# Symmergent Gravity, Seesawic New Physics, and Their Experimental Signatures

Authors: Durmuş Demir
Journal: Advances in High Energy Physics
 (2019)

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## Abstract

The standard model of elementary particles (SM) suffers from various problems, such as power-law ultraviolet (UV) sensitivity, exclusion of general relativity (GR), and absence of a dark matter candidate. The LHC experiments, according to which the TeV domain appears to be empty of new particles, started sidelining TeV-scale SUSY and other known cures of the UV sensitivity. In search for a remedy, in this work, it is revealed that affine curvature can emerge in a way restoring gauge symmetries explicitly broken by the UV cutoff. This emergent curvature cures the UV sensitivity and incorporates GR as symmetry-restoring emergent gravity (symmergent gravity, in brief) if a new physics sector (NP) exists to generate the Planck scale and if SM+NP is Fermi-Bose balanced. This setup, carrying fingerprints of trans-Planckian SUSY, predicts that gravity is Einstein (no higher-curvature terms), cosmic/gamma rays can originate from heavy NP scalars, and the UV cutoff might take right value to suppress the cosmological constant (alleviating fine-tuning with SUSY). The NP does not have to couple to the SM. In fact, NP-SM coupling can take any value from zero to ΛSM2/ΛNP2 if the SM is not to jump from ΛSM≈500GeV to the NP scale ΛNP. The zero coupling, certifying an undetectable NP, agrees with all the collider and dark matter bounds at present. Theseesawic bound ΛSM2/ΛNP2, directly verifiable at colliders, implies that (i) dark matter must have a mass ≲ΛSM, (ii) Higgs-curvature coupling must be ≈1.3%, (iii) the SM RGEs must remain nearly as in the SM, and (iv) right-handed neutrinos must have a mass ≲1000TeV. These signatures serve as a concise testbed for symmergence.


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## Body

## 1. Introduction

The SM, a spontaneously broken renormalizable quantum field theory (QFT) of the strong and electroweak interactions, has shown good agreement with all the experiments performed so far [1, 2]. Its parameters have all been fixed experimentally. This does, however, not mean that it is a complete theory. Indeed, it is plagued by enigmatic problems like destabilizing UV sensitivities [3–7], exclusion of gravity [8], and absence of a dark matter candidate [9], which are impossible to address without new physics beyond the SM (BSM). Schematically, (1)BSM=gravityGR+newQFTbeyondtheSMNPin which the general relativity (GR) structure of gravity is revealed by various experiments [10] and observations [11]. The NP is under way at colliders [2] and dark matter searches [12, 13].Quantum correction to the Higgs boson mass, quadratically sensitive to UV boundary [3–7], exceeds the Higgs boson mass just above the electroweak scale. This means that the SM must stop working below the TeV scale. It does not stop, however. Indeed, the LHC has confirmed the SM [2] is up to multi-TeV energies. This contradiction between the SM loops and the LHC can be eliminated only if the NP in (1) improves the SM in a way without introducing any new interacting particles. The present work approaches this puzzling requirement via proof-by-contradiction; that is, it begins by assuming that the NP is absent (Sections 2 and 3) and, at a later stage, it ends up with NP through the consistency of the induced gravitational constant (Section 4).The GR must be incorporated into the SM [10, 11]. This has been attempted with classical GR [14–16] (despite [17]), quantized GR [18, 19] (despite [20–23]), and emergent GR [24, 25] (see also [26–33]). The present work incorporates curvature into the SM effective action in flat spacetime (Section 2) by building on the nascent ideas proposed in [34–36] and subsequent developments voiced in [37–40]. It incorporates gravity in a way restoring the gauge symmetries broken by the UV cutoff [41, 42] (Section 3.2), in a way elasticating the rigid flat spacetime (Section 3.3), and in a way involving a nontrivial NP sector to induce the gravitational constant (Section 4). This gauge symmetry-restoring emergent gravity,symmergent gravity [38–40] in brief, sets up a framework (Section 4) in which(1)
curvature arises as a manifestation of the elastication of the flat spacetime,(2)
GR emerges along with the restoration of gauge invariance,(3)
gravitational constant necessitates an NP sector,(4)
SM + NP possesses exact Fermi-Bose balance and the induced gravitational constant is suggestive of a trans-Planckian SUSY breaking [43–45], and(5)
the UV boundary can be fixed, in principle, by suppression of the cosmological constant (with no immediate solution for the cosmological constant problem [46, 47] though)so that there arise a number of descriptive signatures with which symmergence can be probed via decisive experiments (Section5):(1)
higher-curvature terms are predicted to be absent. This excludes, for instance,f(R) gravity [48] and agrees well with the current cosmological data [11].(2)
NP scalars with trans-GZK [49, 50] VEVs are predicted to give cause to cosmic rays [51] (digluon) as well as gamma rays [52] (diphoton) via certain Planck-suppressed higher-dimension operators.(3)
symmergence does not necessitate any SM-NP coupling for it to work. This property, not found in the known SM completions (SUSY, extra dimensions, compositeness and others [53, 54]) for which a sizable SM-NP coupling is essential, provides a rationale for stabilizing the electroweak scale against the SM-NP mixing. Indeed, if the SM-NP coupling goes like mH2/mNP2, then the Higgs boson mass mH remains within the allowed limits. This seesawic (seesaw-wise) structure leads to various testable features:
(a)
it is predicted that the heavier the NP the larger the luminosity needed to discover it. This distinctive feature can be probed at present [55, 56] and future colliders [57, 58].(b)
It is also predicted that the right-handed neutrinos [59] must weigh below a 1000 TeV (see also [60, 61]). This bound can be tested at future colliders [62] if not at the near-future SHiP experiment [63, 64].(c)
it turns out that the SM couplings (gauge and nongauge) must run as if NP is absent if the NP lies sufficiently above the electroweak scale. This feature, which rests on the fact that symmergence leaves behind only logarithmic sensitivity to the UV boundary [65, 66], can be tested at present [55, 56] and future colliders [57, 58].(d)
symmergence accommodates both ebony (having only gravitational interactions with the SM) [35, 36, 67, 68] and dark (having seesawic couplings to the SM) matters. They both agree with the current bounds. The latter, thermal dark matter, is predicted to weigh below the electroweak scale. This agrees with current limits and can be tested further in future searches [12, 13, 69].(e)
it is predicted that, in the SM, nonminimal Higgs-curvature coupling equals1.3% at one loop and remains so unless the NP lies near the electroweak scale. This coupling, too small to drive the Higgs inflation [70], can serve as a testbed at collider experiments [71] if not in the astrophysical or cosmological environments.The work is concluded in Section6.


## 2. UV Boundary, Effective SM, and the UV Sensitivity Problems

The NP needed to complete the SM, roughly sketched in (1), can be elucidated only after a complete picture of the SM in regard to its UV boundary and UV sensitivity.


### 2.1. UV Boundary

The Higgs mass-squaredmH2, measured to be mH2LHC≈125GeV2 at the LHC [72], is overwhelmed by the quantum correction [3–7] (2)δmH2=cHΛ2in which Λ is the UV boundary of the SM, and (3)cH=316π2H24mt2-2MW2-MZ2-mH2≈5.14×10-2is the loop factor. The correction (2), a one-loop SM effect, grows quadratically with Λ and exceeds mH2LHC already at Λ=ΛW, where (4)ΛW≈550GeVlies just above the electroweak scale |〈H〉|≈246.22GeV. This low-lying ΛW, which can be changed slightly by incorporating subleading log⁡mH/Λ corrections to (2), is a characteristic feature of the SM spectrum and the experimental result mH2LHC. It implies that the SM must stop working at ΛW. It does not stop, however. Indeed, the LHC experiments show that the SM continues to hold good up to multi-TeV energies without any new field. This contradictory UV overextension is the problem. There is no clear solution. There is even no clear way to search for a solution. There is, however, a possibility that a mechanism, not necessarily unique, might be constructed via proof-by-contradiction [35, 36, 38], that is, by first (5)constructingtheSMeffectiveactionbelowΛWassumingthatthereisnothingbuttheSMallthewayuptoΛU≫ΛLHC,and then (6)revealingthenecessityandstructureoftheNPforincorporationofgravityandneutralizationofthedestabilizingΛUsensitivities.The first step sets up a UV boundary ΛU and reveals SM’s UV sensitivity. The second, on the other hand, uncovers NP via induction of gravity and fixes ΛU in terms of the NP scale.


### 2.2. SM Effective Action

In accordance with (5), integration of the fast modes (fields with energies ≳ΛW) out of the SM spectrum gives an effective action for slow modes ϝ (fields with energies ≲ΛW) [73, 74] (7)Seffη=Streeη,ϝ+δSlogη,ϝ,log⁡ΛWΛU+δSOη,Δ2+δSHη,Δ2+δSVη,Δ2in which ημν is the flat metric, H is the slow Higgs field, Vμ are the slow gauge fields (photon, gluon, W and Z), and finally (8)Δ2=ΛU2-ΛW2is the UV-EW gap. The tree-level SM action Stree and the logarithmic corrections δSlog both lie below ΛW. But, the other three (9)δSO=-∫d4x-η2cOΛW2+∑ϝcϝmϝ2Δ2+cOΔ4(10)δSH=-∫d4x-ηcHΔ2H†H(11)δSV=∫d4x-ηcVΔ2TrVμVμpull the SM off the electroweak scale depending on how large Δ2 is. They tend to destabilize the SM and it is this destabilization that necessitates a neutralization mechanism. Their Wilson coefficients cO,…,V involve only the ratio [3–7] (12)ΛWΛUas the measure of the EW-UV hierarchy.


