Determination of the structure of the $X(3872)$ in $\bar p A$ collisions

Currently, the structure of the $X(3872)$ meson is unknown. Different competing models of the $c\bar c$ exotic state $X(3872)$ exist, including the possibilities that this state is either a mesonic molecule with dominating $D^0 \bar D^{*0} +c.c.$ composition, a $c \bar c q \bar q$ tetraquark, or a $c \bar c$-gluon hybrid state. It is expected that the $X(3872)$ state is rather strongly coupled to the $\bar p p$ channel and, therefore, can be produced in $\bar p p$ and $\bar pA$ collisions at PANDA. We propose to test the hypothetical molecular structure of $X(3872)$ by studying the $D$ or $\bar D^{*}$ stripping reactions on a nuclear residue.


Introduction
The discovery of exotic cc mesons at B-factories and at the Tevatron stimulated interest to explore the possible existence of tetraquark and molecular meson states. The famous X(3872) state has been originally found by BELLE [1] as a peak in π + π − J/ψ invariant mass spectrum from exclusive B ± → K ± π + π − J/ψ decays. Nowadays the existence of the X(3872) state and its quantum numbers J P C = 1 ++ are well established [2]. In particular, radiative decays X(3872) → J/ψγ, X(3872) → ψ ′ (2S)γ [3] point to the positive C-parity of the X(3872). Probably the most intriguing feature is that the mass of the X(3872) is within 1 MeV the sum of the D 0 and D * 0 meson masses. This prompted the popular conception of the X(3872) being a DD * +DD * molecule. Other exotic X,Y,Z states, such as the X(3940) [4], Y (4140) [5], X(4160) [6] (c.f. recent reviews [7,8] for a more complete list), may be interpreted as molecular states of D * D * or D * SD * S . To probe the molecular nature of the X(3872) structure has been difficult. So far, most theoretical calculations have been focused on the description of radiative and isospinviolating decays of the X(3872). For example, the X(3872) → J/ψγ decay can be well understood within the DD * + c.c. molecular hypothesis [9]. On the other hand, the measured large branching fraction B(X(3872) → ψ ′ (2S)γ)/B(X(3872) → J/ψγ) = 3.4 ± 1.4 [3] seems to disfavour the molecular structure and requires a significant pure cc admixture in the X(3872) [10]. The theoretical predictions for the decay rates are, however, quite sensitive to the model details even within various approaches like charmonium or DD * + c.c. molecular models.
In this letter we suggest to test the charm meson molecular hypothesis of the X(3872) structure inpA collisions at PANDA. Assuming that the X(3872) is coupled to the pp channel, we consider the stripping reaction of the D-meson on a nuclear target nucleon such that aD * is produced and vice versa. We show that the distribution of the produced charmed meson in the light cone momentum fraction α with z-axis alongp beam momentum, will be sharply peaked at α ≃ 1 at small transverse momenta which allows to unambiguously identify the weakly coupled DD * + c.c. molecule. Here, k D * and ω D * (k D * ) = (k 2 D * + m 2 D * ) 1/2 are, respectively, the momentum and energy of the producedD * meson in the target nucleus rest frame. Similar studies of hadron-, 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 sections are proportional to the square of the deuteron wave function. For the X(3872) this idea is depicted in Fig. 1 (details follow below).

