NUMERICAL ANALYSIS IN BK EVOLUTION WITH IMPACT PARAMETER · BK with impact parameter General features of solution with impact parameter Saturation scale and diffusion in impact parameter
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NUMERICAL ANALYSIS IN
BK EVOLUTION WITH
IMPACT PARAMETER
Jeffrey Berger
Contents
� BK with impact parameter
� General features of solution with impact parameter
� Saturation scale and diffusion in impact parameter
� Corrected kernel for partial higher-order effects
2
DIS 2011Jeffrey Berger - The Pennsylvania State University
� Corrected kernel for partial higher-order effects
� Running coupling
� Differences in prescriptions for
� Regularization dependence
� Comparison with data
� and
sα
2F LF
Dipole Model3
Nb
r
Photon splits into a color dipole of size r
which interacts at impact parameter b
with the target (nucleon)
Color dipole interacts with partons of
the target through gluon exchange
� This analysis is done in the context of the dipole model of small x scattering. In this regime the evolution of the amplitude can be represented as a dipole cascade.
DIS 2011Jeffrey Berger - The Pennsylvania State University
p p
N(r,b,Y) is the scattering amplitude of
the dipole interaction
[A.H.Mueller, Nucl. Phys B415 373 (1994)]
The BK equation
� Enforces unitarity in the amplitude
� Parent dipole splits into two dipoles of and
� Splitting is determined by the kernel
4
][ 1202011202
201 NNNNNKdY
Ns −−+=
∂
∂∫ 2xα
),,,( YbxNN ijijijij ϑ=
1001 xxx −= 02x 12x
),,( xxxKK =� Splitting is determined by the kernel
� Impact parameter only dependence is in the amplitude
� Angle is the angle between and
� Usually the amplitude is assumed uniform in impact parameter, here we take the full dependences of the amplitude on impact parameter into account
DIS 2011Jeffrey Berger - The Pennsylvania State University
ijϑijx ijb
)(2
1jiij xxb +=
),,( 120201 xxxKK =
Features of BK with impact parameter5
DIS 2011Jeffrey Berger - The Pennsylvania State University
2
12
2
02
2
01
22 xx
xN
z
dzK sc
π
α=
� Leading order kernel used
� Coupling fixed at 1.0=π
α scN
Large contributions at x = 2b6
b
xtarget
� This behavior can be extracted from the
representation in terms conformal eigenfunctions
DIS 2011Jeffrey Berger - The Pennsylvania State University
b
Nontrivial angular dependence.
Peak of the amplitude occurs when
and
bx 2=bx 2||
Impact parameter tails7
� Power-like tails are
generated during the
evolution
� There is a clear ‘ankle’ where dependences of the
amplitude on impact parameter become power-like
DIS 2011Jeffrey Berger - The Pennsylvania State University
evolution
� Initial impact parameter
dependence is
quickly forgotten
22
1b
exeN
−−−=
Towards higher order8
⋅− z
x
xKz
x
xK
xx
xx
01
12
1
01
02
1
1202
12022
+
= z
x
xKz
x
xK
x
zN
z
dzK sc
01
122
1
01
022
12
01
22π
α
LO (solid) vs Modified (dashed)
� Kinematical cut owing to a modification in the energy denominator
� The modified kernel slows the evolution by approximately 30%
� The modified kernel has almost no affect when the impact parameter dependence is neglected due to the saturation of all large dipole sizes.
DIS 2011Jeffrey Berger - The Pennsylvania State University
xxxx 01011202
This kernel reduces to the LO kernel at large
rapidies or when
[L. Motyka and A. M. Stasto, Phys. Rev. D79, 085016 (2009)]
120201 , xxx >>
Saturation Scale9
( )( ) 5.0,,,,1 == YbrN YbQs
θ
( )bSeQYbQY
sssλα2
0
2),( =
Saturation scale was found to have the same
impact parameter dependence at large b
which leads us to a factorized form
( )4
1~
bbS
LO (solid) vs Modified (dashed)
DIS 2011Jeffrey Berger - The Pennsylvania State University
� Saturation is when the parton density becomes large and recombination effects become important� Defined here as the amplitude becomes large and the
nonlinear term becomes important.
