The double logarithmic approximation of deep inelastic leptoproduction in the quasielastic region

Nuclear Physics B232 (1984) 398-412 © North-Holland Publishing Company

THE D O U B L E LOGARITHMIC APPROXIMATION OF DEEP INELASTIC LEPTOPRODUCTION IN THE QUASIELASTIC REGION Th. OHRNDORF 1

Department of Physics, University of Siegen, BRD Received 25 November 1982 (Revised 1 August 1983) We determine the asymptotic behavior of the deep inelastic leptoproduction structure functions in the quasielastic region z ~ 1 in the leading double logarithmic approximation of QCD. It is shown how the asymptotic behavior is related to the ON/OFF limit of the Sudakov form factor. The running coupling constant is incorporated by comparing the double logarithmic approximation with the solution of the renormalization group equation. We compare our result with various suggestions to improve the convergence of the perturbative expansion in the quasielastic region.

1. Introduction A problem which has received considerable attention in perturbative Q C D for the last few years is the presence of large corrections near the boundary of phase space. Such corrections have been discovered in the calculation of first-order corrections to parton scattering amplitudes. For example in the case of leptoproduction [1,2] there are corrections of the form log ( 1 - z ) / ( 1 - z) which become large if z, the parton Bjorken variable is close enough to one, i.e. in the so-called quasielastic region. If moments M~ are used, the structure functions in the quasielastic region are determined by the high moments. A behavior like log (1 - z ) / ( 1 - z ) of the structure functions corresponds to lim,_~ Mn - log 2 n. As one has to expect that higher-order corrections behave like (as log 2 n) L, the perturbative expansion breaks down in that region. Therefore one has to find a way of resumming it. Though this problem is not peculiar to the quasielastic region of deep inelastic scattering, in this paper we shall restrict our considerations to this case. In the literature various suggestions may be found of how to sum the large corrections over all orders of perturbation theory. The first solution of this problem has been proposed by Gribov and Lipatov in their classic paper [3] in the context of spinor electrodynamics. It has been extended to non-abelian gauge theories by Dokshitzer [4]. Subsequently similar solutions have been given by various authors [5-10]. There is a general consensus that the large corrections exponentiate, leading to a strong suppression of the structure functions in the region of interest. Yet there is no agreement what concerns the detailed form of the asymptotic behavior. 1 Supported by "Studienstiftung des deutschen Volkes". 398

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399

Particularly in [11] it was shown that the approaches of ref. [5] and ref. [8] both reproduce the lowest-order correction of [1, 2], but disagree beyond that order. It was emphasized in [11] that no proof has been given so far, which dissolves this unsatisfactory situation. However, lacking a general proof, a leading log calculation going beyond the first order in as would give enough information at least to eliminate one of the two solutions compared in [11]. This is the purpose of the present paper. We have investigated the z--, 1 limit of the leptoproduction structure functions in the leading double logarithmic approximation (LDLA). Although we have carried out this investigation only up to the third order in as our results strongly suggest an extension to all orders. Therefore we shall represent them here in their extended form. The diagrams have been analyzed using a modified version of the standard technique to determine the high-energy behavior of Feynman integrals [12], which has briefly been described in [13]. The Feynman gauge is employed throughout and infrared singularities are regularized by dimensional regularization. We shall work in a completely massless theory. Instead of studying the diagrams of the leptoproduction process directly, it is much more convenient to study those of the forward Compton scattering amplitude and derive the asymptotic behavior therefrom via the optical theorem. The advantage of the Compton scattering amplitude is, that no real corrections have to be taken into account. Let us give a short summary of our results. We find that the z ~ 1 limit of the forward Compton scattering amplitude is essentially dominated by the square of the Sudakov form factor in the O N / O F F limit, which has been determined in [13]. From these grounds we derive the asymptotic behavior of the structure functions. It is in agreement with that given in [4] whereas it disagrees with [5]. In sect. 2 we shall first, after clarifying some notation, define the L D L A and demonstrate how the L D L A of virtual parton-photon scattering and the L D L A of the structure functions are related. Thereafter we shall discuss the structure of the dominant diagrams. The asymptotic behavior of the structure functions is calculated in sect. 3 with a fixed coupling constant. In sect. 4 we compare the asymptotic behavior with the solution of the renormalization group equation (RGE), taking the running coupling constant into account. In sect. 5 we compare our results with those given in [5-10].

