** Journal of Modern Physics** Vol.4 No.12(2013), Article ID:40506,5 pages DOI:10.4236/jmp.2013.412193

A Mechanism for Hadron Molecule Production in Collisions

^{1}Dipartimento di Fisica, Sapienza Università di Roma, Roma, Italy

^{2}Department of Physics, Columbia University, New York, USA

^{3}INFN, Sezione di Pavia, Pavia, Italy

^{4}INFN, Sezione di Roma 1, Roma, Italy

Email: ^{*}antonio.polosa@roma1.infn.it

Copyright © 2013 Angelo Esposito et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received September 17, 2013; revised October 19, 2013; accepted November 16, 2013

**Keywords:** Exotic Spectroscopy; Heavy Quark Phenomenology; Hadron Scattering; Monte Carlo Simulations

ABSTRACT

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.

1. Introduction

The problem of understanding the loosely bound hadron molecule formation in collisions at Tevatron and LHC energies is still open. A recent measurement by the CMS Collaboration [1] basically confirms, at higher energies, older Tevatron results on the prompt production of which were first addressed in [2]. Looking at these new results [1], the questions remain the same as those raised in [2]: how is it possible that a very long lived molecule of a and a meson, with binding energy compatible with zero, could be formed within the bulk of the hadrons ejected in very high energy collisions? Is it the that molecule?

The reply given in [2] to the former question was sharply negative. In that paper, we performed numerical simulations with standard hadronization algorithms (Herwig and Pythia) tuned to fit data on the production of open charm mesons and sought pairs with reasonably low relative momentum in their centre of mass so as to be eligible candidates for becoming molecular loosely bound states. The number of selected pairs allowed to estimate an upper bound on the prompt^{1} production cross section of the which was found to be at least 30 times smaller than the experimental value.

Our analysis was reproduced, with similar results, in [3], where it was also observed that a more appropriate treatment of Tevatron data would rather indicate a discrepancy with theoretical expectations by a factor of 300.

Such a gap did not seem to be unbridgeable to the authors of [3], who resorted to final state interaction (FSI) mechanisms in the system in order to improve the theoretical cross section up to the experimental value. The approach there used was criticised in [4] leaving the controversy somewhat unsolved [5].

2. Molecular

On the other hand, during the last few years, the idea of a molecular, in diverse incarnations [6-13], has been corroborated by the lack of observation of its nearly degenerate charged partners, required by the antagonist tetraquark model [14]. For these reasons we come back here to the problem of the formation in high energy hadron collisions being motivated by a completely different approach. In our view the 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 or component of the would-be-molecule changing the relative momentum in the centre of mass of the pair, , towards lower values—see Figure 1. We can assume the initial total energy of the pair to be positive. However, if gets smaller due to an interaction with the pion, might be found shifted down to some negative— close to zero—value, provided that the 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 would be a genuine, negative energy, bound state of whose lifetime is entirely regulated by the lifetime of the shorter lived component; we would estimate then a total width keV [15]. There are no energetic arguments to stabilize the in the attractive potential.

Such a mechanism is therefore somewhat opposite to the one based on FSI, where the 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 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 , as described in the “Low equation” formalism, see [16]. Resonant scattering can be computed using available data on decay branching fractions (in order to compute the coupling) and averaging the cross section, , with the distribution

of pairs obtained by hadronization algorithms^{2}. It is only when is smaller than some critical value that the resonant scattering into has a non negligible probability to occur. We find a scattering legth of about 4 fm for a total width

MeV (the coupling is a function of the total width) and MeV. The scattering length decreases for smaller values of the total width—see Figure 2. Shifting towards higher values, GeV, decreases to few cents of a fermi.

Figure 1. The elastic scattering of a (or) with a pion among those produced in hadronization could reduce the relative momentum in the centre of mass of the pair.

Figure 2. Scattering length for the process as a function of the total width. The initial pairs are selected with MeV which, in our simulations, represent a few parts over of the total. The error bands account for uncertainties on the data we used.

