Daniel de Florian

Global Analysis of Fragmentation Functions
Daniel de Florian
Dpto. de Física - FCEyN- UBA
Global Analysis of Polarized Parton Distributions in the RHIC Era
BNL - October 8
in collaboration with M. Stratmann and R. Sassot

Disclaimer
We (TH) can compute observables in pQCD with high precision
But about errors ..hmmmmmmm
1.1 ± 0.1 ± 0.2
1.1
±0.1 ± 0.2
1.1 ± � 0.1 2 + 0.2 2

Fragmentation functions
Represent the probability that a parton hadronizes in h
From TH point of view at the same level of pdfs
Relevant any time a hadron is produced in high energy collisions
e+e- : primary “source”
SIDIS : complement DIS to allow flavor separation
pp collisions: signal and “background” for a lot of physics
Global Fit (DSS) that includes all those processes
Heavy Ions
polarized pdfs

In the framework of this workshop:
Fragmentation functions play a very relevant role
Single hadron production in polarized pp collisions great tool to
unveil the gluon distribution : gluons enter at LO
Precise FF needed to perform pdf extraction
Otherwise: “error” in FF propagates to pdf
Trivial example: if gluon FF too small, the mistake in analysis of pp
collisions will result in a gluon pdfs too large to compensate!

parabola and the 1σ uncertainty in any observable would correspond to ∆χ2 = 1. In order to account for unexpected
sources of uncertainty, in modern unpolarized global analysis it is customary to consider instead of ∆χ2 = 1 between
a 2% and a 5% variation in χ2 as conservative estimates of the range of uncertainty.
As expected in the ideal framework, the dependence of χ2 on the first moments of u and d resemble a parabola
(Figures 3a and 3b). The KKP curves are shifted upward almost six units relative to those from KRE, due to the
difference in χ2 of their respective best fits. Although this means that the overall goodness of KKP fit is poorer than
KRE, δd and δu seem to be more tightly constrained. The estimates for δd computed with the respective best fits
are close and within the ∆χ2 = 1 range, suggesting something close to the ideal situation. However for δu, they only
overlap allowing a variation in ∆χ2 of the order of a 2%. This is a very good example of how the ∆χ2 Actual example: analysis of polarized pdfs from SIDIS using
= 1 does not
different FF (see Rodolfo’s talk)
seem to apply due to an unaccounted source of uncertainty: the differences between the available sets of fragmentation
functions.
0.4
0.2
0
-0.2
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
x(!u+!u –
)
x(!d+!d –
)
x!u –
10 -2
x Bj
x!u v
x!d v
x!d –
10 -2
x Bj
x!g –
x!s –
KRE (NLO)
KKP (NLO)
unpolarized
KRE " 2
KRE " min +1
KRE " 2
KRE " min +2%
FIG. 4: Parton densities at Q 2 = 10 GeV 2 , and the uncertainty bands corresponding to ∆χ 2 = 1 and ∆χ 2 = 2%
Using Kretzer or KKP can lead to very
different sea distributions
An interesting thing to notice is that almost all the variation in χ2 comes from the comparison to pSIDIS data.
The partial χ2 value computed only with inclusive data, χ2 pDIS , is almost flat reflecting the fact the pDIS data are
not sensitive to u and d distributions. In Figure 3, we plot χ2 pDIS with an offset of 206 units as a dashed-dotted line.
The situation however changes dramatically when considering δs or δg as shown in Figures 3c and 3f, respectively.
In the case of the variation with respect to δs, the profile of χ2 is not at all quadratic, and the distribution is much
more tightly constrained (notice that the scale used for δs is almost four times smaller than the one used for light
sea quarks moments). The χ2 pDIS corresponding to inclusive data is more or less indifferent within an interval around
10 -2
x Bj
0.4
0.2
0
-0.2
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
8
DdeF, G.Navarro, R.Sassot

