Collinear and TMD densities from Parton Branching Methods
• Ola Lelek 1 Francesco Hautmann 2 Hannes Jung
Voica Radescu 3 Radek Žlebčı́k 1
1
1 Deutsches Elektronen-Synchrotron (DESY)
2 University of Oxford
3 CERN
29.03.2017
Rencontres de Moriond, QCD and High Energy Interactions
La Thuile
1 / 23
Table of Contents
Motivation
Introduction to the method
Collinear PDFs from parton branching method
Results for TMDs
Fit of integrated TMDs for all flavours to HERA DIS data with xFitter
Summary
2 / 23
Collinear and TMD densities from Parton Branching Methods
Motivation
Motivation
3 / 23
Collinear and TMD densities from Parton Branching Methods
Motivation
Advantages of the parton branching method
To be discussed today:
a parton branching method in analogy to a Parton Shower (PS) but used to solve DGLAP
evolution equation.
Parton branching solution reproduces exactly semi-analytical results for collinear PDFs.
Similar codes exist (use similar formalism):
example: evolution code EvolFMC by Cracow group
S. Jadach et al., Markovian Monte Carlo program EvolFMC v.2 for solving QCD evolution equations, Comput.Phys.Commun. 181 (2010) 393-412
4 / 23
Collinear and TMD densities from Parton Branching Methods
Motivation
Advantages of the parton branching method
To be discussed today:
a parton branching method in analogy to a Parton Shower (PS) but used to solve DGLAP
evolution equation.
Parton branching solution reproduces exactly semi-analytical results for collinear PDFs.
Similar codes exist (use similar formalism):
example: evolution code EvolFMC by Cracow group
S. Jadach et al., Markovian Monte Carlo program EvolFMC v.2 for solving QCD evolution equations, Comput.Phys.Commun. 181 (2010) 393-412
Advantages of the parton branching method
A possibility of:
I
studying different orderings (QT -ordering,
(this will be not discussed in detail here),
virtuality ordering, angular ordering)
I
extraction of TMDs (structure of the grid suitable for usage in xFitter)
(this will be discussed).
4 / 23
Collinear and TMD densities from Parton Branching Methods
Motivation
Why Transverse Momentum Dependent PDFs?
Goal:
TMD PDFs for all flavours, all x, µ2 and kT
What is Transverse Momentum Dependent (TMD) Parton Distribution Function (PDF)?
I
TMD PDF is a generalization of a concept of the PDF.
I
TMD: depends not only on x and µ2 but also on kT : TMD(x, µ2 , kT )
5 / 23
Collinear and TMD densities from Parton Branching Methods
Motivation
Why Transverse Momentum Dependent PDFs?
Goal:
TMD PDFs for all flavours, all x, µ2 and kT
What is Transverse Momentum Dependent (TMD) Parton Distribution Function (PDF)?
I
TMD PDF is a generalization of a concept of the PDF.
I
TMD: depends not only on x and µ2 but also on kT : TMD(x, µ2 , kT )
TDMs are important in studies on:
I
resummation of all orders in the QCD coupling to many observables in high-energy hadronic
collisions,
I
nonperturbative information on hadron structure at very low kT ,
I
perturbative region where QCD evolution equations describe processes
I
a proper and consistent simulation of parton showers,
I
multi-scale problems in hadronic collisions,
I
...
Acta Physica Polonica B, Vol. 46 (2015)
5 / 23
Collinear and TMD densities from Parton Branching Methods
Introduction to the method
Introduction to the method
6 / 23
Collinear and TMD densities from Parton Branching Methods
Introduction to the method
DGLAP evolution equation
DGLAP evolution equation for momentum weighted parton density xf (x, µ2 ) = e
f (x, µ2 )
Z
2
X 1
de
fa (x, µ )
x
2
2
=
dzPab αs (µ ), z e
fb
,µ
d ln µ2
z
x
b
(1)
a, b- quark (2Nf flavours) or gluon, x- longitudinal momentum fraction of the proton carried by a parton a,
xi
- splitting variable, µ- evolution mass scale
xi−1
z =
splitting function:
Pab
2
αs (µ ), z
= Dab
2
2
αs (µ ) δ(1 − z) + Kab αs (µ )
1
(1 − z)+
+ Rab
2
αs (µ ), z ,
(2)
R1
R1
0 f(x)g (x)+ dx= 0 (f (x) − f (1))g (x)dx
Rab αs (µ2 ), z has no power divergences (1 − z)−n for z → 1 .
