High Density QCD
M.B. Gay Duati
InstitutodeFsia,UniversidadeFederaldoRioGrande doSul
CaixaPostal 15051,CEP91501-970, PortoAlegre,RS,Brazil
Reeivedon19April,2001
ThedynamisofhighpartonidensityQCDispresentedonsidering, inthedoublelogarithm
ap-proximation,thepartonreombinationmehanismbuiltintheAGLformalism,developedinluding
unitarityorretionsforthenuleonaswellfornuleus. Itisshownthattheseorretionsare
un-der theoretial ontrol. The resulting non linear evolution equation is solved inthe asymptoti
regime,andaomprehensivephenomenologyonerningDeepInelastiSatteringlikeF2,FL,F
2 .
F
2 =lnQ
2
,F A
2 =lnQ
2
,et,ispresented. TheonnetionofourformalismwiththeDGLAPand
BFKLdynamis,andwithotherperurbative(K)andnonperturbative(MV-JKLW)approahesis
analisedindetail. Thephenomenaofsaturationduetoshadowingorretionsandtherelevaneof
thiseetinionphysisandheavyquarkprodutionisemphasized. Theimpliationstoe-RHIC,
HERA-A,andLHCphysisandsomeopenquestionsarementioned.
I Introdution
ThedynamisofthehighdensityQuantum
Chromody-namis (hdQCD)isoneofthepresentmosthalleging
openquestionsinhighenergyphysis. Theintense
the-oretial and experimental ativity towardsthe
under-standingofsmallx(smallfrationofprotonmomentum
arried by the struk parton) QCD takes plae from
Deep InelastiSattering(DIS)at HERA[1℄to heavy
ions ollisions (HIC) at RHIC [2℄. This kinematial
regime willalsobetestedatLHC inanearfuture[3℄.
Important ontribution to the interest of the eld
is due to the puzzling result obtained by HERA at
small-x (x 10 2
) [4℄ for the proton struture
fun-tionF
2 (x;Q
2
). Thisfuntion was observedto inrease
dramatially as x gets smaller (Fig. 1). In the
re-gion of moderate Bjorken x (x 10 2
) the
Opera-tor Produt Expansion (OPE)methods aswell asthe
Renormalization Group Equations (RGE) have been
applied suessfully [5℄. The evolution of quark and
gluondistributionfuntionsgivenbytheDGLAP
equa-tions[6℄isbasedonthesummingoftheleadingpowers
of
s lnQ
2
1,
s
ln(1=x)<<1,
s
<<1,where
s is
thestrongouplingonstant. Theleadingln(1=x)
on-tributions is thease forthe BFKLequation [7℄. The
proedureknownasthedoubleleadinglogarithmi
ap-proximation(DLA)orrespondsinaxialgaugesto
gen-eratethelogarithmsbyladderdiagrams,whoseemitted
gluonshavestronglyorderedtransverseand
longitudi-nal momenta, summing the logs
s lnQ
2
ln(1=x). It
wasshown that the DLA is a ommon limit between
thelineardynamis[8℄.
Theinreasingofthepartondensitiesrequiresa
for-mulationof the QCD at high partoni density, where
unitarity orretions (UC), notonsidered in the
pre-viousdynamisalreadymentioned, areproperlytaken
into aount. In this sense,the small x region, where
thegluondistributionsetsthebehaviorofthemain
ob-servables, providesthe interfae between perturbative
andnonperturbativeQCD,orinotherwords,between
hard and soft physis. Clearly, both experimentalists
and theoretiians are hallenged to desentangle,
mea-sureandformulatethedynamialolletiveeetsthat
aresubjaenttotheobservedresultofinreasingF
2 and
therosssetion
tot
atDIS,asxgetssmaller[4℄. Both
evolutionequations, DGLAP (evolutionin lnQ 2
) and
BFKL (evolutionin ln(1=x)) as representativesof
lin-eardynamis,needontrolinordertorestoreunitarity,
sinetheFroissartlimitrequires
tot
Cteln 2
s[9℄.
Aomprehensivetreatmentshouldenvolveboth
lin-ear and non linear regimes. The main attempts to
develop a formalism for hdQCD are the approahes
of M.Lerran andollaborators (MVJKLW)[10℄, by
Kovhegov (K) [11℄ and by Ayala, Gay Duati and
Levin(AGL)[12,13℄. Derivedindependently,thethree
methods obtainnonlinear evolutionequations forthe
gluondistribution,atthesmallxregion,desribingthe
onsetofhdQCD,althoughonsideringdierentdegrees
offreedom.
InwhatfollowsIwillpresentanintrodutoryreview
ofthesubjetofhdQCD,themainaspetsofthe
formu-lationsto thesubjet,theonnetionsamong themin
theasymptotiregion(x!0), presentthestateofart
of the phenomenology of small x physis and address
someopenquestions.
II The Evolution Equations
ItwillbebrieypresentedtheDGLAPandBFKL
dy-namisand theirpreditionsforthesmall xregime at
DIS. TheDIS is theproess of interation ofalepton
andanuleonexhanginganeletroweakboson
produ-ingmanypartilesatthenalstate,whihisahadroni
stateX. Theproessis
l(k)N(p)!l 0
(k 0
)X ; (1)
wherek,k 0
,pandp 0
arethefourmomentaoftheinitial
and nal lepton, inident nuleon and nal hadroni
system, respetively. Themain variables forthis
pro-ess are Q 2
= q
2
= (k k 0
) 2
, whih is the square
ofthe transferedmomentum, s=(k+p) 2
, thesquare
of the enter of mass energy, W 2
= (q+p) 2
= (p 0
) 2
,
the square of theenter of massenergy of the virtual
boson-nuleon system. The hard sale is given by q 2
(< 0), orresponding to the proess resolution, and
x = Q 2
=2p:q, meaning the virtual boson resolvesthe
hadronistruture,orthepartons,sinex1= p
Q 2
. Arealsouseful thevariables y =q:p=k:p, measuring
theproessinelastiityand=q:p=m
N
,theenergyof
thevirtualbosononetakeninthetargetrestframe.
Intermsofthepartoniontentofthenuleonthe
struturefuntion,whihreetsitsoveralldistribution,
isgivenby
F
2 (x;Q
2
)= X
f e
2
f [q
f
(x;Q 2
)℄; (2)
wherethesumisoveravoursweightedbythe
respe-tivesquaredharges(e 2
f
). Itisthisfuntionthatis
ob-jetof main experimental studies,speially atHERA,
at the small longitudinal nuleon fration of
momen-tum,orsmallx (seeFig. 1)[14℄.
0
0.5
1
1.5
2
2.5
3
3.5
4
10
-3
10
-2
10
-1
1
x
F
2
(x,Q
2
)
0
0.5
1
1.5
2
2.5
3
3.5
4
10
-3
10
-2
10
-1
1
x
F
2
(x,Q
2
)
Figure1
The quark distribution funtion an be shown to
evolveas
q f
(x;Q 2
)
lnQ 2
=
s
2 Z
dx
1
x P
qq (
x
x )q
f
(x;Q 2
); (3)
where P
qq = C
F 1+z
2
1 z j
+
(with self-energy orretions
overthekpropagator)is oneofthesplittingfuntions
P
ij (P
gq =C
F 1+(1 z)
2
z
, et),desribingthetransition
between the quarkstate i to the quark state j, from
fration of momentum x
1
to x. The aboveevolution
refersto thenonsinglet quarksdistributionwheresea
quarksand gluondistributionareunoupled
q
NS (x;Q
2
)q
i (x;Q
2
) q
j (x;q
2
): (4)
Thesinglet quarkdistributionisgivenby
q
S (x;q
2
) X
f
q f
(x;Q 2
)+q f
(x;Q 2
)
; (5)
where the gluon distribution is oupled to the quark
one.
Now theevolutionequations, in thelinear regime,
readforthequarks
q
S (x;Q
2
)
lnQ 2
=
s (Q
2
)
2 Z
1
x
P
qq (
x
x
1 )q
s (x
1 ;Q
2
)
+ P
qg (
x
x
1 )g(x;Q
2
)
; (6)
andforthegluondistribution
g(x;Q 2
)
lnQ 2
=
s (Q
2
)
2 Z
1
x
P
gq (
x
x
1 )q
s (x
1 ;Q
2
)
+ P
gg (
x
x
1 )g(x;Q
2
) : (7)
TheEqs. (6)and(7)wereindependentlyderivedby
Dokshitzer, Gribov and Lipatov, and by Altarelliand
Parisi, known as DGLAP equations in leading order.
