Drag and diffusion coefficients of Bmesons in hot hadronic matter

Santosh K. Das, Sabyasachi Ghosh, Sourav Sarkar, and Jan-e Alam

Theoretical Physics Division, Variable Energy Cyclotron Centre, 1/AF, Bidhan Nagar, Kolkata—700064, India(Received 26 September 2011; published 18 April 2012)

The drag and diffusion coefficients of a hot hadronic medium consisting of pions, kaons and eta using

open beauty mesons as a probe have been evaluated. The interaction of the probe with the hadronic matter

has been treated in the framework of chiral perturbation theory. It is observed that the magnitude of both

the transport coefficients is significant, indicating a substantial amount of interaction of the heavy mesons

with the thermal bath. The results may have significant impact on the experimental observables like the

suppression of single-electron spectra originating from the decays of heavy mesons produced in nuclear

collisions at Relativistic Heavy Ion Collider and LHC energies.

DOI: 10.1103/PhysRevD.85.074017 PACS numbers: 12.38.Mh, 25.75.�q, 24.85.+p, 25.75.Nq

I. INTRODUCTION

The suppression of the transverse momentum (pT) dis-tribution of hadrons produced in nucleus-nucleus relativeto (binary scaled) proton-proton interactions at theRelativistic Heavy Ion Collider (RHIC) [1] has been usedas a tool to understand the properties of matter formed insuch collisions. The large value of the elliptic flow ofhadrons measured at RHIC along with the suppression ofthe high pT hadrons mentioned above indicate that thematter might have been formed in the partonic phasewith liquidlike properties characterized by low value ofshear viscosity (�) to entropy density (s) ratio, �=s with alower bound of �=s� 1=4� [2].

In addition to elliptic flow (v2) and nuclear suppression(RAA) of light hadrons these quantities have also beenmeasured for the single-electron spectra originating fromthe decays of the open charm and beauty mesons producedat RHIC collisions [3,4]. The advantages with heavy me-sons are twofold. First, they contain either a charm or abeauty quark which is produced very early and hence canwitness the evolution of the partonic matter since its in-ception until it reverts to hadronic matter through phasetransition/crossover [5], and second, the heavy quarks donot decide the bulk properties of the latter. Therefore,charm and beauty quarks are considered to be efficientprobes for the characterization of the partonic phase. Inmost of the earlier works [6–18] aimed at extracting theproperties of quark gluon plasma (QGP) by analyzing theRAA and v2 of heavy flavors, the role of the hadronic matterwas ignored. However, for the characterization of QGP theinteractions of heavy flavors with hadronic matter shouldbe taken into consideration and the effects of hadrons mustbe subtracted out from the observables. Though a largeamount of work has been done on the diffusion of heavyquarks in the QGP the diffusion of heavy mesons inhadronic matter has received much less attention so far.Recently the diffusion coefficient of the D meson has beencalculated using heavy meson chiral perturbation theory[19] and also by using the empirical elastic scattering

amplitudes [20] of D mesons with thermal hadrons. TheD-hadron interactions also have been evaluated using Bornamplitudes [21] and unitarized chiral effective D� inter-actions [22]. It has been found that the contributions of theBmeson to the single-electron spectra dominate over thosefrom the D meson for large transverse momentum, pT >5 GeV [23] (see also [24]). Moreover, the future experi-ments are progressing toward precision measurement overa wide range of kinematical variables. In view of this theuse of the B meson as a probe to extract the properties ofmatter at high temperature assumes importance.In the next section we discuss the formalism adopted to

evaluate the drag and diffusion coefficients of the heavyflavored mesons in a hadronic matter consisting of pions,kaons and eta. Results are presented in Sec. III and Sec. IVis dedicated to summary and discussions.

II. FORMALISM

In the present work the drag and diffusion coefficients ofthe Bmeson propagating through a hot hadronic matter areevaluated within the ambit of heavy meson chiral pertur-bation theory ðHM�PTÞ in leading order (LO), next-to-leading order (NLO) and next-to-next-to leading order(NNLO) approximations. We also revisit the transportcoefficients of the D meson in a similar theoretical frame-work. We consider the elastic interaction of the B mesonwith thermal pions, kaons and eta in the temperature (T)range 100–170 MeV. Detailed analysis of the experimentaldata on the hadronic yield in heavy ion collisions showsthat the value of the temperature for the chemical freeze-out of the system produced at RHIC energies is about170 MeV (see [25] for a review). This indicates that theinelastic interactions which are responsible for the changein the number of hadrons become rarer for T below�170 MeV. Thus the contributions of the inelastic colli-sions in evaluating the drag and diffusion coefficients ofthe hadronic matter probed by the heavy flavored mesonscan be ignored.

