S-, P- and D-wave final state interactions and CP violation in B+- --> pi+- pi-+ pi+- decays
J.-P. Dedonder, A. Furman, R. Kaminski, L. Lesniak, B. Loiseau
aa r X i v : . [ h e p - ph ] S e p S -, P - and D -wave ππ final state interactions and CP violation in B ± → π ± π ∓ π ± decays J.-P. Dedonder and B. Loiseau
Laboratoire de Physique Nucl´eaire et de Hautes ´Energies, Groupe Th´eorie,Universit´e Pierre et Marie Curie et Universit´e Paris-Diderot, IN2P3 & CNRS,4 place Jussieu, 75252 Paris, France andA. Furman ul. Bronowicka 85/26, 30-091 Krak´ow, Poland andR. Kami´nski and L. Le´sniak
Division of Theoretical Physics, The Henryk Niewodnicza´nski Institute of NuclearPhysics, Polish Academy of Sciences, 31-342 Krak´ow, PolandWe study CP violation and the contribution of the strong pion-pioninteractions in the three-body B ± → π ± π ∓ π ± decays within a quasi two-body QCD factorization approach. The short distance interaction ampli-tude is calculated in the next-to-leading order in the strong coupling con-stant with vertex and penguin corrections. The meson-meson final stateinteractions are described by pion non-strange scalar and vector form fac-tors for the S and P waves and by a relativistic Breit-Wigner formula forthe D wave. The pion scalar form factor is calculated from a unitary rel-ativistic coupled-channel model including ππ , K ¯ K and effective (2 π )(2 π )interactions. The pion vector form factor results from a Belle Collabora-tion analysis of τ − → π − π ν τ data. The recent B ± → π ± π ∓ π ± BABARCollaboration data are fitted with our model using only three parametersfor the S wave, one for the P wave and one for the D wave. We find notonly a sizable contribution of the S wave just above the ππ threshold butalso under the ρ (770) peak a significant interference, mainly between the S and P waves. For the B to f (1270) transition form factor, we predict F Bf ( m π ) = 0 . ± . f (600), f (980) and f (1400) in terms of the pion non-strange scalar form factor.(1) PACS numbers: 13.25.Hw, 13.75Lb
1. Introduction
Three-body charmless hadronic B meson decays offer one of the besttools for studies of direct CP violation and provide an interesting testingground for strong interaction dynamical models. The present work, part ofa program devoted to the understanding of rare three-body B decays [1, 2,3, 4], is motivated by the recent BABAR Dalitz-plot analysis of the B ± → π ± π ∓ π ± decays [5]. In an isobar model description, the authors of Ref. [5]find evidence for the f (1370) but, within the current experimental accuracy,no significant signal for the f (980). The f (600), not explicitly included inthat analysis, could be part of the non-resonant background. Furthermore,there is a small but visible contribution of the f (1270) resonance [5].Here, the aim is to provide a phenomenological analysis of the B ± → π ± π ∓ π ± decay channels relying on the QCD factorization scheme (QCDF)in the ππ effective mass range from threshold to 1.64 GeV. The focus willbe set on the final state ππ interactions involved since a partial wave anal-ysis of the Dalitz plot should use theoretically and phenomenologically wellconstrained ππ amplitudes.Studies of B decays into two-body and quasi-two-body final states havebeen performed in the QCDF framework [6, 7, 8, 9, 10, 11, 12]. The naivefactorization approach is a useful first order approximation which receivescorrections proportional to the strong coupling constant α s at scales m b and p Λ QCD m b and in inverse powers of the b quark mass m b [13]. In thepresent study, we propose an extension of these results to the three-bodydecays B ± → π ± π + π − .The role of the f (600) (or σ ) in charmless three-body decays of B mesons has been examined by Gardner and Meißner [8] in B → π + π − π decays. Within QCD quasi two-body factorization approach their f (600) π amplitude is described by a unitary pion scalar form factor constrained by ππ scattering and chiral dynamics. This is different from the relativisticBreit-Wigner parametrization used in most experimental analyses and insome theoretical studies, for example in [14]. This has led to improvedtheoretical predictions; the contribution of the f (600) π channel has beenfound to be important in the range of the dominant ρ π intermediate state.However, in recent B → π + π − π Dalitz plot analyses [15, 16] no contribu-tion from B → f (600) π channel has been found. This could be linked tothe present limited statistics in the low effective ππ mass region. Further-more, such a contribution could be hidden in the nonresonant amplitudeintroduced in the experimental analysis. Nevertheless we will show that thecontribution of the S wave is important in the B ± → π ± π + π − decays. Charmless three-body decays of B mesons have also been investigatedby Cheng, Chua and Soni [12] in the framework of quasi two-body factoriza-tion approach using resonant and non-resonant contributions. In particularthey have calculated the B − → π + π − π − branching fractions and CP asym-metries and found a small rate for B − → f (980) π − decay.An achievement in the theory of B decays into two mesons is the confir-mation of the validity of factorization as a leading order approximation. Noproof of factorization has yet been given for the B decays into three mesons.However, three-body interactions are suppressed when specific kinematicalconfigurations with the three mesons quasi aligned in the rest frame of the B meson are considered. This is the case in the effective π + π − mass regionsmaller than 1.64 GeV in the Dalitz plot where most of the π + π − resonantstates are visible. Such processes will be denoted as B ± → π ± [ π + π − ], themesons of the [ π + π − ] pair moving more or less, in the same direction in the B rest frame. Then, it seems reasonable to postulate the validity of factor-ization for this quasi two-body B decay [17] assuming that the [ π + π − ] pairoriginates from a quark-antiquark state.In the factorization approach the B ± → π ± [ π +2 π − ] decay amplitudesare expressed as a superposition of appropriate effective QCD coefficientsand two products of two transition matrix elements. The transition ma-trix elements between the B ± meson and the π ± pion multiplied by thetransition matrix elements between the vacuum and the (cid:2) π +2 π − (cid:3) pion paircorrespond to the first of these products. Here, in the π +2 π − center of massframe, the bilinear quark currents involved force the [ π +2 π − ] pair to be in S or in P state. The second term is associated to the product of the tran-sition matrix elements between the B ± meson and the [ π +2 π − ] pion pairin S , P or D state by the transition matrix elements between the vacuumand the π ± pion. The [ π +2 π − ] S,P transition matrix elements to the vacuumare proportional to the pion scalar and vector form factors. We assumethat the B ± → π +2 π − matrix elements are expressed as products of the B ± → [ π +2 π − ] S,P,D transition form factors by the relevant vertex functiondescribing the decay of the [ π +2 π − ] S,P,D state into the final pion pair. Thevertex functions are in turn assumed to be proportional to the pion scalarform factor for the S wave, to the vector form factor for the P wave and toa relativistic Breit-Wigner formula for the D wave. Here, a single unitaryfunction, namely the pion non-strange scalar form factor, describes then thethree scalar resonances, f (600), f (980) and f (1400) present in the π + π − interaction.In Sec. 2 we present the model used in the analysis. Sec. 3 is devoted tothe construction of the pion scalar and vector form factors. The pertinentobservables and the fitting procedure are described in Sec. 4 while the resultsare discussed in Sec. 5. A summary and some perspectives are outlined in the final Sec. 6. The detailed derivation of the decay amplitudes is presentedin the Appendix A while Appendix B gives the system of equations to besolved to obtain the parameters fixing the low-energy behavior of the pionscalar form factor to be that of one loop calculation in chiral perturbationtheory.
