Running couplings from adiabatic regularization

TL;DR

This paper introduces an extended adiabatic regularization method with an arbitrary mass scale to derive the running of coupling constants, including electric charge, cosmological constant, and gravitational constant, revealing new non-logarithmic effects.

Contribution

The authors develop a robust extension of adiabatic regularization incorporating a mass scale, enabling the derivation of running coupling constants with novel non-logarithmic contributions.

Findings

Derived the running of electric charge in QED.

Renormalized the cosmological and gravitational constants.

Identified new non-logarithmic contributions to running.

Abstract

We extend the adiabatic regularization method by introducing an arbitrary mass scale $\mu$ in the construction of the subtraction terms. This allows us to obtain, in a very robust way, the running of the coupling constants by demanding $\mu$-invariance of the effective semiclassical (Maxwell-Einstein) equations. In particular, we get the running of the electric charge of perturbative quantum electrodynamics. Furthermore, the method brings about a renormalization of the cosmological constant and the Newtonian gravitational constant. The running obtained for these dimensionful coupling constants has new relevant (non-logarithmic) contributions, not predicted by dimensional regularization.

Equations

$$\varphi =\frac{1}{\sqrt{2(2\pi a(t){)}^3}}\int d^{3}k[A_{\vec{k}}{e}^{i\vec{k}\vec{x}}{h}_{\vec{k}}(t)+B_{\vec{k}}^{†}{e}^{-i\vec{k}\vec{x}}{h}_{-\vec{k}}^{\ast }(t)],$$

$${\ddot{h}}_{\vec{k}}+(\frac{(\vec{k}-\vec{A}{)}^2}{a^2}+m^2+\sigma )h_{\vec{k}}=0,$$

$$h_{\vec{k}}\sim \frac{1}{\sqrt{W_{\vec{k}}(t)}}{e}^{-i{\int }^t{W}_{\vec{k}}(t^{\prime })\mathrm{d}{t}^{\prime }}.$$

$$\displaystyle\langle\vec{j}\rangle=\int d^{3}k\vec{j}(\vec{k},t)=\frac{1}{(2\pi)^{3}a^{5}}\int d^{3}k(\vec{k}+\vec{A}(t))|h_{\vec{k}}(t)|^{2}\ .\ \ \ \$$

$$⟨\vec{j}{⟩}_{ren}$$

$${\omega }_{\vec{k}}^{(0)}\equiv \omega =\sqrt{{\vec{k}}^2/a^2+{\mu }^2},$$

$${\omega }^{(1)}$$

$${\omega }^{(2)}$$

$${\omega }^{(3)}$$

$$⟨j^{\beta }{⟩}_{\mathrm{ren}}(\mu )-⟨j^{\beta }{⟩}_{\mathrm{ren}}({\mu }_0)=-2{\delta }_q{\nabla }_{\alpha }{F}^{\alpha \beta },$$

$$\frac{1}{q^2(\mu )}{\nabla }_{\alpha }{F}^{\alpha \beta }=⟨j^{\beta }{⟩}_{ren}(\mu ).$$

$$\frac{1}{q^2({\mu }_0)}{\nabla }_{\alpha }{F}^{\alpha \beta }=⟨j^{\beta }{⟩}_{ren}({\mu }_0).$$

$$\frac{1}{q^2(\mu )}-\frac{1}{q^2({\mu }_0)}=-\frac{1}{48{\pi }^2}\ln \frac{{\mu }^2}{{\mu }_0^2},$$

$$⟨T_{\alpha \beta }{⟩}_{\mathrm{ren}}(\mu )-$$

$${\delta }_G=\frac{-1}{(8\pi {)}^2}\frac{4}{3}\left(1-6\xi \right)(m^2-{\mu }^2-2m^2\ln \frac{m}{\mu })$$

$${\delta }_{{\alpha }^1}=\frac{-1}{(8\pi {)}^2}\frac{2}{9}{\left(1-6\xi \right)}^2\ln \frac{m}{\mu }$$

