ORIGINAL RESEARCH article
Sec. Nuclear Physics
Volume 11 - 2023 | https://doi.org/10.3389/fphy.2023.1250939
Recent progress on in-medium properties of heavy mesons from finite-temperature EFTs
- 1Departament de Física Quàntica i Astrofísica, Facultat de Física, Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, Barcelona, Spain
- 2Theory Center, Thomas Jefferson National Accelerator Facility, Newport News, VA, United States
- 3Institute of Space Sciences (ICE-CSIC), Barcelona, Spain
- 4Institut d’Estudis Espacials de Catalunya (IEEC), Barcelona, Spain
- 5Frankfurt Institute for Advanced Studies, Frankfurt am Main, Germany
Mesons with heavy flavor content are an exceptional probe of the hot QCD medium produced in heavy-ion collisions. In the past few years, significant progress has been made toward describing the modification of the properties of heavy mesons in the hadronic phase at finite temperature. Ground-state and excited-state thermal spectral properties can be computed within a self-consistent many-body approach that employs appropriate hadron-hadron effective interactions, providing a unique opportunity to confront hadronic Effective Field Theory predictions with recent and forthcoming lattice QCD simulations and experimental data. In this article, we revisit the application of the imaginary-time formalism to extend the calculation of unitarized scattering amplitudes from the vacuum to finite temperature. These methods allow us to obtain the ground-state thermal spectral functions. The thermal properties of the excited states that are dynamically generated within the molecular picture are also directly accessible. We present here the results of this approach for the open-charm and open-bottom sectors. We also analyze how the heavy-flavor transport properties, which are strongly correlated to experimental observables in heavy-ion collisions, are modified in hot matter. In particular, transport coefficients can be computed using an off-shell kinetic theory that is fully consistent with the effective theory describing the scattering processes. The results of this procedure for both charm and bottom transport coefficients are briefly discussed.
The discovery in 2003 of the charm-strange mesons
Closely related is the case of the broad structures observed in the Dπ and D∗π invariant mass distributions [16–19] and reported as the
While heavy-quark spin symmetry (HQSS) is responsible for the near degenerate patterns between the open-charm scalars,
A new venue to study the nature of heavy-flavor exotica has recently emerged with relativistic heavy-ion collisions (HICs), where an extremely hot quark-gluon plasma (QGP) is created. At high collision energies, such as those at the Relativistic Heavy-Ion Collider (RHIC) and the Large Hadron Collider (LHC), abundant heavy quark-antiquark pairs are produced in the initial hard scattering between partons. These pairs then propagate through the rapidly expanding and cooling QGP. At a temperature of about Tc = 156 MeV the hadronic medium is eventually formed, and the interactions between the heavy hadrons and the surrounding light mesons occur until the so-called kinetic freeze-out at lower temperatures. This offers an excellent opportunity to test the in-medium properties of the heavy mesons produced, including those of heavy exotica. Furthermore, the novel employment of femtoscopy techniques in pp, pA and AA collisions at the LHC and RHIC to determine the scattering parameters of D mesons with light-flavor hadrons will certainly help probe the hadronic interactions, as well as the effects of the hadronic medium [29–36].
The non-perturbative regime of hot hadronic matter can be consistently treated using effective Lagrangians combined with quantum field theory techniques at finite temperature, often denominated as thermal EFTs. While finite-temperature lattice QCD has been for many years a powerful theoretical source of information on hot QCD matter, thermal EFTs are a complementary tool that enable us to approach the QCD phase transition from the chirally-broken phase of hadrons.
In this work we use thermal EFTs to access the finite-temperature properties of charm and bottom mesons in a hot medium. To this end, we will revisit the calculations that we presented in a series of works on charmed mesons [37–40] and, in addition, we will present extensions of the calculations to the heavy mesons with bottom flavor . We note that, although it will not be discussed here, our findings in the charm sector have been checked against lattice QCD calculations at the level of Euclidean correlators  and that we have also studied the thermal modification of the X(3872) exotic state and its spin-flavor partners when these are assumed to be of molecular nature .
