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 ${\ln }^{2}s$ behavior of $pp$ and $\bar{p}p$ cross sections. By exploiting high quality low energy data, analyticity resolves this ambiguity in favor of cross sections that grow asymptotically as ${\ln }^{2}s$. 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.