Short distance counting rules for low pT fragmentation

Volume 88B, number 1, 2

PHYSICS LETTERS

3 December 1979

SHORT DISTANCE COUNTING RULES FOR LOW PT FRAGMENTATION J.F. GUNION Department of Physics, Umversity of Cahforma, Davis, CA 95616, USA Recezved 16 August 1979

Recent experimental data provides evzdence m favor of a model m whzch hadromc coUiszons are initiated by a factorizing mechamsm, such as gluon exchange. We propose that fast fragments resultmg from such colhsions arise via quantum chromodynamic diagrams in which the required quarks and/or antlquark are produced in "point-like" fashion; these can be shown to be the leadmg diagrams as x, the momentum fraction of the secondary hadron, approaches 1. We obtain good agreement with the observed power law behavzors, dN/dx ~ (1 - x) n, of single particle low PT spectra.

That fragmentation of a fast particle can be studied as a short distance process is already apparent in the calculation of the valence quark distribution for a quark-antiquark state [1]. In fig. 1 we illustrate the reqmred calculation (only the simplest diagram is

shown). As x ~ 1, the gluon and probed fermion propagators are forced far off-shell. In QCD, the fermion traces typically cancel the damping from the off-shell gluon propagators; only the damping due to the offshell quark propagators survives. The same amount of

probe

QCD--> ~ 4

I I

I I

i l-x

I

I

I

>

>

(1 - x)2"1"1

t (a)

I

i I I

>

(1 -x) 2"2"1

f i I

I

I

i i

I

(b) Fzg. 1. (a) The emission of a fast x ~ 1 quark from a q~ state. Large X's mdicate far-off-shell quark lines and i's the associated far-offshell energy denominators. The probability or squared amplitude is shown. (b) The amplitude for fast quark emzssion from a 3-quark state.

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damping is given, therefore, by the simpler ¢4 diagram also shown in fig. I. In this paper we will use the equivalence to discuss the basic x ~ 1 power law of a given QCD diagram. (This procedure does not always treat spin effects precisely, but does yield a good average result ,1 .) Thus, from the ~4 diagram of fig. 1 we may calculate the probability (amplitude squared) for finding a quark (k) with light-cone momentum fraction, x, of the incoming q~ state's momentum, p. We obtain 1

0f yq/.(x)dx -

1

f0

dx d2k± 2(1-

x){[k2+ m2(x)]/(1-x)}

2

(1)

(k± is the momentum o f k transverse to p), where m2(x) = m 2 - x(1 - x ) M 2 and k 2 has been evaluated at the (p - k) 2 pole as k 2 - m 2 = - [k2 + m 2 ( x ) ] / ( 1 - x ) . Note that m2(x) is regular as x -~ 1 and that k 2 1/(1 - x) as x ~ 1. Thus the k 2 of the probed quark is forced far off-mass-shell as x ~ 1, yielding the behavior (at either fixed k± of averaged over low k±),

fq/M x~1 (1 -- X)2" 1-1

(2)

In this k 2 ~ oo limit it has been shown [2], in an axial QCD gauge, that only vertex renormalization and ladder graph corrections to the simplest QCD diagram are important, to leading order in In k 2. They serve to introduce moving coupling constants, gs(k2), at each gluon vertex, and, m some cases, yield "anomalous dimension" powers of In k 2. For the phenomenologxcal apphcations to be discussed here we ignore these logarithmic modifications to the simple power laws. This is a good approximation quantiatively if A 2, the standard scale parameter in %(k 2) ~ lib In k2/A 2, is small. In a similar way the amplitude of fig. lb, and diagrams like it, lead to:

fq/B ~ (1 --x) 3 .

(3)

As in the previous case only off-shell quark propagators (or quark energy denominators) are counted. ,1 The papers quoted in ref. [1] consider the effects of spin for the (presumably) typical deep inelastic probe. They find that the spinless result, fq/M ~ (1 - x), converts to fq/M ( 1 - x) 2 + constant when spin is included. In the deep inelastic case the constant is ~ 1/Q 2 ; in the hadron fragmentation cases discussed shortly the relevant ,,Q2,, is determined by the parameters of the hadron into which a probed quark is absorbed.

