#  Advisory Tool for Managing Failure Cascades in Systems with Wind Power
††thanks: We thank MIT UROP, MITEI, and the NSF EAGER project #2002570 for
funding, and Dan Wu, Xinyu Wu, and Miroslav Kosanic for discussions.

Siyu Liu  Massachusetts Institute of Technology
Cambridge, MA, USA
<EMAIL_ADDRESS>Marija Ilić, IEEE Life Fellow  Massachusetts Institute of
Technology
Cambridge, MA, USA
<EMAIL_ADDRESS>
###### Abstract

This paper concerns the resilience of systems with wind power upon wind
reduction by evaluating the potential of corrective actions, such as
generation and load dispatch, on minimizing the effects of transmission line
failures. Three functions (grid, consumer-centric loss, and resilience impact)
are used to statistically evaluate the criticality of initial contingent
failures and wind reductions. Our model is learned with Monte Carlo, convex
optimization, and adaptive selection, illustrated on the IEEE-30 and IEEE-300
bus systems with both AC and DC models. We highlight the impact of wind
reductions and propose physically implementable solutions.

###### Index Terms:

wind power, cascade failure, influence model

##  I Introduction

Modern power systems are prone to many unpredictable component failures. Past
events have shown that large scale blackouts are typically results of
sequential failures of transmission lines, called failure cascades. Cascades
tend to evolve quickly, leaving only up to 15 minutes for the system operators
to take corrective actions before the failure propagates[5]. Wind power
integration introduces additional unpredictability and potential congestion-
induced failures. Due to the fast-evolving nature of wind power and failure
propagation, it is particularly important to understand the cascade patterns
and their relations to sudden wind reduction, in order to advice system
operators during such extreme events. Failure cascades are extremely
computationally expensive to analyze due to the large number of nonlinear
relations. Most of the early work concerns analysis of cascades and holds a
pessimistic outlook at wind penetration due to its unpredictability[6].

In this paper, we go beyond analysis and explore corrective actions to
minimize the effects of equipment failures that may be exacerbated by wind
penetration. This is done by using probabilistic methods for deriving
statistical information about the most effective corrective actions, such as
generation re-dispatch and preemptive load shed. These are needed to prevent
cascading failures as the actual events occur in on-line operations; they can
also be used to enhance today’s industry manuals. Most of the studies concern
wind penetration risk by solving a DC power flow model, which under-estimates
the effects of failures and does not always provide physically implementable
solutions due to reactive power and voltage constraints [12], [20]-[23]. To
overcome the high complexity of power flow analysis, researchers have also
sought various statistical methods, including random graphs, branching
processes, and flow dynamics models — all typically embed non-trivial
constraint relaxations [16]-[18]. However, very few studies consider
physically-implementable generation re-dispatch [14, 20] or load shedding.
Those that do are only done for the purpose of minimizing grid-centric cost
[9, 10, 11]. Others, including [12] and [13], by implement a constant factor
load shed algorithm without any “smart scheduling.” To the best of our
knowledge, there is so far no statistical model for load shed prediction and
management. To remedy these shortcomings, we adopt a flow-free approach using
the influence model (IM). The IM is a Markovian model that, given the network
profile at each time step, computes the link failure and load shed probability
at all links and buses. It is straightforward to construct and fast to
implement, reducing the prediction tasks to matrix multiplication and
completely eliminating the burden of flow computation. We borrow insights from
[13] and [16] to train the model using Monte Carlo, convex optimization, and
adaptive selection. Our work drastically augments the scope of previous
studies in both methodology, results, and applicability.

In this paper, Section II briefly reviews the IM, and Section III outlines our
simulation process. Section IV proposes three metrics to evaluate various
corrective actions and underlines the significance of our findings for systems
with wind power. Section V summarizes the prediction accuracy and time
complexity of the IM. Section VI demonstrates a comparable effectiveness on a
large-scale system. Finally, Section VII introduces its use as an advisory
tool to system operators.

##  II The Influence Model for Loss Prediction

The IM is a Markovian-like model whose dynamics are described by the state
variable transitions. We propose two IM, first for link failure prediction
through matrix $D,$ and second for load shed prediction through matrix $E.$
Both models operate on a $(N_{br}\times 1)$ network state vector $s$ that
stores the status of all links in binary, where $N_{br}$ is the number of
transmission lines (branches) in the network. Given the $i$-th link,
$s_{i}[t]=1$ indicates that link $i$ is alive at time $t,$ and $s_{i}[t]=0$
indicates that it has failed.

###  II-A Matrix D for link failure prediction

The link failure IM predicts the subsequent network state $s[t+1]$ given the
current state $s[t]$ and trained parameters, which we define as follows.

  * •

Transition probability matrices $A^{01}$ and $A^{11},$ both of size
$(N_{br}\times N_{br}),$ where

$\displaystyle A^{11}_{ji}:=\mathbb{P}(s_{i}[t+1]=1|s_{j}[t]=1),$ (1)
$\displaystyle A^{01}_{ji}:=\mathbb{P}(s_{i}[t+1]=1|s_{j}[t]=0).$ (2)

  * •

The weighted influence matrix $D$ of size $(N_{br}\times N_{br}),$ where the
entry $d_{ij}$ represents the proportional effect from the link $j$ to $i.$ It
can be interpreted as the ratio of the influence from link $j$ to $i$ among
all links over $i.$

  * •

The bisection threshold vector $\epsilon$ of size $(N_{br}\times 1).$ We
determine $\epsilon_{i}$ for each initial failure profile by examining the
$s_{i}$ sequence in all samples.

$A^{01}$ and $A^{11}$ are obtained through a Monte Carlo experiment, and the
$D$ entries are obtained by solving an optimization problem as outlined in
[13].

The transition probabilities from $A^{11}$ and $A^{01}$ weighted by the
influence factors in $D$ gives the prediction $\widetilde{s_{i}}[t+1]:$

$\widetilde{s_{i}}[t+1]=\sum_{j=1}^{N_{br}}d_{ji}\left(A^{11}_{ji}s_{j}[t]+A^{01}_{ji}(1-s_{j}[t])\right),$
(3)

where we predict link remaining healthy when
$\widetilde{s_{i}}[t+1]\geq\epsilon_{i}.$

###  II-B Matrix E for load shed prediction

The link failure IM predicts the $(N\times 1)$ load binary vector $l[t]$,
where $l_{i}[t]=1$ indicates full service, and $l_{i}[t]=0$ indicates load
reduction. The prediction is done based on the network state $s[t]$ and
trained parameters, defined as follows.

  * •

Transition probability matrices $B^{01}$ and $B^{11}$ of sizes $(N_{br}\times
N)$ that define the weighted influences from links to buses, where $N$ is the
number of buses. $B^{01}$ and $B^{11}$ defined as:

$\displaystyle B^{11}_{ji}:=\mathbb{P}(l_{i}[t]=1|s_{j}[t]=1),$ (4)
$\displaystyle B^{01}_{ji}:=\mathbb{P}(l_{i}[t]=1|s_{j}[t]=0).$ (5)

  * •

The weighted influence matrix $E$ of size $(N\times N_{br})$ defines the
weighted influences from links to buses. Each entry $e_{ij}$ denotes the
proportional influence of link $j$ on bus $i.$

  * •

The bisection threshold vector $\delta$ of size $(N\times 1).$ We determine
$\delta_{i}$ for each initial contingency profile by examining all samples
where load shed has occurred at bus $i.$

$B^{01},B^{11},$ and $E$ are obtained through a Monte Carlo experiment and
convex optimization with similar to [13].

The probability of the system being able to serve full load at bus $i$ is
calculated by a weighted sum of the influence from all links, using the
trained parameters in $B^{01},$ $B^{11},$ and $\delta_{i}:$

$\widetilde{l_{i}}[t]=\sum_{j=1}^{N_{br}}e_{ij}\left(B^{11}_{ji}s_{j}[t]+B^{01}_{ji}(1-s_{j}[t])\right),$
(6)

where we predict full service when $\widetilde{l_{i}}[t]\geq\delta_{i}.$

The run time for building the IM is dominated by the optimization step to
obtain $D$ and $E,$ which takes $O(N^{2}N_{br}^{2}).$

##  III Sample Pool Generation

To examine the effects of sudden wind reduction, we first simulate the network
under normal conditions. We introduce a random link failure under normal
conditions where the loading level (base load - wind power) is nominal. If the
failure does not lead to a complete blackout, we continue the simulation by
introducing a wind reduction while the system is operating under $(N-2)$
conditions. The reduction level ranges from $10\%$ to $70\%$ of the base load,
causing the net loading level to rise up to $\times 1.7$ the original loading
level. We analyze the additional network congestion and failures caused by
this load increase. Fig. 1 provides a visualization to this process.

Figure 1: Wind Reduction. Before wind power reduction, the net system load is
at $\times.9$ base load. A $30\%$ wind reduction causes the net system load to
rise to $\times 1.2$ base load, causing the second round of cascade failures.

As there is no standard oracle for assessing failure cascades at present, we
base our experiments on the CFS oracle proposed in [7]. The CFS oracle is
similar to the short-term OPA oracle, except that line outages are treated
deterministically and it does not apply optimal re-dispatch during a failure
cascade. Instead, it only sheds load or curtails generation if system-wide
power mismatch occurs. For samples where no corrective actions are taken, we
follow the CFS oracle exactly, and for sample where corrective actions are
applied, we follow a relaxed version of the CFS oracle by allowing re-dispatch
during the cascade. This relaxation is realistic, as the time between two
failures can be as long as 15 minutes to allow the re-dispatch [5]. In all our
experiments, we initialize the network as fully functional, randomly select
two initial contingencies, and determine the cascade sequence following the
oracle. Long term thermal condition is used when all links are fully
functional, and it changes to short term thermal conditions once failures
occur (which we assume to be $1.05\times$ long term). After each failure, we
solve the DC/AC PF/OPF problem using the MATPOWER Toolbox [8]. Our three sets
of experiments are defined with parameters as follows.

Experiment 1: No corrective action. In this experiment, we simply record the
network status and loading levels at each bus at each step of the cascade
without any corrective actions.

Experiment 2: Generation re-dispatch for full service. We re-dispatch
generation whenever new link failures occur by solving for OPF under both
uniform generation cost and bus-specific generation cost provided by [8]. We
aim to serve all loads in full and only shed load uniformly in scale when
unable to serve in full.

Experiment 3: Generation re-dispatch (smart scheduling). This experiment
resembles Experiment 2, except that instead of aiming for full service, we
find the OPF solution that minimizes cost of shedding load, which we assume to
be either uniform or priority-based. Notably, no links fail in this
experiment, as the optimization step observes link constraints and maximum
service as part of the optimization objective.

Our experiments are done on the IEEE-30 system with initial loading being $c$
times the test case from [8]. The $c$ ranges from $0.9$ to $1.8$ in $0.1$
increments.

##  IV Effect of Corrective Actions

We examine the effects of corrective actions under unexpected wind reduction
under DC and AC models. We propose two loss functions to quantify grid-centric
and consumer-centric loss of each cascade sequence, as well as a resilience
impact function to evaluate the loss given the base load and wind reduction in
Section IV-A – Section IV-E analyze the effects of different corrective
actions assessed by these functions and system-wide structural patterns.

###  IV-A Grid-Centric Loss

For each cascade sample, grid-centric loss is defined as

$G(p)=\sum_{b=1}^{N_{br}}C(b)\cdot e^{-0.2t_{b}},$ (7)

where $G(p)$ is the link failure loss for initial network profile $p,$ $C(b)$
the cost on branch $b,$ proportional to its maximum thermal capacity, and
$t_{b}$ the life time of $b.$ The discounting factor $e^{-0.2t_{b}}$ is to
penalize early failures.

In Experiments 1 & 2, links fail more frequently and earlier on in the cascade
at higher initial loading levels. Even in the only two loading levels where
Experiment 2 only successfully initializes, the loss of link failure is much
greater than that in Experiment 1. This demonstrates that PF models
underestimates failure sizes. There is no observable difference between re-
dispatching with actual or uniform generation cost in Experiments 2 & 3. Fig.
6 illustrates these results.

###  IV-B Consumer-Centric Loss

For each cascade sample, consumer-centric loss is defined with the formula as
follows.

$L(p)=\sum_{l=1}^{N}\sum_{t=1}^{T_{k}-1}C(l)\cdot LS_{l}(t)e^{-0.2t},$ (8)

where $L(p)$ is the load shed loss for initial network profile $p.$ $C(l)$ the
load priority, and $LS_{l}(t)$ the amount of load shed between time steps $t$
and $t+1$ at bus $l.$ The expression is similarly time – discounted by
$e^{-0.2t}.$

Our experiments yield one notable finding. If corrective actions are taken
promptly, we may preserve infrastructure integrity in full without significant
service reduction. In particular, load shed loss is minimized under Experiment
3’s smart scheduling, when we run OPF with cost-based load shed. As such, the
flow on all links are within their capacities and no cascade incurs. As
illustrated in Fig. 6, comparing the load shed across all three experiments,
Experiment 3 reduces the consumer-centric loss to much less than that in
Experiments 1 & 2. The passive, emergency load shed in Experiment 2 incurs the
greatest loss. Whether the cost of generation varies at different buses does
not induce significant difference in the load shed loss.

Figure 2: Link Fail Loss (DC).

Figure 3: Link Fail Loss (AC).

Figure 4: Load Shed Loss (DC).

Figure 5: Load Shed Loss (AC).

Figure 6: Grid-centric and consumer-centric loss over under various corrective
actions for DC and AC models.

###  IV-C Resilience Impact

We propose the following equation to measure network resilience, where
$R(p,\Delta w)$ is network resilience for initial profile $p,$ wind reduction
$\Delta w,$ and $p^{\prime}$ the network profile during the wind reduction
when failure starts propagating.

$\displaystyle R(p,\Delta w)=R^{G}(p,\Delta w)+R^{L}(p,\Delta w),$ (9)
$\displaystyle R^{G}(p,\Delta w)=G(p^{\prime})-G(p)$ (10) $\displaystyle
R^{L}(p,\Delta w)=L(p^{\prime})-L(p),$ (11)

$R^{G}(p,\Delta w)$ and $R^{L}(p,\Delta w)$ correspond to grid-centric and
consumer-centric resilience, respectively.

###  IV-D Corrective Action Analysis

Our experiment finds that, under certain net loading levels, full service is
impossible even without contingencies, as our solver fails to converge. The
problem arises under scenarios at high loading levels or non-uniform shedding
priorities. In Experiment 2, DC OPF and AC OPF fail to converge for loading
levels greater than $1.3\times$ and $1.0\times$ the default loading level,
respectively. This signifies the necessity for smart scheduling. The rest of
this section presents highlights of the analysis where initialization
succeeds.

We find that PF solutions are frequently not physically implementable. This
can be observed in the initial voltages under AC PF in Fig. 7. Bus voltages
drops significantly as loading increases, falling outside the $(0.95,1.05)$
constraint.

Figure 7: Initial bus voltages solved with AC power flow.

In all three experiments, AC models uncover more link failures and greater
load shed than their DC model counterparts. In particular, in Experiment 1,
losses on link failure is only slightly greater in AC than DC models, but this
difference is much greater in Experiment 2. The high levels of losses from AC
solutions renders the DC approximation insufficient to study failure cascades,
as it underestimates the severity of contingencies. Findings about the load
shed losses present an especially optimistic outlook. In Experiment 1 & 2, AC
models render much higher load shed than DC simulations. However, in
Experiment 3, when cost-based flexible load shed is implemented, this load
shed cost is no longer so significant. As AC simulation results are physically
implementable, this result shows that we can serve close to full demand
without causing congestion with optimal re-dispatch for both generation and
load. This is especially promising in practice.

To evaluate the resilience impact $R(p,\Delta w),$ it suffices to know
$G(p),L(p),G(p^{\prime}),L(p^{\prime}).$ We find that DC models significantly
underestimate the impact of wind reduction for both resilience measures. We
present the analysis of one particular scenario — when the initial net load
(load - wind power) is the system load, and examine the effects of different
corrective actions upon wind power reduction up to $70\%$ of the system load.
Our experiments found that, as a result of non-convergence, blackout happens
when wind reduction is $\geq 40\%$ under DC models Experiment 2, and any level
of wind reduction will cause blackout under AC models. Fig. 10 presents the
grid-centric ($R^{G}$) and consumer-centric ($R^{L}$) impact. For both $R^{G}$
and $R^{L},$ all models find the impact to increase drastically at higher
levels of wind reduction. Smart load re-dispatch (as in Exp. 3) minimizes the
impact, reducing it to about one-tenth of the impact when no action is taken
(as in Exp. 1) under both DC and AC models. This result underlines that,
without proactive planning to prevent blackout, wind penetration is risky, as
unexpected wind reduction of as little as $10\%$ of the base load can cause
large-scale congestion, but the risk can be significantly reduced with smart
rescheduling, which ensures near full service and avoids congestion
altogether.