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The drag (�) and diffusion (B0) coefficients of the heavymesons are evaluated using elastic interaction with thethermal hadrons. For the (generic) process, BðpÞ þ hðqÞ !Bðp0Þ þ hðq0Þ (h stands for pion, kaon and eta), the drag �can be calculated by using the following expression [26]:

� ¼ piAi=p2; (1)

where Ai is given by

Ai ¼ 1

2Ep

Z d3q

ð2�Þ3Eq

Z d3p0

ð2�Þ3E0p

Z d3q0

ð2�Þ3E0q

1

gB

�X jMj2ð2�Þ4�4ðpþ q� p0 � q0Þ

� fðqÞð1þ fðq0ÞÞ½ðp� p0Þi�� hhðp� p0Þii; (2)

gB being the statistical degeneracy of the B meson. Thefactor fðqÞ denotes the thermal phase space factor for theparticle in the incident channel and 1þ fðq0Þ is the Boseenhanced final state phase space factor. From Eq. (2) it isclear that the drag coefficient is a measure of the thermalaverage of the momentum transfer, p� p0, weighted bythe interaction through the square of the invariant

amplitude, j M j2.The diffusion coefficient, B0, can be defined as

B0 ¼ 1

4

�hhp02ii � hhðp:p0Þ2ii

p2

�: (3)

Both the drag and diffusion coefficients can be evaluatedfrom a single expression:

hh�ðpÞii ¼ 1

512�4

1

Ep

Z 1

0

Z 1

�1dðcos�cmÞ

�Z 2�

0d�cm

q2dqdðcos�ÞEq

ð1þ fðq0ÞÞfðqÞ

� �ð1=2Þðs; m2p; m

2qÞffiffiffi

sp 1

g

X jMj2 T ðp0Þ (4)

with an appropriate choice ofT ðp0Þ. In Eq. (4) �ðx; y; zÞ ¼x2 þ y2 þ z2 � 2xy� 2yz� 2zx is the triangularfunction.

We start our discussion on the determination of thescattering amplitudes with the Lagrangian of covariantchiral perturbation theory ðC�PTÞ involving the heavy B(or D) mesons [27] given by

LC�PT ¼hDPDPyi�m2BhPPyi�hDP

�DP�y i

þm2B� hP�P�y

iþ ighP�u

Py�PuP�y iþ . . . ;

(5)

where the heavy-light pseudoscalar meson triplet P ¼ðB0; Bþ; Bþ

s Þ, heavy-light vector meson triplet P� ¼

ðB�0 ; B�þ

; B�þs Þ and h. . .i denotes trace in flavor space.

The covariant derivatives are defined as DPa ¼ @Pa �Pb�

ba and DPy

a ¼ @Pya þ �

abP

yb with a, b the SUð3Þ

flavor indices.The vector and axial-vector currents are, respectively,

given by � ¼ 12 ðuy@uþ u@u

yÞ and u ¼ iðuy@u�u@u

yÞ where u ¼ expð i�2F0Þ. The unitary matrix� collects

the Goldstone boson fields and is given by

� ¼ ffiffiffi2

p�0ffiffi2

p þ �ffiffi6

p �� K�

�þ � �0ffiffi2

p þ �ffiffi6

p K0

Kþ K0 � 2�ffiffi6

p

0BBBB@

1CCCCA:

To lowest order in � the vector and axial-vectorcurrents are

� ¼ 1

8F20

½�; @��; u ¼ � 1

F0

@�: (6)

From the first term (or kinetic part of the P fields) ofLC�PT , the matrix elements for the contact diagram in

terms of Mandelstam variables (s, t, u) are obtained as

MBþ�þ ¼ �MBþ�� ¼ � 1

4F2�

ðs� uÞ;

MBþ�0 ¼ MBþ� ¼ MBþKþ ¼ MBþK� ¼ 0;

MBþ �K0 ¼ �MBþK0 ¼ � 1

4F2K

ðs� uÞ:

(7)

These can be represented in the isospin basis as

Mð3=2ÞB� ¼ � 1

4F2�

ðs� uÞ; Mð1=2ÞB� ¼ 1

2F2�

ðs� uÞ;

MB� ¼ 0; Mð1ÞBK ¼ 0; Mð0Þ

BK ¼ � 1

2F2K

ðs� uÞ;

Mð1ÞB �K

¼ �Mð0ÞB �K

¼ � 1

4F2K

ðs� uÞ; (8)

where the isospin of the B� system appears in the super-script. Denoting the threshold matrix elements by T, theseare obtained from (8) and are given by

