Positivity Constraints on Photon Structure Functions

We investigate the positivity constraints for the structure functions of both virtual and real photon. From the Cauchy-Schwarz inequality we derive three positivity conditions for the general virtual photon case, which reduce, in the real photon case, to one condition relating the polarized and unpolarized structure functions. YNU-HEPTh-01-102 CPT-2001/P.4242 KUCP-195 October 2001 e-mail address: [email protected] e-mail address: [email protected] e-mail address: [email protected] The photon structure has been studied through the two-photon processes in e e collisions as well as the resolved photon processes in the electron-proton collider. Based on the perturbative QCD (pQCD), the unpolarized parton distributions in the photon have been extracted from the measured structure function F γ 2 [1]. Recently there has been growing interest in the study of polarized photon structure functions [2, 3]. Especially the first moment of the spin-dependent structure function g 1 has attracted much attention in the literature in connection with its relevance for the axial anomaly [4, 5, 6, 7, 8]. The next-to-leading order QCD analysis of g 1 has been performed in the literature [9, 10, 11]. There exists a positivity bound, |g 1 | ≤ F γ 1 , which comes out from the definition of structure functions, g 1 and F γ 1 , and positive definiteness of the s-channel helicity-nonflip amplitudes. This bound was closely analyzed recently [11]. In the case of virtual photon target, there appear eight structure functions [12, 13, 14], most of which have not been measured yet and, therefore, unknown. In a situation like this, positivity would play an important role in constraining these unknown structure functions. It is well known in the deep inelastic scattering off nucleon that various bounds have been obtained for the spin-dependent observables and parton distributions in a nucleon by means of positivity conditions [15]. In the present paper we investigate the model-independent constraints for the structure functions of virtual (off-shell) and real (on-shell) photon target. We obtain three positivity conditions for the virtual photon case and one condition for the real photon, the latter of which relates the polarized and unpolarized structure functions. Let us consider the virtual photon-photon forward scattering: γ(q) + γ(p) → γ(q) + γ(p) illustrated in Fig.1. The s-channel helicity amplitudes are given by W (ab|ab) = ǫ∗μ(a)ǫ ∗ ρ(b)W μνρτ ǫν(a )ǫτ (b ) , (1) where p and q are four-momenta of the target and probe photon, respectively, ǫμ(a) represents the photon polarization vector with helicity a, and a, a = 0,±1, and b, b = 0,±1. Due to the angular momentum conservation, W (ab|ab) vanishes unless it satisfies the condition a − b = a − b. And parity conservation and time reversal invariance lead to the following properties for W (ab|ab) [16]: W (ab|ab) = W (−a,−b| − a,−b) Parity conservation , 1 = W (ab|ab) Time reversal invariance . (2) Thus in total we have eight independent s-channel helicity amplitudes, which we may take as W (1, 1|1, 1), W (1,−1|1,−1), W (1, 0|1, 0), W (0, 1|0, 1), W (0, 0|0, 0), W (1, 1| − 1,−1), W (1, 1|0, 0), and W (1, 0|0,−1). The first five amplitudes are helicity-nonflip and the rest are helicity-flip. It is noted that s-channel helicitynonflip amplitudes are semi-positive, but not the helicity-flip ones. And corresponding to these three helicity-flip amplitudes, we will obtain three non-trivial positivity constraints. The helicity amplitudes may be expressed in terms of the transition matrix elements from the state |a, b〉 of two virtual photons with helicities a and b, to the unobserved state |X〉 as W (ab|ab) = ∑ X |〈X|a, b〉|, W (ab|ab) = Re ∑ X 〈X|a, b〉〈X|a, b〉 (a 6= a, b 6= b) . (3) Then, a Cauchy-Schwarz inequality [17, 18]