$D\bar{D}^*$ scattering and $\chi_{c1}(3872)$ in nuclear matter

We study the behaviour of the $\chi_{c1}(3872)$, also known as $X(3872)$, in dense nuclear matter. We begin from a picture in vacuum of the $X(3872)$ as a purely molecular $(D \bar D^*-c.c.)$ state, generated as a bound state from a heavy-quark symmetry leading-order interaction between the charmed mesons, and analyze the $D \bar D^*$ scattering $T-$matrix ($T_{D \bar D^*}$) inside of the medium. Next, we consider also mixed-molecular scenarios and, in all cases, we determine the corresponding $X(3872)$ spectral function and the $D \bar D^*$ amplitude, with the mesons embedded in the dense environment. We find important nuclear corrections for $T_{D \bar D^*}$ and the pole position of the resonance, and discuss the dependence of these results on the $D \bar D^*$ molecular component in the $X(3872)$ wave-function. These predictions could be tested in the finite-density regime that can be accessed in the future CBM and PANDA experiments at FAIR.

sections using the molecular approach are consistent with the CDF and CMS measurements by an adequate election of the ultraviolet (UV) cutoff [54], a statement that has been in turn criticized in Ref. [55]. Also, Ref. [56] questioned the production mechanism of the X(3872) shown in Ref. [53], while conjecturing a new mechanism. The controversy has still continued in Ref. [57]. In this latter work, it is shown that the prompt X(3872) cross section at hadron colliders is consistent with those experimentally observed at CDF and CMS. This is concluded thanks to the derivation of a relation between the prompt X(3872) cross section and that of a charm-meson pair, taking into account the threshold enhancement from the X(3872) resonance. More recently, the production rates of promptly produced X(3872) relative to the ψ(2S) as a function of the final state particle multiplicity, obtained recently at LHCb, are explained within a comover interaction model if the X(3872) is a tetraquark [58]. However, this result is again questioned in Ref. [59] as it is argued that the breakup cross section are not well approximated by a geometric cross section inversely proportional to the binding energy of X(3872), as assumed in Ref. [58]. As a consequence, a simple modification of the comover model will give excellent fits to the LHCb data using parameters consistent with X(3872) being a loosely bound charm-meson molecule. Thus, there is still an ongoing debate on the nature of the X(3872) coming from the analysis of pp collisions.
Another possible way to gain some insight about the nature of X(3872) is to analyze its behavior for the extreme conditions present in HiCs at RHIC and LHC energies. The ExHIC Collaboration [60][61][62] has shown that, within the coalescence model, the molecular structure for the X(3872) implies a production yield much larger than for the tetraquark configuration, in particular if one also takes into account the evolution in the hadronic phase [63,64]. This is due to the fact that molecules are bigger than tetraquarks and, hence, the production and absorption cross sections in HiCs are expected to be larger. This was actually shown in Ref. [64], where the time evolution of the X(3872) abundance in the hot hadron gas was obtained, based on all the possible hadronic reactions for the production of X(3782) of Ref. [64,65]. More recently the nature of X(3872) in HiCs has been studied not only within instantaneous coalescence models [66,67], but also using a statistical hadronization model [68] or by means of a thermal-rate equation approach [69]. In those studies it is advocated that the quantitative description for a series of standard HiC observables, such as particle yields or the transverse spectra, might shed some light in the nature of the X(3872).
The studies of the production of X(3872) however do not consider the possible in-medium modification of the hot hadronic phase. Only recently the behaviour of X(3872) in a finite-temperature pion bath has been studied assuming this resonance to be a molecular state generated by the interaction of DD * + c.c. pairs and associated coupled channels [70]. The X(3872) develops a substantial width, of the order of a few tens of MeV, within a hot pionic bath at temperatures 100-150 MeV, whereas its nominal mass moves above the DD * threshold.
In the present work we address the behaviour of X(3872) in a nuclear environment, with the objective of analyzing the finite-density regime that can be accessed in HiCs and the future experiments at FAIR. An early study of the D * s0 (2317) and theorized X(3700) scalar mesons in a nuclear medium was performed in Ref. [71], already showing that the experimental analysis of the properties of those mesons is a valuable test of the nature of the open and hidden charm scalar resonances. More recently, in Ref. [72] the in-medium mass shift of the X(3872) was obtained using QCD sum rules, revealing that the mass of the resonance is considerably affected by the nuclear matter.
We begin here from a picture of the X(3872) as a molecular DD * +c.c. state, generated as a purely molecular bound state from the leading-order interaction of the D andD * mesons, which is constrained by heavy-quark spin symmetry (HQSS) [73][74][75][76]. HQSS predicts that all types of spin interactions vanish for infinitely massive quarks, that is, the dynamics is unchanged under arbitrary transformations in the spin of the heavy quark. As a consequence, open charm pseudoscalar and vector mesons become degenerate in the infinite mass limit. We then implement the changes of the D and D * propagators in nuclear matter in order to obtain the in-medium X(3872) scattering amplitude and the corresponding X(3872) spectral function. Later on, we consider generalizations of the DD * interaction, allowing for scenarios in which the X(3872) is not a purely molecular state, i.e., it can be a compact state, and we also study mixed scenarios. In this way, we extract the modification on the mass and the width of X(3872) in nuclear matter for different scenarios, in view of the forthcoming results on charmed particles in HiCs at CBM in FAIR [77,78]. Moreover, the present study will be also of interest for PANDA, since it is expected that the X(3872) will strongly couple to thepp channel [17], and therefore, this resonance can be produced also inpA collisions [79]. Actually colliding antiprotons on nuclei with PANDA would allow the A−dependence of the production of ψ(2S) and X(3872) near threshold to be compared. This may, after appropriate theoretical study, provide a good way to expose an extended D * D component of the X(3872) state function [80].
