Fragmentation functions of the pion, kaon, and proton in the NLO approximation: Laplace transform approach

0.23 MB·

by  M. Zarei

Fragmentation Functions of pion, kaon and proton at NLO approximation: Laplace
Transform approach
M.Zarei a,∗ F.Taghavi-Shahri a,† S. Atashbar Tehrani b,‡ and M.Sarbishei a§
(a) Department of Physics, Ferdowsi University of Mashhad,
P.O. Box 1436, Mashhad, Iran
(b)Independent researcher,
P.O. Box 1149-8834413 Tehran, Iran
(Dated: January 13, 2016)
UsingrepeatedLaplacetransform,WefindananalyticalsolutionforDGLAPevolutionequations
for extracting the pion, kaon and proton Fragmentation Functions (FFs) at NLO approximation.
6 We also study the symmetry breaking of the sea quarks Fragmentation Functions, Dh(z,Q2) and
q¯
1 simply separated them according totheirmass ratio. Finally,we calculate thetotal Fragmentation
0 Functions of these hadrons and compare them with experimental data and those from global fits.
2 Ourresultsshowagood agreementwith theFFsobtained from globalparameterizations aswellas
n with theexperimental data.
a
J
I. INTRODUCTION with DGLAP evolution equations from a starting dis-
2
1 tribution at a defined energy scale [1, 2].
Understanding the basic internal structure of matter RecentlywehaveusedLaplacetransformandprovided
]
h and the quest for the ultimate constituents has always an analytical method to calculate Polarized Parton Dis-
p beenimportantinhighenergyphysics. Thenucleonsare tribution functions (PPDFs)[3, 4]. In the present paper
- the basic building blocks of atomic nuclei. The internal we will apply this new method introduced by Block et
p
structure of the nucleons determines their fundamental al.[5–10] to calculate pion, kaon and proton fragmenta-
e
h properties and directly affect the properties of the nu- tion functions. Therefore, our main task is finding an-
[ clei. Therefore, understanding how the nucleon is built alytical solutions of DGLAP evolution equations to ex-
intermsofits constituentsisanimportantandchalleng- tract Fragmentation Functions (FFs). To do this, we
1
ing question in modern nuclear physics. use the Laplace transform and find analytical solution
v
5 Informationaboutnucleonstructurecomesfromtwoim- of DGLAP equations for FFs. The initial inputs are se-
1 portant processes: The first one is semi- inclusive deep lected from HKNS code to warranty the correctness of
8 inelastic scattering(SIDIS), whose reactionis as follows: our analytical calculations. Finally, comparison of our
2 l+N →l+h+X, and the second one is semi-inclusive FFswiththosefromglobalfits andalsowithexperimen-
0 hadron reaction like: p+p → h+X . However, both tal data confirms the validity of our calculations.
.
1 of the processes require a knowledge of the parton frag-
The paper is organized as follows. In Section 2 we
0 mentation functions (FFs) which describe the transition
reviewthe method ofanalyticalsolutionofDGLAP evo-
6 parton to hadron: parton→h+X.
1 In general, fragmentation is the QCD process in which lution equations for extracting Fragmentation Functions
: based on the Laplace transform. Then, in Section 3 we
v partons hadronize to colorlesshadrons and the fragmen-
utilize thismethodto calculatethe FragmentationFunc-
i tationfunctions, Dh(z,Q2), representthe probabilityfor
X i tions (FFs) of pion, kaon and proton. We also find a
apartoni tofragmentsintoa particularhadronhcarry-
r ingacertainfractionofthepartonenergyormomentum. simple scenario for studying the symmetry breaking in
a the sea quarks FFs. Finally in section 4 we calculated
They area necessaryingredientin calculationofthe sin-
the total fragmentation functions of pion, kaon and pro-
gle hadron inclusive production in any processes like pp¯,
tonandalsocomparedthemwithavailableexperimental
ep, γp and γγ scattering.
data [11] and those from global fits [12–16].
Fragmentation functions cannot be computed directly
fromperturbativeQCDbecause,transitionbetweencolor
partonsintocolorlesshadronsisasoft/long-distancepro-
cess, leading to divergences in the perturbation theory.
