aa r X i v : . [ h e p - ph ] D ec The charming beauty of the strong interaction
Laura Tolos
Abstract
Charmed and beauty hadrons in matter are discussed within a unitarizedcoupled-channel model consistent with heavy-quark spin symmetry. We analysethe formation of D -mesic states as well as the propagation of charmed and beautyhadrons in heavy-ion collisions from LHC to FAIR energies. One of the main research activities in nuclear and particle physics is the explorationof the Quantum Chromodynamics (QCD) phase diagram for high density and/ortemperature. Up to now the studies have been concentrated for light quarks dueto energy constraints of the experimental setups, but with the upcoming researchfacilities, the goal is to move to the heavy-quark domain, where heavy degrees offreedom, such as charm and beauty, play a crucial role.In order to understand the QCD phase diagram, one needs first to understandthe interaction between heavy hadrons. In particular, the nature of newly discoveredheavy excited states is of major concern, whether they can be described within thestandard quark model and/or better understood as dynamically generated states viahadron-hadron interactions.Given the success of unitarized coupled-channel approaches in the description ofsome of the existing experimental data in the light-quark sector, charmed and beauty
Laura TolosInstitut f¨ur Theoretische Physik, University of Frankfurt, Max-von-Laue-Str. 1, 60438 Frankfurtam Main, Germany,Frankfurt Institute for Advanced Studies, University of Frankfurt, Ruth-Moufang-Str. 1, 60438Frankfurt am Main, Germany,Institute of Space Sciences (CSIC-IEEC), Campus Universitat Aut`onoma de Barcelona, Carrer deCan Magrans, s/n, 08193 Cerdanyola del Vall`es, Spain,e-mail: [email protected] degrees of freedom have been recently incorporated in these models and several ex-perimental states have been described as excited baryon molecules. Some examplescan be found in Refs. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,19, 20, 21, 22, 23, 24, 25, 26]. However, some of these models are not fully consis-tent with heavy-quark spin symmetry (HQSS) [27], which is a QCD symmetry thatappears when the quark masses become larger than the typical confinement scale.Thus, a model that incorporates HQSS constraints has been developed in the pastyears [28, 29, 30, 31, 32, 33, 34, 35].Once the interaction between heavy hadrons has been determined, the study ofthe properties of heavy hadrons in nuclear matter requires the inclusion of nuclearmedium modifications [36]. In this way, it is possible to study the formation of heavymesic states in nuclei [37, 38, 39], as well as the propagation of charmed and beautyin heavy-ion collisions [40, 41, 42, 43, 44], all of these topics matter of the presentpaper.
Recently a predictive model has been developed for four flavors including allground-state hadrons (pseudoscalar and vector mesons, and 1 / + and 3 / + baryons).This scheme reduces to the Weinberg-Tomozawa (WT) interaction when Goldstonebosons are involved and includes HQSS in the sector where heavy quarks appear.In fact, this model is justified due to the results of the SU(6) extension in the three-flavor sector [45] and is based on a formal plausibleness on how the interactionsbetween heavy pseudoscalar mesons and baryons emerge in the vector-meson ex-change picture. The WT potential can be then used to solve the on-shell Bethe-Salpeter equation in coupled channels so as to calculate the scattering amplitudes.The poles of the scattering amplitudes are the dynamically-generated charmed bary-onic resonances (see Ref. [33] for a review).Dynamically generated states with different charm and strangeness are predictedin Refs. [28, 29, 30]. The studies are constrained to the states coming from themost attractive representations of the SU(6) × HQSS scheme. Some of them canbe identified with known states from the PDG [46], by comparing the PDG data onthese states with the mass, width and the dominant couplings to the