P H Y S I C A L R E V I E W D V O L U M E 2 2 , N U M B E R 5 1 S E P T E M B E R 1 9 8 0

Q evolution of multihadron fragmentation functions

U. P. Sukhatme and K. E. Lassila Ames Laboratory-Department of Energy and Department of Physics, Iowa State University, Ames, Iowa 50011

(Received 10 March 1980)

In order to characterize jet structure more completely than is possible with the single-hadron fragmentation functions of partons, we deduce the evolution (in Q 2 , equations for the fragmentation functions of a parton to fragment into an arbitrary number N of hadrons. The solution to the evolution equations is explicitly displayed in terms of multiple moments for the N-hadron fragmentation functions. Our procedure is simpler than using the rules of the jet calculus. We also show how the Q dependence of these fragmentation functions can be understood in a recursive cascade-model framework in which both the momentum and the amount by which a parton is off its energy shell (as measured by Q 2, are reduced at each branching step during jet formation. This reformulation suggests an alternative way for solving the evolution equations in which moment inversion is not required.

1. INTRODUCTION proved useful in character izing jet formation in the nonperturbative regime, one has a unified de-

The experimental study of jets of hadrons pro- scr ipt ion of jets a t any Q2. The final sect ion pro- duced in electron-positron annihilation, neutrino vides a brief summary of this paper . (and antineutrino) reactions, and hadron-hadron collisions has inspired considerable theoretical 11. EVOLUTION EQUATIONS FOR MULTIHADRON research into jet propert ies . F r o m our p resen t FRAGMENTATION FUNCTIONS

level of understanding of partons and quantum chromodynamics (QCD), we believe these hadron jets contain information on the parton confinement mechanism. Such information can be extracted when character izat ion of jet propert ies i s complete. T h i s requ i res m o r e knowledge than that available f rom the single-particle fragmentation functions Dj,,,&, Q2). These functions give the probability of finding in a jet a single hadron h with moment- um fract ion x of that possessed by the jet-initiating p a r t o n j (a quark, antiquark, o r gluon) which i s off m a s s shel l a s specified by Q2. The evolution in Q2 of this probability is given by equations de- duced by Owens' and Uematsu,' which a r e analogous to the Al ta re l l i -Par i s i Q2-evolution equations3 f o r s t ruc ture functions.

T o aid in the complete character izat ion of jets when da ta becomes m o r e complete in the future, we deduce in this work the corresponding evolution equations fo r the fragmentation of parton j into N hadrons. These hadrons c a r r y fractions x,, x,, . . . ,x, of the momentum c a r r i e d by the initial par ton j at s o m e Q2. In Sec. I1 we d i scuss these evolution equations fo r multihadron fragmentation functions i n detail , beginning with a br ief review of the equations fo r single-hadron fragmentation.

In Sec. I11 we presen t explicit solutions of these evolution equations. The relat ion of our work to previous work i s a lso discussed.

In Sec. IV, we show that s ince jet formation pro- ceeds via repeated branching, it is possible to re - formulate the Q2-evolution equations in t e r m s of a recursive cascade model.¶ Since such a model has

Before describing the general c a s e of a parton fragmenting into N hadrons, l e t u s briefly review the formulation of the evolution equations for single-hadron fragmentation functions D,,,. This will es tabl ish the notation and indicate clear ly how the generalization to N hadrons i s made.

The Q2-evolution equation for a par ton i frag- menting into a hadron h with momentum fraction x of that of the initial par ton is "'

where

for N, colors and f flavors, with ff, and A set t ing the s t rength and sca le of the QCD running coupling constant. In Eq. (1) the parton indices i, j take on (2f+ 1) values (q,,q,, . . . , qf, F1,T2,. . . ,%, G ) cor- responding to f quark f lavors and the gluon G . T h e r e is an implied summation over the repeated index j and this summation convention i s used throughout the paper. The P,,(Y) a r e s tandard p a r - ton vertex functions3 for the p rocess i ( p ) - j k p ) , with p the momentum of parton i. Equation (1) is very s i m i l a r to the Altarel l i -Paris i equation for the evolution of s t ruc ture functions. The t e r m s on the right-hand s ide can bes t be visualized pictor- ially a s in Fig. 1 for the c a s e of i =quark fragmen- tation. The two d iagrams depict the initial quark branching into a quark and a gluon, and the ob-

e 2 E V O L U T I O N O F M U L T I H A D R O N F R A G M E N T A T I O N F U N C T I O N S

FIG. 1. Diagrams contributing to single-hadron quark fragmentation. In this and all successive diagrams, heavy lines represent hadrons, the small circle is the fragmentation function, and the thin lines represent par- tons (quarks or gluons).

