Emergent Gravity as the Eraser of Anomalous Gauge Boson Masses, and QFT-GR Concord

The problem of reconciling QFTs with the GR has been under intense study for several decades. The concord between the special relativity and quantum mechanics that led to QFTs seems to be not evident, if not absent, when it comes to the GR and QFTs. This problem has been approached by exploring various possibilities.

The first possibility is to quantize the GR but it is known to be not a renormalizable field theory thooft ; stelle . In general, quantization of gravity is not a straightforward task, and this difficulty has given cause to various alternative approaches qgr-review (string/M theory, loop quantum gravity, asymptotic safety, supergravity, $\dots$). The problem here is actually absence of a renormalizable ultraviolet (UV) completion of the GR (like completion of the Fermi theory by the electroweak theory). Even so, it must be kept in mind that, independent of the details of the UV completion, leading quantum corrections can be reliably calculated as they are determined by the low-energy couplings of the massless fields in the completion donoghue ; holdom . By a similar token, asymptotic safety of the GR is known to lead to a reliable prediction of the Higgs boson mass safety .

The second possibility is to keep the GR classical and make QFT to fit to it qft-cuvred . This route, which may be motivated by the views qgr2 that the gravity could be fundamentally classical, is hindered by the fact that curved spacetime does not allow for preferred states like the vacuum and detectable structures like the particles (positive frequency Fourier components) cgr2 . Indeed, allowing no privileged coordinates, curved spacetimes do not accommodate a unique vacuum state as diffeomorphisms shuffle negative and positive frequency modes and one observer’s vacuum can become another observer’s excited state. Besides, incoming and outgoing particles and their scattering amplitudes are properly defined mainly in symmetric woodard and asymptotically flat hawking spacetimes. These hindering features of the curved geometry have been attempted to overcome, respectively, by promoting the operator product expansion in QFTs to a fundamental status wald-ilk and by introducing microlocal spectrum conditions fredenhagen . These attempts, which leave the mechanism underlying the Newton-Cavendish constant and other couplings to a future quantum gravity theory, have led to axiomatic QFTs wald-sonra in curved spacetime.

The third possibility is to induce the GR from fluctuations of the quantum fields in flat spacetime. (It is possible to consider also curved spacetime but for loop-induced GR to arise outrightly one must find a way of removing at least the Cavendish-Newton constant in the classical action, let alone the problems noted in wald-ilk . One possibility is to invoke classical scale invariance but it removes not only the Cavendish-Newton constant but also all the field masses, including the Higgs mass scale .) This approach, which rests on flat spacetime where QFTs work properly with well-defined vacuum states and particle spectra wald-ilk ; wald ; ashtekar , has the potential to avoid principal difficulties with curved spacetime QFTs wald ; fredenhagen . To this end, Sakharov’s induced gravity sakharov (see also ruzma , and see especially the furthering analysis in visser ) provides a viable route as it starts with flat spacetime QFT, and induces curvature sector at one loop through the curved metric for the fields in the loops. The gravitational constant, viewed as metrical elastic constant of the spacetime sakharov ; ruzma ; demir2015 , emerges in a form proportional to the UV cutoff sakharov ; visser ; demir2017 , with implications for black holes fursaev and Liouville gravity liouville . (Besides the Sakharov’s, various attempts have been made to generate gravity from quantized matter such as entropic gravity emerge-verlinde , entanglement gravity emerge-raamsdonk , analog gravity emerge-analog , broken symmetry-induced gravity adler , Nambu-Jona-Lasinio type gravity terazawa , and agravity strumia , with a critical review in emerge-lit ).

The present work has the same base setup as Sakharov’s induced gravity: A flat spacetime effective QFT with the usual $\log\Lambda_{\wp}$, $\Lambda_{\wp}^{2}$ and $\Lambda_{\wp}^{4}$ loop corrections, $\Lambda_{\wp}$ being the UV cutoff sakharov ; visser . The loop corrections, giving cause to various UV sensitivity problems veltman ; ccp ; ccb , are specific to the flat spacetime demir2019 such that $\Lambda_{\wp}^{2}$ breaks explicitly both the gauge and Poincare (translation) symmetries wigner while $\log\Lambda_{\wp}$ breaks none of them. This symmetry structure paves the way for a covariance relation between $\Lambda_{\wp}^{2}$ and (Poincare-breaking) spacetime curvature, with intact $\log\Lambda_{\wp}$. This novel covariance relation retains the effective QFT with sole $\log\Lambda_{\wp}$ corrections (dimensional regularization) dim-reg2 ; dim-reg3 , and induces the GR in a way removing the anomalous gauge boson masses (which arise from matter loops in exact proportion to $\Lambda_{\wp}$ and which explicitly break charge and color (CCB) symmetries ccb ). The end result is a QFT-GR chime or concord.

