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arxiv: 2607.06317 · v1 · pith:4DYWD3OT · submitted 2026-07-07 · hep-ph

KX(3872) interaction and correlation function

Jing Song , Pedro Brandao , Eulogio Oset This is my paper

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classification hep-ph
keywords correlationfunctioninteractionamplitudeapproximationcorrespondingequationsevaluated
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The pith

Kaon scattering off X(3872) predicts a narrow bound state

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper assumes the exotic hadron X(3872) is a loosely bound molecule of a D and a D* meson, then computes what happens when a kaon scatters off this molecule. The central method is the fixed-center approximation to the Faddeev equations, improved by feeding the resulting amplitude as an optical potential into a Lippmann-Schwinger equation that enforces elastic unitarity. The kaon interacts with both the D and D* constituents of the cluster, and the combined attraction from the KD and KD* channels produces a narrow resonant structure about 50 MeV below the kaon-plus-X(3872) threshold, with a width of roughly 1 MeV. The authors also compute the femtoscopic correlation function, which shows a clear enhancement at low relative momenta, the signature of a strongly attractive interaction that forms a bound state. The scattering length is a = (0.39 - i 0.00) fm and the effective range is r_0 = (1.16 - i 1.66) fm. Because the bound state arises from the kaon coupling to the individual constituents of the X(3872), its existence and properties are directly tied to the molecular interpretation of the X(3872).

Core claim

The paper's central result is the prediction of a narrow KX(3872) bound state at approximately 4315 MeV, about 50 MeV below threshold, with a width of about 1 MeV. This state emerges from the unitarized fixed-center treatment of kaon scattering off the D and D* components of the X(3872) molecule, where the attraction is driven by the D_s0*(2317) and D_s1(2460) poles in the KD and KD* channels respectively. The accompanying correlation function deviates from unity at low momenta in a pattern characteristic of a bound state, and the scattering parameters (a positive real scattering length with negligible imaginary part, and a significant negative imaginary effective range) encode the near-thai

What carries the argument

The fixed-center approximation (FCA) to the Faddeev equations treats the X(3872) as a static cluster of D and D* mesons and sums all sequences in which the kaon scatters off either constituent. The raw FCA amplitude is then used as an optical potential in a Lippmann-Schwinger equation that resums the elastic propagation of the kaon-cluster intermediate state, enforcing elastic unitarity. The two-body inputs are the KD and KD* scattering amplitudes in isospin 0 and 1: the I=0 amplitudes are dominated by the D_s0*(2317) and D_s1(2460) resonances, while the I=1 amplitudes come from vector-meson-exchange potentials. The cluster form factor derives from the molecular wave function of the X(3872),

If this is right

  • If the predicted KX(3872) bound state at ~4315 MeV is observed in three-body invariant mass distributions, it would support the molecular picture of the X(3872).
  • Measurement of the KX(3872) correlation function at low relative momenta would directly test the predicted scattering length and effective range, and deviations from the predicted shape could signal non-molecular components of the X(3872).
  • The same framework can be applied to other exotic molecular candidates, generating a family of predictions for kaon-cluster bound states and their correlation functions that are testable at femtoscopy experiments.
  • The inverse method referenced by the authors could extract the KX(3872) interaction parameters directly from future experimental correlation data, providing a model-independent cross-check of the theoretical predictions.

Load-bearing premise

The fixed-center approximation assumes the D and D* inside the X(3872) are static spectators that do not recoil or rearrange when the kaon scatters off them. Given that the X(3872) binding energy is less than 0.2 MeV, the constituents are barely bound, so treating the molecule as a rigid cluster is the load-bearing premise: if the kaon significantly perturbs the internal D-D* structure, the three-body amplitude could change substantially.

What would settle it

If experimental measurement of the KX(3872) correlation function shows no enhancement at low relative momenta, or if searches for a narrow state near 4315 MeV in the KX channel find nothing, the molecular picture of the X(3872) within this framework would be challenged.

