Anomalous dimensional quark counting of hard processes in QCD

Nuclear PhysicsB223 (1983) 81-103 O North-Holland Publishing Company

ANOMALOUS DIMENSIONAL QUARK COUNTING OF HARD PROCESSES IN QCD I.S. AVALIANI*, V.A. MATVEEV and L.A. SLEPCHENKO *Joint Institute for Nuclear Research, Dubna, USSR

Received 1 June 1982 (Revised 9 December 1982) An approach to the investigation of hard jet and single-particle production asymptotics in hadron collisions is proposed. In the leading order of QCD perturbation theory, the anomalous dimension quark counting (ADQC) rules are derived, which determine the logarithmic corrections to the point-like power asymptotics in terms of anomalous dimensions of the non-singlet and singlet quark and gluon operators. It is argued, that taking the quark distribution and fragmentation functions up to the two-loop order does not destroy universality of the proposed ADQC rules. A parameter-free solution for the effective power exponents in a wide class of hard processes is obtained.

1. Introduction Usually the h a d r o n - h a d r o n reactions at large transverse m o m e n t u m can be explained assuming that the large P r jets arise from hard scattering between constituent particles (partons) in the incoming hadrons [1]. The first application of asythptotically free gauge theory [2] in this direction has been found in ref. [3]. The observation of scale breaking effects in ep,/~p and vp scattering being in agreement with the Q C D predictions suggests that we try to parametrize the constituent distributions in a manner consistent with Q C D and to include it in large PT production calculations. Such attempts are made in refs. [4-6]. In the recent note [7] concerning scale breaking phenomena in the large transverse m o m e n t u m hadron production, the anomalous dimension quark counting rules has been proposed. It has turned out that the scaling violation rate of the corresponding invariant cross sections is related to the number of valence quarks participating in the hard scattering. Note, that these results were derived in the leading order of Q C D restricted to the quark scattering subprocess and the so-called non-singlet part of quark distributions, which dominates in hadron scattering at large XT. On the other hand, the relative importance of the flavor singlet sector of quark and gluon densities, especially at low x is commonly acknowledged from QCD. Because the Q2 dependence is different for different parton distributions, the singlet * Institute for High Energy Physics, Tbilisi State University, Tbilisi, USSR. 81

LS. Avaliani et al. / Hard Processesin QCD

82

contribution can play an important role in derivation of the counting rules for inclusive reactions. The purpose of this work is to extend the method, proposed in ref. [7] to cover the singlet part of the structure functions and all allowed parton subprocesses as well. In sect. 2 we reconsider the 0 2 evolution of the DIS structure functions in the parametrization needed for our purposes. Then, we briefly repeat the results: for the cross-section asymptotics in the non-singlet case [7]. Contributions of the singlet parton densities are presented in sect. 4. Sect. 5 deals with the higher order perturbative corrections to the one-particle (jet) invariant cross sections. Finally, in sect. 7 we discuss the problem of the asymptotic pv 4 law in hadron collisions in terms of the obtained quark counting rules.

2. QZ development of deep inelastic structure functions The structure functions of deep inelastic lepton scattering in the parton model satisfy approximate Bjorken scaling and Q C D then predicts the deviation from exact scaling, which manifests itself in the 0 2 dependence of different parton distributions. This behaviour can be easily understood in terms of the so-called evolution equations for the quark and gluon densities [8]: OlnO 2 -

OG(x, 02) alnO 2 -

2~"

y

pqq

as(O2) f / dy[ 2~r

Y

qi(y'O2)+PqG (y) 2r

pGq

i=1~-'qi(y,O

G(y'O2)

'

2 (y) ] )+Pc~ G ( y , O 2) .

(2.1)

These equations account for the effects of scaling violations in parton structure functions arising in perturbation expansion of renormalizable gauge theory. Here the sum runs over the quark flavors and Pii(z), i, f = q, G are probabilities of the corresponding q,"-~Gq transitions. Eqs. (2.1) are valid strictly for the massless quarks. Let us consider an arbitrary structure function F(x, 02) = ~ xq(x, 02). It can be split into a singlet and non-singlet pieces. By making a separation between valence and " s e a " quark densities (for the SU(f) symmetric sea)

q(x) =qv(x)+qs(X), qv(X)

= u, d,

qs(x) = u, •, d, d, s, g. . . . . .

(2.2)

Eqs. (2.1) reduce to the three integrodifferential equations for the q~(x), q~ (x) and G(x), respectively. The first one is determined by the operators, non-singlets in the flavor symmetry group, and is decoupled from two others. The latter equations are satisfied by the qs and G densities, which do not carry flavor quantum numbers

LS. Avaliani et al. / Hard Processesin QCD

83

and are determined by mixing between flavor singlet and non-singlet parts of the corresponding operators. It is important to note, that solutions of eqs. (2.1) require the knowledge of the definite boundary conditions. If we know the functions q(x) and G(x) for some Q2 = ( ~ (as(Q2)/2zr << 1) then the latter equations enable us to obtain the distributions q(x, 0 2) and G(x, 0 2) for arbitrary values of O2> Q2. Following the prescription of ref. [7] we choose the form of x dependence of the structure functions at large x and fixed Q2 = O2 as dictated by the dimensional quark counting (spectator rule) [9]*

xF(x, Q 2 = Q 2 ) _ ( l _ x ) 2 .

3

x--,1

(2.3)

where n is the minimal number of hard constituents in the hadron:

Mesons ---0--

n=

Baryons nv

n = ns

=

2

----O

n=nv

:

L

~

n = ns~

= 3

5

Fig. 1. Pictorial representation of the bound-state hadron wave functions (quark-gluon content).

and nv, ns, n a are the power law exponents in (2.3) for the valence, sea-quark and gluon distributions, respectively. Solutions of the evolution equations (2.1) for (2.2) with the boundary conditions (2.3) now can be easily obtained, e.g. by Meilin transformations of the corresponding quark and gluon densities 1

Fi(n, Q2) = f0 x"-2 dxFi(x, Q2),

i = v,s, G .

(2.4)

We write down these solutions in appendix A**. For the Q2 development of different parton densities, contributing to the x - 1 region, we have:

xF(x, 0 2) = K (~:)(1 -x)Z"v-3+~eH(x, ~j),

(2.5)

* Up tO 2Ah = 2(hl -hz), where h l and h2 are hadron and quark helicities, respectively. ** See also the results of ref. [10]. In addition in the present paper contributions of the non-leading terms are included, dominating at low values of x.

