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B_l4 decays and the extraction of |V_ub|

The Cabibbo-Kobayashi-Maskawa matrix element |V_ub| is not well determined yet. It can be extracted from both inclusive or exclusive decays, like B -->pi (rho) l nu-bar_l. However, the exclusive determination from B -->rho l nu-bar_l, in particular, suffers from a large model dependence. In this paper, we propose to extract |V_ub| from the four-body semileptonic decay B -->pi pi l nu-bar_l, where the form factors for the pion-pion system are treated in dispersion theory. This is a model-independent approach that takes into account the pi-pi rescattering effects, as well as the effect of the rho meson. We demonstrate that both finite-width effects of the rho meson as well as scalar pi-pi contributions can be considered completely in this way.


I. INTRODUCTION
Precisely determining the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1] plays a very important role in testing the Standard Model. Any deviations from the unitarity of the CKM matrix would be viewed as a sign of new physics. The element |V ub | has been measured from inclusive charmless semileptonic B decay as well as from the exclusive decays B → π(ρ)lν l . For a review on the determination of |V ub |, see Ref. [2]. The value of |V ub | preferred by the current global analysis of CKM data is about 15% smaller than the one from inclusive charmless semileptonic B decays [3-5], a problem unresolved to date. Furthermore, the inclusive determinations of |V ub | are about two standard deviations larger than those obtained from B → πlν, presently with a smaller uncertainty. The value of |V ub | predicted from the measured CKM angle sin 2β, however, is closer to the exclusive result [6], and it should be stressed that various theoretical extractions based on exclusive decays are remarkably consistent among each other [3, [7][8][9][10]. These discrepancies prompted a reexamination of the sources of theoretical uncertainty in the inclusive determination [11,12].
In the present paper, we investigate the four-body semileptonic decay mode B − → π + π − l −ν l (which we will abbreviate as B l4 for short) and propose a method that allows one to extract |V ub | in a model-independent way. As a major step forward to a reliable treatment of the hadron-physics aspects of this decay, we use an approach based on dispersion theory without the need to explicitly match on specific resonance contributions or to separate these from nonresonant background. This presents a significant improvement compared to previous studies of B → ρlν l [13], and should serve as a valuable cross-check for the inclusive determination. In the future the distributions derived below could be used directly in the Monte Carlo generators of the experiments.
We include the kinematic range for invariant masses of the ππ pair below the KK threshold in our analysis and expand the form factors for the full B l4 transition matrix element in ππ partial waves up to P waves; D and higher partial waves have been checked to be negligible at these energies. While this model-independent description of the form factor dependence on the ππ invariant mass is in principle general and holds for arbitrary dilepton invariant masses, in practice we make use of matching to heavy-meson chiral perturbation theory to fix the normalization of the matrix element-a prerequisite for the extraction of |V ub |. This scheme applies in the kinematics where heavy-quark effective field theory is valid, i.e. for very large dilepton invariant masses. We point to Ref. [14] for a lucid illustration of the different effective theories applicable in different kinematic regimes for this decay.
This manuscript is organized as follows. In Sec. II the kinematics for the process of the four-body semileptonic B decay is reviewed, and the form factors for the hadronic transition of B → ππlν l are defined. In Sec. III, we show in detail how to treat these form factors within dispersion theory: the analytic properties are summarized in Sec. III A and the required pole terms calculated in heavy-meson chiral perturbation theory in Sec. III B, before we provide the expressions for the various form factors in the Omnès representation in Sec. III C. We discuss the required matching to leading-order heavy-meson chiral perturbation theory in Sec. III D. Numerical results are discussed in Sec. IV, and we summarize our findings in Sec. V. Some technical details are relegated to the Appendices. The kinematics of the process B − (p B ) → π + (p + )π − (p − )l − (p l )ν l (p ν ) are described in terms of the five variables displayed in Fig. 1 [15][16][17]: (i) the effective mass squared of the pion pair s = (p + + p − ) 2 = M 2 ππ ; (ii) the effective mass squared of the dilepton pair s l = (p l + p ν ) 2 ; (iii) the angle θ π of the π + in the π + π − center-of-mass frame Σ 2π with respect to the dipion line of flight in the B − rest frame Σ B ; (iv) the angle θ l of the charged lepton l in the lepton center-of-mass system Σ lν with respect to the dilepton line of flight in Σ B ; (v) the angle φ between the dipion and dilepton planes.