2. Decay amplitudes
The amplitudes for the non-leptonic decays of the B meson are given asmatrix elements of the effective weak Hamiltonian [6, 7] H eff = G F √ X p = u,c λ p h C O p + C O p + X i =3 C i O i + C γ O γ + C g O g i + h.c., (1)where λ u = V ub V ∗ ud , λ c = V cb V ∗ cd , (2)the V pp ′ ( p ′ = b, d ) being Cabibbo-Kobayashi-Maskawa quark-mixing ma-trix elements. For the Fermi coupling constant G F we take the value1 . × − GeV − . The C i ( µ ) are the Wilson coefficients of the four-quark operators O i ( µ ) at a renormalization scale µ . The O p , are left-handedcurrent-current operators arising from W -boson exchange, O i =3 − are QCDand electroweak penguin operators involving a loop with a u or c quark and a W boson, O γ and O g are the electromagnetic and chromomagnetic dipoleoperators [7].Let p B be the four-momentum of the B ± meson and p that of theisolated π ± . Let then p denote the four-momentum of the π + and p thatof the π − of the interacting [ π + π − ] pair in the B rest frame. One has p B = p + p + p and we introduce the invariants s ij = ( p i + p j ) for i, j = 1 , , i < j . For the B − → π − [ π + π − ] S,P,D amplitude, wework in the center of mass frame of the π + π − pair of pions with respectivefour-momenta p and p (or p and p for the symmetrized amplitudes).These two pions will be either in a relative S , P or D state. In the followingwe derive the amplitudes for the B − → π − [ π + π − ] S,P,D processes. Thetranscription to the B + → π + [ π + π − ] S,P,D processes is straightforward.Applying the QCD factorization formula for the B − → π − [ π + π − ] S,P,D process, the matrix elements of the effective weak Hamiltonian (1) can bewritten as [7] (cid:10) π − ( p ) [ π + ( p ) π − ( p )] S,P,D | H eff | B − ( p B ) (cid:11) = G F √ X p = u,c λ p (cid:10) π − [ π + π − ] S,P,D | T p | B − (cid:11) , (3) to which must be added the symmetrized term h π − ( p )[ π + ( p ) π − ( p )] S,P,D | H eff | B − ( p B ) i . With M ≡ π − and M ≡ [ π + π − ] S,P or M ≡ [ π + π − ] S,P,D while M ≡ π − , one has (cid:10) π − [ π + π − ] S,P,D | T p | B − (cid:11) = h π − [ π + π − ] S,P,D | n a ( M M ) δ pu (¯ ub ) V − A ⊗ ( ¯ du ) V − A + a ( M M ) δ pu ( ¯ db ) V − A ⊗ (¯ uu ) V − A + a ( M M ) X q ( ¯ db ) V − A ⊗ (¯ qq ) V − A + a p ( M M ) X q (¯ qb ) V − A ⊗ ( ¯ dq ) V − A + a ( M M ) X q ( ¯ db ) V − A ⊗ ( ¯ dq ) V + A + a p ( M M ) X q ( − qb ) sc − ps ⊗ ( ¯ dq ) sc + ps + a ( M M ) X q ( ¯ db ) V − A ⊗ e q (¯ qq ) V + A + a p ( M M ) X q ( − qb ) sc − ps ⊗ e q ( ¯ dq ) sc + ps + a ( M M ) X q ( ¯ db ) V − A ⊗ e q (¯ qq ) V − A + a p ( M M ) X q (¯ qb ) V − A ⊗ e q ( ¯ dq ) V − A o | B − i , (4)where a pj are effective QCDF coefficients.In Eq.(4), (¯ q q ) V ∓ A = ¯ q γ µ (1 ∓ γ ) q , (¯ q q ) sc ± ps = ¯ q (1 ± γ ) q and e q denotes the electric charge of the quark q in units of the elementarycharge e . The sum on the index q runs over u and d and the summationover the color degree of freedom has been performed. The notations sc and ps stand for scalar and pseudoscalar, respectively.At next-to-leading order (NLO) in the strong coupling constant α s , thegeneral expression of the a pi quantities in terms of effective Wilson coefffi-cients is [9] a pi ( M M ) = (cid:18) C i + C i ± N C (cid:19) N i ( M ) + C i ± N C C F α s π [ V i ( M )+ 4 π N C H i ( M M ) (cid:21) + P pi ( M ) , (5)where the upper (lower) signs apply when the index i is odd (even), N C isthe number of colors, N C = 3 and C F = ( N C − / N C . The sums over thecolor degree of freedom have been performed in Eq. (4). Note that in theleading-order (LO) contribution N i ( M ) = 0 for M = [ π + π − ] P and i = 6 , N i ( M ) = 1. The NLO quantities V i ( M ) arise from one loop ver-tex corrections, H i ( M M ) from hard spectator scattering interactions and P pi ( M ) from penguin contractions. Here the meson M is the meson whichdoes not include the spectator quark of the B meson. The superscript p in a pi ( M M ) is to be omitted for i = 1, 2, 3, 5, 7 and 9 since the penguincorrections are equal to zero in these cases. In our calculation we shall notinclude the NLO hard scattering corrections nor the annihilation contribu-tions which require the introduction of four phenomenological parameters toregularize end point divergences related to asymptotic wave functions [9].Although we are aware that such contributions might be important, thiswould bring, at this stage of analysis, too many free parameters.In Eq. (4) the symbol ⊗ indicates that the different components of thematrix elements h π − [ π + π − ] S,P,D | T p | B − i are to be calculated in the factor-ized form, (cid:10) π − ( p )[ π + ( p ) π − ( p )] S,P,D | j ⊗ j | B − ( p B ) (cid:11) ≡ (cid:10) [ π + π − ] S,P,D | j | B − (cid:11) (cid:10) π − | j | (cid:11) or (cid:10) π − | j | B − (cid:11) (cid:10) [ π + π − ] S,P | j | (cid:11) , (6)since we neglect B − annihilation contributions which are expected to besmall [6]. Furthermore, as for the hard scattering corrections, their evalu-ation [9] introduces two phenomenological parameters. In Eq. (6) j and j denote the appropriate quark currents entering in Eq. (4). Note that,in our approach, in the evaluation of the long distance matrix element h [ π + π − ] S,P,D | j | B − i , we make the hypothesis that the transitions of B − to the [ π + π − ] S,P,D states go first through intermediate meson resonances R S,P,D which then decay into a π + π − pair. We describe these decays bya vertex function modeled by assuming them to be proportional to thepion scalar or vector form factors or to a relativistic Breit-Wigner formula,respectively. For the short distance part of the decay amplitudes propor-tional to a combination of the effective coefficients a pi ( M M ) it can beseen that for terms coming from the first line of the right hand side of Eq. (6) M ≡ [ π + π − ] S,P,D , M ≡ π − and for those from the second line M ≡ π − while M ≡ [ π + π − ] S,P , the [ π + π − ] D transition to the vacuumbeing zero with the involved bilinear quark current j in Eq. (6) . In thefollowing, when M ≡ [ π + π − ] S,P , we assume that the NLO corrections V i ( M ) and P pi ( M ) are evaluated at the meson resonances R S,P position.Here we take R P ≡ ρ (770) and R S ≡ f (980). A similar approximationhas been applied in Refs. [3, 4] for the [ Kπ ] S,P states with R P ≡ K ∗ (892)and R S ≡ K ∗ (1430).Introducing the following short distance terms, with L ≡ S, P, D andwith R D ≡ f (1270), u ( R L π − ) = λ u (cid:8) a ( R L π − ) + a u ( R L π − ) + a u ( R L π − ) − (cid:2) a u ( R L π − )+ a u ( R L π − ) (cid:3) r πχ (cid:9) + λ c (cid:8) a c ( R L π − ) + a c ( R L π − ) − (cid:2) a c ( R L π − )+ a c ( R L π − ) (cid:3) r πχ (cid:9) , (7) v ( π − R S ) = λ u (cid:2) − a u ( π − R S ) + a u ( π − R S ) (cid:3) + λ c (cid:2) − a c ( π − R S )+ a c ( π − R S ) (cid:3) , (8)and w ( π − R P ) = λ u (cid:26) a ( π − R P ) − a u ( π − R P ) + 32 (cid:2) a ( π − R P ) + a ( π − R P ) (cid:3) + 12 a u ( π − R P ) (cid:27) + λ c (cid:26) − a c ( π − R P ) + 32 (cid:2) a ( π − R P ) + a ( π − R P ) (cid:3) + 12 a c ( π − R P ) (cid:27) , (9)one obtains, from Eqs. (3), (4) and (6), the following S -, P - and D -wavematrix elements X p = u,c λ p (cid:10) π − ( p )[ π + ( p ) π − ( p )] S | T p | B − (cid:11) = X S u ( R S π − ) + Y S v ( π − R S ) , (10) X p = u,c λ p (cid:10) π − ( p )[ π + ( p ) π − ( p )] P | T p | B − (cid:11) = X P u ( R P π − ) + Y P w ( π − R P ) , (11) X p = u,c λ p (cid:10) π − ( p )[ π + ( p ) π − ( p )] D | T p | B − (cid:11) = X D u ( R D π − ) . (12)In Eq. (7) the chiral factor r πχ is given by r πχ = 2 m π / [( m b + m u )( m u + m d )], m u and m d being the u and d quark masses, respectively. The long distancefunctions X S,P,D and Y S,P , evaluated in Appendix A, read X S ≡ (cid:10) [ π + ( p ) π − ( p )] S | (¯ ub ) V − A | B − (cid:11) (cid:10) π − ( p ) | ( ¯ du ) V − A | (cid:11) = − r χ S f π ( M B − s ) F BR S ( m π ) Γ n ∗ ( s ) , (13) Y S ≡ (cid:10) π − ( p ) | ( ¯ db ) sc − ps | B − (cid:11) (cid:10) [ π + ( p ) π − ( p )] S | ( ¯ dd ) sc + ps | (cid:11) = r B M B − m π m b − m d F Bπ ( s ) Γ n ∗ ( s ) , (14) X P ≡ (cid:10) [ π + ( p ) π − ( p )] P | (¯ ub ) V − A | B − (cid:11) (cid:10) π − ( p ) | ( ¯ du ) V − A | (cid:11) = N P f π f R P ( s − s ) A BR P ( m π ) F ππ ( s ) , (15) Y P ≡ (cid:10) π − ( p ) | ( ¯ db ) V − A | B − (cid:11) (cid:10) [ π + ( p ) π − ( p )] P | (¯ uu ) V − A | (cid:11) = ( s − s ) F Bπ ( s ) F ππ ( s ) , (16) X D ≡ (cid:10) [ π + ( p ) π − ( p )] D | (¯ ub ) V − A | B − (cid:11) (cid:10) π − ( p ) | ( ¯ du ) V − A | (cid:11) = − f π √ F BR D ( m π ) r G f D ( s , s ) m R D − s − im R D Γ( s ) , (17)The different quantities entering the above equations are discussed below.The S -wave strength parameter χ S [Eq. (13)] will be fitted togetherwith the correction P -wave parameter N P [Eq. (15]. The deviation of N P from 1 corresponds to the possible variation of the strength of this P -waveamplitude proportional to f π /f R P [compare Eqs. (A.7) and (A.19)].Three scalar-isoscalar f resonances, viz. f (600), f (980) and f (1400),are present in the ππ effective mass range, m ππ , considered here. Since someof them are wide, like f (600), one could have a possible R S dependencein χ S . The transition form factor from B to R S , F BR S ( m π ), could alsodepend on m ππ . However, one expects these dependences to be weakerthan the effective mass dependence of the pion scalar form factor, Γ n ∗ ( s ),in which all these resonances are incorporated. Therefore we assume that χ S and F BR S ( m π ) are constant. This hypothesis will be assessed by thequality of the fit obtained with our model. We shall take R S ≡ f (980) forthe evaluation of F BR S ( m π ) and we use F BR S ( m π ) = 0 .
13 [19].
For the pion decay constant we take f π = 0 . R P decay constant is denoted by f R P and the B -meson mass by M B . Since the π + π − P -wave is largely dominated by the ρ (770) meson we choose f R P = f ρ = 0 .
209 GeV [9]. The quantity B = − h | ¯ qq | i /f π is proportional tothe quark condensate. We calculate it as B ≃ m π / ( m u + m d ). At therenormalization scale µ = m b / m b = 4 . m u = m d =0 .
005 GeV. For the transition form factor between the B meson and R P state we set A BR P ( m π ) = 0 .
37 [20].For the Bπ scalar and vector transition form factors F Bπ ( s ) and F Bπ ( s ),we use the following light-cone sum rule parametrization developed in Ap-pendix A of Ref. [21], viz. F Bπ ( s ) = 0 . − s/s , (18) F Bπ ( s ) = 0 . − s/M B ∗ − . − s/s , (19)with s = 33 .
81 GeV , M B ∗ = 5 .
32 GeV and s = 40 .
73 GeV . Thepion non-strange scalar and vector form factors Γ n ∗ ( s ) and F ππ ( s ) will bediscussed in the next section. Note that [22]Γ n ∗ ( s ) = √ B (cid:10) [ π + π − ] S | ¯ nn | (cid:11) , (20)with ¯ nn = 1 √ uu + ¯ dd ).The transition form factor between the B meson and the R D state F BR D ( m π ) is not well known [23], so it will be taken as a free parame-ter to be fitted. The expressions of the tensor angular distribution factor D ( s , s ) and of the R D mass dependence width Γ( s ), similar to thoseused for the f (1270) contribution in the BABAR Collaboration Dalitz plotanalysis [5], are displayed in Sec. A.5 of the Appendix A. The expression ofthe f (1270) coupling to ππ , G f , is also given there.In summary, from the S -, P - and D -wave matrix elements (10), (11 and(12), we obtain the total symmetrized amplitude for the B − → π + π − π − decay as M − sym ( s , s ) = 1 √ (cid:2) M − S ( s ) + M − S ( s ) + M − P ( s )( s − s )+ M − P ( s )( s − s ) + M − D ( s ) D ( s , s ) + M − D ( s ) D ( s , s ) (cid:3) , (21) with M − S ( s ij ) = G F √ h − χ S f π (cid:0) M B − s ij (cid:1) F BR S ( m π ) u ( R S π − )+ B M B − m π m b − m d F Bπ ( s ij ) v ( π − R S ) (cid:21) Γ n ∗ ( s ij ) , (22) M − P ( s ij ) = G F √ (cid:20) N P f π f R P A BR P ( m π ) u ( R P π − ) + F Bπ ( s ij ) w ( π − R P ) (cid:21) F ππ ( s ij ) , (23)and M − D ( s ij ) = − G F √ u ( R D π − ) f π √ F BR D ( m π ) G f m R D − s ij − im R D Γ( s ij ) . (24)For the fully symmetrized B + → π + π − π + decay amplitude we have M + sym ( s , s ) = 1 √ (cid:2) M + S ( s ) + M + S ( s ) + M + P ( s )( s − s )+ M + P ( s )( s − s ) + M + D ( s ) D ( s , s ) + M + D ( s ) D ( s , s ) (cid:3) , (25)with M + S,P,D ( s ij ) = M − S,P,D (cid:0) s ij , λ u → λ ∗ u , λ c → λ ∗ c , B − → B + (cid:1) . (26)
3. Scalar and vector form factors
As shown in Ref. [24] the full knowledge of strong interaction meson-meson form factors is available if the meson-meson interaction is known atall energies. The calculation of the S - and P -wave amplitudes (22) and(23) requires the values of the scalar and vector Bπ , B ( ππ ) and pion formfactors. The knowledge of the B → π and B → [ ππ ] S,P transition formfactors is needed far below the Bπ and B [ ππ ] S,P scattering region. Onehas then to rely on theoretical models constrained by experiment, as we dohere for the B [ ππ ] S form factor, using the value (see above in the previoussection) determined in Ref. [19]. One could also use covariant light-frontmodel, like that of Ref. [25] or, if available, semi-leptonic decay analysisresults. For the Bπ form factors we take the QCD light-cone sum ruleresults of Ref. [21] recalled above in Eqs. (18) and (19). The special case ofthe pion form factors is developed below. The pion scalar form factor
