Scaling solution for field-dependent gauge couplings in quantum gravity

Often the scale symmetry of the classical action is considered as an approximation, while quantum fluctuations are supposed to lead to a violation of this symmetry. In contrast, for the quantum scale invariant standard model Wetterich (1988b); Shaposhnikov and Zenhausern (2009a) it is the quantum effective action that does not involve any parameter with dimension of mass or length. In this case quantum scale symmetry is an exact global symmetry. If it is broken spontaneously by a non-zero value of the scalar field $\chi$ one predicts the presence of a massless Goldstone boson. Quantum scale symmetry can actually be induced by quantum fluctuations, both for classical actions with and without scale symmetry. Quantum fluctuations generate a flow of couplings or more generally, a flow of coupling functions. Quantum scale symmetry emerges as an exact symmetry for fixed points in the flow of couplings. Those can be related to scaling solutions for couplings functions.

Due to quantum fluctuations the dimensionless couplings become running couplings. The solution of the flow equations for these couplings introduces mass scales as the renormalization scale $\mu$ or the confinement scale $\Lambda_{\text{QCD}}$ in quantum chromodynamics (QCD). If these mass scales are intrinsic parameters, the quantum effects violate scale symmetry. An example is the running electromagnetic coupling in quantum electrodynamics (QED) for electrons coupled to photons. The running gauge coupling is reflected in the gauge invariant quantum effective action by a gauge invariant kinetic term

$$\Gamma_{F}=\frac{1}{4}\int_{x}\sqrt{g}F_{\mu\nu}Z\big{(}-\partial^{2}\big{)}F^{\mu\nu}\ ,\quad F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\ ,$$

(2)

with $\partial^{2}=\partial^{\mu}\partial_{\mu}$. In one loop order one has the approximate form

$$Z=\frac{1}{4\pi\alpha_{0}}-\frac{1}{12\pi^{2}}\ln\left(\frac{-\partial^{2}+4m_{\text{e}}^{2}}{4m_{\text{e}}^{2}}\right)\ ,$$

(3)

with $\alpha_{0}$ the fine structure constant as measured with experiments for which the relevant length scales are much larger than $m_{\text{e}}^{-1}$.

In the presence of scale symmetry the electron mass is field-dependent,

$$m_{\text{e}}=y_{\text{e}}h_{0}=y_{\text{e}}\sqrt{\varepsilon}\chi=h_{\text{e}}\chi\ ,$$

(4)

with $h_{0}$ the expectation value of the neutral component of the Higgs doublet and $y_{e}$ the electron Yukawa coupling. The dimensionless parameter $h_{e}=y_{e}\sqrt{\varepsilon}$ accounts for the exact proportionality between electron mass and the scalar field $\chi$. Inserting eq. (4) into eq. (3) no intrinsic scale appears in $Z$ if $\alpha_{0}$ and $h_{e}$ are constants. The effective action (2) is compatible with quantum scale symmetry.

The running of the gauge coupling appears in the coupling of $A_{\mu}$ to the electrons, which involves the gauge covariant derivative $D_{\mu}$. Eq. (2) uses a normalization of the gauge field for which the covariant derivative $D_{\mu}=\partial_{\mu}-iA_{\mu}$ does not directly involve gauge coupling. Rescaling the gauge field by $A_{\mu}\to Z^{-1/2}A_{\mu}$ shifts the gauge coupling $g$ to the covariant derivative, and one infers a momentum dependent gauge coupling ($-\partial^{2}\to q^{2}$)

$$g^{2}(q)=Z^{-1}(q)\ .$$

(5)

For $q^{2}\gg 4m_{\text{e}}^{2}$ it obeys the usual flow equation

$$q^{2}\frac{\partial}{\partial q^{2}}g^{2}=-Z^{-2}q^{2}\frac{\partial}{\partial q^{2}}Z=\frac{g^{4}}{12\pi^{2}}\ ,$$

(6)

while for $q^{2}\ll 4m_{\text{e}}^{2}$ the flow stops. We could also define the gauge coupling at some arbitrary renormalization scale $\mu$, with $g^{2}(\mu)=\bar{g}^{2}$ taking a fixed value. Consistency with the scale invariant expression (3),(4) requires that $\mu$ is taken proportional to $\chi$. Otherwise $\alpha_{0}$ would depend on $\mu$ for fixed $\bar{g}^{2}$ and therefore induce a violation of quantum scale symmetry.

For QCD we may again define the gauge coupling by $g^{2}(\mu)=\bar{g}^{2}$. The quantum scale invariant standard model Wetterich (1988b); Shaposhnikov and Zenhausern (2009a) requires $\mu\sim\chi$. In this case the confinement scale where $g^{2}(q)$ diverges or turns to non-perturbatively large values is proportional to $\chi$

$$\Lambda_{\text{QCD}}=\eta_{\text{QCD}}\chi\ .$$

(7)

With similar procedures for all dimensionless couplings no parameter with dimension of mass or length is present in the quantum effective action. This realizes the quantum scale invariant standard model. All mass scales are dynamical, given by the value of the cosmon field $\chi$.

In the quantum scale invariant standard model the running of couplings with momentum is directly related to their field-dependence. Indeed, the dimensionless couplings have to be field-dependent – $g^{2}$ depends both on $q^{2}$ and on $\chi^{2}$. Being dimensionless, it is a function of the ratio $q^{2}/\chi^{2}$. Only for this specific $\chi$-dependence of the dimensionless couplings quantum scale symmetry becomes an exact global symmetry in the presence of quantum fluctuations. For any non-zero value of $\chi$ this global scale symmetry or dilatation symmetry is broken spontaneously. One then predicts an exactly massless Goldstone boson.

For couplings depending only on $q^{2}/\chi^{2}$ the renormalization flow with momentum (at fixed $\chi$) is directly linked to the field-dependence of the coupling. This characteristic feature of the quantum scale invariant standard model relates the well known momentum dependence of the couplings to the dependence on scalar fields, and therefore to possible apparent violations of the equivalence principle or to time-varying fundamental constants. Scaling solutions for quantum gravity will introduce an effective infrared cutoff scale $k$. If $\chi/k$ varies in the course of the cosmic evolution, also fundamental parameters can vary. A small scale $k\ll m_{e}$ (say $k\approx 10^{-3}\text{eV}$) will only lead to tiny modifications for the interactions of the particles in the quantum scale invariant standard model. Nevertheless, the Goldstone boson is turned to a pseudo Goldstone boson - the cosmon - with a very small non-zero mass. The interactions mediated by the cosmon need no longer be pure derivative interactions and their effective range is finite.

