Form factors in asymptotic safety: conceptual ideas and computational toolbox

For scalar fields $\newcommand{\p}{{\partial}} \phi$, there is one form factor associated with the kinetic term

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{Sscalar} \Gamma_k^{\rm s,kin}[\phi,g] = \frac{1}{2}\int{\rm d}^d x\sqrt{g} \, \phi\,f_k^{(\phi\phi)}(\Delta)\, \phi \, . \nonumber \end{align} \tag{ 21 }$$

In analogy to computations in flat Minkowski space, we define the wave-function renormalisation of the scalar field according to [12]

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{wavefctren} Z_k^{\rm s} \equiv \left. \frac{\p}{\p p^2} \, \, f_k^{(\phi\phi)}(\,p^2) \, \right|_{p^2 = 0} \, . \nonumber \end{align} \tag{ 22 }$$

The corresponding anomalous dimension is then given by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{anomalousdim} \eta_s(k) \equiv - \p_t \ln Z_k^{\rm s} \, . \nonumber \end{align} \tag{ 23 }$$

One may also extract the zero-momentum behaviour of the form factor. In this way, we define the gap parameter

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{gapparameter} \mu_k^2 \equiv (Z_k^{\rm s})^{-1} \left. \, f_k^{(\phi\phi)}(\,p^2) \right|_{p^2 = 0} \, . \nonumber \end{align} \tag{ 24 }$$

If $\newcommand{\p}{{\partial}} f_k^{(\phi\phi)}(\,p^2)$ is a linear function of the squared momentum and $\newcommand{\p}{{\partial}} \phi$ is in the symmetric phase, $\mu_k$ has the interpretation of the mass of the scalar field, since it corresponds to the pole of the (Wick-rotated) Lorentzian propagator. For a general form factor, this interpretation does not hold however and one has to analyse the pole structure of the (Wick-rotated) form factor in order to gain any insight on the masses of the propagating fields.

To linear order in the spacetime curvature, the scalar sector gives rise to two additional form factors,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{Rphiphi} \Gamma_k^{({\rm R}\phi\phi)}[\phi,g] = \int {\rm d}^dx \sqrt{g} f_k^{({\rm R}\phi\phi)}(\Delta_1,\Delta_2,\Delta_3) \, R \phi \phi \, , \nonumber \end{align} \tag{ 25 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{Ricphiphi} \Gamma_k^{({\rm Ric}\phi\phi)}[\phi,g] = \int {\rm d}^dx \sqrt{g} f_k^{({\rm Ric}\phi\phi)}(\Delta_1,\Delta_2,\Delta_3) \, R^{\mu\nu} (D_\mu D_\nu \phi) \phi \, . \nonumber \end{align} \tag{ 26 }$$

Here $\Delta_i$ is the Laplacian acting on ith field, $\newcommand{\p}{{\partial}} \Delta_1 (R \phi \phi) = (\Delta_1 R) \phi \phi$, see appendix A. An investigation of correlations of this type without form factors in a bi-metric setup has been carried out in [113], and in a Brans–Dicke motivated context in [57, 96, 105, 126]. The monomials (25) and (26) constitute a complete set of form factors at first order in the spacetime curvature. The invariant $\newcommand{\p}{{\partial}} \int {\rm d}^dx \sqrt{g} f_k(\Delta_1,\Delta_2,\Delta_3) \, R^{\mu\nu} (D_\mu \phi) (D_\nu \phi)$ can be mapped to this basis set through integration by parts and the use of the second Bianchi identity. Moreover, any pair of contracted covariant derivatives acting on different fields may be eliminated by means of the identity

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \int \! {\rm d}^dx \sqrt{g} R_1 R_2 (\Delta R_3) \! = \!\!\! \int \! {\rm d}^dx \sqrt{g} \left[ (\Delta R_1) R_2 + R_1 (\Delta R_2) + 2 (D_\mu R_1)(D^\mu R_2) \right] R_3 \, , \nonumber \end{align} \tag{ 27 }$$

where the R_i represent arbitrary tensor fields. These manipulations typically also produce additional curvature tensors by the commutation of covariant derivatives. Since these are of higher order in R, they will not be considered at this stage.

The kinetic term of an abelian gauge field $A_\mu$ with field strength $F_{\mu\nu} = D_\mu A_\nu - D_\nu A_\mu$ reads

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Vkin} \Gamma^{\rm v}_k[A_\mu,g] = \frac{1}{4} \int {\rm d}^dx \sqrt{g} F_{\mu\nu} \, F^{\mu\nu} \, , \nonumber \end{align} \tag{ 28 }$$

where all indices are raised and lowered with the spacetime metric $g_{\mu\nu}$. This term can be generalised to include the form factor $f_k^{\rm v}(\Delta)$,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{vector} \Gamma^{\rm v,kin}_k[A_\mu,g] = \frac{1}{4} \int {\rm d}^dx \sqrt{g} F_{\mu\nu} \, f_k^{\rm v}(\Delta) \, F^{\mu\nu} \, . \nonumber \end{align} \tag{ 29 }$$

Similarly to the scalar case, $f_k^{\rm v}(x)|_{x=0}$ encodes the wave-function renormalisation for the vector fields. In the case of non-abelian gauge fields the connection $D_\mu$ is supplemented by an additional connection piece built from $A_\mu$. A closely related study of a Yang–Mills-gravity system investigating the momentum dependence can be found in [114]; for pure Yang–Mills theory and QCD see e.g. [127, 128].

In d  =  4 spacetime dimensions there is a second interaction monomial constructed from two powers of the field strength tensor contracted with a totally antisymmetric $\newcommand{\e}{{\rm e}} \epsilon$-tensor. Like for the kinetic term (28), one could also generalise this term by including a form factor. In this case it turns out that the flat part is a total derivative, as is the term without form factor, whereas the commutator of the covariant derivatives will give rise to an additional field strength tensor. As a consequence the interaction monomial contains either three powers of the field strength or an additional spacetime curvature tensor and will thus not be considered here.

Our construction of the form factors for fermionic fields builds on the spin-base formalism developed in [129–131]. We start by introducing (spacetime-dependent) Dirac matrices $\gamma_\mu \in {\rm Mat}(d_\gamma \times d_\gamma, \mathbb {C})$, $d_\gamma = 2^{\lfloor d/2 \rfloor}$, satisfying the Clifford algebra

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{clifford} \{\gamma_\mu,\gamma_\nu\} = 2 g_{\mu\nu} \mathbb {1} \, . \nonumber \end{align} \tag{ 30 }$$

Here $\mathbb {1}$ denotes the unit matrix in Dirac space, the curly brackets stand for the anticommutator $\newcommand{\e}{{\rm e}} \{\gamma_\mu,\gamma_\nu\} \equiv \gamma_\mu\gamma_\nu+ \gamma_\nu\gamma_\mu$, and the square half-brackets indicate the floor function. Dirac fermions are then represented by a Grassmann-valued $d_\gamma$-component vector $\newcommand{\p}{{\partial}} \psi$. Fermion bi-linears are formed with the metric $\mathfrak{h}$ on Dirac space, where $\mathfrak{h} \in {\rm Mat}(d_\gamma \times d_\gamma, \mathbb {C})$ is anti-hermitian, $\newcommand{\re}{{\rm Re}} \renewcommand{\dag}{\dagger}\mathfrak{h}^\dagger = - \mathfrak{h}$, and has unit determinant. The conjugate of a Dirac spinor is then defined as $\newcommand{\p}{{\partial}} \newcommand{\re}{{\rm Re}} \newcommand{\e}{{\rm e}} \renewcommand{\dag}{\dagger}\bar{\psi} \equiv \psi^\dagger \mathfrak{h}$. This ensures that $\newcommand{\p}{{\partial}} \newcommand{\re}{{\rm Re}} \renewcommand{\dag}{\dagger} \left(\bar{\psi} \psi \right){}^\dagger = \bar{\psi} \psi$ is real.