This work is organized as follows. In Sec. 2 we present the DD * scattering amplitude and the X(3872) in vacuum and in isospin-symmetric nuclear matter, while showing the open-charm groundstate spectral functions in matter. In Sec. 3 we determine the X(3872) self-energy both in vacuum and in nuclear matter, while connecting the self-energy to the DD * scattering amplitude. We finish by presenting our results in Sec. 4, and conclusions and future outlook in Sec. 5. Finally in Appendix A, we give some details on the approximation used to extend the nuclear medium DD * T −matrix to the complex plane, allowing for the search of poles reported in Sec. 4.

DD * scattering amplitude and X(3872)
To study the X(3872) as a molecular state in the DD * I G (J P C ) = 0 + (1 ++ ) channel, we start by considering the interaction in the particle basis: (2.1) The unitary T -matrix for this basis is written as: with √ s the energy of any of the pairs in the center of mass (c.m.) frame, and the V and G matrices are constructed out of the interaction potential and the two-meson loop functions, respectively. From the leading order HQSS-based Lagrangian, V can be written as a contact interaction [73][74][75][76] is a diagonal matrix (the notation for the matrix elements will be explained below), 2 and the matrix A (satisfying A T = A −1 = A) reads: The G(s) matrix is diagonal, and contains the loop-function for the different two-meson channels in the particle basis, Eq. (2.1), where D Y i and D Y i are the propagators of the two mesons Y i and Y i in the particle basis, Eq. (2.1), and P 2 = s. In terms of the self-energies Π Y (q) of the latter, they can be written as: Note that we will be here only interested in the nuclear medium renormalization of the meson properties. Thus, m Y is the meson mass in the free space, while the self-energy Π Y approaches zero when the nuclear density ρ → 0. Inserting the above representation into Eq. (2.5) and integrating over q 0 leads to: with: where U and W stand for Y i and Y i or Y i and Y i . In the previous equations we have already introduced a sharp momentum cut-off Λ to regularize the UV behavior of the integration over the modulus of q. Specifically, we take Λ = 0.7 GeV.

Vacuum
In vacuum, assuming isospin symmetry, m D ( * )0 = m D ( * +) , the loop functions for the four channels are equal, G(s) = Σ 0 (s)I 4 . The function Σ 0 (s) reduces to a standard loop function regulated via a hard cutoff Λ, G(s, m D , m D * ), and expressions for this can be found in Ref. [81]. The T -matrix diagonalizes in the same way as the kernel matrix V (s), i.e., where: From the eigenvectors of T (and V ) one notices that they are DD * states with well defined isospin I (I = 0, 1) and C-parity (charge-conjugation) quantum numbers. The notation C IC for the low energy constants refers to the potential in each of these channels, with isospin and the charge-conjugation C = +(−) associated with the subindex X(Z). We consider the X(3872) as a J P = 1 + , I C = 0 + state, which is thus associated to the amplitude T 0X , We can thus fix the constant C 0X by requiring the presence of a pole at an energy equal to the X(3872) We will also consider below (see Subsec. 3.2) more general scenarios, in which energy dependence will be actually allowed in the kernel V (s).

Isospin-symmetric nuclear matter
To consider the possible modification of the X(3872) properties in a nuclear medium, we assume that the DD * interaction potentials V d (s) do not change in nuclear matter, 4 and that the T -matrix is modified through the loop functions because of the D ( * ) andD ( * ) self-energies. We still assume isospin symmetry, m D ( * )0 = m D ( * )+ , and S D ( * )+ = S D ( * )0 ≡ S D ( * ) , S D ( * )− = SD( * )0 ≡ SD( * ). However, in general we will have SD( * ) = S D ( * ) in the nuclear environment, since the charmed and anti-charmed meson-nucleon interactions are quite different. 5 In addition, pseudo-scalar-nucleon and vector-nucleon interactions are also different and hence S D = S D * and SD = SD * . We discuss the spectral functions SD( * ) and S D ( * ) in nuclear matter in Sec. 2.4. Consequently, the G-matrix in a nuclear medium of density ρ, G(s; ρ), is no longer proportional to the identity, as opposed to the vacuum case. It reads G(s; ρ) = diag (G DD * (s; ρ), G D * D (s; ρ), G DD * (s; ρ), G D * D (s; ρ)). Hence, the in-medium T -matrix T (s; ρ) cannot be fully diagonalized, and it can only be put in block diagonal form, The 2 × 2 matrix G can be written as: with: and: In other words, defining the states |IC , we have the following matrix elements: The amplitudes T (I) C,C are compactly defined as: where T X (s; ρ) and T Z (s; ρ) are written as in the diagonal case, Equation (2.17) may seem counter intuitive, due to the absence of a δ C,C factor. However, we must bear in mind that, in the presence of nuclear matter, the scattering processes are DD * N → DD * N . Due to the presence of the nucleons, the DD * in the initial and final states do not need to have the same C-parity.
