Structure functions of quarks, gluons and hadrons in quantum chromodynamics

Nuclear Physics B151 (1979) 485-517 © North-Holland Publishing Company

STRUCTURE FUNCTIONS OF QUARKS, GLUONS AND HADRONS IN QUANTUM CHROMODYNAMICS Thomas A. DEGRAND

Department of Physics, University of California, Santa Barbara, California 93106, USA Received 27 November 1978

The Q2 variation of hadronic structure functions and of fragmentation functions in QCD is formulated in terms of Q2 dependent structure functions for the distribution of quarks and gluons in quark and gluon jets. This approach is completely equivalent to the standard moment analysis but is much more useful for obtaining qualitative results. Many old and new results for the behavior of quark and gluon jets and the implications for these results on the evolution of hadrons may be easily derived. It is easy to show, for example, that the ratio of gluon distributions in quark jets to that in gluon jets approaches a Q2 independent constant (~) as x ~ 0, that gluon distributions in gluon jets vanish 9 times as rapidly as quark distributions in a quark jet at large x, Q2, and that at large x the leading hadrons in gluon jet fragmentation are produced via the sequential reaction gluon --*quark --, hadron, so that Dh/g(X, Q2) ~ (1 - x) Dh/q(X, Q2). I present simple analytic expressions for hadronic structure and fragmentation functions and derive counting rules for QCD-basedlarge-p± inclusive jet and hadron cross sections.

1. Introduction In recent years the calculation of scale-violation effects in hadronic structure functions due to the effects of an underlying asymptotically free field theory of colored quarks and gluons, q u a n t u m chromodynamics (QCD), has become a very mature subject *. The literature is filled with numerous computations of Q2 dependent structure functions and of their comparison with data. Similarly, one sees elaborate calculations of inclusive hadroproduction at large transverse m o m e n t u m or inclusive production of high-mass dilepton pairs, calculated using QCD-based nonscaling structure and fragmentation functions. This work has a more modest purpose. Rather than presenting elaborate, detailed calculations of the evolution of structure functions in QCD, my aim will be to provide simple analytic expressions, derivable from the moment equations in QCD (not fits), which can be used to parametrize the behavior of hadronic structure and frag-

* See ref. [1 ] for reviews. 485

T.A. DeGrand / Structure functions in QCD

486

mentation functions, and of reactions involving those structure functions, and which become arbitrarily accurate in limiting regions of phase space. The conventional approach to the calculation of scaling violations has been through the intermediary of moments: one defines the/th moment of a parton distribution F(x, Q2) via 1

M(i, Q2) = f dxx/-1F(x, Q2).

(1.1)

o

Then the evolution of the moment in QCD is evaluated: for non-singlet quark distributions, for example, one finds

M(j, Q2) =M(], Q2) exp(26(/') ~),

(1.2)

where

47rb

~ac(Q2) ] '

o.3)

and ac (/a2) at(Q2) = 1 + bctc(/a2) log(Q2//a 2)

(1.4)

1 -

b log(Q2/A2) '

(1.5)

where b = (11N - 2f)/127r for N colors and f flavors. 66i) is the appropriate anomalous dimension (see below). Finally, one inverts the moment equation to recover the Q2 dependent structure function: _d/

F(x, Q2) = f'~ix-/M(l., Q2).

(1.6)

c

This procedure is not roundabout, since in QCD the different moments evolve independently of each other, and working in terms of moments diagonalizes the problem. This approach is generally fairly easy to implement computationally. Problems arise, however, when one attempts to interpret one's result, which are due to the difficulties inherent in translating statements about the evolution of moments (for which quantitative formulae such as (1.2) exist) into statements about the evolution of structure functions in particular regions o f x . On one level this is an easy procedure: we are all familiar with the qualitative picture of the evolution of structure functions, which, because the moments tend to decrease monotonically at large/', shrink at large x, and (because momentum is conserved) grow at small x. But how fast do they shrink or grow?

ZA. DeGrand/ Structure functions in QCD

487

It would be useful to have semiquantitative expressions for F(x, Q2) which are valid over certain ranges o f x and Q2 and which can be derived as directly as possible from the moment equations (1.2)-(1.6). One would then be able to interpolate the results of calculations of scale-violating structure functions with simple analytic expressions which show explicitly the gross features of asymptotic freedom. A second and more useful application of these quantitative expressions is in the realm of QCD calculations for which the structure functions are ingredients, such as jet and meson production at large transverse momentum in hadron-hadron collisions or in the production of high-mass dilepton pairs. Differential cross sections for these reactions are generally extremely smooth functions of the relevant kinematical variables, and it seems that, possessing approximate quantitative expressions for quark, antiquark and gluon structure functions, one would be able to derive simple analytical expressions which do a good job of reproducing cross sections calculated by long computer programs. The elucidation of simple quantitative expressions for Q2 dependent structure functions and their application to various QCD based reactions is the subject of this note. The general approach to this problem is obtained when one realizes that the moment equation (I .2) involves a product of two terms. The first, the structure function of the parton in the hadron at Q2, is a priori entirely unknown. The second term involves the anomalous dimension of the operator and the ]-dependent projection operators (if necessary) and is uniquely determinable in QCD. This term has a simple physical interpretation, it is the ]th moment of the structure function of the quark in an object which is defined to be a quark at Q2 = Q~. The structure function at Qo2 in this instance is

Cq/q(x, 0?9 =

- x) ,

or, inverting to moments,

M(/,

= i,

whence the Q2 dependence of the moments of the distribution of quarks in a quark beam at higher Q2 is simply

mq/q (], Q2) = e2nff) ~.

(1.7)

It is easy to invert this expression when (1 - x) is small (or when x is small) to obtain an approximate x and Q2 dependent structure function for the (non-singlet in this case) distribution of quarks in a quark beam. Finally, one can invert the moment integral (1.6) and write it as a convolution of a quark distribution seen in the hadron at Q~ and the structure function of the quark (or gluon) in the quark: ldz

rq/p(X, Q2) = f s _ y q / p ( z '

aq/q(X/Z, (Qg, Q')).

(1.8)

9¢

If F(x) and G(x) have simple x-behavior at large x, the convolution integral may be

488

T.A. DeGrand / Structure functions in QCD

easily performed, almost by inspection, to give F(x, Q2) at large x. For example, if F(z) ~ (1 - z) p and GO') ~ (1 - y)q (q is generally Q2 dependent), then Fq/p(x, Q2) = r(1 + p) r(1 + q)(1 -x)P+q+l(1 + 0(1 - x ) ) . I'(2 + p + q)

(1.9)