The perturbative QCD evolution, in the linear setor
isgovernedbyDGLAPequations,moreoverasuitable
non perturbativeiniial ondition, extrated from the
experimentforagivenbosonvirtuality.
It an be shown by suessive derivations that
q
NS
(x;") P
n (
s ")
n
, " = lnQ 2
, whih orresponds
to theemissionofn gluons,showingthat theDGLAP
equations resum the leading lnQ 2
. This an be
un-derstoodasladderdiagrams withastrongordering in
transverse momenta k
?
, i.e., Q 2
0 << k
2
1
<< ::: <<
k 2
n <<Q
2
. ThesaleQ 2
0
istheut,ortransitionvalue
betweenperturbativeandnonperturbativephysis. It
wasshownby Gribov [15℄ thatthis resultis gauge
in-dependentoneoneonsiderstheleadinglogarithm
At small-x the gluons dominate, sine P (0)
gg (z)
2N
z
,andtheparton distributionshavethegeneral
be-havior xp
i (x;Q
2
) x
, > 0. More likely for
ini-tial ondition xp
i (x;Q
2
0
) Const and xp
i (x;Q
2
)
exp p
ln(lnQ 2
)ln1=x, known as double leading
loga-rithm approximation (DLA). From that is lear that
DGLAPpreditstheinreaseofthegluondistribution
funtion,andofthestruturefuntion F
2
withthe
de-reasingofx,whoserelationinthiskinematialregime
isgivenby
F
2 (x;Q
2
)
lnQ 2
=
s (Q
2
)
2 X
f e
2
f
xg(x;Q 2
) (8)
beingequalto 2
s
9
xg(x;Q 2
)forn
f =3.
The DLA implies strong ordering in x and k
T , i.
e., x
1 >> x
2
>> :::: >> x
i 1 >> x
i
>> x and
k
T
1 << k
T
2
<< :::k
T
i 1 << k
T
i
<< Q 2
, the resum
of logs of
s
ln (1=x)lnQ 2
, havingasregionof validity
s
<<1,
s
ln(1=x)<<1and
s
ln(1=x)lnQ 2
1.
The resum of all leading logarithms of Bjorken x,
or the energy, is harateristi of the
Balitsky-Fadin-Kuraev-Lipatov(BFKL)equation. Forverylowx
val-ues the lns beomes largeand
s
ln1=x 1and the
DLA is not valid. The BFKL evolution equation is
proposed for an unintegrated gluon distribution
fun-tioninthetransversemomentumvariable. Itssolution
growsas apower of the enter of mass energy s with
the onsequent violationof the unitaritybound [9℄ at
veryhighenergies. Theurefor thisproblem was not
reahed in the next to leading order alulation [16℄,
andisstillunderresearhforinstane,throughthe
re-summingof allBFKL Pomeronexhanges[17℄,forthe
rosssetionaswellasforthestruturefuntion.
Theamplitudeforthesatteringquark-quark with
one gluon exhange in the t hannel at lowest order
is given by A
0
(s;t) s=t, and for the next order
in
s
the leading terms are given by lns, resulting
A
1
(s;t) A
0
(s;t)lns. One one goes to higher
or-dersthenumberofontributingdiagramsinreasesand
the alulation gains enormously in omplexity [18℄,
and the usual proedure is to introdue an eetive
vertex (Lipatov vertex). It results that the general
termisA
n
(s;t)A
0 (s;t)
n
(t)ln n
(s)=n,where(t)isa
suitable funtion to takeare ofinfrared divergenies,
doile under regularization, for instane, dimensional
regularization.
Clearlythe BFKL evolutionis summing theterms
n
s ln
n
(s), wherelowerorder logarithmsare negleted.
Theresultfortheamplitudeis
A(s;t)=A
0 (s;t)
1
X
n=0
n
(t)ln n
(s)
n!
s (t)
; (9)
with (t)=1+(t). Inthisase,thereisstill the(t)
infrared divergeniestobeured.
Whenjust thesingletontribution inthet-hannel
only Pomeron exhange diagrams are taken into
a-ount,theamplitudeisgivenby
ImA(s;t)
s
= G
2 2
Z
d 2
k
1 d
2
k
2
A (k
1 ;q)
F(y;k
1 ;k
2 ;q)
k 2
2 (k q)
2
B (k
2
;q); (10)
whereG istheolorfatorfortheproessand qisthe
transfered momentum in the t-hannel; the funtions
i
arethe impatfatorssetting theoupling ofF to
theexternalpartilesandnally thefuntion F isthe
perturbativegluonladder. Atleadingorder itonsists
intotheexhangeoftwogluonsbutthesumofallterms
impliesanintegralequationforF,thatisinfrarednite
forareggeizedgluonladder. Thisbehaviorofthe
ker-nelof theBFKL equationis onnetedwith theQCD
Pomeron anditistheresum oftheleadinglogarithms
lns.
The solution of the BFKL equation predits the
steepgrowthofthegluondistribution withdereasing
xaswellasthediusionofthetransversemomenta. As
wellasDGLAPequations,theBFKLequationpredits
thegrowingof theross setionin thesmall-xregime
sinethedynamisofthisobservableisrelatedwiththe
gluondistribution funtion.
From this verybrief disussion on the main issues
ofthelinearformalismsforthedynamisoftheparton
distributions,itgetsleartheneedofformalimproving
inordertoinlude theunitarityorretionspreserving
theFroissartlimit.
This important aspet of high energy physis was
pointed out many years ago by Gribov, Levin and
Ryskin (GLR) in Ref. [19℄. I will present in the
fol-lowingthemainattemptsdevelopedinthereentyears
towardsanon linear dynamis forhigh density QCD,
aswellasthehigh energyphenomenologyprovided.
III The Question
The main question that is addressed one treating
hdQCDishowtoanalytiallyseparatesmallandlarge
distane ontributions to high energy amplitudes in a
properlygaugeinvariantformalism. This orresponds
toestablishthehard andsoft salesfortheproess of
interestanddevelopthephysialmeaningfulmethodto
introduetheunitaryorretions(UC) intotheparton
dynamis.
One thetheoretialneedfor UCis establishedwe
shouldlook fortheirsignaturesanalysing dierent
ob-servables. Besidesomparingthepreditionsofthe
dis-tintformalismsitisrequiredaommonlimitbetween
them, probably to be set by a saturation sale, Q 2
S .
ap-Although someprogress hasbeen made towardstheir
onnetionthereisnoommonanalytisolutionforthe
gluondistributiong(x;Q 2
)forallkinematialrange.
ThephysisofhdQCDshakesthepartonmodeland
theherishedoneptofinoherenethatisbehindthe
standardalulations. Itseemsthatinordertoontrol
theinreasingofthegluonfuntion somegluon
reom-binationmehanismhastotakeplaeastheenergygets
higherandhigher.
This isagoodpointtoremindthat theanalysisof
thestruture funtionshasalreadygivenussome
sur-prises, and the previousimportant one was the EMC
eet[20℄,thathasasamainresultF A
2 =AF
n
2
6=1. This
diereneis not preditedifonerequires omplete
in-oherene of the partons, and reveals the presene of
nulear eets in the struture funtions where they
werenotexpeted. Alargeliteratureisdevotedtothis
phenomena, but it isinterestingto pointoutthat the
shadowing behavior notied in J= prodution ould
benielydesribed[21℄,aswellastheomparisonwith
Drell-Yan proesses [22℄, for the rst data at small-x,
onsidering the reombination approah developed by
Muellerand Qiu [23℄whih is basedonthe GLR
pro-posals.
Twomain aspets are in order, the ontrol of the
inreasingofthegluondistributionfuntionasan
uni-tarityimperativeandtheappearaneofnuleareets
in high energy proesses. Ifthis aspet is relevantfor
xedtargetquarkoniaprodution,itisstrongly
impor-tantforphysisofHERA-A,RHICandLHC with
nu-lei.
