International Journal of Modern Physics a C World Scientific Publishing Company

S is performed using a unitary representation of the ππ and KK final state interactions. Comparison of the theoretical model with the experimental data of the Belle and BaBar Collaborations indicates that charming penguin contributions are necessary to describe the B → f 0 (980)K and B → ρ(770) 0 K decays.


Introduction
Recent experimental results from B factories indicate that charmless hadronic three-body decays are more frequent than two-body ones [1]. Moreover one observes on Dalitz plots a definitive surplus of events at relatively small effective masses. This is a signal of especially strong interactions between hadrons at not too high relative energies. Many resonances are explicitly visible but in general the interference pattern is quite complicated. Knowledge of these final state interactions is important to obtain a precise determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. Weak decay observables give information on hadron-hadron interactions and internal quark or hadronic structure of the produced particles.
Prominent maxima in the π + π − spectra are observed in the B → π + π − K decays in the f 0 (980) region [2][3][4][5][6][7][8][9]. The B + → f 0 (980)K + and B 0 → f 0 (980)K 0 branching ratios are relatively large, and of the order of 10 −5 . Direct * Unité de Recherche des Universités Paris 6 et Paris 7, associée au CNRS and time-dependent CP -violating asymmetries are also measured. The first Belle result on the B + → f 0 (980)K + branching ratio [2] has motivated the study of Chen [10]. In the perturbative QCD approach Chen finds that the non-strange content of the f 0 (980) can be important. According to Cheng and Yang [11], subleading corrections due to intrinsic gluon effects inside B meson may enhance the decay rate of B → f 0 (980)K. B decays into scalar-pseudoscalar or scalar-vector particles have been studied by Minkowski and Ochs with a special emphasis on the presence of the lightest glueball [12]. The π + π − mass spectrum in B → Kππ decays, reported by Belle in 2003, is reproduced by a model amplitude of the coherent sum of f 0 (980), f 0 (1500) and a very broad glueball as a background.
In the present letter we study the B decays into ππK and KKK. We restrict ourselves to the case where the produced ππ or KK pairs interact in isospin zero S-wave from the ππ threshold to about 1.2 GeV. One expects the ππ isospin two S-wave contribution to be small since the upper limit of the branching fraction for the B + → π + π + K − decay is less than 1.8 × 10 −6 [13]. Us-ing the KK/πη branching ratio of a 0 (980) [1] and the upper limit of 2.5 × 10 −6 [14] for the branching ratio of B + → a 0 0 (980)K + , a 0 0 (980) → π 0 η, one can estimate the branching fraction B(B + → a 0 0 (980)K + , a 0 0 (980) → K + K − ) to be smaller than 1 × 10 −6 . This indicates that the KK isospin one S-wave amplitude is suppressed in the B ± → K + K − K ± decays.
Two-pion S-wave rescattering effects have been recently considered by Gardner and Meißner [15]. They study the effect of the f 0 (600) (or σ) resonance on the B 0 decay into π + π − π 0 in the range where the ρ(770)π channel dominates. The σπ channel can play a role in the determination of the CKM angle α from the B 0 → ρπ decays. Gardner and Meißner describe the broad f 0 (600) introducing a scalar form factor constrained by the chiral dynamics of low-energy meson-meson interactions [16]. This scalar form factor is used instead of the commonly applied Breit-Wigner form to improve the description of the broad σ and the understanding of the B → ρπ decays.
We extend the approach of ref. [15] to the f 0 (980) resonance. The four strange and nonstrange ππ and KK scalar form factors are constrained by chiral perturbation theory as developed by Meißner and Oller [16]. Our final state interaction is, however, different from that of [15] and [16]. Here we consider the unitary ππ and KK coupled channel model of [17].
First the B → (ππ) S−wave K and B → (KK) S−wave K decay amplitudes are calculated within the naive factorization approximation [18,19]. Penguin amplitudes interfere destructively which leads to much too small B → f 0 (980)K branching ratios.
