EPJ Web of Conferences 202, 06014 (2019) https://doi.org/10.1051/epjconf/201920206014 CHARM 2018 Parametrizations of three-body hadronic B- and D-decay amplitudes Benoît Loiseau1,∗ , Diogo Boito2 , Jean-Pierre Dedonder1 , Bruno El-Bennich3 , Rafel Escribano4 , Robert Kamiński5 , and Leonard Leśniak5 1 Sorbonne Université, Campus Pierre et Marie Curie, Sorbonne Paris Cité, Université Paris Diderot, et IN2P3-CNRS, LPNHE, Groupe Phénoménologie, Paris, France 2 Instituto de Física de São Carlos, Universidade de São Paulo, SP, Brazil 3 Laboratório de Física Teórica e Computacional, Universidade Cruzeiro do Sul, São Paulo, SP, Brazil 4 Grup de Física Teòrica and IFAE, Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain 5 H. Niewodniczański Institute of Nuclear Physics, PAN, Kraków, Poland Abstract. A short review of our recent work on amplitude parametrizations of three-body hadronic weak B and D decays is presented. The final states are here composed of three light mesons, namely the various charge πππ, Kππ and KK K̄ states. These parametrizations are derived from previous calculations based on a quasi-two-body factorization approach where the two-body hadronic final state interactions are fully taken into account in terms of unitary S - and P- wave ππ, πK and K K̄ form factors. They are an alternative to the isobar-model description and can be useful in the interpretation of CP asymmetries. 1 Introduction 1.1 Motivations: why study three-body hadronic B and D decays? Three-body hadronic B and D decays provide a rich tool to study not only the Standard Model, QCD, CP violation [1] but also hadron physics. The hadron physics, often characterized by two-body resonances and their interferences, affect weak observables and any reliable determination of the later will require a good knowledge of the final state meson-meson interactions. This can be realized by introducing theoretical constraints such as unitarity, analyticity, chiral symmetry and the use of data from reactions other than B and D decays. Basic Dalitz-plot analyzes rely on sums of relativistic Breit-Wigner amplitudes representing the different possible implied resonances to which some non resonant background amplitude is added. The S -wave resonance contributions are often difficult to fit. Can one go beyond this isobar model approach? One can replace the sums of relativistic Breit-Wigner components by parametrizations [2] in terms of unitary two-meson form factors keeping the weak-interaction dynamics govern- ing the flavor-changing process via W-meson exchange. These parametrizations are based on published results and motivated by analyzes of high-statistics present and forthcoming data at BES III, LHCb, Belle II, Super c-tau factory .... Up to now there is no three-body decay factorization theorem but major contributions arise from intermediate resonances such ∗ e-mail: loiseau@lpnhe.in2p3.fr © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). EPJ Web of Conferences 202, 06014 (2019) https://doi.org/10.1051/epjconf/201920206014 CHARM 2018 as ρ(770), K ∗ (892), φ(1020) which allows to describe three-body decays as quasi-two-body ones. For instance, for the three-meson final state of the D0 → KS0 π− π+ decay, one can in- troduce quasi-two-body pairs, [KS0 π+ ]L π− , [KS0 π− ]L π+ , KS0 [π+ π− ]L , two of the three mesons forming a state of angular momentum 0 or 1 with L = S or P, respectively. 1.2 QCD quasi-two-body factorization Decays are mediated by local four-quark operators Oi (µ) forming the weak effective non- renormalizable Hamiltonian Heff . Schematically for B → M1 M2∗ (M2∗ → M3 M4 ) one has GF X hM1 M2∗ |Heff |Bi = √ VCKM Ci (µ)hM1 M2∗ |Oi (µ)|Bi, (1) 2 i where G F is the Fermi decay constant, VCKM the product of Cabibbo-Kobayashi-Maskawa matrix elements and Ci (µ) Wilson coefficients renormalized at scale µ ∼ mb (or mc in D decays). In the factorization approach [3] with the strong coupling α s (µ), i.e. at scale µ, hM1 M2∗ |Oi (µ)|Bi = hM1 |J1ν |BihM2∗ |J2ν |0i  ΛQCD   ! X ν   +hM1 |J3 |0ihM2 |J4ν |Bi 