat pole contributions are dominated by the exchange of vector meson resonances in Figures 1a,b.The different hadronic vertices that enter in decays of charged and neutral P mesons in Figure1, are related by isospin symmetry which will be assumed as a good approximation. We define thestrong P V P (cid:48) vertex as ig PV P (cid:48) ( p − p (cid:48) ) · (cid:15) ( p V ), using the convention V ( p V , (cid:15) V ) → P ( p ) P (cid:48) ( p (cid:48) ). Theweak matrix elements required for our evaluations corresponding to the two vector pole contribu-tions in Figure 1 are (we use the convention of Ref. [22] for the R (cid:48) → P transition) (cid:104) P ( p ) | j µ | R (cid:48) ( ε, p (cid:48) R ) (cid:105) = − iV (cid:48) m R (cid:48) + m (cid:15) µνρσ ε ν p ρ p σR (cid:48) − m R (cid:48) A (cid:48) q (cid:48) · εq (cid:48) q (cid:48) µ − ( m R (cid:48) + m ) A (cid:48) ε β T βµ ( q (cid:48) ) − A (cid:48) ε · q (cid:48) m R (cid:48) + m ( p R (cid:48) + p ) β T βµ ( q (cid:48) ) (7)and (cid:104) R ( ε ∗ , p R ) | j µ | P ( p ) (cid:105) = 2 iVm R + M (cid:15) µνρσ ε ∗ ν p ρ p σR − m R A q · ε ∗ q q µ − ( m R + M ) A ε ∗ β T βµ ( q ) + A ε ∗ · qm R + M ( p + p R ) β T βµ ( q ) (8)for the P → R transition [22]. The weak current is j µ = ¯ q (cid:48) γ µ (1 − γ ) q ; we have assumed thatthe intermediate resonances R ( R (cid:48) ) are vector mesons, with polarization four-vector ε ∗ µ ( ε µ ), suchthat p R · ε ∗ = p R (cid:48) · ε = 0. The primed form factors for the R (cid:48) → P weak transition depend upon q (cid:48) = ( p R (cid:48) − p ) , while those of P → R depend upon q = ( p − p R ) ; owing to energy-momentumconservation q = q (cid:48) = L . In the above expressions T βµ ( q ) ≡ g βµ − q β q µ /q is the hadronic transversetensor ( q β T βµ = 0).As a concrete example, let us consider the B ( p ) → D ( p ) π ( p ) (cid:96) ( p ) ν (cid:96) ( p ) decay, which weassume to be dominated by the B ∗ and D ∗ pole contributions in the region where the Dπ invariantmass is close to the D ∗ resonance. Using the definitions introduced previously, we can computethe form factors defined in Eqs. (3)-(4) and get the following results: − HM B = 2 i (cid:20) − g BB ∗ π D B ∗ ( p − p ) V (cid:48) m B ∗ + m D + g DD ∗ π D D ∗ ( P ) VM B + m D ∗ (cid:21) , (9) − iFM B = g BB ∗ π D B ∗ ( p − p ) (cid:20) A (cid:48) Xm B ∗ + m D + ( m B ∗ + m D ) A (cid:48) X (cid:21) − g DD ∗ π D D ∗ ( P ) (cid:20) A YM B + m D ∗ + ( M B + m D ∗ ) A P · Qm D ∗ (cid:21) , (10) − iGM B = g BB ∗ π D B ∗ ( p − p ) (cid:20) A (cid:48) Xm B ∗ + m D + ( m B ∗ + m D ) A (cid:48) X (cid:21) + g DD ∗ π D D ∗ ( P ) ( M B + m D ∗ ) A , (11) − iRM B = g BB ∗ π D B ∗ ( p − p ) (cid:20) m B ∗ A (cid:48) XL − ( P + Q ) · LL A (cid:48) Xm B ∗ + m D + ( m B ∗ + m D ) A (cid:48) (cid:18) A − X · LL (cid:19)(cid:21) + g DD ∗ π D D ∗ ( P ) (cid:20) m D ∗ A YL + 2 A YM B + m D ∗ P · LL − ( M B + m D ∗ ) A Y · LL (cid:21) . (12)In the above expressions we have defined the four-vectors X β = X + P β + X − Q β + AL β , Y β = Q β − ( P · Q/m D ∗ ) P β , and the Lorentz scalars X = L · X, Y = L · Y, X ± = A/ ±
1, with A = 1 − p B ∗ /m B ∗ [1 + ( P − Q ) · p B ∗ /p B ∗ ], where p B ∗ = P + L − ( P − Q ) /
