The NJL-jet model for quark fragmentation functions

Abstract

A description of fragmentation functions which satisfy the momentum and isospin sum rules is presented in an effective quark theory. Concentrating on the pion fragmentation function, we first explain why the elementary (lowest order) fragmentation process q --> q \pi is completely inadequate to describe the empirical data, although the "crossed" process \pi --> q \bar{q} describes the quark distribution functions in the pion reasonably well. Taking into account cascade-like processes in a generalized jet-model approach, we then show that the momentum and isospin sum rules can be satisfied naturally, without the introduction of ad hoc parameters. We present results for the Nambu--Jona-Lasinio (NJL) model in the invariant mass regularization scheme and compare them with the empirical parametrizations. We argue that the NJL-jet model, developed herein, provides a useful framework with which to calculate the fragmentation functions in an effective chiral quark theory. Comment: 21 pages, 7 figures

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The NJL-jet model for quark fragmentation
functions
W. Bentz∗, T. Ito∗, I.C. Cloët†, A.W. Thomas∗∗and K. Yazaki‡
∗Department of Physics, Tokai University, Hiratsuka-shi, Kanagawa 259-1292, Japan
†Department of Physics, University of Washington, Seattle, WA 98195-1560,USA
∗∗School of Chemistry and Physics, University of Adelaide, Adelaide, SA 5005, Australia
‡Radiation Laboratory, Nishina Accelerator Research Center, RIKEN, Saitama 351-0198, Japan
Abstract. A description of fragmentation functions which satisfy the momentum and isospin sum
rules is presented. We concentrate on the pion fragmentation function, taking into account cascade-
like processesina generalizedjet-modelapproach.Numericalresults obtainedin this NJL-jetmodel
are presented.
Keywords: Semi-inclusive processes, Fragmentation functions, Effective quark theories of QCD
PACS: 13.60.Hb, 13.60.Le, 12.39.Ki
INTRODUCTION AND DEFINITIONS
Quark distribution and fragmentation functions are the basic nonperturbative ingredi-
ents for a QCD-based analysis of hard scattering processes [1, 2]. Distribution func-
tions can be extracted by analyzing inclusive processes, and their description in terms
of effective quark theories of QCD has been quite successful [3]. In recent years there
has been a significant effort to extract the fragmentation functions by analyzing inclu-
sive hadron production (semi-inclusive) processes in e+e−annihilation, deep-inelastic
lepton-nucleon scattering and proton-proton collisions [4, 5]. Because of the importance
of the fragmentation functions many attempts have been made to describe them using
effective quark theories [6]. However, in these earlier attempts only the elementary frag-
mentation process q → qπ was considered, and the result did not satisfy the momentum
and isospin sum rules in a natural way. In a recent publication [7] we have shown that
cascade-type fragmentation chains must be included in order to satisfy these sum rules.
The purpose of this paper is to apply the method of the quark jet-model, as formulated
originally by Field and Feynman [8], to calculate the spin-independent fragmentation
functions to pions in the Nambu–Jona-Lasinio (NJL) model [9], which has proven to be
a successful effective theory of QCD.
The spin-independent fragmentation function for q → h is defined by
Dh
q(z) =
z
12
?dω−
2π
eip−ω−/z∑
n
?p(h),pn|ψ(0)|0?γ+?0|ψ(ω−)|p(h),pn?.
(1)
The field operators refer to a quark of flavour q, the symbol p(h) refers to a hadron
h with momentum p, and pn labels the spectator state. If we introduce the Fourier

