On Possible Implications of Gluon Number
Fluctuations in DIS Data
Wenchang Xiang
Bielefeld University
QCD school in Les Houches,France
March 26, 2008
Kozlov, Shoshi, Xiang
Wenchang Xiang (Uni Bielefeld)
JHEP10(2007)20, arXiv:0707.4142
On Possible Implications of Gluon Number Fluctuations in DIS Data
1 / 15
Outline
“Mean field equations”:
− Kovchegov and B-JIMWLK equations
− Hallmark of “mean field” evolution equations:
Geometrical scaling T (r , Y ) = T (r 2 Qs2 (Y ))
Beyond mean field:
− Gluon number fluctuations or pomeron loops
− Pomeron loop equations
− Hallmark of pomeron loop equations: Diffusive scaling
ln(Q̄s2 (Y )r 2 )
< T (r , Y ) >= T ( √
)
3
2
αs Y / ln (1/αs )
Numercial study DIS data
Summary
Wenchang Xiang (Uni Bielefeld)
On Possible Implications of Gluon Number Fluctuations in DIS Data
2 / 15
Mean field equations
Kovchegov equation:
∂
∂Y
hT iY ∝ ᾱs hT iY − hT iY hT iY
hT iY hT iY ; non-linear evolution, hT iY ≤ 1.
The solution in saturation
region
T (r , Y ) = 1 − C0 exp −C1 (ρ − ρs (Y ))2
ρ = ln(1/r 2 Q02 ), ρs (Y ) = ln(Qs2 (Y )/Q02 )
∂
hT iY 1; linear BFKL equation, ∂Y
hT iY ∝ ᾱs hT iY .
hT i ∼ exp [c ᾱs Y ] −→ unitarity violation!
Solution to BFKL equation with saturation
boundary(T 1 hbut not too samll):
i
2
s (Y ))
T (r , Y ) = C2 exp −λs (ρ − ρs (Y )) − (ρ−ρ
2ᾱχ“ (λ )Y
s
for 1 ρ − ρs (Y ) 2χ“ (λs )ᾱs Y
(ρ−ρs (Y ))2
2ᾱχ“ (λs )Y
Wenchang Xiang (Uni Bielefeld)
−→ violate the Geometrical scaling
On Possible Implications of Gluon Number Fluctuations in DIS Data
3 / 15
Mean field equations
p
Within a restricted window, ρ − ρs (Y ) 2χ“ (λs )ᾱs Y
T (r , Y ) ∼ C2 exp [−λs (ρ − ρs (Y ))]
Geometrical scaling
B-JIMWLK equations:
∂
∂Y
∂
∂Y
hT iY ∝ ᾱs hT iY − hTT iY ,
hTT iY ∝ ᾱs hTT iY − hTTT iY , . . .
Mean field approximation: hTT iY ≈ hT iY hT iY → Kovchegov
equation
Numerical result [Rummukainen, Weigert 04]:
Kovchegov
hT iY
Wenchang Xiang (Uni Bielefeld)
B−JIMWLK
≈ hT iY
On Possible Implications of Gluon Number Fluctuations in DIS Data
4 / 15
Models of amplitude and Geometrical Scaling
Stasto, Golec-Biernat and Kwiecinski
T
GBW
(r , x) = 1 − Exp
Qs2 (x)
Where:
=
Q02
n
−
1
4
Qs2
· (x0 /x)
·r
2
τ = Q 2 R02 (x)
o
σtotγ*p [μb]
GBW Model:
λ
For DIS at HERA
(x ≤ 10−2 and 0.045 < Q 2 < 450GeV 2 ):
λ = 0.29, x0 = 3 · 10−4 , Q0 = 1 GeV .
