D\bar{D}^* scattering and χ_{c1}(3872) in nuclear matter
JJLAB-THY-21-3321 D ¯ D ∗ scattering and χ c (3872) in nuclear matter M. Albaladejo a J. Nieves b L. Tolos c,d,e,f a Theory Center, Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA b Instituto de F´ısica Corpuscular (IFIC), Centro Mixto CSIC-Universidad de Valencia, Institutos de Investi-gaci´on de Paterna, Aptd. 22085, E-46071 Valencia, Spain c Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, 08193 Barcelona, Spain d Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain e Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany f Faculty of Science and Technology, University of Stavanger, 4036 Stavanger, Norway
Abstract:
We study the behaviour of the χ c (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 ¯ D ∗ − c.c. ) state, generatedas a bound state from a heavy-quark symmetry leading-order interaction between the charmed mesons,and analyze the D ¯ D ∗ scattering T − matrix ( T D ¯ D ∗ ) inside of the medium. Next, we consider also mixed-molecular scenarios and, in all cases, we determine the corresponding X (3872) spectral function andthe D ¯ D ∗ amplitude, with the mesons embedded in the dense environment. We find important nuclearcorrections for T D ¯ D ∗ and the pole position of the resonance, and discuss the dependence of theseresults on the D ¯ D ∗ molecular component in the X (3872) wave-function. These predictions could betested in the finite-density regime that can be accessed in the future CBM and PANDA experimentsat FAIR. Contents T -matrix formalism 4 D ¯ D ∗ scattering amplitude and X (3872) 42.2 Vacuum 52.3 Isospin–symmetric nuclear matter 62.4 S D ( ∗ ) and S ¯ D ( ∗ ) in nuclear matter 7 T -matrix formalism. 10 X (3872) self-energy in vacuum and in a nuclear medium 103.2 Extension of the T -matrix formalism and relation with the self-energy formalism 113.3 Extreme cases: P → P → a r X i v : . [ h e p - ph ] F e b Further details on G (eff) ( s ; ρ ) 23 The new quarkonium revolution started in 2003 with the discovery of the X (3872) (recently renamedas χ c (3872) [1]). It was firstly observed in B ± → K ± π + π − J/ψ decays by Belle [2], and subsequentlyconfirmed by BaBar [3], CDF [4–6], D ∅ [7], LHCb [8, 9] and CMS [10]. The spin-parity quantumnumbers J P = 1 ++ were extracted at the 8 σ level in 2013 from the high-statistic measurements of thetwo-pion mode performed in the LHCb experiment [11]. A distinctive feature of the X (3872) is that the ρJ/ψ and ωJ/ψ branching fractions are similar. This points out to an isospin symmetry violation [12],which together with the large disparity between ω and ρ meson widths provides a natural explanationto the observed ρJ/ψ to ωJ/ψ decay ratio [13, 14].The X (3872) is one of the most studied exotic mesons with a c ¯ c content. This state lies extremelyclose to the D ¯ D ∗ threshold, and its (Breit-Wigner) width has been recently measured as Γ =1 . . B -mesons, that include two- (referred as ρJ/ψ as it originatesfrom ρ ) and three-pion (named as ωJ/ψ as it comes from ω ) modes, or Λ b − baryons, as well as incharmonia radiative decays and through lepto- or photo-production. In addition, exhaustive sensitivitystudies for width and line-shape measurements of the X (3872) have been carried out for the reaction p ¯ p → J/ψρ with the PANDA experiment at FAIR [17], and the possibilities of X (3872) photo-production off the nucleon have also been studied [18].In spite of all this experimental progress, the nature of the X (3872) is still elusive. From thepoint of view of constituent quark models, the most natural possibility for the X (3872) is a 2 P c ¯ c charmonium configuration, i.e. , the χ c (2 P ) state. However, the quark model calculations givea value for the mass of this state higher than the experimental one (see, for example, Refs. [19–21]). Moreover, the isospin symmetry violation is difficult to explain using a simple c ¯ c model. Thus,new interpretations have been put forward. On the one hand, this state might be interpreted as acompact diquark and antidiquark (tetraquark) state [22–24]. On the other hand, this state could bean example of a loosely bound hadron molecule (see, for example, Refs. [12–14, 25–27]). The vicinityto the D ¯ D ∗ threshold and the large decay rate to D ¯ D ∗ together with a natural explanation of theisospin symmetry violation have made this interpretation quite popular. Also, other interpretationsinclude hadrocharmonium [28], a mixture between charmonium and exotic molecular states [29–31] orsome relation with a X atom, which is a D ± D ∗∓ composite system with positive charge-conjugationand a mass of ∼ pp reactions or relativistic heavy-ion collisions (HiCs) has become a matter ofrecent interest as the production yield of these exotic hadrons could reflect their internal structure.The high prompt production cross section of the X (3872) measured for pp at CDF [5] and inCMS [10] has cast doubts on its possible interpretation as a D ¯ D ∗ molecule, since it was argued thatthe production of a weakly bound state should be strongly suppressed in high-energy collisions [53].However, this finding has been put into question in Ref. [54], showing that the estimates for the cross A heavy quark-antiquark bound state, characterized by the radial number n , the orbital angular momentum L ,the spin s and the total angular momentum J , is denoted by n s +1 L J . Parity and charge conjugation are given by P = ( − L +1 and C = ( − L + s , respectively. – 2 –ections using the molecular approach are consistent with the CDF and CMS measurements by anadequate election of the ultraviolet (UV) cutoff [54], a statement that has been in turn criticized inRef. [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 latterwork, it is shown that the prompt X (3872) cross section at hadron colliders is consistent with thoseexperimentally observed at CDF and CMS. This is concluded thanks to the derivation of a relationbetween the prompt X (3872) cross section and that of a charm-meson pair, taking into account thethreshold enhancement from the X (3872) resonance. More recently, the production rates of promptlyproduced X (3872) relative to the ψ (2 S ) as a function of the final state particle multiplicity, obtainedrecently 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 sectionare 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 willgive excellent fits to the LHCb data using parameters consistent with X (3872) being a loosely boundcharm-meson molecule. Thus, there is still an ongoing debate on the nature of the X (3872) comingfrom the analysis of pp collisions.Another possible way to gain some insight about the nature of X (3872) is to analyze its behaviorfor the extreme conditions present in HiCs at RHIC and LHC energies. The ExHIC Collaboration[60–62] has shown that, within the coalescence model, the molecular structure for the X (3872) impliesa production yield much larger than for the tetraquark configuration, in particular if one also takesinto account the evolution in the hadronic phase [63, 64]. This is due to the fact that molecules arebigger than tetraquarks and, hence, the production and absorption cross sections in HiCs are expectedto be larger. This was actually shown in Ref. [64], where the time evolution of the X (3872) abundancein 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 withininstantaneous coalescence models [66, 67], but also using a statistical hadronization model [68] or bymeans of a thermal-rate equation approach [69]. In those studies it is advocated that the quantitativedescription 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 modifi-cation of the hot hadronic phase. Only recently the behaviour of X (3872) in a finite-temperature pionbath has been studied assuming this resonance to be a molecular