Estimates of the $X(3872)$ Cross Section at a Hadron Collider

The claim that the $X(3872)$ meson cannot be a charm-meson molecule because its prompt production cross section at hadron colliders is too large is based on an upper bound in terms of a cross section for producing charm-meson pairs. Assuming $X$ is sufficiently weakly bound, we derive an equality between the $X$ cross section and a charm-meson pair cross section that takes into account the threshold enhancement from the $X$ resonance. The cross section for producing $X$ is equal to that for producing $D^{*0} \bar{D}^0$ integrated up to a relative momentum $k_\mathrm{max} = 7.7\,\gamma_X$, where $\gamma_X$ is the binding momentum of $X$. We also derive an order-of-magnitude estimate of the $X$ cross section in terms of a naive charm-meson pair cross section that does not take into account the threshold enhancement, such as that produced by a Monte Carlo event generator. The cross section for producing $X$ can be approximated by the naive cross section for producing $D^{*0} \bar{D}^0$ integrated up to a relative momentum $k_\mathrm{max}$ of order $(m_\pi^2 \gamma_X)^{1/3}$. The estimates of the prompt $X$ cross section at hadron colliders are consistent with the cross sections observed at the Tevatron and the LHC.


I. INTRODUCTION
The discovery of a large number of exotic hadrons containing a heavy quark and its antiquark presents a major challenge to our understanding of QCD [1][2][3][4][5][6][7][8][9][10]. The first of these exotic hadrons to be discovered was the X(3872) meson. It was discovered in 2003 in exclusive decays of B ± mesons into K ± X through the decay of X into J/ψ π + π − [11]. Its existence was quickly verified through inclusive production in pp collisions [12]. The J P C quantum numbers of X were eventually determined to be 1 ++ [13]. Its mass is extremely close to the D * 0D0 threshold, with the difference being only 0.01 ± 0.18 MeV [14]. This suggests that X is a weakly bound S-wave charm-meson molecule with the flavor structure The X can be produced by any reaction that can produce its constituents D * 0D0 and D 0D * 0 . In particular, it can be produced in high energy hadron collisions. The inclusive production of X in pp collisions has been studied at the Tevatron by the CDF [12] and D0 [15] collaborations. The inclusive production of X in pp collisions has been studied at the Large Hadron Collider (LHC) by the LHCb [16], CMS [17], and ATLAS [18] collaborations. At a high energy hadron collider, X is produced by the weak decays of bottom hadrons and by QCD mechanisms that create charm quarks and antiquarks. If X is produced by the weak decays of bottom hadrons, its decay products emerge from a vertex displaced from the collision point. If X is produced by QCD mechanisms, its decay products emerge from the collision point, so these mechanisms are referred to as prompt production. Cross sections for inclusive prompt production of X have been measured by the CDF [12], CMS [17], and ATLAS [18] collaborations.
The substantial prompt production rate of X at hadron colliders has often been used as an argument against its identification as a charm-meson molecule. This argument is based on an upper bound on the cross section for producing X in terms of the cross section for producing the charm-meson pair D * 0D0 integrated up to a maximum relative momentum k max [19]. The estimate for k max in Ref. [19] was approximately the binding momentum γ X of the X. In Ref. [20], it was pointed out that the derivation of the upper bound in Ref. [19] requires k max to be of order the pion mass m π instead of γ X . In this paper, we use the methods of Ref. [20] to derive equalities between the X cross section and D * 0D0 cross sections integrated up to k max . If we take into account the threshold enhancement due to the X resonance, the value of k max is 7.7γ X . If we use a naive D * 0D0 cross section without the threshold enhancement, the value of k max is order (m 2 π γ X ) 1/3 . The resulting estimates of the prompt cross sections are compatible with the measurements at the Tevatron and the LHC.