The first term is entirely adequate at large x. This method of calculating asymptotic freedom corrections to structure functions has several advantages over the usual moment analysis. First, it is exceedingly easy to obtain simple analytic expressions which reproduce the general features of the evolution of structure functions in QCD. Second, it allows one to compare the evolution of different structure functions or of structure functions and fragmentation functions (since all scale violation is contained in q, above). The method is a physical realization of the Kogut-Susskind [2] "partons inside partons" picture : at Qo2 one measures parton distributions in a hadron. At higher Q2, the individual partons evolve independently into the quarks and gluons which are observed at the new momentum scale. Perturbative QCD says nothing about the distribution of partons in a hadron, but it uniquely determines the evolution of a quark (or a gluon) seen at one Q2 into distributions of quarks and gluons as seen at higher Q2. Portions of this program have already appeared in the literature. Early uses of eq. (1.8) are to be found in refs. [3,4]. It is not uncommon to find approximate forms for hadronic structure functions at large x, derived, however without reference to quark and gluon structure functions. Dokshitser, Dyakanov and Troyan [5] have presented calculations of the structure functions of quarks and have applied them in several examples. Cabbibo and Petronzio [6] have studied the Q2 evolution of a three-quark model of the proton. Some of their techniques are similar to those developed here. As this work neared completion, I learned of a development of asymptotic freedom similar to that of this note recently given in lectures by Brodsky [7]. In sect. 2, I will give a more complete discussion of the properties of quark and gluon structure functions, showing expressions for their behavior at small and large x. We will see that a gluon jet "degrades" more quickly than a quark jet, so that a quark jet retains a leading particle at much higher Q 2 than a gluon jet: indeed, at large x Gq/q(X, ~) ~ (1 - x ) 4C2(R)~'-1 , Gg]g(X, ~) ~ (1 - x ) 4C2(G)~-1 , Gg/q(X , ~) ~ Gq]g(X, ~) ~ (1 - x) 4C2(R)t ,

so that at large Q2 the leading particles in the gluon jet are not gluons but quarks. Consequences of these results for fragmentation functions are discussed in sect. 3, one of the most important being that because the leading particles in high-Q 2 gluon jets are quarks, all properties of gluon fragmentation at large Q2 are uniquely deter-

T.A DeGrand / Structure functions in QCD

489

mined by quark fragmentation. Applications to structure functions are to be found in sect. 4, along with a derivation of simple Q2 dependent counting rules which can be used to parametrize large transverse momentum inclusive reactions. Finally, sect. 5 contains some conclusions. It is important that the reader realize that the approach advocated in this note is complimentary to (and entirely equivalent to) the standard techniques in terms of moments. It is, however, an approach which is much more amenable to simple approximation and to computerless computation, and as such will serve as a good check on the trustworthiness of one's long numerical calculation.

2. Properties of quark and gluon structure functions The most convenient starting point for our derivation of quark and gluon structure functions in QCD is via the approach of AltareUi and Parisi [8]. In this formalism, the Q2 evolution of hadron structure functions is given in terms of a set of coupled integro-differential equations. A convenient variable which defines the scale over which the structure functions evolve is the ~-vadable of sect. 1 (eq. (I .3)). For future reference we remind the reader that in SU(N)

~CabcCabc=N, C2(G) =Af~ 1- 1 abc N2 C~(R) = ~

- C2,

T(R) 6ab = Tr(tat b) = ~f~ab 1 , where f i s the number of quark flavors. The Altarelli-Parisi equations take the simple form

f

1 d-~ d q(x' ~) = 1 z~.

tOO/q(X/Z) q(z, ~) + G°/g(X/Z)g(z, ~)],

x

5= d..~g(x, d 1 ~)

i ~dz [GO/q(X/Z) q(z, ~) + G°g/g(X/z)g(z, ~11 ,

(2.1)

x

where q (x, ~) and g(x, ~) are the G-dependent quark and gluon distributions in the hadron and the G~l/'S are the fragmentation functions for parton j into parton i evaluated to lowest order in perturbation theory. By defining moments of structure functions (eq. (1.1)) the convolution formula factorizes and eq. (2.1) become matrix equations 2 d~ \g(j, ~)1

~g(], ~)]"

(2.2)

T.A. DeGrand/ Structure functions in QCD

490

Finally, we define non-singlet (qi - qi) and singlet (E(oa. + qi)) quark flavor combinations which partially diagonalize eq. (2.2), leading to differential equations for moments of the non-singlet quark (qns), singlet quark (qs) and glue distributions in the hadron 1 d - -- qnsU,

~/) = A,,sU)

2 d~/

q,~(J',

~),

1 d

2 dl~ \g(j, ~) ]

LAgq(/) Agg(j)

. J \ g q , ~) ]

The A's are the moments of the lowest-order parton graphs describing the evolution of quarks and gluons. That is, the A (D's may be abstracted from lowest-order calculations performed in a physical gauge using old-fashioned perturbation theory in the infinite m o m e n t u m frame. The appropriate diagrams are shown in fig. 1 and the A's are tabulated in table 1. By "abstracted" I mean that, for example,

G°glq(x,k.L) -

O~c(k~) 1

~

l~ d(x),

1

Agq(j) = f x/-l d(x) dx. o The fragmentation functions G~tlq(X, k t) and Gg°/g(X) must in addition be regulated because the have (1 - x) - t singularities, and 6-functions must be included in the definitions of these quantities to preserve momentum conservation: 1

f dxx[Dq/q(X ) +Og/q(X)] = ] . 0

Fig. 1. Lowest-order graphs for the evolution of quarks and gluons into quarks and gluons, in the Altarelli-Parisi formalism.

T.A. DeGrand / Structure functions in QCD

491

Table 1 Anomalous dimensions Ans(]) =C 2

_

i

1

l+j(]+l)

Agq(fl - C 2+]._+]2 2/(/.2 _ 1)' -

Aqg(]') =

1

2+/+/2

2 1 q + 1)(1+ 2)'

Agg(f) ~ N

[1

+ + /1q_-_

J

= (j+ 1)(]+ 2)

k=2 r~j

Details of these (and other) minor complications are fully discussed in ref. [8] ; I have only included a small recapitulation here for future reference. The A's are also the appropriate anomalous dimensions for the mixing of quark and gluon operators, first derived in refs. [9,10] v& the operator-product expansion. The solution to eq. (2.3) is straightforward: qns(], ~) = qm(], O) exp [2Ans(D ~5] ,

qs0/, ~) = qs(/', 0) GSq/qU, ~) + 2fig(I, O) Gq/gU, ~),

gO, ~) = g(i, o) ag/g(/, ~) + qs(i, o) ag/qO, 0 ,

(2.4)

and we have defined the coefficients of q(/, 0) and g(j, 0) to be moments o f f dependent structure functions for quarks and gluons in quark and gluon beams G~q(], ~) = e 2AnsU) ~,

GO/q(/, ~) =

(2.5)

(Arts - ~,-) e 2x+~ - (Ans - M) e 2x-~ ~_ _ ~_

,

(2.6)

Gq[g(J, ~) = Aqg

e2M-~ _ e2~.-~ ~- - X_ '

(2.7)

Gg/q(/, ~) =Agq

e2X+~ _ e2X-~ ~L~- ~._ '

(2.8)

Gg/g(/', ~) =

(~% -- Ans ) e 2x+~ -- (X_ - Ans) e 2~'-~ ~k+ - ~._

(2.9)

T.A. DeGrand /Structure functions in QCD

492

where )k+ 1 1 _ = ](Ans + Agg) + X/[~(Ans - Agg)] 2 +

2fAgqAqg

(2.10)

are the eigenvalues of the two-by-two mixing matrix for the gluon and singlet quark distributions. This identification is apparent when one defines a patton (quark or gluon) distribution at ~ = 0 to be a &function q (x, 0)

or

g(x, 0) = 8 (1 - x ) ,

all of whose moments are unity, and inserts this distribution into eq. (2.4). It will also prove useful to define "favored" and "unfavored" structure functions for quarks. The favored distribution is the structure function of a quark qi inside a quark of the same flavor

q/ql.], Gfav ~.

