1 Introduction

It is widely accepted that our today’s universe is undergoing the phase of the accelerated expansion which is commonly explained by the presence of the Cosmological Constant (CC) \(\Lambda \). However, the value of \(\Lambda \) required by experiment is in a contradiction with the values emerging from the physics scales associated with known phase transitions in the universe so that severe fine-tuning has to be applied which is at heart of the CC problem. To recall the main aspects of this problem we begin with the Standard Model (SM) formulated on the classical curved background. In order to construct a renormalizable gauge theory in an external gravitational field one starts from the classical action (with \(\varphi \) as the Higgs doublet field)

$$\begin{aligned} S_{\mathrm{vac}}= & {} \int \mathrm{d}^{4}x\sqrt{-g}\,\Biggl \{ \mathcal {L}_{SM}+ \, \xi \,\,\varphi ^{\dagger }\varphi \,R+ a_{1}R^{2}_{\mu \nu \alpha \beta } +a_{2}R^{2}_{\mu \nu } \nonumber \\&+\, a_{3}R^{2} + a_{4}{\square } R-\frac{1}{16\pi G_{vac}} \,(R + 2\Lambda _{vac})\Biggr \}. \end{aligned}$$

The renormalization procedure for the theory (1) consists of the renormalization of the SM matter fields, couplings and masses, non-minimal coupling \(\xi \) and the gravitational couplings \(a_{1,2,3,4},\) \(\,G_\mathrm{vac}\,\) and \(\,\Lambda _\mathrm{vac}\). We are going to work in the low energy domain of the gravitational physics and, for that reason, the short distance effects from the higher derivative terms \(a_{1,2,3,4},\) in (1) are not important for our considerations, and so we start with the usual bare Hilbert–Einstein action with coupling constants \(G_\mathrm{vac},\,\Lambda _\mathrm{vac}\) supplemented with non-minimal coupling \(\xi \):

$$\begin{aligned} S_{HE}= & {} \int \mathrm{d}^{4}x\sqrt{-g}\,\Biggl \{ \, \,\mathcal {L}_{SM}+ \xi ^0 \,\,\varphi ^{\dagger }\varphi \, R \nonumber \\&-\frac{1}{16\pi G_{\mathrm{vac}}^0}(R +2\Lambda _\mathrm{vac}^0)\Biggr \}. \end{aligned}$$

The bare quantities are defined with the superscript “0”. Let us focus on the CC itself which, as we mentioned above, must be renormalized and the connection with experimentally measured value \(\rho _\mathrm{phys}\) is achieved via the renormalization condition (see Eq. (8) below) imposed on the vacuum energy density:

$$\begin{aligned} (\rho ^0_{\Lambda })^\mathrm{vac} = \frac{\Lambda ^0_\mathrm{vac}}{8\pi G^0_\mathrm{vac}} \, \end{aligned}$$

at some energy scale \(\mu \) so that \(\rho _{\Lambda }^\mathrm{vac}(\mu )\). Moreover, in the presence of the dynamical cosmological background characterized by the time-dependent Hubble parameter H(t), \(\rho _{\Lambda }^\mathrm{vac}\) can be dynamical \(\rho _{\Lambda }^\mathrm{vac}(\mu , t)\) that will be reflected in the evolution of \(\rho _{\Lambda }^\mathrm{vac}\) via:

$$\begin{aligned} \frac{\partial \rho _{\Lambda }^\mathrm{vac}(t)}{\mathrm{d}t} = F(H, \rho _\mathrm{matter}, \rho _{\Lambda }^\mathrm{vac}, \ldots ), \end{aligned}$$

which is the important ingredient for cosmological evolution. Currently, there is no consensus on whether \(\rho _{\Lambda }^\mathrm{vac}(\mu , t)\) depends on t. Even if it is time-dependent, to understand the precise form of (4) one may go back to the time-independent RG problem:

$$\begin{aligned} \frac{\partial \rho _{\Lambda }^\mathrm{vac}(\mu )}{d\mu } = \beta _{\rho _{\Lambda }^\mathrm{vac}}(g_i, m_i, \mu , \ldots ), \end{aligned}$$

where \(g_i\) and \(m_i\) are the dimensionless couplings and masses respectively. The \(g_i\) and \(m_i\) are also supplemented with their own Renormalization Group (RG) equations.

Besides \(\rho _{\Lambda }^\mathrm{vac}\), the physical vacuum energy \(\rho _\mathrm{phys}\) consists of several additional parts. One of these parts is ”induced” contribution \(\rho _\mathrm{ind}(\mu )\,\) to the vacuum energy density arising from the vacuum condensates. For example, if \(\varphi _\mathrm{vac}\) is the value of the Higgs field \(\varphi (x)\) which minimizes the Higgs potential \(V(\varphi )\)

$$\begin{aligned} V(\varphi ) = -m^2 \varphi ^{\dagger }\varphi +\frac{\lambda }{2}(\varphi ^{\dagger }\varphi )^2, \end{aligned}$$

the Higgs condensate contribution (at the classical level) to the vacuum energy is

$$\begin{aligned} \rho _\mathrm{ind}(\mu ) =V(\varphi _\mathrm{vac} ) =-\frac{m^4(\mu )}{2 \lambda (\mu )}. \end{aligned}$$

Besides the vacuum and induced terms we may have additional effects from the higher derivative terms in (1) as well as corrections from quantum gravity. Again, these contributions can be classified as coming from purely quantum effects and therefore expected to be \(\mu \)-dependent and some also time-dependent due to the expanding background, and therefore contributing to (4). All in all, the physical value is measured at the cosmological RG scale \(\mu _{c}\), which is experimentally given by \(\mu _{c}=\mathcal {O}(10^{-3})\,\) eV, as

$$\begin{aligned} \rho _\mathrm{phys}=\rho _{\Lambda }^\mathrm{vac}(\mu _c)+\rho _\mathrm{ind}(\mu _c)+ \ldots \, =10^{-47}\, \mathrm{\ GeV}^4. \end{aligned}$$

The problem now is that if we use the experimental Higgs mass \(\,M_{H}= 125\,\)GeV, then the corresponding value \(| \rho _\mathrm{ind}| \simeq 10^{8}\,\mathrm{GeV}^{4}\). In order to keep the QFT consistent with astronomical observations, one has to demand that the parts contributing to the \(\rho _\mathrm{phys}\) should cancel with the accuracy dictated by the current data. For example, if we neglect all the ... terms in (8), the \(\rho _{\Lambda }^\mathrm{vac}\) and \(\rho _\mathrm{ind}\) should cancel with the precision of 55 decimal orders. This is the CC fine-tuning problem [1, 2].

To understand deeper this tuning, one has to take into account the decoupling effects due to massive particles. Clearly, we expect that contribution to the RG running from the particle of mass m should change dramatically, as we go from \(\mu \gg m\) to \(\mu \ll m\) regime. Moreover, requiring the absence (or, at least, reduction) of the tuning may provide a constraint on the spectrum of the particle physics models. In this paper, we deal with the time-independent classical curved background and will derive the RG evolution of \(\rho _{\Lambda }^\mathrm{vac}\) and \(\rho _\mathrm{ind}\) of the form (5) taking into account the decoupling effects due to massive particles by using the mass-dependent RG formalism.