We observe that quantum scale symmetry does not fix the parameter $\varepsilon$ in eq. (1) and therefore is not related to the gauge hierarchy problem. A possible understanding of the naturalness of the tiny value of $\varepsilon$ in terms of an additional “particle scale symmetry” at fixed effective Planck mass is proposed in refs. Wetterich (1984, 1987); Bardeen (1995). This is not the subject of the present work.

Can one get the scale invariant standard model from some consistent quantum field theory that remains valid up to arbitrarily short distances? It has been proposed to employ $\chi$ as an effective ultraviolet cutoff Wetterich (1988b); Shaposhnikov and Zenhausern (2009a, b). This is valid as long as $q^{2}\ll\chi^{2}$, and indeed leads to the quantum scale invariant model. A complete quantum field theory has also to cover the range $q^{2}>\chi^{2}$, however, for which the interpretation of $\chi$ as an effective ultraviolet cutoff is no longer valid. If the present value of $\chi$ is associated to the Planck mass $M$, this momentum range concerns transplanckian physics. Similarly, one wants to understand the field-dependence of couplings for the whole range of $\chi$ from zero to infinity. This is particularly relevant for crossover cosmologies for which $\chi$ starts from zero in the infinite past and reaches infinity in the infinite future Wetterich (2014, 2015, 2022).

A direct way of obtaining the quantum scale invariant standard model is the scaling solution of quantum gravity. The formulation of quantum gravity as a consistent quantum field theory, which remains valid for all distance scales, requires the existence of an ultraviolet fixed point. Quantum gravity formulated in terms of the metric can be either asymptotically safe Weinberg (1979); Reuter (1998); Dou and Percacci (1998); Souma (1999); Lauscher and Reuter (2002); Reuter and Saueressig (2002) or asymptotically free Sen et al. (2022); Stelle (1977); Fradkin and Vilkovisky (1978); Fradkin and Tseytlin (1981). In turn, such a fixed point needs the existence of scaling solutions for the dependence of dimensionless couplings on $\chi^{2}/k^{2}$, with $k$ a suitable renormalization scale. The resulting $\chi$-dependence occurs in addition to the dependence on $q^{2}/\chi^{2}$. We will see that for the running gauge couplings the dependence on $k^{2}/\chi^{2}$ is closely related to the dependence on $q^{2}/\chi^{2}$.

The non-perturbative properties of such a fixed point and the flow towards realistic values can be dealt with by functional renormalization for the effective average action Wetterich (1993). This approach introduces an infrared cutoff scale $k$ such that only fluctuations with momenta $q^{2}>k^{2}$ are included for the computation of the effective action. Instead of a finite number of couplings one follows the flow of whole functions of fields, which amounts to infinitely many couplings. In particular, the existence of an ultraviolet fixed point requires that the functional flow equations admit a scaling solution for which all dimensionless couplings as the gauge couplings depend only on dimensionless ratios as

$$\tilde{\rho}=\frac{\chi^{2}}{k^{2}}\ ,\quad\tilde{q}^{2}=\frac{q^{2}}{\chi^{2}}\ ,$$

(8)

without an additional explicit dependence on $k$.

If for $k\to 0$ the gauge coupling $g^{2}(\tilde{\rho},\tilde{q}^{2})$ reaches a fixed function $g_{*}^{2}(\tilde{q}^{2})$, any dependence on the scale $k$ disappears. This results in dimensionless momentum-dependent and field-dependent couplings, as for eq. (3) for QED,

$$g^{-2}(\tilde{q}^{2})=\frac{1}{4\pi\alpha_{0}}-\frac{1}{12\pi^{2}}\ln\left(\frac{\tilde{q}^{2}+4h_{\text{e}}^{2}}{4h_{\text{e}}^{2}}\right)\ .$$

(9)

Similarly, one obtains for QCD with $\bar{g}^{2}=g^{2}(q^{2}=\chi^{2})$ and $N_{\text{f}}$ flavors of massless quarks

	$\displaystyle g^{-2}(\tilde{q}^{2})=$	$\displaystyle\frac{1}{\bar{g}^{2}}+\frac{B_{F}}{2}\ln(\tilde{q}^{2})\ ,$
	$\displaystyle B_{F}=$	$\displaystyle\frac{1}{16\pi^{2}}\big{(}33-\frac{8N_{\text{f}}}{3}\big{)}\ .$		(10)

The confinement scale can be identified with the momentum for which the one-loop approximation to the running gauge coupling diverges

$$\Lambda_{\text{QCD}}^{2}=\chi^{2}\exp\left(-\frac{2}{B_{F}\bar{g}^{2}}\right)\ ,$$

(11)

realizing eq. (7). We conclude that the limit $k\to 0$ of the scaling solution realizes the quantum scale invariant standard model.

We will next establish that for $\tilde{\rho}=\chi^{2}/k^{2}\gg 1$ the dependence of the gauge couplings on $\tilde{\rho}$ according to the scaling solution is given by the same perturbative $\beta$-functions as for the usual momentum dependence. These $\beta$-functions do not depend on the details of quantum gravity. Quantum gravity is only needed for the existence of the scaling solution. For large $\tilde{\rho}$ and small $\tilde{q}^{2}$ both the running of the gauge couplings with $\tilde{\rho}$ and with $\tilde{q}^{2}$ only involves the fluctuations of an effective particle theory below the Planck mass. This is typically the standard model of particle physics. If beyond standard model effects, for example in the sector of neutrino masses, can be omitted the fifth force mediated by the cosmon-atom coupling becomes calculable.