~) = Gns(], ~) + ~ f (GS(], ~) - Gns(], ~)),

(2.11)

the unfavored distribution is the structure function of any flavor of antiquark or of a quark of a different flavor inside a quark: /q[./, ~) =

--

[GS(], ~) - Gns(], ~)] Gunf ~ql/q/U,~),

i--k~.

(2.12)

(I have chosen to speak of quark structure functions as favored and unfavored rather than as valence or sea, reserving the latter terms for quark distributions in hadrons.) Our promised result is now obtained by inverting eq. (2.4) to obtain a convolution equation for the quark and gluon distributions at x and ~ in terms of quark and gluon distributions at x and ~ = 0 and G-dependent quark and gluon structure functions: 1

qns(X, ~) = f ~ qns(a, O) G~]q(X/g, ~) ,

(2.13)

X

q s(X, 0

=

)dzz [qs(Z,0) V~q/q(X/Z, ~) + 2fg(z, O) Cq/dX/Z, O],

(2.14)

X

1

g(x, ~) = f dz [qs(Z, 0) Gglq(Xlz, ~) + g(z, O) Gg[g(X/2, g

~)] .

(2.1 5)

X

The reader should note that the derivation of these expressions for the Q2 dependence of quark and gluon distributions in a hadron did not need to be carried out in

T.A. DeGrancl/ Structure functions in QCD

493

the context of the Altarelli-Parisi equations, but could have been obtained directly from the mixing matrix for anomalous dimensions which was first computed using the operator-product expansion. For our purposes, the coupled equations are the simplest approach to the desired result. This approach may also be extended to the case of fragmentation functions (following the work of Georgi and Politzer [11], and Owens [12]), for which

o~/.~(~, ~) = f

1

dZz[~. a.//.~(z,

~) D~/n/(x/z, O)

X

+

Gg/q(Z, ~) Dh/g(X/z , 0)] ,

Dh/g(x,~)

"dz ~

=

(2.16)

.=i Gqi]g(z, ~)Dh/qi(X/2, O) [i~_

X

+ Gg/g(Z, ~) Dh/g(X/z, 0)]

(2.17)

I01

G :,t (.9

io-I

1621

o

,

olz

o!4

x

' 0.6

' 0.8

' 1.0

Fig. 2. The structure function xGnS(x, ~), inverted by Yndurain's method, for (a) ~ = 0.05, (b) 0.1, (c) 0.2, (d) 0.3.

494

ZA. DeGrand/ Structure functions in QCD

i(y 2 A x

(..9 X

163

0

0.2

0.4

X

0.6

0.8

1.0

Fig. 3. Same as fig. 2, but xGunf(x, ~).

(followed by the usual non-singlet-singlet separation), where h is any hadron and the Gk//(x, ~)'s are the same quark and gluon structure functions of eqs. (2.5)-(2.9). To discuss the properties of the structure functions G(x, ~), it is necessary to invert their moments, eqs. (2.5)-(2.9). For arbitrary x, ~, this is a procedure which must be carried out numerically. Fortunately, recent theoretical developments have rendered this task quite straightforward: in particular, Yndurain [13] has devised a simple algorithm for expressing "smeared" (locally averaged) structure functions directly in terms of sums of moments. His method is simple and quick. I have used Yndurain's method to evaluate xGnS(x, ~), xGunf(x, ~), xGg/q(X, ~), xGq/g(X, ~) and xag/g(X, ~) for 0.06 ( X ( 0.96 and ~ = 0.05, 0.1,0.2, 0.3: the structure functions are displayed in figs. 2 - 6 . These structure functions display the qualitative features we have come to expect in asymptotic freedom calculations: as Q2 rises their large-x regions shrink and their small-x regions grow. The non-singlet structure functions never develop a dx/x tail, and the x ~ 1 peaking of G ns and Gg/g gradually dies away as ~ rises and soft ghion radiation slows down the leading quark or gluon. Gg/g is seen to fall much more quickly with increasing ~ at large x than G ns or even (at large ~) Gg/q. To quantify these features of the structure functions we now turn to approxi-

495

io-2

,o-al 0

,

i

,

~

0.2

a4

0.6

o.a

I, ,.0

X

Fig. 4. Same as fig. 2, but

xGg/q(X, ~).

io-' A

x-

I 0

Fig. 5. Same as fig. 2, but in the gluon.

xGq/g(X,

I 0,2

! 0.4

X

I 0.6

I 0.8

"1 1.0

~), the distribution of any one flavor of quark or antiquark

496

ZA. DeGrand / Structure functions in QCD iol~-

,,b ,g o, (.9 x

I I,J 2'~-

0

,ox/ ]o t

I

0.2

0.4

X

I

I

0.6

0.8

I

1,0

Fig. 6. Same as fig. 2, but xGg/g(X, l~).

mate, analytic evaluations of the moment inversion integral. These approximate forms will prove to be much more useful than the exact results, since one will be able to use them in convolution to discuss hadronic structure and fragmentation functions. Let us first take small Q2 limit of the structure functions. Writing 1 Q2 ~- ~-~ ac(/~2) l o g - ~ ,

(2.18)

and expanding to lowest order in ~, we find GnS(/",~) ~- 1 +

ore(U2) Q__2.2 27r log U2 ans(/),

Fac~a2), Gunf(] ", ~) ~ L ~ * o g T J

at(u2) Gg/q(j, ~) ~

Q2] 2

AgqAqg ,

Q2 2rr log ~ - [Agq] ,

(2.19)

(2.20)

(2.21)

497

T./h DeGrand / Structure functions in QCD

02

Gq/g(], ~) ~

%(/22) log ~ - A q g ,

(2.22)

2rr

0te(/22), Q2 Gg/g(j, ~) ~ 1 -t" ~ log

7Agg.

(2.23)

The reader will recognize these distributions as precisely the moments of the fragmentation functions obtained from the lowest-order graphs of fig. 1. It is reassuring to see lowest-order perturbation theory recovered in a particular limit of our formafism. Of course, the corrections to eqs. (2.20)-(2.23) are of order (%(/22) log(QZ//~2)) 2, and indeed, an expansion in that variable of our results would be equivalent to a summation of ladder graphs with 1,2, 3 ... rungs, retaining only leading logs, in analogy to the work of refs. [14,5]. Next, we turn to an exposition of approximate expressions for the evolution functions for small 1 - x = e. In that limit the inverse Mellin transform is considerably simplified.

