More loosely bound hadron molecules at CDF?
C Bignamini, B Grinstein, F Piccinini, AD Polosa, V Riquer, C Sabelli
aa r X i v : . [ h e p - ph ] J a n More loosely bound hadron molecules at CDF?
C Bignamini † , B Grinstein ∗ , F Piccinini † , AD Polosa ¶ , V Riquer § and C Sabelli ‡¶ Dipartimento di Fisica Nucleare e Teorica, Universit`a di Pavia, via A. Bassi 6, Pavia, I-27100, Italy ∗ CERN-PH-TH, CH-1211 Geneva 23, Switzerlandand University of California, San Diego, Department of Physics, La Jolla, CA 92093-0315, USA † INFN Pavia, Via A. Bassi 6, Pavia, I-27100, Italy ¶ INFN Roma, Piazzale A. Moro 2, Roma, I-00185, Italy § Fondazione TERA, Via Puccini, Novara, 11 -28100, Italy ‡ Department of Physics, Universit`a di Roma, ‘La Sapienza’, Piazzale A. Moro 2, Roma, I-00185, Italy
In a recent paper we have proposed a method to estimate the prompt production cross sectionof X (3872) at the Tevatron assuming that this particle is a loosely bound molecule of a D anda ¯ D ∗ meson. Under this hypothesis we find that it is impossible to explain the high promptproduction cross section found by CDF at σ ( X (3872)) ∼ ÷
70 nb as our theoretical prediction isabout 300 times smaller than the measured one. Following our work, Artoisenet and Braaten, havesuggested that final state interactions in the D ¯ D ∗ system might be so strong to push the resultwe obtained for the cross section up to the experimental value. Relying on their conclusions weshow that the production of another very narrow loosely bound molecule, the X s = D s ¯ D ∗ s , could besimilarly enhanced. X s should then be detectable at CDF with a mass of 4080 MeV and a promptproduction cross section of σ ( X s ) ∼ ÷ Introduction . In a recent paper [1] we have proposed a method for estimating the prompt production crosssection of X (3872) [2] at the Tevatron making the assumption that X is a loosely bound molecule of D and ¯ D ∗ ,with a binding energy as small as E = − . ± .
40 MeV. The motivation for this study is that, after the Bellediscovery, CDF and D0 confirmed the X (3872) in proton-antiproton collisions [5, 6] and it seems at odds withcommon intuition that such a loosely bound molecule could be produced promptly ( i.e. not from B decay) in ahigh energy hadron collision environment. This was also one of the initial motivations to consider the possibilitythat the X (3872) could be, instead of a molecule, a ‘point-like’ hadron resulting from the binding of a diquarkand an antidiquark [7], following the interpretation proposed by Jaffe and Wilczek [8] of pentaquark baryons(antidiquark-antidiquark-quark).To start let us summarize the content of [1]. Let us suppose that X (3872) is an S -wave bound state of two D mesons, namely a 1 / √ D ¯ D ∗ + ¯ D D ∗ ) molecule (we will use the shorthand notation D ¯ D ∗ ) . Themolecule production cross section will be proportional to the number of D ¯ D ∗ pairs in the event. Thus the X (3872) prompt production cross section at the Tevatron could be written as: σ ( p ¯ p → X (3872)) ∼ (cid:12)(cid:12)(cid:12)(cid:12)Z d k h X | D ¯ D ∗ ( k ) ih D ¯ D ∗ ( k ) | p ¯ p i (cid:12)(cid:12)(cid:12)(cid:12) ≃ (cid:12)(cid:12)(cid:12)(cid:12)Z R d k h X | D ¯ D ∗ ( k ) ih D ¯ D ∗ ( k ) | p ¯ p i (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R d k | ψ ( k ) | Z R d k |h D ¯ D ∗ ( k ) | p ¯ p i| ≤ Z R d k |h D ¯ D ∗ ( k ) | p ¯ p i| ∼ σ ( p ¯ p → X (3872)) max (1)where k is the relative 3-momentum between the D ( p ) , D ∗ ( p ) mesons. ψ ( k ) = h X | D ¯ D ∗ ( k ) i is some nor-malized bound state wave function characterizing the X (3872). R is the integration region where ψ ( k ) issignificantly different from