Within functional renormalization the flow equations for the $k$-dependence of coupling functions $g^{2}(q^{2},\chi^{2},k^{2})$ are evaluated at fixed $q^{2}$ and $\chi^{2}$

$$\partial_{t}g^{-2}=k\partial_{k}g^{-2}\big{|}_{\chi^{2},\tilde{q}^{2}}=\beta_{g^{-2}}\ .$$

(12)

For small $g^{2}$ and $k$ sufficiently below the Planck mass $M$ the $\beta$-function for QED can be approximated by the one-loop expression

$$\beta_{g^{-2}}=B_{F}\,l\left(\frac{q^{2}}{k^{2}},\frac{m_{\text{e}}^{2}}{k^{2}}\right)\ ,\quad B_{F}=-\frac{1}{6\pi^{2}}\ .$$

(13)

The “threshold function” $l$ accounts for the stop of the flow for $k^{2}\ll q^{2}$ or $k^{2}\ll m_{\text{e}}^{2}$, since both $q^{2}$ and $m_{\text{e}}^{2}$ constitute effective infrared cutoffs such that the additional cutoff $~{}\sim k^{2}$ no longer matters. We may approximate this decoupling of “heavy modes” from the flow by

$$l=\frac{k^{2}}{k^{2}+m_{\text{e}}^{2}+q^{2}/4}\ .$$

(14)

The solution

$$g^{-2}=g_{0}^{-2}+\frac{B_{F}}{2}\ln\left(\frac{k^{2}+m_{\text{e}}^{2}+q^{2}/4}{m_{\text{e}}^{2}}\right)$$

(15)

recovers eq. (3) for $k\to 0$. It is one of the advantages of functional renormalization that threshold functions as $l$ are produced automatically and extend directly to $\chi$-dependent masses as $m_{\text{e}}=h_{\text{e}}\chi$.

In the standard model the non-zero masses cut the flow of all couplings except for non-renormalizable couplings related to the masses of neutrinos. The flow of all renormalizable couplings effectively stops for $k$ below the electron mass. As a result, there is only a negligibly tiny difference between the limit $k\to 0$ and some small finite value, say $k\approx 10^{-3}\text{eV}$. Rather realistic cosmology for both inflation and dynamical dark energy obtains for a value of $k$ in the $10^{-3}\text{eV}$ range Wetterich (2022). We will take in the following this value. Since $k$ is the only scale appearing in the scaling solution its value is actually arbitrary, defining the mass units. Only the dimensionless ratio $\tilde{\rho}=\chi^{2}/k^{2}$ matters. Our units are chosen such that for a present ratio $\tilde{\rho}=10^{60}$ one has $\chi=2.44\cdot 10^{18}\text{GeV}$. In runaway cosmologies with ever increasing $\tilde{\rho}$ the huge value of $\tilde{\rho}$ is simply an effect of the large age of the universe in Planck units.

From the flow equation (12) at fixed $\chi$ we can switch to the equivalent equation at fixed $\tilde{\rho}$

$$\big{(}\partial_{t}-2\tilde{\rho}\partial_{\tilde{\rho}}\big{)}g^{-2}\big{|}_{\tilde{\rho},\tilde{q}^{2}}=\beta_{g^{-2}}\ .$$

(16)

The condition for the scaling solution is that the explicit $k$-dependence at fixed $\tilde{\rho}$ vanishes. The condition $\partial_{t}g^{-2}|_{\tilde{\rho},\tilde{q}^{2}}=0$ yields the defining differential equation for the scaling solution.

$$\tilde{\rho}\partial_{\tilde{\rho}}g^{-2}(\tilde{\rho})=-\frac{1}{2}\beta_{g^{-2}}\ .$$

(17)

For $\tilde{\rho}\gg 1$, $\tilde{q}^{2}\ll 1$ it only involves the standard model perturbative $\beta$-function.

In the following we mainly consider the limit $\tilde{q}^{2}\to 0$. The scaling solution for the field-dependent coupling $g^{2}(\tilde{\rho})$ has therefore to obey the non-linear differential equation (17) at $\tilde{q}^{2}=0$. Similar “scaling equations” have to be obeyed for a large coupled system of coupling functions. Away from thresholds, the flow generators ($\beta$-functions) depend only on the dimensionless couplings of the particles that are effectively massless in the corresponding range of $\tilde{\rho}$. Threshold functions similar to eq. (14) account for an effective decoupling if $k$ is smaller than the effective particle mass. For the example of the electron the decoupling takes place at $k^{2}<h_{e}^{2}\chi^{2}$ or $\tilde{\rho}>h_{e}^{-2}$.

As common for such systems of differential equations only few solutions may exist for the whole range $0\leq\tilde{\rho}<\infty$. It is at this point where quantum gravity can become selective, as we will discuss in the second part of this note. Nevertheless, for the flow of the scaling solution in the range $\tilde{\rho}\gg 1$ quantum gravity plays no role. Only the existence of the scaling solution is supposed.

Fundamental scale invariance Wetterich (2021a) postulates that the world is described precisely by a scaling solution, with all relevant parameters for a flow away from the scaling solution set to zero. The requirement of existence of the scaling solution can make this concept rather selective. The absence of relevant parameters makes this scenario highly predictive.

A theory with fundamental scale invariance can be described in terms of scale invariant fields such that the scale $k$ never appears. This holds including the continuum limit. In our case the scale invariant fields are $\tilde{\chi}=\chi/k$ and $\tilde{g}_{\mu\nu}=k^{2}g_{\mu\nu}$.

We may alternatively consider a scenario where the first substantial flow away from the scaling solution would occur at some scale $k_{0}$. There will be not much difference to the exact scaling solution if we set for the latter $k=k_{0}$. Evaluating the exact scaling solution at a fixed $k$, say $k=2\cdot 10^{-3}\text{eV}$, can alternatively be interpreted as a deviation from the scaling solution at $k_{0}=2\cdot 10^{-3}\text{eV}$ due to a relevant parameter. In both cases $k$ or $k_{0}$ only fix the units.

As compared to the scaling solution the relevant parameters can add to quantities with dimension $\text{mass}^{N}$ a contribution $\sim k_{c}^{N}$. For the gauge couplings the effect of the running from $k_{c}$ to zero can be respected.

With $\chi$-dependent particle masses as $m_{\text{e}}=h_{\text{e}}\chi$ and a $\chi$-dependent Planck mass $M=\chi$ one may expect the presence of a fifth force mediated by cosmon exchange. The issue for observational consequences is more subtle, however, since the mass ratio $m_{\text{e}}/M=h_{\text{e}}$ is independent of $\chi$. For a discussion of observations it is best to make a Weyl transformation to the Einstein frame. In the Einstein frame the Planck mass $M=\overline{M}$ is a constant which is introduced only by the variable transformation of the metric $g_{\mu\nu}^{\prime}=\big{(}\chi^{2}/\overline{M}^{2}\big{)}g_{\mu\nu}$. Also the particle masses are constant, $m_{\text{e}}=h_{\text{e}}\overline{M}$, $\Lambda_{\text{QCD}}=\eta_{\text{QCD}}\overline{M}$. Dimensionless couplings as $h_{\text{e}}$ or $g$ remain invariant under a Weyl scaling. The ratio $\tilde{q}^{2}$ does not change its value and is given in the Einstein frame by $\tilde{q}^{2}=q^{2}_{\text{E}}/\overline{M}^{2}$. (Note $q^{2}_{\text{E}}=q_{\mu}q_{\nu}{g^{\prime}}^{\mu\nu}=\big{(}\overline{M}^{2}/\chi^{2}\big{)}q^{2}=\overline{M}^{2}\tilde{q}^{2}$.)