IV The High Density QCD
Ap-proahes
TheleadinglogarithmapproximationDLA(relatedto
DGLAP) and LL(1/x) (related to BFKL) result both
into linear evolutionequations for the gluon
distribu-tionfuntion. Theeet ofsumminglargelogsinhigh
energyregimeimpliestheinreaseofthegluon
distribu-tionfuntion g(x;Q 2
)aswellastherosssetionone
xdereases. However,thisresultviolatestheunitarity
of the sattering matrix, a main theorem of
relativis-ti Quantum Field Theory, the Froissart theorem [9℄,
whihstatestherosssetion annotgrowfasterthan
ln 2
s. TranslatedtotheDIS,thisimpliesinreasing
re-stritions to thestruture funtion and/or totalross
setion,saylowerthanln 2
(1=x)andprovidesa
limita-tioninthexrangetotheappliationoflinearevolutions
inorder togetsuitableresults.
Intuitively we an assoiate xg(x;Q 2
) to the
num-berof gluonsinto the nuleon, n
g
, per rapidityunity,
y = ln(1=x), with transverse size of order 1=Q. In
the hadron-nuleon interation it is the virtual gluon
that probesthenuleonstruture,in analogywiththe
rosssetionis
G
N (x;Q
2
)=
0
xg(x;Q 2
); (11)
where
0 =
G
g!X = Cte
s(Q 2
)
Q 2
, is the total ross
setionofthevirtualgluon,withvirtualityQ 2
,and
nu-leon gluoninteration. Assuming
0
=R
2
HAD ,then
0 xg(x;Q
2
) orresponds to the area oupied by the
gluonsinanuleon. Asx!0,thistransverseareamay
beomparable,orevenbigger,thanR 2
HAD
,following
DGLAPorBFKLpreditionsforsmallx orsmall Q 2
.
Approahing this regime the gluons may begin to
su-perposespatially inthetransversediretionand to
in-terat, behaving not anymore asfree partons. These
interations should slower, or even stop, the intense
growing of the ross setion, xing the limit R 2
HAD
in thesmallx regime.
Introduingthefuntion,withprobabilisti
inter-pretation
=
0
xg(x;Q 2
)
R 2
; (12)
itispossibletoestimateinwhihkinematialregionone
an expet modiationsin the usual evolution
equa-tions. So to say, for << 1 the system stays at x
andQ 2
wheretheusualevolutionequations(linear)are
appliable, governed by individual partoni asades,
withoutinterationsamongtheasades.
As
s
,partons fromdistintasades beginto
interatduetospatialsuperposition.Thisspei
kine-matialregimeortheonsetofthereombination
meh-anism was rst studied by Gribov, Levin and Ryskin
[19℄ almost twentyyears ago, proposing the
introdu-tionofnonlineartermsintotheevolutionequation.
Theregion of ! 1 was addressedmore reently
[12, 13℄ (see Fig. 2) and experiened onsiderable
de-velopment on the theoretial side [10, 11℄, also
moti-vatedbyHERAresultsandthegreatinterestonRHIC
and LHC future data. This is the kinematial regime
that requirestheQCDdynamisforhighpartons
den-sity. Although theoupling onstant
s
is still small,
allowing in priniplethe useof perturbativemethods,
thesystemissodensethatmanifestationofnon-linear
eets are expeted, and theyare required to be
on-sideredinaompleteformalism.
The region of ! 1 orresponds to partons in a
-1.0
0.0
1.0
2.0
3.0
4.0
0.0
5.0
10.0
15.0
20.0
Contour for
κ
= cte for Nucleon
R
2
= 5 GeV
-2
κ = 0.6
= 0.8
= 1.0
= 1.2
= 1.4
HERA 96
ln(1/x)
ln(Q
2
/GeV
2
)
HERA
Figure2
IVI The GLR Formulation
Gribov, Levin and Ryskin [19℄ introdued the
meha-nismofpartonreombinationinperturbativeQCDfor
high density systems,expressing this asunitarity
or-retions inluded in a new evolution equation known
as GLR equation. In terms of diagrams it onsiders
thedominantnon-ladderontributions,ormulti-ladder
graphs,alsodenotedfan diagrams.
The standard QCD evolution is represented by a
asadeof partonideaysinthenuleon. Thephoton
interats with aparton with fration of momentum x
andvirtualityQ 2
,whihisthelastoneofahainwhere
the partonsgetslowerand withbiggervirtuality. The
saleQ
0
setstheinitialvirtualityandatthesametime,
thelimitforperturbativeQCDappliability,andQ 2
is
the higher virtuality of the hain. In the transverse
planethepartonswithlowfrationofmomentumstay
inthelowerpartoftheladderandhavelargetransverse
size;thosewithbiggervirtualityareontheupperpart
oftheladderandtransversallysmaller.
Following DGLAP, the number of partons of low
frationofmomentuminreasesveryrapidly,whih
pi-torially orresponds to bigger density of individuals
in the sameallowedarea, in ontrastwith a more
di-lutedsystematintermediatevaluesofx,farawayfrom
thepossibilityofsuperposition.Thetransitionbetween
theseregimesshouldbeharaterizedbyaritialvalue
x =x
CRIT
. Thesame an be argued throughBFKL
formalism,with thedierenethat in thisasethe
in-reasingof thepartoni distributions,takesplaeat a
xed transverse sale, although theevolution presents
theutuationsinthetransverseplaneduetothe
har-ateristidiusion inBFKL.
Itisimportanttoemphasizethatinbothlinear
dy-namis only thedeayproessesare onsidered in the
partonievolution,howeverweexpetthatthe
anihila-tionmehanismshouldontributeinthelowxregime,
distribution funtion. In the linearapproah we
on-sider one inident and two emergent partons to
on-strut the splitting funtions for the deay proesses.
Nowitistheasetoonsidertwoinidentpartonsand
one emergent one, and to express the reombination
mehanism it is needed a formulationin terms of the
probabilitytoreombinetwoinidentpartons.
As arst approximationoneonsiders the
anihila-tion probability as proportional to the square of the
probability to nd oneinident parton, introduing a
non-linearbehavior.
Taking = xg(x;Q
2
)
R 2
as the gluon density in the
transverseplane,onehasthegeneralbehavior:for
split-ting1!2,theprobabilityisproportionalto
s ,for
anihilation 2 ! 1, the probability is proportional to
2
s
2
=Q 2
; where 1=Q 2
stands for the size of the
pro-dued parton. For x ! 0, inreases and the
anihi-lation proess beomes relevant. Consideringaell of
volumelnQ 2
ln(1=x)in thephasespaeallowsone
towritethemodiationofthepartoni densityas
2
lnQ 2
ln1=x =
s N
2
s
Q 2
2
; (13)
where the oupling in the proess is given by .
Ex-pressing in terms of the gluon distribution the above
equationis
2
xg(x;Q 2
)
lnQ 2
ln1=x =
s N
xg
2
s
Q 2
R 2
[xg℄ 2
: (14)
This equation is the GLR equation [19℄. The
al-ready mentioned work of Mueller and Qiu [23℄ gives
=81=16forN
=3.
Thenon-linearorretions orrespondto alass of
QCDFeynmandiagrams,alled fan diagrams,formed
byagluonladderwithsubsequentsubdivisionsingluon
ladders, where the three ladders vertex is assoiated
withthedeayandonsistsofasumofseveralnon
pla-nardiagrams. Theoverall resultarriesaminus sign,
whih is important in order to ontrol the growing of
theparton distribution onethe fan diagramsbeome
relevant,i.e.,atlowx. Thelowestpartofthediagrams
ouplestothenuleonandtheEq. (14)resumsalllass
ofthediagramsrepresentedinFig. 3.
As is lear from Eq. (14), the non-linearterm
re-duesthegrowingofxg(x;Q 2
)atlowx,inomparison
with the linearequations. It is also predited for the
asymptoti region x ! 0 the saturation of the gluon
distribution,witharitialline betweenthe
perturba-tiveregion and saturationregion, setting its region of
validity (meaning independene of the gluon funtion
with the energy). The subjet of saturation is very
appealing andthere areseveralattempts in the
γ
N
N
G(Q
2
,x)
γ
γ
γ
d)
Figure3
In the asymptoti limit one obtains
xg(x;Q 2
)
GLR
SAT =
16
27
s Q
2
R 2
. Sine the GLR only
inludes therst non-linearterm,although it predits
saturationin theasymptotiregimeitsregionof
valid-itydoesnot extendto veryhigh density wherehigher
ordertermsshouldontributesigniantly.