2. Dominant diagrams Before we proceed to the discussion of the dominant diagrams, let us briefly introduce some notation and define the LDLA. We shall use a decomposition of the forward photon-parton scattering amplitude, respectively its absorptive part, as given e.g. in [2]. The parton structure functions are denoted by Fi(O 2, z), i -- 1, 2. They are related to the corresponding functions of the spin averaged forward photon-parton scattering T~(Q 2, z) via the optical

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theorem Fi( Q2, z) = 1 Im (-iT~(Q 2, z ) ) .

(2.1)

The parton Bjorken variable z is given by _q2 z-

2p- q

.

(2.2)

The incoming parton carries the momentum p and q is the momentum carried by the photon current. Moments are defined by Mn(O 2) =

I0

d z z n r ( o 2, z).

(2.3)

We shall work in d = 4 - 2 e dimensions. Let us now turn to the L D L A . Our aim is to obtain an expansion of the mass singularity free moments MSnUb(Q 2, O 2)

MSUb(O 2, 0 2) =Y~ a p Y~apq(1 + O(as))(log

p

q

( O 2 / O 2 ) ) q ( l o g n) 2p-q ,

(2.4)

where Q2 is the factorization scale. Note that our approximation goes beyond a simple double logarithmic approximation in log 2 n, which would keep in (2.4) only terms with q = 0. Such an approximation breaks down in the region Q 2 / Q 2 ~ n. From the renormalization group, respectively the analysis of mass singularities in hard scattering processes, we know am = 0 for q > p. In order to obtain an expansion of M sub (Q2, Q2) like (2.4) via the optical theorem we have to find the coefficients a m of an expansion of the Compton scattering amplitude T~(Q 2, z) which takes the form -i T~(O 2 , O 2 ) = l_z+ie,~p aP~q a m ( l + O ( a s ) ) x (log

O2/Oz)q(log

[-(1-z)])2P-q

1 ,

(2.5)

.3+

after the removal of the mass singularities occurring as poles at e = 0. The [ ]+ regularization is defined as usual

Io

dzf(z)[g(z)]+ =

dz[f(z)-f(1)]g(z).

(2.6)

The equivalence of (2.5) and (2.4) is readily demonstrated by inserting (2.5) into (2.1) and taking moments. Taking the absorptive part in (2.1) one power of log I(1 - z)[ gets lost. This power is reestablished by taking moments according to

foI d z z ' [

l°gk(-1--z)] L 1-z J+

)k 1 1ogk+, ~ + O ( 1log k+l

.

(2.7)

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401

Q2(1-z)/z )

p

q

Fig. 1. Structure of the dominant diagrams. Now we wish to discuss the structure of the dominant diagrams. As is clear from (2.5), if z goes to 1 all the dominant diagrams have to behave like 1/(1 - z ) modulo logarithms. The moments of contributions which lack such an inverse power of ( 1 - z ) are suppressed by one power of l / n compared to (2.4). Diagrams which are one-particle reducible in a channel with vanishing invariant mass at z = 1 (see fig. 1) clearly behave like 1 / ( l - z ) . Yet these are not the only diagrams which behave in this way. One-particle irreducible diagrams may as well exhibit such a behavior. However, as we shall demonstrate, these diagrams do not contribute to the L D L A . As the irreducible blobs in fig. 1 represent the q u a r k - p h o t o n vertex function, we have a rigorous relation between the O N / O F F version of the Sudakov form factor and the L D L A of the quasielastic limit of the leptoproduction structure functions. The O N / O F F limit of the Sudakov form factor has been determined in [13] in the L D L A . We shall prove the factorization property depicted in fig. 1 explicitly up to the third order in a~. Our strategy is as follows. First we shall eliminate all the one-particle irreducible graphs, which do not behave like 1/(1 - z ) . Then we shall determine an upper bound on the n u m b e r of logarithms for each of the remaining graphs. After introducing Feynman parameters the general form of a Feynman integral of an irreducible graph ~q contributing to the spin averaged forward p h o t o n - p a r t o n scattering amplitude is given by

