Charm-meson Triangle Singularity in ${e^+e^-}$ Annihilation into ${ D^{*0} \bar{D}^0 + \gamma }$

We calculate the cross section for $e^+ e^-$ annihilation into $D^{*0} \bar D^0 +\gamma$ at center-of-mass energies near the $D^{*0} \bar D^{*0}$ threshold under the assumption that $X(3872)$ is a weakly bound charm meson molecule. The Dalitz plot has a $\bar D^{*0}$ resonance band in the squared invariant mass $t$ of $\bar D^0 \gamma$. In the limit as the decay width of the $D^{*0}$ goes to 0, the Dalitz plot also has a narrow band in the squared invariant mass $u$ of $D^{*0} \bar D^0$ from a charm-meson triangle singularity. At the physical value of the $D^{*0}$ width, the narrow band reduces to a shoulder. Thus the triangle singularity cannot be observed directly as a peak in a differential cross section as a function of $u$. It may however be observed indirectly as a local minimum in the $t$ distribution for events with $u$ below the triangle singularity. The minimum is produced by the Schmid cancellation between triangle loop diagrams and a tree diagram. The observation of this minimum would support the identification of $X(3872)$ as a weakly bound charm meson molecule.


I. INTRODUCTION
Since early in this century, a large number of exotic hadrons whose constituents include a heavy quark and its antiquark have been discovered in high energy physics experiments [1][2][3][4][5]. The first of these exotic heavy hadrons to be discovered was the X(3872) meson in 2003 [6]. Its J P C quantum numbers are 1 ++ [7]. Its mass is extremely close to the D * 0D0 threshold, with the difference being less than about 0.2 MeV [8]. These results suggest that X is a weakly bound S-wave charm-meson molecule with the flavor structure There are alternative models for X, including a compact tetraquark state with constituents ccqq and the χ c1 (2P ) charmonium state with constituents cc [1][2][3][4][5]. The nature of X may have important implications for other exotic heavy hadrons. One might have hoped the nature of X could be revealed by its decays. The X has been observed in 7 different decay modes, more than any other exotic heavy hadron. Despite these many decay modes, a consensus on the nature of X has not been achieved.
There may be aspects of the production of X that are more effective at discriminating between models than the decays of X. If X is a weakly bound charm-meson molecule, it can be produced by any reaction that can produce its constituents D * 0D0 and D 0D * 0 . These constituents can be created at short distances of order 1/m π , where m π is the pion mass, and they can subsequently be bound into X at longer distances. The X can also be produced by the creation of D * 0D * 0 , D * 0 D * − , or D * +D * 0 at short distances followed by the rescattering of the charm-meson pair into X and a pion at longer distances [9,10].
One way in which the nature of a hadron can be revealed by its production is through triangle singularities. Triangle singularities are kinematic singularities that arise if three virtual particles that form a triangle in a Feynman diagram can all be on their mass shells simultaneously [11,12]. There have been several previous investigations of the effects of triangle singularities on the production of exotic heavy mesons [13][14][15][16]. Guo recently pointed out that any high-energy process that can create D * 0D * 0 at short distances in an S-wave channel will produce X + γ with a narrow peak near the D * 0D * 0 threshold due to a charmmeson triangle singularity [17]. Any high-energy process that can create an S-wave D * D * pair at short distances can also produce Xπ with a narrow peak near the D * D * threshold due to a charm-meson triangle singularity. Such a narrow peak arises in the production of Xπ at hadron colliders [9] and in decays of B mesons into K + Xπ [10]. It has been suggested that in the decay B → X + Kπ, X is simply a peak from a triangle singularity [18]. The effects of the triangle singularity on the reaction B − → K − + Xπ 0 has recently been reexamined [19]. The contribution of a triangle singularity to the decay X → π + π − π 0 has recently been studied [20]. Guo, Liu, and Sakai have presented a review of triangle singularities in hadronic reactions with an emphasis on exotic heavy hadrons [21].
The quantum numbers 1 ++ of X allow X + γ to be produced by e + e − annihilation into a virtual photon. The virtual photon can create D * 0D * 0 at short distances in a P-wave channel, and the charm-meson pair can subequently rescatter into X + γ. The production of X + γ in e + e − annihilation was first discussed by Dubynskiy and Voloshin [22]. They calculated the absorptive contribution to the cross section from e + e − annihilation into on-shell charm mesons D * 0D * 0 followed by their rescattering into X + γ. The absorptive part of the matrix element was calculated in the limit of zero decay width of the D * 0 . Dubynskiy and Voloshin predicted that the cross section has a narrow peak a few MeV above the D * 0D * 0 threshold whose position and width depend on the X binding energy. As pointed out in Ref. [23], the narrow peak comes from a triangle singularity. In the limit as the X binding energy and the D * 0 width both go to zero, there is a term in the matrix element that diverges logarithmically at a center-of-mass energy near the predicted peak. Thus the D * 0 width may be as important as the X binding energy in determining the shape of the narrow peak. The cross section for e + e − → X + γ in the energy region near the D * 0D * 0 threshold was calculated in Refs. [23,24], taking into account the D * 0 width and including the dispersive as well as the absorptive contributions. For the physical D * 0 width, the position of the narrow peak is fairly insensitive to the X binding energy. The width of the peak is determined by both the binding energy and the D * 0 width.
Since D * 0D0 are constituents of X, the annihilation of e + e − can also produce D * 0D0 + γ. The matrix element for this reaction is the sum of a tree diagram and a pair of loop diagrams with a charm-meson triangle. In the limit of zero D * 0 width, the differential cross section has a double-logarithmic divergence as a function of the D * 0D0 invariant mass in a narrow range of the center-of-mass energy. An interesting aspect of this reaction is a cancellation pointed out by Schmid [25]. In the differential cross section integrated over theD 0 γ invariant mass, the double-logarithmic divergence is canceled by the interference between the tree diagram and the loop diagrams. Anisovitch and Anisovitch showed that the cancellation leaves a single-logarithmic divergence [26].
