Scale invariance beyond criticality within the mean-field analysis of tensorial field theories

The phase structure of TGFTs for Abelian groups $G=$U$(1),\mathbb{R}$ with and without local arguments has been studied in Marchetti:2020xvf and BenGeloun:2015ej ; Benedetti:2015et ; Benedetti:2016db ; BenGeloun:2016kw ; BenGeloun:2018ekd ; Pithis:2020sxm ; Pithis:2020kio ; Geloun:2023ray , using Landau-Ginzburg mean-field theory and the FRG methodology, respectively. In the following, we critically examine the Landau-Ginzburg analysis to show that the resulting effective mass actually vanishes for the modes relevant for the critical behavior.

The specific TGFT model we consider in this section is defined by a real-valued field $\Phi:\mathrm{U}(1)^{4}\times\mathbb{R}^{d_{\mathrm{loc}}}\longrightarrow\mathbb{R}$ with non-local arguments ${\bf\it\theta}=(\theta_{1},\dots,\theta_{4})\in\mathrm{U}(1)^{4}$ and local ones, ${\bf\it x}=(x_{1},\dots,x_{d_{\mathrm{loc}}})\in\mathbb{R}^{d_{\mathrm{loc}}}$.²²2A way to motivate the $d_{\mathrm{loc}}$ local $\mathbb{R}$-valued variables from the TGFT perspective is to introduce discretized scalar fields typically employed as a matter reference frame Li:2017uao ; Oriti:2016qtz . The action governing the field, $S[\Phi]=K[\Phi]+V[\Phi]$, consists of a kinetic and an interaction term, where the former is given by

$$K[\Phi]=\frac{1}{2}\int\limits_{\mathrm{U}(1)^{4}}\differential{{\bf\it\theta}}\int\limits_{\mathbb{R}^{d_{\mathrm{loc}}}}\differential{{\bf\it x}}\Phi({\bf\it\theta},{\bf\it x})\left[\mu-\sum_{c=1}^{4}\Delta_{\theta}^{c}-\Delta_{x}\right]\Phi({\bf\it\theta},{\bf\it x}).$$

(1)

The integration on $\mathrm{U}(1)$ is defined as the integration on the circle $S^{1}$ and $\differential{{\bf\it x}}$ is the standard Lebesgue measure on $\mathbb{R}^{d_{\mathrm{loc}}}$. The Laplace operator $\Delta_{\theta}^{c}$ acts on the variable $\theta_{c}\in\mathrm{U}(1)$ and $\Delta_{x}$ denotes the Laplace operator on $\mathbb{R}^{d_{\mathrm{loc}}}$. The parameter $\mu\in\mathbb{R}$ plays the role of a mass term.

The defining property of TGFTs are the combinatorially non-local interactions in the group variables which we choose in this section to be quartic melonic, pictorially represented in Fig. 1. Explicitly, $V[\Phi]$ is defined as

$$V[\Phi]=\int\differential[8]{\theta}\int\differential{{\bf\it x}}\Phi(\theta_{1},\theta_{2},\theta_{3},\theta_{4},{\bf\it x})\Phi(\theta_{5},\theta_{6},\theta_{7},\theta_{4},{\bf\it x})\Phi(\theta_{5},\theta_{6},\theta_{7},\theta_{8},{\bf\it x})\Phi(\theta_{1},\theta_{2},\theta_{3},\theta_{8},{\bf\it x}),$$

(2)

where the ${\bf\it x}\in\mathbb{R}^{d_{\mathrm{loc}}}$ enter as local variables with point-like interactions and $\theta_{c}$ enter as non-local variables contracted according to the quartic melonic combinatorics. The interaction term can be more compactly denoted by introducing a trace notation Marchetti:2020xvf

$$V[\Phi]=\lambda\int\differential{{\bf\it x}}\Tr_{\gamma}\left[\Phi({\bf\it\theta},{\bf\it x})^{4}\right],$$

(3)

where $\lambda$ is a coupling parameter. The $\theta$-integrations are captured by $\Tr_{\gamma}$, where $\gamma$ denotes the vertex graph Oriti:2014yla , depicted in Fig. 1, which dictates the contraction pattern of the non-local variables. The fourth power indicates that there are four fields $\Phi$ entering the interaction. In this notation, the interactions are straightforwardly generalized by choosing different vertex graphs $\gamma$ with different number of fields $n_{\gamma}$ and it will therefore be heavily utilized in the sections hereafter.

The Landau-Ginzburg method was originally introduced to study phase transitions in local field-theoretic descriptions of lattice systems Kopietz:2010zz ; zinn2021quantum . Their dynamics are captured by the effective action $S[\Phi]$ which is a functional in odd and/or even powers of the field $\Phi$ and its gradient. In general, it is very difficult to exactly compute the partition function $Z$ of such systems. This issue is bypassed by Landau-Ginzburg mean-field theory, which fundamentally assumes that the scales of the system can be separated so that one can average over its microscopic details. Consequently, one works with an effective theory valid from the mesoscopic to the macroscopic regime. Hence, the field $\Phi$ corresponds to an averaged quantity which describes general features of the system like symmetries and the dimensionality of the domain. Its dynamics are typically modelled by the classical action. Thus, averaged-over microscopic details are captured by the field and values of couplings therein. The coarse-graining of microscopically different theories oftentimes leads to the same description on larger scales, referred to as universality.

More precisely, Landau-Ginzburg mean-field theory studies the impact of quadratic fluctuations $\delta\Phi$ over uniform background field configurations $\Phi_{0}$, the latter of which correspond to saddle points of the classical action. One can then solve for the correlation function of those fluctuations and retrieve from it the correlation length $\xi$ which defines the scale beyond which the fluctuations fade away exponentially. It extends from the mesoscale to the macroscale and diverges at criticality. To finally check self-consistency of this approximation, one has to verify that the strength of the fluctuations relative to the background remains small up to the scale set by the correlation length. This is known as the Levanyuk-Ginzburg criterion levanyuk1959contribution ; ginzburg1961some . For local scalar field theories on $\mathbb{R}^{d}$ this allows for the extraction of the upper critical dimension $d_{\text{crit}}$ beyond which this method is valid. For $d<d_{\text{crit}}$ a non-perturbative treatment via the Wilsonian renormalization group formalism is needed instead which would also capture the impact of fluctuations up to the microscale wilson1983renormalization ; dupuis2021nonperturbative . We also refer to Ref. Benedetti:2014gja for a pedagogical discussion of this topic.

As demonstrated in the series of works Pithis:2018eaq ; Marchetti:2020xvf ; Marchetti:2022igl ; Marchetti:2022nrf ; Dekhil:2024djp , this method can be transferred to TGFTs with the main challenge being the hybrid character of such models including local and non-local variables. Hereafter, we summarize the core aspects of this construction, exemplified with the model introduced in the previous section. This will allow us to point out the mechanism of the vanishing effective mass for the relevant modes and to advance the mean-field method to this scenario. The specific construction here will then serve as the basis to generalize the observation to tensorial field theories.

