1

19. Fragmentation Functions in e+ e− , ep, and pp Collisions

19. Fragmentation Functions in e+ e− , ep, and pp Collisions
Revised August 2023 by O. Biebel (Ludwig-Maximilians U.), D. de Florian (ICAS and ICIFI,UNSAM), D. Milstead (Stockholm U.) and W. Vogelsang (Tübingen U.).

19.1 Introduction to fragmentation
Quarks and gluons produced in hard-scattering reactions will ultimately give rise to the colorless
hadronic bound states that may be observed in the detector. The associated hadronization process is described by fragmentation functions Dih (x, µ2 ) (i = q, q̄, g) which are universal functions
representing, in the simplest picture, a measure of the probability density that an outgoing parton
produces a hadron h. Here, x is the fraction of the parton’s momentum transferred to the hadron,and µ is a ‘resolution’ scale known as factorization scale. The Dih (x, µ2 ) may be viewed as the
final-state analogs of the initial-state parton distribution functions (PDFs) addressed in Section 18
of this Review. They are also sometimes referred to as timelike distributions since they are primarily
accessed in e+ e− annihilation via a timelike intermediate boson. (See Refs. [1, 2] for introductory
reviews, and Refs. [3–5] for summaries of experimental and theoretical research in this field).
The cleanest laboratory for the study of fragmentation functions is provided by semi-inclusiveelectron-positron annihilation, e+ e− → γ/Z → h + X. The cross section for this reaction may
be expressed in terms of ‘fragmentation structure functions’ FT,L,A that are directly related to the√fragmentation functions. At center-of-mass (CM) energy s = q 2 we have
1
d 2σh
3
3
3
= (1 + cos2 θ)FTh (x, q 2 ) + sin2 θ FLh (x, q 2 ) + cos θ FAh (x, q 2 ) .
σ0 dx d cos θ
8
4
4(19.1)

Here, q is the four-momentum of the intermediate photon or Z-boson, with q 2 > 0, and x = 2Ph ·q/q 2
with the hadron’s four-momentum Ph is the fragmentation counterpart of the familiar DIS Bjorken√
variable. (Note that x = 2Eh / s ≤ 1 in terms of the energy Eh of the produced hadron in the
CM frame of the electron positron pair.) Furthermore, in the same frame, θ is the hadron’s angle
relative to the electron beam direction. Eq. (19.1) is the most general form for unpolarized inclusive
single-particle production via vector bosons [6]. The fragmentation structure functions FT and FL
represent the contributions from γ/Z polarizations transverse or longitudinal with respect to thedirection of motion of the hadron. The parity-violating term with the asymmetric fragmentation
function FA arises from the interference between vector and axial-vector contributions. Various
normalization factors σ0 are used in the literature, ranging from the total cross section σtot fore+ e− → hadrons, including all weak and QCD contributions, to σ0 = 4πα2 Nc /3s with Nc = 3,
the lowest-order QED cross section for e+ e− → µ+ µ− times the number of colors Nc . LEP1
measurements of the three fragmentation structure functions are shown in Fig. 19.1.
Integration of Eq. (19.1) over all θ yields the total fragmentation structure function F h =FTh + FLh :X 1 dzq2
1 dσ h= F h (x, q 2 ) =Ci z, αs (µ), 2
σ0 dx
µ
x z
i
Z

!

Dih



x 2
,µ
z



.

(19.2)On the right we have written the factorized expression for the structure function in terms of a¯ . . . , g withsum over convolutions of the fragmentation functions Dih for partons i = u, ū, d, d,
perturbative coefficient functions Ci . Since photons and Z bosons do not distinguish between
quarks and antiquarks, e+ e− annihilation primarily constrains the combinations Dqh + Dq̄h . Gluon
fragmentation contributes only at higher order in perturbation theory or by scaling violations.
Corrections to the factorized expression in Eq. (19.2) are suppressed by inverse powers of q 2 . TheyS. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:14am

19. Fragmentation Functions in e+ e− , ep, and pp Collisions

2

500
200
100
50

√s=91 GeV

FT,L(x)20
105LEP FTLEP FL

2
1
0.5
0.2
0.1
0.05

0.02
0.01
0.005

LEP FA

FA(x)

0.002
0.001
0.0005
0.8
0.4
0
-0.4
-0.8
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x1

Figure 19.1: LEP1 measurements of total transverse (FT ), longitudinal (FL ), and asymmetric(FA ) fragmentation structure functions [7]. Data points with relative errors greater than 100% areomitted.

arise from quark and hadron mass terms and from non-perturbative effects. Analogous factorized
31st May, 2024

3

19. Fragmentation Functions in e+ e− , ep, and pp Collisions

expressions as in Eq. (19.2) may be written for each of the structure functions FT,L,A individually.
The fragmentation functions obey the momentum sum rule constraint
XZ 1
h

0

dx x Dih (x, µ2 ) = 1 ,(19.3)

separately for each flavor i. Note that the sum rule involves a sum over all possible produced
hadrons. The dependence of the functions Dih on the factorization scale µ2 will be discussed in thenext section.
Measurements of hadron production in deeply-inelastic lepton-proton scattering and hadronhadron scattering are complementary to those in e+ e− annihilation. The former process, `p →`0 + h + X, is known as semi-inclusive deep-inelastic scattering (SIDIS). Here, in analogy with
Eq. (19.2), the high virtuality of the photon in DIS also permits factorization of the cross section interms of fragmentation functions, PDFs for the incoming proton, and perturbative hard-scatteringcross sections. Likewise, factorization also occurs for pp → h + X at large transverse momentum of
the produced hadron, and for pp → jet(h) + X, where the hadron is part of a fully reconstructedjet. The fragmentation functions contributing to e+ e− → h + X, `p → `0 + h + X, and pp → h + X,
pp → jet(h) + X are universal in the sense that the same functions appear in the factorized
expressions for the three reactions. Modern QCD analyses of fragmentation functions “globally”
take into account experimental data sets for all three types of processes in order to obtain optimal
sets of fragmentation functions.
Electron-positron annihilation has the advantage that there is no hadronic initial state andhence no beam remnant. This is in contrast to `p → `0 + h + X or pp → h + X, which are affectedby hadron remnant contributions associated with the partons of the initial-state hadron(s) which
are collaterally involved in the hard lepton-parton or parton-parton collision. On the other hand,e+ e− → h + X has little sensitivity to Dgh and is insensitive to the charge asymmetries Dqh −Dq̄h . These quantities are best constrained in proton–(anti-)proton and electron-proton scattering,
respectively. Especially the latter provides an environment that allows the study of the influenceof initial-state QCD radiation on the fragmentation process, of the partonic and spin structure of
the hadron target, and of the target remnant system. (See Ref. [27] for a comprehensive review ofthe measurements and models of fragmentation in lepton-hadron scattering).
Moreover, unlike e+ e− annihilation where q 2 = s is fixed by the collider energy, lepton-hadronscattering has two independent scales, Q2 = −q 2 and the invariant mass squared, W 2 ≈ Q2 (1−x)/x,of the hadronic final state, which both can vary by several orders of magnitudes for a given CM
energy, thus allowing the study of fragmentation in different environments by a single experiment.
For example, in photoproduction the exchanged photon is quasi-real (Q2 ≈ 0), leading to processes
akin to hadron-hadron scattering. In DIS (Q2  1 GeV2 ), using factorization, the hadronic fragments of the struck quark can be directly compared with quark fragmentation in e+ e− in a suitable
frame. Results from lepton-hadron experiments quoted in this report primarily concern fragmentation in the DIS regime. Studies performed by lepton-hadron experiments of fragmentation withphotoproduction data containing high transverse momentum jets or particles are also reported,
when these are directly comparable to DIS and e+ e− results.
Fragmentation studies in lepton-hadron collisions are usually performed in one of two frames
in which the target hadron and the exchanged boson are collinear. The hadronic center-of-massframe (HCMS) is defined as the rest system of the exchanged boson and incoming hadron, withthe z ∗ -axis defined along the direction of the exchanged boson. The positive z ∗ direction defines
the so-called current fragmentation region. Fragmentation measurements performed in the HCMS
often use the Feynman-x variable xF = 2p∗z /W , where p∗z is the longitudinal momentum of the
31st May, 2024