The outline of this paper is as follows. In Section II, we describe some universal aspects of weakly bound S-wave molecules and the scattering of their constituents. In Section III, we present experimental upper and lower bounds on the cross sections for the production of X at the Tevatron and the LHC. In Section IV, we discuss the theoretical upper bound on the prompt cross section for producing X at hadron colliders derived in Ref. [19]. In Section V, we derive equalities between the X cross section and a charm-meson pair cross section with and without the threshold enhancement from the X resonance. In Section VI, we summarize our results and discuss their implications.

BOUND S-WAVE MOLECULE
If short-range interactions produce an S-wave bound state extremely close to a scattering threshold, the few-body physics has universal aspects that are determined by the binding momentum γ X of the bound state [21]. The binding energy is γ 2 X /2µ, where µ is the reduced mass of the constituents. The momentum-space wavefunction in the region of the relative momentum k below the inverse range has the universal form ( The low-energy scattering of the constituents also has universal aspects determined by γ X through a simple function of the complex energy E relative to the scattering threshold: This function has a branch cut along the positive E axis and a pole at E = −γ 2 X /2µ. The universal elastic scattering amplitude in the region of relative momentum k below the inverse range is obtained by evaluating this function at energy E = k 2 /2µ + i . The analytic function f X (E) also gives the energy distribution from creation of the constituents at short distances. By the optical theorem, the distribution in the energy E below the energy scale set by the range is proportional to the imaginary part of f X (E): There is a delta-function term at a negative energy from the production of the weakly bound molecule and a theta-function term with positive energy from the production of the constituents of the molecule. The universal energy distribution in Eq. (4) from the creation of the constituents at short distances is illustrated in Fig. 1. The delta-function term below the scattering threshold at E = 0 is represented by a tall rectangle centered at E = −γ 2 X /2µ which would have the correct area if it was multiplied by 10. That area is equal to the area of the energy distribution above the threshold integrated up to the energy 20.2 (γ 2 X /2µ), which corresponds to a relative momentum k max = 4.49 γ X . The energy distribution above the threshold has a maximum at the binding energy γ 2 X /2µ. For E much larger than γ 2 X /2µ, the universal energy distribution in Eq. (4) decreases as E −1/2 . This scaling behavior should continue up to the energy scale set by the range.
The naive energy distribution in the absence of the resonance has a form at low energy that can be obtained from Eq. (4) by omitting the delta-function term and by replacing the denominator γ 2 X + k 2 in the theta-function term by Λ 2 , where Λ is some momentum of order the inverse range: The naive energy distribution is shown as a dashed line in Fig to the area of the naive energy distribution integrated up to the energy (3πγ X Λ 2 ) 2/3 /2µ, which corresponds to a relative momentum k max = (3πγ X Λ 2 ) 1/3 .
If the X(3872) is a weakly bound charm-meson molecule, its constituents are the superposition of charm mesons in Eq. (1). We denote the masses of the charm mesons D 0 and D * 0 by M 0 and M * 0 , respectively. The reduced mass of D * 0D0 is µ = M * 0 M 0 /(M * 0 + M 0 ). The range of the interactions between the charm mesons is 1/m π , where m π is the pion mass. The corresponding energy scale m 2 π /2µ is about 10 MeV. This is comparable to the energy of the D * + D − scattering threshold, which is 8.2 MeV above the D * 0D0 scattering threshold. The present value of the difference E X between the mass of the X and the energy of the D * 0D0 scattering threshold is [14] The central value in Eq. (6) corresponds to a charm-meson pair above the scattering threshold. The value lower by 1σ corresponds to a bound state with binding energy |E X | = 0.17 MeV and binding momentum γ X = 18 MeV. The upper bound on the binding energy with 90% confidence level is |E X | < 0.22 MeV. Some qualitative aspects of the energy distribution illustrated in Fig. 1 have been observed by the Belle collaboration in the decays of B mesons into KD 0D0 π 0 [22]. The D 0D0 π 0 invariant-mass distribution has a peak near the D * 0D0 threshold. The energy resolution was not sharp enough to resolve the contributions from the narrow peak below the D * 0D0 threshold from the X bound state and the peak above the threshold from the D * 0D0 and D * 0D * 0 threshold enhancements. The difference between the fitted curve to the D 0D0 π 0 invariant-mass distribution and the combinatorial background in Ref. [22] has a minimum at a D 0D0 π 0 energy about 11 MeV above the D * 0D0 threshold. If we set Λ 2 /2µ = 11 MeV, we get an estimate for Λ of about 150 MeV. This is consistent with Λ being of order m π .