On the other hand, the scattering length expected for scattering with a shallow bound state is fm (MeV). Such a result, as discussed in [16], is independent on the (unknown) scattering potential.

3. Analysis Method

The binding energy of the is estimated from the mass difference with its constituents MeV. A discrete level at this energy (take the central value) can be accommodated in a square well with a depth of about MeV^{3} and a range fm.

Let be the wave function associated to this level. The average size of the molecule is found to be

fm and a value of MeV is determined. Those pions scattering elastically on or and making the of the pair lower than 50 MeV are able to drop the total energy down to and form a genuine bound state. It is our purpose here to seek such pions and to study numerically their elastic interactions with the or mesons adapting standard hadronization tools such as Herwig and Pythia.

As discussed first in [2], the spectrum of pairs can be represented by a monotonically rising histogram in. 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.

To perform a first qualitative exploration of this phenomenon, we start by generating samples of events in Herwig and Pythia, at Tevatron COM energies (TeV). We list the events containing (resp.) as a function of. The cuts imposed at parton level are: GeV and.

The distributions, where is the difference in azimuthal angles between and, as discussed in [2], are reproduced by choosing the following cuts on the final mesons: open charm meson pairs have and. These cuts allow to reproduce very well CDF data on if a full Quantum Chromodynamics (QCD) generation of events is performed. pairs in the bin are the main would-be-molecule candidates. We observe here that the numerical generation of partially fills the 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 [17], 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 plane, then we randomly choose the meson the pion will interact with (say the), and finally we select the most parallel pion to the non-interacting meson (say the)—see Figure 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 COM by the matrix elements

where the couplings used are, see [18-20]. After the interaction with the pion has taken place in the COM frame, we boost back the in the laboratory (LAB) frame and check if the “new” pair passes the cuts we fixed for the final meson pairs.

We can trace, event by event, the variation of each pair filling a 2D histogram of transition probabilities. Since the interaction with pions can change the and 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 Figure 3.

The open charm mesons might interact with pions more than once before a molecule is formed. Roughly speaking the scattering is proportional to

whereas the decay is “slower” by

^{4}. We assume that a single might

Figure 3. Number of pairs (events) counted with Herwig (upper panel) and Pythia (lower panel) when generating events at TeV with the cuts on partons and hadrons described in the text. The histogram reproduces the shape found in [2]. The histograms named and 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 range.

scatter, on average, with 2 - 3 pions before the relative distances among the flying-out hadrons are such that the interactions are suppressed^{5}.

Therefore, for each pair, we wish to evaluate after n interactions. We do it according to the probability distribution functions (PDF) as extracted from . We build a set of PDFs for each bin in. We assume that the PDFs will be the same for all the interactions, like in a Markov chain. For each event we have a, falling in some particular bin. We randomly extract a according to the distribution and sum 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, , , according to Poissonian distributions with mean values,. We implement the following algorithm: 1) before the -th interaction, we drop out a number of pairs, 2) we produce the new histogram as a result of the interaction with one more pion, 3) after that, we decide to “gain” a number of pairs.

4. Results

The results are showed in Figure 3. The bin we are more interested in is the first one, with MeV. The number of pairs obtained for that bin are reported in Table 1.

As one can see from these plots the feed-down mechanism towards lower relative momentum bins is very effective once the interaction of a or a 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. It is evident here that the elastic scattering with a pion is also causing a net increase of would-bemolecule pairs: it forces a number of pairs to pass the GeV and cuts, which otherwise would be failed.

Table 1. The population of the the MeV bin (pairs), after, interactions.

The results showed in Table 1 are indicating qualitatively that the mechanism described in this letter indeed occurs in numerical simulations of collisions and might play an important role in physical events. For a full determination of prompt production cross sections we need to switch from to the full QCD generation 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.

5. 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 meson pair, 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 one, 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 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.