In the ππ case, the low-energy S wave being known and modeling thehigh-energy part one can rely on the Muskhelishvili-Omn`es equations [26] tobuild up the pion scalar form factors. Their evaluation from these equationshas been discussed in Ref. [27] and followed and developed in Ref. [28]. How-ever here, we shall use another approach, initiated in Ref. [22] and applied,using a different ππ scattering matrix, in Ref. [1]. Extending this last workby introducing three channels and keeping the off-shell contributions, thepion scalar form factor Γ n ∗ ( s ) entering in the S -wave amplitude Eq. (22) ismodeled according to the following relativistic three coupled-channel equa-tions Γ n ∗ i ( s ) = R ni ( E ) + X j =1 R nj ( E ) H ij ( E ) , i = 1 , , , (27)with H ij ( E ) = Z d p (2 π ) T ij ( E, k i , p ) 1 E − q p + m j + iǫ k j + κ p + κ , (28)where E represents the total energy, i.e., in the ππ center of mass, E = √ s and p is the off-shell momentum. In Eqs (27) and (28), the indices i, j =1 , , ππ , K ¯ K and effective (2 π )(2 π ) channels, respectively. Thecenter of mass momenta are k j = q s/ − m j , with m = m π , m = m K and m = m (2 π ) . The T matrix is the corresponding three-channel two-body scattering matrix. Here we use the solution A of the three-coupledchannel model of Refs. [29, 30], where the effective m (2 π ) = 700 MeV. Thefunctions R ni ( E ) are the production functions responsible for the formationof the meson pairs before their scattering. From Eqs. (27) and (28) one cancheck that Im Γ n ∗ i ( s ) = − X j =1 k j √ s π T ∗ ji ( E, k j , k i )Γ n ∗ j ( s ) θ ( √ s − m j ) . (29)This is the same unitary relation as that of the corresponding Muskhelishvili-Omn`es pion scalar form factors constructed in Ref. [28] [see Eq. (28) therein].In Eq. (28) the regulator function ( k j + κ ) / ( p + κ ), which reducesto 1 on-shell ( k j = p ), ensures the convergence of the integral. The rangeparameter κ will be fitted to data. The choice of a separable form forthe interaction yields analytic expressions for the T matrix elements. One introduces a rank-2 separable potential in the ππ channel and a rank-1separable potential in the K ¯ K and in the (2 π )(2 π ) ones. According to theformalism developed in Ref. [31] and applied in Ref. [29] one has for the T matrix elements: T ( E, p, k ) = g ( k ) t ( E ) g ( p ) + g ( k ) t ( E ) g ( p ) + g ( k ) t ( E ) g ( p )+ g ( k ) t ( E ) g ( p ) ,T ( E, p, k ) = g ( k ) t ( E ) g ( p ) + g ( k ) t ( E ) g ( p ) ,T ( E, p, k ) = g ( k ) t ( E ) g ( p ) + g ( k ) t ( E ) g ( p ) , (30)where g ( k ) = r πm π k + β ,g j ( k i ) = r πm i k i + β j , j = 1 , , . (31)The parameters β j , j = 0 , , ,
3, of the separable form of the scattering T matrix are given in Table 1 of Ref. [29] (fit A ).One can extend the expressions of the reduced symmetric t ( E ) matrixelements given in terms of the separable potential parameters in AppendixA of Ref. [31] to the case of Ref. [29] which we use here. The Yamaguchiform [32] of the g ( p ) and g i ( p ) (31) in the T matrix elements (30) leads thefollowing analytic expression for Γ n ∗ i ( s ) in Eq. (27)Γ n ∗ ( s ) = R n ( E ) + R n ( E ) { [ t ( E ) g ( k ) + t ( E ) g ( k )] g ( k ) F ( k )+[ t ( E ) g ( k ) + t ( E ) g ( k )] g ( k ) F ( k ) } + R n ( E )[ g ( k ) t ( E ) + g ( k ) t ( E )] g ( k ) F ( k )+ R n ( E )[ g ( k ) t ( E ) + g ( k ) t ( E )] g ( k ) F ( k ) , (32)where F ( k ) = I , ( k ) g ( k ) h ( k ) ,F ( k ) = I , ( k ) g ( k ) h ( k ) ,F ( k ) = I , ( k ) g ( k ) h ( k ) ,F ( k ) = I , ( k ) g ( k ) h ( k ) , (33)with h i ( k i ) = r πm i k i + κ , i = 1 , , ,h ( k ) = h ( k ) , (34)and I i,j ( k i ) = Z d p (2 π ) g j ( p ) 1 E − q p + m i + iǫ h i ( p ) , (35)where E = 2 q k i + m i , i = 1 , ,
3. The analytical expression for theseintegrals can be found in Appendix A of Ref. [31].As in Ref. [22] one constraints the Γ n ∗ i ( s ) to satisfy the low energy behav-ior given by next-to-leading order one loop calculation in chiral perturbationtheory (ChPT). One writes the expansion at low s asΓ ni ( s ) ∼ = d ni + f ni s, i = 1 , , , (36)with real coefficients, Γ ni ( s ) being real below the ππ threshold. Using theexpressions obtained in NLO in ChPT for the Γ n ∗ i ( s ) given in Refs. [22, 33]one gets, d n = r (cid:20) m π f (2 L r − L r ) + 8 2 m K + 3 m π f (2 L r − L r )+ m π π f + m π π f log m π ν − π f (cid:18) m π m η (cid:19) log m η ν ,f n = r (cid:20) f (2 L r + L r ) − π f (cid:18) m π ν (cid:19) − π f (cid:18) m K ν (cid:19) − m π π f (cid:18) m π − m η (cid:19)(cid:21) , (37)and d n = 1 √ " m η π f log m η ν + 16 m K f (2 L r − L r )+8 6 m K + m π f (2 L r − L r ) + m K π f m η ν ! ,f n = 1 √ " f (2 L r + L r ) − π f m η ν ! − m K π f m η − π f (cid:18) m K ν (cid:19) − π f (cid:18) m π ν (cid:19)(cid:21) , (38) ν being the scale of dimensional regularization and f = f π / √ L rk , k = 4 , , ,
8, we use the recentdeterminations of lattice QCD at ν = 1 GeV as given in Table X of Ref. [34].For f = 92 . d n = 1 . f n = 3 . − , d n = 0 . f n = 1 . − . As in Ref. [28] we assume Γ n (0) = 0 which leadsto d n = 0 and we also assume f n = 0.The real production functions are parametrized as R ni ( E ) = α ni + τ ni E + ω ni E cE , i = 1 , , , (39)the fitted parameter c controling the high energy behavior. The other pa-rameters, α ni , τ ni and ω ni are calculated by requiring that Γ ni ( s ) in Eq. (27)has the low energy expansion Eq. (36). These nine parameters satisfy alinear system of nine equations displayed in Appendix B. Their numericalvalues, depending on the value of the range parameter κ [see Eq. (28)], willbe given in Sec. 5. The pion vector form factor
As for the scalar case one could use the Muskhelishvili-Omn`es equationsto built up the pion vector form factor. This was done in Ref. [3] for the Kπ vector form factor. Here, noting that the knowledge of this form factor isrequired to describe the τ − → π − π ν τ decay, we shall use the phenomeno-logical model of the Belle Collaboration [35]. Fitting their high statisticsdata, they built the pion vector form factor F ππ ( s ) by including the con-tribution of the three vector resonances ρ (770), ρ (1450) and ρ (1700). Herewe use the parameters given in the third column of Table VII of Ref. [35].