I = K F ( #) f ' [ d a ] Z ( a ) C ( a ) ~ =+~/D(a) ~

(2.8)

where K=

i

,

(2.9)

~ = n - 2 L + Le ,

(2.10)

[ d a ] = d a , • • - dan 6 ( a , +" • "+ ~ , , - 1).

(2.11)

L is the number of independent loops of ~q, and n is the number of its internal lines.

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Rules how to calculate the parametric functions Z ( a ) , C ( a ) , and D ( a ) may be found e.g. in [12]. Here we only need some elementary properties of these functions. Z ( a ) is constructed from the numerator of the integrand of the Feynman integral, appearing before the invariant integration has been carried out, simply by replacing the momentum of the ith line by Yi, a linear combination of the external momenta of ~3. Though there are further terms, these can be neglected here. Again, rules how to calculate the momenta Yi for a general graph are given in [12]. The only two things we need to know about these momenta are, that at each vertex they satisfy 4-momentum conservation, and under dilatation of the parameters of subgraph ow~ ~3 up to terms of the order of magnitude of the scaling parameter, the Y~ become the momenta of the reduced graph ~/5¢. q3/5¢ is obtained from ~3 by shrinking all the lines of 5~ into a single point. Here D ( a ) is given by

D(c~) = q2[h(a) + Tg(a)],

(2.12)

,r = ( p + q ) 2 / q 2 .

(2.13)

with

Alternatively we can write n

D(a)=C(a)

• aj(Y~ +ie').

(2.14)

j=l

In order to calculate the leading behavior of I for r<< 1, we take its Mellin transform with respect to ~MT{I}

[

m_ d O

l/u

dT~-' ' 1 ( 7 ) ,

1/u > 1,

(2.15)

eq. (2.15) is calculated to be

M T { I } = K F ( [ ~ - I ) ( q 2) ~

[ d a ] C ( a ) ~ -2÷~,

x F(l)h(a)I-¢(h(~)u + g(~))-t.

(2.16)

A behavior of I like 1/r requires MT{I} to have a pole in the complex /-plane at l = 1. If this pole is of the order n we have

Poles in the /-plane are caused by the vanishing of h ( a ) . h ( a ) vanishes only if a set of Feynman parameters goes to zero simultaneously. This is enforced by the vanishing of p in

ai =pdi,

i = 1. . . . . m ,

~ ~i = 1.

(2.18)

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403

We call a set of parameters which let h ( a ) vanish a scaling 5~. The quantity rne,= r n - 2 L ( S f ) + Z ,

(2.19)

is called the effective length of a scaling, where Z ( a ) - p Z . Setting e to zero, a scaling of length metT induces a pole at l o = p - m ~ t ~ , provided g ( a ) remains finite. We call such a scaling of the type h. If g ( a ) vanishes as well, I behaves like 1/lo, i.e. I - 1/e if p = m~,. Such a scaling is called type g, h. Different scalings can be performed in turn. Having already carried out a certain n u m b e r of them, the possibility of performing a further scaling crucially depends on the proceeding scalings. This is due to the form of the integration measure [ d a ] and the constraint in (2.18) on the transformed variables. The maximum number of logarithms is obviously determined by the maximum number of scalings, which can be performed in succession. To each scaling 5~ corresponds a reduced diagram qd/5e. In order to investigate the reduced diagrams m o r e closely, we choose a frame of reference where q = q - zp ,

q2=p2 =0

and sum over the spins of the outgoing partons by contracting with p. As the parametric function D ( a ) can be expressed as in (2.14), the m o m e n t a Yj of the reduced diagram which corresponds to a scaling ~ of type h, g must all be light-like Yi( ~q/ Se) - p,

