Transcript Lecture III (Seminar)

Lecture IV
Recent progress in
phenomenology (cont’d)
and theory
6. Recent progress in
phenomenology
HERA (Lecture III)
RHIC Au-Au
RHIC deuteron-Au
RHIC physics:
Au-Au collision
CGC at RHIC (Au-Au)
CGC at RHIC (Au-Au)
Most of the produced particles have small momenta less than 1 GeV ~
 Effects of saturation may be visible in bulk quantities
Qs(RHIC)
Multiplicity : pseudo-rapidity & centrality dependences
[Kharzeev,Levin,’01]
 in good agreement with the data
RHIC physics:
deuteron-Au collision
Experimental results
BRAHMS data for deuteron-Au collisions at RHIC
Nuclear modification
factor
- Cronin peak at h=0, suppression at h=3.2
- More enhanced for central at h=0, More suppressed for central at h=3.2
Theoretical analyses based on CGC
Numerical studies
Balitsky-Kovchegov equation
Albacete, Armesto, Kovner,
Salgado, Wiedemann 03
from top to bottom: h=0 to 10
Cronin peak exists at h=0, but rapidly disappears after evolution.
For h>1, the ratio monotonically increases as a function of pt.
Analytical studies
based on the MV model ----- can be used for mid-rapidity (moderate energy)
 Cronin peak
based on the BK equation -- can be used for forward rapidity (high energy)
 High pT suppression
Iancu, Itakura, Triantafyllopoulos, ’04
Cronin effect from MV model
At mid-rapidity, we assume that we can use the MV model.
Simplified, but shows essentially
the same behavior as RpA
Bremsstrahlung at high kt
Quantum evolution
Approximate solutions in each regime
1) CGC
2) BFKL
anomalous dimension
absorptive,
3) DLA
scaling,
scaling violation
(double log approximation)
High pt suppression (I)
- Distinguish three kinematical regimes for proton/nucleus
- Use the approximate solutions in each domain
- Form the ratio as a function of
Large difference btw saturation
scales: Qs(p,y) << Qs(A,y)
BFKL(A)
BFKL(A)
BFKL(p)
DLA(p)
Y=0.75
Y=0.1
Y=1.95
Y=0.7
Y=a
Y=
as sy y
High pt suppression (II)
General arguments
One can show in the linear regime (both p and A) within the saddle
point approximation that the ratio
is ….
1) a decreasing function of rapidity
2) an increasing function of kt
3) a decreasing function of A
where c is the BFKL kernel in the Mellin space
and g is the saddle point.
saturation DLA
c(gp) > c(gA) : proton evolves faster than nucleus.
Proton: far from saturation, fast evolution
Nucleus: already close to saturation, slow evolution
More phenomenological analysis
Jalilian-Marian ’04  used the CGC parametrization
data most forward rapidity y=3.2
hadronize via fragmentation function
Kharzeev, Kovchegov & Tuchin ’04  improved at high mom
CGC at LHC
Obviously, CGC becomes more important in LHC with higher
scattering energy.
√sNN = 14 TeV for pp, 5.5 TeV for PbPb
Rough estimate tells the saturation scale at LHC is increased by
a factor of 3 than that of RHIC.
Qs2(LHC) ~ 3 -- 10 GeV2
(mid) (forward)
Number of gluons in the saturation regime increases.
 Effects of saturation can be more visible.
Phase diagram with numbers (I)
x
From the CGC fit
Qs2(x)=(10-4/x)0.3
CGC Extended
Scaling
~BFKL
10-4
Parton gas
HERA
10-2
Q2
100
103
Phase diagram with numbers (II)
Extended scaling regime
x~10-3
forward rapidity
x~10-2
mid-rapidity
from Dima Kharzeev’s talk at NSAC Subcommittee on
Relativistic Heavy Ions, June 2004
7. Recent progress in theory
Geometric scaling as traveling wave
Physics beyond the BK equation
-- effects of fluctuation, Pomeron loop
-- odderon
Geometric scaling as traveling wave
• Munier & Peschanski ’03
The Balitsky-Kovchegov eq. with reasonable approximation (expansion
around BFKL saddle point) is equivalent to the F-KPP equation.(Fisher,
Kolmogolov, Petrovsky, and Piscounov)
change of variables
F-KPP equation
Logistic equation + spatial derivative
Geometric scaling as traveling wave
Very important because FKPP equation has been investigated over the
many years and understood very well.
- This equation allows a traveling wave solution, which connects the unstable
(u=0) and stable (u=1) fixed points.
u(x,t) = f (x-vt) : “geometric scaling”
- And the velocity of the wave front corresponds to the saturation scale!!!!!
 velocity is essentially determined by the linear part (BFKL)
 precise information about the saturation scale available
Physics beyond the BK equation
WHY??
1. We have been looking at only the first part of Balitsky’s infinite hierarchy,
and even its simplified version. The Balitsky equation
along the path of quark
Assume Factorization + take large Nc limit  Only the dipole operator which is
given as the solution to the BK eq. is relevant.
- Balitsky-Kovchegov eq. = physics of independent dipoles
2. How to justify the factorization <NN>  <N><N>?
Effects of fluctuation? Dipole-dipole correlations?
3. n-gluon exchange? (n Reggeon dynamics a la BKP or Korchemsky)
4. Role of non-dipole operator??
tr(Ux+UwUy+UwUz+Uw)
5. Imaginary part of the dipole scattering amplitude?
So far N(x,y) has been always assumed to be real.
The Langevin approach for the CGC
The Langevin approach is the simplest and most sophisticated
method for the CGC. JIMWLK eq. = Fokker-Planck eq.
After one step of evolution, the gauge field which the dipole feels is given by
The index i is the rapidity step (t = i e).
n(x) is the fluctuation which is given by white noise and generates random gauge
field a(x).
This equation generates everything !
 evolution equations of arbitrary gluonic operators.
The Balitsky eq. from the Langevin eq.
Blaizot, Iancu, Itakura, 04
Diagramatic derivation of the Balitsky equation for tr(U+(x)U(y))
Quadratic w.r.t.
fluctuation
Linear w.r.t. fluctuation
Due to white noise <na(x)>=0, <na(x)nb(y)> ~ dab d(x-y).
Quadratic correlation of the fluctuation gives the nonzero result.
Role of the fluctuation term (I)
Before taking the average, the Balitsky equation has a term linear wrt noise.
This vanishes after taking the average, but is important for the evolution
equations of dipole operators.
Consider one more step of evolution for a single dipole operator tr(U+xUy).
This includes evolution for tr(U+xUz) tr(U+zUy) = Sxz Szy
dipoles
Non-dipoles
Non-dipole operators are created by the linear-noise term of the Balitsky eq.
Role of the fluctuation term (II)
Non-dipole term represents
dipole-dipole interaction !
(Dipole branching
gives just the fan
diagram of Pomeron.)
Dipole branching
Dipole branching
Dipole-dipole
interaction
Evolution of non-dipole operators generates
dipole operators again (but less number of dipoles).
 Eventually generates Pomeron loops !
Pomeron loops !?
1 dipole
2 dipoles
3 dipoles
Dipole branching,
Normal evolution
4 dipoles
2 dipoles + 1 sextupole
3 dipoles + 1 quadrupole
5 dipoles (< 7 dipoles)
Perturbative QCD Odderon
Iancu,Itakura,McLerran,Hatta,in progress
• In QCD, the odderon is a three Reggeized gluon
exchange which is odd under the charge conjugation
cf) BFKL Pomeron
= 2 gluon exchange,
C-even
• What is the relevant operator for the odderon?
- Pomeron = tr(Vx+ Vy) with strong field (saturation)
 2 gluon operator {a(x)-a(y)}2 in weak field limit
(a(x) is the minus component of the gauge field)
- Gauge invariant combination of 3 gluons?
How to construct them?
C-odd operators
• Charge conjugation
• Fermions
mesonic