As long as Pab
αs (µ2 ), z has this structure, the formalism presented today can be applied (LO, NLO, NNLO).
7 / 23
Collinear and TMD densities from Parton Branching Methods
Introduction to the method
DGLAP evolution equation
DGLAP evolution equation for momentum weighted parton density xf (x, µ2 ) = e
f (x, µ2 )
Z
2
X 1
de
fa (x, µ )
x
2
2
=
dzPab αs (µ ), z e
fb
,µ
d ln µ2
z
x
b
(1)
a, b- quark (2Nf flavours) or gluon, x- longitudinal momentum fraction of the proton carried by a parton a,
xi
- splitting variable, µ- evolution mass scale
xi−1
z =
splitting function:
Pab
2
αs (µ ), z
= Dab
2
2
αs (µ ) δ(1 − z) + Kab αs (µ )
1
(1 − z)+
+ Rab
2
αs (µ ), z ,
(2)
R1
R1
0 f(x)g (x)+ dx= 0 (f (x) − f (1))g (x)dx
Rab αs (µ2 ), z has no power divergences (1 − z)−n for z → 1 .
As long as Pab
αs (µ2 ), z has this structure, the formalism presented today can be applied (LO, NLO, NNLO).
Two potential problems for numerical solution: (details in Backup!)
P R1
dzzP
I
presence of the delta function → solved by momentum sum rule
I
integrals separately divergent for (z → 1) → solved by a parameter zM :
c
ca
0
R1
x
αs (µ2 ), z = 0,
Rz
→ xM
7 / 23
Collinear and TMD densities from Parton Branching Methods
Introduction to the method
Sudakov formalism
After some algebraic transformations:
de
fa (x, µ2 )
d ln µ2
=
X Z zM
x
dz
R
Pab
x
X Z zM
2
2
2
αs (µ ), z e
fb
,µ
− e
fa x, µ
dz
z
0
c
R
2
zPca αs (µ ), z
(3)
b
R α (µ2 ), z = R
2
2
1
where Pab
s
ab αs (µ ), z + Kab αs (µ ) 1−z - real part of the splitting function.
8 / 23
Collinear and TMD densities from Parton Branching Methods
Introduction to the method
Sudakov formalism
After some algebraic transformations:
de
fa (x, µ2 )
d ln µ2
=
X Z zM
x
dz
R
Pab
x
X Z zM
2
2
2
αs (µ ), z e
fb
,µ
− e
fa x, µ
dz
z
0
c
R
2
zPca αs (µ ), z
(3)
b
R α (µ2 ), z = R
2
2
1
where Pab
s
ab αs (µ ), z + Kab αs (µ ) 1−z - real part of the splitting function.
Define the Sudakov form factor:

2

X Z zM
02
R
02
d ln µ
dzzPba αs (µ ), z 
ln µ2
0
b
0
Z ln µ2
∆a (µ ) = exp −
(4)
insert it in Eq.(3) and integrate
2
2
2
e
fa (x, µ ) = e
fa (x, µ0 )∆a (µ ) +
Z ln µ2
ln µ2
0
02
d ln µ
Z
∆a (µ2 ) X zM
∆a (µ02 ) b
x
R
dzPab
x
02
02
,µ
.
αs (µ ), z e
fb
z
(5)
8 / 23
Collinear and TMD densities from Parton Branching Methods
Introduction to the method
Iterative solution
Example for a= gluon:
2
2
2
e
fa (x, µ ) = e
fa (x, µ0 )∆a (µ )+
Z ln µ2
ln µ2
0
02
d ln µ
Z
∆a (µ2 ) X zM
∆a
(µ02 )
b
x
R
dzPab
x
02
2
02
αs (µ ), z e
fb
, µ0 ∆b (µ ) + ...
z
Sudakov: probability of evolving from
µ20 to µ2 without any resolvable
branching.
Sudakov: probability of evolving from
µ20 to µ2 without any resolvable
branching.
OR
This problem has an iterative solution which can be easily implemented in the MC code.
