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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 909 (2016) 584–606

www.elsevier.com/locate/nuclphysb

Super-renormalizable or ﬁnite

Lee–Wick quantum gravity

Leonardo Modesto

Center for Field Theory and Particle Physics and Department of Physics, Fudan University, 200433 Shanghai, China

Received 17

May 2016; received in revised form 30 May 2016; accepted 2 June 2016

Available

online 8 June 2016

Editor: Stephan

Stieberger

Abstract

propose a class of multidimensional higher derivative theories of gravity without extra real degrees of

freedom besides the graviton ﬁeld. The propagator shows up the usual real graviton pole in k

=0and extra

complex conjugates poles that do not contribute to the absorptive part of the physical scattering amplitudes.

Indeed, they may consistently be excluded from the asymptotic observable states of the theory making use

of the Lee–Wick and Cutkosky, Landshoff, Olive and Polkinghorne prescription for the construction of a

unitary S-matrix. Therefore, the spectrum consists of the graviton and short lived elementary unstable par-

ticles

that we named “anti-gravitons” because of their repulsive contribution to the gravitational potential

at short distance. However, another interpretation of the complex conjugate pairs is proposed based on the

Calmet’s suggestion, i.e. they could be understood as black hole precursors long established in the clas-

sical

theory. Since the theory is CPT invariant, the conjugate complex of the micro black hole precursor

can be interpreted as a white hole precursor consistently with the ’t Hooft complementarity principle. It

is proved that the quantum theory is super-renormalizable in even dimension, i.e. only a ﬁnite number of

divergent diagrams survive, and ﬁnite in odd dimension. Furthermore, turning on a local potential of the

Riemann tensor we can make the theory ﬁnite in any dimension. The singularity-free Newtonian gravita-

tional

potential is explicitly computed for a range of higher derivative theories. Finally, we propose a new

super-renormalizable or ﬁnite Lee–Wick standard model of particle physics.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

E-mail addresses: lmodesto@fudan.edu.cn, lmodesto1905@icloud.com.

http://dx.doi.org/10.1016/j.nuclphysb.2016.06.004

The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

L. Modesto / Nuclear Physics B 909 (2016) 584–606 585

1. Introduction

We propose a “local” multidimensional gravitational theory compatible with renormalizability

at perturbative level in addition to Lee–Wick [1] and Cutkosky, Landshoff, Olive and Polking-

horne unitarity [2] (CLOP). This work is a generalization of the theory recently proposed in

[3,4]. In the last four years a weakly nonlocal action principle for gravity has been e

xtensively

studied to make up for the shortcomings of the quantization of the Einstein–Hilbert action [5–9].

Research records show that Krasnikov in 1988 and Kuz’min in 1989 proposed a similar theory

[10] following Eﬁmov’s studies in nonlocal interacting quantum ﬁeld theory [11]. Afterwords

Tomboulis extended to gauge interactions the K

uz’min ideas and in 1996 proposed a class of

weakly nonlocal super-renormalizable gauge and gravitational theories [12–14]. Yo u may also

refer to [15,16] about other excellent contributions in nonlocal theories. Recently in [17] it has

been deﬁnitely proved that the theory is actually ﬁnite in any dimension when a local potential

of the Riemann tensor is added. In [18] has been proposed and e

xtensively studied a ﬁnite gen-

eralization of the nonlocal theory for gauge interactions proposed for the ﬁrst time by Tomboulis

[12]. However, the price to pay is that the classical action is weakly nonlocal, although the asymp-

totic polynomial behavior makes the theory ve

ry similar to any local higher derivative theory for

all that concerns the divergent contributions to the quantum effective action.

