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Rescattering and chiral dynamics in B\to \rho\pi decay

We examine the role of B^0(\bar B^0) \to \sigma \pi^0 \to \pi^+\pi^- \pi^0 decay in the Dalitz plot analysis of B^0 (\bar B^0) \to \rho\pi \to \pi^+\pi^-\pi^0 decays, employed to extract the CKM parameter \alpha. The \sigma \pi channel is significant because it can break the relationship between the penguin contributions in B\to\rho^0\pi^0, B\to\rho^+\pi^-, and B\to\rho^-\pi^+ decays consequent to an assumption of isospin symmetry. Its presence thus mimics the effect of isospin violation. The \sigma\pi^0 state is of definite CP, however; we demonstrate that the B\to\rho\pi analysis can be generalized to include this channel without difficulty. The \sigma or f_0(400-1200) ``meson'' is a broad I=J=0 enhancement driven by strong \pi\pi rescattering; a suitable scalar form factor is constrained by the chiral dynamics of low-energy hadron-hadron interactions - it is rather different from the relativistic Breit-Wigner form adopted in earlier B\to\sigma\pi and D\to\sigma\pi analyses. We show that the use of this scalar form factor leads to an improved theoretical understanding of the measured ratio Br(\bar B^0 \to \rho^\mp \pi^\pm) / Br(B^-\to \rho^0 \pi^-).


Introduction
Measurements at SLAC and KEK of the time-dependent CP-violating asymmetry in B(B) → J/ψK s [1,2], yielding sin(2β), have conclusively established the existence of CP violation in the B meson system. The results found are consistent with Standard Model (SM) expectations [3], so that establishing whether the Cabibbo-Kobayashi-Maskawa (CKM) matrix [4] is the only source of CP violation in nature, as in the SM, or not requires the empirical measurement of all the angles of the unitarity triangle.
In this paper we consider the determination of α through a Dalitz plot analysis of the decays B 0 (B 0 ) → ρπ → π + π − π 0 under the assumption of isospin symmetry [5,6]. Ten parameters appear in the analysis, and they can be determined in a fit to the data. Nevertheless, the assumption of ρ dominance in B → 3π decays has no strong theoretical basis [7], so that the contributions from other resonances in the ρπ phase space may be important. We discuss how the isospin analysis can be enlarged to include the σπ channel as well. The σ or f 0 (400 − 1200) "meson" is a broad J = I = 0 enhancement, close to the ρ meson in mass, so that the σπ channel can potentially populate the 3π phase space associated with the ρπ channels. The σπ final state contributes preferentially to the ρ 0 π 0 final state. In the context of the isospin analysis, such contributions are of consequence as they invalidate the underlying assumptions of the isospin analysis and thus mimic the effect of isospin violation.
Our considerations are inspired in part by recent studies of D − → π − π + π − decay: the E791 collaboration find that the pathway D − → π − σ → π − π + π − accounts for approximately half of all D − → π − π + π − decays [8]. Deandrea and Polosa have argued as a consequence that the B → σπ channel contributes significantly to the ρπ phase space in B → ππ + π − and modifies the ratio B(B 0 → ρ ∓ π ± )/B(B − → ρ 0 π − ) to yield better agreement with experiment [9]. The scalar form factor, which describes the appearance of the σ in the π + π − final state, enters as a crucial ingredient in the assessment of the size of these effects. The scalar form factor cannot be determined directly from experiment; nevertheless, ample indirect constraints exist, permitting us to describe its features with confidence [10]. Nevertheless, different approaches, with different dynamical assumptions, yield roughly comparable descriptions of the ππ scattering data, so that the emergence of a favored form of the scalar form factor does not resolve the question of whether the σ is a pre-existing resonance or, rather, a dynamical consequence of ππ interactions in the final-state. We follow Ref. [10] and adopt an unitarized, coupled-channel approach to the final-state interactions (FSI) in the ππ-KK system, and match the resulting scalar form factor to chiral perturbation theory (CHPT) in the regime where the latter is applicable. The resulting form factor, in the ππ channel, discussed in Sec. 5, is strikingly different from the relativistic Breit-Wigner form adopted by the E791 collaboration in their analysis of the σ in D + → π + π + π − decay -the latter form factor is also used in Ref. [9]. The differences are particularly large as √ s → 2M π , so that the relativistic Breit-Wigner form is at odds with CHPT in the precise region where it is applicable, note Fig. 4 in Ref. [10]. This casts doubt on the recent conclusions of Refs. [8,9], prompting new analyses incorporating a suitable scalar form factor.
The generation of the σ resonance via strong rescattering effects, as in the approach we adopt, indicates that OZI-violating effects in the scalar sector are significant. Moreover, the "doubly" OZI-violating form factor 0|ss|ππ is non-trivial as well; such a contribution is needed to fit the ππ and KK invariant mass distributions in J/ψ → φππ(KK) decay [10]. These effects are also needed to explain the branching ratios of the decays of the a 0 (980) and f 0 (980) states into ππ and KK final states [11]. These observations give new insight on rescattering effects in hadronic B decays, generating a new mechanism of factorization breaking in n ≥ 3 particle final states.
The contribution of the B → σπ channel to the B → ρ 0 π phase space can also modify the inferred empirical branching ratios in these channels. Combining the CLEO results [12] Br(B − → ρ 0 π − ) = (10.4 +3. 3 −3.4 ± 2.1) · 10 −6 , Br(B 0 → ρ ± π ∓ ) = (27.6 +8.4 −7.4 ± 4.2) · 10 −6 (2) with the BaBar result [13] Br(B 0 → ρ ± π ∓ ) = 28.9 ± 5.4 ± 4.3 (charge conjugate modes are implied) yields, adding the errors in quadrature and ignoring correlations, This ratio of ratios is roughly 6 if one works at tree level and uses the naive factorization approximation for the hadronic matrix elements [14]. The inclusion of penguin contributions can alter this result, and potentially yield better accord with theory and experiment [15,16,17,18]. However, our focus will parallel that of Ref. [9]: we wish to examine how B → σπ → 3π decay, given a particular scalar form factor, can effectively modify the theoretical prediction of the ratio given in Eq. (3). It is apparent that B → σπ is of greater impact in B → ρ 0 π decay, so that the inclusion of such contributions ought alter the ratio of ratios. We begin by reviewing the isospin analysis in B 0 (B 0 ) → ρπ → π + π − π 0 decay [5,6] in Sec. 2, and discuss its extension to include B 0 (B 0 ) → σπ → π + π − π 0 ) decay in Sec. 3. We proceed by evaluating σ-mediated B → 3π decay in Sec. 4, relegating the ρ-mediated B → 3π decay formulae to App.A. Our analysis employs the scalar and vector form factors discussed in Sec. 5 and Sec. 6, respectively. We conclude with a presentation of our results in Sec. 7 and an accompanying summary.

