Triangle singularity in the B − → K − π 0 X ( 3872 ) reaction and sensitivity to the X ( 3872 ) mass

We have done a study of the B− → K−π0X(3872) reaction by means of a triangle mechanism via the chain of reactions: B− → K−D∗0D̄∗0; D∗0 → π0D0; D0D̄∗0 → X(3872). We show that this mechanism generates a triangle singularity in the π0X(3872) invariant mass for a very narrow window of the X(3872) mass, around the present measured values, and show that the peak positions and the shape of the mass distributions are sensitive to the X(3872) mass, such that a measurement of the reaction can serve to improve on the present values of this mass. In particular, we point out that the X(3872) mass relative to the D0D̄∗0 threshold may be extracted from the asymmetry of the π0X line shape. ∗ shsakai@mail.itp.ac.cn † oset@ific.uv.es ‡ fkguo@itp.ac.cn 1 ar X iv :2 00 2. 03 16 0v 1 [ he pph ] 8 F eb 2 02 0


I. INTRODUCTION
Triangle singularities (TSs) in physical processes, introduced in the 1960s [1,2] and searched for in some reactions without much success [3,4], have undergone a spectacular revival in recent years. The TS in the physical region stems from mechanisms that involve three intermediate particles which can be placed simultaneously on shell while being collinear, and represent a process that can occur classically (Coleman-Norton theorem [5]).
With the outcome of a wealth of experimental data at present, examples of enhancements in cross sections due to TSs have become available. One of the early examples was the explanation of the peak observed by the COMPASS Collaboration in the πf 0 (980) final state at production mode is enhanced compared to the isospin-allowed a 0 (980) mode due to a TS [26].
A detailed list of the proposed reactions has been tabulated in Table 2 of Ref. [21].
Concerning the X(3872) production, an interesting proposal was made to observe this resonance and measure its mass with high precision [27]. The reaction is D * 0D * 0 → γX (3872) where the D * 0D * 0 would be produced by some short-distance source. The reaction develops a TS at γX invariant masses which are very sensitive to the X(3872) mass relative to the D 0D * 0 threshold, i.e. the X(3872) binding energy, and could serve to greatly improve on the present uncertainties in the X(3872) binding energy. When the D * 0D * 0 are in S-wave, the γX line shape is characterized by a D * 0D * 0 threshold cusp and a TS-induced peak above the cusp. As a result, the line shape is extremely sensitive to the X(3872) binding energy. A possible reaction implementing this mechanism with the D * 0D * 0 being in P -wave has been given in Refs. [28,29] with the e + e − → γX(3872) process. A variant of this reaction is also proposed in Ref. [30] with the B 0 → K + Xπ − and B + → K 0 Xπ + processes, where the Xπ system develops a TS. The mechanism for the production is given by B → KD * D * 0 (B → KD * D * 0 ) followed by D * → πD 0 (D * → πD 0 ) andD * 0 D 0 (D * 0D0 ) fusing into the X(3872).
The present work deals again on the B → KXπ reaction with a different emphasis, which is to show the sensitivity of the Xπ invariant mass distribution to the TS position and to the X(3872) mass. Since we aim at high precision in the determination of the TS energy we must improve on approximations done in Ref. [30]. The methods used and input are also different, and we also discuss the need to convolve the results with the mass distribution of the X(3872) to compare with experiment. In addition, we focus on the sensitivity of the π 0 X line shape on the X(3872) binding energy.
This paper is organized as follows. The formalism and some general discussions on the TS are presented in Sec. II. The amplitude for the triangle diagrams is evaluated in Sec. III.
In Sec. IV, we present the numerical results and the related discussions. In particular, we suggest that the X(3872) binding energy may be extracted from the asymmetry of the π 0 X line shape around the TS. A brief summary is given in Sec. V.
The momenta of the corresponding particles are in parentheses.

A. General considerations
We will study the B − → K − π 0 X(3872) reaction which proceeds via the diagrams depicted in Fig. 1.
The picture implicitly uses the fact that the X(3872) couples strongly to D * D + c.c.. This is an observed fact as the D 0D * 0 + c.c. mode consists of a large portion of the X(3872) decays [31][32][33]. Although one does not need assume the X(3872) to be a DD * +c.c. hadronic molecule, a strong coupling is natural under such an assumption. In pictures as Refs. [34,35] where the DD * interaction is studied, the X(3872) results as a dynamically generated resonance from the interaction and the coupling of X(3872) to the DD * +c.c. components can be computed [34][35][36]. In the XEFT effective theory, it is treated as a D 0D * 0 molecule [37].
