Abstract
We study the scattering amplitudes of massive Kaluza-Klein (KK) states of open and closed bosonic strings under toroidal compactification. We analyze the structure of vertex operators for the KK strings and derive an extended massive KLT-like relation which connects the N-point KK closed-string amplitude to the products of two KK open-string amplitudes at tree level. Taking the low energy field-theory limit of vanishing Regge slope, we derive double-copy construction formula of the N-point massive KK graviton amplitude from the sum of proper products of the corresponding KK gauge boson amplitudes. Then, using the string-based massive double-copy formula, we derive the exact tree-level four-point KK gauge boson amplitudes and KK graviton amplitudes, which fully agree with those given by the KK field-theory calculations. With these, we give an explicit prescription on constructing the exact four-point KK graviton amplitudes from the sum of proper products of the corresponding color-ordered KK gauge boson amplitudes. We further analyze the string-based double-copy construction of five-point and six-point scattering amplitudes of massive KK gauge bosons and KK gravitons.
Highlights
The big obstacle to further unification between the gauge forces and gravity force lies in the apparently distinctive natures of Einstein’s generality relativity (GR) including its intricate nonlinearity and perturbative nonrenormalizability
The extensions of conventional double-copy method to massive gauge/gravity theories are generally difficult, because many such theories violate gauge symmetry and diffeomorphism invariance
We study the structure of these color-ordered massive KK gauge boson amplitudes and demonstrate that they can be obtained from the scattering amplitudes of massless zero-mode gauge bosons under proper shifts of the Mandelstam variables
Summary
We consider compactifications of the 26d bosonic string theory with one relatively large extra spatial dimension compactified to a circle and with all other extra spatial dimensions decoupled due to their extremely small radii of O(MP−l1). We study the mass spectra of both KK open and closed strings. We explicitly compute the N -point KK open-string amplitudes under compactification by using the relevant compact photon vertex operators and construct the amplitudes of KK closed-strings by products of two KK open-string amplitudes. The scattering amplitudes of KK closed-strings take the KLT-like form. Taking the low energy field-theory limit α → 0 , we derive the formulas of general N -point amplitudes in the compactified KK field theories, so the KLT-like relation of KK string amplitudes will result in the double-copy formula of the corresponding KK graviton amplitudes
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