2. We have used thenotation D B ∗ ( q ) = q − m B ∗ and D D ∗ ( q ) = q − m D ∗ + im D ∗ Γ D ∗ . It is easy to check that for P → P π(cid:96)ν (cid:96) decays we have to replace B ( ∗ ) → P ( ∗ ) , D ( ∗ ) → P ( P ∗ ) and the corresponding weakform factors and strong coupling constants in the previous expressions. III. DISCUSSIONS ON EXPERIMENTAL OBSERVABLES OF P (cid:96) Several experiments have reported measurements of branching ratios and invariant mass distri-butions of B (cid:96) [24–26] and D (cid:96) [19, 27, 28] decays. The corresponding analysis to extract the CKMmatrix elements from the D, B → V (cid:96)ν (cid:96) observables differ in several ways: ( a ) the form factorsused to model the weak transition, ( b ) the window of the hadronic mass distribution |√ s − m V | chosen to isolate the V vector meson signal, and ( c ) the inclusion of several wave configurationsand resonance contributions in the hadronic system. As an illustration of the second item, thefollowing cuts in the hadronic invariant mass distribution have been used by different experimentsfor B → ππ(cid:96)ν (cid:96) decays: |√ s ππ − m ρ | ≤ Γ ρ [24] ( 2Γ ρ [26]), 0 .
650 GeV ≤ √ s ππ ≤ .
850 GeV [25],and 0 .
60 GeV ≤ √ s ππ ≤ .
00 GeV [29], which prevent a direct comparison of reported valuesfor the branching fractions. In addition, some experiments report values of the combined resultsfrom neutral and charged B meson branching fractions using isospin symmetry. Isospin symmetrybreaking effects should be duly taken into account in analyses when measurements reach the onepercent accuracy. Furthermore, the contributions of the additional pole contribution consideredin this paper become relevant at the few percent level accuracy determinations of CKM matrixelements.In our previous paper [30], we have shown that the effects of the B ∗ pole contribution in B → Dπ(cid:96)ν (cid:96) is negligibly small compared to the D ∗ pole, owing to the very narrow width of the D ∗ resonance which fully dominates the Dπ invariant mass distribution close to the D ∗ mass. Theeffect of the B ∗ pole in the extraction of the ratio R ( D ∗ ) from B (cid:96) ( Dπ ) decays is also negligible[30]. This leads to the interesting question of how large this effect can be for wider resonances andhow it affects the extraction of the CKM matrix elements when using P → V (cid:96)ν decays. Here westudy the effects of the pole diagram of Figure 1(a) in the hadronic invariant mass distribution andthe branching fraction of P (cid:96) decays in the region close to the P resonance and its consequencesfor the extraction of | V qq (cid:48) | .In our calculations we use the following phase convention for pseudoscalar meson states [31]: | π + (cid:105) = − u ¯ d, | π (cid:105) = √ ( u ¯ u − d ¯ d ) , | π − (cid:105) = d ¯ u, | K + (cid:105) = u ¯ s, | K (cid:105) = d ¯ s, | K (cid:105) = − s ¯ d, | K − (cid:105) = s ¯ u, | D + (cid:105) = − c ¯ d, | D (cid:105) = c ¯ u, | D (cid:105) = u ¯ c, | D − (cid:105) = d ¯ c . The convention for B mesons are similar to K mesons under the replacement s → b . With these conventions, isospin symmetry provides thefollowing relations among different couplings: g ρππ = g ρ + π + π = − g ρ π + π − ,g K ∗ Kπ = g K ∗ + K π + = √ g K ∗ + K + π = − g K ∗ K + π − = √ g K ∗ K π ,g D ∗ Dπ = g D ∗ + D π + = −√ g D ∗ + D + π = − g D − ¯ D ∗ π − = −√ g ¯ D D ∗ π ,g B ∗ Bπ = − g B − ¯ B ∗ π − = −√ g ¯ B B ∗ π . (13)For our numerical evaluations we will use the values g ρππ = 5 . ± . g K ∗ Kπ = 3 . ± .