Preliminaries: Isospin analysis of B → ρπ
Let us recall the isospin analysis possible in B → ρπ decay [5,6]. Under the assumption of isospin symmetry, a ρπ final state can have isospin I f = 0, 1, or 2, whereas the B + , B 0 states form an isospin doublet. Thus we can have |∆I| = 1/2, 3/2, or 5/2 transitions in B → ρπ decay, so that we can parametrize the amplitudes which appear by A |∆I|,I f . We have [5,19] and Isospin is merely an approximate symmetry of the SM; nevertheless, our parametrization possesses three independent "isospin" amplitudes, distinguished by I f , to describe the three empirical amplitudes a ij , so that it persists in the presence of isospin breaking as well.
The lowest-dimension operators of the effective, |∆B| = 1 Hamiltonian generate transitions of |∆I| = 1/2 or 3/2 character, so that a |∆I| = 5/2 transition is generated, in this order, through long-distance, isospinbreaking effects in concert with a |∆I| = 1/2 or 3/2, short-distance, weak transition. If we neglect transitions of |∆I| = 5/2 character and, indeed, isospin-violating effects all together, the transitionb → qqd which mediates B → ρπ decay can be realized through a "tree" amplitude with |∆I| = 1/2 or 3/2 or through a "penguin" amplitude with |∆I| = 1/2. Practically, the decay topologies are distinguished by their weak phase, so that the contributions associated with the CKM factors V * ub V ud , e.g., are defined to be tree contributions, regardless of their dynamical origin. The unitarity of the CKM matrix in the SM implies that two combinations of CKM factors suffice in describing b → qqq ′ ; here we associate the combination V * tb V td with the penguin contribution.
The overall weak phase e iβ is without physical impact and can be neglected, because in the SM the weak phase associated with B 0 −B 0 mixing is controlled by q/p = exp(−2iβ) #4 . Thus qā ij /p ∝ exp(−iβ), just as a ij is. Consequently, the isospin analysis in B → ρπ decay determines α. The crucial assumption of the isospin #3 We flip the overall sign of the a 00 amplitude with respect to Ref. [19], to conform with our computation of the amplitudes. #4 Recall that the B mass eigenstates are defined via |B L = p|B 0 + q|B 0 and |B H = p|B 0 − q|B 0 . We assume throughout that the width difference of the two B mass eigenstates is negligible, so that |q/p| = 1.
The overall strong phase in Eq. (8) is trivial, so that with Eq. (9), we have ten parameters in all, which can be determined in an analysis of the Dalitz plot [6,19]. The presence of the σπ final state in the phase space associated with the ρ 0 π 0 channel breaks the relation assumed in Eq. (9), and thus mimics the appearance of isospin violation. In this paper we study how the impact of this additional decay channel can be minimized. It is worth noting, however, that the σπ final state is of definite CP, so that the isospin analysis can be enlarged to include this channel as well -additional observables are also present in this case. Before doing this, let us enumerate the ways in which SM isospin violation can impact the usual B → ρπ analysis, to determine whether the impact of these effects can be reduced as well: i) Isospin violation can generate an additional amplitude, of |∆I| = 5/2 character, as in Eqs. (4,5,6). A |∆I| = 5/2 amplitude can be generated by O(m d − m u ) or O(α) effects in concert with a |∆I| = 3/2 weak transition, or by O(α) effects in concert with a |∆I| = 1/2 weak transition. The O(m d − m u ) term acts as an isovector interaction. We recall that the physical neutral pion state is an admixture of the pseudoscalar octet fields π 0 and η; that is, (π 0 ) phys = π 0 + ǫη with ǫ ∼ O(m d − m u ). Consequently ǫ acts as an I = 1 "spurion" [20], encoding isospin-violating effects so that the matrix elements with the spurion are SU (2) f invariant. Isospin violation is also realized via the B + , B 0 mass difference; such effects are not encoded in the spurion framework, but they are also comparatively trivial.
The impact of these isospin-violating effects can be redressed, at least in part. For example, in B → ρπ decay, the A 5/2,2 amplitude appears in the combination A 5/2,2 + A 3/2,2 throughout Eqs. (4,5,6). Moreover, the two amplitudes share the same weak phase, to a good approximation. This emerges because, unlike K → ππ decay [28], no "|∆I| = 1/2 rule" apparently exists in B → ππ decay -though B(B + → π + π 0 ) has yet to be conclusively determined [29]. Recent theoretical estimates suggest that the magnitude of the ratio of the |∆I| = 1/2 to |∆I| = 3/2 amplitudes in B → ππ decay is roughly 0.3 [30], so that the |∆I| = 5/2 amplitude is driven by an underlying |∆I| = 3/2 weak transition. Strong-interaction isospin violation acts in concert with a |∆I| = 3/2 weak transition to generate a |∆I| = 5/2 amplitude, whereas electromagnetism can generate a |∆I| = 5/2 amplitude from a |∆I| = 1/2 weak transition. The size of strong-interaction isospin violation is typified by the π 0 − η mixing angle ǫ (2) [31], enhancing the extent to which A 5/2,2 and A 3/2,2 share the same weak phase. To the degree that this is true, the phenomenological T ij parameters of Eq. (8) include |∆I| = 5/2 effects as well. Thus we see that the single, crucial assumption of the isospin analysis is that the CKM factor V * tb V td accompanies |∆I| = 1/2 transitions exclusively, for in this case the weak phases of the A 5/2,2 and A 3/2,2 amplitudes are identical. We have shown that isospin-violating contributions built on the |∆I| = 3/2 short-distance, weak transition do not impact the isospin analysis in B → ρπ. However, non-|∆I| = 1/2 penguin effects, be they electroweak penguin contributions or contributions consequent to isospin-violating effects in the hadronic matrix elements of |∆I| = 1/2 operators, present a irreducible hadronic ambiguity from the viewpoint of this analysis.
Empirical information on the all-neutral mode, a 00 , is essential to the extraction of α; however, it is possible to bound the strong-phase uncertainty using bounds on a 00 and its CP-conjugateā 00 [19]. Under the assumptions we have articulated, the bounds on the hadronic uncertainty realized in B → ρπ decay [19] are not modified by the presence of a |∆I| = 5/2 transition.
