Perturbative unitarity of Lee-Wick quantum field theory

Abstract

We study the perturbative unitarity of the Lee-Wick models, formulated as nonanalytically Wick rotated Euclidean theories. The complex energy plane is divided into disconnected regions and the values of a loop integral in the various regions are related to one another by a nonanalytic procedure. We show that the one-loop diagrams satisfy the expected, unitary cutting equations in each region: only the physical degrees of freedom propagate through the cuts. The goal can be achieved by working in suitable subsets of each region and proving that the cutting equations can be analytically continued as a whole. We make explicit calculations in the cases of the bubble and triangle diagrams and address the generality of our approach. We also show that the same higher-derivative models violate unitarity if they are formulated directly in Minkowski spacetime.Comment: 30 pages, 15 figures; v2: more details and comments on generality of approach; v3: minor changes, PR

On the quantum field theory of the gravitational interactions

“…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.…”

“…An example of acceptable solution is (rounding to four decimal places) 14) together with λ C = ξ = α 1 = α 2 = α 4 = α 5 = α 6 = 0. It is easy to check that the unitarity bounds (2.6) are satisfied if η > 0.…”

“…Then, the proof of perturbative unitarity can be worked out by combining the strategy of [14] (to show that the Lee-Wick poles do not propagate through the cuts in the cutting equations), with the strategy of ref. [5] (to show that the gauge degrees of freedom -i.e.…”

“…Viewed as nonanalytically Wick rotated Euclidean theories, such models are perturbatively unitary [14]. Moreover, the new formulation is intrinsically equipped with all that is needed to define the physical amplitudes properly, with no need of ad hoc prescriptions.…”

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.…”