Fragmentation in the phi^3 Theory and the LPHD Hypothesis
aa r X i v : . [ h e p - ph ] N ov Fragmentation in the φ Theoryand the LPHD Hypothesis
Karoly Urmossy ∗ and Jan Rak Wigner RCP of the HAS, 29-33 Konkoly-Thege Miklos Str., Budapest, Hungary,H-1121 University of Jyvaskyla, 9 Survontie Str., Jyvaskyla, Finland, FI-40014
Abstract
We present analytic solution of the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi(DGLAP) equation at leading order (LO) in the φ theory in 6 space-time dimen-sions. If the φ model was the theory of strong interactions, the obtained solutionwould describe the distribution of partons in a jet. We point out that the local parton-hadron duality (LPHD) conjecture does not work in this hypothetical situation. Thatis, treatment of hadronisation of shower partons is essential for the description ofhadron distributions in jets stemming from proton-proton (pp) collisions at √ s = 7TeV and from electron-positron ( e + e − ) annihilations at various collision energies. Weuse a statistical model for the description of hadronisation. Recently, momentum fraction distributions of hadrons in jets stemming from electron-positron ( e + e − ) annihilations and proton-proton ( pp ) collisions have been described bysimple analytic formulas obtained from statisitcal hadronisation models [1, 2, 3]. The ob-tained fragmentation functions (FF) have succesfully been used in a perturbative quantum-chromodynamics (pQCD) improved parton model calculation to obtain the transverse mo-mentum ( p T ) spectrum of charged pions stemming from pp collisions at √ s = 7 TeV [4],assuming log log Q type scale evolution of the parameters of the FFs. This Q scale depen-dence was conjectured based on fits of the newly proposed FFs to AKK-type [5] light-quarkand gluon FFs. However, a solution of the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi(DGLAP) [6, 7, 8] equations and a global fit to measured data is still missing in the case ofthese new FFs. Before addressing the full QCD problem, we examine the situation in the ∗ e-mail: [email protected] φ model in 6 spacetime dimensions, wherethere is only one type of parton.As we shall see, the local parton-hadron duality (LPHD) hypothesis is not sufficientin the φ theory when trying to model strong interactions. In QCD, the distribution ofpartons produced in the branching process inside a jet describes the energy distributionof hadrons stemming from e + e − annihilations (Sec. 7 in [9]). However, in the φ model,parton branching does not produce enough soft ’gluons’ for the description of the lowenergy regime of hadron distributions at parton level. At least that is what the solutionof the DGLAP equation with leading-order (LO) splitting function (presented in Sec. 2)suggests. To solve this problem, we introduce a statistical ’parton-hadronisation’ function,and describe hadronic momentum fraction distributions in jets produced in e + e − and ppcollisions in Sec. 3. φ Theory
In this section, we conjecture that a jet is initiated by an on-shell parton of momentum P init = ( Q, , , Q ), and obtain the longitudinal momentum fraction distribution D ( z, Q )of partons of momenta p = zP init in the jet from the DGLAP equation in the φ theory: dd log Q D (cid:0) z, Q (cid:1) = g Z z dyy P ( y ) D (cid:18) zy , Q (cid:19) . (1)At LO, the coupling g ( Q ) = 1 /β ln( Q / Λ ), and the splitting function [10] is P ( z ) = z (1 − z ) − δ (1 − z ) . (2)Eq. (1) factorizes in Mellin space: dd log Q ˜ D (cid:0) s, Q (cid:1) = g ˜ P ( s ) ˜ D (cid:0) s, Q (cid:1) , (3)where for a function f ,˜ f ( s ) = Z dz z s − f ( z ) , f ( z ) = 12 π ∞ Z −∞ ds z − is ˜ f ( is ) . (4)The sollution of Eq. (3) with t = ln( Q / Λ ) is˜ D ( s, t ) = ˜ D ( s, t ) e ˜ P ( s ) t R t dt ′ g ( t ′ ) = ˜ D ( s, t ) e ˜ P ( s ) b ( t ) , (5)2ith ˜ P ( s ) = 1( s + 2)( s + 1) − , b ( Q ) = 1 β ln (cid:20) ln( Q / Λ )ln( Q / Λ ) (cid:21) . (6)As the initial parton had all of its own momentum, the initial parton distribution D ( z, Q ) = δ ( z − D ( z, Q ) ∼ δ ( z − ∞ X k =1 b k ( Q ) k !