Where is the lightest charmed scalar meson?

The lightest charmed scalar meson is known as the $D_0^*(2300)$, which is one of the earliest new hadron resonances observed at modern $B$ factories. We show here that the parameters assigned to the lightest scalar $D$-meson are in conflict with the precise LHCb data of the decay $B^-\to D^+ \pi^- \pi^-$. On the contrary, these data can be well described by an unitarized chiral amplitude containing a much lighter charmed scalar meson, the $D_0^*(2100)$. We also extract the low-energy $S$-wave $D\pi$ phase of the decay $B^-\to D^+ \pi^- \pi^-$ from the data in a model-independent way, and show that its difference from the $D\pi$ scattering phase shift can be traced back to an intermediate $\rho^-$ exchange. Our work highlights that an analysis of data consistent with chiral symmetry, unitarity, and analyticity is mandatory in order to extract the properties of the ground-state scalar mesons in the singly heavy sector correctly, in analogy to the light scalar mesons $f_0(500)$ and $K_0^*(700)$.

Despite the phenomenological success of this picture in describing the available lattice and LHCb data, the observation that the lightest D * 0 has a mass around 2.1 GeV has not entered the Review of Particle Physics (RPP) [42] yet, which still lists the D * 0 (2300) as the lightest charmed scalar meson and the D 1 (2430) as the corresponding axial-vector meson. In this Letter, we demonstrate that the D * 0 (2300) as in the RPP is not consistent with the most precise data for B − → D + π − π − , contrary to the D * 0 (2100) predicted in UChPT, and conclude that the positive-parity charm-nonstrange meson spectrum in the RPP needs to be revised.
Dπ S-wave phase of B − → D + π − π − .-The decay amplitude in the low-energy region of the Dπ system can be decomposed into S-, P -, and D-waves, Comparison of the predictions of Eq. (5) from UChPT (blue) and a Breit-Wigner parametrization (green) for δ0 with the phase extracted in Ref. [25] (red) and that using Eq. (3) (black). The bands correspond to errors propagated from the input UChPT scattering amplitudes and from the Breit-Wigner resonance parameters.
where A (s) with = 0, 1, 2 correspond to the amplitudes with D + π − in the S-, P -, and D-waves, respectively, s is the c.m. energy squared of the D + π − system, and P (z s ) are the Legendre polynomials with z s the cosine of the helicity angle of the D + π − system, i.e., the angle between the moving directions of the two pions in the D + π − c.m. frame. The angular moments are determined by weighting the data with the Legendre polynomials P (z) [25]. They contain contributions from certain partial waves and their interference terms, and thus the corresponding phase variations. The first few moments are given by [25,27,39] with δ i the phase of A i , i.e., A i = |A i |e iδi . As first proposed in Ref. [27], we use the linear combination P 13 instead of P 1 and P 3 individually, since it only depends on the S-P -wave interference up to = 2 and is particularly sensitive to the S-wave phase motion. For M D + π − < 2.2 GeV, even the D-wave can be neglected, since the narrow tensor resonance D s2 (2460) is sufficiently far away. This can be verified from the data of the angular moments, i.e., Fig. 3 in Ref. [25]. Therefore, in this kinematic regime one obtains cos(δ 0 − δ 1 ) = 3 10 for the S-P phase difference. The P -wave is dominated by the vector resonance D * (2007) 0 below the threshold with a width of less than 60 keV [43,44]. The next vector D * resonance is far above this energy region. Thus, the phase of the P -wave δ 1 can be safely fixed to 180 • for the region we are interested in. The S-wave Dπ phase motion below 2.2 GeV can then be extracted, see For M D + π − < 2.4 GeV, the effect of the ρ meson could be significant via the coupled channel B − → D 0 π 0 π − ; see Fig. 2. This follows directly from the large branching ratio B(B − → D 0 ρ − ) = 1.34%, which is an order larger than B(B − → D + π − π − ) = 0.107% [42]. It is therefore reasonable to assume that the decay B − → D + π − π − is dominated by the process B − → D 0 ρ − → D 0 π 0 π − → D + π − π − . By virtue of soft-pion theorems, one has [45] where p π − (π 0 ) is the momentum of the π − (π 0 ), and F π is the pion decay constant (in the chiral limit). From B(B 0 →D 0 π 0 ) = 2.63 × 10 −4 and B(B − → D 0 π − ) = 4.68 × 10 −3 , one concludes that at low energies for the D + π − (D 0 π 0 ) system, the amplitude of B − → D 0 π 0 π − is much larger than that of B − → D + π − π − . Furthermore, isospin symmetry shows that for the decays B → Dππ with even relative angular momenta between the pions, the amplitude for B − → D + π − π − is larger than that of B − → D 0 π 0 π − by a factor of 2 √ 2 [27,39,46,47]. As in addition even angular momenta here imply isospin I = 2 and therefore nonresonant partial waves, the relative angular momentum of π 0 π − in the decay B − → D 0 π 0 π − is by far dominantly odd in the low-energy regime for D 0 π 0 , and the ρ − plays a crucial role.