on kernels.
The cross sections for the single-inclusive e
e+e- (SIA) single-inclusive annihilation
+ e− annihilation
(SIA) into a specific hadron H,
e + e − → (γ, Z) → H, (6)
at a center-of-mass system (c.m.s.) energy √ s and integrated
over the production angle can be written as
[29, 30]
1 dσ
σtot
H
σ0
= �
dz q ê2 � H
2 F1 (z, Q
q
2 ) + F H L (z, Q2 ) � . (7)
The energy EH of the observed hadron scaled to the beam
energy Q/2 = √ s/2 is denoted by z ≡ 2pH · q/Q2 =
2EH/ √ s with Q being the momentum of the intermediate
γ or Z boson.
σtot = �
ê
q
2 q σ0
�
1 + αs(Q2 �
)
(8)
π
is the total cross section for e + e− → hadrons including
its NLO O(αs) correction and σ0 = 4πα2 (Q2 The cross sections for the single-inclusive e
computed
g spacelike
n functions
m-to-large
olution kerr
as z → 0,
ns are cone
splitting
nant, large
art, which
nctions for
perhaps, to
e evolution
le Q0 < Q.
ortional to
nt. While
z behavior
unique way
ce by intronseparably
corrections
r mass cor-
)/s. The
s ofsums hadron in (7) and (8) run over the nf active quark flavors q,
e factoriza- and the êq are the corresponding appropriate electroweak
ation charges func- (see App. A of Ref. [24] for details).
corrected”
+ e− annihilation
(SIA) into a specific hadron H,
e + e − → (γ, Z) → H, (6)
at a center-of-mass system (c.m.s.) energy √ s and integrated
over the production angle can be written as
[29, 30]
1 dσ
σtot
H
σ0
= �
dz q ê2 � H
2 F1 (z, Q
q
2 ) + F H L (z, Q2 ) � . (7)
The energy EH of the observed hadron scaled to the beam
energy Q/2 = √ s/2 is denoted by z ≡ 2pH · q/Q2 =
2EH/ √ s with Q being the momentum of the intermediate
γ or Z boson.
σtot = �
ê
q
2 q σ0
�
1 + αs(Q2 �
)
(8)
π
is the total cross section for e + e− → hadrons including
its NLO O(αs) correction and σ0 = 4πα2 (Q2 )/s. The
sums in (7) and (8) run over the nf active quark flavors q,
and the êq are the corresponding appropriate electroweak
charges (see App. A of Ref. [24] for details).
To NLO accuracy, the unpolarized “time-like” structure
functions F H 1 and F H L in (7) are given by
H 2 � values of z. On the one hand, the timelike evolution kernels
in (4) develop a strong singular behavior as z → 0,
and, on the other hand, the produced hadrons are considered
to be massless. More specifically, the splitting
(T )
(T )
functions P gq (z) and P gg (z) have a dominant, large
logarithmic piece � ln
�
� �
2 H 2 H 2 2 z/z in their NLO part, which
ultimately leads to negative fragmentation functions for
z ≪ 1 in the course of the Q2 evolution and, perhaps, to
unphysical, negative cross sections, even if the evolution
starts with positive distributions at some scale Q0 < Q.
At small z, also finite mass corrections proportional to
MH/(sz 2 ) become more and more important. While
there are ways to resum the singular small-z behavior
to all orders in αs, there is no systematic or unique way
to correct for finite hadron masses, for instance by introducing
some “re-scaled” variable z ′ in SIA. Inseparably
entwined with mass effects are other power corrections
or “dynamical higher twists”.
Anyway, including small-z resummations or mass corrections
in one way or the other in the analysis of hadron
production rates is not compatible with the factorization
theorem and the definition of fragmentation functions
outlined above. “Resummed” or “mass corrected”
fragmentation functions should not be used with fixed order
expressions for, say, the semi-inclusive deep-inelastic
production of a hadron, eN → e ′ tegrated over the production angle can be
[29, 30]
1 dσ
σtot
HX, discussed in
Sec. II C. Therefore we limit ourselves in our global
analysis to kinematical regions where mass corrections
and the influence of the singular small-z behavior of the
evolution kernels is negligible. It turns out that a cut
z > zmin = 0.05 (0.1) is sufficient for data on pion (kaon)
production.
Finally, conservation of the momentum of the fragmenting
parton f in the hadronization process is summarized
by a sum rule stating that
H
σ0
= �
dz q ê2 � H
2 F1 (z, Q
q
2 ) + F H L (z, Q
The energy EH of the observed hadron scaled t
energy Q/2 = √ s/2 is denoted by z ≡ 2pH
2EH/ √ s with Q being the momentum of the
ate γ or Z boson.
σtot = �
ê
q
2 q σ0
�
1 + αs(Q2 �
)
π
is the total cross section for e + e− → hadron
its NLO O(αs) correction and σ0 = 4πα2 (Q
sums in (7) and (8) run over the nf active qua
and the êq are the corresponding appropriate e
charges (see App. A of Ref. [24] for details).
To NLO accuracy, the unpolarized “timeture
functions F H 1 and F H L in (7) are given b
2F H 1 (z, Q2 ) = �
ê
q
2 �
�DH q q (z, Q 2 ) + D H ¯q
+ αs(Q2 ) � 1
Cq ⊗ (D
2π
H q + DH¯q +C 1 g ⊗ D H �
� 2
g (z, Q ) ,
F H L (z, Q 2 ) = αs(Q2 ) �
ê
2π
q
2 � L
q Cq ⊗ (D H q
+C L g ⊗ D H e
tot q q
h
r The energy EH of the observed hadron scaled to the beam
o energy Q/2 =
n
.
o
e
r
y
-
y
s
rn
-
-
”
ric
n
l
s
e
t
)
-
-
)
� 2
g (z, Q ),
√ s/2 is denoted by z ≡ 2pH · q/Q2 =
2EH/ √ s with Q being the momentum of the intermediate
γ or Z boson.
σtot = �
ê
q
2 q σ0
�
1 + αs(Q2 �
)
(8)
π
is the total cross section for e + e− → hadrons including
its NLO O(αs) correction and σ0 = 4πα2 (Q2 )/s. The
sums in (7) and (8) run over the nf active quark flavors q,
and the êq are the corresponding appropriate electroweak
charges (see App. A of Ref. [24] for details).
To NLO accuracy, the unpolarized “time-like” structure
functions F H 1 and F H L in (7) are given by
2F H 1 (z, Q2 ) = �
ê
q
2 �
�DH q q (z, Q 2 ) + D H ¯q (z, Q2 ) �
+ αs(Q2 ) � 1
Cq ⊗ (D
2π
H q + DH¯q )
+C 1 g ⊗ D H �
� 2
g (z, Q ) , (9)
F H L (z, Q 2 ) = αs(Q2 ) �
ê
2π
q
2 � L
q Cq ⊗ (D H q + D H ¯q )
+C L g ⊗ D H� 2
g (z, Q ), (10)
with ⊗ denoting a standard convolution. The relevant
NLO coefficient functions C1,L of the Q evolution and, perhaps, to
cross sections, even if the evolution
distributions at some scale Q0 < Q.
ite mass corrections proportional to
more and more important. While
esum the singular small-z behavior
here is no systematic or unique way
hadron masses, for instance by introled”
variable z
q,g in the MS scheme can be
found in App. A of Ref. [24].
′ in SIA. Inseparably
effects are other power corrections
