Gravitational Equivalence Theorem and Double-Copy for Kaluza-Klein Graviton Scattering Amplitudes

Abstract

We analyze the structure of scattering amplitudes of the Kaluza-Klein (KK) gravitons and of the KK gravitational Goldstone bosons in the compactified 5d general relativity (GR). Using a general Rξ gauge fixing, we study the geometric Higgs mechanism for the massive spin-2 KK gravitons. We newly propose and prove a gravitational equivalence theorem (GRET) to connect the scattering amplitudes of longitudinal KK gravitons to that of the KK gravitational Goldstone bosons, which formulates the geometric gravitational Higgs mechanism at the scatteringS-matrix level. We demonstrate that the GRET provides a general energy cancellation mechanism guaranteeing the N-point longitudinal KK graviton scattering amplitudes to have their leading energy dependence cancelled down by a large power factor of E2NN≥4 up to any loop level. We propose an improved double-copy approach to construct the massive KK graviton (Goldstone) amplitudes from the KK gauge boson (Goldstone) amplitudes. With these, we establish a new correspondence between the two types of energy cancellations in the four-point longitudinal KK amplitudes at tree level: E4⟶E0 in the KK gauge theory and E10⟶E2 in the KK GR theory.

Massive color-kinematics duality and double-copy for Kaluza-Klein scattering amplitudes

“…The KK compactification can realize a geometric "Higgs" mechanism for mass generation of KK gravitons both at the Lagrangian level [19,20,54] and at the scattering S-matrix level [19,20]. This is shown [19,20] to make the massive KK GR theory free from the van-Dam-Veltman-Zakharov (vDVZ) discontinuity [55,56] and exhibit much better high energy behaviors (as good as that of the massless Einstein gravity). Hence the KK GR theories provide a truly consistent realization of the massive gravity in the effective field theory formulation.…”

“…It has also become a powerful tool for efficiently computing the highly intricate scattering amplitudes of spin-2 gravitons. The massive KK graviton scattering amplitudes are especially involved (with large energy cancellations) [16][17][18][19][20] and the N -point longitudinal KK graviton amplitudes have their leading energy dependence nontrivially cancelled down by a large power factor E 2N (N 4) up to any loop order [19,20]. The first double-copy was constructed by Kawai-Lewellen-Tye (KLT) [21] to link the scattering amplitudes of massless closed strings to the products of scattering amplitudes of massless open strings at tree level.…”

“…But, all these works were to formulate and test double-copy for the scattering amplitudes of massless JHEP03(2023)254 gauge bosons/gravitons [24,25]. Our recent works extended the massless double-copy method to the massive double-copy constructions for the 5d KK gauge/gravity field theories (with orbifold compactification) [19,20], for the compactified 26d KK bosonic string theory plus its field theory limit [33], and for the 3d topologically massive Chern-Simons (CS) gauge/gravity theories [34,35]. In the literatue, some other recent works attempted to generalize the massless double-copy method to the case of massive double-copies, including the 4d massive Yang-Mills (YM) theory versus Fierz-Pauli-like massive gravity [36][37][38][39], the spontaneously broken YM-Einstein supergravity models with adjoint Higgs fields [40], the KK-inspired effective gauge theory with extra global U(1) [41], the 3d Chern-Simons (CS) gauge/gravity theories with or without supersymmetry [42][43][44][45][46][47], and certain massive scalar theories [48,49].…”

“…Extensions of the conventional massless double-copy formalism to the case of massive gauge/gravity theories are generally difficult, because many theories of this kind violate gauge symmetry and diffeomorphism invariance (which include the massive YM theory and the massive Fierz-Pauli gravity [50]). By adding extra non-linear polynomial interaction terms in the literature [36][37][38][39], one could realize the double-copy between the massive YM and Fierz-Pauli gravity theories, but the high energy behavior of the four-point massive graviton amplitudes could be improved to no better than E 6 [51][52][53] and is still much worse than the final energy-dependence of O(E 2 ) in the tree-level massive KK graviton scattering amplitudes [16][17][18][19][20]. The KK compactification can realize a geometric "Higgs" mechanism for mass generation of KK gravitons both at the Lagrangian level [19,20,54] and at the scattering S-matrix level [19,20].…”

“…By adding extra non-linear polynomial interaction terms in the literature [36][37][38][39], one could realize the double-copy between the massive YM and Fierz-Pauli gravity theories, but the high energy behavior of the four-point massive graviton amplitudes could be improved to no better than E 6 [51][52][53] and is still much worse than the final energy-dependence of O(E 2 ) in the tree-level massive KK graviton scattering amplitudes [16][17][18][19][20]. The KK compactification can realize a geometric "Higgs" mechanism for mass generation of KK gravitons both at the Lagrangian level [19,20,54] and at the scattering S-matrix level [19,20]. This is shown [19,20] to make the massive KK GR theory free from the van-Dam-Veltman-Zakharov (vDVZ) discontinuity [55,56] and exhibit much better high energy behaviors (as good as that of the massless Einstein gravity).…”