\det\left[ \cfrac{\delta}{\delta \theta^b}\,\left\{ g^{\mu\nu} \nabla_\mu A^a_\nu \right\}  \right]  = \det\left[ \cfrac{1}{\gs}\,g^{\mu\nu} \, \nabla_\mu \left(\delta^{ab}\nabla_\nu  - \gs \,f^{abc}  \, A^c_\nu \right)\right].
\end{align}

It results in the following action:

\begin{align}
\mathcal{A}_\text{FP} = \int d^4 x \left[  - g^{\mu\nu}\, \nabla_\mu\overline{c}^a \nabla_\nu c^a + \gs \,g^{\mu\nu} \nabla_\mu \overline{c}^a f^{abc} c^b A_\nu^c  \right].
\end{align}

The action's kinetic part is similar to a massless vector field. The section describing the interaction between ghosts, vectors, and gravitons allows for a perturbative expansion.

\begin{align}
\begin{split}
&\int d^4 x \sqrt{-g} \left[ \gs\,\pd_\mu \overline{c}^a \, f^{abc} \, c^b \, A^{c\,\mu} \right]\\
&=\sum\limits_{n=0}^\infty \int \cfrac{d^4 p_1}{(2\pi)^4} \cfrac{d^4 p_2}{(2\pi)^4} \cfrac{d^4 k}{(2\pi)^4}\prod\limits_{i=0}^n \cfrac{d^4 k_i}{(2\pi)^4}\, (2\pi)^4 \delta\left(p_1+p_2+k+\sum k_i\right)\, h_{\rho_1\sigma_1}(k_1)\cdots h_{\rho_n\sigma_n}(k_n)\\
&\hspace{15pt}\times\, i\,\kappa^n\,\gs\,(p_1)_\nu \, f^{abc} \, \overline{c}^a(p_1) \,c^b(p_2) \, \left(\sqrt{-g}\,g^{\mu\nu}\right)^{\rho_1\sigma_1\cdots\rho_n\sigma_n}\,\,A^c_\mu(k).
\end{split}
\end{align}

This expression produced the following rule:

\begin{align}\label{rule_GhGhG}
\nonumber \\
\begin{gathered}
\begin{fmffile}{FR_GhGhG}
\begin{fmfgraph*}(30,30)
\fmfleft{L1,L2}
\fmfright{R1,R2,R3}
\fmf{dbl_wiggly,tension=2}{L1,V}
\fmf{dbl_wiggly,tension=2}{L2,V}
\fmf{gluon,tension=.5}{V,R2}
\fmf{dots_arrow,tension=.5}{R1,V}
\fmf{dots_arrow,tension=.5}{V,R3}
\fmfdot{V}
\fmffreeze
\fmf{dots}{L1,L2}
\fmflabel{$\rho_1\sigma_1$}{L1}
\fmflabel{$\rho_n\sigma_n$}{L2}
\fmflabel{$\mu,c$}{R2}
\fmflabel{$b$}{R1}
\fmflabel{$p_1,a$}{R3}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
\hspace{20pt}=& - \kappa^n\,g_s\,f^{abc} \, (p_1)_\nu \,\left(\sqrt{-g}\,g^{\mu\nu}\right)^{\rho_1\sigma_1\cdots\rho_n\sigma_n}. \\ \nonumber
\end{align}

§.§ General relativity

General relativity is invariant with respect to local transformations spawned by coordinate transformations, so a gauge fixing procedure shall be performed. The perturbative approach describes gravity as small metric perturbations over a flat background. Therefore, it may seem that the theory is reduced to a gauge theory of rank-$2$ symmetric tensor, but this is not the case.

One shall distinguish a geometric theory from a theory of a rank-$2$ symmetric tensor. The difference dictates how to treat the gauge fixing condition. First, let us consider the case of a symmetric $h_{\mu\nu}$ tensor theory that admits the following gauge symmetry:

\begin{align}
\delta h_{\mu\nu} = \pd_\mu \zeta_\nu + \pd_\nu \zeta_\mu .
\end{align}

The $h_{\mu\nu}$ tensor is fundamental to this theory, and a gauge fixing condition can be expressed solely in terms of it. For example, one can use the following gauge fixing condition:

\begin{align}\label{naive_gauge_fixing}
\pd_\mu h^{\mu\nu}  - \cfrac12\,\pd^\nu h = 0\,.
\end{align}

This condition, along with others, determines the composition of the Faddeev-Popov ghosts. Because $h_{\mu\nu}$ is the fundamental object of the theory, the structure of divergences can only be expressed in terms of $h_{\mu\nu}$ alone. All geometric quantities, such as the Riemann tensor, Ricci tensor, scalar curvature, and others, are expressed via small metric perturbations, but the opposite is not true. Operators given in terms of $h_{\mu\nu}$ alone may not represent any geometric quantities. Therefore, in a gauge theory of a rank-$2$ symmetric tensor, one can expect to find divergencies that cannot be described by geometric quantities, which makes it a non-geometric theory.

On the contrary, gauge transformations are generated by coordinate frame transformations within the geometric approach. This has two immediate implications. Firstly, within a geometrical theory, gauge transformations are given by the so-called Lie derivatives:

\begin{align}
\delta g_{\mu\nu} \overset{\text{def}}{=} \mathcal{L}_\zeta g_{\mu\nu} = \nabla_\mu \zeta_\nu + \nabla_\nu \zeta_\mu .
\end{align}

Here, $\mathcal{L}_\zeta$ is the Lie derivative with respect to an arbitrary vector field $\zeta$, which contains the gauge transformation parameters. Secondly, the gauge fixing condition must be expressed in geometrical quantities. Thus, gauge fixing conditions (<ref>) are inconsistent with the geometrical approach and cannot be implemented. Instead, we use the following gauge fixing conditions:

\begin{align}\label{the_gravity_gauge_fixing}
\mathcal{G}^\mu \overset{\text{note}}{=} g^{\alpha\beta} \Gamma^\mu_{\alpha\beta} =0 .
\end{align}

Combined with the perturbative expansion given by equation (<ref>), the gauge fixing conditions represented by equation (<ref>) produce an infinite series.

\begin{align}
\mathcal{G}^\nu = \cfrac{\kappa}{2}\,g^{\mu\nu} g^{\alpha\beta} \left[ \pd_\alpha h_{\beta\mu} + \pd_\beta h_{\alpha\mu} - \pd_\mu h_{\alpha\beta}\right] =  \kappa \left[\pd_\mu h^{\mu\nu} - \cfrac12\,\pd^\nu h \right] + \mathcal{O}(\kappa^2) ,
\end{align}

The infinite expansion defines the ghost sector, and the leading term reproduces naive gauge fixing (<ref>).

The difference between geometrical theories of gravity and a gauge theory of $h_{\mu\nu}$ tensor is marked by the need to use gauge fixing condition (<ref>), and not conditions (<ref>). Proceeding with The Faddeev-Popov prescription, firstly, we shall note that the gauge fixing condition $\mathcal{G}^\mu$ defined by (<ref>) is a vector with mass dimension $+1$. Thus, one shall introduce an additional dimensional parameter in the gauge fixing term:

\begin{align}\label{Hilbert_gauge_fixed}
\mathcal{A}_{\text{H}+\text{gf}} = \int d^4 x \sqrt{-g} \left[ -\cfrac{2}{\kappa^2}\,R + \cfrac{\epsilon}{2\,\kappa^2} \, g_{\mu\nu} \,\mathcal{G}^\mu \mathcal{G}^\nu \right].
\end{align}

Secondly, the Faddeev-Popov ghosts are also vectors. The structure of their action is defined by the variation of the gauge fixing term given by equation (<ref>):

\begin{align}
\delta G^\mu = \mathcal{L}_\zeta \left[ g^{\alpha\beta} \, \Gamma^\mu_{\alpha\beta} \right] = \square \zeta^\mu - 2\, \Gamma^\mu_{\alpha\beta} \, \nabla^\alpha\zeta^\beta + R^\mu{}_\nu \zeta^\nu
\end{align}

with $R_{\mu\nu}$ begin the Ricci tensor. Consequently, the ghost action reads:

\begin{align}
\mathcal{A}_\text{ghost} = \int d^4 x \sqrt{-g} \left[ -g^{\alpha\beta} g^{\mu\nu} \nabla_\alpha \overline{c}_\mu \nabla_\beta c_\nu - 2\,\Gamma^\mu_{\alpha\beta} \,\overline{c}_\mu \,\nabla^\alpha c^\beta + R_{\mu\nu} \, \overline{c}^\mu \,c^\nu \right].
\end{align}

In all other respects, the handling of the Faddeev-Popov ghosts is unchanged.

The standard perturbative expansion generates the interaction rules. The structure of graviton interactions is given by action (<ref>):

\begin{align}
\begin{split}
\mathcal{A}_{\text{H}+\text{gf}} &= \int d^4 x \sqrt{-g} \left[ -\cfrac{2}{\kappa^2}\,R + \cfrac{\epsilon}{2\,\kappa^2}\, g_{\mu\nu}\, g^{\alpha\beta}\, g^{\rho\sigma} \,\Gamma^\mu_{\alpha\beta}\,\Gamma^\nu_{\rho\sigma}\right] = \\
& = \int d^4 x \sqrt{-g} \, g^{\mu\nu} g^{\alpha\beta} g^{\rho\sigma} \left(-\cfrac{2}{\kappa^2}\right) \left[ \Gamma_{\alpha\mu\rho}\Gamma_{\sigma\nu\beta} - \Gamma_{\alpha\mu\nu} \Gamma_{\rho\beta\sigma} - \cfrac{\epsilon}{4} \,\Gamma_{\mu\alpha\beta} \Gamma_{\nu\rho\sigma} \right] .
\end{split}
\end{align}

It admits the following perturbative expansion:

\begin{align}
\begin{split}
\mathcal{A}_{\text{H}+\text{gf}} =& \sum\limits_{n=0}^\infty\int\cfrac{d^4 p_1}{(2\pi)^4} \cfrac{d^4 p_2}{(2\pi)^4} \prod\limits_{i=1}^n \cfrac{d^4 k_i}{(2\pi)^4}\,(2\pi)^4\,\delta\Big(p_1+p_2+\sum k_i \Big) \, h_{\rho_1\sigma_1}(k_1) \cdots h_{\rho_n\sigma_n}(k_n)\\
&\times \left(2\,\kappa^n \right)\left( \sqrt{-g} \, g^{\mu\nu} g^{\alpha\beta} g^{\rho\sigma} \right)^{\rho_1\sigma_1\cdots\rho_n\sigma_n} (p_1)_{\lambda_1} (p_2)_{\lambda_2} \, h_{\mu_1\nu_1}(p_1) h_{\mu_2\nu_2}(p_2) \\
& \times \Bigg[ \left(\Gamma_{\alpha\mu\rho}\right)^{\lambda_1\mu_1\nu_1} \left(\Gamma_{\sigma\nu\beta}\right)^{\lambda_2\mu_2\nu_2} - \left(\Gamma_{\alpha\mu\nu} \right)^{\lambda_1\mu_1\nu_1} \left( \Gamma_{\rho\beta\sigma}\right)^{\lambda_2\mu_2\nu_2} - \cfrac{\epsilon}{4} \left( \Gamma_{\mu\alpha\beta} \right)^{\lambda_1\mu_1\nu_1} \left(\Gamma_{\nu\rho\sigma} \right)^{\lambda_2\mu_2\nu_2}\Bigg].
\end{split}
\end{align}

