Transcript seminar at Nagoya U.

高エネルギーハドロン散乱では何
が起こるのか?
板倉数記
素粒子原子核研究所
高エネルギー加速器研究機構
セミナー＠H研（名古屋大学）
27th October, 2006
Plan
Part 1. カラーグラス凝縮：基本的性質と実験的証拠
Introduction
Unspeakables? / High energy limit of QCD / Color Glass Condensate
Gluon Cascade Linear and nonlinear evolutions
Theoretical framework for the CGC
MV picture / RGE for fast moving hadrons / The BK equation
Experimental evidences
Geometric scaling in DIS / deuteron-Au collisions at RHIC
Part 2. 反応拡散系との関係
Correspondence at mean-field level
BK equation = FKPP equation
More about the reaction-diffusion system
Stochastic FKPP equation / Effects of fluctuation
Back to CGC
K. Itakura (KEK)
October 27th, 2006
Part 1
カラーグラス凝縮：
基本的性質と実験的証拠
K. Itakura (KEK)
October 27th, 2006
Introduction
K. Itakura (KEK)
October 27th, 2006
Introduction (1/7)
Unspeakables in high energy scattering??
LHC完成直前の切迫した状況
小平さんが、スライドに書くのを躊躇したが、口頭で述べずにいられなかった事
（「素粒子物理学の進展 2006」、8月3日 基研） http://higgs.phys.kyushu-u.ac.jp/ppp06/
LHCでの陽子陽子衝突のパートン描像を説明した後、
「さらにですね、こういう分野を真面目にやっている人がかなりseriousに心配している事は
ですね、それを言うとみんなパニックになるのであまり言わないんですけれども､例えば
LHCくらいのエネルギーになると､Small-xのパートンのイベントがかなり多くなります。今ま
での経験でいくと､Small-xにいくと、パートンの密度は異常に上がっていきます。ということ
は、LHCくらいのエネルギーでは､パートンのdistribution functionは、今までのHERAや
Tevatronの単なるエネルギースケールアップで済むのかという恐れがあります。つまり、
RHICで起こっているような多体の衝突がLHCでも起こるかもしれない。そして、パートンの
distribution functionはLHC自体で測定して決める必要が出てくる。(…)業界の中ではそ
ういうことを議論しますが､パニックを生じさせるので､公には誰もしていないという状況で
す。だから、現実はかなり厳しい状況です。」
しばしばLHCにおけるNew physicsを見出すための「バックグラウンド」と
してしか考えられないQCD過程が、実は自明でない物理を孕んでいる！
K. Itakura (KEK)
October 27th, 2006
Introduction (2/7)
Unspeakables in high energy scattering??
他の専門家達の認識
MRST(Martin,Roberts,Stirling,Thorne) hep-ph/0507015
J. Ellis, “From HERA to the LHC” hep-ph/0512070
K. Itakura (KEK)
October 27th, 2006
Introduction (3/7)
High-energy limit of QCD
ハドロンの「ヴァレンス描像」は必ずしも正しくない！
____
baryon ~ qqq
meson ~ qq
（非相対論的クォーク模型）
 ハドロンを見るプローブを指定しないと意味がない
散乱エネルギーが高いとき、ハドロンは単純な少数系でなくなる
 エネルギーの増加とともにグルーオンが生成される
高エネルギーにおけるハドロンの普遍的な描像は
カラーグラス凝縮 （Color Glass Condensate）で与えられる
高密度のグルーオン物質、グルーオン間の非線形相互作用が効く
単純な摂動論では取り扱えない弱結合多体系の典型例
グルオンの多重生成は非平衡過程
K. Itakura (KEK)
October 27th, 2006
Introduction (4/7)
Internal structure of a proton (1/2)
深非弾性散乱 Deep Inelastic Scattering (DIS)
陽子をそれよりも小さなスケールを持つ『硬いプローブ』 (仮想光子)で叩くことで、
陽子の内部構造を知る
g*
陽子のInfinite momentum frameでは
1/Q