19. Fragmentation Functions in e+ e− , ep, and pp Collisions

4
17

200 (b)1/σtot dσ/dx

1/σtot dσ/dx × c(√s)

10
16
(a)
10
15
10
LEP
202
14
GeV
10
LEP
189
13
GeV
10
LEP
183
12
GeV
10
LEP
1
7
2 Ge
V
1011
LEP
1
6
1 Ge
10
V
10
LEP
133
9
G
e
V
10
LEP
, SLC
8
91 G
10
AMY
eV
54 G
7
eV
10
TAS
SO 4
4 Ge
106
TAS
V
SO 3
5
5
G
eV
10
HRS
, MA
RK I
104
TAS
I, TP
SO 2
C 29
3
2 Ge
GeV
10
TAS
V
SO 1
4 Ge
100
TAS
V
SO 1
2
G
e
10
BaB
V
ar 10
.5 G
1
eV
0.1
0.01
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x0.02<x<0.05

100
70
50
30
20

0.05<x<0.1

10
7
5
3
2

0.2<x<0.3

1
0.7
0.5
0.3
0.2

0.4<x<0.6

0.1

0.1<x<0.2

0.3<x<0.4

0.6<x<0.8
TASSO
MARK II
TPC
HRS
AMY

ALEPH
DELPHI
L3
OPAL
SLD

25 50 75 100 125 150 175 200√s [GeV]Figure 19.2: Cross section for e+ e− → h + X for all charged hadrons [8–26], (a) for different CM√√
energies s versus x, and (b) for various ranges of x versus s. For (a) the distributions have been√√√scaled by c( s) = 10i with i ranging from i = 0 ( s = 12 GeV) to i = 13 ( s = 202 GeV).

particle in this frame. As W is the invariant mass of the hadronic final state, xF ranges between−1 and 1.
The Breit system [28, 29] is related to the HCMS by a longitudinal boost such that the timecomponent of q vanishes, i.e, q = (0, 0, 0, −Q). In the parton model, the struck parton then has
the longitudinal momentum Q/2 which becomes −Q/2 after the collision. As compared with the
HCMS, the current fragmentation region of the Breit frame is more closely matched to the partonic
scattering process, and is thus appropriate for direct comparisons of fragmentation functions in DIS
with those from e+ e− annihilation. The variable xp = 2p∗ /Q, where p∗ is the particle’s momentum
in the current region of the Breit frame, is used at HERA for measurements in the Breit frame,
enabling rather direct comparisons of DIS and e+ e− results.

19.2 Scaling violations and QCD corrections
As mentioned, the coefficient functions for the fragmentation structure functions in e+ e− → h + X
are amenable to QCD perturbation theory. For each of the structure functions FT,L,A (x, q 2 ) in
Eq. (19.1) (and hence for the total structure function F h in Eq. (19.2)) the coefficient function has
31st May, 2024

19. Fragmentation Functions in e+ e− , ep, and pp Collisions
10

2

1/N dn/dxp

1/N dn/dxF

5

(a)

10

1

10

10

xp range

ep (ZEUS)
ep+ (H1)
e e-

10

4

10

3

0.02-0.05

102

0.05-0.10

0.0-0.2(x 50)(x 10)(x 5)

0.10-0.20
(x 3)

10
-1

ep (ZEUS) <W>=120 GeV
ep (H1) <W>=117
GeV
2 1/2
µN (E665) <W > =20.5 GeVµp (EMC) <W>=14 GeV

0.20-0.30
0.30-0.40
0.40-0.50

1
10

-2

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
xF

10

-1
-2

0.50-0.70
0.70-1.0

(b)
10

10

2

Q, √s /GeV

Figure 19.3: (a) The distribution 1/N · dN/dxF for all charged particles in DIS lepton-hadronexperiments at different values of W , measured in the HCMS [30–33]. (b) Scaling violations of thefragmentation structure function for all charged particles in the current region of the Breit frame√of DIS [34, 35] and in e+ e− interactions [19, 36]. The data are shown as a function of s for e+ e−
results, and as a function of Q for the DIS results, each within the same indicated intervals of thescaled momentum xp . The data for the four lowest intervals of xp are multiplied by factors 50, 10,
5, and 3, respectively for better visibility.

an expansion of the form

Ca,i

q2
z, αs (µ), 2µ!

= (1 − δaL ) δiq δ(1 − z)αs (µ) (1)q2+ca,i z, 2
2π
µ

!



+

αs (µ) 2 (2)q2ca,i z, 2
2π
µ


!

+ ... ,

(19.4)

where a = T, L, A. At the zeroth order in the strong coupling αs the coefficient functions Cg forgluons vanish, while for (anti-) quarks Ci = gi (s) δ(1 − z) (except for FL for which the leading
contribution is of order αs , as indicated in Eq. (19.4)). Here gi (s) is the appropriate electroweak
coupling. In particular, gi (s) is proportional to the squared charge of the quark i at s  MZ2 , when
weak effects can be neglected. The full electroweak prefactors gi (s) can be found in Ref. [6]. The
first-order QCD corrections to the coefficient functions have been calculated in Refs. [37,38], and thesecond-order terms in [39–41]. Thus, the coefficient functions are known to NNLO, except for FL .
We note that beyond the leading order the coefficient functions, and hence the fragmentation
functions, start to depend on the choice of factorization scheme. The standard choice in theliterature is the MS scheme.
The simplest parton-model approach would predict scale-independent (‘scaling’) x-distributions
for both the structure function F h and the parton fragmentation functions Dih . Perturbative QCD
31st May, 2024

619. Fragmentation Functions in e+ e− , ep, and pp Collisions

corrections lead to logarithmic scaling violations via the evolution equations [42]X 1 dz∂x 2
Dih (x, µ2 ) =Pji z, αs (µ2 ) Djh,µ
2∂ ln µ
z
x z
j


Z



,

(19.5)

where the functions Pij (z, αs (µ2 )) describe the splitting process i → j + X, where parton j carries
the longitudinal momentum fraction z of parton i. Note that for fragmentation the relevant splitting
functions are Pji (rather than Pij as for the PDFs) since Djh represents the fragmentation of the
final parton. Usually the system of evolution equations is decomposed into a 2 × 2 flavor-singlet
sector comprising the gluon and the sum of all quark and antiquark fragmentation functions, andscalar (‘non-singlet’) equations for quark-antiquark and flavor differences.
The splitting functions in Eq. (19.5) have the perturbative expansion
Pji (z, αs ) =αs (0)
P (z) +
2π ji

αs 2 (1)Pji (z) +2παs 3 (2)Pji (z) + . . . ,2π


(19.6)

where the leading-order (LO) functions P (0) (z) [42, 43] are the same as those for the initial-stateparton distributions. The next-to-leading order (NLO) corrections P (1) (z) have been calculated
in Refs. [44–48] (there are well-known misprints in the journal version of Ref. [45]). Ref. [48] also
includes the spin-dependent case. The timelike functions are different from, but related to, theirspacelike counterparts, see also Ref. [49]. The connections between the two sets of functions has(2)
facilitated recent calculations of the next-to-next-to-leading order (NNLO) quantities Pqq (z) and(2)(2)
Pgg (z) in Eq. (19.6) [40, 50]. In the same way, the corresponding off-diagonal quantities Pqg and(2)
Pgq were recently obtained in Ref. [51] with the help of constraints from the momentum sum
rule Eq. (19.3) [50] and of the limit of CA = CF = nf for which QCD becomes supersymmetric.
(2)