r twists”.
g small-z resummations or mass coror
the other in the analysis of hadron
not compatible with the factorizahe
definition of fragmentation func-
. “Resummed” or “mass corrected”
ions should not be used with fixed orsay,
the semi-inclusive deep-inelastic
dron, eN → e ′ HX, discussed in
re we limit ourselves in our global
ical regions where mass corrections
the singular small-z behavior of the
negligible. It turns out that a cut
) is sufficient for data on pion (kaon)
tion of the momentum of the fragn
the hadronization process is sumle
stating that
1
dzzD
0
H i (z, Q2 H
2EH/
) = 1, (5)
√ s with Q being the momentum of the intermediate
γ or Z boson.
σtot = �
ê
q
2 q σ0
�
1 + αs(Q2 �
)
(8)
π
is the total cross section for e + e− → hadrons including
its NLO O(αs) correction and σ0 = 4πα2 (Q2 )/s. The
sums in (7) and (8) run over the nf active quark flavors q,
and the êq are the corresponding appropriate electroweak
charges (see App. A of Ref. [24] for details).
To NLO accuracy, the unpolarized “time-like” structure
functions F H 1 and F H L in (7) are given by
2F H 1 (z, Q2 ) = �
ê
q
2 �
�DH q q (z, Q 2 ) + D H ¯q (z, Q2 ) �
+ αs(Q2 ) � 1
Cq ⊗ (D
2π
H q + DH¯q )
+C 1 g ⊗ D H �
� 2
g (z, Q ) , (9)
F H L (z, Q 2 ) = αs(Q2 ) �
ê
2π
q
2 � L
q Cq ⊗ (D H q + D H ¯q )
+C L g ⊗ D H� 2
g (z, Q ), (10)
with ⊗ denoting a standard convolution. The relevant
NLO coefficient functions C1,L At small z, also finite mass corrections proportional to
MH/(sz
q,g in the MS scheme can be
found in App. A of Ref. [24].
2 ) become more and more important. While
there are ways to resum the singular small-z behavior
to all orders in αs, there is no systematic or unique way
to correct for finite hadron masses, for instance by introducing
some “re-scaled” variable z ′ in SIA. Inseparably
entwined with mass effects are other power corrections
or “dynamical higher twists”.
Anyway, including small-z resummations or mass corrections
in one way or the other in the analysis of hadron
production rates is not compatible with the factorization
theorem and the definition of fragmentation functions
outlined above. “Resummed” or “mass corrected”
fragmentation functions should not be used with fixed order
expressions for, say, the semi-inclusive deep-inelastic
production of a hadron, eN → e ′ HX, discussed in
Sec. II C. Therefore we limit ourselves in our global
analysis to kinematical regions where mass corrections
and the influence of the singular small-z behavior of the
evolution kernels is negligible. It turns out that a cut
z > zmin = 0.05 (0.1) is sufficient for data on pion (kaon)
production.
Finally, conservation of the momentum of the fragmenting
parton f in the hadronization process is summarized
by a sum rule stating that
�
� 1
dzzD
H 0
H i (z, Q2 σtot =
) = 1, (5)
�
ê
q
2 q σ0
�
1 + αs(Q2 �
)
π
is the total cross section for e + e− → hadrons includi
its NLO O(αs) correction and σ0 = 4πα2 (Q2 )/s. T
sums in (7) and (8) run over the nf active quark flavors
and the êq are the corresponding appropriate electrowe
charges (see App. A of Ref. [24] for details).
To NLO accuracy, the unpolarized “time-like” stru
ture functions F H 1 and F H L in (7) are given by
2F H 1 (z, Q2 ) = �
ê
q
2 �
�DH q q (z, Q 2 ) + D H ¯q (z, Q2 ) �
+ αs(Q2 ) � 1
Cq ⊗ (D
2π
H q + DH¯q )
+C 1 g ⊗ D H �
� 2
g (z, Q ) ,
F H L (z, Q 2 ) = αs(Q2 ) �
ê
2π
q
2 � L
q Cq ⊗ (D H q + D H ¯q )
+C L g ⊗ D H� 2
g (z, Q ), (1
with ⊗ denoting a standard convolution. The releva
NLO coefficient functions C1,L densities. For instance, the singlet evolution equatio
schematically reads
d
d ln Q
q,g in the MS scheme can
found in App. A of Ref. [24].
2 � D H (z, Q 2 �
) = ˆP (T )
⊗ D� H �
(z, Q 2 ), (2
where
�D H �
H D
≡ Σ
DH �
, D
g
H �
Σ ≡ (D
q
H q + DH¯q ) (3
and
ˆP (T ) pplicability for fragmentation functions
is severely limited to medium-to-large
he one hand, the timelike evolution kerp
a strong singular behavior as z → 0,
r hand, the produced hadrons are conssless.
More specifically, the splitting
(T )
) and P gg (z) have a dominant, large
�Cross ln section depends on two structure functions
At NLO they can be written as
Fragmentation functions depend on both energy fraction (z) and
energy scale : AP evolution
Notice that SIA can only give information on the sum
� �
(T )
(T )
P qq 2nfP gq
≡ 1 (T ) (T ) . (4
P 2nf
qg P gg
2 z/z in their NLO part, which
to negative fragmentation functions for
se of the Q2 evolution and, perhaps, to
ive cross sections, even if the evolution
ve distributions at some scale Q0 < Q.
finite mass corrections proportional to
e more and more important. While
o resum the singular small-z behavior
s, there is no systematic or unique way
te hadron masses, for instance by introscaled”
variable z ′ in SIA. Inseparably
ass effects are other power corrections
gher twists”.
ing small-z resummations or mass cory
or the other in the analysis of hadron
is not compatible with the factorizathe
definition of fragmentation funcove.
“Resummed” or “mass corrected”
ctions should not be used with fixed orr,
say, the semi-inclusive deep-inelastic
hadron, eN → e ′ at a center-of-mass system (c.m.s.) energy
HX, discussed in
efore we limit ourselves in our global
atical regions where mass corrections
√ s and integrated
over the production angle can be written as
[29, 30]
1 dσ
σtot
H
σ0
= �
dz q ê2 � H
2 F1 (z, Q
q
2 ) + F H L (z, Q2 ) � . (7)
The energy EH of the observed hadron scaled to the beam
energy Q/2 = √ s/2 is denoted by z ≡ 2pH · q/Q2 =
2EH/ √ s with Q being the momentum of the intermediate
γ or Z boson.
σtot = �
ê
q
2 q σ0
�
1 + αs(Q2 �
)
(8)
π
is the total cross section for e + e− → hadrons including
its NLO O(αs) correction and σ0 = 4πα2 (Q2 )/s. The
sums in (7) and (8) run over the nf active quark flavors q,
and the êq are the corresponding appropriate electroweak
charges (see App. A of Ref. [24] for details).
To NLO accuracy, the unpolarized “time-like” structure
functions F H 1 and F H L in (7) are given by
2F H 1 (z, Q2 ) = �
ê
q
2 �
�DH q q (z, Q 2 ) + D H ¯q (z, Q2 ) �
+ αs(Q2 ) � 1
Cq ⊗ (D
2π
H q + DH¯q )
�
�
To NLO accuracy, the unpolarized “time-like” struc

