We present effective-field-theory results with unitarized interactions on the
D-meson transport coefficients in a gas populated by light mesons and baryons
at finite temperature and baryochemical potential. The Fokker-Planck equation
is used to compute the drag force, the relaxation time and the diffusion
coefficients of D mesons for collisions at FAIR. At finite baryochemical
potential, the combined effect of net baryonic density and sizable meson-baryon
interaction makes the D mesons to relax more efficiently than in the case at
zero baryochemical potential. We also describe the connection with the
quark-gluon plasma phase in adiabatic trajectories on the phase diagram at both
zero and finite baryochemical potential.Comment: 5 pages, 4 figures. Contribution to the FAIRNESS 2013 - Workshop for
  young scientists with research interests focused on FAIR physics. 15-21
  September 2013, Berlin (Germany

Abreu, Luciano M.

Cabrera, Daniel

Llanes-Estrada, Felipe J.

Tolos, Laura

Torres-Rincon, Juan M.

Journal of Physics Conference Series

English

arXiv

Published under licence by IOP Publishing Ltd.
Workshop on FAIR Next Generation of ScientistS (FAIRNESS)(2. 2013. Berlin).
JMTR acknowledges the organizers of FAIRNESS 2013 for the hospitality received and the
opportunity to present these results. This work has been financed by grants FPA2010-16963,
FPA2011-27853-C02-01, FPA2011-27853-C02-02 (Spain), BMBF (Germany) under project no.
05P12RFFCQ, EU Integrated Infrastructure Initiative Hadron Physics Project under Grant
Agreement n. 227431 and Grant No. FP7-PEOPLE-2011-CIG under Contract No. PCIG09-
GA-2011-291679. LMA thanks CAPES (Brazil) for partial financial support. DC acknowledges
support from Centro Nacional de Fisica de Partticulas, Astroparticulas y Nuclear (Consolider -
Ingenio 2010) and LT acknowledges support from the Ramon y Cajal Research Programme.We present effective-field-theory results with unitarized interactions on the D-meson transport coefficients in a gas populated by light mesons and baryons at finite temperature and baryochemical potential. The Fokker-Planck equation is used to compute the drag force, the relaxation time and the diffusion coefficients of D mesons for collisions at FAIR. At finite baryochemical potential, the combined effect of net baryonic density and sizable meson-baryon interaction makes the D mesons to relax more efficiently than in the mu(B) = 0 case. We also describe the connection with the quark-gluon plasma phase in adiabatic trajectories on the phase diagram at both zero and finite baryochemical potential.BMBF, AlemaniaCentro Nacional de Fisica de Partîcculas, Astroparticulas y NuclearCAPES, BrasilDepto. de Física TeóricaFac. de Ciencias FísicasTRUEpu

Llanes Estrada, Felipe José

Torres Rincón, Juan M.