zero. The matrix element h D ¯ D ∗ ( k ) | p ¯ p i can be computed using standard matrix-element/hadronization Monte Carlo programs (MC) like Herwig [9] and Pythia [12] . We require our MC toolsto generate 2 → .As for the determination of the region R in (1) we estimate it having in mind a naive gaussian ansatz forthe bound state wave function. It is straightforward to estimate the momentum spread of the gaussian by Such a molecule has the correct 1 ++ quantum numbers of the X (3872). Open charm meson pairs generated with hadronization Monte Carlo are ordered as a function of their relative center-of-mass3-momenta. If more than one D ¯ D ∗ pair is found in the event, we select the pair having the smaller relative 3-momentum k .As a first step we select those pairs which pass the kinematical cuts used in the data analysis made by the CDF collaboration. Configurations with one gluon recoiling from a c ¯ c pair, are those configuration expected to produce two collinear charm quarksand in turn collinear open charm mesons. The parton shower algorithms in Herwig and Pythia treat properly these configurationsat low p ⊥ whereas they are expected to be less important at higher p ⊥ . D mesons. Given that the binding energy E is E ∼ M X − M D − M D ∗ = − . ± .
40 MeV we find that r ∼ . ± . p ∼
12 MeV.Given the very small binding energy we can estimate k to be as large as k ≃ p µ ( − .
25 + 0 . ≃
17 MeV, µ being the reduced mass of D ¯ D ∗ system, or of the order of the center of mass momentum k = p λ ( m X , m D , m ∗ D ) / m X ≃
27 MeV. These considerations imply that we can restrict the integrationregion to a ball R of radius ≃ [0 ,
35] MeV.Keeping k inside R we estimate a σ ( p ¯ p → X (3872)) max which is about 30 times smaller than the mostconservative estimate ( σ ∼ . σ ∼ ÷
70 nb reinforcingthe negative result obtained with our theoretical calculation that would rather be 300 times smaller than themeasured one. This fact would undoubtably put in serious trouble the molecular interpretation of X (3872).In this note we intend to start from the main result discussed in [11] where it is argued that the effect of finalstate interactions in the D ¯ D ∗ system is such that two corrections should be made to our previous calculation: i ) the ball R should be enlarged to include momenta up to Λ ∼
300 MeV; ii ) a correction factor to the crosssection we compute (see (1)) should be considered so that the actual cross section σ ∗ including the full effect offinal state interaction is σ ∗ ( p ¯ p → X (3872)) = σ ( k < Λ) × π p µ |E | Λ (2)Assuming that this is the correct way of discussing the X (3872) production we observe that, besides reconcilingthe experimental result with the theoretical computation for the X , this mechanism should enhance the occur-rence of an hypothetical new molecule, the D s ¯ D ∗ s , which otherwise would be suppressed as one could infer bylooking at data on D s production at Tevatron [14] (as shown in Fig. 1, D s is on average ∼ D ). The same data are used to tune our Monte Carlo (Herwig in this calculation) with respect to D s production as shown in Fig. 1. ôôôôô ô = D * ààààà à = D òòò ò = D s p ¦ H GeV L d Σ (cid:144) dp ¦ H nb (cid:144) G e V L ž È y È £ FIG. 1: Differential charm cross section measured in fully hadronic charm decays using 5 . − at CDF [14]. The errorbars represent the total uncertainty of the measurement. The ratio in the production of D and D s is 4.4 whereas theratios in the production of D ∗ and D s is about 2.2. The dot-dashed lines are the result of the Monte Carlo simulationdone with Herwig rescaling the normalizations of the distributions by a factor K = 1 .