To start off, one computes the classical equations of motion for the field $\Phi$,

$$\left[\mu-\sum_{c=1}^{4}\Delta_{\theta}^{c}-\Delta_{x}\right]\Phi({\bf\it\theta},{\bf\it x})+\lambda\functionalderivative{V[\Phi]}{\Phi({\bf\it\theta},{\bf\it x})}=0,$$

(4)

with

		$\displaystyle\functionalderivative{V[\Phi]}{\Phi(\theta_{1},\dots,\theta_{4},{\bf\it x})}$		(5)
	$\displaystyle=$	$\displaystyle 4\lambda\int\differential{\theta_{5}}\dots\differential{\theta_{8}}\Phi(\theta_{5},\theta_{6},\theta_{7},\theta_{4},{\bf\it x})\Phi(\theta_{5},\theta_{6},\theta_{7},\theta_{8},{\bf\it x})\Phi(\theta_{1},\theta_{2},\theta_{3},\theta_{8},{\bf\it x}).$

The equations of motion are then evaluated on constant field configurations, $\Phi_{0}=\mathrm{const}$, leading to the mean-field equations

$$\mu\Phi_{0}+4\lambda\Phi_{0}^{3}V_{G}^{4}=0,$$

(6)

where $V_{G}=2\pi R$ arises from empty group integrations on $\mathrm{U}(1)$ with $R$ being the radius of the circle. Notice that these factors appear precisely due to the non-localities of the interaction. For $\mu>0$, the mean-field solution is simply given by $\Phi_{0}=0$. For $\mu<0$, the minimum of the theory is instead given by the non-vanishing value

$$\Phi_{0}=\pm\left(\frac{\absolutevalue{\mu}}{4\lambda}\right)^{\frac{1}{2}}V_{G}^{-2}.$$

(7)

To proceed with the Landau-Ginzburg description, one allows for fluctuations around the mean-field solution,

$$\Phi({\bf\it\theta},{\bf\it x})=\Phi_{0}+\delta\Phi({\bf\it\theta},{\bf\it x}),$$

(8)

and linearizes the equations of motion in $\delta\Phi$. This yields

$$\left(\mu-\sum_{c=1}^{4}\Delta_{\theta}^{c}-\Delta_{x}\right)\delta\Phi({\bf\it\theta},{\bf\it x})+\evaluated{\functionalderivative{V[\Phi]}{\Phi({\bf\it\theta},{\bf\it x})}}_{\Phi_{0}+\delta\Phi}=0,$$

(9)

with

		$\displaystyle\evaluated{\functionalderivative{V[\Phi]}{\Phi({\bf\it\theta},{\bf\it x})}}_{\Phi_{0}+\delta\Phi}$		(10)
	$\displaystyle=$	$\displaystyle 4\lambda\Phi_{0}^{2}\int\differential{\theta_{5}}\dots\differential{\theta_{8}}\left(\delta\Phi(\theta_{5},\theta_{6},\theta_{7},\theta_{4})+\delta\Phi(\theta_{5},\theta_{6},\theta_{7},\theta_{8})+\delta\Phi(\theta_{1},\theta_{2},\theta_{3},\theta_{8})\right).$

Inserting the mean-field solution given in Eq. (7), this expression can be re-written as

$$\evaluated{\functionalderivative{V[\Phi]}{\Phi({\bf\it\theta},{\bf\it x})}}_{\Phi_{0}+\delta\Phi}=-\mu\int\differential{\tilde{{\bf\it\theta}}}\chi({\bf\it\theta},\tilde{{\bf\it\theta}})\delta\Phi(\tilde{{\bf\it\theta}},{\bf\it x}),$$

(11)

with $\chi({\bf\it\theta},\tilde{{\bf\it\theta}})$ capturing the non-local character of the interactions which can be derived as the Hessian of the interaction term, as detailed in Marchetti:2020xvf . In this example, it is explicitly given by

$$\chi({\bf\it\theta},\tilde{{\bf\it\theta}})=V_{G}^{-4}+\delta(\theta_{4},\tilde{\theta}_{4})V_{G}^{-3}+\delta(\theta_{1},\tilde{\theta}_{1})\delta(\theta_{2},\tilde{\theta}_{2})\delta(\theta_{3},\tilde{\theta}_{3})V_{G}^{-1}.$$

(12)

Then, the equations of motion at first order in $\delta\Phi$ can be given the form

$$\int\differential{\tilde{{\bf\it\theta}}}\left[\delta({\bf\it\theta},\tilde{{\bf\it\theta}})\left(-\sum_{c}\Delta^{c}_{\tilde{\theta}}-\Delta_{x}\right)+b({\bf\it\theta},\tilde{{\bf\it\theta}})\right]\delta\Phi(\tilde{{\bf\it\theta}},{\bf\it x})=0,$$

(13)

where $\Delta_{\tilde{\theta}}^{c}$ acts on the variable $\theta_{c}$ and with $b({\bf\it\theta},\tilde{{\bf\it\theta}})$ being defined as

$$b({{\bf\it\theta}},\tilde{{\bf\it\theta}})=\mu\left(\delta({\bf\it\theta},\tilde{{\bf\it\theta}})-\chi({\bf\it\theta},\tilde{{\bf\it\theta}})\right).$$

(14)

This determines an effective action for the perturbations $\delta\Phi$, which is given by

$$S_{\mathrm{eff}}[\delta\Phi]=\int\differential{{\bf\it\theta}}\differential{\tilde{{\bf\it\theta}}}\int\differential{{\bf\it x}}\delta\Phi({\bf\it\theta},{\bf\it x})\left[\delta({\bf\it\theta},\tilde{{\bf\it\theta}})\left(-\sum_{c}\Delta_{\tilde{\theta}}-\Delta_{x}\right)+b({\bf\it\theta},\tilde{{\bf\it\theta}})\right]\delta\Phi(\tilde{{\bf\it\theta}},{\bf\it x}).$$

(15)

To compute correlations of fluctuations, it is expedient to perform a Fourier transform on the total domain $\mathrm{U}(1)^{4}\times\mathbb{R}^{d_{\mathrm{loc}}}$. In the non-local variables ${\bf\it\theta}$, the field is a function of periodicity $2\pi R$ allowing for a decomposition in a standard Fourier series. Similarly, the Fourier transform of the local variables is the standard Fourier transform on $\mathbb{R}^{d_{\mathrm{loc}}}$. Overall, it is thus defined as

$$\delta\Phi({\bf\it\theta},{\bf\it x})=\sum_{{\bf\it p}\in\mathbb{Z}^{4}}\int\frac{\differential{{\bf\it k}}}{(2\pi)^{d_{\mathrm{loc}}}}\delta\Phi_{{\bf\it p}}({\bf\it k})\textrm{e}^{i{\bf\it p}{\bf\it\theta}/R}\textrm{e}^{i{\bf\it k}{\bf\it x}}.$$