1 + ∗   rn α s (µ) + O n  , (2) n m b where rn are strong interaction constant factors and |0i the vacuum state. For the leading order the factorization takes place with either weak quark currents J1 , J2 or J3 , J4 . The radiative corrections can be evaluatd to a given order αns (µ). The nonperturbative corrections ΛQCD  to the heavy-quark limit O mb are less reliable for D decays as mc ∼ mb /3; therefore, even though the factorization is more phenomenological for charmed mesons, it can still represent a good starting point. The amplitude hM1 |J1ν |Bi (= hM1 B|J1ν |0i) is a heavy-to-light transition form factor which can be evaluated within light-front and relativistic constituent quark models, light-cone sum rules, continuum functional QCD and lattice QCD (see Appendix A4 of Ref. [2]). Semi- leptonic decay measurements like D0 → π− e+ νe can also allow a phenomenological determi- nation of these form factors. The matrix element hM2∗ |J2ν |0i ∝ hM3 M4 |J2ν |0i, where the M3 M4 resonance, M2∗ , origi- nates from a q̄q pair, corresponds to the M3 M4 form factor. It has been shown, in Ref. [4], that, using dispersion relations and field theory, this form factor can be fully determined, if the M3 M4 strong interaction is known at all energies. These form factors are calculated from Muskhelishvili-Omnès equations [5] using two-body data, unitarity, asymptotic QCD and chiral symmetry constraints at low energies. The term hM1 |J3ν |0i, related to the M1 weak decay constant, is known from experiment, e.g. the pion decay constant, fπ or that of the kaon, fK . It can also be evaluated with lattice- regularized QCD and other nonperturbative approaches. The matrix element hM2∗ |J4ν |Bi ∝ hM3 M4 |J4ν |Bi corresponding to B meson transitions to two-meson pairs via the M2∗ resonance is the biggest uncertainty in our approach. It could be evaluated from semi-leptonic processes: like B0 → K + π− µ+ µ− or D0 → K − π+ µ+ µ− . In the derivation of the amplitude presented here it will be related to the hM2∗ [→ M3 M4 ]|J2ν |0i form factor. Within the soft-collinear effective theory, the amplitude can be factorized in terms of generalized B-to-two-body form factor and two-hadron light-cone distribution amplitude [6]. 2 EPJ Web of Conferences 202, 06014 (2019) https://doi.org/10.1051/epjconf/201920206014 CHARM 2018 1.3 Application to the D+ → K − π+ π+ decay In this process, studied in Ref. [7], the final state π+ π+ interaction can be neglected and the quasi-two-body [K − π+ ]S ,P π+ can be introduced. There is no penguin contribution (loop with W meson) and only the effective Wilson-coefficients a1(2) appear in the quasi-two-body factorized amplitude, GF h[K − π+ ]S ,P π+ |Heff |D+ i = √ cos2 θC a1 h[K − π+1 ]S ,P | s̄γν (1 − γ5 )c|D+ ihπ+2 |ūγν (1 − γ5 )d|0i h 2 + a2 h[K π1 ]S ,P | s̄γν (1 − γ5 )d|0ihπ+2 |ūγν (1 − γ5 )c|D+ i + (π+1 ↔ π+2 ), − + i (3) θC being the Cabbibo angle. The matrix element h[K − π+1 ]S ,P | s̄γν (1 − γ5 )d|0i is given by the Kπ form factors. The term h[K − π+1 ]S ,P | s̄γν (1 − γ5 )c|D+ i is less straightforward to evaluate. Assuming a dominant intermediate resonance R, it can be written as being proportional to the D to R [R → Kπ] transition form factor multiplied by the Kπ form factors. This description is a feature of crucial importance to our proposed parametrizations. In Eq. (3), hπ+2 (p)|ūγν (1 − γ5 )d|0i = −i fπ pν and hπ+2 |ūγν (1 − γ5 )c|D+ i is the Dπ transition form factor. Parametrized amplitudes based on quasi-two-body factorization have been given in Ref. [2] in terms of analytic and unitary meson-meson form factors for final states com- posed of three light mesons, namely the various charge πππ, Kππ and KK K̄ states. For these hadronic three-body decays we have shown, in previous studies, that this approach is phenomenologically successful. Below, we illustrate these parametrizations for the B → Kπ+ π− [8–10] and D0 → KS0 K + K − [11] for meson-meson final states in S wave. Formulae for meson-meson final states in P wave are given in Ref. [2]. 