An uncertainty still remains for the Pqg kernel, which however does not affect the logarithmic
behavior at small and large momentum fractions. With the exception of Ref. [47], all these higherorder results refer to the standard MS scheme with a fixed number nf of light flavors. When thethreshold for the production of a heavier quark flavor is crossed in the course of the scale evolution,
fragmentation functions change. The NLO treatment of these flavor thresholds in the evolution has
been addressed in Ref. [52].
The phenomenological effect of scale evolution is similar in the timelike and spacelike cases: As
the scale increases, one observes a scaling violation in which the x-distribution is shifted towards
lower values. This can be seen from Fig. 19.2 where a set of measurements of the total fragmentation
structure function inpe+ e− annihilation are shown. In particular, the figure on the right exhibits√
the dependence on q 2 = s at fixed values of x. QCD analyses of these data are discussed in
Section 19.5 below.
The NLO coefficient functions for SIDIS, ep → e + h + X, have been presented in Refs. [37, 38]
Corresponding results have also been obtained for the case that a non-vanishing hadron transverse
momentum is required in the HCMS frame [53, 54].
Scaling violations in DIS are shown in Fig. 19.3 for both the HCMS and the Breit frames. In
Fig. 1.3(a) the distribution in terms of xF = 2p∗z /W shows a steeper slope in ep data than for the
lower-energy µp data for xF > 0.15, indicating the scaling violations. At smaller values of xF in
the current jet region, the multiplicity of particles substantially increases with W , owing to theincreased phase space available for the fragmentation process. The EMC data access both thecurrent region and the region of the fragmenting target remnant system. At higher values of |xF |,due to the extended nature of the remnant, the multiplicity in the target region far exceeds that in
the current region. For acceptance reasons the remnant hemisphere of the HCMS is only accessibleby the lower-energy fixed-target experiments.
31st May, 2024

719. Fragmentation Functions in e+ e− , ep, and pp Collisions

Using hadrons from the current hemisphere in the Breit frame, measurements of fragmentation
functions and the production properties of particles in ep scattering have been reported in Refs.
[34, 35, 55–58]. Fig. 19.3(b) compares results from ep scattering and e+ e− experiments; the latter
results have been divided by two as they cover both event hemispheres. The agreement between
the DIS and e+ e− results is fairly good. However, processes in DIS which are not present ine+ e− annihilation, such as boson-gluon fusion and initial-state QCD radiation, can depopulate thecurrent region. These effects become most prominent at low values of Q and xp . Hence, when√compared with e+ e− annihilation data at s = 5.2, 6.5 GeV [59] not shown here, the DIS particle
rates tend to lie below those observed in e+ e− annihilation. A ZEUS study [60] finds that thedirect comparability of the ep data to e+ e− results at low scales is improved if twice the energy
in the current hemisphere of the Breit frame, 2EBcr , is used instead of Q/2 as the fragmentation
scale. Choosing 2EBcr for the fragmentation scale approximates QCD radiation effects relevant at
low scales, as detailed in Ref. [29].

19.3 Fragmentation functions for small particle momenta
The higher-order timelike splitting functions in Eq. (19.6) are singular at small values of x.
They show a double-logarithmic enhancement, with leading terms of the form αsk ln2k−2 x /x at
the kth order of perturbation theory, corresponding to poles αsk (N − 1)1−2k for the Mellin moments
P (k) (N ) =Z 1

dx xN −1 P (k) (x) .

(19.7)

0

Despite large cancellations between leading and non-leading logarithms at non-asymptotic valuesof x, the resulting small-x rise in the timelike splitting functions dwarfs that of their spacelike
counterparts for the evolution of the parton distributions in Section 18 of this Review, see Fig. 1of Ref. [50]. Consequently, in fragmentation the fixed-order approximation to the evolution breaks
down orders of magnitude earlier in x than in DIS.
The pattern of the known coefficients and other considerations suggest that the double-logarithmic terms sum to all-order expressions without any pole at N = 1, such as [61, 62]
LL
Pgg(N ) = −q1
N − 1 − (N − 1)2 · 24 αs /π4(19.8)

for the gluon-to-gluon splitting function at leading logarithmic order. Keeping the first three terms
in the resulting expansion of Eq. (19.5) around N = 1 and taking the Mellin inverse yields a
Gaussian in the variable ξ = ln(1/x) for the small-x fragmentation functions,1
xD(x, q = s) ∝ exp − 2 (ξ − ξp )2 ,
2σ


2



(19.9)

with the peak position and width varying with the energy as [63] (see also Ref. [2])s1
ξp ' ln
4
Λ2




s
ln
Λ2



,

σ ∝



3/4

.

(19.10)

Next-to-leading logarithmic corrections to the above predictions have been calculated [64]. In themethod of Ref. [65], see also Refs. [66,67], the corrections are included in an analytical form known
as the ‘modified leading logarithmic approximation’ (MLLA). Alternatively they can be used tocompute higher-moment corrections to the shape in Eq. (19.9) [68]. The small-x resummation of thecoefficient functions for semi-inclusive e+ e− annihilation and of the timelike spitting functions inthe standard MS scheme was extended in Refs. [69–73] and has reached full next-to-next-to-leading
31st May, 2024

19. Fragmentation Functions in e+ e− , ep, and pp Collisions

8

logarithmic accuracy. Applications of these results to gluon and quark jet multiplicities have beenpresented in Refs. [74].
Fig. 19.4 shows the ξ distribution for charged particles produced in the current region of theBreit frame in DIS and in e+ e− annihilation. Consistently with Eq. (19.9) (the ‘hump backed
plateau’) and Eq. (19.10) the distributions have a Gaussian shape, with the peak position and area
increasing with CM energy (e+ e− ) and Q2 (DIS).

8
7

1/σ dσ/dξ

6
5
4
3

+ −

ee:
LEP 206 GeV
LEP 189 GeV
LEP 133 GeV
LEP 91 GeVTOPAZ 58 GeV
TASSO 44 GeV
TASSO 35 GeV
TASSO 22 GeV

DIS:
H1* 100-8000 GeV2
ZEUS* 80-160 GeV2
ZEUS* 40-80 GeV2
H1* 12-100 GeV2*
2
ZEUS 10-20 GeV

2
1
0

0

1

2
3
ξ=ln(1/xp)

4

5

6

Figure 19.4: Distribution of the normalized fragmentation cross sections in ξ = ln(1/xp ) at several
CM energies (e+ e− ) [10, 11, 16–19, 57, 58, 75–78] and for intervals of Q2 (DIS). At each energy only
one representative measurement is displayed. For clarity some measurements at intermediate CMenergies (e+ e− ) or Q2 ranges (DIS) are not shown. The DIS measurements (∗) have been scaled
by a factor of 2 for direct comparability with the e+ e− results. Fits of simple Gaussian functions
are overlaid for illustration.
The predicted energy dependence of the peak in the ξ distribution (see Eq. (19.10)) is explainedby soft gluon coherence (angular ordering), i.e., the destructive interference of the color wavefunction of low energy gluon radiation, which correctly predicts the suppression of hadron production
at small x. Of course, a decrease at very small x is expected on purely kinematical grounds, but
31st May, 2024

19. Fragmentation Functions in e+ e− , ep, and pp Collisions

9√
this would occur at particle energies proportional to their masses, i.e., at x ∝ m/ s and henceξ ∼ 21 ln s. Thus, if the suppression were purely kinematic, the peak position ξp would vary twice as
rapidly with the energy, which is ruled out by the data in Fig. 19.5. The e+ e− and DIS data agree
well with each other, demonstrating the universality of hadronization and the MLLA prediction.
Measurements of the higher moments of the ξ distribution in e+ e− [19, 78–80] and DIS [58] havealso been performed and show consistency with each other.

