Volume 61B, number 1 PHYSICS LETTERS 1 March 1976

E X P L I C I T I N V E R S I O N O F T H E S C A L I N G F U N C T I O N M O M E N T S

G. EILAM* and M. GLOCK Institut ffir Physik, Universitdt Mainz, 6500 Mainz, West Germany

The scaling function moments are explicitly inverted for both conventional theories and for the non-singlet part in asymptotically free theories.

The q2 dependence of the moments of deep inelastic weak and electromagnetic structure functions is given by [1]

1

f F(x, q2)xn-2 dx = C(n) exp(-anS) (1)

0

where C(n) are unknown. For conventional theories (CT) [1,2] and for the non-singlet part of asymptotically free theories (AF) [3] which are considered here, the anomalous dimensions are

n

a n=A n(n+l) (CT), a n =G -3 n(n+l) l '4 = (AF) (2)

where A, b are constants and G = 4•25 for the four color triplet quark model. Furthermore

s = in q2/q2 (CT) s = In {&(q2o)/a(q2)} (AF) (3)

where q2 is some reference value; the running gauge coupling constant is

~(q2) = a[1 + Be In q2/u2 ] -1 (4)

/a 2 is a scale parameter (taken to be 1 GeV2), B = 25/127r and the initial gauge coupling constant obeys 0.1 ~- a ~< 0.3 [4].

Inversion of eq. (1) requires calculation of the inverse MeUin transform of exp(-ans) [5]

1 T(s,y) = ~ i f dn exp(-ans)Y -n =M -1 (exp(--anS)}, (5)

--ioo

since from the faltung theorem

1 F(x,s)= f F ( y , O ) T(s,y)dy. (6)

X

T(s, y) has not been evaluated explicitly; only approximations for the extreme cases x ~ 1 [6] and x ~ 0 [7] were found (for AF). In this letter we present an explicit expression for T and thus F(x, s) can be directly constructed from an input F(x, O) and a n as given in eq. (2). This is done by expanding the exponent in eq. (5) in powers of s; inverse MeUin transforms can then be found for each term.

For CT our result is

* Permanent address: Physics Department, Technion-Israel Institute of Technology, Haifa, Israel.

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Volume 61B, number 1 PttYSICS LETTFRS 1 March 1976

F(x ,s )=exp(-As) IF(x ,O)+~ (sAb)/22J--IF(f+½) 1 ( ~ - ) 'I j : l f r - ( 2 / : 1). V - f yl/2(-lny)j 1/21j-I/2(-~lny)F ,0 d)

X

(7)

where I is the modified Bessel function of the first kind. For AF we find

1

F(x, s)=exp(3Gs) IF(x, 0 ) ( i - - x ) 4 G s - 4 G s f dy (I v)4Gs-l(F(x, 0)- .vF(~-, 0 ) ) + R] X

(8)

where the first two terms were obtained by summing all orders o f s and the relnainder R is a power series in s (see eq. (14)).

Let us present the proofs of eqs. (7) and (8). (I) Conventional theories: From eqs. (2) and (5) and formula (1) in table 1

T(s,y) = exp( -As) I6(y - 1) 7! - M-' j(1+ . (9)

Now from ref. [8] p. 343 and ref. [9] p. 305, eq. (7) is recovered. Less terms are needed in eq. (7) as s gets small and as x ~ 1 (for estimates see below).

(II) Asymptotically free theories: The / th term in the expansion of T(s, y) includes two types of terms. One type is identical with the above result for CT, with sAb ~ 2Gs. The other type has both inverse Mellin transforms of powers of ~, and of powers of 1/n(n +1) multiplied by powers of ~ where

n 1 ~(n)~-- ~ 1 1_ - t n d t (10) k=~Y=f 1 - t "

0

Using the above integral representation, M -1 ( ~ ) is obtained (formula (5) in table 1) and by a repeated use of faltung theorems as given in the table higher order terms are constructed. Disappearance of infinities in T(s, y) , and cancellation of the remaining infinities in eq. (6) is due to the formal identity

(3) lg(n)

(4) ~ + 1 g(n)

(5) ~'(n)

(6) ~(n) g(n)

Table 1 A useful list of inverse Mellin transforms supplementing the tables in ref. [8], chap. VII. The apparent infinities in the last two rows disappear in eq. (6). ¢, is defined in eq. (10).

