impact of the energy loss spatial profile and shear viscosity to entropy density ratio for the mach...

8/16/2019 Impact of the energy loss spatial profile and shear viscosity to entropy density ratio for the Mach cone vs. head s…

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Impact of the energy loss spatial profile and shear viscosity to entropy density ratio

for the Mach cone vs. head shock signals produced by a fast moving parton in a

quark-gluon plasma

Alejandro Ayala1,4, Jorge David Castaño-Yepes1, Isabel Dominguez2 and Maria Elena Tejeda-Yeomans3,11Instituto de Ciencias Nucleares, Universidad Nacional Aut´ onoma de México,

Apartado Postal 70-543, México Distrito Federal 04510, Mexico.2Facultad de Ciencias F́ısico-Matem´ aticas, Universidad Aut´ onoma de Sinaloa,

Avenida de las Américas y Boulevard Universitarios,Ciudad Universitaria, C.P. 80000, Culiac´ an, Sinaloa, México.

3Departamento de F́ısica, Universidad de Sonora, Boulevard Luis Encinas J. y Rosales,Colonia Centro, Hermosillo, Sonora 83000, Mexico.

4Centre for Theoretical and Mathematical Physics, and Department of Physics,University of Cape Town, Rondebosch 7700, South Africa

We compute the energy and momentum deposited by a fast moving parton in a quark-gluonplasma using linear viscous hydrodynamics with an energy loss per unit length profile proportionalto the path length and with different values of the shear viscosity to entropy density ratio. We showthat when varying these parameters, the transverse modes dominate over the longitudinal ones andthus energy and momentum is preferentially deposited along the head-shock, as in the case of aconstant energy loss per unit length profile and the lowest value for the shear viscosity to entropydensity ratio.

PACS numbers: 25.75.-q, 25.75.Gz, 12.38. Bx

I. INTRODUCTION

Experiments where heavy nuclei are collided at highenergies at the BNL Relativistic Heavy-Ion Collider [1]and the CERN Large Hadron Collider [2] show that inthese reactions the so-called Quark Gluon Plasma (QGP)is formed. The dynamics of the bulk matter can be ac-curately described using viscous hydrodynamics [3]. Oneway to study the properties of the QGP is to considerhow hard scattered partons transfer energy and momen-

tum to this medium.In a given event the particle with the largest mo-

mentum defines the near-side, and the opposite side iscalled the away-side. In practice, one considers thatthe hard scattering happens near the fireball’s surfacein such a way that the away-side patrons deposit en-ergy and momentum to the medium, whereas the near-side ones fragment in vacuum, giving rise to the so-called

jet-quenching [4, 5]. A possible way to characterize themedium is to study azimuthal particle correlations.

These correlations show some interesting features:When the leading and the away-side particles have simi-lar momenta, the correlation shows a suppression of theaway-side peak, compared to proton collisions at thesame energies. However, when the momentum differencebetween leading and away-side particles increases, eithera double peak or a broadening of the away-side peak ap-pears. Neither of these features are present in protoncollisions at the same energies [6].

Explanations based on the emission of sound modescaused by one fast moving parton [7–9], the so-calledMach cones are nowadays considered incomplete, sincethe jet-medium interaction produces also a wake whosecontribution cannot be ignored [10, 11]. Moreover it was

recently shown that it is unlikely that the propagation of a single high-energy particle through the medium leads toa double-peak structure in the azimuthal correlation in asystem of the size and finite viscosity relevant for heavy-ion collisions, since the energy momentum deposition inthe head shock region is strongly forward peaked [12].In addition, the overlapping perturbations in very differ-ent spatial directions wipe out any distinct Mach conestructure, according to the findings of Ref. [13, 14].

Currently, the origin of the double peak/broadening is

described in terms of initial state fluctuations of the mat-ter density in the colliding nuclei. Nevertheless there isalso evidence of a strong connection between the observedaway-side structures and the medium’s path length, ex-pressed through the dependence of the azimuthal corre-lation on the trigger particle direction with respect to theevent plane as measured in away-side correlation studiesperformed by the STAR Collaboration [15]. This connec-tion is made by observing that for selected trigger andassociated particle momenta, the double peak is present(absent) for out-of-plane (in-plane) trigger particle direc-tion. A final-state effect rather than an initial state one,seems more consistent with this observation [16].

