SmallxAsymptotics
oftheGluonHelicity
Distribution
YuriKovchegov
TheOhioStateUniversity
workwithDanPitonyak andMattSievert,
inpreparation
Outline
• Quarkhelicityevolution:re-derivationandreviewintheoperatorform.
• GluonhelicityTMDsatsmallx.
• Small-xevolutionequationsforthegluonhelicityTMDsatlargeNc.
• Solutionoftheevolutionequationsandthesmall-xasymptotics ofthe
gluonhelicitydistribution:
✓ ◆↵ G
1 h
2
G(x, Q ) ⇠
x
with ↵hG =
13
p
4 3
r
↵ s Nc
⇡ 1.88
2⇡
r
↵ s Nc
2⇡
• Forcomparison,quarkhelicityPDFasymptotics is(seeMattSievert’stalk)
✓ ◆↵qh
1
2
q(x, Q ) ⇠
x
with
↵hq
4
p
=
3
r
↵ s Nc
⇡ 2.31
2⇡
r
↵ s Nc
2⇡
2
Howmuchspinisatsmallx?
• E.Aschenaur etal,arXiv:1509.06489 [hep-ph]
• Uncertainties arevery large at small x!
3
QuarkHelicityEvolutionatSmallx
flavor-singletcase
Yu.K.,M.Sievert,arXiv:1505.01176[hep-ph]
Yu.K.,D.Pitonyak,M.Sievert,arXiv:1511.06737[hep-ph],
arXiv:1610.06197[hep-ph],arXiv:1610.06188[hep-ph],
arXiv:1703.05809[hep-ph]
QuarkHelicityObservablesatSmallx
γ∗
x⊥
q
z
σ
k⊥
σ
′
γ∗
y⊥
γ∗
γ∗
σ
y⊥
w⊥
z
q
w⊥
k⊥
x⊥
σ′
Σ
Σ
• Onecanshowthattheg1 structurefunctionandquarkhelicityPDF(Dq)
andTMDatsmall-xcanbeexpressedintermsofthepolarizeddipole
amplitude(flavorsingletcase):
g1S (x, Q2 ) =
Nc Nf
2 ⇡ 2 ↵EM
Nc Nf
q (x, Q ) =
2⇡ 3
S
2
Z1
zi
Z1
dz
z
Z1
zi
dx201
"
1 X
|
2
0
T
0
|2(x2
01 ,z)
+
X
0
|
L
0
|2(x2
01 ,z)
#
G(x201 , z),
1
zi
8 Nc Nf
S
g1L
(x, kT2 ) =
(2⇡)6
dz
z 2 (1 z)
Z
ZzQ2
dx201
G(x201 , z),
2
x01
Z
d2 x01 d2 x00 1 e
1
zs
dz
z
ik·(x01 x00 1 )
x01 · x00 1
G(x201 , z)
2
2
x01 x00 1
• Heresiscms energysquared,zi=L2/s, G(x2 , z) ⌘
01
Z
d2 b G10 (z)
5
PolarizedDipole
• Allflavorsingletsmall-xhelicityobservablesdependononeobject,
“polarizeddipoleamplitude”:
i
h
i EE
1 DD h
pol †
pol †
G10 (z) ⌘
tr V0 V1
+ tr V1 V0
(z)
2Nc
2
Vx ⌘ P exp 4ig
Z1
1
unpolarized3quark
dx+ A (x+ , 0 , x)5
polarizedquark(“polarizedWilsonline”):
eikonal propagation,non-eikonal
spin-dependentinteraction
• Doublebracketsdenoteanobjectwithenergysuppressionscaledout:
DD EE
D E
O (z) ⌘ zs O (z)
6
“PolarizedWilsonline”
Toobtainanexplicitexpressionforthe“polarizedWilsonline”operatordueto
asub-eikonal gluonexchange(asopposedtothesub-eikonal quarksexchange),consider
multipleCoulombgluonexchangeswiththetarget:
p2 − k
p2
σ′
σ
-
k
p1
+
Mostgluonexchangesareeikonal spin-independentinteractions,whileoneofthem
isaspin-dependentsub-eikonal exchange.(cf.Mueller‘90,McLerran,Venugopalan ‘93)
“PolarizedWilsonline”
• AsimplecalculationinA-=0gaugeyieldsthe“polarized
Wilsonline”:
Vxpol =
1
2s
Z1
1
dx P exp
8
<
:
ig
Z1
x
dx0 A+ (x0 , x)
9
=
;
ig r ⇥ Ã(x , x) P exp
⌃
A
(x
,
x)
=
where
⌃
+ Ã(x , x)
2p1
8
>
<
>
:
ig
Zx
dx0 A+ (x0 , x)
1
isthespin-dependentgluonfieldoftheplus-direction
movingtargetwithhelicityS.