Two additional Mandelstam variables are defined as We define the combinations of four vectors P = p + + p − , Q = p + −p − , L = p l +p ν , and make use of the kinematical relations where and the Källén triangle function is given by λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + ac + bc).
We decompose the matrix element in terms of form factors according to where G F = 1.166365 × 10 −5 GeV −2 is the Fermi constant, and we use the convention ǫ 0123 = 1. The first three terms correspond to the axial current part, whereas the last term corresponds to the vector current. The dimensionless form factors F , G, H, and R are analytic functions of three independent variables, e.g. s, s l , and t − u. Their partial-wave expansions for fixed s l read [15,17] where P l (z) are the standard Legendre polynomials and P ′ l (z) = dP l (z)/dz. An alternative set of form factors is given by whose partial-wave expansions directly follow from Eqs. (5) and (6). Note that all partial waves f l , g l , h l , r l (r l ) are functions of s and s l . The lowest angular-momentum ππ state contributing to the form factors F 2 and F 3 is the P -wave state, whereas the form factors F 1 and F 4 start with S waves. For the partial-wave decomposition up to P waves, we can therefore write F 1 = X f 0 (s, s l ) + f 1 (s, s l ) cos θ π + . . . , where the ellipses denote higher partial waves. In the following, we sometimes suppress the dependence on s l in order to ease notation. The decay rate, after integration over the angles φ and θ l , reads with In most of the available phase space (including the kinematic regime where chiral perturbation theory can be applied), the mass of the lepton can be neglected (i.e. z l ≪ 1), and the contribution of F 4 to the decay rate is therefore invisible in particular for B e4 decays, since it is always associated with a factor of z l . We will not analyze the form factor F 4 and its partial wavesr i in the following. Integrating Eq. (9) over cos θ π yields the partial decay rate dΓ/(ds ds l ); neglecting terms of order z l and inserting the partial-wave expansions Eq. (7), we find where the ellipsis denotes the neglected D and higher waves. Interference terms between different partial waves vanish upon angular integration, such that the partialwave contributions to the decay rate can be easily read off.

A. Analytic properties
The principle of maximal analyticity, which states that amplitudes possess no other singularities than those stemming from unitarity and crossing [18], tells us that the partial-wave amplitudes f l , g l , and h l have the following analytic properties.
(i) At fixed s l , they are analytic in the complex s plane, cut along the real axis for s ≥ 4M 2 π and s ≤ 0. The presence of left-hand cuts s ≤ 0 follows from the relations (and equivalent expressions for u), since the form factors F , G, and H have cuts for t, u ≥ (m B + M π ) 2 .
(ii) In the interval 0 ≤ s ≤ 4M 2 π , they are real. (iii) In the interval 4M 2 π ≤ s ≤ 16M 2 π , Watson's theorem [19] is satisfied and therefore the phases of the partial-wave amplitudes (f l , g l , h l ) coincide with the corresponding pion-pion scattering phases. (iv) For the crossed (t and u) channels, due to the lack of experimental information on πB phase shifts, we will approximate the πB interaction by B * pole terms.
In practice, the range of validity of Watson's theorem can be extended to a larger domain, e.g. for the S wave to s ≤ s K = 4M 2 K ≈ 1 GeV 2 , since inelasticities due to four or more pions are strongly suppressed both by phase space and by chiral symmetry. As pointed out e.g. in Refs. [20,21], chiral perturbation theory predicts the inelasticity parameter of the ππ S and P waves to be of order p 8 below the KK threshold, while the corresponding scattering phase shifts are of order p 2 . Phenomenological analyses of the ππ interactions show that final states containing more than two particles start playing a significant role only well above the KK threshold s K [22]. Here we refrain from performing a coupled-channel study, which limits the applicability of our approach to the region below s K . The subtleties associated with the strong onset of inelasticities in the S wave in the vicinity of s K (very close to the f 0 (980) resonance) for scalar form factors of the pion will be briefly discussed in Sec. IV A.