C(x, f

c(j,

--ioo

ioo

~d] exp [je + log GO', ~)] .

f

(2.24)

The transform is sensitive to moments j ~ l[e -~ ~ . In that limit Am(j) ~ C2(~ - 27E -- 2 log j ) , Agg(j)

N ( - g - 27E -- 2 log j) -- ~ T ( R ) ,

X+ ~ A n s + ~ ,

X_ ~ A g g -

8 ,

= 2fAgqAqg(Ans - Agg) -1 ,

Aqg ~ 1 / 2 i ,

Agq ~ C2/]" ,

and 7E = 0.577 ... is Euler's constant. It is now straightforward to evaluate eq. (2.24) via steepest-descent. We fred eAl~

x)4C2~-1

(2.25)

Gg/g(X, ~) -- P(4N~-----S (1 - x ) 4N~-1 ,

(2.26)

Gn~(x' ~) - r(4C2~) (1 eA2

T.A. DeGrand / Structure functions in QCD

498

A 1 = (3 - 47E) C2, A2 = ( ~ - 47E) N -

(2.27)

IT(R).

(2.28)

Both of these expressions reduce to 6 (1 - x) as t ~ 0. At large t the distribution of gluons in the gluon becomes much steeper than the distributions of quarks in the quark. The "softening" of the gluon jet versus the quark jet occurs because the coupling of quarks to gluons involves C2(R), which is much smaller than the coupling of gluons into gluons, measured by N = (72 (G). It is amusing that this result dovetails nicely with the recent calculations of the width of a gluon jet (in a StermanWeinberg [15] sense) by Einhorn, Shizuya, Tye and Weeks [16], who found that gluon jets are broader than quark jets by the same constant of proportionality

C2(G)/C2(R). Also we note that quark jets retain a leading particle character until much higher t values than gluon jets do. If we define the point to where domination of the jet by a leading particle ceases by the value of t at which the exponent of (1 - x) vanishes, we see 1

3

1

to(q/q) = 4 ~ - 1 6 '

to(g/g) = ~

1

= ~~.

With Q~ ~ 1.8 GeV u and a conventional choice of A ~ 0.5 GeV these t's correspond to Q ~ 55 GeV and 3.6 GeV respectively. The evolution of one hard gluon into many soft gluons occurs quickly at a comparitively low value of Q2. Now we turn to the evolution of quarks into gluons, of gluons into quarks and of quarks into antiquarks. For Gq/gand Gg/q, t w o limits are interesting: the first, "small t " , when e 2x+~ and e 2x-~ are comparable, so e 2x-~ ~ e2X+~(1 - 2t (2~+ - X_)), and the second, "large t " , when e 2x+~ > > e 2x-~. In the first case

Gq/g(]', t) ~-

t e2Ans~ /

'

(2.29)

in the second case 1

Gqlq(],

t) --

e 2Ans~

2] (Ans -- Agg)'

(2.30)

SO that

eA~ 1) A ( t , e)(1 - x) 4c2~ Gq/g(X, t) - 2F(4C2 t +

(2.31)

where A (t, e) sm~fl t/2t large ~ 1/A(t) '

(2.32)

TA. DeGrand / Structure functions in QCD

499

where A(e) = 2(N - C2)(log(4C2/e) + "YE) + ~C2 - .-fiN+IX ~T(R) and e = 1 - x. Of course Gg/q(X, ~) = 2C2Gq/g(x, ~) in this limit. (The A(e) factors appearing in eq. (2.32) arise from an approximate evaluation of the steepest-descent integral,

I ~ f d l ' 2~ri exp[je+

4C2 log /1 - log(Am(/) - Agg(j) )1 ,

in which for large ] log(Ans(]) - A g g ( ] ) ) ~ log log ] is neglected with respect to the rest of the expression: the stationary point of the exponent is ]o ~ 46'2/e.) These distributions all vanish as ~ ~ 0 and always decrease monotonically with x. A comparison of (2.26) and (2.31) shows us the surprising result that for 1

~>

4 ( N - 6"2) + (Az - Al)/log(1 - x ) ' the leading partons in a gluon jet are not gluons but quarks! This result is not completely unreasonable when we recall that in lowest-order perturbation theory

G°/g(X) "-" x 2 + (1 - x) 2 ,

G°lg(X)

~

1-x

x

+

x 1- x

+ x(1

- x),

which favors the decay of a gluon into a hard and a soft gluon much more than into two gluons of nearly equal x, while the decay of a gluon into quark pairs favors this unsymmetric decay over a symmetric decay by only a factor of two. Fragmentation of hard quarks persists in higher orders of perturbation theory and eventually comes to dominate the large-x part of the spectrum. The "small ~" approximation works well for ~ < 0.05 or so and the "large ~" approximation is valid above that ~. These approximate forms reproduce the exact moment inversion results quite well for x i> 0.7 or so. The unfavored structure function is well-represented by its small-~ value over the entire range of physically interesting ~.

G'~/q(]', ~) "" 2AgqAqg~ 2 e 2nns~ ,

(2.33)

eAl~

G-~lq(X, ~)

~

C2~2 r(4C2~ + 2) (1 - x) 4c2~+1 .

(2.34)

The reader will note that the small-~ forms of the G's, eq. (2.31) and (2.34), have

T.A. DeGrand / Structure functions in QCD

500

a simple physical interpretation in terms of convolutions: 1

aud~, 0 = f az Z a"=(z,0 aG(x/z, O, x

where G o is the lowest-order structure function computed in eqs. (2.23)-(2.27). Apparently, in this limit, one can think of the structure functions of the quark evolving v/a two steps: soft gluon emission from the quark followed by the evolution of the quark into the desired parton, computed with lowest-order perturbation theory. At small x the moment-inversion equation (2.28) takes the form

xG(x'~)'~"6 f ff~'niexp[j-1¢ +In G(I, ~)1 ,

(2.35)

....

where e = (In(l/x)) - l is very small, and so it is dominated by the j ~ 1 behavior of G(], ~). It is an interesting and complementary exercise to extract the small-x behavior of the quark and gluon structure functions which are non-vanishing there, Gq/g(X, ~), Gg/g(x, ~), Gg/q(X, ~) and G'q/q(X, ~). As ] ~ 1 we find the behavior of the moments: Ans ~ 3C2 (J" - 1 ) ,

Agq

Aqg ~ 2a,

Agg

~ 2c~

j _ ]",

-

-3

_Aft2 ,

aN

j - 1

j - 1

~ N-

"3N2 : / _

5ct.

It is now straightforward to .compute the approximate behavior of the structure functions. As before, two ~ limits are of interest.

j -- 1 e2h+~ N

(2.36)

4C2~

Gg/q0', ~) ~ / _ _ _ e 21 X + ~ ~ - A C2

e2X+~ '

(2.37)

T.A. DeGrand / Structure functions in QCD

501

4N~ e2h+ ~ + 1 ]_ 1

Gg/g(l', ~) ~

big ~ e2X+~ "

(2.38)

The inverse of G = 1 is 5(1 - x), which is of course unimportant for small x. We wiU disregard it here, but will need to retain it if we convolute Gg/g over a distribution with a dx/x singularity. Eqs. (2.37), (2.38) show the familiar result [17]

lim xGg/g(X, ~) x~o

N

9

lim xGg/q(X,~) C2 4' x~O which can also be read off directly by comparing Agg and Agq above. Physically, the radiation of soft gluons from a source is independent of all properties of the source except for its color charge, and a gluon is 9 as "colorful" as a quark. Finally, the singlet distribution is e2~+~ GS(j, ~)~-C2 _~f~2 ~__~ + 1 .

