Fierz-Pauli theory reloaded: from a theory of a symmetric tensor field to
linearized massive gravity

Giulio Gambuti1,a and Nicola Maggiore2,3,b

1 Rudolf Peierls Centre for Theoretical Physics, Oxford University, U.K.

2 Dipartimento di Fisica, Università di Genova, Italy.

3 Istituto Nazionale di Fisica Nucleare - Sezione di Genova, Italy.

Abstract : Modifying gravity at large distances by means of a massive graviton
may explain the observed acceleration of the Universe without Dark Energy. The
standard paradigm for Massive Gravity is the Fierz-Pauli theory, which,
nonetheless, displays well known flaws in its massless limit. The most serious
one is represented by the vDVZ discontinuity, which consists in a disagreement
between the massless limit of the Fierz-Pauli theory and General Relativity.
Our approach is based on a field-theoretical treatment of Massive Gravity:
General Relativity, in the weak field approximation, is treated as a gauge
theory of a symmetric rank-2 tensor field. This leads us to propose an
alternative theory of linearized Massive Gravity, describing five degrees of
freedom of the graviton, with a good massless limit, without vDVZ
discontinuity, and depending on one mass parameter only, in agreement with the
Fierz-Pauli theory.

Keywords: Linearized General Relativity; Massive Gravity; Fierz-Pauli Theory;
Large Distances Modified Gravity.

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction

#### Motivations

It is an observational fact that the Universe is expanding at an accelerated
rate [1, 2]. Within the theory of General Relativity (GR), the cosmological
constant $\Lambda$ explains this phenomenon. In a picture of the Universe seen
as a perfect fluid, the cosmological constant gives rise to an accelerated
expansion by acting as a constant energy density (called Dark Energy)
$\rho\sim\Lambda M_{P}^{2}$ providing a negative pressure. A fine tuning of
the constant $\Lambda$ enables us to match the expansion predicted by GR to
the one which is observed. The estimate from observations is
$\frac{\Lambda}{M_{P}^{2}}\sim 10^{-65}$ [3], where $M_{P}$ is the Planck
mass, while the quantum field theoretical prediction on the cosmological
constant seen as the vacuum energy density gives
$\frac{\Lambda}{M_{P}^{2}}\sim 6\times 10^{54}$. Unfortunately, these two
estimates disagree by about 120 orders of magnitude. This huge tension is
known as the cosmological constant problem [4], which motivates alternative
descriptions for Dark Energy, with the request that any cosmological model
should reproduce an Universe which, at our epoch, is almost perfectly flat and
filled by matter and DE in the ratio of about 3/7, where the DE is effectively
approximated by a constant [5, 6, 7, 8, 9, 10]. Modifying gravity at large
distances by means of a massive graviton is a possible solution to the
cosmological constant problem [11]. In fact, the Yukawa potential for a
massive field at large distances goes like $\propto\frac{1}{r}e^{-\alpha mr}$,
where $m$ is the mass of the field and $\alpha$ a dimensional constant. At
scales comparable to $\frac{1}{\alpha m}$, the exponential factor suppresses
the potential and the strength of interactions as well. This is the reason why
long-range forces are associated with massless bosons and short-range forces
with massive bosons. By means of a Yukawa-like potential, the gravitational
effect of the vacuum energy density is exponentially suppressed at large
scales, thus explaining the disagreement between the quantum field theory
calculation and the observed cosmological value. Clearly, this exponential
suppression of long-range gravitational interactions is constrained by
experimental evidence and, consequently, the mass of the graviton is subject
to restrictive upper limits. Another way of understanding this point is to
remember that a mass acts as a momentum space cutoff of interactions,
effectively damping the effect of low frequency sources. The vacuum energy
density can be considered constant, therefore its contribution to the
expansion of the Universe is greatly reduced by the introduction of a mass for
the graviton, again confirming that in a picture where the graviton is massive
the cosmological constant problem might be fixed by a suitable graviton mass.
Theories in which the graviton is massive are referred to as Massive Gravity
(MG) theories. We nonetheless remark that the introduction of a mass for the
graviton as a potential solution of the cosmological constant problem was a
hope more than a decade ago, but early experience with nonlinear MG (see, for
example [12]) showed that this expectation is not realized in the nonlinear
theory where a cosmological constant bends spacetime in the same way as in GR.
While a large vacuum energy can be canceled by a MG contribution, this still
requires fine tuning and does not amount to a degravitation mechanism.111 We
thank the referee for this remark. In some sense MG is a substantial
modification of GR, even if the mass of the graviton is extremely close to
zero. In fact, when adding a mass, the two degrees of freedom (DOF) carried by
the massless boson become $2S+1$ massive DOF. For the graviton, which is the
spin $S=2$ representation of the Poincaré group, the two massless DOF become
five massive ones.

#### Phenomenological Limits

As previously said, long range forces are usually carried by massless bosons
because the Yukawa exponential factor limits the range of interactions
mediated by massive particles. Nevertheless, choosing a sufficiently small
value of the graviton mass, the interactions at scales of the observable
Universe remain essentially identical to those of ordinary gravity, while only
interactions at greater scales are affected. Indeed, the observed accelerated
rate of the Universe gives an upper limit to the mass of the graviton, which
in [13] is shown to be of order $10^{-34}\;\frac{eV}{c^{2}}$, roughly an order
of magnitude smaller than the Hubble parameter at our epoch. With such a
limit, for interactions at distances much smaller than the radius of the
observable Universe (which is estimated to be about $46.6\times
10^{9}\>\mathrm{ly}$ [14]), the Yukawa exponential drops to one, and the
classical Newton potential $\propto\frac{1}{r}$ is recovered, as desired.
Still, this cosmological upper limit becomes irrelevant when the mass is
acquired through the condensation of some additional scalar field (see [13]).
Another constraint on the mass of the graviton was recently obtained by the
LIGO-Virgo collaboration through the analysis of binary black holes merger
signals, measuring the phase shift between components of different frequencies
of the gravitational waves detected by the interferometers [15]. In this way,
the upper limit for the mass of the graviton is estimated to be $4.7\times
10^{-23}\;\frac{eV}{c^{2}}$, which is several orders of magnitude higher than
the limit derived in [13] by cosmological considerations.

#### State of the Art

In 1939 Fierz and Pauli (FP) proposed a relativistic theory for a massive
particle with arbitrary spin $f$, described by a symmetric rank-$f$ tensor
field [16]. They showed that “in the particular case of spin two, rest-mass
zero, the equations agree in the force-free case with Einstein’s equations for
gravitational waves in GR in first approximation; the corresponding group of
transformations arises from the infinitesimal coordinate transformations”.
This is what is known as the FP theory, which is considered as the standard
paradigm for Linearized MG (LMG). Much later, it has been realized by van Dam,
Veltman and Zakharov [17, 18] that the massless limit of the FP theory
exhibits a discontinuity with GR, known as vDVZ discontinuity. In particular,
in [19] it is shown that the interaction of light with a massive body, like a
star for instance, is 25% smaller in the massless limit of the FP theory than
in GR. Consequently, the angle of deviation of light is also different by 25%
in the two theories, a difference which can be observed by means of the time
delay in the gravitational lensing phenomenon. The problems of the FP theory
at small scales can be solved by the Vainshtein mechanism [20], which
nonetheless requires a nonlinear modification of the FP theory. In principle,
this measurable disagreement between GR and the FP theory of LMG would allow
us to distinguish between a strictly vanishing mass and a very small one. Such
a tension is crucial, since GR has been extensively tested and as a
consequence we might rule out the FP theory as a theory of LMG. It was also
pointed out in [21] that in order for Linearized Gravity (LG) to describe a
pure spin two system, it is necessary to use a particular FP mass term,
referred to as the FP tuning. Otherwise, there will be admixtures of lower
spin, in general with negative-energy, called Boulware-Deser ghost. Moreover,
Boulware and Deser argued that the ghost will reappear if nonlinear extension
of LG are considered. Some possibilities to circumvent these problems have
been proposed in [22, 23, 24]. A detailed review on MG can be found in [25].