Docta Complutense

This content has been downloaded from IOPscience. Please scroll down to see the full text.Download details:IP Address: 147.96.14.16This content was downloaded on 26/06/2014 at 15:26Please note that terms and conditions apply.D-meson diffusion in hadronic matterView the table of contents for this issue, or go to the journal homepage for more2014 J. Phys.: Conf. Ser. 503 012020(http://iopscience.iop.org/1742-6596/503/1/012020)Home Search Collections Journals About Contact us My IOPscienceD-meson diffusion in hadronic matterJuan M Torres-Rincon1,0, Luciano M Abreu2, Daniel Cabrera3,Felipe J Llanes-Estrada4, Laura Tolos1,31 Institut de Ciències de l’Espai (IEEC/CSIC), Campus Universitat Autònoma de Barcelona,Facultat de Ciències, Torre C5, E-08193 Bellaterra, Spain2 Instituto de F́ısica, Universidade Federal da Bahia, 40210-340, Salvador, BA, Brazil3 Frankfurt Institute for Advances Studies. Johann Wolfgang Goethe University,Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany4 Departamento de F́ısica Teórica I, Universidad Complutense, 28040 Madrid, SpainAbstract. We present effective-field-theory results with unitarized interactions on the D-meson transport coefficients in a gas populated by light mesons and baryons at finite temperatureand baryochemical potential. The Fokker-Planck equation is used to compute the drag force,the relaxation time and the diffusion coefficients of D mesons for collisions at FAIR. At finitebaryochemical potential, the combined effect of net baryonic density and sizable meson-baryoninteraction makes the D mesons to relax more efficiently than in the µB = 0 case. We alsodescribe the connection with the quark-gluon plasma phase in adiabatic trajectories on thephase diagram at both zero and finite baryochemical potential.1. IntroductionSome of the most useful tools to study the quark-gluon plasma (QGP) phase –formed in theearly stages of a relativistic heavy-ion collision– are heavy quarks. Due to their large mass(mc,mb  ΛQCD, T ) they can be treated as Brownian particles interacting with lighter partons.Not only the average momentum of the heavy quarks is modified by the collisions, but also themomentum distribution is broadened. The efficiency of these two processes is quantified by thedrag force and the momentum-space diffusion coefficient, respectively.The previous picture also holds after hadronization. The role of the Brownian particleis played by a D or B meson, whereas the thermal bath is now composed of pions, kaons,nucleons and other light hadrons with whom the heavy meson may interact. Thus, the transportcoefficients of D mesons [1, 2] and B mesons [3] become of key importance in the equilibrationof these hadrons. The presence of these coefficients will amount on the modification of the finalheavy meson spectra and affect the flow harmonics vn and the nuclear suppression factor RAA.To study these transport coefficients and quantify the energy loss of heavy quarks in the hotand dense medium we use quantum kinetic theory.We focus on the heavy-ion physics to be produced at the Facility for Antiproton and IonResearch (FAIR) [4], where the collision energy will be lower than that for collisions at theLarge Hadron Collider, for instance. Lower energies typically produce a net baryon density inthe central rapidity region. This fact will imply physics at non-zero baryochemical potentialin the QCD phase diagram, maybe close to the conjectured critical point. For these collisions,0 SpeakerFAIRNESS 2013: FAIR NExt generation of ScientistS 2013 IOP PublishingJournal of Physics: Conference Series 503 (2014) 012020 doi:10.1088/1742-6596/503/1/012020Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd 1the available energy density makes difficult the production of B mesons. Therefore, in thismanuscript we focus our attention to the calculation of D-meson transport coefficients [2].In Sec. 2 we introduce the Fokker-Planck kinetic equation and detail the physicalinterpretation of the D-meson transport coefficients. In Sec. 3 we briefly describe the interactionof D mesons