### 2.3. UV Sensitivity Problems

The power-law quantum corrections in (9), (10), and (11) give cause to serious destabilization problems. They are tabulated in Table 1. The cosmological constant problem (CCP) [46, 47], caused by δSO, exists only when gravity is present. The big hierarchy problem (BHP) [3–7] (gauge hierarchy problem) refers to quadratic UV sensitivities of the Higgs (from δSH) and W/Z (from δSV) masses. The electric charge or color breaking (CCB) [41, 42], on the other hand, arises from the photon and the gluon mass terms in δSV (purely quadratic in Δ). The SM is impossible to make sense in the UV before these problems are satisfactorily resolved.Table 1
Quantum corrections and problems they give cause to. The coefficientscϝ,…,cg are loop factors that depend on ΛW/ΛU. (CCP, with c~O=cO+(2cOΛW2+∑ϝ‍cϝmϝ2)/Δ2, makes sense only in curved geometry, the bold font.)
δSlog δSH δSV δ S O Problem δV ≠ 0 0 0 c ~ O Δ 4 CCP  δmH2 ≠ 0 c H Δ 2 0 0 BHP  δmW2 ≠ 0 0 c W Δ 2 0 BHP  δmZ2 ≠ 0 0 c Z Δ 2 0 BHP  δmγ2 0 0 c γ Δ 2 0 CCB  δmg2 0 0 c g Δ 2 0 CCB


### 2.4. Impossibility of Renormalizing Away CCP, CCB, and BHP

The SM is a renormalizable QFT. If so, why is it not possible to includeδSlog, δSO, δSH, δSV into a renormalization of Stree to get rid of the problems in Table 1? Because Δ is physical. Indeed, ΛU can pertain to gravity or NP sectors [3–7]. Moreover, there is simply no place to hide δmγ,g2=cγ,gΔ2≠0 since (mγ,g2)tree=0. These problems are therefore physical and their solutions entail physical changes on the SM (like inclusion of gravity).


## 2.1. UV Boundary

The Higgs mass-squaredmH2, measured to be mH2LHC≈125GeV2 at the LHC [72], is overwhelmed by the quantum correction [3–7] (2)δmH2=cHΛ2in which Λ is the UV boundary of the SM, and (3)cH=316π2H24mt2-2MW2-MZ2-mH2≈5.14×10-2is the loop factor. The correction (2), a one-loop SM effect, grows quadratically with Λ and exceeds mH2LHC already at Λ=ΛW, where (4)ΛW≈550GeVlies just above the electroweak scale |〈H〉|≈246.22GeV. This low-lying ΛW, which can be changed slightly by incorporating subleading log⁡mH/Λ corrections to (2), is a characteristic feature of the SM spectrum and the experimental result mH2LHC. It implies that the SM must stop working at ΛW. It does not stop, however. Indeed, the LHC experiments show that the SM continues to hold good up to multi-TeV energies without any new field. This contradictory UV overextension is the problem. There is no clear solution. There is even no clear way to search for a solution. There is, however, a possibility that a mechanism, not necessarily unique, might be constructed via proof-by-contradiction [35, 36, 38], that is, by first (5)constructingtheSMeffectiveactionbelowΛWassumingthatthereisnothingbuttheSMallthewayuptoΛU≫ΛLHC,and then (6)revealingthenecessityandstructureoftheNPforincorporationofgravityandneutralizationofthedestabilizingΛUsensitivities.The first step sets up a UV boundary ΛU and reveals SM’s UV sensitivity. The second, on the other hand, uncovers NP via induction of gravity and fixes ΛU in terms of the NP scale.


## 2.2. SM Effective Action

In accordance with (5), integration of the fast modes (fields with energies ≳ΛW) out of the SM spectrum gives an effective action for slow modes ϝ (fields with energies ≲ΛW) [73, 74] (7)Seffη=Streeη,ϝ+δSlogη,ϝ,log⁡ΛWΛU+δSOη,Δ2+δSHη,Δ2+δSVη,Δ2in which ημν is the flat metric, H is the slow Higgs field, Vμ are the slow gauge fields (photon, gluon, W and Z), and finally (8)Δ2=ΛU2-ΛW2is the UV-EW gap. The tree-level SM action Stree and the logarithmic corrections δSlog both lie below ΛW. But, the other three (9)δSO=-∫d4x-η2cOΛW2+∑ϝcϝmϝ2Δ2+cOΔ4(10)δSH=-∫d4x-ηcHΔ2H†H(11)δSV=∫d4x-ηcVΔ2TrVμVμpull the SM off the electroweak scale depending on how large Δ2 is. They tend to destabilize the SM and it is this destabilization that necessitates a neutralization mechanism. Their Wilson coefficients cO,…,V involve only the ratio [3–7] (12)ΛWΛUas the measure of the EW-UV hierarchy.


## 2.3. UV Sensitivity Problems

The power-law quantum corrections in (9), (10), and (11) give cause to serious destabilization problems. They are tabulated in Table 1. The cosmological constant problem (CCP) [46, 47], caused by δSO, exists only when gravity is present. The big hierarchy problem (BHP) [3–7] (gauge hierarchy problem) refers to quadratic UV sensitivities of the Higgs (from δSH) and W/Z (from δSV) masses. The electric charge or color breaking (CCB) [41, 42], on the other hand, arises from the photon and the gluon mass terms in δSV (purely quadratic in Δ). The SM is impossible to make sense in the UV before these problems are satisfactorily resolved.Table 1
Quantum corrections and problems they give cause to. The coefficientscϝ,…,cg are loop factors that depend on ΛW/ΛU. (CCP, with c~O=cO+(2cOΛW2+∑ϝ‍cϝmϝ2)/Δ2, makes sense only in curved geometry, the bold font.)
δSlog δSH δSV δ S O Problem δV ≠ 0 0 0 c ~ O Δ 4 CCP  δmH2 ≠ 0 c H Δ 2 0 0 BHP  δmW2 ≠ 0 0 c W Δ 2 0 BHP  δmZ2 ≠ 0 0 c Z Δ 2 0 BHP  δmγ2 0 0 c γ Δ 2 0 CCB  δmg2 0 0 c g Δ 2 0 CCB


## 2.4. Impossibility of Renormalizing Away CCP, CCB, and BHP

The SM is a renormalizable QFT. If so, why is it not possible to includeδSlog, δSO, δSH, δSV into a renormalization of Stree to get rid of the problems in Table 1? Because Δ is physical. Indeed, ΛU can pertain to gravity or NP sectors [3–7]. Moreover, there is simply no place to hide δmγ,g2=cγ,gΔ2≠0 since (mγ,g2)tree=0. These problems are therefore physical and their solutions entail physical changes on the SM (like inclusion of gravity).


## 3. Incorporation of Curvature into the Effective SM

It is now time to ascertain how curvature can be incorporated into the flat spacetime effective SM in (7) and how that incorporation affects the UV sensitivity problems in Table 1.


### 3.1. How Not to Incorporate: Curvature by Hand

Gravity is incorporated into classical field theories in flat spacetime by first mapping the flat metricημν into a putative curved metric gμν as (13)ημν↪gμνin view of general covariance [75], and then adding curvature of the Levi-Civita connection (14)Γgμνλ=12gλρ∂μgνρ+∂νgρμ-∂ρgμνto make gμν dynamical. The curvature sector, added by hand, ignoring ghosts, can involve all curvature invariants [76]. It can involve, for instance, the Ricci tensor RμνΓg in traced (R=gμνRμνΓg), squared (RμνΓgRμνΓg), or in any other invariant form. This curvature-by-hand method, a standard procedure for classical field theories, leads to (15)Seffg-∫d4x-gM~2R+α~R2+β~RμνRμν+⋯when applied to the SM effective action in (7). The problem with this action is that M~, α~, β~, ⋯ are all inherently incalculable [35, 36, 38, 39]. This is because matter loops have already been used up in forming the flat spacetime effective action Seff, and there have remained thus no loops to induce any extra interaction, with or without curvature. This incalculability constraint, which reveals the difference between classical and effective field theories, renders the tentative action (15) unphysical. It is in this sense that the general covariance [75] is not adequate for incorporating curvature into effective SM. In essence, what is needed is a separate covariance relation between thescales inSeff (say, ΛW2 or Δ2) andcurvature [37] so that gravity can be incorporated in a way involving no arbitrary, incalculable constants.


### 3.2. How to Incorporate: Curvature by Gauge Symmetry Restoration

The gauge partδSVη,Δ2, which must be neutralized for color and electromagnetism to remain exact and electroweak breaking to be spontaneous, poses a vexed problem due to strict masslessness of the photon and the gluon. It can be tackled via neither the Stueckelberg method [77, 78] nor the spontaneous symmetry breaking [79, 80]. It can, nonetheless, be tackled by furthering the nascent ideas proposed in [35, 36] (which attempts at restoring gauge symmetry within the GR with fixed ΛU2+ΛW2) and subsequent advancements voiced in [37–40].


#### 3.2.1.δSVη,Δ2 in a New Light

It proves useful to start with the obvious identity [35–37] (16)δSVη,Δ2=δSVη,Δ2-Iη,V+Iη,Vin which the gauge-invariant kinetic construct (17)Iη,V=∫d4x-ηcV2TrημαηνβVμνVαβis subtracted from and added back to δSV. This construct, involving the loop factor cV and the field strength tensor Vμν, leads to (18)δSVη,Δ2=-Iη,V+∫d4x-ηcVTrVμ-Dμν2+Δ2ημνVν+∫d4x-ηcVTr∂μηαβVαVβμif, at the right-hand side of (16), δSV is replaced with (11), “-I(η,V)” is left untouched, and yet “+I(η,V)” is integrated by-parts to involve Dμν2=D2ημν-DμDν-Vμν where Dμ is gauge-covariant derivative. This recast δSV gets to curved spacetime via (13) to become (19)δSVg,Δ2=-Ig,V+∫d4x-gcVTrVμ-Dμν2+Δ2gμνVν+∫d4x-gcVTr∇μgαβVαVβμwhere Dμ is the gauge-covariant derivative with respect to the covariant derivative ∇μ of the Levi-Civita connection gΓμνλ, and Dμν2=D2gμν-DμDν-Vμν.


