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Resumming large radiative corrections in the high-energyevolution of the Color Glass Condensate
Edmond IancuIPhT Saclay & CNRS
w/ J.D. Madrigal, A.H. Mueller, G.Soyez, and D.N. Triantafyllopoulos
June 19, 2013
10-1
100
101
10-6
10-5
10-4
10-3
10-2
Q2 o
r 4/r
2 [G
eV
2]
x
dataGBW,smallGBW,facrcMV,smallrcMV,fac
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 1 / 33
Introduction
The Color Glass Condensate effective theory is the pQCD descriptionof the wavefunction of a very energetic hadron
non-linear effects associated with the high gluon density
Via appropriate factorization schemes, it controls various high-energyprocesses: DIS, proton-nucleus, nucleus-nucleus (initial conditions)
“Effective theory” : high-energy evolution described by pQCD
Balitisy-JIMWLK hierarchy ≈ BK equation (at large Nc)
non-linear generalizations of the BFKL equation
The non-linear evolution has recently been promoted to NLO
running coupling corrections to BK (Kovchegov, Weigert; Balitsky, 07)full NLO version of the BK equation (Balitsky, Chirilli, 08)JIMWLK evolution at NLO (Kovner, Lublinsky, Mulian, 2013)
Large corrections enhanced by double or single transverse logarithmsBNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 2 / 33
Introduction
Similar corrections were previously identified in the context ofNLO BFKL (Fadin, Kotsky, Lipatov, Camici, Ciafaloni ... 95-98)
Associated with high transverse momenta (weak scattering/low gluondensity) =⇒ cannot be cured by non-linear effects
For NLO BFKL, one has devised powerful resummation schemes ...(Salam, Ciafaloni, Colferai, Stasto, 98-03; Altarelli, Ball, Forte, 00-03)
... which however were formulated in Mellin space
The non-linear evolution equation (JIMWLK, BK) are naturallyformulated in the transverse coordinate space (eikonal approximation)
A new strategy for resumming the large transverse logarithms, bettersuited for non-linear evolution (arXiv:1502.05642)
An all-order resummed (‘collinearly-improved’) version of the BKequation, as a powerful tool for phenomenology (arXiv:1507.03651)
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 3 / 33
Outline
The prototype process at high energy: dipole-hadron scattering
BK equation at leading order (LO)
A glimpse at the NLO corrections
Double transverse logarithms & Time ordering
Adding single transverse logarithms (DGLAP, running coupling)
=⇒ a “collinearly improved” version of BK equation
Fits to the HERA data
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 4 / 33
QCD evolution via Bremsstrahlung
Bremsstrahlung favors soft and collinear emissions:
dP =αsNc
π
dx
x
dk2⊥k2⊥
≡ αs dY dρ
double logarithmic enhancement: Y ≡ ln(1/x), ρ ≡ ln(k2⊥/Q
20)
evolution with increasing Y (decreasing x) : BFKL, BK, JIMWLK
evolution with increasing ρ (or k⊥) : DGLAP
common limit: ‘double logarithmic approximation’ (DLA): (αsY ρ)n
Beyond leading-order (LO) : ‘cross-terms’ between BFKL and DGLAPBNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 5 / 33
Dipole–hadron scattering
A high-energy process interesting for both DIS and pA collisions
σγ∗p(Q2, x) = 2πR2
p
∑f
∫d2r
∫ 1
0dz∣∣Ψf (r, z;Q2)
∣∣2 T (r, x)
x =Q2
2P · q 'Q2
s� 1 (Bjoerken’ x)
T (r, x) : scattering amplitude for a qq color dipole with transverse size r
r2 ∼ 1/Q2 : the resolution of the dipole in the transverse plane
x : longitudinal fraction of a gluon from the target that scattersBNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 6 / 33
Dipole–hadron scattering
A high-energy process interesting for both DIS and pA collisions
Forward quark production in pA collisions
a quark from the proton undergoes transverse momentum broadeningvia rescattering off the nucleus
cross-section = amplitude × complex conjugate amplitude
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 7 / 33
Dipole–hadron scattering
A high-energy process interesting for both DIS and pA collisions
mathematically equivalent (to the accuracy of interest) to the elasticscattering of a dipole (amplitude only)
dσ
dηd2p= xq(x)
1
(2π)2
∫d2xd2y e−ip·(x−y) Sxy
Sxy = 1− Txy : the dipole S-matrix (survival of the singlet state)
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 7 / 33
Dipole–hadron scattering (γ∗p, γ∗A, pA, ...)
