Possible precise measurements of the $X(3872)$ mass with the $e^+e^-\to\pi^0\gamma X(3872)$ and $p\bar p\to\gamma X(3872)$ reactions

It was recently proposed that the $X(3872)$ binding energy, the difference between the $D^0\bar D^{*0}$ threshold and the $X(3872)$ mass, can be precisely determined by measuring the $\gamma X(3872)$ line shape from a short-distance $D^{*0}\bar D^{*0}$ source produced at high-energy experiments. Here, we investigate the feasibility of such a proposal by estimating the cross sections for the $e^+e^-\to\pi^0\gamma X(3872)$ and $p\bar p\to\gamma X(3872)$ processes considering the $D^{*0}\bar D^{*0}D^0/\bar D^{*0}D^{*0}\bar D^0$ triangle loops. These loops can produce a triangle singularity slightly above the $D^{*0}\bar D^{*0}$ threshold. It is found that the peak structures originating from the $D^{*0}\bar D^{*0}$ threshold cusp and the triangle singularity are not altered much by the energy dependence introduced by the $e^+e^-\to\pi^0D^{*0}\bar D^{*0}$ and $p\bar p\to\bar D^{*0}D^{*0}$ production parts or by considering a finite width for the $X(3872)$. We find that $\sigma(e^+e^-\to\pi^0\gamma X(3872)) \times B(X(3872)\to\pi^+\pi^-J/\psi)$ is about 0.03 fb to 0.2 fb with the $\gamma X(3872)$ invariant mass integrated from 4.01 to 4.02 GeV and the c.m. energy of the $e^+e^-$ pair fixed at 4.23 GeV. The cross section $\sigma(p\bar p\to\gamma X(3872))\times B(X(3872)\to\pi^+\pi^-J/\psi)$ is estimated to be of $\mathcal{O}(10~{\rm pb})$. Our results suggest that a precise measurement of the $X(3872)$ binding energy can be done at PANDA.


I. INTRODUCTION
Among many charmonium-like states listed in the Review of Particle Physics (RPP) [1], special attention has been paid to the X(3872). 1 The mass of X(3872) is consistent with the D 0D * 0 threshold energy, m X = (3871.69 ± 0.17) MeV, and only an upper bound is provided for its small width, Γ X < 1.2 MeV [1]. The latest experimental development comes from the LHCb Collaboration that reported precise determinations of the mass and width [2,3]. In particular, a detailed analysis of the X(3872) line shape using the Flatté parametrization [4], which is more proper than the Breit-Wigner (BW) form for states near an S-wave stronglycoupled threshold, is performed in Ref. [2]. The closeness of its mass and the D 0D * 0 threshold invokes the hadronic molecular description of X(3872): the X(3872) is treated as a shallow S-wave bound state of DD * , e.g., in Refs. [5][6][7][8][9][10][11][12][13]. Such a description can successfully explain the large branching ratio of the isospin forbidden X(3872) → π + π − J/ψ relative to the isospin allowed π + π − π 0 J/ψ mode [14], and the strong coupling of the molecular state to its constituents in the molecular description, i.e., X(3872) to DD * , would naturally explain the large branching fractions of the X(3872) to π 0 D 0D0 /D 0D * 0 [1,15,16]. The strong coupling of the X(3872) to the D 0D * 0 in an S-wave implies that there must be a strong cusp exactly at the threshold [17], complicating the line shape analysis. The line shapes of the π + π − J/ψ and/or D 0D * 0 distributions were analyzed with the Flatté parametrization [2,[18][19][20][21] or the effective range expansion [22,23] in which the threshold effect is incorporated by requiring unitarity; however, no conclusive results have not been achieved so far. See, e.g., Refs. [24][25][26][27] and references therein for further information on works related to X(3872), in particular from the hadronic molecular point of view.
