A Mechanism for Hadron Molecule Production in pp p Collisions

The problem of understanding loosely bound hadron molecules prompt production at hadron colliders is still open: how is it possible that meson molecules with binding energy compatible with zero could be formed within the bulk of the hadrons ejected in very high energy collisions? Monte Carlo simulations have been performed in the literature, leading to production cross sections, two orders of magnitude which were smaller than the experimental value. One possible mechanism to reduce this gap could be final state interactions of heavy mesons, but a precise evaluation of such effect is challenged by the presence of pions between the molecular constituents. In this paper, we present a new mechanism by using precisely such comoving pions. Heavy meson pairs can indeed slow down because of elastic scattering with surrounding pions. The number of low-relative-momentum meson pairs increases, thereby enhancing prompt production cross section. In this preliminar simulation, we show that an enhancement of 100 is indeed possible.

In our view the X could rather be the meson-molecule analogue of the stable deuterium. Given the large number of pions produced in the neighbourhood of the open charm meson pairs in momentum phase space, it is plausible that some of those pions could scatter elastically on the D 0 or D * 0 component of the would-be-molecule changing the relative momentum in the centre of mass of the pair, k 0 , towards lower values -see Fig. 1. We can assume the initial total energy E of the pair to be positive. However, if k 0 = |k 0 | gets smaller due to an interaction with the pion, E might be found shifted down to some negative -close to zero -value, provided that the D 0D * 0 pair is under the influence of some (unknown) attractive potential, say a square well potential, similar to the simplest description of deuterium.
In these respects the X would be a genuine, negative energy, bound state of D 0D * 0 whose lifetime is entirely regulated by the lifetime of the shorter lived component D * 0 ; we would estimate then a total width Γ tot (X) ≃ 65 keV [8]. There are no energetic arguments to stabilize the D * in the attractive potential. Such a mechanism is therefore somewhat opposite to the one based on FSI, where the D 0D * 0 pair should rescatter remaining isolated from other hadrons potentially produced close in phase space [3,4].
One more reason to pursue the approach described above is that the resonant scattering D 0D * 0 → X → D 0D * 0 is difficult to be reconciled with the general expectations that can be drawn for the total scattering cross section of two particles allowing a shallow bound state with energy |E| ∼ 0, as described in the 'Low equation' formalism, see [9]. Resonant scattering D 0D * 0 → X → D 0D * 0 can be computed using available data on X decay branching fractions (in order to compute the g XDD * coupling) and averaging the cross section, σ f , with the distribution f (k 0 ) of D 0D * 0 pairs obtained by hadronization algorithms [13]. It is only when k 0 is smaller than some critical value Λ that the resonant scattering into X has a non negligible probability to occur. We find a scattering legth a of about 4 fm for a total width Γ(X) ≃ 1.2 MeV (the g XDD * coupling is a function of the X total width) and Λ = 50 MeV. The scattering length a decreases for smaller values of the total widthsee Fig. 2. Shifting Λ towards higher values, Λ ∼ 10 GeV, a decreases to few cents of a fermi.
On the other hand, the scattering length expected for scattering with a shallow bound state is a = ℏ/ 2µ|E| ≃ 12 fm (E ≃ −0.14 MeV). Such a result, as discussed in [9], is independent on the (unknown) scattering potential. Analysis method . The binding energy of the X is estimated from the mass difference with its constituents E ≃ −0.14 ± 0.22 MeV. A discrete level at this energy (take the central value) can be accommodated in a square well with a depth of about −7 MeV [14] and a range r 0 ≃ 3 fm.
Let ψ 0 be the wave function associated to this level. The average size of the molecule is found to be r 2 ψ0 ≃ 10 fm and a value of k 2 0 ψ0 ≃ 50 MeV is determined. Those pions scattering elastically on D 0 or D * 0 and making the k 0 of the pair lower than 50 MeV are able to drop the total energy down to E and form a genuine D 0D * 0 bound state. It is our purpose here to seek such pions and to study numerically their elastic interactions with the D 0 or D * 0 mesons adapting standard hadronization tools such as Herwig and Pythia.
As discussed first in [2], the spectrum of D 0D * 0 pairs can be represented by a monotonically rising histogram in k 0 . Because of the interaction with pions, pairs with high relative COM (centre of mass) momenta, the majority, could either be pushed to higher momenta or to lower ones. If even a small part of them were rearranged within lower relative momenta, there could be a significant effect of feed-down of pairs towards lower bins, even in the far low energy region below 50 MeV. Populating that region means increasing the formation probability of the loosely bound X.
To perform a first qualitative exploration of this phenomenon, we start by generating samples of pp → cc events in Herwig and Pythia, at Tevatron COM energies ( √ s = 1.96 TeV). We list the events containing D 0D0 * (resp.D 0 D 0 * ) as a function of k 0 . The cuts imposed at parton level are: p part ⊥ > 2 GeV and |y part | < 6. The distributions dσ/d(∆ϕ), where ∆ϕ is the difference in azimuthal angles between D 0 and D * − , as discussed in [2], are reproduced by choosing the following cuts on the final mesons: open charm meson pairs have 5.5 GeV < p ⊥ < 20 GeV and |y| < 1. These cuts allow to reproduce very well CDF data on dσ/d(∆ϕ) if a full Quantum Chromodynamics (QCD) generation of events is performed (cc + gg + gq + qq...). D 0D * 0 pairs in the bin ∆ϕ = [0 • , 18 • ] are the main would-be-molecule candidates. We observe here that the numerical generation of pp → cc partially fills (≈ 10%) the ∆ϕ = [0 • , 18 • ] bin with respect to the full QCD one. In addition, in the central region, which is enforced by the cuts, we have to match our results with those of some Matrix Element Monte Carlo, like Alpgen [10], more than just using shower algorithms. We will present the results of the full QCD simulation, which is much more time consuming, in a future paper.