The rest of the article is organized as follows. In Section 2 we discuss the main ingredients of our thermal EFT approach to address the in-medium properties of open heavy-flavor mesons: The effective hadron-hadron interactions, the use of the imaginary-time formalism to evaluate properties of a system in thermal equilibrium (e.g., thermal corrections to the mass and decay width), and the evolution in real-time to tackle the description of a system out of equilibrium and compute transport coefficients. We will review the key ideas and refer to our previous works for technical details [37–39]. Section 3 presents novel results for bottomed mesons. These include self-energies, spectral functions and transport coefficients, alongside a comparison with selected results from our previous works in the charm sector. We end with a final discussion and conclusions in Section 4.
In this work we use a thermal effective field theory approach that was developed in a series of works [37, 38] for charmed hadrons. It is based on unitarized heavy-meson chiral perturbation theory (HMChPT) combined with thermal field theory techniques using the imaginary-time formalism to address the thermal effects on the properties of heavy mesons in a mesonic medium at finite temperature. The kinetic theory describing the heavy-meson dynamics in the hot medium can be derived using the real-time formalism . The resulting kinetic equation depends on thermal scattering amplitudes and spectral functions. For the calculation of transport coefficients it is sufficient to assume a system near equilibrium and employ equilibrium quantities. In the following we summarize the main steps to compute some relevant quantities at finite temperature.
2.1 Interactions between open heavy-flavor mesons and light mesons
We start by outlining the main features of the interactions between open-heavy flavor mesons, i.e., mesons with one charm or bottom quark
with the subscripts LO and NLO referring to the chiral power counting, and
The tree-level potential for the process H(∗)Φ → H(∗)Φ reads
The partial-wave projection with angular momentum ℓ is then obtained through the relation
where θ is the scattering angle between the initial and final particles in the center of mass, and Pℓ(cos θ) the Legendre polynomial of order ℓ.
2.2 Thermal equilibrium properties
A hadron gas forms once the temperature is turned on. The relative abundance of each hadron species in thermal equilibrium is determined by the corresponding thermal distribution functions, i.e., the Bose-Einstein distribution for mesons and the Fermi-Dirac distribution for baryons. At temperatures T ∼ 100 − 150 MeV light mesons become the primary components of the medium and heavy mesons behave as Brownian particles scattering of the light mesons. We exploit this scenario and employ the HMChPT potential presented above to describe the dynamics of the heavy mesons with the light mesons in the bath.
In order to incorporate the effects of the hot medium, it is necessary to follow the techniques of thermal field theory. There exist two complementary formulations of thermal field theory that can be used to describe a system in thermal equilibrium, the “imaginary-time” and the “real-time” formalisms. In the imaginary-time formalism (ITF), also called Matsubara formalism owing to the pioneering work by Matsubara , time is treated as a purely imaginary quantity and then one performs an analytical continuation from Euclidean to Minkowski spacetime at the end of the calculation. In the real-time formalism, in contrast, the calculation is done in Minkowski spacetime, considering explicitly the evolution in real time [55, 56]. While the latter is capable of describing systems even outside thermal equilibrium, and thus appropriate to address out-of-equilibrium properties and transport coefficients, as we will discuss in the next section, the ITF has the advantage of resembling in a more intuitive way the zero temperature field theory. For instance, the main difference in the form of the propagators and the diagrammatic structure of the perturbative expansion, in the ITF, is the acquisition of thermal weights in the phase-space integrals compared with those at T = 0, as we will show below. An approach based on the ITF has been developed in the recent years to study the properties of open heavy-flavor mesons in hot hadronic matter at vanishing baryonic density [37, 38, 57].