03 and g D ∗ Dπ = 8 . ± .
08 extracted from the experimental widths of resonances [3], and g B ∗ Bπ = 20 . ± . i ) for the B → ρ transitions we rely onLattice calculations of Ref. [33]; ( ii ) the form factors for semileptonic decays of charmed mesons D → K ∗ and D → ρ are taken from experimental data of Refs. [19] and [28], respectively; ( iii ) forthe evaluation of the P ∗ → P form factors, we use the relativistic harmonic oscillator potentialmodel of Refs. [34, 35] (WSB). In this model, the q dependence of all the form factors are assumedto have a monopolar form: F i ( q ) = F i (0)1 − q /m i ( J P ) . (14)The form factors at q = 0 are computed from the overlap of relativistic wave functions in thismodel; the values of pole masses are chosen to correspond to the lightest resonances with appro-priate quantum numbers that allows coupling to the weak currents.In the case of B ∗ → D and B ∗ → π transitions, the form factors at q = 0 have been evaluatedin Ref. [22] using this model. We have checked these values of form factors at q = 0 and haveevaluated within the same model, the form factors corresponding to D ∗ → K and D ∗ → π weaktransitions. The results for the different form factors and the values of pole masses used in ourevaluations are shown in Table I. As long as the P ∗ pole contribution to the decay amplitude issubleading, we should take the numerical contribution due to the P ∗ → P form factors as a goodestimate of their true values. Transition A (cid:48) (0) A (cid:48) (0) A (cid:48) (0) V (cid:48) (0) m (0 − ) m (1 − ) m (1 + ) D ∗ → K D ∗ → π B ∗ → π [22] 0.34 0.38 0.29 0.34 5.27 5.32 5.71 B ∗ → D [22] 0.63 0.66 0.56 0.70 6.30 6.34 6.73TABLE I: Form factors of the weak transition P ∗ → P at q = 0 in the Wirbel-Stech-Bauer model [34, 35].Values of pole masses are given in GeV units. A. Hadronic invariant mass distributions d Γ / d s s [GeV ] ρ + ρ + -B * d Γ / d s s [GeV ] ρ ρ -B * FIG. 2: Invariant-mass distribution of ππ in B → ππ(cid:96) − ¯ ν ( (cid:96) = eµ ) decays. Left (right) panel is for decays ofneutral (charged) B mesons. The solid (dotted) lines describe the dominant ρ ( ρ + B ∗ ) pole contributionsto the hadronic vertex. d Γ / d s s [GeV ] ρ + ρ + -B * d Γ / d s s [GeV ] ρ ρ -B * FIG. 3: Same as Figure 2 for B → ππτ − ¯ ν decays. In Figures 2, 3, 4 and 5 we plot the hadronic invariant mass distributions of B → ππ(cid:96)ν (cid:96) and D → ( Kπ, ππ ) (cid:96)ν (cid:96) decays and compare the single dominant resonance contribution (solid line) withthe full calculation including both poles (dotted line). In the case of B meson decays, we haveplotted separately these distributions for light and heavy τ leptons given the interest for a test oflepton universality. We do not show the corresponding plots for B → Dπ(cid:96)ν (cid:96) decays because theeffect of the additional pole in that case is indistinguishable.A comparison of the left and right panels in each of Figures 2-5 shows important isospin breakingeffects: the full contributions