The motivation for this work is threefold:

  • to derive the leading decoupling effects on the RG running of \(\rho _\mathrm{phys}\). Along the way, we will comment on the inconsistencies of the similar derivation presented in [11], as we demonstrate the importance of considering the RG running of the total vacuum energy \(\rho _{\Lambda }^\mathrm{vac}+\rho _\mathrm{ind}\) since, although \(\rho _{\Lambda }^\mathrm{vac}\) and \(\rho _\mathrm{ind}\) run separately, it is only the sum that exhibits behavior consistent with the Appelquist–Carazzone decoupling theorem [4].

  • to elucidate the implications of these results on the mass spectrum of the SM as well as its extensions. As an outcome, we present a simple phenomenological extension of the SM predicting the Higgs mass correctly.

  • to provide the generalization of these results to the constant curvature spaces important for studies of curvature-induced running of the vacuum energy as well as curvature-induced phase transitions.

The paper is structured as follows. In the next Sect. 2 we briefly discuss the RG running of the CC in the simple \(\phi ^4\)-theory highlighting the necessary RG formalism we use later and also discuss the basic issue of decoupling in the RG running. In Sect. 3 we extend the RG approach to the full SM, in both, mass-independent and mass-dependent RG schemes. Section 4 deals with applications of the derived heavy-mass threshold effects within and beyond the SM and Sect. 6 presents our conclusions. In “Appendices” we provide the technical details, as well as generalize the flat spacetime results to the spaces with constant curvature.

2 RG running of the Cosmological Constant

To prepare for the discussion of the RG dependence of the CC and to setup the formalism, let us consider the skeleton Lagrangian for the real scalar:

$$\begin{aligned} L=\frac{1}{2}m^2 \phi ^2 +\frac{1}{8}\lambda \phi ^4. \end{aligned}$$

The schematic contributions to the one-loop effective potential, up to the four external legs, are shown in Fig. 1.

Fig. 1
figure 1

a The one-loop contributions to the vacuum part of CC are just bubbles of matter fields without external legs; b the one-loop two-point function contributing to the induced part of the CC. c The one-loop four-point function contributing to the induced part of the CC

Correspondingly to this diagrammatic picture and for a general QFT, the renormalized effective potential can be split into two pieces: the \(\phi \)-independent (vacuum) term corresponding to the diagram Fig. 1a and the \(\phi \)-dependent “scalar” term connected with diagrams Fig. 1b, c:

$$\begin{aligned} V(\rho _{\Lambda }^\mathrm{vac},\phi ,m^2,\lambda ,\mu )= & {} V_{\mathrm{vac}}(\rho _{\Lambda }^\mathrm{vac},m^2,\lambda ,\mu )\nonumber \\&+\, V_{\mathrm{scal}}(\phi ,m^2,\lambda ,\mu ), \end{aligned}$$

where the parameter \(\rho _{\Lambda }^\mathrm{vac}=\rho _{\Lambda }^\mathrm{vac}(\mu )\) depends on the vacuum cosmological constant and will be defined in the next section. In order to understand the origin of this splitting, one introduces the functional called the effective action of the vacuum \(\Gamma _{\mathrm{vac}}\). It is part of the full effective action which is left when the mean scalar field \(\phi \) is set to zero: \(\Gamma _{\mathrm{vac}}=\Gamma [\phi =0]\). Thus, it is a pure quantum object which only depends on the set of parameters \(P=m,\lambda ,...\) of the classical theory. At the functional level, the generating functional W for the vacuum-to-vacuum transition amplitude is

$$\begin{aligned} W[J=0]=\mathrm{e}^{i\Gamma _{\mathrm{vac}}}\,=\,\int \mathcal{D}\phi \ \mathrm{e}^{iS[\phi ; J=0]}, \end{aligned}$$

where the source J is set to zero. In this way, the functional \(\Gamma _{\mathrm{vac}}\) is the generator of the proper vacuum-to-vacuum diagrams.

The RG-invariance of the full renormalized effective potential reads (where \(\gamma _m m^2 = \beta _{m^2}\)):

$$\begin{aligned}&\left( \mu \frac{\partial }{\partial \mu }+\beta _{\lambda }\frac{\partial }{\partial \lambda } +\gamma _m m^2 \frac{\partial }{\partial m^2} -\gamma _\phi \phi \frac{\partial }{\partial \phi }+\beta _{\rho _{\Lambda }^\mathrm{vac}}\frac{\partial }{\partial \rho _{\Lambda }^\mathrm{vac}}\right) \nonumber \\&\quad \times V(\rho _{\Lambda }^\mathrm{vac},\phi ,m^2,\lambda ,\mu )\,=\,0. \end{aligned}$$

Using (10), we now show that Eq. (12) is, in fact, a sum of two independent RG equations,

$$\begin{aligned}&\left( \mu \frac{\partial }{\partial \mu } + \beta _{\lambda } \frac{\partial }{\partial \lambda }+\gamma _m m^2 \frac{\partial }{\partial m^2}+\beta _{\rho _{\Lambda }^\mathrm{vac}}\frac{\partial }{\partial \rho _{\Lambda }^\mathrm{vac}}\right) \nonumber \\&\quad \times V_\mathrm{vac}(\rho _{\Lambda }^\mathrm{vac},m^2,\lambda ,\mu ) \,=\,0, \end{aligned}$$
$$\begin{aligned}&\left( \mu \frac{\partial }{\partial \mu } + \beta _{\lambda }\frac{\partial }{\partial \lambda }+\gamma _m m^2 \frac{\partial }{\partial m^2} -\gamma _\phi \phi \frac{\partial }{\partial \phi } \right) \nonumber \\&\quad \times V_\mathrm{scal}(\phi ,m^2,\lambda ,\mu )\,=\,0. \end{aligned}$$

To prove this, notice that from the RG-invariance of the renormalized effective action follows the \(\mu \)-independence of the renormalized functional \(\Gamma _\mathrm{vac}\) and, therefore, we arrive at the first identity (13) for the vacuum part of the effective potential, while the second identity is then the result of the subtraction of (13) from (12). We will illustrate this point later.

The net result is that the vacuum and matter parts of the effective potential are overall \(\mu \)-independent separately and no cancelation between them is expected.

2.1 Vacuum part of the CC

Let us compute the \(V_\mathrm{vac}(\rho _{\Lambda }^\mathrm{vac},m^2,\lambda ,\mu )\) object at the one-loop level. We start from

$$\begin{aligned} S_{HE}=-\frac{1}{16\pi G_\mathrm{vac}^0}\int \mathrm{d}^{4}x\sqrt{-g}\,\{ \, \,R+2\Lambda _\mathrm{vac}^0 \,\} +S_\mathrm{matter}. \end{aligned}$$

As it is well-known the \(\Lambda _\mathrm{vac}^0\)-dependent part has exactly the form of the bare vacuum energy density (3)Footnote 1:

$$\begin{aligned} (\rho ^0_{\Lambda })^\mathrm{vac} = \frac{\Lambda ^0_\mathrm{vac}}{8\pi G^0_\mathrm{vac}}. \end{aligned}$$