The following formula gives the complete expression for the graviton vertex:

\begin{align}\label{Rule_Gravitons}
\nonumber \\
\begin{split}
& \hspace{10pt}
\begin{gathered}
\begin{fmffile}{FR_h}
\begin{fmfgraph*}(30,30)
\fmfleft{L1,L2}
\fmfright{R1,R2}
\fmf{dbl_wiggly}{L1,V}
\fmf{dbl_wiggly}{L2,V}
\fmf{dbl_wiggly}{R1,V}
\fmf{dbl_wiggly}{R2,V}
\fmffreeze
\fmfdot{V}
\fmf{dots}{L1,L2}
\fmflabel{$\mu_1\nu_1,p_1$}{R1}
\fmflabel{$\mu_2\nu_2,p_2$}{R2}
\fmflabel{$\mu_3\nu_3,p_3$}{L1}
\fmflabel{$\mu_n\nu_n,p_n$}{L2}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
\hspace{35pt}= \,i\,2\,\kappa^{n-2}\,\left(\sqrt{-g}\,g^{\mu\nu}g^{\alpha\beta}g^{\rho\sigma}\right)^{\mu_3\nu_3\cdots\mu_n\nu_n} \, (p_1)_{\lambda_1} (p_2)_{\lambda_2} \\ \\
& \times \Bigg[ \left(\Gamma_{\alpha\mu\rho}\right)^{\lambda_1\mu_1\nu_1} \left(\Gamma_{\sigma\nu\beta}\right)^{\lambda_2\mu_2\nu_2} - \left(\Gamma_{\alpha\mu\nu} \right)^{\lambda_1\mu_1\nu_1} \left( \Gamma_{\rho\beta\sigma}\right)^{\lambda_2\mu_2\nu_2} - \cfrac{\epsilon}{4} \left( \Gamma_{\mu\alpha\beta} \right)^{\lambda_1\mu_1\nu_1} \left(\Gamma_{\nu\rho\sigma} \right)^{\lambda_2\mu_2\nu_2}\Bigg] \\
& + \text{permutations}.
\end{split}
\end{align}

The sum goes over all possible permutations of graviton parameters $\{\mu_i\,\nu_i\,p_i\}$.

The ghost action is treated similarly.

\begin{align}
\begin{split}
\mathcal{A}_\text{ghost} =& \int d^4 x \sqrt{-g} \left[ -g^{\alpha\beta} g^{\mu\nu} \nabla_\alpha \overline{c}_\mu \nabla_\beta c_\nu - 2\,\Gamma^\mu_{\alpha\beta} \,\overline{c}_\mu \,\nabla^\alpha c^\beta + R_{\mu\nu} \, \overline{c}^\mu \,c^\nu \right] \\
=& \int d^4 x \sqrt{-g} \left[ -g^{\mu\nu} g^{\alpha\beta} \, \pd_\alpha\overline{c}_\mu \, \pd_\beta c_\nu \right]\\
& + \int d^4 x \sqrt{-g}\, g^{\mu\alpha}g^{\nu\beta}g^{\rho\sigma} \left[ \Gamma_{\beta\rho\alpha} \pd_\sigma \overline{c}_\mu c_\nu - \Gamma_{\alpha\rho\beta}\,\overline{c}_\mu \pd_\sigma c_\nu + \pd_\rho\Gamma_{\sigma\alpha\beta} \,\overline{c}_\mu \,c_\nu - \pd_\alpha \Gamma_{\rho\beta\sigma} \overline{c}_\mu c_\nu \right]\\
& + \int d^4 x \sqrt{-g}\,g^{\mu\alpha} g^{\nu\beta} g^{\rho\sigma} g^{\lambda\tau} \left[ \Gamma_{\rho\alpha\lambda} \Gamma_{\sigma\beta\tau} - \Gamma_{\rho\alpha\beta} \Gamma_{\sigma\lambda\tau} + \Gamma_{\alpha\rho\lambda} \Gamma_{\beta\sigma\tau} \right] \overline{c}_\mu c_\nu .
\end{split}
\end{align}

It has the following perturbative expansion:

\begin{align}
\begin{split}
\mathcal{A}_\text{ghost} =& \sum\limits_{n=0}^\infty\int \cfrac{d^4 p_1}{(2\pi)^4} \cfrac{d^4 p_2}{(2\pi)^4} \prod\limits_{i=1}^n \cfrac{d^4 k_i}{(2\pi)^4} \, (2\pi)^4 \delta\big(p_1+p_2 + \sum k_i\big) h_{\rho_1\sigma_1}(k_1)\cdots h_{\rho_n\sigma_n}(k_n) \, \overline{c}_\mu (p_1) c_\nu(p_2)\\
&\times \kappa^n \,\left(\sqrt{-g}\,g^{\mu\nu} g^{\alpha\beta} \right)^{\rho_1\sigma_1\cdots\rho_n\sigma_n} (p_1)_\alpha (p_2)_\beta\\
+& \sum\limits_{n=1}^\infty\int \cfrac{d^4 p_1}{(2\pi)^4} \cfrac{d^4 p_2}{(2\pi)^4} \prod\limits_{i=1}^n \cfrac{d^4 k_i}{(2\pi)^4} \, (2\pi)^4 \delta\big(p_1+p_2 + \sum k_i\big) h_{\rho_1\sigma_1}(k_1)\cdots         h_{\rho_n\sigma_n}(k_n) \, \overline{c}_\mu (p_1) c_\nu(p_2)\\
&\times \kappa^n (-1)\left(\sqrt{-g}\,g^{\mu\alpha}g^{\nu\beta} g^{\rho\sigma}\right)^{\rho_2\sigma_2\cdots\rho_n\sigma_n} (k_1)_\lambda\left[ (p_1)_\sigma \left(\Gamma_{\beta\rho\alpha}\right)^{\lambda\rho_1\sigma_1}-(p_2)_\sigma \left(\Gamma_{\alpha\rho\beta}\right)^{\lambda\rho_1\sigma_1} \right.\\
&\hspace{210pt}\left. + (k_1)_\rho \left(\Gamma_{\sigma\alpha\beta}\right)^{\lambda\rho_1\sigma_1} - (k_1)_\alpha \left(\Gamma_{\rho\beta\sigma}\right)^{\lambda\rho_1\sigma_1}\right]\\
+& \sum\limits_{n=2}^\infty\int \cfrac{d^4 p_1}{(2\pi)^4} \cfrac{d^4 p_2}{(2\pi)^4} \prod\limits_{i=1}^n \cfrac{d^4 k_i}{(2\pi)^4} \, (2\pi)^4 \delta\big(p_1+p_2 + \sum k_i\big) h_{\rho_1\sigma_1}(k_1)\cdots h_{\rho_n\sigma_n}(k_n) \, \overline{c}_\mu (p_1) c_\nu(p_2)\\
&\times \kappa^n (-1) \left(\sqrt{-g} \,g^{\mu\alpha}g^{\nu\beta} g^{\rho\sigma} g^{\lambda\tau}\right)^{\rho_3\sigma_3\cdots\rho_n\sigma_n}\,(k_1)_{\lambda_1} (k_2)_{\lambda_2} \left[ \left(\Gamma_{\rho\alpha\lambda}\right)^{\lambda_1\rho_1\sigma_1} \left(\Gamma_{\sigma\beta\tau}\right)^{\lambda_2\rho_2\sigma_2} \right.\\
& \hspace{140pt}\left. - \left(\Gamma_{\rho\alpha\beta}\right)^{\lambda_1\rho_1\sigma_1} \left(\Gamma_{\sigma\lambda\tau}\right)^{\lambda_2\rho_2\sigma_2} + \left(\Gamma_{\alpha\rho\lambda}\right)^{\lambda_1\rho_1\sigma_1} \left(\Gamma_{\beta\sigma\tau}\right)^{\lambda_2\rho_2\sigma_2} \right].
\end{split}
\end{align}

The complete expression describing graviton-ghost vertices reads:

\begin{align}\label{Rules_Graviton-Ghosts}
\nonumber \\
\begin{split}
& \hspace{30pt}
\begin{gathered}
\begin{fmffile}{FR_Gh_2}
\begin{fmfgraph*}(30,30)
\fmfleft{L1,L2}
\fmfright{R1,R2}
\fmf{dbl_wiggly}{L1,V}
\fmf{dbl_wiggly}{L2,V}
\fmf{dots_arrow}{R1,V}
\fmf{dots_arrow}{V,R2}
\fmfdot{V}
\fmffreeze
\fmf{dots}{L1,L2}
\fmflabel{$\rho_1\sigma_1,k_1$}{L1}
\fmflabel{$\rho_n\sigma_n,k_n$}{L2}
\fmflabel{$\nu,p_2$}{R1}
\fmflabel{$\mu,p_1$}{R2}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
\hspace{25pt}= i\,\kappa^n \Bigg[ \left(\sqrt{-g} \,g^{\mu\nu}g^{\alpha\beta}\right)^{\rho_1\sigma_1\cdots\rho_n\sigma_n} (p_1)_\alpha (p_2)_\beta \\ \\
&+\Bigg\{ - \left(\sqrt{-g}\,g^{\mu\alpha} g^{\nu\beta}g^{\rho\sigma}\right)^{\rho_2\sigma_2\cdots\rho_n\sigma_n}(k_1)_\lambda \Bigg[ (p_1)_\sigma (\Gamma_{\beta\rho\alpha})^{\lambda\rho_1\sigma_1} - (p_2)_\sigma (\Gamma_{\alpha\rho\beta})^{\lambda\rho_1\sigma_1} \\
& \hspace{150pt}+ (k_1)_\rho (\Gamma_{\sigma\alpha\beta})^{\lambda\rho_1\sigma_1}-(k_1)_\alpha (\Gamma_{\rho\beta\sigma})^{\lambda\rho_1\sigma_1} \Bigg] + \text{permutations} \Bigg\}\\
& + \Bigg\{ -\left(\sqrt{-g}\,g^{\mu\alpha} g^{\nu\beta}g^{\rho\sigma} g^{\lambda\tau} \right)^{\rho_3\sigma_3\cdots\rho_n\sigma_n} \,(k_1)_{\lambda_1} (k_2)_{\lambda_2} \left[ \left(\Gamma_{\rho\alpha\lambda}\right)^{\lambda_1\rho_1\sigma_1} \left(\Gamma_{\sigma\beta\tau}\right)^{\lambda_2\rho_2\sigma_2} \right.\\
& \hspace{70pt}\left. - \left(\Gamma_{\rho\alpha\beta}\right)^{\lambda_1\rho_1\sigma_1} \left(\Gamma_{\sigma\lambda\tau}\right)^{\lambda_2\rho_2\sigma_2} + \left(\Gamma_{\alpha\rho\lambda}\right)^{\lambda_1\rho_1\sigma_1} \left(\Gamma_{\beta\sigma\tau}\right)^{\lambda_2\rho_2\sigma_2} \right] + \text{permutations} \Bigg\} \Bigg].
\end{split}
\end{align}