shift the peak of the distributions to the left (right) of the singledominant pole contribution for decays of neutral (charged) mesons. The origin of this asymmetrylies in the relative signs and different isospin factors for couplings of charged and neutral resonancescoupled to two pseudoscalar mesons. A fit to the P P invariant mass distribution, aiming to extractthe resonance parameters of the P ∗ intermediate state in semileptonic decays, should take intoaccount the two pole contributions. The P ∗ pole contribution in this case, will play the role ofa non-resonant background. A visual inspection of the plots in Figures 2-5 indicates that the P ∗ pole will increase (decrease) the mass of the P ∗ resonance when extracted from neutral (charged)heavy pseudoscalar meson decays with respect to the case where the contribution of Figure 1 isneglected. d Γ / d s s [GeV ] K *+ K *+ -D * d Γ / d s s [GeV ] K *0 K *0 -D * FIG. 4: Invariant-mass distribution of Kπ in D → Kπ(cid:96) − ¯ ν decays. Left (right) panel for decays of ¯ D ( D − )meson. The solid (dotted) lines represent the K ∗ (892) ( K ∗ + D ∗ ) pole contribution. d Γ / d s s [GeV ] ρ + ρ + -D * d Γ / d s s [GeV ] ρ ρ -D * FIG. 5: Same as Figure 4 for D → ππ(cid:96) − ¯ ν decays. B. Branching fractions
We can compute the integrated rates of P → P P (cid:96)ν (cid:96) decays by integrating the hadronicinvariant-mass distributions as shown in Eq. (5). We restrict this integration to the region closeto the mass of the dominant vector resonance R → P P , namely s ± = ( m R ± Γ R / . Our resultsare shown in Table II. We can identify the resulting decay rate with Γ( P → P P (cid:96)ν (cid:96) ) = Γ( P → V (cid:96)ν (cid:96) ) × B ( V → P P ) only in the case that the contribution of Figure 1(a) is neglected (secondcolumn in Table II). When the contribution of diagram in Figure 1(a) is included, the correctformula necessary to extract the branching fraction of the semileptonic P → V transition is: B ( P → V (cid:96)ν (cid:96) ) = τ P · Γ( P → P P (cid:96)ν (cid:96) ) B ( V → P P ) · (1 + δ P ∗ ) , (15)where τ P is the lifetime of the decaying particle and δ P ∗ is the small correction due to subdominantpole contribution.The numerical value of δ P ∗ is obtained from the ratio of the fourth/third columns in Table II (seelast column); this correction can be as large as 15% for B − → π + π − τ − ¯ ν τ decays. Since the effect Channel Γ( P ) Γ( P ∗ + P ∗ ) δ P ∗ D − → K + π − (cid:96) − ¯ ν (cid:96) . ± . . ± .
2) (1.3 %)¯ D → K + π (cid:96) − ¯ ν (cid:96) . ± . . ± .
6) (1.7 %) D − → π + π − (cid:96) − ¯ ν (cid:96) . ± .
16 10.5 %(1.34) (1 . ± .
16) (8.2 %)¯ D → π + π (cid:96) − ¯ ν (cid:96) . ± .
29 2.7 %(2.64) (2 . ± .
29) (1.9 %) B − → π + π − (cid:96) − ¯ ν (cid:96) . ± .
34 9.5 %(9.23) (9 . ± .
33) (6.4 %) B − → π + π − τ − ¯ ν τ . ± .
44 15.2%(4.93) (5 . ± .
43) (10.2 %) B → π + π (cid:96) − ¯ ν (cid:96) . ± .
70 1.8 %(18.59) (18 . ± .
70) (1.7 %) B → π + π τ − ¯ ν τ . ± .
90 3.2 %(9.93) (10 . ± .