IVII The AGL Formulation
This approah developed by Ayala, Gay Duati and
Levin (AGL) [12, 13℄, intents to extend the
perturba-tivetreatmentofQCDuptotheonsetofhighdensity
partons regime, through the alulation of the gluon
distributionwhihisthesolutionofanon-linear
equa-tionthatresumsthemultipleexhangeofgluonladders,
indouble leadinglogarithmapproximation(DLA).
Itis basedonthe developmentofthe Glauber
for-malismfor perturbativeQCD [25℄, onsideringthe
in-terationofthefastestpartons oftheladderswiththe
target, nuleonor nuleus,sineoneofthemain goals
isto obtainthenuleargluondistributionxg A
(x;Q 2
).
We onsidered a virtual probe G
that interats with
the target in the rest frame, through multiple
resat-terings with the nuleons. In this referene framethe
virtualprobe anbe interpreted following the
deom-positionoftheFokstates,anditsinterationwiththe
target oours by the deay of the omponent gg, as
representedin Fig. (4).
Forsmall-xthispairhasalifetimemuhbiggerthan
thenuleus (nuleon) radius and thepairis separated
bythexedtransverseseparationr
t
duringthe
intera-tion,whih isrepresentedbythe exhange ofaladder
ofgluonsstronglyorderedintransversemomentum.
Therosssetionforthisproessisgivenby
G
A
= Z
1
0 dz
Z
d 2
r
t
j
G
t (Q
2
;r
t ;x;z)j
2
gg+A
;(15)
where G
is a olorless virtual probe with virtuality
Q 2
;z is the probe fration of energy arried by the
gluonand G
t
isthewavefuntionofthetransversely
polarizedgluonin theprobe, gg+A
(z;r 2
)is theross
setion of the pair with the target, whih was proven
forperturbativeQCDbyMuellerinRef. [25,26℄
The lower limit estimation of UC is obtained
throughtheinoherentresatteringsof thegluonpair,
withtheonstraintthatonlythefastestpartonsofthe
laddersinteratwiththetarget. Introduingthe
trans-verseimpatparameterb
t
andaprolefuntionforthe
nuleusS(b
t
)weget
G
A
= Z
1
0 dz
Z
1
0 d
2
r
t
Z
d 2
b
t
j
G
t (Q
2
;r
t ;z)j
2
2[1 e 1
2
g g
N (x
0
;4=r 2
t )S(bt)
℄; (16)
where x 0
= x=(zr
t Q
2
), S(b
t
) may be taken as
A
R 2
A e
b
t =R
2
A
foragaussianprole, gg
N =
CA
CF
N ,where
N =
CF
CA (
3
4
s (4=r
2
t ))
2
r 2
t
xg(x;4=r 2
t
), and 4=r 2
t is a
ut for the nonperturbative region. For the virtual
probe with virtuality Q 2
the relation G
A
(x;Q 2
) =
( 4
2
s
Q 2
)xg
A (x;Q
2
)isvalid.
In this approah the gluon pair emission is
de-sribed in DLA of perturbative QCD and from the
Feynmandiagramsoforder n
s
,itshould beextrated
only the terms that ontribute with a fator of order
(
s
ln1=xlnQ 2
=Q 2
0 )
n
. Theinterationofthegluonpair
withthetargetoperatesthroughtheexhangeofa
lad-der whih satises the DGLAP evolution equation in
theDLAlimit.
It is aworking hypothesis that in high energy the
suessiveresatteringsanbetakenasindependent
al-lowingtheemployofGlauberformalism,insuhaway
using the eikonal proedure for a relativisti partile
propagatingin thetarget.
Our master equation for the interation of the gg
pair with the target is known asthe Glauber-Mueller
formulaandis
xg
A (x;Q
2
)= 4
2
Z
1
x dx
0
x 0
Z
1
4=Q 2
d 2
r
t
r 4
t Z
d 2
b
t
2[1 gg
N (x
0
;4=r 2
t )S(b
t
)℄: (17)
Thepitorial representationof thespae-time
evo-lutionofthis formulaisgivenin Fig. (4).
Oneweperformtheimpatparameterintegration
usingagaussianprolefuntionweobtain
xg A (x;Q 2 )= 2R 2 A 2 Z 1 x dx 0 x 0 Z 1=Q 2 0 1=Q 2 d 2 r t r 4 t
C+ln(
G (x 0 ;r 2 t ))+E
1 ( G (x 0 ;r 2 t )) ; (18)
where C is the Euler onstant, E
1
is the exponential
funtion andwherethe
G
funtionwasintroduedas
G (x;r 2 t )= 3 s 2R 2 A r 2 t xg(x;1=r 2 t
): (19)
The expansion of Eq. (18) in terms of
G gives
as the Born term the DGLAP equation in the small
x region, the higherorder terms orresponding to the
unitarityorretionsnaturallyimplementedinthis
for-malism.
Theestimationoftheshadowingeetduetogluon
reombinationanbeimmediatelyobtainedstudingthe
ratio R 1 = xg A (x;Q 2 )=Axg GRV N (x;Q 2
) presented in
Fig. (5), wherewealulate forCaandAu,and
anal-ysed thebehaviorof thisfuntion in termsof ln(1=x),
A 1=3
andlnQ 2
. WeusedtheGRVparametrization[27℄
and adapted the alulation in order to have alarger
domain ofvalidityin xusing
xg(x;Q 2
)=xg
A
[Eq:(18)℄+Axg GRV (x;Q 2 ) A s N Z 1 x Z Q 2 Q 2 0 dx 1 x 1 dQ 02 Q 02 x 0 g(x 0 ;Q 2
); (20)
where Axg GRV
(x;Q 2
0
) is the initial ondition. The
same proedure ould be applied for another global
parametrizationbasedonDGLAP.
AsexpetedtheUCinreasewithA,andgetsmaller
asQ 2
inreasesanditisevidenttheimportaneofthe
eet asx goesto small values. This allowsus to say
that theUCshould beinludedinthealulations
re-lated withthenuleargluondistributionfuntion,and
the obtained funtion xg
A (x;Q
2
) may be used to set
the initial onditions for future experiments. For
in-stanein HERA-A,in proessese
A [28℄thefuntion
xg
A
ould beobtained indiretly and employed as an
initialonditionforthehadronihighenergyproesses
at RHICandLHC.
Thequarksand gluonsdistributionwerealso
anal-ysedforthenuleoninthisformulation[13℄,aswellas
thestruturefuntion F
2
[29℄. Themotivation forthis
0.0
2.0
4.0
6.0
8.0
10.0
0.4
0.6
0.8
1.0
ln(Q
2
)
e)
f)
Ca
0.0
2.0
4.0
6.0
0.4
0.6
0.8
1.0
A
1/3
R
1
R
1
0.0
5.0
10.0
15.0
20.0
0.2
0.4
0.6
0.8
1.0
Q
2
=1.0 Gev
2
Ca
=10. Gev
2
=1.0 Gev
2
Au
=10. Gev
2
R
1
ln(1/x)
a)
0.0
2.0
4.0
6.0
8.0
10.0
0.4
0.6
0.8
1.0
x
B
=10
-5
=10
-4
=10
-3
=10
-2
Au
ln(Q
2
)
0.0
2.0
4.0
6.0
0.4
0.6
0.8
1.0
x
B
=10
-5
=10
-4
=10
-3
=10
-2
Q
2
=1 Gev
2
Q
2
=10 Gev
2
A
1/3
R
1
R
1
R
1
0.0
5.0
10.0
15.0
20.0
0.2
0.4
0.6
0.8
1.0
Q
2
=0.5 Gev
2
Au
=1.0 Gev
2
=10. Gev
2
=100. Gev
2
ln(1/x)
b)
c)
d)
Figure5
ThefreeinterpretationoftheFroissarttheoremfor
hadroni proesses requires a limit for the inreasing
of the ross setion
N and F
2
with the energy so
unitarity is not violated. Conentrating the
disus-sion on the value, = xg(x;Q 2 ) gg =(Q 2 R 2 ) = 3 s xg(x;Q 2 )=2Q 2 R 2
, whihistheprobabilityof
glu-onsinterations inside the partoni asade, and R is
theradiusofthenuleonareaoupiedbythegluons,
we were able to obtain R 2
= 5 GeV 2
, and that
reahes1 at HERA,meaning theeets of shadowing
should be onsidered in the analysis [12, 13, 30℄. In
thenuleonase,followingthesamestepsasbeforewe
obtain xg(x;Q 2 )= 4 2 Z 1 x dx 0 x 0 Z 1 4=Q 2 d 2 r t r 4 t Z 1 0 d 2 b t
2[1 e 1 2 g g N (x 0 ;4=r 2 t )S(b t )
℄; (21)
and requiring the reovering of DGLAP at DLA we
have gg N (x;4=r 2 t )= 3 2 s 4 r 2 t xg(x;4=r 2 t
): (22)
Forthe quarks,onsideringthe satteringof a
vir-tual photon that deays into a quark-antiquark pair,
whihinteratswiththenuleonthroughtheexhange
ofaladderweget
( )= Z 1 0 dz Z d 2 r t j (z;r
t )j 2 qq+N tot ; (23)
where is the wavefuntion of the qqin the virtual
photon[26℄. Weobtain
tot = Z d 2 b t [1 e 1 2 qq(x;rt;bt)
Here
is the opaity funtion that in the
Glauber (or eikonal) approah is equivalent to =
4 2
s (Q
2
)
3Q 2
xg(x;Q 2
)S(b
t
). In doing so we are able to
reprodueDGLAP evolutionfor<1,and guarantee
thevalidityoftheformulationalsoforthekinematial
regionwhere>1.
