Open charm meson in nuclear matter at finite temperature beyond the zero range approximation

The properties of open charm mesons, $D$, $\bar D$, $D_s$ and $\bar D_s$ in nuclear matter at finite temperature are studied within a self-consistent coupled-channel approach. The interaction of the low lying pseudoscalar mesons with the ground state baryons in the charm sector is derived from a $t$-channel vector-exchange model. The in-medium scattering amplitudes are obtained by solving the Lippmann-Schwinger equation at finite temperature including Pauli blocking effects, as well as $D$, $\bar D$, $D_s$ and $\bar D_s$ self-energies taking their mutual influence into account. We find that the in-medium properties of the $D$ meson are affected by the $D_s$-meson self-energy through the intermediate $D_s Y$ loops coupled to $DN$ states. Similarly, dressing the $\bar{D}$ meson in the $\bar{D}Y$ loops has an influence over the properties of the $\bar{D}_s$ meson.


I. INTRODUCTION
Over the past years the properties of charmed hadrons have received a lot of attention in connection with experiments in lepton colliders (CLEO, Belle, BaBar) and hadron facilities (CDF at Fermilab, PHENIX, STAR at RHIC, and the forthcoming PANDA and CBM experiments at FAIR) . The discovery of new resonances with charm content has sparked the interest of not only many experimental but also theoretical research groups in order to find plausible explanations for the nature of the newly found states.
Nuclear medium modifications have been lately incorporated as a second step. The aim is to further investigate on the nature of resonant states, such as Λ c (2595), but also to test the dynamics of charmed hadrons with nucleons and nuclei. The properties of open-charm mesons in nuclear matter can influence the charmonium production in hot dense matter, which might indicate the formation of the quark-gluon plasma phase of QCD at high density and temperature [86]. Another exciting scenario is the possible formation of D-mesic nuclei [87,88] and of exotic nuclear bound states like J/Ψ in nuclei [89][90][91]. From the experimental side, the physics program of the CBM experiment as well as part of the PANDA collaboration at FAIR [92] will be devoted to the properties of open and hidden charm in dense matter. In particular, the physics goal is to extend to the heavy-quark sector the GSI program for in-medium modifications of hadron properties in the light sector, and to provide insight into the charm-nucleus interaction.
Works based on mean-field approaches provided important shifts for the D andD open-charm meson masses [93][94][95][96], which alters the formation of charmonium [97]. Some of those models have been recently revised [98][99][100]. A different perspective is offered by models that, working within coupled-channel unitarized schemes, go beyond mean field and provide the spectral features of the charm mesons in symmetric nuclear matter at zero [74,75,101] and finite temperature [102,103].
Lately, this meson-baryon basis has been extended to incorporate HQS. In this way, not only D-meson but also D * -meson features have been studied [104].
A common feature of the previous models is the use of an interaction kernel in the zero-range approximation (t → 0). This is justified for diagonal amplitudes close to threshold and for nondiagonal transition amplitudes where the masses of mesons and of baryons in the initial and final meson-baryon states differ moderately. However, the charm-exchange processes, for which the difference in masses between the external mesons is comparable with the mass of the charmed vector meson being exchanged, point towards the breakdown of the zero-range approximation.
Charmed baryon resonances have been studied using the full t-dependence of the t-channel vectorexchange driving term in Ref. [105]. Compared to the previous TVME local models, where the t → 0 limit was implemented, the work of Ref. [105] obtained the same amount of resonances but located in general at somewhat higher energies and having larger widths. Some of these resonances could clearly be identifiable with experimentally seen states, such as Λ c (2595), Σ c (2800), Ξ c (2790) and Ξ c (2980).
In the present work, we study the behavior of the dynamically-generated baryonic resonaces in hot dense matter, as well as the spectral features of the open charm mesons (D,D, D s and D s ), within a self-consistent coupled-channel approach that considers the full t-dependent TVME interaction kernel employed in Ref. [105]. We pay a particular attention to the influence that the dressed mesons exert on each other. We find that the simultaneous dressing of the charm mesons (D, D s ) in the C = 1 sector, or the anticharm mesons (D,D s ) in the C = −1 one, affects their in-medium properties in a non-negligible way.