4. Observables and data fitting
Physical observables
The symmetrized B − → π − π +2 π − amplitude (21) depends on the twoeffective ππ masses, m = √ s and m = √ s of the Dalitz plot. In thecenter of mass of π − ( p ) and π + ( p ), the pion momenta fulfill the equations |−→ p | = 12 q m − m π , |−→ p | = |−→ p | , |−→ p | = 12 m rh M B − ( m + m π ) i h M B − ( m − m π ) i , (40)and the cosine of the helicity angle θ between the direction of −→ p and thatof −→ p reads cos θ = 12 |−→ p ||−→ p | (cid:20) − m + 12 (cid:0) M B − m + 3 m π (cid:1)(cid:21) . (41)For fixed values of the effective mass m , the variables cos θ and m areequivalent.The double differential B − → π − π + π − branching fraction is d B − dm d cos θ = 1Γ B m |−→ p ||−→ p | π ) M B (cid:12)(cid:12) M − sym ( s , s ) (cid:12)(cid:12) , (42)where Γ B is the total width of the B − . Since the Dalitz plot is symmetricunder the interchange of m and m , one can limit the integration rangeon m to the values larger than m ; hence, the differential effective massdistribution reads d B − dm = Z cos θ g − d B − dm d cos θ d cos θ, (43) | Γ n | Fig. 1. Modulus of the pion scalar form factor Γ n (solid line), obtained in our fitusing the NLO a pi with κ = 2 GeV and for which the fitted parameter c = (19 . ± .
2) GeV − , compared to that calculated in Ref. [37] using the Muskhelishvili-Omn`es equations (double-dash dot line). The dash-dot line (for c = 15 . − )and the dashed one (for c = 23 . − ) represent the variation of the Γ n moduluswhen c varies within its error band. where cos θ g corresponds to the value of cos θ in Eq. (41) with m = m ,viz., cos θ g = 14 |−→ p ||−→ p | (cid:0) M B − m + 3 m π (cid:1) . (44)The variable m in Eq. (43) is also called the light (or minimal) effectivemass m min while m is the heavy (or maximal) effective mass, m max . The B − → π − π + π − branching fraction is then twice the integral of the differ-ential branching fraction (43) over m . Data fitting
We aim at describing the experimental π + π − distributions obtained bythe BABAR Collaboration in the Dalitz plot analysis of the B ± → π ± π ± π ∓ decays [5]. Two different background distributions, related to the q ¯ q andthe B ¯ B components, are subtracted from Fig. 4 of Ref. [5]. Six light ef-fective π + π − mass distributions are extracted for B + and B − decays with a subdivision of the data into positive and negative values of the cosine ofthe helicity angle θ . For the B + and B − distributions we reject two datapoints corresponding to the π + π − effective masses equal to 485 and 515MeV. Also two points at 470 and 530 MeV for the four mass distributionswith cos θ > θ < B ± → K S π ± .As a by-product of the background subtraction, five data points, with asmall number of events, have negative values with small statistical errors.For these five data points we increase their errors to values correspondingto those of the points lying in a close vicinity. This is done at 1385 MeV forthe B − distribution, at 1475 MeV for the B + one, at 290 and 1610 MeV forthe B − distribution with cos θ > B − one withcos θ < χ fit to the 170 data points corresponding to the sixinvariant mass distributions described above. In addition, we include theexperimental branching ratio for the B ± → ρ (770) π ± , ρ (770) → π + π − decay channel. The theoretical distributions are normalized to the numberof experimental events in the analyzed range from 290 up to 1640 MeV. Inthe fits, done for a fixed value of the range parameter κ entering Eqs. (28)[see Sec. 5], the following four parameters were varied: the productionfunctions R ni ( E ) [Eq. (39)] parameter c , the real S -wave strength parameter χ S , the real P -wave correction parameter N P [Eq. (15)] and the transitionform factor F BR D ( m π ) [Eq. (17)].
5. Results and discussion
In the fits to the selected BABAR data as described in the previoussection, the CKM matrix elements [see Eq. (2)] are calculated with λ =0 . A = 0 . ρ = 0 .
135 and ¯ η = 0 .
349 [18] which leads to λ u =1 . × − − i . × − and λ c = − . × − − i . × − . TheLO contributions of the Wilson coefficients to the a pi Eq. (5) are given inthe second and fourth columns of Table 1. The sum of the leading ordercoefficient plus the next-to-leading order vertex and penguin corrections forthe a pi coefficients, entering into u ( R S,P π − ) [Eq. (7)], v ( π − R S ) [Eq. (8)] and w ( π − R P ) [Eq. (9)], are displayed in columns three and five, respectively. Itcan be seen that the NLO corrections are relatively small except for thecoefficient a which, however, has only a small contribution to the decayamplitude. The corrections are calculated according to Refs. [7] and [9]using the Gegenbauer moments for pions taken from the Table 2 of Ref. [7]and the corresponding moments for the ρ meson from Table 1 of Ref. [36]. Inthe calculation of the coefficients a p ( π − R S ) and a p ( π − R S ), contributing to v ( π − R S ), we apply the method explained in Appendix A of Ref. [11]. Here π + π − light effective mass distributions from the fit to the BABARexperimental data [5], a) for the B − decays and b) for the B + decays. The long-dash line represents the S -wave contribution of our model, the dot line that of the P wave, the short-dash line that of the D wave and the dot-dash line that of theinterference term. The solid line corresponds to the sum of these contributions.Fig. 3. As in Fig. 2 but for the B − decays a) with cos θ < θ > the renormalization scale µ = m b / α s ( m b /
2) = 0 . B + decays.Table 1. Leading order (LO) and next-to-leading order (NLO) coefficients a pi ( R S,P π − ), a pi ( π − R S ) (in parentheses) and a pi ( π − R P ) [see Eq. (5)] entering into u ( R S,P π − ) [Eq. (7)], v ( π − R S )[Eq. (8)] and w ( π − R P ) [Eq. (9)], respectively. TheNLO coefficients are the sum of the LO coefficients plus next-to-leading order ver-tex and penguin corrections. Here the renormalization scale is µ = m b /
2. Thesuperscript p is omitted for i = 1, 2, 3, 5, 7 and 9, the penguin corrections beingzero for these cases. a pi ( R S,P π − ) a pi ( π − R S,P )LO NLO LO NLO a .
039 1 .
071 + i . a . − . − i . a u − . − . − i . − . − . − i . a c − . − . − i . − . − . − i . a u − . − . − i .
017 ( − . − . − i . a c − . − . − i .
004 ( − . − . − i . a . . i . a u . . i . . . i . a c . . i . . . i . a − . − . − i . a u − . . i . − . . i . a c − . . i . − . . i . κ and the high energy cut-off c of the production functions R ni ( E ) Eq. (39) for κ = 2 GeV i α ni τ ni (GeV − ) ω ni (GeV − )1 0 . − . . . − . . .
003 0 . . S -wave Γ n form factor.The other three, χ S , N P and F BR D ( m π ) are related to the strength of the S , P and D amplitudes, respectively. The range κ should be larger than 0.8GeV which is the on-shell pion momentum approximately equal to the halfof the effective m ππ upper limit ∼ .
64 GeV which we used. In our fits wefind that the total χ decreases slowly when κ decreases from the high valueof 5 GeV. Here we fix the range parameter κ to be 2 GeV. We perform twofits for the full S + P + D -wave amplitude calculated with the NLO and withthe LO a pi coefficients. Hereafter the quoted results given inside parenthesescorrespond to the numbers obtained in the second fit. The quoted errors onour results come from the statistical errors in the experimental data.A good overall agreement with BABAR’s data is achieved with c =19 . ± . . ± .
1) GeV − , χ S = − . ± . − . ± .
6) GeV − , N P = 1 . ± .
034 (1 . ± . F BR D ( m π ) = 0 . ± . ± χ is equal to 231.6 (233.5) for the 171 ex-perimental points of the fit. For both fits the branching fraction for the B ± → ρ (770) π ± , ρ (770) → π + π − decay is (8 . ± . × − , to becompared with the BABAR Collaboration determination of (8 . ± . ± . +0 . − . ) × − ≈ (8 . ± . × − from their isobar model analysis [5].Note that for the LO fit we explain essentially the BABAR Collaboration’sresult without significant modification of the P wave normalization, the pa-rameter N P ≈ .