Taking ! 1 and fatorizingthe b
t
dependene
weobtaintheunitaritylimitforthestruturefuntion,
having for the lnQ 2
derivative of F
2 , F
2 =lnQ
2
<
Q 2
R 2
=3 2
. UsingGRV, theunitaritylimitfor HERA
isreahedforQ 2
=Q 2
0
=1 2GeV 2
(y=ln1=x9).
Similarly,forgluonsitisQ 2
=1 2GeV 2
(y=ln1=x
7)forHERA[30℄.
Withtheaimtoobtainanon-linearevolution
equa-tionontainingtheunitarityorretionsthroughthe
in-lusionofalltheinterationsbesidesthefastestparton
from the ladder, wedierentiate our master equation
forthegluonin y=ln1=xand"=lnQ 2
,obtaining
2
xg(y;")
y" =
2Q 2
R 2
[C+ln(
G )+E
1 (
G
)℄; (25)
where DGLAP
G
(x;Q 2
) = Ns
2Q 2
R 2
xg DGLAP
(x;Q 2
) for
alulations.
Intermsof
G
themainevolutionequationis
2
(y;")
y" +
(y;")
y =
N
s
[C+ln(
G )+E
1 (
G )℄:(26)
It should be mentioned that large distane eets are
absorvedin theinitialonditionfortheevolution,and
situating in a onveninet region of Q 2
only short
dis-taneeetsarepresent,meaningaperturbative
alu-lationisreliable.
Equations (25) and (26) were derived in Ref. [12,
13℄, and refered forsimpliity asAGL equation. The
mainpropertiesofthisformulationare:
all ontributions from the diagrams of order
(
s y")
2
areresummed;
in thelimit!0theDGLAPevolutioninDLA
isfullyreovered;
for < 1, and not large, the GLR equation is
reovered;
for
s
y" 1 the equation is equivalent to the
Glauberformalism.
TheUCaredesribedforthe dierentkinematial
regionsof fromstritly perturbativeQCDuptothe
onset of hdQCD. Non-perturbative eets are not
ex-pliitly desribed and this is the objet of a distint
formalismMV JLKM [10℄thatwewillbriey
om-ment in a next subsetion. In Fig. (6) the
ompari-sonbetweenthesolutionsoftheequationsAGL,GLR,
DGLAP and Glauber-Mueller (MOD MF) formula is
presented,wheretheontrolofthegrowingofthegluon
Itwasalsoobtainedtheasymptotisolution>1
oftheAGLequationforxed
s
[12,13℄ aswellasfor
running
s
[31℄. Forhighpartonidensityandy>>y
0
weobtain
asymp
G
(y)=
s N
ylny
s N
y: (27)
0.0
10.0
20.0
30.0
0.0
10.0
20.0
30.0
κ
values
DGLAP
MOD MF
AGL
GLR
Q
2
= 10. GeV
2
κ
y=ln(1/x)
Figure6
Thissolutionisagoodapproximationforverysmall
values of x (O(10 8
)), related with THERA physis
[28℄, regionofaverydensepartonsystem. Intermsof
thegluonfuntiontheasymptotibehavioris
xg(x;Q 2
)= 2N
Q
2
R 2
3 2
ln(1=x); (28)
presentingabehaviorsofterthanpreditedbyDGLAP,
meaningapartial saturation.
Fortherunning
s
theresultis[31℄
xg(x;Q 2
)= "
1+" 2N
Q
2
R 2
3 2
ln(1=x): (29)
where "=lnQ 2
= 2
QCD
. The partial saturationis not
modied, and the main dierene from the previous
resultoours forsmall valuesof". Thisonrms the
expetationthattheUCarealreadyrelevantbeforethe
orretionstoleadingorder[30,32℄.
IVIII The Kovhegov Formulation
The unitarization problem in QCD was addressed as
an extension of the dipoles formalism for the BFKL
equationbyKovhegov[11℄. Thisworkproposesanon
lineargeneralizationofBFKLequation,alsoaddressed
previously in Ref. [33℄ by the use of OPE to QCD
obtainingthe evolutionof Wilson line operators. The
sattering ofadipole(onium - qq)with thenuleonis
desribed byaasade evolutionorrespondingto the
suessivesubdivisionofdipolesfromthefatherdipole.
resummedinordertoobtaintherosssetionofthe
in-teration ofthedipolewith thenuleus. Asaresultit
isderivedtheevolutionequationhavingtheunitarized
BFKLPomeronassolution,intheLL(1=x)
approxima-tion.
The sattering of the onium qq(dipole) with the
nuleus in the rest frame, takes plae through a
as-ade of softgluons, whih one taken in the N
! 1
limitis simpliedbythe suppressionofnon-planar
di-agrams. The gluonsare replaed byqqpairs and the
dipoleMueller'stehniquefortheperturbativeasade
anbeemployed[34℄.
TheKovhegovformulation,as the AGL,is a
per-turbativeQCD alulationand theonsidered dipoles
from the asade interat independently with the
nu-leus. Theonium-oniumfrontalsatteringhastheross
setion = 2ImA,where theamplitude
A= i Z
d 2
x Z
1
0 dz
Z
d 2
x
1 Z
1
0 dz
1
(~x ;z)F(x~
1 ;z
1 );
where(~x ;z)isthesquareoftheoniumwavefuntion,
~
xisthetransverseseparationoftheqqpair,andzisthe
longitudinal fration of momentum of the quark. For
theexhangeof onlytwogluons,withoutgluonladder
evolutionthefuntionF is[26℄
F (0)
(~x ;x~
1 )=
2
s (N
2
1)
N 2
x
2
<
(1+ln ( x
>
x
<
)); (30)
wherex
> (x
<
)isthebiggest(smaller) betweenj~x jand
jx~
1
j. Thetwogluonsapproximationisenergy
indepen-dent, but for high energy the ontributions of order
(
s Y)
n
shouldbeinluded(Y =lns=M 2
is the
rapid-ityandM istheoniummass),sinetheygeneratethe
perturbativeasadeevolution. Thedipole
approxima-tion introdues an arbitrary number of soft gluons in
thesquareoftheoniawavefuntion,andkeepingF
asan exhange of two gluons avoidsto deal with the
reggeizationofthegluonsandtheeetivevertex. The
transverseoordinatesofthequarkandantiquarkofan
ultrarelativistioniumstate in +diretion arex~
0 =0
and~x,andsuessivelyintheevolutionthenext
emit-ted gluonshould be softer. We have p k and k, as
themomenta for thepair, and z
1 =
k +
1
p +
(in light-one
variables[35℄),havingaswavefuntion
(0)
(x
01 ;z
1 )=
Z
d 2
k
1
(2) 2
e i
~
k
1 ~ x
01 (0)
(k
1 ;z
1
); (31)
wherex~
01 =x~
1 ~ x
0
, and (0)
=j (0)
j 2
, keeping
fa-torizationin thisproedure. This allowstoobtainthe
dipoledensity,n,onsideringx
02 >;x
12
>,where
isanultravioletutalsoimpliedbyCintheexpression
below
n (x
01
;x;Y)=xÆ(x x
01 )exp
2
s N
Y ln( x
01
)
+
s N
2
Z
C x
2
01 d
2
x
2
x 2
02 x
2
12 Z
Y
0
dyexp
2
s N
(Y y)ln ( x
01
)
n(x
12
;x;y); (32)
forY =lns=M 2
,whihis representedinFig. (7).