4.5
4

ξp

3.5
3
2.5
2

e+e−:
OPAL
L3
DELPHI
ALEPH
AMY
CELLO
TPC
MARK II
TASSO
BES
DIS:
H1
ZEUS

MLLA QCD, αS(MZ)=0.118
2

1.5
12

3

5 7 1020 30√s [GeV]

50 70 100

200√
Figure 19.5: Evolution of the peak position, ξp , of the ξ distribution with the CM energy s.
The MLLA QCD prediction using αs (s = MZ2 ) = 0.118 is superimposed to the data of Refs.
[10, 12, 15, 19, 56, 57, 76, 77, 80–88].
The average charged-particle multiplicity is another observable sensitive to fragmentation functions for small particle momenta. Perturbative predictions using both NLO [89] and MLLA [90,91]
have been obtained by solving Eq. (19.5) yieldingD2nG (Q )

E"

∝ αsb (Q2 ) · expcαs (Q2 )
p
·
1
+
6a
2
π
4πb0 αs (Q2 )!#,(19.11)√nf
where b = 14 + 10
96π, with b0 = (33 − 2nf )/(12π), cf. Section 9 of this Review, for27 4πb0 , c =nf contributing quark flavors. Higher-order corrections to Eq. (19.11) are known up to next-tonext-to-next-to-leading order (N3 LO), for details and references see [92]. The term proportional toa2 ≈ −0.502+0.0421 nf −0.00036 n2f in Eq. (19.11) is the contribution due to NNLO corrections [93].
The quantity hnG (Q2 )i refers to the average number of gluons, while for hnq (Q2 )i for quarks acorrection factor 1/r is required due to the different color factors in quark and gluon couplings,
so that hnq (Q2 )i = hnG (Q2 )i/r. The correction factor depends only weakly on Q2 ; higher-ordercorrections up to N3 LO on the asymptotic value r = CA /CF = 9/4 [94] are quoted in [92].
Employing the hypothesis of ‘Local Parton-Hadron Duality’ (LPHD) [90], i.e., that the color
charge of partons is balanced locally in phase space and, hence, their hadronization occurs lo31st May, 2024

19. Fragmentation Functions in e+ e− , ep, and pp Collisions

10

40
+ −

e e data_

35p(p) - p data_p(p) - p data, |η|<2.5_

NNLO QCD(αS(MZ2)=0.11840)

25

dn 

〈nch〉 or dη |η|...

30

MARK IIp(p) - p data, |η|=0, ≤0.5 ALEPH, DELPHI
L3, OPAL±e p data
UA5

JADE, TASSO

20

AMY
HRS
TPC
CLEO
ARGUS

10

γγ2, MARK I

S

L

LENA

15

CM

,
AS

TOPAZ,
VENUS

,
DF

AT

,C

A5

U
1,
UA

ISR
MS

AS, C

5

UA1,

H1, ZEUS

UA5,

ATL
ICE,
F, AL

CD

BubbleChambers

0
1

10

10

2

√s or Q [GeV]

10

3

10

4

√
Figure 19.6: Average charged-particle multiplicity hnch i as a function of s or Q for e+ e− andpp annihilations, and pp and ep collisions. The indicated errors are statistical and systematic
uncertainties added in quadrature, except when no systematic uncertainties are given. All NNLO
QCD curves are from Eq. (19.11) with fitted normalization, KLHPD , and offset, n0 , using a fixed√
αs (MZ2 ) = 0.1184 [95] and for e+ e− annihilation data nf = 3, 4, or 5 depending on s, else nf = 3.
e+ e− : Contributions from KS0 and Λ decays included. Data compiled from Refs. [8, 10, 16, 16, 22,
77, 83, 96–106]; e± p : Multiplicities have been measured in the current fragmentation region of theBreit frame. Data compiled from Refs. [35, 57, 58, 60, 107]; p( p) : Measured values above 20 GeV
refer to non-single diffractive (NSD) processes. Central pseudorapidity multiplicities (dn/dη)||η|...
refer to either |η| < 2.5 (CMS: |η| < 2.4) or |η| = 0 (UA5, CMS, ALICE: |η| < 0.5). Data compiled
from Refs. [108–123].

cally such that (Mellin transformed) parton and hadron inclusive distributions directly correspond,
Eq. (19.11) can be applied to describe average charged particle multiplicities obtained in e+ e− annihilation. The equation can also be applied to e± p scattering if the current fragmentation regionof the Breit frame is considered for measuring the average charged-particle multiplicity. Fig. 19.6
shows corresponding data and fits of Eq. (19.11) where apart from an LPHD normalization factor
31st May, 2024

1119. Fragmentation Functions in e+ e− , ep, and pp Collisions

a constant offset has been allowed for, so that hnch (Q)i = KLHPD · hnG (Q)i/r + n0 .
In hadron-hadron collisions beam remnants, e.g. from single-diffractive (SD) scattering where
one colliding proton is negligibly deflected while hadrons related with the other colliding proton are
well-separated in rapidity from the former proton, contribute to the measurement of the hadronmultiplicity from a hard parton-parton scattering, making interpretation of the data more model
dependent. Experimental results are usually given for inelastic processes or for non-single diffractive
processes (NSD). Due to the large beam particle momenta at Tevatron and LHC, not all final stateparticles can be detected within the limited detector acceptance. Therefore, experiments at Tevatron and LHC quote particle multiplicities for limited ranges of pseudo-rapidity η = − ln tan(ϑ/2)
or at central rapidity, i.e. η = 0, as shown in Fig. 19.6.
A universality of the average particle multiplicities in e+ e− and p( p) processes has been reported√
in Ref. [124] when considering an effective collision energy Qeff = s/k in p( p) reduced by a factorof k ≈ 3, plus a constant offset of n0 ≈ 2. A more detailed review is available in Ref. [125].
According to the investigations presented in Ref. [126] the universality of the energy dependenceof average particle multiplicities also applies to hadron-hadron and nucleus-nucleus collisions forboth full and central rapidity multiplicities. Evidence for this universality is given by the good
agreement for the energy dependence of Eq. (19.11) when fit to the p( p) data as shown in Fig. 19.6.

19.4 Fragmentation models
Although the scaling violations can be calculated perturbatively, the actual form of the parton
fragmentation functions is non-perturbative. Perturbative evolution gives rise to a shower of quarks
and gluons (partons). Multi-parton final states from leading and higher order matrix element calculations are linked to these parton showers using factorization prescriptions, also called matching
schemes, see Ref. [127] for an overview.
Phenomenological schemes are then used to model the carry-over of parton momenta and flavor
to the hadrons. Implemented in Monte Carlo event generators (see Section 43 of this Review), theseschemes have been tuned using e+ e− data and provide good description of hadron collisions as well,thus providing evidence of the universality of fragmentation. However, e+ e− mainly fix the quark
jet fragmentation while it provides less constraints for modelling the gluon jet fragmentation.

19.5 Phenomenology of quark and gluon fragmentation functions
The fragmentation functions are solutions to the evolution equations Eq. (19.5), but need to be
specified at some initial scale µ20 (usually around 1 GeV2 for light quarks and gluons, and at m2Q
for heavy quarks). A typical parameterization for a given light hadron is [128, 131–139]
Dih (x, µ20 ) = Ni xαi (1 − x)βi 1 + γi (1 − x)δi

,(19.12)where as indicated the normalization Ni , and the parameters αi , βi , γi and δi depend on the type iof the fragmenting parton. Heavy flavor fragmentation into heavy mesons is discussed in Sec. 19.8
below. The parameters of Eq. (19.12) are obtained by performing global fits to data on various
hadron types for different combinations of partons and hadrons in e+ e− , lepton-hadron and hadronhadron collisions. We note that the choice of parameterization of the fragmentation functions at theinitial scale necessarily introduces a bias since it imposes a certain form of the functions. This bias
is largely avoided in neural network approaches which offer a wide flexibility of the initial functions
and have recently been applied to fragmentation functions as well [130]. Sets of fragmentation
functions are now available for pions, kaons, protons, neutrons, η mesons, Λ baryons, and charged
hadrons [129, 130, 132–144]. They are all at NLO level, except for Refs. [130, 141, 144] which havebeen performed at NNLO level. The latter sets are restricted to the analysis of e+ e− annihilation
data. Recently, data from hadron-hadron collisions have been added in the framework of the neural
31st May, 2024

19. Fragmentation Functions in e+ e− , ep, and pp Collisions

12

10

u(x 10)

1
10
10

10
+

π +π

-2

10

1

c

s

-1

10

g(x 10)

