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Asymptotic Freedom and QCD Scaling Law J iben Sidhanta Department of Physics, Charuchandra College, Calcutta-700029, India

R a j k u m a r Roychoudhury

Electronics Unit, Indian Statistical Institute, Calcutta-700035, India

Z. Naturforsch. 40 a, 450-452 (1985); received November 19, 1984

We present a derivation of the QCD scaling law on the basis of asymptotic freedom

The pioneering investigations [1, 2] in Q C D scal-ing laws were marked by an instructive combina t ion of the techniques of "light cone and renormal iza t ion g roup" (hereafter called L C - R G ) . T h e resulting structure funct ion formulas worked qui te well for large .v but small x posed one serious analytical problem during "inversion". Yet, it was these very works that made the asymptotically free gauge theory suitable for experimental test. Another re-markable development followed when the so-called "anomalous d imens ion" occuring in the m o m e n t equat ion of the preceding papers was related to the Mellin Transform of the quark dis tr ibut ion. The resulting equat ion [3] came to be known as the Q C D master equation.

So far so good. But, then the historical importance of these pattern-sett ing works has almost bl indfolded us about any alternative possibility of deriving the Q C D scaling behaviour without using the results of LC-R G. The present paper derives an analytical expression for nonsinglet structure funct ion which brings out the notable features of Q C D predict ion for both large and small .\\ The only physical assumpt ion required for our work is tha t the strong coupling paramete r becomes insensitive to Q2 varia-tion as it approaches the region of asymptot ic free-dom. The rest is just rigorous appl icat ion of Debye ' s steepest descent method. It is r igour that s implif ies the final formula besides overcoming the t rouble " sub-dominan t contr ibut ions" in case .x is small. We begin with the Q C D master equat ion and employ Debye 's method. It is necessary to stress that unlike in [3], the master equat ion can be derived without usins L C - R G 4 and hence our use of it does not in-

Reprint requests to Prof. R. Roychoudhury, Electronics Unit. Indian Statistical Institute, Calcutta-700035, India.

volve us in L C - R G . In this note we intend to de-monstrate the fact that the scaling behaviour is built in inside the notion of asymptotic f reedom, and other physical ingredients are not required to dis-close that behaviour .

We reiterate that our concern in this brief note is the nonsinglet combina t ion of structure functions. (Work is in progress for the general case which will include the singlet case also.) For nonsinglet struc-ture funct ion the coupled evolution equation (master equat ion) is not required and hence our evolution equat ion is s imple and is given by

dq(x,t) a ( / ) dy

dt \—q(y,Q2)P(x/v), (1)

where t = In Q2/Ql and P{x/y) implies Pq^q(x/y). Taking Meiling t ransform of both sides of (1) we have

d Mq(s, t)

dt = a{t)MAs)MAs,t), (2)

where Mq,Mp are respectively the Mellin trans-forms of q and p def ined by

Mq(s, t) = \dxxsq(x, t),

Mp(s) = \dzzsp(z), o

and

a(t) = q ( p 2 71

(3)

(4)

(5)

In deriving (2) we have used the convolution theo-rem of Mellin t ransform. Equat ion (2) can be solved easily and we obta in

Mq {s, t) = c exp (Mp (5) a t). (9)

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J. Sidhanta and R. Roychoudhury • Asymptotic Freedom and Q C D Scaling Law 451

O u r inversion of (9) by Debye 's me thod begins with In deriving (6) we h a v e used the condit ion of ap-proach to the asympto t ic region by taking

\a(t)dt~at. (7)

If an input dis t r ibut ion q(x, 0) —• (1 — x)d be used in the def ini t ion (3) we get

, (8)

Mq (s, 0) = J d.v xs q (x, 0) = B(s + 1, d + 1) . o

Compar ing (8) and (6) we can replace c by Mq(s, 0) and write

Mq(s, t) = B(s + 1, d + l ) e x p { M A s ) at). (9)

If we take P ( z ) = —

Mp(s) = C2(R)

4 1 + then Mp (5) is given by

1

2 (s 4 - 1 ) (5 + 2 )

