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

Physics Letters B 749 (2015) 35–43

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Physics Letters B

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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 ∗ Received in revised form 29 May 2015 dominating D0 D¯ 0 + 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 X(3872) (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 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 ∗ section is the D(D )-meson stripping cross section. To calculate 1 3 tot tot − d kψ(k)IpD(−k)σ IpD∗ (k)σ ∗ the total Xp cross section within the Glauber theory, we start from 2 pD pD ∗ 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 amplitudes depend only on the momenta of incoming particles and on κ1/2/π the transverse momentum transfer. Thus, we obtain ψ(k) = . (13) κ2 + k2 3 3 d kd q imX ∗ iM(3)(0) = ψ (k − q) Let us now discuss the input parameters of our model. Since 3 ∗ (2π) 2ωD ωD∗ there is no experimental information on Dp and D p interac- iMpD∗ (qt )iMpD(−qt ) tions, we rely on simple estimates in the high-energy limit. For × ψ(k), (5) z small-size qq¯ configurations the color dipole model predicts the 2pp(q + iε) A.B. Larionov et al. / Physics Letters B 749 (2015) 35–43 37

∗ Fig. 2. The amplitude for the process X(3872) + p → D + F where F ≡{F1,...,Fn} is an arbitrary ﬁnal state in the pD interaction. See also caption to Fig. 1.

∗ scaling of the total meson–nucleon cross section with the average 3. D(D¯ ) stripping cross section square of the transverse distance between quark and antiquark in the meson, which is proportional to the square of the Bohr ra- In high energy hadron–deuteron reactions, the main contribu- dius rB = 3/4μαs. Here, μ = mqmq¯ /(mq + mq¯ ) is the reduced mass tion to the fast backward nucleon production (in the deuteron rest with mq and mq¯ being the constituent quark and antiquark masses. frame or equivalently – fast forward in the deuteron projectile The Bohr radii of pion, kaon, D-meson and J/ψ are ordered as case) is given by the inelastic interaction of the hadron with sec- rBπ > rBK > rBD > rBJ/ψ . Hence, we expect that the total meson– ond nucleon of the deuteron [14]. For large nucleon momenta the nucleon cross sections follow the same order. At a beam momen- spectrum is modified as compared to the impulse approximation tum of 3.5 GeV/c (1/2of the momentum of X(3872) formed in (IA) due to the Glauber screening and antiscreening corrections + the pp¯ → X process on the proton at rest) the total π p and [15] since the hadron may interact with both nucleons. In a simi- + + → ∗ + K p cross sections are about 28 mb and 17 mb, respectively [2]. lar way, in calculations of the cross section X p D anything, The J/ψ p cross section is expected to be much smaller, 3.5–6 mb we take into account the IA diagram (Fig. 2a) and the single- (cf. [9] and refs. therein.). We assume the total Dp cross section rescattering diagrams of the incoming proton (Fig. 2b) and of the tot = + outgoing proton or of the most energetic forward going baryon σpD 14 mb, i.e. slightly below the K p total cross section. This choice is in reasonable agreement with effective field theory cal- emerging from the inelastic pD interaction (Fig. 2c). The expres- culations [10]. sions for the invariant matrix elements for the processes (a) and It is well known that at incident energies of a few GeV, (b) in Fig. 2 are straightforward to obtain in the c.m. frame of the molecule state X: the amplitude of meson (nucleon) – nucleon elastic scattering is (to a good approximation) proportional to the product of the ∗ (a) 2mX ωD 3/2 electric form factors of the colliding hadrons (see e.g. [11] and M = (2π) MF;pDψ(k), (14) ω refs. therein). Thus, in the exponential approximation for the D 1/2 t-dependence of the form