se rved hadron h coming from ei ther the quark [Fig. l (a)] o r the gluon [Fig. 1 (b)]. Likewise, Fig. 2 represen ts the two t e r m s on the right-hand s ide of Eq. (1) f o r the gluon fragmentation (i =GI.

FIG. 2 . Diagrams contributing to single-hadron gluon fragmentation.

F o r the fragmentation of a parton into two had- rons, the d iagrams corresponding t o those in Figs. 1 and 2 can readily be sketched for all possible ways in which i, a quark (Fig. 3) o r gluon (Fig. 4), can fragment. Therefore, the evolution equations f o r two-hadron fragmentation functions Di+h,h2 can then b e deduced by analogy with Eq. (1):

where the hadrons h, and h, have momentum fract ions x, and x,, respectively, of that of the initial parton. Also, o u r notation consistently will suppress the symbol f o r summation over repeated parton indices j, b,, b,. The f i r s t t e r m on the right-hand s ide corresponds to a branching i - j (x t ) followed by p a r t o n j f rag- menting into hadrons h, and h, [ s e e Figs. 3(a), ?(b), 4(a), and 4(b)]. The second t e r m corresponds to a branching vertex i - b,(xl) + b,(l - x t ) given by Pi,,l,2(x1) followed by single-particle fragmentation [ s e e Figs. 3(c), 3(d), 4(c), and 4(d)] of each parton to produce the hadrons h, and h,. The l imi t s on the x' inte- grat ion in Eq. (3) come f rom requiring a fragmenting parton to have at least a s much momentum a s the fragmentation products.

As i l lustrated i n Fig. 5, the d iagrams for one- o r two-hadronfragmentation generalize readi ly to N-hadron fragmentation. The evolutionequationsfor theseN-hadronfragmentationfunctions, Di+hl,... , h N ( ~ l , x2, . . . ,zN; y), with hadrons h,, h,, . . . , h, having momentum fract ions x,, x,, . . . , x, of the initial par ton momentum, s t raightforwardly follow:

y4; q<2<h1 q<hl

G 9 G 9 "2 "2

(a ) (b) (c 1 ( d ) FIG. 3. Various contributions to the two-hadron fragmentation function for quarks.

U . P . S U K H A T M E A N D K . E . L A S S I L A

FIG. 4. Various contributions to the two-hadron fragmentation function for gluons.

where

M,,,=&v (N even)

= + & + I ) (N odd). (5

Note that Eq. (4) i s also c o r r e c t f o r the production of N partons, if the labels h,, . . . , h , a r e inter- preted a s parton indices. In fact, at the pavton level both Eq. (4) and the N-parton evolution equa- tions given by ~ i r s c h n e r ~ a r e equivalent-the only difference being that in 9 r procedure the funda- mental branchings P j i , Pi,, , occur f i r s t , not

1 2 last. However, this difference is crucial when one is considering N-hadyon fragmentation func- tions since our Eq. (4) is applicable, but Kirsch- ner 's resul ts a r e not.

To go f rom the parton level to final-state had- rons, it i s necessary to convert the partons into color singlets. Th is could occur , e.g., by fur ther fragmentation, a s in the cascade model,4 o r by r e ~ o m b i n a t i o n . ~ It is preferable to d i scuss frag- mentation functions into hadrons direct ly since these a r e experimental observables .