#### 3.2.2.δSVg,Δ2 in a New “Curvature”

Is there a simple way of killingδSVg,Δ2? Yes, there is. Indeed, δSVg,Δ2 vanishes identically if Δ2gμν is replaced with RμνΓg. This encouraging feature, not to be confused with derivation of gravity from self-interacting spin-2 fields in flat spacetime [81–83], is pitiably problematic because Δ2gμν↪RμνΓg contradicts with ημν↪gμν. If it were not for this contradiction, emergence of curvature from Δ2gμν would solve the CCB [35–39].The contradiction can be avoided by introducing, for instance, a more general map [40] (20)Δ2gμν↪RμνΓin which Rμν(Γ) is the Ricci curvature of a symmetric affine connection Γμνλ (bearing no relationship to the Levi-Civita connection Γgμνλ). This new map removes contradiction because while RμνΓ⟿Δ2gμν fixes the affine connection, gμν⟿ημν does the metric [40]. It throws δSVg,Δ2 in (19) into metric-affine geometry [84–86] to give it the “mass-free” form (21)δSVg,R=-Ig,V+∫d4x-gcVTrVμ-Dμν2+RμνΓVν+∫d4x-gcVTr∇μgαβVαVβμwhose by-parts integration results in (22)δSVg,R,R=∫d4x-gcVTrVμRμνΓ-RμνgVνunder the condition that cV must be held unchanged or, equivalently, the EW-UV hierarchy (23)ΛWΛUmustbeheldunchangedwhile the affine curvature arises as in (20). This preservation is crucial for ensuring that the SM is indeed stabilized at ΛW≪ΛU [3–7].The CCB attains a solution only ifδSVg,R,R in (22) is suppressed. And suppression is seen to necessitate Γμνλ to be close to Γgμνλ. Fortunately, Γμνλ does really lie close to Γgμνλ. Indeed, as already voiced in [40] and as will be derived in equations (45) and (46) in Section 4.3 below, Γμνλ assumes the form (24)Γμνλ=Γgμνλ+OMPl-2H,Vas a solution to its equation of motion. It is clear that Γμνλ differs from gΓμνλ only by Planck-suppressed OMPl-2 terms, which are labeled with H,V to emphasize that they are made up solely of the Higgs and gauge fields. The preconditions for algebraic solutions like (24) have already been revealed in [84, 85, 87, 88], and the analysis in Section 4.3 will follow them.Under the solution (24) for Γμνλ, the Ricci curvature takes the form(25)RμνΓ=Rμνg+OMPl-2H,V and its replacement into the action (22) leads to (26)δSVg,R,R=∫d4x-g0+OMPl-2H,Vas a proof of the fact that the CCB is suppressed up to Planck-suppressed OMPl-2 terms. It implies that color and electromagnetism are restored and spontaneity of the electroweak breaking is ensured modulo Planck-suppressed dimension-6 Higgs and gauge composites.


### 3.3. How Curvature Emerges: Elasticated Flat Spacetime

The flat spacetime is rigid. It remains flat independent of how big the energy it contains is. The quartic quantum correctionδSO in (9), for instance, can no way curve it, even with infinite Δ. But, flat spacetime must develop a certain degree of elasticity if gravity is to be incorporated into the effective SM. The requisite elasticity needs be generated by an extraneous mechanism as it is unlikely to arise from the effective QFT itself. One mechanism as such is Sakharov’s induced gravity [24, 25] in which elasticity emerges from the assumption that the quantum loops have a different metric than the tree-level action. This mechanism generates the gravitational scale as MPl2~Δ2 but leaves behind an O(Δ4) vacuum energy and an O(Δ2) Higgs mass-squared. This means that the CCP and BHP (in Table 1) remain unsolved, and it is necessary to devise an alternative mechanism.In search for a proper mechanism, it proves useful to start with the observation that, in general, the integral(27)∫d4x-detRΓΔ2takes the form(28)∫d4x-gafter integrating out the affine curvature Rμν(Γ). The reason, as was first pointed out in [34], is that the equation of motion for the affine connection(29)∇αΓdetRΓΔ2RΓΔ2-1μν=0,which follows from (27), is solved by(30)RμνΓ=Δ2gμνwith a dynamically induced metric gμν. This solution, the well-known Eddington solution [89–95], is precisely the map (20) imposed by gauge symmetry restoration (partly because curvature scale in (27) is set to Δ2).The meaning of relation (30) is that curvature emerges as elastication of the rigid flat spacetime. To see how this happens, it proves efficacious to focus on the colossal vacuum energy in (9). The metrical action δSO(g), obtained from δSO(η) in (9) via (13), can be mapped into the affine action δSO(Γ) by integrating in RμνΓ via the Eddington solution (30). This “integrating in" stage, illustrated in Figure 1, represents the elastication of the flat spacetime [34]. The problem here is that the affine geometry of δSO(Γ) is devoid of any metrical structure to accommodate δSlog and δSV. The remedy is to integrate out Δ2 to get the metric-affine action δSO(g,Γ) in Figure 1. The metric-affine geometry accommodates all parts of the SM effective action and the resulting framework, which has curvature incorporated into the effective SM, is identical to that obtained via (13) in Section 3.1 followed by (20) in Section 3.2.2. This means that the maps (13) and (20) have a dynamical origin and serve as equivalence relations for incorporating curvature into effective QFTs.Figure 1
The emergence of the affine curvature. The metrical actionδSO(g) can be mapped into the affine action δSO(Γ) after integrating in Rμν(Γ). This affine action, which does not admit any matter sector due to the absence of metric, can be brought into the metric-affine form δSO(g,Γ) by integrating out Δ2. Integrating in and out both rests on the Eddington solution, which is identical to map (20) imposed by gauge symmetry restoration.The emergence mechanism here is similar, at least in philosophy, to that of Sakharov [24, 25] in that curvature is tied to quantum effects. It is based on emergence of the curvature, not that of the spacetime itself, and differs thus from mechanisms like the entropic gravity of Verlinde [26] and entanglement gravity of Raamsdonk [27, 28].


### 3.4. Symmergence Principle

The results of Sections3.2.2 and 3.3 above can be organized as a principle to apply to UV sensitivities of any given QFT. Indeed, it turns out that gravity can be incorporated into a flat spacetime QFT with UV-IR gap Δ and metric ημν by first letting [75] (31)ημν↪gμνand then [40] (32)Δ2gμν↪RμνΓso that if (33)RμνΓ⟿RμνΓgthen CCB gets suppressed and GR emerges as a gaugesymmetry restoring emergent gravity or, briefly,symmergent gravity. This three-stage mainframe will define what shall be calledsymmergence in what follows.


## 3.1. How Not to Incorporate: Curvature by Hand

Gravity is incorporated into classical field theories in flat spacetime by first mapping the flat metricημν into a putative curved metric gμν as (13)ημν↪gμνin view of general covariance [75], and then adding curvature of the Levi-Civita connection (14)Γgμνλ=12gλρ∂μgνρ+∂νgρμ-∂ρgμνto make gμν dynamical. The curvature sector, added by hand, ignoring ghosts, can involve all curvature invariants [76]. It can involve, for instance, the Ricci tensor RμνΓg in traced (R=gμνRμνΓg), squared (RμνΓgRμνΓg), or in any other invariant form. This curvature-by-hand method, a standard procedure for classical field theories, leads to (15)Seffg-∫d4x-gM~2R+α~R2+β~RμνRμν+⋯when applied to the SM effective action in (7). The problem with this action is that M~, α~, β~, ⋯ are all inherently incalculable [35, 36, 38, 39]. This is because matter loops have already been used up in forming the flat spacetime effective action Seff, and there have remained thus no loops to induce any extra interaction, with or without curvature. This incalculability constraint, which reveals the difference between classical and effective field theories, renders the tentative action (15) unphysical. It is in this sense that the general covariance [75] is not adequate for incorporating curvature into effective SM. In essence, what is needed is a separate covariance relation between thescales inSeff (say, ΛW2 or Δ2) andcurvature [37] so that gravity can be incorporated in a way involving no arbitrary, incalculable constants.


## 3.2. How to Incorporate: Curvature by Gauge Symmetry Restoration

The gauge partδSVη,Δ2, which must be neutralized for color and electromagnetism to remain exact and electroweak breaking to be spontaneous, poses a vexed problem due to strict masslessness of the photon and the gluon. It can be tackled via neither the Stueckelberg method [77, 78] nor the spontaneous symmetry breaking [79, 80]. It can, nonetheless, be tackled by furthering the nascent ideas proposed in [35, 36] (which attempts at restoring gauge symmetry within the GR with fixed ΛU2+ΛW2) and subsequent advancements voiced in [37–40].


### 3.2.1.δSVη,Δ2 in a New Light

It proves useful to start with the obvious identity [35–37] (16)δSVη,Δ2=δSVη,Δ2-Iη,V+Iη,Vin which the gauge-invariant kinetic construct (17)Iη,V=∫d4x-ηcV2TrημαηνβVμνVαβis subtracted from and added back to δSV. This construct, involving the loop factor cV and the field strength tensor Vμν, leads to (18)δSVη,Δ2=-Iη,V+∫d4x-ηcVTrVμ-Dμν2+Δ2ημνVν+∫d4x-ηcVTr∂μηαβVαVβμif, at the right-hand side of (16), δSV is replaced with (11), “-I(η,V)” is left untouched, and yet “+I(η,V)” is integrated by-parts to involve Dμν2=D2ημν-DμDν-Vμν where Dμ is gauge-covariant derivative. This recast δSV gets to curved spacetime via (13) to become (19)δSVg,Δ2=-Ig,V+∫d4x-gcVTrVμ-Dμν2+Δ2gμνVν+∫d4x-gcVTr∇μgαβVαVβμwhere Dμ is the gauge-covariant derivative with respect to the covariant derivative ∇μ of the Levi-Civita connection gΓμνλ, and Dμν2=D2gμν-DμDν-Vμν.