meson-baryonchannels.In this work, as an example, the results in the C = , S = , I = Λ c and one Λ ∗ c are obtained. A pole around 2618.8 MeV is iden-tified with the experimental Λ c ( ) resonance, while a second broad Λ c state at2617 MeV shows a similar two-pole pattern as in the Λ ( ) case [47], couplingstrongly to Σ c π . The third spin-1 / Λ c around 2828 MeV cannot be assigned to he charming beauty of the strong interaction 3 I = , J = / I = , J = / I = , J = / I = , J = / [MeV]050100150200 I = , J = / free ρ ,Pauli ρ [MeV] 00.20.40.60.81 I = , J = / |T D*N->D*N | [MeV -2 ] Λ c (2595) Λ c (2660) Λ c (2941) Σ c (2823) Σ c (2902) Σ c (2554) Σ c (2868) q [MeV] S ( q , q = )[ G e V - ] ρ ρ D meson 1600 1700 1800 1900 2000 2100 2200 q [MeV] D* meson
Fig. 1
Left: Charmed baryonic resonances in dense matter (taken from Ref. [48]). Right: The D and D ∗ spectral functions in dense nuclear matter at zero momentum (taken from Ref. [48]). any experimental state. With regards to the spin-3 / Λ c state, this is assigned to Λ c ( ) . The inclusion of dense matter effects modifies the mass and width of the dynamically-generated baryonic states. On l.h.s of Fig. 1 the squared amplitude of D ∗ N - D ∗ N transition is shown for different partial waves as function of the center-of-mass en-ergy for zero total momentum. In this case, the position of the Λ c ( ) is fittedand, as a result, the masses of the other states are slightly modified as compared tothe results reported in the previous section.The SU(6) × HQSS model in the I = , J = / , / I = J = / Λ c ( ) ,( I = J = / Σ c ( ) and Σ c ( ) , ( I = J = / Λ c ( ) ,( I = J = / Λ c ( ) , ( I = J = / Σ c ( ) and ( I = J = / Σ c ( ) resonances.Several medium modifications are considered: no in-medium corrections, the in-clusion of Pauli blocking on the nucleon intermediate states at saturation density ρ = .
17 fm − , and the in-medium solution that takes into account Pauli block-ing as well as the self-consistent inclusion of the D and D ∗ self-energies. Thesestates are modified in mass and width depending on the strength of the coupling tomeson-baryon channels with D , D ∗ and N as well as the closeness to the DN or D ∗ N thresholds.The knowledge of the in-medium modified dynamically generated excited baryonstates is of extreme importance for the determination of the properties of open- Laura Tolos charm mesons, such as D and ¯ D mesons, in the nuclear medium. And the modifica-tion of the properties of open-charm mesons is of crucial value because of the im-plications for charmonium suppression [49] and the possible formation of D -mesonbound states in nuclei [50].The properties of open charm in dense matter have been analysed in differentschemes: QMC schemes [50], QCD sum-rule approaches [51, 52, 53], NJL mod-els [54], chiral effective models in matter [55] or pion-exchange approaches thatincorporate heavy-quark symmetry constraints [56]. Nevertheless, the full spectralfeatures of the open-charm mesons in matter have been obtained in self-consistentunitarized coupled-channel models, where the intermediate meson-baryon channelsare modified including medium corrections [1, 2, 7, 8, 9, 38, 48, 57]. On the r.h.s.of Fig. 1 the D and D ∗ spectral functions are displayed for two densities at zero mo-mentum. The D -meson quasiparticle peak mixes strongly with Σ c ( ) N − and Σ c ( ) N − particle-hole excitations, whereas the Λ c ( ) N − is visible in thelow-energy tail. The D ∗ spectral function incorporates the J = / Σ c ( ) N − and Λ c ( ) N − . For both D and D ∗ mesons, the particle-hole modes smear out with density while the spectralfunctions broaden. D -meson bound states in nuclei Since the work of Ref. [50], there have been speculations about the formation of D -meson bound states in nuclei, which are based on the assumption of an attrac-tive interaction between D -mesons and nucleons. Within the model of Ref. [37], D -mesons bind in nucleus but very weakly (see Fig. 2), in contradistinction to[50]. Moreover, D -mesic