(+ even, -- odd)
baryonic

(+ even, -- odd)
• Gauge fields
any combination of 3 gluons with d-symbol is C-odd.
Intuitive construction of S-matrix
• Dipole-CGC scattering in eikonal approximation
scattering of a dipole in one gauge configuration
stay at the same transverse positions
average over the random
gauge field should be taken
C-odd S-matrix (dipole-CGC scattering)
• Transition from C-even to C-odd dipole states
• Relevant operator
Odipole(x,y) = tr(Vx+ Vy) – tr(Vy+ Vx) = 2i Im tr(Vx+ Vy)
- constructed from gauge fields, but has the same symmetry as for the
fermionic dipole operator M(x,y)-M(y,x)
 anti-symmetric under the exchange of x and y
Odipole(x,y) = - Odipole(y,x)
- imaginary part of the dipole operator tr(Vx+ Vy). Real part of the
scattering amplitude T (S = 1 + iT)
• Weak field expansion  leading order is 3 gluons
- should be gauge invariant combination
Evolution of the dipole odderon (I)
• Non-linear evolution eq. for the odderon operator can
be easily obtained from the Balitsky eq. for tr(V+xVy).
BFKL
- N(x,y) = 1- 1/Nc Re tr(V+xVy) is the usual “scatt. amplitude” (real)
- the whole equation is consistent with the symmetry
Odipole(x,y) = - Odipole(y,x) and N(x,y) = N(y,x)
- becomes equivalent to Kovchegov-Symanowsky-Wallon (2004)
if one assumes factorization <NO>  <N><O>.
- linear part = the BFKL eq. (but with different initial condition)
 reproduces the BKP solution with the largest intercept
found by Bartels, Lipatov and Vacca (KSW,04)
- intercept reduces due to saturation
As N(x,y)  1, Odipole(x,y) becomes decreasing !
Evolution of the dipole odderon
(II)
• The presence of imaginary part (odderon) affects the
evolution equation for the scattering amplitude N(x,y).
Balitsky equation
new contribution!
Open problems
•
Application to Ultra High Energy Cosmic Ray
ideal play ground for CGC : x ~ 10-9 – 10-10
•
Non-equilibrium properties
Langevin equation, Fokker-Planck equation
•
Fluctuations (Balitsky eq. vs BK eq. etc )
•
Impact parameter dependence
•
Phenomenological analysis (RHIC, HERA)
•
BKP equation
•
Exact solution to the BK equation?
Exact solution found in 2+1 dimensions
(n-point function)
Neutrino Nucleon Cross Sections
Contribution of small x partons
Figure from Gluck, Kretzer and Reya, Astropart. Phys.11 (1999) 327
Summary
-- Some of the physics at RHIC are consistent with CGC.
Brahms data on RdA (Cronin effect and high pT suppression)
-- The BK equation is essentially the same as FKPP eq. Geometric scaling
corresponds to the traveling wave solution and its velocity is the saturation
scale.
-- Interesting and rich physics is there if one looks beyond the BalitskyKovchegov equation.
-- Non-dipole operator (sextupole operator with 6 U’s) in the evolution of 2
dipoles appears.
-- This contribution is important since this physically represents the dipoledipole interaction, and eventually leads to dipole fusion, namely, creates
effectively the Pomeron loop.
-- There are still many interesting open problems, and it’s time to join this
activity!!!