Pseudoscalar and vector open charm mesons at finite temperature

Vacuum and thermal properties of pseudoscalar and vector charm mesons are analyzed within a self-consistent many-body approach, employing a chiral effective field theory that incorporates heavy-quark spin symmetry. Upon unitarization of the vacuum interaction amplitudes for the scattering of charm mesons off light mesons in a fully coupled-channel basis, new dynamically-generated states are searched. The imaginary-time formalism is employed to extend the calculation to finite temperatures up to $T=150$ MeV. Medium-modified spectral shapes of the $D$, $D^*$, $D_s$ and $D_s^*$ mesons are provided. The temperature dependence of the masses and decay widths of the nonstrange $D_0^*$ (2300) and $D_1^*$(2430) mesons, both showing a double-pole structure in the complex-energy plane, is also reported, as well as that of the $D_{s0}^0$(2317) and $D_{s1}^*$(2460) resonances and other states not yet identified experimentally. Being the first calculation incorporating heavy-light vector mesons at finite temperature in a self-consistent fashion, it brings up the opportunity to discuss the medium effects on the open charm sector under the perspective of chiral and heavy-quark spin symmetries.


I. INTRODUCTION
The in-medium properties of mesons with charm content have been a matter of high interest over the years (see [1][2][3][4] for reviews). This interest was triggered because of the J/Ψ suppression in heavy-ion collisions, as seen at SPS energies by the NA50 collaboration [5], which was predicted in Ref. [6] as a signature of the existence of the quark-gluon plasma due to color screening. The J/Ψ absorption in hot dense matter could be also modified due to the change of the properties of open-charm mesons in matter in the comover scattering scenario (see, for example, the initial works of Refs. [7][8][9][10]), thus providing a complementary explanation for J/Ψ suppression. Moreover, the possible attraction felt by D mesons in nuclear matter could lead to the formation of open-charm meson bound states in nuclei [11][12][13][14][15].
On the one hand, several theoretical works have addressed the properties of open-charm mesons in nuclear matter. Those works range from phenomenological estimates based on the quark-meson coupling model (see, for example, the initial works of [16,17]), nuclear mean-field calculations in matter [18][19][20], Polyakov-loop extensions of Nambu-Jona-Lasinio models [21], models based on π-exchange implementing heavy-quark symmetries [15], QCD sum-rule (QSR) computations (see [22] for a recent review) to self-consistent unitarized coupled-channel approaches (for a review, see [2]). The full spectral features of the opencharm mesons in nuclear matter emerge naturally from the latter ones. Starting from the exploratory works of Ref. [23,24], the spectral features of open charm in nuclear matter have been studied within t-channel vector-meson exchange unitarized models [25][26][27][28]. Later on, heavy-quark spin symmetry constraints were implemented explicitly in a unitarized coupledchannel approach [29][30][31][32][33], leading to the determination of open charm-nucleon interactions and open-charm spectral functions in nuclear matter [12,13,34].
Recently, a finite-temperature unitarized approach based on a SU (4) effective Lagrangian has been put forward [63], In the present paper we address the properties of open-charm mesons in a hot mesonic bath (mainly formed by pions), within a finite-temperature unitarized approach, following the path of Ref. [63] and extending our previous work in Ref. [64]. As compared to Ref. [63], the dynamics of open charm with light mesons is based on an effective Lagrangian that is expanded up to next-to-leading order in the chiral counting, while keeping leading order in the heavy-quark expansion [50,[65][66][67]. Moreover, the present paper extends our work of Ref. [64], going beyond the analysis of D * 0 (2300) and D s (2317) states and their possible identification as the chiral partners of the D and D s ground states [64].
