Langevin dynamics of heavy quarks in a soft-hard factorized approach

Shuang Li,1, 2, ∗ Fei Sun,1, † Wei Xie,1, ‡ and Wei Xiong1, §

1College of Science, China Three Gorges University, Yichang 443002, China2Key Laboratory of Quark and Lepton Physics (MOE),

Central China Normal University, Wuhan 430079, China(Dated: June 17, 2021)

By utilizing a soft-hard factorized model, which combines a thermal perturbative descriptionof soft scatterings and a perturbative QCD-based calculation for hard collisions, we study theenergy and temperature dependence of the heavy quark diffusion coefficients in Langevin dynamics.The adjustable parameters are fixed from the comprehensive model-data comparison. We findthat a small value of the spatial diffusion coefficient at transition temperature is preferred by data2πTDs(Tc) ' 6. With the parameter-optimized model, we are able to describe simultaneously theprompt D0 RAA and v2 data at pT ≤ 8 GeV in Pb–Pb collisions at

√sNN = 2.76 and

√sNN = 5.02

TeV. We also make predictions for non-prompt D0 meson for future experimental tests down to thelow momentum region.

I. INTRODUCTION

Ultrarelativistic heavy-ion collisions provide a uniqueopportunity to create and investigate the properties ofstrongly interacting matter in extreme conditions of tem-perature and energy density, where the normal matterturns into a new form of nuclear matter, consisting of de-confined quarks and gluons, namely quark-gluon plasma(QGP [1]). Such collisions allow us to study the prop-erties of the produced hot and dense partonic medium,which are important for our understanding of the prop-erties of the universe in the first few milliseconds andthe composition of the inner core of neutron stars [2–4]. Over the past two decades, the measurements withheavy-ion collisions have been carried at the Relativis-tic Heavy Ion Collider (RHIC) at BNL and the LargeHadron Collider (LHC) at CERN [5–7], to search andexplore the fundamental properties of QGP, notably itstransport coefficients related to the medium interactionof hard probes.

Heavy quarks (HQs), including charm and bottom,provide a unique insight into the microscopic propertiesof QGP [8–13]. Due to large mass, they are mainly pro-duced in initial hard scatterings and then traverse theQGP and experience elastic and inelastic scatterings withits thermalized constituents [14, 15]. The transport prop-erties of HQ inside QGP are encoded in the HQ trans-port coefficients, which are expected to affect the distri-butions of the corresponding open heavy-flavor hadrons.The resulting experimental observables like the nuclearmodification factor RAA and elliptic anisotropy v2 of var-ious D and B-mesons are therefore sensitive to the HQtransport coefficients, in particular their energy and tem-perature dependence. There now exist an extensive set

∗ lish@ctgu.edu.cn† sunfei@ctgu.edu.cn‡ xiewei@ctgu.edu.cn§ xiongw@ctgu.edu.cn

of such measurements, which allow a data-based extrac-tion of these coefficients. In this work, we make suchan attempt by using a soft-hard factorized model (seeSec. III) to calculate the diffusion and drag coefficientsrelevant for heavy quark Langevin dynamics.

A particularly important feature of the QGP trans-port coefficients is their momentum and temperature de-pendence, especially how they change within the tem-perature region accessed by the RHIC and LHC exper-iments. For instance the normalized jet transport coef-ficient q/T 3 was predicted to present a rapidly increas-ing behavior with decreasing temperature and developa near-Tc peak structure [16]. The subsequent studies[17–22] seem to confirm this scenario. Another impor-tant transport property, shear viscosity over entropy den-sity ratio η/s, also presents a visible T -dependence witha considerable increase above Tc [23]. Concerning theHQ diffusion and drag coefficients, there are also indi-cations of nontrivial temperature dependence from boththe phenomenological extractions [24–30] and theoreticalcalculations [20, 21, 31–34]. The difference among the de-rived hybrid models is mainly induced by the treatmentof the scale of QCD strong coupling, hadronization andnon-perturbative effects [35, 36]. See Ref. [37–39] for therecent comparisons.

The paper is organized as follows. In Sec. II we intro-duce the general setup of the employed Langevin dynam-ics. Section III is dedicated to the detailed calculation ofthe heavy quark transport coefficients with the factor-ization model. In Sec. IV we show systematic compar-isons between modeling results and data and optimize themodel parameters based on global χ2 analysis. With theparameter-optimized model, the energy and temperaturedependence of the heavy quark transport coefficients arepresented in Sec. V, as well as the comparisons with dataand other theoretical calculations. Section VI containsthe summary and discussion.

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II. LANGEVIN DYNAMICS

The classicial Langevin Transport Equation (LTE) ofa single HQ reads [14]

dxi =pi

Edt (1a)

dpi = −ηDpidt+ Cikρk√dt. (1b)

with i, k = 1, 2, 3. The first term on the right hand sideof Eq. 1b represents the deterministic drag force, F idrag =

−ηDpi, which is given by the drag coefficient ηD(E, T )

with the HQ energy E =√~p 2 +m2

Q and the underlying

medium temperature T . The second term denotes thestochastic thermal force, F ithermal = Cikρk/

√dt, which is

described by the momentum argument of the covariancematrix Cik,

Cik ≡√κT (δik − pipk

~p 2) +√κLpipk

~p 2, (2)

together with a Gaussian-normal distributed randomvariable ρk, resulting in the uncorrelated random mo-mentum kicks between two different time scales

< ~F ithermal(t) · ~Fjthermal(t

′) >ρ

= CikCjkδ(t− t′)(2)=[κT (δij − pipj

~p 2) + κL

pipj

~p 2

]δ(t− t′).

(3)

κT and κL are the transverse and longitudinal momen-tum diffusion coefficients, respectively, which describethe momentum fluctuations in the direction that perpen-dicular (i.e. transverse) and parallel (i.e. longitudinal) tothe propagation. Considering the Einstein relationship,which enforces the drag coefficient starting from the mo-mentum diffusion coefficients as [40]

ηD =κL

2TE+ (ξ − 1)

∂κL∂~p 2

+d− 1

2~p 2

[ξ(√κT +

√κL)2

− (3ξ − 1)κT − (ξ + 1)κL].

(4)

The parameter ξ denotes the discretization scheme ofthe stochastic integral, which typically takes the valuesξ = 0, 0.5, 1, representing the pre-point Ito, the mid-pointStratonovic, and the post-point discretization schemes,respectively; d = 3 indicates the spatial dimension. Inthe framework of LTE the HQ-medium interactions areconveniently encoded into three transport coefficients, i.e.ηD, κT and κL (Eq. 4). All the problems are thereforereduced to the evaluation of κT/L(E, T ), which will bemainly discussed in this work.