### 3.2.2.δSVg,Δ2 in a New “Curvature”

Is there a simple way of killingδSVg,Δ2? Yes, there is. Indeed, δSVg,Δ2 vanishes identically if Δ2gμν is replaced with RμνΓg. This encouraging feature, not to be confused with derivation of gravity from self-interacting spin-2 fields in flat spacetime [81–83], is pitiably problematic because Δ2gμν↪RμνΓg contradicts with ημν↪gμν. If it were not for this contradiction, emergence of curvature from Δ2gμν would solve the CCB [35–39].The contradiction can be avoided by introducing, for instance, a more general map [40] (20)Δ2gμν↪RμνΓin which Rμν(Γ) is the Ricci curvature of a symmetric affine connection Γμνλ (bearing no relationship to the Levi-Civita connection Γgμνλ). This new map removes contradiction because while RμνΓ⟿Δ2gμν fixes the affine connection, gμν⟿ημν does the metric [40]. It throws δSVg,Δ2 in (19) into metric-affine geometry [84–86] to give it the “mass-free” form (21)δSVg,R=-Ig,V+∫d4x-gcVTrVμ-Dμν2+RμνΓVν+∫d4x-gcVTr∇μgαβVαVβμwhose by-parts integration results in (22)δSVg,R,R=∫d4x-gcVTrVμRμνΓ-RμνgVνunder the condition that cV must be held unchanged or, equivalently, the EW-UV hierarchy (23)ΛWΛUmustbeheldunchangedwhile the affine curvature arises as in (20). This preservation is crucial for ensuring that the SM is indeed stabilized at ΛW≪ΛU [3–7].The CCB attains a solution only ifδSVg,R,R in (22) is suppressed. And suppression is seen to necessitate Γμνλ to be close to Γgμνλ. Fortunately, Γμνλ does really lie close to Γgμνλ. Indeed, as already voiced in [40] and as will be derived in equations (45) and (46) in Section 4.3 below, Γμνλ assumes the form (24)Γμνλ=Γgμνλ+OMPl-2H,Vas a solution to its equation of motion. It is clear that Γμνλ differs from gΓμνλ only by Planck-suppressed OMPl-2 terms, which are labeled with H,V to emphasize that they are made up solely of the Higgs and gauge fields. The preconditions for algebraic solutions like (24) have already been revealed in [84, 85, 87, 88], and the analysis in Section 4.3 will follow them.Under the solution (24) for Γμνλ, the Ricci curvature takes the form(25)RμνΓ=Rμνg+OMPl-2H,V and its replacement into the action (22) leads to (26)δSVg,R,R=∫d4x-g0+OMPl-2H,Vas a proof of the fact that the CCB is suppressed up to Planck-suppressed OMPl-2 terms. It implies that color and electromagnetism are restored and spontaneity of the electroweak breaking is ensured modulo Planck-suppressed dimension-6 Higgs and gauge composites.


## 3.2.1.δSVη,Δ2 in a New Light

It proves useful to start with the obvious identity [35–37] (16)δSVη,Δ2=δSVη,Δ2-Iη,V+Iη,Vin which the gauge-invariant kinetic construct (17)Iη,V=∫d4x-ηcV2TrημαηνβVμνVαβis subtracted from and added back to δSV. This construct, involving the loop factor cV and the field strength tensor Vμν, leads to (18)δSVη,Δ2=-Iη,V+∫d4x-ηcVTrVμ-Dμν2+Δ2ημνVν+∫d4x-ηcVTr∂μηαβVαVβμif, at the right-hand side of (16), δSV is replaced with (11), “-I(η,V)” is left untouched, and yet “+I(η,V)” is integrated by-parts to involve Dμν2=D2ημν-DμDν-Vμν where Dμ is gauge-covariant derivative. This recast δSV gets to curved spacetime via (13) to become (19)δSVg,Δ2=-Ig,V+∫d4x-gcVTrVμ-Dμν2+Δ2gμνVν+∫d4x-gcVTr∇μgαβVαVβμwhere Dμ is the gauge-covariant derivative with respect to the covariant derivative ∇μ of the Levi-Civita connection gΓμνλ, and Dμν2=D2gμν-DμDν-Vμν.


## 3.2.2.δSVg,Δ2 in a New “Curvature”

Is there a simple way of killingδSVg,Δ2? Yes, there is. Indeed, δSVg,Δ2 vanishes identically if Δ2gμν is replaced with RμνΓg. This encouraging feature, not to be confused with derivation of gravity from self-interacting spin-2 fields in flat spacetime [81–83], is pitiably problematic because Δ2gμν↪RμνΓg contradicts with ημν↪gμν. If it were not for this contradiction, emergence of curvature from Δ2gμν would solve the CCB [35–39].The contradiction can be avoided by introducing, for instance, a more general map [40] (20)Δ2gμν↪RμνΓin which Rμν(Γ) is the Ricci curvature of a symmetric affine connection Γμνλ (bearing no relationship to the Levi-Civita connection Γgμνλ). This new map removes contradiction because while RμνΓ⟿Δ2gμν fixes the affine connection, gμν⟿ημν does the metric [40]. It throws δSVg,Δ2 in (19) into metric-affine geometry [84–86] to give it the “mass-free” form (21)δSVg,R=-Ig,V+∫d4x-gcVTrVμ-Dμν2+RμνΓVν+∫d4x-gcVTr∇μgαβVαVβμwhose by-parts integration results in (22)δSVg,R,R=∫d4x-gcVTrVμRμνΓ-RμνgVνunder the condition that cV must be held unchanged or, equivalently, the EW-UV hierarchy (23)ΛWΛUmustbeheldunchangedwhile the affine curvature arises as in (20). This preservation is crucial for ensuring that the SM is indeed stabilized at ΛW≪ΛU [3–7].The CCB attains a solution only ifδSVg,R,R in (22) is suppressed. And suppression is seen to necessitate Γμνλ to be close to Γgμνλ. Fortunately, Γμνλ does really lie close to Γgμνλ. Indeed, as already voiced in [40] and as will be derived in equations (45) and (46) in Section 4.3 below, Γμνλ assumes the form (24)Γμνλ=Γgμνλ+OMPl-2H,Vas a solution to its equation of motion. It is clear that Γμνλ differs from gΓμνλ only by Planck-suppressed OMPl-2 terms, which are labeled with H,V to emphasize that they are made up solely of the Higgs and gauge fields. The preconditions for algebraic solutions like (24) have already been revealed in [84, 85, 87, 88], and the analysis in Section 4.3 will follow them.Under the solution (24) for Γμνλ, the Ricci curvature takes the form(25)RμνΓ=Rμνg+OMPl-2H,V and its replacement into the action (22) leads to (26)δSVg,R,R=∫d4x-g0+OMPl-2H,Vas a proof of the fact that the CCB is suppressed up to Planck-suppressed OMPl-2 terms. It implies that color and electromagnetism are restored and spontaneity of the electroweak breaking is ensured modulo Planck-suppressed dimension-6 Higgs and gauge composites.


## 3.3. How Curvature Emerges: Elasticated Flat Spacetime

The flat spacetime is rigid. It remains flat independent of how big the energy it contains is. The quartic quantum correctionδSO in (9), for instance, can no way curve it, even with infinite Δ. But, flat spacetime must develop a certain degree of elasticity if gravity is to be incorporated into the effective SM. The requisite elasticity needs be generated by an extraneous mechanism as it is unlikely to arise from the effective QFT itself. One mechanism as such is Sakharov’s induced gravity [24, 25] in which elasticity emerges from the assumption that the quantum loops have a different metric than the tree-level action. This mechanism generates the gravitational scale as MPl2~Δ2 but leaves behind an O(Δ4) vacuum energy and an O(Δ2) Higgs mass-squared. This means that the CCP and BHP (in Table 1) remain unsolved, and it is necessary to devise an alternative mechanism.In search for a proper mechanism, it proves useful to start with the observation that, in general, the integral(27)∫d4x-detRΓΔ2takes the form(28)∫d4x-gafter integrating out the affine curvature Rμν(Γ). The reason, as was first pointed out in [34], is that the equation of motion for the affine connection(29)∇αΓdetRΓΔ2RΓΔ2-1μν=0,which follows from (27), is solved by(30)RμνΓ=Δ2gμνwith a dynamically induced metric gμν. This solution, the well-known Eddington solution [89–95], is precisely the map (20) imposed by gauge symmetry restoration (partly because curvature scale in (27) is set to Δ2).The meaning of relation (30) is that curvature emerges as elastication of the rigid flat spacetime. To see how this happens, it proves efficacious to focus on the colossal vacuum energy in (9). The metrical action δSO(g), obtained from δSO(η) in (9) via (13), can be mapped into the affine action δSO(Γ) by integrating in RμνΓ via the Eddington solution (30). This “integrating in" stage, illustrated in Figure 1, represents the elastication of the flat spacetime [34]. The problem here is that the affine geometry of δSO(Γ) is devoid of any metrical structure to accommodate δSlog and δSV. The remedy is to integrate out Δ2 to get the metric-affine action δSO(g,Γ) in Figure 1. The metric-affine geometry accommodates all parts of the SM effective action and the resulting framework, which has curvature incorporated into the effective SM, is identical to that obtained via (13) in Section 3.1 followed by (20) in Section 3.2.2. This means that the maps (13) and (20) have a dynamical origin and serve as equivalence relations for incorporating curvature into effective QFTs.Figure 1
The emergence of the affine curvature. The metrical actionδSO(g) can be mapped into the affine action δSO(Γ) after integrating in Rμν(Γ). This affine action, which does not admit any matter sector due to the absence of metric, can be brought into the metric-affine form δSO(g,Γ) by integrating out Δ2. Integrating in and out both rests on the Eddington solution, which is identical to map (20) imposed by gauge symmetry restoration.The emergence mechanism here is similar, at least in philosophy, to that of Sakharov [24, 25] in that curvature is tied to quantum effects. It is based on emergence of the curvature, not that of the spacetime itself, and differs thus from mechanisms like the entropic gravity of Verlinde [26] and entanglement gravity of Raamsdonk [27, 28].


## 3.4. Symmergence Principle

The results of Sections3.2.2 and 3.3 above can be organized as a principle to apply to UV sensitivities of any given QFT. Indeed, it turns out that gravity can be incorporated into a flat spacetime QFT with UV-IR gap Δ and metric ημν by first letting [75] (31)ημν↪gμνand then [40] (32)Δ2gμν↪RμνΓso that if (33)RμνΓ⟿RμνΓgthen CCB gets suppressed and GR emerges as a gaugesymmetry restoring emergent gravity or, briefly,symmergent gravity. This three-stage mainframe will define what shall be calledsymmergence in what follows.


## 4. Incorporation of GR and the Requisite NP

Having set up the symmergence principle, it is now time to determine under what conditions GR arises correctly. It will be found below that it cannot arise without a nontrivial NP sector.


### 4.1. Metric-Affine Gravity

Affine gravity, as inδSO(Γ) in Figure 1, involves only the affine connection [89–95]. Metrical gravity, the GR itself, is based solely on the metric tensor. The metric-affine gravity (MAG), on the other hand, comprises, as in δSO(g,Γ) in Figure 1, both metric and affine connection as two independent dynamical variables [84–88]. The MAG reduces to the so-called Palatini formalism [84, 85] if connection does not appear in the matter sector. It is equivalent to the GR. (In general, affine connection spreads to matter sector via at least the spin connection.)MAG may or may not lead to GR. It depends on couplings of the affine connection. Indeed, solving the equation of motion for affine connection to integrate it out leads to the Levi-Civita connection of the metric plus possible contributions from other fields, including the affine curvature [87, 88, 96]. Then, reduction to the GR can put constraints even on the matter sector. The sections below, starting with Section 4.3, will be devoted to the GR limit of the MAG arising from the SM effective action.