Advantages Very precise data from LEP/SLD
Disadvantages
Heavy quark tagged data
Only fragmentation functions enter (clean process)
SIA data dominated by precise LEP/SLD measurements at MZ
weak scale dependence (bad resolution for g fragmentation)
mostly determine “singlet” distribution (high precision)
Σ = Du + Dū + Dd + D ¯ d + Ds + D¯s + Dc + D¯c + Db + D¯ b
Can not separate
e+e- (SIA) single-inclusive annihilation
not precise at large z (relevant for pp collisions)
D h q (z, Q 2 ) from D h q (z, Q 2 )

Some ansatz needed if only SIA data used, like “linear suppression”
Recent NLO analyses
D h+
q+q(z, Q 2 ) = D h−
q+q(z, Q 2 )
D h+ +h −
q
CGGRW(1994), BFGW(2000)
BKK(1995), KKP(2000), AKK(2005)
KRE(2000)
HKNS(2007)
(z, Q 2 ) = D h+ +h −
q
(z, Q 2 )
D h+
q (z, Q 2 ) = (1 − z) D h+
q (z, Q 2 )
flavor/charge average
L.Bourhis et al.,
Eur.Phys.J C19 (2001) 89.
S. Albino et al.,
Nuc.Phys.B725 (2005) 206.
S. Kretzer, Phys. Rev.D62 (2000) 0540001.
M. Hirai et al., Phys. Rev.D75 (2007) 094009.
use only SIA: no separation or ad-hoc assumption
And some “reasonable” ansatz about charge separation don’t work well