with lighter hadrons, both mesons and baryons. Finally, in Sec. 4 we present ourmain results and conclusions.2. Fokker-Planck equation and transport coefficientsThe collective description of D mesons is encoded in their distribution function f(t, p). Thetemporal evolution of f(t, p) follows a kinetic equation defined in the phase space, which in thecase of a Brownian particle in a thermal bath composed of lighter particles is the Fokker-Planckequation:∂f(t,p)∂t=∂∂pi{Fi(p)f(t,p) +∂∂pj[Γij(p)f(t,p)]}, (1)where i = 1, 2, 3 is the spatial index, Fi is the drag force or friction term accounting for theaverage momentum change, and Γij is the momentum-space diffusion matrix which accountsfor the broadening of the momentum distribution. The computation of these coefficients isperformed through their microscopic expressions in terms of momentum averages,Fi(p) =∫dk w(p,k) ki , Γij(p) =12∫dk w(p,k) kikj , (2)where the interaction measure w(p,k) represents the probability of a D meson with momentump to suffer a collision with a light particle, loosing a momentum k:w(p,k) = gl∫d3q(2π)9nF,B(El(q), T ) [1± nF,B(El(q + k), T )]12ED(p)12El(q)12ED(p− k)× 12El(q + k)(2π)4δ(ED(p) + El(q)− ED(p− k)− El(q + k)) |M|2 , (3)with gl being the spin-isospin degeneration of the light particle, nF,B its equilibrium distributionfunction following Fermi-Dirac or Bose-Einstein statistics, respectively; ED(p) is the energy ofthe D meson with momentum p, El(q) the energy of the light particle with momentum q and|M2| is the average scattering matrix element squared of the binary collision.If we consider an isotropic gas and focus on the static limit (where p → 0) there isonly one independent transport coefficient. For definiteness, we take it to be the drag forceF (p) = Fipi/p2, while the momentum-space diffusion coefficient Γ is related to F through theEinstein relation Γ = FmDT (where mD is the D-meson mass and T the temperature of thethermal medium).In several talks at this workshop some emphasis was put on the concept of relaxation time.This coefficient describes how efficient the relaxation of the non-equilibrium distribution functionis, while it relaxes towards the equilibrium state (Boltzmann’s H theorem). Consider forsimplicity the one dimensional case of Eq. (1) and assume constant transport coefficients Fand Γ. The analytical solution of the kinetic equation isf(t, p) =√F2πΓ(1− e−2Ft)exp[− F2Γ(p− p0e−Ft)21− e−2Ft], (4)where p0 is the initial D-meson momentum. It is immediate to see that F−1 should play therole of an intrinsic time for the relaxation process. In particular, the average momentum reads〈p〉 ≡∫ ∞−∞dp pf(t, p)/∫ ∞−∞dp f(t, p) = p0 e− tτR , (5)FAIRNESS 2013: FAIR NExt generation of ScientistS 2013 IOP PublishingJournal of Physics: Conference Series 503 (2014) 012020 doi:10.1088/1742-6596/503/1/0120202where we have defined the relaxation time τR ≡ 1/F , which provides a characteristic timeaccounting for the exponential decay of the average momentum.We can also introduce the diffusion coefficient in position space Dx. The standard result forthe average square displacement〈(x(t)− x0)2〉 = 2Dxt , (6)shows that Dx is a measure of the “speed” of the diffusion in space. A large Dx implies thatthe D meson can diffusively travel larger distances for a given time. In the static limit thiscoefficient is related to the drag force by the relation Dx = T/(mDF ).All these coefficients depend on the temperature and baryochemical potential. In a heavy-ioncollision both variables change with time along the fireball’s trajectory in the phase diagram.For simplicity, we assume adiabatic trajectories –with a constant entropy per baryon consistentwith the energies expected at FAIR collisions– which give us the implicit relation µB = µB(T ).3. Interaction and effective field theoriesThe interaction between the D mesons and the light hadrons –as it lies in the nonperturbativeand confined regime of QCD– can be described by effective models. The global symmetriesof the QCD Lagrangian should be implemented into the effective models allowing a reliabledescription