PerturbativeQCDdosenotknowanythingaboutexperi-
II. ANALYTICAL SOLUTION OF DGLAP
mentallymeasuredhadrons,butonlyquarks,anti-quarks
EVOLUTION EQUATIONS FOR EXTRACTING
and gluons. Fragmentation Functions can be evolved
FRAGMENTATION FUNCTIONS BASED ON
THE LAPLACE TRANSFORMS
∗ m zarei [email protected]
† f [email protected] The Dokshitzer-Gribov-Lipatov-Altarelli-Parisi
‡ [email protected] (DGLAP) evolution equations [17–19], for the Frag-
§ [email protected] mentation Functions (FFs) can be written as follows
2
[2]: The ⊗ symbol in the above equations refers to the
convolution integral in which the splitting functions in
4π ∂D
ns (z,Q2)=D ⊗ PLO,ns the right-hand side of Eq. (4) are in fact functions of a
αs(Q2)∂ln(Q2) ns qq variable such as x. Using the new variables ν ≡ ln(1)
z z
+αs4(πτ)PqNqLO,ns(cid:21)(z,Q2). (cid:2) (1) ,dewfin≡inglno(fx1z)Dand(z,τQ≡2) =41πF´QQ202(zα,sQ(Q2)′,2)tdhlennQw′2e ahnavdeatlhsoe
ns ns
DGLAPevolutionequationasafunctionofν andτ vari-
4π ∂Ds (z,Q2)=D ⊗ P0 + αs(Q2)P1 (z,Q2) ables as
α (Q2)∂lnQ2 s qq 4π qq
s (cid:18) (cid:19)
α (Q2)
+Dg⊗(cid:18)Pg0q+ s4π Pg1q(cid:19)(z,Q2), (2) ∂∂Fˆτns(v,τ)=ˆ vFˆns(w,τ)e−(v−w) PqLqO,ns(v−w)
0
α (τ) (cid:2)
α 4(Qπ2)∂∂lnDQg2(z,Q2)=Ds⊗ Pq0g + αs4(Qπ2)Pq1g (z,Q2) + s4π PqNqLO,ns(v−w)(cid:21)dw. (6)
s (cid:18) (cid:19)
α (Q2) where
+D ⊗ P0 + s P1 (z,Q2). (3)
g (cid:18) gg 4π qg(cid:19) Fˆns(v,τ)≡Fns(e−v,τ)), (7)
where P0,1 are the leading and next to leading order
ij Becausether.h.sofEq. (6)isanormalconvolutionin-
splittingfunctions. Blocketal. inRefs.[5–7]showedthat
tegral,we canuse the followingpropertyfor the product
using the Laplace transform, one can solve the DGLAP
of Laplace transform :
evolutionequationsdirectlyandextractunpolarizedpar-
ton distribution functions . It is possible to solve an-
alytically the coupled leading and next-to-leading-order v
L Fˆ[w]Hˆ[v−w]dw;s =L[Fˆ[v];s]×L[Hˆ[v];s].
DGLAP evolution equations to extract Fragmentation
ˆ
Functions too. We will give the details here and review (cid:20) 0 (cid:21)
(8)
the method for extracting the Fragmentation Functions
Thenwewillgetasimplesolutionforvalencefragmen-
at NLO approximation.
tation functions in s space:
According to Block’s scenario , by introducing the vari-
able ν ≡ ln(1) into the coupled DGLAP equations,
z
one can turn them into coupled convolution equations fns(s,τ)=eτΦns(s)fns0(s), (9)
in ν space. Now, using a new variable, namely, τ ≡
1 Q2α (Q′2)dlnQ′2, one can use two Laplace trans- in which
4π Q2 s
form´s0from ν space to s space and from τ space to U τ
Φ (s)≡ΦLO(s)+ 2ΦNLO(s), (10)
space. With these two Laplace transforms, the DGLAP ns ns τ ns
evolution equations can be solved iteratively by a set of
convolutionintegralswhicharerelatedtoFragmentation where
Functions at an initial input scale of Q2. Finally, two
0 ΦLO(s)≡L e−vPLO,ns(e−v);s ,
inverseLaplacetransformationswillbackustothe usual ns qq
space (z, Q2)[4, 10]. ΦNnsLO(s)≡L(cid:2) e−vPqNqLO,ns(e−v(cid:3));s . (11)
The Laplace transform(cid:2) of non- singlet spl(cid:3)itting func-
A. Non- Singlet Fragmentation Functions tions, ΦLO(s) and ΦNLO(s) are given in Appendix. A.
ns ns
The τ parameter in Eq. (10) is defined as
2
At the NLO approximation, the fragmentation of va-
lencequarksintohadronsaregivenbyDGLAPevolution
equations as: τ ≡ 1 τα (τ′)dτ′ = 1 Q2α2(Q′2)dlnQ′2,
2 4π ˆ s (4π)2 ˆ s
0 Q2
0
4π ∂D (12)
ns (z,Q2)=D ⊗ PLO,ns
α (Q2)∂ln(Q2) ns qq
s
α (τ) (cid:2) InEq.(9)thefns0(s)functionistheLaplacetransform
+ s PNLO,ns (z,Q2). (4) of valence quark fragmentation functions at initial scale
4π qq
(cid:21) of Q2 = 4.5GeV2. We got them from HKNS code [12].
0
where Finally employing the inverse Laplace transform on Eq.