states show significant widths. No D + -nuclear states arefound since the Coulomb interaction prevents their formation. As for D − or ¯ D ,both mesons bind in nuclei as seen in Fig. 3, though only nuclear states are manifestfor ¯ D in nuclei. The atomic states for D − are less bound as compared to the pureCoulomb levels, whereas the nuclear ones are more bound and might show a signif-icant width, appearing only for low angular momenta [38]. These results are closeto [56], but in contrast to [50] for Pb.As shown in Ref. [39], the experimental detection of D and ¯ D -meson bound statesis complicated. Reactions of the type ( ¯ p , D+N) or ( ¯ p , D+2N) may be indeed pos-sible at PANDA with antiproton beams, as long as formation cross sections are notsuppressed as well as small or even zero momentum transfer reactions are feasible.More successful mechanisms could involve the emission of pions by intermediate D ∗ or ¯ D ∗ , while the resulting open-charm mesons are trapped by the nucleus [39]. he charming beauty of the strong interaction 5 -20-15-10-5 0 5 B ± Γ / [ M e V ] D -nucleus bound states C L= 0 Mg
0, 1 Al
0, 1 Si
0, 1 S
0, 1 Ca
0, 1 Sn
0, 1, 2 Pb
0, 1, 2, 3
L=0L=1L=2L=3
Fig. 2 D -nucleus bound states (taken from [37]). -50-40-30-20-10 0 10 B ± Γ / [ M e V ] D − − nucleus bound states C Ca Sn Pb L=0,1,2,3 0,1,2,3,4 0,1,2,3,4,5,6,7 0,1,2,3,4,5,6,7Coul Atomic, α =1.0Nuclear, α =1.0 -25-20-15-10-5 0 5 B ± Γ / [ M e V ] D– − nucleus bound states C Ca Sn PbL= 0,1 0,1,2,3 0,1,2,3 0,1,2,3 α =1.0 Fig. 3 D − and ¯ D - nucleus bound states (taken from [38]). D -meson propagation in hot matter The transport coefficients of D mesons in the hot dense medium created in heavy-ion collisions offer the possibility to analyse the interaction of D mesons with lightmesons and baryons. Using the Fokker-Planck description, the drag ( F i ) and diffu-sion coefficients ( Γ i j ) of D mesons in hot dense matter can be obtained using aneffective field theory that incorporates both the chiral and HQSS in the meson [58]and baryon sectors [40].The spatial diffusion coefficient D x , that appears in Fick’s diffusion law, is arelevant quantity that involves both the drag and diffusion coefficients. Within anisotropic bath, the spatial diffusion coefficient reads D x = lim p → Γ ( p ) m D F ( p ) , (1)as a function of the scalar F ( p ) and Γ ( p ) coefficients. The interest on the spatialdiffusion coefficient relies on the fact that it might show an extremum around thetransition temperature between the hadronic and QGP phases, as seen previously forthe shear and bulk viscosities [40]. Laura Tolos s/ Bnet
This work Tolos et al. Berrehrah et al. T D x T [MeV]
Fig. 4
The coefficient 2 π T D x around the transition temperature (taken from Ref. [42]). In Fig. 4 the 2 π T D x is shown around the transition temperature following isen-tropic trajectories ( s / n B =ct) from RHIC to FAIR energies. The matching betweencurves in both phases for a given value of s / n B seems to indicate the possible exis-tence of a minimum in the 2 π T D x at the phase transition [42, 59]. The LHCb Collaboration has observed two narrow baryon resonances with beauty,being their masses and decay modes consistent with the quark model orbitally ex-cited states Λ b ( ) and Λ ∗ b ( ) , with J P = / − and 3 / − , respectively [60].The existence of these states is predicted within the unitarized meson-baryoncoupled-channel dynamical model, which implements HQSS and has been pre-sented in Sec. 2. A summary of the predictions is graphically shown in Fig. 5. Withinthat scheme, the experimental Λ ( ∗ ) b states are identified as HQSS partners, explain-ing their approximate mass degeneracy. An analogy is found between the bottom,charm and strange sectors, given that the Λ b ( ) is the bottomed counterpart ofthe Λ ∗ ( ) and Λ ∗ c ( ) states. Moreover, the Λ b ( ) belongs to the two-polestructure similar as the one seen in the case of the Λ ( ) and Λ c ( ) .Mass and decay modes are also predicted for some Ξ b ( / − ) and Ξ b ( / − ) ,that belong to the same SU(3) multiplets as the Λ b ( / − ) and Λ b ( / − ) . Three Ξ b ( / − ) and one Ξ b ( / − ) states are obtained coming from the most attractiveSU(6) × HQSS representations. Two of these states, Ξ b ( . ) and Ξ ∗ b ( . ) ,form a HQSS doublet similar to that of the experimental Λ b ( ) and Λ ∗ b ( ) .Nevertheless, none of these states have been detected yet. he charming beauty of the strong interaction 7