MV Model:
T MV (r , b, x) = 1 − Exp
n
− 14 Qs2 · r 2 · log
1
r 2 Λ2
10
3
10 2
+e
o
Corrections to the region of large Q 2 ;
10
IIM Model(inspired by solution of
BK-equation):
8
>
>
>
<
IIM
T (r , x) =
>
>
>
:
n
2
1 − exp −a ln (b r Qs (x))
N0
r Q
s
2
(x) 2
ln(2/r Qs (x))
λs +
κλY
Two matching parameters;
ZEUS BPT 97
ZEUS BPC 95
2
H1 low Q 95
2
ZEUS+H1 high Q 94-95
1
o
R02 (x) ≡ (x/x0 )λ /Q02
E665
r Qs (x) > 2
x<0.01
2
all Q
r Qs (x) < 2
10
-1
10
-3
10
-2
10
-1
1
10
10
Diffusive corrections;
Wenchang Xiang (Uni Bielefeld)
On Possible Implications of Gluon Number Fluctuations in DIS Data
2
10
3
τ
5 / 15
Geometrical Scaling
Does the Geometrical Scaling is unique possibility to
arrange the HERA data ?
NOT
Does the violation of the geometric scaling come from
BK-diffusion term or from gluon number fluctuations(Pomeron loops) ?
Wenchang Xiang (Uni Bielefeld)
On Possible Implications of Gluon Number Fluctuations in DIS Data
6 / 15
Shortcomings of “mean field equations”: Pomeron loops
Two dipoles scattering off a target [Iancu,Triantafyllopoulos 2005]:
∂
∂Y
{z
|
“mean field equations”
}
|
{z
}
graph missed!
Pomeron loops missed!
Wenchang Xiang (Uni Bielefeld)
On Possible Implications of Gluon Number Fluctuations in DIS Data
7 / 15
Pomeron loop equations
Hierarchy:
∂
hT iY
∂Y
∝
αs [hT iY − hT T iY ]
∂
hT T iY
∂Y
∝
αs
h
hT T iY − hT T T iY + α2s hT iY
i
Stochastic version:
∂
∂Y
TY
∝
αs TY − TY TY +
√
αs T ν
describe Gluon Number Fluctuations / Pomeron Loops;
non-linear evolution, satisfies {s,t}-channel unitarity;
Wenchang Xiang (Uni Bielefeld)
On Possible Implications of Gluon Number Fluctuations in DIS Data
8 / 15
Effects of Pomeron Loops/Gluon Nnumber Fluctuations
Single events
Geometric scaling
T(r,Y)
1
T (r , Y ) = T (r 2 Qs2 (Y ))
ρ = ln 1/r
ρs = ln Q2s (Y )
Average over events
Diffusive scaling
T(r,Y)
1
hT (r , Y )i = F
ln(Q̄s2 (Y ) r 2 )
√
DY
ρ = ln 1/r
Statistical physics ⇐⇒ hdQCD:
hT (ρ − ρs )i =
R
dρs P(ρs ) T (ρ − ρs )
[ Iancu,Mueller,Munier (2004) ]
where
=⇒
P(ρs ) =
√ 1
2πσ 2
2
s i)
exp[ (ρs −hρ
],
2σ 2
σ 2 = hρ2s i − hρs i2 = D Y
shape of hT i becomes flatter with increasing Y
Wenchang Xiang (Uni Bielefeld)
On Possible Implications of Gluon Number Fluctuations in DIS Data
9 / 15
Numerical Evaluation
Fit for the F2 structure function
( ZEUS data in the kinematical range:
F2 (x, Q 2 ) =
2
σT ,L (x, Q ) =
|ΨT |2 =
3 αem
2π 2
|ΨL |2 =
3 αem
2π 2
R
Q2
(σT (x, Q 2 ) + σL (x, Q 2 ))
4π 2 αem
(
2
2
2
dz d r |ψT ,L (z, r , Q )| σdip (x, r )
σdip = 2πR
n
P
P
x ≤ 10−2 and 0.045 GeV2 < Q 2 < 50 GeV2 )
e 2 K 2 (Q
e r ) + m2 K 2 (Q
e r)
ef2 [z 2 + (1 − z)2 ]Q
f
f
0
f 1
f
f
e r)
ef2 4Q 2 z 2 (1 − z)2 K02 (Q
f
o
,
Parameters are fixed via minimization of the χ2
X F mod (p1 , ..., pn ) − F exp 2
i
i
χ2 =
err2i
i
Wenchang Xiang (Uni Bielefeld)
T (r , x)
hT (r , x)i
o
f
n
2
e 2 = z(1−z)Q 2 +m2
Q
f
f
err2 = err2sys +err2sta
On Possible Implications of Gluon Number Fluctuations in DIS Data
10 / 15
Numerical Evaluation
GBW model:
The parameters of the event-by-event T GBW and of the physical hT GBW i amplitude.