Finally, we introduce a few detailed setups of the nu-merical implementation. The space-time evolution of thetemperature field and the velocity field are needed tosolve LTE (Eq. 1a and 1b). Following our previous anal-ysis [27, 41], they are obtained in a 3+1 dimensional vis-cous hydrodynamic calculation [42], with the local ther-malization started at τ0 = 0.6 fm/c, the shear viscosity

η/s = 1/(4π) and the critical temperature Tc = 165 MeV(see details in Ref. [42]). When the medium temper-ature drops below Tc, heavy quark will hadronize intothe heavy-flavor hadrons via a fragmentation-coalescenceapproach. The Braaten-like fragmentation functions areemployed for both charm and bottom quarks [43, 44].An instantaneous approach is utilized to characterizethe coalescence process for the formation of heavy-flavormesons from the heavy and (anti-)light quark pairs. Therelevant coalescence probability is quantified by the over-lap integral of the Winger functions for the meson andpartons, which are defined through a harmonic oscillatorand the Gaussian wave-function [45], respectively. SeeRef. [40] for more details.

III. MOMENTUM DIFFUSION COEFFICIENTSIN A SOFT-HARD FACTORIZED APPROACH

When propagating throughout the QGP, the HQ scat-tering off the gluons and (anti-)partons of the thermal de-confined medium, can be characterized as the two-bodyelementary processes,

Q (p1) + i (p2)→ Q (p3) + i (p4), (5)

with p1 = (E1, ~p1) and p2 are the four-momentum ofthe injected HQ (Q) and the incident medium partonsi = q, g, respectively, while p3 and p4 are for the onesafter scattering. Note that the medium partons aremassless (m2 = m4 ∼ 0) in particular comparing withthe massive HQ (m1 = m3 = mQ in a few timesGeV). The corresponding four-momentum transfer is(ω, ~q ) = (ω, ~qT , qL). The Mandelstam invariants read

s ≡ (p1 + p2)2

t ≡ (p1 − p3)2 = ω2 − q2

u ≡ (p1 − p4)2(6)

with q ≡ |~q | � E1 for small momentum exchange. Thetransverse and longitudinal momentum diffusion coeffi-cients can be determined by weighting the differentialinteraction rate with the squared transverse and longitu-dinal momentum transfer, respectively. It yields

κT =1

2

∫dΓ ~q 2

T =1

2

∫dΓ[ω2 − t−

(2ωE1 − t2|~p1|

)2](7)

and

κL =

∫dΓ q2L =

1

4~p 21

∫dΓ (2ωE1 − t)2. (8)

As the momentum transfer vanishes (|t| → 0), thegluon propagator in the t-channel of the elastic processcauses an infrared divergence in the squared amplitude|M2|t−channel ∝ 1/t2, which is usually regulated by a De-bye screening mass, i.e. t→ t− λm2

D with an adjustableparameter λ [46, 47]. Alternatively, it can be overcome by

3

utilizing a soft-hard factorized approach [48, 49], whichstarts with the assumptions that the medium is thermaland weakly coupled, and then the interactions betweenthe heavy quarks and the medium can be computed inthermal perturbation theory. Finally, this approach al-lows to decompose the soft HQ-medium interactions witht > t∗, from the hard ones with t < t∗. For soft collisionsthe gluon propagator should be replaced by the hard-thermal loop (HTL) propagator [50, 51], while for hardcollisions the hard gluon exchange is considered and theBorn approximation is appropriate. Therefore, the finalresults of κT/L include the contributions from both softand hard components.

As discussed in the QED case [48], µ+ γ → µ+ γ, thecomplete calculation for the energy loss of the energicincident heavy-fermion is independent of the intermedi-ate scale t∗ in high energy limit. However, in the QCDcase, there is the complication that the challenge of thevalidity of the HTL scenario, m2

D � T 2 [49], due to thetemperatures reached at RHIC and LHC energies. Con-sequently, in the QCD case, the soft-hard approach is infact not independent of the intermediate scale t∗ [33, 47].In this analysis we have checked that the calculations forκT/L are not sensitive to the choice of the artificial cutoff

t∗ ∼ m2D.

In the next parts of this section, we will focus on theenergy and temperature dependence of the interactionrate Γ at leading order in g for the elastic process, as wellas the momentum diffusion coefficients (Eq. 7 and 8) insoft and hard collisions, respectively.

A. κT/L in soft region t∗ < t < 0

In soft collisions the exchanged four-momentum is soft,√−t ∼ gT (λmfp ∼ 1/g2T [52]), and the t-channel long-

wavelength gluons are screened by the mediums, thus,they feel the presence of the medium and require the re-summation. Here we just show the final results, and thedetails are relegated to A. The transverse and longitu-dinal momentum diffusion coefficients can be expressedas

κT (E1, T ) =CF g

2

16π2v31

∫ 0

t∗dt (−t)3/2

∫ v1

0

dxv21 − x2

(1− x2)5/2[ρL(t, x) + (v21 − x2)ρT (t, x)

]coth

( x2T

√−t

1− x2)

(9)and

κL(E1, T ) =CF g

2

8π2v31

∫ 0

t∗dt (−t)3/2

∫ v1

0

dxx2

(1− x2)5/2[ρL(t, x) + (v21 − x2)ρT (t, x)

]coth

( x2T

√−t

1− x2),

(10)respectively, with the HQ velocity v1 = |~p|/E1 and thetransverse and longitudinal parts of the HTL gluon spec-

tral functions§ are given by

ρT (t, x) =πm2

D

2x(1− x2)

{[− t+

m2D

2x2(1 +

1− x2

2xln

1 + x

1− x)]2

+[πm2

D

4x(1− x2)

]2}−1(11)

and

ρL(t, x) =πm2Dx

{[ −t1− x2

+m2D(1− x

2ln

1 + x

1− x)]2

+(πm2

D

2x)2}−1

.

(12)

B. κT/L in hard region tmin < t < t∗

In hard collisions the exchanged four-momentum ishard,

√−t & T (λmfp ∼ 1/g4T [52]), and the pQCD

Born approximation is valid in this regime. In analogywith the previous part we give the κT/L results directly,and the detailed aspects of the calculations can be foundin B. The momentum diffusion coefficients reads

κQiT (E1, T ) =1

256π3|~p1|E1

∫ ∞|~p2|min

d|~p2|E2n2(E2)∫ cosψ|max

−1d(cosψ)

∫ t∗

tmin

dt1

a

[− m2

1(D + 2b2)

8~p 21 a

4

+E1tb

2~p 21 a

2− t(1 +

t

4~p 21

)]|M2|Qi

(13)and

κQiL (E1, T ) =1

256π3|~p1|3E1

∫ ∞|~p2|min

d|~p2|E2n2(E2)∫ cosψ|max

−1d(cosψ)

∫ t∗

tmin

dt1

a

[E21(D + 2b2)

4a4

− E1tb

a2+t2

2

]|M2|Qi.