In this paper, we calculate the cross section for e + e − → D * 0D0 + γ at center-of-mass energies near the D * 0D * 0 threshold. The Schmid cancellation of the leading divergence from the triangle singularity is most conveniently revealed by expressing the cross section in a Lorentz-invariant form. In Section II, we repeat the calculation in Ref. [24] of the cross section for e + e − → D * 0D * 0 near the threshold using Lorentz-invariant variables. In Section III, we repeat the calculation in Ref. [24] of the cross section for e + e − → X + γ near the D * 0D * 0 threshold using Lorentz-invariant variables. In Section IV, we calculate the cross section for e + e − → D * 0D0 + γ and identify the triangle singularity. In Section V, we demonstrate the Schmid cancellation of the leading logarithm from that singularity. We point out that the triangle singularity can be observed indirectly through a local minimum in the differential cross section produced by interference associated with the Schmid cancellation. Our results are summarized in Section VI. The nonrelativistic result for the loop amplitude with the triangle singularity is reproduced in Appendix A. The loop integral that produces the triangle singularity is calculated in terms of Lorentz-invariant variables in Appendix B. A pair of spin-1 charm mesons D * 0D * 0 can be produced from the annihilation of e + e − into a virtual photon. The Feynman diagram for this process is shown in Fig. 1. The cross section for this reaction near the threshold was calculated in Ref. [24] using nonrelativistic approximations for the charm mesons. We repeat the calculation here using a relativistic formalism.
The matrix element for e + e − with total momentum Q to produce D * 0 andD * 0 with 4-momenta p andp has the form where √ s is the center-of-mass energy andv and u are the spinors for the colliding e + and e − . The factor of 2M * 0 , where M * 0 is the mass of the D * 0 , compensates for the relativistic normalization of charm meson states. Near the threshold s = 4M 2 * 0 for producing D * 0D * 0 , the charm-meson pair is produced in a P-wave state with total spin either 0 or 2. At energies close enough to threshold that higher partial waves can be ignored, the Lorentz tensor A(Q) µνλσ in Eq. (2) is where g µν Q = g µν − Q µ Q ν /Q 2 . The coefficients A 0 and A 2 are the amplitudes for creating D * 0D * 0 in a P-wave channel with total spin 0 and 2, respectively. The s-dependence of the last term in Eq. (3) is necessary to eliminate interference between the two amplitudes in the differential cross section. The matrix element in Eq. (2) with energy-dependent amplitudes A 0 (s) and A 2 (s) is a good approximation at energies less than about 100 MeV above the threshold [24]. That matrix element with constant amplitudes A 0 and A 2 is a good approximation at energies less than about 10 MeV above the threshold [24]. In this region, s − 4M 2 * 0 is numerically smaller than 4δ 2 , where δ = 142.0 MeV is the mass difference between D * 0 and D 0 . We can therefore consider s − 4M 2 * 0 to be order δ 2 . The differential cross section for producing D * 0D * 0 with scattering angle θ in the centerof-momentum frame is In the region where s − 4M 2 * 0 is order δ 2 , we can simplify the cross section with errors of order (δ/M * 0 ) 2 by setting s = 4M 2 * 0 everywhere except in the factors of (s − 4M 2 * 0 )/s. The cross section reduces to This agrees with the nonrelativistic cross section calculated in Ref. [24]. The cross section integrated over angles is FIG. 2. Cross section for producing D * 0D * 0 as a function of the center-of-mass energy √ s. The values of |A 0 | and |A 2 | are given in Eq. (7). The vertical line is the D * 0D * 0 threshold.
The constant factor |A 0 | 2 + |A 2 | 2 in Eq. (6) can be determined experimentally by measuring the cross section at center-of-mass energies √ s within about 10 MeV of the D * 0D * 0 threshold at 4013.7 MeV. The values of |A 0 | and |A 2 | can be determined separately by measuring the angular distribution and fitting it to the expression in Eq. (5). Estimates of the amplitudes |A 0 | and |A 2 | were presented in Ref. [23]: The cross section for D * 0D * 0 predicted using Eq. (6) with these amplitudes is shown in Fig. 2 for center-of-mass energies up to about 10 MeV above the threshold. The cross section is large enough that it should be relatively easy for the BESIII collaboration to measure |A 0 | and |A 2 |.
The amplitudes |A 0 | and |A 2 | in Eq. (7) were actually obtained by fitting cross sections for e + e − annihilation into D * + D * − to Eq. (6) with M * 0 replaced by the mass M * + of D * + . The justification for approximating the amplitudes for producing D * 0D * 0 by those for producing D * + D * − is that both cross sections are dominated near the threshold by the isospin-0 charmonium resonance ψ(4040). The amplitudes |A 0 | and |A 2 | were determined in Ref. [23] by fitting cross sections obtained by Uglov et al. [27] by analyzing Belle data on e + e − annihilation into pairs of charm mesons [28,29]. The Belle data was also analyzed by Du et al. [30]. Their fit gives a cross section for D * + D * − at √ s = 4.040 GeV that is only 6% smaller than the fit of Uglov et al.. However their ratio 0.81 of the spin-2 and spin-0 cross sections at that energy is significantly smaller than the ratio 2.92 from the fit of Uglov et al.. This suggests that the value |A 0 | 2 + |A 2 | 2 = 290 GeV −2 from Eq. (7) may be considerably more accurate than the value of the ratio |A 2 |/|A 0 |. III. PRODUCTION OF X + γ NEAR THE D * 0D * 0 THRESHOLD If the X(3872) is a weakly bound charm-meson molecule, its constituents are the superposition of charm mesons in Eq. (1). The D * 0 width gives an important contribution to the decay width of X. The full decay width of D * 0 has not been measured. It can be predicted from measurements of the full decay width Γ * + of D * + and the decay branching fractions of D * 0 and D * + using chiral symmetry and isospin symmetry: where M 0 is the mass of D 0 , m 0 and m + are the masses of π 0 and π + , and λ(x, y, z) = x 2 + y 2 + z 2 − 2(xy + yz + zx) is the Källen function. The resulting prediction is Γ * 0 = (55.4 ± 1.5) keV, which agrees with the result in Ref. [17]. It is consistent within errors with the result in Ref. [31]. The corresponding prediction for the radiative decay width of the D * 0 is Γ[D * 0 → D 0 γ] = (19.6 ± 0.9) keV. This is consistent within errors with the result in Ref. [31]. In this paper, we consider the D * 0 width to be accurately predicted, and we take its value to be Γ * 0 = 55 keV. The present value of the difference E X between the mass of X and the energy of the D * 0D0 scattering threshold is [8] The central value corresponds to a charm-meson pair just above the scattering threshold. The value lower by 1σ corresponds to a bound state with binding energy |E X | = 0.17 MeV. The upper bound on |E X | with 90% confidence level is 0.22 MeV. Since the binding energy is so small, we will often use M X as a compact expression for M * 0 +M 0 . We assume in this paper that X is a narrow bound state. The production of the X resonance feature in the case where X is instead a virtual state was considered in the Appendix of Ref. [24] and in Ref. [19]. If X is a narrow bound state, its energy E X and its decay width Γ X can be expressed in terms of the complex binding momentum γ X as where µ = M * 0 M 0 /(M * 0 +M 0 ) is the reduced mass of D * 0D0 . The energy and decay width of the X bound state should be distinguished from the energy and width of the X resonance feature, which has contributions from above the D * 0D0 threshold and may depend on the production process [32]. The first term in Γ X in Eq. (10b) can be interpreted as the partial decay width of the bound state into D 0D0 π 0 and D 0D0 γ. The second term can be interpreted as its partial decay width into other decay modes, such as J/ψ π + π − . To illustrate the possible dependence of production rates on the binding energy and decay width of X, we will consider three choices for E X , Γ X with the physical D * 0 width Γ * 0 = 55 keV: 1. E X = −0.05 MeV, Γ X = Γ * 0 , which corresponds to the real binding momentum γ X = 9.83 MeV, 2. E X = −0.10 MeV, Γ X = Γ * 0 , which corresponds to the real binding momentum γ X = 13.90 MeV, 3. E X = −0.05 MeV, Γ X = 2 Γ * 0 , which corresponds to the complex binding momentum γ X = (10.17 + 2.61i) MeV.