### 4.2. Necessity of NP

Under (13) and (20) or under the emergence mechanism in Figure 1, the Higgs and vacuum sectors of the flat spacetime effective action in (7) transform as (34)δSO+δSH↪∫d4x-g-qμνRμνΓ+cO16gμνRμνΓ2after defining (35)qμν=cO2ΛW2+14∑ϝcϝmϝ2+cH4H†H+cO8gαβRαβΓgμνfor convenience. The right-hand side of (34) forms the curvature sector of the SM effective action in metric-affine geometry [84–88]. The apparent gravitational scale, (cO/2)ΛW2+(1/4)∑ϝcϝmϝ2, is wrong in both sign (cO<0 in the SM) and size (∑ϝcϝmϝ2<ΛW2 in the SM). It is for this reason that an NP sector (made up of entirely new fields ϝ′) must be introduced so that the new gravitational scale (36)cO+cO′ΛW2+12∑f=ϝ,ϝ′cfmf2can come out right thanks to either the NP spectrum (cO′) or the NP scale (mϝ′). In fact, its one-loop value (37)164π2Str1ΛW2+Strm2makes it clear that the NP must have either a crowded (Str1~1035) [38–40] or a heavy (Strm2≳MPl2) bosonic sector.


### 4.3. Incorporation of GR

The problem is to determine how MAG can reduce to GR. This reduction is decided by the dynamics of the affine connection. And part of the total SM + NP action that governs the dynamics(38)∫d4x-g-QμνRμνΓ+116cO+cO′gμνRμνΓ2involves (39)Qμν=12cO+cO′ΛW2+14∑f=ϝ,ϝ′cfmf2+14∑ϕ=H,H′cϕϕ†ϕ+18cO+cO′gαβRαβΓgμν-∑V=V,V′cVTrVμVνas an extension of qμν in (35) to the gauge and NP sectors. It gathers affine curvature terms from (22) (after extending it with the NP gauge bosons Vμ′ with loop factors cV′) and (34) (after augmenting it with the NP vacuum energy with the loop factor cO′ and all of the NP fields with loop factors cϝ′).The equation of motion forΓμνλ, stipulated by stationarity of the action (38) against variations in Γμνλ, turns out to be a metricity condition on Qμν(40)∇ΓλQμν=0and possesses the generic solution [84, 85, 87, 88] (41)Γμνλ=Γgμνλ+12Q-1λρ∇μQνρ+∇νQρμ-∇ρQμνwhich expresses Γμνλ in terms of its own curvature Rμν(Γ), as evidenced by Qμν itself. This dependence on curvature is critical because if it is the case, then (41) acts as a differential equation for Γμνλ, and its solution carries extra geometrical degrees of freedom not found in the Levi-Civita connection Γgμνλ [87, 88]. This means that Rμν(Γ) must disappear from Qμν for GR to be able to symmerge, and it does so if (42)cO+cO′=0or,equivalently,Str1=0or, equivalently, SM + NP has equal bosonic and fermionic degrees of freedom. This Fermi-Bose balance does of course not mean a proper SUSY model since SM-NP couplings do not have to support a SUSY structure [53, 54]. All it does is to trim Qμν in (39) down to (43)Q°μν=M22+14∑ϕ=H,H′cϕϕ†ϕgμν-∑V=V,V′cVTrVμVνin which (44)M2=∑f=ϝ,ϝ′cfmf2=1-loopStrm264π2is the apparent gravitational scale. Now, GR can symmerge because Γμνλ is an auxiliary field (algebraic solution) having no geometrical content beyond Γgμνλ [84, 85, 87, 88]. Indeed, after (43), the affine connection in (41) takes the form (45)Γμνλ=Γgμνλ+12Q°-1λρ∇μQ°νρ+∇νQ°ρμ-∇ρQ°μνto contain only the Levi-Civita connection and the scalars ϕ and vectors Vμ in Q°μν. Given the enormity of M, this expression can always be expanded order by order in 1/M2 to get (46)Γμνλ=Γgμνλ+1M2∇μQ°νλ+∇μQ°νλ-∇λQ°μν+OM-4ϕ,Vusing the notation in (24) for the remainder of the expansion. The higher-order OM-4ϕ,V terms are straightforwardly computed from the exact expression (45) order by order in 1/M2. In accordance with (46), the affine curvature expands as (47)RμνΓ=RμνΓg+1M2∇2μναβQ°αβ+OM-4ϕ,Vwhere ∇2μναβ=∇α∇μδνβ+∇α∇νδμβ-□δμαδνβ-∇ν∇μgαβ [97]. This expansion leads to the reductions (48)∫d4x-gM22gμνRμνΓ⟿∫d4x-gM22Rg+OM-2ϕ,V(49)∫d4x-g∑ϕ=H,H′ϕ†ϕgμνRμνΓ⟿∫d4x-g∑ϕ=H,H′ϕ†ϕRg+OM-2ϕ,V(50)∫d4x-g∑V=V,V′TrVμRμνΓ-RμνgVν⟿∫d4x-g0+OM-2ϕ,Vso that the SM + NP affine action in (38) reduces to (51)∫d4x-g-M22+14∑ϕ=H,H′cϕϕ†ϕRg+OM-2ϕ,Vto yield the Einstein-Hilbert action for nonminimally coupled scalars ϕ=H,H′ [98, 99]. Here, OM-2ϕ,V terms can be constructed from the exact solution (45) order by order in 1/M2. Unlike the generic effective field-theoretic approach [100, 101], the remainder OM-2ϕ,V assumes a specific structure coordinated by (45) through Q°μν in (43).The GR gets properly incorporated if Planck scale is generated correctly. In view of (51), it is given by (52)MPl2=M2+12∑ϕ=H,H′cϕϕ†ϕ=1-loopStrm264π2+12∑ϕ=H,H′cϕ1ϕ†ϕin which vacuum contribution becomes significant when ∑H′cH′〈H′†H′〉~M2. The first part, identical to (44), implies that the bosonic sector of the NP must be either crowded (nb′~1035 and mϝ′≳ΛW) [35, 36, 38–40] or heavy enough (nb′~ much less yet mϝ′≲ΛU) for it to be able to induce the trans-Planckian scale 8πMPl.The NP sector, needed for induction of the gravitational scale as in (36) or (52), can have nontrivial structure. Indeed, given the Fermi-Bose balance of SM+NP and given also that Strm2 in (52) is reminiscent of the mass sum rule in broken SUSY [43–45, 53, 54], it becomes apparent that the SM and NP, with the seesawic couplings in (60), may well be the remnants of a trans-Planckian SUSY broken around 8πMPl. Pivotally, this can provide an explanation for why symmergence must start with flat spacetime SM + NP in that the alleged SUSY theory, whose breaking occurs with an anomalous U(1) factor [43–45] to generate the nonzero sum rule in (52), cannot couple to gravity due to the anomaly. In other words, flat spacetime QFT can be taken as a signature of SUSY. These features can be useful for probing trans-Planckian physics but a complete NP model, which presumably is subject matter of a whole different study, is not an urgency as far as the gravitational constant is concerned.


## 4.1. Metric-Affine Gravity

Affine gravity, as inδSO(Γ) in Figure 1, involves only the affine connection [89–95]. Metrical gravity, the GR itself, is based solely on the metric tensor. The metric-affine gravity (MAG), on the other hand, comprises, as in δSO(g,Γ) in Figure 1, both metric and affine connection as two independent dynamical variables [84–88]. The MAG reduces to the so-called Palatini formalism [84, 85] if connection does not appear in the matter sector. It is equivalent to the GR. (In general, affine connection spreads to matter sector via at least the spin connection.)MAG may or may not lead to GR. It depends on couplings of the affine connection. Indeed, solving the equation of motion for affine connection to integrate it out leads to the Levi-Civita connection of the metric plus possible contributions from other fields, including the affine curvature [87, 88, 96]. Then, reduction to the GR can put constraints even on the matter sector. The sections below, starting with Section 4.3, will be devoted to the GR limit of the MAG arising from the SM effective action.


## 4.2. Necessity of NP

Under (13) and (20) or under the emergence mechanism in Figure 1, the Higgs and vacuum sectors of the flat spacetime effective action in (7) transform as (34)δSO+δSH↪∫d4x-g-qμνRμνΓ+cO16gμνRμνΓ2after defining (35)qμν=cO2ΛW2+14∑ϝcϝmϝ2+cH4H†H+cO8gαβRαβΓgμνfor convenience. The right-hand side of (34) forms the curvature sector of the SM effective action in metric-affine geometry [84–88]. The apparent gravitational scale, (cO/2)ΛW2+(1/4)∑ϝcϝmϝ2, is wrong in both sign (cO<0 in the SM) and size (∑ϝcϝmϝ2<ΛW2 in the SM). It is for this reason that an NP sector (made up of entirely new fields ϝ′) must be introduced so that the new gravitational scale (36)cO+cO′ΛW2+12∑f=ϝ,ϝ′cfmf2can come out right thanks to either the NP spectrum (cO′) or the NP scale (mϝ′). In fact, its one-loop value (37)164π2Str1ΛW2+Strm2makes it clear that the NP must have either a crowded (Str1~1035) [38–40] or a heavy (Strm2≳MPl2) bosonic sector.