Tð3=2ÞB� ¼ �mBm�

F2�

; Tð1=2ÞB� ¼ 2mBm�

F2�

;

TB� ¼ 0; Tð1ÞBK ¼ 0; Tð0Þ

BK ¼ � 2mBmK

F2K

;

Tð1ÞB �K

¼ �Tð0ÞB �K

¼ �mBmK

F2K

:

(9)

One can reproduce these T-matrix elements in the iso-spin basis using the lowest order HM�PT Lagrangian forheavy mesons containing a heavy quark Q and a lightantiquark of flavor a as given below [28]:

LHM�PT ¼ �itrDð �HQa v@H

Qa Þ � itrDð �HQ

a v�ab HQ

b Þþ g

2trDð �HQ

a ��5uab HQb Þ þ . . . ; (10)

DAS et al. PHYSICAL REVIEW D 85, 074017 (2012)

074017-2

where HQa ¼ 1þ6v

2 ðP�a�

þ iPa�5Þ and �HQ

a ¼ ðP�ya� þ

iPya�5Þ 1þ6v

2 and trD denotes trace in Dirac space. In this

formalism, since the factorffiffiffiffiffiffiffimP

pand

ffiffiffiffiffiffiffiffimP�

phave been

absorbed into the Pa and P�a fields, the threshold

T-matrix element ð ~TP�th Þ now has the dimension of scatter-

ing length aP whereas in C�PT we get a dimensionlessT-matrix element ðTP�

th Þ. The relation between these two

T-matrix elements and the scattering length aP is given by

TP�th ¼ mP

~TP�th ¼ 8�ðm� þmPÞaP: (11)

The square of the isospin averaged T-matrix element isgiven by

X jTB�j2 ¼ jTB�j2 þ jTBKj2 þ jTB �Kj2; (12)

where jTB�j2¼ 1ð2þ4Þð2jTð1=2Þ

B� j2þ4jTð3=2ÞB� j2Þ and jTBK= �Kj2¼

1ð1þ3ÞðjTð0Þ

BK= �Kj2þ3jTð1Þ

BK= �Kj2Þ.

III. RESULTS

We evaluate the drag coefficients of the B meson byusing the momentum dependent and momentum indepen-dent matrix elements given by Eqs. (8) and (9), respec-tively. The results are shown by the dash-dotted and dashedlines in Fig. 1. Inspired by the fact that the results for thetwo scenarios are not drastically different in the LO weproceed to evaluate the drag coefficient of heavy mesons

by replacingP jMj2 by

P jTj2 in NLO and NNLO alsowhere the T-matrix elements will be obtained from thescattering lengths.

Liu et al. [29] have obtained the B� scattering lengths(see also [30]) up to NNLO in HM�PT by using thecoupling constant from recent unquenched lattice results

[31]. Using these NLO and NNLO results we estimate theisospin averaged drag coefficients of Bmesons. The resultsare depicted in Fig. 1. The drag coefficient evaluated withNNLO matrix elements increases by 22% compared to theNLO result at T ¼ 170 MeV.We now focus on the temperature dependence of the

drag coefficient of B mesons as shown in Fig. 1. Asmentioned before, � is the thermal average of the squareof the momentum exchanged between the heavy mesonsand the bath particle weighted by the interaction strengththrough the invariant amplitude of the process. Therefore,with the increase in temperature of the thermal bath thekinetic energy of the hadrons increases. Hence the hadronsgain the ability to transfer larger momentum during theirinteraction with the Bmesons—resulting in the increase ofthe drag coefficient. This tendency is observed in Fig. 1quite clearly. The increase of drag with temperature ischaracteristic of a gaseous system. In the case of a liquidthe drag diminishes with T (except for very few cases). Inthis case a significant part of the thermal energy goes intomaking the attraction between the interacting particlesweaker and once this happens the constituents movemore freely resulting in a smaller drag force. Therefore,the variation of the drag with temperature can be used tounderstand the nature of the interaction of the fluid.Since the diffusion coefficient involves the square of the

momentum transfer it is also expected to increase with T.This is seen in Fig. 2. The drag and the diffusion coeffi-cients are related through the Einstein relation as

B0 ¼ MB�T; (13)

where MB is the mass of the B meson. The temperaturedependence of the diffusion coefficient evaluated by usingEqs. (4) and (13) is displayed in Fig. 2. The differencebetween the results obtained from Eq. (4) and the

0.1 0.12 0.14 0.16

T(GeV)

0

0.01

0.02

γ(fm

−1)

B(NNLO)

B(NLO)

B(LO, from Eq. 9)

B(LO, from Eq. 8)

FIG. 1. The variation of drag coefficients with temperature dueto the interaction of B mesons (of momentum 100 MeV) withthermal pions, kaons and eta. The dash-dotted (dashed) lineindicates the results obtained by using the matrix elements ofEq. (8) [T matrix of Eq. (9)]. The solid (dotted) line indicates theresults for the NNLO (NLO) contribution [29].