The article is organized as follows. In Sec. II, we present the formalism. We first revise the model adopted for the free space amplitudes and, next, we describe the modifications that incorporate the medium effects. Our results for the medium modified resonances and for the spectral functions of the open-charm mesons at various densities and temperatures are shown in Sec. III. A summary of our conclusions is presented in Sec. IV.

II. FORMALISM
In this section, we will first review briefly the coupled-channel approach employed in our previous work [105], where we studied open-charm baryon resonances dynamically generated from the freespace interaction of the low-lying pseudoscalar mesons with the ground-state baryons using a t-channel vector-exchange driving force. After that, we will introduce the main sources of medium effects and we will implement them in our coupled-channel formalism.
Since the properties of the D,D, D s andD s mesons in a hot and dense environment will be determined, respectively, from the DN ,DN , D s N andD s N amplitudes, we list in Table I the corresponding set of coupled channels in each of the related isospin (I), strangeness (S) and charm (C) sectors.

A. Free-space coupled-channel approach
The free-space amplitudes, T , which describe the scattering of the pseudoscalar meson fields off the ground-state baryon fields can be obtained by solving the well-known Lippmann-Schwinger equation, which schematically reads  The loop function J is the product of the meson and baryon single-particle propagators, and the scattering kernel V describes the interaction between the pseudoscalar mesons and the ground-state baryons. Following the original work of Hofmann and Lutz [73], we identify a t-channel exchange of vector mesons as the driving force for the S-wave scattering between pseudoscalar mesons in 16-plet and baryons in 20-plet representations. The scattering kernel takes the form (see [73] for details) where the sum runs over all vector mesons of the SU(4) 16-plet, (ρ, K * ,K * , ω, φ, D * , D * s ,D * , D * s , J/Ψ), m V is the mass of the exchanged vector meson, g is the universal vector meson coupling constant, p i , q i , p j and q j are the four momenta of the incoming and outgoing baryon and meson, and the coefficients C (I,S,C) ij;V denote the strength of the interaction in the different (I, S, C) sectors, and meson-baryon channels (i, j). The value of g = 6.6 that reproduces the decay width of the ρ meson [106] has been considered in this work. The S-wave projection of the scattering kernel is easily obtained, and in the center-of-mass (c.m.) frame it takes the analytical form with a, b, α and β being where k i , k j are the initial and final relative momenta, m i , m j , M i , M j are the masses of the incoming and outgoing mesons and baryons, and ing energies, which have been taken to be their on-shell values.
comes from the normalization of the Dirac spinors. We have de- fined Ω(| k|) ≡ ω(| k|) + E(| k|). We note that the zero-range approximation (i.e., t → 0) of the S-wave scattering kernel is obtained by expanding the logarithm of Eq. (3) in the limit b/a → 0 up to the linear term in b/a and setting a = −m 2 V . The interested reader is referred to our previous work of Ref. [105] for a detailed analysis of the validity of the zero-range approximation.
In Eqs. (2) and (3), we have assumed infinitely (zero-width) exchanged vector mesons, because the value of t is never larger than the square of the minimum energy required for the meson to decay. In other words, since the mesons being exchanged in this problem are largely off shell, they will be treated as stable particles.
Once the scattering kernel has been constructed, one can finally write the S-wave projection of the Lippmann-Schwinger equation, where √ s is the total energy in the c.m. frame. The loop function J explicitly reads and F (| k|) is a dipole-type form factor, that has been introduced to regularize the integral. This form is typically adopted in studies of hadron-hadron interactions within the scheme of Lippmann-Schwinger-type equations in the light flavour sector [107]. The value of the cut-off Λ is a free parameter of our model. Given the limited amount of data for charmed baryon resonances, and in order to simplify the analysis, the cut-off Λ is adjusted to 903 MeV/c in order to reproduce the position of the well-known J P = 1/2 − Λ c (2595) having (I, S, C) = (0, 0, 1), and the same value is used for the other sectors explored in this work.