factors, the slope parameters B pM of (b) imX 3 the transverse momentum dependence of the meson–proton cross M = √ d rψ(r) (−z) ∗ 2 + 2 2p p 2ωD ωD section at small t should be proportional to r p r M , where 2 2 2 r p and r M are the mean-squared charge radii of the proton d qt −i(k+q )r × e t MF; M ∗ (q ), (15) 2 ∝ 2 2 p D pD t and meson, respectively. Since r M rBM, the slope parame- (2π) ters should be also ordered as the Bohr radii. Empirical values (b) − ≡ ∗ = + = ± 2 + = where k k . In the case of M we applied the GEA by ex- at plab 3.65 GeV/c are B pπ 6.75 0.12 GeV and B pK D − pressing the propagator of the intermediate proton in the eikonal 4.12 ± 0.12 GeV 2 as fitted at 0.05 ≤−t ≤ 0.44 GeV2 [12]. On the − form and using the coordinate representation with r = r ∗ − r . other hand, B = 3GeV 2 at the comparable beam momenta D D pJ/ψ The explicit form of the amplitude M(c) can be written only for = −2 [11]. We will assume the value B pD 4GeV , since the Bohr specific outgoing states F. However, for the diffractive states in- ∼ ∗ radii of kaon and D-meson differ by 30% only. For the pD including the leading proton, the expression for M(c) can be obtained tot tot ∗ = ∗ = (b) teraction we assume for simplicity σpD σpD and B pD B pD. from the expression for M by replacing (−z) → (z), which ∗ ∗ Our educated guess on the D- and D -meson–nucleon cross reflects the change of the time order of the pD and pD interac- sections and slope parameters should of course be checked exper- tions. Thus, for the diffractive outgoing state F the expression for tot (b) (c) imentally. The empirical information on σpD can be obtained by M + M is given by Eq. (15) with replacement (−z) → 1(ne- measuring the A-dependence of the transparency ratio of D-meson glecting small differences in momenta of incoming and outgoing production in pA¯ reactions at beam momenta beyond the charmo- proton in elementary amplitudes). We assume that the same re- ¯ nium resonance peaks, where the background pp¯ → DD channel placement can be done for any final state F. By summing over all ∗ dominates. The slope parameter B pD can be addressed by measur- states F we then obtain the momentum differential D production ing the transverse momentum spread of D-meson production in (i.e. D-stripping) cross section in the molecule rest frame: ¯ pA reactions. 3 d k ∗ We will further assume that the X(3872) wave function con- ∗ D dσpX→D = 0 ¯ ∗0 + + ∗− + (2 )32 ∗ 4p m tains 86% of D D c.c. and 12% of the D D c.c. component π ωD p X as predicted by the local hidden gauge approach [4]. The bind- ∗ × | (a) + (b) + (c)|2 ing energy of D0 D¯ 0 is likely less than 1MeV [13] and cannot M M M be determined from existing data [2] accurately enough. We set spins and sorts of F 0 ¯ ∗0 + ∗− E D D = 0.5 MeV and E D D = 8MeVin numerical calculations. × 4 (4) + − − b b (2π) δ (pF kD∗ p p p X ) = −1 This corresponds to the range parameters κD0 D¯ ∗0 0.16 fm and 3 3 − d pF1 d pFn + ∗− = 1 × ··· κD D 0.64 fm . With these parameters the total pX cross , (16) ∗ + ∗− 2 32EF 2 32EF tot = 0 ¯ 0 ( π) 1 ( π) n section (10) is σpX 26 and 23 mb for D D and D D com- 2 = 2 ponents, respectively, at the molecule momentum of 7GeV/c in where p X is the four momentum of the molecule (p X mX ). With the proton rest frame. a help of the unitarity relation for the elementary amplitudes [16] 38 A.B. Larionov et al. / Physics Letters B 749 (2015) 35–43

∗0 Fig. 3. The invariant differential cross section of D production in X(3872)p collisions at plab = 7GeV/c. Thick solid line – full calculation according to Eqs. (17)–(19). Thin solid line – the calculation taking into account only IA and screening term of Eq. (18). Dashed line – the calculation with κ = 1in Eq. (17), i.e. only with the IA term. The inset at kt = 0shows the behavior of the differential cross section for a smaller range of α.

