Test of the X(3872) Structure in Antiproton-Nucleus Collisions
aa r X i v : . [ nu c l - t h ] D ec Test of the X (3872) Structure in Antiproton-NucleusCollisions
Alexei Larionov , Mark Strikman and Marcus Bleicher Frankfurt Institute for Advanced Studies (FIAS), D-60438 Frankfurt am Main, Germany National Research Centre ”Kurchatov Institute”, 123182 Moscow, Russia Pennsylvania State University, University Park, PA 16802, USA Institut f¨ur Theoretische Physik, J.W. Goethe-Universit¨at, D-60438 Frankfurt am Main, Germany a) Corresponding author: larionov@fias.uni-frankfurt.de b) [email protected] c) [email protected] Abstract.
The present day experimental data on the X (3872) decays do not allow to make clear conclusions on the dominatingstructure of this state. We discuss here an alternative way to study its structure by means of the two-step ¯ D ∗ (or D ) productionin ¯ pA reactions. If this process is mediated by X (3872), the characteristic narrow peaks of the ¯ D ∗ (or D ) distributions in the lightcone momentum fraction at small transverse momenta will appear. This would unambiguously signal the D ¯ D ∗ + c.c. molecularcomposition of the X (3872) state. Keywords : X (3872); hadronic molecule; D and D ∗ production; ¯ pA reactions. PACS : 14.40.Rt; 14.40.Lb; 25.43. + t Introduction
The c ¯ c containing X (3872) state (will be denoted below as “ X ” for brevity) has been discovered by BELLE [1] as apeak in π + π − J /ψ invariant mass spectrum from B ± → K ± π + π − J /ψ decays. The quantum numbers of X are J PC = ++ as determined by LHCb [2] based on angular correlations in the B + → K + X , X → π + π − J /ψ , J /ψ → µ + µ − decays.The structure of this state is nowadays under extensive discussions. The closeness of the X mass to the two-mesonthreshold D ¯ D ∗ , | m X − m D − m ¯ D ∗ | < X state [3, 4, 5, 6] boundby pion exchange potential . The size of such a molecule, i.e. the root-mean-square distance between components,can be estimated from a binding energy E b as q h r i ¯ DD ∗ ≃ √ a ∼ . − . , (1)where a = p µ E b is a range parameter, µ = m ¯ D m D ∗ / ( m ¯ D + m D ∗ ) is the reduced mass. The lower limit in (1) is obtainedfor the charged components, D − D ∗ + , with E b ≃ D ¯ D ∗ , with E b ≃ . D ∗ mass [7] based on CLEO data results in even smaller binding energy E b < . D ¯ D ∗ + c.c. molecule may be even larger.) Hence, if the X state has the predominant D ¯ D ∗ + c.c. molecular structure, it ismost likely to be a quite extended object with a size larger than the deuteron size. According to the recent theoreticalstudies [8, 9], the radiative decays X → γ J /ψ ( ψ ′ ) are weakly sensitive to the structure of X at large distances. Thedecay channel X → D ¯ D π is more a ff ected by wave function at large distances. However, the actual predictions ofthe model calculations [8] are still quite uncertain due to low energy constants and FSI e ff ects. We disregard the di ff erence between the D and ¯ D states (and similar for the D ∗ and other charmed mesons). Thus, the overbar is dropped inmany places below. IGURE 1.
A schematic view of the D ∗ production ( ¯ D stripping) process induced by the antiproton annihilation on a bound protonin a nuclear target to the X (3872) state assumed to be a ¯ DD ∗ molecule. In this work we further discuss the possibility to explore the structure of X (3872) by using antiproton-nucleusreactions proposed in our recent paper [10]. It is expected that X is strongly coupled to the ¯ pp channel [11] and, thus,can be produced in a ¯ pp → X exclusive reaction. In the case of a nuclear target, the produced X will propagate inthe nuclear residue and possibly experience the stripping reaction on a nucleon, as illustrated in Figure 1. Since therelative motion of the ¯ D and D ∗ in a molecule is slow, the outgoing D ∗ will propagate in a forward direction withmomentum ∼ p lab /
2. In terms of a light cone momentum fraction, α = ω D ∗ + k z ) E ¯ p + m p + p lab , (2)this corresponds to α ≃
1. Here, ω D ∗ ( k ) = ( k + m D ∗ ) / , E ¯ p = ( p + m p ) / , and z axis is chosen along the beammomentum. Model
In order to calculate the process of Fig. 1, we have to know the two main ingredients: the production rate ¯ pp → X ,and the cross section of the process X p → D ∗ .The molecule production rate (see Eq.(12) below) is proportional to the modulus squared of the matrix element.The latter can be expressed via detailed balance as | M X ; ¯ pp | = π (2 J X + m X Γ X → ¯ pp q m X − m p , (3)where an overline means summation over helicity of X and averaging over helicities of ¯ p and p . The partial decaywidth X → ¯ pp has been theoretically estimated in [11] to be Γ X → ¯ pp ≃
30 eV. p F p p k ′ D ∗ k D } ( a ) + p p k ′ D ∗ k ′ D } ( b ) + p p k ′ D ∗ k ′ D ( c ) } p X p F p F p X p X k D ∗ p ′ p k D ∗ FIGURE 2.