(2.20)

independent of r. Therefore such a reduced diagram qS/SP must have the form displayed in fig. 2a. The structure of a reduced diagram belonging to a scaling of type h is slightly more complicated. Now some of the m o m e n t a Yi of the reduced diagram are again parallel to p, while the rest is parallel to q + ( 1 - z)p. Therefore the reduced diagram must have the structure shown in fig. 2b. Now we are ready to identify all those diagrams, which do not behave like 1 / ( 1 - z). For a diagram to diverge like 1 / ( l - z ) , or potentially even stronger, there must exist a scaling of type h of effective length m~. ~

~ 1,

Fig.2a

(2.21)

Fig.2b

Fig. 2. (a) Reduced diagram of a scaling of type h, g. (b) Reduced diagram of a scaling of type h.

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Th. Ohrndorf / Deep inelastic electroproduction

~

B

[3'

Fig. 3. Suppressed contribution.

where n(~g/5e) (L(Cg/0°)) denotes the n u m b e r of lines (independent loops) of (g/Se. It is a simple exercise to show by inserting one additional line into an arbitrary reduced diagram of the kind shown in fig. 2b, that (i) the left-hand side of (2.21) cannot exceed one; (ii) for all the reduced diagrams which can be decomposed as displayed in fig. 3, the left-hand side of (2.21) is smaller than one. The second conclusion can be restated in terms of the original diagram ~d. A graph, in which the two fermion paths propagating from A to B and from A' to B' are connected by a single gluon line, cannot behave like 1/(1 - z). By a single gluon line we mean a genuine gluon line without any vacuum polarisation or self-energy part. This criterion diminishes the n u m b e r of diagrams which remain to be investigated considerably. We can even exclude a larger set of diagrams. All the reduced diagrams, which are obtained by inserting additional lines into a reduced diagram of the form shown in fig. 3 can as well be dropped. We would like to point out that in the process of generating new diagrams we have to treat the three components the three- and the four-gluon vertex consists of separately. We have determined the maximum number of logarithms of all those diagrams up to O ( a 3) which can behave like 1 / ( 1 - z). These diagrams are represented in fig. 4. We have omitted all those diagrams which are obtained by the insertion of a renormalization part into a diagram, already considered at lower order, because such an insertion can increase the number of logarithms at most by two. The leading behavior of each individual diagram is given by

~- I=co(q 2) ~c~ l f ( r , e ) . 7"

(2.22)

The functions f(r, e) are given in table 1. Co and cl are numerical constants varying from graph to graph. Our results clearly show that an irreducible diagram of the Lth order in c~ cannot have more than 2 L - 2 logarithms. We would like to point out, that our result is only valid in the Feynman gauge. Choosing a non-covariant gauge, gauge denominators open an alternative source of a behavior like 1 / ( 1 - z ) . This is for example the case for the famous planar

Th. Ohrndorf / Deep inelastic electroproduction \/ /\

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f ( r , e) 1/e 1/e 1/e 1/e log 2 ( r ) l / e 2 + c 1/e 1/e 1/e

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f ( r , e)

9 10 11 12 13 14 15

log 2 (z) l / e 2 + c I log (r) l / e 3 1/e 4 log (r) l / e 3 l o g ( r ) l / e 2 + c l ( 1 / e 3) 1/e 4 1/e 2 1/e 3

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Th. Ohrndorf / Deep inelastic electroproduction

gauge [15]. As is well-known in this gauge ladder-type diagrams contribute powers of (as log ( 0 2 / O 1 ) log n) L to the moments. Therefore in these gauges our factorization property breaks down. The same thing happens in the Landau gauge.

3. The asymptotic behavior In this section we shall derive the behavior of the leptoproduction structure functions in the quasielastic region. We shall start from the O N / O F F limit of the Sudakov form factor F(Q2, r) and the factorization property of sect. 2. According to the latter T~(Q 2, z) is given by -i T~(Q 2, z) = (1 - z) + ie' F2(Q2' z).