Since the properties of the Clifford algebra depend on the dimension of spacetime, the discussion will focus on four-dimensional (Euclidean signature) spacetimes admitting a spin-structure. In this case the $\gamma_{\mu}$ can be chosen to satisfy the reality property $\newcommand{\re}{{\rm Re}} \renewcommand{\dag}{\dagger}(\gamma_{\mu}){}^\dagger = \mathfrak{h} \gamma_{\mu} \mathfrak{h}^{-1}$. Moreover, the Clifford algebra admits an additional operator $\newcommand{\e}{{\rm e}} \gamma_\ast = \frac{1}{24} \sqrt{g} \, \epsilon_{\mu\nu\rho\sigma} \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma$, where $\newcommand{\e}{{\rm e}} \epsilon$ is the standard Levi-Civita symbol. It satisfies ${\rm tr} \gamma_\ast = 0$, $\newcommand{\re}{{\rm Re}} \renewcommand{\dag}{\dagger}(\gamma_\ast){}^\dagger = \mathfrak{h} \gamma_\ast \mathfrak{h}^{-1}$, and $(\gamma_\ast){}^2 = \mathbb {1}$, which allows to distinguish the left- and right-handed components of the Dirac spinor. Given a set of Dirac matrices satisfying (30) together with the reality properties for the $\gamma_{\mu}$ uniquely determines $\mathfrak{h}$. In order to construct a kinetic term for the fermions, we introduce a covariant derivative $\nabla_\mu$ containing the spin-base connection $\Gamma_{\mu}$²,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \nabla_\mu \psi = \p_\mu \psi + \Gamma_{\mu} \psi \, , \qquad \nabla_\mu \bar{\psi} = \p_\mu \bar{\psi} - \bar{\psi} \Gamma_{\mu} \, . \nonumber \end{align} \tag{ 31 }$$

The spin-base connection $\Gamma_\mu$ is completely determined in terms of the Dirac matrices and the Levi-Civita connection,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Gamma_{\mu} = \sum_{n=1}^4 m_{\mu\rho_1 \cdots \rho_n} \gamma^{\rho_1 \cdots \rho_n} \, , m_{\mu\rho_1 \cdots \rho_n} \equiv \frac{(-1)^{\frac{n(n+1)}{2}} {\rm tr} \left(\gamma_{\rho_1 \cdots \rho_n} \left[ (D_\mu \gamma^\nu) , \gamma_{\nu} \right] \right)}{8 \, n! \, (4 \, (1-(-1)^n) - 2 n)} \, , \nonumber \end{align} \tag{ 32 }$$

where $\newcommand{\p}{{\partial}} D_\mu \gamma^\nu = \partial_\mu \gamma^\nu + \Gamma^\nu{}_{\mu\rho} \gamma^\rho$ and $\gamma^{\rho_1 \cdots \rho_n} = 1/n!(\gamma^{\rho_1} \cdots \gamma^{\rho_n} + \ldots)$ is the completely anti-symmetrised product of n Dirac matrices. The connection ensures that $\newcommand{\p}{{\partial}} \nabla_\mu \psi$ transforms as a covector under general coordinate transformations and as a vector under ${\rm SL}(d_\gamma, \mathbb {C})$ spin-base transformations. As an important property, $\newcommand{\e}{{\rm e}} \not{\nabla} \equiv \gamma^\mu \nabla_\mu$ satisfies the Lichnerowicz relation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{lichnerowicz} \Delta_{\rm D} \equiv (\mathbf i \not{\nabla})^2 = \left(- g^{\mu\nu} D_\mu D_\nu + \frac{1}{4} R \right) \mathbb {1} \, . \nonumber \end{align} \tag{ 33 }$$

Based on these prerequisites, it is now straightforward to introduce the three independent form factors appearing at the level of fermion bi-linears,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \Gamma_k^{\rm D,1}[\bar{\psi}, \psi, g] = \int {\rm d}^4x \sqrt{g} \, \bar{\psi} \, f^{{\rm D},1}_k(\Delta_{\rm D}) \, (\mathbf i \not{\nabla}) \, \psi \, , \nonumber \end{align} \tag{ 34 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \Gamma_k^{\rm D,2}[\bar{\psi}, \psi, g] = \int {\rm d}^4x \sqrt{g} \, \bar{\psi} \, f^{{\rm D},2}_k(\Delta_{\rm D}) \, \gamma_\ast \, \not{\nabla} \, \psi \, , \nonumber \end{align} \tag{ 35 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \Gamma_k^{\rm D,3}[\bar{\psi}, \psi, g] = \int {\rm d}^4x \sqrt{g} \, \bar{\psi} \, f^{{\rm D},3}_k(\Delta_{\rm D}) \, \psi \, , \nonumber \end{align} \tag{ 36 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \Gamma_k^{\rm D,4}[\bar{\psi}, \psi, g] = \int {\rm d}^4x \sqrt{g} \, \bar{\psi} \, f^{{\rm D},4}_k(\Delta_{\rm D}) \, \gamma_\ast \, \psi \, . \nonumber \end{align} \tag{ 37 }$$

The form factors $f^{{\rm D},1}_k(\Delta_{\rm D})$ and $f^{{\rm D},2}_k(\Delta_{\rm D})$ generalises the kinetic term. In particular, linear combinations of $f^{{\rm D},1}_k(0)$ and $f^{{\rm D},2}_k(0)$ define the wave-function renormalisations for the two chiral components of the Dirac field. The form factors $f^{{\rm D},3}_k(\Delta_{\rm D})$ and $f^{{\rm D},4}_k(\Delta_{\rm D})$ generalise the mass terms to momentum-dependent functions. Again the k-dependent mass of the fermion is associated with the roots of the (Lorentzian signature) Dirac equation. In a flat background the relevant equation is

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \bigg[\,f^{{\rm D},1}_k(\Box) (\mathbf i \not{\p})+ f^{{\rm D},2}_k(\Box) \, \gamma_\ast \, \not{\p} - \left(\,f^{{\rm D},3}_k(\Box) + f^{{\rm D},4}_k(\Box) \, \gamma_\ast \right) \bigg] \psi = 0 \, , \nonumber \end{align} \tag{ 38 }$$

so that the construction accommodates scale-dependent mass terms for the right- and left-handed components of the Dirac fermion. Owed to the presence of the scalar curvature term in (33) the form factors $f^{{\rm D},i}_k(\Delta_{\rm D})$ have a non-trivial overlap with scale-dependent functions f _k(R) built from the Ricci scalar. A natural way to disentangle these two sets of functions is to take the flat space limit where the latter are trivial.