In what follows, Sec. 2 explicates the UV sensitivity problems, including the explicit CCB stemming from anomalous gauge boson masses anomaly ; ccb ; demir2016 . Sec. 3 reveals that effective QFTs, unlike classical field theories, necessitate, in getting to curved spacetime, curvature sector to be spanned exclusively by the QFT mass scales. Sec. 4 shows that $\Lambda_{\wp}^{2}$ breaks explicitly both the gauge and Poincare symmetries while $\log\Lambda_{\wp}$ respects both of them, and one takes note of the Poincare affinity between $\Lambda_{\wp}^{2}$-induced broken Poincare (translation) invariance and nonzero spacetime curvature, with intact $\log\Lambda_{\wp}$. Sec. 5 extends the usual general covariance between the flat and curved metrics by introducing a covariance relation between $\Lambda_{\wp}^{2}$ and affine curvature, and shows by a step-by-step analysis that the anomalous gauge boson masses are wiped out (namely, CCB is solved dynamically up to doubly Planck-suppressed terms) by the emergence of curvature. Sec. 6 carries the entire flat spacetime effective QFT into curved spacetime via the extended general covariance, and shows that an intertwined whole of a purely loop-induced GR plus the same QFT with sheer $\log\Lambda_{\wp}$ regularization (equivalent to dimensional regularization cutoff-dimreg ) emerges to form a QFT-GR concord. Sec. 6 discusses some salient aspects of the concord in view of the Starobinsky inflation starobinsky ; Planck , cosmological constant problem (CCP) ccp ; ccp2 , little hierarchy problem little , and induction and field-dependence of the Newton-Cavendish constant non-min ; codata . Sec. 6 also discusses the standard model (SM) and shows how symmergence necessitates a new physics sector demir2016 ; demir2019 and what effects this new physics sector might have on cosmological de ; darksector ; cos-coll , astrophysical cdm ; darksector and collider collider1 ; collider2 phenomena. Sec. 7 concludes the work, and gives prospects for future research by highlighting the salient aspects of the QFT-GR concord.

The UV cutoff $\Lambda_{\wp}$ is a physical scale. It cuts off the loop momenta to lead to finite physical loop corrections. It is not subject to any specific bound (like unitarity bound from graviton exchange) in flat spacetime. Indeed, for flat spacetime QFTs gravity is a completely alien interaction. For incorporating gravity into QFTs, QFTs must be endowed with a mass scale evoking the fundamental scale of gravity or the curvature. The UV cutoff $\Lambda_{\wp}$ is the aforementioned mass scale. It has a physical correspondent in the gravity sector. It therefore is not a regularization scale. Its effects cannot therefore be imitated by the cutoff regularization polchinski or the dimensional regularization dim-reg2 ; dim-reg3 or any other regularization scheme.

In the presence of the UV momentum cutoff $\Lambda_{\wp}$, each coupling in a flat spacetime QFT (of various fields $\psi_{i}$ with masses $m_{i}$) develops a certain $\Lambda_{\wp}$ sensitivity via the matter loops eff-qft ; polchinski ; weinberg-eff . The sensitivity varies with the mass dimension of the operators involved and, in this regard, a systematic analysis of particle masses and vacuum energy proves useful:

1. (1)

   Massless gauge bosons $V_{\mu}$ (like the photon and the gluon) acquire masses

   $\displaystyle\delta M_{V}^{2}=c_{V}\Lambda_{\wp}^{2}$

   (1)

   with a loop factor $c_{V}$ that can involve $\log\Lambda_{\wp}$ at higher loops. These purely loop-induced masses are plain quadratic in $\Lambda_{\wp}$, that is, there exist no logarithmic contributions to $\delta M_{V}^{2}$ (as they live in the transverse part of the $V_{\mu}$ self-energy). It is clear that $\delta M_{V}^{2}$ are anomalous gauge boson masses, which give cause to explicit CCB ccb (see casas for spontaneous CCB), as exemplified in Table 1 for the SM gauge bosons. Here, it must be emphasized that $\delta M_{V}^{2}$ are physical corrections rather than cutoff regularization terms, and their effects therefore are not subject to any (finite or otherwise) renormalization prescription, and the anomalies they give cause to are true anomalies.

2. (2)

   The corrections to scalar masses

   $\displaystyle\delta m_{\phi}^{2}=c_{\phi}\Lambda_{\wp}^{2}+\sum_{i}{c}^{(l)}_{\phi\psi_{i}}m_{i}^{2}\log\frac{m_{i}^{2}}{\Lambda_{\wp}^{2}}$

   (2)

   contain both $\Lambda_{\wp}^{2}$ and $m_{i}^{2}$ contributions. The former, whose loop factor $c_{\phi}$ is given in Table 1 for the SM Higgs boson, gives rise to the big hierarchy problem veltman . The logarithmic $m_{i}^{2}$ contribution, on the other hand, involves a loop factor ${c}^{(l)}_{\phi\psi_{i}}$, and gives cause for the little hierarchy problem little .

3. (3)

   Finally, the shift in the vacuum energy

   $\displaystyle\delta V=c_{\varnothing}\Lambda_{\wp}^{4}+\sum_{i}{c}_{\psi_{i}}m_{i}^{2}\Lambda_{\wp}^{2}+\sum_{i}{c}^{(l)}_{\varnothing\psi_{i}}m_{i}^{4}\log\frac{m_{i}^{2}}{\Lambda_{\wp}^{2}}$

   (3)

   involves quartics and quadratics of both $\Lambda_{\wp}$ and $m_{i}$. The loop factors $c_{\varnothing}$ and ${c}_{\psi_{i}}$ are given in Table 1 for a generic QFT as well as the SM. The shift in the vacuum energy gathers both scales marginally. It causes no problem (like the CCP ccp ; ccp2 ) in flat spacetime.

Classical field theories landau , governed by actions $S_{c}\left(\eta,\psi,\partial\psi\right)$ of various fields $\psi_{i}$ in the flat spacetime of metric $\eta_{\mu\nu}$, are carried into curved spacetime of a metric $g_{\mu\nu}$ by letting

in accordance with general covariance norton , which itself is expressed by the map

such that the Levi-Civita connection

sets the covariant derivative $\nabla_{\mu}$ in (5), and gives rise to the Ricci curvature

as well as the scalar curvature $R(g)=g^{\mu\nu}R_{\mu\nu}({}^{g}\Gamma)$. The curvature sector in (4), added by hand for the curved metric $g_{\mu\nu}$ to be able to gain dynamics, must be of the specific form

for it to be able reproduce the GR at the leading order (with gravitational constant ${\widetilde{G}}$ and curvature couplings ${\tilde{c}}_{i}$). The unknown constants in (8), which are at the same footing as the bare parameters in the QFT sector, cannot in general be avoided or limited simply because curvature sector (kinetic terms for $g_{\mu\nu}$) can contain arbitrary curvature invariants kretschmann (in contrast to gauge theories where renormalizability singles out a specific kinetic term).