}
x
x
x
+
x
0
}
x
=
x
0
1
0
x
1
1
x
Figure7
Thenextstepistoobtainanevolutionequationforthedipoledensityassumingthepropagationofthedipoles
inthetargetisrepresentedbythefuntion
1 (~x ;
~
b),wherebistheimpatparameter,andwhihshouldbeaddedto
thedensityn
2
,equallyonvolutedwith
2 (x~
1 ;
~
b
1 ;x~
2 ;
~
b
2
),et. Assumingnoorrelationamongthedipoles
n (:::)=
1 (x~
1 ;
~
b
1 ):::
1 (x~
n ;
~
b
n
),therosssetionfortheinterationoniumnuleusN(x~
01 ;
~
b
0
;Y)isthen givenby[11℄
N(x~
01 ;
~
b
0 ;Y)=
1
X
i Z
n
i (x
01 ;Y;
~
b
1 ;x~
1 ;:::;
~
b
i ;x~
i )
(x~
1 ;
~
b
1 )
d 2
x
1
2x 2
1 d
2
b
1
:::
(x~
i ;
~
b
i )
d 2
x
i
2x 2
d 2
b
i
Finally,omittingsomestepsofthealulation[11℄theevolutionequationforN(x~
01 ;
~
b
0 ;Y)is
N(x~
01 ;
~
b
0
;Y)= (x~
01 ;
~
b
0 )exp
4
s C
F
ln(
x
01
)Y
+
s C
F
2
Z
Y
0
dyexp
4
s C
F
ln(
x
01
)(Y y)
Z
d
2
x
2 x
2
01
x 2
02 x
2
12
2N(x~
02 ;
~
b
0 +
1
2 ~ x
12 ;y)
N(x~
02 ;
~
b
0 +
1
2 ~ x
12
;y)N(x~
12 ;
~
b
0 1
2 ~ x
20 ;y)
; (34)
d
where x
ij = x
i x
j
, the size of the dipole whose
quarkhastransverseoordinatex
i
, andtheantiquark
x
j , (x~
01 ;
~
b
0
)is thepropagatorof thepair qq through
thenuleus,desribingthemultiple resatteringofthe
dipolewiththenuleonswithinthenuleus. Wedenote
thisequationas theK equation.
Thephysialrepresentationisomparablewiththe
approah Glauber-Mueller sine the inident photon
generatesaqqthatsubsequentlyemitsagluonasade
furtherinterating withthenuleus. AtlargeN
limit
thegluonanberepresentedbyaqqpair,andwean
expet in this limit and DLA that the gluon asade
ouldbeinterpretedasadipoleasade. Although
be-guinningtheformulationswithdistintdegreesof
free-dombothKandAGLresumthemultipleresatterings
in their respetives degrees of freedom, whih allows
to onsidertheyshould oinidein asuitableommon
kinematiallimit,whih wewillshowlateron.
InDLA, wherethe photonsale of momentum Q 2
isbiggerthan 2
QCD
,theKequationsimpliesto
N(x~
01 ;
~
b
0 ;Y)
Y
=
s C
F
x
2
01 Z
1= 2
QCD
x 2
01 d
2
x
02
(x 2
02 )
2
h
2N(x~
02 ;
~
b
0
;Y) N(x~
02 ;
~
b
0
;Y)N(x~
02 ;
~
b
0 ;Y)
i
; (35)
whih is the evolution in transverse size of the
dipoles from x
01
up to 1=
QCD
. Now deriving in
ln(1=x 2
01
2
QCD
)results
2
N(x~
01 ;
~
b
0 ;Y)
Yln(1=x 2
01
2
QCD )
=
s C
F
h
2 N(x~
01 ;
~
b
0 ;Y)
i
N(x~
01 ;
~
b
0
;Y):(36)
settingthatthesuessiveemissionofdipolesgenerates
largertransversesizeforeahhighergeneration.
The linear term reprodues BFKL at low density,
andthequadratitermintroduesUCunitaryzingthe
BFKLPomeronandtheequationreproduesGLRone
weassumeN diretlyrelatedtothegluondistribution
funtion.
IVIV The MV-JKWL Formulation
IntheMV-JKWLformulation[10,36℄averydense
sys-tem is treated in the light-one and onsidering the
light-one gauge (A +
0), x q +
Gluon =Q
+
Nuleon . In
Ref. [10℄thegluonsdistributionforsmallxisproposed
foralargenuleuswherethedegreesoffreedomare
vir-tualquantafromalassialeldgeneratedbytheolor
hargeofthevalenepartons(statisoures). The
ap-proah is originally non-perturbative and the nuleus
is onsidered in the innite momentum frame,
trans-fering the sale of the problem to = 1=R 2
dN=dy,
where N is the density of gluons. For small x and a
large nuleus
s
() is small allowing some
perturba-tivealulationinthiseetivelagrangianformulation
forgluonsondensates.
The density of gluons in momentum spae is
ob-tained in terms of theorrelation of thegluons elds,
in thelightonegauge. Theintrinsiquantum
utu-ations are replaedby alassialaverage onthe olor
harge ensemble. Thegluons distributions at agiven
virtualityQ 2
andxisobtainedfromthedensityof
glu-onsin themomentaspaedN=dq +
q 2
~
q,whihisa
fun-tion of the gluon ondensate <A a
i
(x ;~x )A(x 0
; ~
x 0
) >,
being
xg(x;Q 2
) Z
Q 2
d 2
~ qx
dN
dxq 2
~ q
: (37)
Thegluons distributions, in this framework where
alargenumberof olorhargesgeneratesaQCD
ve-torpotential,isobtainedinlowestorderbysolvingthe
lassialYang-Mills equations,D
F
=j
.
Introduingaregulatorin thevalenepartons
ur-rent singularity by onsidering the olor density a
funtion of rapidity, itwasobtained[39℄an analytial
solutionforthelassialorrelations,withtheproperty
that for high transverse momentum the lassial
glu-ons distribution obeys the Weizsaker-Williams form,
and hasitsbehaviorsoftenedasln(k 2
=(y;k 2
= R 1 y 2 (y;Q 2
)isthesquaredolorhargeperunity
of areaforrapiditybiggerthany.
In the lassial MV the non-linear eets are
in-luded in the harge density solution of Y-M
equa-tions. The quantum orretions areto beonsidered,
andfromRef. [37℄theperturbativeresultforthegluons
distributionuptoseondorderin
s
isgivenby
1 R 2 dN dxd 2 k t = s 1 xk 2 t 1+ 2 s N ln ( k t s
)ln ( 1 x ) ;(38) where = 2 (N 2 1) 2 and 2
is the square ofthe olor
hargeaverage density(perunity of area). The
addi-tional eetofinludingthehardgluonwastreatedin
Ref. [38℄, resultingin thelowdensity limittheBFKL
equation, and for high virtualities the DGLAP
equa-tion. Forhigh densityaompletesolutionwasnotyet
obtained. In Ref. [40℄ JKLW analysed theirevolution
equationin DLAobtainingageneralizationofGLR.
As a summary of the formulations for hdQCD at
present, the Fig. (8) presents their dierent regions
of appliability asfas asin onernedin theln(1=x)
versusQ 2
plane.
Figure8
Themainquestionsatthis pointanbe:
whih isthemost suitableformto introduethe
UC?
anwerelatethedistintformulationsina
om-monlimitanalytially?
whatdowelookattheobservablesasasignature
fortheUC?