The first set of non-trivial form factors in the gravitational sector appears at second order in the spacetime curvature. In this case the basis for the split symmetry invariant form factors can be chosen as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{structure2a} \Gamma_k^{\rm C,}[g] = \frac{1}{16 \pi G_k} \int {\rm d}^dx \sqrt{g} \, C_{\mu\nu\rho\sigma} \, W_k^{\rm C}(\Delta) \, C ^{\mu\nu\rho\sigma} \, , \nonumber \end{align} \tag{ 39 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{structure2b} \Gamma_k^{\rm R}[g] = \frac{1}{16 \pi G_k} \int {\rm d}^dx \sqrt{g} \, R \, W_k^{\rm R}(\Delta) \, R \, . \nonumber \end{align} \tag{ 40 }$$

The choice of these basis terms is distinguished in the sense that the form factors $W_k^{\rm C}(\Delta)$ and $W_k^{\rm R}(\Delta)$ lead to a non-trivial momentum dependence of the transverse-traceless and scalar propagator, respectively, when the background in (14) is chosen as flat Euclidean space. The third potential invariant $\int {\rm d}^dx \sqrt{g} \, R_{\mu\nu} \, R^{\mu\nu}$ does not give rise to an additional form factor. Any term of the structure $\int {\rm d}^dx \sqrt{g} \, R_{\mu\nu} \, \Delta^n \, R^{\mu\nu}$, $n \geqslant 1$ can be mapped to the basis elements and higher order curvature terms by means of the second Bianchi identity (A.6). To demonstrate this, we first note that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\cO}{{{\mathcal O}}} \displaystyle \nonumber D^\alpha D_\alpha R_{\rho\sigma\mu\nu} &= - D^\alpha \left[D_\rho R_{\sigma\alpha\mu\nu} + D_\sigma R_{\alpha\rho\mu\nu}\right] \nonumber \\ & = 2 D_\rho D_{[\mu} R_{\nu]\sigma} - 2 D_\sigma D_{[\mu} R_{\nu]\rho} + \cO(R^2) \, , \nonumber \end{align} \tag{ 41 }$$

where we commuted two covariant derivatives and made use of the contracted Bianchi identity (A.7) in the second step. Contracting with a Riemann tensor, this implies

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\cO}{{{\mathcal O}}} \displaystyle R^{\rho\sigma\mu\nu} D^2 R_{\rho\sigma\mu\nu} = 4 R^{\rho\sigma\mu\nu} D_\rho D_\mu R_{\nu\sigma} + \cO(R^2)\, . \nonumber \end{align} \tag{ 42 }$$

Integrating this equation over spacetime then allows to integrate by parts, so that the covariant derivatives appearing on the right-hand side can again be arranged to act on the Riemann tensor. Again making use of the contracted Bianchi identity establishes that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\cO}{{{\mathcal O}}} \displaystyle \label{curvid} \int {\rm d}^dx \sqrt{g} \left[R^{\rho\sigma\mu\nu} \Delta R_{\rho\sigma\mu\nu} - 4 R^{\mu\nu} \Delta R_{\mu\nu} + R \Delta R\right] = \cO(R^3) \, . \nonumber \end{align} \tag{ 43 }$$

This relation readily extends to higher powers of the Laplacian. This case involves additional commutators when reordering the covariant derivatives before performing the integration by parts. These additional commutators only provide further terms of order $\newcommand{\cO}{{{\mathcal O}}} \cO(R^3)$, so that (43) is correct for all positive powers $\Delta^n$. Using (A.5) in order to eliminate the Riemann tensor in favour of the Weyl tensor leads to a similar relation, albeit with different numerical coefficients,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\cO}{{{\mathcal O}}} \displaystyle \int {\rm d}^dx \sqrt{g} \left[C^{\rho\sigma\mu\nu} \Delta C_{\rho\sigma\mu\nu} - 4 \frac{d-3}{d-2} R^{\mu\nu} \Delta R_{\mu\nu} + \frac{d(d-3)}{(d-1)(d-2)} R \Delta R\right] = \cO(R^3) \, . \nonumber \end{align} \tag{ 44 }$$

This establishes that the monomials (39) and (40) are indeed the only form factors that appear at second order in the curvature, see also [35] for related discussions³. However, in dimensions higher than four, the Ricci-squared term without form factor has to be included, since it cannot be eliminated by the Euler characteristic.

At this stage the following remark is in order. The two monomials (4) spanning the Einstein–Hilbert action do not lend themselves to a generalisation by introducing a non-trivial form factor. Adding a function $f_k(\Delta)$ acting on the Ricci scalar in $\newcommand{\cO}{{{\mathcal O}}} \cO_{\rm R}$ leads to integrands which are total derivatives and thus merely contribute surface terms to the action. This case will not be considered any further at this stage.

We start by generalising the nomenclature introduced in (20) to the scalar-tensor case. In the presence of two fields,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \newcommand{\gb}{{\bar{g}}} \displaystyle \label{vertexex1} \Gamma_k[h,\phi;\gb] = \sum_{m,n} \frac{1}{n! \, m!} \int {\rm d}^dx \sqrt{\gb} \left[\Gamma_k^{(m,n)}[\gb]\right]^{\mu_1\nu_1 \cdots \mu_m \nu_m} h_{\mu_1 \nu_1} \cdots h_{\mu_m \nu_m} \, \phi^n \, . \nonumber \end{align} \tag{ 45 }$$

The vertex entering the monomial with m h-fields and n $\newcommand{\p}{{\partial}} \phi$-fields is denoted by $\newcommand{\gb}{{\bar{g}}} \Gamma_k^{(m,n)}[\gb]$. Moreover, we use a vertical line followed by a string of fields to denote the projection of (45) onto the corresponding string of fields. Notably, the symmetry properties in the vertices $\newcommand{\gb}{{\bar{g}}} \left[\Gamma_k^{(m,n)}[\gb]\right]^{\mu_1\nu_1 \cdots \mu_m \nu_m}$ in terms of their momentum-dependence and tensor structure appear automatically, once they are extracted from (45) using the variational principle.

Order $\newcommand{\cO}{{{\mathcal O}}} \cO(h^0)$. The lowest order vertex describes the propagation of the scalar field in flat space and does not include the graviton fluctuation field,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{Gphiphi} \Gamma_k|_{\phi\phi} = \frac{1}{2} \int {\rm d}^dx \, \phi f^{(\phi\phi)}(\Box) \phi \, . \, \nonumber \end{align} \tag{ 46 }$$

Going to momentum space and taking the functional derivatives with respect to $\newcommand{\p}{{\partial}} \phi$ according to the prescription (A.14) then yields the (symmetrised) two-point vertex

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \Gamma_k^{(0,2)}(\,p^2) = f^{(\phi\phi)}(\,p^2) \, , \nonumber \end{align} \tag{ 47 }$$

where we employed momentum conservation.

Order $\newcommand{\cO}{{{\mathcal O}}} \cO(h^1)$. Once terms containing powers of $h_{\mu\nu}$ are considered the vertices $\newcommand{\gb}{{\bar{g}}} \left[\Gamma_k^{(m,n)}[\gb]\right]^{\mu_1\nu_1 \cdots \mu_m \nu_m}$, $m \geqslant 1$, possess a non-trivial tensor structure. For m  =  1 and n  =  2 this results in four form factors $\newcommand{\p}{{\partial}} f^{(h\phi\phi)}_{{\mathcal T}}$ which are associated with the independent tensor structures

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{eq34} \Gamma_k|_{h\phi\phi} = \int {\rm d}^dx \, \bigg[\,f^{(h\phi\phi)}_{(\bar g)} \, \delta^{\mu\nu} + f^{(h\phi\phi)}_{(11)} \, \p_1^\mu \p^\nu_1 + f^{(h\phi\phi)}_{(22)} \, \p_2^\mu \p_2^\nu + f^{(h\phi\phi)}_{(12)} \, \p_1^\mu \p_2^\nu \bigg] h_{\mu\nu} \phi \phi \, . \nonumber \end{align} \tag{ 48 }$$

The fourth structure deserves a comment. While partial integration allows to write $\newcommand{\p}{{\partial}} \int {\rm d}^dx \, \p_1^\mu \p_2^\nu \, h_{\mu\nu} \phi \phi = - \frac{1}{2} \int {\rm d}^dx \, (\p_1^\mu \p_1^\nu \, h_{\mu\nu}) \phi^2$, identities of this form no longer hold if there is a form factor which contains Laplacians acting on the two $\newcommand{\p}{{\partial}} \phi$-fields with different powers. Thus, in general, the fourth tensor structure must be included.

