Potential Scattering on R3 ⊗ S1

Abstract

In this paper we consider non-relativistic quantum mechanics on a space with an additional internal compact dimension, i.e. R 3 ⊗S 1 instead of R 3 . More specifically we study potential scattering for this case and the analyticity properties of the forward scattering amplitude, T nn (K), where K 2 is the total energy and the integer n denotes the internal excitation of the incoming particle. The surprising result is that the analyticity properties which are true in R 3 do not hold in R 3 ⊗ S 1 . For example, T nn (K), is not analytic in K for ImK > 0, for n such that (|n|/R) > µ, where R is the radius of S 1 , and µ −1 is the exponential range of the potential, V (r, φ) = O(e −µr ) for large r. We show by explicit counterexample that T nn (K) for n = 0, can have singularities on the physical energy sheet. We also discuss the motivation for our work, and briefly the lesson it teaches us. 1

Analyticity properties of scattering amplitude in theories with compactified space dimensions

“…the S 1 compactification. We were motivated to undertake this work from work of Khuri [27] who considered potential scattering with a compact spatial coordinate. He showed the lack of analyticity of the forward scattering amplitude under certain circumstances.…”

“…Khuri [27] identifies a conservation rule: K 2 = k 2 + (n 2 /R 2 ) = k ′2 + (m 2 /R 2 ). Moreover, it is argued that the scattered wave has only (2[KR] + 1) components and those states with (m 2 /(R 2 ) > k 2 + (n 2 /R 2 ) are exponentially damped for large |x| and consequently, these do not appear in the scattered wave.…”

“…We recall that the nonrelativistic theory is invariant only under Galilean transformations whereas QFT's are required to be Lorentz invariant. Khuri [27] encountered a situation, in a nonrelativistic potential model, where the amplitude does not satisfy analyticity in momentum k. The consequences of such a violation of analyticity would not the so serious for scatterings in a nonrelativistic potential model. Whereas, if the amplitude constructed in the frame works of general field theories based on LSZ or Wightman axioms, does not exhibit analyticity then it will raise serious concerns.…”

Analyticity properties and asymptotic behavior of scattering amplitude in higher dimensional theories

“…Indeed, it will be quite interesting to study analyticity properties of scattering amplitude in a higher dimensional theory where some of the spatial coordinates are compactified. It was pointed out, in the context of potential scattering [56,57], that for nonrelativistic theory, the analyticity properties of scattering amplitudes are different from those of a nonrelativistic theory which has no compactified spatial coordinates. It is to be noted that momenta associated with compact directions are discrete.…”

Ultrahigh energy cosmic rays and new particle physics

“…[49]. He showed that analytical properties of the forward scattering amplitude T nn (s, t = 0), which are true in Ê 3 , do not necessarily hold in Ê 3 ⊗ S 1 for n > 0.…”