The energy momentum transferred by the fast travel-ing parton to the medium can be described in terms of linearized viscous hydrodynamics [10, 17–19]. An impor-tant ingredient for this description is the energy loss perunit length dE/dx which enters as the coefficient describ-ing a local hydrodynamic source term. It is known thatthis parameter exhibits a non-trivial dependence on thetraveled path length L. For instance, depending on theinterplay between the evolving density of the mediumduring the collision, the medium’s size and formationlength and the dominating energy loss mechanism (ra-


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diative or collisional), dE/dx could be either constant orproportional to L [20–23].

In a previous work [24], we have explored the conse-quences drawn from assuming that dE/dx is constantand that the shear viscosity to entropy density ratio takeson its lower theoretical value, showing that under suchscenario the Mach cone signal is weaker as compared tothe wake or head-shock. Moreover, we also showed that

under such conditions, the double peak/broadening inazimuthal angular correlations can be better describedby two instead of one parton depositing energy and mo-mentum into the medium. In this work we set out toexplore the consequences of a linear dependence on L of the energy loss per unit length and different values of theshear viscosity to entropy ratio, using the same frame-work. The paper is organized as follows: In Sec. II, weobtain the expression for the local hydrodynamic sourcein Fourier space and, from the solution to the linear vis-cous hydrodynamic equations, we obtain the energy andmomentum deposited by the source into the medium. InSec. III we study different allowed values for the model

parameters, in particular different traveled paths and dif-ferent shear viscosity to entropy density ratios. We showthat the energy and momentum is still preferentially de-posited along the head-shock, as in the case of a constantenergy loss per unit length profile and the lowest value forthe shear viscosity to entropy density ratio. We finallysummarize and conclude in Sec. IV.

II. HYDRODYNAMICAL DESCRIPTION OF

ENERGY LOSS

To describe the interaction between a fast moving par-

ton and the medium, one can resort to linearized vis-cous hydrodynamics. In such a description, the source of energy-momentum is provided by the current producedby the fast moving parton given by

J ν (x, t) =

dE

dx

vν δ 3(x− vt), (1)

where vν is the particle’s four-velocity and dE/dx is theenergy loss per unit length. The current is proportionalto the instantaneous location of the particle which ismodeled by the three dimensional delta function.

We assume that the disturbance induced by the fastmoving parton is small such that the energy-momentumtensor can be written as

Θµν = Θµν 0 + δ Θµν (2)

where δ Θµν is the disturbance generated by the partonand Θµν 0 is the equilibrium energy-momentum tensor of the underlying medium. The tensor’s components satisfy

∂ µδ Θµν = J ν

∂ µΘµν 0 = 0, (3)

where J ν is given in Eq. (1). Equations (3) are solved byconsidering that Θµν consists of a term that describes anisotropic fluid

Θµν 0 = − pgµν + (ǫ + p)uµ0uν 0 , (4)and the disturbance δ Θµν that, to first order in the shearviscosity density η [18] and ignoring bulk viscosity, hasexplicit components given by

δ Θ00 = δǫ,

δ Θ0i = g,

δ Θij = δ ijc2sδǫ −

3

4Γs(∂

igj + ∂ jgi − 23

δ ij∇ · g). (5)

Here we have defined ǫ(t,x) = ǫ0 + δǫ(t,x), with ǫ0 theenergy density of the background fluid and, δǫ and gthe energy and momentum densities associated to thedisturbance, respectively. The vector g is related to thespatial part of the medium’s four-velocity,

u = g

ǫ0(1 + c2s), (6)

where cs is the sound velocity and

Γs ≡ 43

η

ǫ0(1 + c2s) =

4

3

η

s0T (7)

is the sound attenuation length, with s0 the entropy den-sity and T 0 the temperature of the underlying medium.