(𝐴" istheunpolarized eikonal field.)
8
9
>
=
>
;
PolarizedDipoleAmplitude
• Thepolarizeddipoleamplitudeisthendefinedby
1
G10 (z) ⌘
4Nc
Z1
dx
1
D h
i
E
tr V0 [1, 1]V1 [ 1, x ] ( ig) r ⇥ Ã(x , x) V1 [x , 1] + c.c. (z)
withthestandardlight-cone
Wilsonline
Vx [b , a ] = P exp
8
>
<
>
:
ig
Zb
a
dx A+ (x , x)
t
9
>
=
>
;
z
1
0
proton
9
Cross-check:evolution
• Onecancross-checktheoperatorand/ortheevolutionequationwe
derivedforit(seeMatt’stalk):
0
0
2
“c.c.”
k2
k1
1
1
x−
1
x−
2
other LLA diagrams
• Fromthefirsttwographsontherightweget
↵s
(0)
G10 (z) = G10 (z) + 2
⇡
Zz
dz 0
z0
Z
d2 x2 1 DD h b a † i pol ba EE
tr t V0 t V1 U2
+ ...
2
x21 Nc
inagreementwithourearlierworkonquarkhelicityevolution(‘15-’16).
Resummation Parameter
• Forhelicityevolutiontheresummation parameterisdifferentfromBFKL,
BKorJIMWLK,whichresum powersofleadinglogarithms(LLA)
↵s ln(1/x)
• Helicityevolutionresummation parameterisdouble-logarithmic(DLA):
1
↵s ln
x
2
• Thesecondlogarithmofxarisesduetotransversemomentum(or
transversecoordinate)integrationbeinglogarithmicbothinUVandIR.
• Thiswasknownbefore:Kirschner andLipatov ’83;Kirschner ’84;Bartels,
Ermolaev,Ryskin ‘95,‘96;GriffithsandRoss’99;Itakura etal’03;Bartels
andLublinsky ‘03.
11
PolarizedDipoleEvolutionintheLarge-Nc Limit
Inthelarge-Nc limittheequationsclose,leadingtoasystemof2equations:
0
0
G10(z)
∂
∂ ln z
Γ02,21(z)
=
2
z
z
1
1
0
0
Γ02,21(z ′) =
∂
∂ ln z ′
2
G21(z)
G12(z)
Γ03,32(z ′ )
S03(z ′ )
S03(z ′ )
+
−
G32(z ′ )
Γ01,21(z)
−
+
S23(z ′ )
z′
S02(z)
+
S21(z)
3
z′
S02(z)
+
G23(z ′ )
Γ02,32(z ′ )
2
1
1
z
z
G10 (z) =
(0)
G10 (z)
↵ s Nc
+
2⇡
Zz
zi
2
dz 0
z0
Zx10
⇢02
dx221
[2
x221
02, 21 (z
0
) S21 (z 0 ) + 2 G21 (z 0 ) S02 (z 0 )
+ G12 (z 0 ) S02 (z 0 )
02, 21 (z
0
)=
(0)
0
02, 21 (z )
↵ s Nc
+
2⇡
Zz
zi
0
dz 00
z 00
S=foundfromBK/JIMWLK,itisLLA
min{x202Z,x221 z0 /z00 }
⇢00 2
dx232
[2
x232
03, 32 (z
01, 21 (z
00
0
)]
) S23 (z 00 ) + 2 G32 (z 00 ) S03 (z 00 )
+ G23 (z 00 ) S03 (z 00 )
02, 32 (z
00
)]
12
Yourfriendly“neighborhood”dipole
• Thereisanewobjectintheevolutionequation– the neighbordipole.