To compute the thermal corrections to a given quantity, such as the two-meson propagator or the self-energy, the ITF provides some simple rules that basically consist in replacing the zeroth component of the four-momenta of the particles by discrete Matsubara frequencies iωn, with ωn = 2nπ/β for bosons and β = 1/T, and transforming the integration over internal energies into a summation over Matsubara frequencies. Then, by using some established computational techniques based on contour integrals and analytic continuation, the calculations can be done similarly as in the vacuum field theory. For details, we encourage the reader to consult the classical Refs. [58–62].
Using the rules above, the thermal two-meson propagator takes the form
with pμ = (E, p) the momentum in the center of mass of the two-meson system. In the above equation, in addition to the medium corrections arising from the ITF, i.e., the additional weighting factors containing appropriate combinations of Bose-Einstein distribution functions
In the case of zero temperature, it is customary to regularize the vacuum contribution to the two-meson propagator, for example, by introducing a hard cutoff in the three-momentum integration. We use Λ = 800 MeV, a value that corresponds to the scale of the degrees of freedom that were integrated out when constructing the meson-meson interaction amplitude from the effective Lagrangian, and that is consistent with the regularization scheme used in Ref. , from where we adopted the values for the LECs of the NLO potential. This regularization scheme is straightforward to extend to finite temperature. The tree-level potential in Eq. 3 does not change at finite temperature because in the ITF thermal corrections enter in loop diagrams [60, 61].
The thermal effects on the unitarized scattering amplitude
FIGURE 1. (A) Diagrammatic representation of the Bethe-Salpeter equation in coupled channels in Eq. 6. At finite temperature, the
The spectral functions dressing the meson propagators in Eq. 5 take into account the modifications due to the presence of interactions with the medium. At finite temperature the heavy meson retarded propagator is defined by
where mH is the mass of the heavy meson in the vacuum, renormalized by the vacuum contribution of the retarded self-energy ΠH (see Figure 1B). For the purpose of our calculations, using the vacuum propagator for the light meson and thus a δ-type spectral function is a good approximation, as we discussed in our previous works [37, 38].
The light-meson contribution to the self-energy of the heavy meson can be obtained by closing the light-meson line in the corresponding
Using the spectral Lehmann representation for the light meson propagator and the
We note that the self-energy entering in Eq. 7 can only contain thermal corrections after mass renormalization. However, the self-energy computed with Eq. 9 contains both vacuum and thermal corrections. We regularize it by dropping the vacuum contribution, which is identified with the expression obtained when taking the limit T → 0 of Eq. 9. See  for details.
Finally, the spectral function necessary to dress the heavy meson in the two-meson propagator is computed from the imaginary part of the retarded meson propagator,
Eqs 5–10 are interrelated to each other. As a result, solving this set of coupled equations requires an iterative approach until self-consistency is achieved. This process is outlined in Figures 1A–C, where the
2.3 Non-equilibrium properties
Since heavy mesons have large masses compared to the surrounding light mesons their time evolution is typically described by a Fokker-Planck (or Langevin) or a Boltzmann approach. The essential components of these approaches are transport coefficients, which can be calculated from the scattering amplitudes of heavy-flavor mesons with light mesons in the hadronic gas. These transport coefficients are typically derived assuming that the light scattering partners are in thermal equilibrium, and they are often calculated as functions of temperature and momentum.
In Ref.  we calculated the transport coefficients of D mesons in the hadronic phase incorporating medium corrections to the scattering amplitudes. To do so, we extended the kinetic theory of D mesons using the more general Kadanoff-Baym equations, so as to account for thermal and off-shell effects. This off-shell kinetic theory also applies to describe the propagation of
Let us summarize our main results. For a detailed derivation, we recommend to consult our previous work  and references therein. Starting from the Kadanoff-Baym equations and performing a Wigner transform along with a gradient expansion , we arrive to the following form of the off-shell transport equation for the time ordered Green’s function of the heavy meson
The lesser and greater Green’s functions and the self-energies in Eq. 11 are functions of the center-of-mass coordinate X = (t, X) and the four-momentum k = (k0, k) of the external heavy meson. Note that k0 and k are independent variables, although related through the non-equilibrium spectral function
The off-shell collision rate of a heavy meson with energy k0 and momentum k to a final heavy meson with energy
The labels of the momenta correspond to the choice of a generic scattering process H(k) + Φ(k3) → H(k1) + Φ(k2). Each of the two terms of the right-hand side of Eq. 12 can be identified as the collision gain and loss terms respectively. It is important to note that in Eq. 13 there is an implicit sum over the different species Φ and H that can interact with the external off-shell heavy meson.