In Ref. [22], it is discussed that the TS appears when the D * ,D * are placed on shell in the loops of Fig. 1 simultaneously with the D,D * (D, D * ). In addition, in the diagram of Fig. 1a), the D * 0 and π 0 have to have their momentum in the same direction in the π 0 X center-of-mass (c.m.) frame. These conditions lead to two solutions for the momenta of the D * 0 in Fig. 1a) and it is shown that only one (satisfying the Coleman-Norton theorem [5]) leads to a TS in the physical region. These conditions are summarized in a very easy equation: where q on and q a− are the momenta of the intermediate particle that connect the initial and final heavy particles evaluated by putting the D * D * and D * D (orD * D) on shell, respectively.
The expressions for q a− and q on are given in Ref. [22]. Let us take a mass for the X, M X = 3871.70 MeV, inside the currently measured range, 3871.69±0.17 MeV [31], and apply the condition of Eq. (2). We obtain the TS at an π 0 X invariant mass M π 0 X = 4013.84 MeV.
It is interesting to see that the there is a very narrow window of masses where the TS appears. The range is given by (see Eq. (59) in Ref. [21]) and, with Eq. (60) in Ref. [21], correspondingly the TS is located in the range The above ranges can be compared with those for the reaction proposed in Ref. [27].
They can be obtained by simply replacing m π 0 by the zero photon mass: The ranges in Eqs. (3) and (4) are much narrower, but fortunately, the narrow window in Eq. (3) precisely covers where the mass of the X(3872) lies at present, with M X = 3871.69 ± 0.17 MeV [31]. Because the π 0 X line shape depends strongly on the X(3872) binding energy, this gives hopes that investigating precisely where the singularity peak appears in the B → KπX decays one can still induce in retrospective the mass of the X(3872) with much precision. The fact that Eq. (2) has no solution outside the narrow window discussed above does not mean that there is no enhancement in the mass distribution of the πX. By inertia the enhancement remains, even if the intermediate particles in the loops are slightly off shell, but the intensity of the peak and its shape differs from the one in the narrow window of Eq. (2) because in such a case the TS is located in the complex M π 0 X plane even if the D * 0 width is neglected.
We aim at an extraction of the X(3872) binding energy with a high precision for which a very precise calculation for the shape of the π 0 X invariant mass distribution must be done.
This means more precision than the one assumed in Ref. [30], where several kinematical simplifications are done, among them treating the pion nonrelativistically. In order to get a simple analytic expression for the amplitude, the approximation in Ref. [30] amounts to approximate the scalar triangle loop integral by an expression where q is the size of the pion momentum in the πX c.m. frame, and However, such an approximation changes the singularity behavior. The TS is logarithmic, while the above one is certainly not. Furthermore, the square-root branch point at the M π 0 X = 2m D * 0 when Γ D * 0 = 0 is also lost. Therefore, the approximation in Ref. [30] may be used for an estimate of the reaction rate, but is not applicable for a precise description of π 0 X line shape in the vicinity of the TS. In fact, the same approximation has been used to establish the power counting in a nonrelativistic effective field theory [41], and it has been pointed out in Ref. [38] (see Section IV.A.2 therein) that the expansion can only be made in a region away from the TS (for a discussion of the TS in the nonrelativistic formalism, we refer to Ref. [42]).
One may wonder what happens with the diagrams of Fig. 1c) and 1d). One can see using Eq. (2) that they do not develop a singularity in the physical region using the actual mass of the X(3872). Instead we have to go to an X(3872) mass around 3880.00 MeV for the TS to appear at M π 0 X = 4020.54 MeV. It is clear that this singularity will not have any relevance in the region of the actual X(3872) masses and the process will be driven by the contribution of diagrams a) and b) of plus other components that we are not interested in. The cū is the D * 0 and then we can have the K − D * 0D * 0 component but not the K − D * + D * − . To create the K − D * + D * − component we must resort to the internal emission [45] as shown in Fig. 3. Now, the cc component through i cq i q ic in the hadronization process leads to among other components. The K − D * + D * − decay is possible through the internal emission, which is dynamically suppressed by a color factor [45]. Then, it is not surprising to see the experimental results [31] Br We shall use the datum of Eq. (9) to estimate the vertex B − → K − D * 0D * 0 needed for the evaluation of the diagrams of Fig. 1.