Dipole (‘projectile’): large q+, transverse resolution Q2 = 1/r2
Hadron (‘target’): large P−, saturation momentum Q20
Wilson lines : multiple scattering in the eikonal approximation
Sxy =1
Nctr(V †xVy
), V †(x) = P exp
{ig
∫dx+A−a (x+,x)ta
}BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 8 / 33
The target average: CGC
Average over the color fields (or charges) in the target :
〈Sxy〉 =
∫[DA−]W [A−]
1
Nctr(V †xVy
)[A−]
The CGC weight function W [A−] : kind of ‘functional pdf’
B semi-classical treatment of the gluons in the target: high density
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 9 / 33
An instructive model (‘low energy’)
MV model (McLerran, Venugopalan, ’93) : a Gaussian weight function
〈Sxy〉 = e−〈T0(r)〉 ' exp
{−1
4r2Q2
0 ln1
r2Λ2
}B Q2
0 ∝ color charged squared/unit ⊥ area; Λ = IR cutoff (‘confinement’)
B 〈Txy〉 = 1− 〈Sxy〉 ∝ r2 as r → 0 : ‘color transparency’
B 〈Txy〉 ' 1 as r & 1/Qs0 : ‘unitarity’
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 10 / 33
High energy evolution
Probability ∼ αs ln(1/x) to have additional, soft (x� 1), gluons
x ≡ k+/q+ : longitudinal momentum fraction of the emitted gluon
x2 =k+
q+� x1 =
p+
q+� 1 , Y = ln
1
xmin= ln
s
Q20
Leading logarithmic approx: resum (αsY )n with n ≥ 1
strong ordering in x, no special ordering in k⊥BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 11 / 33
High energy evolution
Probability ∼ αs ln(1/x) to have additional, soft (x� 1), gluons
x ≡ k+/q+ : longitudinal momentum fraction of the emitted gluon
Non-linear evolution, due to high gluon density in the target
projectile (dipole) evolution: Balitsky hierarchy, BK equation
target (CGC) evolution: JIMWLK equation (functional)
linear approximation (weak scattering) : BFKL
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 11 / 33
One step in the high energy evolution
‘Real corrections’ : the soft gluon crosses the shockwave
y
x
0 8
z
8ï
t1
2t
0
z
0
z
0
z
‘Virtual corrections’ : evolution in the initial/final state
0
z
0
z
0
z
0
z
Large Nc : the original dipole splits into two new dipoles
y
x
z
y
x
z
y
x
z
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 12 / 33
The BK equation
∂Sxy∂Y
=αs2π
∫d2zMxyz
[SxzSzy − Sxy
]Dipole kernel: BFKL kernel in the dipole picture (Al Mueller, 1990)
Mxyz =(x− y)2
(x− z)2(y − z)2=
[zi − xi
(z − x)2− zi − yi
(z − y)2
]2color transparency : Mxyz ∝ r2, hence Txy ∝ r2 as r → 0
good ‘infrared’ (large z⊥, small k⊥) behavior : dipole
Mxyz 'r2
(z − x)4when |z − x| ' |z − y| � r
‘ultraviolet’ poles (z = x or z = y) cancel between ‘real’ and ‘virtual’
non-linear effects =⇒ unitarity bound : Txy ≤ 1
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 13 / 33
LO BK : numerical solutions
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