We have checked that the term δ G (s; ρ) is small, so we consider throughout this manuscript the limit δ G (s; ρ) → 0. In this limit, T (I) XZ (s; ρ) = 0 [Eq. (2.18c)], and Eq. (2.17) is further diagonalized into DD * C-parity amplitudes, too. We thus find, for the I C = 0 + channel, 6 . (2.20) Note that, from its definition, Σ(s; ρ) can be written more compactly as: where the dependence on the density arises from that of the spectral functions involved in the above equation. We recall that the expressions for f DD * , f D * D are given in Eq. (2.7b). Finally, we note that in the ρ → 0 limit, the vacuum amplitudes are recovered. In principle, given the integral representation in Eq. (2.21), the function Σ(P 0 , | P |; ρ) could be computed for complex values of the energy P 0 . However, we can neither perform its analytical continuation into the lower half of the complex plane, nor define the second Riemann sheet for finite densities. This is because it would require to know the meson spectral functions S U,W for complex values of its arguments, which cannot be computed within the standard scheme that will be presented below, see Subsec. 2.4. Nevertheless, as discussed below in Subsec. 4.2, we will derive a reasonable approximation for the in-medium loop-function Σ(P 0 , | P |; ρ) of Eq. (2.15), which will allow for a meaningful extension of the isoscalar T -matrix to the complex plane and the search for poles also in nuclear matter.

S D ( * ) and SD( * ) in nuclear matter
The spectral functions of D ( * ) andD ( * ) in symmetric nuclear matter are obtained following a unitarized self-consistent procedure in coupled channels, as described in Refs. [82,83] for the D ( * ) meson and in Ref. [84] forD ( * ) meson. In the following we present the main features.
The s-wave transition charmed meson-nucleon kernel of the Bethe-Salpeter equation (BSE) is derived from an effective Lagrangian that implements HQSS [85][86][87]. HQSS is an approximate QCD symmetry that treats on equal footing heavy pseudoscalar and vector mesons, such as charmed and bottomed mesons [78,[82][83][84][88][89][90][91][92][93][94]. The effective Lagrangian accounts for the lowest-lying pseudoscalar and vector mesons as well as 1/2 + and 3/2 + baryons. It reduces to the Weinberg-Tomozawa (WT) interaction term in the sector where Goldstone bosons are involved and incorporates HQSS in the sector where heavy quarks participate. Thus, it is a SU(6)×HQSS model, that is justified in view of the reasonable semi-qualitative outcome of the SU(6) extension [95] and on a formal plausibleness on how the SU(4) WT interaction in the heavy pseudoscalar meson-baryon sectors comes out in the vector-meson exchange picture (see for instance Refs. [96,97]).
This extended WT meson-baryon potential in the coupled meson-baryon basis with total charm C, strangeness S, isospin I and spin J, is given by where √ t is the center of mass (C.M.) energy of the meson-baryon system; E i and M i are, respectively, the C.M. on-shell energy and mass of the baryon in the channel i; and f i is the decay constant of the meson in the i-channel. Symmetry breaking effects are introduced by using physical masses and decay constants. The D CSIJ ij are the matrix elements coming from the group structure of the extended WT interaction.
The amplitudes in nuclear matter, t ρ,CSIJ (R 0 , R ) with R = (R 0 , R ) the total meson-baryon four-momentum (t = R 2 ), are obtained by solving the on-shell BSE using the previously described potential, v CSIJ ( √ t): where the diagonal g ρ CSIJ (R) loop-matrix accounts for the charmed meson-baryon loop in nuclear matter [82,84]. We focus in the non-strange S = 0 and singly charmed C = 1 sector, where DN and D * N are embedded, as well as the C = −1 one, withDN andD * N . 7 The D(D) and D * (D * ) self-energies in symmetric nuclear matter, Π(E, q ; ρ), are obtained by summing the different isospin transition amplitudes for D(D)N and D * (D * )N over the nucleon Fermi distribution, p F . For the D(D) we have In the above equations, R 0 = q 0 + E N ( p ) and R = ( q + p ) are the total energy and momentum of the meson-nucleon pair in the nuclear matter rest frame, and (q 0 , q ) and (E N , p ) stand for the energy and momentum of the meson and nucleon, respectively, in that frame. Those self-energies are determined self-consistently since they are obtained from the in-medium amplitudes which contain the meson-baryon loop functions, and those quantities themselves are functions of the self-energies. The D(D) and D * (D * ) spectral functions are then defined from the in-medium D(D) and D * (D * ) meson propagators: The D(D) and D * (D * ) spectral functions are shown in Fig. 1 as function of the meson energy E = q 0 for zero momentum q = 0 and two different densities, ρ = 0.5ρ 0 and ρ = ρ 0 . Apart from the quasiparticle peak, obtained from E 2 qp = q 2 + m 2 + ReΠ(E qp ( q ), q ), with m the meson mass, these spectral functions show a rich structure as a result of the presence of several resonance-hole excitations. The masses and widths of these resonances were obtained in Refs. [88][89][90].
The D meson spectral function is depicted in the upper left-hand side panel. As described in Ref. [82], the D meson quasiparticle peak moves to lower energies with respect to the free mass position as density increases. Moreover, several resonant-hole excitations appear around the quasiparticle peak. In the low-energy tail of the D spectral function, we observe the Λ c (2556)N −1 and Λ c (2595)N −1 excitations, whereas Σ * c N −1 excitations appear on the right-hand side of the quasiparticle peak. With regards to the D * meson spectral function shown in Ref. [82] and depicted here in the righthand side panel, the quasiparticle peak moves to higher energies with density and fully mixes with the sub-threshold J = 3/2 Λ c (2941) state, while the mixing of J = 1/2 Σ c (2868)N −1 and J = 3/2 Σ c (2902)N −1 is seen on the left-hand side of the peak. Other dynamically-generated particle-hole states appear for higher and lower energies.  Finally, theD andD * spectral functions are shown in the lower left-hand side panel and lower right-hand side one, respectively. In both cases, the spectral functions show a rich structure due to the presence of several resonance-hole states. Note that those resonant states have a pentaquark-like content and have to be taken with caution.