In this work we perform a detailed analysis of all strange-isospin channels in the pseudoscalar and vector open-charm sectors in vacuum and at finite temperature using a unitarized approach based on a chiral effective field theory that implements heavy-quark spin symmetry at leading order. We start by analyzing the pole structure in vacuum of the dynamically-generated D * (2300) and D * s0 (2317) states as well as D * 1 (2430) and D * s1 (2460). We then extend our calculations to finite temperature by means of a self-consistent manybody approach and study the thermal dependence of the D, D * , D s and D * s mesons together with that of D * (2300), D * s0 (2317), D * 1 (2430) and D * s1 (2460) states. The paper is organized as follows. In Sec. II we present the details of the effective La-

II. D-MESON INTERACTION WITH LIGHT MESONS
In this section we give details on the effective field theory used to describe the dynamics of D and D * mesons, and their interactions with light mesons: π, K,K and η mesons (we do not consider η mesons in this work because of their larger mass).
In the first part we present the effective Lagrangian involving the heavy and light mesons, which is based on chiral and heavy-quark symmetries and expanded up to next-to-leading order (NLO) in the chiral counting and at leading order (LO) in heavy-quark mass counting [50,[65][66][67]. We describe how to fix the unknown low-energy constants from recent lattice-QCD calculations in the same sector [68]. Then, we provide the values of the treelevel scattering amplitudes for the different processes considered in this work. Finally, we describe how to construct unitarized amplitudes by solving a Bethe-Salpeter equation in a coupled-channel basis, at both T = 0 and T = 0. For the latter, we work within the imaginary-time formalism (ITF) requiring self-consistency of the heavy-meson self-energy.
We summarize in Table I the degrees of freedom (both heavy and light mesons) used in this work together with their vacuum masses. Interactions between heavy and light mesons are described in terms of an effective Lagrangian for the degrees of freedom described in Table I. It is based on chiral symmetry (involving the physics of the pseudo-Goldstone bosons at low energies) as well as heavyquark spin-flavor symmetry for the charmed mesons (both pseudoscalar D and vector D * mesons). In the heavy-mass counting we work at leading order (LO), whereas in chiral power counting we also consider NLO terms, The LO Lagrangian contains the kinetic terms and interactions of the D mesons as well as self-interactions of the pseudo-Goldstone bosons. For the latter, the pure light-meson sector is described by the standard chiral perturbation theory (ChPT) [70], whose Lagrangian is not explicitly written here. The LO Lagrangian reads where D denotes the antitriplet of 0 − D-mesons, D = D 0 D + D + s , and similarly for the vector 1 − states, is the unitary matrix of pseudo-Goldstone bosons in the exponential representation: and f π is the pion decay constant, f π = 92.4 MeV. We note that in the matrix representation of the SU (3) meson octet we have already identified the η 8 with the physical η by neglecting the η − η mixing.
The tree-level amplitudes are extracted from the LO+NLO Lagrangian and they have already been given in the cited references. For convenience we reproduce them here once more. For a binary scattering involving a charm meson with incoming channel i and outgoing channel j the amplitudes read  [66,71] by fits to the scattering lengths calculated on the lattice, simultaneously also to lattice finite-volume energy levels in [68]. We have taken the values of the LECs from the Fit-2B in [68] for which the full amount of lattice data available and the physical value of f π are considered, and which is the preferred fit of the authors according    to large N c arguments. The values are shown in Table IV for the sake of completeness, for both pseudoscalar and vector charmed mesons, where we have considered h i =h i at LO in the heavy quark expansion but used the different physical values ofM D andM D * in the determination of h 4 , h 5 ,h 4 andh 5 . The difference between our unitarized amplitudes and those in [68] lies in the regularization procedure, as explained in the next section.
To conclude this section we comment on the extraction of the S-wave component of these tree-level scattering amplitudes. The S-wave projected amplitude is computed as where t(s, cos θ) is given in terms of s, cos θ and the meson masses cf. [68], u = i m 2 i − s − t and P L (cos θ) is the Legendre polynomial of order L normalized to 1 −1 dxP L (x)P L (x) = 2δ LL /(2L + 1). In particular, for the S-wave projection we only need P L=0 = 1. The analysis of higher partial waves is left for future studies.    [68]) for the interaction of pseudoscalar (first row) and vector (second row) charmed mesons with the light mesons.