The standard procedure derives propagators for ghosts and gravitons. The following expression gives the ghost propagator.

\begin{align}
\begin{gathered}
\begin{fmffile}{FR_Ghost_Propagator}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R}
\fmf{dots}{L,R}
\fmflabel{$\mu$}{L}
\fmflabel{$\nu$}{R}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
\hspace{15pt} = i \, \cfrac{\eta_{\mu\nu}}{k^2}\,.
\end{align}

The graviton propagator contains the gauge fixing parameter $\epsilon$. The propagator corresponds to the part of the microscopic action quadratic in perturbations:

\begin{align}
\int d^4 x \sqrt{-g} \left[ -\cfrac{2}{\kappa^2}\, R + \cfrac{\epsilon}{2\,\kappa^2} ~\mathcal{G}_\mu \mathcal{G}^\mu \right] = \int d^4 x \left[  -\cfrac12\,h^{\mu\nu} \mathcal{D}_{\mu\nu\alpha\beta}(\epsilon)\, \square h^{\alpha\beta}  \right] +\okappa{1}.
\end{align}

The Nieuwenhuizen operators providee a good mean to express the $\mathcal{D}$ operator in the momentum representation <cit.>

\begin{align}
\mathcal{D}_{\mu\nu\alpha\beta} (\epsilon) = \cfrac{3 \epsilon -8}{4}\, P^0_{\mu\nu\alpha\beta} + \cfrac{\epsilon}{2}\,P^1_{\mu\nu\alpha\beta} + P^2_{\mu\nu\alpha\beta} - \cfrac{\epsilon}{4} ~ \overline{P}^0_{\mu\nu\alpha\beta} - \cfrac{\epsilon}{4} ~ \overline{\overline{P}}^0_{\mu\nu\alpha\beta} \,.
\end{align}

Operators $P^0$ and $P^2$ are gauge invariant, making the operator non-invertible unless $\epsilon \not =0$. The inverse operator reads:

\begin{align}
\mathcal{D}^{-1}_{\mu\nu\alpha\beta}(\epsilon) = -\cfrac12\,P^0_{\mu\nu\alpha\beta}+\cfrac{2}{\epsilon}\, P^1_{\mu\nu\alpha\beta} + P^2_{\mu\nu\alpha\beta} - \cfrac{3\,\epsilon -8}{2\,\epsilon}~\overline{P}^0_{\mu\nu\alpha\beta} - \cfrac{1}{2} ~\overline{\overline{P}}^0_{\mu\nu\alpha\beta} .
\end{align}

The given formula expresses the graviton propagator in any arbitrary gauge.

\begin{align}
\begin{gathered}
\begin{fmffile}{FR_Graviton_Propagator}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R}
\fmf{dbl_wiggly}{L,R}
\fmflabel{$\mu\nu$}{L}
\fmflabel{$\alpha\beta$}{R}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
\hspace{20pt} = i ~ \cfrac{ \mathcal{D}^{-1}_{\mu\nu\alpha\beta}(\epsilon) }{k^2} \, .
\end{align}

We can discuss the general case, but using $\epsilon = 2$ in practical applications is easier. With this value, the operator $\mathcal{D}^{-1}$ becomes much simpler in form:

\begin{align}
\mathcal{D}^{-1}_{\mu\nu\alpha\beta}(2) = \cfrac12\left[ \eta_{\mu\alpha} \eta_{\nu\beta} + \eta_{\mu\beta} \eta_{\nu\alpha} - \eta_{\mu\nu} \eta_{\alpha\beta} \right].
\end{align}

§ FEYNGRAV

FeynGrav is a package for Wolfram Mathematica extending FeynCalc functionality <cit.>. FeynCalc provides tools to study both tree and loop-level amplitudes. At the same time, there are many packages further extending its functionality <cit.>, which makes it a platform for different computational tools of high energy physics. Because of these reasons, FeynCalc was chosen to implement the perturbative quantum gravity framework described above.

This subsection is split into two parts to discuss two different aspects of FeynGrav. The first subsection discussed the general features of the package and its published versions. The second subsection discussed the commands implemented in the latest available version.

§.§ Implementation

We shall begin with a discussion of the existing versions of FeynGrav. There are a few published versions of the package, each extending its functionality and improving performance. The package is constantly developing, and the latest version is publically available at <cit.>. We shall briefly discuss the features of all these versions for completeness.

The original version 1.0 of FeynGrav was published in <cit.>. This version only considered matter with massless spin $0$, $1$, $2$, and $1/2$, and did not include the $SU(N)$ Yang-Mills model. The recursive relations discussed in this paper were yet to be discovered during the development of this version. Consequently, the package used a less effective algorithm to generate all perturbative expansions. The package's applicability was also limited since the gauge-fixing algorithm was not considered in detail.

The second version published in <cit.> presented a significant update. This version added interaction rules for massive matter with spins $0$, $1$, $1/2$, and implemented rules for the $SU(N)$ Yang-Mills. The gauge-fixing procedure was fully addressed, so the implemented interaction rules became applicable at any perturbation theory level. However, the recursive algorithm still needed to be discovered, so the package used the same algorithm for perturbative expansions.

The recursive relations were discovered after the publication <cit.> and were implemented in the latest FeynGrav version 2.1. The discovery of the recursive algorithm allowed expressions for the interaction rules to be generated more efficiently, so the corresponding libraries were updated. This version also implemented the polarisation tensors for gravity, discussed below. Lastly, minor misprints were corrected, and the sample file was significantly updated and improved.

The interaction rules for the Horndeski theory still need to be implemented in any existing version of FeynGrav. Because of their shared length and complexity, their implementation takes time and is expected to be published soon.

Let us discuss the package structure that remains similar in all the versions. The package addressed a few different challenges, consisting of a few semi-independent modules. The main computational challenge is the generation of the perturbative expansion terms. The package's core addresses this problem but operates separately from the main file. In turn, the main file of the package only imports the interaction rules in the FeynCalc environment, allowing a user to operate with them.

The above recursive relations are implemented in a series of sub-packages in a separate folder. Each package provides a tool to calculate a separate family of tensors and can be used independently within FeynCalc. As discussed above, it is essential to introduce tensors with particular symmetry for the Lorentz indices. With this symmetry imposed, a typical tensor with $2n$ pair of Lorentz indices will have approximately $2^n \, n!$ terms. Because of this large number of terms, a computation of an interaction rule involving many particles can take significant time. To soften this issue, the interaction rules are calculated separately.

The interaction rules discussed above are implemented in a single separate sub-package. Because such rules only require information about perturbative expansions, this sub-package depends on sub-packages describing perturbative expansions. It shall be run independently from FeynGrav to generate libraries containing the final expressions for the interaction rules.

The main package file is independent of these sub-packages directly since it only imports the existing libraries and places them in the FeynCalc environment. Because of this package structure, the final user does not have to perform computationally heavy calculations of the interaction rules, significantly improving the package performance. When FeynGrav imports the rules, FeynCalc performs index contractions and other operations that constrain FeynGrav's performance.

§.§ Interaction rules

The scalar field kinetic and potential energy interaction rules are implemented with commands “GravitonScalarVertex” and “GravitonScalarPotentialVertex”. The first command takes four arguments: the array of graviton indices, two momenta of scalar fields, and the scalar field mass. The second command takes two arguments: the array of graviton indices and the scalar field coupling coupling. FeynGrav also contains a command realising the scalar field propagator. The command is “ScalarPropagator” and takes two arguments: the scalar field momentum and mass. Table <ref> presents examples of these commands' usage.

Interaction rules for the Proca field are implemented with a single command “GravitonMassiveVectorVertex”. The name is chosen for the sake of naming consistency. The command takes six arguments: the array of graviton indices, the Lorentz index and momentum of the first vector particle, the Lorentz index and momentum of the second vector particle, and the Proca field mass. The package also has the Proca propagator implementation. The command “ProcaPropagator” implements the propagator and takes four arguments: two Lorentz indices, the momentum and mass of the Proca field. Table <ref> presents examples of these commands' usage.

The interaction rules for a single massless vector field are implemented with two commands, “GravitonVectorVertex” and “GravitonVectorGhostVertex”. The command “GravitonVectorVertex” takes five arguments: the array of Lorentz indices and momenta of gravitons, the Lorentz index and momentum of the first vector particle, the Lorentz index and momentum of the second particle. The command “GravitonVectorGhostVertex” takes three arguments: the array of Lorentz indices and momenta of gravitons, and the momenta of the Faddeev-Popov ghost. Table <ref> presents examples of these commands. Let us also note that FeynCalc already provides the propagators required to treat a massless scalar field.