An isospin analysis of B → ππ decay also permits the extraction of sin(2α) from the mixing-induced CP asymmetry in B → π + π − [32]. In this case, in contrast, the |∆I| = 5/2 amplitude cannot be combined with the |∆I| = 3/2 amplitude. With A |∆I|,I f , we have As the case of B → ρπ, three isospin amplitudes describe three empirical amplitudes, so that we expect our parametrization to persist in the presence of isospin violation. The lowest-dimension operators of the effective weak Hamiltonian generate transitions of |∆I| = 1/2 and |∆I| = 3/2 character, so that in the absence of isospinviolating effects in the hadronic matrix elements, A 1/2,0 → A 0 and A 3/2,2 → A 2 , and two amplitudes suffice to describe the three transitions. The isospin analysis in B → ππ relies on the relation (b +− −b 00 )/ √ 2−b +0 = 0 [32]. The right-hand side of this relation is proportional to A 5/2,2 , so that the required relation is broken by amplitudes of |∆I| = 5/2 character. With such effects the isospin analysis can fail to determine the true value of sin(2α) [25]. The small B → π 0 π 0 rate makes the full isospin analysis difficult to effect, so that bounds on the hadronic uncertainty in the extraction of sin(2α) have also been constructed [33,34,35,36]. The presence of the |∆I| = 5/2 amplitude, as well as that of electroweak penguins, imply that the bounds can underestimate the size of the hadronic uncertainty [25]. However, bounds which rely on the neutral B modes exclusively, such as Eq. (83) of Ref. [33], contain the same linear combination of |∆I| = 3/2 and |∆I| = 5/2 amplitudes throughout -so that our arguments concerning the |∆I| = 5/2 amplitude in B → ρπ decay are germane here as well. We conclude, to the extent the |∆I| = 3/2 and |∆I| = 5/2 amplitudes share the same weak phase, that such bounds are insensitive to the |∆I| = 5/2 amplitude and yield more reliable bounds on the hadronic uncertainty.

Extension of the isospin analysis: Inclusion of the σπ channel
The B → σπ channel has definite properties under CP, so that it can be included in the B → ρπ analysis as well. Defining a σ 00 = A(B 0 → σπ 0 ), we have e iβ a σ 00 = T 00 σ e −iα + P 00 σ .
T 00 σ and P 00 σ are unrelated to the parameters of Eq. (8), so that we gain four additional hadronic parameters. However, more observables are present as well. Including the scalar channel, we now have A 3π ≡ A(B 0 → π + π − π 0 ) = f + a +− + f − a −+ + f 0 a 00 + f σ a σ 00 , where f σ is the form factor describing σ → π + π − . It is worth noting that any discernable presence of the B → σπ channel in the B → ρπ phase space falsifies the notion that the "nonresonant" background can be characterized by a single, constant phase across the Dalitz plot [37]. For further discussion of the treatment of nonresonant contributions, specifically in D → 3π decay, see Ref. [38] note also Ref. [7].
We note that λ 3π ≡ qĀ 3π /pA 3π , where we have definedĀ 3π ≡ A(B 0 → π + π − π 0 ), and ∆m ≡ M H − M L . Different observables are possible. For example, we can consider untagged observables, for which the identity of the B meson at t = 0 is unimportant, so that Γ(B 0 (t) → π + π − π 0 ) + Γ(B 0 (t) → π + π − π 0 ) ∝ (1 + |λ 3π | 2 ), or we can consider time-integrated, tagged observables, containing Γ(B 0 (t) → π + π − π 0 ) − Γ(B 0 (t) → π + π − π 0 ), which are sensitive to (1 − |λ 3π | 2 ). The products f i f * j contained therein are distinguishable through the Dalitz plot of this decay and thus the coefficients of these functions are distinct observables [6]. Were we to neglect the σπ channel, nine distinct, untagged observables exist, so that all the hadronic parameters save one would be determinable from the untagged data, for which greater statistics will be available [19]. If we enlarge the analysis to include the σπ 0 channel, the additional interferences possible imply that there are now sixteen distinct, untagged observables. Moreover, there are fifteen, rather than eight, tagged, time-integrated observables as well. Nevertheless, it would seem that the additional hadronic parameters associated with the σπ 0 final state can be extracted from untagged data alone. Of course the practicability of the procedure relies on the amount of data eventually collected; moreover, the observables are highly correlated.
The effective, |∆B| = 1 Hamiltonian for b → dqq decay is given by where λ q ≡ V qb V * qd with V ij an element of the CKM matrix. The Wilson coefficients C i and operators O i are detailed in Ref. [40], though we shall interchange 10 . The contributions with i = 3 . . . 6 correspond to strong penguin effects, whereas those with i = 7 . . . 10 are characterized by electroweak penguin effects. The Wilson coefficients with i = 7 . . . 10 are numerically smaller than those with i = 3 . . . 6, and penguin effects are not CKM-enhanced in b → dqq decay, so that we shall neglect the terms with i = 7 . . . 10 all together.
The decay amplitude for B → M 1 M 2 , where M 1 and M 2 are mesons, is given by The requisite matrix element contains terms of the form We adopt the naive factorization approximation to effect estimates of the hadronic matrix elements. To wit, we separate O i into a product of factorized currents, j 1 ⊗ j 2 , and evaluate M 1 |j 1 |B M 2 |j 2 |0 , so that the operator matrix element becomes a product of a form factor and a decay constant. Such a treatment, albeit simple, is incomplete. The amplitude A(B → M 1 M 2 ) is related to a physical observable and as such must be µ-independent, though the C i therein do depend on µ. Evidently the µ dependence of the operator matrix elements compensates to yield a µ independent result. In the naive factorization approximation, we have replaced the operator matrix element by a product of a form factor and decay constant. These quantities are themselves physical observables and thus are without µ dependence, so that the overall µ dependence of the computed amplitude remains. Effecting this approximation, however, allows us to realize a clear connection to earlier work [9,16], for our purpose is to illustrate the impact of using a scalar form factor consistent with low-energy constraints.
In the naive factorization approximation, we can replace the effective Hamiltonian by the sum of products of factorized currents, so that H eff = T 1,2 + T 3,4 + T 5,6 , where and We define a i ≡ C i + C i+1 /3 for i odd and a i ≡ C i + C i−1 /3 for i even; note, too, that q ∈ u, d, s, c if µ m b . We now specifically consider B → σπ transitions; the relevant formulae for B → ρπ are detailed in App. A. By "σ," we always mean a two-pion state with total isospin zero and in a relative S-wave state, (ππ) S , understanding its dynamical origin in the strong pionic FSI for these quantum numbers -see the following section for a more detailed discussion. The matrix elements involving the σ are where q ≡ p B − p σ and σ|q ′ γ µ (1 + γ 5 )q ′ |0 , noting q ′ ∈ u, d, s, vanishes by C-invariance. It is just such a suppression mechanism that prompts the authors of Ref. [41] to argue that non-factorizable effects ought be relatively enhanced in two-body B decays to final states with scalar mesons, as the factorization contribution is itself small. Nevertheless, as our focus is the scalar form factor itself, we proceed with our estimates.