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LS. Avaliani et al. / Hard Processes in Q C D

(q~(x) ) F(x, O 2) = G(x) , qs(x) 1 F ( 2 n v - 2 +r~) 2(

lt (x, ~) =

(l-x) F ( 2 n v - l + r()[ln [ 1 / ( 1 - x)]+ qt(2n,,- l + r~) + C ] (1 - - X )

1 +Sr~:

F ( 2 n ~ - 1 + 9r~)[ln (1/[1 - x]) + q~(2n~- 1 + 9r~:) + C] ~r~(1 - x ) 2

r(2nv+r~)[ln (1/[1 - x ] ) + !F(2nv+r~)+C] where ~¢= In [as(O2)/as(OZ)], f = quark,

r = 16/(33 - 2 f ) ,

21-2/

flavor n u m b e r ,

2 2 r(~i K (~) = [as(OO)/as(O )]

y E = 0.5772 . . . .

C = ~'E - - , 20 ~/E)

~F(2nv - 2).const,

gt(z)=dlnF(z)/dz

are Euler's number and the digamma function, respectively. Note, that in the framework of leading order Q C D perturbation theory, we can consider the O z development of the parton fragmentation functions on equal footing. The difference between F(x, O 2) and D(z, O 2) which arises beyond the leading order corresponds to the breakdown of the Gribov-Lipatov reciprocity relation [11].

3. Anomalous dimension quark counting: non-singlet case In this section we repeat the arguments of ref. [7] and give a general formula called in the following the anomalous dimension quark counting rule. It determines the leading log corrections to the canonical point-like asymptotics [9] of an arbitrary particle reaction at large transferred momentum in terms of active and passive quarks of the hadrons participating in the relevant reaction. The invariant cross section for the large-pT hadron (jet) production in two-hadron collisions AB -~ C(J) +. • . reads O'[AB __)jetX) ~ a~ , ,c

MindXa

Mbin dx~Va/A(Xa)V(x,)

x (g/~r)6 (g + f + tT)(d(~/df)ab~. o- =-E do'/ d3p .

~ dxc (3.1)

I.S. Avalianiet al. / HardProcessesin QCD

85

w h e r e (dd-/df)ab~ is the hard parton scattering cross section a, b, c = q, C:1, G and in the jet k i n e m a t i c case = (Pa 4- 10b) 2 ~-XaXbS, min Xa =xl/(1

l' = (loa--Pc) 2 ~ X a t , rain Xb = X a X 2 / ( X a - - X 1 ) ,

-x2),

t~ = ( p b - - l o c ) Z ~ x b U , Xl-

H

X 2 --

S

t S

In the leading log a p p r o x i m a t i o n the initial valence quark distribution function (non-singlet part of (2.5) in the x -~ 1 region has the following form F a / A ( X a , 0 2 ) : C AVK( JAk()~ ) ( 1 - - X a ) A - 1 '

F(x)=-F~S(x)'

(3.2)

where

A=A+rC~=2(nA-1)+rCs,

~ = - l o g c~s(O2),

KA(~) = exp {r~( 3 - yE)}F(A) and nA=--na is the valence q u a r k n u m b e r in h a d r o n A. Inserting this expression (and similarly the Fb/B function) in the hard scattering ansatz (3.1), we find in the jet p r o d u c t i o n case a~') 2 exp {2r~:(~-TE)} o-(AB ~ jetX) = ( ~ T / F(A)F(B) C A C B F ( A ) F ( B ) ( X l ,

j=(1-Xl-X2)

[

x2)2j,

= £1Ildudv "u ~ lvR 16 (1-u-v)+(l_Xl)(l_x2)Uv

[1

l-x2

u

1

l XlX2] 1 -x

1

1

V

= (1 - - X 1 - ~(A, B) (1 --X2)A(1 --Xl) B XFl

(

1-x,-x2

B'*-2'4'A+B;(1-x~)(1-x2)'

l~xl~x2~ l-x, /'

w h e r e ~ (c~,/3) is E u l e r ' s beta function and F l (o~, ~,/3 ', T ; X, y ) is the h y p e r g e o m e t r i c function of two variables [12] (see a p p e n d i x B). 1 In the case of 0 = 90 ° , Xl = x 2 =sxq2

CACBF(A)F(B)F(A +B) ~'A+R 1,r,Lc~stpT)j, 2,1alIA+B, 2/,a+~,,,

cr(AB--, jet) = (c~/,J

~-'}@,

where (~

4

=XT(1--

1 2XT)FI ,

F1 -~Fx(B, A -2, 4, A

1 e ' = (1--XT)/(1 --SXr), + B ; e ' / ( 1 - 21XT), e ' ) .

(3.4)

(3.3

l.S. Avaliani et al. / Hard Processes in OCD

86

Thus, in the limit xv-~ 1 we have o'(AB ~ jet) = o'0

CACBF(A)F(B)

E A+B

I[~2.S(p2T)]2D(A+B)

,

(3.5)

F ( A + B)

where as)22A+B

e = 1 --XT

=

A ( B ) = 2(n~a(m - 1),

1-2px/~/s,

+log e ) ,

D(n)=d(n)-r(1/n

2n(n + l)

d ( n ) = - d Ns = r

=

'

is the one-loop anomalous dimension of the non-singlet quark distribution function. For the arbitrary hard asymptotic process the following general formula is valid [7] EdO'(AB_+CX) d3p

• 2

s

1

/ a s ~ e p r e 2\1liD(S,,) ~p2} F ( S p ) IC~s~T)J

2rln2

(3.6)

Here, H is the total number of active quarks, which coincides with the total number of hadrons in the reaction and S o = 2npassivc = ~ i - h a d . . . . 2 ( n i - 1 ) is twice the total number of passive quarks (spectators) belonging to the hadrons which participate in the reaction. Some illustrative examples of this formula: E d~ ( a ) 2 ( 1 - - x ) A I[1ogOe]_ID(A}, d 3 k - Q-~ F(A) e

A

5-A

Fig. 2. D e e p inelastic scattering e A ~ e + . • . ; Sp = A = 2(n A - l ), H = 1.

