LETTER~ AX, NUOVO OIMENTO VOL. 28, ~. 13 26 LugHo 1980

Quark Mass Effect on Fragmentation Functions

and Charmed-Hadron Productions in e+e- Anvihilation.

T. KOBAYASttI and N. YA~AZAKX

Institute ]or Physics, University o] Tsukuba - Ibaraki 305, Japan

(r icevuto il 22 l~Iaggio 1980)

I t is known tha t hard-sca t te r ing processes can be wr i t t en in a form (1) factor ized into a cont r ibut ion of a soft (long distance) dynamics and t h a t of a hard (short distance) dynamics . The effect of the hard dynamics is calculable in pe r tu rba t ive QCD (2), while the so-called confinement dynamics will p lay an essential role to solve the effect of the soft dynamics .

I n order to make our discussion clear, we shall discuss a hadron product ion in e+e - annihi la t ion, which m a y be wr i t t en in terms of a f r agmen ta t ion funct ion /~m repre- sent ing the f ragmenta t ion of a pa r ton i into a hadron h wi th m o m e n t u m fract ion x. The f r agmenta t ion funct ion is described in a factor ized form as

(1)

1

where i and j denote par tons (quarks, an t iquarks wi th f f iavours and gluon), v~m(y ) represents a f ragmenta t ion func t ion of a pa r ton j into a hadron h wi th a m o m e n t u m fract ion y a t Q~ = Ft ~ s tanding for t he soft dynamics and Gji the f r agmenta t ion func- ~ o n of a p a t t o n i into a pa r ton j s tanding for the hard dynamics . The f r agmenta t ion funct ion /)hi satisfies the Al tare l l i -Par is i - l ike equa t ion (8.4) and the Q2-dependenee for

(1) Yu. L. DOKSHITSER, D. I. I)fYAKONOV and S. I. TROlrAIg: Report No. SLAC-TRANS-0183 (June 1978); R. K. ELLIS, H. GEORGI, ~t~. MACHA]~K, H. D. POLITZER and G. G. Ross : Phys. Left. B, 78, 281 (1978); Nucl. Phys. B, 152, 285 (1979); S. LIBBY and G. STERMAN: Phys. Let~. B, 78, 618 (1978); Phys. Rev. D, 18, 3252, 4737 (1978); D. AMATI, R. PETRONZIO and G. VEI~EZIAI~O: _~ucl. Phys. B, 140, 54 (1978); 146, 29 (1978); Y. KAZAW~ and Y. P. YAo: Phys. Rev. Lett., 41, 611 (1978); Phys. Rev. D, 19, 3111, 3121 (1979); A. H. J.~CIf.~LLER: Phys. Re~). D, 18, 3705 (1978); S. GUPTA and A. H. M~bLER: PhyS. _l:~eV. D, 20, 118 (1979). (I) R'. GEORGI and. H. ~). POLITZER: Phys. Rev. D, 9, 416' (1974); D. J. GROSS and F. WILCZEK: Phys. ReV. D, 9, 980 (1974). (a) G. ALTARELLI and G. PARISI: NUCL Phys. B, 126, 298 (1977). (') J. F. OWENS: Phys. Left. B, 76, 85 (1978); T. UEMATSU: PhyS. Left. B, 79, 97 (1978).

450

QUARK MASS EFFECT ON FI~AGMENTATION FUNCTIONS ETC. 451

the n-th moment of Gjl is derived in per turbat ive QCD as

_ 2 f + I

(2) 8--t G,,(n, Q~) = ~_. G,k(n, Q~) A~(n) ,

where t = (16/(33-- 2])) In (~(Q~)/~(Q~)) with ~8(Q 2) = (12:t /(33-- 2]))(lnQ2/A2) -~ and A ~ 500 MeV, and Al:i(n) is the well-known Altarell i-Parisi function (~).

The value of #3 is theoretically not very clear but we may est imate its order to b e

2 ~ ~t~ ~ k~ = (0.3--0.5) (GeV/c) 2 (k~ = transverse momentum of hadrons) for light partons = light quarks (u, d, s, ~, d, s) and gluons, whereas

~ 2 ~ 2 2 2

is est imated for charmed quarks (c, c) with mass m~. This est imation is supported by the fact that the rat io / ~ : g(e+e--+hadrons)/a(e+c---,~t+~ -} in experiments already has a flat region before charmed-part icle productions and the value of R in this region is consistent with tha t obtained from the l ight-quark dominance. The impor tant point in the above discussion is t ha t ~m(x) representing a soft dynamics is determined at a certain renormalization point of Q2, which may depend essentially on the mass of the i-quark. Then i t is expected tha t the symmetry breaking by heavy-quark masses will be seen in hadron productions v ia ~ i - We shall discuss this symmetry-breaking effects in this letter.