$${\delta }_{\Lambda }=\frac{1}{(8\pi {)}^2}(-3m^4+4m^2{\mu }^2-{\mu }^4-4m^4\ln \frac{\mu }{m})$$

$${\delta }_q=\frac{1}{3(4\pi {)}^2}\ln \frac{\mu }{m}.$$

$$⟨T_{\alpha \beta }{⟩}_{\mathrm{ren}}-⟨{\tilde{T}}_{\alpha \beta }{⟩}_{\mathrm{ren}}=$$

$$⟨T_{\alpha \beta }{⟩}_{ren}(\mu )+T_{\alpha \beta }^{EM}(\mu )=\frac{-G_{\alpha \beta }}{8\pi G(\mu )}-\Lambda (\mu )g_{\alpha \beta }-{\alpha }^1(\mu )H_{\alpha \beta }^{(1)},$$

$$\Lambda (\mu )-\Lambda ({\mu }_0)=\frac{1}{(8\pi {)}^2}({\mu }^4-{\mu }_0^4-4m^2({\mu }^2-{\mu }_0^2)+2m^4\ln \frac{{\mu }^2}{{\mu }_0^2})$$

$$\frac{1}{16\pi G(\mu )}-\frac{1}{16\pi G({\mu }_0)}=\frac{(\frac{1}{6}-\xi )}{(4\pi {)}^2}({\mu }_0^2-{\mu }^2+m^2\ln \frac{{\mu }^2}{{\mu }_0^2})$$

$${\alpha }^1(\mu )-{\alpha }^1({\mu }_0)=\frac{1}{(4\pi {)}^2}(\frac{1}{6}-\xi {)}^2\ln \frac{{\mu }^2}{{\mu }_0^2}.$$

$$G(\mu )=\frac{G({\mu }_0)}{1+aG({\mu }_0)({\mu }^2-{\mu }_0^2)}.$$

$$\Lambda (\mu )-\Lambda ({\mu }_0)=\frac{-1}{96{\pi }^2}[48m^3({\mu }_0-\mu )+36m^2({\mu }^2-{\mu }_0^2)$$

$$-16m({\mu }^3-{\mu }_0^3)+3({\mu }^4-{\mu }_o^4)+6m^4\ln \frac{{\mu }^2}{{\mu }_0^2}]$$

$$\frac{1}{16\pi G(\mu )}-\frac{1}{16\pi G({\mu }_0)}=$$

$$\frac{1}{32{\pi }^2}[4m(\mu -{\mu }_0)-({\mu }^2-{\mu }_0^2)-m^2\ln \frac{{\mu }^2}{{\mu }_0^2}],$$

$$(\frac{1}{q^{2}(m_{j})}+\frac{1}{6\pi^{2}}\ln\frac{m_{j}^{N-1}}{m_{1}m_{2}\hat{m_{j}}m_{N}})\nabla_{\alpha}F^{\alpha\beta}=\sum_{i}\langle j_{i}^{\beta}\rangle_{ren}(m_{i})\ ,\ \ \ \ \$$

$$[\frac{1}{8\pi G(m_j)}-f_j(m_k)]G_{\alpha \beta }+[\Lambda (m_j)-g_j(m_k)]g_{\alpha \beta }=$$

$$-\sum _{i=1}^N⟨T_{\alpha \beta }{⟩}_{ren}(m_i)$$

$$\bar{G}(k)=\frac{1}{(-k^2+m^2)}+\frac{(1/6-\xi )R}{(-k^2+m^2{)}^2}+....$$

$${\bar{G}}_{\mu }(k)=\frac{1}{(-k^2+{\mu }^2)}+\frac{({\mu }^2-m^2+(1/6-\xi )R)}{(-k^2+{\mu }^2{)}^2}+....$$

Full text

Keywords: Adiabatic renormalization, running couplings, semiclassical Maxwell-Einstein equations.