In the standard QFT, the loop-divergent terms in the vacuum density are absorbed by the bare cosmological constant term \((\rho ^0_{\Lambda })^\mathrm{vac}\) of the Hilbert–Einstein action. For this, we split the bare term \((\rho ^0_{\Lambda })^\mathrm{vac} \) as

$$\begin{aligned} (\rho ^0_{\Lambda })^\mathrm{vac} = \rho _{\Lambda }^\mathrm{vac}(\mu )+\delta \rho _{\Lambda }^\mathrm{vac}, \end{aligned}$$

where the counterterm \(\delta \rho _{\Lambda }^\mathrm{vac}\) depends on the regularization and the renormalization scheme. Specifically, the one-loop effects encoded in \({\bar{V}}_\mathrm{vac}^{(1)}\) modify this relation as follows:

$$\begin{aligned} V_\mathrm{vac}(\rho _{\Lambda }^\mathrm{vac},m^2,\lambda ,\mu )= (\rho ^0_{\Lambda })^\mathrm{vac}+{\bar{V}}_\mathrm{vac}^{(1)} =\rho _{\Lambda }^\mathrm{vac}(\mu )+\delta \rho _{\Lambda }^\mathrm{vac}+{\bar{V}}_\mathrm{vac}^{(1)}. \end{aligned}$$

Now, starting from (11), the variation of vacuum-to-vacuum transition amplitude with mass m leads to the (Euclidean) Green’s function at coincident points [3]

$$\begin{aligned} \frac{\partial }{\partial m^2} \log W[J=0]= & {} -\frac{1}{2} \int \mathrm{d}^n_E x \Delta _E(0) \quad \text {with} \quad \Delta _E(0)\nonumber \\= & {} \int \frac{\mathrm{d}^n_E k}{(2\pi )^n}\frac{1}{k^2+m^2}=\frac{(m^2)^{\frac{n}{2}-1}}{(4\pi )^{n/2}}\Gamma (1-\frac{n}{2}).\nonumber \\ \end{aligned}$$

In terms of Feynman diagrams this is just the vacuum bubble shown in Fig.1a. Integrating this equation and adding \((\rho ^0_{\Lambda })^\mathrm{vac}\) we obtain the vacuum energy density (18) as:

$$\begin{aligned} W[J=0]= & {} \mathrm{e}^{- \int \mathrm{d}^n_E x \ V_\mathrm{vac}^{}} \quad \text {with} \quad V_\mathrm{vac}^{}\nonumber \\= & {} (\rho ^0_{\Lambda })^\mathrm{vac}+{\bar{V}}_\mathrm{vac}^{(1)} =(\rho ^0_{\Lambda })^\mathrm{vac}\nonumber \\&+\, \frac{m^n}{(4\pi )^{n/2}}\frac{1}{n}\Gamma \left( 1-\frac{n}{2}\right) \,. \end{aligned}$$

The pole of the Gamma function in four dimensions \(\Gamma (1-\frac{n}{2})\sim 2/(n-4)\) so for \(n\rightarrow 4\):

$$\begin{aligned} {\bar{V}}_\mathrm{vac}^{(1)} = \frac{m^4}{64\pi ^2}\,\left( \frac{2}{n-4} -\ln \frac{4\pi \mu ^2}{m^2}+\gamma _E-\frac{3}{2}\right) . \end{aligned}$$

Equation (21) is divergent and needs a subtraction. If we adopt the \(\overline{MS}\) subtraction scheme, the counterterm \(\delta \rho _{\Lambda }^\mathrm{vac}\) gets fixed in such a way that the renormalized vacuum energy density at 1-loop is

$$\begin{aligned} V_\mathrm{vac}(\rho _{\Lambda }^\mathrm{vac},m^2,\lambda ,\mu )\,= & {} \,\rho _{\Lambda }^\mathrm{vac}(\mu )+ \delta \rho _{\Lambda }^\mathrm{vac}+{\bar{V}}^{(1)}_\mathrm{vac} \,\nonumber \\= & {} \,\rho _{\Lambda }^\mathrm{vac}(\mu )+\frac{m^4}{64\pi ^2}\, \,\left( \ln \frac{m^2}{\mu ^2}-\frac{3}{2}\right) .\nonumber \\ \end{aligned}$$

This is the result for \(V_\mathrm{vac}(m^2,\lambda ,\rho _{\Lambda }^\mathrm{vac},\mu )\) at 1-loop. Notice that it is a pure quantum object that (to one-loop order) depends only on the parameter m of the classical Lagrangian and does not depend (to this order) on \(\lambda \).

It is clear from (22) that the cosmological constant is renormalized according to:

$$\begin{aligned} (\rho ^0_{\Lambda })^\mathrm{vac} =\mu ^{n-4} \bigg (\frac{m^4}{2(4\pi )^2}\frac{1}{n-4}+ \rho _{\Lambda }^\mathrm{vac}(\mu )\bigg ) \qquad \end{aligned}$$


$$\begin{aligned} \mu \frac{\partial \rho _{\Lambda }^\mathrm{vac}}{\partial \mu }\equiv \beta _{\rho _{\Lambda }^\mathrm{vac}} =\frac{m^4}{32\pi ^2}. \nonumber \\ \end{aligned}$$

This is the expression for \(\beta _{\rho _{\Lambda }^\mathrm{vac}}\) calculated in the \(\overline{MS}\) scheme.

In writing this equation we used the renormalized mass because the RG equations must involve only finite (renormalized) quantities. However, so far, we only computed the vacuum bubble in the free theory, where renormalized and bare masses are the same. In the interacting theory, we have to take care about the renormalization of the mass m itself. The leading mass correction is based on the correction to the scalar propagator shown in Fig. 1b and after the standard calculation we arrive at:

$$\begin{aligned} m_0^2 = m^2 \bigg ( 1- \frac{3\lambda }{(4\pi )^2}\frac{1}{n-4} \bigg ) \quad \Longrightarrow \quad \mu \frac{\partial m^2}{\partial \mu }=m^2 \frac{3\lambda }{(4\pi )^2}. \nonumber \\ \end{aligned}$$

The addition of the interactions modifies the renormalization of the cosmological constant according to the two-loop diagram shown in Fig. 2.

Fig. 2
figure 2

The two-loop contribution to the cosmological constant in \(\phi ^4\) theory

which leads to:

$$\begin{aligned} V_\mathrm{vac}^{}= & {} (\rho ^0_{\Lambda })^\mathrm{vac}+\frac{m_0^n}{(4\pi )^{n/2}}\frac{1}{n}\Gamma \left( 1-\frac{n}{2}\right) \nonumber \\&\quad +\frac{3\lambda _0}{8}\Bigg [\frac{m_0^{n-2}}{(4\pi )^{n/2}}\Gamma \left( 1-\frac{n}{2}\right) \Bigg ]^2. \end{aligned}$$

To put this expression in the renormalized form we have to replace the bare mass by the renormalized one using (25), while we can use renormalized quartic coupling \(\lambda _0=\lambda \) to this order. One obtains:

$$\begin{aligned} V_\mathrm{vac}= & {} (\rho ^0_{\Lambda })^\mathrm{vac}+\frac{m^4}{2(4\pi )^{n/2}}\frac{\mu ^{n-4}}{n-4}\bigg ( 1- \frac{3\lambda }{(4\pi )^2}\frac{1}{n-4} \bigg ) \nonumber \\&+\,\mathrm{finite} \end{aligned}$$

where we observe that the leading term is written in terms of the renormalized mass. It is clear from (27) that the cosmological constant is then renormalized according to:

$$\begin{aligned} (\rho ^0_{\Lambda })^\mathrm{vac}&=\mu ^{n-4} \bigg [\frac{m^4}{2(4\pi )^2}\frac{1}{n-4}\bigg ( 1- \frac{3\lambda }{(4\pi )^2}\frac{1}{n-4} \bigg )+ \rho _{\Lambda }^\mathrm{vac}(\mu )\bigg ] \;\nonumber \\&\Longrightarrow \quad \beta _{\rho _{\Lambda }^\mathrm{vac}} =\frac{m^4}{32\pi ^2}\Bigg [1+\mathcal {O}\Big (\frac{\lambda }{16\pi ^2}\Big )\Bigg ]. \end{aligned}$$

Note again that there is no correction to the RG to the leading order in \(\lambda \). Basically, each of the two bubbles in Fig. 2 acts as a mass correction to the other one and gets reabsorbed into the renormalized mass.