B. Unitarization and self-consistent propagators
We summarize here the unitarization procedure which allows us to fulfill exact unitarity in all scattering amplitudes. Starting from the tree-level amplitudes V ij (s) we construct the unitarized ones T ij (s) by solving a Bethe-Salpeter equation for the 2-body problem in a full coupled-channel basis. We distinguish the cases at T = 0 and T = 0 as their methodologies and ulterior analyses differ considerably.

Unitarization in vacuum
In the on-shell factorization approach [72,73], the Bethe-Salpeter equation for the unitarized amplitude at T = 0 reads: where i, j represent the incoming and outgoing channels (see Table II), and we sum over the possible intermediate channels l. The two-body propagator function in vacuum is the loop with p µ = (E, p). We make explicit that at T = 0 the loop function is given as a function of the Mandelstam variable s = p 2 . This function should be regularized in vacuum, for which one can employ a hard momentum cut-off Λ, and the corresponding expression of the loop with the phase-space factor σ = Alternatively, the loop integral can be calculated in dimensional regularization (DR): G l DR (s; a l (µ)) = 1 16π 2 a l (µ) + ln where a l (µ) is the subtraction constant at the regularization scale µ for channel l and q l = σ/(2 √ s) is the on-shell three-momentum of the meson in the loop. If one demands that both regularization procedures give the same value of the loop function at threshold, the following expression for the subtraction constants is obtained: for a given µ. Notice that the running of a l (µ) cancels the explicit µ dependence in Eq. (10), so the loop function does not depend on the regularization scale.
In Ref. [68] the loop function is regularized with DR and the subtraction constants are considered as fit parameters together with the LECs. In a different approach, here we use the cut-off regularization scheme so as to follow the same convention as for T > 0, where the loop function can only be regularized by limiting the integrals to | q | < Λ. Using Eq. (11) we find that the subtraction constants in [68] correspond to small and unrealistic values of the cut-off Λ for certain channels, and therefore we ignore these values and fix a constant cut-off for all the channels (Λ = 800 MeV). We have checked that the values of the subtraction constants are compatible with those specified in [68], with µ = 1 GeV.
In the on-shell approximation [72] the unitarized scattering amplitude has an algebraic solution which, in the general coupled-channel case, is a matrix equation.
Notice that the internal propagators of the loop function should include self-energy corrections due to interactions in vacuum. This effect dresses (and renormalizes) the mass of the mesons. At T = 0 the dressed masses are simply the physical ones given in Table I, and there is no need to perform a self-consistent procedure. This is different in the T = 0 case.
The unitarization process leads to the potential emergence of poles in the resummed amplitude T ij (s) at the zeros of the denominator of Eq. (12). These poles correspond to states that are dynamically generated by the attractive coupled-channel meson-meson interactions.
The characterization of these states requires to analytically continue the T -matrix to the complex energy plane, where the search for poles should be performed in the correct Riemann sheet (RS). The loop function in Eq. (8) is a multivalued function with two RSs above threshold. To select a particular RS one needs to add a contribution to the imaginary part [74], where the subindices I and II denote the first and second RSs, respectively. The same result follows from changing the sign of the momentum q l in Eq. (10) or σ in Eq. (9) and taking the phase prescription of the logarithms ln z = ln |z| + iθ as 0 ≤ θ < 2π.
The analytic structure of G l provides the unitarized amplitude T ij (s) with a set of 2 n RSs, where n is the number of coupled channels. We define the second RS of the T (s) amplitude as the one which is connected to the real energy axis from below and is obtained by using G l This prescription provides the pole position and half-width closer to the values obtained from a Breit-Wigner parametrization of the associated amplitude [74]. However, not too far from a threshold channel to which a resonance couples strongly, the pole might appear in a RS for which the transition from G l I to G l II is not applied and yet the resonance has a visible effect in the real-axis amplitudes. This is the case for the second, higher energy poles associated to the D * 0 (2300) and the D * 1 (2430), as will be discussed in Sect. III. In the complex √ s ≡ √ z plane, the real and imaginary parts of the pole positions √ z p , give the mass and the half width of the dynamically generated states, respectively: The poles located on the real axis of the physical RS, below the lowest threshold, correspond to bound states, those on an unphysical sheet at a necessarily complex energy correspond to resonances, and virtual states are poles that lie on the real axis of an unphysical sheet, below the lowest threshold. Resonance poles that are located on the unphysical sheet closest to the physical sheet (the second RS) are the ones that, together with bound states, are more likely to generate structures in the scattering amplitude. Therefore it is common to call resonances only the resonant poles in the second sheet and generalize the term "virtual state" to resonant poles in any other unphysical sheet, which can still yield to structure and cusps near the thresholds.