(14)The integrations limits and the short notations are shownin Eq. B8-B14.

C. Complete results in soft-hard scenario

Combining the soft and hard contributions to the mo-mentum diffusion coefficients via

κT/L(E, T ) = κsoftT/L(E, T ) + κhardT/L (E, T )

= κsoftT/L(E, T ) +∑i=q,g

κhard−QiT/L(15)

§The spectral function involves only the low frequency excitations,namely the Landau cut, while the quasiparticle excitations is irrel-evant in this regime. See Eq.A8 for more details.

4

while κsoftT/L(E ≡ E1, T ) is given by Eq. 9 and 10, and

κhard−QiT/L is expressed in Eq. 13 and 14 for a given inci-

dent medium parton i = q, g. Adopting the post-pointdiscretization scheme of the stochastic integral, i.e. ξ = 1in Eq. 4, the drag coefficient ηD(E, T ) can be obtainedby inserting Eq. 15 into Eq. 4.

IV. DATA-BASED PARAMETEROPTIMIZATION

Following the strategies utilized in our previouswork [40, 53], the two key parameters in this study, theintermediate cutoff t∗ and the scale µ of running coupling(Eq. A14a), are tested within a wide range of possibilityand drawn constrains by comparing the relevant charmmeson data with model results. We calculate the cor-responding final observable y for the desired species ofD-meson. Then, a χ2 analysis can be performed by com-paring the model predictions with experimental data

χ2 =

N∑i=1

(yDatai − yModel

i

σi

)2

. (16)

In the above σi is the total uncertainty in data points,including the statistic and systematic components whichare added in quadrature. n = N − 1 denotes the degreeof freedom (d.o.f) when there are N data points used inthe comparison. In this study, we use an extensive setof LHC data in the range pT ≤ 8 GeV: D0, D+, D∗+

and D+s RAA data collected at mid-rapidity (|y| < 0.5)

in the most central (0 − 10%) and semi-central (30 −50%) Pb–Pb collisions at

√sNN = 2.76 TeV [54, 55] and√

sNN = 5.02 TeV [56], as well as the v2 data in semi-central (30− 50%) collisions [57–59].

We scan a wide range of valus for (t∗, µ): 1 ≤ |t∗|

m2D≤ 3

and 1 ≤ µπT ≤ 3. A total of 20 different combina-

tions were computed and compared with the experimen-tal data. The obtained results are summarized in Tab. I.The χ2 values are computed separately for RAA and v2as well as for all data combined. To better visualize theresults, we also show them in Fig. 1, with left panelsfor RAA analysis and right panels for v2 analysis. In

both panels, the y-axis labels the desired parameters, |t∗|

m2D

(upper) and µπT (lower), and x-axis labels the χ2/d.o.f

within the selected ranges¶. The different points (filledgry circles) represent the different combinations of pa-

rameters ( |t∗|

m2D

, µπT ) in Tab. I, with the number on top of

each point to display the relevant “Model ID” for thatmodel. A number of observations can be drawn fromthe comprehensive model-data comparison. For the RAA,

¶χ2/d.o.f is shown in the range 0.5 < χ2/d.o.f < 2.5 for bettervisualization.

several models achieve χ2/d.o.f ' 1 with |t∗|m2

D. 1.5 and

widespread values of µ: 1 ≤ µπT ≤ 3. This suggests that

RAA appears to be more sensitive to the intermediate cut-off while insensitive to the scale of coupling constant. Forthe v2, it clearly shows a stronger sensitivity to µ, whichseems to give a better description (χ2/d.o.f ' 1.5− 2.0)of the data with µ

πT = 1. It is interesting to see that RAA

data is more powerful to constrain |t∗|m2

D, while v2 data is

more efficient to nail down µπT . Taken all together, we

can identify a particular model that outperforms oth-ers in describing both RAA and v2 data simultaneouslywith χ2/d.o.f = 1.3. This one will be the parameter-optimized model in this work: |t∗| = 1.5m2

D and µ = πT .

Model ID|t∗|/m2

D µ/πT χ2/d.o.f χ2/d.o.fTotal

(Cutoff) (Scale) (RAA) (v2)1 1.00 1.00 1.02 5.08 1.612 1.00 1.50 2.46 6.08 2.983 1.00 2.00 1.71 6.11 2.354 1.00 3.00 0.83 4.20 1.325 1.50 1.00 1.16 2.10 1.306 1.50 1.50 3.90 9.82 4.767 1.50 2.00 5.38 10.90 6.188 1.50 3.00 4.85 9.12 5.479 2.00 1.00 2.44 1.54 2.3110 2.00 1.50 2.99 7.11 3.5811 2.00 2.00 6.81 10.19 7.3012 2.00 3.00 8.91 8.19 8.8013 2.50 1.00 4.04 1.61 3.6814 2.50 1.50 2.08 6.78 2.7715 2.50 2.00 6.59 8.51 6.8716 2.50 3.00 11.80 12.04 11.8317 3.00 1.00 5.46 1.46 4.8818 3.00 1.50 1.52 8.05 2.4719 3.00 2.00 6.27 9.88 6.7920 3.00 3.00 12.94 15.21 13.27

TABLE I. Summary of the adjustable parameters in thiswork, together with the relevant χ2/d.o.f obtained for RAA

and v2.

V. RESULTS

In this section we will first examine the t∗ dependenceof κT/L (Eq. 15) for both charm and bottom quark.Then, the relevant energy and temperature dependenceof κT/L(E, T ) will be discussed with the optimized pa-rameters. For the desired observables we will perform thecomparisons with the results from lattice QCD at zeromomentum limit, as well as the ones from experimentaldata in the low to intermediate pT region.

A. Energy and temperature dependence of thetransport coefficients

In Fig. 2, charm (left) and bottom quark (right) κT(thin curves) and κL (thick curves) are calculated, with

5

0.5 1 1.5 2

D2|t*

|/m

0

1

2

3

1 234

5 6 78

9 10 11 12

1314 15 16

1718 19 20

dataAARUsing (a)

d.o.f/2χ0.5 1 1.5 2

Tπ/µ

0

1

2

3

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

(b)1.5 2

D2|t*

|/m0

1

2

3

1 234

5 6 78

9 10 1112

13 14 15 16

17 18 19 20

data2vUsing (c)

d.o.f/2χ1.5 2

Tπ/µ

0

1

2

3

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

(d)

FIG. 1. Comparision of χ2/d.o.f based on the experimen-tal data of RAA (left) and v2 (right). The model predictionsare calculated by using various combinations of parameter|t∗|/m2

D (upper) and µ/πT (lower), which are represented asχ2/d.o.f (x-axis) and the desired parameter (y-axis), respec-tively. See the legend and text for details.

m2D ≤ |t∗| ≤ 2.5m2

D and µ = πT , at the temperature T =0.40 GeV (upper) and the energy E = 10.0 GeV (lower).κT and κL are, as expected, identical at zero momentumlimit (E = mQ), while the latter one has a much strongerenergy dependence at larger momentum. Furthermore,κT/L(E, T ) behave a mild sensitivity to the intermediatecutoff t∗ [33]. Because the soft-hard approach is strictlyspeaking valid when the coupling is small m2

D � T 2 [49],thus, the above observations support the validity of thisapproach within the temperature regions even though thecoupling is not small.