2.2 Decoupling effects

By definition, the RG equation (28) holds in the region \(\mu \gg m\) and to go to the opposite regime \(\mu \ll m\) would require to take into account: (1) the contribution of heavy particles at the energies near their mass, (2) the residual effects from the heavy particles at energies well below their mass.

It is well-known that the decoupling of heavy particles does not hold in a mass-independent scheme like the \(\overline{MS}\), and for this reason they must be decoupled by hand using the sharp cut-off procedure or some of the mass-dependent schemes. The quantum effects of the massive particles are, in principle, suppressed at low energies by virtue of the Appelquist–Carazzone theorem [4], so that in the region below the mass of the particle its quantum effects become smaller. At this point we need the relation between the IR and the UV regions which would require to extend the Wilson RG for the quantitative description of the threshold effects, and to apply a mass-dependent RG formalism.

On purely dimensional grounds, in the regime \(\mu \ll m\) one expects the corrections to the CC of the type \(\mu ^2 m^2\). These corrections can be seen from the fact that in a mass-dependent subtraction scheme a heavy mass m enters the \(\beta \)-functions through the dimensionless combination \(\mu /m\), so that the CC, being a dimension-4 quantity, is expected to have the \(\beta \)-function corrected as follows:

$$\begin{aligned} \beta \bigg (m_{\mathrm{light}},\frac{\mu }{m}\bigg )=a\,m_{\mathrm{light}}^{4}+b\,\left( \frac{\mu }{m}\right) ^{2}m^{4}+ c \,\left( \frac{\mu }{m}\right) ^{4}m^{4} +\cdots \end{aligned}$$

where a,b and c are some coefficients, \(m_\mathrm{light}\) is some light mass \(m_\mathrm{light} \ll \mu \), and the dots stand for terms suppressed by higher order powers of \(\mu /m \ll 1\). Equipped with the necessary RG formalism and expectation of decoupling behavior based on the dimensional analysis, we will show how one can deal with the decoupling effect in the full SM and how to calculate explicitly the coefficients abc for any model.

3 RG running of the Cosmological Constant in the Standard Model

Before discussing the mass-dependent RG schemes relevant for decoupling, let us recall the results in the usual \(\overline{MS}\) scheme.

3.1 Mass-independent (\(\overline{MS}\)) scheme

The renormalized effective potential of the SM, V, can be written in the ’t Hooft–Landau gauge and the \(\overline{MS}\) scheme as [5, 6]

$$\begin{aligned} V(\rho _{\Lambda }^\mathrm{vac},\phi , m^2,\lambda _i,\mu )\equiv V_0 + V_1 + \cdots , \end{aligned}$$

where \(\lambda _i\equiv (g,g',\lambda ,h_t)\) runs over all dimensionless couplings and \(V_0\), \(V_1\) are the tree level potential and the one-loop correction respectively, namely

$$\begin{aligned} V_0=-{\displaystyle \frac{1}{2}}m^2\phi ^2 + {\displaystyle \frac{1}{8}}\lambda \phi ^4, \end{aligned}$$
$$\begin{aligned} V_1={\displaystyle \sum _{i=1}^5}{\displaystyle \frac{n_i}{64\pi ^2}} M_i^4(\phi )\left[ \log {\displaystyle \frac{M_i^2(\phi )}{\mu ^2}} -c_i\right] +\rho _{\Lambda }^\mathrm{vac}, \end{aligned}$$

withFootnote 2

$$\begin{aligned} M_i^2(\phi )=\kappa _i\phi ^2-\kappa _i^m m^2, \end{aligned}$$

and coefficients \(n_i\), \(\kappa _i\), \(\kappa _i^m\), and \(c_i\) defined in Table 1.

Table 1 Contributions to the effective potential (32) from the SM particles \(W^{\pm }\), \(Z^0\), top quark t, Higgs \(\phi \) and the Goldstone bosons \(\chi _{1,2,3}\)

\(M_i^2(\phi )\) are the tree-level expressions for the background-dependent masses of the particles that enter in the one-loop radiative corrections, namely \(M_1\equiv m_W\), \(M_2\equiv m_Z\), \(M_3\equiv m_t\), \(M_4\equiv m_\mathrm{Higgs}\), \(M_5\equiv m_\mathrm{Goldstone}\). The parameter \(\rho _{\Lambda }^\mathrm{vac}(\mu )\) is the SM analogue of the renormalized cosmological constant \(\rho _{\Lambda }^\mathrm{vac}(\mu )\) for the real scalar field discussed in the previous section. Repeating the procedure as before, we split the effective potential into two pieces: the \(\phi \)-independent (vacuum) term and the \(\phi \)-dependent “scalar” term

$$\begin{aligned} V(\rho _{\Lambda }^\mathrm{vac},\phi ,m^2,\lambda _i,\mu )= & {} V_\mathrm{vac}(\rho _{\Lambda }^\mathrm{vac},m^2,\lambda _i,\mu ) \nonumber \\&+\, V_\mathrm{scal}(\phi ,m^2,\lambda _i,\mu ). \end{aligned}$$

Various pieces satisfy the RG equations (12) and (14, 13) with \(\lambda \rightarrow \lambda _i\) and these equations are valid for any value of \(\phi \).

However, for the extremum value \(\phi =\langle \phi \rangle \) defined via \(\frac{\partial V_\mathrm{scal}(\phi )}{\partial \phi }\Big |_{\langle \phi \rangle }=0\), the term containing anomalous dimension of the Higgs \(\gamma _\phi \) drops out and (14) reads:

$$\begin{aligned} \Bigg (\mu \frac{\partial }{\partial \mu } + \beta _{\lambda _i}\frac{\partial }{\partial \lambda _i} +\gamma _m m^2 \frac{\partial }{\partial m^2} \Bigg )V_{scal}(\langle \phi \rangle ,m^2,\lambda ,\mu )\,=\,0. \nonumber \\ \end{aligned}$$

Using the tree-level potential (31), it is useful to define parameter

$$\begin{aligned} \rho _{\mathrm{ind}}(\mu )\equiv V_0 (\langle \phi \rangle )=-\frac{m^4(\mu )}{2\lambda (\mu )}. \end{aligned}$$

The running of this parameter reads:

$$\begin{aligned} \beta _{\rho _\mathrm{ind}}\equiv \mu \frac{\partial \rho _\mathrm{ind}(\mu )}{\partial \mu }=\rho _\mathrm{ind}(\mu ) \ \Big (2\gamma _m -\frac{\beta _{\lambda }}{\lambda } \Big ). \end{aligned}$$