Then we consider some QCD factorization corrections [20] calculated by de Groot, Cottingham and Whittingham [21]. These corrections are not sufficient to obtain agreement with experiment. Further contributions are needed. Here we include the longdistance contributions which have been considered in [21] to improve their fit to hadronic charmless strange and non-strange two-body B-decay data. These amplitudes, called charming penguin terms, originate from enhanced charm quark loops [22]. They could, for instance, correspond to weak decays of B to intermediate D ( * ) s D ( * ) states followed by transitions to f 0 (980)K final states via cc annihilations. Their addition allows us to obtain a good agreement with the measured B → f 0 (980)K branching fractions.
In sect. 2 we describe our weak decay amplitudes supplemented by the scalar form factors. Our model for the final state interactions is given in sect. 3. Results of calculations and comparison with available data are presented in sect. 4. In sect. 5 we give some conclusions and final remarks.

Amplitudes for the B → ππK and B → KKK decays
We shall write the model amplitudes for the following decays: Here by (ππ) S and (KK) S we mean π + π − or π 0 π 0 and K + K − or K 0 K 0 pairs in isospin zero S-wave. In the approximation used in our approach the effective weak Hamiltonian H is replaced by the sum of products of factorized currents [19]. We introduce some of the QCD factorization corrections and the charming penguin amplitudes considered in [21]. Then the B − → (π + π − ) S K − decay amplitude is where G F is the Fermi coupling constant and χ is a constant which will be estimated from the properties of the f 0 (980) decay. The functions Γ n 1 (m ππ ) and Γ s 1 (m ππ ) are the non-strange and strange pion scalar form factors depending on the effective pion-pion mass m ππ . Furthermore the functions P (m ππ ) and Q(m ππ ) defined as (2) f K is the kaon decay constant. In eq. (3) B 0 is related to the vacuum quark condensate: f π being the pion decay constant equal to 92.4 MeV. We use the formula B 0 = m 2 π /(2 m), where m is the average mass of the light quarks u and d. We put m = 5 MeV and following ref. [19] we take m s = 0.122 GeV and m b = 4.88 GeV. The functions U and V in (1) depend on the combinations of the coefficients a i [18][19][20][21] and on the products of the CKM matrix elements λ u = V ub V * us and λ t = V tb V * ts : where the chiral factor r = 2M 2 K /[(m b +m u )(m s + m u )]. In the numerical calculations we set m u = m and the coefficients a 1 , a u 4 , a c 4 , a u 6 and a c 6 , evaluated at the scale µ = 2.1 GeV, are taken from table III of [21]. These coefficients take into account some QCD factorization corrections. We do not include small corrections coming from hard gluon exchanges with spectator-quark, from annihilation terms and from electroweak penguin diagrams. The charming penguin contribution can be parametrized as where m is m ππ or M K , F π is the B → π transition form factor calculated at the zero m 2 π limit and P GIM 1 , P 1 are complex parameters. Determination of these parameters has been done in [21][22][23][24] by fitting some charmless two-body B-decay data.
The neutral B 0 → (π + π − ) S K 0 S amplitude is similar to the charged one: with a 1 = 0 and m u → m d .
The amplitudes for B-decays into three kaons read: and Here m KK is the K + K − effective mass and m KK is the effective mass with the second possible K + K − combination. Γ n 2 (m KK ) and Γ s 2 (m KK ) are the non-strange and strange kaon scalar form factors. Decay amplitudes with (π 0 π 0 ) S and (K 0 K 0 ) S final states are the same as those with (π + π − ) S and (K + K − ) S pairs, respectively. If one uses the a i coefficients [19] a u 4 = a c 4 = a 4 and a u 6 = a c 6 = a 6 , then our formula (1) with C(m) = 0 has the same algebraic structure as the B − → σπ − amplitude of ref. [15] (see their eq. (25)). The above equalities for a i are valid in naive factorization but not in QCD factorization. A particular feature of the two penguin contributions to the b → s transition is the near cancellation of these two terms in eq. (4) due to a c 4 ≈ a c 6 and r ≈ 1. This has been pointed out by Chernyak in his estimation of the scalar production in B decays [25] and also by Gardner and Meißner [15].