The success with classical field theories prompts an important question: Can general covariance carry also effective QFTs into curved spacetime? This question is important because effective QFTs, obtained by integrating out all high-frequency quantum fluctuations of the QFT fields, are reminiscent of the classical field theories in view of their long-wavelength field spectrum (with loop-corrected tree-level couplings). The answer lies in the effective action (leaving out higher-dimension operators) demir2019 ; demir2017

in which $S_{c}\left(\eta,\psi\right)$ is the classical action,

is the logarithmic action composed of the $\log\Lambda_{\wp}$ parts of (3) and (2),

is the vacuum plus scalar mass action gathering $\Lambda_{\wp}^{4}$ and $\Lambda_{\wp}^{2}$ parts of (3) and (2), and

is the anomalous gauge boson mass action formed by the purely quadratic corrections in (1).

If the effective QFTs described by (9) are really like the classical field theories then the general covariance map in (5) take them into curved spacetime as

in parallel with the transformation of the classical action in (4), with the curvature sector defined in (8). The problem with this transformation is that the parameters ${\widetilde{G}}$, ${\tilde{c}_{2}}$, ${\tilde{c}_{3}}$, $\cdots$ in the curvature sector are all bare, and thus, they are not at the same footing as the loop-corrected constants in the effective QFT. They remain bare since matter loops have already been used up in forming the flat spacetime effective action $S_{eff}\left(\eta,\psi,\Lambda_{\wp}\right)$ in (9), and there have remained thus no loops (quantum fluctuations) to induce or correct any interaction like (8) or not. This means that the curvature sector parameters bear no sensitivity to $\Lambda_{\wp}$. This discord between the two sectors implies that the parameters ${\widetilde{G}}$, ${\tilde{c}_{2}}$, ${\tilde{c}_{3}}$, $\cdots$ are in fact all incalculable demir2016 ; demir2019 . This problem, which reveals the difference between the classical field theories and the effective QFTs, can be overcome if the curvature sector in effective QFTs is also loop-corrected or loop-induced. Namely, curvature sector must arise from the effective QFT itself during the map of $S_{eff}\left(\eta,\psi,\Lambda_{\wp}\right)$ into curved spacetime. This requirement falls outside the workings of the general covariance map in (5) as it involves only the flat metric $\eta_{\mu\nu}$ in $S_{eff}\left(\eta,\psi,\Lambda_{\wp}\right)$. This means that it is necessary to construct a whole new transformation rule for taking effective QFTs into curved spacetime. In this regard, this first stage would be the determination of what parameters to transform other than the flat metric: Field masses $m_{i}$? The UV cutoff $\Lambda_{\wp}$? Some other scale in $S_{eff}\left(\eta,\psi,\Lambda_{\wp}\right)$? The second stage would be the determination of the transformation rule itself (presumably some extension of the general covariance). The question of what to transform other than $\eta_{\mu\nu}$ will be answered in Sec. 4 below by revealing the symmetry properties of different UV sensitivities in $S_{eff}\left(\eta,\psi,\Lambda_{\wp}\right)$. The transformation rule itself, on the other hand, will be determined in Sec. 5 by requiring that the transformation must be able to erase the anomalous gauge boson masses in (1).

The UV cutoff $\Lambda_{\wp}$ is not just a mass scale. It is more than that. To see why, it suffices to scrutinize the two distinct roles it plays in shaping the effective QFTs:

1. (1)

   The $\Lambda_{\wp}^{2}$ correction is additive as ensured by the scalar masses (2). It breaks gauge symmetries explicitly as proven by the gauge boson masses (1). It breaks also Poincare (translation) symmetry wigner ; wald as it restricts loop momenta $\ell_{\mu}$ into the range $-\Lambda_{\wp}^{2}\leq\eta_{\mu\nu}\ell^{\mu}\ell^{\nu}\leq\Lambda_{\wp}^{2}$. Thus, $\Lambda_{\wp}^{2}$ has an affinity for spacetime curvature as both of them break the Poincare symmetry.

2. (2)

   The $\log\Lambda_{\wp}$ correction is always multiplicative. It does not alter the symmetry structure of the quantity it multiplies. The field masses $m_{i}$, for instance, respect both the gauge and Poincare symmetries and so do the logarithmic corrections $\delta m^{2}_{i}\propto m^{2}_{i}\log\Lambda_{\wp}$. (The scalar masses in (2) set an example with $m_{\phi}^{2}$ being the Casimir invariants of the Poincare group wigner .) The same is true for all the QFT couplings. In parallel with this, as follows from the item (1) above, $\Lambda_{\wp}^{2}$ breaks both the gauge and Poincare symmetries and so does the $\Lambda_{\wp}^{2}\log\Lambda_{\wp}$ (which can arise at higher loops). It thus turns out that $\log\Lambda_{\wp}$ respects both the gauge and Poincare symmetries. It can have therefore no affinity for (Poincare-breaking) spacetime curvature.

The two distinct roles played by $\Lambda_{\wp}$ are contrasted in Table 2. The main lesson is that $\log\Lambda_{\wp}$ must remain intact under a possible correspondence between $\Lambda_{\wp}^{2}$ and curvature on the basis of their Poincare affinity.