X(3872)-proton cross section
For brevity, R stands for X(3872) and the bar, which can be seen over the D * or D, will be dropped in many cases below. The most important ingredients of our calculations are the total Rp cross section and the momentum differential Rp → D * (D)X cross section. In the molecular picture, the latter cross section is the D(D * )-meson stripping cross section. To calculate the total Rp cross section within the Glauber theory, we start from the graphs shown in Fig. 1 which assume the DD * composition of R. It is convenient to perform calculations in the DD * molecule center-of-mass (c.m.) frame with proton momentum p p directed along z-axis. The invariant forward scattering amplitudes of the first two processes are where m R = ω D + ω D * is the mass of the molecule and ω D (ω D * ) is the energy of D (D * )-meson. (The different assumptions on the momentum dependence of meson energies Figure 1: Processes contributing to the forward scattering amplitude of a proton on the DD * molecule. Wavy lines denote the pD and pD * elastic scattering amplitudes. Straight lines are labeled with particle's four-momenta. The blobs represent the wave function of the molecule. discussed in the next section have practically no effect on the Rp cross section.) The molecule wave function in momentum space is defined as where k is the D * momentum in the DD * c.m. frame, with the normalization condition d 3 k|ψ(k)| 2 = 1. For the calculation of the third and forth processes in Fig. 1 we apply the generalized eikonal approximation (GEA) [11,12] which assumes the nonrelativistic motion of D and D * inside the molecule. In this approximation, the propagator of the intermediate proton depends only on the z-component of momentum transfer q ≡ k D * − k ′ D * , while the pD and pD * elastic scattering amplitudes depend only on the momenta of incoming particles and on the transverse momentum transfer. Thus, we obtain Therefore, The optical theorem for the proton-molecule forward scattering amplitude is Substituting (4) (0) and using the parameterization of the elementary amplitudes in the usual form as with ) 2 ] 1/2 being the Moeller flux factor we obtain the following expression for the proton-molecule total cross section: where the normalized flux factors are defined as I pD ( * ) (k) ≡ I pD ( * ) (k)/p p ω D ( * ) . In the small binding energy limit the molecule wave function decreases rapidly with increasing momentum k and becomes negligibly small at k ≪ B −1/2 pD . In this case one can set B pD = B pD * = 0 and perform the Taylor expansion of the flux factors in k z in Eq. (10). Then, for the S-state molecule with accuracy up to the linear terms in k z /m D and assuming that m D ≃ m * D , σ tot pD * ≃ σ tot pD we obtain the formula in line with previous calculations of the proton-deuteron total cross section [13]. We choose the wave function of a DD * molecule as the asymptotic solution of the Schroedinger equation at large distances: where the range parameter κ = √ 2µE b depends on the reduced mass µ = m D m D * /(m D + m D * ) and on the binding energy E B of the molecule. The corresponding momentum space wave function is Let us now discuss the input parameters of our model. Since there is no experimental information on Dp and D * p interactions, we rely on simple estimates in the high-energy limit. For small-size qq configurations the color dipole model predicts the scaling of the total meson-nucleon cross section with the average square of the transverse distance between quark and antiquark in the meson, which is proportional to the square of the Bohr radius r B = 3/4µα s . Here, µ = m q mq/(m q + mq) is the reduced mass with m q and mq being the constituent quark and antiquark masses. The Bohr radii of pion, kaon, and D-meson and J/ψ are ordered as r Bπ > r BK > r BD > r BJ/ψ . Hence, we expect that the total meson-nucleon cross sections follow the same order. At a beam momentum of 3.5 GeV/c (1/2 of the momentum of R formed in thepp → R process on the proton at rest) the total π + p and K + p cross sections are about 28 mb and 17 mb, respectively [2]. The J/ψp cross section is expected to be much smaller, 3.5 − 6 mb (c.f. [14] and refs. therein.). We assume the total Dp cross section σ tot pD = 14 mb, i.e. slightly below the K + p total cross section. This choice is in reasonable agreement with effective field theory calculations [15].
It is well known that at incident energies of a few GeV, the amplitude of meson (nucleon) -nucleon elastic scattering is (to a good approximation) proportional to the product of the electric form factors of the colliding hadrons (see e.g. [16] and refs. therein). Thus, in the exponential approximation for the t-dependence of the form factors, the slope parameters B pM of the transverse momentum dependence of the meson-proton cross section at small t should be proportional to r 2 p + r 2 M , where r 2 p and r 2 M are the mean-squared charge radii of the proton and meson, respectively. Since r 2 M ∝ r 2 BM , the slope parameters should be also ordered as the Bohr radii. Empirical values at p lab = 3.65 GeV/c are B pπ + = 6.75 ± 0.12 GeV −2 and B pK + = 4.12 ± 0.12 GeV −2 as fitted at 0.05 ≤ −t ≤ 0.44 GeV 2 [17]. On the other hand, B pJ/ψ = 3 GeV −2 at the comparable beam momenta [16]. We will assume the value B pD = 4 GeV −2 , since the Bohr radii of kaon and D-meson differ by ∼ 30% only. For the pD * interaction we assume for simplicity σ tot pD * = σ tot pD and B pD * = B pD .
Our educated guess on the D-and D * -meson-nucleon cross sections and slope parameters should of course be checked experimentally. The empirical information on σ tot pD can be obtained by measuring the A-dependence of the transparency ratio of D-meson production inpA reactions at beam momenta beyond the charmonium resonance peaks, where the backgroundpp →DD channel dominates. The slope parameter B pD can be addressed by measuring the transverse momentum spread of D-meson production inpA reactions.