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FIGURE 1.
quark lines with momentum k are connected by a γ+.
Cut diagram for the elementary fragmentation function dπ
q(z). Here p−= zk−and the two
decomposition of the “good” light-cone quark field operator, we obtain the expression
Dh
q(z)dz =1
6dp−
?
d2p⊥∑
α
?k(α)|a†
h(p)ah(p)|k(α)?
?k(α)|k(α)?
.
(2)
Here p−= zk−for some fixed k−> 0. The creation and annihilation operators refer
to the hadron h, and k(α) labels a quark state of flavour q with momentum k and spin-
color α. The result (2) can be interpreted as the light-cone momentum distributionof the
hadron h in the quark q. This interpretation provides a natural link to models describing
the hadron cloud around constituent quarks, like the familiar pion cloud model.
The momentum and isospin sum rules obtained from Eq. (2) are
∑
h
?1
0
dzzDh
q(z) = 1,
∑
h
?1
0
dzthDh
q(z) =tq.
(3)
The condition which lies at the basis of these sum rules is that the initial quark state
is an eigenstate of the momentum and isospin operators, expressed solely in terms of
hadrons. The physical content of these sum rules is that 100% of the initial quark light-
cone momentum (k−) and isospin (tq) are transferred to the hadrons. (Note that the
definition in Eq. (1) implies an average over the isospin of the soft quark remainder
of a fragmentation chain.)
ELEMENTARY FRAGMENTATION FUNCTION
The elementary fragmentation function for the pion is represented in Fig. 1 as a cut
diagram, and is expressed by
dπ
q(z) =1
2
?1+τπτq
?zg2
π
?
d2p⊥
(2π)3
p2
⊥+M2z2
?p2
⊥+M2z2+(1−z)m2π
?2.
(4)
Here M is the constituent quark mass, gπis the pion-quark coupling constant defined via
the residue of the q¯ q t-matrix at the pion pole, and (τu,τd) = (1,−1), (τπ+,τπ0,τπ−) =
(1,0,−1). This elementary fragmentation function satisfies the following relation:
?1
0
dzdπ
q(z) =1
3
?1+τπτq
?(1−ZQ) ⇒
?1
0
dz∑
τπ
dπ
q(z) = 1−ZQ,
(5)

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where ZQis the residue of the quark propagator in the presence of the pion cloud. (The
corresponding Feynman diagram for the quark self energy is obtained by connecting the
pion lines in Fig. 1.)
Because ZQis interpreted as the probability to find a bare constituent quark without
the pion cloud, Eq. (5) indicates that the elementary fragmentation function is normal-
ized to the number of pions per quark. This is expected from our discussions in relation
to Eq.(2). Because typical values of ZQin models based on constituent quarks are be-
tween 0.8 and 0.9, we see from Eq. (5) that the momentum sum rule?1
Ref. [4] found a pion momentum sum of ≃ 0.74. From this we can anticipate that the
elementary fragmentation function dπ
Although a description of fragmentation functions using only the elementary frag-
mentation processes does not violate any conservation law, it is completely inadequate
for the following reasons: Firstly, there is a large probability (ZQ) that the initial quark
does not fragment. Secondly, if it does fragment the momentumfraction 1−ZQis shared
between the quark remainder and the pion. Both points are in contradiction to the usual
assumption of complete hadronization, which is expressed by the momentum sum rule
in Eq. (3).
0dzz∑τπdπ
q(z)
will be much smaller than typical empirical values. For example, the NLO analysis of
qwill be very small compared to the empirical one.
GENERALIZED PRODUCT ANSATZ FOR QUARK CASCADES
We first introduce an auxiliary quantity dQ
mentation of a quark q to another quark Q. (The variable η is the light cone momentum
fraction of Q w.r.t. to the initial quark q.) This function, which is essentially the same as
the distribution function of Q inside q, is expressed as follows1:
q(η), which describes the elementary frag-
6dQ
q(η) = ZQδ(η −1)+dπ
q(1−η) ≡ ZQδ(η −1)+(1−ZQ)F(η).
(6)
The first term on the r.h.s. of Eq.(6) refers to the process where the initial quark does
not emit a pion. The second term, which refers to the pion emission process with
probability 1−ZQ, is obtained by replacing η → 1−η in the elementary fragmentation
function dπ
By definition, the function F(η) is normalized to 1.
Because the function dQ
we make the following product ansatz for the total fragmentation function Dπ
q(η) of the previous section (apart from a similar replacement for isospin).
q is the splitting function for the elementary process q → Q,
q(z):
Dπ
q(z) =
?1
0
dη1
?1
0
dη2...
?1
0
dηN×6d(η1)·6d(η2)...6d(ηN)
?
N
∑
m=1
δ(z−zm)
?
.
(7)
1For simplicity we do not write out the explicit isospin dependence on the r.h.s. of Eq.(6). The full
expressions can be found in Ref.[7].