-5 0 5 10 15 20 25
T(ρ
,Y)
ρ=log(1/r2)
LO, α- s=0.25
Y=0
Y=4
Y=8
Y=12
Y=16 0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20
dlo
g[Q
2 s(Y
)]/d
Y
Y
speed, α- s=0.25
LO
T (ρ, Y ) as a function of ρ = ln(1/r2Q20) with increasing Y
color transparency at large ρ (small r) : T (r, Y ) ∝ r2Q20 = e−ρ
unitarity (‘black disk limit’) at small ρ (large r) : T (r, Y ) ' 1
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 14 / 33
LO BK : numerical solutions
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
-5 0 5 10 15 20 25
T(ρ
,Y)
ρ=log(1/r2)
LO, α- s=0.25
Y=0
Y=4
Y=8
Y=12
Y=16 0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20
dlo
g[Q
2 s(Y
)]/d
Y
Y
speed, α- s=0.25
LO
T (ρ, Y ) as a function of ρ = ln(1/r2Q20) with increasing Y
saturation momentum Qs(Y ): T (r, Y ) = 0.5 when r = 2/Qs(Y )
saturation exponent: λs ≡ d lnQ2s(Y )
dY ' 4.88αs ' 1 for Y & 10
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 14 / 33
Next-to-leading order
Two successive emissions strongly ordered in p+ (or x) : ∼ (αsY )2
two iterations of BK : part of the LO evolution
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 15 / 33
Next-to-leading order
Two successive emissions strongly ordered in p+ (or x) : ∼ (αsY )2
two iterations of BK : part of the LO evolution
‘NLO’ : any effect of O(α2sY )
the prototype: two successive emissions, one soft and one non-soft
Caution: two strongly-ordered emissions contribute to NLO as well
in fact, they give the largest NLO corrections (see below)
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 15 / 33
Some NLO graphs
NLO graphs too can be ‘real’ or ‘virtual’
They can involve quark loops as wellBNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 16 / 33
BK equation at NLOBalitsky, Chirilli (arXiv:0710.4330 [hep-ph]) : Nf = 0, large Nc
dSxy
dY=αs
2π
∫d
2z
(x−y)2
(x−z)2(y−z)2
(SxzSzy − Sxy
){1 +
+ αs
[b ln(x−y)
2µ
2 − b(x−z)2 − (y−z)2
(x−y)2ln
(x−z)2
(y−z)2
+67
36−π2
12−
1
2ln
(x−z)2
(x−y)2ln
(y−z)2
(x−y)2
]}
+α2s
8π2
∫d2u d2z
(u−z)4
(SxuSuzSzy − SxuSuy
){−2 +
(x−u)2(y−z)2 + (x−z)2(y−u)2 − 4(x−y)2(u−z)2
(x−u)2(y−z)2 − (x−z)2(y−u)2ln
(x−u)2(y−z)2
(x−z)2(y−u)2
+(x−y)2(u−z)2
(x−u)2(y−z)2
[1 +
(x−y)2(u−z)2
(x−u)2(y−z)2 − (x−z)2(y−u)2
]ln
(x−u)2(y−z)2
(x−z)2(y−u)2
}
blue : leading-order (LO) terms
red : NLO terms enhanced by (double or single) transverse logarithms
black : pure α2s effects (no logarithms)
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 17 / 33
Large transverse logarithms
Running coupling corrections: proportional to b = 1112 − 1
6NfNc
taken care off via ‘standard’ prescriptions; see below
Collinear logarithms: ratios of widely separated dipole sizes
the double-logarithmic correction is already manifest
−1
2ln
(x−z)2
(x−y)2ln
(y−z)2
(x−y)2' −1
2ln2 (x−z)2