of the hadronic interaction. We take the chiral and heavy-quark-spin symmetries tobe the fundamental guides in order to construct our effective Lagrangian (therefore, it results ina double expansion). In addition to these symmetries, our third principle is the exact satisfactionof the unitarity condition of the scattering matrix.To account for the D meson–light meson scattering we introduce pions, kaons, antikaonsand η mesons. Details on the effective theory based on chiral and heavy-quark-spin expansionscan be found in Refs. [1, 2, 5]. Using Feynman rules we construct the perturbative amplitude(or potential) V for each scattering channel. In the meson-baryon sector we also construct theeffective Lagrangian based on heavy-quark-spin symmetry which accounts for the interaction ofD mesons with nucleons and ∆ baryons. Details on the effective theory and the computation ofthe potential V can be found in [2, 6, 7, 8, 9].In both sectors we apply a unitarization method [10] (based on a Bethe-Salpeter (BS)equation) to extend the interaction description up to temperatures T ' 140 MeV withoutloosing physical sense. In addition, we are able to dynamically generate resonant states likethe D0(2400) in the meson–meson sector and the Σc(2800) and Λc(2595) in the meson–baryonsector, which are automatically taken into account in the interaction. The scattering matrixelement in Eq. (3) is related to the perturbative amplitude as:M(s) ∝ V (s)1−G(s)V (s), (7)where G represent the diagonal matrix of the two intermediate propagators in the BS equation.4. Results and ConclusionsWe present our main results taken from Ref. [2]. In Fig. 1 we show the drag coefficientF (p = 100 MeV) for a D meson as a function of the temperature when the baryochemicalpotential is set to zero. The largest contribution comes from the pions, as these mesons populatethe gas at temperatures below the hadronization one. The kaons, antikaons and η mesonscontribute appreciably only at high temperatures T > 100 MeV, being the baryon contribution(N + ∆) negligible at µB = 0.The position-space diffusion coefficient Dx is shown at µB = 0 in Fig. 2, when all hadronsare included in the calculation. We normalize this coefficient by the thermal wavelength 2πTFAIRNESS 2013: FAIR NExt generation of ScientistS 2013 IOP PublishingJournal of Physics: Conference Series 503 (2014) 012020 doi:10.1088/1742-6596/503/1/0120203T (MeV)40 60 80 100 120 140F (1/fm)0.010.020.03∆+N+η+K+K+πη+K+K+ππFigure 1. Drag force F (p) for a Dmeson with momentum p = 100 MeV as afunction of temperature for several speciesin the thermal bath at µB = 0 (LHCcollisions). Clearly the dominant contributioncomes from pions as they are the mostabundant in the bath. The contributionof baryons is practically negligible. Thediffusion coefficient in momentum space isrelated to this coefficient via the Einsteinrelation Γ = FmDT .to construct an adimensional number. The calculation for the QGP (Rapp and Hees) is a T -matrix calculation from Ref. [11] and the lattice-QCD computation (Banerjee, Datta, Gavai andMajumdar) is taken from Ref. [12].T (MeV)0 100 200 300 400x T Dπ2 010203040This workRapp and van HeesBanerjee, Datta, Gavai and MajumdarFigure 2. Diffusion coefficient multiplied bythe thermal wavelength 2πT as a function oftemperature, around the crossover tempera-ture at µB = 0. The solid line represents ourresults for hadronic matter with all species in-cluded. The dashed line shows the result ofRef. [11] for a quark-gluon plasma. Dots rep-resent the result from the lattice-QCD calcu-lation extracted from Ref. [12].Turning to the µB 6= 0 case –relevant for the physics at FAIR– we consider three characteristicadiabatic trajectories in the phase diagram for different initial collision energies (√s ≈ 5− 40AGeV [4]). In the left panel of Fig. 3 we sketch our trajectories on the phase diagram. Theycorrespond to fixed entropy per baryon of 10 − 30 [13]. In the right panel we plot our resultsin the hadron gas (at low temperature) and the results of a perturbative QGP at finite quarkchemical potential based on the