The amplitude for the process X (3872) + p → D ∗ + F where F ≡ {F , . . . , F n } is an arbitrary final state in the pD interaction. Wavy lines denote the elastic scattering amplitudes. Straight lines are labelled with particle’s four-momenta. The blobrepresents the wave function of the molecule. The amplitude of the D -stripping process with arbitrary final states is shown in Fig. 2. We take into account theimpulse approximation (IA) graph (a) and the graphs where either incoming (b) or outgoing (c) proton (or the mostenergetic forward product of the inelastic pD interaction) rescatters elastically on the D ∗ meson. The di ff erential crossection of D ∗ production due to the D stripping from the molecule X in the collision with a proton can be written inthe molecule rest frame as d σ pX → D ∗ d k = σ tot pD I pD ( − k ) | ψ ( k ) | κ , (4)where σ tot pD is the total pD interaction cross section, I pD ( k ) = [( E p ω D − p p k z ) − ( m p m D ) ] / p p ω D (5)is the Moeller flux factor (normalized to 1 for D at rest), ψ ( k ) is the wave function of the molecule. κ is a factor takinginto account the screening and antiscreening corrections: κ = − σ tot pD ∗ I pD ∗ ( k ) Z d q t (2 π ) ψ ∗ ( k + q t ) ψ ∗ ( k ) e − ( B pD + B pD ∗ ) q t / + ( σ tot pD ∗ I pD ∗ ( k )) Z d q t d q ′ t (2 π ) ψ ( k + q t ) ψ ∗ ( k + q ′ t ) | ψ ( k ) | e − [ B pD ∗ ( q t + q ′ t ) + B pD ( q ′ t − q t ) ] / , (6)where we used the expression for the elementary pD elastic scattering amplitude M pD ( q t ) = ip p ω D I pD ( k D ) σ tot pD e − B pD q t / , (7)with q t being the transverse momentum transfer. (Expressions for for the flux factor I pD ∗ ( k ) and for the amplitude M pD ∗ ( q t ) of pD ∗ scattering are given by Eqs.(5),(7) with replacement D → D ∗ .)In the summation over final states F we used the unitarity relation [12]: M fi − M ∗ i f = X F d p F E F (2 π ) · · · d p F n E F n (2 π ) i (2 π ) δ (4) ( p F − p f ) M ∗F f M F i , (8)where ’ i ’ and ’ f ’ are the elastic scattering states of the pD system. In the impulse approximation κ = k t < ∼ . / c. The second (negative) term in Eq.(6) is the screeningcorrection due to the interference of the IA amplitude (a) with the rescattering amplitudes (b) and (c) of Fig. 2. Thethird (positive) term is the antiscreening correction due to the modulus squared of the sum of (b) and (c) amplitudes.The total cross sections of pD and pD ∗ interactions are estimated as σ tot pD ≃ σ tot pD ∗ ≃ σ tot π + p ( p lab / / ≃
14 mbbased on the color dipole model and comparison of the mesonic radii. (Here, p lab = / c is the antiproton beammomentum for the on-shell X production in the ¯ pp → X process.) The slope parameters of the pD and pD ∗ scatteringare estimated as B pD ≃ B pD ∗ ≃ B pK + with B pK + = − as follows from the comparison of the radii of the D , D ∗ and K mesons [10]. The total X p cross section is close to the sum of the pD and pD ∗ cross sections with a screeningcorrection depending on the molecule wave function. In calculations, we use σ tot Xp =
26 (23) mb for the D ¯ D ∗ ( D + D ∗− )component [10].For the molecule wave function we adopt the asymptotic solution of a Schroedinger equation at large distances, ψ ( k ) = a / /π a + k , (9)normalized as R d k | ψ ( k ) | =
