Projection of Wilson coefficients from deep inelastic leptoproduction data

Nuclear Physics 0 North-Holland

B122 (1977) 412-428 Publishing Company

PROJECTION OF WILSON COEFFICIENTS FROM DEEP INELASTIC LEPTOPRODUCTION

DATA *

S. WANDZURA Joseph Henry Laboratories, Princeton Received (Revised

University, Princeton, New Jersey 08540

18 December 1976 15 February 1977)

We show in detail how the techniques of Nachtmann can be applied to the operator product expansion of two-vector (or vector minus axial vector) current in order to extract Wilson coefficients from the deep inelastic leptoproduction data. Included are new results for the G1 and G2 (spin dependent) and Wj (parity violating) structure functions.

1. Introduction

Asymptotically free gauge theories [l] (AFGT) of the strong interactions predict approximate scaling for the Wilson coefficients [2] appearing in the operator product expansion of current products. In the limit of infinite space-like momentum transfer, these coefficients approach simple moments of dimensionless structure functions. At finite momentum transfer, however, a non-zero target mass modifies this relationship, and the techniques introduced in ref. [5] must be used to find the Wilson coefficients from the structure function data. It has recently been proposed that, target masses having thus been taken into account, remaining mass corrections, due to finite quark and renormalization point masses could be calculated in perturbation theory [3]. Although this idea now appears to be overly optimistic [4], it might still be thought that since AFGT make their most direct statements about Wilson coefficients, their extraction could prove worthwhile. Accordingly, we gave the details of Nachtmann’s [5] methods applied to deep inelastic leptoproduction, including spin dependent and parity violating terms. The main results of this paper are eqs. (39)-(48) and (54)-(.59), expressing the Wilson coefficient in terms of measurable quantities.

*Work

supported

in part by the Eastman

Kodak

412

Corporation.

S. Wandzura / Deep inelastic leptoproduction

413

2. Electroproduction The forward Compton amplitude for scattering of a photon of momentum

q (q2 = Q2 < O) off a nuclear target of momentum p (p2 = M2) and spin s (p . s = O, s2 = - l ) i s

[7]

T u v ( p , q , s ) =i~

uv

4 xeiq.x(p,s I * e . m . T {J~ (X)Jve . m . (0)3[p, s)

Q2]

-~

pv

°-~-qv T2(o, Q 2)

+ iM euvxo qXs° UI(V, Q2) +/__ M euvx°qX(vsa - q " sp°)U2(v' Q2),

(1)

where v = p - q. The absorptive part Wuv = Im Tuv has invariant amplitudes W1, 2 = Im T1,2 ,

(2)

G1, 2 -- Im U1, 2 ,

(3)

which are measured in deep inelastic electroproduction. Making the usual assumptions about their Regge behavior, these amplitudes satisfy the dispersion relations:

TI(v, Q2) = TI(0, Q2) + 2 /

v2 do' V'( v'2 __ 02) Wl(V' ' Q2),

(4)

02/2

T2(v, Q2) =

f

v' dv'

v, 2 _ v2 W2(v, Q2),

(5)

Q2/2

v' dr'

U1(v, Q2)=2 / 0,2_02 GI(V',Q2), 02/2 vdv' U2(V, Q2) = 2 f u,2 v2 G2(u', Q2 ) . 02/2

(6)

(7)

The operator production expansion for the time ordered product of currents is [7,8] T* {J~'m'(x)Je'm'(o)] = (ggv ~p OP - ~ ~v)

~ ~ (~)(x 2 - ie) J=O, 2.... i

414

S. Wandzura / Deep inelastic leptoproduction CS#l'"uJ(O)+(guvOaO~+g#~gu~OpaP X X#l ... x #j,.,,

× J=2,4 ....

i

-- i etzuho 0 h

~ ~ J=l, 3 .... i

+ 2i euvpoOoOx

(i)l'X : --

~

J=2, 4 ....