(2.40)

This "small ~" form of the result should be accurate over the range of ~ which is physically interesting. These moment equations are most easily inverted when one recalls the integral form for Bessel functions: d]

1

=[y]l--p

exp[]e 1 +

where y = x/16N In(l/x), 1/e = In(1/x), and In 0') is the Bessel function of imaginary argument. It is a simple matter to read off the structure functions:

~,ig ~ ~ xGg/g(X,

e-~

4N~ In

~) =-~2xGg/q(X' ~) ~ big ~

xGSq/q(X, ~) "" 4~2C2 e - ~ Io(y) . At large y, In(y) ~ e Y / x / ~ y .

/20'),

(2.42)

4 ~ e-~Io(y) In

/rI ( y ) ,

(2.43) (2.44)

502

i0 I

~=O.I

\ \

io-~

,

0

0.2

0.4

0.6

0.8

1.0

x unf

Fig. 7. Exact and approximate expressions for (a) xG~q(X, ~), (b) (xG-~]q(X, ~)) X 10 and (c) xGg/q(X, ~) at ~ = 0.1. Straight lines are the exact expressions, short dashes the approximate expressions (or the small ~ approximations, when applicable) and long dashes, large ~ approximarion. iO I

\

~--0.1

oo IO-I ~

0

o I

0.2

I

0.4

0

I6

I

0.8

I

1.0

-

X

Fig. 8. Exact and approximate expressions for (a) xGq/g(X, Same labels as for fig. 7.

t~) and (b) XGglg(X, ~) at ~ = 0.1.

T.A. DeGrand / Structure functions in QCD

503

Figs. 7 and 8 compare the exact and approximate (large and small x) expressions for the structure functions of quarks and gluons at ~ = 0.1.

3. Quark, gluon and hadron fragmentation The asymptotic states of QCD are of course not quarks and gluons, but hadrons, and the distributions of quarks and gluons in hadrons provide the ~ = 0 "bare" distributions which are modified by asymptotic freedom. We first discuss fragmentation functions. Quark or gluon beams are produced in reactions such as, for example, e+e - ~ q ~ hadrons, p - p ~ / l - q X -+/a-hadron X (quarks); and (gluons) T -+ 3 gluons, r/v -+ 2 gluons or p~ ~ p+p- at large Pi (balanced by a gluon jet). In QCD, the properties of gluon and quark jets are quite similar at large ~, so we may discuss them together. First, at large ~, the momentum balance between quarks and gluons equilibrates in the jet to a value which is independent of jet type [9]. This may be easily read off the moment equations (2.3). In a quark jet qns(~ = 0) = qs(~ = 0) = 1, g(~ = 0) = 0 and qns(~) = e x p ( - ] C 2 ~ ) , qs(~) -

g(~)

4C2 e -x~ + f 4C2 + f

(3.1) ,

_ 4(72(1 - e -;~) 4C2 + f

(3.2)

(3.3)

In a gluon jet (q(~ = 0) = 0,g(~ = 0) = 1) qns(~) = O, qs(~) g(~)

f(1 - e -x~) , 4(72 + f

_ 4C2 + f e -x~ 46'2 + f

(3.4) (3.5) (3.6)

Here all q and g are shorthand for fd x dxf(x, ~), and x=s 2

~c2 + J .

The "time constant" for equilibration between quarks and gluons, X - l , is much shorter than the "time constant" for the non-singlet distribution to fall to zero, ~C2. This is to be expected, since we found in sect. 2 that the non-singlet structure function was very flat. Thus at moderate ~ quark jet evolution is distinguished from gluon jet evolution by the presence of a strongly leading fast parton (the valence

T.A. DeGrand/ Structurefunctions in QCD

504

quark) in addition to the cloud of soft gluons and quarks (which carry much of the momentum of the jet). The ~-dependent quark and gluon fragmentation functions are Dh/qi(X'

~) = ~ C1Z[Gfav(z, ~)Dh/qi(X/Z'

O)

Z

x

+ ~ Gunf(g, ~)Dh/qk(X/2, k~i

O) + Gg/q(Z, ~ ) D h / g ( X / Z , 0)]

,

(3.7)

ldz

Dh/g(X, ~) = f -'7 [Gg/g(Z, ~) Dh/g(X/Z, O) x

+ .~= Gqi[g(g, ~) Dh/qi(X/Z, 0)]

.

( 3.8)

At large x we know experimentally that Dh/q(X, 0) is large only for "favored" fragmentation: u ~ rr+, and not u ~ rr-, for example. Similarly, we have seen that at large x for all ~ the largest G-dependent fragmentation function is Gfav(x, ~). Thus for large x, ~ it is possible to neglect all fragmentation but favored fragmentation and still have a good representation for the fragmentation function. A convenient (though not a unique) parametrization for favored fragmentation at SPEAR energies at large x is

XDh/q(X, O) '~ 0.5(1

-- X).

(3.9)

With that distribution one can give a simple prediction for the behavior of the quark fragmentation function at large x, Q2:

Dh/q(X, ~) ~ f dz Gfav(g ' ~) Dh/q(X/Z, O) x

Z

= 0.5 e AI~ (1 - x) 1+4C21~ r (2 + 4(72 ~) '

(3.10)

The steepening of the inclusive hadronic spectrum may be visible by the upper range of PEP or PETRA, where Q ~ 30 and 4C2~ ~ 1 (with A = 0.5 GeV and Q~ = 1.8 GeV2). At large x, ~, we saw that Gg/q(X, ~) becomes comparable in size to Gg/g(X, ~), rising linearly with ~ at very small ~, and as x ~ 1 it vanishes much more slowly than Gg/g. Thus at large x, ~, gluon fragmentation into hadrons is dominated by the

T.A. DeGrand / Structure functions in QCD

505

sequential decay gluon -~ quark -> hadron: ldz

f z

O)

x

=

~

0.5 eAl~(1 -- X) 2+4C2~

i = favored

['(3 + 4C2~ ) A(e)

'

(3.11)

with the ~ = 0 quark fragmentation function given above. For mesons, the sum is equivalent to an overall multiplicative factor o f 2: for instance, g ~ zr+ proceeds via g -* u~, u -+ 7r+, or g -> dd, d --> rr+. At large x, ~, gluon fragmentation functions are one power o f (1 - x) steeper than quark fragmentation functions, apart from a very weak (log(l/1 - x)) -1 behavior at x = 1. This important result has long been part o f the counting-rule folklore, but as far as I know, a derivation from QCD has not been given until now. Moreover, the constant o f proportionality is known as well, so that gluon fragmentation at large Q2 is known uniquely in terms o f the quark fragmentation functions at low Q2.

10-2_

10- 3

.?

io -4

0

C

10-~

i

,

0.8

li.O

x Fig. 9. The gluon fragmentation function at ~ = 0.13. (a)

Dh/q/g(X)~th Dhlq of eq. (3.9).

(b) Dhlg/g withg = 2 and (c) g = 1, where g is defined in the text.

