Transcript 幻灯片 1

Relativistic nuclear collision
in pQCD and corresponding
dynamic simulation
Ben-Hao Sa
China Institute of Atomic Energy
2015/7/18
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INTRODUCTION
HADRON-HADRON COLLI. IN pQCD
DYNAMIC SIMULATION FOR hh
COLLI. (PYTHIA MODEL)
NUCLEUS-NUCLEUS COLLI. IN pQCD
DYNAMIC SIMULATION FOR
NUCLEUS-NUCLEUS COLLI. (PACIAE
MODEL)
LONGITUDINAL SCALING
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INTRODUCTION
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RHIC, hottest physical frontier in particle
and nuclear physics
• Primary goal of RHIC:
Study properties of extremely high
energy and high density matter
• Explore phase transition from HM to
QGM, QGP transition
• Evidences for sQGP, existed, however,
it is still debated

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•
The ways studying RHIC:
 Perturbative QCD (pQCD)
 Phenomenologic model (eg. NJL)
 Hydrodynamic
 Dynamical simulation:
– Hadron cascade model:
PYTHIA,RQMD,HIJING,VENUS,
QGSM, HSD, LUCIAE
(JPCIAE), AMPT, uRQMD,
etc.
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– Parton and hadron cascade model:
PCM (VNI),
AMPT (string melting),
PACIAE
Zhe Xu & C. Greiner
– Better parton and hadron cascade
model, required by present RHIC
experiments
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HADRON-HADRON
COLLI. IN pQCD
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1. Cross section of hadron production
in hadron-hadron ( a + b ) colli.
hh colli. = superposition of
parton-parton colli.
d
1
E 3 (a  b  h  x; s, pT ,cm )  
d p
i, j 
1


ximin
1
dxi

dx j fi / a ( xi , Q 2 ) f j / b ( x j , Q 2 ) Dkh ( zk , Q 2 )
x min
j
1 d

(ij  kl; s , t )
zk dt
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d
(ij  kl ; s , t ) : cross section of sub-process
dt
f i / a : parton (i ) distribution function in
hadron (a)
xi : momentum fraction taking by i
from a
Q2  4 pT2 : scale of scattering
Dkh : fragmentation function of k to h
x1 x2
zk  
xi x j
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u 1 xT
t 1
x1   
, x2    xT h
s 2 h
s 2
 cm
pT
xT  2
, h  tan( )
2
s
xi x2
x1
min
min
xi 
, xj 
1  x2
xi  x1
2. cross section of partonic sub-process
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Subprocess cross section expresses as


d
2
(ij  kl : s , t ) 
| M (ij  kl ) | 
ij kl
dt
s
s
2
s
2
2
s
2
(after average and sum over initial
and final states)
LO pQCD cross section of seven
contributed processes and two processes
with photon are:
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4 s2  u2
4 s2  u2 s2  t 2 8 s2
 qi q j qi q j   qi q j qi q j 
,  qi qi qi qi  ( 2  2 ) 
,
2
9 t
9 t
u
27 ut
4 s2  u2 t 2  u2 8 u2
32 t 2  u 2 8 t 2  u 2
 qi qi qi qi  ( 2  2 ) 
,  qi qi  gg 

,
2
9 t
s
27 ts
27 ut
3 s
1 t 2  u2 3 t 2  u2
4 s2  u2 s2  u2
 gg qi qi 

,  qi g qi g  
 2 ,
2
6 ut
8 s
9 us
t
9
ut us st
 gg  gg  (3  2  2  2 );
2
s t u
 em 2 8 u t
 em 2 1 t s
 qi qi  g 
ei (  ),  qi g qi  
ei (  )
s 9 t u
s 3 s t
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Mandelstum variables:
p1
p3
p2
p4
s  ( p1  p2 ) , t  ( p1  p3 ) , u  ( p1  p4 )
2
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2
3. Parton distribution function (PDF) and
fragmentation (decay) function (PFF)
• Can’t calculate from first principle
• There are a lot of parameterizations
based on the experimental data
of lepton-hadron deep inelastic
 
e
scatterings (for PDF) and/or of the e
annihilations (for PFF)
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• Most simplest PDF (without
depen.) at
large x region is something like
Q2 , pT
xu ( x) ~ (1  x)3 , xd ( x) ~ (1  x) 4 , xu ( x) ~ (1  x)10 ,
xd ( x) ~ (1  x) , s( x)  s ( x)  0.1(1  x)
7
8
• Most simple PFF is some thing like
Dg
0
4 (1  z )3