(16)

As a result, the effective action in Eq. (15) is represented as

$$S_{\mathrm{eff}}[\delta\Phi]=(2\pi R)^{4}\sum_{{\bf\it p}}\int\frac{\differential{{\bf\it k}}}{(2\pi)^{d_{\mathrm{loc}}}}\delta\Phi_{-{\bf\it p}}(-{\bf\it k})\left(\frac{1}{R^{2}}{\bf\it p}^{2}+{\bf\it k}^{2}+b_{{\bf\it p}}\right)\delta\Phi_{{\bf\it p}}({\bf\it k}),$$

(17)

where the spectrum of the Laplace operator on $\mathrm{U(1)}$ enters as $p_{c}^{2}/R^{2}$ and $b_{{\bf\it p}}$ is the bi-local function $b({\bf\it\theta},\tilde{{\bf\it\theta}})$ in Fourier representation. It is in this representation that the role of $b_{{\bf\it p}}$ as effective mass is most apparent,

$$b_{{\bf\it p}}=\mu(1-\chi_{{\bf\it p}}).$$

(18)

Notice that in quartic local field theories, the effective mass is a constant and it is simply given by $b=2\absolutevalue{\mu}$ Kopietz:2010zz ; zinn2021quantum ; Benedetti:2014gja . However, as explained in detail in Marchetti:2020xvf , the non-local interactions that enter the expression of the effective mass through the Hessian contribution $\chi_{{\bf\it p}}$ take a specific form that explicitly depends on the representation labels ${\bf\it p}$. Explicitly, for the present type of quartic interactions, we find

$$\chi_{{\bf\it p}}=\prod_{c=1}^{4}\delta_{p_{c},0}+\prod_{d\neq 4}\delta_{p_{d},0}+\delta_{p_{4},0},$$

(19)

with the Kronecker-$\delta$ on $\mathrm{U}(1)$-momenta defined as

$$\delta_{p,p^{\prime}}=\frac{1}{(2\pi R)}\int\differential{\theta}\textrm{e}^{i(p-p^{\prime})\theta/R}.$$

(20)

We observe from Eq. (19) that the effective mass $b_{{\bf\it p}}$ contains products of projections onto zero modes where the momenta $p$ are set to zero. To elucidate the consequences of this structure, we expand the effective action in terms of zero modes, i.e., we split the $p$-sums into a contribution where $p=0$ and the rest with $p\neq 0$ Marchetti:2020xvf . As a result,

	$\displaystyle S_{\mathrm{eff}}[\delta\Phi]$	$\displaystyle=(2\pi R)^{4}\sum_{s=0}^{4}\sum_{(c_{1}\dots c_{s})}\sum_{{\bf\it p}_{4-s}}\int\frac{\differential{{\bf\it k}}}{(2\pi)^{d_{\mathrm{loc}}}}\delta\Phi_{-{\bf\it p}_{4-s}}(-{\bf\it k})$		(21)
		$\displaystyle\times\left(\frac{1}{R^{2}}\sum_{c=c_{s+1}}^{c_{4}}p_{c}^{2}+{\bf\it k}^{2}+b_{c_{1}\dots c_{s}}\right)\delta\Phi_{{\bf\it p}_{4-s}}({\bf\it k}),$

where $s$ is the number of zero modes, $c_{1}\dots c_{s}$ are the slots where the zero modes are injected and ${\bf\it p}_{4-s}=(p_{c_{s+1}},\dots p_{c_{4}})$ is a short-hand notation for the remaining $4-s$ non-zero momenta. The effective mass $b_{{\bf\it p}}$ evaluated on $s$ zero modes in the slots $c_{1}\dots c_{s}$ is denoted by $b_{c_{1}\dots c_{s}}$ which is constant in the remaining $4-s$ variables.

At this point, we make the main observation of this work: The effective mass $b_{c_{1}\dots c_{s}}$ vanishes for particular zero mode injections. It is furthermore only positive if $s=4$ and negative in all the remaining cases. Let us look at the three cases separately in the following.

First, if $s=4$ zero modes are considered, it follows immediately from Eq. (19) that $\chi_{(0,0,0,0)}=3$. As a result, the effective mass is given by $b_{c_{1}\dots c_{s}}=2\absolutevalue{\mu}$ which corresponds to the result obtained in local field theory. The effective action for this configuration therefore corresponds to a stable parabola opened upwards.

Second, for $s<4$, there exist configurations of zero modes for which $\chi_{{\bf\it p}}=0$ and thus $b_{c_{1}\dots c_{s}}=-\absolutevalue{\mu}<0$. Investigating Eq. (19) closely, one finds that these correspond to $s<3$ with $p_{4}\neq 0$, so for instance $(p_{1},p_{2},0,p_{4})$ or $(0,0,p_{3},p_{4})$. We denote the set of these configurations for a particular number of zero modes $s$ as $\bar{\mathcal{O}}_{s}$. In this case, the effective action takes the form of an unstable parabola that is opened downwards. The consequences of this negative effective mass on the correlations depend on the specific properties of the non-local domain. Since we consider in this section $\mathrm{U}(1)$ which is compact, $b_{c_{1}\dots c_{s}}<0$ does not bear physical consequences, as demonstrated above. However, in Sec. 2.6 we study the non-compact limit $R\rightarrow\infty$ and show that $b<0$ leads to an oscillating correlation function. We argue that such configurations should be excluded when studying the critical behavior.

Lastly, there are configurations for $s_{0}\leq s<4$ for which the effective mass vanishes, i.e. $b_{c_{1}\dots c_{s}}=0$. Here, $s_{0}$ is the number below which the effective mass cannot vanish which is characteristic for the type of interactions considered. For the quartic melonic interaction one has $s_{0}=1$. The effective mass vanishes if $\chi=1$ which, by following Eq. (19), holds for any configuration with $s=3$ or $s<3$ and $p_{4}=0$. We denote the set of these configurations for a particular number of zero modes $s$ as $\mathcal{O}_{s}$. In these instances, the theory becomes effectively massless for any finite value of $\mu<0$ in the broken phase and we therefore observe the emergence of scale- or even conformal symmetry on the residual domain $\mathrm{U}(1)^{4-s}\times\mathbb{R}^{d_{\mathrm{loc}}}$. We elaborate on these symmetries in Sec. 4.3. This feature is distinctive for the non-local interactions of the theory and we argue that this holds for general tensorial field theories, representing the main novelty of the present article.