Thelasttwoquestions,I willbriey addressin the
rest of this presentation followingour personal
ontri-V Connetion Among the F
or-mulations
The AGL equation was originally obtained from the
Glauber-Mueller approah, but it an be also derived
from the dipole representation [41℄. We obtained the
ross setion for the virtual probe G
with the
nu-leus G A = R 1 0 dz R d 2 rt j G t j 2 gg+A
, that an be
expressed by meansof the dipoles qqone weremind
gg+A = (C A =C F ) qq+A
. Now in order to estimate
the UC the resatterings of the qq pair into the
nu-leus should be onsidered, having in mind that [30℄
qq N = C F CA (3 s (4=r 2 t )=4) 2 r 2 t xg N (x;4=r 2 t
), where xg
N
is the nuleon gluon distribution. The wave
fun-tion G
alulated in [26, 30℄ is suh that j G t j 2 = 1 z(1 z) [( 2 K 0 (t t ) K 1 (r t )=r t ) 2 + 1=r t (K 1 (r t )) 2 ℄, where 2 = Q 2
z(1 z), and the K
i
are the modied
Besselfuntions. Forsmallz andr
t
<<1weobtain
xg A (x;Q 2 )= 4 C A C F Z 1 x dx 0 x 0 Z 1 4=Q 2 d 2 r t r 4 t 2 h 1 e 1 2 qq N S(b t ) i : (39)
From this equationweanobtainthe AGL
equa-tion in the dipole representation by dierentiating in
y=ln1=xandlnQ 2 = 2 QCD having 2 xg(x;Q 2 )
y lnQ 2 = 2 QCD =C 0 Q 2 Z d 2 b t h 1 e 1 2 qq N S(b t ) i ;(40)
validinDLA,onsideringeahgluonoftheasadeas
aqq inthehigh N
limit,andwhere C 0 =2C A = 2 C F .
For a entral ollision and S
?
= R
2
, and S(0) =
A=R 2 2 xg(x;Q 2 )
ylnQ 2 = 2 QCD =DQ 2 1 e 2 s 2 N S ? Q 2 xg A ; (41) for N
= 3, C
F = N
=2 at high N
, and where
D=
NCFS?
3
.
The GLR for a ylindrial nuleus is immediately
obtained from Eq. (41) by its expansion up to
se-ond order in xg
A
. Again, for small UC only the rst
termontributeswhihreproduesDGLAPintheDLA
limit. ThoseresultsareinRef. [41℄
Now, theEq. (34)is theKequation that in DLA,
wherethesaleofmomentumofthephotonQ 2
ishigher
thanthesaleofmomentumofthenuleus 2
QCD ,
sim-pliesas
N(x~
01 ;
~
b
0 ;Y)
Y = s C F x 2 01 Z 1= 2 QCD x 2 01 dx 2 02 (x 2 02 ) 2 h
2N(x~
02 ;
~
b
0
;Y) N 2
(x~
02 ;
~
b
0 ;Y)
i
Thisequationonsiderstheevolutionofthedipoles
from x
01
up to 1=
QCD
in the transverse diretion.
NowderivingEq. (42)in ln(1=(x 2
01
2
QCD
))weget
2
N(x~
01 ;
~
b
0 ;Y)
Yln(1=x 2
01
2
QCD )
=
s C
F
[2 N℄N: (43)
We should relate now the funtion N(~x
01 ;
~
b
0 ;Y)
withthegluondistributionfuntion. Forthat we
on-sider the struture funtion F
2
for the nuleus as
ob-tained in [11℄, following [25℄, and analized in [29℄ for
b
t
=0,whihis
F A
2 (x;Q
2
)= Q
2
4 2
em R
2 Z
dz Z
d 2
r
t
j j
2
2 "
1 e
s C
F
2
N 2
S
? r
2
t
Axg(x;1=r 2
t )
#
: (44)
This estimates the UC for the nulear struture
funtion,for entral ollisions in theDLAlimitin the
Glauber-Muellerapproah.
Considering the unitarity orretions due to the
multiple resatteringof the qq pairs with the distint
nuleons into the nuleus, from the just obtained
ex-pressionforF A
2
,itresultstherelation
N(x~
01 ;
~
b
0
=0;Y)=2 "
1 e 2
s C
F
2
N 2
S
? x
2
01
Axg(x;1=x 2
01 )
#
;(45)
where x
01 = x
0 x
1 = r
t
, Y = ln(s=Q 2
) = ln(1=x)
establishing aonnetion between theross-setion of
the qq pair and the gluon struture funtion in DLA
limit.
We are in good terms to verify the onnetion
among the Kand AGL formulations sinewealready
obtainedthe AGL equation in thedipole formulation,
Eq. (40),the Kequation in theDLA limit, Eq. (42),
therosssetionofthepairthroughthedipoledensity
fromKandthenuleargluondistributionfuntion,Eq.
(45).
HavingEq. (45)inEq. (41)andforx
01
2=Q,as
in[10℄,weobtain
2
xg
A (x;Q
2
)
yln(Q 2
= 2
QCD )
=DQ 2
1 e 2s
2
N
s
? Q
2 xga
; (46)
resultalreadyobtained,andthatgivesGLRasalimit.
Our omparison has physial meaning for dipoles
with small transverse sizes and for the above
onne-tionamongN andxg
A
[Eq. (45)℄.
In Refs. [38, 39℄ it was applied the Wilson
renor-malizationgrouptothemodelofMLerranand
Venu-gopalan. The non-linear evolution equation then
ob-taineddealswiththeweightfuntionoftheolorharge
densities,validatleadingorder
s
andfordensitiesup
to 1= . The omplete analytial solution is not yet
obtainedbut somelimits aredisussed. Atlow
densi-tiesBFKL is reovered,and thenat DLA at largeQ 2
DGLAPisreovered. Inthework[40℄is proposed the
equation
2
xg(x;Q 2
;b
t )
y"
=Q 2
1 1
x
exp(1=)E
1 (1=)
;(47)
where =N
(N
1)=2and(x;Q 2
;b
t )=2
s =(N
1)Q 2
xg(x;Q 2
;b
t ).
Forlarge,afatorizedb
t
dependeneand
onsid-eringaentral ollisionweobtainforthisequation
2
xg(x;Q 2
)
y"
=R 2
Q 2
; (48)
whihsolutionis
xg(x;Q 2
)=R 2
Q 2
ln(1=x); (49)
presentingthe sameQ 2
andx behaviorasthe
asymp-toti solution for AGL. The main point is the partial
saturationof the gluon distribution presented in both
formulations in the asymptoti region. A onnetion
among those two formulations in a more broad
kine-matialregionisstillanopenquestion.
Theasymptoti behaviorof the struture funtion
alsorequiredourattention. Consideringtherelationof
andxg(x;Q 2
)weanwrite[29℄
F
2 (x;Q
2
)= 2
s
9 Z
Q 2
Q 2
0 dQ
2
Q 2
xg(x;Q 2
) (50)
whihisaleadingtwistequation,withlimited
applia-tionforhighdensities,duetohighertwisttermsrelated
withtheUC.
UsingthesolutionofAGLintheasymptotiregime
asinput in theaboveequation we obtainF
2 (x;Q
2
)'
s
3
R 2
Q 2
ln(1=x), whih again presents partial
satura-tion, meaning the Froissart limit is not violated [44℄.
Analogous resultwasobtainedby Kovhegov[42℄
em-ploying the solution of the K equation [11℄. We
ob-tainedthat theasymptotibehaviorofF
2
isageneral
harateristithatappearstobeindependentfrom the
approahthatisused[31℄.
Assuming the asymptoti behavior of the gluon
funtion is xg(x;Q 2
) = 2Q 2
R 2
=3
s
, it implies
satu-rationforF
2 (R
2
Q 2
)forverysmallx. Howeverthis
resultshould betakenwith autionsineit isvalid in
akinematialregionwherehigherorderinthepartoni
densityarenotsigniative. Thesubjetofsaturation
isatrikyoneanditseemswearefarfromestablishing
itsfeatures in asolid theoretialbasis[24℄. Important
ontributions to these hallenging aspets of hdQCD
are to befound in Mueller[24℄ forthe theoretial
dis-ussionandGole-BiernatandWustho[24℄fora
In[43,44℄wewereabletoshowthat
F
2 (x;Q
2
) = R
2
2 2
X
i e
2
i Z
1=Q 2
0
1=Q 2
d 2
r
t
r 4
t [C
+ ln
q +E
1 (
q
)℄ ; (51)
where
q
=4=9
g
. Fromthatwean estimatetheUC
for F
2
in the DLA limit. For large
q
, and using the
asymptotisolutionofAGL,weobtained[31℄
F
2 (x;Q
2
)' R
2
Q 2
3 2
ln[ 4
s
3
ln (1=x)℄; (52)
when highertwisttermsare onsidered in F
2
. Thisis
a softer behavior, but in both asesthere is no
viola-tionoftheFroissartlimit. Thisaboveequationwasnot
studied inKorMV-JKLWapproahes.
