Thermal Relaxation of Charm in Hadronic Matter

The thermal relaxation rate of open-charm ($D$) mesons in hot and dense hadronic matter is calculated using empirical elastic scattering amplitudes. $D$-meson interactions with thermal pions are approximated by $D^*$ resonances, while scattering off other hadrons ($K$, $\eta$, $\rho$, $\omega$, $K^*$, $N$, $\Delta$) is evaluated using vacuum scattering amplitudes as available in the literature based on effective Lagrangians and constrained by realistic spectroscopy. The thermal relaxation time of $D$-mesons in a hot $\pi$ gas is found to be around 25-50\,fm/$c$ for temperatures $T$=150-180\,MeV, which reduces to 10-25\,fm/$c$ in a hadron-resonance gas. The latter values, argued to be conservative estimates, imply significant modifications of $D$-meson spectra in heavy-ion collisions. Close to the critical temperature ($T_c$), the spatial diffusion coefficient ($D_s$) is surprisingly similar to recent calculations for charm quarks in the Quark-Gluon Plasma using non-perturbative $T$-matrix interactions. This suggests a possibly continuous minimum structure of $D_s$ around $T_c$.


I. INTRODUCTION
After the initial discovery of a new state of matter in high-energy nuclear collisions at the Relativistic Heavy Ion Collider (RHIC), the focus is now shifting to quantifying the properties of what is believed to constitute a strongly coupled Quark-Gluon Plasma (sQGP). Basic quantities characterizing the medium are its thermal spectral functions and transport properties. In heavy-ion collisions (HICs), the former are most directly studied in the electromagnetic (vector) channel via the thermal emission of lepton pairs (cf. Ref. [1] for a recent review). The latter, however, are best studied using observables with small but controlled deviations from thermal equilibrium. Thus, heavy quarks (charm (c) and bottom (b)), whose thermal equilibration time is expected to be of the order of the QGP lifetime in HICs, are a promising tool to quantify flavor transport, and eventually deduce general properties of the QGP as formed in these reactions [2].
The large masses of c and b quarks (m c,b ) enable us to assess the modifications of their momentum spectra in HICs via a diffusion process in an evolving background medium as formulated, e.g., within a Fokker-Planck equation [3] (typical early temperatures of the medium produced at RHIC, T ≃ 250 MeV [4], are well below m c,b ≃ 1.5, 4.5 GeV). A reliable determination of the heavy-quark (HQ) transport coefficients in the QGP depends on several components. First and foremost these are microscopic calculations of the thermal HQ relaxation rate in the QGP [3,[5][6][7][8][9][10][11][12][13][14] (see Ref. [2] for a review). Second, the coefficients need to be implemented into a realistic bulk medium evolution (see, e.g., Ref. [15] for a recent discussion). Third, heavy-flavor (HF) interactions in evolution phases other than the QGP have to be evaluated, i.e., in the so-called pre-equilibrium phase as well as in the hadronic phase. The former is of a relatively short duration, ∆τ pre 1 fm/c, and is sometimes mimicked by reducing the formation time of the QGP. The duration of the hadronic phase is substantially longer, ∆τ had ≃ 5-10 fm/c. Its relevance for HF phenomenology is further augmented by the fact that the hadronic medium inherits the full momentum anisotropy from the QGP, believed to be close to the finally observed one. Thus, even a rather weak coupling of HF hadrons to hadronic matter can lead to noticable contributions to their elliptic flow. Furthermore, if the QGP realizes a minimum in its viscosity-toentropy-density ratio, η/s, close to the (pseudo-) critical temperature, T c ≃ 170 MeV, a hadronic liquid close to T c should possess similar properties. This is usually referred to as a "quark-hadron duality", as suggested, e.g., in calculations of thermal dilepton emission rates [1].
Charm diffusion in hadronic matter has received little attention to date (see Ref. [16] for a very recent estimate using heavy-meson chiral perturbation theory). Its potential relevance has been noted in Ref. [2] based on calculations of D-meson spectral functions in nuclear matter using effective hadron Lagrangians [17,18], as well as in a hot pion gas [19]. In the present paper, we augment these works to evaluate charm diffusion in hadronic matter. Since the latter features many resonances at temperatures approaching T c , we not only utilize Dπ and DN interactions but also scattering amplitudes off excited hadrons (K, η, ρ, ω, K * , ∆), as constructed in the literature using effective Lagrangians and constrained by charm-resonance spectroscopy. In this sense we provide a lower estimate of the diffusion coefficient, based on existing elastic D-hadron amplitudes.
Our paper is organized as follows. In Sec. II we "reconstruct" microscopic models for Dπ scattering in a hot pion gas (via s-channel resonances; Sec. II A), for D scattering off strange and vector mesons (Sec. II B), and off baryons (Sec. II C), by parameterizing pertinent scatter-ing amplitudes. In Sec. III these are applied to calculate thermal D-meson relaxation rates and diffusion coefficients in hot hadronic matter at vanishing chemical potential, first in a pion gas (Sec. III A) and then in a resonance gas (Sec. III B), and finally including chemical potentials as appropriate for heavy-ion collisions at RHIC (Sec. III C). Conclusions are given in Sec. IV.