In this paper we w

ant to expand and specialize the seminal paper [20] about a general local

super-renormalizable gravitational theory capitalizing what we learned in quasi-polynomial or

weakly nonlocal theories. Actually, many results can be exported directly to the theory here

proposed making a proper replacement of the nonlocal form factor in [17] with the local form

factor that we are going to properly deﬁne later in this paper

The theory here proposed fulﬁlls a synthesis of minimal requirements: (i)

Einstein–Hilbert

action should be a good approximation of the theory at a much smaller energy scale than the

Planck mass; (ii) the theory has to be super-renormalizable or ﬁnite at quantum level; (iii) the

theory has to be unitary, with no other real poles in the propag

ator in addition to the graviton;

if we require other poles neither real nor complex, then the theory will prove non-polynomial

or weakly nonlocal. The outcome of previous studies is a nonlocal classical theory of gravity

perturbatively super-renormalizable at quantum level. On the footprint of the nonlocal action

we propose here a “local” theory that holds the same properties, bu

t showing up extra complex

conjugate poles besides the graviton.

Studies of higher deri

vative theories date back to quadratic gravity proposed in 1977 by Stelle

[19]. This theory is renormalizable and asymptotically free, but unfortunately it violates unitarity

showing up a real ghost state in the spectrum. In this paper we go behind the Stelle’s action intro-

ducing a ﬁnite number of ex

tra higher derivative operators to make the theory even more conver-

gent: super-renormalizable or ﬁnite. However, we do not blindly introduce all the possible opera-

tors to a ﬁxed order in the number of derivatives of the metric tensor. We actually consider a class

of local theories that avoid extra real poles in the propag

ator. Looking at the above list of require-

ments (i)–(iii), the news with respect to the previous work on non-polynomial theories sits in the

third point. We indeed do not exclude the possibility of complex conjugate mass poles, which do

not prevent us from constructing a unitary local theory of gra

vity in the Lee–Wick formalism [1].

Lee and Wick argued that, as long as all ghost degrees of freedom in the interacting theory have

complex energies, one obtains a unitary theory by constraining the physical subspace to be ex-

actly the one for the states that have real energy. In gravity we end up with a classical theory with

an e

xtended spectrum in which the graviton is free to propagate on long distances while a bunch

of other virtual elementary particles can only intrinsically live for a short amount of timeblandly

586 L. Modesto / Nuclear Physics B 909 (2016) 584–606

[30]. It is well known that in quantum electro-dynamics a photon can get converted into e

−

pairs, or more complicated channels, only when it interacts with matter, but when radiated into

a “perfect vacuum” it will travel on indeﬁnitely distances as a stable particle. In ﬁeld theory this

is described by a gauge independent pole at k

= 0in the transverse photon propagator, which

ﬁxes the photon free ﬁeld equations to 2A

= 0. By the contrast, we here have a ﬁnite number

of short lived particles (named “anti-gravitons”) that rapidly convert themselves into gravitons.

The dispersion relation for these particles must show off a ﬁnite lifetime through gauge indepen-

dent complex poles in the propagator, and the free equations of motion are [2 +(A +iB

)]φ =0

(where A, B ∈ R). In particle physics a Lee–Wick extension of the standard model has been

proposed to avoid quadratic divergences in the Higgs mass and hence no hierarchy puzzle [22].

In this theory, and generalizations, the classical action has a real ghost pole that, at one loop, is

shifted out the real ax

es into a complex ghost pair. In gravity a similar Lee–Wick unitarization of

the Stelle’s theory [19] was evoked in [25–27] to remove the real ghost from the asymptotically

free quadratic gravity [28]. Indeed, at one loop the real ghost pole splits into a pair of comple

conjugate poles. In this paper we go beyond four derivatives and following the seminal papers

[29,31] we propose a theory in which a ﬁnite number of complex conjugate poles, or unstable

particles, are already present in the classical action [32]. Let us give here a taste of the theory,

=−





|g|2κ

−2



R −2

+Gγ (2)Ric +V



, (1)

where γ (2) is a polynomial (of the d’Alembertian operator 2) constructed so as to avoid extra

real poles in the propagator besides the graviton, G is the Einstein tensor, Ric is the Ricci tensor,

and V is a potential at least cubic in the curvature tensor.

At classical le

vel the solutions are stable when Lee and Wick appropriate boundary conditions

are imposed [1,2]. More recently a mathematically well deﬁned prescription has been deﬁned

in [33]. However, microcausality is violated.