Correlations of the fluctuations are captured by the two-point function

$$C({\bf\it\theta},{\bf\it x})=\langle\delta\Phi({\bf\it 0},{\bf\it 0})\delta\Phi({\bf\it\theta},{\bf\it x})\rangle$$

(22)

which is defined as the inverse kinetic kernel of the effective action above. In Fourier space, the correlator is simply given by the multiplicative inverse of the kinetic kernel of the effective action in Eq. (17). This yields the real space correlator defined as an integral

$$C({\bf\it\theta},{\bf\it x})=\frac{1}{(2\pi R)^{4}}\sum_{{\bf\it p}}\int\frac{\differential{{\bf\it k}}}{(2\pi)^{d_{\mathrm{loc}}}}\frac{\textrm{e}^{i{\bf\it p}{\bf\it\theta}/R}\,\textrm{e}^{i{\bf\it k}{\bf\it x}}}{\frac{1}{R^{2}}{\bf\it p}^{2}+{\bf\it k}^{2}+b_{{\bf\it p}}}.$$

(23)

To determine the behavior of fluctuations in the local and non-local variables separately, it has been suggested in Marchetti:2020xvf to define local and non-local correlation functions, obtained by integrating out the complementary set of variables. For local variables, we thus define

$$C({\bf\it x})=\int\differential{{\bf\it\theta}}C({\bf\it\theta},{\bf\it x}),$$

(24)

which amounts to setting the $\mathrm{U}(1)$-momenta to zero,

$$C({\bf\it x})=\int\frac{\differential{{\bf\it k}}}{(2\pi)^{d_{\mathrm{loc}}}}\frac{\textrm{e}^{i{\bf\it k}{\bf\it x}}}{{\bf\it k}^{2}+b_{{\bf\it 0}}}.$$

(25)

Here, $b_{{\bf\it 0}}$ is the effective mass evaluated on four zero modes which, according to Eq. (19), results in $b_{{\bf\it 0}}=2\absolutevalue{\mu}$. Following (GradshteynBook, , 6.566, Formula 2.), this integral explicitly evaluates to

$$C({\bf\it x})=\frac{2^{d}\pi^{\frac{d}{2}}}{(2\pi)^{d}\ell^{d-2}}\left(\sqrt{b_{{\bf\it 0}}}\ell\right)^{\frac{d-2}{2}}K_{\frac{d-2}{2}}(\sqrt{b_{{\bf\it 0}}}\ell),$$

(26)

with $d\equiv d_{\mathrm{loc}}$, $\ell\equiv\absolutevalue{{\bf\it x}}$ and $K_{\alpha}(z)$ the modified Bessel function of the second kind GradshteynBook . The result corresponds precisely to the correlation function of a local field theory on $\mathbb{R}^{d_{\mathrm{loc}}}$ which is consistent with the fact that we have integrated out all the non-local variables. From the asymptotic behavior of the correlation function at large distances $\ell\gg 1$,

$$C({\bf\it x})\underset{\ell\gg 1}{\longrightarrow}\textrm{e}^{-\sqrt{b_{{\bf\it 0}}}\ell}$$

(27)

the local correlation length is identified as $\xi_{\mathrm{loc}}^{-2}=b_{{\bf\it 0}}$.

To obtain a correlation function in the non-local variables, the local variables are integrated out,

$$C({\bf\it\theta})=\int\differential{{\bf\it x}}C({\bf\it\theta},{\bf\it x}).$$

(28)

Since U$(1)$ is a compact group, no long-range behavior can be determined. Thus, a correlation length (if existent), can only be given by the system size, where $\xi_{\mathrm{nloc}}=\frac{\pi}{\sqrt{6}}R<2\pi R$ was computed explicitly in Marchetti:2020xvf via the second-moment method. An expansion of $C({\bf\it\theta})$ in zero modes shows

$$C({\bf\it\theta})=\frac{1}{(2\pi R)^{4}}\left[\frac{1}{b_{{\bf\it 0}}}+\sum_{s=0}^{3}\sum_{(c_{1}\dots c_{s})}\sum_{{\bf\it p}_{4-s}}\frac{\prod\limits_{c=c_{s+1}}^{c_{4}}e^{ip_{c}\theta_{c}/R}}{\frac{1}{R^{2}}\sum_{c}p_{c}^{2}+b_{c_{1}\dots c_{s}}}\right]\underset{\mu\rightarrow 0}{\longrightarrow}\frac{1}{(2\pi R)^{4}b_{{\bf\it 0}}},$$

(29)

which is dominated by the first term in the limit $\mu\rightarrow 0$, even if the effective mass $b_{c_{1}\dots c_{s}}$ vanishes. Notice that this is a direct consequence of the presence of the Laplace operator, which introduces a preference for small values of ${\bf\it p}^{2}$.

The Landau-Ginzburg analysis is completed by validating the self-consistency of the mean-field approach which is the case if the fluctuations $\delta\Phi$ remain small relative to the mean-field solution $\Phi_{0}$ in the considered regime. The relative size of fluctuations is quantified by the Levanyuk-Ginzburg-$Q$, defined as

$$Q=\frac{\int_{\Omega_{\xi}}\differential{{\bf\it\theta}}\differential{{\bf\it x}}\expectationvalue{\delta\Phi({\bf\it 0},{\bf\it 0})\delta\Phi({\bf\it\theta},{\bf\it x})}}{\int_{\Omega_{\xi}}\differential{{\bf\it\theta}}\differential{{\bf\it x}}\Phi_{0}^{2}}.$$

(30)

The integration domain $\Omega_{\xi}\subset\mathrm{U}(1)^{4}\times\mathbb{R}^{d_{\mathrm{loc}}}$ is restricted by the local and non-local correlation lengths $\xi_{\mathrm{loc}}$ and $\xi_{\mathrm{nloc}}$, respectively, which have been extracted above. Finally, Landau-Ginzburg mean-field theory is said to be valid near criticality if $Q\ll 1$ in the limit $\mu\rightarrow 0$.