9 / 23
Collinear and TMD densities from Parton Branching Methods
Introduction to the method
Iterative solution
Example for a= gluon:
2
2
2
e
fa (x, µ ) = e
fa (x, µ0 )∆a (µ ) +
Z ln µ2
Sudakov: probability of evolving from
µ20 to µ2 without any resolvable
branching.
OR
ln µ2
0
02
d ln µ
Z
∆a (µ2 ) X zM
∆a
(µ02 )
b
x
R
dzPab
x
02
2
02
αs (µ ), z e
fb
, µ0 ∆b (µ )+...
z
Sudakov: probability of evolving from
µ20 to µ2 without any resolvable
branching.
OR
OR
This problem has an iterative solution which can be easily implemented in the MC code.
9 / 23
Collinear and TMD densities from Parton Branching Methods
Introduction to the method
Iterative solution
Example for a= gluon:
2
2
2
e
fa (x, µ ) = e
fa (x, µ0 )∆a (µ ) +
Z ln µ2
Sudakov: probability of evolving from
µ20 to µ2 without any resolvable
branching.
ln µ2
0
d ln µ
02
Z
∆a (µ2 ) X zM
∆a
(µ02 )
b
x
R
dzPab
x
02
2
02
αs (µ ), z e
fb
, µ0 ∆b (µ ) + ...
z
Sudakov: probability of evolving from
µ20 to µ2 without any resolvable
branching.
...
OR
OR
OR
This problem has an iterative solution which can be easily implemented in the MC code.
9 / 23
Collinear and TMD densities from Parton Branching Methods
Collinear PDFs from parton branching method
Collinear PDFs from parton branching method
10 / 23
Collinear and TMD densities from Parton Branching Methods
Collinear PDFs from parton branching method
LO comparison with semi analytical methods
xf(x)
xf(x)
Initial distribution: e
fb0 (x0 , µ20 ) - from QCDnum
The evolution performed with parton branching method up to a given scale µ2 .
Obtained distribution compared with a pdf calculated at the same scale by semi analytical method (QCDnum)
10
102
1
10
10−1
1
10−2
10−1
ratio
−4.5
−4
10−4
−3.5
−3
−3
10
gluon
10−3
µ2 = 10 GeV2
µ2 = 1000 GeV2
µ2 = 100000 GeV2
10−4
−5
101.1
−5
1.05
1
0.95
0.9
−5
10
10−2
down
µ2 = 10 GeV2
µ2 = 1000 GeV2
µ2 = 100000 GeV2
10−4
−2.5
−2
10−2
−1.5
−1
10−1
−0.5
0
1
−5
101.1
−5
1.05
1
0.95
0.9
−5
10
ratio
10−3
−4.5
−4
10−4
−3.5
−3
−3
10
−2.5
−2
10−2
−1.5
−1
10−1
Upper plots: collinear pdfs from the parton branching method
x
Lower plots: ratios of the pdfs from a parton branching method and pdfs from QCDnum.
−0.5
0
1
x
Very good agreement with the results coming from semi analytical methods (QCDnum).
11 / 23
Collinear and TMD densities from Parton Branching Methods
Collinear PDFs from parton branching method
NLO comparison with semi analytical methods
xf(x)
xf(x)
Initial distribution: e
fb0 (x0 , µ20 ) - from QCDnum
The evolution performed with parton branching method up to a given scale µ2 .
Obtained distribution compared with a pdf calculated at the same scale by semi analytical method (QCDnum)
10
102
1
10
10−1
1
10−1
10−2
−4
10−4
−3.5
−3
−3
10
µ2 = 10 GeV2
µ2 = 1000 GeV2
µ2 = 100000 GeV2
10−4
−2.5
−2
10−2
−1.5
−1
10−1
−0.5
0
1
−5
101.1
−5
1.05
1
0.95
0.9
−5
10
ratio
ratio
−4.5
gluon
10−3
µ2 = 10 GeV2
µ2 = 1000 GeV2
µ2 = 100000 GeV2
10−4
−5
101.1
−5
1.05
1
0.95
0.9
−5
10
10−2
down
10−3
−4.5
−4
10−4
−3.5
−3
−3
10
−2.5
−2
10−2
−1.5
−1
10−1
Upper plots: collinear pdfs from the parton branching method
x
Lower plots: ratios of the pdfs from a parton branching method and pdfs from QCDnum.