B. Heavy-meson chiral perturbation theory
In the process B − → π + π − l −ν l , u-channel contributions contain pole terms, while t-channel contributions do not. We obtain the pole terms by computing the leadingorder diagrams (b) and (c) of Fig. 2 in the framework of heavy-meson chiral perturbation theory [23][24][25]. Let us briefly review the heavy-meson chiral Lagrangian. Define the heavy-meson field and its conjugate as where P * aµ is the field operator that annihilates a P * a meson with velocity v, satisfying v µ P * aµ = 0, and P a annihilates a P a meson of velocity v. For the B meson family, we have which have dimension [mass] 3/2 . The light pseudoscalar Goldstone boson fields are organized in f π ≃ 92.2 MeV is the pion decay constant [26]. Based on these building blocks, the leading-order Lagrangian describing the interactions of the B family and the Goldstone bosons reads [23] Determining the coupling g = g B * Bπ = g B * B * π , using heavy-quark symmetry, from the partial decay width for D * + → D 0 π + leads to g = g D * Dπ = 0.58 ± 0.07, with the error given by the uncertainty in the width of the D * + . This is in surprisingly good agreement with the most recent lattice simulations, which find g B * Bπ = 0.516 ± 0.052 [27] and g B * Bπ = 0.569 ± 0.076 [28] (we have added different error sources in quadrature for simplicity in both cases). In the present analysis, we stick to the experimental number extracted from D * + decays for illustration. The dominant parts of the B l4 amplitude will depend on g in a very simple manner (being directly proportional either to g or to g 2 ), thus suggesting a straightforward strategy towards an extraction of |V ub | via lattice calculations of g B * Bπ . To improve on the analytic properties of the amplitudes calculated in heavy-meson chiral perturbation theory, we include the effect of the B * -B mass splitting, , in the propagators, which in the heavy-meson approximation are of the form i 2v · k for the pseudoscalar B meson, where k is the small residual momentum of the propagating B or B * . We do not otherwise include heavy-quarksymmetry-breaking effects, and stick to Eq. (17) for the determination of the interaction vertices.
The left-handed current L νa =q a γ ν (1 − γ 5 )Q, with q a denoting a light and Q the heavy quark, is written in chiral perturbation theory as where the ellipsis denotes terms with derivatives, factors of the light-quark mass matrix m q , or factors of 1/m Q .
Computing the trace, one can write it explicitly as f B is the B meson decay constant; averaging the most recent lattice calculations with 2 + 1 dynamical quark flavors leads to the very precise value f B = 190.5 ± 4.2 MeV [29]. The whole B l4 decay amplitude is proportional to f B , such that any uncertainty on this parameter directly translates into a contribution to the error in the extraction of |V ub |. We briefly discuss the chiral power counting of the B l4 amplitudes and form factors. If we denote soft pion momenta, or derivatives acting on the pion field, by p generically, the current of Eq. (19) is O(p 0 ), and so we expect to be the leading-order amplitude resulting from the diagrams in Fig. 2. Equation (4) then suggests the leading contributions to the form factors F , G, H, and R to be of chiral orders p −1 , p −1 , p −2 , and p 0 , respectively (remember that the dilepton momentum L µ is large, of order m B ); the alternative form factors F 1 and F 4 both are O(p 0 ).