II. D-MESON SCATTERING AND WIDTH IN HOT HADRONIC MATTER
In this section we recapitulate basic elements of the D-hadron scattering amplitudes and apply them to calculate pertinent D-meson widths in hadronic matter. We only employ known amplitudes from the literature for the most abundant hadrons in a hot medium and combine them into a lower-limit estimate for the D-meson width (and D-meson relaxation rate in Sec. III). We first focus on pion-gas effects, followed by interactions with strange and vector mesons, as well as hot nuclear matter.

A. Hot Pion Gas
Following Ref. [19] D interactions in the pion gas are dominated by its chiral partner, D * 0 (2308), by the vector D * (2010) and by the tensor meson D * 2 (2460). Phenomenological control over their interaction vertices has become possible due to new observations of D-meson resonances by the BELLE Collaboration [20]. Especially, the large width of the D * 0 (2308), Γ tot 0 = 276 ± 99 MeV, attributed to S-wave pion decay, leads to a large DπD * 0 coupling constant. With the constituent-quark model assignment of isospin I=1/2 for D-states, the pertinent forward scattering amplitudes have been parameterized in Breit-Wigner form as where √ s and k denote the center-of-mass energy and 3momentum in the Dπ collision, respectively, and j is the resonance spin. The total resonance decay width Γ tot , is assumed to be saturated by the partial one, Γ Dπ 0,1 , for D * and D * 0 , while for the narrow state D * 2 (Γ tot 2 = 45.6 ± 12.5 MeV) a branching fraction estimated by the particle data group is employed [21]. The resulting total Dπ cross section is displayed in the upper panel of Fig. 1.
The D-meson self-energy in a pion gas can now be obtained by the standard procedure of closing the inand outgoing pion lines of the forward Dπ amplitude with a thermal pion propagator. In the narrow-width approximation for the pion propagator, the D-meson selfenergy takes the form where f B is the thermal Bose function and E π the onshell pion energy. Alternatively, one can obtain the collision rate (or on-shell width), Γ=−ImΠ(p 2 D =m 2 D , T )/m D from the Boltzmann equation as where γ π =3 denotes the spin-isospin degeneracy factor of the in-medium particle (pion), and p, q and p ′ , q ′ are the momenta of in-and outgoing particles, respectively. Equations (2) and (3) are related via the optical theorem (we have checked their consistency). In the following we employ the latter since it is close to the form of the thermal relaxation rate discussed in Sec. III below.
The scattering width of a D-meson at rest in a pion gas is displayed in the lower panel of Fig. 1. For T =150-180 MeV we find Γ D =20-40 MeV, where the chiral part-ner of the D provides the largest contribution through Swave Dπ scattering. For constant resonance widths, we find close agreement with the results plotted in Fig. 2 of Ref. [19], while for energy-momentum dependent widths (as quoted in Ref. [19]) our results displayed in Fig. 1 turn out to be slightly smaller (by ca. 20%).