We will further assume that the X(3872) wave function contains 86% of D 0D * 0 + c.c. and 12% of the D + D * − + c.c. component as predicted by the local hidden gauge approach [9]. The binding energy of D 0D * 0 is likely less than 1 MeV [8] and can not be determined from existing data [2] accurately enough. We set MeV in numerical calculations. This corresponds to the range parameters κ D 0D * 0 = 0.16 fm −1 and κ D + D * − = 0.64 fm −1 . With these parameters the total pR cross section (10) is σ tot pR = 25 and 23 mb for D 0D * 0 and D + D * − components, respectively, at the molecule momentum of 7 GeV/c in the proton rest frame.

D(D * ) stripping cross section
In high energy hadron-deuteron reactions, the main contribution to the fast backward nucleon production (in the deuteron rest frame or equivalently -fast forward in the deuteron projectile case) is given by the inelastic interaction of the hadron with second nucleon of the deuteron [18]. For large nucleon momenta the spectrum is modified as compared to the impulse approximation due to the Glauber screening and antiscreening corrections [19]  since the hadron may interact with both nucleons. In a similar way, in calculations of the Rp → D * X cross section, we take into account the impulse approximation diagram (IA, Fig. 2a) and the single-rescattering diagrams of the incoming proton ( Fig. 2b) and of the outgoing proton or of the most energetic forward going baryon emerging from the inelastic pD interaction (Fig. 2c). The expressions for the invariant matrix elements for the processes (a) and (b) in Fig. 2 are straightforward to obtain in the c.m. frame of the molecule state R: where k ≡ k ′ D * . In the case of M (b) we applied the GEA by expressing the propagator of the intermediate proton in the eikonal form and using the coordinate representation with r = r D * − r D . The explicit form of the amplitude M (c) can be written only for specific outgoing states X. However, for the diffractive states including the leading proton, the expression for M (c) can be obtained from the expression for M (b) by replacing Θ(−z) → Θ(z), which reflects the change of the time order of the pD * and pD interactions. Thus, for the diffractive outgoing state X the expression for M (b) + M (c) is given by Eq. (15) with replacement Θ(−z) → 1 (neglecting small differences in momenta of incoming and outgoing proton in elementary amplitudes). We assume that the same replacement can be done for any final state X. By summing over all states X we then obtain the momentum differential D * production (i.e. D-stripping) cross section in the molecule rest frame: where P is the four momentum of the molecule (P 2 = m 2 R ). With a help of the unitarity relation for the elementary amplitudes [20] the sum over spin states and sorts of X and the integration over phase space volume can be reduced to the products of the imaginary parts of elastic scattering amplitudes. This leads to the following expression for the momentum differential cross section in the molecule rest frame: The first term in the r.h.s. of Eq. (18) is the pure IA contribution. The second and third terms are, respectively, the screening and antiscreening corrections (see Eqs. (8a) and (8b) in [19]). The D * meson is assumed to be on its vacuum mass shell, ω D * (k) = m 2 D * + k 2 , while the energy of the D meson is calculated from energy conservation, ω D (−k) = m R − ω D * (k). (The condition ω D > 0 constrains the maximum momentum of the emitted D * , k < 3.3 GeV/c. Above this value our model looses its applicability.) In the case of the D-meson production one has to exchange I pD ↔ I pD * , σ tot pD ↔ σ tot pD * and B pD ↔ B pD * in Eqs. (17), (18). In this case the on-shell condition is applied to the D-meson, while the D * energy is determined by energy conservation. It is convenient to express the differential invariant D * production cross section (17) in terms of the relative fraction α of the light cone momentum of the DD * molecule carried by the D * : where α = 2(ω D * (k) − k z )/m R . Figures 3 and 4 show the differential cross section of D * 0 and D * − production from X(3872) collisions at 7 GeV/c with proton at rest as a function of α for several values of transverse momentum k t . At k t = 0, the cross section has a sharp maximum at α ≃ 2m D * /m R ≃ 1.04 and is almost unaffected by the screening and antiscreening corrections. With increasing k t , the width of α-distribution increases while the screening and antiscreening corrections to the IA term become important. This is expected since the large-k t component of the molecule wave function corresponds to small transverse separation between D and D * . The corrections become large for α ≃ 1 and large transverse momenta as can be directly seen from the structure of the integrands in Eq. (18). Indeed, α ≃ 1 corresponds to k z ≃ 0 in the molecule rest frame. Then at finite transverse momentum transfer q t the ratio ψ * (k t + q t )/ψ * (k t ) is less than unity at k t = 0 and asymptotically tends to unity with growing k t . Due to the extremely narrow wave function of the D 0 D * 0 molecule in momentum space, the screening and antiscreening corrections are sharply peaked at α ≃ 1.1 and develop structures in the α-dependence of the cross section at large transverse momenta. In the case ofpA reactions these structures are slightly smeared out due to the nucleon Fermi motion (see Fig. 6 below).