In view of the conclusion arrived in Sec. 3, spacetime curvature must emerge from within $S_{eff}\left(\eta,\psi,\Lambda_{\wp}\right)$ in order not to hinge on arbitrary, incalculable constants as in (13). This condition can be taken to imply that there must exist some covariance relation between the mass scales in $S_{eff}\left(\eta,\psi,\Lambda_{\wp}\right)$ and spacetime curvature. To determine if there exists such a relation, it proves effectual to focus first on the gauge sector, whose anomaly action (12) breaks gauge symmetries explicitly ccb . It continues to break in curved spacetime if carried there via (13). But the gauge anomaly (12) is a pure $\Lambda_{\wp}^{2}$ effect and, in view of the Poincare affinity between $\Lambda_{\wp}^{2}$ and curvature (as revealed in Sec. 4, Table 2), it is legitimate to ask a pivotal question: Is it possible to carry the effective QFT in (9) into curved spacetime in a way erasing the anomalous gauge boson mass term (12)? It will take a set of progressive steps to find out but the answer will turn out to be “yes” demir2016 ; demir2019 :

Step 1. The starting point of investigation is the self-evident identity

involving the gauge-invariant kinetic construct

whose second line, obtained via by-parts integration of the first line, consists of a surface term (the total divergence) and inverse propagator $D_{\mu\nu}^{2}=D^{2}\eta_{\mu\nu}-D_{\mu}D_{\nu}-iV_{\mu\nu}$ with $D^{2}=\eta^{\mu\nu}D_{\mu}D_{\nu}$ such that $D_{\mu}=\partial_{\mu}+iV_{\mu}$ is the gauge-covariant derivative, $V_{\mu}=V_{\mu}^{a}T^{a}$ is the gauge field with gauge group generators $T^{a}$, and $V_{\mu\nu}=V^{a}_{\mu\nu}T^{a}$ is the field strength tensor with the components $V^{a}_{\mu\nu}=\partial_{\mu}V^{a}_{\nu}-\partial_{\nu}V^{a}_{\mu}+if^{abc}V^{b}_{\mu}V^{c}_{\nu}$ ($f^{abc}$ are structure constants). Now, the identity (14) can be put into the equivalent form

after, at the right-hand side of (14), $\delta S_{V}$ is replaced with its expression in (12), $``+I_{V}"$ is replaced with its expression in (16), and yet $``-I_{V}"$ is left untouched (kept as in (15)).

Step 2. Now, the rearranged gauge boson anomalous mass action in (17) gets to curved spacetime via the general covariance map (5) to take there the “curved” form

in which $\nabla_{\mu}$ is the spacetime covariant derivative with respect to the Levi-Civita connection (6), ${\mathcal{D}}_{\mu}=\nabla_{\mu}+iV_{\mu}$ is the gauge-covariant derivative with respect to $\nabla_{\mu}$ so that ${\mathcal{D}}^{2}=g^{\mu\nu}{\mathcal{D}}_{\mu}{\mathcal{D}}_{\nu}$ and ${\mathcal{D}}_{\mu\nu}^{2}={\mathcal{D}}^{2}g_{\mu\nu}-{\mathcal{D}}_{\mu}{\mathcal{D}}_{\nu}-iV_{\mu\nu}$.

Step 3. Now, a closer look at the action (18) reveals a crucial property:

1. (1)

   if $\Lambda_{\wp}^{2}g_{\mu\nu}$ were replaced with the Ricci curvature $R_{\mu\nu}\left({}^{g}\Gamma\right)$, and

2. (2)

   if $c_{V}$ (which can involve $\log\Lambda_{\wp}$ at higher loops) were held intact under (1)

then ${{\delta S_{V}\left(g,\Lambda_{\wp}^{2}\right)}}$ would reduce to zero identically. To see this, one first replaces $\Lambda_{\wp}^{2}g_{\mu\nu}$ with $R_{\mu\nu}\left({}^{g}\Gamma\right)$ in (18) to get

and then integrates (19) by parts using $\left[{\mathcal{D}}_{\mu},{\mathcal{D}}_{\nu}\right]=R_{\mu\nu}\left({}^{g}\Gamma\right)+iV_{\mu\nu}$ to arrive at

which reduces to zero identically, as claimed above. This result holds provided that $\Lambda_{\wp}^{2}g_{\mu\nu}$ is replaced with $R_{\mu\nu}\left({}^{g}\Gamma\right)$ in (18) and provided that this replacement leaves $c_{V}$ (in fact, $\log\Lambda_{\wp}$) intact. It is striking that the conditions (1) and (2) above are, respectively, the first and the second rows of Table 2 in Sec. 4. This accord between the erasure of the anomalous gauge boson masses and the Poincare structure of the UV sensitivities of the QFT can be taken as a confirmation of the applied method.

It seems all fine. But actually there is a serious inconsistency problem here. Indeed, in the flat limit ($g_{\mu\nu}\leadsto\eta_{\mu\nu}$) curvature remains nonzero ($R_{\mu\nu}\left({}^{g}\Gamma\right)\leadsto\Lambda_{\wp}^{2}\eta_{\mu\nu}$). If it were not for this contradiction metamorphosis of $\Lambda_{\wp}^{2}g_{\mu\nu}$ into curvature (confirmed by Table 2) would completely erase the anomalous gauge boson mass (12) and solve the CCB demir2016 ; demir2019 .