The numerator of $Q$ is computed by extending the local integration to the whole of $\mathbb{R}^{d_{\mathrm{loc}}}$ justified by the exponential suppression at large local distances $\ell\gg 1$. The $\theta$-integrations are bounded by the constant non-local correlation length $\xi_{\mathrm{nloc}}<2\pi R$. As a result, we obtain

$$\int_{[0,\xi_{\mathrm{nloc}}]^{4}}\differential{{\bf\it\theta}}C({\bf\it\theta})=\sum_{s=0}^{4}\frac{\xi_{\mathrm{nloc}}^{s}}{(2\pi R)^{4}}\sum_{(c_{1}\dots c_{s})}\int\differential{{\bf\it\theta}_{4-s}}\sum_{\{p_{c}\}}\frac{\prod_{c}\textrm{e}^{ip_{c}\theta_{c}}}{\frac{1}{R^{2}}\sum_{c}p_{c}^{2}+b_{c_{1}\dots c_{s}}}.$$

(31)

Independent of $b_{c_{1}\dots c_{s}}$ being negative, zero, or positive, the dominant contribution is given by the $s=4$ case. Consequently, the numerator of $Q$ is given by

$$\int_{[0,\xi_{\mathrm{nloc}}]^{4}}\differential{{\bf\it\theta}}C({\bf\it\theta})=\left(\frac{\xi_{\mathrm{nloc}}}{2\pi R}\right)^{4}\frac{1}{b_{{\bf\it 0}}}.$$

(32)

Together with the denominator of $Q$,

$$\int_{\Omega_{\xi}}\differential{{\bf\it\theta}}\differential{{\bf\it x}}\Phi_{0}^{2}=\frac{\absolutevalue{\mu}}{4\lambda}\left(\frac{\xi_{\mathrm{nloc}}}{2\pi R}\right)^{4}\xi_{\mathrm{loc}}^{d_{\mathrm{loc}}},$$

(33)

we finally obtain

$$Q\sim\lambda\xi_{\mathrm{loc}}^{4-d_{\mathrm{loc}}}.$$

(34)

In the limit $\absolutevalue{\mu}\rightarrow 0$, or equivalently $\xi_{\mathrm{loc}}\rightarrow\infty$, the Ginzburg-$Q$ is small if the local dimension satisfies $d_{\mathrm{loc}}>4$, thus exactly reproducing the behavior of local field theories. These results are supported by Geloun:2023ray where in the infrared, a Gaussian fixed point is found if the local dimension takes values above the critical dimension, irrespective of the number of the group copies. In other words, the effective dimension of the model reduces to that of the local theory.

The non-local correlation length $\xi_{\mathrm{nloc}}$ and the radius of U$(1)$ do not enter $Q$ as they refer to the compact variables that do not affect the critical behavior of the theory. In particular, the effective mass $b_{c_{1}\dots c_{s}}$ does not appear in the final expression of $Q$ which reflects two defining properties of the model:

1. 1.

   The group domain is compact, therefore not allowing for long-range correlations. Thus, a phase transition, characterized by a transition between long-range and short-range correlations cannot occur for the non-local variables. This is in agreement with standard results from local field theory strocchi2005symmetry ; Benedetti:2014gja ; zinn2021quantum . It is therefore also expected that for the non-Abelian $G=\mathrm{SU}(2)$, which would be a basic ingredient to relate to loop quantum gravity Oriti:2014yla , the same behavior would be obtained.

2. 2.

   The kinetic term we considered includes a Laplace operator which leads to a dominance of low-spin contributions. This can be seen explicitly in the non-local correlation where the $s=4$ zero mode term dominates. Notice that this is different from the tensor field theory case treated in Sec. 3.2 where no such Laplacian is present and where the vanishing effective mass therefore plays a distinguished role in charting the phase structure.

Exploring the physical consequences of a negative or vanishing effective mass has been obstructed by the two points listed above. In the next section, we therefore study in detail the non-compact limit of the present $\mathrm{U}(1)$-model and extend the Landau-Ginzburg method to vanishing effective masses. This is intended to serve as a guiding example for the more general cases of TGFTs on non-compact group domains and tensor field theories, considered in Secs. 3.1.1 and 3.2.

In this section, we study the non-compact limit of the theory defined in the previous section. More explicitly, we study the limit of infinite radius of $\mathrm{U}(1)$, $R\rightarrow\infty$, resulting in a tensorial theory defined on $\mathbb{R}^{4}\times\mathbb{R}^{d_{\mathrm{loc}}}$.

Clearly, the behavior of the local correlation function $C({\bf\it x})$ is unaffected by the non-compact limit, showing an asymptotic exponential suppression with a correlation length $\xi_{\mathrm{loc}}^{-2}=b_{{\bf\it 0}}$.

In contrast, the non-local correlation function given in Eq. (29) explicitly depends on the variables ${\bf\it\theta}$. In the non-compact limit, we perform a continuum approximation of the discrete sums over the $p_{c}$,

$$\frac{1}{(2\pi R)}\sum_{p}f(p/R)\longrightarrow\int\frac{\differential{\tilde{p}}}{2\pi}f(\tilde{p}),$$

(35)

with $\tilde{p}=p/R$, where the above approximation is valid for any function $f$. In this case, the non-local correlation function in Eq. (29) reads as

$$C({\bf\it\theta})=\sum_{s=0}^{4}\sum_{(c_{1},\dots,c_{s})}C_{c_{1}\dots c_{s}}({\bf\it\theta}_{4-s})=\sum_{s=0}^{4}\sum_{(c_{1},\dots,c_{s})}\frac{1}{(2\pi R)^{s}}\int\frac{\differential{\tilde{{\bf\it p}}}_{4-s}}{(2\pi)^{4-s}}\frac{\prod_{v}\textrm{e}^{i\tilde{p}_{c_{v}}\theta_{c_{v}}}}{\sum_{v}\tilde{p}_{c_{v}}^{2}+b_{c_{1}\dots c_{s}}}.$$

(36)

Notice that the residual non-local correlation function $C_{c_{1}\dots c_{s}}$ characterizing the $4-s$ zero modes contribution is similar to the local correlation function in Eq. (25) with the difference being that the effective mass here is given by $b_{c_{1}\dots c_{s}}$ and not by $b_{{\bf\it 0}}$. As a result, the explicit form of the effective mass $b_{c_{1}\dots c_{s}}$ evaluated on $s$ zero modes is important. The values that $b_{c_{1}\dots c_{s}}$ takes are the same as those discussed in Sec. 2.4 and are unaffected by the non-compact limit which is only performed in the $4-s$ residual variables ${\bf\it p}_{4-s}$.

At four zero modes, the non-local correlation function is a constant that scales as $b_{{\bf\it 0}}^{-1}$, which is consistent with the result of the previous section.

For $s<4$ zero modes and negative effective mass, that is $(c_{1}\dots c_{s})\in\bar{\mathcal{O}}_{s}$, the non-local correlation function $C({\bf\it\theta})$ exhibits an oscillatory behavior with polynomial decay in the regime of large distances, $\absolutevalue{{\bf\it\theta}}\gg 1$. This behavior is reminiscent of anti-ferromagnets which exhibit a vanishing vacuum expectation value and long-range correlations at all scales $\mu$. The corresponding effective action is given by a downward opened parabola which leads to an unstable Gaussian as the partition function. In conclusion, these modes do not affect the critical behavior around $\mu\rightarrow 0$, characterized by a transition between short-range and long-range correlations. As we discuss below, results from an FRG analysis of the present model Geloun:2023ray justify this exclusion a posteriori.