−0.5
0
1
x
Very good agreement with the results coming from semi analytical methods (QCDnum).
12 / 23
Collinear and TMD densities from Parton Branching Methods
Collinear PDFs from parton branching method
Cross check for different zM
Comparison of the results for different zM values .
Upper plot: collinear pdfs from a MC method
Lower plot: ratios of the pdfs from a MC method and pdfs from
QCDnum.
There is no dependence on zM as long as zM
large enough.
Here results at NLO, at LO the same conclusion.
13 / 23
Collinear and TMD densities from Parton Branching Methods
Results for TMDs
Results for TMDs
14 / 23
Collinear and TMD densities from Parton Branching Methods
Results for TMDs
kT dependence
Parton branching method: for every branching µ2 is generated and available.
I
I
How to connect µ with QT of the emitted and kT of the propagating parton?
−
→
QT - ordering: Q 2T ,n = µ2 .
−
→
virtuality ordering: Q 2T ,n = (1 − z)µ2
In this method kT
−
→
−
→
−
→
k T ,n = k T ,n−1 − Q T ,n
kT contains the whole history of the evolution.
is treated properly from the beginning of the evolution- no extra reshuffling at
the end is required.
15 / 23
Collinear and TMD densities from Parton Branching Methods
Results for TMDs
TMD PDFs from different kT definition at LO
Reminder: for collinear PDFs there was no zM dependence.
What about zM dependence for TMDs?
→
−
2
QT - ordering: Q 2
T ,n = µ
→
−
2
virtuality ordering: Q 2
T ,n = (1 − z)µ
large z - soft gluons!
QT - ordering: for every zM value we obtain different TMD
→ not physical behaviour, QT - ordering shouldn’t be used
Virtuality ordering: no zM dependence (because of the (1 − z) term soft gluons suppressed)
16 / 23
Collinear and TMD densities from Parton Branching Methods
Fit of integrated TMDs for all flavours to HERA DIS data with xFitter
Fit of integrated TMDs for all flavours to HERA DIS data with xFitter
17 / 23
Collinear and TMD densities from Parton Branching Methods
Fit of integrated TMDs for all flavours to HERA DIS data with xFitter
Procedure of the fit to the HERA 1+2 F2 data
I
A kernel Aba is determined from the parton branching method from a toy starting
distribution: f0,b = δ(1 − x).
xFitter chooses a starting distribution A0,b and performs a convolution of the kernel
Aba with the starting distribution A0,b to obtain a parton density
!
Z ∞
0
2 Z
x0 2
dkT
0
0 x
b
2
2
e
dx A0,b (x ) Aa
, kT , µ
fa (x, µ ) =
2
kT
x
x
0
|
{z
}
1
(6)
e
fa (x,µ2 ,k 2 )
T
I
Obtained parton density e
fa (x, µ2 ) is fitted to the F2 data and χ2 is calculated.
Data: arXiv:1506.06042v3, Abramowicz, H. and others.
I
2 e+p → e+X (NC)
1.5
0.5
The procedure is repeated with the new starting distributions A0,b many times to
minimize χ2 .
A very good χ2 /ndf ∼ 1.2 is obtained for 3.5 < Q 2 < 30000 GeV2 and x > 4 · 10−5 .
0
Theory/Data
I
σred
Goal: TMD PDF sets for all flavours, all x, Q 2 and kT
−0.5
HERA1+2 Data Q2 = 150
δ uncorrelated
δ total
1.05
0.01
0.1
0.01
0.1
1
0.95
x
18 / 23
Collinear and TMD densities from Parton Branching Methods
Fit of integrated TMDs for all flavours to HERA DIS data with xFitter
TMDs from fits - comparison of LO and NLO TMDs
TMDs with experimental uncertainties from the fit.
Comparison of the LO and NLO TMDs from the fit.
19 / 23
Collinear and TMD densities from Parton Branching Methods
Fit of integrated TMDs for all flavours to HERA DIS data with xFitter
TMDs from the fit at NLO
From parton branching method we can obtain TMDs for all flavours
At small kT (no branching or just a few branchings), the difference in the quark TMDs comes from initial distributions.
At large kT (many branchings) TMDs for quarks the same.