Step 4. The inconsistency above can be remedied by introducing a more general map affine ; demir2019

in which ${\mathbb{R}}_{\mu\nu}(\Gamma)$ is the Ricci curvature of a symmetric affine connection $\Gamma^{\lambda}_{\mu\nu}$. (Here, $\Gamma^{\lambda}_{\mu\nu}$ and ${\mathbb{R}}_{\mu\nu}(\Gamma)$ have, respectively, nothing to do with ${}^{g}\Gamma^{\lambda}_{\mu\nu}$ in (6) and $R_{\mu\nu}({}^{g}\Gamma)$ in (7), as shown contrastively in Table 3.) The metamorphosis of $\Lambda_{\wp}^{2}g_{\mu\nu}$ into ${\mathbb{R}}_{\mu\nu}(\Gamma)$ goes parallel with the metamorphosis of $\eta_{\mu\nu}$ into $g_{\mu\nu}$ as a correspondence between physical quantities in the flat and curved spacetimes, and removes the inconsistency since the two maps, (5) and (21), involve independent dynamical variables. In fact, affine curvature can well be the substance that fixes the vacuousity kretschmann of general covariance. In view of this fixture, it proves efficacious to introduce the extended general covariance (EGC)

by combining the affine curvature map in (21) with the general covariance map in (5) on the effective action $S_{eff}$ in (9). The underlying symmetry structure is given in Table 2 in Sec. 4. The EGC reduces to the usual general covariance when $\Lambda_{\wp}$ is absent.

Step 5. The EGC map in (22) takes the action (18) into

which reduces to

for the same reason that (19) reduced to (20). This resultant action, which shows metamorphosis of the anomalous gauge boson masses in (12) into curvature terms, is a truly metric-affine action affine ; damianos in that it involves both the affine and metrical curvatures (see Table 3). The “$\delta S_{V}$” is no longer a gauge boson mass term. It is in this sense that the anamalous gauge boson masses get erased. The fate of the anomaly (the CCB), as will be analyzed below, is determined by the dynamics of the affine connection $\Gamma^{\lambda}_{\mu\nu}$.

Step 6. It is the curvature sector that decides on if ${\mathbb{R}}_{\mu\nu}(\Gamma)$ comes close to $R_{\mu\nu}\left({}^{g}\Gamma\right)$ to suppress the gauge anomaly, that is, the action (24). It originates from (11) plus (12) by way of the EGC in (22), and takes the compact form

after utilizing (24) and introducing

as a disformal metric disformal typical of the metric-affine geometry affine ; damianos ; shimada . (Transmutations of various objects from geometry to geometry are given in Table 3.) The metric-affine curvature sector (25), which involves not a single incalculable constant, is precisely the structure anticipated at the end of Sec. 3 after realizing problems with the by-hand curvature sector in (8). The EGC in (22) seems to have done the job.

Step 7. The affine dynamics, which follows from the requirement that the curvature sector (25) must remain stationary against variations in $\Gamma^{\lambda}_{\mu\nu}$, takes the compact form (affine covariant derivative ${}^{\Gamma}\nabla$ defined in Table 3)

after replacing the affine curvature affine ; damianos

in the curvature sector in (25). The solution of the equation of motion (27)

is a first order nonlinear partial differential equation for $\Gamma^{\lambda}_{\mu\nu}$ since ${\mathbb{Q}}_{\mu\nu}$ involves not only the scalars $\phi$ and vectors $V_{\mu}$ but also the affine curvature ${\mathbb{R}}_{\mu\nu}(\Gamma)$ in (28). This means that $\Gamma^{\lambda}_{\mu\nu}$ from (29) can have degrees of freedom beyond $\phi$, $V_{\mu}$ and ${}^{g}\Gamma^{\lambda}_{\mu\nu}$. Nevertheless, a short glance at ${\mathbb{Q}}_{\mu\nu}$ reveals that it has in it the inverse Newton-Cavendish constant

which is set by the masses $m_{i}$ of the QFT fields $\psi_{i}$ and vacuum expectation values $\langle\phi\rangle$ of the QFT scalars. It is the largest known mass scale (the Planck scale $M_{Pl}=(8\pi G_{N})^{-1/2}$), and its enormity enables one to expand $({\mathbb{Q}}^{-1})^{\mu\nu}$ as

so that $\Gamma^{\lambda}_{\mu\nu}$ in (29) becomes

and ${\mathbb{R}}_{\mu\nu}\left(\Gamma\right)$ in (28) takes the form

where $\left(\nabla^{2}\right)_{\mu\nu}^{\alpha\beta}=\nabla^{\alpha}\,\nabla_{\mu}\delta^{\beta}_{\nu}+\nabla^{\beta}\,\nabla_{\mu}\delta^{\alpha}_{\nu}-\Box\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}-\nabla_{\mu}\,\nabla_{\nu}g^{\alpha\beta}+(\mu\leftrightarrow\nu)$ with $\Box=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$. In the expansions (32) and (33), iteration of ${\mathbb{R}}_{\mu\nu}(\Gamma)$ order by order in $G_{N}$ reveals that dependencies on ${\mathbb{R}}_{\mu\nu}(\Gamma)$ reside always in the remainder (one higher-order in $G_{N}$). This means that $\Gamma^{\lambda}_{\mu\nu}$ and ${\mathbb{R}}_{\mu\nu}(\Gamma)$ get effectively integrated out of the dynamics to leave behind only the scalars $\phi$, vectors $V_{\mu}$ and the Levi-Civita connection ${}^{g}\Gamma^{\lambda}_{\mu\nu}$. This solution of ${\mathbb{R}}_{\mu\nu}(\Gamma)$ causes the action (24) to vanish

up to an anomaly-plagued ${\mathcal{O}}\!\left(G_{N}\right)$ remainder

which is an all-order derivative interaction following from (31). It never generates any mass for $\phi$ and $V_{\mu}$. In effect, anomalous gauge boson masses have been completely erased. The remnant anomaly effects in (35), which go like $G_{N}E^{2}$ at boson-boson (say, photon-photon, Higgs-gluon, $\dots$) collisions of energy $E$, are too tiny to leave any detectable signatures at current collider experiments. They become important though at scales near $G_{N}^{-1/2}$ as then the power series expansion in (31) fails, the affine connection remains unintegrated-out, and the fifth-force type effects start coming into play will ; 5th .