B. Strange and Vector Meons
In a hot meson gas, the next abundant species after the pions are the strangeness carrying Goldstone bosons and the light and strange vector mesons.
For Dη scattering we also adopt Ref. [22], where the S-wave (I, S)=(1/2,0) amplitude is governed by a narrow state of mass 2413 MeV just below threshold.
The evaluation of DV scattering (V =ρ, ω, K * ) requires to go beyond the chiral Lagrangian. This has been done in Ref. [27] starting from SU (4) flavor symmetry and then implementing chiral breaking terms. This framework, properly unitarized, recovers the resonance poles computed with the chiral Lagrangians, but extends to axialvector states coupling to S-wave DV interactions (in particular D 1 (2420), D ′ 1 (2427), D s1 (2460), D s1 (2536)). A convenient Breit-Wigner parametrization of the elastic coupling of DV to the dynamically generated resonances has been quoted as where g DV is the dimensionful coupling constant and s R the complex resonance-pole position. We include the three I=1/2 resonance couplings to Dρ and Dω from Tab. 7 in Ref. [27], two I=0 and two I=1 resonances with DK * coupling (Tabs. 5 and 4 in [27], respectively) and one I=0 state coupling to DK * (Tab. 8 in [27]). As a representative, the isospin I=1/2 Dρ cross section is shown in the upper panel of Fig. 1.
The lower panel of Fig. 1 shows the temperature dependence of mesonic contributions to the D-meson scattering width as calculated from the above amplitudes. The width from anti-/kaons is the next largest contribution after the pion. The effect of vector mesons is smaller but significant, especially for the K * . The total D width in a hot meson gas reaches ∼80 MeV around T ≃180 MeV, which should be a lower limit since several channels have still been neglected, e.g., higher partial waves (except for pions) and inelastic channels (e.g. DK * ↔ D s π).

C. Hot Nuclear Matter
To evaluate D scattering off baryons we follow the same strategy as for mesons, employing vacuum scattering amplitudes. More elaborate many-body calculations for D-mesons in nuclear matter are available [18,28], but our procedure keeps consistency with the mesonic sector and allows for an estimate of the systematic error due to in-medium effects (e.g., selfconsistency of selfenergy and in-medium scattering amplitudes).
We start from Ref. [18] where the T -matrix results of an effective DN interaction with coupled channels have been parameterized in analogy to Eq. (4) as where m R is the resonance mass and Γ R the width. The T -matrix, Eq. (5), is related to the invariant scattering amplitude through M(s) = N (s)T (s) with N (s)=(s + m 2 The key states are the dynamically generated I=0 Λ c (2595) and I=1 Σ c (2620) S-wave bound states. The former is experimentally well established, while the latter is ca. 180 MeV too deep compared to the empirical Σ c (2800) state. However, the I=1 DN cross section above threshold (shown in the upper panel of Fig. 2) is quite comparable to the results of a recent meson-exchange model calculation [29] in which the Σ c is generated at its experimental mass (in fact, at threshold the I=1 scattering length in Ref. [29] is significantly larger than for the Σ c (2620) state of Ref. [18]). In the I=0 channel, the scattering lengths of Refs. [18] and [29] agree well. The corresponding cross section is also shown in the upper panel of Fig. 2.
The DN scattering amplitude can be inferred from DN due to C-symmetry of strong interactions. We have found no evidence in the literature for (multi-quark) resonances in this system and adopt the D − N elastic S-wave amplitude calculated in Ref. [18]. The pertinent scattering and relaxation rates are about a factor of ∼3 smaller than from DN scattering.
The only available calculation of D∆ we are aware of has been conducted in Ref. [30], within the same framework as our DN amplitudes are based on. In the only available I=1 S-wave channel the parametrization, Eq. (5), reflects a rather deep bound state (m R =2613 MeV, Γ R ≃0, g=8.6). The cross section is shown in the upper panel of Fig. 2. Unlike the DN case, the D∆ system is predicted to support a shallow I=1 bound state (m R =2867 MeV, Γ R ≃0, g=5.8). As a result, the contribution of D∆ scattering to the D-meson width and relaxation rate is about half of that from D∆ scattering.
The lower panel of Fig. 2 shows that the width of a D-meson at rest from scattering off thermally excited nucleons and ∆'s at vanishing chemical potential (µ B =0) is comparable to that of light vector mesons (cf. lower panel of Fig. 2). When addingN and∆ contributions, the baryon-induced D-meson width computed here amounts to ca. 15 MeV at T ≃180 MeV. Note again that we have neglected higher partial waves as well as higher excited resonances including strange anti-/baryons.
It is instructive to compare our nucleon-induced width to a selfconsistent many-body calculation [17]. From Fig. 6 in Ref. [17] we read off Γ D ≃100(80) MeV at T =100(150) MeV and ̺ N =̺ 0 , vs. Γ D ≃75(65) MeV in our approach. This indicates that neglecting in-medium effects does not lead to an overestimate in our calculation.