In Table II,  In the other C = −1 case, having (I = 1 2 , S = −1), we find a pole just below the D s N threshold. The remaining cases have C = 1 and, although they were deeply analyzed in Ref. [105], we briefly comment here a few essential characteristics that will be useful for our discussion of the in-medium results in the next section. In the (I = 0, S = 0) sector, apart from the Λ c (2595) resonance to which we fit the model, there is another very narrow one at 2805 MeV, just below the threshold for DN states but coupling very little to them. We also predict two narrow resonances in the (I = 1, S = 0) sector at 2551 and 2804 MeV, right below the thresholds of the channels to which they couple more strongly, namely πΣ c and DN , respectively. In the (I = 1 2 , S = 1) case, we predict a cusp-like structure placed at the threshold of KΣ c , the channel that shows the largest coupling to this state.

B. Medium effects
The are two main sources of medium effects to consider: one is a consequence of the Pauli exclusion principle, that prevents the scattering of two nucleons into states which are already occupied. The other is related to the fact that the properties of all mesons and baryons are modified in the medium due to their interactions with the Fermi sea of nucleons. Pauli blocking and finite temperature effects can be incorporated in the coupled-channel equations by simply replacing the free nucleon propagator by the in-medium one, where (p 0 , p ) is the total four-momentum of the nucleon in the nuclear matter rest frame, n N ( p, ρ, T ) is the usual Fermi-Dirac distribution function, and E N (| p |) is the on-shell energy of the nucleon.
The nuclear medium effects on the mesons can be incorporated by including their corresponding self-energies, Π m (q 0 , q, ρ, T ), in the meson propagator being (q 0 , q ) the four-momentum of the meson. This is done in practice through the corresponding Lehmann representation of the meson propagator where S m(m) (ω, q, ρ, T ) is the spectral function of the meson m(m): We note here that in this work only the D,D, D s andD s mesons have been dressed by self-energy insertions. Mesons π, K, η, η ′ and η c have not been dressed, as done e.g., in Refs. [75,103,104].
The reason is that the states containing these mesons couple weakly to the DN and D s N ones and, therefore, it is expected that approximating the π, K, η, η ′ spectral functions by the free-space ones, i.e., delta functions, will not influence much the in-medium properties of the D and D s mesons. We emphasize, however, that the present work addresses for the first time the simultaneous dressing of the D and D s mesons in the charm C = 1 sector, and that of theD andD s mesons in the charm The loop function for the free case given by Eq. (6) must now be replaced by the one including the medium and temperature effects on the baryon and meson propagators, as given by Eqs. (8) and (10). Using the Imaginary Time (or Matsubara) Formalism [108] we obtain: where are the total energy, total momentum, and relative momentum of the meson-baryon pair in the nuclear matter rest frame, n is the Fermi distribution of the baryon and f is the Bose enhancement factor of the meson. In practice, given the nuclear densities and temperatures explored in the present work, we can set f = 0 for all mesons and n = 0 for all baryons except for nucleons. One might argue that the Bose enhancement factor for the pions should not be ignored. However, as tested in Ref. [103], the DN amplitudes are insensitive to this factor due to the reduced coupling to πΣ c states resulting from the heavy mass of the meson exchanged in the transition potential.

III. RESULTS AND DISCUSSION
We will start discussing our results for the C = 1 mesons, D and D s . First of all, we note that their in-medium properties will be influenced by the charm C = 1 baryonic resonances that couple significatively to DN and D s N . From the results of our previous work [105], summarized in Table II,  strength distribution which, as we will see, shows a quasiparticle peak at a lower energy than in free space and a pronounced peak at even lower energies related to Λ c (2595)N −1 excitations.
The reduced in-medium DN threshold opens decay channels for the Σ c (2804) which, therefore, broadens considerably. As for the Λ c (2595), its position makes it very sensitive to the low energy strength of the D spectral function and, together with the larger coupling to DN states, explains why the resonance acquires such a large amount of attraction.