"'" X U j - 2 v(i)

C~)~)(X2

~

--

ie)x,1

g~uOUO

_g~uOvO~ )

"J-2(0)

c--~'°#l .-./~J- 1(0 ) ... X ~ j _ 1 "-"(i)

cO~i)(x2 - ie)xul ""xuj-1 ~(i)xauz... uJ-l(O )

i

(8)

We have chosen the convention that the Wilson coefficients are labeled by the spin of the operator. The index i labels different operators of the same spin and symmetry properties. The operators 0 ul ...uj and 0 'aul ...uj - 1 transform according to the (71 j , l j ) representation of 0 ( 4 ) and O XaUl ... uj - 1 transforms according to the ( ½ ( J - 1), l ( j + 1)) representation [9]. In AFGT, the leading (Q2 _+ ~ ) operators O and ©' are twist two [1,7]: O~)1 """J(x) = P ( j ) ~ l l ...... ul ujj( l • ) J - l - - ~(X)~[ r i D v2 . . . O

vJ)kj 2 ~J(x),

(9)

or

"'"

- P ( J ) V l ...v,r

2

Tr F v' ~(x)D v2 ... B t)J-I F~J(x) ,

f-~'°#I""#J-I(x)=P alal""#J-l(i)J "-"(i) --(J) Vl...p J

~g(x),),5,),VlDV2

--.

~vj)kJ ~-~(x),

(10)

(11)

or

o,OUl ...us-1/v.'~ =/9 a#l . . . u j - 1 ( i ) s (i) ~'~l * (J) U l " ' v J 2 <.-*.p

~+

X ~ vjc~fi'r TrFfiv(x)D 1 ... D V J - 2 F ~ J - l ( x ) ,

(12)

and the leading 0 are twist three [7] : (~ hOUl "-/~J-1 =/~

(i)

h°/'tl ""#S--1 (i) J

-(J) ~ u I ... uj _ 1

X ~ ( x ) T s T a D ~ D vl ...D vJ-1 Xj ~(x) 2

(13)

415

S. Wandzura /Deep inelastic leptoproduction or O~Ot'tl ""#J-I (i)

X

= /~

~'O/'/1""#S--I

- - ( J ) p r y I ... v j _ 1

~ r ~ v Tr F~v(x)/~ vl

. . .

(i)S+l

~g-1

F~(x),

(14)

where D u is the covariant derivative O, + i g r a A g acting on gluons and 0 u + igoaA~ acting on quarks, Fuv = (OUA v - OVAUa - g f a a c A bu A cv) r a , Oa and "/'a, are the appropriate representation matrices of the color gauge group, ~j are the fermion representation matrices of the flavor group, and P(j) and/~(j) are the projection operators defined in appendix A. Taking the expectation value of eq. (8), Fourier transforming, using the contractions found in appendix C, and using the recursion relations for Gegenbauer polynomials, we identify the invariant amplitudes as Tl -

1 Q)J ~M J=0,2 ....

Cj(1)(T~) e

4riM J=2,4 .... J ( J -

J(Q2)

(j(j_

1)

1)Q1)(n)_ 2Cs(2)2(n))e,J(Q2) , (15)

' T2

47rM J=2,4 ....

1

=_ U1

8Cj(3_)2 (9,J(Q2),

G

1rMQ2 J=l, 3 ....

'(U'

{ Cj(2__)l(,)7) + 4,q CJ3)2 (rt) } c'/)J(Q 2)

1 G 1 [ i M ~ J 4 c ( 3 ) r..,, + zrMQ ~ s=2,4,... J ( J + 1) k Q ] s - 2 v " ~-DJ(Q2) '

I U2

Z;

~

iMJ

(16)

(17)

<3>

7rMQ2 J=1,3 ....