r2if |z − x| ' |z − y| � r
the single logs are still hidden: needs to perform the integral over u
1/Qs � |z − x| ' |z − y| ' |z − u| � |u− x| ' |u− y| � r
all dipoles are relatively small (� 1/Qs): weak scattering
... but such that their sizes are strongly increasing:
=⇒ logarithmic phase-space for the intermediate gluon at u
N.B. Collinear logs are important at weak scattering (dilute target)
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 18 / 33
Unstable numerical solution
Keeping just the collinear logarithms =⇒
dT (r)
dY= αs
∫ 1/Q2s
r2
dz2 r2
z4
(1− 1
2αs ln2 z
2
r2− 11
12αs ln
z2
r2
)T (z)
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
-5 0 5 10 15 20 25
T(ρ
,Y)
ρ=log(1/r2)
DLA at NLO, α- s=0.25
Y=0
Y=4
Y=8
10−5 10−4 10−3 10−2 10−1 100
rΛQCD
−0.10
−0.05
0.00
0.05
0.10
0.15
0.20
0.25
(∂yN)/
N
Qs,0/ΛQCD = 4Qs,0/ΛQCD = 13
Qs,0/ΛQCD = 19Qs,0/ΛQCD = 26
Left: LO BK + the double collinear logarithmRight: full NLO BK (Lappi, Mäntysaari, arXiv:1502.02400)
The main source of instability: the double collinear logarithmBNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 19 / 33
Double logarithms in the QCD evolution
Where do the double collinear logs come from ?
Bremsstrahlung naturally introduces double-logarithmic corrections ...
dP =αsNc
π
dx
x
dk2⊥k2⊥
= αs dY dρ
... but an energy logarithm times a collinear one !
Hint: From a constraint on the energy phase-space ...
Y > ρ =⇒ αsY ρ −→ αs(Y − ρ)ρ = αsY ρ− αsρ2
... which is introduced by time orderingBNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 20 / 33
Time ordering
To get double logs, successive emissions must be strongly ordered in ...
in longitudinal momentum : q+ � p+ � k+ · · · � q+0
in transverse momentum/size: Q2 � p2⊥ � k2
⊥ · · · � Q2s
in lifetime: τp = p+/p2⊥ � τk = k+/k2
⊥ (factorization)
Both p+ and p2⊥ are decreasing ⇒ potential conflict with time ordering
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 21 / 33
Time ordering
To get double logs, successive emissions must be strongly ordered in ...
p+
p2⊥>
k+
k2⊥=⇒ p+
k+>
p2⊥k2⊥
=⇒ ∆Y ≡ lnp+
k+> ∆ρ ≡ ln
p2⊥k2⊥
This condition enters perturbation theory via energy denominatorsBNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 21 / 33
Two successive emissions: p+ � k+
Light-cone (time-ordered) perturbation theory: physics is transparent
mixed F representation (p+, t ≡ x+), with p− = p2⊥/2p
+ = 1/τp
the time integrals yield energy denominators
1∑i∈interm
k−i − P−0' 1
p− + k−
x
y
z, k+
z, k+
u, p+
u, p+pp
k k
p
k k
(a) (b)
t1 t2
⌧1
⌧2
1
p+u2 + k+z2=
1
p+u2p+u2
p+u2 + k+z2' 1
p+u2Θ(p+u2 − k+z2)
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 22 / 33
Two successive emissions: p+ � k+
Light-cone (time-ordered) perturbation theory: physics is transparent