Recently, a possible way to precisely determine the X(3872) binding energy, which is defined as the difference between the D 0D * 0 threshold and the X(3872) mass 2 was proposed in Ref. [28]. This can be done by measuring the γX(3872) distribution instead of the X(3872) line shape in its decay products like π + π − J/ψ or D 0D * 0 . Consider a triangle diagram for the transition of an S-wave D * 0D * 0 pair, produced at short distances in some high-energy experiment, into γX(3872). The D * 0 (D * 0 ) subsequently decays 1 In this paper, the χ c1 (3872) in the RPP [1] is denoted by X(3872) or merely X, and Z c (4020) or Z c stands for the X(4020) in the RPP. 2 A negative δ corresponds to a mass above the threshold and thus a resonant state in this paper.
into γD 0 (γD 0 ), and the X(3872) is produced by merging the D 0D * 0 +D 0 D * 0 pair at the last step. The process thus proceeds via a D * 0D * 0 D 0 triangle loop. This loop can have a triangle singularity (TS) due to the simultaneous on-shellness of all three intermediate mesons, which leads to a peak in the γX(3872) distribution just above the D * 0D * 0 threshold. With the Landau equation [29] or with a simple equation for the TS position derived with a refined formulation [30], one sees that the TS position is sensitive to the X(3872) mass: the TS is located at 4015.14 MeV with δ = −180 keV and 4015.64 MeV with δ = −50 keV. For the X(3872) mass within (3871.69 ± 0.17) MeV [1], the TS appears in the range of m γX ∈ [4015. 17, 4016.40] MeV which can be obtained by using Eqs. (55) and (60) in Ref. [17]. While the TS, at which the amplitude diverges logarithmically, is turned into a finite peak due to the width of the internal particles, the peak originating from the TS of the D * D * D loop should be still clear thanks to the tiny width of the D * 0 , which is only 55.3 ± 1.4 keV [28,31]. Then, one expects that the X(3872) binding energy can be determined well with the precise measurement of the TS peak in the γX(3872) distribution.
The role of the TS stemming from the D * D * D loop on the X(3872) production has been studied in some papers. The e + e − → γX(3872) transition is studied in Refs. [32,33]. In Ref. [34], the energy dependence of the Z c (4050) 0 → γX(3872) branching fraction is studied.
One can see the difference of the energy dependence by changing the X(3872) binding energy.
In addition to the radiative reactions, decays emitting a pion with the D * 0D * 0 D 0 loop has also been considered [34][35][36]. While the TS appears in a smaller range of the πX(3872) energy compared with the γX(3872) case, the asymmetry of the πX(3872) line shape may be used to extract the X(3872) binding energy. The decay process B → (J/ψπ + π − )Kπ with the J/ψπ + π − produced by the D 0D * 0 rescattering considering the D * +D * 0 D 0 /D * − D * 0D0 loop is studied in Ref. [37]. For more works related to the TS, we refer to Ref. [17].
In this paper, we investigate two promising reactions in which the proposal of precisely measuring the X(3872) binding energy by virtue of the TS mechanism may be realized: the e + e − → π 0 γX(3872) and pp → γX(3872) reactions. In these reactions, the D * D * pair can be produced in an S wave. In the case of the e + e − collisions, the isovector resonance Z c (4020) seen in the D * D * distribution of the e + e − → π 0 (D * D * ) 0 process [38] is expected to be a good source of the S-wave D * D * pair, and high-statistics data can be expected for the pp reaction by the PANDA experiment at the Facility for Antiproton and Ion Research (FAIR) in the near future.
FIG. 1. Triangle diagram contributing to the e + e − → π 0 γX(3872) process considered here. This paper is organized as follows. In Sec. II, the formalism for calculating the e + e − → π 0 γX(3872) and pp → γX(3872) amplitudes is provided with the effect of the X(3872) width is taken into account. The results of our calculation, the γX(3872) invariant mass distributions in these reactions and the estimated cross sections, are given in Sec. III. A brief summary is given in Sec. IV. Detailed expressions of the amplitudes used in Sec. II are relegated to Appendix A.

II. FORMALISM
A. e + e − → π 0 γX(3872) First, we consider the e + e − → π 0 γX(3872) amplitude with the D * 0D * 0 D 0 /D * 0 D * 0D0 loops. The diagram is given in Fig. 1. Only the neutral D * D * D/D * D * D loops are accounted for the process because we focus on the TS peak of the γX(3872) invariant mass distribution near the D * 0D * 0 threshold and the X(3872) appears near the D 0D * 0 threshold as a narrow peak. As found in Ref. [38], the (D * D * ) 0 distribution of e + e − → π 0 (D * D * ) 0 at the c.m.