Figure 8: Grid-Centric.

Figure 9: Consumer-Centric.

Figure 10: Resilience Impact.

###  IV-E System-Wide Structures

A few interesting system-wide influence structures arise from our $D$ and $E$
matrices. Heatmaps of selected scenarios for $D$ and $E$ matrices are shown in
Fig. 17, where darker colors denote higher influence levels. The $E$ matrices
display a sparse structure. When no corrective actions are taken, the $D$
matrix has a sparse structure under DC models and linear structure under AC
models (Fig. 17, Fig. 17). However, when corrective actions are taken, the $D$
matrices display a linear structure, where the pair-wise influence values are
high in particular columns (Fig. 17, Fig. 17). This suggests a uni-
directional, strong influence from one to many other links.

This linear structure has significant practical value. Identifying the high-
influence links in $D$ can be extremely informative to the operators: when
critical links fail, there is higher value to execute scheduling according to
the scheme in Experiment 3 to preserve infrastructure integrity.

Figure 11: $D$ matrix for DC PF, $1.6\times$ loading

Figure 12: $D$ matrix for AC PF, $1.6\times$ loading

Figure 13: $D$ matrix for DCOPF, $1\times$ loading

Figure 14: $D$ matrix for ACOPF, $1\times$ loading

Figure 15: $E$ matrix for DC PF, $1.6\times$ loading

Figure 16: $E$ matrix for AC PF, $1.6\times$ loading

Figure 17: $D,E$ matrix structures.

##  V Prediction Accuracy

Both link failure and load shed prediction reach high prediction accuracy,
$>90\%$ for most cases and $>80\%$ for all. There is no significant difference
between the training and testing sets, which verifies that our model does not
overfit. We find no significant difference between the DC and AC models. The
IM framework provide significant reduction in computational time, especially
for the AC models, which requires solving nonlinear equations. While flow-
based solutions become more inefficient at higher loading level, the
computational cost is identical for all cases under our method. To demonstrate
that our proposed framework scheme captures the failure cascade features, we
compare it to two prediction methods that do not depend on the IM: the
uniform, deterministic prediction and the randomized prediction. The IM
methods gave better performance than both random and uniform predictions with
no significant difference across different loading levels as shown by the mean
error rates in Table II and Table II.

| IM | Rand. | Unif.
---|---|---|---
exp1 | 0.038 | 0.188 | 0.109
exp2 | 0.019 | 0.093 | 0.049
exp3 | 0.000 | 0.094 | 0.049
TABLE I: Link Failure Prediction Error Rates.

| IM | Rand. | Unif.
---|---|---|---
exp1 | 0.214 | 0.318 | 0.255
exp2 | 0.043 | 0.082 | 0.043
exp3 | 0.014 | 0.026 | 0.014
TABLE II: Load Shed Prediction Error Rates.

##  VI Tests in Large-Scale IEEE 300 Bus System

We verify that our methodology is accurate and scalable by experimenting on
the IEEE300 system[8]. In this system, smart scheduling in Experiment 3 can
reduce the cost of load shed by as much as $90\%$ at all loading levels, while
completely avoiding all link failures. Prediction with the IM model yields
even higher accuracy, with $>95\%$ for most cases, for both link failure and
load shed, with the exception of load shed prediction in Experiment 1, which
also produced $>75\%$ accuracy (the latter requires further assessment). The
influence model has significant computation cost advantage. Table III shows an
example at the default loading level on 1000 samples. The influence model can
predict link failures and load shed within 1/10 of the time required for
running PF or OPF 111Tested with MATLAB R2022a on Intel(R) Core(TM) i5-1135G7
<EMAIL_ADDRESS>Processor with 8GB installed memory.. Further studies are needed
to assess the economic advantage of our adaptive approach of rescheduling
generation upon contingencies.

| Simulation (s) | Training (s) | Prediction (s)
---|---|---|---
exp1 | 169.77 | 611.50 | 15.40
exp2 | 183.35 | 305.63 | 10.05
exp3 | 246.23 | 332.68 | 6.76
TABLE III: Time Cost at Default Loading Level.

##  VII Conclusion: Advisory Tool for Operators

Based on our study of (1) the impact of wind reduction and effects of
corrective actions during a failure cascade and (2) the resilience impact
measures and risks of wind reduction, we propose an advisory tool for
operators to assess contingency criticality, predict losses, and strategize
for loss minimization with smart rescheduling. The tests have shown promising
robust results. The advisory tool comprises:

###  VII-A Wind Reduction Risk Assessment

The IM helps us determine the most critical links and initial contingencies,
determined by a combination of criticality values as well as expected
$G(p),L(p),$ and $R(p,\cdot)$ where profile $p$ embeds the initial
contingencies. Grid and consumer-centric criticality values are computed with
equations as follows.

$\displaystyle
C^{D}(j)=\sum_{i=1,\dots,N_{br}}d_{ij}(A^{11}_{ji}-A^{01}_{ji}),$ (12)
$\displaystyle C^{E}(j)=\sum_{i=1,\dots,N}e_{ij}(B^{11}_{ji}-B^{01}_{ji}),$
(13)

where $j$ is the link index, and $i$ enumerates over all links for $C^{D}(j)$
or all buses for $C^{E}(j)$. With this information, system operators can
identify the impact of sudden wind reduction when there is a pre-existing link
failure in the network. As greater values of the resilience impact signify
greater cost, this tool informs operators of upcoming risks.

###  VII-B Cascade Management

The failure cascade and load shed prediction given by our models can inform
operators the best course of action for loss minimization. Operators may also
use this tool to predict the impact of wind reduction and deploy load
reduction.

## References

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FT/UCM 214-2023

IFT-UAM/CSIC-23-81

The one-loop unimodular graviton propagator in any dimension

Jesus Anero†, Carmelo P. Martin†† and Eduardo Velasco-Aja†††

$\dagger$,$\dagger\dagger\dagger$Universidad Autońoma de Madrid (UAM),
Departamento de Física Teórica and Instituto de Física Teórica (IFT-UAM/CSIC),
C/ Nicolás Cabrera 13-15 Universidad Autońoma de Madrid Cantoblanco, Madrid
28049, SPAIN

$\dagger\dagger$Universidad Complutense de Madrid (UCM), Departamento de
Física Teórica and IPARCOS, Facultad de Ciencias Físicas, 28040 Madrid, Spain

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

For unimodular gravity, we work out, by using dimensional regularization, the
complete one-loop correction to the graviton propagator in any space-time
dimension. The computation is carried out within the framework where
unimodular gravity has Weyl invariance in addition to the transverse
diffeomorphism gauge symmetry. Thus, no Lagrange multiplier is introduced to
enforce the unimodularity condition. The quantization of the theory is carried
out by using the BRST framework and there considering a large continuous
family of gauge-fixing terms. The BRST formalism is developed in such a way
that the set of ghost, anti-ghost and auxiliary fields and their BRST changes
do not depend on the space-time dimension, as befits dimensional
regularization. As an application of our general result, and at $D=4$, we
obtain the renormalized one-loop graviton propagator in the dimensional
regularization minimal subtraction scheme. We do so by considering two
simplifying gauge-fixing choices.

Keywords: Models of quantum gravity, unimodular gravity, renormalization.

##  1 Introduction.

If one applies the effective field theory formalism to General Relativity, one
concludes that the value of the Cosmological Constant should be some hundred
of orders of magnitude larger than its measured value. This a part of the so-
called Cosmological Constant problem [1]. Unimodular gravity furnishes a
solution [2, 3, 4, 5, 6] to this part of the problem since in that gravity
theory the vacuum energy does not gravitate: in unimodular gravity the metric
always has determinant equal to $-1$ –see [7, 8], for introductions.

There are several approaches to define unimodular gravity as a quantum field
theory, e.g., [9, 10, 11, 12, 13, 14, 15, 16, 17]. It is not known whether
they yield the same quantum theory, even when the background metric is
Minkowski, for they involve different sets of ghosts. This is an open problem,
as it is their equivalence to General Relativity when the Cosmological
Constant is set to zero.

In this paper we shall adhere to the formulation of unimodular gravity
introduced in [18, 19], which was put into BRST form in [11]. The starting
point of this formulation is to solve first the unimodularity constraint ${\rm
det}\hat{g}_{\mu\nu}(x)=-1$ –$\hat{g}_{\mu\nu}$ being the unimodular metric in
$D$-dimensional space-time- by introducing an unconstrained tensor field
$g_{\mu\nu}(x)$ such that

$\hat{g}_{\mu\nu}(x)=\dfrac{g_{\mu\nu}(x)}{|g|^{1/D}(x)}.$

Then, the path integral of the theory is defined by using the standard linear
splitting $g_{\mu\nu}(x)=\bar{g}_{\mu\nu}+h_{\mu\nu}$, along with BRST
techniques. $\bar{g}_{\mu\nu}$ is the background field and $h_{\mu\nu}$ is the
graviton field, which is integrated over in the path integral. Of course, one
may use any other type of splitting –e.g., the splitting induced by the
exponential parametrization– to define the quantum theory. Let us stress that
in general the linear splitting cannot be used in a given formulation of
unimodular gravity which the unimodularity constraint by using a Lagrange
multiplier [10, 12, 15, 17], say $\lambda(x)$. Indeed, as shown in [20], UV
divergent contributions which are quadratic in $\lambda(x)$, arise at one and
higher loop levels. As also shown in [20] such a problem with the use of a
Lagrange multiplier never happens if the parametrization of the metric in
terms of the graviton field, $h_{\mu\nu}$, is such that the unimodularity
condition is equivalent to $h_{\mu\nu}$ being traceless –an instance of such
parametrization is the exponential parametrization. Because we prefer to have
any kind of sensible splitting at our disposal, we favour in this paper the
formulation introduced in [18, 19]. Another reason may be found in that this
formulation, which has Weyl invariance, finds nice accommodation within the
string theory framework [21]. Further, the formulation in question also fits
in the gauge/gravity duality framework [22].

One of the most important quantities in a quantum field theory of gravity is
the complete one-loop graviton propagator. Indeed, this quantity gives rise,
in any space-time dimension, to a highly non-trivial one-loop contribution to
the scattering of two particles; which in turn yields a part of the quantum
correction to Newton’s Law [23]. It is also advisable to compute that one-loop
contribution for large families of gauge-fixing choices, since the
cancellation of the gauge dependence when computing a physical quantity is a
non-trivial check that our computation has been done correctly. In the General
Relativity case, such computations were done long ago in [24, 25] and just
last year [26]. These types of computations –perhaps because they are far more
involved than in General Relativity– have never been done for unimodular
gravity. The final goal of this paper is to remedy this state of affairs and
compute the complete one-loop correction to the graviton propagator for a very
large continuous family of gauge-fixing terms in any number of space-time
dimensions. To do so we shall adapt the BRST formalism in [11] to the case of
a Minkowski background and generalize the choice of gauge-fixing term. Then,
we shall work out the free propagators and the interaction vertices involving
three fields. This, in the case at hand, is a very laborious task, let alone
the computation of the one-loop contributions, that we have been able to do
thanks to invaluable help from the symbolic manipulation systems ${\it FORM}$
[27], ${\it xAct}$[28] and ${\it Mathematica}$[29]. Let us point out that our
BRST ghost, anti-ghost and auxiliary field system and their transformations do
not depend on the space-time dimension so that we can keep $D$ arbitrary in
our computations. This independence on the space-time dimension does not hold
for the BRST quantization approach of reference [17].

The layout of the paper is as follows. In section 2, we carry out the BRST
quantization of our theory, introducing a continuous large family of gauge-
fixing terms. The free propagators, relevant vertices and Feynman rules are
displayed in section 3. Section 4 is devoted to the computation, in
dimensional regularization, of the one-loop contribution to the graviton
propagator in any number of dimensions. In this section two choices of gauge
parameters that greatly simplify the final result are also introduced.
Comments and conclusions can be found in section 5. We include three
appendices. In Appendix A, we give formulae leading to the renormalized, at
$D=4$, one-loop graviton propagator in the minimal subtraction scheme of
dimensional regularization; here some choices of gauge parameters are put in
place. The value of each non-vanishing 1PI Feynman diagram contributing to the
one-loop graviton propagator is given in Appendix B. In Appendix C, we show
how our general free gravity propagator yields Newton’s law and discuss how
this fact demands that two gauge-dependent coefficients of the free propagator
satisfy a precise linear equation.

##  2 BRST quantization of unimodular gravity with Weyl invariance.

The classical action of our formulation of unimodular gravity reads

$S_{\text{\tiny{UG}}}\,=\,-2\int d^{D}x\,R[\hat{g}_{\mu\nu}],$ (2.1)

where $R[\hat{g}]$ is the Ricci scalar for the unimodular metric
$\hat{g}_{\mu\nu}$. The metric $\hat{g}_{\mu\nu}$ is defined in terms of the
unconstrained tensor field $g_{\mu\nu}$ as follows:

$\hat{g}_{\mu\nu}(x)=\frac{g_{\mu\nu}(x)}{|g|^{1/D}(x)}.$ (2.2)

$g_{\mu\nu}$ is a tensor field under transverse diffeomorphisms and it is the
field variable that will be used, being unconstrained, to quantize the
classical theory. $D$ is the space-time dimension.

The action in (2.1) as a functional of $g_{\mu\nu}$ has got two gauge
symmetries. These gauge symmetries are transverse diffeomorphisms and Weyl
transformations; which, in infinitesimal form, read

$\begin{array}[]{l}{\delta_{\rm\tiny{TD}}g_{\mu\nu}(x)=\nabla_{\mu}\epsilon^{T}_{\nu}(x)+\nabla_{\nu}\epsilon^{T}_{\mu}(x),\quad\epsilon^{T}_{\mu}(x)=g_{\mu\nu}\epsilon^{T\,\nu}(x),\quad\partial_{\mu}\epsilon^{T\,\mu}(x)=0,}\\\\[4.0pt]
{\delta_{W}g_{\mu\nu}(x)=2\epsilon(x)g_{\mu\nu}(x).}\end{array}$ (2.3)

In the previous definitions $\epsilon^{T\,\mu}(x)$ and $\epsilon(x)$ are
infinitesimal vector and scalar fields, respectively, and $\nabla_{\mu}$ is
the covariant derivative for $g_{\mu\nu}$.

With the goal of formulating a field theory of gravitons propagating on
Minkowski space-time, we shall consider the standard linear splitting of
$g_{\mu\nu}$ in (2.2):

$g_{\mu\nu}(x)=\eta_{\mu\nu}+h_{\mu\nu}(x).$

The field $h_{\mu\nu}$ is the graviton field.

To quantize the classical theory with action in (2.1) and gauge symmetries in
(2.3) we shall use the Batalin-Vilkovisky BRST formalism [30], for the vector
field $\epsilon^{T\,\mu}$ above is constrained by transversality. This
constraint makes the transverse diffeomorphism transformations in (2.3) a
first-stage gauge reducible transformations, so that, according to [30], the
following set of fields is to be introduced to set up the BRST formalism for
the transverse diffeomorphism gauge symmetry:

$\begin{array}[]{l}{h_{\mu\nu}^{(0,0)},\,c^{(1,1)\,\mu},\,b^{(1,-1)\,\mu},\,f^{(0,0)\,\mu},\,\phi^{(0,2)},}\\\\[4.0pt]
{\pi^{(1,-1)},\,\pi^{\prime(1,1)},\,\bar{c}^{(0,-2)}\quad\text{and}\quad
c^{\prime(0,0)}.}\end{array}$ (2.4)

On the other hand, the Weyl transformations in (2.3) give rise to an
irreducible gauge symmetry which demands the use of the following ghost, anti-
ghost and auxiliary fields

$c^{(1,1)},\,b^{(1,-1)}\quad\text{and}\text{\quad}f^{(0,0)},$ (2.5)

to formulate the BRST formalism. In the previous equations the first entry,
$n$, of the superscript pair $(n,m)$ in the fields is the Grassmann number;
the second entry, $m$, is the ghost number.