Writing eq. (15) in the form

	$\displaystyle g^{-2}=$	$\displaystyle g_{0}^{-2}+\frac{B_{F}}{2}\ln\left(1+\frac{1}{h_{\text{e}}^{2}\tilde{\rho}}+\frac{\tilde{q}^{2}}{4h_{\text{e}}^{2}}\right)$
	$\displaystyle=$	$\displaystyle g_{0}^{-2}+\frac{B_{F}}{2}\ln\left(1+\frac{\overline{M}^{2}k^{2}}{m_{\text{e}}^{2}\chi^{2}}+\frac{q_{\text{E}}^{2}}{4m_{\text{e}}^{2}}\right)\ ,$		(18)

a dependence on $\chi$ remains only for $k\neq 0$. (The second equation uses the fixed value $m_{\text{e}}=h_{\text{e}}\overline{M}$.) For the exact quantum scale invariant standard model for $k=0$ only the derivative couplings of a Goldstone boson remain.

For a present value of $\chi$ equal to $\overline{M}$ the cosmon coupling to atoms is suppressed by a tiny factor $k^{2}/m_{\text{e}}^{2}\approx 10^{-17}$. The quantum scale invariant standard model is a very good approximation, predicting the absence of non-derivative atom cosmon couplings. Our scenario predicts with high precision the absence of an apparent violation of the equivalence principle or a time variation of fundamental couplings in the present cosmological epoch. The only loophole could be a violation of quantum scale symmetry in the beyond standard model sector, related to an effective dependence of the ratio neutrino mass over electron mass, $m_{\nu}/m_{e}$, on $\tilde{\rho}$ Wetterich (2019).

In earlier epochs of cosmology when $\tilde{\rho}$ assumed smaller values the time variation of $g^{2}$ with varying $\tilde{\rho}$ and the corresponding fifth force are more substantial. This happens for $\tilde{\rho}<h_{\text{e}}^{-2}$ Wetterich (2022). In this range the difference of the fine structure constant from the present value is given according to eq. (III) by

	$\displaystyle\Delta\alpha$	$\displaystyle=\alpha(\tilde{\rho})-\alpha_{0}=\Big{\{}\big{[}1+\frac{\alpha_{0}}{3\pi}\ln\big{(}h_{e}^{2}\tilde{\rho}\big{)}\big{]}^{-1}-1\Big{\}}\alpha_{0}$
		$\displaystyle\approx-\frac{\alpha_{0}^{2}}{3\pi}\ln\big{(}h_{e}^{2}\tilde{\rho}\big{)}\ .$		(19)

For an estimate of $\tilde{\rho}$ in the radiation dominated cosmological epoch we observe that according to the scaling solution the scalar potential divided by the fourth power of the effective Planck mass obeys for $\chi\to\infty$ Pawlowski et al. (2019)

$$\frac{U}{M^{4}}=\frac{u_{\infty}k^{4}}{\chi^{4}}\ ,\quad u_{\infty}=\frac{5}{256\pi^{2}}\ .$$

(20)

This ratio is the same in all metric frames. Eq. (20) determines the order of magnitude also for finite large $\chi/k$ such that we can employ eq. (20) replacing $\mu_{\infty}\to d_{u}\mu_{\infty}$ with $d_{u}$ of the order one. This also includes the case of a relevant parameter which adds to $U/M^{4}$ a part $\lambda k^{4}/\chi^{4}$ with $\lambda$ of the order $u_{\infty}\,$.

Let us assume that in the radiation dominated era cosmology one has a small fraction $\Omega_{e}$ of early dark energy. This implies in the Einstein frame

$$U=f_{e}\Omega_{e}\rho_{c}=3f_{e}\Omega_{e}M^{2}H^{2}=f_{e}\Omega_{e}c_{T}T^{4}\ ,$$

(21)

with $f_{e}=(1-w_{e})/2$ involving the equation of state $w_{e}$ of early dark energy. For a cosmic scaling solution where dark energy assumes a constant fraction of the radiation energy density one has $f_{e}=1/3$. If the kinetic energy of the scalar is negligible one finds $f_{e}=1$. The critical energy density $\rho_{c}$ is related to the temperature $T$ by $\rho_{c}=c_{T}T^{4}$. Eq. (20) results in

$$\tilde{\rho}=\left(\frac{d_{u}u_{\infty}}{f_{e}\Omega_{e}c_{T}}\right)^{\frac{1}{2}}\frac{M^{2}}{T^{2}}\ .$$

(22)

For large enough $T$ eq. (III) implies a relative change of the fine structure constant

$$\frac{\Delta\alpha}{\alpha}=\frac{2\alpha}{3\pi}\bigg{[}\ln\left(\frac{T}{m_{e}}\right)+\frac{1}{4}\ln\left(\frac{f_{e}\Omega_{e}c_{T}}{d_{u}u_{\infty}}\right)\bigg{]}\ .$$

(23)

This formula becomes valid if the square bracket exceeds one. In this range $\Delta\alpha/\alpha$ exceeds a value $2\alpha/(3\pi)\approx 1.55\cdot 10^{-3}$. For smaller $T$ the relative change of $\alpha$ is reduced according to

	$\displaystyle\frac{\Delta\alpha}{\alpha}$	$\displaystyle=\frac{\alpha}{3\pi}\ln\left(1+\sqrt{\frac{f_{e}\Omega_{e}c_{T}}{d_{u}u_{\infty}}}\frac{T^{2}}{m_{e}^{2}}\right)$
		$\displaystyle\approx 7.7\cdot 10^{-4}\ln\left(1+12.9-\sqrt{\frac{f_{e}\Omega_{e}g_{\text{eff}}}{d_{u}}}\frac{T^{2}}{m_{e}^{2}}\right)\ .$		(24)

Here we employ eq. (20) for $u_{\infty}$ and $c_{T}=(\pi^{2}/30)g_{\text{eff}}$, with $g_{\text{eff}}=29/4$ for $T\lesssim m_{e}$ and $g_{\text{eff}}=43/4$ for $T\gtrsim m_{e}$. To our knowledge eq. (III) is the first example of a quantitative estimate of the variation of the fine structure constant from a well defined setting of a fundamental theory.