energies
√ s = 4.23 and 4.26 GeV can be described well by including a resonance with J P = 1 + , and the (D * D * ) 0 pair is predominantly produced by the resonance around the D * D * threshold. Here, we also assume that the Z c (4020) is the J P = 1 + exotic state which can decay into an S-wave D * D * pair. The πZ c (4020) pair is produced by the ψ(4230) resonance, which is seen in some hidden-and open-charm productions [1] and would be needed to describe the dependence of the cross section on the e + e − c.m. energy because the e + e − → πD * D * cross section at √ s = 4.26 GeV is smaller than that of √ s = 4.23 GeV [38].
We use the central values of the mass and width of the ψ(4230) given in the RPP [1], m ψ = (4220 ± 15) MeV and Γ ψ = (60 ± 40) MeV. Note that, while the width of the ψ(4230) is not fixed well, the γX(3872) invariant mass distribution at a given √ s, which will be considered in this work, is not affected by the details of the ψ(4230) properties.
The S-wave transition amplitude of the D 0D * 0 → X(3872) transition is written as and the coupling constant ofD 0 D * 0 → X(3872) is the same. We estimate the coupling constant g 4 with two different ways for the X(3872) mass above or below the D 0D * 0 threshold. When the X(3872) mass is below the D 0D * 0 threshold, the coupling constant can be evaluated assuming the X(3872) be an S-wave D 0D * 0 molecule [41][42][43], with µ D 0D * 0 and δ being the D 0D * 0 reduced mass and the X(3872) binding energy given by Eq. (1), respectively. In Eq. (6), g X is the coupling constant of X(3872) to the DD * pair of J P C = 1 ++ , and g 4 and g X are related with g 4 = g X /2 [36]. When the X(3872) mass is above the D 0D * 0 threshold, g 4 can be obtained by using the X(3872) → D 0D * 0 branching The coupling constant g 4 as a function of the X(3872) binding energy, δ. The black and red lines in δ > 0 and δ < 0 correspond to the cases with the X(3872) mass below and above the D 0D * 0 threshold, and g 4 is evaluated with Eqs. (6) and (7), respectively. ratio [44]; using Eq. (5), we have In this work, the mass of X(3872) is treated as a parameter, and it will be changed to see the difference of the γX(3872) invariant mass distribution. The width of X(3872), Γ X , is currently not known and the upper bound is provided [1]. Here, Γ X is assumed to be 100 keV, which is the value expected from the calculation of the X(3872) → π 0 D 0D0 partial width in the hadronic molecular picture [8,45,46] and the X(3872) → π 0 D 0D0 branching ratio [1,15,16]. The coupling constant g 4 as a function of δ is shown in Fig. 2. Note that the values of g 4 from both the δ > 0 and δ < 0 sides are similar if we neglect the part with δ in the vicinity of 0. In that special region, the absolute value of the imaginary of the pole position cannot be approximated by half the width computed using Eq. (7). Furthermore, the coupling of the X(3872) to the charged and neutral DD * can be computed from the residue of the coupled-channel D 0D * 0 -D + D * − T -matrix. It is found that the couplings of the X(3872) to D 0D * 0 and to D + D * − are approximately the same [14,45], and are consistent with the values shown in Fig. 2 (see also the discussion in Ref. [36]). For an estimate of the cross sections, in the following we will show the results with δ = ±50, ±180 keV and use the coupling shown in Fig. 2. Then, with the amplitudes Eqs. (2), (4), and (5), the e + e − → π 0 γX(3872) production p pD * 0 /D * 0 (−l) amplitude considering the D * 0D * 0 D 0 andD * 0 D * 0D0 triangle loops in Fig. 1 is given by with The factor of 2 in the above equation comes from the same contribution from the chargeconjugated loops. The library LoopTools is used for the evaluation of the one-loop integral [47]. The width of the particles are taken into account by replacing the mass of D * 0 With the e + e − → π 0 γX(3872) amplitude in Eq. (8), the γX(3872) invariant mass distribution is given by where p π 0 and p e are given by the expressions below Eq. (3) changing m 2 D * D * to m 2 γX , and . Ω π 0 and Ω γ are the solid angles of the π 0 in the e + e − c.m. frame and of the photon in the γX(3872) c.m. frame, respectively.