3 December 1979

It is important to note that the present situation is different from that defined by a deep inelastic probe of, say, off-shell momentum Q2. There the probe itself defines a second off-shell scale and gluons are radiated from the quark line probed. This radiation results in a change m the (1 - x) power law as leen m deep inelastic scattering. The above two results for valence quarks are relatively unambiguous, modulo the logarithmic corrections mentioned. However, when one considers sea quarks two natural possibilities arise: (1)H The sea quark originates as an intrinsic part o f the hadron bound state or, in other words, as a member of a "hadronized" higher Fock state of the hadron; the minimal diagram is shown in fig. 2a for a meson primary. (2)p L The sea quark is created in a "point-like" manner as shown in fig. 2b; here the created quark (and antiquark) do not reinteract with the rest of the meson's components - the bound state momentum is not redistributed so that all four components (quarks and antiquarks) share it equally. Both diagrams are inevitably present in QCD but the point-like diagram dominates as x ~ 1. However, a large normalization for the Fock state used in (1)a could make the hadronized contribution temporarily dominant. The latter choice is, in fact, the basis of standard quark counting rules [ 1 ] for sea-quark and gluon distributions. The comparisons for a sea quark in a meson or baryon are:

fqsea/M(X) ~ (1 --x) 5 (I -- x) 3

fqsea/B(X) ~ (1 -- X) 7 (1 -- x) 5

(1)H , (2)p L , (1)H ,

(4)

(2)p L .

These results may be summarized in terms of spectator counting rules. We divide the spectators (unprobed quarks) into those which originated from the hadron Fock state (n H in number) and those which participated in the point-like pair creation process (npL in number). We have , 2 , for the diagrams we will discuss: ,2 The point-like sea-quark amplitude was considered by Gunion, ref. [1]. The general rule was first discussed by Blankenbecler et al. [3]. (Note that gluons would not be counted among the spectators even if present.) The extension to include spin effects is partly studied in ref. [2] and is also under consideration by Brodsky and Gunion. 151

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I

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3 December 1979

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11 - xl 2-3 -1

( 1 - x) 2.2 -1

(a) (b) Fig. 2. (a) The emission of a fast sea-quark, qsea, from a q ~ state based on a h a d r o m z e d qqqseaqsea non-valence F o c k state wave function. (b) The emission o f a fast sea quark from a q ~ state using point-like pair creation for qseaqsea. × and I as in fig. la.

f(x) ~ (1 -- x) 2nH+nPL-1.

(5)

Thus, for example, in fig. 2a n H = 3, npL = 0 while in fig. 2b n H = 1, npL = 2 -- only one of the final quarks in the latter case was not part of the point-like creation process. There is, in fact, some evidence from deep inelastic scattering that the sea quark distribution in a proton has power behavior as low as (1 - x ) 5 when Q2 is small [4] ; it may be the additional Q2 controlled radiation which increases the power to (1 - x ) 7 at large Q2.

t3 The importance o f this physical situation for tests o f QCD was considerbd by Brodsky and Gunion [6 ]. See also ref. [7]. Other approaches appear in ref. [8].

(5 -* 5 + [4CF/(11 - ~-NF) ] × In [(In Q2/A2)/(ln
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The unknown values of A 2 and (k 2) determine the amount of increase [5] .) Here we present evidence, based on experimental studies of low PT fragmentation in hadron collisions, which favors the point-like diagrams. We thus consider the extension of the preceding cO... counting rules to fragmentation of a secondary hadron from a primary hadron ,3. Again we are faced with the two choices just considered. Fig. 3a shows a diagram typical of the hadronized, (1)H , way of obtaining a n o from arr +; fig. 3b shows a point-like, (2)pL, mecha-

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rr°

(1 - x) 2.1-1 (b)

Fig. 3. (a) The emission o f a fast n o from a hadronlzed 4 - c o m p o n e n t Fock state of a 7r÷. (b) T h e emission o f a fast 7to from a 7r+ using point-like creation to o b t a m t h e requtred ~ antiquark.

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nism. In both cases the original valence u quark transmits all its momentum to the 7r0 . The (1 - x ) suppressions arise from the necessity to transmit all but a fraction (1 - x) of the remaining momentum of the rr+ to a sea fi quark. Pictorially, fig. 3 shows that the rr+'s original valence u quark is simply appended (for "free") to the diagrams which would describe fragmentation of a sea quark from a single initial valence quark. Thus the generalized counting rule, (5), still applies - spectators are those quarks which are not part of the emitted 7r0. Obviously p ~ 7r+ fragmentation can be obtained analogously. We have the following results:

rr+~lr0 ~ ( l - x ) 3

p-,..tr

+

(2)pL ,

~(1-x)5

(1)a,

(1 -- x ) 3

(6)

(2)a L .

For p ~ 7r- the minimal diagrams require use of the proton's d valence quark. As is well known (see Jackson and Ferrar, ref. [1 ]), the SU(6) spin structure of the proton leads to a numerical suppression of the d quark in the x ~ 1 limit. We will phenomenologically (see Gunion, ref. [1 ]) incorporate this suppression by adding 1 unit to our powers in such a case. Thus

(l--x) 4

> > >

> < < < <

< < < (a)

(b)

Fig. 4. Diagrams for the colhsion o f two protons. (a) Quarkexchange as the hadronic collision mechanism; (b) Gluon exchange as the hadronic collision mechanxsm.

