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Evolution Kernels for Light-Ray Operators Twist 2 and Twist 3 Contributions.pdf

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Evolution Kernels for Light-Ray Operators Twist 2 and Twist 3 Contributions

a r X i v : h e p - p h / 9 7 0 5 4 5 9 v 1 2 9 M a y 1 9 9 7 Evolution Kernels for Light-Ray Operators : Twist 2 and Twist 3 Contributions Johannes Blu?mlein?, Bodo Geyer? and Dieter Robaschik? ?DESY – Zeuthen, Platanenallee 6, D – 15735 Zeuthen, Germany ? Naturwissenschaftlich–Theoretisches Zentrum der Universita?t Leipzig, Augustusplatz 10, D–04109 Leipzig, Germany Abstract. The general evolution kernels of the twist 2 light–ray operators for unpolarized and polarized deep inelastic scattering are calculated inO(αs). From these evolution kernels a series of special evolution equations can be derived, among them the Altarelli-Parisi equations and the evolution equation for the meson wave function. In the case of twist 3 the results of Balitzki and Braun are confirmed. INTRODUCTION The study of the Compton amplitude for scattering of a virtual photon off a hadron is one of the basic tools in QCD to understand the short–distance behavior of the theory. The Compton amplitude for the general case of non– forward scattering is given by Tμν(p+, p?, Q) = i ∫ d4xeiqx〈p2|T (Jμ(x/2)Jν(?x/2))|p1〉, (1) where p+ = p2 + p1, p? = p2 ? p1 = q1 ? q2 and q = (q1 + q2)/2. The time– ordered product in eq. (1) can be represented in terms of the operator product expansion. Here we use the representation derived in ref. [1,2] T (Jμ(x/2)Jν(?x/2)) ≈ ∫ +∞ ?∞ dκ? ∫ +∞ ?∞ dκ+[Ca(x 2, κi, μ 2)Sμν ρσx?ρO a σ(κix?, μ 2) +Ca,5(x 2, κi, μ 2)εμν ρσx?ρO a 5,σ(κix?, μ 2)], (2) with Sμνρσ = gμνgρσ ? gμρgνσ ? gμσgνρ and εμνρσ denoting the Levi–Civita symbol. The light–like vector x? = x + r(x.r/r.r) [√ 1? x.xr.r/(x.r)2 ? 1 ] is c? 1997 American Institute of Physics 1 2related to x and a subsidiary four–vector r, Ca and Ca5 denote the respec- tive coefficient functions. Different kinematic situations in the Bjorken region are described by different matrix–elements of the involved light–ray operators Oaσ(κ+x?, κ?x?, μ 2) and Oa5,σ(κ+x?, κ?x?, μ 2). Let us illustrate this with the help of the non–si