(9)[10], we can derive the valence quark fragmentation
Dh (z,Q2)=Dh(z,Q2)−Dh(z,Q2) (5) functions in (z,Q2) space.
ns q q¯
3
12 10 3 3
10 u 8 d u s
+π2)(z,QD 468 ADHKSKSKNS +π2D(z,Q) 46 Q2=MNLz2O GeV2 +2k(z,Q)D21 ADHKSKSKNS +k2D(z,Q)21 Q2=MNLz2O GeV2
2 Model 2 Model
0 0 0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
z z z z
10 12 3 3
2)Q 68 u 2Q)108 d 2Q)2 u 2Q)2 d
+π(z,D 4 +πD(z, 46 +k(z,D1 +kD(z,1
2 2
00 0.2 0.4z0.6 0.8 1 00 0.2 0.4z0.6 0.8 1 00 0.2 0.4z0.6 0.8 1 00 0.2 0.4z0.6 0.8 1
8 8 3 3
s c s c
2Q)6 2Q)6 2Q)2 2Q)2
+π(z,D24 +πD(z,24 +k(z,D1 +kD(z,1
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0
z z 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
z z
10
+π2(z,Q)D2468 b +π2D(z,Q) 2468 g +2k(z,Q)D231 b +k2D(z,Q)21 g
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0
z z 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
z z
Figure 1: Pion fragmentation functions at Q2 =M2 and
z Figure 2: Kaon fragmentation functions at Q2 =M2and
Comparison with AKK,DSS and HKNSglobal fits. z
Comparison with AKK,DSSand HKNSglobal fits.
B. Singlet and gluon Fragmentation Functions
written in terms of ν and τ variables, which have been
defined in previous section. Then we will arrive at:
The coupled NLO DGLAP evolutionequationsfor ex-
tracting the singlet and gluon fragmentation functions
∂Fˆ v
are given as follows s(v,τ)= Fˆ (w,τ) Hˆ (v−w)
∂τ ˆ s qq
0
4π ∂D α (Q2) (cid:16)
αs(Q2)∂lnQs2(z,Q2)=Ds⊗(cid:18)Pq0q + s4π Pq1q(cid:19)(z,Q2) +αs4(πτ)Hˆq1q(v−w) dw
+Dg⊗(cid:18)Pg0q+ αs4(Qπ2)Pg1q(cid:19)(z,Q2), (13) +ˆ0vGˆ(w,τ) Hˆgq((cid:19)v−w)
(cid:16)
α (τ)
+ s Hˆ1 (v−w) dw, (16)
4π ∂D α (Q2) 4π gq
g (z,Q2)=D ⊗ P0 + s P1 (z,Q2) (cid:19)
α (Q2)∂lnQ2 s qg 4π qg
s (cid:18) (cid:19)
α (Q2) ∂Gˆ v
+Dg⊗ Pg0g+ s4π Pq1g (z,Q2). (14) ∂τ (v,τ)=ˆ Fˆs(w,τ) Hˆqg(v−w)
(cid:18) (cid:19) 0 (cid:16)
α (τ)
where the singlet fragmentation function is defined as + s Hˆ1 (v−w) dw
4π qg
(cid:19)
v
+ Gˆ(w,τ) Hˆ (v−w)
Dh(z,Q2)= [Dh(z,Q2)+Dh(z,Q2)] (15) ˆ gg
s q q¯ 0
(cid:16)
q=u,d,s,c,b α (τ)
X + s Hˆ1 (v−w) dw, (17)
4π gg
Using the convention zD (z,Q2) ≡ F (z,Q2) and (cid:19)
s s
zD (z,Q2) ≡ G(z,Q2), these coupled equations can be in which we use the definitions:
g
4
2 1.5 10000
P2D(z,Q)01..551 ADHMKSKoSudKNeSl P2D(z,Q)0.51 Q2=MNLdz2O GeV2 H2(z,Q) 100 QN2L=OMz2 π+
F 1
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
z z
0.01
2Q) 1 u 2Q) 1 d 0.01 0z.1 1
z, z, 10000
P(0.5 P(0.5
D D SLD K+
Model
00 0.2 0.4z0.6 0.8 1 00 0.2 0.4z0.6 0.8 1 2Q) 100 HKNS
1 H(z,
0.8 s 1 c F 1
2Q)0.6 2Q)
z, z,
P(0.4 P(0.5 0.01
D D 0.01 0.1 1
0.2 z
0 0 10000
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
z z
P
1 b 1 g 100
P2D(z,Q)000...468 P2D(z,Q)000...468 H2F(z,Q) 1
0.2 0.2
0 0
0 0.2 0.4z0.6 0.8 1 0 0.2 0.4z0.6 0.8 1 0.001.01 0.1 1
z
Figure 3: Proton fragmentation functions at Q2=M2and
z
Figure 4: Total fragmentation functions of pion, kaon and
Comparison with AKK,DSS and HKNSglobal fits.
proton and comparison with experimentaldata from SLD
[11] at Q2=M2. We also compared our results with HKNS
z
global fit.