In the case of $s_{0}\leq s<4$ zero modes that are injected at arguments $(c_{1},\dots,c_{s})\in\mathcal{O}_{s}^{\gamma}$, the effective mass vanishes. Instead of computing the correlation function by direct integration, we introduce a regularization of the effective mass, given by³³3Notice that the introduction of a regularization for the effective mass is similar to working with a vanishing effective mass and introducing cutoffs for the $Q$-integration. Because such cutoffs carry a dimension, a similar relation in terms of the parameter $\mu$ must be given in this case. The resulting Ginzburg-$Q$ turns out to be the same.

$$b=\lim\limits_{\epsilon\rightarrow 0^{+}}\epsilon\absolutevalue{\mu}.$$

(37)

We regularize the effective mass with the inclusion of $\mu$ in order to have a dimensionless small regulator $\epsilon$. As a result, the non-local correlation function exhibits an asymptotic exponential behavior in the $4-s$ residual variables ${\bf\it\theta}_{4-s}\in\mathbb{R}^{4-s}$ and we extract the correlation length $\xi_{\mathrm{nloc}}^{-2}=\epsilon\absolutevalue{\mu}$. In the limit $\epsilon\rightarrow 0$, the non-local correlation length diverges even for finite $\mu<0$.

As the remaining step of this section, we compute the Ginzburg-$Q$ as defined in Eq. (30) in the non-compact limit. In this case, the integration range for computing $Q$ is given by $\Omega_{\xi}=[-\xi_{\mathrm{nloc}},\xi_{\mathrm{nloc}}]^{4}\times[-\xi_{\mathrm{loc}},\xi_{\mathrm{loc}}]^{d_{\mathrm{loc}}}\subset\mathbb{R}^{4}\times\mathbb{R}^{d_{\mathrm{loc}}}$. For the numerator of Eq. (30), we find

$$\int_{\Omega_{\xi}}\differential{{\bf\it\theta}}\differential{{\bf\it x}}C({\bf\it\theta},{\bf\it x})=\int\differential{{\bf\it\theta}}C({\bf\it\theta})=\sum_{s=s_{0}}^{4}\left(\frac{\xi_{\mathrm{nloc}}}{2\pi R}\right)^{s}\sum_{(c_{1},\dots,c_{s})\in\mathcal{O}_{s}^{\gamma}}\frac{1}{b_{c_{1}\dots c_{s}}}.$$

(38)

where we took into consideration that the local correlation function exhibits an asymptotic exponential suppression. For the denominator, we insert the background mean-field solution in the large-$R$ limit, yielding

$$\int\differential{{\bf\it\theta}}\differential{{\bf\it x}}\Phi_{0}^{2}=\left(\frac{\xi_{\mathrm{nloc}}}{2\pi R}\right)^{4}\xi_{\mathrm{loc}}^{d_{\mathrm{loc}}}\frac{\absolutevalue{\mu}}{4\lambda}.$$

(39)

Combing the obtained expressions for the denominator and numerator, the parameter $Q$ evaluates to

$$Q\sim\lambda\xi_{\mathrm{loc}}^{-d_{\mathrm{loc}}}\sum_{s=s_{0}}^{4}\left(\frac{\xi_{\mathrm{nloc}}}{2\pi R}\right)^{s-4}\sum_{(c_{1},\dots,c_{s})\in\mathcal{O}_{s}^{\gamma}}\frac{1}{b_{c_{1}\dots c_{s}}}.$$

(40)

Taking the limit of large radius $R$ before the limit $\epsilon,\mu\rightarrow 0$, the sum over the number of zero modes is dominated by the smallest summand, here being the $s_{0}$-term, which in particular implies that $b_{c_{1}\dots c_{s}}=\epsilon\absolutevalue{\mu}$. As elaborated in Sec. 2.4, for the quartic melonic interaction one has $s_{0}=1$. After absorbing the radius into the coupling $\lambda$, we define the re-scaled coupling as

$$\bar{\lambda}=(2\pi R)^{4-s_{0}}\lambda.$$

(41)

The Ginzburg-$Q$ parameter expressed in terms of $\epsilon$ and $\mu$ is finally given by

$$Q\sim\bar{\lambda}\epsilon^{-1+\frac{4-s_{0}}{2}}\absolutevalue{\mu}^{-\frac{1}{2}\left(4-d_{\mathrm{loc}}-(4-s_{0})\right)}.$$

(42)

It is important to mention that if the following inequalities

$$4-d_{\mathrm{loc}}-(4-s_{0})\leq 0,\qquad-1+\frac{4-s_{0}}{2}\geq 0$$

(43)

are satisfied, the limits taken in $\epsilon$ and $\mu$ commute and $Q\rightarrow 0$ in the limits $\epsilon,\mu\rightarrow 0$. A critical dimension of the total domain $d_{\mathrm{loc}}+(4-s_{0})|_{\mathrm{crit}}=4$ is identified. This is in agreement with the results for local quartic field theories which is expected since for a fixed number of zero modes $s_{0}$, the theory is effectively local on the domain $\mathbb{R}^{4-s_{0}}\times\mathbb{R}^{d_{\mathrm{loc}}}$. Moreover, these results agree with the FRG analysis conducted in Geloun:2023ray , where in the infrared a Gaussian fixed point is found if the dimension of the total domain, $d_{\mathrm{loc}}+(4-s_{0})$, is above the critical dimension.

The computation of the Ginzburg-$Q$ in the non-compact limit concludes this section. Along the lines of a melonic TGFT on $\mathrm{U}(1)^{4}\times\mathbb{R}^{d_{\mathrm{loc}}}$ we have presented an example for two novelties of the present work. First, the effective mass is non-positive for $s<4$ zero modes. Terms of negative effective mass are neglected in studying the critical behavior for the reasons given above, such that the relevant contributions to the correlation function and the Ginzburg-$Q$ are those of vanishing effective mass. Second, we have provided a regularization method to advance the Landau-Ginzburg analysis to the case of vanishing effective mass. In the next two sections, we show that the arguments developed in this section generalize to general tensorial field theories. For the sake of performing explicit computations, a set of exemplary models is studied in the upcoming sections.

In the following, we generalize the mean-field results of the previous section to fields of arbitrary rank $r$ and living on any Lie group $G$ and consider a single but arbitrary combinatorially non-local interaction. The field is thus defined as $\Phi:G^{r}\times\mathbb{R}^{d_{\mathrm{loc}}}\longrightarrow\mathbb{R}$ with the non-local variables now collectively denoted as ${\bf\it g}=(g_{1},\dots,g_{r})\in G^{r}$. The kinetic term is the same as in Eq. (1) with the $\mathrm{U}(1)$-integrations and Laplacians replaced by the corresponding objects on $G^{r}$.

To define the single but arbitrary interaction term $V[\Phi]$, we introduce a vertex graph $\gamma$ as in Ref. Oriti:2014yla , in which every vertex $v\in\gamma$ represents a field and every edge a non-local variable. Thus, $\gamma$ captures the contraction pattern of the non-local variables. In Fig. 1, we encountered such a vertex graph for the quartic melonic interaction at rank $r=4$. Further examples of $r=4$ vertex graphs are shown in Fig. 2 and more general graphs are depicted in Tab. 1.