If we assume that the decay B − → D + π − π − is dominated by the process in Fig. 2, the Dπ S-wave part of the triangle diagram can be estimated by the integral whereP (s) is the production amplitude for The evaluation of Eq. (5) depends on the asymptotic behavior of the integrand, which is divergent in general. We may estimate Eq. (5) using a cutoff at where q max ≈ 1 GeV (another way is to introduce a form factor, e.g., e −(s−s th )/s0 with s 0 = O(1 GeV) [48]). We evaluate Eq. (5) by employing both the Dπ scattering amplitude from UChPT [19] and that of a Breit-Wigner (BW) parametrization of the D * 0 (2300) for comparison, despite the deficiencies of the latter discussed in Ref. [39]; see also Ref. [49].
The results with q max = 1 GeV are shown in Fig. 1, where the solid blue band and the green dashed band correspond to the Dπ scattering amplitudes from UChPT and BW, respectively. The obtained phase describes the data perfectly for the UChPT amplitude, while the BW one fails. We have checked that the obtained phases are insensitive to a variation of the cutoff in a reasonable region, q max ∈ [0.8, 1.2] GeV.
Khuri-Treiman formalism.-While Eq. (5) provides a reasonable estimation of the S-wave decay amplitude with a clear underlying physical picture, it does not respect three-body unitarity. In order to check if the conclusion formulated above is robust, we cure this deficiency by employing the Khuri-Treiman equations [50], which are based on two-body elastic phase shifts and explicitly generate the crossed-channel rescattering between final-state particles. The formulas are constructed from dispersion relations for the related crossed scattering processes and then analytically continued to the decay region, referring to the continuation of the triangle graph [51].
We can write amplitudes for A +−− (B − → D + π − π − ) and A 00− (B − → D 0 π 0 π − ) in terms of single-variable functions according to a reconstruction theorem [47,52], where the subindex and superindex I of the singlevariable amplitudes F I represent the angular momentum and isospin, respectively, and only the I < 3/2 and ≤ 2 terms are taken into account. The Mandelstam variables The corresponding angles are given by . Since we are interested in the s-channel process, we use the index A (B) to label the two-body channels corresponding to D + π − and D 0 π 0 . The partial-wave decomposition for the decay amplitudes A A reads with b A I, denoting Clebsch-Gordan coefficients. By comparing with Eq. (1), it is easy to obtain A (s) = (2 We have the following partial-wave unitarity relation for elastic rescattering: where δ I (s) is the elastic final-state scattering phase shift.
The discontinuities of f I and those of the single-variable functions κ F I coincide on the right-hand cut by construction. Thus, one has where Ω I (s) = exp s/π ∞ s th ds δ I (s )/[s (s − s)] is the Omnès function [53], Q I (s) is a polynomial at least of degree (n − 1) (see discussion below), and the number of subtractions n is chosen to guarantee the convergence of the dispersion integral.
The inhomogeneityF I is determined by the partialwave decomposition of Eq. (6) as the projection of the crossed-channel amplitudes onto the considered channel. Around the Dπ threshold in the s-channel, √ t ∼ 5 GeV, there is no resonance in the t-channel, and thus the interaction is supposed to be very weak. The only possible significant crossed-channel effect is from the ρ meson through B − → D 0 π 0 π − . The resulting inhomogeneity for the S-wave s-channel amplitude is [47] For technical details regarding this integral, see Ref. [54] and the Supplemental Material [55]. The full solution for the decay amplitudes can be obtained by solving a set of coupled integral equations in terms of a few linearly independent complex subtraction constants contained in Q I (s), which cannot be determined a priori in the framework of dispersion theory. Since we are only interested in the Dπ low-energy regime and especially in its S-wave, based on the large branching ratio of B − → D 0 ρ − , it is reasonable to approximate F 1 1 (u) in Eq. (6) by a BW function for the ρ meson. In this case, F 1 1 (u) behaves as u −1 for u → ∞, thusF I (s) in Eq. (12) approaches a constant as s → ∞. The number of the subtractions n in Eq. (11) is then determined by the asymptotic behavior of the scattering phase δ I (s). For the BW phase and that of UChPT taken from Ref. [19], one single subtraction is sufficient. Moreover, the phase of UChPT, as well as that of the BW, is unreliable at high energies. Thus, the dispersion integral will be evaluated up to a cutoff Λ, and the effect of cutting off the integral may be absorbed into the polynomial Q I (s). Explicitly, for an integral For