In Ref. [75], where the TVME in the t → 0 limit is employed, a similar behavior is observed for the Λ c (2595). The repulsive shift with respect to the free space position due to Pauli blocking effects is compensated by the attractive self-consistent dressing of the D meson. However, the shift is smaller in Ref. [75], as it can be seen from MeV below the free D-meson mass. In addition, each resonance leaving a signature in the selfenergy produces a resonant-hole excitation peak in the spectral function, located at a somewhat different value of energy due to the complex structure of the self-energy. The common behavior is that the resonance-hole modes in the spectral function get displaced such that they move further away from the quasiparticle peak. In the case of Pauli blocking, we can clearly distinguish three of such modes, associated to Σ c (2551)N −1 , Λ c (2595)N −1 and Σ c (2804)N −1 excitations. When meson dressing is incorporated, only the Λ c (2595)N −1 excitation mode is clearly visible. The Σ c (2804)N −1 mode merges with the quasi-particle peak, and the Σ c (2551)N −1 one is no longer visible in the spectral function as compared to the Λ c (2595)N −1 mode. A similar behavior has been observed in Refs. [75,103]. In contrast, in Ref. [101], the Σ c (2804)N −1 mode appears at a much lower energy and mixes with the Λ c (2595)N −1 one, while the quasiparticle peak of the D meson experiences a repulsive shift of 32 MeV. It is also worth mentioning that in the SU(8)-inspired model of Ref. [104] the quasiparticle peak appears at slightly lower energies than the free mass but the D-meson spectral function shows a completely different shape due to the different resonant-hole composition of the D-meson self-energy.
The imaginary part of the D s self-energy, displayed in the upper right panel of Fig. 2 shows only a small enhancement at around 2 GeV. This is a reflection of the enhanced cusp found in the (I = 1/2, S = 1, C = 1) amplitude at the KΣ c threshold [105]. This structure generates a small but non-negligible amount of strength in the D s spectral function to the right of the quasi-particle peak, which barely moves from its free location. This is in contrast to Ref. [101], where a resonance is generated dynamically 75 MeV below the D s N threshold, and the corresponding resonance-hole state in the spectral function appears on the left-hand side of the quasiparticle peak.
In spite of the featureless aspect of the D s spectral function in our model, this relocation of strength from the quasi-particle peak to higher energies diminishes the size of the D s Y loops involved in the coupled-channel problem. Therefore, the simultaneous dressing of the D and D s mesons in our self-consistent coupled-channel model produces a less bound Λ c (2595) resonance in nuclear matter, as already shown in Fig. 1. From Fig. 2 we can see that the corresponding Λ c (2595)N −1 excitation mode of the D-meson spectral function appears approximately 40 MeV higher in energy than when only the D-meson dressing is considered. The in-medium properties of the C = −1 mesons,D andD s , will be determined by the behavior of the correspondingDN andD s N amplitudes in the nuclear medium. In Fig. 3 we display the imaginary part of theD s N amplitude at normal nuclear matter saturation density and zero temperature as a function of the center-of-mass energy P 0 , for various approximations: free (dotted- . This is the most interesting of the two C = −1 cases since theD s N system develops in free space a subthreshold bound state at 2906 MeV that couples significatively toD s N states. Therefore, this pole will be very sensitive to the medium effects. Indeed, when only Pauli blocking effects are considered, the pole moves about 40 MeV towards higher energy as expected. We observe very drastic changes when the dressing of theD andD s mesons is incorporated. The reason is that, as we will see, the in-medium quasiparticle peak of theD s meson experiences a strong attraction. This moves the in-medium threshold forD s N states below the position of the resonance, making its decay possible and quite probable due to the significant coupling to these states. TheD andD s self-energies and spectral functions are shown in Fig. 4 (Fig. 3). The dressing of theD s meson smears this structure in such a way that one barely sees any trace of it in the corresponding spectral function. Moreover, the delta-like quasi-particle peak, appearing 60 MeV below the freeD s mass when only Pauli blocking effects are considered, moves to slightly lower energies when theD s meson is dressed. Considering the additional dressing of theD meson in the relatedDY loops produces a substantial change in thē D s self-energy. This is easy to understand from the results of Table II, where we see that the pole at 2906 MeV couples also very strongly toDY states. The loss of attraction in the region of the quasiparticle peak moves it towards a higher energy, and ends up being 50 MeV below the free mass and merging with the resonant-hole strength. Our findings differ again quite strongly from those of Ref. [101], which are dominated by an exotic coupled-channel molecule at 2780 MeV [73], which is the equivalent to the pole at 2906 MeV found in the model of Ref. [105] and used in the present work. As a consequence, the spectral function for theD s meson found in Ref. [101] shows two distinct peaks, the quasi-particle one located about 10 MeV above the freeD s mass, and a narrow resonance-hole mode located 150 MeV below.   behavior in all spectral functions is that finite temperature moves the quasiparticle peak towards its free location. This is a reflection of the reduced size of the self-energy, because, being built up from an average over the smeared thermal Fermi distribution, involves higher momentum components for which the meson-nucleon interaction is weaker. Except for a few cases, increasing the temperature gives rise to wider quasiparticle peaks because of the increase of collisional width. However, the opposite effect is seen for the D meson in Fig. 6. As already discussed in Ref. [103], this is due to the fact that the strength under this peak also receives contributions from Σ c (2804)N −1 hole excitations, which are washed out by temperature as any other resonant-hole mode. Consequently, the peak of the D-meson spectral function becomes narrower and more symmetric as temperature increases, similarly to Ref. [103].
