Implications of $B\to χ_{c1}πK$ data for an isovector $G$-odd $D^\ast\bar D^\ast$ molecular virtual state

Abstract: Establishing the near-threshold spin-isospin multiplet spectrum of $D^{(\ast)}\bar D<sup>{(\ast)}$ systems is central to testing molecular interpretations of the $X(3872)$, $Z_c(3900)$, and $Z_c(4020)$. The isovector channels are particularly important in this context, as the associated structures would have a clear exotic character. In this work, we analyze the BaBar and Belle data on $B\toχ<em>{c1}πK$ to search for possible isovector $D^\ast\bar D<sup>\ast$ molecules with $J<sup>{PC}=0<sup>{++}$ and $2<sup>{++}$. The three-body decay amplitude includes an effective nonresonant term, intermediate kaon resonances, and $χ</em>{c1}π$--$D^\ast\bar D<sup>\ast$ coupled-channel rescattering. For each data set, the $χ<em>{c1}π$ and $Kπ$ invariant-mass distributions are fitted simultaneously under three scenarios: without rescattering and with either $J=0$ or $J=2$ rescattering. We then analytically continue the fitted coupled-channel $T$ matrices to search for poles. For the case of $J<sup>{PC}=0<sup>{++}$, the BaBar and Belle fits yield virtual poles at $-2.15^{+2.10}</em>{-6.41}$ MeV and $-18.40<sup>{+8.58}_{-13.70}$ MeV, respectively, relative to the $D^\ast\bar D<sup>\ast$ threshold. For $J<sup>{PC}=2<sup>{++}$, the corresponding virtual poles are located at $-3.42<sup>{+2.37}_{-4.29}$ MeV and $-4.26<sup>{+1.34}_{-1.69}$ MeV, respectively. The tensor virtual-state pole is consistent with a prediction from chiral effective field theory, which also predicts the existence of $W_{c1}$, an isospin partner of $X(3872)$. Since the invariant-mass distributions alone cannot distinguish total spin $J$, we also predict angular distributions. The $\cosθ<em>{χ</em>{c1}π}$ distribution provides a direct spin discriminator, while the $\cosθ_{Kπ}$ distribution further tests the rescattering contribution. The measurements of these predictions would help establish the $D^{(\ast)}\bar D<sup>{(\ast)}$ molecular multiplet spectrum in the future.

• The paper shows that unitary χc1π–D*D̄* coupled-channel fits improve descriptions of BaBar and Belle data and produce virtual-state poles in both the 0++ and 2++ sectors.
• The paper finds mutually consistent 2++ pole masses near 4014 MeV that align with effective field theory predictions, while the 0++ poles show greater uncertainty and tension with most bound-state models.
• The paper proposes using the χc1π and Kπ angular distributions to distinguish spin assignments, with a rapid rise toward cosθχc1π=1 favoring 2++ and a near-flat distribution favoring 0++.

This paper performs a data-driven search for isovector, $G$-odd $S$-wave $D^\ast\bar D^\ast$ molecular states with $J^{PC}=0^{++}$ and $2^{++}$ using the BaBar and Belle measurements of $B\to\chi_{c1}\pi K$. The analysis combines a unitary coupled-channel rescattering amplitude with a systematic pole search, and proposes angular observables capable of discriminating the two spin assignments.

Motivation and context

The molecular interpretation of near-threshold hidden-charm states such as $X(3872)$, $Z_c(3900)$, and $Z_c(4020)$ implies, via heavy-quark spin symmetry, a full $S$-wave multiplet of $S$0 structures. The isovector sector is particularly constraining because its members are manifestly exotic. At leading order in a heavy-meson contact EFT, the isovector $S$1 $S$2 and $S$3 $S$4 channels share the same low-energy-constant combination $S$5, while the $S$6 $S$7 channel is governed by the independent combination $S$8. Consequently, a confirmed isovector tensor pole near the $S$9 threshold would provide an indirect test of the predicted $D^\ast\bar D^\ast$0 virtual state, the isospin partner of the $D^\ast\bar D^\ast$1, whose neutral signal is difficult to isolate from the much stronger $D^\ast\bar D^\ast$2 contribution (Zhang et al., 2024).