III. CROSS SECTIONS FOR X AT HADRON COLLIDERS
In this section, we summarize experimental results on the inclusive prompt cross sections for X(3872) at the Tevatron and the LHC. We use them to obtain upper and lower bounds on the cross sections.
Within months of the discovery of the X in B meson decays by the Belle Collaboration in September 2003 [11], its existence was confirmed by the CDF Collaboration through inclusive production of X in pp collisions at the Tevatron [12]. The X was observed in the discovery decay mode J/ψ π + π − . Some of the X events were produced from decays of bottom hadrons, with the ultimate decay products µ + µ − π + π − emerging from a displaced vertex. The majority of X events were produced promptly, with the ultimate decay products emerging from the primary collision vertex. The prompt X events presumably come from QCD production mechanisms. The CDF Collaboration has reported ratios of the prompt cross sections for X and ψ(2S) modulo the ratio ψ / X of the efficiencies for observations of ψ(2S) and X in their J/ψ π + π − decay modes [23]. The ratio ψ / X is presumably closer to 1 than to 2 or 1/2. In Ref. [19] (BGP 2 S), measurements of the prompt cross sections for X and ψ(2S) by the CDF Collaboration were used to obtain an estimate of the product σ Br of the prompt cross section for X and its branching fraction into J/ψ π + π − in the region with rapidity |y| < 0.6 and transverse momentum p T > 5 GeV: A loose lower bound on the cross section for X can be obtained by using Br < 1. The inclusive production of the X in pp collisions at the LHC has been studied by the LHCb [16], CMS [17], and ATLAS [18] collaborations. The CMS collaboration measured the product σ Br of the prompt cross section for X and its branching fraction into J/ψ π + π − for X with |y| < 1.2 and 10 GeV < p T < 30 GeV at center-of-mass energy 7 TeV [17]: The prompt fraction was measured to be about 74% in the range 10 GeV < p T < 50 GeV [17]. In Ref. [24], we derived upper and lower bounds on the branching fraction Br for the X bound state to decay into J/ψ π + π − : The loose lower bound Br > 4% is derived from a recent measurement by the BaBar collaboration of the inclusive branching fraction for B + into K + plus the X resonance feature [25]. The upper bound Br < 33% is derived from measurements of branching ratios of J/ψ π + π − over other short-distance decay modes of the X. Given the results for σ Br in Eqs. (7) and (8), a constraint on the branching fraction into J/ψ π + π − gives constraints on the prompt cross sections. A lower bound on the prompt cross section σ can be estimated by decreasing the central value of σ Br by 1 standard deviation and then dividing it by the upper bound on Br in Eq. (9). A loose upper bound on σ can be estimated by increasing the central value of σ Br by 1 standard deviation and then dividing it by the lower bound on Br in Eq. (9). Using the estimate in Eq. (7), the bounds on the prompt cross section at the Tevatron with |y| < 0.6 and p T > 5 GeV are Using the measurement in Eq. (8), the bounds on the prompt cross section at the LHC with |y| < 1.2 and 10 GeV < p T < 30 GeV are In both Eqs. (10) and (11), the loose upper bound is more than 10 times larger than the lower bound. In Ref. [26], ranges of prompt cross sections σ for X at the Tevatron and at the LHC were obtained by considering branching fractions in the range 2.7% < Br < 8.3% [27]. The lower ends of their ranges were about 5 times larger than our lower bounds on the cross sections in Eqs. (10) and (11). The differences come primarily from two sources. First, the lower end of their range for σ was obtained from the central value of σ Br rather than from the value lower by 1 standard deviation. Second, the upper end of their range for Br was about 4 times smaller than our upper bound in Eq. (9). The smaller upper bound on Br was obtained by assuming that measurements of the branching fraction of X into D 0D * 0 are dominated by the decay of the X resonance into D 0D0 π 0 and D 0D0 γ below the D 0D * 0 threshold and have a negligible contribution from the threshold enhancement in the production of D 0D * 0 above the threshold. This assumption is contradicted by measurements of the width of the X from the D 0D * 0 decay mode, which are significantly larger than the upper bound on the width obtained by the Belle collaboration from the J/ψ π + π − decay mode [28].