On the one hand, the spectral function forD stems from the self-energy ofD displayed in Ref. [84]. The position of the quasiparticle peak ofD is located below theD mass and below the Θ c (2805)N −1 excitation. The C = −1 pentaquark-like resonance Θ c (2805) was a theoretical prediction of Ref. [89]. This corresponds to a pole in the free space amplitude of the sector I = 0,J = 1/2 (a weakly bound state) that strongly couples toDN andD * N , also found in Ref. [98], though it has not been observed yet.
The upper energy tail of theD spectral function shows also the contribution of I = 1 resonanthole states. On the other hand, theD * spectral function depicts the contribution of several I = 0 and I = 1 resonant-hole states close to the quasiparticle peak, that is located slightly above to 2 GeV. All these pentaquark-like states are described in Ref. [89].

Self-energy formalism and extension of the T -matrix formalism.
From this section on, and since we focus on the I C = 0 + channel, where the X(3872) is located, for DD * we mean the appropriate combination of states, (DD * − D * D ) √ 2, with even C−parity and coupled to zero isospin.

X(3872) self-energy in vacuum and in a nuclear medium
We shall now discuss the self-energy formalism for the X(3872). Let us consider a "pre-existing" state with bare massm and bare coupling squared to each of the four channelsĝ 2 /4. (The isospin related factor 1/4 is included for convenience.) The free-space bare propagator∆(q 2 ) is: Upon resumation of the contributions in Fig. 2, the dressed propagator reads: This renormalizes the mass and coupling of the state in the medium, with ρ the nuclear-matter density as in the previous sections, and the derivative taken with respect to q 2 = s. These equations are also true in particular for the ρ = 0 case, so that we can relate the bare mass and coupling to the vacuum ones, m 0 and g 0 : This allows in turn to rewrite the in-medium mass and coupling, m(ρ) and g(ρ), in terms of the physical ones in vacuum: .
Note that m 2 (ρ) is in general a complex quantity, its imaginary part being originated by that of Σ[m 2 (ρ); ρ]. 8 We can also rewrite the in-medium X(3872) propagator as: which defines the X(3872) self-energy in a nuclear medium, Π X (q 2 ; ρ). We can now rewrite Eqs. (3.5) as: Once the X(3872) propagator or self-energy are known, one can also define the X(3872) spectral function, S X (q 2 ; ρ), (3.9) The quasi-particle peak energy, E qp , is defined from the equation:

Extension of the T -matrix formalism and relation with the self-energy formalism
We now seek for a relation between the T -matrix and the self-energy formalism introduced in the previous subsection. We consider the in-medium T -matrix, , with a potential V somewhat more general than a simple constant, C 0X . Specifically, we allow for a term linear in the Mandelstam variable s, and write: , which is the same constant term that was previously considered, see Eq. (2.11). Hence, with this potential, the free-space amplitude T 0 (s) has a pole at s = m 2 0 , with coupling g 0 given by: According to the Weinberg compositeness condition [99], the factor −g 2 0 Σ 0 (m 2 0 ) represents the DD * component in the X(3872) wave function. Hence, the linear term in the potential is chosen so as to set this probability equal to P 0 .
If there is a pole of the amplitude T (s; ρ) at m 2 (ρ), then: from where one obtains the same equation for m 2 (ρ) than that obtained in Eq. (3.5a) within the self-energy formalism. Analogously, the in medium coupling g(ρ) would be given by: Note that this alternate definition of V (s) can also be written as: i.e., the kernel has a "bare" pole at the "bare" mass squaredm 2 . Then we would obtain: which allows to recover Eq. (3.5b). Equation (3.16a) should be a good approximation to (3.11a) for s in the neighborhood of m 2 0 if the factor Σ 0 (m 2 0 )(1 − P 0 )/P 0 is sufficiently small. Indeed, it has been considered also in Eq. (3.14). Hence, we find equivalence between the self-energy formalism (Subsec. 3.1) and the T -matrix formalism(s) presented here.
3.3 Extreme cases: P 0 → 0 and P 0 → 1 Let us briefly discuss here the extreme molecular or compact state scenarios, which correspond to P 0 → 1 or P 0 → 0, respectively. We start by considering the case when P 0 → 0. In this case one has g 0 = 0, i.e., the state does not couple to the two-meson channel. Physically, one would say that the interaction does not renormalize the bare state. Indeed, one sees also thatĝ = g 0 = g(ρ) = 0, and thatm = m 0 = m(ρ). This case is nonphysical, since it would require V (s = m 0 ) ∼ 1/P 0 → ∞.
