Lee-Wick量子场论中的质点
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Massive poles in Lee-Wick quantum field theory
John F. Donoghue
*
Department of Physics, University of Massachusetts Amherst, Massachusetts 01003, USA
Gabriel Menezes
†
Department of Physics, University of Massachusetts Amherst, Massachusetts 01003, USA
and Departamento de Física, Universidade Federal Rural do Rio de Janeiro,
23897-000 Serop´edica, RJ, Brazil
(Received 14 January 2019; published 29 March 2019)
Most discussions of propagators in Lee-Wick theories focus on the presence of two massive complex
conjugate poles in the propagator. We show that there is, in fact, only one pole near the physical region or,
in another representation, three polelike structures with compensating extra poles. The latter modified
Lehmann representation is useful caculationally and conceptually only if one includes the resonance
structure in the spectral integral. We treat both the photon propagator in Lee-Wick electrodynamics and the
spin-two propagator in quadratic gravity.
DOI: 10.1103/PhysRevD.99.065017
I. INTRODUCTION
Lee and Wick have formulated a type of theory which is
finite, yet yields all the usual predictions at low energy
[1–3]. They endow new fields with a negative metric, with
the result that the propagation of these fields cancels off the
high-energy divergences of usual field theory. In rather
simplistic terms, it is similar to including the Pauli-Villars
regulators as the dynamical fields. For example the electro-
magnetic propagator is modified at tree level via
iD
Fμν
ðqÞ¼−ig
μν
1
q
2
−
1
q
2
− Λ
2
¼ −ig
μν
−Λ
2
q
2
ðq
2
− Λ
2
Þ
¼ −ig
μν
1
q
2
ð1 −
q
2
Λ
2
Þ
ð1Þ
The fact that the propagator goes asymptotically like q
−4
implies that loop integrals are not divergent. However, the
massive field appears with negative norm—it is a ghost
field. Once interactions are introduced, this dangerous
feature is alleviated because the massive field decays into
the light particles in the theory, such that it is not an
asymptotic state in the spectrum. With some prescriptions
for the treatment of loop integrals [4,5], the theory appears
to be consistent and unitary, although there is a microscopic
violation of causality on small scales [6–8]. This Lee-Wick
mechanism for dealing with theories with quartic propa-
gators is thought to be an important ingredient for many
other higher derivative theories, including that of quadratic
gravity [9–13]. Recent attempts to treat quadratic gravity as
a fundamental quantum field theory [14–27] also have to
address a negative norm state that appear in the graviton
propagator and the analogy with Lee-Wick theories is quite
close. It is, therefore, important to understand the under-
lying physics of Lee-Wick theories.
In these theories, when the massive state decays, the state
develops a width. In most of the literature, the treatment
involves a pair of poles that appear at the positions which
are complex conjugates of each other, q
2
¼ M
2
¼ m
2
p
þ iγ
and q
2
¼ M
2
¼ m
2
p
− iγ, with m
2
p
; γ both real. However,
when explicit calculations are needed, one finds that there
is only one pole. A representation with three poles—with
two of them compensating—is also valid and useful. To our
knowledge, the compensation of the latter poles was first
noted in the context of an OðNÞ model by Grinstein,
O’Connell and Wise [8]. The purpose of this paper is to
provide a clear discussion of this issue and to highlight the
importance of the spectral integral in the latter representation.
II. LOCATION OF THE POLE
IN THE PROPAGATOR
Both the usual photon and the heavy Lee-Wick particle
couple to the electromagnetic current, and so the interaction
can be described by a combined propagator. We start with a
simple representation of this taken from the literature,
which is quite intuitive and which captures the essence of
*
†
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI. Funded by SCOAP
3
.
PHYSICAL REVIEW D 99, 065017 (2019)
2470-0010=2019=99(6)=065017(10) 065017-1 Published by the American Physical Society
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