Associated with this graph is a vertex set $\mathcal{V}_{\gamma}$ with cardinality $\absolutevalue{\mathcal{V}_{\gamma}}=n_{\gamma}$ that corresponds to the power of the fields in the interaction. We assume every $v\in\mathcal{V}_{\gamma}$ to have the same valency (given by the rank $r$) and we exclude loops, i.e. edges that start and end at the same vertex. Furthermore, we introduce the set $\mathcal{A}_{v}$ as the set of vertices $v^{\prime}$ adjacent to $v$. These structures, as we will see below, are employed when computing the Hessian of the action within the Landau-Ginzburg approach. Given such a graph $\gamma$, the interaction term is then given by

$$V_{\gamma}[\Phi]=\lambda\int\differential{{\bf\it x}}\Tr_{\gamma}\left[\Phi({\bf\it g},{\bf\it x})^{n_{\gamma}}\right],$$

(44)

where the trace denotes the integration over $\frac{rn_{\gamma}}{2}$ elements in $G$ according to $\gamma$. In Eq. (2), an explicit example for this notation has been given for the case of a quartic melonic interaction at $r=4$.

To appreciate the implications of Eq. (44), we illustrate its quantum geometric interpretation in the context of TGFTs. Fundamental excitations of a rank-$r$ TGFT are interpreted as $(r-1)$-simplices which constitute the fundamental building blocks of discretized geometries. In this picture, interactions govern the combinatorial gluing of such elements to generate $r$-dimensional cellular complexes. Simplicial interactions, as depicted on the very right of Fig. 2, establish a direct connection of TGFTs to quantum gravity approaches such as LQG, spin foam models, simplicial gravity or dynamical triangulations. Tensor-invariant interactions, such as the three left diagrams in Fig. 2, arise from colorizing the group fields Gurau:2009tw and integrating out all but one color Bonzom:2012hw . This type of interactions is heavily studied in tensor models and tensor field theories as the generated Feynman graphs are bijective to cellular pseudo-manifolds Gurau:2010nd ; Gurau:2011xp ; Bonzom:2012hw .

Notice that $V_{\gamma}$ consists of a single interaction. A straightforward generalization consists of a sum of multiple interaction terms, all of which enter the action with a different coupling $\lambda_{\gamma}$. As it has been shown in Marchetti:2020xvf ; Marchetti:2022nrf , the mean-field description admits a simple inclusion of multiple interactions as long as their power in fields, $n_{\gamma}$, is equal. We discuss such an extension briefly in Sec. 3.1.3. In the most generic case of multiple interactions capturing different degrees $n_{\gamma}$, the mean-field equations are possibly complicated polynomial equations to which a solution might be difficult to find. In the remainder of this work we focus on the simpler case and give emphasis when going beyond it.

Following the steps detailed in Sec. 2.3, one computes the classical equations of motions,

$$\left[\mu+-\sum_{c=1}^{r}\Delta^{c}_{g}-\Delta_{x}\right]\Phi({\bf\it g},{\bf\it x})+\lambda\sum_{v\in\mathcal{V}_{\gamma}}\Tr_{\gamma/v}\left[\Phi({\bf\it g},{\bf\it x})^{n_{\gamma}-1}\right]=0,$$

(45)

where the sum over vertices arises from the product rule when taking the functional derivative of $V_{\gamma}[\Phi]$. Here, $\Tr_{\gamma/v}$ denotes the contraction of those non-local variables which are not affected by the functional derivative of the field at vertex $v$. Evaluating the equations of motion on constant field configurations, $\Phi_{0}=\mathrm{const.}$, its solutions are given by $\Phi_{0}=0$ for $\mu>0$ and

$$\Phi_{0}=\left(\frac{\absolutevalue{\mu}}{\lambda n_{\gamma}}\right)^{\frac{1}{n_{\gamma}-2}}\mathrm{V}_{G}^{-\frac{r}{2}},$$

(46)

for $\mu<0$. In the example of $G=\mathrm{U}(1)$, $V_{G}=2\pi R$ has been given as the volume of the circle. Now, $V_{G}$ denotes the result of an empty $G$-integral,

$$\int_{G}\differential{g}=\mathrm{V}_{G}.$$

(47)

If $G$ is non-compact, volume factors necessarily diverge. Instead of restricting to compact $G$, we regularize such divergences via a cut-off $L$ in the non-compact coordinates on $G$ and take the limit of $L\rightarrow\infty$ at the end of the computation. It is in this way that physically interesting models such as the spacelike Barrett-Crane model defined on $G=\text{SL$(2,\mathbb{C})$}$ with additional constraints, which we will discuss later in Sec. 3.1.2, are contained within the ensuing analysis.⁴⁴4In Marchetti:2022igl ; Marchetti:2022nrf , the non-compact domain $G=\text{SL$(2,\mathbb{C})$}$ is regularized to Spin$(4)$ and the non-compact limit is taken at the end of the computation, similar to the previous section for the groups $\mathbb{R}$ and $\mathrm{U}(1)$. This procedure turns out to be equivalent to our proposal of working with regularized volume factors from the outset.

Introducing fluctuations around the mean-field solution, $\Phi({\bf\it g},{\bf\it x})=\Phi_{0}+\delta\Phi({\bf\it g},{\bf\it x})$, the equations of motion are linearized in $\delta\Phi$ which yields

$$\int\differential{\tilde{{\bf\it g}}}\left[\delta({\bf\it g},\tilde{{\bf\it g}})\left(-\sum_{c}\Delta^{c}_{\tilde{g}}-\Delta_{x}\right)+b({\bf\it g},\tilde{{\bf\it g}})\right]\delta\Phi(\tilde{{\bf\it g}},{\bf\it x})=0,$$

(48)

where $b({\bf\it g},\tilde{{\bf\it g}})=\mu(\delta({\bf\it g},\tilde{{\bf\it g}})+\chi({\bf\it g},\tilde{{\bf\it g}}))$ with $\chi$ obtained from linearizing the interaction term in $\delta\Phi$. Notice that $\chi$ is computed in precisely the same way as shown in Sec. 2.3 and is given as the sum of products of $\delta$-functions on the group together with corresponding volume factors. Examples of $\chi$ in Fourier space for different vertex graphs are displayed in Tab. 1.

