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 prompt1 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 ![](https://www.scirp.org/html/3-7501539\7901e136-258b-4e45-8a96-c9c933b1ac9d.jpg)
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 algorithms2. 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.
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
MeV3 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
![](https://www.scirp.org/html/3-7501539\e67222a9-4be5-46c2-be2e-386474089b66.jpg)
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
scatter, on average, with 2 - 3 pions before the relative distances among the flying-out hadrons are such that the interactions are suppressed5.
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.
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.
NOTES
1i.e. not produced in B decays but at the hadron collision vertex.
2![](https://www.scirp.org/html/3-7501539\2b4c4f0c-c157-4dcd-9038-16ca254ca6b6.jpg)
3–20 MeV in the case of deuterium.
4We might say that
where we used the
reduced mass for
. On the other hand
where
is the decay momentum. Thus
.