6. Acknowledgements

A. P. thanks E. Braaten for stimulating discussion.

REFERENCES

- S. Chatrchyan, et al. Journal of High Energy Physics, Vol. 1304, 2013, p. 154. http://dx.doi.org/10.1007/JHEP04(2013)154
- C. Bignamini, B. Grinstein, F. Piccinini, A. D. Polosa and C. Sabelli, Physical Review Letters, Vol. 103, 2009, Article ID: 162001. http://dx.doi.org/10.1103/PhysRevLett.103.162001
- P. Artoisenet and E. Braaten, Physical Review Letters, Vol. D81, 2010, Article ID: 114018. http://dx.doi.org/10.1103/PhysRevLett.103.162001
- C. Bignamini, B. Grinstein, F. Piccinini, A. D. Polosa, V. Riquer and C. Sabelli, Physics Letters B, Vol. 684, 2010, pp. 228-230. http://dx.doi.org/10.1016/j.physletb.2010.01.037
- P. Artoisenet and E. Braaten, Physical Review D, Vol. 83, 2011, Article ID: 014019. http://dx.doi.org/10.1103/PhysRevD.83.014019
- F. E. Close and P. R. Page, Physics Letters B, Vol. 628, 2005, p. 215. http://dx.doi.org/10.1016/j.physletb.2005.09.016
- E. Braaten and M. Kusunoki, Physical Review D, Vol. 69, 2004, Article ID: 074005. http://dx.doi.org/10.1103/PhysRevD.69.074005
- F. E. Close and P. R. Page, Physics Letters B, Vol. 578, 2004, p. 119. http://dx.doi.org/10.1016/j.physletb.2003.10.032
- N. A. Tornqvist, Physics Letters B, Vol. 590, 2004, p. 209. http://dx.doi.org/10.1016/j.physletb.2004.03.077
- E. S. Swanson, Physics Reports, Vol. 429, 2006, p. 243. http://dx.doi.org/10.1016/j.physrep.2006.04.003
- S. Fleming, M. Kusunoki, T. Mehen and U. van Kolck, Physical Review D, Vol. 76, 2007, Article ID: 034006. http://dx.doi.org/10.1103/PhysRevD.76.034006
- E. Braaten and M. Lu, Physical Review D, Vol. 76, 2007, Article ID: 094028. http://dx.doi.org/10.1103/PhysRevD.76.094028
- E. Braaten and M. Lu, Physical Review D, Vol. 77, 2008, Article ID: 014029. http://dx.doi.org/10.1103/PhysRevD.77.014029
- L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, Physical Review D, Vol. 71, 2005, Article ID: 014028. http://dx.doi.org/10.1103/PhysRevD.71.014028
- E. Braaten and M. Lu, Physical Review D, Vol. 76, 2007, Article ID: 094028. http://dx.doi.org/10.1103/PhysRevD.76.094028
- S. Weinberg, “Lectures on Quantum Mechanichs,” Cambridge University Press, Cambridge, 2013.
- M. L. Mangano, M. Moretti, F. Piccinini, R. Pittau and A. D. Polosa, Journal of High Energy Physics, Vol. 0307, 2003. http://dx.doi.org/10.1088/1126-6708/2003/07/001
- R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Physics Reports, Vol. 281, 1997, p. 145. http://dx.doi.org/10.1016/S0370-1573(96)00027-0
- A. Deandrea, N. Di Bartolomeo, R. Gatto, G. Nardulli and A. D. Polosa, Physical Review D, Vol. 58, 1998, Article ID: 034004. http://dx.doi.org/10.1103/PhysRevD.58.034004
- A. Deandrea, R. Gatto, G. Nardulli and A. D. Polosa, Physical Review D, Vol. 59, 1999, Article ID: 074012. http://dx.doi.org/10.1103/PhysRevD.59.074012

NOTES

^{*}Corresponding author.

^{1}i.e. not produced in B decays but at the hadron collision vertex.

2

^{3}–20 MeV in the case of deuterium.

^{4}We might say that

where we used the reduced mass for. On the other hand where is the decay momentum. Thus.