• ThisisspecificfortheDLAevolution.Gluonemissionmayhappeninone
dipole,but,duetotransversedistanceordering,may’know’aboutanother
dipole:
0
0
3, z ′′
2
1
2, z ′
1
x221 z 0
x232 z 00
• Wedenotetheevolutionintheneighbordipole02by
0
(z
)
02, 21
13
Large-Nc Evolution
• InthestrictDLAlimit(S=1)andatlargeNc weget(hereG isanauxiliary
functionwecallthe‘neighbour dipoleamplitude’)
G(x210 , z) =
G
(0)
(x210 , z)
↵ s Nc
+
2⇡
Zz
1
x210 s
(x210 , x221 , z 0 )
=
(0)
(x210 , x221 , z 0 )
↵ s Nc
+
2⇡
2
dz
z0
0
Zz
1
Zx10
1
z0 s
0
⇤
dx221 ⇥
2
2
0
2
0
(x
,
x
,
z
)
+
3
G(x
,
z
)
10
21
21
x221
min
dz 00
z 00
n
0
x210 ,x221 zz00
Z
o
1
z 00 s
x210 s
⇤
dx232 ⇥
2
2
00
2
00
(x
,
x
,
z
)
+
3
G(x
,
z
)
10
32
32
x232
• TheinitialconditionsaregivenbyBorn-levelgraphs
x0
x1
1−z
x0
x1
z
(x210 , x221 , z) = G(0) (x210 , z)
G(0) (x210 , z) =
↵s2 CF
Nc
h
⇡ CF ln
zs
⇤2
z
0
0
(0)
1−z
x0
x1
2 ln(zs x210 )
i
0
x0
x1
1−z
z
0
1−z
z
x0
x1
1−z
z
0
14
Scaling
• Onecansolvethehelicity
evolutionequationsnumerically:
⌘=
r
↵ s Nc
zs
ln 2
2⇡
⇤
s10 =
r
↵ s Nc
1
ln 2 2
2⇡
x10 ⇤
• Thesolutioniswell
approximatedby
G(s10 , ⌘) / e2.31 (⌘
s10 )
• Thismotivatedustolookforthesolutioninthefollowingscalingform:
G(s10 , ⌘) = G(⌘
(s10 , s21 , ⌘ 0 ) = (⌘ 0
s10 )
s10 , ⌘ 0
s21 )
15
AnalyticSolutionandIntercept
• The(dominantpartofthe)scalingsolutionis
1 p4 ⇣
G(⇣) ⇡ e 3
3
✓
◆
0
⇣
⇣
0
1 p4 ⇣
p
0
(⇣, ⇣ ) ⇡ e 3
4e 3
3
3
✓
◆
0
⇣ ⇣
p
0
= G(⇣ ) 4e 3
3
• Thecorrespondinghelicityinterceptis
↵hq
4
p
=
3
r
↵ s Nc
⇡ 2.3094
2⇡
r
↵ s Nc
2⇡
• Thisisincompleteagreementwiththenumericalsolution!
↵hq
⇡ 2.31
r
↵ s Nc
2⇡
seeMatt’stalk
16
GluonHelicityTMDs
DipoleGluonHelicityTMD
• NowletusrepeatthecalculationforgluonhelicityTMDs.