Nucleosynthesis occurs in a range where $T/m_{e}\approx 0.2$. For $f_{e}/d_{u}\approx 1$ this yields in the range relevant for nucleosynthesis the approximate value ($g_{\text{eff}}=29/4$)

$$\frac{\Delta\alpha}{\alpha}\approx 10^{-3}\sqrt{\Omega_{e}}\left(\frac{5T}{m_{e}}\right)^{2}\ .$$

(25)

There are severe upper bounds on $\Omega_{e}\$. For a cosmic scaling solution one expects $\Omega_{e}$ to be similar to later stages of radiation domination, where one finds $\Omega_{e}<10^{-2}$ Gómez-Valent et al. (2021). Direct bounds from the effective number of massless particles during nucleosynthesis are similar, albeit slightly weaker. If the scalar field has settled to a value for which the potential becomes relevant only after nucleosynthesis the value of $\Omega_{e}$ is smaller than for the cosmic scaling solution. We conclude $\Delta\alpha/\alpha\lesssim 10^{-4}$, unless a kination epoch ends only very close to nucleosynthesis.

A variation of the fine structure constant affects the relative element abundances $Y_{i}$ obtained from primordial nucleosynthesis according to

$$\frac{\Delta Y_{i}}{Y_{i}}=c_{i}\frac{\Delta\alpha}{\alpha}\ ,$$

(26)

with $c_{i}=(3.6,1.9,-11)$ for $i=(\text{D},{}^{4}\text{He},{}^{7}\text{Li})$ Dent et al. (2007). A variation $\Delta\alpha/\alpha\lesssim 10^{-4}$ seems to be too small for an observable modification of the primordial element abundances. Due to the strong dependence of $\Delta\alpha/\alpha$ on $T$ a more detailed quantitative estimate would need to follow explicitly the time-variation of $\alpha$ during nucleosynthesis. Similar considerations can be made for the field dependence of other couplings, as the Yukawa coupling of the electron, which affects the ratio of electron to proton mass.

So far we have assumed the existence of the scaling solution for $\tilde{\rho}>\tilde{\rho}_{\text{c}}$, with $\tilde{\rho}_{\text{c}}\gg 1$ such that the metric fluctuations in quantum gravity can be neglected. The question is what happens for $\tilde{\rho}<\tilde{\rho}_{\text{c}}$? The answer needs an extension of the flow equation to quantum gravity. Functional renormalization has found for the flow of the gauge coupling without a scalar field a gravity-induced anomalous dimension $B_{g}$ Daum et al. (2010); Folkerts et al. (2012); Harst and Reuter (2011); Christiansen and Eichhorn (2017); Christiansen et al. (2018); Eichhorn and Versteegen (2018); Daum et al. (2011); de Brito and Eichhorn (2022); Eichhorn and Held (2022) see also Robinson and Wilczek (2006); Pietrykowski (2007); Toms (2007); Ebert et al. (2008); Tang and Wu (2010); Toms (2010),

$$\partial_{t}g^{2}=-B_{g}g^{2}-B_{F}g^{4}\ ,$$

(27)

with

$$B_{g}=2C_{g}\frac{k^{2}}{M^{2}(k)}=\frac{C_{g}}{w}\ ,\quad w=\frac{M^{2}(k)}{2k^{2}}\ .$$

(28)

More details will be presented in later parts of this note. For the UV-fixed point $w$ approaches a constant $w_{*}$ and therefore $B_{g}$ becomes constant. On the other hand, the gravitational contribution decreases rapidly for $k^{2}\ll M^{2}(k)$.

The flow equation (27) is easily extended to the presence of a scalar field by having $M^{2}$ depending on $\chi^{2}$ and on $k$. A typical form of the scaling solution for $w$ is given by Henz et al. (2013, 2017); Wetterich and Yamada (2019)

$$w=w_{0}+\frac{1}{2}\tilde{\rho}\ ,$$

(29)

where we have taken a normalization of the scalar field such that for $\chi\to\infty$ one has $M^{2}=\chi^{2}$. According to eq. (17) the scaling solution for the gauge coupling has to obey

$$\tilde{\rho}\partial_{\tilde{\rho}}g^{2}=\frac{C_{g}g^{2}}{2w_{0}+\tilde{\rho}}+\frac{B_{F}g^{4}}{2}\ .$$

(30)

The UV-limit corresponds to $\tilde{\rho}\to 0$, or $k\to\infty$ at fixed $\chi$. For the UV-completion of the scaling solution we therefore need to consider the region where $\tilde{\rho}$ can be neglected in the denominator of the first term in eq. (30). For this regime both $B_{g}=C_{g}/w_{0}$ and $B_{F}$ are constant. The constant $B_{F}$ arises from the fluctuations of the gauge bosons and charged particles. It includes contributions from all particles that are effectively massless in the transplanckian regime for $\tilde{\rho}\to 0$. Typically one expects massless particles beyond the ones present in the standard model as an effective low energy theory.

Depending on the sign of $B_{g}$ and $B_{F}$ we distinguish four scenarios:

1. (i)

   $B_{g}>0$, $B_{F}>0$: In this case the gauge coupling is asymptotically free. It reaches zero for $\tilde{\rho}\to 0$. A family of scaling solutions can be characterized by the value

   $$g^{2}(\tilde{\rho}_{d})=g^{2}_{d}\ ,\quad\tilde{\rho}_{d}\ll 1\ .$$

   (31)

   Here $\tilde{\rho}_{d}$ replaces the usual renormalization scale in case of a scaling solution. Its value can be chosen freely as long as $\tilde{\rho}_{d}\ll 1$. For increasing $\tilde{\rho}>\tilde{\rho}_{d}$ the gauge coupling increases. For the gauge bosons of the standard model the value of $g_{d}^{2}$ has to be small enough such that at the decoupling of the gravity fluctuations for $\tilde{\rho}\approx 2w_{0}$ the gauge couplings are not too large. Observation requires $g^{2}(\tilde{\rho}=20w_{0})\approx 0.3$. If $g_{d}^{2}$ remains below the resulting upper bound, the flow can be continued from $\tilde{\rho}=20w_{0}$ towards large $\tilde{\rho}$, matching the discussion above for $\tilde{\rho}\gg 2w_{0}$ where only the part $\sim B_{F}$ contributes to the flow.