The scattering amplitude can be expanded in a Laurent series around the pole position where g i is the coupling of the resonance or bound state to the channel i and g i g j is the residue around the pole. Therefore, from the residue of the different components of the T -matrix around the pole one can extract the coupling constants to each of the channels, The concept compositeness of shallow bound states was formulated by Weinberg in Ref. [75], applied to narrow resonances close to threshold in [76,77], subsequently extended to the pole position of a resonance by analytical continuation in [78][79][80], and appropriately transformed to obtain a probabilistic interpretation in [81], where it was shown that the expression gives the amount of i th channel meson-meson component in the dynamically generated state.

Finite temperature case
For T = 0 we need to account for several modifications, both in the methodology and in the final analysis of the dynamically-generated states. We use the ITF, where the time dimension is Wick rotated and compactified in the range [0, β = 1/T ], where T is the temperature of the system. Any integration over the energy variable is therefore transformed into a summation over the so-called Matsubara frequencies, q 0 → iω n = 2nπT i for bosons (see [82][83][84][85][86] for details of the formalism).
The tree-level amplitudes remain unmodified, but the two-body loop function in the Bethe-Salpeter equation is now modified as compared to the vacuum case. It reads Before performing the Matsubara summation over ω n we introduce the Lehmann representation for the propagators in terms of the spectral function, where the subindex M denotes the meson (D or Φ) and in the second equality we have separated the particle and antiparticle parts. Using delta-type spectral functions, By keeping generic spectral functions S D (ω, q ) and S Φ (ω , p − q ) the Matsubara summation using Cauchy's residue theorem gives the following expression for the loop function where f (ω, T ) = 1/ (exp(ω/T ) − 1) is the equilibrium Bose-Einstein distribution function.
and SM (−ω, q; T ) = −S M (ω, q; T ) and analytically continue the external Matsubara frequency to real energies iω m → E + i so as to write the expression above in the following compact way: where the integrals over energy extend from −∞ to +∞.
At finite temperature the meson masses are dressed by the medium. The effects of finite temperature in the unitarized scattering amplitudes are readily obtained by solving Eq. (7) with finite temperature loops containing dressed mesons. Notice that in the ITF, as the thermal corrections enter in loop diagrams [84,85], the tree-level scattering amplitudes remain the same as in vacuum (with the zeroth component of the four-momentum expressed as a bosonic Matsubara frequency).
The spectral function of the heavy meson is computed from the imaginary part of its retarded propagator, where the heavy-meson self-energy follows from closing the light meson line in the T-matrix.
For temperatures below T c the largest contribution will come from pions, as the abundance of heavier light mesons, i.e. kaons and eta mesons, is suppressed by the Bose factors. We note that the contribution of a kaonic bath can be relevant for temperatures close to T c .
In order to study this contribution, in Sec. III we will analyze the modification of open charm mesons in the presence of a kaonic bath by taking into account the corresponding Bose-Einstein distribution.
The pion contribution to the heavy-meson self-energy is computed in the ITF: where q = p − q is the three-momentum of the pion. It is convenient to use the Lehman representation for the pion propagator introduced in Eq. (19) as well as for the T-matrix, and, following the same procedure as for the loop function described above, the expression obtained for the heavy-meson self-energy reads with ω π = q 2 + m 2 π . This expression can be condensed in the following, after the analytical continuation iω n → ω +iε. Note that we only consider the Bose-Einstein distribution for pions at finite temperature while neglecting other possible medium modifications. In appendix B we show that, since the pion mass slightly varies for temperatures below T = 150 MeV, the modification of the properties of the charm mesons is mild as the self-energy of open charm is barely affected.