• WestartwiththedefinitionofthegluondipolehelicityTMD:
2i SL
g1G (x, kT2 ) =
xP+
Z
d⇠ d2 ⇠ ixP + ⇠
e
(2⇡)3
ik·⇠
D
P, SL |✏ij
T
h
tr F
+i
(0) U
[+]†
[0, ⇠] F
U^[+]
• HereU[+] andU[-] arefutureand
pastWilsonlinestaples (hence
thename’dipole’TMD,
F.Dominguezetal’11– looks
likeadipolescatteringona
proton):
+j
(⇠) U
[ ]
i
[⇠, 0] |P, SL
E
⇠ + =0
t
j
z
i
U^[−]
proton
DipoleGluonHelicityTMD
• Atsmallx,thedefinitionofdipolegluonhelicityTMDcanbemassaged
into
Z
Z
2
g1G dip (x, kT2 ) =
8i Nc SL
g 2 (2⇡)3
i ij
d2 x10 eik·x10 k?
✏T
d2 b10 Gj10 (zs =
Q
)
x
• Hereweobtainanewoperator,whichisatransversevector(A-=0
gauge):
Gi10 (z)
1
⌘
4Nc
Z1
1
dx
D h
i
E
i
tr V0 [1, 1]V1 [ 1, x ] ( ig) Ã (x , x) V1 [x , 1] + c.c. (z)
t
i ij
• Notethatcanbethoughtof
k?
✏T
i
asatransversecurlactingon Gi10 (z)
Ãi (x , x) different
andnotjuston--
z
fromthepolarizeddipoleamplitude!
1
0
proton
DipoleTMDvsdipoleamplitude
• Notethattheoperatorforthedipole gluonhelicityTMD
1
Gi10 (z) ⌘
4Nc
Z1
dx
1
D h
i
E
i
tr V0 [1, 1]V1 [ 1, x ] ( ig) Ã (x , x) V1 [x , 1] + c.c. (z)
isdifferentfromthepolarizeddipole amplitude
1
G10 (z) ⌘
4Nc
Z1
1
dx
D h
i
E
tr V0 [1, 1]V1 [ 1, x ] ( ig) r ⇥ Ã(x , x) V1 [x , 1] + c.c. (z)
• WeconcludethatthedipolegluonhelicityTMDdoesnot
dependonthepolarizeddipoleamplitude!(Hencethe
‘dipole’namemaynotevenbevalidforsuchTMD.)
• Thisisdifferentfromtheunpolarized gluonTMDcase.
Dictionary
• Weseemtohavetwooperators:
• Quarkhelicity
1
G10 (z) ⌘
4Nc
Z1
dx
1
D h
i
E
tr V0 [1, 1]V1 [ 1, x ] ( ig) r ⇥ Ã(x , x) V1 [x , 1] + c.c. (z)
• Gluonhelicity
1
Gi10 (z) ⌘
4Nc
Z1
1
dx
D h
i
E
i
tr V0 [1, 1]V1 [ 1, x ] ( ig) Ã (x , x) V1 [x , 1] + c.c. (z)
GluonHelicityTMDs:Small-xEvolution
EvolutionEquation
• Toconstructevolutionequationfortheoperator𝐺 $
governingthegluonhelicityTMDweresum similar
(tothequarkcase)diagrams:
0
0
2
1
i
“c.c.”
k2
k1
1
i
x−
1
x−
2
i
other LLA diagrams
i
Large-Nc Evolution:Diagrams
• Atlarge-Nc theequationsare
0
0
Γ20 , 21(z ′ s)
∂
∂Y
Gi10 (zs)
2
=
2
+
z′
z′
z
z
1
0
i
+
“c.c.”
+
“c.c.”
i
1
0
“c.c.”