state generated by the interactionof D ¯ D ∗ + 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 itsnominal mass moves above the D ¯ D ∗ threshold.In the present work we address the behaviour of X (3872) in a nuclear environment, with theobjective of analyzing the finite-density regime that can be accessed in HiCs and the future experimentsat FAIR. An early study of the D ∗ s (2317) and theorized X (3700) scalar mesons in a nuclear mediumwas performed in Ref. [71], already showing that the experimental analysis of the properties of thosemesons 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, revealingthat 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 D ¯ D ∗ +c.c. state, generated as apurely molecular bound state from the leading-order interaction of the D and ¯ D ∗ mesons, which isconstrained by heavy-quark spin symmetry (HQSS) [73–76]. HQSS predicts that all types of spininteractions vanish for infinitely massive quarks, that is, the dynamics is unchanged under arbitrarytransformations in the spin of the heavy quark. As a consequence, open charm pseudoscalar and vectormesons 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– 3 –nd the corresponding X (3872) spectral function. Later on, we consider generalizations of the D ¯ D ∗ interaction, allowing for scenarios in which the X (3872) is not a purely molecular state, i.e. , it can be acompact state, and we also study mixed scenarios. In this way, we extract the modification on the massand the width of X (3872) in nuclear matter for different scenarios, in view of the forthcoming resultson charmed particles in HiCs at CBM in FAIR [77, 78]. Moreover, the present study will be also ofinterest for PANDA, since it is expected that the X (3872) will strongly couple to the ¯ pp channel [17],and therefore, this resonance can be produced also in ¯ pA collisions [79]. Actually colliding antiprotonson nuclei with PANDA would allow the A − dependence of the production of ψ (2 S ) and X (3872) nearthreshold to be compared. This may, after appropriate theoretical study, provide a good way to exposean extended D ∗ ¯ D component of the X (3872) state function [80].This work is organized as follows. In Sec. 2 we present the D ¯ D ∗ scattering amplitude and the X (3872) in vacuum and in isospin-symmetric nuclear matter, while showing the open-charm ground-state spectral functions in matter. In Sec. 3 we determine the X (3872) self-energy both in vacuumand in nuclear matter, while connecting the self-energy to the D ¯ D ∗ scattering amplitude. We finish bypresenting 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 D ¯ D ∗ T − matrix to thecomplex plane, allowing for the search of poles reported in Sec. 4. T -matrix formalism D ¯ D ∗ scattering amplitude and X (3872) To study the X (3872) as a molecular state in the D ¯ D ∗ I G ( J P C ) = 0 + (1 ++ ) channel, we start byconsidering the interaction in the particle basis: { D ¯ D ∗ , D ∗ ¯ D , D + ¯ D ∗− , D ∗ + D − } . (2.1)The unitary T -matrix for this basis is written as: T − ( s ) = V − ( s ) − G ( s ) , (2.2)with √ s the energy of any of the pairs in the center of mass (c.m.) frame, and the V and G matricesare constructed out of the interaction potential and the two-meson loop functions, respectively. Fromthe leading order HQSS-based Lagrangian, V can be written as a contact interaction [73–76] V ( s ) = A − V d ( s ) A , (2.3)where V d ( s ) = diag( e C Z , e C X , e C Z , e C X ) is a diagonal matrix (the notation for the matrix elementswill be explained below), and the matrix A (satisfying A T = A − = A ) reads: A = 12 +1 +1 +1 +1+1 − − − − − − . (2.4)The G ( s ) matrix is diagonal, and contains the loop-function for the different two-meson channels inthe particle basis, Eq. (2.1), G i ( s ) = i Z d q (2 π ) D Y i ( P − q ) D Y i ( q ) . (2.5) Note that the e C low-energy constants here are dimensionless, while those introduced in [73–76] have dimensions offm . This is because here we adopt relativistic D ( ∗ ) − meson propagators and non-relativistic kinematics was used in theprevious works. – 4 –here 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 = s . In terms of the self-energies Π Y ( q ) of the latter, they can be written as: D Y ( q ) = 1( q ) − ω Y ( ~q ) − Π Y ( q , ~q ) = Z ∞ dω (cid:18) S Y ( ω, | ~q | ) q − ω + iε − S ¯ Y ( ω, | ~q | ) q + ω − iε (cid:19) (2.6)with ω Y ( ~q ) = q m Y + ~q . Note that we will be here only interested in the nuclear medium renormal-ization of the meson properties. Thus, m Y is the meson mass in the free space, while the self-energyΠ Y approaches zero when the nuclear density ρ →
0. Inserting the above representation into Eq. (2.5)and integrating over q leads to: G i ( P , ~P ) = 12 π Z ∞ dΩ f Y i Y i (Ω , | ~P | ) P − Ω + iε − f Y i Y i (Ω , | ~P | ) P + Ω − iε , (2.7a)with: f UW (Ω , | ~P | ) = 14 π Z Λ0 d ~q Z Ω0 d ωS U ( ω, | ~P − ~q | ) S W (Ω − ω, | ~q | ) , (2.7b)where U and W stand for Y i and Y i or Y i and Y i . In the previous equations we have already introduceda sharp momentum cut-off Λ to regularize the UV behavior of the integration over the modulus of ~q .Specifically, we take Λ = 0 . In vacuum, assuming isospin symmetry, m D ( ∗ )0 = m D ( ∗ +) , the loop functions for the four channels areequal, G ( s ) = Σ ( s ) I . The function Σ ( s ) reduces to a standard loop function regulated via a hardcutoff Λ, G ( s, m D , m D ∗ ), and expressions for this can be found in Ref. [81]. The T -matrix diagonalizesin the same way as the kernel matrix V ( s ), i.e. , T − ( s ) = A − T − d ( s ) A , (2.8)where: T − d ( s ) = diag (cid:16) e C − Z − Σ ( s ) , e C − X − Σ ( s ) , e C − Z − Σ ( s ) , e C − X − Σ ( s ) (cid:17) . (2.9)From the eigenvectors of T (and V ) one notices that they are D ¯ D ∗ states with well defined isospin I ( I = 0 ,
1) and C -parity (charge-conjugation) quantum numbers. The notation e C I C for the low energyconstants 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 X , T − X ( s ) = e C − X − Σ ( s ) . (2.10)We can thus fix the constant e C X by requiring the presence of a pole at an energy equal to the X (3872)mass m , e C X = 1 / Σ ( m ) . (2.11)We will also consider below (see Subsec. 3.2) more general scenarios, in which energy dependence willbe actually allowed in the kernel V ( s ). Note that, due to the regularization procedure, we should actually write C e C X (Λ) = Σ ( m ; Λ). For the sake ofbrevity, we omit this dependence on the UV cutoff throughout the manuscript. – 5 – .3 Isospin–symmetric nuclear matter To consider the possible modification of the X (3872) properties in a nuclear medium, we assumethat the D ¯ D ∗ interaction potentials V d ( s ) do not change in nuclear matter, and that the T -matrixis modified through the loop functions because of the D ( ∗ ) and ¯ D ( ∗ ) self-energies. We still assumeisospin symmetry, m D ( ∗ )0 = m D ( ∗ )+ , and S D ( ∗ )+ = S D ( ∗ )0 ≡ S D ( ∗ ) , S D ( ∗ ) − = S ¯ D ( ∗ )0 ≡ S ¯ D ( ∗ ) . However,in general we will have S ¯ D ( ∗ ) = S D ( ∗ ) in the nuclear environment, since the charmed and anti-charmedmeson–nucleon interactions are quite different. In addition, pseudo-scalar-nucleon and vector-nucleoninteractions are also different and hence S D = S D ∗ and S ¯ D = S ¯ D ∗ . We discuss the spectral functions S ¯ D ( ∗ ) and S D ( ∗ ) in nuclear matter in Sec. 2.4. Consequently, the G -matrix in a nuclear medium ofdensity ρ , G ( s ; ρ ), is no longer proportional to the identity, as opposed to the vacuum case. It reads G ( s ; ρ ) = diag ( G D ¯ D ∗ ( s ; ρ ) , G D ∗ ¯ D ( s ; ρ ) , G D ¯ D ∗ ( 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, T − ( s ; ρ ) = V − ( s ) − G ( s ; ρ ) = A (cid:16) V − d ( s ) − A G ( s ; ρ ) A (cid:17) A , (2.12)with A G ( s ; ρ ) A = e G ( s ; ρ ) 00 e G ( s ; ρ ) ! . (2.13)The 