Edo" ( c ~ ) 2 ( 1 - x ) a+c 1 d3k - ~ F ( A + C ) (log 0 2)-2°(a+c~, e

C

Fig.

3. D e e p

inelastic

semi-inclusive

'~ J

,

h a d r o n electroproduction 2 ( n A + no--2), H = 2 .

eAse+C+..."

sp=A+C

=

I.S. Avaliani et al. / Hard Processes in QCD

87

E &r (as 2 (1 __x)A+B+C-1 2,1-3D(A+B+C) d3p "\~TT) F(A+B+C) [C~s~PT)] r

C

,

c'P - ~

S'Z~ + B +-C

Fig. 4. I n c l u s i v e h a d r o n p r o d u c t i o n a t l a r g e p T i n A + B - + C + . . . ; s p = A + B + C = 2 ( n A + n B + n c H=3.

3),

Finally, we stress that the logarithmic exponents in (3.6) are determined by the anomalous dimensions of the moments of non-singlet quark distribution and decay functions of the participating hadrons. The number of the corresponding moments (indices of d~ s) are related to the number of valence quarks constituting these hadrons. (See table 1). 4. Quark counting for singlet distributions Now we proceed with the calculation of the more general case of jet and single particle production in hadron collision at large PT. TABLE 1

Anomalous dimensions d ( n ) =- d . NS of the non-singlet quark operators, corresponding to the different hard processes (different spectator numbers Sp) Process K

A+B-~C+.

( # - e± '~ k_g±,

(' + rr -+ (' +. • (+{-+rr+.. f+p~f+... ( + ~ - - + ~ + rr + ( + p - + g+~r rr +p-+iet, g, ( 2 + . • • (+p-~ (+p+. • . p + p - + j e t , (, ( { + - - • rr +p--, rr + . . • p+p-+~r+". rr +p-+ rr +rr +. p + p - + p + . •. rr + p ~ r r + p + . p+p~Tr+p+.

d{Sp)' $p = 2K.

d [ 2 ( 2 - 1)] d [ 2 ( 2 - 1)] d [ 2 ( 3 - 1)] d[2(4 - 2)] d[2(5 - 2 ) ] d[2(5 - 2 ) ] d[2(6-2)] d[2(6 - 2)] d[2(7 - 3)] d[2(8-3)] d[2(9-4)] d[2(9-3)] d[2(10-4)] d[2(10-4)]

Hd(Sp)

ld(2) ld(2) ld(4) 2d(4) 2d(6) 2d(6) 2d(8) 2d(8) 3d(8) 3d(lO) 4d(lO) 3d(12) 4d(12) 4d(12)

88

LS. Avaliani et aL / Hard Processes in QCD

In what follows we shall consider the contribution of the singlet parton distribution as well as examine all the fundamental parton subprocesses qiqi->qiqi,

q/G->q/G,

G G -->qicli

qiCqi~qi(ti,

qi(li ~ G G ,

GO-> G G

which can contribute in the leading order in as to the invariant cross section (3.1). Now instead of the (Fa, Fb, Dc) we substitute the functions Z,kF~FiDK, where the indices i,/', K run over the flavor f of quarks (anti-quarks) and gluons

O2),

Fi/H(X, ( ~ 2 ) ~ x q i ( x ,

i = V, S, G ,

and the final jet consists of the possible combinations of q(q) and G in accordance with the initial if state. The derivation of the valence quark%ontribution to the o-(AB-->C(/)+X) was performed in sect 3. We proceed now with the analysis of the non-diagonal terms VG, V S etc in leading order of perturbation theory. Inserting in (3.1) the corresponding singlet components of the parton distributions (2.5) and carrying out the integration in the large pv(xt) region and 0 = 90 °, we obtain the following. 4.l. JET PRODUCTION A+B~J+X S 1 F. P

o'(AB ~ J = O'o ~ F ~ o ) aq(AB)q~A/q~

(4.1)

where t

t

2s2,~A+BI1

o'o = tO's/PT) Z

2x2z

x - 2 r In 2

k~XT) [OLs)

The sum runs over the fundamental subprocesses V, S, G. Three-component vectors 4'r~ entering eq. (4.1) contain information on the scale breaking of parton distributions and leads to the logarithmic deviation from the canonical p~4 law o " = C.Y(H)[o~s(p~-)]D'~o'E(~),

t¢"vl

H = A, B

D~(Sp)

[OLs] k~(H + r ~ + 1 ) + /

\

D(Sp)

~ ( H +94rC~+ 1)+/J

(3/40)r~c (4.2)

where CH are the normalization constants, and the exponents D(sp), DS(sv) are determined by the corresponding anomalous dimensions d ( d ~) of the non-singlet

LS. Avaliani et al. / Hard Processes in O C D

89

and singlet operators

D(n)=d(n)-r(l+log

e) ,

e = 1--XT,

D~ (n ) = d~ (n ) - ¼r(l + log e) , 3 d(n)-~d2 "s z -r ~-

d'(n)=-d~ =-r

+O(1/K2),

(33~62f

J 9 ~ l / K ) + O ( 1 / n 2)

(4.3)

The components of the d matrix which enter into eq. (4.1) correspond to the definite combinations of the partonic subprocess cross sections [13]

(dd'/df)q = rrag/s , A22ij(O), for 0 = 90 °. [ VAVB

aii(AB)=~

SAVB

\ GAVB

VASB

VAGB 1

SASB

SAGB],

GASB

GAGB /

(4.4)

where the matrix elements VV, G G . . . . correspond to the subprocesses q q ~ qq, GG--, G G . . . . etc, and are, in general, process dependent. For example:

(i) quark jet trigger AB --, q ((1)+ X VV

d(AB)=

f 2nBp55n__~B ~,9

2nAp 4fp ll0f 9

55hA

9 llOf

9

(4.5)

7f 24

32

p = 2~+~f =~[16+63f] •

(4.5)

Note, that the singlet components of the parton distributions contribute in this case only a few percent to the invariant cross section at large PT when compared with the dominant valence quark contribution, and increases for the small XT values.