(3.1)

Let us recall the O N / O F F limit of the Sudakov form factor

1"( Q2, r) = exp {- C(1 - r-~)},

(3.2)

where

(l-z)

(p+q)2 7"=

- -Z ,

q2

(3.3)

C = C F - ~ (JJ,2/~)2)e ~ .

(3.4)

/.£2 is an arbitrary scale which has to be introduced in d dimensions in order to keep the coupling constant dimensionless. Inserting (3.1) and (3.2) into the right-hand side of the optical theorem (2.1) and expanding l"(O 2, r) in powers of as we obtain

Fi(O2, z)=e -2c { 6 ( l - z ) -

1

~ ~.v(2C)L(1

_ Z ) _ e L _ 1 sin (~Le)}

,

(3.5)

L=I

for the structure function. The e--> 0 limit of (3.5) is not finite, there are poles at e = 0 which have to be interpreted as mass singularities. According to the theorem on the factorization of mass singularities [16] a finite structure function F~ub (Q2, 0 2, z) can be defined as follows F,(Q 2, z) = f l dYFo(Q2, l / e , y)F~Ub(Q 2, O 1, x/y) ; dz Y

(3.6)

or after taking moments M,(Q2) = Mno ( Qo, 2 1/ e)M~Ub ( Q 2, Q2) .

(3.7)

All the mass singularities are absorbed into F°(Q2, 1/e, y) respectively M ° (Q2, 1/e) which is associated with the bare fragmentation function. Eq. (3.5) is easily cast

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into a convolution like (3.6) by first expanding ( l - z ) -~L-~ = log

1) -eL--I

(3.8)

(1 + O ( 1 - z ) ) ,

and subsequently substituting the identity [(log (1/z)) a+b 1 .][ (1/y))b-t] = - - F ~ + b - ) --- ' 1] (log t

F--~ f[~[(log(y/z))a

ifa#-b (3.9)

F--~

_]

ifa=-b,

~6(1-z),

into (3.5) with a = - Z o - e L and b=Zo, where Z o = Z o ( Q 2, l / e ) is a subtraction constant introduced to remove the mass singularities. It is of the order as/e. The identity (3.9) is easily shown either by direct evaluation (a # - b ) or by taking moments (a = - b ) . We obtain jC'sub(4~)2 0 2 , z ) = e --' "~ '

2C

L~0~.w (2C)L (log(1/z)) -z°-~L V(-Zo-eL)

'

F o (Q2, l / e , z) - (log (1/z)) zo-'

(3.1o) (3.11)

r(Zo) Before we can take e to zero, we must regularize the singularity at z = 1. This is done by the [ ]+ regularization being defined by (2.6). Using the property

dzF~sub ( O ,2O 2 , z)=

dzf°(O2,1/s,z)=l,

(3.12)

we get

F?ub(O:, O 2, Z)= [F~ub(O:, Q 2, z ) ] +

+ 6(1-

z) ,

(3.13)

F° ( O 2, 1/ e, z ) = [ F ° ( O 2, l/e, z ) ] + + 8 ( 1 - z ) .

(3.14)

Now we can let e go to zero. To leading double logarithmic accuracy we can set

1 F ( - Z o - eL)

-(-Zo-eL)+"

" ,

(3.15)

in (3.10). Therefore it becomes FSUb(o2, Oo2, Z) =

exp{2C[(1-z)-~-l]-Zolog

(l-z)}

+6(l-z). -t-

(3.16)

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408

Expanding the exponential in powers of e (3.16) takes the form

F~Ub(Q2,Q2, z)=

0

exp

{os

GF-~ log (1-- z)

x(log(Q2/ix2)+½10g(1-z)-l/e-Zozr/a~CF)}] ++ 6(1 -

z).