BDSS21
JAF20
NNFF1.0-

2x Di(x, MZ)
-1

x

10

-1

x

10

-1

-2

1

Figure 19.7: Comparison of up, strange, charm and gluon NLO fragmentation functions for
π + + π − at the mass of the Z. The different lines correspond to the results of the analyses
performed in Refs. [128–130].
network approach at NLO accuracy for charged hadrons [145]. It is noteworthy that the NNLO
effects lead to an improvement in the theoretical description of the data in e+ e− annihilation.
Data from e+ e− annihilation present the cleanest experimental source for the measurement offragmentation functions, but cannot be used to disentangle quark from antiquark fragmentation.
Since the bulk of the e+ e− annihilation data is obtained at the mass of the Z-boson, where theelectroweak couplings are roughly the same for the different partons, it provides the most precise
determination of the flavor-singlet combination of quark and antiquark fragmentation functions.
Flavor-tagged results [146], distinguishing between the light quark, charm and bottom contributions are of particular value for flavor decomposition, even though those measurements cannot be
unambiguously interpreted in perturbative QCD.
The most relevant source for quark-antiquark (and also flavor) separation is provided by SIDIS
data. Semi-inclusive measurements are usually performed at much lower scales than for e+ e−annihilation. The inclusion of SIDIS data in global fits allows for a wider coverage in the evolutionof the fragmentation functions, resulting at the same time in a stringent test of the universality ofthe distributions. Charged-hadron production data in hadronic collisions also have sensitivity to(anti-)quark fragmentation functions. A recent analysis [129] performs a simultaneous extractionof parton distributions and fragmentation functions based on SIDIS data.
The gluon fragmentation function Dgh (x) can be extracted, in principle, from the longitudinalfragmentation structure function FL in Eq. (19.2), as the coefficient functions CL,i for quarks andgluons are comparable at order αs . However at NLO, i.e., including the O(αs2 ) coefficient functions(2)
CL,i [39], quark fragmentation is dominant in FL over a large part of the kinematic range, reducing
the sensitivity to Dgh . This distribution could be determined also by analyzing the scale evolutionof the fragmentation functions. This possibility is limited by the lack of sufficiently precise data at
31st May, 2024

13

19. Fragmentation Functions in e+ e− , ep, and pp Collisions

1/σhad dσ/dx

300
100
30
10
3
1
0.3
0.1
0.03
0.01
30

(a)

π± (√s = 91 GeV)
π± (√s = 29 GeV)
π± (√s = 10 GeV)(b)

1/σhad dσ/dx

10
3
1
0.3
0.1
0.03

K± (√s = 91 GeV)
K± (√s = 29 GeV)
K± (√s = 10 GeV)

0.01
30

(c)

1/σhad dσ/dx

10
3
1
0.3
_

0.1
0.03

p,p_ (√s = 91 GeV)
p,p_ (√s = 29 GeV)
p,p (√s = 10 GeV)

0.01
0.005 0.01 0.02

0.05 0.1
xp = p/pbeam

0.2

0.5

Figure 19.8: Scaled momentum spectra of (a) π ± , (b) K ± , and (c) p, p at
GeV [24, 26, 85, 147, 148].

1√

s = 10, 29, and 91

energy scales away from the Z-resonance and the dominance of the quark contributions at medium
and large values of x. In e+ e− annihilation, Dgh can also be deduced from the study of three-jet
events in which the gluon jet is identified, for example, by tagging the other two jets with heavy
31st May, 2024

1/N dn/dxF

1/N dn/dz

19. Fragmentation Functions in e+ e− , ep, and pp Collisions

14

(a)
1

µp π0 (EMC)
ep π0 (HERMES)
ep 1/2(π++π-) (HERMES)

10

-1

10
-2

10
-3

1/N dn/dxF

101

ep K0 (H1) <W>=138 GeV
0µN K (E665) <W>=17.1 GeVµp K+ (EMC) <W2>1/2=11.4 GeV1

ep 1/2(Λ+Λ) (H1) <W>138 GeVµp Λ <W2>1/2=11.4 GeV
2 1/2µp Λ <W > =11.4 GeV

-1

-1

10

10
-2

10

(b)
-2

-3

10

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
xF

10

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1zFigure 19.9: (a) 1/N · dn/dxF for identified strange particles in DIS at various values of W
[149–151]. (b) 1/N · dn/dz for measurements of pions in fixed-target DIS experiments [152–154].

a) π0

10 9

Ed3σ/dp3 [ pb/GeV2 ]10

WA70 (√s =23 GeV)
E706 (√s =31.5 GeV)
AFS (√s =63 GeV)
R806 (√s =62.8 GeV)PHENIX (√s =62 GeV)PHENIX (√s =200 GeV)
UA2 (√s =540 GeV)
ALICE (√s =7 TeV) x 0.1

8

10 6

b) (h++h-)/2

10 9

UA1 (√s =200 GeV)
UA1 (√s =500 GeV)
UA1 (√s =900 GeV)

10 8

CDF (√s =630 GeV)
CDF (√s =1800 GeV)
CMS (√s =7 TeV)

10 7

10 4
10 6
10 2

10 5

1

10

10 4
-2

2

4

6

8

10

2

4

6

8

10

pT [ GeV]

pT [ GeV]

Figure 19.10: Selection of inclusive (a) π 0 and (b) charged-hadron production data from pp[118, 155–160] and pp̄ [114, 161, 162] collisions.

quark decays. To leading order, the measured distributions of x = Ehad /Ejet for particles in gluon
jets can be identified directly with the gluon fragmentation function Dgh (x).
31st May, 2024

1519. Fragmentation Functions in e+ e− , ep, and pp Collisions
Table 19.1: Classification of spin- and transverse-momentum dependent
quark fragmentation functions. For simplicity we have left out the ubiquitous label for flavor i of the fragmenting quark and for hadron species h.
Each of the functions carries the argument (x, x2 kT2 ) (plus dependence on afactorization scale), where xkT = phT is the hadron’s transverse momentum.
λ and Λ are the quark’s and hadron’s helicities, respectively,
and ~sT and~~
ST are their transverse spin vectors. We have defined ~a × b ≡ a1 b2 −a2 b1 .
Finally, mh is the mass of the produced hadron.
quark polarization
long. polarizedtransv. polarizedhadron pol.

unpolarized

unpol.

D–

long. pol.

–λ Λ GL~kT ·~sT⊥
mh Λ HL~kT ·S~T⊥
mh λ GT~T ) HT + kT ·ST kT ·~sT H ⊥(~sT · S
T
mh
mh





transv. pol.

~kT ×S
~T
mh



DT⊥

~kT ×~sT
mh



~

H⊥

~ ~

Data for p( p) → h + X provide much more direct constraint on Dgh . At variance with e+ e−annihilation and SIDIS, here gluon fragmentation contributes already at the lowest order in the
coupling constant. At large x & 0.5, where information from e+ e− is sparse, data from hadronic
colliders significantly improve extractions of Dgh [131, 132, 138, 140]. Recent LHC data has been
included in the NLO analyses [128, 138, 139] of pion-fragmentation functions; see Sec.(17.7) for
more details. Note that these analyses are currently the only ones that ‘globally’ incorporate
available data from all sources, e+ e− → h + X, ep → e0 h + X and pp → h + X.
We note that recently a ‘hybrid’ type of high-pT jet/hadron observable has also been consideredboth theoretically [163–169] and experimentally [170–177]. It is defined by an identified specifichadron found inside a fully reconstructed jet. This gives rise to a same-side hadron-jet momentum
correlation that may be addressed using perturbative methods. One of several relevant kinematicalvariables (see [168] for an overview) is zh ≡ (~pTh · p~Tjet )/(pTjet )2 , where p~Th and p~Tjet are the transverse
momenta of the hadron and the jet, respectively. The observable provides an alternative window
on fragmentation functions in a more exclusive setting, enabling novel tests of the universality offragmentation functions. Varying zh and/or the hadron species, one can map out the fragmentation
functions ‘locally’ as functions of x. This is in contrast to the single-inclusive observable pp → h+X,
which inevitably samples over a broad range of x. Although hadron-in-jet data are not yet routinely
included in analyse of fragmentation functions, a ‘proof-of-principle’ analysis does exist [178] thatshows the potential of the observable in providing constraint on fragmentation functions.
A comparison of recent NLO fits of fragmentation functions for π + +π − obtained by the updated
version of BDSS21 [128], JAF20 [129] and NNPDF1.0 [130] is shown in Fig. 19.7. Differencesamong the functions for these sets are visible, especially for the gluon fragmentation function overthe full range of x and for the quark functions at large momentum fractions. The differences
are even larger for other species of hadrons like kaons and protons [131, 132, 136, 140]. Recent
analyses [128–130, 136, 138, 139, 179, 180] estimate the uncertainties involved in the extraction offragmentation functions.
Photonic fragmentation functions play a relevant role in the theoretical understanding of in31st May, 2024