5
z
• Total fractional momentum carried by
q, q , g
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1
 x ch arg e   dxx[u ( x)  d ( x)  s( x) 
0
u ( x)  d ( x)  s ( x)]  0.5
(u ( x)  f u / p ( x))
1
 dxxf
g/ p
( x)  0.5
0
• Approximately 3/5 of parton momentum
goes to pions and the rest to kaon and
baryon pair.
• As gluon is a flavor isosinglet its
momentum equally distributes among
  ,  , 0
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DYNAMIC SIMULATION FOR
HADRON-HADRON COLLI.
(PYTHIA MODEL)
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• Sketch for pp simulation in PYTHIA
Remnant
Initial state
p radiation
Rescattering ?
fi p (x)
Parton distribution
function
p
f j (x)
p
Remnant
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…
fk
d
dtˆ
h
fl
Decay
Final state
radiation
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Hadronization
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• Differences from pQCD are:
– Monte Carlo simulation instead
of analytic calculation
– There is additions of initial and
final states QCD radiations
– String fragmentation instead of rule
played by fragmentation function in
pQCD
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– Semihard interactions between other
partons of two incoming hadrons
(multiple interaction)
– Addition of soft QCD process such as
diffractive, elastic, and non-diffractive
(minimum-bias event)
– Remnant may have a net color charge
to relate to the rest of final state
– Multiple string fragmentation
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Multiple String
Fragmentation
q0
q0
q0
q1
q1
q0
q 0 q1
q1
q2
q2
q0
 
…
– Hadron rescattering (?) and decay
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We have expended PYTHIA 6.4 including
parton scattering and then hadronization
(both string fragmentation and coalecense)
and hadron rescattering. We are please if
you are interested to use it
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NUCLEUS-NUCLEUS COLLI.
IN pQCD
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A) Hadron production cross section in
nucleus-nucleus (A+B) collision is
calculated under assumptions of
– Nucleus-nucleus collision is a
superposition of nucleon-nucleon
collision
– A+B reaction system is assumed to be a
continuous medium
B) Convolution method
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``Skecth of AB collision projected to
transverse plane”
(beam, i. e. z axis, is perpendicular to page)
A
bA
oA
b-bB+bA
b
bB
oB
B
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–The cross section can be expressed as
d AB
d pp
Eh 3   dbA dz A  A (bA , z A )dbB dzB  B (b  bB  bA , zB ) Eh 3
d ph
d ph
d pp
  dbA dbB  A (bA )B (b  bB  bA ) Eh 3
d ph
 A (bA )   dz A  A (bA , z A ),