#### Strategy

The approach adopted in this paper is based on a field-theoretical treatment
of LG. We consider LG as a gauge theory of a symmetric rank-2 tensor field,
the gauge symmetry being the infinitesimal diffeomorphism invariance [26].
Before adding a mass term to the action, which breaks the gauge symmetry, the
massless theory should be well defined, in the sense that a well defined
partition function must exist. In gauge field theory this is achieved by gauge
fixing the action. This procedure (gauge fixing the massless action first,
adding a breaking mass term after) leads to a gauge theory of LMG describing
five DOF for the massive graviton, with a good massless limit and without the
vDVZ discontinuity. In the FP theory, instead, the mass term is added directly
to the invariant action and effectively plays the double and unnatural role of
both mass term and gauge fixing. Basically, this is the reason why the FP
theory displays an ill-defined massless limit, which shows itself in a
divergent propagator and, physically, in the vDVZ discontinuity with GR. We
stress that any theory of LMG should display a good massless limit since the
phenomenological limits on the mass of the graviton are such that the mass
term must be seen as a small perturbation of massless LG and, equivalently, as
a small breaking of diffeomorphism invariance.

#### Summary

This paper is organized as follows: in Section 2 the gauge fixed massive
theory is presented. We also review the FP theory and the problems related to
its massless limit which motivate this work, and we remark that, curiously,
the original 1939 paper by Fierz and Pauli does not exactly describe what is
generally known as the FP theory. In Section 3 we show, in a gauge independent
way, that the propagating DOF of the ten components of the rank-2 symmetric
tensor describing the massive graviton are indeed five, and depend on one mass
parameter only, in agreement with the FP theory. In Section 4 we compute the
propagator of the theory, we show that its massless limit is regular and that
the same observable which is used to unveil the vDVZ discontinuity presents an
unique pole in the same mass parameter which characterizes the five DOF of the
theory. In Section 5 we show the absence of the vDVZ discontinuity, for any
gauge choice. Our results are summarized and discussed in the concluding
Section 6.

List of acronyms: DOF: Degree(s) of Freedom; EOM: Equation(s) of Motion; FP:
Fierz-Pauli; GR: General Relativity; LG: Linearized Gravity; LMG: Linearized
Massive Gravity; MG: Massive Gravity; vDVZ: van Dam, Veltman and Zacharov.

##  2 The massive action

The weak field expansion of GR around the flat Minkowskian background
$\eta_{\mu\nu}=diag(-1,1,1,1)$ is given by the LG action

$S_{LG}[h]=\int\mathrm{d^{4}x}\;\;\Big{[}\;\frac{1}{2}h\partial^{2}h\;-\;h_{\mu\nu}\partial^{\mu}\partial^{\nu}h\;-\;\frac{1}{2}h^{\mu\nu}\partial^{2}h_{\mu\nu}\;+\;h^{\mu\nu}\partial_{\nu}\partial^{\rho}h_{\mu\rho}\;\Big{]}\;,$
(2.1)

where $h_{\mu\nu}(x)$ is a symmetric rank-2 tensor field representing the
graviton, and $h(x)\equiv\eta_{\mu\nu}h^{\mu\nu}(x)$ is its trace. The action
$S_{LG}[h]$ (2.1) is the most general functional invariant under the
infinitesimal diffeomorphism transformation

$\delta
h_{\mu\nu}(x)=\partial_{\mu}\xi_{\nu}(x)+\partial_{\nu}\xi_{\mu}(x)\;,$ (2.2)

where $\xi_{\mu}(x)$ is a local vector parameter. The transformation (2.2)
represents the gauge symmetry of the action $S_{LG}$ (2.1).

###  2.1 The Fierz-Pauli theory

The most general mass term which can be added to the invariant action $S_{LG}$
(2.1), respecting Lorentz invariance and power counting, is

$S_{m}[h;m_{1}^{2},m_{2}^{2}]=\frac{1}{2}\int\mathrm{d}^{4}x\
(m_{1}^{2}h_{\mu\nu}h^{\mu\nu}+m_{2}^{2}h^{2})\;,$ (2.3)

where $m_{1}^{2}$ and $m_{2}^{2}$ are massive parameters. The presence of a
mass term breaks the diffeomorphism invariance (2.2), as usual in any gauge
field theory. It can be shown (see for instance [25]) that the action

$S=S_{LG}+S_{m}\;.$ (2.4)

describes the propagation of five DOF only if

$m_{1}^{2}+m_{2}^{2}=0\ ,$ (2.5)

otherwise a sixth ghost mode with negative energy appears [21], and the theory
does not describe a massive graviton. The choice (2.5) is generally referred
to as FP tuning, and the FP theory is defined by the action

$S_{FP}[h;m_{1}^{2}]\equiv S_{LG}[h]+S_{m}[h;m_{1}^{2},-m_{1}^{2}]\;.$ (2.6)

Following [11], we now show that the theory described by the action $S_{FP}$
(2.6) does indeed display five DOF. The Equations of Motion (EOM) obtained
from (2.6) read

$\frac{\delta S_{FP}}{\delta
h^{\mu\nu}}=\partial^{2}h_{\mu\nu}-\partial_{\alpha}\partial_{\mu}{h^{\alpha}}_{\nu}-\partial_{\alpha}\partial_{\nu}{h^{\alpha}}_{\mu}+\eta_{\mu\nu}\partial_{\alpha}\partial_{\beta}h^{\alpha\beta}+\partial_{\mu}\partial_{\nu}h-\eta_{\mu\nu}\partial^{2}h-m_{1}^{2}(h_{\mu\nu}-\eta_{\mu\nu}h)=0\;,$
(2.7)

which, saturated with $\partial^{\mu}$, yield the constraint

$\partial^{\mu}h_{\mu\nu}-\partial_{\nu}h=0\;.$ (2.8)

Plugging (2.8) into (2.7) we get

$\partial^{2}h_{\mu\nu}-\partial_{\mu}\partial_{\nu}h-m_{1}^{2}(h_{\mu\nu}-\eta_{\mu\nu}h)=0\;.$
(2.9)

Saturating (2.9) with $\eta^{\mu\nu}$ we find

$h=0\;,$ (2.10)

which, together with (2.8), implies

$\partial^{\mu}h_{\mu\nu}=0\;.$ (2.11)

Therefore, the EOM (2.7) imply the following set of equations

$\displaystyle(\partial^{2}-m_{1}^{2})h_{\mu\nu}(x)=0\;$ (2.12)
$\displaystyle\partial^{\mu}h_{\mu\nu}(x)=0\;$ (2.13) $\displaystyle
h(x)=0\;.$ (2.14)

Eq. (2.12) is the Klein-Gordon equation for the field $h_{\mu\nu}(x)$, while
(2.13) and (2.14) represent five constraints (transversality and
tracelessness) which reduce the ten independent components of $h_{\mu\nu}$ to
five. These five components carry the five massive DOF of the graviton.

###  2.2 Problems with the Fierz-Pauli theory

The propagator of the FP theory (2.6) is

$G^{FP}_{\mu\nu,\alpha\beta}(p)=\frac{2}{p^{2}+m_{1}^{2}}\left[\frac{1}{2}(P_{\mu\alpha}P_{\nu\beta}+P_{\nu\alpha}P_{\mu\beta})-\frac{1}{3}P_{\mu\nu}P_{\alpha\beta}\right]\;,$
(2.15)

where $P_{\mu\nu}$ is the transverse massive projector defined as

$P_{\mu\nu}=\eta_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{m_{1}^{2}}\;.$ (2.16)

A crucial remark is that the propagator (2.15) exists only thanks to the
presence of the mass term (2.3) on the FP point (2.5). As a consequence of
this fact, it is apparent from (2.16) that the FP theory has a divergent
massless limit. Moreover, the massless limit of the FP theory is flawed by the
vDVZ discontinuity [17, 18], which basically consists in the fact that the
correlator involving two energy-momentum tensors, computed in the FP theory in
the limit $m_{1}^{2}\rightarrow 0$, does not match the GR prediction. We
sketch here the proof (for details see [11, 19]). The gravitational
interaction between two non relativistic energy-momentum tensors
${T_{\mu\nu}^{(1)}}$ and ${T^{(2)}}_{\mu\nu}$ (which are conserved, $i.e.$
$p^{\nu}\tilde{T}_{\mu\nu}^{(1)}=p^{\nu}\tilde{T}_{\mu\nu}^{(2)}=0$) is
described by the introduction in the action of an interaction term of the type

