Analyticity as a Robust Constraint on the LHC Cross Section

It is well known that high energy data alone do not discriminate between asymptotic ln s and ln s behavior of pp and p̄p cross sections. By exploiting high quality low energy data, analyticity resolves this ambiguity in favor of cross sections that grow asymptotically as ln s. We here show that two methods for incorporating the low energy data into the high energy fits give numerically identical results and yield essentially identical tightly constrained values for the LHC cross section. The agreement can be understood as a new analyticity constraint derived as an extension of a Finite Energy Sum Rule. High precision low energy data represent a powerful constraint on the high energy behavior of hadronic cross sections via duality[1, 2]. The low energy data can be separated into two energy regimes, the resonance region and a region with energies in excess of a laboratory energy ν0 where the resonances average into a featureless cross section in the sense of duality. These data represent powerful constraints on asymptotic fits to high energy data. Igi and Ishida[1] realized these constraints using a Finite Energy Sum Rule (FESR) which numerically averages the resonances, while Block and Halzen[2] simply required that the high energy amplitudes fit both the experimental cross sections and their derivatives at the transition energy ν0. Both methods discriminate between a ln s and ln s asymptotic behavior of the asymptotic cross section, conclusively favoring the latter. They appear to be more selective than conventional fitting techniques[3]. In this note we will show that the constraints of Block and Halzen[2] derive from analyticity[4], as does the FESR(2) of Igi and Ishida[1]. The purpose of this note is to show that they are in fact equivalent, as confirmed by fitting the two apparently very different methods to a common data set of pp and p̄p cross sections[6]. Following Block and Cahn[7], we describe the high energy data in terms of real analytic amplitudes