(18) 7rMQ2 J=2,4 .... J ( J + 1)

{2C53-)'0/) + 2Cj(3-)3(~)] Q)J(Q2),

where r/= iu/MQ and we have defined the following matrix elements and Fourier

S. Wandzura /Deep inelastic leptoproduction

416 transforms:

(p, Si O~i~ "'laJ(o) ip, S} = (i) J ~ C(i) {p~l ...17t'zJ] j , M

(p, slC-~'°Ul'"UJ-l(O)]p,s)=(i)J-la(i){s°pUl ~'(i)

"'"

(p, s I ,-,~x°ul "'ua- l(O)lp, s)= (i)J dy) {PXs° P

(19)

pUJ-11j

(20)

'

...pUg-i] MS ,

(21)

G j ( Q Z ) = i n 2 j + 2 [ ~dQ2 d ]'~J ~i c(l)• f d4x eiq.x e ~ ) ( x 2

ie),

(22)

~ , J ( Q 2 ) = iQ2J

-- i6) ,

(23)

cZ)J(Q2)=~

~dQ2]

Gi c(j )

d4x eiq .x ~,y~ 2

~(i)tx

~i a(i)d d4xeiq'xc-l)~i)(x2-i¢)

cDS(Q2)=n2J+a[ "~ ~dQd~--~ 2] s - ' ~i. ay).fd4x e,q" x q)~)(x 2

Jodd,

ie)

(24)

Yeven. (25)

(Notice that ~(Q2), ~ , ( Q 2 ) and C~(Q2)have canonical dimension zero.) The above expansions for T2 and Uz are in orthogonal sets of polynomials, but those for T1 and UI are not. We can, however, project out the Wilson coefficients * by looking at the combinations

rrM s=o,2 ....

1

26)

2rrM s=2, 4....

U1

o +My U2 - - 7rMQ - - 2 J=l, 3....

1

* It should be kept in mind that although we refer to the G's and c-/),s as Wilson coefficients, they are actually sums of products of Wilson coefficients and (unknown) matrix elements.

S. Wandzura / Deep inelastic leptoproduction

417

1T, f-~-IJ(Jc(2)(rl) 2nMQ 2 J=2, 4. . . . J(J+ 1)\ ~ / (27)

_ ( j + 2)C(~2 07)] c~ J(Q2) .

In eqs. (15) (18) and (26), (27), a Gegenbauer polynomial of negative order is to be taken as zero. We can now project out the Wilson coefficients by using (i) the dispersion relations (4)-(7), (ii) the orthogonality of the Cj(x) [11] 1

dr/(1

r/2)x-J/2 c(X) 77 C (x) 77

( J + 2X) j ( ) j, ( ) = ~2n Fc2(x)a!

(28)

--1

and (iii) the integral [11,12] 1

f ,7"d,7(1-,:)~-'/:(~

,)-'cj(X)(~)

--1

,-)~.-1~m(~2 _ 7r

1)(X-I)/2[~

(~2

1)ll2]J+X

- ~

F(J+2~,) F(X)F(J+ X + 1)

_~ + (~2 _ 1),/2], ×F

X, 1 - X ; J + X + I ,

2~--_]-)175

]

(29)

where F is the hypergeometric function (which truncates to a polynomial in its final argument for integer X. Thus eqs. (16), (26), (18) and (21) give

I dx

~

-IJ+l

}

/ ~-11+ X/1~M2x21Q23 2Fl(x' Q2)_(~-x+4X~)F2(x ' Q2) = _ (gJ(Q2)

dx

g1 (~ ,J(Q2),

2x

.-]J+ 1

J = 2, 4 .....

(30)

(j2 + 2J + 3)+ 3(J + 1)X/1 + 4M2x2/Q 2

0

4M2x 2]

+ J(J + 2 ) ~ - - ]

I dx I

f-~0

2x

F2(x, Q2)= _ ~ ( j + 2)(J + 3 ) C ' J ( Q 2 ) ,

-~s+2

1 +X/1 +4M2x2/Q2.]