The self-energy needs to be regularized as it also contains the vacuum contribution. In order to do so, we separate the vacuum and matter parts. The vacuum contribution is identified with the expression surviving in the limit T → 0. One can see from Eq. (25) that in this limit the Bose factors in the statistical weights of the numerators cancel, and only the terms with a factor 1 remain. Our prescription for the regularization is to take these terms to zero at finite temperature, as they where effectively included in the renormalization of the D-meson mass at T = 0.
This set of equations is solved iteratively until self-consistency is obtained. The procedure is sketched in Figures 1a, 1b and 1c. The T -matrix amplitude is represented by a red blob, whereas the perturbative amplitude V (s) is denoted with a blue dot. Figure 1a shows the of not only the ground-state mesons but also the dynamically-generated states.
A. Scattering amplitudes and dynamically-generated states at T = 0 In this section we focus on the T = 0 case and distinguish the scalar D and vector D * cases separately. Notice that, since we are working at LO in the heavy-mass expansion, these two sectors are not mixed, but are related by heavy-quark spin symmetry.
1. J = 0 case: Interactions and D * (2300) and D * s0 (2317) We start by analyzing the D and D s interaction with non-charmed pseudoscalar mesons for two different sectors given by total strangeness S = 0 (Fig. 2), corresponding to the Dπ, Dη and D sK coupled-channels calculation, and S = 1 (Fig. 3), built from the DK and D s η channels. We focus on those two sectors since we obtain several resonant states, among them two that can be identified with the experimental D * 0 (2300) and D * s0 (2317), as we will discuss later on. Our results for the strangeness S = 1 and isospin I = 0 sector are shown in Fig. 3. In occurs below the m D + m K threshold and therefore leads to a bound sate in the real axis at T = 0, as we will see.
As discussed in the previous plots, apart from threshold effects, the different structures that appear in the scattering amplitudes correspond to the presence of poles or dynamically generated states that appear due to the attractive coupled-channel meson-meson interactions. The five poles that we find in the J = 0 sector at T = 0 are summarized in Table V.  In (S, I) = (−1, 0) sector, we find one virtual state, as the pole lies below the lowest threshold but it appears in the unphysical (-) RS. In the (S, I) = (0, 1 2 ) sector we find two poles that correspond to the D * 0 (2300) state. This double pole structure of the D * 0 (2300) is well documented [69], being our results compatible with those given in Refs. [68,87].
For the position of the lower pole, we find that the real part lies between the Dπ and Dη thresholds and it has a sizable imaginary part, which is a consequence of the large value of the coupling of the generated resonance to the Dπ channel, to which it can decay. The mass of the higher pole is above the last threshold, that is, the D sK one, and also has a large decay width, as it couples sizably to the channels opened for its decay. However, this pole appears in the (−, −, +) RS, with this RS being only connected to the real axis between the Dη and the D sK thresholds. In fact, for different values of the parameters [68,87] this pole appears between the Dη and D sK thresholds or even below the Dη threshold. Moreover, it is worth noticing that the lower pole qualifies mainly as a Dπ state, as indicated by the large value of the compositeness, whereas the higher one is essentially D sK system, although we should note that this case does not correspond to a canonical resonance in the sense that the associated pole does not reside in the RS that is directly accessible from the physical one.