1. The molecule composition is given by 86% of the D ¯ D ∗ + c.c. contribution, 12%of the D + D ∗− + c.c. contribution, and 2% of the D + s D ∗− s + c.c. contribution, as it follows from the local hidden gaugecalculations [13]. We neglect the small D + s D ∗− s + c.c. component in calculations.In order to calculate the di ff erential cross section of D ∗ ( D ) production in ¯ pA interactions we apply the generalizedeikonal approximation [14, 15]. This method is based on the Feynman graph representation of the multiple scatteringprocess and on the three assumptions: nonrelativistic motion of nucleons in the initial and final nuclei; no energytransfer in the multiple soft scatterings; no longitudinal momentum transfer in elementary amplitudes. By keeping theleading order (absorptive) term in the scattering expansion, i.e. neglecting the product terms in the matrix elementquared with the same nucleons-scatterers in the direct and conjugated matrix elements, we obtain the Glauber-typeexpression for the di ff erential cross section: α d σ ¯ pA → D ∗ d α d k t = v − p Z d r e − σ tot¯ pN z R −∞ dz ρ ( b , z ) Z d p t d Γ → X ¯ p ( r ) d p t G p → D ∗ X ( α, k t − α p t ) × ∞ Z z dz e − σ tot XN z R z dz ρ ( b , z ) ρ ( b , z ) e − σ tot D ∗ N ∞ R z dz ρ ( b , z ) . (10)Here, G p → D ∗ X ( α, k t ) ≡ ω D ∗ d σ Xp → D ∗ d k = α d σ Xp → D ∗ d α d k t (11)is the invariant cross section of D ∗ production (or D -stripping), d Γ → X ¯ p ( r ) d p t = | M X ; ¯ p | v ¯ p (2 π ) p E n p ( r ; p t , ∆ m X ) (12)is the in-medium width of ¯ p with respect to the production of X with transverse momentum p t , v ¯ p = p lab / E ¯ p is theantiproton velocity, ∆ m X = m p + E + E ¯ p E − m X p lab (13)is the longitudinal momentum of the struck proton obtained from the condition that the produced X is on the massshell, i.e. ∆ m X = p z , ( p ¯ p + p ) = m X . The quantity n p ( r ; p t , ∆ m X ) in Eq.(12) is the proton phase space occupationnumber. We apply a model where the local Fermi distribution is complemented with a high-momentum tail due to theshort range proton-neutron correlations [16]: n p ( r ; p ) = (1 − P ) Θ ( p F − p ) + π P ρ p | ψ d ( p ) | Θ ( p − p F ) ∞ R p F d p ′ p ′ | ψ d ( p ′ ) | , (14)where p F ( r ) = (3 π ρ p ( r )) / is the local Fermi momentum of protons, ρ p ( r ) is the proton density, P ≃ .
25 is theproton fraction above Fermi surface, and ψ d ( p ) is the deuteron wave function. Results
Figure 3 shows the invariant di ff erential cross section of D ∗ and D production (10) as a function of the light conemomentum fraction α defined by Eq.(2) at the two di ff erent values of the transverse momentum. At k t =
0, the crosssection reveals sharp peaks at α = m ∗ D / m X = .
04 for D ∗ and α = m D / m X = .
96 for D . The peaks are much higherand narrower for D ∗ and D as compared to D ∗± and D ± . This is due to larger probability to find the charge neutral D ¯ D ∗ + c.c. configuration in the molecule and due to its smaller binding energy. With increasing transverse momentumthe peaks gradually become smeared. It is, therefore, important that the transverse momentum of the outgoing D ∗ ( D )is small enough, k t < ∼ . / c, in order the stripping signal to be visible.The major background for the X -mediated D ∗ (or D ) production is given by the direct process ¯ pN → D ¯ D ∗ + c.c.on the bound nucleon. The cross section of the ¯ pp → D ∗ ¯ D process has been estimated in [11] from dimensionalcounting considerations based on the measured ¯ pp → K ∗− K + cross section. Using the result of ref. [11] as an input,we have calculated the background cross section of D ∗ production. As one can see from Fig. 4, the background crosssection is much broader distributed in α than the signal, i.e. the X -mediated cross section.The binding energy of the molecule is the most crucial parameter which strongly influences the height and thewidth of the α -distribution for the signal cross section. This is also quantified in Fig. 4, where the calculations areshown for the three di ff erent values of the molecule binding energy. We observe that such a small binding energy like E b ∼ . ∆ α ∼ . .00.51.01.50.5 1.0 1.5 α d σ / d α d k t ( nb c / G e V ) α p-(7 GeV/c) + Ar → D*(D)k t =0 GeV/cD* /100D /100D* ± D ± α d σ / d α d k t ( nb c / G e V ) α k t =0.3 GeV/cD* D D* ± D ± FIGURE 3.
