Is the X(3872) a bound state ?

TL;DR

This paper questions whether the X(3872) is a bound state by analyzing experimental data and theoretical cancellations, suggesting it may be a half-bound state rather than a true bound state.

Contribution

It introduces a novel cancellation effect in the $1^{++}$ channel and compares experimental production data to challenge the bound state interpretation of X(3872).

Findings

Cancellation in the $1^{++}$ channel affects invariant mass analysis.

Experimental production rates are consistent with a half-bound state.

The bound state nature of X(3872) is strongly questioned.

Abstract

All existing experimental evidence of the bound state nature of the $X(3872)$ relies on considering its decay products with a finite experimental spectral mass resolution which is typically $\Delta m \ge 2 $MeV and much larger than its alleged binding energy, $B_X=0.00(18)$MeV. On the other hand, we have found recently that there is a neat cancellation in the $1^{++}$ channel for the invariant $D \bar D^*$ mass around the threshold between the continuum and bound state contribution. This is very much alike a similar cancellation in the proton-neutron continuum with the deuteron in the $1^{++}$ channel. Based on comparative fits of experimental cross section deuteron and $X(3872)$ prompt production in pp collisions data with a finite $p_T$ to a common Tsallis distribution we find a strong argument questioning the bound state nature of the state but also explaining the large observed production rate likely consistent with a half-bound state.

Equations

$$N(M)=\sum _i\theta (M-M_i^B)+\frac{1}{\pi }\sum _{\alpha =1}^n[{\delta }_{\alpha }(M)-{\delta }_{\alpha }(M_{\mathrm{th}})].$$

$$O_{\Delta m}\equiv {\int }_{m-\Delta m/2}^{m+\Delta m/2}dM\rho (M)O(M).$$

$$\rho (M)=\frac{dN(M)}{dM}=\sum _i\delta (M-M_i^B)+\frac{1}{\pi }\sum _{\alpha =1}^n{\delta }_{\alpha }^{\prime }(M),$$

$$O{|}_{M^B\pm \Delta m}=O(M^B)+\frac{1}{\pi }{\int }_{M_{\mathrm{tr}}}^{M_{\mathrm{tr}}+\Delta m/2}dM{\delta }_{\alpha }^{\prime }(M)O(M).$$

$$\frac{d^{3}N}{d^{3}p}=\frac{gV}{(2\pi {)}^3}{\left(1+(q-1)\frac{E(p)}{T}\right)}^{-\frac{q}{q-1}}\stackrel{q\to 1}{\to }\frac{gV}{(2\pi {)}^3}{e}^{-\frac{E(p)}{T}}$$

$$\frac{1}{2\pi p_T}\frac{d\sigma (m)}{d{p}_T}=\mathcal{N}\int dyE(p_T,y){\left[1+\frac{q-1}{T}E(p_T,y)\right]}^{\frac{q}{1-q}}$$

$$\frac{{\sigma }_{m=M_X\pm \Delta m}}{{\sigma }_{M_X}}\sim \frac{{\mathcal{N}}_{\Delta m}}{{\mathcal{N}}_X}.$$

Full text

2.39.Pn, 14.40.Lb, 14.40.Rt

1 Introduction

The early possibility of loosely bound states near the charm threshold first envisaged in Ref. [1] seems to be confirmed now by the wealth of evidence on the existence of the $X(3782)$ state with binding energy $B_{X}=M_{D}+M_{\bar{D}^{*}}-M_{X}=0.00(18)$MeV [2] and which has triggered a revolution by the proliferation of the so-called X,Y,Z states (for reviews see e.g. [3, 4]). In the absence of electroweak interactions this state has the smallest known hadronic binding energy. However, since this state is unstable, all the detection methods of the $X(3872)$ are based on looking for its decay channels spectra such as $X\to J/\psi\pi^{+}\pi^{-}$ where the mass resolution never exceeds $\Delta m\sim 1-2$MeV [5, 6, 7, 8] (see e.g. [9] for a pictorial display on spectral experimental resolution). Therefore it is in principle unclear if one can determine the mass of the $X(3872)$ or equivalently its binding energy $\Delta B_{X}\ll\Delta m$ with such a precision, since we cannot distinguish sharply the initial state.