By equating the terms with the different powers of \(\phi \) :

$$\begin{aligned} \mu \frac{\mathrm{d}V}{\mathrm{d} \mu }\sim \big ( \ \phi ^4 [ \ldots ]+ m^2 \phi ^2[....]+ m^4[\ldots ]\ \big )=0 , \end{aligned}$$

it is straightforward to check that the requirement (12) applied to the full one-loop effective potential (30) leads to

$$\begin{aligned} {\displaystyle \frac{1}{8}}\beta _{\lambda }-{\displaystyle \frac{1}{2}} \gamma _\phi \lambda = {\displaystyle \sum _i}{\displaystyle \frac{n_i\kappa _i^2}{32\pi ^2}} \ , \end{aligned}$$
$$\begin{aligned} {\displaystyle \frac{1}{2}}\gamma _{m}-\gamma _\phi = {\displaystyle \sum _i}{\displaystyle \frac{n_i\kappa _i\kappa _i^{m}}{16\pi ^2}} \ , \end{aligned}$$
$$\begin{aligned} {\displaystyle \mu \frac{\partial \rho _{\Lambda }^\mathrm{vac}}{\partial \mu }}=m^4{\displaystyle \sum _i}{\displaystyle \frac{n_i(\kappa _i^{m})^2}{32\pi ^2}} \ , \end{aligned}$$

up to two-loop corrections. The first two equations come from \(\phi ^4 [...]\) and \(m^2 \phi ^2[....]\) terms respectively and they belong to (14) and the last condition came from \(m^4[...]\) and satisfy (13). These equations show explicitly that the vacuum \(V_\mathrm{vac}\) and scalar \(V_\mathrm{scal}\) parts of the full effective potential satisfy independent RG equations.

Subtracting (39) from (40) appropriately, we reconstruct 37

$$\begin{aligned} \beta _{\rho _\mathrm{ind}}=\rho _\mathrm{ind}\Bigg ({\displaystyle 2\gamma _{m}}-{\displaystyle \frac{\beta _{\lambda }}{\lambda }}\Bigg )= m^4\Bigg ({\displaystyle \sum _i}{\displaystyle \frac{n_i\kappa _i^2}{8\pi ^2\lambda ^2}} - {\displaystyle \sum _i}{\displaystyle \frac{n_i\kappa _i\kappa _i^{m}}{8\pi ^2\lambda }} \Bigg ). \nonumber \\ \end{aligned}$$

Combining (42) and (41) we finally obtain

$$\begin{aligned} \boxed {\displaystyle \mu \frac{\partial (\rho _{\Lambda }^\mathrm{vac}+ \rho _\mathrm{ind})}{\partial \mu }=m^4\Bigg ({\displaystyle \sum _i}{\displaystyle \frac{n_i\kappa _i^2}{8\pi ^2\lambda ^2}}- {\displaystyle \sum _i}{\displaystyle \frac{n_i\kappa _i\kappa _i^{m}}{8\pi ^2\lambda }} +{\displaystyle \sum _i}{\displaystyle \frac{n_i(\kappa _i^{m})^2}{32\pi ^2}}\Bigg ) ={\displaystyle \sum _{i}^{}}{\displaystyle \frac{n_i}{32\pi ^2}} M_i^4(\langle \phi \rangle ) ,} \end{aligned}$$

where we used

$$\begin{aligned} M_i^2(\langle \phi \rangle )=\kappa _i\langle \phi \rangle ^2-\kappa _i^m m^2 = m^2 \left( \kappa _i \frac{2 }{\lambda } - \kappa _i^m \right) \end{aligned}$$

and \(\langle \phi \rangle ^2 = 2 m^2/\lambda \). Equation (43) is the central equation valid in the UV regime of massless and massive theories, theories with the spontaneous symmetry breaking (SSB) and without. This equation defines, in a compact form, the total running of the implicit \(\mu -\)dependences on the l.h.s. by balancing them with the explicit \(\mu -\)dependences on the r.h.s [8].

3.2 Mass-dependent scheme

Now, following the discussion above, we may generalize approach to the mass-dependent RG scheme. As we discussed above the decoupling of heavy particles does not hold in a mass-independent \(\overline{MS}\) scheme and here we recall how to get around this problem.

The basic issue can be seen in the computations of 2\(\rightarrow \) 2 scattering amplitude in a simple \(\phi ^4\)-theory:

$$\begin{aligned} V_{\phi ^4}=-{\displaystyle \frac{1}{2}}m^2\phi ^2 + {\displaystyle \frac{1}{8}}\lambda \phi ^4 \end{aligned}$$

which is just the potential of (31) limited to one real scalar. The exemplary scattering amplitude is shown in Fig. 3, where \(p=p_1+p_2\) is the total incoming momenta. Expanding in terms of the external momentum p, it is only the term \(p=0\) which is divergent since every power of p effectively gives one less power of k for large k.

Fig. 3
figure 3

a The one-loop contributions to 4-point function in \(\phi ^4\) theory

Computing the logarithmically-divergent integral using, for example, dimensional regularization, we obtain:

$$\begin{aligned} A(p^2)= & {} -\frac{1}{32 \pi ^2}\int _0^1 \mathrm{d}x \ \Bigg (\frac{2}{\epsilon } -\gamma _E+\log (4\pi )\nonumber \\- & {} \log [m^2-x(1-x)p^2]\Bigg ) \end{aligned}$$

where we see explicitly that \(p^2\)-terms are finite.

In the \(\overline{MS}\) renormalization scheme, one chooses counterterms ( c.t.) in such a way as to remove the divergent \(2/\epsilon \) pole and scale independent number \(-\gamma _E+\log (4\pi ) \) and therefore, by construction, the counterterms are mass-independent. Also one introduces the arbitrary mass parameter \(\mu _{\overline{MS}}\) to make equation dimensionally correct so that finally:

$$\begin{aligned} A(p^2)_{\overline{MS}}= & {} A(p^2)+\mathrm{c.t.}\nonumber \\= & {} \frac{1}{32 \pi ^2}\int _0^1 \mathrm{d}x \ \log \Bigg (\frac{m^2-x(1-x)p^2}{\mu _{\overline{MS}}^2}\Bigg ). \end{aligned}$$

From the RG equation applied to the 4-point function:

$$\begin{aligned}&G_{\overline{MS}}^{(4)}(m^2,\lambda ,\mu _{\overline{MS}})\sim -3i\lambda +9(-i\lambda )^2\nonumber \\&\quad \left[ i A(s)_{\overline{MS}}+i A(t)_{\overline{MS}}+i A(u)_{\overline{MS}}\right] \end{aligned}$$

one now derives the \(\beta \)-function of the theory:

$$\begin{aligned}&\Bigg (\mu _{\overline{MS}}\frac{\partial }{\partial \mu _{\overline{MS}}} + \beta _{\lambda }^{\overline{MS}}\frac{\partial }{\partial \lambda } \Bigg )G^{(4)}(m^2,\lambda ,\mu _{\overline{MS}})\,=\,0\, \qquad \nonumber \\&\quad \Longrightarrow \qquad \beta _{\lambda }^{\overline{MS}}= \frac{9\lambda ^2}{16\pi ^2} . \end{aligned}$$

In a mass-dependent renormalization scheme, the counterterms are mass-dependent and can be chosen, for example, to subtract from (46), in addition to the divergent pole and scale-independent numbers, also the \(\log [m^2-x(1-x)p^2]\) evaluated at the \(p^2=-\mu ^2\) where \(\mu \) is yet another arbitrary scale. After this additional finite subtraction, (46) will be replaced by the corresponding expression in the momentum subtraction scheme (MOM) as

$$\begin{aligned} A(p^2)_{{\text{ mom }}}= & {} A(p^2)+ \mathrm{c.t.}\nonumber \\= & {} \frac{1}{32 \pi ^2}\int _0^1 \mathrm{d}x \ \log \Bigg (\frac{m^2-x(1-x)p^2}{m^2+x(1-x)\mu ^2}\Bigg ) \ . \end{aligned}$$