(3.17) In order to subtract the mass singularity we choose

Zo=-CF 7as le( \Q.22' ~] - ~

(3.18)

Consequently our final result for the frozen coupling constant is

f~uU(o2, Q2, z) =

Cv

%

1

"B" 1 - - Z

(log

(Q2/Q2)-log (1 - z ) )

xexp(CF~lOg(1-z)[log(Q2/Q2)+½1og(1-z)])]+ +3(l-z).

(3.19)

The moments M ° (Qo2, I / e ) and MSUb(Q2, Q2) are most conveniently derived from (3.10) and (3.11) by using the identity

f dzz"

log

F(a)=n -a

( (1)) 1+O

.

(3.20)

We immediately obtain M , ( Q 2) = exp { - 2 C(1 - n~)}, M ~ b ( O 2,

Qg)=exp{CF~log(1)(log(QZ/O2)+½1ogl)}, 2 m ,0 (Qo, l / e ) = exp { - Z o log n}.

(3.21a) (3.21b) (3.22)

At this stage it is instructive to compare (3.2) and (3.21a). The asymptotic behavior of the moments is just given by the square of the O N / O F F limit of the Sudakov form factor with r replaced by 1/n. The reader should bear in mind that the terms which have been dropped in going from (3.2) to (3.21a) are down by (log n) -1. Therefore the absence of logarithmic corrections to (3.2) in the limit e + 0 does not imply that there are no logarithmic corrections to the e + 0 limit of (3.21a) or vice versa. The first term in the exponential of (3.21b) - l o g ( l / n ) l o g (02/02) is usually obtained as a solution of the R G E whereas the second term has to be associated

Th. Ohrndorf/ Deepinelasticelectroproduction

4(19

with the coefficient function. Note we obtain the n-->co limit of the anomalous dimensions yo(n) lira ~/o(n) = 8 Cv log n

(3.23)

n~ce

solely from the O N / O F F Sudakov form factor. A relation of this kind has been discovered previously in [19].

4. Incorporation of the running coupling constant According to the classification scheme of leading and subleading logarithms presented in sect. 2 the variation of the coupling constant is a sub-leading effect. This implies that in the L D L A only a fixed coupling constant can occur. Yet the variation of the coupling constant in QCD is a phenomenon too important to be neglected. Fortunately we know that we can use the R G E to perform a systematic resummation of logarithms in 02. We shall take these effects into account by comparing (3.21) with the solution of the RGE. The formulation of the R G E in d dimensions as required here may be found in [17, 18]. The /3-function in d dimensions el(g, e) is given by

~(g, e)=,8(g)-ge.

(4.1)

We refer to the/3(g) -->0 limit of the running coupling constant as the fixed coupling constant c7~(O2). It is given by

c~(O 2) = a~(p. 2)

_

.

(4.2)

Setting

y(n,

c~) = E= \ 4 n - ] y' , ( n ) ,

(4.3)

and /3(g) = -g3/3o,

(4.4)

the solution of the R G E takes the form

M•b(o 2,0~)

\a,(O~)] x

[ 1 '~ry,(n)

]

(4.5)

where the Cn(a~(O2)) are the moments of the coefficient function of the operator product expansion. In the limit/3o--> 0, M,,(O ~, l / e ) becomes

Mn(O2,1/e)=exp{~e[(~)yo(n)+"']}C,,(6L~(02)).

(4.6)

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Comparing (4.6) with (3.21 ) we can fix yo(n) and the coefficient function C, ( ff~(O 2)). Inserting both back into (4.5) we obtain a renormalization group improvement of the L D L A

Msub(oL ""~

2 Oo)

=

exp, [

2rr

Cvlog 2 n j \ ~ ]

(4.7)

We may obviously use the renormalization group and the factorization property of sect. 2 to improve the O N / O F F limit of the Sudakov form factor as well. Our result is

f 2Cv v(oZ, r)=exPi-~logrlog((~°/41r) ~s(O2) +6)/ +Cv O/s(O2) 4~-- (log T)2]. ) (4.8) Unfortunately (4.6) and (3.21) cannot be compared without expanding ( 1 - n ~) in (3.21) in powers of e. Therefore the ON limit of the Sudakov form factor as discussed in [13] cannot be obtained as the r ~ 0 limit of (4.8) as it was possible before the incorporation of the running coupling constant.