Next we discuss the opposite case P 0 → 1. This situation would correspond to the pure hadronmolecular case, for which V (s) = 1/Σ 0 (m 2 0 ) is constant, independent of s. The search of a pole in the nuclear-medium T -matrix would lead to Σ[m 2 (ρ); ρ] = V −1 = Σ 0 (m 2 0 ). Actually in this limiting case, which cannot have a pole on the real axis, since Σ(s; ρ) is a complex magnitude, and if there exists a pole, it will be located at a complex value √ s = m(ρ) ∈ C (see Subsec. 4.2), and the coupling from the residue will be given by g 2 (ρ) = −1/Σ [m 2 (ρ); ρ]. These results would also make sense to the first of the Eqs. (3.18): since the denominator 1 − P 0 tends to zero, the numerator must also vanish, finding thus Σ[m 2 (ρ); ρ] − Σ 0 (m 2 0 ) = 0. Physically, taking P 0 → 1 means that the state is a purely molecular one. The "pre-existing" component is null, and thus one can neither think about the bare mass nor about the bare coupling. Indeed, for P 0 → 1, the factor 1/(1 − P 0 ) diverges and so it does 1/ 1 + g 2 0 Σ 0 (m 2 0 ) , and henceĝ → ∞ andm → ∞, in Eq. (3.4). For simplicity, in the discussion above, we have not considered the pathological case in which the bound state in vacuum is placed exactly at threshold. In that case Σ 0 (m 2 0 ) diverges, and this singular behaviour needs to be taken into account.

In medium modification of the amplitudes
We now discuss the results that we obtain in a nuclear medium for the DD * amplitude 9 |T (E; ρ)| 2 (cf. Eq. (2.20), using a very general energy dependent potential instead of just a constant C 0X ), the X(3872) self-energy Π X (E; ρ) [cf. Eq.  Fig. 3 we show the real (solid lines) and imaginary (dot-dashed lines) parts of the loop function Σ(s; ρ) computed for different densities ρ in the range 0 ρ ρ 0 , where ρ 0 is the normal nuclear density, ρ 0 = 0.17 fm −3 . We see that the sharp DD * threshold observed in the 9 We recall here again that we are working on the I C = 0 + channel, where the X(3872) is located, and that for DD * we mean the appropriate combination of states, (DD * − D * D ) √ 2, with even C−parity and coupled to zero isospin.
vacuum case (ρ = 0) is progressively smoothed out for increasing densities, being almost inappreciable for ρ = ρ 0 . This is due to the width acquired by the D,D, D * andD * mesons in the nuclear medium. We also notice that the real part of the loop function is smaller in magnitude for increasing densities. Naively, this would imply that the effect of the medium is to generate repulsion in the DD * interaction, in the sense that a more attractive potential would be necessary to compensate this change of the loop function. However, this repulsive effect is not clear, because the imaginary part of Σ(E; ρ) is also large, and below threshold, it turns out that |ImΣ(E; ρ)| |Re (Σ(E; ρ) − Σ 0 (E))|. Within the present approach, the DD * T −matrix in the nuclear environment is determined from the X(3872) mass and its DD * probability (m 0 and P 0 ) in the vacuum (ρ = 0). As we work on the isospin limit, m D ( * ) = (m D ( * )+ + m D ( * )0 ) /2, we cannot consider the physical X(3872) mass. We instead take a binding energy B = 2 MeV with respect to the DD * threshold, m 0 = m D + m D * − B. Throughout this manuscript, we will study the in-medium effects for different molecular probabilities P 0 , that enter into the calculation of the amplitude through the potentials V A (s) or V B (s), Eqs. (3.11) and (3.16), respectively. Indeed, in the lower plots of Fig. 3, we show these interaction kernels computed for different values of P 0 . Both types of interactions give the same pole position at m 2 0 and probability P 0 (alternatively, the same coupling g 0 ) for the vacuum T -matrix, although they have different analytical properties (V A (s) has a zero, while V B (s) presents a bare pole) and, hence, they might produce differences in the medium T -matrix, as we will discuss below. In the lower panels of Fig. 3 we observe, on the one hand, that for values of P 0 above P 0 = 0.8 both kernels are very similar in the energy region explored. This is due to the fact that the zero of V A (s) and the bare pole of V B (s) are far from the energies considered. On the other hand, for lower values of P 0 , e.g. P 0 = 0.2, both potentials are quite different, because the zero of V A (s) and the bare pole V B (s) come closer to the energy region of interest. Therefore, one should expect that they lead to significantly different in medium T -matrices.
Once discussed the in-medium modified DD * loop function and the energy-dependent potential, in Fig. 4 we show, for different nuclear densities and molecular probabilities P 0 = 1, 0.9 and 0.8, the squared modulus of the amplitudes T (E; ρ), normalized to be one at the maximum E max (top panels), the inverse of the X(3872) propagator, ∆ −1 (E; ρ) (medium panels), and the spectral function, S X (E; ρ) (bottom panels), conveniently scaled by Z 0 = (1 − P 0 ) and Z −1 0 , respectively. The calculations are performed using the potential V A (s), introduced in Eqs. (3.11), though, as shown above, for these high-molecular component scenarios the V B (s)−type interaction, cf. Eqs. (3.16), leads to very similar predictions, with differences that would be difficult to appreciate in the plots.
Focusing first on the squared amplitudes, it can be seen that the density behaviour is qualitatively different for the three examined probabilities. Thus, while for P 0 = 0.8 the maximum of the squared modulus is shifted to the right when the density grows (towards higher DD * c.m. energies), it however moves to the left in the purely molecular (P 0 = 1) scenarios. The results for P 0 = 1 stem from the energy and density behaviour of the factor Σ 0 (m 2 0 ) − Σ(E; ρ) in Eq. (3.19), by taking into account the in-medium two-meson loop function Σ(E; ρ) depicted in Fig. 3. For the P 0 = 0.8 case, the energy dependence of the V A potential, shown in the left-bottom plot of Fig. 3, leads to the mild shift towards higher energies of the maximum as the density increases. The position of the peak hardly changes in the intermediate P 0 = 0.9 case, displayed in the second-column plot, but as expected, the width of the in-medium X(3872) peak significantly increases with density.