SIDIS
Advantages : allows flavor/charge separation
larger z
smaller Q2 , improves scale coverage in evolution : Dg
Hermes and Compass kinematics
Disadvantage : would introduce “dependence” on pdfs (but unpolarized pdfs
very well constrained from DIS in the same kinematical range)
non-perturbative corrections at small Q2 The cross section for the semi-inclusive deep-inelastic
production of a hadron, eN → e
?
′ HX, is proportional to
certain combinations of both the parton distributions of
the nucleon N and the fragmentation functions for the
hadron H. It can be written in factorized form in a way
very similar to the fully inclusive DIS case [24, 29–31]:
dσH =
dx dy dzH
2 πα2
Q2 � 2 (1 + (1 − y) )
2 F
y
H 1 (x, zH, Q 2 )
+ 2(1 − y)
F
y
H L (x, zH, Q 2 �
) , (11)
with x and y denoting the usual DIS scaling variables
(Q2 = sxy), and where [29, 30] zH ≡ pH · pN/pN · q
with an obvious notation of the four-momenta, and with
−q2 ≡ Q2 . Strictly speaking, Eq. (11) and the variable
zH only apply to hadron production in the current
fragmentation region. This is usually ensured by a cut
xF > 0 on the Feynman-variable representing the fractional
longitudinal c.m.s. momentum. If necessary, target
fragmentation could be accounted for by transforming to
the variable [31, 32] zH → z ≡ EH
EN (1−x) , the energies EH,
EN defined in the c.m.s. frame of the nucleon and the virtual
photon, and by introducing the so-called “fracture
functions” [32].
The structure functions F H 1 and F H L in (11) are given
at NLO by
2F H 1 (x, zH, Q 2 ) = �
e
q,q
2 �
q q(x, Q 2 )D H q (zH, Q 2 )
+ αs(Q2 �
)
q ⊗ C
2π
1 qq ⊗ D H D. Hadron-Hadron Collisions
The single-inclusive production of a hadron H at high
transverse momentum pT in hadron-hadron collisions is
also amenable to QCD perturbation theory. Up to corrections
suppressed by inverse powers of pT , the differential
cross section can be written in factorized form as [25, 33]
d
EH
q
3σ dp3 =
H
�
fa ⊗ fb ⊗ dˆσ
a,b,c
c ab ⊗ DH c , (14)
where the sum is over all contributing partonic channels
a + b → c + X, with dˆσ c ab the associated partonic cross
section. dˆσ c ab can be expanded as a power series in the
strong coupling αs and the O(α3 certain combinations of both the parton distributions of
the nucleon N and the fragmentation functions for the
hadron H. It can be written in factorized form in a way
very similar to the fully inclusive DIS case [24, 29–31]:
dσ
s) NLO corrections are
available [25, 33]. As always, the factorized structure (14)
forces one to introduce into the calculation scales of the
order of the hard scale in the reaction – but not specified
further by the theory – that separate the short- and longdistance
contributions. We have suppressed the explicit
dependence on these renormalization and factorization
scales in Eq. (14), for details, see, e.g., Ref. [25].
In studies and quantitative analyzes of hadronic cross
sections, NLO corrections are of particular importance
and generally indispensable in order to arrive at a firm
theoretical prediction for (14). Since NLO corrections
are known to be significant, LO approximations usually
significantly undershoot the available data. In addition,
hadronic reactions suffer from much enhanced theoretical
uncertainties than the reactions described above due to
the presence of more non-perturbative, scale dependent
functions. The dependence on the unphysical factoriza-
H
=
dx dy dzH
2 πα2
Q2 � 2 (1 + (1 − y) )
2 F
y
H 1 (x, zH, Q 2 )
+ 2(1 − y)
F
y
H L (x, zH, Q 2 �
) , (11)
with x and y denoting the usual DIS scaling variables
(Q2 = sxy), and where [29, 30] zH ≡ pH · pN/pN · q
with an obvious notation of the four-momenta, and with
−q2 ≡ Q2 . Strictly speaking, Eq. (11) and the variable
zH only apply to hadron production in the current
fragmentation region. This is usually ensured by a cut
xF > 0 on the Feynman-variable representing the fractional
longitudinal c.m.s. momentum. If necessary, target
fragmentation could be accounted for by transforming to
the variable [31, 32] zH → z ≡ EH
EN (1−x) , the energies EH,
EN defined in the c.m.s. frame of the nucleon and the virtual
photon, and by introducing the so-called “fracture
functions” [32].
The structure functions F H 1 and F H L in (11) are given
at NLO by
2F H 1 (x, zH, Q 2 ) = �
e
q,q
2 �
q q(x, Q 2 )D H q (zH, Q 2 )
+ αs(Q2 �
)
q ⊗ C
2π
1 qq ⊗ D H q
+q ⊗ C 1 gq ⊗ D H The single-inclusiv
transverse momentum
also amenable to QCD
tions suppressed by in
cross section can be w
d
EH
g
� �
3σ dp3 =
H
where the sum is ove
a + b → c + X, with
section. dˆσ c tional longitudinal c.m.s. momentum. If necessary, target
fragmentation could be accounted for by transforming to
Distributions the in variable x and z[31,
32] zH → z ≡
ab can be
strong coupling αs a
available [25, 33]. As
forces one to introdu
order of the hard scal
further by the theory
distance contribution
dependence on these
scales in Eq. (14), for
In studies and qua
sections, NLO correc
and generally indispe
theoretical prediction
are known to be sign
significantly undersho
hadronic reactions su
uncertainties than th
the presence of more
functions. The depen
tion and renormaliza
and quantified at NL
As will be discus
EH
EN (1−x) , the energies EH,
EN defined in the c.m.s. frame of the nucleon and the virtual
photon, and by introducing the so-called “fracture
functions” [32].
The structure functions F H 1 and F H L in (11) are given
at NLO by
2F H 1 (x, zH, Q 2 ) = �
e
q,q
2 �
q q(x, Q 2 )D H q (zH, Q 2 )
+ αs(Q2 �
)
q ⊗ C
2π
1 qq ⊗ D H q
+q ⊗ C 1 gq ⊗ D H g
+g ⊗ C 1 qg ⊗ DH �
q (x, zH, Q 2 �
) ,(12)
F H L (x, zH, Q 2 ) = αs(Q2 ) �
e
2π
q,q
2 �
q q ⊗ C L qq ⊗ D H q
+q ⊗ C L gq ⊗ DH g
+g ⊗ C L qg ⊗ D H �
q (x, zH, Q 2 further by the theor
distance contributio
dependence on thes
scales in Eq. (14), fo
In studies and qu
sections, NLO corre
and generally indisp
theoretical predictio
are known to be sig
significantly undersh
hadronic reactions s
uncertainties than t
the presence of mor
functions. The dep
tion and renormaliz
and quantified at N
As will be disc
hadronic cross sectio
tion functions is the
fragmentation funct
gX processes for h
transverse momenta
tion at very high z.
and K
), (13)
± provide add
aration of the DH i .
“effective charge”
at LO
xF > 0 on the Feynman-variable representing the frac-
}
order of the hard sca

Transverse momentum distribution
hard scale
Hadron-Hadron collisions
Advantages : allows flavor/charge separation form hadron measurements
several subprocesses
much larger z
large contribution from Dg
analysis of data allows to have FF that work at RHIC and
can be used in polarized/heavy ion collisions!
Disadvantage : Problems for fixed target experiments, use only colliders
Otherwise Threshold resummation needed (not in first approach)
larger TH uncertainty (scale dependence)
DdeF, W.Vogelsang