3 ∗ 3 ∗ ∗ the sum over spin states and sorts of F and the integration over d σpX→D d σpX→D p→D ω ∗ = α ≡ G (α, k ), (19) D 3 2 X t phase space volume can be reduced to the products of the imag- d k dαd kt inary parts of elastic scattering amplitudes. This leads to the fol- z where α = 2(ωD∗ (k) − k )/mX . Figs. 3 and 4 show the differen- ∗ ∗± lowing expression for the momentum differential cross section in tial cross section of D 0 and D production from X(3872) col- the molecule rest frame: lisions at 7GeV/c with proton at rest as a function of α for several values of transverse momentum kt . At kt = 0, the cross 3 ∗ d σpX→D ∗ tot 2 section has a sharp maximum at α 2mD /mX 1.04 and is al- = σ IpD(−k)|ψ(k)| κ , (17) d3k pD most unaffected by the screening and antiscreening corrections. 2 ∗ With increasing kt , the width of α-distribution increases while d qt ψ (k + qt ) − + ∗ 2 tot (B pD B pD )q /2 κ = 1 − σ ∗ IpD∗ (k) e t the screening and antiscreening corrections to the IA term be- pD (2π)2 ψ∗(k) come important. This is expected since the large-kt component tot 2 ∗ ( ∗ I ∗ (k)) 2 2 + + of the molecule wave function corresponds to small transverse σpD pD d qtd q ψ(k qt )ψ (k q ) ∗ + t t separation between D and D . The corrections become large for 4 2 4 | k |2 ( π) ψ( ) α 1 and large transverse momenta as can be directly seen from −[B ∗ (q2+q 2)+B (q −q )2]/2 the structure of the integrands in Eq. (18). Indeed, 1 cor- × e pD t t pD t t . (18) α responds to kz 0in the molecule rest frame. Then at ﬁnite ∗ + ∗ The ﬁrst term in the r.h.s. of Eq. (18) is the pure IA contribu- transverse momentum transfer qt the ratio ψ (kt qt )/ψ (kt ) is = tion. The second and third terms are, respectively, the screening less than unity at kt 0 and asymptotically tends to unity with and antiscreening corrections (see Eqs. (8a) and (8b) in [15]). growing kt . Due to the extremely narrow wave function of the ∗ 0 ∗0 The D meson is assumed to be on its vacuum mass shell, D D molecule in momentum space, the screening and antis- 2 creening corrections are sharply peaked at α 1.1 and develop ω ∗ (k) = m ∗ + k2, while the energy of the D meson is cal- D D structures in the α-dependence of the cross section at large trans- − = − ∗ culated from energy conservation, ωD ( k) mX ωD (k). (The verse momenta. In the case of pA¯ reactions these structures are condition ωD > 0 constrains the maximum momentum of the ∗ slightly smeared out due to the nucleon Fermi motion (see Fig. 6 emitted D , k < 3.3GeV/c. Above this value our model looses its below). applicability.) In the case of the D-meson production one has to I ↔ I ∗ tot ↔ tot ↔ ∗ ∗ exchange pD pD , σpD σpD∗ and B pD B pD in Eqs. (17), 4. D and D production off nucleus (18). In this case the on-shell condition is applied to the D-meson, ∗ ∗ while the D energy is determined by energy conservation. In antiproton–nucleus interactions, we focus on the D (or D) ∗ It is convenient to express the differential invariant D produc- meson production in the two-step process pp¯ → X, XN → ∗ tion cross section (17) in terms of the relative fraction α of the D (D) + anything. Similar to the case of Xp interactions, we ap- ∗ ∗ light cone momentum of the DD molecule carried by the D : ply the Glauber theory to calculate the differential cross sections A.B. Larionov et al. / Physics Letters B 749 (2015) 35–43 39

∗± Fig. 4. Same as Fig. 3 but for D production.

∗ ∗ Fig. 5. The amplitude of the process pA¯ → D F(A − 2) . The wave functions of the initial and ﬁnal nuclei are denoted as ψA and ψA−2, respectively. The mass numbers are shown as subscripts. Wavy lines represent elastic scattering amplitudes on nucleons. “F” stands for the arbitrary ﬁnal state particles in the semi-inclusive process ∗ ∗ X2 → D F. The summation is performed over all possible sets of nucleon scatterers {n1}, {n2} and {n3} for the p¯ , X and D , respectively.