mixed F representation (p+, t ≡ x+), with p− = p2⊥/2p
+ = 1/τp
the time integrals yield energy denominators
1∑i∈interm
k−i − P−0' 1
p− + k−
x
y
z, k+
z, k+
u, p+
u, p+pp
k k
p
k k
(a) (b)
t1 t2
⌧1
⌧2
Integrate out the harder gluon (p+,u) to double-log accuracy:
αs
∫ z2
r2
du2
u2
∫ q+
k+
dp+
p+Θ(p+u2 − k+z2) = αsY ρ−
αsρ2
2
the expected LO contribution αsY ρ
NLO contribution αsρ2 to the kernel for emitting a softer gluon (k+, z)BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 22 / 33
The double logarithmic approximation (DLA)
Enforce time-ordering in the ‘naïve’ DLA limit of BFKL =⇒ DLA 2.0
∂Txy∂Y
=αs2π
∫d2zMxyz
[Txz + Tzy − Txy
]large daughter dipoles: 1/Q0 � |z − x| ' |z − y| � r
=⇒ Mxyz 'r2
(z − x)4
T (r) ∝ r2 =⇒ Txz ' Tzy � Txy : only ‘real’ terms matter
∂T (Y, r2)
∂Y= αs
∫ 1/Q20
r2
dz2
z2
r2
z2T (Y, z2)
∂T (Y, ρ)
∂Y= αs
∫ ρ
0
dρ1 e−(ρ−ρ1) T (Y − ρ+ ρ1, ρ1)
introduce time-ordering =⇒ non-local in Y
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 23 / 33
The double logarithmic approximation (DLA)
Enforce time-ordering in the ‘naïve’ DLA limit of BFKL =⇒ DLA 2.0
∂Txy∂Y
=αs2π
∫d2zMxyz
[Txz + Tzy − Txy
]large daughter dipoles: 1/Q0 � |z − x| ' |z − y| � r
=⇒ Mxyz 'r2
(z − x)4
T (r) ∝ r2 =⇒ Txz ' Tzy � Txy : only ‘real’ terms matter
∂T (Y, ρ)
∂Y= αs
∫ ρ
0
dρ1 e−(ρ−ρ1) T (Y, ρ1)
introduce logarithmic variable ρ ≡ ln(1/r2Q20)
∂T (Y, ρ)
∂Y= αs
∫ ρ
0
dρ1 e−(ρ−ρ1) T (Y − ρ+ ρ1, ρ1)
introduce time-ordering =⇒ non-local in YBNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 23 / 33
The double logarithmic approximation (DLA)
Enforce time-ordering in the ‘naïve’ DLA limit of BFKL =⇒ DLA 2.0
∂Txy∂Y
=αs2π
∫d2zMxyz
[Txz + Tzy − Txy
]large daughter dipoles: 1/Q0 � |z − x| ' |z − y| � r
=⇒ Mxyz 'r2
(z − x)4
T (r) ∝ r2 =⇒ Txz ' Tzy � Txy : only ‘real’ terms matter
∂T (Y, ρ)
∂Y= αs
∫ ρ
0
dρ1 e−(ρ−ρ1) T (Y, ρ1)
introduce logarithmic variable ρ ≡ ln(1/r2Q20)
∂T (Y, ρ)
∂Y= αs
∫ ρ
0
dρ1 e−(ρ−ρ1) T (Y − ρ+ ρ1, ρ1)
introduce time-ordering =⇒ non-local in YBNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 23 / 33
DLA 2.0
The importance of time-ordering has since long been recognized
coherence effects, kinematical constraint, choice of rapidity scale ...Ciafaloni (88), CCFM (90), Lund group (Andersson et al, 96),Kwiecinski et al (96), Salam (98) ... Motyka, Stasto (09), G. Beuf (14)
This is more than a prescription: it is a systematic approximation
resums powers of αsY ρ and αsρ2 to all orders
more precisely: all terms of order αnsY kρ2n−k, n ≥ 1 and 0 ≤ k ≤ n
However, in order to be useful, this should also include
the terms of order (αsY )n, n ≥ 1 (‘BFKL’)
the non-linear effects expressing saturation (‘BK’)
To that aim, one would need a ‘genuine’ evolution equation, local in Y
Does it exist ? Not a priori clear !