In summary, the EGC (22) has converted the anomalous gauge boson mass term (12) into the metric-affine action (24), and integration of the affine curvature out of the dynamics has killed (24) to leave behind only the doubly Planck-suppressed CCB-plagued remainder in (35). The key determinant is the EGC. It has enabled curvature to symmerge, that is, emerge in a way restoring gauge symmetries (albeit with an ${\mathcal{O}}\left(G_{N}\right)$ tiny breaking anomaly ; ccb ).

It is now time to carry the entire flat spacetime effective QFT in (9) into curved spacetime. To this end, the EGC in (22), which has arrived on the scene as if a deus ex machina, provides the requisite transformation rules (in parallel with Table 2).

The power-law part of the flat spacetime effective action in (9), which has turned into the metric-affine curvature sector in (25) via the EGC in (22), leads to the GR action

after integrating out ${\mathbb{R}}_{\mu\nu}(\Gamma)$ from (25) via its solution in (33). Each and every coupling ($G_{N},c_{\phi},c_{\varnothing},\dots$) in this action is a bona fide quantum effect. In fact, the UV sensitivity problems revealed in Sec. 2 are seen to have all disappeared en route to (36). They have disappeared through the EGC as follows:

1. (a)

   EGC suppressed the CCB anomaly ; demir2016 ; ccb by mapping the anomalous gauge boson masses in (1) into the ${\mathcal{O}}\left(G_{N}\right)$ derivative interactions in (35),

2. (b)

   EGC eliminated the big hierarchy problem veltman by converting the $\Lambda_{\wp}^{2}$ part of the scalar masses in (2) into non-minimal coupling $c_{\phi}/4$ between the scalars $\phi$ and the curvature scalar $R(g)$ non-min (This does not mean that the hierarchy problem is solved because logarithmic corrections in (2) are not dealt with yet.), and

3. (c)

   EGC prevented occurrence of the CCP ccp ; ccp2 by transfiguring the $\Lambda_{\wp}^{2}$ and $\Lambda_{\wp}^{4}$ parts of the vacuum energy (3) into the Einstein-Hilbert and quadratic curvature terms fR , respectively (This does not mean that the CCP is solved because logarithmic corrections in (3) are not dealt with yet.).

The resolution of these notorious power-law UV sensitivity problems is one important difference between symmergent gravity and Sakharov’s induced gravity sakharov ; visser .

The logarithmic part $\delta S_{l}\left(\eta,\psi,\log\Lambda_{\wp}\right)$ plus the classical part $S_{c}\left(\eta,\psi\right)$ of the flat spacetime effective action (9) gets to curved spacetime

since $\log\Lambda_{\wp}$ remains unchanged under the EGC map in (22). In essence, the logarithmic UV sensitivity is equivalent to the dimensional regularization cutoff-dimreg . Indeed, the formal equivalence $\log\Lambda_{\wp}^{2}\equiv{1}/{\epsilon}-\gamma_{E}+1+\log 4\pi\mu^{2}$ can always be used to translate (37) into dimensional regularization scheme in $4+\epsilon$ dimensions with Euler-Mascheroni constant $\gamma_{E}\approx 0.577$ and renormalization scale $\mu$. The removal of the $1/\epsilon$ pieces, for instance, corresponds to minimal subtraction scheme renormalization of (37) dim-reg2 ; dim-reg3 .

The EGC images in (36) and (37) of the flat spacetime effective action in (9) combine to form the intertwined whole

which describes matter by the regularized QFT and geometry by the GR. It is the QFT-GR concord sought for. Its primary features are as follows:

1. 1.

   The fact that a QFT-GR concord is formed can be understood via scattering amplitudes. To this end, as an illustration, it can prove useful to take a glance at $f_{1}f_{2}\rightarrow\phi_{1}\phi_{2}\phi_{3}$ scattering – coannihilation of two fermions $f_{1}$ and $f_{2}$ into three scalars $\phi_{1}$, $\phi_{2}$, $\phi_{3}$. It rests on the field operator structure

   	$\displaystyle\int d^{4}x\sqrt{-g(x)}\int d^{4}y\sqrt{-g(y)}\,{\overline{f}}_{1}(x)f_{2}(x)\Bigg{\{}h_{f_{1}f_{2}\phi}\Delta_{\phi}(x,y)\lambda_{\phi\phi_{1}\phi_{2}\phi_{3}}$
   	$\displaystyle+\frac{{\widetilde{\lambda}}_{f_{1}f_{2}\phi_{1}\phi_{2}\phi_{3}}}{\Lambda_{QFT}^{2}}\delta^{(4)}(x,y)\Bigg{\}}{\phi}_{1}(y){\phi}^{\dagger}_{2}(y){\phi}_{3}(y)$		(39)

   which has a non-contact part mediated by the scalar propagator $\Delta_{\phi}$, and a contact part suppressed by the QFT scale $\Lambda_{QFT}$ (and hence left out of the effective action (9)). The Yukawa coupling $h$, the quartic coupling $\lambda$, and the scalar mass $m_{\phi}$ in $\Delta_{\phi}$ are not tree-level objects but rather loop-level objects corrected by the flat spacetime loops. Likewise, the higher-order coupling ${\widetilde{\lambda}}$ originates from the flat spacetime loops. The field operators (like $\phi_{1}$ and $f_{2}$), on the other hand, are actually mean fields (${\bar{\phi}}_{1}$ and ${\bar{f}}_{2}$) averaged over their quantum fluctuations while forming the flat spacetime effective action in (9). In symmergence, therefore, operator structures like (39) attain the mean-field form