The comparison of the first and second choices shows the effect of doubling the binding energy of X. The comparison of the first and third choices shows the effect of doubling the width of X. We will also sometimes consider zero decay width for X, which requires the unphysical D * 0 width Γ * 0 = 0: 4. E X = −0.05 MeV, Γ X = 0, which corresponds to γ X = 9.83 MeV, 5. E X = 0, Γ X = 0, which corresponds to γ X = 0. The X can be produced in e + e − annihilation through the creation of D * 0D * 0 by a virtual photon followed by the rescattering of the charm-meson pair into X + γ. The Feynman diagrams for this process are shown in Fig. 3. The cross section for this reaction near the D * 0D * 0 threshold was calculated in Ref. [24] using nonrelativistic approximations for the charm mesons. We repeat the calculation here using a relativistic formalism. We first describe each of the vertices in Fig. 3 that involve charm mesons.
At center-of-mass energies near the D * 0D * 0 threshold, the vertex for the virtual photon with momentum Q = p +p and Lorentz index µ to create D * 0 andD * 0 with momenta p andp and with Lorentz indices λ and σ is e( The vertex for the radiative transition of D * 0 with momentum p and index α to D 0 by emitting a photon with momentum q and index β is The transition magnetic moment eν can be determined from the radiative decay width of D * 0 : Using the predicted radiative width Γ[D * 0 → D 0 γ] = (19.6 ± 0.9) keV, the factor ν in the transition magnetic moment is determined to be ν = 0.92 GeV −1 .
The binding of D * 0D0 or D 0D * 0 into X can be described within an effective field theory called XEFT [33,34]. The relativistic generalization of the vertex for the coupling of D * 0D0 or D 0D * 0 to X with momentum P can be expressed as where µ and ν are the Lorentz indices of the spin-1 charm meson and X. In the rest frame of X, the vertex in Eq. (12) reduces to the nonrelativistic vertex i(πγ X /µ 2 ) 1/2 δ mn [35] multiplied by a factor (8M X M * 0 M 0 ) 1/2 that compensates for the relativistic normalization of states.
The matrix element for e + e − → X + γ is the sum of the two diagrams with charm-meson loops in Fig. 3. The loop integral produces a triangle singularity [23]. The singularity arises from the integration region where the three charm mesons whose lines form triangles in the diagrams in Fig. 3 are all on their mass shells simultaneously. The two charm mesons that become constituents of X are both on their mass shells in the limit where the binding energy is 0. The two spin-1 charm mesons can be both on their mass shells if the decay width Γ * 0 is 0. The photon energy can be tuned so that the charm mesons before and after the radiative transition are both on their mass shells. The matrix element has a logarithmic branch point from the triangle singularity at a complex value of s that becomes real in the limit where E X , Γ X , and Γ * 0 are all 0. That real branch point is determined in Appendix B: It can be expressed more concisely as s = (M * 0 /M 0 )M 2 X . The predicted center-of-mass energy for the triangle singularity is √ s = 4016.4 MeV, which is about 2.7 MeV above the D * 0D * 0 threshold. A nonrelativistic approximation for s was obtained in Ref. [24]. If we set M 0 = M * 0 − δ, s can be expanded in powers of δ. The expansion of the nonrelativistic approximation for s agrees with the relativistic result in Eq. (13) through third order in δ, but it disagrees at fourth order.
The nonrelativistic loop integral from the diagrams in Fig. 3 was calculated in Ref. [24]. The integral is ultraviolet convergent, and it can be reduced to a scalar function given analytically in Appendix A. The relativistic loop integral from the diagrams in Fig. 3 is quadratically ultraviolet divergent. The integral depends on the center-of-mass energy and the invariant mass √ u of the charm-meson-pair system that recoils against the photon. In the region near the triangle singularity where s − 4M 2 * 0 and u − M 2 X are order δ 2 , the loop integral can be expanded in powers of δ/M * 0 . The leading terms in the expansion are ultraviolet finite. In Appendix B, the relativistic loop integral is reduced to the Lorentzscalar function F (s, u) defined by the ultraviolet-convergent momentum integral in Eq. (B6). For s − 4M 2 * 0 and u − M 2 X both of order δ 2 , the integral is dominated by momentum scales of order δ and smaller, and the leading term scales as 1/(M X δ). In the matrix element for producing X + γ, F (s, u) is evaluated at u = M 2 X − iM X Γ X , which is the position of the complex pole in the elastic scattering amplitude for the charm mesons D * 0 andD 0 .
