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The Open String Regge Trajectory and Its Field Theory Limit

We study the properties of the leading Regge trajectory in open string theory including the open string planar one-loop corrections. With SU(N) Chan-Paton factors, the sum over planar open string multi-loop diagrams describes the 't Hooft limit N\to\infty. Our motivation is to improve the understanding of open string theory at finite α' as a model of gauge theories. SU(N) gauge theories in D space-time dimensions are described by requiring open strings to end on a stack of N Dp-branes of space-time dimension D=p+1. The large N leading trajectory α(t)=1+α' t+Σ(t) can be extracted, through order g^2, from the s\to-\infty limit, at fixed t, of the four open string tree and planar loop diagrams. We analyze the t\to0 behavior with the result that Σ(t)\sim-Cg^2(-α' t)^{(D-4)/2}/(D-4). This result precisely tracks the 1-loop Reggeized gluon of gauge theory in D>4 space-time dimensions. In particular, for D\to4 it reproduces the known infrared divergences of gauge theory in 4 dimensions with a Regge trajectory behaving as -\ln(-α^\prime t). We also study Σ(t) in the limit t\to-\infty and show that, when D<8, it behaves as α^\prime t/(\ln(-α^\prime t))^γ, where γ>0 depends on D and the number of massless scalars. Thus, as long as 4<D<8, the 1-loop correction stays small relative to the tree trajectory for the whole range -\infty<t<0. Finally we present the results of numerical calculations of Σ(t) for all negative t.