The pp → γX(3872) amplitude is considered in this part. The diagram of the pp → γX(3872) transition with the D * 0D * 0 D 0 /D * 0 D * 0D0 loops is shown in Fig. 3. model. With the effective Lagrangian for the pΛ c D * 0 coupling [49], the pp →D * 0 D * 0 transition amplitude with the Λ c exchange is written as where u andv are the spinors of the proton and antiproton, and a form factor F p,D * Λc is introduced. For the parameter g v , we take the value in Refs. [49,50] obtained by using the SU(4) model, g v = −5.20. For the form factor F p,D * Λc , we use The form factor like Eq. (11) is used, e.g., in Refs. [43,51], and the cut off is typically set to be around Λ = 2 GeV. Here, since the aim is to get an order-of-magnitude estimate of the cross section for the pp → γX(3872), it suffices to take a value used in the literature, and we take Λ = 2.0 GeV. The dependence of our results on this parameter will be checked.
We are interested in the manifestation of the TS in the γX(3872) invariant mass distribution. As shown in Ref. [52], the TS emerges when the process can occur classically, i.e., the internal particles of the loop are simultaneously placed on shell and all the momenta are collinear. At this time, the exchanged Λ c in theD * 0 D * 0 production is far away from on shell. Then, Eq. (10) can be approximated by taking the leading term of the expansion in powers of 1/m Λc . The pp →D * 0 D * 0 production amplitude is reduced to Because the internal particles are close to on shell in the vicinity of the TS energies, the 4-momentum transfer (p − k) 2 in F 2 p,D * Λc can be approximated by where the spatial momentum of theD * 0 is ignored because the TS energy is close to the D * 0D * 0 threshold.
The part of the triangle loop in Fig. 3 is the same as the e + e − → π 0 γX(3872) reaction given in Sec. II A. The pp → γX(3872) amplitude with the D * 0D * 0 D 0 loop is written as and theD * 0 D * 0D0 loop gives −iM The details of M where M ISI is a factor to take into account the pp initial-state interaction (ISI). In Ref. [49], this factor |M ISI | 2 is about 0.25 at √ s = 5 GeV and moderately increases along with √ s.
Here we treat M ISI as a constant and take |M ISI | 2 = 0.2 for an estimation of the ISI effect.

C. Width effect of the X(3872)
To take into account the width of the X(3872), the cross sections need to be convolved with the spectral function of the X(3872). 4 The spectral function may be parametrized using either the BW or the Flatté form. Although the latter is more proper for analyzing the X(3872) line shape in order to extract the pole position, they do not make much difference when convolved with cross sections. We consider both forms in the following.
First, we take the BW amplitude with a constant width for the X(3872) spectral function, As mentioned above, the X(3872) width Γ X = 100 keV is used in this calculation and the X(3872) mass will be changed within ±180 keV with respect to the D 0D * 0 threshold energy, which covers the range of the uncertainty of the X(3872) mass given in Ref. [1].
For comparison, the results smeared with a spectral function of the Flatté type will also be provided. In this case, the spectral function is given by [2,18] Γ X (m X ) =g(k 1 + k 2 ) + Γ X,ρ (m X ) + Γ X,ω (m X ) + Γ X0 , with Γ ρ and Γ ω being the widths of the ρ and ω mesons, respectively. The nonrelativistic momenta k 1,2 are analytically continued below the threshold. In the case with the Flatté amplitude, due to the scaling property [54] which hinders a determination all free parameters, here we only make a qualitative discussion on the line shape of the γX(3872) distribution expecting the magnitude to be of the same order as that in the BW case. We make use of the Flatté parameters, m X0 , Γ X0 , g, f ρ , and f ω from Ref. [2] which fixes m X0 and fits the other parameters to the data.