The momentum dependence of the form factors is conveniently captured by the squares of the momenta associated with the three fields,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{momdep3pt} f^{(h\phi\phi)}_{{\mathcal T}} = f^{(h\phi\phi)}_{{\mathcal T}}(\,p_1^2,p_2^2,p_3^2) \, . \nonumber \end{align} \tag{ 49 }$$

The fact that the combinations $\newcommand{\e}{{\rm e}} y_{ij} \equiv p_{i\mu}\, p_j^\mu$, $i,j = 1, \cdots, m+n$, $i \not = j$ do not appear as independent arguments follows from momentum conservation at the vertex, equation (A.13), which for any three-point vertex gives

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{mom3pt} p_1^\mu + p_2^\mu + p_3^\mu = 0 \, . \nonumber \end{align} \tag{ 50 }$$

This entails

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle p_3^2 = (\,p_{1\mu} + p_{2\mu})(\,p_{1}^{\mu} + p_{2}^{\mu}) = p_1^2 + p_2^2 + 2 y_{12} \, , \nonumber \end{align} \tag{ 51 }$$

which allows to express y ₁₂ in terms of the $p_i^2$. Identities for remaining y _ij can be obtained along the same lines, yielding

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle y_{12} = \frac{p_3^2 - p_1^2 - p_2^2}{2} \, , \quad y_{13} = \frac{p_2^2 - p_1^2 - p_3^2}{2} \, , \quad y_{23} = \frac{p_1^2 - p_2^2 - p_3^2}{2} \, . \nonumber \end{align} \tag{ 52 }$$

These identities allow to eliminate the dependence of the $\newcommand{\p}{{\partial}} f_{{\mathcal T}}^{(h\phi\phi)}$ on the y _ij in favour of a dependence on the squared momenta, justifying (49).

We close the discussion with the following remarks. The classification of the form factors related to higher-order vertices follows the same pattern as the one for the three-point vertices. Firstly, one determines the independent arguments of $\newcommand{\p}{{\partial}} f_{{\mathcal T}}^{(h^m\phi^n)}$ using momentum conservation at the vertex. Secondly, one determines the independent tensor structures providing a suitable basis for the expansion (45). Notably, the number of tensor structures proliferates rather quickly. The vertex with m  =  n  =  2 is discussed in detail in appendix D. In this case there are 35 independent tensor structures which all come with their own form factor,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{4ptsf} f_{{\mathcal T}}^{(hh\phi\phi)} = f_{{\mathcal T}}^{(hh\phi\phi)}(\,p_1^2,p_2^2,p_3^2,y_{12},y_{13},y_{23}) \, . \nonumber \end{align} \tag{ 53 }$$

Momentum conservation at the vertex implies that any form factor appearing in a four-point vertex can depend on six different combinations of the incoming momenta. Systematically eliminating $p_4^\mu$ by imposing momentum conservation at the vertex, $p_4^\mu = - (\,p_1^\mu + p_2^\mu + p_3^\mu)$, leads to the arguments appearing in (53).

The vertex expansion can readily be generalised to an arbitrary background $\newcommand{\gb}{{\bar{g}}} \gb_{\mu\nu}$. In this case, there is the additional complication that operator structures acting on the same field no longer commute. For example, the Laplacian $\newcommand{\Delb}{{\bar{\Delta}}} \Delb_1$ no longer commutes with $\newcommand{\Db}{{\bar{D}}} \Db_{1\mu} \Db_i^\mu$, $i \not = 1$. This raises the need to impose some convention on how the operators are ordered. This is particularly relevant for vertex functions of higher order, as, e.g. the four-point vertex discussed in appendix D where the form factors depend on a subset of both $\newcommand{\Delb}{{\bar{\Delta}}} \Delb_i$ and $\newcommand{\Db}{{\bar{D}}} \Db_{i\mu} \Db_j^\mu$. In this case one may impose that all $\newcommand{\Db}{{\bar{D}}} \Db_{i\mu} \Db_j^\mu$ are to the left of all $\newcommand{\Delb}{{\bar{\Delta}}} \Delb_i$ and $\newcommand{\Delb}{{\bar{\Delta}}} \Delb_j$. Different choices for the operator ordering are equivalent up to terms of order $\newcommand{\Rb}{{\bar{R}}} {\mathcal O}(\Rb)$.

At this stage it is natural to ask about the relation between the split symmetry invariant form factors introduced in section 3.1.1 and the vertex expansion of the previous section. In order to clarify this question we expand the split symmetry invariant action (21) in terms of graviton fluctuations in a flat background $\newcommand{\gb}{{\bar{g}}} \gb_{\mu\nu} = \delta_{\mu\nu}$. Following the notation introduced in section 3.2.1, it is convenient to give the results in terms of projected components appearing in the expansion (45). At zeroth order in the h-field, the projection yields

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{zerovertex} \Gamma_k^{\rm s,kin}|_{\phi\phi} = \frac{1}{2}\int{\rm d}^d x \, \, \phi f^{(\phi\phi)}(\Box)\, \phi \, . \nonumber \end{align} \tag{ 54 }$$

The terms linear in h originate from expanding $\sqrt{g}$ and the form factor respectively. The former follow from

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\cO}{{{\mathcal O}}} \displaystyle \sqrt{g} \simeq 1 + \frac{1}{2}h + \frac{1}{8} h^2 - \frac{1}{4} h^{\mu\nu} h_{\mu\nu} + \cO(h^3) \, , \nonumber \end{align} \tag{ 55 }$$

where $\newcommand{\gb}{{\bar{g}}} \newcommand{\e}{{\rm e}} h \equiv \gb^{\mu\nu} h_{\mu\nu}$. The latter arise from expanding the Laplacian acting on scalar fields in powers of h,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \Delta \, \phi \simeq \left[\Box + \mathbb d_1 + \mathbb d_2 + \cdots \right] \phi \, . \nonumber \end{align} \tag{ 56 }$$

The explicit computation gives

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \mathbb d_1 = h_{\mu\nu} \p^\mu \p^\nu + (\p^\mu h_{\mu\nu}) \p^\nu - \frac{1}{2} (\p_\alpha h) \p^\alpha \, , \label{eq:d1scalarm} \nonumber \end{align} \tag{ 57 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \mathbb d_2 &= -{h_\mu}^\alpha h_{\alpha\nu} \p^\mu \p^\nu - h^{\alpha\beta}(\p_\beta h_{\alpha\mu}) \p^\mu - {h_\mu}^\beta (\p^\gamma h_{\beta\gamma}) \p^\mu \nonumber \\ &\qquad + \frac{1}{2} h^{\alpha\beta} (\p_\mu h_{\alpha\beta}) \p^\mu + \frac{1}{2} h^{\mu\nu} (\p_\mu h)\p_\nu \, . \label{eq:d2scalarm} \nonumber \end{align} \tag{ 58 }$$

Combining these basic expansions with the computational techniques introduced in appendix C allows to generalise these results to functions of the Laplacian. This results in the following expansion coefficient:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \nonumber & \Gamma_k^{\rm s, kin}|_{h\phi\phi} = \frac{1}{2}\int{\rm d}^d x \, \, \Big[ \frac{1}{2} \, f^{(\phi\phi)}(\Box_2) h \phi \phi \nonumber \\ \label{skinhpp} & \qquad + \int_0^\infty \!\!\!\! {\rm d} s \, \tilde f^{(\phi\phi)}(s) \sum_{j=0}^\infty \frac{(-s)^{\,j+1}}{(\,j+1)!} \sum_{l=0}^{\,j} {j \choose l} (-1)^l (\Box^{\,j-l} \phi) \, \mathbb d_1 \, \Box^l \, {\rm e}^{-s\Box}\phi \Big] \, . \nonumber \end{align} \tag{ 59 }$$