Let us introduce the BRST operators, $s_{\rm\tiny{TD}}$, and
$s_{\rm\tiny{W}}$, defining the BRST transformations associated to the
transverse diffeomorphisms and Weyl transformations in (2.3), respectively.
Following ref. [11], we define the action of $s_{\rm\tiny{TD}}$ and
$s_{\rm\tiny{W}}$ on the fields in (2.4) and (2.5) as follows

$\begin{array}[]{l}{s_{\rm\tiny{TD}}\eta_{\mu\nu}=0,}\\\\[4.0pt]
{s_{\rm\tiny{TD}}h_{\mu\nu}=\partial_{\mu}c^{T}_{\nu}+\partial_{\nu}c^{T}_{\mu}+c^{\rho
T}\partial_{\rho}h_{\mu\nu}+h_{\rho\nu}\partial_{\mu}c^{T\rho}+h_{\rho\mu}\partial_{\nu}c^{T\rho},}\\\\[4.0pt]
{s_{\rm\tiny{TD}}c^{(1,1)\,\mu}=\frac{1}{\Box}\left(c^{\rho
T}\partial_{\rho}c^{T\mu}\right)+\partial^{\mu}\phi^{(0,2)},}\\\\[4.0pt]
{s_{\rm\tiny{TD}}b^{(1,-1)\,\mu}=f^{(0,0)\,\mu},\quad
s_{\rm\tiny{TD}}f^{(0,0)\,\mu}=0,\quad
s_{\rm\tiny{TD}}\phi^{(0,2)}=0,}\\\\[4.0pt]
{s_{\rm\tiny{TD}}\pi^{(1,-1)}=0,\quad
s_{\rm\tiny{TD}}\pi^{\prime(1,1)}=0,\quad
s_{\rm\tiny{TD}}\bar{c}^{(0,-2)}=\pi^{(1,-1)},\quad
s_{\rm\tiny{TD}}c^{\prime(0,0)}=\pi^{\prime(1,1)},}\\\\[4.0pt]
{s_{\rm\tiny{TD}}c^{(1,1)}=c^{T\rho}\partial_{\rho}c^{(1,1)},\quad
s_{\rm\tiny{TD}}b^{(1,-1)}=c^{T\rho}\partial_{\rho}b^{(1,-1)},\,s_{\rm\tiny{TD}}f^{(0,0)}=c^{T\rho}\partial_{\rho}f^{(0,0)},}\\\\[8.0pt]
{s_{\rm\tiny{W}}\eta_{\mu\nu}=0,}\\\\[4.0pt]
{s_{\rm\tiny{W}}h_{\mu\nu}=2c^{(1,1)}\left(\eta_{\mu\nu}+h_{\mu\nu}\right),}\\\\[4.0pt]
{s_{\rm\tiny{W}}c^{(1,1)\,\mu}=0,\quad s_{\rm\tiny{W}}b^{(1,-1)\,\mu}=0,\quad
s_{\rm\tiny{W}}f^{(0,0)\,\mu}=0,\quad
s_{\rm\tiny{W}}\phi^{(0,2)}=0,}\\\\[4.0pt]
{s_{\rm\tiny{W}}\pi^{(1,-1)}=0,\quad s_{\rm\tiny{W}}\pi^{\prime(1,1)}=0,\quad
s_{\rm\tiny{W}}\bar{c}^{(0,-2)}=0,\quad
s_{\rm\tiny{W}}c^{\prime(0,0)}=0,}\\\\[4.0pt]
{s_{\rm\tiny{W}}c^{(1,1)}=0,\quad s_{\rm\tiny{W}}b^{(1,-1)}=f^{(0,0)},\quad
s_{\rm\tiny{W}}f^{(0,0)}=0.}\end{array}$ (2.6)

In the previous equations indices are raised and lowered by using
$\eta_{\mu\nu}$ and the following definition is assumed

$c^{T\,\mu}=\left(\delta^{\mu}_{\nu}\Box-\partial^{\mu}\partial_{\nu}\right)c^{(1,1)\,\nu}.$

Note that both $s_{\rm\tiny{TD}}$ and $s_{\rm\tiny{W}}$ have got both
Grassmann number and ghost number equal 1.

Using the results

$\partial_{\mu}\left(c^{\rho
T}\partial_{\rho}c^{T\mu}\right)=0,\quad\partial_{\mu}\left[\frac{1}{\Box}\left(c^{\rho
T}\partial_{\rho}c^{T\mu}\right)\right]=0$

and the definitions in (2.6), one may show that

$s_{\rm\tiny{TD}}^{2}=0,\quad s_{\rm\tiny{W}}^{2}=0\quad\text{and
that}\quad\\{s_{\rm\tiny{TD}},s_{\rm\tiny{W}}\\}=0.$

Next, we define the full BRST operator, $s$, as follows

$s=s_{\rm\tiny{TD}}\,+\,s_{\rm\tiny{W}}.$ (2.7)

We are now ready to introduce the BRST invariant action $S_{\rm BRST}$ of our
unimodular gravity theory with all the gauge symmetries in (2.3) fixed:

$S_{\rm BRST}\,=\,S_{\rm UG}\,-\,\int\,d^{D}x\,sX.$ (2.8)

Note that $S_{\rm UG}$ is given in (2.1) and the integration domain is
Minkowski space-time. $X$ is a local functional of the fields in (2.4) and
(2.5), and their derivatives, such that ${\it i})$ it has Grassmann number
equal to 1 and ghost number equal to -1, and ${\it ii})$ the differential
operator in the contribution to $S_{\rm BRST}$ which is quadratic in the
fields is invertible –this way free-propagators will exist. The operator $s$
is defined in (2.7).

Our choice of $X$ will be quadratic in the fields but general enough as to
introduce a large continuous gauge parameter dependence in most coefficients
of free propagator of the graviton –see (3.12) and (3.13). Actually our choice
of $X$ guarantees that one of the coefficients of two of the most involved
terms, namely,

$\frac{1}{(p^{2})^{2}}\left(\eta_{\mu_{1}\mu_{3}}p_{\mu_{2}}p_{\mu_{4}}+\eta_{\mu_{1}\mu_{4}}p_{\mu_{2}}p_{\mu_{3}}+\eta_{\mu_{2}\mu_{3}}p_{\mu_{1}}p_{\mu_{4}}+\eta_{\mu_{2}\mu_{4}}p_{\mu_{1}}p_{\mu_{3}}\right)\quad\text{and}\quad\frac{1}{(p^{2})^{3}}p_{\mu_{1}}p_{\mu_{2}}p_{\mu_{3}}p_{\mu_{4}},$

can always be set to zero by a suitable choice of the gauge parameters.
Agreement with Newton’s law makes it impossible for the existence of a gauge-
fixing term such that the coefficients of both terms vanish –see Appendix C.

Our $X$ is the following large family of functions:

$X\,=\,X_{\rm\tiny{TD}}+\,X_{\rm\tiny{W}},$ (2.9)

where

$\begin{array}[]{l}{X_{\rm\tiny{TD}}=b_{\mu}\left[\gamma_{1}\partial^{\nu}h_{\nu}^{\mu}+\gamma_{2}\partial^{\mu}h+\rho_{1}f^{\mu}\right]+\bar{c}\left[\alpha_{2}\partial^{\mu}c_{\mu}+\frac{1}{2}\Box\pi^{\prime}\right]+c^{\prime}\left[\alpha_{1}\partial^{\mu}b_{\mu}-\frac{1}{2}\Box\pi\right],}\\\\[4.0pt]
{X_{\rm\tiny{W}}=\partial_{\mu}b\,\partial^{\mu}\left(f-\alpha
h\right).}\end{array}$ (2.10)

$h$ stands for the contraction $h^{\mu}_{\mu}$. The fields in (2.10) are the
fields in (2.4) and (2.5) without the superscript pair $(m,n)$ so as to make
the expressions more readable. $X_{\rm\tiny{TD}}$ carries out the gauge-fixing
of the transverse diffeomorphisms. $X_{\rm\tiny{W}}$ is needed to gauge-fix
the invariance under Weyl transformations.
$\gamma_{1},\gamma_{2},\alpha_{1},\alpha_{2}$ and $\alpha$ are the gauge-
fixing parameters defining the family of gauge-fixing terms in $S_{\rm BRST}$.
These gauge-fixing parameters can have any value which yields well-defined
free propagators. We shall see below that this is achieved if
$\gamma_{1},\alpha_{1},\alpha_{2}$ and $\alpha$ never vanish. Finally, notice
that $X$ in (2.9) is reminiscent of the gauge-fixing terms used in references
[24, 25, 26] for General Relativity and generalizes quite a lot the gauge-
fixing term used in [11] in the unimodular gravity case.

We shall see that all the free propagators obtained from $S_{\rm BRST}$ as
defined by (2.8) and (2.9) are massless. Hence, to compute the one-loop
correction to the graviton propagator we may drop any contribution to $S_{\rm
BRST}$ involving more than three fields. Indeed, the vertices involving four
fields give rise to tadpole contributions, which vanish in dimensional
regularization, and one cannot draw a one-loop propagator Feynman diagram with
vertices having more than four legs. Accordingly, let us express $S_{\rm
BRST}$ as follows

$S_{\rm BRST}\,=\,S_{2}^{(\rm even)}+S_{2}^{(\rm odd)}+S_{3}^{(\rm
even)}+S_{3}^{(\rm odd)}+S_{4}.$

$S_{2}^{(\rm even)}$ and $S_{2}^{(\rm odd)}$ are the quadratic contributions
with fields having even and odd Grassmann number, respectively. $S_{3}^{(\rm
even)}$ is the three-field contribution with fields having only even Grassmann
number, $S_{3}^{(\rm odd)}$ the contribution involving three fields some of
them having odd Grassmann number and $S_{4}$ the summand with more than three
fields. A little algebra, with a lot of help from ${\it xAct}$, yields

$\begin{array}[]{l}{S_{2}^{(\rm even)}=-\frac{1}{2}\int
d^{D}x\Big{\\{}h_{\mu\nu}\bar{\Box}h^{\mu\nu}-2h^{\mu\lambda}\partial_{\mu}\partial_{\nu}h^{\nu}_{\lambda}-\frac{D+2}{D^{2}}h\bar{\Box}h+\frac{4}{D}h^{\mu\nu}\partial_{\mu}\partial_{\nu}h}\\\\[4.0pt]
{\phantom{S_{2}}-2h^{\mu\nu}\Big{[}\frac{\gamma_{1}}{2}\left(\partial_{\nu}f_{\mu}+\partial_{\mu}f_{\nu}\right)+\gamma_{2}\eta_{\mu\nu}\partial^{\lambda}f_{\lambda}\Big{]}+2\rho_{1}f_{\mu}f^{\mu}+2\alpha_{1}c^{\prime}\partial^{\mu}f_{\mu}-2f\Box
f+2\alpha h\Box f\Big{\\}},}\\\\[14.0pt] {S_{2}^{(\rm odd)}=\int
d^{D}x\Big{\\{}b_{\mu}\Big{[}\gamma_{1}\Box
c^{T\,\mu}+2\left(\gamma_{1}+D\gamma_{2}\right)\partial^{\mu}c\Big{]}-b_{\mu}\alpha_{1}\partial^{\mu}\pi^{\prime}-\pi\alpha_{2}\partial^{\mu}c_{\mu}-\pi\Box\pi^{\prime}-2D\alpha\partial_{\mu}b\partial^{\mu}c\Big{\\}},}\\\\[14.0pt]
{S_{3}^{(\rm even)}=\int
d^{D}x\Big{\\{}\frac{D+2}{D^{2}}\partial_{\mu}hh^{\rho\sigma}\partial^{\mu}h_{\rho\sigma}-\frac{D+2}{2D^{3}}h\partial_{\mu}h\partial^{\mu}h+\frac{2}{D^{2}}h\partial_{\mu}h\partial_{\rho}h^{\mu\rho}-\frac{1}{2}h^{\mu\nu}\partial_{\mu}h_{\rho\sigma}\partial_{\nu}h^{\rho\sigma}}\\\\[4.0pt]
{\phantom{S_{3}^{(\rm even)}=\int
d^{D}x\Big{\\{}}+\frac{D+2}{2D^{2}}h^{\mu\nu}\partial_{\mu}h\partial_{\nu}h-\frac{2}{D}\partial^{\mu}hh^{\rho}_{\sigma}\partial^{\sigma}h_{\rho\mu}-\frac{2}{D}h^{\mu\nu}\partial_{\mu}h\partial_{\rho}h_{\nu}^{\rho}-\frac{1}{D}h\partial^{\mu}h_{\rho\sigma}\partial^{\sigma}h^{\rho}_{\mu}}\\\\[4.0pt]
{\phantom{S}+\frac{1}{2D}h\partial_{\mu}h^{\rho\sigma}\partial^{\mu}h_{\rho\sigma}-\frac{2}{D}h^{\mu\nu}\partial_{\rho}h_{\mu\nu}\partial_{\sigma}h^{\rho\sigma}+2h^{\mu\nu}\partial_{\nu}h_{\rho\sigma}\partial^{\sigma}h^{\rho}_{\mu}+h^{\mu\nu}\partial^{\rho}h_{\nu\sigma}\partial^{\sigma}h_{\mu\rho}-h^{\mu\nu}\partial_{\rho}h_{\nu\sigma}\partial^{\rho}h_{\mu}^{\sigma}\Big{\\}}}\\\\[4.0pt]
{\text{and}}\\\\[4.0pt] {S_{3}^{(\rm odd)}=\int
d^{D}x\Big{\\{}-\gamma_{1}\partial_{\nu}b_{\mu}\left(c^{\rho
T}\partial_{\rho}h^{\mu\nu}+h^{\rho\nu}\partial^{\mu}c^{T}_{\rho}+h^{\rho\mu}\partial^{\nu}c^{T}_{\rho}\right)-\gamma_{2}\partial^{\mu}b_{\mu}\left(c^{\rho
T}\partial_{\rho}h+2h_{\rho\lambda}\partial^{\lambda}c^{T\rho}\right)}\\\\[4.0pt]
{\phantom{S_{3}^{(\rm odd)}=\int
d^{n}x\Big{\\{}}-2\gamma_{1}c\,h^{\mu\nu}\partial_{\nu}b_{\mu}-2\gamma_{2}c\,h\partial^{\mu}b_{\mu}+2\alpha\Box
b\,c\,h+\alpha\partial_{\rho}b\,c^{T\rho}\Box h+\alpha\Box
b\,c^{T\rho}\partial_{\rho}h}\\\\[4.0pt] {\phantom{S_{3}^{(\rm odd)}=\int
d^{n}x\Big{\\{}}+2\alpha\Box
b\,\partial^{\lambda}c^{T\rho}h_{\rho\lambda}-\partial_{\rho}b\,c^{T\rho}\Box
f-\Box b\,c^{T\rho}\partial_{\rho}f\Big{\\}}.}\end{array}$ (2.11)

Furnished with the action $S_{\tiny{\rm BRST}}$ above, one constructs the
quantum theory by using the standard path integral method.

##  3 Propagators, Vertices and Feynman rules.

In this section we shall display the free propagators, the three-field
vertices and the corresponding Feynman rules of our unimodular gravity theory.