I Introduction

Quantum field theories are intrinsically plagued with ultraviolet divergences, which need to be first isolated with the help of a regularization method and then removed to produce finite results. This is the so-called renormalization procedure, which was developed to prove the perturbative consistency of quantum electrodynamics (QED) and led to very significant predictions, like the anomalous magnetic moment of the electron. Another remarkable prediction is the effective running of the electric charge, tied to vacuum polarization phenomena. The renormalization procedure typically introduces an arbitrary mass scale $\mu$, reflecting the inherent ambiguity of any scheme of renormalization Hooft2 ; Hooft1 . Demanding that the bare electric charge be independent of the renormalization scale $\mu$ leads to the effective running of the electric charge in QED: $e^{2}(\mu^{2})=e^{2}(\mu_{0}^{2})/(1-\frac{e^{2}(\mu_{0}^{2})}{12\pi^{2}}\ln\frac{\mu^{2}}{\mu_{0}^{2}})$. The choice of the reference scale $\mu_{0}$ at which one defines the gauge coupling will not influence physical predictions.

The paradigm of QED as a quantum field theory model was generalized in two directions. On the one hand, the abelian gauge symmetry of electrodynamics was generalized to non-abelian symmetries to construct a very successful model of the electroweak and strong interactions. On the other hand, quantum field theory in Minkowski space was extended to curved spacetime, and the particle creation phenomena by the changing metric of an expanding universe was discovered parker66 ; parker68 (see also parker-toms ; birrell-davies ; fulling ; Waldbook ; Hollands ). Pairs of particles are created out of the vacuum in a non-perturbative way. Physical consistency demands that this process should be compatible with the covariant conservation of the stress-energy tensor of the quantized matter field. However, the formal expression for $\langle T^{\mu\nu}\rangle$ is, as expected, afflicted by ultraviolet divergences (UV). General covariance strongly restricts the way one could construct the subtraction terms for renormalization. A physically well motivated method of regularization and renormalization, known as adiabatic regularization, was introduced in parker-fulling . It was based on the adiabatic definition of particle states in an expanding universe, obtained by solving the equation for the modes in the adiabatic limit, and using a WKB-type asymptotic expansion for scalar fields parker-toms ; fulling ; birrell-davies . The method has been widely used in many relevant applications in cosmology Birrell ; Bunch80 ; Anderson-Parker . The adiabatic regularization method can also be adapted to deal with a quantized field in time varying electric backgrounds Cooper1 , quantized Dirac fields in an expanding universe with a Yukawa coupling rio1 ; Yukawa ; eduardo , and quantized fields in the presence of a time-varying electric field in an expanding spacetime ferreiro-navarro ; pla . These generalizations are important to account for backreaction effects in the (non-perturbative) Schwinger pair creation phenomena Schwinger , in the lab Dunne ; Ritus and also in astrophysics and cosmology PR , and the particle creation producing the reheating of the universe after inflation KLS .

However, all the above improvements of the adiabatic method implicitly assume that the renormalization scale is fixed at the mass of the quantized field. In this work we show how an arbitrary renormalization mass scale $\mu$, playing a role somewhat similar to the conventional unit of mass $\mu$ of dimensional regularization Hooft1 ; Hooft2 , can be naturally incorporated into the adiabatic method. We also study the physical consequences of this novel proposal.

To simplify the discussion we first restrict our analysis to a charged scalar field living in a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime, where particle creation can also be induced by the presence of a background electric field. We will show how the adiabatic renormalization method predicts the running of the electric charge, exactly in the amount predicted by more conventional methods, such as dimensional regularization. Moreover, we will find significant corrections to the running of the cosmological constant, $\Lambda_{c}$, and the Newtonian gravitational constant, $G$. The running obtained for these couplings has new relevant contributions, not predicted by dimensional regularization. They are linked to the intrinsic quartic and quadratic UV divergences of the stress-energy tensor. For the extra (dimensionless) coupling, associated with the term quadratic in the curvature, our results agree with the slow logarithmic running predicted by dimensional regularization. The method can also be applied to quantized Dirac fields. We report here our main results. We use units for which $c=1=\hbar$.