5. Comparison In this section we shall compare our results with those obtained previously by other authors. In [3, 4] the asymptotic behavior, in the region where the double logarithms are unimportant, has been determined in the single logarithmic approximation via a careful analysis of ladder diagrams. It has been claimed that the double logarithms arising in the quasielastic region are simply summed up by taking into account the correct kinematic limits of each ladder cell momentum integral. This manipulation leads to -.-, , M~'ub(o2, O 2 ) = e x p

y"[ l+y2 ] [ °2/I-Y) dk2 O/s(k2)'~ 2Cv

, dy

[.1-~-J+dog

k2

4~r J '

(5.1)

As the difference between (5.1) and (4.7) is only subleading both results are compatible. An alternative solution has been given in [8]. It is obtained from (5.1) by choosing as the upper limit of the dk 2 integral 0 2 instead of O 2 ( l - y ) and changing the argument of the coupling constant from k 2 into k 2 ( 1 - y ) . This implies that there is no difference between the standard single logarithmic approximation and the L D L A if/3(g) is zero. Our calculation does not support this conclusion. The origin of this discrepancy is immediately revealed. In [8] the modified version of (5.1) was obtained as the solution of a modified Altarelli-Parisi equation. Its modification amounted to the same replacement of the coupling constant as used above. This modified version of the Altarelli-Parisi equation was integrated with the boundary condition M~n~b(Q)2, 0o2) = 1. Yet as at least part of the double logarithmic effects

Th. Ohrndoff / Deep inelastic electroproduction

411

have to be associated with the coefficient function, the correct boundary condition is M Sub(02, 02) = Cn(as(02)). Though this replacement is not important if we are only interested in the variation of Mn with 02, it has to be well understood, if e.g. patton distributions of different processes are compared. A further solution has been proposed in [5]. In contrast to (5.1), where the correct kinematic limits of each ladder cell momentum has been taken into account, in [5] only the limit of the transversal integration of the most virtual cell has been changed, compared to the standard single logarithmic approach. It has been claimed that this procedure results in an additional contribution to the structure function 6q(z, 02)

F(Q 2, z)-q(z, Q2)+6q(z,

Q2),

(5.2)

where q(z, Q2) is the usual single logarithmic parton distribution. In order to compare the results of [5] with our own we have calculated the fixed coupling constant limit of the quantity 6q(z, Qz) +q(z, Q2) q(z, O z) (5.3) If we insert

6q(z, O 2) into

(5.3) as has been given by [5] we obtain

~q(z, O2)+q(z, Q2)_l + }~ 1 CFa~log2(l_z) q(z, O 2) i=2 77"

(5.4)

On the other hand using our own result (3.19) we get

6q(z, Q2)+q(z, Q 2 ) { log(l-z)} q(z, O2) - 1 log(O2/O2 )

exp

{as 2 _z)} Cv~log (1 .

(5.5)

The discrepancy is obvious. An important point are the subleading logarithms. In [9] subleading logarithms have been analyzed. It has been claimed that all the large logarithms (as log n) i i ~>2 which may potentially contribute to the anomalous dimension y(n) are absent. Yet in that paper subleading corrections to the coefficient functiott have not been discussed. Due to the intimate relation of both kinds of logarithms it is not unlikely that the subleading corrections to the coefficient function are considerably reduced as well. We would like to thank Prof. L. Schiilke for his constant interest and support. References [1] G. Altarelli, R.K. Ellis and G. Martinelli, Nucl. Phys. B 143 (1978) 521; B 146 (1978) 544; J. Kubar-Andre and F.E. Paige, Phys. Rev. D19 (1979) 221; K. Harado, T. Kaniko and N. Sakai, CERN preprint TH 2619 (1979) [2] G. Altarelli, R.K. Ellis and G. Martinelli, Nucl. Phys. B 157 (1979) 461 [3] V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438

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