Again, \(\mu \)-dependence will determine the beta function of the theory through the RG-equation and we obtain:

$$\begin{aligned}&\Bigg (\mu \frac{\partial }{\partial \mu } + \beta ^{{\text{ mom }}}_{\lambda }\frac{\partial }{\partial \lambda } \Bigg )G_{{\text{ mom }}}^{(4)}(m^2,\lambda ,\mu )\,=\,0\, \quad \nonumber \\&\Longrightarrow \quad \beta _{\lambda }^{{\text{ mom }}}= \frac{9\lambda ^2}{16\pi ^2}\int _0^1 \frac{x(1-x)\mu ^2}{m^2+x(1-x)\mu ^2} \mathrm{d}x \end{aligned}$$

which in the \(\mu \ll m\) region, reproduces the decoupling behavior \(\mu ^2/m^2\) we discussed in (29).

Now, we need to generalize the above derivation in the mass-dependent scheme for the simple \(\phi ^4\)-theory to the full SM including the loops of the WZt and Goldstones. In Appendix A, we show that the appropriate generalization of (39) is given by:

$$\begin{aligned} \left[ {\displaystyle \frac{\beta _{\lambda }}{8}}-{\displaystyle \frac{\gamma _\phi \lambda }{2}}\right] _{{\text{ mom }}}= {\displaystyle \sum _i}{\displaystyle \frac{n_i\kappa _i^2}{32\pi ^2}}\int _0^1 \frac{x(1-x)\mu ^2 \mathrm{d}x}{(M_{\mathrm{phys} }^2)_i+x(1-x)\mu ^2} \nonumber \\ \end{aligned}$$

in agreement with [9]. For the single real scalar case discussed above, we have to 1-loop \(\gamma _\phi =0\) and, from Table 1 we have \(n_i=1\) and \(\kappa _i=3\lambda /2\) so that we reproduce (51). Notice that when performing the sum over Goldstones, the parameter \((M_{\mathrm{phys} }^2)_i\) becomes the physical mass of the vector boson corresponding to the Goldstone of type i.

Similarly, in “Appendix A” we also show that the generalization of (40) is given by:

$$\begin{aligned} \left[ {\displaystyle \frac{1}{2}}\gamma _{m}-\gamma _\phi \right] _{{\text{ mom }}}= {\displaystyle \sum _i}{\displaystyle \frac{n_i\kappa _i\kappa _i^{m}}{16\pi ^2}} \int _0^1 \frac{x(1-x)\mu ^2\mathrm{d}x}{(M_{phys }^2)_i+x(1-x)\mu ^2}. \nonumber \\ \end{aligned}$$

We therefore conclude, that in this mass-dependent scheme, the corresponding MOM expression for the \(\overline{MS}\) running of \(\rho _\mathrm{ind}\) in (42) takes the following form:

$$\begin{aligned} \mu \frac{\partial \rho _\mathrm{ind}(\mu )}{\partial \mu }_{|\mathrm MOM}= & {} \rho _\mathrm{ind}(\mu ) \ \Big (2\gamma _m -\frac{\beta _{\lambda }}{\lambda } \Big )\nonumber \\= & {} \frac{ \langle \phi \rangle ^4 }{32\pi ^2}\sum _{i} n_i \big [\kappa _i^2- \kappa _i \kappa _i^m \lambda \big ]\nonumber \\&\times \int _0^1 \frac{x(1-x)\mu ^2 \mathrm{d}x}{(M_\mathrm{phys }^2)_i+x(1-x)\mu ^2}. \end{aligned}$$

Now it remains to derive the vacuum part, Eqs. (24) and (41), in the mass-dependent scheme. To accomplish that, one starts from the simple observation that the expression for the unrenormalized vacuum density (21) can be brought to the following form:

$$\begin{aligned} {\bar{V}}_\mathrm{vac}^{(1)} = - \frac{m^4}{64\pi ^2}\,\left( \frac{A_0(m)}{m^2} + \frac{1}{2}\right) . \end{aligned}$$

In above, \(A_0(m)\) is the one-point Passarino–Veltman function with the properties given in the “Appendix A”. Now, using the relation

$$\begin{aligned}&\mu \frac{\partial A_0 (M)}{\partial \mu }_{|\mathrm MOM}= M^2 \mu \frac{\partial B_0 (0,M,M)}{\partial \mu }_{|\mathrm MOM}\nonumber \\&=M^2 \mu \frac{\partial B_0 (p,M,M)}{\partial \mu }_{|\mathrm MOM} = 2 M^2 \int _0^1 \frac{x(1-x)\mu ^2 \mathrm{d}x}{M^2+x(1-x)\mu ^2} \nonumber \\ \end{aligned}$$

and the fact that \(V_\mathrm{vac}(m^2,\lambda ,\rho _{\Lambda }^\mathrm{vac},\mu )\) satisfies the RG equation (13) we obtain for the running of the vacuum part (41) in the MOM schemeFootnote 3

$$\begin{aligned} {\displaystyle \mu \frac{\partial \rho _{\Lambda }^\mathrm{vac}}{\partial \mu }}_{|\mathrm{MOM}}=m^4 { \sum _i}{ \frac{n_i(\kappa _i^{m})^2}{32\pi ^2}}\int _0^1 \frac{x(1-x)\mu ^2 \mathrm{d}x}{(M_\mathrm{phys }^2)_i+x(1-x)\mu ^2}. \end{aligned}$$

The essence of (56) is to ensure that once the finite subtraction (defining the MOM scheme) was made for \(\beta _{\rho _\mathrm{ind}}\), the same finite subtraction is made for \(\beta _{\rho _{\Lambda }^\mathrm{vac}}\).

Putting everything together, and using again \(\langle \phi \rangle ^2 = 2 m^2/\lambda \), we achieve the generalization of (43)

$$\begin{aligned} \boxed {{\displaystyle \mu \frac{\partial (\rho _{\Lambda }^\mathrm{vac}+\rho _\mathrm{ind})}{\partial \mu }}_{|\mathrm MOM}={\displaystyle \sum _{i}^{}}{\displaystyle \frac{n_i}{32\pi ^2}} (M_{phys }^4)_i \int _0^1 \frac{x(1-x)\mu ^2 \mathrm{d}x}{(M_\mathrm{phys }^2)_i+x(1-x)\mu ^2}} , \end{aligned}$$

to the mass-dependent scheme. The Eq. (58) is the master equation describing the running of CC in any regime, non-decoupling and decoupling one, which is valid both in the UV and the IR regime. In the UV regime we recover (43), while in the IR regime we obtain the running of \(\rho _{\Lambda }^\mathrm{vac}+ \rho _\mathrm{ind}\) including the decoupling effects associated with the mass thresholds as

$$\begin{aligned}&\int _0^1 {\displaystyle \frac{x(1-x)\mu ^2 \mathrm{d}x}{(M_\mathrm{phys }^2)_i+x(1-x)\mu ^2}}\nonumber \\&= \left\{ \begin{array}{lll} \quad \qquad \qquad 1 \qquad \qquad \qquad \mu ^2\gg (M_\mathrm{phys }^2)_i\\ {\displaystyle \frac{\mu ^2}{6(M_\mathrm{phys }^2)_i}-\frac{\mu ^4}{30 (M_\mathrm{phys }^4)_i} \qquad \mu ^2\ll (M_\mathrm{phys }^2)_i}\\ \end{array}\right. , \end{aligned}$$

which will be used in the following to show the consequences of such running.