IV. UPPER BOUND ON CROSS SECTION FOR X
In this section, we present the upper bound on the inclusive prompt cross section for producing X(3872) in Ref. [19]. We explain why the derivation of the upper bound requires the charm-meson pair cross section to be integrated up to a relative momentum of order m π instead of order γ X , as apparently assumed in Ref. [19].
If X is a charm-meson molecule with the flavor structure in Eq. (1), the inclusive cross section for producing X can be expressed in terms of the same amplitudes as those in the inclusive cross sections for producing D * 0D0 and D 0D * 0 [19]. The inclusive cross sections for producing D * 0D0 and D 0D * 0 with small relative momentum k in the charm-meson-pair rest frame and the inclusive cross section for producing X can be expressed as where µ is the reduced mass of D * 0D0 . The sums over y are over all the additional particles that can be produced. The amplitudes that appear in the cross section for X in Eq. (12c) are the charge-conjugation-even superpositions of the amplitudes for producing D * 0D0 + y and D 0D * 0 + y. The momentum-space wavefunction for the X in Eq. (12c) is normalized so (d 3 k/(2π) 3 ) |ψ X (k)| 2 = 1. The differential phase space dΦ (D * D )+y is that for a composite particle denoted by (D * D ) with mass M * 0 + M 0 plus the additional particles y. The mass of X is sightly smaller than M * 0 + M 0 and the invariant mass of a charm-meson pair is larger than M * 0 + M 0 , but the differences in the phase space integrals are negligible. Factors of 3 from the sums over the spin states of D * 0 orD * 0 or X are absorbed into the amplitudes A. The phase-space integrals in Eqs. (12) are over the 3-momenta of the additional particles y, but the cross sections remain differential in the 3-momentum P of (D * D ). Thus the D * 0D0 and D 0D * 0 cross sections in Eqs. (12a) and (12b) are differential in both P and k, while the X cross section in Eq. (12c) is differential only in P .
In the expression for the X cross section in Eq. (12c), there are interference terms between the amplitudes for producing D * 0D0 + y and D 0D * 0 + y. The interference terms are positive for some sets of additional final-state particles y and negative for others. In high-energy hadron collisions, there are dozens or even hundreds of additional particles. The sum over the many additional particles y gives cancellations that suppress the interference terms. The X cross section in Eq. (12c) then reduces to the sum of a D * 0D0 term and a D 0D * 0 term. At large transverse momentum, the hadronization of a cc pair is equally likely to produce D * 0D0 and D 0D * 0 , because the probability of a light quark or antiquark from a colliding hadron to become a constituent of one of the charm mesons is very small. The D * 0D0 term and the D 0D * 0 term should therefore be equal, and the expression for the X cross section can be reduced to The authors of Ref. [19] (BGP 2 S) derived a theoretical upper bound on the cross section for producing X in terms of a cross section for producing the charm meson pair D * 0D0 . To derive their upper bound, BGP 2 S first restricted the integral over the relative momentum in Eq. (13) to a region |k| < k max in which ψ X (k) differs significantly from 0. They then applied the Schwarz inequality to that integral: where the primes on the integrals indicate restrictions to |k| < k max . The first factor on the right side of Eq. (14) is the probability for the constituents of the X to have relative momentum less than k max . Since this probability is less than 1, they obtained the inequality The validity of this inequality hinges on the validity of restricting the integral in Eq. (13) to the region |k| < k max . In Ref. [19], BGP 2 S did not give an unambiguous prescription for the maximum momentum k max in the inequality in Eq. (15). They quoted the difference E X between the mass of the X and the energy of the D * 0D0 scattering threshold at that time as E X = −0.25±0.40 MeV. The central value corresponds to binding momentum γ X = 22 MeV, and the value lower by 1σ corresponds to γ X = 35 MeV. In Ref. [19], BGP 2 S chose k max in the inequality in Eq. (15) to be 35 MeV. In a subsequent paper Ref. [29], whose authors included most of those of Ref. [19], an updated estimate k max = 20 MeV was given. This is close to the binding momentum γ X = 18 MeV from the value of E X that is 1σ below the central value in Eq. (6). The choice for k max in both papers is consistent with the assumption that k max is approximately γ X , although this assumption was not stated explicitly in Ref. [19].