II Adiabatic regularization with an arbitrary mass scale $\mu$

The main idea of the paper can be understood by considering a quantized charged scalar field living in a FLRW spacetime with metric $ds^{2}=dt^{2}-a^{2}(t)d\vec{x}^{2}$, and coupled to an homogeneous electric field. The complex scalar field is assumed to obey the Klein-Gordon equation $(D_{\mu}D^{\mu}+m^{2}+\xi R)\phi=0$, where $D_{\mu}\phi=(\nabla_{\mu}+iA_{\mu})\phi$. $\nabla_{\mu}$ is the covariant derivative associated with the dynamical metric. $R$ is the Ricci scalar. For our purposes it is enough to assume that the vector potential is of the form $A_{\mu}=(0,-A(t),0,0)$. The electric charge $q$ has been reabsorbed in the definition of $A_{\mu}$. Therefore, it only appears in the Maxwell Lagrangian for the pure electromagnetic field $-\frac{\sqrt{-g}}{4q^{2}}F_{\mu\nu}F^{\mu\nu}$, where $F_{\mu\nu}$ is the field strength. The quantized field can be expanded in modes of definite $3$-momentum $\vec{k}$

[TABLE]

where the time-dependence is fixed by the wave equation

[TABLE]

and $\sigma=(6\xi-3/4)\dot{a}^{2}/a^{2}+(6\xi-3/2)\ddot{a}/a$. One-particle states can only be well-defined in the adiabatic limit of slow expansion and weak electric field, where

[TABLE]

$W_{\vec{k}}$ is obtained by a WKB-type asymptotic expansion in powers of $A(t)$, $\dot{a}(t)$; $\dot{A}(t)$, $\ddot{a}(t)$; etc. [Note that, as explained in Refs. ferreiro-navarro ; pla , we have taken into account that $A(t)$, as well as $\dot{a}(t)$, are of adiabatic order 1.] Therefore, $W_{\vec{k}}=\omega^{(0)}_{\vec{k}}+\omega^{(1)}_{\vec{k}}+\omega^{(2)}_{\vec{k}}+\cdots$  . The leading term, of zero adiabatic order, can be naturally taken as $\omega^{(0)}_{\vec{k}}\equiv\omega=\sqrt{\vec{k}^{2}/a^{2}+m^{2}}$. The next to leading term, of adiabatic order 1, is given by $\omega^{(1)}_{\vec{k}}=-Ak_{x}/(a^{2}\omega)$. A well-defined recursion relation univocally defines the higher-order terms (for details see ferreiro-navarro ).

A very important advantage of the above expansion is that it allow us to identify the UV divergences emerging in the formal expectation values of non-linear operators. For instance, the formal vacuum expectation value of the current is given by

[TABLE]

The ultraviolet divergences of the electric current $\langle j^{\mu}\rangle$ can be learnt from the corresponding adiabatic expansion. The mode expansion for $h_{k}(t)$ is plugged in $j^{\mu}(\vec{k},t)$ to generate an adiabatic series. One finds UV divergences up to the third adiabatic order. The minimal number of terms in this series should be subtracted from $j^{\mu}(\vec{k},t)$ to cancel out all UV divergences. Therefore, the physical expectation values of the current are obtained by subtracting up to and including the third adiabatic order. This will give

[TABLE]

where $(|h_{\vec{k}}(t)|^{2})^{(0-n)}=(\omega_{\vec{k}}^{(0)})^{-1}+(W_{\vec{k}}^{-1})^{(1)}+\cdots+(W_{\vec{k}}^{-1})^{(n)}$. One also obtains $\langle j^{0}\rangle_{ren}=0$, which corresponds to the fact that no total charge is created by the external electric field. We have to stress that the above subtraction scheme is also acting as a regularization procedure. There is no need to introduce any cut-off or regularization parameter, as in any other regularization method (for instance, $\epsilon=4-d$, in dimensional regularization). Hence the traditional adiabatic regularization name used to refer to the whole procedure parker-toms ; birrell-davies ; fulling .