Q2 = qT2 : transverse resolution
x =p+/P+ : longitudinal mom. fraction
transverse
ここで P+ = (P0 + P3)/21/2
longitudinal
1/xP+
Pm = (P+, P--, PT)
陽子は「パートン」という点状の構成要素から成る
x ~ Q2/(Q2+W2) 散乱エネルギー（W2）を上げる→小さい x に行く
K. Itakura (KEK)
October 27th, 2006
Introduction (5/7)
Internal structure of a proton (2/2)
陽子が単純に３つのvalence
quarkからなれば、全運動量はそ
の３つに分配されるだろう
1/3
1/3
1/3
図は本来の20分の１
しかし、実際には
小さいｘ（高エネルギー）では
グルーオンだらけ！
P
Distr.
1/3
1
x
 高エネルギー
x ~ Q2/(Q2+W2)
素朴なヴァレンス描像は、実際の散乱では（見る場所によっては）役立たない。
高エネルギーでは、陽子は高密度グルーオン状態じゃないか!
Introduction (6/7)
Universality of hadronic cross section
Hadronic cross section at high energy (total cross sec. for pp)
 ab  Z ab  B ln2 (s / s0 )  
Including cosmic ray data
of AKENO and Fly’s eye
S1/2
10
102
103
104GeV
Recent PDG  consistent with ln2 s.
[COMPETE Collab.]
-- saturating unitarity (Froissart) bound
--
The coefficient B is universal (B=0.308mb) for pp, p p, p p, etc…..
K. Itakura (KEK)
October 27th, 2006
Introduction (7/7)
Color Glass Condensate
High-energy limit of QCD is the Color Glass Condensate (CGC)!!
COLOR : A matter made of gluons with colors.
GLASS : Almost “frozen” random color source creates gluon fields
CONDENSATE: High density. Occupation number ~ O( 1/as )
Many people contributed to establish the modern picture of the gluon saturation (Gribov,Levin,Ryskin,
Mueller,Qiu,McLerran,Venugopalan,Jalilian-Marian,Iancu,Weigert,Leonidov,Kovner,Balitsky,Kovchegov)
Multiple gluon
production
higher energy
Dilute gas
A new “semi-hard” scale:
“saturation scale” QS > LQCD
= typical transverse size of
gluons at saturation
CGC: high density gluons
Saturation scale = typical transverse momentum of gluons
 weak coupling aS(QS) << 1
Weakly interacting many body system of gluons (cf: CS, QGP)
K. Itakura (KEK)
October 27th, 2006
Gluon Cascade
K. Itakura (KEK)
October 27th, 2006
Gluon Cascade (1/3)
Linear evolution: BFKL equation
BFKL evolution : multiple soft gluon production Balitsky-Fadin-Kuraev-Lipatov
Leading log resummation
Growth of gluon number per energy (rapidity) increase
dn(Y )
  n(Y )
 : growth rate
dY
BFKL equation:
linear evolution  rapid growth  Unitarity violation!!
K. Itakura (KEK)
October 27th, 2006
Gluon Cascade (2/3)
Solution to the BFKL equation
• Coordinate space representation
t = ln 1/x
rapidity
• Mellin transform + saddle point approximation