There is work done on the B → KD ( * )D( * ) reactions, parameterizing the amplitudes and making a fit to experimental data, relating different decays [46,47]. Since we are only concerned about the B − → K − D * 0D * 0 decay we can make the analysis for this channel alone using the information of Eq. (9). For an order-of-magnitude estimate of the reaction rate, we neglect the contribution from possible intermediate resonances, which is irrelevant for the TS. Since we are interested in the π 0 X line shape in the immediate vicinity of the D * 0D * 0 threshold, we choose the amplitude involving the lowest angular momentum between It is unnecessary to take the 0 component in such a case as shown in Ref. [48] (see Appendix of this reference). We have in this case with where λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2yz − 2zx. Then we determine C 2 as Let us emphasize that this value is only for an estimate of the reaction rate, and does not play any role in the π 0 X line shape.
The value of 55.4 keV is similar to the one obtained in Ref. [49] and the one used in Ref. [50], Here, we make connection with Refs. [34,36] where, referring to couplings, or wave functions at the origin, the X(3872) is written as with our phase convention for the doublets (D + , −D 0 ), (D 0 , D − ) (same for D * ), and CD + = D − , CD * + = −D * − for the C-parity operator. The wave function of Eq. (19) has I G (J P C ) = 0 + (1 ++ ), as by the PDG [31]. In Ref. [36], the coupled channels are studied considering mass differences between the neutral and charged channels, and the couplings to the individual channels are very close to what we obtain from Eq. (19). The coupling of the X(3872) to D * 0D0 is given by where by g X we mean the coupling of the X(3872) to the whole combination of Eq. (19). 1 One can make an estimate of g X by using the Weinberg compositeness condition [51][52][53] with the normalization required here [53] 2 where s is the square of the mass of the X(3872), µ the reduced mass of D * 0 and D 0 , and δ X the binding energy of the X(3872) with respect to the D * 0D0 threshold defined in Eq. (1).
We should note that Eq. (21) is valid for bound states. We shall also consider the cases where the D * 0D0 pair is not bound. It can be computed as the residue of the scattering T -matrix in the same way as the bound state case with the X(3872) mass used as an input.
In this case, we should recall that the D * + D − and D * − D + components are bound by about 8 MeV and when one works with coupled channels, as in Refs. [34,36], there is basically no difference whether the D * 0D0 channel is bound or unbound and the coupling can be equally calculated in either case. All this said, for the evaluations we shall work with a coupling close to the values found in Eq. (22) and in Ref. [36]. Once again we should note that the approximations that we have done only affect the strength, but not the position of the TS or the π 0 X line shape.
We have now all the ingredients needed for the evaluation of the triangle diagrams of Fig. 1 which we do in the next section.

III. EVALUATION OF THE TRIANGLE DIAGRAMS
We are now in a condition to evaluate the diagrams of Figs. 1a) and 1b). First, we should note that the two diagrams give exactly the same contribution, since, in Fig. 1b), there are two minus signs relative to Fig. 1a) from the D * 0 → π 0 D 0 (D * 0 → π 0D0 ) andD * 0 D 0 → X 2 A similar relationship of the coupling and binding energy is used in Ref. [54] to estimate the coupling of (D * 0D0 → X) vertices. Hence, we just have to evaluate the diagram of Fig. 1a). The amplitude (including the factor 2) is given by with where the evaluation has been done in the π 0 X(3872) c.m. frame, where P = 0.
In the next step, we evaluate (t T ) i of Eq.
where E D * 0 = q 2 + m 2 D * 0 , ED * 0 = q 2 + m 2D * 0 , E D 0 = ( q + k) 2 + m 2 D 0 , and θ is the angle between q and k: q · k = qk cos θ. The q 0 integral was done using Cauchy's theorem by picking up the pole at q 0 = ED * 0 − i in the lower half of the complex q 0 plane. Here, W denotes P 0 = M π 0 X . The external π 0 momentum and energy, k and k 0 are given by We take into account the width of D * 0 by the following replacement of the D * 0 andD * 0 energies in the integrand of Eq. (29): Because k is the only vector quantity which is not integrated, the vector integral qD ∆ should be proportional to k [22,55]: Then, The integrations of q and cos θ are evaluated numerically. In practice, whilet T in Eq. (35) is convergent, we introduce a momentum cutoff q max = 600 MeV for the q integration which does not affect the TS position and just gives tiny difference in the magnitude.