Actually, in the second row of plots of Fig. 4, we see that the energy dependence of ImΠ X (E; ρ) for finite density clearly departs from the sharp step-function shape obtained in vacuum, with ImΠ X (E; ρ) becoming an increasingly smoother function of E, as the density grows. We moreover observe nonvanishing values below the free-space threshold, which increase with the density, due to the appearance of new many-body decay channels, like DD * N → DD * N , driven by the self-energies of the (anti)charmed mesons embedded in the nuclear medium. Above the free-space threshold, ImΠ X (E; ρ) decreases when the density grows. This behaviour can be inferred from the imaginary part of Σ (E; ρ) shown in the top plot of Fig. 3.
We should also note that ImΠ X (E; ρ) strongly depends on P 0 , and it behaves as g 2 0 /[1+g 2 0 Σ 0 (m 2 0 )] ∝ P 0 /(1 − P 0 ), as deduced from Eq. (3.7). Looking now at real part of ∆ −1 (E; ρ), we observe that for P 0 = 1, there is not quasi-particle solution (Eq. (3.10)) for densities higher than about one tenth of the normal nuclear matter density, with an increasingly flatter E−dependence of Re[∆ −1 (E; ρ)] as the density grows. Hence, the behavior exhibited in the P 0 = 1 case in left-top plot for the modulus squared of the amplitude, with the maximum displaced to the left with increasing densities, can be correlated to the growth of ImΠ X (E; ρ), both with the density and the c.m. energy. On the contrary, for P 0 = 0.8, we find solutions for the quasi-particle equation for all densities, at energies above threshold that move away of it as the density increases.
The spectral function plotted in the bottom panels of Fig. 4 is determined by Im[∆(E; ρ)], and its dependence on E, ρ and the molecular probability P 0 can be deduced from the discussion above on the real and imaginary parts of ∆ −1 (E; ρ) in the second-row panels of this figure. We should make here two remarks. First, we observe that the typical delta-function shape expected for the spectral function of a narrow state in the free space gets diluted as the density grows. This is due to the enhancement of the X(3872) width with density. Second, we find that, for purely molecular case (P 0 = 1), the features of the modulus squared of the T −matrix (top-left plot) can not be inferred from the spectral function S X (E; ρ). This situation slowly changes as the molecular probability decreases. Indeed, for P 0 = 0.8, we observe already some resemblances between |T (E; ρ)| 2 and S X (E; ρ). Nevertheless, the squared amplitude |T (E; ρ)| 2 is the observable that elucidates the properties of the X(3872) in the medium, especially in cases of high (dominant) molecular components in its vacuum structure.
Next, in Figs. 5 and 6 we consider smaller molecular components, P 0 = 0.4 and P 0 = 0.2. As we discussed in Fig. 3, for these probabilities, the V A (s) [Eq. (3.11)] and V B (s) [Eq. (3.16)] potentials, despite leading to the same mass (m 0 ) and DD * coupling (g 0 ) of the X(3872) in the free space, considerably differ in the region of interest for the present study. Hence, the corresponding T −matrices are different, even in the free space. Those deduced from V A show the zero that this potential has below m 0 . As the molecular probability decreases, this zero gets closer to the X(3872) vacuum mass, since the slope of V A (s) grows (in absolute value) as 1/P 0 . The position of the zero is independent of the nuclear density, being, however, the dependence of the amplitude on the density clearly visible, both for energies below and above the energy, E 0 , for which the potential and scattering amplitude vanish. Density effects for energies lower (higher) than E 0 become more (less) relevant for the P 0 = 0.2 case than for the P 0 = 0.4 one.
In sharp contrast to the results stemming from V A (s), when |T (E; ρ)| 2 is computed using the V B (s) interaction, we see little structure beyond the peak induced by the bare pole present in the potential. The effects due to the medium dressing are small for P 0 = 0.4 and already quite difficult to disentangle for P 0 = 0.2. Hence, experimental input on |T (E; ρ)| 2 , especially for energies below E 0 , might shed light into the dynamics of the interacting DD * pair that could be difficult to infer from their scattering in the free space.
In Figs. 5 and 6 we also show the inverse propagator ∆ −1 (E; ρ) and the spectral function S X (E; ρ). These quantities do not depend on the type of potential employed -V A (s) or V B (s)-, since they are determined by the vacuum X(3872) and the in-medium two-meson loop function Σ(E; ρ) given in Fig. 3. In what respects to the Im[∆ −1 (E; ρ)], the results here are the same as those discussed above in Fig 4, scaled down by the corresponding factor P 0 /(1 − P 0 ). On the other hand, the plots for real part of ∆ −1 (E; ρ) show that, for small molecular components, there is always a quasi-particle solution very close to m 0 , and very little affected by the nuclear matter density. Finally, the spectral function S X (E; ρ) embodies the main features of |T (E; ρ)| 2 when the potential V B is used. However, it does not account for the medium modifications observed in the T − matrix below E 0 when V A is employed.