A
(bA )dbA  1
 A (bA )
: normalized thickness function of
nucleus A
– Phenomenological considerations for:
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• Nuclear shadowing
• Multiple scattering (ela. diffractive,…)
• Jet quenching (energy lose)
d AB
d pp
Eh 3   dbAdz A  A (bA , z A )dbB dzB  B (b  bB  bA , zB )Eh 3
d ph
d ph
 S (...)  M (...)  Q(...)
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C) Glauber method
(Glauber theory with nn inelastic cross
replaced by pQCD nn cross section)
–
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t (b )db : probability having a nn colli.
within transverse area db when nucleon a
passes c at impact parameter b
t (b ) : thickness nucleon a
db
function of nn
b
collision
c
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 t (b )db  1 (total probability  1)
– A (bA , zA )d bAd zA: probability finding a nucleon in
volume dbAdz A in nucleus A at
which is normalized as
 db dz
A
A
(bA , zA ) ,
 A (b , z A )  1
A
– Probability for occurring an inela. nn
colli. when nucleus A passes B at an
impact parameter b is
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  (b , z )db dz  (b , z )db dz t (b  b  b )
  db db  (b ) (b )t (b  b  b )  T (b )
T (b ) : thickness function of A  B colli.  T (b )db  1
A
A
A
A
B
A
A
A
A
B
B
B
B
B
B
A
B
B
A
in
B
in
in
– Probability for occurring n inela. colli.
Probability having
combinations
is
an inela. Colli.
AB
  n
p(n, b )    s (1  s) ABn , s  T (b ) in
 n 
as there can be up to A  B collisions
– Total probability for occurring an event
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of nucleus-nucleus inela. colli. at impact
parameter b is
d AB AB
AB
  p(n, b )  1  (1  s)
db
n 1
– Total cross section of above event is
AB
 AB   db [1  (1  s ) ]
– If one use pQCD p+p cross section
instead of  in in above equations
one has pQCD inela. cross section for
(A+B) colli.
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DYNAMIC SIMULATION OF
NUCLEUS-NUCLEUS COLLI.
(PACIAE MODEL)
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Overview for PACIAE model:
– In PACIAE model
• Nucleus-nucleus colli. is decomposed
into nucleon-nucleon (nn) colli.
• nn colli. is described by PYTHIA, where
nn colli. is decomposed into partonparton colli. described by pQCD
– The PACIAE constructs a huge building
using block of PYTHIA & plays a role like
convolution in nucleus-nucleus cooli. in
pQCD
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– The PACIAE model is composed of
(1) Parton initialization
(2) Parton evolution
(3) Hadronization
(4) Hadron evolution
four parts
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



(1) Parton initialization
Nucleon in colliding nucleus is
distributed due to Woods-Saxon ( r ) and
4 (solid angel) distributions
Nucleon is given beam momentum
Nucleon moves along straight line
nn collision happens if their least
approaching distence
d min
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
their collision time is then calculated
Particle (nucleon) list
order # of particle
four momenta
.
.
.
.
.
.
and nn collision time list
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order # of colliding pair collis. time
.
.
.
.
.
.
.
.
.
are constructed
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


A nn collision with least colli. time,
selected in colli. time list, executed
by PYTHIA with fragmentation
switched off
Consequence of nn collision is a
configuration of q (q ) and g ( if
diquark (anti-diquark) is forced
splitting into qq (qq) randomly)
Nucleon propagate along straight
line in time interval equal to difference
between last and current colli. times
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Update particle list, i. e. move out colliding
particles and put in produced particles
 Update colli. (time) list:
Move out colli. pairs which constituent
involves colliding particle
Add colli. pairs with components one
from colliding nucleon and another from
particle list
 Next nn colli. is selected in updated colli.
list, processes above are repeated until
nn colli. list is empty



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(2) Parton evolution (scattering)
• Only 2→2 process, considered for parton
scattering and LO pQCD cross
section,employed.
• If LO pQCD differential cross section
denotes as
d s ij® kl
dt
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s
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s
å
ij ® kl
40
• For process of q1q2 ® q1q2 , for instance
2 2
4 s u

2

9 t
q1q2 q1q2
• That has to be regularized as
2 2
4 s u
 

2 2
9
q q q q
t   
by introducing color screening mass 
1 2
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• Total cross section of sub-process