02 being close to 1. For the NLO fit, N P ≈ . ± . N P − ≈
25% with the average 20% error of theexperimental branching ratio.The CP average total branching fraction of the B ± → π ± π ∓ π ± decayscalculated in the NLO fit is equal to (15 . ± . × − to be compared tothe measured value of (cid:0) . ± . ± . +0 . − . (cid:1) × − (table III of Ref. [5]).The branching fraction for the S wave equals to (2 . ± . × − and thatfor the D wave is (2 . ± . × − . The latter value is larger than thebranching fraction for the f (1270) π ± , (0 . ± . ± . +0 . − . ) × − , deter-mined in Ref. [5]. In the experimental analysis the two resonances, namely f (1270) and f (1370), overlap to a large extent, which makes their sep-aration difficult and some part of the branching fraction obtained for oneresonance could have been attributed to the other one. The isobar modelanalysis of Ref. [5] gives (2 . ± . ± . +0 . − . ) × − for the branching frac- tion of f (1370) π ± . Then, the sum of the branching fractions for the tworesonances equals to 3 . × − . This value compares well with the branch-ing fraction of 3 . × − obtained by integrating our distribution in the m ππ range between 1.0 and 1.64 GeV in which both f (1270) and f (1370)give their dominant contributions. In our model the D -wave contribution isdominant in this range. Let us note that the value we obtain for the transi-tion form factor F BR D ( m π ) is 29% larger than the value 0.076 given in Table1 of Ref. [23] for the ISGW2 model. The S -wave contribution representshere as much as 15% of the total branching fraction. This contribution isof the same order as that of the ρ (1450) and ρ (1700) which also represents15 % of the total P -wave contribution.Before comparing our effective mass distributions to the experimentalones, we now give our result for the pion scalar form factor Γ n ( s ). With thefixed value of κ = 2 GeV used in the fits, one obtains for the α ni , τ ni and ω ni , i = 1 , ,
3, entering into Eq. (39), the values given in Table 2. Then, inFig. 1, we show the modulus of the pion scalar form factor obtained usingthe NLO coefficients a pi for the fitted value of the parameter c = (19 . ± . − together with its envelope when c varies within its error band. It isalso compared to that of the scalar form factor calculated by Moussallam [37]solving the Muskhelishvili-Omn`es equations [26] with a high-energy ansatzstarting at 2 GeV and the same low-energy three coupled-channel scatteringT-matrix as in our model (see Sec. 3.1). However, in his calculation theoff-diagonal matrix elements T ( E, k i , p ) and T ( k i , E, p ) are set to zeroin the unphysical region E < m = 1 . n ∗ ( s ) moduliis quite similar. It can be seen in Fig. 1 that, within our model, the neededΓ n ∗ ( s ) is relatively well constrained. If we fix κ = 3 GeV then the fit toBABAR data gives c = (30 . ± .
6) GeV − , χ S = ( − . ± .
9) GeV − witha total χ of 234.1. In this range of variation of the strongly correlated κ and c parameters, we have checked that the scalar form factor varies smoothly.The corresponding values of the strength parameter χ S , being very close to −
20 GeV − , are not sensitive to these variations. For κ = 3 GeV the valuesof the branching fractions for the different ππ waves stay within the errorbands of those for the κ = 2 GeV case.The threshold behavior of our pion form factor is governed by the chiralperturbation expansion Eq. (36). These ChPT constraints, not explicitlyincluded in Moussallam’s case, lead to Γ n ∗ ( s ) moduli of both approaches todiffer only slightly near the ππ threshold. Above the ππ threshold, there isa maximum corresponding to the f (600) resonance, then close to 1 GeVa characteristic dip due to the f (980) and finally, below the spike at 1.4GeV related to the opening of the third channel, there is some enhance- ment generated by the f (1400) present in the ππ three-channel model usedhere [29, 30]. The third threshold energy equal to 1.4 GeV is a parameterrepresenting twice the mass of the effective two-pion mass m (2 π ) used toaccount for the four pion decays of scalar mesons (see Ref. [29]). Thus, innature there is no such sharp energy behavior. These characteristic featuresof the pion scalar form factor Γ n ( s ) are essential to obtain a good fit of theexperimental effective mass distributions of the B ± to 3 π decays.The results of the fit on the experimental distributions, obtained usingthe NLO coefficients a pi in the B ± → π ± π ∓ π ± amplitudes, are displayed inFigs. 2, 3 and 4. The ρ (770)-resonance contribution dominates the π + π − spectrum, but that of the S -wave is non negligible. As seen, the S -wavepart is sizable near 500 MeV which is related to the contribution of thescalar resonance f (600), not explicitely included in the BABAR Dalitzplot analysis [5]. In the 1 GeV range the f (980) resonance is not observedas a peak in the π + π − spectrum. This fact is easily explained in our modelsince the decay amplitudes are proportional to the pion scalar form factorwhich has a dip near 1 GeV as seen in Fig. 1. Around 1.3 GeV there is amaximum coming from the contribution of the f (1270) resonance. Near 1.4GeV the f (1400) scalar resonance [29, 30] gives only a tiny enhancementin the distributions.Figure 2 exhibits a small CP asymmetry, the B − and B + effective massdistributions being very close. Summing the number of experimental eventsin the m π + π − range between 290 and 1640 MeV one finds 616 events for the B − decay and 606 for that of the B + . This leads to a CP asymmetry of(0 . ± . . ± . − . ± . CP asymmetry A CP = (cid:0) . ± . ± . +2 . − . (cid:1) % for the total sample of π ± π ∓ π ± events [5]. For the particular decay mode, namely for the B ± decayinto ρ (770) π ± , ρ (770) → π + π − , the isobar model analysis gives A CP = (cid:0) ± ± +2 − (cid:1) %, while from our model we get 3 . ± .
2% ( − . ± . a pi coefficients are quite smallas it could have been expected.Figures 3 and 4 show a spectacular feature, namely that the interfer-ence term of the S , P and D waves is quite important under the ρ (770) maximum. Here the S - P interference dominates. The sign of this inter-ference term depends on the sign of cos θ , so the ρ peak is reduced for thenegative values of cos θ and enhanced for the positive values. This is a clearindication that the π + π − effective mass distribution cannot be reproducedwithout the S -wave contribution. If we try to fit the data without the S - wave amplitude then we obtain a poor fit with χ = 316 .
3. In this casethe effective mass distributions are not well described below 600 MeV andalso under the ρ maximum. One striking feature is that the interferenceterms allow an extremely good representation of the separate cos θ < θ > B + decays (Fig. 4) and yield for the full spectrum[Fig. 2b)] a χ /point of 1.07. The fit of the separate B − spectra (Fig. 3)is less satisfactory whereas that of the full spectrum [Fig. 2a)] is almostperfect with a χ /point of 1.2.