Global FIT
Advantages : Constrain FF with almost all available data
Check of pQCD framework
Precise determination of distributions and estimation of
uncertainties
Disadvantage : more work required !
Feasible if Mellin technique used (Marco’s talk)
Tension between different observables requires careful analysis
Complicated observables:
without simple NLO interpretation
involving MC
“averaged” bins
large scale dependence
Weights

fragmentation functions for
LO and NLO global fits available
DSS global analysis
π + , π − , π 0 , K + , K − , K 0 , K 0 , p, p, n, n, h + , h −
SIA data includes: TPC, TASSO, SLD, ALEPH, DELPHI, OPAL +
“flavor” tag
SIDIS data from : HERMES, EMC
pp data from : PHENIX, STAR, BRAHMS, CDF, UA1, UA2
Checks with other data sets: STAR, pT distributions at HERA (H1)
Estimation of uncertainties using Lagrange multipliers
residual
h + = π + + K + + p + res +

nt of experimental
Technical details
D H i (z, Q 2 0) = Ni z αi βi (1 − z) � � δi 1 + γi(1 − z)
Q 2 0 = 1 GeV 2
u, d, s, g
Q 2 0 = m 2 Flexible parametrization
ALYSIS
to-large z. Accordingly, additional power terms in z, emphasizing
the small z region, have little or no impact on
at initial scale
ls of our analysis. the fit and are not pursued further. The initial scale µ0
e of parametriza- for the Q
t of experimental
Q c, b
wewith determine the
inimization. αs andWe ΛQCD from MRST
f Mellin moments
the cross sections
how we assess unntation
Try to avoid
functions
Isospin symmetry assumptions
r technique.
on functions are
] and have cho-
2-evolution is taken to be µ0 = 1 GeV in our
analysis.
Since the initial fragmentation functions (15) at scale
µ0 should not involve more free parameters than can be
extracted from data, we have to impose, however, certain
relations upon the individual fragmentation functions
for pions and kaons. We have checked in each case
that relaxing these assumptions indeed does not significantly
improve the χ2 of the fit of presently available
data to warrant any additional parameters. In detail, for
{u, ū, d, ¯ d} → π + we impose isospin symmetry for the
sea fragmentation functions, i.e.,
D π+
ū = D π+
of experimental
e determine the
nimization. We
Mellin moments
e cross sections
w we assess untation
functions
technique.
n functions are
and have choiz
d , (16)
but we allow for slightly different normalizations in the
αi βi (1 − z) to
e µ0 for the Q2- A cross section
ion on D π+ +π −
q+¯q
t assumptions it
“valence” from
r instance, Dπ+ analysis.
Since the initial fragmentation functions (15) at scale
µ0 should not involve more free parameters than can be
extracted from data, we have to impose, however, certain
relations upon the individual fragmentation functions
for pions and kaons. We have checked in each case
that relaxing these assumptions indeed does not significantly
improve the χ
u
erious limitation
2 of the fit of presently available
data to warrant any additional parameters. In detail, for
{u, ū, d, ¯ d} → π + we impose isospin symmetry for the
sea fragmentation functions, i.e.,
D π+
ū = D π+
d , (16)
but we allow for slightly different normalizations in the
q + ¯q sum:
D π+
d+ ¯ d = NDπ+ u+ū . (17)
For strange quarks it is assumed that
D π+
s = D π+
¯s = N ′ D π+
ū
(18)
with N ′ we determine the
inimization. We
f Mellin moments
the cross sections
how we assess unentation
functions
er technique.
ion functions are
0] and have cho-
Niz
independent of z.
αi βi (1 − z) to
ale µ0 for the Q2- SIA cross section
ation on D π+ +π −
q+¯q
ut assumptions it
or “valence” from
for instance, Dπ+ Since the initial fragmentation functions (15) at scale
µ0 should not involve more free parameters than can be
extracted from data, we have to impose, however, certain
relations upon the individual fragmentation functions
for pions and kaons. We have checked in each case
that relaxing these assumptions indeed does not significantly
improve the χ
u
serious limitation
tained fragmenta-
2 of the fit of presently available
data to warrant any additional parameters. In detail, for
{u, ū, d, ¯ d} → π + we impose isospin symmetry for the
sea fragmentation functions, i.e.,
D π+
ū = D π+
d , (16)
but we allow for slightly different normalizations in the
q + ¯q sum:
D π+
d+ ¯ d = NDπ+ u+ū . (17)
For strange quarks it is assumed that
D π+
s = D π+
¯s = N ′ D π+
ū
(18)
with N ′ independent of z.
It is worth noticing that assuming N = N ′ f the SIA cross section
nformation on D
= 1 [7, 10]
in Eqs. (17) and (18), respectively, SIA data alone allow
π+ +π −
q+¯q
Without assumptions it
ored” or “valence” from
tion, for instance, Dπ+ u
is is a serious limitation
the obtained fragmentacompare
to a wealth of
charged pions and kaons
ollisions [21]. In Ref. [7]
Dπ+ u = (1 − z) was as-
This was later shown to
ed pion multiplicities in
a LO combined analysis
so Fig. 4 and discussions
determine for the first
unctions for quark and
ll as gluons from data.
ental information from
on scattering data, we
input distribution than
i [1 + γi(1 − z) δi
For strange quarks it is assumed that
D
]
,
π+
s = D π+
¯s = N ′ D π+
ū
(18)
with N ′ independent of z.
It is worth noticing that assuming N = N ′ = 1 [7, 10]
in Eqs. (17) and (18), respectively, SIA data alone allow
to distinguish between favored and unfavored fragmentation
functions in principle. We shall scrutinize the compatibility
of these assumptions with SIDIS and hadronic
scattering data in Sec. IV F. At any rate, their impact
on the assessment of uncertainties of fragmentation functions
is highly non trivial.
For charged kaons we fit DK+ u+ū and DK+ s+¯s independently
to account for the phenomenological expectation that the
formation of secondary s¯s pairs, which is required to form
a |K + 〉 = |u¯s〉 from a u but not from an ¯s quark, should
be suppressed. Indeed, we find from our fit, see Sec.
IV below, that DK+ s+¯s > DK+ u+ū in line with that expectation.
For the unfavored fragmentation the data are
unable to discriminate between flavors and, consequently,
we assume that all distributions have the same functional
form:
D K+
ū = D K+
s = D K+
d = D K+
Normalizations for different experiments (if not included in syst.)
Allowing for possible breaking
of SU(3) of sea and SU(2) in
favored distributions
unless data can not discriminate
for unfavored fragmentations
¯d . (19)
are to a wealth of
D d+ ¯ d = ND u+ū . (17)
′