Next, one may exploit the separation of scales between the meson masses to arrive to an off-shell Fokker-Planck equation for
with Δij = δij − kikj/k2, and the transport coefficients
and the transverse and longitudinal momentum diffusion coefficients read.
The angle brackets
where f (Ei; T) is the equilibrium occupation number, i.e., the Bose-Einstein distribution function, and
The formalism described in the previous section provides a framework to compute the in-medium properties of heavy mesons. Here we present our results for D and
For temperatures T ≲ 150 MeV, pions give the largest contribution to the medium corrections. This is because they are the lightest mesons and therefore the most abundant species in the thermal bath. Unless otherwise stated, in the calculations presented in this work we only consider the thermal effects due to pions and neglect the contribution of the heavier kaons and eta mesons. We note that the results in the charm sector were already published in our previous works [37–39] and are reproduced here for the sake of comparison between the bottom and charm sectors.
We start with the discussion of the self-consistent results of the self-energy of the ground-state heavy mesons in a pionic medium at finite temperature, displayed as a function of the energy in Figure 2, for zero three-momentum and scaled by the mass of the heavy meson in vacuum. We show the results for three different values of the temperature of the medium, T = 80, 120 and 150 MeV, in different line styles.
FIGURE 2. Real and imaginary parts of the pion contribution to the self-energy of the ground-state heavy mesons at several temperatures (see legend). Panels in the two top rows correspond to charmed mesons, and panels in the two bottom rows to bottomed mesons. Different columns correspond to states with angular momentum and strangeness (J, S) = (0, 0), (0, 1), (1, 0), (1, 1), in this order.
The real part of the self-energy is related to the thermal correction to the mass. This is evident from the expression of the heavy-meson retarded propagator at finite temperature in Eq. 7. In the quasiparticle approximation, which we will see is well-grounded for both D and
The imaginary part of the self-energy relates to the thermal width acquired by the heavy meson due to interactions with pions within the medium. The panels in the second and fourth rows of Figure 2 show the imaginary part of the self-energy of the charmed and bottomed mesons, respectively, over their respective mass in vacuum. The insets provide a zoom in the region E ≈ [mH − 2mπ, mH + 2mπ]. Similar features are observed in all the panels. The imaginary part of the self-energy is essentially zero for energies below mH − 2mπ, above which it starts decreasing mildly. This initial drop at around mH is exclusively caused by the presence of the thermal medium, which allows for the absorption of two thermal pions. These absorption processes make it possible for the scattering amplitude to be non-zero even below the two-meson threshold due to the so-called Landau cut [58, 67] of the two-meson propagator (see also our discussion in Refs. [38–40]). Our calculations show that also the self-energy of the heavy meson can reveal the effects of the Landau cut, thanks to the self-consistency of our approach. This effect becomes more relevant at higher temperatures, i.e., T ≳ 100 MeV, where the pion density is larger. A substantially larger drop takes place at mH + 2mπ, which is the energy where the heavy meson at rest can emit two pions. This later growth of the magnitude of the imaginary part of the self-energy takes place at similar rates for all temperatures, since the emission of two pions is also possible in vacuum for a large enough energy of an off-shell heavy meson, and it is related to the unitary cut of the propagator. As a result of the combination of the Landau and unitary cut effects, and by virtue of the relation between Im ΠH and the thermal decay width, we expect the ground-state spectral functions to broaden as the temperature of the thermal medium increases. Similarly as it happened for the real parts, the magnitude of Im ΠH/mH at a given temperature is similar in size when comparing results for pseudoscalar and vector mesons, and it is somewhat larger in the bottom sector than in the charm sector. As a reminder, the real and imaginary parts of the self-energy are connected by analyticity constraints, which means that the oscillations seen in the real part are fully determined by the structure of the imaginary part that we have described in detail here.