A close reexamination of the above description of the adiabatic regularization procedure unravels an intrinsic ambiguity in the definition of the leading term in the adiabatic expansion of $W_{\vec{k}}(t)$. The leading and fundamental term in the adiabatic expansion, $\omega^{(0)}_{\vec{k}}$, can be indeed defined in a slightly more general way without spoiling the consistency of the overall renormalization scheme. One can take

[TABLE]

where $\mu$ is an arbitrary mass scale, instead of the choice $\sqrt{\vec{k}^{2}/a^{2}+m^{2}}$. The higher order adiabatic terms are univocally recalculated as (using (2), (3), and (7))

[TABLE]

where $\alpha^{2}\equiv m^{2}-\mu^{2}$. The new terms, proportional to $\alpha^{2}$, serve to remove UV divergences, in accordance with the new definition of $\omega^{(0)}_{\vec{k}}$, while maintaining locality and general covariance. Note that $\alpha^{2}$ should be regarded as a parameter of adiabatic order 2. The more conventional adiabatic method is recovered when the mass scale $\mu$ is fixed at the physical mass of the quantized field, i.e., $\mu=m$ and hence $\alpha=0$. The vacuum expectation values for the stress-energy tensor in Minkowski space, and in the absence of additional external fields, are then predicted to be zero. Otherwise, for a generic one $\mu$, the result is $\langle T_{\mu\nu}\rangle_{ren}=\frac{1}{(8\pi)^{2}}\left(-3m^{4}+4m^{2}\mu^{2}-\mu^{4}-4m^{4}\ln{\left(\frac{\mu}{m}\right)}\right)g_{\mu\nu}$. More technical details will be given elsewhere.

III Renormalization of the electric current and the running of the electric charge

The introduction of the mass scale $\mu$ leads to an inherent ambiguity in the adiabatic renormalization scheme, as also happens in dimensional regularization. It is natural to compare the renormalized current at two different scales: $\langle j^{\beta}\rangle_{\rm ren}(\mu)-\langle j^{\beta}\rangle_{\rm ren}(\mu_{0})=\langle j^{\beta}\rangle^{(0-3)}(\mu_{0})-\langle j^{\beta}\rangle^{(0-3)}(\mu)$. By using the above adiabatic expansion we find (we rewrite the result in covariant terms)

[TABLE]

with $\delta_{q}=\frac{1}{3(4\pi)^{2}}\ln{(\frac{\mu}{\mu_{0}})}$. The semiclassical Maxwell equations should take the form, irrespective of the value of renormalization parameter $\mu$,

[TABLE]

Therefore, we must also have

[TABLE]

Demanding now physical equivalence between (12) and (13), and using (11), one obtains the running of the electric charge

[TABLE]

in full agreement with the result obtained within perturbative scalar QED in Minkowski space (using, for instance, dimensional regularization and the modified minimal subtraction scheme Srednikybook ). Note that, for getting the above result, there has been no need to assume a generic form for the electromagnetic background. It has been enough to use a background potential of the form $A_{\mu}=(0,-A(t),0,0)$. We also remark that (14) has been obtained without using any perturbative expansion in the coupling constant $q$. There are no corrections to the running of the gauge coupling coming from classical gravity.

IV Renormalization of the stress-energy tensor and the running of the gravitational couplings

The formal vacuum expectation value of the stress-energy tensor $\langle T_{\mu\nu}\rangle$ possesses UV divergences up to adiabatic order 4, inclusive. Therefore, the physical expectation values $\langle T_{\mu\nu}\rangle_{ren}$ are obtained by subtracting up to and including the fourth adiabatic order. Comparing now $\langle T_{\mu\nu}\rangle_{ren}(\mu)$ and $\langle T_{\mu\nu}\rangle_{ren}(\mu_{0})$ we find (for simplicity we take here $\mu_{0}=m$)

[TABLE]

with

[TABLE]

$H^{(1)}_{\mu\nu}$ is the conserved curvature tensor obtained by functionally differentiating the quadratic curvature lagrangian $R^{2}$ with respect to the metric. The extra term $\delta_{\alpha^{1}}H^{(1)}_{\mu\nu}$ implies the existence of a modification of general relativity due to quantum effects, as first pointed out in Ref. Utiyama for asymptotically flat spacetimes. Here there is no need to introduce the additional conserved tensor, $H^{(2)}_{\mu\nu}$, coming from the lagrangian $R_{\mu\nu}R^{\mu\nu}$. This is because, in a FLRW spacetime, $H^{(1)}_{\mu\nu}$ and $H^{(2)}_{\mu\nu}$ are not independent. As long as we treat the gravitational field as a classical background, no terms of higher order in the curvature are required.