16

19. Fragmentation Functions in e+ e− , ep, and pp Collisions

clusive photon production in (leptonic and hadronic) high energy processes. In the sprit of theanalogy between parton fragmentation functions and parton distribution functions, also photonic
fragmentation functions are analogous to the photon structure function F2γ and to the proton’s
photonic parton distributions (see review on structure functions in Section 18 of this Review).
Since photons have a pointlike coupling to quarks [181], the corresponding fragmentation functions
obey inhomogeneous evolution equations and are generally decomposed into a perturbative and a
non-perturbative component [135, 182, 183]. The hadronic part, sometimes approximated by the
Vector Meson Dominance Model, can in principle be obtained by performing a global analysis to theavailable prompt photon production data [7,12,15,19–21,85,147,184,185], although in practice thishas not been done. We note that also the cross section for photons produced in fully reconstructedjets has been proposed [186] as a new tool for obtaining access to photon fragmentation functions,in analogy to the hadron-in-jet cross section discussed above.

19.6 Identified particles in e+ e− and semi-inclusive DIS
There is a great wealth of measurements of e+ e− fragmentation into identified particles. A
collection of references for data on fragmentation into identified particles is provided in Table 53.1of this Review. As a representative example, Figure 19.8 shows differential charged-hadron spectra
as functions of the scaled hadron momentum at several CM energies.
Quantitative results of studies of scaling violations in e+ e− fragmentation have been reported
in [7, 21, 187, 188]. Scaling violations may be used to extract a value of αs ; the values obtained areconsistent with the world average (see review on QCD in Section 9 of this Review).
Many studies have been made of production of identified particles in lepton-hadron scattering, although fewer particle species have been measured than in e+ e− collisions. References[149, 150, 152–154, 189–191] and [151, 192–197] are representative of the data from fixed target
and ep collider experiments, respectively. QCD calculations performed at NLO provide an overall
good description of the HERA data [33, 34, 58, 197–199], both for SIDIS [200] and for the hadrontransverse momentum distribution [53, 201] in the kinematic regions in which the calculations are
predictive. Threshold corrections to SIDIS have been computed at NNLO [202] and even at oneorder higher [203]. Very recently, the full set of NNLO QCD corrections to the corresponding
coefficient functions were obtained in [204, 205].
Fig. 19.9(a) compares lower-energy fixed-target and HERA data on strangeness production,
showing that the HERA spectra have substantially increased multiplicities, albeit with statisticalprecision that is insufficient to study scaling violations. The fixed-target data show that the Λrate substantially exceeds the Λ rate in the remnant region, owing to the conserved baryon number
from the baryon target. Fig. 19.9(b) shows 1/N · dn/dz for neutral and charged pion production,
where z is defined as the ratio of the pion energy to that of the exchanged boson, both measured in
the laboratory frame. Results are shown from the HERMES and the EMC experiments, where the
HERMES data have been evolved to hQ2 i = 25 GeV2 at NLO QCD, in order to be comparable with
the EMC data. Each of the experiments uses various kinematic cuts to ensure that the measured
particles lie in the region that is expected to be associated with the struck quark. In the DIS
kinematic regime accessed at these experiments, and over the range in z shown in Fig. 19.9, the z
and xF variables have similar values [30]. The precision data on identified particles can be used in
the study of the quark flavor content of the proton [179, 221, 222].
Data on identified particle production can aid the investigation of the universality of jet fragmentation in e+ e− and DIS. The strangeness suppression factor γs , as derived principally fromtuning the Lund string model [223] within JETSET [224], is typically found to be around 0.3 ine+ e− experiments [75], although values closer to 0.2 [225] have also been obtained. A numberof measurements of so-called V 0 -particles (K 0 , Λ0 ) and the relative rates of V 0 ’s and inclusively
31st May, 2024

19. Fragmentation Functions in e+ e− , ep, and pp Collisions

10

(a)

1
charged particles

10-113000 GeV pp

2-2

1/NNSD 1/(2πpT) d N/dηdpT [GeV ]17

107000 GeV pp-22360 GeV pp_

1960 GeV pp

10-3

900 GeV pp_

900 GeV pp

10

_

-4

500 GeV pp_

200 GeV pp

10
10

1/NNSD d2N/dydpT [GeV-2]

10

-5
-6

-7

0.1

0.2

0.3 0.4 0.50.6 0.8 1

10

2

(b)

3

4

5 6

8 10
pT [GeV]

(c)

(d)

1

10

10

10

-1

-2

-3

+

+

−

(π +π )/2
13000 GeV pp
7000 GeV pp2760 GeV pp2360 GeV pp
900 GeV pp
200 GeV pp

0.1 0.2

0.5_−(K +K )/2
13000 GeV pp
7000 GeV pp2760 GeV pp2360 GeV pp
900 GeV pp
200 GeV pp

12 3
pT [GeV]0.2

0.5(p+p)/2
13000 GeV pp
7000 GeV pp2760 GeV pp2360 GeV pp
900 GeV pp
200 GeV pp

12 3
pT [GeV]

0.2

0.5

1

2 3
pT [GeV]Figure 19.11: (a) Selection of inclusive charged-particle transverse momentum spectra [114, 117,
119,121,161,206,207], normalized to the non-single diffractive cross section (NSD). (b)-(d) Selectionof identified charged-particle transverse momentum spectra [208–213] normalized to the NSD crosssection. All spectra are scaled to the NSD cross-section using measurements of total, inelastic,elastic, single, or non-single diffractive cross sections from [214–218, 218–220]. The overall normalization uncertainty of about 3% is not shown. Superimposed are fits of the Tsallis distribution in
Eq. (19.13).

produced charged particles have been performed at HERA [151, 192, 226] and fixed target experi31st May, 2024

18

19. Fragmentation Functions in e+ e− , ep, and pp Collisions

ments [149]. These typically favour a stronger suppression (γs ≈ 0.2) than usually obtained frome+ e− data, although values close to 0.3 have also been obtained [227, 228].
However, when comparing the description of QCD-based models for lepton-hadron interactions
and e+ e− collisions, it is important to note that the overall description by event generators ofinclusively produced hadronic final states is more accurate in e+ e− collisions than in lepton-hadroninteractions [229]. Predictions of particle rates in lepton-hadron scattering are affected by uncertainties in the modelling of the parton composition of the proton and photon, the extended target
remnant, and initial and final-state QCD radiation. Furthermore, the tuning of event generatorsfor e+ e− collisions is typically based on a larger set of parameters and uses more observables [75]than are used when optimizing models for lepton-hadron data [230].

