Transverse structure function in the factorisation method

We study the structure function gT = g1+g2, using the factorisation method. It turns out that gT has a much simpler operator structure as compared to g2, in spite of the fact that, like g2, gT has a twist-three component not suppressed by powers of Q . We demonstrate factorisation of the hadronic tensor into hard and soft parts for the case of gT , even though g2 does not admit of such a demonstration. We show that the first moment of the gluonic contribution to gT vanishes, and discuss possible physical applications. [email protected] [email protected] [email protected] The recent transversely polarised deep inelastic scattering (DIS) experiments [1] have opened up new avenues to explore the polarised structure of the proton. With more data, better accuracy and new proposed experiments [2], it would soon be possible to extract the twist-three contribution for the first time. Though extraction of the twist-three contributions is in general quite difficult[4, 5], in these experiments it is possible to kinematically eliminate the leading twist contribution [6]. The polarised structure of the proton is characterised by two structure functions g1(x,Q ) and g2(x,Q ) which can be measured in a polarised lepton-proton DIS experiment l(k)P (p) → l(k)X(px). The spin dependent part of the proton tensor is parametrised as W̃μν(x,Q ) = i p · q ǫμνλσ q λ { s ( g1(x,Q ) + g2(x,Q ) ) − q · s p · q p g2(x,Q ) } , (1) where sμ is the spin vector of the proton and is normalised as s 2 = −M2 with s · p = 0, M being the target mass. The spin-dependent cross-section [6] is given by d∆σ(α) dxdydφ = e 4π2Q2 { cosα {[ 1− y 2 − y 2 4 (κ− 1) ] g1(x,Q )− y 2 (κ− 1)g2(x,Q) } − sinα cosφ √√√(κ− 1) ( 1− y − y 2 4 (κ− 1) )[ y 2 g1(x,Q ) + g2(x,Q ) ]  , (2) where y = p · q/p · k, κ = 1 + 4xM/Q, φ is the azimuthal angle and α is angle between the spin vector s and the incoming lepton momentum k. In a longitudinally polarised experiment (α = 0), the dominant contribution comes from the structure function g1(x,Q ) while g2(x,Q ) is suppressed by a factor M/Q, thus enabling the extraction of g1(x,Q ). The longitudinally polarised DIS process has been studied quite extensively [5, 7] and there is a considerable amount of data on g1(x,Q ) [8]. In contrast, the extraction of g2(x,Q ) requires transversely polarised proton (α = 90) and further this cross-section is suppressed by a factor M/ √ Q2 relative to the longitudinal case. Hence the extraction of g2(x,Q ) is much more complicated as compared to g1(x,Q ). Recently, experimental information on g2(x,Q ) has become available [1], but the data have large errors and do not provide a definite answer to the question of the validity of the sum-rules associated with g2(x,Q ), like the Burkhardt-Cottingham (BC) sum-rule [9], the Wandzura-Wilczek (WW) sum-rule [10], or the recently proposed Efremov-Leader-Teryaev (ELT) sum-rule [11]. contribution etc. In transversely polarised DIS experiments, the asymmetry that is measured is the virtual photon absorption asymmetry A2(x,Q ) = √ Q2 ν g1(x,Q ) + g2(x,Q ) F1(x,Q) , (3) where F1(x,Q ) is the spin-averaged structure function. We see from Eq. 3 that the asymmetry is proportional not to g2 alone, but to gT (x,Q ) = g1(x,Q )+g2(x,Q ). In this letter, we show that the quantity gT admits of a much simpler description than does g2. We suggest that this may help in going some way towards a fuller understanding of the transverse spin structure of the nucleon. We begin by discussing the free field theory analysis in order to elucidate the operator structure of the structure functions and demonstrate the importance of the target mass term in maintaining gauge invariance of the hadronic tensor. We then discuss the first moment of gT (x,Q ) and its relation to the spin content of the proton, and study the gluonic contribution to the first moment, using the Factorisation Method (FM). g1(x,Q ). The hadronic tensor Wμν(p, q, s) has the form [15] Wμν(p, q, s) = 1 4π ∫ dξ e 〈ps| [Jμ(ξ), Jν(0)] |ps〉c , (4) Retaining the dominant contribution in the light-cone limit, ξ → 0, identified as the most singular part of the time-ordered product of these currents on the light-cone, we find W̃μν(p, q, s) = i 4π2 ǫμνλρ ∫ dξ e ξ δ(ξ) ǫ(ξ0) 〈ps| :O A(ξ, 0) : |ps〉c , (5) where O A(ξ, 0) = ψ̄(ξ)γγ5ψ(0) + ψ̄(0)γγ5ψ(ξ) , (6) To arrive at the above result we used iS(ξ, 0) = −〈0|T (ψ(ξ)ψ̄(0))|0〉 , (7) = − i 2π2 6ξ (ξ2 − iǫ)2 +O(m) , (8) where order m terms are neglected. The importance of these terms will be shown later. To find the dominant contribution coming from these operators, we still have to make them local and then pick up the dominant part. Hence,