Small-x physics
2- The saturation regime of QCD and the Color Glass Condensate
Cyrille Marquet
Columbia University
The saturation regime of QCD
gluon kinematics
recombination cross-section
gluon density per unit area
• gluon recombination in the hadronic wave function
recombinations important when
• the BFKL approximation
summing terms leads to a growth of the gluon densityin the hadronic wave function but high-density effects are missing
when the gluon density is large enough, gluon recombination is important
the saturation regime: for with
magnitude of Qs x dependencethis regime is non-linear,yet weakly coupled
the idea of the CGC is to take into account this effect via strong classical fields
the CGC is moving in the direction and the gauge isthe current is
solving Yang-Mills equations gives
The Color Glass Condensate
gggggqqqqqqgqqq .........hadron
separation between the long-lived high-x
partons and the short-lived low-x gluons
CGC wave function
CGC][hadron xD
the BFKL equation is recovered in the low-density regime
short-lived fluctuations
• effective description of a high-energy hadron/nucleus
][][2
fDf xx in practice we deal with CGC averages such as
we are interested in the evolution of with x, which sums both
2][x
Outline of the second lecture
• The evolution of the CGC wave functionthe JIMWLK equation and the Balitsky hierarchy
• A mean-field approximation: the BK equationsolutions: QCD traveling wavesthe saturation scale and geometric scaling
• Beyond the mean field approximationstochastic evolution and diffusive scaling
• Computing observables in the CGC frameworksolving evolution equation vs using dipole models
The evolution of theCGC wave function:
the B-JIMWLK equations
The JIMWLK equationJalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner
22][][
)/1ln( x
JIMWLKx H
xd
d
2][x• a functional equation for the x evolution of :
with
the JIMWLK equation gives the evolution of the wave function for small enough x
the equivalent Balitsky equations are obtained by consideringthe scattering of simple test projectiles (dipoles) off the CGC
the Wilson lines sum powers of
adjoint representation
Scattering off the CGC
their interaction should conserve their spin, polarisation, color, momentum …
in the high-energy limit, the eigenstates are simple when the partons
transverse momenta are Fourier transformed to transverse coordinates
x, y : transverse coordinates
in the large-Nc limit, further simplification: the eigenstates are only made of dipoles
),( yxother quantum numbers aren’t explicitely written, the
eigenvalues depend only on the transverse coordinates),();...;,( 11 NN yxyx
• eigenstates ?
xyxy ST 1][][
2 SDS xCGC
),();...;,(...),();...;,(ˆ1111 11 NNCGCNN nn
SSS yxyxyxyx yxyx
• eigenvalues
scattering amplitude off the (dense) target CGC
the interaction with the CGC conserves transverse position
Dipoles as test projectiles
))()((1
1][ uvuv FFc
WWTrN
T
][][][nn11nn11
2 vuvuvuvu TTDTT xx
u : quark space transverse coordinatev : antiquark space transverse coordinate
the dipole:qq
JIMWLK equation → evolution equation for the dipole correlators
scattering of the quark:
• dipoles are ideal projectiles to probe small distances
dependence kept implicit in the following
The Balitsky hierarchy
an hierarchy of equation involving correlators with more and more dipoles
Balitsky (1996)
YYYYYYTfTTTTT
zdT
dY
d)(
)()(
)(
2 22
22
uvzvuzuvzvuzuv vzzu
vu
2
1)()(
cYYY N
OTfTTTfTTdY
dzvuzzvuzzvuz
• equations for dipoles scattering of the CGC
the BFKL kernel
BFKL saturation
general structure:
we will now derive the first equation of the hierarchy
solving the B-JIMWLK equations gives , …which can then be used to compute observables
YTT zyxz
YTxy
• in the large Nc limit, the hierarchy is restricted to dipoles
• the valence component (using the mixed space):
Recall the dipole wave function
)();()(~22 yxyx qqyxddd
x : quark space transverse coordinatey : antiquark space transverse coordinate
other degrees of freedom are not explicitely written
qqd
such that only perturbative
sizes r << 1/ QCD are included )(~ r
22 )()( zyzy
zxzx
)();(),,(1)(~
2/122222 yxzyxyx qqzdNCgyxddd cFS
),();();(),,(2 agqqTzdg a
S zyxzyx
from QCD at order gS
colorindex
gqqqqd
the dipole rapidity:
• the dipole dressed with one gluon
Elastic scattering of the dipole
)();())()(()();(ˆ yxxyyx qqWWTrqqS FF
scattering of the quarkscattering of the antiquark
for the qqg component:
let’s compute the elastic scattering amplitude Ael(Y ) in the
dipole-CGC collision where is the total rapidityY = YCGC + Yd
• the dipole scattering is described by Wilson lines
)(xFW )(yFW )(zAW
][][)(~)(2222 xyyx TDyxddYiA Yel
• in the frame in which and
))()((1
1][ xyxy FFc
WWTrN
T Y
Tyxdd xyyx 222 )(~
Frame independence of Ael(Y )
CGCYel TyxddYiA xyyx222 )(~)(
using the following identity to get rid of the adjoint Wilson line
))()((2
1))()(())()((
2
1)())()(( xyxzzyzxy FF
cFFFF
abA
bF
aF WWTr
NWWTrWWTrWTWTWTr
= 1/2 -1/(2Nc)gqq qqqq qq
• in the frame in which andgqqqqd