Here $\newcommand{\p}{{\partial}} \tilde{f}^{(\phi\phi)}(s)$ is the inverse Laplace transform of the form factor $\newcommand{\p}{{\partial}} f^{(\phi\phi)}(\Box)$. Remarkably, all sums and the Laplace transform can be performed explicitly. Labelling the Laplacians acting on h, the first and second scalar field by $\Box_1$, $\Box_2$ and $\Box_3$, respectively, one has

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \nonumber & \int_0^\infty {\rm d} s \, \tilde f^{(\phi\phi)}(s) \sum_{j=0}^\infty \frac{(-s)^{\,j+1}}{(\,j+1)!} \sum_{l=0}^{\,j} {j \choose l} (-1)^l \Box^{\,j-l}_2 \, \Box^l_3 \, {\rm e}^{-s\Box_3} h \phi \phi \nonumber \\ \nonumber & \quad = \int_0^\infty {\rm d} s \, \tilde f^{(\phi\phi)}(s) \sum_{j=0}^\infty \frac{(-s)^{\,j+1}}{(\,j+1)!} \left(\Box_2 - \Box_3 \right)^{\,j} \, {\rm e}^{-s\Box_3} h \phi \phi \nonumber \\ \nonumber & \quad = \left(\Box_2 \, - \Box_3 \right)^{-1} \int_0^\infty {\rm d} s \, \tilde f^{(\phi\phi)}(s) \left({\rm e}^{-s(\Box_2 - \Box_3)} - 1 \right) \, {\rm e}^{-s\Box_3} h \phi \phi \nonumber \\ & \quad = \left(\Box_2 \, - \Box_3 \right)^{-1} \, \left(\,f^{(\phi\phi)}(\Box_2) - f^{(\phi\phi)}(\Box_3) \right) h \phi \phi \, . \nonumber \end{align} \tag{ 60 }$$

Substituting this result into (59) then yields the final form of the coefficient $\newcommand{\p}{{\partial}} \Gamma_k^{\rm s,kin}|_{h\phi\phi}$:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \nonumber & \Gamma_k^{\rm s,kin}|_{h\phi\phi} = \frac{1}{2}\int{\rm d}^d x \, \, \Big[ \frac{1}{2} \, \delta^{\mu\nu} f^{(\phi\phi)}(\Box_2) \nonumber \\ \label{skinhpp2} & + \frac{f^{(\phi\phi)}(\Box_2) - f^{(\phi\phi)}(\Box_3)}{\Box_2 \, - \Box_3} \, \left(\p^\mu_3 \p^\nu_3 + \p^\mu_1 \p^\nu_3 - \frac{1}{2} \delta^{\mu\nu} \p_{1\alpha} \p^\alpha_3 \right) \Big] h_{\mu\nu} \,\phi \phi \, . \nonumber \end{align} \tag{ 61 }$$

A series expansion of the second line shows that the coefficient is finite also on the locus $\Box_2 \, - \Box_3 = 0$.

The expansion of the monomials (25) and (26) in metric fluctuations in a flat background starts at order h. Thus they do not contribute to $\newcommand{\p}{{\partial}} \Gamma_k|_{\phi\phi}$. The terms $\newcommand{\p}{{\partial}} \Gamma^{({\rm R}\phi\phi)}_k|_{h\phi\phi}$ and $\newcommand{\p}{{\partial}} \Gamma_k^{({\rm Ric}\phi\phi)}|_{h\phi\phi}$ are found by replacing the curvature tensors by their leading coefficients in h. They read

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{expRphiphi} \Gamma^{{\rm R}\phi\phi}_k|_{h\phi\phi} = \int {\rm d}^dx f^{({\rm R}\phi\phi)}(\Box_1,\Box_2,\Box_3) \left[\Box_1 \delta^{\mu\nu} + \p_1^\mu \p_1^\nu \right] h_{\mu\nu}\phi\phi \, , \nonumber \end{align} \tag{ 62 }$$

and

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \nonumber \Gamma^{{\rm Ric}\phi\phi}_k|_{h\phi\phi} = & \frac{1}{2} \int {\rm d}^dx \, f^{({\rm Ric}\phi\phi)}(\Box_1,\Box_2,\Box_3) \, \Big[ (\Box_1 + \Box_2-\Box_3) \, \p^\mu_1 \, \p^\nu_2 \nonumber \\ \label{expRicphiphi} & \qquad - \frac{1}{4} (\Box_1 + \Box_2-\Box_3)^2 \, \delta^{\mu\nu} + \Box_1 \, \p^\mu_2 \, \p^\nu_2 \Big] h_{\mu\nu}\phi\phi \, . \nonumber \end{align} \tag{ 63 }$$

The structure of the four-point vertex can be analysed along the same lines. Since the intermediate steps and results are bulky, they have been relegated to appendix D. At this stage, it is sufficient to remark that split-symmetric actions containing more than one curvature tensor do not contribute to the $\newcommand{\p}{{\partial}} (h\phi\phi)$-vertex when expanded around a flat background. Thus (61)–(63) capture all contributions originating from a split-symmetric action.

We are now in the position to discuss the relation between the split symmetry invariant interaction monomials and the vertex expansion discussed in sections 3.1.1 and 3.2.1, respectively. In general, any split-symmetric action will give rise to an infinite tower of interaction vertices $\Gamma_k^{(m,n)}$ when one expands in graviton fluctuations h around a background $\newcommand{\gb}{{\bar{g}}} \gb$. This tower then entails that there must be a relation between the form factors appearing in the split-symmetric action and the vertex expansion⁴. We illustrate this general property for the form factor appearing in the scalar kinetic term (21). Combining (54) and (61) gives the expansion of the split-symmetric scalar kinetic term $\newcommand{\p}{{\partial}} \Gamma_k^{\rm s,kin}[\phi,g]$ in a flat background spacetime up to terms of second order in h:

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \newcommand{\cO}{{{\mathcal O}}} \displaystyle \label{skinexp} & \Gamma_k^{\rm s,kin} = \frac{1}{2}\int{\rm d}^d x \, \, \Big\{\phi f^{(\phi\phi)}(\Box)\, \phi + \Big[ \frac{1}{2} \, \delta^{\mu\nu} f^{(\phi\phi)}(\Box_2) \nonumber \\ \nonumber & + \frac{f^{(\phi\phi)}(\Box_2) - f^{(\phi\phi)}(\Box_3)}{\Box_2 \, - \Box_3} \, \big(\p^\mu_2 \p^\nu_2 + \p^\mu_1 \p_2^\nu - \frac{1}{2} \delta^{\mu\nu} \p_{1\alpha} \p^\alpha_2 \big) \Big] h_{\mu\nu} \,\phi \phi + \cO(h^2) \Big\} \, . \nonumber \end{align} \tag{ 64 }$$

Here we used the symmetry in the two $\newcommand{\p}{{\partial}} \phi$-fields to exchange the indices 2 and 3. By comparing (64) to (46) and (48), we see that split symmetry entails a specific relation between the form factors appearing in the vertex expansion. The $\newcommand{\p}{{\partial}} (\phi\phi)$-vertex receives contributions from the scalar kinetic term only, and anticipating this we deliberately chose the same name for the two functions.