###  3.1 Free propagators.

The free propagators and Fourier transforms of the fields with even Grassman
number are denoted by the following symbols:

$\begin{array}[]{l}{\langle
h_{\mu_{1}\mu_{2}}(x)h_{\mu_{3}\mu_{4}}(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}iG^{(hh)}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(p),}\\\\[4.0pt]
{\langle
h_{\mu_{1}\mu_{2}}(x)f_{\mu_{3}}(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}\,iG^{(hf_{\mu})}_{\mu_{1}\mu_{2}\mu_{3}}(p),}\\\\[4.0pt]
{\langle
h_{\mu_{1}\mu_{2}}(x)f(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}iG^{(hf)}_{\mu_{1}\mu_{2}}(p),\;\langle
h_{\mu_{1}\mu_{2}}(x)c^{\prime}(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}\delta\,e^{-ip(x-y)}\,iG^{(hc^{\prime})}_{\mu_{1}\mu_{2}}(p),}\\\\[4.0pt]
{\langle
f_{\mu_{1}}(x)f_{\mu_{2}}(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}iG^{(f_{\mu}f_{\nu})}_{\mu_{1}\mu_{2}}(p),\;\langle
f_{\mu_{1}}(x)f(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}iG^{(f_{\mu}f)}_{\mu_{1}}(p),}\\\\[4.0pt]
{\langle
f_{\mu_{1}}(x)c^{\prime}(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}iG^{(f_{\mu}c^{\prime})}_{\mu_{1}}(p),\;\langle
f(x)f(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}iG^{(ff)}(p),}\\\\[4.0pt]
{\langle
f(x)c^{\prime}(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}iG^{(fc^{\prime})}(p),\;\langle
c^{\prime}(x)c^{\prime}(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}iG^{(c^{\prime}c^{\prime})}(p).}\end{array}$
(3.12)

The values of these propagators are obtained from $S^{(\rm even)}_{2}$ in
(2.11) by using standard path integral methods. A very lengthy computation
yields

$\begin{array}[]{l}{G^{(hh)}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(p)=}\\\\[2.0pt]
{\frac{1}{2}\frac{1}{p^{2}}\left(\eta_{\mu_{1}\mu_{3}}\eta_{\mu_{2}\mu_{4}}+\eta_{\mu_{1}\mu_{4}}\eta_{\mu_{2}\mu_{3}}\right)-\frac{D^{2}\alpha^{2}-2D+4}{D^{2}(D-2)\alpha^{2}}\frac{1}{p^{2}}\eta_{\mu_{1}\mu_{2}}\eta_{\mu_{3}\mu_{4}}+\frac{2}{D-2}\frac{1}{(p^{2})^{2}}\left(\eta_{\mu_{3}\mu_{4}}p_{\mu_{1}}p_{\mu_{2}}+\eta_{\mu_{1}\mu_{2}}p_{\mu_{3}}p_{\mu_{4}}\right)}\\\\[4.0pt]
{\quad\quad\quad\quad\quad-\frac{\gamma_{1}^{2}-4\rho_{1}}{2\gamma_{1}^{2}}\frac{1}{(p^{2})^{2}}\left(\eta_{\mu_{1}\mu_{3}}p_{\mu_{2}}p_{\mu_{4}}+\eta_{\mu_{1}\mu_{4}}p_{\mu_{2}}p_{\mu_{3}}+\eta_{\mu_{2}\mu_{3}}p_{\mu_{1}}p_{\mu_{4}}+\eta_{\mu_{2}\mu_{4}}p_{\mu_{1}}p_{\mu_{3}}\right)}\\\\[4.0pt]
{\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-\frac{4(\gamma_{1}^{2}+2(D-2)\rho_{1})}{(D-2)\gamma_{1}^{2}}\frac{1}{(p^{2})^{3}}p_{\mu_{1}}p_{\mu_{2}}p_{\mu_{3}}p_{\mu_{4}},}\\\\[14.0pt]
{G^{(hf_{\mu})}_{\mu_{1}\mu_{2}\mu_{3}}(p)=-\frac{i}{\gamma_{1}}\frac{1}{(p^{2})^{2}}\Big{[}p^{2}\left(\eta_{\mu_{1}\mu_{3}}p_{\mu_{2}}+\eta_{\mu_{2}\mu_{3}}p_{\mu_{1}}\right)-2p_{\mu_{1}}p_{\mu_{2}}p_{\mu_{3}}\Big{]},}\end{array}$
(3.13)
$\begin{array}[]{l}{\phantom{\frac{1}{2}\frac{1}{p^{2}}\left(\eta_{\mu_{1}\mu_{3}}\eta_{\mu_{2}\mu_{4}}+\eta_{\mu_{1}\mu_{4}}\eta_{\mu_{2}\mu_{3}}\right)-\frac{D^{2}\alpha^{2}-2D+4}{D^{2}(D-2)\alpha^{2}}\frac{1}{p^{2}}\eta_{\mu_{1}\mu_{2}}\eta_{\mu_{3}\mu_{4}}+\frac{2}{D-2}\frac{1}{(p^{2})^{2}}\left(\eta_{\mu_{3}\mu_{4}}p_{\mu_{1}}p_{\mu_{2}}+\eta_{\mu_{1}\mu_{2}}p_{\mu_{3}}p_{\mu_{4}}\right)}}\\\\[4.0pt]
{G^{(hf)}_{\mu_{1}\mu_{2}}(p)=\frac{1}{D\alpha}\frac{1}{p^{2}}\eta_{\mu_{1}\mu_{2}},}\\\\[14.0pt]
{G^{(hc^{\prime})}_{\mu_{1}\mu_{2}}(p)=\frac{(D^{2}\alpha^{2}+2D-4)\gamma_{1}+2(D-2)D\gamma_{2}}{(D-2)D^{2}\alpha^{2}\alpha_{1}}\frac{1}{p^{2}}\eta_{\mu_{1}\mu_{2}}-\frac{D}{D-2}\frac{\gamma_{1}}{\alpha_{1}}\frac{1}{(p^{2})^{2}}p_{\mu_{1}}p_{\mu_{2}},}\\\\[14.0pt]
{G^{(f_{\mu}f_{\nu})}_{\mu_{1}\mu_{2}}(p)=0,\quad
G^{(f_{\mu}f)}_{\mu_{1}}(p)=0,\quad
G^{(f_{\mu}c^{\prime})}_{\mu_{1}}(p)=-\frac{i}{\alpha_{1}}\frac{1}{p^{2}}p_{\mu_{1}},}\\\\[14.0pt]
{G^{(ff)}(p)=0,\quad
G^{(fc^{\prime})}(p)=\frac{1}{D}\frac{\gamma_{1}+D\gamma_{2}}{\alpha\alpha_{1}}\frac{1}{p^{2}},}\\\\[14.0pt]
{G^{(c^{\prime}c^{\prime})}(p)=-\frac{(D^{3}\alpha^{2}-D^{2}\alpha^{2}-2D+4)\gamma_{1}^{2}-4(D-2)D\gamma_{1}\gamma_{2}-2(D-2)D^{2}(\gamma_{2}^{2}+\alpha^{2}\rho_{1})}{(D-2)D^{2}\alpha^{2}\alpha_{1}^{2}}\frac{1}{p^{2}}.}\end{array}$
(3.14)

Next, the following symbols

$\begin{array}[]{l}{\langle
c_{\mu_{1}}(x)b_{\mu_{2}}(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}iG^{(c_{\mu}b_{\nu})}_{\mu_{1}\mu_{2}}(p),\;\langle
c(x)b_{\mu_{1}}(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}\,iG^{(cb_{\mu})}_{\mu_{1}}(p),}\\\\[4.0pt]
{\langle
c^{T}_{\mu_{1}}(x)b_{\mu_{2}}(y)\rangle_{0}=(\Box_{x}\delta_{\mu_{1}}^{\rho}-\partial^{x}_{\mu_{1}}\partial_{x}^{\rho})\langle
c_{\rho}(x)b_{\mu_{2}}(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}iG^{(c^{T}_{\mu}b_{\nu})}_{\mu_{1}\mu_{2}}(p),}\\\\[4.0pt]
{\langle\pi^{\prime}(x)b_{\mu_{1}}(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}\,iG^{(\pi^{\prime}b_{\mu})}_{\mu_{1}}(p),}\\\\[4.0pt]
{\langle
c_{\mu_{1}}(x)b(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}iG^{(c_{\mu}b)}_{\mu_{1}}(p),\;\langle
c(x)b(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}\,iG^{(cb)}(p),}\\\\[4.0pt]
{\langle\pi^{\prime}(x)b(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}\,iG^{(\pi^{\prime}b)}(p),}\\\\[4.0pt]
{\langle
c_{\mu_{1}}(x)\pi(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}iG^{(c_{\mu}\pi)}_{\mu_{1}}(p),\;\langle
c(x)\pi(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}\,iG^{(c\pi)}(p),}\\\\[4.0pt]
{\langle\pi^{\prime}(x)\pi(y)\rangle_{0}=\int\\!\\!\dfrac{d^{D}p\\!}{(2\pi)^{D}}e^{-ip(x-y)}\,iG^{(\pi^{\prime}\pi)}(p),}\end{array}$
(3.15)

stand for the free-propagators and Fourier transform of the fields with an odd
Grassmann number. From $S_{2}^{(\rm odd)}$ in (2.11), one gets

$\begin{array}[]{l}{G^{(c_{\mu}b_{\nu})}_{\mu_{1}\mu_{2}}(p)=\frac{1}{\gamma_{1}}\frac{1}{{(p^{2})^{3}}}\left(p^{2}\eta_{\mu_{1}\mu_{2}}-p_{\mu_{1}}p_{\mu_{2}}\right)+\frac{1}{\alpha_{1}\alpha_{2}}\frac{1}{p^{2}}\,p_{\mu_{1}}p_{\mu_{2}},\quad
G^{(cb_{\mu})}_{\mu_{1}}(p)=-i\frac{\gamma_{1}+D\gamma_{2}}{D\alpha\alpha_{1}\alpha_{2}}\frac{1}{p^{2}}\,p_{\mu_{1}}}\\\\[4.0pt]
{G^{(c^{T}_{\mu}b_{\nu})}_{\mu_{1}\mu_{2}}(p)=-\dfrac{1}{\gamma_{1}(p^{2})^{2}}(p^{2}\eta_{\mu_{1}\mu_{2}}-p_{\mu_{1}}p_{\mu_{2}}),}\\\\[4.0pt]
{G^{(\pi^{\prime}b_{\mu})}_{\mu_{1}}(p)=-\frac{i}{\alpha_{2}}\frac{1}{p^{2}}\,p_{\mu_{1}},\quad
G^{(c_{\mu}b)}_{\mu_{1}}(p)=0,\quad
G^{(cb)}(p)=-\frac{1}{2D\alpha}\frac{1}{p^{2}},\quad
G^{(\pi^{\prime}b)}(p)=0}\\\\[4.0pt]
{G^{(c_{\mu}\pi)}_{\mu_{1}}(p)=-\frac{i}{\alpha_{1}}\frac{1}{p^{2}}\,p_{\mu_{1}},\quad
G^{(c\pi)}(p)=-\frac{\gamma_{1}+D\gamma_{2}}{D\alpha\alpha_{1}}\frac{1}{p^{2}},\quad
G^{(\pi^{\prime}\pi)}(p)=0.}\end{array}$ (3.16)

###  3.2 Three-field vertices.

We now come to the three-field vertices. By a Fourier transformation, one can
recast $iS_{3}^{(\rm even)}$ in (2.11), into the form,

$iS_{3}^{(\rm
even)}=\int\prod_{i=1}^{3}\frac{d^{D}p_{i}}{(2\pi)^{D}}(2\pi)^{D}\delta(p_{1}+p_{2}+p_{3})\dfrac{1}{3!}\,V^{\mu_{1}\nu_{1},\mu_{2}\nu_{2},\mu_{3}\nu_{3}}{(p_{1},p_{2},p_{3})}h_{\mu_{1}\nu_{1}}(p_{1})h_{\mu_{2}\nu_{2}}(p_{2})h_{\mu_{3}\nu_{3}}(p_{3}),$

where the three-graviton vertex reads

$\begin{array}[]{l}{V^{\mu_{1}\nu_{1},\mu_{2}\nu_{2},\mu_{3}\nu_{3}}(p_{1},p_{2},p_{3})=}\\\\[4.0pt]
{i\,3!\,\mathcal{S}\Big{\\{}-\frac{D+2}{D^{2}}\Big{[}p_{1}p_{3}\eta^{\mu_{1}\nu_{1}}\eta^{\mu_{2}\mu_{3}}\eta^{\nu_{2}\nu_{3}}\Big{]}+\frac{D+2}{2D^{3}}\Big{[}p_{2}\cdot
p_{3}\eta^{\mu_{1}\nu_{1}}\eta^{\mu_{2}\nu_{2}}\eta^{\mu_{3}\nu_{3}}\Big{]}-\frac{2}{D^{2}}\Big{[}p^{\mu_{3}}_{2}p^{\nu_{3}}_{3}\eta^{\mu_{1}\nu_{1}}\eta^{\mu_{2}\nu_{2}}\Big{]}}\\\\[4.0pt]
{+\frac{1}{2}\Big{[}p^{\mu_{1}}_{2}p^{\nu_{1}}_{3}\eta^{\mu_{2}\mu_{3}}\eta^{\nu_{2}\nu_{3}}\Big{]}-\frac{D+2}{2D^{2}}\Big{[}p^{\mu_{1}}_{2}p^{\nu_{1}}_{3}\eta^{\mu_{2}\nu_{2}}\eta^{\mu_{3}\nu_{3}}\Big{]}+\frac{2}{D}\Big{[}p^{\mu_{3}}_{1}p^{\nu_{2}}_{3}\eta^{\mu_{1}\nu_{1}}\eta^{\mu_{2}\nu_{3}}\Big{]}}\\\\[4.0pt]
{+\frac{2}{D}\Big{[}p^{\mu_{1}}_{2}p^{\mu_{3}}_{3}\eta^{\mu_{2}\nu_{2}}\eta^{\nu_{1}\nu_{3}}\Big{]}+\frac{1}{D}\Big{[}p^{\nu_{3}}_{2}p^{\nu_{2}}_{3}\eta^{\mu_{1}\nu_{1}}\eta^{\mu_{2}\mu_{3}}\Big{]}-\frac{1}{2D}\Big{[}p_{2}p_{3}\eta^{\mu_{1}\nu_{1}}\eta^{\mu_{2}\mu_{3}}\eta^{\nu_{2}\nu_{3}}\Big{]}}\\\\[4.0pt]
{+\frac{2}{D}\Big{[}p^{\nu_{3}}_{2}p^{\mu_{3}}_{3}\eta^{\mu_{1}\mu_{2}}\eta^{\nu_{1}\nu_{2}}\Big{]}-2\Big{[}p^{\nu_{1}}_{2}p^{\nu_{2}}_{3}\eta^{\mu_{3}\mu_{1}}\eta^{\mu_{2}\nu_{3}}\Big{]}-\Big{[}p^{\nu_{3}}_{2}p^{\nu_{2}}_{3}\eta^{\nu_{1}\mu_{2}}\eta^{\mu_{1}\mu_{3}}\Big{]}+\Big{[}p_{2}\cdot
p_{3}\eta^{\nu_{1}\mu_{2}}\eta^{\nu_{2}\nu_{3}}\eta^{\mu_{3}\mu_{1}}\Big{]}\Big{\\}}.}\end{array}$
(3.17)

Note that $\mathcal{S}$ is a shorthand for a double symmetrization, namely, a
symmetrization with regard to each pair on indices $\mu_{1}\nu_{1}$,
$\mu_{2}\nu_{2}$ and $\mu_{3}\nu_{3}$, first, and, then, a symmetrization with
regard to all pairs
$(p1,\mu_{1}\nu_{1}),(p_{2},\mu_{2}\nu_{2}),(p_{3},\mu_{3}\nu_{3})$.
Symmetrization of $F[1,2,3,...,n]$ is carried out with the formula

$\sum_{\Sigma}\,\frac{1}{n!}\,F[\Sigma(1),\Sigma(2),...,\Sigma(n)],$

where the sum runs over all permutations –generally denoted by $\Sigma$– of
$(1,2,...,n)$.