one obtains
our two expressions for Ael(Y )
should be identical, this
requirement in the limit
Y - YCGC dY 0 gives
the first Balitsky equation
• frame independence:
CGCCGCCGCCGC YYYYCGC TTTTT
zdYY zyxzxyzyxzyzzx
yx22
22
)()(
)(
2)(
A mean field approximation: the Balitsky-Kovchegov equation
The BK equation
YYYTTTT zyxzzyxz the BK equation is a closed equation for obtained by assuming
YTxy
YYYYYY
TTTTTzd
TdYd
zyxzxyzyxzxy yzzxyx
22
22
)()()(
2
let’s consider impact-parameter independent solutions
as gets close to 1 (the stable fixed point of the equation), the non-linear term becomes
important, and , saturates at 1
with increasing Y, the unitarization scale get bigger
• solutions: qualitative behavior
• obtain by neglecting correlations
at small Y, is small, and the quadratic term can be neglected, the equation
reduces then to the linear BFKL equation and rises exponentially with Y
r = dipole size
Coordinate vs momentum space
both equation are in the same universality class as the F-KPP equation
unintegrated gluon distributiondipole scattering amplitude
• let’s go to momentum space
coordinate space momentum space
due to conformal invariance, the linear part of the equation is the same for and
genuine saturation: no real saturation:
linear BFKL equation linear BFKL equation
The F-KPP equation• same features as the BK equation
the precise forms of the space derivatives and of the non-linear term doesn’t matter
Fisher, Kolmogorov, Petrovsky, Piscounov
when expanding to second order,
the equations are the same (in momentum space)
in spite of small difference, same universality class
• dictionary F-KPP → BK
time derivative space derivative linear term meaning
exponential growth
non linear term
meaning saturation
F-KPP and BK asympotic (in t or Y) solutions are the same: traveling waves
these equations belong to the same universality class
position:
Traveling wave solutions
the speed of the wave is determined
only by the linear term of the equation
• what is a traveling wave
saturation
region
wave traveling at speed v,
independently of the initial condition
provided it is steep enough
there the initial condition
has not been erased yet
BK solutions: same rapidity evolution
quantitative features can be derived
• the formation of the traveling wave
Recall the BFKL solutions
initial condition in Mellin space
• the minimal speed
• a superposition of waves with speeds
is obtained for and its value is
this will be the speed of the
asymptotic traveling wave
• the saturation exponent
for the traveling wave to form, the initial condition
must feature
Y0
Y >Y0
QCD traveling waves• the initial condition is steep enough Munier and Peschanski (2004)
• asymptotic solutions of BK
in QCD
same features for exceptin the saturation region
• the saturation scale
with then
called geometric scaling
Numerical solutions
sub-asymptotic corrections (≡ geometric scaling violations) are also known:
• numerical simulations confirm the results
for the scattering amplitude for the saturation scale
Running coupling corrections• running coupling corrections to the BK equation
taken into account by the substitution
Kovchegov
Weigert
Balitsky
• consequences
similar to those first obtained by the simpler substitution
running coupling corrections slow down the increase of Qs with energy
also confirmed by numerical simulations, howeverthis asymptotic regime is reached for larger rapidities
Beyond the mean fieldapproximation
Recall the Balitsky hierarchy• equations for dipole correlators ][][][
nn11nn11
2 vuvuvuvu TTDTT xx
it describes the so-called dense regime where
YYYYYYTfTTTTT
zdT
dY
d)(
)()(
)(
2 22
22
uvzvuzuvzvuzuv vzzu
vu
21)()( cYYY
NOTfTTTfTTdY
d zvuzzvuzzvuz
general structureat large Nc
the BK equation is agood approximation
the equation for is fine, its dilute limit is the BFKL equation, however the other equationsare incomplete, they do not agree with Mueller’s dipole model in the limit
incorrect results for have consequences for
YTxy
1Y
TT zyxz
• still there is something missing
describing properly the dilute regime is also important
the B-JIMWLK equations do not describe it properly
YTxy
A new hierarchy of equations• the hierarchy must be completed
BFKL fluctuations (included to recover Mueller’sdipole model in the dilute limit) important when
saturationimportant when
Iancu and Triantafyllopoulos (2005)
this QCD evolution is equivalent to a stochastic process
• reformulation into a Langevin problem
average over therealisations of the stochastic process
the correlators can all be obtained from a singlestochastic quantity which obeys a Langevin equation
average over theCGC wave function
The stochastic F-KPP equation
high-energy QCD evolution = stochastic process in the universalityclass of reaction-diffusion processes, of the sF-KPP equation
Iancu, Mueller and Munier (2005)
• the QCD Langevin equation
),,,(),()()()(
)()(ln
21622
222
222222 YTvudzdd Y
S zvuvuzuvu
zyuxuyzx
vu
deterministic part of the equation: BK !