2 × e G can be written as: e G ( s ; ρ ) = Σ( s ; ρ ) δ G ( s ; ρ ) δ G ( s ; ρ ) Σ( s ; ρ ) ! , (2.14)with: Σ( s ; ρ ) = G D ¯ D ∗ ( s ; ρ ) + G D ∗ ¯ D ( s ; ρ )2 , (2.15)and: δ G ( s ; ρ ) = G D ¯ D ∗ ( s ; ρ ) − G D ∗ ¯ D ( s ; ρ )2 . (2.16)In other words, defining the states | I Ci , we have the following matrix elements: D I C (cid:12)(cid:12)(cid:12) ˆ T ( s ; ρ ) (cid:12)(cid:12)(cid:12) I C E = δ I,I T ( I ) C , C ( s ; ρ ) (2.17)The amplitudes T ( I ) C , C are compactly defined as: h T ( I ) XX ( s ; ρ ) i − = h T ( I ) X ( s ; ρ ) i − − T ( I ) Z ( s ; ρ ) δ G ( s ; ρ ) , (2.18a) h T ( I ) ZZ ( s ; ρ ) i − = h T ( I ) Z ( s ; ρ ) i − − T ( I ) X ( s ; ρ ) δ G ( s ; ρ ) , (2.18b) h T ( I ) XZ ( s ; ρ ) i − = h δ G ( s ; ρ ) T ( I ) X ( s ; ρ ) T ( I ) Z ( s ; ρ ) i − − δ G ( s ; ρ ) , (2.18c)where T X ( s ; ρ ) and T Z ( s ; ρ ) are written as in the diagonal case, h T ( I ) X ( s ; ρ ) i − = e C − IX − Σ( s ; ρ ) , (2.19a) h T ( I ) Z ( s ; ρ ) i − = e C − IZ − Σ( s ; ρ ) . (2.19b)Equation (2.17) may seem counter intuitive, due to the absence of a δ C , C factor. However, we mustbear in mind that, in the presence of nuclear matter, the scattering processes are D ¯ D ∗ N → D ¯ D ∗ N .Due to the presence of the nucleons, the D ¯ D ∗ in the initial and final states do not need to have thesame C -parity. This approximation is justified because they are short range (contact) interactions. Note for example that a ¯ DN resonance would imply a pentaquark-like structure. – 6 –e have checked that the term δ G ( s ; ρ ) is small, so we consider throughout this manuscript thelimit 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 D ¯ D ∗ C -parity amplitudes, too. We thus find, for the I C = 0 + channel, T − ( s ; ρ ) = e C − X − Σ( s ; ρ ) . (2.20)Note that, from its definition, Σ( s ; ρ ) can be written more compactly as:Σ( P , | ~P | ; ρ ) = 14 π Z ∞ dΩ (cid:18) P − Ω + iε − P + Ω − iε (cid:19) × (cid:16) f D ¯ D ∗ (Ω , | ~P | ) + f D ∗ ¯ D (Ω , | ~P | ) (cid:17) (2.21)where the dependence on the density arises from that of the spectral functions involved in the aboveequation. We recall that the expressions for f D ¯ D ∗ , f D ∗ ¯ D are given in Eq. (2.7b). Finally, we note thatin the ρ → P , | ~P | ; ρ ) could becomputed for complex values of the energy P . However, we can neither perform its analytical continu-ation 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 itsarguments, which cannot be computed within the standard scheme that will be presented below, seeSubsec. 2.4. Nevertheless, as discussed below in Subsec. 4.2, we will derive a reasonable approximationfor the in-medium loop-function Σ( P , | ~P | ; ρ ) of Eq. (2.15), which will allow for a meaningful extensionof the isoscalar T -matrix to the complex plane and the search for poles also in nuclear matter. S D ( ∗ ) and S ¯ D ( ∗ ) in nuclear matter The spectral functions of D ( ∗ ) and ¯ D ( ∗ ) in symmetric nuclear matter are obtained following a unitarizedself-consistent procedure in coupled channels, as described in Refs. [82, 83] for the D ( ∗ ) meson and inRef. [84] for ¯ D ( ∗ ) meson. In the following we present the main features.The s -wave transition charmed meson–nucleon kernel of the Bethe-Salpeter equation (BSE) isderived from an effective Lagrangian that implements HQSS [85–87]. HQSS is an approximate QCDsymmetry that treats on equal footing heavy pseudoscalar and vector mesons, such as charmed andbottomed mesons [78, 82–84, 88–94]. The effective Lagrangian accounts for the lowest-lying pseu-doscalar and vector mesons as well as 1 / + and 3 / + baryons. It reduces to the Weinberg-Tomozawa(WT) interaction term in the sector where Goldstone bosons are involved and incorporates HQSS inthe sector where heavy quarks participate. Thus, it is a SU(6) × HQSS model, that is justified in viewof the reasonable semi-qualitative outcome of the SU(6) extension [95] and on a formal plausiblenesson how the SU(4) WT interaction in the heavy pseudoscalar meson-baryon sectors comes out in thevector-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 v CSIJij ( √ t ) = D CSIJij √ t − M i − M j f i f j s E i + M i M i s E j + M j M j , (2.22)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 themeson in the i -channel. Symmetry breaking effects are introduced by using physical masses and decayconstants. The D CSIJij are the matrix elements coming from the group structure of the extended WTinteraction. Since from now on-wards the focus will be exclusively on this channel, we will omit the 0 X subindex for simplicity. – 7 –he amplitudes in nuclear matter, t ρ,CSIJ ( R , ~R ) with R = ( R , ~R ) the total meson-baryonfour-momentum ( t = R ), are obtained by solving the on-shell BSE using the previously describedpotential, v CSIJ ( √ t ): t ρ,CSIJ ( R ) = h − v CSIJ ( √ t ) g ρCSIJ ( R ) i − v CSIJ ( √ t ) , (2.23)where the diagonal g ρCSIJ ( R ) loop-matrix accounts for the charmed meson–baryon loop in nuclearmatter [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 = − DN and ¯ D ∗ N . The D ( ¯ D ) and D ∗ ( ¯ D ∗ ) self-energies in symmetric nuclear matter, Π( E, ~q ; ρ ), are obtained bysumming the different isospin transition amplitudes for D ( ¯ D ) N and D ∗ ( ¯ D ∗ ) N over the nucleon Fermidistribution, p F . For the D ( ¯ D ) we haveΠ D ( ¯ D ) ( q , ~q ; ρ ) = Z p (cid:54) p F d p (2 π ) h t ρ, , / D ( ¯ D ) N ( R , ~R ) + 3 t ρ, , / D ( ¯ D ) N ( R , ~R ) i , (2.24)while for D ∗ ( ¯ D ∗ )Π D ∗ ( ¯ D ∗ ) ( q , ~q ; ρ ) = Z p (cid:54) p F d p (2 π ) " t ρ, , / D ∗ ( ¯ D ∗ ) N ( R , ~R ) + t ρ, , / D ∗ ( ¯ D ∗ ) N ( R , ~R )+ (2.25)23 t ρ, , / D ∗ ( ¯ D ∗ ) N ( R , ~R ) + 2 t ρ, , / D ∗ ( ¯ D ∗ ) N ( R , ~R ) . In the above equations, R = q + E N ( ~p ) and ~R = ( ~q + ~p ) are the total energy and momentumof the meson-nucleon pair in the nuclear matter rest frame, and ( q , ~q ) and ( E N , ~p ) stand for theenergy and momentum of the meson and nucleon, respectively, in that frame. Those self-energies aredetermined self-consistently since they are obtained from the in-medium amplitudes which contain themeson-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: D ρD ( ¯ D ) ,D ∗ ( ¯ D ∗ ) ( q , ~q ) = (cid:16) ( q ) − ~q − m − Π D ( ¯ D ) ,D ∗ ( ¯ D ∗ ) ( q ) (cid:17) − ,S D ( ¯ D ) ,D ∗ ( ¯ D ∗ ) ( q , ~q ) = − π Im D ρD ( ¯ D ) ,D ∗ ( ¯ D ∗ ) ( q ) (for q > . (2.26)The D ( ¯ D ) and D ∗ ( ¯ D ∗ ) spectral functions are shown in Fig. 1 as function of the meson energy E = q for zero momentum ~q = 0 and two different densities, ρ = 0 . ρ and ρ = ρ . Apart fromthe quasiparticle peak, obtained from E = ~q + m + 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-holeexcitations. The masses and widths of these resonances were obtained in Refs. [88–90].The D meson spectral function is depicted in the upper left-hand side panel. As described inRef. [82], the D meson quasiparticle peak moves to lower energies with respect to the free mass positionas 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 − and Λ c (2595) N − excitations, whereas Σ ∗ c N − 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 right-hand side panel, the quasiparticle peak moves to higher energies with density and fully mixes withthe sub-threshold J = 3 / c (2941) state, while the mixing of J = 1 / c (2868) N − and J = 3 / c (2902) N − is seen on the left-hand side of the peak. Other dynamically-generated particle-holestates appear for higher and lower energies. Note that D denotes D + and D , whereas ¯ D indicates D − and ¯ D . – 8 – . . . . E (MeV) S D ( E , ~ q = ) (cid:2) G e V − (cid:3) ρ = 0 . ρ ρ = 0 . ρ ρ = 1 . ρ . . . . . E (MeV) S D ∗ ( E , ~ q = ) (cid:2) G e V − (cid:3) ρ = 0 . ρ ρ = 0 . ρ ρ = 1 . ρ .
84 1 .
86 1 .
88 1 . . E (MeV) S D ( E , ~ q = ) (cid:2) G e V − (cid:3) ρ = 0 . ρ ρ = 0 . ρ ρ = 1 . ρ . .