Existing theoretical predictions for these isovector channels diverge widely. One-boson-exchange models with natural cutoffs generally disfavor bound states, particularly in the $D^\ast\bar D^\ast$3 channel, while contact EFT calculations yield bound, virtual, or resonant poles depending on the input used to fix the low-energy constants. A comparatively constrained prediction comes from chiral EFT with both contact couplings fixed by the $D^\ast\bar D^\ast$4 pole and its isospin-breaking decay ratios, which gives an isovector $D^\ast\bar D^\ast$5 virtual pole at $D^\ast\bar D^\ast$6 MeV (Zhang et al., 2024), and an independent contact EFT analysis constrained by $D^\ast\bar D^\ast$7 and $D^\ast\bar D^\ast$8 data gives $D^\ast\bar D^\ast$9 MeV (Baru et al., 2021).

Experimentally, the situation is unsettled. Belle reported $J^{PC}=0^{++}$0 in $J^{PC}=0^{++}$1 with a significance above $J^{PC}=0^{++}$2 (0806.4098), but BaBar found no evidence for it (Collaboration et al., 2011) using a Breit-Wigner parameterization. The paper correctly notes that such a parameterization is inappropriate for a near-threshold molecule because it omits the threshold dynamics and violates unitarity; a unitary coupled-channel amplitude is required. The related $J^{PC}=0^{++}$3 status is likewise ambiguous, with LHCb's recent $J^{PC}=0^{++}$4 analysis of $J^{PC}=0^{++}$5 finding no need for the structure (collaboration et al., 3 Sep 2025).

Amplitude framework

The three-body amplitude comprises three coherent contributions: an effective nonresonant term, intermediate kaon resonances $J^{PC}=0^{++}$6, $J^{PC}=0^{++}$7, and $J^{PC}=0^{++}$8, and a $J^{PC}=0^{++}$9–$2^{++}$0 coupled-channel rescattering term. The two-channel $2^{++}$1 matrix is obtained from a Lippmann-Schwinger equation with a potential containing only the $2^{++}$2 coupling $2^{++}$3 and the $2^{++}$4 elastic contact $2^{++}$5; the $2^{++}$6 elastic potential is set to zero on the grounds that the $2^{++}$7 carries no light valence quarks. The $2^{++}$8 channel is $2^{++}$9 wave, the $B\to\chi_{c1}\pi K$0 channel $B\to\chi_{c1}\pi K$1 wave, with a monopole regulator at $B\to\chi_{c1}\pi K$2 GeV fixed throughout. The channel-$B\to\chi_{c1}\pi K$3 weak-production source is neglected based on factorization arguments: it is color-suppressed and further suppressed by the $B\to\chi_{c1}\pi K$4-wave loop factor $B\to\chi_{c1}\pi K$5.

Three fit schemes are defined: Scheme I (continuum plus kaon resonances only), Scheme II (plus $B\to\chi_{c1}\pi K$6 rescattering), and Scheme III (plus $B\to\chi_{c1}\pi K$7 rescattering). No molecular pole is imposed a priori; poles are identified only by analytic continuation of the fitted $B\to\chi_{c1}\pi K$8 matrices. The BaBar and Belle data sets are fitted independently and simultaneously in the $B\to\chi_{c1}\pi K$9 and $X(3872)$0 distributions.

Invariant-mass fits

Both data sets show an excess near the $X(3872)$1 threshold at $X(3872)$2 MeV that the reflection contributions alone cannot reproduce. The reduced $X(3872)$3 values improve from 1.15 to 0.90 (Scheme II) and 0.97 (Scheme III) for BaBar, and from 2.24 to 1.69 and 2.16 for Belle. The Belle data favor the $X(3872)$4 assignment, since the $X(3872)$5 rescattering produces an asymmetric threshold peak with an extended high-mass tail that improves the description over a broader region, whereas the $X(3872)$6 contribution is more localized; the BaBar data do not distinguish the two hypotheses. This preference is a fit-quality observation within the present amplitude framework, not a significance claim, and the paper states plainly that neither spin assignment can be excluded from the invariant-mass spectra alone.