111. SOLUTION OF THE EVOLUTION EQUATIONS

A. Single-hadron fragmentation functions

Let u s f i r s t review how the evolution equations f o r single-particle fragmentation [ E ~ . (111 a r e solved. The moments of the single-particle frag- mentation functions and of the anomalous dimen- s ions a r e defined, respectively, by

Di.+,(n; Y)= i l d ~ ~ " - l ~ i + h ( ' ; Y), (6 )

AYi ' ~ l d x i * . r " - l ~ ~ ~ ( x ) . (7)

Explicit expressions f o r the moments of these vertex functions P j i b ) a r e

1 A;,@, 3 z = 6ij.4!q= h i j N c 2 - 2Nc (-:+- - j=2

n 2 + n + 2 AiiG = 2n(n + 1 ) & + 2 ) '

NC2- 1 (n2+tz+2) (8) AZa,=ABv= 2*TC n(nZ-l) 9

f 1 2 A ; ~ = N , [- -+- 2 -- - 2 C;].

6 n ( n - l ) + ( n + l ) ( n + 2 ) 3N, 1-23

As written, Eqs. (8) a r e fo r integer n , but the analytic continuation to noninteger values follows f rom

F o r l a t e r u s e we presen t (Fig. 6) graphs of the n dependence of the anomalous dimensions [ E ~ s . @)I .

Taking moments of Eq. (1) gives

and

~ ~ . + , ( n ; Y) =[e 'Y-Yo)Anlj i~j .+h(n; Yo), (1 1 )

where Dj,,(n; Yo) is the boundary value of the frag- mentation function. This solution is valid f o r any mat r ix A. However, s ince A i s flavor independent [ s e e Eq. (811, the mat r ix exp[(Y - Y,l4" i s par - ticularly simple8 ( y =Y - Yo),

The quantities u:, X:, and A", appear in Eqs. (12), s ince A, and A,, a r e eigenvalues of the anomalous- dimension mat r ix A , and a, appear in the eigen- vectors . They a r e plotted in Fig. 7 and a r e given

by

CT: = (A& -Aze) -- -A,"J2+ 8fArGABo]1'2 4fA 'Aa , 3

Combining Eqs. (11) and (12) gives explicit known expressions f o r the gluon ( G ) , singlet (S), and nonsinglet (NS) fragmentation functions," 218-10

e.g., fo r a specific quark (qi),

22 - Q 2 E V O L U T I O N O F M U L T I H A D R O N F R A G M E N T A T I O N F U N C T I O N S 1187

Dqi,,(n; Y) = e A & ( Y - Y o ) ~ qi+hc; y o ) + D ~ + h ( n ; Yo] ( e * c ( ~ - ~ o ) - C ~ ! ( ~ - ~ o ) 2f (a: - 5:) )

The fragmentation functions in x space can be obtained by moment i n v e r s i ~ n . ~ * ' ~ ' " Note that Eq. (14) be- comes considerably s i m p l e r a t l a r g e Y. F o r example, if n i s l a rge (i.e., x - I), then A: =A:, > A1 (see Figs. 6 and 7) and at l a r g e Y the t e r m s involving e&Al(y -Yo)] do not contribute. This leads to the x - 1 l imi t s of q, G, NS, and S fragmentation functions discussed in Refs. 1, 5, and 12.

B. Multihadron fragmentation functions

The solution of the evolution equations (3) and (4) f o r the two-hadron and multihadron fragmentation func- tions c a n a l so be obtained by taking moments. In general , the moments of an N-part ic le fragmentation function a r e defined by

Before proceeding to the solution f o r the multihadron fragmentation functions, we f i r s t examine the two- hadron functions. We take moments of Eq. (3), using N = 2 in Eq. (15), to obtain a differential equation f o r the double moments,

d - dY Di-hlh2hl, n,; Y) = G ( n 1 + n 2 ) j i ~ j + h l , 2 ~ l , a,; y ) + pblb2* n l n z Db1+h1671; Y)Db2-h2(n2; 9

a r e knoun functions ( see Table 111 in Ref. 5). These double-moment equations (16) a r e a s e t of f i r s t -o rder , l inear-differential equations. Hence, the moments of the two-hadron fragmentation functions a r e

D,+fil,2bl,n2; y ) = l ; d y [ e ( y - y ' * n l + n ~ .Pb n i n f ' D b l + h l ~ l ; ~ ) D b z - h z ~ z ~ ~ )

+ ~ e ( ~ - ~ o ) ~ n l + n 'I jiDj-~,n,blt nzi YO). (1 8)

Note that Eq. (18) provides explicit expressions f o r quark and gluon fragmentation into two hadrons, s ince the mat r ix exp[ (Y - Y,)A~'+"~] is given by Eqs. (12) and D,,,h, y ) by Eq. (11).