90) (2.7 %) B → D + π (cid:96) − ¯ ν (cid:96) . ± . B → D + π τ − ¯ ν τ . ± . − (10 − ) GeV for D ( B ) meson decays. The dominant (full)pole contribution is shown in the second (third) column. The quoted uncertainties arise from uncertaintiesin form factor inputs. Within parenthesis we have indicated the results obtained in the narrow widthapproximation (see text), except for the last two rows that does not change. Of course, this is true in the case that experiments have removed the contributions of excited resonances in the P P system or that they are well separated from the dominant resonance region. P ∗ pole is to increase the decay rates compared to the cases where it is neglected,the values extracted for | V qq (cid:48) | will be decreased by δ P ∗ / P → V (cid:96)ν .For comparison, we also show in Table II the results obtained in the narrow width approximationfor the dominant resonant contribution (figures within parenthesis). We have implemented thislimit by replacing the propagator of the P ∗ resonance as follows:1 (cid:12)(cid:12) s − m R + im R Γ R (cid:12)(cid:12) → πm R Γ R δ ( s − m R ) . (16)Using this approximation in the integrand of Eq. (5), the integration over the five-dimensionalphase-space, reduces to an integration over four dimensions. As it can be observed, the corre-sponding results change only slightly compared to the ones obtained by integrating over the finiterange ( m R − Γ R / ≤ s ≤ ( m R + Γ R / , except for B → π + π − (cid:96)ν (cid:96) decays, where the largestvariations are obtained. IV. EFFECTS ON THE EVALUATION OF CKM MATRIX ELEMENTS
As discussed in Refs. [12–14], the value of | V ub | is increased if one uses the four-body B → ( ππ ) ρ (cid:96)ν (cid:96) ( B (cid:96) ) decays , instead of the corresponding three-body B → ρ(cid:96)ν (cid:96) decay in itsdetermination. This happens owing to the dynamics of the strong interactions manifested asrescattering effects and orbital angular configurations of the ππ system different from L = 1. Inthis section we consider the additional modification of the | V qq (cid:48) | mixing owing to strong interactionsin the initial state of P (cid:96) decays described in this paper. As a general trend, those effects tends todecrease the value of the CKM matrix element extracted from P (cid:96) decays.As an illustrative example let us estimate the effect on the extraction of | V ub | due to the ad-ditional pole contribution in the case of charmless B → ( ππ ) ρ (cid:96)ν decays as measured by the Bellecollaboration in Ref. [26]. A rigorous procedure should include a fit to the measured q = s distribution in order to determine the free constants of a given form factor model and then extractthe value of | V ub | from the measured branching fraction. Instead, we estimate the effect of the B ∗ pole contribution using Eq. (15), which is equivalent to the formula given in Ref. [26] in theabsence of the δ P ∗ term: | V ub | = (cid:115) B ( B → ( ππ ) ρ (cid:96)ν (cid:96) ) τ B · ∆ ζ · (1 + δ P ∗ ) B ( ρ → ππ ) . (17)Here, B ( B → ( ππ ) ρ (cid:96)ν (cid:96) ) is the measured branching fraction and ∆ ζ = (cid:82) d Γ / | V ub | the normalized(to the squared | V ub | quark mixing matrix element) rate integrated over the s ± = ( m ρ ± ρ ) window, and τ B is the B meson lifetime. The quantity ∆ ζ depends of the model used to describethe form factors of the B → ρ transition. Using the model of Ref. [33], as done by Belle in The notation ( P P ) V means that the invariant mass of the pair of pseudoscalar mesons is taken in a small windowaround the V meson mass. ζ = (13 . ± .
9) ps − for the range |√ s − m ρ | < ρ . As a checkof our calculation, using the narrow width approximation and the value of | V ub | as in Ref. [26], wereproduce the value ∆ ζ = (16 . ± .
5) ps − as reported in that reference for the form factor modelof [33].Using the branching fractions B ( B − → ( ππ ) ρ (cid:96) − ¯ ν (cid:96) ) = (1 . ± . ± . × − and B ( ¯ B → ( ππ ) ρ + (cid:96) − ¯ ν (cid:96) ) = (3 . ± . ± . × − as reported in [26] for |√ s − m ρ | < ρ , and using δ B ∗ = 2 . . B ∗ pole correction in the same range of the invariant mass of π + π − ( π + π )system, we get | V ub | = (cid:40) (3 . ± . × − from B − decay(3 . ± . × − from ¯ B decay , (18)where the B ∗ pole effects mainly affects the decays of the charged B meson. The weighted averageof the above results is | V ub | = (3 . ± . × − , which is closer to the determination obtainedfrom B → π(cid:96)ν decays | V ub | = (3 . ± . × − as reported by the PDG [3]. Let us mention thatusing the narrow width approximation as in Ref. [26], the effect becomes smaller; using the sameinput data, and the corresponding values of the B ∗ pole correction, δ B ∗ = 1 . . | V ub | = (3 . ± . × − for the average from B − and ¯ B decays.The effect of the additional pole considered in this paper will be also non-negligible for improvedmeasurements of Cabibbo-suppressed D → ππ(cid:96) − ¯ ν (cid:96) decays. The invariant mass distribution of the ππ system measured by CLEO [28] for | m ππ − m ρ | ≤
150 MeV is dominated by the ρ (770) resonance.By assuming a monopolar form of the different form factors and assuming | V cd | = 0 . ± . V (0) = 0 . ± . +0 . − . , A (0) = 0 . ± . +0 . − . and A (0) = 0 . ± . ± .