D * and D production off nucleus
In antiproton-nucleus interactions, we focus on the D * (or D) meson production in the two-step processpp → R, RN → D * (D)X. Similar to the case of Rp interactions, we apply the Glauber theory to calculate the differential cross sections of the D * production in antiproton-nucleus interactions. We start from the multiple scattering diagram shown in Fig. 5 which can be evaluated within the GEA. We will assume that the nucleus can be described within the independent particle model disregarding the c.m. motion corrections (c.f. [21]). The incoming antiproton, intermediate molecular state R and outgoing D *meson are allowed to rescatter on nucleons elastically an arbitrary number of times. The D * production cross section is proportional to the product of the sum of the amplitudes of Fig. 5 and their conjugated. The R state is formed on a proton 1, while the D * is produced in the collision of R with a nucleon 2. The nucleons 1 and 2 are fixed in the direct and conjugated amplitudes while the sets of other nucleon scatterers are arbitrary. The leading order contribution is given by the product term without elastic rescatterings. Nuclear absorption corrections are accounted for by summing all possible product terms with non-overlapping sets of nucleon scatterers. This gives the following expression for the momentum differential cross section of D * production on a nucleus: where is the in-medium width ofp with respect to production of R with transverse momentum p 1t ; vp = p lab /Ep is the antiproton velocity; n p (r 1 ; p 1t , ∆ 0 R ) is the proton occupation number. The longitudinal momentum ∆ 0 R of the proton 1 is obtained from the condition of on-shell production of the state R in the processp 1 → R: The proton occupation number is taken as the depleted Fermi distribution supplemented by high-momentum tail due to short-range quasideuteron correlations [22,23,21]: where p F (r) = (3π 2 ρ p (r)) 1/3 is the proton Fermi momentum, P 2 = 0.25 is the proton fraction above the Fermi surface, ρ p (r) is the proton density, and ψ d (p) is the deuteron wave function. In Eq. (20), the nuclear absorption is given by the survival probabilities of the antiproton, the molecule, and the D * : where ρ = ρ p + ρ n is the nucleon density. We use the two-parameter Fermi distributions of protons and neutrons [14]. As usual in the Glauber theory, Eqs.(24)-(26) neglect the Fermi motion of nucleon scatterers. In a similar way, in writing Eq. (20) we neglected the Fermi motion of nucleon 2 since the elementary cross section (19) depends only weakly on the proton momentum (via the flux factors, screening-and antiscreening contributions) and is in leading order proportional to the square of the molecule wave function. However, the transverse Fermi motion of proton 1 is taken into account in Eq. (20) in the high-energy approximation (c.f. [19]). The latter implies that the light cone momentum fraction α can be expressed in the target nucleus rest frame according to Eq.(1) where the Fermi motion of the proton 1 is still neglected. (We have numerically checked that using the exact Lorentz transformation to the c.m. frame of R to evaluate the invariant cross section ω D * instead of using the infinite momentum frame in Eq. (20) which conserves α and assumes Galilean transformation for k t produces indistinguishable results.) Thepp → R matrix element in Eq.(21) is a major uncertainty in our calculations. Its modulus squared can be formally expressed in terms of the partial decay width Γ R→pp as where the overline means averaging over antiproton and proton helicities and summation over the helicity of R. There is no experimental data on the partial decay width Γ X(3872)→pp .