The results for the individual diagrams of Fig. 2 are given in Appendix A. In order to ensure that we do not miss any effects of the nontrivial analytic structure of triangle graphs, resulting from the B * pole terms once rescattering between the two outgoing pions is taken into account, we keep the full relativistic form of the denominator part of the propagator. The latter is connected with the above heavy-meson approximation Eq. (18) by [30] where Written in terms of s and s l , the pole terms can then be easily identified as using the abbreviations All pole contributions start to contribute at the expected leading chiral orders. We note, though, that , and can be neglected; they are indeed partially an artifact of the translation of the heavy-meson formalism back into relativistic kinematics in the calculation of Ref. [24]. We will use the contributions ∝ α in the partial waves f i later on to illustrate potential higher-order effects, although these are neither complete nor necessarily dominant amongst the subleading contributions (cf. the discussion of the scaling behavior of higher-order terms in the current in Ref. [31]). For the purpose of the (s-channel) partialwave projections to be performed later, the u-channel pole can be written in terms of s and cos θ π , Finally, the remaining, nonpole, parts of the amplitude can also be extracted from the expressions in Appendix A. There are nonvanishing contributions to the form factor F 1 only, which in view of the required partialwave expansion we write as M 0 (s) χPT and M 1 (s) χPT are found to be of chiral orders p 0 and p, respectively, and therefore suppressed by one order compared to the pole terms [31], as explained in Appendix A. We will use these expressions in Sec. III D to match the polynomial parts of the dispersive representations of the corresponding amplitudes, but again rather in order to illustrate potential uncertainties due to subleading effects: these contributions are not complete even at the chiral order at which they occur.
To conclude this section, we point out that in order for the chiral counting scheme to work consistently, we have to assume the lepton invariant mass squared s l to be large, of the order of m 2 B . This limits the kinematic range of applicability of our approach to match the dispersive representation derived in the following to heavy-meson chiral perturbation theory.

C. Omnès representation
Having fixed the tree-level decay amplitude and in particular the pole terms, we proceed to analyze the effects of pion-pion rescattering using dispersion relations. This will give access to the s dependence of the decay form factors (roughly up to 1 GeV, as detailed in Sec. III A) in a model-independent way. We will resort to the formalism based on Omnès representations as introduced in Ref. [32]. For its application to the closely related process of K l4 decays, see Refs. [33,34]. Note, however, that everything discussed in the following is to be understood at fixed s l : dispersion theory as applied here does not allow us to improve on the form factor dependence on the dilepton invariant mass, beyond what the chiral representation in the previous section includes. We emphasize once more that the dispersive aspect of our analysis is in principle independent of the matching to heavy-meson chiral perturbation theory: the validity of any theoretical description of the different form factors in the soft-pion limit (s ≈ 0) can be extended at least to the whole kinematic region of elastic ππ scattering with this method.
We may write an alternative form of the partial-wave expansion Eq. (8) for the pole-term-subtracted amplitudes, neglecting terms beyond P waves, Here and in the following we suppress the dependence on s l , which is kept fixed. The additional factor of X 2 in the definition of M 1 avoids the introduction of kinematic singularities at the zeros of X (in particular at the limit of the physical decay region s = (m B − √ s l ) 2 ). The functions M 0 , M 1 , U 1 , and V 1 defined this way possess right-hand unitarity branch cuts as their only nontrivial analytic structure and no poles. Since the pole terms which allows us to write The real "hat functions"M 0 (s),M 1 (s),Û 1 (s), andV 1 (s) are the partial-wave projections of the pole terms given in Eqs. (22)- (23), which explicitly read where the Q l (y) are Legendre functions of the second kind, Explicitly, the first three of these read We have projected onto the partial waves of F 2 and F 3 [whose partial-wave expansions proceed in derivatives of Legendre polynomials-see Eq. (7)] using Note that, in order to show that the partial-waveprojected pole terms above indeed are real everywhere along the right-hand cut, i.e. for all s ≥ 4M 2 π , care has to be taken about the correct analytic continuation. For example, X, only defined unambiguously in the physical decay region in Eq. (3), is continued according to [35,36] (where the last range is of no practical relevance for our dispersive integrals). Furthermore, in the range of (m B − √ s l ) 2 < s < (m B + √ s l ) 2 , the argument y of the Legendre functions of the second kind becomes purely imaginary; the lowest one can be expressed as Q 0 (y) = i(π/2 − arctan |y|). In particular, no singularities arise at the zeros of X, s = (m B ± √ s l ) 2 . Physically, the reality of the pole terms is based on the fact that the B * cannot go on its mass shell in any kinematic configuration.