0  d ijkl

 ij sˆ    ˆdt

s
dt
k ,l
(4)
at high energy
• Using above cross sections parton
scattering can be simulated by MC
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(3) Hadronization
• Partons begin to hadronize when their
interactions have ceased (freeze-out).
• Hadronized by:
— Fragmentation model :
 Field-Feynman model (IF)
 Lund siring fragmentation model
— Coalescence model
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• Ingredients of coalescence model:
Field-Feynman parton generation
mechanism is applied to deexcite
energetic parton and increase parton
multiplicity like multiple fragmentation
of string in Lund model
The gluons are forcibly splitting into
qq pair randomly
There is a hadron table composed of
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Field-Feynman parton
generation mechanism
q0
Original quark jet
1
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q1
q1 q2
q2 q3
q3 q4
q4
q5
q5
…
Created quark pairs from vacuum
(if mother with enough energy)
3
5
7
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11 …
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B
mesons & baryons, made of u, d, s, &
c quarks
Meson: pseudoscalar and vector
mesons, B , B0 , B* , and g
Baryon: SU(4) multiplets of baryons
0
L
and b
Two partons, coalesce into a meson, three
partons into a baryon (anti-baryon), due to
their flavor, momentum, and spatial
coordinate and according to valence quark
0
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structure of hadron
 If coalescing partons can form either a
pseudoscalar or a vector meson, such
+
as ud can form either a r or a p +,
a principle of less discrepancy between
invariant mass of coalescing partons
and mass of coalesced hadron
invoked to select one from them
The same for baryon.
 Three momentum conservation is
required
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Phase space requirement
3
16p 2
h
D r 3D p3 =
9
g
h3
: Volume occupied by a single hadron
g
in phase space
g = 4: Spin and parity degeneracies
D r: relative distance between coalescing partons
for meson
D p: relative momentum between coalescing
partons for meson
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(4) Hadron evolution (rescattering)
 Consider only rescattering among
p, n, p , k, L, S , D, r (w), J Y, y '
 FOR simplicity,  inel  tot  0.85 is assumed
at high energ
 Assume
 pp   pn   nn   n   
 Usual tow-body collision model, employed
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LOGITUDINAL SCALING
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• Longitudinal scaling (rapidity scaling):
– Eg. dNch / dh(h' ), h' = h - ybeam ,
independent to beam energy
– A kind of limiting fragmentation ansatz (1969)
– First observed by BRAHMS (2001), then
PHOBOS (2003-2005)
– Using PACIAE to confront with that
• Model parameters are fixed, except b in
Lund string fragmentation, b is assumed
approximately proportion to sNN
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sNN GeV
b
200
6
theo.
Nch
exp .
Nch
5001
1 30
5
3977
62.4
2
2788
19.6
0.12
1588
5060 ± 250 4170 ± 210 2845 ± 142 1680 ± 100
• Results
– Charged particle transverse momentum
distribution
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– The dNch / dh(h)
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– Longitudinal (rapidity) scaling
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• v2 longitudinal scaling
– v2 together with jet quenching is an
evidence of sQGP
– Important and widely studied observable,
did not well introduced
– Give a exact deduction starting from
invariant cross section as follows
E
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d 3s
3
d p
µ E
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d 3N
d3 p
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Transferring into momentum cylindrical system,
substituting pz by y, and using dy / dpz = 1 / E
we have density function
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If distribution function N can separated
then multiplicity density function reads
where superscript on N is omitted
If proper normalization is introduced as follow
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the study of v2(y) should be started from
multiplicity azimuthal density function
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If above density function is isotropic then
above azimuthal density function reads
if
is 2p periodic and even function, above
density function can be expended as
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or
(1)
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It is obvious, < > means an average first over
particles in an event and then average over
all events if multiple events are generated .
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If azimuthal density distribution is isotropic then
because of
above azimuthal density function reduced to
so the anisoptropic effects are in
rather than
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p
63
The basic paper (PR, C58(1998)1671) starts
(2)
and then gave a statement
Reavtion plane: impact parameter vector in
px –py plane and pz axis
: measured with respect to reaction plane
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Reaction angle
: angle between reaction
plane and px axis, introduced for extracting
elliptic flow in experiment
In theory the impact parameter vector can be
fixed at px axis, so
 reaction plane is just the px –pz plane
 reaction angle =0
 is consistent with the definition before
 Eq. (2), different from (1) in normal. factor
and integrals over y and pT which make
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meaning of average more transparent
 As azimuthal density function reduces to
in isotropic azimuth, anisotropic effect is
referred by
rather than by
 Because, in that paper it is mentioned,
no possible, factor
in vn
was abserbed
.
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Conclusions:
1. The average, < >, should be first over
particles in an event and then over events,
rather than “over all particles in all events”
THE “over all particles in all events”
without the weight of event total multiplicity
is not correct in physics.
2. Anisotropic effects should be studied by
1
rather then
p
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SPECIFIC HEAT IN HM & QGM
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• Singularity behavior of specific heat,
relevant to phase transition
• Confusing status at present:
Specific heat of charged pions=
60 ±100, from experimental charged
pion transverse momentum distribution
in Pb+Pb colli. at 158A GeV
Specific heat=1.66, from simulated +
p
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transverse mass distribution in Pb+ Pb
colli. at 158A GeV by JPCIAE
(a hadron and string cascade model)
A specific heat of 13.2 was found
for pions in a pure statistical model
QCD matter (QGM) specific heat,
found to be larger than an ideal
gas of ～30 in thermodynamic potential
of pQCD
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 However, in a pure gauge theory,
specific heat of QGM is lower than
ideal gas of ～21
• To cleaning up, a parton and hadron cascade
model, PACIAE, used to study specific heat