   	$\displaystyle\int d^{4}x\sqrt{-g(x)}\int d^{4}y\sqrt{-g(y)}\,{\overline{\bar{f}}}_{1}(x){\bar{f}}_{2}(x)\Bigg{\{}h_{f_{1}f_{2}\phi}\Delta_{\phi}(x,y)\lambda_{\phi\phi_{1}\phi_{2}\phi_{3}}$
   	$\displaystyle+\frac{{\widetilde{\lambda}}_{f_{1}f_{2}\phi_{1}\phi_{2}\phi_{3}}}{\Lambda_{QFT}^{2}}\delta^{(4)}(x,y)\Bigg{\}}{\bar{\phi}}_{1}(y){\bar{\phi}}^{\dagger}_{2}(y){\bar{\phi}}_{3}(y)$		(40)

   whose fields (like ${\tilde{\phi}}_{1}$ and ${\tilde{f}}_{2}$) are solutions of the associated wave equations in the curved spacetime of the metric $g_{\mu\nu}$. This means that scattering and decay rates can be analyzed as interactions of the relativistic wavefunctions in curved spacetime cl-scat-th . In symmergence, therefore, QFT-GR concord is achieved not by quantizing gravity but by grounding on the flat spacetime effective QFT demir2016 ; demir2019 .

   In the traditional approach of curved spacetime QFTs qft-cuvred , attempts to determine scattering amplitudes get stuck already at the operator stage in (39) simply because it is not possible to construct the $|{\rm in}\rangle$ and $|{\rm out}\rangle$ Fock states wald ; ashtekar . Symmergence is immune to all these problems simply because it is based on flat spacetime effective QFTs.

2. 2.

   The coupling $c_{\phi}/4$ in the GR action $S_{\rm GR}(g,\phi)$ is a loop factor expected to be a few $\%$ ($1.3\%$ in the SM). It couples scalar curvature $R(g)$ to scalar fields $\phi$, and gives cause thus to Newton-Cavendish constant to vary with $\phi$ non-min . For high field values near $M_{GUT}\sim 10^{-2}M_{Pl}$, $G_{N}$ gets rescaled to $G_{N}(1-10^{-4}c_{\phi})\sim G_{N}(1-10^{-6})$ since $\phi^{\dagger}\phi-\langle\phi^{\dagger}\phi\rangle\sim M_{GUT}^{2}$. This variation in $G_{N}$ remains well within the experimental uncertainty in $G_{N}$ codata . The agreement with data gets better and better with lower and lower field values. For higher field values, however, variation in $G_{N}$ exceeds the experimental range (by three orders of magnitude at $|\phi|\sim M_{Pl}$). This means that $|\phi_{max}|\sim M_{GUT}$ is the largest field swing allowed.

3. 3.

   The Newton-Cavendish constant in $S_{\rm GR}(g,\phi)$

   $\displaystyle\frac{1}{G_{N}}=4\pi\left(\sum\limits_{i}c_{\psi_{i}}m_{i}^{2}+c_{\phi}\langle\phi^{\dagger}\phi\rangle\right)\xrightarrow{\rm 1-loop}\frac{{\rm str}[m^{2}]}{8\pi}+4\pi c^{(1)}_{\phi}\langle\phi^{\dagger}\phi\rangle$

   (41)

   must agree with empirical data will for gravity to symmerge correctly. It constrains the QFT mass spectrum as ${\rm str}\left[m^{2}\right]\sim M_{Pl}^{2}$ (barring flat directions in which $\langle\phi\rangle\gg m_{\phi}$). Its one-loop form reveals that the QFT particle spectrum must be dominated by bosons either in number or in mass or in both. It thus turns out that the Newton-Cavendish constant can be correctly induced in a QFT having

   1. (i)

      either a light spectrum with numerous more bosons than fermions (for instance, $m_{b}\sim m_{f}\sim{\rm TeV}$ with $n_{b}-n_{f}\sim 10^{32}$),

   2. (ii)

      or a heavy spectrum with few more bosons than fermions (for instance, $m_{b}\sim m_{f}\lesssim M_{Pl}$ with $n_{b}-n_{f}\gtrsim 5$),

   3. (iii)

      or a sparse spectrum with net boson dominance.

   The QFT is best exemplified by the SM, which is fully confirmed by the LHC experiments and their priors. Its spectrum yields $G_{N}\sim-({\rm TeV})^{-2}$, which is unphysical in both sign and size. It is because of this inadequacy of the SM spectrum that symmergence predicts existence of new physics beyond the SM (BSM). The BSM, whose spectrum adds to (41) to correct the SM result, has no obligation to couple to the SM in a specific scheme and strength. It can thus form a completely decoupled black sector demir2019 ; black or a weakly coupled dark sector demir2019 ; darksector , with distinctive signatures for collider searches keremle , dark matter searches cemle , and other possible phenomena demir2019 .

4. 4.

   The quadratic curvature term in the GR action $S_{\rm GR}(g,\phi)$ can facilitate Starobinsky inflation starobinsky ; Planck ; irfan since (at one loop)

   $\displaystyle c_{\varnothing}=-\frac{(n_{b}-n_{f})}{128\pi^{2}}$

   (42)

   acquires right sign and size for $n_{b}-n_{f}\approx 10^{13}$. In case this constraint is not met, inflation can also be realized with the scalar fields $\phi$ in the spectrum Higgs-inflation ; bauer .