For the π + we have and With these relations, Eqs. (19,20,21), and the equations of motion for the quark fields, we have and where we have replaced the u, d quark masses withm and set M π 0 = M π ± = M π and M B ± = M B 0 ,B 0 = M B , as we neglect isospin-violating effects. We adopt the usual phase conventions for the flavor wave functions π + , π 0 , π − = ud, (uū − dd)/ √ 2, dū, and adopt analogous relations for the ρ mesons as well. In the context of the B → ρπ analysis, the decayB 0 → σπ, specifically its penguin contributions, modify Eq. (9), the assumption crucial to the analysis. Thus it is important to make an assessment of the size of penguin effects in this decay. The terms containing a 6 , the scalar penguin contribution, are formally 1/m b suppressed, but can be chirally enhanced: the numerical factor M 2 π /(mm b ) ∼ 0.6 is only modestly less than unity. The second term proportional to a 6 contains σ|dd|0 , where we anticipate, in the vicinity of the σ resonance, is the scalar form factor, which we detail in the next section, and s = (p + + p − ) 2 . The parameter B 0 is related to the vacuum quark condensate. Neglecting small terms of second order in the quark masses, B 0 = − 0|qq|0 /F 2 π , where F π , the π 0 decay constant, is f π / √ 2. Commensurately, we can simply set B 0 ≡ M 2 π /(2m) to realize our numerical estimates. Note that Γ σππ describes the σ → π + π − form factor. With our conventions, B 0 > 0, so that the two a 6 contributions are of the same sign. Using the parameters of Ref. [9] and Γ σππ = Γ n * 1 χ with, as we shall determine, χ = 20.0 GeV −1 , we find that the a 6 term containing f π to be roughly a factor of four larger. The σ|dd|0 term, present in the penguin contributions in B → σπ, slightly enhances the a 6 contribution and its subsequent cancellation with the a 4 contribution, as a 4 and a 6 are of the same sign. The same cancellation occurs inB 0 → ρ 0 π 0 (see App. A for a compilation of the relevant formulae), so that penguin contributions inB 0 → σπ 0 can be expected to be crudely comparable to those inB 0 → ρ 0 π 0 . Modifications of Eq. (9) can thus be expected to occur.
In the previous section we determined how the isospin analysis in B → ρπ could be extended to include B → σπ. Eqs. (25) and (26), however, are related by isospin symmetry, so that it is useful to determine whether additional constraints on P 00 σ in Eq. (13) can be realized. Parametrizing the amplitudes in terms of A |∆I|,I f , we have and so that two isospin amplitudes appear for the two empirical amplitudes. Although the |∆I| = 1/2 amplitudes are simply related, we see that no useful constraint emerges, as the tree amplitude in B + → σπ + , which includes |∆I| = 1/2 and 3/2 amplitudes, gives rise to two additional, undetermined hadronic parameters.
We have illustrated that the penguin relation essential to the isospin analysis in B → ρπ, Eq. (9), can be broken through the presence of the B → σπ decay channel. In our numerical estimates, however, we neglect penguin contributions, in order to retain a crisp comparison with earlier work [9], for our purpose is to illuminate the impact of the σ → π + π − form factor. With such an approximation, a computation of the Wilson coefficients in leading order in α s suffices [42], so that C 1 (µ) = 1.124 and C 2 (µ) = −0.271 at µ = m b = 4.8 GeV as per Ref. [43], to yield a 1 = 1.034 and a 2 = 0.104. In contrast, Ref. [9] uses the "fitted" values C 1 (m b ) = 1.105 and C 2 (m b ) = −0.228, to yield a 1 = 1.029 and a 2 = 0.140. These are very similar to the Wilson coefficients in next-to-leading-order QCD, after Ref. [44], used in the B → ρπ analysis of Ref. [16]. For definiteness, we shall adopt these last values, detailed in Sec. 7, in our numerical analysis. The values of a 1 are quite similar, whereas those of a 2 differ by tens of percent. Generally, we expect our numerical predictions for decay channels controlled by a 2 to be less robust, as the scale dependence of a 2 , as illustrated in Table III of Ref. [43], is severe. Note that it persists to a significant degree in the next-to-leading order treatment of Ref. [30] as well.
Nevertheless, let us proceed to consider numerical estimates for B → σπ decay. We reconstruct the σ meson from the (π + π − ) S final state, so that we have where Γ σππ is the σ → π + π − form factor introduced previously. The scalar form factor contains the meson loop function and thus a regularization scale µ r at which it is evaluated. In our approach, this scale dependence is disjoint from that associated with the renormalization of the operators of the effective, weak Hamiltonian, so that it is chosen for convenience and is quite independent of µ. It may seem untoward to graft two very different calculations, namely, of A(B → σπ) and of Γ σππ , to yield A(B → (σ → π + π − )π). In a holistic treatment one might hope to recast a resonance and its subsequent decay products in terms of a single, complex hadron distribution amplitude, φ(x, µ), which describes the non-perturbative dynamics. The analysis of the decay amplitude could then proceed via standard pQCD techniques [45,46]. In this manner the µ r dependence of which we have spoken is connected, albeit loosely, to the scale dependence of φ(x, µ). Nevertheless, the explicit "QCD factorization" analysis of Ref. [30] shows that the scale dependence of the hadron distribution functions is trivial in an NLO analysis in α s , so that the consistency issues to which we have alluded are beyond the scope of current calculations. Fig. 1. (The corresponding contributions to B → ρπ decay are illustrated in Fig. 2.) The two contributions add coherently, so that the branching ratio for B − (p B ) → π + (p + )π − (p 1 )π − (p 2 ) is enhanced through the presence of two identical pions in the final state. InB 0 decay this does not occur, and we haveB 0 (p B ) → π + (p + )π − (p 1 )π 0 (p 2 ). Thus we find In Ref. [9], the σ → π + π − vertex function is chosen to be where the running width Γ σ (x) is defined as We shall adopt, however, the definition where the normalization χ is fixed to be identical to that of Eq. (32), namely Using the parameters g σπ + π − , M σ , and Γ σ of Ref. [9], that is, M σ = 478 ± 24 MeV and Γ σ = 324 ± 41 MeV as reported by the E791 collaboration in D + → 3π, and g σππ = 2.52 GeV, we find χ = 20.0 GeV −1 . Alternatively, χ = 2/3B 0 / σ|dd|0 , so that this procedure determines σ|dd|0 as well.
The σ meson is a broad I = J = 0 enhancement, close to the ρ meson in mass, so that B → σπ decay can contribute to the allowed phase space of B → ρπ decay as well. To ascertain the impact of the B → σπ channel to B → ρπ decay, we combine the decay channels at the amplitude level and then integrate over the relevant three-body phase space to determine the effective B → ρπ branching ratio. For B(p B ) → π + (p + )π − (p 1 )π(p 2 ) decay, we define where the primed variables refer to the momenta in the rest frame of the π + (p ′ where "ρ eff " is determined by (M ρ − δ) 2 ≤ t ≤ (M ρ + δ) 2 , β ′ i refers to the velocity of particle i in the primed frame, and M is the sum of the amplitudes of interest #5 . Typically δ ∼ (1 − 2)Γ ρ .