III. THERMAL RELAXATION IN HADRONIC MATTER
The standard expression for the thermal relaxation rate of a particle (D) in a heat bath in terms of its scattering amplitude on medium particles (h) reads [3] with p ( q) and p ′ ( q ′ ) being the D-meson (h) momentum before and after the interaction, respectively. The form of this expression is very similar to the one for the total width, Eq. (3). The latter can be expressed in terms of the average defined above as Γ = 1 . In Sec. III A we evaluate A(p, T ) for D-mesons in a pion gas (h=π) and in Sec. III B for all other hadrons whose amplitudes have been constructed in the previous section. In Sec. III C we evaluate the relaxation rate at finite baryon and meson chemical potentials as applicable to the hadronic phase below the chemical freeze-out temperature in heavy-ion collisions at RHIC.

A. Pion Gas
The thermal relaxation rates for a D-meson at rest due to scattering off pions in a thermal gas in chemical equilibrium (µ π =0), using the the scattering amplitude of Eq. (1), is displayed in Fig. 3 as a function of temperature. From T =100 MeV to 180 MeV the rate increases by about a factor of 7, basically following the increase in pion density from 0.2 to 1.4̺ 0 . Its magnitude at T=180 MeV, A ≃ 1/25 fm, is small but not negligible. When replacing the Dπ amplitude in Eq. (3) by one yielding a constant S-wave cross section of σ S Dπ =7-10 mb, the pertinent band for the relaxation rate essentially covers the result of our microscopic calculations. The latter are closer to the upper end of the band at lower T but to the lower end at higher T 150 MeV. This reflects the increased ther- mal motion of pions probing higher √ s in the amplitudes where the latter decrease.