$S_{int}=\lambda\int\mathrm{d}^{4}x\;h_{\mu\nu}T^{\mu\nu}\;,$ (2.17)

where $T_{\mu\nu}$ denotes a generic energy-momentum tensor, coupled to
$h_{\mu\nu}(x)$ through a constant $\lambda$, which we call $\lambda_{LG}$ and
$\lambda_{FP}$ for LG and FP theory, respectively. Therefore, the interaction
strength between ${T_{\mu\nu}^{(1)}}$ and ${T_{\mu\nu}^{(2)}}$ can be computed
by contraction with the propagator of the graviton. In LG, the propagator
$G_{LG}^{\mu\nu,\alpha\beta}$ is obtained from the LG action (2.1) after a
gauge fixing, as done in [11]. The FP propagator
$G_{FP}^{\mu\nu,\alpha\beta}$, on the other hand, is given by (2.15). In the
non relativistic limit, only the 00-components of the energy-momentum tensors
are non negligible. Two cases are considered: in the first,
${T_{\mu\nu}^{(1)}}$ and ${T_{\mu\nu}^{(2)}}$ are associated with massive
objects and therefore have non vanishing trace. In the second case,
${T_{\mu\nu}^{(1)}}$ still has a non vanishing trace whereas
${T_{\mu\nu}^{(2)}}$ is traceless, representing, for instance, electromagnetic
radiation ($e.g.$ light). Concerning the first case, the interaction strength
in LG is

$\lambda^{2}_{LG}\>\tilde{T}^{(1)}_{\mu\nu}G_{LG}^{\mu\nu,\alpha\beta}\tilde{T}^{(2)}_{\alpha\beta}=\lambda^{2}_{LG}\>\tilde{T}^{(1)}_{00}\tilde{T}^{(2)}_{00}\frac{1}{p^{2}}\;,$
(2.18)

while, in the second, $i.e.$ when ${T_{\mu\nu}^{(2)}}$ is traceless, we have

$\lambda^{2}_{LG}\>\tilde{T}_{\mu\nu}^{(1)}G_{LG}^{\mu\nu,\alpha\beta}\tilde{T}^{(2)}_{\alpha\beta}=\lambda^{2}_{LG}\>\tilde{T}^{(1)}_{00}\tilde{T}^{(2)}_{00}\frac{2}{p^{2}}\;.$
(2.19)

On the other hand, the interaction strengths corresponding to (2.18) and
(2.19) obtained using the FP propagator (2.15), in the massless limit
$m_{1}\rightarrow 0$ are, respectively

$\lambda^{2}_{FP}\>\tilde{T}_{\mu\nu}^{(1)}G_{FP}^{\mu\nu,\alpha\beta}\tilde{T}^{(2)}_{\alpha\beta}=\frac{4}{3}\lambda^{2}_{FP}\>\tilde{T}^{(1)}_{00}\tilde{T}^{(2)}_{00}\frac{1}{p^{2}}$
(2.20)

and

$\lambda^{2}_{FP}\>\tilde{T}_{\mu\nu}^{(1)}G_{FP}^{\mu\nu,\alpha\beta}\tilde{T}^{(2)}_{\alpha\beta}=\lambda^{2}_{FP}\>\tilde{T}_{00}^{(1)}\tilde{T}_{00}^{(2)}\frac{2}{p^{2}}\;.$
(2.21)

The vDVZ discontinuity arises from the observation that it is not possible to
choose an unique FP coupling $\lambda_{FP}$ which matches both (2.18) with
(2.20) and (2.19) with (2.21): the value
$\lambda^{2}_{FP}=\frac{3}{4}\lambda^{2}_{LG}$ matches the first pair but not
the second. The interaction strength between a massive body and an
electromagnetic wave in the massless limit of the FP theory turns out to be
$\frac{3}{4}$ of the LG prediction, hence the discontinuity.

###  2.3 Back to the origins

In [27] Fierz studied the general relativistic theory of force-free particles
with any (integer or half-integer) spin $f$, described by symmetric “world”
tensors $A_{ik...l}(x)$222 In this subsection we respectfully maintain the
notation used in the original paper by Fierz and Pauli in 1939. In particular,
the indices $(i,j)$ refer to spacetime, contrarily to the rest of the paper,
where greek letters are used.. The starting point in [27]. The starting point
in [27] is the request that the fields $A_{ik...l}(x)$ satisfy the massive
wave equation (which Fierz called of the “Schrödinger-Gordon type”)

$\partial^{2}A_{ik...l}=\kappa^{2}A_{ik...l}\ ,$ (2.22)

where $\kappa$ is a constant with the dimension of the inverse of a length (to
which corresponds the mass $m=\frac{\hbar\kappa}{c}$). In addition, the fields
$A_{ik...l}(x)$ are asked to satisfy the additional constraints (called
“secondary conditions”)

$\displaystyle A_{ii...l}$ $\displaystyle=$ $\displaystyle 0$ (2.23)
$\displaystyle\partial_{i}A_{ik...l}$ $\displaystyle=$ $\displaystyle 0\ .$
(2.24)

As explained in [27], the secondary conditions (2.23) and (2.24) are
introduced in order to ensure that only particles with the spin $f$ and not
also those of smaller spins could be assigned to the tensor field, and that if
the field $A_{ik...l}(x)$ describes a particle with spin $f$ satisfying the
massive “Schrödinger-Gordon” wave equation (2.22), then the number of linearly
independent plane waves is $2f+1$, which differ by the orientation of the
spin. Shortly after, in [16] Fierz and Pauli applied the general formalism
described in [27] to the particular case of spin $f=2$, simply taken as an
example. Hence, they looked for the theory of a symmetric tensor field
$A_{ij}(x)$ satisfying the massive wave equation

$\partial^{2}A_{ij}=\kappa^{2}A_{ij}\ ,$ (2.25)

together with the secondary conditions

$\displaystyle A_{ii}$ $\displaystyle=$ $\displaystyle 0$ (2.26)
$\displaystyle\partial_{i}A_{ij}$ $\displaystyle=$ $\displaystyle 0\ .$ (2.27)

Imposing the condition of tracelessness (2.26) by hand, they derived the above
equations (2.25) and (2.27) from the EOM of the following Lagrangian

${\cal
L}_{orig}=\kappa^{2}A_{ij}A^{ij}+\partial_{l}A_{ij}\partial^{l}A^{ij}+a_{1}\partial_{i}A^{ik}\partial_{j}{A^{j}}_{k}+a_{2}\kappa^{2}C^{2}+a_{3}\partial_{i}C\partial^{i}C+\partial_{i}A^{ij}\partial_{j}C\;,$
(2.28)

where an additional scalar field $C(x)$ is introduced, and $a_{i},\ i=1,2,3$
are dimensionless constant parameters. They showed that the desired results
(2.25) and (2.27) are obtained if

$\displaystyle a_{1}=-2\;;$ (2.29) $\displaystyle
2a_{3}=a_{2}=-\frac{3}{4}\;.$ (2.30)

In (2.28) the introduction of the scalar field $C(x)$ is an artifice which
enables one to derive the transversality condition (2.27). In a more modern
language, we would call this scalar field a Nakanishi-Lautrup Lagrange
multiplier [28, 29] introduced to implement the gauge condition (2.27). It is
therefore interesting (and surprising !) to remark that, in the original
article [16], Fierz and Pauli wrote an action for the graviton which included
both a mass term and a gauge fixing term, implemented by means of the Lagrange
multiplier $C(x)$. It appears that the problem of treating the mass term as a
gauge fixing was absent in the original formulation. Moreover, the massless
sector of the lagrangian $L_{orig}$ (2.28) does not coincide with the LG
action $S_{LG}$ (2.1), since the trace $A_{ii}(x)$ (a.k.a. $h(x)$) is set to
zero a priori. As a consequence, the only mass parameter appearing in (2.28)
is the one coupled to $A_{ij}A^{ij}$ (a.k.a. $h_{\mu\nu}h^{\mu\nu}$), which
therefore corresponds to $m_{1}^{2}$ in (2.3). It seems that, despite the fact
that the original FP approach to LMG was realized in term of a gauge fixed
action, later developments left the gauge fixing term behind, leaving room for
the divergent massless limit and for the vDVZ discontinuity.

In this paper we adopt a conservative policy, recovering the original FP
approach to LMG, with a few important differences.