∗ of the D production in antiproton–nucleus interactions. We start lowing expression for the momentum differential cross section of ∗ from the multiple scattering diagram shown in Fig. 5 which can be D production on the nucleus: evaluated within the GEA. We will assume that the nucleus can be 3 ∗ d σpA¯ →D −1 described within the independent particle model disregarding the α = v d3r P ¯ (b , −∞, z ) 2 p¯ 1 p,surv 1 1 c.m. motion corrections (cf. [17]). The incoming antiproton, inter- dαd kt ∗ 2 1→X d ∗ mediate molecular state X and outgoing D -meson are allowed to p¯ p→D α × d2 p G (α, k − p ) rescatter on nucleons elastically an arbitrary number of times. The 1t 2 X t 1t ∗ d p1t 2 D production cross section is proportional to the product of the ∞ sum of the amplitudes of Fig. 5 and their conjugated. The X state ∗ × dz P (b , z , z )ρ(b , z ) is formed on a proton 1, while the D is produced in the collision 2 X,surv 1 1 2 1 2 of X with a nucleon 2. The nucleons 1 and 2 are fixed in the direct z1 and conjugated amplitudes while the sets of other nucleon scat- × PD∗,surv(b1, z2, ∞), (20) terers are arbitrary. The leading order contribution is given by the where product term without elastic rescatterings. Nuclear absorption cor- → d2 1 X | |2 rections are accounted for by summing all possible product terms p¯ M X;p¯ 1 v p¯ 0 = np(r1; p1t , ) (21) with non-overlapping sets of nucleon scatterers. This gives the fol- d2 p 2 2 mX 1t (2π) 4plab E1 40 A.B. Larionov et al. / Physics Letters B 749 (2015) 35–43 is the in-medium width of p¯ with respect to production of X with where the overline means averaging over antiproton and proton transverse momentum p1t ; v p¯ = plab/E p¯ is the antiproton velocity; helicities and summation over the helicity of X. There is no ex- ; 0 = − ¯ np(r1 p1t , m ) is the proton occupation number; E1 mN B perimental data on the partial decay width X(3872)→pp. (The re- X ¯ + → ¯ + with B = 8.6MeVbeing the nucleon binding energy in 40Ar nu- cent LHCb data on pp invariant mass spectra from B ppK ¯ cleus. The longitudinal momentum 0 of the proton 1 is ob- decays [21] do not allow to clearly identify X(3872) in the pp mX tained from the condition of on-shell production of the state X decay channel due to statistical limitations.) In the present calcula- in the process p¯ 1 → X: tions, we will use the value X(3872)→pp¯ 30 eV as suggested by theoretical estimates [22]. This value is about two times smaller 2 + 2 + − 2 → ¯ m E 2E p¯ E1 m than χc1(1P) pp. However, one should note that, in the molec- 0 = N 1 X . (22) ¯ mX ular picture, the decay of the X(3872) to the pp state requires 2plab the production of only two qq¯ pairs, and not three qq¯ pairs as in The nucleon occupation numbers are taken as the depleted Fermi the ordinary charmonium decay to the pp¯ channel. Thus, the par- distributions supplemented by high-momentum tail due to short- tial decay width of the X(3872) into the pp¯ channel may be even range quasideuteron correlations (SRCs) [18–20]: larger than that of the χc1(1P) state [22]. Formula (20) has a simple physical interpretation if we express 3 2 nq(r; p) = (1 − P2,q) (p F ,q − p) the integral d r1 as d b1 dz1. The factor Pp¯ ,surv(b1, −∞, z1) =−∞ 2 | |2 − is the probability that the incoming from z antiproton with π P2,qρq ψd(p) (p p F ,q) = + ∞ , q = p,n (23) impact parameter b1 will reach the point z z1. The combination 2 2 1→X 2 2 2 (dz1/v p¯ )d p1td ¯ /d p1t is the X(3872) formation probability dp p |ψd(p )| p 2 p F ,q within the transverse momentum element d p1t when the p¯ is passing the longitudinal element dz1. The factor PX,surv(b1, z1, z2) where p (r) =[3π 2ρ (r)]1/3 are the nucleon Fermi momenta, F ,q q is the probability that the molecule will reach the point z = z2. → ∗ P2,p = 0.25 and P2,n = P2,p Z/N are the proton and neutron frac- 2 p D α The combination dz2(dα/α)d kt G (α, kt − p1t )ρ(b1, z2) is tions above the Fermi surface, ρ (r) are the nucleon densities, and ∗ X 2 q the probability that a D will be produced in the kinematical ψd(p) is the deuteron wave function. In Eq. (20), the nuclear ab- 2 element dαd kt when the X(3872) is passing the longitudinal