TMDs sets available soon!
→ check on TMDplotter and TMDlib
http://tmdplotter.desy.de/
http://https://tmdlib.hepforge.org/doxy/html/index.html
20 / 23
Collinear and TMD densities from Parton Branching Methods
Summary
Summary
21 / 23
Collinear and TMD densities from Parton Branching Methods
Summary
Summary
New approach to solve coupled gluon and quark DGLAP evolution equation with a parton
branching method was shown.
Advantages:
I
it reproduces exactly semi-analytical solution for collinear PDFs (results consistent with
QCDNum),
moreover:
I
extraction of TMD PDFs possible and done,
fit to F2 Hera data at LO and NLO was performed within xFitter,
TMDs sets for all flavours with uncertainties were obtained from the fit,
I
options to study different orderings for collinear and TMD PDFs available within this
framework.
Prospects:
I
TMD sets released soon,
I
application in measurements,
I
long term goal: direct usage in PS matched calculation.
22 / 23
Collinear and TMD densities from Parton Branching Methods
Summary
Thank you!
23 / 23
Collinear and TMD densities from Parton Branching Methods
Back up
Back up
1/8
Collinear and TMD densities from Parton Branching Methods
Back up
DGLAP evolution equation
DGLAP evolution equation
DGLAP evolution equation for momentum weighted parton density xf (x, µ2 ) = e
f (x, µ2 )
X
de
fa (x, µ2 )
=
d ln µ2
b
Z
x
1
x 2
dzPab αs (µ2 ), z e
fb
,µ
z
(7)
a, b- quark (2Nf flavours) or gluon, x- longitudinal momentum fraction of the proton carried by a parton a,
xi
- splitting variable, µ- evolution mass scale and
xi−1
z =
a structure of a splitting function:
Pab αs (µ2 ), z = Dab αs (µ2 ) δ(1 − z) + Kab αs (µ2 )
1
+ Rab αs (µ2 ), z ,
(1 − z)+
(8)
R1
R
R
f (x)g (x)+ dx = 01 f (x)g (x)dx − 01 f (1)g (x)dx
0 Dab αs (µ2 ) = δab da αs (µ2 ) , Kab αs (µ2 ) = δab ka αs (µ2 ) ,
Rab αs (µ2 ), z contains logarithmic terms in ln(1 − z) and has no power divergences (1 − z)−n for z → 1 .
2/8
Collinear and TMD densities from Parton Branching Methods
Back up
DGLAP evolution equation
DGLAP evolution equation
DGLAP evolution equation for momentum weighted parton density xf (x, µ2 ) = e
f (x, µ2 )
X
de
fa (x, µ2 )
=
d ln µ2
b
Z
x
1
x 2
dzPab αs (µ2 ), z e
fb
,µ
z
(7)
a, b- quark (2Nf flavours) or gluon, x- longitudinal momentum fraction of the proton carried by a parton a,
xi
- splitting variable, µ- evolution mass scale and
xi−1
z =
a structure of a splitting function:
Pab αs (µ2 ), z = Dab αs (µ2 ) δ(1 − z) + Kab αs (µ2 )
1
+ Rab αs (µ2 ), z ,
(1 − z)+
(8)
R1
R
R
f (x)g (x)+ dx = 01 f (x)g (x)dx − 01 f (1)g (x)dx
0 Dab αs (µ2 ) = δab da αs (µ2 ) , Kab αs (µ2 ) = δab ka αs (µ2 ) ,
Rab αs (µ2 ), z contains logarithmic terms in ln(1 − z) and has no power divergences (1 − z)−n for z → 1 .