19.7 Fragmentation in hadron-hadron collisions
An extensive set on high-transverse momentum (pT ) single-inclusive hadron data has beencollected in h1 h2 → hX scattering processes, both at high energy colliders and fixed-target experiments [155–162, 185, 231–253]. Fig. 19.10 shows the invariant cross sections Ed3 σ/dp3 for a√compilation of neutral-pion and charged-hadron production data for energies in the range s ≈ 23
- 7000 GeV.
The differential cross section for high-transverse momentum hadron production has been computed to NLO accuracy in perturbative QCD [254]. The NLO corrections are typically large andcan even double the prediction for the cross section at fixed-target energies. Nevertheless, the
NLO calculations significantly under-predict the cross-section for several fixed-target energy data
sets [249, 255, 256]. Different strategies have been developed to ameliorate the theoretical description at fixed-target energies. A possible phenomenological approach involves the introduction ofa non-perturbative intrinsic partonic transverse momentum [160, 249, 257, 258]. Furthermore, theresummation of the dominant higher order corrections at threshold produces an enhancement of
the theoretical calculation that significantly improves the description of the data [259, 260].
Data collected at high energy colliders are either included in global fit analyses or used as atest for the universality of fragmentation functions. A certain tension has been observed between
data sets from RHIC and the LHC [261]. The tension can be largely resolved [128] by taking
properly into account the theoretical scale dependence in the global analysis, allowing factorization
and renormalization scales to variate according to the kinematics of each experiment within aconventional range.
Transverse momentum distributions can usually be fit by power laws [262]. An approach todescribe the low pT particle spectra is the Tsallis distribution [263–265], which is based on a nonextensive generalization of the Boltzmann-Gibbs statistics. The functional form [266]d2 N
dN(n − 1)(n − 2)
mT − m0 −n= pT
1+
dpT dydy nT (nT + m0 (n − 2))nT(19.13)is frequently used to fit the transverse momentum spectra, where dN/dy is the particle’s multiplicity,
T and n are fit parameters of the Tsallis distribution, m0 is the either the mass of the mostabundant
q particle, i.e. the pion for inclusive spectra, or the mass of an identified particle, andmT = p2T + m20 . The parameter n is related to the non-extensive parameter q = n/(n − 1) of
the original Tsallis formula [267], and T is connected to the temperature in the Boltzmann-Gibbs
statistics. The Tsallis distribution has been very successfully fit to measured transverse momentum
distributions of both inclusive charged particles and identified particle spectra for hadron-hadroncollisions, see for example [268–270], for collisions of heavy nuclei, see for example [271], and also
for e+ e− collisions, see for example [272]. The energy dependence of the fitted Tsallis parameters
31st May, 2024

19. Fragmentation Functions in e+ e− , ep, and pp Collisions

19

has also been investigated in detail, see [266, 273]. Fig. 19.11 shows examples of hadron production
data in pp and pp̄ collisions compared to Tsallis distributions.
Hadron production provides a critical observable for probing the high energy-density matterproduced in heavy-ion collisions. Measurements at colliders show a suppression of inclusive hadronyields at high transverse momentum for AA collisions compared to pp scattering, indicating theformation of a dense medium opaque to quark and gluons, see e.g. [274].

19.8 Heavy quark fragmentation
It was recognized very early [275] that a heavy flavored meson should retain a large fractionof the momentum of the primordial heavy quark, and therefore its fragmentation function should
be much harder than that of a light hadron. In the limit of a very heavy quark, one expects thefragmentation function for a heavy quark to go into any heavy hadron to be peaked near x = 1.
When the heavy quark is produced at a momentum much larger than its mass, one expects
important perturbative effects, enhanced by powers of the logarithm of the transverse momentum
over the heavy quark mass, to intervene and modify the shape of the fragmentation function. In
leading logarithmic order (i.e., including all powers of αs log(mQ /pT )), the total (i.e., summed over
all hadron types) perturbative fragmentation function is simply obtained by solving the leading
evolution equation for fragmentation functions, Eq. (19.5), with the initial condition due to thefinite mass of the heavy quark given by DQ (x, µ2 ) µ2 =m2 = δ(1 − x) and Di (x, µ2 ) µ2 =m2 = 0 forQQ

i 6= Q (here Di (x, µ2 ), stands for the probability to produce a heavy quark Q from parton i with afraction x of the parton momentum).
16

3.5
D0 D*

12

(a)
CLEO
BELLE

ALEPH 91 GeVOPAL 91 GeVSLD 91 GeV
DELPHI 91 GeV

2.5

10
8

2

1.5

6
4

1

2

0.5

0

(b)

3
1/σ dσ/dxB

sB dσ/dxp (GeV2 nb)

140

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
xp=p/pmax

100

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
xB

1

Figure 19.12: (a) Efficiency-corrected inclusive cross-section measurements for the production of√D0 and D∗+ in e+ e− measurements at s ≈ 10.6 GeV, excluding B decay products [276] [277]. (b)√Measured e+ e− fragmentation function of b quarks into B hadrons at s ≈ 91 GeV [278].
Several extensions of the leading logarithmic result have appeared in the literature. Next-toleading-log (NLL) order results for the perturbative heavy quark fragmentation function have beenobtained in [279]. The resummation of the dominant logarithmic contributions at large x was
performed in [280] to next-to-leading-log accuracy. Fixed-order calculations of the fragmentation
function at order αs2 in e+ e− annihilation have appeared in [281] while the initial condition for the
perturbative heavy quark fragmentation function has been extended to NNLO in [282].
Inclusion of non-perturbative effects in the calculation of the heavy-quark fragmentation function is done by convoluting the perturbative result with a phenomenological non-perturbative form.
This form follows from the simple kinematical consideration that the formation of a hadron by attaching light quarks/anti-quarks to the heavy quark will slightly decelerate the heavy quark. Thus,
31st May, 2024

20

19. Fragmentation Functions in e+ e− , ep, and pp Collisions

its shape will show a peak that becomes increasingly centered next to x = 1 the higher the quark
mass. Among the most popular parameterizations we have the following:−2
11Peterson et al. [283] : Dnp (x) ∝
1− −
,
x
x 1−x
Kartvelishvili et al. [284] : Dnp (x) ∝ xα (1 − x) ,

1 − x (2 − x)C
Collins & Spiller [285] : Dnp (x) ∝
×
+
x
1−x


1
C −2
2
(1 + x )×
1− −
x 1−x
Colangelo & Nason [286] : Dnp (x) ∝ (1 − x)α xβBowler [287] : Dnp (x) ∝ x



−(1+bm2h,⊥ )(19.14)(19.15)(19.16)(19.17)×

bm2h,⊥(1 − x) exp −xaBraaten et al. [288] : (see Eqs. (31), (32) in [288])!

(19.18)(19.19)where , C , a, bm2h,⊥ , α, and β are non-perturbative parameters that depend on the heavy hadronconsidered. The parameters entering the non-perturbative forms are fitted together with somemodel of hard radiation, which can be either a shower Monte Carlo, a leading-log or NLL calculation(which may or may not include Sudakov resummation), or a fixed order calculation. In [281], forexample, the Peterson et al. [283]  parameter for charm and bottom production is fitted fromthe measured distributions of Refs. [289, 290] for charm, and of [291] for bottom. If the leadinglogarithmic approximation (LLA) is used for the perturbative part, one finds c ≈ 0.05 and b ≈0.006; if a second order calculation is used one finds c ≈ 0.035 and b ≈ 0.0033; if a NLL improved
fixed order O(αs2 ) calculation is used instead of NLO O(αs ) one finds c ≈ 0.022 and b ≈ 0.0023.
The larger values found in the LL approximation are consistent with what is obtained in thecontext of parton shower models [292], as expected. The  parameter for charm and bottom scales
roughly with the inverse square of the heavy flavor mass. This behavior can be justified by several
arguments [275, 293, 294]. It can be used to relate the non-perturbative parts of the fragmentation
functions of charm and bottom quarks [281, 286, 295].
A more conventional approach [296] involves the introduction of a unique set of heavy quark
fragmentation functions of non-perturbative nature that obey the usual massless evolution equations in Eq. (19.5). Finite mass terms of the form (mQ /pT )n are kept in the corresponding shortdistance coefficient function for each scattering process. Within this approach, the initial condition
for the perturbative fragmentation function provides the term needed to define the correct subtraction scheme to match the massless limit for the coefficient function (see e.g. [297]). Such an
implementation is in line with the variable flavor number scheme introduced for parton distributions
functions, as described in Section 18 of this Review.
High statistics data for charmed-meson production near the Υ resonance (excluding decay products of B mesons) have been published [276, 277]. They include results for D and D∗ , Ds (seealso [298, 299]) and Λc . Shown in Fig. 19.12(a) are the CLEO and BELLE inclusive cross-sections
times branching ratio B, sBdσ/dxp , for the production of D0 and D∗+ . The variable xp approximates the light-cone momentum fraction x, but is not identical to it. The two measurements are
consistent with each other.
The branching ratio B represents D0 → K − π + for the D0 results and for the D∗+ the product31st May, 2024