Using the Peter-Weyl decomposition for compact $G$ or the Plancherel decomposition for non-compact $G$, functions in $L^{2}(G)$ are decomposed into unitary irreducible representations of $G$ with representation matrices $D^{(j)}(g)$ which are often referred to as Wigner matrices Ruehl1970 . The details of the labels $j$, in particular whether they are discrete or continuous, depend on the specific properties of the group. Taking the example of the previous section with $G=\mathrm{U}(1)$ or $\mathbb{R}$, the $j$ correspond either to the discrete momenta $p\in\mathbb{Z}$ on the circle or the continuous momenta $p\in\mathbb{R}$, respectively. In Sec. 3.1.2, we consider $G=\text{SL$(2,\mathbb{C})$}$ where the $j$ are given by real numbers denoted as $\rho\in\mathbb{R}$. Together with the Fourier transform on $\mathbb{R}$, the fluctuation $\delta\Phi\in L^{2}(G^{r}\times\mathbb{R}^{d_{\mathrm{loc}}})$ is expanded as

$$\delta\Phi({\bf\it g},{\bf\it x})=\;\;\mathclap{\displaystyle\int}\mathclap{\textstyle\sum}\;\;\;\!\!{\raisebox{-7.11317pt}{\scalebox{0.8}[0.8]{${\bf\it j}$}}}\;\int\frac{\differential{{\bf\it k}}}{(2\pi)^{d_{\mathrm{loc}}}}\delta\Phi_{{\bf\it j}}({\bf\it k})\prod_{c=1}^{r}D_{G}^{(j_{c})}(g_{c})\textrm{e}^{i{\bf\it k}{\bf\it x}},$$

(49)

where the details of the measure on the variables ${\bf\it j}$ depends again on the details of $G$. In this decomposition, the effective action takes the form⁵⁵5Here, $-{\bf\it j}$ denote the dual labels to ${\bf\it j}$, which is a consequence of $\delta\Phi$ being real-valued.

$$S_{\mathrm{eff}}[\delta\Phi]=\;\;\mathclap{\displaystyle\int}\mathclap{\textstyle\sum}\;\;\;\!\!{\raisebox{-7.11317pt}{\scalebox{0.8}[0.8]{${\bf\it j},{\bf\it k}$}}}\;\delta\Phi_{{\bf\it j}}({\bf\it k})\left[\sum_{c}\lambda_{g}(j_{c})+{\bf\it k}^{2}+b_{{\bf\it j}}\right]\delta\Phi_{-{\bf\it j}}(-{\bf\it k}),$$

(50)

with $\lambda_{g}(j_{c})$ the eigenvalues of the Laplace operator $-\Delta^{c}_{g}$ acting on the Wigner matrices $D^{(j_{c})}(g_{c})$. The effective mass in Fourier representation is given by $b_{{\bf\it j}}=\mu\left(1-\chi_{{\bf\it j}}\right)$ with the function $\chi_{{\bf\it j}}$ being of the general form

$$\chi_{{\bf\it j}}=\sum_{s=0}^{r}\sum_{(c_{1}\dots c_{s})}\tilde{\chi}^{\gamma}_{c_{1}\dots c_{s}}\prod_{c=c_{1}}^{c_{s}}\delta_{j_{c},j_{0}},$$

(51)

with $\tilde{\chi}$ combinatorial coefficients that depend on the graph structure encoded in $\gamma$ being are either zero or one Marchetti:2020xvf . The $\delta_{j_{c},j_{0}}$ represents a Kronecker-$\delta$ on $G$, singling out the value $j_{c}=j_{0}$ for which $\lambda_{g}(j_{0})=0$. Following the designation of the previous section, we refer to this value as the zero mode, obtained from the projection

$$\frac{1}{\mathrm{V}_{G}}\int\differential{g}D_{G}^{(j)}(g)=\delta_{j,j_{0}}.$$

(52)

For $G$ being a compact Lie group, the label $j_{0}$ corresponds to the trivial representation which is part of the decomposition of functions on $L^{2}(G)$ and is associated with the constant function. In the case of non-compact manifolds, such as $\mathbb{R}$ or SL$(2,\mathbb{C})$, the constant function is not square integrable, and therefore, the trivial representation is not part of the decomposition of the $L^{2}$-space into unitary irreducible representations. Thus, defining the symbol $\delta_{j,j_{0}}$ requires either a regularization via compactification, as proposed in Marchetti:2020xvf ; Marchetti:2022igl ; Marchetti:2022nrf , or a careful extension of the $L^{2}$-space to the space of hyperfunctions Ruehl1970 ; hormander2015analysis . The latter contains in particular constant field configurations which correspond to the trivial representation. For a TGFT model describing $4d$ Lorentzian quantum gravity with $G=\text{SL$(2,\mathbb{C})$}$ this is applied in Refs. Marchetti:2020xvf ; Marchetti:2022igl ; Marchetti:2022nrf ; Dekhil:2024djp .

The sum/integrals over the labels $j$ can be split into zero mode contributions

	$\displaystyle S_{\mathrm{eff}}[\delta\Phi]$	$\displaystyle=\sum_{s=0}^{r}\sum_{(c_{1}\dots c_{s})}\;\;\mathclap{\displaystyle\int}\mathclap{\textstyle\sum}\;\;\;\!\!{\raisebox{-7.11317pt}{\scalebox{0.8}[0.8]{${\bf\it j}_{r-s}$}}}\;\int\frac{\differential{{\bf\it k}}}{(2\pi)^{d_{\mathrm{loc}}}}\delta\Phi_{{\bf\it j}_{r-s}}({\bf\it k})$		(53)
		$\displaystyle\times\left[\sum_{c=c_{s+1}}^{c_{r}}\lambda_{g}(j_{c})+{\bf\it k}^{2}+b_{c_{1}\dots c_{s}}\right]\delta\Phi_{-{\bf\it j}_{r-s}}(-{\bf\it k}),$

where $b_{c_{1},\dots c_{s}}$ is the effective mass with $s$ zero modes injected into the arguments $(c_{1},\dots c_{s})$. At $s=r$, the effective mass evaluates to $b_{c_{1}\dots c_{s}}=\absolutevalue{\mu}(n_{\gamma}-2)$ which is consistent with the results of local field theories. For $s<r$, there exist zero mode injections at $(c_{1},\dots c_{s})\in\bar{\mathcal{O}}_{s}^{\gamma}$ for which $b_{c_{1}\dots c_{s}}<0$. In particular, if the function $\chi$ vanishes for such configurations, one finds $b_{c_{1}\dots c_{s}}=-\absolutevalue{\mu}$. For $s_{0}\leq s<r$ zero modes injected at $(c_{1},\dots c_{s})\in\mathcal{O}_{s}^{\gamma}$, the effective mass vanishes. Notice that for every graph $\gamma$, one can define a characteristic number $s_{0}$ below which $b_{c_{1}\dots c_{s}}<0$ Marchetti:2020xvf . For rank $4$, we find for instance $s_{0}=0$ for double-trace melonic, $s_{0}=1$ for melonic, $s_{0}=2$ for necklace and $s_{0}=3$ for simplicial interactions.