For the linear approximation the dynamical descrip-tion of the disturbance is given by the first of Eqs. (3),whose explicit components can be written as

∂ 0δǫ + ∇ · g = J 0,∂ 0g

i + ∂ jδ Θij = J i. (8)

These equations can be readily solved in momentumspace. We define the Fourier transform pair f (x, t) andf (k, ω) as

f (x, t) = 1

(2π)4

d3k

dω eik·x−iωtf (k, ω). (9)

Using Eq. (9) into Eqs. (8), together with Eqs. (5), weobtain

−iωδǫ + ik · g = J 0,−iωgi + ic2skiδǫ +

3

4Γs(k

2gi + k i

3 (k · g)) = J i. (10)

If we decompose g into its longitudinal and transverseparts, with respect to the Fourier mode k, in the form

g = gL + gT , (11)

with the definition of longitudinal and transverse compo-nents of any vector σ given by

σL ≡ (σ · k)k2

k, (12)

σT ≡ σ − σL, (13)


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we can solve Eqs. (10) for each of the g modes as well asfor the energy density δǫ, which gives

δǫ(k, ω) = ik · J(k, ω) + J 0(k, ω)(iω − Γsk2)

ω2 − c2sk2 + iΓsωk2 , (14)

gL(k, ω) = i ωk2k · J(k, ω) + c2sJ 0(k, ω)

k

ω2 − c2sk2 + iΓsωk2 , (15)

gT (k, ω) = g − gL = iJT (k, ω)

ω + i 34Γsk2 . (16)

In a recent study [24], dE/dx was taken as constant. Un-der such assumption it was found that the longitudinalsignal is weaker than the transverse one and that sincethe former is mostly directed along the perpendicular di-rection of motion of the source whereas the latter is for-ward peaked, the energy-momentum was preferentiallydeposited along the direction of motion of the hard par-ton. Nevertheless, it is known that depending on the sizeand treatment of the scattering properties of the medium,dE/dx can depend on the traveled path. Let us thereforeconsider a simple scenario where the length dependence

of dE/dx is linear, namely let us takedE

dx

= Cz, (17)

where we have explicitly considered that the particle’sdirection of motion is along the ẑ direction and intro-duced the dimensionful proportionality constant C whichis fixed later on. We thus write explicitly the current as

J ν (x, t) = Czvν δ 3(x− vt), (18)whose Fourier transform can be written as

J ν (k, ω) = −2iCvπvν ∂ ∂ω

δ (ω − k · v). (19)When considering the effect of the derivative of the delta

function in Eq. (19) for the integrations that lead to theenergy and momentum components deposited into themedium, we can generically write

dω F (ω) ∂

∂ωδ (ω − k · v) = − ∂

∂ωF (ω)

ω=k·v

. (20)

Also, the dependence on ω of this function is a productof the form F (ω) = e−iωtf (ω), thus

∂

∂ωF (ω) = −ite−iωtf (ω) + e−iωt ∂

∂ωf (ω). (21)

Therefore the total energy or momentum deposited intothe medium can be expressed in terms of two contribu-tions: The first term in Eq. (21) which corresponds to theone computed in Ref. [24] as if the energy per unit lengthwas constant, multiplied by the time interval t, and thesecond one in that equation, which corresponds to a newcontribution stemming from the derivative of the func-tion multiplying the exponential in the integrands. Wewrite these generic contributions as

F (x, t) = vC F0 (x, t) t + F̃ (x, t)

. (22)

In order to make the analysis more transparent, letus consider that t represents a parameter that accountsfor the time during which the parton travels trough themedium. For a hydrodynamical description, we requirethat this time is large enough compared to the soundattenuation length. Thus, it is convenient to expressthis time in units of Γs, introducing a dimensionless phe-nomenological quantity κ, given by

Ct = C κt

3Γs2v

≡ C κκ, (23)where κ = (3Γs/2v)

−1t is a characteristic time scale givenin units of the sound attenuation length and C κ is a di-mensionless free parameter that will be fixed by requiringthat the total energy and momentum deposited withinthe medium by the fast moving parton is the same as inthe case of a constant dE/dx. With this definition theenergy and momentum deposited into the medium canbe written as

F(x

, t) = C κv

κF0(x

, t) + 2v

3Γs F̃

(x

, t)

. (24)