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19. Fragmentation Functions in e+ e− , ep, and pp Collisionsof the branching fractions for D∗+ → D0 π + and D0 → K − π + . Given the high precision of
CLEO’s and BELLE’s data, a superposition of different parametric forms for the non-perturbative
contribution is needed to obtain a good fit [52]. Older studies are reported in Refs. [290, 300, 301].
Charmed meson spectra on the Z peak have been published by OPAL and ALEPH [302, 303].
Charm quark production has also been extensively studied at HERA by the H1 and ZEUS
collaborations. Measurements have been made of D∗± , D± , and Ds± mesons and the Λc baryon.
See, for example, Refs. [304, 305].
Experimental studies of the fragmentation function for b quarks, shown in Fig. 19.12(b), havebeen performed at LEP and SLD [278, 291, 306]. Commonly used methods identify the B meson
through its semileptonic decay or based upon tracks emerging from the B secondary vertex. Heavy
flavor contributions from gluon splitting are usually explicitly removed before fitting for the fragmentation functions. The studies in [278] fit the B spectrum using a Monte Carlo shower model
supplemented with non-perturbative fragmentation functions yielding consistent results.
The experiments measure primarily the spectrum of B mesons. This defines a fragmentation
function that includes the effect of the decay of higher mass excitations, like the B ∗ and B ∗∗ . In
the literature (cf. details in Ref. [307]), there is sometimes ambiguity in what is defined to be thebottom fragmentation function. Instead of using what is directly measured (i.e., the B meson
spectrum), in some cases corrections are applied to account for B ∗ or B ∗∗ production.
Heavy-flavor production in e+ e− collisions is the primary source of information for the roleof fragmentation effects in heavy-flavor production in hadron-hadron and lepton-hadron collisions.
The QCD calculations tend to underestimate the data in certain regions of phase space. Someexperimental results from LHC summarized in [308] show such deviations e.g. at high transverse
jet momentum and also at low di-jet separation angles, see [309] for details, and were alreadytheoretically investigated in [310].
Both bottomed- and charmed-meson spectra have been measured at the Tevatron with unprecedented accuracy [311]. The measured spectra are in good agreement with QCD calculations(including non-perturbative fragmentation effects inferred from e+ e− data [312]).
The HERA collaborations have produced a number of measurements of beauty production; see,for example, Refs. [304, 313–316]. As for the Tevatron data, the HERA results are described wellby QCD-based calculations using fragmentation models optimized with e+ e− data.
Besides degrading the fragmentation function by gluon radiation, QCD evolution can also gener√
ate soft heavy quarks, increasing in the small x region as s increases. Several theoretical studies
are available on the issue of how often bb̄ or cc̄ pairs are produced indirectly via a gluon splitting mechanism [317–319]. Experimental results from studies on charm and bottom production
via gluon splitting, given in [303, 320–324], yield weighted averages of ng→cc = 3.05 ± 0.45% andng→bb = 0.277 ± 0.072%, respectively. The production of bottom-antibottom quark pairs via gluon
splitting has also been investigated at hadron colliders, see for example [325–327].

19.9 Spin-dependent and transverse-momentum dependent fragmentation functions
The fragmentation functions we have considered so far apply to the spin-averaged case in which
the polarization of the produced hadron is not observed, or the hadron has spin-0. We have also only
considered ‘collinear’ fragmentation functions Dih (x, µ2 ) which carry only one kinematical variable,
the momentum fraction x. New insights into fragmentation and hadronization become available
when also the dependence of fragmentation functions on the spin of the produced hadron and/or
its relative transverse momentum with respect to the fragmenting parton are considered. In thelatter case, one refers to the fragmentation functions as ‘transverse-momentum dependent (TMD)’
fragmentation functions.
31st May, 2024

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19. Fragmentation Functions in e+ e− , ep, and pp Collisions

Staying first with collinear fragmentation functions, two types of spin-dependent fragmentation
functions to spin-1/2 hadrons can be considered. The helicity-dependent fragmentation function
measures the transfer of longitudinal spin from the fragmenting parton to the hadron [38,328–331].
It is given by+−∆Dih (x, µ2 ) ≡ Dih+ (x, µ2 ) − Dih+ (x, µ2 ) ,(19.20)where the superscripts ± refer to the helicities of the parton and hadron. Λ hyperons are ideallysuited for measurements of the ∆Dih , thanks to their self-analyzing weak decay Λ → πp. Measurements of the longitudinal spin transfer to Lambda hyperons have been presented in e+ e− (on the
Z resonance), `p, and pp scattering in Refs. [332–338]. One may readily extend Eq. (19.20) to thecase of transverse polarization of hadrons and quarks [339], where the corresponding fragmentation functions are known as ‘transversity’ fragmentation functions. There are also measurements
constraining these fragmentation functions [333, 340, 341]. A recent phenomenological analysis waspresented in [342].
If the transverse-momentum (kT ) dependence of fragmentation functions is considered, there
are eight types of leading-twist functions, defined by the correlations among the hadronic andpartonic spin vectors and transverse-momentum vectors they represent. (For review, see [5]). We
note that the eight fragmentation functions given in the table below exist separately for eachquark and antiquark flavor, and a similar set may be introduced for gluons. Upon integration over
the transverse momentum kT the collinear unpolarized, helicity, and transversity fragmentation
functions are reproduced.
The various fragmentation functions may be obtained from spin asymmetries and angular distributions in hadron production processes. There is a large body of precision data by now ontransverse-momentum distributions in e+ e− annihilation [343] and SIDIS [190, 344] that provide
constraints on the unpolarized TMD fragmentation functions Dih , which have been analyzed theoretically, partly also including TMD evolution effects and high orders of perturbation theory [345–352].
Besides the unpolarized functions D most of the attention in experiment and theory has been on
the function H ⊥ which describes the production of unpolarized (or spin-0) hadrons by transversely
polarized quarks. This function is known as the ‘Collins function’ [353]. Its importance also derives
from the fact that it may be used to probe the quark transversity PDF of the nucleon [354] which
gives the probability of finding a transversely polarized quark with its spin aligned or anti-alignedwith the spin of a transversely polarized nucleon. The transversity function is chiral-odd, andtherefore not accessible through measurements of inclusive lepton-hadron scattering. The Collins
effect in semi-inclusive DIS, on the other hand, provides an avenue for accessing transversity.
The Collins fragmentation function is chiral-odd and T-odd, leading to a characteristic single-spin
asymmetry in the azimuthal angular distribution of the produced hadron in the hadron scattering
plane. A number of SIDIS [355–366] and e+ e− experiments [367–371] have performed measurementsof the Collins effect, for charged pions and kaons. These have been analyzed theoretically [372,373],
leading to an extraction of the nucleon’s transversity distributions [373]. The Collins effect has also
been studied in pp scattering, where one considers azimuthal transverse single-spin asymmetries for
distributions of hadrons inside jets [177, 374, 375].
In the context of extractions of transversity PDFs also fragmentation functions for same-sidepairs of hadrons with small invariant mass, dihadrons, have been introduced and studied [376–384].
Compared to the Collins effect, dihadron fragmentation functions have the advantage that they may
be defined purely in collinear factorization. The relevant spin-dependent dihadron fragmentation
function exploits a correlation between the transverse polarization of the fragmenting quark and therelative momentum of the two hadrons. In SIDIS with a transversely polarized hadron beam, thedihadron cross section then contains a specific modulation in the azimuthal orientation of the plane
31st May, 2024

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19. Fragmentation Functions in e+ e− , ep, and pp Collisions

containing the momenta of the two hadrons. The coefficient of this modulation is a product of thespin-dependent dihadron fragmentation function and the target’s transversity PDF. The dihadronfragmentation functions may be separately extracted from measurements in e+ e− annihilation, andthe Belle experiment has presented data [385] that have been analyzed theoretically [386, 387]. In
lepton scattering, HERMES [388] and COMPASS [389, 390] have reported data sensitive to thespin-dependent dihadron fragmentation functions, and recently the STAR experiment at RHIC haspresented data in the azimuthal distribution of π + π − pairs produced in pp scattering with one
transversely polarized proton [391]. The results have been successfully used for the extraction oftransversity PDFs [387, 392–394].
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