Let us formulate our idea in the Q2-evolution equation of QCD in the case tha t there exist partons wi th some different renormalization points. For simplicity, we consider the problem in the model with. l ight par tons and charm quarks. The extension to the system including more heavy quarks (b, t . . . . ) is straightforward and the same argument can also be extended to other hadronic production processes.

We follow the following procedures:

1) In the region p~ <<Q~<< 4m~ we adopt the renormalization group equation with three fiavours.

2) In the region 4m~ << Q2 the same equation but with four fiavours is adopted.

3) As a simple example we pu t /~ ~--4m~ and two fragmentat ion functions /)m evaluated in the procedures 1) and 2) for the l ight pa t t ens coincide with each other at Q~ ~ 4m~o.

This procedure may be par t ly supported by dual i ty arguments tha t in e+e--annihila - tion into hadrons the to ta l cross-section smeared over some finite-energy range closely imitates, even in the resonance region, the smooth, asymptot ic cross-section given, for example, by par tou model (~). Therefore, the procedure 3) as an init ial condition is not so unreasonable, if we use eq. (2) far from the resonance region (Q2<< 4m~ or q~ >> 4m~).

When we recognize #~ ~ 4m~ >>/~, we may conclude tha t charmed haflron (hc) production is unable to be included in v~m for i ~ light pat tens , while ordinary hadrons can be produced from all partons. For example, a fight quark with Q2>> 4m~o can

(6) See, for example , K. ISHIKAWA and J. J. SAKURAI: Z . P/byS., 1, 117 (1979), and references c i ted therein.

4 ~ 2 T. KOBAYASHI and N. YAMAZAKI

produce charmed hadrons via only the process of

light quark --~ gluon + light quark

I-+ charmed quark + its ant iquark

charmed hadrons.

The mult ipl ic i ty of charmed hadrons will be proport ional to tha t of charm quarks produced in hard process and equal contribution of charm quarks to D and D production leads us to the relation for Q2>> 4m~

(3) [da(D~176 0 ( ~ )

in the e+e - annihilation. Provided tha t ~hc,o is not very different from ~m, we also expect tha t eharmed-badron product ion is suppressed for Q2>> 4m~, where the S U4 symmet ry in the par ton level is almost recovered,

(4) da(D+)/da(~ +) ~�89 .

That is in hadron productions the S U 4 symmetry is bad ly broken even at large Q2. These relations should be compared with the results implied in ref. (6), where for Q~ > 4mo ~

[da(D ~ - -da(D+)] / [da(D ~ + da(D+)] = ~ > 0 (5)

with a > 0 and

(6) da(D+)/da(~+) = 1 + 0 ( 1 )

are expected since the S U 4 symmet ry is postulated for the fragmentat ion funct ions/)hi . The difference d~(D ~ - - d ~ ( D +) arises from the contribution of the nonsiglet compo- nent in ref. (e), while tha t contribution vanishes in the leading order in our model.

In comparison with the e+e--annihilation experiments (7) for 6GeV/e~V ' (~ -~ 7 GeV/c, the definition of scaling variable is important . In other words the charmed-

hadron mass effect is still large at those Q2, so the conclusion depends on the definition of the scaling variable z, i.e. z=2p/v /~ "~ ( p = h a d r o n momentum) or z = 2 E / ~ / ~ ( E = h a d r o n energy). Relations (4) and (5) are consistent for the choice z = 2p/%/~, whereas rela- t-ion (6) seems to be bet ter for z = 2E/~ /~ . Anyway, in order to see the val id i ty of (3) and (4) we need experiments a t higher energies, where the difference between two definitions disappears.

One of us (NY) acknowledges the financial support from the F~jukai .

(*) J . DIA8 DE I)EUS a n d N. SAKAI: Phys. Left. B, 86, 321 (1979). (v) G. J . FELDMAN: Proceedings of the X I X International Con]erenee on High-Energy Physics ( T o k y o , 1978) , p . 824.