To optimize the selection of events, we choose the 10 most complanar pions to the D 0D * 0 plane, then we randomly choose the meson the pion will interact with (say the D 0 ), and finally we select the most parallel pion to the non-interacting meson (say theD * 0 ) -see Fig. 1. In physical events, we expect such a pion to be the most effective one to the phenomenon we are describing.
The elastic interactions with the pions are regulated in the πD 0 COM by the matrix elements π(p)D(q)|D * (P, η) = g πDD * η · p π(p)D * (q, λ)|D * (P, η) = g πD * D * M D * ǫ αβγδ λ α η β p γ q δ where the couplings used are g πDD * ≈ 11, g πD * D * ≈ 17, see [11]. After the interaction with the pion has taken place in the COM D 0 π frame, we boost back the D 0 in the laboratory (LAB) frame and check if the 'new' D 0D * 0 pair passes the cuts we fixed for the final meson pairs. We can trace, event by event, the variation k 0 → k ′ 0 of each D 0D * 0 pair filling a 2D histogram of transition probabilities P (k 0 , ∆k 0 ). Since the interaction with pions can change the p ⊥ and y of the molecule, a pair might fail the strict meson cuts before the interaction and pass them after it (a 'gained' would-be-molecule) and viceversa (a 'lost' one): see Fig. 3.
The open charm mesons might interact with pions more than once before a molecule is formed. Roughly speaking the πD 0 → πD 0 scattering is proportional to g 4 πDD * whereas the D * → Dπ decay is 'slower' by g 2 πDD * [15]. We assume that a single D 0 (D * 0 ) might scatter, on average, with 2 ÷ 3 pions before the relative distances among the flying-out hadrons are such that the interactions are suppressed [16].
Therefore, for each pair, we wish to evaluate k (n) 0 after n interactions. We do it according to the probability distribution functions (PDF) as extracted from P (k 0 , ∆k 0 ). We build a set of PDFs P i (∆k 0 ) for each bin i in dσ/dk 0 . We assume that the PDFs will be the same for all the interactions, like in a Markov chain. For each event we have a k (n) 0 , falling in some particular bin i. We randomly extract a ∆k 0 according to the distribution P i (∆k 0 ) and sum |k thus producing a new histogram. We must also take into account the 'lost' and 'gained' would-be-molecules. In each iteration, we generate the number of 'lost' and 'gained' ones, l (n) , g (n) , according to Poissonian distributions with mean values l (1) , g (1) . We implement the following algorithm: i) before the n-th interaction, we drop out a number l (n) of pairs, ii) we produce the new histogram as a result of the interaction with one more pion, iii) after that, we decide to 'gain' a g (n) number of pairs.
Results. The results are showed in Fig. 3. The bin we are more interested in is the first one, with k 0 < 50 MeV. The number of pairs obtained for that bin are reported in Table I. As one can see from these reproduces the shape found in [2]. The histograms named 1π and 3π are related to the elastic scattering of open charm mesons with one or three pions selected as described above. In the insects we report a broader k0 range.
plots the feed-down mechanism towards lower relative momentum bins is very effective once the interaction of a D 0 or a D * 0 with a pion from the hadronization is taken into account. The effect gets magnified if successive interactions are allowed (up to three). In the insects we show a broader range in k 0 . It is evident here that the elastic scattering with a pion is also causing a net increase of would-be-molecule pairs: it forces a number of pairs to pass the p ⊥ > 5 GeV and |y| < 0.6 cuts, which otherwise would be failed.   The results showed in Table I are indicating qualitatively that the mechanism described in this letter indeed occurs in numerical simulations of pp collisions and might play an important role in physical events. For a full determination of prompt production cross sections we need to switch from pp → cc to the full QCD generation pp → cc + gg + gq + qq... which is a harder task in terms of numerical computation, yet, from the exploration here reported, we have a clear clue on what to expect.
Conclusions. We have presented a new mechanism to explain the prompt formation of loosely bound open charm meson molecules at hadron colliders as induced by elastic scattering with comoving pions. Simplified numerical simulations show that pions produced in hadronization might be effective at decresing the relative momentum in the center of mass of the D meson pair which, if under the influence of an attractive potential, might therefore be found at some small negative energy, like in a shallow bound state in a potential well. Such a bound state will have a lifetime which is as long as the D * 0 one, Γ ∼ 65 keV, still well below actual experimental resolution. With the results of the full numerical simulations we will provide expected prompt cross sections for the production of the X(3872) at the LHC.
Considering the known limits of the available hadronization models, the results of numerical simulations have to be taken as compelling but qualitative descriptions of the suggested mechanism. We believe that several more investigations in this direction are possible.
Acknowledgements. A.P. wishes to thank E. Braaten for stimulating discussion.