ZA. DeGrand/ Structure functions in QCD

506

A comparison o f figs. 5,6 will show the reader that at moderate x, Dqlg(X, ~) > > Dg/g(X, ~) only for very large Q2 (~ > 0.2). The results of this section may still be applicable to the T(9.4) ifDh/g(X, 0) is itself much steeper than Dh/q(X, 0), for then the favored fragmentation channel will still be gluon ~ quark ~ hadron. For instance, we may parametrize

xOh/g(X, 0) ~ c(1 + g)(1 - x) g ,

(3.12)

so that

Dh/g/g(x' ~) larg'ex,~ceA2~ F(2P(1 + gX1 - x) g+4N~ + g + 4Ng)

(3.13)

A comparison Of Dh/q/g and Dh/g/q at the T(9.4) (~ = 0.13) with c = 0.25 a n d g = 1 or 2 is shown in fig. 9. We see that if at ~ = 0 gluon fragmentation is intrinsically steeper than quark fragmentation, T fragmentation is dominated by Dhlqlg; if ~ = 0 gluon and quark fragmentation are equal, the two sources ofhadrons are roughly comparable at large x. At any rate, it will be worthwhile to compare the inclusive hadron spectrum at the T. to the inclusive spectrum at the ~k: the former should show a steeper large-x falloff than the latter. The real test will come, of course, if a new q~ bound state, with a mass greater than 20 GeV or so, is discovered. Detailed QCD calculations of the G-evolution of quark and gluon jets v/a moment analyses are now being carried out [12]. The reader is referred to them for calculations of quark and gluon fragmentation functions for all x and ~.

4. Qualitative behavior of hadronic structure functions Of course, the acme of QCD perturbation theory is the calculation of the evolution o f hadronic structure functions as seen in deep inelastic scattering, and many detailed studies of this subject have been carried out *. This is not the place for another long calculation: rather, I will discuss the qualitative effects o f asymptotic freedom structure functions and then apply these results to the derivation of simple counting rules for use, for example, in large-p± scattering.

4.1. Structure functions The discussion here will be brief, since many of the results I obtain are quite well known in the literature. At ~ = 0 one may parametrize the valence, sea and glue distributions in the proton as

x V(x, O) ~ x P(1 - x) v , * A partial summary includes the papers of refs. [18,9,10].

(4.1 a)

ZA. DeGrand / Structure functions in QCD

507

with f V(x, 0) dx -- 3 , or alternatively at large x , (4.1b)

x V ( x , O) ~ Vo(1 + oX1 - x) ° , and x S ( x , 0) = So(I + s)(1 - x) s ,

(4.2)

xG(x, O) = Go(1 + g ) ( 1

(4.3)

-

x)g.

So and Go are the fractions of the proton's m o m e n t u m carried by the sea and by gluons. As Q2 increases, the structure functions evolve. At large x, it is easy to perform the convolution integrals and extract the behavior of the structure functions. The valence distribution evolves v/a the favored quark structure function F(2 + v) e AI~ (1 - x) u+4c2~ V(x, ~) lar~ x Vo P(1 + v + 4C2~)

(4.4)

The gluon distribution at large x receives contributions from two sources: the glue itself evolves without quark interactions 1-'(2 + g) e A2~ Gg(x, ~) ~ G O 1-'(1 + g + 4Ar~) (1 - x) g+4N~ ,

(4.5)

and from bremsstrahlung from the fast (valence) quarks: Gv(x, ~) ~ Vo A (x, ~) C2 e Al ~

P (2 + v) (1 - x) 0+4C2 ~+1 , r ( 2 + v + 4C2~)

(4.6)

where A (x, g) is given in sect. 2. The sea evolves from favored evolution from itself Ss (x, ~) "~ So

F(2 + s) eAl~(1 -- x) s+4c2~ , P(1 + s + 4C2~)

(4.7)

from evolution of the glue into q~ pairs Sg(x, ~) ~ 2T(R) Go r ( 2 r+( 2g ++ g) 4C2~) e AI ~A(x, ~)(1 - x) g+4c2~÷l

(4.8)

and from unfavored evolution of the valence distribution Sv(x, ~) ~ Iio 2fC2~ 2 e A1 ~

P(2 + v) (1 - x) ~+4c2~+2 . 1-'(~ + v + 4C2~)

(4.9)

At large x, ~ the non-valence distributions will be dominated by the evolution of

T.A. DeGrand / Structure functions in QCD

508

the valence partons, so that

G(x, ~) V(x, ~)

(1 - x ) ,

(4.10)

S(x, ~) (1 - x ) : , V(x, ~)

(4.11)

results which should be quite familiar to the reader. To extract the behavior of the hadron structure functions at small x, it is most economical to invert the product of the hadron and parton moments, rather than to perform the convolution directly. In this range o f x the ] = 1 moments are probed: inverting eqs. (4.1)-(4.3) we find S(/)

So(1 s____...~), + ]-1

V(]) "~ 3 .

G(/') ~ G ° ( 1 +g) , ]-1

(4.12)

One can then take the product of these moments with the small-x parton structure function moments and use eq. (2.41) to read off the behavior of the structure functions: gluons from the valence quarks:

Gv(x, ~) = 3Gglq(X, ~) ;

(4.13)

gluons from gluons:

xGg(x,~)-xGg(x, 0)sm~/i ~ Go(1 + g)4Ar'~ e - ° ~ ( 4 N ~ ( l n l ) 3 ) - l l 2 I i ( y ) b i ~ Go(1

+g)Io(g) e - ~

;

(4.14)

and gluons from the sea (negligible compared to gluons from gluons): 4 So(1 + s) Gg(x, ~) ; Cs(x, ~) = ~ Co(1 + g)

(4.15)

singlet quarks arise from gluons:

XSg(X, ~) sma'll ~ G0(1 + g) 4f~ e - ~ l o ( y ) big"~ Go(1 +g) f-~.e-a~(4N~(ln~3)U2'II(y);

31V

\

\

XI

(4.16)

(This is presumably the source at high Q2 of flavors which are absent in the proton at low Q2: charm and b quarks.) and from the singlet distribution:

T.A. DeGrand / Structure functions in Q CD

509

> o

o~-4

-6

I 0.6

I

I 0.8

I

I 1.0

x

Fig. 10. Proton valence quark distribution for (a) ~ = 0, (b) ~ = 0.1, (c) ~ = 0.2.

and from the valence distribution:

Sv(x, ~) = 6fGunf(x, ~) .

(4.18)

(One must n o t forget to include the constant terms o f the m o m e n t s o f Gglg and G s, since the bare glue a n d sea have dx/x tails.) It is o f interest to graph the approximate structure functions at large and small x.

2K,

o -,

\,, \

/1

6

0

I 0.2

I

I 0.6

I

I 1.0

X

Fig. 11. Glue distribution for ~ = 0, 0.1. Straight lines are XGg(X); dashed lines, xGv(x).

T.A. DeGrand / Structure functions in QCD

510

o _'..<

2

~-d-4

I 0.2

I x

I 0.6

t

I 1.0

Fig. 12. Sea distribution for ~ = O, O.1. Straight lines are xSs(x), short dashes xSg(x), long

dashes xSv(x).