1 foo g(n)Y -n dn g(n) f(y) = ~ . - 1 ~

(1) const, const. 6(y - 1) (2) exp(an) eas(y - e a)

1 dt f f(t) ~- Y

1

y f f(t) ~2 Y

1 d t 8(y 1) f

1 - t v

f(y)

Y 1 - y

dt ~ f(t) dt o - f ~ - Y y t - y t

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Volume 61B, number 1 PHYSICS LETTERS 1 March 1976

[ f d, l k 1 dt F( 0 1 - t J = k f l - - ~ i k j l d - ~ z z ] 0

Two terms that appear in M -1 ( 5 / )

1 #(4Gs)JF(x,O)lnJ(l-x) (j>O)

and

k -1

(11)

(12)

1 1 ( 4 G s ) / f f d y l n J - l ( 1 - Y ) ( F ( x , 0 ) - Y F ( v O ) ) l _ y / - - " ' , ( / ~ 1 ) (13)

X

sum to yield the first two terms in eq. (8). The remainder is then a power series in s, given by

1 14(2Gs)2(( l_y) ln( l_y)_~}lny) l dY +O(s31 R = f [2Gs(1 --y)--l(2Gs)2((1 + y ) l n y + 2 ( 1 - - y ) ) + ~

X

Again less terms are needed in R as x -> 1. Higher order terms in R can be found by using faltung theorems as given in the table; these terms, starting from

/ = 3, include the dilogarithm function [10]. To estimate the convergence in eqs. (7) and (8) we choose F(x, 0) = x 1/2(1 - x) 3 and q2 = 5 GeV 2. Then for

CT with b = 6 ,A = 0.25 [11] only the first order term in the sum is needed for x > 0.8 and q2 < 1000 GeV 2, two are needed fo~ x > 0.3 and q2 < 200 GeV 2, five for x > 0.01 and q2 < 300 GeV 2 etc. Thus the series con- verges fast. For CT one has the advantage of a closed form in eq. (6), which can be integrated analytically (I/_ 1/2 is given by a finite sum [12]). Thus asymptotic expressions are easy to obtain.

In AF the convergence is even better since if a = 0.3 [4] then s = 0.6 for q2 ~ 1000 GeV 2. Therefore the ex- pansion parameter is 2Gs < 0.19 for q2 < 1000 GeV 2 (to be compared with sAb < 7.95 for CT); for such s values the remainder given in eq. (15) is accurate enough. Note that we do not use the asymptotic form Sas = In (ln q2/ In qo 2) since s gives better agreement with experiment; furthermore there are substantial differences between s and Sas even if both q2 and q2 are large [4].

There is a numerical advantage over Parisi's integro-differential equation [5] for F(x, s) which corresponds to treating the first two terms in T(s, y). Using Parisi's equation F(x, s) has to be successively constructed and inte- grations are required for each infinitesimal step in s. Here, only integrations are needed to obtain F(x, s) directly from F(x, 0). Note that here, as in Parisi's equation, to determine F(x, s) only F(~, 0) for x ~< ~ <~ 1 is needed.

We would like to thank E. Reya for helpful discussions. One of us (G.E.) thanks M. Kretzschmar and the mem- bers of the Mainz group for their hospitality.

R e f e r e n c e s

[1] N. Christ, B. Hasslacher and A. Mueller, Phys. Rev. D6 (1972) 3543. [2] G. Parisi, Phys. Lett. 43B (1973) 207;

K. Wilson, Phys. Rev. D8 (1973) 2911. [3] D.J. Gross and F. Wilczek, Phys. Rev. D9 (1974) 416. [4] H.D. Politzer, Phys. Rev. D9 (1974) 2174;

H. Fritzsch and P. Minkowski, Nucl. Phys. B76 (1974) 365; G. Eilam, M. Gliick and E. Reya, Comparing asymptotically free theories with experiment, preprint MZ-TH 75/6, Sep. 1975.

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Volume 61B, number 1 PHYSICS LETTERS 1 March 1976

[5] G. Parisi, Phys. Lett. 50B (1974) 367; G. Parisi in ref. [2].

[6] D.J. Gross, Phys. Rev. Lett. 32 (1974) 1071. [7] A. De Rfijula et al., Phys. Rev. D10 (1974) 1649. [8] E. Erd~lyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Tables of integral transforms, Vol. I (McGraw Hill, 1954). [9] W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and theorems for the special functions of mathematical physics

(Springer-Verlag, (1966) [10] Ref. [9] p. 33. [11] Wu-ki Tung, Phys. Rev. Lett. 35 (1975) 490. [12] I.S. Gradshteyn and I.M. Rhyzhik, Tables of integrals, series and products (Academic Press, 1965) p. 967.

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