The matrix element for e + e − → X + γ can be calculated most conveniently by first reducing the charm-meson loops to an effective vertex for the coupling of the virtual photon to X and a real photon given in Eq. (B11). The matrix element for producing X with momentum P and γ with momentum q is The differential cross section for producing X + γ with scattering angle θ is In the region where s − 4M 2 * 0 is order δ 2 , we can simplify the cross section with errors of order (δ/M * 0 ) 2 by setting s = 4M 2 * 0 everywhere except in the factors of (s − M 2 X )/s and in the function F . The cross section reduces to where M X = M * 0 + M 0 except in the second argument of the loop amplitude F , which depends sensitively on E X through that argument. The cross section for producing X + γ integrated over angles is The factor in square brackets in Eq. (17) that depends on A 0 and A 2 differs from the factor |A 0 | 2 + |A 2 | 2 in the cross section for producing D * 0D * 0 in Eq. (6) by a multiplicative factor that depends on the complex ratio A 2 /A 0 . If |A 0 | 2 + |A 2 | 2 has been determined but |A 2 |/|A 0 | has not, we should allow for all possible complex values of A 2 /A 0 . For the value of |A 0 | 2 + |A 2 | 2 determined by Eq. (7), the multiplicative factor can range from 0.34 to 1.31. If |A 0 | 2 and |A 2 | 2 have both been determined, we must still allow for all possible phases of A 2 /A 0 . For the values of |A 0 | and |A 2 | in Eq. (7), the multiplicative factor can range from 0.36 to 1.10, and it is equal to 0.73 if the phase of A 2 /A 0 is ±i.
The cross section for X + γ predicted using Eq. (17) is shown in Fig. 4 for center-ofmass energies up to about 10 MeV above the D * 0D * 0 threshold. The absolute values of the amplitudes A 0 and A 2 are given in Eq. (7). For purposes of illustration, we choose the complex phase of A 2 /A 0 to be ±i. The cross section is shown for the first three choices of binding energy and width of X enumerated after Eq. (10). The cross section has a narrow peak near the energy for the triangle singularity predicted by Eq. (13). For |E X | = 0.05 MeV, Γ X = Γ * 0 , the peak in the cross section is at 4016.1 MeV, which is only 0.3 MeV lower than the prediction from Eq. (13). Doubling the binding energy |E X | has very little effect on the position of the peak in the cross section, but it increases its height by about 7%. Doubling the width Γ X moves the peak to an energy higher by about 0.2 MeV and increases its height by about 10%. The peak in the cross section in Fig. 4 is large enough that it may be observable by the BESIII collaboration. Its energy is in a range not covered by previous measurements of the cross section for e + e − → X + γ by the BESIII collaboration [36,37].
In Ref. [24], the cross section for e + e − → X + γ was calculated using nonrelativistic approximations. The result is consistent with that in Eq. (17) to within the accuracy of the  [24], that function was evaluated at the negative energy U = E X . It should more appropriately be evaluated at the complex energy U = E X − iΓ X /2. The difference has a negligible effect on the numerical results in Ref. [24].
The origin of the peak in the cross section in Fig. 4 is the logarithmic branch point of the loop amplitude F (s, (17) at a complex value of s near s in Eq. (13). An analytic approximation to the logarithmic term in F (s, u) can be obtained by setting W = √ s − 2M * 0 and U = E X − iΓ X /2 in the nonrelativistic loop amplitude F (W, U ) in Eq. (A3). In the limit as the D * 0 width Γ * 0 , the binding energy E X , and the width Γ X all go to 0, the branch point approaches the real s axis and the loop amplitude has a logarithmic divergence proportional to log |s − s |. However the cross section goes to 0 in this limit because of the factor of |γ X | in Eq. (17). The divergence from the triangle singularity is illustrated in Fig. 5, which shows σ/|γ X | as a function of √ s for the values of E X , Γ X numbered 1, 4, and 5 after Eq. (10). The curve for |E X | = 0, Γ X = 0 has a log 2 (|s−s |) divergence. The curve for |E X | = 0.05 MeV, Γ X = 0 shows that the divergence is reduced to a narrow peak by the suppression from the binding energy only. The curve for |E X | = 0.05 MeV, Γ X = Γ * 0 shows that the D * 0 width provides some small additional suppression of the peak. Since e + e − annihilation can produce X(3872) recoiling against a photon, it can also produce its constituents D * 0D0 recoiling against a photon. The virtual photon from e + e − annihilation creates the charm mesons D * 0 andD * 0 , and the real photon is produced by a subsequent radiative transitionD * 0 →D 0 . The final state D * 0D0 +γ can either be produced directly through the tree diagram in Fig. 6 or through the loop diagrams in Fig. 7. The tree diagram is enhanced by theD * 0 resonance. The loop diagrams have a charm-meson triangle. The loop diagrams in Fig. 7 involve the amplitudes for the scattering of D * 0D0 or D 0D * 0 into D * 0D0 . A Lorentz-covariant expression for the vertex for either of those amplitudes is where P is the total momentum of the charm-meson pair, µ and ν are the Lorentz indices of the spin-1 charm mesons, γ X is the X binding momentum, and λ(x, y, z) is the Källen function defined after Eq. (8). If we set Γ * 0 = 0, the second term in the denominator reduces to −ik rel , where k rel is the relative 3-momentum of D * 0D0 in their center-of-momentum frame.
In that frame, the vertex in Eq. (18)   The loop integrals for the diagrams in Fig. 7 produce a triangle singularity from the region where the three charm mesons that form the loop are all on their mass shells. The triangle singularity produces a logarithmic branch point in F (s, u) along a line of complex values of s near 4M 2 * 0 and u near M 2 X . In the limit Γ * 0 → 0, the line approaches real values of s and u. In Appendix B, the condition for the triangle singularity is deduced from the on-shell conditions for the three charm mesons in the triangle. The range of s for the triangle singularity is 4M 2 * 0 < s ≤ s , where s is given in Eq. (13). The solution for u as a function of s is The function u (s) is illustrated in Fig. 8. As s increases from 4M 2 * 0 to s , u decreases from 2(M 2 * 0 + M 2 0 ) to (M * 0 + M 0 ) 2 . Numerically, as √ s increases from 4013.7 MeV to 4016.4 MeV, u decreases from 15.010 GeV 2 to 14.990 GeV 2 . The triangle singularity appears only in narrow intervals of s and u with lengths (M * 0 /M 0 )δ 2 and δ 2 , respectively. If we set M 0 = M * 0 − δ and take s − 4M 2 * 0 to be order δ 2 , the solution for u at the triangle singularity in Eq. (19) can be expanded in powers of δ. The expansion to order δ 2 can be expressed as In Fig. 8, this approximation is compared to the exact result in Eq. (19). The expansion to order δ 3 would be difficult to distinguish from the exact result.
