Conical Flow induced by Quenched QCD Jets

Jorge Casalderrey-Solana,

Edward Shuryak and Derek Teaney, hep-ph/0411315

SUNY Stony Brook

Outline• Basic Ingredients: Hydrodynamics Thermalization of energy loss• Assumption: small perturbations due to

energy loss• Solution to the linearized problem: Conical shock waves• Possible experimental confirmation• Conclusions.

Hydrodynamics

• (local) Energy-momentum and baryon number conservation.

• At mid rapidity (neglecting nB)

• Ideal case (=0) provides a remarkable description of radial and elliptic flows at RHIC

• The viscosity at RHIC seems to be close to its minimal conjectured bound.

0 T

Jet Quenching and Energy Loss• High pt particles lose energy in the medium Radiative losses (main effect) Collision losses Ionization losses (bound states)

• From the hydrodynamical point of view, the different mechanisms may be only distinguished by the deposition process (what mode they excite)

• We study this modification through hydrodynamics.

• Similar ideas have been discussed by H. Stoeker (nucl-th/0406018)

fm

GeV

dx

dE2 Shuryak+Zahed,

hep-ph/0406100

Basic Assumptions• The deposited energy thermalizes at a scale:

• Minimal value >> point-like . s will be the only scale of the “source”

• Outside of the “source”, the modification of the properties of the medium is small

• Thus, linearized hydrodynamic description is valid:

3

4

pes

1)4( Ts 1)( jetP

0 T

depositedE <<depositedV

Summing the Spherical WavesParticle moving in the static medium with velocity v

00 vtx

0xxX

After the disturbance is thermalized

ijij TcT 002 00: T ii Tg 0:

Given the symmetries of the problem, we need to specify:

),(),( 0 XRtX

),(),(),( 0 XRgvXRgtXg ov

Adding all displacements we obtain the Mach cone

The different terms lead to different excitations of the medium

Two (linear) Hydro Modes

03

1

4

3 22

kgkgkkicg st

tl gkgg ˆ

0 lt ikg

Sound waves (propagating) Diffuson (not propagating)

),(4

3),( 2 tkgktkg

t tst

By defining the system decouples:

After Fourier transformed (space coordinates)0 gkit

022 lslt gkikcg

),(),( 0 ZRtX

vXRgtXg v

),(),( 0

),( XRgo

Excitations: Sound DiffusonYes

Yes

Yes

No

Yes

No

Flow Picture

0),( ZR 0),( XRgv

0),( XRgo

0),( ZR 0),( XRgv

0),( XRgoxv xv

Diffuson: Matter moving mainly along the jet direction

Sound motion along Mach direction.

<= RHIC

• Flow of the background medium modifies the shape and angle of the cone (Satarov et al.)

• c2s is not constant though system evolution:

csQGP= , cs= in the resonance gas and cs~0 in the mixed phase.

p/e() = EoS along fixed nB/s lines

Considerations about Expansion

•Distance traveled by sound is reduced Mach direction changes

2.031

33.0)(1

sf

avs cdc

(Hung,E. Shuryak hep-ph/9709264)

• = 1.23 rad =71o

Spectrum• Cooper-Fry with equal time freeze out

f

t

ffff

z

T

vp

T

T

T

E

T

E

T

up

ptz

dVedVepddp

dN

0

2

•At low pt~Tf

)cos(4

0

2

p

P

T

PE

T

EVe

pddp

dN dep

f

tdep

f

T

E

ptz

f

z

• Pt >>the spectrum is more sensitive to the “hottest points” (shock and regions close to the jet)

•If the jet energy is enough to punch through, fragmentation part on top of “thermal” spectrum

Two Particle Correlations

t

zppttz ddppdp

dN

Qc

0

1:)(

t

zppttz dppdp

dNQ

0:

•Normalized correlation function:

•The cone is also observed in the spectrum

3

1arccos

s=1/4T

Is such a sonic boom already observed?

M.Miller, QM04

Flow of matter normal to the Mach cone seems to be observed!

+/-1.23=1.91,4.37

(1/N

trig)d

N/d

()

STAR Preliminary

cGeVp

cGevpassocT

trigT

/42.0

/64

Conclusions

• We have used hydrodynamics to follow the energy deposited in the medium.

• Finite cs leads to the appearance of a Mach cone (conical flow correlated to the jet)

• Depending on the initial conditions,the direction of the cone is reflected in the final particle production.

Outlook

•Systematic study of initial conditions

•Role of non-linearities (mixing the modes)

•Precise effect of changing speed of sound as

well as the expanding media

• Realistic simulation of collision geometry

• Three particle correlations.

Problems that need to be addressed (on progress):

Swinging the back jetAssume a boost invariant medium and a yj-distribution for the backjet P(yj) (flat). Boosting by yj we assume a particle distribution:

**3

coscos cjpj pfPdypd

dNE

After boosting back to the lab frame

2

*2*

***

coscos

1cos

coscosc

c

jjc

yy

*

*

cos

coscosh

cjy

Now we integrate over yj:

2

*2*

3

coscos

1cos

1

cos

cos

cc

cTp pPf

pd

dNE

p*

*c

*c

Swinging the back jet (II)If we simply rotate the jet axis (Vitev):

22*2 tancottan

tan

cottan

2/322

2/122

22

*

cottan1

cottan

cossin

1

,

,sin

And use

sinsinsin

**

Pdd

dN

2*22 tantancos

1

cd

dNIntegrating over

*

z

y

x

I. Vitev hep-ph/0501255

However long tails may fill up the cone.

How to observe it?

• the direction of the flow is normal to the Mach cone, defined entirely by the ratio of the speed of sound to the speed of light

• Unlike the (QCD) radiation, the angle is not shrinking (1/ with the increase of the momentum of the jet but is the same for all jet momenta

• At high enough pt a punch through is expected, filling the cone