The scalar form factor we adopt describes the f 0 (980) → π + π − vertex function as well, albeit with a new normalization factor, as Γ f0ππ (t) f 0 |dd|0 = π + (p + )π − (p 1 )|dd|0 = 2/3B 0 Γ n * 1 (t), in the vicinity of the f 0 (980) resonance. The appearance of the f 0 (980) resonance in the π + π − channel is complicated by the opening of the KK threshold; the f 0 (980) → π + π − form factor is decidedly not of Breit-Wigner form. It should be noted that the unitarization procedure we employ here neglects the ηη channel, though, as we discuss in the next section, this has no impact on the description of ππ scattering below s ≃ 1.1 GeV. Note, too, that although multiparticle final states, particularly the 4π state, can contribute, they are demonstrably small for s ≤ 1.4 GeV [47,48]. The form factor(s) we adopt can be tested though the shape of the f 0 (980) contribution in B → f 0 (980)π → 3π, as well as through that in B → f 0 (980)π → K + K − π.

Coupled-channel pion and kaon scalar form factors
In the preceding section, we encountered the non-strange, scalar form factor of the pion, Γ n 1 (t). Such scalar form factors play a unique role in strong-interaction physics because they measure the strength of the quark mass term H QCD m = m uū u + m dd d + . . ., i.e., the explicit chiral-symmetry-breaking term in QCD. However, since no scalar-isoscalar sources exist, these form factors cannot be determined directly but, rather, must be inferred indirectly from hadron-hadron scattering data. The most prominent example in this context is the pionnucleon sigma term, which can be extracted from the analytically-continued and Born-term-subtracted isoscalar S-wave amplitude. Similarly, the pion scalar form factor can be obtained from meson-meson scattering data, with the added complication of channel couplings above theKK threshold at √ s ≃ 1 GeV. At lower energies, the pion scalar form factor can be calculated in CHPT at one- [49] and two-loop accuracy [50,51]. The oneloop representation fails at surprisingly low energies, a consequence of the strong pionic FSI present in this #5 For completeness, we note that u = M 2 channel [52]. This is also signalled by the very large pion scalar radius, r 2 S π ≃ 0.6 fm 2 , which is sizeably bigger than the pion vector radius governed by the rho mass r 2 V π ≃ 6/M 2 ρ ≃ 0.4 fm 2 , indicating a smaller breakdown scale in the scalar-isoscalar channel. In fact, these FSI are so strong that they can dynamically generate the σ meson, with no need of a genuine (pre-existing) quark model state [53]. Furthermore, extending such an analysis to three flavors, one finds that all the scalar mesons with mass below 1.1 GeV can be dynamically generated, so that the genuine quark model nonet would have its center of gravity at about 1.3 GeV, see e.g. Ref. [54] and references therein. Irrespective of this admittedly controversial assignment of the scalars, for further recent discussion see Ref. [55], the pion scalar form factor cannot be represented simply in terms of a scalar meson with a certain mass and width. Although the description of the low-lying scalar states in terms of dynamically generated, rather than "pre-existing," states may be controversial and thus subject to ongoing discussion, let us stress that the scalar form factor itself is not. The form factor which emerges for the chiral unitary approach adopted here is quite comparable to that which emerges from the dispersion analysis of Ref. [56]. Consequently, when one has a source with vacuum quantum numbers coupled to a two-pion state, it can be very misleading to use a simple Breit-Wigner parametrization, albeit with a running width. We shall display this graphically after discussing the scalar form factor and its construction, to be used in calculating the branching ratios to which we have alluded. We note in passing that this casts doubt on the extraction of the σ meson properties from D → 3π decays, note Ref. [8]. This is quite in contrast to the pion vector form factor and the ρ meson, for which such a description works to good accuracy.
We now summarize how to calculate the scalar form factor in Eq. (34), following Ref. [10], to which we refer in all details. Of course, one could also use the numerical result of the dispersion analysis of [56] for the scalar form factor. Note that the band found there overlaps with the band of Ref. [10] if one allows for parameter variations within known bounds. As a first step, we prefer to work in the framework of the chiral unitary approach, as it yields a form factor which is more convenient for applications. As we have noted, we need the scalar form factor of the pion for momentum transfers larger than 1 GeV. Consequently, we cannot work with the scalar form factor as computed in CHPT, but must invoke some resummation technique, as well as account for the channel coupling between the ππ and the KK systems. The resummation method is constrained only by unitarity and thus is not entirely model-independent. However, it can be strongly constrained by requiring that the so-constructed form factors match to the CHPT expressions, in the region where CHPT is applicable. Due to the channel coupling, we have to consider transition matrix elements of the non-strange and strange scalar quark bilinears between two meson states of isospin zero, namely ππ andKK, and the vacuum. We exclude the ηη channel, as it does not affect the phase shifts or pion/kaon decays and transitions -it only plays a role in describing the inelasticity of I = J = 0 ππ scattering above 1.1 GeV, the physical ηη threshold. For a detailed discussion of this point, we refer to Refs. [53,57,58]. The pertinent matrix elements are given in terms of four scalar form factors #6 , where B 0 is a measure of the vacuum quark condensate, B 0 = − 0|qq|0 /F 2 π , with F π ≃ 93 MeV the pion decay constant (strictly speaking its value in the limit of vanishing quark masses). In Eq.(38), the following notation is employed. The superscript s(n) refers to the strange/non-strange quark operator whereas the subscript 1(2) denotes pions and kaons, respectively. Furthermore,nn = (ūu +dd)/ √ 2. The pion scalar form factors Γ n 1 (s) and Γ s 1 (s) are calculated in Ref. [49,59] through one loop in CHPT and Γ n 1 (s) through two loops in Refs. [50,51]. The scalar kaon form factors at next-to-leading order in CHPT were first explicitly given in [10]. Since it is central to our discussion, we give the explicit expression for Γ n 1 (s) [59] Γ n 1 (s) = with f (s) and f (s) given by #6 Here "ππ" and "KK" states denote the linear combinations of physical ππ and KK states, respectively, with zero total isospin.