In the elastic regime, the right-hand cut of the partial waves f i (i = 0, 1), g 1 , h 1 for s > 4M 2 π is given by discontinuity equations relating them to the elastic ππ partial-wave amplitudes t i i (s), i = 0, 1, 1 according to 1 We use this somewhat unusual notation owing to the fact that we only consider S and P waves, and no isospin I = 2 is allowed.
where we have expressed t i i (s) in terms of the corresponding phase shift δ i i (s) in the usual way. Analogous equations hold for g 1 and h 1 . Equation (34) implies Watson's theorem: the phase of the partial wave equals the elastic phase shift. From Eqs. (27) and (28), one finds and similarly for U 1 (s), V 1 (s). Equation (35) demonstrates that the hat functions constitute inhomogeneities in the discontinuity equations. The solution is given by [32] M i (s) = Ω i i (s) P n−1 (s) where P n−1 (s) is a subtraction polynomial of degree n−1, and the Omnès function is defined as [37] Ω I l (s) = exp .
The standard Omnès solution P n−1 (s)Ω i i (s) of the homogeneous discontinuity equation (M i = 0), valid for form factors without any left-hand pole or cut structures, is modified by a dispersion integral over the in-homogeneitiesM i , which in the present case are given by the partial-wave-projected pole terms.
The minimal order of the subtraction polynomial is dictated by the requirement of the dispersive integral to converge. First we note that, if the phase δ I l (s) asymptotically approaches a constant value cπ, then the corresponding Omnès function falls off asymptotically ∼ s −c . We will assume both ππ input phases to approach π for large energies, such that Ω 0 0 (s), Ω 1 1 (s) ∼ 1/s for large s. A more problematic question concerns the behavior of the hat functions for large s. In principle, this is entirely determined by the partial-wave-projected B * pole terms as given in Eq. (29). However, as we have decided to include the relativistic pole graphs, these explicitly contain the scale m B , and the asymptotic behavior is only reached for √ s ≫ m B -far too high a scale, given that we realistically know the pion-pion phase shifts only up to well below 2 GeV, and that we presently neglect all inelastic contributions, which set in above 1 GeV. We can formally remedy this problem by just considering the large-s behavior of the heavymeson approximation of the pole terms, 2 in which m B 2 Remember that we made use of the relativistic pole terms mainly to ensure the correct analytic properties at low energies, i.e. in the near-threshold region. only features parametrically as a prefactor; being aware that corrections to the heavy-meson approximation scale like √ s/m B , which is not a very small quantity in the region of 1 GeV √ s 2 GeV, say. In the heavy-meson approximation, i.e., at leading order in an expansion of 1/m B , the inhomogeneities of Eq. (29) behave according toM Together with the large-s behavior of the Omnès functions, we conclude that the representation for M 1 (s) requires at least two subtractions, while for M 0 (s), U 1 (s), and V 1 (s), one subtraction each seems to be sufficient. However, looking at the behavior of the various hat functions in the low-energy region in Fig. 3 (for a special value of s l = (m B − 1 GeV) 2 ), we note that the falling ofM 0 (s),Û 1 (s), andV 1 (s) barely seems to set in in the kinematical region s 1 GeV 2 where we have to assume the spectral function to be saturated, whileM 1 (s) even grows at those energies instead of approaching a constant value. It seems therefore advisable to oversubtract all the dispersive representations once, such as to allow for two subtraction constants each for M 0 (s), U 1 (s), and V 1 (s), and three for M 1 (s). This way, inelastic contributions at higher energies that we do not take into account explicitly should also be more effectively suppressed. The complete set of dispersion relations of the Omnès type therefore reads The subtraction constants are a priori unknown, and need to be determined either by further theoretical input, or by fitting to experimental data. It is easy to check that the functions M 0 (s), . . ., V 1 (s) themselves do not satisfy Watson's theorem; however, taking into account Eq. (28), the partial-wave amplitudes f 0 (f 1 , g 1 , h 1 ) do; i.e., their phases equal the elastic scattering phases δ 0 0 (δ 1 1 ). We add a few further remarks concerning Fig. 3. All of the partial-wave-projected pole terms display singular behavior of square-root type at s = 0 (suppressed as s 3/2 in the case ofM 1 (s); note that alsoM 0 (s) has a squareroot singularity, which is hard to discern in Fig. 3 due to the axis scaling). These left-hand singularities obviously carry over to the partial waves: close to the ππ threshold, the partial-wave amplitudes cannot be represented by simple scalar or vector form factors.