The density effects observed in the spectral functions are also clearly understood. In general, we find that the self-energy roughly doubles its size when going from nuclear matter at normal nuclear matter saturation density to a system which is two times denser. This is consistent with the low density limit behavior and points at a weak density dependence of the in-medium meson-nucleon amplitude in this density region. This is the reason why, in general, the quasiparticle peak of the spectral functions at 2ρ 0 are found approximately twice further away from the free space position and are twice wider than in the case of ρ 0 .

IV. SUMMARY AND CONCLUSIONS
We have studied the properties of open charm mesons, D,D, D s andD s , in nuclear matter at finite temperature within a self-consistent coupled-channel approach which uses, as meson-baryon interaction, a full t-dependent vector meson exchange driving force.
The in-medium scattering amplitudes are obtained by solving the Lippmann-Schwinger equation at finite temperature including Pauli blocking effects, as well as D,D, D s andD s self-energies, paying a particular attention to their mutual influence.
We have analyzed how our dynamically generated resonances are affected by density and temperature. As in other similar approaches, the resonances that couple strongly to intermediate states involving nucleons, move upwards in energy when Pauli blocking effects are considered, as a consequence of the loss of phase space. When the self-consistent dressing of the charm mesons is incorporated, the resonances gain attraction again.
We have seen that dressing the D s meson has a non-negligible effect on the DN amplitude and on the properties of the D meson. Therefore, we conclude that a simultaneous in-medium treatment of both mesons, as the one attempted in the present work, is necessary. Similarly, the in-medium properties of theD s andD mesons are interrelated and must be also considered together.
The spectral functions of the D andD s mesons are quite rich. At T = 0 MeV and normal nuclear matter density one finds a quasiparticle peak located below the corresponding free meson mass by about 50 MeV, as well as strength associated to resonant-hole excitations which, in the particular case of the D meson, is clearly visible as a narrow Λ c (2595)N −1 excitation peak.
In general, increasing the temperature has the effect of moving the quasiparticle peak towards its free location making it wider, as a consequence of a milder meson-baryon interaction and a larger amount of collisions. The exception found for the D-meson is naturally explained in terms of the mixing of the quasi-particle peak with a resonant-hole mode.
For the densities explored, up to twice nuclear matter normal saturation density, we have found that the density effects follow the linear behavior expected for the low density regime: the selfenergy roughly doubles its size when going from nuclear matter at normal saturation density to a system which is two times denser, indicating a mild density dependence of the in-medium mesonbaryon interaction amplitudes.
The enormous computational effort of the present work, which uses a coupled-channel formalism, an interaction that goes beyond the t → 0 limit, and the simultaneous consideration of the inmedium D and D s (D s andD) meson self-energies, has prevented us from incorporating the coupling to states involving vector-mesons. We are aware that, given the availability of models that permit dealing with these important degrees of freedom, our approach should be extended to the vector mesons such that it also includes, for instance, the D * N and D * s Y channels in the C = 1, S = 0 sector. We hope that, by first identifying which channels play a relevant role and which ones might be omitted, we can make progress toward this goal in the nearby future.