95 2 2 .
05 2 . E (MeV) S D ∗ ( E , ~ q = ) (cid:2) G e V − (cid:3) ρ = 0 . ρ ρ = 0 . ρ ρ = 1 . ρ Figure 1 . The D (upper left-hand side), ¯ D (lower left-hand side), D ∗ (upper right-hand side) and ¯ D ∗ (lowerright-hand side) spectral functions as function of the meson energy E and zero momentum ~q = 0 for twodensities ρ = 0 . ρ (green lines) and ρ = ρ (blue lines). Finally, the ¯ D and ¯ D ∗ spectral functions are shown in the lower left-hand side panel and lowerright-hand side one, respectively. In both cases, the spectral functions show a rich structure due tothe presence of several resonance-hole states. Note that those resonant states have a pentaquark-likecontent and have to be taken with caution.On the one hand, the spectral function for ¯ D stems from the self-energy of ¯ D displayed in Ref. [84].The position of the quasiparticle peak of ¯ D is located below the ¯ D mass and below the Θ c (2805) N − excitation. The C = − 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 / DN and ¯ D ∗ N , also found in Ref. [98], though it has not been observedyet. The upper energy tail of the ¯ D spectral function shows also the contribution of I = 1 resonant-hole states. On the other hand, the ¯ D ∗ spectral function depicts the contribution of several I = 0 and– 9 – XD + D ∗− X XD − D ∗ + X XD ¯ D ∗ X X ¯ D D ∗ Figure 2 . Contributions to the X (3872) self-energy in nuclear matter. Circles represent the X (3872) couplingsto the meson pairs, and the squares the interaction of the charm mesons with nuclear matter. I = 1 resonant-hole states close to the quasiparticle peak, that is located slightly above to 2 GeV. Allthese pentaquark-like states are described in Ref. [89]. T -matrix formalism. From this section on, and since we focus on the I C = 0 + channel, where the X (3872) is located,for D ¯ D ∗ we mean the appropriate combination of states, ( D ¯ D ∗ − D ∗ ¯ D ) √
2, with even C− parity andcoupled 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” statewith bare mass ˆ m and bare coupling squared to each of the four channels ˆ g /
4. (The isospin relatedfactor 1 / q ) is:ˆ∆ − ( q ) = q − ˆ m + iε . (3.1)Upon resumation of the contributions in Fig. 2, the dressed propagator reads:∆ − ( q ; ρ ) = ˆ∆ − ( q ) − ˆ g Σ( q ; ρ ) . (3.2)This renormalizes the mass and coupling of the state in the medium, m ( ρ ) = ˆ m + ˆ g Σ[ m ( ρ ); ρ ] , (3.3a) g ( ρ ) = ˆ g − ˆ g Σ [ m ( ρ ); ρ ] . (3.3b)with ρ the nuclear-matter density as in the previous sections, and the derivative taken with respect to q = s . These equations are also true in particular for the ρ = 0 case, so that we can relate the baremass and coupling to the vacuum ones, m and g :ˆ m = m − g g Σ ( m ) Σ ( m ) , (3.4a)ˆ g = g g Σ ( m ) . (3.4b)This allows in turn to rewrite the in-medium mass and coupling, m ( ρ ) and g ( ρ ), in terms of thephysical ones in vacuum: m ( ρ ) = m + g g Σ ( m ) h Σ[ m ( ρ ); ρ ] − Σ ( m ) i , (3.5a) g ( ρ ) = g − g (cid:2) Σ [ m ( ρ ); ρ ] − Σ ( m ) (cid:3) . (3.5b)– 10 –ote that m ( ρ ) is in general a complex quantity, its imaginary part being originated by that ofΣ[ m ( ρ ); ρ ]. We can also rewrite the in-medium X (3872) propagator as:∆ − ( q ; ρ ) = q − m − g g Σ ( m ) (cid:16) Σ( q ; ρ ) − Σ ( m ) (cid:17) ≡ q − m − Π X ( q ; ρ ) , (3.6)Π X ( q ; ρ ) = g g Σ ( m ) (cid:16) Σ( q ; ρ ) − Σ ( m ) (cid:17) , (3.7)which defines the X (3872) self-energy in a nuclear medium, Π X ( q ; ρ ). We can now rewrite Eqs. (3.5)as: m ( ρ ) = m + Π X [ m ( ρ ); ρ ] , (3.8a) g ( ρ ) = ˆ g − Π X [ m ( ρ ); ρ ] = g − Π X ( m ; ρ = 0)1 − Π X [ m ( ρ ); ρ ] . (3.8b)Once the X (3872) propagator or self-energy are known, one can also define the X (3872) spectralfunction, S X ( q ; ρ ), S X ( q ; ρ ) = − π Im∆( q ; ρ ) = − π ImΠ X ( q ; ρ ) (cid:2) q − m − ReΠ X ( q ; ρ ) (cid:3) + [ImΠ X ( q ; ρ )] . (3.9)The quasi-particle peak energy, E qp , is defined from the equation: E − m − ReΠ( E ; ρ ) = 0 . (3.10) 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 theprevious subsection. We consider the in-medium T -matrix, T − ( s ; ρ ) = V − ( s ) − Σ( s ; ρ ), with apotential V somewhat more general than a simple constant, e C X . Specifically, we allow for a termlinear in the Mandelstam variable s , and write: V ( s ) = 1Σ ( m ) + Σ ( m )Σ ( m ) 1 − P P ( s − m ) (3.11a)= ˆ g m − ˆ m − ˆ g ( m − ˆ m ) (cid:16) s − m (cid:17) ≡ V A ( s ) . (3.11b)Note that V A ( m ) = 1 / Σ ( m ), which is the same constant term that was previously considered, seeEq. (2.11). Hence, with this potential, the free-space amplitude T ( s ) has a pole at s = m , T ( s ) ’ g s − m + · · · , (3.12)with coupling g given by: 1 g = d T − ( s )d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s = m = − Σ ( m ) P . (3.13)According to the Weinberg compositeness condition [99], the factor − g Σ ( m ) represents the D ¯ D ∗ component in the X (3872) wave function. Hence, the linear term in the potential is chosen so as toset this probability equal to P . Even assuming that in the free-space the X (3872) is bound, and therefore Σ ( m ) is real, the in-medium self-energymight acquire an imaginary part since new many-body decay modes, induced by the quasi-elastic interactions of the D ( ∗ ) and ¯ D ( ∗ ) with nucleons, are open. – 11 –f there is a pole of the amplitude T ( s ; ρ ) at m ( ρ ), then:0 = T − [ m ( ρ ); ρ ] = V − [ m ( ρ )] − Σ[ m ( ρ ); ρ ] (3.14) ’ Σ ( m ) + 1 + g Σ ( m ) g (cid:16) m ( ρ ) − m (cid:17) − Σ[ m ( ρ ); ρ ] , from where one obtains the same equation for m ( ρ ) than that obtained in Eq. (3.5a) within theself-energy formalism. Analogously, the in medium coupling g ( ρ ) would be given by:1 g ( ρ ) = d T − ( s ; ρ )d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s = m ( ρ ) = " Σ[ m ( ρ ); ρ ]Σ ( m ) g Σ ( m ) g − Σ [ m ( ρ ); ρ ] . (3.15)This latter equation does not give exactly the same result than Eq. (3.5b) because of the factor betweenthe square brackets. If that factor is taken as 1, one recovers Eq. (3.5b).Alternatively, we could have made the linear expansion in 1 /V ( s ) instead of in V ( s ) [Eq. (3.11a)],thus getting: V − ( s ) = Σ ( m ) − Σ ( m ) 1 − P P ( s − m ) ≡ V B ( s ) . (3.16a)Note that this alternate definition of V ( s ) can also be written as: V B ( s ) = ˆ g s − ˆ m . (3.16b) i .e., the kernel has a “bare” pole at the “bare” mass squared ˆ m . Then we would obtain:1 g ( ρ ) = d T − ( s ; ρ )d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s = m ( ρ ) = 1 + g Σ ( m ) g − Σ [ m ( ρ ); ρ ] , (3.17)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 if the factor Σ ( m )(1 − P ) /P is sufficiently small. Indeed, it hasbeen 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.Note finally that taking into account the relation P = − g Σ ( m ), Eqs. (3.5) can be cast as: m ( ρ ) = m − P − P Σ[ m ( ρ ); ρ ] − Σ ( m )Σ ( m ) , (3.18a) g ( ρ ) = − [ m ( ρ ); ρ ] + − P P Σ ( m ) . (3.18b) P → P → Let us briefly discuss here the extreme molecular or compact state scenarios, which correspond to P → P →
0, respectively.We start by considering the case when P →
0. In this case one has g = 0, i.e. , the state does notcouple to the two-meson channel. Physically, one would say that the interaction does not renormalizethe bare state. Indeed, one sees also that ˆ g = g = g ( ρ ) = 0, and that ˆ m = m = m ( ρ ). This case isnonphysical, since it would require V ( s = m ) ∼ /P → ∞ .Next we discuss the opposite case P →
1. This situation would correspond to the pure hadron-molecular case, for which V ( s ) = 1 / Σ ( m ) is constant, independent of s . The search of a pole in thenuclear-medium T -matrix would lead to Σ[ m ( ρ ); ρ ] = V − = Σ ( m ). Actually in this limiting case, T ( s ; ρ ) = 1Σ ( m ) − Σ( s ; ρ ) (3.19)– 12 –