X
de
fa (x, µ2 )
=
2
d ln µ
b
1
1
x 2
e
+ Rab αs (µ2 ), z
fb
,µ +
(1 − z)
z
x
Z
X
1
1
e
−
fb x, µ2
dz Kab αs (µ2 )
− Dab αs (µ2 ) δ(1 − z)
(1 − z)
0
b
Z
dz
Kab αs (µ2 )
(9)
2/8
Collinear and TMD densities from Parton Branching Methods
Back up
DGLAP evolution equation
DGLAP evolution equation
DGLAP evolution equation for momentum weighted parton density xf (x, µ2 ) = e
f (x, µ2 )
X
de
fa (x, µ2 )
=
d ln µ2
b
Z
x
1
x 2
dzPab αs (µ2 ), z e
fb
,µ
z
(7)
a, b- quark (2Nf flavours) or gluon, x- longitudinal momentum fraction of the proton carried by a parton a,
xi
- splitting variable, µ- evolution mass scale and
xi−1
z =
a structure of a splitting function:
Pab αs (µ2 ), z = Dab αs (µ2 ) δ(1 − z) + Kab αs (µ2 )
1
+ Rab αs (µ2 ), z ,
(1 − z)+
(8)
R1
R
R
f (x)g (x)+ dx = 01 f (x)g (x)dx − 01 f (1)g (x)dx
0 Dab αs (µ2 ) = δab da αs (µ2 ) , Kab αs (µ2 ) = δab ka αs (µ2 ) ,
Rab αs (µ2 ), z contains logarithmic terms in ln(1 − z) and has no power divergences (1 − z)−n for z → 1 .
X
de
fa (x, µ2 )
=
2
d ln µ
b
1
1
x 2
e
+ Rab αs (µ2 ), z
fb
,µ +
(1 − z)
z
x
Z
X
1
1
e
−
fb x, µ2
dz Kab αs (µ2 )
− Dab αs (µ2 ) δ(1 − z)
(1 − z)
0
b
Z
dz
Kab αs (µ2 )
Two potential problems for numerical solution:
I presence of the delta function,
I integrals separately divergent for z → 1.
(9)
2/8
Collinear and TMD densities from Parton Branching Methods
Back up
DGLAP evolution equation
Momentum sum rule
To get rid of the delta function:
P R1
2
We use momentum sum rule
c 0 dzzPca αs (µ ), z = 0:
3/8
Collinear and TMD densities from Parton Branching Methods
Back up
DGLAP evolution equation
Momentum sum rule
To get rid of the delta function:
P R1
2
We use momentum sum rule
c 0 dzzPca αs (µ ), z = 0:
X
de
fa (x, µ2 )
=
d ln µ2
b
1
1
x 2
e
+ Rab αs (µ2 ), z
,µ +
fb
(1
−
z)
z
x
Z
X
1
1
e
− Dab αs (µ2 ) δ(1 − z) +
−
fb x, µ2
dz Kab αs (µ2 )
(1 − z)
0
b
Z
1
X
−e
fa (x, µ2 )
dzzPca αs (µ2 ), z =
Z
dz
Kab αs (µ2 )
c
=
1
XZ
b
1
x 2
e
+ Rab αs (µ2 ), z
fb
,µ +
1−z
z
Z
1
X 1
dz z Kca αs (µ2 )
+ Rca αs (µ2 ), z
−e
fa x, µ2
1−z
0
c
dz
x
0
Kab αs (µ2 )
(10)
3/8
Collinear and TMD densities from Parton Branching Methods
Back up
DGLAP evolution equation
Momentum sum rule
To get rid of the delta function:
P R1
2
We use momentum sum rule
c 0 dzzPca αs (µ ), z = 0:
X
de
fa (x, µ2 )
=
d ln µ2
b
1
1
x 2
e
+ Rab αs (µ2 ), z
,µ +
fb
(1
−
z)
z
x
Z
X
1
1
e
− Dab αs (µ2 ) δ(1 − z) +
−
fb x, µ2
dz Kab αs (µ2 )
(1 − z)
0
b
Z
1
X
−e
fa (x, µ2 )
dzzPca αs (µ2 ), z =
Z
dz
Kab αs (µ2 )
c
=
1
XZ
b
1
x 2
e
+ Rab αs (µ2 ), z
fb
,µ +
1−z
z
Z
1
X 1
dz z Kca αs (µ2 )
+ Rca αs (µ2 ), z
−e
fa x, µ2
1−z
0
c
dz
x
0
Kab αs (µ2 )
(10)
We got rid of the delta function,
both pieces of the equation written in the same way.
Virtual and non-resorvable pieces still included.