Diagram      Command
\begin{gathered}
\begin{fmffile}{Table_Illustration_Scalar_0}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R}
\fmf{dashes}{L,R}
\fmflabel{~$p,m$}{R}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}

\begin{gathered}
\begin{fmffile}{Table_Illustration_Scalar_1}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R1,R2}
\fmf{dbl_wiggly}{L,V}
\fmf{dashes}{R1,V}
\fmf{dashes}{R2,V}
\fmfdot{V}
\fmflabel{$\rho_1\sigma_1$}{L}
\fmflabel{$p_1,m$}{R1}
\fmflabel{$p_2,m$}{R2}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonScalarVertex$[\{\rho_1 , \sigma_1\},p_1,p_2,m]$

\begin{gathered}
\begin{fmffile}{Table_Illustration_Scalar_2}
\begin{fmfgraph*}(30,30)
\fmfleft{L1,L2}
\fmfright{R1,R2}
\fmf{dbl_wiggly}{L1,V}
\fmf{dbl_wiggly}{L2,V}
\fmf{dashes}{R1,V}
\fmf{dashes}{R2,V}
\fmfdot{V}
\fmflabel{$\rho_1\sigma_1$}{L1}
\fmflabel{$\rho_2\sigma_2$}{L2}
\fmflabel{$p_1,m$}{R1}
\fmflabel{$p_2$}{R2}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonScalarVertex$[\{\rho_1 , \sigma_1,\rho_2,\sigma_2\},p_1,p_2,m]$

\begin{gathered}
\begin{fmffile}{Table_Illustration_Scalar_3}
\begin{fmfgraph*}(30,30)
\fmfleft{L1}
\fmfright{R1,R2,R3}
\fmf{dbl_wiggly,tension=3}{L1,V}
\fmf{dashes}{R1,V}
\fmf{dashes}{R2,V}
\fmf{dashes}{R3,V}
\fmfdot{V}
\fmflabel{$\rho_1\sigma_1$}{L1}
\fmflabel{$\lambda_3$}{R2}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonScalarPotentialVertex$[\{\rho_1 , \sigma_1\},\lambda_3]$

\begin{gathered}
\begin{fmffile}{Table_Illustration_Scalar_4}
\begin{fmfgraph*}(30,30)
\fmfleft{L1,L2}
\fmfright{R1,R2,R3,R4}
\fmf{dbl_wiggly,tension=2}{L1,V}
\fmf{dbl_wiggly,tension=2}{L2,V}
\fmf{dashes}{R1,V}
\fmf{dashes}{R2,V}
\fmf{dashes}{R3,V}
\fmf{dashes}{R4,V}
\fmfdot{V}
\fmflabel{$\rho_1\sigma_1$}{L1}
\fmflabel{$\rho_2\sigma_2$}{L2}
\fmflabel{$\lambda_4$}{R2}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonScalarPotentialVertex$[\{\rho_1 , \sigma_1, \rho_2, \sigma_2\},\lambda_4]$

\begin{gathered}
\begin{fmffile}{Table_Illustration_Proca_0}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R}
\fmf{photon}{L,R}
\fmflabel{$\mu$}{L}
\fmflabel{$\nu ~~ p,m$}{R}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}

\begin{gathered}
\begin{fmffile}{Table_Illustration_Proca_1}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R1,R2}
\fmf{dbl_wiggly}{L,V}
\fmf{photon}{R1,V}
\fmf{photon}{R2,V}
\fmfdot{V}
\fmflabel{$\rho_1\sigma_1$}{L}
\fmflabel{$p_1,\lambda_1,m$}{R1}
\fmflabel{$p_2,\lambda_2,m$}{R2}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonMassiveVectorVertex$[\{\rho_1 , \sigma_1\},\lambda_1,p_1,\lambda_2,p_2,m]$

\begin{gathered}
\begin{fmffile}{Table_Illustration_Vector_1}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R1,R2}
\fmf{dbl_wiggly}{L,V}
\fmf{photon}{R1,V}
\fmf{photon}{R2,V}
\fmfdot{V}
\fmflabel{$\rho_1\sigma_1,k_1$}{L}
\fmflabel{$\lambda_1, p_1$}{R1}
\fmflabel{$\lambda_2, p_2$}{R2}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonVectorVertex$[\{\rho_1 , \sigma_1, k_1\},\lambda_1,p_1,\lambda_2,p_2]$

\begin{gathered}
\begin{fmffile}{Table_Illustration_Vector_2}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R1,R2}
\fmf{dbl_wiggly}{L,V}
\fmf{dots}{R1,V}
\fmf{dots}{R2,V}
\fmfdot{V}
\fmflabel{$\rho_1\sigma_1,k_1$}{L}
\fmflabel{$p_1$}{R1}
\fmflabel{$p_2$}{R2}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonVectorGhostVertex$[\{\rho_1 , \sigma_1, k_1\},p_1,p_2]$

Examples of interaction rules for the scalar, Proca, and massless vector fields.

The interaction rules for Dirac fermions are implemented with a single command “GravitonFermionVertex”. The command takes four arguments: the array of graviton Lorentz indices, the momentum of the in-going fermion line, the momentum of the out-going fermion line, and the fermion mass. The fermion propagator has already been implemented in FeynCalc. Table <ref> provides an example of this interaction rule.

The $SU(N)$ Yang-Mills model implementation is sophisticated and involves several commands. The command “GravitonGluonVertex implements a coupling of a few gluons to gravity. The command's first argument is an array of Lorentz indices and momenta. The other arguments describe gluons Lorentz indices, momenta, and $SU(N)$ indices. The command can describe a coupling of two, three, and four gluons to gravity. Gravitational coupling of quarks kinetic energy matches the expression for the coupling of a Dirac fermion. The coupling of the quark-gluon interaction energy is described by the “GravitonQuarkGluonVertex” command. It takes only three arguments: the array of graviton Lorentz indices, the quark-gluon vertex Lorentz index, and the colour index. Lastly, two commands are responsible for gravitational coupling to the Faddeev-Popov ghosts of the Yang-Mills theory. The command “GravitonYMGhostVertex” describes the coupling to the ghost itself. The command takes five arguments: the array of graviton Lorentz indices, the momentum and colour index of the first ghost, and the momentum and colour index of the second ghost. The command “GravitonGluonGhostVertex” corresponds to the gravitational coupling of a vertex describing the interaction between two ghosts and one gluon. The command's arguments are the array of graviton Lorentz indices, the Lorentz indices, momenta, and colour indices of other particles. Table <ref> lists examples of these commands.

Diagram      Command

\begin{gathered}
\begin{fmffile}{Table_Illustration_Fermion_1}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R1,R2}
\fmf{dbl_wiggly}{L,V}
\fmf{fermion}{R1,V}
\fmf{fermion}{V,R2}
\fmfdot{V}
\fmflabel{$\rho_1\sigma_1$}{L}
\fmflabel{$p_1,m$}{R1}
\fmflabel{$p_2$,m,m}{R2}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonFermionVertex$[\{\rho_1 , \sigma_1\},p_1,p_2,m]$

\begin{gathered}
\begin{fmffile}{Table_Illustration_Gluon_1}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R1,R2}
\fmf{dbl_wiggly,tension=2}{L,V}
\fmf{gluon}{R1,V}
\fmf{gluon}{R2,V}
\fmfdot{V}
\fmflabel{$\rho_1\sigma_1,k_1$}{L}
\fmflabel{$\lambda_1,p_1,a_1$}{R1}
\fmflabel{$\lambda_2,p_2,a_2$}{R2}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonGluonVertex$[\{\rho_1 , \sigma_1, k_1\},p_1,\lambda_1,a_1,p_2,\lambda_2,a_2]$

\begin{gathered}
\begin{fmffile}{Table_Illustration_Gluon_2}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R1,R2,R3}
\fmf{dbl_wiggly,tension=3}{L,V}
\fmf{gluon}{R1,V}
\fmf{gluon}{R2,V}
\fmf{gluon}{R3,V}
\fmfdot{V}
\fmflabel{$\rho_1\sigma_1,k_1$}{L}
\fmflabel{$\lambda_1,p_1,a_1$}{R1}
\fmflabel{$\lambda_2,p_2,a_2$}{R2}
\fmflabel{$\lambda_3,p_3,a_3$}{R3}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonGluonVertex$[\{\rho_1 , \sigma_1, k_1\},p_1,\lambda_1,a_1,p_2,\lambda_2,a_2,\lambda_3,p_3,a_3]$

\begin{gathered}
\begin{fmffile}{Table_Illustration_Gluon_3}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R1,R2,R3,R4}
\fmf{dbl_wiggly,tension=4}{L,V}
\fmf{gluon}{R1,V}
\fmf{gluon}{R2,V}
\fmf{gluon}{R3,V}
\fmf{gluon}{R4,V}
\fmfdot{V}
\fmflabel{$\rho_1\sigma_1,k_1$}{L}
\fmflabel{$\lambda_1,p_1,a_1$}{R1}
\fmflabel{$\lambda_2,p_2,a_2$}{R2}
\fmflabel{$\lambda_3,p_3,a_3$}{R3}
\fmflabel{$\lambda_4,p_4,a_4$}{R4}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonGluonVertex$[\{\rho_1 , \sigma_1, k_1\},p_1,\lambda_1,a_1,p_2,\lambda_2,a_2,\lambda_3,p_3,a_3,\lambda_4,p_4,a_4]$

\begin{gathered}
\begin{fmffile}{Table_Illustration_Quark-Gluon_1}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R1,R2,R3}
\fmf{dbl_wiggly,tension=3}{L,V}
\fmf{fermion}{R1,V}
\fmf{gluon}{R2,V}
\fmf{fermion}{V,R3}
\fmfdot{V}
\fmflabel{$\rho_1\sigma_1,k_1$}{L}
\fmflabel{$\lambda, a$}{R2}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonQuarkGluonVertex$[\{\rho_1, \sigma_1\}, \lambda, a]$

\begin{gathered}
\begin{fmffile}{Table_Illustration_YM_Ghost_1}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R1,R2}
\fmf{dbl_wiggly,tension=2}{L,V}
\fmf{dots}{R1,V}
\fmf{dots}{R2,V}
\fmfdot{V}
\fmflabel{$\rho_1\sigma_1,k_1$}{L}
\fmflabel{$p_1,a_1$}{R1}
\fmflabel{$p_2,a_2$}{R2}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonYMGhostVertex$[\{\rho_1 , \sigma_1\}, p_1, a_1, p_2, a_2]$

\begin{gathered}
\begin{fmffile}{Table_Illustration_YM_Ghost_2}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R1,R2,R3}
\fmf{dbl_wiggly,tension=3}{L,V}
\fmf{dots}{R1,V}
\fmf{gluon}{R2,V}
\fmf{dots}{R3,V}
\fmfdot{V}
\fmflabel{$\rho_1\sigma_1,k_1$}{L}
\fmflabel{$p_2,a_2$}{R1}
\fmflabel{$\lambda_1,p_1,a_1$}{R2}
\fmflabel{$p_3,a_3$}{R3}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonGluonGhostVertex$[\{\rho_1 , \sigma_1\},p_1, \lambda_1, a_1, p_2, \lambda_2, a_2, p_3, \lambda_3, a_3]$

Examples of interaction rules for the scalar, Proca, and massless vector fields.