(ii) gluon jet trigger AB -~ G +X c~(AB) =

' VV 56

27nB

56 2~r/A 1121`

27j

55 9~nA

~of

(4.6)

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I.S. Avaliani et al. / Hard Processes in Q C D

(iii) quark-gluon jet trigger A B ~ q((l)G + X 110

2n AK 4fK

VV fi(AB)=[' 2 n u K \110n~/9

220.e 9 j

-~-n A \ 2~f 1,

(4.7)

243 + ! ¢ ] 4 24j,

K =2[10+21f], where, the valence VV quark scattering contributions qq-~ qq distinguish for the different initial particles VV: (f = 4) (i) q(q) jet

(ii) G jet

1360 27 1360 27 2969 476

pp "+iX p~ ~ ] X rr p ~ i X 37+p

jX

(iii) q(q)G jet 136(I 27 164(I 27 100027 532

0 280 27

27-112 56

27

27

(4.8)

27

In contrast with the case of quark jet production, the gluon trigger component in hadron collisions dominates. 4.2. SINGLE-HADRON PRODUCTION A + B ~ C + X v.~.AqoB--C o-(AB--,C)=o-oL~7~--~ai, kt, A ntJ C ,) w i iu'k. ilk

(4.9)

~ p]

All quantities entering eq. (4.9)*are defined by (4.1)-(4.3), and elements of the three-dimensional matrix aqk(ABC) have the following form:

vvv vvs vv% vsv vGv

vss vGs

vsGj:

vGG/

/vvv vsv vGv

ssv svs svo) / svv SSV SGV

SSS SGS

GVV

GVS

VVG 56 ~n A

55 ffffA

55 9-flA

GVG' 1

,

(4.10)

56

2nBp

~-~rlB

4fp

~/,

SSG = ~ 2ncp SGG \55nc/9

(4.11)

/ GVV

GSV Gss

6s6j=~55~/9

GGV

GGG]

GGS

VVS 2nAp

\,7nc/48

llOg, 9J ~SJ¢

11o ~43~J 4

(4.12)

f

where the elements VVV . . . . etc, label the corresponding elementary subprocesses, for example VVV: VGV: VGG:

q+q-*q+X, q+G~q+X, q+G~G+X.

* Assuming the validityof the reciprocityrelation [11].

(4.13)

LS. Avaliani et al. / Hard Processes in O C D

91

Note, that the elements V V V , S V V . . . . etc. depend in general on the type of hadrons A, B, C participating in the inclusive reaction A + B-~ C + X and on the quark flavours. In particular, for the case of two-proton collisions, and for f = 4: p p -~

VVV

SVV = VSV

7r -

w°

9a4 27 1480

416

944

68(I 27 404

W

122

110

55

55

275

2~

27

VGV = GVV

15

7r +

27

9

9

p 256 9~

6

9~

0 614

2-7 0

and independently on the final states VVG(pp)

= O,

VVS(pp)-

1360

27 •

The matrix elements aiik for the ~'±p and pl5 scattering are given in table 2. Note, that the spectator number S v (twice of the passive constituent number in the nonsinglet case) b e c o m e s So

= 2

rti - i

+

Y~ A S p .

1

i=1

Here, H is the number of hadrons in the process, n; is the valence quark number in the ith hadron (n = 2 for mesons; n = 3 for baryons),

TABLE 2 The elements p~) -',

VVV SVV GVV

v~rs

=

rr +

of matrices (4.10)-(4.12)

VVV VSV VGV

rr o

620

620

620

2~-

38

944 27 55 9

1480 2~ i 1o ~

404 ~ 55 6

122 275 ~

27

2--7-

n-+

~.

272 272

~27 27 0

VVV VSV VGV

0

~

P

568

140 R08 27

412 27~ 1480 27

296

448 110

55

110

27

N

9

~

9

728

27

136 2~

32

448

272

82078

2~ 0

V V S = ~7~__, V V G = ~ , S V V = S V V ( p p ) ,

9-

944 2~ 55 9

GVV = OVV(pp).

Ir

] ~m

38 614 2-r

VGV = VGV(pp).

~"

~

P

¢}

V V S = 296_, V V G = --,'~2 S V V = S V V ( p p ) , rr p

ff

rr -

t36o 2 7 , V V G = ~s~, V S V = V S V ( p p ) ,

~" p

f o r p15 a n d ¢r±p s c a t t e r i n g

15

s59 GVV = GVV(pp).

p

202

134

944

1480 27

27

27 559

3

~91°

92

LS. A v a l i a n i et al. / H a r d Processes in Q C D

ASp is an additional passive quark from the non-valence states, ASp : 1(2) in the case of the gluon ("sea" quark), respectively. Thus, Sp runs over the following values A + B-->jet + X VV VG VS, G G GS SS

A+B~C+X

Sp Sp + 1 Sp + 2 Sp + 3 Sp + 4

VVV VGV VSV, V G G GSV, G G G SSV, GGS SSG SSS

Sp Sp + 1 Sp + 2 Sp + 3 Sp + 4 Sp + 5 Sp + 6. (4.15)

Thus, in the leading order of QCD perturbation theory, taking into account the non-valence hadron constituents, modifies the hard scattering cross section and leads at large pT(XT)to the power series expansion in e = 1 - x v [AB--*jet'~ o'~ AB -->C )

=

(0~S)2 2rln2+HD(Sr,)ES p p4

1 2H

~ C'kek(C~s)a(sr'k~ , k :o

(4.16)

where Sp=A+B and H = 2 for AB -->jet + X, A(B)=2n~ (B~-2; Sp=A+B+C and H = 3 , for A B ~ C + X and C 0 = l , A(Sp,k=O)=O, A(Sp,k=l)= 2[D(Sp+ 1 ) - D ( S p ) ] , etc. Like the non-singlet case [7], exponents of as(pT2) characterizing the devitation of inclusive cross sections from the scaling f o r m pT 4 are determined by the anomalous dimension of the structure function moments. Furthermore, the moments number (index of anomalous dimension) becomes the physical meaning of the spectator number. According to eq. (4.16) the effective exponent An, which corresponds to the scaling deviation of inclusive cross sections: p T n ( n = 4+An) increases with Sp i.e. together with the complexity of hadrons participating in the reaction (number of non-valence passive constituents). Note, that this analysis becomes more significant for the lower XT values, a n d / o r for the gluon initiated processes.