The uncertainty bands forM i (s), i = 0, 1, in Fig. 3 indicate the effect of the (incomplete) higher-order contribution ∝ α in Eq. (29), suppressed by 1/m B and found to be surprisingly small. We do not include the uncertainty due to the overall scaling with the coupling constant g, which translates directly into an uncertainty of a projected extraction of |V ub |, but does not (at this order) affect the shape of the distributions. The inhomogeneities scale with g according toM i (s), The dispersive method using inhomogeneities as described above has by now been used for a variety of lowenergy processes, such as η → 3π [32,38], ω/φ → 3π [39], K → ππ [40], K l4 [33,34], γγ → ππ [41,42], or γπ → ππ [43,44]. In several of those cases, the inhomogeneities (given in terms of hat functions), which incorporate left-hand-cut structures, and the amplitudes given in terms of Omnès-type solutions with a right-hand cut only are calculated iteratively from each other, until convergence is reached. In our present analysis, the ansatz is comparably simpler, as the left-hand cut is approximated by pole terms, whose partial-wave projections then determine the inhomogeneities. This is closely related to the method of Ref. [41] for γγ → ππ, where the lefthand structures are approximated by Born terms and resonance contributions to γπ → γπ.

D. Matching the subtraction constants
We need to consider two essentially different contributions to the subtraction constants in the representation Eq. (40), writing them formally as and similar decompositions for the a ′ i , b i , and c i . We discuss the contributionsâ i etc. first. We argue in Appendix B that for inhomogeneities of essentially constant (M 0 ,Û 1 ,V 1 ) or approximately linear (M 1 ) behavior over a large part of the kinematical region of interest, the coefficients of the highest power in the subtraction polynomials (a 1 , a ′ 2 , b 1 , and c 1 ) need to be adjusted in order to provide a reasonable high-energy behavior. 3 These coefficients are given by the derivative of the corresponding Omnès function at s = 0, multiplied with the constant/the derivative of the inhomogeneity in question. Obviously, the hat functions are not exactly constant/linear: to the contrary, they include square-root singularities at s = 0 due to the left-hand cut. There is, therefore, necessarily an uncertainty due to the choice of a "matching point" s m at which to evaluate these "constants," We choose s m = M 2 ρ , due to the expected strong enhancement of the distribution at the ρ resonance peak. Here,Ω I l (0) = dΩ I l (s)/ds| s=0 . All other subtraction constants do not receive "hat" contributions.
The second contribution to the subtraction constants, dominantly to those of low polynomial order in s, stems from matching to the nonpole part of the chiral amplitude Eq. (25), which yields (for fixed s l ) a polynomial contribution in s. In this exploratory study we use the leadingorder expressions only. We expect the chiral expansion 3 This can be corroborated to some extent by arguments from Brodsky-Lepage quark counting rules [45] and soft-collinear effective theory [46], albeit in kinematic regions with completely different scaling of s l with respect to m 2 B (taken as fixed and not particularly large here). Assuming the large-s behavior of the different form factors and partial waves is independent thereof, we indeed need to require the leading powers in s to cancel between the dispersion integrals over the inhomogeneities and the subtraction polynomial.
to converge best at the sub-threshold point s = 0, as opposed to, e.g., the ππ threshold [47].
As we match the dispersive representation Eq. (40) to the leading chiral tree-level amplitude, which does not contain any rescattering or loop corrections, we identify the subtraction constantsā 0−1 ,ā ′ 0−2 by setting the scattering phases to zero, i.e., Ω i i (s) ≡ 1, and the dispersive integrals over the inhomogeneities vanish. At s = 0, we find from Eq. (25) The term ∝ā ′ 2 s 2 , stemming from the expansion of X 2 , is chirally suppressed and could as well be neglected. F 2 and F 3 at leading order coincide with their pole terms, thus the matching implies that the parametersb i andc i vanish.