{1 + ( J + 1)x/1 +4M2x2/Q 2]

(31)

418

X Wandzura/ Deep inelastic leptoproduction X {gl(X, Q2)+g2(x, Q 2 ) t - - 4 1 J ( J + 3) M 2

8

d+l

Q2

1J+2

J

Q) J+ 1(Q2)

@J(QZ)-I(j+2)@J-I(Q2) J = 1,3 .... ,

(32)

(the second term of the right-hand side is absent for J = 1), and 1 +x/1 +4MZxZ/Q

f~-

(j2 + 4 J + 6 ) + 3 ( J + 2 ) x / 1

+4M2xZ/Q 2

0

4M2x21 1 J(J +(j+ l)(J+3)__~__lgz(x, Q2)=

+ 3)(J + 4) @j+j

J+l

,

(Q2)

-~(J+3)(J+4)c~J(Q 2 ) + ~ J(J+ 1)(J+4)M_~2 (Z)j+2(Q 2 J+2 Q2 )

J = 2 , 4 .....

(33) where

Q2 X = -

(34)

FI(x, Q2)=MWI(V, Q2),

(3s)

2v '

F2(x ' Q2) = _~Wz(v, Q2),

0

(36)

gl(x, Q2) = MvC.(u, Q2) ,

(37)

O2

gz(x, Q2) = M G2(°' Q2).

(38)

Solving these, the Wilson coefficients are C j(Q2) =/j1,j(Q2)

e,J(Q2) =/.12,j(Q2)

1 2) -- ~M~TI(O, Q2)(~j,o , gld2,.l(Q

(39)

(4O)

,

,1=, J(J+ 1) )] + (J+ 2)(J + 2n - l) vJ+2n-l(Q2

~

L(J~-)(2+2n) oJ+2"(Q2) J odd,

(41)

S. Wandzura/ Deep inelasticleptoproduction

(_ M2 -F

~J(QZ) =

J + 2n

2s

QZ] LJ+ 2n + 1 °J+2n+l(Q2)

rt=l

+--

419

vj+ 2n(Q 2

J even,

(42)

where 2X

J+ 1

X 2FI(x, Q2) -

~]F2(x,Q

2)

J=0,2,...,

U2, o = 0,

(44)

/a2'J--(J+2)(J+3)

o ~

(j2+2J+3)

l+~/l+4M2x2/Q

4M2x 2I

+ 3(J+ 1),,/i +4M~x~/O~+J(J+ 2) ~ / F 2 ( x ,

&), J = 2, 4 .....

Vo = O,

OJ -

(43)

(45) (46)

4

J +2

j~ dx i~0

-~

2x

-]J+l

+ ~/1 + 4M~x2/Q2J

X (1 + ( J + 1)~/1 +4MZx2/Q2t (gl(x, Q2)+g2(x, Q2)] J = 1, 3 .....

8

I( J 2 + 4 J + 6 )

vJ = - ( j + l)(J+ 3)(J+4) o -~ l +~/l +4M2x2/Q2 . 4M2x 2 ]

(47)

+ 3(J + 2 ) ( 1 + 4MZx2/Q 2 + (J+ 1)(J+ 3 ) - ~ j g 2 ( x ,

Q2)

J = 2, 4 .....

(48)

420

S.