In most analyses up to now (see however [10]) the bound state nature is assumed rather than deduced. In fact, the molecular interpretation has attracted considerable attention, since for a loosely bound state many properties are mainly determined by its binding energy [4] and characterized by a line shape in production processes [11]. However, we have noticed recently a neat and accurate cancellation between the would-be X(3872) bound state and the $D\bar{D}^{*}$ continuum which has a sizable impact on the occupation number at finite temperature [12, 13]. This reduction stems from a cancellation in the density of states in the $1^{++}$ channel and potentially blurs any detected signal where a superposition of $1^{++}$ states is at work. Such a circumstance makes us questioning in the present letter the actual character of the state. We will do so by analyzing the $p_{T}$ distribution of the $X(3872)$ in high energy production experiments and folding the expected distribution with the actual mass distribution corresponding to the $1^{++}$ spectrum via the level density within the accessible experimental resolution. For our argument a qualitative and quantitative comparison with a truely weakly bound state such as the deuteron, $d$, will be most enlightening. As a matter of fact, the similarities between $d$ and $X(3872)$ have been inspiring [14, 15, 16]. Compared to the $X(3872)$ the main difference is that the deuteron is detected directly by analyzing its well defined track and/or stopping power. Actually, the production of loosely bound nuclei and anti-nuclei, including $d$,$\bar{d}$, ${}^{3}{\rm He}_{\Lambda}$, etc. in ultra-high pp collisions is a remarkable and surprissing experimental observation in recent years [17] and so far poorly understood [18].

The cancellation echoes a similar effect on the deuteron pointed out by Dashen and Kane in their discussion on the counting of states in the hadron spectrum in a coarse grained sense [19] which we review in some detail in the next section. In section 3 we analyze the consequences in a production process. Finally, in section 4 we draw our the conclusions and provide an outlook for future work.

2 Dashen and Kane cancellation mechanism

In order to illustrate the Dashen-Kane mechanism [19] we introduce the cumulative number of states with invariant CM mass $\sqrt{s}$ below $M$ in a given channel with fixed $J^{PC}$ quantum numbers. This involves the $J^{PC}$ spectrum which contains bound states and continuum states with threshold $M_{\rm th}$ and is given as

[TABLE]

where the index $i$ runs over the $M_{i}^{B}$ bound states and $\alpha$ over the $n$ coupled channels. Here we have separated bound states $M_{n}^{B}$ explicitly from scattering states written in terms of the eigenvalues of the S-matrix, i.e. $S=U{\rm Diag}(e^{2i\delta_{1}},\dots,e^{2i\delta_{n}})U^{\dagger}$, with $U$ a unitary transformation for $n$-coupled channels and $\delta_{i}(M)$ the eigenphaseshifts for the channel $i$ at CM invariant mass $\sqrt{s}=M$. This definition fulfills $N(0)=0$. In the single channel case, and in the limit of high masses $M\to\infty$ one gets $N(\infty)=n_{B}+\frac{1}{\pi}[\delta(\infty)-\delta(M_{\rm th})]=0$ due to Levinson’s theorem. While the origin of the bound state term is quite obvious, the derivation of the continuum term is a bit subtle but standard and can be found in many textbooks on statistical mechanics dealing with the quantum virial expansion (see e.g. [20, 21]). In potential scattering it can be best deduced by confining the system in a large spherical box which quantizes the energy and relates the energy shift due to the interaction to the phase-shift and then letting the volume of the system go to infinity [19].