Figures

Figures reproduced from arXiv: 2607.06317 by Eulogio Oset, Jing Song, Pedro Brandao.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

We investigate the $K X(3872)$ interaction and the corresponding correlation function, assuming the $X(3872)$ to be a molecular state of $D \bar D^*$ and $D^* \bar D$ with isospin $I=0$ and positive $C$-parity. The interaction is treated within the fixed-center approximation (FCA) to the Faddeev equations, in which the $X(3872)$ is taken as a cluster of its constituents and the kaon interacts with the $D^*$ and $D$ components. The three-body scattering amplitude is evaluated using the Fixed Center Approximation (FCA) to the Faddeev equations, improved by taking the FCA amplitude as an optical potential which is later unitarized by means of the Lippmann-Schwinger equation. We find a narrow resonant structure about 50~MeV below the $K^+ X(3872)$ threshold with a width of approximately 1~MeV, and determine the $K X(3872)$ scattering length $a = (0.39 - i\,0.00)$~fm and effective range $r_0 = (1.16 - i\,1.66)$~fm. The corresponding correlation function is evaluated and shows a clear deviation from unity at low momenta, characteristic of a strongly attractive interaction leading to a bound state. These predictions are tied to the molecular nature of the $X(3872)$ and can be measured experimentally via measurements of the $K X(3872)$ correlation function and three-body invariant mass distributions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 6 minor

Summary. This manuscript investigates the $KX(3872)$ interaction within the unitarized fixed-center approximation (FCA) to the Faddeev equations, assuming the $X(3872)$ is a $Dbar D^*$ molecular state. The FCA amplitude, constructed from known two-body $KD$ and $KD^*$ interactions, is used as an optical potential in a Lippmann-Schwinger equation to enforce elastic unitarity. The authors predict a narrow bound state approximately 50 MeV below the $K^+X(3872)$ threshold with a width of about 1 MeV, extract the scattering length and effective range, and compute the corresponding femtoscopic correlation function. The work is motivated by the recent ALICE measurements of $pf_1(1285)$ correlations and extends a similar formalism to the $X(3872)$ system.

Significance. The paper provides a concrete, falsifiable prediction for the $KX(3872)$ correlation function and a bound state at $sim 4315$ MeV, tied to the molecular nature of the $X(3872)$. The unitarized FCA formalism is internally consistent and preserves elastic unitarity by construction (Eq. 22). The two-body inputs (couplings, cutoffs) are taken from established chiral unitary approaches. The correlation function prediction is timely given ongoing experimental efforts. The approach is not parameter-free (cutoffs $q_{max}$ and couplings are external inputs), but the three-body prediction emerges from the dynamics rather than being fitted to it.

major comments (3)
  1. The source radius $R$ in Eq. (31) is never specified in the text, yet it directly determines the shape of the correlation function in Fig. 5. Without this value, the central prediction of the paper is not reproducible. Please state the value of $R$ used and justify the choice (e.g., from typical pp or pA collision sources).
  2. The FCA treats the $X(3872)$ as a rigid cluster, but the $KD$ and $KD^*$ attractions that generate the $D_{s0}^*(2317)$ and $D_{s1}(2460)$ poles involve binding of 45-70 MeV, which is 200-350 times larger than the $<0.2$ MeV $Dbar D^*$ binding of the $X(3872)$ itself. This raises a quantitative concern: when the kaon scatters off a constituent inside the $X(3872)$, the attraction may be strong enough to rearrange the system into $D_{s0}^* D^*$ or $D_{s1} D$ channels rather than $KX(3872)$. The FCA does not include these rearrangement channels. The authors cite Ref. [37] as evidence of robustness, but that work also uses the FCA, so it is a consistency check within the same approximation, not an independent validation. The authors should at minimum discuss this limitation quantitatively and assess whether the absence of rearrangement channels could shift the predicted bound state energy.
  3. The width of the predicted state is estimated as ~1 MeV from the half-height width of Im[T^{tot}] (Fig. 4), but the $D_{s0}^*(2317)$ and $D_{s1}(2460)$ input widths are set to 0.1 MeV (Appendix A). The relationship between the input two-body widths and the output three-body width is not discussed. Is the 1 MeV width driven by the phase space available in the FCA loops, or is it sensitive to the assumed input widths? Please clarify the mechanism generating the width.
minor comments (6)
  1. The abstract states the width is 'approximately 1 MeV' but the text does not provide a systematic uncertainty or sensitivity analysis. A brief comment on the precision of this estimate would help.
  2. In Eq. (9), the Heaviside functions $Theta(q_{max}^{(1)} - q_1^*)$ and $Theta(q_{max}^{(2)} - q_2^*)$ are introduced but the values of $q_{max}^{(1)}$ and $q_{max}^{(2)}$ are only given in Appendix A. A forward reference would improve readability.
  3. The scattering length $a = (0.39 - i 0.00)$ fm has a vanishingly small imaginary part. The text attributes this to 'weak inelasticity,' but given that the $KD$ and $KD^*$ channels are below threshold at the $KX$ threshold, it would be useful to state explicitly which channels could contribute to Im[a] and why they are suppressed.
  4. Figures 3 and 4 show the amplitude but the axes are not labeled with physical units or clear scale markers. Please ensure axis labels are readable in the final version.
  5. The phrase 'reduced masss' in Eq. (25) contains a typo.
  6. The claim that the width 'stems from the suppressed decay channels due to the molecular nature of the system' (Sec. III) is vague. Please specify which decay channels are relevant and how they enter the formalism.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful reading and constructive comments. All three major comments are well-taken. We will (1) add the missing source radius value and its justification, (2) add a quantitative discussion of the rearrangement-channel limitation inherent to the FCA, and (3) clarify the mechanism generating the predicted width. We briefly defend the internal consistency of the approach while acknowledging the stated limitations.