As pointed out in Ref. [28], for determining the X(3872) binding energy from the γX(3872) line shape, the X(3872) needs to be reconstructed from decay modes other than the π 0 D 0D0 one; otherwise, one has to consider the tree-level contribution of D * 0D * 0 → π 0 D 0D0 , which has a subtle interference with the triangle diagrams and cannot be treated as a smooth background near the TS energies [55][56][57][58]. In Ref. [33], the e + e − → γD * 0D0 process is studied, and it is found that the D 0D * 0 distribution with a fixed √ s is completely dominated by the tree-level contribution, which increases rapidly at the TS energy. In this work, we consider the π + π − J/ψ mode for reconstructing the X(3872). Then, we make the convolution as follows: with F being dσ e + e − ,π 0 γX /dm γX or σ pp,γX . The coupling constant g 4 is kept fixed to the value evaluated at the central value m X in the spectral function, and the X(3872) mass appearing in the loop amplitude is changed tom X in the convolution. Γ X,ρ (m X ) is the π + π − J/ψ part of the X(3872) decay width. In the BW case, the Γ X,ρ is given by the X(3872) → π + π − J/ψ partial width with the branching ratio (4.1 ± 1.3)% extracted by the BaBar Collaboration [44], and Eq. (17) is used for Γ X,ρ in the Flatté case [2,18]. The integration range for the convolution with the Flatté amplitude is chosen to be ±400 keV from the D 0D * 0 threshold which is twice of the half-maximum width of the peak. See Fig. 5 for a plot of (Γ X,ρ /|D X | 2 )/(2πN ).
Finally, the parameters used in this calculation are summarized in Table I.

III. RESULTS
A. e + e − → π 0 γX(3872) First, we show the γX(3872) invariant mass distribution in the e + e − → π 0 γX(3872) reaction, where X(3872) decays further into π + π − J/ψ, denoted by d σ e + e − ,π 0 γX /dm γX (here and in the following, we use σ to denote cross sections convolved with the X(3872) spectral     1)). The plot of the γX(3872) distribution of the e + e − → π 0 γX(3872) cross section normalized to the value at m γX = m D * 0 + mD * 0 with δ = 180 keV is also given in the right panel of Fig. 6 to make the comparison of the line shapes easier.
The distribution d σ e + e − ,π 0 γX /dm γX , which involves the X(3872) decay into the π + π − J/ψ mode, is the order 0.01 pb/GeV within δ = ±180 keV. As one can see in the left panel of Fig. 6, the magnitude is bigger with larger δ. This is because the D 0D * 0 → X(3872) coupling is bigger with larger δ except for the vicinity of δ = 0, |δ| < 50 keV (see Fig. 2 for g 4 as a function of δ). In the right panel of Fig. 6, one can see that the peak of the TS looks masses convolved with the BW distribution Eq. (15). The e + e − c.m. energy is fixed to be √ s = 4.23 GeV, and the X(3872) → π + π − J/ψ branching fraction has been taken into account. Right: The e + e − → π 0 γX(3872) cross section normalized with the value at m γX = m D * 0 + mD * 0 of δ = 180 keV. In both panels, the vertical line is the D * 0D * 0 threshold.
are dictated by the TS whose location can be easily obtained using the master formula in Ref. [30]. On the other hand, the peak around m γX = 4.016 GeV with δ > 0 is a remnant of the TS because the TS is in the complex plane even when the D * 0 width is neglected in this case.
Other than the TS peak, one can see a cusp of the D * 0D * 0 threshold slightly below m γX = 4.014 GeV as a consequence of the S-wave production of D * 0D * 0 . The two relevant singularities, the cusp at the D * 0D * 0 threshold and the peak caused by the TS, fix the line shape, and the peak is sensitive to the binding energy as can be seen from the figure.
As studied in Ref. [53], even after considering the energy resolution, the shapes can still be distinguished for different binding energies. The distribution shows slightly increasing behavior along with increasing m γX . This is because of the Z c (4020) resonance included in the D * D * production mechanism. Yet, its inclusion does not change the TS peak structures in the γX(3872) distribution.
Notice that for the e + e − → γX(3872) cross section [32], three is no D * 0D * 0 threshold cusp as the D * 0D * 0 pair is produced in P wave in that case, and only the TS peak can be seen in the γX(3872) distribution.
To see the impact of the smearing of the cross section with the X(3872) mass distribution, we show in Fig. 7 the γX(3872) invariant mass distribution of δ = 50 keV with and without smearing. The cross section without smearing is given by Eq. (9) multiplied by the X(3872) → π + π − J/ψ branching fraction to compare with the smeared distribution. As one can see in Fig. 7, the TS peak position is slightly shifted to a lower energy by the smearing.