Lastly, the following commands describe the gravitational sector. The command “GravitonPropagator” implements the graviton propagator with “FeynAmpDenominator” functions from FeynCalc. This command shall be used for loop calculations. The command “GravitonPropagatorAlternative” implements the graviton propagator with the simple scalar products in the denominators. This command shall only be used in three-level calculations. Two more commands, “GravitonPropagatorTop” and “GravitonPropagatorTopFAD”, generate the numerator of the graviton propagator alone. The first one uses simple scalar products, while the second one uses “FeynAmpDenominator”. All these commands take five arguments: four Lorentz indices and the momenta of a graviton. The command “GravitonVertex” corresponds to the $n$-graviton vertex and takes $3 n$ arguments: two Lorentz indices and the momentum of each graviton. The Faddeev-Popov ghosts are vectors, so their propagator receives an additional Minkowski metric in the numerator. One can use the command for the Yang-Mills ghost propagator and manually multiply it on the flat metric. Because of this, no new command corresponding to this propagator is present in FeynGrav. Lastly, the rules for graviton coupling to the corresponding Faddeev-Popov ghosts are implemented with “GravitonGhostVertex”. It takes five arguments: the array of Lorentz indices and momenta of the gravitons, and Lorenz indices and momenta of ghosts. Table <ref> presents these rules.

Diagram      Command
\begin{gathered}
\begin{fmffile}{Table_Illustration_Graviton_0}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R}
\fmf{dbl_wiggly}{L,R}
\fmflabel{$\mu\nu$}{L}
\fmflabel{$\alpha\beta ~~ p$}{R}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonPropagator$[\mu, \nu, \alpha, \beta, p]$

\begin{gathered}
\begin{fmffile}{Table_Illustration_Graviton_1}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R1,R2}
\fmf{dbl_wiggly}{L,V}
\fmf{dbl_wiggly}{V,R1}
\fmf{dbl_wiggly}{V,R2}
\fmflabel{$\rho_1,\sigma_1,p_1$}{L}
\fmflabel{$\rho_2,\sigma_2,p_2$}{R1}
\fmflabel{$\rho_3,\sigma_3,p_3$}{R2}
\fmfdot{V}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonVertex$[\rho_1, \sigma_1, p_1, \rho_2, \sigma_2, p_2, \rho_3, \sigma_3, p_3]$

\begin{gathered}
\begin{fmffile}{Table_Illustration_Graviton_2}
\begin{fmfgraph*}(30,30)
\fmfleft{L1,L2}
\fmfright{R1,R2}
\fmf{dbl_wiggly}{L1,V}
\fmf{dbl_wiggly}{L2,V}
\fmf{dbl_wiggly}{V,R1}
\fmf{dbl_wiggly}{V,R2}
\fmflabel{$\rho_1,\sigma_1,p_1$}{L1}
\fmflabel{$\rho_2,\sigma_2,p_2$}{L2}
\fmflabel{$\rho_3,\sigma_3,p_3$}{R1}
\fmflabel{$\rho_4,\sigma_4,p_4$}{R2}
\fmfdot{V}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonVertex$[\rho_1, \sigma_1, p_1, \rho_2, \sigma_2, p_2, \rho_3, \sigma_3, p_3, \rho_4, \sigma_4, p_4]$

\begin{gathered}
\begin{fmffile}{Table_Illustration_Graviton_Ghost}
\begin{fmfgraph*}(30,30)
\fmfleft{L}
\fmfright{R1,R2}
\fmf{dbl_wiggly}{L,V}
\fmf{dots}{V,R1}
\fmf{dots}{V,R2}
\fmflabel{$\rho_1,\sigma_1,k_1$}{L}
\fmflabel{$\lambda_1, p_1$}{R1}
\fmflabel{$\lambda_2, p_2$}{R2}
\fmfdot{V}
\end{fmfgraph*}
\end{fmffile}
\end{gathered}
GravitonGhostVertex$[\{\rho_1 , \sigma_1, k_1\},\lambda_1, p_1, \lambda_2, p_2]$

Examples of interaction rules for gravitons.

FeynGrav also implements a few additional tools. Firstly, the package provides a realisation of the standard gauge projectors:

\begin{align}
\theta_{\mu\nu}(p) & = \eta_{\mu\nu} - \cfrac{ p_\mu p_\nu}{p^2} \, , & \overline{\theta}_{\mu\nu} & = \cfrac{p_\mu p_\nu}{p^2} \, .
\end{align}

They are realised with commands “GaugeProjector”, “GaugeProjectorBar”, “GaugeProjectorFAD”, and “GaugeProjectorBarFAD”. The first two commands realise these projectors with the simple scalar products, while the former two commands use “FeynAmpDenominator”.

Secondly, the package provides a realisation of the Nieuwenhuizen operators <cit.>. These operators $P^0_{\mu\nu\alpha\beta}(p)$, $P^1_{\mu\nu\alpha\beta}(p)$, $P^2_{\mu\nu\alpha\beta}(p)$, $\overline{P^0}_{\mu\nu\alpha\beta}(p)$, $\overline{\overline{P^0}}_{\mu\nu\alpha\beta}(p)$ generalise the standard projectors for the spin-$2$ systems. Their features were discussed in detail in a previous publication <cit.>. Commands “NieuwenhuizenOperator0”, “NieuwenhuizenOperator1”, “NieuwenhuizenOperator2”, “NieuwenhuizenOperator0Bar”, and “NieuwenhuizenOperator0BarBar” implement these operators using only the simple scalar product. The set of command “NieuwenhuizenOperator0FAD”, $\cdots$, “NieuwenhuizenOperator0BarBarFAD” implements them with “FeynAmpDenominator”.

Lastly, the package implements the polarisation tensors for gravitons. Following the standard procedure, the polarisation tensor $\epsilon_{\mu\nu}^{\pm}$ for gravitons is defined through the polarisation vector $\epsilon_\mu^\pm$ for the standard vector field:

\begin{align}
\epsilon_{\mu\nu}^{\pm} = \epsilon^\pm_\mu \epsilon^\pm_\nu .
\end{align}

The command “PolarizationTensor” implements this definition. The command takes three arguments: two Lorentz indices and one four-momentum. The command directly multiplies polarisation vectors implemented in FeynCalc. Due to the definitions used in FeynCalc, the implemented polarisation tensor is neither traceless nor transverse – the command “SetPolarizationTensor” the tensors traceless and transverse.

§.§ Examples

Several publications have used FeynGrav <cit.>. These publications discuss the problems in detail, so we will only provide a brief discussion. We also focus on two vivid examples.

The first illustrative example of FeynGrav usage is the calculation of $2\to 2$ graviton scattering cross sections at the leading order of perturbation theory. The on-shell cross section was calculated long ago without using any contemporary computational tools <cit.>, which makes such calculations notoriously complicated. To be exact, in the original article <cit.>, the author operated with expressions symmetries in a certain way to make the calculation more compact. Nonetheless, even the basic description of the calculations takes a few pages in the original publication.

FeynGrav extremely simplifies such calculations. Firstly, one shall fix all the momenta on the mass shell and define all relations for scalar products of polarisation tensors and momenta since FeynCalc does not do this automatically. Secondly, one calculates the matrix element for the $s$-channel amplitude:

\begin{align}
\nonumber \\
\begin{split}
& \hspace{40pt}
\begin{gathered}
\begin{fmffile}{Examples_Pic_1}
\begin{fmfgraph*}(30,30)
\fmfleft{L1,L2}
\fmfright{R1,R2}
\fmf{dbl_wiggly}{L1,V1,R1}
\fmf{dbl_wiggly}{L2,V2,R2}
\fmf{dbl_wiggly}{V1,V2}
\fmfdot{V1,V2}
\fmflabel{$\mu_1\nu_1,p_1$}{L1}
\fmflabel{$\mu_2\nu_2,p_2$}{R1}
\fmflabel{$\mu_3\nu_3,p_3$}{L2}
\fmflabel{$\mu_4\nu_4,p_4$}{R2}
\end{fmfgraph*}
\end{fmffile}
\end{gathered} \hspace{35pt} = \text{PolarizationTensor}[\mu_1,\nu_1,p_1] \text{PolarizationTensor}[\mu_2,\nu_2,p_2] \\ \\
& \times \text{ComplexConjugate}[\text{PolarizationTensor}[\mu_3,\nu_3,p_3]] \text{ComplexConjugate}[\text{PolarizationTensor}[\mu_4,\nu_4,p_4]]\\
& \times \text{GravitonVertex}[\mu_1,\nu_1,p_1,\mu_2,\nu_2,p_2,\alpha_1,\beta_1,p_1 \!+\! p_2] \text{GravitonVertex}[\alpha_2,\beta_2,p_1 \!+\! p_2,\mu_3,\nu_3,p_3,\mu_4,\nu_4,p_4] \\
& \times \text{GravitonPropagatorAlternative}[\alpha_1,\beta_1,\alpha_2,\beta_2,p_1 \!+\! p_2].
\end{split}
\end{align}

An average computer not designed for intensive computational tasks can complete this element in less than five minutes. One obtains the expressions for $t$ and $u$-channels similarly. The following expression gives the contact four-graviton interaction:

\begin{align}
\nonumber \\
\begin{split}
& \hspace{40pt}
\begin{gathered}
\begin{fmffile}{Examples_Pic_2}
\begin{fmfgraph*}(30,30)
\fmfleft{L1,L2}
\fmfright{R1,R2}
\fmf{dbl_wiggly}{L1,V}
\fmf{dbl_wiggly}{L2,V}
\fmf{dbl_wiggly}{R1,V}
\fmf{dbl_wiggly}{R2,V}
\fmflabel{$\mu_1\nu_1,p_1$}{L1}
\fmflabel{$\mu_2\nu_2,p_2$}{R1}
\fmflabel{$\mu_3\nu_3,p_3$}{L2}
\fmflabel{$\mu_4\nu_4,p_4$}{R2}
\fmfdot{V}
\end{fmfgraph*}
\end{fmffile}
\end{gathered} \hspace{40pt} = \text{PolarizationTensor}[\mu_1,\nu_1,p_1] \text{PolarizationTensor}[\mu_2,\nu_2,p_2] \\ \\
& \times \text{ComplexConjugate}[\text{PolarizationTensor}[\mu_3,\nu_3,p_3]] \text{ComplexConjugate}[\text{PolarizationTensor}[\mu_4,\nu_4,p_4]]\\
& \times \text{GravitonVertex}[\mu_1,\nu_1,p_1,\mu_2,\nu_2,p_2,\mu_3,\nu_3,p_3,\mu_4,\nu_4,p_4] .
\end{split}
\end{align}

This expression takes less than one minute to be computed.