el- sorption is given by the survival probabilities of the antiproton, the ∗ ement dz2. Finally, the factor PD∗,surv(b1, z2, ∞) is the probability molecule, and the D : ∗ that the D will escape from the nucleus. In the spirit of the ⎧ ⎫ ⎨ z1 ⎬ eikonal approach, all particles propagate parallel to the beam di- P −∞ = − tot rection. (For example, we assumed that the transverse momentum p¯ ,surv(b1, , z1) exp σpp¯ dzρ(b1, z) , (24) ⎩ ⎭ of the molecule, p1t , does not influence its trajectory.) The integra- −∞ ⎧ ⎫ tion over z2 can be taken with the explicit forms of the survival ⎨ z2 ⎬ probabilities Eqs. (25), (26). As a result Eq. (20) takes the following P b z z = exp − tot dz b z (25) simple form: X,surv( 1, 1, 2) ⎩ σpX ρ( 1, )⎭ , z1 3 ⎧ ⎫ d σpA¯ →D∗ 1 ⎨ ∞ ⎬ α = d3r P ¯ (b , −∞, z ) 2 tot − tot 1 p,surv 1 1 tot dαd kt v p¯ (σ σ ∗ ) P ∗ b z ∞ = exp − ∗ dz b z (26) pX pD D ,surv( 1, 2, ) ⎩ σpD ρ( 1, )⎭ , 2 1→X z d ∗ 2 2 p¯ p→D α × d p1t G (α, kt − p1t ) d2 p X 2 where ρ = ρp + ρn is the total nucleon density. We use the two- 1t parameter Fermi distributions of protons and neutrons [9]. As ×[PD∗,surv(b1, z1, ∞) − PX,surv(b1, z1, ∞)] . usual in the Glauber theory, Eqs. (24)–(26) neglect the Fermi motion of nucleon scatterers. In a similar way, in writing Eq. (20) (28) we neglected the Fermi motion of nucleon 2 since the elementary cross section (19) depends only weakly on the proton momentum In Fig. 6 we display the differential cross sections of charmed (via the flux factors, screening- and antiscreening contributions) meson production in antiproton collisions with argon nucleus ∗ and is in leading order proportional to the square of the molecule at 7GeV/c. The D and D cross sections are peaked at α = ∗ = wave function. However, the transverse Fermi motion of proton 1 2mD /mX 0.96 and α 2mD /mX 1.04, respectively, and be- is taken into account in Eq. (20) in the high-energy approximation have in similar way as a function of α and kt . The widths of ∗0 0 (cf. [15]). The latter implies that the light cone momentum frac- α-dependence of the D and D cross sections are much smaller ∗± tion α can be expressed in the target nucleus rest frame according and the peak values are much larger as compared to the D and ± ∗ to Eq. (1) where the Fermi motion of the proton 1 is still ne- D cross sections. The α-dependence of D and D production in glected. (We have numerically checked that using the exact Lorentz pA¯ collisions is dominated by the elementary cross section (cf. transformation to the c.m. frame of X to evaluate the invariant Figs. 3, 4). However, a closer look reveals significant differences 3 ∗ d σ → ∗ cross section ω ∗ pX D instead of using the infinite momen- between D production on a nucleus and on a proton due to the D d3k tum frame in Eq. (20) which conserves α and assumes Galilean Fermi motion. These are better visible in the ratio of the two cross sections depicted in Fig. 7. At kt = 0the ratio has a minimum transformation for kt produces indistinguishable results.) ∗ The pp¯ → X matrix element in Eq. (21) is one of the major un- at α 2mD /mX because in this case the contribution from tar- certainties in our calculations. Its modulus squared can be formally get protons with finite transverse momentum p1t is suppressed by α 2 2 the factor |ψ(kt − p1t )| /|ψ(kt )| . However, with increasing kt expressed in terms of the partial decay width X→pp¯ as 2 this factor becomes larger than unity for comoving proton 1. This 4π(2 J + 1)m2 → ¯ leads to the observed local maximum in the α-dependence for | |2 = X X X pp M X;p¯ 1 , (27) kt 0.1–0.4GeV/c. At large kt or for large deviations of α from 2 − 2 mX 4mN unity the ratio tends to the constant value. A.B. Larionov et al. / Physics Letters B 749 (2015) 35–43 41

∗0 0 ∗± ± ¯ 40 Fig. 6. The invariant differential cross sections of D , D , D and D production in p Ar collisions at plab = 7GeV/c. The calculations are done using full cross section ∗ ∗0 0 X(3872)p → D (D) in Eq. (28) including the IA term as well as screening and antiscreening corrections (see Eqs. (17)–(19)). For kt = 0, the cross sections of D and D production are divided by a factor of 100.