## 4.3. Incorporation of GR

The problem is to determine how MAG can reduce to GR. This reduction is decided by the dynamics of the affine connection. And part of the total SM + NP action that governs the dynamics(38)∫d4x-g-QμνRμνΓ+116cO+cO′gμνRμνΓ2involves (39)Qμν=12cO+cO′ΛW2+14∑f=ϝ,ϝ′cfmf2+14∑ϕ=H,H′cϕϕ†ϕ+18cO+cO′gαβRαβΓgμν-∑V=V,V′cVTrVμVνas an extension of qμν in (35) to the gauge and NP sectors. It gathers affine curvature terms from (22) (after extending it with the NP gauge bosons Vμ′ with loop factors cV′) and (34) (after augmenting it with the NP vacuum energy with the loop factor cO′ and all of the NP fields with loop factors cϝ′).The equation of motion forΓμνλ, stipulated by stationarity of the action (38) against variations in Γμνλ, turns out to be a metricity condition on Qμν(40)∇ΓλQμν=0and possesses the generic solution [84, 85, 87, 88] (41)Γμνλ=Γgμνλ+12Q-1λρ∇μQνρ+∇νQρμ-∇ρQμνwhich expresses Γμνλ in terms of its own curvature Rμν(Γ), as evidenced by Qμν itself. This dependence on curvature is critical because if it is the case, then (41) acts as a differential equation for Γμνλ, and its solution carries extra geometrical degrees of freedom not found in the Levi-Civita connection Γgμνλ [87, 88]. This means that Rμν(Γ) must disappear from Qμν for GR to be able to symmerge, and it does so if (42)cO+cO′=0or,equivalently,Str1=0or, equivalently, SM + NP has equal bosonic and fermionic degrees of freedom. This Fermi-Bose balance does of course not mean a proper SUSY model since SM-NP couplings do not have to support a SUSY structure [53, 54]. All it does is to trim Qμν in (39) down to (43)Q°μν=M22+14∑ϕ=H,H′cϕϕ†ϕgμν-∑V=V,V′cVTrVμVνin which (44)M2=∑f=ϝ,ϝ′cfmf2=1-loopStrm264π2is the apparent gravitational scale. Now, GR can symmerge because Γμνλ is an auxiliary field (algebraic solution) having no geometrical content beyond Γgμνλ [84, 85, 87, 88]. Indeed, after (43), the affine connection in (41) takes the form (45)Γμνλ=Γgμνλ+12Q°-1λρ∇μQ°νρ+∇νQ°ρμ-∇ρQ°μνto contain only the Levi-Civita connection and the scalars ϕ and vectors Vμ in Q°μν. Given the enormity of M, this expression can always be expanded order by order in 1/M2 to get (46)Γμνλ=Γgμνλ+1M2∇μQ°νλ+∇μQ°νλ-∇λQ°μν+OM-4ϕ,Vusing the notation in (24) for the remainder of the expansion. The higher-order OM-4ϕ,V terms are straightforwardly computed from the exact expression (45) order by order in 1/M2. In accordance with (46), the affine curvature expands as (47)RμνΓ=RμνΓg+1M2∇2μναβQ°αβ+OM-4ϕ,Vwhere ∇2μναβ=∇α∇μδνβ+∇α∇νδμβ-□δμαδνβ-∇ν∇μgαβ [97]. This expansion leads to the reductions (48)∫d4x-gM22gμνRμνΓ⟿∫d4x-gM22Rg+OM-2ϕ,V(49)∫d4x-g∑ϕ=H,H′ϕ†ϕgμνRμνΓ⟿∫d4x-g∑ϕ=H,H′ϕ†ϕRg+OM-2ϕ,V(50)∫d4x-g∑V=V,V′TrVμRμνΓ-RμνgVν⟿∫d4x-g0+OM-2ϕ,Vso that the SM + NP affine action in (38) reduces to (51)∫d4x-g-M22+14∑ϕ=H,H′cϕϕ†ϕRg+OM-2ϕ,Vto yield the Einstein-Hilbert action for nonminimally coupled scalars ϕ=H,H′ [98, 99]. Here, OM-2ϕ,V terms can be constructed from the exact solution (45) order by order in 1/M2. Unlike the generic effective field-theoretic approach [100, 101], the remainder OM-2ϕ,V assumes a specific structure coordinated by (45) through Q°μν in (43).The GR gets properly incorporated if Planck scale is generated correctly. In view of (51), it is given by (52)MPl2=M2+12∑ϕ=H,H′cϕϕ†ϕ=1-loopStrm264π2+12∑ϕ=H,H′cϕ1ϕ†ϕin which vacuum contribution becomes significant when ∑H′cH′〈H′†H′〉~M2. The first part, identical to (44), implies that the bosonic sector of the NP must be either crowded (nb′~1035 and mϝ′≳ΛW) [35, 36, 38–40] or heavy enough (nb′~ much less yet mϝ′≲ΛU) for it to be able to induce the trans-Planckian scale 8πMPl.The NP sector, needed for induction of the gravitational scale as in (36) or (52), can have nontrivial structure. Indeed, given the Fermi-Bose balance of SM+NP and given also that Strm2 in (52) is reminiscent of the mass sum rule in broken SUSY [43–45, 53, 54], it becomes apparent that the SM and NP, with the seesawic couplings in (60), may well be the remnants of a trans-Planckian SUSY broken around 8πMPl. Pivotally, this can provide an explanation for why symmergence must start with flat spacetime SM + NP in that the alleged SUSY theory, whose breaking occurs with an anomalous U(1) factor [43–45] to generate the nonzero sum rule in (52), cannot couple to gravity due to the anomaly. In other words, flat spacetime QFT can be taken as a signature of SUSY. These features can be useful for probing trans-Planckian physics but a complete NP model, which presumably is subject matter of a whole different study, is not an urgency as far as the gravitational constant is concerned.


## 5. Observational and Experimental Tests

This section collects various predictions and experimentally testable features from both the gravity and NP sectors. They can play a crucial role in revealing and testing the physics of symmergence.


### 5.1. CCB Is Suppressed

Nullification of the gauge part in (50) ensures that color and electromagnetism are restored and electroweak breaking is rendered spontaneous up to doubly Planck-suppressed terms involving ϕ and Vμ. This is important for assuring neutralization of the CCB in accordance with the symmergence principle. To that end, the exact form of the affine connection in (45), which is never derivative-free, ensures that gauge boson masses are impossible to arise at any order in 1/M2. Indeed, iteration of (51) up to next order (53)∫d4x-g-M22+14∑ϕ=H,H′cϕϕ†ϕRg+1M2Q°μν∇2μναβQ°αβ+OM-4ϕ,Vexemplifies that what arises at each order involves derivatives of ϕ and Vμ.


### 5.2. The UV Boundary Might Have a Say in CCP

The total vacuum energy at one loop(54)VΛU=Vtreeϕ+164π2Strm4log⁡m2ΛU2-12is composed of the tree-level potential plus the loop corrections involving the UV scale ΛU. It does not have to be small. In fact, it can readily exceed the current observational bounds [11] to give cause to the CCP [46, 47]. Needless to say, the CCP arises from not the Δ4 sensitivity in (11) (which is neutralized by symmergence as in (34)) but the energies released by the QCD, electroweak, and possible NP transitions as well as the zero-point energies of all the SM + NP fields. The question of how these distinct energy sources, making up (54), can conspire to yield the observed value Vobs=2.57×10-3eV4 [11] is what the CCP is all about [46, 47]. Indeed, the empirical equality(55)VΛU0=Vobsgives a fix on the UV boundary in that the solution ΛU=ΛU0 expresses ΛU in terms of the SM and NP parameters. This solution can hardly make any sense unless the aforementioned energy sources are put in relation by some governing rule. Indeed, a correlation rule, which might be a remnant of the trans-Planckian SUSY above 8πMPl [102–104], can prevent fine-tuning and render ΛU0 physical.Symmergent gravity and the NP it necessitates agree with all the existing bounds thanks to their prediction that gravity is Einstein (as in [11]), missing matter can be ebony (as in [12, 13]), and non-SM interactions are not a necessity (as in [2]). In case of experimental discoveries, however, they can be probed with conclusive tests via the dark matter and SM-NP couplings.


### 5.3. Higher-Curvature Terms Are Absent

Symmergence leads uniquely to Einstein gravity. It excludes all higher-curvature terms. This is an important result since extinction of higher-curvature terms is impossible to guarantee in a theory in which general covariance is the only symmetry. To this end, the question of why it is notf(R) gravity [48] but just the GR has an answer in symmergence. And current observations [11] seem to have already confirmed the symmergence.


### 5.4. Higgs-Curvature Coupling Is Predicted

The nonminimal Higgs-curvature coupling [98, 99], symmergence of the Higgs quadratic UV sensitivity in (10), is a model-dependent loop factor. It has the numerical value(56)cH4≈1.29×10-2as follows from (3). This prediction is specific to the SM spectrum. It changes by an amount ∝λHϝ′2 in the presence of the NP. This change, as ensured by the bound (60), can be significant only if the NP lies close to the electroweak scale. This means that the questions of if the underlying mechanism is symmergence and if the NP is heavy or not can be conclusively probed by measuring cH.Obviously,cH is too small to facilitate the Higgs inflation [70]. It can, nevertheless, give cause to observable effects in strong gravitational fields (e.g., early universe and black hole horizons) or Planckian-energy particle scatterings [71].The nonminimalH′-curvature coupling [98, 99] is expected to have similar features. Indeed, cH′ in (51) can be directly computed with a concrete NP model. It may differ significantly from cH. In fact, what drives cosmic inflation could well be H′ not H [35, 36, 38].


### 5.5. Couplings Run Nearly as in the SM

Symmergence leaves behind only logarithmic sensitivity to the UV boundary. This remnant sensitivity, with all gauge symmetries restored, can naturally be interpreted in the language of dimensional regularization. Indeed, the formal correspondence [35, 36, 38, 40, 65, 66](57)log⁡ΛU0ΛW2=1ϵ+log⁡ΛU0μ2 expresses all amplitudes in terms of the renormalization scale μ after subtracting 1/ϵ terms in MS¯ scheme. Their independence from μ leads to the usual RGEs. The important point here is that dimensional regularization arises as a result, not as an arbitrarily chosen regularization method. The testable feature is that the SM couplings and masses must run as in the SM at all scales unless the NP lies close to ΛW. It can be directly tested at present [55, 56] and future [57, 58] colliders.