Some plots
SIA: still works very well
within the global fit
Large errors at z>0.5
“inclusive”
heavy quark tagged
✓ extrapolation to smaller z
10 4
10 3
10 2
10
1
10 -1
10 -2
10 -3
10 -4
10 3
10 2
10
1
10 -1
10 -2
10 3
10 2
10
10 -1
1
10 -2
10 -3
10 -4
10 -5
z>0.05 (0.1 for Kaons/Protons)
not
fitted
not
fitted
THIS FIT
KRE
AKK
10 -1
1 d! "
! tot dz charm
1 d! "
THIS FIT
KRE
AKK
! tot dz bottom
10 -1
1 d! "
! tot dz
(× 0.1)
(× 0.1)
(× 10)
(× 100)
(× 0.01)
(× 10)
(× 10)
z
z
1
1
TPC
SLD
ALEPH
DELPHI
OPAL
10 -1
TPC (charm)
SLD (charm)
TPC (bottom)
SLD (bottom)
DELPHI (bottom)
10 -1
z
z
1
1
0.4
0.2
0
-0.2
-0.4
0.4
0.2
0
-0.2
-0.4
0.2
0.1
0
-0.1
-0.2
-0.2
-0.4
0
0.6
0.4
0.2
0.6
0.4
0.2
0
-0.2
0.5
0
-0.5
0.5
0
-0.5
0.5
0
-0.5
0.5
0
-0.5
0.5
0
-0.5
(data - theory)/theory
(data - theory)/theory

SIDIS
ad-hoc charge separation
from Kretzer fails
large z covered
1
10 -1
1
10 -1
1/NDIS dN " /dzdQ 2
+
THIS FIT
KRE
1/NDIS dN " /dzdQ 2
-
not fitted
Q 2 [GeV]
Hermes
z - bin
0.25 - 0.35
0.35 - 0.45
0.45 - 0.60
0.60 - 0.75
z - bin
0.25 - 0.35
0.35 - 0.45
0.45 - 0.60
0.60 - 0.75
1 10
" +
0.25 ! z ! 0.35
1 10
" -
Q 2 [GeV]
0.35 ! z ! 0.45
0.45 ! z ! 0.60
0.60 ! z ! 0.75
0.2
0.1
0
-0.1
-0.2
0.2
0.1
0
-0.1
-0.2
0.2
0.1
0
-0.1
-0.2
0.2
0.1
0
-0.1
-0.2
(data - theory)/theory

10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
10 -7
10 -8
10 -9
0.5
0
-0.5
THIS FIT
KRE
AKK
scale uncertainty
d 3 ! "
dp 3 E [mb / GeV 2 0
]
PHENIX data
(preliminary)
(data - theory)/theory
0 5 10 15 20
Neutral pions at Phenix
pp data
p T [GeV]
1
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
1.5
1
0.5
0
-0.5
d 3 ! "
dp 3
E [mb / GeV 2 +
]
THIS FIT
KRE
scale uncertainty
BRAHMS data
# = 2.95
1 2 3 4 1 2 3 4 5
p T [GeV]
µF = µR = pT
d 3 ! "
dp 3
E [mb / GeV 2 -
]
(data - theory)/theory
BRAHMS data
# = 2.95
Charged pions (at large z)
p T [GeV]

1
10 -1
10 -2
10 -3
10 -4
10 -5
0.5
0
-0.5
THIS FIT
KRE
AKK
E
scale uncertainty
(data - theory)/theory
d 3 ! K
dp 3
[mb / GeV 2 ]
1 2 3 4 5
0
S
STAR data
p T [GeV]
pp data
Kaons Protons
1
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
10 -7
1
0.5
0
-0.5
E
d 3 ! p
dp 3
THIS FIT
HKNS
scale uncertainty
[mb / GeV 2 ]
STAR data
1 2 3 4 5 6 7 1 2 3 4 5 6 7
p T [GeV]
E
d 3 ! p
dp 3
(data - theory)/theory
[mb / GeV 2 ]
STAR data
p T [GeV]
Good agreement with all RHIC data and visible differences with other sets

Typically
Large χ from a few isolated data points (small z in SIA, and some SIDIS and pp)
2
very precise SIA
Also tension between experiments (like Delphi at large z)
χ 2
χ 2 /dof ∼ 2
grows (~25%) for LO fits : mostly from pp data
where NLO corrections are very large
Pions + kaons + protons almost saturate charged hadrons:
residual only sizable for HQ