Hˆ0(v)≡e−vP0(e−v),Hˆ1(v)≡e−vP1(e−v), (18)
ij ij ij ij where
f(s,τ)≡L[Fˆ (v,τ);s], g(s,τ)≡L[Gˆ(v,τ);s] (22)
Fˆ (v,τ)≡F (e−v,τ), Gˆ(v,τ)≡G(e−v,τ), (19) s
s s
The Laplace transform of singlet and gluon split-
At NLO approximation we need two Laplace trans- ting functions, ΦLO,NLO(s) are given in Appendix. A.
formstodecoupletheDGLAP equationsintotwosimple f,g
The second Laplace transform from τ space to U space
equationsthatcanbesolvediteratively. ThefirstLaplace
changes Eq. (20) and Eq. (21) into two simple linear
transform from ν space to s space changes the DGLAP
algebraic equations as
evolution equation to the first order coupled differential
equations as
UF(s,U)−f (s)=
0
α (τ)
∂∂fτ(s,τ)= ΦLfO(s)+ α4s(πτ)ΦNf LO(s) f(s,τ) ΦLfO(s)F(s,U)+ΦNf LO(s)L[ s4π f(s,τ);U]
(cid:18) (cid:19) α (τ)
α (τ) +ΘLO(s)G(s,U)+ΘNLO(s)L[ s g(s,τ);U],(23)
+ ΘLO(s)+ s ΘNLO(s) g(s,τ), (20) g g 4π
g 4π g
(cid:18) (cid:19)
UG(s,U)−g (s)=
0
∂g α (τ)
∂τ(s,τ)= ΦLgO(s)+ s4π ΦNg LO(s) g(s,τ) ΦLO(s)G(s,U)+ΦNLO(s)L[αs(τ)g(s,τ);U]
(cid:18) (cid:19) g g 4π
+ ΘLfO(s)+ αs4(πτ)ΘNf LO(s) f(s,τ), (21) +ΘLfO(s)F(s,U)+ΘNf LO(s)L[αs4(πτ)f(s,τ);U].(24)
(cid:18) (cid:19)
5
where we have which are denoted by f (s) and g (s) respectively, their
0 0
evolved solutions in the Laplace s space are given by [9]
F(s,U)≡L[f(s,τ);U], G(s,U)≡L[g(s,τ);U],
(25)
f(s,τ)=k (s,τ)f (s)+k (s,τ)g (s)
ff 0 fg 0
g(s,τ)=k (s,τ)g (s)+k (s,τ)f (s) (32)
gg 0 gf 0
∂f
L (s,τ);U =UF(s,U)−f (s),
0
∂τ where the k’s in Eq. (32) have been introduced in
(cid:20) (cid:21)
Refs. [5]. These function are given in Appendix. B for
∂g
L (s,τ);U =UG(s,U)−g0(s), (26) the first iteration. The initial inputs are selected from
∂τ
(cid:20) (cid:21) HKNS code [12] at initial scale of Q2 = 4.5GeV2. Fi-
0
To simplify the NLO calculations we use an excellent nally with an inverse laplace transform one can derive
approximation relation a(τ) = αs4(πτ) ≈ a0 + a1e−b1τ, tshpeacsein[1g0le]t. aItndshgoluulodnbferangomteednttahtaiotnoufurnicntiitoianlsiinnpu(zts,Qar2e)
where a0 = 0.0037,a1 = 0.025,b1 = 10.7 [5]. There-
quoted from HKNS code to confirm the validity of our
fore we write the Laplace transform of L[αs(τ)f(s,τ);U]
4π analyticalsolutions. If we reach to the acceptable agree-
and L[αs(τ)g(s,τ);U] which are needed in Eq. (24) and ment between our FFs and FFs obtained by global fits
4π
Eq. (25) as andalsowiththosefromexperimentaldata,thenwecan
be sure that our analytical solution for FFs are correct.
Inthenextwork,thismethodisemployedtoyieldusthe
1
αs(τ) initial inputs via global fit to experimental data.