Lastly, one constructs the complete scattering amplitude accounting for all contributions. It recovers the well-known expressions for chiral amplitudes in $d=4$:

\begin{align}
\mathcal{M}(++++) &= i\, \cfrac{\kappa^2}{4}\,\cfrac{s^4}{s\,t\,u} \, , & \mathcal{M}(+-+-) & = i\, \cfrac{\kappa^2}{4}\,\cfrac{u^4}{s\,t\,u} \, , & \mathcal{M}(+--+) & = i\, \cfrac{\kappa^2}{4}\,\cfrac{t^4}{s\,t\,u} \, ,
\end{align}

\begin{align*}
\mathcal{M}(++--) &= \mathcal{M} (+++-) = 0 .
\end{align*}

This example vividly shows that such complicated yet essential calculations are significant with FeynGrav. Moreover, the paper <cit.> presents an explicit expression for the scattering amplitude expressed in terms of graviton chiralities.

The second example is the calculation of the one-loop graviton-scalar vertex operator. Since a scalar field interacts with gravity, the interaction vertex receives correction at the loop level. The one-loop vertex operator describes these corrections:

\begin{align}
\nonumber \\
\begin{gathered}
\begin{fmffile}{Example_Vertex_0}
\begin{fmfgraph*}(30,30)
\fmftop{T1,T2}
\fmfbottom{B}
\fmf{dashes}{T1,V,T2}
\fmf{dbl_wiggly}{B,V}
\fmfv{decor.shape=circle,decor.filled=shaded,decor.size=15}{V}
\fmflabel{$\mu\nu,k$}{B}
\fmflabel{$p_1$}{T1}
\fmflabel{$p_2$}{T2}
\end{fmfgraph*}
\end{fmffile}
\end{gathered} \hspace{20pt} = i\, \Gamma_{\mu\nu}(p_1,p_2,k,m)
\begin{gathered}
\begin{fmffile}{Example_Vertex_1}
\begin{fmfgraph}(30,30)
\fmftop{T1,T2}
\fmfbottom{B}
\fmf{dbl_wiggly,tension=2}{B,VB}
\fmf{dashes,tension=2}{T1,VT1}
\fmf{dashes,tension=2}{T2,VT2}
\fmf{phantom}{VB,VT2,VT1,VB}
\fmfdot{VB,VT1,VT2}
\fmffreeze
\fmf{dashes}{VT1,VT2}
\fmf{dbl_wiggly}{VT1,VB,VT2}
\end{fmfgraph}
\end{fmffile}
\end{gathered}
\begin{gathered}
\begin{fmffile}{Example_Vertex_2}
\begin{fmfgraph}(30,30)
\fmftop{T1,T2}
\fmfbottom{B}
\fmf{dbl_wiggly,tension=2}{B,VB}
\fmf{dashes,tension=2}{T1,VT1}
\fmf{dashes,tension=2}{T2,VT2}
\fmf{phantom}{VB,VT2,VT1,VB}
\fmfdot{VB,VT1,VT2}
\fmffreeze
\fmf{dbl_wiggly}{VT1,VT2}
\fmf{dashes}{VT1,VB,VT2}
\end{fmfgraph}
\end{fmffile}
\end{gathered}
\begin{gathered}
\begin{fmffile}{Example_Vertex_3}
\begin{fmfgraph}(30,30)
\fmftop{T1,T2}
\fmfbottom{B}
\fmf{dashes,tension=2}{T1,VT,T2}
\fmf{dbl_wiggly,left=1}{VT,VB,VT}
\fmf{dbl_wiggly,tension=2}{B,VB}
\fmfdot{VT,VB}
\end{fmfgraph}
\end{fmffile}
\end{gathered}
\begin{gathered}
\begin{fmffile}{Example_Vertex_4}
\begin{fmfgraph}(30,30)
\fmftop{T1,T2}
\fmfbottom{B}
\fmf{dashes}{T1,V,T2}
\fmf{dbl_wiggly,tension=2}{V,B}
\fmffreeze
\fmf{phantom}{V,VV,T1}
\fmffreeze
\fmf{dbl_wiggly,left=1}{V,VV}
\fmfdot{V,VV}
\end{fmfgraph}
\end{fmffile}
\end{gathered}
\begin{gathered}
\begin{fmffile}{Example_Vertex_5}
\begin{fmfgraph}(30,30)
\fmftop{T1,T2}
\fmfbottom{B}
\fmf{dashes}{T1,V,T2}
\fmf{dbl_wiggly,tension=2}{V,B}
\fmffreeze
\fmf{phantom}{V,VV,T2}
\fmffreeze
\fmf{dbl_wiggly,right=1}{V,VV}
\fmfdot{V,VV}
\end{fmfgraph}
\end{fmffile}
\end{gathered}
\begin{gathered}
\begin{fmffile}{Example_Vertex_6}
\begin{fmfgraph}(30,30)
\fmftop{T1,T,T2}
\fmfbottom{B}
\fmf{dashes,right=.3}{T1,V,T2}
\fmf{dbl_wiggly,tension=2}{V,B}
\fmffreeze
\fmf{phantom,tension=2}{T,VT}
\fmf{phantom}{V,VT}
\fmffreeze
\fmf{dbl_wiggly,left=.7}{V,VT,V}
\fmfdot{V}
\end{fmfgraph}
\end{fmffile}
\end{gathered} . \\ \nonumber
\end{align}

In contrast to the previous example, this expression was only calculated with FeynGrav because it involved hundreds of terms. The expression itself is too long to be presented in a printed ford. It is published in an open-access repository <cit.>. The obtained expression allows one to study the low energy limit of a gravitational scattering of two scalars <cit.>:

\begin{align}
\cfrac{d\sigma}{d\Omega} \Bigg|_\text{low energy} = \cfrac14\, \left(G_\text{N} \,\mu\, m_1\, m_2\right)^2 \cfrac{1}{p^4\, \sin^4\frac{\theta}{2}}\left[ 1 + 32\, \pi^2 \,G_\text{N} \, p (m_1+m_2) \sin\cfrac{\theta}{2}  +\mathcal{O}\left( G_\text{N}^2\right)\right].
\end{align}

In this expression for the differential cross section, $p$ is the centre of mass scattering momentum, $\theta$ is the scattering angle, $m_1$ and $m_2$ are masses of the scattered particles, and $\mu$ is the reduced mass.

§ SUMMARY AND FURTHER DEVELOPMENT

The paper reviews a recently developed theoretical framework for efficient computation of Feynman rules for perturbative quantum gravity. The perturbative approach to quantum gravity allows one to study quantum gravitational effects within the standard quantum field theory framework. The resulting theory is effective, so it cannot be indefinitely extended in the ultraviolet region and treated similarly to other renormalisable theories. Due to its effective nature, the theory admits infinite interaction terms, all parametrised with a single gravitational coupling. The discussed theoretical framework provides a universal way to compute such interaction rules at any order of perturbation theory.

Perturbative quantum gravity admits the factorisation theorem applicable to any gravity model within Riemann geometry. The theorem states that the action splits into two parts in a broad class of gravity models. One part does not involve derivatives and is constituted of three independent factors that are infinite series. The other part involves derivatives, but it is a finite expression that can be calculated explicitly.

Certain recursive relations exist for the factors generating infinite series. They provide a way to efficiently calculate their contribution at any order of perturbation theory. However, the number of terms in each contribution grows faster than the factorial number of particles involved in the interaction. This feature makes manual calculations of such factors challenging.

FeynGrav provides a tool to operate with the theory at the practical level. It implements the discussed theoretical framework within FeynCalc, which possesses a broad spectrum of tools to operate with expressions within the quantum field theory.

The latest version implements the interaction rules for massless and massive scalars, for scalar field potential interaction, for massive and massless Dirac fermions, for Proca field, for massless vector field, for $SU(N)$ Yang-Mills model, and general relativity. The package also implements auxiliary tools such as graviton polarisation tensors, gauge projectors, and their generalisation for the spin-$2$ case.

The package is rapidly developing and is expected to receive a new update soon. The update will be devoted to the Horndeski gravity, the most general scalar-tensor gravity admitting second-order field equations and the minimal coupling to the matter degrees of freedom.

Further development of the package aims in two directions: to improve the performance and to account for other relevant gravity models. The discovery of the recursive algorithm discussed in this paper significantly increased performance. At the same time, the same relations directly indicate the limits of this existing approach. The number of terms in an $n$ particle interaction vertex grows faster than $n!$, which poses an additional computation challenge. Although such behaviour is expected for an effective theory considered perturbatively, we will strive to improve the computational algorithm even further to push the perturbation theory to its limits.

Consideration of more sophisticated yet relevant models of gravity is another challenge. The Horndeski gravity, the most studied extension of general relativity, is studied in this paper, and its implementation is to enter the next version of FeynGrav. There are few options for other models of gravity that would be relevant to the contemporary research field. The most straightforward generalisation would be massive gravity, but the contemporary observational data imposes strong constraints on the graviton mass. Another possible way to extend FeynGrav is to implement Beyond Horndeski theories, which introduce non-minimal couplings between the scalar field and matter. Lastly, it may be fruitful to consider some supersymmetric expressions and examine whether they can be treated similarly.

In conclusion, the theoretical framework discussed in the paper and the FeynGrav package provide a valuable and efficient tool to study perturbative quantum gravity, with further development aimed at improving performance and incorporating other relevant gravity models.

§ ACKNOWLEDGMENT

This work was supported by the Institute for Basic Science Grant IBS-R018-Y1.