To conclude, in Figs  and S X (E; ρ), respectively, for all molecular probabilities considered above in Figs. 4-6. We also give the quasi-particle energies, E qp , obtained by solving Re[∆ −1 (E; ρ)] = 0 when they exist. For low values of P 0 (Fig. 8), we provide separately E max obtained from V A (s) or of V B (s) potentials. The results in these two figures reinforce the conclusions previously outlined. Indeed, we graphically see for the highest values of P 0 , the appreciable difference between E max and E spe , with even an opposite density slope in the P 0 = 1 case. In Fig. 7, we only observe for P 0 = 0.8 some resemblances between the maxima of |T (E; ρ)| 2 and S X (E; ρ), with quasi-particle energies well separated from both of them and exhibiting a significantly larger sensitivity with density. Medium effects are much smaller in Fig. 8, where results for P 0 = 0.2 and P 0 = 0.4 are collected. Some differences between E max obtained from V A or V B potentials are visible, even for the lowest of the molecular probabilities, for densities close to ρ 0 . The quasi-particle and spectral-function energies are closer, and for P 0 = 0.2 become indistinguishable from E max computed using V B . This supports that, in this case, one is dealing with a compact state little affected by the dressing of the meson loops in the medium.

Poles in the complex plane
As already mentioned, the integral representation of Eq. (2.21) for the in-medium loop function Σ(s; ρ) is not well suited for its continuation into the whole complex plane. The rich dynamical structure of the spectral functions S D ( * ) and SD( * ) shown in Fig. 1 is washed out by the Ω and ω integrations implicit    Figure 7. Positions E max and E spe of the maxima of |T (E; ρ)| and S X (E; ρ), respectively, as a function of the nuclear matter density. From left to right, the three plots show the cases P 0 = 1, 0.9, and 0.8. In the latter case, we also give the quasi-particle energy,   in Eq. (2.21) (see also Eq. (2.7)). Thus, almost no trace of the several peaks present in Fig. 1 can be distinctly appreciated in the resulting loop functions Σ(s, ρ) depicted in Fig. 3 for several densities. Actually, the latter are essentially equivalent to the loop function of a two-meson system regulated via a hard cutoff Λ, but evaluated with complex masses. Hence, we make the following approximation: with m (eff) D ( * ) complex valued, and the superscript "(eff)" is included to remark that these are densitydependent effective masses, and do not correspond to the pole positions associated to the D ( * ) and D ( * ) peaks in Fig. 1. Additional details, including a discussion on the accuracy of the approximation, can be found in Appendix A.
By means of the approximation in Eq. (4.1) we can now compute the in-medium T DD * (s; ρ) in the whole complex plane, for the different medium densities ρ and vacuum probabilities P 0 , and search for poles in the complex plane. We find a pole on the first Riemann sheet of the amplitude (as defined in Appendix A), off the real axis. This does not represent any violation of the analyticity properties of the complete-system scattering T −matrix, because of the effective procedure used to take into account the many body channels of the type DD * N → DD * N . In the present scheme, they are not explicitly considered in the coupled-channel space and only their effects on DD * → DD * are included through the in-medium charmed-meson self-energies.
The pole position depends on the nuclear medium density ρ and on the value chosen for the parameter P 0 , the X(3872) molecular probability in the vacuum. The pole position is represented in Fig. 9 for different values of P 0 and ρ, with each of the colors associated to a particular density, and both V A (s) (left) and V B (s) (right) free space DD * −potentials considered in this work. For each density, the zigzag lines represent the loop function G (eff) (s; ρ) right hand cut: √ s ∈ C Im p 2 (s, m  D * ), and that the effect is large or small depending on whether the in-vacuum probability P 0 is close to 1 or to 0, respectively. We also observe some dependence of the pole position, which as expected grows as the molecular content P 0 deviates from 1, on the used DD * interaction in the free space, namely V A (s) (Eq. (3.11), left plot of Fig. 9) or V B (s) (Eq. (3.16), right plot of Fig. 9).
In our amplitudes, the vacuum molecular probability P 0 is a free parameter that we have varied to explore different scenarios. We can define the quantity P ρ , which generalizes to the nuclear medium the formula for the vacuum probability [cf. Eq. (3.13)]. Since the pole position is in general complex, so will be this quantity. Therefore, in general, it will not be possible to interpret it as a probability. In Fig. 10, we show P ρ for different nuclear densities as a function of the vacuum probability P 0 . This figure complements the results of Fig. 9. We observe for 11 Note that, as previously discussed, there is little difference between the results obtained with VA(s) or VB(s) when P0 is close to one. Therefore, the argument presented here with VA(s) can be readily applied to the case of VB(s).
Re P ρ 0.30ρ 0 Im P ρ 0.50ρ 0 P 0 0.75ρ 0 1.00ρ 0  Fig. 10), but the qualitative behaviour is very similar. In the intermediate regions, far from the end points P 0 = 0 and P 0 = 1, the imaginary part of P ρ can be sizeable, and for most of these values it increases with the density. In general, the effect of the nuclear medium in this intermediate P 0 region is to decrease both the real part and the modulus of P ρ with respect to its original value P 0 . However, we see that for both ends P 0 → 0 or P 0 → 1, we have that Im P ρ 0, and P ρ P 0 . We thus see that in these cases the X(3872) state can be said to conserve its original nature in the nuclear medium.