In the particular case of the deuteron, which is a neutron-proton $1^{++}$ state bound by $B_{d}=2.2{\rm MeV}$, the cancellation between the continuum and discrete parts of the spectrum was pointed out by Dashen and Kane long ago [19]. (see also [22, 23] for an explicit picture and further discussion within the resonance gas model framework). The opening of new channels and the impact of confining interactions was discussed in Ref. [24]. In the $1^{++}$ channel, the presence of tensor force implies a coupling between the ${}^{3}S_{1}$ and ${}^{3}D_{1}$ channels. While the partial wave analysis of NN scattering data and the determination of the corresponding phase-shifts is a well known subject [25], we note that a similar analysis in the $D\bar{D}^{*}$ case is at present in its infancy. In our first model determination in Ref. [12, 13] the mixing has an influence for larger energies than those considered here. Therefore, in order to illustrate how the cancellation comes about, we consider a simple model which works sufficiently accurately for both the deuteron and the $X(3872)$ by just considering a contact (gaussian) interaction [26] in the ${}^{3}S_{1}$-channel and using efective range parameters to determine the corresponding phase-shift in the $d$ and $X(3872)$ channels [27, 12] respectively.

The result for $N(M)$ in both $d$ or $X$ cases depicted in Fig. 2 display a similar pattern for the $np$ or $D\bar{D}^{*}$ invariant masses respectively. The sharp rise of the cumulative number is followed by a strong decrease generated by the phase-shift. For larger invariant masses $M$ several effects appear, and in particular the nuclear core (see e.g. [23]) or composite nature of the $X(3872)$ and its $c\bar{c}$ content becomes manifest (see eg. [28]).

\figcaption

Cumulative number in the $1^{++}$ channel for the deuteron (solid) and $X(3872)$ (dashed) as a function of the invariant mass $M$ respect to np and $D\bar{D}^{*}$ values respectively. We divide by spin degeneracy.

An immediate consequence of this effect trivially follows from Eq. (1) for an observable depending on the invariant mass function $O(M)$. The corresponding measured quantity for a bin in the range $(m-\Delta m/2,m+\Delta m/2)$ becomes

[TABLE]

where $\rho(M)$ is the density of states, defined as

[TABLE]

where $\delta^{\prime}_{\alpha}(M)$ denotes the derivative of the phase shift with respect to the mass.

In the single channel case, with phase shift $\delta_{\alpha}(M)$, and if the resolution is much larger than the binding energy $\Delta m\gg|B|\equiv|M_{B}-M_{\rm tr}|$ one has

[TABLE]

which, on view of Fig. 2 and for a smooth observable $O(M)$, points to the cancellation, anticipated by Dashen and Kane [24]. The effect was explicitly seen in the $np$ virial coefficient at astrophysical temperatures, $T\sim 1-10$ MeV [29]. We have recently shown [12] how this cancellation can likewise be triggered for the $X(3872)$ ocupation number at quark-gluon crossover temperatures $T\sim 100-200$MeV. This will be relevant in relativistic heavy ion collisions when X-production yields are measured, because the partition function involves a Boltzmann factor, $\sim e^{-\sqrt{p^{2}+m^{2}}/T}$ with the density of states, Eq. (3) and the measured yields reproduce remarkably the predictions occupation numbers in the hadron resonance gas model [30].

Therefore, given these tantalizing similarities a comparative study of the deuteron and $X(3872)$ production rates to ultra-high energies in colliders provides a suitable callibration tool in order to see the effects of the Dashen-Kane cancellation due to the finite resolution $\Delta m$ of the detectors signaling the $X(3872)$ state via its decay products and decide on its bound state character. Here we propose to study the effect in the observed tranverse momentum ($p_{T}$) distributions.

3 $X(3872)$ production abundance

While the theory regarding the shape of transverse momentum distribution is not fully developped (see e.g. Ref. [31] for an early review, Ref. [32] for a historical presentation), we will rest on phenomenological ansatze which describe the data. On the one hand, the asymptotic $p_{T}$-spectrum [33] provides a production rate $1/p_{T}^{8}$ based on quark-quark scattering. Hagedorn realized that an interpolation between the power correction and a thermal Boltzmann $p_{T}$-distribution would work [34]. A thermodynamic interpretation for non-extensive systems [35] of the rapidity distribution was proposed by Tsallis [36] and first applied to high energy phenomena in Refs. [37, 38], namely the differential occupation number is given by

[TABLE]

where $E(p)=\sqrt{p^{2}+m^{2}}$, $V$ is volume of the system, $T$ the temperature and $g$ the degrees of freedom and, as indicated, the limit $q\to 1$ produces the Boltzmann distribution. We use here the form obtained by the maximum Tsallis entropy principle [39].