G21(z ′ s)
z
i
1
+
0
Γ20 , 21(z ′ s)
2
+
+
z′
z
1
i
2
Γ21 , 20(z ′ s)
z
1
0
i
0
z
Γi10 , 21(z ′ s)
+
2
+
2
′
z′
Gi12 (z ′ s)
z
1
z′
i
z
1
i
Large-Nc Evolution:Diagrams
• and
0
0
0
Γ30 , 31(z ′′ s)
∂
∂Y
Γi10 , 21(z ′ s)
3
+
3
=
+
z ′′
“c.c.”
z ′′
2
2
z′
1
2
z′
i
z′
i
1
G30 (z ′′ s)
i
1
0
0
Γ30 , 31(z ′′ s)
+
+
3
3
z
z ′′
2
“c.c.”
+
“c.c.”
2
z′
z′
i
1
+
′′
Γ31 , 30(z ′′ s)
i
1
0
0
Γi10 , 31(z ′′ s)
3
+
z ′′
3
+
z ′′
2
z′
1
Gi30 (z ′′ s)
i
1
i
Large-Nc Evolution:Equations
• Thisresultsinthefollowingevolutionequations:
Gi10 (zs)
=
i (0)
G10 (zs)
Zz
↵ s Nc
+
2⇡ 2
dz 0
z0
⇤2
s
↵ s Nc
2⇡ 2
Zz
dz 0
z0
⇤2
s
+
↵ s Nc
2⇡
dz
z0
=
i (0)
G10 (z 0 s)
0
j h
✏ij
T (x20 )?
d x2 ln
x21 ⇤
x220
Zx10
1
z0 s
↵ s Nc
+
2⇡ 2
Zz
0
↵ s Nc
2⇡ 2
0
dz 00
z 00
⇤2
s
↵ s Nc
+
2⇡ 2
Zz
0
1
x2
10 s
Z
dz 00
z 00
dz 00
z 00
Z

z0
min x210 , x221 z 00
Z
1
z 00 s
+ G21 (z s)
gen
00
30 , 31 (z s)
i
1
gen
00
30 , 31 (z s)
dx231 h i 00
G13 (z s)
x231
i
i
j h
✏ij
T (x31 )?
d x3 ln
x31 ⇤
x231
i
0
10 , 21 (z s)
j h
✏ij
T (x30 )?
d x3 ln
x31 ⇤
x230
1
2
0
+
2
gen
0
20 , 21 (z s)
gen
0
20 , 21 (z s)
gen
0
21 , 20 (z s)
dx221 h i 0
G12 (z s)
x221
⇤2
s
Zz
j h
✏ij
T (x21 )?
d x2 ln
x21 ⇤
x221
1
2
1
2
2
Zz
1
x2
10 s
i
0
10 21 (z s)
Z
Z
+
+ G31 (z 00 s)
gen
00
31 , 30 (z s)
i
00
10 , 31 (z s)
i
.
i
i
Large-Nc Evolution:Equations
• Here
gen (x20 , x21 , z
0
x21 ) (x20 , x21 , z 0 s) + ✓(x21
s) = ✓(x20
x20 ) G(x20 , z 0 s)
isanobjectwhichweknowfromthequarkhelicityevolution,asthe
lattergivesusGandG.
• Notethatourevolutionequationsmixthegluon(Gi)andquark(G)
small-xhelicityevolutionoperators:
↵ s Nc
i (0)
Gi10 (zs) = G10 (zs) +
2⇡ 2
Zz
⇤2
s
↵ s Nc
2⇡ 2
Zz
⇤2
s
↵ s Nc
+
2⇡
Zz
1
x2
10 s
dz 0
z0
Z
dz
z0
Z
j h
✏ij
T (x21 )?
d x2 ln
x21 ⇤
x221
2
1
j h
✏ij
T (x20 )?
