A new formulation of Lee-Wick quantum field theory

Abstract

The Lee-Wick models are higher-derivative theories that are claimed to be unitary thanks to a peculiar cancelation mechanism. In this paper, we provide a new formulation of the models, to clarify several aspects that have remained quite mysterious, so far. Specifically, we define them as nonanalytically Wick rotated Euclidean theories. The complex energy plane is divided into disconnected regions, which can be related to one another by a well-defined, albeit nonanalytic procedure. Working in a generic Lorentz frame, the models are intrinsically equipped with the right recipe to treat the pinchings of the Lee-Wick poles, with no need of external ad hoc prescriptions. We describe these features in detail by calculating the one-loop bubble diagram and explaining how the key properties generalize to more complicated diagrams. The physical results of our formulation are different from those of the previous ones. The unusual behaviors of the physical amplitudes lead to interesting phenomenological predictions.

On the quantum field theory of the gravitational interactions

“…Although the CLOP or other ad hoc prescriptions have been advocated in such investigations, some conclusions may survive once those prescriptions are removed in favor of the formulation of ref. [11].…”

“…Consider a Feynman diagram and integrate the loop energies by means of the residue theorem, as usual. We recall [11,14] that the nonanalytic behavior of the Wick rotation is due to the pinching of LW poles and that the LW pinching conditions equate a linear combination p 0 of the external energies to a sum of frequencies ω i (k, p), where k denotes the loop space momenta. Now, the ultraviolet divergences are studied by keeping the external momenta p fixed and letting k tend to infinity.…”

“…A good check that (5.2) is well defined can be made by calculating the bubble diagram explicitly with the technique explained in ref. [11] and then take the limit E → 0. The…”

“…On the other hand, the Wick rotation from Euclidean space is obstructed when the free propagators have poles in the first or third quadrants of the complex energy plane. The obstruction can actually be overcome by a nonanalytic procedure, which leads to a new formulation [11] of an interesting subclass of higher-derivative theories, the Lee-Wick (LW) models [12,13].…”

Agravity up to infinite energy

“…Since this symmetry is anomalous, conformal gravity cannot be the complete theory: going to lower energy the conformal model of the graviton 1 This can maybe be avoided introducing an infinite series of higher derivative terms [12], but the resulting gravity theories contain infinite free parameters and are not known to be renormalizable. 2 The ample literature of 'ghosts' was critically reviewed in [14]; for later work, see [15][16][17][18][19][20][21][22], where it was proposed that a four-derivative variable q(t) contains two canonical degrees of freedom (d.o.f. ), q 1 = q and q 2 =q, with opposite time-reflection parity, such that usual T -even representation (q 1 |x = x|x and p 1 |x = i d dx |x ) must be combined with the T -odd representation (q 2 |y = iy|y and p 2 |y = d dy |y ) obtaining consistent results (positive energy, normalizable wave functions, Euclidean continuation), although the interpretation of the resulting negative norm is unclear.…”