∗ ¯ 40 Fig. 7. The ratio of D production cross sections in p Ar and X(3872)p collisions at plab = 7GeV/c for several values of transverse momentum kt as a function of light cone ∗ ∗± momentum fraction α. The ratio is normalized at unity for α = 0.5. Panel (a) – D 0. Panel (b) – D . ∗ 5. Uncertainty and background X(3872) and D D¯ production in pp¯ collisions are almost the same. The background cross section can be calculated as (derivation is The uncertainty of our calculations can be read from Fig. 8. similar to that of Eq. (28)) ∗ The unknown binding energy of the D0 D¯ 0 molecule is the main source of uncertainty as a weaker binding produces a narrower 3 bg d σ ¯ → ∗ 2 α-distribution and vice versa. As a consequence of the partial can- α pA D = ∗ 2 3 z dαd kt (2π) E1 plabk cellation between the survival probabilities of D and molecule the N =n,p D ∗ 1 reduction of the pD and pD cross sections from 14 mb to 7mb ∗0 2 ∗ leads to the reduction of the peak of D production cross section 2 qp¯ 1(ε)ε dσp¯ 1→D D (ε) × d p1t by ∼ 15% only. Of course, on the top of these two effects there is ∗ qDD (ε) d an uncertainty due to the experimentally unknown width X→pp¯ which enters the cross section (28) as an overall multiplication fac- × 3 P −∞ ; 0 d r1 p¯ ,surv(b1, , z1)n1(r1 p1t ,ε) tor. The qualitative behavior of the α-distribution is not changed by varying the model parameters. × PD∗,surv(b1, z1, ∞)[1 − PD,surv(b1, z1, ∞)] , The major background is given by the direct process pN¯ → ∗ ∗ D D¯ , DD¯ on the bound nucleon, because the thresholds of (29) 42 A.B. Larionov et al. / Physics Letters B 749 (2015) 35–43 = z 2 + 2 with ωD kD mD . The longitudinal momentum of the nu- 0 cleon, , is given by Eq. (22) with mX replaced by ε. ε ∗ For the D 0 production on the proton, the near-threshold ∗ = S-wave cross section is σpp¯ →D D (ε) 2σpp¯ →D∗0 D¯ 0 (ε), where the direct (non-resonant) cross section σpp¯ →D∗0 D¯ 0 (ε) has been taken from Ref. [22] which is the only estimate of the discussed cross section available in the literature. (We included the factor of 2 ∗ ∗ ∗ since our D 0 includes both physical states, D 0 and D¯ 0.) The estimate of [22] was obtained by using dimensional counting consider- ∗ ations to express the cross section of pp¯ → D 0 D¯ 0 at high energies ∗− + in terms of the cross section of pp¯ → K K which is known in ∗ the limited energy range. As the next step, the pp¯ → D 0 D¯ 0 cross section was extrapolated in [22] towards the threshold and mul- tiplied by the S-wave fraction f L=0 9.3% which is regarded by the author of [22] himself as “a crude extrapolation”. Thus, we feel that the near-threshold estimate of [22] can be considered as an ∗ order of magnitude estimate. In the case of the D 0 production on ∗ − the neutron, the only possible channel is pn¯ → D 0 D . Thus, we ∗ = assume σpn¯ →D D (ε) σpp¯ →D∗0 D¯ 0 (ε). The result of calculation using Eq. (29) is shown in Fig. 8. The ∗ dependence of the background cross section on the pD and pD cross sections is quite modest and follows the tendency of the ∗ signal cross section. Thus, at kt = 0, the sharp peak of D production at α = 1.04 due to the stripping reaction is clearly visible on the smooth background. The peak is almost not influenced by intramolecular screening and antiscreening effects. Moreover, we expect that the elastic rescattering of antiproton and produced par- ∗ ticles on the nucleons will practically not change the D¯ and D spectra at small kt [17]. Finally, the influence of the SRCs – which are always included in default calculations – is mainly in the reduction of the nucleon occupancies at small momenta. This results in 15% reduction of the background at α 1(Fig. 8a) and the corresponding rescaling of the signal cross section (not shown). The signal-to-background ratio at the peak stays almost constant as the mass number of the target nucleus varies between 20 and 208. The shape of the α-dependence of the signal cross section practically does not vary with mass number in that region. ∗ The mass dependence of the total D 0 production