( k − k − X j =0 ( k − j )! j !( k − − j )! z ln k − − j (cid:20) z (cid:21) h ( − j +( − k z i . (7)As can be seen in Fig. 1 (top panels), the distribution of partons in the jet Eq. (7)describes hadron momentum fraction distributions only in an intermediate range in thecase of jets stemming from e + e − and pp collisions. This effect might have been predictedfrom the terms in the gluon-to-gluon splitting function in QCD P gg ( z ) ∼ − zz + z − z + z (1 − z ) + c δ (1 − z ) , (8)not present in Eq. (2), and which enhance the production of very soft and very high- z gluons. In this section, we introduce a hadronisation function d ( z ) to describe the probability ofa parton stemming from the branching proccess to produce some hadrons. This case, thehadron distribution in a jet becomes dNdz = Z z dyy D ( y, Q ) d (cid:18) zy (cid:19) . (9)Eq. (5) with initial condition D ( z, Q ) = δ ( z −
1) (that is, ˜ D ( s, Q ) = 1) shows thatthe number of branchings in the parton evolution process has Poissonian distribution, as˜ D ∼ P ( ˜ P b ) k /k !. In z -space, products of ˜ P -s are convolutions, thus Eq. (9) can be writtenas dNdz = ∞ X k =0 b k ( Q ) k ! k Y j =1 Z dy j P ( y j ) d ( y k +1 ) δ ( y · · · y k +1 − z ) . (10)If the splitting function P ( y ) had a single peak at some y ∗ , Eq. (10) could be approximatedby dNdz ≈ ∞ X k =0 [ b ( Q ) P ( y ∗ )] k k ! y ∗ k d (cid:18) zy ∗ k (cid:19) . (11)3 Q pp, Cal 0.168 ± ± ± ± e + e − ± ± b ( Q ) parameter of Eq. (13) obtained from fits shown inFig. 1.To choose a simple model for the hadronisation function d ( z ), we conjecture that thisprocess is dominantly determined by the phasespace of the produced hadrons. Argumentssupporting such a conjecture can be found in [11, 12, 13, 14, 15]. Furtheremore, we assumethat hadrons are collinear to their parent parton, and we neglect masses. Thisway, themomentum fraction distribution of one hadron out of n hadrons stemming from the sameparton becomes a one-dimensional microcanonical distribution [1, 2, 3]: d n ( z ) ∼ (1 − z ) n − . (12)Though the maximum of the splitting function at LO in the φ theory (Eq. (2)) at y ∗ = 1 / dNdz ≈ ∞ X k =0 b k ( Q ) k !2 k (cid:16) − k z (cid:17) n − . (13)Fig. 1 (central panels) shows that Eq. (13) provides a reasonably good fit of dN/dz and dN/dx distributions of hadrons in jets of various energy, stemming from pp [16, 17] and e + e − [18, 19, 20, 21, 22, 23, 24] collisions. As expected, the model fails to reproduce hadronspectra at very low z , due to the lack of soft ’gluon’ radiation in the φ theory. Besides,at z ≈
1, the curve of Eq. (13) becomes uneven, as an artefact of the replacement of thecontinuous integral in Eq. (9) by the integrand taken at y ∗ = 1 / n = 5 has been used, while the obtained values ofthe b ( Q ) are shown in Fig. 1 (bottom panel) with Q = √ s for e + e − and Q = P jet for ppcollisions. The scale dependence of the b parameter was fitted by the LO φ theory resultEq. (6) with Λ = 0 . β and Q coincide in case of jetsstemming from e + e − and pp collisions with calorimetric jet reconstruction [16]. However,these values differ in the case of pp data with track-based jet reconstruction [17]. Acknowledgement
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C27 , 467 (2003), hep-ex/0209048.6igure 1: Momentum fraction distributions of charged hadrons in jets stemming from pp( left ), and e + e − ( right ) collisions. Data are compared to calculated results for partonsEq. (7) ( top ) and final state hadrons Eq. (13) ( center ). Obtained values of the b ( Q )parameter are fitted with Eq. (6) ( bottombottom