L[ f(s,τ);U]= a F(s,U +b ),
j j
4π
j=0
X
1
α (τ)
s
L[ g(s,τ);U]= a G(s,U +b ),b =0 (27)
4π j j 0 III. PION, KAON AND PROTON
Xj=0 FRAGMENTATION FUNCTIONS
Now we define
In this section, we present the results of partons frag-
mentation functions of pion, kaon and proton. As we
Φ (s)≡ΦLO(s)+a ΦNLO(s), did in the lastsections, we cancalculate the non-singlet,
f f 0 f
singletandgluonFragmentationFunctions usinganalyt-
Φ (s)≡ΦLO(s)+a ΦNLO(s), (28)
g g 0 g ical solution of DGLAP evolution equations in Laplace
space(s,τ). Then, with aninverselaplacetransformthe
valence , singlet and gluon Fragmentation Functions in
Θ (s)≡ΘLO(s)+a ΘNLO(s), (z,Q2) space are obtained. In this connection we need
f f 0 f
to use the flavor symmetries between different kinds of
Θ (s)≡ΘLO(s)+a ΘNLO(s), (29)
g g 0 g fragmentation functions in pion, kaon or proton at scale
of Q2 as it follows: [12]:
Finally, in s and U space, we arrive at the follow-
ingtwocoupledalgebraicequationsforsingletandgluon
fragmentationfunctionswhichcanbesolvedbyiteration
method described in [4, 5]: Dπ+(z,Q2)=Dπ+(z,Q2)6=Dπ+(z,Q2)
u¯ d s
Dπ+(z,Q2)=Dπ+(z,Q2)
u d¯
[U −Φf(s)]F(s,U)−Θg(s)G(s,U)=f0(s) Dπ+(z,Q2)=Dπ+(z,Q2)
s s¯
+a1 ΦNf LO(s)F(s,U +b1)+ΘNg LO(s)G(s,U +b1) , Dπ+(z,Q2)=Dπ+(z,Q2)
c c¯
(30)
(cid:2) (cid:3) Dπ+(z,Q2)=Dπ+(z,Q2) (33)
b ¯b
−Θ (s)F(s,U)+[U −Φ (s)]G(s,U)=g (s)
f g 0
+a1 ΘNf LO(s)F(s,U +b1)+ΦNg LO(s)G(s,U +b1) , DuK¯+(z,Q2)6=DdK+(z,Q2)6=DsK+(z,Q2)
(cid:2) (cid:3)(31) DdK+(z,Q2)=DdK¯+(z,Q2)
DK+(z,Q2)=DK+(z,Q2)
With the initial input functions for singlet (sumof va- c c¯
lence and sea quarks) and gluon sectors of distributions, DK+(z,Q2)=DK+(z,Q2) (34)
b ¯b
6
quarks A B
u¯ 0.25 4651
Dp+(z,Q2)6=2Dp+(z,Q2) d¯ 0.25 2325.5
u d s 0.45 107.33
Dp+(z,Q2)6=Dp+(z,Q2)6=Dp+(z,Q2) c 0.95 8.7893
u¯ d¯ s b 2.1 2.6577
Dp+(z,Q2)=Dp+(z,Q2)
s s¯
Table I: Parameters A and B in thesea quark
Dcp+(z,Q2)=Dcp¯+(z,Q2) fragmentation functions.
Dp+(z,Q2)=Dp+(z,Q2) (35)
b ¯b
evolution equations are correct and these solutions are
correctlyusedto calculatethe FragmentationFunctions.
A. Symmetry breaking in the sea quarks
Inthe nextsectionwe willcalculatetotalFragmentation
Fragmentation Functions
Functions of pion, kaon and proton to test our calcula-
tions with experimental data.
The total sea quarks fragmentation function is calcu-
lated as follows
IV. TOTAL FRAGMENTATION FUNCTIONS
OF PION, KAON AND PROTON
D (z,Q2)−D (z,Q2)=D (z,Q2) (36)
s ns q¯
Where D (z,Q2) is In this section we intend to calculate the total hadron
q¯
fragmentation function for pion, kaon and proton. We
Dq¯(z,Q2)=2Du¯(z,Q2)+2Dd¯(z,Q2)+2Ds(z,Q2) use FFs obtained in the previous section at Q2 = Mz2.
+2D (z,Q2)+2D (z,Q2), (37) The experiments showed that at this value of Q2, the
c b
interaction between electron - positron accurse via weak
Now to investigate the symmetry breaking of sea quarks interaction. Inthisregionthetotalhadronfragmentation
fragmentation functions we use the fact that heavier function is given as follows [20, 21]
sea quarks can produce hadrons with higher probability.
