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.

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 γ 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 γ 1 has been performed in the literature [9,10,11]. There exists a positivity bound, |g γ 1 | ≤ F γ 1 , which comes out from the definition of structure functions, g γ 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.
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 Then, a Cauchy-Schwarz inequality [17,18] which holds for an arbitrary real number α, leads to a positivity bound for the helicity amplitudes: Writing down explicitly, we obtain the following three positivity constraints: In terms of the eight independent amplitudes introduced by Budnev, Chernyak and Ginzburg [12], the above three conditions can be rewritten as where T and S refer to the transverse and longitudinal photon, respectively, and the superscripts "τ " and "a" imply the relevance to the helicity-flip amplitudes and polarized ones, respectively. For the real photon, p 2 = 0, the number of independent helicity amplitudes reduces to four. They are W (1, 1|1, 1), W (1, −1|1, −1), W (0, 1|0, 1), and W (1, 1| − 1, −1) , which are related to four structure functions W γ i as follows [12,13,14,19]: where the last one is the polarized structure function and usually denoted by g γ For the real photon case we have only one constraint, i.e., the first inequality (5), which is rewritten as It is interesting to recall that the polarized structure function W γ 4 of the real photon satisfies a remarkable sum rule [4,5,6,7,8] The integral of |W γ 3 | is, therefore, bounded from above by the first moment of W γ 1 , Now let us examine whether the inequality (12) is actually satisfied or not by the structure functions obtained in the simple parton model (PM). By evaluating the box (a massive quark-loop) diagrams with p 2 = 0, ignoring the power correction of m 2 /Q 2 with quark mass m, the photon structure functions have been obtained as follows: where x = Q 2 /(2p · q), α = e 2 /4π, the QED coupling constant, and δ γ = 3 i , with N f , the number of the active flavors. Using these expressions, we examine the constraint (12) numerically and find that it is satisfied almost all allowed region of x except near the limit x → x max = 1/(1 + 4m 2 /Q 2 ). However, the violation of the inequality near x max is an artifact, since the limiting procedures of Q 2 → ∞ and x → x max are not exchangeable. In fact, the exact PM calculation of W γ i 's with Q 2 kept finite gives where β = 1 − 4m 2 x Q 2 (1−x) . The above results are in accord with the cross sections for the γγ → e + e − (µ + µ − ) process obtained by Budnev et al. [20]. Also the expression of W γ 4 | PM is consistent with the result of Ref. [21], where polarized gluon structure functions were considered. It is noted that since β → 0 for x → x max , all W γ 1 | PM , W γ 3 | PM , and W γ 4 | PM vanish at x = x max . Using these exact PM results in (16), we find numerically that the inequality (12) is indeed satisfied for all allowed region of x . Moreover, once expressed as functions of x and β, the helicity-nonflip amplitudes W (1, 1|1, 1)| PM and W (1, −1|1, −1)| PM are easily shown to be non-negative for 0 ≤ β < 1, and 0 ≤ x < 1, as they should be. On the other hand, the helicity-flip amplitude W (1, 1| − 1, −1)| PM turns out to be negative.
As stated earlier, in the case of virtual photon, p 2 = −P 2 = 0, there appear eight structure functions (four of them are new) and we have derived three positivity constraints on these functions. But up to now little attention has been paid to the virtual photon case and, therefore, we have slight knowledge of the new photon structure functions. In this situation it is worthwhile to investigate these new structure functions in the simple PM and examine that the three positivity constraints (8)-(10) actually hold [22].
Especially, in the kinematical region, Λ 2 ≪ P 2 ≪ Q 2 , where the mass squared of the target photon (P 2 ) is much bigger than the QCD scale parameter (Λ 2 ), some of the photon structure functions are predictable in pQCD entirely up to the nextleading-order (NLO), since the hadronic component on the photon can also be dealt with perturbatively. Following this strategy, the virtual photon structure functions, unpolarized F γ 2 (x, Q 2 , P 2 ) and F γ L (x, Q 2 , P 2 ) [23] and polarized g γ 1 (x, Q 2 , P 2 ) [10], were studied up to the NLO. Since F γ 1 ≡ (F γ 2 − F γ L )/x = 2W γ 1 and g γ 1 = 2W γ 4 , it is also interesting to see if the inequality (12) is satisfied by the pQCD results for the above kinematical region [22]. The virtual photon structure function W γ 3 (x, Q 2 , P 2 ) is expected to be given by the same expression as the PM result (15) up to O(1/ ln(Q 2 /Λ 2 )), since there exist no twist-2 quark operators contributing to W γ 3 [19,24]. So far we have only considered the constraints on the structure functions. Now our argument can be extended to the quark contents of the photon, for which we can also write down inequalities involving various distributions. Following Ref. [17], let us define the helicity amplitudes given by where all the suffices, a, b, a ′ and b ′ , refer to the helicities of the quarks and virtual photons, and O's denote bilinear quark operators. One also has to sum over all intermediate states X. Then we can derive, in a similar fashion based on the Cauchy-Schwarz inequality, In our present case, the above helicity amplitudes become nothing but the fol-lowing quark distributions: where q ± γ (q 0 γ ) denotes the longitudinally (transversely) polarized quark distribution inside the photon, and h q γ is the (chirality-odd) transversity distribution, the photon analog of h q 1 for the nucleon case. Here we note that the photon structure function F γ 1 (g γ 1 ) can be expressed as a sum over the active quark (or anti-quark) distributions Hence we have the following positivity condition for the transversity distribution h q γ , |h q γ | ≤ ( This is an extension of the inequality obtained for the nucleon case [15,17,18] 1 . The transversity distribution h q γ of the photon could be measured by the semi-inclusive process in the two-photon reactions provided by the future polarized e + e − collision experiments.
In summary we have investigated the model-independent positivity constraints for the photon structure functions which could be studied in future experiments.
We also discussed a positivity bound for the quark distributions relevant for the spin-dependent semi-inclusive process in two-photon reactions. We expect these bounds would provide uselful constraints for studying the yet unknown polarized and unpolarized photon structures.  Virtual photon-photon forward scattering with momenta q(p) and helicities a(b) and a ′ (b ′ ).