$S_{3}^{(\rm odd)}$ as defined in (2.11) gives rise to the following three-
field interaction terms in momentum space

$\begin{array}[]{l}{iS_{3}^{(\rm
odd)}=\int\prod_{i=1}^{3}\frac{d^{D}p_{i}}{(2\pi)^{D}}(2\pi)^{D}\delta(-p_{1}+p_{2}+p_{3})}\\\\[4.0pt]
{\Big{[}V_{1}^{\mu_{1},\mu_{2},\mu_{3}\nu_{3}}(p_{1},p_{2},p_{3})\,b_{\mu_{1}}(p_{1})c_{\mu_{2}}^{T}(p_{2})h_{\mu_{3}\nu_{3}}(p_{3})+V_{2}^{\mu_{1},\mu_{3}\nu_{3}}(p_{1},p_{2},p_{3})\,b_{\mu_{1}}(p_{1})c(p_{2})h_{\mu_{3}\nu_{3}}(p_{3})+}\\\\[4.0pt]
{\phantom{\Big{[}}V_{3}^{\mu_{3}\nu_{3}}(p_{1},p_{2},p_{3})\,b(p_{1})c(p_{2})h_{\mu_{3}\nu_{3}}(p_{3})+V_{4}^{\mu_{2},\mu_{3}\nu_{3}}(p_{1},p_{2},p_{3})\,b(p_{1})c^{T}_{\mu_{2}}(p_{2})h_{\mu_{3}\nu_{3}}(p_{3})+}\\\\[4.0pt]
{\phantom{\Big{[}}V_{5}^{\mu_{2}}(p_{1},p_{2},p_{3})\,b(p_{1})c^{T}_{\mu_{2}}(p_{2})f(p_{3})\Big{]},}\end{array}$

with

$\begin{array}[]{l}{V_{1}^{\mu_{1},\mu_{2},\mu_{3}\nu_{3}}(p_{1},p_{2},p_{3})=-i\gamma_{1}\Big{[}\frac{1}{2}\left(p_{1}^{\nu_{3}}p_{3}^{\mu_{2}}\eta^{\mu_{1}\mu_{3}}+p_{1}^{\mu_{3}}p_{3}^{\mu_{2}}\eta^{\mu_{1}\nu_{3}}\right)+\frac{1}{2}\left(p_{1}^{\nu_{3}}p_{2}^{\mu_{1}}\eta^{\mu_{2}\mu_{3}}+p_{1}^{\mu_{3}}p_{2}^{\mu_{1}}\eta^{\mu_{2}\nu_{3}}\right)}\\\\[4.0pt]
{\quad\quad\quad\quad+\frac{1}{2}p_{1}\cdot
p_{2}\left(\eta^{\mu_{1}\mu_{3}}\eta^{\mu_{2}\nu_{3}}+\eta^{\mu_{1}\nu_{3}}\eta^{\mu_{2}\mu_{3}}\right)\Big{]}-i\gamma_{2}\Big{[}p_{1}^{\mu_{1}}p_{3}^{\mu_{2}}\eta^{\mu_{3}\nu_{3}}+p_{1}^{\mu_{1}}p_{2}^{\mu_{3}}\eta^{\mu_{2}\nu_{3}}+p_{1}^{\mu_{1}}p_{2}^{\nu_{3}}\eta^{\mu_{2}\mu_{3}}\Big{]},}\\\\[14.0pt]
{V_{2}^{\mu_{1},\mu_{3}\nu_{3}}(p_{1},p_{2},p_{3})=\gamma_{1}\left(p_{1}^{\nu_{3}}\eta^{\mu_{1}\mu_{3}}+p_{1}^{\mu_{3}}\eta^{\mu_{1}\nu_{3}}\right)+2\gamma_{2}p_{1}^{\mu_{1}}\eta^{\mu_{3}\nu_{3}},}\\\\[14.0pt]
{V_{3}^{\mu_{3}\nu_{3}}(p_{1},p_{2},p_{3})=-2i\alpha
p_{1}^{2}\eta^{\mu_{3}\nu_{3}},}\\\\[14.0pt]
{V_{4}^{\mu_{2},\mu_{3}\nu_{3}}(p_{1},p_{2},p_{3})=\alpha
p_{1}^{\mu_{2}}p_{3}^{2}\eta^{\mu_{3}\nu_{3}}-\alpha
p_{1}^{2}p_{3}^{\mu_{2}}\eta^{\mu_{3}\nu_{3}}-\alpha
p_{1}^{2}\left(p_{2}^{\mu_{3}}\eta^{\mu_{2}\nu_{3}}+p_{2}^{\nu_{3}}\eta^{\mu_{2}\mu_{3}}\right),}\\\\[14.0pt]
{V_{5}^{\mu_{2}}(p_{1},p_{2},p_{3})=-p_{1}^{\mu_{2}}p_{3}^{2}+p_{3}^{\mu_{2}}p_{1}^{2}.}\end{array}$
(3.18)

###  3.3 Feynman rules.

The results in the previous subsections lead to the Feynman rules in. figs. 1
and 2 . The reader should bear in mind the definitions in (3.13), (3.14),
(3.16), (3.17) and (3.18), when consulting those rules.

$\tensor{h}{{}_{\mu_{1}}{}_{\nu_{1}}}(p)$$f(-p)$$p$$=i\,G^{(hf)}_{\mu_{1}\nu_{1}}(p)$

$\tensor{h}{{}_{\mu_{1}}{}_{\nu_{1}}}(p)$$\tensor{h}{{}_{\mu_{2}}{}_{\nu_{2}}}(-p)$$p$$=i\,G^{(hh)}_{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p)$

$c^{T}_{\mu_{1}}(p)$$\tensor{b}{{}_{\mu_{2}}}(p)$$p$$=i\,G^{(c^{T}_{\mu}b_{\nu})}_{\mu_{1}\mu_{2}}(p)$

$c(p)$$\tensor{b}{{}_{\mu_{1}}}(p)$$p$$=i\,{G}^{(cb_{\nu})}_{\mu_{1}}(p)$

$c(p)$$b(p)$$p$$=i\,\tensor{G}{{}^{(cb)}}(p)$

Figure 1: Free propagators.

$p_{3}$$\tensor{h}{{}_{\mu_{3}}{}_{\nu_{3}}}(p_{3})$$p_{2}$$\tensor{h}{{}_{\mu_{2}}{}_{\nu_{2}}}(p_{2})$$p_{1}$$\tensor{h}{{}_{\mu_{1}}{}_{\nu_{1}}}(p_{1})$$=V^{\mu_{1}\nu_{1},\mu_{2}\nu_{2},\mu_{3}\nu_{3}}(p_{1},p_{2},p_{3})$

$p_{3}$$\tensor{h}{{}_{\mu_{3}}{}_{\nu_{3}}}(p_{3})$$p_{2}$$c^{T}_{\mu_{2}}(p_{2})$$p_{1}$$b_{\mu_{1}}(p_{1})$$=V^{\mu_{1},\mu_{2},\mu_{3}\nu_{3}}_{1}(p_{1},p_{2},p_{3})$

$p_{3}$$\tensor{h}{{}_{\mu_{3}}{}_{\nu_{3}}}(p_{3})$$p_{2}$$c(p_{2})$$p_{1}$$b_{\mu_{1}}(p_{1})$$=V^{\mu_{1},\mu_{3}\nu_{3}}_{2}(p_{1},p_{2},p_{3})$

$p_{3}$$\tensor{h}{{}_{\mu_{3}}{}_{\nu_{3}}}(p_{3})$$p_{2}$$c(p_{2})$$p_{1}$$b(p_{1})$$=V^{\mu_{3}\nu_{3}}_{3}(p_{1},p_{2},p_{3})$

$p_{3}$$\tensor{h}{{}_{\mu_{3}}{}_{\nu_{3}}}(p_{3})$$p_{2}$$c^{T}_{\mu_{2}}(p_{2})$$p_{1}$$b(p_{1})$$=V^{\mu_{2},\mu_{3}\nu_{3}}_{4}(p_{1},p_{2},p_{3})$

$p_{3}$$f(p_{3})$$p_{2}$$c^{T}_{{\mu_{2}}}(p_{2})$$p_{1}$$b(p_{1})$$=V^{\mu_{2}}_{5}(p_{1},p_{2},p_{3})$

Figure 2: Three-field Vertices.

##  4 The one-loop contribution to the graviton propagator.

Let $iG^{(hh\,(1))}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(p)$ denote the one-loop
contribution to the graviton propagator. Then, we have

$\begin{array}[]{l}{iG^{(hh\,(1))}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(p)=iG^{(hh)}_{\mu_{1}\mu_{2}\rho_{1}\rho_{2}}(p)\,\Gamma^{(hh)}_{\rho_{1}\rho_{2}\rho_{3}\rho_{4}}(p)\,iG^{(hh)}_{\rho_{3}\rho_{4}\mu_{3}\mu_{4}}(p)+}\\\\[4.0pt]
{\phantom{iG^{(hh\,(1))}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(p)=}iG^{(hf)}_{\mu_{1}\mu_{2}}(p)\,\Gamma^{(fh)}_{\rho_{3}\rho_{4}}(p)\,iG^{(hh)}_{\rho_{3}\rho_{4}\mu_{3}\mu_{4}}(p)+iG^{(hh)}_{\mu_{1}\mu_{2}\rho_{1}\rho_{2}}(p)\,\Gamma^{(hf)}_{\rho_{1}\rho_{2}}(p)\,iG^{(hf)}_{\mu_{3}\mu_{4}}(p).}\end{array}$
(4.19)

where $G^{(hh)}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(p)$ and
$G^{(hf)}_{\mu_{1}\mu_{2}}(p)$ have been given in (3.13) and (3.14),
respectively. $\Gamma^{(hh)}_{\rho_{1}\rho_{2}\rho_{3}\rho_{4}}(p)$ is the sum
of the values of all the 1PI diagrams in fig. 4,
$\Gamma^{(fh)}_{\rho_{3}\rho_{4}}(p)$ is given the value of the 1PI diagram
referred to as Diagram 8 in fig. 4(b). Also in fig. 4(b), the 1PI diagram
called Diagram 9 yields $\Gamma^{(hf)}_{\rho_{1}\rho_{2}}(p)$. The value of
each 1PI Feynman diagram in fig. 4 can be found in Appendix B. In the current
section, we shall display the value of
$\Gamma^{(hh)}_{\rho_{1}\rho_{2}\rho_{3}\rho_{4}}(p)$,
$\Gamma^{(hf)}_{\rho_{1}\rho_{2}}(p)$ and
$\Gamma^{(fh)}_{\rho_{1}\rho_{2}}(p)$ that, for arbitrary complex values of
$D$, we have obtained by using dimensional regularization.

$q$$p+q$$\tensor{h}{{}_{\mu_{1}}{}_{\mu_{2}}}$$\tensor{h}{{}_{\rho_{1}}{}_{\rho_{2}}}$$\tensor{h}{{}_{\mu_{3}}{}_{\mu_{4}}}$$\tensor{h}{{}_{\rho_{3}}{}_{\rho_{4}}}$$p$$p$
(a) Diagram 1 .

$p$$p$$q$$p+q$$\tensor{h}{{}_{\mu_{1}}{}_{\mu_{2}}}$$\tensor{h}{{}_{\rho_{1}}{}_{\rho_{2}}}$$\tensor{h}{{}_{\mu_{3}}{}_{\mu_{4}}}$$\tensor{h}{{}_{\rho_{3}}{}_{\rho_{4}}}$$\tensor{b}{{}_{\alpha}}$$c^{T}_{\beta}$$c^{T}_{\gamma}$$\tensor{b}{{}_{\delta}}$
(b) Diagram 2 .

$q$$p+q$$\tensor{h}{{}_{\mu_{1}}{}_{\mu_{2}}}$$\tensor{h}{{}_{\rho_{1}}{}_{\rho_{2}}}$$\tensor{h}{{}_{\mu_{3}}{}_{\mu_{4}}}$$\tensor{h}{{}_{\rho_{3}}{}_{\rho_{4}}}$$\tensor{b}{{}_{\alpha}}$$c$$c$$\tensor{b}{{}_{\beta}}$$p$$p$
(c) Diagram 3 .

$q$$p+q$$\tensor{h}{{}_{\mu_{1}}{}_{\mu_{2}}}$$\tensor{h}{{}_{\rho_{1}}{}_{\rho_{2}}}$$\tensor{h}{{}_{\mu_{3}}{}_{\mu_{4}}}$$\tensor{h}{{}_{\rho_{3}}{}_{\rho_{4}}}$$\tensor{b}{{}_{\alpha}}$$c^{T}_{\beta}$$c$$\tensor{b}{{}_{\delta}}$$p$$p$
(d) Diagram 4 .

$q$$p+q$$\tensor{h}{{}_{\mu_{1}}{}_{\mu_{2}}}$$\tensor{h}{{}_{\rho_{1}}{}_{\rho_{2}}}$$\tensor{h}{{}_{\mu_{3}}{}_{\mu_{4}}}$$\tensor{h}{{}_{\rho_{3}}{}_{\rho_{4}}}$$\tensor{b}{{}_{\alpha}}$$c$$c^{T}_{\gamma}$$b$$p$$p$
(e) Diagram 5 .

$q$$p+q$$\tensor{h}{{}_{\mu_{1}}{}_{\mu_{2}}}$$\tensor{h}{{}_{\rho_{1}}{}_{\rho_{2}}}$$\tensor{h}{{}_{\mu_{3}}{}_{\mu_{4}}}$$\tensor{h}{{}_{\rho_{3}}{}_{\rho_{4}}}$$b$$c$$c^{T}_{\alpha}$$\tensor{b}{{}_{\beta}}$$p$$p$
(f) Diagram 6

$q$$p+q$$\tensor{h}{{}_{\mu_{1}}{}_{\mu_{2}}}$$\tensor{h}{{}_{\rho_{1}}{}_{\rho_{2}}}$$\tensor{h}{{}_{\mu_{3}}{}_{\mu_{4}}}$$\tensor{h}{{}_{\rho_{3}}{}_{\rho_{4}}}$$\tensor{b}{{}_{\alpha}}$$c^{T}_{\beta}$$c$$b$$p$$p$
(g) Diagram 7

Figure 3: Non-vanishing contributions to
$\Gamma^{(hh)}_{\rho_{1}\rho_{2}\rho_{3}\rho_{4}}(p)$.

$q$$p+q$$\tensor{h}{{}_{\mu_{1}}{}_{\mu_{2}}}$$\tensor{h}{{}_{\mu_{3}}{}_{\mu_{4}}}$$\tensor{b}{{}_{\alpha}}$$c^{T}_{\beta}$$f$$c$$b$$p$$\tensor{h}{{}_{\rho_{3}}{}_{\rho_{4}}}$$p$
(a) Diagram 8

$q$$p+q$$\tensor{h}{{}_{\mu_{1}}{}_{\mu_{2}}}$$\tensor{h}{{}_{\mu_{3}}{}_{\mu_{4}}}$$\tensor{h}{{}_{\rho_{1}}{}_{\rho_{2}}}$$f$$b$$c$$c^{T}_{\alpha}$$\tensor{b}{{}_{\beta}}$$p$$p$
(b) Diagram 9

Figure 4: Non-vanishing contributions to $\Gamma^{(fh)}_{\rho_{3}\rho_{4}}(p)$
and $\Gamma^{(hf)}_{\rho_{1}\rho_{2}}(p)$.

.