• Asymptotic solution at high energy
t}
K. Itakura (KEK)
Dominant energy dep.
is given by exp{w aS
w = 4October
ln 2 =2.8
27th, 2006
Gluon Cascade (3/3)
Non-linear evolution: BK equation
What is missing in the BFKL dynamics?
Rapid growth = “population explosion”
 feed back effect reduces the speed of growth
When the gluon density becomes high,
produced gluons start to interact with each other!
dn(Y )
  n(Y )
dY

 n 2 (Y )
Logistic equation + transv. dep
 Balitsky-Kovchegov eq.
vs
ggg (increase)
rapid increase
gg  g (recombination)
saturation
Evolution becomes nonlinear  saturation
K. Itakura (KEK)
October 27th, 2006
Theoretical framework for the CGC
- Effective description of a fast moving hadron
- Its change under the increase of scattering energy
 Renormalization group equation
- Evolution equations
K. Itakura (KEK)
October 27th, 2006
Theoretical framework (1/9)
Effective theory of a fast moving hadron
McLerran-Venugopalan picture
Small x partons
(mostly gluons)
Small longitudinal mom. p+
 large LC energy ELC
 short life time ~ 1/ ELC
ELC
p2
p

, x 
2p
P
Large x partons
(mostly quarks)
Large longitudinal mom. p+
 small LC energy ELC
 long life time ~ 1/ ELC
1. During the short life time of small-x gluons, motion of large x partons is frozen.
2. At each time of the “life time”, configuration of large x partons will be different.
3. Small-x gluons have long wavelength in depth (longitudinal) direction,
allowing multiple “coherent” interactions with large x partons.
Small-x gluons can be treated as classical radiation field
created by static random color source on the transverse plane.
need average over
m a
m a

D F
  r ( x , x )
random color source r
Stochastic Yang-Mills equation
 weight function Wx[r]

K. Itakura (KEK)

October 27th, 2006
Theoretical framework
(2/9)
Effective theory of a fast moving hadron
• Observables are evaluated with average over the random source.
O[ r ] x   [ Dr ] O[ r ] Wx [ r ]
• Color neutrality must be ensured by the weight function. r a x  0
• Realized by the Gaussian distribution as the simplest example
(referred to as the McLerran-Venugopalan model, taken as the initial condition).
 1 2 r a ( x ) r a ( x ) 
Wx 0 [ r ]  Ν exp  d x

2
2
m


r a ( x ) : color charge distr. in transverse 2 dim. space
• Effective action for the small-x gluon field is given with (static) gauge
invariant source term.
S[ A, r ]  
K. Itakura (KEK)


1 4
i
a
m
3
 
d
x
F
F

d
x
T
r
r
(
x
)
T
exp
ig
dx
A
m a



4
gNc

October 27th, 2006
Theoretical framework (3/9)
Renormalization group equation
Higher energy
1
r
x0
x1
0
Wx0[r]
Only weight func.
gets modified at
ln (1/x) accuracy
 RG for W
r
Integrate
A
x
A
x0 > x1
weight function for random source
The JIMWLK equation
Wx1[r]
(Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner)
rapidity
WY [ r ] 1  2

WY    WY  , Y  ln1 / x0

Y
2 rr
r
 ,  : One and two point functions of r
K. Itakura (KEK)
October 27th, 2006
Theoretical framework (4/9)
Stochastic description of “evolution”(1/2)
Can change the variable from r to a = A+ in the LC gauge
• A functional Fokker-Planck equation
t = ln 1/x0 time
Wt[a]  probability density
• Normalization of Wt[a] is preserved under the RGE evolution
 Da Wt [a ]  1
at any t
• Can derive the Langevin equation which gives the JIMWLK eq.
 a highly non-linear equation for a with a white noise.
 can be understood as a Brownian motion in the space of Wilson lines.
( see Blaizot et al. Nucl. Phys. A713 (2003) 441. )
K. Itakura (KEK)
October 27th, 2006
Theoretical framework (5/9)
Stochastic description of “evolution”(2/2)
• Evolution of observable operators  the JIMWLK equation
Wt [a ]

O[a ] t   Da O[a ]
t
t
• The simplest and most important example:
2-point function of Wilson lines  S-matrix of the scattering of a color dipole
random gauge field
 The Balitsky equation : evolution equation for tr(Vx+Vy)
K. Itakura (KEK)
October 27th, 2006
Theoretical framework (6/9)
The basic equations for CGC
tr(Vx+Vy) is the relevant operator (Vx+ is in the fund. rep.)
The Balitsky equation evolution equation for energy increase
In general, CGC generates infinite series of evolution equations.
The Balitsky-Kovchegov equation
Nt(x,y) = 1 - (1/Nc)<tr(Vx+Vy)>t is the scattering amplitude
Balitsky eq.