Further author information: (Send correspondence to K.K.)
K.K.: E-mail<EMAIL_ADDRESS>
# Large format imaging spectrograph for the Large Submillimeter Telescope
(LST)

Kotaro Kohno  Institute of Astronomy, School of Science, The University of
Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan  Research Center for the
Early Universe, School of Science, The University of Tokyo, 7-3-1 Hongo,
Bunkyo, Tokyo 113-0033, Japan  Ryohei Kawabe  National Astronomical
Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  The
Graduate University for Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka,
Tokyo 181-8588, Japan  Yoichi Tamura  Division of Particle and Astrophysical
Science, Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan
Akira Endo  Faculty of Electrical Engineering, Mathematics and Computer
Science, Delft University of Technology, Delft, The Netherlands  Kavli
Institute of NanoScience, Faculty of Applied Sciences, Delft University of
Technology, Delft, The Netherlands  Jochem J. A. Baselmans  SRON – Netherlands
Institute for Space Research, Utrecht, The Netherlands  Faculty of Electrical
Engineering, Mathematics and Computer Science, Delft University of Technology,
Delft, The Netherlands  Kenichi Karatsu  SRON – Netherlands Institute for
Space Research, Utrecht, The Netherlands  Faculty of Electrical Engineering,
Mathematics and Computer Science, Delft University of Technology, Delft, The
Netherlands  Akio K. Inoue  Department of Physics, School of Advanced Science
and Engineering, Faculty of Science and Engineering, Waseda University, 3-4-1
Okubo, Shinjuku, Tokyo 169-8555, Japan  Waseda Research Institute for Science
and Engineering, Faculty of Science and Engineering, Waseda University, 3-4-1
Okubo, Shinjuku, Tokyo 169-8555, Japan  Department of Environmental Science
and Technology, Faculty of Design Technology, Osaka Sangyo University, 3-1-1
Nakagaito, Daito, Osaka 574-8530, Japan  Kana Moriwaki  Department of Physics,
School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo
113-0033, Japan  Natsuki H. Hayatsu  Department of Physics, School of Science,
The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  Naoki
Yoshida  Department of Physics, School of Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  Kavli Institute for the Physics
and Mathematics of the Universe (WPI), UT Institutes for Advanced Study, The
University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan
Research Center for the Early Universe, School of Science, The University of
Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  Yuki Yoshimura  Institute
of Astronomy, School of Science, The University of Tokyo, 2-21-1 Osawa,
Mitaka, Tokyo 181-0015, Japan  Bunyo Hatsukade  Institute of Astronomy, School
of Science, The University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015,
Japan  Hideki Umehata  RIKEN Cluster for Pioneering Research, 2-1 Hirosawa,
Wako, Saitama 351-0198, Japan  Institute of Astronomy, School of Science, The
University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan  Tai Oshima
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan  Tatsuya Takekoshi  Kitami Institute of Technology, 165 Koen-
cho, Kitami, Hokkaido 090-8507, Japan  Institute of Astronomy, School of
Science, The University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan
Akio Taniguchi  Division of Particle and Astrophysical Science, Graduate
School of Science, Nagoya University, Nagoya 464-8602, Japan  Pamela D.
Klaassen  UK Astronomy Technology Centre, Royal Observatory Edinburgh,
Blackford Hill, Edinburgh EH9 3HJ, UK  Tony Mroczkowski  European Southern
Observatory (ESO), Karl-Schwarzschild-Strasse 2, Garching 85748, Germany
Claudia Cicone  Institute of Theoretical Astrophysics, University of Oslo,
P.O. Box 1029, Blindern, 0315 Oslo, Norway  Frank Bertoldi  Max-Planck-
Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany
Argelander-Institut für Astronomie, University at Bonn, Auf dem Hügel 71,
D-53121 Bonn, Germany  Helmut Dannerbauer  Instituto de Astrofísica de
Canarias, E-38205 La Laguna, Tenerife, Spain  Universidad de La Laguna,
Departamento de Astrofísica, E-38206 La Laguna, Tenerife, Spain  Tomoka Tosaki
Joetsu University of Education, Yamayashiki-machi, Joetsu, Niigata 943-8512,
Japan

###### Abstract

We present a conceptual study of a large format imaging spectrograph for the
Large Submillimeter Telescope (LST) and the Atacama Large Aperture
Submillimeter Telescope (AtLAST). Recent observations of high-redshift
galaxies indicate the onset of earliest star formation just a few 100 million
years after the Big Bang (i.e., $z$ = 12–15), and LST/AtLAST will provide a
unique pathway to uncover spectroscopically-identified “first forming
galaxies” in the pre-reionization era, once it will be equipped with a large
format imaging spectrograph. We propose a 3-band (200, 255, and 350 GHz),
medium resolution ($R$ = 2,000) imaging spectrograph with $\sim$1.5 M
detectors in total based on the KATANA concept (Karatsu et al. 2019), which
exploits technologies of the integrated superconducting spectrometer (ISS) and
a large-format imaging array. A 1-deg2 drilling survey (3,500 hr) will capture
a large number of [O iii] 88 $\mu$m (and [C ii] 158 $\mu$m) emitters at $z$ =
8–9, and constrain [O iii] luminosity functions at $z>12$.

###### keywords:

galaxy formation and evolution, fine structure lines ([O iii] , [C ii] ),
kinetic inductance detectors (KIDs), integrated superconducting spectrometer
(ISS), DESHIMA/MOSAIC/KATANA, imaging spectrograph, The Large Submillimeter
Telescope (LST), The Atacama Large Aperture Submillimeter Telescope (AtLAST)

##  1 INTRODUCTION

Recent multi-wavelength surveys of distant galaxies have revealed that the
majority of the star-forming activities at a redshift $z=$ 1–3, where the
cosmic star formation rate densities peak, is obscured by dust, emphasizing
importance of the study of dust-obscured activities in the universe. However,
the roles of the dust-obscured star-formation beyond the redshift of $z>$ 4–8
remain unclear, because different measurement techniques result in different
indications, as shown in Figure 1. For instance, ALMA observations of Lyman
break galaxies in Hubble Ultra Deep Field suggest that the dust obscured star
formation plays rather minor roles among such star-forming galaxies selected
by rest-frame ultraviolet (UV) radiation[1, 2]. On the other hand, Herschel
observations of “red” SPIRE sources suggest elevated star-formation rate
densities up to $z\sim 6$ [3], which is also supported by recent ALMA
observations [4]. Latest ALMA studies also demonstrate the presence of
sub/millimeter-selected galaxies without any significant counterpart seen in
the optical and near-infrared: HST-dark galaxies [5, 6, 7, 8, 9]. Although
such a class of galaxies has been known from the beginning of the discovery of
the submillimeter galaxies (SMGs), recent ALMA observations unveil much
fainter sub/millimeter sources ($S_{\rm 1mm}\sim$ a few 10–100 $\mu$Jy), which
were unable to isolate before due to the source confusion limit of
submillimeter-wave survey facilities like SCUBA/SCUBA2 on JCMT 15-m telescope
and AzTEC on ASTE 10-m telescope. And more importantly, such faint
sub/millimeter galaxies are much more ubiquitous[6] and therefore responsible
for the bulk of the cosmic infrared background-light (CIB)[5, 10, 11]. The
importance of HST-dark but IRAC-detected (a.k.a. H-dropout) galaxies as a key
tracer of the early phases of massive galaxy formation at $z\sim$ 4–6 has been
discussed [6, 12], but the difficulty of obtaining spectroscopic redshifts for
such HST-dark galaxies hampers the physical characterization of these sources.
After pioneering spectral scan observations of the Hubble Deep Field North
using IRAM Plateau de Bure Interferometer [13], ALMA and NOEMA have been used
to search for millimeter-wave line emitters[14, 15, 16, 17, 18, 19, 20]
without any priors in the optical/near-infrared bands, but the total surveyed
area has been (and will be) limited to tiny patches of the sky because of the
narrowness of the ALMA/NOEMA field-of-view. All of this progress motivates us
to design a wide-area spectroscopic survey of dust-enshrouded galaxies in a
more systematic way [21].

---
Figure 1:  The redshift evolution of the cosmic star-formation rate density,
illustrating two inconsistent trends of measurements at $z=4-8$, i.e., “dust-
rich, actively star-forming universe” (suggested by Herschel \- FIR,
long-$\gamma$-ray bursts, and cosmic infrared background-light (CIB) lensing
analysis with Planck) and “dust-poor, calm early universe” (suggested by
observations of Lyman break galaxies with dust-extinction corrections).

Recent discoveries of star-forming galaxies at $z$ = 8–11[22, 23, 24] and
candidate passive galaxies at $z\sim 6$[25] indicate the onset of the earliest
star formation just a few 100 million years after the Big Bang, i.e., $z\sim$
12–15. This prompts us to establish a methodology to uncover a statistically
large number of spectroscopically-identified, “first forming galaxies” during
the epoch of reionization (EoR) and beyond, i.e., the pre-reionization
era/cosmic dawn. Although both ALMA and JWST/NIRSpec can detect [O iii] 88
$\mu$m and [C iii] 1907Å+ C iii] 1909Å lines, respectively, at $z>$ 10–15 with
a reasonable observing time, they are not optimized for a wide-area survey to
uncover such candidate sources. A $>$100 deg2 near-infrared imaging survey at
$\lambda$ = 2–5 $\mu$m with a modest depth ($\sim$27 $m_{\rm AB}$) would be
the optimum option to find spectroscopic follow-up targets at $z\sim 15$
(Inoue et al.) but the Roman Space Telescope will have no survey capability at
$\lambda>2$ $\mu$m.

Here we argue that next-generation large submillimeter single-dish telescopes
optimized for wide-area, broad-band spectral coverage surveys, such as the
Large Submillimeter Telescope (LST)[26]111https://www.lstobservatory.org and
the Atacama Large Aperture Submillimeter Telescope
(AtLAST)[27]222https://atlast-telescope.org, will provide a unique pathway for
that purpose once these telescopes are equipped with a large format imaging
spectrograph[28]. In this paper, we present a conceptual study of a 3-band
imaging spectrograph, which specifically aims for the [O iii] 88 $\mu$m and
the [C ii] 158 $\mu$m tomography at $z$ = 4–8 (with [C ii] ) and $z$ = 8–16
(with [O iii] ), based on the KATANA concept (Karatsu et al.
2019)333https://agenda.infn.it/event/15448/contributions/95630/ using the
technologies of the integrated superconducting spectrometer (ISS)[29, 30] and
a large-format imaging array like A-MKID[31]. Our goal here is not to propose
a very detailed specification of the observing instrument, but to provide a
conceivable observing instrument case based on a set of science requirements;
it will give some insights and implications for further investigations of the
observing instrument and technologies for LST/AtLAST. A brief overview and
updates of the LST project are also given.

##  2 The Large submillimeter telescope

The LST project was originally discussed as the planning of a next-generation
telescope for sub/millimeter-wavelengths that could inherit both the large
collecting area of the Nobeyama Radio Observatory (NRO) 45-m telescope[32] and
the submillimeter capabilities of Atacama Submillimeter Telescope Experiment
(ASTE) 10-m telescope [33]. The current major specifications of the LST, along
with the conceptual design and key science behind have been summarized in
Kawabe et al. (2016)[26]. In brief, the LST will be a 50-m diameter high-
precision (45 $\mu$m rms) telescope for wide-area imaging and spectroscopic
surveys with a field-of-view of $>$0.5 deg $\phi$ primarily focusing on the
70–420 GHz frequency range, with a capability for high frequencies up to 1 THz
using an inner high-precision surface. A novel concept of a millimeter
wavefront sensor that allows real-time sensing of the surface, which will be a
key to establish “millimetric adaptive optics (MAO)”, has been proposed by
Tamura, Y., et al.[34]. Statistical approaches to efficient atmospheric noise
removal for submillimeter-wave spectroscopy have been proposed and implemented
by Taniguchi, A., et al. for e.g., NRO 45-m telescope[35] and DESHIMA on
ASTE[36].