simplicity, we neglect the I = 3/2 contribution since it contains no resonances. Therefore, for the S-wave amplitude at low energies, Eq. (11) can be written as The constants g 0 and g 1 have to be fixed by data. We fit P 0 , P 13 , and P 2 of the decay B − → D + π − π − up to 2.4 GeV, which is below the Dη and D sK thresholds. To describe the angular moments, one needs the explicit amplitudes for the Dπ P -and D-waves. The P -wave can be safely parametrized as δ 1/2 1 (s) = π θ s−M 2 D * 0 as discussed below Eq. (3). Consequently, the dispersion integral (11) for the P -wave can be neglected since sin π = 0. The D-wave is dominated by the resonance D 0 2 (2460) with a width of 47.5 MeV [42], which is above the region we are interested in. Thus, the D-wave phase is close to 0 below 2.4 GeV, and the corresponding dispersion integral can be neglected as well. Therefore, for the P -(D-)wave amplitudes F 1/2 1,2 , we can use the same BW forms as those in the LHCb analysis [25], which is equivalent to the corresponding Omnès function multiplied by a polynomial. In the isobar model used in Ref. [25], complex factors are introduced for each resonance BW function. Without crossed-channel effects, these factors become real according to Watson's theorem [56]. For the P -and D-waves, as discussed above, the Omnès representation should be a good approximation in the energy region we are interested in, and the normalization factor is real. We also consider a complex normalization and find the results unchanged.
For the Dπ S-wave, we employ both the scattering phase shifts from UChPT [19], which contains the D * 0 (2100), and the BW for the D * 0 (2300). The fit results are shown in Fig. 3, where the blue and green bands correspond to the best fits from UChPT and the BW, respectively. While UChPT describes the data very well with χ 2 /d.o.f. = 1.2, the BW fails to reproduce the data with χ 2 /d.o.f. = 2.0. The difference of these two values is significant from the statistical point of view: the corresponding p-values are 0.1 for the UChPT fit and 3 × 10 −5 for the BW fit, respectively. Thus, the former can be accepted as a good description of the data, while the latter is highly disfavored [57]. The error bands correspond to the 1σ uncertainties propagated from the input phases. The borders of the band of P 13 for the BW are plotted in dotted and dashed curves to make it evident that the data for P 0 and P 13 cannot be described by the BW phase. With the fitted parameters g 0 and g 1 , we obtain the S-wave phase of the decay amplitude for Fig. 4, where the results corresponding to UChPT and the BW are plotted as blue and green bands, respectively. As expected, UChPT describes the S-wave phase extracted using Eq. (3) and that obtained in Ref. [25] well up to 2.4 GeV. For the BW one, although the error band is broad, either the low-energy or the high-energy region cannot be described.
Conclusion.-The existence of the D * 0 (2300) as given in the RPP is the starting point of many theoretical analyses (see, e.g., Refs. [58][59][60][61]). The results obtained in this Letter show that the D * 0 (2300), whose resonance parameters were obtained using the BW parametrization from the Belle [17] and BaBar [62] analyses, is in conflict with the much more precise LHCb data for B − → D + π − π − , which, however, can be well reproduced by the UChPT amplitude containing the D * 0 (2100). We expect that the D 1 (2430) as given in the RPP [42] will also be in conflict with future high-quality data of B − → D * + π − π − from LHCb [63] and Belle-II, and that the lightest D 1 meson is the D 1 (2250) predicted by UChPT [26,27].
The D * 0 is analogous to the more famous f 0 (500) and K * 0 (700), whose masses have been significantly shifted from earlier versions of the RPP due to improved data and improved theoretical analyses-for recent discussions see Refs. [64][65][66][67][68] and the review on scalar mesons in the RPP [42]. We expect a similar change in all systems emerging from the scattering of a pion off an isospinnonsinglet hadron. The lightest resonance in that case should not be extracted from data using the usual BW form-a parametrization accounting for chiral symmetry and coupled channels is mandatory. The Dπ S-wave phase extracted model-independently here provides valuable information for further understanding matter-field-Goldstone-boson scattering and the structure of positiveparity heavy hadrons. Note added.-Recently, a lattice calculation also concluded that the D * 0 mass should be lower than the RPP value [69]. The authors found a mass of (2196 ± 64) MeV with a pion mass of 239 MeV, only (77 ± 64) MeV above the Dπ threshold. Thus, our conclusion receives a strong support from lattice QCD calculations.