In order to illustrate the relative importance of the (partial-wave-projected) pole terms relative to the subtraction polynomial-that is, the decompositions M i (s)+ M i (s) on tree level, for i = 0, 1-we also show these for s l = (m B − 1 GeV) 2 in Fig. 3. We verify the expected dominance of the pole terms/the hat functions in f 0 (s) and f 1 (s), as suggested by power counting arguments. For the uncertainty bands of the polynomial corrections with mixed dependence on g, we have varied this coupling within its assumed uncertainty, g = 0.58 ± 0.07.
Remember that g 1 (s) and h 1 (s) consist of B * pole terms only at leading order: this pole dominance should have very favorable consequences for the reliability of the form factor prediction, as the pole contributions are essentially fixed by the coupling constant g (as well as f B ) beyond the chiral expansion; the latter affects only the precision of the polynomial contribution. Next-toleading-order corrections to the residues of the pole terms seem to have surprisingly little effect.

A. Scattering phase input
The ππ phase shifts are known to sufficient accuracy in the region s s 0 ≡ (1.4 GeV) 2 (cf. Refs. [48,49]). In order to ensure the assumed asymptotic behavior δ 0 0 (s), δ 1 1 (s) → π for s → ∞, we continue the phases beyond s 0 according to the prescription [50] There is a further subtlety concerning the S-wave phase shift: as we have discussed in Sec. III A, the elastic approximation breaks down at the KK threshold s K with the occurrence of the f 0 (980) resonance. Both the phase of the partial wave arg t 0 0 (s) and, e.g., the phase of the nonstrange scalar form factor of the pion arg F S π (s) differ significantly from δ 0 0 (s) in this region: they quickly drop and then roughly follow the energy dependence of δ 0 0 (s) again, with δ 0 0 (s)− arg t 0 0 (s) ≈ δ 0 0 (s)− arg F S π (s) ≈ π [51]. Therefore a single-channel approximation to the pion scalar form factor only works for s < s K if a phase of the form of either arg t 0 0 (s) or arg F S π (s) are used as input to the Omnès function instead of δ 0 0 (s). We use such a form factor phase taken from Ref. [52]. Obviously, we cannot provide a reliable description of pion-pion rescattering effects where the inherent two-channel nature of the problem becomes important, hence our dispersive description is confined to below s K .
With the phase shift input thus continued formally up to infinity, the Omnès integrals can be fully performed. We have checked that different continuation prescriptions from the one given in Eq. (44) above s 0 have very little impact on the physics at low energies, i.e., below 1 GeV.
In order to ensure numerically stable results, we perform the dispersion integrals over the inhomogeneities Eq. (40) up to √ s = 3 GeV. This upper limit of the integration does not have any real physical significance: it merely represents an attempt to sum up the high-energy remainder of the integral to reasonable approximation, and does not mean we pretend to understand ππ interactions at such scales.

B. Subtraction constants, spectrum
We illustrate the results of our discussion for a sample value of s l = (m B − 1 GeV) 2 , which means the kinematically allowed range in the invariant mass of the pion pair extends to √ s = 1 GeV. Evaluating the (nonvanishing) subtraction constants obtained from matching to the nonpole, polynomial parts of the chiral tree-level amplitude, Eq. (43), we find where we have displayed the scaling with g explicitly and shown the range of parameters in the F 1 partial waves due to the higher-order corrections discussed above.
For demonstration, we plot the partial decay rate in Fig. 4 for the dilepton invariant mass s l = (m B −1 GeV) 2 . We find that the S-wave contribution leads to a significant enhancement of the spectrum at low ππ invariant masses, beyond what might be considered ρ dominance. The near-threshold dominance of the S wave was already pointed out in Ref. [31] in the context of heavymeson chiral perturbation theory. Concerning the different P waves, we find that the kinematical prefactor X 2 /m 4 B strongly suppresses the partial wave h 1 or the form factor F 3 for the values of s l considered here. Of the other two, g 1 yields a contribution to the differential rate roughly twice as large as f 1 .

V. DISCUSSION AND SUMMARY
We wish to emphasize that matching to chiral perturbation theory at leading order can only be considered an estimate, and mainly serves for illustration purposes here. Higher-order corrections are expected to be significant.