One of the recent major milestones of the LST project is the master plan 2020
(MP2020) led by the Science Council of Japan (SCJ), which aims to set the list
of high priority large academic research projects in Japan. The LST project
was invited to give a presentation at two MP2020 symposia in September 2018
and January 2019, which were organized by the astronomy and astrophysics sub-
committee of SCJ, to discuss the progress of the project since the previous
master plan activity (MP2017) and the status of the international
collaboration including the EU-led AtLAST project. A support letter from the
AtLAST community was highly appreciated. Now the LST is formally listed as one
of the large academic projects in the astronomy and astrophysics field in
MP2020, which was announced in January 2020. During these activities, a merger
between the AtLAST and LST projects has been intensively discussed. After the
success of the 3.5 M Euro ERC program for the AtLAST design study in
2021–2024[27], we anticipate the merger in 2024 where the design study will
end. There are some apparent inconsistencies of telescope specifications such
as the target surface accuracy and the field of view, which are expected to be
discussed further during the AtLAST design study in coming years. A next step
forward in the LST would be a proposal to National Astronomical Observatory of
Japan (NAOJ) to launch a study group of the LST under a support from the
community in Japan. Figure 2 displays the LST timelines and milestones, along
with those of the AtLAST.

---
Figure 2:  LST project timeline and milestones.
---
Figure 3:  Summary of the major specifications of DESHIMA, MOSAIC, and the
proposed 3-band imaging spectrograph based on the KATANA concept, which will
exploit technologies of the integrated superconducting spectrometer (DESHIMA)
and a large-format imaging array (like A-MKID). Pictures of DESHIMA and
A-MKID, along with the proposed array configuration of MOSAIC, are inserted.

##  3 From DESHIMA to KATANA: 3-band imaging spectrograph

We propose a 3-band (200, 255, and 350 GHz), medium resolution ($R$=2,000)
imaging spectrograph with $\sim$1.5 M detectors in total, which is anticipated
to become available in the early 2030s. The key technologies of the the KATANA
concept (Karatsu et al. 2019) are a dual-polarization sensitive broadband
antenna, an on-chip filter-bank spectrometer, and NbTiN-Al hybrid microwave
kinetic inductance detectors (MKIDs) to readout the spectral channels, which
have been successfully demonstrated by the recent DESHIMA 1.0 on ASTE campaign
[37, 29, 38]. DESHIMA 2.0, which will have a much wider frequency coverage
with a better optical efficiency, will be deployed in ASTE for full science
operation in mid-2021. MOSAIC, a multi-pixel version of DESHIMA, has been
proposed for the Large Millimeter Telescope Alfonso Serrano
(LMT)444http://lmtgtm.org and discussed during the Guillermo Haro 2018
Workshop. Figure 3 presents a summary of major specifications of DESHIMA,
MOSAIC, and the proposed 3-band imaging spectrograph with the KATANA concept.
Here “KATANA” is for the existing facilities like ASTE and LMT, whereas KATANA
2.0 is for the future survey telescopes in the early 2030s, i.e., the
LST/AtLAST. Figure 4 summarizes the frequency ranges and corresponding
redshifts of [O iii] and [C ii] , along with the conceptual view of a fiducial
1-deg2 drilling survey.

One of the key requirements for the instrument is a spectral resolution $R$ of
2,000 (a velocity resolution $dv$ of 150 km s-1), because the target sources
at $z>8$ are expected to be less massive and therefore have a narrow line
width of $\sim$100 km s-1, as demonstrated by reported [O iii] 88 $\mu$m line
emitters at $z$ = 7–9 ($dv$ = a few 10–150 km s-1)[39, 23, 24]. This is
currently the limiting factor of the spatial pixels because we need $>$100
spectral channels to have a redshift width $dz$ of around unity for each band
(Figure 4). The line mapping speed of the proposed configurations was computed
based on the achievements with the DESHIMA 1.0 along with the reasonable
assumptions of the telescope surface accuracy (45 $\mu$m rms) and a
precipitable water vapor (0.5 mm). We set a 5$\sigma$ line sensitivity of 1
mJy (peak flux) or 0.15 Jy km s-1 at Band-3, which corresponds to a [O iii] 88
$\mu$m line luminosity of $4\times 10^{8}$ $L_{\odot}$ at $z\sim$ 8–9, based
on the estimation of the [O iii] 88 $\mu$m line luminosity functions from the
UV luminosity functions[40, 41] and their extrapolation (Inoue et al.).
Theoretical simulations[42] also support the validity of the target line
sensitivity.

We find that a 1-deg2 survey spending 3,500 hrs with the proposed 3-band
configuration will uncover a statistically large number of [O iii] 88 $\mu$m
line emitters at $z$ = 8–9. A fraction of them will be [O iii] - [C ii] dual
line emitters because [C ii] can also be bright at $z\sim 8$ galaxies[43].
Furthermore, Band-2 is designed to cover the same redshift range in the [N ii]
122 $\mu$m line. Although the nitrogen line must be significantly weaker than
[O iii] and [C ii] , stacking may work to assess the average ISM properties,
such as metallicity, radiation field, and gas density[44]. The expected number
of sources for the redshift bins of $\sim 12$ and 16 is highly uncertain at
this stage, but we will be able to put a meaningful observational constraint
on the [O iii] 88 $\mu$m luminosity functions at $z\sim 12$ and 16. A wider
survey with this depth will be necessary for better statistics in any case,
implying for the necessity for more detectors to provide a higher line mapping
speed.

---
Figure 4:  The proposed 1 deg2 drilling survey using the KATANA 3-band imaging
spectrograph, which is designed to blindly uncover [O iii] 88 $\mu$m line
emitters at 3 specific redshift ranges. The lowest frequency band also
corresponds to a [C ii] 158 $\mu$m redshift range of $z$ = 8.3–9.0, allowing
to detect [O iii] - [C ii] dual line emitters at this redshift range. With the
observing time of 3,500 hrs, we will reach a 5$\sigma$ line sensitivity of
0.5–1.0 mJy (peak flux) or 0.074–0.15 Jy km s-1, which are translated into a
[O iii] 88 $\mu$m line luminosity of $\sim(4-5)\times 10^{8}$ $L_{\odot}$.
Note that with this line sensitivity we can detect both [O iii] and [C ii]
lines of MACS0416_Y1 at $z=8.31$ [24, 43].

##  4 Technical considerations

###  4.1 ISS option

One of the technical issues is how to realize $R$ = 2,000 with the ISS
technology. DESHIMA1.0 has a coplanar waveguide (CPW) signal line coupled to
planar filters, but suffers from large radiation losses. Microstrip filters
will eliminate such losses, but then we need transparent deposited
dielectrics. The development will require a better understanding of the noise
behavior of a deposited dielectric, which has two level systems (TLS)
noise[45].

Another challenge is how to implement $>$1,000 spatial pixels which are
coupled with a broad-band antenna. Further investigations on the quasi-optics
as well as the chip structure will be necessary to have a better design for
very large arrays.

The line mapping speed with the proposed 3-band configuration, which requires
$\sim$1.5 M detectors in total, seems to be minimal to achieve the science
goal with a reasonable observing time (for $z=12$ and 16, we need an even
wider surveys with this depth). It implies that the cost of the proposed
imaging spectrograph can be comparable to the telescope itself. What is the
anticipated cost per channel in 2030s?

###  4.2 Heterodyne option

High spectral resolution ($R\sim 10^{7}$ or $dv$ $\sim$0.03 km s-1) is
available, but such high $R$ may not be mandatory for the proposed high-
redshift galaxy survey using emission lines; a moderate $R$ (a few 1,000) can
work. Nevertheless, the heterodyne receivers are indeed an attractive option
as demonstrated by a 275–500 GHz heterodyne receiver with an instantaneous IF
bandwidth of 4–21 GHz[46], which exploits the high-current-density
superconductor-insulator-superconductor (SIS) junctions and related
technologies. It is therefore conceivable to develop a dedicated moderate-
resolution digital spectrometer system to be connected to such a novel wide-
band heterodyne receiver.

Another technical challenge is implementation of $\sim 10^{3}$ pixel
heterodyne receiver array, where power consumption of a large number of
cryogenic low-noise amplifiers and complexity of the structure can be an
issue, although possible solutions have been proposed and investigated[47,
48]. A heterodyne receiver array with $\sim$1,000 pixels per band has been
considered for the AtLAST, and the primary limiting factor is likely to be
cost[49].

##  5 Summary and outlook

Successive detection of the [O iii] 88 $\mu$m line in $z$ = 7–9 galaxies along
with the mounting evidence for the earliest star formation at $z\sim$12–15
galaxies motivate us to establish a pathway to find suitable targets for ALMA
spectroscopy at these redshift range. We propose and discuss a 3-band imaging
spectroscopy survey with the KATANA concept (Karatsu et al. 2019) for
LST/AtLAST by exploiting the technologies of the integrated superconducting
spectrometer (ISS) and a large-format imaging array. A 1-deg2 drilling survey
(3,500 hrs) with the proposed 3-band configuration (covering 200, 255, and 350
GHz band, 1,000–3,000 spatial pixels per band, $R$ = 2,000 spectroscopy,
yielding $\sim$1.5 M detectors in total) will capture a statistically large
number of [O iii] line emitters at $z\sim$ 8–9 (with some [O iii] - [C ii]
dual line detection along with [N ii] 122 $\mu$m constraint by stacking), and
a significant chance of uncovering [O iii] emitter candidates up to at
$z\sim$12 and $\sim$16\. Development of low-loss films, which requires better
understanding of noise behavior of dielectrics, is necessary for realizing $R$
= 2,000 with ISS. How to implement $>$1,000 spatial pixels with a broad-band
antenna is another challenge. A large format heterodyne array may also become
an option given the rapid progress of SIS receiver technologies, but the cost
remains a big issue.

###### Acknowledgements.

This research was supported by the Netherlands Organization for Scientific
Research NWO (Vidi grant no. 639.042.423, NWO Medium Investment grant no.
614.061.611 DESHIMA), the European Research Council ERC (ERC-CoG-2014 \-
Proposal no. 648135 MOSAIC), the Japan Society for the Promotion of Science
JSPS (KAKENHI grant nos. JP25247019 and JP17H06130), National Astronomical
Observatory of Japan NAOJ ALMA Scientific Research (grant no. 2018-09B), NAOJ
Research Coordination Committee, National Institutes of Natural Sciences
(grant no. 1901-0102), and the Grant for Joint Research Program of the
Institute of Low Temperature Science, Hokkaido University.

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