Balitsky-Kovchegov eq. (ignore fluctuation)
<tr(V+V) tr(V+V)>
<tr(V+V)> <tr(V+V)> valid for large nucleus
K. Itakura (KEK)
October 27th, 2006
Theoretical framework (7/9)
The Balitsky-Kovchegov equation
Consequences
Y ~ ln s rapidity
• Txy Y : scattering amplitude of a color dipole ~ gluon number
• Derived from QCD in leading log accuracy (asln 1/x) in the mean-field
approximation
• BFKL + non-linear term
 Txy Y saturates (unitarizes) at fixed b = (x+y)/2 : Txy Y  1
• Saturation scale Qs(Y ) increases with rapidity Y : Qs2(Y ) ~ elY
• Geometric Scaling  amplitude Txy Y is a function of (x-y)Qs(Y )
[Levin,Tuchin,Motyka, etc, etc]
• Approximate scaling persists even outside of CGC regime [Iancu,KI,McLerran]
 discovery of new regime
K. Itakura (KEK)
October 27th, 2006
Saturation scale
Theoretical framework
(8/9)
1/QS(x) : transv. size of gluons when the transv. plane of a hadron is filled by gluons
1) r ・ ~ 1
2) when the unitarity effects set in
R
TY (r  1 / Qs)  1
Boundary between CGC and non-saturated regimes
- Energy and nuclear A dependences
LO BFKL
NLO BFKL
A dependence is modified in running coupling case
[Gribov,Levin,Ryskin 83, Mueller
99 ,Iancu,Itakura,McLerran’02]
[Triantafyllopoulos, ’03]
[Al Mueller ’03]
- Similarity between HERA (x~10-4, A=1) and RHIC (x~10-2, A=200)
QS(HERA) ~ QS(RHIC)
K. Itakura (KEK)
October 27th, 2006
Theoretical framework (9/9)
1/x in
log scale
QS2(x) ~ 1/xl: grows as x  0
Non-perturbative (Regge)
Higher energies 
Emerging picture
CGC
LQCD2
K. Itakura (KEK)
Extended
scaling
regime
QS4(x)/LQCD2
Parton gas
BFKL
BK
DGLAP
Fine transverse resolution 
Q2 in log scale
October 27th, 2006
Experimental evidences
K. Itakura (KEK)
October 27th, 2006
Experimental evidences (1/5)
Geometric Scaling
DIS cross section x,Q) depends only on Qs(x)/Q at small x
[Stasto,Kwiecinski,Golec-Biernat 01]
Natural interpretation in CGC
Qs(x)/Q=(1/Q)/(1/Qs) : # of overlapping times
1/Q: gluon size
Total
cross
section
Qs(x)/Q=1
=
Once transverse area is filled with gluons, the only
relevant variable is “number of covering times”.
 Geometric scaling!!
Saturation scale from the data
consistent with theoretical results
Geometric scaling persists even outside of CGC!!
 “Scaling window”
K. Itakura (KEK)
[Iancu,Itakura,McLerran,’02]
October 27th, 2006
Experimental evidences (2/5)
DIS at HERA
The simplest and cleanest process  precise information about CGC
No nuclear enhancement
 need very small x to see CGC
DIS at small x : color dipole formalism
intuitively transparent formula for the total cross section and F2
Y : LC wavefunction of a virtual photon (known)
Need the “dipole cross section” Effects of saturation
 dipole ( x, r )  2  d 2b N x (r , b)
Golec-Biernat & Wusthoff model ---- a simple parametrization
[Golec-Biernat,Wusthoff, Bartels, Kowalski,Teaney]
The CGC fit (based on solutions to the BK eq)
K. Itakura (KEK)
[Iancu,KI,Munier]
October 27th, 2006
Experimental evidences (3/5)
CGC fit to the HERA data
Fit for F2 data in x < 0.01 & 0.045 < Q2 <45
[Iancu, Itakura, Munier,’03]
GeV2
- Based on analytic solutions
to the BK equation Including
geometric scaling and its
violation, saturation effects.
- Only 3 parameters
[proton radius, x0 and l0.25
for Qs2(x)=(x0/x)l GeV2]
- Good agreement with data
- The same fit works well for
vector meson production,
diffractive F2, [Forshaw et al ’04 ]
FL [Goncalves,Machado’04]
K. Itakura (KEK)
October 27th, 2006
Experimental evidences
(4/5)
Deuteron-Au collisions at RHIC
Going forward in p(d)-A collision corresponds
to probing nuclear wavefunction at smaller x
Nuclear modification factor (Brahms)
RdAu
1 dNd  Au / d 2 pt dh