   At this level there are continuous families of scaling solutions parametrized by different values of $g_{d}^{2}$. Obtaining phenomenologically acceptable values of $g^{2}(\tilde{\rho}=20w_{0})$ does not require relevant parameters for deviations from the scaling solution. One rather has to pick one out of the family of scaling solutions. What is not guaranteed, however, is that all members of the family of scaling solutions result in a viable scaling solution for the coupled system of many coupling functions.

2. (ii)

   $B_{g}>0$, $B_{F}<0$: In this case the gauge coupling remains asymptotically free. Its flow exhibits now an infrared stable fixed point

   $$g_{*}^{2}=-\frac{B_{g}}{B_{F}}\ .$$

   (32)

   For $g_{d}^{2}<g_{*}^{2}$ the gauge coupling $g^{2}$ increases for increasing $\tilde{\rho}$, approaching rapidly the value $g_{*}^{2}$. On the other hand, for $g_{d}^{2}>g_{*}^{2}$ the gauge coupling decreases towards $g_{*}^{2}$. For a consistent scaling solution we should be able to choose $\tilde{\rho}_{d}$ arbitrarily close to one. This implies a bound

   $$0\leq g^{2}(\tilde{\rho}=2w_{0})\lesssim g_{*}^{2}\ .$$

   (33)

   For small enough $\tilde{\rho}_{d}$ arbitrarily large values of $g_{d}^{2}$ are mapped at $\tilde{\rho}=2w_{0}$ to the fixed point value $g^{2}(2w_{0})\approx g_{*}^{2}$. This follows from the dominance of the term $B_{F}$ for large $g^{2}$, according to

   $$\tilde{\rho}\partial_{\tilde{\rho}}g^{-2}=B_{g}g^{-2}+B_{F}\ .$$

   (34)

   The restricted range (33) is a first example for the predictivity of scaling solutions. The infrared stable fixed point for $B_{g}>0$, $B_{F}<0$ has been discussed for abelian gauge theories in ref. Daum et al. (2010), and for grand unified theories in refs. Eichhorn et al. (2020, 2018).

3. (iii)

   $B_{g}<0$, $B_{F}>0$: For this case the gauge coupling is asymptotically safe instead of asymptotically free. At the UV-fixed point the gauge couplings differ from zero, taking the value (32). The fixed point (32) becomes infrared unstable. Unless $g_{d}^{2}$ is chosen very close to $g_{*}^{2}$ the coupling $g^{2}(\tilde{\rho}=2w_{0})$ is either zero or it diverges before $\tilde{\rho}=2w_{0}$ is reached. Both cases are unacceptable. Only the possibility of tuning $g_{d}^{2}$ very close to $g_{*}^{2}$ remains for very small $\tilde{\rho}_{d}$.

4. (iv)

   $B_{g}<0$, $B_{F}<0$: For this situation the theory is trivial in the gauge sector. Any positive coupling $g_{d}^{-2}\geq 0$ at $\tilde{\rho}_{d}\to 0$ is mapped to $g^{2}(\tilde{\rho}=2w_{0})=0$. This prediction of the scaling solution is not compatible with observation.

From the point of view of consistency the requirement of existence of the scaling solution leads to predictions in two cases: $g^{2}(2w_{0})<g_{*}^{2}$ for case (ii), and $g^{2}(2w_{0})=0$ for case (iv). Further predictivity is gained if we insist on avoiding tuning of parameters. Assume at $\tilde{\rho}_{d}\ll 1$ some arbitrary value $g_{d}^{2}$ not extremely close to zero and not extremely close to $g_{*}^{2}$. For a sizable $B_{g}$ this value changes quickly with $\tilde{\rho}$. For small $g^{2}$ it obeys

$$g^{2}(\tilde{\rho})=g_{d}^{2}\left(\frac{\tilde{\rho}}{\tilde{\rho}_{d}}\right)^{B_{g}/2}\ ,$$

(35)

while in the presence of a second fixed point $g_{*}^{2}$ one has for small $g^{2}-g_{*}^{2}$

$$g^{2}(\tilde{\rho})=g_{*}^{2}+\big{(}g_{d}^{2}-g_{*}^{2}\big{)}\left(\frac{\tilde{\rho}}{\tilde{\rho}_{d}}\right)^{-B_{g}/2}\ .$$

(36)

Any value of $g_{d}^{2}$ away from the fixed point is driven rapidly towards the infrared stable fixed point, $g_{*}^{2}=-B_{g}/B_{F}$ for case (ii) or $g_{*}^{2}=0$ for the cases (iii) and (iv). For case (i), as well as for the case (iii) with $g_{d}^{2}>g_{*}^{2}$, the gauge coupling diverges before $\tilde{\rho}$ reaches the value $2w_{0}$.

We conclude that a particularly robust situation arises for case (iii) with a predicted value

$$g^{2}(\tilde{\rho}=2w_{0})=-\frac{B_{g}}{B_{F}}\ .$$

(37)

Comparison of this prediction with the value $g^{2}\approx 0.3$ compatible with observation requires knowledge of $B_{g}$ and $B_{F}$. The constant $B_{F}$ can be determined by the usual one-loop formula in the absence of gravity. It involves the number of massless gauge bosons, fermions and scalars for transplanckian physics. If $B_{g}$ is computed reliably, compatibility with observation restricts the particle content of transplanckian physics.

Previous computations have mainly found positive $B_{g}$. The issue of the sign is complex, however, since two diagrams contribute with opposite sign. Typically, one also finds parameter ranges for which $B_{g}$ is negative. The results obtained so far show substantial differences both for the absolute magnitude of $B_{g}$ and the boundary in parameter space separating positive and negative $B_{g}$. In view of the importance of $B_{g}$ for restricting the transplanckian particle content we compute this quantity in the next part of this note by employing the gauge invariant formulation of the functional flow equation Wetterich (2016). This formulation separates clearly between physical fluctuations and gauge fluctuations. Furthermore, the contribution of the gauge fluctuations combined with the regularized Faddeev-Popov determinant or ghost sector yields a universal measure contribution which does not depend on the form of the effective average action $\Gamma_{k}$. A clear separation of the different modes provides for a simple physical picture of the flow and is expected to be particularly robust.