6. Summary and outlook
The present paper is a continuation of our efforts [1, 2, 3, 4] in con-straining theoretically the meson-meson final state strong interactions inhadronic charmless three-body B decays. If the strong interaction ampli-tudes are sufficiently well understood then one can improve the precision ofthe weak interaction amplitudes extracted from these reactions.Our theoretical model for the B ± → π ± π ∓ π ± is based on the applicationof the QCD factorization [6, 7, 9, 13] to quasi two-body processes in whichonly two of the three produced pions interact strongly, forming either an S -, P - or D -wave state. One assumes that the third pion, being fast inthe B -meson decay frame, does not interact with this pair. This hypothesisis mainly valid in a limited range of the π + π − effective mass, here takenbetween the ππ threshold and 1.64 GeV.The short-distance interaction part of the decay amplitudes describesthe flavor changing processes b → u ¯ ud and b → d ¯ dd . It is proportional toCabibbo-Kobayashi-Maskawa matrix elements multiplied by effective coef-ficients calculable in the perturbative QCD formalism. This short-distanceamplitude is multiplied by a long-distance contribution expressed in termsof two products. The first one is the product of the pion decay constant bythe B → ππ transition matrix element and the second one is the product ofthe pion form factor by the B → π transition form factor. The parametriza-tion [Eqs. (18), (19)] of the scalar and vector B to π transition form factorsfollow from the light-cone sum rule study of Ref. [21].The effective Wilson coefficients are calculated to next-to-leading orderin the strong coupling constant. They include vertex and penguin correc-tions but neither hard-scattering ones nor annihilation contributions sincethese last two terms contain unknown phenomenological parameters relatedto amplitude divergences [9]. We find that these vertex and penguin cor-rections are small in comparison to the leading order term (see Table 1).However, they allow to generate some non-zero CP asymmetries.We then assume the B to ππ transition matrix element to be equal tothe product of the B to intermediate meson transition form factor by the decay amplitude of this meson into two pions being either in S , P or D wave.The next step is to suppose the latter decay amplitude to be proportionalto the pion non-strange scalar or vector form factor depending on the wavestudied. For the S wave the proportionality factor is given by a fittedparameter χ S and for the P wave it is related to the inverse of the ρ decayconstant. For the limited range of the effective ππ mass, from ππ thresholdto 1 .
64 GeV, the B → ππ transition form factors are taken as constantsgiven by the B → f (980) [19] and by the B → ρ (770) [20] transition formfactors at q = m π . The decay amplitude for the ππ D wave is described bya relativistic Breit-Wigner formula and the not well known B to f (1270)transition form factor is fitted. We find F Bf ( m π ) = 0 . ± . ππ , K ¯ K and effective (2 π )(2 π )scattering T matrix of Refs. [29, 30]. This form factor depends on twofitted parameters: the first one κ insures the convergence of the involvedintegrals and the second one, c , controls the high-energy behavior of theproduction functions accountable for the meson pair formation. The pionvector form factor takes into account the contribution of the ρ (770), ρ (1450)and ρ (1700), and follows from the parametrization of the Belle Collaborationin their study of the semi-leptonic τ − → π − π ν τ decays. For the P -waveamplitude we introduce a fitted correction factor N P .We obtain a good fit to the ππ effective mass distributions of the BABARCollaboration data of the B ± → π ± π ∓ π ± decays [5]. The value of thebranching fraction for the B ± → ρ (770) π ± decays, (8 . ± . ± . +0 . − . ) × − , is well reproduced with the correction factor N P close to 1. This showsthat the QCD factorization gives the right strength of the B to ρπ decay am-plitude. The π + π − spectra are dominated by the ρ (770) resonance but, atlow effective mass, the S -wave contribution is sizable. Here the f (600) reso-nance manifests its presence. Furthermore one observes a strong interferenceof the S and P waves in the event distributions for cos θ > θ < f (980) is not directly visible as a peak, since the pion scalar formfactor has a dip near 1 GeV. The surplus of events in the π + π − effective massclose to 1.25 GeV is well described by the contribution of the f (1270) res-onance. The branching fraction for B ± → f (1270) π ± , f (1270) → π + π − decay is found to be of (2 . ± . × − . At 1.4 GeV, the tiny maximum ofthe S -wave distribution comes from the scalar resonance f (1400) [29, 30].Our model yields a unified description of the contribution of the threescalar resonances f (600), f (980) and f (1400) in terms of one function:the pion non-strange scalar form factor. This reduces strongly the numberof needed free parameters to analyze the Dalitz plot. The functional formof our S -wave amplitude [Eq. (22)], proportional to Γ n ∗ ( s ), could be used inDalitz-plot analyses and the table of Γ n ∗ ( s ) values can be sent upon request. The strong interaction phases of the decay amplitudes are constrainedby unitarity and meson-meson data. Their determination should help inthe extraction of the weak angle phase γ or φ equal to arg( − λ ∗ u /λ ∗ c ). Ofcourse new experimental data with better statistics would be welcome. Oneexpects B ± → π ± π ∓ π ± events from the Belle Collaboration, and probably,in the near future, from LHCb and from the near term super B factories.The authors are obliged to Bachir Moussallam for providing them thevalues of his pion scalar form factor Γ n ( s ) and to Gagan Bihari Mohantyfor useful comments on the BABAR data. We are very grateful to MariaR´o˙za´nska, Bachir Moussallam, Eli Ben-Haim and Jos´e Ocariz for helpfuldiscussions. This work has been supported in part by the Polish Ministryof Science and Higher Education (grant No N N202 248135) and by theIN2P3-Polish Laboratories Convention (project No 08-127). Appendix ALong-distance functions X S,P,D and Y S,P
Appendix A.1
The function X S from the S -wave amplitudeproportional to BR S transition matrix element From Eq. (13) the function X S reads X S ≡ (cid:10) [ π + ( p ) π − ( p )] S | (¯ ub ) V − A | B − (cid:11) (cid:10) π − | ( ¯ du ) V − A | (cid:11) = G nR S π + π − ( s ) (cid:10) R S | (¯ ub ) V − A | B − (cid:11) (cid:10) π − | ( ¯ du ) V − A | (cid:11) , (A.1)where the vertex function G nR S π + π − ( s ) describes the R S decay into a[ π + π − ] S pair. The B to R S transition matrix element reads (see e.g.Eq. (B6) of Ref. [12]) h R S ( p + p ) | ¯ uγ µ (1 − γ ) b | B − ( p B ) i = i ( (cid:20) ( p B + p + p ) µ − M B − s m π p µ (cid:21) F BR S ( m π )+ M B − s m π p µ F BR S ( m π ) ) , (A.2)where F BR S ( m π ) and F BR S ( m π ) are the BR S scalar and vector form fac-tors, respectively. The pion decay constant f π is defined as h π − ( p ) | ¯ dγ µ (1 − γ ) u | i = if π p µ . (A.3) The product of Eqs. (A.2) and (A.3) yields (cid:10) R S | (¯ ub ) V − A | B − (cid:11) (cid:10) π − | ( ¯ du ) V − A | (cid:11) = − ( M B − s ) f π F BR S ( m π ) . (A.4)The vertex function G nR S π + π − ( s ), as in Ref. [2], is modeled by (cid:10) [ π + π − ] S | ¯ nn | (cid:11) = G nR S π + π − ( s ) h R S | ¯ nn | i . (A.5)An effective scalar decay constant f nR S can be introduced with h R S | ¯ nn | i = m R S f nR S . (A.6)From Eqs. (A.5), (20) and (A.6) one obtains G nR S π + π − ( s ) = r χ S Γ n ∗ ( s ) = r √ B m R S f nR S Γ n ∗ ( s ) , (A.7)with χ S = √ B m R S f nR S . (A.8)The effective scalar decay constant has a role comparable to the R P decayconstant as can be seen comparing Eqs. (A.7) and (A.19). The product ofEqs. (A.7), (A.2) and (A.3) gives X S = − r χ S f π ( M B − s ) F BR S ( m π ) Γ n ∗ ( s ) . (A.9) Appendix A.2