Final state interactions
In the B decays considered above, one should include in the final states the ππ → ππ or the KK → KK rescattering and the ππ → KK or the KK → ππ transitions. The (ππ) S and (KK) S pairs are formed from the uu, dd and ss pairs. The four scalar form factors appearing in eqs. (1) and (8) are defined [16] as where m is the effective ππ or KK mass, nn = (uu + dd)/ √ 2 and |0 > denotes the vacuum state. The final state interactions, which satisfy the unitarity constraints, are incorporated in the following formulae: where |k i and |k j represent the wave functions of two mesons in the momentum space and the indices i, j = 1, 2 refer to the ππ and KK channels, respectively. The center of mass channel momenta are given by k 1 = m 2 /4 − m 2 π and k 2 = m 2 /4 − m 2 K . The matrix T is the twobody scattering matrix. Here we use the solution A of the ππ and KK coupled channel model [17]. As we restrict ourselves to m ππ 1.2 GeV, the third effective (2π)(2π) coupled channel considered in this model, with a threshold around 1.4 GeV, has a small effect. The functions G i (m) are the free Green's functions defined in [17] and R n,s i (m) are the production functions responsible for the initial formation of the meson pairs prior to rescattering. The production functions have been derived by Meißner and Oller in the one-loop approximation of the chiral perturbation theory [16]. Using their eqs.
where m is in GeV.
If one considers only the on-shell contributions in eq. (11) then the scalar form factors can be written in terms of the phase shifts δ ππ (m), δ KK (m) and of the inelasticities η(m): Below the KK threshold η(m) = 1 and Particularly interesting is the m ππ range where the phase shifts δ ππ are close to 180 • . Then one expects a maximum in |Γ n, s 1 |. As we shall see in the next section, this is the case for the production of the f 0 (980) resonance. Note that the Γ n, s 1 is zero when δ ππ = π/2.
The off-shell contributions to the form factors are very much model dependent and are not considered in the following calculations.

Results
The amplitudes for the B → (ππ) S K decays considered in sect. 2 depend only on the effective mass m ππ . Integrating on the Dalitz plot over the kinematically allowed range of m πK , one obtains the differential B → ππK decay distribution where k 1 and p K = E 2 K (m ππ ) − m 2 K are the pion and kaon momenta in the ππ center of mass denotes the decay amplitude given by eq. (1) or (7). Dividing dΓ/dm ππ by the appropriate B + or B 0 total width Γ B one obtains the differential branching ratio dB/dm ππ .
Before presenting our results we fix the constants which appear in the formulae for the decay amplitudes (1), (7), (8) and (9). The masses of pions, kaons, B-mesons and their life times, the values of the Fermi coupling constant G F and the kaon decay constant f K = 0.1598 GeV are taken from [1]. For the kaon mass we use the average of the charged and neutral ones. Following the results obtained in [26] and applied in [15] we use F The B → K transition form factor is approximated by a constant equal to 0.39 which is close to the number F B→K 0 (0) = 0.379 quoted by Bauer, Stech and Wirbel in table 14 of [18]. This approximation is justified since we consider a relatively narrow range of the effective masses m ππ . These masses are much smaller than the mass of the heavy excited B meson used in polar models of the transition form factor. So we where m f0 is the f 0 (980) mass. The value 0.379 quoted in [18] as well as a more recent value of 0.33, obtained by Ball and Zwicky in [27], are within the limits given by Beneke and Neubert in table 1 of [28] for F B→K 0 (0) = 0.34 ± 0.05. Using the Wolfenstein representation [29], the CKM matrix elements are written in a form accurate to the level of λ 6 [30]. The values of the parameters, taken from the CKMfitter Group, are: λ ≡ V us = 0.2265, ρ = 0.189, η = 0.358 and A = 0.801 [31].
In this paper we do not attempt to make our own fits to data by adjusting the constants P GIM 1 and P 1 in the function C(m) of (6). We present the results obtained with the charming penguin amplitudes determined in [21] and [24]. The theoretical curves shown in figs. 2 to 5 correspond to the first set of amplitudes.
In figs. 2 and 3 we show a comparison of the ππ effective mass distributions for the B ± → π + π − K ± decays calculated in our model with the results obtained by BaBar [5] and Belle [7].