At this point we should remark that expression (15) is compatible with the ambiguities in the quantization of the stress-energy tensor found in the algebraic approach to QFT in curved spacetime Waldbook ; Hollands ; Wald-Hollands01 . To be more precise, any two local and covariant procedures of renormalization of the stress-energy tensor should differ at most in a linear combination of conserved local terms: $a_{1}g_{\mu\nu}+a_{2}G_{\mu\nu}+\alpha_{1}H_{\mu\nu}^{(1)}+\alpha_{2}H_{\mu\nu}^{(2)}$. In a FLRW spacetime, $H_{\mu\nu}^{(2)}$ is proportional to $H_{\mu\nu}^{(1)}$, hence $\alpha_{2}$ can be reabsorbed into $\alpha_{1}$. Moreover, since we have an additional external field (the electromagnetic background), the ambiguity should also include the electromagnetic stress-energy tensor. Therefore, given two prescriptions to renormalize the stress-energy tensor, denoted by $\langle T_{\alpha\beta}\rangle_{\rm ren}$ and $\langle\tilde{T}_{\alpha\beta}\rangle_{\rm ren}$, the difference for the expected stress-energy tensor is parametrized by the following linear combination

[TABLE]

The constant parameters $a_{1},a_{2},a_{3}$ and $\alpha_{1}$ are not constrained within the axiomatic approach.

We can identify now $\langle\tilde{T}_{\alpha\beta}\rangle_{\rm ren}$ with the standard adiabatic prescription to renormalize the stress-energy tensor $\langle\tilde{T}_{\alpha\beta}\rangle_{\rm ren}\equiv\langle T_{\alpha\beta}\rangle_{\rm ren}(m)$, and $\langle T_{\alpha\beta}\rangle_{\rm ren}$ with our modified adiabatic prescription (parametrized by the mass scale $\mu$): $\langle T_{\alpha\beta}\rangle_{\rm ren}\equiv\langle T_{\alpha\beta}\rangle_{\rm ren}(\mu)$. Therefore, the constant and finite parameters $a_{1},a_{2},a_{3}$ and $\alpha_{1}$ naturally acquire a dependence on the scale $\mu$. This dependence can be obtained by direct computation and the result is given by the above expressions for $\delta_{\Lambda}$, $\delta_{G}$, $\delta_{q}$ and $\delta_{\alpha_{1}}$, respectively. Furthermore, as we will see now, this implies a natural running for the gravitational coupling constants.

The semiclassical Maxwell-Einstein equations are given by (12) together with

[TABLE]

with $T_{\alpha\beta}^{EM}=\frac{1}{q^{2}(\mu)}\left(\frac{1}{4}F_{\sigma\rho}F^{\sigma\rho}g_{\alpha\beta}-F_{\alpha}^{\ \rho}F_{\beta\rho}\right)$. The coupling $\Lambda$ is related to the cosmological constant $\Lambda_{c}$ by the relation $\Lambda=\Lambda_{c}/(8\pi G)$. Enforcing that the above equations be independent of the scale $\mu$, we obtain, using the above results for $\delta_{G}$ and $\delta_{\Lambda}$, the running of the Newton gravitational constant $G$ and $\Lambda$. The running of $q$ can also be obtained, and coincides with the result (14), derived directly from the renormalization of the electric current. We find

[TABLE]

[TABLE]

[TABLE]

We observe that the logarithmic terms in the above expressions coincide exactly with the running predicted by dimensional regularization (see parker-toms , section 6.7). However, we find a corrected behavior for the running of $G$ and $\Lambda$, derived from (15). The polynomial terms in the mass scales are associated to the quadratic and quartic UV divergences of the stress-energy tensor of the matter field. These contributions to the running of dimensionful constants are not captured, as expected, by a mass-independent subtraction scheme, like dimensional regularization with minimal subtraction. For the dimensionless coupling constants (electric charge $q$ or $\alpha_{1}$) the adiabatic method and dimensional regularization lead to the same running.