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 24 / 33
Getting local
T (Y, ρ) = T (0, ρ) + αs
∫ ρ
0dρ1 e−(ρ−ρ1)
∫ Y−ρ+ρ1
0dY1 T (Y1, ρ1)
For Y ≥ ρ, the solution T (Y, ρ) to the above equation coincides withthe solution T (Y, ρ) to the following problem:
T (Y, ρ) = T (0, ρ) + αs
∫ Y
0dY1
∫ ρ
0dρ1 e−(ρ−ρ1)KDLA(ρ− ρ1)T (Y1, ρ1)
with the following, all-orders, kernel:
KDLA(ρ) =J1(2√αsρ2
)√αsρ2
= 1− αsρ2
2+
(αsρ2)2
12+ · · ·
... and the following, all-orders, initial condition:
T (0, ρ) = T (0, ρ)−√αs∫ ρ
0dρ1 e−(ρ−ρ1) J1
(2√αs(ρ− ρ1)2
)T (0, ρ1)
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 25 / 33
Adding the single transverse logarithms
Recall the NLO equation with all the single logs
dT (r)
dY= αs
∫dz2
r2
z4
{1− αs
(1
2ln2 z
2
r2+
11
12lnz2
r2− b ln r2µ2
)}T (z)
The double-logarithm is already included within KDLA(ρ) XThe collinear single-log comes from the DGLAP regime:
one soft (x� 1) emission + one non-soft (x ∼ 1) one
coefficient A1 = 11/12 related to the DGLAP anomalous dimension:
γ(ω) =
∫ 1
0
dz zω[Pgg(z) +
CF
NcPqg(z)
]=
1
ω−A1 +O
(ω,
Nf
N3c
)can be resummed by including the A1 piece of γ(ω) into KDLA(ρ) X
The running coupling log is resummed by replacing αs → αs(r2) X
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 26 / 33
Extending to BFKL/BK
In the DLA regime (strong ordering in k⊥) we have so far obtained
∂T (Y, ρ)
∂Y= αs(ρ)
∫ ρ
0
dρ1 e−(1+αsA1)(ρ−ρ1)KDLA(ρ− ρ1) T (Y, ρ1)
LLA: daughter dipoles can also be comparable/smaller than the parentmatch onto the known result at NLLA:
KDLA ' 1− αs2
ln2 z2
r2→ 1− αs
2ln
(x−z)2
(x−y)2ln
(y−z)2
(x−y)2
restore the virtual and the non-linear terms
extend the prescription for RC using guidance from DGLAP
dTxydY
=
∫d2z
2παs(rmin)
(x−y)2
(x−z)2(z−y)2
(Txz + Tzy − Txy − TxzTzy
)×[
(x−y)2
min{(x−z)2, (y−z)2}
]±αsA1
KDLA
(ρ(x,y, z)
)BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 27 / 33
Numerical solutions
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
-5 0 5 10 15 20 25
T(ρ
,Y)
ρ=log(1/r2)
DLA at NLO, α- s=0.25
Y=0
Y=4
Y=8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
-5 0 5 10 15 20 25
T(ρ
,Y)
ρ=log(1/r2)
DLA resum, α- s=0.25
Y=0
Y=4
Y=8
Y=12
Y=16
Fixed coupling αs = 0.25, double collinear logs alone
left: expanded to NLOright: resummed to all orders
The resummation stabilizes & slows down the evolutionBNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 28 / 33
Numerical solutions
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20
dlo
g[Q
2 s(Y
)]/d
Y
Y
speed, α- s=0.25
LO
DLA at NLO
DLA resum
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
-5 0 5 10 15 20 25
T(ρ
,Y)
ρ=log(1/r2)
DLA resum, α- s=0.25
Y=0
Y=4
Y=8
Y=12
Y=16
Fixed coupling αs = 0.25, double collinear logs alone
left: saturation exponent λs ≡ d lnQ2s(Y )/dY
LO: λs ' 4.88αs ' 1 (for Y & 10)
DLA resummed: λs ' 0.5
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 28 / 33