We consider the production of D * 0 ,D 0 , and γ with 4-momenta p 1 , p 0 , and q and with total 4-momentum Q = p 1 + p 0 + q. A convenient set of Lorentz scalars is the center-of-mass energy squared s = Q 2 and the squares u = (p 1 + p 0 ) 2 and t = (p 0 + q) 2 of the invariant masses of D * 0D0 andD 0 γ, respectively. The matrix element from the tree diagram in Fig. 6 is The loop integral from the loop diagrams in Fig. 7 can be simplified in the region near the triangle singularity where s − 4M 2 * 0 and u − M 2 X are both order δ 2 . In Appendix B, the leading term in an expansion in powers of δ/M * 0 is reduced to the function F (s, u) defined by the Lorentz-invariant momentum integral in Eq. (B6). The charm-meson loop can be reduced to an effective vertex for the coupling of the virtual photon to D * 0 ,D 0 , and a real photon that is given in Eq. (B12). The matrix element from the loop diagrams in Fig. 7 is where P = p 0 + p 1 . Our matrix elements M tree in Eq. At a given center-of-mass energy √ s, the square of the matrix element summed over spins is a function of u, t, and two angles. The differential cross section integrated over angles and summed over final spins is The angular brackets indicate the average over angles. The matrix elements in Eqs. (21) and (22) both depend on the amplitudes A 0 and A 2 in the factor A(Q) µνλσ . Their absolute values are given in Eq. (7). For purposes of illustration, we choose the complex phase of A 2 /A 0 to be ±i. The matrix element M loop also depends on the binding momentum γ X through the denominator in Eq. (22). The scatter plot for the differential cross section in the u − t plane is called a Dalitz plot. The boundary of the Dalitz plot is the curve defined by the equation  To illustrate the behavior of the differential cross section as a function of u and t, we choose the center-of-mass energy  Fig. 9. The boundary of the Dalitz plot is actually a little fuzzy on the scale in Fig. 9, because the D * 0 has a nonzero width. This effect is illustrated by showing three boundaries corresponding to setting the D * 0 mass equal to M * 0 − Γ * 0 , M * 0 , and M * 0 + Γ * 0 . To generate events populating the Dalitz plot, we first generate random dots in the region of the u − t plane shown in Fig. 9. For each point, we generate a random number between 0 and 1. We keep the dot in the plot if dσ/dudt at that point is larger than the product of the random number and the maximum value of dσ/dudt in the region. The probability of keeping the dot is the ratio of dσ/dudt to its maximum value. The density of dots is therefore proportional to the differential cross section. The ratio of the number of dots kept to the number of dots generated is about 1%. The resulting Dalitz plot at a center-of-mass energy 1 MeV above the D * 0D * 0 threshold is shown in Fig. 9. There is an obvious horizontal band near t = M 2 * 0 from theD * 0 resonance. The triangle singularity would be expected to produce a narrow vertical band in the Dalitz plot near the value u = 14.993 GeV 2 predicted by Eq. (19). There is no narrow vertical band near that value of u. However the density of points to the left of u is significantly larger than that just to the right of u . The effects of the triangle singularity on the Dalitz plot would be more distinct if the D * 0 width Γ * 0 were smaller. TheD * 0 resonance band would be narrower in proportion to Γ * 0 , and the number of events in the resonance band would be larger in proportion to 1/Γ * 0 . The Dalitz plot at a center-of-mass energy 1 MeV above the D * 0D * 0 threshold is shown in Fig. 10 for the limiting case Γ * 0 → 0. Since dσ/dudt diverges at t = M 2 * 0 , events were generated only in the region t < 4.026 GeV 2 . A faint narrow vertical band is visible along the triangle-singularity line at u = 14.993 GeV 2 predicted by Eq. (19). A more obvious feature is that the density of points to the left of u is much larger than that just to the right of u . 14 FIG. 11. Differential cross section dσ/dudt for producing D * 0D0 + γ as a function of u at √ s = 4014.7 MeV and t = 4.0209 GeV 2 . The binding energy and width of X are |E X | = 0.05 MeV, Γ X = Γ * 0 (solid red curve), |E X | = 0.05 MeV, Γ X = 0 (dashed purple curve), and |E X | = 0, Γ X = 0 (dotted blue curve). The vertical line is at the predicted value u = 14.993 GeV 2 for the triangle singularity.
In Fig. 11, we illustrate the differential cross section dσ/dudt as a function of u in the region near the predicted triangle singularity at u for √ s = 4014.7 MeV. We choose thē D * 0 γ invariant mass-squared to be its value at the left-most point of the Dalitz plot in Fig. 9: t = 4.0209 GeV 2 . We compare dσ/dudt for the values of E X , Γ X numbered 1, 4, and 5 after Eq. (10). For E X = 0, Γ * 0 = 0, dσ/dudt diverges at u = u , with a shape consistent with a log 2 divergence. For |E X | = 0.05 MeV, Γ X = 0, dσ/dudt also diverges at u = u but the peak is a little narrower. For |E X | = 0.05 MeV, Γ X = Γ * 0 , dσ/dudt decreases monotonically with u out to beyond the triangle singularity. The suppression from the physical D * 0 width is large enough that there is not even a local maximum near u .

V. SCHMID CANCELLATION
The effects of the triangle singularity on the cross section for e + e − annihilation into D * 0D0 + γ are not obvious in the Dalitz plot in Fig. 9 or in the differential cross section in Fig. 11 for the physical value of the D * 0 width. The effects of a triangle singularity can be suppressed by nonzero widths of particles in the triangle. They can also be suppressed by cancellations between diagrams. In the limit Γ * 0 → 0, the triangle singularity in e + e − annihilation into D * 0D0 + γ produces a log 2 divergence in the differential cross section dσ/dudt along the triangle-singularity line u = u (s) with t below theD * 0 resonance. Schmid pointed out that in the differential cross section dσ/du integrated over t, the triangle singularity is suppressed by interference between the loop diagrams in Fig. 7 and the tree diagram in Fig. 6 [25]. The cancellation is between the loop contribution along the trianglesingularity line in the region of t below the resonance band and the interference contribution from the region of t in the resonance band. Anisovich and Anisovich pointed out that the divergence in dσ/du is not completely cancelled, but the log 2 divergence is reduced to a single logarithm [26]. Dalitz plot distributions in the presence of triangle singularities have been studied in detail by Szczepaniak [15]. The effects on the Schmid cancellation from the decay widths of particles in the triangle loop have been considered by Debastiani, Sakai and Oset [38].