From the already obtained results it follows the
identity[45℄
F
2 (x;Q
2
)
lnQ 2
=F
2 (x;Q
2
); (53)
as an important signature of the asymptoti regime
of QCD for dense systems. It is relevant to mention
that for thesameenter ofmass energythis regimeis
reahedfor nuleus forsmallerpartonidensities than
in thenuleonase,sine
A =A
1=3
N .
VI Phenomenology
From the Glauber-Mueller formalism for high dense
partoni systemswasdemonstrated theAGL equation
and its asymptotibehavior. It wasalso obtainedthe
nuleonandthenulear gluondistributionfuntion as
well as the respetive struture funtions and
deriva-tives. This formulation inorporates the UC required
bytheFroissartbound,throughanonlineardynamis.
Inthissetionthebehaviorofthemainobservables
obtainedin epollisions,andrelevantforeAollisions,
willbeanalysedwiththegoaltoshedsomelightinthe
subjetofUC.
For epwestudiedthebehavioroftheproton
stru-ture funtion F
2
, its derivative F2
lnQ 2
, the harmed
omponentofthestruturefuntionF
2
,andthe
longi-tudinal distributionfuntion,F
L [43℄.
ThereisalargeamountofdatafromHERAto
mo-tivateadetailedstudyoftheseobservablesdiretly
on-neted with thegluondistributionfuntion. As
previ-ously demonstrated thegluon distribution is modied
inaunitarityorretedformulation,meaningthose
ob-servablesshould beaeted.
Wewere lukytoshowthat the F
2
lnQ 2
,theF
2 are
learlymodied. Also,theeA analysisprovides
strik-ing results for the nulear struture funtion F A
and
itsderivative,asanimportantsignatureoftheUC
or-retions. These resultsare importantsinethey area
preditionbothforHERA-Aandfore-RHIC,inwhih
ahighdensepartonsystemshouldbeformed.
TheinreasingofF
2
inHERAinthesmallxregion
(10 2
>x>10 5
) isobservedevenfor small
virtuali-ties(Q 2
1GeV 2
). TakingF
2 x
,forsmallxdata
is ompatible with =0:15 (Q 2
=0:85 GeV 2
) up to
=0:4(Q 2
=20GeV 2
). ThisisdesribedbyDGLAP
withsuitableinput initial onditionforQ 2
and
distri-butionsbydierentgroups[27, 47℄. Itwill ondut to
theideaUCarenotobservableintheHERA
kinemati-alrange. Wehaveshownthatthestruturefuntionis
tooinlusiveinthegluonfuntiontolearlyexpliitate
theUC.
Wearrivedat adierent onlusionapplying AGL
to F
2
lnQ 2
,F
L andF
2
,allobservablesdiretlyassoiated
withthegluonfuntion.
ThederivativeofF
2 is
F
2
lnQ 2
= R
2
Q 2
2 2
X
i e
2
i
[C+ln(
q )+E
1 (
q
)℄; (54)
that we solvedusing thesame proedure asEq. (20).
Theusual parametrizations[27, 47℄donotinlude the
UCforthegluonsexpliitly. WeuseEq. (20)forA=1
asinput,andweobtaintheorretionsfromboth
se-tors quark and gluons. The last one gains in
impor-taneasQ 2
inreases. InFig. (9)theresultsfor F2
lnQ 2
arepresentedfor R 2
=5GeV 2
. Fortheomplete
dis-ussionwerefer to [48℄. The UC forboth setors are
ableto desribe properlythe datainluding the
turn-over. Our onlusions is this is a good observable to
evidentiatethe preseneof theUC.This questionwas
addressedalso in [49℄ alulating the suppression
fa-torsseparately.
We believe that the UC should be extrated from
datarelatedtoobservablesthatarediretlydependent
ofthegluonfuntion. Thelongitudinalstruture
fun-tion F
L
is a diÆult measurement requiring distint
valuesof the enter of massenergy, meaning dierent
energybeams. An alternativeis toonsider the
radia-tionofahardphotonfromtheinidenteletron,
redu-ing theenter of mass energy. If this should be done
weanstudyF
L (x;Q
2
10
-4
10
-3
10
-2
10
-1
x
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
dF
2
/dlogQ
2
ZEUS
GRV94(LO)
Quark + Gluon Sector
Gluon Sector
Quark Sector
R
2
= 5 GeV
-2
Figure9
Expressed onsidering the quarks transverse
mo-menta due to gluon radiation, the longitudinal
stru-turefuntionreads
F
L (x;Q
2
)=
s (Q
2
)
2 x
2 Z
1
x dy
y 3
8
3 F
2 (y;Q
2
)
+4 X
f e
2
f (1
x
y
)yg(y;Q 2
) 3
5
; (55)
wherey =Q 2
=sxandthedependeneonthegluon
dis-tribution is expliit, meaning this funtion should be
sensitive to unitarity orretions in HERA
kinemati-al region. Our results for small x region are in Fig.
(10)[43℄omparedwith theH1data [50℄. Althoughit
seems to be a good observable to evidentiate the UC
theavailabledata donotallowanydeniteonlusion
fornow.
0.00010
0.00020
0.00030
0.00040
0.00050
0.0
0.2
0.4
0.6
0.8
1.0
H1
GRV(94)LO
GM
F
L
x
Figure10
A problablymorepromissingobservableistherate
R
F = F
2 (x;Q
2
)=F
2 (x;Q
2
), where F
2
is the harmed
omponentof thestruture funtion. Consideringthe
approah ofboson-gluonin order toreate the pair
weobtainedintheGlauber-Muellerformalismtheratio
R .
This ratiois presented in Fig. (11)[43℄ as a
fun-tionofln(1=x). Thereisstrongmodiationoftheratio
oneUCareinludedin thealulation. Weurgedata
in this observable. The suppression is muh stronger
thanin theF
2
ase,andweexpetalowerprodution
ofquarkharmforsmallx,andthisisrelatedwiththe
produtionof J= whih isproportionaltothesquare
ofthegluonfuntion.
0.0
5.0
10.0
15.0
20.0
0.0
0.1
0.2
0.3
0.4
0.0
5.0
10.0
15.0
20.0
0.0
0.1
0.2
0.3
0.4
0.0
5.0
10.0
15.0
20.0
0.0
0.1
0.2
0.3
0.4
0.0
5.0
10.0
15.0
20.0
0.0
0.1
0.2
0.3
0.4
2.5
5.0
10.0
20.0
ln(1/x)
ln(1/x)
R
F
ln(1/x)
ln(1/x)
R
F
R
F
R
F
Figure11
Finally, one of our most striking results onerns
eAphysis,andisrelatedwiththehighdensepartoni
system in the nulear medium. The nulear
shadow-ing is a hallenge for hdQCD and mainly important
for HERA-A, RHIC and LHC physis. We estimated
how the nulear struture funtion and its derivative
aremodiedbytheeetsofhighpartonidensitiy.
Theshadowingorretionsto F A
2
are assoiatedto
theresatteringsoftheqqin thenuleonsintothe
nu-leus, beingdependenton the nuleongluon
distribu-tion funtion. Here also we separate the two ases:
quarksetor,where thegluondistribution isnot
mod-ied byUC,andquark+gluonsetor,wherenowthe
gluon distribution is modied a la Glauber-Mueller.
The results are presented in Fig. (12)[51℄ as a ratio
R
1 = F
A
2 =AF
N
2
, showing that for small x the gluon
setor ontribution should be inluded, and promote
saturation.
We obtain that the suppression due to the
shad-owing in F A
2
is proportionally smaller than in xg
A in
aperturbativeframework,in adierentresult that in
[19℄ wheresoftphysisisthemain issue. Asanew
re-sultthesaturationoftheratioisattainedat HERA-A
regionwhenthegluonsetorisinluded. Thepresene
ofsaturationintheperturbativeregion(Q 2
>1GeV 2
)
denotes the large shadowing orretions in the gluon