Therefore,thefractionofdifferentkindofseaquarkscan
1 dσh 1
beproportionaltotheirmassratio. Forexamplewehave =FH(z,Q2)= [2FH(z,Q2)+FH(z,Q2)]
Du¯ ≃ mu. As an example, if we want to calculate the c σtot dz qe2q 1 L
Dc mc
quark fragmentation function, we have (41)
P
b
where we have
m m
D (z,Q2)=2 uD (z,Q2)+2 dD (z,Q2)
q¯ c c
m m
c c
m m 2F (z,Q2)= e2[DH +DH](z,Q2)
+2 sD (z,Q2)+2D (z,Q2)+2 bD (z,Q2),(38) 1 q q
m c c m c q
c c X
α
+ s[C1⊗(DH +bDH)+C1⊗DH](z,Q2), (42)
2π q q q g g
D (z,Q2)
D (z,Q2)≃ q¯ (39)
c (2mmuc +2mmdc +2mmcs +2+2mmcb) FLH(z,Q2)= 2απs e2[CqL⊗(DqH +DqH)+CgL⊗DgH],
This leads to the following general relation: Xq
b (43)
D (z,Q2)= Dq¯(z,Q2), (40) and e2q is the Electroweak charge that is defined as
quark BA
e2 =be2−2e χ (Q2)V V +χ (Q2)(1+V2)(1+V2),
whereB is the massratioanditis constantparameter q q q 1 e q 2 e q
for eachkind of sea quark. The free parameter A should (44)
be extracted from experimental data, however we have b
and the Electroweak parameters are defined as
usedHKNScodeforextractingthe seaquarksFFs to be
sureaboutouranalyticalsolutions. Theresultsarelisted
inTable1. Theresultsforallfragmentationfunctionsfor 1 s(s−M2)
pion, kaon and proton at Q2 = Mz2 are shown in figures χ1(s)= 16sin2θW cos2θW (s−MZ2)2+ZMZ2Γ2Z,
1, 2 and 3 respectively. We also compared our FFs with
1 s2
those from global fits of HKNS, AKK and DSS groups χ (s)= .
[12–16]. They is good agreement between them. The 2 256sin4θW cos4θW (s−MZ2)2+MZ2Γ2Z
results show that our analytical solutions for DGLAP (45)
7
comparedour resultwith those fromHKNS globalfitad
alsowithdatafromSLDexperiment[11]. Theagreement
between data and our model is quite reasonable and it
V =−1+4sin2θ ,
e W means our analytical solutions are correct.
8
V =+1− sin2θ ,
u W
3
4 V. CONCLUSIONS AND REMARKS
V =−1+ sin2θ . (46)
d W
3
We utilized the Laplace transform technique to calcu-
late the Laplace transform of splitting functions and ex-
TheWilsoncoefficientsusedinEq. (42)andEq. (43)are
tracttheFragmentationFunctionsofpion,kaonandpro-
defined as follows [22],
ton at NLO approximation. This technique makes this
facilitythattheanalyticalsolutionfortheFragmentation
Functions (FFs) are obtained more strictly by using the
related kernels and we can control the calculations in a
ln(1−z) 3 1
Cq1(z)=CF (1+z2) 1−z − 2(1−z) better way.
" (cid:18) (cid:19)+ + We also found a simple approach to study the symme-
1+z2 3 3 9 try breaking in the sea quarks FragmentationFunctions.
+2 ln(z)+ (1−z)+ π2− δ(1−z) ,
1−z 2 2 2 Ourresultsarecomparedwiththosefromglobalfitsand
(cid:18) (cid:19) (cid:21) also with experimental data which indicate good agree-
(47)
ments between them.
We have also used the HKNS code for initial input frag-
mentation functions to be sure about our solutions for
DGLAP evolution equations. In a new work, we are at-
tempting to determine the initial input Fragmentation
1+(1−z)2 1−z Functions by Laplace transform technique via a global
C1(z)=2C ln(z2(1−z))−2 (4,8)
g F z z fit. To do this, we have to used available data for total
(cid:20) (cid:21)
fragmentation functions and also multiplicity data.
ACKNOWLEDGMENT
This work is supported by Ferdowsi University of
CqL(z)=CF, (49) Mashhad under grant 2/32653(1394/01/25). F. Taghavi
Shahri thanks to professor Firooz Arash and professor
AbolfazlMirjaliliforreadingthemanuscriptandfortheir
useful comments.
(1−z)
CL(z)=4C . (50) APENDIX A
g F z
We presenthere the resultsfor the Laplacetransforms
The total fragmentation functions of pion, kaon and ofsplittingfunctions,denotedbyΦLO,NLOandΘLO,NLO
proton at the Q2 = M2 scale are shown in Fig. (4). We at the NLO approximation.