2 Parametrized amplitudes for the B → Kπ+ π− decays 2.1 Parametrization of the B → K[π± π∓ ]S amplitudes Let us label the momenta as B(pB ) → K(p1 )π+ (p2 )π− (p3 ) with s12 = (p1 + p2 )2 , s13 = (p1 + p3 )2 , s23 = (p2 + p3 )2 and s12 + s13 + s23 = m2B + m2K + 2m2π . As can be seen from Eq. (1) of Ref. [8] the B → K[π+ π− ]S amplitude can be parametrized in terms of three complex parameters, bSi , i = 1, 2, 3, for the different charged states B = B± , K = K ± and B = B0 ( B̄0 ), K = K 0 (K̄ 0 ) or KS0 . For the B− decays one has AS (s23 ) ≡ hK − [π+ π− ]S |Heff |B− i = bS1 MB2 − s23 F0n ππ ππ     (s23 ) + bS2 F0BK (s23 ) + bS3 F0s (s23 ), (4) where F0BK (s) is the B to K transition form factor (see Refs. [2, 8]). The non-strange scalar ππ form factor F0n (s) contains the contributions of f0 (500), f0 (980) and f0 (1400). Several models are compared in Fig. 8 of Ref. [12]. Although there are large differences, it has been checked by the authors that, with the fitted form factor to obtain the lowest χ2 for D0 → KS0 π+ π− , the main conclusions achieved for the B± → π+ π− π± in Ref. [13] were un- changed (see Ref. [12] for explanations). The modulus of the Moussallam pion scalar form factor [14], calculated by solving the Muskhelishvili-Omnès equation [5], is close to that of the form factor obtained in Ref. [12], notably below 1 GeV. A plot of the strange scalar form ππ factor F0s (s), which receives the contribution of the f0 (980) and f0 (1400), can be found in Fig. 6 of Ref. [15]. It has been calculated using the Muskhelishvili-Omnès approach. 3 EPJ Web of Conferences 202, 06014 (2019) https://doi.org/10.1051/epjconf/201920206014 CHARM 2018 In terms of the original amplitude [8] one has1 , F0B→ f0 (980) (m2K ) being the B to f0 (980) transition form factor evaluated at m2K [2], GF h i bS1 = √ χ fK F0B→ f0 (980) (m2K ) U − C̃ , (5) 2   where C̃ = fπ Fπ λu PGI 1 +λt P1 , λu = Vub Vus , λt = Vtb Vts , F π is the Bπ form factor at mπ = 0, M ∗ ∗ 2 GI M P1 , P1 complex charming penguin parameters and U is a short-distance contribution given in terms of CKM matrix element multiplied by effective Wilson coefficients. The fitted pa- rameter χ represents the strength of the non-strange pion form factor contribution, further- more. Its value can be estimated from the f0 (980) decay properties [8]. A summary of the models for the scalar-isoscalar pion form factor can be found in Appendix A4 of Ref. [2] and, as just noted above, see also the recent determination of Ref. [15] and the review talk [16]. 2.2 Parametrization of the B → [Kπ± ]S π∓ amplitudes In terms of the two complex parameters cS1 , cS2 (see Eq. (68) of Ref. [10]) one has  F0Bπ (s12 )F0Kπ (s12 ) AS (s12 ) ≡ hπ− [K − π+ ]S |Heff |B− i = cS1 + cS2 s12 , (6) s12 where F0Kπ (s) (contribution of K0∗ (800) or κ and of K0∗ (1430), see e.g. Fig. 7 of Ref. [12]) and F0Bπ (s) are the Kπ and Bπ scalar form factors, respectively. This parametrization has been used with success in the amplitude analysis [17] of the Dalitz-plot distribution of the LHCb B̄ → KS0 π+ π− data. One has [10] au10 (S ) ac10 (S ) " ! !# GF cS1 = √ (MB − mπ )(mK − mπ ) λu a4 (S ) − 2 2 2 2 u + c4 + λc a4 (S ) − u c + c4 , (7) c 2 2 2 where λc = Vcb Vcs ∗ i (S ), i = 4, 10 are the leading order effective Wilson coefficients . The au(c) including vertex and penguin corrections. The cu(c) 4 are free fitted parameters simulating non- perturbative and higher order contributions to the penguin diagrams. Models for the F0Kπ (s) form factor are described in Ref. [2], see also some complementary aspects in Ref. [16]. 3 D0 → KS0 [K + K − ]S and D0 → [KS0 K ± ]S K ∓ parametrized amplitudes The [K + K − ] pairs can have isospin 0 or 1 but the [KS0 K ± ] ones have isospin 1. The f0 (980), f0 (1400), a0 (980)0 and a0 (1450)0 contribute to the following parametrized amplitude       A0S ,0 (s23 ) = hS1 m2D0 − s23 F0n K K̄ (s23 ) + hS2 m2K 0 − s23 F0s K K̄ (s23 ) + hS3 m2D0 − s23 G0K K̄ (s23 ). (8) where s23 is the energy squared of the K + K − pair while s12 is associated to the KS0 K − pair and s13 to the KS0 K + one. The decay amplitude associated with the a0 (980)− and a0 (1450)− resonances, can be parametrized as:   A0S ,− (s12 ) = hS4 + hS5 s12 G0K K̄ (s12 ). (9) The amplitude carrying contributions from a0 (980)+ and a0 (1450)+ ] reads 1 The interested reader will find, in Appendix B of Ref. [2], the corresponding relations for the other parameters. 