χ2
χ2 /d.o.f
x0 (×10−4 )
λ
R(fm)
D
T GBW
266.22
1.74
4.11
0.285
0.594
0
hT GBW i
173.39
1.14
0.0546
0.225
0.712
0.397
IIM model:
The parameters of the event-by-event T IIM and of the physical hT IIM i amplitude.
χ2
χ2 /d.o.f
x0 (×10−4 )
λ
R(fm)
D
T IIM
150.45
0.983
0.5379
0.252
0.709
0
hT IIM i
122.62
0.807
0.0095
0.198
0.812
0.325
Wenchang Xiang (Uni Bielefeld)
On Possible Implications of Gluon Number Fluctuations in DIS Data
11 / 15
Model independent approach
In case fluctuations are important in the range of HERA data,
one finds the diffusive scaling behavior:
!
hT (r , Y )i = T̄ (r , Y ) = T̄
ln( 21 2 )
r Q
√ s
DY
Quality factor[Gelis,Peschanski,Soyez,Schoeffel]:
O(λ) =
P
(σi −σi−1 )2
i (τi −τi−1 )2 +ε2 ,
4π 2 αem F2 (x, Q 2 )/Q 2 ,
with σ =
and ε = 1/n
√
τ = ln(1/r 2 Qs2 )/ DY
We got λ = 0.215 for the input-values 0.01 ≤ D ≤ 0.7,
which seems tell us that gluon number fluctuations are relevent
in the range of HERA data.
Wenchang Xiang (Uni Bielefeld)
On Possible Implications of Gluon Number Fluctuations in DIS Data
12 / 15
F2
Geometrical versus Diffusive Scaling - small Q 2
Q2 =0.11
0.4
Q2 =0.15
Q2 =0.2
Q2 =0.25
0.2
0
Q2 =0.3
Q2 =0.4
Q2 =0.5
Q2 =0.65
0.5
0
-6
10
-3
10
Wenchang Xiang (Uni Bielefeld)
-6
10
-3
10
-6
10
-3
10
-6
10
On Possible Implications of Gluon Number Fluctuations in DIS Data
-3
10
13 / 15
F2
Geometrical versus Diffusive Scaling - large Q 2
2
1
Q2=18
Q2=20
Q2=22
Q2=25
Q2=27
Q2=35
Q2=45
Q2=60
Q2=70
Q2=90
Q2=120
Q2=150
0
2
1
0
2
1
0
2
1
Q2 =200
Q2 =250
Q2 =350
Q2 =450
0
-4
10
-1
10
Wenchang Xiang (Uni Bielefeld)
-4
10
-1
10
-4
10
-1
10
10-4
On Possible Implications of Gluon Number Fluctuations in DIS Data
10-1
X
14 / 15
Summary
Description of the DIS data is improved once gluon number fluctuations
are taken into account.
This outcome seems to indicate the evidence of geometric scaling
violations, and a possible implication of gluon number fluctuations,
in the DIS data.
However, looking only on HERA data one cannot exclude the possibility
that the scaling violations may also come from the diffusion part of the
solution to the BK-equation.
For the technical details:
Wenchang Xiang (Uni Bielefeld)
Kozlov, Shoshi, Xiang
arXiv:0707.4142
On Possible Implications of Gluon Number Fluctuations in DIS Data
15 / 15