###  2.4 Adding masses to the gauge fixed action

One of the reasons of the divergent massless limit of the FP theory is that
the mass term $S_{m}$ (2.3) serves as gauge fixing as well, in the sense that
its presence is necessary to define the propagator (2.15), as shown in Section
2.2. The standard way to proceed in gauge field theory, instead, is first to
gauge fix the invariant action, in order to have a well defined partition
function $Z[J]$. This latter generates all the correlation functions of the
theory, starting from the 2-points green function, a.k.a. the propagator. Only
after having obtained that, the fields of the theory can be given masses
through various procedures. In LMG this is achieved by adding to the gauge
fixed action the mass term $S_{m}$ (2.3) [30]. So, let us first proceed by
gauge fixing the invariant action $S_{LG}[h]$ (2.1). The gauge field is
represented by a rank-2 symmetric tensor $h_{\mu\nu}(x)$. Hence, the most
general covariant gauge fixing is

$\partial_{\mu}h^{\mu\nu}+\kappa\partial^{\nu}h=0\ ,$ (2.31)

which, by means of the usual Faddeev-Popov ($\Phi\Pi$) exponentiation [31],
yields the gauge fixing action term

$S_{gf}[h;k,\kappa]=-\frac{1}{2k}\int\mathrm{d^{4}x}\
\Big{[}\partial_{\mu}h^{\mu\nu}+\kappa\partial^{\nu}h\Big{]}^{2}\;.$ (2.32)

Notice that the gauge fixing term (2.32) depends on two gauge parameters $k$
and $\kappa$, which play different roles. In fact, $k$ determines how the
gauge fixing condition (2.31) is enforced. It can be seen as a kind of
$primary$ gauge fixing parameter, which corresponds to the standard gauge
fixing parameter of Yang-Mills theory: $k=0$ corresponds to the Landau gauge,
for instance. On the other hand, the parameter $\kappa$ fine-tunes the class
of gauge fixing identified by $k$. It plays a $secondary$ role. As an example,
the harmonic, or Lorenz, gauge is obtained with the choice $\kappa=-1/2$.
Hence, it makes sense to talk about harmonic-Landau gauge, for instance,
meaning by that the choice $k=0$ and $\kappa=-1/2$. Once the action
$S_{LG}[h]$ (2.1) has been gauge fixed by the gauge fixing term (2.32), we can
add the mass term (2.3), so that our starting point for a theory of LMG is
given by the action

$S_{LMG}=S_{LG}+S_{gf}+S_{m}\;.$ (2.33)

The action (2.33) is the starting, rather than the arrival, point, because the
road ahead of us is still long. We have indeed to face the problems which
affect the FP theory: in particular the massless limit and the absence of the
vDVZ discontinuity. But, first, we have to deal with the main feature of LMG:
the five DOF which must characterize a spin-2 massive particle. Our comparison
is the FP theory, which reaches this goal with one mass parameter only,
because of the FP tuning (2.5). The action (2.33), instead, depends on two
masses, for now. This will be done in the next Section.

##  3 Degrees of freedom

A realistic theory of MG needs five propagating massive DOF. The easiest way
to see this is to notice that the massive graviton is a spin $S=2$ particle,
which displays $2S+1$ independent components. Hence, given that the graviton
is described by a symmetric rank-2 tensor $h_{\mu\nu}(x)$, only five out of
its ten components correspond to physical DOF. Therefore, a necessary
condition for a gauge theory of a massive rank-2 symmetric tensor to be
promoted to a theory of LMG is to recover the five linear equations
represented by the constraints of tracelessness (2.26) (or (2.14)) and of
transversality (2.27) (or (2.13)), in order to lower the number of independent
components of $h_{\mu\nu}(x)$ from ten to five. The realization of this
necessary condition, in the framework of a well defined gauge field theory of
a symmetric rank-2 tensor, is the main aim of this paper. To reach our goal,
we shall now consider the EOM of the action (2.33) and we shall manage to
restrict the mass parameters $m^{2}_{1}$ and $m^{2}_{2}$ to the cases in which
we can find enough constraints to ensure the propagation of five massive DOF.
Moreover, in order to deal with a physically consistent theory, we must also
require that the propagating DOF do not depend on the gauge parameters $k$ and
$\kappa$, allowing instead, of course, a dependence on the mass parameters
$m^{2}_{1}$ and $m^{2}_{2}$.
The action $S_{LMG}$ (2.33) in momentum space reads:

$S_{LMG}=\int\mathrm{d^{4}p}\
\tilde{h}_{\mu\nu}\;\Omega^{\mu\nu,\alpha\beta}\;\tilde{h}_{\alpha\beta}\;,$
(3.1)

where the kinetic operator $\Omega$ is

$\Omega_{\mu\nu,\alpha\beta}=\frac{1}{2}\left[m_{2}^{2}-\left(1+\frac{\kappa^{2}}{k}\right)p^{2}\right]\eta_{\mu\nu}\eta_{\alpha\beta}+\frac{1}{2}\left(1-\frac{\kappa}{k}\right)\left(\eta_{\mu\nu}e_{\alpha\beta}+\eta_{\alpha\beta}e_{\mu\nu}\right)p^{2}+\\\\[10.0pt]
\frac{1}{2}(p^{2}+m_{1}^{2})\mathcal{I}_{\mu\nu,\alpha\beta}-\frac{1}{4}\left(1+\frac{1}{2k}\right)\left(e_{\mu\alpha}\eta_{\nu\beta}+e_{\nu\alpha}\eta_{\mu\beta}+e_{\mu\beta}\eta_{\nu\alpha}+e_{\nu\beta}\eta_{\mu\alpha}\right)p^{2}\;,$
(3.2)

$\mathcal{I}$ is the rank-4 tensor identity

$\mathcal{I}_{\mu\nu,\rho\sigma}=\frac{1}{2}(\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho})$
(3.3)

and $e_{\mu\nu}$ is the transverse projector

$e_{\mu\nu}=\frac{p_{\mu}p_{\nu}}{p^{2}}\ .$ (3.4)

From the action $S_{LMG}$ (3.1) we get the momentum space EOM

$\displaystyle\frac{\delta S}{\delta\tilde{h}_{\mu\nu}}=\;$
$\displaystyle-\left(1+\frac{\kappa^{2}}{k}\right)\eta^{\mu\nu}p^{2}\tilde{h}\;+\;\left(1-\frac{\kappa}{k}\right)p^{\mu}p^{\nu}\tilde{h}\;+\;\left(1-\frac{\kappa}{k}\right)\eta^{\mu\nu}p^{\alpha}p^{\beta}\tilde{h}_{\alpha\beta}\;$
$\displaystyle+p^{2}\tilde{h}^{\mu\nu}-\left(1+\frac{1}{2k}\right)\big{(}p^{\mu}p^{\alpha}{\tilde{h}^{\nu}}_{\alpha}+p^{\nu}p^{\alpha}{\tilde{h}^{\mu}}_{\alpha}\big{)}\;+\;m_{1}^{2}\,\tilde{h}^{\mu\nu}+m_{2}^{2}\,\eta^{\mu\nu}\tilde{h}=0\;.$
(3.5)

In order to study the propagating DOF, we saturate the EOM (3.5) with
$\eta_{\mu\nu}$ and $e_{\mu\nu}$ (3.4), to get

$\displaystyle\boldsymbol{\eta_{\mu\nu}}:$ $\displaystyle\
\left[(m_{1}^{2}+4m_{2}^{2})-\left(2+\frac{\kappa}{k}(1+4\kappa)\right)p^{2}\right]\>\tilde{h}\;+\;\left(2-\frac{1}{k}(1+4\kappa)\right)\>p^{2}\>e_{\mu\nu}\tilde{h}^{\mu\nu}=0$
(3.6) $\displaystyle\boldsymbol{e_{\mu\nu}}:$ $\displaystyle\
\left[m_{2}^{2}-\frac{\kappa}{k}(1+\kappa)p^{2}\right]\>\tilde{h}\;+\;\left[m_{1}^{2}-\frac{1}{k}(1+\kappa)p^{2}\right]\>e_{\mu\nu}\tilde{h}^{\mu\nu}=0\;.$
(3.7)

From these two equations we deduce that, if $m^{2}_{1}\neq 0$ (we will
consider the case $m^{2}_{1}=0$ later), the only solution is