Finally, we obtain the B − → K − π 0 X(3872) decay amplitude given by the triangle diagrams in Fig. 1a) and 1b) as and the π 0 X invariant mass distribution is written as with We should note that the X(3872) does not have a precise mass since it has a width, and, hence, it has a mass distribution (spectral function) that has to be taken into account for a realistic comparison with experiment (see related work in Ref. [56]). With a spectral function of the X(3872), ρ X (m), where M X and Γ X are the nominal mass and width of X(3872), the π 0 X mass distribution convoluted with the X(3872) spectral function is given by

IV. RESULTS AND DISCUSSION
In Fig. 4, we show the plot of dΓ B − →K − π 0 X /dM π 0 X /Γ B − as a function of the π 0 X(3872) invariant mass, M π 0 X , given in Eq. (42) with δ X = ±150, ±100, ±50, and 0 keV and Γ X = 100 keV. In Fig. 5, we show the plot of the same results but normalized to the same value at the peak with the maximum of the δ X = 0 keV case. The plot in smaller range of We should note that not only the peak position but also, more importantly, the asymmetric line shapes of the distributions are rather different for different M X values. For instance, by looking at Fig. 6, the differences in shape between δ X = 50 keV and −50 keV are perfectly In Fig. 6, the sensitivity of the peak of the TS to the X(3872) binding energy is clear. We can see that changes in the binding energy of 100 keV revert into similar changes in the position of the TS peak in M π 0 X , but, even more striking, the shapes are evidently different. The interesting thing is that one does not need to measure the X(3872) nor the π 0 . It is sufficient to know that a π 0 has been produced but its precise measurement is unnecessary. Technically, it is not even necessary to measure the π 0 , but its simultaneous detection together with the K − reduces drastically the background. One measures the K − energy and determines Thus, a precise measurement of the K − energy for a B − at rest for the case where the peak appears is all that is needed to determine M X , establishing the correspondence of the peak observed and M X in Fig. 6.
We should note that the convolution with the spectral function of the X(3872) considering its width smooths the peaks and changes the shapes. We have seen that if we take Γ X = 1.2 MeV, which is the current upper limit, the dependency of the peak position of the TS with M X would be softened. However, there are good reasons to think that the width of the X(3872) would not be larger than 100 keV, which makes our predictions realistic: the D 0D0 π 0 mode consists of 40% branching fraction of the X(3872) decays [31,57], and its partial decay width is expected to be about 40 keV [37,58,59]. The rates obtained are small, but B − branching ratios of the order of 10 −7 are already recorded [31]. One should also note that the B + → KπX(3872) has been measured (not the specific TS peaks) with ratios (1.06 ± 0.31) × 10 −5 for Br(B + → K 0 π + X(3872)) × Br(X(3872) → J/ψπ + π − ) [60] and (7.9 ± 1.3 ± 0.4) × 10 −6 for Br(B 0 → K + Xπ − ) × Br(X(3872) → J/ψπ + π − ) [60].
We have mentioned above the sensitivity of the shape above the peak to the X binding energy. In order to make this point more clear, we define the following asymmetry: with M max being the π 0 X invariant mass where the π 0 X line shape takes its maximal value and δ a small range, say, 2 MeV. The dependence of this asymmetry on the X binding energy δ X is shown in Fig. 10. We find that this magnitude is very sensitive to changes of δ X . Since one has more statistics in an integrated distribution, this magnitude could turn out to be the ideal one to determine the X mass with precision, and the X(3872) binding energy may be extracted by measuring this asymmetry.

V. CONCLUSIONS
We have made a study of the B − → K − π 0 X(3872) decay considering a triangle mechanism with a primary B − → K − D * 0D * 0 decay followed by D * 0 → π 0 D 0 and fusion of D 0D * 0 to give the X(3872). The triangle diagram of this mechanism develops a triangle singularity in a very narrow window of X(3872) masses, between 3871.68 MeV and 3971.93 MeV, producing a peak in the π 0 X invariant mass around 4013.75 MeV. The results of our investigation point out the high sensitivity of the peak position and, in particular, the line shape of the π 0 X mass distribution of this process to the X(3872) binding energy. The shape predicted in this work is very useful to determine the X(3872) mass, and we show that even with 20% uncertainties in the measured points of the mass distribution one could discriminate 100 keV in the X(3872) mass relative to the D 0D * 0 threshold. To stress this property, we introduced a new magnitude, meaning the asymmetry of the distribution and found that it is very sensitive to the X(3872) mass, and benefits with larger statistics since it involves integrated mass distributions. The novelty in the reaction proposed is that it establishes a one-to-one correspondence between the X(3872) binding energy and the asymmetry in the line shape of the π 0 X mass distribution. Technically, one does not need to measure the π 0 .
The π 0 X invariant mass would be determined from the energy of the emitted kaon, which can be measured with high precision. Yet, the detection of the pion, albeit without the need for precision, would serve to reduce the background.
The obtained branching ratios are in the limit of the smallest present rates reported by the PDG, and should be accessible in new rounds of measurements or future updates of the present facilities. With a more precise measurement of the X(3872) binding energy and its width, both of which are intimately connected to the structure of the X(3872), new insights into the nature of the X(3872), which is the object of much debate, are foreseen.