Daniel de Florian

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