4 EPJ Web of Conferences 202, 06014 (2019) https://doi.org/10.1051/epjconf/201920206014 CHARM 2018  S F0DK (s13 )   A0S ,+ (s13 ) = h6 + h7 (mK − s13 ) G0K K̄ (s13 ). S 2  (10) s13 K K̄ Models for the F0n(s) (s) form factors entering Eq. (8) have been derived in Ref. [18, 19] (see their Figs. 1) solving three coupled channels viz. ππ, K K̄ and 4π (effective 2π-2π or σσ or ρρ ...) and imposing chiral symmetry constraints. The F0s K K̄ (s) form factor has also been calculated in a dispersive approach in Ref. [15] (see their Fig. 7). In Eqs. (9) and (10), the scalar-isovector G0K K̄ (s) form factor, built in Ref. [20] from a unitary S -wave coupled channel (ηπ, K K̄) model, is plotted in their Fig. 7. This model, derived from the Muskhelishvili-Omnès equation [5], imposes the presence of the a0 (980) and a0 (1450) and includes asymptotic QCD and chiral symmetry constraints. Models for the transition form factor F0DK (s) in Eq. (10) can be found in Ref. [2]. The above complex hSi coefficients are given in terms of the original amplitudes in Appendix B of Ref. [2]. 4 Concluding remarks Alternatives to isobar Dalitz-plot model for weak D, B decays into various πππ, Kππ and KK K̄ charge states have been presented in Ref. [2]. Let us recall that isobar parametrizations do not respect unitarity and extraction of strong CP phases should be taken with caution. Furthermore S -wave resonance contributions are hard to fit. Our parametrizations, although not fully three-body unitary, are based on a sound theo- retical application of QCD factorization to a hadronic quasi-two-body decay. They assume that final three-meson state are preceded by intermediate resonant states which is justified by phenomenological and experimental evidence. Analyticity, unitarity, chiral symmetry plus correct asymptotic behavior of the two-meson scattering amplitude in S and P waves are im- plemented via analytical and unitary S - and P-wave ππ, πK and K K̄ form factors entering in hadronic final states of our amplitude parametrizations. These parametrized amplitudes can be readily used adjusting parameters in a least-square fit to the Dalitz plot for a given decay channel and employing tabulated form factors as func- tions of momentum squared or energy. The reproduction of the Dalitz-plot data might require some adjustment of the meson-meson form factors. The addition of phenomenological am- plitudes (contributions of higher interacting waves, in particular D waves or J=2 resonances), and possible three-body rescattering effects may be necessary. We have exemplified here expressions for the B → Kπ+ π− [8–10] and D0 → 0 + − KS K K [11] for meson-meson final states in S wave. In Ref. [2] one can find other explicit amplitude expressions for meson-meson final states in S and P wave for B± → π+ π− π± , B → K π+ π− , B± → K + K − K ± , D+ → π− π+ π+ , D+ → K − π+ π+ , D0 → KS0 π+ π− . Previous studies have shown that this approach is successful. In addition, expressions for D0 → KS0 K + K − are also given in Ref. [2]. We have derived preliminary parametrized am- plitudes for the B± → K + K − π± decays [1, 21] and for the B0 → KS0 K + K − process presently analyzed by the LHCb collaboration. B.L. thanks A. Bondar and S. Eidelman for their kind invitation to present this contribution to this CHARM2018 international workshop. B.E. and B.L. are grateful to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and its partial support during the completion of this work. References [1] R. Aaij et al. [LHCb Collaboration], Measurements of CP violation in the three-body phase space of charmless B± decays, Phys. Rev. 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