### 5.6. SM-NP Coupling Is Not a Necessity

Emergence of the gravitational scale does not necessitate any SM-NP coupling. The SM and NP do not have to interact. The NP sector can therefore come in three different kinds:(1)
Ebony NP. This kind of NP has no nongravitational couplings to the SM [67, 68]. It forms an insular sector (composed of, for instance, high-rank non-Abelian gauge fields and fermions [35, 36, 38]). In view of the present searches, which seem all negative, this ebony, pitch-dark NP shows good agreement with all the available data [12, 13].(2)
Dark NP. This type of NP is neutral under the SM but couples to it via Higgs, hypercharge and lepton portals [35, 36, 59, 105]. Indeed, at the renormalizable level, scalars H′, Abelian vectors Vμ′, and fermions N′ can couple to the SM via (58)Sint=∫d4x-gλHH′2H†HH′†H′+λBZ′BμνZ′μν+λHN′L¯HN′+h.c.in which the couplingsλHH′2, λBZ′, λHN′ can take, none to significant, a wide range of values with characteristic experimental signals [12, 13, 69].(3)
Visible NP. In this case, the NP is partly or wholly charged under the SM (e.g., H′ in (58) can be an SU(2) doublet) [35, 36]. It couples to the SM with SM coupling strengths, and it seems to have already been sidelined by the current bounds [2, 106].The NP sector, whose subsectors are depicted in Figure2, can be of any composition (ebony to dark), can have any couplings (none to significant), and can lie anywhere (from ΛW to ΛU) as long as the gravitational scale in (36) comes out right. Nonnecessity of any sizable NP-SM couplings is what distinguishes the NP of the symmergence from SUSY, extra dimensions, composites models and others [53, 54] where a sizable SM-NP coupling is a necessity. It is this sizable coupling of theirs that make such NP models sidelined under the LHC bounds [2].Figure 2
The three kinds of the NP sectors.


### 5.7. Electroweak Stability Restricts SM-NP Coupling

The Higgs part of (34) makes it clear that the quadratic UV sensitivity in flat spacetime changes to Higgs-curvature coupling in metric-affine spacetime. This solves the BHP [3–7]. In the presence of NP, however, the Higgs sector is destabilized by not only the BHP but also the SM-NP coupling [35, 36, 38, 107–109]. Indeed, the SM-NP interactions in (58), for instance, lead to a new Higgs mass correction (in addition to δSH in (10)) (59)δSH′=∫d4x-g∑ϝ′c~Hλϝϝ′2mϝ′2log⁡mϝ′ΛUH†Hwith a loop factor c~H. This is sensitive not to ΛU but mϝ′. The problem is that the heavier the NP, the larger the shift in the Higgs mass and the stronger the destabilization of the electroweak scale. The more serious problem is that symmergence can do nothing about it. In fact, no UV completion can do anything about it. The reason is that problem is caused by coupling between two scale-split QFTs and there can hardly exist any solution other than suppression of the coupling itself. In SUSY, extra dimensions and compositeness SM-NP coupling is a must for them to work [53, 54, 107–109]. In symmergence, however, SM-NP coupling is not a necessity [35, 36, 38–40]. It works with any coupling strengths such as the seesawic ones (60)λϝϝ′2≲mH2mϝ′2with which the Higgs mass shift in (59) falls below the Higgs mass to ensure stability of the electroweak scale. This seesawic structure, explicated in Figure 3, implies that the heavier the NP, the weaker its coupling to the SM.Figure 3
The allowed and disallowed regions according to the electroweak stability bound in (60).


### 5.8. The NP Can Be Probed in Cosmic Rays

The Planck-suppressedϕ and Vμ composites in (53) do not produce any mass terms. They can produce, however, sizable signals if the NP scalars H′ develop large VEVs (compared to MPl). This effect, as depicted in Figure 4, leads to observable effects. The H′ decays into two gluons of GZK energy [49, 50], for instance, occuring at a rate (61)Γgg≈cH′10-22cg10-22mH′1011GeV58.08min-1with 〈H′〉≃mH′. These gluons can partake in ultrahigh energy cosmic rays [51] upon hadronization. The diphotons, produced similarly, can contribute to diffuse gamma-ray background [52]. These events, which weaken at low 〈H′〉, are cosmic probes of symmergence.Figure 4
The NP scalarH′ can have sizable decay rates at large VEV.


### 5.9. The NP Can Be Probed in Colliders

One immediate implication of the bound (60) is that the heavier the NP, the larger the luminosity needed to probe them at colliders. This can prove important in designing accelerators and detectors, as implied by the exemplary analyses in the subsections below.


#### 5.9.1. NP Scalars and Vectors

One way to see if the underlying model is symmergent or not is to measure scalar and vector masses and then determine if their production cross sections comply with the seesawic couplings in (60). For instance, mH′ can be measured from either of HH→H′→HH in Figure 5(a) or HH→H′H′ in Figure 5(b) such that the ratio of their cross sections(62)σHH→H′→HHσHH→H′H′≃mH2mH′22acts as an efficient probe of symmergence. This procedure works also for the Z′ scatterings in Figures 5(c) and 5(d). These 2→2 scatterings (and various other channels) can be directly tested at present [55, 56] and future [57, 58] colliders.Figure 5
Basic search channels for NP scalars ((a) and (b)) and NP vectors ((c) and (d)).


#### 5.9.2. Right-Handed Neutrinos

Symmergence has testable implications also for right-handed neutrinos. Indeed, if the bound (60) is to be respected and if the active neutrinos are to have correct masses (mν≲1eV), then the right-handed neutrinos must have masses [59–61](63)mN′≲1000TeVwhich can be probed at future experiments (based presumably on Higgs factories and accelerator neutrinos [62]) if not indirectly at the near-future SHiP experiment [63, 64]. For mν≲0.1eV [110], as revealed with recent cosmological data, the bound in (63) increases by an order of magnitude.


### 5.10. Dark Matter Weighs below Weak Scale

Symmergence has candidates for missing matter in both the ebony and dark NP sectors. The question of which one is preferred by nature can be answered only by observations (on, for instance, less-baryonic galaxies like cosmic seagull [111]).The dark NP can be probed directly [12, 13, 69]. The scalar field H′ in (58), for instance, develops a vanishing VEV, 〈H′〉=0, and becomes stable if it is real and has odd Z2 parity with respect to H. It is stable but its density is depleted by not only the expansion of the Universe but also the coannihilations into the SM fields as H′H′→W+W-,ZZ,t¯t,⋯. The coannihilations into W/Z are Goldstone-equivalent to H′H′→HH in Figure 5(b) so that H′ scalars attain the observed relic density [12, 13, 112] if(64)λHH′4GeVmH′2≈10-7 or, impliedly, (65)mH′≲367GeVafter using the seesawic structure in (60) for λHH′. This bound on mH′, which makes sense barely as it lies beneath ΛW, implies that thermal dark matter cannot weigh above the electroweak scale. (Nonthermal dark matter, belonging presumably to the ebony NP, is free from this bound.) This means that symmergence can be probed by measuring the dark matter mass [12, 13, 69] and contrasting it with (65) or, more safely, with ΛW.


## 5.1. CCB Is Suppressed

Nullification of the gauge part in (50) ensures that color and electromagnetism are restored and electroweak breaking is rendered spontaneous up to doubly Planck-suppressed terms involving ϕ and Vμ. This is important for assuring neutralization of the CCB in accordance with the symmergence principle. To that end, the exact form of the affine connection in (45), which is never derivative-free, ensures that gauge boson masses are impossible to arise at any order in 1/M2. Indeed, iteration of (51) up to next order (53)∫d4x-g-M22+14∑ϕ=H,H′cϕϕ†ϕRg+1M2Q°μν∇2μναβQ°αβ+OM-4ϕ,Vexemplifies that what arises at each order involves derivatives of ϕ and Vμ.


## 5.2. The UV Boundary Might Have a Say in CCP

The total vacuum energy at one loop(54)VΛU=Vtreeϕ+164π2Strm4log⁡m2ΛU2-12is composed of the tree-level potential plus the loop corrections involving the UV scale ΛU. It does not have to be small. In fact, it can readily exceed the current observational bounds [11] to give cause to the CCP [46, 47]. Needless to say, the CCP arises from not the Δ4 sensitivity in (11) (which is neutralized by symmergence as in (34)) but the energies released by the QCD, electroweak, and possible NP transitions as well as the zero-point energies of all the SM + NP fields. The question of how these distinct energy sources, making up (54), can conspire to yield the observed value Vobs=2.57×10-3eV4 [11] is what the CCP is all about [46, 47]. Indeed, the empirical equality(55)VΛU0=Vobsgives a fix on the UV boundary in that the solution ΛU=ΛU0 expresses ΛU in terms of the SM and NP parameters. This solution can hardly make any sense unless the aforementioned energy sources are put in relation by some governing rule. Indeed, a correlation rule, which might be a remnant of the trans-Planckian SUSY above 8πMPl [102–104], can prevent fine-tuning and render ΛU0 physical.Symmergent gravity and the NP it necessitates agree with all the existing bounds thanks to their prediction that gravity is Einstein (as in [11]), missing matter can be ebony (as in [12, 13]), and non-SM interactions are not a necessity (as in [2]). In case of experimental discoveries, however, they can be probed with conclusive tests via the dark matter and SM-NP couplings.


## 5.3. Higher-Curvature Terms Are Absent

Symmergence leads uniquely to Einstein gravity. It excludes all higher-curvature terms. This is an important result since extinction of higher-curvature terms is impossible to guarantee in a theory in which general covariance is the only symmetry. To this end, the question of why it is notf(R) gravity [48] but just the GR has an answer in symmergence. And current observations [11] seem to have already confirmed the symmergence.


## 5.4. Higgs-Curvature Coupling Is Predicted

The nonminimal Higgs-curvature coupling [98, 99], symmergence of the Higgs quadratic UV sensitivity in (10), is a model-dependent loop factor. It has the numerical value(56)cH4≈1.29×10-2as follows from (3). This prediction is specific to the SM spectrum. It changes by an amount ∝λHϝ′2 in the presence of the NP. This change, as ensured by the bound (60), can be significant only if the NP lies close to the electroweak scale. This means that the questions of if the underlying mechanism is symmergence and if the NP is heavy or not can be conclusively probed by measuring cH.Obviously,cH is too small to facilitate the Higgs inflation [70]. It can, nevertheless, give cause to observable effects in strong gravitational fields (e.g., early universe and black hole horizons) or Planckian-energy particle scatterings [71].The nonminimalH′-curvature coupling [98, 99] is expected to have similar features. Indeed, cH′ in (51) can be directly computed with a concrete NP model. It may differ significantly from cH. In fact, what drives cosmic inflation could well be H′ not H [35, 36, 38].