d x2 ln
x21 ⇤
x220
2
2
0
dz 0
z0
Zx10
1
z0 s
1
dx221 h i 0
G12 (z s)
x221
gen
0
20 , 21 (z s)
i
0
10 , 21 (z s)
i
+
gen
0
20 , 21 (z s)
0
+ G21 (z s)
gen
0
21 , 20 (z s)
i
i
InitialConditions
• Initialconditionsforthisevolutionaregivenbythelowestordert-channel
gluonexchanges:
0
0
1
1
i
b
Z
d
2
i (0)
b G10 (z)
=
Z
2
d b
i
b
i (0)
10,21 (z)
=
↵s2 CF
1
2
⇡ ✏ij xj10 ln
Nc
x10 ⇤
• Notethattheseinitialconditionshavenolns,unliketheinitialconditions
forthequarkevolution:
G
(0)
(x210 , z)
=
(0)
(x210 , x221 , z)
=
↵s2 CF
2
⇡ ln(zs x210 )
Nc
Large-Nc Evolution:PowerCounting
• ThekernelmixingGi orGi withGandG isLLA:
Gi10 (zs)
=
i (0)
G10 (zs)
↵ s Nc
+
2⇡ 2
Zz
⇤2
s
↵ s Nc
2⇡ 2
Zz
⇤2
s
↵ s Nc
+
2⇡
Zz
1
x2
10 s
dz 0
z0
Z
dz
z0
Z
j h
✏ij
T (x21 )?
d x2 ln
x21 ⇤
x221
2
1
j h
✏ij
T (x20 )?
d x2 ln
x21 ⇤
x220
2
2
0
dz 0
z0
Zx10
1
z0 s
1
dx221 h i 0
G12 (z s)
x221
gen
0
20 , 21 (z s)
i
0
10 , 21 (z s)
i
+
gen
0
20 , 21 (z s)
0
+ G21 (z s)
gen
0
21 , 20 (z s)
i
i
LLA
DLA
• But,theinitialconditionsforGandG haveanextralns,makingthetwo
termscomparable(order-𝛼&' in𝛼& 𝑙𝑛' 𝑠~1 DLApowercounting).
GluonHelicityTMDs:Small-xAsymptotics
Large-Nc EvolutionEquations:Solution
• Tosolvetheequations,firstdecomposetherelevantobjectasfollows:
Z
d2 b Gi10 (z) = xi10 G1 (x210 , z) + ✏ij xj10 G2 (x210 , z)
Z
d2 b i10 (z) = xi10 1 (x210 , z) + ✏ij xj10 2 (x210 , z)
• ItturnsoutthatonlyG2 andG2 contributetoevolutionandtothegluon
helicityTMD.
Large-Nc EvolutionEquations:Solution
• Pluggingintheanalyticsolutionforthequarkhelicityoperators(Matt’s
talk),weget
G2 (x210 , zs) =
(0)
G2 (x210 , zs)
↵ s Nc
2⇡
Zz
1
x2
10 s
2
2
0
2 (x10 , x21 , z s)
=
↵ s Nc
2⇡
0
1
x2
10 s
p4
3
2
dz 0
z0
(0)
G2 (x210 , z 0 s)
Zz
↵ s Nc
3⇡
dz 00
z 00
Zx10
1
z0 s
dx221
x221
↵ s Nc
3⇡

1
q
↵s Nc
2⇡
p ↵s N c
2⇡
ln
1
x10 ⇤
2
2
0
2 (x10 , x21 , z s),
p4
3
1
q
↵s Nc
2⇡
z0
min x210 , x221 z 00
Z
zsx210
4
p
3
1
z 00 s
dx231
x231
z 0 sx210
4
p
3
p ↵s N c
2⇡
2
2
00
2 (x10 , x31 , z s)
ln
1
x10 ⇤
Large-Nc EvolutionEquations:Scaling
• Justlikeinthequarkhelicityevolutioncase,theequations
simplifyoncewerecognizethefollowingscalingproperty:
G2 (x210 , zs) = G2
2
2
0
2 (x10 , x21 , z s)
=
2
r
r
↵ s Nc
ln(zsx210 )
2⇡
!