The invariant differential production rate, $d^{3}N/(d^{2}p_{T}dy)\equiv E_{p}d^{3}N/d^{3}p$ with $y=\tanh^{-1}(E_{p}/p_{z})$ the rapidity, has the asymptotic matching corresponds to $q=1.25$ [40]. While the thermodynamic interpretation is essential to link the degrees of freedom $g$ with the production rate [41], we note that we have checked [42] that a Tsallis distribution describes accurately the results from particle Monte Carlo generators such as PYTHIA [43, 44]. This distribution has also been applied recently by the ALICE collaboration to $d$-production [45] in $pp$ collisions.

We show next that the $X(3872)$, $\Psi(2S)$ and deuteron prompt production cross sections can be described with the same Tsallis distribution:

[TABLE]

with $E(p_{T},y)=\sqrt{p_{T}^{2}+m^{2}}\cosh y$ and ${\cal N}$ a normalization factor. Obviously, a direct comparison requires similar $p_{T}$ values as possible for both $d$, $\Psi(2S)$ and $X(3872)$; the closest ones come from ALICE [45] and CMS [46, 47] respectively. The ATLAS data for $X(3872)$ [48] confirm a power law behaviour in $p_{T}$ but extend over a much larger range than the available $d$ and hence are not used in this study. The deuteron data is given in invariant differential yields $d^{2}N/(2\pi p_{T}dp_{T}dy)$, hence the inelastic $pp$ cross section at $\sqrt{s}=7$ TeV, $\sigma_{\rm inel}^{pp}=73.2\pm 1.3$ mb, as measured by TOTEM [49] has been used to transform it into differential cross section.

\tabcaption

Best fit of parameters for Tsallis distribution. The X data used from CMS [47] is multiplied by the branching fraction ${\cal B}_{X}\equiv{\cal B}(X\to J/\psi\pi^{+}\pi^{-})$. Correlation between $q$ and $T$ parameters is practically $-1$ ($r=-0.9992$).

$X(3872)+d$ $X(3872)+\Psi(2S)+d$

$\ln({\cal N}_{X}{\cal B}_{X})$ $41.4\pm 0.4$ $41.4\pm 0.4$

$\ln({\cal N}_{d})$ $40.35\pm 0.09$ $40.35\pm 0.09$

$\ln({\cal N}_{\Psi})$

$44.3\pm 0.2$

$q$ $1.122\pm 0.001$ $1.122\pm 0.001$

$T$ [MeV] $7.017\pm 0.07$ $7.018\pm 0.07$

$N_{d}$ $(2.02\pm 0.02)\cdot 10^{-4}$ $(2.01\pm 0.02)\cdot 10^{-4}$

$N_{X}{\cal B}_{X}$ $(9\pm 3)\cdot 10^{-6}$ $(9\pm 3)\cdot 10^{-6}$

$N_{\Psi}$

$(2.2\pm 0.3)\cdot 10^{-4}$

$\langle p_{T}\rangle_{d}$ $1.102\pm 0.007$ $1.102\pm 0.007$

$\langle p_{T}\rangle_{X}$ $2.249\pm 0.015$ $2.249\pm 0.015$

$\langle p_{T}\rangle_{\Psi}$ $2.142\pm 0.014$ $2.142\pm 0.014$

$N_{X}{\cal B}_{X}/N_{d}$ $0.046^{+0.016}_{-0.013}$ $0.046^{+0.015}_{-0.013}$

$N_{\Psi}/N_{d}$

$1.09^{+0.16}_{-0.17}$

On a phenomenological level we perform two fits: One including $d$ and $X(3872)$ data and another one adding the $\Psi(2S)$ data. In both cases the ${\cal N}_{d,X,[\Psi]}$, $q$ and $T$ are fitted by minimizing the corresponding $\chi^{2}$ function with Minuit [50]. The experimental error in the $x$-axis has been incorporated in the $\chi^{2}$ via a MonteCarlo procedure with $5000$ runs, where the $p_{T}$ value of each experimental data has been randomly shifted within the experimental range with an uniform distribution. Due to the scarcity of X data, we assume the production rate is mainly driven by the deuteron. That way, an initial minimization of the $q$, $T$ and ${\cal N}_{d}$ is done, and the resulting best-fit values of $q$ and $T$ are employed to fix ${\cal N}_{X}$ and ${\cal N}_{\Psi}$.

