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Determination of the Structure of the X(3872) in P̄a Collisions

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Determination of the Structure of the X(3872) in P̄a Collisions Physics Letters B 749 (2015) 35–43 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Determination of the structure of the X(3872) in pA¯ collisions ∗ A.B. Larionov a,b, , M. Strikman c, M. Bleicher a,d a Frankfurt Institute for Advanced Studies (FIAS), D-60438 Frankfurt am Main, Germany b National Research Centre “Kurchatov Institute”, 123182 Moscow, Russia c Pennsylvania State University, University Park, PA 16802, USA d Institut für Theoretische Physik, J.W. Goethe-Universität, D-60438 Frankfurt am Main, Germany a r t i c l e i n f o a b s t r a c t Article history: Currently, the structure of the X(3872) meson is unknown. Different competing models of the cc¯ Received 11 February 2015 exotic state X(3872) exist, including the possibilities that this state is either a mesonic molecule with 0 ¯ ∗0 Received in revised form 29 May 2015 dominating D D + c.c. composition, a ccq¯ q¯ tetraquark, or a cc¯-gluon hybrid state. It is expected that Accepted 19 July 2015 the X(3872) state is rather strongly coupled to the pp¯ channel and, therefore, can be produced in pp¯ and Available online 22 July 2015 pA¯ collisions at PANDA. We propose to test the hypothetical molecular structure of X(3872) by studying Editor: V. Metag ¯ ∗ the D or D stripping reactions on a nuclear residue. Keywords: © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license 3 X(3872) (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . pA¯ reactions Charmed meson production 1. Introduction cant pure cc¯ admixture in the X(3872) [5]. The theoretical pre- dictions for the decay rates are, however, quite sensitive to the The discovery of exotic cc¯ mesons at B-factories and at the model details even within various approaches like charmonium or ¯ ∗ Tevatron stimulated interest to explore the possible existence of D D + c.c. molecular models. tetraquark and molecular meson states. The famous X(3872) state In this letter we suggest to test the charm meson molecular + − has been originally found by BELLE [1] as a peak in π π J/ψ hypothesis of the X(3872) structure in pA¯ collisions at PANDA. ± ± + − invariant mass spectrum from exclusive B → K π π J/ψ de- Assuming that the X(3872) is coupled to the pp¯ channel, we con- cays. Nowadays the existence of the X(3872) state and its quantum sider the stripping reaction of the D-meson on a nuclear target ++ ∗ numbers J PC = 1 are well established [2]. In particular, radia- nucleon such that a D¯ is produced and vice versa. We show that → → tive decays X(3872) J/ψγ , X(3872) ψ (2S)γ [3] point to the distribution of the produced charmed meson in the light cone the positive C-parity of the X(3872). Probably the most intrigu- momentum fraction α with z-axis along p¯ beam momentum, ing feature is that the mass of the X(3872) is within 1MeV the ∗ sum of the D0 and D 0 meson masses. This prompted the popular z ¯ ∗ + ¯ ∗ 2(ωD∗ (kD∗ ) + k ∗ ) conception of the X(3872) being a D D DD molecule. α = D , (1) To probe the molecular nature of the X(3872) structure has E p¯ + mN + plab been difficult. So far, most theoretical calculations have been fo- cused on the description of radiative and isospin-violating decays will be sharply peaked at α 1at small transverse momenta of the X(3872). For example, the X(3872) → J/ψγ decay can which allows to unambiguously identify the weakly coupled ¯ ∗ ∗ be well understood within the D D + c.c. molecular hypothe- ¯ + ∗ ∗ ∗ = 2 + 2 1/2 D D c.c. molecule. Here, kD and ωD (kD ) (kD∗ mD∗ ) ∗ sis [4]. On the other hand, the measured large branching frac- are, respectively, the momentum and energy of the produced D¯ → → = ± tion B(X(3872) ψ (2S)γ )/B(X(3872) J/ψγ ) 3.4 1.4 [3] meson in the target nucleus rest frame. Similar studies of hadron–, seems to disfavor the molecular structure and requires a signifi- lepton–, and nucleus–deuteron interactions at high energy have been proposed long ago to test the deuteron structure at short distances as in the spectator kinematics the n-or p-stripping cross Corresponding author. * sections are proportional to the square of the deuteron wave func- E-mail addresses: larionov@fias.uni-frankfurt.de (A.B. Larionov), [email protected] (M. Strikman), [email