Ncoll dN p p / d 2 pt dh
d
q, g
x1 
pt h
e
s
x2 
pt h
e
s
g
If RdAu=1, d-Au is just
a summation of pp
Au
(up to iso-spin effect)
h(h-+h+)/2
Cronin enhancement
at h=0, suppression at
h=3.2
Lots of studies in the CGC framework
• Qualitative behaviors consistent with predictions of CGC.
• Cronin peak  multiple Glauber-Mueller scattering (MV model)
• High pt suppression  due to mismatch between “evolution speeds”
of proton & nucleus. Nucleus grows only slowly due to saturation.
• Quantitative results also available
K. Itakura (KEK)
October 27th, 2006
Experimental evidences (5/5)
pt spectrum in the CGC approaches
Kharzeev-Kovchegov-Tuchin ’04
Jalilian-Marian ’04
quark+gluon production
Valence quark
distribution
+ KKT param.
+ FF(LOKKP)
+ nonpert. Cronin
quark production
LO GRV98
for deuteron
+ IIM param.
(the CGC fit)
+ FF(LOKKP)
+ K factor
DumitruHayashigakiJalilian-Marian ’05
x- and DGLAP evolution
quark + gluon production
DGLAP for deuteron
+ FF(LO KKP)
+ LO CTEQ5 with K factor
+ KKT param.
K. Itakura (KEK)
October 27th, 2006
Phase diagram with numbers
From the CGC fit
Qs2(x)~(10-4/x)0.25
proton
x in log
x A1/3 ~ 6
nucleus (A~200)
x2 in log
Extended
CGC Scaling
~BFKL
LHC
10-4
10-4 y=4
Parton gas
y=2
HERA
10-2
10-2
y=0
y=0
RHIC
100
103
Q2 in log
A1/3 x100
103
k2 in log
CGC at LHC
LHC
√sNN = 14 TeV for pp, 5.5 TeV for PbPb
For the same pt, Qs2(LHC) is increased by a factor of 3 than Qs2 (RHIC).
Qs2(LHC) ~ 3 -- 10 GeV2
mid forward
Number of gluons in the saturation regime increases.
Predictions
 Effects of saturation will be more visible!!
Multiplicities in PbPb and p-Pb
[Kharzeev,Levin,Nardi]
RpA (red dashed line)
[Kharzeev,Kovchegov,Tuchin]
History of Small-x physics
as a summary of part 1
1960
1970
1980
1990
2000
QCD
S-matrixの理論
Reggeon &Pomeron
“Soft” Pomeron
ハドロン断面積がエネル
ギーとともに、ゆっくりと増加
する
 ~ s0.08
J.C.Collins “Introduction
to Regge theory and high
energy physics” (1977)
BFKL(’76-’78)
LO in as ln 1/x
GLR ~1983
MV 1994
CGC 2000
CGCのプロトタイプ
新しい見方の導入
定式化:JIMWLK, BK,...
’93 HERA F2 at small-x
 “hard” Pomeron
Soft Pomeronより 速い増加
~
s0.3
Forshaw & Ross “Quantum
Chromodynamics and the
Pomeron” (1997)
G.Salam, hep-ph/9910492.
非常によい講義録
K. Itakura (KEK)
NLO < 2000
’01 HERA 幾何的スケーリング
 Saturation scale
’04 RHIC 前方 dAu
 Quantum evolution
’01 COMPETE Collab.
 ~ ln2 s Froissart上限
Iancu & Venugopalan,
hep-ph/0303204 (review)
K. Itakura, hep-ph/0511031,
板倉：日本物理学会誌59（2004）148
October 27th, 2006
Part 2
反応拡散系との関係
K. Itakura (KEK)
October 27th, 2006
Plan (reprise)
Part 1. カラーグラス凝縮：基本的性質と実験的証拠
Introduction
Unspeakables? / High energy limit of QCD / Color Glass Condensate
Gluon Cascade Linear and nonlinear evolution
Theoretical framework for the CGC
MV picture / RGE for fast moving hadrons / The BK equation
Experimental evidences
Geometric scaling in DIS / deuteron-Au collisions at RHIC
Part 2. 反応拡散系との関係
Correspondence at mean-field level
BK equation = FKPP equation
More about the reaction-diffusion system
Stochastic FKPP equation / Effects of fluctuation
Back to CGC
K. Itakura (KEK)
October 27th, 2006
Correspondence at mean-field level
K. Itakura (KEK)
October 27th, 2006
Mean-field level (1/4)
BK equation ~ FKPP equation
Munier & Peschanski (2003~)
Within a reasonable approximation, the BK equation in momentum space
is rewritten as the F-KPP equation (Fisher, Kolmogorov, Petrovsky, Piscounov)
where t ~ Y, x ~ ln kt2 and u(t, x) ~ <T (k)>Y .
More precisely,…
The BK equation in momentum space
expand  around its saddle point g=1/2 to second order
K. Itakura (KEK)
October 27th, 2006
Mean-field level (2/4)
Global behavior w/o spatial dependence
Population dynamics
N(t) : (normalized) polulation density
When N<< 1 [Malthus 1798]
d
N (t )  N (t )
dt
t
N
(
t
)