↵ s Nc
ln(z 0 sx210 ),
2⇡
r
↵ s Nc
ln(z 0 sx221 )
2⇡
!
• Theequationsbecome
G2 (⇣) =
2 (⇣, ⇣
0
)=
1
2
1
2
r
r
↵s Nc p4 ⇣
e 3
6⇡
Z⇣
d⇠
0
↵s Nc p4 ⇣
e 3
6⇡
Z⇣
0
Z⇠
d⇠ 0
2 (⇠, ⇠
0
),
0
0
d⇠
Z⇠
0
d⇠ 0
2 (⇠, ⇠
0
)
Z⇣
⇣0
d⇠
Z⇣
0
0
d⇠ 0
2 (⇠, ⇠
0
)
Large-Nc EvolutionEquations:Solution
• Theseequationscanbesolvedintheasymptotichigh-energyregion
usingacombinationofODEsolvingandLaplacetransform,yielding
r
p
1 2 ↵s Nc 19 3 13
p ⇣
G2 (⇣
1) =
e4 3 ,
3
⇡
64
"p
#
r
p
p
p
1 2 ↵ s Nc
3 p4 ⇣ 43 ⇣ 0 3 3 p4 ⇣ 0 43 ⇣
0
1, ⇣
1) =
e 3
+
e 3
2 (⇣
3
⇡
4
64
• Thesmall-xgluonhelicityinterceptis
13
↵hG = p
4 3
r
↵ s Nc
⇡ 1.88
2⇡
r
↵ s Nc
2⇡
• Weobtainthesmall-xasymptotics ofthegluonhelicitydistributions:
p ↵s N c
✓ ◆ 13
p
2⇡
4 3
1
G dip
2
2
G(x, Q ) ⇠ g1L (x, kT ) ⇠
x
ScalingSolutionCross-Check
2
= f (s21 s10 )
• Onecancheckthescalingpropertyofouranalytic
G2
solutioninthenumericalsolutionofourequations:
Γ2 (s10 ,s21 ,η)/G2 (s10 ,η)
7
8
9
10
● s21 -s10 =0.1
■ s21 -s10 =0.5
◆ s21 -s10 =0.8
1.3
η
Γ2 (s10 ,s21 ,η)/G2 (s10 ,η)
1.05
◆
● η=7.5
1.04
◆
■
■ η=8.5
1.2
◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆
◆
●
■
◆ η=9.5
1.03
●
■
◆
1.02
1.1
◆
●
■
■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
●
◆
■
1.01
1.0
1.00◆
●
■
⌘=
◆
●
■
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
r
s10 =
●
■
◆
●
■
◆
0.05
●
■
◆
●
■
◆
0.10
0.15
0.20
0.25
0.30
↵ s Nc
zs
ln 2
2⇡
⇤
r
↵ s Nc
1
ln 2 2
2⇡
x10 ⇤
G2 (s10 , ⌘) = G2 (⌘
2 (s10 , s21 , ⌘
0
)=
2 (⌘
0
s10 )
s10 , ⌘ 0
s21 )
0.35
(s21 -s10 )
Conclusions
• Wethusconcludethatthesmall-xasymptotics ofgluon
helicity(atlargeNc)is
✓ ◆↵ G
1 h
2
G(x, Q ) ⇠
x
with ↵hG =
13
p
4 3
r
↵ s Nc
⇡ 1.88
2⇡
r
↵ s Nc
2⇡
whilethequarkhelicityasymptotics is(Matt’stalk)
✓ ◆↵qh
1
2
q(x, Q ) ⇠
x
with
↵hq
4
p
=
3
r
↵ s Nc
⇡ 2.31
2⇡
r
↵ s Nc
2⇡
• Swell…
• Mayuseourapproachtoconstrainthequarkandgluonspin
(andOAM)atsmallx(inprogress,along-termgoal).