Wandzura/ Deep inelastic leptoproduction

The moments/11, J and t.12,j correspond to/~l,n and/In ]./1,n + 3/22,n of Nachtmann [5] and t o f 1 + i f 4 and (fln + ~f4) _ 3f3(n + 1)/(n - 1) of Baluni and Eichten [6]. Although the momentum integration for ~q, o may not converge, and Tl (0, Q2) is not measured in deep inelastic scattering, the expression for Qo is included for completeness. The infinite sums in eqs. (41) and (42) can be pulled under the moment integrals (47) and (48) and expressed analytically, but the results are not useful for numerical processing of the data as they involve small differences of large functions. =

3. Neutrino scattering The forward scattering amplitudes analogous to (1) for unpolarized inelastic neutrino and antineutrino scattering are

TUv ( ' )"( p ,-a, a ) : i~l ~ ~ f

uv (P , q ) : ~ il

d4x

f

eiq .X(p, siT* (j~weak(x)J~veak(o)] IP, S),

d4x eiq'x(p, slT* {J2"eak(x)J?uweak(o)]lP, s)

: T(2( p, - q ) . The absorptive part of

(49)

(50)

Tuv is expanded as

+ 21142i euuxeq XpO W}~(v, Q2) + terms ~qu, qv

(51)

where terms which do not contribute to the cross section for scattering of massless leptons have been dropped. If we write the OPE as [13] T* {j?weak . (X) jweak v ( 0) )

J=o

i

s. Wandzura/ Deep inelasticleptoproduetion + (guuOo,O~ +guagvoOpa °) ~

421

~ e ~ i ) ( x 2 - ie)

J=2

i

C-i ~#'x I .../JS--2 ( 0 )

X X # l ... X U j _ 2 ,a(i )

- i c~o ~ ~

J=l

~ O(o(x ~ - ic)x, 1 ...x,,_, o ~ ' " ~ ' - ' ( o ) i

+ terms ~a u, ~ ,

(52)

define the momentum space Wilson coefficients G(Q 2) and e'(Q 2) as before, and / d \J-1

OS(Q2):iQ2a{d~)

?cJi)fd4xeiq'x~S(x2-ie).

(53)

We obtain CJ(Q2)

=/21, g

(Q2) - gu2,J(Q , 2 ) - ~Mrr{T(v)(o, Q2)+ rl(-a)(O, Q2)] 6j, o ' (54)

~,J(Q2) = Uz,j(Q2),

(55)

~J(Q2) =/13,j(Q2),

(56)

where

2x

i

+ (_)j Fl(V-)(x,

2xM2 {F(V)(x, Q2) + (_)JF2(V-)(x,Q2)~

(57)

6x + ~ - (57a)

/'/2,0 =/'/2, 1 = 0 ,

~2,J- - ( j + 2 ) ( j + 3) ° x-3

+X/I+4M2x2/Q

× /(J ~ + 2 J + 3 ) + 3(J+ 1)X/1 +4M2x2/Q 2 t

422

S. Wandzura / Deep inelastic leptoproduction + J ( J + 2) 4M2x21 Q2 ] {F2(V)(x, Q2) + (_)JF2(VJ(x, Q2))

//3,J--J+2o

x-2

J ~> 2 ,

(58)

1 +~/I +2M2x2/Q

X {1 + (J + 1)x/l + 4M2x2/Q 23 { ( - ) J + ' Fa(V)(x, Q2) + Fa(~)(x ' Q2)} J ~> 1 ,

(59)

where 0

F3(x ' Q2) = _~ W3(v ' Q2) = I m mY--T3(v ' QZ) .

(60)

I would like to thank D. Soper, S. Gottlieb and F. Wilczek for many helpful discussions.

Appendix A Tensors belonging to the (l ( j

1), ½(J + 1}j representation o f O(4)

One term of the operator product expansion of T* {Ju(x)J,,(O)) contains operators of mixed symmetry, which transform according to the (½(J - 1), ~(J + 1)) representation of 0 (4) [9]. As discussed in ref. [9] such a tensor M x°ul "'"uj - 1 satisfies (i) M x°**l "" " J - l is antisymmetric in (X, o); (ii) M x°u~ "'"" J - 1 is symmetric in the/a's; (iii)MXOUl ...ug 1 + MOlalhla2...#J-I + M#I~a#2...#J-1 = 0; (iv) MX°Ul "'" uj - 1 is traceless. Using techniques similar to those found in ref. [9] we may construct the orthogonal projection operator onto the subspace of such tensors, which we denote by/~(j). The result is ""vg-1