It is interesting to relate the non-logarithmic contributions to the running of the Newton constant with the results of the asymptotic safety approach to quantum gravity review (see also polyakov ). The running obtained within quantum gravity is often encapsulated by the expression

[TABLE]

This result coincides with the one predicted by (23) for negligible mass, with $a=(\xi-1/6)/\pi$.

V Running couplings, Dirac fields, and backreaction equations

The details of the scaling behavior depend on the type of field. For a Dirac field $\psi$ the computations are more involved, since in this case the adiabatic expansion of the fermionic modes requires a generalization of the WKB-type ansatz rio1 ; Yukawa ; eduardo ; ferreiro-navarro . Our calculations yield

[TABLE]

[TABLE]

while for the electric charge we have the standard result for a Dirac fermion $q^{-2}(\mu)-q^{-2}(\mu_{0})=-\frac{1}{12\pi^{2}}\ln\frac{\mu^{2}}{\mu_{0}^{2}}$.

Let us now consider $N$ charged fields $\psi_{i}$ with masses $m_{i}$. For simplicity we take all fields with the same electric charge. In this case the semiclassical Maxwell equations should read $q^{-2}(\mu)\nabla_{\alpha}F^{\alpha\beta}=\sum_{i=1}^{N}\langle j_{i}^{\beta}\rangle_{ren}(\mu)$. As above, we demand these equations be independent of $\mu$. Hence, the running of the coupling is now found $q^{-2}(\mu)-q^{-2}(\mu_{0})=-\frac{N}{12\pi^{2}}\log\frac{\mu^{2}}{\mu_{0}^{2}}$. The final form of the effective semiclassical equations turns out to be

[TABLE]

where the currents $\langle j_{i}^{\beta}\rangle_{ren}$ have been renormalized at their respective natural mass scales $m_{i}$. This is the origin of the second term inside the parenthesis on the left-hand-side in (28). It is easy to check that the above equations are indeed independent of the reference mass $m_{j}$. These equations are univocally fixed once we give the masses and the experimental value of the charge at the reference scale, $q^{2}(m_{j})$ ($m_{j}$ can the taken as the smallest value of the masses).

Following the argument that leads to (28) one can also determine the form of the semiclassical Einstein equations for $N$ species of fields with different masses. The effective semiclassical Einstein equations take the form

[TABLE]

where $f_{j}(m_{k})$ and $g_{j}(m_{k})$ are functions of the masses that can be derived, for Dirac fields, from (V) and (V). The equations are also independent of the reference scale $m_{j}$ used. Since the observed value of $(8\pi G)^{-1}$ is very large, the running does not significantly modify the value of $(8\pi G)^{-1}(m_{j})$, unless very large masses are added to the spectrum. On the contrary, the value of the observed effective cosmological constant depends, in an involved way, on the pattern of masses. It also depends on the value of $\Lambda(m_{j})$, where $m_{j}$ is a reference mass. In a more realistic scenario one should include the whole field content of the Standard Model.

In the case of electrodynamics we have empirical data for $\alpha(m_{e})$ (using, for instance, the Josephson effect or Thomson scattering). Hence we can predict the effective value for $\alpha$ in the semiclassical Maxwell equations for very strong electric fields, capable to create different species of charged particles. Unfortunately, and in sharp contrast to the case of electrodynamics, we do not have direct data for $\Lambda(m_{j})$ and this makes it extremely difficult to produce specific observable predictions for the effective cosmological constant.