),(),(),(),(),()()(
)(2
),( 22
22
yzzxyxyzzxyzzx
yxyx YYYYYY TTTTT
zdT
dYd
noise
the reduction to one dimension introduces the noise strength parameter
noise
r = dipole size
• the stochastic F-KPP equation
A stochastic saturation scale
equivalent to a stochastic saturation scale Qs
the dispersion coefficient: D
• the stochastic evolutionthe Langevin evolution generates, at each rapidity, a stochastic ensemble of solutions for
the average speed of the waves:
(for ) DYSS 22 Q/Qln
DYDYP S
)Q/Q²(ln
exp1
)Q(ln22
S2SQs is a stochastic variable distributed
according to a Gaussian probability law:
corrections to the Gaussian law for improbable fluctuationsalso known, picture confirmed by numerical simulations C.M., Soyez and Xiao
average saturation scale:
• the probability distribution of Qs
analytic expressions for v and Dare known only for very small
A new scaling law
: the diffusion is negligible and )(Q)( YrTrT SY
one recovers geometric scaling
DYYrTrT SY)(Qln)( 22: the diffusion is important and
new regime: diffusive scaling
• the average dipole scattering amplitude
very high energies
The diffusive scaling regime
in the diffusive scaling regime:
the amplitudes are dominated by events thatfeature the hardest fluctuation of
in average the scattering is weak, yet saturation is the relevant physics
• important properties
• running coupling corrections
• conclusion at the moment
push the diffusive scaling regime up to asymptotically high energies
Dumitru et al.
using the mean field approximation is good enough for phenomenology
Computing observablesin the CGC framework
The CGC averages][][
2 SDS x
CGC
x
the energy evolution of cross-sections is encoded in the evolution of2
][x
in the CGC framework, any cross-section is determined by colorlesscombinations of Wilson lines, averaged over the CGC wave function
22][][
)/1ln( x
JIMWLKx H
xd
d
• the Wilson line correlators
Balitsky/BK equation for the correlators
this wave function is mainly non-perturbative, but its evolution is known
xFF WWTr ))()(( xy
this is the most common average, (for instance it determines deep inelastic scattering)its Fourier transform is the unintegrated gluon distribution, it obeys the BK equation
• the 2-point function or dipole amplitude
for 2-particle correlations
• more complicated correlators
for instance
The strategy
the same correlators enter in the formulation of very different processesthe more exclusive the final state is, the more complicated the correlators are
• what should be done in principle
- express the cross-sections in terms of correlators of Wilson lines
use appropriate initial conditionsinclude non-perturbative effects such as the impact parameter dependencerunning coupling effects should also be included
- use B-JIMWLK or BK equation to compute the correlators
many processes have been studied in the CGC framework using dipole models
• what is done in practice
solving evolution equations with running coupling is not the favored approachinstead one uses parametrizations with features inspired from theoretical results
it makes it easier to deal with impact parameter dependence of the dipole amplitude
famous dipole models: GBW, IIM, IPsat
Outline of the third lecture
• The hadron wave functionsummary of what we have learned
• The saturation modelsfrom GBW to the latest ones
• Deep inelastic scattering (DIS)the cleanest way to probe the CGC/saturationallows to fix the model parameters
• Diffractive DIS and other DIS processesthese observables are predicted
• Forward particle production in pA collisionsand the success of the CGC picture at RHIC
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