z
8 1 1
ΦLO(s)=4− + +2(ψ(s+1)+γ ) (51)
f 3 s+1 s+2 E
(cid:18) (cid:19)
16 2 2 2
ΘLO (s)= n − + , (52)
g f
3 s s+2 s+3
(cid:18) (cid:19)
1 2 2
ΘLO (s)= − + , (53)
f
s+1 s+2 s+3
33−2n 1 2 1 1
ΦLO (s)= f +12 − + − −ψ(s+1)−γ , (54)
g E
3 s s+1 s+2 s+3
(cid:18) (cid:19)
8
C 2 2 4 π2 1.9968 2
ΦNLO =C − A +C − + − − −
nsqq F 2 F (s+1)3 (s+1)2 s+1 3(s+1) (s+2)3 (s+2)2
(cid:18) (cid:19)(cid:18)
3.3246 3.9404 7.1312 3.602 5.8861 2.6484 3.9432 1.2696
+ + − − + + + −
s+2 (s+3)3 s+3 (s+4)3 s+4 (s+5)3 s+5 (s+6)3
14.24 0.2796 20.43 19.77 13.05 6.286 1.997 0.3076
− + + − + + + −
s+6 (s+7)3 s+7 s+8 s+9 s+10 s+11 s+12
4 ln(4) ψ(s +1) ψ(s+1) ψ′(s +1) ψ′(s+1)
−2 − − 2 + 2 + 2 − 2
(s+1)3 (s+1)2 (s+1)2 (s+1)2 2s+2 2(s+1)
(cid:18) (cid:19)
0.9984 16 12s s s+1
− + +(s+2)ln(16)−2(s+2)ψ( +1)+2(s+1)ψ( )
(s+2)3 (s+1)2 (s+1)2 2 2
(cid:18)
s s+1 1.9702 164
+(s+2)2ψ′( +1)−(s+2)2ψ′( ) − +
2 2 (s+3)3 (s+1)2(s+2)2
(cid:19) (cid:18)
284s 188s2 60s3 8s4
+ + + −
(s+1)2(s+2)2 (s+1)2(s+2)2 (s+1)2(s+2)2 (s+1)2(s+2)2
s s+1 s
4(s+3)ln(2)−2(s+3)ψ( +1)+2(s+3)ψ( )+(s+3)2ψ′( +1)−
2 2 2
s+1 1.801 2176 4392s
(s+3)2ψ′( ) − +
2 (s+4)3 (s+1)2(s+2)2(s+3)2 (s+1)2(s+2)2(s+3)2
(cid:19) (cid:18)
3504s2 1408s3 288s4
+ + +
(s+1)2(s+2)2(s+3)2 (s+1)2(s+2)2(s+3)2 (s+1)2(s+2)2(s+3)2
24s5 s s+1
+ +4(s+4)ln(2)−2(s+4)ψ( +1)+2(s+4)ψ( )+
(s+1)2(s+2)2(s+3)2 2 2
s s+1 1.3242 57328
(s+4)2ψ′( +1)−(s+4)2ψ′( ) − +
2 2 (s+5)3 (s+1)2(s+2)2(s+3)2(s+4)2
(cid:19) (cid:18)
146144s 162160s2
+ +
(s+1)2(s+2)2(s+3)2(s+4)2 (s+1)2(s+2)2(s+3)2(s+4)2
103728s3 42144s4
+ +
(s+1)2(s+2)2(s+3)2(s+4)2 (s+1)2(s+2)2(s+3)2(s+4)2
11160s5 1880s6
+ +
(s+1)2(s+2)2(s+3)2(s+4)2 (s+1)2(s+2)2(s+3)2(s+4)2
184s7 8s8
+ −4(s+5)ln(2)
(s+1)2(s+2)2(s+3)2(s+4)2 (s+1)2(s+2)2(s+3)2(s+4)2
s s+1 s s+1
−2(5+s)ψ( +1)+2(5+s)ψ( )+(s+5)2ψ′( +1)−(s+5)2ψ′( ) −
2 2 2 2
(cid:19)
0.6348 s s+7 s s+7
ln(16)−2ψ( +4)+2ψ( )+(s+6)ψ′( +4)−(s+6)ψ′( ) +
(s+6)2 2 2 2 2
(cid:18) (cid:19)
0.1398 s s+9 s s+9
ln(16)+2ψ( +4)−2ψ( )−(s+7)ψ′( +4)+(s+7)ψ′( )
(s+7)2 2 2 2 2
(cid:18) (cid:19)(cid:19)
9
2 2 2 22 4
ΦNLO =C T − − − + + ψ′(s+1) +
nsqq F f 3(s+1)2 9(s+1) 3(s+2)2 9(s+2) 3
(cid:18) (cid:19)
5 5 5 5 3 5
C2 + − + + +
F (s+1)3 (s+1)2 s+1 (s+2)3 (s+2)2 s+2
(cid:18)
2 1
− γ + ψ(s+1)−(s+1)ψ′(s+2)
(s+1)2 E s+1
(cid:18) (cid:19)
2 1
− γ + ψ(s+2)−(s+2)ψ′(s+3)
(s+2)2 E s+2
(cid:18) (cid:19)
1
