Table Of ContentDihadron Fragmentation Functions in the
NJL-jet model
A. Casey, A. W. Thomas, H. H. Matevosyan
CoEPPandCSSM,SchoolofChemistryandPhysics,UniversityofAdelaide,5005
2 Abstract. TheNJL-jetmodelprovidesaframeworkforcalculatingfragmentationfunctionswith-
1
out introducing ad hoc parameters. Here the NJL-jet model is extended to investigate dihadron
0
fragmentationfunctions.
2
Keywords: Dihadronfragmentationfunctions,NJL-jetmodel
n
PACS: 12.39.Ki,13.60.Hb,13.60.Le
a
J
6
INTRODUCTION
]
h
Deep inelastic scattering (DIS) has proven to be an invaluable source of information
p
- aboutthestructureofthenucleon[1],providinginsightintopartondistributionfunctions.
p
Semi-inclusivedeep-inelasticscattering(SIDIS)hasextendedourunderstandingfurther,
e
h allowingfurtheraccesstothetransversestructureofnucleons.
[
With a deeper understanding of the fragmentation functions, these studies will con-
1 tinuetoprovideuswithvaluableinformationonnucleonstructure.Fragmentationfunc-
v
tions are an important theoretical tool in the investigation of scattering reactions. This
1
5 has led to the development and study of the NJL-jet model [2, 3, 4]. The NJL-jet model
3 builds on the Field-Feynman Quark-jet model (FFQJM)[5], by using an effective chiral
1
quarkmodelframeworkinwhichcalculationsofbothquarkdistributionandfragmenta-
.
1
tionfunctionscanbeperformedwithoutintroducingadhocparameters.Pionfragmenta-
0
2 tion functions in the NJL-jet model were calculated in Ref. [2]. The NJL-jet model was
1 then extended to include strange quark contributions, and kaon fragmentation functions
:
v were obtained in Ref. [3] and in Ref. [4] the fragmentation functions to vector meson
i
X andnucleon-antinucleonswerecalculated.
r
a
FRAGMENTATION FUNCTIONS AND THE NJL-JET MODEL
In Fig. 1, it is shown how a quark can produce a cascade of hadrons, producing what is
interpreted as jet events in Deep-Inelastic Scattering (DIS). It is important to note that
within the model the emitted hadrons do not interact with the other hadrons produced
in the quark jet. An integral equation for the quark cascade process shown in Fig. 1 is
derivedintheNJL-jetmodelofRef.[2].Theintegralequationforthetotalfragmentation
functionis:
(cid:90) 1 dy z
Dm(z)dz=dˆm(z)dz+∑ dˆQ( )Dm(y)dz, dˆQ(z)=dˆm(1−z)| (1)
q q y q y Q q q m=qQ¯
z
Q
) )
(z 1 (z 2
h 1 h 2
q Q Q’ Q’’
FIGURE1. QuarkCascade
The first term on the right hand side is the driving function and represents the prob-
ability of creating a meson m carrying momentum fraction z to z+dz from the initial
quark(i.e.h =m).Thesecondtermrepresentstheprobabilityofcreatingthemeson,m,
1
further in the quark decay chain(i.e. h = m). The method used to solve this equation
2
approximatesintegralsassumsoverdiscretizedvaluesofz,sothatDm(z)anddˆm(z)can
q q
be written as vectors and the integral term (without Dm(z)) can be written as a matrix,
q
where the elements are their values at z=z. In index form the elements of the vectors
i
can be written as Dm = f +g Dm . Writing in vector form and rearranging to solve for
qi i ij qj
(cid:126)Dm,weobtain(cid:126)Dm =(I−g)−1(cid:126)f,where f istakentobethedrivingfunctionandgisthe
q q
matrixformoftheintegrandwithoutthefragmentationfunction.
DIHADRON FRAGMENTATION FUNCTIONS
The extension to dihadron fragmentation functions in the NJL-jet model is now consid-
h ,h
ered. Dihadron fragmentation functions D 1 2(z ,z ) correspond to the probability of
q 1 2
observing two hadrons, h and h , with light-cone momentum fractions z and z , re-
1 2 1 2
spectively, fragmenting from the initial quark q. An illustration of how a quark cascade
can produce two observed hadrons in the NJL-jet model is shown in Fig. 1, where h
1
andh aretheobservedhadrons.
2
h ,h
The integral equation for the dihadron fragmentation function D 1 2(z ,z ) con-
q 1 2
structedfromFieldandFeynman’sEqs(2.43a)-(2.43d)ofRef.[5]is:
Dh2( z2 ) Dh1( z1 )
Dh1,h2(z ,z ) = dˆh1(z ) q1 1−z1 +dˆh2(z ) q2 1−z2 (2)
q 1 2 q 1 1−z q 2 1−z
1 2
(cid:90) 1 dη z z
+∑ dˆQ(η)Dh1,h2( 1, 2), q→h +q ; q→h +q ,
η2 q Q η η 1 1 2 2
z +z
Q 1 2
where dˆh(z) and dˆQ(η) are the elementary splitting functions of the quark q to the
q q
corresponding hadron h and quark Q. The first term corresponds to the probability of
producinghadronh fromquarkqatthefirststepinthecascade,followedbyhadronh
1 2
eitherdirectlyafterwardsorfurtherdownthecascadechain.Similarforthesecondterm
with hadrons h and h switched in order. The third term corresponds to the probability
1 2
ofbothhadronsbeingproducedafterthefirststepofthecascade.