We can now use Eqs. (14)–(16) to obtain the space-timesolutions for δǫ(x, t) and g(x, t). Using Eq. (9) and afterintegration in ω , the new contributions are

g̃T (x, t) =

d3k

(2π)3eik·(x−vt)

×v − (k · J)k

k2

1

(k · v + i 34Γsk2)2, (25)

g̃L(x, t) = −g̃L1(x, t) + g̃L2(x, t), (26)with

g̃L1(x,t) =

d3k

(2π)3ek·(x−vt)

× (k · v)kk2 [(k · v)2 − c2sk2 + iΓs(k · v)k2]

, (27)

g̃L2(x, t) =

d3k

(2π)3kek·(x−vt)

×

(k·v)2

k2 + c2s

2k · v + iΓsk2

[(k · v)

2

− c2sk

2

+ iΓs(k · v)k2

]2 , (28)

and

δ ̃ǫ(x, t) = δ ̃ǫ1(x, t) − δ ̃ǫ2(x, t) (29)with

δ ̃ǫ1(x, t) = −

d3k

(2π)3ek·(x−vt)

(k · v)2 − c2sk2 + iΓs(k · v)k2,

(30)


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4

α0.5 1

1.5 2

2.5 3

3.5 4

β -5

-4-3

-2-1

01

23

45

, 7 5 )

β ,

α (

g T z

I0

0.005

0.01

0.015

0.02

(a)

α0.5 1

1.5 2

2.5 3

3.5 4

β -5

-4-3

-2-1

01

23

45

, 1 0 0 )

β

, α (

g T z

I 0

0.005

0.01

0.015

0.02

(b)

α0.5 1 1.5

2 2.5 3 3.5

4

β -5-4

-3-2

-10

12

34

5

, 7 5 )

β , α ( ∈ δ I

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

(c)

α0.5 1 1.5

2 2.5 3 3.5

4

β -5-4

-3-2

-10

12

34

5

, 1 0 0 )

β , α ( ∈ δ I

-0.005

0

0.005

0.01

0.015

0.02

0.025

(d)

FIG. 1. (Color online) Three dimensional plots (surfaces and contours) for I gTz and I gδǫ as functions of α, β and κ forη/s = 1/4π. The plots are shown starting from a minimum value of αmin = 0.5 and for values (left to right) of κ = 75, 100.

δ ̃ǫ2(x, t) = i

d3k

(2π)3ek·(x−vt)

×

2ik · v − Γsk2

2k · v + iΓsk2

[(k · v)2 − c2sk2 + iΓs(k · v)k2]2 .

(31)

In order to compute the integrals in Eqs. (25)–(31) weuse cylindrical coordinates with kz directed along the di-rection of motion v of the fast parton. Let us look indetail at the computation of the z -component of g̃T . Af-ter carrying out the angular integration we get

(g̃T )z = v

∞

0

dkT (2π)2

∞

−∞

dkzeikz(z−vt)

× k3T J 0(kT xT )

(kzv + i34Γsk

2)2(k2T + k2z)

= 2πiv ∞0

dkT (2π)2 k

3T J 0(kT xT )

× [Res1 + Res2] , (32)where J 0 is a Bessel function and xT =

y2 is the dis-

tance from the parton along the transverse direction (di-rected along the ŷ axis, in the geometry we are using) andRes1 and Res2 represent the residues at the two poles inthe integrand of Eq. (32). To carry out the contour in-tegration we close the contour on the lower half kz-planein order to ensure causal motion (z − vt


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minα

1 1.2 1.4 1.6 1.8 2

0

I

-10

0

10

20

30 π= 1/40

/ sη(a)

minα

1 1.2 1.4 1.6 1.8 2

I

-10

0

10

20

30 π= 1/40

/ sη= 75,κ (b)

minα

1 1.2 1.4 1.6 1.8 2

I

-10

0

10

20

30 π= 1/40

/ sη= 90,κ (c)

minα

1 1.2 1.4 1.6 1.8 2

I

-10

0

10

20

30 π= 1/40

/ sη= 100,κ (d)

FIG. 2. Integrals of the functions I gTz (κ), I gTy (κ), I gLz(κ), I gLy (κ), I gδǫ (κ), I gz(κ) and I gy (κ), defined in Eqs. (46)-(47), overthe domain αmin < α