$\displaystyle\tilde{h}$ $\displaystyle=0$ (3.8) $\displaystyle
e^{\mu\nu}\tilde{h}_{\mu\nu}$ $\displaystyle=0\;,$ (3.9)

since (3.6) and (3.7) form a homogeneous system of two linear equations which
has a non trivial solution only if the determinant of the coefficients matrix
vanishes. For $m^{2}_{1}\neq 0$ this determinant cannot vanish, independently
of the choice of $\kappa$, $k$ and $m_{2}^{2}$. Substituting (3.8) and (3.9)
into the EOM (3.5) we get

$(p^{2}+m_{1}^{2})\tilde{h}_{\mu\nu}-\left(1+\frac{1}{2k}\right)\Big{(}p_{\mu}p^{\alpha}\tilde{h}_{\nu\alpha}+p_{\nu}p^{\alpha}\tilde{h}_{\mu\alpha}\Big{)}=0\;,$
(3.10)

which, saturated with $p^{\nu}$ and using again (3.9), yields

$\left(m_{1}^{2}-\frac{1}{2k}p^{2}\right)p^{\nu}\tilde{h}_{\mu\nu}=0\;.$
(3.11)

The above equation is satisfied if

$p^{2}=2km_{1}^{2}$ (3.12)

or

$p^{\nu}\tilde{h}_{\mu\nu}=0\;,$ (3.13)

but the requirement that the physical masses should not depend on the gauge
parameters, implies that the only allowable solution of (3.11) is (3.13).
This, together with (3.8), gives the five constraints

$\displaystyle\tilde{h}$ $\displaystyle=0$ (3.14) $\displaystyle
p^{\mu}\tilde{h}_{\mu\nu}$ $\displaystyle=0\;,$ (3.15)

which ensure the propagation of five DOF. Conditions (3.14) and (3.15)
inserted into the EOM (3.5) give the massive propagation of $h_{\mu\nu}$

$(p^{2}+m_{1}^{2})\tilde{h}_{\mu\nu}=0\;,$ (3.16)

which, interestingly, does not depend on the mass parameter $m_{2}$. If, on
the other hand, $m^{2}_{1}=0$, the two equations (3.6) and (3.7) have non
trivial solutions only if

$\kappa=\,-1\ \ ;\ \ k=\,-\frac{3}{2}\;,$ (3.17)

which, plugged back into (3.6) and (3.7) with $m^{2}_{1}=0$, yield

$h=0\;.$ (3.18)

Substituting $m^{2}_{1}=0$ and $h=0$ in the EOM (3.5) we notice that all the
dependence on the mass parameters vanishes. The resulting EOM would therefore
describe the propagation of a massless field or, alternatively, the
propagation of a field with a mass dependent on the gauge parameters $k$ and
$\kappa$. Both cases do not represent acceptable descriptions of a LMG theory,
therefore we conclude that it must be

$m_{1}^{2}\neq 0\ .$ (3.19)

To summarize, we showed that the gauge fixed action $S_{LMG}$ (2.33), with the
mass term $S_{m}$ (2.3), displays five DOF, provided that $m^{2}_{1}\neq 0$,
for any value of $m_{2}^{2}$ and for arbitrary gauge parameters $k$ and
$\kappa$.

##  4 Propagators

The momentum space propagator $G_{\mu\nu,\alpha\beta}(p)$ is defined by the
condition

${\Omega_{\mu\nu}}^{\alpha\beta}G_{\alpha\beta,\rho\sigma}=\mathcal{I}_{\mu\nu,\rho\sigma}\;,$
(4.1)

where $\Omega_{\mu\nu,\alpha\beta}(p)$ is the kinetic operator introduced in
(3.2) and $\mathcal{I}_{\mu\nu,\rho\sigma}$ is the rank-4 tensor identity
(3.3). In order to find $G_{\mu\nu,\alpha\beta}(p)$ it is convenient to
introduce the rank-2 projectors $e_{\mu\nu}$ (3.4) and

$d_{\mu\nu}\equiv\eta_{\mu\nu}-e_{\mu\nu}$ (4.2)

which are idempotent and orthogonal

$e_{\mu\lambda}{e^{\lambda}}_{\mu}=e_{\mu\nu},\ \mathrm{}\
d_{\mu\lambda}{d^{\lambda}}_{\nu}=d_{\mu\nu},\ \mathrm{}\
e_{\mu\lambda}{d^{\lambda}}_{\nu}=0\>.$ (4.3)

With these rank-2 projectors we can construct a basis formed by five rank-4
tensors which we collectively denote

$X_{\mu\nu,\alpha\beta}\equiv(A,B,C,D,E)_{\mu\nu,\alpha\beta}$ (4.4)

with the symmetry properties

$X_{\mu\nu,\alpha\beta}=X_{\nu\mu,\alpha\beta}=X_{\mu\nu,\beta\alpha}=X_{\alpha\beta,\mu\nu}\;.$
(4.5)

In terms of the projectors $e_{\mu\nu}$ and $d_{\mu\nu}$ the operators
$X_{\mu\nu,\alpha\beta}$ read

$\displaystyle A_{\mu\nu,\alpha\beta}$
$\displaystyle=\frac{d_{\mu\nu}d_{\alpha\beta}}{3}$ (4.6) $\displaystyle
B_{\mu\nu,\alpha\beta}$ $\displaystyle=e_{\mu\nu}e_{\alpha\beta}$ (4.7)
$\displaystyle C_{\mu\nu,\alpha\beta}$
$\displaystyle=\frac{1}{2}\left(d_{\mu\alpha}d_{\nu\beta}+d_{\mu\beta}d_{\nu\alpha}-\frac{2}{3}d_{\mu\nu}d_{\alpha\beta}\right)$
(4.8) $\displaystyle D_{\mu\nu,\alpha\beta}$
$\displaystyle=\frac{1}{2}\left(d_{\mu\alpha}e_{\nu\beta}+d_{\mu\beta}e_{\nu\alpha}+e_{\mu\alpha}d_{\nu\beta}+e_{\mu\beta}d_{\nu\alpha}\right)$
(4.9) $\displaystyle E_{\mu\nu,\alpha\beta}$
$\displaystyle=\frac{\eta_{\mu\nu}\eta_{\alpha\beta}}{4}\;,$ (4.10)

and have the following properties:

  * •

decomposition of the identity $\mathcal{I}_{\mu\nu,\alpha\beta}$ (3.3) :

$A_{\mu\nu,\alpha\beta}+B_{\mu\nu,\alpha\beta}+C_{\mu\nu,\alpha\beta}+D_{\mu\nu,\alpha\beta}=\mathcal{I}_{\mu\nu,\alpha\beta}\
;$ (4.11)

  * •

idempotency :

$X_{\mu\nu}^{\ \ \rho\sigma}X_{\rho\sigma,\alpha\beta}=X_{\mu\nu,\alpha\beta}\
;$ (4.12)

  * •

orthogonality of $A$, $B$, $C$ and $D$ :

$X_{\mu\nu,\alpha\beta}{{X^{\prime}}^{\alpha\beta}}_{\rho\sigma}=0\ \
\mbox{if}\ (X,X^{\prime})\neq E\ \mbox{and}\ X\neq X^{\prime}\ ;$ (4.13)

  * •

contractions with $E$ :

$\displaystyle A_{\mu\nu,\alpha\beta}{E^{\alpha\beta}}_{\rho\sigma}$
$\displaystyle=\frac{d^{\mu\nu}\eta_{\rho\sigma}}{4}$ (4.14) $\displaystyle
B_{\mu\nu,\alpha\beta}{E^{\alpha\beta}}_{\rho\sigma}$
$\displaystyle=\frac{e^{\mu\nu}\eta_{\rho\sigma}}{4}$ (4.15) $\displaystyle
C_{\mu\nu,\alpha\beta}{E^{\alpha\beta}}_{\rho\sigma}$
$\displaystyle=D_{\mu\nu,\alpha\beta}{E^{\alpha\beta}}_{\rho\sigma}=0\ .$
(4.16)

The kinetic operator $\Omega_{\mu\nu,\alpha\beta}(p)$ (3.2) can be written in
terms of the rank-4 projectors (4.6)-(4.10) :

$\Omega_{\mu\nu,\alpha\beta}=tA_{\mu\nu,\alpha\beta}+uB_{\mu\nu,\alpha\beta}+vC_{\mu\nu,\alpha\beta}+zD_{\mu\nu,\alpha\beta}+wE_{\mu\nu,\alpha\beta}\;,$
(4.17)

where, after a lenghty but straightforward calculation, the coefficients are
given by

