

DESY 94-149

August 1994

Small x Contributions to the Structure

Function FL(x; Q2)

Johannes Bl"umlein DESY-Institut f"ur Hochenergiephysik, Zeuthen,

Platanenallee 6, D-15735 Zeuthen, Germany

Abstract The gluon contributions to FL(x; Q2) in O(ffs) are calculated taking into account the transverse momentum of the initial state parton. In comparison with collinear factorization FL(x; Q2), is not affected at large x but takes smaller values in the small x range. The onset of the k? effect is a function of Q2.

Contribution to the Proceedings of the International Conference QCD '94, Montpellier, France, 7-13th July 1994, to appear in Nucl. Phys. B (Proc. Suppl.).

Small x Contributions to the Structure Function FL(x; Q2) J. Bl"umleina aDESY-Zeuthen, Platanenallee 6, D-15735 Zeuthen, Germany

The gluon contributions to FL(x; Q2) in O(ffs) are calculated taking into account the transverse momentum of the initial state parton. In comparison with collinear factorization, FL(x; Q2) is not affected at large x but takes smaller values in the small x range. The onset of the k? effect is a function of Q2.

1. Introduction In the small x range a novel behaviour of nucleon structure functions is expected. Among possible dynamical effects are those due to non strong k? ordering [1] and screening [2]. Their description requires a generalization the factorization of the hadronic matrix elements. The k? dependence of the parton distributions can no longer be neglected in the hard scattering cross sections and K2 = \Gamma k2? dependent parton distributions must be used [3], i.e.

jMj2 , Z

_2

dK2^oe(x; K2; _2) \Omega @xG(x; K

2)

@K2 (1)

instead of the collinear relation jMj2 , ^oe(x; K2 = 0; _2) \Omega xG(x; _2): (2) The present paper aims on finding a consistent solution of the above problem without any approximations of the x and k? behaviour of the coefficient functions. In this way former investigations [4,5] are extended. We aim on a general formulation of the gluon contribution to structure functions which offers the possibility to unfold the k? dependence of the gluon distribution at small x. Theoretical predictions of its small x behaviour can thus be directly compared with the data and test the k? dependence.

2. k? Factorization The K2 integral in (1) extends to K2 = 0. However, a perturbative definition of a gluon distribution is only possible at suitably large virtualities.

Therefore, we use [6]

F gL(x; Q2) = Z

1

x

dzz fg;0

L (z) xz G(z; Q20)

+Z

1

x

dzz Z

K2max

Q20

dK2fgL(z; K

2

Q2 )

xz @G(x=z; K2)

@K2

\Theta `(K2max \Gamma Q20)

(3)

where K2max = Q2(1 \Gamma z)=z. We introduced a scale Q0 for which we demand that Q20 !! Q2. (3) is equivalent to (1) up to terms ofO

((Q20=Q2)n). Note that eq. (3) contains the gluon distribution G(x; K2) only at virtualities in the perturbative range.

3. FL(x; Q2) in O(ffs) For the gluonic contribution to FL(x; Q2) eq. (3) the coefficient function takes the following form

fgL(z; K

2

Q2 ) =

2 ss ffs(Q

2)

Nf=2X

q=1 \Gamma

e2qu + e2qd\Delta

\Theta ( 164z ` Q

2

K2 '

2

G(0;4)1L (!; fi)

+ z 116 Q

2

K2 ^G

(2;4) 2L (!; fi)

+ G(1;5)3L (!; fi) log fififi 1 \Gamma !1 + ! fififi *) (4) with ! = p1 \Gamma 4K2z=Q2, i = 4K2z=Q2, cos fi = (1 \Gamma i=2)=p1 \Gamma zi, and G(a;b)iL (!; fi) =P

b k=a g

(L) ki (fi)=!

k. The coefficients gki are:

g(L)01 (fi) = 52 + 3 cos2 fi \Gamma 32 cos4 fi

g(L)11 (fi) = 4 cos fi \Gamma 12 cos3 fi g(L)21 (fi) = 3 \Gamma 18 cos2 fi + 15 cos4 fi g(L)31 (fi) = \Gamma 12 cos fi + 20 cos3 fi

g(L)41 (fi) = \Gamma 32 + 15 cos2 fi \Gamma 352 cos4 fi g(L)22 (fi) = 4 \Gamma 12 cos2 fi g(L)23 (fi) = \Gamma 24 cos fi + 40 cos3 fi g(L)24 (fi) = \Gamma 3 + 30 cos2 fi \Gamma 35 cos4 fi

g(L)31 (fi) = 34 + 12 cos2 fi + 34 cos4 fi g(L)32 (fi) = 2 cos fi \Gamma 6 cos3 fi g(L)33 (fi) = 72 \Gamma 15 cos2 fi + 152 cos4 fi g(L)34 (fi) = \Gamma 18 cos fi + 30 cos3 fi g(L)35 (fi) = \Gamma 94 + 452 cos2 fi \Gamma 1054 cos4 fi (5)

In the limit K2 ! 0 one obtains the well-known result [9]

fg;0L (z) = 2ss ffs(Q2)

Nf=2X

q=1 \Gamma

e2qu + e2qd\Delta z2(1 \Gamma z): (6)

4. Numerical Results Figure 1 shows the logarithmic derivative of the gluon distribution dxG(x; Q2)=d log Q2 for different sets of parton parametrizations in the MS scheme. The most recent results, CTEQ2M and MRSA, were determined using the data measured at HERA, and do practically coincide, while earlier ones show some variation at small x. We will refer to the CTEQ2 parametrization [8] as an input in the following.