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FIGURE 2.
Graphical representation of Eq.(8).
Here zm=η1η2...ηm−1(1−ηm) for m>1, and z1=1−η1. We introduced a parameter
N, which is the maximum number of pions which can be produced by the initial quark.
To see the physical meaning of this ansatz, we use Eq.(6) to rewrite Eq.(7) identically
as follows:
Dπ
q(z) =
N
∑
k=1
P(k)
?1
0
dη1
?1
0
dη2...
?1
0
dηk×F(η1)·F(η2)...F(ηk)
?
k
∑
m=1
δ(z−zm)
?
.
(8)
This expression is represented graphically by Fig. 2. The binomial probability distribu-
tion that k pions are produced for a maximum of N pions is given by
P(k) =
?
N
k
?
ZN−k
Q
(1−ZQ)k,
(9)
and satisfies the normalization condition∑N
N → ∞ the binomial distribution (9) becomes a normalized Gauss (normal) distribution
with the same mean number and variance as the original binomial distribution. In order
to see whether the momentum sum rule in (3) is satisfied by the fragmentation function
(8), we note that in each elementary fragmentation process, a fraction α ≡?zF(z)? is left
to the quark, where ?...? means an integration over z. Therefore the momentum fraction
left to the final quark remainder after emission of a maximum of N pions is given by
k=0P(k)=1. It is well known that in the limit
1−
?1
0
dz∑
τπ
zDπ
q(z) =
N
∑
k=0
P(k)αk.
(10)
In order to satisfy the momentum sum rule in (3), this should vanish. It is easy to see
that (10) vanishes only in the limit N → ∞, because in this limit P(k) becomes a normal
distribution with mean value ?k? = N(1−ZQ) → ∞, and then the functions P(k) and αk
have zero overlap. Therefore, in the limit that the maximum number of mesons which
can be produced by the fragmenting quark is assumed to be infinite, 100% of the initial
quark momentum is transfered to the mesons, and the sum rule in (3) is satisfied.
In the actual calculation it is not necessary to evaluate the products (8) explicitly,
because a compact integral equation for the fragmentation function can be derived[7].
NUMERICAL RESULTS AND DISCUSSIONS
In this Section we present the numerical results for the fragmentation function in the
NJL-jet model, which was developed in the previous section. We will use the same

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LO
0
0.2
0.4
0.6
0.8
1.0
1.2
z Dπ+
u
00.20.40.60.81.0
z
NJL-elementary
NJL-jet (Q2
NJL-jet (Q2= 4GeV2)
Empirical (Q2= 4GeV2)
0= 0.18GeV2)
FIGURE 3.
tationfunction,andthe dottedline is the full fragmentationfunctionin the NJL-jet model.Thesolid line is
the result after LO evolution to Q2= 4 GeV2, and the dashed line is the empirical NLO result of Ref. [4],
evolved to Q2= 4 GeV2.
FavouredfragmentationfunctionzDπ+
u(z). The dash-dottedline is the elementaryfragmen-
LO
0
0.2
0.4
0.6
0.8
z Dπ+
¯ u
00.20.40.60.81.0
z
NJL-elementary
NJL-jet (Q2
NJL-jet (Q2= 4GeV2)
Empirical (Q2= 4GeV2)
0= 0.18GeV2)
FIGURE 4.
model. The solid line is the result after LO evolution to Q2= 4 GeV2, and the dashed line is the empirical
NLOresult ofRef. [4],evolvedto Q2=4 GeV2.Note that theelementaryfragmentationfunctionvanishes
for this case because of charge conservation.
Unfavoured fragmentation function zDπ+
¯ u(z). The dotted line is the result in the NJL-jet
regularization scheme as in Ref. [10], namely the invariant mass, or Lepage-Brodsky
(LB) [11] regularization scheme, with the same values of the constituent quark mass
(M = 300 MeV) and the equivalent 3-momentum cut-off (Λ3= 670 MeV), which is
determined by reproducing the experimental pion decay constant.
As usual, we will associate a low energy renormalization scale (Q2
results and evolve them in Q2by using the QCD evolution equations. We will use the
value Q2
empirical distribution functions at a high energy scale Q2= 4 GeV2. For the evolution
of the fragmentation functions we limit ourselves to the leading order (LO) in αs. For
this purpose, we use the Q2evolution code of Ref. [12] at LO for the distribution
functions, and perform the transformation of the kernels as explained in Ref.[4] or
Ref.[7]. (Unfortunately, a next-to-leading (NLO) evolution code for the fragmentation
functions is not yet publicly available. In this paper we do not attempt a quantitative
comparison with the empirical functions, therefore we leave the NLO calculation to a
future work.)
0) to our NJL
0= 0.18 GeV2, which was determined in Ref. [10] from a comparison with the