(The recent LHCb data onpp invariant mass spectra from B + → ppK + decays [24] do not allow to clearly identify X(3872) in the pp decay channel due to statistical limitations.) In the present calculations, we will use the value Γ X(3872)→pp ≃ 30 eV as suggested by theoretical estimates [25]. This value is about two times smaller than Γ χ c1 (1P )→pp . However, one should note that, in the molecular picture, the decay of the X(3872) to the pp state requires the production of only two qq pairs, and not three qq pairs as in the ordinary charmonium decay to the pp channel. Thus, the partial decay width of the X(3872) into the pp channel may be even larger than that of the χ c1 (1P ) state [25]. Formula (20) has a simple physical interpretation if we express the integral d 3 r 1 as d 2 b 1 dz 1 . The factor Pp ,surv (b 1 , −∞, z 1 ) is the probability that the incoming from z = −∞ antiproton with impact parameter b 1 will reach the point z = z 1 . The combination (dz 1 /vp)d 2 p 1t d 2 Γ 1→R p /d 2 p 1t is the probability that the molecule R will be formed within the transverse momentum element d 2 p 1t when thep is passing the longitudinal element dz 1 . The factor P R,surv (b 1 , z 1 , z 2 ) is the probability that the molecule will reach the point z = z 2 . The combination dz 2 (dα/α)d 2 k t G 2→D * R (α, k t − α 2 p 1t )ρ(b 1 , z 2 ) is the probability that a D * will be produced in the kinematical element dαd 2 k t when the molecule R is passing the longitudinal element dz 2 . Finally, the factor P D * ,surv (b 1 , z 2 , ∞) is the probability that the D * will escape from the nucleus. In the spirit of the eikonal approach, all particles propagate parallel to the beam direction. (For example, we assumed that the transverse momentum of the molecule, p 1t , does not influence its trajectory.) The integration over z 2 can be taken with the explicit forms of the survival probabilities Eqs. (25), (26). As a result Eq.(20) takes the following simple form: In Fig. 6 we display the differential cross sections of charmed meson production in antiproton collisions with lead nucleus at 7 GeV/c. The D + cross section is peaked at α ≃ 2m D /m R = 0.96 and behaves similar to the D * − cross section as a function of α and k t . The width of α-dependence of the D * 0 cross section is much smaller and the peak value is much larger as compared to the D * − and D + cross sections. The α-dependence of D * and D production inpA collisions is dominated by the elementary cross section (c.f. Figs. 3,4). However, a closer look reveals significant differences between D * production on a nucleus and on a proton due to the Fermi motion. These are better visible in the ratio of the two cross sections depicted in Fig. 7. At k t = 0 the ratio has a minimum at α ≃ 2m D * /m R because in this case the contribution from target protons with finite transverse momentum p 1t is suppressed by the factor |ψ(k t − α 2 p 1t )| 2 /|ψ(k t )| 2 . However, with increasing k t this factor becomes larger than unity for comoving proton 1. This leads to the observed local maximum in the α-dependence for k t ≃ 0.1 − 0.4 GeV/c. At large k t or for large deviations of α from unity the ratio tends to the constant value.

Discussion and conclusions
We proposed the idea of D(D * ) stripping from the X(3872) state, to investigate if X(3872) has a molecular structure. However, a major background is given by the direct processpp → DD * +c.c. in the target nucleus, because the thresholds of X(3872) and DD * production inpp collisions are almost the same. Due to the two-body final state, at small k t thepp → DD * cross section on the proton at rest has two peaks at Since the cross sectionpp → DD * grows with √ s close to threshold, the backgroundpA →D * X cross section at small k t will be widely distributed in α around α = 1 due to Fermi motion. Thus, the sharp peaks of stripping reactions at α = 1.04 forD *and at α = 0.96 for D-production should be clearly visible on the smooth background. As we have seen, these peaks are not influenced by intramolecular screening and antiscreening effects. Moreover, we expect that the elastic rescattering of antiproton and produced particles on the nucleons will practically not change theD * and D spectra at small k t [21]. There is another possible source of narrow peaks in α-distributions ofD * and D. The BELLE collaboration [26] has found a significant near-threshold enhancement in the D * 0D0 invariant mass spectrum from B → D * 0D0 K decays. We note that this does not exclude the existence of the D * 0D0 bound state. (One similar example is Λ(1405) which lies about 30 MeV below K − p threshold and can be treated as a K − p quasibound state although it strongly influences the K − p → Σ ± π ∓ and K − p → Σ 0 π 0 cross sections at small beam momenta [27,28].) But it is also possible that X(3872) is a resonance coupled to the D * 0D0 + c.c. channel. If such a resonance state is produced in peripheralpA collisions, it will decay far away from the nucleus, since the width of X(3872) is less than 1 MeV. The resulting α-distributions of D * 0 andD 0 will be also sharply peaked near α ≃ 1 at small k t . However, in this case both decay products can be detected. This gives a clear experimental condition for rejecting such decay events. In contrast, the stripping events would contain only one meson, D * 0 orD 0 , in the same kinematical region.
In conclusion, we have demonstrated that the spectra ofD * and D in the light cone momentum fraction at small transverse momenta allow to test the hypothetical DD * molecular structure of the X(3872) produced inpA collisions at threshold. We propose to search the narrow peak inD * or D production at α ≃ 1 and small k t as an unambiguous signal of the DD * molecular state formation inpA collisions in PANDA experiment at FAIR.