5. This value is in very goodagreement with the K factor found in [1] ( K = 1 .
8) using data on dσ/d ∆ φ , ∆ φ being the angle between D and D ∗− mesons produced at CDF within some definite cuts in rapidity and transverse momentum. The X s (1 ++ ) molecule should exist as a partner with strange light quarks of the X (3872). One could expectit to be a more compact molecule with respect to the X (3872) as η particle exchange forces would be at work.This enlarges the spread ∆ p in the relative momentum and naturally makes the ball R larger. We will postulatea binding energy for X s as small as that of the X (3872) and its mass is expected to be M X s = 4080 MeV. X s could decay into J/ψππ with a narrow width because of its mass and flavor content (it cannot decay into K + K − J/ψ (via φ ) because of phase space; it cannot either decay to J/ψf (980) because of quantum numbers),or in D s D s γ . Using the Herwig hadronization algorithm to compute σ ( k < Λ) in (2) we obtain σ ∗ ( p ¯ p → X s (4080)) = 1 ÷ Which corresponds to a k of the Gaussian at ∼
27 MeV and a spread of +12 MeV. D * H k <
300 MeV L x =
50 MeV x =
100 MeV < x relative to D or D * Σ H nb L FIG. 2: The cross section integrated in bins containing n = 0 , , , ... extra hadrons having a relative momentum k < x MeV with respect the D or the D ∗ composing the X (3872) molecule. Following [11] we assume that the moleculeis formed in S -wave with a relative k in the center of mass of D and D ∗ as large as 300 MeV. where the value of 3 nb is found pushing the Λ value up to 600 MeV (following some considerations on thepossible values of the Λ cutoff made in [11]). We obtain definitely similar results using Pythia [12].Such numbers should put the X s (4080) molecule in the conditions to be observed at CDF. We would findrather surprising that no such state is found assuming that the mechanism (2) is correct thus we encouragesearches of this resonance.On the other hand we cast some doubts on the possibility that final state interactions can indeed play sucha pivotal role as described in [11]. First of all we remind that Watson formulae [13] used in [11] are valid for S -wave scattering, whereas a relative three-momentum k of 300 MeV indicates that higher partial waves shouldbe taken into account.Most importantly, we have verified in our MC simulations that as the relative momentum k in the center ofmass of the molecule is taken to be up to 300 MeV, then other hadrons (on overage more than two) have arelative momentum k <
100 MeV with the D or the D ∗ constituting the molecule (see Fig. 2). On the otherhand the Migdal-Watson theorem for final state interactions requires that only two particles in the final stateparticipate to the strong interactions causing them to rescatter. In other words the extra hadrons involved inthe process do necessarily interfere in an unknown way with the mesons assumed to rescatter into an X (3872).This is particularly true as one further enlarges the dimensions of the momentum ball R as required in [11].Tetraquarks with a [ cs ][¯ c ¯ s ] might also occur, and one expects the lightest of this family to be a scalar at about3930 MeV, as estimated in [15]. Computing the prompt production cross section is an harder task though. Thiswould require some specific model for the fragmentation of partons into diquarks allowing to extract from dataa ratio of the production rate of [ cs ] and [ cq ] diquarks. In turn this would allow, for example, to estimatethe prompt production cross section of the X s under the hypothesis that the X (3872) produced at CDF is atetraquark. A simple model of parton to diquark fragmentation could be drawn along the lines discussed in [16]where the case of light diquarks was treated. Yet we prefer to postpone such estimate as soon as the first dataon exotic hadron production will be available from LHCb and ALICE.In this note we show that starting from the results discussed in [11] we should expect an enhancement in theprompt production cross section of an hypothetical new X s (4080) molecular loosely bound resonance constitutedby a D s ¯ D ∗ s pair. We estimate such cross section to be between 1 and 3 nb at the Tevatron. On the other handwe cast some doubts on the applicability of the Watson theorem for final state interactions in the calculation athand. We show that in the hadronization shower the number of hadrons in a momentum volume R ( k ) tends togrow with k whereas the final state interactions formulae used in [11] (see [13]) should involve only two hadronsat a time. Acknowledgments
We wish to thank Marco Rescigno for his indispensable hints on CDF data. The work of one of us (B.G.) issupported in part by the US Department of Energy under contract DE-FG03-97ER40546.3