J

2 ( J + 1) [{(g~P(J) ev~

J-I ~ [~(.7~ai~ ,p

2 ( J + 1) 2 i=J

_

....

va-l"

° ° / " l ' " [ # i ] ' " # J - l ) _ ( ) k ~ o ) 1 _ {¢~+->~]

c~ (J)t3v1 ...uj_ 1

1 x o" /.tl...bt J 1] __ (~k ,_~ O)] J + 1 [(ge~g~P(J-1)Vl ""VJ-l"

S. Wandzura / Deep inelastic leptoproduction

423

J-I

1

I3

2 ( J + 1) i=1

[rr-h~#ip akI,/~o~,~j3

Ul"'[tsil'"laJ-l°)--(X'>,O) 1 -- {O~o~]]

( J - 1 ) v 1 ... v J _ 1

J-I 1

J(J+ 1)

~ i=1

[ { ( g X ~ Z i g O g ~ f 3 , e ( j _ l ) VUl...l~zil l . . . v j _ ~. . # J - l f l ' )

()k +_>.0-)] _ {~ <-+ ~)]

J-I 1

~

2J(J + 1) i~j

[((g

34Ji #j p #l...[#il...Itsjl...gJ-l°O') go~ g(3fl' ( J - 1 ) v 1 . . . v J _ 1

J-I

1 ~ Ui#i h lal...luil~..l#il...gJ-lO[3' ) g [ ( ( g a g ~ ' P ( J - l ) v 1 ...vj_ - 2J(J + 1) i
{ aX b° cUl

hOtZl ...l~J-1 ~C~bt3 cU 1 ... ZVJ--1 ... Z I 2 J - I ] M J ~ --(d) oo3v1 ... V j _ l u

(A.2)

and a totally symmetric traceless tensor as [9] "-" #J ( a u l "'" ZIAJ]J -- P ( J ) #1 v 1 ... uj aul "'" zuJ ,

(A.3)

Appendix B Sample calculations o f contractions appearing in the operator product expansion In order to project Wilson coefficients, we want to express the invariant amplitudes as expansions in orthogonal sets of polynomials, in particular Gegenbauer polynomials of some index. Contractions are computed using: (a) the expressions for P(j) found in appendix A or a recursion relation for P(j) from ref. [9]; (b) removal of a factor qu by differentiation; (c) recursion relations for the Gegenbauer polynomials [ 10] and (d) Nachtmann's [5] basic contraction: qtz I "'" qt~J ( P tal "'" P ~zj] j = ( _ 1 iMQ)J CO)Q?),

(B. 1)

S. Wandzura /Deep inelastic leptoproduction

424 whe re

iv

03.2)

rl = MQ ' and C(1) is a Gegenbauer polynomial of index one. To illustrate this procedure, we show the computational details of two o f the contractions which arise:

qUl "'" QUJ-1 {papUl ... p # J - 1 3 j

- J Oqc~ qgl "'" q g j ( p U l ... puj~ 1 J

( _ ~ i M Q ) j ( _ j C ( , ) ( r ? ) + 7 ? C O ) , ( r D } +pC~(_~iMQ)~

_ 1 ~q~

-JL " ( - ~ i M Q ) J

~

i

•

2C(~2(r~)+P ( ~ t M Q )

J--I

CSa_~l(n) 1 ,

J

, ~J

(B.3)

q~tl "" quj {S~zlpg2 ... pgJ} j J =q~l "'quj

1

j

s~i{P~l "'" [ptai]'''p~J]J-1

q g~iU]s~ {p~p~l ... [ptSi] ... [plaj] . . . p ~ t j l j _ l | 3 i
= q" s ( ~ i M Q )

J - ' C(')_,07) + l q .