In figure 2 the gluonic contributions to FL(x; Q2) using either eq. (6) or (3) are compared. At large x coinciding results are obtained, but at small x the collinear approach yields larger values for FL. Setting _2 = Q2(1\Gamma z)=z, the kinematical upper limit of the K2 integral (3), instead of _2 = Q2, in (6) leads to a lowering of FL(x; Q2) already. Figure 2 shows that with rising Q2, the effect due to finite k? emerges at smaller values of x. As expected, the onset of small x effects is Q2 dependent.

The separation scale Q20 required in (3) affects FL(x; Q2) very weakly as long as Q2 AE Q20, which

we assume. This is illustrated in figure 3. The effect of this choice of scale is comparable to that of Q020, the starting point of QCD evolution.

0 2 4 6 8 10 12

10-4 10-3 10-2 10-1 1x dxG(x,Q 2 )/dlog Q 2

Fig. 1 Logarithmic slope of the gluon momentum distribution vs x for different parton parametrizations [7].

0 0.1 0.2 0.3 0.4 0.5 0.6

10-4 10-3 10-2 10-1 x FLg (x,Q 2 )

Fig. 2 Comparison of the gluonic contributions to FL(x; Q2) in the collinear case and k? factorization. The steeper lines are at Q2 = 104 GeV2, the others at Q2 = 20 GeV2.

In figure 4, the O(ffs) result using k? factorization is compared with results of a O(ff2s) calcu lation in the collinear approach [10]. The gluon contribution to FL in the collinear approach is diminished by about 10% by the O(ff2s) term for x , 10\Gamma 4 and Q2 , O(20 GeV2). The O(ffs) value of FL using k? factorization is somewhat smaller than the O(ff2s) value in the collinear approach. Note that the results are nearly equal in the range x , 10\Gamma 4. The quark contribution to FL in O(ff2s) [10] amounts to , 10% at x , 10\Gamma 4.

0 0.1 0.2 0.3 0.4 0.5 0.6

10-4 10-3 10-2 10-1 x FLg (x,Q 2 )

Fig. 3 Dependence of F gL(x; Q2) on the choice of the separation scale Q20.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

10-4 10-3 10-2 10-1 x FL(x,Q 2 )

Fig. 4 Comparison of the O(ff2s) calculation [10] with the result obtained in the k? factorization scheme.

5. Conclusions A representation of k? factorization which is consistent with perturbative QCD has been given. The gluon contribution to the structure function FL(x; Q2) was calculated using k? factorization without using any approximations of the Mellin convolution or the x dependence of the coefficient functions, unlike some earlier investigations. The contributions to the structure functions obtained are positive in the whole x range.

The derived coefficient functions approach those found using mass factorization in the limit K2 ! 0. The numerical value obtained in k? factorization for suitably `large' values of x approach the result which ignores the k? dependence of the coefficient functions. This has been an expectation in the parton model [3]. There is no fixed onset (e.g. x , 10\Gamma 2 [5]) of the small x effects observed. Deviations from the collinear result become smaller with rising Q2 at constant x. The effect of the separation scale Q0 is found to be subleading.

The k? dependence of the coefficient function and gluon distribution results into smaller values of FL in O(ffs) in the small x range. Quite similar values are obtained for FL in O(ff2s) [10] using mass factorization.

For discussions I would like to thank to W. van Neerven, J. Botts, and S. Catani.

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Y. Balitzkii and L. Lipatov, Sov. J. Nucl. Phys. 28 (1978) 882. 2. L. Gribov et al., Nucl. Phys. B188 (1981) 555. 3. S. Drell and T. Yan, Ann. Phys. 66 (1971) 578. 4. E. Levin and M. Ryskin, Sov. J. Nucl. Phys. 53

(1991) 657. 5. A.J. Askew et al., Phys. Rev. D47 (1993) 3775. 6. J. Bl"umlein, J. Phys. G19 (1993) 1623;

J. Bl"umlein, DESY report to appear. 7. H. Plothow-Besch, CERN Progr. Lib. W5051. 8. J. Botts et al., MSU-HEP 93/24 (1993). 9. A. Zee et al. Phys. Rev. D10 (1974) 2881. 10. E.B. Zijlstra and W.L. van Neerven, Nucl. Phys.

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