The conclusions of Ref. [19] were challenged in Ref. [20], which argued that the appropriate choice of k max in the upper bound in Eq. (15) is of order m π instead of order γ X . If short-range interactions produce an S-wave bound state close to a scattering threshold, the momentum-space wavefunction in the momentum region below the inverse range has the universal form in Eq. (2). The normalization integral of the probability density |ψ X (k)| 2 is dominated by k of order γ X . However the integral over k in Eq. (12c), whose integrand has only one factor of ψ X (k), is not dominated by k of order γ X . It has significant contributions from the region extending up to k of order m π , which is where the wavefunction ψ X (k) begins to fall faster than 1/k 2 . Thus the derivation of the upper bound in Ref. [19] requires k max to be of order m π instead of order γ X , as was apparently assumed in Ref. [19].
In Ref. [19], BGP 2 S estimated the cross section for charm-meson pairs with relative momentum k at the Tevatron using the event generators Herwig and Pythia to produce hadronic final states from 2 → 2 parton processes, primarily gg → gg. This extremely inefficient method gave distributions at small k with the behavior k dk. Their estimate for the theoretical upper bound on the prompt cross section for X at the Tevatron obtained by inserting k max = 35 MeV into Eq. (15) was 0.07 nb using Herwig and 0.11 nb using Pythia. These cross sections are about 30 times smaller than the loose lower bound of 3.1 nb given by the right side of Eq. (7). BGP 2 S concluded that if the X was a weakly bound charm-meson molecule, its formation from charm mesons at the rate observed at the Tevatron would be unlikely. Given that their D * 0D0 cross section scaled as k 2 max , the value of k max would have to be larger than about 280 MeV for the D * 0D0 cross section calculated using Pythia to be above the lower bound on the X cross section at the Tevatron in Eq. (10).