820 3840 3860 3880 3900 3920 3940 − − − − E (MeV) Σ ( E ; ρ ) ρ = 0 ρ = 0 . ρ ρ = 0 . ρ ρ = 0 . ρ ρ = 0 . ρ ρ = 1 . ρ − − . − − . . E (MeV) − V A ( E ) P = 0 . P = 0 . P = 0 . − − . − − . . E (MeV) − V B ( E ) P = 0 . P = 0 . P = 0 . Figure 3 . Top panel: The loop function Σ( E ; ρ ), with E = s , for different densities ρ in the range 0 (cid:54) ρ (cid:54) ρ asa function of the center-of-mass energy of the D ¯ D ∗ pair. The solid (dashed) lines stand for the real (imaginary)parts. Bottom panels: Two different parameterizations of the energy-dependent potential V ( s ). On the left[right] plot, V A ( s ) [ V B ( s )], as given in Eq. (3.11) [Eq. (3.16)] . which cannot have a pole on the real axis, since Σ( s ; ρ ) is a complex magnitude, and if there existsa pole, it will be located at a complex value √ s = m ( ρ ) ∈ C (see Subsec. 4.2), and the couplingfrom the residue will be given by g ( ρ ) = − / Σ [ m ( ρ ); ρ ]. These results would also make sense to thefirst of the Eqs. (3.18): since the denominator 1 − P tends to zero, the numerator must also vanish,finding thus Σ[ m ( ρ ); ρ ] − Σ ( m ) = 0. Physically, taking P → P →
1, the factor 1 / (1 − P ) diverges and so it does1 / (cid:0) g Σ ( m ) (cid:1) , and hence ˆ g → ∞ and ˆ m → ∞ , in Eq. (3.4).For simplicity, in the discussion above, we have not considered the pathological case in which thebound state in vacuum is placed exactly at threshold. In that case Σ ( m ) diverges, and this singularbehaviour needs to be taken into account. – 13 – . . . . | T ( E ; ρ ) / T ( E m a x ; ρ ) | P = 1 00 . . . . P = 0 . . . . . P = 0 . − . − . . . . − Z ∆ − ( E ; ρ )( M e V ) − . − . . . . − . − . . . . E (MeV) Z − S X ( E ; ρ )( M e V − ) . ρ . ρ . ρ . ρ . ρ E (MeV)0 . ρ . ρ . ρ . ρ . ρ E (MeV)0 . ρ . ρ . ρ . ρ . ρ Figure 4 . Top panels: Squared modulus of the amplitudes T ( E ; ρ ), normalized to be one at the maximum, E max as a function of the energy of the D ¯ D ∗ -pair in the c.m. frame. The amplitudes are computed with Eq. (2.20),using the energy dependent potential of Eq. (3.11) instead of the constant e C X . Middle panels: Real (solidlines) and imaginary parts (dashed lines) of the inverse of the propagator ∆( E ; ρ ) (Eq. (3.6)) multiplied by Z = (1 − P ), as a function of the energy of the D ¯ D ∗ -pair in the c.m. frame. Bottom panels: Spectral functionof the X (3872) (Eq. (3.9)) multiplied by Z − , as a function of the energy of the D ¯ D ∗ -pair in the c.m. frame.From left to right, the three columns show the cases P = 1, 0 .
9, and 0 .
8. For these high molecular probabilities,the numerical differences due to the use of V A ( s ) or of V B ( s ) potentials [Eqs. (3.11) and (3.16), respectively]are very small. The different colors in each figure represent calculations performed at different nuclear densities0 (cid:54) ρ (cid:54) ρ . We now discuss the results that we obtain in a nuclear medium for the D ¯ D ∗ amplitude | T ( E ; ρ ) | ( cf. Eq. (2.20), using a very general energy dependent potential instead of just a constant e C X ), the X (3872) self-energy Π X ( E ; ρ ) [cf. Eq. (3.7)] (or, equivalently, the inverse of the propagator ∆ − ( E ; ρ )[cf. Eq. (3.6)]), and its spectral function S X ( E ; ρ ) [cf. Eq. (3.9)]. Note that we use the energy, E , of the D ¯ D ∗ -pair in the c.m. frame, with s = E . In order to compute all these quantities, we need the energy-dependent potential V ( s ) [ cf. (3.11) or (3.16)] and the in-medium modified D ¯ D ∗ loop function Σ( s ; ρ )[cf. Eq. (2.21)]. In the upper plot of Fig. 3 we show the real (solid lines) and imaginary (dot-dashedlines) parts of the loop function Σ( s ; ρ ) computed for different densities ρ in the range 0 (cid:54) ρ (cid:54) ρ , where ρ is the normal nuclear density, ρ = 0 .
17 fm − . We see that the sharp D ¯ D ∗ threshold observed in the We recall here again that we are working on the I C = 0 + channel, where the X (3872) is located, and that for D ¯ D ∗ we mean the appropriate combination of states, ( D ¯ D ∗ − D ∗ ¯ D ) √
2, with even C− parity and coupled to zero isospin. – 14 –acuum case ( ρ = 0) is progressively smoothed out for increasing densities, being almost inappreciablefor ρ = ρ . This is due to the width acquired by the D , ¯ D , D ∗ and ¯ D ∗ 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 D ¯ D ∗ interaction,in the sense that a more attractive potential would be necessary to compensate this change of theloop function. However, this repulsive effect is not clear, because the imaginary part of Σ( E ; ρ ) is alsolarge, and below threshold, it turns out that | ImΣ( E ; ρ ) | (cid:38) | Re (Σ( E ; ρ ) − Σ ( E )) | .Within the present approach, the D ¯ D ∗ T − matrix in the nuclear environment is determined fromthe X (3872) mass and its D ¯ D ∗ probability ( m and P ) in the vacuum ( ρ = 0). As we work onthe isospin limit, m D ( ∗ ) = ( m D ( ∗ )+ + m D ( ∗ )0 ) /
2, we cannot consider the physical X (3872) mass. Weinstead take a binding energy B = 2 MeV with respect to the D ¯ D ∗ threshold, m = m D + m D ∗ − B .Throughout this manuscript, we will study the in-medium effects for different molecular probabilities P , 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 com-puted for different values of P . Both types of interactions give the same pole position at m andprobability P (alternatively, the same coupling g ) for the vacuum T -matrix, although they havedifferent analytical properties ( V A ( s ) has a zero, while V B ( s ) presents a bare pole) and, hence, theymight produce differences in the medium T -matrix, as we will discuss below. In the lower panels ofFig. 3 we observe, on the one hand, that for values of P above P = 0 . 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 , e.g. P = 0 . V A ( s ) and the bare pole V B ( s ) come closer tothe energy region of interest. Therefore, one should expect that they lead to significantly different inmedium T -matrices.Once discussed the in-medium modified D ¯ D ∗ loop function and the energy-dependent potential,in Fig. 4 we show, for different nuclear densities and molecular probabilities P = 1 , . T ( E ; ρ ), normalized to be one at the maximum E max (top panels),the inverse of the X (3872) propagator, ∆ − ( E ; ρ ) (medium panels), and the spectral function, S X ( E ; ρ )(bottom panels), conveniently scaled by Z = (1 − P ) and Z − , respectively. The calculations areperformed using the potential V A ( s ), introduced in Eqs. (3.11), though, as shown above, for thesehigh-molecular component scenarios the V B ( s ) − type interaction, cf. Eqs. (3.16), leads to very similarpredictions, 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 qualitativelydifferent for the three examined probabilities. Thus, while for P = 0 . D ¯ D ∗ c.m. energies), it howevermoves to the left in the purely molecular ( P = 1) scenarios. The results for P = 1 stem from theenergy and density behaviour of the factor (cid:12)(cid:12) Σ ( m ) − Σ( E ; ρ ) (cid:12)(cid:12) in Eq. (3.19), by taking into accountthe in-medium two-meson loop function Σ( E ; ρ ) depicted in Fig. 3. For the P = 0 . V A potential, shown in the left-bottom plot of Fig. 3, leads to the mild shift towardshigher energies of the maximum as the density increases. The position of the peak hardly changes inthe intermediate P = 0 . 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 ) forfinite 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 non-vanishing values below the free-space threshold, which increase with the density, due to the appear-ance of new many-body decay channels, like D ¯ D ∗ N → D ¯ D ∗ 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 )– 15 –hown in the top plot of Fig. 3.We should also note that ImΠ X ( E ; ρ ) strongly depends on P , and it behaves as g / [1+ g Σ ( m )] ∝ P / (1 − P ), as deduced from Eq. (3.7). Looking now at real part of ∆ − ( E ; ρ ), we observe that for P = 1, there is not quasi-particle solution (Eq. (3.10)) for densities higher than about one tenth ofthe normal nuclear matter density, with an increasingly flatter E − dependence of Re[∆ − ( E ; ρ )] asthe density grows. Hence, the behavior exhibited in the P = 1 case in left-top plot for the modulussquared of the amplitude, with the maximum displaced to the left with increasing densities, can becorrelated to the growth of ImΠ X ( E ; ρ ), both with the density and the c.m. energy. On the contrary,for P = 0 .
