1. Ben-Hao Sa China Institute of Atomic Energy
Relativistic nuclear collision in pQCD and corresponding dynamic simulation
Ben-Hao Sa
China Institute of Atomic Energy
2019/4/17
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2. HADRON-HADRON COLLI. IN pQCD DYNAMIC SIMULATION FOR hh
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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3. INTRODUCTION
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4. 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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5. 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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6. 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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7. HADRON-HADRON
COLLI. IN pQCD
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8. 1. Cross section of hadron production in hadron-hadron ( + ) colli.
hh colli. = superposition of
parton-parton colli.
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9. : cross section of sub-process
: parton ( ) distribution function in
hadron ( )
: momentum fraction taking by
from
: scale of scattering
: fragmentation function of to
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10. 2. cross section of partonic sub-process
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11. Subprocess cross section expresses as
(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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13. Mandelstum variables:p1 p3 p2 p4
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14. 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
scatterings (for PDF) and/or of the
annihilations (for PFF)
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15. Most simplest PDF (without depen.) at large x region is something like
Most simple PFF is some thing like
Total fractional momentum carried by :
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16. 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
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17. DYNAMIC SIMULATION FOR HADRON-HADRON COLLI. (PYTHIA MODEL)
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18. … Sketch for pp simulation in PYTHIA p p Remnant
Initial state radiation p
Rescattering ? … h
Parton distribution
function
Decay p
Final state radiation
Remnant
Hadronization
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19. 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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20. 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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21. … Hadron rescattering (?) and decay Multiple String Fragmentation
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22. 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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23. NUCLEUS-NUCLEUS COLLI.
IN pQCD
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24. nucleus-nucleus (A+B) collision is calculated under assumptions of
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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25. ``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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26. The cross section can be expressed as
: normalized thickness function of
nucleus
Phenomenological considerations for:
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27. Multiple scattering (ela. diffractive,…) Jet quenching (energy lose)
Nuclear shadowing
Multiple scattering (ela. diffractive,…)
Jet quenching (energy lose)
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28. : probability having a nn colli.
C) Glauber method
(Glauber theory with nn inelastic cross
replaced by pQCD nn cross section)
: probability having a nn colli.
within transverse area when nucleon
passes at impact parameter
: thickness
function of nn
collision
nucleon a c
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29. Probability for occurring an inela. nn
: probability finding a nucleon in volume in nucleus A at , which is normalized as
Probability for occurring an inela. nn
colli. when nucleus A passes B at an impact parameter is
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30. Probability for occurring n inela. colli. is
as there can be up to collisions
Total probability for occurring an event
Probability having
an inela. Colli.
combinations
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31. of nucleus-nucleus inela. colli. at impact parameter is
Total cross section of above event is
If one use pQCD p+p cross section
instead of in above equations
one has pQCD inela. cross section for
(A+B) colli.
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32. NUCLEUS-NUCLEUS COLLI. (PACIAE MODEL)
DYNAMIC SIMULATION OF
NUCLEUS-NUCLEUS COLLI.
(PACIAE MODEL)
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33. 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 parton-parton 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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34. The PACIAE model is composed of (1) Parton initialization
(2) Parton evolution
(3) Hadronization
(4) Hadron evolution
four parts
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35. (1) Parton initialization Nucleon in colliding nucleus is
distributed due to Woods-Saxon ( ) and
4 (solid angel) distributions
Nucleon is given beam momentum
Nucleon moves along straight line
nn collision happens if their least approaching distence
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36. their collision time is then calculated Particle (nucleon) list
order # of particle four momenta
and nn collision time list
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37. order # of colliding pair collis. time . . .
are constructed
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38. 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 and g ( if
diquark (anti-diquark) is forced
splitting into randomly)
Nucleon propagate along straight
line in time interval equal to difference
between last and current colli. times
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39. 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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40. (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
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41. For process of , for instance
That has to be regularized as
by introducing color screening mass
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42. Total cross section of sub-process (4) at high energy
Using above cross sections parton scattering can be simulated by MC
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43. (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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44. 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
pair randomly
There is a hadron table composed of
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45. Field-Feynman parton generation mechanism … Original quark jet
Created quark pairs from vacuum
(if mother with enough energy) …
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46. mesons & baryons, made of u, d, s, & c quarks
Meson: pseudoscalar and vector
mesons, and
Baryon: SU(4) multiplets of baryons
and
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
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47. Three momentum conservation is required
structure of hadron
If coalescing partons can form either a pseudoscalar or a vector meson, such
as can form either a or a ,
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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48. Phase space requirement
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49. Consider only rescattering among
(4) Hadron evolution (rescattering)
Consider only rescattering among
FOR simplicity, is assumed
at high energ
Assume
Usual tow-body collision model, employed
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50. LOGITUDINAL SCALING
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51. Longitudinal scaling (rapidity scaling):
Eg ,
independent to beam energy
A kind of limiting fragmentation ansatz (1969)
First observed by BRAHMS (2001), then
PHOBOS ( )
Using PACIAE to confront with that
Model parameters are fixed, except b in
Lund string fragmentation, b is assumed
approximately proportion to
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52. Charged particle transverse momentum distribution
Results
Charged particle transverse momentum distribution
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53. 2019/4/17
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54. The
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55. Longitudinal (rapidity) scaling
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56. 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
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57. substituting pz by y, and using we have density function
Transferring into momentum cylindrical system,
substituting pz by y, and using
we have density function
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58. then multiplicity density function reads
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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59. multiplicity azimuthal density function
the study of v2(y) should be started from
multiplicity azimuthal density function
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60. above azimuthal density function reads
If above density function is isotropic then
above azimuthal density function reads
if is periodic and even function, above
density function can be expended as
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61. or
(1)
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62. 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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63. above azimuthal density function reduced to
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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64. and then gave a statement
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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65. 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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66. 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 was abserbed
in vn .
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67. 1. The average, < >, should be first over
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
rather then
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72. SPECIFIC HEAT IN HM & QGM
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73. 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
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74. 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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75. specific heat of QGM is lower than ideal gas of ～21
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 ( ) in an unified framework
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76. Heat capacity, , is the quantity of heat
needed to raise the temperature of a
system by one unit of temperature (e. g.
one GeV)
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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77. Specific heat, : heat capacity per particle which composes the system
Fitting the measured (calculated) particle transverse momentum distribution
to an exponential distribution
temperature, T, extracted event-by-event
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78. If reaction system (fireball), equilibrium,
event-by-event temperature fluctuation
obeys
: mean (equilibrium) temperature
: temperature variance
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79. Comparing above temperature distribution
to the general Gaussian distribution
one finds following expression for heat
capacity
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80. 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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81. Transverse momentum distribution of HM ( ) and QGM ( ) systems are
sum of their constituents with weight of
their multiplicity
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82. Heat capacity of HM & QGM is obtained
Temperature of HM and QGM systems is obtained by fitting above transverse momentum distribution to an exponential distribution, within event-by-event, respectively
Heat capacity of HM & QGM is obtained
HM specific heat, for instance, reads
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84. 2019/4/17
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85. QGM in initial partonic stage and HM in
final hadronic stage, seem to be in
equilibrium
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86. ,a measure of temperature fluctuation
T , increase with
T in “HM v. QGM”>T in “HM” reflecting effect of initial partonic state
,decreases with
,a measure of temperature fluctuation
The higher temperature the lower fluctuation
in “HM v. QGM”, a bit larger than in “HM”, attributed to competition between temperature and multiplicity fluctuation
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87. 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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88. Thank you !!!
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