At the level of the $\newcommand{\p}{{\partial}} (h\phi\phi)$-vertices, the split-symmetric expansions (61)–(63) induce the following form factors associated with the tensor structures (48):

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \nonumber f_{(\bar g)}^{(h\phi\phi)} &= \frac{1}{8} \left[\,f^{(\phi\phi)}(\,p_2^2) + f^{(\phi\phi)}(\,p_3^2) - p_1^2 \frac{f^{(\phi\phi)}(\,p_2^2)-f^{(\phi\phi)}(\,p_3^2)}{p_2^2-p_3^2} \right] + p_1^2 f^{({\rm R}\phi\phi)} \nonumber \\ & \label{split1} - \frac{1}{8} (\,p_1^2 + p_2^2- p_3^2)^2 f^{({\rm Ric}\phi\phi)} \, , \nonumber \end{align} \tag{ 65 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{split2} f_{(11)}^{(h\phi\phi)} = \, f^{({\rm R}\phi\phi)} \, , \nonumber \end{align} \tag{ 66 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{split3} f_{(22)}^{(h\phi\phi)} = \frac{1}{2} \frac{f^{(\phi\phi)}(\,p_2^2) - f^{(\phi\phi)}(\,p_3^2)}{p_2^2 - p_3^2} + \frac{1}{2} p_1^2 \, f^{({\rm Ric}\phi\phi)} \, , \nonumber \end{align} \tag{ 67 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{split4} f_{(12)}^{(h\phi\phi)} = \frac{1}{2} \frac{f^{(\phi\phi)}(\,p_2^2) - f^{(\phi\phi)}(\,p_3^2)}{p_2^2 - p_3^2} + \frac{1}{2} (\,p_1^2 + p_2^2- p_3^2) \, f^{({\rm Ric}\phi\phi)} \, . \nonumber \end{align} \tag{ 68 }$$

To ease the notation we have suppressed the arguments of all functions which depend on all three squared momenta. This establishes that split symmetry enforces relations between the four independent form factors appearing in the vertex expansion. In other words, extracting the parts of the $\newcommand{\p}{{\partial}} h\phi\phi$-vertices which can be completed into split-symmetric actions requires contributions from all four tensor structures with the corresponding momentum-dependent form factors being fixed in terms of the two free functions $\newcommand{\p}{{\partial}} f^{({\rm R}\phi\phi)}$ and $\newcommand{\p}{{\partial}} f^{({\rm Ric}\phi\phi)}$ by the relations (65)–(68). This also establishes that there are two combinations of tensor structures which cannot be completed into split-symmetric monomials. At the practical level this suggests that the amount of split symmetry breaking induced by the regulator and gauge fixing terms can be quantified by these equations. This provides a much more straightforward way to check how strongly the full diffeomorphism symmetry is broken than the evaluation of the non-trivial Nielsen identity.

Similar relations for the form factors associated with vertices of higher order $\newcommand{\p}{{\partial}} f_{{\mathcal T}}^{(h^m\phi^2)}$, $m \geqslant 2$, can be obtained along the same lines. Their explicit construction requires classifying all split-symmetric invariants containing up to m powers of the curvature tensor. Subsequently, this set of actions is expanded in the fluctuation field up to mth order. The result is then compared to the classification of tensor structures involving m h-fields and two scalars. Given that $\newcommand{\p}{{\partial}} f_{{\mathcal T}}^{(hh\phi\phi)}$ already gives rise to 35 independent tensor structures, it is clear that already the next order of relations will be very involved. We relegate partial results to appendix D.

We close this section with a conceptual remark. In general the effective average action $\Gamma_k$ does not reduce to a functional of only one metric in the limit $k \to 0$. The reason is that the Nielsen identity is still non-trivial even in this limit, in particular because the gauge-fixing term is still present. Our discussion in terms of form factors then might serve as an approximate solution to the Nielsen identity which does not include the breaking induced by gauge fixing, and fix an infinite number of couplings. In the specific example of the $\newcommand{\p}{{\partial}} h\phi\phi$-vertices, split symmetry restoration (up to the non-trivial part of the Nielsen identity) at k  =  0 corresponds to imposing the boundary conditions (65)–(68), thereby fixing two linear combinations of the form factors introduced in (48).

The beta functions in the gravitational sector are explicitly given by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{betag} \partial_tg = \left(d-2+\eta_N\right)g \, , \nonumber \end{align} \tag{ 80 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{betalambda} \partial_t\lambda = g\left(L_1(\lambda) +L_3[\,f] \right) +g\eta_NL_2(\lambda) -(2-\eta_N)\lambda . \nonumber \end{align} \tag{ 81 }$$

The anomalous dimension $\newcommand{\e}{{\rm e}} \eta_N$ can be cast into the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \eta_N = \frac{g\left(B_1(\lambda)+B_3[\,f]\right)}{1-gB_2(\lambda)} . \nonumber \end{align} \tag{ 82 }$$

The functions $B_1(\lambda)$, $B_2(\lambda)$, $L_1(\lambda)$ and $L_2(\lambda)$ only depend on the gravitational sector, and were first derived in [39]. Explicitly, they are given by

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle B_1(\lambda) =& \frac{1}{3}(4\pi)^{1-d/2}\bigg(d(d+1)\Phi^1_{d/2-1}[\ell_{-2\lambda}] -4d\Phi^1_{d/2-1}[\ell_0] \nonumber \\& -6d(d-1)\Phi^2_{d/2}[\ell_{-2\lambda}] -24\Phi^2_{d/2}[\ell_0] \bigg) \, ,\nonumber \end{align} \tag{ 83 }$$

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle B_2(\lambda) = -\frac{1}{6}(4\pi)^{1-d/2}\bigg(d(d+1)\tilde{\Phi}^1_{d/2-1}[\ell_{-2\lambda}] -6 d\tilde{\Phi}^2_{d/2}[\ell_{-2\lambda}] \bigg) \, , \nonumber \end{align} \tag{ 84 }$$

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle L_1(\lambda) = (4\pi)^{1-d/2}\bigg(d(d+1)\Phi^1_{d/2}[\ell_{-2\lambda}] -4d\Phi^1_{d/2}[\ell_0] \bigg) \, , \nonumber \end{align} \tag{ 85 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle L_2(\lambda) = -\frac{1}{2}(4\pi)^{1-d/2}d(d+1)\tilde{\Phi}^1_{d/2}[\ell_{-2\lambda}] \, . \nonumber \end{align} \tag{ 86 }$$

The functionals B₃ and L₃ are novel, and read

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle B_3[\,f] = \frac{2}{3}(4\pi)^{1-d/2}\left(\Phi^{1}_{d/2-1}[\,f] -\frac{1}{2}\eta_s \,\tilde{\Phi}^{1}_{d/2-1}[\,f] \right) , \nonumber \end{align} \tag{ 87 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle L_3[\,f] = (4\pi)^{1-d/2}\left(\Phi^{1}_{d/2}[\,f] -\frac{1}{2}\eta_s \,\tilde{\Phi}^{1}_{d/2}[\,f] \right) . \nonumber \end{align} \tag{ 88 }$$

In these expressions, we have conveniently used the generalised threshold functionals

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Phi^{\,p}_n[\,f] = \frac{1}{\Gamma(n)}\int_0^\infty{\rm d} zz^{n-1}\frac{r(z)-z r'(z)}{(\,f(z)+r(z))^{\,p}} , \nonumber \end{align} \tag{ 89 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tilde{\Phi}^{\,p}_n[\,f] = \frac{1}{\Gamma(n)}\int_0^\infty{\rm d} zz^{n-1}\frac{r(z)}{(\,f(z)+r(z))^{\,p}} . \nonumber \end{align} \tag{ 90 }$$

These functionals reduce to the threshold functionals as defined in [39] for a function $\newcommand{\e}{{\rm e}} \ell_w$ of the form $\newcommand{\e}{{\rm e}} \ell_w(z) = z +w$:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Phi^{\,p}_n(w)=\Phi^{\,p}_n[\ell_w] ,\quad \tilde{\Phi}^{\,p}_n(w)=\tilde{\Phi}^{\,p}_n[\ell_w]. \nonumber \end{align} \tag{ 91 }$$

The flow equation for f  is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{structuremaster} \left(1-\frac{1}{2}\eta_s\right)\,f(q^2) +\frac{1}{2}\partial_tf(q^2) -q^2f'(q^2) = \mathcal{K}_1 +\mathcal{K}_2 +\mathcal{K}_3 , \nonumber \end{align} \tag{ 92 }$$

where the $\mathcal{K}_i$ correspond to the three Feynman diagrams in figure 2. The diagram consists structurally of a momentum and angular integral over a number of graviton and scalar propagators, and the derivative of the regulator, connected by their vertex functions. The structural part of the diagrams reads