The branching fraction B(B ± → f 0 (980)K ± , f 0 (980) → π + π − ) = (9.2 ± 1.2 +2.1 −2.6 )× 10 −6 obtained by BaBar [5] can be reproduced in our model for the value of χ = 35.0 GeV −1 . This value is close to our estimation given above. The Belle Collaboration has reported a slightly smaller value of the branching ratio . Effective π + π − mass distribution in B ± → π + π − K ± decays. The BaBar data are taken from [5]. The solid line results from our model. are compatible within their error bars. The average value given by the Heavy Flavor Averaging Group (HFAG) is equal to (8.49 +1.35 −1.26 )×10 −6 [33]. For this value the constant χ is 33.5 GeV −1 when we use the charming penguin amplitudes of [21].
In fig. 2 the normalization of the theoretical curve to the data in the f 0 (980) range is based on the total number of events seen by BaBar for m ππ from 0.9 to 1.1 GeV, multiplied by a correction factor of 0.92 being a ratio of the branching ratios 8.49 × 10 −6 (HFAG's value) and 9.2 × 10 −6 (BaBar's value).
In fig. 3 the solid curve is normalized at m ππ = 976 MeV. This corresponds to the maximum of the background subtracted mass distribution. We calculate it from fig. 9e of [7] as 133 events/50 MeV. This number is obtained by taking into account two factors. The first factor, equal to 92%, follows from the fraction of the f 0 (980) components in the full spectrum resulting from the solution 1 of the Belle model, called Kππ − C 0 [7]. The second factor, equal to 1.125, comes from the ratio of the above mentioned branching ratios: 8.49 × 10 −6 and 7.55 × 10 −6 .   [7] π + π − effective mass distribution for the B ± → π + π − K ± decays with our model. The dashed line corresponds to the B + → π + π − K + decays, the dotted-dashed line is for the B − → π + π − K − decays and the solid line is the average for the B + and B − decays.
One can see from fig. 2 that our model describes quite well the π + π − spectrum measured by BaBar in vicinity of f 0 (980). The model depicts also a very pronounced maximum seen by Belle near 1 GeV (fig. 3). This maximum is attributed to the f 0 (980) resonance. At lower ππ masses near 500 MeV one can notice a broad theoretical maximum which we can relate to the σ or f 0 (600) meson. Experimental data in this mass range are not in disagreement with this feature of our model although the present errors are too large to draw a definite conclusion supporting the evidence of f 0 (600) in the B ± → π + π − K ± decay. Also the preliminary data of the BaBar Collaboration [6] below 600 MeV show some enhancement of the π + π − K ± events over the background. Let us remark that both collaborations have not included the σ meson in their fits to data. Integrating the ππ spectrum between the π + π − threshold and 700 MeV we find the average branching ratio for the B + → σK + and B − → σK − decays equal to 3.9 × 10 −6 . A surplus of events near 0.8 GeV is due to the B ± → ρ 0 K ± decay which are not taken into account in this model since we concentrate ourselves on the S-wave π + π − events.
In figs. 2 and 3 we have shown the averaged π + π − spectra of the two decays: B + → π + π − K + and B − → π + π − K − . They have been calculated for the charming penguin amplitudes C(m) of eq. (6) fitted in ref. [21] by P 1 = (0.068 ± 0.007) exp[i(1.32 ± 0.10)] and P GIM 1 = (0.32 ± 0.14) exp[i(1.0 ± 0.27)]. Note that the strength of the P 1 contribution to C(m) is about ten times larger than that of P GIM 1 . If we neglect both of them the branching ratios are smaller by a factor of about 4.
Using these long distance contributions we have found a very pronounced direct CP asymmetry in the π + π − spectra. There are many more decays of B + into π + π − K + than of B − into π + π − K − (see fig. 3) This asymmetry for the above choice of parameters is even higher than the direct CP asymmetry recently found in the B 0 → K + π − and B 0 → K − π + decays [34,35]. The charge asymmetry is defined as If we integrate dΓ/dm ππ between 0.9 and 1.1 GeV then A CP = −0.52 ± 0.12. The errors come from the uncertainties of the charming penguin amplitudes determined in [21]. It would be very useful to confront this number with a future experimental determination of this asymmetry in the B ± → f 0 (980)K ± decays. The charge CP violating asymmetry is very sensitive to the magnitude and the phase of the charming penguin contribution. Using the different approach of [22] and the fits presented in [24] for the B → Kπ decays the value P 1 = (0.08 ± 0.02) exp[−i(0.6 ± 0.5)] has been obtained, while P GIM 1 has not been determined. With this value of P 1 and with P GIM 1 = 0, one obtains a good agreement with the HFAG branching fraction for χ = 23.5 GeV −1 . Then the charge asym-metry is positive and equal to 0.20 ± 0.20. We see that the different charming penguin amplitudes fitted to data are not in mutual agreement. However, the two analyses are based on different data sets. In [21] 18 different branching ratios have been fitted and in [24] the fit has been performed on 8 observables for the B → Kπ decays.