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6

α0.5 1

1.5 2

2.5 3

3.5 4

β -5

-4-3

-2-1

01

23

45

, 5 0 )

β ,

α (

g T z

I0

0.005

0.01

0.015

0.02

(a)

α0.5 1

1.5 2

2.5 3

3.5 4

β -5

-4-3

-2-1

01

23

45

, 7 0 )

β ,

α (

g T z

I0

0.005

0.01

0.015

0.02

(b)

α0.5 1 1.5

2 2.5 3 3.5

4

β -5-4

-3-2

-10

12

34

5

, 5 0 )

β , α ( ∈ δ I

-0.01

0

0.01

0.02

0.03

0.04

(c)

α0.5 1 1.5

2 2.5 3 3.5

4

β -5-4

-3-2

-10

12

34

5

, 7 0 )

β , α ( ∈ δ I

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

(d)

FIG. 3. (Color online) Three dimensional plots (surfaces and contours) for I gTz and I gδǫ as functions of α, β and κ forη/s = 1.5/4π. The plots are shown starting from a minimum value of αmin = 0.5 and for values (left to right) of κ = 50, 70.

Note that the variables α and β represent the distancefrom the source in the parton direction of motion andin the transverse direction, respectively, in units of thesound attenuation length, whereas ξ is the transverse mo-mentum in units of the inverse of the sound attenuation

length.

Putting all together, the integral in Eq (32) becomes

(g̃T )z = 1

v 1

4π 2v

3Γs 1

α2 + β 2− ∞

0

dss− α(s + 1)

(s + 1)2 (s + 2)

× J 0

β

s(s + 2)

e−αs

≡

1

4π

2v

3Γs

Ĩ gTz . (36)

In a similar fashion we get


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7

minα

1 1.2 1.4 1.6 1.8 2

0

I

-10

0

10

20

30 π= 1/40

/ sη(a)

minα

1 1.2 1.4 1.6 1.8 2

I

-10

0

10

20

30 π= 1.5/40

/ sη= 50,κ (b)

minα

1 1.2 1.4 1.6 1.8 2

I

-10

0

10

20

30 π= 1.5/40

/ sη= 60,κ (c)

minα

1 1.2 1.4 1.6 1.8 2

I

-10

0

10

20

30 π= 1.5/40

/ sη= 70,κ (d)

FIG. 4. Integrals of the functions I gTz(κ), I gTy (κ), I gLz (κ), I gLy (κ), I gδǫ (κ), I gz (κ) and I gy (κ), Eqs. (46)-(47), over the domainαmin < α < 6, −5 < β


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8

α0.5 1

1.5 2

2.5 3

3.5 4

β -5

-4-3

-2-1

01

23

45

, 3 5 )

β ,

α (

g T z

I0

0.005

0.01

0.015

0.02

(a)

α0.5 1

1.5 2

2.5 3

3.5 4

β -5

-4-3

-2-1

01

23

45

, 5 5 )

β ,

α (

g T z

I0

0.005

0.01

0.015

0.02

(b)

α0.5 1 1.5

2 2.5 3 3.5

4

β -5-4

-3-2

-10

12

34

5

, 3 5 )

β , α ( ∈ δ I

-0.1

-0.05

0

0.05

0.1

0.15

0.2

(c)

α0.5 1 1.5

2 2.5 3 3.5

4

β -5-4

-3-2

-10

12

34

5

, 5 5 )

β , α ( ∈ δ I

-0.01

0

0.01

0.02

0.03

0.04

(d)

FIG. 5. (Color online) Three dimensional plots (surfaces and contours) for I gTz and I gδǫ as functions of α, β and κ forη/s = 2/4π. The plots are shown starting from a minimum value of αmin = 0.5 and for values (left to right) of κ = 35, 55.

action, the quantity c2s/v2 is small, since for a fast mov-

ing (massless) parton v ≃ 1 and for a relativistic gas,cs ≃

1/3. Therefore, we can expand the integrand in

Eq. (38) in this parameter. To first order in c2s/v2, we

get

(g̃L1)z = 1

v

dkT (2π)2

dkz

kT k2zJ 0(xT kT )eikz(z−vt)(k2T + k

2z )