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathcal{K}_1 = -32g\int {\rm d} \mu[\eta_N] \,\mathcal{V}_1(\,p,q,x)\left(G_0^{{\rm grav}}(\,p^2)\right)^2 \, , \nonumber \end{align} \tag{ 93 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathcal{K}_2 = 32g\int {\rm d} \mu[\eta_s] \,\mathcal{V}_2(\,p,q,x) G_0^{{\rm grav}}(\mathfrak{s})\left(G_0^{{\rm scalar}}(\,p^2)\right)^2 \, , \nonumber \end{align} \tag{ 94 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathcal{K}_3 = 32g\int {\rm d} \mu[\eta_N] \,\mathcal{V}_3(\,p,q,x)\left(G_0^{{\rm grav}}(\,p^2)\right)^2 G_0^{{\rm scalar}}(\mathfrak{s}) , \nonumber \end{align} \tag{ 95 }$$

where we have introduced the regularised integrals

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \int {\rm d} \mu[\eta] \equiv \frac{(4\pi)^{d/2}}{\Gamma(d/2)}\int_0^\infty{\rm d} p \int_{-1}^1{\rm d} x \, (1-x^2)^{\frac{d-3}{2}}\left[ r\left(\,p^2\right)-\frac{1}{2}\eta r\left(\,p^2\right) - p^2r'\left(\,p^2\right) \right] \, , \nonumber \end{align} \tag{ 96 }$$

for $\newcommand{\e}{{\rm e}} \eta \in \{\eta_s, \eta_N\}$. The regularized dimensionless graviton and matter propagators are

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle G_0^{{\rm grav}}(z) = \left(z+r(z)-2\lambda\right)^{-1} \, , \nonumber \end{align} \tag{ 97 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Gscalar} G_0^{{\rm scalar}}(z)=\left(\,f(z)+ r(z) \right)^{-1} \, . \nonumber \end{align} \tag{ 98 }$$

Zoom In Zoom Out Reset image size

We have also introduced the Mandelstam variable $\mathfrak{s}=p^2+q^2+2pqx$. The vertex functions $\mathcal{V}_i$ are given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathcal{V}_1 = -\frac{1}{8}d(d+1)\,f(q^2) +\frac{1}{2}f'(q^2)q^2 \left(d -\frac{\mathfrak{s}-p^2+\frac{d-4}{d-2}q^2}{\mathfrak{s}-q^2} \right) \nonumber \\ +\frac{1}{2} \frac{f\left(\mathfrak{s}\right)-f(q^2)}{\mathfrak{s}-q^2} q^2\left[ 1 +\frac{\mathfrak{s}-p^2+\frac{d-4}{d-2}q^2}{\mathfrak{s}-q^2} \right] \, ,\nonumber \end{align} \tag{ 99 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathcal{V}_2 = -\frac{d}{2}\frac{1}{d-2}\left(\,f(\,p^2)\right)^2 +\frac{1}{2}\left(\mathfrak{s}+\frac{d+2}{d-2}p^2-q^2\right)\,f(\,p^2)\frac{f(\,p^2)-f(q^2)}{p^2-q^2} \nonumber \\ -\left(\frac{\mathfrak{s}}{2} +\frac{1}{d-2}p^2-q^2\right) \left(\frac{f(\,p^2)-f(q^2)}{p^2-q^2} \right)^2p^2 \, ,\nonumber \end{align} \tag{ 100 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \nonumber \mathcal{V}_3 = -\frac{d}{2}\frac{1}{d-2}\left(\,f(\mathfrak{s})\right)^2 +\frac{1}{2}\left(\,p^2+\frac{d+2}{d-2}\mathfrak{s}-q^2\right)\,f(\mathfrak{s})\frac{f(\mathfrak{s}) - f(q^2)}{\mathfrak{s}-q^2} \nonumber \\ -\left(\frac{p^2}{2}+\frac{1}{d-2}\mathfrak{s}-q^2\right) \left(\frac{f(\mathfrak{s})-f(q^2)}{\mathfrak{s}-q^2}\right)^2\mathfrak{s} . \nonumber \end{align} \tag{ 101 }$$

Note that the vertices $\mathcal{V}_2$ and $\mathcal{V}_3$ are related by the exchange $\mathfrak{s} \leftrightarrow p^2$, consistent with the fact that the diagrams differ only in the insertion of the regulator $\newcommand{\p}{{\partial}} \partial_t \mathcal{R}_k$ and a relabeling of momenta.

We remark that the left-hand side of the beta function is a function of q², whereas on the right-hand side also odd powers of q occur. We can properly symmetrise the expression for the right-hand side by the observation that odd powers of x integrate to zero.

From this equation, we derive a separate equation for $\newcommand{\e}{{\rm e}} \eta_s$. This is done by taking two derivatives with respect to q and consecutively setting q  =  0. Then using the requirement that $f'(0)=1$ gives the following implicit equation for $\newcommand{\e}{{\rm e}} \eta_s$:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \eta_s = -\left.\frac{{\rm d}^2}{{\rm d} q^2}\left(\mathcal{K}_1+\mathcal{K}_2+\mathcal{K}_3 \right)\right|_{q=0, f'(0)=1} \, . \nonumber \end{align} \tag{ 102 }$$

The explicit expression for $\newcommand{\e}{{\rm e}} \eta_s$ is rather lengthy and can be found in the supplementary notebook fpsolver.nb.

We study the asymptotic properties of the fixed point solution by the assertion that for large momentum q², the function f  behaves like $f(q^2) \sim f_\infty q^{2\alpha}$, where $f_\infty, \alpha>0$. Inserting this into the fixed point equation gives a consistent equation for $\alpha<2$. In this range, the diagram $\mathcal{K}_2$ is always sub-leading.

Explicitly, the equation for $\alpha$ reads

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{eq:alpha} &-\frac{1}{2}\eta_s-(\alpha-1) \nonumber \\&\qquad\quad=\frac{4g (d-3)(d-2\alpha)(d-2(\alpha-1))}{(4\pi)^{d/2} \, (d-2)} \left[ \Phi^2_{d/2}[\ell_{-2\lambda}]-\frac{1}{2}\eta_N \tilde{\Phi}^2_{d/2}[\ell_{-2\lambda}]\right]. \nonumber \end{align} \tag{ 103 }$$

We can solve this quadratic equation for the asymptotic exponent $\alpha$ in terms of $\newcommand{\e}{{\rm e}} \eta_s$, g and $\lambda$. Expanding in g, we infer that $\alpha$ is of the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \alpha = 1-\frac{1}{2}\eta_s+\mathcal{O}(g) , \nonumber \end{align} \tag{ 104 }$$

which is in agreement with the classical expectation that the fall-off behaviour of the propagator is determined by the anomalous dimension and receives quantum corrections due to gravity. Interestingly, we note that the ${\mathcal O}(g)$ corrections in (103) in d  =  3 dimensions vanish exactly.

Another analytic property that we can derive is the small-momentum behaviour of f . Upon evaluation at q  =  0, the fixed point equation reduces to the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:zeromomentum} \left(1-\frac{1}{2}\eta_s\right)\,f(0) = \mathcal{L}_0(g,\lambda,f)\,f(0) \, . \nonumber \end{align} \tag{ 105 }$$

We conclude that this equation is satisfied if $f(0)=0$, or if $\newcommand{\e}{{\rm e}} \mathcal{L}_0 = 1-\frac{1}{2}\eta_s$. Given a numerical solution, we can check whether one or both of these conditions holds.