Technical details
The inhomogeneity for the S-wave s-channel amplitude iŝ where Σ 0 = M 2 B + M 2 D + 2M 2 π , and the integral end points and path need to be understood with the prescription M 2 B → M 2 B + i . The inhomogeneities give the contributions from the crossed channels, i.e., the left-hand cut contributions. The integration end points u ± (s) contain the only nontrivial M B -dependent pieces. The continuation following the M 2 The amplitude F 1 1 (u) is parameterized by a Breit-Wigner (BW) amplitude of the ρ − meson, since, as explained in the main text, the process B − → D 0 π 0 π − is assumed to proceed through the intermediate decay where the p u and q u are the momenta of the D 0 and π + (π 0 ) in the π 0 π − rest frame, respectively. The ρ resonance lineshape is modelled by a relativistic BW function R 1 (u) given by with a mass-dependent decay width defined as where q 0 u is the value of q u when √ u = M ρ , and Γ ρ is the full width. The Blatt-Weisskopf barrier factors X L (x) are given by where x = | q|r BW or | p|r BW , and x 0 is the value of x evaluated at √ u = M ρ . The barrier radius, r BW , is taken to be 4.0 GeV −1 to be consistent with Ref. [25].
For the P -(D-)wave amplitudes F 1/2 1,2 (s), we use the same BW amplitudes as those in the LHCb analysis [25], i.e., where q 0 is the value of q evaluated at the resonance mass √ s = m J with m J the mass of the P -or D-wave resonance and Γ 0 J the corresponding width. The P -wave resonance D * is outside the kinematically allowed region. Therefore, its mass m 1 is modified to an effective mass m eff 1 as in the LHCb analysis [25], where m max and m min represent the upper and lower limits of the kinematically allowed range.

Parameters
The masses and widths of the P -and D-wave resonances used in the calculation are listed in Table I and are the same as those used in the LHCb analysis [25]. The UChPT amplitudes for the Dπ interaction and corresponding low energy constants (LECs) used in this work are taken from Ref. [19]. For the BW parameterization of the S-wave Dπ amplitude, the amplitude of Eq. (20) with J = 0 is employed. The parameters determined in the best fits to data for the Khuri-Treiman formalism, e.g., Eq. (14), with input from UChPT and the BW amplitudes, respectively, are collected in Table II, where only the statistical uncertainties are given. Each partial wave has an overall normalization factor, e.g., c 1 and c 2 for P -and D-waves, respectively. Since we fit to the unnormalized angular moments, the units of the fitted parameters contain some arbitrary overall normalization factors. To make it explicit that the requirement of chiral symmetry has a sizeable impact on the determination of resonance parameters using the BW parameterization in the isobar model as argued in Ref. [39], here we show two different fits: • model I: the S-wave Dπ is parameterized as a relativistic BW amplitude, Eq. (20); • model II: the S-wave Dπ is parameterized as a BW amplitude times an E π factor, as required by the chiral symmetry of QCD: with E π the energy of the pion in the Dπ rest frame. Note that in the isobar model, the crossed-channel effects are not explicitly considered and the coefficients c J are complex to describe the relative contribution of each intermediate process.
Here, we do not use the resonance parameters of D * 0 in the Review of Particle Physics (RPP), but determine the BW mass and width by fitting to the high quality LHCb data in Ref. [25]. The best fits to the angular moments are shown in Fig. 5, where the orange (with χ 2 /d.o.f. = 1.8) and the blue curves (χ 2 /d.o.f. = 1.1) correspond to model I and model II, respectively. For model I, the fit results in the D * 0 BW mass and width parameters of (2319 ± 9) MeV and (420 ± 25) MeV, respectively. The fitted mass is consistent with the RPP average mass of (2300 ± 19) MeV. However, the width is much larger than the RPP average value, i.e., (274 ± 40) MeV. Note that in Ref. [25], a quasi-model-independent approach is used to describe the D + π − S-wave. The D + π − S-wave is extracted using cubic splines to describe the magnitude and phase variation. Accordingly, in this reference no pole parameters for the D * 0 are given. When the E π factor is considered, i.e., in model II, the fit quality is significantly improved with a much smaller χ 2 . The fitted parameters are given in Table III with only statistical uncertainties. In addition, the fit results in a BW mass of (2206±4) MeV, which is much lower than that given in the RPP, while the width is (341 ± 17) MeV. This clearly shows that the BW parameters for a broad resonance can be affected significantly by the chiral symmetry constraint, as argued in Ref. [39]. Nevertheless, the modification of the BW in model II only applies in a small energy region before the coupled-channel effects become important, and thus is neither practical nor systematic. The resonance parameters would get further modified after the Dη and D sK coupled channels are taken into account. The UChPT formalism used in the main text provides a theoretical framework satisfying both chiral symmetry and coupled-channel unitarity. [GeV] 0.420 ± 0.025 0.341 ± 0.017