� Numbers are consistent with analytical estimates
( )bSeQYbQs 0),( =
LO Modified
4.4 3.6 (2.5 )s
λ 1.0=sα 2.0=sα
( )4
~b
bS
[S. Munier and R. B. Peschanski, Phys. Rev. D69, 034008 (2004)]
[A. H. Mueller and D. N. Triantafyllopoulos, Nucl. Phys. B640, 331]
A Second Saturation Scale10
( )( ) 5.0,,,,1 == YbrN YbQsL
θ
Equation has two solutions now! Same Parameterization
( )bSeQYbQ L
Y
LsLsLsλα−= 2
0
2),(
LO (solid) vs Modified (dashed)
DIS 2011Jeffrey Berger - The Pennsylvania State University
� Larger dipole sizes have slightly different
saturation scale exponents
� More thinking to be done on this result…
LO Modified
6.0 5.8 (5.2 )sL
λ 1.0=sα 2.0=sα
Diffusion in impact parameter11
( ) 5.0,,, == YbBrN s θ
Growth of the black disk corresponds to growth
of the cross section
( )rFeBYrBYsBsλα22
),( = YsBeλσ 2≈
LO (solid) vs Modified (dashed)
� Increasing energy causes the dense region of the dipole cascade to expand in impact parameter space
� Size of the dense or ‘black’ region characterized by a radius of this black disk
� Fast increase in is partially due to the lack of scale in the solution currently
DIS 2011Jeffrey Berger - The Pennsylvania State University
LO Modified
2.6 2.2 (2.0 )sB
λ 1.0=sα 2.0=sα
( )rFeBYrBY
sssBsλα2
0
2),( = YsBe
λσ 2≈
Running coupling
� Several different prescriptions for running coupling
� Balitsky
� Kovchegov
12
−+
−+= 1
)(
)(11
)(
)(1
2
)(2
02
2
12
2
12
2
12
2
02
2
02
2
12
2
02
2
01
2
2
01
x
x
xx
x
xxx
xxN
z
dzK
s
s
s
ssc
α
α
α
α
π
α
( ) ( ) ( ) ( )⋅
−−−− 23
523
5
44
11ee
Ndzγγ
ααxx
[I. Balitsky, Phys. Rev. D75, 014001 (2007)]
� Kovchegov
-Weigert
� Parent Dipole
� Minimum Dipole
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( )2
12
2
02
2
01
2
2
01
2 xx
xxN
z
dzK sc
π
α=
( ) ( ) ( ) ( )( )
( )
⋅
−+=−−
−−−−
2102
23
5
220
3
220
3
212
23
5
220
23
5
;,
4
44
2
12
2
20
4
2
12
4
2
20
22
11
2xxxR
es
x
esx
es
x
esx
es
c
xxxx
N
z
dzK
γ
γγ
α
αααα
π2012 xx
( )( )2
12
2
02
2
01
2
2
02
2
12
2
01
2
,,min
xx
xxxxN
z
dzK sc
π
α=
[Y. V. Kovchegov and H. Weigert, Nucl. Phys. A784, 188 (2007]
Results with running coupling13
Fixed (solid) vs Running (Balitsky, dashed) Miniumum Prescription (solid) vs Balitsky Prescription (dashed)
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� IR regularization of the kernel is important due to large dipole evolution
� Balitsky’s running coupling is well slower than the minimum dipole prescription
Adding mass parameter
� Full cut with theta function
� Splitting the theta function
� Bessel function cut
14
( ) ( )2
1212
021
2
12
2
02
2
01
222
2xx
xx
xN
z
dzK
mm
sc −−= θθπ
α
( ) ( ) ( ) ( )
−−
⋅−−+−= 2
0212
121
2
12
2
02
12022
121
2
12
2
021
2
02
22222 2
11
2xx
xx
xxx
xx
x
N
z
dzK
mmmm
sc θθθθπ
α
� Bessel function cut
� Running coupling with theta function
� Modified kernel with theta function
DIS 2011Jeffrey Berger - The Pennsylvania State University
( ) ( )2
0212
121
2
02
2
12
2
12
2
12
2
02
2
02
2
12
2
02
2
01
2
2
01221
)(
)(11
)(
)(1
2
)(xx
x
x
xx
x
xxx
xxN
z
dzK
mm
s
s
s
ssc −−
−+
−+= θθ
α
α
α
α
π
α
( ) ( )2
0212
121
01
121
01
021
1202
1202
01
122
1
01
022
12
01
222
2
2xxz
x
xKz
x
xK
xx
xxz
x
xKz
x
xK
x
zN
z
dzK
mm
sc −−
⋅−
+
= θθ
π
α
( ) ( ) ( ) ( )
⋅−+=
1202
1202
12102112
2
102
2
12
2
22 xx
xxmxKmxKmxKmxK
mN
z
dzK sc
π
α
15
� The prescription by Balitsky
for running coupling has
unusual properties
� Slower than expected from
2F
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� Slower than expected from
the momentum space analysis
� Extremely sensitive to the
form of regularization of
� Closeness to the data should
perhaps be regarded as
accidental at this time
Fixed coupling kernels evolve too fast unless
coupling is artificially low
Minimum dipole prescription is also too fast
)(2
xsα
16
LFF &2
2F � In general the slope is too steep to fit the data
� Data is underestimated due to lack of
contribution from large dipole sizes
Need a separate contribution due to these large,
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� data is not very
discriminatory due to large
error bars
LFLF
� Need a separate contribution due to these large,
non-perturbative dipoles
Conclusions
� Solving the BK equation with impact parameter is
crucial – many features are left out otherwise!
