We present the results of a calculation of deep inelastic electron-photon scattering at a linear collider for very high virtuality of the intermediate gauge boson up to NLO in perturbative QCD. The real photon is produced unpolarized via the Compton back s

4 For the scattering of an e? with the photon, the charged current structure functions F2;CC and F3;CC at the lowest order are respectively given by, F2;CC= x u(x; Q2 )+ c(x; Q2 )+d(x; Q2 )+ s(x; Q2 ) F3;CC= u(x; Q2 )+ c(x; Q2 )?d(x; Q2 )? s(x; Q2 ): (11) For e+ scattering, one has to make the following replacements: u(x; Q2 )+ c(x; Q2 ) ! u(x; Q2 )+ c(x; Q2 ) and d(x; Q2 )+ s(x; Q2 ) ! d(x; Q2 )+ s(x; Q2 ). The CKM matrix is approximated by the unity matrix, avour mixing e ects can here be neglected. At the next-to-leading order, corrections proportional to s log(Q2 ) have to be included in all the above leading-order expressions. To the terms proportional to the parton distributions themselves, terms involving convolution of these with quark (Cq ) and gluon (Cg ) coe cient functions have to be taken into account. For example, for the light avours, F2;NC given at leading order in eq.( 9) becomes in the MS scheme at the nextto-leading order

4. Results and conclusionAs mentioned before, we shall present our results for xed values of Q2, namely Q2= min min 10000 GeV2 and Q2= 1000 GeV2 . Since min the leading and next-to-leading order results are found to be very close to each other, all results will be given at the next-to-leading order level only. The NLO corrections being at most of the percent level indicates furthermore that the obtained results are perturbatively stable. In Figs. 2 and 3 we present the di erent contributions to the total NC and CC cross sections as a function of the electron-positron p center of mass energy, the latter varying between s= 200 GeV p and s= 2000 GeV for virtualities of the intermediate gauge boson equal to 10000 GeV2 and 1000 GeV2 respectively. The cross sections being inversely proportional to Q4, their value is dominated by the smallest Q2 values. As can be seen in these gures, for Q2= 10000 GeV2, the largest cross section, min the total charged current cross section is of O(pb) while for Q2= 1000 GeV2, the dominant neumin tral current cross section is about 10 times larger. For this latter choice of Q2 the charged curmin rent cross section is approximately a third of the neutral current cross section. By these high virtualities of the intermediate gauge boson, the exchange of a W-boson gives always rise to signi cantly high cross sections. The contribution arising from

the exchange of a Z-boson is the smallest for both choices of Q2 . min The di erential cross p sections with respect to x, which are shown for s= 500 GeV in Figs. 4 and 5 di er not only in magnitude but also in shape. This can be understood as follows. For a given value of x, the allowed phase space regions in the (x; Q2 ) plane corresponding to the two values of Q2 chosen here, di er signi cantly from min each other. As can be seen in Fig. 1, this results in an enhanced importance of the small x region for smaller Q2 . min Finally, we also present the di erential cross section with respect to Q2 for Q2 values varying between 1000 GeV2 and the electron-positron p center-of-mass energy squared s, for s= 500 GeV. As can be seen in Fig. 6 the neutral

F2;NC= x+ 2

X

s (Q2 ) h

q

e2 q(x; Q2 )+ q (x; Q2 )^q q(x; Q2 )+ q (x; Q2 )i

Cq

+Cg g(x; Q2 )+ C;2:

(12)

At this order, we use the beyond-leadinglogarithmic (BLL) GRV 5] massless parton distributions q(x; Q2 ) and g(x; Q2 ). These are given in the DIS factorization scheme, de ned by, (13) q(x; Q2 )DIS= q(x; Q2 )MS+ 2 C;2: As a consequence, in this DIS factorization scheme the direct term C;2 is absent from the expression of F2;NC . The gluon distribution g(x; Q2 ) remains una ected by the change of fac-

torization scheme. The precise expressions of the other neutral and charged structure functions at the next-to-leading order will be given in 6].