+4 (ψ(s+1)+γ )ψ′(s+1)− ψ′′(s+1) −3ψ′(s+1)+4ψ′′(s+1)
E
2
(cid:18) (cid:19) (cid:19)
1 5 53 π2 1
+C C − + + + −
A F (s+1)3 6(s+1)2 18(s+1) 6(s+1) (s+2)3
(cid:18)
5 187 π2 67 1
+ − + − (ψ(s+1)+γ )+ π2
6(s+2)2 18(s+2) 6(s+2) 9 E 3
67 π2 11
(ψ(s+1)+γ )+2 − (ψ(s+1)+γ )− ψ′(s+1)−ψ′′(s+1)
E E
18 6 3
(cid:18) (cid:19) (cid:19)
10
40 4 28 146 4 52 94
ΦNLO =C T − + + − + + + +
q F f 9s (s+1)3 3(s+1)2 9(s+1) (s+2)3 3(s+2)2 9(s+2)
(cid:18)
16 112 4 7 3 1 π2
+ + ψ′(s+1) +C2 + − − +
3(s+3)2 9(s+3) 3 F (s+1)3 (s+1)2 s+1 3(s+1)
(cid:19) (cid:18)
3.0032 1 8.3246 3.9404 7.1312 3.602 5.886 2.6484
+ + + − − + +
(s+2)3 (s+2)2 s+2 (s+3)3 s+3 (s+4)3 s+4 (s+5)3
3.9432 1.2696 14.2478 0.2796 20.4376 19.7727 13.056 6.2862
+ − − + + − + −
s+5 (s+6)3 s+6 (s+7)3 s+7 s+8 s+9 s+10
1.9971 0.3075 8 2ln(4) 2ψ(s +1) 2ψ(s+1) ψ′(s +1)
+ − − + + 2 − 2 − 2 +
s+11 s+12 (s+1)3 (s+1)2 (s+1)2 (s+1)2 s+1
ψ′(s+1) 0.9984 16 12s s
2 − + +(s+2)ln(16)−2(s+2)ψ( +1)+
(s+1)2 (s+2)3 (s+1)2 (s+1)2 2
(cid:18)
s+1 s s+1 1.9702 164
2(s+2)ψ( )+(s+2)2ψ′( +1)−(s+2)2ψ′( ) −
2 2 2 (s+3)3 (s+1)2(s+2)2
(cid:19) (cid:18)
284s 188s2 60s3 8s4
+ + + + −
(s+1)2(s+2)2 (s+1)2(s+2)2 (s+1)2(s+2)2 (s+1)2(s+2)2
s s+1 s s+1
4(s+3)ln(2)−2(s+3)ψ( +1)+2(s+3)ψ( )+(s+3)2ψ′( +1)−(s+3)2ψ′( )
2 2 2 2
(cid:19)
1.801 2176 4392s 3504s2
− + + +
(s+4)3 (s+1)2(s+2)2(s+3)2 (s+1)2(s+2)2(s+3)2 (s+1)2(s+2)2(s+3)2
(cid:18)
1408s3 288s4 24s5
+ + +
(s+1)2(s+2)2(s+3)2 (s+1)2(s+2)2(s+3)2 (s+1)2(s+2)2(s+3)2
s s+1 s s+1
4(s+4)ln(2)−2(s+4)ψ( +1)+2(s+4)ψ( )+(s+4)2ψ′( +1)−(s+4)2ψ′( )
2 2 2 2
(cid:19)
1.3242 57328 146144s
− + +
(s+5)3 (s+1)2(s+2)2(s+3)2(s+4)2 (s+1)2(s+2)2(s+3)2(s+4)2
(cid:18)
162160s2 103728s3
+ +
(s+1)2(s+2)2(s+3)2(s+4)2 (s+1)2(s+2)2(s+3)2(s+4)2
42144s4 11160s5
+ +
(s+1)2(s+2)2(s+3)2(s+4)2 (s+1)2(s+2)2(s+3)2(s+4)2
1880s6 184s7
+ +
(s+1)2(s+2)2(s+3)2(s+4)2 (s+1)2(s+2)2(s+3)2(s+4)2
8s8 s s+1
−4(s+5)ln(2)−2(s+5)ψ( +1)+2(s+5)ψ( )
(s+1)2(s+2)2(s+3)2(s+4)2 2 2
s s+1 2 1
+(s+5)2ψ′( +1)−(s+5)2ψ′( ) − (γ + +ψ(s+1)−(s+1)ψ′(s+2))
2 2 (s+1)2 E s+1
(cid:19)
2 1 0.6348 s
− (γ + +ψ(s+2)−(s+2)ψ′(s+3))− ln(16)−2ψ( +4)+
(s+2)2 E s+2 (s+6)2 2
(cid:16)
s+7 s s+7 0.1398 s
2ψ( )+(s+6)ψ′( +4)−(s+6)ψ′( ) + ln(16)+2ψ( +4)−
2 2 2 (s+7)2 2
(cid:19) (cid:16)
s+9 s s+9
2ψ( )−(s+7)ψ′( +4)+(s+7)ψ′( ) +
2 2 2
(cid:19)
1
4 (ψ(s+1)+γ )ψ′(s+1)− ψ′′(s+1) −3ψ′(s+1)+4ψ′′(s+1) +
E
2
(cid:18) (cid:19) (cid:19)