In Ref. [20], the cross section for charm-meson pairs with small relative momentum k at the Tevatron was estimated using the event generator Pythia to produce hadronic final states from the 2 → 3 parton process gg → ccg. The distribution had the behavior k 2 dk in the region k < m π . This behavior should be more accurate than the behavior k dk obtained in Ref. [19], because the event generator, which has not been tuned to reproduce distributions in k, plays a smaller role in generating the distributions. Their estimate for the charm-meson-pair cross section at the Tevatron integrated up to a relative momentum k max was The subscript "naive" emphasizes that the cross section calculated using an event generator does not take into account the effects of the X resonance. The theoretical upper bound in Eq. (15) is greater than the loose lower bound of 3.1 nb given by the right side of Eq. (7) if k max is greater than about 160 MeV. The authors of Ref. [20] concluded that the upper bound in Eq. (15) with k max of order m π was compatible with the observed prompt cross section for X at the Tevatron. The upper and lower bounds on the X cross section at the Tevatron in Eq. (10) take into account the bounds on the branching fraction for X → J/ψ π + π − in Eq. (9). The value of k max would have to be larger than 220 MeV for the naive D * 0D0 cross section to be above the lower bound on the X cross section at the Tevatron in Eq. (10). This value of k max is also compatible with the scale m π . The conclusions of Ref. [19] were also challenged in Ref. [26]. They used an effective field theory with ultraviolet cutoff Λ in which X is treated as a charm-meson molecule. They calculated the inclusive prompt cross sections for producing X in pp collisions at the Tevatron and in pp collisions at the LHC using the event generators Herwig and Pythia to calculate the production rate of D * D at short distances and using the effective field theory to calculate the formation rate of the X at long distances. With Λ = 100 MeV, their cross sections using Pythia were 0.05 nb at the Tevatron and 0.04 nb at the LHC, which are much smaller than the results from CDF and CMS in Eqs. (7) and (8). Their cross sections were compatible with the results from CDF and CMS for Λ in the range from 500 MeV to 1000 MeV. For these large ultraviolet cutoffs, the contributions from the charged-charmmeson-pair channels D * + D − and D + D * − were larger than those from the D * 0D0 and D 0D * 0 channels by about a factor of 2.
The analysis in Ref. [26] was rejected in Ref. [29], whose authors included most of those of Ref. [19]. They argued that k max must be determined "independently of any educated guesses on the explicit form" of ψ X (k). They did not address the issue that their derivation of the upper bound with k max approximately equal to γ X fails for the explicit wavefunction in Eq. (2).

V. ESTIMATE OF CROSS SECTION FOR X
In this section, we derive an equality between the X(3872) cross section and a charmmeson-pair cross section that takes into account the threshold enhancement produced by the X resonance. We also present an order-of-magnitude estimate of the X cross section in terms of a naive charm-meson-pair cross section that does not take into account the threshold enhancement.
Expressions for the cross sections for producing charm meson pairs and for producing X in Eqs. (12) that take into account the X resonance were presented in Ref. [20]. The cross sections were expressed in factored forms, with long-distance factors that involve the binding momentum γ X and with short-distance factors that involve only momentum scales of order m π or larger. The amplitude for producing D * 0D0 + y in Eq. (12a) can be decomposed into charge-conjugation even (C = +) and charge-conjugation odd (C = −) components. The C = + component is enhanced by the X resonance. If the nonresonant C = − component is neglected, the amplitude for producing D * 0D0 + y can be expressed as a product of the C = + component of a short-distance amplitude and a resonance factor that depends on γ X : The expression for the corresponding amplitude A D 0D * 0 +y (k) is identical. The short-distance amplitudes A s.d. D * 0D0 +y and A s.d. D 0D * 0 +y are independent of the momentum if k is small compared to m π . The constant Λ in the numerator of the resonance factor should be of order m π . The only dependence on the small momentum γ X is in the denominator of the resonance factor. Since Λ γ X , the absolute value of the resonance factor is approximately 1 at k = Λ, so Λ can be interpreted as the momentum scale where the amplitude becomes comparable in magnitude to the amplitude in the absence of the resonance. The resonance factor in Eq. (17) produces a threshold enhancement in the cross section. The differential cross section dσ/dE in the kinetic energy E of D * 0D0 in the D * 0D0 center-of-momentum (CM) frame should have a local minimum above the threshold enhancement. A simple physical interpretation of Λ is that the kinetic energy E at the local minimum is roughly Λ 2 /2µ.