\figcaption

Comparison between the prompt production cross section in $pp$ collisions of $X(3872)$ (blue), the deuteron (green) and the $\Psi(2S)$ (red). $\Psi(2S)$ data from CMS [46]. The $X(3872)$ data from CMS [47] is multiplied by the branching fraction ${\cal B}(X\to J/\psi\pi\pi)$. Deuteron data in $pp$ collisions are taken from ALICE [45]. The lines are Tsallis distributions fitted to each data set, with the same $q$ and $T$ parameters. We fit both $X(3872)$, $\Psi(2S)$ and deuteron data. The shadowed bands represent the statistical $68\%$ confidence level (CL) obtained from the fit.

\figcaption

In red, the $X(3872)$ vs deuteron prompt production ratio as a function of the branching fraction ${\cal B}(X\to J/\psi\pi^{+}\pi^{-})$ when fitting to $X(3872)$, $\Psi(2S)$ and deuteron data. The shadowed band represent the statistical $68\%$ confidence level (CL) obtained from the fit. The green band shows the constrains of the recent analysis of C. Li et al from BESIII data [51] ${\cal B}_{X}\equiv{\cal B}(X\to J/\psi\pi^{+}\pi^{-})=4.5^{+2.3}_{-1.2}$.

The results can be found in Tab. 3, and the final production fit at Fig. 3 for the two considered calculations, one with the $X(3872)$ and the deuteron, and another for the $X(3872)$, the $\Psi(2S)$ and the deuteron. They are both compatible, as expected, as the $X$ to $\Psi(2S)$ production ratio, measured at CMS, is almost constant [47]. The $X/d$ production ratio is $0.046^{+0.016}_{-0.013}$ for $X+d$ fit (and practically the same value for X+$\Psi$+d fit), dependent on the value of the branching fraction. Note that we do not have the pure cross section for the $X$, as it is multiplied by the unmeasured branching fraction which has been recently constrained in an analysis of BESIII data by C. Li et al [51] to be ${\cal B}_{X}\equiv{\cal B}(X\to J/\psi\pi^{+}\pi^{-})=4.5^{+2.3}_{-1.2}\%$. This value is consistent with the PDG lower- ${\cal B}_{X}>3.2\%$ [52], and upper bound ${\cal B}_{X}<6.6\%$ at $90\%$ C.L. [53]. The uncertainty comes from the most recent value of ${\cal B}_{X}\cdot(B^{-}\to K^{-}X(3872))<2.6\times 10^{-4}$ at $90\%$ C.L. [54]. We note that in a recent paper, Esposito et al. [55] consider the wider range $8.1^{+1.9}_{-3.1}\%$.

Consequently, we can study the ratio of the X/d occupation numbers as a function of the ${\cal B}_{X}$ branching fraction. In Fig. 3 we see the results. Considering the error bars, the experimental constrains give ratios between $0.3$ and $1.9$ for $N_{X}/N_{d}$.

In our previous fits above we have neglected the role played by the finite resolution of the detectors, $\Delta m$, which we dicuss next. Ref. [47] uses a $\pm 2\sigma$ window around the $X(3872)$ mass, with $\sigma=5-6$ MeV, to select the $X(3872)$ events in the $J/\psi\pi\pi$ invariant mass spectrum. That means that the branching fraction ${\cal B}(X\to J/\psi\pi^{+}\pi^{-})$, as measured by CMS is averaged in the $\left[M_{X}-2\sigma,M_{X}+2\sigma\right]$ energy window, which includes the continuum. As a consequence of the energy window, there are many decays that can be affected, those involving the $\bar{D}^{0}D^{0\,*}$ channel.