1.4
1.2
1
0.8
0.6
0.4
0.2
2
1.5
1
0.5
2
1.5
1
0.5
0
+
zD ! zDi (z)
Q 2 = 10 GeV 2
10 -1
u + u
u
s + s
Distributions (pions)
1
zD ! zDi (z)
Q 2 = 10 GeV 2
THIS FIT / KRE THIS FIT / KRE
D "
THIS FIT / AKK
z
+
THIS FIT / AKK
10 -1
Large differences visible in the gluon at large z : explains pp
gluon
c + c
Large differences in unfavored distributions : explains SIDIS and pp
For pions u fragmentation smaller than AKK : required by SIDIS and
compensated in SIA by larger s
Similar singlet
z
1
10
1
10
1
10
1
10 -1
10 -2
1
0.5
0
-0.5
large rapidity at STAR
THIS FIT
KRE
AKK
scale uncertainty
(data - theory)/theory
d 3 ! "
dp 3 E [µb / GeV 2 0
]
STAR data
#$% = 3.3
#$% = 3.8
-1
1 2 3 4
p T [GeV]

0.7
0.6
0.5
0.4
0.3
0.2
0.1
3
2
1
3
2
1
0
10 -1
u + u
u
s + s
THIS FIT / KRE
D !
Distributions (Kaons)
THIS FIT / AKK
+
zD K zDi (z)
Q 2 = 10 GeV 2
z
1
gluon
c + c
THIS FIT / KRE
THIS FIT / AKK
10 -1
Smaller u required by SIDIS
VERY different gluon by pp
+
zD K zDi (z)
Q 2 = 10 GeV 2
Similar singlet (larger dominant s)
_
Issues for K- (s and u in proton at large x)
z
1
10
1
10
1
1
10 -1
10 -2
10 -3
10 -4
10 -5
0.5
0
-0.5
THIS FIT
KRE
AKK
E
scale uncertainty
(data - theory)/theory
d 3 ! K
dp 3
[mb / GeV 2 ]
1 2 3 4 5
1 +
0.25 ! z ! 0.35
10 -1
1
10 -1
10 -2
1/N DIS dN K /dzdQ 2
THIS FIT
KRE
1/NDIS dN K /dzdQ 2
-
not fitted
Q 2 [GeV]
z - bin
0.25 - 0.35
0.35 - 0.45
0.45 - 0.60
z - bin
0.25 - 0.35
0.35 - 0.45
0.45 - 0.60
1 10
0
S
K +
0.35 ! z ! 0.45
0.45 ! z ! 0.60
STAR data
Q 2 [GeV]
p T [GeV]
K -
1 10
0.4
0.2
0
-0.2
-0.4
0.4
0.2
0
-0.2
-0.4
0.4
0.2
0
-0.2
-0.4
(data - theory)/theory

0.3
0.25
0.2
0.15
0.1
0.05
2
1.5
1
0.5
2
1.5
1
0.5
0
zD p zDi (z)
THIS FIT / HKNS
10 -1
D !
THIS FIT / AKK
D !
Distributions (protons)
u + u
u * 2
s + s
Q 2 = M 2
Q 2 = MZ z
1
THIS FIT / HKNS
THIS FIT / AKK
10 -1
gluon
c + c
b + b
z
Differences with HKNS also sizeable : gluons, unfavored and large z (pp)
1
10
1
10
1
1
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
10 -7
1
0.5
0
-0.5
E
d 3 ! p
dp 3
THIS FIT
HKNS
scale uncertainty
[mb / GeV 2 ]
STAR data
1 2 3 4 5 6 7 1 2 3 4 5 6 7
p T [GeV]
E
d 3 ! p
dp 3
(data - theory)/theory
Similar singlet, but large diff. between HKNS and AKK (using same data)
In general differences look much larger than for pdf fits :
comparison between GLOBAL fits
To “estimate” uncertainties using different sets (like MRST vs CTEQ),
more global fits needed
[mb / GeV 2 ]
STAR data
p T [GeV]

Uncertainties
Use Lagrange multipliers technique to estimate uncertainties (from exp.
errors) on some observables
∆χ 2 n
Φ(λi, {aj}) = χ 2 ({aj}) + �
i
λi Oi({aj})
See how fit deteriorates when FFs forced to give different prediction for Oi
should be parabolic if data set can determine the observable
(otherwise monotonic o flat)
We study truncated moments:
� 1
0.2
z D H i (z, Q = 5 GeV) dz

# 2
# 2
860
855
850
845
860
855
850
845
"+
! g
"+
! u+u –
! u – "+
"+
! c+c –
0.9 1 1.1 0.9 1 1.1
!/! 0
!/! 0
u < 5%
s ~ 10 %
∆χ 2 = 15(∼ 2%)
"+
! s+s –
"+
! –
b+b
0.9 1
!/! 0
1.1
dominated by z~0.2
Uncertainties
#$ 2
#$ i
15
10
5
0
-5
"+
! u+u –
0.98 1 1.02
#$ 2
#$ TOTAL
#$ 2
#$ e+e-
#$ 2
#$ HERMES
#$ 2
#$ OPAL
#$ PHENIX
2
#$ STAR
2
#$ BRAHMS
2
Tension
Individual profiles
!/! 0
"+
! g
"+
! s+s –
0.9 1 1.1
Complementarity
Constrained parabola as a result of global fit
15
10
5
0
-5
15
10
5
0
-5
Precision
RHIC

Conclusions
FFs are a fundamental tool to describe HEP observables within pQCD
NLO (and LO) fragmentation functions from a global fit: electronpositron,
lepton-nucleon and hadron-hadron scattering
Charge and flavor separation from data (no ad-hoc assumptions)
For this workshop: ffs work in the kinematic range relevant for
polarized pdfs extraction: RHIC, Hermes, Compass
First “global” fit fully developed using Mellin techniques : it can be done!