To do so, we take o,g and s to be given by their counting rule values, 3, 5 and 7, Go ~ 0.4, So ~ 0.1, and Vo = 1.26 (a large-x approximation to the ~ = 0 BurasGaemers [19] valence structure function). Typical curves for renormalization group improved valence, glue, and sea distributions are shown in figs. 1 0 - 1 2 , for ~ = 0.1. By only moderate ~, the large-x glue distribution is dominated by bremsstrahlung from the valence quarks. Equally rapidly, the sea comes to be dominated by gluon and valence evolution into q~ pairs. Only by ~ ~ 0.2 is the large-x sea distribution due primarily to valence evolution. By that point the valence distribution has steepened by one power of (1 - x), with this choice of o, g and s, and the sea has softened to (1 - x) 6. At higher ~, the three distributions lock into their asymptotic values V : G : S =1 : ( 1 - x ) : ( 1 - x ) 2. Finally, it is of interest to compare our approximate formulas to more accurate calculations. The work of Buras and Gaemers [19] is particularly relevant, as they give fits to the valence, glue and sea structure functions of the form (note the (1 x) 1° bare sea and glue) -

V(x, ~) ~ xP(~)(l - x) Q(~),

xG(x, ~) ~ (1 - x) 1°+~(~) , xS(x,

~) ~ (1 - x ) ~°+~'~) .

ZA. DeGrancl/ Structure functions in QCD

511

They fit Q(~) = 3 + 5C2~; the simple asymptotic formula gives Q = 3 + 4C2~, rather good agreement. The values o f ~ and ~"they quote involve coefficients of C2~ which are enormous, over 10, and grow with ~, in contrast to the small coefficient (4) of the asymptotic formula. This discrepancy is spurious and is due to the small-x behavior of the quark and gluon structure functions. At small x the functional form ofxG(x) or xS(x) is not (1 - x) n, it is (log(I/x)) k expx/A log(I/x). The effective (1 - x) power of such a distribution becomes infinite as x vanishes and remains large for x ~ 0.1 to 0.2, the range o f x over which the gluon and sea distributions are not van±shingly small and the range o f x (presumably) for which the powers and ~'were fit. One can see this result explicitly in the small-x region of figs. 11 and 12. A better approach for future detailed calculations of asymptotic freedom corrections to structure functions would be to fit effective powers only at very large x, and to fit the glue and sea structure functions to functions of log(l/x) at small x.

4.2. Large-px hadronic scattering and counting rules As a final example of the utility of our fragmentation formalism, let us consider large transverse momentum scattering in QCD. This field has also received much attention of late [20], and it is once again not my intention to attempt a detailed calculation. Again, my aim is more modest, to provide simple analytic expressions which account qualitatively or serniquantitatively for the behavior of the differential cross sections. These expressions will be given in the form of counting rules: Ed__~p ( d o AB ~ CX)-~ (1 --pN xay~ (log p~)Z, a(~) ,

(4.19)

at 0e.m. = 90 °, X = 2pJ./X/S. In the parton model N and F are integers and L = 0; in QCD F may be a function of 1 lo [%(Q~)X

and L = - 2 . These effects are the result of scale violations in the structure function and in the running coupling constant of the subprocess scattering cross section. (It is odd that QCD-based counting rules have not been given before, only the results of elaborate computer programs, when one considers their utility in interpreting large-p± physics for 2 < p ± < 8 GeV/c. Presumably, that is because so far all experiments at large enough p± to raise QCD above the CIM background [21 ] have involved small x±, and counting rules are most trustworthy at 1 - x± small.) Counting rules are derived by taking the standard parton convolution formula, 1

/"

CX)= abe Ef

1

0

1

O ~

6 a / A ( X a ) Gb[B(Xb) Dclc(Z)

512

T.A. DeGrand / Structure functions in QCD

X ~_ d e 7r

^

~tt (ab ~ cX) 6(g + t + a) ~ = XaXbS , f = Xat/Z Iff = XbU/Z

S : (PA + PB) 2 ,

t = (PA -- PC) 2 ,

(4.20)

u : (PB - Pc) 2 ,

scaling all external invariants in terms ofp~, and taking the limit x±-+ 1 to extract the leading power of (1 - xi). With the parametrizations d o = frO(Q2 ) s _ N ( _ t / s ) _ T(_U/S ) - U dt

(4.21)

with Q2 some combination of s, t and u, which govern the point at which % is measured, and x G i / l ( x ) = (1 + gi) Gi/l(1 -- x ) gi ,

(4.22)

we find for no final-state fragmentation [22] do E-~p-p (AB ~ cX)

~.F D (P.~) XR'~I

pN

~

Ga/AGb/B

F(2 +ga) F(2 + g b ) 2 p j F(2 + ga + gb)

'

(4.23)

where e = 1 - XR

XR = 2 IPcl/X/s,

,

F=ga+gb+

X F = X R COS 0 ,

p =ga +gb + T + U + 2 - 2 N ,

1 ,

and j=L 1 Jo / dw(1

+ w)ga(1

-

W)gbf 1 + XF+CW1T+I+gb_N L 1 + XF-- e]

--1

f l -- X F -- e W 7 U+l+ga-N D(Q 2) D(p~)

X L l _ XF_ e J

(4.24) '

1

J0 = /

dw(1

+ w)ga(1

-

w) gb •

(4.25)

-1

The useful approximate form of (4.22) is attained by evaluating J at e = 0, J = 1. The evaluation of the integral in QCD differs from that for a scaling theory in that ga and gb are ~ dependent, and ~ is a function of the invariants of the subprocess, 5, i or t2. However, this dependence is very weak, ~log log i = log log Xat, for instance, so as long as x a and xb are bounded, ~ does not depend on them very much

T.A. DeGrand / Structure f u n c t i o n s in QCD

513

at all. Now in eq. (4.20) 1 - e + XF Xa = 1 + X F + e W ,

1 -- e -- X F

(4.26)

1 - - X F - - e.W '

Xb-

and ga, gb are large numbers (>3). Thus the integral is dominated by w ~ 0 or (Xv = O) (Xa) ~ (xb) ~ 1 - e = x ± = 2 p x / ~ / s , which is non-zero. Further, the bounds on the integral, - 1 < w < 1, insure that neither Xa or x b ever vanishes. We can evaluate hadron scattering using eq. (4.22), taking ga and gb to be functions of a ~ variable which is a function of the kinematical invariants (s, t, u) of the reaction. Similarly, the Q2 governing % is nearly a Q2 determined by the external kinematics of the problem: Q2 ~ p~, times (nearly) irrelevant factors on the order of unity. If particle c also fragments, with xDc/c(x,

(4.27)

~) = d(1 + ge)(1 - x ) gc ,

one can show by similar arguments that

do

E~

(AB

-+

cX)

E do

da p (AB-~CX)

largex±

0cm = 90 °

degc+ 1 I"(2 + ga + gb) 1"(2 + gc) I"(2 + ga + gb + gc)

(4.28)

To illustrate this procedure, consider the reactions pp ~ jet X, 7r°X mediated by vector gluon exchange between valence quarks. The appropriately symmetrized subprocess cross section, summing over flavors while taking into account towards and away side jet triggers and beam-target symmetrization, is [23] d t s 2 d ° ( j e t ) e f f =47rOtAEs2+u2

L--7~ 4

do

7ro.

sZ+t 2

.2

5_2s~1

9 3 tu J '

1 do

-~'( )eff =-2 ~(jet)eft.