protected] tion. For the X(3872) this idea is depicted in Fig. 1 (details follow (M. Bleicher). below). http://dx.doi.org/10.1016/j.physletb.2015.07.045 0370-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 36 A.B. Larionov et al. / Physics Letters B 749 (2015) 35–43 3 3 d kd q imX ∗ iM(4)(0) = ψ (k − q) 3 (2π) 2ωD ωD∗ iMpD∗ (qt )iMpD(−qt ) × ψ(k). (6) z 2pp(−q + iε) Therefore, iM(3)(0) + iM(4)(0) 3 2 d kd qt mX ∗ = ψ (k − q )iM ∗ (q ) 2 t pD t (2π) 4ωD ωD∗ pp × iMpD(−qt )ψ(k). (7) The optical theorem for the proton–molecule forward scattering amplitude is = tot ImM(0) 2ppmX σpX . (8) Fig. 1. Processes contributing to the forward scattering amplitude of a proton on = (1) + (2) + (3) + (4) ∗ ∗ Substituting M(0) M (0) M (0) M (0) M (0) and us- the DD molecule. Wavy lines denote the pD and pD elastic scattering ampli- ing the parameterization of the strong interaction scattering am- tudes. Straight lines are labeled with particle’s four-momenta. The blobs represent the wave function of the molecule. plitudes in the usual form as − 2 tot B (∗) qt /2 M (∗) (q ) = 2iI (∗) (k (∗) ) ∗ e pD , (9) 2. X(3872)-proton cross section pD t pD D σpD( ) ∗ z 2 2 1/2 with I (∗) (k (∗) ) =[(E ω (∗) − p k ∗ ) − (m m (∗) ) ] being For brevity, the bar, which can be seen over the D or D, will pD D p D p D( ) p D be dropped in many cases below. The charge conjugated states are the Moeller flux factor we obtain the following expression for the implicitly included in the calculated cross sections. proton–molecule total cross section: The most important ingredients of our calculations are the to- tot = 3 | |2[I − tot + I ∗ tot ] tal Xp cross section and the momentum differential cross section σpX d k ψ(k) pD( k)σpD pD (k)σpD∗ ∗ Xp → D (D) + anything. In the molecular picture, the latter cross ∗ 1 section is the D(D )-meson stripping cross section. To calculate − 3 I − totI ∗ tot d kψ(k) pD( k)σpD pD (k)σpD∗ the total Xp cross section within the Glauber theory, we start from 2 ∗ the graphs shown in Fig. 1 which assume the DD composition 2 d qt ∗ − ∗ + 2 ∗ (B pD B pD)q /2 of X(3872). It is convenient to perform calculations in the DD × ψ (k − qt )e t , (10) (2π)2 molecule center-of-mass (c.m.) frame with proton momentum pp I ≡ directed along z-axis. The invariant forward scattering amplitudes where the normalized flux factors are defined as pD(∗) (k) of the first two processes are I (∗) (k)/p ω (∗) . In the small binding energy limit the molecule pD p D wave function decreases rapidly with increasing momentum k and mX iM(1)(0) = d3k |ψ(k)|2iM (0), (2) −1/2 pD becomes negligibly small at k B pD . In this case one can set ωD B pD = B pD∗ = 0 and perform the Taylor expansion of the flux fac- mX z (2) 3 2 ∗ tors in k in Eq. (10). Then, for the S-state molecule with accuracy iM (0) = d k |ψ(k)| iMpD (0), (3) ω ∗ z ∗ D up to the linear terms in k /mD and assuming that mD mD , tot tot ∗ where mX = ωD + ωD∗ is the mass of the molecule and ωD (ωD∗ ) σpD σpD we obtain the formula ∗ is the energy of D (D )-meson. (The different assumptions on the tot tot σ ∗ σ momentum dependence of meson energies discussed in the next tot tot tot pD pD −2 σ = σ ∗ + σ − r DD∗ , (11) section have practically no effect on the Xp cross section.) The pX pD pD 4π molecule wave function in momentum space is defined as in line with previous calculations of the proton–deuteron total d3r cross section [8]. = −ikr ∗ ψ(k) e ψ(r), (4) We choose the wave function of a DD molecule as the asymp- (2π)3/2 ∗ ∗ totic solution of the Schroedinger equation at large distances: where k is the D momentum in the DD c.m. frame, with the − normalization condition d3k|ψ(k)|2 = 1. κ e κr ψ(r) = , (12) For the calculation of the third and forth processes in Fig. 1 we 2π r apply the generalized eikonal approximation (GEA) [6,7] which as- √ ∗ = sumes the nonrelativistic motion of D and D inside the molecule. where the range parameter κ 2μEb depends on the reduced = ∗ + ∗ In this approximation, the propagator of the intermediate pro- mass μ mDmD /(mD mD ) and on the binding energy E B of ton depends only on the z-component of momentum transfer the molecule. The corresponding momentum space wave function ∗ ≡ ∗ − is q kD kD∗ , while the pD and pD elastic scattering ampli- tudes depend only on the momenta of incoming particles and on κ1/2/π the transverse momentum transfer.
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