5. 5.

   The vacuum energy in $S_{\rm QFT}(g,\psi,\log\Lambda_{\wp})$

   	$\displaystyle V(\Lambda_{\wp})$		$\displaystyle=V\left(\langle\phi\rangle\right)+\sum_{i}{c}^{(l)}_{\varnothing\psi_{i}}m_{i}^{4}\log\frac{m_{i}^{2}}{\Lambda_{\wp}^{2}}$		(43)
   			$\displaystyle\xrightarrow{\rm 1-loop}V^{(1)}\left(\langle\phi\rangle\right)+\frac{1}{64\pi^{2}}{\rm str}\left[{m^{4}}\log\frac{m^{2}}{\Lambda_{\wp}^{2}}\right]$

   gathers together the relevant $\log\Lambda_{\wp}$ corrections in (10) in the minimum of the scalar potential $V(\phi)$ at $\phi=\langle\phi\rangle$. Its empirical value is $V_{emp}=\left(2.57\times 10^{-3}\ {\rm eV}\right)^{4}$ ccp2 , and the QFT vacuum must reproduce this specific value. This constraint puts severe restrictions on the UV cutoff and other parameters of the QFTs. This is what the CCP ccp is all about. In Sakharov’s induced gravity sakharov ; visser , for instance, induction of the Newton-Cavendish constant fixes $\Lambda_{\wp}$ to a Planckian value, and this fix leads to an ${\mathcal{O}}\left(M_{Pl}^{4}\right)$ vacuum energy. In symmergence, however, $\log\Lambda_{\wp}$ is not fixed (as in Table 2), and it can be fixed in a way suppressing the vacuum energy. Indeed, in view mainly of the one-loop value in (43), the rough concordance

   $\displaystyle\Lambda_{\wp}^{2}\sim{\rm str}\left[m^{2}\right]\sim M_{Pl}^{2}$

   (44)

   suppresses the logarithms and induces the Newton-Cavendish constant consistently. (This becomes clear especially when ${\rm str}\left[m^{2}\right]$ is saturated by one large mass.) It involves a severe fine-tuning, and thus, it certainly is not a solution to the CCP. But it might be sign of an underlying symmetry principle or a dynamical theory of the Poincare breaking scale $\Lambda_{\wp}$. Symmergence cannot go beyond (44). (Clearly, the CCP can be approached by other methods like degravitation mechanisms ccp3 ; ccp4 .)

6. 6.

   Light scalars $\phi_{L}$ in $S_{\rm QFT}(g,\psi,\log\Lambda_{\wp})$ have their masses shifted as

   $\displaystyle\delta m_{\phi_{L}}^{2}={c}^{(l)}_{\phi_{L}\psi_{H}}m_{\psi_{H}}^{2}\log\frac{m_{\psi_{H}}^{2}}{\Lambda_{\wp}^{2}}$

   (45)

   via their couplings ${c}^{(l)}_{\phi_{L}\psi_{H}}$ to heavy fields $\psi_{H}$, as defined in (2) as well as (10). These logarithmic corrections are of a new kind because they are sensitive to field masses not to the UV cutoff. And they are crucial because heavier the $\psi_{H}$ larger the $\delta m_{\phi_{L}}^{2}$ and stronger the destabilization of the light sector of the QFT. Symmergence cannot solve this problem (EGC in (22) leaves $\log\Lambda_{\wp}$ untouched) but provides a viable way to avoid it. The thing is that induction of the Newton-Cavendish constant in (41) is the only constraint on the QFT field spectrum and it does not require any special coupling scheme or strength among the fields. Namely, induction of the Newton-Cavendish constant is immune to with what strengths the QFT fields are coupled. This immunity enables QFTs to maintain their two-scale structure at the loop level by having their heavy and light sectors coupled in a way keeping them scale-split. The implied coupling

   $\displaystyle\left|{c}^{(l)}_{\phi_{L}\psi_{H}}\right|\lesssim\frac{m_{\phi_{L}}^{2}}{m_{\psi_{H}}^{2}}$

   (46)

   is see-sawish in nature and capable of stabilizing the light sector, as ensured by (45). This implies that only those QFTs having see-sawish couplings can withstand loop-induced heavy-light mixing. And symmergence allows the see-sawish couplings demir2019 .

   The importance of the see-sawish couplings is best revealed by examining the realistic case of the SM. The SM needs be extended for various empirical and conceptual reasons beyond . The extension, a BSM sector, is formed by superpartners in supersymmetry, Kaluza-Klein modes in extra dimensions, and technifermions in technicolor. Each of these BSM sectors couples to the SM with the SM couplings themselves as otherwise their underlying symmetries get broken. And this means that they never go to the see-sawish regime in (46) and, as a result, the SM Higgs sector gets destabilized by these heavy BSM sectors. Indeed, even the Planck-scale supersymmetry (generating no quadratic correction by its nature and merging with the SM through an intermediate-scale singlet sector) destabilizes the Higgs boson mass as in (45) quiros . It is for this non-see-sawish nature of theirs that all these BSM sectors have already been sidelined by the LHC experiments for certain mass ranges.

   The BSM sector of the symmergence, required by the Newton-Cavendish constant in (41), differs from superpartners, Kaluza-Klein levels and technifermions by its congruence to the see-sawish couplings in (46). It is different in that it contains only those fields which enjoy the see-sawish regime, and such fields may conveniently be termed as symmergeons. In fact, new physics searches beyond the TeV domain must assume a BSM sector that does not destabilize the SM Higgs sector and, in this respect, possible discoveries at future experiments collider1 may fit to the symmergeons.