CollectingthefirsttwotermsofEq.(2)together,atermanalogoustothedrivingfunc-
tionofthesinglehadronfragmentationfunctionisfoundforthedihadronfragmentation
function. Due to the discretization method used to solve the equation, the division by η
causes the values of Dh1,h2(z1,z2) obtained to not correspond to the discrete values in
Q η η
thechosenregionforz andz .Tofixthisissue,thevaluesofz andz werediscretized
1 2 1 2
in the region [0,1] and new variables were set as ξ = z1 and ξ = z2. Values for ξ and
1 η 2 η 1
ξ were discretized in the same way as z and z in the region [0,1]. Rewriting the inte-
2 1 2
gral part of Eq.(2) in terms of these new variables, the dihadron fragmentation function
becomes:
Dh2( z2 ) Dh1( z1 )
Dh1,h2(z ,z ) = dˆh1(z ) q1 1−z1 +dˆh2(z ) q2 1−z2 (3)
q 1 2 q 1 1−z q 2 1−z
1 2
(cid:90) z1 (cid:90) z2
+∑ z1+z2 dξ z1+z2 dξ δ(z ξ −z ξ )dˆQ(z /ξ )Dh1,h2(ξ ,ξ )
1 2 2 1 1 2 q 1 1 Q 1 2
z z
Q 1 2
Mathematica was used to solve for the single hadron and dihadron fragmentation
functions. The number of points used for the dihadron cases was 200, while the single
hadron case was 500 points. Fig. 2 shows the results for the dihadron fragmentation
function Dπ+π−(z ,z ) solved for 50, 100, 150 and 200 points at z = 0.1(left) and
u 1 2 1
z = 0.5(right). For the z = 0.5 case there is very good agreement for each choice
1 1
of the number of points. For the z =0.1 case, sufficient convergence of the function is
1
obtained by 200 points, and so this is the number of points chosen to be used. To solve
for the delta function in the integral term the values of z and z are looped over first,
1 2
and then ξ is looped over(z and z share the same range of values as ξ and ξ , and
1 1 2 1 2
are discretized in the same way). This leaves ξ with a value obtained from the delta
2
function that may not be one of the discretised values. The values of the DFFs at ξ
2
wereobtainedusinglinearinterpolationfromneighbouringdiscretevalues.
0. 0.1 0.2 0.3 0.4 0.5
0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.1
(cid:45)Πz,z1200..0068 (cid:72)(cid:76)Numbe115r500o00fPoints 00..0068(cid:45)Πz,z12000...456 (cid:72)(cid:76)Numbe115r500o00fPoints 000...456
(cid:43)Π 0.04 200 0.04(cid:43)Π 0.3 200 0.3
Du Du0.2 0.2
z20.02 0.02z20.1 0.1
0. 0. 0. 0.
0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0. 0.1 0.2 0.3 0.4 0.5
z z
2 2
FIGURE 2. Dihadron fragmentation functions at the model scale for h =π+, h =π− for varying
1 2
numberofzpointsat(left)z =0.1and(right)z =0.5
1 1
RESULTS
Using the data obtained from solving for the dihadron fragmentation functions, it is
h h
possible to compare the solutions (z D 1 2) to the driving function (the first two terms
2 q
of Eq. 2). In Fig. 3 it can be seen that the driving functions contribute almost all of the
DFF.Thefigureontheleftshowsthatforaninitialupquark,thereisaslightdifference
betweentheDFFandthedrivingfunctionatlowvaluesofz foralowvalueofz (=0.1).
2 1
The strange quark has a zero value driving function, and therefore is completely made
up of the integral term. The difference between the DFFs and the driving functions for
each case is of the same order of magnitude as the strange quark DFF. This difference
is only visible for the initial up quark case when the driving function becomes small,
whichoccursatlowvaluesofz .
1
0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0. 0.1 0.2 0.3 0.4
5 5 0.8 0.8
s x1000 s x1000
0.7 0.7
4 d 4 d
(cid:76) 0.6 (cid:76) 0.6
2 2
z,z1 3 D(cid:72)f:d (cid:76) 3 z,z1 0.5 D(cid:72)f:d (cid:76) 0.5
(cid:45)Π (cid:72)u (cid:45)Π 0.4 (cid:72) u 0.4
(cid:43) (cid:43)
ΠDq 2 Df:u 2 ΠDq 0.3 Df:u 0.3
z2 1 1 z2 0.2 0.2
0.1 0.1
0 0 0. 0.
0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0. 0.1 0.2 0.3 0.4
z z
2 2
FIGURE3. Dihadronfragmentationfunctionsatthemodelscaleforh =π+,h =π−at(left)z =0.1
1 2 1
and(right)z =0.5
1
CONCLUSIONS
In this document, results have been presented for dihadron fragmentations calculated
within the NJL-jet model. The difference between the driving functions and the full
DFFs was shown to become apparent only at low z . The integral term for each case is
1
of the same order of magnitude, though they are not equal. The difference is visible for
theinitialupquarkcaseonlywherethedrivingfunctionbecomessmall.
ACKNOWLEDGMENTS
This work was supported by the Australian Research Council(Grant No. FL0992247)
andbytheUniversityofAdelaide.
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3. Matevosyan, Hrayr H. and Thomas, Anthony W. and Bentz, Wolfgang, Phys.Rev. D83, 074003
(2011),arXiv:1011.1052[hep-ph].
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