VI Conclusions and final comments

In this work we have generalized the adiabatic regularization method to accommodate, in a natural way, an arbitrary renormalization mass scale $\mu$. The overall renormalization method maintains covariance and locality and also retains its practical advantage for numerical calculations. We have shown how the renormalization group flow for coupling constants emerges within the adiabatic method. Concerning the scaling behavior of the electric charge, we have reproduced the well-known fact (14) in a new way, and without invoking any perturbative argument.

One of the main advantages of the adiabatic approach is that it determines the effective semiclassical Maxwell equations for several quantized matter fields with different masses, as stated in (28). For instance, in a scenario where the electric field is strong enough to create electron-positron (and muon-antimuon) pairs, the semiclassical (backreaction) Maxwell equations are $\alpha^{-1}\nabla_{\sigma}F^{\sigma\beta}=\langle j_{e}^{\beta}\rangle_{ren}(m_{e})+\langle j_{\mu}^{\beta}\rangle_{ren}(m_{\mu})$. The inverse of the effective fine structure constant $\alpha\equiv e^{2}/4\pi$ in the above equations is found to be, according to (28), $\alpha^{-1}=\alpha^{-1}(m_{e})+\frac{2}{3\pi}\ln(m_{e}/m_{\mu})\approx 135.9$, instead of the conventional value, $\alpha^{-1}(m_{e})\approx 137.0$ .

Furthermore, the adiabatic method allows us to compute, in a very direct way, the running of the gravitational constants. Here the adiabatic method gives results that differ from those found with dimensional regularization, although it agrees exactly as far as the logarithmic pieces of the running constants are concerned parker-toms . We have found additional contributions which are quadratic and quartic in the mass scale $\mu$. For instance, for very large $\mu$ the running found for $G(\mu)$ fits well with the predictions of some quantum gravity approaches review ; polyakov .

The reason of the discrepancy with dimensional regularization could be traced back to the fact that dimensional regularization is not sensitive to the non-logarithmic UV divergences of the stress-energy tensor. On the contrary, the adiabatic subtractions are constructed to fit all types of UV divergences, while maintaining general covariance and locality. We also note that this is not a solely property of the adiabatic regularization method. Let us remark that the conventional adiabatic renormalization method (with $\mu=m$) has been proved Anderson-Parker ; rio to be equivalent to the DeWitt-Schwinger point-splitting method. In momentum space the latter is based on the Bunch-Parker adiabatic expansion of the two-point function. For a scalar field one has Bunch-Parker ; parker-toms

[TABLE]

Our discussion suggests, as a by-product, that one can also introduce an arbitrary mass scale to modify the above expansion, mimicking the introduction of the mass scale $\mu$ into the adiabatic regularization method. Therefore, one has the generalized expansion

[TABLE]

We have checked that this new $\mu$-dependent expansion is equivalent to the adiabatic expansion in the modes introduced in this work. The predictions of the expansion (31) for the running of the couplings should be then equivalent to those presented in this work.

Finally, we note that there is no universal agreement in the literature on the actual renormalization flow for the gravitational coupling constants, nor on the gravitational corrections to the running of the gauge couplings Robinson-Wilczek ; Toms ; Ander-Donoghue , and the physical interpretation of the running Ander-Donoghue ; Sola . We may expect that, future empirical inputs at high curvatures or energies, either from very early cosmology or localized strong gravity events (as such detected by the LIGO-Virgo collaboration), may test the theoretical running of the gravitational couplings.

Acknowledgments.– We thank I. Agullo, J. F. Barbero G., A. Del Rio, F. Torrenti and E.J.S. Villaseñor for useful comments on the manuscript. A. F. thanks S. Antusch for his hospitality at the Physics Department of the University of Basel while part of this work was carried out. This work has been supported by research Grants No.  FIS2017-84440-C2-1-P; No.  FIS2017-91161-EXP; No.  SEJI/2017/042 (Generalitat Valenciana); COST action CA15117 (CANTATA), supported by COST (European Cooperation in Science and Technology); No.   SEV-2014-0398. A. F. is supported by the Severo Ochoa Ph.D. fellowship, Grant No. SEV-2014-0398-16-1, and the European Social Fund. Most of the computations have been done with the help of Mathematica.