A very involved computation –a computation not feasible without the invaluable
help from FORM– leads to the following result

$\begin{array}[]{l}{\Gamma^{(hh)}_{\rho_{1}\rho_{2}\rho_{3}\rho_{4}}(p)=}\\\\[2.0pt]
{P_{1}(p)\,(p^{2})^{2}\left(\eta_{\rho_{1}\rho_{3}}\eta_{\rho_{2}\rho_{4}}+\eta_{\rho_{1}\rho_{4}}\eta_{\rho_{2}\rho_{3}}\right)+P_{2}(p)\,(p^{2})^{2}\eta_{\rho_{1}\rho_{2}}\eta_{\rho_{3}\rho_{4}}+P_{3}(p)\,p^{2}\left(\eta_{\rho_{3}\rho_{4}}p_{\rho_{1}}p_{\rho_{2}}+\eta_{\rho_{1}\rho_{2}}p_{\rho_{3}}p_{\rho_{4}}\right)+}\\\\[4.0pt]
{P_{4}(p)\,p^{2}\left(\eta_{\rho_{1}\rho_{3}}p_{\rho_{2}}p_{\rho_{4}}+\eta_{\rho_{1}\rho_{4}}p_{\rho_{2}}p_{\rho_{3}}+\eta_{\rho_{2}\rho_{3}}p_{\rho_{1}}p_{\rho_{4}}+\eta_{\rho_{2}\rho_{4}}p_{\rho_{1}}p_{\rho_{3}}\right)+P_{5}(p)\,p_{\rho_{1}}p_{\rho_{2}}p_{\rho_{3}}p_{\rho_{4}},}\end{array}$
(4.20)

where

$\displaystyle
P_{1}(p)=\frac{i(-p^{2})^{D/2-2}\pi^{(1-D)/2}\Gamma(1-\frac{D}{2})\Gamma(\frac{D}{2})}{4^{3+D}\Gamma\bigl{(}\frac{D+3}{2}\bigr{)}}\Bigg{\\{}\frac{88+D\biggl{(}26+D\Bigl{(}D\bigl{(}24+(13-4D)D\bigr{)}-127\Bigr{)}\biggr{)}}{(D-2)^{2}}+$
$\displaystyle+32\frac{\frac{\gamma_{1}^{4}(\gamma_{1}+D\gamma_{2})^{2}}{\alpha_{1}^{2}\alpha_{2}^{2}}+(1+D)\bigl{(}\frac{(5-2D)D\gamma_{1}^{2}}{D-2}+(8-6D)\rho_{1}{}\bigr{)}}{D^{2}\alpha^{2}\gamma_{1}^{2}}-\frac{4(D-2)(1+D)\bigl{(}D(15+D)-28\bigr{)}\rho_{1}^{2}}{\gamma_{1}^{4}}+$
$\displaystyle+\frac{4(1+D)\biggl{(}48+D\Bigl{(}D\bigl{(}161+D(4D-47)\bigr{)}-182\Bigr{)}\biggr{)}\rho_{1}{}}{(D-2)\gamma_{1}^{2}}-\frac{16\gamma_{1}(\gamma_{1}+D\gamma_{2})}{\alpha\,\alpha_{1}\,\alpha_{2}}\Bigg{\\}},$
$\displaystyle
P_{2}(p)=\frac{i\left(-p^{2}\right)^{D/2-2}\Gamma(1-\frac{D}{2})\Gamma(\frac{D}{2})}{4^{3+D}\pi^{D/2}}\Bigg{\\{}-\frac{2^{5+D}\Gamma\left(\frac{D}{2}\right)}{\left(D-1\right)D\,\Gamma\left(D-2\right)}+$
$\displaystyle+\frac{\pi^{1/2}}{\left(\gamma_{1}^{2}\,D^{2}\left(D-2\right)\alpha\,\alpha_{1}\,\alpha_{2}\right)^{2}\Gamma\left(\frac{D+3}{2}\right)}\Bigg{[}16\left(D-2\right)^{2}D^{3}\left(4+3D\right)\alpha\,\alpha_{1}\alpha_{2}\gamma_{1}^{5}\left(\gamma_{1}+D\gamma_{2}\right)+$
$\displaystyle+32\left(D\left(D-2\right)\gamma_{1}^{3}\left(\gamma_{1}+D\gamma_{2}\right)\right)^{2}-\alpha_{1}^{2}\alpha_{2}^{2}\Bigg{[}\gamma_{1}^{4}\Big{(}32\left(D-2\right)\left(1+D\right)\left(D\left(76+\left(D-28\right)D\right)-32\right)+$
$\displaystyle+\alpha^{2}\Big{(}D\left(640+D\left(D\left(2856+D\left(1668+D\left(D\left(1371+D\left(D\left(4D-37\right)-100\right)\right)-3382\right)\right)\right)-3424\right)\right)+$
$\displaystyle+512\Big{)}\Big{)}+4\left(D-2\right)\left(1+D\right)\Bigg{(}\Bigg{(}D\Big{(}128+D\Big{(}3040+D(D\left(3748+D\left(D\left(79+3D\right)-1006\right)\right)-$
$\displaystyle-5576)\Big{)}\Big{)}-512\Bigg{)}\alpha^{2}-16\left(D-2\right)\Big{(}D\left(28+\left(D-4\right)D\right)-32\Big{)}\Bigg{)}\gamma_{1}^{2}\rho_{1}{}+$
$\displaystyle+4(\alpha\rho_{1})^{2}\left(D-2\right)^{3}\left(1+D\right)\Bigg{(}D\Big{(}64+D\Big{(}1168+D\Big{(}D(636+D\left(9D-133\right))-1404\Big{)}\Big{)}\Big{)}-256\Bigg{)}\Bigg{]}\Bigg{]}\Bigg{\\}},$

$\displaystyle
P_{3}(p)=\frac{i(-p^{2})^{D/2-2}\Gamma(1-\frac{D}{2})\Gamma(\frac{D}{2})}{4^{3+D}\pi^{D/2}}\Bigg{\\{}\frac{2^{4+D}\Gamma(\frac{D}{2})}{(D-1)\Gamma(D-2)}+$
$\displaystyle+\frac{\pi^{1/2}}{D^{3}((D-2)\alpha\,\alpha_{1}\,\alpha_{2}\gamma_{1}^{2})^{2}\Gamma\bigl{(}\frac{D+3}{2}\bigr{)}}\Bigg{[}-16(D-2)^{2}D^{2}\bigl{(}2+D(2+D)\bigr{)}\alpha\,\alpha_{1}\,\alpha_{2}\gamma_{1}^{5}(\gamma_{1}+D\gamma_{2})-$
$\displaystyle-32D(2+D)(\gamma_{1}^{3}(D-2)(\gamma_{1}+D\gamma_{2}))^{2}+\alpha_{1}^{2}\alpha_{2}^{2}\Bigg{[}\Bigg{[}32(D-2)(1+D)\Bigl{(}D\bigl{(}76+D(5D-38)\bigr{)}-32\Bigr{)}+$
$\displaystyle+\Bigg{[}512+D\Bigg{(}640+D\Big{(}D\left(3128+D\Biggl{(}1296+D\biggl{(}D\Bigl{(}1747+D\bigl{(}D(4D-5)-302\bigr{)}\Bigr{)}-3400\biggr{)}\Biggr{)}\right)-$
$\displaystyle-3552\Big{)}\Bigg{)}\Bigg{]}\alpha^{2}\Bigg{]}\gamma_{1}^{4}-4(D-2)(1+D)\gamma_{1}^{2}\rho_{1}{}\Bigg{(}16(D-2)\Bigl{(}D\bigl{(}36+(D-10)D\bigr{)}-32\Bigr{)}+$
$\displaystyle+(D-4)\alpha^{2}\Biggl{(}D^{2}\biggl{(}776+D\Bigl{(}D\bigl{(}836+D(17D-213)\bigr{)}-1312\Bigr{)}\biggr{)}-128\Biggr{)}\Bigg{)}+$
$\displaystyle+4(D-2)^{3}(1+D)\alpha^{2}\rho_{1}^{2}\left(D\Biggl{(}64+D\biggl{(}1168+D\Bigl{(}D\bigl{(}666+D(9D-131)\bigr{)}-1460\Bigr{)}\biggr{)}\Biggr{)}-256\right)\Bigg{]}\Bigg{]}\Bigg{\\}},$
$\displaystyle
P_{4}(p)=-\frac{i(-p^{2})^{D/2-2}\pi^{(1-D)/2}\Gamma(1-\frac{D}{2})\Gamma(\frac{D}{2})}{4^{3+D}\left(D(D-2)\alpha\gamma_{1}^{2}\right)^{2}\,\alpha_{1}\,\alpha_{2}\Gamma\bigl{(}\frac{D+3}{2}\bigr{)}}\Bigg{\\{}16\alpha\gamma_{1}^{5}(D-2)^{2}\left(\gamma_{1}+\gamma_{2}D\right)D(D(D-2)-2)-$
$\displaystyle-32\alpha_{1}\alpha_{2}\gamma_{1}^{2}(D-2)^{2}\Bigg{(}\frac{\gamma_{1}^{4}D\left(\gamma_{1}+\gamma_{2}D\right){}^{2}}{\alpha_{1}^{2}\alpha_{2}^{2}}+(D+1)\Big{(}\frac{\gamma_{1}^{2}D(2D-5)}{D-2}+2(3D-4)\rho_{1}\Big{)}\Bigg{)}-$
$\displaystyle-\alpha^{2}\alpha_{1}\alpha_{2}D^{2}\Bigg{[}4(D+1)(D(D+15)-28)(D-2)^{3}\rho_{1}^{2}+{\gamma_{1}^{4}}D\Bigg{(}D\Big{(}D(D(4D-13)-24)+127\Big{)}-$
$\displaystyle-26\Bigg{)}-\gamma_{1}^{2}4(D+1)\Bigg{(}D\Big{(}D\Big{(}D(4D-47)+161\Big{)}-182\Big{)}+48\Bigg{)}(D-2)\rho_{1}-88{\gamma_{1}^{4}}\Bigg{]}\Bigg{\\}},$

$\displaystyle
P_{5}(p)=-\frac{i(-p^{2})^{D/2-2}\pi^{(1-D)/2}\Gamma(1-\frac{D}{2})\Gamma(\frac{D}{2})}{4^{3+D}(D-2)(D\alpha\,\alpha_{1}\,\alpha_{2}\gamma_{1}^{2})^{2}\Gamma\bigl{(}\frac{D+3}{2}\bigr{)}}\Bigg{\\{}(4D-8)^{2}D\bigl{(}D(D-4)-4\bigr{)}\alpha\alpha_{1}\alpha_{2}\gamma_{1}^{5}(\gamma_{1}+D\gamma_{2})-$
$\displaystyle-32((D-2)\gamma_{1}^{3}(\gamma_{1}+D\gamma_{2}))^{2}D+\alpha_{1}^{2}\alpha_{2}^{2}\Bigg{[}\gamma_{1}^{4}\Bigg{[}32(D-2)(1+D)\bigl{(}16+5(D-4)D\bigr{)}+$
$\displaystyle+\alpha^{2}\Bigg{[}D\Bigg{(}D\left(1792+D\Biggl{(}D\biggl{(}D\Bigl{(}1303+D\bigl{(}D(27+4D)-408\bigr{)}\Bigr{)}-1162\biggr{)}-840\Biggr{)}\right)-448\Bigg{)}-$
$\displaystyle-256\Bigg{]}\Bigg{]}-4(D-2)(1+D)\Bigg{(}16(D-2)\bigl{(}8+(D-8)D\bigr{)}+\Biggl{(}(D-4)D\biggl{(}16-D\Bigl{(}360-$
$\displaystyle-D\bigl{(}434+D(29D-199)\bigr{)}\Bigr{)}\biggr{)}-256\Biggr{)}\alpha^{2}\Bigg{)}\gamma_{1}^{2}\rho_{1}{}+4(D-2)^{3}(1+D)\Biggl{(}128+$
$\displaystyle+D\biggl{(}32+D\Bigl{(}D\bigl{(}444+D(9D-113)\bigr{)}-512\Bigr{)}\biggr{)}\Biggr{)}\alpha^{2}\rho_{1}^{2}\Bigg{]}\Bigg{\\}}.$

For $\Gamma^{(fh)}_{\rho_{1}\rho_{2}}(p)$, we have obtained the following
value

$\Gamma^{(fh)}_{\rho_{1}\rho_{2}}(p)=Q_{1}(p)\,(p^{2})^{2}\eta_{\rho_{1}\rho_{2}}+Q_{2}(p)\,p^{2}p_{\rho_{1}}p_{\rho_{2}},$
(4.21)

where

$\begin{array}[]{l}{Q_{1}(p)=i\,\dfrac{2^{-(2+D)}\pi^{-D/2}(-p^{2})^{-2+D/2}\Gamma\left(1-D/2\right)(\Gamma\left(D/2\right))^{2}}{\alpha(-1+D)D\Gamma\left(D-2\right)}}\\\\[4.0pt]
{Q_{2}(p)=-i\,\dfrac{2^{-(2+D)}\pi^{-D/2}(-p^{2})^{-2+D/2}\Gamma\left(1-D/2\right)(\Gamma\left(D/2\right))^{2}}{\alpha(-1+D)\Gamma\left(D-2\right)}.}\end{array}$
(4.22)

Not surprisingly,
$\Gamma^{(hf)}_{\rho_{1}\rho_{2}}(p)=\Gamma^{(fh)}_{\rho_{1}\rho_{2}}(p)$.

###  4.1 Two simplifying gauges

Let us present two gauge choices for which
$\Gamma^{(hh)}_{\rho_{1}\rho_{2}\rho_{3}\rho_{4}}(p)$ simplifies greatly.
These gauge choices simplify as much as possible the involved tensor structure
of the free-graviton propagator given in (3.12) and (3.13). Let us first point
out that the last two summands -the most involved ones– of that tensor
structure cannot be set to zero at the same time. Actually, this is needed to
obtain Newton’s law as seen in Appendix C.

Let us set $\alpha^{2}=\dfrac{(2D-4)}{D^{2}}$ and
$\rho_{1}=\dfrac{1}{4}\gamma_{1}^{2}$. This choice sets to zero the second and
third summands of $G^{(hh)}_{\mu_{1},\mu_{2},\mu_{3},\mu_{4}}(p)$ in (3.13).
Let us further impose $\gamma_{2}=-\dfrac{\gamma_{1}}{D}$. Then,
$P_{1}(p),P_{2}(p),P_{3}(p)$ and $P_{4}(p)$ in (4.20) read

$\begin{array}[]{l}{P_{1}(p)=i\,\left(256+D(-864+D(-260+D(549+5D(-40+3D))))\right)\,\Phi(p,D),}\\\\[8.0pt]
{P_{2}(p)=-i\,\dfrac{1}{D^{4}}(-4096+D(-3072+D(26112+D(-19200+D(-15664+D(19812}\\\\[4.0pt]
{\phantom{P_{2}(p)=-i\,\dfrac{1}{D^{4}}(-4096+D}+D(-7520+D(1247+3D(-44+3D)))))))))\,\Phi(p,D),}\\\\[8.0pt]
{P_{3}(p)=i\dfrac{1}{D^{3}}(-4096+D(-3072+D(26624+D(-21888+D(-12912}\\\\[4.0pt]
{\phantom{P_{3}(p)=i\dfrac{1}{D^{3}}\quad\quad}+D(21932+D(-10202+3D(709+D(-70+3D)))))))))\,\Phi(p,D),}\\\\[8.0pt]
{P_{4}(p)=-i\,(256+D(-864+D(-260+D(549+5D(-40+3D)))))\,\Phi(p,D),}\\\\[8.0pt]
{P_{5}(p)=-i\dfrac{1}{D^{2}}(2048+D(2560+D(-11776+3D(1472}\\\\[4.0pt]
{\phantom{P_{2}(p)=-i\,\dfrac{1}{D^{4}}}+D(2544+D(-2284+D(689+D(-80+3D))))))))\,\left(D-2\right)\Phi(p,D),}\end{array}$

where

$\Phi(p,D)=\dfrac{4^{-(4+D)}\pi^{(1-D)/2}(-p^{2})^{-2+D/2}\Gamma\left(1-D/2\right)\Gamma\left(D/2\right)}{\left(D-2\right)\Gamma\left(3+D/2\right)}.$

The second gauge choice sets to zero the last summand of
$G^{(hh)}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(p)$ in (3.13) and reads

$\alpha^{2}=\dfrac{(2D-4)}{D^{2}},\,\rho_{1}=-\dfrac{1}{2(D-2)}\gamma_{1}^{2}\quad\text{and}\quad\gamma_{2}=-\dfrac{\gamma_{1}}{D}.$

For this gauge choice, $P_{1}(p),P_{2}(p),P_{3}(p)$ and $P_{4}(p)$ in (4.20)
are given by the following expressions

$\begin{array}[]{l}{P_{1}(p)=-i\,(-32+D(74+D(51+D(-37+6D))))\Psi(p,D),}\\\\[8.0pt]
{P_{2}(p)=-i\,\dfrac{1}{D^{4}}(-512+D(-640+D(3040}\\\\[4.0pt]
{\phantom{P_{2}(p)=-i\,\dfrac{1}{1}}+D(-984+D(-2468+D(1498+D(-213+D(-13+2D))))))))\,\Psi(p,D),}\\\\[8.0pt]
{P_{3}(p)=i\,\dfrac{1}{D^{3}}(-512+D(-640+D(3104+D(-1304}\\\\[4.0pt]
{\phantom{P_{3}(p)=i\,\dfrac{1}{D^{3}}(-512}+D(-2256+D(1816+D(-431+D(23+2D))))))))\,\Psi(p,D),}\\\\[8.0pt]
{P_{4}(p)=i\,(-32+D(74+D(51+D(-37+6D))))\,\Psi(p,D),}\\\\[8.0pt]
{P_{5}(p)=-i\,\dfrac{1}{D^{2}}(256+D(448+D(-1296}\\\\[4.0pt]
{\phantom{P_{5}(p)=-i\,\dfrac{1}{D^{2}}}+D(-16+D(986+D(-449+D(51+2D)))))))\,\left(D-2\right)\Psi(p,D),}\end{array}$

where

$\Psi(p,D)=\dfrac{2^{-(5+2D)}\pi^{(1-D)/2}(-p^{2})^{-2+D/2}\Gamma\left(1-D/2\right)\Gamma\left(D/2\right)}{\left(D-2\right)\Gamma\left(3+D/2\right)}.$

##  5 Comments and conclusions.

We have spelt out a BRST formalism where quantum corrections, in any number of
dimensions and around a Minkowski background, can be computed for unimodular
gravity with Weyl invariance. We have applied the formalism to the one-loop
computation of the complete graviton propagator for a large family of gauge-
fixing terms and in any number of dimensions. Our main conclusion is that
computing quantum gravitational corrections in unimodular gravity is feasible.
Of course, the computations are very involved due to the fact that the free
graviton propagator contains higher-order poles.