2. The theory

The class of theories we are going to propose can be read out from the “non-polynomial”

theories recently introduced and extensively studied in [5,12,17]. We here focus on a general

local action compatible with unitarity [1,31] and super-renormalizability or ﬁniteness,

=−2κ

−2

√



R +R γ

(2)R +Ric γ

(2)Ric +Riem γ

(2)Riem +V



, (2)

where we have distinguished the operators linear and quadratic in the curvature tensor from the

higher in curvature operators. We can rewrite the theory making use of a more compact notation

introducing a tensorial form factor, namely

=−2κ

−2

√

(

R +Riem γ (2)Riem +V

)

≡−2κ

−2

√



R +R

μνρσ

γ(2)

μνρσ

αβγ δ



≡−2κ

−2

√



R +R

μνρσ

μρ

αγ

νσ

βδ

(2) +g

μρ

αγ

νβ

σδ

(2)

μα

νβ

ργ

σδ

(2)]R

αβγ δ



. (3)

Deﬁnitions and notations. The deﬁnitions used in this paper are: the metric tensor g

μν

has signature (+ −...−) ; the

curvature tensor R

νσρ

=−∂



νσ

+..., the Ricci tensor R

μν

μρ ν

, and the curvature scalar R = g

μν

. We also

use the notation R for the Riemann tensor when the indexes are suppressed.

L. Modesto / Nuclear Physics B 909 (2016) 584–606 587

The theory consists of a kinetic local operator quadratic in the curvature, three polynomials

(2), γ

(2), and a local potential V made of the following three sets of operators,

V =

N+2



j=3



k=3



(j)

k,i



∇

2(j−k)



n+N+1



j=N+3



k=3



(j)

k,i



∇

2(j−k)



n+N+2



k=3



k,i



∇

2(n+N+2−k)



where the third set of operators are called killers because they are crucial in making the theory

ﬁnite in any dimension.  is an invariant mass scale and the indices, c

(j)

k,i

, d

(j)

k,i

, s

k,i

are running

or not coupling constants, while the tensorial structure has been neglected. The last set of op-

erators with front coefﬁcients s

k,i

are technically called “killers” and are crucial in making the

theory ﬁnite. The capital Nis deﬁned to be the following function of the spacetime dimension D:

2N +4 = D, while n is a positive integer, i.e. n ∈N

. Moreover, 2 = g

μν

∇

is the covariant

box operator. The polynomials γ

(2) are:

(2) =−

N+n+1



i=1



i−1

−4γ

(2) =

D −2

4(D −1)

N+n+1



i=1



i−1

4(D −1)

N+n+1



i=1



i−1

+4γ

, (4)

where z := −2/

, n ∈N

, and we implicitly introduced the following two polynomials,

P(z)=1 +

N+n+1



i=1

,Q(z)=1 +

N+n+1



i=1

,n∈N

. (5)

The reason of this particular choice of the polynomials γ

(2) will be clear in the next subsection

when we explicitly evaluate the propagator for the theory (2).

2.1. Propagator

Splitting the spacetime metric g

μν

into the ﬂat Minkowski background and the ﬂuctua-

tion h

μν

deﬁned by g

μν

= η

μν

+ κ

μν

, we can expand the action (2) to the second order

in h

μν

. The outcome of this expansion together with the usual harmonic gauge ﬁxing term

reads [40] L

lin

+ L

= 1/2h

μν

μν,ρσ

ρσ

, where the operator O is made of two terms,

one coming from the linearization of (2) and the other from the following gauge-ﬁxing term,

= ξ

−1

∂

μν

ω(−2



)∂

ρμ

(ω(−2



) is a weight functional [19,39]). The d’Alembertian

operator in L

lin

and the gauge ﬁxing term must be conceived on the ﬂat spacetime. Inverting the

operator O [40], we ﬁnd the two-point function in the harmonic gauge (∂

μν

=0),

−1

ξ(2P

(1)

(0)

)

ω(k

/

)

(2)

P(k

/

)

−

(0)

Q(k

/

)

(

D −2

)

. (6)

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