$\displaystyle t$
$\displaystyle=\left(\frac{3\kappa}{2k}-1\right)p^{2}+\frac{1}{2}m_{1}^{2}$
(4.18) $\displaystyle u$
$\displaystyle=-\frac{1}{4k}\left(2\kappa+1\right)p^{2}+\frac{1}{2}m_{1}^{2}$
(4.19) $\displaystyle v$ $\displaystyle=\frac{1}{2}(p^{2}+m_{1}^{2})$ (4.20)
$\displaystyle z$ $\displaystyle=-\frac{1}{4k}p^{2}+\frac{1}{2}m_{1}^{2}$
(4.21) $\displaystyle w$
$\displaystyle=-\frac{2\kappa}{k}(1+\kappa)p^{2}+2m_{2}^{2}\;.$ (4.22)

Similarly, we can expand the propagator $G_{\mu\nu,\alpha\beta}(p)$ :

$G_{\mu\nu,\alpha\beta}=\text{
}\hat{t}A_{\mu\nu,\alpha\beta}+\hat{u}B_{\mu\nu,\alpha\beta}+\hat{v}C_{\mu\nu,\alpha\beta}+\hat{z}D_{\mu\nu,\alpha\beta}+\hat{w}E_{\mu\nu,\alpha\beta}\;,$
(4.23)

where $\hat{t}$, $\hat{u}$, $\hat{v}$, $\hat{z}$ and $\hat{w}$ are functions
of the momentum $p$ and depend on the gauge parameters $k$ and $\kappa$
appearing in $S_{gf}$ (2.32) and on the masses $m^{2}_{1}$ and $m^{2}_{2}$ of
the mass term $S_{m}$ (2.3). Solving the defining equation (4.1), we find :

$\displaystyle\hat{t}=$
$\displaystyle\;\frac{2(1+\kappa)(1+4\kappa)p^{2}-2k(m_{1}^{2}+4m_{2}^{2})}{\text{DN}(m_{1},m_{2},k,\kappa,p^{2})}$
(4.24) $\displaystyle\hat{u}=$
$\displaystyle\;\frac{2\big{[}\kappa(1+4\kappa)+2k\big{]}p^{2}-2k(m_{1}^{2}+4m_{2}^{2})}{\text{DN}(m_{1},m_{2},k,\kappa,p^{2})}$
(4.25) $\displaystyle\hat{v}=$ $\displaystyle\;\frac{2}{p^{2}+m_{1}^{2}}$
(4.26) $\displaystyle\hat{z}=$ $\displaystyle\;\frac{-4k}{p^{2}-2km_{1}^{2}}$
(4.27) $\displaystyle\hat{w}=$
$\displaystyle\;\frac{8km_{2}^{2}-8\kappa(1+\kappa)p^{2}}{\text{DN}(m_{1},m_{2},k,\kappa,p^{2})}\;,$
(4.28)

where we used the shorthand notation for the denominator

DN $\displaystyle(m_{1},m_{2},k,\kappa,p^{2})\equiv$
$\displaystyle-2(1+\kappa)^{2}p^{4}+\Big{[}(1+2\kappa+4\kappa^{2}+2k)m_{1}^{2}+(3+2k)m_{2}^{2}\Big{]}p^{2}-4km_{1}^{2}m_{2}^{2}-km_{1}^{4}\;.$
(4.29)

The propagator (4.23) displays poles that depend on the gauge parameters $k$
and $\kappa$ and might even be tachyonic. This does not come as a surprise,
given that the theory describes the dynamics of a rank-2 symmetric tensor
field, and the pole structure of its propagator is more complicated than the
usual scalar, spinor or vector cases. Neither it should be seen as a problem,
since in the previous Section we proved that only five of the ten components
of $h_{\mu\nu}$ represent independent DOF, which satisfy the massive wave
equation of the Klein-Gordon type (3.16) which depends on the mass $m_{1}^{2}$
only, in agreement with the FP theory. Hence, looking at the whole propagator
is neither helpful nor correct in order to identify the physical pole in this
case, since we may allow for non physical poles located in non physical
sectors of the propagator. Rather, what should be done in order to select the
physical pole, is to look to the observables related to the propagator (4.23).
An important example is the one already considered in the analysis of the vDVZ
discontinuity: the scattering amplitude of light and a massive object,
mediated by the gravitational interaction, which is responsible for the
observed time delay in gravitational lensing. Formally, this observable can be
traced back to the more general interaction amplitude of two conserved energy-
momentum tensors $T^{(1)}_{\mu\nu}$ and $T^{(2)}_{\mu\nu}$, of which the one
corresponding to light (which in the following we choose to be
$T^{(2)}_{\mu\nu}$) is traceless :

$\tilde{T}^{(1)}_{\mu\nu}G^{\mu\nu,\alpha\beta}\tilde{T}^{(2)}_{\alpha\beta}\;.$
(4.30)

Substituting the propagator (4.23) into (4.30), we get

$\tilde{T}^{(1)}_{\mu\nu}\Big{(}\hat{t}A^{\mu\nu,\alpha\beta}+\hat{u}B^{\mu\nu,\alpha\beta}+\hat{v}C^{\mu\nu,\alpha\beta}+\hat{z}D^{\mu\nu,\alpha\beta}+\hat{w}E^{\mu\nu,\alpha\beta}\Big{)}\tilde{T}^{(2)}_{\alpha\beta}\
.$ (4.31)

The above expression contains contractions of the tensor projectors
(4.6)-(4.10) with $\tilde{T}^{(1)}_{\mu\nu}$ and $\tilde{T}^{(2)}_{\mu\nu}$.
These are greatly simplified thanks to the fact that the energy-momentum
tensors are conserved:
$p^{\nu}\tilde{T}_{\mu\nu}^{(1)}=p^{\nu}\tilde{T}_{\mu\nu}^{(2)}=0$. In
particular:

$\displaystyle\tilde{T}^{(1)}_{\mu\nu}A^{\mu\nu,\alpha\beta}\tilde{T}^{(2)}_{\alpha\beta}=\frac{1}{3}\tilde{T}^{(1)}_{\mu\nu}\eta^{\mu\nu}\eta^{\alpha\beta}\tilde{T}^{(2)}_{\alpha\beta}$
(4.32)
$\displaystyle\tilde{T}^{(1)}_{\mu\nu}B^{\mu\nu,\alpha\beta}\tilde{T}^{(2)}_{\alpha\beta}=0$
(4.33)
$\displaystyle\tilde{T}^{(1)}_{\mu\nu}C^{\mu\nu,\alpha\beta}\tilde{T}^{(2)}_{\alpha\beta}=\frac{1}{2}\tilde{T}^{(1)}_{\mu\nu}(\eta^{\mu\alpha}\eta^{\nu\beta}+\eta^{\mu\beta}\eta^{\nu\alpha}-\frac{2}{3}\eta^{\mu\nu}\eta^{\alpha\beta})\tilde{T}^{(2)}_{\alpha\beta}$
(4.34)
$\displaystyle\tilde{T}^{(1)}_{\mu\nu}D^{\mu\nu,\alpha\beta}\tilde{T}^{(2)}_{\alpha\beta}=0$
(4.35)
$\displaystyle\tilde{T}^{(1)}_{\mu\nu}E^{\mu\nu,\alpha\beta}\tilde{T}^{(2)}_{\alpha\beta}=\frac{1}{4}\tilde{T}^{(1)}_{\mu\nu}\eta^{\mu\nu}\eta^{\alpha\beta}\tilde{T}^{(2)}_{\alpha\beta}\
.$ (4.36)

Eqs. (4.32)-(4.36) can be further simplified by using the fact that
$\tilde{T}_{\mu\nu}^{(2)}$ is traceless
($\eta^{\mu\nu}\tilde{T}_{\mu\nu}^{(2)}=0$), which means that the only non-
vanishing contraction left is

$\tilde{T}^{(1)}_{\mu\nu}C^{\mu\nu,\alpha\beta}\tilde{T}^{(2)}_{\alpha\beta}=\frac{1}{2}\tilde{T}^{(1)}_{\mu\nu}(\eta^{\mu\alpha}\eta^{\nu\beta}+\eta^{\mu\beta}\eta^{\nu\alpha})\tilde{T}^{(2)}_{\alpha\beta}\
.$ (4.37)

Therefore the scattering amplitude (4.30) reduces to

$\tilde{T}^{(1)}_{\mu\nu}\Big{(}\hat{v}\;\mathcal{I}^{\mu\nu,\alpha\beta}\Big{)}\tilde{T}^{(2)}_{\alpha\beta}\;,$
(4.38)

according to which only the pole contained in the coefficient $\hat{v}$ (4.26)
plays a physical role. Reassuringly, that pole is

$p^{2}=-m_{1}^{2}\ ,$ (4.39)

which confirms that the theory describes a graviton with mass $m^{2}_{1}$, in
agreement with the massive wave equation (3.16), which was obtained by
studying the EOM deriving from the action $S_{LMG}$ (2.33), and also with the
FP theory, (2.12). Notice that the same argument may be repeated to show that
also the scattering amplitude between two radiation-like objects, both
described by conserved and traceless energy-momentum tensors, isolates (4.39)
as the unique physical pole of the propagator (4.23).