In the following subsections the predictions for the two penguin amplitudes are given without any readjustment of the constants χ.

B
In fig. 4 we show the comparison of the model predictions for the neutral B decays with the BaBar results [4]. Here the experimental background contribution is added to the theoretical part calculated from eq. (7). The average branching ratio for the B 0 and B 0 decays into (π + π − ) S K 0 S , for the π + π − mass between 0.85 and 1.1 GeV, equals to 2.93 × 10 −6 . Twice this value compares well with the experimental determination of the BaBar Collaboration B(B 0 → f 0 (980)K 0 ) × B(f 0 (980) → π + π − ) = (6.0 ± 0.9 ± 0.6±1.2)×10 −6 . Our curve in fig. 4 is normalized to the total number of events multiplied by the ratio of 2 × 2.93/6.0 = 0.98. The branching ratio diminishes by a factor of 18 if the charming penguins amplitudes are omitted. This was expected due to the near cancellation, mentioned in section 2, between the two penguin diagram contributions. The absence of tree diagram in neutral B decays explains the difference with the charged B decays where the branching ratio drops by a factor of 4 if the charming penguin terms are not present.
The direct CP violation asymmetry between the decay of B 0 and B 0 into (π + π − ) S K 0 S defined as is much smaller than the asymmetry (19) for the B ± → (π + π − ) S K ± decays. It amounts to 0.01 ± 0.10 when calculated with the charming penguin parameters of [21] in the m ππ range between 0.85 and 1.1 GeV. The reason of its smallness is due to the absence of the tree diagram contribution  . Effective π + π − mass distribution for the B 0 → π + π − K 0 decays. The BaBar data and the dashed line corresponding to the background, are taken from [4]. The solid line is our model result.
The BaBar [3] and Belle [8] values for this asymmetry, A = 0.24 ± 0.31 ± 0.15 and A = −0.39 ± 0.27 ± 0.08, respectively, have large experimental errors and agree with our result. The HFAG average is −0.14 ± 0.22 [33]. In fig. 5 we compare the π + π − spectrum presented by K.-F. Chen on behalf of the Belle Collaboration for the B 0 → π + π − K 0 S decays [9] with our calculation when the background is subtracted. Our curve is normalized to the number of 94 events attributed to the f 0 (980) K 0 S decay in the m ππ range between 0.89 and 1.088 GeV [8]. The Belle experimental determination of the branching ratio for the B 0 → π + π − K 0 S decay is not yet available. As in the B ± → π + π − K ± case, shown in fig. 3, we predict some σ contribution of the decay amplitude in the low π + π − range below the position of ρ(770) enhancement visible in the data. The zero value of our spectrum near m ππ =0.8 GeV, also present in fig. 3, comes from δ ππ = π/2 (see eq. (15)).
We have also calculated the time-dependent CP asymmetry in the neutral B 0 decays into π + π − K 0 S : The time dependence of this asymmetry can be approximated as where ∆m is the difference between the masses of the heavy and light B meson eigenstates.
In the f 0 (980) mass range between 0.85 and 1.1 GeV the asymmetry parameter S equals to −0.63 ± 0.09. This result corresponds to the charming penguin parameters of [21]. The BaBar result is S = −0.95 +0. 32 −0.23 ± 0.10 [3] and the Belle number S = +0.47 ± 0.41 ± 0.08 [8]. The two experimental results are not in agreement with each other but their experimental errors are large. Our result agrees better with the BaBar value. The HFAG [33] gives the average S = −0.39 ± 0.26 which is in agreement with our prediction of −0.63. The value of S for the charming penguin amplitudes of [24], considered in section 4.1, is −0.77.