Pole analysis and comparison with theory

Analytic continuation of the fitted $X(3872)$7 matrices to the $X(3872)$8 Riemann sheet yields virtual-state poles in all four fits:

Data set	Sector	$X(3872)$9 [MeV]	$Z_c(3900)$0 [MeV]
BaBar	$Z_c(3900)$1	$Z_c(3900)$2	$Z_c(3900)$3
Belle	$Z_c(3900)$4	$Z_c(3900)$5	$Z_c(3900)$6
BaBar	$Z_c(3900)$7	$Z_c(3900)$8	$Z_c(3900)$9
Belle	$Z_c(4020)$0	$Z_c(4020)$1	$Z_c(4020)$2

The deeper Belle $Z_c(4020)$3 pole reflects the broader excess above threshold in that data set, consistent with the more localized BaBar cusp. For the $Z_c(4020)$4 sector, the two independently determined poles agree within uncertainties and are close to the chiral EFT prediction of $Z_c(4020)$5 MeV (Zhang et al., 2024) and the contact EFT value $Z_c(4020)$6 MeV (Baru et al., 2021). If the threshold structure is indeed $Z_c(4020)$7, the agreement supports the heavy-quark spin symmetry relation and, indirectly, the predicted $Z_c(4020)$8 pole. By contrast, the $Z_c(4020)$9 result is in tension with most theoretical calculations that predict a bound scalar state, which the paper identifies as a substantive contradiction; the only compatible prediction is the virtual pole of $S$0 MeV (Ji et al., 2022), whose large uncertainty reaches the near-threshold region.

The paper emphasizes that the negative pole energies cannot be read as binding energies: these are virtual-state poles below the threshold on the unphysical sheet, while the physical-axis spectrum exhibits a cusp-like enhancement.

Angular-distribution predictions

Since the invariant-mass spectra cannot discriminate $S$1, the paper predicts angular distributions using the fitted amplitudes. The $S$2 distribution is the direct spin discriminator: in the narrow window $S$3 GeV, the $S$4 amplitude predicts an approximately flat distribution beyond the $S$5 peak, whereas the $S$6 amplitude predicts a local minimum followed by a rapid rise toward $S$7. Critically, both the BaBar-fitted and Belle-fitted amplitudes independently produce this same qualitative contrast, making the positive-endpoint rise a robust signature of the $S$8 assignment within this framework. A flat distribution alone, however, would not separate the $S$9 hypothesis from Scheme I.

The $S$00 distribution serves a different purpose: it maps the narrow threshold cusp into a broader angular feature via the one-to-one kinematic correlation between $S$01 and $S$02 at fixed $S$03. The mapped structure shifts from approximately $S$04 to $S$05 as the $S$06 integration window moves from $S$07–$S$08 to $S$09–$S$10 GeV, providing a detection strategy that does not require fine $S$11 binning. The clearest signal appears in the middle window $S$12 GeV.

Limitations and open questions

Several caveats bear directly on the results. The analysis neglects the $S$13 $S$14 rescattering contribution associated with $S$15, justified by the absence of structure near the $S$16 threshold in the data; this is an assumption rather than a demonstration. The channel-$S$17 production source and the $S$18 contribution are dropped on factorization and loop-suppression grounds, and the $S$19 elastic potential is set to zero. The cutoff is fixed at $S$20 GeV without a systematic variation study, so the sensitivity of the pole positions to the regulator is not quantified. The Belle fit quality remains marginal ($S$21 at best), and the paper explicitly concedes that the present data precision is insufficient to claim the establishment of either $S$22 or $S$23. The pole-position uncertainties for the $S$24 sector, particularly the Belle result with a lower bound extending to roughly $S$25 MeV, are large enough that the scalar pole position remains weakly constrained. Finally, the angular predictions are untested; no measured angular distributions for this decay mode currently exist for comparison.

Conclusion

This work demonstrates that both BaBar and Belle $S$26 data prefer the inclusion of a unitary $S$27–$S$28 coupled-channel rescattering contribution, and the analytically continued $S$29 matrices consistently yield near-threshold virtual-state poles in both the $S$30 and $S$31 isovector sectors. The $S$32 pole positions from the two independent data sets are mutually consistent and agree with existing EFT predictions, whereas the $S$33 result conflicts with the majority of bound-state predictions in the literature. The proposed $S$34 and $S$35 observables provide concrete, model-independent-in-form tests that Belle II and LHCb measurements could perform to establish the spin assignment and, by extension, the isovector $S$36 molecular multiplet spectrum.