Here,J P P (s) (P = π, K, η) is the standard meson loop function [49], and µ, in this section, is the scale of dimensional regularization. The quantities µ P in Eq.(39) are given by Furthermore, the L r i (µ) are scale-dependent, renormalized low-energy constants. We use here the same values as in Ref. [10]. Since the combination 2L r 6 − L r 4 multiplies M 2 K , the normalization of Γ n 1 (s) is sensitive to the precise value of these low-energy constants, which are only poorly known. This motivates our choice for the normalization of Γ σππ (s), given in Eq. (35). The one-loop representation can not be trusted beyond √ s ≃ 400 MeV, as it begins to diverge from the form factor extracted from ππ scattering data. To go to higher energies, one therefore has to study the constraints that unitarity imposes on the scalar form factors. The imaginary part of any scalar form factor is given by the appropriate meson-meson scattering T -matrix, so that the starting point for any unitary resummation scheme is exactly this scattering T -matrix [60], where s denotes the center-of-mass energy squared and K(s) can be obtained from the lowest order CHPT Lagrangian, e.g., K(s) 11 = (s − M 2 π /2)/F 2 π . This T -matrix not only describes meson-meson scattering data but also, after gauging, photon decays and transitions, as reviewed in Ref. [11]. In Eq.(42), the diagonal matrix g(s) is nothing but the familiar scalar loop integral given here in dimensional regularization for the MS scheme, and σ i (s) = 1 − 4M 2 i /s. In what follows, we will set the regularization scale µ = 1.08 GeV. Of course, observables do not depend on this choice, and we could choose another value for µ. However, the original investigation of meson-meson scattering transitions and decays by Oller and Oset [60] uses a three-momentum cut-off in the pion loop function. Translated to dimensional regularization, this gives the stated value of µ. For energies above the threshold of the state i, unitarity implies the following relation between the form factors and the isospin-zero scattering T -matrix, employing an obvious matrix notation with where p i (s) = s/4 − M 2 i is the modulus of the c.m. three-momentum of the state i. Substituting Im Γ(s) with (Γ(s) − Γ(s) * )/(2i) and T (s) with the expression of Eq. (42) and using the properties of the matrices g(s) and K(s), one can express Γ(s) as: where R(s) is a vector of functions free of any singularity. We remark that this procedure of taking the final state interactions into account is based on the work of Ref. [61]. We also wish to stress that Eq. (46) can be applied to any K-matrix without unphysical cut contributions. As the final step, one fixes the functions in the vector R(s) by requiring matching of Eq. (46) to the next-to-leading order (one loop) CHPT ππ and KK scalar form factors. This matching ensures that for energies where CHPT is applicable, these form factors fulfill all requirements given by chiral symmetry and the underlying power counting. The matching procedure thus determines the vector R(s), as detailed in Ref. [10]. For completeness, we give the expression for R n 1 pertinent to the scalar form factor Γ n 1 (s), where we have used the Gell-Mann-Okubo relation 3M 2 η = 4M 2 K − M 2 π . This representation of the scalar form factors is valid from threshold up to energies of about 1.2 GeV. This range could be extended to higher energies by including multi-particle states. In fact, as we have noted before, in the u-channel one encounters larger values of √ s. Therefore, we simply match our representation at √ s 0 = 1.2 GeV to the following asymptotic forms We have checked that the final results are insensitive to this choice of the matching point. The asymptotic form of the real part of the scalar form factor follows from quark counting rules [62] in the crossed channel; it has also been found in the dispersion analysis of the Higgs decay into two pions [56]. For continued contact with Ref. [9] we match to the asymptotic form of the vertex function of Eq. (32); a more precise treatment, if it were warranted, would involve smoothly letting K(s) → 0 and solving for the form factors in a manner consistent with unitarity -for further discussion, see Ref. [56]. In Fig. 3 we display the scalar form factor Γ n 1 (s) for √ s ≤ 1.2 GeV. In the low-energy region, the modulus of the form factor has it maximum at √ s ≃ 0.46 GeV, very close to the central value of the σ meson mass deduced by the E791 collaboration from analyzing the D → 3π data. In fact, since we wish to examine the consequences of using the more general vertex, Eq.(34), as compared to the choice Eq.(32), used by Ref. [9], we have fixed the normalization constant χ such that Re Γ n 1 (s) has the same value as the Breit-Wigner representation at √ s = 0.478 GeV. This amounts to setting χ = 20.0 GeV −1 . The peak at √ s ≃ 1 GeV is due to the f 0 (980) and the opening of theKK channel. Also shown in Fig. 3 is the scalar form factor generated using the Breit-Wigner form with a running width, adopted in Refs. [8,9]. The differences between this form factor and that deduced from the low-energy effective field theory of QCD are marked. In particular, in Fig. 3 we see that the Breit-Wigner representation of Γ σππ is deficient in that i) Im Γ σππ (s) has a different shape as s approaches physical threshold, s → 4M 2 π and ii) Re Γ σππ (s) does not possess a unitarity cusp at s = 4M 2 π . It is also of the wrong sign in this limit. Moreover, the shapes of the two form factors are very different above √ s ≃ 0.5 GeV -this has particular consequence for the B → ρπ analysis.

Vector form factor
Thus far we have considered the σ meson contribution to B → π + π − π decay. In this section we turn to the ρ meson contribution, B(p B ) → ρπ → π + (p + )π − (p 1 )π(p 2 ), which presumably dominates for t ≃ M 2 ρ . In analogy to Eq. (29), the amplitude for B → π + π − π decay as mediated by the ρ resonance, A ρ (B → π + π − π), is the product of a B → ρπ amplitude and a ρ → ππ vertex function, Γ ρππ . We give the relevant formulae for B → ρπ → π + π − π decay in App. A. In this section we focus on the construction of Γ ρππ , generating a form which is consistent with all known theoretical constraints. We detail our procedure, as the form of Γ ρππ is important to the goals of the B → ρπ analysis: it drives the size of the interference between ρ states produced in different regions of the Dalitz plot. Our vertex function differs from that of Ref. [16], as the latter adopt a Breit-Wigner form. We also compare our form with that adopted in Ref. [7]. At this point we should mention that the numerical differences are not large -simply because the pion vector form factor can be described fairly well by a Breit-Wigner form. We also note that a unitarized version of the vector form factor starting from tree-level CHPT, including resonance fields, has been presented in Ref. [63], based on the methods described in Sec. 5. It could be used equally well as the parametrization employed here.
The vector form factor F ρ (s) can be directly determined from e + e − → π + π − data in the ρ resonance region. General theoretical constraints guide its construction: charge conservation requires F ρ (0) to be unity, and timereversal invariance and unitarity lead to the identification of the phase of F ρ (s) with the l = 1, I = 1 ππ phase shift, δ 1 1 (s), in the region where ππ scattering is elastic, s (M π + M ω ) 2 . Moreover, F ρ (s) is an analytic function in the complex s plane, with a branch cut along the real axis beginning at the physical threshold s = 4M 2 π . Below the two-pion cut at s = 4M 2 π , the vector form factor is real. Furthermore, at small s the form factor can be computed in CHPT, as detailed in Refs. [50,51]. All these constraints are captured by the Muskhelishvili-Omnès (MO) integral equation [64]. For s (M π + M ω ) 2 , its solution can be written [65] #7 #7 The solution of the MO equation with inelastic unitarity, important for s (Mπ + Mω) 2 , has been discussed in Ref. [66].