B → KKK decays
We have calculated the (K + K − ) S spectra and the branching ratios for the B + → (K + K − ) S K + and B − → (K + K − ) S K − decays. For the charming penguin amplitudes of [21] one obtains a large direct CP violating asymmetry of −0.44 ± 0.12 in the K + K − mass range between the threshold and 1.1 GeV. The average branching ratio for the above mass range equals to (1.8 ± 0.4) × 10 −6 . This value is below the upper limit of 2.9 × 10 −6 found by Garmash et al. for the branching fraction B(B + → f 0 (980)K + , f 0 (980) → K + K − ) [7]. The theoretical (K + K − ) S spectrum is flat in the range between 1.0 and 1.2 GeV and agrees well with the experimental distribution shown in fig. 13d of [7]. As in the case of the B ± → π + π − K ± decays the asymmetry for the B ± → (K + K − ) S K ± process strongly depends on the charming penguin amplitude. With the amplitudes of [24] considered in section 4.1, the asymmetry is positive and equal to 0.29 ± 0.21, the average branching ratio being (1.7 ± 0.7) × 10 −6 . In these calculations we neglect the symmetrized form factors Γ n,s 2 ( m KK ) in eq. (8) assuming that for small masses of m KK the m KK masses are sufficiently high and that the Γ 2 form factors decrease rapidly with increasing m KK .
Within our model one finds the same asymmetries for the B 0 → K 0 S process provided that Table 1 Average branching fractions B in units of 10 −6 , asymmetries A CP , A and S of our model compared to the average values of HFAG [33]. The m ππ mass ranges for the B ± → f 0 (980)K ± and for the B 0 → f 0 (980)K 0 decays are (0.9, 1.1) GeV and (0.85, 1.1) GeV, respectively. The upper limit of the (K + K − ) S or K 0 S K 0 S S effective mass is 1.1 GeV. The model errors come from the uncertainties of the charming penguin amplitudes C(m) (eq. (6)) determined in the fits of [21]

Summary and outlook
We have analyzed the charged and neutral three-body B meson decays into the π + π − K, K + K − K and K 0 S K 0 S K 0 S systems. Our primary aim is a construction of decay amplitudes including the π + π − final state interactions in a rather wide range of effective masses between the ππ threshold and 1.2 GeV. The model is based on a factorization approximation with some QCD corrections and contains the dominant charming penguin terms. Using a single amplitude, we are able to describe simultaneously the production of two scalar-isoscalar resonances f 0 (600) and f 0 (980). The B decay amplitudes to the ππK and KKK states are connected as the coupling between the ππ and KK channels above 1 GeV is incorporated in our model. No adjustable free parameters, like arbitrary phase factors between contributions of different resonances, are needed. We have obtained a good agreement with most of the recent BaBar and Belle data for the ππ effective mass distributions, the branching ratios and the time-dependent CP violating asymmetries. Our results are summarized in table 1. These numbers depend only weakly on the choice of the renormalization scale, the corresponding quark masses m b and m s , and the value of F B→K 0 (0). The changes are smaller than the errors in the determination of the charming penguin amplitudes.
If we use the charming penguin amplitudes determined in [21] then the direct CP violation asymmetry in the charged B decays to ππK is strongly negative (∼ −0.5). However, for the charming penguin amplitudes taken from [24] this asymmetry is positive (∼ +0.2). A similar difference is found for the B ± → K + K − K ± reactions. Future independent measurements of the B ± → π + π − K ± and B ± → K + K − K ± asymmetries will be crucial for a decisive test of the phase of the long distance contributions. Let us stress the importance of charming penguin amplitudes. If we omit them then the average branching ratio for the B 0 → π + π − K 0 S decay is too small by a factor of 18 and that of the B ± → π + π − K ± mode by a factor of 4. Even an ad hoc adjustment of the constant χ to fit the experimental charged B decays does not allow one to explain the neutral B decays. However, when the charming penguin amplitudes are included, we get a good agreement for both channels for χ values close to the estimation based on the f 0 (980) properties.
The model presented in this paper can be extended to larger effective ππ mass range, in particular to the range where the f 0 (1500) is important. One can also include the final state interactions between one of the pions and the kaon in the B or D decays to the ππK system. Especially interesting are the π − K + or π + K − subsystems where scalar and vector resonances can play an important role.