200 300 400 500 Λ b Λ ∗ b Ξ b Ξ ∗ b ∆ M R [ M e V ] Λ b ππΣ b π Σ ∗ b πΞ b ππ Ξ ’ b π Ξ ∗ b πΞ b π Ξ ’ b Ξ b Exp
Fig. 5
Summary of the new predicted Λ ( ∗ ) b and Ξ ( ∗ ) b states (red lines). We also show the experimen-tally observed Λ b ( ) and Λ b ( ) states (black dots) and some relevant hadronic thresholds(blue dotted lines). These results have been taken from [31]. ¯ B meson and Λ b propagation in hot matter The physical observables in heavy-ion collisions, such as particle ratios, R AA or v ,are strongly correlated to the behavior of the transport properties of heavy hadrons.The transport properties depend on the interactions of the heavy particles with thesurrounding medium, and these are described by means of effective theories thatincorporate the heavy degrees of freedom.Following the initial works of Refs. [58, 61, 62, 63, 64], the effective interactionof heavy mesons, such as D [40] and ¯ B [41] mesons, with light mesons and baryonshas been obtained by exploiting chiral and heavy-quark symmetries (see Sec. 2.1 forthe case of baryons with charm). With these interactions, the heavy-meson transportcoefficients are obtained as a function of temperature and baryochemical potentialof the hadronic bath using the Fokker-Planck equation, as described in Sec. 2.4.In Fig. 6 the spatial diffusion coefficients D x , multiplied by the thermal wavenum-ber (2 π T ), for ¯ B and also Λ b are presented, as derived in Eq. 1. On the l.h.s the spatialdiffusion coefficient for ¯ B is shown for different isentropic trajectories. The resultsare quite independent of the entropy per baryon as long as it is high enough, thatis, the collision energy is sufficiently high. Thus, these results can be taken as pre-diction for the hadronic medium created at high energy collisions (like those at theRHIC or the LHC), independently of the precise value of the entropy per baryon ofthe trajectory. Although the relaxation time is smaller with larger baryonic density,the ¯ B meson can hardly relax to the equilibrium. Moreover, on the r.h.s of Fig. 6 theanalogous coefficient for Λ b is shown, but only for µ B =
0. It is obtained that theoutcome for the spatial diffusion coefficient from the Fokker-Planck formalism isin good agreement to the one coming from the solution of the Boltzmann-Uehling-Uhlenbeck transport equation. Moreover, a similar behavior for both the ¯ B and Λ b spatial diffusion coefficients is observed at µ B =
0, due to the comparable mass and
Laura Tolos
T (MeV)
100 110 120 130 140 x T D π =0 B µ =300 B s/n =100 B s/n =30 B s/n T (MeV)
80 100 120 140 160 T x D π Fokker-Planck eq.Boltzmann-Uehling-Uhlenbeck eq.
Fig. 6
Spatial diffusion coefficient, D x , multiplied by the thermal wavenumber (2 π T ), for ¯ B meson(left, taken from [41] ) and Λ b (right, taken [65]). While the ¯ B spatial diffusion coefficient is shownfor different isentropic trajectories, only the µ B = Λ b . cross sections [65]. The phenomenological implications of these findings in heavy-ion collisions have been analysed in [66]. Acknowledgements
The author warmly thanks Santosh K. Das, Carmen Garcia-Recio, JuanNieves, Lorenzo L. Salcedo, Olena Romanets and Juan M. Torres-Rincon for their collabora-tion and discussions that has made this work possible. She also acknowledges support from theHeisenberg Programme of the Deutsche Forschungsgemeinschaft under the Project Nr. 383452331,the Ram´on y Cajal research programme, FPA2013-43425-P and FPA2016-81114-P Grants fromMINECO, and THOR COST Action CA15213.
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