The factorization formula for the D * 0D0 cross section can be obtained simply by inserting the amplitude in Eq. (17) into Eq. (12a). The factorization formula for the X cross section cannot be obtained so simply. If the universal wavefunction in Eq. (2) is inserted into Eq. (12c), the momentum integral is logarithmically ultraviolet divergent. The factorization formula for the X cross section can be obtained instead by requiring the sum of the cross sections for producing X and the cross sections for producing D * 0D0 and D 0D * 0 integrated over k to be consistent with the optical theorem in Eq. (4). The resulting factorization formulas for the inclusive cross sections are The differential cross sections for D * 0D0 and D 0D * 0 in Eqs. (18a) and (18b) should be good approximations up to relative momentum k of about Λ. The short distance factors in Eqs. (18) can be eliminated to obtain an expression for the D * 0D0 cross section in terms of the X cross section: This relation is analogous to relations between cross sections for the production of the deuteron bound state with large momentum transfer and cross sections for the production of proton-neutron pairs [30]. The integral of this D * 0D0 cross section over the region |k| < k max is There is a value of k max such that the integrated cross section is equal to that for X: k max = 7.73 γ X . The resulting equality between the X cross section and a D * 0D0 cross section is The right side of this equality can also be expressed as the sum of the D * 0D0 and D 0D * 0 cross sections integrated up to a smaller maximum value of k: The equalities in Eqs. (21) and (22) are equivalent provided γ X is sufficiently small. As a reasonable condition for the validity of the equality, we require k max to be smaller than the inverse range m π . If this condition is applied to the equalities in Eqs. (21) and (22), it requires the binding energy |E X | to be less than about 0.2 MeV and 0.5 MeV, respectively. The equality in Eq. (21) is consistent with the upper bound in Eq. (15) for any value of k max greater than 7.73 γ X . However the derivation of the upper bound in Eq. (15) requires k max to be order m π , because the derivation must allow for the possibility that the wavefunction for X has the form in Eq. (2). The equality in Eq. (21) is incompatible with the upper bound in Eq. (15) if k max is taken to be approximately γ X , as was apparently assumed in Ref. [19].
The equality between the cross sections for X and D * 0D0 in Eq. (21) assumes the D * 0D0 cross section has the threshold enhancement from the X resonance. If an event generator such as Herwig or Pythia is used to estimate the charm-meson-pair cross section, the equality in Eq. (21) cannot be used because the event generator is not informed about the resonance. The naive cross section for producing D * 0D0 can be obtained from Eq. (12a) by replacing the amplitude A D * 0D0 +y (k) by the short-distance amplitude A s.d.
D * 0D0 +y : This naive D * 0D0 cross section integrated over the region |k| < k max scales like k 3 max , in agreement with the charm meson cross section calculated using an event generator in Ref. [20]. In the expression for the X cross section in Eq. (18c), the sum over the many additional particles y give cancellations that suppress the interference terms between the amplitudes A s.d.
D * 0D0 +y and A s.d. D 0D * 0 +y . In a high energy hadron collider, the production rates for D * 0D0 and D 0D * 0 at large transverse momentum should be equal, because the light quarks in the charm mesons are unlikely to come from the colliding hadrons. The D * 0D0 and D 0D * 0 contributions should therefore be equal, so the cross section reduces to dσ[X(3872)] = 1 flux y dΦ (D * D )+y A s.d.
The short distance factor in the expression for the naive D * 0D0 cross section in Eq. (23) can then be eliminated in favor of the X cross section using Eq. (24): Note that this is larger by a factor of 2 than the cross section obtained from the equality in Eq. (19) by replacing γ 2 X + k 2 in the denominator by Λ 2 . If the naive D * 0D0 cross section in Eq. (25) is integrated over the region |k| < k max , there is a value of k max for which the integrated cross section is equal to the X cross section: This result can be used to obtain an order of magnitude estimate of the X cross section using naive charm-meson-pair cross sections calculated using a Monte Carlo event generator.
Since Λ is order m π , this estimate is compatible with the upper bound in Eq. (15) with k max of order m π .
In Ref. [20], the estimate of the X cross section in terms of a naive charm-meson-pair cross section was expressed in the form The naive D * 0D0 cross section integrated up to a relative momentum k max that was calculated using the Pythia event generator in Ref. [20] scales as k 3 max . The estimate in Eq. (27) therefore differs from that in Eq. (26) only by a multiplicative factor of 2. Thus the estimate in Eq. (26) is essentially just a convenient repackaging of the estimate from Ref. [20] in Eq. (27).