In Fig. 3, charm quark κT (left) and κL (middle) areevaluated with the optimized parameters, |t∗| = 1.5m2

Dand µ = πT , including both the soft (dotted blue curves)and hard contributions (dashed black curves), at fixedtemperature T = 0.40 GeV (upper) and at fixed energyE = 10.0 GeV (lower). It is found that the soft com-ponents are significant at low energy/temperature, whilethey are compatible at larger values. The combined re-sults (solid red curves) are presented as well for compar-ison. With the post-point scheme (ξ = 1 in Eq. 15), thedrag coefficients (right) behave (1) a nonmonotonic de-pendence, in particular on the temperature, which is inpart due to the improved treatment of the screening insoft collisions, and in part due to the procedure of infer-ring ηD from κT/L [15]; (2) a weak energy dependence atE & 2mc = 3 GeV. Similar conclusions can be drawn forbottom quark, as shown in Fig. 4.

Figure 5 presents the transport coefficient of charmquark, q = 2κT , at fix momentum p = 10 GeV (solid red

curve). It can be seen that q/T 3 reaches the maximumnear the critical temperature, and then followed by adecreasing trend with T , providing a good description ofthe light quark transport parameter (black circle points).The results from various phenomenological extractionsand theoretical calculations, including a phenomenolog-ical fitting analysis with the Langevin-transport withGluon Radiation (LGR; dotted blue curve [40]), a LOcalculation with a Linearized Boltzmann Diffusion Model(LIDO‡; dashed black curve [60]), a nonperturbativetreatment with Quasi-Particle Model (QPM in Catania;dot-dashed green curve [61]), a novel confinement withsemi-quark-gluon-monopole plasma approach (CUJET3;shadowed red band [18, 20, 21]), are displayed as wellfor comparison. Similar temperature dependence can beobserved except the CUJET3 approach, which shows astrong enhancement near Tc regime.

The scaled spatial diffusion coefficient describes thelow energy interaction strength of HQ in medium [63],

2πTDs = limE→mQ

2πT 2

mQ · ηD(E, T ), (17)

and it can be calculated by substituting Eq. 4 into Eq. 17.The obtained result for charm quark (mc = 1.5 GeV) isdisplayed as the solid red curve in Fig. 6. It is found thata relatively strong increase of 2πTDs(T ) from crossovertemperature Tc toward high temperature. Meanwhile,the v2 data prefers a small value of 2πTDs near Tc,2πTDs(Tc) ' 3 − 6§, which is close to the lattice QCDcalculations [64–68]. The relevant results from other the-oretical analyses, such as LGR [40], LIDO [60], Cata-nia [61] and CUJET3¶ [18, 20, 21], show a similar trendbut with much weaker temperature dependence.

B. Comparison with experimental data: RAA and v2

Figure 7 shows the RAA of D0 (a), D+ (b), D∗+ (c)and D+

s (d) in the most central (0 − 10%) Pb–Pb col-lisions at

√sNN = 2.76 TeV, respectively. The calcu-

lations are done with FONLL initial charm quark spec-tra and EPS09 NLO parametrization for the nPDF inPb [27], and the pink band reflects the theoretical uncer-tainties coming from these inputs. It can be seen that,within the experimental uncertainties, the model calcu-lations provide a very good description of the measuredpT-dependent RAA data for various charm mesons. Con-cerning the results in Pb–Pb collisions at

√sNN = 5.02

TeV, as shown in Fig. 8, a good agreement is found be-tween the model and the measurement at pT . 6 GeV/c,while a slightly larger discrepancy observed at larger pT.

‡LIDO results are shown with only elastic scattering channels.§This range is estimated from the testing models as displayed in theright panels of Fig. 1.

¶CUJET3 results are obtained by performing the energy interpola-tion down to E = mQ.

6

E (GeV)5 10 15 20

/fm)

2 (

GeV

κ

0

2

4

6

2

D: |t*|=1.0 mTκ

2

D: |t*|=1.5 mTκ

2

D: |t*|=2.0 mTκ

2

D: |t*|=2.5 mTκ

2

D: |t*|=1.0 mLκ

2

D: |t*|=1.5 mLκ

2

D: |t*|=2.0 mLκ

2

D: |t*|=2.5 mLκ

Charm: pQCD+HTL

Tπ = 1.0 µ

T = 0.40 GeV

(a)

E (GeV)5 10 15 20

/fm)

2 (

GeV

κ

0

1

2

3

4 2

D: |t*|=1.0 mTκ

2

D: |t*|=1.5 mTκ

2

D: |t*|=2.0 mTκ

2

D: |t*|=2.5 mTκ

2

D: |t*|=1.0 mLκ

2

D: |t*|=1.5 mLκ

2

D: |t*|=2.0 mLκ

2

D: |t*|=2.5 mLκ

Bottom: pQCD+HTL

Tπ = 1.0 µ

T = 0.40 GeV

(c)

T (GeV)0.2 0.3 0.4 0.5 0.6

/fm)

2 (

GeV

κ

0

2

4

6

2

D: |t*|=1.0 mTκ

2

D: |t*|=1.5 mTκ

2

D: |t*|=2.0 mTκ

2

D: |t*|=2.5 mTκ

2

D: |t*|=1.0 mLκ

2

D: |t*|=1.5 mLκ

2

D: |t*|=2.0 mLκ

2

D: |t*|=2.5 mLκ

Charm: pQCD+HTL

Tπ = 1.0 µ

E = 10.0 GeV

(b)

T (GeV)0.2 0.3 0.4 0.5 0.6

/fm)

2 (

GeV

κ

0

1

2

3

4 2

D: |t*|=1.0 mTκ

2

D: |t*|=1.5 mTκ

2

D: |t*|=2.0 mTκ

2

D: |t*|=2.5 mTκ

2

D: |t*|=1.0 mLκ

2

D: |t*|=1.5 mLκ

2

D: |t*|=2.0 mLκ

2

D: |t*|=2.5 mLκ

Bottom: pQCD+HTL

Tπ = 1.0 µ

E = 10.0 GeV

(d)

FIG. 2. (Color online) Comparison of the momentum diffusion coefficients κT (thin black curves) and κL (thick red curves), forcharm (left) and bottom quarks (right), displaying separately the results based various testing parameters: m2

D ≤ |t∗| ≤ 2.5m2D

and µ = πT .