In (58) we derived the running of \(\rho _{\Lambda }^\mathrm{vac}+\rho _\mathrm{ind}\) valid in the regime of decoupling of heavy particles \(\mu ^2\ll M_i^2(\langle \phi \rangle )\) while for the light particles \((m_\mathrm{light}^2)_j \ll \mu ^2\), we can simply use (43). Working in the region where \((m_\mathrm{light}^2)_j \ll \mu ^2\ll M_i^2(\langle \phi \rangle )\), we may combine these asymptotic results to obtain:

$$\begin{aligned}&\mu \frac{\partial (\rho _{\Lambda }^\mathrm{vac}+\rho _\mathrm{ind})}{\partial \mu }={\displaystyle \sum _{j}^{}}{\displaystyle \frac{n_j (m_\mathrm{light}^4)_j}{32\pi ^2}}+ \frac{1}{32\pi ^2}{\displaystyle \sum _i}n_i (M_\mathrm{phys }^4)_i\nonumber \\&\times \int _0^1 \frac{x(1-x)\mu ^2 \mathrm{d}x}{(M_\mathrm{phys }^2)_i+x(1-x)\mu ^2}\nonumber \\ =&{\displaystyle \sum _{j}^{}}{\displaystyle \frac{n_j (m_\mathrm{light}^4)_j }{32\pi ^2}} +\frac{\mu ^2}{12 (4\pi )^2}\Bigg [-12 M_t^2+6 M_{W}^2+3 M_Z^2+M_H^2 \Bigg ] \nonumber \\&+ \frac{\mu ^4}{30 (4\pi )^2} +\mathcal {O}\ \Bigg (\frac{\mu ^6}{(M_\mathrm{phys}^2)_i}\Bigg ) \end{aligned}$$

The above expression is exactly of the form of (29) and proves the expected decoupling behavior in the effective theories. The light masses \(m_\mathrm{light}\) may be, again, generated by the Higgs vev \(m_\mathrm{light} (\langle \phi \rangle )\) such as a mass for, say, charm quark, or may be a new mass parameters in the SM Lagrangian related, for example, to the neutrino masses. As the \(\mu \)-scale slides down the energy, more and more SM masses will migrate from the \(m_\mathrm{light}^4\)-term to the inside of the brackets in the \(\mu ^2\)-term.

Notice that, when performing the sum over Goldstones, both beta functions for \(\rho _{\Lambda }^\mathrm{vac}\) (57) and \(\rho _\mathrm{ind}\) (54) have the term \(\sim M_H^4(2/M_W^2+1/M_Z^2)\) but since it comes with the opposite sign it cancels in (60). This demonstrates the importance of considering the RG running of the total \(\rho _{\Lambda }^\mathrm{vac}+\rho _\mathrm{ind}\) parameter rather than RG running of these contributions separately.Footnote 4

The \(\mu ^2 M_i^2\) term in the running of \(\rho _{\Lambda }^\mathrm{vac}+ \rho _\mathrm{ind}\) provides the leading RG effect due to the heavy SM particlesFootnote 5 and we may demand it to vanish as to reduce the fine-tuning in the physical value of the CC at the \(\mu _{c}=\mathcal {O}(10^{-3})\,\) eV. This requirement, however, leads to the SM prediction \(m_H \approx 550\) GeV, inconsistent within the experimental value of \(m_H\approx 125\) GeV.

As discussed in [12], heavy mass terms \(\mu ^2 M_i^2\) may also affect nucleosynthesis if we choose \(\mu \sim T\), because they would induce vacuum energy density \(\sim (T^2 M_i^2)/(4\pi )^2\) much bigger than the energy density of radiation \(\rho _\mathrm{rad}\) at the typical energy of nucleosynthesis \(T\sim 10^{-4}\) GeV. On the other hand, the \(m_\mathrm{light}^4\)- and \(\mu ^4\)-terms obey the constraint \(\rho _\mathrm{rad}<\rho _{\Lambda }^\mathrm{vac}+\rho _\mathrm{ind}\) in the energy interval relevant for nucleosynthesis. To avoid the problem, either we have to use alternative choice \(\mu \sim H\) or, again, sufficient amount of fine-tuning should be arranged among the various \(\mu ^2 M_i^2\) terms. Since, as we saw, the heavy SM spectrum does not have this tuning, our results imply that SM has to be extended or \(\mu \sim H\) choice is preferred over the \(\mu \sim T\) [12].

4 Massless theories

We now apply our result (58) to the massless theories which, as we will see, will give us new insights.

4.1 Massless Standard Model

In the massless limit of (60) \(m=0\) ( i.e. all the terms with \(\kappa _i^m\) absent), from (57) we have \(\rho _{\Lambda }^\mathrm{vac}=\) const and only \(\rho _\mathrm{ind}\) runs with \(\mu \). In this case, the \(\mu ^2\)-term can be related to the Veltman condition as we now show.

In the massless theory at the tree-level \(\langle \phi \rangle =0\), which means that the tree-level mass of the Higgs is zero and the electroweak symmetry needs to be broken radiatively. For this to happen, we need to balance the tree-level potential against the 1-loop contribution, so that for consistent perturbative expansion we have to impose the value of the Higgs quartic couplings at the electroweak scale to be parametrically given as \(\lambda \sim (g^4, g^{\prime ^4}, y_t^4)\). This allows us to neglect the \(\lambda \)-terms in (60) associated with the Higgs and Goldstones and we obtain (\(i=W, Z, t\) and neglecting the light masses \(m_\mathrm{light}\)):

$$\begin{aligned}&\mu \frac{\partial \rho _\mathrm{ind}}{\partial \mu }=\frac{ \langle \phi \rangle ^4 }{32\pi ^2}\sum _{i} n_i \kappa _i^2\ \int _0^1 \frac{x(1-x)\mu ^2 \mathrm{d}x}{(M_{\mathrm{phys} }^2)_i+x(1-x)\mu ^2}\nonumber \\&= \frac{\mu ^2}{12 (4\pi )^2}[-12 M_t^2+ 6 M_{W}^2+ 3 M_Z^2] + \frac{\mu ^4}{ 20 (4\pi )^2} \nonumber \\&=\frac{\mu ^2\langle \phi \rangle ^2}{12 (4\pi )^2}\frac{\partial }{\partial \phi ^2}\sum _i n_i M_i^2(\phi ) + \frac{\mu ^4}{ 20 (4\pi )^2}, \end{aligned}$$

where in the last line we used the fact that in the massless theory with only one background field \(\phi \), any mass can be written as \(M_i^2(\phi )=\kappa _i \phi ^2\). Notice that the \(\mu ^2\)-proportional term is nothing but the generalization of the well known Veltman condition i.e. the requirement of the absence of the quadratic divergence.Footnote 6 for the Higgs mass (cancellation of the prefactor of the \(\phi ^2\)-term). This means that in the massless case the fine-tuning problem of the Higgs mass is linked to the fine-tuning problem of the Cosmological Constant value Footnote 7