\figcaption

The relative occupation number with and without finite resolution $\Delta m=2\sigma$ as a function of the $X(3872)$ binding energy. Using $M_{X}=M_{D}+M_{\bar{D}^{*}}-\gamma_{X}^{2}/(2\mu_{D,\bar{D}^{*}})$ we take the positive branch for the virtual state ($\gamma_{X}>0$) and the negative branch for the bound state ($\gamma_{X}<0$) for different $\sigma$ values.

In fact, the distribution obtained from Eq. 5 depends on the mass, and hence its observed value undergoes formula 4, reflecting the finite resolution. Similar to the finite temperature case [12] we have checked that the Tsallis $p_{T}$-shape is basically preserved for $p_{T}\gg\Delta m$, but the occupation number is modified for $\Delta m\gg B$.

For definiteness we use $\Delta m=2\sigma$, as CMS measures the $X(3872)$ in a $\pm 2\sigma$ region around the central value of the X mass. The net effect is summarized in a ratio, which we find to be practically independent of the transverse momentum $p_{T}$ for the Tsallis distribution shape,

[TABLE]

This formula will allow us to set values for the relative occupation numbers due to the finite resolution. We take $M_{X}=M_{D}+M_{\bar{D}^{*}}-\gamma_{X}^{2}/(2\mu_{D,\bar{D}^{*}})$ as a parameter by looking at the poles of the $D\bar{D}^{*}$ S-matrix in the ${}^{3}S_{1}-^{3}D_{1}$ channel [12]. Therefore, while in the limit of $\Delta m\to 0$ we should expect the ratio $N_{X}/N_{d}\to 1$, $1/2$ or [math] for a bound ($\gamma_{X}>0$), half-bound ($\gamma_{X}=0$) or unbound (actually virtual, ($\gamma_{X}<0$)) state, for increasing and finite $\Delta m$ the value lies somewhat in between and the different situations can be hardly distinguished. However, as seen in Fig. 3 the numerical value $N_{\Delta m}/N_{X}\sim 0.5-0.6$ is rather stable for a reasonable range of $B_{X}$ and $\sigma$ values. If we re-interpret $N_{X}$ as $N_{X,\Delta m}$ this falls remarkably in the bulk of Fig. 3 where $N_{X,\Delta m}/N_{d}\sim 0.5$ implies $N_{X}/N_{d}\sim 1$. Thus, unlike expectations, we do not find the production rate to change dramatically due to binding energy effects due to $\Delta m$; the $p_{T}$ shape will likewise not depend on this (unlike expectations [55]). In a recent and insightful paper Kang and Oller have analyzed the character of the $X(3872)$ in terms of bound and virtual states within simple analytical parameterizations [10]. While the Dashen-Kane cancellation has not been explicitly identified, it would be interesting to see if their trends can be reproduced by more microscopic approaches.

4 Conclusions

Theoretically, it is appealing an scenario where the $X(3872)$ is a half-bound state (zero binding energy) corresponding to the so-called unitarity limit, characterized by scale invariance [56]. In this case, the phase-shift becomes $\delta=\pi/2$ around threshold, and the occupation number becomes a half of that of the bound state. Our analysis shows that the large production rate of the $X(3872)$ at finite $p_{T}$ does not depend strongly on the details of the binding since the experimental bin size is much larger than the binding energy. We also find striking and universal shape similarities with the $\psi(2S)$ and deuteron production data via a common Tsallis distribution. A more direct check of our predicted mild suppression might be undertaken if all production data would be within the same $p_{T}$ values. Finally, we note that in order to envisage a clear fingerprint of the $X(3872)$ binding character a substantial improvement on the current experimental resolution of its decay products would be required.

Acknowledgements.

One of us (E.R.A.) thanks Airton Deppman for discussions on Tsallis distributions. This work is partly supported by the Spanish Ministerio de Econom a y Competitividad and European FEDER funds (grants FPA2016-77177-C2-2-P and FIS2017-85053-C2-1-P) and Junta de Andalucía (grant FQM-225).