where P (s) is a real polynomial and the Omnès function, Ω(s), contains all the phase information, where δ 1 1 (s) is the phase shift of I = 1, L = 1 scattering. In the Heyn-Lang parametrization [65], Ω(s) is approximated by the quotient of two analytic functions, which contain polynomial pieces and the one-pion-loop expression for the ρ self-energy function. P (s) is chosen to be of third order in s in Ref. [65]. We use here a recent update of the pion form factor [21], based on the Heyn-Lang parametrization. Specifically, we use the parameter set of "solution B" of Ref. [21], reflecting a fit to the e + e − → π + π − data in the elastic region, subject to the constraint that the model reproduces the empirical ππ scattering length in the I = 1, L = 1 channel, a 1 1 = (0.038 ± 0.0.002) M π −3 [67]. In what follows, we neglect the presence of the ω resonance, or effectively ρ 0 − ω mixing. The latter is an important isospin-violating effect visible in the e + e − → π + π − data in the close vicinity of s = M 2 ω -the fits of Ref. [21] do include it. To realize the vertex function Γ ρππ (s), we define where, as described in App. A, the electromagnetic coupling constant of the ρ meson, f ργ , is f ργ = 0.122 ± 0.001 GeV 2 [68]. Ref. [16] adopts a Breit-Wigner form for the vertex function, namely with the parameters g ρ = 5.8 and Γ ρ = 150 MeV -we use M ρ = 769.3 MeV [69]. The two forms are compared in Fig. 4 -the Breit-Wigner form offers a reasonable description of the vector form factor, though differences can be seen. In particular, the imaginary part of the Breit-Wigner form does not vanish below physical threshold, as it ought. This deficiency can be repaired by giving the Breit-Wigner form a running width, i.e., where p(s) = s/4 − M 2 π . This form, modulo the proportionality constant, is adopted by Ref. [7]. Fig. 5 compares Eqs. (51) and (53) -the two forms are really very similar, though the forms differ slightly as s → 4M 2 π . The phase of the form factors, namely tan −1 ((Im Γ ρππ )/(Re Γ ρππ )), is plotted in Fig. 6. Unitarity and time-reversal invariance dictates that the phase be that of I = 1, L = 1 scattering; the phase shifts from the data of Refs. [70,71] are shown for comparison. The forms of Eq. (51) and Eq. (53) confront the phase shift data nicely. The agreement of the latter form is a particular surprise, as it contains only two free parameters. Apparently the form of the imaginary part is precisely captured by the ππ branch cut, so that the phase is accurately determined by an arctan prescription to unitarize the amplitude. This is also realized in the CHPT analysis including an explicit ρ meson, for details see Ref. [72].
In the application to follow, we need to evaluate Γ ρππ (s) for s > (M π + M ω ) 2 , so that at √ s = (M π + M ω ) ≃ 920 MeV, the form factor of Eq. (51) is matched to the Breit-Wigner form of Eq. (52), yielding Re F ρ (s) ∼ a ′ /s and Im F ρ (s) ∼ b ′ /s 2 for large s. We now turn to a discussion of our numerical results in ρ and σ-mediated B → 3π decay.

Results and discussion
First, we must collect parameters. For crisp comparison with Refs. [9,16], we adopt the parameters used therein but reiterate them here for convenience. For meson masses and widths we use M B = 5.279 GeV, M π = 139.57 MeV, M ρ = 769.3 MeV, Γ ρ = 150 MeV, M σ = 478 MeV, and Γ σ = 324 MeV. We neglect the B + , B 0 lifetime difference and use τ B = 1.6 · 10 −12 sec. For quark masses, we use m b = 4.6 GeV andm = 6 MeV. As for the CKM matrix elements, we adopt the Wolfenstein parametrization [73], retaining terms of O(λ 3 ) in the real part and of O(λ 5 ) in the imaginary part, using A = 0.806, ρ = 0.05, η = 0.36, and λ = 0.2196. For the Wilson coefficients, we use C 1 = 1.100, C 2 = −0.226, C 3 = 0.012, C 4 = −0.029, C 5 = 0.009, and C 6 = −0.033, after Ref. [44]. For form factors and coupling constants we use F  Table 1. In the treatment of Ref. [16], the branching ratios of B → M 1 M 2 decay and its charge conjugate are identical, even with penguin contributions, as no strong phase between the amplitudes of differing weak phase has been included. Table 1: Branching ratios (in units of 10 −6 ) for B → ρπ and B → σπ decay, computed at tree level. The numbers in parentheses include penguin contributions as well, after Ref. [16]. The first row of numbers compute the branching ratios using two-body phase space. Although Br(ρ → π + π − ) ≃ 1, Br(σ → π + π − ) ≃ 2/3, so that the numbers in brackets reflect the branching ratio times 2/3. The rows labelled "3-body" compute B → ρπ → 3π and B → σπ → π + π − π decay, integrating over the entire three-body phase space. "BW" denotes the use of the form factors of Refs. [9,16], Eqs. (32,52), whereas "RW" denotes the use of the vector form factor of Ref. [7], Eq. (53). Finally, " * " denotes the use of the form factors we have advocated. Proceeding to treat B → ρπ → 3π and B → σπ → π + π − π decay, we follow the ρ and σ intermediate states to their ππ final states. We realize the transition amplitudes as per Eqs. (A.4), (A.5), or (29), and integrate over the three-body phase space as per Eq. (37), computing the integral in t over [2M π , M B − M π ]. With this procedure, the branching ratios for B → ρ − π + , B → ρ + π − , and B → ρ 0 π 0 become identical; we simply report the final result in the B → ρ − π + column. In treating B − → π + π − π − decay, we divide the total rate by 1/2, to compensate for integrating over equivalent configurations. Noting that Br(σ → π + π − ) ≃ 2/3, the quantity in brackets in the first row includes the factor of 2/3, for comparison with the three-body results. Comparing the two-body branching ratios with those computed by integrating over the entire three-body phase space, it is evident that the branching ratios do not agree. The deviations can be attributed to both interference effects and finite-width effects. As an example of the former, both the diagrams illustrated in Figs. 1 and 2, as well as those with p 1 ↔ p 2 , contribute to B − → π + π − π − decay. Clearly the interference of these diagrams is not included when the B − → ρ 0 π − or B − → σπ − process is treated as a two-body decay. As an illustration of latter, note that the couplings g ρ and g σππ are typically chosen so that they reproduce the ρ → ππ and σ → π + π − decay rates, namely and For the meson masses and widths we have used, these formulae yield g σππ = 2.53 and g ρ = 6.03, respectively. Adopting these couplings in place of those used in Refs. [9,16] does reduce the discrepancy. Note that it is a "finite width" effect in that reducing the numerical width of the ρ or σ meson, in concert with Eqs. (55,56), reduces the discrepancy between the two-and three-body treatments. It is worth noting, however, for the physical values of the meson widths, that there is no one fixed coupling g σππ or g ρ which removes the discrepancy entirely -the needed coupling in any given case is sensitive to the form factor chosen, as well as to the masses of the other particles in the final state. The former is apparent from a comparison with the vector form factor of Ref. [7], Eq. (53), for which we use g ρ = 5.8 as well. The normalization issue of which we speak is particularly relevant for the comparison of theoretical branching ratios, for B → V P decay, e.g., to experiment. It is present regardless of the form factor used. That is, in the case of the vector form factor we adopt, Eq. (51), the determination of f ργ can also be modified by finite width effects. The sign and size of the mismatch between the two-and three-body phase space calculations can be quite sensitive to the choice of form factor, as illustrated by the scalar case. We set the normalization of the form factor of Sec. 5, denoted by " * " in the Tables, to that of the form factor of Ref. [9], Eq. (32). This is the only manner in which the parameters M σ , Γ σ enter our analysis. Were we to determine the normalization so that the two-and three-body computations of thē B 0 → σπ 0 → π + π − π 0 branching ratio yield identical results, the effective impact of the σ in the ρπ phase space would be reduced by some 10%. The numbers in parentheses include penguin contributions as well, after Ref. [16]. The form factors are defined as in Table 1.  In Table 2 we report the B → ρπ → 3π branching ratios, computed in the manner of Ref. [16]. Our numerical results differ slightly from theirs (note that B 0 ↔B 0 in their Table III). The upshot is that our estimate of R with penguin contributions is ∼ 5.1, rather than the 5.5 they estimate. We show the branching ratios computed for differing vector form factors; these differing choices have little impact on the resulting branching ratios, or on R.