Proceeding in a s i m i l a r manner we can solve Eqs. (4) in the general c a s e f o r the N-hadron fragmentation functions. The resul t (after s o m e manipulations) is

where evolution equations. This procedure, in our opin- ion, i s considerably s impler than using the ma- nL =nl + n 2 + * . * + n M , n, =n,+,+-*.+n,. (20) chinery of the jet c a l c u l ~ s . ~

This notation is obviously suggested by Fig. 5, and Eq. (19) can b e readily checked by the r e a d e r f o r N = 3 . IV. CONNECTION BETWEEN THE EVOLUTION

EQUATIONS AND A CASCADE MODEL In Eq. (19) we have presented the solutions f o r

N-particle fragmentation functions by a s t raight- The c u r r e n t view of jet formation is that one forward generalization of the single-particle Q2- s t a r t s with a jet-initiating parton i which has a

U . P . S U K H A T M E A N D K . E . L A S S I L A

FIG. 5. Pictorial representation of a typical contribu- tion to the N-hadron fragmentation function of a parton i.

of lower momentum and l e s s f a r off the energy FIG. 7. The n dependence of the quantities a, and the shel l (lower Y). Eventually, a point Y = Yo is eigenvalues A* defined in the text.

reached13 where one e n t e r s the nonperturbative

la rge momentum P and is f a r f r o m i t s m a s s shel l -1 2

stage of jet evolution and the partons convert into observable hadrons. This l as t s tage, which is not understood f rom a fundamental viewpoint, can be well-described phenomenologically by a recursive cascade model based on repeated branching (or breakups) quark - meson+quark , gluon- meson + gluon, and gluon - quark + a n t i q ~ a r k . ~

Since the perturbat ive QCD stage of jet evolution is a branching process , i t mus t be possible to descr ibe the p r o c e s s in a cascade-model f rame- work. T h i s connection can be made manifest by integrating the evolution equations f o r single-had- ron fragmentation functions [ E ~ . (I)] for Y 2 Yo. Thus,

\ \

where

(large Y o r Q2). Initially, per turbat ive QCD is -1 4 I I I I I I I I I

applicable and the parton branches repeatedly via 01'02 4 6 8 10 12 14 16 18 20 22 24

the ver t i ces P, ,k ) . T h i s produces many partons n

n FIG. 6. The n dependence of the QCD anomalous di-

mensions A,,, Aac, AGq, AGG.

The second t e r m in Eq. (21) shows that the pertur- bative QCD stage of jet evolution is a cascade pro- c e s s in which both the momentum and the anqount off the energy shel l a r e reduced at each breakup, s ince ~ , ~ b , y ) may b e identified a s the probability f o r i(p, Y) - j &, y ). F o r Y < Yo, one can use the s tandard cascade model in which only the moment- u m is reduced in each breakup (via the momentum- sharing f ~ n c t i o n ) . ~ Thus, one has a unified way of describing jet formation in both the perturbat ive and nonperturbative regimes of jet formation. This should be no s u r p r i s e , if jet evolution is indeed a branching process .

Note that in the leading-logarithm approxima- tion, the vertex P, ,&, y ) is part icular ly s imple s ince i t depends only on the variable x [ s e e Eq. (22)]. Going beyond this approximation will in general produce a y dependence.14 However, if the branching charac te r i s t i cs of jet formation re- main t rue, then integral Eq. (21) will s t i l l be valid, and can be used f o r determining jet propert ies .