E (GeV)5 10 15 20

/fm)

2 (

GeV

Tκ

0

0.5

1

1.5

pQCD

HTL

pQCD+HTL

Charm2

D|t*| = 1.5 m

Tπ = 1.0 µ

T = 0.40 GeV

(a)

E (GeV)5 10 15 20

/fm)

2 (

GeV

Lκ

0

2

4

6pQCD

HTL

pQCD+HTL

Charm2

D|t*| = 1.5 m

Tπ = 1.0 µ

T = 0.40 GeV

(c)

E (GeV)5 10 15 20

)-1

(fm

Dη

0

0.2

0.4

0.6pQCD

HTL

pQCD+HTL

Charm2

D|t*| = 1.5 m

Tπ = 1.0 µ

T = 0.40 GeV

(e)

T (GeV)0.2 0.3 0.4 0.5 0.6

/fm)

2 (

GeV

Tκ

0

0.5

1

1.5

pQCD

HTL

pQCD+HTL

Charm2

D|t*| = 1.5 m

Tπ = 1.0 µ

E = 10.0 GeV

(b)

T (GeV)0.2 0.3 0.4 0.5 0.6

/fm)

2 (

GeV

Lκ

0

2

4

6pQCD

HTL

pQCD+HTL

Charm2

D|t*| = 1.5 m

Tπ = 1.0 µ

E = 10.0 GeV

(d)

T (GeV)0.2 0.3 0.4 0.5 0.6

)-1

(fm

Dη

0

0.2

0.4

0.6pQCD

HTL

pQCD+HTL

Charm2

D|t*| = 1.5 m

Tπ = 1.0 µ

E = 10.0 GeV

(f)

FIG. 3. (Color online) Charm quark κT (left) and κL (middle) are shown at fixed temperature T = 0.40 GeV (upper) andat fixed energy E = 10.0 GeV (lower), contributed by the soft (dotted blue curves) and hard collisions (dashed black curves).The combined results (solid red curves) are shown as well for comparison. The derived drag coefficient ηD (right; Eq. 4) areobtained with the post-point scenario.

Figure 9 presents the elliptic flow coefficient v2 of non-strange D-meson (averaged D0, D+, and D∗+) in semi-central (30−50%) Pb–Pb collisions at

√sNN = 2.76 TeV

(a) and√sNN = 5.02 TeV (b). Within the uncertainties

of the experimental data, our model calculations describewell the anisotropy of the transverse momentum distribu-tion of the non-strange D-meson. The sizable v2 of thesecharm mesons, in particular at intermediate pT ' 3 − 5

GeV, suggests that charm quarks actively participate inthe collective expansion of the fireball.

In Fig. 10, the pT-differential RAA of non-promptD0 mesons are predicted with the parameter-optimizedmodel, and shown as a function of pT in central (0−10%)Pb–Pb collisions at

√sNN = 5.02 TeV. Comparing with

prompt D0 RAA (solid curve), at moderate pT (pT ∼5 − 7), a less suppression behavior is found for D0 from

7

E (GeV)5 10 15 20

/fm)

2 (

GeV

Tκ

0

0.5

1

pQCD

HTL

pQCD+HTL

Bottom2

D|t*| = 1.5 m

Tπ = 1.0 µ

T = 0.40 GeV

(a)

E (GeV)5 10 15 20

/fm)

2 (

GeV

Lκ

0

1

2

3

pQCD

HTL

pQCD+HTL

Bottom2

D|t*| = 1.5 m

Tπ = 1.0 µ

T = 0.40 GeV

(c)

E (GeV)5 10 15 20

)-1

(fm

Dη

0

0.1

0.2

pQCD

HTL

pQCD+HTL

Bottom2

D|t*| = 1.5 m

Tπ = 1.0 µ

T = 0.40 GeV

(e)

T (GeV)0.2 0.3 0.4 0.5 0.6

/fm)

2 (

GeV

Tκ

0

0.5

1

pQCD

HTL

pQCD+HTL

Bottom2

D|t*| = 1.5 m

Tπ = 1.0 µ

E = 10.0 GeV

(b)

T (GeV)0.2 0.3 0.4 0.5 0.6

/fm)

2 (

GeV

Lκ

0

1

2

3

pQCD

HTL

pQCD+HTL

Bottom2

D|t*| = 1.5 m

Tπ = 1.0 µ

E = 10.0 GeV

(d)

T (GeV)0.2 0.3 0.4 0.5 0.6

)-1

(fm

Dη

0

0.1

0.2

pQCD

HTL

pQCD+HTL

Bottom2

D|t*| = 1.5 m

Tπ = 1.0 µ

E = 10.0 GeV

(f)

FIG. 4. Same as Fig. 3 but for bottom quark.

cT/T1 1.5 2 2.5 3

)c (

p=10

GeV

/3

/ T

q

0

10

20

30HTL+pQCD

LGR

LIDO

Catania

CUJET3

JET Collaboration

Charm

FIG. 5. (Color online) Transport coefficient, q/T 3(T ),of charm quark from the various calculations, including:the soft-hard factorized approach (solid red curve), theLGR model with data optimized parameters (dotted bluecurve [40]), a Bayesian ayalysis from LIDO (dashed blackcurve [60]), a quasi-particle model from Catania (dot-dashedgreen curve [62]), CUJET3 (shadowed red band [18, 20, 21])and JET Collaboration (black circle points [19]) at p = 10GeV.

B-hadron decays (dashed curve), reflecting a weaker in-medium energy loss effect of bottom quark (mb = 4.75GeV), which has larger mass with respect to that ofcharm quark (mc = 1.5 GeV). We note that the futuremeasurements performed for the non-prompt D0 RAA

and v2, are powerful in nailing down the varying rangesof the model parameters (see Sec. IV), which will largelyimprove and extend the current understanding of the in-medium effects.

cT/T1 2 3

sT

Dπ2

0

20

40

Charm, HTL+pQCD

Charm, LGR

Charm, LIDOCharm, Catania

Charm, CUJET3

D-meson

Banerjee 2011

Ding 2012

Kaczmarek 2014Brambilla 2020

Altenkort 2020

FIG. 6. (Color online) Spatial diffusion constant 2πTDs(T )of charm quark from the sof-hard scenario (solid red curve),LGR (dotted blue curve [40], LIDO (dashed black curve [60]),Catania (dot-dashed green curve [61]), CUJET3 (shadowedred band [18, 20, 21]) and lattice QCD calculations (blackcircle [64], blue triangle [65], pink square [66], red invertedtriangle [67] and green plus [68]). The result for D-meson(long dashed pink curve [69]) in the hadronic phase is shownfor comparison.

