Parton Production in nn and AA collisiParton Production in nn and AA collisionon

신기량안동대학교 물리학과

Sept. 3, 2005 APCTP

0. Introduction

• QM05:– We have a parton system which is not just a bunch o

f partons but has collective effects: flow, jet suppression and very dense.

– Which means that the classical parton cascade may have only very limited application! -> quantum Boltzmann or PC+Hydro

– Before we move on, we need to understand very carefully the partonic system: ‘How many parton’, ‘How are they evolving’, ‘How are they breaking up’, ‘How are the particles formed’

1. Particle production

• Depends on the colliding particles:– ep– nn– AA

• Depends on the available energy:– KeV– MeV– GeV– TeV

CROSS SECTION:CROSS SECTION:

1) Physical meaning:

a.

b.

2) Fermi Golden Rule:

( , ) ( , )sin

d b dbd D d D

d d

1d dNdN Ld

d L d

1 2 3 4 ... n

3 3322 3 4

2 2 22 2 23 41 2 1 2

4 41 2 3

...(2 ) 2 (2 ) 2 (2 ) 24 ( ) ( )

(2 ) ( ... )

n

n

n

cd p cd pcd pSd M

E E Enp p m m c

p p p p

3. ep Collisions

Depending on energy:– Movie at KeV:– Movie at MeV:– Movie at GeV:– Movie at TeV:

• ep at KeV: see only electric field: Rutherford scattering, Mott formula

• ep at MeV: Rosenbluth formula(Elastic scattering)

2

22 22 2

) ( ) cos22 sin ( / 2)

mott

dmc p

d p

����������������������������

22

2 2)

2 sin ( / 2)Rutherford

d e

d mv

22

42

,4

21 3 3 1 1 3

3 521 2 2 2 2 2 22 2 2

'

8

2 ( ) ( )

( )( ) ( ) ( )

eelectron proton

electron

proton

d EM

d McE

gM L K

q

L p p p p g mc p p

K KKK K g p p q q p q p q

Mc Mc Mc

Thus,2

2 21 22

') 2 sin ( / 2) cos ( / 2)

24 sin ( / 2)Rosenbluth

d EK K

d ME E

• ep at GeV: Inelastic collisions2

2

2

42

,4

52 41 2 2 2 2 2 22 2 2

'

'

( )( ) ( ) ( )

eelectron proton

proton

d EM

dE d cq E

gM L W

q

WW WW W g p p q q p q p q

Mc Mc Mc

2

2 2 2 21 22

2 ( , )sin ( / 2) ( , ) cos ( / 2)' 2 sin ( / 2)

dW q x W q x

dE d E

2

...2

qwhere x

q p

<Bjorken function>

<Callan & Gross relation>

21 1

22

2 22

( , ) ( ),

( , ) ( )2

MW q x F x

qW q x F x

Mc x

1 22 ( ) ( )xF x F x

• ep at TeV:Hard particles: More than 80%, comming from elasti

c collision between e-partons.Soft particles: relatively small number.-> the parton distribution of nucleon is very importa

nt.

At RHIC energy, about half of them are hard particles and the other half soft particles.

4. nn, AA scattering

• pp scattering at MeV:– Mostly elastic collision

• pp scattering at GeV:– Inelastic collision with lots of soft particles

• pp scattering at TeV:– Elastic collisions among partons with minijets

Models

• Schwinger mechanism:• Inside-Outside cascade:

• Classical String Model:• Dual Parton Model:

• Parton-Parton elastic scattering (factorization):• CGC shattering(Yang-Mills equation):

Toy models

• Schwinger mechanism:-two infinite plates with charges(condensers):-constant electric field between them-vacuum persistance probability: P_V-particle production=1-P_V

Assume the constant color electric field(flux) between q and q_bar.If field energy > threshold energy, (qq_bar) pair produced.->Production rate: quark field equation in the potential -> Klein-Gordon eq

uation -> Scheodinger equation to identify the effective potential -> sea quark tunnels the potential to leave antiquark -> WKB approximation gives the tunneling rate.

• String model(Lund model):– Meson: q-q_bar connected with string in yo-yo state.– Suppose a string connected to quark and antiquark separating eac

h other

• See Figure• Quark-antiquark born at V_i• V_i = (x,t)-> (t+x,t-x) =• Verticies are on the proper time: At high energy, the produced particle

s are independent of rapidity• The probability element: for example, 2 vertices production

– V_0: – V_1: Where

( , ), ,y y rapidity

0 0 0 0 0( , )dP y d dy 2 2 2

1 01 01 01 01( , ( )dP f z M g M dz dM

0 1 0( ) / ,z p p p 201 0 1 1 0( )( )M p p p p

figure1

- The solution:

- FRITOF etc

2 2 201 01( ) ( )g M c M m

2 /2 (1 )( , )

abm zz

f z m N ez

( ) a bC e

• Dual Resonance Model:– Consider H-H collision at high energy. The exchange of a particle

of mass M, spin J:

– Sum of J: Veneziano scattering amplitude

– For fixed t, expand in s. Each term corresponds to a t-channel Feynman diagram with exchange of a particle:

• Regge trajectory -> Reggeon– Same for s-channel:

JA s

2

'0

( ( )) ( ( ))( , )

( ( ) ( ))

... ( ) .

s tA s t g

s t

where x x

' 20 JJ M

• figure2

Symmetrize over possible channels:

• Basic properties of HH collisions at high energy:– Inelastic processes with no quantum number flow=dominant– Forward scattering amplitude ~ purely imaginary– Total cross section ~ constant.