5. Anomalous dimension quark counting beyond the leading order In this section we shall discuss the next to leading order corrections to inclusive jet and single-particle production cross sections at large transverse momentum. A lot has been learned the last time about higher order QCD corrections to deepinelastic scattering [14]. These corrections turn out to improve the agreement of the theory with the experimental data. There are however still many problems to be solved. In particular, the open question remains, especially the uncertainties in

LS. Avaliani et al. / Hard Processesin QCD

93

form and magnitude of higher order corrections in analysis of inclusive hadron initiated processes in QCD 1-14-17]. It was shown in preceding sections that the asymptotics of the hard hadronic cross sections are related with the anomalous log p2 counting, controlled in some universal way in leading order QCD perturbation theory by the number of active and passive constituents in hadrons. Here, we shall put particular emphasis on the higher order corrections in structure functions of DIS and the validity of proposed anomalous-dimension quark counting rules beyond the leading order. In what follows we restrict ourselves to the 0 = 90 ° case and neglect the next-loop corrections to the hard scattering cross sections, which do not affect the universality of the latter. Let us consider now the two-loop corrections to the evolution of the quark distribution and decay functions [18, 19]. For the non-singlet moments (2.4) on the two-loop level, we have

F~(n, O ~) = F : (n, '~':"L~s~o)J

L1

4-,,-

J'

where d, = y~/2[3o is the anomalous one-loop dimension (fl0 = 11-32f) and the two-loop correction has the form* t~ (1) 1 R2(n) T~°) PlYn 47r 2/30 2[3Z0+C.(I) Co+C110gn+~log2n (5.2) where fll = 1 0 2 - ~ f ,

Co=-1.18,

C1 = 0.66.

Note, that the running coupling constant ~,(O 2) is defined now as [20] ~2), ,,'~2, 41r [ ~1 lOgl~g (-O-~'~l°g (02/A2)] O~s ~w )=as(O2)=flolog(O2/A2 ) 1 -j.

(5.3)

Notice, that the two-loop calculations depend in general on the choice of the renormalization scheme and violate the reciprocity relation of [11]. We must take into account a difference arising in higher orders between the functions F~22)(x, O 2) and D(2~(z, 02). Taking into account the boundary conditions (2.3) and inverting the moments** (5.1), (5.2) we receive for the valence quark distribution in hadron A:

xF~(x'O2)-

CAKA(~) _X),~ 1 [2 F(~,) (1 exp{c~s.a'(A)} l + a r r a ' l o g 2 ( 1 - x )

]

'

(5.4) * Here and through the paper in scheme MS and f = 4. ** See also ref. [21].

I.S. Avaliani et al. / Hard Processesin QCD

94

where a '(A) = C o - (2/3~-)[ gt 2(A) + qt,(.~)], -A = A - a ~ ( O 2 ) b ( A ) = 2(n A - 1) + r ~ - a s b ( A ) ,

b(A)=Cl+(4/3~r)gt(~,),

,~:= - l o g a~(O2).

The factor KA(~) is defined by (3.2), but the constants Co(C~) are given in the MS scheme by eq. (5.2), and gt(z) = d In F(z)/dz and gt'(z) are Euler's digamma function and its derivative, respectively. Note, that the evolution development up to O(a~) of the structure function can be represented in the form

(

4

0 log 0 2 - a~4r - ~ 2 a (A) + ~

gr (N)

)

(5.5)

with " b o u n d a r y " condition

A I o ~ o ~ , = A = 2(hA-- 1) , where a (A) = C0 + C1 ~ (A) + 3 ~ [ ~ 2 ( ~ ) _ gt,(~)], guarantees in the final result, square dependence on the anomalous dimension d,Y,s To calculate the jet production cross section (3.1) in the second order, we rewrite now (5.4) in the following form: F(2} 2C~s, a/A ~ [Xa, O 2) = Fa/A(Xa, 0 2) e ~'~(O2)[a(~)-b(~) log(l x~.)][ 1 + ~ log 2 ,[I, - Xa) ] (5.6) and Fa/A(Xa, 0 2) is the usual one-loop result (3.2). Inserting this expression into eq. (3.1) and carrying out the integrations, we obtain for x - 1 and 0 = 90°: o-(2~(AB ~ JX) = o-(I'(AB ~ JX){1 + 2as(pZ)R (A, B) + O(c~ 2s)},

(5.7)

where cr ~1}~ E &r/d3p (AB + JX) is given by (3.4) and R ( A , B) ~- C o - ~ 1_ [ ~ 2 ( ~ ) + gt 2(1~) + ~ , ( N ) + gr'(I~)] .372.,

3~" -- [CI -I- 4 ~ tF(A + I3)] loge'+3@[log2e'+I(A,B)] 3~e ' = (1 - XT)/(1 -- ½XT),

(5.8)

LS. Avaliani et al. / Hard Processes in OCD

95

and the function I(A, B): I(A,

-½ 11 VrT[7~12-T~T-I1dVVfi

~ t A , D,FI

1(1 - V) N-1 , [1--e'v/t,--2~/~)jL, e'Vj 2

[

× l°g2 V + l ° g 2 1--e'V/(1--}XT)

]

'

reduces to 2I(A, B) = [gt ( ~ ) _ gt(A + ~)]2 + [gt ( ~ ) _ ~ ( A + B)] 2 + gt,(A) + ~'(B) - 2 gt,(A + B) + O(o~~).

(5.9)

Collecting the factors (5.8) and (5.9), we obtain o'I2'(AB ~ J X ) : o'It'(AB ~ J X ) / [ 1 + 2 ~2a~(p l o ~-)g

2 2e]

x exp {2as(p2)[a (A + B ) - b (A + B) log 2e ]}.

(5.10)

Performing similar calculations in the case of a single-particle cross section, we can formulate the general rule: crlZ~(AB _~ C X ) = o.~l,(AB ~ CX)exp {c~sA(~p)}[ 1 +2c~31rH log 2 2e] , A(Sp) = [a (Sp)- b (Sp)log 2e ] +}~-HT o.la~(AB~CX)

(o.02

(5.11)

2 r l n 2 8 S p 1 V , 2,nHD(Sp} 4 F ( S p ) l-C~stP T)J '

pT

where Sp = Sp+Hr~ and all the other quantities in these expressions are defined 2 above and the factor exp [5~'c~sHT] arises due to the difference in higher order between the quark distribution and fragmentation functions. This fact is related with the transition from the spacelike to timelike values of variable 0 2 [22], and HT is the total number of active quarks in the subprocesses, where q2= _ 0 2 > 0 , . A precise definition of the constituent structure functions generally speaking depends crucially on the choice of the factorization/renormalization prescriptions and could affect the magnitude of the higher-order corrections. During the calculations of the two-loop corrections we used: prescription A (of ref. [15]), the MS renormalization scheme with the corresponding A values, and a definite choice for the momentum transfer scale 0 2 = O2(~, [, a). Thus, taking into account the next-to-leading QCD corrections (at least to the non-singlet quark distribution and fragmentation functions) does not violate the universal character of the anomalous dimension counting rules [7]. * F o r m u l a (5.10) r e p r e s e n t s the p a r t i c u l a r case of eq. (5.11) for Sp = A + B, H = 2 a n d Hq: = 0.