of HM (represented by    ) and QGM
( u  d  g ) in an unified framework
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• Heat capacity,
Cv ,
is the quantity of heat
needed to raise the temperature of a
system by one unit of temperature (e. g.
one GeV)
 S 
 E 
Cv  T 
 


T

T

V

V , N
where T, V, N, S and E are, respectively,
the temperature, volume, number of
particles, entropy, and internal energy of
system
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• Specific heat, cv : heat capacity per
particle which composes the system
• Fitting the measured (calculated) particle
transverse momentum distribution
1 dN
P  pt  
pt dpt
to an exponential distribution
 pt 
PT ( pt )  A exp   
 T
temperature, T, extracted event-by-event
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• If reaction system (fireball), equilibrium,
event-by-event temperature fluctuation
obeys
Cv ( T ) 2
P (T ) ~ exp[ 
]
2
2 T
T : mean (equilibrium) temperature
T  T  T : temperature variance
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• Comparing above temperature distribution
to the general Gaussian distribution
1
1 (x)
P( x) 
exp[
]
2
2 
2
2
one finds following expression for heat
capacity
T  T
1

2
Cv
T
2
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• Three kinds of simulations
– Default (complete) simulations
labeled by “HM v. QGM”
– Simulation ended at partonic scattering,
labeled by “QGM”
– Pure hadronic cascade simulation
labeled by “HM ”
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
Transverse momentum distribution of HM





(
) and QGM ( u  d  g ) systems are
sum of their constituents with weight of
their multiplicity
P  pt  HM 
P  pt QGM
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M 
M   M 
P  pt   
M 
M   M 
P  pt  
Mg
Mu
Md

P  pt u 
P  pt d 
P  pt g
Mu  M d  M g
Mu  M d  M g
Mu  M d  M g
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• Temperature of HM and QGM systems is
obtained by fitting above transverse
momentum distribution to an exponential
distribution, within pt  1 event-by-event,
respectively
• Heat capacity of HM & QGM is obtained
• HM specific heat, for instance, reads
Cv
cv 
M   M 
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QGM in initial partonic stage and HM in
final hadronic stage, seem to be in
equilibrium
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• T , increase with SNN
• T in “HM v. QGM”>T in
“HM” reflecting effect of
initial partonic state
• cv ,decreases with SNN
cv ,a measure of
temperature fluctuation
The higher temperature
the lower fluctuation
• cv in “HM v. QGM”, a bit
larger than cv in “HM”,
attributed to competition
between temperature and
multiplicity fluctuation
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• CONCLUSIONS
(a) HM specific heat excitation function
resulting from “HM v. QGM” simulations is
close to the one from “HM ” simulations
(b) That indicates QGM specific heat, hard
to survive the hadronization
(c) There is no peak structure in “QGM”,
“HM v. QGM”, & “HM” specific heat
excitation functions in studied energy
region
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Thank you !!!
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