The invariant di ff erential cross sections of D ∗ , D , D ∗± and D ± production in ¯ p Ar collisions at p lab = / c vslight cone momentum fraction α at k t = k t = . / c (right panel). For k t =
0, the cross sections of D ∗ and D production are divided by a factor of 100. α d σ / d α d k t ( nb c / G e V ) α p-(7 GeV/c) + Ar → D* , k t =0 GeV/cE b =0.2 MeVE b =0.5 MeVE b =1 MeVbgr./3 0 50 100 150 200 1.0 1.1 FIGURE 4.
The α -dependence of D ∗ production at k t = p Ar collisions at p lab = / c. The signal cross section due to D -stripping from the intermediate X is shown for the di ff erent binding energies, E b , of the D ¯ D ∗ molecule. The background crosssection is divided by a factor of 3. The inset shows a narrower region of α . The X (3872) state is the lightest exotic c ¯ c state. There are several exotic states containing a c ¯ c pair which are notfit in the charmonium systematics, e.g. charge-neutral ones, X (3940), Y (4140), X (4160), Y (4260), Y (4360), and thecharged ones, Z c (3900) , Z c (4020) (cf. [17, 18, 19]). The charged states are likely to be the compact tetraquarks [19].However, the neutral ones have possible molecular structures which can also be tested in ¯ pA reactions in a similarway as X (3872). In particular, the 1 −− state Y (4360) may be the bound state of the D ∗ ¯ D + c.c. with a binding energyof 67 MeV [19]. In this case, the α -distribution of the D ∗ and D at k t = Γ Y (4360) → ¯ pp / Γ tot Y (4360) = − , with the total width Γ tot Y (4360) = ff erence of D ∗ and D mesons is large, ∼
414 MeV, the peaks of D ∗ and D distributions in α are well separated. Due to the large binding energy of Y (4360) state, the peaks are much smoother than in the caseof X (3872). Assuming the same shape of the α -dependence of the background as for X (3872) such peaks would bevisible as the bumps in the di ff erential production cross section of D ∗ (at α ≃ .
9) and D (at α ≃ .
1) at k t = Conclusions and outlook
We have demonstrated that the possible D ¯ D ∗ + c.c. molecular structure of X (3872) manifests itself in the sharp peaksof exclusive D ∗ or D production at α ≃ X on the nucleon and are well visible on the smooth background dueto the direct production of charmed mesons in ¯ pN collision. α d σ / d α d k t ( nb c / G e V ) α p-(9.2 GeV/c) + Ar → D (D *0 ), k t =0 GeV/cD D* FIGURE 5.
The α -dependence of D ∗ + ¯ D ∗ and D + ¯ D production at k t = p Ar collisions at p lab = . / c due to thestripping reaction with intermediate Y (4360) state. Other possible structures of X , e.g. charmonium, tetraquark or c ¯ c -gluon hybrid, should produce more flat α -distributions of D ∗ and D due to more violent production mechanisms in XN collisions. Most likely, in these casesthe charmed mesons will be uniformly distributed in the available phase space volume in the XN center-of-massframe. Thus, the proposed observable, i.e. the light cone momentum fraction distributions of D ∗ and D at small k t , should be very sensitive to the hypothetical molecular structure of X state and, probably, of the other exotic c ¯ c candidates. Similar processes can be considered to investigate the possible molecular structures of other hadrons. Forexample, the assumed K ¯ K molecule composition of a (980) and f (980) mesons could be tested in a two-step process γ ( π ) N → f N , f N → ¯ K ( K ) + anything ( f ≡ a (980) , f (980)).The experimental studies of such processes are possible at PANDA, J-PARC, JLab and COMPASS. ACKNOWLEDGMENTS
The work of AL has been financially supported by HIC for FAIR within the framework of the LOEWE program.
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