VI. SUMMARY

In this work we have used a soft-hard factorized modelto investigate the heavy quark momentum diffusion coef-ficients κT/L in the quark-gluon plasma in a data-drivenapproach. In particular we’ve examined the validity ofthis scenario by systematically scanning a wide range ofpossibilities. The global χ2 analysis using an extensiveset of LHC data on charm meson RAA and v2 has allowedus to constrain the preferred range of the two parame-ters: soft-hard intermediate cutoff |t∗| = 1.5m2

D and thescale of QCD coupling constant µ = πT . It is foundthat κT/L have a mild sensitivity to t∗, supporting the

8

2 4 6 8

AA

R

0

0.5

1

1.5 Pb-Pb @2.76 TeV, |y|<0.50-10%

0D

(a)

2 4 6 80

0.5

1

1.5 *+D

(c)

2 4 6 80

0.5

1

1.5 Data

Model

+D

(b)

)c (GeV/T

p2 4 6 8

0

0.5

1

1.5s+D

(d)

FIG. 7. (Color online) Comparison between experimentaldata (red box [54, 55]) and soft-hard factorized model cal-culations (solid black curve with pink uncertainty band) forthe nuclear modification factor RAA, of D0 (a), D+ (b), D∗+

(c) and D+s (d) at mid-rapidity (|y| < 0.5) in central (0−10%)

Pb–Pb collisions at√sNN = 2.76 TeV.

2 4 6 8

AA

R

0

0.5

1

1.5 Pb-Pb @5.02 TeV, |y|<0.50-10%

0D

(a)

2 4 6 80

0.5

1

1.5 *+D

(c)

2 4 6 80

0.5

1

1.5 Data: JHEP10(2018)174

Data: 1910.01981Model

+D

(b)

)c (GeV/T

p2 4 6 8

0

0.5

1

1.5s+D

(d)

FIG. 8. Same as Fig. 7 but for Pb–Pb collisions at√sNN =

5.02 TeV. The data (solid [56], open [70]) are shown for com-parison.

validity of the soft-hard approach when the coupling isnot small. With this factorization model, we have cal-culated the transport coefficient q, drag coefficient ηD,spatial diffusion coefficient 2πTDs, and then comparedwith other theoretical calculations and phenomenologi-cal extractions. Our analysis suggests that a small value2πTDs ' 6 appears to be much preferred near Tc. Fi-nally we’ve demonstrated a simultaneous description ofcharm meson RAA and v2 observables in the range pT ≤ 8GeV. We’ve further made predictions for bottom mesonobservables in the same model.

We end with discussions on a few important caveats inthe present study that call for future studies:

)c (GeV/T

p1 2 3 4 5 6 7 8

2v

0

0.1

0.2

0.3

0.4Data

Model

Pb-Pb @2.76 TeV, |y|<1.0, 30-50%, Average (a)

)c (GeV/T

p1 2 3 4 5 6 7 8

2v

0

0.1

0.2

0.3

0.4ALICE Data: Average

0CMS Data: D

Model

Pb-Pb @5.02 TeV, |y|<1.0, 30-50% (b)

FIG. 9. (Color online) Comparison between experimentaldata (red [57], black [58] and blue boxes [71]) and model cal-culations (solid black curve with pink uncertainty band) forthe elliptic flow v2 of non-strange D-meson at mid-rapidity(|y| < 0.5) in semi-central (30 − 50%) Pb–Pb collisions at√sNN = 2.76 TeV (a) and

√sNN = 5.02 TeV (b).

)c (GeV/T

p2 4 6 8

AA

R

0

0.5

1

1.5

Model Data

Model

0Pb-Pb @5.02 TeV, 0-10%, |y|<0.5, D

non-prompt prompt

FIG. 10. Comparison of non-prompt (dashed blue curve) andprompt D0 (solid black curve) RAA calculations as a functionof pT with the measured values (point [56]), in the central0 − 10% Pb–Pb collisions at

√sNN = 5.02 TeV. See legend

and text for details.

• In the framework of Langevin approach, the heavyflavor dynamics is encoded into three coefficient,κT , κL and ηD, satisfying Einstein’s relationship. Itmeans that two of them are independent while thethird one can be obtained accordingly. The finalresults therefore depend on the arbitrary choice ofwhich of the two coefficients are calculated with theemployed model [15, 38]. For consistency, we cal-

9

culate independently the momentum diffusion coef-ficients κT/L with the factorization approach, andobtain the drag coefficient via ηD = ηD(κT , κL)(see Eq. 4). A systematic study among the differ-ent options may help remedy this situation.

• According to the present modeling, the result-ing spatial diffusion coefficient exhibits a relativelystrong increase of temperature comparing with lat-tice QCD calculations, in particular at large T , asshown in Tab. II. It corresponds to a larger relax-ation time and thus weaker HQ-medium couplingstrength. This comparison can be improved by (1)determining the key parameters based on the up-coming measurements on high precision observables(such as RAA, v2 and v3) of both D and B-mesonsat low pT; (2) replacing the current χ2 analysis witha state-of-the-art deep learning technology.

T = Tc T = 2Tc T = 3Tc

This work ∼6 ∼22 ∼31LQCD (median value) [64, 67] ∼5 ∼10 ∼13

TABLE II. Summary of the different models for 2πTDs atdesired temperature values.

• It is realized [72] that the heavy quark hadro-chemistry, the abundance of various heavy flavorhadrons, provides special sensitivity to the heavy-light coalescence mechanism and thus plays an im-portant role to understand the observables like thebaryon production and the baryon-to-meson ratio.A systematic comparison including the charmedbaryons over a broad momentum region is there-fore crucial for a better constraining of the modelparameters, as well as a better extraction of theheavy quark transport coefficients in a model-to-data approach.

• The elastic scattering (2 → 2) processes betweenheavy quark and QGP constituents are dominatedfor heavy quark with low to moderate transversemomentum [41]. Thus, here we consider only theelastic energy loss mechanisms to study the ob-servables at pT ≤ 8 GeV. The missing radiative(2↔ 3) effects may help to reduce the discrepancywith RAA data in the vicinity of pT = 8 GeV, asmentioned above (see Fig. 8). With the soft-hardfactorized approach, it would be interesting to in-clude both elastic and radiative contributions in asimultaneous best fit to data in the whole pT region.We also plan to explore this idea in the future.

ACKNOWLEDGMENTS

The authors are grateful to Andrea Beraudo and Jin-feng Liao for helpful discussions and communications.We also thank Weiyao Ke and Gabriele Coci for providing

the inputs as shown in Fig. 5 and 6. This work is sup-ported by the Hubei Provincial Natural Science Founda-tion under Grant No.2020CFB163, the National ScienceFoundation of China (NSFC) under Grant Nos.12005114,11847014 and 11875178, and the Key Laboratory ofQuark and Lepton Physics Contracts Nos.QLPL2018P01and QLPL201905.