The differential cross section for e + e − annihilation into D * 0D0 + γ averaged over angles is given in Eq. (23). The matrix elements M tree and M loop are given in Eqs. (21) and (22). The full expression for dσ/du dt is very complicated. To exhibit the Schmid cancellation, it is helpful to simplify the cross section in the region of s, u, and t near the triangle singularity. If s − 4M 2 * 0 and u − M 2 X are both order δ 2 , the kinematic constraints on the Dalitz plot require t − M 2 * 0 to also be order δ 2 . It is convenient to introduce subtracted variables: In the region where s − , u − , and t − are all order δ 2 , the equation for the boundary of the Dalitz plot in Eq. (24) The function D(s − , u − , t − ) is positive inside the boundaries of the Dalitz plot. The triangle singularity condition in Eq. (20) reduces at leading order in δ to To simplify the differential cross section, we set M 0 = M * 0 − δ and we take s − , u − , and t − to be order δ 2 . We also take γ X to be order δ and M * 0 Γ * 0 to be order δ 2 . We then expand dσ/dudt in powers of δ. The leading terms from the tree diagram, the interference, and the loop diagrams are Given that M X F (s, u) is order 1/δ, all three terms are order δ 0 . For t − close to 0, the tree contribution is enhanced by the Breit-Wigner resonance factor. The loop and interference contributions may be enhanced along the triangle singularity line The integrals over t of the terms in Eqs. (27) can be evaluated analytically to obtain the differential cross section dσ/du. The Schmid cancellation can be exhibited by adding the interference and loop contributions: In the limit Γ * 0 → 0 and γ X → 0, there is a cancellation between a logarithm in F (s, u) * whose argument has real part √ u − + √ s − − δ and the corresponding explicit logarithm in the second factor inside the real part. This is the cancellation pointed out by Schmid [25]. The cancellation leaves a single logarithm of √ u − + √ s − − δ from the factor of F (s, u), in agreement with the analysis of Anisovich and Anisovich [26]. That logarithm gives a divergence in the limit Γ * 0 → 0.
An expression for the loop amplitude F (s, u) that is accurate up to corrections of order δ/M * 0 is given in Eq. (B10). The contribution to the differential cross section from loops plus interference in Eq. (28) can be simplified by taking the limits Γ * 0 → 0 and γ X → 0. As √ u − approaches the triangle singularity at δ − √ s − , the diverging term is where x = √ s − /δ. There is a single logarithm that diverges at the triangle singularity. Its coefficient is negative for x between 0 and 0.782 and positive for x between 0.782 and 1. The diverging negative contribution to the cross section is not a problem, because the tree contribution also has diverging terms. It has terms that diverge as 1/Γ * 0 as Γ * 0 → 0 as well as terms that diverge logarithmically as u approaches the triangle singularity. The integral over t of the tree contribution to the differential cross section in Eq. (27a) can be simplified by taking the limit Γ * 0 → 0. As √ u − approaches the triangle singularity at δ − √ s − , the diverging terms are The sum of the two contributions to dσ/du in Eqs. (29) and (30)   In Fig. 12, we illustrate the Schmid cancellation by showing the differential cross section dσ/du integrated over t as a function of u for the physical D * 0 width Γ * 0 = 55 keV. We choose the center-of-mass energy √ s = 4014.7 MeV, which is 1 MeV above the D * 0D * 0 threshold and near the middle of the triangle-singularity region. The triangle singularity is predicted by Eq. (19) to be at u = 14.993 GeV 2 . The tree contribution to dσ/du and the contribution from loops plus interference are shown in Fig. 12 as well as their sum. The tree contribution is strongly suppressed below u . It is smaller than the contribution from loops plus interference for u < 14.9915 GeV 2 . The tree contribution increases dramatically as u increases past u , because theD * 0 resonance becomes kinematically accessible. The contribution from loops plus interference is negative for u > 14.9923 GeV 2 , and it has a small peak near the triangle singularity. Since s − 4M 2 * 0 /δ = 0.631, which is less than 0.782, this negative peak is qualitatively consistent with our limiting expression in Eq. (29), which has a negative logarithmic divergence. Our limiting expression for the tree contribution in Eq. (30) includes a term with a larger positive logarithmic divergence. For the physical D * 0 width, a term from the tree contribution with a positive peak whose height is comparable to that of the negative peak from loops plus interference in Fig. 12 would not be visible. It would be completely overwhelmed by the rapidly increasing contribution from the opening up of theD * 0 resonance.
In Fig. 13, we show the differential cross section dσ/du integrated over t as a function of u in the limit Γ * 0 → 0 for √ s = 4014.7 MeV. The tree contribution is infinitely large for u > u , because it includes an integral over t of a Breit-Wigner resonance with zero width. The contribution from loops plus interference is negative near the triangle singularity, in agreement with the limiting expression in Eq. (29). It diverges at u = u , with a shape consistent with a negative single log. The limiting expression for the tree contribution to dσ/du in Eq. (30) suggests that it also has a term that diverges at u = u , with a shape consistent with a positive single log. Its effects are not visible for u < u , because other terms in the tree contribution are growing even more rapidly.
For the physical value of the D * 0 width, the triangle singularity is not directly observable as a peak in dσ/dudt at a value of t below theD * 0 resonance band or as a peak in dσ/du integrated over t. It is possible however that the triangle singularity could be observed indirectly in some other way. In the Dalitz plot in Fig. 9, the density of points is significantly larger to the left of the triangle-singularity line than just to the right of that line. However the density of points to the left of the triangle-singularity line decreases as t approaches thē D * 0 resonance band. This decrease in the density of points for t near the resonance band arises from an interference effect associated with the triangle singularity. This effect can be quantified by calculating dσ/dt integrated over u < u (s) as a function of t. This function is shown in Fig. 14 for √ s = 4014.7 MeV. The tree contribution behaves as |t − |/(t 2 − + M 2 * 0 Γ 2 * 0 ) near the upper kinematic endpoint at t = M 2 * 0 , which is the center of theD * 0 resonance band. The factor of |t − | comes from the integration over u. The tree contribution increases to a sharp peak near t − = −M * 0 Γ * 0 , before decreasing to 0 at t − = 0. The contribution from loops plus interference is negative for t larger than about 4.024 GeV 2 . It decreases to a minimum near 4.027 GeV 2 , before increasing sharply to 0. The cancellation between the tree contribution and the contribution from loops plus interference produces a local minimum near 4.025 GeV 2 .