Conclusions
In this work we have studied the behaviour of the χ c1 (3872), also known as X(3872), in dense nuclear matter. The X(3872) appears in the vacuum as a pole in the DD * scattering amplitudes, which are parametrized in a quite general form. The in-medium effects have been incorporated by dressing the DD * loop functions with the corresponding spectral functions of the charmed mesons. As a result, the DD * amplitudes, when the charmed mesons are embedded in the nuclear medium, have been determined for energies around the nominal X(3872) mass. The X(3872) spectral function has been also obtained for densities ranging up to that of nuclear matter saturation.
For the kernel of the DD * scattering, we have used two possible energy-dependent potentials, each of them depending on two free parameters. Imposing that the vacuum amplitude has a pole in the physical Riemann sheet, these two parameters allow to fix the nominal X(3872) mass and its coupling to the DD * channel, or, alternatively, the mass and the molecular probability P 0 . Therefore, both types of interactions allow for the study of the X(3872) as either a pure hadron-molecule state or a genuine quark state, as well as intermediate possibilities, in terms of P 0 . However, both types of interactions have different analytical properties, which can give rise to different scattering amplitudes at finite density.
Using these two models for the interaction, we have explored the connection between the inmedium behaviour of the X(3872) and its nature. In the case of the X(3872) being mostly a molecular state, both interaction potentials behave similarly and lead to equivalent results for the in-medium amplitudes. In this case, we have found that the DD * amplitudes strongly depend on the density. The width of the X(3872)-peak significantly grows when the density is increased, while its position moves to higher energies, as the molecular component is lowered. The X(3872) spectral function follows the imaginary part of the X(3872) self-energy, that increases with density due to the appearance of new many-body decay channels in matter. On the other hand, when smaller molecular components are considered, the DD * amplitudes depend on the choice of the energy-dependent potential, specially for energies below the free-space X(3872) mass. Hence, the experimental input on the amplitudes at finite density might shed light into the dynamics of the DD * interaction in the case of a state with a large genuine constituent quark component. Moreover, in this case, the X(3872) spectral function, which is independent of the potential employed, is very little affected by the density.
The in-medium DD * loop functions strongly depend on the interaction of D, D * ,D andD * with nuclear matter. However, one can reasonably approximate them by a standard loop function evaluated with complex, effective masses of the D ( * ) andD ( * ) mesons. This fact allows for an analytical continuation of the loop function, and hence of the scattering amplitude, to the whole complex plane and to the second Riemann sheet. In turn, this allows for the search of the pole associated to the X(3872) in the nuclear medium. For finite density, the pole is found in the first Riemann sheet, but in the complex energy plane. However, this does not represent any violation of the analyticity properties of the T −matrix, because, in the present scheme, the DD * N → DD * N many-body channels are not explicitly considered in the coupled-channel space, since their effects on DD * → DD * are included via the in-medium charmed-meson self-energies. The behaviour of the X(3872) pole with density is moreover fully in line with the change in matter of the squared modulus of the T −matrix amplitudes for real energies. Complex poles for the X(3872) produced inside of a nuclear medium are collected in Fig. 9, for different densities and free-space molecular probabilities. In the light of these results, we conclude that for the nuclear matter saturation density and molecular components of the order of 60% for the X(3872), the many-body modes considered in this work provide widths for this resonance of around 30-40 MeV, and more modest mass-shifts (repulsive) with a maximum of 10 MeV. This latter outcome contradicts the results obtained in the QCD-sum-rule calculation carried out in Ref. [72] and based on a diquark-antidiquark picture for the X(3872). Indeed, in the approach of Ref. [72], the massshift due to the nuclear matter is negative and is about 25% (∼ 800-900 MeV) when the saturation density is used. Therefore, any experimental analyses on the in-medium properties of X(3872) and comparison of those with the results of the present study can increase our knowledge of the X(3872) and help us gain useful information on the not well-known structure of this exotic state.
In this work we have studied the contribution of the dominant DD * channel to the X(3872) dynamics. In the future, we aim at extending our calculation to a more realistic situation by incorporating also coupled channels involving hidden-charm mesons, such as J/ψ π. Also, the results presented in this manuscript are based on a specific model for the D ( * ) N andD ( * ) N interactions, which determine the in-medium modifications of the DD * loop functions. Different or more elaborate models for these amplitudes could also be employed in the formalism we have derived here. In any case, our results indicate a very different behaviour with density of the DD * amplitudes and the X(3872) spectral function depending on the nature of the X(3872). Thus, experiments that can access the nuclear finite-density regime, such as HiCs like CBM or those with fixed nuclear targets such asp-nuclei in PANDA, are necessary and complementary to the spectroscopic analyses so as to discern the nature of X(3872).
In this Appendix we give further details on the approximation made in Sec. 4.2, and on the analytical properties of the loop function employed. The approximation is: where G(s, m 1 , m 2 ) can be computed using the explicit formulas given for instance in Ref. [81] (see in particular the erratum), regulated with a momentum cutoff of 0.7 GeV. In addition, we take for the density dependent effective masses 2 with respect to the vacuum masses. Being this an effective representation, we find that this ansatz is enough to approximate the original loop function, Σ(s, ρ). In Fig. 11 we show in the left (right) panel the imaginary (real) part of the loop function Σ(s, ρ) together with the approximation determined by Eq. (4.1), computed with the parameters ∆m(ρ) and Γ(ρ) collected in Table 1. The latter are chosen so as to approximately match the original loop functions Σ(s; ρ) for the different densities considered in this work. As can be seen, the approximation works reasonably well.