This tendency is the same for different δ values.
The black solid line is the plot with the best-fit parameters of the Flatté analysis in Ref. [2], and the gray band is given by the parameter uncertainties (the statistical and systematic errors are summed in quadrature). The shape looks similar to the line of δ = 50 keV in Fig. 6, but the peak position is slightly lower, about 4.015 GeV, and the peak is more obscure. This is a consequence of the smearing: the original position of the TS peak without smearing would be about 4.016 GeV as in the case of δ > 0 of Fig. 6 because the X(3872) is seen in the π + π − J/ψ distribution as a peak at the D 0D * 0 threshold (see Fig. 5). In the smearing with the Flatté distribution, the cross section is averaged in the range of m X ∈ m D 0 + m D * 0 ± 400 keV to cover the X(3872) peak region in Fig. 5. 5 Hence, the smearing effect is larger than the smearing with the BW distribution with Γ X = 100 keV.
The parameter uncertainties give relatively large uncertainties in the magnitude as seen from the gray band in Fig. 8, but the peak position is not changed.
Let us make a comment on the uncertainties of the order of magnitude. The uncertainty of the e + e − → π 0 (D * D * ) 0 , which is used to fix the parameter g 0 g 1 g 2 , is about 10%, and the uncertainty of the D * → γD part would be a few percent referring to the relative errors of the D * + full width and the D * branching ratios [1]. Then, the uncertainties of the cross section is expected be about 10% which mainly comes from the e + e − → π 0 D * 0D * 0 input.
Integrating the differential cross section in Fig The plot of the pp → γX(3872) cross section, σ pp,γX (note that the X(3872) → π + π − J/ψ branching fraction has been taken into account as before), as a function of the pp c.m. energy, √ s, is given in the left panel of Fig. 9, and the right panel of Fig. 9 is the plot with all line shapes normalized to that of δ = 180 keV at the D * 0D * 0 threshold as the right panel of Fig. 6.
The γX(3872) invariant mass distribution of the pp → γX(3872) process is qualitatively the same as the e + e − → π 0 γX(3872) case, since the singularities are the same. The cross section increases with larger δ, and the peak structure looks more significant with δ < 0. in Ref. [17], a high-precision measurement of the X(3872) binding energy is foreseen even after further smearing due to the energy resolution is taken into account [53]. In particular, such a smearing effect at PANDA will be very small since the energy resolution can reach the level of 100 keV [59,60]. 6

IV. SUMMARY
In this paper, we have estimated the cross sections for the production of γX(3872) from a short-distance D * 0D * 0 source. A measurement of the γX(3872) line shape was proposed to achieve an unprecedented precision in determining the X(3872) binding energy [28]. We focused on two processes in this paper: e + e − → π 0 γX(3872) and pp → γX(3872). The γX(3872) invariant mass distributions for these two processes were computed, which clearly show a special peak sandwiched between the D * 0D * 0 threshold and the triangle singularity of the D * 0D * 0 D 0 /D * 0 D * 0D0 loops. The obtained line shapes with different X(3872) binding energies can be distinguished from each other in both the e + e − and pp processes: the peak is more narrow when the X(3872) mass is above the D 0D * 0 threshold. Convolving the distributions with the spectral function of the X(3872) does not change the conclusion, and the effect of smearing is marginal considering a width of 100 keV for the X(3872).
In the e + e − → π 0 γX(3872) reaction, the Z c (4020) resonance is introduced, and it is found that this resonance does not essentially change the peak structure caused by the TS. For the c.m. energy of the e + e − pair fixed at 4.23 GeV, with inputs from the BESIII measurements of the e + e − → π 0 (D * D * ) 0 [38], we find that the cross section σ(e + e − → π 0 γX(3872)) × B(X(3872) → π + π − J/ψ) is about 0.03 fb to 0.2 fb with the γX(3872) 6 The beam energy resolutions for the high luminosity and high resolution modes of the High Energy Storage Ring are 167.8 keV and 33.6 keV, respectively [59,60].
The pp → γX(3872) amplitude Eq. (12) with the particle momenta assigned as in Fig. 3 is reduced to Adding these two terms, we get