= ~1q " s ( - ~ i M Q ) J - 1 C(jz)_, .

s(_½iMQ)j

, C(2)3(r~ )

(B.4)

Appendix C Table o f con tractions We give below a table of contractions which appear in matrix elements of operator product expansions. In this table p2 = M 2 ' q2 = _ Q 2 , p . s = 0, r/= i p . q/MQ,

425

S. Wandzura / Deep inelastic leptoproduetion a = --~iMQ,

and all Gegenbauer polynomials of negative order are taken to be zero:

qul "-" quj

{p~l ... PUJ]s = aSCjO)(r/) ,

qtal "'" q u J - 1 {pC~pUl ... p U j -

= JI-Fq~ a s 2C~20/)+ [Q:

(C.1)

l]j

Cj~,(rl)]

p~a s-'

(C.2)

qul "'" quJ--2 {pap~pUl ... p U J - 2 ] j

_

J(J-

1

FgU~

1) L Q a

s.~(3)

~q

~ ..q

a -t~)_ z(r 0 * - ~ -

aJ

8Cj(3_)4(r~)+ papt~ a J - 2 2C(3)z(r/)

+ P~q~Qt P~ aJ-' 4CJ3_)3(r/)l,_]

(C.3 )

qul "'" q u J - 3 {pC~p~ pT pUl ... plaJ- 3] j

= J(J -

1

1)(J - 2)

fgC~¢q'y + gaTq~ + g~37qa

+ gC~p7 + gOe~ Q2pO + g ~ +p~p~pVaJ-3

Q4

pa a y - I

aJ

8C(3-)4(r/)

~ 4C(3-)3(r/) + qO~q~q7

6C(4_)3(r~)+ qaq~p7

+ p a p # q ~ +paq~p,~ + qe, pOp,y a J - 2

Q2

~J

48C(4-)6('r/)

+ qap~qV + p a q ~ q 7 a J - I Q4

12Cj(4-)4('o)] ,

qul "'" quJ { sul pU2 ... pUJ) j = 1 q " s a J - l C(2)_1(r/),

qul "'" quJ-1 {s~p~I "'" ~"

aJ

24C(4-)s(r/)

(c.4)

(c.5)

j2

+q'~ q ~ a"-' 4cJ~(,7) +p° q. sa '-~ 2c~(,7)q

(c.6)

426

S. Wandzura / Deep inelastic leptoproduction q'~ "'" quJ-2 {sap#pUt "'" p U J - Z ) j

+ ~o~¢3 ~

+

j2(j_

aJ-I 24Cj(~5 (r/) + pC~p¢ q ' s

S_°~q ~ + q°e S~3aj -

02

i) ga~

a J - l 4Cj(~3(r/)

a J - 3 6Cj(4)307)

1 4CJ3_)3(rl) + (sap~ + p a s ~ ) a J - 2 2Cs(3_)2(r/)

+(paqe+qap~)~aJ-2

12C(4_,)4(~)] ,

(c.7)

q ~ q u l "'" (lgj-1 {s~'p°Pul "'" P u J - I } M j

_

J(J+l 1) Ip~q sa J-~ ~-Q'),(?)_

- Q~(,~)I

q •S j

+ SkglJ{1jC52)(~ ) _ ~(S + 2)Cj(Z__)z(r/)] + qx ~ - - a

(3) ] {-4C)_2(r/)] , (C.8)

qul "'" q u j - l {s~'p°Pul "'" P u J - 1)Mj _

1 I(sXpO _ p X s O ) a J - I _J + 1 C(2) (r?'~ J ( J + 1) 2 ,t-l'- "

+

S~,q o _ qXs ° • _ Q2 a J ( j + 2)Cj(2!2(7~)+ (p2~qO _ ql~.pa)_~_~_dJ-I 2Cj(3)3(~)] .