N
e
0

population explosion
When N~1 [Verhulst 1838]
The Logistic equation
Gluon dynamics
NY: gluon density
The BFKL eq. [’75~]
Multiple gluon emission
NY  exp(wPY ), wP  4a S ln 2
unitarity violation
The BK eq. [’99~]
Gluon recombination
d
N (t )   ( N (t )  N (t ) 2 )
nonlinear
dt
Exponential growth is tamed by the
nonlinear term  saturation !
Initial condition dependence disappears
at late time  universal !
stable
unstable
Time
(energy)
Mean-field level (3/4)
Traveling wave
FKPP eq.
= “reaction” + “diffusion”
Famous equation in non-equilibrium statistical physics with broad range of
application: pattern formulation, expansion of epidemic, chemical reaction, etc
Reaction : logistic growth (ggg, vs ggg)
Diffusion : expansion of stable region
 Traveling wave solution
u = 1: stable
t
t’ > t
u = 0:unstable
K. Itakura (KEK)
October 27th, 2006
Mean-field level (4/4)
Saturation scale and geometric scaling
For a “traveling wave” solution, one can define the position
of a “wave front” x(t) = v(t)t .
 x(t) ~ ln Qs2(Y) Saturation scale !
“Boundary” btw dilute and saturated regimes
Main part of QS(Y) is determined irrespective
of initial conditions (selection of front velocity)
l
Q ( x)  (1/ x)  e
2
S
saturated
R
dilute
lY
At late time, the shape of a traveling wave is preserved,
and the solution is only a function of x – vt.
 x - v(t)t ~ ln k2/Qs2(Y) Geometric scaling !!
Asymptotic solution u ~ exp{ -g (x-vt)}
QS(Y) from the data consistent with theoretical results.
K. Itakura (KEK)
October 27th, 2006
More about the reaction-diffusion system
K. Itakura (KEK)
October 27th, 2006
The Reaction-Diffusion system (1/9)
Definition
Dynamics in 1 dimension
N
i
Splitting AAA and merging AAA occur at each site
Diffusion : Hopping to the right and left
N : allowed number of particles at each site
Equation for n(i): the number of particles at site i
mean-field approximation  FKPP equation
K. Itakura (KEK)
October 27th, 2006
The Reaction-Diffusion system (2/9)
Exact implementation: Master equation
ni(t) = number of particles at site i , {ni} = (n0 , n1, n2 …..) configuration
P({ni}; t ) : probability to have a particular configuration {ni} at time t
Change of P({ni}; t )
1) Diffusion (hopping to the right i  i+1, and to the left i  i –1 ) at rate D/h2
D
h2

D
h2
 (n
i
i , j 
 1) P (...ni  1, n j  1,...;t )  (n j  1) P(...ni  1, n j  1,...;t )
 n P({n}; t )  n P({n}; t )
i
i , j 
j
gain

h
loss
2) Splitting (AAA) at each site i at rate ls
lS  (ni 1) P(...,ni 1,...;t )  ni P({n}; t )
i
gain
loss
3) Merging (AAA) at each site i at rate lm
n (n  1)
 ni (ni  1)