3.2 Spectral functions
The spectral function for the ground-state heavy mesons follows the standard definition in terms of the retarded propagator—see Eqs 7, 10—, in which the self-energy is responsible for the thermal corrections with respect to the vacuum propagator. In Figure 3 we show the energy dependence of the spectral function of the charmed mesons (top panels) and the bottomed mesons (bottom panels) at rest, at the same temperatures as for the self-energy described above. The vertical solid lines depict the corresponding value of the mass in vacuum. From these plots, the drop of the mass and the increase of the width anticipated from analyzing the self-energies become manifest. This is evident as the maximum of the spectral function shifts towards lower energies and it becomes wider with increasing temperature.
FIGURE 3. Spectral function of the ground-state charmed mesons (top panels) and bottomed mesons (bottom panels) at several temperatures (see legend). The column description is the same as in Figure 2. Vertical lines depict the values in vacuum (T = 0).
In the quasiparticle approximation, the spectral function admits a Lorentzian shape peaked at the quasiparticle energy Ek(T) (with M(T) ≡ Ek at rest) and a spectral width γk(T) ≪ Ek(T). For the spectral functions in Figure 3, which are narrow, the quasiparticle approximation is indeed justified. Figure 4 shows the values of the mass (left panels) and the decay width (right panels) as a function of the temperature determined by analyzing the position and the width of the peak of the spectral functions. For the charmed mesons, we find a reduction in mass
FIGURE 4. Temperature evolution of the mass (left panels) and the width (right panels) of ground-state charmed mesons (top) and bottomed mesons (bottom). Data points in the top left panel correspond to lattice QCD calculations from Ref. .
We remind that in the self-consistent calculations of the self-energies and the spectral functions, only the impact of the thermal pions is taken into account, arguing the contribution of other light mesons to be presumably suppressed. The contribution of each of the light mesons in the medium to the thermal width of the heavy meson can be easy analyzed from its definition in terms of the retarded self-energy in the quasiparticle approximation,
with zk ≈ 1. Since Im Π(Ek, k; T) is given by the integration over the imaginary part of the unitarized scattering amplitude,
FIGURE 5. Contribution to the averaged thermal width of the D meson (left panel) and the
The process of unitarization of the scattering amplitude of Eq. 6 leads to the emergence of two poles in the sectors with strangeness S = 0 and isospin I = 1/2 that correspond to the two-pole structure of the
FIGURE 6. Imaginary part of the diagonal elements of the scattering amplitudes in the charm sector (top panels) and in the bottom sector (bottom panels) at several temperatures. Different columns correspond to states with angular momentum, strangeness, and isospin (J, S, I) = (0, 0, 1/2), (0, 1, 0), (1, 0, 1/2), (1, 1, 0), in this order. Vertical lines in the sectors with strangeness depict the energy location of the bound states in vacuum.
In the cases with zero strangeness, the proximity of the position of the resonances to channel thresholds gives rise to complicated structures. However, one can still clearly recognize a gradual evolution of the peaks and widths as T increases. The strange sectors are more straightforward. The T = 0 delta-type spectral function (i.e., bound state) acquires a non-zero width, and the shift and widening of the peak is comparable to that of the ground states. Nevertheless, an increase in strength is visible on the right-hand side of the distributions. This asymmetry can be explained by the fact that the channel threshold is not sharp anymore due to the widening of the D(∗) or
3.3 Transport coefficients
Now, we discuss the results of the transport coefficients for a heavy meson propagating through the hadronic medium. We will specifically focus on the comparison between D and
FIGURE 7. Transport coefficients of the
The 2 ↔ 2 scatterings are described by the λ = λ′ terms in Eq. 17, while the 1 ↔ 3 processes take λ = −λ′ This fact can easily be grasped by looking at the signs of the energy conservation delta. In Ref.  we reported that the number-violating processes contribute very little to the transport coefficients, while the 2 ↔ 2 collisions make the leading contributions. Among the latter, the case λ = λ′ = + corresponds to the standard term in which the binary collision is taking place at an energy corresponding to the sum of the incoming energies k0 + E3 (cf. Eq. 17). However the case λ = λ′ = −, corresponds to a binary collision in which the scattering matrix is evaluated at the energy difference k0 − E3. For typical energies around the quasiparticle masses, this difference probes the kinematic region below the two-particle threshold. For interactions computed in vacuum, this region has a vanishing
To compare the results between D and
Finally we plot in Figure 8 the so-called spatial diffusion coefficient,
where we express it with calligraphic font not to confuse it with Ds mesons.