8, we find solutions for the quasi-particle equation for all densities, at energies abovethreshold 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 itsdependence on E, ρ and the molecular probability P can be deduced from the discussion above on thereal and imaginary parts of ∆ − ( E ; ρ ) in the second-row panels of this figure. We should make heretwo remarks. First, we observe that the typical delta-function shape expected for the spectral functionof a narrow state in the free space gets diluted as the density grows. This is due to the enhancement ofthe X (3872) width with density. Second, we find that, for purely molecular case ( P = 1), the featuresof 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 . | T ( E ; ρ ) | and S X ( E ; ρ ). Nevertheless, the squaredamplitude | T ( E ; ρ ) | 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 . P = 0 .
2. As wediscussed 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 ) and D ¯ D ∗ coupling ( g ) of the X (3872) in the free space,considerably differ in the region of interest for the present study. Hence, the corresponding T − matricesare different, even in the free space. Those deduced from V A show the zero that this potential hasbelow m . 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 . The position of the zero is independent ofthe nuclear density, being, however, the dependence of the amplitude on the density clearly visible,both for energies below and above the energy, E , for which the potential and scattering amplitudevanish. Density effects for energies lower (higher) than E become more (less) relevant for the P = 0 . P = 0 . V A ( s ), when | T ( E ; ρ ) | is computed using the V B ( s ) interaction, we see little structure beyond the peak induced by the bare pole present in thepotential. The effects due to the medium dressing are small for P = 0 . P = 0 .
2. Hence, experimental input on | T ( E ; ρ ) | , especially for energies below E ,might shed light into the dynamics of the interacting D ¯ D ∗ pair that could be difficult to infer fromtheir scattering in the free space.In Figs. 5 and 6 we also show the inverse propagator ∆ − ( 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 aredetermined by the vacuum X (3872) and the in-medium two-meson loop function Σ( E ; ρ ) given inFig. 3. In what respects to the Im[∆ − ( E ; ρ )], the results here are the same as those discussed abovein Fig 4, scaled down by the corresponding factor P / (1 − P ). On the other hand, the plots for realpart of ∆ − ( E ; ρ ) show that, for small molecular components, there is always a quasi-particle solutionvery close to m , 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 ) | when the potential V B is used. However, it doesnot account for the medium modifications observed in the T − matrix below E when V A is employed.To conclude, in Figs. 7 and 8, we show the positions E max and E spe of the maxima of | T ( E ; ρ ) | – 16 –
800 3825 3850 3875 3900 392500 . . . .
81 [ V ( s ) = V A ( s )] E (MeV) | T ( E ; ρ ) / T ( E m a x ; ρ ) | . . . .
81 [ V ( s ) = V B ( s )] E (MeV) | T ( E ; ρ ) / T ( E m a x ; ρ ) | − . − . . E (MeV) − Z ∆ − ( E ; ρ )( M e V ) E (MeV) Z − S X ( E ; ρ )( M e V − ) . ρ . ρ . ρ . ρ . ρ P = 0 . Figure 5 . Top plots: Squared modulus of the amplitude T ( E ; ρ ), normalized to be one at the maximum, E max , as a function of the center-of-mass energy of the D ¯ D ∗ pair, for a vacuum molecular probability P = 0 . V A ( s ) in Eq. (3.11) (left plot) or V B ( s )in Eq. (3.16) (right plot). Bottom plots: Inverse of the propagator ∆( E ; ρ ) (left) and the spectral function S X ( E ; ρ ) (right) for P = 0 .
4, and multiplied by Z = (1 − P ) and Z − , respectively. Neither the propagator,nor the spectral function depend on the kernel V ( s ), since they are determined by the vacuum X (3872) andthe in-medium two-meson loop function Σ( E ; ρ ). and S X ( E ; ρ ), respectively, for all molecular probabilities considered above in Figs. 4–6. We also givethe quasi-particle energies, E qp , obtained by solving Re[∆ − ( E ; ρ )] = 0 when they exist. For low valuesof P (Fig. 8), we provide separately E max obtained from V A ( s ) or of V B ( s ) potentials. The resultsin these two figures reinforce the conclusions previously outlined. Indeed, we graphically see for thehighest values of P , the appreciable difference between E max and E spe , with even an opposite densityslope in the P = 1 case. In Fig. 7, we only observe for P = 0 . | T ( E ; ρ ) | and S X ( E ; ρ ), with quasi-particle energies well separated from both of them andexhibiting a significantly larger sensitivity with density. Medium effects are much smaller in Fig. 8,where results for P = 0 . P = 0 . 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 .The quasi-particle and spectral-function energies are closer, and for P = 0 . E max computed using V B . This supports that, in this case, one is dealing with a compact statelittle affected by the dressing of the meson loops in the medium. 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 thespectral functions S D ( ∗ ) and S ¯ D ( ∗ ) shown in Fig. 1 is washed out by the Ω and ω integrations implicit– 17 –
800 3825 3850 3875 3900 392500 . . . .
81 [ V ( s ) = V A ( s )] E (MeV) | T ( E ; ρ ) / T ( E m a x ; ρ ) | . . . .
81 [ V ( s ) = V B ( s )] E (MeV) | T ( E ; ρ ) / T ( E m a x ; ρ ) | − . − . . . E (MeV) − Z ∆ − ( E ; ρ )( M e V ) E (MeV) Z − S X ( E ; ρ )( M e V − ) . ρ . ρ . ρ . ρ . ρ P = 0 . Figure 6 . Same as Fig. 5, but for P = 0 . . . . . ρ/ρ E ( M e V ) P = 1 E max E spe E th . . . . ρ/ρ P = 0 . E max E spe E th . . . . ρ/ρ P = 0 . E max E spe E qp E th Figure 7 . Positions E max and E spe of the maxima of | T ( E ; ρ ) | and S X ( E ; ρ ), respectively, as a function ofthe nuclear matter density. From left to right, the three plots show the cases P = 1, 0 .
9, and 0 .
8. In thelatter case, we also give the quasi-particle energy, E qp , obtained by solving Re[∆ − ( E ; ρ )] = 0. For these highmolecular probabilities, the numerical differences due to the use of V A ( s ) or of V B ( s ) potentials [Eqs. (3.11) and(3.16), respectively] are very small. The black dashed-double dotted line represents the vacuum D ¯ D ∗ threshold,whereas the empty circle at ρ = 0 is the X (3872) vacuum mass m of the X (3872). – 18 – . . . . ρ/ρ E ( M e V ) P = 0 . E max [ V ( s ) = V A ( s )] E max [ V ( s ) = V B ( s )] E qp E spe E th . . . . ρ/ρ P = 0 . E max [ V ( s ) = V A ( s )] E max [ V ( s ) = V B ( s )] E qp E spe E th Figure 8 . Same as Fig. 7, but for small molecular components, P = 0 . .