0.1 0.12 0.14 0.16

T(GeV)

0

0.005

0.01

0.015

0.02

B0(

GeV

2 /fm

)

FIG. 2. Variation of diffusion coefficient as a function oftemperature. The solid line indicates the variation of the diffu-sion coefficient with temperature obtained from Eqs. (3) and (4).The momentum of the Bmeson is taken as 100 MeV. The dashedline stands for the diffusion coefficient obtained from theEinstein relation [Eq. (13)].

DRAG AND DIFFUSION COEFFICIENTS OF B MESONS . . . PHYSICAL REVIEW D 85, 074017 (2012)

074017-3

Einstein’s relation is about 6%–7% at T ¼ 170 MeV forthe B meson momentum, p ¼ 100 MeV. This small dif-ference illustrates the validity of the Einstein relation in thelow momentum (nonrelativistic) domain.

The energy loss of a B meson moving through a had-ronic system may be estimated from the relation

� dE

dx¼ �p: (14)

The magnitude of � obtained in the present calculationreveals that the B mesons dissipate significant amount ofenergy in the medium. This might have crucial consequen-ces on quantities such as the nuclear suppression factor ofsingle electrons originating from the decays of heavymesons.

We also evaluate the D meson drag and diffusion coef-ficients using the interactions of D mesons with thermalhadrons discussed in Refs. [27,29] in LO, NLO and NNLOapproximations. The results are displayed in Fig. 3. In theLO approximation the drag is similar for both the cases.However, for NLO and NNLO, the drag coefficient eval-uated using the T-matrix elements obtained from the scat-tering lengths of Ref. [29] is slightly higher than thatobtained from Ref. [27].

The drag of D mesons in hot hadronic matter hasrecently been studied by using different approaches.

While empirical scattering cross sections were used inRef. [20], the authors of Ref. [22] used unitarized chiraleffective D� interactions to evaluate the drag. We ob-serve that the magnitude of the drag of the D mesonobtained in the present work is similar to that obtainedin Refs. [20,22]. The smaller value in the present case isdue to the lower values of the D meson-hadron crosssections.

IV. SUMMARYAND DISCUSSIONS

In summary we have evaluated the drag and diffusioncoefficients of open beauty mesons interacting with ahadronic background composed of pions, kaons and eta.It is found that the values of both the transport coefficientsincrease with temperature. The magnitude of the dragcoefficient of the B meson indicates that while evaluatingthe suppression of the high pT single electrons originatingfrom the decays of B mesons the effects of hadrons shouldbe taken into account. Within the same formalism, thetransport coefficients of the D meson has been calculated.TheDmeson drag coefficient is found to be lower than thevalues obtained in Refs. [20,22].Some comments on the effects of the exclusions of

inelastic channels and nonperturbative processes on thedrag and diffusion coefficients are in order here. Theinelastic channels do contribute to the scattering matrixthrough coupled channels via loops in the unitarizationprocedure. However, in a thermal background such as ina heavy ion collision the scattering amplitude is weightedby phase space factors which essentially control the rate ofreactions. The fact that heavy ion collisions undergochemical freeze-out at about 170 MeV means that thenumber-changing reactions will certainly be inhibited be-low this temperature. The presence of resonances makesthe evaluation of scattering amplitudes a nonperturbativeproblem. Unitarization of scattering amplitudes, preferablywith loops evaluated with thermal propagators, will cer-tainly improve the reliability of our results at higher en-ergies. However, in the absence of any information aboutB� mesons so far, this exercise will however be far lessconstrained than the charm sector where the masses andwidths of the excited states are known.

ACKNOWLEDGMENTS

S. K.D. and J. A. are partially supported by DAE-BRNSProject Sanction No. 2005/21/5-BRNS/2455.

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T (GeV)

0

0.01

0.02

0.03γ(

fm−1

)D(NNLO)D(NLO)D(LO)

0.1 0.12 0.14 0.16 0.1 0.12 0.14 0.16T (GeV)

0

0.01

0.02

0.03

D(NNLO)D(NLO)D(LO)

FIG. 3. The variation of drag coefficients of D mesons withtemperature due to interaction with thermal pions, kaons and etain LO, NLO and NNLO approximations for interactions of Dwith thermal hadrons taken from Refs. [27] (left panel) and [29](right panel).

DAS et al. PHYSICAL REVIEW D 85, 074017 (2012)

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