read point-by-point responses

  1. Referee: The source radius R in Eq. (31) is never specified in the text, yet it directly determines the shape of the correlation function in Fig. 5. Without this value, the central prediction of the paper is not reproducible. Please state the value of R used and justify the choice (e.g., from typical pp or pA collision sources).

    Authors: The referee is correct that the source radius R is not stated in the manuscript, and we apologize for this omission. The value used in the calculation is R = 1 fm, which is consistent with the source sizes employed in the femtoscopic studies of analogous molecular systems, including the pf1(1285) correlation function (Ref. [28]) and the Kf1(1285) system (Ref. [32]). This value is representative of the emission source in pp collisions at LHC energies for particles in the charm sector, as extracted from ALICE measurements. We will add this value and its justification to the revised manuscript, in the paragraph preceding Eq. (31) and in the caption of Fig. 5. revision: yes

  2. Referee: The FCA treats the X(3872) as a rigid cluster, but the KD and KD* attractions that generate the Ds0*(2317) and Ds1(2460) poles involve binding of 45-70 MeV, which is 200-350 times larger than the <0.2 MeV Dbar D* binding of the X(3872) itself. This raises a quantitative concern: when the kaon scatters off a constituent inside the X(3872), the attraction may be strong enough to rearrange the system into Ds0* D* or Ds1 D channels rather than KX(3872). The FCA does not include these rearrangement channels. The authors cite Ref. [37] as evidence of robustness, but that work also uses the FCA, so it is a consistency check within the same approximation, not an independent validation. The authors should at minimum discuss this limitation quantitatively and assess whether the absence of rearrangement channels could shift the predicted bound state energy.

    Authors: We agree with the referee that the rearrangement channels (Ds0* D* and Ds1 D) are a legitimate concern and that Ref. [37] provides a consistency check within the FCA rather than an independent validation. We will add a dedicated discussion of this limitation in the revised manuscript. To address the quantitative aspect: the key point is that the FCA optical potential already incorporates the full KD and KD* scattering amplitudes, including the Ds0*(2317) and Ds1(2460) poles, as the driving two-body dynamics. The rearrangement channels would appear as explicit coupled channels in a full Faddeev calculation, but their effect is partially captured through the energy dependence of the two-body amplitudes t1 and t2 evaluated at the sub-energies of Eqs. (18)-(20). That said, we acknowledge that the FCA does not include the explicit propagation of Ds0* D* or Ds1 D intermediate states, which could in principle shift the bound state energy. We estimate the potential impact as follows: the binding of the predicted state (~50 MeV) is comparable to the binding in the analogous systems studied in Refs. [29, 30, 32], where the FCA results were consistent with expectations. The fact that the earlier non-unitarized FCA calculation of Ref. [37] found a bound state at a similar energy provides an internal consistency check, though we agree it is not an independent validation. We will add a paragraph in Section III discussing this limitation, noting that a full coupled-channel Faddeev calculation including rearrangement channels would be needed to quantify the shift, and that our prediction should be understood as the result within the stated FCA framework. revision: partial

  3. Referee: The width of the predicted state is estimated as ~1 MeV from the half-height width of Im[T^tot] (Fig. 4), but the Ds0*(2317) and Ds1(2460) input widths are set to 0.1 MeV (Appendix A). The relationship between the input two-body widths and the output three-body width is not discussed. Is the 1 MeV width driven by the phase space available in the FCA loops, or is it sensitive to the assumed input widths? Please clarify the mechanism generating the width.