Numerical solutions
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20
dlo
g[Q
2 s(Y
)]/d
Y
Y
speed, α- s=0.25
LO
DLA at NLO
DLA resum
DLA+SL resum
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
-5 0 5 10 15 20 25
T(ρ
,Y)
ρ=log(1/r2)
DLA+SL resum, α- s=0.25
Y=0
Y=4
Y=8
Y=12
Y=16
Further slowing down when including the single collinear logs
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 29 / 33
Numerical solutions
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 5 10 15 20
dlo
g[Q
2 s(Y
)]/d
Y
Y
speed, β0=0.72, smallest
LO
DLA at NLO
DLA resum
DLA+SL resum
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
-5 0 5 10 15 20 25
T(ρ
,Y)
ρ=log(1/r2)
DLA+SL resum, β0=0.72, smallest
Y=0
Y=4
Y=8
Y=12
Y=16
... and even more so after also using a running coupling
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 29 / 33
Fitting the HERA data (1)
Use numerical solutions to collinearly-improved running-coupling BKequation using initial conditions which involve free parameters
a similar strategy as for the DGLAP fits
Various choices for the initial condition at Y = Y0 :
GBW : T (Y0, r) =
{1− exp
[−(r2Q2
0
4
)p]}1/p
rcMV : T (Y0, r) =
{1− exp
[−(r2Q2
0
4αs(CMVr)
[1 + ln
(αsat
αs(CMVr)
)])p]}1/p
One loop running coupling with scale µ = 2Cα/r :
αs(r) =1
b0 ln[4C2
α/(r2Λ2)
] , with r = min{|x−y|, |x−z|, |y−z|}
Up to 5 free parameters: Rp (proton radius), Q0, p, Cα, (CMV)BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 30 / 33
Fitting the HERA data (2)
3 light quarks + charm quark, all treated on the same footing
quark masses are not fit parameters, but they are varied to test thesensitivity of the fit
good quality fits for mu,d,s = 50÷ 140 MeV and mc = 1.3 or 1.4 GeV
The most recent HERA data for the reduced photon-protoncross-section (combined analysis by ZEUS and H1)
small Bjoerken x: x ≤ 0.01
not very high Q2 : Q2 < Q2max with Q2
max = 50÷ 400 GeV2
Good quality fits: χ2 per point around 1.1-1.2
Very discriminatory: the fits favor
rcMV initial condition (pQCD + saturation)
physical prescriptions for RC: smallest-dipole, FAC
physical values for the free parameters
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 31 / 33
Fitting the HERA data (2)
3 light quarks + charm quark, all treated on the same footing
quark masses are not fit parameters, but they are varied to test thesensitivity of the fit
good quality fits for mu,d,s = 50÷ 140 MeV and mc = 1.3 or 1.4 GeV
The most recent HERA data for the reduced photon-protoncross-section (combined analysis by ZEUS and H1)
small Bjoerken x: x ≤ 0.01
not very high Q2 : Q2 < Q2max with Q2
max = 50÷ 400 GeV2
Good quality fits: χ2 per point around 1.1-1.2
Very discriminatory: the fits disfavorfixed coupling MV, GBW at high Q2
Balitsky prescriptions for RC
‘anomalous dimension’ γ > 1 in the initial condition
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 31 / 33