Formulae (C. I)-(C.4) may be generalized to

(C.9)

qul "'" q u j - n {P al "'" p % pUl ... pUS-n } a ln/2l

n-2t

- ( " - ")! ~3 ~3 2"-='-"'"r("- e)r J]

1=0 m=o l ! m ! ( n - 2 1 - m)!

X {g al a2 ... g a21- I ee2lpC~21+1 ... pC~2l+m qa21+m + 1 ... qCb*)Sn

a y-m p(n-l+ 1) (-n$ X Q 2 ( n - l - m ) ~J-2n+21+mk't.; '

(c.lo)

S. Wandzura / Deep inelastic leptoproduction

427

where S n denotes s y m m e t r i z a t i o n o f the n indices. This may be proved by i n d u c t i o n using the m e t h o d s described above. A similar expression generalizing (C.6) and (C.7) can be found: q u l "" q u j

n {salpa2

... p a n p U l

ln/21n

21 2 n _ 2 1 _ m n ! ( n _ l +

I=O m=o

JJ!

l!rn!(n - 2l-

× (g ala2 ...ga2l--la2lpa2l+l

×

...pUJ-n]s

l)!

m)!

...pa2l+mqa21+m+l

...qan]s n

a J - m - 1 t . ( n -•+2) Q2 (n - m -1) "-'J-2n +21+m - 1(?'/)

[(n-O/2] J J!

1:0

n-21-1

m

2 n-2l-m-1 l!m!(n

n ! ( n -- l)!

- 2l-m

- 1)!

X {ga I c~2 ... gC~2l- 1 a21 pC~2l+ 1 ... pC~21+rnqa21+m + 1 ... q a n - 1 sCtn ] Sn

a J-m-1 X Q2(n-m-l-

r(n-l+l) r~a 1) ~ J - 2 n + 2 l + m + 1 k,t)

(C.11) ,

h o w e v e r the generalization o f (C.8) and (C.9) is m o r e difficult and has n o t been attempted.

References [1] D.J. Gross and F.A. Wilczek, Phys. Rev. D8 (1973) 3633; D9 (1974) 980; H. Georgi and H.D. Politzer, Phys. Rev. D9 (1974) 416. [2] K.G. Wilson, Phys. Rev. 179 (1969) 1499. [3] H. Georgi and H.D. Politzer, Phys. Rev. D14 (1976) 1829; A. De Rujula, H. Georgi and H.D. Politzer, Harvard preprint HUTP-76/A155 (1976). [4] D.J. Gross, S.B. Treiman and F.A. Wilczek, Princeton preprint (1976); R. Barbieri, J. Ellis, M. Gaillard and G. Ross, CERN preprint (1976); R. Ellis, G. Parisi and R. Petronzio, Univ. di Roma preprint (1976). [5] O. Nachtmann, Nucl. Phys. B63 (1973) 237; B78 (1974) 455. [6] V. Baluni and E. Eichten, Phys. Rev. Letters 37 (1976) 1181; Phys. Rev. D14 (1976) 3045. [7] N. Christ, B. Hasslacher and A.H. Mueller, Phys. Rev. D6 (1972) 3543; M.A. Ahmed and G.G. Ross, Phys. Letters 56B (1975) 385; K. Sasaki, Kyoto Univ. preprint KUNS 318 (1975). [8] R.L. Heimann, Nucl. Phys. B65 (1973) 429.

428

S. Wandzura / Deep inelastic leptoproduetion

[9] A.H. Guth and D.E. Soper, Phys. Rev. D12 (1975) 1143. [10] I.S. Gradshteyn and I.W. Ryzhik, Tables of series, integrals and products (Academic Press, New York, 1965) p. 1030. [11] See ref. [10], p. 827. [12] A. Erdelyi et al., Higher transcendental functions, vol. 1 (McGraw-Hill, New York, 1953) p. 137. [13] M.A. Ahmed and G.G. Ross, CERN preprint TH.2138 (1976).