The exact flow equationWetterich (1993); Reuter and Wetterich (1994),

$$k\partial_{k}\Gamma_{k}=\frac{1}{2}\mathrm{tr}\Big{\{}\big{(}\Gamma_{k}^{(2)}+R_{k}\bigr{)}^{\!-1}k\partial_{k}R_{k}\Big{\}}-\delta_{k}\ ,$$

(40)

involves the second functional derivative $\Gamma_{k}^{(2)}$ of the effective average action, and $R_{k}$ is a suitable infrared cutoff function. For the gauge invariant formulation of the flow equation Wetterich (2016) the first term in eq. (40) involves only the contributions from the physical fluctuations. The appropriate projection is realized by a particular “physical gauge fixing”, and $R_{k}$ acts only on the physical fluctuations. In this formulation $\delta_{k}$ is a measure contribution with a similar structure as the first term, see below. The flow of $Z=g^{-2}$ can be extracted by evaluating $\partial_{k}\Gamma_{k}$ for non-zero gauge fields and a flat space metric.

The flow equation depends on the field-dependent propagator for the physical fluctuations in the presence of the cutoff, $(\Gamma_{k}^{(2)}+R_{k})^{-1}$. Since it involves the second functional derivative of the effective average action eq. (40) is a functional differential equation. The approximation consists in evaluating $\Gamma_{k}^{(2)}$ for the particular truncation (38). For the computation of $\Gamma_{k}^{(2)}$ for field configurations with non-zero gauge fields $\bar{A}_{\mu}^{z}$ and flat geometry we expand

$$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\;,\quad A_{\mu}^{z}=\bar{A}_{\mu}^{z}+a_{\mu}^{z}\ .$$

(41)

We work in euclidean space, $\eta_{\mu\nu}=\delta_{\mu\nu}$, and may continue analytically to Minkowski space at the end. We also consider a scalar field $\chi=\overline{\chi}+\delta\chi$. Since we do not compute here the effect of scalar fluctuations we can omit $\delta\chi$ and simply consider constant $\chi$.

For an evaluation of the inverse propagator $\Gamma_{k}^{(2)}$ in the presence of the gauge fields $\bar{A}_{\mu}^{z}$ we expand the truncation (38) in quadratic order in $a_{\mu}^{z}$ and $h_{\mu\nu}$

$$\Gamma_{k,2}=\Gamma_{a}+\Gamma_{h}+\Gamma_{hg}+\Gamma_{F1}+\Gamma_{F2}\ .$$

(42)

The term quadratic in the gauge field fluctuations

$$\Gamma_{a}=\int_{x}\frac{Z}{2}a_{\mu}^{z}(\mathcal{D}_{T})_{zy}^{\mu\nu}a_{\nu}^{y}\ ,$$

(43)

involves the operator

$$(\mathcal{D}_{T})_{zy}^{\mu\nu}=-(D^{\rho}D_{\rho})_{zy}\eta^{\mu\nu}+(D^{\mu}D^{\nu})_{zy}+2f^{u}{}_{zy}F_{u}^{\mu\nu}\ ,$$

(44)

with

$$D_{\mu}a_{\nu}^{z}=\partial_{\mu}a_{\nu}^{z}+f_{uv}{}^{z}A_{\mu}^{u}a_{\nu}^{v}\ .$$

(45)

Here and in the following we omit the bar on the “background gauge fields”.

For the terms quadratic in the metric fluctuations one has ($h=h_{\mu}^{\mu}\,$, $\,\partial^{2}=\partial^{\mu}\partial_{\mu}$)

$$\Gamma_{h}=\frac{1}{8}\int_{x}\Big{\{}-M^{2}(h^{\mu\nu}\partial^{2}h_{\mu\nu}-h\partial^{2}h)+V(h^{2}-2h_{\mu\nu}h^{\mu\nu})\Big{\}}\ ,$$

(46)

and

$$\Gamma_{hg}=-\frac{1}{4}\int_{x}M^{2}(\partial_{\rho}h_{\nu}^{\rho}-\partial_{\nu}h)\partial_{\mu}h^{\mu\nu}\ .$$

(47)

The first term $\Gamma_{h}$ concerns the physical metric fluctuations, while the second $\Gamma_{hg}$ involves the gauge fluctuations.

Furthermore, there are contributions $\sim F^{2}$

	$\displaystyle\Gamma_{F2}=\int_{x}$	$\displaystyle\frac{Z}{4}\Big{[}F_{\mu\nu}^{z}F_{z\rho\sigma}h^{\mu\rho}h^{\nu\sigma}$
		$\displaystyle+F_{\mu\nu}^{z}F_{z\rho}{}^{\nu}(2h_{\tau}^{\mu}h^{\tau\rho}-hh^{\mu\rho})$
		$\displaystyle+\frac{1}{8}F_{\mu\nu}^{z}F_{z}^{\mu\nu}(h^{2}-2h_{\rho\tau}h^{\rho\tau})\Big{]}\ .$		(48)

Finally, $\Gamma_{F1}$ mixes the gauge field and metric fluctuations

$$\Gamma_{F1}=\int_{x}\frac{Z}{4}(D_{\mu}a_{\nu}^{z}-D_{\nu}a_{\mu}^{z})(hF_{z}^{\mu\nu}-4h^{\mu\rho}F_{z\rho}{}^{\nu})\ .$$

(49)

We consider a restricted set of “background field strengths” $F_{\mu\nu}$ for which the covariant derivative vanishes

$$D_{\rho}F_{\mu\nu}^{z}=0\ .$$

(50)

This simplifies the mixing term, using partial integration

$$\Gamma_{F1}=\int_{x}ZF_{z}^{\mu\rho}\Big{(}\partial_{\mu}h_{\rho}^{\nu}-\frac{1}{2}\partial_{\mu}h\delta_{\rho}^{\nu}\Big{)}a_{\nu}^{z}\ .$$

(51)

For an extraction of the flow of $Z$ it is sufficient to evaluate the flow of the effective action for constant gauge field strength $F_{z}^{\mu\nu}$. This will further simplify the flow equation.