cross section due to the stripping reaction can be well approximated by formula 0.46 σ ∗0 = 16 pb · A . With the high luminosity mode at PANDA, D − − L = 2 · 1032 cm 2 s 1, the estimated production rate due to the ∗ stripping reaction is about 60 D 0 events per hour. ∗0 ¯ 40 Fig. 8. The α-dependence of D production at kt = 0in p Ar collisions at plab = tot = 7GeV/c. Panel (a) shows calculations with default cross section σpD 14 mb. The 0 ¯ ∗0 6. Discussion and conclusions signal cross section (28) is shown for the different binding energies Eb of the D D molecule. The background cross section (29) is shown with SRCs (default calcula- ¯ ∗ tion) and without SRCs. Panel (b) shows calculations with default Eb = 0.5MeVfor We proposed the idea of D(D ) stripping from the X(3872) tot tot = tot the different σpD as indicated. It is always assumed that σpD∗ σpD. Insets show state, to investigate if X(3872) has a molecular structure from ∗ the narrower region of α. The background cross section is divided by a factor of 3. the narrow peaks in α-distribution of D¯ and D at α 1. Other microscopic models of X(3872), e.g. tetraquark or cc¯-gluon hy- ∗ where brid, would lead to the flat α-spectrum of D¯ (D). In such models, there are no primordial hadronic components in X(3872) to be 1/2 [ − 2 + 2 + 2] “released” or “knocked-out”. Thus, the momentum distribution of α 2 (2 α)m ∗ αm 2kt ¯ ∗ ¯ ∗ ≡ ( , k = k − p ) = D D D (D) in the process XN → D (D) would be dominated by phase ε ε α t t 1t − 2 α(2 α) space of the final state particles. There is, however, another possible source of narrow peaks in (30) ∗ α-distributions of D¯ and D. The BELLE Collaboration [23] has ∗0 ¯ 0 ∗ 2 2 1/2 found a significant near-threshold enhancement in the D D in- is the c.m. energy of D and D ; q ¯ (ε) = (ε /4 − m ) and ∗ p1 p variant mass spectrum from B → D 0 D¯ 0 K decays. We note that 2 2 2 2 2 2 1/2 ∗ =[ + − ∗ − ] ∗0 ¯ 0 qDD (ε) (ε mD mD ) /4ε mD are the c.m. momenta this does not exclude the existence of the D D bound state. (One ¯ ∗ − of the colliding pN pair and of the produced DD pair, respec- similar example is (1405) which lies about 30 MeV below K p z − tively. The longitudinal momentum kD of D-meson in the nucleus threshold and can be treated as a K p quasibound state although − ± ∓ − rest frame is calculated from relation it strongly influences the K p → π and K p → 0π 0 cross sections at small beam momenta [24,25].) But it is also possible z ∗0 ¯ 0 2(ωD + k ) that X(3872) is a resonance coupled to the D D + c.c. channel. 2 − = D , (31) α If such a resonance state is produced in peripheral pA¯ collisions, it E p¯ + mN + plab A.B. Larionov et al. / Physics Letters B 749 (2015) 35–43 43 will decay far away from the nucleus, since the width of X(3872) Acknowledgements ∗ is less than 1MeV. The resulting α-distributions of D 0 and D¯ 0 will be also sharply peaked near α 1at small kt . However, in this M.S.’s research was supported by the US Department of Energy case both decay products can in principle be detected. This gives a Office of Science, Office of Nuclear Physics under Award No. DE- clear experimental signature for distinguishing such decay events. FG02-93ER40771. This work was supported by HIC for FAIR within In contrast, the stripping events would contain only one meson, the framework of the LOEWE program. ∗ D 0 or D¯ 0, in the same kinematical region. The stripping reaction can be also considered for other produc- References tion channels of X(3872), e.g. in proton-, electron- and photon- induced reactions on nuclei. Other exotic X, Y , Z states, such as [1] S.K. Choi, et al., Belle Collaboration, Phys. Rev. Lett. 91 (2003) 262001. the X(3940) [26], Y (4140) [27], X(4160) [28] (cf. recent reviews [2]. K.A Olive, et al., Particle Data Group, Chin. Phys. C 38 (2014) 090001. [29,13] for a more complete list), may be interpreted as molecular [3] B. Aubert, et al., BaBar Collaboration, Phys. Rev. Lett. 102 (2009) 132001. ∗ ∗ ∗ ∗ ¯ ¯ [4] F. Aceti, R. Molina, E. Oset, Phys. Rev. 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