Therefore, as discussed in Ref. [81], the physical interpretation of Eq. (17) as a probabilistic compositeness is not valid for this resonance, a fact that is corroborated by the value larger than one obtained in this case. In the (S, I) = (1, 0) sector we find only one pole, which lies on the real axis below the DK threshold, that is in the (+, +) RS. It is identified with the D * s0 (2317) resonance, and has sizable couplings to both DK and D s η, as given by the compositeness. With the present model, the pole mass turns out to be smaller than that of the experimental resonance, but small variation of the parameters can easily accommodate this state to the observed position, in line with similar models in the literature that have advocated this resonance to be mostly a DK hadronic molecule (see Ref. [88] and references therein). In the (S, I) = (1, 1) sector, there is a resonance in the (-,+) RS with a large width, as it couples strongly to D s π states to which it can decay. This resonance cannot be identified with any of the PDG states known up to now. At T = 0 we also find five poles in the J = 1 amplitudes, which are summarized in Table VI. As seen, a double pole structure, which can be identified with the D * 1 (2430) resonance listed in the PDG [69], is obtained in the (S, I) = (0, 1 2 ) sector, one pole being mostly a D * π state and the other mostly a D * sK one. A resonance coupling mostly to D * K

Determination of masses and widths at T = 0
In order to obtain a quantitative description of the thermal dependence of the masses and widths of the open-charm ground states, one should analyze the behavior of the corresponding spectral functions with temperature. The mass change with temperature can be extracted from the position of the peak of the spectral function at different temperatures, that is, from the position of the so-called quasiparticle peak, ω qp , obtained from: whereas the variation of the width with temperature can be obtained from the thermal spectral function at half height. However, the determination of the behavior of the mass and width with temperature of the dynamically-generated states, such as D * (2300) and D * s0 (2317) as well as D * 1 (2430) and D * s1 (2460), is rather delicate. The calculation of the poles in the complex plane at finite temperature, while performing the self-consistency program, is computationally very expensive and unfeasible. In addition, the analytic continuation to the different RSs should be performed, not from the real energy axis, but from the imaginary Matsubara frequencies, and it is not clear how to perform this technically. Therefore, we employ the method described in the following to obtain the particle properties on the real axis, through fits of the imaginary part of the unitarized scattering amplitudes at finite temperature shown in For isolated resonances close to the real energy axis and not close to any threshold we simply use a Breit-Wigner form. But in the case of resonances that interact with the background of another resonance (in the coupled-channel case) we use a Breit-Wigner-Fano shape [90]. This can be used for the lower pole in the double pole structure of the D * 0 (2300). Indeed, we have checked that the obtained mass and width of the fit at T = 0 are in very good agreement with the values of the pole mass and the width in Table V. The Breit-Wigner-Fano-type distribution provides a simple parametrization to describe the distorted lower resonance at finite temperature: where q is the Fano parameter measuring the ratio of resonant scattering to background scattering amplitude. In the absence of background the value of q goes to infinite and Eq. (28) becomes the usual Breit-Wigner distribution.
For resonances close to a threshold we fit to a Flatté-type distribution [91]. In particular for the higher pole in the double pole structure we first subtract the background and then, we use a generalized Flatté parametrization with three coupled-channels: where ρ i stands for the phase space of the i th channel, The resonance width is given by where the phase spaces have been evaluated at the resonance mass. In our case the subindices correspond to 1 ≡ Dπ, 2 ≡ Dη and 3 ≡ D sK . In order to avoid an ill behavior of the fit due to the large amount of free parameters, the value of g 1 is imposed to vary linearly from its lowest value at T = 0 to the highest one at T = 150 MeV.
The Breit-Wigner-Fano distribution is also used for isolated resonances at high temperatures, if they become wide enough to be affected by threshold effects. Our results for the masses and widths of the ground-state mesons D and D s as well as the dynamically generated states D * (2300) and D * s0 (2317) are shown in Fig. 8 for a pionic bath (solid lines) and when the medium is populated by both pions and kaons (dashed lines).