computation of Ref. [14].Finally, we show the relaxation time for each adiabatic trajectory plus the result at µB = 0as a function of temperature in Fig. 4. Notice that the presence of baryons at finite µB enhancesthe D-meson stopping and the relaxation time is accordingly reduced. Therefore collisions atFAIR put the D mesons closer to the equilibrium state than at the LHC. These relaxation timesshould be compared to the fireball’s lifetime which is of the order of tf ∼ 10 fm. In any case, theD mesons cannot totally relax through collision as τR > tf , confirming that D mesons shouldbe sensible to some properties of the initial QGP.In summary, we have presented our latest results on the transport coefficients of D mesonsthat describe the energy loss and diffusion of these states in a hadronic medium at finitetemperature and baryochemical potential. Focusing on FAIR physics, we have obtained that thecollisions with baryons make the D mesons more thermally relaxed than the case at vanishingbaryochemical potential. We have obtained a nice connection with the results of QGP at hightemperature, providing an indication for a possible minimum of 2πTDx at both the crossovertemperature (at µB = 0) and at the first-order transition temperature (at FAIR collisions).FAIRNESS 2013: FAIR NExt generation of ScientistS 2013 IOP PublishingJournal of Physics: Conference Series 503 (2014) 012020 doi:10.1088/1742-6596/503/1/0120204 (MeV)Bµ0 200 400 600 800 1000 1200T (MeV)0100200300400=30Bs/n=20Bs/n=10Bs/nQ G PHGT (MeV)0 50 100 150 200 250 300 350 400x T Dπ2 010203040=30Bs/n=20Bs/n=10Bs/nFigure 3. Left panel: Adiabatic trajectories for different initial conditions at FAIR collisions.Right panel: Dx as a function of temperature for the three adiabatic trajectories.T (MeV)100 110 120 130 140 (fm)Rτ20406080100=0Bµ=30Bs/n=20Bs/n=10Bs/nFigure 4. Relaxation time for the adiabatictrajectories as a function of temperature.The trajectories with larger baryonic density(those with smaller entropy per baryon) makethe D meson to have a lower relaxationtime and hence, to follow a more efficientrelaxation via baryonic collisions.AcknowledgmentsJMTR acknowledges the organizers of FAIRNESS 2013 for the hospitality received and theopportunity to present these results. This work has been financed by grants FPA2010-16963,FPA2011-27853-C02-01, FPA2011-27853-C02-02 (Spain), BMBF (Germany) under project no.05P12RFFCQ, EU Integrated Infrastructure Initiative Hadron Physics Project under GrantAgreement n. 227431 and Grant No. FP7-PEOPLE-2011-CIG under Contract No. PCIG09-GA-2011-291679. LMA thanks CAPES (Brazil) for partial financial support. DC acknowledgessupport from Centro Nacional de Fisica de Part́ıculas, Astropart́ıculas y Nuclear (Consolider -Ingenio 2010) and LT acknowledges support from the Ramón y Cajal Research Programme.References[1] Abreu L M, Cabrera D, Llanes-Estrada F J, Torres-Rincon J M 2011 Annals Phys. 326 2737[2] Tolos L and Torres-Rincon J M 2013 Phys. Rev. D 88 074019[3] Abreu L M, Cabrera D and Torres-Rincon J M 2013 Phys. Rev. D 87 034019[4] Friman B et al 2001 Lect. Notes Phys. 814, 1[5] Guo F K, Hanhart C and Meissner U G, 2009 Eur. Phys. J. A 40 171[6] Mizutani T and Ramos 2006 Phys. Rev. C 74 065201[7] Garcia-Recio C et al 2009 Phys. Rev. D 79 054004[8] Garcia-Recio C, Nieves J, Romanets O, Salcedo L L and Tolos L 2013 Phys. Rev. D 87 074034[9] Romanets O et al 2012 Phys. Rev. D 85 114032[10] Oller J A and Oset E, 1997 Nucl. Phys. A 620 438[11] Rapp R and van Hees H 2010 Quark Gluon Plasma 4 (World Scientific) 111[12] Banerjee D, Datta S, Gavai R and Majumdar P 2012 Phys. Rev. D 85 014510[13] Bravina L V et al. 2008 Phys. Rev. C 78 014907[14] Moore G D and Teaney D 2005 Phys. Rev. C 71 064904FAIRNESS 2013: FAIR NExt generation of ScientistS 2013 IOP PublishingJournal of Physics: Conference Series 503 (2014) 012020 doi:10.1088/1742-6596/503/1/0120205

D-meson diffusion in hadronic matter

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D-meson diffusion in hadronic matter

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