BackupSlides
Large-Nc Evolution:Equations
• Thisresultsinthefollowingevolutionequations:
Zz
↵ s Nc
(0)
G2 (x210 , zs) = G2 (x210 , zs) +
2⇡ 2
⇤2
s
↵ s Nc
2⇡ 2
Zz
⇤2
s
Zz
↵ s Nc
2⇡
1
x2
10 s
2
2
0
2 (x10 , x21 , z s)
=
Z
dz 0
z0
↵ s Nc
2⇡ 2
dz
z0
dz 00
z 00
⇤2
s
↵ s Nc
2⇡
Zz
0
1
x2
10 s
x10 · x21 h
d x2 ln
x21 ⇤ x210 x221
1
2
x10 · x20 h
d x2 ln
x21 ⇤ x210 x220
1
2
0
Zx10
1
z0 s
dx221
x221
↵ s Nc
+
2⇡ 2
Zz
⇤2
s
0
Z
2
2
0
gen (x20 , x21 , z s)
2
2
0
gen (x20 , x21 , z s)
+
+
G(x221 , z 0 s)
2
2
0
gen (x21 , x20 , z s)
i
i
2
(0)
G2 (x210 , z 0 s)
Zz
dz 0
z0
Z
dz 00
z 00
2
2
0
2 (x10 , x21 , z s)
0
dz 00
z 00
Z
x10 · x31 h
d x3 ln
x31 ⇤ x210 x231
1
2
x10 · x30 h
d x3 ln
x31 ⇤ x210 x230
1
2

z0
min x210 , x221 z 00
Z
1
z 00 s
dx231
x231
2
2
00
gen (x30 , x31 , z s)
2
2
00
gen (x30 , x31 , z s)
2
2
00
2 (x10 , x31 , z s)
+
+
G(x231 , z 0 s)
2
2
00
gen (x31 , x30 , z s)
i
i
Large-Nc Evolution:Equations
• Thisresultsinthefollowingevolutionequations:
↵ s Nc
i (0)
Gi10 (zs) = G10 (zs) +
2⇡ 2
Zz
⇤2
s
↵ s Nc
2⇡ 2
Zz
dz 0
z0
⇤2
s
↵ s Nc
+
2⇡ 2
Zz
dz 0
z0
1
x2
10 s
i
0
10 21 (z s)
=
Z
i (0)
G10 (z 0 s)
↵ s Nc
2⇡ 2
0
⇤2
s
↵ s Nc
+
2⇡ 2
Zz
0
1
x2
10 s
j h
✏ij
T (x21 )?
d x2 ln
x21 ⇤
x221
1
2
j h
✏ij
T (x20 )?
d x2 ln
x21 ⇤
x220
Z
x210 h i 0
d x2 2 2 G12 (z s)
x21 x20
2
↵ s Nc
+
2⇡ 2
dz 00
z 00
Z
1
2
Zz
⇤2
s
Zz
dz 0
z0
Z
dz 00
z 00
0
dz 00
z 00
Z
gen
0
20 , 21 (z s)
i
0
10 , 21 (z s)
Z
x210 h i 00
d x3 2 2 G13 (z s)
x31 x30
2
+ G21 (z s)
i
i
i
1
j h
✏ij
T (x30 )?
d x3 ⇥ ln
x31 ⇤
x230
0
gen
0
21 , 20 (z s)
+
j h
✏ij
T (x31 )?
d x3 ln
x31 ⇤
x231
2
1
2
gen
0
20 , 21 (z s)
gen
00
30 , 31 (z s)
gen
00
30 , 31 (z s)
i
00
10 , 31 (z s)
i
+
00
+ G31 (z s)
gen
00
31 , 30 (z s)
i
i