• Out of Regge trajectory, satisfies these.– => Pomeranchuk trajectory: Pomeron

4

( ) ' ( )2 ( )

( , ) ( , ) ( , )

(1 ) ( )

(1 ( )) sin( ( ))

i t tt

A A s t A t u A u s

e sg e

t t

0 1

Elastic scattering between partons:

• Simple and Clear!• Inclusive gluon production:

where

Pros: Pretty good with minijetsCons: no soft particles

2 2 2 2 21 1 1 2 2 2 1 2 1 22 2 2

16 1( ) ( , ) ( , ) ( )

1c

s tt c t

NdF x k F x k k k p d k d k

dyd p N p

������������������������������������������

1/ 2

2 2 2 21 2

exp( )

max( , , )

t

t

px y

s

k k p

• Minijets:

2 22 1 / 1 2 / 2

2( ) ( , ) ( , ) ( , , )

jett

ij kli A t j B tijklt

pdNKT b dy x f x p x f x p s t u

dydp s

• Unintegrated gluon distribution:– BFKL: solution of BFKL equation

– Kimber-Martin-Ryskin: DGLAP evolution

1/ 2 22 2 22 0

2 ''''0

''

0

ln ( / )( , ) exp ,

2 ln(1/ )2 ln(1/ )

4 ln 2, 28 (3), 3 / , (3) 1.202,

1.19, 0.15

t tt

s s s s

k k qCf x k

x q r xx

where

C r

2 2

2 22

212 2 2 2 2

''

( , ) ( , ) ,

( )( , , ) ( , ) ( ) '( , )

2

Q k

sa a aax

a

F x k xg x QQ

k x xf x k T k P z a k dz

z z

– Kharzeev-Levin: based on CGC

– GRV98: we have the code!

2 20

2 22 2

0 2

22 2 0

0

.......... ..( , )

... ..

( ) 1( ) , 170

s

ss

s

f if k QF x k Q

f if k Qk

where

xQ x GeV f mb

x

- CGC distribution: Triantafyllopoulos

2

2

2 2

2 2

2 2

02 2

1ln .......................... .

1( , ) ln ........ .

4 ln .......

st s

s t

s tt t

s t s

s ts t s

t s

Qif k Q

k

Q kk y if k Q

k Q

Q kI y k Q

k Q

• GRV98 & EKS98+minijets:

P_T distribution rapidity distribution

• DGLAP evolution:(DGLAP evolution:(Dokshitzer-Gribov-Lipatov-Altarelli-Parisi)Dokshitzer-Gribov-Lipatov-Altarelli-Parisi)

• Modified DGLAP:Modified DGLAP:

221 2 2

2,

( )( , )( , ( )) ( , ),

ln 2

: , ..ker ,

& : ..

sab sx

b q g

ab

a x xdzP z b

z

where

P b ac branching nal

a b parton distrstribution

221 12 2 2

2 0,

( )( , )[ ( , ( )) ( , ) ( , ) ( )]

ln 2s

ab s baxb q g

a x xdzP z b a x d P

z

• BFKL equation:(BFKL equation:(Balitsky-Fadin-Kuraev-Lipatov)Balitsky-Fadin-Kuraev-Lipatov)

Where

2 '2 2 222

2 '4 4 1/ 2'2 20

( , ) ( , ) ( , ) ( , )

ln(1/ ) (4 )g t g t g t g tA s t

tt t tt t

f x k f x k f x k f x kC dkk

x k k kk k

2 22 2

20( , ) ( , )t

a tt

dka x f x k

k

Yang-Mills Equation:

• Yang-Mills equation of motion: • Krasnitz, Nara and Venugopalan; Lappi

• Solve the equation and identify the number density:

,

1 2

,

( ) ( )

a aD F J

where

J x x

12 2

12

4 42

1 2

exp( / ) 1 ..( / 1.5)1

log(4 / ) ............( / 1.5)

0.137, 0.0087,

0.0358 , 0.465

glue t eff t s

s t s t t s

s eff s

a k m T kf

g a k k k

where

a a

m T

• KNV:

energy distribution

Number distribution rapidity distribution

• Pros and cons:– Good: based on the first principle(QCD)– Bad: infinite flat plates, only gluons,

no information on space

5. Summary

• Needs lots more work on particle production.• Of cause the work is one of great problems to

challenge.

• Need a new method to solve physics problems? I.e. we have a finite proton-proton cross section but we cannot calculate that in current formalism.