The Fit in tables
init RC sing. χ2 per data point parameterscdt. schm logs σred σccred FL Rp[fm] Q0[GeV] Cα p CMVGBW small yes 1.135 0.552 0.596 0.699 0.428 2.358 2.802 -GBW fac yes 1.262 0.626 0.602 0.671 0.460 0.479 1.148 -rcMV small yes 1.126 0.578 0.592 0.711 0.530 2.714 0.456 0.896rcMV fac yes 1.222 0.658 0.595 0.681 0.566 0.517 0.535 1.550GBW small no 1.121 0.597 0.597 0.716 0.414 6.428 4.000 -GBW fac no 1.164 0.609 0.594 0.697 0.429 1.195 4.000 -rcMV small no 1.097 0.557 0.593 0.723 0.497 7.393 0.477 0.816rcMV fac no 1.128 0.573 0.591 0.703 0.526 1.386 0.502 1.015
init RC sing. χ2/npts for Q2max
cdt. schm logs 50 100 200 400GBW small yes 1.135 1.172 1.355 1.537GBW fac yes 1.262 1.360 1.654 1.899rcMV small yes 1.126 1.172 1.167 1.158rcMV fac yes 1.222 1.299 1.321 1.317GBW small no 1.121 1.131 1.317 1.487GBW fac no 1.164 1.203 1.421 1.622rcMV small no 1.097 1.128 1.095 1.078rcMV fac no 1.128 1.177 1.150 1.131
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 32 / 33
The Fit in plots
00.05
0.10.15
0.2 0.045m
red
0.065 0.085 0.1 0.15
00.10.20.30.40.5
10-6 10-5 10-4
0.2
mre
d
x10-6 10-5 10-4
0.25
x10-6 10-5 10-4
0.35
x10-6 10-5 10-4
0.4
x10-6 10-5 10-4
0.5
x
00.10.20.30.40.50.60.70.80.9
0.65
mre
d
0.85 1.2 1.5 2
00.20.40.60.8
11.2
10-5 10-4 10-3
2.7
mre
d
x10-5 10-4 10-3
3.5
x10-5 10-4 10-3
4.5
x10-5 10-4 10-3
6.5
x10-5 10-4 10-3
8.5
x
00.20.40.60.8
11.21.4 10
mre
d
12 15 18 22
00.20.40.60.8
11.21.41.6
10-3 10-2
27
mre
d
x10-3 10-2
35
x10-3 10-2
45
x
r2
1.131.221.101.13
hs0.220.240.200.23
HERA datasmall, DL+SLfac, DL+SLsmall, DLfac, DL
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 33 / 33
The Fit in plots
theory
/data
0.9
0.95
1
1.05
1.1 0.045,0.065 0.085 0.1 0.15 0.2
theory
/data
x
0.9
0.95
1
1.05
1.1
10-6
10-5
10-4
0.25
x10
-610
-510
-4
0.35
x10
-610
-510
-4
0.4
x10
-610
-510
-4
0.5
x10
-610
-510
-4
0.65
theory
/data
0.9
0.95
1
1.05
1.1 0.85 1.2 1.5 2 2.7
theory
/data
x
0.9
0.95
1
1.05
1.1
10-5
10-4
10-3
3.5
x10
-510
-410
-3
4.5
x10
-510
-410
-3
6.5
x10
-510
-410
-3
8.5
x10
-510
-410
-3
10
theory
/data
0.9
0.95
1
1.05
1.112 15 18 22 27
theory
/data
x
0.940.960.98
11.021.041.06
10-3
10-2
35
x10
-310
-2
45 χ2
1.131.221.101.13
λs
0.220.240.200.23
HERA datasmall, DL+SLfac, DL+SLsmall, DLfac, DL
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 33 / 33
The Fit in plots
10-1
100
101
10-6
10-5
10-4
10-3
10-2
Q2 o
r 4/r
2 [G
eV
2]
x
dataGBW,smallGBW,facrcMV,smallrcMV,fac
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 33 / 33
The Fit in plots
10-1
100
101
10-6
10-5
10-4
10-3
10-2
Q2 o
r 4
/r2 [
Ge
V2]
x
dataGBW,smallGBW,facrcMV,smallrcMV,fac
10-5
10-4
10-3
10-2
10-1
100
10-2
10-1
100
101
DL+SL resummedT
(r,Y
=0
)
r [GeV-1
]
GBW,smallGBW,facrcMV,smallrcMV,fac
Rather stable predictions for the saturation line and the shape of theinitial amplitude
BNL, Long Island, July 2015 Collinearly-improved CGC Edmond Iancu 33 / 33