2. We show separately E max obtained from V ( s ) = V A ( s ) or V ( s ) = V B ( s ) [Eqs. (3.11) and (3.16), respectively]. in Eq. (2.21) (see also Eq. (2.7)). Thus, almost no trace of the several peaks present in Fig. 1 can bedistinctly 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 viaa hard cutoff Λ, but evaluated with complex masses. Hence, we make the following approximation:Σ( s ; ρ ) ’ G (eff) ( s ; ρ ) ≡ G ( s, m (eff) D ( ρ ) , m (eff) D ∗ ( ρ )) , (4.1)with m (eff) D ( ∗ ) complex valued, and the superscript “(eff)” is included to remark that these are density-dependent 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 D ¯ D ∗ ( s ; ρ ) in thewhole complex plane, for the different medium densities ρ and vacuum probabilities P , and search forpoles in the complex plane. We find a pole on the first Riemann sheet of the amplitude (as defined inAppendix A), off the real axis. This does not represent any violation of the analyticity properties ofthe complete-system scattering T − matrix, because of the effective procedure used to take into accountthe many body channels of the type D ¯ D ∗ N → D ¯ D ∗ N . In the present scheme, they are not explicitlyconsidered in the coupled-channel space and only their effects on D ¯ D ∗ → D ¯ D ∗ are included throughthe in-medium charmed-meson self-energies.The pole position depends on the nuclear medium density ρ and on the value chosen for theparameter P , the X (3872) molecular probability in the vacuum. The pole position is representedin Fig. 9 for different values of P 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 D ¯ D ∗ − potentials considered in this work. For eachdensity, the zigzag lines represent the loop function G (eff) ( s ; ρ ) right hand cut: √ s ∈ C . h Im p ( s, m (eff) D ( ∗ ) , m (eff) D ) = 0 i and h Re p ( s, m (eff) D ( ∗ ) , m (eff) D ) > i (4.2)extending to the right and starting at the branch point, √ s = ( m (eff) D ( ∗ ) + m (eff) D ), where p ( s ) = 0. Inaddition, p ( s ) is defined in the Appendix. The dotted lines extending to the left represent the segmentsin which Im p ( s ) = 0 and Re p ( s ) <
0, where the density-dependent loop functions are thus real, Because of the limited range in Re √ s explored in Fig. 9, the curves in which Im p ( s ) = 0 (the zigzag and dottedlines) look like straight lines, parallel to the real axis, although in general they are not, and have some curvature. – 19 –
870 3875 3880 3885 3890 − − − − P = 0 P = 1 [case V ( s ) = V A ( s )] Re √ s (MeV) I m √ s ( M e V ) ρ = 0 0 . ρ . ρ . ρ . ρ . ρ − − − − P = 0 P = 1 [case V ( s ) = V B ( s )] Re √ s (MeV) I m √ s ( M e V ) ρ = 0 0 . ρ . ρ . ρ . ρ . ρ Figure 9 . Complex pole position of the X (3872) for different nuclear densities ( ρ ) and vacuum molecularprobabilities ( P ). Results in the left and right plots have been obtained using amplitudes computed with V ( s ) = V A ( s ) [ cf. Eq. (3.11)] and V ( s ) = V B ( s ) [ cf. Eq. (3.16)], respectively. The dashed curves show thecontinuous variation of the pole position with P , and the points represent steps in the probability ∆ P = 0 . G (eff) ( s ; ρ ) function (see text and Appendix A for further details). Im G (eff) ( s ; ρ ) = 0. The dashed lines show the continuous variation of the pole position with P , wherethe points represent steps in the probability ∆ P = 0 .
1. When P → i.e. , when the X (3872)molecular component tends to vanish, the coupling of the X (3872) to the D ∗ ¯ D channel tends to zero,and therefore, in this case, the pole remains at the original position in vacuum, independently of thenuclear density. On the other end, when P → i.e. , when the X (3872) tends to be a purely molecularstate, the pole appears to the left of the effective complex threshold, exactly in the segment whereIm p ( s ) = 0. This happens because, in this limit, the derivative term of the kernel V A ( s ) [ cf. Eq. (3.11)]vanishes, and V A ( s ) is just a real constant. Therefore the pole, solution of [1 − V A ( s ) G (eff) ( s ; ρ ) = 0],should also satisfy Im G (eff) ( s ; ρ ) = 0. We also see in Fig. 9 that the in-medium X (3872) pole positionsatisfies (cid:12)(cid:12) Im √ s P (cid:12)(cid:12) (cid:54) (cid:12)(cid:12)(cid:12) Im (cid:16) m (eff) D + m (eff) D ∗ (cid:17)(cid:12)(cid:12)(cid:12) , i.e. , the X (3872) width is always smaller than the sum ofthe D and ¯ D ∗ effective widths. One can say that the pole position is dragged by the effective threshold( m (eff) D + m (eff) D ∗ ), and that the effect is large or small depending on whether the in-vacuum probability P is close to 1 or to 0, respectively. We also observe some dependence of the pole position, which asexpected grows as the molecular content P deviates from 1, on the used D ¯ D ∗ interaction in the freespace, 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 is a free parameter that we have variedto explore different scenarios. We can define the quantity e P ρ , e P ρ = − g ( ρ ) d G (eff) ( s ; ρ )d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s = m ( ρ ) , (4.3)which generalizes to the nuclear medium the formula for the vacuum probability [ cf. Eq. (3.13)]. Sincethe pole position is in general complex, so will be this quantity. Therefore, in general, it will not bepossible to interpret it as a probability. In Fig. 10, we show e P ρ for different nuclear densities as afunction of the vacuum probability P . This figure complements the results of Fig. 9. We observe for Note that, as previously discussed, there is little difference between the results obtained with V A ( s ) or V B ( s ) when P is close to one. Therefore, the argument presented here with V A ( s ) can be readily applied to the case of V B ( s ). – 20 – . . . . . . . . P e P ρ = − g ( ρ ) ˙ G ( s ; ρ ) [ c a s e V ( s ) = V A ( s ) ] . ρ Re e P ρ . ρ Im e P ρ . ρ P . ρ . ρ . . . . . . . . P e P ρ = − g ( ρ ) ˙ G ( s ; ρ ) [ c a s e V ( s ) = V B ( s ) ] . ρ Re e P ρ . ρ Im e P ρ . ρ P . ρ . ρ Figure 10 . Dependence of the quantity e P ρ [ cf. Eq. (4.3)] with the vacuum molecular probability P for differentdensities. The solid (dashed) lines represent the real (imaginary) part of e P ρ . The left and right plots correspondto the cases V ( s ) = V A ( s ) and V ( s ) = V B ( s ), respectively. this magnitude some quantitative differences between the results obtained with V A ( s ) (Eq. (3.11),left plot of Fig. 10) or V B ( s ) (Eq. (3.16), right plot of Fig. 10), but the qualitative behaviour is verysimilar. In the intermediate regions, far from the end points P = 0 and P = 1, the imaginary part of e P ρ can be sizeable, and for most of these values it increases with the density. In general, the effect ofthe nuclear medium in this intermediate P region is to decrease both the real part and the modulusof e P ρ with respect to its original value P . However, we see that for both ends P → P →
1, wehave that Im e P ρ ’