The local minimum in the differential cross section dσ/dt integrated over u < u (s) is insensitive to the binding energy and the width of X. This is illustrated in Fig. 15, which shows the cross section for the first three choices of E X , Γ X enumerated after Eq. (10). For |E X | = 0.05 MeV, Γ X = Γ * 0 , the shallow minimum in dσ/dt is at t = 4.0254 GeV 2 . The contrast defined by the ratio of the value at the local maximum at t = 4.0206 GeV 2 and the value at the local minimum is 5.8. The narrow peak near the upper endpoint of t is almost unaffected by changing either |E X | or Γ X by a factor of 2. Doubling either |E X | or Γ X decreases the position of the local minimum in t by 0.0002 GeV 2 . Doubling |E X | decreases the contrast between the local maximum and the local minimum to 5.6, while doubling Γ X decreases the contrast to 3.9.

VI. SUMMARY
If X(3872) is a weakly bound charm-meson molecule, a charm-meson triangle singularity produces a narrow peak in the cross section for e + e − annihilation into X + γ just above the D * 0D * 0 threshold [23]. The peak appears near a value of the center-of-mass energy √ s given by s in Eq. (13). The cross section is shown as a function of √ s in Fig. 4 for several values of the binding energy |E X | and the width Γ X of X. The observation of the narrow peak in the cross section would support the identification of X as a weakly bound charm-meson molecule.
In this paper, we have studied the effects of the charm-meson triangle singularity on the cross section for e + e − annihilation into D * 0D0 + γ just above the D * 0D * 0 threshold. The matrix element is the sum of the matrix element M tree in Eq. (21) from the tree diagram and the matrix element M loop in Eq. (22) from the loop diagrams. The differential cross section dσ/dudt is a function of the squared invariant masses u of D * 0D0 and t ofD 0 γ as well as the squared center-of-mass energy s. The matrix element M loop has a factor of the loop amplitude F (s, u), which depends on the D * 0 width Γ * 0 , and it also has a denominator that depends on the complex binding momentum γ X of X. approximation to the loop amplitude is given analytically in Eq. (A3) of Appendix A. If the width Γ * 0 of the D * 0 is sufficiently small, the triangle singularity produces a narrow peak in the differential cross section dσ/dudt as a function of u near the value u (s) in Eq. (19), which requires t to be below theD * 0 resonance band, for s in the narrow range between the D * 0D * 0 threshold and s in Eq. (13).
The effects of the triangle singularity are particularly dramatic in the limit Γ * 0 → 0. The differential cross section dσ/dudt has a term proportional to log 2 |u − u (s)| that diverges at u = u (s). As pointed out by Schmid, there is a cancellation of the log 2 divergence in dσ/du integrated over t [25]. The cancellation is between |M loop | 2 along the triangle singularity line and the interference term 2 Re[M loop M * tree ] in the region near the resonance band. The exact cancellation of the log 2 terms requires both Γ * 0 → 0 and γ X → 0. The Schmid cancellation leaves a term in dσ/du proportional to log |u−u (s)|, as pointed out by Anisovich and Anisovich [26]. The position of that divergence coincides with the dramatic increase in dσ/du proportional to 1/Γ * 0 from theD * 0 resonance becoming kinematically accessible.
For the physical value of the D * 0 width, the triangle singularity is not directly observable as a peak in a differential cross section. It is not observable as a peak in dσ/dudt, because the suppression from the D * 0 width reduces the peak in u near u (s) to a shoulder that drops off beyond u (s), as illustrated in Fig. 11. The triangle singularity is not observable as a peak in dσ/du integrated over t, because the term with a peak near u = u (s) is overwhelmed by the rapidly increasing tree contribution, as illustrated in Fig. 12.
The effects of the charm-meson triangle singularity may be observable indirectly as a local minimum in dσ/dt integrated over u < u (s). The differential cross section is shown as a function of t in Fig. 15 for several values of |E X |, Γ X . The local minimum comes from destructive interference between |M loop | 2 + 2 Re[M loop M * tree ] and |M tree | 2 that is related to the Schmid cancellation. The observation of this local minimum would provide additional support for the identification of X as a weakly bound charm-meson molecule.
The loop integral in Eq. (B1) produces a triangle singularity determined by the following conditions. First, the three charm mesons that form the loop are all on their mass shells: k 2 = M 2 * 0 , and (Q − k) 2 = M 2 * 0 , and where u = P 2 and θ is the angle between the 3-momenta P and k. In the e + e − CM frame, we have P 0 = (s + u)/(2 √ s), k 0 = √ s/2, |P | = (s − u)/(2 √ s), and |k| = s − 4M 2 * 0 /2, where s = Q 2 . A second condition is that the 3-momenta P − k and k of the two charm mesons are parallel in the e + e − CM frame: cos θ = 1. With this value of cos θ, Eq. (B2) can be reduced to Solving for u gives the expression for its value u (s) at the triangle singularity in Eq. (19). This is a necessary condition for the triangle singularity, but it is not sufficient. The final condition for the triangle singularity is that the velocity of the spin-0 charm meson must be greater than or equal to that of the spin-1 meson in order for it to overtake that meson so they can scatter: This gives an upper bound on u in terms of s: The minimum value of u is (M * 0 + M 0 ) 2 . Inserting this value into Eq. (B5) and solving for s gives the upper endpoint s ≤ s of the triangle-singularity region, where s is given in Eq. (13). The differential cross sections for e + e − → X + γ in Eq. (15) and e + e − → D * 0D0 + γ in Eq. (27) are expressed in terms of a Lorentz-invariant loop amplitude F (s, u). The nonrelativistic approximation to this amplitude is given as a loop integral in Eq. (A2). A Lorentz-invariant generalization of that amplitude can be obtained by replacing the nonrelativistic propagators in Eq. (A2) by the appropriate relativistic propagators and replacing the multiplicative factor of q · k by the appropriate Lorentz-invariant generalization: where Q 2 = s, P 2 = u, and Q.q = (s − u)/2. This can be expressed as an integral over two Feynman parameters: The nonrelativistic loop amplitude F (W, U ) in Eq. (A2), with W = √ s − 2M * 0 and U = √ s − (M * 0 +M 0 ), gives an excellent approximation to the function F (s, u) in the triangle singularity region. In the region where √ s is less than 10 MeV above 2M * 0 and √ u is less than 10 MeV above M * 0 +M 0 , the numerical differences seem to be consistent with errors of order (δ/M * 0 ) 2 .