(1)H,

(1 - x)

P-> rr- ~ ( l - x ) 6

3 December 1979

(1)H'

(7)

At first sight forward (low PT) hadron production in np, Kp and pp collisions provides the ideal testing ground. However, one must be careful to account for the required interaction between the colliding hadrons. There are many possibilities [6-9] ,4. For our counting rule applications nearly all of them can be characterized in terms of one of two basic QCD mechanisms: quark exchange [6-8] or gluon exchange [9], illustrated for pp collisions in fig. 4. In quark exchange the forward and backward systems are dissimilar while in gluon exchange they are the same. If we examine a 7r~ in the forward direction and a lr~ in the backward direction, the above differences between quark and gluon exchange lead to the following predictions for

(2)p L .

How do we observe these hadronic fragmentations 9.

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4:4 For an example o f a non-QCD interaction approach see Capella et al., ref. [8].

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I (1 - x) 2 " 2 " 1 (1 - x)" 1

(1 - x ) T M

(s)

(b)

K"

U

Fig. 5. Point-like mechanism diagrams for: (a) ~r+--* p; (b) n + ~ K - . All spectators are "point-like" in both cases.

153

GLUON EXCHANGE + P O I N T - L I K E SEA

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n FRAGMENTATION

Fig. 6. A summary of expertmental results [13] and theoret]cal predictions (as obtained here) for all available single particle frag. mentations. Here sohd lines indicate predictions for gluon exchange and point-like pair creation. Broken lines indicate that one unit has been added to the naive point-like prediction because a proton's d quark is used in the fast fragmentation.

Volume 88B, number 1,2

PHYSICS LETTERS

the correlation function;

dN/dx 1 dx 2 R (Xl, x2) = (dN/dxl) (dN/dx2) = (1 - Xl)4 + (1 - x2) 4 ,

quark exchange,

= 1,

gluon exchange,

(8)

and depends only on the interaction mechanism. Recent data [10] rule out the quark-exchange posslbility;R is measured to be unity over a wide range o f x 1 and x 2 values. Given gluon exchange we may distinguish between (1)H and (2)p L by comparing predictions with data for pp -> lr+. All experiments * s agree on a power slightly larger than 3, in agreement with the point-like approach and certainly much below the power of 5 predicted by the hadronized wave-function approach. We thus turn to an examination of additional predictions based on the combination of: (i) gluon exchange as the hadromc interaction mechanism; and (ii) pointlike pair creation as the mechanism for producing sea quarks. Before presenting an overall summary of such predictions we single out two further examples. First is the case rr+ ~ p. The diagram of fig. 5a yields [11]. zr+ -* p ~ (1 - x ) 2 .

(9)

This result is correctly obtained from the spectator counting rule (5), (n H = 0, npL = 3). The second example is the case of rr+-* K - fragmentation. The K has no valence quarks in common with the rr+; we will term this "exotic" fragmentation in analogy to the t-channel triple Regge terminology. One of the minimal diagrams is illustrated in fig. 5b. Eq. (5) applies with n H = 0 and npL = 4 yielding zr+--> K - ~ (1 - x ) 3 .

(10)

This result is very different from (1 - x) 7 as would have been predicted by the original quark counting roles based on a hadronized wave function. Also the hadronized counting rules would have predicted that line reversal, p -+ rr+ ~ lr+-+ p, should hold.

,s See ref. [101, for example. A more complete hsting will appear shortly. The power depends shghtly (and randomly) on the PT value (of the meson) at which it is measured. The theory predicts approximate PT independence of the power for low PT"

3 December 1979

In fact experimental data appears to support both line reversal violation and minimal suppression for exotic fragmentations. This is clearly demonstrated in fig. 6 , 6 which presents the theoretical predictions based on gluon exchange plus pomt-like sea creation for the various fragmentations (the proton d quark rule is also incorporated), and compares the predicted powers to experiment [13]. While in a few cases there is some disagreement between different experiments (e.g. K e --> he), It is clear that in general the agreement between theory and experiment is quite good. The only clear exceptions are p -+ ~ and p -+ A where the experimental powers appear to lie above the rather low powers predicted by the present approach for such exonc fragmentations. Of course, a variety of additional tests of this approach are possible. In particular it is important to extend these counting rules to the production of pairs of particles. These and other applications will be considered elsewhere [11 ]. From single particle fragmentation, however, it appears that there is substantial support for using the point-like QCD diagrams, with minimal suppression as x --> 1, to explain fast forward fragmentation in hadronic reactions; fragmentation acts as a perturbative probe of the hadronic state structure and can be treated by extensions of the currently available leading log technologies. m

I would like to thank the Department of Energy and the A.P. Sloan Foundation for support. I would also like to thank H. Sens, S. Brodsky, A. Capella, and R. Hwa for helpful conversations. I am grateful to DESY and CERN for hospitality during the course of this investigation. ,6 A figure like this, except that it was prepared for "quark exchange + hadronized wave function" predictions, appears in ref. [12].