We compute the B → σπ branching ratios in Table 3. In this case our computed branching ratios, for B − → σπ − decay, are a factor of two larger with the same form factors and parameters input, as our formula, Eq. (30), differs from theirs by a factor of √ 2. Thus the impact of the σ in the ρ 0 π − phase space is rather larger than that estimated in Ref. [9]. Updating the scalar form factor to use what we feel is its best estimate, we find that the values of R are smaller still. Interestingly, the computed values of R are comparable to the empirical results, albeit the errors are large. [An additional contribution to the phenomenological value of R, realized through a diagram mediated by the a − 1 meson, is proposed in Ref. [75].] Turning to B → σπ 0 decay, we see that the contribution of the σ meson to B 0 (B 0 ) → ρπ decay is much smaller -with the scalar form factor we advocate, the effect is some 10%. Interestingly the σ has a tremendous impact on B − → ρ 0 π − decay, and a relatively modest one onB 0 → ρ 0 π 0 decay. Let us emphasize that we have realized our numerical analysis at tree level. It is the relative size of the penguin contributions inB 0 → σπ 0 andB 0 → ρ 0 π 0 decay which is of relevance to the isospin analysis to extract α. The presence of the σπ 0 final state in the ρ 0 π 0 phase space can break the assumed relationship, Eq. (9), between the penguin contributions in ρπ and thus mimic the effect of isospin violation -alternatively we can expand the ρπ analysis to include the σπ channel. Nevertheless, we expect our estimates to be crudely indicative of the importance of these effects -quantitatively, however, differences may exist. It is worth noting that the σπ 0 and ρ 0 π 0 contributions can, to some measure, be distinguished. Certainly the σπ 0 and ρπ 0 contributions behave differently under the cut on the invariant mass of the π + π − pair, recalling Eq. (37). Moreover, making a cut on the helicity angle θ, defined in Eq. (36), ought also be helpful in separating the ρ 0 and σ contributions. This is illustrated in Figs. 7 and 8. The ρ 0 π contributions roughly follow a cos 2 (θ) distribution, whereas the σπ contributions are quite flat, save for the bump resulting from the Γ σππ (u) term in Eq. (30). Cutting on the helicity angle θ should also help disentangle the contributions from some of the B * resonances, discussed in Ref. [16]. The contributions of B * resonances to the ρπ channels should be included in a more refined analysis, but they will not alter the conclusions drawn here. Table 3: Effective branching ratios (in units of 10 −6 ) for B → σπ and B → ρπ decay, computed at tree level. The form factors are defined as in Table 1.

Summary
In this paper, we have scrutinized the role of the σ meson in B → ρπ → 3π decay, understanding its dynamical origin in the strong pion-pion final state interactions in the scalar-isoscalar channel. The presence of the σπ 0 contribution in the ρ 0 π 0 phase space is important in that it can break the assumed relationship between the penguin amplitudes, Eq. (9), consequent to an assumption of isospin symmetry. In this, then, its presence mimics the effect of isospin violation. The salient results of our investigation can be summarized as follows: i) We have considered how SM isospin violation can impact the analysis to extract α in B → ρπ decay. Under the assumption that |∆I| = 3/2 and |∆I| = 5/2 amplitudes share the same weak phase, the presence of an additional amplitude of |∆I| = 5/2 character, induced by isospin-violating effects, does not impact the B → ρπ analysis in any way. This is in contradistinction to the isospin analysis in B → ππ. Thus the isospin-violating effects of importance are those which can break the assumed relationship between the penguin contributions, Eq. (9).
ii) The scalar form factor can be determined to good precision by combining the constraints of chiral symmetry, analyticity, and unitarity. The form factor we adopt describes the appearance of the f 0 (980) as well, so that the shape of the f 0 (980) contribution in B → f 0 (980)π → 3π, e.g., should serve as a test of our approach. We emphasize that the resulting scalar form factor is very different from the commonly used Breit-Wigner form with a running width. This is in stark contrast to the vector form factor, which is dominated by the ρ resonance. In that case, one can construct simple forms that fit the theoretical and empirical constraints.
iii) We have pointed out that the two-versus three-body treatments of the decays B → ρπ, B → σπ can lead to differing results due to finite-width and interference effects. iv) Remarkably, the impact of the σπ channel on the ratio R, cf. Eq. (3), is huge. The numbers we find for R are in agreement with the empirical ones, given its sizeable experimental uncertainty. This underscores the suggestion made, as well as improves the calculations done, in Ref. [9]. Our analysis is based on consistent scalar and vector form factors.
v) On the other hand, the impact of the σπ channel on the B → ρπ isospin analysis is merely significant.
Varying the cuts on the ππ invariant mass and helicity angle θ should be helpful in disentangling the various contributions.
vi) We have shown that one can expand the isospin analysis to include the σπ channel because it has definite properties under CP. This may be necessary if varying the cuts in the ππ invariant mass and helicity angle θ are not sufficiently effective in suppressing the contribution from the σπ 0 channel in the ρ 0 π 0 phase space.
This work is merely a first step in exploiting constraints from chiral symmetry, analyticity, and unitarity in the description of hadronic B decays. In particular, the contribution of the "doubly" OZI-violating strange scalar form factor and its phenomenological role in factorization breaking ought be investigated.