They are summarized as follows: 1. In a pionic medium the ground-state D mass shows a sizable decrease of ∆m D ∼ 40 MeV at the largest temperature T = 150 MeV. This reduction is twice larger than that observed in [60], where a more phenomenological approach for the D-meson propagator is used. Our reduction, however, is smaller than the one shown in Ref. [92], where a non-unitarized ChPT is considered. In contrast, the SU (4) effective approach of [63] shows no significant modification of the mass of the ground state. On the other hand, the mass of the two poles of the D * 0 (2300) change less rapidly with temperature compared to the ground state, moving downwards and distancing from each other with increasing temperature. As a consequence, we cannot conclude that masses of opposite parity states become degenerate for the temperatures studied, as discussed in Ref. [64].  With regards to the thermal evolution of the masses and widths in the J = 1 case for the ground-state mesons D * and D * s as well as the dynamically generated states D * 1 (2430) and D * s1 (2460), we observe a clear parallelism in their behavior with that obtained for the J = 0 states, and which is shown in Fig. 9. Again, this is due to the fact that interactions of light mesons with pseudoscalar open-charm ones is related by heavy-quark spin symmetry to those with vector open-charm ground states. Thus, our conclusions are similar to the ones presented for J = 0, namely: 1. In a medium with pions, the D * mass shows a sizable decrease of ∆m D * ∼ 40 MeV at the largest temperature T = 150 MeV, similar to the D mass shift. As for two poles that formed the D * 1 (2430), their masses decrease less rapidly with temperature compared to the ground state, distancing from each other as temperature increases, in an analogous manner as for the two poles of the D * 0 (2300). As a consequence, also in the J = 1 case, we cannot conclude that masses of opposite parity states become degenerate with temperature, at least for the range of temperatures studied here. significantly, in particular for the D s . The effect of the kaonic and pionic medium on the masses and widths of the dynamically generated states is, however, rather moderate. We notice that the modifications of the J = 1 states are similar to the J = 0 sector due to heavy-quark spin symmetry.
Our results provide the first systematic approach to the thermal effects on open charm mesons below the crossover temperature. In the future we will explore the effects of mediummodified light mesons in the self-consistent calculation, and the extension to bottom flavor, exploiting the heavy-quark flavor symmetry. On the other hand, we will apply our results of the D-meson spectral functions to the calculation of transport coefficients like the heavyflavor diffusion coefficient. This will constitute an extension of our previous results [51-53, 93, 94] incorporating off-shell effects in the kinetic approach.      (5) in the sectors with charm C, strangeness S and charge Q in the particle basis.  In this work we have neglected the medium modifications of the light mesons and used vacuum spectral functions for them, in both the T −matrix calculation as well as in the D-meson self-energy corrections. This approximation-which should be reasonable at low temperatures-was implemented in Ref. [64], where we based our assumption on the pion mass modifications given in Refs. [95,96]. We leave a more thorough study of the thermal modification of light mesons into our self-consistent approach for the future. In this appendix we present a validity check using a medium-modified pion mass.
To address the correction of the pion self-energy due to the thermal bath, we have applied the methodology of [95]. As opposed to our calculation for heavy mesons, the method in [95] is not self consistent but based on the one-loop correction to the meson self-energy in the dilute limit. We have computed the real part of the pole of the pion propagator, whose self-energy is corrected by the thermal medium producing a modified dispersion relation, where ω p = p 2 + m π (T = 0) is the vacuum dispersion relation (with m π (T = 0) = 138 MeV), f (ω p , T ) is the Bose-Einstein distribution function, T ππ (s) is the (isospin averaged) forward amplitude of the ππ → ππ process and s = (p + q) 2 the Mandelstam variable.
T ππ (s) is calculated using the unitarized scattering amplitudes coming from SU (3) chiral perturbation theory Lagrangian [97,98]. The unitarization approach used in [97,98] is similar (but not equal) to ours. In particular, the scattering amplitudes from [98] have no corrections due to the temperature, but this is consistent with the one-loop approximation for the pion self-energy.
In this appendix we neglect the pion width-which is also generated due to temperature effects-so we can still use Dirac delta spectral functions peaked at ω(p). We define the thermal pion mass as the value m π (T ) = ω(p = 0; T ) and plot it in Fig. 10 up to T = 150 MeV. At this temperature the pion mass is m π (T = 150MeV) = 120 MeV.
We have run our code for the D-meson self-energy at T = 150 MeV using this reduced pion mass (the pion decay will be added in a future study). We find that the mass of the ground-state D ( * ) and D  In conclusion, for low temperatures T 150 MeV it is acceptable to neglect the medium effects of the light mesons. For the largest temperature considered T = 150 MeV, the effects of a medium-modified pion are noticeable, but still small. In the future, we plan to incorporate the medium-modified spectral functions (with both mass and decay width depending on temperature), to decide whether the widening of the pion (and also the modification of the other light mesons) can produce a significant change on the properties of heavy-flavor mesons at intermediate temperatures.