P(...,ni  1,...;t )  i i
P({n}; t )
2
2


lm  
i
K. Itakura (KEK)
October 27th, 2006
The Reaction-Diffusion system (3/9)
“Field-theory” representation
Second quantization method
Doi ’76 , Peliti ’85, Cardy & Tauber ’98
a useful technique for a system with creation and annihilation of particles
Introduce bosonic operators
ai+ creates a particle at site i.
[ai , aj+] =  ij
[ai , aj] = [ai+, aj+] = 0
(Non-hermite) “Hamiltonian” is defined for a “probability vector”
  t (t )  H (t ) ,
(t )   P({n}; t ) {n}
{n}
For AAA and AAA, the master equation is exactly reproduced by
H 
D
h2


 (ai  a j )(ai  a j )  ls  (ai ) 2 ai  ai ai 
i , j 
i
lm
2
 a

i
ai  (ai ) 2 ai
2
2

i
Can construct a “field theory” by using coherent state path-integral

S   dxdt f ( t  D 2 )f  g s (f f  f 2f )  g m (f f 2  f 2f 2 )

where f is a coherent state eigen-value of annihilation operator, ai fi  fi fi

n

a
ai ~ fi
i
i
g =l , g =l h/2, particle number (density) :
s
s
m
K. Itakura (KEK)
m
October 27th, 2006
The Reaction-Diffusion system (4/9)
Deterministic & stochastic F-KPP equations

S   dxdt f ( t  D 2 )f  f ( g sf  g mf 2 )  f 2 ( g sf  g mf 2 )

Introduce a Gaussian noise h(x,t) as an auxiliary field
to resolve the last “fluctuation” term