FIGURE 8. Off-shell spatial diffusion coefficient of the
In the left panel of this figure we show our result for this coefficient at low temperatures for D mesons (solid red line) and for
In this paper we have obtained the properties of mesons with open heavy-flavor at finite temperature using an effective field theory based on chiral and HQSF symmetries within the imaginary-time formalism. The interaction of these pseudoscalar and vector open heavy-flavor ground-state mesons with light mesons (π, K,
With this methodology, we have obtained the self-energies and, hence, the corresponding spectral functions of open-charm (D(∗),
From the behavior of the spectral functions, we have quantified the thermal dependence of the masses and the decay widths of the open heavy-flavor ground states. We have observed a generic downshift of the thermal masses with temperature, as large as of a few tens of MeV at T = 150 MeV in a pionic bath, while the decay widths increase with temperature up to values of some tens of MeV at T = 150 MeV. Compared to recent lattice QCD simulations for open-charm ground states , a similar trend can be determined although a systematic shift is seen as a heavy non-physical pion mass is used in the lattice.
As a byproduct of the unitarization, we have also obtained the two-pole
And, finally, we have computed the transport coefficients for D and
The diffusion coefficient of
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
All authors contributed equally to the paper. GM wrote the first draft of the manuscript. JT-R, LT, and ÀR wrote sections of the manuscript. All authors contributed to the article and approved the submitted version.
This research has been supported from the projects CEX2019-000918-M, CEX2020-001058-M (Unidades de Excelencia “María de Maeztu”), PID2019-110165GB-I00 and PID2020-118758GB-I00, financed by the Spanish MCIN/AEI/10.13039/501100011033/, as well as by the EU STRONG-2020 project, under the program H2020-INFRAIA-2018-1 grant agreement no. 824093. GM acknowledges support from the FPU17/04910 Doctoral Grant from the Spanish Ministerio de Universidades, and U.S. DOE Contract No. DE-AC05-06OR23177, under which Jefferson Science Associates, LLC, operates Jefferson Lab. LT and JT-R acknowledge support from the DFG through projects no. 411563442 (Hot Heavy Mesons) and no. 315477589—TRR 211 (Strong-interaction matter under extreme conditions). LT also acknowledges support from the Generalitat Valenciana under contract PROMETEO/2020/023.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Keywords: effective hadron theories, chiral symmetry, heavy-quark spin-flavor symmetry, D mesons, B mesons, finite temperature, transport coefficients
Citation: Montaña G, Ramos À, Tolos L and Torres-Rincon JM (2023) Recent progress on in-medium properties of heavy mesons from finite-temperature EFTs. Front. Phys. 11:1250939. doi: 10.3389/fphy.2023.1250939
Received: 30 June 2023; Accepted: 07 August 2023;
Published: 05 September 2023.
Edited by:Kanchan Khemchandani, Federal University of São Paulo, Brazil
Reviewed by:Francesco Giovanni Celiberto, Bruno Kessler Foundation (FBK), Italy
Alejandro Ayala, National Autonomous University of Mexico, Mexico
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*Correspondence: Glòria Montaña, firstname.lastname@example.org