##  6 Acknowledgements.

We are thankfull to Professor Enrique Álvarez Vázquez for plenty of
illuminating blackboard discussions. One of us (EVA) is grateful for
informative email exchange with M. Herrero-Valea and acknowledges partial
financial support by the Spanish MINECO through the Centro de excelencia
Severo Ochoa Program under Grant CEX2020-001007-S funded by
MCIN/AEI/10.13039/501100011033 as well as to the European Union’s Horizon 2020
research and innovation programme under the Marie Sklodowska-Curie grant
agreement No 860881-HIDDeN and also byGrant PID2019-108892RB-I00 funded by
MCIN/AEI/ 10.13039/501100011033 and by “ERDF A way of making Europe”.

##  Appendix A  Results when D is close to 4.

Here we shall display the values of
$\Gamma^{(hh)}_{\rho_{1}\rho_{2}\rho_{3}\rho_{4}}(p)$ and
$\Gamma^{(fh)}_{\rho_{1}\rho_{2}}(p)$ in the vicinity of $D=4$. We shall
choose the gauges in the subsection 4.1. Let us begin with the gauge choice
$\alpha^{2}=\dfrac{(2D-4)}{D^{2}},\,\rho_{1}=\dfrac{1}{4}\gamma_{1}^{2}$ and
$\gamma_{2}=-\dfrac{\gamma_{1}}{D}$. For this gauge choice,
$P_{1}(p),\,P_{2}(p),\,P_{3}(p)$ and $P_{4}(p)$ in (4.20) read

$\begin{array}[]{l}{P_{1}(p)=-i\,\dfrac{21}{320\pi^{2}}\big{\\{}\dfrac{1}{D-4}+\dfrac{1}{2}{\rm
Ln}\left(-\frac{p^{2}}{4\pi
e^{-\gamma}\mu^{2}}\right)\big{\\}}+i\,\dfrac{813}{12800\pi^{2}}+O(D-4),}\\\\[8.0pt]
{P_{2}(p)=i\,\dfrac{37}{960\pi^{2}}\big{\\{}\dfrac{1}{D-4}+\dfrac{1}{2}{\rm
Ln}\left(-\frac{p^{2}}{4\pi
e^{-\gamma}\mu^{2}}\right)\big{\\}}-i\,\dfrac{13691}{230400\pi^{2}}+O(D-4),}\\\\[8.0pt]
{P_{3}(p)=-i\,\dfrac{11}{480\pi^{2}}\big{\\{}\dfrac{1}{D-4}+\dfrac{1}{2}{\rm
Ln}\left(-\frac{p^{2}}{4\pi
e^{-\gamma}\mu^{2}}\right)\big{\\}}+i\,\dfrac{3779}{57600\pi^{2}}+O(D-4),}\\\\[8.0pt]
{P_{4}(p)=i\,\dfrac{21}{320\pi^{2}}\big{\\{}\dfrac{1}{D-4}+\dfrac{1}{2}{\rm
Ln}\left(-\frac{p^{2}}{4\pi
e^{-\gamma}\mu^{2}}\right)\big{\\}}-i\,\dfrac{813}{12800\pi^{2}}+O(D-4),}\\\\[8.0pt]
{P_{5}(p)=-i\,\dfrac{41}{240\pi^{2}}\big{\\{}\dfrac{1}{D-4}+\dfrac{1}{2}{\rm
Ln}\left(-\frac{p^{2}}{4\pi
e^{-\gamma}\mu^{2}}\right)\big{\\}}+i\,\dfrac{1169}{28800\pi^{2}}+O(D-4).}\end{array}$

When the gauge parameters satisfy
$\alpha^{2}=\dfrac{(2D-4)}{D^{2}},\,\rho_{1}=-\dfrac{1}{2(D-2)}\gamma_{1}^{2}\quad\text{and}\quad\gamma_{2}=-\dfrac{\gamma_{1}}{D}$,
we have

$\begin{array}[]{l}{P_{1}(p)=-i\,\dfrac{31}{1920\pi^{2}}\big{\\{}\dfrac{1}{D-4}+\dfrac{1}{2}{\rm
Ln}\left(-\frac{p^{2}}{4\pi
e^{-\gamma}\mu^{2}}\right)\big{\\}}+i\,\dfrac{1967}{115200\pi^{2}}+O(D-4),}\\\\[8.0pt]
{P_{2}(p)=i\,\dfrac{17}{960\pi^{2}}\big{\\{}\dfrac{1}{D-4}+\dfrac{1}{2}{\rm
Ln}\left(-\frac{p^{2}}{4\pi
e^{-\gamma}\mu^{2}}\right)\big{\\}}-i\,\dfrac{1583}{115200\pi^{2}}+O(D-4),}\\\\[8.0pt]
{P_{3}(p)=-i\,\dfrac{1}{80\pi^{2}}\big{\\{}\dfrac{1}{D-4}+\dfrac{1}{2}{\rm
Ln}\left(-\frac{p^{2}}{4\pi
e^{-\gamma}\mu^{2}}\right)\big{\\}}+i\,\dfrac{53}{3200\pi^{2}}+O(D-4),}\\\\[8.0pt]
{P_{4}(p)=i\,\dfrac{31}{1920\pi^{2}}\big{\\{}\dfrac{1}{D-4}+\dfrac{1}{2}{\rm
Ln}\left(-\frac{p^{2}}{4\pi
e^{-\gamma}\mu^{2}}\right)\big{\\}}-i\,\dfrac{1967}{115200\pi^{2}}+O(D-4),}\\\\[8.0pt]
{P_{5}(p)=-i\,\dfrac{19}{160\pi^{2}}\big{\\{}\dfrac{1}{D-4}+\dfrac{1}{2}{\rm
Ln}\left(-\frac{p^{2}}{4\pi
e^{-\gamma}\mu^{2}}\right)\big{\\}}-i\,\dfrac{209}{3200\pi^{2}}+O(D-4).}\end{array}$

$Q_{1}(p)$ and $Q_{2}(p)$ in (4.22) have the following Laurent expansion at
$D=4$:

$\begin{array}[]{l}{Q_{1}(p)=i\,\dfrac{1}{384\alpha\pi^{2}}\big{\\{}\dfrac{1}{D-4}+\dfrac{1}{2}{\rm
Ln}\left(-\frac{p^{2}}{4\pi
e^{-\gamma}\mu^{2}}\right)\big{\\}}-i\,\dfrac{13}{4608\alpha\pi^{2}}+O(D-4),}\\\\[8.0pt]
{Q_{2}(p)=-i\,\dfrac{1}{96\alpha\pi^{2}}\big{\\{}\dfrac{1}{D-4}+\dfrac{1}{2}{\rm
Ln}\left(-\frac{p^{2}}{4\pi
e^{-\gamma}\mu^{2}}\right)\big{\\}}+i\,\dfrac{5}{576\alpha\pi^{2}}+O(D-4).}\end{array}$

In the previous result $\gamma$ stands for the Euler’s constant and $\mu$ for
the dimensional regularization scale.

Let us finally point out that in the minimal subtraction scheme of dimensional
regularization the renormalized one-loop correction to the graviton propagator
is obtained by removing first the single pole at $D=4$ from the $P_{i}(p)$’s
and $Q_{i}(p)$’s above, then, inserting the resulting renormalized
$P_{i}(p)$’s and $Q_{i}(p)$’s in the corresponding formulae in (4.20) and
(4.21) and, finally, substituting the so renormalized
$\Gamma^{(hh)}_{\rho_{1}\rho_{2}\rho_{3}\rho_{4}}(p)$ and
$\Gamma^{(fh)}_{\rho_{1}\rho_{2}}(p)$ in (4.19), as $D$ goes to 4.

##  Appendix B Value of each Feynman diagram.

Each diagram in Figure 3 is given by the following general tensor structure

$\begin{array}[]{l}{R_{1}(p)\,(p^{2})^{2}\left(\eta_{\rho_{1}\rho_{3}}\eta_{\rho_{2}\rho_{4}}+\eta_{\rho_{1}\rho_{4}}\eta_{\rho_{2}\rho_{3}}\right)+R_{2}(p)\,(p^{2})^{2}\eta_{\rho_{1}\rho_{2}}\eta_{\rho_{3}\rho_{4}}+R_{3}(p)\,p^{2}\left(\eta_{\rho_{3}\rho_{4}}p_{\rho_{1}}p_{\rho_{2}}+\eta_{\rho_{1}\rho_{2}}p_{\rho_{3}}p_{\rho_{4}}\right)+}\\\\[4.0pt]
{R_{4}(p)\,p^{2}\left(\eta_{\rho_{1}\rho_{3}}p_{\rho_{2}}p_{\rho_{4}}+\eta_{\rho_{1}\rho_{4}}p_{\rho_{2}}p_{\rho_{3}}+\eta_{\rho_{2}\rho_{3}}p_{\rho_{1}}p_{\rho_{4}}+\eta_{\rho_{2}\rho_{4}}p_{\rho_{1}}p_{\rho_{3}}\right)+R_{5}(p)\,p_{\rho_{1}}p_{\rho_{2}}p_{\rho_{3}}p_{\rho_{4}},}\end{array}$

where the value of each $R_{i}(p)$, $i=1...5$, depends on the diagram. Below
we give the value of $R_{i}(p)$, $i=1..5$, for each diagram in such figure.

Diagram 1 in figure 3.

$\begin{array}[]{l}{R_{1}(p)=\frac{i\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{4^{3+D}(-2+D)^{2}D^{2}\alpha^{2}\gamma_{1}^{4}\Gamma\bigl{(}\tfrac{D+3}{2}\bigr{)}}f_{1}[D,\alpha,\rho_{1}\gamma_{1}],}\end{array}$

with

$\displaystyle
f_{1}[D,\alpha,\rho_{1}\gamma_{1}]=D\Bigg{\\{}-320-32D\bigl{(}1+D(-7+2D)\bigr{)}+$
$\displaystyle+D\Bigg{[}120+D\biggl{(}34+D\Bigl{(}-167+D\bigl{(}42+(11-4D)D\bigr{)}\Bigr{)}\biggr{)}\Bigg{]}\alpha^{2}\Bigg{\\}}\gamma_{1}^{4}+$
$\displaystyle+4(-2+D)(1+D)\Bigg{\\{}-128+160D-48D^{2}+$
$\displaystyle+D^{2}\biggl{(}48+D\Bigl{(}-182+D\bigl{(}161+D(-47+4D)\bigr{)}\Bigr{)}\biggr{)}\alpha^{2}\Bigg{\\}}\gamma_{1}^{2}\rho_{1}-$
$\displaystyle-4(-2+D)^{3}D^{2}(1+D)\bigl{(}-28+D(15+D)\bigr{)}\alpha^{2}\rho_{1}^{2};$

$\begin{array}[]{l}{R_{2}(p)=-\frac{i\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{4^{3+D}(-2+D)^{2}D^{4}\alpha^{2}\gamma_{1}^{4}\Gamma\bigl{(}\tfrac{D+3}{2}\bigr{)}}f_{2}[D,\alpha,\rho_{1}\gamma_{1}],}\end{array}$

with

$\displaystyle
f_{2}[D,\alpha,\rho_{1}\gamma_{1}]=\Bigg{\\{}32(-2+D)(1+D)\Bigl{(}-32+D\bigl{(}76+(-28+D)D\bigr{)}\Bigr{)}+$
$\displaystyle+\Bigg{[}512+640D+D^{2}\Big{[}-3424+D\Big{(}2856+D\big{(}1700+$
$\displaystyle+D\left[-3326+D\left(1363+D(-150+D(-19+4D))\right)\right]\big{)}\Big{)}\Big{]}\Bigg{]}\alpha^{2}\Bigg{\\}}\gamma_{1}^{4}+$
$\displaystyle+4(-2+D)(1+D)\Bigg{\\{}-16(-2+D)\Big{[}-32+D\bigl{(}28+(-4+D)D\bigr{)}\Big{]}+$
$\displaystyle+\Bigg{[}-512+128D+3040D^{2}+$
$\displaystyle+D^{3}\left[-5576+D\Bigl{(}3748+D\bigl{(}-1006+D(79+3D)\bigr{)}\Bigr{)}\right]\Bigg{]}\alpha^{2}\Bigg{\\}}\gamma_{1}^{2}\rho_{1}+$
$\displaystyle+4(-2+D)^{3}(1+D)\Bigg{\\{}-256+64D+$
$\displaystyle+D^{2}\Big{(}1168+D\left[-1404+D\left(636+D(-133+9D)\right)\right]\Big{)}\Bigg{\\}}\alpha^{2}\rho_{1}^{2};$

$\displaystyle
R_{3}(p)=-\frac{i\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{4^{3+D}(-2+D)^{2}D^{3}\alpha^{2}\gamma_{1}^{4}\Gamma\bigl{(}\tfrac{D+3}{2}\bigr{)}}f_{3}[D,\alpha,\rho_{1}\gamma_{1}],$

with

$\displaystyle
f_{3}[D,\alpha,\rho_{1}\gamma_{1}]=\Bigg{\\{}-32(-2+D)(1+D)\Bigl{(}-32+D\bigl{(}76+D(-38+5D)\bigr{)}\Bigr{)}-$
$\displaystyle-\Bigg{[}512+640D+D^{2}\Bigg{(}-3552+D\Big{[}3128+D\Big{(}1344+$
$\displaystyle+D\biggl{(}-3408+D\left[1695+D\bigl{(}-268+D(-11+4D)\bigr{)}\right]\biggr{)}\Big{)}\Big{]}\Bigg{)}\Bigg{]}\alpha^{2}\Bigg{\\}}\gamma
1^{4}+$
$\displaystyle+4(-2+D)(1+D)\Bigg{\\{}16(-2+D)\Bigl{(}-32+D\bigl{(}36+(-10+D)D\bigr{)}\Bigr{)}+$
$\displaystyle+(4-D)\Bigg{[}128-D^{2}\Biggl{(}776+D\big{[}-1312+D\bigl{(}836+D(17D-213)\bigr{)}\big{]}\Biggr{)}\Bigg{]}\alpha^{2}\Bigg{\\}}\gamma
1^{2}\rho_{1}-$ $\displaystyle-4(-2+D)^{3}(1+D)\Bigg{\\{}-256+64D+$
$\displaystyle+D^{2}\biggl{(}1168+D\Bigl{(}-1460+D\bigl{(}666+D(-131+9D)\bigr{)}\Bigr{)}\biggr{)}\Bigg{\\}}\alpha^{2}\rho_{1}^{2};$

$\displaystyle
R_{4}(p)=-\frac{i\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{4^{3+D}(-2+D)^{2}D^{2}\alpha^{2}\gamma_{1}^{4}\Gamma\bigl{(}\tfrac{D+3}{2}\bigr{)}}f_{4}[D,\alpha,\rho_{1}\gamma_{1}],$

with

$\displaystyle
f_{4}[D,\alpha,\rho_{1}\gamma_{1}]=-D\Bigg{\\{}32(-2+D)(1+D)(-5+2D)+$
$\displaystyle+D\Biggl{[}-152+D\biggl{(}6+D\Bigl{(}183+D\bigl{(}-80+D(5+2D)\bigr{)}\Bigr{)}\biggr{)}\Biggr{]}\alpha^{2}\Bigg{\\}}\gamma
1^{4}+$ $\displaystyle+4(-2+D)(1+D)\Bigg{\\{}-128+160D-48D^{2}+$
$\displaystyle+D^{2}\Biggl{(}48+D\bigl{(}-182+D\bigl{(}161+D(-47+4D)\bigr{)}\bigr{)}\Biggr{)}\alpha^{2}\Bigg{\\}}\gamma
1^{2}\rho_{1}-$
$\displaystyle-4(-2+D)^{3}D^{2}(1+D)\bigl{(}-28+D(15+D)\bigr{)}\alpha^{2}\rho_{1}^{2};$

$\displaystyle
R_{5}(p)=-\frac{i\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{4^{3+D}(-2+D)^{2}D^{2}\alpha^{2}\gamma_{1}^{4}\Gamma\bigl{(}\tfrac{D+3}{2}\bigr{)}}f_{5}[D,\alpha,\rho_{1}\gamma_{1}],$

with

$\displaystyle
f_{5}[D,\alpha,\rho_{1}\gamma_{1}]=\Bigg{\\{}32(-2+D)(1+D)\bigl{(}16+5(-4+D)D\bigr{)}+$
$\displaystyle+\Bigg{[}-256+D\Big{(}-448+D\big{[}1920+D\Big{(}-872+D\big{(}-1266+D\big{(}1375+$
$\displaystyle+D\bigl{(}-426+D(29+4D)\bigr{)}\big{)}\big{)}\Big{)}\big{]}\Big{)}\Bigg{]}\alpha^{2}\Bigg{\\}}\gamma_{1}^{4}-$
$\displaystyle-4(-2+D)(1+D)\Bigg{\\{}-256(1+\alpha^{2})+(-4+D)D\Biggl{(}16(-6+D)+$
$\displaystyle+\biggl{(}16+D\Bigl{(}-360+D\bigl{(}434+D(-199+29D)\bigr{)}\Bigr{)}\biggr{)}\alpha^{2}\Biggr{)}\Bigg{\\}}\gamma
1^{2}\rho_{1}+$
$\displaystyle+4(D-2)^{3}(1+D)\Bigg{\\{}128+32D+D^{2}\Bigl{(}-512+D\bigl{(}444+D(9D-113)\bigr{)}\Bigr{)}\Bigg{\\}}\alpha^{2}\rho_{1}^{2}.$

Diagram 2 in figure 3.