Ultimately, the subtraction constants that influence the shape ought to be determined by fits to experimental data; they can be thought of as parametrizing a "background polynomial," beyond the dominant pole terms, albeit with completely correct rescattering corrections, obeying Watson's theorem. The necessary theoretical normalization of the form factors is essentially provided at s = M 2 ρ , via Eq. (42); its stability under higher-order corrections still merits further investigation in order to provide a theoretical uncertainty for |V ub | extracted from B l4 decays.
To summarize, we have provided a description of the form factors for the decay B − → π + π − l −ν l using dispersion theory, which should lead to an improved method to measure |V ub |. Pion-pion final-state interactions have been included nonperturbatively in the elastic approximation, while left-hand-cut structures in the πB interaction are approximated by B * pole terms. We stress that our formalism allows, for the first time, to use the full information for ππ invariant masses below 1 GeV, without the need to refer to particular parametrization for selected resonances such as the ρ (770) [or the f 0 (980)]; it allows for a full exhaustion of the corresponding spectra. Improved experimental data to allow for such an analysis to be performed in practice is therefore highly desirable.
As an outlook concerning theoretical improvement, we have hinted at the possibility to extend the present analysis to lower values of the dilepton invariant mass s l , beyond the range of applicability of heavy-meson chiral perturbation theory, but still making use of dispersion relation for the dependence on the dipion invariant mass s. One promising constraint could be obtained from soft-pion theorems [56], which relate linear combinations of B l4 form factors at s = M 2 π , but arbitrary s l , to B → πlν (B l3 ) form factors at same s l . Given reliable phenomenological information on the form factors for B l3 , this may provide precisely (part of) the matching information needed to extend the dispersive method of this paper to lower values of s l .
Identifying the contributions to the individual decay form factors, we find for these as the leading-order (LO) results From these, it is then straightforward to identify the pole contributions given in Eq. (22), as well as the nonpole pieces of Eq. (25). It is obvious that all diagrams (a)-(d) are formally of O(p 0 ) in terms of soft pion momenta. Note, however, that all pieces proportional to p µ B = P µ + L µ are effectively suppressed: the part ∝ L µ enters the form factor R, which is suppressed by the small lepton mass and neglected throughout the main text, while the part ∝ P µ leads to a chiral suppression by one order (and is at least partially an artifact of the heavy-meson approximation anyway). As a consequence, the only leading contributions are given by the amplitudes B (1) and C (2) in the above, and hence the B * pole graphs. This was already pointed out in Ref. [31].
However, in the spirit of the solution discussed in the main text, it should also be possible to treat this constant as an inhomogeneity and reconstruct the same solution from the corresponding formalism. Our solution is then of the form f (s) = A + Ω(s) a + a ′ s + s 2 π ∞ 4M 2 π A sin δ(s ′ )ds ′ |Ω(s ′ )|s ′ 2 (s ′ − s) , (B2) where we have chosen the minimal number of subtractions (two) required to make the dispersion integral converge. Note that the subtraction constants a, a ′ are not a priori fixed from the tree-level input; we can set a = 0 by requiring the normalization of the amplitude at s = 0 to match the tree-level input. The integral in Eq. (B2) can be performed explicitly, using a dispersive representation of the inverse of the Omnès function whereΩ(0) = dΩ(s)/ds| s=0 . As a result, we find f (s) = Ω(s) A + a ′ − AΩ(0) s .
Therefore, Eq. (B1) is reproduced if we choose a = 0, a ′ = AΩ(0). We essentially apply the same requirement on the high-energy behavior as in Eq. (B1): terms that do not vanish for large s are only canceled for this specific choice of a ′ .
More generally, if we match to a tree-level amplitude of the form A s n , demanding the same leading behavior near s = 0 such that all subtraction terms ∝ s m≤n can be put to zero, the solution using this tree-level input as an inhomogeneity, A s n + Ω(s) a ′ s n+1 + s n+2 π ∞ 4M 2 π A s ′ n sin δ(s ′ )ds ′ |Ω(s ′ )|s ′ n+2 (s ′ − s) , (B5) agrees with the "canonical" solution A s n Ω(s), with the "correct" high-energy behavior, only if a ′ = AΩ(0).