In this section, we present the techniques that we used in our numerical study of the fixed point equation. The fixed point equation is a non-linear integro-differential equation, and cannot be solved exactly. Therefore, we resort to pseudo-spectral numerical methods [38, 98, 132, 133] to obtain an approximate solution.

The solving algorithm is implemented as follows. First, in order to obtain a bounded function on a compact domain, we rescale the function f  as

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle f(z)=(1+z)^{\alpha_{\rm num}}\tilde{f}\left(\frac{z-L}{z+L} \right) , \nonumber \end{align*}$$

where we take the compactification scale L to be 1. From the previous paragraph, we see that setting $\alpha_{\rm num} = 2$ is sufficiently large that the function $\tilde{f}$ is bounded. The function $\tilde{f}$ is now expanded in Chebyshev polynomials,

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \tilde{f}(s) = \sum_{i=0}^{N-1}a_iT_i(s) , \nonumber \end{align*}$$

where T_i denotes the ith Chebyshev polynomial of the first kind. Evaluation of $\tilde{f}$ and its derivatives is efficiently achieved by a Clenshaw algorithm. By evaluating the fixed point equation on the Gauss–Chebyshev grid of degree N, we obtain N independent algebraic equations for the a_i.

The constraints for $f'(0)=1$ and $\newcommand{\e}{{\rm e}} \eta_s$ are implemented by replacing the equations of the first two collocation points. Together with the gravitational beta functions and the constraint $\newcommand{\e}{{\rm e}} \eta_N=2-d$, this yields a set of N  +  4 equations for the N  +  4 parameters $\newcommand{\e}{{\rm e}} \{g, \lambda, \eta_N, \eta_s\} \cup \{a_i\}_{i=0}^{N-1}$. Solving this set of equations is achieved by a Newton–Raphson algorithm.

Finally, in order to concretely evaluate the flow equation, we have to choose a regulator function. We choose a regulator of exponential type,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle r(z) = \exp(-c z) . \nonumber \end{align} \tag{ 106 }$$

As advocated in [34], this regulator allows for fast numerical evaluation. The parameter c allows to scan for a range of regulators. In the following, we fix c  =  1.

The described algorithm is executed in Mathematica. Integration over the momentum and angles is done using Mathematica's NIntegrate routine, with GaussKronrodRule as Method option. We have run the algorithm for $N=20,30,40,50$. Starting from $N \approx 30$ the resulting solutions are stable and robust with respect to increasing N. Figures 3 and 4 show the results obtained for N  =  50 coefficients and 32 digits numerical precision. In figure 3, we plot the absolute values of the coefficients a_i. We see that the coefficients converge algebraically to zero. Following [38, chapter 2.12], the error of the fixed point solution is estimated as the number of coefficients times the absolute value of the last retained coefficient. This gives an error estimate for f  of the order 10⁻⁶.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We will now discuss the properties of the fixed point solution. We find the fixed point values

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle g_\ast=0.349 ,\qquad \lambda_\ast=0.300 ,\qquad \eta_s^\ast=-0.176 ,\qquad \alpha=0.949 . \nonumber \end{align} \tag{ 107 }$$

We see that the asymptotic power $\alpha$ is very close to one. Indeed, as is shown in figure 4 the solution only deviates a tiny amount from the Gaussian function $f_{\mathrm{Gaussian}}(z)=z$. However, $f_{\mathrm{Gaussian}}$ is not an exact solution of the fixed point equation. Moreover, we can insert the ansatz $f_{\mathrm{Gaussian}}$ into the right-hand side of (92) to obtain a one-loop approximation for f . Solving the linear differential equation gives however a solution that either has a singularity at zero momentum q, or has a root at a positive value of q. Both features are undesirable.

Evaluating f  directly at zero momentum, we find that f  vanishes up to our numerical precision of $\sim10^{-6}$. Thus the fixed point value of the (dimensionless) gap parameter (24) vanishes,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mu_\ast = 0 \, . \nonumber \end{align} \tag{ 108 }$$

In combination with the positivity properties of f  shown in figure 4, this entails that the scalar propagator at the fixed point corresponds to a single massless scalar degree of freedom. This is also confirmed by checking (105); we find

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:L0} \mathcal{L}_0-\left(1-\frac{1}{2}\eta_s\right) = -0.582 . \nonumber \end{align} \tag{ 109 }$$

Since this is significantly different from zero, we conclude that the fixed point solution is gapless. This supports the conjecture [113] that the scalar sector of the gravity-matter fixed point possesses a shift symmetry, i.e. it is invariant under $\newcommand{\p}{{\partial}} \phi \mapsto \phi + c$ with c being constant.

In order to study the stability of the fixed point, we construct the stability matrix as discussed in section 2. At this stage, the following conceptual remark is in order. The relation (98) (with the regulator $r(z)$ set to zero) shows that the form factor f  is closely related to the propagator of the scalar field,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \left(G^{\rm scalar}\right)^{-1} \propto f(\,p^2) \propto \mathcal{Z}^s(\,p^2) \, (\,p^2 + \mu^2) \, . \nonumber \end{align} \tag{ 110 }$$

Since $\mu_\ast = 0$, the momentum-dependent wave-function renormalisation $\mathcal{Z}^s(\,p^2)$ is depicted in the lower panel of figure 4. Since ${\mathcal Z}^s_\ast(\,p^2) > 0$ is positive, it may be absorbed in a field redefinition without affecting the pole structure of the propagator. This suggests to generalise (23) to a the momentum-dependent anomalous dimension $\newcommand{\p}{{\partial}} \newcommand{\e}{{\rm e}} \eta_s^\ast(\,p^2) \equiv - \p_t \ln \mathcal{Z}^s_\ast(\,p^2)$. It is then natural that there are no stability coefficients associated with deformations of ${\mathcal Z}^s_\ast(\,p^2)$.

Owed to this observation, a stability analysis should be limited to the three critical exponents associated with the couplings $\vec {u} = \{g,\lambda,\mu^2\}$ [33]. As an approximation, we perturb the fixed point solution $\vec {u}^\ast = \{g_\ast,\lambda_\ast, f_\ast(z) \}$ by perturbations of the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \vec {u} = \vec {u}^\ast + \sum_{I=1}^3 \delta \vec {u}_I \, {\rm exp}(- \theta_I t) \, . \nonumber \end{align} \tag{ 111 }$$

Here the $\delta \vec {u}_I$ are the (momentum-independent) eigenvectors associated with the critical exponents $\theta_I$. An exact treatment takes the variation of the scalar wave-function renormalisation into account, which we neglect here. Substituting this ansatz into the flow equations (80), (81) and (92) and linearising in the perturbations then yields

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{c1} \theta_{1,2} = 1.64 \pm 4.14 \mathbf i \, , \qquad \theta_3 = 1.16 \, . \nonumber \end{align} \tag{ 112 }$$

We note that, by the vanishing of the gap parameter at the fixed point, the critical exponent $\theta_3$ is related to (109) by a factor of minus two, see also (92). Thus the NGFP comes with three relevant directions, acting as a UV attractor for $g, \lambda$ and the gap parameter $\mu$. The complex pair of critical exponents $\theta_{1,2}$ is characteristic for studying gravity and gravity-matter systems in the present setup. It already occurs in the approximation (75) where one finds

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{{\partial}} \displaystyle \label{c2} \theta_{1,2} = 1.64 \pm 4.12 \mathbf i \, . \nonumber \end{align} \tag{ 113 }$$

This indicates that our analysis provides the first study of the gravity-matter fixed point identified in [12, 31, 84, 101] at the level of self-consistent form factors. Comparing the critical exponents (112) and (113) furthermore suggests that the properties of the fixed point are essentially determined by the gravitational contributions [101]. Including a self-consistent form factor in the scalar sector has a very mild effect on the stability coefficients.