e
2
f f f 2

  Dh e
1
 h 2 hf 2(f f 2 )
2
EOM  Stochastic F-KPP equation (after rescaling)
(t   2 )  (   2 )  2(   2 ) / N h ( x, t )  0
h( x, t )h( x' , t ' )   ( x  x' ) (t  t ' ) Gaussian noise
In the limit N =gs/gm infinity, reduces to the FKPP equation
K. Itakura (KEK)
October 27th, 2006
The Reaction-Diffusion system (5/9)
Effects of fluctuation: numerical result
Two major modifications to deterministic FKPP
solution:
Solutions at two different time,
for many events
Enberg, Golec-Biernat, Munier PRD72 (05)
1. Front velocity becomes slower.
2. The shape of the traveling wave does not change a lot,
but the position of the front becomes stochastic (Gaussian).
Space-time uncorrelated noise induces quite nontrivial response
K. Itakura (KEK)
October 27th, 2006
The Reaction-Diffusion system (6/9)
Reduction of front velocity
Discreteness becomes important when the number of particles are few.
 At the tail of a traveling wave : (x,t) ~ 1/N << 1
 large effect: Diffusion controls the propagation because linear growth
does not work without “seeds”.
 The velocity of a traveling wave is reduced.
Brunet, Derrida, PRE ’97
FKPP with a cutoff
K. Itakura (KEK)
October 27th, 2006
The Reaction-Diffusion system (7/9)
Stochastic front position
1. Treat fluctuation as perturbation to the cutoff FKPP equation
0(z=x-vt) : unperturbed solution, scaled  cutoff FKPP
 (z,t): perturbed, not scaled, to be determined
Linearized equation (linear response analysis)
2. Equation for  can be solved by introducing Green’s function
where
Green function is associated with fluctuation of the (cutoff) FKPP equation.
K. Itakura (KEK)
October 27th, 2006
The Reaction-Diffusion system (8/9)
Stochastic front position
3. Stability analysis of the (cutoff) FKPP equation
Spectrum of the fluctuation around the FKPP solution
Dominant fluctuation around 0  zero mode
Zero mode is due to the translational invariance.
induces the shift of the solution!!
4. This zero mode couples to the external noise term
Change of the front position X(t) due to the noise is
proportional to the noise
t
d
X (t )  C 2  dt ' dz ' e vz'
0
dz
( z ' ) 2( 0   02 ) / N h ( z ' , t ' )
can easily compute the diffusion coefficient < X(t)2>=Dt
K. Itakura (KEK)
October 27th, 2006
The Reaction-Diffusion system (9/9)
Stochastic front position
Comments
1. Already done some years ago… ex) Rocco, et al. PRE65(2001)012102
Exactly the same formula for the displacement X !!
2. Not the same as the diffusion coefficient obtained by the
other method [Brunet, Derrida, Mueller, Munier, PRE73(2006)056126]
! Our result shows different asymptotic behavior D~1/ln6N
 because of different definition of the front?? [Panja ’03]
3. Possible to treat the front position as a collective coordinate
in the field theory representation [work in progress]
(cf: Zero-mode fluctuations of a soliton are treated as collective coordinates)
 can go beyond the linear response theory??
K. Itakura (KEK)
October 27th, 2006
Back to CGC
K. Itakura (KEK)
October 27th, 2006
Back to CGC (1/3)
Effects of fluctuation on the BK eq.
“Mean-field BK equation = Deterministic FKPP equation ”
Equation representing “full” reaction-diffusion dynamics with A AA and
AAA is not the FKPP equation, but the stochastic FKPP equation.
Need to find an equation representing the full dynamics of
both the Pomeron merging and splitting. (BK contains the effects
of splitting only partially)
1. Effects of fluctuation is significant when gluon (dipole) number is small
(high transverse momentum).
T (r )
Y
 a S2 n(r , Y )  a S2
2. Add 3 Pomeron vertex (Pomeron splitting) which becomes important in
the dilute regime (in scattering of 2 dipoles)
splitting
K. Itakura (KEK)
October 27th, 2006
Back to CGC (2/3)
Effects of fluctuation on the BK eq.
3. Construct the evolution equation so that the BK + noise term can correctly
reproduce the evolution equation of the 2 dipoles  Stochastic BK eq.
[Iancu-Triantafyllopoulos]
+
4. Stochastic BK reduces to Stochastic FKPP in the diffusive approximation.
5. Saturation scale becomes slowly increasing due to diffusion at the edge,
stochastic variable due to fluctuation term in sFKPP eq.
[Iancu,Mueller,Munier]
6. The stochastic saturation scale is induced by the zero mode fluctuation of
the BK equation, which is related to the scale transformation in 2
dimensional transverse momentum.
7. The “field theory” representation of the reaction-diffusion dynamics is very
similar to the Reggeon field theory.
K. Itakura (KEK)
October 27th, 2006
Back to CGC (3/3)
Effects of fluctuation on the BK eq.
Dipole-pair scattering
K. Itakura (KEK)
October 27th, 2006
Summary of Part 2
• At very high energy, a proton (in fact, any hadrons) looks as
the Color Glass Condensate, a densely saturated gluonic
system. This is a weakly interacting many body state.
• Its dynamics is essentially equivalent to the reactiondiffusion dynamics. The BK equation ~ the FKPP equation.
Rich information from the statistical physics is available.
• In particular, the effects of fluctuation beyond the meanfield BK picture have been recognized to be significant in
dilute regime (at high transverse momentum)
• Slowly-growing and stochastic saturation scale is obtained.
K. Itakura (KEK)
October 27th, 2006
Outlook: problem list
• LHC : Need to compute all the possible observables measured
in LHC. This involves pp, pA, and AA collisions.
• Ultra high-energy cosmic ray : The ideal laboratory for small-x
physics (x~10-9 ). CGC picture is vital for collisions btw the
primary cosmic ray (~proton) and nuclei in the atmosphere.
• Thermalization : CGC provides initial condition for the heavy
ion collisions. But the matter just after the collision should also
be described by strong gauge fields (called “glasma”) that are
very similar to CGC. Need to know how glasma evolves into
thermally equilibrium plasma (QGP). Fluctuations?
Instabilities? Unruh effects?
• Reggeon field theory from CGC: Need to extend the reactiondiffusion picture to include the impact parameter dependence.
K. Itakura (KEK)
October 27th, 2006