## 5.5. Couplings Run Nearly as in the SM

Symmergence leaves behind only logarithmic sensitivity to the UV boundary. This remnant sensitivity, with all gauge symmetries restored, can naturally be interpreted in the language of dimensional regularization. Indeed, the formal correspondence [35, 36, 38, 40, 65, 66](57)log⁡ΛU0ΛW2=1ϵ+log⁡ΛU0μ2 expresses all amplitudes in terms of the renormalization scale μ after subtracting 1/ϵ terms in MS¯ scheme. Their independence from μ leads to the usual RGEs. The important point here is that dimensional regularization arises as a result, not as an arbitrarily chosen regularization method. The testable feature is that the SM couplings and masses must run as in the SM at all scales unless the NP lies close to ΛW. It can be directly tested at present [55, 56] and future [57, 58] colliders.


## 5.6. SM-NP Coupling Is Not a Necessity

Emergence of the gravitational scale does not necessitate any SM-NP coupling. The SM and NP do not have to interact. The NP sector can therefore come in three different kinds:(1)
Ebony NP. This kind of NP has no nongravitational couplings to the SM [67, 68]. It forms an insular sector (composed of, for instance, high-rank non-Abelian gauge fields and fermions [35, 36, 38]). In view of the present searches, which seem all negative, this ebony, pitch-dark NP shows good agreement with all the available data [12, 13].(2)
Dark NP. This type of NP is neutral under the SM but couples to it via Higgs, hypercharge and lepton portals [35, 36, 59, 105]. Indeed, at the renormalizable level, scalars H′, Abelian vectors Vμ′, and fermions N′ can couple to the SM via (58)Sint=∫d4x-gλHH′2H†HH′†H′+λBZ′BμνZ′μν+λHN′L¯HN′+h.c.in which the couplingsλHH′2, λBZ′, λHN′ can take, none to significant, a wide range of values with characteristic experimental signals [12, 13, 69].(3)
Visible NP. In this case, the NP is partly or wholly charged under the SM (e.g., H′ in (58) can be an SU(2) doublet) [35, 36]. It couples to the SM with SM coupling strengths, and it seems to have already been sidelined by the current bounds [2, 106].The NP sector, whose subsectors are depicted in Figure2, can be of any composition (ebony to dark), can have any couplings (none to significant), and can lie anywhere (from ΛW to ΛU) as long as the gravitational scale in (36) comes out right. Nonnecessity of any sizable NP-SM couplings is what distinguishes the NP of the symmergence from SUSY, extra dimensions, composites models and others [53, 54] where a sizable SM-NP coupling is a necessity. It is this sizable coupling of theirs that make such NP models sidelined under the LHC bounds [2].Figure 2
The three kinds of the NP sectors.


## 5.7. Electroweak Stability Restricts SM-NP Coupling

The Higgs part of (34) makes it clear that the quadratic UV sensitivity in flat spacetime changes to Higgs-curvature coupling in metric-affine spacetime. This solves the BHP [3–7]. In the presence of NP, however, the Higgs sector is destabilized by not only the BHP but also the SM-NP coupling [35, 36, 38, 107–109]. Indeed, the SM-NP interactions in (58), for instance, lead to a new Higgs mass correction (in addition to δSH in (10)) (59)δSH′=∫d4x-g∑ϝ′c~Hλϝϝ′2mϝ′2log⁡mϝ′ΛUH†Hwith a loop factor c~H. This is sensitive not to ΛU but mϝ′. The problem is that the heavier the NP, the larger the shift in the Higgs mass and the stronger the destabilization of the electroweak scale. The more serious problem is that symmergence can do nothing about it. In fact, no UV completion can do anything about it. The reason is that problem is caused by coupling between two scale-split QFTs and there can hardly exist any solution other than suppression of the coupling itself. In SUSY, extra dimensions and compositeness SM-NP coupling is a must for them to work [53, 54, 107–109]. In symmergence, however, SM-NP coupling is not a necessity [35, 36, 38–40]. It works with any coupling strengths such as the seesawic ones (60)λϝϝ′2≲mH2mϝ′2with which the Higgs mass shift in (59) falls below the Higgs mass to ensure stability of the electroweak scale. This seesawic structure, explicated in Figure 3, implies that the heavier the NP, the weaker its coupling to the SM.Figure 3
The allowed and disallowed regions according to the electroweak stability bound in (60).


## 5.8. The NP Can Be Probed in Cosmic Rays

The Planck-suppressedϕ and Vμ composites in (53) do not produce any mass terms. They can produce, however, sizable signals if the NP scalars H′ develop large VEVs (compared to MPl). This effect, as depicted in Figure 4, leads to observable effects. The H′ decays into two gluons of GZK energy [49, 50], for instance, occuring at a rate (61)Γgg≈cH′10-22cg10-22mH′1011GeV58.08min-1with 〈H′〉≃mH′. These gluons can partake in ultrahigh energy cosmic rays [51] upon hadronization. The diphotons, produced similarly, can contribute to diffuse gamma-ray background [52]. These events, which weaken at low 〈H′〉, are cosmic probes of symmergence.Figure 4
The NP scalarH′ can have sizable decay rates at large VEV.


## 5.9. The NP Can Be Probed in Colliders

One immediate implication of the bound (60) is that the heavier the NP, the larger the luminosity needed to probe them at colliders. This can prove important in designing accelerators and detectors, as implied by the exemplary analyses in the subsections below.


### 5.9.1. NP Scalars and Vectors

One way to see if the underlying model is symmergent or not is to measure scalar and vector masses and then determine if their production cross sections comply with the seesawic couplings in (60). For instance, mH′ can be measured from either of HH→H′→HH in Figure 5(a) or HH→H′H′ in Figure 5(b) such that the ratio of their cross sections(62)σHH→H′→HHσHH→H′H′≃mH2mH′22acts as an efficient probe of symmergence. This procedure works also for the Z′ scatterings in Figures 5(c) and 5(d). These 2→2 scatterings (and various other channels) can be directly tested at present [55, 56] and future [57, 58] colliders.Figure 5
Basic search channels for NP scalars ((a) and (b)) and NP vectors ((c) and (d)).


### 5.9.2. Right-Handed Neutrinos

Symmergence has testable implications also for right-handed neutrinos. Indeed, if the bound (60) is to be respected and if the active neutrinos are to have correct masses (mν≲1eV), then the right-handed neutrinos must have masses [59–61](63)mN′≲1000TeVwhich can be probed at future experiments (based presumably on Higgs factories and accelerator neutrinos [62]) if not indirectly at the near-future SHiP experiment [63, 64]. For mν≲0.1eV [110], as revealed with recent cosmological data, the bound in (63) increases by an order of magnitude.


## 5.9.1. NP Scalars and Vectors

One way to see if the underlying model is symmergent or not is to measure scalar and vector masses and then determine if their production cross sections comply with the seesawic couplings in (60). For instance, mH′ can be measured from either of HH→H′→HH in Figure 5(a) or HH→H′H′ in Figure 5(b) such that the ratio of their cross sections(62)σHH→H′→HHσHH→H′H′≃mH2mH′22acts as an efficient probe of symmergence. This procedure works also for the Z′ scatterings in Figures 5(c) and 5(d). These 2→2 scatterings (and various other channels) can be directly tested at present [55, 56] and future [57, 58] colliders.Figure 5
Basic search channels for NP scalars ((a) and (b)) and NP vectors ((c) and (d)).


## 5.9.2. Right-Handed Neutrinos

Symmergence has testable implications also for right-handed neutrinos. Indeed, if the bound (60) is to be respected and if the active neutrinos are to have correct masses (mν≲1eV), then the right-handed neutrinos must have masses [59–61](63)mN′≲1000TeVwhich can be probed at future experiments (based presumably on Higgs factories and accelerator neutrinos [62]) if not indirectly at the near-future SHiP experiment [63, 64]. For mν≲0.1eV [110], as revealed with recent cosmological data, the bound in (63) increases by an order of magnitude.


## 5.10. Dark Matter Weighs below Weak Scale

Symmergence has candidates for missing matter in both the ebony and dark NP sectors. The question of which one is preferred by nature can be answered only by observations (on, for instance, less-baryonic galaxies like cosmic seagull [111]).The dark NP can be probed directly [12, 13, 69]. The scalar field H′ in (58), for instance, develops a vanishing VEV, 〈H′〉=0, and becomes stable if it is real and has odd Z2 parity with respect to H. It is stable but its density is depleted by not only the expansion of the Universe but also the coannihilations into the SM fields as H′H′→W+W-,ZZ,t¯t,⋯. The coannihilations into W/Z are Goldstone-equivalent to H′H′→HH in Figure 5(b) so that H′ scalars attain the observed relic density [12, 13, 112] if(64)λHH′4GeVmH′2≈10-7 or, impliedly, (65)mH′≲367GeVafter using the seesawic structure in (60) for λHH′. This bound on mH′, which makes sense barely as it lies beneath ΛW, implies that thermal dark matter cannot weigh above the electroweak scale. (Nonthermal dark matter, belonging presumably to the ebony NP, is free from this bound.) This means that symmergence can be probed by measuring the dark matter mass [12, 13, 69] and contrasting it with (65) or, more safely, with ΛW.


## 6. Conclusion

Needless to repeat, symmergence propounds a novel framework in which notorious problems of the SM can be consistently addressed. It incorporates GR into the SM with a seesawic NP sector and can be probed conclusively via various experimental tests ranging from collider searches to dark matter. Highlighted in Figure6 are the salient features of the symmergent GR plus seesawic NP setup. It is clear that symmergence with a seeswic NP leads to a natural and viable setup with various testable predictions.Figure 6
Fundamental aspects of the symmergent GR plus seesawic NP setup.The mechanism can be furthered in various aspects. First, the trans-Planckian SUSY, its breaking mechanism, and its possible role in solving the CCP need be explored properly. Second, the MAG with quadratic curvature terms (38) can give GR-like geometry in small curvature limit and conformal affine geometry in high curvature limit so that a complete solution [40] can shed light on strong gravity regime (like black holes). Third, Qμν in the MAG action (38) is suggestive of bimetric gravity [113], whose investigation may provide alternative views of symmergence. Fourth, Qμν in the MAG action (38) is reminiscent of the disformal curvature couplings [114, 115], whose exploration may shed light on the cosmological implications of symmergence. These four and various other aspects need be studied in detail to have a complete picture of symmergence. Experimental tests are of paramount importance to decide if symmergence is realized in nature or not.



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*Source: 4652048-2019-08-22.xml*