$\displaystyle
R_{1}(p)=\frac{i\bigl{(}-4+(-5+D)D\bigr{)}\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{2^{5+2D}\Gamma\bigl{(}\tfrac{3+D}{2}\bigr{)}},$
$\displaystyle
R_{2}(p)=\frac{i\bigl{(}4+D(11+9D)\bigr{)}\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{2^{5+2D}\Gamma\bigl{(}\tfrac{3+D}{2}\bigr{)}},$
$\displaystyle
R_{3}(p)=\frac{iD\bigl{(}-6+(-5+3D)D\bigr{)}\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{2^{5+2D}\Gamma\bigl{(}\tfrac{3+D}{2}\bigr{)}},$
$\displaystyle
R_{4}(p)=\frac{i\bigl{(}8+(-4+D)(-1+D)D\bigr{)}\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{2^{5+2D}\Gamma\bigl{(}\tfrac{3+D}{2}\bigr{)}},$
$\displaystyle
R_{5}(p)=\frac{i(-2+D)\Bigl{(}16+D\bigl{(}12+(-5+D)D\bigr{)}\Bigr{)}\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{2^{5+2D}\Gamma\bigl{(}\tfrac{3+D}{2}\bigr{)}}.$

Diagram 3 in figure 3.

$\displaystyle
R_{1}(p)=\frac{i\gamma_{1}^{2}(\gamma_{1}+D\gamma_{2})^{2}\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{2^{1+2D}D^{2}\alpha^{2}\alpha_{1}^{2}\alpha_{2}^{2}\Gamma\bigl{(}\tfrac{D+3}{2}\bigr{)}},$
$\displaystyle
R_{2}(p)=\frac{i\gamma_{1}^{2}(\gamma_{1}+D\gamma_{2})^{2}\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{2^{1+2D}D^{2}\alpha^{2}\alpha_{1}^{2}\alpha_{2}^{2}\Gamma\bigl{(}\tfrac{D+3}{2}\bigr{)}},$
$\displaystyle
R_{3}(p)=-\frac{i(2+D)\gamma_{1}^{2}(\gamma_{1}+D\gamma_{2})^{2}\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{2^{1+2D}D^{2}\alpha^{2}\alpha_{1}^{2}\alpha_{2}^{2}\Gamma\bigl{(}\tfrac{D+3}{2}\bigr{)}},$
$\displaystyle
R_{4}(p)=\frac{i\gamma_{1}^{2}(\gamma_{1}+D\gamma_{2})^{2}\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{2^{1+2D}D\alpha^{2}\alpha_{1}^{2}\alpha_{2}^{2}\Gamma\bigl{(}\tfrac{D+3}{2}\bigr{)}},$
$\displaystyle
R_{5}(p)=\frac{i\gamma_{1}^{2}(\gamma_{1}+D\gamma_{2})^{2}\left(D-2\right)\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{2^{1+2D}D\alpha^{2}\alpha_{1}^{2}\alpha_{2}^{2}\Gamma\bigl{(}\tfrac{D+3}{2}\bigr{)}}.$

Diagrams 4 and 5 in figure 3.

Diagrams 4 and 5 have the same $R_{i}(p)$, $i=1..5$, which are given next:

$\displaystyle
R_{1}(p)=-i\frac{\gamma_{1}(\gamma_{1}+D\gamma_{2})\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{2^{3+2D}\alpha\alpha_{1}\alpha_{2}\Gamma\bigl{(}\tfrac{D+3}{2}\bigr{)}},$
$\displaystyle
R_{(}p)=i\frac{(4+3D)\gamma_{1}(\gamma_{1}+D\gamma_{2})\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{2^{3+2D}D\alpha\alpha_{1}\alpha_{2}\Gamma\bigl{(}\tfrac{D+3}{2}\bigr{)}},$
$\displaystyle
R_{3}(p)=-i\frac{\bigl{(}2+D(2+D)\bigr{)}\gamma_{1}(\gamma_{1}+D\gamma_{2})\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{2^{3+2D}D\alpha\alpha_{1}\alpha_{2}\Gamma\bigl{(}\tfrac{D+3}{2}\bigr{)}},$
$\displaystyle
R_{4}(p)=-i\frac{\bigl{(}-2+(-2+D)D\bigr{)}\gamma_{1}(\gamma_{1}+D\gamma_{2})\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{2^{3+2D}D\alpha\alpha_{1}\alpha_{2}\Gamma\bigl{(}\tfrac{D+3}{2}\bigr{)}},$
$\displaystyle
R_{5}(p)=-i\frac{\bigl{(}-4+(-4+D)D\bigr{)}\gamma_{1}(\gamma_{1}+D\gamma_{2})\left(D-2\right)\pi^{\frac{1-D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\Gamma(\tfrac{D}{2})}{2^{3+2D}D\alpha\alpha_{1}\alpha_{2}\Gamma\bigl{(}\tfrac{D+3}{2}\bigr{)}}.$

Diagrams 6 and 7 in figure 3.

Diagrams 6 and 7 have the same $R_{i}(p)$, $i=1..5$, whose values read:

$\displaystyle R_{1}(p)=0,$ $\displaystyle
R_{2}(p)=-i\frac{\pi^{-\frac{D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\left(\Gamma(\tfrac{D}{2})\right)^{2}}{2^{2+D}D(-1+D)\Gamma(-2+D)},$
$\displaystyle
R_{3}(p)=i\frac{\pi^{-\frac{D}{2}}(-p^{2})^{-2+\frac{D}{2}}\Gamma(1-\tfrac{D}{2})\left(\Gamma(\tfrac{D}{2})\right)^{2}}{2^{3+D}(-1+D)\Gamma(-2+D)},$
$\displaystyle R_{4}(p)=0,$ $\displaystyle R_{5}(p)=0.$

##  Appendix C  Newton’s Law and gauge-fixing.

In our formulation of unimodular gravity the minimal action,$S_{\phi}$, of a
massive scalar field, $\phi$, coupled to gravity in 4 dimensions reads

$\begin{array}[]{l}{S_{\phi}=\dfrac{1}{2}\int
d^{4}x\,\hat{g}^{\mu\nu}\partial_{\mu}\phi(x)\partial_{\nu}\phi(x)-m^{2}(\phi(x))^{2}=}\\\\[8.0pt]
{\phantom{S_{\phi}}=\dfrac{1}{2}\int
d^{4}x\,\eta^{\mu\nu}\partial_{\mu}\phi(x)\partial_{\nu}\phi(x)-m^{2}(\phi(x))^{2}-\dfrac{1}{2}\int
d^{4}x\,\hat{h}_{\mu\nu}(x)T^{\mu\nu}(x)+O((h_{\mu\nu})^{2})}\end{array}$

where $\hat{g}_{\mu\nu}=\dfrac{g_{\mu\nu}}{|g|^{1/4}}$ –with
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, $T^{\mu\nu}(x)$ is the Energy-Momentum
tensor

$T^{\mu\nu}(x)=\partial^{\mu}\phi\partial^{\nu}\phi-\dfrac{1}{2}\eta^{\mu\nu}(\partial_{\lambda}\phi\partial^{\lambda}\phi-m^{2}\phi^{2}),$

and

$\hat{h}_{\mu\nu}=h_{\mu\nu}-\dfrac{1}{4}h\,\eta_{\mu\nu},\quad
h=h^{\rho}_{\rho}.$

Following reference [23], we shall obtain the Newtonian potential by
considering two very massive scalar particles, with masses $m_{1}^{2}$ and
$m_{2}^{2}$, and computing the static limit of the amplitude of the tree-level
on-shell process in which those particles exchange one virtual graviton. In
our case, the amplitude, ${\cal A}$, in question is given by

${\cal
A}=-\dfrac{i}{4}T^{1}_{\mu_{1}\mu_{2}}(p_{1},p^{\prime}_{1})\,\langle\hat{h}^{\mu_{1}\mu_{2}}(k)\hat{h}^{\mu_{3}\mu_{4}}(-k)\rangle_{0}\,T^{2}_{\mu_{3}\mu_{4}}(p_{2},p^{\prime}_{2})$

where $k=p_{1}-p^{\prime}_{1}=p^{\prime}_{2}-p_{2}$,
$p_{i}^{2}=(p^{\prime}_{i})^{2}=m_{i}^{2}$, $i=1,2$ and
$T^{i}_{\mu_{1}\mu_{2}}(p_{i},p^{\prime}_{i})$ is the lowest order
contributions to the on-shell energy-momentum tensor, i.e.,

$T^{i}_{\mu_{1}\mu_{2}}(p_{i},p^{\prime}_{i})=p_{i\,\mu_{1}}p^{\prime}_{i\,\mu_{2}}+p_{i\,\mu_{2}}p^{\prime}_{i\,\mu_{1}}+\dfrac{1}{2}\,k^{2}\eta_{\mu_{1}\mu_{2}}.$

The Green function
$\langle\hat{h}^{\mu_{1}\mu_{2}}(k)\hat{h}^{\mu_{3}\mu_{4}}(-k)\rangle_{0}$ is
obtained in an obvious way from the graviton free propagator $\langle
h_{\mu_{1}\mu_{2}}(k)h_{\mu_{3}\mu_{4}}(-k)\rangle_{0}$ in (3.12) and (3.13).

For very massive particles in the static limit, we have
$k^{\mu}=(0,\vec{k})^{\mu}$ and
$T^{i}_{\mu_{1}\mu_{2}}(p_{i},p^{\prime}_{i})=2m_{i}^{2}\eta_{\mu_{1}0}\eta_{\mu_{2}0}$,
$i=1,2$, so that the Newtonian potential, $V_{\rm Nw}(\vec{k})$, in momentum
space is given by

$V_{\rm Nw}(\vec{k})=\dfrac{1}{2m_{1}\,2m_{2}}{\cal
A}=-i\dfrac{1}{4}m_{1}m_{2}\langle\hat{h}^{00}(k)\hat{h}^{00}(-k)\rangle_{0}.$
(C.23)

Of course, a correct quantum theory of gravity must yield

$V_{\rm Nw}(\vec{k})\,=\,-\dfrac{1}{8}\dfrac{m_{1}m_{2}}{\vec{k}^{2}},$ (C.24)

which is the Fourier transform of the Newtonian potential found in any
elementary textbook in the unit system $8\pi G=1$.

Let us see what restriction agreement between (C.23) and (C.24) imposes on the
free graviton propagator $\langle
h_{\mu_{1}\mu_{2}}(k)h_{\mu_{3}\mu_{4}}(-k)\rangle_{0}$. Let us introduce the
following general ansatz for the free graviton propagator

$\begin{array}[]{l}{\langle
h_{\mu_{1}\mu_{2}}(k)h_{\mu_{3}\mu_{4}}(-k)\rangle_{0}=}\\\\[4.0pt]
{i\,\dfrac{A_{1}}{k^{2}}\left(\eta_{\mu_{1}\mu_{3}}\eta_{\mu_{2}\mu_{4}}+\eta_{\mu_{1}\mu_{4}}\eta_{\mu_{2}\mu_{3}}-\eta_{\mu_{1}\mu_{2}}\eta_{\mu_{3}\mu_{4}}\right)+i\,\dfrac{A_{2}}{k^{2}}\eta_{\mu_{1}\mu_{2}}\eta_{\mu_{3}\mu_{4}}+i\,\dfrac{A_{3}}{(k^{2})^{2}}\left(\eta_{\mu_{3}\mu_{4}}k_{\mu_{1}}k_{\mu_{2}}+\eta_{\mu_{1}\mu_{2}}k_{\mu_{3}}k_{\mu_{4}}\right)}\\\\[4.0pt]
{+i\,\dfrac{A_{4}}{(k^{2})^{2}}\left(\eta_{\mu_{1}\mu_{3}}k_{\mu_{2}}k_{\mu_{4}}+\eta_{\mu_{1}\mu_{4}}k_{\mu_{2}}k_{\mu_{3}}+\eta_{\mu_{2}\mu_{3}}k_{\mu_{1}}k_{\mu_{4}}+\eta_{\mu_{2}\mu_{4}}k_{\mu_{1}}k_{\mu_{3}}\right)+i\,\dfrac{A_{5}}{(k^{2})^{3}}k_{\mu_{1}}k_{\mu_{2}}k_{\mu_{3}}k_{\mu_{4}}.}\end{array}$
(C.25)

$A_{i}$, $i=2..5$ are constants. We shall assume that

$A_{1}=\dfrac{1}{2}$

so that the contribution

$i\,\dfrac{A_{1}}{p^{2}}\left(\eta_{\mu_{1}\mu_{3}}\eta_{\mu_{2}\mu_{4}}+\eta_{\mu_{1}\mu_{4}}\eta_{\mu_{2}\mu_{3}}-\eta_{\mu_{1}\mu_{2}}\eta_{\mu_{3}\mu_{4}}\right)$

to the propagator can be interpreted as coming from the bit of the graviton
field which contains the physical graviton polarizations and the corresponding
creation and annihilation operators.

Using (C.25), one readily deduces that

$\langle\hat{h}^{00}(k)\hat{h}^{00}(-k)\rangle_{0}=-\dfrac{i}{4\vec{k}^{2}}\big{[}3+\dfrac{1}{4}A_{5}+A_{4}\big{]},$

if $k^{\mu}=(0,\vec{k})^{\mu}$. Inserting the previous result in (C.23) yields

$V_{\rm
Nw}(\vec{k})=-\dfrac{m_{1}m_{2}}{16\,\vec{k}^{2}}\big{[}3+\dfrac{1}{4}A_{5}+A_{4}\big{]}.$

Hence,

$A_{4}+\dfrac{1}{4}A_{5}=-1,$ (C.26)

for the Newtonian potential must be given by (C.24).

We thus conclude that both $A_{4}$ and $A_{5}$ in the free graviton propagator
in (C.25) cannot vanish at the same time, regardless of the Lorentz covariant
gauge-fixing term that one uses. Notice that for our general class of gauge-
fixing terms introduced in section 2, we have –see (3.13)–

$A_{4}=-\dfrac{\gamma_{1}^{2}-4\rho_{1}}{2\gamma_{1}^{2}},\quad
A_{5}=-2\dfrac{\gamma_{1}^{2}+4\rho_{1}}{\gamma_{1}^{2}},$

which satisfy (C.26).

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