The function Y S from the S -wave amplitudeproportional to Bπ transition matrix element From Eq. (14) one has Y S ≡ (cid:10) π − | ( ¯ db ) sc − ps | B − (cid:11) (cid:10) [ π + ( p ) π − ( p )] S | ( ¯ dd ) sc + ps | (cid:11) = (cid:10) π − | ¯ db | B − (cid:11) (cid:10) [ π + ( p ) π − ( p )] S | ¯ dd | (cid:11) . (A.10)From the Dirac equations satisfied by b ( p B ) and ¯ d ( p ) one obtains (cid:10) π − ( p ) (cid:12)(cid:12) ¯ d ( p ) b ( p B ) (cid:12)(cid:12) B − ( p B ) (cid:11) = (cid:28) π − ( p ) (cid:12)(cid:12)(cid:12)(cid:12) ¯ d ( p ) γ · ( p B − p ) m b − m d b ( p B ) (cid:12)(cid:12)(cid:12)(cid:12) B − ( p B ) (cid:29) . (A.11) The B to π transition matrix element (cid:10) π − | ( ¯ db ) V − A | B − (cid:11) , entering into theabove expression, can be written as (see e.g. Eq. (5) of Ref. [3]) h π − ( p ) | ¯ dγ µ (1 − γ ) b | B − ( p B ) i = (cid:20) ( p B + p ) µ − M B − m π q q µ (cid:21) F Bπ ( q ) + M B − m π q q µ F Bπ ( q ) , (A.12)where F Bπ ( q ) and F Bπ ( q ) are the Bπ scalar and vector form factors,respectively and q = p B − p = p + p . Using Eqs. (A.12) and (20) inEq. (A.10), yields Y S = r B Γ n ∗ ( s ) M B − m π m b − m d F Bπ ( s ) . (A.13) Appendix A.3
The function X P from the P -wave amplitudeproportional to BR P transition matrix element From Eq. (15) one has for the function X P (see Eq. (3.1) of Ref. [12]) X P ≡ (cid:10) [ π + ( p ) π − ( p )] P | (¯ ub ) V − A | B − (cid:11) (cid:10) π − | ( ¯ du ) V − A | (cid:11) = G nR P π + π − ( s ) √ ǫ · ( p − p ) (cid:10) R P | (¯ ub ) V − A | B − (cid:11) (cid:10) π − | ( ¯ du ) V − A | (cid:11) , (A.14)where the R P decay into a [ π + π − ] P pair is described by the vertex function G nR P π + π − ( s ). Here ǫ represents the polarization vector of the P -wavemeson R P . The factor 1 / √ R P represents the ρ (770) . As seen from e.g. Eq. (B6) of Ref. [12] or Eq. (24) of Ref. [6], (cid:10) R P ( p + p ) | (¯ ub ) V − A | B − ( p B ) (cid:11) = − i m R P ǫ ∗ · p B p p A BR P ( p )+other terms . (A.15)The “other terms” do not give any contribution when multiplying this ma-trix element by that given in Eq. (A.3). Plugging this expression intoEq. (A.14) one has a product of polarization vectors and the sum overthe three possible polarization eigenvalues of the state R P should be done.From X λ =0 , ± ǫ λµ ( p ) ǫ λ ∗ ν ( p ) = − ( g µν − p µ p ν p ) , (A.16) one obtains X λ =0 , ± ǫ λ · ( p − p ) ǫ λ ∗ · p B = − p · ( p − p ) . (A.17)Then X P = N P f π f R P ( s − s ) A BR P ( m π ) F ππ ( s ) . (A.18)Above, as shown in Ref. [3] for the K ∗ (892) → ( Kπ ) P decay case [see theirEq. (D9)], we have parametrized the R P π + π − vertex function in terms ofthe pion vector form factor F ππ ( s ). One has G R P π + π − ( s ) = N P √ m R P f R P F ππ ( s ) , (A.19) f R P being the charged R P decay constant. Above we have introduced aparameter N P to take into account the possible deviation of the strength ofthe P wave, here proportional to 1 /f R P . Appendix A.4
The function Y P from the P -wave amplitudeproportional to the Bπ transition matrix element From Eq. (16) Y P ≡ (cid:10) π − | ( ¯ db ) V − A | B − (cid:11) (cid:10) [ π + ( p ) π − ( p )] P | (¯ uu ) V − A | (cid:11) . (A.20)The pion vector form factor is defined by (see e.g. Eq. (36) of Ref. [6]) h R P | (¯ uu ) V − A | i = h [ π + ( p ) π − ( p )] P | ¯ uγ µ (1 − γ ) u | i = − ( p − p ) µ F ππ ( q ) . (A.21)The minus sign arises from the definition of the form factor F ππ ( q ) whichcontains a plus sign for a ( ¯ dd ) V − A current [similar to Eq. (A.12], then as ρ = 1 / √ u ¯ u − d ¯ d ), there will be a minus sign for a (¯ uu ) V − A current. Theproduct of Eqs. (A.12) and (A.21) gives Y P = − p · ( p − p ) F Bπ ( q ) F ππ ( q ) = ( s − s ) F Bπ ( q ) F ππ ( q ) . (A.22) Appendix A.5
The function X D from the D -wave amplitudeproportional to BR D transition matrix element From Eq. (17) one has X D ≡ (cid:10) [ π + ( p ) π − ( p )] D | (¯ ub ) V − A | B − (cid:11) (cid:10) π − ( p ) | ( ¯ du ) V − A | (cid:11) = 1 √ G R D π + π − ( s ) X λ = − ǫ αβ p α p β D R λD ( p D ) | (¯ ub ) V − A | B − E(cid:10) π − ( p ) | ( ¯ du ) V − A | (cid:11) , (A.23)with p D = p + p . The factor of 1 / √ R D [the meson f (1270)]. The R D decay into a [ π + π − ] D pair isdescribed by the vertex function G R D π + π − ( s ). Here ǫ αβ ( λ ) represents thepolarization tensor of the f (1270) and λ is its spin projection (see Ref. [38],p. 147). Taking Eq. (A3) for (cid:10) π − ( p ) | ( ¯ du ) V − A | (cid:11) and Eq. (4) of Ref. [23]for the transition matrix element (cid:10) R λD ( p D ) | (¯ ub ) V − A | B − (cid:11) we obtain X D = − f π √ G R D π + π − ( s ) F BR D ( m π ) X λ = − ǫ αβ ( λ ) p α p β ǫ ∗ µν ( λ ) p νB p µ . (A.24)To be consistent with the choice of normalization of Eq. (A2), we havemultiplied by i the right hand side of Eq. (4) in Ref. [23]. One can showthat (see Eqs. (7.7) and (7.8) of Ref. [38], p. 73) D ( s , s ) ≡ X λ = − ǫ αβ ( λ ) p α p β ǫ ∗ µν ( λ ) p νB p µ = 13 ( |−→ p ||−→ p | ) − ( −→ p ·−→ p ) , (A.25) −→ p and −→ p being the momenta of the π − ( p ) and the π + ( p ) in the rest frameof π + ( p ) and π − ( p ). One obtains , with m = √ s , −→ p · −→ p = 14 ( s − s ) , |−→ p | = 12 q m − m π , |−→ p | = |−→ p | , |−→ p | = 12 m rh M B − ( m + m π ) i h M B − ( m − m π ) i , (A.26)which allows to express Eq. (A.25) in terms of s and s . The vertexfunction entering into Eq. (A.23) is parametrized as being proportional toa relativistic Breit-Wigner resonance formula, we write G R D π + π − ( s ) = r G f m R D − s − im R D Γ( s ) , (A.27)where (see Ref. [38], p.147) G f = m f s π Γ f ππ q f , Γ f ππ = 0 .
848 Γ f (A.28)and the mass-dependent width Γ( s ) can be expressed as (see Eq. (7) ofRef. [5]), Γ( s ) = Γ f (cid:18) |−→ p | q f (cid:19) m f m X ( |−→ p | ) X ( q f ) . (A.29)Here Γ f is the total width of the f (1270) resonance, m f its mass and q f is the pion momentum in the f c.m. system. The Blatt-Weisskopf barrierform factor is given by [5] X ( z ) = 1( zr BW ) + 3( zr BW ) + 9 , (A.30)where the meson radius parameter r BW = 4 (GeV/c) − . Finally one has X D = − √ f π F BR D ( m π ) r G f D ( s , s ) m R D − s − im R D Γ( s ) . (A.31) Appendix BLinear system of equations for α ni , τ ni and ω ni The linear system of nine equations satisfied by the nine production functionparameters α ni , τ ni and ω ni , i = 1 , ,
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