The estimate of the X cross section in Eq. (26) depends on the combination (Λ 2 γ X ) 1/3 of unknown parameters. If we use the lower bound on the X cross section at the Tevatron in Eq. (10) and the naive estimate of the D * 0D0 cross section at the Tevatron in Eq. (16), we get the lower bound (Λ 2 γ X ) 1/3 > 100 MeV. The order-of-magnitude estimate of this combination of parameters obtained by inserting Λ = m π and γ X = 18 MeV is (Λ 2 γ X ) 1/3 ∼ 70 MeV. We conclude that the estimate of the X cross section at the Tevatron in Eq. (27) is compatible with the observed cross section for some value of Λ of order m π if the binding energy of the X is roughly 0.17 MeV.
There does not seem to be any calculation using event generators in the literature of the naive cross section for producing D * 0D0 with small relative momentum at the LHC. If there was such a calculation for a single small value of k max , the cross section as a function of k max could be obtained simply by assuming it scales as k 3 max . Eq. (26) would then give an estimate of the cross section for producing X at the LHC that could be compared with the measured value in Eq. (11). Given the large uncertainty in k max = (3πΛ 2 γ X ) 1/3 and the fact that the estimate of the cross section scales as k 3 max , there is little doubt that the estimate would be compatible with the observed cross section.

VI. SUMMARY AND DISCUSSION
We have discussed the inclusive prompt production of the X(3872) at high energy hadron colliders under the assumption that the X is a weakly bound charm-meson molecule with the particle content in Eq. (1). We considered the production of X through the creation of its constituents D * 0D0 or D 0D * 0 at short distances of order 1/m π or smaller. The formation of the X proceeds on longer distance scales of order 1/γ X , where γ X is the binding momentum.
The theoretical upper bound on the cross section for producing X in Eq. (15) was derived in Ref. [19]. It is given by the D * 0D0 cross section integrated up to relative momentum k max . The authors did not give any clear prescription for k max , but their numerical value for k max was consistent with it being approximately equal to γ X . In Ref. [20], it was pointed out that the derivation of the upper bound in Eq. (15) actually requires k max to be of order m π . A specific example of a wavefunction for which the derivation of the upper bound requires k max γ X is the universal wavefunction for a weakly bound molecule in Eq. (2). This failure of the derivation of their upper bound with k max approximately equal to γ X has never been addressed by the authors of Ref. [19].
Assuming the binding energy of the X is sufficiently small, we used the methods of Ref. [20] to derived the equality in Eq. (21) between the X cross section and the D * 0D0 cross section integrated up to relative momentum k max = 7.73 γ X . This equality takes into account the threshold enhancement in the charm-meson-pair cross section associated with the X resonance. The condition for the validity of this equality is that the binding energy |E X | is less than about 0.2 MeV.
The equality in Eq. (21) is not applicable if the charm-meson-pair cross section is estimated using a naive method that is not informed about the resonance, such as a Monte Carlo event generator. We used the methods of Ref. [20] to derive the order-of-magnitude estimate for the X cross section in Eq. (26). It is expressed as the naive D * 0D0 cross section integrated up to a relative momentum k max of order (m 2 π γ X ) 1/3 . The resulting estimate for the prompt cross section for X at the Tevatron using the naive D * 0D0 cross section in Eq. (16) is compatible with the experimental lower bound on the prompt X cross section given in Eq. (10). Given the large uncertainty in k max and the k 3 max scaling of the naive cross section, there is little doubt that the corresponding estimate of the cross section at the LHC would be compatible with the measured value in Eq. (11). We conclude that the prompt cross sections for X at the Tevatron and at the LHC are compatible with the identification of X(3872) as a weakly bound charm-meson molecule.