Appendix A: Derivation of the interaction rate andmomentum diffusion coefficients in soft collisions

In the QED case, µ + γ → µ + γ, one can calculaterelevant interaction rate with small momentum trans-fer by using the imaginary part of the muon self-energyΣ(p1) [73]

Γ(E1, T ) = − 1

2E1nF (E1)Tr

[(p/1 +m1)ImΣ(p1)

].

(A1)where, p1 = (E1, ~p1) and m1 are the four-momentum andmass of the injected muon, respectively §. The trace termin Eq. A1 was calculated with a resummed photon prop-agator, which is very similar with the one in QCD ¶. It isrealized [74, 75] that, in the QCD case, the contributionsto Γ can be obtained from the corresponding QED cal-culations by simply substitution: e2 → CF g

2, where, e(g) is the QED (QCD) coupling constant and CF = 4/3is the quark Casimir factor. It yields [74]

Tr[(p/1 +m1)ImΣ(p1)

]= −2CF g

2(1 + e−E1/T )

∫q

∫dω nB(ω)

E21

E3AB,

(A2)

with

A ≡ (1− ω + ~v1 · ~q2E1

)ρL(ω, q)+[~v 21 (1− (v1 · q)2)

− ω − ~v1 · ~qE1

]ρT (ω, q)

(A3)

B ≡ nF (E3)δ(E1 − E3 − ω)− nF (E3)δ(E1 + E3 − ω)(A4)

where, ~v1 = ~p1/E1 denotes the HQ velocity; nB/F (E) =

(eE/T ∓ 1)−1 indicates the thermal distributions forBosons/Fermions and nB/F ≡ 1 ± nB/F accounts forthe Bose-enhancement or Pauli-blocking effect. Here we

have used the short notation∫q≡∫

d3~q(2π)3 for phase

space integrals. Taking |~p1| and m1/3 = mQ are bothmuch greater than the underlying medium temperature,

§The notations for the injected muon are same with the ones for theinjected HQ in the elastic process.

¶The structure of the QED HTL-propagator follows Eq. A10 butwith opposite signs [48].

10

i.e. |~p1|,m1/3 � T , thus, E1/3 � T and nF (E3) is ex-ponentially suppressed and can be dropped. Moreover,the first δ funciton in Eq. A4 can be simplified sinceE3 =

√(~p1 − ~q)2 +m2

1 ≈ E1 − ~v1 · ~q. Concerning thesecond δ funciton, it cannot contribute for ω less than oron the order of T [74], which will be deleted in this work.Finally, Eq. A3 and A4 can be reduced to

A ≡ ρL(ω, q) + ~v 21

[1− (v1 · q)2

]ρT (ω, q) (A5a)

B ≡ δ(ω − ~v1 · ~q). (A5b)

By substituting Eq. A5a and A5b into Eq. A2, one gets

Tr[(p/1 +m1)ImΣ(p1)

]= −2CF g

2E1

∫q

∫dω nB(ω)δ(ω − ~v1 · ~q ){

ρL(ω, q) + ~v 21

[1− (v1 · q)2

]ρT (ω, q)

}.

(A6)

Equation A1 can be rewritten as

Γ(E1, T )

= CF g2

∫q

∫dω nB(ω)δ(ω − ~v1 · ~q ){

ρL(ω, q) + ~v 21

[1− (v1 · q)2

]ρT (ω, q)

} (A7)

with the transverse and longitudinal spectral functionsare given by the imaginary part of the retarded propaga-tor

ρT/L(ω, q) ≡ 2 · ImDRT/L(ω, q). (A8)

We note that, in the weak coupling limit, a consistentmethod is to use the HTL resummed propagators, whichis contributed by the quasiparticle poles and the Landaudamping cuts [51]. In this analysis, we mainly focus onthe low frequency excitation (|ω| < q), where the Landaudamping is dominant and the quasiparticle excitation isirrelevant. The resulting spectral function is thereforedenoted by ρ in Eq. A8.

The retarded propagator in Eq.A8 reads

DRT/L(ω, q) ≡ ∆T/L(ω + iη, q), (A9)

which is defined by setting q0 = ω + iη (η → 0+),i.e. the real energy, for the dressed gluon propagator∆T/L(q0, q) [51]

∆T (q0, q) =−1

(q0)2 − q2 −ΠT (x)

∆L(q0, q) =−1

q2 + ΠL(x).

(A10)

The medium effects are embedded in the HTL gluon self-energy

ΠT (x) =m2D

2

[x2 + (1− x2)Q(x)

]ΠL(x) = m2

D

[1−Q(x)

] (A11)

where, x = q0/q; Q(x) is the Legendre polynomial ofsecond kind

Q(x) =x

2lnx+ 1

x− 1(A12)

and m2D is the Debye screening mass squared for gluon

m2D = g2T 2

(1 +

Nf6

). (A13)

The coupling constant, g, is quantified by the two-loopQCD beta-function [76]

g−2(µ) = 2β0ln(µ

ΛQCD) +

β1β0ln[2ln(

µ

ΛQCD)]

(A14a)

β0 =1

16π2(11− 2

3Nf ) (A14b)

β1 =1

(16π2)2(102− 38

3Nf ) (A14c)

where, πT ≤ µ ≤ 3πT and ΛQCD = 261 MeV. Nf is thenumber of active flavors in the QGP. Finally, for space-like momentum, Eq. A8 can be expressed as

ρT (ω, q) =πωm2

D

2q3(q2 − ω2)

{[q2 − ω2

+ω2m2

D

2q2(1 +

q2 − ω2

2ωqlnq + ω

q − ω)]2

+[πωm2

D

4q3(q2 − ω2)

]2}−1(A15)

ρL(ω, q) =πωm2

D

q

{[q2 +m2

D(1− ω

2qlnq + ω

q − ω)]2

+(πωm2

D

2q

)2}−1,

(A16)

with which Eq. A7 is computable.The momentum diffusion coefficients can be calculated

by subsituting Eqs. A7, A15 and A16 back into Eq. 7 and8, respectively, and then performing the angular integral,yielding¶

κT (E1, T ) =CF g

2

8π2v1

∫ qmax

0

dq q3∫ v1q

0

dω (1− ω2

v21q2

)

[ρL(ω, q) + (v21 −

ω2

q2)ρT (ω, q)

]coth

ω

2T(A17)

and

κL(E1, T ) =CF g

2

4π2v1

∫ qmax

0

dq q

∫ v1q

0

dωω2

v21[ρL(ω, q) + (v21 −

ω2

q2)ρT (ω, q)

]coth

ω

2T(A18)

¶Note that the spectral functions (Eq. A15 and A16) are odd,and the resulting ρT/L(−ω, q) = −ρT/L(ω, q) are used to obtainEq. A17 and A18.