×

1

k2z + iΓskzv

(k2T + k2z)

+ (k2T + k

2z)

k2z + iΓskzv

(k2T + k2z)2

c2sv2

. (39)

To perform the integral it is convenient to introduce thevariable r related to ξ by r = 43ξ . Once again, in order todescribe causal motion (z−vt


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9

minα

1 1.2 1.4 1.6 1.8 2

0

I

-10

0

10

20

30 π= 1/40

/ sη(a)

minα

1 1.2 1.4 1.6 1.8 2

I

-10

0

10

20

30 π= 2/40

/ sη= 35,κ (b)

minα

1 1.2 1.4 1.6 1.8 2

I

-10

0

10

20

30 π= 2/40

/ sη= 45,κ (c)

minα

1 1.2 1.4 1.6 1.8 2

I

-10

0

10

20

30 π= 2/40

/ sη= 55,κ (d)

FIG. 6. Integrals of the functions I gTz (κ), I gTy (κ), I gLz(κ), I gLy (κ), I gδǫ (κ), I gz(κ) and I gy (κ), defined in Eqs. (46)-(47), overthe domain αmin < α


10/13


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11

Figure 2 shows the integral defined in Eq. (46) and thedifferent components of the integrals defined in Eq. (47),integrated over the domain αmin = 0.5, 1, αmax = 6and −5 < β < 5 for several values of κ. The figure alsoshows I δǫ0 and the components of I

g0 which correspond

to a constant energy-loss per unit length. Note that thehierarchy of momentum deposition is the same in bothcases. This means that the momentum is preferentially

deposited also in the forward direction for this value of η/s0.The value η/s0 = 1/4π, corresponds to a universal

lower bound for all relativistic quantum field theories inthe strongly coupled limit [18]. However, we can testthe sensitivity to the momentum deposition when vary-ing η/s0 [18, 25–27]. Since Γs is proportional to η/s0, atraveled path length L in the range 7 fm < L < 10 fm,corresponds to different values of κ than for the previ-ously discussed case where we took η/s = 1/4π. Note

that since δ̃ǫ is intrinsically negative, if we require thatδǫ = κδǫ0 − |δ̃ǫ| > 0 then not all values of κ are allowed.To find a restriction involving κ and η/s0, note that

κ > |δ̃ǫ|

δǫ0≈ 30 (49)

therefore

η

s0<

vtmin2

δǫ0

δ̃ǫT 0 ≈ 2.5 1

4π, (50)

where tmin is the minimum time that we consider for thefast moving parton to have traveled in the medium. Fordefinitiveness we take this parameter to be tmin = 7 fm,given that the maximum time corresponds to twice thenuclear radius which for led nuclei is of order 10 fm.

Figure 3 shows the integral defined in Eq. (46) and the

component I gTz of the integrals defined in Eq. (47) asfunctions of α and β . The plots are shown in the range0.5 < α < 4 and −5 < β < 5 for several values of κ.The normalization constant C κ is computed also fromthe requirement in Eq. (48). There is not much of a dif-ference between the three dimensional surfaces in Fig. 3and those in Fig. 1. This means that the spatial distri-bution of energy and momentum are very much alike forthe cases with η/s = 1/4π, 1.5/4π. Figure 4 shows thecomparison between the integrals of Eqs. (46) and (47),with respect to α and β , with the case corresponding toa constant energy-loss per unit length for η/s = 1.5/4π.Note that the hierarchy of strengths for the momentumcomponents for the case with η/s = 1.5/4π is maintainedwith respect to the case with η/s = 1/4π. The only sig-nificant change comes from the energy deposition whichis 30% smaller in the latter case.

For completeness, we also study the case with η/s =2/4π. Figure 5 shows the three dimensional plots corre-sponding to Eqs. (46) and (47) for several values of κ.Figure 6 shows the comparison between all componentsof these integrals and I δǫ0 and the components of I

g0 which

correspond to a constant energy-loss per unit length forthe case η/s = 1/4π. The normalization constant C κ is

computed also with the requirement in Eq. (48). Notethat the energy deposition decreases about 60% with re-spect to the case with η/s = 1/4π but the hierarchyof strengths between the momentum modes remains thesame as the case with η/s = 1/4π.