For the setting of the gauge invariant flow equation Wetterich (2016) we split the metric fluctuations into physical fluctuations and gauge fluctuations Wetterich (2017)

$$h_{\mu\nu}=t_{\mu\nu}+\frac{1}{3}\tilde{P}_{\mu\nu}\sigma+a_{\mu\nu}\ .$$

(52)

In momentum space the physical graviton fluctuation is traceless and transverse, and the physical scalar fluctuation $\sigma$ in the metric is multiplied by a projector $\tilde{P}_{\mu\nu}$,

$$q^{\nu}t_{\mu\nu}=0\;,\quad t_{\mu}^{\mu}=0\;,\quad\tilde{P}_{\mu\nu}=\eta_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\ .$$

(53)

The gauge fluctuations of the metric $a_{\mu\nu}$ are treated separately. Their contribution, together with the regularized Faddeev-Popov determinant or associated ghost fluctuations, yields the universal measure contribution in the functional flow equation. The same holds for the longitudinal gauge field fluctuations which correspond to the gauge degrees of freedom of the local gauge symmetry. The combined measure contributions are denoted by $\delta_{k}$ in eq (40).

The measure contribution for the metric does not depend on the gauge fields. It therefore does not contribute to the flow equation for $Z$. The measure contribution for the gauge fields is the standard one in flat space Reuter and Wetterich (1994); Wetterich (2018). The simple separate treatment of the measure contribution is an important technical advantage of the gauge invariant formulation of the functional flow equation. In our truncation an identical result can be obtained in a standard background gauge fixing procedure with ghosts, using a “physical gauge fixing”. For the gauge fields this is Landau gauge, while for the metric the gauge fixing term $(D^{\mu}h_{\mu\nu})^{2}$ is multiplied by a coefficient that is taken to infinity. As a result, the propagator matrix becomes block diagonal in the physical and gauge fluctuations. This gauge fixing operates an effective projection of the propagator on the physical degrees of freedom. For the part of the physical fluctuations we can simply impose the “physical gauge fixing”

$$\partial^{\mu}h_{\mu\nu}=0\;,\quad D^{\mu}a_{\mu}^{z}=0\ .$$

(54)

As a result, $\Gamma_{hg}$ in eq. (47) can be omitted.

In the sector of physical fluctuations we can write the matrix $\Gamma_{k}^{(2)}+R_{k}$ in block form

$$\Gamma_{k}^{(2)}+R_{k}=\begin{pmatrix}P^{(h)}+P^{(h,F)}&,&P^{(ha)}\\ P^{(ha)\dagger}&,&P^{(a)}\end{pmatrix}\ .$$

(55)

In momentum space one has for the physical metric fluctuations

	$\displaystyle P_{\mu\nu\rho\sigma}^{(h)}=\frac{M^{2}}{4}\bigg{[}$	$\displaystyle\Big{(}q^{2}-\frac{2V}{M^{2}}\Big{)}P_{\mu\nu\rho\sigma}^{(t)}$
		$\displaystyle-2\Big{(}q^{2}-\frac{V}{2M^{2}}\Big{)}P_{\mu\nu\rho\sigma}^{(\sigma)}+\tilde{R}_{k,\mu\nu\rho\sigma}^{(h)}\bigg{]}\ ,$		(56)

where the projectors on the graviton fluctuations and scalar metric fluctuations read

	$\displaystyle\widehat{P}_{\mu\nu\rho\sigma}^{(t)}$	$\displaystyle=\frac{1}{2}(\tilde{P}_{\mu\rho}\tilde{P}_{\nu\sigma}+\tilde{P}_{\mu\sigma}\tilde{P}_{\nu\rho})-\frac{1}{3}\tilde{P}_{\mu\nu}\tilde{P}_{\rho\sigma}\ ,$
	$\displaystyle\widehat{P}_{\mu\nu\rho\sigma}^{(\sigma)}$	$\displaystyle=\frac{1}{3}\tilde{P}_{\mu\nu}\tilde{P}_{\rho\sigma}\;,\quad\widehat{P}^{(t)}\widehat{P}^{(\sigma)}=0\ .$		(57)

The projector on the physical metric fluctuations reads

$$\widehat{P}_{\mu\nu\rho\sigma}^{(f)}=\widehat{P}_{\mu\nu\rho\sigma}^{(t)}+\widehat{P}_{\mu\nu\rho\sigma}^{(\sigma)}.$$

(58)

For the transverse gauge boson fluctuation block one has

$$P_{\;\;zy}^{(a)\mu\nu}=Z\Big{\{}(\mathcal{D}_{T})_{zy}^{\mu\nu}+\tilde{R}_{k\,,\;\,zy}^{(a)\,\mu\nu}\Big{\}}\ .$$

(59)

The off-diagonal term is linear in $F_{\mu\nu}$. For constant $F_{\mu\nu}$ it reads in momentum space

$$P_{\,z}^{(ha)\mu\nu,\rho}=-\frac{i}{2}q_{\sigma}ZF_{z}{}^{\sigma\tau}\big{(}\delta_{\tau}^{\mu}\eta^{\nu\rho}+\delta_{\tau}^{\nu}\eta^{\mu\rho}-\delta_{\tau}^{\rho}\eta^{\mu\nu}\Big{)}\ .$$

(60)

Here we have assumed that the infrared cutoff $R_{k}$ is block diagonal. Finally, one has

	$\displaystyle P^{(hF)\mu\nu\rho\sigma}=$	$\displaystyle\frac{Z}{4}\Big{\{}F_{z}^{\mu\rho}F^{z\nu\sigma}+F_{z}{}^{\mu\sigma}F^{z\nu\rho}$
		$\displaystyle+F_{z}{}^{\mu}{}_{\tau}F^{z\sigma\tau}\eta^{\nu\rho}+F_{z}^{\nu}{}_{\tau}F^{z\sigma\tau}\eta^{\mu\rho}$
		$\displaystyle+F_{z}{}^{\mu}{}_{\tau}F^{z\rho\tau}\eta^{\nu\sigma}+F_{z}{}^{\nu}{}_{\tau}F^{z\rho\tau}\eta^{\mu\sigma}$		(61)
		$\displaystyle-F_{z}^{\rho}{}_{\tau}F^{z\sigma\tau}\eta^{\mu\nu}-F_{z}{}^{\mu}{}_{\tau}F^{z\nu\tau}\eta^{\rho\sigma}$
		$\displaystyle+\frac{1}{4}F_{z}{}^{\tau\lambda}{}F^{z\tau\lambda}\big{(}\eta^{\mu\nu}\eta^{\rho\sigma}-\eta^{\mu\rho}\eta^{\nu\sigma}-\eta^{\mu\sigma}\eta^{\nu\rho}\big{)}\Big{\}}\ .$