11

with v1 ≡ |~v1|. The maximum momentum exchange isqmax =

√4E1T in the high-energy limit [77].

Next, we implement the further calculations by per-forming a simple change of variables,

t = ω2 − q2 < 0 q2 =−t

1− x2

x =ω

q< v1 ω = x

√−t

1− x2,

(A19)

resulting in

dtdx =

∣∣∣∣ ∂(t, x)

∂(q, ω)

∣∣∣∣dqdω = 2(1− x2)dqdω. (A20)

Using Eq. A19 and A20, one arrives at Eq. 9-12. Similarresults can be found in Ref. [33, 78].

Appendix B: Derivation of interaction rate andmomentum diffusion coefficients in hard collisions

For two-boday scatterings, the transition rate is de-fined as the rate of collisions with medium parton i, whichchanges the momentum of the HQ (parton i) from ~p1 (~p2)to ~p3 = ~p1 − ~q (~p4 = ~p2 + ~q),

ωQi(~p1, ~q, T ) =

∫p2

n(E2)n(E3)n(E4)vreldσQi(~p1, ~p2 → ~p3, ~p4).

(B1)Typically, one can assume n(E3) = 1 by neglecting thethermal effects on the HQ after scattering. The differ-ential cross section summed over the spin/polarizationand color of the final partons and averaged over those ofincident partons,

vreldσQi(~p1, ~p2 → ~p3, ~p4)

=1

2E1

1

2E2

d3~p3(2π)32E3

d3~p4(2π)32E4

|M2|Qi(2π)4

δ(4)(p1 + p2 − p3 − p4)

(B2)

where, vrel = (√

(p1 · p2)2 − (m1m2)2)/(E1E2) is the rel-ative velocity between the projectile HQ and the targetparton. The interaction rate for a given elastic processreads

ΓQi(E1, T ) =

∫d3~q ωQi(~p1, ~q, T )

(B1,B2)=

1

2E1

∫p2

n(E2)

2E2

∫p3

1

2E3

∫p4

n(E4)

2E4

|M2|Qi(2π)4δ(4)(p1 + p2 − p3 − p4)(B3)

The momentum diffusion coefficients can be obtained byinserting Eq. B3 into Eq. 7 and 8, yielding

κQiT (E1, T )

=1

2E1

∫p2

n(E2)

2E2

∫p3

1

2E3

∫p4

n(E4)

2E4

~q 2T

2

θ(|t| − |t∗|)|M2|Qi(2π)4δ(4)(p1 + p2 − p3 − p4)

=1

256π4|~p1|E1

∫ ∞|~p2|min

d|~p2|E2n(E2)

∫ cosψ|max

−1d(cosψ)∫ t∗

tmin

dt

∫ ωmax

ωmin

dωn(ω + E2)√

G(ω)

[ω2 − t− (2E1ω − t)2

4~p21

]|M2|Qi

(B4)and

κQiL (E1, T )

=1

2E1

∫p2

n(~p2)

2E2

∫p3

1

2E3

∫p4

n(~p4)

2E4

(2E1ω − t2|~p1|

)2θ(|t| − |t∗|)|M2|Qi(2π)4δ(4)(p1 + p2 − p3 − p4)

=1

512π4|~p1|3E1

∫ ∞|~p2|min

d|~p2|E2n(E2)

∫ cosψ|max

−1d(cosψ)∫ t∗

tmin

dt

∫ ωmax

ωmin

dωn(ω + E2)√

G(ω)(2E1ω − t)2|M2|Qi

(B5)Note that,

(1) for hard collisions the momentum exchange is con-strained by imposing θ(|t|−|t∗|) in the first equalityof Eq. B4 and B5;

(2) the tree level matrix elements squared includes thecontributions from the various channels, which aregiven in Ref. [79]:

• for Q+ q → Q+ q (t-channel only)

|M2|Qq(s, t)

= 2Nf · 2Nc · g44

9

(m21 − u)2 + (s−m2

1)2 + 2m21t

t2

(B6)

• for Q + g → Q + g (t, s and u-channel com-bined)

|M2|Qg(s, t)

= 2(N2c − 1)g4

[2

(s−m21)(m2

1 − u)

t2

+4

9

(s−m21)(m2

1 − u) + 2m21(s+m2

1)

(s−m21)2

+4

9

(s−m21)(m2

1 − u) + 2m21(m2

1 + u)

(m21 − u)2

+1

9

m21(4m2

1 − t)(s−m2

1)(m21 − u)

+(s−m2

1)(m21 − u) +m2

1(s− u)

t(s−m21)

− (s−m21)(m2

1 − u)−m21(s− u)

t(m21 − u)

](B7)

12

Concerning the degeneracy factors, in Eq. B6, 2Nfreflects the identical contribution from all lightquark and anti-quark flavors, and 2Nc indicates thesumming, rather than averaging, over the helici-ties and colors of the incident light quark, whilein Eq. B7, the factor 2(N2

c − 1) denotes the sum-ming over the polarization and colors of the inci-dent gluon. The running coupling constant takesg(µ)2 = 4παs(µ), which is given by Eq. A14a withthe scale µ =

√−t.

(3) with the help of δ-function, we can reduce the in-tegral in Eq. B4 and B5 from 9-dimension (9D)to 4D in the numerical calculations, by transform-ing the integration variables from (~p2, ~p3, ~p4) to(|~p2|, cosψ, t), where ψ is the polar angle of ~p2; ityields the resutls as shown in the second equalityof Eq. B4 and B5; the relevant limits of integrationtogether with the additional notations are summa-rized below:

|~p2|min =|t∗|+

√(t∗)2 + 4m2

1|t∗|4(E1 + |~p1|)

(B8)

cosψ|max = min

{1,E1

|~p1|− |t

∗|+√

(t∗)2 + 4m21|t∗|

4|~p1| · |~p2|

}(B9)

tmin = − (s−m21)2

s(B10)

ωmax/min =b±√D

2a2with (B11)

a =s−m2

1

|~p1|(B12)

b = − 2t

~p 21

[E1(s−m2

1)− E2(s+m21)]

(B13)

c = − t

~p 21

{t[(E1 + E2)2 − s

]+ 4~p 2

1 ~p22 sin

2ψ

}(B14)

D = b2 + 4a2c = −t[ts+ (s−m2

1)2]·(

4E2sinψ

|~p1|

)2

(B15)

G(ω) = −a2ω + bω + c (B16)

(4) then, one can follow the procedure of Ref. [48] forthe analytical evaluation of ω integral. The ob-tained results for κT and κL are shown in Eq. 13and 14, respectively.

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