##  5 Absence of the vDVZ discontinuity

In Section 2.2 we reviewed the vDVZ discontinuity, which concerns the mismatch
between the gravitational interaction of two energy-momentum tensors in the
pure massless theory described by the action $S_{LG}$ (2.1) and in the
massless limit of FP theory $S_{FP}$ (2.6). In [32] it has been shown that in
a particular gauge (namely the $k=-\frac{1}{2}$, $\kappa=-\frac{1}{2}$
harmonic one) the theory described by the action $S_{LMG}$ (2.33) is free of
the vDVZ discontinuity. In this Section we generalize the result of [32] to
all possible gauge choices, $i.e.$ for every value of the gauge parameters $k$
and $\kappa$. With the propagator $G_{\mu\nu,\alpha\beta}$ (4.23) derived from
the gauge fixed action $S_{LMG}$ (2.33), hence for generic $k$ and $\kappa$
gauge parameters, we can compute the interaction amplitudes between two non-
relativistic conserved energy-momentum tensors ${T_{\mu\nu}^{(1)}}$ and
${T_{\mu\nu}^{(2)}}$ and compare them to the corresponding LG amplitudes,
(2.18) and (2.19). As explained in Section 2.2, the interaction of the
graviton field with an external energy-momentum tensor is obtained by means of
the interaction term $S_{int}$ (2.17). To distinguish the gauge fixed theory
$S_{LMG}$ (2.33) from the LG and FP theories, we denote its coupling constant
with $\lambda_{LMG}$. Following the steps described in [19], if neither
${T_{\mu\nu}^{(1)}}$ nor ${T_{\mu\nu}^{(2)}}$ are traceless, which corresponds
to the scattering between two massive bodies, the resulting amplitude in the
massless limit ($m^{2}_{1},m^{2}_{2})\rightarrow 0$ of the theory is

$\lambda^{2}_{LMG}\>\tilde{T}^{(1)}_{\mu\nu}G^{\mu\nu,\alpha\beta}\tilde{T}^{(2)}_{\alpha\beta}=\lambda^{2}_{LMG}\>\tilde{T}^{(1)}_{00}\tilde{T}^{(2)}_{00}\frac{1}{p^{2}}\;,$
(5.1)

whereas, if ${T_{\mu\nu}^{(2)}}$ is traceless, corresponding to the scattering
between light and a massive body, like in the gravitational lensing, we have

$\lambda^{2}_{LMG}\>\tilde{T}_{\mu\nu}^{(1)}G^{\mu\nu,\alpha\beta}\tilde{T}^{(2)}_{\alpha\beta}=\lambda^{2}_{LMG}\>\tilde{T}^{(1)}_{00}\tilde{T}^{(2)}_{00}\frac{2}{p^{2}}\;.$
(5.2)

It is straightforward to see that if we set

$\lambda_{LMG}=\lambda_{LG}$ (5.3)

the LG amplitudes (2.18) and (2.19) exactly match (5.1) and (5.2)
respectively, which proves the absence of the vDVZ discontinuity in a gauge
independent way. We stress that, although the massive propagator
$G_{\mu\nu,\alpha\beta}$ (4.23) depends on the gauge parameters $k$ and
$\kappa$ through the coefficients (4.24)-(4.28), the massless limit of the
interaction amplitudes (5.1) and (5.2) is gauge independent and coincides with
the LG prediction. This is related to the fact that the massless limit of the
action $S_{LMG}$ (2.33) is the LG gauge fixed action

$S_{LG}+S_{gf}\ ,$ (5.4)

where $S_{LG}$ and $S_{gf}$ are given by (2.1) and (2.32), respectively.
Hence, in the massless limit, the propagators of the two theories coincide.
This is the basic reason of the absence of the vDVZ discontinuity. In other
words, the vDVZ discontinuity is a direct consequence of the structure of the
FP action, where the mass term $S_{m}$ (2.3) is added directly to the
invariant action $S_{LG}$ (2.1), therefore acting effectively as a gauge
fixing term. Without the mass term, the invariant action does not have a
propagator. Therefore, the mass parameters, which also play the role of gauge
fixing parameters, cannot be physical. In the massless limit, the FP theory is
not dynamical ($i.e.$ does not have a propagator), and therefore it is not
surprising that a discontinuity arises. Throughout this paper, we repeatedly
claimed that the vDVZ discontinuity may be related to the nature of the FP
mass term as a gauge fixing term. The massive gravity propagator in (2.15) is
not well defined in the zero mass limit due to the lack of gauge fixing in the
massless theory. However, the vDZV discontinuity appears in the analysis of
the propagator contracted with energy-momentum tensors like in (4.30). This
quantity is gauge invariant in linearized GR (for conserved sources). Hence
one might expect that the ill defined terms in the zero mass limit of massive
FP propagator drop out of equations like (4.30) which contribute to the vDVZ
discontinuity (this quantity indeed remains finite in the zero mass limit).
The apparent contradiction between this observation and the explanation
provided in this paper is explained by observing that when we say that the
mass term $S_{m}$ (2.3) at the FP point (2.5) has the role of a gauge fixing
in the FP theory, we are only referring to the fact that it enables us to
invert the quadratic part of the action. In fact, not all quadratic terms
which make it possible to find a propagator are also legitimate gauge fixing
terms. The gauge fixing procedure is much more than this. Indeed, it consists
in limiting the functional integral in order to choose one representative for
each gauge orbit (modulo Gribov copies). This is realized through the highly
non trivial $\Phi\Pi$ trick. In the case of an abelian theory such as
linearized gravity, where the ghost sector decouples, the $\Phi\Pi$ gauge
fixing is realized by adding to the action the square of a functional
$F^{\mu}[h_{\alpha\beta}]$, where $F$ must carry the same number of Lorentz
indices as the gauge parameter of the theory, defined in our case in (2.2).
Since the mass term cannot be written as the square of any such functional, it
follows that $S_{m}$ is not a true gauge fixing term and therefore no
contradiction arises when comparing the linearized GR and FP propagators.

##  6 Summary of results and Discussion

In this paper we presented a gauge field theory of a massive symmetric tensor
field describing a massive spin-2 particle. We summarize here the main
features which allow to interpret this theory as an alternative to the FP
theory of LMG.

