parton production in nn and aa collision 신기량 안동대학교 물리학과 sept. 3, 2005 apctp

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.