� N � 0 for large dipole sizes
� Amplitude enhanced at x = 2b with peaks at
Power tails in impact parameter
17
( ) 1,1cos −+=θ
� Power tails in impact parameter
� Second wavefront develops evolving to larger dipole size
� Running coupling prescriptions slow the evolution more
than expected, bringing us surprisingly close to the
data, however there is a large sensitivity to
regularization as well as unexpected behavior.
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More work to be done!
� More kinematical constraints implemented at the
kernel level.
� Does this slow the evolution more and lead to a better
fit of the data?
18
� Exclusive diffractive production of
� Impact parameter dependence corresponds to
momentum transfer
� Numerical solution of full NLO Kernel?
DIS 2011Jeffrey Berger - The Pennsylvania State University
ψJ
Thank You
19
Thank You
DIS 2011Jeffrey Berger - The Pennsylvania State University
Special Thanks to : My advisor Anna Stasto as well as Henry Kowalski for discussions and
use of his code and Emil Avsar for interesting discussions.
Saturation Scale – B dependence20
( )( ) 5.0,,,,1 == YbrN YbQs
θ
Saturation scale was found to have the same
impact parameter dependence at large b
which leads us to a factorized form
DIS 2011Jeffrey Berger - The Pennsylvania State University
� Large impact parameters yield similar slopes with
similar dependences
( )bSeQYbQY
sssλα2
0
2),( =
which leads us to a factorized form
Angular Dependence21
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� Angular dependence only comes in when x = 2b
� Enhancements when ( ) 1,1cos −+=θ
� Naïve analysis leads us to believe the equivalence
of the minimum dipole size coupling and Balitsky’s
� Numerical analysis reveals this not to be true
Unusual slowness of the coupling22
DIS 2011Jeffrey Berger - The Pennsylvania State University
When one daughter dipole is small there are regions where one prescription
dominates when [left] the minimum dipole size method dominates
while when [right] the Balitsky prescription for running coupling
dominates, however these regions are not equal in BK.
( ) 1cos +=θ
( ) 1cos −=θ
Surprising behaviors of Balitsky’s kernel23
( )( )211
2
22ln
1)(
µα
+=
Λ x
sb
x
Using a factor to regularize the coupling or a µ
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Increasing the decreases the
coupling but in the case of the
Balitsky kernel this increases
the amplitude
Using a factor to regularize the coupling or a
sharp cutoff was found to change the amplitude by
much more than expected (a factor of 2 or more in
some cases), indicating a great sensitivity to the
specific form the coupling takes.
µ
µ
Impact Parameter is so importiant!24
� Impact parameter corresponds to momentum
transfer, neglecting impact parameter is equivalent
to setting momentum transfer � 0
� With BFKL this is self consistent
DIS 2011Jeffrey Berger - The Pennsylvania State University
� With BFKL this is self consistent
� Only linear terms (two pomeron vertex)
� This assumption with BK causes problems
� Nonlinear term (three pomeron vertex)
� Momentum transfer cannot stay zero
without altering the interaction
P=0
P=0
P=0
P=+p
P=-p
Conformal Symmetry?25
� LO Kernel is conformally invariant
� Expect evolution in small dipole and large dipole
directions to be the same
� Additional angular dependence? Numerics say no
DIS 2011Jeffrey Berger - The Pennsylvania State University
� Additional angular dependence? Numerics say no
dice
� Need full higher order corrections?
ϕθ
0x
xb
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