(4.29) (4.30)

Using the large-x valence distributions of sect. 3, we have -=-de -+jet X) ~ a2c(p~) V2 e2Al ~ (F(O + 2) 2(u+4C2~))2 E dOp (PP p4 P(2 + 20 + 8C2~ ) × ~ ~(123 _ x_O2v+l+sc2~ ,

(4.31)

T.A. DeGrand / Structure functions in QCD

514

E do ,, - - ~ fro.R,.~ ,~o~(p~) V2 d e3A1 ~ (r(o+ 2) 2(°+4C21~))2 d-~ptPP "', p4 r(3 + 2v + 12C2~) X r(3

+ 4C2~) 95-~'( 1~"2 3

__ Xj.)2o+3+12C2~

(4.32)

These approximate forms for the cross sections at x/s = 53 GeV are shown in fig. 13. Also plotted is the result o f a numerical integration o f (4.20) for jet cross sections using Buras-Gaemers [19] valence structure functions, where the Q2 governing the coupling constant has been taken to be (~h~) 1/3, and the inclusive jet and rr° calculation o f Horgan and Scharbach [20], which includes qq ~ qq, qg ~ qg, and gg -+ gg subprocesses. Agreement between the approximate and exact results is extremely good, especially for P_L> 10 GeV/c. It would appear that eq. (4.21) is a most economical way to represent the results o f large-p parton model calculations. The other components of pp scattering: gg -+ gg, gq ~ gq, may also be computed. For pp --*jet X, these two subprocesses yield sums o f terms for Eda/dap, since the glue distribution at high g evolves from b o t h the non-singlet quark and glue distributions at low ~.

-5

pp --~ J e t X , ' r r * X

\ b

\

o~k~N\~

4"~= 5 3 GeV

0c.m.=90*

NJ

~ 0

c

i

E

% o -II

-13

Fig. 13. Edo/d3p

I 4

I

I I I 8 12 P.I. (GeV/c)

I

I 16

for the reaction pp - * j e t X, pp ~ lr0X at ~/s = 53 GeV, with the QCD subpro-

cess qq -* qq. (a) Jet cross section calculated by a numerical integration of eq. (4.20), (b) jet cross section using the counting rule formula, eq. (4.31), (c) 7r0 cross section using eq. (4.32). The long-and-short dashed curves are the jet and ~r0 predictions of ref. [24], which include the subprocesses qq -* qq, qg --, qg, and gg -* gg.

Z A. DeGrand/Structure functions in QCD

515

At large x the exponent of(1 - x) for gg --> gg -->jet rises from 2(g+ 4NIj) + 1 at low ~ to 2(v + 4C2~ + 1) + 1 at high ~, and the exponent for qg -+ qg from v + g + 4 ( N + C2) ~ + 1 to 2(v + 4C2~) + 2, so these contributions are unimportant at large x±. Gluon final states and less important than quark final states at large X_L, for producing fast mesons, because the proton has many more fast quarks than fast gluons and because (recall sect. 3) gluon fragmentation is steeper than quark fragmentation. So pp -+ 7r°X is dominated by qq -+ qq scattering at much smaller x± than pp -+jet is, as we see comparing my curves in fig. 13 to those of Horgan and Scharbach. Similarly, counting rules for other reactions may be easily obtained. For instance, in p~ Drell-Yan production of a massive dilepton pair, valence quarkvalence antiquark annihilation dominates the cross section and

~ e•Filp(X, ~) FPl~(X, ~) 11[2 - ~ -do ~ - Y (PP"-+/a+U- X) [y=o = 41r32 i=u,u,d

L

3'

(4.33)

where x 2 = r = M2u~/s. Drell-Yan production for pp collisions is a sum of three terms, since the large-~ antiquark distribution in the proton arises from the evolution of low-~ valence, sea and glue distributions. At very large ~ the exponent of (1 - x) is 2V+ 2 + 8C2~, since there the fast sea is due to unfavored emission from valence quarks. The reader can doubtless imagine many more applications of the counting rules.

5. Conclusions

In previous sections, I have outlined a simple alternate formalism in terms of Q2 dependent structure functions for quarks and gluons in quarks and gluons which is useful for the calculation of QCD effects in hadronic structure and fragmentation functions. It is completely equivalent to the usual moment analysis of asymptotic freedom evolution of parton distributions, but is much easier to use in performing qualitative calculations in certain regions of phase space. Thus the method discussed in this paper should be of particular use as an adjunct to detailed numerical calculations of QCD effects, as preliminary estimates to determine the magnitude of a reaction, and as an aid in the derivation of simple analytic expressions which reproduce the major features of the more exact calculations. The analysis of parton distributions in hadrons in terms of parton distributions in quarks and gluons can also be used to derive old and new results by itself, without the need to carry out elaborate numerical calculations. The best examples of that kind of result shown in this work are the large-~ behavior of the gluon fragmen-

516

T.~ DeGrand / Structure functions in QCD

tation function, Dh/g(x, ~) larg~ex, (1 - x) Dh/q(X, ~) ,

a result often conjectured by phenomenologists but never before derived, the (related) behavior of the quark and gluon structure functions at large x, aq]q(X, ~) ~

( l - x ) 4C2/~-1 ,

Gg/g(X, ~) ~ (1

- x) 4N~-I ,

etc.,

and the G-dependent counting rules which so simply parametrize large-p± reactions. The study of quark and gluon structure functions carried out in this paper could be extended in several directions. First, only spin-averaged structure functions have been computed. It is a simple matter (since the approximate anomalous dimensions and projection operators are well-known [25,8]) to repeat the analysis for the evolution of polarized quark and gluon distributions. I have not chosen to do that here because most of the QCD reactions which one imagines in the near future involve spin-averaged quantities. Nevertheless, we have recently seen on the one hand experimental measurements of polarized deep inelastic scattering [26] and on the other, theoretical calculations of large-p± scattering of polarized protons [27], so that the computation is certainly worth performing. The main theoretical difficulty here lies in formulating a good model for the distribution in x of partons of various spin states at ~ = 0 (el. ref. [28]). Second, the quark and gluon structure functions discussed here are derived from a moment analysis which involves only the lowest-order evaluation of the/~ and 3' functions. As higher-order contributions to 13and 7 are now being computed [29], it will be again a fairly straightforward exercise to calculate the structure functions of quarks and gluons, including higher-order effects. But both of these extensions are merely teChnical. Neither obscure the simplicity inherent to a discussion of hadronic structure function evolution in QCD in terms of the evolution of the constituent quarks and gluons. It is well-known that in QCD the quarks and gluons seen at some Q2 evolve independently of each other into the quarks and gluons seen at a different Q2. A formalism which describes this independent evolution as directly as possible in terms of x-dependent distributions of partons in hadrons and partons in partons would seem to be particularly economical. The results obtained in this note show that it is a formalism which is extremely easy to work with, as well, since in many cases the necessary convolution integrals can be performed almost by inspection. I would like to thank Bob Sugar and Doug Toussaint for many valuable conversations and Stan Brodsky for informing me of his calculations prior to publication. This work was supported by the National Science Foundation.

Z A DeGrand / Structure functions in QCD

517

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