B. Hadron Resonance Gas
A hot hadron gas in equilibrium is characterized by an increasing abundance of resonances with rising temperature. For example, at T =180 MeV, the density of baryons plus antibaryons is above ̺ 0 and that of mesons with masses below 2 GeV is above 3̺ 0 . To improve the estimate of D-meson diffusion in a pion gas for a more realistic hadron-resonance gas, we include rescattering on all particles which at T =180 MeV and µ h =0 have a density at least 0.1̺ 0 , i.e., π, K, η, ρ, ω and K * (892) in the meson sector and anti-/nucleons, and ∆(1232),∆(1232) in the anti-/baryon sector, using all scattering amplitudes of Sec. II. The resulting D-meson friction coefficient in hadronic matter at vanishing chemical potentials increases substantially over the pion-gas result, by about a factor of ∼2(3) at T =150(180) MeV, see Fig. 4. The individual contributions of K +K and N ,N are compared to constant-cross-section calculations in Fig. 3, indicating that a "constituent" light-quark cross section of 3-4 mb is compatible with our lower-limit estimates. A quantitative decomposition of the individual hadron contributions to kinetic D-meson relaxation at T =180 MeV is given in Tab. I. Anti-/kaons provide the next-to-leading contributions after the pions, while vector mesons, nucleons and Deltas play a smaller but non-negligible role.
As an estimate of medium effects in our vacuum Dh amplitudes we have performed a calculation for A where we have introduced an in-medium broadening of Γ med R =200 MeV into the Breit-Wigner parameterizations. The final result for A changes by less than 5%.
In Fig. 5 we display the spatial D-meson diffusion coefficient, D s = T /(m D A(p = 0, T )), in hadronic matter. When normalized to the thermal wavelength, 1/(2πT ), this quantity decreases with T , reaching a value of ∼5 at T =180 MeV. Again, this is surprisingly close to Tmatrix results for charm quarks in the QGP [14], and, together with those results, suggests a minimum across the hadron-to-quark transition.

C. RHIC Conditions
In relativistic HICs the chemical freeze-out of hadron ratios [31] at a temperature of T chem ≃170 MeV is significantly earlier than thermal freeze-out of the light hadrons at T fo ≃100 MeV. Therefore, to conserve the observed particle ratios in the hadronic evolution, effective chemical potentials are required, reaching appreciable values even at RHIC energies [32], e.g., µ π (T =100 MeV)≃80 MeV. We implement the chemical potentials into the thermal hadron distribution functions and recalculate the Dmeson equilibration rate, Eq. (6). As a result, the latter is enhanced at temperatures below T chem , staying above 1/(25 fm) for T ≥130 MeV (cf. Fig. 4), implying noticeable modifications of D-meson spectra in the hadronic phase of nuclear collisions at RHIC. For example, if the hadronic evolution lasts for ∆τ had ≃5 fm, the expected modification amounts to ca. (1 − exp[A ∆τ had ]) ≃ 20%.

IV. SUMMARY AND CONCLUSION
We have evaluated kinetic transport of D-mesons in hot hadronic matter by elastic scattering off the 10 most abundant hadron species. The interaction strength with mesons and baryons has been estimated from existing microscopic models for D-hadron scattering, constrained by chiral symmetry and vacuum spectroscopy. In a pion gas at T =100 MeV, Dπ resonance scattering leads to a relaxation rate which is substantially smaller than what has been found in a recent calculation using heavy-meson chiral perturbation theory. Yet, when extrapolating our full results to temperatures in the vicinity of T c , the relaxation rate reaches ca. 0.1/fm, translating into a spatial diffusion coefficient of D s ≃ 5/(2πT ). This is comparable to non-perturbative T -matrix calculations of charmquark relaxation in the QGP. On the one hand, this suggests a rather smooth evolution of charm transport through T c , i.e., a kind of "duality" of hadronic and quark-based calculations, reminiscent of what has been found for dilepton emission rates. On the other hand, it implies that quantitative calculations of D-meson spectra in heavy-ion collisions have to account for hadronic diffusion. This insight is reinforced once chemical freezeout is implemented into the evolution of the hadronic phase (via effective chemical potentials), with an estimated modification of D-meson observables of at least 20%. The apparent agreement of hadron-and quarkbased approaches, when extrapolated to around T c , is encouraging, especially since the magnitude of the transport coefficient is compatible with the phenomenology of current heavy-flavor observables at RHIC. Our findings thus pave the way for an improved theoretical accuracy which will be needed to take advantage of upcoming precision measurements at RHIC and LHC.
Note added. Two subsequently submitted papers have also addressed hadronic D-meson diffusion. In Ref. [33] the use of unitarized chiral effective Dπ interactions leads to relaxation rates in a pion gas in close agreement with our results. In Ref. [34] D-hadron interactions have been evaluated using Born amplitudes, leading to relaxation rates significantly larger than our results.