In order to further study the energy-momentum de-position, we proceed as in Ref. [28], defining the energydensity and momentum flux angular distributions as

dI δǫdθ

= 2πR2 sin θI δǫ, (51)

and

dI gdθ

= 2πR2 sin θ R̂ · Ig= 2πR2 sin θ (|gz| cos θ + |gy| sin θ) , (52)

respectively, where R is the distance vector from thesource measured from the forward direction.

Figure 7 shows the angular distribution for energy den-sity (a) and momentum flux (b), for different values of

distances to the source R in units of sound attenuationlength, for η/s = 1/4π and κ = 75. Note that both angu-lar distributions peak for angles close the source, whichstrengthens the conclusion that energy and momentumdeposition is in the forward direction. The energy den-sity increases and the momentum flux decreases with thedistance to the source. This can be understood from thefact that the energy density contains an extra power of R with respect to the momentum flux.

IV. SUMMARY AND CONCLUSIONS

In summary, we have studied the energy-momentumdeposition produced by a fast moving parton travelingin a medium modeled by linear viscous hydrodynam-ics. The energy loss per unit length dE/dx has beentaken as proportional to the traveled length. We foundthat the transverse modes still dominate the momentumdeposition and therefore this case is similar to the onewhere dE/dx is taken as independent of the traveled pathlength. This situation is also maintained when the shearviscosity to entropy ratio η/s0 is increased from its the-oretical lower bound. The only significant change comesfrom the energy deposition which decreases as η/s in-creases. Therefore, the momentum is forward peaked asin the case with constant energy loss per unit length aswell as for the case of the lower value of η/s previouslystudied [24]. We conclude that for the cases where dE/dxis constant or proportional to the path length, as well asfor larger than the lower theoretical bound values of η/s,the energy and momentum are preferentially depositedalong the direction of motion of the traveling parton.Therefore, for the cases studied, the conical emission of particles is suppressed with respect to the forward emis-sion, which means for instance that it is unlikely thatthe propagation of a single fast moving parton leads to


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12

(rad)θ

1.6 1.8 2 2.2 2.4 2.6 2.8 3

θ

/ d ∈ δ

d I

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

R=2.0

R=3.0

R=4.0

(a)

(rad)θ

1.6 1.8 2 2.2 2.4 2.6 2.8 3

θ

/ d g

d I

0

0.01

0.02

0.03

0.04

0.05

0.06 R=2.0

R=3.0

R=4.0

(b)

FIG. 7. (Color online) Angular distribution of (a) energy density dI δǫ/dθ and (b) momentum flux dIg/dθ over an angular range[π/2, π] at distances R = 2.0, 3.0 and 4.0 in units of the sound attenuation length Γs for η/s0 = 1/4π and κ = 75.

the appearance of a double peak structure in azimuthalangular correlations in heavy-ion collisions.

We also point out that it is easy to generalize thepresent studies to the case where dE/dx ∝ Ln with nan integer larger than 1. In such a situation, the Fouriertransform of J ν (x, t) ∝ δ (n) (ω − k · v), where n is thenth-derivative of the delta function. Thus, the expres-

sions for the energy and momentum deposition becomepolynomials of degree n in t, where the coefficient of tn

corresponds to the strength of the term F0(x, t) whichthen becomes the dominant component for the range 7fm < L < 10 fm. Therefore the hierarchy of the modesretain the general features already seen in the case of a

constant energy loss per unit length profile.

ACKNOWLEDGMENTS

We acknowledge useful conversations with J. Jalilian-Marian during the initial stages of this work. Supportfor this work has been received in part from CONACyT-

México under grant number 128534, from PAPIIT-UNAM under grant number IN101515 and from Pro-grama de Intercambio UNAM-UNISON and Programa Anual de Cooperaci´ on Académica UAS-UNAM . M. E.T.-Y. acknowledges support from the CONACyT-Méxicosabbatical grant number 232946.

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impact of the energy loss spatial profile and shear viscosity to entropy density ratio for the mach...

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