    Authors: We thank the referee for raising this point, which requires clarification in the manuscript. The ~1 MeV width of the predicted three-body state is not directly inherited from the input two-body widths (0.1 MeV for Ds0* and Ds1). Rather, it is generated by the imaginary part of the FCA loop functions G0 and Gc (Eqs. 9 and 16), which arise from the phase space available when the kaon and the cluster constituents propagate on shell in the intermediate states. The two-body amplitudes t1 and t2 are complex due to their own coupled-channel structure (the KD and KD* systems have open channels below threshold), and this complexity feeds into the optical potential. The width of the three-body state is then determined by the interplay between the imaginary part of the optical potential and the phase space of the KX(3872) elastic channel encoded in Gc. Since the predicted state is below the KX(3872) threshold, the width is not from elastic KX decay but from the virtual coupling to sub-threshold channels through the optical potential. We will add a clarifying paragraph in Section III explaining this mechanism, and we will note that the width is not sensitive to the precise input values of the Ds0* and Ds1 widths (0.1 MeV), since the dominant contribution comes from the loop function imaginary parts rather than from the Breit-Wigner widths of the two-body poles. revision: yes

Circularity Check

0 steps flagged

No significant circularity: the KX(3872) bound-state prediction emerges from external two-body inputs (Ds0*, Ds1 poles, vector-exchange potentials), not from a fitted three-body parameter renamed as a prediction.

full rationale

The central prediction—a bound state at ~4315 MeV—arises from the three-body FCA dynamics driven by the KD and KD* two-body amplitudes (Appendix A), which are constructed from the Ds0*(2317) and Ds1(2460) poles and hidden-gauge vector-exchange potentials. These are external inputs from chiral unitary / hidden-gauge approaches, not quantities fitted to the KX(3872) system itself. The cutoff q_max = 288 MeV (Eq. 10) is fitted to reproduce the X(3872) mass in Ref. [16] (a prior work by overlapping authors), but this parameterizes the internal DD* wave function of the cluster, not the KX interaction output. The prediction of a KX bound state is therefore not equivalent to its inputs by construction. The self-citations to Refs. [29]-[34] for the unitarized FCA formalism are methodological references; the formalism itself (Lippmann-Schwinger resummation of an optical potential) is a standard, independently verifiable technique, not a self-citation that defines the result. The correlation function (Eq. 29) follows from the scattering amplitude via standard femtoscopy formalism. No step in the derivation chain reduces to its own inputs by construction. The reader's concern about the FCA validity is a correctness/approximation risk, not a circularity issue. Score 2 reflects minor self-citation load (q_max from Ref. [16], formalism from Refs. [29]-[34]) that is not load-bearing for the central prediction's independence from its inputs.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

No new particles, forces, or entities are postulated. The bound state predicted is a dynamical consequence of known interactions, not a new entity introduced by fiat. The D*_s0(2317) and D_s1(2460) are known resonances used as inputs.

free parameters (5)
  • q_max (DD* loop) = 288 MeV
    Cutoff for the DD* loop function, determined in Ref. [16] by fitting to reproduce the X(3872) mass. Used in the cluster form factor Fc(q), Eq. (10-11).
  • q_max (KD loop) = 689 MeV
    Cutoff for the KD and KD* loop functions in the two-body amplitudes, Eq. (A9). Taken from Refs. [35, 51].
  • g_Ds0,KD = 9600 MeV
    Coupling of D*_s0(2317) to KD channel, taken from Refs. [35, 52]. Used in the Breit-Wigner parametrization, Eq. (A12).
  • g_Ds1,KD* = 12107 MeV
    Coupling of D_s1(2460) to KD* channel, taken from Ref. [46]. Used in the Breit-Wigner parametrization, Eq. (A3).
  • R (source radius) = not stated in paper
    Gaussian source radius in the correlation function, Eq. (31). Required for the correlation function calculation but its value is not given in the text.
axioms (5)
  • domain assumption X(3872) is a D D* molecular state with I=0 and positive C-parity
    Stated in the abstract and Sec. I. This is the foundational assumption of the entire calculation. While supported by experimental evidence, it is not established beyond doubt (tetraquark alternatives exist, Ref. [20]).
  • ad hoc to paper Fixed-center approximation: the D and D* constituents are static during kaon scattering
    Encoded in Eqs. (5)-(9). The cluster is treated as a rigid bound state. Given the X(3872) binding energy < 0.2 MeV, this is a strong assumption.
  • domain assumption D* to D-bar* transition is suppressed (negligible double-scattering between components)
    Sec. II, Eq. (22): the transition requires charmed meson exchange and is taken as highly suppressed per Ref. [42]. This justifies averaging the two components independently.
  • ad hoc to paper Binding energy distributed between D* and D in proportion to their masses
    Eq. (20), the parameter xi = Mc/(MD* + MD_bar). This is a modeling choice for how the kaon energy is partitioned.
  • domain assumption Local hidden gauge approach with vector meson exchange for KD and KD* interactions
    Appendix A, Eqs. (A7-A8, A13-A14). Standard framework in this group's prior work but an assumption about the underlying dynamics.

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Reference graph

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