96

I.S. A v a l i a n i et al. / H a r d Processes in Q C D

Note, that in addition to these calculations the second loop corresponds to the expansion of the effective as exponent in anomalous dimension series (as) °, D

= O(Sp)

+ Co~sD2(Sp) +.

6. Effective power

and px 4

• •

problem

The A D Q C rules give the algorithm for the calculation of the hadron hard scattering asymptotics up to one-loop logarithmic Q C D corrections to the canonical point-like P x 4 law. Let us consider an arbitrary cross section for the large pr jet (single hadron) production 4

rn

2

~r (AB ~ C) - p T @ (Xx)a s (P x),

(6.1)

where m = 2 - 2r In 2 + l i d (Sp) controls the magnitude of Q C D scale violation in hadron collisions. Calculation of the effective power o-(AB ~ C) - p T ' " " ,

(6.2)

is rather simple because the invariant cross section (6.1) has the form of the structure function m o m e n t s (2.4), satisfying the evolution equation 0 log

4 2 [pTcr(n, pT)] _

t~s(p 2)

0 log p2

2~

d(n)

(6.3)

and thus, the effective power (6.2) can be expressed in terms of the hadron structure corrections to the hard scattering of quarks in Q C D . For phenomenological purposes we exploit the following definition of nea at fixed values of the xv variable:

neff(Xv =

~(sl,p~ ~)/,

4S~

fixed) = - l o g ~r (s2, p ~ ) ) / J ° g x/s2'

2p~; 2p~ ~

(6.4)

XT= 4Sl -- 4 S 2 "

which corresponds to the two different m e a s u r e m e n t s at e n e r g y - m o m e n t u m $1, p ~ and s2, p ~ respectively. According to this definition the total effective power at large xx values is Fleff(XT)

=4-2[2-2r

log 2 +

HD(sv)]R(sl) \$21

(6.5)

L$. Avaliani et al. / Hard Processes in QCD

F

7~

A : 01GeV/c

97

S =

(Gev2~

s ~

~~--C~

~ o

~

~o,

J LL

x~ -

c

A : O.SOeV/c

/

S[GeV2} .102 ! i

:

'

j

01

03

05 XT

07

Og

Fig. 5. Effective PT power neff(XT) (See eqs. (6.5) and (6.6)) of the invariant cross section for ~.o meson production in pp collisions as a function of XT calculated for two QCD scales A=0.1 GeV/c and A = 0.5 GeV/c, respectively. where

D (Sp) = d (Sp) - r(~p+ lOg ( 1 - xT)) , a n d t h e q u a r k r e s o l u t i o n f u n c t i o n at e n e r g i e s Sh S2 is

R{,Sl~, = log (a~(s 1)/c~(s2)) \$2 /

log

($1/$2)

(6.6)

Thus, we o b t a i n a p a r a m e t e r - f r e e s o l u t i o n of the p~r4 p r o b l e m in h a d r o n collisions*. (The o n l y p a r a m e t e r in eqs. (6.5)-(6.6) is t h e r u n n i n g c o n s t a n t scale A ~ g , which is fixed in D I S ) . N o t e , that this s o l u t i o n is e x p r e s s e d in t e r m s of t h e p e r t u r b a t i v e l o g a r i t h m i c c o r r e c t i o n s to the d i m e n s i o n a l p o w e r a s y m p t o t i c s . T h e m a g n i t u d e of t h e s e s h o r t d i s t a n c e c o r r e c t i o n s is g o v e r n e d b y the q u a r k c o n t e n t of t h e h a d r o n s in t h e r e a c t i o n . T h e d e v i a t i o n f r o m t h e pSr4 p o i n t - l i k e b e h a v i o u r is l a r g e r for a l a r g e r total n u m b e r of p a s s i v e c o n s t i t u e n t s of the h a d r o n s . See for i l l u s t r a t i o n the t a b l e 3 a n d figs. 5 a n d 6**. N o t i c e the i n t e r e s t i n g c o n s e q u e n c e of t h e s e results a d o p t e d to the different cross s e c t i o n ratios. In t h e case w h e n the h a d r o n s , p a r t i c i p a t i n g in h a r d s c a t t e r i n g r e a c t i o n s * Note, we do not consider here the transverse momentum smearing effects, which have little effect on the results, obtained in the large-pT and xT regions. ** Note, that we use the 0 2 = -?" variable.

98

I.S. Avaliani et al. / Hard Processes in OCD TABLE 3

The Pv power n~r defined for fixed XT by Edcr/d3p (pp--> r/~, j e t + X ) - - p T ''''~ (see eqs. (6.5) and (6.6)) p p ~ n-°+X \

- xT

S GeV2

10 z 103 10'~ 10 s

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

5.23 5.97 4.99 5.41 4.81 5.08 4.69 4.88

5.29 6.05 5.03 5.47 4.85 5.12 4.72 4.91

5.35 6.15 5.08 5.53 4.89 5.18 4.76 4.96

5.41 6.26 5.13 5.61 4.93 5.24 4.79 5.01

5.50 6.39 5.20 5.71 4.98 5.31 4.84 5.06

5.59 6.55 5.28 5.82 5.05 5.39 4.90 5.14

5.72 6.75 5.38 5.96 5.13 5.51 4.97 5.23

5.90 7.04 5.52 6.17 5.25 5.66 5.07 5.36

6.21 7.53 5.77 6.52 5.46 5.93 5.24 5.58

pp->jet + X eV2~ G~

T

10 2

103 104 105

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

5.14 6.34 4.85 5.37 4.67 4.96 4.54 4.74

5.07 5.99 4.82 5.27 4.66 4.91 4.55 4.72

5.06 5.88 4.82 5.24 4.66 4.91 4.56 4.72

5.07 5.84 4.83 5.24 4.68 4.92 4.57 4.74

5.10 5.86 4.87 5.27 4.70 4.95 4.60 4.76

5.15 5.90 4.9l 5.32 4.74 4.99 4.63 4.80

5.22 5.99 4.96 5.39 4.79 5.06 4.67 4.85

5.32 6.14 5.05 5.51 4.86 5.15 4.73 4.93

5.51 6.43 5.20 5.72 4.99 5.32 4.84 5.07

The two values of n~ff at each energy correspond to ,4 = 0.1 GeV/c and A = 0.5 GeV/c, (lower value) respectively. differ by some definite number

of q u a r k s , it is u s e f u l t o f o r m u l a t e

quark interval rule. E.g. for the one-quark

the following

differentiated processes:

H = 2

Sp

H = 3

Sp

nnr ~jet

4,

~rrr -~ 7r

6,

~rp-> j e t

6,

7rp--> n-

pp-~ jet

8,

p p - ~ n" pp - p, etc

8, 10, 12,

we have 2 R [or (Sp + 2 ) ] _ e t J (sp+15(sp+2)

[Ces(p 2)] aD(spm

X T -- f i x e d

'

(6.7)

'

where dD(Sp)=D(Sp+2)-D(Sp)=r

[ 2Sp+3 2 ] 2r 8 [ ( S p + - - - 1 ) ( S p + 2 ) ~-8 p ( S p + 2 ) - S o =--8pYF,

(6.8)

99

[.S. Avaliani et al. / Hard Processes in OCD

55

S =10C6eV2 ~ p ~ T T x

]

5 z..5

, 01

,

, 03

,

1 05 zT

,

t @7

i

0

19

Fig. 6. Comparative curves, as in Fig. 5, for jet and single rr production in pp collisions. a n d "/v = 4 is t h e q u a r k a n o m a l o u s d i m e n s i o n ( F e y n m a n g a u g e ) . A s a n i l l u s t r a t i o n of this r u l e c o n s i d e r h e r e t h e f o l l o w i n g ratios: beam ratio

/ p p -+ ~r \ /,,~.p~Tr//{ p p + p'~

particle ratio

/\ p p ~

particle/jet ratio

rr//

(

1 --XT) 2 v / 2,13"y F 9-_~ l astPTlJ , (1

-- XT)

2

11 • 12

2

12

[O:s(PT)] s "YF

( p p ~ 7r '] _ (1 -- XT)2+4,~ \pp-+ jet/ -9-: 1-O [as(p2)]~

(6.9)

T h e s e r a t i o s c a n b e t e s t e d e x p e r i m e n t a l l y in m e a s u r e m e n t s at d i f f e r e n t e n e r g i e s a n d t r a n s v e r s e m o m e n t u m . W e p r o c e e d n o w w i t h s o m e n u m e r i c a l e s t i m a t e s of t h e effective p o w e r e x p o n e n t h e , in t h e r a n g e of a c c e s s i b l e e n e r g i e s a n d t r a n s v e r s e m o m e n t a . T h e r e s u l t s of t h e c a l c u l a t i o n s (with t h e s u i t a b l e c h o i c e of t h e A scale m a g n i t u d e a n d 0 2 = 02(~, ?, 4) d e p e n d e n c e ) a r e in fair a g r e e m e n t w i t h t h e c o r r e s p o n d i n g e x p e r i m e n t a l v a l u e s a n d are listed in t a b l e 4. D u e to t h e p a r a m e t e r l e s s n a t u r e of t h e s e p r e d i c t i o n s , a full p h e n o m e n o l o g i c a l a n a l y s i s of e x p e r i m e n t a l d a t a will b e v e r y d e s i r a b l e . TABLE

4 th

The PT power n teh defined for fixed XT by Edo'/d3p(AB ~ CX) - p T""" exp

th*l

AB ~ C + X

",/S. GeV

XT

n e,

n etT

Ref.

7rrr -+ rrX ~'p -+ "rrX pp-+ jetX pp-+ rrX pp~ ~rX

10.3 13.7 13.7-24.7 13.7 62

0.35 0.32 0.35 0.32 0.35

6.2± 1.0 7.46±0.15 6.2±0.3 7.93+0.13 6.1±0.3

6.5 7.6 6.4 8.0 6.0

[23] [23, 24] [25] [23,24] [26]

Experimental data are taken from the refs. [23-26]. , Q2 = 2g~,~/(g2+ f2 + t~2), .1 = 0 . 5 G e V / c .

100

LS. Avaliani et al. / Hard Processes in O C D

7. Concluding remarks Starting from the Q C D evolution of the quark and gluon distribution (fragmentation) functions and exploiting the spectator counting rules for the initial x dependence of F(x, 0 g) and D (z, 0 2) at 0 2 = O0z we obtain in a leading log approximation the A D Q C rules for any arbitrary inclusive hadron reactions at large pT involving all the possible Q C D hard scattering subprocesses and singlet distributions. Analysis of the next-to-the-leading-order Q C D corrections to the quark evolution does not destroy the universality of these rules. On this base, Q C D gives a unique p a r a m e t e r - f r e e prediction (pT "eft solution) for a wide class of hard processes. The authors are very indebted to Prof. A. N. Tavkhelidze for his constant interest in this work and valuable remarks, and to Drs. K.G. Chetyrkin, N.V. Krasnikov, A.N. Kvinikhidze, R.M. Muradyan, A.V. Radyushkin, D. Robaschik and A.N. Sissakian for useful discussions.

Appendix A Let us assume that the initial valence, " s e a " quark and gluon distributions at 0 2 = 0 2 and large x have the form:

Xqv(X, 0 2) = Cv(1 - - x ) 2 n ' - 3

~

(~8) -'~ C o ( 1 - x ) 2n'' 1 ,

xG(x,

xqs(x, O 2) = Cs(1 - x ) 2~V+' .