We proceed to verify analytically that the relativistic loop amplitude F (s, u) in Eq. (B6) can be reduced to the nonrelativistic loop amplitude F (W, U ) in Eq. (A2). We will verify this only up to errors first order in δ/M * 0 . It is convenient to change the integration variables in Eq. (B7) to t and ζ defined by x = 1 2 (1 + ζ)(1 − t) and y = 1 2 (1 + ζ)t. In the region where s − 4M 2 * 0 and u − M 2 X are both order δ 2 , the leading term in the expansion of F (s, u) in powers of δ/M * 0 comes from the region where ζM X is order δ. If we keep only the leading terms in the numerator and in the denominator, the integral reduces to where M X = M * 0 +M 0 . We have reintroduced the width of the D * 0 through the substitution M 2 * 0 − i → M 2 * 0 − iM * 0 Γ * 0 and treated M * 0 Γ * 0 as order δ 2 . Since the leading contributions to the integral in Eq. (B8) come from the region where M X ζ is order δ, the endpoints of the integral over ζ can be extended to ±∞. The analytic result for the integral over ζ then becomes relatively simple: where s 0 = 4M 2 * 0 − 4iM * 0 Γ * 0 and u 0 = (M * 0 + M 0 ) 2 − 2iM * 0 Γ * 0 . The final integral over t can also be evaluated analytically: If we make the substitutions s → (2M * 0 + W ) 2 and u → (M * 0 + M 0 + U ) 2 , this agrees to leading order in δ/M * 0 with the nonrelativistic loop amplitude F (W, U ) in Eq. (A2) in the region where s − 4M 2 * 0 and u − (M * 0 +M 0 ) 2 are order δ 2 . The analytic approximation to F (s, u) in Eq. (B10) has errors that are first order in δ/M * 0 , so it is not useful as a quantitative approximation. The nonrelativistic amplitude F (W, U ) in Eq. (A3) is a much better approximation.
We proceed to show that the tensor loop integral in Eq. (B1) can be reduced to the scalar loop integral in Eq. (B6) in the region where s − 4M 2 * 0 and u − M 2 X are both order δ 2 . We will verify this only to leading order in δ/M * 0 . In a frame where q µ is order δ, the tensor loop integral in Eq. (B1) can be expanded in powers of δ/M * 0 with M 0 = M * 0 − δ. The loop integral in Eq. (B1) is quadratically ultraviolet divergent. Up to corrections suppressed by δ/M * 0 , the tensor g λρ − k λ k ρ /M 2 * 0 in the numerator of Eq. (B1) can be replaced by g λρ − Q λ Q ρ /Q 2 and then pulled outside the loop integral. This reduces the numerator of the loop integral in Eq. (B1) to k ν (Q − k) α , so the ultraviolet divergence is only logarithmic. That loop integral is a tensor function of the 4-momenta Q and q with indices ν and α. It can be decomposed into a linear combination of the 5 tensors g να , Q ν Q α , Q ν q α , q ν Q α , and q ν q α with coefficients that are functions of Q 2 and Q.q. Tensors with a factor of Q ν contract to 0 with the γ * -to-D * 0D * 0 vertex. Tensors with a factor of q α contract to 0 with theD * 0 -to-D 0 γ vertex. The only tensors that remain are g να and q ν Q α . The coefficient of g να is suppressed compared to that of q ν Q α by a factor of order δ 2 . Since q ν is order δ, the g να term is suppressed compared to the q ν Q α term by a factor of order δ.
The reduction of the tensor loop integral in Eq. (B1) to the scalar loop integral in Eq. (B6) can be completed by showing that the numerator factor k ν (Q − k) α can be replaced by the numerator factor in Eq. (B6) multiplied by 1 2 q ν Q α /(Q.q) 2 up to corrections suppressed by factors of δ/M * 0 . In the tensor reduction of the momentum integral of the k ν (Q − k) α term, the scalar integral coefficient of q ν Q α term is very different from half the scalar momentum integral in Eq. (B6). For example, its numerator includes terms quadratic in the loop momentum k. However after introducing Feynman parameters and integrating over the loop momentum k, the Feynman parameter integral differs from that in Eq. (B7) only by a factor of 1 − x − y in the numerator. After the change of variables to t and ζ as in Eq. (B8), that factor becomes 1 2 (1 − ζ). Since the leading contribution comes from the region where ζM X is order δ, that factor reduces to 1 2 . This completes the demonstration that the tensor loop integral can be reduced to the loop amplitude F (s, u) in Eq. (B6) up to corrections suppressed by at least one power of δ/M * 0 .
In the sum of the two triangle diagrams, the triangle of charm-meson propagators and the three attached vertices can be represented by a Lorentz-covariant effective vertex. The effective vertex for the coupling of a virtual photon with momentum Q = P +q to X with momentum P and a real photon with momentum q is −4ie 2 ν(2M X ) 3/2 √ πγ X F (Q 2 , P 2 ) A(Q) µνλσ q ν σαβτ Q α q β , where µ, λ, and τ are the Lorentz indices of the virtual photon, X, and real photon, respectively. The effective vertex for the coupling of a virtual photon with momentum Q = P + q to D * 0 andD 0 with total momentum P and a real photon with momentum q is −8iπe 2 νM X √ 4M * 0 M 0 −γ X − i λ 1/2 (P 2 , M 2 * 0 −iM * 0 Γ * 0 , M 2 0 )/ 2 √ P 2 ×F (Q 2 , P 2 ) A(Q) µνρσ q ν g ρλ − P ρ P λ P 2 σαβτ Q α q β , where µ, λ, and τ are the Lorentz indices of the virtual photon, D * 0 , and real photon, respectively.