04 were derived from the measured branching fractions and invariantmass distributions [28].In order to estimate the effect of the D ∗ pole contribution in the determination of | V cd | we canuse the same form factors and branching fractions measured in [28] using Eq. (17). Since Ref. [28]uses a different resonant shape of the ππ invariant mass than ours, for the purposes of estimatingthe effect of the D ∗ pole contribution we will use our results in the narrow width approximation(in this case ∆ ζ = (4 . ± . × s − ) and the values of δ D ∗ are given in Table II. By includingthe ρ and D ∗ poles, we obtain | V cd | ρ + D ∗ = 0 . ± . , (19)from the average of D − and D semileptonic decays. For comparison, the value obtained byincluding only the ρ meson resonance is | V cd | = 0 . ± . D ∗ pole shifts downwards the value of this CKM matrix element by 2.7%. Although this effect issmall compared to the statistical uncertainty of current measurements, it will become relevant inanalyses of improved measurements in the future.Let us emphasize that the aim of our evaluations is to estimate the shift produced by theadditional pole contribution in the determination of | V ub | and | V cd | mixing matrix elements. More2refined analysis which includes the effects of the s -wave ππ system and more precise measurementsof the D ( B ) → ππ(cid:96) − ¯ ν (cid:96) branching fractions would allow to assess correctly the size of the additional D ∗ ( B ∗ ) pole contribution. Conversely, by using precise measurements of P (cid:96) observables combinedwith the most reliable determinations of the quark mixing elements would allow to test the formfactor models describing the dominant B → V weak transition as well as to understand someunderlying dynamics of the hadronic system. V. SUMMARY AND CONCLUSIONS
Precise determinations of the b -quark mixing matrix elements are necessary to find possiblesources of CP violation beyond the CKM mechanism. One way to reach this goal is to combinemixing values extracted from different decays of b -flavored hadrons. Solving current discrepanciesbetween the most precise determinations of | V qb | ( q = u, c ) from exclusive and inclusive channels,combined with more precise calculations of form factors and refined measurements at b -factorieswill provide a consistency test of the SM and look for possible effects of new physics. In this paperwe have studied the P → P P (cid:96)ν (cid:96) semileptonic decays of B and D mesons by modelling the weakhadronic matrix element of the P → P P transition with two poles contributions. The well known P → V ( → P P ) (cid:96)ν (cid:96) , with V a resonant vector meson pole, gives the dominant contribution forinvariant masses of the P P within a small window around the V meson mass. A subleadingtree-level additional pole contribution is identified which becomes relevant for decay observablesat the few-percent level.We have considered the effects of the subleading pole contribution in the invariant mass dis-tribution of the meson pair in the D → ( Kπ, ππ ) (cid:96)ν (cid:96) and B → ( Dπ, ππ ) (cid:96)ν (cid:96) semileptonic decays.This correction shifts the invariant mass distribution differently for decays of charged and neutralheavy mesons owing mainly to different isospin factors of the strong vertex involved in each case.We have also evaluated the correction in the branching fractions of these four-body decays inducedby this subleading pole. We have illustrated how these corrections affects the determination ofthe | V ub | matrix element extracted by Belle [26] from B → ππ(cid:96)ν (cid:96) decays, using a window of ± ρ around the ρ peak in the ππ invariant mass. The shift in the value of | V ub | is not significant com-pared to current experimental uncertainties, althought it becomes in better agreement with thedetermination based on B → π(cid:96)ν (cid:96) decays. Similar considerations can be applied to the extractionof CKM matrix elements from other four-body decays of B and D mesons. Analysis of improveddata expected in future measurements of these semileptonic decays must consider the effect of theadditional pole contributions discussed in this paper. Acknowledgements
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