(A.1)

The solutions of the evolution equations for an arbitrary O 2 receive then the following form: (i) Valence quarks (1 - - X ) 2n'-3+rE Xqv(X, 0 2) = Cv(~)

F(2nv- 2 +r£)

Cv(4~) = Cv exp {r~:(~- yE)}F(2nv- 2), r

16 33-2f'

A = YE(ii)

~ = In

[o&(Q2)/adQ2)],

21-2f 20

(A.2)

"Sea" quarks Xqs(X , 0 2) = Fqsv q-FqsG q-Fqsq~

(1 _x)an,. 1+r¢ Gsv = Gsv(~')

F(2nv+r~)[ln [1/(1 - x ) ] + qt(2nv+r~:) +A] '

Cq~v(~:) = C,3or~ exp {r~(~ - yE)}F(2nv -- 2),

(A.3)

1.S. Avaliani et al. / Hard Processes in OCD

101

(1 - - X ) 2n'+r~

Fq,G = Cq~o(~:) F (2nv + 1 + r~:)[ln [ 1/( 1 - x )] + gt (2n ~ + 1 + r~:) + A ] (1 - x ) 2~'+~

-Cq,G(cs)F(2n,+l+grse)[ln[1/(l_x)]+g,(2n,.+l+gres)+A], yv_)}F(2nv)

CqsG(s~) = ~]~CG exp {r~:(3 1

(A.4)

9

(1 - x ) 2~'+1+~ 3

Cq~qs(~) = Cq~ exp {r~(~- ~F)}F(2nv + 2). (ii)

(A.5)

Gluons xG ix,

(~2) = FGV + FGqs + FGG ,

(1 __X)2,,, 2+r( FGv = CGV(()

F(2nv-

1 + rs~)[ln [1/( 1 - x )] + gt ( 2 n v - 1 + rs~) + A ] ( 1 - x ) 2"v 2 + ~

- Coy(C)

F ( 2 n v - 1 + 9r~:)[ln [1/(1 - x ) ] + ~ ( 2 n v - 1 + 9r~:) + A ] ' C c v ( se) = }Cv exp {r~:(] -

(A.6)

yE)}F(2nv -- 2 ) ,

Cov(~)=2Cvexp{r~@-94yE)}F(2nv-2), (1 -- X) 2n~+2+rff

FGq~ = Coq~(~:) F ( 2 n v + 3 + rsC)[ln [1/(1 - x )] + qt (2n~ + 3 + r~:) + A ] (1 --X) 2n'+a+9rg~

F ( 2 n , , + 3 +9r()[ln [1/(1 - x ) ] + qt(2nv + 3 +gr~) +A ' CGq~(~) =

4

"~fCq~exp

(A.7)

3

{r~(a- yE)}F(2nv + 2),

CGq~(s~)=4fCqsexp{r~(1-gyE)}F(anv+2) (1 - x ) 2"'-1+]~ FoG = CGG(~:)

F(2n~ + ]r~) ' 1

9

Neglecting the n o n - l e a d i n g (at x ~ 1) t e r m s in ( A I . I - 8 ) we receive the set (2.5).

LS. Avaliani et al. / Hard Processes in QCD

102

Appendix B (i) Hypergeometric function of two variables F(y) f[ u~-lv ~'-1 du dv Fl(a'/3'fl" Y;x' Y ) - F ( f l ) F ( / 3 ' ) F ( y - / 3 - f l ' ) - _ ( 1 - u - v ) ~+~' v + l ( 1 - - U x - - v y )

'~

F(y) f duu'* l ( 1 - u ) ~ 1 F(a)F(y -c~) Jo (1 - ux)~(1 - u y ) e' =E m,.

(y),.+~m !n!

F(a+n)

x y ,

(a).---

F(a)

'

(B.1)

1

I

)

dzz ~ 1(1-z) t~-llogz=~(a,/3)[~(a)-~(a+/3)],

(B.2)

1 I)

d z z ~ - l ( 1 - z ) ~ 1 log2 z = ~ ( a , / 3 ) { [ ~ ( a ) -

~(a

_[_/3)]2 q_ 1/~,(OL)-

1/tt(OL _÷_/3)} ,

where ~ ( a , / 3 ) is the Euler's beta function, and ~(z) =

d log F ( z )

dz

,

~"(z) =

d 2 log F(z) dz 2

References [1] S. Berman, J. Bjorken and J. Kogut. Phys. Rev. D4 (1971) 3388 [2] D. Gross and F. Wilczek. Phys. Rev. D8 (1973) 3633; D9 (1974) 980; H. Georgi, H.D. Politzer, Phys. Rev. D9 (1974) 416; G. Parisi. Phys. Lett. 43B (1973) 207; 50B (1974) 367 [3] R.F. Cahalan, K.A. Geer, J. Kogut and L. Susskind, Phys. Rev. D l l (1975) 1199 [4] R.C. Hwa, A.J. Spiessbach and M.J. Teper, Phys. Rev. Lett. 36 (1976) 1418; A.P. Contogouris and R. Gaskell, Nucl. Phys. B126 (1977) 157; D. Duke, Phys. Rev. D16 (1977) 679 [5] R.D. Field, R.P. Feynman and G.C. Fox, Nucl. Phys. B128 (1977) 1; R.P. Feynman and R.D. Field, Phys. Rev. D15 (1977) 2590 [6] V.A. Matveev, L.A. Slepchenko and A.N. Tavkhelidze, JINR, E2-11580, Dubna (1978); E211894, Dubna (1978), presented by V.A. Matveev in Proc. 19th Int. Conf. on High energy physics (Tokyo, 1978) p. 224-226 [7] V.A. Matveev, L.A. Slepchenko and A.N. Tavkhelidze, Phys. Len. B100 (1981) 75 [8] G. Altarelli and G. Parisi, Nucl. Phys. B126 (1977) 298; K.J.Kim and K. Schilcher, Phys. Rev. D17 (1978) 2800; Yu.L. Dokshitzer, D.I. Dyakonov and S.I. Troyan, Phys. Reports 58 (1980) 271 [9] V.A. Matveev, R.M. Muradyan and A.N. Tavkhelidze. Len. Nuovo Cim. 7 (1973) 719; S.J. Brodsky and G. Farrar, Phys. Rev. Lett. 31 (1973) 1153; J. Gunion, Phys. Rev. D10 (1974) 242 [10] C. Lopez and F.J. Yundurian, Nucl. Phys. B171 (1980) 231; F. Martin, Phys. Rev. D19 (1979) 1382 [1l] V.N. Gribov and L.N. Lipatov. Sov. J. Nucl. Phys. 15 (1972) 781, 1218 [12] H. Bateman, Higher transcendental functions, vol. 1 (McGraw-Hill, New York, 1953) 224-247

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