Finally, l e t u s note that it is possible to solve Eq. (21) by successive iteration. This procedure is convergent, and often convenient, since the problem of taking and inverting moments is avoided. The solution is

where

This method is quite suitable f o r numerical work

22 - Q' E V O L U T I O N O F M U L T I H A D R O N F R - A G R I E N T A T I O N F U N C T I O N S 1189

since the multiple integrations involved can be partons produced in the perturbat ive QCD stage well handled by Monte Car lo techniques.15 of jet evolution a r e recombined at QO2. In such

a scheme, the Q2 evolution of D,,, i s quite differ- V. SUMMARY AND DISCUSSION ent f rom Eq. (14), s ince i t i s governed by the two-

In this paper we have presented the Q2-evolution equations f o r the fragmentation functions descr ib - ing parton (quark o r gluon) fragmentation into N hadrons. These equations were solved in t e r m s of multiple moments of the N-hadron fragmentation functions. Such functions a r e , in fact, measurable quantities which can s e r v e to charac te r ize prop- e r t i e s of jets and complement the information available f rom the usual (single-hadron) fragmen- tation functions. Data present ly in existence do not have sufficient s ta t is t ical significance o r range

parton distributidns [ E ~ . (18)lwith the s i b s c r i p t h, interpreted a s discussed in Sec. 11.

We have also demonstrated that the Q 2 depen- dence of the multihadron fragmentation functions could be readily understood in t e r m s of a recur - s ive cascade model f o r jet formation. Each s tep in the branching process degrades both momentum and the amount by which a parton is off i t s energy shel l a s measured by Q2. Consequently, i terat ion of these cascade-type equations is suggested a s a pract ical alternative to moment inversion.

in Q2 to t es t whether o r not the multihadron frag- ACKNOWLEDGMENTS

mentation functions evolve according to o u r QCD predictions; it is likely that future da ta may.I6 The authors would l ike to thank Dr. L. M. Jones, Nevertheless , our new Q2 dependence, e.g., f o r Dr. R. Orava, Dr. G. Thomas, and Dr. M. Puhala two-particle fragmentation functions, is of im- f o r useful discussions. This work was supported mediate theoret ical interest . We note the rele- by the U. S. Department of Energy under Contract vance to the Q2 variat ion displayed by Jones et al.7 No. W-7405-eng-82, Office of Basic Energy Sci- in the i r calculation of D,,, i n a model in which ences (HK-02-01-01).

'J. F. Owens, Phys. Lett. E, 85 (1978). 'T. Uematsu, Phys. Lett. z, 97 (1978). 3 ~ . Par i s i , in Weak Interactions and Neutrino Physics,

proceedings of the XIth Rencontre de-Moriond, Flaine- Haute-Savoie, 1976, edited by J. T r a n Thanh ~ z n (CNRS, P a r i s , 1976); G. Altarel l i and G . Par i s i , Nucl. Phys. z, 298 (1977).

4 ~ . Sukhatme, Phys. Lett. m, 478 (1978); 2. Phys. C2, 321 (1979); R. Field and R. Feynman, Nucl. Phys. z, 1 (1978).

5 ~ . Konishi, A. Ukawa, and G . Veneziano, Phys. Lett. m, 243 (1978); Nucl. Phys. w, 45 (1979).

6 ~ . Kirschner, Phys. Lett. 8 2 , 266 (1979). IL. Jones, K. Lassi la , and D. Willen, Argonne-Illinois-

Iowa State Repor t No. ANL-HEP-79-32, 1979 iunpub- l ished).

8 ~ . Willen, Phys. Rev. D z , 1781 (1980). 'WU Chi-Min, CERN Report No. TH2769, 1979 (unpub-

lished). 'OM. Puhala, Iowa State University r e p o r t (in prepara-

tion). "F. Yndurain [Phys. Lett. m, 68 (1978)1, d i scusses

moment inversion using Bernstein polynomials. "T. DeGrand, Nucl. Phys. M, 485 (1979). 1 3 ~ . Amati and G . Veneziano, Phys. Lett. m, 87

(1979). 14see, for example, G . Altarelli, R. E l l i s , G . Marti-

nelli, and S. Pi , Nucl. Phys. z, 301 (1979). ' 5 ~ u c h a technique has been used for s t ruc ture functions

by W. Furmanski and S. Pokorsbi, CERN Report No. TH2685, 1979 (unpublished).

1 6 ~ . Orava, private communication.