Determination of the Bjorken Sum and Strong Coupling from Polarized Structure Functions

We present a NLO perturbative analysis of all available data on the polarized structure function g1(x,Q ) with the aim of making a quantitative test of the validity of the Bjorken sum rule, of measuring αs, and of deriving helicity fractions. We take particular care over the small x extrapolation, since it is now known that Regge behaviour is unreliable at perturbative scales. For fixed αs we find that if all the most recent data are included gA = 1.18±0.09, confirming the Bjorken sum rule at the 8% level. We further show that the value of αs is now reasonably well constrained by scaling violations in the structure function data, despite the fact that it cannot yet be reliably fixed by the value of the Bjorken sum: our final result is αs(mZ) = 0.120 +0.010 −0.008 . We also confirm earlier indications of a sizeable positive gluon polarization in the nucleon. CERN-TH/96-345 December 1996 Royal Society University Research Fellow Much experimental and theoretical work has been devoted in recent years to polarized deep inelastic scattering [1]. Reasonably precise data on the polarized structure functions of proton [2]-[5] and deuteron [5]-[8] have been collected down to values of x near and below x = 0.01 for Q > 1 GeV. Very recently precise results on the neutron structure function from scattering on He targets have also become available [9, 10]. The calculation of the kernels for the perturbative QCD evolution of polarized parton distributions has recently been completed to next-to-leading order (NLO) [11], thus reaching the same level of accuracy as in the unpolarized case. Experience obtained from the small x behaviour of unpolarized structure functions observed at HERA [12] is now sufficient to indicate at least qualitatively the sort of behaviour we might expect for polarized structure functions at small x. The purpose of this paper is to take advantage of all this accumulated knowledge and experience in order to extract from the data on polarized structure functions the polarized parton densities and their moments for comparison with theoretical expectations. In particular we will show that when all the data are included it is now possible to make a reliable test of the Bjorken sum rule [13]. We will also investigate to what extent one can use the data to make an accurate determination of αs. For an experimental verification of the Bjorken sum rule one has to extract from the data the first moment of the difference of polarized up and down quark densities at some convenient value of Q. Data taken at all kinematically accessible values of x and Q, and on all available targets, contain information relevant for the reconstruction of polarized parton densities at a given Q and ought therefore to be included. The complete NLO evolution kernels [11] can be used to reduce to the same Q data measured at different Q for each x. Since the evolution equations [15] for partons at a given x and Q depend only on the values of the parton densities at larger values of x and the same Q, the necessary correction can only be performed through a general fit to all the data, which yields a set of polarized parton densities obeying the correct evolution equations [16, 17]. However in order to perform a fit one must start with a particular ansatz for the parton densities at some reference Q0. Clearly the results of the fit will depend to some extent on the starting ansatz one adopts, and this dependence will induce an error in the computed first moments, and in particular in the Bjorken sum. Here we will devote special attention to this issue. Once the data are reduced to a common Q for all x values, an extrapolation to unmeasured values at small and large x is needed in order to obtain the first moment. The extrapolation at small x is especially important [18]. In most of the existing analyses, including those in the experimental papers, it has been performed by assuming a simple power behaviour based on Regge theory [19]. This leads to a rather small contribution to first moments from the small x region, since the expected extrapolation is at most flat. But two main considerations are now severely undermining these attempts. First of all the data at small x for g1 indicate In the past this has been done by assuming that the polarization asymmetries are Q independent [2, 14], but this approximation is questionable given the current precision of the data. Note that the behaviour at small x of the input ansatz for the parton densities at Q20 is not relevant for the evolution correction, which only depends on x values larger than the smallest measured one. On the contrary the integration at small x that completes a given moment is very much dependent on the small x behaviour of the input distributions, as we shall see.