  1. 1.

finite massless limit
The theory described by the action $S_{LMG}$ (2.33) yields the propagator
$G_{\mu\nu,\alpha\beta}$ (4.23) which is regular in the limits
($m_{1}^{2},m_{2}^{2})\rightarrow 0$ ;

  2. 2.

five massive DOF
In Section 3 we showed that only five of the ten components of the massive
symmetric tensor $h_{\mu\nu}(x)$ actually represent DOF, as required for a
massive graviton . The remarkable fact is that the mass associated with the
propagation of the five DOF is $m^{2}_{1}$, while the value of $m^{2}_{2}$ is
irrelevant, under this respect. Although obtained in a quite different way,
this result is in agreement with the FP theory, which is characterized by one
mass parameter only ;

  3. 3.

gauge independence
In gauge field theory, any claim concerning physical properties should not
depend on a particular gauge choice. In our case, this requirement is met in
the determination of the physical DOF of the theory: none of the steps leading
to the five constraints represented by (3.14) and (3.15) rely on particular
values of the gauge parameters $k$ and $\kappa$ ;

  4. 4.

continuity with LG
Any candidate theory for LMG must display a finite massless limit, which
implies a non divergent propagator, and, in this limit, it should also provide
physical predictions in agreement with GR. Physically, LMG should represent
only a small correction of GR, therefore the mass term (2.3) should be seen as
a perturbation. In particular, the effects of a mass of the graviton should
become relevant only at very large distances, while at smaller scales GR
predictions must be restored. As discussed in Section 2.2, a serious flaw of
the FP theory of LMG is the vDVZ discontinuity [17, 18]. This well known issue
is fixed by the Vainshtein mechanism [11, 20, 25], which recovers the non
linearities of GR in order to shield the effect of the extra scalar DOF
introduced by the Stückelberg formalism [33] in the massless limit of the FP
theory. Our gauge fixed theory of LMG restores the continuity with GR in a
more natural way, without the introduction of additional fields, cutoffs or
non linearities of the theory. In [32] it has been shown that in the
particular $(k=\kappa=-\frac{1}{2})$ harmonic gauge, the theory described by
the action $S_{LMG}$ (2.33) does not display the vDVZ discontinuity. In
Section 5 this result has been generalized to any gauge choice. The fact that
the gauge fixed massive theory described by the action $S_{LMG}$ (2.33) is not
affected by the vDVZ discontinuity with GR encourages to believe that the way
of approaching LMG presented in this paper is correct. Moreover, although the
absence of the vDVZ discontinuity is a specific test and it does not prove in
general that every measurable quantity is continuous with GR, the way the
theory itself was constructed might imply this result. In fact, the evaluation
of any observable in quantum field theory needs a gauge fixing in order to
define a propagator, which is guaranteed by the $\Phi\Pi$ procedure, adopted
in this paper. The key feature of our approach is that the massive action
$S_{LMG}$ (2.33) in the zero-mass limit becomes exactly the $\Phi\Pi$ gauge
fixed LG action, $i.e.$ the one we would have used to calculate the propagator
and every other quantity in massless LG as well. Therefore, while it does not
come as a surprise that the gravitational coupling between non relativistic
matter and light turns out to be continuous with GR, we may also infer that
every other observable is, indeed, continuous with LG, in the massless limit.
Moreover, the striking difference between the FP theory of LMG, where the vDVZ
discontinuity is present, and the massive $\Phi\Pi$ gauge fixed theory
described in this paper, where it is absent, gives evidence of the fact that,
as anticipated, the FP theory is not a sub-case of our theory despite the fact
that our approach contains the FP tuning (2.5) ;

  5. 5.

role of the mass parameters
We stress as a remarkable fact that, although by means of a completely
different approach, we recover here the main result of the FP theory, $i.e.$
that only the mass parameter $m^{2}_{1}$ associated to $h^{\mu\nu}h_{\mu\nu}$
in the action $S_{m}$ (2.3) appears in the propagation of the physical DOF
through the massive wave equation (3.16), which is the starting request (2.25)
of the original FP theory [16]. Therefore, the mass of the graviton should be
identified by $m^{2}_{1}$.

## Acknowledgments

We gratefully acknowledge Gianluca Gemme for his phenomenological hints and
enlightening observations.

## References

  * [1] A. G. Riess et al. [Supernova Search Team], Astron. J. 116 (1998), 1009-1038 doi:10.1086/300499.
  * [2] S. Perlmutter et al. [Supernova Cosmology Project], Astrophys. J. 517 (1999), 565-586 doi:10.1086/307221.
  * [3] P. Ade et al. [Planck], Astron. Astrophys. 594 (2016), A13 doi:10.1051/0004-6361/201525830.
  * [4] S. Weinberg, Rev. Mod. Phys. 61 (1989), 1-23 doi:10.1103/RevModPhys.61.1 .
  * [5] C. Wetterich, Nucl. Phys. B 302 (1988), 668-696 doi:10.1016/0550-3213(88)90193-9.
  * [6] R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. 80 (1998), 1582-1585 doi:10.1103/PhysRevLett.80.1582.
  * [7] C. Armendariz-Picon, V. F. Mukhanov and P. J. Steinhardt, Phys. Rev. Lett. 85 (2000), 4438-4441 doi:10.1103/PhysRevLett.85.4438.
  * [8] T. Damour and A. M. Polyakov, Nucl. Phys. B 423 (1994), 532-558 doi:10.1016/0550-3213(94)90143-0.
  * [9] L. Amendola, Phys. Rev. D 62 (2000), 043511 doi:10.1103/PhysRevD.62.043511.
  * [10] M. Bonici and N. Maggiore, Eur. Phys. J. C 79 (2019) no.8, 672 doi:10.1140/epjc/s10052-019-7198-1.
  * [11] K. Hinterbichler, Rev. Mod. Phys. 84 (2012) 671 doi:10.1103/RevModPhys.84.671.
  * [12] S. F. Hassan and R. A. Rosen, JHEP 07 (2011), 009 doi:10.1007/JHEP07(2011)009.
  * [13] G. D’Amico, C. de Rham, S. Dubovsky, G. Gabadadze, D. Pirtskhalava and A. Tolley, Phys. Rev. D 84 (2011), 124046 doi:10.1103/PhysRevD.84.124046.
  * [14] J. Gott, III, M. Juric, D. Schlegel, F. Hoyle, M. Vogeley, M. Tegmark, N. A. Bahcall and J. Brinkmann, Astrophys. J. 624 (2005), 463 doi:10.1086/428890.
  * [15] B. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. D 100 (2019) no.10, 104036 doi:10.1103/PhysRevD.100.104036.
  * [16] M. Fierz and W. Pauli, Proc. Roy. Soc. Lond. A 173 (1939) 211. doi:10.1098/rspa.1939.0140.
  * [17] H. van Dam and M. J. G. Veltman, Nucl. Phys. B 22 (1970) 397. doi:10.1016/0550-3213(70)90416-5.
  * [18] V. I. Zakharov, JETP Lett. 12 (1970) 312 [Pisma Zh. Eksp. Teor. Fiz. 12 (1970) 447].
  * [19] V. A. Rubakov and P. G. Tinyakov, Phys. Usp. 51 (2008) 759 doi:10.1070/PU2008v051n08ABEH006600.
  * [20] A. Vainshtein, Phys. Lett. B 39 (1972), 393-394 doi:10.1016/0370-2693(72)90147-5.
  * [21] D. Boulware and S. Deser, Phys. Rev. D 6 (1972), 3368-3382 doi:10.1103/PhysRevD.6.3368.
  * [22] C. de Rham and G. Gabadadze, Phys. Rev. D 82 (2010), 044020 doi:10.1103/PhysRevD.82.044020.
  * [23] C. de Rham, G. Gabadadze and A. J. Tolley, Phys. Rev. Lett. 106 (2011), 231101 doi:10.1103/PhysRevLett.106.231101.
  * [24] S. Hassan and R. A. Rosen, Phys. Rev. Lett. 108 (2012), 041101 doi:10.1103/PhysRevLett.108.041101.
  * [25] C. de Rham, Living Rev. Rel. 17 (2014), 7 doi:10.12942/lrr-2014-7.
  * [26] A. Blasi and N. Maggiore, Class. Quant. Grav. 34 (2017) no.1, 015005 doi:10.1088/1361-6382/34/1/015005.
  * [27] M. Fierz, Helv. Phys. Acta 12 (1939), 3-37.
  * [28] B. Lautrup, Kong. Dan. Vid. Sel. Mat. Fys. Med. 35 (1967) no.11 NORDITA-214.
  * [29] N. Nakanishi, Prog. Theor. Phys. 35 (1966), 1111-1116 doi:10.1143/PTP.35.1111.
  * [30] A. Blasi and N. Maggiore, Eur. Phys. J. C 77 (2017) no.9, 614 doi:10.1140/epjc/s10052-017-5205-y.
  * [31] L. D. Faddeev and V. N. Popov, Phys. Lett. 25B (1967) 29. doi:10.1016/0370-2693(67)90067-6.
  * [32] G. Gambuti and N. Maggiore, Phys. Lett. B 807 (2020), 135530 doi:10.1016/j.physletb.2020.135530.
  * [33] E. C. G. Stueckelberg, Helv. Phys. Acta 11, 225 (1938). doi:10.5169/seals-110852.