- 2 i//(s + 2) - 2 y (10)

where c2(R) is a Cas imi r operator for the adjoint representa t ion G of the colour group, y/(s) the logar i thmic g a m m a funct ion and y the Euler con-stant def ined by

y = lim 1 1 ,

+ — + . . . 4 log n ( 1 1 )

2 . 0

1.0

0 . 5

0.2

0.1

0 . 0 5

•XF, 0.0 6 <_X <0-1

10 20 50 Q?[GeV2]

0.5 <x <0.6

XR

_1 I I I L 10 100 200 20 50

Q2 [GeV2] Fig. 1. Comparison of xF3(x, Q2) with CDHS data given in Fig. 13 ([4]) continuous curves represents present work. Above small x (= 0.1), below large .v (= 0.5).

q(x, t) = 7 [ j exp [/?/(*, x ) ] £ ( s + \,d+ 1) ds] , 2 n i

where

/ ( s , x ) = M . , ( s ) - ( s + l ) In x

a t

(12)

(13)

and /? = a t . The contour c of (12) is such that on a par t c0 of it

the Debye condit ions hold. Wi thou t dwell ing on analyticity details we just point out that unl ike in the nonrigorous versions we use a full expansion of /(5) about s 0 , the saddle point

f ( s ) = f ( s 0 ) - u 2 , (14)

whereby the Debye condit ions u is real and where Ref (5) registers a s teep fall with the rise of u. F rom above it is easy to see that (14) allows a good ap-proximat ion of (12) in the following form.

? ( * , / ) = exp ( A / ( j 0 ) ) (15) 2 n i ds

• f exp ( - h u2) Z? (5+ 1, d+ 1) —— (u) du . - 0 0 du

Using the power series expansion

1 ds 00

-—:B(s + \,d+\) — (u) = X fl* w 2711 du „ = 0

we obtain f rom (15) 00 00

q (x, t) = exp (h f(s0)) X an I exp ( - h u2) u" dn n = 0 - x

(16)

= exp (hf(s0)) X alnF(n + 1/2)- / r ( " + 1 / 2 ) . n = 0

(17) But scaling law exper iments imply large t (t = In Q2/Qo), i.e. large h, and the factor / r ( " + 1 / 2 ) of (17) indicates that the lowest order app rox ima t ion is good enough for the present purpose . To the lowest order, we have f rom (17)

q (x, 0 = fl0 1/2 exp (a t Mp (s0)) (1 /x)*>+1, (18)

where s0 given by the equat ion

J 1 0 . ÖS Is = so

(19)

Since the higher values of n in (17) are fo rb idden in the method adopted in the present work, the s imple looking formula (18) is not open to ques t ion re-

452 J. Sidhanta and R. Roychoudhury • Asymptotic Freedom and QCD Scaling Law 452

garding un i fo rm convergence in n of h igher order terms in Wilson Coeff ic ients and anomalous d imen-sions. This explains why (18) is valid even for small x. If q be taken to represent the dis t r ibut ion of quark minus an t iquark , (18) itself represents x F 3 .

To compare our results with exper iment we cal-culate q ( x , t ) according to the fo rmula given in (18). In Fig. 1 our theoretical calculation is compared with the exper imenta l result for small x (x = .1).

According to (18) a slow rise of .xF3 is predicted with the rise of t when x is small. This is consistent with the exper imenta l f indings. A close look at the

[1] D. J. Gross, Phys. Rev. Lett. 32,1071 (1974). [2] A. de Rujula, S. L. Glashow, H. D. Politzer, S. B. Trie-

man. F. Wilczek, and A. Zee, Phys. Rev. D10, 1649 (1974).

formula (18) would reveal that for relatively large x our formula predicts a decrease of x F3 with in-creasing Q2. As can be seen in Fig. 1 this is entirely in agreement with the experimental results. As men-tioned earlier, in deriving our results we did not use the results of L C - R G and hence we conclude that the scaling behaviour is built in inside the not ion of asymptotic f reedom. The fact that earl ier deriva-tions L C - R G had given similar results points out an inner consistency of the formal ism an d particularly indicates the coherence of R G equa t ion and asymp-totic f reedom conjecture.

[3] G. Altarelli and G. Pariri, Nucl. Phys. B126, 298 (1977).

[4] F. E. Close, Th. 2594 CERN 15 December 1978, pp. 13, 14.