References [1] R. Blankenbecler and S.J. Brodsky, Phys. Rev. D10 (1974) 2973; J.F. Gunion, Phys. Rev. D10 (1974) 242; G.R. Farrar, Nucl. Phys. B77 (1974) 429; G.R. Farrar and D.R. Jackson, Phys. Rev. Lett. 35 (1975) 1416. [2] G.P. Lepage and S.J. Brodsky, SLAC-PUB,Phys. Rev. Lett., to be published, 155

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[3] [4] [5] [6] [7] [8]

[9]

[10]

[11]

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PHYSICS LETTERS

see also S.J. Brodsky and G.P. Lepage, SLAC-PUB-2294; related results are derived by G. Parlsi, CERN preprint (1979); A. Duncan, Columbia Univ. preprint (1979). R. Blankenbecler, S.J. Brodsky and J.F. Gunion, Phys. Rev. D12 (1975) 3469. See the rewew by E. Gabathuller, EPS Intern. Conf. on High energy physics (Geneva, Switzerland, June 1979). See for example, W.R. Frazer and J.F. Gunion, Phys. Rev. D19 (1979) 2447. S.J. Brodsky and J.F. Gunion, Phys. Rev. D17 (1978) 848. S.J. Brodsky and J.F. Gunion, in: Proc. VIIth Intern. Colloquium on Multiparticle reactions (Tutzing, 1976). H. Goldberg, Nucl. Phys. B44 (1972) 149; S. Pokorski and L. Van Hove, Acta Phys. Pol. B5 (1974) 229; and CERN prepfints; W. Ochs, Nucl. Phys. B l l 8 (1977) 397; K.P. Das and R.C. Hwa, Phys. Lett. 68B (1977) 459; T.A. De Grand and H.I. Miettinen, Phys. Rev. Lett. 40 (1978) 612; A. Capella, U. Sukhatme and J. Tran Thanh Van, LPTPE 79/23 (June 1979); B. Anderson, G. Gust~son and C. Peterson, Phys. Lett. 69B (1977) 221. F.E. Low, Phys. Rev. D12 (1975) 163; S. Nussinov, Phys. Rev. Lett. 34 (1975) 1286; J.F. Gunion and D.E. Soper, Phys. Rev. D15' (1977) 2617; S.J. Brodsky and J.F. Gunion, Phys. Rev. D19 (1979) 1005. M.M. Block et al., Amsterdam-Louvain-North Western eollab., Correlations between high momentum mesons in pp ~ ~ t X at x ~ -- 63 GeV, submitted to EPS meeting (Geneva, 1979). J.F. Gunion, in preparation.

3 December 1979

[12] R. Diebold, Intern. Conf. on High energy physics (Tokyo, 1978). [131 D. Cutts et al., B r o w n - C E R N - F N A L - I N F N - M I T coilab., submitted to Phys. Rev. Lett.; J. SaudraLx et al., SACLAY-DPh DE 7803 (1978); Nucl. Phys. B149 (1979) 189; R.T. Edwards et al., Phys. Rev. D18 (1978) 76; K. B6ckmann, Symp. on Hadron structure and multiparticle production (Kazimierz, 1977); W. Lockman et al., Phys. Rev. Lett. 41 (1978) 680; J. Singh et al., Nucl. Phys. B140 (1978) 189 (also ref. [101); Bodier et al., referenced m Bourqum below; Hungerbfihler et al., referenced m Bourquin below; M. Bourqmn et al., Print-78-0804 (CERN) (p ~ ~--); the above were summarized by R. Diebold, ref. [ 12 ] ; the following results were submitted to the EPS Intern. Conf. on High energy physics (Geneva, 1979) or are recent preprints: J.R. Johnson et al., FNAL preprint (1978); Brussels-CERN-Genova-Mons-Nijmegen - S e r p u k h o v Tel Aviv collab., submitted to EPS meeting (K÷p -~ ,r-); Aachen- Berlin-CERN -Cracow - L o n d o n - V i e n n a Warsaw collab, submitted to EPS meeting (K-p interactions); B. Alper et al., Amsterdam-CERNhCracow-Munich Oxford-Rutherford collab., submitted to EPS meeting (inclusive ~ production in K ÷, K-, ,r÷, *r-, p beams); I.V. Ajmenko et al., Serpukhov-Brussels-MonsPHNHE-CEN collab., submitted to EPS meeting (K÷p interactm ns); E.A. De Woff et al., Belgium-Serpukhov collab., submitted to EPS meeting (K÷p interactions).