4.2 Massless Standard Model with extra massless real scalar

Let us now consider the simplest extension of the SM by adding one extra massless real scalar S:

$$\begin{aligned} V_0 = V_0^{SM}+ \lambda _{HS} \Phi ^\dagger \Phi \, S^2 + \frac{\lambda _S}{4} S^4 \end{aligned}$$

so that contribution from the Higgs background to the mass of the scalar S is given by \(M_S^2(\phi )=\lambda _{HS} \phi ^2\). In this model, the solution to the Veltman condition (61) readsFootnote 8

$$\begin{aligned}&12 M_t^2- 6 M_{W}^2-3 M_Z^2-M_S^2=0 \ \ \Longrightarrow \ \ \lambda _{HS}(\mu )\\&\quad = 6y_t^2(\mu )-\frac{9}{4}g^2(\mu )-\frac{3}{4}g^{\prime 2}(\mu ) \mathop {\approx }\limits ^{\mu \approx v_{EW}} 4.8 . \end{aligned}$$

Working in the parameter space of the model where \(\langle S \rangle \)=0, see [14] for details, leads to the scalar mass \(M_S=\sqrt{\lambda _{HS}}\ v_{EW} \approx 550\) GeV which we already noticed above in the massive version of the SM where the role of the scalar S was played by the Higgs. Remarkably, with the mass of the scalar S satisfying the Veltman condition, we correctly predict the one-loop induced Higgs mass from the Coleman–Weinberg potential

$$\begin{aligned} M^2_{H}= & {} \frac{3}{8\pi ^2} \left[ \frac{1}{16}\big (3g^4+2 g^2 g'^2+g'^4\big )-y_t^4+\frac{1}{3} \lambda _{HS}^2\right] v_{EW}^2 \quad \nonumber \\&\Longrightarrow \quad M_H \approx 125 \ \text {GeV}. \end{aligned}$$

This provides an interesting example of how the demand for the absence of leading RG effects in the running of the \(\rho _\mathrm{ind}\) due to the heavy particles may provide the hints on the possible extensions of the SM. Moreover, in this model there is no problem with nucleosynthesis for either of the choices for the RG scale \(\mu \sim T\) or \(\mu \sim H\).

5 Standard Model in the constant curvature space

For our final generalization of (58), we work with the full renormalized version of the Hilbert-Einstein action (2) containing additional coupling constants \(\kappa =(16\pi G)^{-1}=M_{pl}^2/2\), and non-minimal coupling \(\xi \). We consider the Standard Model in the constant curvature space \(R_{\mu \nu }=(R/4)g_{\mu \nu }\) and working in the linear curvature approximation in “Appendix B” we show that appropriate generalization of (43) is given by

$$\begin{aligned} {\displaystyle \mu \frac{\partial (\rho _{\Lambda }^\mathrm{vac}+\rho _\mathrm{ind}+\kappa R)}{\partial \mu }}={\displaystyle \sum _{i}^{}}{\displaystyle \frac{n_i}{32\pi ^2}} \mathcal {M}_i^4(\langle \phi \rangle ) \end{aligned}$$


$$\begin{aligned} \mathcal {M}_i^2(\langle \phi \rangle )=\kappa _i\langle \phi \rangle ^2-\kappa _i^m m^2 +\bigg (\kappa _i^{R}-\frac{1}{6}\bigg ) R \ , \end{aligned}$$

with parameters \(\kappa _i^R\) defined in Table 2 in “Appendix B”. Generalizing to the mass-dependent scheme we obtain (see “Appendix B” for details):

$$\begin{aligned}&{\displaystyle \mu \frac{\partial (\rho _{\Lambda }^\mathrm{vac}+\rho _\mathrm{ind}+\kappa R)}{\partial \mu }}={\displaystyle \sum _{i}^{}}{\displaystyle \frac{n_i}{32\pi ^2}} \mathcal {M}_i^4(\langle \phi \rangle )\nonumber \\&\times \int _0^1 \frac{x(1-x)\mu ^2 \mathrm{d}x}{\mathcal {M}^2_i(\langle \phi \rangle )+x(1-x)\mu ^2}\nonumber \\&={\displaystyle \sum _{j}^{}}{\displaystyle \frac{n_j (\mathcal {M}_{\mathrm{light}}^4)_j }{32\pi ^2}}+\frac{\mu ^2}{12 (4\pi )^2}\nonumber \\&\times \Bigg [-12 \widetilde{m}_t^2+6 \widetilde{m}_{W}^2+3 \widetilde{m}_Z^2+\widetilde{m}_H^2+\frac{7}{3} R\Bigg ]+\,\frac{\mu ^4}{30 (4\pi )^2},\nonumber \\ \end{aligned}$$

where masses \(\widetilde{m}_i^2\) have corrections from the non-minimal Higgs coupling \(\xi \)

$$\begin{aligned} \widetilde{m}^2_i\equiv M_i^2(\langle \phi \rangle )-2\kappa _i\frac{ \xi R}{\lambda }\ \end{aligned}$$

with \(M_i^2(\langle \phi \rangle )\) defined in (44). The result (68) generalizes effective theory expansion (29) to the constant curvature space

$$\begin{aligned} \beta \bigg (m_{\mathrm{light}},\frac{\mu }{m}\bigg )=a_1\,m_{\mathrm{light}}^{4}+b_1\, \mu ^{2}m^{2} + c_1 \,\mu ^{4} + d_1 \mu ^{2}R +\cdots \end{aligned}$$

which also appears via explicit calculations on the expanding cosmological background where vacuum energy is dynamical [15]. The result (68) also generalizes the flat space result to possibility of, for example, curvature-induced running of the vacuum energy and curvature-induced phase transitions [16,17,18,19,20,21].

6 Conclusions

We revisited the decoupling effects associated with heavy particles in the RG running of the vacuum energy using the mass-dependent renormalization scheme. We derived the universal one-loop beta function of the vacuum energy \(\rho _{\Lambda }^\mathrm{vac}+\rho _\mathrm{ind}\), arising from the Higgs vacuum and the Cosmological Constant term in the entire energy range, valid in the UV and in the IR regime. We have shown that although \(\rho _{\Lambda }^\mathrm{vac}\) and \(\rho _\mathrm{ind}\) run separately, it is only the sum \(\rho _{\Lambda }^\mathrm{vac}+\rho _\mathrm{ind}\) that exhibits behavior consistent with the decoupling theorem.

At the energy scale lower than the mass of the particle, the leading term in the RG running of \(\rho _{\Lambda }^\mathrm{vac}+\rho _\mathrm{ind}\) is proportional to the square of the mass of the heavy particle which leads to the enhanced RG running and, consequently, severe fine-tuning problem with the measured value of the Cosmological Constant. We show that the condition of absence of this leading effect is not satisfied in the SM, while in the massless theories, where Higgs mass is generated radiatively via Coleman–Weinberg mechanism, this constraint formally coincides with Veltman condition. In a simple extension of the SM with addition of one massless real scalar the condition of absence of leading effect in \(\beta _{\rho _\mathrm{ind}}\) allowed us to predict the radiative Higgs mass correctly.

Finally, we also provided the generalization to the constant curvature space in the linear curvature approximation finding the effective field theory expansion that also appears via explicit calculations on the expanding cosmological background. In view of this, our results also might have impact on the models based on the dynamical cosmological constant which were confronted with new cosmological observations in [22, 23] with the results being still inconclusive [24, 25].