0, and e P ρ ’ P . We thus see that in these cases the X (3872) state can be said toconserve its original nature in the nuclear medium. In this work we have studied the behaviour of the χ c (3872), also known as X (3872), in dense nuclearmatter. The X (3872) appears in the vacuum as a pole in the D ¯ D ∗ scattering amplitudes, which areparametrized in a quite general form. The in-medium effects have been incorporated by dressing the D ¯ D ∗ loop functions with the corresponding spectral functions of the charmed mesons. As a result,the D ¯ D ∗ amplitudes, when the charmed mesons are embedded in the nuclear medium, have beendetermined for energies around the nominal X (3872) mass. The X (3872) spectral function has beenalso obtained for densities ranging up to that of nuclear matter saturation.For the kernel of the D ¯ D ∗ 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 inthe physical Riemann sheet, these two parameters allow to fix the nominal X (3872) mass and itscoupling to the D ¯ D ∗ channel, or, alternatively, the mass and the molecular probability P . Therefore,both types of interactions allow for the study of the X (3872) as either a pure hadron-molecule stateor a genuine quark state, as well as intermediate possibilities, in terms of P . However, both types ofinteractions have different analytical properties, which can give rise to different scattering amplitudesat finite density.Using these two models for the interaction, we have explored the connection between the in-medium behaviour of the X (3872) and its nature. In the case of the X (3872) being mostly a molecularstate, both interaction potentials behave similarly and lead to equivalent results for the in-mediumamplitudes. In this case, we have found that the D ¯ D ∗ amplitudes strongly depend on the density. Thewidth of the X (3872)-peak significantly grows when the density is increased, while its position movesto higher energies, as the molecular component is lowered. The X (3872) spectral function follows the– 21 –maginary part of the X (3872) self-energy, that increases with density due to the appearance of newmany-body decay channels in matter. On the other hand, when smaller molecular components areconsidered, the D ¯ D ∗ amplitudes depend on the choice of the energy-dependent potential, speciallyfor energies below the free-space X (3872) mass. Hence, the experimental input on the amplitudes atfinite density might shed light into the dynamics of the D ¯ D ∗ interaction in the case of a state witha 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 D ¯ D ∗ loop functions strongly depend on the interaction of D , D ∗ , ¯ D and ¯ D ∗ with nuclear matter. However, one can reasonably approximate them by a standard loop functionevaluated with complex, effective masses of the D ( ∗ ) and ¯ D ( ∗ ) mesons. This fact allows for an analyticalcontinuation of the loop function, and hence of the scattering amplitude, to the whole complex planeand 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 inthe complex energy plane. However, this does not represent any violation of the analyticity propertiesof the T − matrix, because, in the present scheme, the D ¯ D ∗ N → D ¯ D ∗ N many-body channels are notexplicitly considered in the coupled-channel space, since their effects on D ¯ D ∗ → D ¯ D ∗ are includedvia the in-medium charmed-meson self-energies. The behaviour of the X (3872) pole with density ismoreover fully in line with the change in matter of the squared modulus of the T − matrix amplitudesfor real energies. Complex poles for the X (3872) produced inside of a nuclear medium are collectedin Fig. 9, for different densities and free-space molecular probabilities. In the light of these results, weconclude 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 ofaround 30-40 MeV, and more modest mass-shifts (repulsive) with a maximum of 10 MeV. This latteroutcome contradicts the results obtained in the QCD-sum-rule calculation carried out in Ref. [72] andbased on a diquark-antidiquark picture for the X (3872). Indeed, in the approach of Ref. [72], the mass-shift due to the nuclear matter is negative and is about 25% ( ∼ X (3872) andcomparison 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 D ¯ D ∗ channel to the X (3872) dy-namics. In the future, we aim at extending our calculation to a more realistic situation by incorporatingalso coupled channels involving hidden-charm mesons, such as J/ψ π . Also, the results presented inthis manuscript are based on a specific model for the D ( ∗ ) N and ¯ D ( ∗ ) N interactions, which determinethe in-medium modifications of the D ¯ D ∗ loop functions. Different or more elaborate models for theseamplitudes could also be employed in the formalism we have derived here. In any case, our results indi-cate a very different behaviour with density of the D ¯ D ∗ amplitudes and the X (3872) spectral functiondepending on the nature of the X (3872). Thus, experiments that can access the nuclear finite-densityregime, such as HiCs like CBM or those with fixed nuclear targets such as ¯ p -nuclei in PANDA, arenecessary and complementary to the spectroscopic analyses so as to discern the nature of X (3872). Acknowledgments
We thank E. Oset for valuable discussions at an early stage of this project, and for a careful reading ofthe manuscript. M.A. work supported by the U.S. Department of Energy, Office of Science, Office ofNuclear Physics under contract DE-AC05-06OR23177. L.T. acknowledges support from the DeutscheForschungsgemeinschaft (DFG, German research Foundation) under the Project Nr. 411563442 (HotHeavy Mesons), the CRC-TR 211 ’Strong-interaction matter under extreme conditions’- project Nr.315477589 - TRR 211, and the THOR COST Action CA15213. This research has been also supportedby the Spanish Ministerio de Econom´ıa y Competitividad, Ministerio de Ciencia e Innovaci´on and the– 22 –uropean Regional Development Fund (ERDF) under contracts FPA2016-81114-P, FIS2017-84038-C2-1-P, PID2019-105439G-C22 and PID2019-110165GB-I00, by Generalitat Valenciana under contractPROMETEO/2020/023 and by the EU STRONG-2020 project under the program H2020-INFRAIA-2018-1, grant agreement no. 824093.
A Further details on G (eff) ( s ; ρ ) In this Appendix we give further details on the approximation made in Sec. 4.2, and on the analyticalproperties of the loop function employed. The approximation is:Σ( s ; ρ ) ’ G (eff) ( s ; ρ ) ≡ G ( s, m (eff) D ( ρ ) , m (eff) D ∗ ( ρ )) , (4.1)where G ( s, m , m ) can be computed using the explicit formulas given for instance in Ref. [81] (see inparticular the erratum ), regulated with a momentum cutoff of 0 . m (eff) D ( ρ ) = m D + ∆ m ( ρ ) − i Γ( ρ )2 , (A.1a) m (eff) D ∗ ( ρ ) = m D ∗ + ∆ m ( ρ ) − i Γ( ρ )2 . (A.1b)with m D ( ∗ ) , the vacuum masses, and ∆ m ( ρ ) and Γ( ρ ) real quantities. We note that in the m (eff) D and m (eff) D ∗ definitions we have forced a common shift ∆ m ( ρ ) − i Γ( ρ )2 with respect to the vacuum masses.Being this an effective representation, we find that this ansatz is enough to approximate the originalloop function, Σ( s, ρ ). In Fig. 11 we show in the left (right) panel the imaginary (real) part of theloop function Σ( s, ρ ) together with the approximation determined by Eq. (4.1), computed with theparameters ∆ m ( ρ ) and Γ( ρ ) collected in Table 1. The latter are chosen so as to approximately matchthe original loop functions Σ( s ; ρ ) for the different densities considered in this work. As can be seen,the approximation works reasonably well. ρ/ρ ∆ m ( ρ ) (MeV) − Γ( ρ )2 (MeV)0 .
10 +0 . − . .
30 +0 . − . . − . − . . − . − . . − . − . Table 1 . Parameter values used to determine the effective masses m (eff) D ( ρ ) and m (eff) D ∗ ( ρ ) [Eqs. (A.1)] for differentnuclear densities ρ . The loop function G (eff) ( s ; ρ ) can be continued analytically to the whole complex plane, and thesecond (or nonphysical) Riemann sheet is defined as: G (eff)II ( s ; ρ ) = G (eff) ( s ; ρ ) + i p [ s, m (eff) D ( ρ ) , m (eff) D ∗ ( ρ )]4 π √ s ,p ( s, m , m ) = (cid:2) s − ( m + m ) (cid:3) (cid:2) s − ( m − m ) (cid:3) √ s (A.2)In Fig. 12 and for ρ = ρ /
2, we show in blue (red) the physical (nonphysical) Riemann sheet of thefunction G (eff) ( s ; ρ ) in the √ s − complex plane. The cut of G (eff) ( s ; ρ ) lies on a curve for the variable √ s , given in Eq. (4.2) of the main text, which in the free space ( ρ →
0) is the usual right hand cut, √ s > ( m D + m D ∗ ) on the real axis, with √ s = ( m D + m D ∗ ) the branch point. For finite density andtherefore complex masses, this branch point moves from the real axis into the complex plane, and thecut does not lie in the real axis either. – 23 –
800 3840 3880 3920 3960 − − − − √ s (MeV) I m Σ ( s , ρ ) . ρ . ρ . ρ . ρ Σ( s ; ρ )1 . ρ G (eff) ( s ; ρ ) − − . − − . − − . √ s (MeV) R e Σ ( s , ρ ) . ρ . ρ . ρ . ρ Σ( s ; ρ )1 . ρ G (eff) ( s ; ρ ) Figure 11 . The original loop function Σ( s ; ρ ) (solid lines), shown in Fig. 3, compared with the approximatedone, G (eff) ( s ; ρ ), obtained from Eq. (4.1) (dashed lines). The imaginary and real parts of both functions, as afunction of the c.m. energy of the D ¯ D ∗ pair are displayed in the left and right plots, respectively. − − − − −
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