3/8
Collinear and TMD densities from Parton Branching Methods
Back up
DGLAP evolution equation
Divergence for z → 1
To avoid divergence when z → 1 a cut off must be introduced:
1
XZ
b
dz
x
1
x 2
e
+ Rab αs (µ2 ), z
fb
,µ +
1−z
z
Z
X 1
1
−e
fa x, µ2
dz z Kca αs (µ2 )
+ Rca αs (µ2 ), z
1−z
0
c
X Z zM
1
x 2
e
→
dz Kab αs (µ2 )
+ Rab αs (µ2 ), z
fb
,µ +
1−z
z
x
b
Z
1
X zM
dz z Kca αs (µ2 )
−e
fa x, µ2
+ Rca αs (µ2 ), z
1−z
0
c
Kab αs (µ2 )
It can be shown that terms
R1
zmax
(11)
skipped in the integral in eq. (11) are of order O(1 − zmax ) .
4/8
Collinear and TMD densities from Parton Branching Methods
Back up
DGLAP evolution equation
Divergence for z → 1
To avoid divergence when z → 1 a cut off must be introduced:
1
XZ
b
dz
x
1
x 2
e
+ Rab αs (µ2 ), z
fb
,µ +
1−z
z
Z
X 1
1
−e
fa x, µ2
dz z Kca αs (µ2 )
+ Rca αs (µ2 ), z
1−z
0
c
X Z zM
1
x 2
e
→
dz Kab αs (µ2 )
+ Rab αs (µ2 ), z
fb
,µ +
1−z
z
x
b
Z
1
X zM
dz z Kca αs (µ2 )
−e
fa x, µ2
+ Rca αs (µ2 ), z
1−z
0
c
Kab αs (µ2 )
It can be shown that terms
R1
zmax
(11)
skipped in the integral in eq. (11) are of order O(1 − zmax ) .
Different choices of zmax :
I zmax - fixed
I zmax - can change dynamically with the scale:
angular ordering: zmax = 1 − QQ0
2
or virtuality ordering: zmax = 1 − QQ0
In this presentation: results from fixed zmax .
4/8
Collinear and TMD densities from Parton Branching Methods
Back up
Sudakov formalism
Sudakov form factor
X
de
fa (x, µ2 )
=
d ln µ2
b
zM
Z
x
X
x 2
R
fa x, µ2
,µ −e
dz Pab
αs (µ2 ), z e
fb
z
c
zM
Z
R
dz zPca
αs (µ2 ), z
0
(12)
R
where Pab
1 - real part of the splitting function.
αs (µ2 ), z = Rab αs (µ2 ), z + Kab αs (µ2 )
1−z
5/8
Collinear and TMD densities from Parton Branching Methods
Back up
Sudakov formalism
Sudakov form factor
X
de
fa (x, µ2 )
=
d ln µ2
b
zM
Z
x
X
x 2
R
fa x, µ2
,µ −e
dz Pab
αs (µ2 ), z e
fb
z
c
zM
Z
R
dz zPca
αs (µ2 ), z
0
(12)
R
where Pab
1 - real part of the splitting function.
αs (µ2 ), z = Rab αs (µ2 ), z + Kab αs (µ2 )
1−z
Defining the Sudakov form factor:
∆a (µ2 ) = exp
X
de
fa (x, µ2 )
=
2
d ln µ
b
zM
Z
x
Z
ln µ2
−
ln µ20
d ln µ02
X
b
zM
Z
0
!
R
dzzPba
αs (µ02 ), z
x 2
d∆a (µ2 )
1
R
αs (µ2 ), z e
fb
dzPab
,µ +e
fa x, µ2
,
2
z
∆a (µ ) d ln µ2
(13)
(14)
5/8
Collinear and TMD densities from Parton Branching Methods
Back up
TMDs from fits at LO
TMDs from the fit at LO
From parton branching method we can obtain TMDs for all flavours
At small kT (no branching or just a few branchings), the difference in the quark TMDs comes from initial distributions.
At large kT (many branchings) TMDs for quarks the same.
6/8
Collinear and TMD densities from Parton Branching Methods
Back up
TMDs from fits at LO
integrated TMD from parton branching method and HERA pdf
7/8
Collinear and TMD densities from Parton Branching Methods
Back up
TMDs from fits at LO
Role of the limit in kT integration
Comparison of int TMDs integrated up to a diffent kT values
The integral over kT has to be performed up to a value higher than the evolution scale to obtain collinear PDF which agrees well
with the HERA pdf.
8/8